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Nonlinear Dynamics in Optical Systems

Spectra of second harmonic generation and sum frequency generation for multimode fields

Jerzy S. Krasiński, Donald F. Heller, Jay R. Ackerhalt, Y. B. Band

Published Jan 1, 1990• 0 citations
Publisher

Abstract

The theory of three wave mixing was formulated in a classic paper by Armstrong, Bloembergen, Ducuing, and Pershan over 25 years ago.1 A theoretical description of three wave mixing for multimode time-dependent input fields (as opposed to cw single frequency time independent input fields) has recently been presented.2 Analytic solutions were developed for die time-dependent intensities of the output fields. These solutions differ dramatically from those for single-mode fields. New frequencies in the sum-frequency output spectrum and in the input-field output spectrum are created and grow in magnitude as the input intensities are increased. The output spectrum broadens under conditions of strong nonlinear coupling, ultimately reaching the phasematched bandwidth limit. Here, we review the results presented in Ref. 2 for sum frequency generation (SFG) and explicidy compare these results with those for second harmonic generation (SHG). We show that SHG cannot be described as a limit of the SFG formalism for multimode fields. The SFG formulation for the interaction of photons close to frequency ω1 with photons close to frequency ω2 producing photons close to frequency ω 3= ω 1+ ω 2 does not allow ω1(ω2) photons to interact with each other to produce photons close to frequency 2ω1(2ω2). This self interaction is crucial for SHG and must be incorporated. Therefore, a separate formulation is used to describe SHG with multimode input fields.

DOI
10.1364/nldos.1990.dmmpcps497

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The theory of three wave mixing was formulated in a classic paper by Armstrong, Bloembergen, Ducuing, and Pershan over 25 years ago.1 A theoretical description of three wave mixing for multimode time-dependent input fields (as opposed to cw single frequency time independent input fields) has recently been presented.2 Analytic solutions were developed for die time-dependent intensities of the output fields. These solutions differ dramatically from those for single-mode fields. New frequencies in the sum-frequency output spectrum and in the input-field output spectrum are created and grow in magnitude as the input intensities are increased. The output spectrum broadens under conditions of strong nonlinear coupling, ultimately reaching the phasematched bandwidth limit. Here, we review the results presented in Ref. 2 for sum frequency generation (SFG) and explicidy compare these results with those for second harmonic generation (SHG). We show that SHG cannot be described as a limit of the SFG formalism for multimode fields. The SFG formulation for the interaction of photons close to frequency ω1 with photons close to frequency ω2 producing photons close to frequency ω 3= ω 1+ ω 2 does not allow ω1(ω2) photons to interact with each other to produce photons close to frequency 2ω1(2ω2). This self interaction is crucial for SHG and must be incorporated. Therefore, a separate formulation is used to describe SHG with multimode input fields.
Analytic solutions are developed for the time-dependent intensity that results from the three-wave mixing of multimode fields. These differ dramatically from the solutions for single-mode (time-independent) fields. New frequencies in the output spectrum are created as input intensities increase. Under conditions of strong nonlinear coupling the output spectrum broadens, ultimately reaching the phase-match bandwidth limit, and the dynamics can become chaotic. A new technique for enhancing mixing efficiencies is presented.
We calculate the spectrum of second-harmonic generation (SHG) for multimode input fields. We show that SHG cannot be described as the degenerate limit of sum-frequency generation (SFG) for multimode fields, because the dynamical equations describing SFG do not properly account for this degeneracy. We consider SHG for amplitude-modulated as well as frequency-modulated fundamental fields. The bandwidth of the second harmonic generated from an amplitude-modulated fundamental field depends on the fundamental input intensity and the conversion strength of the nonlinear medium. Additional frequencies that are not contained in the set of frequencies ${\mathrm{\ensuremath{\omega}}}_{1\mathit{i}}$+${\mathrm{\ensuremath{\omega}}}_{1\mathit{j}}$ will be created at intermediate conversion strengths. However, the growth of the spectral bandwidth is much less pronounced than in SFG. For very high conversion, the spectrum of the second harmonic narrows with increased conversion strength. For frequency-modulated input fields, the second-harmonic spectrum is independent of both the fundamental input intensity and the conversion strength.
We analyze the dynamics of sum-frequency generation (SFG) and difference-frequency generation (DFG) for broadband input fields. We demonstrate that amplitude modulation of the input fields significantly affects conversion efficiency of SFG and DFG, but frequency modulation does not affect conversion efficiency. Analytic results for SFG and DFG output-field intensities are thereby available for arbitrary fields. Self- and cross-phase modulation can affect the dynamics in the limit of very high field strengths, but these effects are of higher nonlinearity and therefore usually negligible. The quantum efficiency of DFG is shown to be generally higher than SFG, but the energy efficiency is much lower if the difference frequency is significantly less than the input frequencies. Optimal SFG quantum conversion efficiency for such systems approaches that of second-harmonic generation only if the quantum fluence (total number of photons) in the two input beams are equal. Optimal DFG quantum conversion efficiency occurs when the number of photons of frequency ${\mathrm{\ensuremath{\omega}}}_{2}$ is small yet sufficient for significant stimulation. A recently developed method for obtaining efficient SFG for multimode input fields is theoretically analyzed. The method involves using an arrangement with two or more nonlinear mixing crystals with a time-delay line situated between the crystals that delays one of the fundamental fields relative to the other. The efficiency in the second crystal depends on the cross-correlation function of the two fundamental fields upon leaving the first nonlinear crystal. The time-delay method is not effective for multimode DFG.
We calculate the spectrum of third-harmonic generation (THG) for multimode fundamental input fields. We consider input fields from (1) a laser with gain on several cavity modes, (2) pulses with short temporal duration, and (3) frequency-modulated input fields. The effects of phase modulation in the nonlinear medium on the spectrum of the THG field are considered. Using the estimates by Miles and Harris [IEEE J. Quantum Electron. QE-9, 470 (1973)] for the nonlinear susceptibilities ${\mathrm{\ensuremath{\chi}}}^{(3)}$(-3\ensuremath{\omega};\ensuremath{\omega},\ensuremath{\omega},\ensuremath{\omega}) and ${\mathrm{\ensuremath{\chi}}}^{(3)}$(-\ensuremath{\omega};\ensuremath{\omega},-\ensuremath{\omega},\ensuremath{\omega}) in alkali-metal vapors, we find that phase modulation can profoundly affect the efficiency and the spectrum of the THG. Generally, the self-phase modulation of the fundamental will tend to reduce the phase matching of an intense fundamental field and the third harmonic and thereby lower the conversion efficiency. New frequencies not contained in the set of frequencies ${\mathrm{\ensuremath{\omega}}}_{1\mathit{i}}$+${\mathrm{\ensuremath{\omega}}}_{1\mathit{j}}$+${\mathrm{\ensuremath{\omega}}}_{1\mathit{k}}$, where {${\mathrm{\ensuremath{\omega}}}_{1\mathit{i}}$} are frequencies in the fundamental laser spectrum, are created even at relatively low conversion due to phase modulation. Effects of cross phase modulation are also studied.
We propose non-phase-matched sum frequency generation (SFG) as a method for characterizing broadband multimode light. Both the central wavelength and the bandwidth are in this case not limited by the phase matching condition. As an example, we consider bright squeezed vacuum (BSV) generated through high-gain parametric down conversion (PDC). In the spectrum of SFG from BSV, we observe the coherent peak and the incoherent background. We show that the ratio of their widths is equal to the number of frequency modes in BSV, which in the case of low-gain PDC gives the degree of frequency entanglement for photon pairs. By generating the sum frequency in the near-surface region of a nonlinear crystal, we increase the SFG efficiency and get rid of the modulation caused by chromatic dispersion, known as Maker fringes. This allows one to use non-phasematched SFG as a wavelength-independent autocorrelator. Furthermore, we demonstrate efficient non-phase-matched three- and four-frequency summation of broadband multimode light, hardly possible under phase matching. We show that the latter contains the coherent peak while the former does not.
The nonlinear interaction of two intense optical waves with three energy levels of an atomic system is analyzed theoretically. The frequencies of the applied fields are assumed to be on or near two of the resonances involving the three levels. Particular attention is given to the generation of power at the difference frequency of the two applied fields where their frequency separation lies in the far-infrared range of the spectrum. In the limit of large input fields where atomic transitions are driven to saturation, the generated difference-frequency power is no longer proportional to the product of the two input powers. It is shown that, in this limit, the total rate of emission of photons at the difference frequency may, in principle, approach a value given by rNT\frac{\mathrm{rN}}{T}TrN​, where NNN is the total number of atoms, TTT is the average time of coherent interaction of atoms and the radiation field, and rrr is a factor which in favorable cases approaches unity at low temperatures. Also, a number of additional effects are encountered due to coherent double-quantum transitions among various levels which may be ignored only for weak input powers. These effects are highly dependent on the modes of relaxations of atomic levels and the details of the line shape. The results are applied to several cases, in which the resonance condition yields large efficiency in nonlinear frequency mixing.