$Light refracting through a prism creating a spectrum$

Total internal reflection and fiber optics

PHYS 310 · Geometric Optics

Total internal reflection occurs when light tries to pass from higher to lower refractive index at a large angle. This lesson explains critical angle, evanescent fields, and fiber optic guidance.

Key equations

n_1>n_2n_1\sin\theta_1=n_2\sin\theta_2\theta_2=90^\circn_1\sin\theta_c=n_2\sin 90^\circ\sin\theta_c=\frac{n_2}{n_1}\theta_1>\theta_cn_{core}>n_{clad}NA=\sqrt{n_{core}^2-n_{clad}^2}

Learning objectives

Define the critical angle.
Apply the condition for total internal reflection.
Explain why total internal reflection requires light to go from higher to lower index.
Describe evanescent fields conceptually.
Explain how optical fibers guide light.

Refraction away from the normal

When light travels from a higher-index medium into a lower-index medium, it bends away from the normal. For example, light traveling from glass into air has

$n_1>n_2$

Snell's law gives

$n_1sin heta_1=n_2sin heta_2$

As the incident angle $heta_1$ increases, the refracted angle $heta_2$ increases even more.

Eventually, the refracted ray would need to emerge at $90^circ$ to the normal. This defines the critical angle.

Critical angle

At the critical angle $heta_c$ ,

$heta_2=90^circ$

$n_1sin heta_c=n_2sin 90^circ$

Since $sin 90^circ=1$ ,

$sin heta_c= rac{n_2}{n_1}$

This formula applies only when $n_1>n_2$ . If light tries to go from lower index to higher index, total internal reflection does not occur.

Total internal reflection

For incident angles greater than the critical angle,

$heta_1> heta_c$

there is no transmitted ray in the usual geometric sense. Instead, all the light reflects back into the higher-index medium. This is total internal reflection.

The reflection is not due to a metallic coating. It is a wave effect caused by boundary conditions when no propagating refracted wave is allowed in the lower-index medium.

Evanescent field

Even during total internal reflection, the electromagnetic field slightly penetrates into the lower-index medium. This non-propagating field is called an evanescent field. Its amplitude decays rapidly with distance from the boundary.

If another material is brought very close, light can tunnel across the gap. This is called frustrated total internal reflection.

Fiber optics

An optical fiber guides light by total internal reflection. A typical fiber has a core with refractive index $n_{core}$ surrounded by cladding with slightly lower index $n_{clad}$ :

$n_{core}>n_{clad}$

Light entering within an allowed range of angles reflects repeatedly at the core-cladding boundary and remains trapped in the core.

Numerical aperture

The range of input angles accepted by a fiber is described by numerical aperture. For a step-index fiber in air, an idealized expression is

$NA=sqrt{n_{core}^2-n_{clad}^2}$

A larger numerical aperture means the fiber can accept light over a wider cone of angles.

Communication through fibers

Fiber optic communication sends information using pulses or modulated light signals. Fibers are useful because they have low loss, high bandwidth, immunity to electromagnetic interference, and small size.

Signals can travel long distances, though they may require amplification or regeneration. Dispersion can spread pulses, limiting data rates if not controlled.

Other applications

Total internal reflection appears in binoculars, prisms, endoscopes, decorative lighting, sensors, and medical instruments. Diamonds sparkle partly because their high refractive index gives a small critical angle, making internal reflections likely.

The big idea

Total internal reflection occurs when light in a higher-index medium reaches a lower-index boundary at an angle above the critical angle. The critical angle satisfies $sin heta_c=n_2/n_1$ . This phenomenon allows optical fibers to guide light efficiently and underlies many optical technologies.

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