Diffraction

Question 1

What is diffraction?

Answer 1

The tendency of light (or any wave) to bend around objects

Question 2

Why is it hard to see diffraction

Answer 2

Poor source temporal or spatial coherence masks the diffraction ripples

Question 3

What is required to properly see diffraction

Answer 3

A point source
because any off axis rays will blur the shadow

Question 4

When passing a wave through a slit, how does the slit size affect the diffraction pattern

Answer 4

as the size of the slit reduces towards the wavelength of the wave, the diffraction pattern will be more pronounced

Question 5

Where does fresnel diffraction occur

Answer 5

in the near field, close to the slit

Question 6

Where does Fraunhofer diffraction occur

Answer 6

In the far field, far from the slit

Question 7

what is the distinction between interference and diffraction

Answer 7

interference involves the production of two or more separate beams
diffraction occurs naturally when a single wave is limited in some way

Question 8

What is the Huygens-Fresnel Principle

Answer 8

every unobstructed point in a wavefront at a given instant serves as a source of spherical secondary waves.
amplitude of the optical field at any point beyond is the superposition of all these wavelets

Question 9

What approach will be used to solve diffraction problems

Answer 9

Scalar Wave Approximation

Question 10

What is the scalar wave approximation

Answer 10

discard the fact that light consists of two oscillating vector fields and imagine the wave in terms of a single scalar variable with angular frequency w and wave vector k

Question 11

What assumptions do we make about the aperture and the observation region for diffraction

Answer 11

• assume the aperture region is much smaller than the observation region
• inside the aperture, the field and its spatial derivative are the same as if the screen were not present
• outside the aperture (in the shadow of the screen), the field and its spatial derivative are zero

Question 12

What is the problem with Kirchoff’s diffraction assumptions

Answer 12

they can be shown to yield zero field everywhere

Question 13

What is the field in the observation plane for diffraction given by

Answer 13

E(x1,y1) at a distance z from the aperture plane is given by a convolution
E(x1,y1) = double integral over A(x0,y0) = h(x1-x0,y1-y0)E(x0,y0)dx0dy0
h(…)=(1/i𝜆)(exp(ikr01)/r01)
r01 = sqrt(z^2 + (x0-x1)^2 + (y0-y1)^2)

Question 14

Why can we not approximate r01 in the exp of h(…)=(1/i𝜆)(exp(ikr01)/r01)

Answer 14

it gets multiplied by k, which is big and so small changes in r01 can make a big difference

Question 15

In Fresnel diffraction when do we approximate r01, abd what is the approximation

Answer 15

in the denominator it is approximated by z

Question 16

What is the second approximation for Fresnel diffraction simplification

Answer 16

𝜀 ≪ 1, sqrt(1 + 𝜀) ≈ 1 + 𝜀/2 where 𝜀 is the x’s and y’s

Question 17

What does the Fresnel integral yield

Answer 17

the light wave electric field at the distance z from the screen

Question 18

What is the fresnel integral

Answer 18

𝐸 (𝑥1, 𝑦1) = exp (𝑖𝑘𝑧)/(𝑖𝜆𝑧) exp [𝑖𝑘 (𝑥1^2 + 𝑦1^2/2𝑧] double integral over 𝐴(𝑥0,𝑦0) exp {𝑖𝑘 [(−2𝑥0𝑥1 − 2𝑦0𝑦1)/(2𝑧) + (𝑥0^2 + 𝑦1^2)/(2𝑧)]} 𝐸(𝑥0, 𝑦0) 𝑑𝑥0𝑑𝑦0

Question 19

What do we typically assume about the wave incident on the aperture for diffraction

Answer 19

It is a plane wave with exp(i(wt-kz)) constant wrt x0 and y0

Question 20

What is the final fresnel integral if we ignore the factors in front and explicitly write the aperture function

Answer 20

𝐸 (𝑥1, 𝑦1) ∝ double integral over -∞ to ∞ exp {𝑖𝑘 [(−2𝑥0𝑥1 − 2𝑦0𝑦1)/(2𝑧) + (𝑥0^2 + 𝑦1^2)/(2𝑧)]} A(𝑥0, 𝑦0) 𝑑𝑥0𝑑𝑦0

Question 21

What does Fresnel theory predict for the pattern if a wave encounters a stop

Answer 21

it develops a hole (shadow of circular obstacle) which fills from the centre first as the beam diffracts, at quite a high irradiance with a bright spot - called poisson’s spot

Question 22

What if the diffraction from an array of slits

Answer 22

the talbot effect says an array of slits produces a beam pattern which alternates between 2 fringe patterns