1. Solving Wave Equation by Beam Propagation Method and Ray Tracing

2. Equation for Propagating Wave in Longitudinally Varying Waveguide
From
For “slow” varying waveguide, we have
Hence

3. Equation for Propagating Wave in Longitudinally Varying Waveguide
Unlike in solving the purely guided wave, we should allow a longitudinally dependent field envelope to reflect the longitudinally varying feature of the waveguide:
where
Hence the 2nd order derivative of the field envelope on z is negligible in comparing with its 1st order derivative – this is known as the slow-varying envelope approximation and is a general approach in treating a slow-varying structure or a function with weak dependence on its variable.
The wave equation for the envelope of the transverse E-field is therefore obtained as:

4. Equation for Propagating Wave in Longitudinally Varying Waveguide
Or, it can be expressed as:
where

5. Equation for Propagating Wave in Longitudinally Varying Waveguide
Similarly, we have:
where

6. Equation for Propagating Wave in Longitudinally Varying Waveguide
Under the semi-vectorial approximation, the full vectorial operators are reduced to:
As such, all equations become decoupled and each E- or H- field component can be solved independently.

7. Equation for Propagating Wave in Longitudinally Varying Waveguide
Under the weak confinement approximation, the operators are further reduced to:
The wave equation for either E- or H- field takes the same form as:
- scalar wave equation

8. Equation for Propagating Wave in Longitudinally Varying Waveguide
The operator on the left hand side can always be treated under the Pade approximation:
with
By letting i=1, we obtain the paraxial scalar wave equation (paraxial full vectorial and semi-vectorial wave equations can be obtained similarly by replacing the operator on the right hand side with the corresponding full vectorial and semi-vectorial operators, respectively):

9. Equation for Propagating Wave in Longitudinally Varying Waveguide
By letting i=2, we obtain the 1st order wide angle scalar wave equation (1st order wide angle full vectorial and semi-vectorial wave equations can be obtained similarly by replacing the operator on the right hand side with the corresponding full vectorial and semi-vectorial operators, respectively):

10. Beam Propagation Method
In dealing with the equation in the form of
we can always identify a small step Δz, at which the solution to it can be formally expressed as:

11. Explicit Scheme Letting α=0, we obtain the explicit BPM scheme
In the explicit scheme, the numerical treatment is a simple marching process along z: for a given initial field distribution at a starting point z0, the right hand side can be computed by taking (numerical) derivatives directly to obtain the field distribution at z0+Δz; by iterating this process, the entire field distribution along z can readily be obtained.

12. Implicit Scheme
For 0<α<1, we obtain the implicit BPM scheme (where α=0.5 is the Crank-Nicholson’s scheme that preserves the unitarity of the entire exponential operator):
In the implicit scheme, a full numerical treatment is needed by discretizing the field in the cross-sectional plane (x, y) through, e.g., the finite-difference scheme: for a given initial field distribution at a starting point z0, seeking the solution of the field distribution at z0+Δz becomes solving a system of linear equations with the field values at a set of grid points as unknowns; once the field distribution is obtained at z0+Δz, this process will be repeated by updating z0 with z0+Δz and z0+Δz with z0+2Δz …; as such, the entire field along z can be obtained.

13. Implicit Scheme
Assuming there are N and M grid points in x and y direction, respectively, after discretization under the FD scheme, the original (paraxial, scalar) wave equation will be in the form of
where the coefficient matrix A and B are in NM×NM, the field distribution at z0+Δz (as unknown) and at z0 (given) are vectors in NM.
Since the right hand side of the above equation can be treated as a given vector in NM, the solution to this problem can be found by inverting matrix A:

14. A Few Remarks about BPM
BPM is a initial value problem (in explicit scheme) or mixed initial-boundary value problem (in implicit scheme), hence there is the stability issue as we have encountered in the FDTD algorithm. (On the contrary, the mode solver, identified as an eigen value problem, is a boundary value problem and is unconditionally stable.)
Through von-Neumann analysis, we find that the stability criteria is given by α≤0.5, independent of the longitudinal marching step size Δz or the grid sizes (Δx, Δy) in the cross-section.
Numerical dissipation – error introduced to the amplitude of the field distribution in the marching process; zero numerical dissipation obviously requires α=0.5, however, this usually leads to the unstable problem in practice; a compromise can be made be choosing, e.g., α=0.5+δ, where δ is a very small number for minimizing the numerical dissipation but is sufficient to avoid the instability. For a given α, a smaller marching step size Δz obviously gives a smaller numerical dissipation.

15. A Few Remarks about BPM
Numerical dispersion – phase error introduced by the discretization; von-Neumann analysis shows that, independent of α and Δz, the numerical dispersion is only related to the grid sizes (Δx, Δy) in the cross-section, as well as the selection of the propagation constant β.
Boundary condition – PML
Initial condition – source excitation at a starting point z0
Other (explicit) BPM methods:
Multiple step BPM
Series expansion BPM

16. A Few Remarks about BPM
Apply BPM to uniform waveguide for solving mode
Bi-directional BPM
Time domain BPM – by applying the slow-varying envelope approximation in terms of the time variable (t) in treating time domain wave equations

17. A Summary on Numerical Solution Techniques
Solution Method
Varieties
Numerical problem
Remarks
Maxwell’s equations (mixed I/B problem)
FDTD
Dispersive media
Marching (iteration)
Extendable for solving the Schrodinger equation by mapping the real and imaginary wave functions to E and H fields, respectively
Wave equations (mixed I/B problem)
FFT-BPM
FD-BPM
FD-TD, explicit and implicit, split-step
Marching (iteration) and matrix Inversion
Through mode matching, can be treated as eigen value problems
Eigen value equations (B problem)
FD Mode- Solver
FE, collocation
Matrix diagonalization
Through extension in (t) or (z), can be treated as wave propagation problems
Poisson’s equation (B problem)
FD Solver
Matrix inversion

18. Ray Tracing Method X Y Z x y z O o P0 sf s0
A given ray is specified by the coordinates (X0, Y0, Z0) of a point P0 through which the ray passes and by its direction cosines (K, L, M) in a reference coordinate system O-(X, Y, Z)
A surface S is specified by an equation F(x, y, z)=0 referred to a coordinate system o-(x, y, z). The orientation of the o-system relative to the O-system and the coordinates of the origin of the o-system are given in the O-system.

19. Ray Tracing Steps
Transform the ray-point coordinates (X0, Y0, Z0) and direction cosines (K, L, M) into their values in the o-system.
Find the point of the intersection of the ray with the surface S.
Find the change in direction of the ray refraction, reflection, or diffraction at the corner point or in the case of a grating, at the surface S.
Transform the new ray-point coordinates and direction cosines back to the O-system (optional*).
Repeat 1-4 for succeeding surfaces in sequence.
* Step 4 can obviously be omitted if the coordinate system associated with the next surface is referred to the system associated with S (o) instead of to the reference system (O).

20. Coordinate Transformation from Reference to Object System
The specified ray through P0 (X0, Y0, Z0) oriented along (K, L, M) in the O-system can be expressed in the o-system, the transformation is:
- coordinates of the origin of the o-system measured in the O-system

21. Find the Intersection Point
Parametric equation for ray
- distance along the ray measured from
It is convenient to determine firstly the intersection of the ray with the z=0 plane – we can easily find
as the required distance, and
as the intersection point at z=0

22. Find the Intersection Point
With now s measured from
we have
By substituting this expression into the equation that specifies the surface S, i.e., F(x, y, z)=0, we should be able to find s and to obtain the intersection point.
However, an analytical solution to the above root searching problem may not exist, we will, therefore, follow the Newton-Raphson method to find the required s through a numerical iteration approach:
The iteration can be started from s1=0 and terminated by |sf-sf-1|<δ.

23. Find the Intersection Point
Hence we find the intersection point:
And the surface normal direction at the intersection point specified by:

24. Chang in Direction of the Ray by Refraction
Following Snell’s law:
we can write:
- a multiplier to be determined
Since
we then find: Or
For n=n’, there is no interface, b=0, and the orientation shouldn’t change, we should choose:
as the solution to the direction change by refraction.

25. Chang in Direction of the Ray by Refraction
(k’, l’, m’) for n’<n
(k, l, m)
(ks, ls, ms)
(k’, l’, m’) for n’>n
It is quite obvious that 0<a<n/n’, with a=0 and a=n/n’ corresponding to the grazing and normal incidence, respectively.
If b>a2, TIR happens.
For n’>n, b<0
the new direction has an inward bending
For n’<n, b>0
the new direction has an outward bending

26. Chang in Direction of the Ray by Reflection
In this case, we just need to take the other solution by letting n’=n, b=0
Hence or

27. Chang in Direction of the Ray by Diffraction
Refer to:
R. Minkowski, Astrophys. J. 96, 305 (1942).
T. di Francia, Contributed article on the Ronchi Test, “Optical image evaluation,” NBS Circ. 526, 165 (1954).
J. Guild, The Interference Systems of Crossed Diffraction Gratings, Oxford University Press, New York, (1956).
G. Spencer and M. Murty, “General ray-tracing procedure,” J. OSA, 52(6), (1961).

28. Coordinate Transformation from Object to Reference System
The specified ray through P (x, y, z) oriented along (k, l, m) in the o-system can be expressed in the O-system, the transformation is:
- coordinates of the origin of the o-system measured in the O-system
By noticing the unitarity of the rotational matrix R, with the property that its inverse is the same as its transpose, we find:

29. Minor Project 2 Topics (choose one to work with)
Find the mode of a SOI waveguide with the Si core thickness in 220nm, width in 300nm. The surrounded media is air and the operating wavelength is 1550nm. (Mode Solver)
Find the reflectivity and transmissivity of a 1D SOI waveguide Bragg grating: the Si core thickness is 340nm; the grating depth is 240nm; there are 100 grating pitches with a period of 260nm and a duty cycle of 50%. The surrounded media is air. (Mode Matching)
Find the field distribution along a SOI horn waveguide: the Si core has a thickness of 340nm and is etched on both side with a residue Si thickness in 100nm; its width on the narrow and wide end are 150nm and 450nm, respectively; its length is 30,000nm. The surrounded media is air and the operating wavelength is 1550nm. The incident wave is launched from the narrow end with 100% matching to the local fundamental mode. (BPM)
Design a micro-lens endoscope system to send a Gaussian beam in 650nm with a waist of 650nm for 2m, the lens has an aperture in 2mm and focal length in 1cm. Calculate the power loss. (Ray Tracing)