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Physical Review A

Nonlinear-optical properties of two-level systems with permanent dipole moments

Raanan Bavli, Donald F. Heller, Y. B. Band

Published Apr 1, 1990• 52 citations
Publisher

Abstract

We calculate the transition rates and the linear and nonlinear-optical properties of a homogeneously broadened two-level system with permanent dipole moments in the presence of two pump fields. We present absorption and emission spectra for the system in the steady state, without making the rotating-wave approximation. Expressions for the nonlinear susceptibilities for wave mixing are derived.

DOI
10.1103/physreva.41.3960

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Nonlinear absorption and dispersion in a two-level system with permanent dipole moments

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We calculate the third-order nonlinear absorption and dispersion (self-phase-modulation and cross-phase modulation) of a homogeneously broadened two-level molecular system with permanent dipole moments interacting with (1) an optical field of frequency \ensuremath{\omega}, and (2) two optical fields of frequency ${\mathrm{\ensuremath{\omega}}}_{1}$ and ${\mathrm{\ensuremath{\omega}}}_{2}$, i.e., the susceptibilities ${\mathrm{\ensuremath{\chi}}}^{(3)}$(-\ensuremath{\omega};\ensuremath{\omega},-\ensuremath{\omega},\ensuremath{\omega}) and ${\mathrm{\ensuremath{\chi}}}^{(3)}$(-${\mathrm{\ensuremath{\omega}}}_{1}$;${\mathrm{\ensuremath{\omega}}}_{1}$,-${\mathrm{\ensuremath{\omega}}}_{2}$,${\mathrm{\ensuremath{\omega}}}_{2}$). Sharp features in the nonlinear absorption and dispersion appear whenever one- or two-photon resonance conditions are met. Only features corresponding to one-photon resonance arise if the difference between the permanent dipole moments of the levels, \ensuremath{\Delta}\ensuremath{\mu}, vanishes. The ${\mathrm{\ensuremath{\chi}}}^{(3)}$(-\ensuremath{\omega};\ensuremath{\omega},-\ensuremath{\omega},\ensuremath{\omega}) spectrum has two features, one at frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{b}\mathit{a}}$ dominated by contributions proportional to ${\mathrm{\ensuremath{\mu}}}_{\mathit{a}\mathit{b}}^{4}$, where ${\mathrm{\ensuremath{\mu}}}_{\mathit{a}\mathit{b}}$ is the transition dipole moment of the two levels with transition frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{b}\mathit{a}}$, and the other at frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{b}\mathit{a}}$/2 dominated by contributions proportional to (\ensuremath{\Delta}\ensuremath{\mu}${)}^{2}$${\mathrm{\ensuremath{\mu}}}_{\mathit{a}\mathit{b}}^{2}$. Sharp features in ${\mathrm{\ensuremath{\chi}}}^{(3)}$(-${\mathrm{\ensuremath{\omega}}}_{1}$;${\mathrm{\ensuremath{\omega}}}_{2}$,-${\mathrm{\ensuremath{\omega}}}_{2}$,${\mathrm{\ensuremath{\omega}}}_{1}$) appear when ${\mathrm{\ensuremath{\omega}}}_{2}$ is near ${\mathrm{\ensuremath{\omega}}}_{\mathit{b}\mathit{a}}$,${\mathrm{\ensuremath{\omega}}}_{1}$,${\mathrm{\ensuremath{\omega}}}_{1}$+${\mathrm{\ensuremath{\omega}}}_{\mathit{b}\mathit{a}}$, and \ensuremath{\Vert}${\mathrm{\ensuremath{\omega}}}_{1}$-${\mathrm{\ensuremath{\omega}}}_{\mathit{b}\mathit{a}}$\ensuremath{\Vert}.

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The interaction of two continuous wave electric fields (lasers) with a two-level molecule, that has a nonzero difference d, between the permanent dipoles of the two states involved in a multiphoton two-color transition, is studied in the rotating wave approximation and by using exact methods. Analytical rotating wave approximation results for the time-dependent populations of the molecular states and for the resonance profiles are derived and used to help discuss the effects of permanent dipole moments in the two-color absorption problem. Several examples of multiphoton, two-color, two-level resonance profiles calculated in the rotating wave approximation and by using exact techniques, are employed for illustrative purposes. They are used to help quantify the conditions for the validity of the two-color rotating wave approximation, which are considerably more restrictive than the analogous conditions for the one-color rotating wave approximation with d≠0, and to illustrate the interpretive and predictive nature of the rotating wave approximation results. For example, there are many multiphoton, two-color transitions that are available if a molecule possesses permanent dipoles (d≠0). The analytic rotating wave approximation expression for the molecule-two laser coupling can often be used to reliably estimate the field parameters of the two lasers that are required to optimize a given transition relative to others.

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We consider the Four-Wave mixing (FWM) signal and the effects of the pump beam on the nonlinear optical responses of a two-level system where intramolecular coupling is modelled by two coupled harmonic electronic potentials, displaced both in their equilibrium minimum position and energy. We investigate the modifications introduced in the FWM signal, considering a simple representation for the vibrational structure and including molecular electronic states with nonzero permanent dipole moments. The critical quantities for this analysis are the transition and permanent dipole moments, which depend strongly on the intramolecular coupling. Our results show how both absorption and dispersion are affected by the vibronic coupling.

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We calculate the transition rates and the linear and nonlinear-optical properties of a homogeneously broadened two-level system with permanent dipole moments in the presence of two pump fields. We present absorption and emission spectra for the system in the steady state, without making the rotating-wave approximation. Expressions for the nonlinear susceptibilities for wave mixing are derived.
We calculate the third-order nonlinear absorption and dispersion (self-phase-modulation and cross-phase modulation) of a homogeneously broadened two-level molecular system with permanent dipole moments interacting with (1) an optical field of frequency \ensuremath{\omega}, and (2) two optical fields of frequency ${\mathrm{\ensuremath{\omega}}}_{1}$ and ${\mathrm{\ensuremath{\omega}}}_{2}$, i.e., the susceptibilities ${\mathrm{\ensuremath{\chi}}}^{(3)}$(-\ensuremath{\omega};\ensuremath{\omega},-\ensuremath{\omega},\ensuremath{\omega}) and ${\mathrm{\ensuremath{\chi}}}^{(3)}$(-${\mathrm{\ensuremath{\omega}}}_{1}$;${\mathrm{\ensuremath{\omega}}}_{1}$,-${\mathrm{\ensuremath{\omega}}}_{2}$,${\mathrm{\ensuremath{\omega}}}_{2}$). Sharp features in the nonlinear absorption and dispersion appear whenever one- or two-photon resonance conditions are met. Only features corresponding to one-photon resonance arise if the difference between the permanent dipole moments of the levels, \ensuremath{\Delta}\ensuremath{\mu}, vanishes. The ${\mathrm{\ensuremath{\chi}}}^{(3)}$(-\ensuremath{\omega};\ensuremath{\omega},-\ensuremath{\omega},\ensuremath{\omega}) spectrum has two features, one at frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{b}\mathit{a}}$ dominated by contributions proportional to ${\mathrm{\ensuremath{\mu}}}_{\mathit{a}\mathit{b}}^{4}$, where ${\mathrm{\ensuremath{\mu}}}_{\mathit{a}\mathit{b}}$ is the transition dipole moment of the two levels with transition frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{b}\mathit{a}}$, and the other at frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{b}\mathit{a}}$/2 dominated by contributions proportional to (\ensuremath{\Delta}\ensuremath{\mu})2{)}^{2})2${\mathrm{\ensuremath{\mu}}}_{\mathit{a}\mathit{b}}^{2}$. Sharp features in ${\mathrm{\ensuremath{\chi}}}^{(3)}$(-${\mathrm{\ensuremath{\omega}}}_{1}$;${\mathrm{\ensuremath{\omega}}}_{2}$,-${\mathrm{\ensuremath{\omega}}}_{2}$,${\mathrm{\ensuremath{\omega}}}_{1}$) appear when ${\mathrm{\ensuremath{\omega}}}_{2}$ is near ${\mathrm{\ensuremath{\omega}}}_{\mathit{b}\mathit{a}}$,${\mathrm{\ensuremath{\omega}}}_{1}$,${\mathrm{\ensuremath{\omega}}}_{1}$+${\mathrm{\ensuremath{\omega}}}_{\mathit{b}\mathit{a}}$, and \ensuremath{\Vert}${\mathrm{\ensuremath{\omega}}}_{1}$-${\mathrm{\ensuremath{\omega}}}_{\mathit{b}\mathit{a}}$\ensuremath{\Vert}.
Properties of the resonance peaks associated with the non-degenerate four-wave mixing signal in a two-level molecular system with non-zero permanent dipole moments are studied in the space defined by the pump ( omega 1 ) and probe ( omega 2 ) frequencies. An analytical expression for the Fourier components of the nonlinear homogeneous macroscopic polarization are obtained, where the terms responsible for the oscillation (at frequency omega 3 =2 omega 1 - omega 2 ) of the populations and coherences are also identified. Topological properties of the resulting nonlinear spectra are studied in frequency space using three-dimensional curves and contour plots. We have additionally identified the dominant contributions of the coherence and/or population that determine the intensity and the spectral linewidth. The critical quantities for the present analysis are the transition and permanent dipole moments when the rotating-wave approximation (RWA) is not considered.
The inclusion of the permanent dipole moments and the solvent on the optical conventional Bloch equations (OCBE) allowed us to obtain analytical expressions for the optical properties of a two-level molecular system. We employed the methodology developed by Colmenares et al.1, in which they model the collisional effect of the solvent through a stochastical function, ξ(t) = ω0 + σ(t), so the OCBE become a set of coupled integro-differential stochastical equations that we solved, up to third order in the incident field, employing the perturbation theory. Once obtained the analytical expressions for the density matrix elements, macroscopic polarization and effective susceptibility of the system, we studied the optical properties derived in the frequency space, inside and outside the rotating wave approximation.
We calculate the electric-field-induced second-harmonic-generation (EFISH) susceptibility of a homogeneously broadened two-level system with permanent dipole moments. This susceptibility has contributions from the second-harmonic-generation (SHG) susceptibility of the same system without the presence of a dc field and the four-wave-mixing (FWM) component of the susceptibility ${\ensuremath{\gamma}}_{\mathit{e}}$(-2\ensuremath{\omega};\ensuremath{\omega},\ensuremath{\omega},0). The magnitude and phase of the various contributions to EFISH arising from SHG and from FWM depend on the frequency of the fundamental optical field, the transition dipole moment ${\mathrm{\ensuremath{\mu}}}_{\mathit{a}\mathit{b}}$, the permanent dipole moment of the ground-state level ${\mathrm{\ensuremath{\mu}}}_{\mathit{a}\mathit{a}}$, and the difference of the ground- and excited-state dipole moments \ensuremath{\Delta}\ensuremath{\mu}. We present non-rotating-wave-approximation results for both the quasi-steady-state in systems where the dephasing time T2{\mathit{T}}_{2}T2​ is much shorter than the population decay time T1{\mathit{T}}_{1}T1​, so the polarization is in a steady state but the population is not, and for the complete steady state. Extraction of the SHG susceptibility from EFISH measurements is considered.
The interaction of two continuous wave electric fields (lasers) with a two-level molecule, that has a nonzero difference d, between the permanent dipoles of the two states involved in a multiphoton two-color transition, is studied in the rotating wave approximation and by using exact methods. Analytical rotating wave approximation results for the time-dependent populations of the molecular states and for the resonance profiles are derived and used to help discuss the effects of permanent dipole moments in the two-color absorption problem. Several examples of multiphoton, two-color, two-level resonance profiles calculated in the rotating wave approximation and by using exact techniques, are employed for illustrative purposes. They are used to help quantify the conditions for the validity of the two-color rotating wave approximation, which are considerably more restrictive than the analogous conditions for the one-color rotating wave approximation with d≠0, and to illustrate the interpretive and predictive nature of the rotating wave approximation results. For example, there are many multiphoton, two-color transitions that are available if a molecule possesses permanent dipoles (d≠0). The analytic rotating wave approximation expression for the molecule-two laser coupling can often be used to reliably estimate the field parameters of the two lasers that are required to optimize a given transition relative to others.
We consider the Four-Wave mixing (FWM) signal and the effects of the pump beam on the nonlinear optical responses of a two-level system where intramolecular coupling is modelled by two coupled harmonic electronic potentials, displaced both in their equilibrium minimum position and energy. We investigate the modifications introduced in the FWM signal, considering a simple representation for the vibrational structure and including molecular electronic states with nonzero permanent dipole moments. The critical quantities for this analysis are the transition and permanent dipole moments, which depend strongly on the intramolecular coupling. Our results show how both absorption and dispersion are affected by the vibronic coupling.