Wave Optics and Diffraction Part 1 Flashcards
What is the general exponential function for a stationary wave and what is its trig function equal to?
exp(-ikx) = coskx + isinkx
What is the wavenumber, k, equal to?
k = 2π/λ = ω/c
How do we know if a wave is right or left travelling?
- 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
What is the function for a right-travelling wave?
u(x,t) = A*exp(-i(kx-ωt))
How do we check that a wave function is a solution to the wave equation? What can we use this for?
Plug it into the wave equation. Can use this to find equations for variables (wave equation can only hold if these relationships are true)
How do we write a wave function in more than one dimension?
u(r,t) = G*exp(-i(kr-ωt))
What can we do to understand the wavenumber k?
- 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
What is another way of writing the k vector?
k(hat) * |k|
What do both the terms mean in the equation for k(hat)*r?
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
What do we get from taking a fourier transform of a right-moving 1D wave?
- 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)
What are 3 real world phenomena with wave-like properties?
- Acoustics
- Optics
- Quantum Mechanics
In interference, when is it destructive or constructive?
Destructive if 2 waves have phase difference (n+1/2)λ, constructive if nλ.
What is Huygens principle of secondary sources?
Consider slits to have point sources inside them, and the interference is the superposition to the waves coming from all of these sources.
What are two equations for the intensity (power per unit area) of a point source and how can these be used?
- I = 4πr^2 = const = u^2
- Can use this to show that u ∝ 1/r
How can we start to find a solution to the 3d wave equation of the form u(r,t) = v(r,t)/r?
Use wave equation but instead of d^2u/dx^2, have ∇^2 u in 1d
What is ∇^2 u equal to?
= 2/r * du/dr + d^2u/dr^2
What do we find ∇^2 is equal to after subbing in u(r,t)?
1/r * d^2v/dr^2
What is the finding after subbing in ∇^2 to the wave equation?
Takes same form as normal wave equation.
What is the solution to the new wave equation in 3D?
v(r,t) = Aexp(ik(r/ct))+Bexp(ik(r+ct)), and u(r,t = v(r,t)/r
What do we do to the B term in the solution to the 3D wave equation and why?
Set it equal to zero as we only want to look at waves travelling away from aperture not towards.
Which two coordinate systems do we need to consider?
One on the aperture screen (y1,y2,y2) and one of the detector screen (x1,x2,x3)
What are the position vectors y and x for the two screens and why?
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
What is the equation for the amplitude of light at x and t from y, and what is r equal to?
1/r * exp(ik(r-ct))
r = x-y = |x-y|
How do we add together all the separate sources amplitudes?
Do a sum over i and keep the x values the same but A and y have i values (change for each source)
What is a good way of starting the sum to find the phase difference?
- Only have 2 point sources at y1 and y2, y1 = y and y2 = 0
- l1 = |x-y|
- l2 = |x-0| = |x|
