Laser Theory: Come On, How Hard Can It Be

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Laser TheoryCome On, How HardCan It Be ?
Graeme HirstSTFC Central Laser Facility

• Cockcroft Institute Laser Lectures
• April 2008

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Lecture 3 Plan

• Stimulated emission and laser gain
• Rate equations and gain saturation
• Linewidth
• Laser cavities
• Spectrum (axial modes)
• Gaussian beams (transverse modes)
• Modelocking
• Conclusions

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Elements of a Laser
Pump
Optics
Gain medium
All lasers contain a medium in which optical gain
can beinduced and a source of energy which pumps
this medium Many also contain optical elements
whichmodify the laser beam
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Einstein A and B Coefficients
The formal similarity of the curve of the
chromatic distributionof black-body radiation
and the Maxwell velocity-distribution istoo
striking to be hidden for long
Consider a gas of two-level atomsin equilibrium
with photons. Einstein showed that the black
bodyspectrum and the Maxwell distributioncould
be reconciled by adding to thefamiliar processes
of spontaneousemission and absorption a
thirdinduced radiation process.
Opening lines of A Einstein, On the Quantum
Theory of Radiation, Physikalische Zeitschrift
18 121 (1917)
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Laser Gain
Neglecting other loss processes, changes to the
number ofcoherent photons, n, will be controlled
by the balancebetween absorption and stimulated
emission
Where N1 and N2 are the populations of the
respective levels Since B21 B12 this simplifies
to
Laser gain therefore requires N2 gt N1 i.e.
population inversion, which is a non-equilibrium
condition
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Population Dynamics
R2
Consider a two-level gain mediumin the presence
of light Neglecting thermal excitation thefull
rate equations for the levelpopulations are
N2
g2
n
B12
A21, B21
R1
N1
g1
R2
g (inverse lifetime) characterises relaxations to
all other levels NB these equations do not
fullydescribe the evolution of n
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Achieving Inversion
Considering the conditions for steady-state
inversionin the absence of lasing (these are
easiest to achieve)
, n ? 0
?
,
Inversion requires N2/N1 gt 1 and depends on a
combinationof
selective pumping
and
a favourable lifetime ratio
Even with the best selective pumping, i.e. R10,
the equationfor N1 shows that cw inversion
requires g1/A21 gt 1 i.e. thethe lower level must
empty faster than its being filled. Without this
the only possibilitywill be transient lasing.
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Gain Saturation
The steady-state solutions of the rate equations
with lasing(i.e. with n?0) and with negligible
R1 can be shown to be
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Gain Saturation
Saturation occurs when stimulated emission
becomescomparable with spontaneous emission
Efficient, low-noise laser operation, requiring
stimulated emission to dominate spontaneous
emission, dependson strong saturation and,
therefore, operation at low gain Any loss
processes can then become very wasteful
CW lasers, or those where the interpulse period
is less than1/(g2A21), are characterised by
Isat ? hn(g2A21)/s W/m2 In pulsed lasers Esat ?
hn/s J/m2 (for emission cross-section s) Isat and
Esat depend only on the lasing species and not,
forexample, on the pumping rate, inversion
density etc.
Saturation suppresses ASE butdistorts pulse
shapes
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Linewidth
In fact emission and absorption depend on photon
frequency andare characterised by a linewidth

• Line broadening types include
• Natural, from finite lifetime (H)
• Phonon, from lattice vibrations (H)
• Collisional, in gases (H)
• Strain, from static lattice inhomogeneities (I)
• Doppler, in gases (I)
• and the laser levels mayalso be closely
  spacedmanifolds with more orless inter-level
  coupling

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Linewidth Effects
A laser beam of intensity I (W/m2), propagating
in thez direction through a medium with gain
coefficient g (m-1)grows in intensity as I I0
exp(gz) Since g depends on wavelength, this
process will increase theintensity for
wavelengths near line-centre faster than for
thosein the wings, leading to gain-narrowing of
the spectrum
In inhomogeneously broadened media a laser beam
at onewavelength will not saturate the whole
population inversion The part that remains can
support lasing at other wavelengths,making
single-wavelength operation hard to achieve
Inhomogeneously broadened media tend to saturate
as1/(1I/Isat)½ rather than 1/(1I/Isat)
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Oversimplifications
The analysis so far reproduces the basic features
of reallasers. However in reality

• Many more energy levels and pathways are involved

• Levels have degeneracy which must be accounted
  for

• Optical losses (in addition to lower-to-upper
  levelabsorption) need to be included

• Pulsed lasers and transient effects in cw ones
  are verycommon and merit separate treatment
  (although the cwresults are surprisingly
  relevant)

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Optical Cavities
In the absence of external optics the light from
a laser mediumwill be more or less bright ASE
with the following properties

• SPECTRUM set by the mediums spectral gain
  profile,perhaps subject to gain-narrowing and
  effects ofinhomogeneous broadening

• TEMPORAL PROFILE spiky (incoherent) with spike
  widthscorresponding to the Fourier transform of
  the spectrum

• TRANSVERSE PROFILE divergence (coherence)limited
  by the physical extent of the gain medium -
  long,thin media may yield quite low-divergence
  beams

• POLARISATION random unless crystal effects
  causethe gain to be polarisation sensitive

Optics can improve all ofthese parameters
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Cavity Elements
Optical cavities take control of a laser by
feeding back verymuch more light than is present
from optical noise
If the feedback is high enough(i.e. cavity
losses are lowenough) there will be
netround-trip gain andoscillation. The
intracavitylight will saturatethe gain until
itjust balancesthe loss. Intensities
insidecavities can be very high indeed.
L
R2
R1
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Intracavity Fields
Consider the electric field in an x-y plane
inside the cavity.After one round trip the field
will be transformed fromE(x0, y0) to E(1)(x, y)
by a propagation integral of the form
The kernel K(x,y,x0,y0) describes the whole
transformation,including free-space Huygens
propagation, the effects ofapertures, optics etc
except for the axial phase change e-j2kL The
equation can be solved numericallyfor particular
cases
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Intracavity Fields
Consider the eigenmodes of the cavity i.e. those
fielddistributions Enm(x, y) which satisfy
where the eigenvalue e-j2kLgnm is a complex
numberdescribing the overall phase and amplitude
changeexperienced by the n,mth mode for each
round trip.
Without laser gain the amplitude of gnm will be
less than onebecause of diffraction losses
around the mirrors edges(diffraction
redistributes energy from the sharp-edged
fielddistribution immediately after reflection
into a larger areawhen the beam returns).
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Spectral Control
In practice the phase, f, of gnm is almost
independentof the photon wavenumber, k, for
different modes. Thisallows the overall phase of
e-j2kLgnm to be separated fromthe amplitude. The
solutions of (1) then need to satisfy
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Transverse Profile
It is, perhaps, intuitive that themodes
wavefronts will be spherical,since this will
allow them to matchexactly the surfaces of the
sphericalmirrors and thus be reflected back
along their incoming paths
This wave has the undesirable property that the
fieldamplitude is independent of x and y, so the
wavefrontsextend to infinity away from the
optical axis, whichis unphysical given the
finite sizesof the mirrors
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Gaussian Beams
A better choice of spherical wave turns out to be
of the sameform, but with a complex radius of
curvature, q(z), where
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Gaussian Beams

• This beam is the lowest order (TEM00) eigenmode
  of the propagation integral for empty
  spherical-mirror cavities (in fact for the
  stable sub-set of such cavities)

• There is an infinite series of
  higher-order Hermite-Gaussian modes, making up
  a complete basis set which can be used
  to analyse any field distribution in the cavity

• In real cavities different transverse modes
  experience different gains and losses. Mode
  discrimination can quickly lead to few-mode or
  single-mode operation.

• The sometimes surprising properties of Gaussian
  laser beams (forms of R(z), positions of beam
  waists etc) persist outside the optical cavity
  remember this when transporting them !

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Temporal Control
For applications needing high peak power it is
necessary tooperate the laser in a pulsed
mode. Simply pulsing the pump may be sufficient
if, for example,the cw power limit is a thermal
one. But repetition rates arelikely to be low
and each pulse will need to start from noise.
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Temporal Control
Modelocking If the laser output is tobe a
regular train of veryshort pulses
(d-functions)then the spectrum mustalso tend
towards acomb of phase-lockedd-functions (short
pulsesneed bandwidth)
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Modelocking
Active modelocking A loss-modulator put intothe
cavity will introducesidebands on each of
theaxial modes with an offsetequal to the drive
frequencyand a fixed relative phase.
If wmod c/2L then the sideband radiation can
compete withnoise and take control of the
phase of adjacent modes.
In the time domain the modulator can be seen as a
shutterwhich opens once per cavity round trip.
Light which arrivesjust when the shutter opens
will be preferentially amplified.Modulation
depth need not be large butfrequency matching is
critical.
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Modelocking
Passive modelocking Nonlinear opticalprocesses
can lowercavity losses for highintensity
pulses,forming shutterswhich
automaticallyopen at the correct time.
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Modelocking
Passive modelocking Nonlinear opticalprocesses
can lowercavity losses for highintensity
pulses,forming shutterswhich
automaticallyopen at the correct time.
Intracavity dispersion can stretch a pulse making
it lesseffective at opening the shutter.
Dispersion compensationis critical if the
shortest pulses are to be generated. In the
frequency domain dispersion changes the
axialmode spacing, preventing lockingacross the
full spectrum.
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Polarisation
This can be controlled easily using intracavity
elementsand lasers will often be linearly
polarised

• because the laser gain is polarisation-sensitive
  (crystals)

• because optics in the cavity have particularly
  low lossfor one polarisation (Brewster windows,
  dielectric mirrors )

• because the cavity contains elements which use
  polarisation(Pockels cell, AO modulator,
  intracavity doubler, diffractiongrating, Lyot
  filter )

• to avoid problems in beam propagation outside the
  cavity

• to facilitate nonlinear optics outside the cavity

Polarisation ratios of gt1001 are typical
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Conclusions

• Lasers consist of an optical gain medium and a
  pumpsource. Extra optics can modify the laser
  beams properties.

• Gain requires population inversion which is
  achieved by acombination of selective pumping
  and favourable lifetimes.

• Laser gain is saturable. Saturation is needed for
  efficiencybut can distort beams spatial and
  spectral profiles.

• Lasers have linewidth.

• Laser cavity modes influence the beams spectral
  and temporal properties (axial) and spatial
  profile (transverse).

• Gaussian beams have distinctive properties and
  cannotbe reliably treated with simple geometric
  optics.

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Thank you !