Slowly Varying Envelope Approximation


Suppose Φ is an electric or magnetic component of the optical electromagnetic field. This component is a periodic (harmonic) function of position, it changes most rapidly along the optical axis, z , and has a period that is on the order of the optical wavelength. The slowly varying approximation involves replacing the quickly varying component, Φ , with a slowly varying one, φ

Φ( x, y, z ) = φ( x, y, z ) exp ( –jkn0 z )                                      (1)

where = 2π ⁄ λ , and n0  is known as the Reference Index. If the light is travelling mostly parallel to the z axis (paraxial approximation), and is monochromatic (wavelength λ ), then it should be possible to select a reference index n0  which makes φ a slowly varying function in all three directions, x , y , and z . If φ is slowly varying, the requirements on the mesh to represent derivatives by finite differences are relaxed. It is possible to choose fewer mesh points to improve the speed of the calculation without compromising the accuracy too much. This is the main reason why BPM can do accurate calculations of light propagation using step sizes many times larger than the optical wavelength, especially in the z  direction.

Equation 1 can also be interpreted to give some indication about what will be required from the mesh, it must be sufficiently fine to approximate spatial derivatives of φ accurately by using finite differences. Since φ varies much more slowly than Φ , significant savings in calculation time can be obtained. On the other hand, if the problem cannot be reduced so simply, then the φ will need to vary more quickly, and the changes must be taken up by making the mesh finer (smaller Δx , Δy , and shorter propagation step Δz ). For many practical waveguides, the refractive index contrast in the transverse plane ( x y plane) is small, which means the variation of the field Φ in the transverse plane is slow, compared to the wavelength. On the other hand, sometimes there are large index contrasts in the transverse plane, and a finer mesh in x , y , or both axes might be needed. Another case in which the effectiveness of Equation 1 is compromised is when some of the light deviates from the direction of the z axis. This is another case in which faster variation in φ is unavoidable, and sometimes a finer mesh or Wide Angle methods are necessary.

In any case, it is recommended to experiment with several different meshes applied to the same problem, and to compare the results. Usually, the same results are found in all cases, but failing this, the results can be analyzed for trends. Eventually a suitable range of mesh parameters can be found which gives consistent results. In OptiBPM, it is very easy to apply many different meshes to the same problem and compare the results.

Sometimes a single value of reference index will not reduce the variation everywhere, but will follow the optical field well in one region of the device. Another region may follow a different Reference Index. For example, in a star coupler, a waveguide connects to a free propagation region. The reference index for the waveguide is the waveguide’s modal index, but the reference index for the coupler would be the slab waveguide’s modal index. The best solution in this case is to define one reference index for one region and a different one for another region. This is easily done in OptiBPM.