An optical fiber is a flexible, transparent fiber devised to transmit light between the two ends of the fiber. Complete transmission of light is achieved through total internal reflection. This problem aims to calculate the minimum index of refraction n of the optical fiber necessary to obtain total internal reflection for every possible incidence angle.
(a) Express sinθ, where the angle θ is defined in the figure above, in terms of the incidence angle α and the index of refraction n of the optical fiber. Evaluate this function for n=1.5 and α=70∘. Take the index of refraction of air to be 1.
(b) The condition on n for total internal reflection of all beams entering the fiber is achieved when θ=90∘ for all values of α. Determine the smallest value of n that satisfies that condition.
To solve this problem, we need to understand the concept of total internal reflection and the relationship between angles and indices of refraction in optical fibers.
In an optical fiber, total internal reflection occurs when light traveling within the fiber encounters an interface with a medium of lower refractive index at an angle greater than the critical angle. At angles less than the critical angle, some of the light is refracted out of the fiber, resulting in loss of light transmission.
(a) To express sinθ in terms of the incidence angle α and the index of refraction n, we can use Snell's Law, which relates the angles and indices of refraction of two media.
Snell's Law states that n₁sin(α) = n₂sin(θ), where n₁ and n₂ are the indices of refraction of the two media, and α and θ are the angles of incidence and refraction, respectively.
For total internal reflection to occur, the angle of refraction θ must be 90 degrees. Therefore, sinθ = 1.
Substituting into Snell's Law, we get:
n₁sin(α) = n₂sin(θ)
⇒ n₁sin(α) = n₂(1)
Since we want to calculate sinθ, we need to express n₂ in terms of n, the index of refraction of the fiber. Since the fiber is surrounded by air, n₂ = 1 (refractive index of air).
Therefore, the equation becomes:
n₁sin(α) = n(1)
To evaluate this function for n = 1.5 and α = 70 degrees, we substitute the values:
1.5sin(70) = 1(1)
⇒ 1.5 x 0.9397 = 1
Hence, sinθ = 0.9397.
(b) To determine the smallest value of n that satisfies the condition θ = 90 degrees for all values of α, we need to find the critical angle. The critical angle is the angle of incidence at which light is refracted along the interface, resulting in the angle of refraction being 90 degrees.
The critical angle can be calculated using the equation:
sin(critical angle) = (n₁/n₂)
Substituting n₁ = n (as the index of the fiber) and n₂ = 1 (refractive index of air):
sin(critical angle) = n/1
⇒ sin(critical angle) = n
We want sin(critical angle) to be equal to or greater than 1 (since sin values range from -1 to 1). Therefore, the smallest value of n that satisfies the condition θ = 90 degrees for all values of α is 1.