4.4. FresnelFresnel andand FraunhoferFraunhofer diffractiondiffraction Remind!Remind!Diffraction Diffraction underunder paraxialparaxial approx.approx. Remind!Remind!Huygens-Fresnel Huygens-Fresnel principleprinciple “Every unobstructed point of a wavefront, at a given instant in time, serves as a source of secondary wavelets (with the same frequency as that of the primary wave). The amplitude of the optical field at any point beyond is the superposition of all these wavelets (considering their amplitude and relative phase).” Huygens’s principle: By itself, it is unable to account for the details of the diffraction process. It is indeed independent of any wavelength consideration. Fresnel’s addition of the concept of interference Remind!Remind!After After thethe Huygens-FresnelHuygens-Fresnel principleprinciple Fresnel’s shortcomings : He did not mention the existence of backward secondary wavelets, however, there also would be a reverse wave traveling back toward the source. He introduce a quantity of the obliquity factor, but he did little more than conjecture about this kind. Gustav Kirchhoff : Fresnel-Kirhhoff diffraction theory A more rigorous theory based directly on the solution of the differential wave equation. He, although a contemporary of Maxwell, employed the older elastic-solid theory of light. He found K(χ) = (1 + cosθ )/2. K(0) = 1 in the forward direction, K(π) = 0 with the back wave. Arnold Johannes Wilhelm Sommerfeld : Rayleigh-Sommerfeld diffraction theory A very rigorous solution of partial differential wave equation. The first solution utilizing the electromagnetic theory of light. Remind!Remind!Huygens-Fresnel Huygens-Fresnel PrinciplePrinciple revisedrevised byby 11stst RR –– S S solutionsolution 1 exp( jkr ) U P = U P 01 cosθ ds H-F principle ()0 ∫∫ ()1 jλ ∑ r01 1 amplitude ∝ ∝ ν λ 1 phase : (lead the incident phase by 90o ) j U P = h P , P U P ds ( 0 ) ∫∫ ( 0 1 ) ( 1 ) ∑ 1 exp( jkr01 ) h()P0 , P1 = cosθ Impulse response function jλ r01 Each secondary wavelet has a directivity pattern cosθ Huygens-FresnelHuygens-Fresnel PrinciplePrinciple z cos θ = r01 1 exp ( jkr ) r01 U P = U P 01 cos θ ds ()0 ∫∫ ()1 jλ ∑ r01 z exp ( jkr ) U x, y = U ξ ,η 01 dξdη () ∫∫ () 2 jλ ∑ r01 Aperture (ξ,η) Screen (x,y) 2 2 2 r01 = z + ()()x − ξ + y −η Note! So far we have used only two assumptions (R-S solution) : Scalar wave equation + ( r01 >> λ ) FresnelFresnel (paraxial)(paraxial) ApproximationApproximation 2 2 ⎡ 1 ⎛ x −ξ ⎞ 1 ⎛ y −η ⎞ ⎤ r01 ≈ z⎢1+ ⎜ ⎟ + ⎜ ⎟ ⎥ ⎣⎢ 2 ⎝ z ⎠ 2 ⎝ z ⎠ ⎦⎥ jkz ∞ e ⎧ k 2 2 ⎫ U ()x, y = ∫∫U ()ξ,η exp⎨ j [()()x −ξ + y −η ]⎬ dξdη jλz −∞ ⎩ 2z ⎭ 7 Since k is 10 order, the quadratic terms in r01 must be considered in the exponent. jkz k 2 2 ∞ k 2 2 2π e j ()x + y ⎧ j ()ξ +η ⎫ − j ()xξ + yη U()x, y = e 2z ∫∫⎨U()ξ,η e 2z ⎬e λz dξdη jλz −∞⎩ ⎭ jkz k 2 2 k 2 2 e j ()x + y ⎧ j ()ξ +η ⎫ U (x, y) = e 2z F ⎨U ()ξ,η e 2z ⎬ jλz ⎩ ⎭ f X =x / λz, fY = y / λz Fresnel diffraction integral FresnelFresnel (near-field)(near-field) diffractiondiffraction This is most general form of diffraction – No restrictions on optical layout • near-field diffraction k 22 ⎧ j ()ξη+ ⎫ • curved wavefront Uxy(, )≈ F ⎨ U()ξη , e2z ⎬ – Analysis somewhat difficult ⎩⎭ Curved wavefront (diverging wavelets) Screen z Fresnel Diffraction PositivePositive phasephase andand NegativeNegative phasephase Wavefront G y emitted y earlier k exp()jkr01 exp( j2παy) ⎡ k ⎤ exp j ()x2 + y 2 θ ⎢ 2z ⎥ ⎣ ⎦ z=0 z Wavefront z diverging emitted later Wavefront emitted exp()− jkr01 later exp(− j2πα y) ⎡ k 2 2 ⎤ exp⎢− j ()x + y ⎥ ⎣ 2z ⎦ z=0 Wavefront converging emitted earlier AccuracyAccuracy ofof thethe FresnelFresnel ApproximationApproximation 3 π 2 2 2 z 〉〉 [()()x −ξ + y −η ] 4λ max • Accuracy can be expected for much shorter distances for U(,)ξ ηξλ smooth & slow varing function; 2x−=≤ D 4 z D2 z ≥ Fresnel approximation 16λ ImpulseImpulse ResponseResponse andand TransferTransfer FunctionsFunctions ofof FresnelFresnel diffractiondiffraction jkz ∞ e ⎧ k 2 2 ⎫ U ()x, y = ∫∫U ()ξ,η exp⎨ j [()()x −ξ + y −η ]⎬ dξdη jλz −∞ ⎩ 2z ⎭ ∞ U ()x, y = ∫∫U ()(ξ ,η h x − ξ , y − η ) dξdη − ∞ This convolution relation means that Fresnel diffraction is linear shift-invariant. e jkz ⎡ jk ⎤ h()x, y = exp ()x2 + y 2 jλz ⎣⎢2z ⎦⎥ Impulse response function of Fresnel diffraction ImpulseImpulse ResponseResponse andand TransferTransfer FunctionsFunctions ofof FresnelFresnel diffractiondiffraction Hff( XY, ) ≡ F {} hxy(, ) ⎧⎫e jkz ⎡⎤π =+=−+expjxy22 ejkz exp ⎡ jzffπλ 2 2⎤ F ⎨⎬⎢⎥()⎣ (XY )⎦ ⎩⎭jzλλ⎣⎦ z Remind! Transfer function of wave propagation phenomenon in free space ⎡ z 22⎤ εε()()ffzXY,;=−− ff XY ,;0exp2 j πλλ 1()() f X f Y ⎣⎢ λ ⎦⎥ ⎡ z 22⎤ Hf()XY,exp21 f=−− jπλλ()() f X f Y ⎣⎢ λ ⎦⎥ Under the paraxial approximation, the transfer function in free space becomes 22 22()λλf ()f 11−−≈−−()()λλff XY XY 22 We confirm again that 2π the Fresnel diffraction jz H ff =−+e λ exp ⎡ jπλz ff22⎤ is paraxial approximation ()XY, ⎣ () X Y⎦ FresnelFresnel DiffractionDiffraction betweenbetween ConfocalConfocal Spherical Spherical SurfacesSurfaces 2 2 2 r01 = z + ()()x −ξ + y −η xξ yη r ≈ z − − 01 z z jkz ∞ 2π e − j ()xξ + yη U ()x, y = ∫∫U ()ξ,η e λz dξdη jλz −∞ e jkz = F{}U ()ξ,η jλz f X =x /λz, fY = y / λz The Fresnel diffraction makes the exact Fourier transforms between the two spherical caps. In summary, Fresnel diffraction is … FraunhoferFraunhofer ApproximationApproximation 2 2 xξ yη k(ξ +η )max ⎡ ⎤ z〉〉 rz01 ≈−−1 2 ⎣⎢ z z ⎦⎥ k j ( x2 + y2 ) e jkze 2 z ∞ ⎡ 2π ⎤ U ()x, y = ∫∫U ()ξ,η exp⎢− j ()xξ + yη ⎥dξdη jλz −∞ ⎣ λz ⎦ k j (x2 + y2 ) e jkze 2z = F{}U ()ξ,η jλz f X =x / λz, fY = y / λz 2D2 z > : anttena designer' s formular λ FraunhoferFraunhofer (far-field)(far-field) diffractiondiffraction • Specific sort of diffraction – far-field diffraction ∞ ⎡⎤2π Ux(),,expy ≈−+∫∫U ()ξ ηξηξη⎢⎥j ()x y dd – plane wavefront −∞ ⎣⎦λz – Simpler maths Plane waves ⎛⎞r ⎧1 ra≤ Circular aperture circ⎜⎟= ⎨ ⎝⎠a ⎩1 ra> Circular Aperture Intensity Single slit Circular (sinc ftn) aperture 0 λ/D 2λ/D 3λ/D −3λ/D −2λ/D −λ/D Sin θ Airy Disc • Similar pattern to single slit – circularly symmetrical • First zero point when ½kD sinθ = 3.83 λ – sinθ = 1.22 /D – Central spot called “Airy Disc” • Every optical instrument • microscope, telescope, etc. – Each point on object • Produces Airy disc in image. (Appendix : Step and repeat function) Two-dimensional array of rectangular apertures Two-dimensional array with irregular spacing randomized Diffraction from an array of finite size Diffraction from an array of finite size (2) : for large C unit cell (a x l) c Double-slit diffraction b (a+b)/2 (a-b)/2 a b ⎛ E −(a−b) 2 ⎞ ⎛ E (a+b) 2 ⎞ E = ⎜ L e ikssinθ ds⎟ + ⎜ L e ikssinθ ds⎟ R ⎜ ∫−(a+b) 2 ⎟ ⎜ ∫(a−b) 2 ⎟ ⎝ ro ⎠ ⎝ ro ⎠ EL 1 ik(−a+b)sinθ / 2 ik(−a−b)sinθ / 2 ik(a+b)sinθ / 2 ik(a−b)sinθ / 2 = []e()()()()− e + e − e ro ik sinθ 1 1 β ≡ 2 kbsinθ , α ≡ 2 ka sinθ EL b iα iβ −iβ −iα iβ −iβ 2E Lb sin β ER = []e (e − e ) + e (e − e ) = cosα ro 2iβ ro β ⎛ ε oc ⎞ 2 I = ⎜ ⎟E R ; irradiance ⎝ 2 ⎠ 2 2 2 ⎛ ε c ⎞⎛ 2E b ⎞ ⎛ sin β ⎞ ⎛ sin β ⎞ o ⎜ L ⎟ 2 2 = ⎜ ⎟⎜ ⎟ ⎜ ⎟ cos α = 4Io ⎜ ⎟ cos α ⎝ 2 ⎠⎝ ro ⎠ ⎝ β ⎠ ⎝ β ⎠ Double-slit diffraction 2 2 ⎛ sin β ⎞ ⎛ ε c ⎞⎛ E b ⎞ 2 o ⎜ L ⎟ I = 4Io ⎜ ⎟ cos α, Io = ⎜ ⎟⎜ ⎟ ⎝ β ⎠ ⎝ 2 ⎠⎝ ro ⎠ 2 2 ⎡πa sinθ ⎤ The interference term : cos α = cos ⎢ ⎥ - - - - - - - - - (1) ⎣ λ ⎦ interference maxima : pλ = a sinθ, p = 0, ± 1, ± 2, " 2 ⎛ πbsinθ ⎞ ⎜ sin ⎟ The diffraction envelope term : ⎜ λ ⎟ - - - - - - - - - (2) ⎜ πbsinθ ⎟ ⎜ ⎟ ⎝ λ ⎠ diffraction minima : mλ = bsinθ, m = 0, ± 1, ± 2, " ⎛ p ⎞ Condition for missing orders, or satisfying (1) & (2) : a = ⎜ ⎟b ⎝ m ⎠ Multiple-slit diffraction N / 2 E []−(2 j−1)a+b) 2 [](2 j−1)a+b) 2 E = L e ikssinθ ds + e ikssinθ ds R ∑ {∫∫[]−(2 j−1)a−b) 2 [](2 j−1)a−b) 2 } ro j=1 2 2 ⎛ sin β ⎞ ⎛ sin Nα ⎞ I = Io ⎜ ⎟ ⎜ ⎟ ⎝ β ⎠ ⎝ sinα ⎠ 2 ⎛ sin Nα ⎞ The interference term : ⎜ ⎟ - - - - - - - - - (1) ⎝ sinα ⎠ principal maxima : α → mπ , or mλ = a sinθ, m = 0, ± 1, ± 2, " sin Nα N cos Nα magnitude : lim = lim = ± N α →mπ sinα α →mπ cosα pπ pλ minima : α = , or = a sinθ, p = 0, ± 1, ± 2, " N N ⇒ principal maxima occur for p = 0, ± N, ± 2N, " minima occur for p = all other integers Examples of Fraunhofer Diffraction Thin sinusoidal amplitude grating ⎡1 m ⎤ ⎛ ξ ⎞ ⎛ η ⎞ tA = ⎢ + cos(2πfoξ )⎥rect⎜ ⎟rect⎜ ⎟ ⎣2 2 ⎦ ⎝ 2ω ⎠ ⎝ 2ω ⎠ Thin sinusoidal amplitude grating ⎡1 m ⎤ 1 m m FT + cos(2πfoξ ) = δ ( f X , fY ) + δ ( f X + fo , fY ) + δ ( f X − fo , fY ) ⎣⎢2 2 ⎦⎥ 2 4 4 2 2 2 ⎡ A ⎤ 2 ⎛ 2ωy ⎞⎧ 2 ⎛ 2ωx ⎞ m 2 ⎛ 2ω ⎞ m 2 ⎛ 2ω ⎞⎫ I(x, y) = ⎢ ⎥ sin c ⎜ ⎟⎨sin c ⎜ ⎟ + sin c ⎜ (x + foλz)⎟ + sin c ⎜ (x − foλz)⎟⎬ ⎣2λz ⎦ ⎝ λz ⎠⎩ ⎝ λz ⎠ 4 ⎝ λz ⎠ 4 ⎝ λz ⎠⎭ η = 0.25 m=1 0 2 η+1 = m /16 ≤ 6.25% 2 η−1 = m /16 ≤ 6.25% Thin sinusoidal phase grating ⎡ m ⎤ ⎛ ξ ⎞ ⎛ η ⎞ t A ()ξ ,η = exp ⎢ j sin ()2πf 0ξ ⎥rect ⎜ ⎟rect ⎜ ⎟ ⎣ 2 ⎦ ⎝ 2w ⎠ ⎝ 2w ⎠ ⎧ ⎡ m ⎤⎫ ∞ ⎛ m ⎞ FT⎨exp⎢ j sin(2πfoξ)⎥⎬ = ∑ Jq ⎜ ⎟δ ( f X − qfo , fY ) ⎩ ⎣ 2 ⎦⎭ q=−∞ ⎝ 2 ⎠ 2 ∞ ⎛ A ⎞ 2 ⎛ m ⎞ 2 ⎡2w ⎤ 2 ⎛ 2wy ⎞ I()x, y ≈ ⎜ ⎟ ∑ J q ⎜ ⎟sinc ⎢ ()x − qf0λz ⎥sinc ⎜ ⎟ ⎝ λz ⎠ q=−∞ ⎝ 2 ⎠ ⎣ λz ⎦ ⎝ λz ⎠ 2⎛ m ⎞ ηq = Jq ⎜ ⎟ ⎝ 2 ⎠ m/2=4 2 ⎛ m ⎞ η±1,max = J±1⎜ = 1.8⎟ = 33.8% ⎝ 2 ⎠ 2 ⎛ m ⎞ 2 ⎛ m ⎞ η0,min = J0 ⎜ = 2.4⎟ = 0, where J±1⎜ = 2.4⎟ = 27% ⎝ 2 ⎠ ⎝ 2 ⎠ Fresnel Diffraction by Square Aperture Fresnel Diffraction from a slit of width D = 2a.