April 03LASERS 51

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

April 03LASERS 51

Interference from multiple aperturesBright fringes when OPD=nλ

dnLx λ

=source

two slitsscreen

OPD

d

L

x

Complete destructive interference halfway between

40

position on screen

Inte

nsity

source threeequally spacedslits screen

d

OPD 1

OPD 2 40

position on screen

Inte

nsity

OPD 1=nλ, OPD 2=2nλall three waves interfere constructively

OPD 2=nλ, n oddouter slits constructively interferemiddle slit gives secondary maxima

April 03LASERS 51

Diffraction from multiple apertures• Fringes not sinusoidal for

more than two slits• Main peak gets narrower

– Center location obeys sameequation

• Secondary maxima appearbetween main peaks– The more slits, the more

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

2 slits

3 slits

4 slits

5 slits

April 03LASERS 51

Reflection and transmission gratings

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

Inputwave

Huygenswavelets

screen

path length toobservation point

opaque

transmittingopening

Inputwave

wavelets

screen

path length toobservation point

absorbing reflecting

Transmission grating Reflection grating

April 03LASERS 51

Grating equation – transmission grating with normal incidence

• Θd is angle of diffracted ray• λ is wavelength• l is spacing between slits• p is order of diffraction

Θdinput

Diffracted light

lp

dλθ =sin

Except for not making a small angle approximation, this is identical to formula for location of maxima in multiple slit problem earlier

April 03LASERS 51

Diffraction gratings – general incidence angle

• Grating equationlp

idλθθ =− sinsin

l=distance between grooves (grating spacing)

Θi=incidence angle (measured from normal)

Θd=diffraction angle (measured from normal)

p=integer (order of diffraction)

Θi

Θd

• 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)

April 03LASERS 51

Intensities of orders – allowed orders• Diffraction angle can be found only for

certain values of p– If sin(Θd) is not

between –1 and 1,there is no allowed Θd

• Intensity of other ordersare different dependingon wavelength, incidence angle,and construction of grating

• Grating may be blazed to makea particular order more intense thanothers– angles of orders unaffected by blazing

inputbeam

strong diffractedorder

weak diffractedorder

Blazed grating

April 03LASERS 51

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–– Grating equation can be written in terms of grating

constant

lg 1

=

( ) λpgid =Θ−Θ sin)sin(

April 03LASERS 51

Diffraction grating - applications

• Spectroscopy– Separate colors, similar to

prism

• Laser tuning– narrow band mirror– Select a single line of

multiline laser– Select frequency in a

tunable laser

• Pulse stretching and compression– Different colors travel

different path lengths

grating

Θ

Littrow mounting – input and output angles identical

( )dλ

=Θsin2

grating

1storder

2ndorder

negativeorders

two identicalgratings

April 03LASERS 51

Fabry-Perot InterferometerInput

Reflected field

Transmitted

field

Partially reflecting mirrors

transmitted through first mirror Beam is partially reflected and

partially transmitted at each mirror

All transmitted beams interfere with each other

All reflected beams interfere with each other

OPD depends on mirror separation

• Multiple beam interference – division of amplitude– As in the diffraction grating, the lines become narrow as

more beams interfere

April 03LASERS 51

Fabry-Perot Interferometer

• 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

trans

mis

sion

0

1

frequency or wavelength

free spectral range,

fsrLinewidth=fsr*finesse

April 03LASERS 51

Diffraction at an aperture—observations

Light through aperture on screen downstream

Aperture

• 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

April 03LASERS 51

Pattern on screen at various distances

25 mm from screen, bright fringes just inside edges

250 mmlight penetrates into shadow region

2500 mmpattern doesn’t closely resemble 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.

Intermediate fieldNear Field2.5mm

Immediately behind screen

April 03LASERS 51

Huygens-Fresnel diffraction

• 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

Point source

screen with aperture

observing screen

Wavelets generated in hole

April 03LASERS 51

Fresnel zones

• Incident wave propagating to right• What is the field at an observation point a

distance of b away?• Start by drawing a sphere with radius

b+λ/2• Region of wave cut out by this sphere is

the first Fresnel zone• All the Huygens wavelets in this first

Fresnel zone arrive at the observation point approximately in phase

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

incidentwavefront

observationpoint

b

λ/2b +First Fresnel zone

April 03LASERS 51

Fresnel’s zones – continued • Divide incident wave into

additional Fresnel zones bydrawing circles with radii,b+2λ/2, b+3λ/2, etc.

• Wavelets from any one zoneare approximately in phaseat observation point– out of phase with wavelets from a

neighboring zone• Each zone has nearly same area• Field at observation point due to second Fresnel zone

is A2, etc.

incidentwavefront

observationpoint

b

b b +λ +λ/2

• All zones must add up to the uniform field that we must have at the observation point

April 03LASERS 51

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 phaseamplitudes slowly decrease

So far this is a complex way of showing an obvious fact.

April 03LASERS 51

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

• This only applies tointensity on axis

incidentwavefront

observationpoint

b

b b +λ +λ/2

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

April 03LASERS 51

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 andlight spots alongaxis– circular fringesoff the axis

April 03LASERS 51

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 as before, we find that the intensity on axis (in the geometrical shadow) is just what it would be in the absence of the obstacle

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

incidentwavefront

observationpoint

b

b+λ/2

April 03LASERS 51

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,

but it continues to get larger

April 03LASERS 51

How far is the far field?

zAF number, Fresnel

wavelengthaperture of areaA

screenobservingtoaperturefrom distancez

λ

λ

=

==

=

Fresnel number characterizes importance of diffraction in any 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

April 03LASERS 51

Imaging and diffraction

• 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)

screen with aperture

observing screen at image of plane P

Lens Image of aperture

Diffraction pattern at some plane, P

April 03LASERS 51

Imaging and far-field diffractionscreen with aperture

f

observing screen

Lens

• 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.

April 03LASERS 51

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

April 03LASERS 51

Fraunhofer diffraction at a slit• Traditional (pre laser)

setup– source is nearly

monochromatic

• Condenser lens collects light

• 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

f2

Lightsource

Condenserlens

smallsource slit

Collimatinglens

Focusinglens

Observationscreen

f1

Diffractingslit

April 03LASERS 51

Fraunhofer diffraction by slit—zeros• Wavelets radiate in all

directions– Point D in focal plane is at

angle Θ from slit, D=Θf– Light from each wavelet

radiated in direction Θ arrives at D

• Distance travelled is different for each wavelet

• Interference between the light 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

field radiated bywavelets at angle

λ/2

f

D

Slit width = d

Θ

Θ

= λ fd

λ

April 03LASERS 51

Fraunhofer diffraction by slit—complete pattern

• Evenly spaced zeros• Central maximum brightest, twice as wide as

others

slit

Diffraction pattern, short exposure time

Diffraction pattern, longer exposure time

April 03LASERS 51

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

position on screen

Inte

nsity

Three-slit interference pattern with single-slit diffraction included

April 03LASERS 51

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

April 03LASERS 51

Resolution of optical systems• Same optical system

as shown previously without diffracting slit – produces image of

source slit on observing screen

– magnification f2/f1

f2

Lightsource

Condenserlens

smallsource slit

Collimatinglens

Focusinglens

Observationscreen

f1

• 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 03LASERS 51

Resolution of optical systems (cont.)• With two source

slits we can ask the question, will we see two images on the observation screen or just a diffraction pattern? f2

Lightsource

Condenserlens

Collimatinglens

Focusinglens

Observationscreen

f1

Diffractingslit

screen withtwo source slits

• 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

Main lobe of pattern due to one slit

Rayleigh criterion-images are just resolved if minimum of one coincides with peak of neighbor

April 03LASERS 51

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

/#22.122.1distance resolvable minimum fDfR λλ

===f= focal lengthD=diameter of aperture stopR= 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, then system is 2x diffraction limited, or 2 XDL

April 03LASERS 51

Resolution of spots and Rayleigh limit

A A A

Well resolved Rayleigh limit 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

April 03LASERS 51

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

0

,angle half divergence fieldfar wπλθ =

0

,radius beam fieldfar wzw

πλ

=

April 03LASERS 51

Diffraction losses in laser resonators2a

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 laser resonator is not 1, multiply by nL

aFλπ 2

=