Diffraction. Interference with more than 2 beams. Diffraction gratings. Diffraction by an aperture. Diffraction of a laser beam

Transcription

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

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

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

4 Reflection and transmission gratings Transmission grating many closely spaced slits Reflection grating many closely spaced reflecting regions Input wave opaque transmitting opening Huygens wavelets Input wave screen path length to observation point wavelets screen Transmission grating path length to observation point absorbing reflecting Reflection grating

5 Grating equation transmission grating with normal incidence input Θ d Diffracted light sinθ = d p l λ Θ d is angle of diffracted ray λ is wavelength l is spacing between slits p is order of diffraction Except for not making a small angle approximation, this is identical to formula for location of maxima in multiple slit problem earlier

6 Diffraction gratings general Grating equation incidence angle sinθ sinθ = d i pλ l l=distance between grooves (grating spacing) Θ i =incidence angle (measured from normal) Θ d =diffraction angle (measured from normal) p=integer (order of diffraction) Θ d Θ i 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)

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

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 ( Θ ) pgλ d ) sin i =

9 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 Littrow mounting input and output angles identical Θ two identical gratings 1st order negative orders grating 2sin 2nd order ( Θ) grating = λ d

10 Fabry-Perot Interferometer Input Reflected field transmitted through first mirror Transmitted field Partially reflecting mirrors 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

11 transmission 1 0 Fabry-Perot Interferometer free spectral range, fsr 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 frequency or wavelength Linewidth= fsr*finesse

12 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

13 Pattern on screen at various distances 2.5mm Near Field Intermediate field Immediately behind screen 25 mm from screen, bright fringes just inside edges 250 mm light penetrates into shadow region 2500 mm pattern 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.

14 Huygens-Fresnel diffraction screen with aperture observing screen Point source 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 Wavelets generated in hole Non-zero in illuminated area

15 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, A 1 incident wavefront b + λ/2 First Fresnel zone b observation point

16 Fresnel s zones continued Divide incident wave into additional Fresnel zones by drawing circles with radii, b+2λ/2, b+3λ/2, etc. b +λ/2 b +λ 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 A 2, etc. All zones must add up to the uniform field that we must have at the observation point observation point

17 Adding up contributions from Fresnel zones A 1, the amplitude due to the first zone and A 2, the amplitude from the second zone, are out of phase (destructive interference) A 2 is slightly smaller than A 1 due to area and distance The total amplitude if found by adding contributions of all Fresnel zones A=A 1 -A 2 +A 3 -A 4 + minus signs because the amplitudes are out of phase amplitudes slowly decrease So far this is a complex way of showing an obvious fact.

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 This only applies to intensity on axis b b +λ +λ/2 b observation point incident wavefront Blocking two Fresnel zones gives almost zero intensity on axis!!

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

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 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) incident wavefront b b+λ/2 observation point

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, but it continues to get larger

22 z A = = λ = area of wavelength How far is the far field? distance from aperture to observing screen aperture Fresnel number, F = A λz 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

23 screen with aperture Imaging and diffraction Lens Image of aperture observing screen at image of 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) Diffraction pattern at some plane, P

24 Imaging and far-field diffraction screen with aperture Lens observing screen 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.

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

26 Fraunhofer diffraction at a slit Traditional (pre laser) setup source is nearly monochromatic Condenser lens collects light Light source Condenser lens Source slit creates point source produces spatial coherence at the second slit small source slit Collimating lens 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 f1 Diffracting slit f2 Focusing lens Observation screen

27 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 field radiated by wavelets at angle 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 λ/2 λ Slit width = d Θ f Θ D = λ f d

28 Fraunhofer diffraction by slit complete slit pattern Diffraction pattern, short exposure time Diffraction pattern, longer exposure time Evenly spaced zeros Central maximum brightest, twice as wide as others

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 Intensity Three-slit interference pattern with single-slit diffraction included position on screen

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=2.44λf 2 /d where d= diameter of aperture Note: this is 2.44λf/# angle=1.22λ/d

31 Resolution of optical systems Same optical system as shown previously without diffracting slit produces image of source slit on observing screen magnification f 2 /f 1 Light source Condenser lens small source slit Collimating 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 f1 f2 Observation 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

32 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? Main lobe of pattern due to one slit Light source Condenser lens Observation screen with screen two source slits Collimating lens Diffracting slit f1 Focusing lens Rayleigh criterion-images are just 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 f2

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 1.22λ f minimum resolvable distance = R = = 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. D If geometrical spot size is 2 times size of diffraction spot, then system is 2x diffraction limited, or 2 XDL

34 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

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 TEM 00 laser beam has a Gaussian rather than uniform intensity pattern no edge to measure from so we use 1/e 2 radius, w w o 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 = λz far field beam radius, w = π w 0 λ πw 0

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