University of Leicester
Department of Physics and Astronomy
Lecture Notes
Lasers and Quantum Optics

Dr. R. Willingale

Dec 7, 2007

Contents

1  Books
2  Introduction
    2.1  Coherence - quantum cooperation
    2.2  Polarization - angular momentum
    2.3  Fields and photons
    2.4  Classical field theory
    2.5  Quantum field theory
    2.6  Properties of photons
3  Lasers
    3.1  Stimulated emission - cloning photons
    3.2  Population inversion
    3.3  Pumping - 3 and 4 level systems
    3.4  Optical feedback - resonant cavities
    3.5  Line broadening
    3.6  Laser modes
    3.7  The hallmarks of laser activity
4  Types and classes of laser
    4.1  Doped insulator lasers
    4.2  Semiconductor lasers
    4.3  Atomic gas lasers
    4.4  Ion lasers
    4.5  Molecular lasers- the carbon dioxide laser
    4.6  Liquid dye lasers
    4.7  The free electron laser
    4.8  The properties of laser light
5  Optical wave guides - fibre optics
    5.1  Planar wave guides - a simple wave model
    5.2  The uncertainty principle
    5.3  Polarization states
    5.4  Rays in a cylindrical cladded fibre
    5.5  Fibre optics
    5.6  Step index and graded index fibres
    5.7  Dispersion in fibre optics
    5.8  Losses in fibre optics
    5.9  Communication using fibre optics
    5.10  A communication example
6  Polarization
    6.1  The linear polarizer
    6.2  The law of Malus
    6.3  Dichroic crystals
    6.4  Complex refractive index
    6.5  The wire grid polarizer
    6.6  Dichroic sheet - Polaroid
    6.7  Polarization by scattering - dipole scattering
    6.8  Polarization by reflection
7  Optical anisotropy
    7.1  Birefringence - double refraction
    7.2  Optical activity
    7.3  Retarders
8  Induced optical anisotropy
    8.1  Faraday rotation - magnetically induced optical activity
    8.2  Liquid crystals
    8.3  The Kerr and Pockels effects
9  Nonlinear optics
    9.1  Solitons
    9.2  Frequency doubling and mixing
10  What are photons?
    10.1  Quantization of the free EM field
    10.2  Quantum states of the EM field

1  Books

2  Introduction

There are 4 underlying forces at work in physics
Gravity is only important in large scale phenomena.
The strong and weak nuclear forces only come into play in nuclear physics when we deal with the very microscopic world and high quantum energies. e.g. radioactivity
The electrostatic force between 2 protons is 1038 > gravitational attraction. The electromagnetic interation between electrons (charge) and photons dominates in almost all commonly observed phenomena. Therefore classical electromagnetism or quantum electrodynamics are at the heart of much of physics.
Light is electromagnetic and the physics of light, optics, is central to many physical situations. We are interested in how light behaves and ultimately we want to answer the question "What is light?" In order to study light we must concern ourselves with a large number of subjects, for example:
There are 2 threads in the course:

2.1  Coherence - quantum cooperation

Coherence is associated with the wave like nature of light. It manifests itself in interference and/or diffraction effects.
Temporal coherence and the harmonic content of the wave at a fixed point in space.
Perfect temporal coherence f(t) = A exp(iw0t)
Take Fourier transform F(w) = Cd(w-w0)
Only 1 frequency is present and the wave is continuous in time. In practice the length of the wave is finite, a pulse of length Dt. This leads to a spread in frequency:
Df » 1/Dt = Dw/2p
For a Gaussian wavepacket we have:
f(t)=A exp(iw0t) exp(-t2/2Dt2)
F(w)=Bexp(-Dt2 (w0-w)2/2)
If have successive wave packets arriving randomly then:
C(t)=åm=1n Am exp(iw(t-tm))exp(-(t-tm)2/2Dt2)
and the resulting amplitude is µ n1/2.
If all the same length and frequency then still get the same bandwidth.
Can express a bandwidth or coherence time using a coherence length. This amounts to taking a snapshot of the packet as it passes and measuring its length. If the coherence time is Dt then length of packet in direction of travel must be cDt = Dx.
The concept of coherence length is important in amplitude splitting interference devices such as the Michelson Interferometer.
We can also study the coherence of light perpendicular to the direction of travel. This is known as lateral coherence or spatial coherence and is important in wavefront splitting interference such as seen from Young's slits.
The combination of temporal and lateral coherence define a coherence volume.
We characterise or quantify coherence using correlation functions. Such correlations are found by taking time averages of the wave. This removes fluctuations due to the discrete wave trains or packets.
When we calculate interference or diffraction patterns we are finding correlation functions. We combine component complex amplitudes arising from both the temporal (frequency) and spatial (wavelength) characteristics of the travelling wave.
Likewise when we measure interference or diffraction patterns the form of the distribution tells us something about the coherence of the light. Different degrees of coherence are illustrated in Figs. 1, 2,3 and 4.
lect4313_fig1.ps
Figure 1: Perfect coherence
lect4313_fig2.ps
Figure 2: Equal temporal and lateral coherence
lect4313_fig3.ps
Figure 3: High lateral and low temporal coherence
lect4313_fig4.ps
Figure 4: High temporal and low lateral coherence

2.2  Polarization - angular momentum

The polaraization of light is described using different polarization states:
Natural light has a random polarization as a function of time.
Plane polarized light can be described as a linear combination of right and left states P=L+R. The electric field components of the P-state are shown in Fig. 5.
lect4313_fig5.ps
Figure 5: P-state of polarization
Circular light can be described as the summation of 2 orthogonal P states out of phase by p/2. The electric field components of the R-state are shown in Fig. 6.
lect4313_fig6.ps
Figure 6: R-state of polarization
The polarization of light is important in reflection, scattering, birefringence, optical activity.
Retarders and polarizers can be used to convert between states,
P « R or L
What happens to electrons exposed to circular light? Using the 2 P state picture we see that the electron is driven sinusoidally in 2 orthogonal directions but with a p/2 phase difference. Therefore a free electron will move in a circle with angular frequency w. Thus angular momentum is imparted to the electron.
The power delivered is the rate of energy dissipation dE/dt and this is equal to the product of the torque T and w
dE/dt = T w
But T=dL/dt is the rate of change of angular momentum. Therefore assuming all the light (photon) energy is absorbed by the electron we can integrate giving
L=E/w
R or L states have associated angular momenta described by vectors. The R state rotates clockwise looking towards the source and is represented by [R\vec] pointing away from the source. The L state is represented by [L\vec] a vector pointing towards the source (opposite to the [k\vec] wave vector).

2.3  Fields and photons

A light beam has coherence that depends on the nature and geometry of the source and region of propogation. The coherence depends on the bandwidth Dw which in turn depends on the energy bandwidth DE.
A light beam has a polarization and carries angular momentum.
A light beam carries energy - irradiance - associated with the square of the amplitude of the electric field vector.
How are these properties tied up with the emission and absorption processes?

2.4  Classical field theory

Described by Maxwell's equations. Charge, stationary and moving, generates electric and magnetic fields which in turn impose forces on other charge (Lorentz force). The classical E-M field is physical, it is not just a mathematical trick to calculate the forces between charges.
Consider a parallel plate capacitor charged up with a vacuum between the plates. There is an E-field in the gap. Now imagine the plates are moved apart. This requires work to be done against the attractive force between the 2 sheets of charge on the plates. Afterwards the plates etc. look exactly the same. The only change is that there is a greater volume of vacuum between the plates. Yet work has been done and more energy is stored in the system. This energy is in the electric field between the plates. The field contains energy and has an equivalent mass given by E=mc2. There is no ether to support the field and it has energy (or mass) as real as any other stuff.

2.5  Quantum field theory

All interactions involve the creation and annihilation of particles. Forces come about through the exchange of lumps of energy or quanta of the field in question. The field itself is quantized. Classically we describe interference by summing electric field amplitudes and finding the intensity by calculating |amp|2.
In QFT we consider the probabilities of particles arriving at some point from a source. These probabilities (or probability densities) are calculated from wave functions Prob=|f|2. We sum the probabilites in the following way f = f1+f2+f3+¼. Then
Prob=|f1+f2+f3+¼|2
NOT Prob=|f1|2+|f2|2+|f3|2+¼
This process naturally leads to interference. Consider a beam reflecting off a plane mirror. We must sum the probabilites over all possible paths in a similar way to the application of Fermat's principle. In the summation it is only contributions near to the classical reflection point that add up in phase. the contributions from the various points over the mirror surface stack together like a Cornu spiral. Contributions from remote areas of the mirror surface or an aperture are small and have little effect on the resultant probability. We only start to see interference effects when the dimension of the mirror or aperture become comparable to the wavelength of the functions fn.
So our classical E-M field becomes a quantized field which describes the probability of detecting a photon.

2.6  Properties of photons

They carry energy E=hn
They carry momentum P=(E2-m02c4)1/2/c
But m0=0, zero rest mass, so P=E/c=h/l
They carry angular momentum |L|=E/w = ±(h/2p)
where ± refer to the R and L polarization states or photon spin.
So emission of photons puts energy, momentum and angular momentum into the quantum field in lumps and absorption removes them from the field in lumps.
Is the quantum field real?
Of course. A photon detector placed in such a field (light beam), receives energy, momentum and angular momentum from the field. The detector feels the beam!
Coherence is quantum cooperation. A very large number of photons are in the same state. This is possible because photons are bosons with spin ±1. They are force carriers.
Interference. Photons can only interfere with themselves. Different photons NEVER interfere. So different sources of photons don't interfere. Interference between sources is done by splitting the field from a single source. You can set up Young's slits with such a low intensity of light that at any time only 1 photon is interacting with the slits. It still works.
Polarization. When a photon is detected it can be in a R or L state (spin ±1). Therefore you can extract angular momentum from a circularly polarized beam.

3  Lasers

3.1  Stimulated emission - cloning photons

A photon is emitted when an electron in an atom undergoes a transition between 2 energy levels.
n = DE/h=(E2-E1)/h
This can occur in 2 ways:
We normally only see spontaneous emission. This requires just an atom rather than an atom plus a photon and consequently the transition probability is usually much higher.
The opposite process, absorption, always requires a photon plus an atom and it is really the reverse of stimulated emission.
Stimulated emission was demonstrated to exist by Einstein in 1917. It can be regarded as a resonance phenomenon. We can derive relationships between the processes by considering a simple system in thermal equilibrium.
In equilibrium the average number of upward transitions must equal the average number of downward transitions. Let N1 be the number density of atoms in the low state and N2 be the number density in the high state. Let rn be the energy density of the light at frequency n.
rn=Nnhn where Nn is the number of density of photons per unit frequency interval.
There are 3 transition rates:
In equilibrium N1rnB12=N2rnB21+N2A21
or rn=[(A21/B21)/((B12N1/B21N2)-1)]
In thermal equilibrium Boltzman statistics give:
Nj=[(gjN0exp(-Ej/kT))/(åigiNiexp(Ei/kT))]
where N0 is the total population density and Nj is the occupancy and gj is the degeneracy of the jth level. Using this we get:
N1/N2=(g1/g2)exp(hn/kT)
But rn is also given by the Planck distribution for black-body radiation:
rn=[(8phn3)/(c3)][1/(exp(hn/kT)-1)]
So equating the 2 expressions for rn gives us
g1B12=g2B21 and A21/B21=8phn3/c3
These are known as the Einstein relations and A12, B12 and B21 are called the Einstein coefficients. The ratio of spontaneous to stimulated emission is given by
R=A21/rnB21=exp(hn/kT)-1
For example in a hot filament at 2000K with n » 5×1014 then R=1.5×105. Stimulated emission is completely swamped by spontaneous emission.
In order to boost the stimulated component we must increase N2/N1 and rn. This is achieved in the LASER process - Light Amplification by Stimulated Emission of Radiation.

3.2  Population inversion

Imagine we have a collimated, monochromatic beam passing through an absorbing medium containing just 1 relevant transition E2 «E1 with n12=(E2-E1)/h.
If a is the absorption coefficient
dI(x)/dx=-aI(x) Þ I=I0exp(-ax)
Ignoring scattering losses and spontaneous emission which is isotropic
-dNn/dt=N1rnB12-N2rnB21
so -dNn/dt=(g2N1/g1-N2)rnB21
If n is the refractive index In=rnc/n=Nnhn12 c/n
DNn(x)=Dx [(dIn)/dx] n/hn12 c
but Dt=Dxn/c so we get
dNn/dt=[(dIn)/dx][1/(hn12)] = -aIn/hn12
dNn/dt=-arnc/nhn12
Therefore a is given by
a = (g2N1/g1-N2)B21h n12 n/c
If a is negative N2 > g2N1/g1 and the beam intensity will increase exponentially I=I0exp(kx) where
k = -a is called the small signal gain.
To get positive gain we require a population inversion. Such an inversion requires energy from a pumping process. The pumping produces nonthermal equilibrium.

3.3  Pumping - 3 and 4 level systems

Because B21=B12 we must use a third level to get N2 > g2N1/g1.
Pumping is into an energy level above the top level of the LASER transition. This level must decay rapidly to fill the top level. Often the pumped levels consist of a band which give a high level of absorption to a wide spectrum.
The 3 level system is inefficient because the lower level of the LASER transition is always populated.
A 4 level system is much more efficient. In this case the population in E1 will be low providing E1-E0 > kT.
The 3 and 4 level systems are illustrated in Figs. 7 and 8. In each case the left-hand diagram shows the distribution at thermal equilibrium (Boltzman) and the right-hand diagram shows the distribution with pumping.
lect4313_fig7.ps
Figure 7: 3 level laser system
lect4313_fig8.ps
Figure 8: 4 level laser system
In order to achieve and maintain a population inversion we must excite electrons into the short lifetime levels above the upper LASER level. There are a number of ways this can be done.

3.4  Optical feedback - resonant cavities

The LASER scheme described so far amplifies the light beam but the gain per unit length in the active medium is usually rather small. To get large gains we need a long path all of which is pumped. A neat way of getting this is to fold the beam on itself by reflection between 2 mirrors. This traps the beam in a resonant cavity often called a Fabry Perot Resonator.
Different forms of cavity are used for different applications:
These are shown in Fig. 9.
lect4313_fig9.ps
Figure 9: Laser cavity geometries
In each case one mirror is made highly relecting while the other is made to leak so that the beam penitrates to the ouside world. In many lasers tuning is achieved using filters and/or quarter wave stacks on the mirrors to select specific wavelengths.
The power output is limited by a number of loss mechanisms:
A laser cavity has a characteristic life time. Let the loss (output beam) at one end of the cavity be:
DW = -W (1-R)
where W is the energy density and R is the reflectivity.
The time for 1 round trip up and down is:
Dt = [2 L/v]
where L is the length of the cavity and v is the velocity.
So we have the rate of loss of energy density as:
[(DW)/(Dt)] = -W(1-R)[v/2 L]
Integrating we get:
ln(W) = ln(Wo) -(1-R) [vt/2L]
So the time constant of the cavity is:
tc = [2L/(v(1-R))]
W=Wo exp( [(-t)/(tc)])
If the gain is switched off then the output decays exponentially. This can control the coherence of the output beam.
e.g. R=0.99, L=5mm, tc ~ 3.3 ns
Dn = [1/(tc)] = 3×108 Hz.
or correlation length:
ctc = 1 m
If the frequency is 3×1014 Hz (optical) then the Q factor is given by:
Q=[(n)/(Dn)] ~ 106.

3.5  Line broadening

So far we have ignored the fact that the spectral lines or transitions have a finite width, they are not truly monochromatic.
The small signal gain must be written as:
k(ns)=(N2-N1g2/g1)B21hnsng(ns)/c
where g(n) is the line profile function and the beam is monochromatic with frequency ns.
The shape of g(n) arises from the following:
If all atoms yield the SAME centre frequency then a Lorentzian profile results. This is the case for natural broadening.
g(n)=[(Dn)/(2p)][1/((n-n0)2+(Dn/2)2)]
If each atom yields a different centre frequency as in the case of Doppler broadening, then a Gaussian profile results.
g(n)=[2/(Dn)]Ö{ln(2)/p}exp(-ln(2)([(n-n0)/(Dn/2)])2)
Note I(n,x)=I(n,0)exp(k(n)x)
Since k(n) µ g(n) we do NOT get the line profile on amplification. The non-linear response boosts the centre frequencies and produces spectral narrowing.
Typical transmission and emission profiles for a single transition are shown in Fig. 10.
lect4313_fig10.ps
Figure 10: Transition profiles of transmission and emission

3.6  Laser modes

In the resonant cavity formed by the mirrors we get a standing wave pattern of electric field intensity.
We know pl/2=l where p is an integer and l is the optical path length of the cavity. Therefore we get a series of frequencies given by:
np=pc/2l
Each p defines an axial mode of the cavity. The separation of the modes is therefore
dn = c/2l
Only those modes which lie within the peak of the gain profile will be amplified and maintained. Note that the peaks are much narrower than the Fabry-Perot resonance response because of the gain.
Fig. 11 illustrates how the combination of the gain profile and cavity modes give rise to a series of closely packed excited modes.
lect4313_fig11.ps
Figure 11: Excited laser modes within the gain profile
The width of the mode peaks is usually described in terms of a Q factor.
Q=[(2p(energy stored))/((energy dissipated per cycle))]=n/Dn
Typically Q=108 which is very large compared with, say, 100 for an electrical oscillator.
The Dn » 1MHz compared with 109 Hz for a Fabry-Perot cavity.
If care is taken to reduce losses the Q can be very high.
The electric field vector is always transverse so we get so called TEM's, transverse electric modes.
The modes are labelled by the number of minima in the electric field intensity seen when scanning up or across the cavity. So we get TEM00, TEM01 etc..
The TEM00 mode adjusts itself so that the mirror surfaces are surfaces of constant phase.
The TEM00 mode can be selected out using a suitable aperture in the centre of the cavity producing a so called uniphase mode.
Fig. 12 shows the intensity distribution of TEM modes at the centre of a laser cavity.
lect4313_fig12.ps
Figure 12: Intensity of TEM modes at centre of laser cavity
Fig. 13 illustrates the TEM00 mode. The value of w0 on the diagram is usually defined to be the width of the mode such that the field amplitude has dropped to 1/e of its maximum value.
lect4313_fig13.ps
Figure 13: TEM00 mode
Uniphase operation gives very high spectral purity.
Multimode operation can give high power.

3.7  The hallmarks of laser activity

Laser activity can be distinguished from amplified spontaneous emission, superluminescence or similar phenomena by:

4  Types and classes of laser

4.1  Doped insulator lasers

The active medium is solid, usually crystalline, containing impurities often introduced by doping.
Historically this was the first class of laser, using ruby crystals.
A more modern example is Nd:YAG which consists of ythrium aluminium garnet (Y3Al5O12) with neodymium Nd3+ impurity in ythrium sites. It is this impurity which does the work.
Fig. 14 is a simplified energy level diagram of a Nd:YAG system. Essentially a 4 level system with a laser transition of l = 1.06mm (1.17eV).
lect4313_fig14.ps
Figure 14: The Nd:YAG energy level diagram
Pumping is by optical flash, using a light pulse of duration 1ms. Fig. 15 shows a schematic of the complete optical system.
lect4313_fig15.ps
Figure 15: Schematic of the optical system in a doped insulator laser
The laser starts 0.5ms after the pumping flash starts and the output consists of a series of 1ms pulses separated by 1ms. This is because the energy loss due to the LASER action is faster than the pumping. Such pulsing behaviour reduces the coherence of the laser light because the spikes are unrelated.
The efficiency is only about 0.1%. By contrast the related Nd:Glass laser produces 3 times the power of Nd:YAG but has a broader line width.

4.2  Semiconductor lasers

They are basically a diode junction. In order to get LASER action there needs to be a region where BOTH excited electron states and holes (vacant electron states) are present.
This is achieved using heavily doped n and p material and applying a forward bias to the junction. Fig. 16 shows the energy levels either side of the diode junction.
lect4313_fig16.ps
Figure 16: Energy levels in a semiconductor laser (a) in equilibrium and (b) with forward bias
Under forward bias the active region is only thin, for example in GaAs at room temperature 1 to 3 mm. This thickness is controlled by the diffusion length of the electrons. The geometry of the active region is shown in Fig. 17.
lect4313_fig17.ps
Figure 17: The active region of a semiconductor laser
No external mirrors are used. The faces are cleaved or ground perpendicular to the junction. Using Fresnel's equations the normal incidence reflectivity is determined by the refractive index. For example n=3.6 for GaAs which gives a reflectance of 0.32.
The active junction region has a slightly different n so acts as a waveguide which contains the laser light. A schematic view of the complete semiconductor laser is given in Fig. 18.
lect4313_fig18.ps
Figure 18: A semiconductor laser
The pumping energy comes from the diode current. LASER action is induced above a certain threshold current density.
The principle loss mechanism is scattering by optical inhomogeneities in the active volume.
Because the active region is thin the exit beam spreads due to diffraction. The beam fans out perpendicular to the junction plane.
q = l/t, if t=3mm and l = 0.84mm then q = 19°.

4.3  Atomic gas lasers

The common He-Ne laser. Ne provides the energy levels and He provides pumping by electron and atomic collisions. Pumping is by a two stage process:
e1+He=He*+e2 an electron looses energy to He atom.
He*+Ne=Ne*+He excited He collides with a Ne atom.
So pumping is by d.c.discharge, 2 to 4 kV across a tube of gas at 10 torr.
A complete system is shown in Fig. 19 and the energy level diagram for the He-Ne laser is shown in Fig. 20.
lect4313_fig19.ps
Figure 19: The atomic gas laser
lect4313_fig20.ps
Figure 20: The energy level diagram of the He-Ne laser
Low power but high collimation, polarization selected by Brewster window and Dl extreemly small. The Fresnel reflectivities utilized in a Brewster window are shown in Fig. 19.

4.4  Ion lasers

The principle of operation is similar to atomic gas lasers but high discharge currents are used to strip the atoms of electons to form ions.
An example is the Ar ion laser. Powerful Continuous Wave operation can be achieved by complicated discharge geometry to maximize the pumping power.
Several watts CW output or upto 1 kilowatt in microsecond pulses can be generated.
High discharge current 15-50 A. Pump to 4P states, 35eV above ground state by multiple collisions. Transitions correspond to 4P-4S, 514.5nm and 488nm. Use Brewster window at ends of gas tube to isolate a single polarisation with minimum reflection losses.
The construction of a typical argon ion laser is shown in Fig. 21.
lect4313_fig21.ps
Figure 21: The construction of an argon ion laser
Used to pump dye lasers (see below).

4.5  Molecular lasers- the carbon dioxide laser

Very important in technological and industrial applications.
Use the vibrational modes of the molecule. Get the output in the IR. For CO2 laser the main transition is at a wavelength of 10.6mm. The vibrational modes of the molecule are shown in Fig. 22 and the energy level diagram for the CO2 system is shown in Fig. 23.
lect4313_fig22.ps
Figure 22: The vibrational modes of a CO2 molecule
lect4313_fig23.ps
Figure 23: The energy level diagram of a CO2 laser
Again use d.c. discharge to pump but because the coupling between the energy levels is so high this type of laser can be very efficient, upto 30%.
Can easily obtain 100 watts CW from a laser 1 metre long.
Using gas dynamic pumping can get an incredible 100kW of CW power.
Compress and heat a mixture of nitrogen and carbon dioxide. In this state a large amount of energy is stored in vibrational modes of the N2. If the gas is then allowed to expand fast into a low pressure region the temperature drops and some energy is transfered by resonant collisions into the (001) state of the CO2 thus creating a population inversion.

4.6  Liquid dye lasers

These contain an organic dye in a solvent.
Such dyes can be excited by absorption of short wavelengths and fluoresce by emitting at longer wavelengths.
There are a large number of electronic energy levels in bands. Therefore get a large number of possible LASER transitions and such lasers are tunable.
Pumping is done optically using radiation from another laser, for example a Ar ion laser.
A schematic of the energy level diagram of a liquid dye laser is given in Fig 24. The output spectra from various dyes are shown in Fig. 25. The optical elements of a tunable laminar flow dye laser are shown in Fig 26.
lect4313_fig24.ps
Figure 24: Schematic energy level diagram of a liquid dye laser
lect4313_fig25.ps
Figure 25: Output spectra from various dyes
lect4313_fig26.ps
Figure 26: The optical layout of a tunable flow dye laser
Use laminar flow in a thin layer to prevent build up of absorption losses (absorption T1 to T2).
The small signal gain is relatively high because in the liquid (dense) state.

4.7  The free electron laser

Unlike the other types of laser there is no medium which contains bound electrons.
The free electrons are stripped from atoms in an electron gun and accelerated to relativistic velocities.
The beam of electrons is injected through an undulator - a periodic array of magnetic dipoles. An initial radiation field from a seed laser or spontaneous emission in the undulator is amplified by the interaction of the electrons with the electromagnetic radiation field. As the electrons pass through the undulator they are accelerated up and down in the transverse direction by the magnetic field. They move in a sinusoidal path and emit Synchrotron Radiation.
lect4313_fig27.ps
Figure 27: Basic components of a FEL
lect4313_fig28.ps
Figure 28: The undulator at the SRC University of Wisconsin Madison
The radiation field grows exponentially and this is accompanied by pronounced longitudinal density bunching of the electrons.
The wavelength of the laser is not limited by specific transition energies but depends on the accelerator energy and the undulator properties. VUV and X-ray wavelengths are possible.
Can use mirror cavities in optical but for VUV and X-ray wavelengths mirrors don't reflect at normal incidence and so one pass through a long undulator must be used. In their rest frame the electrons emit dipole radiation but because they a travelling at relativistic velocity the radiation in the laboratory frame is beamed towards the forward direction with open angle:

1
g
 = mec2
Ee
The deflection of the electrons from the forward direction is comparable to this opening angle so that the radiation generated by electrons along a period of the undulator, lu overlaps (interferes constructively) giving the first harmonic radiation wavelength:

lph= lu
2g2
 (1+Krms2)
where Krms is the ratio of the average deflection angle of the electrons to the typical opening angle of the synchrotron radiation. If Bu is the rms magnetic field in the undulator:

Krms= eBulu
2pme c
The interference condition means that in travelling one period along the undulator the electrons slip one period of the radiation wavelength (because the electromagnetic field moves faster than the electrons).
lect4313_fig29.ps
Figure 29: The electron orbit
The electrons interact with their own spontaneous emission. Electrons that are oscillating in phase with the EM wave are retarded while those out of phase are accelerated. This leads to electron bunching which in turn amplifies the EM field and the electrons then radiate in phase with the radiation.
To get exponential amplification of this spontaneous emission you need a very monochromatic electron beam, high electron density, precise magnetic field and many periods in the undulator.
In the beginning, with a bunch of Ne electrons, the spontaneous emission power µ Ne. Later with bunching the power µ Ne2 and the stimulated process dominates.
The exponential gain is expressed using a gain length Lg. The power increases as:

P(x)=P0 exp(2x/Lg)
where Lg » lu/(4pr).
r = Dw/w is the FEL amplifier bandwidth or ratio of output radiation power to electron beam power. This parameter depends on the quality of the electron beam and undulator and is typically 10-3 to 10-4.
The radiation has very high lateral coherence and is plane polarized.
The temporal coherence is limited by the factor r above. This is due to the inital shot-noise and the way the electron bunching occurs.
The VUV FEL under construction at the TESLA Test Facility (TTF) at DESY (Deutsches Electronen-Synchrotron) uses the ßelf-amplified spontaneous emission" (SASE) mode. It will deliver sub-picosecond radiation pulses with gigawatt peak powers down to a wavelength of 6 nm. So far down to 80 nm.

4.8  The properties of laser light

To greater or lesser extent laser light is:
The intensity depends on the pumping power and the efficiency of the LASER mechanism. There is always a tradeoff between the intensity and the coherence and/or collimation.
The collimation is set by diffraction.
q = Kl/D where K » 1 and D is the effective aperture diameter.
Typical angular beam widths are:
Type milli radians
He-Ne 0.5
Ar 0.8
CO2 2
Dye 2
Nd:Glass 5
GaAs 20 by 200
The coherence depends on the number of modes excited and the duration of the pulses. Typical coherence values expressed as lengths are:
Type coherence length m
He-Ne (single mode) upto 1000
He-Ne (multimode) 0.2
GaAs 1×10-3
Nd:Glass 2×10-4

5  Optical wave guides - fibre optics

Light can be transmitted from A to B using a transparent dielectric fibre or light pipe.
The principle has been known since 1847 but it is only with the recent development of very thin cladded dielectric fibres that light pipes have become useful. So called fibre optics now play an important role in communications systems.
The problems that need to be overcome are:
If the diameter of the fibre is large compared with the wavelength, l, then the propogation process can be described by geometric (ray) optics.
If the diameter is the same order as l then we must use wave (physical) optics to describe the behaviour. In this case the fibre acts as a wave guide.
Fibre optics utilize Total Internal Reflection (TIR) at the interface between 2 media, where the light is travelling in the denser medium. This is illustrated in fig. 30.
lect4313_fig30.ps
Figure 30: A ray in a planar wave guide n1 > n2
The critical angle of reflection occurs when qt=90° and the transmitted ray travels parallel to the surface. Using Snell's law at the critical angle we get:
sinqi=nc/nf where nf > nc and the light is incident from nf.
So the critical angle is given by qc=sin-1(nc/nf)

5.1  Planar wave guides - a simple wave model

If the diameter of the fibre, D, is small we must include the effects of interference as the light propogates. To do this we will consider a simple planar waveguide like a microscope slide.
A plane wave is propogating at reflection angle qr down a planar guide of width d. Points A and C lie on a common wavefront of all the components travelling in the direction A to B. See fig. 31.
lect4313_fig31.ps
Figure 31: The interference condition in a planar waveguide
The phase difference introduced by reflection A to B to C must be 2p if we are to achieve propogation. Under this condition all the reflected wavefronts interfere constructively.
(AB+BC)2pnf/lo-2f = 2pm where m is an integer and f is the phase change suffered at each reflection.
AB+BC=BC(cos2qr+1)=2BCcos2qr=2Dcosqr
Therefore we get propogation if:
2pnfDcosqr/lo-f = pm
Unfortunately f = f(qr) so we can't solve this equation easily.
However rearranging gives:
m=2Dnfcosqm/lo-f/p
where qm is the reflection angle for the mth mode.
But since we have TIR sinqm > nc/nf or
cosqm < Ö{1-(nc/nf)2} so
m £ (2Dnf/lo)Ö{1-(nc/nf)2}-f/p
Since f £ p we can estimate the maximum number of modes that can propogate in the wave guide.
If we define V=(pD/lo)Ö{n2f-n2c} then
m £ (2V-f)/p
V is called the normalised thickness of the film. If 2V < f then there is no propogation.
As you might expect, the larger the normalised thickness V, the greater the number of modes which can propogate.
Fig. 32 shows a ray injected into the and face of the waveguide.
lect4313_fig32.ps
Figure 32: Injection of a ray into the waveguide. a is the injection angle.
In Fig. 32 if q < qc then we get no TIR and hence no propogation. This defines an aimax.
sinqc=nc/nf==cos(90-q) = Ö{1-sin2(90-q)}
But from Snell's law sin(90-q)=nosina/nf so
(nc/nf)2=1-(no/nf)2sin2aimax or
nosinaimax=Ö{nf2-nc2}
nosinaimax is called the numerical aperture. It determines the speed or light gathering power of the system.

5.2  The uncertainty principle

The argument above shows that light will not propogate down a wave guide if the thickness (or more strictly the normalised thickness) is too small. From the quantum mechanical viewpoint this says that photons will not propogate in the guide if the dimensions are too small. Why not?
Using the de Broglie relationship the momentum of a photon is:
P=[(h nf)/(lo)]
At the limit when m=1 and cosqm=Ö{1-(nc/nf)2} we get
1 £ [ D P/h] 2 cosqm-[(f)/(p)]
But P cosqm=Py is the transverse component of the momentum and D=Dy is the uncertainty in the position of a photon across the guide so we only get propogation if:
Dy Py ³ (1+[(f)/(p)])[h/2]
The photons obey the Uncertainty Principle.

5.3  Polarization states

We must consider two polarization states when calculating values of qm. If the E vector is in the plane of reflection we get transverse magnetic (TM) modes and if H is in the plane of reflection we get transverse electric (TE) modes. The phase change at TIR depends on the polarization state and therefore the propogation angles are different for the TM and TE modes. The mode number m is usually incorporated into the nomenclature so we get TMm etc..
The electric field amplitude at some position in the guide can be calculated by summing the components reflected from each interface. The effective amplitude in the guide is given by:
2Eocos(pm/2-(pm +f)y/D) where y is the distance from the centre line and D is the width of the guide.
Each mode produces a standing wave pattern across the guide.
When we have TIR at a dense to rare interface there is still field penitration into the rare medium. We get an evanescent wave which decays exponentially:
E(y)=Eoexp(-(2pncy/lo)Ö{nf2sin2qr/nc2-1})
where qr is the reflection angle. This expression tells us how thick the cladding must be to give a good reflection.
The electric field amplitudes across the guide are illustrated in Fig. 33.
lect4313_fig33a.ps
Figure 33: Electric field profiles across a planar waveguide
lect4313_fig33b.ps
The path lengths of rays associated with each mode depends on qm and so different modes travel at different velocities. This gives rise to inter-modal dispersion. The paths are illustrated in fig. 34.
lect4313_fig34.ps
Figure 34: Paths for different modes in a planar waveguide
We can calculate the time delay between the extreme modes corresponding to qmin=qc (m large) and qmax=90° (m small).
The velocity is csinq/nf so
Dt=(1/sinqc-1/sin90°)L nf/c
Dt=(nf-nc)L nf/(c nc)
To minimize the effects of this delay we must reduce the number of modes. We can reduce D so we get just a small number of possible modes.
Note at qc the velocity is given by c/nc. It is controlled by the refractive index of the cladding only.
The above ray model only gives us a qualitative picture of what is happening. If we go the whole way and solve Maxwell's equations for the EM fields in the guide we find that the mode field varies as:
expi(wt - bz) where z is the axis along the centre of the guide and b is known as the mode propogation constant.
You should recall that for such a wave the phase velocity is given by w/b and the group velocity is given by dw/db.
The phase and group velocity are plotted as a function of b in fig. 35.
lect4313_fig35.ps
Figure 35: Phase and group velocity

5.4  Rays in a cylindrical cladded fibre

A cladded fibre has a core refractive index nf and an outer cladding refractive index nc which is slightly less than the core. Then rays travelling down the fibre nearly parallel to the axis will suffer TIR.
Consider a meridional ray in the fibre. Such a ray is coplanar with the axis of the cylinder as shown in Fig. 36.
lect4313_fig36.ps
Figure 36: Meridional ray in cylindrical fibre
If qi is the injection angle wrt the axis and qt is the transmission angle wrt the axis inside the fibre then over a fibre length L the path length is given by:
l=L/cosqt=L/Ö{1-sin2qt}
The number of reflections in length L with diameter D is:
N » l/(D/sinqt) to within ±1
So if D=50mm and qt=30°, N=11500 per metre. Therefore there are potentially a large number of reflections and possible large losses due to imperfections in the surface.

5.5  Fibre optics

A practical fibre optic consists of a very thin core of doped silica (very pure glass) surrounded by a cladding of silica with slightly smaller refractive index. This is then encased in protective coatings of silicone and buffer materials.
To find the modes of propogation in the fibre we should solve Maxwell's equations in cylindrical polar coordinates with the appropriate boundary conditions at the core cladding interface. It is found that there are TM and TE modes similar to the planar waveguide case but 2 integers are now required to label the modes, TEml. Such transverse modes correspond to meridional rays. However we can also get skew rays which spiral down the cylindrical core and these lead to HE and EH modes in which both E and H have transverse components. The ray picture of these modes is illustrated in Fig. 37.
lect4313_fig37.ps
Figure 37: Skew ray in cylindrical fibre
These modes can be approximated by linearly polarized modes with radial and azimuthal structure in the fibre. These are designated LPlm where the first subscript refers to the azimuthal state and the second to the radial state. The simplest mode has no azimuthal structure LP01. The intensity of modes across the fibre are illustrated in Fig. 38.
lect4313_fig38.ps
Figure 38: Modes patterns in cylindrical fibre
We can define a normalised thickness for a cylindrical fibre
V=(2pa/lo)Ö{nf2-nc2}=(2pa/lo)(N.A.)
where a is the radius of the core and N.A. is the numerical aperture.
If V < 2.4 then only the LP01 mode can propogate and we get a single mode fibre.
a < (2.4lo/2pn)/(N.A.)
For example if nf=1.53 and nc=1.5 if lo=1mm then a < 1.27mm. Single mode fibres are very thin!
Thicker fibres that can propogate many modes are called multimode fibres. The total number of modes N » V2/2 (note now proportional to V2 because the fibre is cylindrical rather than planar). The crossections of different types of fibre are shown in Fig. 39.
lect4313_fig39.ps
Figure 39: Construction of optical fibres a) Wide-band graded-index multimode, b) Step-index single-mode and c) large-core plastic-clad optical.

5.6  Step index and graded index fibres

In a step index fibre the transition from core fibre index nf to cladding index nc is sharp.
In a graded index fibre the core fibre index nf varies radially through the core with a peak on the axis and a smooth or graded drop to the cladding nc at the core radius a.
These are illustrated in Fig. 40.
lect4313_fig40.ps
Figure 40: The index profiles of step index and graded index fibres
Both types of fibre propogate 3 kinds of rays:
These are illustrated in Fig. 41.
lect4313_fig41.ps
Figure 41: Types of ray in cylindrical fibres
In a step index fibres the meridional and helical rays have larger optical path lengths and hence are much slower than the central ray mode.
In a graded index fibre the meridional and helical rays (modes) spend less time in regions of high index. Therefore despite the longer physical path length the corresponding modes tend to have velocities which are larger and closer to the central ray mode. That is graded index fibres exhibit reduced intermodal dispersion.

5.7  Dispersion in fibre optics

Intermodal dispersion arises from the different path lengths for propogation modes.
Profile dispersion arises from the fact that D = (nf-nc)/nf is a function of l. Real sources of light always have a finite bandwidth and so D varies over the bandwidth of the pulse. This dispersion depends on both nc and nf.
Material dispersion arises because nf depends on l so even if the wave is confined to the core we still suffer dispersion from the finite bandwidth.
vg=dw/dk=-l2dv/dl but
v=c/(ln) so
vg=(1+(l/n)dn/dl)c/n
If there is a spread of wavelengths Dl then the corresponding spread in group velocity will be
Dvg=Dldvg/dl
So if we initially have a very short pulse after a distance L it will be spread in time by
Dt=L(1/v1-1/v2)=-LDvg/vg2
Substituting for vg and Dvg gives us an expression for this time delay. In practice the 2nd derivative term d2n/dl2 dominates and we find
Dt » |-(LlDl/c)(d2n/dl2)|
For silica this passes through zero for l = 1.3mm Fig. 42 shows the 2nd derivative term for silica.
lect4313_fig42.ps
Figure 42: 2nd derivative term in silica
Waveguide dispersion arises from the fact that the mode velocity depends on b(l). This comes from the solution to Maxwell's equations in a cylindrical waveguide.

5.8  Losses in fibre optics

Bending alters the TIR conditions at the fibre-cladding interface and at some critical radius of curvature the fibre will leak badly for a given mode. Another way of viewing this is to consider the wavefronts for a particular mode. At a bend wavefronts at large radii must travel faster than the speed of light to maintain phase. As this is not possible these wavefronts are radiated away.
The absorption constant associated with a bend is given by
aR=Cexp(-R/Rc) where R is the radius of the bend and Rc is a critical radius. In practice
Rc=a/(N.A.)2
and heavy losses don't occur before the fibre snaps. Bending losses are illustrated in Fig. 43.
lect4313_fig43.ps
Figure 43: Bending losses occur when wavefronts are unable to maintain phase continuity.
Intrinsic scattering and absorption occurs because the material is inhomogeneous. If the refractive index varies over a length scale of » l/10 then get Rayleigh scattering.
Impurities in the glass cause absorption. The pure silica contains traces of Fe3+, Cu2+ and OH- along with the doping material GeO2.
Joins are a major source of losses. Fresnel losses arise because of reflections at any air-glass interfaces. Losses obviously result if the fibres and not aligned to a common axis. Diffraction losses occur at any exit and entrance apertures.
RF=(nf-no)2/(nf+no)2
The reflection losses are reduced by excluding the air using a fluid with refractive index matched to the fibre material.

5.9  Communication using fibre optics

The potential advantages of fibre optics for use in communication systems are considerable:
However they are not without their problems:
The present technology can achieve ~ 600 M bits/sec. This is pushing the fibre, the source and the detector technology. We must work at 1.3 mm to reduce attenuation and dispersion losses in silica fibres and hence the sources and detectors must work at this wavelength.

5.10  A communication example

A typical telephone conversation has a bandwidth of at least 0-4kHz i.e. fmax=4kHz.
For digital encoding we require 8×2×4000 = 64 kbits/sec using 1 byte per sample and two samples per period (the Nyquist rate).
Therefore a single link can transmit up to 8000 speach channels allowing for coding overheads, error trapping and so forth. Not bad for 1 fibre. However the limit set by the carrier frequency and dispersion in the fibre is > 109!

6  Polarization

6.1  The linear polarizer

Such a device produces a P-state from an unpolarized beam. Natural light ® linear light.
We can model natural light as 2 orthogonal P-states which are incoherent, i.e. the phase difference between the states keeps changing. The linear polarizer removes or absorbs 1 of these states. The axis of the linear polarizer is defined by the plane of polarization of the resultant P-state.
The effectiveness of a linear polarizer is indicated by the extinction ratio, the ratio of the min:max transmission (intensity) for a plane polarized beam.

6.2  The law of Malus

Suppose we have 2 linear polarizers in series as illustrated in Fig. 44.
lect4313_fig44.ps
Figure 44: Using a polarizer and analyser
The first is called the polarizer and produces plane polarized light from an unpolarized beam. The second is called the analyser.
Let the angle between the axes of the polarizer and analyser be q.
The P-state produced by the polarizer can be resolved into P-states || and ^ to the axis of the analyser:
E||=Ecosq and E^=Esinq.
The analyser will remove the E^ component leaving amplitude Ecosq. Hence the intensity is:
I(q)=Io2cos2q
This is the law of Malus named after Etienne Malus who published this relationship in 1809.

6.3  Dichroic crystals

Crystals which acted as linear polarizers were known to Newton. Materials which preferentially absorb one P-state are said to be dichroic because the behaviour is dependent of colour (wavelength) and the crystals appear to have different colours depending on the illumination and direction of viewing.
Dichroism arises from a complex refractive index which depends on the direction of propogation - anisotropy of the refractive index.
Examples of naturally occuring dichroic crystals are tourmaline and herapathite. Within such crystals there is a specific direction (or directions) know as the optic axis (or optic axes).
Uniaxial crystals contain just 1 optic axis.
If the E vector is ^ to this axis then the polarization state is stongly absorbed. So if the crystal is cut with a face || to the optic axis and illuminated ^ to that face it acts as a linear polarizer.
The efficiency of the polarizer depends on the wavelength. e.g. if tourmaline is illuminated by natural light normal to the principal optic axis it looks green (transmitted light) but if illuminated parallel to the same axis it looks pearly black. (2 colours - dichroic). The mineral hypersthene can look pink or green when illuminated by plane polarized white light.
This behaviour is caused by anisotropy in the electronic bonding structure of the crystal. Stiff bonds corresponding to a large restoring force per unit displacement have high resonance frequencies and weaker bonds have low resonance frequencies. Furthermore the bonds have different damping constants. This leads to different refractive indices for different directions in the crystal. Dichroic behaviour arises when the damping (absorption) is particularly strong for some directions.

6.4  Complex refractive index

The phase velocity is given by v=dw/dk and the refractive index by n=c/v. Therefore:
k=wn/c
An harmonic travelling wave is given by:
A expi(wt- kz)=Aexpi(wt - wn z/c)
If the refractive index is complex then:
n=nR+inI
If we substitute this into the functional form of the harmonic travelling wave we get:
A expi (wt - wnR z/c) exp( wnI z/c)
The first exponential term propogates at v=c/nR and the second exponential term produces an exponential decay (aborption) or increase (amplification) depending on the sign of nI. The absorption constant is a = 2wnI/c m-1. (Factor of 2 because energy flux is proportional to amplitude squared).

6.5  The wire grid polarizer

The simplest form of linear polarizer appeals directly to the electromagnetic nature of light. It consists of a grid of parallel conducting wires with a spacing d order of the wavelength.
The E vector parallel to the wires is strongly attenuated because currents are induced. It is tempting to imagine that the E vector slips between the wires. On the contrary it is the plane of polarization perpendicular to the wires that is transmitted.
If d < l/2 then reflection is strong.
If d >> l then transmission of both polarizations is high.
Such devices are most useful at large wavelengths (m-waves) because of fabrication problems at optical wavelengths.

6.6  Dichroic sheet - Polaroid

Synthetic dichroic material, dichroic sheet or polaroid was invented by Herbert Land in 1928.
He first used synthetic herapathite (named after W. Herapath who discovered it).
Polaroid J-sheet is made from finely ground, needle like herapathite crystals aligned on a substrate. The crystals are ground small to reduce the scattering but J-sheet still looks milky.
Polaroid H-sheet invented by Land in 1938 is the molecular analog of the wire grid. Polyvinyl alcohol is stretched in 1 direction to form a sheet in which the chain molecules are aligned. It is then doped (soaked) with iodine ink to make the molecules conducting. The high transmission is perpendicular to the stretching direction. H-sheet does not suffer from scattering but is not very good at the blue end of the spectrum.
Modern developments e.g. K-sheet and HR-sheet improve the spectral range.
Linear Polarizer Extinction ratio
calcite 10-7
polaroid 10-3 to 10-5
tourmaline 10-3
wire grid 10-2

6.7  Polarization by scattering - dipole scattering

From EM theory the radiation E-field from an oscillating dipole at a distance r >> l is:

E= k2[P]sinq
4peor
^
q
 
where [P] is the dipole moment at the retarded time t¢=t-r/c.
[P]=qloexpiw(t-r/c) where q is the charge and lo is the maximum separation. c.f. the previous expression for a current element in a half wavelength dipole. In that case we expressed the field in terms of a current which introduced a factor of iw.
The geometry is shown in Fig. 45.
lect4313_fig45.ps
Figure 45: Radiation pattern from an oscillating dipole
Similarly the magnetic field strength of the radiation field is:

H=- ck2
4pr
 [P]Ù
^
r
 
which is perpendicular to E.
So the Poynting vector N=EÙH is proportional to k4.
Thus the intensity of scattering from a simple dipole is µ 1/l4. This is called Rayleigh Scattering. It occurs when the dimensions of the scatterer are small compared with the wavelength of the EM radiation.
Rayleigh scattering gives us blue sky and red sunsets. Fine particles in the atmosphere scatter strongly at the blue end of the spectrum so viewing perpendicular to the solar radiation we see blue.
The radiation scattered through 90° is plane polarized because the E vector is in the [^(q)] direction wrt the dipole. The geometry of the scattering is shown in Fig. 46.
lect4313_fig46.ps
Figure 46: Scattering of P-state by a molecule.

6.8  Polarization by reflection

Fresnel's equations derived from EM theory give us the amplitude reflection coefficients from an interface between 2 media.
R^=r2^=[Eor/Eoi]^2
R||=r2||=[Eor/Eoi]||2

r^= nicosqi-ntcosqt
nicosqi+ntcosqt

r||= ntcosqi-nicosqt
ntcosqi+nicosqt
If we substitute for the refractive indices using Snell's law:

r^=- sin(qi-qt)
sin(qi+qt)

r||= tan(qi-qt)
tan(qi+qt)
When tan(qi+qt)®¥, r||® 0
qt=90-qi so sinqt=cosqi
but nisinqi=ntsinqt so:
tanqB=nt/ni where qB is Brewster's angle.
If the E vector is in the plane of reflection (parallel) then no reflection occurs at qi=qB.

7  Optical anisotropy

7.1  Birefringence - double refraction

If a material is optically anisotropic, the refractive index n varies with direction and birefringence or double refraction can occur.
The best known example of a birefringent material is the crystaline form of CaCO3, calcite. The structure is shown in Fig. 47.
lect4313_fig47.ps
Figure 47: Structure of calcite
Within the crystal structure the carbonate groups lie in planes which are perpendicular to the optic axis.
The O atoms either appear in planes or distributed depending on the viewing direction wrt the crystal optic axis.
Calcite cleaves along 3 planes to form rhomboid crystals. However the optic axis is not perpendicular to any of the planes.
If a light rays impinges normal to any face 2 rays (beams) are generated in the crystal:
These rays are shown in Fig. 48.
lect4313_fig48.ps
Figure 48: e-ray and o-ray in calcite
The o-ray behaves as expected. Away from normal incidence refracted wavefronts are constructed using Huygens' principle in the usual way.
The e-ray behaviour is more complicated. The phase velocity is different parallel and perpendicular to the optic axis.
n||=1.486 and n^=1.658
The wavelets are ellipsoidal NOT spherical. The ray does not move perpendicular to the wavefronts!
The ray is defined by the direction of energy propogation. i.e. the Poynting vector, S.
S ^E
k ^D
but E is NOT || to D which means the polarization is NOT || to E. The wavelet construction for the o-ray and e-ray are shown in Fig. 49.
lect4313_fig49.ps
Figure 49: Wavelets of the e-ray and o-ray
Birefringent materials can be utilized to make very good linear polarizers. There are a number of cunning arrangements used to separate the o-ray and e-ray. The extinction ratio achieved from calcite can be » 10-7.

7.2  Optical activity

In the above disccussion of birefringence we might expect a ray propogating parallel to the optic axis, an o-ray, to exit unchanged. However in 1811 Arago discovered that quartz could rotate the plane of polarization of a P-state beam. This behaviour is known as optical activity. It is illustrated in Fig. 50.
lect4313_fig50.ps
Figure 50: Optical activity in quartz
Such rotation takes place is some crystals, e.g. quartz, and some solutions, e.g. sugar solutions. Some rotate to the left and some to the right.
This behaviour is dependant on wavelength. It is quantified as a specific rotation, rotation per unit length in the material.
It is due to the chiral nature, handedness, of either the crystal structure or the molecules themselves.
Molecules in solution can have a chiral characteristic independent of their orientation - c.f. a box of right handed screws. They are right handed screws whatever and exhibit mirror or reflection symmetry.
Optical activity arises because the refrective index is different for the circular  and L states.
P=Â+L but the wavelength is slightly different for the component circular states so P rotates.
In a liquid there is no optic axis and the rotation proceeds whatever the direction. However in an optically active crystal the polarization state that can propogate unchanged depends on the direction.
Schematically for any direction in an optically active crystal we can get 2 refractive indices and 2 propogation states of polarization. We can plot the indices as a function of direction to generate normal surfaces. These surface are shown schematically in Fig. 51.
lect4313_fig51.ps
Figure 51: Wave propogation in an anisotropic medium - normal or index surfaces, (a) a positive and (b) a negative uniaxial crystal.
Along the optic axis the polarization states are  and L and in the case of quartz the normal surfaces don't meet. So in the direction of the optic axis these states propogate at different speeds hence we see optical activity. Note the separation of the normal surfaces on the diagram is exaggerated. It should be » 0.6% of the radius normal to the axis and » 0.005% parallel to the axis.

7.3  Retarders

Propogation perpendicular to the optic axis the polarization states are 2 orthogonal P-states with the E vector parallel and perpendicular the axis. For a positive uniaxial material, e.g. quartz, the state in which E is parallel to the optic axis is slow. For a negative uniaxial material like calcite this is the fast direction. So for propogation perpendicular to the optic axis 1 P-state is retarded wrt the other.
The path difference is given by:
D = d(|no-ne|) where d is the distance through the crystal.
Therefore the phase difference is:

d = 2p
l
 d(|no-ne|)
Retarders are characterised by the phase difference they introduce between the P-states with E vector perpendicular and parallel to the optic axis:
Note that retardation plates have no effect if linear light is incident perpendicular or parallel to the optic axis since only 1 mode is excited. You only get a phase difference between 2 modes if there are 2 modes.
A half-wave plate is illustrated in Fig. 52.
lect4313_fig52.ps
Figure 52: Half-wave plate

8  Induced optical anisotropy

8.1  Faraday rotation - magnetically induced optical activity

In 1845 Michael Faraday discovered that the propogation of light through a medium could be influenced by the application of a strong magnetic field. At the time this was a strong indication that there was an intimate connection between light an electromagnetism.
If the applied B field is parallel to the direction of propogation then the plane of polarization of linear light is rotated by an angle:
b = VBd
where V is the Verdet constant and d is the path length. V is a property of the medium and varies with temperature and the frequency of the light.
The theoretical treatment of the Faraday effect involves the quantum mechanical theory of the molecular electron energy levels in the presence of a magnetic field. A naive classical picture involves the magnetic force on the bound electrons. Such forces act radially on the orbiting electrons. The interaction alters the electric dipole moment of the electron and hence changes the dielectric constant, e.
Under the influence of a beam of circular light the electrons are forced to circulate by the varying E field.
Thus in the presence of an applied B field and a circularly polarized light beam the electrons orbit at slightly different radii depending on the direction of the B field wrt the direction of propogation. We get two refractive indices nR and nL.
The circular components of a plane polarized beam travel at slightly different speeds and so the plane of polarization rotates - optical activity is induced by the magnetic field.
The plane of polarization is rotated in the same direction as the current required to generate the applied field if the Verdet constant is positive (diamagnetic).
Note the handedness of the rotation depends on the direction of the beam. This is different from normal optical activity.
Note also that in the case of magnetic materials the rotation angle b depends on the magnetization in the material not the applied field.
The Faraday effect can be utilized in a modulator. An infra read modulator can use YIG (magnetic yttrium iron garnet + gallium). This is illustrated in Fig. 53.
lect4313_fig53.ps
Figure 53: Faraday effect modulator
The specimen is saturated by a constant B field perpendicular to the axis and the parallel component is modulated using an axial coil. The rotation b depends on the modulated parallel field and the analyser samples according to the Law of Malus. Hence the laser beam is amplitude modulated.
The Faraday effect can also be used to produce an optical isolator - a device which transmits light in one direction but not the other. A Faraday rotating medium is placed between two polarizers set at 45° to each other. The magnetic field is set to give a rotation of 45° so that a P-state beam is transmitted without loss. However if the beam is reversed the plane of polarization is rotated in the opposite direction (The B-field now points in the opposite direction wrt the propogation direction, see above) and therefore hits the second polarizer at 90° and is blocked.
There are also the Voigt (in vapour) and Cotton-Mouton (in liquid) magneto-optical effects in which the B field is applied perpendicular to the propogation direction and birefringence is induced in the material.

8.2  Liquid crystals

The passive digital liquid crystal display on your wrist watch is a fine example of useful optics.
The liquid crystal state is a phase occupied by many organic materials over a small range of temperature.  As the temperature increases the material passes from solid, through a liquid crystal phase (often with a milky appearance) to a clear liquid.
The material consists of rod like molecules that assume certain orientations with each other in the L.C. state.
A director is used to describe the preferred orientation in the L.C. state. This is a unit vector in the frame of reference of each molecule.
Three types of ordering occur, nematic, cholesteric and smectic.
Liquid crystals have the following properties:
The ordering is shown schematically in Fig. 54.
lect4313_fig54.ps
Figure 54: Order in liquid crystals, (a) nematic and (b) cholersteric.
We can use the above properties in what is called a twisted nematic cell. Liquid crystal is sandwiched between parallel plates.
The surface ordering property is used to twist the directors through 90° as they pass from one side of the cell to the other.
If a voltage is now applied between the plates such that E > Ec directors in the bulk of the cell are forced parallel to the applied field.
When V=0 a plane polarized beam passing across the cell will be rotated through 90°. When V > Ecd a plane polarized beam is not effected as it passes through the cell. This is shown in Fig. 55.
lect4313_fig55.ps
Figure 55: Operation of liquid crystals
We can place such a cell between two polarizers aligned to the ordering direction defined by the plate surfaces. A mirror is placed on one side of the cell.
Incident, unpolarized light is polarized on entry to the cell. If V=0 the plane of polarization is rotated so that it is aligned with the other polarizer when it exits the cell. The mirror reflects the light and the beam travels back though the system. So if V=0 we see a good reflection.
If V > Ecd is applied the plane of polarization is not rotated by the liquid crystal and the beam cannot penitrate to the mirror so we see very little reflection.
So we can control the reflectivity using an applied voltage. In practice an A.C. waveform, 25® 1000Hz, is used and the cell reacts to the r.m.s. voltage. This gives the cell a much longer lifetime.
The action of an LCD cell is shown in Fig. 56.
lect4313_fig56.ps
Figure 56: Action of Liquid Crystal Display
The now familiar LCD displays are:
The behaviour of the LC to an external field is an example of an electro-optic effect.

8.3  The Kerr and Pockels effects

The first electro-optic effect was discovered by John Kerr in 1875.
Birefringence can be induced by an electric field. Refractive indices n|| and n^ are seen parallel and perpendicular to the field.
The difference in refractive index is given by:
Dn=lo K E2
where K is the Kerr constant for the medium. Note the quadratic dependence on E. The effect is used in a Kerr Cell which can be used as an optical shutter.
If the applied voltage V=0 then there is no transmission through the crossed polarizers. A V is applied birefringence is induced and the cell transmits.
Such shutters can be used to Q switch lasers.
Such switches can operate at a frequency of » 1010Hz.
Kerr Cells act as a variable wave plate. Such a cell is illustrated in Fig. 57.
lect4313_fig57.ps
Figure 57: Kerr cell
Df = 2pK l V2/d2
For example using the liquid nitrobenzene in a cell d=10mm and l=50mm, if V=3×104V get a half wave plate.
Carl Pockels discovered a linear electro-optic effect in 1893. In is only seen in certain crystals that lack a centre of symmetry (and which are also piezoelectric).
A Pockels cell can be constructed in a similar way to the Kerr cell. They require only a tenth of the voltage of that needed for the Kerr cell and don't contain toxic liquid. The electric field is applied perpendicular (transverse) or parallel (longitudinal) to the direction of propogation depending on the crystal type. They have switching times of less than 10ns and can modulate at frequencies upto 25GHz. A longitudinal cell is shown in Fig. 58. The phase retardance is given by:
lect4313_fig58.ps
Figure 58: Pockels cell
Df = 2pno3 r63 V/lo
where r63 is the electro-optic constant in m/V, no is the refractive index, V is the applied potential and lo is the vacuum wavelength in metres.
An example of a suitable crystal is ammonium dihydrogen phosphate (NH4H2PO4) ADP for which r63=8.5×10-12 m/V, no=1.52 at lo=546.1 nm and the voltage for a half wave plate is Vl/2=9.2 kV.

9  Nonlinear optics

Glass and similar dielectric materials have a dielectric constant or refractive index which is nonlinear. That is it depends on the electric field intensity. For example in fused quartz this is caused by the Kerr effect which has a response time of 10-15 seconds.
SiO2 has a nonlinear refractive index of the form

n(w)=no(w)+ic(w) +n2|E|2
where no=1.5, n2=3×10-22 (V/m)-2 and c represents absorption.

9.1  Solitons

The transverse electric field of an EM-wave pulse travelling in an optical fibre has the form

E(x,r,t)=R(r) Re{f(x,t)exp(i(kox-wot))}
where x is along the fibre, R(r) is a radial eigenfunction and the phase velocity is given by wo/ko=c/no(wo). The function f(x,t) is a complex envelope which is assumed to vary slowly compared with the carrier. Integrating over the radial coordinate the classical equation of motion of the envelope can be derived from Maxwell's equations and is given by

i f
t
 + iwo¢ f
x
 + inof+ 1
2
 wo" 2f
x2
 + awo n2
no
 |f|2f = 0
where wo¢=wo/ko, wo"=2wo/ko2, no=c(wo)wo/no and a is approximately unity but depends on the radial profile of the electric field.
The second term gives the main group velocity of the envelope, the third term describes absorption, the fourth term describes further dispersion and the fifth term is the nonlinearity.
If the absorption term is zero and wo" > 0 (anomalous dispersion) and the nonlinearity is positive n2 > 0 this wave equation has stationary pulse solutions of the form

f(x,t)=Essech( t-to-x/vg
t
 )exp(i(kx-Wt))
where t is the pulse width, to is the pulse centre and vg is the pulse speed. The shape of the pulse does not change with time, that is there is no dispersion. The nonlinearity has effectively cancelled the effects of dispersion. The width of the pulse is inversely proportional to the maximum field intensity.

Es2= nowo"
awo n2 vg2t2
Note that the wavenumber and frequency of the carrier are modified (shifted) slightly by k and W.
Using the values for the refractive index of SiO2 if t = 3 ps and fo=wo/(2p)=4×1014 Hz then in a fibre of cross-section 10mm2 the power in the pulse is 90 mW.
Such a pulse is called an optical soliton. It was predicted to be possible theoretically in 1973 and first demonstrated in 1980.
In 1988 soliton pulses were transmitted 4000 km error free using the Raman effect to overcome losses (provide optical gain).
In 1991 the record was extended to 14000 km using optical fibre amplifiers. Sections of fibre were doped with erbium and pumped to energise the optical pulses.
A so called solitary wave was first observed by J. Scott Russell on the Edinburgh- Glasgow canal in 1834. The equation which describes such a water wave was discovered by Korteweg and de Vries in 1895 and is referred to as the KdV equation.

u
t
 + v0 ( u
x
 + r u u
x
 + q 3 u
x3
 )=0
(1)
This equation was originally analysed in the context of the irrotational two-dimensional flow of an incompressible inviscid fluid bounded above by a free surface and below by a horizontal plane. In other words waves in water of finite depth.
Soliton solutions to the KdV equation and optical solitons have the remarkable property that they interact strongly when they overlap but after the interaction they remain intact. They behave rather like particles.

9.2  Frequency doubling and mixing

In the linear, isotropic case the polarization in a dielectric is proportional to the electric field:
P=eoceE
and the electric displacement is given by:
D=eoE+P=eo(1+ce)E=eoerE
so the susceptibility is related to the permittivity:
ce=er-1
In a non-linear dielectric the polarization includes higher order terms of susceptibility:
P=eo(c1E+c2E2...)
Therefore the E field from a light wave Eocoswt induces a polarization of the form:
P=eo(c1Eocoswt+c2(Eocoswt)2...)
or
P=eo(c1Eocoswt+[1/2]c2Eo2(cos2wt+1)...)
This polarization is oscillating at harmonic frequencies 2w, 3w etc. as well as the fundamental w. The dielectric radiates at these higher frequencies. The second harmonic is readily produced in non-linear dielectrics - frequency doubling. Unfortunately dielectrics are also dispersive so that the different frequencies travel at different velocities and interfere. This limits the thickness of dielectric that can be used and hence the efficiency. However birefringent materials produce two waves in different directions and the direction of illumination can be chosen to reduce the interference leading to much higher efficiencies - e.g. potassium dihydrogen phosphate KDP can give a 30% yield in frequency doubling.
If two laser beams of different frequency w1 and w2 propogate in a non-linear dielectric then the non-linearity couples the two waves producing the sum and difference frequencies w1-w2 and w1+w2. This is called mixing.
In frequency doubling and mixing photons add (join) or subtract to form new photons. These process need 2 photons and a molecule all close together so they only occur with high probability when the photon flux is very large (or equivalently when the E field is very large).

10  What are photons?

How do we go from the concept of a classical electromagnetic field to the idea of photons? What is the relationship between the field and photons? How are photons related to measurements of the E and B fields? What are the properties of the quantized EM field?

10.1  Quantization of the free EM field

We will not be able to go through all the logical/mathematical steps however I hope I can give you an idea of how quantization of the EM field is achieved.
We start with differential equations for A (the vector potential), E and B which come directly from Maxwell's equations:

Ñ2A(r,t)- 1
c2
 2
t2
 A(r,t)=0

Ñ.A(r,t)=0

E(r,t)=-
t
 A(r,t)

B(r,t)=Ñ×A(r,t)
We want to find the Hamiltonian (Classical) equations of motion. To do this we expand A(r,t) as a three-dimensional Fourier Transform in a box of side L (which we can later let tend to ¥):

A(r,t)= 1
eo1/2L3/2
å
k 
 Ak(t)eik.r
Substituting this into the differential equation above gives:

1
eo1/2L3/2
å
k 
 (-k2- 2
t2
 )Ak(t)eik.r=0
If we let w = ck then

( 2
t2
 +w2)Ak(t)=0
The general solution for Ak(t) is therefore:

Ak(t)=cke-iwt+c-k*eiwt
All this is similar to describing the amplitude of an oscillating string in terms of harmonics. However the thing we are describing is a complex vector field A(r,t) which exists over a three-dimensional volume rather than just a scalar amplitude along a one-dimensional string. Therefore the coefficients are complex vectors.
The coefficient vectors ck can be resolved into 2 orthogonal components (polarizations) chosen such that the divergence condition above is satisfied:

ck= 2
å
s=1 
 cksaks
Thus A(r,t) is expressed as a sum of vector mode functions (normal modes of vibration):

akseik.r
with complex amplitudes:

uks(t)=ckse-iwt
Each mode is labelled with a wave vector k and polarization index s.
There is no stuff which is vibrating as in the case of a sound wave, however you might like to think of the vibrating A(r,t) as a vibration of the vacuum itself!
The energy in the electromagnetic field is given by:

H= 1
2
 ó
õ

L3 
 [eoE2(r,t)+ 1
mo
 B2(r,t)]d3r
This reduces to a simple sum over all modes:

H=2
å
k 
å
s 
 w2|uks(t)|2
In order to quantize the field we have to write this energy in Hamiltonian form using real canonical variables:

qks(t)=[uks(t)+uks*(t)]

pks(t)=-iw[uks(t)-uks*(t)]
such that

t
 qks(t)=pks(t)

t
 pks(t)=-w2qks(t)
this gives:

H= 1
2
å
k 
å
s 
 [pks2(t)+w2qks2(t)]
This has the form of the energy of a system of independent harmonic oscillators, one for each ks mode.
The equations of motion of the the EM field in terms of real canonical variables which we were trying to find are:

H
pks
 = qks
t

H
qks
 =- pks
t
Now we have reached the point where we can apply the postulates of quantum mechanics. This procedure was first done by Dirac in 1927. It is called second quantization or field quantization. Second because the first quantization was achieved for one particle and in that procedure all the forces on that particle were wrapped up in fields (EM, gravitational or whatever) which remained classical and continuous. In second quantization these fields are quantized. We end up with equations which describe the equations of motion of many particles.
We set up Hilbert space operators which correspond to the classical dynamical variables. For a particle you would set up position and momentum operators corresponding to the so called conjugate variables in the equations of motion. In the case of the EM field qks(t) and pks(t) are conjugate variables and the fields A(r,t), E(r,t), B(r,t) which can be expressed as linear functions of qks(t) and pks(t) are also dynamical variables. Note that the position r and time t are NOT dynamical variables of the equations of motion of the field, they are just parameters. All the expansions and equations of motion which were true for the classical dynamical variables are also true for the corresponding Hilbert space operators.
For each pair of canonically conjugate variables qks(t) and pks(t) there are a pair of operators which do not commute:

[
^
q
 
ks 
 (t),
^
p
 
k¢s¢ 
 (t)]=i(h/2p)dkk¢3dss¢

[
^
q
 
ks 
 (t),
^
q
 
k¢s¢ 
 (t)]=0

[
^
p
 
ks 
 (t),
^
p
 
k¢s¢ 
 (t)]=0
Then we can write the Hamiltonian operator:

^
H
 
 = 1
2
å
k 
å
s 
 [
^
p
 
 2
ks 
 (t)+w2
^
q
 
 2
ks 
 (t)]
The eigenvalues of this operator are (nks+1/2)(h/2p)w where n=0,1,2...
i.e. they are the eigenvalues of a harmonic oscillator. Even in the lowest state nks=0 there is a zero point energy contribution [1/2](h/2p)w. As nks increases so the excitation of the ks state increases. Each increment corresponds to a photon and nks is the occupancy of the state. Unlike electrons (fermions) we can have any number of photons (bosons) in a particular state.

10.2  Quantum states of the EM field

The state of the EM field which is an eigenstate of energy or photon number is called a Fock state. If you could observe such a state you would find it contained a particular number of photons and a given amount of energy. This would be like the atomic orbital state for an electron. If we observe atoms they all contain a particular number of electrons. A stable atom is a quantum state of energy for electrons. In fact you can set up an operator which gives the photon number rather than energy in a given state [^n]ks and a Fock state |nks > is an eigenstate of this operator.
So what quantum state do we find in a coherent laser beam? Well not surprisingly this is a close approximation to the so called coherent state denoted by |n > . Such a state is the closest approximation quantum mechanics can give to the classical plane wave coherent beam of light. However this state is NOT an eigenstate of [^H] the energy operator or the number operator [^n]ks.
So when you count photons in a laser beam you don't see a constant value, it fluctuates. In fact the probability of seeing n photons in a coherent state is given by the Poisson distribution. That is the coherent state is a superposition of an infinite spectrum of Fock states. When you count photons there is a finite probability that the number you see is anything from 0 to ¥! Of course the mean number you see depends on the intensity of the beam.
You can't say there is a definite number of photons in a laser beam just as you can't say that an electron (or some other particle) has a definite momentum. Actually there is no observable which corresponds to the coherent state. You can try and count photons or measure its phase but both these properties fluctuate. The coherent state is a superposition of phase states or number states.
Photon-number squeezing is a process by which the photon-number fluctuations can be reduced below the expected Poisson statistics level. In such a process the suppression of fluctuations is accompanied by a corresponding increase in phase uncertainty.

File translated from TEX by TTH, version 3.74.
On 7 Dec 2007, 08:18.