Total Internal Reflection, Optical Fibres and Dispersion

Start with the direction of incidence. Call the incident material's refractive index n1 and the other material's index n2. These numbers describe optical propagation, not mass density. Total internal reflection requires n1 > n2.

Measure incidence from the normal, perpendicular to the surface. As incidence increases toward the critical angle θc, the transmitted ray approaches the boundary direction. At the critical limit its angle from the normal is 90 degrees. Snell's law then gives sin θc = n2 ÷ n1. Use inverse sine to find θc.

Above θc, an ideal isolated boundary between transparent, nonabsorbing materials has no propagating transmitted ray carrying light across it. An evanescent field can extend into the lower-index material, so saying that no field exists there would be inaccurate.

An angle measured from the surface must be converted before comparison. For example, 35 degrees from the surface is 55 degrees from the normal. A larger normal angle is closer to grazing incidence. At exactly θc the refracted direction is the 90-degree limit; total internal reflection is the above-critical case.

A step-index fibre places a higher-index core inside lower-index cladding. In the ray model, a guided ray strikes the core boundary above the critical angle. The index step permits repeated total internal reflection, but it does not make every launch direction suitable.

A multimode fibre can guide different paths or modes with different travel times. Parts of one pulse can arrive at different times even while remaining guided. This is modal dispersion. Chromatic dispersion is a separate timing effect: different wavelength components have different pulse travel times.

If pulses spread enough to overlap, the receiver can no longer separate neighbouring signals at the original data rate. Single-mode operation removes modal dispersion, but chromatic dispersion can remain. A graded-index multimode design can reduce modal timing differences; it does not automatically remove chromatic dispersion.

Signal loss is another measurement. Absorption, scattering and excessive bending can reduce the light reaching the receiver. A broader output pulse does not by itself prove that light escaped through the cladding, nor does it prove that no loss occurred.

White light contains many wavelengths. A transparent prism can send them in different directions because its refractive index varies with wavelength. Apply Snell's law separately to each wavelength to find its direction. The prism separates colours already present in the light; it does not create them by changing the source frequency.

For ordinary transparent glass with normal dispersion across visible wavelengths, blue has a higher index than red and a greater total deflection through the prism. That ordering belongs to the stated material and wavelength range, not to every possible optical medium.

The formula c ÷ n gives phase speed, the speed of a fixed phase of a wave. A pulse contains a range of frequencies, and its envelope travels at group velocity. Wavelength-dependent pulse travel times can broaden a fibre pulse. Phase-speed ranking alone is not a complete rule for pulse arrival order.

Prism dispersion separates wavelengths by direction. Chromatic dispersion in a fibre can separate pulse components by arrival time. Both depend on wavelength, but the measured effects differ.