1. Diffraction
• Interference with more than 2 beams
– 3, 4, 5 beams
– Large number of beams
• Diffraction gratings
– Equation
– Uses
• Diffraction by an aperture
– Huygen’s principle again, Fresnel zones, Arago’s spot
– Qualitative effects, changes with propagation distance
– Fresnel number again
– Imaging with an optical system, near and far field
– Fraunhofer diffraction of slits and circular apertures
– Resolution of optical systems
• Diffraction of a laser beam
LASERS 51 April 03

2. Interference from multiple apertures
L
Bright fringes when OPD=nλ
40
x
nLλ
x=
Intensity
d
source OPD
d
two slits position on screen
screen Complete destructive interference halfway between
OPD 1 OPD 1=nλ, OPD 2=2nλ
OPD 2 all three waves interfere constructively
40
d
Intensity
source
three position on screen
equally spaced
slits screen
OPD 2=nλ, n odd
outer slits constructively interfere
middle slit gives secondary maxima
LASERS 51 April 03

3. Diffraction from multiple apertures
• Fringes not sinusoidal for
more than two slits 2 slits
• Main peak gets narrower
– Center location obeys same 3 slits
equation
• Secondary maxima appear 4 slits
between main peaks
– The more slits, the more 5 slits
secondary maxima
– The more slits, the weaker the
secondary maxima become
• Diffraction grating – many slits, very narrow spacing
– Main peaks become narrow and widely spaced
– Secondary peaks are too small to observe
LASERS 51 April 03

4. Reflection and transmission gratings
• Transmission grating – many closely spaced slits
• Reflection grating – many closely spaced reflecting regions
Input screen
wave
path length to
observation point
Input
opaque Huygens wave
transmitting wavelets
opening wavelets
path length to
observation point
screen absorbing reflecting
Transmission grating Reflection grating
LASERS 51 April 03

5. Grating equation – transmission
grating with normal incidence
Diffracted
Θd
light
pλ
input sin θ d =
l
• Θd is angle of diffracted ray Except for not making a
small angle approximation,
• λ is wavelength this is identical to formula
for location of maxima in
• l is spacing between slits multiple slit problem earlier
• p is order of diffraction
LASERS 51 April 03

6. Diffraction gratings – general
incidence angle
• Grating equation pλ
sin θ d − sin θ i =
l
l=distance between grooves (grating spacing)
Θi
Θi=incidence angle (measured from normal) Θd
Θd=diffraction angle (measured from normal)
p=integer (order of diffraction)
• Same formula whether it’s a transmission or reflection
grating
– n=0 gives straight line propagation (for transmission grating) or
law of reflection (for reflection grating)
LASERS 51 April 03

7. Intensities of orders – allowed orders
• Diffraction angle can be found only for
certain values of p strong diffracted weak diffracted
– If sin(Θd) is not order order
input
between –1 and 1, beam
there is no allowed Θd
• Intensity of other orders
are different depending
on wavelength, incidence angle,
and construction of grating
• Grating may be blazed to make Blazed grating
a particular order more intense than
others
– angles of orders unaffected by blazing
LASERS 51 April 03

8. Grating constant (groove density) vs.
distance between grooves
• Usually the spacing between grooves for a grating
is not given
– Density of grooves (lines/mm) is given instead
1
– g=
l
– Grating equation can be written in terms of grating
constant
sin(Θ d ) − sin (Θ i ) = pgλ
LASERS 51 April 03

9. 2nd
Diffraction grating - applications order
1st
order
• Spectroscopy grating
– Separate colors, similar to negative
prism orders
• Laser tuning Littrow mounting – input
– narrow band mirror and output angles identical
– Select a single line of λ
Θ 2 sin (Θ ) =
multiline laser d
– Select frequency in a grating
tunable laser
• Pulse stretching and
compression
– Different colors travel
different path lengths two identical
LASERS 51 gratings April 03

10. Fabry-Perot Interferometer
Input transmitted through
first mirror Beam is partially reflected and
partially transmitted at each
mirror
Transmitted All transmitted beams interfere
with each other
Reflected field
All reflected beams interfere with
field Partially each other
reflecting OPD depends on mirror
mirrors separation
• Multiple beam interference – division of amplitude
– As in the diffraction grating, the lines become narrow as
more beams interfere
LASERS 51 April 03

11. Fabry-Perot Interferometer
1 free
spectral
transmission
range, Linewidth=
fsr fsr*finesse
0 frequency or wavelength
• Transmission changes with frequency
– Can be very narrow range where transmission is high
• Width characterized by finesse
• Finesse is larger for higher reflectivity mirrors
– Transmission peaks are evenly spaced
• Spacing called “Free spectral range”
• Controlled by distance between mirrors, fsr=c/(2L)
• Applications
– Measurement of laser linewidth or other spectra
– Narrowing laser line
LASERS 51 April 03

12. Diffraction at an aperture—observations
Light
Aperture through
aperture on
screen
downstream
• A careful observation of the light transmitted by an
aperture reveals a fringe structure not predicted by
geometrical optics
• Light is observed in what should be the shadow region
LASERS 51 April 03

13. Pattern on screen at various distances
Near Field Intermediate field
2.5mm
25 mm from screen, 250 mm 2500 mm
Immediately
bright fringes just light penetrates pattern doesn’t
behind screen
inside edges into shadow closely resemble
region mase
Far field – at a large enough distance
shape of pattern no longer changes but
it gets bigger with larger distance.
Symmetry of original mask still is
evident.
LASERS 51 April 03

14. Huygens-Fresnel diffraction
screen observing
with screen
aperture
Point Wavelets
source generated in
hole
• Each wavelet illuminates the observing screen
• The amplitudes produced by the various waves at the
observing screen can add with different phases
• Final result obtained by taking square of all amplitudes
added up
– Zero in shadow area
– Non-zero in illuminated area
LASERS 51 April 03

15. Fresnel zones
• Incident wave propagating to right
• What is the field at an observation point a b + λ/2
observation
distance of b away? First Fresnel point
zone
• Start by drawing a sphere with radius
b+λ/2
• Region of wave cut out by this sphere is b
the first Fresnel zone
• All the Huygens wavelets in this first incident
wavefront
Fresnel zone arrive at the observation
point approximately in phase
• Call field amplitude at observation point
due to wavelets in first Fresnel zone, A1
LASERS 51 April 03

16. Fresnel’s zones – continued
• Divide incident wave into
additional Fresnel zones by
drawing circles with radii, observation
b+2λ/2, b+3λ/2, etc. b +λ/2 b +λ point
• Wavelets from any one zone
b
are approximately in phase
at observation point
– out of phase with wavelets from a
neighboring zone
incident
• Each zone has nearly same area wavefront
• Field at observation point due to second Fresnel zone
is A2, etc.
• All zones must add up to the uniform field that we must
have at the observation point
LASERS 51 April 03

17. Adding up contributions from Fresnel
zones
• A1, the amplitude due to the first zone and A2, the amplitude
from the second zone, are out of phase (destructive
interference)
– A2 is slightly smaller than A1 due to area and distance
• The total amplitude if found by adding contributions of all
Fresnel zones
A=A1-A2+A3-A4+…
minus signs because the amplitudes are out of phase
amplitudes slowly decrease
So far this is a complex way
of showing an obvious fact.
LASERS 51 April 03

18. Diffraction from circular apertures
• What happens if an aperture the diameter of the
first Fresnel zone is inserted in the beam?
• Amplitude is twice as high
as before inserting aperture!!
– Intensity four times as large
observation
• This only applies to b +λ/2 b +λ point
intensity on axis
b
incident
wavefront
Blocking two Fresnel zones gives almost zero
intensity on axis!!
LASERS 51 April 03

19. Fresnel diffraction by a circular aperture
• Suppose aperture size and observation distance chosen so
that aperture allows just light from first Fresnel zone to pass
– Only the term A1 will contribute
– Amplitude will be twice as large as case with no aperture!
• If distance or aperture size changed so two Fresnel zones are
passed, then there is a dark central spot
– alternate dark and
light spots along
axis
– circular fringes
off the axis
LASERS 51 April 03

20. Fresnel diffraction by circular obstacle—
Arago’s spot
• Construct Fresnel zones just as
before except start with first zone
beginning at edge of aperture
• Carrying out the same reasoning b observation
as before, we find that the point
intensity on axis (in the
geometrical shadow) is just what b+λ/2
it would be in the absence of the
obstacle
• Predicted by Poisson from
incident
Fresnel’s work, observed by wavefront
Arago (1818)
LASERS 51 April 03

21. Character of diffraction for different
locations of observation screen
• Close to diffracting screen (near field)
– Intensity pattern closely resembles shape of aperture, just like
you would expect from geometrical optics
– Close examination of edges reveals some fringes
• Farther from screen (intermediate)
– Fringes more pronounced, extend into center of bright region
– General shape of bright region still roughly resembles
geometrical shadow, but edges very fuzzy
• Large distance from diffracting screen (far field)
– Fringe pattern gets larger
– bears little resemblance to shape of aperture (except symmetries)
– Small features in hole lead to larger features in diffraction pattern
– Shape of pattern doesn’t change with further increase in distance,
LASERS 51 it continues to get larger
but April 03

22. How far is the far field?
z = distance from aperture to observing screen
A = area of aperture Fresnel number
λ = wavelength characterizes importance
A of diffraction in any
Fresnel number, F =
λz situation
• A reasonable rule: F<0.01, the screen is in the far
field
– Depends to some extent on the situation
• F>>1 corresponds to geometrical optics
• Small features in the aperture can be in the far
field even if the entire aperture is not
• Illumination of aperture affects pattern also
LASERS 51 April 03

23. screen Imaging and diffraction observing
with Lens Image of aperture
aperture screen at
image of
plane P
Diffraction pattern
at some plane, P
• Image on screen is image of diffraction pattern at P
– Same pattern as diffraction from a real aperture at image location
except:
• Distance from image to screen modified due to imaging equation
• Magnification of aperture is different from magnification of diffraction
pattern
• Important: for screen exactly at the image plane there is no
diffraction (except for effects introduced by lens aperture)
LASERS 51 April 03

24. Imaging and far-field diffraction
screen Lens observing
with screen
aperture
f
• Looking from the aperture, the observing screen
appears to be located at infinity. Therefore, the
far-field pattern appears on the screen even though
the distance is quite finite.
LASERS 51 April 03

25. Fresnel and Fraunhofer diffraction
• Fraunhofer diffraction = infinite observation distance
– In practice often at focal point of a lens
– If a lens is not used the observation distance must be large
– (Fresnel number small, <0.01)
• Fresnel diffraction must be used in all other cases
• The Fresnel and Fraunhofer regions are used as synonyms
for near field and far field, respectively
– In Fresnel region, geometric optics can be used for the most part;
wave optics is manifest primarily near edges, see first viewgraph
– In Fraunhofer region, light distribution bears no similarity to
geometric optics (except for symmetry!)
– Math in Fresnel region slightly more complicated
• mathematical treatment in either region is beyond the scope of this course
LASERS 51 April 03

26. Fraunhofer diffraction at a slit Observation
small
• Traditional (pre laser) Light source slit screen
source Collimating Diffracting
setup lens slit
– source is nearly
monochromatic
• Condenser lens collects
light f1 f2
Condenser
Focusing
lens
lens
• Source slit creates point source
– produces spatial coherence at the second slit
• Collimating lens images source back to infinity
– laser, a monochromatic, spatially coherent source, replaces all this
• second slit is diffracting aperture whose pattern we want
• Focusing lens images Fraunhofer pattern (at infinity) onto
screen
LASERS 51 April 03

27. Fraunhofer diffraction by slit—zeros
• Wavelets radiate in all
directions field radiated by
– Point D in focal plane is at wavelets at angle Θ
angle Θ from slit, D=Θf
D= λ
– Light from each wavelet f
radiated in direction Θ arrives Θ d
λ/2
at D λ
• Distance travelled is different for
each wavelet Slit
• Interference between the light width = d
from all the wavelets gives the f
diffraction patter
– Zeros can be determined easily
• If Θ=λ/d, each wavelet pairs with one exactly out of phase
– Complete destructive interference
– additional zeros for other multiples of λ, evenly spaced zeros
LASERS 51 April 03

28. Fraunhofer diffraction by slit—complete
pattern
slit
Diffraction pattern,
short exposure time
Diffraction pattern,
longer exposure time
• Evenly spaced zeros
• Central maximum brightest, twice as wide as
others
LASERS 51 April 03

29. Multiple slit diffraction
• In multiple slit patterns discussed earlier, each slit
produces a diffraction pattern
• Result: Multiple slit interference pattern is
superimposed over single slit diffraction pattern
Three-slit interference
pattern with single-slit
diffraction included
Intensity
position on screen
LASERS 51 April 03

30. Fraunhofer diffraction by other apertures
• Rectangular aperture
– Diffraction in each direction is
just like that of a slit
corresponding to width in that
direction
– Narrow direction gives widest
fringes
• Circular aperture
– circular rings
– central maximum brightest
– zeros are not equally spaced
– diameter of first zero=2.44λf2/d
where d= diameter of aperture
– Note: this is 2.44λf/#
– angle=1.22λ/d
LASERS 51 April 03

31. Resolution of optical systems Observation
small screen
• Same optical system Light
source source slit
Collimating
as shown previously lens
without diffracting slit
– produces image of
source slit on
observing screen f1 f2
Condenser
– magnification f2/f1 lens Focusing
lens
• We’ve assumed before that the source slit is very small,
let’s not assume that any more
– each point on source slit gives a point of light on screen
– if we put the diffracting aperture back in, each point gives rise to
its own diffraction pattern, of the diffracting slit
– ideal point image is therefore smeared
LASERS 51 April 03

32. Resolution of optical systems (cont.)
• With two source screen with
Observation
Light screen
slits we can ask the source
two source slits
Collimating
question, will we see lens Diffracting
slit
two images on the
observation screen
or just a diffraction
pattern? Condenser
f1 f2
Focusing
Main lobe of lens
lens
pattern due to Rayleigh criterion-images are just
one slit resolved if minimum of one
coincides with peak of neighbor
• Answer: If the spacing between the images is larger
than the diffraction pattern, then we see images of two
slits, i.e. they are resolved. Otherwise they are not
distinguishable and we only see a diffraction pattern April 03
LASERS 51

33. Resolution of optical systems (cont.)
• Limiting aperture is usually a round aperture stop, so
Rayleigh criterion is found using diffraction pattern of a
round aperture
1.22λf
minimum resolvable distance = R = = 1.22λf /#
D
f= focal length
D=diameter of aperture stop
R= distance spots which are just resolved
Diffraction Limited System: Resolution of an optical system
may be worse than this due to aberrations, ie not all rays
from source point fall on image point. An optical system for
which aberrations are low enough to be negligible
compared to diffraction is a diffraction limited system.
If geometrical spot size is 2 times size of diffraction spot,
LASERS 51 then system is 2x diffraction limited, or 2 XDL April 03

34. Resolution of spots and Rayleigh limit
A
Well resolved
A
Rayleigh limit
A
Slightly closer, are you
sure it’s really two spots?
• At the Rayleigh limit, two spots can be
unambiguously identified, but spots only slightly
closer merge into a blur
LASERS 51 April 03

35. Diffraction of laser beams
• Till now, disscussion has been of uniformly illuminated
apertures
– mathematical diffraction theory can treat non-uniform
illumination and even non-plane waves
• A TEM00 laser beam has a Gaussian rather than uniform
intensity pattern
– no edge to measure from so we use 1/e2 radius, w
– wo is radius where beam is smallest (waist size)
– relatively simple formulae for diffraction apply both in near field
(Fresnel) and far field (Fraunhofer) zones
– only far field result will be presented here
λ
far field divergence half angle,θ =
πw0
λz
far field beam radius, w =
πw0
LASERS 51 April 03

36. Diffraction losses in laser resonators
2a
L
• Light bounces back and forth between mirrors
• Spreads due to diffraction as it propagates
• Some diffracted light misses mirror and is not fed back
• Resonator Fresnel Number measures diffraction losses
If index of refraction in
πa 2
F= laser resonator is not 1,
λL multiply by n
LASERS 51 April 03