Analysis of TE and TM modes in DFB lasers by two-dimensional theory

The two-dimensional theory of a distributed feedback (DFB) laser (which was previously presented and applied to the analysis of the laser threshold conditions for the transverse-electric (TE) mode in a simple three-layer waveguide structure) is developed to treat both TE and transverse-magnetic (TM) modes in a four-layer waveguide structure with a thin grating layer, which more closely reflects actual DFB laser structure. The differences between TE and TM modes for the dispersion relations and the laser threshold conditions are clarified. The effects of the waveguide structure (including grating layer thickness, refractive indexes of layers, coupling constant, and corrugation period) on the threshold gains and the gain differences between the two longitudinal modes on both sides of the Bragg frequencies are studied in detail for both TE and TM modes     

Analysis of radiation mode effects on oscillating properties of DFBlasers

The effects of radiation mode on the oscillating properties of distributed feedback (DFB) lasers with second-order corrugations are analyzed for designing a new type of DFB laser. A formulation based on the transfer matrix technique is applied to calculating Streifer's ζ-terms added to the coupled-wave equations. These terms represent the effects of radiation and evanescent modes. This formulation greatly simplifies the analysis of distributed resonance along multilayered waveguide structures with arbitrary-shaped second-order corrugations. The effects of vertical resonance are also incorporated into the formulation. Various types of DFB lasers with phase-shifted second-order corrugations are analyzed using this method. It is found that the phase shift and the blaze of the corrugations greatly affect the longitudinal mode selectivity. A new phase-shift DFB laser structure with two complementary blazing regions connected at the shift is proposed. It is demonstrated that this structure has small radiation loss resulting in low-threshold performance despite employing second-order corrugations     

Polarization optical bistability induced by TM mode injection to aDFB laser with complex coupled coefficient

A novel method is proposed to produce an optical bistability by using a dynamically stable complex coupled DFB (CC-DFB) laser with TM mode injection. In this paper polarization optical bistabilities are analyzed in detail using coupled mode equations and rate equations for the CC-DFB lasers considering the longitudinal hole burning and carrier dependent complex coupling coefficients. Several parameters reflecting the physical features of a complex coupled DFB laser are discussed. It is shown that for a CC-DFB laser the polarization bistability induced by the TM mode injection is much stronger for the antiphase of complex coupling than that for the in-phase. In addition, the influences of initial coupling condition for gain grating structure on the optical bistability are also investigated considering both cases of the antiphase and in-phase     

Improved coupled mode analysis of corrugated waveguides and lasers

Corrugated waveguides and lasers in resonant and non-resonant situations are analyzed by an improved coupled mode theory based on a set of the coupled mode equations for guided modes and radiation continuum. The distributed feedback (DFB) coefficient and the radiation loss coefficient are given in closed forms. The formulation can be applicable to arbitrarily shaped gratings and multilayer waveguide structures. The accuracy of the theory is examined by comparing it with Tamir's exact calculation for a nonresonant situation and also with Streifer's one for a DFB structure. Reasonable accuracy is obtained by the proper choice of the unperturbed waveguide parameter. The dependence of the two coefficients on the grating depth, the grating period, the guide layer thickness, and the refractive index difference between core and cladding layers is obtained for all Bragg orders up to the fourth, and for four typical grating shapes, namely, for rectangular, sinusoidal, symmetric triangular, and sawtooth gratings. Both the threshold gain of DFB lasers utilizing higher order Bragg reflection and the output coupling efficiency of grating beam couplers are also calculated for these parameters. A new multilayer structure for controlling the radiation loss is proposed and analyzed. This structure is suitable for the suppression of the radiation loss in DBR reflectors as well as for the improvement of the output coupling efficiency in grating beam couplers.     

Analysis, design, and fabrication of GaAlAs/GaAs DFB lasers withmodulated stripe width structure for complete single longitudinal modeoscillation

Complete single longitudinal mode operation of a GaAlAs/GaAs distributed-feedback (DFB) laser has been accomplished by making use of a modulated stripe width (MSW) structure, which is characterized by its simple configuration and versatility, compared with equivalent schemes. The coupled-mode equations are transformed into the characteristic equation for the DFB laser with arbitrary index modulation which has a form convenient for numerical analysis. Two specific examples of the stripe width modulation, namely, stepped and symmetric linear modulations, are then analyzed by making use of the equations. Optimum design of the stripe shape is subsequently carried out by assuming a particular waveguide structure. A GaAlAs/GaAs DFB laser having stepped-width modulation was fabricated by using a double-channel planar buried heterostructure     

Analysis of distributed feedback semiconductor lasers bytwo-dimensional theory

The two-dimensional theory of distributed feedback (DFB) lasers to take account of the planar waveguide structure which was presented in a previous paper (see ibid., vol.26, no.3, p.467-72, 1990) is applied to the analysis of the DFB laser threshold conditions with respect to the effects of the waveguide structure and the facet reflection. The asymmetric properties of the transverse functions of the coupled modes with respect to the Bragg frequency in the dispersion relations are found to be enhanced by the asymmetric index waveguide structure and by the asymmetric facet reflectivity. Therefore, the resulting confinement factor differences in the grating layer between the two adjacent lasing modes on both sides of the Bragg frequency give large threshold gain differences     

Analysis of TE/TM mode selectivity in DFB lasers with a phase-shift region

In phase-shift distributed feedback (DFB) lasers, there is limited suppression of the TM mode despite extremely large submode suppressibility. The TE/TM mode selectivity of a DFB laser structure with a nonuniform waveguide region as the phase shifter is analyzed. Calculations of the threshold gain difference between the TE and TM modes are performed using the effective index method and the coupled-wave theory. It is found that the TM mode suppressibility can be doubled by optimizing the dimensions of the phase-shift region. This structure overcomes the TM mode problem.<>     

Propagation losses at 10.6 µm in hollow-core rectangular waveguides for distributed feedback applications

Eigenvalue equations for the TE and TM modes propagating in a rectangular hollow-core waveguide are derived. The solutions to the eigenvalue equation are used to determine the theoretical losses for the lowest even order mode propagating in a rectangular waveguide whose cross-sectional dimensions are suitable for distributed feedback at 10.6 μm. Waveguide materials such as gold, BeO, glass, and germanium are analyzed. It is shown that by using gold for the top and bottom waveguide regions with BeO for the side walls, loss as small as 1.19 dB/m can be obtained with a cross-sectional dimension 0.1 mm × 0.8 mm. Using external discharge pumping the authors have created an active medium with gain in excess of 17 dB/m in a 0.08 mm hollow-core slab waveguide. Loss calculations indicate the feasibility of succesfully constructing a conventional electric discharge waveguide laser for DFB operation at 10.6 μm. Using a BeO-glass combination with cross-sectional dimensions 0.1 mm × 1 mm, a loss of 2.75 dB/m is calculated. It is also shown that a TEA waveguide laser could be built using the side walls for electrodes. This would result in a DFB waveguide laser at 10.6 μm with 2.75 dB/m loss and a cross section 0.1 mm × 2 mm. These results indicate that with good fabrication techniques and with the application of distributed feedback, it may be possible to construct a CO2waveguide laser with a significantly reduced waveguide cross section.     

Two-dimensional theory of distributed feedback semiconductor lasers

A theory which takes account of the two dimensional waveguide structure of the distributed-feedback (DFB) laser is presented. Laser threshold conditions in the case of no external reflections are calculated for a three-layer model in which one cladding layer has a periodically changing dielectric constant. In contrast with the coupled-wave theory the threshold conditions are found to be asymmetric with respect to the Bragg frequency. The longitudinal mode which lies below and nearest the Bragg frequency has the lowest threshold gain. The difference between the threshold gains of the two adjacent longitudinal modes straddling the Bragg frequency has a maximum as a function of the coupled strength of the grating     

Spatial hole burning effects in distributed feedback lasers

The effects of spatial hole burning in a steady-state distributed feedback (DFB) laser are examined by numerically solving the coupled mode equations that describe the system. An approximate solution for the gain above threshold is derived and compared to the exact solution. It is shown that the self-induced grating that arises due to spatial hole burning significantly reduces the mode discrimination of index-coupled DFB lasers. This makes it difficult for these lasers to maintain single-longitudinal-mode behavior above threshold. However, it is found in addition that bulk-modulated (gain-coupled) DFB lasers do not lose their mode selectivity above threshold, indicating that these lasers may be better choices for narrow-linewidth operation     

Laterally coupled strained MQW ridge waveguide distributed-feedbacklaser diode fabricated by wet-dry hybrid etching process

The fabrication and the characteristics of the laterally coupled GaInAsP-InP quantum-well ridge waveguide distributed-feedback (DFB) lasers are presented. The electron beam (EB) lithography and the wet and dry hybrid etching technique have been used to fabricate the deep grating structures for the DFB lasers on and beside the sidewalls of the narrow ridge waveguide. The threshold current was 18.5 mA at 20°C, and the sidemode suppression ratios (SMSRs) were ensured to be more than 40 dB for as-cleaved devices with various cavity lengths. The continuous-wave output powers of over 15 mW/facet have been observed, while transverse and longitudinal modes have remained in single mode at this output level     

Selection and modulation of high-order transverse modes invertical-cavity surface-emitting lasers

A theoretical proposal to double the pulse repetition frequency by alternating current modulation of two orthogonal high-order transverse modes of a vertical-cavity surface-emitting laser (VCSEL) is presented. Static and dynamic characteristics of weakly index-guided VCSELs in a multitransverse mode regime are analyzed. An efficient model that takes into account all the transverse modes supported by the waveguide is developed. Mode partition noise in a current modulated VCSEL is studied, taking into account the azimuthal degree of freedom. Different transverse modes can be excited with a probability that is numerically calculated. For high injection currents, modes that are not favored in the steady state can be excited with higher probability. The excitation probability is similar for azimuthally orthogonal modes. This symmetry can be broken by selecting a particular high-order transverse mode by using azimuthal-dependent current profiling. This selection can be achieved over current ranges as wide as ten times the threshold current. Current modulation of this transverse mode is then analyzed. Alternate current modulation of two orthogonal high-order transverse modes is also studied. Alternating modulation of these modes can double the pulse repetition frequency obtained by modulating just one high-order transverse mode, without increasing injected current density levels. This current-induced spatial switching leads to high-frequency beam steering in the laser azimuthal direction     

部分填充双各向异性介质的圆柱形波导中混合模特性研究

本文首先引入一种新型双各向异性波导结构;给出了这类复杂波导中横向场分量的纵向场表示式以及纵向场分量满足的一组耦合模方程;重点分析了具有不同电磁参数的双各向异性介质中心加载时,金属圆柱形波导中混合模的分裂效应,色散特性和场分布。结果表明,由于多个电磁参数的引入,双各向异性波导具有一般各向异性和双各向同性波导结构所没有的新奇特性。     

Simplified coupled-wave equations for cylindrical waves in circulargrating planar waveguides

A simplified form of the coupled-wave equations for cylindrical waves in circular grating distributed feedback (DFB) planar waveguides is presented. The coupling coefficients can be expressed as the product of a cylindrical-wave factor and the corresponding coupling coefficient between planar waveguide modes. Under the large radius approximation (βr≫1), which covers most practical cases, the interpolarization coupling between TE and TM cylindrical waves approaches zero. The coupled-wave equations can then be reduced to two independent families, one for TE- and the other for TM-cylindrical waves     

Modeling of distributed feedback semiconductor lasers withaxially-varying parameters

A numerical model that is capable of predicting important laser characteristics such as the threshold gain and the gain margin between the main and side modes for a distributed-feedback (DFB) semiconductor laser of arbitrary complexity is described. The method consists of solving the coupled-mode equations with axially varying parameters iteratively until the boundary conditions at the two facets are satisfied. The numerical model is applied to two DFB laser structures. In the case of a multiple-phase-shift DFB laser the results show that such devices can have a more uniform axial distribution than that of a conventional quarter-wave-shifted DFB laser while maintaining sufficient gain margin between the main and side modes. In the case of a dual-pitch DFB laser it is shown that the incorporation of a slightly different grating period (~0.1%) over a small section can provide a gain margin that is comparable to that achieved in quarter-wave-shifted DFB lasers     

Numerical simulation of a pulsed laser pumped distributed-feedback waveguided dye laser by coupled-wave theory

A dynamic model of pulsed laser pumped distributed-feedback (DFB) waveguided dye laser based on a coupled-wave theory is described. Due to the periodical distribution of the intensities of pump source and stimulated emission along the waveguide, the rate equations of the population densities are turned into the equations of the Fourier coefficients. Coupled-wave equations of optical fields are used to simulate the laser oscillation. Besides the temporal evolution of the output intensity, the spectra can also be obtained by the Fourier transform of the optical fields. Two different configurations of the waveguided dye laser, prefabricated DFB (mainly index coupling), first- and second-order holographic DFB (dynamic gain-coupling), are considered in the model. The simulation shows that: 1) the temporal waveforms of the holographic DFB consist of sharp spikes; 2) the broadened spectral widths resulted from the possible nonuniformities in propagation constant or grating period are less than 50 pm except for the second-order holographic DFB; and 3) strong parasitic oscillations can be observed in the second-order holographic DFB with terminal reflection. These results and the comparisons of some of them to the experiments are reported.     

Optimized overmoded TE01-to-TM11 mode converters for high-power millimeter wave applications at 70 and 140 GHz

Mode coupling in bent, oversized, smooth-wall circular waveguides was studied by means of numerical integration of coupled-mode differential equations in order to optimize high-power TE01-to-TM11 mode transducers at 70 GHz and 140 GHz. Such mode transformers are used in the mode conversion sequence TEOn to TE01 to TM11 to HE11 for generating the almost perfectly linearly polarized Gaussian-like HE11 mode from circular electric TEOn gyrotron modes. This quasi-optical HE11 hybrid mode is in many respects ideal for electron cyclotron resonance heating (ECRH) of magnetically confined plasmas in thermonuclear fusion research and for other technical applications. Curvature and ellipticity coupling as well as ohmic attenuation of 6 coupled modes (TE01, TM11, TE11, TE12, TE21, TM21) are included in the coupling matrices. Integral expressions were used for deriving the coupling coefficients for arbitrary modes in bent, smooth-wall waveguide. Lowest level of unwanted spurious modes together with highest transmission efficiency (shortest arc length) is achieved with sinusoidal curvature distribution instaed of constant curvature. The calculated conversion efficiencies of 98.0% at 70 GHz and 95.2% at 140 GHz (interior waveguide diameter D=27.8 mm for 200 kW transmission lines) are in excellent agreement with the measured values of (97.6±0.4)% and (95±1)%, respectively.     

Determination of the Photon Lifetime for DFB Lasers

The photon lifetime of the distributed feedback (DFB) laser is obtained through the relations between cavity photon flux and power associated with the electric fields. The solution of the coupled wave equations provides the propagation parameters and the amplitude gain coefficients for which numerical and approximate solutions are obtained. Using the photon lifetime the threshold condition of the DFB laser within a fixed mirror cavity is determined. The evaluation of both Fabry-Peacuterot and DFB modes is shown by the calculation, using two different photon lifetimes, of all modal concentrations, with the Fermi energy as the independent variable. The current density is also evaluated using the Fermi energy and threshold current is identified as the value when the Fermi energy clamps     

Mutual injection locking of a fibre laser and a DFB semiconductorlaser

A fibre laser and a distributed feedback (DFB) laser are coupled through a 50:50 fibre coupler in such a way that the two lasers mutually injection-lock. The DFB laser forces the fibre laser to oscillate in a nearly single longitudinal mode. At the same time, the linewidth of the DFB laser is reduced to below 1.5 MHz because of mutual injection-loading