Analysis of optical fiber directional coupling based on theHE11 modes. II. The nonidentical-core case

For pt.I see ibid., vol.8, no.6, p.823-31 (1990). The analysis of the directional coupling between two single-mode optical fiber cores based on the exact HE₁₁ modes is extended to the cases involving nonidentical fibers. The coupled-mode theory in the vectorial form is used, and analytical expressions for the coupling coefficients and the butt coupling coefficient appearing in the conventional and the coupled-mode equations are presented. The accuracy of the conventional and new theories as applied to the two-core system is examined by comparing the coupled-mode predictions with the exact numerical values for different core-radius ratios, waveguiding strengths, polarization states, and core separations. It is shown that the errors in the coupled-mode theories increase as the two cores become more dissimilar. As long as the dissimilarity between the cores is kept small, the coupled-mode calculations using the HE₁₁ modes in predicting the coupling strength can be of satisfactory accuracy even when the individual guides are not weakly guiding ones. It is found that the new theory may give relatively large errors in the touching-core case with distinctly different core radii     

Random coupling theory of single-mode optical fibers

A random coupling theory is developed for analyzing the propagation characteristics of the polarization state of light in single-mode optical fibers (including conventional and polarization maintaining fibers) under random disturbances. The basic idea is that the disturbances that a fiber suffers in practice continuously change with time and space, so time-varying coupling will occur along the fiber between two linearly polarized modes HE₁₁ that may propagate in the fiber. A coupled-mode equation of single-mode fibers under random disturbances is derived and solved rigorously with few assumptions. A random coupled-mode equation is derived considering time and space variation. Analytic solutions are obtained and used for analyzing the effect of random birefringence, polarization dispersion, polarization fluctuation, and evolution of the degree of polarization in single-mode fibers and for characterizing fiber properties     

Propagation of lower-order modes in a radially anisotropiccylindrical waveguide with liquid crystal cladding

The propagation behavior of the four lower-order modes, HE₁₁ , TE₀₁, TM₀₁, and HE₂₁, in a radially anisotropic cylindrical waveguide with liquid crystal cladding is studied both theoretically and experimentally. The cylindrical waveguide is a doubly-clad fiber with an isotropic core and inner cladding and a radially anisotropic outer cladding made of nematic liquid crystal. Theoretically, the propagation and decay constants for the TE₀₁ and TM₀₁ modes are obtained by solving the wave equations exactly, while those for the HE₁₁ and HE ₂₁ modes are derived using perturbation techniques under the weakly guiding approximation. It is predicted that in such a structure the guided TE₀₁ mode can be separated from the leaky HE₁₁, TM₀₁, and HE₂₁ modes. The theoretical results show good agreement with the experimental observations for a 3 cm long fiber cell with a 5 μm inner cladding radius     

Experimental coupling efficiency of shaping mirrors matching a168-GHz gyrotron output wave to the HE11 mode

This paper presents an experiment in which the phase- and amplitude-flattened output of a 168 GHz gyrotron was converted into the HE₁₁ mode, the basic transmitting mode in corrugated waveguides, by means of an external matching box (MBOX) comprising two curved-surface mirrors. In estimating the coupling efficiency between the gyrotron output wave and the HE₁₁ mode, an improved method was proposed in which the reconstructed phase of the gyrotron output wave at four distances are averaged. From the phase reconstruction with averaging, it was found that 76% of the gyrotron output coupled into the HE₁₁ mode in a corrugated waveguide, while a coupling efficiency of 85% was calculated for an ideal gyrotron output. A detailed discussion on the MBOX performance as well as the accuracy of phase reconstruction shows that this low coupling efficiency is due to the nonideality of the actual gyrotron output and that designing the MBOX mirrors based on the actual measurement at the gyrotron window can improve the coupling efficiency     

Investigation of mode coupling due to ohmic wall losses inovermoded uniform and varying-radius circular waveguides by the methodof cross sections

The effect of ohmic wall losses on mode coupling in overmoded varying-radius circular waveguides is investigated. Mode coupling and multimode propagation in uniform lossy-wall circular waveguides is also discussed. The expressions for the coupling coefficients are given by line integrals of the power-normalized fields of the normal modes along the boundary of the waveguide cross section. Numerical results are presented for the case of propagation of an HE₁₁-like mode excitation in a uniform smooth lossy-wall circular waveguide     

Coupling Between Dissimilar Waveguides

Investigators have used coupled-mode theory to analyze the coupling between identical waveguides; in such cases the coupling coefficients are found to be identical. If the waveguides differ, the coupling coefficients are asymmetrical and difficult to evaluate by strictly theoretical methods. An alternate approach to this case is considered in the present work. A pair of coupled-mode equations is first developed from a consideration of the permissible fields within the device. This clarifies the relationship between the coupled-mode theory and the more general classical electromagnetic theory by giving a careful definition of the coupled and the normal modes of a coupled structure. It is shown that the coupled-mode equations are an exact representation of the waveguide fields, although for engineering purposes it is often convenient to use approximate values of the coefficients of these equations. The mutual coupling coefficients are obtained from a two transmission-line model of the structure, with the actual coupling mechanism represented by a mutual impedance common to the two lines. For dissimilar lines, the ratio of the coupling coefficients is found to be equat to the ratio of the characteristic impedances. For the cases considered, this is the same as the ratio of the propagation constants of the uncoupled lines, which permits the coupling coefficients to be determined from relatively simple measurements. The adequacy of the theory has been confirmed by a series of experiments.     

Improved coupled-mode theory of anisotropic optical waveguides

A coupled-mode equation for anisotropic waveguide systems of arbitrary cross section and general dielectric distribution is derived. Numerical results comparing the exact calculations to those of the method of Hardy et al. (Opt. Lett., vol.11, 742-4, 1986) show that the same accuracy can be obtained not only for TE, but also for TM mode coupling in the case of anisotropic waveguides, and the improved coupled-mode theory is applicable to the situation when moderately strong coupling occurs under the condition where the edge-to-edge separation of two coupled guides D₂ is about 0.1 μm     

Computational modeling of diffused channel waveguides using adomain integral equation

A domain integral equation is proposed for the computational modeling of diffused channel waveguides. In the modeling the propagation properties and the field distributions of the lower order guided modes are computed. The method is used to design the channel waveguides that are realized by an ion-exchange process in glass substrates. In particular, the method is applied to the design of ion-exchanged waveguides with low fiber/chip coupling losses. The aim is to realize modal distributions in the channel waveguides that closely match the rotationally symmetric field distributions of the HE₁₁ fiber mode. Some technological aspects of the realization of such ion-exchange waveguides are indicated, and various numerical results relevant to the design process are presented     

Optical bistability in reflective fiber gratings

Optical bistability in a nonlinearly reflective fiber grating through the mode coupling between the LP₀₁ and counter propagation LP₀₂ modes (i,e., the reflective LP₀₁&rlhar2;LP₀₂ mode converter) is investigated by using the coupled-mode theory. Both the transmissive and the reflective properties of this nonlinear device are analyzed, which show that the output-versus-input relation exhibits the optical bistability. The switching power and the bistable hysteresis performances are also discussed. In addition, the comparison between the nonlinearly reflective mode converter (LP₀₁&rlhar2;LP₀₂) and the nonlinear fiber Bragg grating reflector (i,e., the LP₀₁ and counter-propagation LP₀₁ modes) is also presented. It is shown that the former has much lower switching power than the latter     

Design of short high-power TE11-HE11 modeconverters in highly overmoded corrugated waveguides

A theoretical parametric study of TE₁₁ to HE₁₁ mode conversion in highly oversized, circumferentially corrugated circular waveguides with different inner diameters is presented for various frequencies in the range of 28 to 140 GHz. The depth of the annular slots is tapered gradually from one half to one quarter wavelength. Computer-aided optimization of converter length, shape of corrugations, and nonlinear slot depth variation has been achieved with a scattering matrix code employing the modal field matching techniques (modular analysis concept). Relatively short mode transducers with matched converter lengths are feasible. The optimum conversion lengths are L₁≈0.5λ_B (short converter) and L₂≈1.2λ_B (long converter), where λ_B is the beat wavelength of the TE₁₁ and TM₁₁ modes in the corresponding smooth-walled circular waveguide. The predicted performances of short and long mode transducers are practically identical. Experimental results taken at 70 GHz (i.d.=27.79 mm) are in excellent agreement with the theoretically predicted performance data. In all cases the HE₁₁ output mode purity is 99% to 99.5%. The maximum cross-polarization and input-reflection levels are below -29 dB and -50 dB, respectively     

Variational coupled-mode theory of optical couplers

A scalar coupled-mode theory is developed from the variational principle. The formulation is variational in the sense that the mode parameters are adjustable and can be optimized subject to the coupling between the guides. The theory is applied to slab and fiber couplers. In comparison with the conventional scalar coupled-mode theory in which only the amplitudes of the modes in the individual guides are adjustable, the variational scalar coupled-mode theory predicts more accurate propagation constants and field patterns of the normal modes of the couplers     

Exact coupled-mode theories for multilayer interference coatingswith arbitrary strong index modulations

The analysis of multilayered interference coatings by coupled-mode theory is considered to be an approximation useful for small index, gain, and/or loss modulations. In this paper, we show that an exact analysis of a multilayer coating with coupled-mode theory is possible by redefining the coupling and detuning coefficients. We derive the correct coefficients for the case of a Bragg mirror consisting of layers with arbitrary high and low refractive indexes. A detailed comparison with coupled-mode theories using the standard coupling and detuning coefficients is presented     

Experimental characterization of coupled waveguides by normal-modeexcitation

We present a novel method for measuring the coupling coefficient of a symmetric directional coupler, based on the direct excitation of the normal modes of the entire structure. The output-power distribution at different coupling lengths was related to the experimentally measured normal modes in order to find the coupling coefficient. The device was alternatively characterized by means of the individual-waveguide modes according to coupled-mode theory in the strong-coupling regime. Both methods were implemented on single-mode directional couplers in Ti:LiNbO ₃. The validity and accuracy of both approaches are discussed     

Resonance in cylindrical-rectangular and wraparound microstripstructures

A rigorous analysis of the resonance frequency problem of both the cylindrical-rectangular and the wraparound microstrip structure is presented. The problem is formulated in terms of a set of vector integral equations. Using Galerkin's method to solve the integral equations, the complex resonance frequencies are studied with sinusoidal basis functions which incorporate the edge singularity. The complex resonance frequencies are computed using a perturbation approach. Modes suitable for resonator or antenna applications are investigated. The edge singularity of the patch current is shown to have no significant effect on the accuracy of the results. It is shown that the HE₁₀ modes of the cylindrical-rectangular and wraparound patches are more appropriate for resonator applications. The HE₀₁ and TE ₀₁ modes of the cylindrical-rectangular and wraparound patches, respectively, are efficient radiating modes     

Full-wave analysis of the image hybrid dielectric/HTS resonator

An analysis of the image hybrid dielectric/high-temperature superconductor (HTS) resonator is carried out. A full-wave radial mode-matching method is used to obtain the electromagnetic fields inside the resonator for single TE₀₁ and dual HE₁₁ modes. Measured resonant frequencies and quality factors of these modes are compared with numerical results of the analysis. The resonator power-handling capability is estimated from the field at the surface of the HTS film, assuming a certain value for the critical field of the HTS film     

Ray-optical determination of the coupling coefficients of gratingwaveguide by use of the rigorous coupled-wave theory

The ray-optics approach based on the rigorous coupled wave theory, called rigorous ray-optics method (RROM), is developed for the calculation of backward coupling coefficients of grating waveguide devices. The coupling coefficients of several grating structures, such as rectangular, sinusoidal, triangular, and trapezoidal shapes, are evaluated by the RROM, and they are compared with those obtained by two conventional methods of the ray-optics method (ROM) and the coupled-mode method (CMM). In the case of rectangular gratings, the coupling coefficients are evaluated in more detail by varying grating depth and duty-cycle. We have found that the RROM gives us more exact solutions for the backward coupling coefficients of even arbitrary shapes of diffractive grating waveguides than the other two conventional methods     

Modeling and analysis of fiber-optic mode transducers: single fiberwith periodic perturbations

The fiber-optic LP₀₁&lrarr2;LP₁₁ mode transducers are analyzed by a scalar coupled-mode theory with vector correction. The authors deal with fiber-optic mode transducers made of a single fiber with periodic perturbations due to microbends, acoustic waves, or a photoinduced index grating. Both the couplings caused by the index perturbations and by the vector property of the fields (polarization effect) are taken into account in the analysis. Approximate analytical solutions to the coupled-mode equations are obtained. The power exchange among the modes along the fiber and spectral properties of the mode transducers are examined. The functions of the mode transducers used as wavelength filters and polarizers are studied     

Polarization-selective mode coupling in two-mode Hi-Bi fibers

We propose and demonstrate a mode coupler which converts either of the LP₀₁ polarization states in a two-mode high-birefringence (Hi-Bi) fiber to the LP₁₁ mode with the same polarization. We use coupled-mode theory to develop design rules based on the polarization splitting of the beat length between the two lower-order modes. The device can be operated either as a narrow-band device in a region with large intermodal group delay difference or as a broadband device in a region with zero group delay difference. We use this novel device as a key component in a two-mode Hi-Bi fiber polarizer. In this configuration either of the polarization eigenstates can be selected and transmitted with an adjustable extinction ratio which can be as large as 30 dB. The coupling loss in the transmitted state of polarization can be less than 0.2 dB. In the broadband polarizer we demonstrate -20 dB extinction over 42 nm with a potential for considerable improvement. We also propose and experimentally investigate an increased differential group delay obtained by propagating one polarization state in the LP₁₁ mode instead of in the LP₀₁ mode. The largest differential group delay measured in this configuration is 14.5 ps/m which is seven times larger than the differential group delay between the polarization modes. We discuss several possible uses in fiber sensors and measure the transmission of a proposed two-coupler configuration     

Performance and modeling of proton exchanged LiTaO3branching modulators

Proton-exchanged LiTaO₃ branching modulators are demonstrated experimentally and modeled using a normal mode coupled-mode theory. The normal modes are approximated by ordinary coupled-mode theory parameters which are experimentally obtained. Agreement between the model and the experimental results is demonstrated. Crosstalk in these linear branch devices is found to be linearly proportional to the voltage-length product, a result which leads to an analytic expression for the output power