Theoretical Competition-Problem No. 2
Optical fiber
x
x
n =1
0 n
2
O
a n
1
O z θ z
i
An optical fiber consists of a cylindrical core of radius a, made of a transparent
material with refraction index varying gradually from the value n=n on the axis to n = n
1 2
(with 1<n <n ) at a distance a from the axis, according to the formula
2 1
2 2
n=n(x)= n 1−α x
1
where x is the distance from the core axis and α is a constant. The core is surrounded by a
cladding made of a material with constant refraction index n . Outside the fiber is air, of
2
refractive index n .
0
Let Oz be the axis of the fiber, with O - the center of the fiber end.
Given n =1.000; n =1.500; n = 1.460, a = 25 µm.
0 1 2
1. A monochromatic light ray enters the fiber at point O under an incident angleθ, the
i
incident plane being the plane xOz.
a. Show that at each point on the trajectory of the light in the fiber, the refractive index n
and the angle θ between the light ray and the Oz axis satisfy the relationship ncosθ=C
where C is a constant. Find the expression for C in terms of n and θ. [1.0 points]
1 i
1
b. Use the result found in 1.a. and the trigonometric relation cosθ= ( 1+tan2 θ )− 2 , where
dx
tanθ= = x' is the slope of the tangent to the trajectory at point (x, z), derive an equation
dz
for x'. Find the full expression for α in terms of n , n and a. By differentiating the two sides
1 2
of this equation versus z, find the equation for the second derivative x''. [1.0 points]
c. Find the expression of x as a function of z , that is x = f(z), which satisfies the above
equation. This is the equation of the trajectory of light in the fiber. [1.0 points]
d. Sketch one full period of the trajectories of the light rays entering the fiber under two
different incident anglesθ. [1.0 points]
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2. Light propagates in the optical fiber.
a. Find the maximum incident angleθ , under which the light ray still can propagate
iM
inside the core of the fiber. [1.5 points]
b. Determine the expression for coordinate z of the crossing points of a light ray with Oz
axis for θ ≠ 0. [1.5 points]
i
3. The light is used to transmit signals in the form of very short light pulses (of negligible
pulse width).
a. Determine the time τ it takes the light to travel from point O to the first crossing point
with Oz for incident angle θ ≠ 0 and θ ≤θ .
i i iM
The ratio of the coordinate z of the first crossing point and τis called the propagation
speed of the light signal along the fiber. Assume that this speed varies monotonously with θ.
i
Find this speed (called v ) for θ =θ .
M i iM
Find also the propagation speed (called v ) of the light along the axis Oz.
0
Compare the two speeds. [3.25 points]
b. The light beam bearing the signals is a converging beam entering the fiber at O under
different incident angles θ with 0≤θ ≤θ . Calculate the highest repetition frequency f of
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the signal pulses, so that at a distance z = 1000 m two consecutive pulses are still separated
(that is, the pulses do not overlap). [1.75 points]
Attention
1. The wave properties of the light are not considered in this problem.
2. Neglect any chromatic dispersion in the fiber.
3. The speed of light in vacuum is c = 2.998×108 m/s
4. You may use the following formulae:
• The length of a small arc element ds in the xOz plane is
dx
2 ds
dx
ds =dz 1+
dz dz
dx 1 bx
•∫ = Arcsin
2 2 2 b a
a −b x
bx
a2Arcsin
•∫
x2dx
=−
x a2 −b2x2
+ a
a2 −b2x2 2b2 2b3
•Arcsinxis the inverse function of the sine function. Its value equals the less angle the
sine of which is x. In other words, ify = Arcsinx then siny = x.