Wave Optics and Diffraction Part 1

Question 1

What is the general exponential function for a stationary wave and what is its trig function equal to?

Answer 1

exp(-ikx) = coskx + isinkx

Question 2

What is the wavenumber, k, equal to?

Answer 2

k = 2π/λ = ω/c

Question 3

How do we know if a wave is right or left travelling?

Answer 3

• Rearrange exponent by taking out k
• Phase of wave Ф = k(x-ct) = const
• Rearrange to find x = const + ct, so is moving is +ve x-direction
• Left travelling wave has opposite sign

Question 4

What is the function for a right-travelling wave?

Answer 4

u(x,t) = A*exp(-i(kx-ωt))

Question 5

How do we check that a wave function is a solution to the wave equation? What can we use this for?

Answer 5

Plug it into the wave equation. Can use this to find equations for variables (wave equation can only hold if these relationships are true)

Question 6

How do we write a wave function in more than one dimension?

Answer 6

u(r,t) = G*exp(-i(kr-ωt))

Question 7

What can we do to understand the wavenumber k?

Answer 7

• Imagine surface of constant phase, Ф = kr-ωt = const + 2nπ
• Wave is family of planes separated by λ = 2π/|k|
• Sub in k = k(hat) * |k| and rearrange for k(hat)*r
• Find that k(hat)*r = d, the distance from origin to closest point on plane

Question 8

What is another way of writing the k vector?

Answer 8

k(hat) * |k|

Question 9

What do both the terms mean in the equation for k(hat)*r?

Answer 9

The first part is d0, the distance to surface of plane at t=0, the second part is the distance after time t, moving away are speed c

Question 10

What do we get from taking a fourier transform of a right-moving 1D wave?

Answer 10

• u(k,ω) tells us how much of each wavenumber and frequency we have in real life space signal
• u(k’,ω’) = 𝛿(k’+k)*𝛿(ω’+ω)

Question 11

What are 3 real world phenomena with wave-like properties?

Answer 11

• Acoustics
• Optics
• Quantum Mechanics

Question 12

In interference, when is it destructive or constructive?

Answer 12

Destructive if 2 waves have phase difference (n+1/2)λ, constructive if nλ.

Question 13

What is Huygens principle of secondary sources?

Answer 13

Consider slits to have point sources inside them, and the interference is the superposition to the waves coming from all of these sources.

Question 14

What are two equations for the intensity (power per unit area) of a point source and how can these be used?

Answer 14

• I = 4πr^2 = const = u^2

- Can use this to show that u ∝ 1/r

Question 15

How can we start to find a solution to the 3d wave equation of the form u(r,t) = v(r,t)/r?

Answer 15

Use wave equation but instead of d^2u/dx^2, have ∇^2 u in 1d

Question 16

What is ∇^2 u equal to?

Answer 16

= 2/r * du/dr + d^2u/dr^2

Question 17

What do we find ∇^2 is equal to after subbing in u(r,t)?

Answer 17

1/r * d^2v/dr^2

Question 18

What is the finding after subbing in ∇^2 to the wave equation?

Answer 18

Takes same form as normal wave equation.

Question 19

What is the solution to the new wave equation in 3D?

Answer 19

v(r,t) = Aexp(ik(r/ct))+Bexp(ik(r+ct)), and u(r,t = v(r,t)/r

Question 20

What do we do to the B term in the solution to the 3D wave equation and why?

Answer 20

Set it equal to zero as we only want to look at waves travelling away from aperture not towards.

Question 21

Which two coordinate systems do we need to consider?

Answer 21

One on the aperture screen (y1,y2,y2) and one of the detector screen (x1,x2,x3)

Question 22

What are the position vectors y and x for the two screens and why?

Answer 22

y = (y1, y2, 0) since the origin lies in centre of aperture so the “z” part is zero

x = (x1, x2, D), since the detector screen is a distance D from the origin

Question 23

What is the equation for the amplitude of light at x and t from y, and what is r equal to?

Answer 23

1/r * exp(ik(r-ct))

r = x-y = |x-y|

Question 24

How do we add together all the separate sources amplitudes?

Answer 24

Do a sum over i and keep the x values the same but A and y have i values (change for each source)

Question 25

What is a good way of starting the sum to find the phase difference?

Answer 25

• Only have 2 point sources at y1 and y2, y1 = y and y2 = 0
• l1 = |x-y|
• l2 = |x-0| = |x|