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 revised by 1stst R – S solution revisedλ by 1 R – S solution 1 exp(jkr ) U ()P = U ()P 01 cosθ ds H-F principle 0 ∫∫ 1 j ∑ r01 1 amplitude ∝ λ ∝ ν 1 phase : (lead the incident phase by 90o ) j

U (P )= h(P , P )U (P )ds 0 ∫∫ 0 1 1 ∑ λ 1 exp(jkr01 ) h()P0 , P1 = cosθ Impulse response function j r01 Each secondary wavelet has a directivity pattern cosθ Huygens-FresnelHuygens-Fresnel PrinciplePrinciple

z cos θ = rλ01

1 exp ( jkr ) r01 U P = U P 01 cos θ ds ()0 ∫∫ ()1 j λ∑ r01 ξ η

z exp ( jkr ) U ()x, y = U (), 01 dξdη ∫∫ 2 j ∑ r01 Aperture (ξ,η) Screen (x,y) ξ 2 2 2 r01 = z + ()()x − + y −η

Note! So far we have used only two assumptions (R-S solution) :

Scalar wave equation + ( r01 >> λ ) Fresnel (paraxial) Approximation Fresnel (paraxial)ξ Approximation λ 2 2 ⎡ 1ξ ⎛ηx − ⎞ 1 ⎛ y −η ⎞ ⎤ r01 ≈ z⎢1+ ⎜ ⎟ + ⎜ ⎟ ⎥ ⎣⎢ 2 ⎝ z ⎠ 2 ⎝ ξ z ⎠ ⎦⎥

jkz ∞ η e ⎧ k 2 2 ⎫ ξ U ()x, y = λ ∫∫U (), exp⎨ j [()()x − + y − ]⎬ d dη j z −∞ ⎩ 2z ⎭ ξ η 7 Since k is 10 order, the quadratic terms in r01 mustξ be considered in the exponent. η π jkz k ∞ k 2 ξ e j ()x2 + y2 ⎧ j ()2 + 2 ⎫ − j λ ()x + yη 2z 2zξ z ξ U()x, y = e ⎨U(), e η ⎬e d dη j z ∫∫ λ −∞⎩ ⎭ jkz k k e j ()x2 + y2 ⎧ j ()2 + 2 ⎫ U (x, y) = e 2z U ()ξ,η e 2z F ⎨ ⎬ λ j z ⎩ ⎭ f X =x / z, 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 2 max −=≤ ] 2

η − y + 2

ξ ring function; x D z − Fresnel approximation Fresnel x ()() [

2 λ λ

π D 4 16 ≥ 〉〉 3 z z

ηξλ

Accuracy can be expected for much shorter distances Accuracy can be expected for ξ (,) 2 4 • for U slow va smooth & Accuracy of the Fresnel Approximation Accuracy of the Fresnel Approximation λ ImpulseImpulse ResponseResponseξ η andand TransferTransfer FunctionsFunctions of Fresnel diffraction of Fresnel diffractionξ

jkz ∞ η e ⎧ k 2 2 ⎫ ξ U()x, y = U(), exp⎨ j [()()x − + y − ]⎬ d dη j z ∫∫ ⎩ 2z ⎭ −∞ ξ η ξ ∞ η U ()x, y = ∫∫U ()(, h x − , y − dξ )dη − ∞ 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 =+=−+λλexpjxy22 ejkz exp ⎡ jzff 2 2⎤ F ⎨⎬⎢⎥()⎣ XY⎦ ( ) ⎩⎭jz⎣⎦ z

Remind! Transfer function of wave propagation phenomenon in free space πλ ⎡ z 22⎤ εε()()ffzXY,;=−− ff XY ,;0exp2 πλλ j 1()() f X f Y ⎣⎢ ⎦⎥ ⎡ z 22⎤ Hf()XY,exp21 f=−−⎢ jπλλ()() f X f Y⎥ ⎣ λ ⎦ Under the paraxialλλ approximation,λ the transfer function in free space becomes

22 22()λλf ()f 11−−≈−−()()ff 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

ξ 2 2 2 r01 = z + ()()x − + y −η xξ yη r ≈ z − − 01 z z λ ξ η π jkz ∞ 2 ξ e − j λ ()x + yη U ()x, y = ∫∫U (), e z dξdη j z −∞ jkz e λ λ = F{}U ()ξ,η j z f X =x / z, fY = y / λz

The Fresnel diffraction makes the exact Fourier transforms between the two spherical caps. In summary, Fresnel diffraction is … FraunhoferFraunhofer ApproximationApproximation

ξ k( 2 +η 2 ) λ ⎡ xξ yη ⎤ max ξ z〉〉 ηrz01 ≈−−⎢1 ⎥ 2 ⎣ π z z ⎦

k 2 2 ξ j ( x + y ) λ jkz 2 z ∞ η e e ⎡ 2 ⎤ ξ U ()x, y = ∫∫U (), exp⎢− j ()x + y ⎥d dη j z −∞ ⎣ z ⎦

k j (x2 + y2 ) e jkze 2zλ = F{}U ()ξ,η λ j z f X =x / z, 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 −∞ ⎣⎦z – Simpler maths λ

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 θ β θ b

⎛ Eθ −(a−b) 2 ⎞ ⎛ E (a+b) θ2 ⎞ E = ⎜ L α e ikssinθ ds⎟ + ⎜ L e ikssin ds⎟ R ⎜ ∫−(a+b) 2 ⎟ ⎜ ∫(a−b) 2 ⎟ ⎝ ro θ ⎠ ⎝ ro ⎠ β θ ε EL 1 ()()()()ik(−a+b)sin / 2 ik(−a−b)sin / 2 ik(a+b)sin / 2 ik(a−b)sin / 2 = α[]e − e + e − e r ik sin β o θ 1 1 β ≡ 2 kbsin , ≡ 2 ka sin α ε β θ EL b i i −i −i i −iβ 2E Lβb sin ER = []e (e − e ) + e (e − e ) = cos r 2i r β o β o α ⎛ c ⎞ I = ⎜ o ⎟E 2 ; irradianceβ 2 R α ⎝ ⎠ β 2 2 2 ⎛ c ⎞⎛ 2E b ⎞ ⎛ sin ⎞ ⎛ sinβ ⎞ o ⎜ L ⎟ 2 2 = ⎜ ⎟⎜ ⎟ ⎜ ⎟ cos = 4Io ⎜ ⎟ cos α ⎝ 2 ⎠⎝ ro ⎠ ⎝ ⎠ ⎝ ⎠ β β Double-slitα diffraction ε 2 2 ⎛ sin ⎞ ⎛ c ⎞⎛ E b ⎞ 2 o ⎜ L ⎟ I = 4Io ⎜ ⎟ cos , Io = ⎜ ⎟⎜ ⎟ ⎝ ⎠ α ⎝ 2 ⎠⎝ ro ⎠ π 2 θ2 ⎡ a sin ⎤ The interference term : cos = cos ⎢ ⎥ ------(1) λ ⎣ ⎦ λ interference maxima : p = a sinθ, p = 0, ± 1, ± 2, " 2 ⎛π bsin ⎞ ⎜ sin θ ⎟ The diffraction envelope term : ⎜ λ ⎟ ------(2) ⎜π bsin ⎟ ⎜ θ ⎟ ⎝ λ ⎠ diffraction minima : mλ = bsinθ, m = 0, ± 1, ± 2, " ⎛ p ⎞ Condition for missing orders, or satisfying (1) & (2) : a = ⎜ ⎟b ⎝ m ⎠ Multiple-slit diffractionθ

N / 2 E []−(2 j−1)a+b) 2 [](2 j−1)a+b) 2 E = L { e ikssin ds + e ikssinθ ds} R r ∑ ∫∫[]−(2 j−1)a−b) 2 [](2 j−1)a−b) 2 o βj=1 2 2 ⎛ sinβ ⎞ ⎛ sin Nα ⎞ I = Io ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ sinα ⎠

α 2 ⎛ sin N ⎞ The interference term : ⎜α ⎟ ------(1) α ⎝ sin ⎠ π principal maxima α: → m , or m = a sinθ, m = 0, ± 1, ± 2, " π λ sinαN N cos N magnitude : lim = αlim = ± N α π →m sinα →mπ cos p pλ α minima : = , or = a sinθ, p = 0, ± 1, ± 2, " N N α ⇒ principal maxima occur for p = 0, ± N, ± 2N, " minima occur for p = all other integers Examples of Fraunhofer Diffraction

Thin sinusoidal amplitude grating

π ξ ξ ⎡1 m ⎤ ⎛ ω ⎞ ⎛η ⎞ tA = ⎢ + cos(2 fo )⎥rect⎜ ⎟rect⎜ ⎟ ⎣2 2 ⎦ ⎝ 2 ⎠ ⎝ 2ω ⎠ π ξ δ Thin sinusoidal amplitude grating δ λ ω ⎡1 m λ ⎤ 1ω m m FT ⎢ + cos(2 fo )⎥ = ( f X , fY ) + ( f X + fo , fY ) + δ ( f X − fo , fY ) ⎣2 2 ⎦ 2 λ 4 4 ω λ λ ω 2 2 2 ⎡ A ⎤ 2 ⎛ 2 y ⎞⎧ 2 ⎛ 2 x ⎞ m 2 ⎛ 2 ⎞ m 2 ⎛ 2λ ⎞⎫ I(x, y) = ⎢ ⎥ sin c ⎜ ⎟⎨sin c ⎜ ⎟ + sin c ⎜ (x + fo z)⎟ + sin c ⎜ (x − foλz)⎟⎬ ⎣2 z ⎦ ⎝ z ⎠⎩ ⎝ z ⎠ 4 ⎝ z ⎠ 4 ⎝ z ⎠⎭

η = 0.25 m=1 0 2 η+1 = m /16 ≤ 6.25%

2 η−1 = m /16 ≤ 6.25% ξ η

Thin sinusoidal phase gratingπ ξ ξ ⎡ m ⎤ ⎛ ⎞ ⎛ η ⎞ t A (), = exp ⎢ j sin ()2 f 0 ⎥rect ⎜ ⎟rect ⎜ ⎟ ⎣ 2 ⎦ ⎝ 2w ⎠ ⎝ 2w ⎠ π ξ λ ⎧ ⎡ m ⎤⎫ ∞ ⎛ m ⎞ FT⎨exp⎢ j sin(2 fo )⎥⎬ = ∑ Jq ⎜ ⎟δ ( f X − qfo , fY ) ⎣ 2 ⎦ q=−∞ ⎝ 2 ⎠ ⎩ ⎭ λ λ 2 ∞ ⎛ A ⎞ 2 ⎛ m ⎞ 2 ⎡2w ⎤ 2 ⎛ 2wy ⎞ I()x, y ≈ ⎜ ⎟ ∑ J q ⎜ ⎟sinc ⎢ ()x − qf0 z ⎥sinc ⎜ ⎟ ⎝ z ⎠ q=−∞ ⎝ 2 ⎠ ⎣ z ⎦ ⎝ λz ⎠

2⎛ m ⎞ ηq = Jq ⎜ ⎟ ⎝ 2 ⎠ m/2=4

η 2 ⎛ m ⎞ ±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. (a) Shaded 2w 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 x = ( λ / D ) 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.

NF = w/ λz : Fresnel number

ξ η Talbot Images 1 πλ t (), = []1+ mcos()2πξ / L A 2 π

1 ⎡ ⎛ z ⎞ ⎛ 2 x ⎞ 2 2 ⎛ 2πx ⎞⎤ I()x, y = ⎢1+ 2mcos⎜ 2 ⎟cos⎜ ⎟ + m cos ⎜ ⎟⎥ 4 ⎣ ⎝ L ⎠ ⎝ L ⎠ ⎝ L ⎠⎦ Appendix : Shah function (Impulse train function) Appendix : Sampling, Step and Repeat Function