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Journal of physics. A, Proceedings of the Physical Society. General

Reflection of short pulses in linear optics

J. Chris Eilbeck

Published Sep 1, 1972• 42 citations
Publisher

Abstract

Two methods of calculating the reflected wave generated by a short optical pulse falling on a linear dielectric medium are given. As examples the cases of an input sech pulse modulating a resonant carrier wave and an input delta function are calculated. At atomic densities greater than about 10 18 atoms cm -3 an appreciable amount of the energy of the sech pulse is reflected. It is suggested that any nonlinear theory which ignores reflection may break down at these densities.

DOI
10.1088/0305-4470/5/9/008

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Two methods of calculating the reflected wave generated by a short optical pulse falling on a linear dielectric medium are given. As examples the cases of an input sech pulse modulating a resonant carrier wave and an input delta function are calculated. At atomic densities greater than about 10 18 atoms cm -3 an appreciable amount of the energy of the sech pulse is reflected. It is suggested that any nonlinear theory which ignores reflection may break down at these densities.
Refraction of a double-frequency optical pulse at the interface between two linear dielectrics containing a thin film of resonant atoms is considered. The dynamics of the system is treated based on the model of three-level atoms with the V -type energy diagram. Refraction of short and long (compared with the Rabi oscillation period) pulses is analyzed with allowance for local-field effects. In both cases, the shape of the pulse corresponding to one carrier wave frequency is shown to depend on the energy of the pulse with the other carrier wave frequency. In particular, both the frequency and the depth of the amplitude modulation arising in the refracted pulse are changed.
A study is made of the problem of transmission of light pulses by a boundary of a nonlinear medium under conditions of coherent interaction with matter. It is shown that in the case of a semiinfinite nonlinear medium the reflection coefficient is governed by the linear characteristics of the media in contact so that the boundary-value problem reduces to the Cauchy problem. In another case, when a nonlinear medium is a thin film at the interface between two linear media, it is found that such a film is an almost perfect mirror reflecting weak pulses but it becomes transparent to pulses of sufficiently high intensity and the mirror then modulates the amplitude of such pulses.
The optical density of a resonant medium significantly affects the output signal in time resolved nonlinear optical interactions. The resonant interaction with a medium reshapes any propagating short pulse, and the mutual interaction of several such pulses, as in four-wave mixing, must be treated self-consistently. We present a theoretical framework for the proper handling of the propagation of all pulses, input as well as generated, in the small area limit. For optically thick resonant absorbers, negative time delay signals are observed, and the apparent decay rate of the induced polarization is faster than the rate observed for thin samples. We experimentally measure degenerate four-wave mixing in an atomic medium as an example, and demonstrate the quality of the theoretical model by the excellent fit to measured signals over several orders of magnitude. The improved understanding enables us to provide a simple, but surprisingly accurate, estimate for the apparent decay rate in homogeneously broadened optically thick media: If the absorption is given by \ensuremath{\alpha}, the propagation length is L, and the transverse relaxation time is T2{\mathit{T}}_{2}T2​, the apparent decay rate 2/Ta{\mathit{T}}_{\mathit{a}}Ta​ of a time resolved four-wave mixing signal is given by 2/Ta{\mathit{T}}_{\mathit{a}}Ta​=(2/T2{\mathit{T}}_{2}T2​)(1+\ensuremath{\alpha}L/2).
A more accurate description of the 2 pi pulse of self-induced transparency is obtained. This is an exact solution of an approximate form of the Maxwell-Bloch equations in which only backscattering is neglected; the equations are valid at densities less than about 10 18 atoms cm -3 . At such densities the 2 pi sech pulse of McCall and Hahn (1967, 1969) remains a good approximate well into the picosecond range. The more accurate solution is chirped by a factor proportional to the square of the ratio of the spectral width of the pulse to its carrier wave frequency. The stability of this pulse solution and other N soliton solutions of the approximate Maxwell-Bloch equations is demonstrated.
In this book, we focus on the stochastic nature of pulses to achieve a more realistic description of ultra-short optical pulses affected by noise. In order to simulate ultra-short pulses, various stochastic pulse models were applied. The effects of pulse amplitude and phase fluctuations on its propagation in various linear and non-linear media were studied based on optical coherence theory. The response of the medium atoms to the applied stochastic pulses was investigated for the particular resonant case in which the pulse frequency nearly coincides with the atom resonance frequency. Analytical and numerical propagation methods were applied to explore evolutions of ultra-short stochastic pulses in the vicinity of optical resonance.
We consider the propagation of extremely short pulses through a dielectric thin film containing resonant atoms (two-level atoms) with permanent dipole. Assuming that the film width is less than the field wavelength, we can solve the wave equation and reduce the problem to a system of generalized Bloch equations describing the resonant atoms. We compute the stationary solutions for a constant irradiation of the film. Superimposing a small amplitude linear wave we calculate the reflection and transmission coefficients. From these, one can then deduce the different parameters of the model. We believe that this technique could be used in experiments to obtain the medium atomic and relaxation parameters.