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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Quantum electronics analysis of second-harmonic generation in optical waveguides

Xiaohan Sun, Mingde Zhang

Published Apr 6, 1993• 0 citations
Publisher

Abstract

In this paper we study the quantum-theoretical description of second-harmonic generation (SHG) in optical waveguides. The interaction operator of three waveguide modes, by means of which the equations for spatially slowly varying annihilation and creation operators of the field strength amplitudes can be determined, is obtained by using the mode fields in the waveguides. The equations for spatially slowly varying annihilation and creation operators of the field are derived. We seek approximate solutions of the operator equations, and obtain the conversion efficiency of SHG.

DOI
10.1117/12.141895

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In this paper we study the quantum-theoretical description of second-harmonic generation (SHG) in optical waveguides. The interaction operator of three waveguide modes, by means of which the equations for spatially slowly varying annihilation and creation operators of the field strength amplitudes can be determined, is obtained by using the mode fields in the waveguides. The equations for spatially slowly varying annihilation and creation operators of the field are derived. We seek approximate solutions of the operator equations, and obtain the conversion efficiency of SHG.
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.
A quantum mechanical analysis of the propagation of coupled modes in integrated optical waveguides is given. The modal orthonormalization property on a cross-section of an optical waveguide, the vector structure of the guided optical modes and the reversal-time symmetry are taken into account to derive the quantum momentum operator and Heisenberg's equations giving a quantum-consistent formulation of the coupled mode propagation as a function of forward and backward creation and annihilation operators.
English: In this project has been designed and implemented a setup to produce the nonlinear process of the second harmonic generation (SHG) in Bragg reflection waveguides (BRW). Starting with the adjustment of a tunable single frequency laser at 1550 with an external cavity configuration. Two diferent types of SHG have been observed, we measured their spectrum and its shifting caused by temperature variations. Finally, it has also been observed the spatial shape of the Bragg mode guided along the BRW, which is the responsible for reaching the phase-matching condition between all the waves involved in this non-linear process. The results obtained in this project will be used in a future experiments, specially, in the implementation of a monolithic integrated source of entangled photon pairs generated by using the spontaneous parametric down-conversion (SPDC) process.
Theoretical analysis of squeezed light generation by phasematched SHG, applicable generally to various SHG configurations and appropriate for mutual comparison, is presented. The analysis is based on analytical solutions of the linearized differential equations in space coordinate and the input-output equations for the amplitude fluctuations, and calculations of squeezing spectra using the covariance matrixes. Formulations describing the squeezing characteristics in terms of the device parameters are derived. Typical numerical results are also given to illustrate the features of traveling-wave SHG, singly-resonant SHG, and doubly-resonant SHG configurations including cases where propagation losses are involved. The results, which are consistent with those in previous work, provide many insights for design and application of SHG devices for squeezed-light generation.
It is theoretically and experimentally demonstrated that optical second-harmonic generation (SHG) is possible to be observed in atomic media irradiated with an intense laser pulse. The SHG is induced by a static electric field spontaneously generated in the interaction, and any special condition is not required for the SHG. The experiment to observe the SHG has been made using atomic sodium vapor and a mode-locked Nd-doped yttrium aluminum garnet laser system (pulse duration, 28 psec; output energy \ensuremath{\lesssim}50 mJ). The second harmonic (SH) polarization, spatial beam pattern, and dependences of SH signal on the sodium density and on the fundamental intensity have been examined. The experimental results agree well with the theoretical predictions and numerical results.
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.