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Japanese Journal of Applied Physics

Optical Transmission in an Excited Two-Level Medium

Yoshihiro Ohtsuka

Published May 1, 1971• 1 citation
Publisher

Abstract

Transmission characteristics in an excited absorbing medium are discussed on the basis of the photon transport-equation and the rate equations. An increasing power of the C W laser beam passing through the medium is considered when this medium is excited at the same time by a different pulse of radiation. A typical example of a square excitation-pulse states that the transmittance recovers to its original value after it once reaches a maximum value. The time response of the medium depends upon the total number of absorbing centers M, the absorption cross-section σ, and the relaxation time τ as well as the intensity of radiation sources I. The decay time of the increased transmittance is proportional to τ/σM if the quantity (1-M/2τI) is kept constant. As a numerical example the relaxation time of SF6 is estimated to be 2×10-6 s at a pressure of 3.7 Torr in comparison with an experimental result.

DOI
10.1143/jjap.10.630

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Transmission characteristics in an excited absorbing medium are discussed on the basis of the photon transport-equation and the rate equations. An increasing power of the C W laser beam passing through the medium is considered when this medium is excited at the same time by a different pulse of radiation. A typical example of a square excitation-pulse states that the transmittance recovers to its original value after it once reaches a maximum value. The time response of the medium depends upon the total number of absorbing centers M, the absorption cross-section σ, and the relaxation time τ as well as the intensity of radiation sources I. The decay time of the increased transmittance is proportional to τ/σM if the quantity (1-M/2τI) is kept constant. As a numerical example the relaxation time of SF6 is estimated to be 2×10-6 s at a pressure of 3.7 Torr in comparison with an experimental result.
Numerical integration of a system of equations describing the coherent interaction between radiation and a two-level medium (e.g., a semiconductor) is employed to analyze the effect of relaxation processes on the energy characteristics and dynamics of radiation pulses of duration τpΣ T '2 (T '2 is the irreversible polarization relaxation time). It is shown that the interaction of these pulses with the medium is strongly dependent on their area on entry to the medium and the attenuation of the pulses as a result of relaxation processes decreases with increasing pulse input power. This may be utilized for probing the medium in the resonant absorption range for short polarization relaxation times, in solution 42.65.Gv.
We theoretically study the propagation through a resonant absorbing medium of a time-dependent perturbation modulating the amplitude of a continuous wave (cw). Modeling the medium as a two-level system and linearizing the Maxwell-Bloch equations for the perturbation, we establish an exact analytical expression of the transfer function relating the Fourier transforms of the incident and transmitted perturbations. It directly gives the gain and the phase shift undergone in the medium by a harmonic modulation. For the case of a pulse modulation, it enables us to determine the transmission time of the pulse center of mass (group delay), evidencing the relative contributions of the coherent and incoherent (population) relaxations. We show that the group delay has a negative value (fast light) fixed by the coherent effects when the cw intensity is small compared to the saturation intensity and becomes positive (slow light) when this intensity increases, before attaining a maximum that cannot exceed the population relaxation time. The analytical results are completed by numerical determinations of the shape of the transmitted pulses in the different regimes.
We model the response of a molecular medium to an intense pulse of light. We begin by reducing the Maxwell-Bloch equations for a multilevel molecular model system to a set of simplified propagation equations. We thereby obtain expressions for the frequency, temperature, and field intensity dependence of the ground and excited electronic state absorption cross sections, and the emission cross section, as well as the temperature dependence of the decay rate of the excited state. The phenomena of saturation, excited-state absorption, optically induced dichroism and birefringence, vibronic energy transfer, induced molecular alignment, decay of the matter-field interaction energy, and rotational diffusion are considered within the general formalism for describing the propagation of an intense pulse of light in the molecular medium. We find that the cross section for absorption (emission) of a photon is not identical to the cross section for molecular excitation (deexcitation) via the matter-field interaction. This is explained in terms of an energy balance between the matter, radiation, the matter-radiation interaction, and the bath responsible for the broadening of the molecular transitions. The inequality of these cross sections determines whether the system supplies or draws energy from the thermal bath. The rise in the vibrational temperature of molecules subjected to an intense short pulse is shown to significantly affect propagation of the pulse. The alignment of the molecular system resulting from saturation of the ground-state transition by a polarized laser pulse and the effects of the orientational dependence of the matter-field interaction including effects of rotational diffusion are determined. As an example of the formalism we consider pulse propagation in a Rhodamine 6G solution. We fit the parameters of the model to determine the frequency dependence of the ground- and excited-state absorption cross sections and the emission cross section and compare with experimentally determined cross sections. We describe the propagation of a pulse whose temporal width is comparable to the rotational diffusion time of the molecules in the dye solution and calculate polarization pulse probe signals.
A theoretical study has been made of the transmission of a monochromatic, well-collimated beam of radiation normally incident on idealized material containing only two energy levels. The nonlinear partial differential equations governing the variation of population and photon density in space and time have been solved exactly for arbitrary initial conditions under the assumption that spontaneous emission and thermal relaxation from the excited state can be neglected. These conditions are satisfied in practice when the radiation is in the form of a pulse whose duration is short compared with the characteristic times of these relaxation processes. Two cases are considered in detail: (a) when all atoms are in the ground state and the material absorbs; and (b) when there is a population inversion and the material amplifies. In case (a), the radiation, with a characteristic velocity, ``bores'' its way through an optically dense substance emerging delayed in time and leaving the material in a perfectly transparent or saturated state. In case (b), an incoming pulse is amplified and sharpened—to a degree determined by the gain of the medium. In addition to these cases, the dependence of the apparent spontaneous emission lifetime on the size and population distribution of the material is mentioned.
As a high-intensity monochromatic light pulse passes through a material containing two-level systems, saturable absorption or amplification of the radiation may occur. The resultant ``shaping'' of the intensity-vs-time curve as a function of distance into the sample is the subject of this paper. A theory is developed based on the Einstein rate equations, and pulse-shape functions are derived which depend on the absorption and loss coefficients of the material, the relaxation time of the systems, the input-pulse intensity and width, the initial ``state-of-pump'', and number density of absorbers in the sample.
The phenomena which take place during the time of propagation of a monochromatic pulse of a resonance radiation through optically dense on the transition frequency medium containing resonance absorbing centers are considered. Due to the action of the pulse of the resonance radiation the saturation of absorption in the layer of the medium takes place. As a result the absorption decreases in the layer and the radiation penetrates into the next layer. Thus translucent region continuously moves deeper into the medium and the process is wavelike.When the density of the initial photon flux is constant and the distribution of absorbing centers is homogeneous the absorption coefficient and the photon density in the depth of the media depends only on the variable $t\phantom{\rule{0.15em}{0ex}}\ensuremath{-}\phantom{\rule{0.15em}{0ex}}x/v$. The velocity of translucency wave v may be by several orders of magnitude less than the velocity of light. The mechanism of the saturation can be various. The small value of the loss of photons in the transluced region of the medium is the necessary condition for the rise of translucent waves. The condition of realization of translucent wave is fulfilled for a large number of condensed states having intense absorption lines. These conditions are the most favourable in the case of presence of methastable levels between the levels 1 and 2 (type-ruby system). The effect of the translucent waves is general for various ranges of spectrum of electromagnetic radiation and may be realized in astrophysical scale.