Optical Wave Equation Research Articles

The exact solutions for the fractional Riemann wave equation in quantum mechanics and optics

In this paper, the fractional Riemann wave equation with M-truncated derivative (FRWE-MTD) is considered. The Jacobi elliptic function method and the modified extended tanh function method are applied to acquire new elliptic, rational, hyperbolic, and trigonometric functions solutions. Moreover, we expand some earlier studies. The obtained solutions are important in explaining some exciting physical phenomena, since the Riemann wave equation is used in various fields, including quantum mechanics, optics, signal processing, and general relativity. Also, this equation is used to describe the propagation of waves in various dispersive systems, where wave motion is affected by the medium through which it travels. Several 3D and 2D graphs are shown to demonstrate how the M-truncated derivative affects the exact solutions of the FRWE-MTD.

A review: Rise of PT-symmetry for laser applications

In quantum mechanics, with the groundbreaking discovery of parity-time symmetry (PT-symmetry), Bender and Boettcher unfold the nature of the non-Hermitian Hamiltonian to exhibit real eigenvalues. This opened a new doorway to efficiently explore all the previously ignored quantum non-Hermitian systems. The analogy between the Schrödinger equation in quantum mechanics and the paraxial wave equation in optics spurred researchers to apply the concept of PT-symmetry to optical systems. On exploring the PT-symmetric optics experimentally, by engineering loss and gain in the system, a lot of applications have been reported. It includes single-mode lasers, coherent perfect absorber lasers (CPA lasers), exceptional point sensors (EPS), spectral and spatial filters, etc. In addition to linear theoretical models on PT-symmetry, nonlinear theoretical models and the corresponding experimental results have also been reported in the last few decades. This review article begins with the fundamental idea of PT-symmetry for beginners. Subsequently, we elucidate its advancements in the field of optics. Besides, we present the recent developments of various laser systems using PT-symmetry. Thus, this review intends to focus on novel theoretical and experimental works on PT-symmetry.

Simulation of CZTSSe kesterite thin film photovoltaics with TiO2 or CdS/ZnO: Comparing the electrical and thermal properties

The optical, electrical and thermal properties of two Cu2ZnSn4SxSe4−x (CZTSSe) based thin film solar cells with TiO2 and CdS/ZnO electron transport layer (ETL) were investigated through 2D and 3D mapping simulations. The simulations compared the optical photogeneration rates, current-voltage characteristics of the two cells, Shockley-Read-Hall (SRH) recombination rate and the electric-field distribution at the interface of CZTSSe and ETL materials by solving the coupled drift-diffusion, Poisson, and rate equations. The maximum non-radiative SRH recombination is 2.5 × 1028 1/m3s at CSSs/TiO2 interface which is half of it at CZTSSe/CdS/ZnO interface (5 × 1028 1/m3. s). This is because of the electric-field spike at the CdS/ZnO/ITO interface which enhances the electron passivation at this interface and in turn improves the open-circuit voltage of the CZTSSe/CdS/ZnO based solar cell to 0.55 V. The heat generation in the two cells was also simulated by solving the heat equation coupled with electrical and optical wave equations. SRH recombination heat is higher in CZTSSe/CdS/ZnO cell (1010 W/m3) compared to cell with CZTSSe/TiO2 (0.4 × 1010 W/m3) due to less average heat conductivity of CdS/ZnO. The conductive heat flux is higher at the junction of CZTSSe/TiO2 (5 × 105 W/m2) which is over 10 times more than the heat conduction flux at the junction of CZTSSe/CdS/ZnO (2 × 104 W/m2). The practical outcome of this simulation is that although CdS/ZnO is a better ETL material for electrical characteristics of the cell, the thermal stability of the cell is better with TiO2 layer. This might be a guiding outcome for practical realization of CZTSSe solar cell structures with high thermal stability.

Bilinear Bäcklund transformation, Lax pair and interactions of nonlinear waves for a generalized (2 + 1)-dimensional nonlinear wave equation in nonlinear optics/fluid mechanics/plasma physics

In this paper, outcomes of the study on the Backlund transformation, Lax pair, and interactions of nonlinear waves for a generalized (2 + 1)-dimensional nonlinear wave equation in nonlinear optics, fluid mechanics, and plasma physics are presented. Via the Hirota bilinear method, a bilinear Backlund transformation is obtained, based on which a Lax pair is constructed. Via the symbolic computation, mixed rogue–solitary and rogue–periodic wave solutions are derived. Interactions between the rogue waves and solitary waves, and interactions between the rogue waves and periodic waves, are studied. It is found that (1) the one rogue wave appears between the two solitary waves and then merges with the two solitary waves; (2) the interaction between the one rogue wave and one periodic wave is periodic; and (3) the periodic lump waves with the amplitudes invariant are depicted. Furthermore, effects of the noise perturbations on the obtained solutions will be investigated.

Comparison of Lambert W function with Adomian decomposition method for wave propagation in saturable absorption medium

We present a comparative study to examine the performances of both the Lambert W function method and Adomian decomposition method (ADM) for solving the nonlinear optical beam propagations in a saturable absorption (SA) medium for two distinct SA models. The SA-I model describes a fast saturable absorber (in which the relaxation time is much shorter than the pulse duration of the incident laser beam and no excited state absorption occurs) for a three-level system, while the SA-II model depicts a four-level system or two-photon absorption saturation. We derive the explicit analytic solutions based on the Lambert W function as well as ADM without using thin-sample approximation, and without weak saturation limit. The two different approaches to solve the nonlinear optical wave equation for two distinct SA models are compared and discussed in detail, and show excellent agreement with each other. Finally, to validate the theoretical analysis, we provide some experimental data for nonlinear optical transmission in methyl orange-doped PVA films, revealing excellent agreement with the theory of the SA-II model.

3D non-paraxial kernel for two-point correlation modelling in optical and quantum interference at the micro and nano-scales

Optical and quantum interference at the micro and nano-scales are of growing interest. Its accurate description bases on the non-paraxial propagation of the spatial correlation, in which the physical observable, determined by the square modulus of the optical and quantum wave functions, is expressed as a modal expansion on a 3D non-paraxial geometric kernel, with the spatial correlation as coefficient. The kernel plays the main role of the model and is deduced from the optical wave equation in free-space as well as from the Schrödinger equation for particle propagation in field-free regions. Two features are analyzed in detail, i.e. the physical implications on the wave and particle interference due to the 3D spatial modulations provided by the local and non-local components of the kernel at the micro and nano-scales, and the decay of the kernel terms with the propagation distance which leads to a novel criterion for the kernel accuracy. The interference modeling is implemented on a matrix algorithm and is illustrated by some examples with nano-structured masks.

Field theory of monochromatic optical beams: II. Classical and quantum paraxial fields

This work is the second part of an investigation aiming at the study of optical wave equations from a field-theoretic point of view. Here, we study classical and quantum aspects of scalar fields satisfying the paraxial wave equation. First, we determine conservation laws for energy, linear and angular momentum of paraxial fields in a classical context. Then, we proceed with the quantization of the field. Finally, we compare our result with the traditional ones.

Surface acoustic waves for acousto-optic modulation in buried silicon nitride waveguides.

We theoretically investigate the use of Rayleigh surface acoustic waves (SAWs) for refractive index modulation in optical waveguides consisting of amorphous dielectrics. Considering low-loss Si3N4 waveguides with a standard core cross-section of 4.4×0.03 μm2 size, buried 8-μm deep in a SiO2 cladding, we compare surface acoustic wave generation in various different geometries via a piezo-active, lead zirconate titanate film placed on top of the surface and driven via an interdigitized transducer (IDT). Using numerical solutions of the acoustic and optical wave equations, we determine the strain distribution of the SAW under resonant excitation. From the overlap of the acoustic strain field with the optical mode field, we calculate and maximize the attainable amplitude of index modulation in the waveguide. For the example of a near-infrared wavelength of 840 nm, a maximum shift in relative effective refractive index of 0.7x10-3 was obtained for TE polarized light, using an IDT period of 30-35 μm, a film thickness of 2.5-3.5 μm, and an IDT voltage of 10 V. For these parameters, the resonant frequency is in the range of 70-85 MHz. The maximum shift increases to 1.2x10-3, with a corresponding resonant frequency of 87 MHz, when the height of the cladding above the core is reduced to 3 μm. The relative index change is about 300 times higher than in previous work based on non-resonant proximity piezo-actuation, and the modulation frequency is about 200 times higher. Exploiting the maximum relative index change of 1.2×10-3 in a low-loss, balanced Mach-Zehnder modulator should allow full-contrast modulation in devices as short as 120 μm (half-wave voltage length product = 0.24 Vcm).

Electromagnetic-continuum-induced nonlinearity

A nonrelativistic Hamiltonian describing interaction between a mechanical degree of freedom and radiation pressure is commonly used as an ultimate tool for studying system behavior in opto-mechanics. This Hamiltonian is derived from the equation of motion of a mechanical degree of freedom and the optical wave equation with time-varying boundary conditions. We show that this approach is deficient for studying higher order nonlinear effects in an open resonant opto-mechanical system. Opto-mechanical interaction induces a large mechanical nonlinearity resulting from a strong dependence of the power of the light confined in the optical cavity on the mechanical degrees of freedom of the cavity due to coupling with electromagnetic continuum. This dissipative nonlinearity cannot be inferred from the standard Hamiltonian formalism.

On a System of Nonlinear Optical Wave Equations in Two-Spatial Dimension

We show that for a system of nonlinear weakly dispersive wave equations in two-spatial dimension, which is a model in nonlinear optics, the local L2 norm of its solutions decays to zero as time approaches infinity.

Spontaneous generation of singularities in paraxial optical fields.

In nonrelativistic quantum mechanics, the spontaneous generation of singularities in smooth and finite wave functions is a well understood phenomenon also occurring for free particles. We use the familiar analogy between the two-dimensional Schrödinger equation and the optical paraxial wave equation to define a new class of square-integrable paraxial optical fields that develop a spatial singularity in the focal point of a weakly focusing thin lens. These fields are characterized by a single real parameter whose value determines the nature of the singularity. This novel field enhancement mechanism may stimulate fruitful research for diverse technological and scientific applications.

Subwavelength anti-diffracting beams propagating over more than 1,000 Rayleigh lengths

Researchers take advantage of the nonlinear response of a dipolar glass and observe non-paraxial propagation of a sub-micrometre-sized beam over 103 diffraction lengths. Propagating light beams with widths down to and below the optical wavelength require bulky large-aperture lenses and remain focused only for micrometric distances1,2. Here, we report the observation of light beams that violate this localization/depth-of-focus law by shrinking as they propagate, allowing resolution to be maintained and increased over macroscopic propagation lengths. In nanodisordered ferroelectrics3,4 we observe a non-paraxial propagation of a sub-micrometre-sized beam for over 1,000 diffraction lengths, the narrowest visible beam reported to date5,6,7,8. This unprecedented effect is caused by the nonlinear response of a dipolar glass, which transforms the leading optical wave equation into a Klein–Gordon-type equation that describes a massive particle field9. Our findings open the way to high-resolution optics over large depths of focus, and a route to merging bulk optics into nanodevices.

Optical wave equation for light scattering in micro- and nanostructured media and its application

An optical wave equation for scattered light in materials with macroscopic spatial inhomogeneities is presented and analyzed. The equation is applicable to a variety of linear and nonlinear optical interactions in media with intrinsic, induced, or engineered inhomegeneities. While it is valid regardless of the spatial scale of these inhomogeneities, the strongest effect is expected from inhomogeneities of a subwavelength scale size typical for nanostructured optical media.

Modulation analysis of nonlinear beam refraction at an interface in liquid crystals

A theoretical investigation of solitary wave refraction in nematic liquid crystals is undertaken. A modulation theory based on a Lagrangian formulation of the governing optical solitary wave equations is developed. The resulting low-dimensional equations are found to give solutions in excellent agreement with full numerical solutions of the governing equations, as well as with previous experimental studies. The analysis deals with a number of types of refraction from a more to a less optically dense medium, the most famous being the Goos-H\anchen shift upon total internal reflection.

Environmentally induced effects and dynamical phase transitions in quantumsystems

The dynamics of open quantum systems is determined by singularities in thecontinuum of scattering wavefunctions which are related to avoided and true crossingsof eigenvalue trajectories of a non-Hermitian Hamiltonian. The phases of theeigenfunctions are not rigid in approaching the crossing points so that environmentallyinduced spectroscopic redistribution processes may take place and a dynamicalphase transition may occur. Due to the formal equivalence between the quantummechanical Schrödinger equation and the optical wave equation in symmetric lattices, the dynamics of the system is determined also in this latter case byavoided and true crossings of eigenvalue trajectories of the non-Hermitian Hamiltonian. Incontrast to the eigenvalues characterizing an open quantum system, the eigenvaluesdescribing the symmetric optical lattice are real as long as the influence of the environment (lattice) onthe optical wave equation is small. In the regime of avoided level crossings, the symmetry isbroken, the eigenvalues become complex and a dynamical phase transition occurs, similaras in the open quantum system. These results can be proven experimentally by tracingthe phase rigidity of the eigenfunctions of the Hamiltonian and its correlationwith the transparency. The redistribution processes in the regime of avoided levelcrossings allow us to design systems with desired properties in a broad parameterrange.

A new nonlinear-wave-equation formalism for stimulated Brillioun scattering

A revised formalism of SBS is given based on a new optical nonlinear wave equation which explicitly accounts for the macroscopic spatial inhomogeneity resulting from the induced acoustic wave in the medium. The new equation applies to other scattering phenomena, and more generally to optical wave propagation and interaction in nanostructured media for which characteristic spatial scale lengths of material structures (existing or optically induced) are smaller than the optical wavelength.

Application of Post’s formula to optical pulse propagation in dispersive media

In this paper we consider the propagation of optical pulses in dielectric media with nontrivial dispersion relations. In particular, we implement Post’s Laplace transform formula to invert in time the Fourier–Laplace space coefficients which arise from the joint space solution of the optical dispersive wave equation. Due to the inefficiency of a direct application of this formula, we have considered and present here two more efficient implementations. In the first, the Gaver–Post method, we utilize the well known Gaver functionals but store intermediate calculations to improve efficiency. The second, the Bell–Post method, involves an analytic reformulation of Post’s formula such that knowledge of the dispersion relation and its derivatives are sufficient to invert the coefficients from Laplace space to time. Unlike other approaches to the dispersive wave equation which utilize a Debye–Lorentzian assumption for the dispersion relation, our algorithm is applicable to general Maxwell-Hopkinson dielectrics. Moreover, we formulate the approach in terms of the Fourier–Laplace coefficients which are characteristic of a dispersive medium but are independent of the initial pulse profile. They thus can be precomputed and utilized when necessary in a real-time system. Finally, we present an illustration of the method applied to optical pulse propagation in a free space and in two materials with Cole-type dispersion relations, room temperature ionic liquid (RTIL) hexafluorophosphate and brain white matter. From an analysis of these examples, we find that both methods perform better than a standard Post-formula implementation. The Bell–Post method is the more robust of the two, while the Gaver–Post is more efficient at high precision and Post formula approximation orders.

Ultra wide band (UWB) of optical fiber Raman amplifiers in advanced optical communication networks

In the present paper, we have analyzed numerically the ultra wide band (UWB) optical fiber Raman amplifiers in advanced optical communication networks. Moreover, we have developed the numerical transmission and propagation techniques based on mathematical laboratory programming to solve the coupled optical wave equations to obtain design parameters, operational characteristics, and its implementation in the light wave modulated envelop propagation equations for transmission of multi Gbit/sec signals in dense wavelength division multiplexing optical communication systems. We have investigated numerically and parametrically the soliton transmission technique to handle the bit rates and products either per link or per channel in all pumping direction configurations to upgrade optical network efficiency in order to support ultra high capacity of number of users. Also, we have concentrated on the fiber Raman optical amplifier is quickly emerging as an important part of ultra long-distance, high-capacity, and high-speed optical communication systems. Key words: Raman amplification technique, advanced optical networks, ultra wide-band, dense wavelength division multiplexing (DWDM), space division multiplexing (SDM).

Solutions to Maxwell's equations using spheroidal coordinates

Analytical solutions to the wave equation in spheroidal coordinates in the short wavelength limit are considered. The asymptotic solutions for the radial function are significantly simplified, allowing scalar spheroidal wave functions to be defined in a form which is directly reminiscent of the Laguerre–Gaussian solutions to the paraxial wave equation in optics. Expressions for the Cartesian derivatives of the scalar spheroidal wave functions are derived, leading to a new set of vector solutions to Maxwell's equations. The results are an ideal starting point for calculations of corrections to the paraxial approximation.

Spinor-electron wave guided modes in coupled quantum wells structures by solving the Dirac equation

A quantum analysis based on the Dirac equation of the propagation of spinor-electron waves in coupled quantum wells, or equivalently coupled electron waveguides, is presented. The complete optical wave equations for Spin-Up (SU) and Spin-Down (SD) spinor-electron waves in these electron guides couplers are derived from the Dirac equation. The relativistic amplitudes and dispersion equations of the spinor-electron wave-guided modes in a planar quantum coupler formed by two coupled quantum wells, or equivalently by two coupled slab electron waveguides, are exactly derived. The main outcomes related to the spinor modal structure, such as the breaking of the non-relativistic degenerate spin states, the appearance of phase shifts associated with the spin polarization and so on, are shown.