Complex Coupling Coefficient Research Articles

Improved implementation of optical space domain reflectometry for characterizing the complex coupling coefficient of strong fiber Bragg gratings

Optical space domain reflectometry (OSDR) implemented with a novel interferometric characterization method is demonstrated to be an accurate technique for characterizing the complex coupling coefficient of strong fiber Bragg gratings. A theoretical model is also presented, incorporating the effect of the heat perturbation shape, which accurately predicts the measurement behavior. It is shown that the measurement accuracy and spatial resolution are dramatically improved by removing the effect of the heat perturbation shape on the reconstructed profile using a deconvolution technique. The improvement in accuracy is illustrated by the excellent agreement between a weak grating reconstructed with OSDR and the same grating reconstructed with optical frequency domain reflectometry and a layer-peeling method. Reconstruction of a strong grating with an integrated coupling coefficient, qL=8.14, demonstrates the utility of this technique.

The use of real or complex coupling coefficients for lossy piezoelectric materials

Two competing approaches for calculating coupling coefficients for lossy piezoelectric materials, one producing a real result and the other a complex result, are compared and analyzed. It is found that the complex coupling coefficient suffers from mathematical difficulties, which the real coupling coefficient does not exhibit. Moreover, it is pointed out that a prediction made by the complex coupling coefficient theory conflicts with experiment while the corresponding real coupling coefficient theory prediction does not. When a coupling coefficient of interest has been computed from the real coupling coefficient theory using piezoelectric equations having intensive independent variables, the resulting expression has the same algebraic form as the corresponding static coupling coefficient formula. Moreover, only the real parts of the piezoelectric, elastic, and dielectric material properties appear.

Diagonal dominance of damping and the decoupling approximation in linear vibratory systems

A common approximation in the analysis of non-classically damped systems is to ignore the off-diagonal elements of the modal damping matrix. This procedure is termed the decoupling approximation. It is generally believed that errors due to the decoupling approximation should be negligible if the modal damping matrix is diagonally dominant. In addition, the errors are expected to decrease as the modal damping matrix becomes more diagonally dominant. It is shown numerically in this paper that, over a finite range, errors due to the decoupling approximation can increase monotonically at any specified rate while the modal damping matrix becomes more diagonally dominant with its off-diagonal elements decreasing continuously in magnitude. An explanation for these unexpected drifts of decoupling errors is provided with the use of complex coupling coefficients. Small off-diagonal elements in the modal damping matrix are not sufficient to ensure small errors due to the decoupling approximation. Any error-criterion based solely upon the diagonal dominance of the modal damping matrix would not be accurate.

TWO-WAY COUPLED-MODE SOLUTIONS IN THE COMPLEX HORIZONTAL WAVENUMBER PLANE

A coupled-mode formalism based on complex Airy layer mode solutions is presented. It is an extension into the complex horizontal wavenumber plane of the companion article [Stotts, J. Acoust. Soc. Am.111 (2002) 1623–1643], referred to here as the real horizontal wavenumber version, which accounted for general ocean environments but was restricted to normal modes on the real horizontal wavenumber axis. A formulation of the expressions for both trapped and continuum complex coupling coefficients is developed to dramatically reduce computer storage requirements and to make the calculation more practical. The motivation of this work is to demonstrate the numerical implementation of the derivations and to apply the method to an example benchmark. Differences from the real horizontal wavenumber formalism are highlighted. The coupled equations are solved using the Lanczos method [Knobles, J. Acoust. Soc. Am.96 (1994) 1741–1747]. Comparisons of the coupled-mode solution to a parabolic equation solution for the ONR shelf break benchmark validate the results.

Rapid characterization of the ultraviolet induced fiber Bragg grating complex coupling coefficient as a function of irradiance and exposure time

We report the application of optical frequency domain reflectometry and a discrete-layer-peeling inverse scattering algorithm to the spatial characterization of the UV induced complex coupling coefficient during fiber Bragg grating growth. The fiber grating is rapidly characterized using this technique to give irradiance dependent growth as a function of exposure time, thereby providing the complete characterization of the coupling coefficient in the form of a "growth surface," which is related to the fiber's photosensitivity. We compare measurements of fiber Bragg grating growth in SMF-28 when exposed to continuous wave 244 nm irradiation from 0 to 90 W cm(-2) for exposure times up to 3230 s with a selection of other fibers including high germanium concentration fiber and erbium doped fiber.

A Comprehensive Analysis of Quadrature Signal Synthesis in Cross-Coupled RF VCOs

This paper presents a linear model for cross-coupled quadrature voltage-controlled oscillators (QVCOs) together with a simple generalized proof for the condition of quadrature oscillation. The analysis provides insight into the oscillation mechanism and the previously observed frequency shift when the magnitude and phase of the coupling signal are deliberately modified by a complex coupling coefficient Kc. It is demonstrated that the steady-state oscillation condition GMRp=1 holds only if GM is replaced by an effective large-signal transconductance GMeff that is a function of the coupling transconductance G Mc and the imaginary part of Kc. Closed-form expressions for the phase imbalance and amplitude error in presence of mismatch between the LC tank circuits as well as device transconductances are also derived. We introduce the new concept of quadrature resistance/quadrature conductance (Rquad/Gquad), an incremental element synthesized by the coupling transistors, and show that its magnitude is responsible for the frequency shift and quadrature oscillation. A one-port model of the QVCO is studied and the loading effect of Gquad on the tank is examined. Particularly, it is shown that Gquad degrades the open-loop quality factor and worsens the phase noise. The analysis in this paper can be applied to most QVCO topologies presented in the literature

Detection of non-paraxial optical fields by optical fiber tip probes

The optical response of a single-mode uncoated fiber tip to a 3D polarized field, including longitudinal components, is investigated. The 3D field is produced by an opportune superposition of TE and TM plane waves. The contribution of the different field components to the detected signals was discriminated by integrating the scanning probe microscope with a multi-heterodyne detection technique. A simple coupling model for the tip is introduced. The longitudinal field component was assumed to couple to the transverse fiber modes through complex coupling coefficients. Coupling coefficients were obtained by fitting the parameters of the model to the experimental data. These results demonstrate that the longitudinal components of the field are coupled by this probe with an efficiency approximately equal to that of the transverse polarization components.

Gain clamping in semiconductor optical amplifiers with second-order index-coupled DFB grating

A systematic investigation of gain-clamped semiconductor optical amplifiers (GC-SOAs) based on the second-order index-coupled DFB gratings is carried out by way of simulation. In particular, we focus on the main effects of the radiation loss caused by the first-order diffraction of the gratings on the amplifier performance. The magnitude of the total complex coupling coefficient is the main factor to determine the level of gain clamping. We demonstrate that a high-performance GC-SOA can be realized by using purely loss-coupled second-order DFB gratings with more relaxed tolerance on grating strength and period. It is shown that, in the presence of weak reflection-related coupling, the parasitic radiation loss associated with the second-order grating always helps to expand the linear amplification region and to reduce the longitudinal spatial hole burning along the cavity. Further, we demonstrate through comparison that the GC-SOAs have higher saturation power and much shorter carrier lifetime than the conventional SOAs. An improved design by longitudinal variation of the grating duty cycle is proposed such that the noise performance of the amplifier can be enhanced without much sacrifice on the linear amplification regime.

Dynamics of coupled self-pulsating semiconductor lasers

We introduce and analyze a model for the dynamics of two coupled self-pulsating semiconductor lasers. We investigate the role of the complex coupling coefficient in the static and dynamic properties of the device. We find conditions for the emergence of coherent laser pulses, in which the two lasers display synchronous coherent self-pulsations (self-pulsating super modes). Nonlinear dynamics and two different routes to chaos are also individuated and discussed.

Spectral—threshold characteristics of a distributed-feedback laser with the sinusoidal modulation of coupling coefficient

The spectral-threshold characteristics of a distributed-feedback (DFB) laser with the sinusoidal modulation of the complex coupling coefficient are studied within the framework of the linear theory of coupled modes. The analysis is performed in a broad range of amplitudes of the coupling coefficient for phase, amplitude, and amplitude — phase gratings providing distributed feedback in the active medium of the laser. It is shown that the eigenmode spectrum of the DFB laser on a phase grating with a large-scale modulation of the coupling coefficient is similar to the eigenmode spectrum of the DFB laser on an amplitude grating with a constant coupling coefficient. The DFB laser under study is promising for obtaining stable single-frequency lasing and can find applications in devices in integrated optics.

The geometry underlying photorefractive two-beam coupling

We cast the theory of photorefractive two-beam coupling in geometrical terms and derive a closed form solution to the problem of two-beam coupling with time-dependent fields that evolve in such a way as to give rise to a stationary index of a refraction grating. The geometrical approach treats the optical field amplitudes in terms of vectors whose spatial dynamics is governed by a Hamiltonian. A density matrix formed from the optical fieldvectors is shown to have particularly simple dynamics. We derive a formula for a transfer matrix that accommodates complex coupling coefficients and that can be used to calculate the instantaneous output of the photorefractive grating given any input. The geometrical picture of photorefractive two-beam coupling is pedagogically developed through several specific examples.

Local coupling-coefficient characterization in fiber Bragg gratings.

We propose a new method for characterizing the local parameters of fiber Bragg gratings. This method combines measurement of the complex impulse response by optical low-coherence reflectometry and reconstruction of the complex coupling coefficient by layer peeling. Application of the method to a nonhomogeneous grating shows that the local coupling coefficient can be precisely determined with an axial resolution below 20 microm and a maximum error of less than 5% for amplitude and phase, respectively.

Couplage de deux microlasers dans une galette de vanadate d'yttrium dopé au néodyme

In this paper we study the coupling of two microlasers. This system displays instabilities in the intermediate regime between locked and independent states of emission. The frequency and the amplitude of these instabilities exhibit a strong asymmetry between positive and negative detunings when the pumping rates of the two lasers are different These results can be accounted for by coupled equations with complex coupling coefficient. We describe a method to measure this coefficient and to study its dependence versus the separation between lasers.

Modeling coupled microchip lasers requires complex coupling coefficients

Two coupled microlasers display instabilities in the intermediate regime between locked and independent states of oscillations. The frequencies and amplitudes of these oscillations exhibit strongly asymmetric behaviors for positive and negative detunings. Our experimental findings cannot be accounted for by coupled-equation models with purely real or imaginary coupling coefficients but are well described if complex coupling coefficients are introduced. We describe a method to determine experimentally the coupling coefficient and study its dependence as a function of the separation between lasers.

Effect of excess quantum noise on output power in distributed feedback lasers

The effect of the mode non-orthogonality on the mean output of a distributed feedback laser is analysed taking saturation effects fully into account. A stochastic approach based on the Fokker—Planck equation is used. Numerical results for a DFB laser with non-vanishing end reflectivity and a complex coupling coefficient are obtained. Particularly, it has been shown that for small coupling strength because of strong field non-uniformity, the light emission is dominated by the excess spontaneous emission and the significant mean output power level can be observed.

Reconstruction of fiber grating refractive-index profiles from complex Bragg reflection spectra

Reconstruction of the refractive-index profiles of fiber gratings from their complex Bragg reflection spectra is experimentally demonstrated. The amplitude and phase of the complex reflection spectrum were measured with a balanced Michelson interferometer. By integrating the coupled-mode equations, we built the relationship between the complex coupling coefficient and the complex reflection spectrum as an iterative algorithm for reconstructing the index profile. This method is expected to be useful for reconstructing the index profiles of fiber gratings with any apodization, chirp, or dc structures. An apodized chirped grating and a uniform grating with a depression of index modulation were used to demonstrate the technique.

Measurement of the modulus and phase of the linear coupling coefficient by analysis of the transverse beam profile

We study the dynamics of transverse oscillations near the linear coupling resonance excited by a pair of skew quadrupoles at the Laborat\'orio Nacional de Luz S\'{\i}ncrotron UVX electron storage ring through the analysis of the beam profile. Transverse coherent oscillations were excited with a fast kicker and the profile of the oscillating beam was observed by focusing visible synchrotron radiation from a bending magnet onto a fast charge-coupled device camera. Using a single resonance approximation, we calculated the border of the time-averaged transverse beam profile as a function of the complex coupling coefficient $\ensuremath{\kappa}$, which characterizes the distribution of coupling fields along the storage ring. A least-squares fit of the calculated beam profile border to the experimentally obtained isointensity contours provided a new method to determine both the modulus and the phase of $\ensuremath{\kappa}$. The values obtained for the modulus are in good agreement with those from the conventional normal mode tune separation technique, and the values obtained for the phase of $\ensuremath{\kappa}$ agree with calculations based on the model lattice and the known skew quadrupole distribution.

Non-reciprocal emission with high single-mode operation in multisection complex-coupled distributed feedback cavities

We investigate theoretically the threshold properties of complex-coupled distributed feedback (CC-DFB) lasers in which the index and gain gratings are purposely out of phase by π/2. Such a phase difference between the two coupling constants is liable to produce a non-reciprocity in emission and leads to the same modal selectivity as in purely gain-coupled or, on the contrary, purely-index-coupled Bragg reflectors. Furthermore, by distributing non-uniformly the whole complex coupling coefficient and by varying the ratio of gain and index coupling along the cavity, the threshold properties of these particular devices can be strongly modified with respect to conventional complex-coupled structures: single-mode operation can be notably enhanced while maintaining a low threshold gain, the intra-cavity field profile of the lasing mode (which is related to spatial hole burning) is found to be well affected and the ratio of output optical powers emitted by each facet (the non-reciprocity) can be chosen with almost no effects on the threshold net gain and gain margin of the device.

Influence of air loading and dissipation on plate transducer amplitude

Air-loaded plate transducers driven piezoelectrically are used in a variety of modern applications such as optical modulators, microactuators, and certain types of resonant acoustic sensors and frequency control elements. For these uses it is necessary or desirable to know the maximum mechanical amplitudes obtainable, and how these are limited by the various internal dissipation mechanisms and the loading of the ambient fluid. Internal conduction, dielectric, and acoustic viscosity losses are incorporated, and, by introducing of a complex piezoelectric coupling coefficient, changes in critical frequencies and amplitudes are traced with variations in type and size of dissipation mechanism, and the ambient loading. The analysis also shows that the unavoidable interfacial stresses existing between the crystal plate surfaces and the exciting electrodes can become exceedingly large at the frequencies commonly used for frequency control of cellular communications. Because the various devices covered usually form components of an overall circuit design, a number of alternative equivalent electrical circuits are also provided whereby the physics of these structures may be rendered and readily interpreted in network terms.

Diode-laser-to-waveguide butt coupling: extremely short external cavity

An ultrashort composite cavity (</=10 mum in length) laser diode was used to verify experimentally a butt-coupling model that considers interference inside the external cavity and its influence on laser operation. The fine structure of an experimental coupled power versus laser-to-fiber separation curve is explained when a complex coupling coefficient is introduced into the model. The importance of the phase term of this coefficient and regions of validity of the modified butt-coupling model are also discussed.