what is effective index method which is used to solve modes in rectangular waveguide? I actually want to know the basics of effective index methods for solving the optical modes in slab/rectangular waveguides.
 A: The effective index method is an analytical method applicable to complicated waveguides such as ridge waveguides and diffused waveguides.
The ridge waveguide, such as shown in Figure Below, is difficult to analyze with simple method since the waveguide structure is too complicated to deal with by the division of waveguide. In order to analyze ridge waveguides, you should use numerical methods, such as the finite element method and finite difference method. 

The wave equation for $E_x$ is given by:
$$\frac{\partial^2H_y}{\partial x^2}+\frac{\partial^2 H_y}{\partial y^2}+[k^2n^2(x, y)-\beta^2]H_y=0$$
The basic assumption of the effective index method is that the electromagnetic field can be expressed, with the separation of variables, as:
$$H_y(x,y)=X(x)Y(y)$$
Substituting this above and divided by XY
$$\frac{1}{X}\frac{d^2 X}{dx^2}+\frac{1}{Y}\frac{d^2 Y}{dy^2}+[k^2n^2(x, y)-\beta^2]=0$$
Here we add to and subtract from the equation above the independent value of $k^2n_{eff}^2$
and separate the equation into two independent equations:
$$\frac{1}{Y}\frac{d^2 Y}{dy^2}+[k^2n^2(x, y)-k^2n_{eff}^2(x)=0$$$$\frac{1}{X}\frac{d^2X}{dx^2}+[k^2n_{eff}^2(x)-\beta^2]=0$$
Now we have two equation with one variable that can be applied to x and y boundary and in this way discontinuity of refractive index can be treated simpler.

