Soviet Journal of Quantum Electronics

Theory of resonance two-photon absorption

Z.A. Demidenko

Published Jul 31, 1979• 1 citation

Abstract

A theory of resonance two-photon absorption, which describes this effect as a combination of two-stage and two-photon processes, is developed in a unified mathematical formalism. Expressions for the transition probabilities and kinetic equations are derived, the absorption coefficient is determined for different and identical frequencies, and the shift of the maximum of the resonance two-photon absorption line is obtained. Investigations are made of the intensity dependences of these expressions and of the kinetics of resonance two-photon transitions. The results are compared with available experimental data.

DOI: 10.1070/qe1979v009n07abeh009190

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Soviet Journal of Quantum Electronics

Theory of resonance two-photon absorption

Z.A. Demidenko

Published Jul 31, 1979• 1 citation

Publisher

Abstract

DOI: 10.1070/qe1979v009n07abeh009190

Related papers

Saturation Effects in Resonant Two-Photon Processes

B. R. Mollow

Resonant two-photon absorption is discussed for the case in which the exciting field may be intense enough and near enough to resonance to cause appreciable saturation. A single pair of levels in an otherwise quite general atom is assumed to satisfy the appropriate resonance condition, and equations are derived governing the time evolution of the associated 2 \ifmmode\times\else\texttimes\fi{} 2 submatrix of the full atomic density matrix. Atomic relaxation is treated explicitly. The analysis, which treats both inversion-symmetric and inversion-nonsymmetric cases within the same general formalism, is developed systematically, the necessary approximations receiving careful discussion. The results are analogous to those for the corresponding one-photon process, but with a shift in the resonance frequency proportional to the intensity of the exciting field. This frequency shift is not identical to the one calculated by time-independent (second-order) perturbation theory, and becomes large when the energy of an intermediate state of the atom approaches the mean of the energies of the active states.Resonant two-photon absorption is discussed for the case in which the exciting field may be intense enough and near enough to resonance to cause appreciable saturation. A single pair of levels in an otherwise quite general atom is assumed to satisfy the appropriate resonance condition, and equations are derived governing the time evolution of the associated 2 \ifmmode\times\else\texttimes\fi{} 2 submatrix of the full atomic density matrix. Atomic relaxation is treated explicitly. The analysis, which treats both inversion-symmetric and inversion-nonsymmetric cases within the same general formalism, is developed systematically, the necessary approximations receiving careful discussion. The results are analogous to those for the corresponding one-photon process, but with a shift in the resonance frequency proportional to the intensity of the exciting field. This frequency shift is not identical to the one calculated by time-independent (second-order) perturbation theory, and becomes large when the energy of an intermediate state of the atom approaches the mean of the energies of the active states.

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Two-photon transitions have provided the basis for numerous useful nonlinear interactions with radiation. In this paper we calculate the absorption and coupling coefficients for one-, two-, and three-wave mixing in two-photon homogeneously broadened media. The resulting field equations form the basis for understanding saturation spectroscopy, phase conjugation,1 and single-mode instabilities2 in lasers and optical bistability in two-photon media. We assume that the wave frequencies are not resonant with any intermediate state j and that the probe and conjugate waves cannot saturate the two-photon transition. This allows us to use the two-photon two-level model and to transfer much of our knowledge of the one-photon two-level systems to the two-photon case. Two major differences occur between the two models. First, dynamic Stark shifts of the active levels play an important role in generating asymmetric spectra and nonzero dispersion (≃ω s T1) at line center. Second, high pump intensities do not bleach the absorption coefficient at the Rabi sidebands.Two-photon transitions have provided the basis for numerous useful nonlinear interactions with radiation. In this paper we calculate the absorption and coupling coefficients for one-, two-, and three-wave mixing in two-photon homogeneously broadened media. The resulting field equations form the basis for understanding saturation spectroscopy, phase conjugation,1 and single-mode instabilities2 in lasers and optical bistability in two-photon media. We assume that the wave frequencies are not resonant with any intermediate state j and that the probe and conjugate waves cannot saturate the two-photon transition. This allows us to use the two-photon two-level model and to transfer much of our knowledge of the one-photon two-level systems to the two-photon case. Two major differences occur between the two models. First, dynamic Stark shifts of the active levels play an important role in generating asymmetric spectra and nonzero dispersion (≃ω s T1) at line center. Second, high pump intensities do not bleach the absorption coefficient at the Rabi sidebands.

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Albert Gold, John P. Hernandez

The theory of one- and two-photon transitions between a given pair of impurity levels of the same parity is developed. As is well known, lattice vibrations assist the otherwise forbidden one-photon transition, increasing its intensity with temperature. The two-photon transition rate is shown to have a complementary decrease with increasing temperature. A measurement of both temperature dependences determines the effective frequency and coupling constant for the interaction. The formalism is applied to predicting the two-photon transition rate for the $4{d}^{10}\ensuremath{-}4{d}^{9}5s$ transition of the silver ion in NaCl:Ag. For an experiment using a ruby laser and an ultraviolet beam the maximum absorption coefficient predicted for the ultraviolet beam is $3.2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}48}\mathrm{FN}$ ${\mathrm{cm}}^{\ensuremath{-}1}$ at low temperatures, where $F$ is the flux (photons/ ${\mathrm{cm}}^{2}$ sec) of the laser and $N$ is the ${\mathrm{Ag}}^{+}$ concentration. This should be within reach of present experimental techniques.The theory of one- and two-photon transitions between a given pair of impurity levels of the same parity is developed. As is well known, lattice vibrations assist the otherwise forbidden one-photon transition, increasing its intensity with temperature. The two-photon transition rate is shown to have a complementary decrease with increasing temperature. A measurement of both temperature dependences determines the effective frequency and coupling constant for the interaction. The formalism is applied to predicting the two-photon transition rate for the $4{d}^{10}\ensuremath{-}4{d}^{9}5s$ transition of the silver ion in NaCl:Ag. For an experiment using a ruby laser and an ultraviolet beam the maximum absorption coefficient predicted for the ultraviolet beam is $3.2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}48}\mathrm{FN}$ ${\mathrm{cm}}^{\ensuremath{-}1}$ at low temperatures, where $F$ is the flux (photons/${\mathrm{cm}}^{2}$ sec) of the laser and $N$ is the ${\mathrm{Ag}}^{+}$ concentration. This should be within reach of present experimental techniques.

Physical ReviewSep 13, 196532 citations

Two-Photon Absorption and Field Correlation Functions

B. R. Mollow

The probability of an atom undergoing a transition to a given final state by means of two-photon absorption is expressed for arbitrary field states in terms of the second-order normally-ordered field correlation function, evaluated at the position of the atom. In the case of stationary fields, an expression for the absorption rate is found which reduces to a particularly simple form for narrow-bandwidth fields near resonance. The effect of field statistics is illustrated by comparing the absorption rate $w_{2}^{}{}_{}{}^{(\mathrm{ch})}$ for chaotic or Gaussian light to the rate $w_{2}^{}{}_{}{}^{(l)}$ for laser light. The rate for laser light is calculated within the context of a particular model based on the assumption of fixed field amplitude and random frequency modulation, and the chaotic light to which it is compared is assumed to have the same (Lorentzian) power spectrum. When the width ${\ensuremath{\kappa}}_{f}$ of the final atomic level is much greater than the bandwidth $b$ of the field, we obtain the previously derived result $\frac{w_{2}^{}{}_{}{}^{(\mathrm{ch})}}{w_{2}^{}{}_{}{}^{(l)}}=2$ . When ${\ensuremath{\kappa}}_{f}\ensuremath{\ll}b$, on the other hand, the ratio between the two rates depends on the (mean) frequency of the field, and assumes the (maximum) value 4 when the field is exactly on resonance.The probability of an atom undergoing a transition to a given final state by means of two-photon absorption is expressed for arbitrary field states in terms of the second-order normally-ordered field correlation function, evaluated at the position of the atom. In the case of stationary fields, an expression for the absorption rate is found which reduces to a particularly simple form for narrow-bandwidth fields near resonance. The effect of field statistics is illustrated by comparing the absorption rate $w_{2}^{}{}_{}{}^{(\mathrm{ch})}$ for chaotic or Gaussian light to the rate $w_{2}^{}{}_{}{}^{(l)}$ for laser light. The rate for laser light is calculated within the context of a particular model based on the assumption of fixed field amplitude and random frequency modulation, and the chaotic light to which it is compared is assumed to have the same (Lorentzian) power spectrum. When the width ${\ensuremath{\kappa}}_{f}$ of the final atomic level is much greater than the bandwidth $b$ of the field, we obtain the previously derived result $\frac{w_{2}^{}{}_{}{}^{(\mathrm{ch})}}{w_{2}^{}{}_{}{}^{(l)}}=2$. When ${\ensuremath{\kappa}}_{f}\ensuremath{\ll}b$, on the other hand, the ratio between the two rates depends on the (mean) frequency of the field, and assumes the (maximum) value 4 when the field is exactly on resonance.

Physical ReviewNov 1, 1968212 citations

F

is the flux (photons/

{\mathrm{cm}}^{2}

sec) of the laser and

N

is the

{\mathrm{Ag}}^{+}

concentration. This should be within reach of present experimental techniques.

w_{2}^{}{}_{}{}^{(\mathrm{ch})}

for chaotic or Gaussian light to the rate

w_{2}^{}{}_{}{}^{(l)}

for laser light. The rate for laser light is calculated within the context of a particular model based on the assumption of fixed field amplitude and random frequency modulation, and the chaotic light to which it is compared is assumed to have the same (Lorentzian) power spectrum. When the width ${\ensuremath{\kappa}}_{f}$ of the final atomic level is much greater than the bandwidth

b

of the field, we obtain the previously derived result

\frac{w_{2}^{}{}_{}{}^{(\mathrm{ch})}}{w_{2}^{}{}_{}{}^{(l)}}=2

. When ${\ensuremath{\kappa}}_{f}\ensuremath{\ll}b$, on the other hand, the ratio between the two rates depends on the (mean) frequency of the field, and assumes the (maximum) value 4 when the field is exactly on resonance.