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Journal of the Optical Society of America

Laws of optics at high irradiance I Steady-state theory of a saturable slab*

James D. Macomber, William H. Thomason

Published Apr 1, 1977• 3 citations
Publisher

Abstract

The ancient laws developed by Snell, Bouguer, Lambert, Fresnel, and Beer, describing the reflection, absorption, and transmission of light by a plane slab of absorbing materials, are applicable only at low irradiances. We have developed new laws that apply at irradiances sufficiently high to produce partial optical saturation of the chromophoric atoms, ions, or molecules of the slab. The radiation field is treated in accordance with the characteristic matrix method of Abelès, based upon Maxwell’s equations. The chromophores are treated in accordance with the optical analogs to the Bloch equations, based upon quantum-statistical mechanics. Our results are presented in the form of an algorithm, by means of which the optical properties are initially estimated assuming no saturation. These properties are then used to compute the field intensities within the medium, and these field intensities are employed in the computation of estimates for the saturated properties. Successive iterations of this procedure are used until the calcuated fields and properties are self-consistent. We show for the first time how to calculate the influence of optical saturation upon the reflectance and angle of refraction of an absorbing sample, and how the method of characteristic matrices can be employed in a problem in nonlinear optics. In addition, the effects of the divergence and finite width of the incident beam, of the cell windows, and the interaction of the forward- and backward-traveling waves to produce a spatial variation in the saturation within the sample, are all treated correctly. The results may therefore be of use to those who wish to design a practical optical memory for a computer, using bistable saturable Fabry-Perot interferometers.

DOI
10.1364/josa.67.000453

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The ancient laws developed by Snell, Bouguer, Lambert, Fresnel, and Beer, describing the reflection, absorption, and transmission of light by a plane slab of absorbing materials, are applicable only at low irradiances. We have developed new laws that apply at irradiances sufficiently high to produce partial optical saturation of the chromophoric atoms, ions, or molecules of the slab. The radiation field is treated in accordance with the characteristic matrix method of Abelès, based upon Maxwell’s equations. The chromophores are treated in accordance with the optical analogs to the Bloch equations, based upon quantum-statistical mechanics. Our results are presented in the form of an algorithm, by means of which the optical properties are initially estimated assuming no saturation. These properties are then used to compute the field intensities within the medium, and these field intensities are employed in the computation of estimates for the saturated properties. Successive iterations of this procedure are used until the calcuated fields and properties are self-consistent. We show for the first time how to calculate the influence of optical saturation upon the reflectance and angle of refraction of an absorbing sample, and how the method of characteristic matrices can be employed in a problem in nonlinear optics. In addition, the effects of the divergence and finite width of the incident beam, of the cell windows, and the interaction of the forward- and backward-traveling waves to produce a spatial variation in the saturation within the sample, are all treated correctly. The results may therefore be of use to those who wish to design a practical optical memory for a computer, using bistable saturable Fabry-Perot interferometers.
The theory of the effect of optical saturation upon the reflectance of a plane slab (irradiated by a linearly polarized monochromatic electromagnetic wave of constant amplitude at normal incidence) is developed. The absorbing centers within the slab are described by means of the density matrix, and the electromagnetic field is described by Maxwell's equations. For simplicity's sake, it is assumed that only two quantum states (characterized by a transition dipole moment μ and relaxation times T1 and T2) at exact resonance with the electromagnetic wave are necessary to describe the absorption process. The coupling between waves travelling in opposite directions is provided by Fresnel reflection at the boundaries of the slab. Only the steady-state solutions are found.
Propagation of laser light through a saturable absorber is theoretically studied. Computed steady state solutions of the Maxwell equations describing the unidimensional propagation of a plane monochromatic wave without introducing the slowly-varying envelope approximation are presented showing how saturation effects can influence the absorption of the field. At a certain range of refractive index and extintion coefficients, computed solutions display a very susprising behaviour, and a self-reflected wave appears inside the absorber. This can be useful for a new kind of biestable device, similar to a standard bistable cavity but with the back mirror self-induced by the light.
Reverse-saturable absorbers are of considerable interest for optical limiting. Using the electric dipole perturbation, we derived the rate equation for a five-level system describing reverse-saturable absorbers. Traditional theories for the propagating laser beam in these materials are expressed in terms of the optical intensity. However, with the introduction of high-power short-pulsed lasers, the propagation of light in these materials may be subject to nonlinear phenomena such as self-focusing and self-phase modulation. Furthermore, conventional theories treat the laser light as a continuous wave or as a very broad temporal pulse in which dispersive effects are neglected. In order to incorporate these other nonlinear or dispersive effects, and therefore determine their influence in reverse-saturable absorbers, we derived an equation for the propagation of the electromagnetic field, rather than the intensity, coupled to the rate equations for a five-level system. We also coupled our theory to experimentally measurable parameters for these materials and detailed the various physical approximations used to obtain the rate equations.
A comprehensive model of intense laser light propagation in a fully saturable medium is presented. This model employs the Maxwell-Bloch equations for a two-level atomic system. Light propagating through a medium creates intensity-dependent index of refraction gradients that reduce to a Kerr nonlinearity only for a weak field. When the laser detuning is positive, self-induced index gradients cause the beam to self-focus. Stationary filaments occur when self-focusing exactly balances diffraction. There exists a unique set of two-dimensional fundamental (and higher mode) solutions for which the filament transverse profile (not Gaussian in general) depends on the following quantities: density of saturable absorbers, laser detuning from resonance, laser power, and natural linewidth. (In contrast the Kerr solution does not have a unique filament diameter.) Emphasis will be placed on propagation of Gaussian incident beams; deviations from the stationary solution in either shape or power generally give rise to oscillating spatial patterns, including donutlike profiles, or beam breakup.
The major portion of this monograph discusses how given pulses of laser radiation affect individual atoms. This chapter inverts that relationship, describing how the matter alters the fields. Those field changes provide quantitative measures of the quantum-state changes produced in the atoms. Their description therefore is an adjunct to Chap. 19. Basically the incident radiation produces excitation which, in turn, alters the various multipole moments of the atoms. When viewed as a macroscopic sample of matter, such changes alter the electric polarization field P and the magnetization field M of the matter through which the radiation must pass. The Maxwell equations, see App. C.1, provide the needed description of how the P and M fields alter the electric and magnetic fields E and B . The combination of the Maxwell equations for the fields and the Schrödinger, Bloch, or Liouville equations for the atoms provide the tools needed to construct a self-consistent description of radiation passing through matter – atoms responding to a pulsed field and traveling waves being modified by the resulting atomic changes [All87; Ale92; Muk99; Die06; Vit01a]. The present chapter, drawing on [Sho90, Chap. 12] and App. C, discusses this theory. Incoherent radiation passing through matter typically undergoes exponential attenuation in accord with eqn. (6.9). A measure of the incremental change of intensity in distance L by absorption coefficient κ is the optical depth κ L . This parameter appears in rateequation treatments of incoherent light attenuation.