The Interaction of Radiation and Matter: Semiclassical
Theory (cont.)
VI. Semiclassical
LASER Theory (pdf)[1]
LASER SelfConsistency Equations
Lamb's theory of laser operation provides a very powerful means
for interpreting and predicting complex time dependent behavior without
invoking all the intricacy of a full quantum mechanical theory. It is semiclassical,
selfconsistent theory in the following sense:
Suppose that the field in the laser is represented in the form

[ VI1 ] 
where, for example,
in a simple a cavity laser
standing
wave 
[ VI2a ] 
and in a ring laser
traveling
wave 
[ VI2b ] 
With this representation for the field the induced
polarization can be expressed as

[ VI3 ] 
where
is the complex, slowly varying component of the polarization of the nth
mode. If we take the wave equation in the form [2]

[ VI4 ] 
where second term is included as a means to account for cavity
losses. From Equation [ VI1 ] assuming that
,
, and
are slowly varying functions of time[3]

[ VI5a ] 
where
are the eigenfrequencies of the cold resonatoreigenmodes.

[ VI5b ] 

[ VI5c ] 

[ II5d ] 
which reduces in the SVAP
approximation to

[ VI5d' ] 
and from Equation [ VI3 ]

[ VI6 ] 
If these representations are to be valid when substituted into
Equation [ VI4 ], the following equations must hold:

[ VI7 ] 
We adjust the fictional conductivity to account
for the loss of energy or time decay of the given mode  viz. we
take
where
is the "Q" of the mode. Equating real and imaginary parts of Equation [
VI7 ] we obtain Lamb's master laser selfconsistency
equations  viz.

[ VI8a ] 

[ VI8b ] 
or

[ VI8b' ] 
The Polarization of the Medium:
The simplest application of Lamb's theory assumes that the
laser medium consists of an ensemble of twolevel atoms with a single welldefined
transition
which is homogeneously broaden. Possible "pump" and "decay"
processes are indicated schematically in the following figure:
The model describes, in effect, a so called fourlevel
homogeneously broaden laser where both levels relaxed to some presumed
ground
state and are "pumped" via higher energy excited states.
is defined as the density matrix operator at a place
and a time
which is associated with atoms excited to the state
( where
) at a time
and
as the rate at which atoms are excited to the state
(per atom, per second per unit volume). The macroscopic polarization is,
thus, given by

[ VI9 ] 
If we project this expression on to the modes of the laser
cavity and compare with Equation [ VI3 ], we see that

[ VI10a ] 
where
is the mode normalization factor. To facilitated the integration of this
equation, we rewrite it in terms of the population matrix operator
 viz.

[ VI10b ] 
where
. 
[ VI11 ] 
The equation of motion of this population matrix is found by
differentiating its defining equation  i.e.
. 
[ VI12 ] 
By definition ,
so that first term on the righthandside is replaced by the matrix
If we interpret the
components of the second term on the righthandside of Equation [ VI12
], as the time derivatives of the pure state density matrix
components given by Equations [ III14a ] and [ III14b ] of this set of
lectures notes,[4] the component
equations of motion for the population matrix
become

[ VI13a ] 

[ VI13b ] 

[ VI13c ] 
Formal integration of Equation [ VI13c ] yields

[ VI14 ] 
Single Mode Operation:
For a singlemode standing wave field in the
rotatingwave
approximation

[ VI15 ] 
so that

[ VI16 ] 
The integration can be done simply, if the changes in ,,
and
are negligible in a time
so that

[ VI17 ] 
where
represents the so called complex Lorentzian denominator. Substituting
this result into Equations [ VI13a ] and [ VI13b ] we find the previously
discussed rate equation approximation for the population components

viz.

[ VI18 ] 
with the rate constant is given by

[ VI19 ] 
where
represents the dimensionless Lorentzian function.
For steady state  i.e.

[ VI20 ] 
where
is the unsaturated population difference and .
Combining Equations [ VI10b ], [ VI17 ] and [ VI20 ] we obtain

[ VI21 ] 
Although this result can be integrated exactly,^{5}
there is more insight to be gained by expanding the saturation denominator
in
powers of
 viz.

[ VI22a ] 
To second order in
we obtain

[ VI22b ] 
where

[ VI23a ] 
and

[ VI23b ] 
is the dimensionless mode intensity.
To complete the selfconsistency loop, we combine our
expression for the complex mode polarization, Equation [ VI22b ], and
the selfconsistency conditions, Equations [ VI8 ].
From Equation [ VI8a ].

[ VI24 ] 
Multiplying this equation by ,
we obtain

[ VI25a ] 
This nonlinear equation of motion may be written to advantage
as

[ VI25b ] 
where the linear gainis
measured by

[ VI26a ] 
and nonlinear saturation
by

[ VI26b ] 
From Equation [ VI8b' ], we see that
Thus

[ VI27 ] 
where the linear mode pulling effectis
measured by

[ VI28a ] 
and nonlinear pushing effectby

[ VI28b ] 
From Equation [ VI25a ], we find the "atresonance"threshold
condition to be:

[ VI29 ] 
where
is the required population inversion at threshold.
For the general steady state condition, we have

[ VI30a ] 
or

[ VI30b ] 
SINGLEMODE, STANDINGWAVE LASER INTENSITY
The time dependent intensity buildup
follows from integrating Equation [ VI25b ] to obtain

[ VI29 ] 
TRANSIENT INTENSITY BUILDUP
Multimode Operation:
See multimode gain curve 1.
See multimode gain curve 2.
See a note on van der Pol oscillators
When we consider the possibility that two or more laser modes may be
simultaneously excited or oscillating, things get quite a bit more complicated
and we must refine our analysis. In particular, we must treat the time
dependence of the population difference
with greater care since variations in the population difference tend to
modulate and couple the possible modal excitations.
Reflection on the form of Equations [ VI13a ] and [ VI13b
] in the absence of modal excitation, suggests that we take

[ VI31 ] 
as the zerothorder approximation to the time dependence of
the population difference. With finite excitation of one or more modes
we may write in the rotatingwave approximation

[ VI32 ] 
and use Equation [ VI14 ] to obtain the following firstorder
approximation to the time dependence of the offdiagonal component of the
population matrix:

[ VI33 ] 
In carrying out this integration we have, again, assumed that
changes in ,
and ,
are negligible in a time .
Using this result and Equation [ VI13a ] we obtain the time derivative
of the secondorder approximation to the time dependence of the population
of the upper level  viz.

[ VI34 ] 
The key point here is that the population
has pulsations at the intermode beat frequencies!
Integrating we find

[ VI35 ] 
and since
we obtain the secondorder approximation to the population difference as

[ VI36 ] 
Substituting this expression into the formal integral of Equation
[ VI14 ], we find, directly, the thirdorder approximation to the offdiagonal
component of the population matrix

[ VI37 ] 
Using Equation [ VI10b ] and invoking Equations [ VI8 ] the
general selfconsistency conditions may be written [5]

[ VI38 ] 

[ VI39 ] 
with the coefficients summarized in the following table for
a standing wave configuration:
Coefficients

Significance


Linear net gain


Linear mode pulling


Self saturation


General saturation term


Stationary coefficient


Crosssaturation


Relative phase angle 

Self mode pushing


Cross pushing


Firstorder factor


Thirdorder factor


Spatial factor


Lorentzian denominator


Dimensionless Lorentzian 
TwoMode Operation:
For two modes the amplitude determining equations  i.e. Equation [ VI38 ]  reduce to

[ VI40 ] 
and the frequency determining equations  i.e. Equation
[ VI39 ]  reduce to

[ VI41 ] 
Multiplying the amplitude equations by
we obtain the equations of motion for the dimensionless intensities

[ VI42 ] 
Stability of Possible Steady state
Solutions:
Stability criterion for stationary solutions: If
and ,
and
as .
Solution 1:
so that

[ VI43a ] 
or

[ VI43b ] 
Stability requires that
remain negative. If
but
one says that
inhibits the oscillation of
(mode inhibition). If
becomes large to overcome the inhibiting effect of
, the solution becomes unstable and
builds up.
Solution 2:
so that

[ VI44 ] 
which has the solution

[ VI45 ] 
where the coupling constant .
Substituting this solution into Equation [ VI42 ] we see that

[ VI46a ] 
In matrix form

[ VI46b ] 
The solutions will be stable if the eigenvalues of this equation
are both negative.

[ VI47 ] 

[ VI48 ] 
Case 1: Weak coupling where :
Stable
since
both eigenvalues negative
Case 2: Strong coupling:
Unstable
since
one eigenvalue is positive
[1] An adaptation or interpretation of Lamb's "semiclassical"
or "selfconsistent" laser theory as first presented in
Phys. Rev. 134,
A1429 (1964) and refined in countless other treatments. In these lecture
notes we drawn extensively on M. Sargent III, M. O. Scully and W. E. Lamb,
Jr., Laser Physics, AddisonWesley (1974) and P. Meystre and M.
Sargent III, Elements of Quantum Optics, SpringerVerlag (1992).
[2] In this formulation we are assuming that
[3] The so called slowlyvarying
amplitude and phase approximation(SVAP) is used extensively
in treating problems in laser dynamics. In the SVAP approximation it is
assumed that
and
[4] Including damping, Equations [ III16a ] and [
III16b ] of this set of lecture notes would have the form
[5] Exactly
where
and
.
[5] It is important to note, that use of Equation [ VI10b
] requires evaluation of the following integral :
Thus, the coefficients in Equations [ VI38] and [ VI39 ] are sensitive
to the particulars of the laser mode configuration. Since the modal excitations
cannot change rapidly , the only integrals of importance are those for
which
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This page was prepared and is maintained
by R. Victor Jones, jones@deas.harvard.edu
Last updated March 16, 2000