Undepleted Pump Approximation Research Articles

Four-wave mixing analysis of Brillouin dynamic grating in a polarization-maintaining fiber: theory and experiment

We investigate the Brillouin dynamic grating generation and detection process in polarization-maintaining fibers for the case of continuous wave operation both theoretically and experimentally. The four interacting light waves couple together through the material density variation due to stimulated Brillouin scattering. The four coupled equations describing this process are derived and solved analytically for two cases: moving fiber Bragg grating approximation and undepleted pump and probe waves approximation. We show that the conventional grating model oversimplifies the Brillouin dynamic grating generation and detection process, since it neglects the coupling between all the four waves, while the four-wave mixing model clearly demonstrates this coupling process and it is verified experimentally by measuring the reflection of the Brillouin dynamic grating. The trends of the theoretical calculation and experimental results agree well with each other confirming that the Brillouin dynamic grating generation and detection process is indeed a four-wave mixing process.

Moderate-gain Brillouin amplification: An analytical solution below pump threshold

Highly accurate closed-form analytical formula, based on the undepleted pump approximation (UPA), is derived for pump, Stokes and the gain of a Brillouin fiber amplifier (BFA) operating in the moderately depleted pump regime. The material loss of the medium, which is a strong effect in optical fibers, is taken into account. To define validity ranges of the solution, the threshold power of BFA is accurately determined. The obtained results are numerically and experimentally validated. The presented analysis can be used to accurately predict the performance of moderate-gain BFAs.

Continuation Finite Element Simulation of Second Harmonic Generation in Photonic Crystals

AbstractA computational study on the enhancement of the second harmonic generation (SHG) in one-dimensional (1D) photonic crystals is presented. The mathematical model is derived from a nonlinear system of Maxwell’s equations, which partly overcomes the shortcoming of some existing models based on the undepleted pump approximation. We designed an iterative scheme coupled with the finite element method which can be applied to simulate the SHG in one dimensional nonlinear photonic band gap structures in our previous work. For the case that the nonlinearity is strong which is desirable to enhance the conversion efficiency, a continuation method is introduced to ensure the convergence of the iterative procedure. The convergence of our method is fast. Numerical experiments also indicate the conversion efficiency of SHG can be significantly enhanced when the frequencies of the fundamental and the second harmonic wave are tuned at the photonic band edges. The maximum total conversion efficiency available reaches more than 50% in all the cases studied.

Properties of periodic multicrystal configurations in walk-off-compensating second harmonic generation of ultrashort pulses

This work designs a four-platelet periodic multicrystal configuration in the second harmonic generation of ultrashort pulses as a new walk-off-compensating device. It theoretically investigates a proposed active and a typical passive compensating scheme with the undepleted-pump approximation. The result shows that the angular and spectral band-widths are proportional to the number of crystal pairs as expected, but the temperature tunability is basically unaltered owing to inter-plate pulse interference. At the same time, an analysis reveals that a misuse of the phase mismatch factor is responsible for a historic controversy about pulse interference. A real design of an ultraviolet second harmonic generation (262.5 nm) is considered in a passive periodic β-Barium Borate-calcite configuration, where the inter-plate pulse interference is found to form an azimuthal tuning restriction and to lower plate length tolerance. A subsequent numerical simulation with pump depletion is in good accordance with theoretical prediction.

Modulation instability of plane electromagnetic beams in centro-symmetric crystals due to two-photon photorefractive effects

This paper presents an investigation of the modulation instability (MI) of a broad optical beam in biased centro-symmetric two-photon photorefractive media under undepleted pump approximation. From the evolution equation of the perturbation field, we obtain the expression for growth rate of the sidebands. The occurrence of MI has been found to be constrained by the system parameters. The MI gain function, its maximum value, critical frequency and bandwidth have been evaluated in the unstable region.

Maximization of Gain in Slow-Light Silicon Raman Amplifiers

We theoretically study the problem of Raman gain maximization in uniform silicon photonic-crystal waveguides supporting slow optical modes. For the first time, an exact solution to this problem is obtained within the framework of the undepleted-pump approximation. Specifically, we derive analytical expressions for the maximum signal gain, optimal input pump power, and optimal length of a silicon Raman amplifier and demonstrate that the ultimate gain is achieved when the pump beam propagates at its maximum speed. If the signal’s group velocity can be reduced by a factor of 10 compared to its value in a bulk silicon, it may result in ultrahigh gains exceeding 100 dB. We also optimize the device parameters of a silicon Raman amplifier in the regime of strong pump depletion and come up with general design guidelines that can be used in practice.

Parametric interaction and compression of optical pulses in field of high-power pump wave

It is shown that, in the undepleted pump approximation, the nonlinear problem of the three-wave parametric interaction under the conditions of phase mismatch can be reduced to a problem of linear interaction of the signal and idle waves with a strong low-frequency pump wave. The whole wave packet formed by the two waves breaks down to partial pulses whose dynamics is controlled by effective dispersion parameters. This can be accompanied by the compression of either one of the pulses that comprises the wave packet or the whole wave packet.

Spectral compression and group delay of optical pulses in silicon Raman amplifiers

We present a general study of pulse amplification in silicon Raman amplifiers, with special emphasis on the effects of spectral compression and group delay due to the Raman gain dispersion. We use the undepleted-pump approximation to analytically calculate the dynamics of an arbitrary pulse spectra and find the temporal profile of the pulse at the amplifier's output. We show that cw-pumped silicon waveguides are extremely inefficient in amplification of subpicosecond optical pulses but provide large net gains and controllable group delays for pulses with widths of ∼10 ps.

Quadripartite continuous-variable entanglement via quadruply concurrent down-conversion

We investigate an intra-cavity coupled down-conversion scheme to generate quadripartite entanglement using concurrently resonant nonlinearities. We verify that quadripartite entanglement is present in this system by calculating the output fluctuation spectra and then considering violations of optimized inequalities of the van Loock-Furusawa type. The entanglement characteristics both above and below the oscillation threshold are considered. We also present analytic solutions for the quadrature operators and the van Loock-Furusawa correlations in the undepleted pump approximation.

Exact iterative solution of second harmonic generation in quasi-phase-matched structures

A versatile and accurate approach that combines a numerical iteration technique and a transfer-matrix method (TMM) is developed to solve the general problem of second harmonic generation (SHG) with pump depletion in quasi-phase-matched (QPM) nonlinear optical structures. We derive the iterative formulae from the nonlinear coupled wave equations and obtain the intensity distribution of fundamental wave and second harmonic wave by TMM. The approach shows quick numerical convergence of iteration and maintains perfect conservation of total energy. The simulation results show that the model coincides with the one under undepleted pump approximation very well when the SHG efficiency is small (well below 15%) and agrees very well with the effective nonlinear susceptibility model in handling general SHG problems even when the conversion efficiency is high up to 100%. Our method is applicable to general nonlinear optical structures, such as periodic, quasi-periodic, and aperiodic QPM structures, photonic crystals, and micro-cavities that might involve complicated modulation on the linear and nonlinear susceptibility.

Effect of photoconductivity and dielectric constant of the photorefractive materials on two-beam coupling gain and phase-shifts for a single unidirectional photorefractive ring resonator

Photoconductive dependence of two-beam coupling between the pump beam and the signal beam in photorefractive materials have been analyzed in case of non-degenerate wave mixing under the undepleted pump approximation method. During the two-wave mixing in photorefractive materials, steady state amplification of the signal beam and oscillation characteristics of a single unidirectional ring resonator has been studied. The domination of the two-beam coupling gain over the combined absorption and resonator losses such as Fresnel reflections from the crystal and imperfect mirrors builds up unidirectional oscillation. The buildup of such an oscillation leads to a saturation of the gain, which can be explained in terms of the photorefractive phase-shift. The existence of this phase-shift between the photorefractive index grating and the illumination intensity pattern, which is of characteristic of the photorefractive effect, leads to an energy transfer between the two beams. For a single unidirectional ring resonators, the effects of photoconductivity of the materials, two-beam energy coupling coefficient, dielectric constant, crystal thickness, and material's absorption coefficient on amplification of the two-beam coupling gain and photorefractive phase-shifts of the signal beam have also been studied in detail. It has been found that amplification of the signal beam and phase-shift can be enhanced by taking the photorefractive crystal having higher photoconductivity and lower dielectric constant, which improves performance of the resonators.

Nonlinear Silicon Photonics: Analytical Tools

Since the recent demonstration of chip-scale, silicon-based, photonic devices, silicon photonics provides a viable and promising platform for modern nonlinear optics. The development and improvement of such devices are helped considerably by theoretical predictions based on the solution of the underlying nonlinear propagation equations. In this paper, we review the approximate analytical tools that have been developed for analyzing active and passive silicon waveguides. These analytical tools provide the much needed physical insight that is often lost during numerical simulations. Our starting point is the coupled-amplitude equations that govern the nonlinear dynamics of two optical waves interacting inside a silicon-on-insulator waveguide. In their most general form, these equations take into account not only linear losses, dispersion, and the free-carrier and Raman effects, but also allow for the tapering of the waveguide. Employing approximations based on physical insights, we simplify the equations in a number of situations of practical interest and outline techniques that can be used to examine the influence of intricate nonlinear phenomena as light propagates through a silicon waveguide. In particular, propagation of single pulse through a waveguide of constant cross section is described with a perturbation approach. The process of Raman amplification is analyzed using both purely analytical and semianalytical methods. The former avoids the undepleted-pump approximation and provides approximate expressions that can be used to discuss intensity noise transfer from the pump to the signal in silicon Raman amplifiers. The latter utilizes a variational formalism that leads to a system of nonlinear equations that governs the evolution of signal parameters under the continuous-wave pumping. It can also be used to find an optimum tapering profile of a silicon Raman amplifier that provides the highest net gain for a given pump power.

Enhancement of second harmonic generation in one-dimensional nonlinear photonic-crystal microcavities

A numerical iteration technique starting from the analytical solution of the fundamental wave and the second harmonic wave propagation equations in undepleted-pump approximation is presented for analysis of second harmonic generation in one-dimensional nonlinear photonic-crystal microcavities under pump light incidence at an arbitrary angle, accounting for pump depletion. Numerical results are in good agreement with experimental observation and theoretical predication by transfer matrix method. Analysis of the conversion efficiency of second harmonic generation in nonlinear photonic-crystal microcavities shows that dramatic enhancement of second harmonic generation occurs in the microcavities which have the optimal numbers of distributed Bragg reflector layers. The enhancement of nonlinear interactions is ascribed to mode resonance and localization of high intensity fundamental field, corresponding to the defect mode within the forbidden band of the photonic-crystal microcavities.

Computing for second harmonic generation in one-dimensional nonlinear photonic crystals

A numerical study of second harmonic generation (SHG) in one-dimensional nonlinear photonic crystals based on full nonlinear system of equations, implemented by a combination of the method of finite elements and fixed-point iterations, is reported. This model is derived from a nonlinear system of Maxwell’s equations, which partly overcomes the known shortcoming of some existing models relied on the undepleted-pump approximation. We derive a general solution of SHG in one-dimensional nonlinear photonic crystals structures. The convergence of our method is fast. Numerical simulations also show the conversion efficiency of SHG can be significantly enhanced when the frequencies of the fundamental wave are located at the photonic band edges or are assigned to the designed defect states.

Continuous-wave Raman amplification in silicon waveguides: beyond the undepleted pump approximation

We present approximate analytical expressions describing nonlinear interaction of two optical waves copropagating inside a silicon waveguide in the presence of linear losses, stimulated Raman scattering, and free-carrier absorption. Our approach avoids the undepleted-pump approximation, which we show to be inadequate to describe accurately the Raman-gain process. Based on our calculations, we propose a new generalized definition for the effective length and show that it provides better insight into the impact of nonlinear absorption on Raman amplification in silicon waveguides.

Geometrical representation of sum frequency generation and adiabatic frequency conversion

We present a geometrical representation of the process of sum frequency generation in the undepleted pump approximation, in analogy with the known optical Bloch equations. We use this analogy to propose a technique for achieving both high efficiency and large bandwidth in sum frequency conversion using the adiabatic inversion scheme. The process is analogous with rapid adiabatic passage in NMR, and adiabatic constraints are derived in this context. This adiabatic frequency conversion scheme is realized experimentally using an aperiodically poled potassium titanyl phosphate (KTP) device, where we achieved high efficiency signal-to-idler conversion over a bandwidth of $140\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$.

Pair correlations of scattered atoms from two colliding Bose-Einstein condensates: Perturbative approach

We apply an analytical model for anisotropic, colliding Bose-Einstein condensates in a spontaneous four-wave-mixing geometry to evaluate the second-order correlation function of the field of scattered atoms. Our approach uses quantized scattering modes and the equivalent of a classical, undepleted pump approximation. Results to lowest order in perturbation theory are compared with a recent experiment and with other theoretical approaches.

Fourier modal method for the analysis of second-harmonic generation in two-dimensionally periodic structures containing anisotropic materials

The rigorous Fourier modal method for crossed anisotropic gratings is extended to the analysis of second-harmonic generation in two-dimensionally periodic structures. The method takes the undepleted-pump approximation that uncouples the calculations of the fundamental and the second-harmonic fields. The effectiveness of the method is verified by solving a one-dimensional problem, which has been analyzed by two previously developed methods, and by comparing the simulation results of the L-shaped gold nanoparticle arrays with the previous experimental measurements.

Modeling second-harmonic generation by use of mode expansion

We present an accurate and efficient method of modeling second-harmonic generation in two-dimensional structures by use of eigenmode expansion. By using the undepleted-pump approximation we uncouple the calculations for the fundamental and second harmonic. Expansion of the field in eigenmodes gives rise to a linear matrix formalism. The method includes reflections and is especially suited for periodic structures. Several examples, including a two-dimensional photonic-crystal-cavity device, are studied.

Second-Harmonic Generation in the Undepleted-Pump Approximation

We study electromagnetic diffraction by a dielectric object surrounded by a nonlinear thin layer. The geometry of the problem is two-dimensional, and the incident wave is transverse magnetic polarized. We derive a first-order expansion of the fundamental field and then derive the leading term of the second-harmonic field. Our approach is based on layer potential techniques through integral representation formulas of the fields.