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IEEE Journal of Quantum Electronics

High harmonic generation induced by permanent dipole moments

Óscar G. Calderón, R. Gutiérrez-Castrejón, J.M. Guerra

Published Jan 1, 1999• 22 citations
Publisher

Abstract

The interaction between radiation and a two-level molecular system in the presence of permanent dipole moments is analyzed. It is found that the nonzero difference between permanent dipoles induces a polarization that oscillates at frequencies /spl omega/, 2/spl omega/, 3/spl omega/, etc., where /spl omega/ is the frequency of the electromagnetic field resonant with the two-level system. An analysis of the propagation equation of the harmonics shows that even harmonics are amplified while odd harmonics are absorbed.

DOI
10.1109/3.737618

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Journal of Physics B Atomic Molecular and Optical Physics•Dec 19, 2011•20 citations
The interaction between radiation and a two-level molecular system in the presence of permanent dipole moments is analyzed. It is found that the nonzero difference between permanent dipoles induces a polarization that oscillates at frequencies /spl omega/, 2/spl omega/, 3/spl omega/, etc., where /spl omega/ is the frequency of the electromagnetic field resonant with the two-level system. An analysis of the propagation equation of the harmonics shows that even harmonics are amplified while odd harmonics are absorbed.
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}.
This paper presents an analytical study of high-order harmonic generation (HHG) by quantum systems possessing permanent dipole moments (PDMs), such as atoms/molecules placed in a static electric field or polar molecules. HHG by conventional, non-PDM systems typically demonstrated a plateau in the envelope of harmonics followed by a steep fall in harmonic intensities when the harmonic number exceeds a certain threshold. This paper shows that the HHG by PDM systems results in a significant extension of the plateau to higher frequencies and a slower decline of intensities at frequencies higher than the end of the plateau . Moreover, there occurs a substantial growth of the total, summed intensity of all components of the scattering spectrum . Other physically interesting distinctive features of the HHG by PDM systems are: the presence of both odd and even harmonics of the laser frequency in the scattering spectrum; a triplet structure of both odd and even harmonics; and the appearance of a component at the doubled Rabi frequency (i.e. at a frequency much lower than the laser frequency). The results are obtained by using the adiabatic basis for the analysis of a multi-quantum resonance in a two-level system possessing PDMs.
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 radiative renormalization technique is used to analyze two-level systems with permanent dipole moments under the influence of a single-frequency applied field. The advantage of radiative renormalization over the usual perturbative or harmonic balance techniques lies in the computational ease with which the density matrix operators can be determined to any order in applied field. We characterize the absorption and dispersion of a typical two-level system within this framework and find that although the absorption saturates as expected with increasing incident field strength, additional oscillatory behavior is evident at extremely high field strengths. The structure and period of these oscillations is dependent upon the magnitude of the permanent dipole moment. Our general formalism can easily be extended to the study of other nonlinear optical phenomena. \textcopyright{} 1996 The American Physical Society.
A molecular system having two energy levels separated by\hbar\Omegaand coupled to an electromagnetic field by a transition dipole moment will resonantly absorb energy at all odd sub-harmonics of the energy level separation\omega =\Omega,\Omega/3,\Omega/5.... These correspond to 1, 3, 5,... quantum absorption processes. Part of this absorbed energy is emitted back into the field at harmonics of\omegaand part is dissipated into the lattice degrees of freedom. It is therefore possible to drive the molecular system with a field of frequency\omega \approx \Omega/m(n odd), and obtain radiation at a frequencym\omega \approx \Omega. Here we present results of a theoretical study of these resonant multi-quantum conversion processes. Two characteristic regions of operation are predicted: one previously discussed by Fontana, Pantell and Smith3in which the molecular efficiency\varepsilon= (power converted ton \omega)/ (power abosrbed at\omega) =\frac{1}{2}under conditions of maximum power converted per unit volume; and another region (not previously reported) in which maximum power conversion per unit volume can occur with molecular efficiencies approaching unity.
The theory of coherent radiation emission in two-level quantum systems with permanent dipole moments at the multiphoton resonant excitation by moderately strong laser fields is presented. A simple analytic formula for the expectation value of a dipole moment operator is obtained. The spectrum corresponding to harmonic generation and low-frequency radiation on the generalized Rabi frequency is investigated both analytically and numerically. The effect of dynamic Stark shift on a dipole spectrum under direct multiphoton resonant excitation is considered.