Optics Flashcards
radio wavelength
> 1m
microwave wavelength
between 1m and 1mm
visible light wavelength
400-700nm
UV wavelength
100-400nm
IR wavelength
between 780 nm and 1 mm
x rays and gamma rays wavelength
< 10nm
direction of energy flow is given by
the poynting vector
S=1/u0 ExB
wave equation
describes its position in space
E(t,z) = Asin(wt-kz+Φ)
polarisation
direction of the electric field oscilation of a light beam
circular polarisation
E-field direction is processing around circle
elliptical polarisation
somewhere in between linear and circular
random polarisation
ordinary light source consists of a very large number of randomly orientated atomic emitters
result is unpredictable, rapidly changing polarisation
(unpolarised)
in general, light is neither completely…
polarised or unpolarised
(partially polarised)
a polariser is
an instrumnet that selects only a specific polarisation from incoming randomly polarised light
4 fundamental mechanisms that polarisers are based on
dichroism (by absorption)
reflection
scattering
birefringence
polarisation by absorption (dichroism)
selective absorption of one of the two orthogonal E field polarisation components
wire-grid polariser: polarisation parallel to wires is absorbed
wire spacing needs to be much smaller than wavelength
polarised filter, stretched polymer chains
stretched sheet of polyvinyl alcohol, get long alligned molecules
sheet dipped in iodine solution, iodine attaches to plastic molecules
conduction of electrons of iodine can then move along those molecules as if they were long wires
dichroism in crystals
some crystals have a strong anisotropy in their crystal structure
light that is perpendicular to their optical axis is absorbed
leads to different colours depending on viewing angle
Malus’ law
E parallel = EcosΦ
I=Imaxcos^2Φ
polarisation by reflection
by reflection from a higher refractive index
unpolarised light incident at polarising angle, reflected light 100% polarised perpendicular to the plane of incidence
transmitted light is partially polarised parallel to plane of incidence
derivation of Brewster’s angle
start with snell’s law
require θp+ θt=90 degrees
θt=90- θp
ni sin θp = nt sin θt
ni sin θp = nt cos θp
sin θp/cos θp = nt/ni
tan θp=nt/ni
rearrange for θp
s polarisation
not parallel
p polarisation
parallel
at Brewster’s angle, only get
s polarisation
Brewster’s angle
angle of incidnece where reflected polarised ray and refracted (slightly polarised) ray are perpendicular
polarisation by birefringence
splitting the light as the parallel and perpendicular polarisations have different refractive indices
ordinary ray
linear polarisation that is perpendicular to the optical axis which sees ‘n0’
extraordinary ray
linear polarisation at 90 degrees that is parallel to the optical axis, this polarisation sees ‘ne’
waveplates
rotation of the polarisation by a birefringent crystal
light is normal incident
there is a phase difference as light travels at a different speed
thicker material = more difference in phase
half waveplate (HWP)
lambda/2
the thickness of the crystal produces a half-wave shift between the two polarisations
phase change = pi
rotates the polarisation
quarter waveplate (QWP)
lambda/4
the thickness of the crystal produces a half-wave shift between the two polarisations
phase change = pi/2
produces elliptical or circular polarised light
3d glasses
viewer wears glasses with opposite polarising filters for each eye
each filter passes only light similarly polarised and blocks the opposite polarisation
two different ways of making 3d glasses
- linearly polarised at +/- 45 degrees
viewers must keep level head as tilting causes images to bleed over - circular polarised right/left (quarter waveplate and linear polariser
viewers can tilt head and still maintain left/right separation
polarisation by scattering
electric charges in air molecules oscillate in the direction of the E field of the incident light from the sun that produce scattered light
scattered light reaches observer
air molecules scatter blue light more than red, sky looks blue overhead
red sunset because blue has been scattered away
phasor length
amplitude
phasor angle
phase
wave intensity
I = |E|^2 = Ae^iwt . Ae^-iwt
multiplied by its complex conjugate
superposition of waves derivation
start with E=A1e^iw1t + A2e^iw2t
collect the e^iwt together and e^-1wt together
use foil to multiply out
take A1A2 out as common factor and change so signs are same in brackets
sub in 2cos(w1t-w2t)
superposition of waves derivation - if A1=A2
use 2cos^2theta = 1 + cos2theta
get I0=cos^2(delta/2)
superposition - if w1t=w2t
no phase difference
get max value
superposition - if w1t = -w2t
4A^2cos^2wt
superposition - if w1t = w2t + pi
zero
constructive interference
when two waves are in phase
for equal intensities I=A^2, the resulting intensity is (2A^2)=4I
destructive interference
when two waves are out of phase
for equal intensities, I=A^2, the resulting intensity is zero
location of minima in intensity
delta = (2m+1)pi
path diff = (m+1/2)lambda
location of maxima in intensity
delta = 2mpi
path diff = mlambda
Huygens principle
each point on a wave front acts as a point source of new waves
the envelope of secondary waves forms new wave front and so on
applications of Huygens principle
refraction
young’s double slits
young’s double slits
each slit becomes a new emitter producin a new wavefront
the interference pattern of the beams produces a sinusoidal pattern
two waves E1=A1e^itheta1 and E2=A2e^itheta2 are said to be coherent if
their phase difference (theta 1 - theta 2) is constant
oscillating in step
temporal coherence
measure of purity of the wave
different wavelength = low coherence
same wavelength = high coherence
coherence time tc
the time over which the phase of a wave at a given point remains predictable
coherence length lc
=tc c
length over which the phase of the wave remains predictable
coherence length of lasers
lasers have very narrow linewidth leading to lc»10km
spatial coherence
choose two points on same wavefront at t=0
for any t>0 the if phase difference between two points stays constant, wave has spatial coherence between those points
temporal coherence can be quantified by
measuring variations in phase difference between points on a line radiating out from the source
spatial coherence can be quantified by
measuring variations in phase difference between points at the same distance from the source, but separated laterally
coherent light
light that has properties of temporal and spatial coherence
interferometer
device using interference to measure lengths and/or indices of refraction
Mach Zehnder inferometer
laser (or other light source)
goes to square of 2 beam splitters and 2 mirrors
ends at screen
simple inferometer
source goes to beam splitter in middle
straight on or up to mirror
back to middle and down to detector
inferometer delta=
2|L1-L2|
beam splitter
usually a semi-silvered mirror
some light reflecting, some transmitted
Michelson inferometer
source eg sodium lamp
in middle to beam spliiter (half silvered surface) and compensation plate
fixed mirror straight on
moveable mirror up
telescope/eye down
purpose of the compensation plate
match the optical path lengths of the two beams
beam 2 travels 3 times through BS glass while beam 1 only travels ones
compensation plate is identical to the BS glass
why is mirror 2 movable
can be moved an accurately measurable distance to allow changes in the path difference to be measured
inferometer - ether
at the time thought light travelling through ether
as earth orbits sun, should see change as earth orbits
no results from rotating in the ehter so ruled out
mechanical wave on rope - heavy rope tied to light rope
transmitted wave continues
reflected wave moves back and undergoes no phase change
waves travel slower on thick ropes than on thin ones
mechanical wave on rope - light rope tied to heavier rope or a rigid support
reflected wave undergoes a half-cycle phase shift
EM wave propagating in optical materials - if transmitted wave moves faster than the incident wave
reflected wave undergoes no phase change
EM wave propagating in optical materials - if the incident and transmitted waves have the same speed
there is no reflection
EM wave propagating in optical materials - if the transmitted wave moves slower than the incident wave
the reflected wave undergoes a half-cycle phase shift
if n1<n2
reflected light has phase change of pi
if n1=n2
no interface so no reflection
if n1>n2
no phase change
Lloyds mirror
two sources, real (s) and virtual (s’) become analogues to the two slits in Young’s double slits experiment
dark fringe at bottom, bounced off mirror so pi phase change and destructive interference
Lloyds mirror phase difference
delta = 2pi/lambda dsintheta - pi
michelson inferometer fringe pattern
circles
going out d
d prop to theta
fringe formation for parallel mirrors - the path difference between beams 1 and 2
delta = 2dcostheta
fringe formation for parallel mirrors - why is there a pi phase difference between beams
counting phase flips during reflections
beam 1 undergoes two reflections with n1<n2
beam 2 only 1
overall pi phase change
fringe formation with phase flip - although total path difference is 2dcostheta the total phase difference is
delta = 2pi/lambda(2dcostheta) - pi
=2pi/lambda (2dcos theta-lambda/2)
fringe formation with phase flip - looking at superposition of two electric field
I12=I0cos^2(delta/2)
fringe formation - condition for constructive interference
2dcostheta=lambda(m+1/2)
fringe formation - condition for destructive interference
cocentric dark rings
2dcostheta=lambda m
fringe formation - the physical path difference ‘d’ can changed by
moving the mirror
dark port condition
if d=0 then I12=0 regardless of theta or m
corresponds to darkness across the whole field
misalignment fringes - fringe spacing
D=lambda / sin thi
approx lambda / thi at small angles
misalignment fringes - tilt angle
theta = 2 theta_mirror
if a source of white light is used, fringes will only be seen if
the path length difference is smaller than a few wavelengths
(coherence length approx 1 micro metre)
compensation plate is necessary to get
level of precision in arm length difference
the fringes for a given colour are more widely spaced the greater the wavelength and the fringes for different colours only coincide for d=0
An optical thin film can be engineered to make
a coating that reduces reflection from a surface, filters
wavelength of light or makes highly reflecting surfaces
The optical thin films will have a thickness
similar to
a single wavelength of light for which the coating is being designed.
To calculate the interference effects of a thin film we consider
r beam of light from a source that is
incident on to the surface of a thin film with refractive index n
thin films - Part of the beam is reflected at the
angle of incidence θ, and a part is
refracted at an angle θ
′ according to Snell’s law
