Showing papers on "Coupled mode theory published in 2002"

Polymer micro-ring filters and modulators

Abstract: Micro-ring wavelength filters and resonant modulators using polymer materials at 1300 nm and 1550 nm are analyzed, designed, and demonstrated. The rings are integrated with vertically coupled input and output waveguides. The devices are fabricated using optical lithography. Filters with a finesse of 141 and free spectral range of 5 nm at 1300 nm and finesse of 117 with a free spectral range (FSR) of 8 nm at 1550 nm are demonstrated. Ring resonators with a Q as high as 1.3 /spl times/ 10/sup 5/ at 1300 nm are demonstrated. The filters can be temperature tuned at the rate of 14 GHz//spl deg/C. Resonant ring modulators, which use an electrooptic polymer, are demonstrated. The resonance wavelength voltage tunes at the rate of 0.82 GHz/V. The modulators have a bandwidth larger than 2 GHz. Using the resonant modulator, and open eye diagram at 1 Gb/s is demonstrated.

Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals.

Abstract: We prove that an adiabatic theorem generally holds for slow tapers in photonic crystals and other strongly grated waveguides with arbitrary index modulation, exactly as in conventional waveguides. This provides a guaranteed pathway to efficient and broad-bandwidth couplers with, e.g., uniform waveguides. We show that adiabatic transmission can only occur, however, if the operating mode is propagating (nonevanescent) and guided at every point in the taper. Moreover, we demonstrate how straightforward taper designs in photonic crystals can violate these conditions, but that adiabaticity is restored by simple design principles involving only the independent band structures of the intermediate gratings. For these and other analyses, we develop a generalization of the standard coupled-mode theory to handle arbitrary nonuniform gratings via an instantaneous Bloch-mode basis, yielding a continuous set of differential equations for the basis coefficients. We show how one can thereby compute semianalytical reflection and transmission through crystal tapers of almost any length, using only a single pair of modes in the unit cells of uniform gratings. Unlike other numerical methods, our technique becomes more accurate as the taper becomes more gradual, with no significant increase in the computation time or memory. We also include numerical examples comparing to a well-established scattering-matrix method in two dimensions.

Fabrication of coupled mode photonic devices in glass by nonlinear femtosecond laser materials processing

Abstract: Coupled mode devices are fabricated in transparent glasses by nonlinear materials processing with femtosecond laser pulses Using the direct output of an extended cavity femtosecond laser, without the need for a laser amplifier, single mode waveguides can be rapidly fabricated with well controlled parameters A variety of photonic waveguide devices are demonstrated Directional couplers with various interaction lengths and coupling coefficients are fabricated and their coupling properties are characterized Measurements demonstrate coupled mode behavior consistent with theory An unbalanced Mach-Zehnder interferometer is also fabricated and demonstrated as a spectral filter

Coupled resonator optical waveguides

Abstract: Properties of the recently introduced family of coupled resonator optical waveguides (CROWs) are reviewed, particularly with reference to CROWs designed as planar waveguides in two-dimensional photonic crystal slabs to enhance nonlinear interactions and develop novel all-optical information processing devices. Topics covered include: pulse propagation both in the nondispersive approximation and to all orders of dispersion, and the coupled mode theory of nonlinear optics with pulses in CROWs and its applications to second-harmonic generation and wave coupling via field-induced refractive-index gratings. We also review recent experimental progress in the fabrication and characterization of CROWs, and applications of the CROW concept to fiber gratings and microwave waveguides.

Long-period gratings in planar optical waveguides

Abstract: We present a theoretical analysis of light propagation in a four-layer planar waveguide that consists of a long-period grating (LPG) having a period of the order of 100 µm. By means of the coupled-mode theory, we show that such a structure is capable of coupling light from the fundamental guided mode to the cladding modes at specific wavelengths (resonance wavelengths) and thus results in sharp rejection bands in the transmission spectrum of the waveguide. Our numerical results show that the resonance wavelengths as well as the transmission spectrum can be significantly changed with the waveguide and grating parameters. A waveguide-based LPG should provide a useful approach to the design of a wide range of integrated-optic devices, including wavelength-tunable filters, switches, and environmental sensors.

Coupling between photonic crystal waveguides

Abstract: A calculation procedure for evaluation of the coupling length of two parallel coupled channel waveguides in a planar photonic crystal is proposed. The first step of the calculation is evaluation of the band structure of a photonic crystal containing two coupled linear defects. The eigenvalue corresponding to eigenstates localized in the linear defect (the waveguide) is split due to the coupling. This splitting is treated within the coupled-mode theory that yields a simple relation between the splitting and the coupling length. The MIT photonic bands code is used to evaluate the coupling between channel waveguides in silicon./sup 1/ These calculations show that in contrast to the finite-difference time-domain approach, the method is effective for three-dimensional light propagation.

Geometric variations in high index-contrast waveguides, coupled mode theory in curvilinear coordinates.

Abstract: Perturbation theory formulation of Maxwell’s equations gives a theoretically elegant and computationally efficient way of describing small imperfections and weak interactions in electro-magnetic systems. It is generally appreciated that due to the discontinuous field boundary conditions in the systems employing high dielectric contrast profiles standard perturbation formulations fail when applied to the problem of shifted material boundaries. In this paper we developed a novel coupled mode and perturbation theory formulations for treating generic non-uniform (varying along the direction of propagation) perturbations of a waveguide cross-section based on Hamiltonian formulation of Maxwell equations in curvilinear coordinates. We show that our formulation is accurate and rapidly converges to an exact result when used in a coupled mode theory framework even for the high index-contrast discontinuous dielectric profiles. Among others, our formulation allows for an efficient numerical evaluation of induced PMD due to a generic distortion of a waveguide profile, analysis of mode filters, mode converters and other optical elements such as strong Bragg gratings, tapers, bends etc., and arbitrary combinations of thereof. To our knowledge, this is the first time perturbation and coupled mode theories are developed to deal with arbitrary non-uniform profile variations in high index-contrast waveguides.

Generalized coupled-mode theory for χ (2) interactions in finite multilayered structures

Abstract: Using a first-order multiple-scale expansion approach, we derive a set of coupled-mode equations that describe both forward and backward second-harmonic generation and amplification processes in nonlinear, one-dimensional, multilayered structures of finite length. The theory is valid for index modulation of arbitrary depth and profile. We derive analytical solutions in the undepleted pump regime under different pumping circumstances. The model shows excellent agreement with the numerical integration of Maxwell’s equations.

Analysis and design of periodic structures for microstrip lines by using the coupled mode theory

Abstract: In this work, the coupled mode theory is formulated to analyze distributed periodic structures for microstrip lines. After it, several reasonable approximations are introduced giving rise to analytical solutions for the problem. The obtained results, both numerical and analytical, are checked against measurements showing very good agreement. The proposed method is very attractive for the study of these devices since it avoids the time-consuming full-wave electromagnetic simulations customarily employed and provides analytical solutions that are very useful for analysis and synthesis purposes.

Grating-assisted coupling of optical fibers and photonic crystal waveguides.

Abstract: We propose and analyze a highly efficient method of coupling light from optical fibers to two-dimensional photonic crystal waveguides. Efficient coupling is achieved by positioning of a tapered fiber parallel to the linear defect, where the photonic crystal’s cladding functions as a grating coupler and provides field confinement as well. Numerical simulations indicate that better than 90% transmission is possible with a full width at half-magnitude bandwidth of 12 nm. It is shown that one can increase the bandwidth by increasing the field overlap between the two waveguides.

Polymeric optical spot-size transformer with vertical and lateral tapers

Abstract: The design, fabrication, and characterization of a polymeric optical spot-size transformer with vertical and lateral tapers is reported. The vertical taper is formed by utilizing the planarization properties of a polymer, while the lateral taper is defined by photolithography. An optimization method for the taper shape is described based on fundamental results from coupled local mode theory. A total fiber to waveguide insertion loss of 0.6 dB has been measured with the integration of the transformer, an improvement of 2.1 dB. The spatial alignment tolerance for 1 dB excess loss was measured to ±2.7μm, consistent with theory.

Multislit interpretation of cascaded fiber gratings

Abstract: The analogy between the transmission spectrum of a cascaded long-period grating (LPG) and the diffraction pattern of a multislit is presented. Both of them can be expressed with the exact same equation. Only the phase parameters that describe the interference are different. The spectrum of the cascaded LPG and/or a single LPG has been derived into a single analytic equation by diagonalizing the matrix that describes the system. The diagonalizing method that gives the analytic solution of the cascaded device is introduced in detail. Experimentally and theoretically, it is presented that the device composed of N equally spaced identical LPGs has a series of regularly spaced main-loss peaks enveloped by a slowly varying curve determined by the spectrum of the single LPG. Between the adjacent main peaks, there exists N-2 side peaks. With the multislit analogy, the complicated mode coupling in an LPG and/or cascaded LPGs can be explained with the interference among the beams coupled by the series of gratings from an undepleted core mode into a cladding mode. The recoupling back to the core mode does not need to be considered but embedded in the phases of the coupled beams. The analogy holds for any strength of gratings. The phase parameter is proved to be the eigenvalue of the system unit matrix composed of an LPG and the grating-free region between LPGs. The spectral properties of the cascaded LPG are analyzed in detail. As the diffraction pattern of the multislit is enveloped by the curve determined by the shape of a single slit, the interference of the cascaded LPG is enveloped by the curve determined by the specification of a single LPG. The fringe-enveloping curve is also derived in a closed analytic form. As the finesse of the multislit becomes high as the number of slits increases, the finesse of the cascaded LPG increases with the number of the LPGs. The free spectral range decreases with the separation between elements in both cases.

Enhanced Čerenkov second-harmonic generation in a planar nonlinear waveguide that reproduces a one-dimensional photonic bandgap structure

Abstract: Second-harmonic generation in the Cerenkov configuration is investigated under conditions for which the use of a linear grating fabricated on top of the waveguide reproduces a photonic bandgap structure. The fundamental mode of the guide at the fundamental frequency is tuned at the photonic band-edge resonance, thus producing great confinement and enhancement of the electromagnetic field inside the structure. The conversion efficiency achieved in both the forward and the backward directions is at least 1 order of magnitude greater than that of a conventional Cerenkov emission in a waveguide of the same length. An analysis of the tolerances of the grating period on the conversion efficiency is presented.

Transmission and reflection analysis of functional coupled cavity components

Abstract: This paper contributes to the ongoing discussion within the photonic crystal community by providing essential insight into the limiting conditions of the coupled cavity waveguiding mechanism. A theoretical and numerical description of coupled defects in PBG crystals is applied to quantify the conditions under which reflections occur within coupled cavity photonic crystal systems. We present an analysis of coupled cavity systems that form a straight and bent waveguide, a Y-shaped symmetric power splitter, and a waveguide incorporating two bends. The method is based on a weak interaction approach; the actual configuration of the defects (chain, lattice, bend, splitter, or anything else) enters the equations as a linear coupling between neighboring defects. The strength of this method is that many solutions of this system are known analytically, and that the band structure as well as the transmission and reflection response of the system can be determined.

Mode couplings in superstructure fiber Bragg gratings

Abstract: The spectral characteristics of superstructure fiber Bragg gratings are analyzed numerically based on the coupled mode theory, simultaneously taking into account the counterdirectional guided mode coupling, codirectional and counterdirectional claddings mode coupling The theoretical results were compared with experimental data and very good agreement was obtained

Rigorous derivation of coupled mode equations for short, high-intensity grating-coupled, co-propagating pulses

Abstract: We present a rigorous derivation of coupled mode equations for the grating-assisted coupling of short, co-propagating pulses. Since this is a copropagating geometry, the required grating period is much longer than the wavelength of the light and is typically 0.1–10 mm. This can exceed the spatial extent of the pulses substantially, and the validity of the usual assumptions that lead to coupled mode theory is therefore not guaranteed. Here we show rigorously that the standard nonlinear coupled mode equations can be used to describe these experiments and we give expressions for the parameters.

Acousto-optic modulation in magnetised diffusive semiconductors

Abstract: The modulation of an intense electromagnetic beam induced by the acousto-optic (AO) effect has been analysed in a transversely magnetised semiconductor-plasma medium. The effect of carrier diffusion on the threshold field and gain profile of the modulated wave has been extremely investigated using coupled mode theory. The origin of the AO interaction is assumed to lie in the induced nonlinear diffusion current density of the medium. By considering the modulation process as a four wave parametric interaction an expression for effective third-order AO susceptibility describing the phenomena has been deduced. The modulation is greatly modified by propagation characteristics such as dispersion and diffraction due to dielectric relaxation of the acoustic mode. The threshold pump field and the steady state growth rates are estimated from the effective third-order polarisation in the plasma medium. Analytical estimation reveals that in the presence of enhanced diffusion due to excess charge carriers the modulated beam can be effectively amplified in a dispersionless acoustic wave regime. The presence of an external dc magnetic field is found to be favourable for the onset of diffusion induced modulational amplification of the modulated wave in heavily doped regime.

Modeling of nonlinear pulse propagation in periodic and quasi-periodic binary long-period fiber gratings

Abstract: A generalized transfer-matrix method is used to model nonlinear pulse propagation in a binary long-period fiber grating (LPFG). Two interface matrices are used to describe power coupling at the heterointerfaces, as in the linear case. Nonlinear phase shifts and pulse dispersion through the two basic regions are modeled by coupled nonlinear Schrodinger equations. Based on the generalized transfer-matrix model, a local intensity-dependent detuning parameter is introduced with which we investigate the general conditions for complete switching. Nonlinear switching in a quasi-periodic Fibonacci LPFG is also studied, and it is shown that complete switching can be achieved in such a quasi-periodic grating.

Photonic-bandgap planar hollow waveguide

Abstract: A combination of a diffraction grating and a mirror integrates a hollow waveguide and a photonic-bandgap structure into a compact optical element, offering a simple new structure for various applications in nonlinear and ultrafast optics. The main features of transmission spectra observed in experiments performed with such waveguide structures are qualitatively interpreted in terms of the coupled-mode theory. Localization of light near the surface of a metal-coated grating in lowest-order TM modes in the created waveguide enhances effects related to the photonic-bandgap structure.

Gap Solitons in Nonlinear Polyatomic Chains

Abstract: Dynamics of gap solitons in various kinds of polyatomic chains are analyzed comprehensively. First, we consider a case that the periodic modulation of the anharmonic lattice parameters is small (shallow grating) and obtain coupled mode models between the forward and backward propagating waves at the Bragg wavenumber. Depending on the period of the chain N (>1) and on the nonlinearity, we derive four types of coupled mode equations. Moving localized solutions for gap solitons are obtained analytically. It is found that owing to the quadratic nonlinearity, static dc waves should be taken into account, which leads to the concept of “ dynamical rectification ”. The theoretical results for gap solitons are checked by numerical simulations. Secondly, we consider the case of large modulation of the lattice parameters (deep grating). We develop a theory of the nonlinearity-induced carrier-wave modulations of the lattice mode dynamics (Bloch wave) and derive the effective nonlinear Schrodinger equation. Numerical ...

Coupled-mode theory of waveguides with conducting interfaces

Abstract: In this paper, the coupling coefficient between two adjacent waveguides with conducting interfaces is calculated analytically. It is shown that the presence of interface charges, their density being controlled via a transverse voltage, leads to the control of the coupling. TO derive the coupling coefficient, the basic Coupled Mode Theory (CMT) with paraxial approximation is improved to include the effect of conducting interfaces. The analysis is performed independently for TE and TM polarizations. Several direct applications of this effect such as multiplexer, coupler and Mach/Zhender modulator/switch and programmable grating are introduced

A simple analysis of a multimode three optical slab waveguides directional coupler

Abstract: A new algorithm is presented to analyze three optical slab waveguide directional couplers to obtain the power transferred and field distribution through slabs. Using the effective index method, the coupler is treated as three couplers, each consisting of two slabs and the effect of the remainder waveguide is taken into account. The coupling coefficient between two slabs is derived by approximate coupled mode theory. The power transferred is evaluated by using a coupled wave differential equation system and Laplace transform. The transverse field distribution is calculated by using the power contained through each waveguide. Two coupled waveguides can be isolated by using an intermediate waveguide, which applies the isolation condition with higher values of the propagation constant than that of the original waveguides. A simple calculation of the intermediate waveguide is derived. The effect of coupler parameters on power transferred is applied. The obtained results are compared with other published data and these results can be used as design guide curves.

Design of guided-wave grating-assisted tunable filters for telecommunications systems

Abstract: In this paper, a number of critical aspects in the design and performance of guided-wave grating-assisted tunable filters for dense telecommunications systems are investigated by using a rigorous numerical approach. It is shown how highly narrow filters, with the potential to be tunable over a large number of channels (> 50), can be accurately designed by using well-selected apodization windows.

Wavelength tunability of laterally coupled semiconductor optical amplifier and semiconductor laser diode

Abstract: A novel tunable laser consisting of a laterally coupled semiconductor optical amplifier and a semiconductor laser is proposed. This structure exhibits a high degree of frequency selectivity, wide tunability with electrically controlled wavelength, and relatively high side-mode suppression achieved by the additional short electrode in the semiconductor laser. The proposed structure is simulated using the coupled mode theory and the modified transfer matrix method (TMM). This simulation yields a tuning range of up to 48 nm with a side-mode suppression ratio of approximately 35 dB.

Nonlinear Localized Modes in Photonic Crystals

Abstract: We give a brief overview of nonlinear localized modes in photonic crystals. We explain how photonic crystals can potentially be important in making small scale active devices which operate in an all optical way. Two models to approach nonlinear photonic crystals, the coupled mode theory and the discrete lattice theory using a Green's function, are explained.

Electro-Optic Intensity Modulator for Broadband Optical Communications

Abstract: An LiNbO 3 electro-optic grating modulator, composed of coplanar microstrip electrodes with periodic small fins and an airbridge structure, is presented. The small fins slow down the microwave propagating speed, while the effect of the airbridge is the reduction of the effective index. The modulator design has good performance in terms of wavelength selection and high-speed characteristics. The microwave and optical analysis are carried out using the three-dimensional finite difference method and coupled mode equations.

Wavelength Tunability of Laterally Coupled Semiconductor Optical Amplifier and Semiconductor Laser Diode : Optics and Quantum Electronics

Abstract: A novel tunable laser consisting of a laterally coupled semiconductor optical amplifier and a semiconductor laser is proposed. This structure exhibits a high degree of frequency selectivity, wide tunability with electrically controlled wavelength, and relatively high side-mode suppression achieved by the additional short electrode in the semiconductor laser. The proposed structure is simulated using the coupled mode theory and the modified transfer matrix method (TMM). This simulation yields a tuning range of up to 48 nm with a side-mode suppression ratio of approximately 35 dB.

Enhanced Cerenkov SHG in planar nonlinear waveguide reproducing a 1-D PBG

Abstract: Summary form only given. Guided infrared modes can be coupled with a radiation mode according to the Cerenkov configuration. The guided infrared modes creates a nonlinear polarization which has a faster phase velocity than that of a free wave at the harmonic frequency in the material, the usual case with normally dispersive materials. In this case, the harmonic radiation created by the nonlinear polarization radiates into the substrate at an angle that assures conservation of the k-vector component parallel to the interface, automatically providing phase-matching. Enhancement in the nonlinear conversion efficiency can be increased if a "longitudinal" field confinement is achieved. This can be obtained by a suitable grating design on a waveguide which "simulates" a 1-D photonic band gap structure (PBG). The advantages that can derive from the use of a medium with periodic linear properties have been recently considered in the study of the photonic band gap (PBG) structures. Not only is exact phase-matching achievable, but a further enhancement of the conversion efficiency is possible by tuning the fundamental field near the photonic band edge, at a transmission resonance. There, the waves experience confinement, which translates into a large density of modes (DOM), and the field is enhanced inside the PBG structure leading to enhanced available nonlinear gain. In the present work, the Cerenkov SHG process enhanced by a PBG waveguide is investigated through a coupled mode theory, which permits both linear and nonlinear mode coupling. Our theory shows how the band edge tuning conditions and the phase-matching requirements can be satisfied at the same time.