Question

An optical fiber with an index of refraction of 1.50 is used to transport light of wavelength \(400 . \mathrm{nm}\). What is the critical angle for transporting light through this fiber without loss? If the fiber is immersed in water? In oil?

Short answer

Explain how the critical angle changes for each situation. Answer: The critical angle for an optical fiber with an index of refraction of 1.50 is not applicable when immersed in air, as the light will always be trapped inside the fiber (total internal reflection). When the optical fiber is immersed in water, the critical angle is approximately 80.5°. When the fiber is immersed in oil, the critical angle is approximately 78.5°. The critical angle decreases as the index of refraction of the surrounding medium increases (from air to water to oil), due to the change in the relationship between the indices of refraction and the resulting angles of incidence and refraction according to Snell's Law.

Step-by-step solution

Recall the formula for Snell's Law and the concept of the critical angle

Snell's Law states that the product of the index of refraction of a material and the sine of the angle of incidence (θ1) is equal to the product of the index of refraction of the other material and the sine of the angle of refraction (θ2): n1 * sin(θ1) = n2 * sin(θ2) The critical angle (θc) is the angle of incidence when the angle of refraction is 90°. At this angle, light is refracted at the boundary between the two materials rather than passing into the second material.

Calculate the critical angle for the optical fiber in air

To find the critical angle for the optical fiber in air, we need to use the index of refraction of air (n1 = 1) and the index of refraction of the optical fiber (n2 = 1.50). Since we are looking for the critical angle, the angle of refraction (θ2) is 90°. Snell's Law in this case is: 1 * sin(θc) = 1.50 * sin(90°) sin(θc) = 1.50 / 1 θc = arcsin(1.50) (but this gives an error, as the arcsin function accepts values from -1 to 1 only) Since the index of refraction of the optical fiber is greater than that of air, the light will always be trapped inside the fiber (total internal reflection) and there will not be any loss.

Calculate the critical angle for the optical fiber in water

To find the critical angle for the optical fiber in water, we need to use the index of refraction of water (n1 = 1.33) and the index of refraction of the optical fiber (n2 = 1.50). The angle of refraction (θ2) is again 90°. Snell's Law in this case is: 1.33 * sin(θc) = 1.50 * sin(90°) sin(θc) = 1.50 / 1.33 θc = arcsin(1.50 / 1.33) θc ≈ 80.5° The critical angle for transporting light through the optical fiber without loss when the fiber is immersed in water is approximately 80.5°.

Calculate the critical angle for the optical fiber in oil

To find the critical angle for the optical fiber in oil, we need to use the index of refraction of oil (n1 ≈ 1.45) and the index of refraction of the optical fiber (n2 = 1.50). The angle of refraction (θ2) is again 90°. Snell's Law in this case is: 1.45 * sin(θc) = 1.50 * sin(90°) sin(θc) = 1.50 / 1.45 θc = arcsin(1.50 / 1.45) θc ≈ 78.5° The critical angle for transporting light through the optical fiber without loss when the fiber is immersed in oil is approximately 78.5°.

Key concepts

Snell's Law

Snell's Law is the scientific principle that describes how light bends, or refracts, as it travels from one medium to another. The law can be expressed with the equation:
\[\begin{equation} n1 \times \text{sin}(\theta1) = n2 \times \text{sin}(\theta2) \end{equation}\]
where 'n1' and 'n2' represent the indices of refraction of the first and second media, and '\theta1' and '\theta2' are the angles of incidence and refraction, respectively.

This fundamental concept in optics helps predict the path light will take when it encounters a boundary between different materials, which is essential for the design of lenses, glasses, and, as in our exercise, optical fibers. The ability of Snell's Law to determine the bending of light at various angles and through different substances makes it a cornerstone of understanding optical phenomena.

Total Internal Reflection

Total internal reflection (TIR) is an optical event that occurs when light strikes the boundary of a less dense medium at an angle greater than the critical angle. At this point, all the light is reflected back into the denser medium, and none passes through the boundary. This phenomenon is described mathematically by Snell's Law but requires a critical caveat: total internal reflection only happens when light travels from a medium with a higher index of refraction to one with a lower index.

In optical fibers, TIR is harnessed to transmit light signals over long distances without significant loss. The exercise demonstrates how knowing the critical angle, which delineates the onset of TIR, is pivotal to preventing light loss in fiber optic cables, especially when the external environment changes to other substances like water or oil.

Index of Refraction

The index of refraction, denoted as 'n', is a dimensionless number that describes how fast light travels in a medium relative to the speed of light in a vacuum. Materials with a higher index of refraction slow down light more than those with a lower index. The index of refraction is not only a critical player in calculating the bending of light, as seen in Snell's Law, but also determines the critical angle for total internal reflection.

For example, the optical fiber in our exercise has an index of refraction of 1.50, which indicates how much it slows down the light relative to air (with an index close to 1). This property impacts how light is guided through the fiber and where it could potentially escape, based on the surrounding medium's refractive index.

Wavelength of Light

Wavelength is the distance between successive peaks of a wave, which in the case of light, dictates its color when it is in the visible spectrum. Typically measured in nanometers (nm), the wavelength of light is a critical factor in many optical phenomena and can influence the index of refraction.

In our textbook problem, although light of a 400 nm wavelength is mentioned, the critical angles calculated are independent of the wavelength. However, in real-world applications, the wavelength can affect the attenuation and dispersion within an optical fiber, which are important considerations in the design and use of fiber optic systems. It's useful to understand how wavelength interacts with materials to fully grasp the complexities of light transmission.