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PA3230
Interaction of Radiation and Matter
Lectures 9-12
LASERS
Professor R. Willingale
Department of Physics and Astronomy
University of Leicester
November 2015
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
Books
• Buzz words are Optics, Photonics, Lasers
• Optics and Photonics An Introduction, F.G.Smith and
T.A.King, Wiley
• Optoelectronics: An introduction, Second edition.
J.Wilson and J.F.B.Hawkes, Prentice Hall
• There are many more excellent books on optics,
photonics and lasers in the library!
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Syllabus
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Coherence
– Temporal coherence
– Lateral coherence
– Coherence volume
Fields and photons
Lasers
– Stimulated emission – cloning photons – revision from previous lectures
– Population inversion – revision from previous lectures
– Pumping – 3 and 4 level systems
– Optical feedback – resonant cavities
– Line broadening
– Laser modes
– Hallmarks of laser activity
Types of laser – examples
– Atomic gas laser
– Semiconductor laser
Properties of laser light
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Coherence
• Lasers are sources of coherent radiation
• Coherence is associated with the wave like nature of light
• Coherence manifests itself in
– Interference
– Diffraction
• Temporal coherence
– In the direction of travel (wave vector k)
– Important in amplitude splitting - Michelson Interferometer
• Lateral coherence
– In directions perpendicular to k
– Important in wave front splitting interference – Young’s slits
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Temporal Coherence
• Consider the harmonic content of the light wave at a fixed point
in space
• Perfect temporal coherence implies amplitude as
– f(t)=Aexp(-iω0t)
– Take Fourier transform F(ω)=C.δ(ω-ω0)
– Only 1 frequency present and wave continous in t
• In reality length of wave (duration of oscillation) is finite
– Duration Δt gives spread of frequencies Δf~1/Δt~Δω/2π
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Gaussian wavepackets
• Example of finite light wave – a Gaussian wavepacket
– f(t)=Aexp(iω0t).exp(-t2/2Δt2)
– Take Fourier transform F(ω)=B.exp(-Δt2(ω-ω0)2/2)
• Simple statistical model of light – many (n) wavepackets
arriving randomly at times tm
– C(t)=∑ Am.exp(iω0(t-tm)).exp(-(t-tm)2/2Δt2)
– If all packets are same length and same frequency then get
same bandwidth Δf
– The resulting average amplitude is proportional to √n
• Express coherence as bandwidth Δf or coherence time Δt or
coherence length Δx=cΔt
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Lateral Coherence – Coherence Volume
• Coherence moving along the wavefronts - lateral
• Also known as spatial coherence
• The temporal and lateral coherence are defined using
correlation functions that average over time
• Coherence volume is the volume defined by the
combination of temporal coherence expressed as a
coherence length combined with lateral coherence in 2
independent directions perpendicular to k
• Observation of interference or diffraction patterns
provides a way of measuring the coherence of the light
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Perfect coherence
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Equal lateral and temporal coherence
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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High lateral low temporal coherence
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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High temporal low lateral coherence
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Fields and Photons
• Field - Light beam in a volume
– coherence depends on the nature and geometry of the
source and the region of propogation – Δω is spread in
angular frequency
– Carries energy – irradiance – associated with the square of
the electric field amplitude
– Polarized – associated with angular momentum
• Photons
– Energy E=hν=ħω
– Momentum P=E/c=h/λ
– Angular momentum E/ω = ±ħ – photon spin
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Emission and Absorption
• Emission of photons puts energy, momentum and
angular momentum into the field
• Absorption of photons takes energy, momentum and
angular momentum out of the field
• Coherence is quantum cooperation
– A large number of photons are in the same state of
energy, momentum and angular momentum
– Interference of photons - only ever get interference
of photons with themselves! Photons that originate
from different sources never interfere
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Emission of photons
• A photon is emitted when an electron in a atom undergoes a
transition between 2 energy levels
– ν=ΔE/h=(E2-E1)/h
– Degeneracy of levels g1 and g2
• Can occur in 2 ways
– Spontaneous emission – random – coefficient A21 governed by lifetime 1/A21=𝛕21
– Stimulated – in presence of stimulating photon frequency
ν=ΔE/h – coefficient B21
– Coefficient B12 for photoelectric absorption
• Einstein relations – g1B12 = g2B21 and A21/B21=8ᴨhν3/c3
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Einstein Relations depend on...
• If you define the A21, B21 and B12 using energy density
– N1B12ρν = N2A21 + N2B21 ρν
– then get
g1B12 = g2B21 and A21/B21=8ᴨhν3/c3
• If you define the A21, B21 and B12 using mean specific intensity
– N1B12jν = N2A21 + N2B21 jν
– then get
g1B12 = g2B21 and A21/B21=2hν3/c2
• Both of the above occur in the literature
– Optics/Laser books tend to use energy density
– Theoretical radiation books tend to use specific intensity
• Difference is a factor of c/4π in the definition of B21 and B12
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Lasers – Population Inversion
• Consider a collimated (1 wavevector k), monochromatic (1
frequency ν21) light beam passing through an absorbing
medium containing 1 relevant transition E2<->E1 where
ν21=(E2-E1)/h
• Photon number density of light beam Nν
• Energy density of light beam ρν =Nνhν
• Absorption coefficient α given by
– dI(x)/dx=-αI(x)  I(x)=I0.exp(-αx)
• Ignore spontaneous emission because isotropic
• Ignore scattering out of beam because very low
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Population inversion – small signal gain
• N1 electrons in state 1, N2 electrons in state 2
• -dNν/dt = (g2N1/g1-N2)ρνB21
• If n=c/(dx/dt) is refractive index of medium
– Iν=ρνc/n = Nνhν21c/n
– dNν/dx=dIν/dx.n/hν21c
• So dNν/dt=dIν/dx.1/hν21=-αIν/hν21
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dNν/dt=-αρνc/nhν21
• Therefore α=(g2N1/g1-N2)B21hν21n/c
• If N2>g2N1/g2 α is negative, κ=-α is the small signal gain
– exponential increase in intensity I=I0.exp(κx)
• Light Amplification by Stimulated Emission of Radiation
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Pumping – 3 and 4 level systems
• Because B21=B12 we must use a 3rd level to get N2>g2N1/g1
• Pump electrons into an energy level (or levels) higher that the
upper LASER transition level
• Electrons must decay rapidly from the pumped levels into the
upper LASER transition level
• The 3 level system is inefficient because the lower LASER
transition level is always populated
• A 4 level system has a level E0 below the lower LASER
transition level
• A 4 level system is more efficient providing E1-E0>kT
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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3 level system
Thermal equilibrium
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
Pumped
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4 level system
Thermal equilibrium
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
Pumped
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Pumping
• Optical pumping – use an intense flash of white light.
Use reflectors to concentrate as much of this light into
the LASER beam volume
• Electric current – Power source use directly in
semiconductor lasers
• D.C. discharge – used in gas lasers
• Gas dynamic discharge – uses the thermodynamic
properties of a gas – push gas through a compressionexpansion cycle
• Lasers pumping other lasers
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Optical feedback – resonant cavities
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The gain per unit length (small signal gain) in the active pumped laser medium is
usually rather low
Need a long path length to obtain large gain
Fold the beam back on itself by reflection
– 2 mirrors form a resonant cavity
– often called a Fabry-Perot Resonator
Different forms of cavity are used for different applications
– Plane-plane – produces a large mode volume but difficult to make the mirrors flat
and parallel
– Large r1 large r2 – maximum power (large volume)
– Confocal – very easy to align
– Hemispherical – can get high coherence
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Laser cavity geometries
• 1 mirror highly reflecting
• 1 mirror has small
transmission – laser beam
enters the outside world
• Can modify mirrors using
filters and/or coatings to
select specific λ’s
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Limits on power/gain
• Transmission by output mirror
• Absorption and scattering at the mirrors
• Absorption in the laser medium and other electron
transitions
• Optical scattering in the laser medium – inhomogeneities
in density and composition
• Diffraction in the cavity – mirror apertures
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Characteristic lifetime of cavity
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Loss of energy density ρν at the output mirror of cavity
– Δρν =-ρν(1-R) where R is mirror reflectivity
Time for round trip up-down cavity length L
– Δt=2L/v
where v is velocity in cavity
Rate of loss in energy density Δρν /dt=- ρν(1-R)v/2L
Integrating we get
log(ρν)=log(ρν0)-(1-R)vt/2L
Time constant of the cavity tc=2L/(v(1-R))
If gain is switched off then output decays exponentially ρν = ρν0 exp(-t/tc)
e.g. R=0.99, L=5 mm, tc=3.3 ns
– Δν=1/tc=3 x 108 Hz or correlation length ctc = 1m
– Effective number of round trips in cavity is ~100
– If frequency ν=3 x 1014 Hz (visible light) Q factor ν/Δν ~ 106
Laser with cavity is really an oscillator rather than just an amplifier
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Line broadening
• Above we ignored the fact that spectral lines or electron transitions
have a finite width/spread in energy
• The finite width is described by a line profile function g(ν)
• We must re-write the small signal gain as
– κ(ν)=(N2-N1g2/g1)B21 hν n g(ν)/c
• The shape and width of g(ν) arises from
– Doppler broadening – physical motion of the atoms
– Collision (pressure) broadening – a 3rd interrupting particle
– Natural broadening – life time or damping of the transition
• g(ν) is normalised – integral over ν is 1 – probability density
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Broadening profile
• Change in intensity given by I(ν,x)=I(ν,0)exp(κ(ν)x)
• If all atoms give the same centre frequency then a Lorentzian
profile – natural broadening
– g(ν)=(Δν/2π)/((ν-ν0)2+(Δν/2)2)
• If each atom gives a different centre frequency – e.g. Doppler
broadening – then a Gaussian profile
– g(ν)=(2/Δν)(log(2)/π)1/2 exp(-log(2)(ν-ν0)2/(Δv/2)2)
• κ(ν) depends on g(ν)
– We do NOT get the line profile on amplification
– Gain boosts th central frequencies - spectral narrowing
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Transmission and emission profiles
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Laser modes
• In the resonant cavity we get a standing wave pattern of the
electric field intensity
• If the optical path length of the cavity is L then
– pλ/2=L where p is an integer
• We get a series of frequencies given by
– νp=p(c/2L)
– Each p defines an axial mode of the cavity
– The separation between modes is constant δν=c/2L
• Only modes which lie within the peak of the gain profile will
be amplified and maintained
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Excited laser modes
Gain profile
Cavity modes
Excited modes – very sharp
peaks because of spectral
narrowing
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Q factor
• The width of the mode peaks is usually described in terms of a
Q factor
– Q=2π(energy stored)/(energy dissipated per cycle)
– Q= ν/Δν
• Typically Q=108 – very large compared with ~100 for an
electrical oscillator
• Δν≃106 Hz compared with 109 Hz for a Fabry-Perot cavity
• If reduce losses then can increase Q further
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Transverse Electric Modes
• The electric vector in always
transverse (across the cavity
• So called TEM’s labelled using the
number of minima in the electric field
scanning across the cavity
• TEM00, TEM01,…
• TEM00
– mirror surfaces are surfaces of
constant phase
– Can be selected using an aperture
in the centre of the cavity
– Produces a so-called uniphase
mode
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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TEM00 mode
ω0 is defined as width of mode – where field amplitude drops by a
factor of 1/e of maximum
• Uniphase operations gives high spectral purity
• Multimode operation can give high power
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Hallmarks of Laser Activity
• Laser activity can be distinguished from amplified
spontaneous emission, superluminesence…
– A threshold in output energy as a function of input or
pumping energy – laser action above threshold
– Strong polarization in output beam
– Spatial coherence – measured by diffraction and
beam speckle
– Significant spectral line narrowing
– Existence of laser cavity resonances or modes
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Atomic Gas Laser
• Common He-Ne laser
– Ne provides the energy levels and He provides pumping by
electron and atomic collisions
– Pumping is in 2 stages
• e1+He=>He*+e2
electron looses energy to atom
• He*+Ne=>Ne*+He excited He collides with Ne atom
• Pumping is by d.c. discharge, 2 to 4 kV across tube of gas mixture
at pressure ~10 torr
• Low power but high collimation
• Polarization selected using a Brewster window
• Δλ very small
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Atomic Gas Laser
Hemispherical cavity shown
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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He-Ne Laser energy level diagram
633 nm is the
well known red
coherent laser
light
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Temperature Broadening in gas
• In a gas laser the atom/molecules mass M are moving speed vx
– (1/2)Mvx2=(1/2)kT  vx=(kT/M)1/2
• Frequency of the transition is Doppler shifted for each
atom/molecule
– ν21ʹ=ν21(1±vx/c) 
Δν/ν = vx/c
• Δν = ν(vx/c)
• But λ=c/ν 
Δλ=-Δνc/ν2
• Δλ=vx/ν

Δλ/λ=vx/c
• For Ne atomic mass is M=20.2 and T=400 K, λ=634 nm
– Δλ=8.6x10-4 nm  FWHM of gain curve 2.36Δλ= 1.9x10-3 nm
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Semiconductor lasers
• Diagrams show electron
energy levels across a
semiconductor diode
junction
• (a) equilibrium
• (b) forward bias voltage
applied
• To get laser action need a
region where BOTH
excited electron states
and holes (vacant electron
states) are present
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Semiconductor lasers
• Use heavily doped n and p material to increase the
number of carriers
• Apply forward bias voltage to create a thin active region
• Thickness of active region controlled by the diffusion
length of electrons
– E.g. for GaAs at room temperature 1-3 μm thick
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Semiconductor laser cavity
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No external mirrors 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
– E.g. n=3.6 for GaAs gives reflectance of 0.32
The active junction region has slightly different refractive index so acts as a wave
guide to contain laser light
Pumping energy comes from the diode current
Laser action starts above a threshold current density
The principle loss mechanism is scattering by optical inhomogeneities in active
volume
Because the active region is so thin the exit beam spreads due to diffraction
– Beam fans out perpendicular to junction plane
– E.g. θ=λ/t if t=3 μm and λ=0.84 μm then θ=19 degrees
– Can obtain a parallel beam using a cylindrical lens
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Semiconductor laser
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Gain vs. Loss
• The threshold conditions for laser action depend on
– Small signal gain in medium
– Loss which is the output beam (mirror reflectivities)
– Propogation losses in medium
• If the cavity mirror reflectivities are R1 and R2, κ is the small
signal gain, γ is the linear loss coefficient and L is the length
of the cavity for 1 round trip in cavity
– G=(final irradiance/initial irradiance)=R1R2exp(2(κ-γ)L)
• Threshold condition for laser action G=1
– κth=γ+1/(2L) log(1/(R1R2))
• Pumping must achieve a small signal gain >κth
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Properties of Laser light
• To greater or lesser extent laser light is:
– Very intense
– Highly collimated
– Highly coherent
– Highly polarized
– Continuous or very short pulses
• The intensity depends on the pumping power and the
efficiency of the Laser mechanism
– Tradeoff between intensity, coherence, collimation
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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Properties of Laser light
• Collimation set by diffraction
– θ = Kλ/D where K≈1 and is an effective aperture diameter
• Coherence length Lc depends on number of modes excited and
duration of pulses
• Typical angular beam widths and coherence lengths are:
Type
milli radians
Lc m
He-Ne
0.5
Single mode Upto 1000,
multimode 0.2
Ar
0.8
CO2
2
Dye
2
Nd:Glass
5
2 x 10-4
GaAs
20 by 200
1 x 10-3
PA3230 Lectures 9-12, Prof. R. Willingale, November 2015
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