Nonlinear Transfer Matrix Method Research Articles

Investigation of the nonlinear optical frequency conversion in ultrathin franckeite heterostructures

Layered franckeite is a natural superlattice composed of two alternating layers of different compositions, SnS2- and PbS-like. This creates incommensurability between the two species along the planes of the layers, resulting in spontaneous symmetry-break periodic ripples in the a-axis orientation. Nevertheless, natural franckeite heterostructure has shown potential for optoelectronic applications mostly because it is a semiconductor with 0.7 eV bandgap, air-stable, and can be easily exfoliated down to ultrathin thicknesses. Here, we demonstrate that few-layer franckeite shows a highly anisotropic nonlinear optical response due to its lattice structure, which allows for the identification of the ripple axis. Moreover, we find that the highly anisotropic third-harmonic emission strongly varies with material thickness. These features are further corroborated by a theoretical nonlinear susceptibility model and the nonlinear transfer matrix method. Overall, our findings help to understand this material and propose a characterization method that could be used in other layered materials and heterostructures to assign their characteristic axes.

Defect-doped one-dimensional nonlinear photonic crystals exhibit coherent perfect absorption with bistable properties

In this paper, the bistable phenomenon with the coherent perfect absorption properties are studied through the nonlinear transfer matrix method in one-dimensional nonlinear photonic crystals doped with a defect. The focus is on the influences of the relative phase of coherent light on the bistable coherent absorption intensity, and the tuning of the bistable threshold by parameters such as angle and wavelength of incidence. The simulated results show that the coherent absorption intensity is very sensitive to the relative phase of coherent incidence. As the phase decreases, the absorption intensity also decreases, varying from 1 to −1 when the phase change is within 180°. The tuning of the relative phase to the absorption intensity provides great convenience for the application of bistable, and parameters such as the incident angle can also achieve sensitive tuning of the bistable threshold. The larger the incident angle, the larger the threshold, and the more obvious the bistable phenomenon at TE mode. The application of nonlinear optical devices has been expanded in the fields of filters, optical switches, optical power threshold devices, etc.

Bistable absorption in a 1D photonic crystal with a nanocomposite defect layer.

We investigate the nonlinear absorption properties of a defective one-dimensional photonic crystal at the near-infrared range using the nonlinear transfer matrix method. The defect is a nanocomposite layer containing vanadium dioxide nanoparticles sandwiched between two nonlinear dielectric layers. The linear absorption spectrum of the designed structure has three resonant absorption lines at the bandgap region of the photonic crystal. We can reconfigure the structure in the linear regime from nearly transparent to absorbent or vice versa in multiple resonant wavelengths by adjusting the temperature. Moreover, the system shows absorptive bistability by adjusting the intensity and incident angle of the input light. We discuss the tunability of the nonlinear absorption in detail. In the nonlinear regime, we show that, besides the temperature, the structure can be reconfigured from absorbent to transparent and vice versa by adjusting the incident optical power and the incident angle. We validate the results by examining the electric field distribution throughout the structure.

Tunable Tamm states at the interface of a 1D graphene-based photonic crystal and a nonlinear dielectric slab

We investigate the dispersion of the Tamm states at the interface of a 1D graphene-based photonic crystal and a nonlinear dielectric slab. We use the nonlinear transfer matrix method to obtain the dispersion of the Tamm states in both the graphene induced and Bragg gaps in the terahertz region. We explore the spectral tunability of the surface modes by adjusting parameters such as the thickness of the nonlinear slab, the intensity of the surface modes, and the chemical potential of graphene layers. We reveal that the penetration depth of the graphene induced bandgap modes in the uniform environment is several times greater than that of the conventional Bragg gap modes. Also, it is shown that we can improve the penetration depth of the surface modes by adjusting the thickness of the nonlinear slab, the intensity of the Tamm states, or the chemical potential of the graphene layers.

Solving scattering problems in the half-line using methods developed for scattering in the full line

We reduce the solution of the scattering problem defined on the half-line [0,∞) by a real or complex potential v(x) and a general homogeneous boundary condition at x=0 to that of the extension of v(x) to the full line that vanishes for x<0. We find an explicit expression for the reflection amplitude of the former problem in terms of the reflection and transmission amplitudes of the latter. We obtain a set of conditions on these amplitudes under which the potential in the half-line develops bound states, spectral singularities, and time-reversed spectral singularities where the potential acts as a perfect absorber. We examine the application of these results in the study of the scattering properties of a δ-function potential and a finite barrier potential defined in [0,∞), discuss optical systems modeled by these potentials, and explore the configurations in which these systems act as a laser or perfect absorber. In particular, we derive an explicit formula for the laser threshold condition for a slab laser with a single mirror and establish the surprising fact that a nearly perfect mirror gives rise to a lower threshold gain than a perfect mirror. We also offer a nonlinear extension of our approach which allows for utilizing a recently developed nonlinear transfer matrix method in the full line to deal with finite-range nonlinear scattering interactions defined in the half-line.

Giant Second-Harmonic Generation in Cantor-like Metamaterial Photonic Superlattices.

We present a nonlinear transfer matrix method for studying the second-harmonic generation (SHG) in nonperiodic metamaterial photonic superlattices. A large enhancement of up to 5 orders of magnitude in SHG efficiency was observed for superlattices made with a Cantor-like quasiperiodic assembly of a nonlinear material and a metamaterial. The enhancement was found to depend much more on the electric field amplitude along the structure because of self-similarity effects than on the amount of nonlinear material, which opens the way to design superlattices for tailored applications in broad-band tunable lasers.

High-power piezoelectric vibration model considering the interaction between nonlinear vibration and temperature increase

For high-power piezoelectric devices, nonlinear vibrations and related increases in temperature are critical problems produced by large internal stress and strain. Such nonlinear vibrations have been studied by some researchers; however, the related increase in temperature has not been taken into consideration, because it is a complicated phenomenon. In this study, the mechanism underlying the interaction between nonlinear vibration and temperature increase in a piezoelectric transducer under high-power conditions was clarified. For this purpose, cubic terms of the mechanical strain, the nonlinear transfer matrix method, and the heat conduction equation were combined. Additionally, it is necessary to utilize the distributed parameter model because the temperature increase is not uniform. The calculation results obtained using the measured temperature dependence of material constants verified that the temperature increase significantly degrades the piezoelectric vibration. It is expected that the proposed model will prove indispensable in the development of piezoelectric materials for high-power piezoelectric devices.

Modeling of enhanced spontaneous parametric down-conversion in plasmonic and dielectric structures with realistic waves

A numerical study of the enhancement of the spontaneous parametric down-conversion in plasmonic and dielectric structures is considered. The modeling is done using a nonlinear transfer-matrix method which is extended to include vacuum fluctuations and realistic waves (e.g. Gaussian beam). The results indicate that in the case of short-range surface plasmon polaritons, the main limiting factor of the enhancement is the short length of the coherent buildup. In the case of long-range surface plasmon polaritons or dielectric guided waves, the very narrow resonances are the main limiting factor instead.

Terahertz bistability and multistability in graphene/dielectric Fibonacci multilayer.

Here, we benefit from the strong nonlinear response of graphene and the rich variety of resonances provided by a graphene/dielectric Fibonacci multilayer to realize bistability and multistability in the terahertz (THz) frequency range. Toward this pursuit, we employ the nonlinear transfer matrix method. We examine the suitability of resonances in the Fibonacci multilayer for the bi/multistability purposes and determine the proper working point. We report various switching up/down manners via single or stepwise jumps between states of the same or different contrasts upon increasing followed by decreasing the intensity of the incident wave. We show that graphene samples of high quality are preferred for bi/multistable switching in terms of reducing the switch-up/-down thresholds and widening the multistable region. We also explore the possibility of tuning the bi/multistable behavior via the frequency and angle of the incident wave as well as the graphene Fermi level. We envision precious applications in THz switching, realizing logic gates, and so on for this system.

New generation of one-dimensional photonic crystal cavities as robust high-efficient frequency converter

An effective method is proposed to design finite one-dimensional photonic crystal cavities (PhCCs) as robust high-efficient frequency converter. For this purpose, we consider two groups of PhCCs which are constructed by stacking m nonlinear (LiNbO3) and n linear (air) layers with variable thicknesses. In the first group, the number of linear layers is less than the nonlinear layers by one and in the second group by two. The conversion efficiency is calculated as a function of the arrangement and thicknesses of the linear and nonlinear layers by benefiting from nonlinear transfer matrix method. Our numerical simulations show that for each group of PhCCs, there is a structural formula by which the configurations with the highest efficiency can be constructed for any values of m and n (i.e. any number of layers). The efficient configurations are equivalent to Fabry–Pérot cavities that depend on the relationship between m and n and the mirrors in two sides of these cavities can be periodic or nonperiodic. The conversion efficiencies of these designed PhCCs are more than 5 orders of magnitude higher than the perfect ones which satisfy photonic bandgap edge and quasi-phase matching. Moreover, the results reveal that conversion efficiencies of Fabry–Pérot cavities with non-periodic mirrors are one order of magnitude higher than those with periodic mirrors. The major physical mechanisms of the enhancement are quasi-phase matching effect, cavity effect induced by dispersive mirrors, and double resonance for the pump and the harmonic fields in defect state. We believe that this method is very beneficial to the design of high-efficient compact optical frequency converters.

Kolakoski sequence as an element to radiate giant forward and backward second harmonic signals

We propose a novel type of aperiodic one-dimensional photonic crystal structures which can be used for generating giant forward and backward second harmonic signals. The studied structure is formed by stacking together the air and nonlinear layers according to the Kolakoski self-generation scheme in which each nonlinear layer contains a pair of antiparallel 180° poled LiNbO3 crystal layers. For different generation stages of the structure, conversion efficiencies of forward and backward second harmonic waves have been calculated by nonlinear transfer matrix method. Numerical simulations show that conversion efficiencies in the Kolakoski-based multilayer are larger than the perfect ones for at least one order of magnitude. Especially for 33rd and 39th generation stages, forward second harmonic wave are 42 and 19 times larger, respectively. In this paper, we validate the strong fundamental field enhancement and localization within Kolakoski-based multilayer due to periodicity breaking which consequently leads to very strong radiation of backward and forward second harmonic signals. Following the applications of analogous aperiodic structures, we expect that Kolakosi-based multilayer can play a role in optical parametric devices such as multicolor second harmonic generators with high efficiency.

Controllable single-mode random laser using stimulated Raman gain.

The present study demonstrates numerically a novel approach to perform a controllable single-mode random laser, using stimulated Raman gain. Due to the narrow linewidth of the Raman line shape, only one of the modes of the passive random system can lase. The robust control on the emission spectrum was achieved through the selection of any desired quasi-modes, by adjusting the pump wavelength in order to place the center of Raman line shape on desired quasi-modes. This approach was proved using a developed nonlinear transfer matrix method for a 1D Raman random system. The proposed method includes the Raman gain saturation and the frequency-dependent index of refraction.

Composite patterns formed by paraxial vortex-beams propagation in one-dimensional multilayer plasma photonic crystals

Optical properties of twisted electromagnetic beam propagating in one-dimensional binary dielectric-plasma photonic crystal are investigated. Reflection and transmission spectra by using nonlinear Transfer Matrix Method of Degenerate Four-Wave Mixing are calculated. These characteristics are studied in low intensity, high temperature range of plasma, large Rayleigh range and neglecting the pumps depletion. In our calculation, the standard slowly varying field approximation is employed. The results show that the composite vortex patterns in the left of photonic crystal structure is dependent on the beam vortex charge number, the incident beam frequency and plasma temperature. Also, it is indicated that reflection and transmission may be more than unity for the sake of amplifying through the nonlinear medium. In fact, this structure can be exploited for characterization of phase conjugation in multilayer optical structures and optoelectronic.

Optical bistability in one-dimensional doped photonic crystals with spontaneously generated coherence

We investigate optical bistability in a multilayer one-dimensional photonic crystal where the central layer is doped with $\ensuremath{\Lambda}$-type three-level atoms. We take into account the influence of spontaneously generated coherence when the lower atomic levels are sufficiently close to each other, in which case Kerr-type nonlinear response of the atoms is enhanced. We calculate the propagation of a probe beam in the defect mode window using the numerical nonlinear transfer matrix method. We find that Rabi frequency of a control field acting on the defect layer and the detuning of the probe field from the atomic resonance can be used to control the size and contrast of the hysteresis loop and the threshold of the optical bistability. In particular we find that at the optimal spontaneously generated coherence, a three orders of magnitude lower threshold can be achieved relative to the case without the coherence.

HIGH EFFICIENT ALL-OPTICAL DIODE ACTION THROUGH NONLINEAR FINITE ONE-DIMENSIONAL PHOTONIC CRYSTAL COATED WITH A METAL FILM

Light propagation in a finite nonlinear one-dimensional photonic crystal coated with a metal film is theoretically investigated through the nonlinear transfer matrix method and the finite-difference time-domain (FDTD) method. It is found that the layered structure exhibits both pronounced unidirectionality and high transmission. This leads to all-optical diode actions. Compared with other similar studies, the designed structure has some advantages. First, the structure only with seven periods is much simpler than presented similar structures; second, the structure has both high transmittance and large transmission contrast; third, the perfect performance of AOD is robust to the metal loss.

All-optical diode actions through a coupled system of Tamm plasmon-polariton and nonlinear cavity mode

Light propagation in a coupled system of Tamm plasmon-polariton and nonlinear cavity mode is theoretically investigated through the nonlinear transfer matrix method. It is found that the asymmetric layered structure exhibits both pronounced unidirectionality and high transmission. This leads to all-optical diode actions. Compared with other similar studies, the designed structure is much simple only with seven periods. The unique feature is that the direction of on-off can be reversed depending on the working frequencies. The effect of metal loss is also considered in this study.

The nonreciprocal of second harmonic generation with dielectric/antiferromagnetic/dielectric structure in Voigt geometry

We calculate the second harmonic generation (SHG) from structures with dielectric/antiferromagnetic(AF)/dielectric in Voigt geometry. The calculations are performed with a nonlinear transfer matrix method and examples of nonreciprocal SHG are given. In particular, we study the nonreciprocal SHG in four different configurations when the direction of an applied magnetic field and the order of dielectric films are reversed. The nonreciprocal SHG can be realized in a resonance zone of AF, including THz frequency range. The increasing of the angle of incidence wave can enhance the effect of nonreciprocal SHG. In terms of applications, AF may be of interest in signal processing in the THz.

Analysis of three-wave mixing in one-dimensional nonlinear multilayer structures with pump depletion

We present a general theoretical approach that combines the iterative technique with the nonlinear transfer matrix method to solve the general problems of three-wave mixing with pump depletion in nonlinear multilayer structures. The theoretical model is directly derived from nonlinear coupled wave equations, shows fast numerical convergence and maintains conservation of total energy perfectly for general structures. We have employed this approach to discuss second harmonic generation in photonic bandgap crystals with slow light effect and obtained a conversion efficiency of about 25% for both the forward and backward second harmonic waves. The detailed wave evolution process and energy conversion mechanism of sum frequency generation and difference frequency generation in perfect and imperfectly quasiphase-matching structures have been systematically investigated and their subtle and strong dependence on the initial conditions of incident waves has been revealed. The developed theoretical approach can be very useful for efficient and quantitative analysis of various nonlinear optical problems and help to design optimum nonlinear multilayer structures for specific high-efficiency nonlinear three-wave mixing processes.

All-optical diode with photonic multilayers based on asymmetric light localization

An all-optical diode (AOD) with structure (AB)m(BA)n(BBAA)k is proposed based on asymmetric light localization, and its optical bistability are numerically investigated by the nonlinear transfer matrix method. Research results show that the behavior of the AOD strongly depends on the period number m, n, and k, the transmission direction of the AOD is related to the values of m and n, while k affects the transmission contrast of the AOD. It is a significant reference for the design of all-optical signal processing devices.

Properties of defect modes in one-dimensional photonic crystals containing two nonlinear defects

Transmission properties of one-dimensional photonic crystal with two nonlinear defects were investigated based on the nonlinear transfer matrix method. From the inspection of the optical bistabilities around different defect modes, it was found that the threshold of bistability for the bonding mode is always lower than that for the anti-bonding mode. When intensity of the incident wave increases bistability could happen at any wavelength larger than the resonance wavelength of the linear anti-bonding mode in the gap. The threshold of bistability increases initially with the wavelength, then decreases sharply in reaching the resonance wavelength of the linear bonding mode. The coupling behavior of the two defects was also discussed, and it was found that the threshold of bistability increases with the distance between the two defects. A flat spectrum similar to that formed in the photonic crystal molecules was obtained under certain incident intensity.