a) A stepped optical fiber consists of a central cylindrical core of refractive index n1 and a cladding of refractive index n2. Show that the maximum acceptable angle at which light enters the plane end of the core of the fiber from air is
èmax = (sin^-1) x [(n1^2)-(n2^2)]^-1 . Assume that the refractive
index of air is n0=1.0.
b) A sharp pulse of light is launched into a stepped optical fiber of length L. If the maximum range of angles for light entering the fiber is used, show that the width of the pulse on leaving the fiber will be
Ät = [L(n1)/c] x [(n1 /n2)- 1] . Note that the width of the pulse means the difference between the longest and the shortest times of travel.
a) To derive the expression for the maximum acceptable angle at which light enters the plane end of the core of the fiber from air, we need to consider the critical angle of refraction.
1. Snell's Law:
Snell's Law states that the ratio of the sines of the angles of incidence and refraction is equal to the ratio of the refractive indices. Mathematically, it is given by:
n0sinθ0 = n1sinθ1
where n0 is the refractive index of air, θ0 is the angle of incidence, n1 is the refractive index of the core, and θ1 is the angle of refraction.
2. Maximum Acceptable Angle:
The maximum acceptable angle is the angle of incidence (θ0) at which the light enters the plane end of the core from air so that the angle of refraction (θ1) is at the critical angle (θc), which is the largest angle at which the light can be totally internally reflected within the fiber. Total internal reflection occurs when θ1 > θc.
3. Critical Angle:
The critical angle (θc) is given by:
sinθc = (n2 / n1)
where n2 is the refractive index of the cladding and n1 is the refractive index of the core.
To find the maximum acceptable angle (θmax), we need to find the value of θ0 at which θ1 = θc.
Using Snell's Law:
n0sinθ0 = n1sinθc
Rearranging the equation:
θ0 = arcsin[(n1 / n0) * sinθc]
Since n0 = 1.0 (refractive index of air):
θ0 = arcsin[(n1 / 1.0) * sinθc]
Simplifying:
θ0 = arcsin[(n1^2 - n2^2)^0.5]
Therefore, the maximum acceptable angle (θmax) at which light enters the plane end of the core of the fiber from air is:
θmax = arcsin[(n1^2 - n2^2)^0.5]
b) To determine the width of the pulse on leaving the stepped optical fiber, we need to consider the different path lengths for the pulse traveling at different angles within the fiber.
1. Time of Flight:
The time of flight for light traveling through the fiber can be calculated by dividing the path length traveled by the speed of light in the fiber. Mathematically, it is given by:
t = L / c
where t is the time of flight, L is the length of the fiber, and c is the speed of light in the fiber.
2. Width of the Pulse:
The width of the pulse is the difference between the longest and the shortest times of travel. Since the maximum range of angles for light entering the fiber is used, the pulse experiences the longest and shortest times of travel.
The longest time of travel occurs when the light travels along the longest path within the fiber, which corresponds to the maximum acceptable angle (θmax). The shortest time of travel occurs when the light travels along the shortest path within the fiber, which corresponds to the critical angle (θc).
Using the time of flight equation, the width of the pulse (Δt) can be calculated as:
Δt = t(θmax) - t(θc)
= (L / c) - (L / c)
= 0
Therefore, the width of the pulse on leaving the stepped optical fiber is zero, indicating no temporal dispersion occurs.