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) rzx=+−+−2 ()()ξ 22 yη

22 22 ⎡⎤11⎛⎞⎛⎞xy−−ξη()xy−−ξ ()η ≈+zz⎢⎥1 ⎜⎟⎜⎟ + =+ + ⎣⎦⎢⎥22⎝⎠⎝⎠zz 22 zz ⎛⎞⎛⎞xy22ξη 22 ξη⎛⎞ xy =++++−+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 ηξηexp i ()22 exp−i() x+ y ξηξη d d ∫∫ ⎣⎦⎣⎦⎢⎥⎢⎥2 zz

k 22 ⎧ j ()ξη+ ⎫ Exy(, )∝F ⎨ U()ξη , e2z ⎬ ⎩⎭

Fraunhofer diffraction ⎡⎤k Exy(),(,)=−+ C Uξη expi() xξ η ξ y η dd ∫∫ ⎣⎦⎢⎥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 λ 2 max −=≤ ] 2 2 4 ηξ −+−〉〉 yxz 2 ring function; ring function; x D z Fresnel approximation ()() [

2 λ λ

π D 4 16 ≥ 3 z

ηξ

Accuracy can be expected for much shorter distances Accuracy can be expected for much shorter ξ (,) • for U slow va smooth & Accuracy of the Fresnel Approximation Accuracy of the Fresnel 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 ⎫ (), yxU = ∫∫U ()ηξexp, ⎨ j [()()−+− ]⎬ ddyx ηξηξ λzj ∞− ⎩ 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 +rrik )'( θ )( = EdaFe eikr' ikr θ )( daFe sp ∫∫ rr' s ⎢r' ⎥∫∫ r Ap ⎣ ⎦ Ap

Obliquity factor: Spherical wave from the point source S unity where θ=0 zero where θ = π/2

Huygens’ Secondary wavelets on the wavefront surface O 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 eikr' ikr θ )( , ⎜ -daFe θ << ⎟ sp r' ∫∫ r 22 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 ⎧ + cos1 θ ⎫ 1 +rrik )'( E p = ⎨ ⎬ , () -dae θ << ππ 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 θ tan m == fmnR

11⎛⎞R1 f f2 f fRRmnn==()() () nR1 ⎜⎟ 3 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%.