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4. Fresnel and Fraunhofer Diffraction

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13. Fresnel diffraction Remind! Diffraction regimes Fresnel-Kirchhoff diffraction formula E exp(ikr) E PF= 0 ()θ dA ()0 ∫∫ irλ ∑ z r Obliquity factor : F ()θθ= cos = r zE exp (ikr ) E x,y = 0 ddξ η () ∫∫ 2 irλ ∑ Aperture (ξ,η) Screen (x,y) 22 rzx=+−+−2 ()()ξ yη 22 22 ⎡⎤11⎛⎞⎛⎞xy−−ξη()xy−−ξ ()η ≈+zz⎢⎥1 ⎜⎟⎜⎟ + =+ + ⎣⎦⎢⎥22⎝⎠⎝⎠zz 22 zz ⎛⎞⎛⎞xy22ξη 22⎛⎞ xy ξη =++++−+z ⎜⎟⎜⎟⎜⎟ ⎝⎠⎝⎠22zz 22 zz⎝⎠ zz E ⎡⎤k Exy(),expexp=+0 ()ikz i() x22 y izλ ⎣⎦⎢⎥2 z ⎡⎤⎡⎤kk22 ×+−+∫∫ exp⎢⎥⎢⎥ii()ξ ηξ exp()xyη ddξη ∑ ⎣⎦⎣⎦2 zz E ⎡⎤k Exy(),expexp=+0 ()ikz i() x22 y r izλ ⎣⎦⎢⎥2 z ⎡⎤⎡⎤kk ×+−+exp iiξ 22ηξexp xyη dξdη ∫∫ ⎢⎥⎢⎥() () Aperture (ξ,η) Screen (x,y) ∑ ⎣⎦⎣⎦2 zz ⎡⎤⎡⎤kk22 =+−+Ci∫∫ exp ⎢⎥⎢⎥()ξ ηξexp ixydd()η ξη ∑ ⎣⎦⎣⎦2 zz Fresnel diffraction ⎡⎤⎡⎤kk E ()xy,(,)=+ C Uξ ηξηξηξηexp i ()22 exp−i() x+ y d d ∫∫ ⎣⎦⎣⎦⎢⎥⎢⎥2 zz k 22 ⎧ j ()ξη+ ⎫ Exy(, )∝F ⎨ U()ξη , e2z ⎬ ⎩⎭ Fraunhofer diffraction ⎡⎤k Exy(),(,)=−+ C Uξη expi() xξ y η dd ξ η ∫∫ ⎣⎦⎢⎥z =−+CU(,)expξ ηξθηθξη⎡⎤ ik sin sin dd ∫∫ ⎣⎦()ξη Exy(, )∝F { U(ξ ,η )} Fresnel (near-field) diffraction 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 (parabolic wavelets) Screen z Fresnel Diffraction Accuracy of the Fresnel Approximation 3 π 2 2 2 z〉〉[()() x −ξ + y η −] 4λ max • Accuracy can be expected for much shorter distances for U(,)ξ ηξ smooth & slow varing function; 2x−=≤ D 4λ z D2 z ≥ Fresnel approximation 16λ In summary, Fresnel diffraction is … 13-7. Fresnel Diffraction by Square Aperture Fresnel Diffraction from a slit of width D = 2a. (a) Shaded 2w2a area is the geometrical shadow of the aperture. The dashed line is the width of the Fraunhofer diffracted beam. (b) Diffraction pattern at four axial positions marked by the arrows in (a) and corresponding to the Fresnel numbers NF=10, 1, 0.5, and 0.1. The shaded area represents the geometrical shadow of the slit. The dashed lines at = ( λ / Dx ) d represent the width of the Fraunhofer pattern in the far field. Where the dashed lines coincide with the edges of the geometrical shadow, the Fresnel number NF=0.5. 2 NazF = /λ : Fresnel number jkz ∞ e ⎧ k 2 2 ⎫ U() x, = y ∫∫U,ξ() η exp⎨ j x()[−ξ + y () η − ]⎬ d ξ d η jλ z − ∞ ⎩ 2z ⎭ 2 Δ=vNF = a/λ z : Fresnel number Fresnel diffraction from a wire Fresnel diffraction from a straight edge From Huygens’ principle to Fresnel-Kirchhoff diffraction Huygens’ principle Every point on a wave front is a source of secondary wavelets. i.e. particles in a medium excited by electric field (E) re-radiate in all directions i.e. in vacuum, E, B fields associated with wave act as sources of additional fields New wavefront Construct the wave front tangent to the wavelets Secondary wavelet r = c ∆t ≈ λ secondary wavelets Given wave-front at t What about –r direction? (π-phase delay when the secondary Allow wavelets to evolve wavelets, Hecht, 3.5.2, 3nd Ed) for time ∆t Huygens’ wave front construction Incompleteness of Huygens’ principle Fresnel’s modification Î Huygens-Fresnel principle Huygens-Fresnel principle P O 1 ⎡ 1 ⎤ 1 EE= eik(') Fr+ r ()θ da= Eeikr' eikr F()θ da p s∫∫ rr' s ⎢r' ⎥∫∫ r Ap ⎣ ⎦ Ap factorObliquity: Spherical wfrom the point ave source S wunityere hθ=0 zero where θ = π/2 Osurface won the avefront w Secondaryavelets gens’Huy Kirchhoff modification 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. 1 1 ⎛ π π ⎞ EE= eikre' Fikr( )θ da ,⎜ < -θ < ⎟ p sr' ∫∫ r 2 2 Ap ⎝ ⎠ Gustav Kirchhoff : Fresnel-Kirchhoff 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 F(θ) = (1 + cosθ )/2. F(0) = 1 in the forward direction, F(π) = 0 with the back wave. Fresnel-Kirchhoff diffraction formula Fresnel-Kirchhoff diffraction integral − ikEs 1⎧ + cosθ ⎫ 1 ik(') r+ r E p = ⎨ ⎬e da , ()π< - θ < π 2π ∫∫ 2 rr' Ap ⎩ ⎭ 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. 1 eikr E = E cosθ da p iλ ∫∫ O r Ap This final formula looks similar to the Fresnel-Kirchhoff formula, therefore, now we call this the revised Fresnel-Kirchhoff formula, or, just call the Fresnel-Kirchhoff diffraction integral. : Fresnel Zones Obliquity factor: unity where χ=0 at C Spherical wave from source Po zero where χ=π/2 at high enough zone index Huygens’ Secondary wavelets on the wavefront surface S Z2 Z3 Z1 λ/2 : Fresnel Zones Z2 Z3 The average distance of successive zones from P differs by λ/2 -> half-period zones. Thus, the contributions of the zones to the disturbance at P alternate in sign, Z1 (1/2 means averaging of the possible values, more details are in 10-3, Optics, Hecht, 2nd Ed) For an unobstructed wave, the last term ψn=0. Therefore, one can assume that the complex amplitude of Whereas, a freely propagating spherical wave from the source Po to P is 1⎛⎞ exp(iks ) = ⎜⎟ isλ ⎝⎠ : Diffraction of light from circular apertures and disks (a) The first two zones are uncovered, 1 (consider the point P at the on-axis P) 2 (b) The first zone is uncovered if point P is placed father away, 1 : Babinet principle (c) Only the first zone is covered by an opaque disk, 1 1 ≈ 2ψ1 P R R Diffraction patterns from Variation of on-axis irradiance circular apertures Fresnel diffraction from a circular aperture Poisson spot Babinet principle At complementary At screen screen without screen ψS ψCS Amplitude of ψ Amplitude of ψ { S} { CS } ψSCSUN+=ψψ Phase of {ψS } Phase of {ψCS } : Straight edge Damped oscillating At the edge Monotonically decreasing 13-6. The Fresnel zone plate The average distance of successive zones from P differs by λ/2 -> half-period zones. Thus, the contributions of the zones to the disturbance at P alternate in sign, RN Assume R4 plane wavefronts R3 R2 O λ R1 rN+ 0 2 r0 P 2 2 ⎡ 2 ⎤ 222⎛⎞λλλn ⎛⎞ Rrnrrnn =+⎜⎟00 −=0⎢ +⎜⎟⎥ Rnrr≈ λ >> λ 24rr n 00( ) ⎝⎠ ⎣⎢ 00⎝⎠⎦⎥ Z2 Z3 If the even zones Z (n=even) are blocked 1 ψ ()P =ψψψ135+++ Bright spot at P It acts as a lens! Fresnel zone-plate lens RN Rnrrn ≈ 00λ ( >> λ ) R4 R3 2 Rn R2 λ r0 = R O 1 rN0 + nλ 2 2 R1 r0 frn===(1) 10 λ P Fresnel zone-plate lens has multiple foci. n 1 Rn Rr==λ R n 0122nn Rn mλ ()∼Rmnmsinθλ=⇒ sin θm tan θ m == fmnR 11⎛⎞R f fRR== nR 1 f3 2 f mnn()() ()1 ⎜⎟ 1 mmλ ⎝⎠2 n λ R2 f = 1 m mλ Fresnel zone-plate lens Binary zone plate: The areas of each ring, both light and dark, are equal. It has multiple focal points. For soft X-ray focusing Sinusoidal zone plate: This type has a single focal point. Fresnel lens: This type has a single focal point. Focusing efficiency approaches 100%..
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