Graded-index Planar Waveguides Research Articles

Electromagnetic waves in graded-index planar waveguides

The propagation of guided TE and TM modes through graded-index planar waveguides is reported in this paper. Both a real-valued and complex-valued position-dependent refractive index are supposed for a film layer. It is possible to set various refractive index profiles based on five distribution parameters. For the position-dependent refractive index, the governing equations are transformed to Heun’s differential equation, an exact local solution expressed in terms of local Heun functions. The general nature of these functions is demonstrated based on four degenerate cases of Heun’s equation. The calculation of guided modes requires evaluation of the general solution in the interval containing two regular singular points. For this purpose, the generalized Heun function is introduced and employed in general solutions to the governing equations. The applicability of the generalized solutions is demonstrated by the calculation of guided modes for both the real-valued and complex-valued refractive index.

Resonant-light diffusion in a disordered atomic layer

Light scattering in dense media is a fundamental problem of many-body physics, which is also relevant for the development of optical devices. In this work we investigate experimentally light propagation in a dense sample of randomly positioned resonant scatterers confined in a layer of sub-wavelength thickness. We locally illuminate the atomic cloud and monitor spatially-resolved fluorescence away from the excitation region. We show that light spreading is well described by a diffusion process, involving many scattering events in the dense regime. For light detuned from resonance we find evidence that the atomic layer behaves as a graded-index planar waveguide. These features are reproduced by a simple geometrical model and numerical simulations of coupled dipoles.

Bidirectional Mode Slicing and Re-Combining for Mode Conversion in Planar Waveguides

Photonics integrated circuits and planar lightwave circuits are crucial in future high capacity optical communication networks. Such circuits can perform complicated and sophisticated signal processing functions’ on-chip level. However, they should have high bandwidth in order to handle ultrafast data flow and thus achieve high-capacity transmission. One way to achieve such high bandwidth is using few-mode waveguides. Therefore, it becomes mandatory to find a way to convert from current conventional single-mode planar waveguides into few-mode waveguides, and moreover to selectively excite the desired modes within the converted multimode waveguides. In this paper, a novel silica-glass planar waveguide converter is numerically demonstrated for the first time. This device can convert in a bidirectional fashion between single-mode and two, three, or four modes planar waveguides, in addition, to selectively excite the desired mode. The device principle of operation mainly depends on spatial modes slicing and re-combining to achieve the desired higher or lower order modes. That could be accomplished by cascading stages of V-shape and M-shape graded-index planar waveguides. The graded-indices allow for flexible and simple geometrical designs that can split or combine fundamental as well as higher order modes. The finite-difference time-domain numerical method is utilized to verify the device operation and evaluate its performance for each excited mode in bidirectional directions. The device shows good performance over the entire C-band wavelength range with the worst-case magnitude of insertion-loss ≅ 3 dB, polarization-dependent loss ≅ 0.6 dB, and return-loss ≅ 21 dB. It also shows a worst-case cross-talk of ≅ −24.1 dB among the excited high-order modes in the forward direction, and a worst-case mode-rejection ratio of ≅ −23 dB for non-excited modes in the reverse direction. Moreover, the device shows reasonable performance tolerance to variations in V-shape and M-shape waveguide design parameters.

Propagation of p-polarized waves in a linearly graded index film surrounded by negative index materials

The theory of step-index waveguides is well-established. Most practical slab waveguide structures have a graded-index profile. The basic properties of graded-index planar waveguide structures are similar to those of step-index waveguides with subtle differences. The most common types of graded-index slab waveguides are linearly and exponentially graded-index profiles. We here treat linearly graded-index slab waveguide. In this work, a three-layer waveguide structure with linearly graded-index film is considered. We assume three structures: the first structure comprises a left-handed material (LHM) cladding, the second structure contains a LHM substrate layer and the third has a LHM cladding and substrate. Closed-form expressions for electric and magnetic fields and the characteristic equation are derived. The three normalized parameters: the asymmetry coefficient (a), the normalized film thickness (V) and the normalized guide index (b) are used to study the dispersion properties of the proposed slab waveguide structure.

Dynamics and trajectory of nonautonomous rogue wave in a graded-index planar waveguide with oscillating refractive index

We study rogue waves in a graded-index planar waveguide with oscillating refractive index. We find that an additional refractive index can be used to manipulate the trajectory of the rogue wave without changing its shape evolution characters. The density distribution profile of rogue wave with the highest peak can be kept well through manipulating the graded-index term and nonlinear coefficient. Furthermore, the trajectories of these nonautonomous rogue waves still look like an “X” shape. These results provide possibilities to manipulate rogue wave in nonautonomous nonlinear systems.

Graded-index planar waveguide solar concentrator

Planar waveguides are useful to transport, concentrate and distribute light uniformly over large dimensions. Their capacity to collect and gather light efficiently over a large distance is interesting for many applications, like backlighting and solar concentration. For these reasons, the possibility of making them even more efficient could be of considerable interest for the community. The observation of the ray path inside a graded-index (GRIN) fiber inspired the development of a similar technology inside planar waveguides. In this Letter, we show that it has the potential to dramatically increase the efficiency of planar waveguide-based solar concentrators or backlighting using GRIN planar waveguides.

GRIN planar waveguide concentrator used with a single axis tracker.

It is generally accepted that small to medium level concentrators could be used as cost-competitive replacements for tracked solar panels. The objective is to design a system that can reach a good level of sun concentration with only one sun-tracking axis and is cheap to fabricate. As the most critical parameter for all concentrator designs, optical efficiency needed improvement to reduce the cost of power produced by our system. By using a graded-index planar waveguide with an index profile similar to SELFOC fiber, the ray's path can be controlled. Also, the concentrator can be fabricated in a single block, which reduces Fresnel reflections. Overall, the optical efficiency can be improved by as much as 33% compared to the same system made with a homogeneous waveguide. Furthermore, the ability to cost-effectively fabricate the concentrator by molding can be preserved, making it possible to reduce the cost of the solar power produced.

High Resolution through Graded-Index Microoptics

By solving Helmholtz equations, relationships to describe propagating modes in an arbitrary graded-index planar waveguide are derived. We show that in the quadratic- and secant-index waveguides a minimal mode width is 0.4λ/n, where λ is the wavelength in free space and n is the refractive index on the fiber axis. By modeling in FullWAVE, we show that the high-resolution imaging can be achieved with half-pitch graded-index Mikaelian microlenses (ML) and Maxwell’s “fisheye” lenses. It is shown that using a 2D ML, the point source can be imaged near the lens surface as a light spot with the full width at half maximum (FWHM) of 0.12λ. This value is close to the diffraction limit for silicon (n=3.47) in 2D media FWHM=0.44λ/n=0.127λ. We also show that half-pitch ML is able to resolve at half-maximum two close point sources separated by a 0.3λ distance.

Approach for determining the refractive-index profile of graded-index planar waveguides

We measured the modal indices of a planar waveguide made from GeSeSb glass sandwiched between SiO(2) and air by using the prism-coupling technique. Based on the measured indices of the TE modes, we determined the position of the turning point corresponding to each mode of the waveguide by using the inverse WKB method. Using the modified fitting criterion introduced previously [Appl. Opt. 33, 3227 (1994)], we accurately determined the spatial profile of the refractive index for such a waveguide. Such a graded-index profile is probably caused by compositional variation of the GeSeSb guiding layer.

An interferometric method for refractive index profiling of planar gradient index waveguides

In this paper, we present an interferometric method to determine the refractive index profile in graded-index planar waveguides. To begin, part of the soda-lime glass substrate surface was coated with aluminum and then was immersed in molten AgNO 3 to fabricate a planar waveguide on the uncoated part of the surface. After the ion exchange, the coating was removed. Then, the sample was polished obliquely along the boundary between the ion-exchanged and non-ion-exchanged regions, to form a wedge. Thereafter, it was placed inside a Mach–Zehnder interferometer. The fringe pattern was analyzed using the well-known Fourier method and the refractive index profile was determined. The sample preparation, data analysis and experimental results are presented.

Second-Order Optical Nonlinearity Induced in Thermally Poled PbO–SiO2 Mixed Film on Fused Silica

We developed a lead oxide PbO film of ∼170 nm thickness by RF sputtering on a fused silica substrate. When this film sample is annealed in an oven at 650 °C for 4 min, its physical thickness measured using a surface profiler decreases to ∼163 nm, but its optical film thickness determined using a spectrophotometer increases to 382 nm. This fact implies that PbO will diffuse into the fused silica substrate and form a PbO–SiO2 mixed film that is a graded-index planar waveguide with two transverse electric (TE) modes at 633 nm wavelength. Although the Maker fringe patterns of both fused silica and PbO–SiO2 mixed film samples thermally poled at 6 kV, 275 °C and 30 min look similar, we speculate that second-order nonlinearity exists in poled PbO–SiO2 mixed film because of the higher refractive index contrast at the air-to-film interface. A nonlinear optical coefficient of approximately 1 pm/V in poled PbO–SiO2 mixed film is calculated using a double-step nonlinear profile and grid search fitting technique. The induced second-order nonlinearity is stable for at least several months.

Refractive-Index Profiling of Buried Planar Waveguides by an Inverse Wentzel–Kramer–Brillouin Method

We propose an inverse Wentzel-Kramer-Brillouin (WKB) method for the construction of the refractive-index profiles of buried graded-index planar waveguides from the measured effective indexes of the guided modes. Our method involves experimental determination of the location of the peak index in the buried profile, measurement of the effective indexes in air, and an index-matching liquid, and application of a searching algorithm to find the best-fit profile through inverting the WKB equations. We present numerical examples and experimental results to demonstrate the performance of the method.

Design and simulation of a novel arrayed-waveguide grating

A novel arrayed-waveguide grating (AWG) based on unbent waveguides is proposed. Two graded-index planar waveguides (GISLAB) are used as input and output planar waveguides, respectively, and the arrayed waveguides are replaced by unbent waveguides, which are photosensitive waveguides exposed to UV light twice. Next, the 2-D semivectorial beam propagation method is adopted to simulate the light transmitting process in the device, and the simulation shows that this device and the design are feasible.

Index profiling of anisotropic graded-index planar waveguides from effective indices

We present a method for recovering the refractive-index profile of an anisotropic graded-index waveguide from the effective indices by using a cubic spline interpolation function. The first and second derivatives of cubic splines are continuous to ensure a smooth index profile, which is consistent with practical graded-index distributions. A straightforward iteration with an exact dispersion equation to verify the interpolated profile makes this method easy and reliable for application. This approach is proved by numerical analysis of several typical index distributions and experimental examples showing that the refractive-index profiles in anisotropy can be reconstructed close to the exact profile. Waveguides with both more modes (more than four guiding modes) and fewer modes (two to four) can be universally profiled with good accuracy.

Construction of refractive-index profiles of planar waveguides with additional information obtained from surface plasmon resonance

A surface plasmon resonance (SPR) is excited between a metal film and a graded-index planar waveguide. After the propagation constant of the SPR is measured, the refractive index near the surface of the waveguides, which is difficult to obtain by conventional techniques, is determined experimentally. With this nondestructive technique, combined with the inverse analytical transfer matrix method, the planar waveguide can be profiled to a high degree of accuracy.

Construction of refractive‐index profiles of graded‐index planar waveguides from effective indexes measured with different mode types and external refractive indexes

AbstractA method is presented whereby the refractive‐index profile (RIP) can be reconstructed from sets of effective indexes measured with different mode types and external refractive indexes. Single‐mode waveguide RIP characterization can be easily realized using an index‐matching liquid at a single wavelength. This method is described and demonstrated via numerical examples and experimental results. The method has the advantages of convenient operation and accurate results for the characterization of graded‐index planar waveguides that support both mode types and contain known modal birefringence. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 45: 342–345, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20818

Modeling of holographic gratings in gradedindex photorefractive planar waveguides

A numerical model is presented for the evaluation of the dielectric permittivity tensor changes as induced by guided modes during the formation of holographic gratings in arbitrary photorefractive graded-index planar waveguides. Comparisons among lithium niobate waveguides with different cuts and technology are shown.

Simple algebraic recursion formulas for calculation of propagation characteristics of graded-index waveguides

Using the perturbation method with the aid of the hypervirial theorem and Hellmann–Feynman theorem in quantum mechanics, simple algebraic recursion formulas for calculating propagation constants of graded-index planar optical waveguides in successive orders of approximation are derived. Algebraic formulas for calculation of the modal field functions are also presented. Numerical results for typical graded-index waveguides (Gaussian, exponential, complementary error function and symmetric Epstein profiles) demonstrate that the present method can be applied to an accurate calculation of the propagation characteristics to an arbitrary order of perturbation approximation.

Analytical approximations for fundamental‐mode field and dispersion equation of planar waveguides through the Stevenson–Padé approach

A novel method for the systematic derivation of analytical approximations of the fundamental-mode dispersion equation and field distribution of graded-index planar waveguides with uniform cladding is presented. The method is based on a Stevenson–Padé approach, recently introduced by the authors for the numerical analysis of graded-index optical fibers. A number of examples are presented in order to illustrate the power and accuracy of the method. © 2000 John Wiley & Sons, Inc. Microwave Opt Technol Lett 27: 158–162, 2000.

Refractive-index profiling of graded-index planar waveguides from effective indexes measured with different external refractive indexes

We extend the well-known inverse Wentzel-Kramer-Brillouin (WKB) method for refractive-index profiling of graded-index planar waveguides. We demonstrate, with numerical examples and experimental results, that the refractive-index profile of a graded-index planar waveguide can be calculated from effective indexes measured with different external refractive indexes. With this technique, single-mode and two-mode waveguides can be profiled easily to a good accuracy.