Fresnel and Fraunhofer Diffraction

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

Interference from multiple apertures

L Bright fringes when OPD=nλ
40
d
OPD x x nLλ

two slits = d Intensity

source

position on screen

screen Complete destructive interference halfway between

OPD 1 OPD 1=nλ, OPD 2=2nλ
OPD 2 all three wave4s0 interfere constructively

d

Intensity

source three position on screen
equally spaced
LASERS 51 slits screen OPD 2=nλ, n odd
outer slits constructively interfere
middle slit gives secondary maxima

April 03

Diffraction from multiple apertures

• Fringes not sinusoidal for 2 slits
more than two 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

Reflection and transmission gratings

• Transmission grating – many closely spaced slits
• Reflection grating – many closely spaced reflecting regions

Input screen
wave
Input path length to
opaque Huygens wave observation point
transmitting
opening wavelets wavelets

path length to absorbing reflecting
observation point
screen Reflection grating

Transmission grating

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Grating equation – transmission

grating with normal incidence

Diffracted

input Θd light sin θ d = pλ

l

• Θd is angle of diffracted ray Except for not making a
• λ is wavelength small angle approximation,
this is identical to formula

• l is spacing between slits for location of maxima in
• p is order of diffraction multiple slit problem earlier

LASERS 51 April 03

Diffraction gratings – general

incidence angle
pλ
• Grating equation sin θ d − sinθi =
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

Intensities of orders – allowed orders

• Diffraction angle can be found only for

certain values of p strong diffracted weak diffracted
order order
– If sin(Θd) is not
between –1 and 1, input
there is no allowed Θd beam

• 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

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
Ggra=ti1lng
– equation can be written in terms of grating
–

constant

sin(Θd ) − sin(Θi ) = pgλ

LASERS 51 April 03

Diffraction grating - applications 2nd
order

• Spectroscopy 1st grating
order
– Separate colors, similar to negative
prism orders

• Laser tuning Littrow mounting – input
and output angles identical
– narrow band mirror

– Select a single line of Θ 2 sin(Θ) = λ
multiline laser
d
– Select frequency in a
tunable laser grating

• Pulse stretching and
compression

– Different colors travel two identical April 03
different path lengths gratings

LASERS 51

Fabry-Perot Interferometer

Input transmitted through

Reflected first mirror Beam is partially reflected and
field partially transmitted at each
mirror

Transmitted All transmitted beams interfere

field with each other

Partially All reflected beams interfere with
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

Fabry-Perot Interferometer

1 free

transmission spectral

range, Linewidth=
fsr fsr*finesse

0 frequency or wavelength April 03

• 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
LASER–S 5N1 arrowing laser line

Diffraction at an aperture—observations

Aperture Light
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

Pattern on screen at various distances

Near Field Intermediate field

2.5mm

Immediately 25 mm from screen, 250 mm 2500 mm
behind screen bright fringes just light penetrates pattern doesn’t
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

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

LASERS–51Non-zero in illuminated area April 03

Fresnel zones

• Incident wave propagating to right

• What is the field at an observation point a b +λ/2 observation
distance of b away? point
First Fresnel
• Start by drawing a sphere with radius zone
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
Fresnel zone arrive at the observation wavefront
point approximately in phase

• Call field amplitude at observation point
due to wavelets in first Fresnel zone, A1

LASERS 51 April 03

Fresnel’s zones – continued

• Divide incident wave into b +λ/2 b+λ observation
additional Fresnel zones by point
drawing circles with radii,
b+2λ/2, b+3λ/2, etc.

• Wavelets from any one zone b
are approximately in phase
at observation point

– out of phase with wavelets from a incident
neighboring zone wavefront

• Each zone has nearly same area

• 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

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

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 b +λ/2 b+λ observation
b point
• This only applies to
intensity on axis

incident
wavefront

Blocking two Fresnel zones gives almost zero
intensity on axis!!

LASERS 51 April 03

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

Fresnel diffraction by circular obstacle—
Arago’s spot

• Construct Fresnel zones just as b observation
before except start with first zone
beginning at edge of aperture point

• Carrying out the same reasoning b+λ/2
as before, we find that the
intensity on axis (in the incident
geometrical shadow) is just what wavefront
it would be in the absence of the
obstacle

• Predicted by Poisson from
Fresnel’s work, observed by
Arago (1818)

LASERS 51 April 03

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,
LASERSbu51t it continues to get larger
April 03

How far is the far field?

z = distance from aperture to observing screen

A = area of aperture Fresnel number
characterizes importance
λ = wavelength of diffraction in any
situation
Fresnel number, F = A

λz

• 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

screen Imaging and diffraction observing
with screen at
aperture Lens Image of aperture image of
plane P

Diffraction pattern

• Image on scatrseomene pilasnei,mP age 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

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

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

Fraunhofer diffraction at a slit Observation
small screen
• Traditional (pre laser) Light source slit
source
setup Collimating Diffracting
lens slit
– source is nearly

monochromatic

• Condenser lens collects f1 f2
light Focusing
lens
Condenser
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

LASsEcRrSe5e1n April 03

Fraunhofer diffraction by slit—zeros
• Wavelets radiate in all
directions
field radiated by
– Point D in focal plane is at
angle Θ from slit, D=Θf wavelets at angle Θ

– Light from each wavelet Θ D = λf
radiated in direction Θ arrives f d
at D λ/2

• Distance travelled is different for λ

each wavelet Slit
• Interference between the light width = d
from all the wavelets gives the
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

Fraunhofer diffraction by slit—complete
pattern

slit

Diffraction pattern,
short exposure time

Diffraction pattern,
longer exposure time

• Evenly spaced zeros April 03

• Central maximum brightest, twice as wide as
others

LASERS 51

Intensity 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

position on screen

LASERS 51 April 03

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

– wdihaemreetder=odfiafimrsettzeerroof=a2p.4e4rtλufr2e/d

– Note: this is 2.44λf/#

– angle=1.22λ/d

LASERS 51 April 03

Resolution of optical systems Observation
screen
• Same optical system Light small
as shown previously source source slit
without diffracting slit
Collimating
lens

– produces image of

source slit on

observing screen f1 f2

– magnification f2/f1 Condenser Focusing
lens 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 April 03

LASERS 51

Resolution of optical systems (cont.)

• With two source Light screen with Observation
slits we can ask the source two source slits screen
question, will we see
two images on the Collimating
lens Diffracting

slit

observation screen

or just a diffraction

pattern? f1 f2

Main lobe of Condenser Focusing
pattern due to lens lens

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

LASdEiRsSti5n1guishable and we only see a diffraction pattern April 03

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 = D = 1.22λf /#

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

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

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

• AintTenEsMity00plaatsteerrnbeam has a Gaussian rather than uniform

– 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

far field divergence half angle,θ = λ
πw0
far field beam radius, w λz
= πw0

LASERS 51 April 03

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

F = πa 2 If index of refraction in
λL laser resonator is not 1,
multiply by n

LASERS 51 April 03