Optics Flashcards
spatial frequency of a wave
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.
form of plane wave equation
E = E0 e^(i(kx x + Ky y + Kz z))
form of paraxial spherical wave equation
E = E0 (e^(ikz)/ikz) e^((ikx^2)/2z)
if propogating along z-axis, source point at origin
thin lens equation
1/s1 + 1/s2 = 1/f
s2 = image distance
s1 = object distance
beam radius
(at distance z)
w = ∆θz
∆θ = angular spread = λ/πw0
behaviour of light within fresnel zones
eg. (1) for aperature equal to first zone; (2) for aperature equal to first 2 zones
(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.
angular spacing between adjacent phasors
2πud
d = slit slacing
u = spacing = x/λz
(x=position of first zero, z=distance to observation plane)
output field general equation for plane waves
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)
outout intensity general equation
I is proportional to |E|^2 = EE*
eg. two paths at beamsplitter:
I = (I0/2)(1+cos(2k(L2 - L1)
what equation represents power in a michelson interferometer
∆L = ∆P/P * I/∆I
power = intensity * area , ∆P/P = ∆I/I
why are young’s double slits worse than the michelson interferometer for gravitational waves
- 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
fraunhofer approximation
z > > x’
paraxial distance in fraunhofer approximation,
resulting paraxial wave
rp = Z + x^2/2z - 2xx’/2z + x’^2/2z
fraunhofer: z»_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
fraunhofer approximation of paraxial spherical wave for more slit
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))
intensity of paraxial wave, fraunhofer
I proportional to N^2
where N = number of phasors
I = I0/kz
peak I = N^2 I = N^2(I0/kz)
time averaged intensity
(1/2) c (epsilon)0 E0
spatial frequency of k
kx = 2(pi)u
ky = 2(pi)v
spatial frequency of plane wave travelling at an angle
u,v = |k|sin(theta)
for two slits seperated by d, what is the intenisty and and position of minima in the far field
I = 4Iscos^2(kdx/2z)
minima @ x = (n + (1/2))(lamda)z/d
exponential trig rules
(e^ix + e^-ix)/2 = cos(x)
(e^ix - e^-ix)/2i = sin(x)
equation for width of central maxima on interference pattern
w=2(lamda)/D
equation for position of first zero along x
x = (lamda)z/a
x = (lamda)f/a
how do fresnel zones work
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
how to use wave vector when it is a set of coords {k,k,k}
y-z-x aces
x,y,z pints plotted
give an angle, fringes are observed at angle to propogation direction
fraction of intensity of a beam
area of beam : area of telescope
(ratio)
rayleigh range
zR = π(w0)^2/λ
if no light scattering is observed along the x axis at z=0, which is the initial polarisation state of the laser?
linearly polarised along x
