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Physical review. B, Solid state

Ultrashort-Pulse Generation by Q -Switched Lasers

J. A. Fleck

Published Jan 1, 1970• 185 citations
Publisher

Abstract

We have undertaken the complete temporal description of pulsed emission by a homogeneously broadened laser, including the effects of spontaneous emission, the detailed geometry of the laser cavity, and the variation of atomic polarization and level populations over wavelength distances. The model is based on traveling-wave equations which are derived from Maxwell's equations and solved in conjunction with boundary conditions imposed at the cavity mirrors. Thus, any direct assumptions concerning the nature of the laser's longitudinal mode structure is avoided. Variations in polarization and population over wavelength distances are treated by means of expansions in spatial Fourier series, having as fundamental a half optical wavelength. The Fourier series are truncated after the first harmonic. The treatment differs from earlier work in that the dephasing of the dipole moment is treated exactly without a rate-equation approximation. Spontaneous emission is simulated both as to spectrum and Gaussian character by including in the dipole equations stochastic shot-noise sources. The model equations are solved numerically, and results include the details of $Q$-switched pulse evolution from noise for both passive and active switching. In the case of an actively switched laser, the two-photon fluorescence intensity pattern has been calculated. It reveals a well-defined structure of subsidiary intensity maxima, even though subcavities are not assumed in the calculation. The pattern can be correlated directly with the emission pulse structure, and should vary from shot to shot. No single point in the pattern is suitable for a peak-to-background ratio determination. However, if the background is averaged over a distance in the fluorescing medium equal to twice the separation between cavity mirrors, the peak-to-background ratio would be \ensuremath{\approx} 1.6, indicating a highly uncorrelated spectrum.

DOI
10.1103/physrevb.1.84

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We have undertaken the complete temporal description of pulsed emission by a homogeneously broadened laser, including the effects of spontaneous emission, the detailed geometry of the laser cavity, and the variation of atomic polarization and level populations over wavelength distances. The model is based on traveling-wave equations which are derived from Maxwell's equations and solved in conjunction with boundary conditions imposed at the cavity mirrors. Thus, any direct assumptions concerning the nature of the laser's longitudinal mode structure is avoided. Variations in polarization and population over wavelength distances are treated by means of expansions in spatial Fourier series, having as fundamental a half optical wavelength. The Fourier series are truncated after the first harmonic. The treatment differs from earlier work in that the dephasing of the dipole moment is treated exactly without a rate-equation approximation. Spontaneous emission is simulated both as to spectrum and Gaussian character by including in the dipole equations stochastic shot-noise sources. The model equations are solved numerically, and results include the details of QQQ-switched pulse evolution from noise for both passive and active switching. In the case of an actively switched laser, the two-photon fluorescence intensity pattern has been calculated. It reveals a well-defined structure of subsidiary intensity maxima, even though subcavities are not assumed in the calculation. The pattern can be correlated directly with the emission pulse structure, and should vary from shot to shot. No single point in the pattern is suitable for a peak-to-background ratio determination. However, if the background is averaged over a distance in the fluorescing medium equal to twice the separation between cavity mirrors, the peak-to-background ratio would be \ensuremath{\approx} 1.6, indicating a highly uncorrelated spectrum.
A stochastic, semiclassical model is developed for a multimode, homogeneously broadened laser with rapid-dipole dephasing (e.g., semiconductor or dye lasers). The theory is valid for systems with a free spectral range that is large compared to the decay rates of the population inversion and the field. Population dynamics including temporal beating effects and relaxation oscillations, spatial-hole burning, coherent-wave mixing, and quantum noise are all included self consistently. A principal restriction of the model, when applied to coupled-cavity systems, is that the modes satisfying the boundary conditions of the gain medium must form a subset of the modes defined by the cavity and that only modes in this subset are above threshold. For typical systems, the model is valid for pump rates up to several times threshold. The effects on the emission linewidth of coherent-wave mixing and the number of modes included in the simulation at various pump rates are discussed.
Recent advances in laser technology have led to the production of ultrashort electromagnetic pulses of very high peak power. In this paper we develop a theory which describes the transient behavior of such pulses in a laser amplifier. The effects of atomic coherence and inhomogeneous broadening, necessary for a complete treatment of the problem, are included. In the present development, the degree of homogeneous and inhomogeneous broadening is variable. The two-level active atoms are characterized by a phase memory time T2{T}_{2}T2​ and an inhomogeneous frequency distribution (which leads to a reversible decay time T2∗T_{2}^{}{}_{}{}^{*}T2​​∗). The field (Maxwell) and atomic (Schr\"odinger) equations are coupled in a self-consistent manner as in Lamb's theory of an optical maser. The resulting equations are solved analytically in certain limits, while the general case is integrated numerically. The emphasis is placed on the situation in which T2∗T_{2}^{}{}_{}{}^{*}T2​​∗ and the pulse width are much shorter than T2{T}_{2}T2​. Thus our main effort is devoted to the physics of the inhomogeneously broadened system. The theory itself, however, contains the effects of both forms of broadening. The basic difference between inhomogeneous and homogeneous broadening is discussed. In the former case, the decay of radiation is caused by the dephasing of dipoles, and is reversible. The atomic memory is retained during the dephasing, and an internal reflection of the atomic coordinates, as is exemplified in the "photon echo" process, will cause the dipoles to rephase. The phonon interruptions or atomic collisions that bring about the homogeneous broadening are random processes which lead to an irreversible destruction of the atomic phase memory. The implications of the reversible versus irreversible decay processes are seen to have important consequences in the problem of pulse amplification.
Paraxial Maxwell-Bloch theory is used to describe amplified spontaneous emission (ASE) in strongly inverted media. Counterpropagation of waves coupled through gain saturation and diffraction, and pumping by a laser beam which propagates transverse to the amplifier are included in the treatment. The reactive part of the medium polarization is calculated by population-rate equations in the limit of rapid dephasing of the atomic dipoles. The spontaneous emission noise source needed for the numerical simulation of Maxwell's equations is derived from microscopic theory by coarse-graining, thereby establishing the compatibility of the spherically symmetric microscopic source with paraxial propagation theory. The three coupled Maxwell equations for the counterpropagating ASE waves and the transversely propagating pump wave are solved numerically in time and two spatial dimensions. This simulation is used to calculate the longitudinal profile of the transverse fluorescence, which measures the saturation of the gain, the longitudinal profile of the transmitted pump wave, which measures the saturation of pump absorption, and the transverse coherence function of the radiation emitted from the ends of the amplifier.
This chapter concludes the discussion about temporal instabilities and self-oscillations, considering the instabilities that arise in the systems analyzed in Chapter 13, with the exception of optical bistability, which has been discussed separately in Chapter 11. Section 25.1 focuses on the laser with an injected signal. As already mentioned in Section 13.1, in this system self-pulsing instabilities are rather ubiquitous, and on exploring the parameter space one meets an extremely rich variety of oscillatory behaviors. For example, under extremely high-gain conditions, breathing, spiking and chaos have been identified over various ranges of the driving field strength [294]. In this section we show some results that arise under parametric conditions that are more accessible experimentally. We illustrate especially the coexistence, for the same parametric values, of different oscillatory behaviors that are reached by starting from different initial conditions. On the other hand, Section 25.2 deals with a laser with a saturable absorber. In this system even the full scenario of stationary intensity solutions in the single-mode model is very complex. For the sake of simplicity, in this section we focus on a simpler rate-equation model that is obtained from the single-mode model of Eqs. (13.13)–(13.17) by adiabatically eliminating the atomic polarization of both active and passive medium. As in Chapter 13, we assume that the atomic transition frequencies are equal and also coincide with a cavity frequency. In this way all the variables in the model can be assumed real, and the linearization around the stationary solution described in Section 13.2 leads to a cubic eigenvalue equation. The spontaneous oscillations which arise from this instability are called repetitive passive Q-switching in the literature. Section 25.3 considers the stability of the stationary solutions of the degenerate optical parametric oscillator described in Section 13.4 and outlines the scenario of period doubling and chaos that arises under appropriate parametric conditions.
Propagation and amplification of an ultrashort laser pulse in a two-level medium is numerically investigated with the use of the finite-difference time-domain (FDTD) procedure simultaneously solving Maxwell and Schrodinger equations with no assumptions characteristic of slowly varying envelope approximation. Comparison of the results of FDTD numerical simulations with the predictions of the pulse area theorem has demonstrated that the developed numerical procedure provides an adequate description of pulse evolution in a two-level medium. Amplification of ultrashort light pulses in a two-level medium is analyzed. Two methods of improving the gain of ultrashort pulses in a two-level medium by modulating the spatial distribution of dipole moments of resonant transitions and using initially chirped frequency-detuned pulses are explored.
A stochastic, semiclassical model is developed for a multimode, homogeneously broadened laser with rapid dipole dephasing, appropriate for semiconductor, Ti:sapphire, or dye lasers. The theory self-consistently incorporates population dynamics including temporal beating effects and relaxation oscillations, spatial hole burning, coherent-wave mixing, and quantum noise. The model is valid for single- and compound-cavity lasers in which the mode frequencies are well defined. We pay particular attention to finding a useful mode basis in the case that the gain medium does not completely fill the cavity. This situation can lead to coupled-cavity effects. For typical systems the model is valid for pump rates up to several times threshold and is tractable for numerical simulations. The theoretical development described in this paper is applied to an experimental system in a companion paper (J. Opt. Soc. Am. B14, 180 (1997)].