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LASER DYNAMICS: THE LASER CAVITY EQUATIONS

This is a continuation from the previous tutorial - More on unstable resonators

In earlier tutorials we analyzed the steady-state behavior of laser cavities using simplified interferometer models, with a few extensions into transient behavior, as in the cavity build-up equations.

The objective in this tutorial is to give a more complete and systematic derivation of the combined cavity and atomic equations of motion for a real multimode laser. We will then use these equations of motion in later tutorials to analyze dynamic phenomena such as spiking, \(\text Q\)-switching, mode locking, and injection locking in lasers.

 

 

1.  DERIVATION OF THE LASER CAVITY EQUATIONS 

In this opening section we derive the basic laser-cavity equation of motion, using a so-called "Slater normal mode approach". This approach is independent of the specific form or details of the actual laser cavity.

Like essentially every other derivation of the laser-cavity equations in the literature or in other textbooks, this analysis will make certain assumptions about the lossless and orthonormal character of the cavity modes—assumptions which are not, in fact, valid for real laser cavities. 

Despite this fundamental weakness in the starting approximations, the approach presented here is the standard approach used in the laser field. We reproduce this standard derivation here partly because it will permit students to compare results with the standard laser literature; partly because this derivation also gives many useful insights; but mostly because the final results of this derivation are still essentially correct despite the weakness of the initial assumptions. 

 

The Vector Wave Equation

We begin the analysis as usual with Maxwell's equations for the real vector electromagnetic fields in the cavity, namely, 

\[\tag{1}\nabla\times\boldsymbol{\varepsilon}(r,t)=-\frac{\partial b(r,t)}{\partial t}\quad\text{and}\quad\nabla\times h(r,t)=j(r,t)+\frac{\partial d(r,t)}{\partial t}.\]

We add to these the "constitutive relations"

\[\tag{2}b(r,t)=\mu_0[h(r,t)+m_a(r,t)],\quad d(r,t)=\epsilon\varepsilon(r,t)+p_a(r,t),\] 

and also an assumed ohmic loss relation

\[\tag{3}j(r,t)-\sigma\varepsilon(r,t).\]

The field quantities \(\varepsilon(r,t)\), \(b(r,t)\), \(j(r,t)\), and so on, are all real functions of space and time at this point. The dielectric constant e includes the dielectric permeability of the host crystal lattice or any other dielectric material inside the laser cavity, but does not include the resonant laser transition itself. The polarizations \(p_a(r,t)\) and \(m_a(r,t)\) then represent any electric-dipole or magnetic-dipole atomic transitions (e.g., laser transitions) that may be present.

The conductivity a represents ohmic losses inside the laser cavity, extended to include scattering and coupling losses as well. 

We then proceed by taking the curl of the first Maxwell equation and combining it with the other equations, plus the vector identity \(\nabla\times\nabla\times\varepsilon\equiv\nabla(\nabla\cdot\varepsilon)-\nabla^2\varepsilon\).

We also assume that \(\nabla\cdot\varepsilon\equiv 0\) inside a closed cavity with no free charges present. The result, after some rearrangement, is the full vector wave equation

\[\tag{4}\frac{\partial^2\boldsymbol{\varepsilon}(r,t)}{\partial t^2}+\frac{\sigma}{\epsilon}\frac{\partial\boldsymbol{\varepsilon}(r,t)}{\partial t}-\frac{1}{\mu_0\epsilon}\nabla^2\varepsilon(r,t)=-\frac{1}{\epsilon}\left[\frac{\partial p_a(r,t)}{\partial t^2}+\frac{\partial}{\partial t}\nabla\times m_a(r,t\right].\]

This is the basic equation for calculating how the cavity fields \(\varepsilon(r,t)\) on the lefthand side of Equation 4 will be driven or excited by any electric and magnetic atomic polarizations \(p_a(r,t)\) and \(m_a(r,t)\) appearing on the right-hand side.

 

Normal Mode Expansion

We now make the crucial assumption that we can write the fields inside any laser cavity as an expansion in a set of normal modes or eigenmodes \(u_n(r)\) in the form 

\[\tag{5}\boldsymbol\varepsilon(r,t)=\sum_n E_n(t)u_n\boldsymbol{(r)},\] 

where the expansion coefficients \(E_n(t)\) are scalar functions of time only. The normal modes \(u_n\boldsymbol{(r)}\) in this expansion are assumed to be solutions of Laplace's equation 

\[\tag{6}\left[\nabla^2+k_n^2\right]u_n(r)=0\]

which satisfy the boundary conditions of the particular cavity being analyzed, without the atoms being present. These cavity modes are therefore the eigenmodes of the cavity, which can exist—that is, which can satisfy both the wave

 

 

FIGURE 1.  Resonant eigenmodes of a closed rectangular cavity.

equation and the boundary conditions—only for certain discrete values of the separation constant or eigenvalue \(k_n\). This eigenvalue can in turn be written as 

\[\tag{7}k_n\equiv\omega_n\sqrt{\mu_0\epsilon}=\omega_n/c,\] 

where the \(\omega_n's\) are the resonance frequencies of the cavity (without laser atoms). The modes un and frequencies un are thus the resonant eigenmodes and eigenfrequencies of the empty cavity, leaving out any effects of the atomic polarizations \(p_a\) or \(m_a\). 

We will also assume that these modes are orthogonal, and that their amplitudes \(u_n(r)\) can always be normalized so that they satisfy the orthonormality relation 

\[\tag{8}\int\int\int_\text{cavity}u_n(r)\cdot u_m(r)dr=V_c\times\delta_{nm},\]

where the integral \(dr\equiv dx\;dy\;dz\) is over the entire cavity volume. An arbitrary normalization factor Vc is included on the right-hand side of this equation. The value of this normalization factor can be arbitrarily chosen, since it merely determines a scale factor for the amplitudes of the cavity modes \(u_n\).

The most elementary choice, for example, would be simply to set \(V_c=1\).

If, however, this arbitrary normalization factor \(V_c\) is chosen to be more or less equal to the volume of the cavity—or, more precisely, to the mode volume occupied by electromagnetic fields in the laser cavity—then the normal mode functions \(u_n(r)\) themselves will be dimensionless and will have magnitudes of order unity, i.e.,

\[\tag{9}|u_n(r)\approx1|\quad V_c\approx\text{actual}\;\text{cavity}\;\text{volume}.\]

With this choice the mode amplitude coefficients \(E_n(t)\) will have the dimensions of electric field and magnitudes more or less equal to the actual field \(\boldsymbol{\varepsilon}(r,t)\) inside the laser cavity.

 

Eigenmode Example: Closed Rectangular Cavity

Let us take a moment to look at some elementary examples of cavity eigen-modes \(u_n(r)\), since this may help to understand the nature of typical cavity eigenmodes. 

First of all, it is well-known from microwave theory that closed cavities with lossless boundary conditions will always possess just such a complete set of lossless and orthogonal normal modes for the electromagnetic fields within the cavity.

 

 

FIGURE 2.  Hermite-gaussian eigenmodes of an open-sided, stable optical resonator.

For example, in a closed rectangular cavity with perfectly conducting walls (Figure 1) there is a set of x polarized transverse-electric \(\text{(TE)}\) modes which can be written as 

\[\tag{10}u_{mnq}(r)=\sqrt{8}\cos\left(\frac{m\pi x}{a}\right)\sin\left(\frac{n\pi y}{b}\right)\sin\left(\frac{q\pi z}{d}\right)x,\]

where \(x\) is a unit vector in the x direction, and we have made the arbitrary choice that \(V_c=abd\), the cavity volume. The associated eigenvalues are given by 

\[\tag{11}k^2_{mnq}=\left(\frac{\omega_{mnq}}{c}\right)^2=\left(\frac{m\pi}{a}\right)^2+\left(\frac{n\pi}{b}\right)^2+\left(\frac{q\pi}{d}\right)^2.\] 

(We will use multiple mode indices, like \(mnq\), where they are useful to identify separate spatial coordinates or axes; but will condense this back to just a single overall index n when we are talking about a general mode expansion or about a single mode in a general expansion.)

The field pattern for a typical mode of this type is shown in Figure 1. These \(x\)-polarized \(\text{TE}\) modes, plus a similar set of \(y\)-polarized \(\text{TE}\) modes and a set of \(\text{TM}\) modes, can be shown to provide a complete set for expanding any electromagnetic field distribution within such a rectangular cavity.

 

Second Example: Stable Optical Cavity Eigenmodes

As one example of the eigenmodes in an optical or laser cavity, we might consider the transverse and longitudinal eigenmodes in a stable two-mirror optical resonator of length \(\text L\) with mirrors large enough to have negligible diffraction losses, as illustrated in Figure 2. These eigenmodes will be given, to a very good approximation, by 

\[\tag{12}u_{mnq}(r)\approx H_m\left(\frac{\sqrt{2}x}{w(z)}\right)H_n\left(\frac{\sqrt{2}y}{w(z)}\right)e^{-r^2/w^2(z)}\sin\left(kz+\psi(z)+\frac{\pi r^2}{\lambda R(z)}\right)\]

with eigenvalues given by

\[\tag{13}k^2_{mnq}=\left(\frac{\omega_{mnq}}{c}\right)^2\approx\frac{\pi}{L}\left[q+(n+m+1)\frac{\cos^{-1}(g_1g_2)^{1/2}}{\pi}\right].\]

The index \(q\) (typically a very large integer) identifies the axial or longitudinal mode number, whereas the indices \(m\) and \(n\) (typically small integers) identify different transverse mode patterns associated with each longitudinal mode.

The gaussian beam parameters \(g_1,g_2,w(z),R(z)\) and \(\psi(z)\) in Equations 12 and 13 are given by stable gaussian mode formulas that are discussed elsewhere

in this tutorial, and the normalization constant in front of the eigenmode expression has been left out for simplicity.

The fundamental weakness mentioned at the beginning of this section is, however, that real laser cavities like Figure 2 are in fact neither closed nor lossless. Real laser cavities usually have a significant power loss out of the cavity through (or past the edges of) at least one end mirror.

In addition they normally have open sides, which unavoidably allow some energy to leak out to infinity.

The transverse and axial eigenmodes of conventional laser cavities, therefore, as normally calculated in optical-resonator theory ( OPTICAL BEAMS AND RESONATORS ), are not lossless (against diffraction and coupling losses).

In addition, they generally do not satisfy the orthogonality relation given in Equation 8 (they obey instead a more general biorthogonality relation derived in Orthogonality Properties of Optical Resonator Modes).

Finally, there are some fundamental mathematical difficulties in even proving that such modes can exist in an open-sided laser cavity, much less that they will form a complete set.

The ideal Hermite-gaussian functions written in Equation 12 are in fact lossless and do obey the orthogonality property given in Equation 8. The functions in Equation 12 are, however, only approximations, though usually very good approximations, to the exact modes of such an open-sided laser cavity.

The exact eigenmodes in a stable laser cavity with finite diameter mirrors will differ slightly from these Hermite-gaussian modes, primarily out near the mirror edges, because of the diffraction effects associated with these edges.

The Hermite-gaussian modes will be an excellent approximation over most of the resonator volume, so long as the "spillover" losses at the mirror edges are small.

All these mathematical difficulties with real laser cavity modes are generally ignored in conventional laser analyses, and we will ignore them in the present section also.

This is not as criminal an act as it might seem—the general form of the derivation, and the final equations resulting from it, are still essentially correct, despite these formal weaknesses in the analytical approach.

 

Cavity Mode Equation of Motion

The next important step in our derivation is to substitute the normal mode expansion given in Equation 5 into the vector wave equation 4. From this point on let us also keep only the electric polarization pa and drop the magnetic polarization ma, since we are most often interested in electric-dipole atomic transitions. Equation 4 then reduces to the form 

\[\tag{14}\sum_n\left[\frac{\partial^2E_n(t)}{\partial t^2}+\frac{\sigma}{\epsilon}\frac{\partial E_n(t)}{\partial t}+\omega^2_n E_n(t)\right]u_n(r)=-\frac{1}{\epsilon}\frac{\partial^2p_a(r,t)}{\partial t^2}.\]

Suppose the atomic polarization term \(p_a\) on the right-hand side of this equation happens to be zero. Each term of the left-hand summation can then be independently set to zero, and the free decay of each cavity mode will be given by independent solutions of the form

\[\tag{15}E_n(t)=R_eE_0\;\text{exp}\left[-\frac{\sigma}{2\epsilon}t+j\hat{\omega}_nt\right]=R_eE_0\;\text{exp}\left[-\frac{\gamma_{0n}}{2}t+j\hat{\omega}_nt\right].\]

We introduce the notation \(\gamma_{0n}\) to represent the decay rate for the cavity-mode energy due to internal losses. The exact resonant frequency of the cavity mode is then given by 

\[\tag{16}\hat{\omega}_n\equiv\sqrt{\omega^2_n-(\gamma_{0n}/2)^2.}\]

Since the cavity loss rate \(\gamma_{0n}\) is normally many orders of magnitude smaller than the oscillation frequency, we will ignore the very small distinction between \(\hat{\omega}_n\) and \(\omega_n\) from here on.

In the general situation the driving polarization pa on the right-hand side will not be zero. We can then multiply both sides of Equation 14 by any one particular mode function \(u_n(r)\); integrate over the cavity volume; and make use of the orthonormality relation 5 for the cavity modes. If we do this, we obtain a separate equation of motion for the amplitude of that one particular cavity mode, namely,

\[\tag{17}\frac{d^2E_n(t)}{dt^2}+\gamma_{0n}\frac{dE_n(t)}{dt}+\omega^2_nE_n(t)=-\frac{1}{\epsilon}\frac{d^2P_n(t)}{dt^2},\]

The amplitude of this eigenmode is driven or excited by a polarization term \(P_n(t)\) which is given by the overlap integral

\[\tag{18}P_n(t)\equiv\frac{1}{V_c}\int\int\int_\text{cavity}p_a(r,t)\cdot u_n(r)dr.\]

A separate cavity-mode equation, with a separate polarization driving term like this, can then be written for each separate normal mode of the cavity. Each such equation has the form of a second-order differential equation, with a characteristic decay rate \(\gamma_{0n}\), resonance frequency \(\omega_n\), and driving polarization \(P_n(t)\).

In many practical lasers, multiple cavity modes will oscillate simultaneously. We must then write a separate cavity equation like Equation 17 for each such mode. When we include inhomogeneous transitions, saturation, and similar effects, different cavity modes \(E_n\) may become coupled to each other, as we shall see later.

In mathematical terms this coupling occurs through the polarization terms \(P_n\), i.e., the polarization term \(P_n\) for the \(n\)-th mode may include contributions proportional to the amplitudes of other modes \(E_m\), as well as the mode \(E_n\) itself.

In the simplest situation, however, a laser may operate in only one oscillation mode; and only one cavity equation need then be written. In this situation we will often use the notation \(\omega_c\) rather than \(\omega_n\) for the cavity resonant frequency; and we will also drop the subscript \(n\) on all other quantities in the equations.

 

The Polarization Driving Term

The polarization driving term \(P_n(t)\) on the right-hand side of Equation 17 corresponds to the expansion coefficient for the \(n\)-th eigenmode \(u_n(r)\) in an eigenmode expansion of \(p_a(r,t)\). That is, if we write \(p_a(r,t)\) as 

\[\tag{19}p_a(r,t)=\sum_nP_n(t)u_n(r),\]

then \(P_n(t)\) as defined in Equation 18 is just the expansion coefficient in this expansion. In many situations the atomic polarization \(p_a(r,t)\) will itself be coherently produced by the cavity-mode field \(\varepsilon(r,t)\) acting on the atoms in a laser medium. In this situation the polarization \(p_a(r,t)\) will have more or less the same spatial pattern as the cavity field \(u_n(r)\), at least within the laser medium.

Suppose for example that a laser medium with linear susceptibility \(\chi\) fills part of the laser cavity, as in Figure 3. We can then write the polarization in this laser medium as \(p_a(r,t)\approx\chi\epsilon\varepsilon(r,t)\approx\chi\epsilon E_n(t)u_n(r)\). (Writing the polarization

 

 

FIGURE 3.  Overlap integral between cavity fields and laser atoms.

 

 

FIGURE 4.  A situation where the cavity fields and the atomic polarization have nearly zero overlap.

in this form leaves out the phase shift associated with a complex susceptibility \(\tilde{\chi}\), which we cannot handle properly until we convert to sinusoidal signals.) The driving polarization then becomes 

\[\tag{20}P_n(t)\approx\frac{\chi\epsilon E_n(t)}{V_c}\int\int\int_\text{atoms}u_n(r)\cdot u_n(r)dr\approx\eta_c\chi\epsilon E_n(t),\]

where \(eta_c\) is a filling factor \(\leq 1\) given by

\[\tag{21}\eta_c\equiv\frac{1}{V_c}\int\int\int_\text{atoms}u_n(r)\cdot u_n(r)dr.\]

Because the volume of integration in these integrals extends only over the atoms, and not the full cavity volume, the filling factor has a value \(\eta_c\leq 1\) which reduces the effective value of \(\chi\) on the right-hand side of Equation 20.

There can even be situations, as illustrated in Figure 4, where the laser atoms or the driving polarization may be located at a node of the electric field. The field then produces no polarization in the material; but even if it did, because of the lack of spatial overlap between the polarization \(p_a(r,t)\) and the cavity mode pattern \(u(r,t)\), this polarization would not couple back into the cavity.

 

Cavity Lumped Equivalent Circuits

Some people find it helpful to interpret each of the separate resonant modes in a microwave or optical cavity as represented by a lumped resonant electrical circuit, as shown in Figure 5. The circuit equation for this lumped resonant circuit is 

\[\tag{22}i_a(t)=C\frac{dv(t)}{dt}+G_0v(t)+\frac{1}{L}\int v(t)dt,\]

 

 

FIGURE 5.  A lumped-element resonant circuit corresponding to a single resonant cavity mode.

which we can differentiate once more and rewrite as 

\[\tag{23}\frac{d^2v(t)}{dt^2}+\gamma_0\frac{dv(t)}{dt}+\omega^2_cv(t)=\frac{1}{C}\frac{di_a(t)}{dt}.\]

Then this equation obviously has the same basic form as the cavity equation 17 for a single cavity mode. 

Noticing the identity in form between the cavity equation 17 and the circuit equations 23 is the important point here, rather than making an exact connection between the two systems.

We can set up at least two relations between the lumped circuit and the real cavity parameters, however, by writing 

\[\tag{24}\gamma_{0n}=G_0/C\quad\text{and}\quad\omega^2_n\equiv\omega^2_c=1/LC.\]

Specifying the actual decay rate \(\gamma_{0n}\) and the actual cavity frequency \(\omega_c\) thus determines any two of the lumped circuit elements \(G_0\), \(\text L\) and \(\text C\) in terms of the third. The value of the third element is still arbitrary.

We can go further if we wish, however, and relate the cavity-mode amplitude \(E_n(t)\) and the equivalent circuit voltage \(v(t)\) in a convenient way as follows. The stored electrical energy in the cavity electric fields can be written as

\[\tag{25}U_\text{cavity}=\frac{\epsilon}{2}\int\int\int_\text{cavity}|\varepsilon(r,t)|^2dr=\frac{1}{2}\epsilon V_cE^2_n(t)\]

while the stored electrical energy in the lumped circuit can be written as

\[\tag{26}U_\text{circuit}=\frac{1}{2}Cv^2(t).\]

We will make these two energies numerically equal if we make the identification that

\[\tag{27}v(t)\equiv-\left(\frac{\epsilon V_c}{C}\right)^{1/2}E_n(t).\] 

The minus sign in this relation is arbitrary; we insert it merely because the conventional relation between a field \(\text E\) and the voltage \(v\) across a distance \(d\) is \(E=-v/d\).

If we put this into the circuit equation 23, then the circuit equation becomes

\[\tag{28}\frac{d^2E_n(t)}{dt^2}+\gamma_{0n}\frac{dE_n(t)}{dt}+\omega^2_nEn(t)=-\left(\frac{1}{\epsilon CV_c}\right)^{1/2}\frac{di_a(t)}{dt}.\]

But this has exactly the same form as the cavity equation 17 provided we make the final identification

\[\tag{29}i_a(t)\equiv\left(\frac{CV_c}{\epsilon}\right)^{1/2}\frac{dP_n(t)}{dt}.\]

If \(V_c\) has dimensions of volume (which we have said is a convenient choice for this parameter) then \(P_n(t)\) will have the dimensions of electric polarization, or electric dipole moment per unit volume. The first time derivative of \(P_n(t)\) will have dimensions of polarization current density, or current per unit area. Since the square-root factor in Equation 29 has the dimensions (in mks) of area, the quantity \(i_a(t)\) has exactly the proper dimensions of a current.

 

Conclusion

The net result of all this is that a single cavity mode containing a driving polarization \(p_a(r,t)\) can be made formally identical to a simple lumped shunt resonant circuit, with the atomic polarization term \(P_n(t)\) represented by an equivalent shunt current generator \(i_a(t)\). There are five parameters describing the lumped circuit \((G_0,L,C,v,\text{and}\;i_a)\), but only four connecting relations between the lumped circuit and the real cavity.

Hence, there is an arbitrary scale factor in this relationship, which can be thought of as an arbitrary choice of impedance level for the lumped equivalent circuit. That is, given specified values of the parameters of the real optical cavity, we can arbitrarily choose \(C=1\) \(fd\), say, or perhaps \(1/G_0= 50\Omega\), or any other convenient choice for the lumped circuit. All the other lumped circuit parameters will then be uniquely determined. 

Equivalent resonant circuits such as these can be extremely useful, both for purposes of visualization and for understanding the resonance and transient behavior of cavities, and the coupled-mode effects in multimode cavities.

 

 

2.  EXTERNAL SIGNAL SOURCES 

A real resonant cavity will normally have some form of coupling to the external world, whether through a partially transmitting end mirror, a coupling hole, or perhaps a partially reflecting beam splitter inside the laser cavity. The cavity modes may then be driven or excited by external signals sent into the cavity through one of these output coupling ports.

In this section we develop an analysis of external coupling and external signal injection, using the lumped circuit model as an analytical approach. 

 

Wave Analysis of External Coupling and External Signal Injection

Analyzing the external coupling or external signal injection into a resonant cavity is not always simple or straightforward in a full vector field analysis or in a normal-mode formulation. Figure 6 shows, for example, some of the typical cavity coupling methods used for microwave resonant cavities. 

One analytical approach often used in microwave problems like these is to treat the coupling hole in a waveguide cavity, or the coupling loop or coupling probe from a coaxial line, as a lumped delta-function polarization. That is, one writes the induced polarization due to the coupling element in the form \(p_e(r,t)=P_e(t)\delta(r-r_e)\) or \(m_e(r,t)=M_e(t)\delta(r-r_e)\), where re is the point at which the coupling occurs, and the amplitudes \(P_e(t)\) or \(M_e(t)\) are related to the amplitude of the external driving signal. 

This polarization due to the external coupling is then added to the righthand side of the wave equation 4 as an additional driving term for the cavity fields. Additional analysis is required to connect the amplitude of the driving polarization terms \(P_e(t)\) or \(M_e(t)\) to the amplitude of the externally applied signal.

 

 

FIGURE 6.  Microwave cavity coupling methods.

 

 

FIGURE 7.  One possible optical cavity coupling method.

As an optical example of the same approach, suppose a very thin, partially reflecting dielectric beamsplitter with a small reflection coefficient and a large transmission coefficient is put inside a laser cavity to serve as an external coupling element, as shown in Figure 8. (Such a very thin beamsplitter, if made from a dielectric film, is sometimes called a pellicle.)

Suppose a wave coming from an external signal source is sent into this beamsplitter at the correct angle so that a fraction of the external signal wave will be reflected along the cavity axis and will couple into the cavity mode.

At the same time, the cavity mode fields propagating back and forth along the cavity axis will be partially reflected into the external coupling direction, and will be coupled out of the laser cavity (in two opposite directions, going both upward and downward, with this particular coupling method). Reflection from this pellicle this provides both external coupling and external signal injection to the cavity. 

From another viewpoint, however, we can say that the external signal fields \(\varepsilon_e(r,t)\), in passing through the dielectric beamsplitter, will create a dielectric polarization which we could call \(p_e(r,t)\) within the volume of the beamsplitter, as shown in Figure 8. The magnitude of this polarization depends on the dielectric constant and thickness of the beamsplitter, The presence of this externally driven polarization inside the laser cavity will then provide an additional

 

 

FIGURE 8.  Detailed view of the optical coupling method.

polarization driving term on the right-hand side of the wave equation 4, or of the cavity equation, which will excite or drive the cavity mode amplitude. 

Tilting the pellicle to the proper angle to reflect the external wave exactly along the cavity axis ensures that the polarization \(p_e(r,t)\) in the pellicle volume will have exactly the optimum overlap integral with the cavity mode \(u_n{r)\). Note also that the incident wave should have a transverse mode pattern and wavefront curvature which just matches the cavity transverse mode we want to excite. 

At the same time, the fields \(\varepsilon(r,t)\) of the cavity mode, in passing through the same pellicle, will induce a similar dielectric polarization in the pellicle volume which will, in part, radiate out of the laser cavity, giving the external coupling out of the cavity illustrated in Figure 7.

The same pellicle necessarily acts both to couple the external signal into the cavity and to couple internal energy from the cavity mode out of the cavity. Hence, there is a fundamental connection between the decay rate for external coupling or loss out of the cavity, and the coupling strength for coupling of an external signal into the cavity.

 

How to Calculate External Cavity Coupling and Signal Injection

Given a specific cavity geometry and pellicle geometry, we could calculate both the external coupling and the signal injection for a cavity model like Figure 7. This calculation would be somewhat subtle, however, and would require us to assume a specific cavity and coupling geometry (although the results would in fact turn out to be independent of the specific geometry).

To analyze external cavity coupling in a manner which does not depend on the specific cavity geometry, we will follow instead a (possibly) simpler approach, in which we use the lumped equivalent circuit developed in the previous section as an analog model for analyzing an eternally coupled cavity. 

That is, we will derive a general formula for external coupling and external signal injection effects in a resonant system using the lumped circuit model with an external signal source; and we will then assume that this same formula will apply equally well to a resonant optical cavity mode (as indeed it does). The

 

 

FIGURE 9.  Circuit model to represent external cavity coupling

resulting equation will be correct for essentially all types of optical cavities and coupling methods, even though it has been derived from a low-frequency lumped-circuit model. 

 

Lumped Circuit Analysis of External Coupling

In order to follow this line of attack, we first recall that in a real optical cavity with round-trip internal power losses \(2a_0p\) and with one partially transmitting end mirror having power reflectivity \(R\equiv r^2\), the circulating energy in the cavity will decay in each round trip with a total energy decay rate \(\gamma_c\) given by

\[\tag{30}\gamma_c=\gamma_0+\gamma_e=2a_0c+\frac{1}{T}1n(1/R).\] 

where \(\text T\) is the round-trip transit time in the cavity. We are following the same convention as in earlier chapters by using the subscript o (for "ohmic") to refer to internal cavity losses such as absorption or scattering losses; the subscript e to refer to external coupling effects; and the subscript c to refer to total cavity losses including both of these.

Now, an analogous external coupling can be added to the lumped circuit model of Figure 5 by adding an "external" conductance \(G_e\) across the lumped circuit, as in Figure 9, so that the total decay rate in the lumped circuit becomes

\[\tag{31}\gamma_c=\gamma_0+\gamma_e=\frac{G_0+G_e}{C}.\]

For the decay rates in the lumped circuit and the real cavity to remain matched, the additional or external conductance should have a value given by

\[\tag{32}G_e=\gamma_eC,\]

just as \(G_0\) was given by \(\gamma_0C\). The conductance \(G_e\) then represents the external loading or the external coupling to the real optical cavity.

 

 

FIGURE 10.  Circuit model to represent external coupling plus an external signal source.

 

External Signal Source in the Optical Cavity 

Next suppose an externally generated signal wave of amplitude \(\varepsilon_e(t)\) is sent into the laser cavity from some outside signal source (perhaps another laser), as shown in Figure 10, using \(\varepsilon_e(t)\) to indicate the amplitude of the impinging wave outside the cavity mirror. We do not know, yet, how much of this wave will be reflected at the input and how much will be coupled into the cavity.

We also assume that \(\varepsilon_e(t)\) represents only that part of this signal that is properly transverse mode-matched so as to couple into the cavity mode under consideration. (Any remaining part of the incident signal that is not transversely mode matched to the cavity mode in question will have no effect on that mode, but will presumably couple into other transverse modes of the cavity.) 

A lumped-circuit analog to such an external signal can then be obtained by adding an external current source \(i_e(t)\) in parallel with the external conductance \(G_e\), as shown in the lower part of Figure 10. We can relate the amplitude of this external current source \(i_e(t)\) in the lumped circuit to the amplitude of the external wave \(\varepsilon_e(t)\) in the real laser cavity by the following argument.

Suppose we normalize the external wave amplitude \(\varepsilon_e(t)\) of this incident wave such that the instantaneous incident optical power being carried by this incident optical wave (in the mode-matched portion) is given by

\[\tag{33}P(t)\equiv2\varepsilon^2_e(t).\]

We include the factor of 2 in this equation because \(\varepsilon_e(t)\) will normally represent a quasi-sinusoidal wave, which we will write in the form \(\varepsilon_e(t)\equiv R_e\tilde{E}_e(t)e^{j\omega t}\); and we can then write the time-averaged power being carried by the external wave in the particularly simple form

\[\tag{34}P_\text{av}=|\tilde{E}|^2.\]

In electrical circuit jargon, \(P_\text{av}\) represents the available power from the external source, i.e., the maximum power that this wave could deliver to a totally absorbing and reflectionless surface.

The particular normalization used in Equations 33 and 34 is quite arbitrary, and could be chosen differently. Obviously this particular normalization means that \(\varepsilon_e(t)\) must be interpreted as an instantaneous normalized wave amplitude, rather than an optical E field.

In particular, \(E_n(t)\) and \(\varepsilon_e(t)\) will have different dimensions, because we have chosen to relate them in different ways to the real optical \(E\) fields inside and outside the cavity.

 

External Signal Source in the Lumped Circuit

Now, in the lumped circuit analog of Figure 10, the available power from the source current generator \(i_e{t}\) in parallel with the source conductance or external conductance \(G_e\) is the power which this external source could deliver to an optimum load conductance \(G_\text{load}=G_e\). (This corresponds to a signal generator delivering power to a matched or nonreflecting load.) This available power from the parallel combination of \(i_e(t)\) and \(G_e\) can be written as 

\[\tag{35}P(t)=\frac{i^2_e(t)}{4G_e}\quad\text{or}\quad P_\text{av}=\frac{1}{8G_e}|\tilde{I}_e|^2,\]

where we again assume that \(i_e(t)\) will probably be a quasi-sinusoidal signal in the form \(i_e(t)=Re\tilde{I}_e(t)e^{j\omega t}\).

If we want the external signal source in the real cavity and the external signal source in the lumped circuit model to have the the same available power, so that the lumped-circuit model will continue to be a good representation for the cavity, then we must match up Equations 33, 34, and 35 such that the lumped current source \(i_e(t)\) will be related to the normalized external wave amplitude \(\varepsilon_e(t)\) by

\[\tag{36}i_e(t)\equiv-(8G_e)^{1/2}\varepsilon_e(t).\] 

The minus sign is entirely arbitrary and is inserted only for later convenience.

 

General Equation for an Externally Excited Cavity

When the external coupling and external signal source are added, Equation 23 for the lumped equivalent circuit is expanded to 

\[\tag{37}\frac{d^2v}{dt^2}+\left(\frac{G_0+G_e}{C}\right)\frac{dv}{de}+\left(\frac{1}{LC}\right)v=\frac{1}{C}\left[\frac{di_a(t)}{dt}+\frac{di_e(t)}{dt}\right].\] 

If we convert this lumped circuit equation to optical-cavity quantities by using all the definitions that we have developed above, then the final result for the lumped circuit is

\[\tag{38}\frac{d^2E_n}{dt^2}+\gamma_c\frac{dE_n}{dt}+\omega^2_cE_n=-\frac{1}{\epsilon}\frac{d^2P}{dt^2}+\left(\frac{8\gamma_e}{\epsilon V_c}\right)^{1/2}\frac{d\varepsilon_e}{dt}.\]

This was derived as a lumped-circuit equation, but it is now expressed entirely in terms of optical-resonator quantities. That is, all of the factors like \(L,\;G\) and \(C\) that are specific to the lumped circuit model have canceled out, and only parameters like \(\gamma_e\), \(\gamma_c\), \(\omega_c\) and \(V_c\)that apply equally well to the optical cavity are left. We can therefore take Equation 38 as representing the general form of the equation of motion for an arbitrary optical cavity mode having internal losses, resonant atoms, external coupling, and an externally injected signal.

 

 

FIGURE 11.  Circuit model to represent an optical cavity with multiple external coupling ports.

This equation, applied to any arbitrary form of external coupling into an optical or laser cavity, is the primary result of this section. 

A real optical cavity may have output coupling at several different points, the simplest example being two partially transmitting mirrors with power reflectivities \(R_1\) and \(R_2\) at the two ends of a laser cavity, as illustrated in Figure 11.

Each such output coupling port must then be represented by a separate external conductance \(G_{e1}\) and \(G_{e2}\), as illustrated in the lumped equivalent circuit. If necessary separate current sources \(i_{e1}(t)\) and \(i_{e2}(t)\) must also be added to represent external signals coming into both of these ports.

The complete equation for the cavity mode in the two-port case then takes on the form 

\[\tag{39}\frac{d^2E_n}{dt^2}+\gamma_c\frac{dE_n}{dt}+\omega^2_cE_n=-\frac{1}{\epsilon}\frac{d^2P}{dt^2}+\left(\frac{8\gamma_{e1}}{\epsilon V_c}\right)^{1/2}\frac{d\varepsilon_{e1}}{dt}\left(\frac{8\gamma_{e2}}{\epsilon V_c}\right)^{1/2}\frac{d\varepsilon_{e2}}{dt},\]

where \(\gamma_c=\gamma_0+\gamma_{e1}+\gamma_{e2}\); the magnitude of each external conductance is related to the contribution of that output coupling port to the total external-coupling decay rate in the form \(\gamma_{e,i}\equiv G_{e,i}/C\); and the external current sources are related to the incident external wave amplitudes by \(i_{e,i}(t)=-(8G_{e,i})^{1/2}\varepsilon_{e,i}(t)\).

We emphasize again that the lumped circuit model and the cavity equation of motion that we have developed here represent only one cavity resonant mode— that is, if multiple cavity modes are active, each separate transverse and axial laser cavity mode needs a separate mode equation and a separate lumped-circuit model.

If several modes are being excited or are oscillating in a real laser, a separate lumped circuit and a separate equation is required for each such mode. Each mode will then have separate mode amplitudes \(E_n\) and will in general have different values of internal loss \(\gamma_{0n}\), external coupling \(\gamma_{en}\), resonance frequency \(\omega_n\), and so forth.

An external signal \(\varepsilon_e(t)\) must generally have a sinusoidal frequency close to some one cavity frequency \(\omega_c\) in order to excite a significant response in that mode.

In a general multimode analysis, the various frequency components contained in a possibly broadband external signal \(\varepsilon_e(t)\) will in general excite separately each of the different laser-cavity modes with which different frequency components are in resonance.

To see just how an external signal will excite the cavity mode, and where the input power will go, the reader may want to consider the problems at the end of this section.

 

 

3.  COUPLED CAVITY-ATOM EQUATIONS

The analysis developed in the preceding two sections, combined with the analyses in earlier chapters, has now given us a full set of coupled cavity and atomic equations that can be used to solve almost any practical laser problem. The purpose of this section and the following section is to combine these equations into a single set, using a unified and consistent notation, and then to show how the resulting set of equations can be simplified and approximated in various useful ways.

These simplifications will include the slowly varying envelope approximation \(\text{(SVEA)}\) and the phase-amplitude form of the equations, as first introduced by Lamb, and a simplified set of coupled cavity and atomic rate equations. 

 

Atomic Polarization Equation

The atomic polarization \(p_a(r,t)\) at any point inside a laser cavity will depend on the signal field \(\varepsilon(r,t)\) at that point and perhaps also, as we will see later, on various nonlinear and/or modulation effects occurring inside the cavity. In the simplest situation, for a two-level electric-dipole model as derived in earlier chapters, the atomic polarization at every point r is coupled to the signal field by the resonant dipole equation of motion 

\[\tag{40}\frac{\partial^2 P_a(r,t)}{\partial t^2}+\Delta\omega_a\frac{\partial p_a(r,t)}{\partial t}+\omega^2_aP_a(r,t)=k\Delta N(r,t)\varepsilon(r,t),\]

with the constant \(k\) given by

\[\tag{41}k\equiv\frac{3^*\epsilon\lambda^3\omega_a\gamma_{rad}}{4\pi^2}.\]

For simplicity we will ignore the tensor nature of the atomic response from here on, since the tensor properties complicate the algebra with little or no gain in physical significance.

Suppose we dot-multiply both sides of Equation 40 by one particular mode function \(u_n{r)\) and integrate, using the orthonormality properties. The result is a scalar time-varying equation of motion

\[\tag{42}\frac{d^2P_n(t)}{dt^2}+\Delta\omega_a\frac{dP_n(t)}{dt}+\omega^2_aP_n(t)=\frac{k}{V_c}\int\int\int_\text{atoms}\Delta N(r,t)\varepsilon(r,t)\cdot u_n(r)dr,\]

where \(Pn(t)\) is the same polarization driving term as introduced in the cavity equations 17 and 18.

Equation 42 is still quite general, since it involves the total field \(\varepsilon(r,t)\) on the right-hand side. Note also that we must allow for the possibility that the population difference \(\Delta\mathcal{N}(r,t)\) as well as the field \(\varepsilon(r,t)\) may both be functions of both space and time inside the laser cavity. Suppose, however, that only a single cavity mode \(\varepsilon(r,t)=E_n(t)u_n(r)\) is excited, or is oscillating. The righthand side of Equation 42 can then be written as

\[\tag{43}kE_n(t)\times\frac{1}{V_c}\int\int\int\Delta N(r,t)u_n(r)\cdot u_n(r)dr\equiv\eta_n\Delta N(t)E_n(t),\]

where \(\Delta\mathcal{N}(t)\) is an average population difference density given by

\[\tag{44}\Delta\mathcal{N}(t)\equiv\frac{1}{V_c}\int\int\int_\text{atoms}\Delta N(r,t)dr.\]

This quantity is evaluated as if the total number of atoms \(\int\int\int\Delta N(r,t)dr\) were uniformly distributed over the cavity volume \(V_c\) with a uniform density \(\Delta\mathcal{N}(t)\); but then this average density must be multiplied by a filling factor \(\eta_n\) for the \(n\)-th mode given by

\[\tag{45}\eta_n\equiv\frac{\int\int\int\Delta N(r,t)|u_n(r)|^2dr}{\int\int\int\Delta N(r,t)dr}.\]

This dimensionless filling factor is always \(\leq1\) and, together with \(\Delta\mathcal{N}(t)\), takes account of the fact that the atoms may not be uniformly distributed inside the laser cavity.

In terms of these quantities the polarization equation (42) for any one resonant mode simplifies to

\[\tag{46}\frac{d^2P_n(t)}{dt^2}+\Delta\omega_a\frac{dP_n(t)}{dt}+\omega^2_aP_n(t)=k_n\Delta\mathcal{N}(t)E_n(t),\]

where \(k_n\equiv\eta_nk\). This equation represents the simplest but yet still fairly general form of the atomic polarization equation for the n-th cavity mode. The filling factor \(\eta_n\) describes how well the field pattern \(u_n(r)\) overlaps the distribution \(\Delta N(r,t)\) of atoms inside the cavity—for example, \(\eta_n\rightarrow0\) if atoms are located only where the fields are not, and \(\eta_n\rightarrow1\) if the atoms are located optimally where the fields are. The numerical value of \(\eta_n\) can in fact most conveniently be absorbed into the definition of the constant \(k_n\equiv\eta_nk\), as we will do from now on.

 

Atomic Population Equation

Besides the polarization equation for \(P_n(t)\), we also need an equation of motion for the population difference \(\Delta\mathcal{N}(t)\). To derive this, suppose we define the zero level of energy for a two-level atomic system halfway between the lower and upper energy levels \(E_1\) and \(E_2\), as we can always do. Then the energy density \(U_a(r,t)\) (per unit volume) associated with the energy level populations in the collection of atoms is 

\[\tag{47}U_a(r,t)=[N_2(r,t)-N_1(r,t)]\times\hslash\omega/2=-\frac{1}{2}\Delta N(r,t)\hslash\omega.\]

The rate of change of this energy density due to stimulated transitions caused by the cavity fields must be just the power density delivered by the signal fields to the atoms.

But the power per unit volume delivered to the atoms at any point \(r\) by the field \(\varepsilon(r,t)\) acting on the atomic polarization \(p_a(r,t)\) is \(\varepsilon\cdot\partial p_a/\partial t\).

Hence in the vicinity of any point \(r\) the equation of motion, or rate equation, for the atomic population difference \(\Delta N\), with relaxation included, may be written as

\[\tag{48}\frac{d\Delta N(r,t)}{dt}+\frac{\Delta N(r,t)-\Delta N_0(r)}{T_1}=-\left(\frac{2^*}{\hslash\omega}\right)\varepsilon(r,t)\cdot\frac{\partial p_a(r,t)}{\partial t},\]

where \(T_1\) is the effective population relaxation time for the two levels involved; and the factor \(2^*\) depends upon whether the population "bottlenecks" in the lower level (if so, \(2^*=2\)), or whether atoms relax very rapidly out of the lower level to still lower levels (if so, \(2^*=1)\).

Suppose we now expand both \(\varepsilon(r,t)\) and \(p_a(r,t)\) in the cavity eigenmodes, and then integrate both sides of Equation 48 over the cavity volume. After using the orthogonality properties of the eigenmodes \(u_n(r)\), plus the definition of cavity-averaged atomic density \(\Delta\mathcal{N}(t)\) just given in Equation 44, we find that Equation 48 reduces to

\[\tag{49}\frac{d\Delta\mathcal{N}(t)}{dt}+\frac{\Delta\mathcal{N}(t)-\Delta\mathcal{N}_0}{T_1}=-\frac{2^*}{\hbar\omega}\sum_nE_n(t)\frac{dP_n(t)}{dt},\]

where the right-hand side is the sum over all cavity modes that are significantly excited. This is the simplest general form of the atomic population equation. 

 

Final Result: The "Neoclassical Equations"

Suppose finally that only a single cavity mode is significantly excited, as can often happen in real laser systems. The complete cavity-plus-atom equations of motion for the laser can then be reduced to just three coupled differential equations connecting the cavity mode amplitude \(E(t)\), the atomic polarization amplitude \(P(t)\), and the population difference \(\Delta\mathcal{N}(t)\) relevant to that particular cavity mode. (We can drop the subscripts \(n\) for simplicity). 

These equations consist of the cavity equation, the atomic polarization equation, and the population difference equation, or

\[\tag{50}\begin{align}\frac{d^2E(t)}{dt^2}+\gamma_c\frac{dE(t)}{dt}+\omega^2_cE(t)&=-\frac{1}{\epsilon}\frac{d^2P(t)}{dt^2}+\left(\frac{8\gamma_e}{\epsilon V_c}\right)^{1/2}\frac{d\varepsilon_e}{dt}\\\frac{d^2P(t)}{dt^2}+\Delta\omega_a\frac{dP(t)}{dt}+\omega^2_aP(t)&=k\Delta\mathcal{N}(t)E(t)\\\frac{d\Delta\mathcal{N}(t)}{dt}+\frac{\Delta\mathcal{N}(t)-\Delta\mathcal{N}_0}{T_1}&=-\frac{2^*}{\hbar\omega}E(t)\frac{dP(t)}{dt}.\end{align}\]

These three equations are used as the starting point for a great many laser analyses in the literature. They are sometimes called the "neoclassical formulation of laser theory" based on a series of theoretical papers by Jaynes, since they represent the simplest form of semiclassical quantum theory (i.e., the atoms are quantized, but the electromagnetic field is not) as applied to a single cavity mode and a two-level atomic system.

They can in principle be solved with any given initial conditions, taking into account an external driving field \(\varepsilon_e(t)\). Self-consistent solutions to Equations 50 can bring out nearly every significant classical and quantum feature of laser dynamics and of quantum electronic phenomena (except for noise phenomena), at both small and large signal levels.

Note that all the variables in Equations 50 are functions of time only, since all spatial variations have been eliminated by using the normal mode formulation. The essential parameters in Equations 50 are then the cavity decay rate \(\gamma_c\), the atomic homogeneous linewidth \(\Delta\omega_a\), the population recovery time \(T_1\), and the coupling coefficient \(k\) between the cavity signal and the atoms.

Note also that the arbitrary normalization constant \(V_c\) disappears from Equations 50, except for its role in establishing the amplitude of the externally injected signal term.

Equations 50 can be extended to multi-cavity-mode problems by writing separate cavity and polarization equations for the amplitudes \(E_n\) and \(P_n\) of each significant cavity mode; and then restoring the summation over these modes on the right-hand side of Equation 49. This still does not include, however, one potentially important aspect of multimode behavior, namely, the cross-coupling or cross-saturation effects between modes.

If multiple modes are excited, interference effects between modes can cause the degree of saturation of \(\Delta N(r,t)\) to be significantly different at peaks or nulls of the multimode field pattern.

This effect can be difficult to include in the preceding analysis, since it shows up primarily as a complicated intensity-dependent change in the filling factor rjn for each different normal mode, caused by changes in the spatial distribution of \(\Delta N(r,t)\).

More complicated analytical approaches must be adopted when spatially varying saturation effects become important.

 

 

4.  ALTERNATIVE FORMULATIONS OF THE LASER EQUATIONS

The three coupled cavity-plus-atomic equations in Equation 50 can be manipulated and simplified into several other useful forms. Two of these forms involve the slowly varying envelope approximation, and the phase-amplitude form of the coupled equations, both of which are developed in this section. 

 

Slowly Varying Envelope Approximation

Suppose we consider a single-mode laser, and assume, as is almost always reasonable, that the field amplitude \(E(t)\) and the polarization \(P(t)\) will both be quasi-sinusoidal quantities, with slowly varying amplitudes and phases referenced to some carrier frequency \(\omega\). We can then write the \(ac\) signal quantities in the slowly varying sinusoidal forms 

\[\tag{51}E(t)=\frac{1}{2}[\tilde{E}(t)e^{j\omega t}+c.c.],\quad\text{and}\quad\varepsilon_e(t)=\frac{1}{2}[\tilde{E}_e(t)e^{j\omega t}+c.c.],\]

and also write the atomic polarization in the same form

\[\tag{52}P_n(t)=\frac{1}{2}[\tilde{P}(t)e^{j\omega t}+c.c.].\]

The quantities \(\tilde{E}(t)\), \(\tilde{P}(t)\) and \(\tilde{E}_e(t)\) are thus the slowly time-varying complex phasor amplitudes of these nearly sinusoidal quantities. 

Suppose we introduce these expressions into the second-order differential equations for \(E(t)\) and \(P(t)\), and pick out only the \(e^{e\omega t}\) terms on each side of the equations. The equation for \(E(t)\), for example, with the external signal term left off for simplicity, then takes on the expanded form

\[\tag{53}\begin{align}\frac{d^2\tilde{E}(t)}{dt^2}&+[2j\omega+\gamma_c]\frac{d\tilde{E}(t)}{dt}+[j\omega\gamma_c+\omega^2_c-\omega^2]\tilde{E}(t)\\&=-\frac{1}{\epsilon}\left[\frac{d^2\tilde{P}(t)}{dt^2}+2j\omega\frac{d\tilde{P}(t)}{dt}-\omega^2\tilde{P}(t)\right]\left(\frac{8\gamma_e}{\epsilon V_c}\right)^{1/2}\left[\frac{d\tilde{E}_e(t)}{dt}+j\omega\tilde{E}(t)\right].\end{align}\]

We can eliminate or simplify a number of the terms in this equation, and in the corresponding equation for \(P(t)\), by making the slowly varying envelope approximation or \(\text{SVEA}\).

This approximation assumes that the time variations of the phasor amplitudes \(\tilde{E}(t)\), \(\tilde{E}_e(t)\) and \(\tilde{P}(t)\) will all be slow compared to the optical carrier frequency \(\omega\), and that the cavity decay rate \(\gamma_c\) and the atomic linewidth \(\Delta\omega_a\) will also be small compared to \(\omega\), \(\omega_a\), or \(\omega_c\).

In taking derivatives of the complex phasor amplitudes, therefore, as in Equation 53, we can order all of the terms according to the magnitudes

\[\tag{54}\left(\frac{d^2}{dt^2},\gamma_c\frac{d}{dt},\Delta\omega_a\frac{d}{dt}\right)\ll\left(\omega\frac{d}{dt},\omega\gamma_c,\omega\Delta\omega_a\right)\ll(\omega^2,\omega^2_a,\omega^2_c),\]

where \(\omega\) in the middle term means either \(\omega\), \(\omega_c\) or \(\omega_a\). We then drop all the terms of second-order smallness (i.e., \(d^2/dt^2\) or \(\gamma_cd/dt)\) with respect to terms of first-order or zero-order time variation. We also make the same resonance approximation as in earlier chapters, namely,

\[\tag{55}\omega^2-\omega^2_a=(\omega+\omega_a)(\omega-\omega_a)\approx2\omega(\omega-\omega_a).\]

Finally, we can argue that the whole polarization term on the right-hand side of Equation 53 is inherently of first-order smallness, since this term basically represents a comparatively small gain and/or frequency pulling effect on the cavity due to the atoms.

We can therefore drop both the \(d^2\tilde{P}/dt^2\) and the \(2j\omega\; d\tilde{P}/dt\) terms on the right-hand side as being of second or even third-order smallness, and keep only the \(\omega^2\tilde{P}\) term on this side.

If we put all these approximations — which are in fact quite mild approximations — into Equations 50, the result is a pair of first-order but now complex equations for the phasor amplitudes \(\tilde{E}(t)\) and \(\tilde{P}(t)\), plus a modified population equation for \(\Delta\mathcal{N}(t)\), in the form

\[\tag{56}\begin{align}&\frac{d\tilde{E}(t)}{dt}+[\gamma_c/2+j(\omega-\omega_c)]\tilde{E}(t)=-j\frac{\omega}{2\epsilon}\tilde{P}(t)+\left(\frac{2\gamma_e}{\epsilon V_c}\right)^{1/2}\tilde{E}_e(t)\\&\frac{d\tilde{P}(t)}{dt}+[\Delta\omega_a/2+j(\omega-\omega_a)]\tilde{P}(t)=-j\frac{k}{2\omega}\Delta\mathcal{N}(t)\tilde{E}(t)\\&\frac{d\Delta\mathcal{N}(t)}{dt}+\frac{\Delta\mathcal{N}(t)-\Delta\mathcal{N}_0}{T_1}=-j\frac{2^*}{4\hbar}[\tilde{E}(t)\tilde{P}^*(t)-\tilde{E}^*(t)\tilde{P}(t)].\end{align}\]

These three equations are essentially as accurate as the second-order real amplitude equations 50, but they are considerably easier to work with. Hence these simplified SVEA equations are also very widely used.

 

Discussion of the Slowly Varying Envelope Approximation

Despite its name, the "slowly varying" envelope approximation actually allows for quite fast variations in the complex phasor amplitudes \(\tilde{E}(t)\) and \(\tilde{P}(t)\). These quantities may have, for example, rapid rates of change compared to the atomic linewidth \(\Delta\omega_a\) or to any of the various decay rates \(\gamma\).

The \(\text{SVEA}\) merely says that the amplitudes and phases of all quantities vary slowly with respect to the optical carrier frequency \(\omega\) itself—and this is virtually always true.

This means that the \(\text{SVEA}\) equations can still accurately describe coherent transients, Rabi frequency behavior, and other large-signal or fast-pulse effects.

They cannot, however, in their ordinary form, describe harmonic generation or similar effects occurring at totally different optical frequencies. 

The reader should also note that in setting up the \(\text{SVEA}\) equations, the choice of the reference frequency or carrier frequency \(\omega\) to use in expanding \(E(t)\) and \(P(t)\), as in Equations 51 and 52, is essentially arbitrary.

As an elementary example to demonstrate this, suppose we use some arbitrary value of \(\omega\) in writing these quantities, and then solve for the free decay of the cavity fields and the atomic polarization, ignoring the coupling terms on the right-hand sides of Equations 56. The solution for the cavity signal phasor will be

\[\tag{57}\tilde{E}(t)=\tilde{E}(0)\text{exp}[-(\gamma_c/2)t-j(\omega-\omega_c)t].\]

This says that the real cavity-mode amplitude will then oscillate and decay in the form

\[\tag{58}\begin{align}E(t)&=\tilde{E}(t)\;\text{exp}[j\omega t]\\&=\tilde{E}(0)\;\text{exp}[-(\gamma_c/2)t-j(\omega-\omega_c)t+j\omega t]\\&=\text{exp}[-(\gamma_c/2)t+j\omega_ct].\end{align}\]

In other words, the final solution for \(E(t)\) will turn out to oscillate at the correct frequency \(\omega_c\) regardless of the choice of the reference frequency \(\omega\) used in the initial \(\text{SVEA}\) expansions.

The reader can similarly demonstrate that the free oscillation of \(P(t)\) will end up at the correct frequency ua, independent of the initial choice of \(\omega\).

In the more general situation where we are solving some real laser problem, we can choose any arbitrary value of u anywhere near \(\omega_c\) (or \(\omega_a)\) to set up the \(\text{SVEA}\) expansion in Equations 51 and 52; and the \(\text{SVEA}\) equations will automatically take this arbitrary choice of \(\omega\) into account through the \(\omega\) — \(\omega_a\) and \(\omega\) - \(\omega_c\) terms in Equations 56.

Suppose an external signal with some definite frequency \(\omega_e\) is being applied to a laser. Then it will probably make the most sense to choose the reference frequency \(\omega\) to be equal to the applied signal \(\omega_e\).

Suppose on the other hand that the cavity is oscillating as a free-running laser oscillator with no injected signal, and the cavity frequency \(\omega_a\) and the atomic transition frequency ua are not tuned to be equal to each other.

Then, the actual oscillation frequency will lie somewhere between these two frequencies and will be initially unknown.

The correct procedure in this case is to write all the equations using an initially unknown value of \(\omega\), and then solve these equations to calculate what the proper value of \(\omega=\omega_{osc}\).

 

Phase-Amplitude Equations of Motion

It can be useful in certain situations to break the complex \(\text{SVEA}\) equations of motion for \(\tilde{E}(t)\) and \(\tilde{P(t)}\) into separate equations of motion for the amplitudes and the phases of these quantities. A particularly useful way of doing this was introduced in an important early laser analysis by Willis Lamb, Jr., and this form has since been widely copied. 

In this modification the complex signal phasors in the \(\text{SVEA}\) equations are separated into their magnitudes and phases in the forms

\[\tag{59}\tilde{E}(t)\equiv E_c(t)e^{j\phi_c(t)}\quad\text{and}\quad\tilde{E}_e(t)\equiv E_e(t)e^{j\phi e(t)}.\]

The atomic polarization \(\tilde{P}(t)\) is then written in the slightly more subtle form

\[\tag{60}\tilde{P}(t)\equiv[C(t)-jS(t)]\times e^{j\phi_c(t)},\]

The \(C(t)\) and \(S(t)\) functions in this equation evidently give the in-phase and quadrature components—that is, the "cosine" and "sine" parts—of the sinusoidal polarization \(\tilde{P}(t)\).

These in-phase and quadrature components are measured, however, not with respect to some absolute sine wave, but with respect to the instantaneous phase angle \(\phi_c(t)\) of the sinusoidal cavity Geld \(\tilde{E}(t)\) (which may itself be varying substantially with time).

The \(C(t)\) and —\(jS(t)\) parts thus correspond, at least in the linear steady state, to the \(\mathcal{X}'\) or reactive and the \(\mathcal{X}''\) or dissipative parts of the atomic susceptibility, respectively.

The \(\text{SVEA}\) equations from 56 now break up into five coupled purely real equations. These consist of the cavity phase and amplitude equations, namely,

\[\tag{61}\frac{dE_c(t)}{dt}+\frac{\gamma_c}{2}E_c(t)=-\frac{\omega}{2\epsilon}S(t)+\left(\frac{2\gamma_e}{\epsilon V_c}\right)^{1/2}\;E_e(t)\cos[\phi_c(t)-\phi_e(t)]\]

and

\[\tag{62}\frac{d\phi_c(t)}{dt}+\omega-\omega_c=-\frac{\omega}{2\epsilon}\frac{C(t)}{E_c(t)}-\left(\frac{2\gamma_e}{\epsilon V_c}\right)^{1/2}\frac{E_e(t)}{E_c(t)}\sin[\phi_c(t)-\phi_e(t)],\] 

plus two equations for the real and imaginary parts of the atomic polarization, namely,

\[\tag{63}\begin{align}&\left(\frac{d}{dt}+\frac{\Delta\omega_a}{2}\right)C(t)+\left(\frac{d\phi_c(t)}{dt}+\omega-\omega_a\right)S(t)=0\\&\left(\frac{d}{dt}+\frac{\Delta\omega_a}{2}\right)S(t)-\left(\frac{d\phi_c(t)}{dt}+\omega-\omega_a\right)C(t)=-\frac{k}{2}\Delta\mathcal{N}(t)E_c(t),\end{align}\] 

and finally the population equation

\[\tag{64}\frac{d\Delta\mathcal{N}(t)}{dt}+\frac{\Delta\mathcal{N}-\Delta\mathcal{N}_0}{T_1}=-\frac{1}{\hbar}E_c(t)S(t).\]

These five equations, though somewhat complicated in appearance, contain a large amount of useful information.

 

Discussion of the Phase-Amplitude Equations

We can see immediately, for example, that the cavity signal amplitude \(E_c(t)\) is driven only by the \(S(t)\) or quadrature component of the polarization \(\tilde{P}(t)\)—which is the part proportional to \(\mathcal{X}"\).

Similarly, the time-varying phase or frequency expression \(d\phi_c(t)/dt+\omega\)  is affected only by the in-phase polarization component C(£), or the part proportional to \(\mathcal{X}'\).

The final or population equation also makes clear that power transfer from field to atoms involves only the field amplitude \(E_c(t)\) times the quadrature component \(S(t)\) and not the reactive component \(C(t)\) of the polarization. 

We can also see that the quantity \(d\phi_c(t)+\omega\) always appears in combination in these equations. This is another manifestation of the arbitrary nature for the choice of the reference frequency \(\omega\) used in making the slowly varying envelope approximation in Equations 51 and 52.

Choosing an incorrect (or inappropriate) value of \(\omega\) for the initial expansion simply shows up as a compensating linear phase variation of \(\phi_c(t)\) with time, which is the same thing as a frequency correction \(d\phi_c(t)/dt\) added to \(\omega\). (Note also that when we speak of steady-state solutions to a laser problem, we generally mean that the time derivatives \(d/dt\equiv 0\) for all amplitude quantities; but steady state can still mean \(d\phi_c(t)/dt=\) const, or \(\phi_c(t)=\) const \(\times\;t\) for the phase variation in a steady-state solution.)

The phase-amplitude equations in the form given in Equations 61 to 64 were extensively used by Lamb and other authors in laser analyses and in modelocking calculations in earlier years.

They seem to have become less popular in recent years, as compared to the complex \(\text{SVEA}\) equations. We will use the cavity phase-amplitude equations later on, however, in the study of laser injection locking.

 

 

5.  CAVITY AND ATOMIC RATE EQUATIONS

In the simplest limit of all, the coupled cavity-plus-atomic equations derived in the preceding sections can be reduced to a pair of coupled cavity and atomic rate equations which we will now derive.

These rate equations, although valid only within certain approximations, provide a simple and very useful way of analyzing many significant laser phenomena with more than adequate accuracy, including spiking, \(\text{Q}\)-switching, mode locking (in most situations), laser power output, and laser threshold behavior. 

 

Cavity Photon Number and Cavity Energy Equation

As a first step we express the electromagnetic signal energy contained in a laser-cavity mode in units of \(\hbar\omega\), which is to say, we consider the number of photons \(n(t)\) in the cavity as defined by 

\[\tag{65}n(t)\equiv\frac{\text{cavity}\;\text{energy}}{\hbar\omega}=\frac{\epsilon}{2\hbar\omega}\int\int\int|\varepsilon(r,t)|^2dr.\]

Note that in discussing the number of photons here, we are not implicitly assigning any "billiard-ball" particle properties to photons. We are merely expressing the signal energy in the cavity in the convenient units of \(\hbar\omega\).

For a single cavity mode with electric field given by \(\varepsilon(r,t)=E_n(t)u_n(r)\) and \(E(t)=\frac{1}{2}[\tilde{E}(t)e^{j\omega t}+c.c]\), the cavity energy oscillates at frequency \(2\omega\) back and forth between the electric and magnetic fields.

To obtain the total or time-averaged energy in the cavity, therefore, we must evaluate the integral in Equation 65 using either the peak value of \(E_n(t)\) (corresponding to those instants in the cycle when all the energy in the cavity is in electric form), or else use twice the time-averaged value of \(E\)-field integral (to account for electric power plus magnetic stored energies).

From either argument the cavity photon number in a given cavity mode is given by

\[\tag{66}n(t)=\frac{\epsilon V_c}{2\hbar\omega}|E_n(t)|^2_\text{peak}=\frac{\epsilon V_c}{2\hbar\omega}|\tilde{E}(t)|^2.\]

If we differentiate Equation 66 with respect to time and use the \(\text{SVEA}\) cavity equation (56) for the cavity field \(\tilde{E}(t)\), we can show that the rate of change of \(n(t)\) is given by

\[\tag{67}\frac{dn(t)}{dt}+\gamma_cn(t)=j\frac{V_c}{4\hbar}[\tilde{E}(t)\tilde{P^*}(t)-\tilde{E}^*(t)\tilde{P}(t)].\]

The \(\gamma_c\) term obviously represents photon decay due to cavity losses plus output coupling, whereas the term on the right-hand side gives the time-averaged energy flow (in photon units) delivered by the atomic polarization to the cavity fields.

The external signal term has been dropped because external driving signals to a cavity cannot in general be properly accounted for in a purely rate equation analysis.

 

Atomic Population Equation

If we multiply the average population density \(\Delta\mathcal{N}(t)\) by the cavity volume Vc, we get the total number of atoms \(\Delta N(t)\equiv V_c\Delta\mathcal{N}(t)\) in the cavity. The atomic population equation for this quantity then becomes 

\[\tag{68}\frac{d\Delta N(t)}{dt}+\frac{1}{T_1}[\Delta N(t)-\Delta N_0]=-j\frac{2^*V_c}{4\hbar}[\tilde{E}(t)\tilde{P}^*(t)-\tilde{E}^*(t)\tilde{P}(t)].\]

The right-hand side of this equation has exactly the same form as in the cavity rate equation 67, as it must, since the right-hand side of each equation represents energy flow from the atoms to the fields, or vice versa.

The only difference is the factor of \(2^*\), which we have explained earlier. Note also that the cavity equation for \(n(t)\) is not really an independent addition to our earlier set of three coupled Equations 56, since it contains no new information over and above the cavity equation for \(En(t)\) or \(\tilde{E}(t)\).

 

Conversion to Coupled Rate Equations

In many real situations the atomic linewidth \(\Delta\omega_a\), which gives the dephasing rate or decay rate for any coherent atomic polarization \(\tilde{P}(t)\), is much faster than the decay rate for the cavity fields, i.e., \(\Delta\omega_a\gg\gamma_c\).

Even if this is not true, it may still be true that the linewidth Auja is large compared to the rate of change \(|d\tilde{E}/dt|\) of the cavity field phasor \(\tilde{E}\).

If either of these conditions holds true, this means physically that the transient response time \(T_2\equiv 1/\Delta\omega_a\) with which the polarization \(\tilde{P}(t)\) will follow any variation in \(\Delta N(t)\tilde{E}(t)\) will be much faster than the rate of variation of the quantity \(\tilde{E}(t)\) itself.

In other words, the complex polarization \(\tilde{P}(t)\) will follow the complex cavity field \(\tilde{E}(t)\) with negligible delay on the time scale of variations in \(\tilde{E}(t)\). 

Within this approximation, which is essentially the linear-susceptibility or rate-equation approximation, we can assume that \(\tilde{P}(t)\) will be related to \(\tilde{E}(t)\) by the steady-state solution to the \(\text{SVEA}\) polarization equations, or

\[\tag{69}\tilde{P}(t)\approx-j\frac{2k}{\Delta\omega_a}\frac{1}{1+2j(\omega-\omega_a)/\Delta\omega_a}\Delta\mathcal{N}(t)\tilde{E}(t).\]

If we put this result into the right-hand sides of the two rate equations 67 and 68, we will find that both right-hand sides have the form

\[\tag{70}j\frac{V_c}{4\hbar}[\tilde{E}(t)\tilde{P}^*(t)-\tilde{E}^*(t)\tilde{P}(t)]=-\frac{kV_c}{2\hbar\Delta\omega_a}\frac{\Delta N(t)}{[1+2(\omega-\omega_a)/\Delta\omega_a]^2}|\tilde{E}(t)|^2.\]

Given the connection between the photon number \(n(t)\) and \(|\tilde{E}(t)|^2\), we can simplify this further into the form

\[\tag{71}j\frac{V_c}{4\hbar}\left[\tilde{E}(t)\tilde{P}^*(t)-\tilde{E}^*(t)\tilde{P}(t)\right]=-K\Delta N(t)n(t),\]

where the constant \(K\) appearing in Equation 71 is the same rate-equation \(K\) value derived in earlier chapters (note that this constant includes a lorentzian lineshape dependence on the frequency \(\omega\)).

Hence we obtain, finally, the pair of particularly simple coupled first-order differential equations, namely,

\[\tag{72}\begin{align}&\frac{dn(t)}{dt}+\gamma_cn(t)=-K\Delta N(t)n(t)\\&\frac{d\Delta N(t)}{dt}+\frac{\Delta N(t)-\Delta N_0}{T_1}=-2*K\Delta N(t)n(t).\end{align}\]

These are the coupled rate equations connecting the cavity photon number \(n(t)\) and the effective number of absorbing atoms \(\Delta N(t)\) in the cavity.

These equations state the physically reasonable fact that—within the rate-equation approximation—the rates of change of \(n(t)\) and \(\Delta N(t)\) will be proportional to their respective relaxation terms, plus a stimulated transition term which is proportional to the product of the number of photons \(n(t)\) acting on the population difference \(\Delta N(t)\).

These rate equations are very widely useful for a broad range of laser problems, including spiking, \(Q\)-switching, laser amplitude modulation, and laser mode-locking, where the underlying rate-equation approximation remains valid.

Note that these rate equations discard all knowledge of the phase angles of either the atomic polarization or the cavity field, except for the implicit assumption that \(\tilde{P}(t)\) remains coherently driven by \(\tilde{E}(t)\). The benefit gained, on the other hand, is simplification to a pair of real, first-order, only slightly nonlinear equations to be solved.

We will make extensive use of these equations in subsequent sections.

 

An Alternative Formulation: Radiative Damping

There is also an opposite limiting situation, considered by Tang and others, in which the cavity mode has a very rapid energy-decay rate and hence a wide bandwidth, whereas the atomic resonance transition is very sharp or narrow, so that the relevant condition is \(\gamma_c\gg\Delta\omega_a\).

In this limit we can assume that the cavity field \(\tilde{E}(t)\) follows the polarization \(\tilde{P}(t)\) on a nearly instantaneous basis, rather than the opposite condition as assumed in the preceding paragraphs.

In physical terms, the atomic polarization serves as a kind of flywheel, dragging the cavity field along with it, rather than the cavity fields predominantly forcing the atomic oscillations to follow as in the usual rate-equation limit. 

In this limit we can solve the \(\text{SVEA}\) cavity equation of motion in steady-state form, with an atomic polarization present but with no externally applied signal, to get the on-resonance relationship

\[\tag{73}\tilde{E}(t)\approx-j\frac{\omega}{\gamma_c\epsilon}\tilde{P}(t).\]

This is equivalent to assuming that the cavity fields are generated by radiation from the atomic polarization and not by anything else, particularly not by any externally applied signal. 

Putting this result into the \(\text{SVEA}\) polarization equation 56 then yields the modified equation 

\[\tag{74}\frac{d\tilde{P}(t)}{dt}+\left[\frac{\Delta\omega_a}{2}+\frac{k\omega\Delta N}{2\gamma_c\epsilon}+j(\omega-\omega_a)\right]\tilde{P}(t)=0.\]

For the noninverted population situation, \(\Delta N>0\), this says that the usual decay or dephasing rate \(\Delta\omega_a/2\) for the polarization \(\tilde{P}(t)\) is increased by an additional term proportional to \(\Delta N\) and to the coupling factor \(k\) between the polarization and the cavity.

This additional term represents essentially a speeded-up decay of the polarization because \(\tilde{P}(t)\) radiates energy into the cavity fields, from where this energy decays more or less instantaneously into the cavity losses.

Hence this effect, which occurs only in this special limiting case, is commonly called radiation damping from the atoms into the cavity.

This special form of radiation damping represents an increased or extended form of the free-space radiative decay rate \(\gamma_\text{rad}\) associated with radiating atoms that we discussed in an early tutorial.

That is, we can recall that for atoms in free space, \(\gamma_\text{rad}\) is the physical mechanism by which the atoms radiate away energy to their surroundings. When such atoms are placed inside a sufficiently high-\(Q\) cavity, the polarization \(p(r,t)\) associated with the oscillating atomic dipoles sees a significantly different radiation impedance, because the cavity in effect responds to the radiated fields and reflects them back at the atoms.

The radiative decay rate can then be increased because the dipoles are radiating into a cavity impedance which is better matched than simply radiating into free space.

This kind of increased radiative damping is not usually of significance in practical lasers, and we will not have occasion to solve any practical laser problems using this alternative approximation. Significant increased radiative damping can occur in special situations, however, both in certain optical spectroscopy experiments and in low-frequency magnetic resonance experiments (where the concept of radiation damping first originated).

Frequency standards and atomic clocks are another example where an atomic transition with a very narrow resonance line may interact with a resonant cavity with a substantially wider cavity linewidth, so that cavity-increased radiative damping can be significant.

 

 

 

 

 

 


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