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Mode competition in a two-mode laser

Charles E. Greninger

Published Jan 1, 1971• 0 citations
OpenAlex

Abstract

In this thesis we develop a Lamb-type theory for a two-mode, standing wave, homogeneously broadened laser. The theory differs from Lamb's in that the coupled equations of motion for the atomic system are solved with a matrix perturbation theory that converges best for large, nearly equal mode amplitudes rather than with a perturbation theory that is a power series in the field which converges poorly for large mode amplitudes. The theory has an exact matrix solution when certain symmetries in mode amplitude and mode detuning occur. Asymmetries in these parameters may then be treated by a perturbation theory that converges best when the mode amplitudes are large and nearly equal. The theory makes it possible to solve for the macroscopic polarization at certain cavity positions, and it is found that the polarization consists of an infinite number of harmonic sidebands spaced at multiples of the mode difference frequency from atomic resonance. The projection of the first order polarization onto the two spatial modes gives the first order driving functions for the mode amplitude equations. Two investigations are made: 1) we expand the driving functions in powers of the mode amplitudes to compare with Lamb theory, and 2) we program the mode amplitude equations on a digital computer without expanding in a power series in mode amplitudes to plot out the evolution of the amplitudes for strong fields. The computer shows that even for strong fields, the stronger mode quenches the weaker mode in a time dependent on the laser parameters. Although the theory has been developed for use with longitudinal modes, it may also be used for transverse modes. While the losses of the various longitudinal modes are approximately equal, higher order transverse modes arc known to have higher losses than lower order transverse modes. When the losses of the various modes are taken to be unequal, the matrix theory is expected to give different results than in the case for longitudinal modes.

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Physical Review A•Oct 1, 1976•15 citations
In this thesis we develop a Lamb-type theory for
a two-mode, standing wave, homogeneously broadened laser. The
theory differs from Lamb's in that the coupled equations of motion
for the atomic system are solved with a matrix perturbation theory
that converges best for large, nearly equal mode amplitudes rather
than with a perturbation theory that is a power series in the field
which converges poorly for large mode amplitudes.
The theory has an exact matrix solution when
certain symmetries in mode amplitude and mode detuning occur.
Asymmetries in these parameters may then be treated by a
perturbation theory that converges best when the mode amplitudes
are large and nearly equal. The theory makes
it possible to solve for the macroscopic polarization at certain
cavity positions, and it is found that the polarization consists of
an infinite number of harmonic sidebands spaced at multiples of the
mode difference frequency from atomic resonance.
The projection of the first order polarization
onto the two spatial modes gives the first order driving functions
for the mode amplitude equations. Two investigations are made: 1)
we expand the driving functions in powers of the mode amplitudes to
compare with Lamb theory, and 2) we program the mode amplitude
equations on a digital computer without expanding in a power series
in mode amplitudes to plot out the evolution of the amplitudes for
strong fields. The computer shows that even for strong fields, the
stronger mode quenches the weaker mode in a time dependent on the
laser parameters. Although the theory has been
developed for use with longitudinal modes, it may also be used for
transverse modes. While the losses of the various longitudinal
modes are approximately equal, higher order transverse modes arc
known to have higher losses than lower order transverse modes. When
the losses of the various modes are taken to be unequal, the matrix
theory is expected to give different results than in the case for
longitudinal modes.
A Lamb semiclassical theory is developed for multimode, mode-locked laser operation. The equality of frequency spacings between adjacent modes causes population pulsations generated in the nonlinear atomic response to be integer multiples of the adjacent-mode spacing. This allows the atomic response to be written in terms of Fourier series, thereby reducing coupled differential equations of motion to algebraic equations ultimately expressible in terms of a continued fraction. A unidirectional ring configuration is assumed so that the more complicated spatial dependence of standing waves is avoided. Reductions are made to the rateequation limit and to the special two-mode cases of bidirectional ring operation and single-mode, standing-wave, gas-laser operation. Numerical analysis of the equations is motivated. Specific results are given in the following (companion) paper.
A high-intensity single-mode gas laser is analyzed by several mathematical methods. Exact numerical results are best obtained by solving for the coefficients of a Fourier series solution by using a backward recurrence scheme or a successive convergents method for evaluating a continued fraction. Numerical integration techniques can also be used to convert the periodic boundary conditions to initial conditions and then solve the resulting initial-value problem. Direct analytic integration leads to exact solutions in terms of Bessel functions for the special case of zero detuning, equal decay rates, and neglecting of collision broadening. For general single-mode operation, an improved analytic approximation [the second recursion relation approximation (2RCRA)] is obtained by taking an additional term in the recursion relations for the Fourier coefficients. This approximation predicts the existence of one of the secondary resonances in graphs of the spatially averaged population inversion and quadrature polarization coefficient densities and matches the exact solution from the position of the first resonance and outward independent of the intensity. The 2RCRA also matches the exact detuning curve except at small detunings. The first-order ("rate equation") approximation, by comparison, does not predict any secondary resonances and does not match the exact solution at higher intensities. A physical interpretation for the appearance of the secondary resonances is presented in terms of multiphoton interactions and a prediction for the positions of the resonances for the special case discussed above is obtained. As predicted by Greenstein the exact detuning curves show quantitatively that the Lamb dip decreases and disappears as the intensity increases for high excitations.
A modified perturbation theory is developed for the calculation of the time-dependent wave function describing the propagation of an atom in a two-mode laser field. With the mode frequencies nearly equal, a first approximation is obtained that reproduces and extends known results, for both resonant and nonresonant transitions. This lowest-order solution sums terms that describe the absorption of a photon from one mode and emission into the other, leaving the total number of photons unchanged and shifting the momentum of the atom accordingly. Corrections are generated by expansion of a resolvent from which the nearly degenerate states have been projected out. These approximate solutions are used to construct asymptotic states in a formulation of atom-atom scattering in the two-mode field. In close analogy with the soft-photon approximation, a ``soft-pair'' approximation is developed based on the dominance of those asymptotic interactions in which the transfer of energy from the field to the atom is small, allowing one to relate the scattering amplitude in the presence of the field to the field-free amplitude. When the atomic ground state and an excited state are closely coupled by the field, the asymptotic wave function can have mixed parity allowing for long-range contributions to the effective potential. In addition to an enhancement of forward-scattering probabilities, this leads to a threshold anomaly that is analyzed here, in the resonant case, with the aid of a modified effective-range theory.
This paper extends the calculations of the semiclassical Lamb theory of a Doppler-broadened gas laser (optical maser) to arbitrary intensities for the case of single-mode operation. The coupled equations of the classical electromagnetic field and an ensemble of two-level atoms are set up, and a solution is obtained in the form of a continued fraction. This is used to compute intensity-detuning curves and atomic population inversion densities as functions of both the velocity and the position in the laser. When the laser is tuned to resonance, the velocity dependence of the inversion density shows a previously unknown fine structure consisting of a "bump" in the bottom of the hole. The calculations are compared to results obtained from a perturbation expansion in powers of the field and exact results known for atoms with zero velocity. The response of the laser output to a slow modulation and the buildup of oscillations are also discussed.
The semiclassical interaction of a two-level system with a multimode electromagnetic field is discussed in detail. The equations of motion for the macroscopic density matrix are transformed into a single integral equation for the population density, with a complex kernel which can take into account both running- and standing-wave configurations. The iterative solution to this integral equation, which corresponds to an expansion in terms of powers of the electromagnetic field amplitudes, turns out to be divergent when the field amplitudes are greater than some critical value. In the case in which the kernel is a periodic or quasiperiodic function of time, a general solution to the integral equation is found, and this solution turns out to be unique and convergent for every value of the electromagnetic (e.m.) field amplitude. This solution is expressed through a continued-fraction expansion, whose terms are matrices, which can be readily obtained from the kernel. This solution is a generalization of the continued-fraction expansion usually met in the strong-signal semiclassical theory of lasers. Some particular cases are then treated in detail; among them, the solution to the multimode operation in laser devices is given in which the excited level lifetime is long compared with the period of the beating term between two adjacent modes. Although the theory presented is directed towards the steady-state regime in multimode lasers, it may also treat the transient regime of laser operations, or it may be extended to include phenomena where a strong e.m. field, made up of several modes equally spaced in frequency, interacts with a two-level system.