Diffraction

Question 1

Fraunhofer

Answer 1

Diffraction is observed in the image plane of the source

Question 2

Fresnel

Answer 2

Diffraction is observed close to the diffracting object

wavefront are significantly curved

Question 3

Fraunhofer Diffraction from a single slit

Answer 3

Ep ∝ [-1/iksin(θy) exp{-ikysin(θy)} ] a/2 -a/2

=> Ep ∝ asinc(ka/2 sin(θy)

Question 4

Kirchoff Integral Theorem KIT

Answer 4

Ψp = 1/4π ( ∫ S) (Ψ∇ [exp{ikr}/r] - exp{ikr}/r ∇Ψ) * d∑͢

Question 5

∇ [exp{ikr}/r] =

Answer 5

= -ik exp{ikr}/r n͢

Question 6

∇Ψ =

Answer 6

= ik∇Ψn͢

Question 7

Reevaluating KIT

Answer 7

Ψp = 1/4π ( ∫ Q) (-ikΨ[exp{ikr}/r]n͢ - exp{ikr}/r ikΨn͢’ ) * d∑͢

Ψp = -1/4π ( ∫ Q) ([exp{ikr}/r]ikΨ ( n͢ + n͢’)) * d∑͢

Ψp = -i/λ ( ∫ Q) ([exp{ikr}/r]Ψ (( n͢ + n͢’)/2)) * d∑͢

Question 8

Calculating Fraunhofer diffraction pattern

Answer 8

Ψp = -i/λ ( ∫ Q) Ψ 1/r exp{-ik(xθₓ + yθᵧ)} d∑

Ψp ∝ ( ∫ aperture) A(x,y) exp{-ik(xθₓ + yθᵧ)} d∑

Question 9

Fraunhofer diffraction diagram

Answer 9

see notes

Question 10

first zeros occur for a single slit when

Answer 10

sin(kθa/2) = 0

or when kθa / 2 = πn

θ = nλ/a

Question 11

distance from the centre of the pattern of the observation screen

Answer 11

y = fθ

Question 12

wavefront for a slit of finite width

Answer 12

Ψ = A a sinc(kθa/2)

Question 13

first zeros occur for a double slit when

Answer 13

when kθd / 2 = (2n+1)π/2

Question 14

d = x Δθ

Answer 14

angular resolution

Question 15

diffraction pattern of fraunhofer diffraction of circular aperture

Answer 15

is an airy disc

Question 16

rayleigh criterion

Answer 16

is for two-point sources to be resolved, the bright peak in the image from one source should be no closer than the first minimum

Question 17

fresnel diffraction diagram

Answer 17

see notes

Question 18

fresnel variables

Answer 18

dΣ - vector area element pointing inwards

i - phase offset

e^i(kr)/r - gives amplitude a

r - is the distance from the area element to point P

n’ - unit vector in the direction of propogation

(n+n’/2) - obliquity factor

Question 19

fresnel improvements over huygens

Answer 19

correctly includes wavelength dependence

provides correct prediction of phase

obliquity factor removes a backward propogating wave from the spherical Huygens wavelets.

Question 20

When applied to the diffraction of light, the Fresnel-Kirchhoff result is
not complete in its description of the diffraction process.

Answer 20

The Fresnel-Kirchhoff treatment calculates diffraction of a scalar field. Light,
being an electromagnetic wave, is a vector quantity. The principal omission is that
the scalar treatment cannot include consideration of polarisation effects.

Question 21

proof of I(kx,ky)

Answer 21

A’(kx,ky) =(b/2 ∫-b/2) (a/2 ∫ -a/2) e^-i(xkx+yky) dx dy

seperate integrals to solve

Question 22

fraunhoffer diffraction pattern

Answer 22

square diffraction pattern.

Question 23

diffraction pattern from full aperture

Answer 23

see notes