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Optics Part II: Diffraction and Geometrical Optics

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Solo Hermelin
Solo Hermelin

This document provides a table of contents for a document on optics. It outlines topics like Maxwell's equations, electromagnetic wave equations, geometrical optics, diffraction, and more. Key figures in the development of these topics are also mentioned, such as Maxwell, Helmholtz, Kirchhoff, and Fresnel. The document derives equations like the scalar Helmholtz differential equation and provides an overview of Fresnel-Kirchhoff diffraction theory.

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1 OPTICS Part II SOLO HERMELIN Updated: 16.01.10http://www.solohermelin.com
2 Table of Content SOLO OPTICS Maxwell’s Equations Boundary Conditions Electromagnatic Wave Equations Monochromatic Planar Wave Equations Spherical Waveforms Cylindrical Waveforms Energy and Momentum Electrical Dipole (Hertzian Dipole) Radiation Reflections and Refractions Laws Development Using the Electromagnetic Approach IR Radiometric Quantities Physical Laws of Radiometry Geometrical Optics Foundation of Geometrical Optics – Derivation of Eikonal Equation The Light Rays and the Intensity Law of Geometrical Optics The Three Laws of Geometrical Optics Fermat’s Principle (1657) O P T I C S P a r t I
3 Table of Content (continue) SOLO OPTICS Plane-Parallel Plate Prisms Lens Definitions Derivation of Gaussian Formula for a Single Spherical Surface Lens Using Fermat’s Principle Derivation of Gaussian Formula for a Single Spherical Surface Lens Using Snell’s Law Derivation of Lens Makers’ Formula First Order, Paraxial or Gaussian Optics Ray Tracing Matrix Formulation O P T I C S P a r t I
4 Table of Content (continue) SOLO OPTICS Optical Diffraction Fresnel – Huygens’ Diffraction Theory Complementary Apertures. Babinet Principle Rayleigh-Sommerfeld Diffraction Formula Extensions of Fresnel-Kirchhoff Diffraction Theory Phase Approximations – Fresnel (Near-Field) Approximation Phase Approximations – Fraunhofer (Near-Field) Approximation Fresnel and Fraunhofer Diffraction Approximations Fraunhofer Diffraction and the Fourier Transform Fraunhofer Diffraction Approximations Examples Resolution of Optical Systems Optical Transfer Function (OTF) Point Spread Function (PSF) Modulation Transfer Function (MTF) Phase Transfer Function (PTF) Relations between Wave Aberration, Point Spread Function and Modulation Transfer Function Other Metrics that define Image Quality – Srahl Ratio Other Metrics that define Image Quality - Pickering Scale Other Metrics that define Image Quality – Atmospheric Turbulence Fresnel Diffraction Approximations Examples
5 SOLO OPTICS Continue from OPTICS Part I
6 Optical DiffractionSOLO Augustin Jean Fresnel 1788-1827 In 1818 Fresnel, by using Huygens’ concept of secondary wavelets and Young’s explanation of interface, developed the diffraction theory of scalar waves. P 0P Q 1x 0x 1y 0y   Fr  Sr    r  O '  Screen Image plane Source plane 0O 1O Sn1   - Screen Aperture Sn1 - normal to Screen FrPP  0 SrQP  0 rQP   From a source P0 at a distance from a aperture a spherical wavelet propagates toward the aperture:    Srktj S source Q e r A tU   ' '  According to Huygens Principle second wavelets will start at the aperture and will add at the image point P.                   dre rr A Kdre r U KtU rrktj S sourcerkttjQ P S 2/2/' ',',   where:  ',K obliquity or inclination factor    SSS nrnr 11cos&11cos' 11            0',0 max0',0   K K Obliquity factor and π/2 phase were introduced by Fresnel to explain experiences results. Fresnel Diffraction Formula Fresnel took in consideration the phase of each wavelet to obtain: Fresnel – Huygens’ Diffraction Theory Fresnel –Kirchoff Diffraction Formula See full development in P.P. “Diffraction” Table of Content
7 SOLO Fresnel-Kirchhoff Diffraction Theory In 1882 Gustav Kirchhoff, using mathematical foundation, succeeded to show that the amplitude and phases ascribed to the wavelets by Fresnel, by enhancing the Huyghen’s Principle, were a consequence of the wave nature of light. HBED    & For an Homogeneous, Linear and Isotropic Medium where are constant scalars, we have , t E t D H t t H t B E ED HB                           Since we have also tt      t D H      t B E      For Source less Medium                               0& 0 2 2 2 DED EEE t E E       02 2 2     t E E    Maxwell Equations are   eJ t D HA        mBGM   )(   mJ t B EF          eDGE   James C. Maxwell (1831-1879) Gustav Robert Kirchhoff 1824-1887 Optical Diffraction
8 SOLO Fresnel-Kirchhoff Diffraction Theory 0 1 2 2 2 2          U tv Scalar Differential Wave Equation For a monochromatic wave of frequency f ( ω = 2πf ) a solution is:               tjPjPUPtPUtPU  expexpRecos,  Define the phasor       PjPUPU  exp U v U tv 2 2 2 2 2 1         2 2  v f v k   022  UkPhasor Scalar Differential Wave Equation This is the Scalar Helmholtz Differential Equation Hermann von Helmholtz 1821-1894 Boundary Conditions for the Helmholtz Differential Equation: • Dirichlet (U given on the boundary) • Neumann (dU/dn given on the boundary) Johann Peter Gustav Lejeune Dirichlet 1805-1859 Franz Neumann 1798-1895  1 0 11 2 2 2 2 2 2 2 2 2             vE tvt E v E    Vector Differential Wave Equation Optical Diffraction
9 To find the solution of the Scalar Helmholtz Differential Equation we need to use the following: • Scalar Green’s Identity       SV dSGUUGdVGUUG 22 • Green’s Function     SF SF FS rr rrkj rrG       exp ; This Green’s Function is a particular solution of the following Helmholtz Non-homogeneous Differential Equation:      SFFSFSS rrrrGkrrG   4;; 22 SOLO Fresnel-Kirchhoff Diffraction Theory provided that and are continuous in volume V UUU 2 ,,  GGG 2 ,,  Free-Space Green’s Function  n i iSS 1  iS nS dV dSnS  1 V Fr  Sr  F 0r SF rrr   PositionSourcerS  PositionFieldrF    022  Uk Scalar Helmholtz Differential Equation Optical Diffraction
10 SOLO • Scalar Green’s Identities       SV dSGUUGdVGUUG 22 Let start from the Gauss’ Divergence Theorem    SV dSAdVA  Karl Friederich Gauss 1777-1855 where is any vector field (function of position and time) continuous and differentiable in the volume V bounded by the enclosed surface S. Let define . A  UGA     UGUGUGA 2   Then        S Gauss VV dSUGdVUGUGdVUG 2        S Gauss VV dSGUdVGUUGdVGU 2 Subtracting the second equation from the first we obtain First Green’s Identity Second Green’s Identity We have GEORGE GREEN 1793-1841 Fresnel-Kirchhoff Diffraction Theory To find a general solution of the Scalar Helmoltz Differential Equation we need to use the  n i iSS 1  iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr   If we interchange with we obtainG U Optical Diffraction
11 Integral Theorem of Helmholtz and Kirchhoff       F V sF V SS rUdVUrrUGkUGkdVGUUG   442222   Using:      SFFSFSS rrrrGkrrG   4;; 22  n i iSS 1  iS nS dV dSnS  1 V Fr  Sr  F 0r SF rrr   PositionSourcerS  PositionFieldrF  SOLO Fresnel-Kirchhoff Diffraction Theory     0,22  SFS rrUk  From the left side of the Second Scalar Green’s Identity we have:                 SS SS dS n G U n U GdSGUUG     SF SF FS rr rrkj rrG       exp ;Using: we obtain:                                  S SF SF SF SF F dS rr rrkj n U n U rr rrkj rU      expexp 4 1  This is the Integral Theorem of Helmholtz and Kirchhoff that enables to calculate as function of the values of and on the enclosed surface S.nU  /UU Note: This Theorem was developed first by H. von Helmholtz in acoustics. Hermann von Helmholtz 1821-1894 Gustav Robert Kirchhoff 1824-1887 From the right side of the Second Scalar Green’s Identity, using we have:dS n U dSnUdSU SSS     1 Scalar Helmholtz Differential Equation Optical Diffraction
12 Sommerfeld Radiation Conditions SOLO Fresnel-Kirchhoff Diffraction Theory                              SS S F dS n G U n U G dS n G U n U GrU 1 4 1 4 1    P Fr  Sr  r   1S S R Screen Aperture d  Sn1  Sn1 since the condition that the previous integral be finite is:     R Rkj rrG SFS exp ;    Consider the surface of integration  SSS 1 1S - on the screen S - hemisphere with radius R   Gkj R Rkj R kj n G          exp1                        dRUkj n U GdS n G U n U G S 2   1 exp limlim   R Rkj RGR RR 0lim           Ukj n U R R This is Sommerfeld Radiation Conditions  - on the aperture Optical Diffraction
13 is known as optical disturbance. Being a scalar quantity, it cannot accurately represent an electromagnetic field. However, the square of this scalar quantity can be regarded as a measure of the irradiance at a given point. U Sommerfeld Radiation Conditions (continue) SOLO Fresnel-Kirchhoff Diffraction Theory                              SS S F dS n G U n U G dS n G U n U GrU 1 4 1 4 1    0lim           Ukj n U R R This is Sommerfeld Radiation Conditions This implies that: 0 4 1             S dS n G U n U G  and the Integral of Helmholtz and Kirchhoff becomes:                1 4 1 S F dS n G U n U GrU   P Fr  Sr  r   1S S R Screen Aperture d Sn1  Sn1 0P Q Arnold Johannes Wilhelm Sommerfeld 1868 - 1951 Optical Diffraction
14 The Kirchhoff Boundary Conditions SOLO Fresnel-Kirchhoff Diffraction Theory Kirchhoff assumed the following boundary conditions:                dS n G U n U GrU F 4 1 1. The field distribution and its derivative , across the aperture , are the same as in the absence of the screen. U nU  /  2. On the shadowed part of the screen and0 1 S U 0/ 1  S nU The Integral of Helmholtz and Kirchhoff becomes: The field at point P is the superposition of the aperture values 0 U 0/   nU Note: Moreover, mathematically the condition implies0/&0 11  SS nUU 0/&0   nUU However, if the dimensions of the aperture are large relative to the wavelength λ, the integral agrees well with the experiment. P Fr  Sr  r   1S S R Screen Aperture 0P 0,0 1 1     S S n U U     n U U , Q  Sn1  Sn1 S  Kirchhoff boundary conditions are not physical since the presence of the screen changes field values on the aperture and on the screen. Optical Diffraction
15 Fresnel-Kirchhoff Diffraction Formula SOLO Fresnel-Kirchhoff Diffraction Theory                dS n G U n U GrU F 4 1 The Integral of Helmholtz and Kirchhoff: Assume that the aperture is illuminated by a single spherical wave:     S Ssource S r rkjA rU exp                                   SS S Ssource S S S Ssource SSSS S nr r rkjA r kj n r rkjA nrU n rU 11 exp1 1 exp 1       SF SF FS rr rrkj rrG       exp ;                                      S S SF SF S rrr SFSS FS nr r rkj r kj n rr rrkj nrrG n rrG SF 11 exp1 1 exp 1, ,      P Fr  Sr  r   1S S R Screen Aperture 0P 0,0 1 1     S S n U U     n U U , Q  Sn1  Sn1 S  Optical Diffraction
16 Fresnel-Kirchhoff Diffraction Formula SOLO Fresnel-Kirchhoff Diffraction Theory                dS n G U n U GrU F 4 1 The Integral of Helmholtz and Kirchhoff: Assume that the aperture is illuminated by a single spherical wave, and: Srr,               SS S SsourceS nr r rkjA j n rU 11 exp2        r rkj rrG FS exp ;                 S FS nr r rkj j n rrG 11 exp2,    Srr k 1 , 12                                            dS nrnr rr rrkjA jrU SSS s ssource F 2 1111 exp       S Ssource S r rkjA rU exp   P Fr  Sr  r   1S S R Screen Aperture 0P 0,0 1 1     S S n U U     n U U , Q  Sn1  Sn1 S  Optical Diffraction
17 Fresnel-Kirchhoff Diffraction Formula SOLO Fresnel-Kirchhoff Diffraction Theory                                                                   dSK rr rrkj A dS nrnr rr rrkjA jrU S s s source SSS s ssource F     , 2 exp 2 1111 exp                                  SSSS S SSS S nrnr nrnr K 11cos&11cos 2 coscos 2 1111 ,    1. Obliquity or Inclination Factor:     0,0&10,0   SS KK 2. Additional phase π/2 3. The amplitude is scaled by the factor 1/λ (not found in Fresnel derivation) P Fr  Sr  r   1S S R Screen Aperture 0P 0,0 1 1     S S n U U     n U U , Q  Sn1  Sn1 S  We recovered Fresnel Diffraction Formula with: Optical Diffraction
18 Reciprocity Theorem of Helmholtz SOLO Fresnel-Kirchhoff Diffraction Theory                                                                   dSK rr rrkj A dS nrnr rr rrkjA jrU S s s source SSS s ssource F     , 2 exp 2 1111 exp We can see that the Fresnel-Kirchhoff Diffraction Formula is symmetrical with respect to r and rS, i.e. point source and observation point. Therefore we can interchange them and obtain the same relation. This result is called Reciprocity Theorem of Helmholtz. P Fr  Sr  r   1S S R Screen Aperture 0P 0,0 1 1     S S n U U     n U U , Q  Sn1  Sn1 S  Hermann von Helmholtz 1821-1894 Note: This is similar to Lorentz’s Reciprocity Theorem in Electromagnetism. Optical Diffraction
19 Huygens-Fresnel Principle SOLO Fresnel-Kirchhoff Diffraction Theory                           dSK rr rrkj A rU S s s source F    , 2 exp  The Fresnel Diffraction Formula can be rewritten as:         dS r rkj QVrU F exp where:     s s S source r rkj K A QV         2 exp ,    The interpretation of this formula is that each point of a wavefront can be considered as the center of a secondary spherical wave, and those secondary spherical waves interfere to result in the total field, is known as the Huygens-Fresnel Principle. P Fr  Sr  r   1S S R Screen Aperture 0P 0,0 1 1     S S n U U     n U U , Q  Sn1  Sn1 S  Table of Content Optical Diffraction
20 SOLO Consider a diffracting aperture Σ. Suppose that the aperture is divided into two portions Σ 1 and Σ 2 such that Σ = Σ1 + Σ2. The two aperture Σ1 and Σ2 are said to be complementary. Complementary Apertures. Babinet Principle From the Fresnel Diffraction Formula:                     21 dS r rkj QVdS r rkj QV dS r rkj QVrU F expexp exp P Fr  1Sr  1 r  2 1S Screen Apertures 0P 1 Q Sn1 2Sr  2r 1 2Q We can see that the result is the added effect of all complimentary apertures. This is known as Babinet Principle. The result can be very helpful when Σ is a very complicated aperture, that can be decomposed in a few simple apertures. Table of Content Optical Diffraction
21 SOLO The Kirchhoff Diffraction Formula is an approximation since for zero field and normal derivative on any finite surface the field is zero everywhere. This was pointed out by Poincare in 1892 and by Sommerfeld in 1894. The first rigorous solution of a diffraction problem was given by Sommerfeld in 1896 for a two-dimensional case of a planar wave incident on an infinitesimally thin, perfectly conducting half plane. This solution is not given here. Arnold Johannes Wilhelm Sommerfeld 1868 - 1951 Jules Henri Poincaré 1854-1912 Sommerfeld, A. : “Mathematische Theorie der Diffraction”, Math. Ann., 47:317, 1896 translated in english as “Optics, Lectures on Theoretical Physics”, vol. IV, Academic Press Inc., New York, 1954 Rayleigh-Sommerfeld Diffraction Formula Optical Diffraction
22 SOLO Rayleigh-Sommerfeld Diffraction Formula Let start from the Helmholtz and Kirchhoff Integral: P Fr  Sr  r   1S S R Screen Aperture 0P 0,0 1 1     S S n U U     n U U , Q  Sn1  Sn1 S  0'PFr'          SSSSS nnrrr 112'                 1 4 1 S F dS n G U n U GrU   Suppose that the Scalar Green Function is generated not only by P0 located at , but also by a point P’0 located symmetric relative to the screen at         SSSSS nnrrr 112'  Sr  G       SF SF SF SF SF rr rrkj rr rrkj rrG ' 'expexp ,_                  SF SF SF SF SF rr rrkj rr rrkj rrG ' 'expexp ,            or: We have 11 ,, ' SSFSSF rrrr             SSSFSSSF nrrnrr 1'1 11 ,,  0 1,   S G   011 exp1 2 11 ,, _                         S S S nr r rkj r kj n G 0 1,      S n G  0 exp 2 1 1 , ,          S S r rkj G Optical Diffraction
23 SOLO Rayleigh-Sommerfeld Diffraction Formula 1. Start from the Helmholtz and Kirchhoff Integral: P Fr  Sr  r   1S S R Screen Aperture 0P 0,0 1 1     S S n U U     n U U , Q  Sn1  Sn1 S  0'PFr'          SSSSS nnrrr 112'                 1 4 1 S F dS n G U n U GrU         SF SF SF SF SF rr rrkj rr rrkj rrG ' 'expexp ,_           Choose 0 1,   S G   011 exp1 2 11 ,, _                         S S S nr r rkj r kj n G On the shadowed part of the screen and0 1 S U 0/ 1  S nU                             dSnr r rkj rU j dS n G UrU SS k r kj F 11 exp 4 1 /2 1 _    This is Rayleigh-Sommerfeld Diffraction Formula of the first kind SF rrr   Arnold Johannes Wilhelm Sommerfeld 1868 - 1951     S Ssource S r rkjA UrU exp    John William Strutt Lord Rayleigh (1842-1919)                dSnr r rkj r rkjAj rU S S Ssource F 11 expexp   we obtain: Optical Diffraction
24 SOLO Rayleigh-Sommerfeld Diffraction Formula 2. Start from the Helmholtz and Kirchhoff Integral: P Fr  Sr  r   1S S R Screen Aperture 0P 0,0 1 1     S S n U U     n U U , Q  Sn1  Sn1 S  0'PFr'          SSSSS nnrrr 112'                 1 4 1 S F dS n G U n U GrU         SF SF SF SF SF rr rrkj rr rrkj rrG ' 'expexp ,            Choose On the shadowed part of the screen and0 1 S U 0/ 1  S nU SF rrr  0 1,      S n G  0 exp 2 1 1 , ,          S S r rkj G                   dS n U r rkj dS n U GrU F exp 2 1 4 1       S Ssource S r rkjA UrU exp                  SS S SsourceS nr r rkjA j n rU 11 exp2                   dSnr r rkj r rkjAj rU SS S Ssource F 11 expexp   For we obtain: This is Rayleigh-Sommerfeld Diffraction Formula of the second kind Table of Content Optical Diffraction
25 P 0P Q 1x 0x 1y 0y   Sr'  Sr    r  O S  Screen Image plane Source plane 0O 1O Sn1   - Screen Aperture Sn1 - normal to Screen 1r 0r  SSS rn cos11  cos11  rnS z Sn1 'r  Fr  FrPP  0 SrQP  0 rQP   SrOP '0   '1 rOO   SOLO          dS r rkj r rkjAj rU S S Ssource F 2 coscosexpexp    Start with Fresnel-Kirchhoff (or Rayleigh-Sommerfeld) Diffraction Formula 1. If the inclination factor is nearly constant over the aperture   constKK S  , Extensions of Fresnel-Kirchhoff Diffraction Theory         dS r rkj r rkjAKj rU S Ssource F expexp           dS r rkj rU Kj rU SF exp  2. Replace the incident point source wavefront with a general waveform   S S r rkjexp  Sinc rU  3. Characterize the aperture by a transfer function τ to model amplitude or phase changes due to optic system           dS r rkj rrU j rU SSF exp   Table of Content Optical Diffraction
26 SOLO Phase Approximations – Fresnel (Near-Field) Approximation Fresnel Approximation or Near Field Approximation can be used when aperture dimensions are comparable to distance to source rS or image r.          dS r rkj r rkjAj rU S S Ssource F 2 coscosexpexp    Start with Fresnel-Kirchhoff Diffraction Formula If the inclination factor is nearly constant over the aperture   constKK S  ,              dS r rkj rU Kj dS r rkj r rkjAKj rU S S Ssource F expexpexp   P Q 1x 1y     r  O  Screen Image plane 1O Sn1  1r  z Sn1 'r  rQP   '1 rOO   P 0P Q 1x 0x 1y 0y   Sr'  Sr    r  O S  Screen Image plane Source plane 0O 1O Sn1   - Screen Aperture Sn1 - normal to Screen 1r 0r  SSS rn cos11  cos11  rnS z Sn1 'r  Fr        1 ''2 ' ' 1''2' ' 2 2 1 2 1 2 11 2/1 2 2 1 2/1 11 0 1 2 1                           r r k r r r r r rrrrrrr rrr x x                                    '2 exp ' 'expexp 2 1 r r kj r rkj r rkj         2 max 2 1 2 1 ' '2 exp ' 'exp rrk dS r r kjrU r rkjKj rU SF                   Augustin Jean Fresnel 1788-1827 Optical Diffraction
27 SOLO Phase Approximations – Fraunhofer (Near-Field) Approximation Fraunhofer Approximation or Far Field Approximation can be used when aperture dimensions are very small comparable to distance to source rS or image r.          dS r rkj r rkjAj rU S S Ssource F 2 coscosexpexp    Start with Fresnel-Kirchhoff Diffraction Formula If the inclination factor is nearly constant over the aperture   constKK S  ,              dS r rkj rU Kj dS r rkj r rkjAKj rU S S Ssource F expexpexp   P Q 1x 1y     r  O  Screen Image plane 1O Sn1  1r  z Sn1 'r  rQP   '1 rOO   P 0P Q 1x 0x 1y 0y   Sr'  Sr    r  O S  Screen Image plane Source plane 0O 1O Sn1   - Screen Aperture Sn1 - normal to Screen 1r 0r  SSS rn cos11  cos11  rnS z Sn1 'r  Fr              ' exp ' 'expexp 1 r r kj r rkj r rkj           2 max 22 1 1 ' 2 ' exp ' 'exp rr k dS r r kjrU r rkjKj rU SF                      1 '2' ' '2' ' ' 2 1''2' ' 2 22 11 2 22 11 2 11 2/1 2 2 1 2 1 2/1 11 0 1 2 1                           r rk r r r r r r r r r rr rrrrrrr rrr x x              Optical Diffraction
28 0P Q 0x 0y  Sr'  Sr    O S ScreenSource plane 0O  Sn10r  SrQP  0 S rOP '0   SOLO Fresnel and Fraunhofer Diffraction Approximations Fresnel Approximations at the Source                                   S S SS S S xx x SS S S SSS SS r r rr r r rr r r rrr rr '2 '1 '2' ' ' '' ' 21' '2' ' 2 282 11 2/12 2 2/122 2                         S S S S S S S r r kjrkj r rkj r rkj '2 '1 exp'1exp ' 'expexp 2 2       S S r rkj ' 'exp    Srkj '1exp            S S r r kj '2 '1 exp 2 2   Spherical wave centered at P0. Lowest order approximation to the phase of a spherical wavefront Planar wave propagating in directionSr'1 P 0P Q 1x 0x 1y 0y   Sr'  Sr    r  O S  Screen Image plane Source plane 0O 1O Sn1   - Screen Aperture Sn1 - normal to Screen 1r 0r  SSS rn cos11  cos11  rnS z Sn1 'r  Fr  Optical Diffraction
29 SOLO Fresnel and Fraunhofer Diffraction Approximations P Q 1x 1y     r  O  Screen Image plane 1O Sn1  1r  z Sn1 'r  rQP   '1 rOO                                         ''2 ' ' 2 1' '2' ' 1 22 1 2 11 2/1 2 2 1 2 1 2/1 11 0 1 2 1 r r r r r r rr r rrrrrr rrr x x                                                        ' exp '2 exp '2 exp ' 'expexp 1 22 1 r r kj r kj r r kj r rkj r rkj   Fresnel Approximations at the Image plane P 0P Q 1x 0x 1y 0y   Sr'  Sr    r  O S  Screen Image plane Source plane 0O 1O Sn1   - Screen Aperture Sn1 - normal to Screen 1r 0r  SSS rn cos11  cos11  rnS z Sn1 'r  Fr    ' 'exp r rkj     '1exp rkj            '2 '1 exp 2 2 r r kj   Spherical wave centered at O. Lowest order approximation to the phase of a spherical wavefront Planar wave propagating in direction'1r Optical Diffraction
30 SOLO Fresnel and Fraunhofer Diffraction Approximations (1st way)            dS r rkj r rkjAj rU SK S S Ssource F        , 2 coscosexpexp Fresnel Approximation                                            dS r rrr r r kjrrkjrrkj rr rrkjKAj rU S S S S Ssource F '2 '1 '2 '1 exp'1'1exp'1exp '' ''exp 2 1 2 1 2 2 1      Fraunhofer Approximation      1 '2 '1 '2 '12 2 1 2 1 2 2     S S k r rrr r r     or S MAX rr ',' 2                 dSrrkjrrkj rr rrkjKAj rU S S Ssource F    '1'1exp'1exp '' ''exp 1 If     1 '2 '1 '2 '1 exp 2 1 2 1 2 2                         S S r rrr r r kj   we obtain Augustin Jean Fresnel 1788-1827   constKK S  , Start with '1'1 rrq S   P 0P Q 1x 0x 1y 0y   Sr'  Sr    r  O S  Screen Image plane Source plane 0O 1O Sn1   - Screen Aperture Sn1 - normal to Screen 1r 0r  SSS rn cos11  cos11  rnS z Sn1 'r  Fr  FrPP  0 SrQP  0 rQP   SrOP '0   '1 rOO   Optical Diffraction
31 SOLO Fresnel and Fraunhofer Diffraction Approximations (2nd way) Fresnel Approximation                      dS r rr kjrrU r rkjj rU SSF '2 exp ' 'exp 11      Fraunhofer Approximation    '1 '2 2 max 2 1 22 1 2 r r r r k        If we obtain Augustin Jean Fresnel 1788-1827 Start with           dS r rkj rrU j rU SSF exp   - aperture optical transfer function Sr   - disturbance at the aperture SrU                        dS r r kjrrU r rkjj rU SSF ' exp ' 'exp 1                                          ' exp '2 exp ' 'expexp 1 2 1 2 r r kj r r kj r rkj r rkj   P Q 1x 1y     r  O  Screen Image plane 1O Sn1  1r  z Sn1 'r  rQP   '1 rOO      1' rrr                                       2 11 2/1 2 11 2/1 11 0 1 2 '2 1' ' 1''2' r rr r r rr rrrrrrr                                    dS r r kj r r kjrrU r rkjj rU SSF ' exp '2 exp ' 'exp 1 2 1 2                                        '2' ' ' 2 1''2' 22 11 2 112/1 2 2 1 2 1 2/1 11 0 1 2 r r r r r r rr rrrrrrr x x   Optical Diffraction
32 SOLO Fresnel and Fraunhofer Diffraction Approximations Augustin Jean Fresnel 1788-1827 1x 1y  max D Screen 1O 1r  z  2 D R  Fresnel Region Fraunhofer Region  2 D R  R  O    '1 '2 2 max 2 1 22 1 2 r r r r k         Fraunhofer Approximation If Optical Diffraction
33 SOLO Fraunhofer Diffraction and the Fourier Transform                       dS r r kjrrU r rkjj rU SSF ' exp ' 'exp 1          11 1 ' 2 ' yx rr r k                               ddyx r jrrU r rkjj rU SSF 11 ' 2 exp ' 'exp  The integral is the two dimensional Fourier Transform of the field within the aperture    SS rrU                 fFTddkkjfkkF yxyx   exp, 2 1 :, 2           SSF rrUFT r rkjj rU    2 2 ' 'exp Therefore P 0P Q 1x 0x 1y 0y   Sr'  Sr    r  O S  Screen Image plane Source plane 0O 1O Sn1   - Screen Aperture Sn1 - normal to Screen 1r 0r  SSS rn cos11  cos11  rnS z Sn1 'r  Fr  FrPP  0 SrQP  0 rQP   SrOP '0   '1 rOO   Two Dimensional Fourier Transform Optical Diffraction
34 SOLO Fraunhofer Diffraction Approximations Examples Rectangular Aperture P Q 1x 1y     r  O  Screen Image plane 1O Sn1 1r  z Sn1 'r 1 2 1 2                                                   1 1 1 1 11 0 2 11 2 10 ' 2 exp ' 2 exp '2 'exp ' exp ' exp '2 exp ' 'exp                 dy r jdx r j r rkjUkj dd r y kj r x kj r r kj r rkjUj rU k F           elsevere U rrU SS 0 & 21110    For a Rectangular Aperture Therefore                             dS r r kjrrU r r kj r rkjj rU SSF ' exp '2 exp ' 'exp 1 2 1                                                11 11 1 1 1111 1 ' 2 ' 2 sin 2 ' 2 ' 2 exp ' 2 exp ' 2 exp 1 1                     x r x r x r j x r jx r j dx r j                                           11 11 1 1 1111 1 ' 2 ' 2 sin 2 ' 2 ' 2 exp ' 2 exp ' 2 exp 1 1                     y r y r y r j y r jy r j dy r j                                   11 11 11 11 4/ 11 0 ' 2 ' 2 sin ' 2 ' 2 sin ' 'exp2              y r y r x r x r r rkjUkj rU A F  Optical Diffraction
35 SOLO Fraunhofer Diffraction Approximations Examples Rectangular Aperture (continue – 1) Since U stands for scalar field intensity (E or H), the irradiance I is given by where < > is the time average and * is the complex conjugate.                                  11 11 11 11 0 ' 2 ' 2 sin ' 2 ' 2 sin ' 'exp8              y r y r x r x r Ar rkjUkj rU F         FFF rUrUrI  ~ Therefore     2 11 11 2 2 11 11 2 ' 2 ' 2 sin ' 2 ' 2 sin 0                                          y r y r x r x r IrI F  I (0) is the irradiance at O1 (x1 = y1 = 0). Hecht pg.466 Optical Diffraction
36 SOLO Fraunhofer Diffraction Approximations Examples Single Slit Aperture Let substitute in the rectangular aperture ξ1 → 0 where < > is the time average and * is the complex conjugate.                                  11 11 11 11 0 ' 2 ' 2 sin ' 2 ' 2 sin ' 'exp8              y r y r x r x r Ar rkjUkj rU F         FFF rUrUrI  ~ Therefore     2 11 11 2 ' 2 ' 2 sin 0                      y r y r IrI F  I (0) is the irradiance at O1 (x1 = y1 = 0). to obtain the single (vertical) slit diffraction                    11 11 0 ' 2 ' 2 sin ' 'exp2        y r y r Ar rkjUkj rU FSLITSINGLE  Since U stands for scalar field intensity (E or H), the irradiance I is given by Hecht, pg. 453 Hecht, pg. 456 Optical Diffraction
37 SOLO Fraunhofer Diffraction Approximations Examples Single Slit Aperture (continue)     2 11 11 2 ' 2 ' 2 sin 0                      y r y r IrI F  I (0) is the irradiance at O1 (x1 = y1 = 0). Hecht, pg. 456 Hecht 455 Define: 11 ' 2 :      y r     2 2 sin 0    II  The extremum of I (β) is obtained from:       0 sincossin2 0 3        I d Id The results are given by: minimum,3,2,0sin   maximum tan The solutions can be obtained graphically as shown in the figure and are: ,4707.3,4590.2,4303.1   Optical Diffraction
38 SOLO Fraunhofer Diffraction Approximations Examples Double Slit Aperture                                                        d r x kjd r x kj r r kj r rkjUj rU ba ba ba ba F 2/ 2/ 1 2/ 2/ 1 2 10 ' exp ' exp '2 exp ' 'exp P Q 1x 1y     r  O  Screen Image plane 1O Sn1 1r  z Sn1 'r  1 b b a                            dS r r kjrrU r r kj r rkjj rU SSF ' exp '2 exp ' 'exp 1 2 1                                                                 ax r j bx r bx r b x r j bax r jbax r j dx r j ba ba 1 1 1 1 112/ 2/ 1 ' exp ' ' sin 1 ' 2 ' exp ' exp ' 2 exp                                                                           ax r j bx r bx r b x r j bax r jbax r j dx r j ba ba 1 1 1 1 112/ 2/ 1 ' exp ' ' sin 1 ' 2 ' exp ' exp ' 2 exp                                                ax r bx r bx r br r kj r rkjUj rU F 1 1 12 10 ' cos ' ' sin 2 '2 exp ' 'exp                                   ax r bx r bx r IrI F 1 2 2 1 1 2 ' cos ' ' sin 0               FFF rUrUrI  ~                  elsevere babababaU rrU SS 0 2/2/&2/2/0    Optical Diffraction
39 SOLO Fraunhofer Diffraction Approximations Examples Double Slit Aperture (continue -= 1) Hecht p.458                2 2 2 12 2 1 12 cos sin 0 ' cos ' ' sin 0 I a r x b r x b r x IrI F                        P Q 1x 1y     r  O  Screen Image plane 1O Sn1 1r  z Sn1 'r  1 b b a The factor (sin β/ β)2 that was previously found as the distribution function for a single slit is here the envelope for the interference fringes given by the term cos2γ. Bright fringes occur for γ = 0,±π ,±2π,… The angular separation between fringes is Δγ = π. Optical Diffraction
40 Hecht 459 SOLO Fraunhofer Diffraction Approximations Examples Double Slit Aperture (continue – 2) Optical Diffraction
41 SOLO Fraunhofer Diffraction Approximations Examples Multiple Slit Aperture P 1y r  Image plane 1O 1r  Q     O  Screen  Sn1 'r  b b a a b b a b a The Aperture consists of a large number N of identical parallel slits of width b and separation a.                                   1 0 2/ 2/ 1 2 10 ' exp '2 exp ' 'exp N k bak bak F d r x kj r r kj r rkjUj rU                                 dS r r kjrrU r r kj r rkjj rU SSF ' exp '2 exp ' 'exp 1 2 1                                                                     akx r j bx r bx r b x r j b kax r j b kax r j dx r j bka bka 1 1 1 1 112/ 2/ 1 ' 2 exp ' ' sin 1 ' 2 2' 2 exp 2' 2 exp ' 2 exp                                                                                                                                           ax r aNx r bx r bx r br r kj r rkjUj ax r j aNx r j bx r bx r br r kj r rkjUj akx r j bx r bx r br r kj r rkjUj rU N k F 1 1 1 12 10 1 1 1 12 10 1 0 1 1 12 10 ' sin ' sin ' ' sin 1 '2 exp ' 'exp ' 2 exp1 ' 2 exp1 ' ' sin 1 '2 exp ' 'exp ' 2 exp ' ' sin 1 '2 exp ' 'exp                                    elsevere NkbkabkaU rrU SS 0 1,,1,02/2/0    Optical Diffraction
42 SOLO Fraunhofer Diffraction Approximations Examples Multiple Slit Aperture (continue – 1) P 1y r  Image plane 1O 1r  Q     O  Screen  Sn1 'r  b b a a b b a b a The Aperture consists of a large number N of identical parallel slits of width b and separation a.                                        ax r aNx r bx r bx r br r kj r rkjUj rU F 1 1 1 12 10 ' sin ' sin ' ' sin 1 '2 exp ' 'exp                             22 2 2 2 1 22 1 2 2 1 1 2 sin sinsin 0 ' sin ' sin ' ' sin 0 N N I ax r N aNx r bx r bx r IrI F                                       FFF rUrUrI  ~ Optical Diffraction
43 SOLO Fraunhofer Diffraction Approximations Examples Multiple Slit Aperture (continue – 2) Hecht p.462 Hecht p.463                   22 2 2 2 1 22 1 2 2 1 1 2 sin sinsin 0 ' sin ' sin ' ' sin 0 N N I ax r N aNx r bx r bx r IrI F                                Optical Diffraction
44 SOLO Fraunhofer Diffraction Approximations Examples Multiple Slit Aperture (continue – 2)                   22 2 2 2 1 22 1 2 2 1 1 2 sin sinsin 0 ' sin ' sin ' ' sin 0 N N I ax r N aNx r bx r bx r IrI F                                Sears p.222 Hecht p. 462 Sears p.236 Interference Irradiation for 1, 2, 3 and 4 slits as function of observation angle. Diffraction Pattern for 1, 2, 3, 4 and 5 slits. Optical Diffraction
45 SOLO Resolution of Optical Systems According to Huygens-Fresnel Principle, a differential area dS, within an optical Aperture, may be envisioned as being covered with coherent secondary point sources. z y   Z Y  R  q  sincos  yz  sincos qYqZ Differential area dS, coordinates Image , coordinates  dddS    dSe r E dE rktiA          The spherical wave that propagates from dS to Image is where           22/122/1222 /1/21 RZzYyRRZzYyRzZyYXr    2/1222 ZYXR                 RkaqJkaqRe R E ddee R E dSee R E dEE RktiA a RkpqiRktiA Aperture RzZyYkiRktiA Aperture // 1 0 2 0 cos//                                The spherical wave at Image, for a Circular Aperture, is Optical Diffraction
46 SOLO Resolution of Optical Systems z y   Z Y  R  q      RkaqJkaqRe R E E RktiA // 1          where           2 0 cos 2 dve i uJ vuvmi m m Bessel Function (of the first kind) E. Hecht, “Optics” The spherical wave at Image, for a Circular Aperture, is Optical Diffraction
47 SOLO Resolution of Optical Systems z y   Z Y  R  q Irradiance     EEHEHESI EH    2 1 2 1 2 1             2 1 2 1 2 22 / /2 0 / /22 2 1               Rkaq RkaqJ I Rkaq RkaqJ R aE EEI A        Daaak RkaquuJ n Rq      22.1 2 22.1 2 83.383.3 sin83.3/0 sin/ 11   D nnn   44.22  E. Hecht, “Optics” Circular Aperture Optical Diffraction
48 SOLO Resolution of Optical Systems z y   Z Y  R  qDistribution of Energy in the Diffraction Pattern at the Focus of a Perfect Circular Lens E. Hecht, “Optics” Ring f/(λf#) Peak Energy in ring Illumination (%) Central max 0 1 83.9 1st dark ring 1.22 0 1st bright ring 1.64 0.017 7.1 2nd dark 2.24 0 2nd bright 2.66 0.0041 2.8 3rd dark 3.24 0 3rd bright 3.70 0.0016 1.5 4th dark 4.24 0 4th bright 4.74 0.00078 1.0 5th dark 5.24 0 Optical Diffraction
49 SOLO Fraunhofer Diffraction Approximations Examples Circular Aperture Hecht p.469 Optical Diffraction
50 SOLO Resolution of Optical SystemsAiry Rings In 1835, Sir George Biddell Airy, developed the formula for diffraction pattern, of an image of a point source in an aberration-free optical system, using the wave theory. E. Hecht, “Optics” Optical Diffraction
51 SOLO Resolution of Optical Systems E. Hecht, “Optics” Optical Diffraction
52 Rayleigh’s Criterion (1902) The images are said to be just resolved when the center of one Airy Disk falls on the first minimum of the Airy pattern of the other image. The minimum resolvable angular separation or angular limit is: D nnn   44.22  Sparrow’s Criterion At the Rayleigh’s limit there is a central minimum Or saddle point between adjacent peaks. Decreasing the distance between the two point sources cause the central dip to grow shallower and ultimately to disappear. The angular separation corresponding to that configuration is the Sparrow’s Limit. SOLO Resolution of Optical Systems Optical Diffraction
53 Resolution – Diffraction Limit Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,University of California, Berkley Optical Diffraction
54 Diffraction limit to resolution of two close point-object images: best resolution is possible when the two are of near equal, optimum intensity. As the two PSF merge closer, the intensity deep between them rapidly diminishes. At the center separation of half the Airy disc diameter - 1.22λ/D radians (138/D in arc seconds, for λ=0.55μ and the aperture diameter D in mm), known as Rayleigh limit - the deep is at nearly 3/4 of the peak intensity. Reducing the separation to λ/D (113.4/D in arc seconds for D in mm, or 4.466/D for D in inches, both for λ=0.55μ) brings the intensity deep only ~4% bellow the peak. This is the conventional diffraction resolution limit, nearly identical to the empirical double star resolution limit, known as Dawes' limit. With even slight further reduction in the separation, the contrast deep disappears, and the two spurious discs merge together. The separation at which the intensity flattens at the top is called Sparrow's limit, given by 107/D for D in mm, and 4.2/D for D in inches (λ=0.55μ). Optical Diffraction
55 SOLO Fresnel Diffraction Approximations Examples Rectangular Aperture                      dS r rr kjrrU r rkjj rU SSF '2 exp ' 'exp 11      define Augustin Jean Fresnel 1788-1827 P Q 1x 1y     r  O  Screen Image plane 1O Sn1 1r  z Sn1 'r 1 2 1 2                                                 2 1 2 1 ' 2 2 exp ' 2 2 exp '2 'exp '2 exp '2 exp ' 'exp 2 1 2 10 2 2 1 2 10                 d r y jd r x j r rkjUkj dd r y kj r x kj r rkjUj rU k F           elsevere U rrU SS 0 & 21110    For a Rectangular Aperture         d r d r x ' 2 ' 2 : 2 12                    2 1 2 1 2 2 1 2 exp 2 ' ' 2 2 exp          dj r d r x j    212111 ' 2 & ' 2       x r x r Therefore         d r d r y ' 2 ' 2 : 2 12       212111 ' 2 & ' 2       y r y r                 2 1 2 1 2 2 1 2 exp 2 ' ' 2 2 exp          dj r d r y j Optical Diffraction
56 SOLO Fresnel Diffraction Approximations Examples Rectangular Aperture (continue – 1) Augustin Jean Fresnel 1788-1827 P Q 1x 1y     r  O  Screen Image plane 1O Sn1 1r  z Sn1 'r 1 2 1 2                                         2 1 2 1 2 1 2 1 220 2 1 2 10 2 exp 2 exp 2 'exp ' 2 2 exp ' 2 2 exp ' 'exp                    djdj rkjUj d r y jd r x j r rkjUj rU F  Define Fresnel Integrals                             0 2 0 2 2 sin: 2 cos: dS dC      SjCdj       0 2 2 exp               2 1 2 1 2 'exp0      SjCSjC rkjUj rU F   Using the Fresnel Integrals we can write     5.0 SC Optical Diffraction
57 SOLO Fresnel Diffraction Approximations Examples Rectangular Aperture (continue – 2) Augustin Jean Fresnel 1788-1827 Hecht p.499 Optical Diffraction
58 SOLO Fresnel Diffraction Approximations Examples Augustin Jean Fresnel 1788-1827 P Q 1x 1y     r  O  Screen Image plane 1O Sn1 1r  z Sn1 'r 1 2 1 2 Optical Diffraction
59 SOLO Fresnel Diffraction Approximations Examples Cornu Spiral Fresnel Integrals are defined as                   uu duuuSduuuC 0 2 0 2 2 sin:& 2 cos:     uSjuCduuj u       0 2 2 exp      5.0 SC Marie Alfred Cornu professor at the École Polytechnique in Paris established a graphical approach, for calculating intensities in Fresnel diffraction integrals. The Cornu Spiral is defined as the plot of S (u) versus C (u) duuSd duuCd               2 2 2 sin 2 cos       duSdCd  22 Therefore u may be thought as measuring arc length along the spiral. “Méthode nouvelle pour la discussion des problèmes de diffraction dans le cas d’une onde cylindrique”, J.Phys.3 (1874), 5-15,44-52 Optical Diffraction
60 SOLO Fresnel Diffraction Approximations Examples Cornu Spiral (continue – 1)                   uu duuuSduuuC 0 2 0 2 2 sin:& 2 cos:     uSjuCduuj u       0 2 2 exp      5.0 SC The Cornu Spiral is defined as the plot of S (u) versus C (u) duuSdduuCd              22 2 sin& 2 cos      duSdCd  22                     2 2 2 2 tan 2 cos 2 sin u u u Cd Sd    Therefore every point on the curve makes the angle with the real ( C ) axis. 2 2 u  The radius of curvature of Cornu Spiral is The tangent vector of Cornu Spiral is SuCuT 1 2 sin1 2 cos 22                   u SuCuu udTdSdCdTd    1 1 2 cos1 2 sin 1 / 1 / 1 22 22                         showing that the curve spirals toward the limit points.        2 1 2 2 cos u u duu         2 1 2 2 sin u u duu         2 1 2 2 exp u u duuj  Table of Content Optical Diffraction
61 SOLO Fresnel Diffraction Approximations Examples Augustin Jean Fresnel 1788-1827 Circular Aperture Hecht p.491 Hecht p.492 Optical Diffraction
62 SOLO Fresnel Diffraction Approximations Examples Augustin Jean Fresnel 1788-1827 Circular Obstacles Optical Diffraction
63 SOLO Fresnel Diffraction Approximations Examples Augustin Jean Fresnel 1788-1827 Fresnel Zone Plate Optical Diffraction
64 SOLO Fresnel Diffraction Approximations Examples Augustin Jean Fresnel 1788-1827 Fresnel Diffraction by a Slit Hecht p.504 a Fresnel Diffraction Hecht p.504 b Optical Diffraction
65 SOLO Fresnel Diffraction Approximations Examples Augustin Jean Fresnel 1788-1827 Semi-Infinite Opaque Screen Hecht p.506 a Hecht p.506 Optical Diffraction
66 SOLO Fresnel Diffraction Approximations Examples Augustin Jean Fresnel 1788-1827 Semi-Infinite Opaque Screen Hecht p.506 a Hecht p.507 Optical Diffraction
67 SOLO Optics Optical Transfer Function (OTF)
68 Point Spread Function (PSF) The Point Spread Function, or PSF, is the image that an Optical System forms of a Point Source. The PSF is the most fundamental object, and forms the basis for any complex object. PSF is the analogous to Impulse Response Function in electronics.    2 , yxPFTPSF  The PSF for a perfect optical system (with no aberration) is the Airy disc, which is the Fraunhofer diffraction pattern for a circular pupil. SOLO Optics
69 Point Spread Function (PSF) As the pupil size gets larger, the Airy disc gets smaller. SOLO Optics
70 Convolution      yxIyxOyxPSF ,,,          yxIyxOFTyxPSFFTFT ,,,1  Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,University of California, Berkley SOLO Optics
71 Modulation Transfer Function (MTF) Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,University of California, Berkley SOLO Optics
72 Modulation Transfer Function (MTF) The Modulation Transfer Function (MTF) indicates the ability of an Optical System to reproduce various levels of details (spatial frequencies) from the object to image. Its units are the ratio of image contrast over the object contrast as a function of spatial frequency.   3.57 a fcutoff Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,University of California, Berkley SOLO Optics
73 Modulation Transfer Function (MTF) Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,University of California, Berkley SOLO Optics
74 Phase Transfer Function (PTF) • PTF contains information about asymmetry in PSF • PTF contains information about contrast reversals (spurious resolution) Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,University of California, Berkley SOLO Optics
75 Relations between Wave Aberration, Point Spread Function and Modulation Transfer Function • The Point Spread Function (PSF) is the Fourier Transform (FT) of the pupil function • The Modulation Transfer Function (MTF) is the amplitude component of the FT of the PSF • The Phase Transfer Function (PTF) is the phase component of the FT of the PSF • The Optical Transfer Function (OTF) composed of MTF and PTF can also be computed as the autocorrelation of the pupil function.               yxWi eyxPFTyxPSF , 2 ,,        iiyx yxPSFFTAmplitudeffMTF ,,       iiyx yxPSFFTPhaseffPTF ,,        yxyxyx ffPTFiffMTFffOTF ,exp,,  SOLO Optics
76 Relations between Wave Aberration, Point Spread Function and Modulation Transfer Function             yxWi eFTyxPSF , 2 ,        iiyx yxPSFFTAmplitudeffMTF ,,       iiyx yxPSFFTPhaseffPTF ,,  Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”, University of California, Berkley SOLO Optics Ideal Optical System
77 Relations between Wave Aberration, Point Spread Function and Modulation Transfer Function             yxWi eFTyxPSF , 2 ,        iiyx yxPSFFTAmplitudeffMTF ,,       iiyx yxPSFFTPhaseffPTF ,,  Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”, University of California, Berkley SOLO Optics Real Optical System
78 Relations between Wave Aberration, Point Spread Function and Modulation Transfer Function             yxWi eFTyxPSF , 2 ,        iiyx yxPSFFTAmplitudeffMTF ,,       iiyx yxPSFFTPhaseffPTF ,,  Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”, University of California, Berkley SOLO Optics Real Optical System
79 FIGURE : PSF and MTF for nominal wavefront errors of selected aberrations resulting in ~0.80 Strehl (~1/14 wave RMS), and the error doubled. (a) the effect of 1/4 and 1/2 wave P-V wavefront error of defocus on the PSF intensity distribution (left) and image contrast (right). Doubling the error nearly halves the peak diffraction intensity, but the average contrast loss nearly triples (evident from the peak PSF intensity). (b) 1/4 and 1/2 wave P-V of spherical aberration. While the peak PSF intensity change is nearly identical to that of defocus, wider energy spread away from the disc results in more of an effect at mid- to high- frequency range. Central disc at 1/2 wave P-V becomes larger, and less well defined. The 1/2 wave curve indicates ~20% lower actual cutoff frequency in field conditions. http://www.telescope-optics.net/ SOLO Optics
80 FIGURE : PSF and MTF for nominal wavefront errors of selected aberrations resulting in ~0.80 Strehl (~1/14 wave RMS), and the error doubled. (c) 0.42 and 0.84 wave P-V wavefront error of coma. Both, intensity distribution (PSF) and contrast transfer change with the orientation angle, due to the asymmetric character of aberration. The worst effect is along the axis of aberration (red), or length-wise with respect to the blur (0 and π orientation angle), and the least is in the orientation perpendicular to it (green). (d) 0.37 and 0.74 wave P-V of astigmatism. Due to the tighter energy spread, there is less of a contrast loss with larger, but more with small details, compared to previous wavefront errors. Contrast is best along the axis of aberration (red), falling to the minimum (green) at every 45° (π/4), and raising back to its peak at every 90°. The PSF is deceiving here: since it is for a linear angular orientation, the energy spread is lowest for the contrast minima. http://www.telescope-optics.net/ SOLO Optics
81 (e) Turned down edge effect on the PSF and MTF. The P-V errors for 95% zone are 2.5 and 5 waves as needed for the initial 0.80 Strehl (the RMS is similarly out of proportion). Lost energy is more evenly spread out, and the central disc becomes enlarged. Odd but expected TE property - due to the relatively small area of the wavefront affected - is that further increase beyond 0.80 Strehl error level does almost no additional damage. f) The effect of ~1/14 and ~1/7 wave RMS wavefront error of roughness, resulting in the peak intensity and contrast drop similar to those with other aberrations. Due to the random nature of the aberration, its nominal P-V wavefront error can vary significantly for a given RMS error and image quality level. Shown is the medium- scale roughness ("primary ripple" or "dog biscuit", in amateur mirror makers' jargon) effect. (g) 0.37 and 0.74 wave P-V of wavefront error caused by pinching having the typical 3-sided symmetry (trefoil). The aberration is radially asymmetric, with the degree of pattern deformation varying between the maxima (red MTF line, for the pupil angle θ=0, 2π/3, 4π/3), and minima (green line, for θ=π/3, π, 5π/3); (the blue line is for a perfect aperture). Other forms do occur, with or without some form of symmetry. FIGURE : PSF and MTF for nominal wavefront errors of selected aberrations resulting in ~0.80 Strehl (~1/14 wave RMS), and the error doubled.
82 h) 0.7 and 1.4 wave P-V wavefront error caused by tube currents starting at the upper 30% of the tube radius. The energy spreads mainly in the orientation of wavefront deformation (red PSF line, to the left). Similarly to the TE, further increase in the nominal error beyond a certain level has relatively small effect Contrast and resolution for the orthogonal to it pattern orientation are as good as perfect (green MTF line). (i) Near-average PSF/MTF effect of ~1/14 and ~1/7 wave RMS wavefront error of atmospheric turbulence. The atmosphere caused error fluctuates constantly, and so do image contrast and resolution level. Larger seeing errors (1/7 wave RMS is rather common with medium-to-large apertures) result in a drop of contrast in the mid- and high- frequency range to near-zero level. FIGURE : PSF and MTF for nominal wavefront errors of selected aberrations resulting in ~0.80 Strehl (~1/14 wave RMS), and the error doubled. http://www.telescope-optics.net/ SOLO Optics
83 Other Metrics that define Image Quality Strehl Ratio Strehl, Karl 1895, Aplanatische und fehlerhafte Abbildung im Fernrohr, Zeitschrift für Instrumentenkunde 15 (Oct.), 362-370. Dr. Karl Strehl 1864 -1940 One of the most frequently used optical terms in both, professional and amateur circles is the Strehl ratio. It is the simplest meaningful way of expressing the effect of wavefront aberrations on image quality. By definition, Strehl ratio - introduced by Dr. Karl Strehl at the end of 19th century - is the ratio of peak diffraction intensities of an aberrated vs. perfect wavefront. The ratio indicates image quality in presence of wavefront aberrations; often times, it is used to define the maximum acceptable level of wavefront aberration for general observing - so-called diffraction-limited level - conventionally set at 0.80 Strehl. SOLO Optics
84 The Strehl ratio is the ratio of the irradiance at the center of the reference sphere to the irradiance in the absence of aberration. Irradiance is the square of the complex field amplitude u 0 E E Strehl  2 uE   dxdyyxWjUu )),(2exp(0  Other Metrics that define Image Quality Strehl Ratio Expectation Notation   dxdy dxdyyxu uu ),( SOLO Optics
85 Derivation of Strehl Approximation  2 0 21 W E E Strehl  ),(2 0 yxWj eUu     22 0 ),(2 2 1 ),(21 yxWyxWjUu     2 0 2 00 ),(2 2 1 ),(2 yxWUyxWUjUu   series expansion     2 0 22 0 2 0 ),(2),(2 yxWEyxWEEE   multiply by complex conjugate 2222 ),(),(),(),( yxWyxWyxWyxWW  wavefront variance: SOLO Optics
86  2 0 21 W E E Strehl  22 2 ),(),( yxWyxWWWW  where W is the wavefront variance:  2 2 W eStrehl  Another approximation for the Strehl ratio is Strehl Approximation Diffraction Limit 8.0Strehl A system is diffraction-limited when the Strehl ratio is greater than or equal to 0.8 Maréchal’s criterion: This implies that the rms wavefront error is less than /13.3 or that the total wavefront error is less than about /4. SOLO Optics
87 Other Metrics that define Image Quality Strehl Ratio dl eye H H RatioStrehl  Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,University of California, Berkley SOLO Optics
88 Other Metrics that define Image Quality Strehl Ratio   2m n Crms when rms is small  2 2 2 1 rmsStrehl          SOLO Optics
89 Other Metrics that define Image Quality FIGURE 34: Pickering's seeing scale uses 10 levels to categorize seeing quality, with the level 1 being the worst and level 10 near-perfect. Its seeing description corresponding to the numerical seeing levels are: 1-2 "very poor", 3 "poor to very poor", 4 "poor", 5 "fair", 6 "fair to good" 7 "good", 8 "good to excellent", 9 "excellent" and 10 "perfect". Diffraction-limited seeing error level (~0.8 Strehl) is between 8 and 9. Pickering 1 Pickering 2 Pickering 3 Pickering 4 Pickering 5 Pickering 6 Pickering 7 Pickering 9 Pickering 10Pickering 8 William H. Pickering (1858-1938) SOLO Optics
90 Other Metrics that define Image Quality FIGURE: Illustration of a point source (stellar) image degradation caused by atmospheric turbulence. The left column shows best possible average seeing error in 2 arc seconds seeing (ro~70mm @ 550nm) for four aperture sizes. The errors are generated according to Eq.53-54, with the 2" aperture error having only the roughness component (Eq.54), and larger apertures having the tilt component added at a rate of 20% for every next level of the aperture size, as a rough approximation of its increasing contribution to the total error (the way it is handled by the human eye is pretty much uncharted territory). The two columns to the right show one possible range of error fluctuation, between half and double the average error. The best possible average RMS seeing error is approximately 0.05, 0.1, 0.2 and 0.4 wave, from top to bottom (the effect would be identical if the aperture was kept constant, and ro reduced). The smallest aperture is nearly unaffected most of the time. The 4" is already mainly bellow "diffraction-limited", while the 8" has very little chance of ever reaching it, even for brief periods of time. The 16" is, evidently, affected the most. The D/ro ratio for its x2 error level is over 10, resulting in clearly developed speckle structure. Note that the magnification shown is over 1000x per inch of aperture, or roughly 10 to 50 times more than practical limits for 2"-16" aperture range, respectively. At given nominal magnification, actual (apparent) blur size would be smaller inversely to the aperture size. It would bring the x2 blur in the 16" close to that in 2" aperture (but it is obvious how a further deterioration in seeing quality would affect the 16" more). Eugène Michel Antoniadi (1870 –, 1944) The scale, invented by Eugène Antoniadi, a Greek astronomer, is on a 5 point system, with one being the best seeing conditions and 5 being worst. The actual definitions are as follows: I. Perfect seeing, without a quiver. II. Slight quivering of the image with moments of calm lasting several seconds. III. Moderate seeing with larger air tremors that blur the image. IV. Poor seeing, constant troublesome undulations of the image. V. Very bad seeing, hardly stable enough to allow a rough sketch to be made. Image Degradation Caused by Atmospheric Turbulence SOLO Optics
91      iiyx yxPSFFTAmplitudeffMTF ,,              yxWi eFTyxPSF , 2 ,   Point Spread Function SOLO Optics
92             yxWi eFTyxPSF , 2 ,        iiyx yxPSFFTAmplitudeffMTF ,,  Point Spread Function SOLO Optics
93      iiyx yxPSFFTAmplitudeffMTF ,,              yxWi eFTyxPSF , 2 ,   Point Spread Function SOLO Optics
94 SOLO converging beam = spherical wavefront parallel beam = plane wavefront Image Plane Ideal Optics ideal wavefront parallel beam = plane wavefront Image Plane Non-ideal Optics defocused wavefront ideal wavefrontparallel beam = plane wavefront Image Plane Non-ideal Optics aberrated beam = iregular wavefront diverging beam = spherical wavefront aberrated beam = irregular wavefront Image Plane Non-ideal Optics ideal wavefront Optical Aberration See full development in P.P. “Optical Aberration”
95 SOLO converging beam = spherical wavefront parallel beam = plane wavefront Image Plane Ideal Optics P' Optical Aberration converging beam = spherical wavefront Image Plane Ideal Optics diverging beam = spherical wavefront P P' An Ideal Optical System can be defined by one of the three different and equivalent ways: All the rays emerging from a point source P, situated at a finite or infinite distance from the Optical System, will intersect at a common point P’, on the Image Plane. 1 All the rays emerging from a point source P will travel the same Optical Path to reach the image point P’. 2 The wavefront of light, focused by the Optical System on the Image Plane, has a perfectly spherical shape, with the center at the Image point P. 3 Ideal Optical System
96 SOLO ideal wavefrontparallel beam = plane wavefront Image Plane Non-ideal Optics aberrated beam = iregular wavefront diverging beam = spherical wavefront aberrated beam = irregular wavefront Image Plane Non-ideal Optics ideal wavefront Optical Aberration Real Optical System An Aberrated Optical System can be defined by one of the three different and equivalent ways: The rays emerging from a point source P, situated at a finite or infinite distance from the Optical System, do not intersect at a common point P’, on the Image Plane. 1 The rays emerging from a point source P will not travel the same Optical Path to reach the Image Plane 2 The wavefront of light, focused by the Optical System on the Image Plane, is not spherical. 3
97 Optical Aberration W (x,y) is the path deviation between the distorted and reference Wavefront. SOLO Optical Aberration
98 SOLO Optical Aberration Display of Optical Aberration W (x,y) Rays Deviation1 Optical Path Length Difference2 wavefront shape W (x,y)3 x y x x y  yxW , y  yxW , Red circle denotes the pupile margin. Arrows shows how each ray is deviated as it emerges from the pupil plane. Each of the vectors indicates the the local slope of W (x,y). The aberration W (x,y) is represented in x,y plane by color contours. Wavefront Error Optical Distance Errors Ray Errors The Wavefront error agrees with Optical Path Length Difference, But has opposite sign because a long (short) optical path causes phase retardation (advancement). Aberration Type: Negative vertical coma Reference
99 SOLO Optical Aberration Display of Optical Aberration W (x,y) Advanced phase <= Short optical path Retarded phase <= Long optical path Reference Ectasia x y y  yxW , x x y  yxW , Ray Errors Optical Distance Errors Wavefront Error
100 Optical AberrationSOLO Real Imaging Systems Start from the idealized conditions of Gaussian Optics. Object Gaussian Image planeExit Pupil (ExP) Optics  00 ,0, zxP   gg zxP ,0,' iz gz y x z Gaussian Image  zyxQ ,,  0,0,0O n We have an Ideal Optical System with the center of the Exit Pupil (ExP) at point O (0,0,0). The Optical Axis (OA) passes through O in the z direction. Normal to OA we defined the Cartezian coordinates x,y. (x,z) is the tangential (meridional) plane and (y,z) the sagittal plane defined by P and OA.  00 ,0, zxP  Object  0,0,0O Center of ExP  gg zxP ,0,' Gaussian Image gzz  Gaussian Image plane 'POP Chief Ray 'PQP General Ray  zyxQ ,, The Gaussian Image is obtained from rays starting at the Object P that passing through the Optics intersect at P’.   ':' QPnPQnPQPpathOptical  General Ray Aberrations
101 SOLO Real Imaging Systems 'POP Chief Ray 'PQP General Ray  zyxQ ,, For an idealized system all the optical paths are equal.   ':' QPnPQnPQPpathOptical  General Ray     '' '' OPnPOnPOP QPnPQnPQP                 2/1222/12 0 2 0 2/1222 2/12 0 22 0 gg gg zxnzxn zzyxxn zzyxxn    Object Gaussian Image planeExit Pupil (ExP) Optics  00 ,0, zxP   gg zxP ,0,' iz gz y x z Gaussian Image  zyxQ ,,  0,0,0O n Optical Aberration Aberrations (continue – 1)
102 SOLO Real Imaging Systems Object Gaussian Image planeExit Pupil (ExP) Optics  00 ,0, zxP   gg zxP ,0,' iz Gaussian Spherical Waves center P gz y x z Gaussian Image  zyxQ ,,  0,0,0O Gaussian Spherical Waves center P' For homogeneous media (n = constant) the velocity of light is constant, therefore the rays starting/arriving from/to a point are perpendicular to the spherical wavefronts. Optical paths from P:      2/12 0 22 0),( zzyxxnQPV       2/1222 )',( gg zzyxxnPQV  Optical paths to P’: Rays from P:              2/12 0 22 0 00 ,,),( ˆ , 1 zzyxx zzzyyxxx QPV n s zyxQP       Rays to P’:              2/1222 ,,)',( ˆ ', 1 gg gg zyxPQ zzyxx zzzyyxxx PQV n s       Optical Aberration Aberrations (continue – 2)
103 SOLO Real Imaging Systems Departures from the idealized conditions of Gaussian Optics in a real Optical System are called Aberrations  00 ,0, zxP  Object  0,0,0O Center of ExP  gg zxP ,0,' Gaussian Image gzz  Gaussian Image plane The aberrated image of P in the Gaussian Image plane is  gii zyxP ,," Define the Reference Gaussian Sphere having the center at P’ and passing through O: 022222  gg zzxxzyx P” is the intersection of rays normal to the Aberrated Wavefront that passes trough point O (OP” is a Chief Ray). Choose any point on the Aberrated Wavefront. The Ray intersects the Reference Gaussian Sphere at Q (x, y, z). Q "PQ Object Gaussian Image planeExit Pupil (ExP) Optics  00 ,0, zxP   zyxQ ,,  gg zxP ,0,'  gii zyxP ,," iy iz Reference Gaussian Sphere center P' Aberrated Wavefront center P"  0,0,0O gz y x Q z Gaussian Image Aberrated Image ChiefRay Chief Ray Optical Aberration Aberrations (continue – 3)
104 SOLO Real Imaging Systems Object Gaussian Image planeExit Pupil (ExP) Optics  00 ,0, zxP   zyxQ ,,  gg zxP ,0,'  gii zyxP ,," iy iz Reference Gaussian Sphere center P' Aberrated Wavefront center P"  0,0,0O gz y x Q z Gaussian Image Aberrated Image ChiefRay Chief Ray Choose any point on the Aberrated Wavefront. The Ray intersects the Reference Gaussian Sphere at Q (x, y, z). Q "PQ    QPVQPVW ,,  By definition of the wavefront, the optical path length of the ray starting at the object P and ending at is identical to that of the Chief Ray ending at O. Q Therefore the Wave Aberration is defined as the difference in the optical paths from P to Q V (P,Q) to that from P to  QPVQ ,, Define the Optical Path from P(x0,0,-z0) to Q (x,y,z) as:         zyxQ zxP raydnQPV ,, ,0, 00 :, Since by definition:    OPVQPV ,,              zyxQWOzxPVzyxQzxPVW ,,0,0,0,,0,,,,,0, 0000  Since Q (x,y,z) is constraint on the Reference Guidance Sphere: we can assume that z is a function of x and y, and 022222  gg zzxxzyx             0,0,0,,0,,,,,,0,, 0000 OzxPVyxzyxQzxPVyxW  Optical Aberration Aberrations (continue – 4)
105 SOLO Real Imaging Systems Object Gaussian Image planeExit Pupil (ExP) Optics  00 ,0, zxP   zyxQ ,,  gg zxP ,0,'  gii zyxP ,," iy iz Reference Gaussian Sphere center P' Aberrated Wavefront center P"  0,0,0O gz y x Q z Gaussian Image Aberrated Image ChiefRay Chief Ray Given the Wave Aberration function W (x,y) the Gaussian Image P’(xg,0,zg) of P and the point Q (x,y,z) on the Reference Guidance Sphere we want to find the point P”(xi,yi,zg) 022222  gg zzxxzyx             0,0,0,,0,,,,,,0,, 0000 OzxPVyxzyxQzxPVyxW  Solution:           Q x z z yxzyxQPV x yxzyxQPV x yxW           ,,,,,,,,,          2/1222 ,, ,, zzyyxx zzyyxx n z V y V x V gii gii               Compute relative to Q by differentiating relative to x:022222  gg zzxxzyx Q x z          gi g g gi xx R n zz xx zz R n xx R n x yxW       ''' ,   x yxW n R xx gi    ,' In the same way:   y yxW n R yi    ,' The ray from Q to P” is given by (see ): Forward to a 2nd way Optical Aberration Aberrations (continue – 5)
106 SOLO Real Imaging Systems Defocus Aberration Consider an optical system for which the object P, the Gaussian image P’ and the aberrated image P” are on the Optical Axis. The Gaussian Reference Sphere passing through O (center of ExP) has the center at P’. The Aberrated Wavefront Sphere passing through O (center of ExP) has the center at P”. Consider a ray ( on the Aberrated Wavefront Sphere) that intersects the Gaussian Reference Sphere at Q, that is at a distance r from the Optical Axis. Q"PQ     UBBnQQnQQVrW cos/, The Wave Aberration is defined as           12 22 1 22 2 cos '"'" cos RRrRrR U n PPBPPB U n rW Reference Sphere center at P2 O QQ "P'P Aberrated Wavefront center at P1 Exit Pupil (ExP) B r 1R 2R B U Gaussian image plane Optical Axis R Image plane Optical Aberration
107 SOLO Real Imaging Systems Defocus Aberration (continue – 1) Reference Sphere center at P2 O QQ "P'P Aberrated Wavefront center at P1 Exit Pupil (ExP) B r 1R 2R B U Gaussian image plane Optical Axis R Image plane Let make the following assumptions:           12 22 1 22 2 cos '"'" cos RRrRrR U n PPBPPB U n rW 21,1cos RRrU                                                           4 1111 2 82 1 82 1 '"'" cos 4 3 2 3 1 2 21 124 1 4 2 1 2 14 2 4 2 2 2 2 r RR r RR n RR R r R r R R r R r Rn PPBPPB U n rW 1 1682 11 32  x xxx x  Assume: RRRRRR  2112 &   2 2 2 r R Rn rW  we have: Δ R is called the Longitudinal Defocus. Optical Aberration
108 SOLO Real Imaging Systems Defocus Aberration (continue – 2) For a circular exit pupil of radius a we have:   22 2 # 8  dA f Rn W    a R f 2 :# F number: Define: a r : Therefore Where is the peak value of the Defocus Aberration 2 # 8 : f Rn Ad   Reference Sphere center at P2 O QQ "P'P Aberrated Wavefront center at P1 Exit Pupil (ExP) B r 1R 2R B U Gaussian image plane Optical Axis R Image plane a Optical Aberration
109 http://129.7.217.162/VOI/WavefrontCongress/2006/presentations/1ROORDAprincip.pdf    22 , yxAyxW d  Optical Aberration Wave Aberration: Defocus SOLO Real Imaging Systems Defocus Aberration (continue – 3)
110http://129.7.217.162/VOI/WavefrontCongress/2006/presentations/1ROORDAprincip.pdf Optical Aberration Wavefront Errors for Defocus
111 SOLO Real Imaging Systems Wavefront Tilt Aberration Reference Sphere centered at P1 O  zyxQ ,,Q 1P Abrrrated Wavefront center at P2 Exit Pupil (ExP) B r B U ix 2P  Gaussian image plane Tangential plane 1R R R Assume an optical system that has one ore more optical elements tilted and/or decentered. The object P is on the Optical Axes (OA), therefore the Gaussian image P1 is also on OA. Therefore the Gaussian Reference Sphere that passes trough ExP center O has it’s center at P1. P2 is the aberrated image on the Gaussian image plane (that contains P1) is a distance xi from OA. The Aberrated Wavefront that passes through O has it’s center at P2. Therefore for small P1P2 the two surfaces are tilted by an angle β. Consider the ray where:2QPQ  zyxQ ,, on the Gaussian Reference Sphere 02 1 222  xRzyx Q on the Aberrated Wavefront Sphere centered at P2 and radius R. cos1 RR   12 ,0, RxP i the aberrated image  RRxi  sin     sin,cos, rryx  Optical Aberration
112 SOLO Real Imaging Systems Wavefront Tilt Aberration (continue – 1) Reference Sphere centered at P1 O  zyxQ ,,Q 1P Abrrrated Wavefront center at P2 Exit Pupil (ExP) B r B U ix 2P  Gaussian image plane Tangential plane 1R R R We have x W n R Rxi         QQnQQVrW  , The Wave Aberration is n x W     cos 0 rnxnxd x W W x      For a circular exit pupil of radius a we have: a r :    coscos, 1BanW  where: anB :1 Optical Aberration
113 SOLO Real Imaging Systems Departures from the idealized conditions of Gaussian Optics in a real Optical System are called Aberrations Monochromatic Aberrations Chromatic Aberrations • Monochromatic Aberrations Departures from the first order theory are embodied in the five primary aberrations 1. Spherical Aberrations 2. Coma 3. Astigmatism 4. Field Curvature 5. Distortion This classification was done in 1857 by Philipp Ludwig von Seidel (1821 – 1896) • Chromatic Aberrations 1. Axial Chromatic Aberration 2. Lateral Chromatic Aberration Optical Aberration
114 SOLO Real Imaging Systems Optical Aberration
115 SOLO Real Imaging Systems Optical Aberration
116 SOLO Real Imaging Systems Optical Aberration
117 SOLO Real Imaging Systems Seidel Aberrations Consider a spherical surface of radius R, with an object P0 and the image P0’ on the Optical Axis. n 'n CB R 0P '0P  ,rQ 0V r z  s 's Chief Ray General Ray AS Enp Exp The Chief Ray is P0 V0 P0’ and a General Ray P0 Q P0’. The Wave Aberration is defined as the difference in the optical path lengths between a General Ray and the Chief Ray.          snsnQPnQPnPVPQPPrW  '''''' 00000000 On-Axis Point Object The aperture stop AS, entrance pupil EnP, and exit pupil ExP are located at the refracting surface. Optical Aberration
118 SOLO Real Imaging Systems Seidel Aberrations (continue – 1) n 'n CB R 0P '0P  ,rQ 0V r z  s 's Chief Ray General Ray AS Enp Exp          2 2 22 11 R r RrRRz Define:   2 2 11 2 2 R r xxf R r x       2/1 1 2 1 '   xxf     2/3 1 4 1 "   xxf     2/5 1 8 3 '"   xxf Develop f (x) in a Taylor series            0"' 6 0" 2 0' 1 0 32 f x f x f x fxf 1 1682 11 32  x xxx x  Rr R r R r R r R r Rz           5 6 3 42 2 2 1682 11 On Axis Point Object From the Figure:   222 rzRR  02 22  rRzz Optical Aberration
119 SOLO Real Imaging Systems Seidel Aberrations (continue – 2) n 'n CB R 0P '0P  ,rQ 0V r z  s 's Chief Ray General Ray AS Enp Exp From the Figure:           2/1 2 2/12 2 2/12222/122 0 212 2 22           z s sR sszsR rsszzrszQP rzRz                   2 4 2 2 1 1682 11 2 1 1 32 z s sR z s sR s x xxx x                                2 3 42 4 2 3 42 2 82 822 1 82 1 3 42 R r R r s sR R r R r s sR s R r R r z                                4 2 2 22/122 0 11 8 111 8 111 2 1 r sRssRR r sR srszQP                                4 2 2 22/122 0 1 ' 1 '8 11 ' 1 8 11 ' 1 2 1 ''' r RssRsR r Rs srzsPQ In the same way: On Axis Point Object Optical Aberration
120 SOLO Real Imaging Systems Seidel Aberrations (continue – 3) n 'n CB R 0P '0P  ,rQ 0V r z  s 's Chief Ray General Ray AS Enp Exp                             4 2 2 2 0 11 8 111 8 111 2 1 r sRssRR r sR sQP                             4 2 2 2 0 1 ' 1 '8 11 ' 1 8 11 ' 1 2 1 '' r RssRsR r Rs sPQ Therefore:       4 22 2 42 000 11 ' 11 ' ' 8 1 82 ' ' ' '''' r sRs n sRs n R rr R nn s n s n snsnQPnQPnrW                                    Since P0’ is the Gaussian image of P0 we have   R nn s n s n     ' ' ' and:   44 22 0 11 ' 11 ' ' 8 1 rar sRs n sRs n rW S                      On Axis Point Object Optical Aberration
121 SOLO Real Imaging Systems Seidel Aberrations (continue – 4) Off-Axis Point Object Consider the spherical surface of radius R, with an object P and its Gaussian image P’ outside the Optical Axis. The aperture stop AS, entrance pupil EnP, and exit pupil ExP are located at the refracting surface. Using ''~ 00 CPPCPP  the transverse magnification     s n s n nn s s n s n nn s Rs Rs h h Mt             ' ' ' ' ' ' ' ''  sn sn nn s s nn nn s s nn Mt      ' ' ' ' ' ' ' ' n 'n CB R 0P '0P  ,rQ 0V r z  s 's Chief Ray GeneralRay AS Enp Exp 'P Undeviated Ray P  h 'h  V Optical Aberration
122 SOLO Real Imaging Systems Seidel Aberrations (continue – 5) Off-Axis Point Object The Wave Aberration is defined as the difference in the optical path lengths between the General Ray and the Undeviated Ray.                  4 0 4 0 '''' '' VVVQa PVPPPVPVPPQP PVPPQPQW S    For the approximately similar triangles VV0C and CP0’P’ we have: CP CV PP VV ''' 0 0 0 0  '' ' '' ' 0 0 0 0 hbh Rs R PP CP CV VV    Rs R b   ' :                      22 11 ' 11 ' ' 8 1 sRs n sRs n aS n 'n CB R 0P '0P  ,rQ 0V r z  s 's Chief Ray GeneralRay AS Enp Exp 'P Undeviated Ray P  h 'h  V Optical Aberration
123 SOLO Real Imaging Systems Seidel Aberrations (continue – 6) Off-Axis Point Object Wave Aberration.        4 0 4 '' VVVQaPVPPQPQW S  Define the polar coordinate (r,θ) of the projection of Q in the plane of exit pupil, with V0 at the origin.  cos'2'cos2 222 0 2 0 2 2 hbrhbrVVrVVrVQ  '0 hbVV            442222 4 0 4 'cos'2' '' hbhbrhbra VVVQaPVPPQPQW S S        cos'4'2cos'4cos'4';, 33222222234 rhbrhbrhbrhbrahrW S  P r 'y 'x n 'n CB R 0P '0P  ,rQ 0V r z  s 's Chief Ray GeneralRay AS Enp Exp 'P Undeviated Ray P  h 'h  V Optical Aberration
124 SOLO Real Imaging Systems Seidel Aberrations (continue – 7) General Optical Systems    cos''cos'cos'';, 33222222234 rhbCrhbCrhbCrhbCrChrW DiFCAsCoSp  A General Optical Systems has more than on Reflecting or Refracting surface. The image of one surface acts as an object for the next surface, therefore the aberration is additive. We must address the aberration in the plane of the exit pupil, since the rays follow straight lines from the plane of the exit pupil. The general Wave Aberration Function is: 1. Spherical Aberrations CoefficientSp C 2. Coma CoefficientCoC 3. Astigmatism CoefficientAsC 4. Field Curvature CoefficientFCC 5. Distortion CoefficientDiC where: n 'n C O 0P '0P  ,rQ 0V r  s 's Chief Ray GeneralRay Exit Pupil Exp 'P Undeviated Ray P  h 'h  ~ Optical Aberration
125 Spherical Aberations B 4 1  r 4 4 1 rBW  O W ( a ) Coma cos' 3 rhFW  W 'hF O ( b ) r  Astigmatism cos' 22 rhCW  W ( c )  2 'hC O r Curvature of Field 2 ' 2 1 hD r 22 ' 2 1 rhDW  O W ( d ) Distortion 3 'hE cos'3 rhEW  W ( e ) r O  SOLO Real Imaging Systems Seidel Aberrations (continue – 8)    cos''cos'cos'';, 32222234 rhCrhCrhCrhCrChrW DiFCAsCoSp  Optical Aberration
126 OpticsSOLO Real Imaging Systems – Aberrations (continue – 1) 1. Spherical Aberrations Longitudinal variation of focus with aperture (the distance of the parallel rays from the Optical Axis. n 'n C A F’ R Paraxial Focus Spherical Aberration
127 SOLO Real Imaging Systems Graphical Explanation of Coma Blur 1 1 2 2 3 3 4 4 Optical Axis 1 Meridional (Tangential) Plane P Image Plane Tangential Rays 1 O Lens A Tangential Rays 1 Chief Ray 1 1 1 2 2 3 3 4 4 Optical Axis 1 Sagittal Plane P Image Plane Sagittal Rays 2 O Lens A 2 Sagittal Rays 2 Chief Ray 2 1 1 2 2 3 3 4 4 Optical Axis 1 P Image Plane Skew Rays 3 O Lens A 2 3 Skew Rays 3 Chief Ray 3 1 1 2 2 3 3 4 4 Optical Axis 1 P Image Plane Skew Rays 4 O Lens A 2 3 4 Skew Rays 4 Chief Ray 4 2. Coma Optical Aberration
128 SOLO Real Imaging Systems 1 2 3 4 P Image Plane O SC SC ST CC 3 Coma Blur Spot Shape Tangential Coma Sagittal Coma 1 1 2 2 3 3 4 4 Optical Axis 1 P Image Plane O Lens A 2 3 4 Coma Image Pattern Chief Ray Tangential Plane Sagittal Plane Graphical Explanation of Coma Blur (continue – 1) 2. Coma (continue – 1) Optical Aberration 1 '1 24 '4 '2 3 '3 0  60 1 Corresponding points on 11 1 33 4 4 2 2 '1 '1 '2 '2 '3'3 '4 '4 0 Points on lens '1 1 1 '1 S P,O
129 OpticsSOLO Real Imaging Systems – Aberrations (continue – 3) 3. Astigmatism Meridional plane Sagittal plane Primary image Secondary image Circle of least confusion Object point Optical System Chief ray SF TF Ray in Sagittal plane Ray in Meridional plane
130 SOLO Real Imaging Systems – Aberrations 4. Field Curvature   222 '';, rhbChrW FCFC  Optical Aberration
131 4. Field Curvature SOLO Optical Aberration
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics
Optics Part II: Diffraction and Geometrical Optics

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Optics Part II: Diffraction and Geometrical Optics

  • 1. 1 OPTICS Part II SOLO HERMELIN Updated: 16.01.10http://www.solohermelin.com
  • 2. 2 Table of Content SOLO OPTICS Maxwell’s Equations Boundary Conditions Electromagnatic Wave Equations Monochromatic Planar Wave Equations Spherical Waveforms Cylindrical Waveforms Energy and Momentum Electrical Dipole (Hertzian Dipole) Radiation Reflections and Refractions Laws Development Using the Electromagnetic Approach IR Radiometric Quantities Physical Laws of Radiometry Geometrical Optics Foundation of Geometrical Optics – Derivation of Eikonal Equation The Light Rays and the Intensity Law of Geometrical Optics The Three Laws of Geometrical Optics Fermat’s Principle (1657) O P T I C S P a r t I
  • 3. 3 Table of Content (continue) SOLO OPTICS Plane-Parallel Plate Prisms Lens Definitions Derivation of Gaussian Formula for a Single Spherical Surface Lens Using Fermat’s Principle Derivation of Gaussian Formula for a Single Spherical Surface Lens Using Snell’s Law Derivation of Lens Makers’ Formula First Order, Paraxial or Gaussian Optics Ray Tracing Matrix Formulation O P T I C S P a r t I
  • 4. 4 Table of Content (continue) SOLO OPTICS Optical Diffraction Fresnel – Huygens’ Diffraction Theory Complementary Apertures. Babinet Principle Rayleigh-Sommerfeld Diffraction Formula Extensions of Fresnel-Kirchhoff Diffraction Theory Phase Approximations – Fresnel (Near-Field) Approximation Phase Approximations – Fraunhofer (Near-Field) Approximation Fresnel and Fraunhofer Diffraction Approximations Fraunhofer Diffraction and the Fourier Transform Fraunhofer Diffraction Approximations Examples Resolution of Optical Systems Optical Transfer Function (OTF) Point Spread Function (PSF) Modulation Transfer Function (MTF) Phase Transfer Function (PTF) Relations between Wave Aberration, Point Spread Function and Modulation Transfer Function Other Metrics that define Image Quality – Srahl Ratio Other Metrics that define Image Quality - Pickering Scale Other Metrics that define Image Quality – Atmospheric Turbulence Fresnel Diffraction Approximations Examples
  • 6. 6 Optical DiffractionSOLO Augustin Jean Fresnel 1788-1827 In 1818 Fresnel, by using Huygens’ concept of secondary wavelets and Young’s explanation of interface, developed the diffraction theory of scalar waves. P 0P Q 1x 0x 1y 0y   Fr  Sr    r  O '  Screen Image plane Source plane 0O 1O Sn1   - Screen Aperture Sn1 - normal to Screen FrPP  0 SrQP  0 rQP   From a source P0 at a distance from a aperture a spherical wavelet propagates toward the aperture:    Srktj S source Q e r A tU   ' '  According to Huygens Principle second wavelets will start at the aperture and will add at the image point P.                   dre rr A Kdre r U KtU rrktj S sourcerkttjQ P S 2/2/' ',',   where:  ',K obliquity or inclination factor    SSS nrnr 11cos&11cos' 11            0',0 max0',0   K K Obliquity factor and π/2 phase were introduced by Fresnel to explain experiences results. Fresnel Diffraction Formula Fresnel took in consideration the phase of each wavelet to obtain: Fresnel – Huygens’ Diffraction Theory Fresnel –Kirchoff Diffraction Formula See full development in P.P. “Diffraction” Table of Content
  • 7. 7 SOLO Fresnel-Kirchhoff Diffraction Theory In 1882 Gustav Kirchhoff, using mathematical foundation, succeeded to show that the amplitude and phases ascribed to the wavelets by Fresnel, by enhancing the Huyghen’s Principle, were a consequence of the wave nature of light. HBED    & For an Homogeneous, Linear and Isotropic Medium where are constant scalars, we have , t E t D H t t H t B E ED HB                           Since we have also tt      t D H      t B E      For Source less Medium                               0& 0 2 2 2 DED EEE t E E       02 2 2     t E E    Maxwell Equations are   eJ t D HA        mBGM   )(   mJ t B EF          eDGE   James C. Maxwell (1831-1879) Gustav Robert Kirchhoff 1824-1887 Optical Diffraction
  • 8. 8 SOLO Fresnel-Kirchhoff Diffraction Theory 0 1 2 2 2 2          U tv Scalar Differential Wave Equation For a monochromatic wave of frequency f ( ω = 2πf ) a solution is:               tjPjPUPtPUtPU  expexpRecos,  Define the phasor       PjPUPU  exp U v U tv 2 2 2 2 2 1         2 2  v f v k   022  UkPhasor Scalar Differential Wave Equation This is the Scalar Helmholtz Differential Equation Hermann von Helmholtz 1821-1894 Boundary Conditions for the Helmholtz Differential Equation: • Dirichlet (U given on the boundary) • Neumann (dU/dn given on the boundary) Johann Peter Gustav Lejeune Dirichlet 1805-1859 Franz Neumann 1798-1895  1 0 11 2 2 2 2 2 2 2 2 2             vE tvt E v E    Vector Differential Wave Equation Optical Diffraction
  • 9. 9 To find the solution of the Scalar Helmholtz Differential Equation we need to use the following: • Scalar Green’s Identity       SV dSGUUGdVGUUG 22 • Green’s Function     SF SF FS rr rrkj rrG       exp ; This Green’s Function is a particular solution of the following Helmholtz Non-homogeneous Differential Equation:      SFFSFSS rrrrGkrrG   4;; 22 SOLO Fresnel-Kirchhoff Diffraction Theory provided that and are continuous in volume V UUU 2 ,,  GGG 2 ,,  Free-Space Green’s Function  n i iSS 1  iS nS dV dSnS  1 V Fr  Sr  F 0r SF rrr   PositionSourcerS  PositionFieldrF    022  Uk Scalar Helmholtz Differential Equation Optical Diffraction
  • 10. 10 SOLO • Scalar Green’s Identities       SV dSGUUGdVGUUG 22 Let start from the Gauss’ Divergence Theorem    SV dSAdVA  Karl Friederich Gauss 1777-1855 where is any vector field (function of position and time) continuous and differentiable in the volume V bounded by the enclosed surface S. Let define . A  UGA     UGUGUGA 2   Then        S Gauss VV dSUGdVUGUGdVUG 2        S Gauss VV dSGUdVGUUGdVGU 2 Subtracting the second equation from the first we obtain First Green’s Identity Second Green’s Identity We have GEORGE GREEN 1793-1841 Fresnel-Kirchhoff Diffraction Theory To find a general solution of the Scalar Helmoltz Differential Equation we need to use the  n i iSS 1  iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr   If we interchange with we obtainG U Optical Diffraction
  • 11. 11 Integral Theorem of Helmholtz and Kirchhoff       F V sF V SS rUdVUrrUGkUGkdVGUUG   442222   Using:      SFFSFSS rrrrGkrrG   4;; 22  n i iSS 1  iS nS dV dSnS  1 V Fr  Sr  F 0r SF rrr   PositionSourcerS  PositionFieldrF  SOLO Fresnel-Kirchhoff Diffraction Theory     0,22  SFS rrUk  From the left side of the Second Scalar Green’s Identity we have:                 SS SS dS n G U n U GdSGUUG     SF SF FS rr rrkj rrG       exp ;Using: we obtain:                                  S SF SF SF SF F dS rr rrkj n U n U rr rrkj rU      expexp 4 1  This is the Integral Theorem of Helmholtz and Kirchhoff that enables to calculate as function of the values of and on the enclosed surface S.nU  /UU Note: This Theorem was developed first by H. von Helmholtz in acoustics. Hermann von Helmholtz 1821-1894 Gustav Robert Kirchhoff 1824-1887 From the right side of the Second Scalar Green’s Identity, using we have:dS n U dSnUdSU SSS     1 Scalar Helmholtz Differential Equation Optical Diffraction
  • 12. 12 Sommerfeld Radiation Conditions SOLO Fresnel-Kirchhoff Diffraction Theory                              SS S F dS n G U n U G dS n G U n U GrU 1 4 1 4 1    P Fr  Sr  r   1S S R Screen Aperture d  Sn1  Sn1 since the condition that the previous integral be finite is:     R Rkj rrG SFS exp ;    Consider the surface of integration  SSS 1 1S - on the screen S - hemisphere with radius R   Gkj R Rkj R kj n G          exp1                        dRUkj n U GdS n G U n U G S 2   1 exp limlim   R Rkj RGR RR 0lim           Ukj n U R R This is Sommerfeld Radiation Conditions  - on the aperture Optical Diffraction
  • 13. 13 is known as optical disturbance. Being a scalar quantity, it cannot accurately represent an electromagnetic field. However, the square of this scalar quantity can be regarded as a measure of the irradiance at a given point. U Sommerfeld Radiation Conditions (continue) SOLO Fresnel-Kirchhoff Diffraction Theory                              SS S F dS n G U n U G dS n G U n U GrU 1 4 1 4 1    0lim           Ukj n U R R This is Sommerfeld Radiation Conditions This implies that: 0 4 1             S dS n G U n U G  and the Integral of Helmholtz and Kirchhoff becomes:                1 4 1 S F dS n G U n U GrU   P Fr  Sr  r   1S S R Screen Aperture d Sn1  Sn1 0P Q Arnold Johannes Wilhelm Sommerfeld 1868 - 1951 Optical Diffraction
  • 14. 14 The Kirchhoff Boundary Conditions SOLO Fresnel-Kirchhoff Diffraction Theory Kirchhoff assumed the following boundary conditions:                dS n G U n U GrU F 4 1 1. The field distribution and its derivative , across the aperture , are the same as in the absence of the screen. U nU  /  2. On the shadowed part of the screen and0 1 S U 0/ 1  S nU The Integral of Helmholtz and Kirchhoff becomes: The field at point P is the superposition of the aperture values 0 U 0/   nU Note: Moreover, mathematically the condition implies0/&0 11  SS nUU 0/&0   nUU However, if the dimensions of the aperture are large relative to the wavelength λ, the integral agrees well with the experiment. P Fr  Sr  r   1S S R Screen Aperture 0P 0,0 1 1     S S n U U     n U U , Q  Sn1  Sn1 S  Kirchhoff boundary conditions are not physical since the presence of the screen changes field values on the aperture and on the screen. Optical Diffraction
  • 15. 15 Fresnel-Kirchhoff Diffraction Formula SOLO Fresnel-Kirchhoff Diffraction Theory                dS n G U n U GrU F 4 1 The Integral of Helmholtz and Kirchhoff: Assume that the aperture is illuminated by a single spherical wave:     S Ssource S r rkjA rU exp                                   SS S Ssource S S S Ssource SSSS S nr r rkjA r kj n r rkjA nrU n rU 11 exp1 1 exp 1       SF SF FS rr rrkj rrG       exp ;                                      S S SF SF S rrr SFSS FS nr r rkj r kj n rr rrkj nrrG n rrG SF 11 exp1 1 exp 1, ,      P Fr  Sr  r   1S S R Screen Aperture 0P 0,0 1 1     S S n U U     n U U , Q  Sn1  Sn1 S  Optical Diffraction
  • 16. 16 Fresnel-Kirchhoff Diffraction Formula SOLO Fresnel-Kirchhoff Diffraction Theory                dS n G U n U GrU F 4 1 The Integral of Helmholtz and Kirchhoff: Assume that the aperture is illuminated by a single spherical wave, and: Srr,               SS S SsourceS nr r rkjA j n rU 11 exp2        r rkj rrG FS exp ;                 S FS nr r rkj j n rrG 11 exp2,    Srr k 1 , 12                                            dS nrnr rr rrkjA jrU SSS s ssource F 2 1111 exp       S Ssource S r rkjA rU exp   P Fr  Sr  r   1S S R Screen Aperture 0P 0,0 1 1     S S n U U     n U U , Q  Sn1  Sn1 S  Optical Diffraction
  • 17. 17 Fresnel-Kirchhoff Diffraction Formula SOLO Fresnel-Kirchhoff Diffraction Theory                                                                   dSK rr rrkj A dS nrnr rr rrkjA jrU S s s source SSS s ssource F     , 2 exp 2 1111 exp                                  SSSS S SSS S nrnr nrnr K 11cos&11cos 2 coscos 2 1111 ,    1. Obliquity or Inclination Factor:     0,0&10,0   SS KK 2. Additional phase π/2 3. The amplitude is scaled by the factor 1/λ (not found in Fresnel derivation) P Fr  Sr  r   1S S R Screen Aperture 0P 0,0 1 1     S S n U U     n U U , Q  Sn1  Sn1 S  We recovered Fresnel Diffraction Formula with: Optical Diffraction
  • 18. 18 Reciprocity Theorem of Helmholtz SOLO Fresnel-Kirchhoff Diffraction Theory                                                                   dSK rr rrkj A dS nrnr rr rrkjA jrU S s s source SSS s ssource F     , 2 exp 2 1111 exp We can see that the Fresnel-Kirchhoff Diffraction Formula is symmetrical with respect to r and rS, i.e. point source and observation point. Therefore we can interchange them and obtain the same relation. This result is called Reciprocity Theorem of Helmholtz. P Fr  Sr  r   1S S R Screen Aperture 0P 0,0 1 1     S S n U U     n U U , Q  Sn1  Sn1 S  Hermann von Helmholtz 1821-1894 Note: This is similar to Lorentz’s Reciprocity Theorem in Electromagnetism. Optical Diffraction
  • 19. 19 Huygens-Fresnel Principle SOLO Fresnel-Kirchhoff Diffraction Theory                           dSK rr rrkj A rU S s s source F    , 2 exp  The Fresnel Diffraction Formula can be rewritten as:         dS r rkj QVrU F exp where:     s s S source r rkj K A QV         2 exp ,    The interpretation of this formula is that each point of a wavefront can be considered as the center of a secondary spherical wave, and those secondary spherical waves interfere to result in the total field, is known as the Huygens-Fresnel Principle. P Fr  Sr  r   1S S R Screen Aperture 0P 0,0 1 1     S S n U U     n U U , Q  Sn1  Sn1 S  Table of Content Optical Diffraction
  • 20. 20 SOLO Consider a diffracting aperture Σ. Suppose that the aperture is divided into two portions Σ 1 and Σ 2 such that Σ = Σ1 + Σ2. The two aperture Σ1 and Σ2 are said to be complementary. Complementary Apertures. Babinet Principle From the Fresnel Diffraction Formula:                     21 dS r rkj QVdS r rkj QV dS r rkj QVrU F expexp exp P Fr  1Sr  1 r  2 1S Screen Apertures 0P 1 Q Sn1 2Sr  2r 1 2Q We can see that the result is the added effect of all complimentary apertures. This is known as Babinet Principle. The result can be very helpful when Σ is a very complicated aperture, that can be decomposed in a few simple apertures. Table of Content Optical Diffraction
  • 21. 21 SOLO The Kirchhoff Diffraction Formula is an approximation since for zero field and normal derivative on any finite surface the field is zero everywhere. This was pointed out by Poincare in 1892 and by Sommerfeld in 1894. The first rigorous solution of a diffraction problem was given by Sommerfeld in 1896 for a two-dimensional case of a planar wave incident on an infinitesimally thin, perfectly conducting half plane. This solution is not given here. Arnold Johannes Wilhelm Sommerfeld 1868 - 1951 Jules Henri Poincaré 1854-1912 Sommerfeld, A. : “Mathematische Theorie der Diffraction”, Math. Ann., 47:317, 1896 translated in english as “Optics, Lectures on Theoretical Physics”, vol. IV, Academic Press Inc., New York, 1954 Rayleigh-Sommerfeld Diffraction Formula Optical Diffraction
  • 22. 22 SOLO Rayleigh-Sommerfeld Diffraction Formula Let start from the Helmholtz and Kirchhoff Integral: P Fr  Sr  r   1S S R Screen Aperture 0P 0,0 1 1     S S n U U     n U U , Q  Sn1  Sn1 S  0'PFr'          SSSSS nnrrr 112'                 1 4 1 S F dS n G U n U GrU   Suppose that the Scalar Green Function is generated not only by P0 located at , but also by a point P’0 located symmetric relative to the screen at         SSSSS nnrrr 112'  Sr  G       SF SF SF SF SF rr rrkj rr rrkj rrG ' 'expexp ,_                  SF SF SF SF SF rr rrkj rr rrkj rrG ' 'expexp ,            or: We have 11 ,, ' SSFSSF rrrr             SSSFSSSF nrrnrr 1'1 11 ,,  0 1,   S G   011 exp1 2 11 ,, _                         S S S nr r rkj r kj n G 0 1,      S n G  0 exp 2 1 1 , ,          S S r rkj G Optical Diffraction
  • 23. 23 SOLO Rayleigh-Sommerfeld Diffraction Formula 1. Start from the Helmholtz and Kirchhoff Integral: P Fr  Sr  r   1S S R Screen Aperture 0P 0,0 1 1     S S n U U     n U U , Q  Sn1  Sn1 S  0'PFr'          SSSSS nnrrr 112'                 1 4 1 S F dS n G U n U GrU         SF SF SF SF SF rr rrkj rr rrkj rrG ' 'expexp ,_           Choose 0 1,   S G   011 exp1 2 11 ,, _                         S S S nr r rkj r kj n G On the shadowed part of the screen and0 1 S U 0/ 1  S nU                             dSnr r rkj rU j dS n G UrU SS k r kj F 11 exp 4 1 /2 1 _    This is Rayleigh-Sommerfeld Diffraction Formula of the first kind SF rrr   Arnold Johannes Wilhelm Sommerfeld 1868 - 1951     S Ssource S r rkjA UrU exp    John William Strutt Lord Rayleigh (1842-1919)                dSnr r rkj r rkjAj rU S S Ssource F 11 expexp   we obtain: Optical Diffraction
  • 24. 24 SOLO Rayleigh-Sommerfeld Diffraction Formula 2. Start from the Helmholtz and Kirchhoff Integral: P Fr  Sr  r   1S S R Screen Aperture 0P 0,0 1 1     S S n U U     n U U , Q  Sn1  Sn1 S  0'PFr'          SSSSS nnrrr 112'                 1 4 1 S F dS n G U n U GrU         SF SF SF SF SF rr rrkj rr rrkj rrG ' 'expexp ,            Choose On the shadowed part of the screen and0 1 S U 0/ 1  S nU SF rrr  0 1,      S n G  0 exp 2 1 1 , ,          S S r rkj G                   dS n U r rkj dS n U GrU F exp 2 1 4 1       S Ssource S r rkjA UrU exp                  SS S SsourceS nr r rkjA j n rU 11 exp2                   dSnr r rkj r rkjAj rU SS S Ssource F 11 expexp   For we obtain: This is Rayleigh-Sommerfeld Diffraction Formula of the second kind Table of Content Optical Diffraction
  • 25. 25 P 0P Q 1x 0x 1y 0y   Sr'  Sr    r  O S  Screen Image plane Source plane 0O 1O Sn1   - Screen Aperture Sn1 - normal to Screen 1r 0r  SSS rn cos11  cos11  rnS z Sn1 'r  Fr  FrPP  0 SrQP  0 rQP   SrOP '0   '1 rOO   SOLO          dS r rkj r rkjAj rU S S Ssource F 2 coscosexpexp    Start with Fresnel-Kirchhoff (or Rayleigh-Sommerfeld) Diffraction Formula 1. If the inclination factor is nearly constant over the aperture   constKK S  , Extensions of Fresnel-Kirchhoff Diffraction Theory         dS r rkj r rkjAKj rU S Ssource F expexp           dS r rkj rU Kj rU SF exp  2. Replace the incident point source wavefront with a general waveform   S S r rkjexp  Sinc rU  3. Characterize the aperture by a transfer function τ to model amplitude or phase changes due to optic system           dS r rkj rrU j rU SSF exp   Table of Content Optical Diffraction
  • 26. 26 SOLO Phase Approximations – Fresnel (Near-Field) Approximation Fresnel Approximation or Near Field Approximation can be used when aperture dimensions are comparable to distance to source rS or image r.          dS r rkj r rkjAj rU S S Ssource F 2 coscosexpexp    Start with Fresnel-Kirchhoff Diffraction Formula If the inclination factor is nearly constant over the aperture   constKK S  ,              dS r rkj rU Kj dS r rkj r rkjAKj rU S S Ssource F expexpexp   P Q 1x 1y     r  O  Screen Image plane 1O Sn1  1r  z Sn1 'r  rQP   '1 rOO   P 0P Q 1x 0x 1y 0y   Sr'  Sr    r  O S  Screen Image plane Source plane 0O 1O Sn1   - Screen Aperture Sn1 - normal to Screen 1r 0r  SSS rn cos11  cos11  rnS z Sn1 'r  Fr        1 ''2 ' ' 1''2' ' 2 2 1 2 1 2 11 2/1 2 2 1 2/1 11 0 1 2 1                           r r k r r r r r rrrrrrr rrr x x                                    '2 exp ' 'expexp 2 1 r r kj r rkj r rkj         2 max 2 1 2 1 ' '2 exp ' 'exp rrk dS r r kjrU r rkjKj rU SF                   Augustin Jean Fresnel 1788-1827 Optical Diffraction
  • 27. 27 SOLO Phase Approximations – Fraunhofer (Near-Field) Approximation Fraunhofer Approximation or Far Field Approximation can be used when aperture dimensions are very small comparable to distance to source rS or image r.          dS r rkj r rkjAj rU S S Ssource F 2 coscosexpexp    Start with Fresnel-Kirchhoff Diffraction Formula If the inclination factor is nearly constant over the aperture   constKK S  ,              dS r rkj rU Kj dS r rkj r rkjAKj rU S S Ssource F expexpexp   P Q 1x 1y     r  O  Screen Image plane 1O Sn1  1r  z Sn1 'r  rQP   '1 rOO   P 0P Q 1x 0x 1y 0y   Sr'  Sr    r  O S  Screen Image plane Source plane 0O 1O Sn1   - Screen Aperture Sn1 - normal to Screen 1r 0r  SSS rn cos11  cos11  rnS z Sn1 'r  Fr              ' exp ' 'expexp 1 r r kj r rkj r rkj           2 max 22 1 1 ' 2 ' exp ' 'exp rr k dS r r kjrU r rkjKj rU SF                      1 '2' ' '2' ' ' 2 1''2' ' 2 22 11 2 22 11 2 11 2/1 2 2 1 2 1 2/1 11 0 1 2 1                           r rk r r r r r r r r r rr rrrrrrr rrr x x              Optical Diffraction
  • 28. 28 0P Q 0x 0y  Sr'  Sr    O S ScreenSource plane 0O  Sn10r  SrQP  0 S rOP '0   SOLO Fresnel and Fraunhofer Diffraction Approximations Fresnel Approximations at the Source                                   S S SS S S xx x SS S S SSS SS r r rr r r rr r r rrr rr '2 '1 '2' ' ' '' ' 21' '2' ' 2 282 11 2/12 2 2/122 2                         S S S S S S S r r kjrkj r rkj r rkj '2 '1 exp'1exp ' 'expexp 2 2       S S r rkj ' 'exp    Srkj '1exp            S S r r kj '2 '1 exp 2 2   Spherical wave centered at P0. Lowest order approximation to the phase of a spherical wavefront Planar wave propagating in directionSr'1 P 0P Q 1x 0x 1y 0y   Sr'  Sr    r  O S  Screen Image plane Source plane 0O 1O Sn1   - Screen Aperture Sn1 - normal to Screen 1r 0r  SSS rn cos11  cos11  rnS z Sn1 'r  Fr  Optical Diffraction
  • 29. 29 SOLO Fresnel and Fraunhofer Diffraction Approximations P Q 1x 1y     r  O  Screen Image plane 1O Sn1  1r  z Sn1 'r  rQP   '1 rOO                                         ''2 ' ' 2 1' '2' ' 1 22 1 2 11 2/1 2 2 1 2 1 2/1 11 0 1 2 1 r r r r r r rr r rrrrrr rrr x x                                                        ' exp '2 exp '2 exp ' 'expexp 1 22 1 r r kj r kj r r kj r rkj r rkj   Fresnel Approximations at the Image plane P 0P Q 1x 0x 1y 0y   Sr'  Sr    r  O S  Screen Image plane Source plane 0O 1O Sn1   - Screen Aperture Sn1 - normal to Screen 1r 0r  SSS rn cos11  cos11  rnS z Sn1 'r  Fr    ' 'exp r rkj     '1exp rkj            '2 '1 exp 2 2 r r kj   Spherical wave centered at O. Lowest order approximation to the phase of a spherical wavefront Planar wave propagating in direction'1r Optical Diffraction
  • 30. 30 SOLO Fresnel and Fraunhofer Diffraction Approximations (1st way)            dS r rkj r rkjAj rU SK S S Ssource F        , 2 coscosexpexp Fresnel Approximation                                            dS r rrr r r kjrrkjrrkj rr rrkjKAj rU S S S S Ssource F '2 '1 '2 '1 exp'1'1exp'1exp '' ''exp 2 1 2 1 2 2 1      Fraunhofer Approximation      1 '2 '1 '2 '12 2 1 2 1 2 2     S S k r rrr r r     or S MAX rr ',' 2                 dSrrkjrrkj rr rrkjKAj rU S S Ssource F    '1'1exp'1exp '' ''exp 1 If     1 '2 '1 '2 '1 exp 2 1 2 1 2 2                         S S r rrr r r kj   we obtain Augustin Jean Fresnel 1788-1827   constKK S  , Start with '1'1 rrq S   P 0P Q 1x 0x 1y 0y   Sr'  Sr    r  O S  Screen Image plane Source plane 0O 1O Sn1   - Screen Aperture Sn1 - normal to Screen 1r 0r  SSS rn cos11  cos11  rnS z Sn1 'r  Fr  FrPP  0 SrQP  0 rQP   SrOP '0   '1 rOO   Optical Diffraction
  • 31. 31 SOLO Fresnel and Fraunhofer Diffraction Approximations (2nd way) Fresnel Approximation                      dS r rr kjrrU r rkjj rU SSF '2 exp ' 'exp 11      Fraunhofer Approximation    '1 '2 2 max 2 1 22 1 2 r r r r k        If we obtain Augustin Jean Fresnel 1788-1827 Start with           dS r rkj rrU j rU SSF exp   - aperture optical transfer function Sr   - disturbance at the aperture SrU                        dS r r kjrrU r rkjj rU SSF ' exp ' 'exp 1                                          ' exp '2 exp ' 'expexp 1 2 1 2 r r kj r r kj r rkj r rkj   P Q 1x 1y     r  O  Screen Image plane 1O Sn1  1r  z Sn1 'r  rQP   '1 rOO      1' rrr                                       2 11 2/1 2 11 2/1 11 0 1 2 '2 1' ' 1''2' r rr r r rr rrrrrrr                                    dS r r kj r r kjrrU r rkjj rU SSF ' exp '2 exp ' 'exp 1 2 1 2                                        '2' ' ' 2 1''2' 22 11 2 112/1 2 2 1 2 1 2/1 11 0 1 2 r r r r r r rr rrrrrrr x x   Optical Diffraction
  • 32. 32 SOLO Fresnel and Fraunhofer Diffraction Approximations Augustin Jean Fresnel 1788-1827 1x 1y  max D Screen 1O 1r  z  2 D R  Fresnel Region Fraunhofer Region  2 D R  R  O    '1 '2 2 max 2 1 22 1 2 r r r r k         Fraunhofer Approximation If Optical Diffraction
  • 33. 33 SOLO Fraunhofer Diffraction and the Fourier Transform                       dS r r kjrrU r rkjj rU SSF ' exp ' 'exp 1          11 1 ' 2 ' yx rr r k                               ddyx r jrrU r rkjj rU SSF 11 ' 2 exp ' 'exp  The integral is the two dimensional Fourier Transform of the field within the aperture    SS rrU                 fFTddkkjfkkF yxyx   exp, 2 1 :, 2           SSF rrUFT r rkjj rU    2 2 ' 'exp Therefore P 0P Q 1x 0x 1y 0y   Sr'  Sr    r  O S  Screen Image plane Source plane 0O 1O Sn1   - Screen Aperture Sn1 - normal to Screen 1r 0r  SSS rn cos11  cos11  rnS z Sn1 'r  Fr  FrPP  0 SrQP  0 rQP   SrOP '0   '1 rOO   Two Dimensional Fourier Transform Optical Diffraction
  • 34. 34 SOLO Fraunhofer Diffraction Approximations Examples Rectangular Aperture P Q 1x 1y     r  O  Screen Image plane 1O Sn1 1r  z Sn1 'r 1 2 1 2                                                   1 1 1 1 11 0 2 11 2 10 ' 2 exp ' 2 exp '2 'exp ' exp ' exp '2 exp ' 'exp                 dy r jdx r j r rkjUkj dd r y kj r x kj r r kj r rkjUj rU k F           elsevere U rrU SS 0 & 21110    For a Rectangular Aperture Therefore                             dS r r kjrrU r r kj r rkjj rU SSF ' exp '2 exp ' 'exp 1 2 1                                                11 11 1 1 1111 1 ' 2 ' 2 sin 2 ' 2 ' 2 exp ' 2 exp ' 2 exp 1 1                     x r x r x r j x r jx r j dx r j                                           11 11 1 1 1111 1 ' 2 ' 2 sin 2 ' 2 ' 2 exp ' 2 exp ' 2 exp 1 1                     y r y r y r j y r jy r j dy r j                                   11 11 11 11 4/ 11 0 ' 2 ' 2 sin ' 2 ' 2 sin ' 'exp2              y r y r x r x r r rkjUkj rU A F  Optical Diffraction
  • 35. 35 SOLO Fraunhofer Diffraction Approximations Examples Rectangular Aperture (continue – 1) Since U stands for scalar field intensity (E or H), the irradiance I is given by where < > is the time average and * is the complex conjugate.                                  11 11 11 11 0 ' 2 ' 2 sin ' 2 ' 2 sin ' 'exp8              y r y r x r x r Ar rkjUkj rU F         FFF rUrUrI  ~ Therefore     2 11 11 2 2 11 11 2 ' 2 ' 2 sin ' 2 ' 2 sin 0                                          y r y r x r x r IrI F  I (0) is the irradiance at O1 (x1 = y1 = 0). Hecht pg.466 Optical Diffraction
  • 36. 36 SOLO Fraunhofer Diffraction Approximations Examples Single Slit Aperture Let substitute in the rectangular aperture ξ1 → 0 where < > is the time average and * is the complex conjugate.                                  11 11 11 11 0 ' 2 ' 2 sin ' 2 ' 2 sin ' 'exp8              y r y r x r x r Ar rkjUkj rU F         FFF rUrUrI  ~ Therefore     2 11 11 2 ' 2 ' 2 sin 0                      y r y r IrI F  I (0) is the irradiance at O1 (x1 = y1 = 0). to obtain the single (vertical) slit diffraction                    11 11 0 ' 2 ' 2 sin ' 'exp2        y r y r Ar rkjUkj rU FSLITSINGLE  Since U stands for scalar field intensity (E or H), the irradiance I is given by Hecht, pg. 453 Hecht, pg. 456 Optical Diffraction
  • 37. 37 SOLO Fraunhofer Diffraction Approximations Examples Single Slit Aperture (continue)     2 11 11 2 ' 2 ' 2 sin 0                      y r y r IrI F  I (0) is the irradiance at O1 (x1 = y1 = 0). Hecht, pg. 456 Hecht 455 Define: 11 ' 2 :      y r     2 2 sin 0    II  The extremum of I (β) is obtained from:       0 sincossin2 0 3        I d Id The results are given by: minimum,3,2,0sin   maximum tan The solutions can be obtained graphically as shown in the figure and are: ,4707.3,4590.2,4303.1   Optical Diffraction
  • 38. 38 SOLO Fraunhofer Diffraction Approximations Examples Double Slit Aperture                                                        d r x kjd r x kj r r kj r rkjUj rU ba ba ba ba F 2/ 2/ 1 2/ 2/ 1 2 10 ' exp ' exp '2 exp ' 'exp P Q 1x 1y     r  O  Screen Image plane 1O Sn1 1r  z Sn1 'r  1 b b a                            dS r r kjrrU r r kj r rkjj rU SSF ' exp '2 exp ' 'exp 1 2 1                                                                 ax r j bx r bx r b x r j bax r jbax r j dx r j ba ba 1 1 1 1 112/ 2/ 1 ' exp ' ' sin 1 ' 2 ' exp ' exp ' 2 exp                                                                           ax r j bx r bx r b x r j bax r jbax r j dx r j ba ba 1 1 1 1 112/ 2/ 1 ' exp ' ' sin 1 ' 2 ' exp ' exp ' 2 exp                                                ax r bx r bx r br r kj r rkjUj rU F 1 1 12 10 ' cos ' ' sin 2 '2 exp ' 'exp                                   ax r bx r bx r IrI F 1 2 2 1 1 2 ' cos ' ' sin 0               FFF rUrUrI  ~                  elsevere babababaU rrU SS 0 2/2/&2/2/0    Optical Diffraction
  • 39. 39 SOLO Fraunhofer Diffraction Approximations Examples Double Slit Aperture (continue -= 1) Hecht p.458                2 2 2 12 2 1 12 cos sin 0 ' cos ' ' sin 0 I a r x b r x b r x IrI F                        P Q 1x 1y     r  O  Screen Image plane 1O Sn1 1r  z Sn1 'r  1 b b a The factor (sin β/ β)2 that was previously found as the distribution function for a single slit is here the envelope for the interference fringes given by the term cos2γ. Bright fringes occur for γ = 0,±π ,±2π,… The angular separation between fringes is Δγ = π. Optical Diffraction
  • 40. 40 Hecht 459 SOLO Fraunhofer Diffraction Approximations Examples Double Slit Aperture (continue – 2) Optical Diffraction
  • 41. 41 SOLO Fraunhofer Diffraction Approximations Examples Multiple Slit Aperture P 1y r  Image plane 1O 1r  Q     O  Screen  Sn1 'r  b b a a b b a b a The Aperture consists of a large number N of identical parallel slits of width b and separation a.                                   1 0 2/ 2/ 1 2 10 ' exp '2 exp ' 'exp N k bak bak F d r x kj r r kj r rkjUj rU                                 dS r r kjrrU r r kj r rkjj rU SSF ' exp '2 exp ' 'exp 1 2 1                                                                     akx r j bx r bx r b x r j b kax r j b kax r j dx r j bka bka 1 1 1 1 112/ 2/ 1 ' 2 exp ' ' sin 1 ' 2 2' 2 exp 2' 2 exp ' 2 exp                                                                                                                                           ax r aNx r bx r bx r br r kj r rkjUj ax r j aNx r j bx r bx r br r kj r rkjUj akx r j bx r bx r br r kj r rkjUj rU N k F 1 1 1 12 10 1 1 1 12 10 1 0 1 1 12 10 ' sin ' sin ' ' sin 1 '2 exp ' 'exp ' 2 exp1 ' 2 exp1 ' ' sin 1 '2 exp ' 'exp ' 2 exp ' ' sin 1 '2 exp ' 'exp                                    elsevere NkbkabkaU rrU SS 0 1,,1,02/2/0    Optical Diffraction
  • 42. 42 SOLO Fraunhofer Diffraction Approximations Examples Multiple Slit Aperture (continue – 1) P 1y r  Image plane 1O 1r  Q     O  Screen  Sn1 'r  b b a a b b a b a The Aperture consists of a large number N of identical parallel slits of width b and separation a.                                        ax r aNx r bx r bx r br r kj r rkjUj rU F 1 1 1 12 10 ' sin ' sin ' ' sin 1 '2 exp ' 'exp                             22 2 2 2 1 22 1 2 2 1 1 2 sin sinsin 0 ' sin ' sin ' ' sin 0 N N I ax r N aNx r bx r bx r IrI F                                       FFF rUrUrI  ~ Optical Diffraction
  • 43. 43 SOLO Fraunhofer Diffraction Approximations Examples Multiple Slit Aperture (continue – 2) Hecht p.462 Hecht p.463                   22 2 2 2 1 22 1 2 2 1 1 2 sin sinsin 0 ' sin ' sin ' ' sin 0 N N I ax r N aNx r bx r bx r IrI F                                Optical Diffraction
  • 44. 44 SOLO Fraunhofer Diffraction Approximations Examples Multiple Slit Aperture (continue – 2)                   22 2 2 2 1 22 1 2 2 1 1 2 sin sinsin 0 ' sin ' sin ' ' sin 0 N N I ax r N aNx r bx r bx r IrI F                                Sears p.222 Hecht p. 462 Sears p.236 Interference Irradiation for 1, 2, 3 and 4 slits as function of observation angle. Diffraction Pattern for 1, 2, 3, 4 and 5 slits. Optical Diffraction
  • 45. 45 SOLO Resolution of Optical Systems According to Huygens-Fresnel Principle, a differential area dS, within an optical Aperture, may be envisioned as being covered with coherent secondary point sources. z y   Z Y  R  q  sincos  yz  sincos qYqZ Differential area dS, coordinates Image , coordinates  dddS    dSe r E dE rktiA          The spherical wave that propagates from dS to Image is where           22/122/1222 /1/21 RZzYyRRZzYyRzZyYXr    2/1222 ZYXR                 RkaqJkaqRe R E ddee R E dSee R E dEE RktiA a RkpqiRktiA Aperture RzZyYkiRktiA Aperture // 1 0 2 0 cos//                                The spherical wave at Image, for a Circular Aperture, is Optical Diffraction
  • 46. 46 SOLO Resolution of Optical Systems z y   Z Y  R  q      RkaqJkaqRe R E E RktiA // 1          where           2 0 cos 2 dve i uJ vuvmi m m Bessel Function (of the first kind) E. Hecht, “Optics” The spherical wave at Image, for a Circular Aperture, is Optical Diffraction
  • 47. 47 SOLO Resolution of Optical Systems z y   Z Y  R  q Irradiance     EEHEHESI EH    2 1 2 1 2 1             2 1 2 1 2 22 / /2 0 / /22 2 1               Rkaq RkaqJ I Rkaq RkaqJ R aE EEI A        Daaak RkaquuJ n Rq      22.1 2 22.1 2 83.383.3 sin83.3/0 sin/ 11   D nnn   44.22  E. Hecht, “Optics” Circular Aperture Optical Diffraction
  • 48. 48 SOLO Resolution of Optical Systems z y   Z Y  R  qDistribution of Energy in the Diffraction Pattern at the Focus of a Perfect Circular Lens E. Hecht, “Optics” Ring f/(λf#) Peak Energy in ring Illumination (%) Central max 0 1 83.9 1st dark ring 1.22 0 1st bright ring 1.64 0.017 7.1 2nd dark 2.24 0 2nd bright 2.66 0.0041 2.8 3rd dark 3.24 0 3rd bright 3.70 0.0016 1.5 4th dark 4.24 0 4th bright 4.74 0.00078 1.0 5th dark 5.24 0 Optical Diffraction
  • 49. 49 SOLO Fraunhofer Diffraction Approximations Examples Circular Aperture Hecht p.469 Optical Diffraction
  • 50. 50 SOLO Resolution of Optical SystemsAiry Rings In 1835, Sir George Biddell Airy, developed the formula for diffraction pattern, of an image of a point source in an aberration-free optical system, using the wave theory. E. Hecht, “Optics” Optical Diffraction
  • 51. 51 SOLO Resolution of Optical Systems E. Hecht, “Optics” Optical Diffraction
  • 52. 52 Rayleigh’s Criterion (1902) The images are said to be just resolved when the center of one Airy Disk falls on the first minimum of the Airy pattern of the other image. The minimum resolvable angular separation or angular limit is: D nnn   44.22  Sparrow’s Criterion At the Rayleigh’s limit there is a central minimum Or saddle point between adjacent peaks. Decreasing the distance between the two point sources cause the central dip to grow shallower and ultimately to disappear. The angular separation corresponding to that configuration is the Sparrow’s Limit. SOLO Resolution of Optical Systems Optical Diffraction
  • 53. 53 Resolution – Diffraction Limit Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,University of California, Berkley Optical Diffraction
  • 54. 54 Diffraction limit to resolution of two close point-object images: best resolution is possible when the two are of near equal, optimum intensity. As the two PSF merge closer, the intensity deep between them rapidly diminishes. At the center separation of half the Airy disc diameter - 1.22λ/D radians (138/D in arc seconds, for λ=0.55μ and the aperture diameter D in mm), known as Rayleigh limit - the deep is at nearly 3/4 of the peak intensity. Reducing the separation to λ/D (113.4/D in arc seconds for D in mm, or 4.466/D for D in inches, both for λ=0.55μ) brings the intensity deep only ~4% bellow the peak. This is the conventional diffraction resolution limit, nearly identical to the empirical double star resolution limit, known as Dawes' limit. With even slight further reduction in the separation, the contrast deep disappears, and the two spurious discs merge together. The separation at which the intensity flattens at the top is called Sparrow's limit, given by 107/D for D in mm, and 4.2/D for D in inches (λ=0.55μ). Optical Diffraction
  • 55. 55 SOLO Fresnel Diffraction Approximations Examples Rectangular Aperture                      dS r rr kjrrU r rkjj rU SSF '2 exp ' 'exp 11      define Augustin Jean Fresnel 1788-1827 P Q 1x 1y     r  O  Screen Image plane 1O Sn1 1r  z Sn1 'r 1 2 1 2                                                 2 1 2 1 ' 2 2 exp ' 2 2 exp '2 'exp '2 exp '2 exp ' 'exp 2 1 2 10 2 2 1 2 10                 d r y jd r x j r rkjUkj dd r y kj r x kj r rkjUj rU k F           elsevere U rrU SS 0 & 21110    For a Rectangular Aperture         d r d r x ' 2 ' 2 : 2 12                    2 1 2 1 2 2 1 2 exp 2 ' ' 2 2 exp          dj r d r x j    212111 ' 2 & ' 2       x r x r Therefore         d r d r y ' 2 ' 2 : 2 12       212111 ' 2 & ' 2       y r y r                 2 1 2 1 2 2 1 2 exp 2 ' ' 2 2 exp          dj r d r y j Optical Diffraction
  • 56. 56 SOLO Fresnel Diffraction Approximations Examples Rectangular Aperture (continue – 1) Augustin Jean Fresnel 1788-1827 P Q 1x 1y     r  O  Screen Image plane 1O Sn1 1r  z Sn1 'r 1 2 1 2                                         2 1 2 1 2 1 2 1 220 2 1 2 10 2 exp 2 exp 2 'exp ' 2 2 exp ' 2 2 exp ' 'exp                    djdj rkjUj d r y jd r x j r rkjUj rU F  Define Fresnel Integrals                             0 2 0 2 2 sin: 2 cos: dS dC      SjCdj       0 2 2 exp               2 1 2 1 2 'exp0      SjCSjC rkjUj rU F   Using the Fresnel Integrals we can write     5.0 SC Optical Diffraction
  • 57. 57 SOLO Fresnel Diffraction Approximations Examples Rectangular Aperture (continue – 2) Augustin Jean Fresnel 1788-1827 Hecht p.499 Optical Diffraction
  • 58. 58 SOLO Fresnel Diffraction Approximations Examples Augustin Jean Fresnel 1788-1827 P Q 1x 1y     r  O  Screen Image plane 1O Sn1 1r  z Sn1 'r 1 2 1 2 Optical Diffraction
  • 59. 59 SOLO Fresnel Diffraction Approximations Examples Cornu Spiral Fresnel Integrals are defined as                   uu duuuSduuuC 0 2 0 2 2 sin:& 2 cos:     uSjuCduuj u       0 2 2 exp      5.0 SC Marie Alfred Cornu professor at the École Polytechnique in Paris established a graphical approach, for calculating intensities in Fresnel diffraction integrals. The Cornu Spiral is defined as the plot of S (u) versus C (u) duuSd duuCd               2 2 2 sin 2 cos       duSdCd  22 Therefore u may be thought as measuring arc length along the spiral. “Méthode nouvelle pour la discussion des problèmes de diffraction dans le cas d’une onde cylindrique”, J.Phys.3 (1874), 5-15,44-52 Optical Diffraction
  • 60. 60 SOLO Fresnel Diffraction Approximations Examples Cornu Spiral (continue – 1)                   uu duuuSduuuC 0 2 0 2 2 sin:& 2 cos:     uSjuCduuj u       0 2 2 exp      5.0 SC The Cornu Spiral is defined as the plot of S (u) versus C (u) duuSdduuCd              22 2 sin& 2 cos      duSdCd  22                     2 2 2 2 tan 2 cos 2 sin u u u Cd Sd    Therefore every point on the curve makes the angle with the real ( C ) axis. 2 2 u  The radius of curvature of Cornu Spiral is The tangent vector of Cornu Spiral is SuCuT 1 2 sin1 2 cos 22                   u SuCuu udTdSdCdTd    1 1 2 cos1 2 sin 1 / 1 / 1 22 22                         showing that the curve spirals toward the limit points.        2 1 2 2 cos u u duu         2 1 2 2 sin u u duu         2 1 2 2 exp u u duuj  Table of Content Optical Diffraction
  • 61. 61 SOLO Fresnel Diffraction Approximations Examples Augustin Jean Fresnel 1788-1827 Circular Aperture Hecht p.491 Hecht p.492 Optical Diffraction
  • 62. 62 SOLO Fresnel Diffraction Approximations Examples Augustin Jean Fresnel 1788-1827 Circular Obstacles Optical Diffraction
  • 63. 63 SOLO Fresnel Diffraction Approximations Examples Augustin Jean Fresnel 1788-1827 Fresnel Zone Plate Optical Diffraction
  • 64. 64 SOLO Fresnel Diffraction Approximations Examples Augustin Jean Fresnel 1788-1827 Fresnel Diffraction by a Slit Hecht p.504 a Fresnel Diffraction Hecht p.504 b Optical Diffraction
  • 65. 65 SOLO Fresnel Diffraction Approximations Examples Augustin Jean Fresnel 1788-1827 Semi-Infinite Opaque Screen Hecht p.506 a Hecht p.506 Optical Diffraction
  • 66. 66 SOLO Fresnel Diffraction Approximations Examples Augustin Jean Fresnel 1788-1827 Semi-Infinite Opaque Screen Hecht p.506 a Hecht p.507 Optical Diffraction
  • 68. 68 Point Spread Function (PSF) The Point Spread Function, or PSF, is the image that an Optical System forms of a Point Source. The PSF is the most fundamental object, and forms the basis for any complex object. PSF is the analogous to Impulse Response Function in electronics.    2 , yxPFTPSF  The PSF for a perfect optical system (with no aberration) is the Airy disc, which is the Fraunhofer diffraction pattern for a circular pupil. SOLO Optics
  • 69. 69 Point Spread Function (PSF) As the pupil size gets larger, the Airy disc gets smaller. SOLO Optics
  • 70. 70 Convolution      yxIyxOyxPSF ,,,          yxIyxOFTyxPSFFTFT ,,,1  Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,University of California, Berkley SOLO Optics
  • 71. 71 Modulation Transfer Function (MTF) Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,University of California, Berkley SOLO Optics
  • 72. 72 Modulation Transfer Function (MTF) The Modulation Transfer Function (MTF) indicates the ability of an Optical System to reproduce various levels of details (spatial frequencies) from the object to image. Its units are the ratio of image contrast over the object contrast as a function of spatial frequency.   3.57 a fcutoff Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,University of California, Berkley SOLO Optics
  • 73. 73 Modulation Transfer Function (MTF) Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,University of California, Berkley SOLO Optics
  • 74. 74 Phase Transfer Function (PTF) • PTF contains information about asymmetry in PSF • PTF contains information about contrast reversals (spurious resolution) Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,University of California, Berkley SOLO Optics
  • 75. 75 Relations between Wave Aberration, Point Spread Function and Modulation Transfer Function • The Point Spread Function (PSF) is the Fourier Transform (FT) of the pupil function • The Modulation Transfer Function (MTF) is the amplitude component of the FT of the PSF • The Phase Transfer Function (PTF) is the phase component of the FT of the PSF • The Optical Transfer Function (OTF) composed of MTF and PTF can also be computed as the autocorrelation of the pupil function.               yxWi eyxPFTyxPSF , 2 ,,        iiyx yxPSFFTAmplitudeffMTF ,,       iiyx yxPSFFTPhaseffPTF ,,        yxyxyx ffPTFiffMTFffOTF ,exp,,  SOLO Optics
  • 76. 76 Relations between Wave Aberration, Point Spread Function and Modulation Transfer Function             yxWi eFTyxPSF , 2 ,        iiyx yxPSFFTAmplitudeffMTF ,,       iiyx yxPSFFTPhaseffPTF ,,  Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”, University of California, Berkley SOLO Optics Ideal Optical System
  • 77. 77 Relations between Wave Aberration, Point Spread Function and Modulation Transfer Function             yxWi eFTyxPSF , 2 ,        iiyx yxPSFFTAmplitudeffMTF ,,       iiyx yxPSFFTPhaseffPTF ,,  Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”, University of California, Berkley SOLO Optics Real Optical System
  • 78. 78 Relations between Wave Aberration, Point Spread Function and Modulation Transfer Function             yxWi eFTyxPSF , 2 ,        iiyx yxPSFFTAmplitudeffMTF ,,       iiyx yxPSFFTPhaseffPTF ,,  Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”, University of California, Berkley SOLO Optics Real Optical System
  • 79. 79 FIGURE : PSF and MTF for nominal wavefront errors of selected aberrations resulting in ~0.80 Strehl (~1/14 wave RMS), and the error doubled. (a) the effect of 1/4 and 1/2 wave P-V wavefront error of defocus on the PSF intensity distribution (left) and image contrast (right). Doubling the error nearly halves the peak diffraction intensity, but the average contrast loss nearly triples (evident from the peak PSF intensity). (b) 1/4 and 1/2 wave P-V of spherical aberration. While the peak PSF intensity change is nearly identical to that of defocus, wider energy spread away from the disc results in more of an effect at mid- to high- frequency range. Central disc at 1/2 wave P-V becomes larger, and less well defined. The 1/2 wave curve indicates ~20% lower actual cutoff frequency in field conditions. http://www.telescope-optics.net/ SOLO Optics
  • 80. 80 FIGURE : PSF and MTF for nominal wavefront errors of selected aberrations resulting in ~0.80 Strehl (~1/14 wave RMS), and the error doubled. (c) 0.42 and 0.84 wave P-V wavefront error of coma. Both, intensity distribution (PSF) and contrast transfer change with the orientation angle, due to the asymmetric character of aberration. The worst effect is along the axis of aberration (red), or length-wise with respect to the blur (0 and π orientation angle), and the least is in the orientation perpendicular to it (green). (d) 0.37 and 0.74 wave P-V of astigmatism. Due to the tighter energy spread, there is less of a contrast loss with larger, but more with small details, compared to previous wavefront errors. Contrast is best along the axis of aberration (red), falling to the minimum (green) at every 45° (π/4), and raising back to its peak at every 90°. The PSF is deceiving here: since it is for a linear angular orientation, the energy spread is lowest for the contrast minima. http://www.telescope-optics.net/ SOLO Optics
  • 81. 81 (e) Turned down edge effect on the PSF and MTF. The P-V errors for 95% zone are 2.5 and 5 waves as needed for the initial 0.80 Strehl (the RMS is similarly out of proportion). Lost energy is more evenly spread out, and the central disc becomes enlarged. Odd but expected TE property - due to the relatively small area of the wavefront affected - is that further increase beyond 0.80 Strehl error level does almost no additional damage. f) The effect of ~1/14 and ~1/7 wave RMS wavefront error of roughness, resulting in the peak intensity and contrast drop similar to those with other aberrations. Due to the random nature of the aberration, its nominal P-V wavefront error can vary significantly for a given RMS error and image quality level. Shown is the medium- scale roughness ("primary ripple" or "dog biscuit", in amateur mirror makers' jargon) effect. (g) 0.37 and 0.74 wave P-V of wavefront error caused by pinching having the typical 3-sided symmetry (trefoil). The aberration is radially asymmetric, with the degree of pattern deformation varying between the maxima (red MTF line, for the pupil angle θ=0, 2π/3, 4π/3), and minima (green line, for θ=π/3, π, 5π/3); (the blue line is for a perfect aperture). Other forms do occur, with or without some form of symmetry. FIGURE : PSF and MTF for nominal wavefront errors of selected aberrations resulting in ~0.80 Strehl (~1/14 wave RMS), and the error doubled.
  • 82. 82 h) 0.7 and 1.4 wave P-V wavefront error caused by tube currents starting at the upper 30% of the tube radius. The energy spreads mainly in the orientation of wavefront deformation (red PSF line, to the left). Similarly to the TE, further increase in the nominal error beyond a certain level has relatively small effect Contrast and resolution for the orthogonal to it pattern orientation are as good as perfect (green MTF line). (i) Near-average PSF/MTF effect of ~1/14 and ~1/7 wave RMS wavefront error of atmospheric turbulence. The atmosphere caused error fluctuates constantly, and so do image contrast and resolution level. Larger seeing errors (1/7 wave RMS is rather common with medium-to-large apertures) result in a drop of contrast in the mid- and high- frequency range to near-zero level. FIGURE : PSF and MTF for nominal wavefront errors of selected aberrations resulting in ~0.80 Strehl (~1/14 wave RMS), and the error doubled. http://www.telescope-optics.net/ SOLO Optics
  • 83. 83 Other Metrics that define Image Quality Strehl Ratio Strehl, Karl 1895, Aplanatische und fehlerhafte Abbildung im Fernrohr, Zeitschrift für Instrumentenkunde 15 (Oct.), 362-370. Dr. Karl Strehl 1864 -1940 One of the most frequently used optical terms in both, professional and amateur circles is the Strehl ratio. It is the simplest meaningful way of expressing the effect of wavefront aberrations on image quality. By definition, Strehl ratio - introduced by Dr. Karl Strehl at the end of 19th century - is the ratio of peak diffraction intensities of an aberrated vs. perfect wavefront. The ratio indicates image quality in presence of wavefront aberrations; often times, it is used to define the maximum acceptable level of wavefront aberration for general observing - so-called diffraction-limited level - conventionally set at 0.80 Strehl. SOLO Optics
  • 84. 84 The Strehl ratio is the ratio of the irradiance at the center of the reference sphere to the irradiance in the absence of aberration. Irradiance is the square of the complex field amplitude u 0 E E Strehl  2 uE   dxdyyxWjUu )),(2exp(0  Other Metrics that define Image Quality Strehl Ratio Expectation Notation   dxdy dxdyyxu uu ),( SOLO Optics
  • 85. 85 Derivation of Strehl Approximation  2 0 21 W E E Strehl  ),(2 0 yxWj eUu     22 0 ),(2 2 1 ),(21 yxWyxWjUu     2 0 2 00 ),(2 2 1 ),(2 yxWUyxWUjUu   series expansion     2 0 22 0 2 0 ),(2),(2 yxWEyxWEEE   multiply by complex conjugate 2222 ),(),(),(),( yxWyxWyxWyxWW  wavefront variance: SOLO Optics
  • 86. 86  2 0 21 W E E Strehl  22 2 ),(),( yxWyxWWWW  where W is the wavefront variance:  2 2 W eStrehl  Another approximation for the Strehl ratio is Strehl Approximation Diffraction Limit 8.0Strehl A system is diffraction-limited when the Strehl ratio is greater than or equal to 0.8 Maréchal’s criterion: This implies that the rms wavefront error is less than /13.3 or that the total wavefront error is less than about /4. SOLO Optics
  • 87. 87 Other Metrics that define Image Quality Strehl Ratio dl eye H H RatioStrehl  Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,University of California, Berkley SOLO Optics
  • 88. 88 Other Metrics that define Image Quality Strehl Ratio   2m n Crms when rms is small  2 2 2 1 rmsStrehl          SOLO Optics
  • 89. 89 Other Metrics that define Image Quality FIGURE 34: Pickering's seeing scale uses 10 levels to categorize seeing quality, with the level 1 being the worst and level 10 near-perfect. Its seeing description corresponding to the numerical seeing levels are: 1-2 "very poor", 3 "poor to very poor", 4 "poor", 5 "fair", 6 "fair to good" 7 "good", 8 "good to excellent", 9 "excellent" and 10 "perfect". Diffraction-limited seeing error level (~0.8 Strehl) is between 8 and 9. Pickering 1 Pickering 2 Pickering 3 Pickering 4 Pickering 5 Pickering 6 Pickering 7 Pickering 9 Pickering 10Pickering 8 William H. Pickering (1858-1938) SOLO Optics
  • 90. 90 Other Metrics that define Image Quality FIGURE: Illustration of a point source (stellar) image degradation caused by atmospheric turbulence. The left column shows best possible average seeing error in 2 arc seconds seeing (ro~70mm @ 550nm) for four aperture sizes. The errors are generated according to Eq.53-54, with the 2" aperture error having only the roughness component (Eq.54), and larger apertures having the tilt component added at a rate of 20% for every next level of the aperture size, as a rough approximation of its increasing contribution to the total error (the way it is handled by the human eye is pretty much uncharted territory). The two columns to the right show one possible range of error fluctuation, between half and double the average error. The best possible average RMS seeing error is approximately 0.05, 0.1, 0.2 and 0.4 wave, from top to bottom (the effect would be identical if the aperture was kept constant, and ro reduced). The smallest aperture is nearly unaffected most of the time. The 4" is already mainly bellow "diffraction-limited", while the 8" has very little chance of ever reaching it, even for brief periods of time. The 16" is, evidently, affected the most. The D/ro ratio for its x2 error level is over 10, resulting in clearly developed speckle structure. Note that the magnification shown is over 1000x per inch of aperture, or roughly 10 to 50 times more than practical limits for 2"-16" aperture range, respectively. At given nominal magnification, actual (apparent) blur size would be smaller inversely to the aperture size. It would bring the x2 blur in the 16" close to that in 2" aperture (but it is obvious how a further deterioration in seeing quality would affect the 16" more). Eugène Michel Antoniadi (1870 –, 1944) The scale, invented by Eugène Antoniadi, a Greek astronomer, is on a 5 point system, with one being the best seeing conditions and 5 being worst. The actual definitions are as follows: I. Perfect seeing, without a quiver. II. Slight quivering of the image with moments of calm lasting several seconds. III. Moderate seeing with larger air tremors that blur the image. IV. Poor seeing, constant troublesome undulations of the image. V. Very bad seeing, hardly stable enough to allow a rough sketch to be made. Image Degradation Caused by Atmospheric Turbulence SOLO Optics
  • 91. 91      iiyx yxPSFFTAmplitudeffMTF ,,              yxWi eFTyxPSF , 2 ,   Point Spread Function SOLO Optics
  • 92. 92             yxWi eFTyxPSF , 2 ,        iiyx yxPSFFTAmplitudeffMTF ,,  Point Spread Function SOLO Optics
  • 93. 93      iiyx yxPSFFTAmplitudeffMTF ,,              yxWi eFTyxPSF , 2 ,   Point Spread Function SOLO Optics
  • 94. 94 SOLO converging beam = spherical wavefront parallel beam = plane wavefront Image Plane Ideal Optics ideal wavefront parallel beam = plane wavefront Image Plane Non-ideal Optics defocused wavefront ideal wavefrontparallel beam = plane wavefront Image Plane Non-ideal Optics aberrated beam = iregular wavefront diverging beam = spherical wavefront aberrated beam = irregular wavefront Image Plane Non-ideal Optics ideal wavefront Optical Aberration See full development in P.P. “Optical Aberration”
  • 95. 95 SOLO converging beam = spherical wavefront parallel beam = plane wavefront Image Plane Ideal Optics P' Optical Aberration converging beam = spherical wavefront Image Plane Ideal Optics diverging beam = spherical wavefront P P' An Ideal Optical System can be defined by one of the three different and equivalent ways: All the rays emerging from a point source P, situated at a finite or infinite distance from the Optical System, will intersect at a common point P’, on the Image Plane. 1 All the rays emerging from a point source P will travel the same Optical Path to reach the image point P’. 2 The wavefront of light, focused by the Optical System on the Image Plane, has a perfectly spherical shape, with the center at the Image point P. 3 Ideal Optical System
  • 96. 96 SOLO ideal wavefrontparallel beam = plane wavefront Image Plane Non-ideal Optics aberrated beam = iregular wavefront diverging beam = spherical wavefront aberrated beam = irregular wavefront Image Plane Non-ideal Optics ideal wavefront Optical Aberration Real Optical System An Aberrated Optical System can be defined by one of the three different and equivalent ways: The rays emerging from a point source P, situated at a finite or infinite distance from the Optical System, do not intersect at a common point P’, on the Image Plane. 1 The rays emerging from a point source P will not travel the same Optical Path to reach the Image Plane 2 The wavefront of light, focused by the Optical System on the Image Plane, is not spherical. 3
  • 97. 97 Optical Aberration W (x,y) is the path deviation between the distorted and reference Wavefront. SOLO Optical Aberration
  • 98. 98 SOLO Optical Aberration Display of Optical Aberration W (x,y) Rays Deviation1 Optical Path Length Difference2 wavefront shape W (x,y)3 x y x x y  yxW , y  yxW , Red circle denotes the pupile margin. Arrows shows how each ray is deviated as it emerges from the pupil plane. Each of the vectors indicates the the local slope of W (x,y). The aberration W (x,y) is represented in x,y plane by color contours. Wavefront Error Optical Distance Errors Ray Errors The Wavefront error agrees with Optical Path Length Difference, But has opposite sign because a long (short) optical path causes phase retardation (advancement). Aberration Type: Negative vertical coma Reference
  • 99. 99 SOLO Optical Aberration Display of Optical Aberration W (x,y) Advanced phase <= Short optical path Retarded phase <= Long optical path Reference Ectasia x y y  yxW , x x y  yxW , Ray Errors Optical Distance Errors Wavefront Error
  • 100. 100 Optical AberrationSOLO Real Imaging Systems Start from the idealized conditions of Gaussian Optics. Object Gaussian Image planeExit Pupil (ExP) Optics  00 ,0, zxP   gg zxP ,0,' iz gz y x z Gaussian Image  zyxQ ,,  0,0,0O n We have an Ideal Optical System with the center of the Exit Pupil (ExP) at point O (0,0,0). The Optical Axis (OA) passes through O in the z direction. Normal to OA we defined the Cartezian coordinates x,y. (x,z) is the tangential (meridional) plane and (y,z) the sagittal plane defined by P and OA.  00 ,0, zxP  Object  0,0,0O Center of ExP  gg zxP ,0,' Gaussian Image gzz  Gaussian Image plane 'POP Chief Ray 'PQP General Ray  zyxQ ,, The Gaussian Image is obtained from rays starting at the Object P that passing through the Optics intersect at P’.   ':' QPnPQnPQPpathOptical  General Ray Aberrations
  • 101. 101 SOLO Real Imaging Systems 'POP Chief Ray 'PQP General Ray  zyxQ ,, For an idealized system all the optical paths are equal.   ':' QPnPQnPQPpathOptical  General Ray     '' '' OPnPOnPOP QPnPQnPQP                 2/1222/12 0 2 0 2/1222 2/12 0 22 0 gg gg zxnzxn zzyxxn zzyxxn    Object Gaussian Image planeExit Pupil (ExP) Optics  00 ,0, zxP   gg zxP ,0,' iz gz y x z Gaussian Image  zyxQ ,,  0,0,0O n Optical Aberration Aberrations (continue – 1)
  • 102. 102 SOLO Real Imaging Systems Object Gaussian Image planeExit Pupil (ExP) Optics  00 ,0, zxP   gg zxP ,0,' iz Gaussian Spherical Waves center P gz y x z Gaussian Image  zyxQ ,,  0,0,0O Gaussian Spherical Waves center P' For homogeneous media (n = constant) the velocity of light is constant, therefore the rays starting/arriving from/to a point are perpendicular to the spherical wavefronts. Optical paths from P:      2/12 0 22 0),( zzyxxnQPV       2/1222 )',( gg zzyxxnPQV  Optical paths to P’: Rays from P:              2/12 0 22 0 00 ,,),( ˆ , 1 zzyxx zzzyyxxx QPV n s zyxQP       Rays to P’:              2/1222 ,,)',( ˆ ', 1 gg gg zyxPQ zzyxx zzzyyxxx PQV n s       Optical Aberration Aberrations (continue – 2)
  • 103. 103 SOLO Real Imaging Systems Departures from the idealized conditions of Gaussian Optics in a real Optical System are called Aberrations  00 ,0, zxP  Object  0,0,0O Center of ExP  gg zxP ,0,' Gaussian Image gzz  Gaussian Image plane The aberrated image of P in the Gaussian Image plane is  gii zyxP ,," Define the Reference Gaussian Sphere having the center at P’ and passing through O: 022222  gg zzxxzyx P” is the intersection of rays normal to the Aberrated Wavefront that passes trough point O (OP” is a Chief Ray). Choose any point on the Aberrated Wavefront. The Ray intersects the Reference Gaussian Sphere at Q (x, y, z). Q "PQ Object Gaussian Image planeExit Pupil (ExP) Optics  00 ,0, zxP   zyxQ ,,  gg zxP ,0,'  gii zyxP ,," iy iz Reference Gaussian Sphere center P' Aberrated Wavefront center P"  0,0,0O gz y x Q z Gaussian Image Aberrated Image ChiefRay Chief Ray Optical Aberration Aberrations (continue – 3)
  • 104. 104 SOLO Real Imaging Systems Object Gaussian Image planeExit Pupil (ExP) Optics  00 ,0, zxP   zyxQ ,,  gg zxP ,0,'  gii zyxP ,," iy iz Reference Gaussian Sphere center P' Aberrated Wavefront center P"  0,0,0O gz y x Q z Gaussian Image Aberrated Image ChiefRay Chief Ray Choose any point on the Aberrated Wavefront. The Ray intersects the Reference Gaussian Sphere at Q (x, y, z). Q "PQ    QPVQPVW ,,  By definition of the wavefront, the optical path length of the ray starting at the object P and ending at is identical to that of the Chief Ray ending at O. Q Therefore the Wave Aberration is defined as the difference in the optical paths from P to Q V (P,Q) to that from P to  QPVQ ,, Define the Optical Path from P(x0,0,-z0) to Q (x,y,z) as:         zyxQ zxP raydnQPV ,, ,0, 00 :, Since by definition:    OPVQPV ,,              zyxQWOzxPVzyxQzxPVW ,,0,0,0,,0,,,,,0, 0000  Since Q (x,y,z) is constraint on the Reference Guidance Sphere: we can assume that z is a function of x and y, and 022222  gg zzxxzyx             0,0,0,,0,,,,,,0,, 0000 OzxPVyxzyxQzxPVyxW  Optical Aberration Aberrations (continue – 4)
  • 105. 105 SOLO Real Imaging Systems Object Gaussian Image planeExit Pupil (ExP) Optics  00 ,0, zxP   zyxQ ,,  gg zxP ,0,'  gii zyxP ,," iy iz Reference Gaussian Sphere center P' Aberrated Wavefront center P"  0,0,0O gz y x Q z Gaussian Image Aberrated Image ChiefRay Chief Ray Given the Wave Aberration function W (x,y) the Gaussian Image P’(xg,0,zg) of P and the point Q (x,y,z) on the Reference Guidance Sphere we want to find the point P”(xi,yi,zg) 022222  gg zzxxzyx             0,0,0,,0,,,,,,0,, 0000 OzxPVyxzyxQzxPVyxW  Solution:           Q x z z yxzyxQPV x yxzyxQPV x yxW           ,,,,,,,,,          2/1222 ,, ,, zzyyxx zzyyxx n z V y V x V gii gii               Compute relative to Q by differentiating relative to x:022222  gg zzxxzyx Q x z          gi g g gi xx R n zz xx zz R n xx R n x yxW       ''' ,   x yxW n R xx gi    ,' In the same way:   y yxW n R yi    ,' The ray from Q to P” is given by (see ): Forward to a 2nd way Optical Aberration Aberrations (continue – 5)
  • 106. 106 SOLO Real Imaging Systems Defocus Aberration Consider an optical system for which the object P, the Gaussian image P’ and the aberrated image P” are on the Optical Axis. The Gaussian Reference Sphere passing through O (center of ExP) has the center at P’. The Aberrated Wavefront Sphere passing through O (center of ExP) has the center at P”. Consider a ray ( on the Aberrated Wavefront Sphere) that intersects the Gaussian Reference Sphere at Q, that is at a distance r from the Optical Axis. Q"PQ     UBBnQQnQQVrW cos/, The Wave Aberration is defined as           12 22 1 22 2 cos '"'" cos RRrRrR U n PPBPPB U n rW Reference Sphere center at P2 O QQ "P'P Aberrated Wavefront center at P1 Exit Pupil (ExP) B r 1R 2R B U Gaussian image plane Optical Axis R Image plane Optical Aberration
  • 107. 107 SOLO Real Imaging Systems Defocus Aberration (continue – 1) Reference Sphere center at P2 O QQ "P'P Aberrated Wavefront center at P1 Exit Pupil (ExP) B r 1R 2R B U Gaussian image plane Optical Axis R Image plane Let make the following assumptions:           12 22 1 22 2 cos '"'" cos RRrRrR U n PPBPPB U n rW 21,1cos RRrU                                                           4 1111 2 82 1 82 1 '"'" cos 4 3 2 3 1 2 21 124 1 4 2 1 2 14 2 4 2 2 2 2 r RR r RR n RR R r R r R R r R r Rn PPBPPB U n rW 1 1682 11 32  x xxx x  Assume: RRRRRR  2112 &   2 2 2 r R Rn rW  we have: Δ R is called the Longitudinal Defocus. Optical Aberration
  • 108. 108 SOLO Real Imaging Systems Defocus Aberration (continue – 2) For a circular exit pupil of radius a we have:   22 2 # 8  dA f Rn W    a R f 2 :# F number: Define: a r : Therefore Where is the peak value of the Defocus Aberration 2 # 8 : f Rn Ad   Reference Sphere center at P2 O QQ "P'P Aberrated Wavefront center at P1 Exit Pupil (ExP) B r 1R 2R B U Gaussian image plane Optical Axis R Image plane a Optical Aberration
  • 109. 109 http://129.7.217.162/VOI/WavefrontCongress/2006/presentations/1ROORDAprincip.pdf    22 , yxAyxW d  Optical Aberration Wave Aberration: Defocus SOLO Real Imaging Systems Defocus Aberration (continue – 3)
  • 111. 111 SOLO Real Imaging Systems Wavefront Tilt Aberration Reference Sphere centered at P1 O  zyxQ ,,Q 1P Abrrrated Wavefront center at P2 Exit Pupil (ExP) B r B U ix 2P  Gaussian image plane Tangential plane 1R R R Assume an optical system that has one ore more optical elements tilted and/or decentered. The object P is on the Optical Axes (OA), therefore the Gaussian image P1 is also on OA. Therefore the Gaussian Reference Sphere that passes trough ExP center O has it’s center at P1. P2 is the aberrated image on the Gaussian image plane (that contains P1) is a distance xi from OA. The Aberrated Wavefront that passes through O has it’s center at P2. Therefore for small P1P2 the two surfaces are tilted by an angle β. Consider the ray where:2QPQ  zyxQ ,, on the Gaussian Reference Sphere 02 1 222  xRzyx Q on the Aberrated Wavefront Sphere centered at P2 and radius R. cos1 RR   12 ,0, RxP i the aberrated image  RRxi  sin     sin,cos, rryx  Optical Aberration
  • 112. 112 SOLO Real Imaging Systems Wavefront Tilt Aberration (continue – 1) Reference Sphere centered at P1 O  zyxQ ,,Q 1P Abrrrated Wavefront center at P2 Exit Pupil (ExP) B r B U ix 2P  Gaussian image plane Tangential plane 1R R R We have x W n R Rxi         QQnQQVrW  , The Wave Aberration is n x W     cos 0 rnxnxd x W W x      For a circular exit pupil of radius a we have: a r :    coscos, 1BanW  where: anB :1 Optical Aberration
  • 113. 113 SOLO Real Imaging Systems Departures from the idealized conditions of Gaussian Optics in a real Optical System are called Aberrations Monochromatic Aberrations Chromatic Aberrations • Monochromatic Aberrations Departures from the first order theory are embodied in the five primary aberrations 1. Spherical Aberrations 2. Coma 3. Astigmatism 4. Field Curvature 5. Distortion This classification was done in 1857 by Philipp Ludwig von Seidel (1821 – 1896) • Chromatic Aberrations 1. Axial Chromatic Aberration 2. Lateral Chromatic Aberration Optical Aberration
  • 117. 117 SOLO Real Imaging Systems Seidel Aberrations Consider a spherical surface of radius R, with an object P0 and the image P0’ on the Optical Axis. n 'n CB R 0P '0P  ,rQ 0V r z  s 's Chief Ray General Ray AS Enp Exp The Chief Ray is P0 V0 P0’ and a General Ray P0 Q P0’. The Wave Aberration is defined as the difference in the optical path lengths between a General Ray and the Chief Ray.          snsnQPnQPnPVPQPPrW  '''''' 00000000 On-Axis Point Object The aperture stop AS, entrance pupil EnP, and exit pupil ExP are located at the refracting surface. Optical Aberration
  • 118. 118 SOLO Real Imaging Systems Seidel Aberrations (continue – 1) n 'n CB R 0P '0P  ,rQ 0V r z  s 's Chief Ray General Ray AS Enp Exp          2 2 22 11 R r RrRRz Define:   2 2 11 2 2 R r xxf R r x       2/1 1 2 1 '   xxf     2/3 1 4 1 "   xxf     2/5 1 8 3 '"   xxf Develop f (x) in a Taylor series            0"' 6 0" 2 0' 1 0 32 f x f x f x fxf 1 1682 11 32  x xxx x  Rr R r R r R r R r Rz           5 6 3 42 2 2 1682 11 On Axis Point Object From the Figure:   222 rzRR  02 22  rRzz Optical Aberration
  • 119. 119 SOLO Real Imaging Systems Seidel Aberrations (continue – 2) n 'n CB R 0P '0P  ,rQ 0V r z  s 's Chief Ray General Ray AS Enp Exp From the Figure:           2/1 2 2/12 2 2/12222/122 0 212 2 22           z s sR sszsR rsszzrszQP rzRz                   2 4 2 2 1 1682 11 2 1 1 32 z s sR z s sR s x xxx x                                2 3 42 4 2 3 42 2 82 822 1 82 1 3 42 R r R r s sR R r R r s sR s R r R r z                                4 2 2 22/122 0 11 8 111 8 111 2 1 r sRssRR r sR srszQP                                4 2 2 22/122 0 1 ' 1 '8 11 ' 1 8 11 ' 1 2 1 ''' r RssRsR r Rs srzsPQ In the same way: On Axis Point Object Optical Aberration
  • 120. 120 SOLO Real Imaging Systems Seidel Aberrations (continue – 3) n 'n CB R 0P '0P  ,rQ 0V r z  s 's Chief Ray General Ray AS Enp Exp                             4 2 2 2 0 11 8 111 8 111 2 1 r sRssRR r sR sQP                             4 2 2 2 0 1 ' 1 '8 11 ' 1 8 11 ' 1 2 1 '' r RssRsR r Rs sPQ Therefore:       4 22 2 42 000 11 ' 11 ' ' 8 1 82 ' ' ' '''' r sRs n sRs n R rr R nn s n s n snsnQPnQPnrW                                    Since P0’ is the Gaussian image of P0 we have   R nn s n s n     ' ' ' and:   44 22 0 11 ' 11 ' ' 8 1 rar sRs n sRs n rW S                      On Axis Point Object Optical Aberration
  • 121. 121 SOLO Real Imaging Systems Seidel Aberrations (continue – 4) Off-Axis Point Object Consider the spherical surface of radius R, with an object P and its Gaussian image P’ outside the Optical Axis. The aperture stop AS, entrance pupil EnP, and exit pupil ExP are located at the refracting surface. Using ''~ 00 CPPCPP  the transverse magnification     s n s n nn s s n s n nn s Rs Rs h h Mt             ' ' ' ' ' ' ' ''  sn sn nn s s nn nn s s nn Mt      ' ' ' ' ' ' ' ' n 'n CB R 0P '0P  ,rQ 0V r z  s 's Chief Ray GeneralRay AS Enp Exp 'P Undeviated Ray P  h 'h  V Optical Aberration
  • 122. 122 SOLO Real Imaging Systems Seidel Aberrations (continue – 5) Off-Axis Point Object The Wave Aberration is defined as the difference in the optical path lengths between the General Ray and the Undeviated Ray.                  4 0 4 0 '''' '' VVVQa PVPPPVPVPPQP PVPPQPQW S    For the approximately similar triangles VV0C and CP0’P’ we have: CP CV PP VV ''' 0 0 0 0  '' ' '' ' 0 0 0 0 hbh Rs R PP CP CV VV    Rs R b   ' :                      22 11 ' 11 ' ' 8 1 sRs n sRs n aS n 'n CB R 0P '0P  ,rQ 0V r z  s 's Chief Ray GeneralRay AS Enp Exp 'P Undeviated Ray P  h 'h  V Optical Aberration
  • 123. 123 SOLO Real Imaging Systems Seidel Aberrations (continue – 6) Off-Axis Point Object Wave Aberration.        4 0 4 '' VVVQaPVPPQPQW S  Define the polar coordinate (r,θ) of the projection of Q in the plane of exit pupil, with V0 at the origin.  cos'2'cos2 222 0 2 0 2 2 hbrhbrVVrVVrVQ  '0 hbVV            442222 4 0 4 'cos'2' '' hbhbrhbra VVVQaPVPPQPQW S S        cos'4'2cos'4cos'4';, 33222222234 rhbrhbrhbrhbrahrW S  P r 'y 'x n 'n CB R 0P '0P  ,rQ 0V r z  s 's Chief Ray GeneralRay AS Enp Exp 'P Undeviated Ray P  h 'h  V Optical Aberration
  • 124. 124 SOLO Real Imaging Systems Seidel Aberrations (continue – 7) General Optical Systems    cos''cos'cos'';, 33222222234 rhbCrhbCrhbCrhbCrChrW DiFCAsCoSp  A General Optical Systems has more than on Reflecting or Refracting surface. The image of one surface acts as an object for the next surface, therefore the aberration is additive. We must address the aberration in the plane of the exit pupil, since the rays follow straight lines from the plane of the exit pupil. The general Wave Aberration Function is: 1. Spherical Aberrations CoefficientSp C 2. Coma CoefficientCoC 3. Astigmatism CoefficientAsC 4. Field Curvature CoefficientFCC 5. Distortion CoefficientDiC where: n 'n C O 0P '0P  ,rQ 0V r  s 's Chief Ray GeneralRay Exit Pupil Exp 'P Undeviated Ray P  h 'h  ~ Optical Aberration
  • 125. 125 Spherical Aberations B 4 1  r 4 4 1 rBW  O W ( a ) Coma cos' 3 rhFW  W 'hF O ( b ) r  Astigmatism cos' 22 rhCW  W ( c )  2 'hC O r Curvature of Field 2 ' 2 1 hD r 22 ' 2 1 rhDW  O W ( d ) Distortion 3 'hE cos'3 rhEW  W ( e ) r O  SOLO Real Imaging Systems Seidel Aberrations (continue – 8)    cos''cos'cos'';, 32222234 rhCrhCrhCrhCrChrW DiFCAsCoSp  Optical Aberration
  • 126. 126 OpticsSOLO Real Imaging Systems – Aberrations (continue – 1) 1. Spherical Aberrations Longitudinal variation of focus with aperture (the distance of the parallel rays from the Optical Axis. n 'n C A F’ R Paraxial Focus Spherical Aberration
  • 127. 127 SOLO Real Imaging Systems Graphical Explanation of Coma Blur 1 1 2 2 3 3 4 4 Optical Axis 1 Meridional (Tangential) Plane P Image Plane Tangential Rays 1 O Lens A Tangential Rays 1 Chief Ray 1 1 1 2 2 3 3 4 4 Optical Axis 1 Sagittal Plane P Image Plane Sagittal Rays 2 O Lens A 2 Sagittal Rays 2 Chief Ray 2 1 1 2 2 3 3 4 4 Optical Axis 1 P Image Plane Skew Rays 3 O Lens A 2 3 Skew Rays 3 Chief Ray 3 1 1 2 2 3 3 4 4 Optical Axis 1 P Image Plane Skew Rays 4 O Lens A 2 3 4 Skew Rays 4 Chief Ray 4 2. Coma Optical Aberration
  • 128. 128 SOLO Real Imaging Systems 1 2 3 4 P Image Plane O SC SC ST CC 3 Coma Blur Spot Shape Tangential Coma Sagittal Coma 1 1 2 2 3 3 4 4 Optical Axis 1 P Image Plane O Lens A 2 3 4 Coma Image Pattern Chief Ray Tangential Plane Sagittal Plane Graphical Explanation of Coma Blur (continue – 1) 2. Coma (continue – 1) Optical Aberration 1 '1 24 '4 '2 3 '3 0  60 1 Corresponding points on 11 1 33 4 4 2 2 '1 '1 '2 '2 '3'3 '4 '4 0 Points on lens '1 1 1 '1 S P,O
  • 129. 129 OpticsSOLO Real Imaging Systems – Aberrations (continue – 3) 3. Astigmatism Meridional plane Sagittal plane Primary image Secondary image Circle of least confusion Object point Optical System Chief ray SF TF Ray in Sagittal plane Ray in Meridional plane
  • 130. 130 SOLO Real Imaging Systems – Aberrations 4. Field Curvature   222 '';, rhbChrW FCFC  Optical Aberration
  • 131. 131 4. Field Curvature SOLO Optical Aberration