# OptiBPM Tutorials

## Wide Angle (WA), Pade(1,1)

Using the recurrence formula Equation 118, we get: Here a = 2k0nref . Using Equation 119 into Equation 131, we get: Thus, and, where Thus, we can get the unknown field φ l + 1  at z + Δz from the known field φ l  solving the triagonal linear system: If we consider FEM we get, Thus, we can…

## Wide Angle (WA), Pade(2,2)

Using recurrence formula Equation 118 for Padé(2,2) we get: Here  a = 2k0nref . Using Equation 144 into Equation 143, we get: From Equation 145, we get: Thus, and, where: Thus, the unknown field φ l + 1  at z + Δz is related to the known field φ l at z as follows: Multistep Method In order to…

## Wide Angle (WA), Pade(3,3)

Using recurrence formula Equation 118 for Padé (3,3), we get: Using Equation 145 into Equation 169, we get: Finally, using Equation 171 into Equation 169, we get: Here a = 2k0nre f . From Equation 172, we get: Thus, and where Thus, the unknown field φ l + 1  at z + Δz is related to the known…

## Wide Angle (WA), Pade(4,4)

Using recurrence formula Equation 118 for Padé (4,4) we get: Using Equation 191 into Equation 190 we get: From Equation 192, we get: Thus, and here Thus, the unknown field φ l + 1  at z + Δz is related to the unknown field φ l at z as follows: Multistep Method In order to solve Equation…

## References

           M.D. Feit and J.A. Fleck, Jr.: Light Propagation in Graded-Index Optical Fibers, Appl. Opt. 17, (1978): 3990-3998.            M.D. Feit and J.A. Fleck, Jr.: Analysis of Rib Waveguides and Couplers by the Propagating Beam Method, J. Opt. Soc. Am. A 7, (1990): 73-79.            D. Yevick and B. Hermansson: Efficient Beam Propagation Techniques, IEEE J.…

## Conformal Mapping Regions Introduction

Conformal mapping region in BPM is used to simulate curved optical waveguides. The method uses conformal mapping in the complex plane to transform a curved waveguide in ( x, y )  coordinates into a straight waveguide with a modified refractive index in new ( u, v ) coordinates (Figure 1). It can be used to treat…

## Conformal Mapping Regions References

 M. Heiblum and J.H. Harris: Analysis of Curved Optical Waveguides by Conformal Mapping, IEEE J. Quant. Electron. 11, (1975): 75-83.  S.J. Garth: Modes on a Bent Optical Waveguide, IEEE Proc. J. 134, (1987): 221-229.  P.L. Fan, M.L. Wu, and C.T. Lee: Analysis of Abrupt Bent Waveguides by the Beam Propagation Method and the Conformal…

## Titanium Diffusion in Lithium Niobate

The Titanium diffused waveguides in Lithium Niobate, or the Ti:LiNbO3 waveguides, are formed by the diffusion of the Titanium dopant into the Lithium Niobate host. To form a waveguide, a stripe of Titanium is deposited on the LiNbO3 substrate. For a given stripe width, which we identify with the waveguide width, the amount of Titanium…

## Magnesium Diffusion in Lithium Niobate

The diffusion of Magnesium dopant into the Lithium Niobate host induces negative index changes. The process starts by deposition of a stripe Magnesium source, usually the oxide of Magnesium, onto the Lithium Niobate crystal. The sample is then heated for several hours, similar to the Titanium diffusion process. Formally, the resulting refractive index distribution can be…

## Proton Exchange Process in Lithium Niobate

Proton exchange in Lithium Niobate involves a replacement of Lithium ions (Li+) by hydrogen ions, or protons (H+). The replacement causes a change in refractive index, thus forming a waveguide. Proton exchange is one of the methods used for forming optical waveguides in Lithium Niobate, LiNbO3, as well as in Lithium Tantalate, LiTaO3. The waveguide…