Planar Dielectric Waveguides
| Previous Topic | Return to Optical Fibers, Waveguides, and Lasers Menu | Next Topic |
Planar dielectric waveguides
Planar (slab) waveguides are the basis of waveguides used in integrated optoelectronics. The same mathematical ideas can be applied (with minor modifications) to circular waveguides.
The waveguide consists of a semi-infinite slab of dielectric materials with thickness d and refractive index n1 (the core) that is sandwiched between two regions (the cladding) both of refractive index n2, and where n1>n2.
A ray of light may propagate down the core provided that total internal reflection occurs at the core/cladding interface.
this requires that:
- <math>90\deg >\theta_i > \theta_c\,\!</math>
Where
- <math>\theta_i\,\!</math> is the internal ray angle (from now on written as theta)
In fact there are ‘infinite” number of rays, all slightly displaced from each other, also propagating down the guide. The dotted line that is perpendicular to the wave lines is the wavefront of the propagating beam. The rays represent lines drawn normally to the plane wavefronts.
The wavefront FC intersects tow the upwardly travelling portions of the same ray at points A and C. Therefore the phase at C and A must be the same or differ by a multiple of 2pi.
Otherwise there would be destructive interference between out-of-phase waves and the light will not propagate. It also requires very specific angles theta above the critical angle.
Consider the phase difference between A and C
There are two factors -the path length of AB + BC -the phase change due to reflection at B and C
We write the phase change resulting from reflection simply as delta(theta) For perpendicular radiation phi (theta) is 2 psi, for parallel radiation phi(theta) = 2 delta.
Total phase change
- <math>(AB + BC) \frac {2\pi n_1 }{\lambda_0} - 2 \phi (\theta)\,\!</math>
Where
- <math>Lambda_0\,\!</math> is the wavelength of light in the medium
To determine the path of the light from a to b to c using trigonometry:
- <math>AB = BC cos 2 \theta\,\!</math>
Thus
- <math>AB + BC = BC(1+ cos2 \theta)\,\!</math>
Since
cos 2theta = 2 cos^2theta -1
- <math>AB + BC = 2 BC cos^2 \theta\,\!</math>
- <math>BD = BC cos \theta = d\,\!</math> that is the thickness of the slab
So that
- <math>AB +BC = 2d cos \theta\,\!</math>
The thickness of the slab determines the number of modes or angles that light will propagate.
In order for the mode to propagate the total phase change must be a multiple of 2pi:
- <math>
\frac {4\pi n_1d cos \theta } {\lambda_0} -2 \phi (\theta) = 2m \pi\,\!</math>
- <math>\frac {2\pi n_1d cos \theta} {\lambda_0} -\phi (\theta) = m \pi\,\!</math>
Where
m is an integer
So for each value of m there will be an angle theta_m that satisfies the equation
Each value of theta m (>theta c) has a distinct distribution of electric field across the guide. This distribution is known as a mode. Depending on the mode there may a distribution that is centered in the core or may have 2 spots, 4 spots etc.
When \theta m is = the \theta c : the mode is at cut-off
If
\theta m < theta c : the mode is below cutoff resulting in rapid attenuation and light will not be propagated.
If
\theta m > \theta c : the mode is above cut-off which can propagate