## Single Grating Formula

A grating can be represented by the formula that combines a grating shape function,

an average index modulation function, a period chirp function, and an apodization

function:

where

*n _{0}* – waveguide refractive index

*Δn *– index modulation amplitude

θ -grating tilt angle

*ƒ[Λ(z)/cosθ, z]* – shape function

*Δn _{0}(z) *– average index modulation function

*Λ(z) *– period chirp function

*Λ(z) *– apodization function

P – Photosensitivity profile of waveguide

In Equation 15, the average index modulation function *Δn _{0}* is applied only in the

region of the grating, i.e. in layers with non-zero photosensitivity. Layers with very

small photosensitivity will respond to average index modulation, but the grating will be

very weak there.

If the User wants to apply the Average Index change to a layer that does not contain

the grating, he should enter a very small number (.01) for the Photosensitivity of that

layer.

Detailed definitions are given below.

## Grating Shape

In OptiGrating, the Grating Shape is the profile of index modulation within one grating

period Λ.

The default grating shapes are:

- Rectangle

- Sine

Other grating shapes can be supplied from a user file or defined by user functions.

## Average Index Modulation

In OptiGrating, the Average Index Modulation is the average index change along the

grating length. The default average index modulations are listed below. The symbol

delta (Δ) is the total index change along the grating.

- Uniform

- Linear

Other average index modulations can be supplied from a user file or defined by user

functions.

## Grating Period Chirp

Grating Period Chirp is the distance-dependent grating period modulation. The

default Period Chirp options are listed below, where the constant chirp term is the

grating period in the middle of the grating, L is the grating length, and the symbol delta

*(Δ)* stands for the total chirp.

- No chirp (constant period):

- Linear:

- Quadratic:

- Square root:

- Cubic root:

Other chirp forms can be supplied from a user file or defined by user functions.

## Grating Apodization

The Grating Apodization is the slowly varying envelope of the grating profile.

The default apodization options are listed below, where s is the taper parameter and

L is the grating length.

- Uniform (no apodization):

- Gaussian:

- Hyberbolic tangent:

Other chirp forms can be supplied from a user file or defined by user functions.

## Phase Shift

The Phase Shift element is a piece of the current waveguide structure without the

grating.

The Phase Shift element can be defined alternatively by one of the two parameters:

- Phase – phase change accumulated in the Phase Shift element
- Length – length of the element

## Multiple Gratings

You can simulate a collection of gratings and phase shifts in one waveguide device.