Undepleted Pump Approximation Research Articles

Highly squeezed states in ring resonators: Beyond the undepleted-pump approximation

Highly squeezed states in ring resonators: Beyond the undepleted-pump approximation

Scattering matrix for chiral harmonic generation and frequency mixing in nonlinear metasurfaces

We generalize the concept of optical scattering matrix (S-matrix) to characterize harmonic generation and frequency mixing in planar metasurfaces in the limit of undepleted pump approximation. We show that the symmetry properties of such nonlinear S-matrix are determined by the metasurface symmetries at the macroscopic and microscopic scale. We demonstrate that for description of degenerate frequency mixing processes such as optical harmonic generation, the multidimensional S-matrix can be replaced with a reduced two-dimensional S-matrix. We show that for metasurfaces possessing specific point group symmetries, the selection rules determining the transformation of the reduced nonlinear S-matrix are simplified substantially and can be expressed in a compact form. We apply the developed approach to analyze chiral harmonic generation in nonlinear metasurfaces with various symmetries including rotational, inversion, in-plane mirror, and out-of-plane mirror symmetries. For each of those symmetries, we confirm the results of the developed analysis by full-wave numerical calculations. We believe our results provide a new paradigm for engineering nonlinear optical properties of metasurfaces which may find applications in active and nonlinear optics, biosensing, and quantum information processing.

Fundamental limits on radiative χ(2) second harmonic generation.

Recent advances in fundamental performance limits for power quantities based on Lagrange duality are proving to be a powerful theoretical tool for understanding electromagnetic wave phenomena. To date, however, in any approach seeking to enforce a high degree of physical reality, the linearity of the wave equation plays a critical role. In this manuscript, we generalize the current quadratically constrained quadratic program framework for evaluating linear photonics limits to incorporate nonlinear processes under the undepleted pump approximation. Via the exemplary objective of enhancing second harmonic generation in a (free-form) wavelength-scale structure, we illustrate a model constraint scheme that can be used in conjunction with standard convex relaxations to bound performance in the presence of nonlinear dynamics. Representative bounds are found to anticipate features observed in optimized structures discovered via computational inverse design. The formulation can be straightforwardly modified to treat other frequency-conversion processes, including Raman scattering and four-wave mixing.

Photorefractive four-wave mixing cannot be treated perturbatively. Revisiting the undepleted-pump approximation

We theoretically consider photorefractive degenerate four-wave mixing oscillators and show that, for this particular system, the linear stability analysis technique leads to incorrect results. The reason for this astonishing failure lies in the unphysical predictions of the undepleted-pump approximation, which appear naturally during the linearization process of the full model. As a consequence, photorefractive four-wave mixing does not seem to permit a perturbative treatment that allows the derivation of mean-field models, and one is forced to use the full model in order to make sensible predictions.

Equations describing the direct relation between laser phase and Stokes light for SBS pumped by phase-modulated laser in optical fibers

Phase modulation is one of the most important methods to increase the threshold of the stimulated Brillouin scattering (SBS) in optical fibers. The mainstream theoretical model used to predict the SBS threshold is the transient three-wave coupling model. However, it is a general model and does not establish a straight relation between the laser phase and the Stokes light. In this paper, undepleted pump approximation is applied to the transient three-wave coupling model. Equations describing the evolution of the Stokes light in both time domain and frequency domain are derived. They establish a direct relation between the laser phase and Stokes light. All terms in the equations have clear physical interpretations. The equations are validated numerically by comparing with the original coupling model as well. It is found that the results match well when the Stokes average reflectivity is not greater than 1% which is usually used to define the SBS threshold.

Second Harmonic Generation in Generalized Ferroelectric Superlattices

The goal of the work presented in this investigation is to achieve the efficient parametric interaction of narrow beams (in particular second harmonic generation) using a nonlinear (ferroelectric-LiTaO3) photonic crystal tuned to self-collimation (nondiffractive) regimes. A numerical study of second harmonic generation (SHG) in one-dimensional nonlinear photonic crystals based on a 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 that partly overcomes the known shortcoming of some existing models that rely on the undepleted pump approximation. We derive a general solution of SHG in one-dimensional nonlinear photonic crystals structures. Numerical simulations also show that 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.

Conversion of spin and orbital angular momentum in the third harmonic generation process in the bulk of an isotropic medium

The interaction between spin and orbital components of the angular momentum of an elliptically polarized beam with helicoidal phase profile and arbitrary axially symmetric intensity profile in the third-harmonic generation process is investigated. The third-harmonic beam is generated in an isotropic medium with cubic nonlinearity within undepleted pump approximation. The conservation of the total projections of both the spin angular momentum and orbital angular momentum components in four-wave mixing is demonstrated for interacting waves.

Atomic twin beams and violation of a motional-state Bell inequality from a phase-fluctuating quasicondensate source

We investigate the dynamics of atomic twin beams produced from a phase-fluctuating source, specifically a 1D Bose gas in the quasi-condensate regime, motivated by the experiment reported in Nature Physics 7, 608 (2011). A short-time analytic model is constructed, which is a modified version of the undepleted pump approximation widely used in quantum and atom optics, except that here we take into account the initial phase fluctuations of the pump source as opposed to assuming long-range phase coherence. We use this model to make quantitative and qualitative predictions of how phase-fluctuations of the source impact the two-particle correlations of scattered atom-pairs. The model is benchmarked against detailed numerical simulations using stochastic phase-space methods, and is shown to validate the intuitive notion that the broadening of momentum-space correlation functions between atoms scattered from a quasi-condensate is driven by the broadened momentum width of the source compared to a true phase coherent condensate. Finally, we combine these theoretical tools and results to investigate the effect phase fluctuations of the twin-beam source can have on a proposed demonstration of a violation of a Bell inequality, which intrinsically relies on phase-sensitive pair correlations.

Interconversion of orbital and spin angular momenta of light beams in the sum-frequency generation process from the surface of the isotropic chiral medium

We study the interaction between spin and orbital components of the angular momentum of electromagnetic waves in the sum-frequency generation process on a surface of the nonlinear isotropic chiral medium within undepleted pump approximation. Both bulk and near-surface responses of the medium are taken into account. Classical and quantum explanations of particular features of three-wave mixing on the surface of a nonlinear medium are presented.

Path-integral description of quantum nonlinear optics in arbitrary media

We present a method, based on Feynman path integrals, to describe the propagation and properties of the quantised electromagnetic field in an arbitrary, nonlinear medium. We provide a general theory, valid for any order of optical nonlinearity, and we then specialise the the case of second order nonlinear processes. In particular, we show, that second-order nonlinear processes in arbitrary media, under the undepleted pump approximation, can be described by an effective free electromagnetic field, propagating in a vacuum, dressed by the medium itself. Moreover, we show,that the probability of such processes to occur is related to the biphoton propagator, which contains informations about the structure of the medium, its nonlinear properties, and the structure of the pump beam.

Spinor Bose-Einstein condensate interferometer within the undepleted pump approximation: Role of the initial state

Most interferometers operate with photons or dilute, non-condensed cold atom clouds in which collisions are strongly suppressed. Spinor Bose-Einstein condensates (BECs) provide an alternative route toward realizing three-mode interferometers; in this realization, spin-changing collisions provide a resource that generates mode entanglement. Working in the regime where the pump mode, i.e., the m=0 hyperfine state, has a much larger population than the side or probe modes (m=+1 and m=-1 hyperfine states), f=1 spinor BECs approximate SU(1,1) interferometers. We derive analytical expressions within the undepleted pump approximation for the phase sensitivity of such an SU(1,1) interferometer for two classes of initial states: pure Fock states and coherent spin states. The interferometer performance is analyzed for initial states without seeding, with single-sided seeding, and with double-sided seeding. The validity regime of the undepleted pump approximation is assessed by performing quantum calculations for the full spin Hamiltonian. Our analytical results and the associated dynamics are expected to guide experiments as well as numerical studies that explore regimes where the undepleted pump approximation makes quantitatively or qualitatively incorrect predictions.

Enhancement of efficiency of second-harmonic generation from MoS2 monolayers in 1D Fibonacci photonic crystals

In this article, the increase of the efficiency of second-harmonic generation (SHG) from MoS2 monolayers embedded in different one-dimensional Fibonacci photonic crystal structures is investigated. The systems contain ZnS, SiO2, and MoS2 layers with the same thickness of ZnS and SiO2 films, which are arranged on the basis of different Fibonacci sequences. The transfer matrix method is used to calculate the forward and backward SHG efficiencies in the undepleted pump approximation. The thicknesses of the ZnS and SiO2 films are changed to obtain the maximum frequency conversion in different structures. Tuning the thicknesses causes the second-harmonic waves generated in each MoS2 monolayer to interfere constructively. Also, the fundamental and second-harmonic wavelengths are both located at the photonic band gap edges, where the density of electromagnetic modes and the nonlinear interaction time are enhanced. Our results show that the SHG efficiencies were increased in quasiperiodic photonic crystals with respect to the periodic structure with the same number of MoS2 layers. In Fibonacci photonic crystal structures there are more geometrical parameters that can be tuned to obtain the highest efficiencies. Furthermore, we used a Bragg mirror at one side of each structure to control the propagation direction of the generated second-harmonic waves and to decrease the backward waves, which led to enhancement of the forward waves.

New transfer-matrix method for frequency conversion in nonlinear multilayered structures based on coupled-amplitude equations

Based on complete coupled-amplitude equations, a new transfer-matrix method for noncollinear frequency conversion in nonlinear multilayered structures is proposed in consideration of oblique incidence and optical anisotropy. The analytic result under the undepleted pump approximation is derived for a general three-wave interaction. Its numerical model is also developed into the transfer-matrix shooting method, suitable for the depleted pump case, and is verified by the shooting method upon coupled-wave equations through a second-harmonic generation process in a hypothetical structure. Analysis of the computation time in the numerical model demonstrates that the efficiency is greatly improved compared with that of the classical shooting method.

Giant enhancement of second harmonic generation efficiency from MoS2 mono layers embedded in 1D photonic crystals

In this paper an enhancement of second harmonic generation efficiency which radiated from MoS2 monolayers embedded in one-dimensional photonic crystal structures is studied. The system contains air, SiO2 and MoS2 layers in periodic manner with the same thickness of air and SiO2 layers. The transfer matrix method is used for calculating the forward and backward second harmonic generated wave efficiencies in undepleted pump approximation. Our results show the giant enhancement of second harmonic generation efficiencies up to seven orders of magnitude only with $N=40$ MoS2 monolayers in the system. The results are obtained by tunning the thickness of air and SiO2 layers at about 1284 nm for fundamental wave of $\lambda = 810$ nm wavelength. Choosing the above thickness causes the second harmonic waves, generated in each MoS2 monolayers, to interference constructively. Both of the fundamental and second harmonic wavelengths are located at the photonic band gap edges where the density of electromagnetic modes and the nonlinear interaction time are enhanced. These two mechanisms help us to improve the second harmonic generation efficiencies. Increasing the segments numbers enhanced the overall thickness of MoS2 nonlinear layers which affect the phase matching conditions and decreased the SH efficiencies.

Analytic description of four-wave mixing in silicon-on-insulator waveguides

We develop an analytic description of continuous-wave four-wave mixing in silicon-on-insulator (SOI) waveguides including linear loss, two-photon absorption, and free-carrier absorption. Under the undepleted pump approximation, the pump equation decouples from the signal and idler equations and becomes a nonlinear differential equation that we solve analytically without further approximations. The signal and idler equations have no known solutions for arbitrary pump power evolution, but we calculate approximate field expressions based on a Magnus expansion, which has been used to study time-ordering effects in quantum optics. Lastly, we show that the phase-matching condition changes through the waveguide and that this explains the shape of the wavelength-conversion-efficiency spectrum in SOI waveguides and why it differs from that of highly nonlinear silica fibers.

Time-reversed and reciprocal second-harmonic generation with pump depletion

The time-reversed second-harmonic generation in one-dimensional nonlinear photonic crystals has been theoretically studied without the undepleted pump approximation. A simple criterion has been deduced which determines the energy flow. Based on it, two kinds of structures with different symmetries are presented to realize the nonlinear time reversal effect. A completely reciprocal nonlinear response is also found in the same process. Furthermore, a multi-section-cascaded structure is proposed to realize the nonlinear time reversal at any given position.

High-efficiency frequency upconversion of 1.5 μm laser based on a doubly resonant external ring cavity with a low finesse for signal field

A doubly resonant external ring cavity with a low finesse for the signal field is used to improve the frequency upconversion efficiency of a weak 1583 nm signal laser to 636 nm by mixing with a resonance power enhanced 1064 nm pump laser in a 25 mm periodically poled lithium niobate crystal. The process of frequency upconversion is described and optimized by the doubly resonant cavity-enhanced sum frequency generation theory under the condition of undepleted pump approximation. By selecting the suitable reflectivity of the signal input mirror and the incident pump power, a cavity-enhanced frequency conversion efficiency of 94.6% was obtained for signal powers up to 25 mW with an input pump power of 780 mW.

Determination of the stimulated raman scattering threshold for a pump pulse of arbitrary width

A theoretical solution to the problem of determining the stimulated Raman scattering (SRS) threshold has been found within the undepleted pump approximation for a pump pulse of arbitrary width, which distinguishes it from the known solutions for the limiting cases of very short (highly transient SRS) and very long (quasi-steady-state SRS) pump pulses with respect to the oscillation dephasing time of the SRS medium. The general formula of the theoretical estimate of SRS threshold, in dependence of not only the pump radiation intensity and the SRS interaction length but also the pump-pulse width, is obtained based on the found solution. The theoretical estimate of the SRS threshold has been shown to be in good agreement with the experimental results on the excitation of picosecond SRS in crystals, which justifies the new express method for estimating the SRS gain in experimental measurements of the picosecond SRS threshold.

Bipartite entanglement in continuous-variable tripartite systems

In the field of continuous-variable tripartite entanglement, the systems utilised can be either asymmetric or symmetric. It is therefore of interest to examine the differences in the entanglement properties of these two types of system, using two examples that are known to produce tripartite entanglement. We examine one asymmetric and one fully symmetric Gaussian continuous-variable system in terms of their tripartite and bipartite entanglement properties. We first treat pure states and are able to find analytic solutions using the undepleted pump approximation for the Hamiltonian models. Our symmetric system exhibits perfect tripartite correlations, but only in the unphysical limit of infinite squeezing. For more realistic squeezing parameters, the two systems exhibit both tripartite and bipartite entanglement. Secondly we treat the more experimentally reasonable situation where the interactions take place inside optical cavities and we are dealing with mixed states. In these cases, where the criteria for genuine tripartite entanglement are more stringent, we find that tripartite entanglement is still available, although over smaller bandwidths than three-mode inseparability. In general, the spectral results are consistent with the analytical solutions. We conclude that none of the outputs are completely analogous to either GHZ or W states, but there are parameter regions of the Hamiltonian dynamics where they produce T states as introduced by Adesso et al. [1,2]. In the intracavity cases, both bipartite entanglement and tripartite inseparability are always present, with genuine tripartite entanglement appearing as the pumping rate is increased. The qualitative differences in the output states for different interaction parameters indicate that continuous-variable tripartite quantum information systems offer a versatility not found in two-mode bipartite systems.

Adiabatic second-harmonic generation.

Adiabatic three-wave mixing processes enable broadband, efficient, and robust frequency conversion by slowly varying the phase mismatch between the interacting waves along the interaction region. Up until now, this method was mainly used in the case in which one of the waves was undepleted. Here we experimentally study fully nonlinear adiabatic processes by implementation in type I and type II second-harmonic generation processes, where the undepleted pump approximation does not hold. Using quasi-phase-matched interaction in chirped gratings, we obtain conversion efficiency approaching 60% and 80%, with corresponding wide thermal acceptance bandwidths of >100°C and 30°C, respectively. The transition between the depleted and undepleted pump regimes is also studied by varying the input polarization angle in the type II process; thus we also test current theory with arbitrary initial conditions. The results are in excellent agreement with analytic predictions for the fully nonlinear adiabatic process.