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 x nLλ d x =

d Intensity source OPD two slits position on screen screen Complete destructive interference halfway between

OPD 1 OPD 1=nλ, OPD 2=2nλ OPD 2 all three waves40 interfere constructively

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 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 Reflection and transmission gratings

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

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

path length to observation point screen absorbing reflecting Transmission grating Reflection grating

LASERS 51 April 03 Grating equation – transmission grating with normal incidence Diffracted light Θ pλ input d sinθ = d l

Except for not making a • Θd is angle of diffracted ray small angle approximation, • λ is wavelength this is identical to formula • l is spacing between slits for location of maxima in multiple slit problem earlier • p is order of diffraction

LASERS 51 April 03 Diffraction gratings – general incidence angle pλ • Grating equation sinθ − sinθ = d i l l=distance between grooves (grating spacing) Θi Θ =incidence angle (measured from normal) i 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 – If sin(Θ ) is not order order d 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 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(Θd ) − sin(Θi )= pgλ

LASERS 51 April 03 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 2sin()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 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 field Reflected 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 Fabry-Perot Interferometer 1 free spectral range, Linewidth= fsr fsr*finesse transmission 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 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 Pattern on screen at various distances Near Field Intermediate field 2.5mm

2500 mm Immediately 25 mm from screen, 250 mm pattern doesn’t behind screen bright fringes just light penetrates 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 – Non-zero in illuminated area LASERS 51 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? First Fresnel point • 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 additional Fresnel zones by

drawing circles with radii, observation b +λ b+2λ/2, b+3λ/2, etc. b +λ/2 point • Wavelets from any one zone are approximately in phase b 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 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 observation b +λ • This only applies to b +λ/2 point intensity on axis b

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 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 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, LASERSbut 51 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 λ = wavelength characterizes importance A Fresnel number, F = of diffraction in any 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 Imaging and diffraction screen 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 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 small Observation • Traditional (pre laser) Light source slit screen source setup CollimatingDiffracting lens slit – source is nearly monochromatic • Condenser lens collects light f1 f2 Condenser Focusing lens • Source slit creates point source lens – 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 LASERSscreen 51 April 03 Fraunhofer diffraction by slit—zeros • Wavelets radiate in all directions field radiated by – Point D in focal plane is at wavelets at angleΘ angle Θ from slit, D=Θf

– Light from each wavelet D= λf 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 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 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 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 Resolution of optical systems Observation Light small screen • Same optical system source slit source 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 Resolution of optical systems (cont.)

• With two source Observation screen with screen Light two source slits slits we can ask the source Collimating lens Diffracting question, will we see slit two images on the observation screen or just a diffraction pattern? f1 f2 Condenser lens Focusing Main lobe of 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

LASERSdistinguishable 51 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 = =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 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

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

•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 = πw λz 0 far field beam radius, w = 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 πa2 If index of refraction in F = laser resonator is not 1, λL multiply by n

LASERS 51 April 03