Mode coupling in tilted planar waveguide gratings

Three different grating tilting formulas for predicting the optimum grating tilt angle for strong mode coupling in planar waveguide gratings are derived. The optimum tilt angles obtained by the ray-optics approach deviate ~1 degrees for transmissive mode coupling and ~1.5 degrees for reflective mode coupling from those computed by the coupled-mode approach. The coupled-mode analysis and ray-optics analysis of the tilted planar waveguide gratings show that the transmissive planar waveguide gratings should be tilted more than 84 degrees for strong TE(0)-to-TE(mu) mode coupling near 1550 nm.     

Polarization-mode coupling in birefringent fiber gratings

Polarization-mode coupling in birefringent fiber gratings is analyzed. The general expression for coupling coefficient components is also derived. It indicates that the polarization-mode coupling between any two linearly polarized (LP) core modes is possible by appropriately adjusting the grating parameters such as the grating tilt angle, the grating length, the orientation of the grating plane, the grating period, the birefringence, and the birefringent axis. It is analytically found that the complete LP01x-to-LP01y mode coupling and LP01x-to-LP11y mode coupling occur when fiber is pressed periodically. The LP01x-to-LP11y mode coupling in the linearly birefringent gratings created by pressing a two-mode fiber with a groove plate with a period of 80 microm at a tilt angle between 81.5 degrees and 83.5 degrees has also been experimentally demonstrated. The resonant LP01x-to-LP11y mode coupling in the birefringent gratings and their experimental transmission spectra were reasonably well predicted by the coupled-mode analysis.     

Fiber mode conversion with tilted gratings in an optical fiber

The mode coupling in double-sided tilted gratings between the LP01 and LP11 modes is analyzed for three different excitation conditions. It is found that a number of modes, including various LP11q modes with different polarization states and TE01, TM01, and HE21 modes, can be formed through the mode conversion process in the gratings and with a linear combination of four different LP11 modes. Efficient single-sided and double-sided tilted gratings are demonstrated. The tilted gratings presented exhibit LP01-to-LP11 mode coupling over a spectral width as wide as 160 nm. The transmission spectra of the tilted fiber gratings agree well with the spectrum predicted by the coupled-mode analysis.     

Polarization mode coupling in circularly birefringent gratings

Polarization mode coupling in circularly birefringent gratings is analyzed. It is numerically found that efficient LP01x-LP02y mode coupling (where LP is linear polarization) is possible in a 50-cm-long circularly birefringent fiber grating formed in a terbium-doped borosilicate glass fiber and that complete LP01x-LP02y and LP01x-LP03y mode couplings result after a few-centimeter-long circularly birefringent grating that is formed in a bismuth-substitute iron garnet waveguide. Various parameters of polarization mode coupling in a number of circularly birefringent gratings are also computed.     

Electronically reconfigurable superimposed waveguide long-period gratings

The perturbation to the refractive index induced by a periodic electric field from two systems of interdigitated electrodes with the electrode-finger period l is analyzed for a waveguide with an electro-optically (EO) active core-cladding. It is shown that the electric field induces two superimposed transmissive refractive-index gratings with different symmetries of their cross-section distributions. One of these gratings has a constant component of an EO-induced refractive index along with its variable component with periodicity l, whereas the second grating possesses only a variable component with periodicity 2l. With the proper waveguide design, the gratings provide interaction between a guided fundamental core mode and two guided cladding modes. Through the externally applied electric potential, these gratings can be independently switched ON and OFF, or they can be activated simultaneously with electronically controlled weighting factors. Coupling coefficients of both gratings are analyzed in terms of their dependence on the electrode duty ratio and dielectric permittivities of the core and cladding. The coupled-wave equations for the superimposed gratings are written and solved. The spectral characteristics are investigated by numerical simulation. It is found that the spectral characteristics are described by a dual-dip transmission spectrum with individual electronic control of the dip depths and positions. Within the concept, a new external potential application scheme is described in which the symmetry of the cross-sectional distribution of the refractive index provides coupling only between the core mode and the cladding modes, preventing interaction of the cladding modes with each another. This simple concept opens opportunities for developing a number of tunable devices for integrated optics by use of the proposed design as a building block.     

Leaky-mode loss of the second propagating mode in single-mode fibers with index well profiles

Radiation loss characteristics are calculated for the first two LP modes of a double-clad lightguide structure in which the inner cladding forms a low refractive-index well between the core and the outer cladding. The higher-order mode is defined to be effectively cut off at wavelengths where the LP(11) mode power is sufficiently attenuated to ensure negligible modal noise due to interference between LP(01) and LP(11) modes. Calculations apply for various deposited cladding thicknesses and findings can be easily extrapolated if specifications are changed for system requirements on the minimum unspliced fiber length or the extinction ratio between P(11) and P(01) mode powers, Re = 10 log P(11)/P(01). The latter is specified at points such as splice joints where modal noise is generated. Theoretical results are correlated with measurements in order to determine the effective cutoff wavelength for fibers in which leaky-mode loss mechanisms dominate. The procedure should be extremely useful for determining the shortest usable wavelength for single-mode lightwave systems.     

Control of the characteristics of a long-period grating by cladding etching

We have analyzed the effects of the fiber cladding radius on the characteristics of long-period fiber gratings. By etching the cladding of a common single-mode fiber, we verified the characteristics experimentally. When by etching we reduce the cladding radius of a common single-mode fiber on which a long-period refractive-index modulation has been imposed, the coupling strength of the core and the cladding modes increases. In addition, the difference in the propagation constants (for a fixed wavelength) between the core mode and the cladding modes increases; hence the resonant transmission dip wavelengths shift to longer wavelengths. The proposed method can be useful in making and detuning long-period fiber grating filters.     

Propagation and coupling of hybrid modes in twisted fibers

The first-order paraxial approximation is used to obtain the distributions of the electric and magnetic fields for the core and cladding hybrid fiber modes. The coupling coefficients of these modes are found for fibers subject to twist. The longitudinal electric field component determines the mode coupling in twisted fibers. It is shown that in the first-order paraxial approximation the cladding hybrid modes propagating in a twisted fiber rotate along the direction of the twist at the same rate as the core mode, independently of the azimuthal and radial mode numbers. Four hybrid modes constituting one linearly polarized mode have different longitudinal components, and the corresponding cladding-mode resonances of a long-period fiber grating undergo different shifts owing to different mode self-coupling coefficients. This results in the removal of mode degeneracy and splitting of resonances of long-period gratings in twisted fibers.     

Characteristics of a long-period fiber grating with reduced cladding for refractive index sensing

The sensitivity to surrounding refractive index (SRI) of a long-period fiber grating (LPFG) can be effectively improved by decreasing the cladding radius. When the cladding is reduced, a three-layer model is necessary to evaluate the effective refractive index (ERI) of the core mode. A variation of SRI can induce a greater resonant wavelength shift when the core mode is coupled to a higher-order cladding mode. However, as the cladding is reduced further, the highest-order cladding mode would be cut off, i.e. the number of cladding modes that a given fiber structure can support would be less; thus, the higher-order cladding modes that can be used for higher sensitivity are limited. Hence, the implementation of high sensitivity for SRI sensing with cladding-reduced LPFGs is dependent on the proper combination of cladding radius and cladding mode order. Based on the vector coupled-mode theory, the transmission spectrum and sensitivity are numerically analyzed with respect to the cladding radius, which shows that the SRI sensitivity of the HE₁₂ mode with cladding radius a ₂?=?20?µm is 32 times as high as that with a ₂?=?62.5?µm and the SRI resolution is available to the order of 10^?7.     

Shear mode coupling and tilted grain growth of A1N thin films in BAW resonators.

Polycrystalline A1N thin films were deposited by RF reactive magnetron sputtering on Pt(111)/Ti electrode films. The substrates were tilted by an angle ranging from 40 degrees to 70 degrees with respect to the target normal. A low deposition temperature and a high sputter gas pressure were found ideal for tilted growth. The resulting grain tilt angle amounts to about half the substrate tilt angle. For coupling evaluation, 5 GHz solidly mounted resonator structures have been realized. The tilted grain A1N films exhibited a permittivity in the 9.5-10.5 range and loss tangent of 0.3%. Two shear modes as well as the longitudinal mode could be clearly identified. The coupling coefficient k2(eff) of the fundamental thickness shear mode (TS0) was found to be about 0.5%, which is compatible with a c-axis tilt of about 6 degrees.     

Measurement of duty cycles of photoresist grating masks made on top of multilayer dielectric stacks

We present a novel and nondestructive method for measuring the duty cycles (ratio of ridge width to period) of submicrometer rectangular photoresist gratings made on top of multilayer dielectric stacks. The method exploits the fact that the effective index of the leaky mode that has a strong evanescent tail in the cladding changes with the duty cycle of the grating situated at the interface between the top dielectric layer and the cladding. By comparing measured coupling angles of the leaky mode with a theoretical or experimentally calibrated relationship between coupling angles and duty cycle, one can determine the duty cycle of the grating. This method is applicable even when the grating period is less than the measurement wavelength. It is simple because it does not require any power measurement. Most importantly, it is virtually independent of groove depth. The physical principle of the method and the results of experimental verification are presented.     

Characteristics of short-period blazed fiber Bragg gratings for use as macro-bending sensors

The characteristics of short-period blazed fiber Bragg gratings for use as macro-bending sensors are discussed. This sensor is able to detect macro bending with the transmitted power variation of the first side mode in the blazed fiber Bragg grating. Since an incident ray experiences different variations of tilt angles with respect to bending direction, the blazed fiber Bragg grating has different coupling efficiencies of the first side mode, which can be reduced considerably in the case of twisted blazed fiber Bragg gratings.     

Coupling characteristics of cladding modes in tilted optical fiber bragg gratings

We have studied both theoretically and experimentally the effect of grating tilting on the coupling between the fundamental core mode and the cladding modes in an optical fiber Bragg grating. The coupling is shown to be very sensitive on the tilting angle. It is also shown that tilting angle has to be minimized in fibers with designs to suppress the coupling between the fundamental core mode and the cladding modes. We have also studied the single, strong loss peak accompanying the Bragg reflection peak in depressed-cladding fibers, thus showing a good agreement between behavior that is measured and that is predicted theoretically.     

Measurement of pressure and temperature sensitivities of a Bragg grating imprinted in a highly birefringent side-hole fiber

The sensing characteristics of a Bragg grating imprinted in a specially developed highly birefringent side-hole fiber were investigated in detail. We showed that such a grating has almost identical sensitivities to temperature for both linearly polarized modes LP01(x) and LP01(y) (approximately 5.9 pm/K) and significantly different sensitivities to hydrostatic pressure (-1.93 pm/MPa for LP01(x) and +5.37 pm/MPa for LP01(y) mode). The sensitivity differences versus mode polarization are so high that this effect can be employed for simultaneous sensing of temperature and pressure by interrogation of the wavelength shifts at LP01(x) and LP01(y) modes. Applying interferometric methods, we also measured the sensitivity of the host side-hole fiber to temperature and hydrostatic pressure for each polarization mode. Our results show that there is good agreement between the normalized sensitivities of the host side-hole fiber and those of the grating for the same polarization modes.     

Polarization-independent wavelength filter based on dual asymmetric long-period gratings in channel waveguides

Channel waveguides with asymmetric long period gratings exhibiting coupling from the fundamental (symmetric) core mode to the first-order (antisymmetric) cladding mode were analyzed. Using this geometry, we show that it is possible to realize polarization-dependent and polarization-independent operations at desired wavelengths. We have chosen the device parameters to demonstrate its operation in the C+L band. Sensitivity of the coupling behavior to random variation in various parameters such as core and cladding dimensions, grating period, etc., has been analyzed. The proposed structure should be interesting from the point of view of realizing waveguide devices such as filters, polarizers, etc., based on waveguide gratings.     

Transfer-matrix method based on a discrete coupling model for analyzing uniform and nonuniform codirectional fiber grating couplers

A simple and efficient transfer-matrix method based on a discrete coupling model is presented to analyze uniform and nonuniform fiber grating couplers between copropagating core and cladding modes. Uniform and piecewise-uniform long-period gratings were fabricated by a point-by-point arc discharge technique. Their measured transmission spectra were compared with the transmission spectra calculated by the presented method.     

Analysis of a long-period grating inscribed in the core of a fibre with inverse parabolic-index cladding

The operation of a long-period grating inscribed into the core of a single-mode fibre with the cladding having an inverse parabolic-index profile is theoretically investigated. Two-dimensional bidirectional mode expansion and propagation method is used to simulate mutual interaction of rotationally symmetric LP modes in the fibre. Numerical analysis has shown that the inverse parabolic-index profile of the cladding helps increase the sensitivity of a fibre as a refractometric sensor.     

Littrow angle method to remove alignment errors in grating pulse compressors

An alignment method for pulse-compression gratings that obviates the need to place the gratings at normal incidence to remove grating-tip error is proposed. Grating-tip and groove-orientation errors are removed using two alignment wavelengths in a manner analogous to a laser-beam pointing and centering procedure entirely at the respective Littrow angles for the two wavelengths. By choosing wavelengths with Littrow angles close to the use angle of the grating, the residual tip and groove-orientation errors that may be introduced when the grating mount is tilted to its use angle are reduced. This method has greatly facilitated the alignment of the OMEGA Extended Performance (EP) large-aperture pulse compressors, thereby reducing residual pulse-front tilt caused by nonparallel gratings. OMEGA EP is a high-energy, petawatt-class laser at the University of Rochester's Laboratory for Laser Energetics. A numerical simulation of the alignment procedure is presented.     

Nonlinear Dispersion Coefficients of a Cylindrical Optical Waveguide

Abstract

The nonlinear carrier wave dispersion relation at weak nonlinearity is derived for the linearly polarized (LP) modes of a step-index optical fibre that has both a nonlinear core and a nonlinear cladding. The calculation begins with the exact equations for the nonlinear fibre and the nonlinear shift (coefficient), from its linear value, of the propagation wave-number is given in closed analytical form. The nonlinear coefficient is completely general and accounts both for the nonlinearity and the structure of the guided mode. Some numerical results for the LP₀₁ mode are presented which show that significant deviations occur from the conventionally accepted (averaged) nonlinear coefficient.