**Graded-Index Fibers**

Guiding light rays with multiple lenses or mirrors is possible, however this may result in great loss of optical power due to refraction in a system if there are many lenses or mirrors. Using total internal reflection however, rays may be transmitted over long distances without these losses. Glass fibers are the primary choice for guiding light in this manner using total internal reflection. Glass fibers consist of a glass wire with a cladding. The refractive index of the outer cladding will be smaller than the glass core. This allows for a consistent total internal reflection throughout the wire.

A graded-index material (GRIN) has a refractive index that varies throughout the material. When a ray moves through a graded-index material, the variance in refractive index causes the ray to bend and curve according to how the graded index is laid out.

The path of an optical ray in graded-index material is determined by Fermat’s principle, which states that the path of a ray is the integral of the refractive index (a function of position on the material) between two points, equated to zero. The ray equation can solve this problem, however for simplification, a paraxial approach is taken to give the paraxial ray equation.

Ray Equation:

Paraxial Ray Equation:

A graded index glass fiber is modeled below:

**Matrix Optics**

A paraxial ray is described by a coordinate and angle. Using this approximation, the output paraxial ray going through a system can be written in matrix form:

,

An optical system can be modeled using the 2×2 ABCD matrix. Matrices of systems may also be cascaded to describe the effect of many systems on a ray.

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