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Physical Review A

Sum- and difference-frequency generation for broadband input fields

Y. B. Band, Czesław Radzewicz, Jerzy S. Krasiński

Published Jan 1, 1994• 11 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.49.517

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We calculate the sum- and difference-frequency-generation (SFG and DFG) susceptibilities, ${\mathrm{\ensuremath{\chi}}}^{(2)}$(-(${\mathrm{\ensuremath{\omega}}}_{1}$+${\mathrm{\ensuremath{\omega}}}_{2}$);${\mathrm{\ensuremath{\omega}}}_{1}$,${\mathrm{\ensuremath{\omega}}}_{2}$) and ${\mathrm{\ensuremath{\chi}}}^{(2)}$(-(${\mathrm{\ensuremath{\omega}}}_{1}$-${\mathrm{\ensuremath{\omega}}}_{2}$);${\mathrm{\ensuremath{\omega}}}_{1}$,-${\mathrm{\ensuremath{\omega}}}_{2}$), of a homogeneously broadened two-level system with permanent dipole moments in the presence of pump fields at frequencies ${\mathrm{\ensuremath{\omega}}}_{1}$ and ${\mathrm{\ensuremath{\omega}}}_{2}$. Both SFG and DFG susceptibilities have two maxima at frequencies corresponding to the one- and two-photon resonances of the two-level system. Knowledge of the magnitude of one of the maxima of either SFG or DFG is sufficient for determining the magnitude of the second peak of that process and also the two peaks of the other process. In both processes the peak at the two-photon resonance has dominant contributions arising from off-diagonal elements of the density matrix. The interference between these contributions is constructive in SFG and destructive in DFG. Therefore, SFG is more efficient at the two-photon resonance than DFG. An expression for the ratio between these peaks is obtained. The peak of the spectrum of each process corresponding to the one-photon resonance has two main contributions, one of which is diagonal and the other off diagonal. In DFG the interference between the two contributions is constructive and in SFG the interference is destructive. Hence, DFG is more efficient in the vicinity of the one-photon resonance. However, since the off-diagonal contributions to SFG and DFG on the one-photon resonance are equal in magnitude (and differ by a phase difference of \ensuremath{\sim}\ensuremath{\pi}) the efficiencies of the two processes are equal when the duration of the pulses is comparable to or shorter than ${\mathit{T}}_{1}$.

Physical Review A•May 1, 1991•16 citations
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 \ensuremathω2{\mathrm{\ensuremath{\omega}}}_{2}\ensuremathω2​ 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.
A new method of efficient sum-frequency generation for broadband input fields is theoretically analyzed and experimentally demonstrated. The method involves using an arrangement with two or more nonlinear mixing crystals, with a time-delay line situated between the crystals, for one of the fundamental fields relative to the other. The delay line temporally shifts the fundamental fields, one relative to another, by a time longer than their coherence time. The improvement in efficiency for sum-frequency generation using this method is much higher than for difference-frequency generation.
In this thesis, high-efficiency frequency conversion Sum Frequency Generation (SFG) is theoretically analyzed and experimentally demonstrated. Our approach to SFG consists of two input phase conjugated broadband signal and idler beams undergoing SFG. By using the signal waveform as a gating beam, a specific mode may be recognized and separated out, providing a promising method for mode selection. The signal and idler are produced by a monochromatic pump and a broadband signal through Difference Frequency Generation (DFG). To obtain good mode selectivity, high conversion efficiency of the selected mode is required. We study the factors that influence the conversion efficiency of the selected mode, like the length of the waveguide and the power of the gating beam. The highest conversion efficiency of the mode achieved is over 90%. By breaking the correlation between mode and gating beams, we find around 40% conversion efficiency of the mismatched mode.
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.
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.
Sum-frequency generation in nonlinear optics is a key technology enabling optical frequency conversion. However, how to most effectively enhance the frequency conversion efficiency in nonlinear crystals remains a fundamental problem; shorter crystal length and lower pump intensity while maintaining high-efficiency frequency conversion are critical for enhancing device integration potential and improving device performance in practical sum-frequency generation (SFG) device design. This study employed the Pontryagin maximum principle to investigate the SFG process. A cost functional was formulated that complies with the minimum constraint during SFG, enabling the derivation of optimal control parameters. A central finding revealed an inverse proportionality between the coupling coefficient and the crystal length: for a given crystal length, the pump intensity in the SFG is minimized, and vice versa. The optimal evolution trajectory follows a geodesic path on the Bloch sphere surface, representing the minimal-length evolutionary path from the signal field to the sum-frequency field. This geodesic behavior also indicates that the loss and absorption of the optical field in the crystal medium vanish during SFG. Furthermore, the crystal modulation period for the SFG is determined using the optimal phase-matching function. These results offer valuable insights for the design of compact optical chips and nonlinear optical devices.
We calculate the sum- and difference-frequency-generation (SFG and DFG) susceptibilities, ${\mathrm{\ensuremath{\chi}}}^{(2)}$(-(${\mathrm{\ensuremath{\omega}}}_{1}$+${\mathrm{\ensuremath{\omega}}}_{2}$);${\mathrm{\ensuremath{\omega}}}_{1}$,${\mathrm{\ensuremath{\omega}}}_{2}$) and ${\mathrm{\ensuremath{\chi}}}^{(2)}$(-(${\mathrm{\ensuremath{\omega}}}_{1}$-${\mathrm{\ensuremath{\omega}}}_{2}$);${\mathrm{\ensuremath{\omega}}}_{1}$,-${\mathrm{\ensuremath{\omega}}}_{2}$), of a homogeneously broadened two-level system with permanent dipole moments in the presence of pump fields at frequencies ${\mathrm{\ensuremath{\omega}}}_{1}$ and ${\mathrm{\ensuremath{\omega}}}_{2}$. Both SFG and DFG susceptibilities have two maxima at frequencies corresponding to the one- and two-photon resonances of the two-level system. Knowledge of the magnitude of one of the maxima of either SFG or DFG is sufficient for determining the magnitude of the second peak of that process and also the two peaks of the other process. In both processes the peak at the two-photon resonance has dominant contributions arising from off-diagonal elements of the density matrix. The interference between these contributions is constructive in SFG and destructive in DFG. Therefore, SFG is more efficient at the two-photon resonance than DFG. An expression for the ratio between these peaks is obtained. The peak of the spectrum of each process corresponding to the one-photon resonance has two main contributions, one of which is diagonal and the other off diagonal. In DFG the interference between the two contributions is constructive and in SFG the interference is destructive. Hence, DFG is more efficient in the vicinity of the one-photon resonance. However, since the off-diagonal contributions to SFG and DFG on the one-photon resonance are equal in magnitude (and differ by a phase difference of \ensuremath{\sim}\ensuremath{\pi}) the efficiencies of the two processes are equal when the duration of the pulses is comparable to or shorter than T1{\mathit{T}}_{1}T1​.