Fibre Optics and Endoscopy

The total internal reflection principle

Total internal reflection is the optical event in which light waves reach the boundary between two media. The light is not refracted to the opposite medium but is completely reflected back to the initial medium instead. This phenomenon occurs when the opposite medium has a higher refractive index than the initial medium.

We can calculate the refractive index of a medium with this equation:

\[n = \frac{c}{v}\]

Here, c is the speed of light in a vacuum (3.00 ⋅ 10⁸m/s), and v is the phase velocity of light in the medium.

For this phenomenon to occur, the angle of incidence at the boundary between two mediums must be greater than the critical angle. Let’s say that a light beam travels from a medium that has a refractive index of n₁ to a second medium with a refractive index of n₂, and n₂ ≤ n₁. We can then find the critical angle (θ_c) with this equation:

\[\theta_c = \arcsin \Big(\frac{n_2}{n_1} \Big)\]

When the angle of incidence is lower than the critical angle

If the angle of incidence (θ₁) is lower than the critical angle, then the wave will refract to the opposite medium with an angle of θ₂ (yellow arrows in the illustration below). We can now find the relation between these angles with this equation:

\[n_1 \cdot \sin \theta_1 = n_2 \sin \theta_2\]

When the angle of incidence is equal to the critical angle

If the angle of incidence (θ₁) is equal to the critical angle (θ_c), then the wave will continue on its way in alignment with the boundary (purple arrows in the illustration).

When the angle of incidence is greater than the critical angle

If the angle of incidence is greater than the critical angle, total internal reflection occurs, and the wave will reflect back to the initial medium with the same angle it had initially (red arrows in the illustration).

Fibre optic endoscopy procedure

When using endoscopes with fibre optics, a light source in the tip of the endoscope illuminates the passageways. The light is reflected from the passageways back into this tip. Light is guided by a step-index structure inside the optical fibre that induces total internal reflection and directs coupled light from one end to the other without loss. For this to happen, the refractive index of the fibre core material (n₁) must be greater than that of the fibre cladding material (n₂).

Light with an incidence angle greater than the maximum acceptance angle θ_a will be partially refracted outside of the fibre. Light with an incidence angle less than the maximum acceptance angle θ_a will be partially refracted inside the fibre. After several encounters with the fibre core-cladding interface, the residual partly reflected light is finally lost.

The requirement for 100% internal reflection at the core-cladding contact determines the maximum acceptance angle θ_a. To determine the acceptance angle θ_a, remember that to find the critical angle θ_c, we use the equation:

\[\sin(\theta_c) = \frac{n_2}{n_1}\]

Thus, assuming the fibre is in the air with a refractive index of 1, the following equation helps us find the acceptance angle θ_a. Note that θ_c' = 90° - θ_c.

\[\sin (\theta_a) = \frac{n_1}{1} \cdot \sin(\theta'_c)\]

In conclusion, the light wave, which reflects from the passageways of the human body and enters the optical fibres in the endoscope, travels through the optical fibres with total internal reflection. As a result, the light exits the optical fibres in the same way it entered them (with the same angle). The image is then transferred for a viewer to examine.

What are the applications of endoscopy?

Endoscopy is currently used in a variety of applications in medical physics. While endoscopy is mainly used in surgeries to allow surgeons to visualise the interior of the human body while operating, other uses include looking into the passageways of the human body to determine any abnormalities. Here are some other uses of endoscopy:

• Panendoscopy for plastic surgery (triple endoscopy).
• Orthopaedic surgeries, such as knee surgeries and hand surgery.
• Examining the foetus and the amnion during pregnancy.
• Examining the female reproductive system.
• Looking into the interior of the ear.