Optics

Question 1

spatial frequency of a wave

Answer 1

u = 1/vT (v=speed, T=time)
u = 1/d (d=spacing)

how many waves per distance aka u=0.01 m^-1 corresponds to 1 wave evry 1/0.01 metres aka every 100m.

Question 2

form of plane wave equation

Answer 2

E = E0 e^(i(kx x + Ky y + Kz z))

Question 3

form of paraxial spherical wave equation

Answer 3

E = E0 (e^(ikz)/ikz) e^((ikx^2)/2z)
if propogating along z-axis, source point at origin

Question 4

thin lens equation

Answer 4

1/s1 + 1/s2 = 1/f

s2 = image distance
s1 = object distance

Question 5

beam radius

Answer 5

(at distance z)

w = ∆θz
∆θ = angular spread = λ/πw0

Question 6

behaviour of light within fresnel zones
eg. (1) for aperature equal to first zone; (2) for aperature equal to first 2 zones

Answer 6

(1) Ra = sqrt(zλ)
all the light interferes constructivly, end up with a maximum on axis
(2) Ra = sqrt(2zλ)
light from second zone is out of phase and exactly cancels the contribution from the first giving zero intensity on axis.

Question 7

angular spacing between adjacent phasors

Answer 7

2πud

d = slit slacing
u = spacing = x/λz
(x=position of first zero, z=distance to observation plane)

Question 8

output field general equation for plane waves

Answer 8

E = E0/sqrt(2) (e^(ikr))
eg. for two paths at beamsplitter:
neglect phase chnges on reflection at splitter
E = (1/sqrt(2))(E0/sqrt(2))(e^i2kL1+e^i2kL2)

Question 9

outout intensity general equation

Answer 9

I is proportional to |E|^2 = EE*
eg. two paths at beamsplitter:
I = (I0/2)(1+cos(2k(L2 - L1)

Question 10

what equation represents power in a michelson interferometer

Answer 10

∆L = ∆P/P * I/∆I
power = intensity * area , ∆P/P = ∆I/I

Question 11

why are young’s double slits worse than the michelson interferometer for gravitational waves

Answer 11

• young’s inefficient for laser power, most of wavefront isn’t used
• from slit, optical paths form a diamond, path difference is not sensitive to a deformation of the diamond

Question 12

fraunhofer approximation

Answer 12

z > > x’

Question 13

paraxial distance in fraunhofer approximation,
resulting paraxial wave

Answer 13

rp = Z + x^2/2z - 2xx’/2z + x’^2/2z
fraunhofer: z&raquo_space;x’ so x’^2/2z tends to 0 and can be neglected

two phasor sum, paraxial wave:
E = (Es/ikr)e^(ikr) (e^(ikxd/2z) + e^(-ikxd/2z))

d/2 is slit position, so could lose the 2 depending on question

Question 14

fraunhofer approximation of paraxial spherical wave for more slit

Answer 14

middle slit at x=0 therefore = 1 (because e^0)
k = 2π/λ
at second slit, k = 4π/λ, and so on.
should have a +ve and a -ve term for each slit

eg.
5 slits, sapcing d, centred around x=0:
E = (E0e^(ikr)/sqrt(ikz)) (e^(i4πdx/λ) + e^(i2πdx/λ) + 1 + e^(-i2πdx/λ) + e^(-i4πdx/λ))

Question 15

intensity of paraxial wave, fraunhofer

Answer 15

I proportional to N^2
where N = number of phasors

I = I0/kz
peak I = N^2 I = N^2(I0/kz)

Question 16

time averaged intensity

Answer 16

(1/2) c (epsilon)0 E0

Question 17

spatial frequency of k

Answer 17

kx = 2(pi)u
ky = 2(pi)v

Question 18

spatial frequency of plane wave travelling at an angle

Answer 18

u,v = |k|sin(theta)

Question 19

for two slits seperated by d, what is the intenisty and and position of minima in the far field

Answer 19

I = 4Iscos^2(kdx/2z)
minima @ x = (n + (1/2))(lamda)z/d

Question 20

exponential trig rules

Answer 20

(e^ix + e^-ix)/2 = cos(x)
(e^ix - e^-ix)/2i = sin(x)

Question 21

equation for width of central maxima on interference pattern

Answer 21

w=2(lamda)/D

Question 22

equation for position of first zero along x

Answer 22

x = (lamda)z/a
x = (lamda)f/a

Question 23

how do fresnel zones work

Answer 23

a circular aperture can be considered to be made up of a series of circular fresnel zones.
At a distant point z on this axis, x-rays from alternate fresnel zones contribute with opposite phases to total electric field.
A zone plate consists of rings that block or phase shift the out-of-phase components such that all transmitted light has the same phase - leading to construcitve interference & focussing effect

Question 24

how to use wave vector when it is a set of coords {k,k,k}

Answer 24

y-z-x aces
x,y,z pints plotted
give an angle, fringes are observed at angle to propogation direction

Question 25

fraction of intensity of a beam

Answer 25

area of beam : area of telescope

(ratio)

Question 26

rayleigh range

Answer 26

zR = π(w0)^2/λ

Question 27

if no light scattering is observed along the x axis at z=0, which is the initial polarisation state of the laser?

Answer 27

linearly polarised along x