The Interaction of Radiation and Matter: Semiclassical Theory (cont.)

LASER Self-Consistency 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, self-consistent theory in the following sense:

Suppose that the field in the laser is represented in the form

[ VI-1 ]

where, for example,

in a simple a cavity laser

standing wave

[ VI-2a ]

and in a ring laser

traveling wave

[ VI-2b ]

With this representation for the field the induced polarization can be expressed as

[ VI-3 ]

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

[ VI-4 ]

where second term is included as a means to account for cavity losses. From Equation [ VI-1 ] assuming that  ,  , and  are slowly varying functions of time[3]

[ VI-5a ]

where  are the eigenfrequencies of the cold resonatoreigenmodes.

[ VI-5b ]

[ VI-5c ]

[ II-5d ]

which reduces in the SVAP approximation to

[ VI-5d' ]

and from Equation [ VI-3 ]

[ VI-6 ]

If these representations are to be valid when substituted into Equation [ VI-4 ], the following equations must hold:

[ VI-7 ]

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 [ VI-7 ] we obtain Lamb's master laser self-consistency equations -- viz.

[ VI-8a ]

[ VI-8b ]

or

[ VI-8b' ]

The Polarization of the Medium:

The simplest application of Lamb's theory assumes that the laser medium consists of an ensemble of two-level atoms with a single well-defined 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 four-level 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

[ VI-9 ]

If we project this expression on to the modes of the laser cavity and compare with Equation [ VI-3 ], we see that

[ VI-10a ]

where  is the mode normalization factor. To facilitated the integration of this equation, we re-write it in terms of the population matrix operator -- viz.

[ VI-10b ]

where

.

[ VI-11 ]

The equation of motion of this population matrix is found by differentiating its defining equation -- i.e.

.

[ VI-12 ]

By definition , so that first term on the right-hand-side is replaced by the matrix

If we interpret the  components of the second term on the right-hand-side of Equation [ VI-12 ], as the time derivatives of the pure state density matrix components given by Equations [ III-14a ] and [ III-14b ] of this set of lectures notes,[4] the component equations of motion for the population matrix  become

[ VI-13a ]

[ VI-13b ]

[ VI-13c ]

Formal integration of Equation [ VI-13c ] yields

[ VI-14 ]

Single Mode Operation:

For a single-mode standing wave field in the rotating-wave approximation

[ VI-15 ]

so that

[ VI-16 ]

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

[ VI-17 ]

where  represents the so called complex Lorentzian denominator. Substituting this result into Equations [ VI-13a ] and [ VI-13b ] we find the previously discussed rate equation approximation for the population components -- viz.

[ VI-18 ]

with the rate constant is given by

[ VI-19 ]

where  represents the dimensionless Lorentzian function.

For steady state -- i.e. 

[ VI-20 ]

where  is the unsaturated population difference and . Combining Equations [ VI-10b ], [ VI-17 ] and [ VI-20 ] we obtain

[ VI-21 ]

Although this result can be integrated exactly,⁵ there is more insight to be gained by expanding the saturation denominator in powers of  -- viz.

[ VI-22a ]

To second order in  we obtain

[ VI-22b ]

where

[ VI-23a ]

and

[ VI-23b ]

is the dimensionless mode intensity.

To complete the self-consistency loop, we combine our expression for the complex mode polarization, Equation [ VI-22b ], and the self-consistency conditions, Equations [ VI-8 ].

From Equation [ VI-8a ].

[ VI-24 ]

Multiplying this equation by , we obtain

[ VI-25a ]

This nonlinear equation of motion may be written to advantage as

[ VI-25b ]

where the linear gainis measured by

[ VI-26a ]

and nonlinear saturation by

[ VI-26b ]

From Equation [ VI-8b' ], we see that

Thus

[ VI-27 ]

where the linear mode pulling effectis measured by

[ VI-28a ]

and nonlinear pushing effectby

[ VI-28b ]

From Equation [ VI-25a ], we find the "at-resonance"threshold condition to be:

[ VI-29 ]

where  is the required population inversion at threshold.

For the general steady state condition, we have

[ VI-30a ]

or

[ VI-30b ]

The time dependent intensity build-up follows from integrating Equation [ VI-25b ] to obtain

[ VI-29 ]

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 [ VI-13a ] and [ VI-13b ] in the absence of modal excitation, suggests that we take

[ VI-31 ]

as the zeroth-order approximation to the time dependence of the population difference. With finite excitation of one or more modes we may write in the rotating-wave approximation

[ VI-32 ]

and use Equation [ VI-14 ] to obtain the following first-order approximation to the time dependence of the off-diagonal component of the population matrix:

[ VI-33 ]

In carrying out this integration we have, again, assumed that changes in , and , are negligible in a time . Using this result and Equation [ VI-13a ] we obtain the time derivative of the second-order approximation to the time dependence of the population of the upper level -- viz.

[ VI-34 ]

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

[ VI-35 ]

and since  we obtain the second-order approximation to the population difference as

[ VI-36 ]

Substituting this expression into the formal integral of Equation [ VI-14 ], we find, directly, the third-order approximation to the off-diagonal component of the population matrix

[ VI-37 ]

Using Equation [ VI-10b ] and invoking Equations [ VI-8 ] the general self-consistency conditions may be written [5]

[ VI-38 ]

[ VI-39 ]

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
	Cross-saturation
	Relative phase angle
	Self mode pushing
	Cross pushing
	First-order factor
	Third-order factor
	Spatial factor
	Lorentzian denominator
	Dimensionless Lorentzian

                                                                                               
Two-Mode Operation:

For two modes the amplitude determining equations -- i.e. Equation [ VI-38 ] - reduce to

[ VI-40 ]

and the frequency determining equations -- i.e. Equation [ VI-39 ] -- reduce to

[ VI-41 ]

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

[ VI-42 ]

Stability of Possible Steady state Solutions:

Stability criterion for stationary solutions: If  and , and  as .

Solution 1:  so that

[ VI-43a ]

or

[ VI-43b ]

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

[ VI-44 ]

which has the solution

[ VI-45 ]

where the coupling constant . Substituting this solution into Equation [ VI-42 ] we see that

[ VI-46a ]

In matrix form

[ VI-46b ]

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

[ VI-47 ]

[ VI-48 ]

Case 1: Weak coupling where : Stable since both eigenvalues negative
Case 2: Strong coupling: Unstable since one eigenvalue is positive

[4] Including damping, Equations [ III-16a ] and [ III-16b ] of this set of lecture notes would have the form

[5] Exactly

and

.

Thus, the coefficients in Equations [ VI-38] and [ VI-39 ] 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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