W. P. Brown

Optical resonators with phase-conjugate mirrors

The operating characteristics of optical resonators with phase-conjugate mirrors are discussed. Degenerate four-wave mixing is assumed to be the process giving rise to the phase-conjugated phase fronts. We have found that these resonators are unconditionally stable, have no longitudinal mode conditions that invoke the length of the cavity, and have smooth transverse mode structures. Computer simulation results pertaining to the properties of the lowest-order modes and the effect of mirror errors on these modes are also discussed.

Fourth Moment of a Wave Propagating in a Random Medium

The fourth statistical moment of a scalar wave propagating in a random medium is investigated. Starting with a partial differential equation obtained by various authors in a multiple-scattering approximation, we discuss the general properties of the fourth moment of an initially plane wave and then present numerical results for a two-dimensional plane wave. Results are presented for the variance and covariance of irradiance scintillations. Both results are shown to agree well with experiment. In particular, the variance exhibits the experimentally observed saturation phenomenon, and the covariance results indicate that the correlation length for irradiance scintillations is not proportional to (λz)12 in the saturation region and that the aperture-averaging effect is less than that predicted by results based on the Born or Rytov approximations.

Absorption and depletion effects on degenerate four-wave mixing in homogeneously broadened absorbers

The coupled equations satisfied by the pump, probe, and signal waves in a four-wave interaction in a two-level saturable absorber are derived, taking into account the effects of absorption and depletion. Numerical solutions of these equations are presented in two limits; one in which the probe-wave amplitude is small compared with that of the pump waves and another in which the probe- and pump-wave amplitudes are comparable. The results indicate that absorption effects are important when the wave amplitudes are less than the saturation amplitude of the absorption process. The importance of the depletion effects depends on the type of four-wave interaction. The most significant effects occur when the undepleted theory predicts probe-wave reflectivities exceeding unity.

Moment Equations for Waves Propagated in Random Media

We show that the partial differential equations obtained by Tatarski, Beran, and Ho for the first, second, and fourth statistical moments of a wave propagating in a random medium are special cases of a general partial differential equation satisfied by the mth moment when the refractive-index inhomogeneities are sufficiently weak. In addition, we derive moment equations that apply regardless of the strength of the inhomogeneities. The results obtained in this case are in the form of difference relations satisfied for the moments.

Validity of the Rytov Approximation in Optical Propagation Calculations

The applicability of the Rytov approximation to the calculation of the characteristics of optical propagation in a weakly inhomogeneous random medium is investigated. The condition that the mean square value of the second term in the associated perturbation expansion be smaller than that of the first is adopted as a criterion for the validity of the Rytov approximation. It is shown that there is a very severe range limitation on the validity of the Rytov approximation for optical propagation in the lower portions of the earth’s atmosphere. Furthermore, comparison of the limiting form for large x/kl2 of the validity condition derived in this paper with the condition on the validity of the Born approximation obtained independently by Mintzer, and by Kay and Silverman reveals that the Rytov and Born approximations have the same domain of validity in this limiting case. The equivalence of the validity conditions for the Rytov and Born approximations contradicts the statements of Tatarski and Chernov who contend that the Rytov approximation is superior to the Born approximation. It is conjectured that the Rytov and Born approximations have the same domain of validity for all x/kl2.

Second Moment of a Wave Propagating in a Random Medium

The Bethe–Salpeter equation for the second moment of a wave propagating in a random media is solved in the ladder approximation. The solution obtained applies to propagation problems in which the wavelength is much smaller than the scale of the inhomogeneities, and the product of the rms index fluctuation, the wavenumber, and a characteristic scale length of the inhomogeneities is also much smaller than unity. The field is assumed given on an initial plane and to have appreciable amplitude over a domain many wavelengths in extent in this plane. It is not required, however, that this domain be larger than the scale of the inhomogeneities. Quantitative results are given for a gaussian beam. We show that the ensemble-averaged distribution of irradiance remains gaussian and give the e-folding radius of this distribution. The two-point correlation function of the field in such a beam is shown to depend only on the distance between the points, not on their relative position within the beam. The coherence properties of the beam are expressed in terms of a coherence length. We also consider the ensemble-averaged interference pattern of the field from multiple apertures and verify that well-defined interference fringes occur only when the apertures are separated by a distance smaller than the coherence length of the individual beams. The solution of the Bethe–Salpeter equation obtained here satisfies an equation derived by Beran. We show that the local-independence assumption used by Beran is valid for sufficiently weak inhomogeneities.

Validity of the Rytov Approximation

The limitations on the applicability of the Rytov approximation are examined in this paper. It is shown that (1) the singular behavior of the perturbation series of which the Rytov approximation is the first term cannot be removed by adding a “constant” as suggested by the proponents of Rytov’s method; (2) the validity of Rytov approximation is limited to situations in which the mean-square value of the logarithmic amplitude fluctuation is much smaller than unity; (3) the MTF data of Coulman do not provide a good test on the validity of the Rytov approximation; (4) experimental data on the range dependence of intensity fluctuations do not agree with the predictions of the Rytov approximation; and (5) in problems involving propagation near the zenith, the validity of the Rytov approximation is limited to the near field of the turbulent layers at the surface of the earth and near the tropopause.

Coupled cw dye lasers using intracavity four-wave mixing.

An experimental investigation of laser-oscillator frequency locking by a four-wave interaction was conducted. An experiment was performed that successfully demonstrated the locking of two cw dye lasers using this concept with Na vapor serving as the nonlinear (coupling) medium.

Propagation of laser beams having an on-axis null in the presence of thermal blooming

The propagation of focused beams having an on-axis irradiance null is considered in the presence of thermal blooming. Two cases are treated: (a) beam profiles that have an irradiance zero in the beam center at the focal plane as well as the transmitter aperture; (b) beam profiles that have an on-axis irradiance null only at the transmitter. It is demonstrated that none of the beam profiles considered in case (a) has a meaningful advantage over a Gaussian beam profile. Some of the case (b) profiles do produce a larger bloomed irradiance in the focal plane, particularly when the initial intensity distribution is very uniform in its nonzero regions. Addition of a simple central obscuration to a “filled” irradiance distribution is found to have no advantage, however, for the cases considered.

Coherent coupling of CW laser oscillators using intracavity four-wave mixing

A theoretical study of frequency locking laser oscillators using an intracavity four-wave interaction is discussed. An experimental demonstration that successfully locked two CW dye lasers using this concept was performed, along with intracavity four-wave mixing studies, with sodium vapor serving as the nonlinear (mixing) element. >