If the wavelength is larger than the aperture, the wave will spread out at a large angle. [Picture P445] . Distance l S

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Vincent Austin
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1 Chapter 10 Diffraction 10.1 Preliminary Considerations Diffraction is a deviation of light from rectilinear propagation. t occurs whenever a portion of a wavefront is obstructed. Hecht; 11/8/010; 10-1 No significant distinction between interference and diffraction. Superposition of a few waves, many waves Huygens-Fresnel Principle Every point on a wavefront serves as a source of spherical secondary wavelets with the same frequency. The amplitude of the optical field at a later time is the superposition of all the secondary wavelets. f the wavelength is larger than the aperture, the wave will spread out at a large angle. [Picture P445] A. Opaque Obstructions There is no optical field beyond it. The incident wave and the electron-oscillator fields superimpose to yield zero field beyond the screen. A screen = Apertured screen + Small disk Diffraction field + Electron-oscillator field Zero field Unpaired electron-oscillator at the rim Not negligible near the aperture. B. Fraunhofer-Fresnel Diffraction A point source at infinity Aperture Screen Distance l O. Distance l S. Diffraction pattern. Plane wave. l S Short dist. ntermediate Long dist. The diffraction pattern Same shape as the aperture. Geometrical optics applies. ecognizable aperture shape with fringes. Fresnel or Near-field diffraction. Smooth pattern with no resemblance to the aperture. Fraunhofer or Far-field diffraction. educing the wavelength. Fraunhofer diffraction Fresnel diffraction Geometrical optics The point source closer to the aperture. Not plane wave at the aperture. Fresnel diffraction even if l S Fraunhofer diffraction when > a / λ : Smaller distance of l O and l S. a: aperture size.

2 Fraunhofer diffraction can be obtained by lenses. (S and P are equivalently at infinity) Hecht; 11/8/010; 10- P and S are on the focal points of lenses L 1 and L, respectively. P is an image of S n general, an image is a Fraunhofer diffraction pattern C. Several Coherent Oscillators A linear array of N identical coherent point sources Electric field at an infinite distance, P ( 1 ) ikr ( t) N E ω ω = Eo ( r1) e + Eo ( r ) e Eo ( rn ) e ikr t ikr ( ωt) Assuming same wave amplitudes at P ikr t i i i N 1 = o : δ = kd sin ( ) ( 1 ω ) δ δ δ 1 ( )... ( ) E E r e e e e iωt i kr1 + ( N 1 ) δ / sin Nδ / ik ( ωt) sin Nδ/ E = Eo r e e Eo ( r) e sin δ/ sin δ/ 1 = ( N 1) dsinθ + r1, the distance from the center to P ( ) θ The intensity at P : The principal maxima : = o [ Nkd θ ] [ kd θ ] sin ( /)sin sin ( /)sin = on for ( /)sin sin m kd θ = mπ d θ = mλ f there is a phase shift between adjacent sources of δ = kd sinθ + ε, the principal maxima occur at dsin θ = mλ ε/ k m

An idealized line source Each point emits a spherical wavelet E E = o cos ( kr ωt) r Hecht; 11/8/010; 10-3 The electric field from an infinitesimal segment Δ yi E o = cos ( ω ) N E Δ i kri t yi : D

3 An idealized line source Each point emits a spherical wavelet E E = o cos ( kr ωt) r Hecht; 11/8/010; 10-3 The electric field from an infinitesimal segment Δ yi E o = cos ( ω ) N E Δ i kri t yi : D is the array length ri D : N is total number of sources The total field at P = N EL Ei cos ( kri ωt) Δyi r i = 1 i 1 lim D N where E ( E N) L o L D / ( kr ωt ) D / cos E = E dy r (1) From the Problem 9.15 y r = ysinθ + cos θ +... The third term can be ignored if the phase Fraunhofer condition k ( D ) / cos θ is very small () nsert () into (1) D / EL E = cos k( ysinθ) ωt dy D / sin β ' E = ELD β ' cos ( k ωt ) : β ' ( kd ) / sinθ For D >> λ, E becomes a circular wave in xz-plane (No diffraction and no wave in vertical direction) t looks like a spherical wave coming from the center that is cut by the horizontal plane. The irradiance sin β ' 0 0 sinc β ' ( θ ) = ( ) ( ) β ' (3)

$10. Fraunhofer Diffraction Goto Section D first then come back here Hecht; 11/8/010; 10-4 A. The Single Slit From Eq. (4) of Sec D.$ = = θ rradiance sinα sin β = α β ( θ ) ( o) Very narrow slit, b 0.

= = θ rradiance sinα sin β = α β ( θ ) ( o) Very narrow slit, b 0.

4 10. Fraunhofer Diffraction Goto Section D first then come back here Hecht; 11/8/010; 10-4 A. The Single Slit From Eq. (4) of Sec D. ( xyz,, ) ik ( ωt) A E e y= b/ a/ ik ( Yy Zz )/ E + = e dzdy y= b/ z= a/ ( XY,, Z) ik ( ωt) lbeae sin sin E α β = α β where α kbz kly kl =, sin β = = θ rradiance sinα sin β = α β ( θ ) ( o) Very narrow slit, b 0. The irradiance sin β ' ( θ ) = ( 0) ( 0) sinc β β ' Compare this with the previous result Eq. (3). ( ) 0 θ = for β =± π, ± π... Destructive interference whenever the path difference between the top and the bottom rays is mλ, or bsinθ = mλ.

(Yy + Zz )/ dzdy l / a b / b / a +b / ik ( Zz )/ + e dz e ik ( Zz )/ dz b/ a b /

$θ y The irradiance sin β sin α (θ ) = 4 o cos γ β α Diffraction. nterference.$

5 Hecht; 11/8/010; 10-5 B. The Double Slits Waves from each slit will interfere (x, y, z ) (X,Y, Z ) The total electric field at P using Eq. (4) l / b / E =C e ik (Yy + Zz )/ l / a +b / dzdy + C l / b / sin β E = lc β e ik (Yy + Zz )/ dzdy l / a b / b / a +b / ik ( Zz )/ + e dz e ik ( Zz )/ dz b/ a b / sin β E = lbce iγ β sin α cos γ α : β = kly : α= kbz kb kaz ka = sin θ, γ = = sin θ y The irradiance sin β sin α (θ ) = 4 o cos γ β α Diffraction. nterference. : o is the irradiance from each slit An interference maximum and a diffraction minimum correspond to the same θ Missing order For a = mb, there are m bright fringes within the central diffraction peak

$Hecht; 11/8/010; 10-6 C. Diffraction by Many Slits The total electric field at P l/ b/ l/ a+ b/ l/ a+ b/ l/ ( N 1) a+ b/ ( ) ( ) ( ).$ .. e + + + + β α ikza / ikza ( / ) ikza ( / ) ( N 1) i N sin β sinα sin Nγ E lbce γ = β α sinγ where kly kbz kb kaz ka β =, α = = sinθ, γ = = sinθ The irradiance ( 0) sin β sinα sin Nγ (

.. e + + + + β α ikza / ikza ( / ) ikza ( / ) ( N 1) i N sin β sinα sin Nγ E lbce γ = β α sinγ where kly kbz kb kaz ka β =, α = = sinθ, γ = = sinθ The irradiance ( 0) sin β sinα sin Nγ (

6 Hecht; 11/8/010; 10-6 C. Diffraction by Many Slits The total electric field at P l/ b/ l/ a+ b/ l/ a+ b/ l/ ( N 1) a+ b/ ( ) ( ) ( )... ( ) E = C F z dzdy + C F z dzdy + C F z dzdy + + C F z dzdy l/ b/ l/ a b/ l/ a b/ l/ ( N 1) a b/ sin β sinα E = lbc 1 e e... e β α ikza / ikza ( / ) ikza ( / ) ( N 1) i N sin β sinα sin Nγ E lbce γ = β α sinγ where kly kbz kb kaz ka β =, α = = sinθ, γ = = sinθ The irradiance ( 0) sin β sinα sin Nγ ( θ ) = N β α sinγ (0), peak of the total at the screen center o, peak irradiance by one slit The principal maxima for sinγ = 0, γ = 0, ± π, ± π... π π 3π (N-1) minima for sin Nγ = 0, γ =±, ±, ±... N N N 3π 5π A subsidiary maxima at γ =±, ±... N N The irradiance of the first subsidiary maximum 1 (0) 3π

z a/ Ae = = ik ( Yy Zz )/ E + = e dydz y= b/ z= a/ (4) ik ( t) abe Ae sin ' sin ' E ω α β = α' β : α ' = kaz /, β ' = kby / The irradiance sin α' sin β ' Y (, Z) =

7 D. The ectangular Aperture Hecht; 11/8/010; 10-7 y z b a ds r x P(X,Y,Z) The differential electric field at P from a differential source ikr ( ωt) e de = ( EAdS ) r adial distance EAdS ( ) ( ) ( ) ( ) ( ) r = X + Y y + Z z = 1 + y + z / Yy + Zz / 1 Yy + Zz / n the Fraunhofer region The total electric field at P ik ( ωt) E y b/ z a/ Ae = = ik ( Yy Zz )/ E + = e dydz y= b/ z= a/ (4) ik ( t) abe Ae sin ' sin ' E ω α β = α' β : α ' = kaz /, β ' = kby / The irradiance sin α' sin β ' Y (, Z) = (0) α' β ' (0) The maxima along β ' axis : = for β (0) The maxima along α ' axis : = for α ' m ' m 3π 5π β ' m =±, ±,... and α ' = 0 3π 5π α ' m =±, ±,... and β ' = 0

$= = 1/ 1 u 0 : Airy pattern λ λ The radius of the first dark ring : r 1 = 1. θhalf, D=a, aperture dia. a D Diffraction at the focal point of a lens : q1 1.$

8 E. The Circular Aperture Hecht; 11/8/010; 10-8 The electric field at P ik ( ωt) ik ( ωt) a π A ik ( Yy + Zz )/ EAe i ( k ρq / ) cos( φ Φ) E e E = e ds e ρdρdφ Aperture ρ= 0φ= 0 Φ = 0 due to axial symmetry z = ρ cos φ, y = ρsinφ = πj0 ( kρ q/ ), Bessel function of zero order Z = qcos Φ, Y = qsin Φ ik ( ωt) A E π a J1 E e kaq = kaq The irradiance ( / ) ( / ) 1 1 E A J kaq A (0) J kaq = kaq / kaq / (0) = E A / Using q/ = sinθ ( ka θ ) J sin ka sinθ 1 = (0) A since ( ) J u / u = = 1/ 1 u 0 : Airy pattern λ λ The radius of the first dark ring : r 1 = 1. θhalf, D=a, aperture dia. a D Diffraction at the focal point of a lens : q1 1. f λ D

9 Hecht; 11/8/010; 10-9 F. esolution of maging Systems ayleigh criterion ( Δϕ )min = 1.λ /D ( Δl )min = 1.λ f / D : Two spots can be resolved when one Airy disk falls on the first minimum of the second Airy disk. : Limit of resolution esolving power is defined as 1/ ( Δϕ )min or 1/ ( Δl )min G. The Zeroth-Order Bessel Beam One solution of the differential wave equation is i (k z ωt ) : Bessel beam E (r,θ, z, t ) J o (k r ) e & The irradiance (r, θ, z, t ) J o (k r ) : ndependent of z No diffraction Not possible to create the perfect Bessel beam due to its infinite extent.

10 Hecht; 11/8/010; H. The Diffraction Grating A diffraction grating produces periodic alterations in the phase and/or amplitude of the incident wave. Transmission amplitude grating A multiple-slits modulates the amplitude. Transmission phase grating uling of parallel notches on a flat glass plate. (Periodic change of the refractive index) z Grating equation Constructive interference condition a sin θm = m λ : Normal incidence a ( sin θm sin θi ) = m λ : Oblique incidence The main disadvantage of these gratings Most of the incident energy is wasted in the specular reflection y Blazed grating can shift energy from the zeroth order to a higher order θi and θm are measured from the normal to the grating plane not to the groove surface. Grating Spectroscopy Two and Three Dimensional Gratings eflection phase grating uling on a reflective surface

Lecture 11: Introduction to diffraction of light

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