(II) Two rays A and B travel down a cylindrical optical...

(II) Two rays $A$ and $B$ travel down a cylindrical optical fiber of diameter $d=75.0 \mu \mathrm{m},$ length $\ell=1.0 \mathrm{km},$ and index of refraction $n_{1}=1.465 .$ Ray A travels a straight path down the fiber's axis, whereas ray $\mathrm{B}$ propagates down the fiber by repeated reflections at the critical angle each time it impinges on the fiber's boundary. Determine the extra time $\Delta t$ it takes for ray $B$ to travel down the entire fiber in comparison with ray A (Fig. $58 ),$ assuming $(a)$ the fiber is surrounded by air, (b) the fiber is surrounded by a cylindrical glass "cladding" with index of refraction $n_{2}=1.460 .$

Key Concepts

Geometric Path Length and Time Delay

In optical fibers, the actual path length a light ray travels depends on its angle of propagation relative to the fiber axis. Rays that zig-zag at the critical angle travel a longer path compared to those moving straight along the axis, leading to a measurable delay in their arrival time. This concept is key in understanding the temporal dispersion effects in fiber optic systems.

Critical Angle

The critical angle is the minimum angle of incidence for which total internal reflection occurs at the interface between two media with differing refractive indices. It is derived from the refractive indices of the media and is central in designing optical fibers, as it determines the range of angles over which light can be guided efficiently within the fiber.

Total Internal Reflection

Total internal reflection is the phenomenon whereby light, when traveling from a medium with a higher refractive index to a medium with a lower refractive index, is completely reflected back into the original medium if it strikes the interface at an angle greater than a certain critical threshold. This principle is essential for keeping the light within the fiber core over long distances.

Refractive Index and Light Propagation

The refractive index of a material quantifies how much the speed of light is reduced inside the medium compared to a vacuum. This factor is critical in determining how light propagates in optical fibers, as variations in refractive index between the core and cladding enable the confinement of light and influence the light’s travel speed along the fiber.

Optical Fiber

An optical fiber is a cylindrical dielectric waveguide that transmits light by confining it through the mechanism of total internal reflection. Its performance depends on parameters such as core diameter, length, and the refractive indices of both the core and surrounding material (cladding or air). This concept is fundamental in understanding how light signals propagate for telecommunications and data transfer applications.

Transcript

00:01 Okay, in this problem we have a fiber optic cable with diameter d equal to 750 micrometers and has a length of one kilometer and we have two rays traveling down this fiber optic cable one travels just along the axis.

00:16 We call this ray a and the second as is in the case of a real fiber optic cable and it's bouncing off the walls of the cable these multiple deflections occur before it gets the end and we're asking to it's asking to find the change in time it takes for both these rays to reach the end of the cable.

00:36 So we're assuming that the medium of a cable has an index of refraction of 1 .465 in both cases.

00:42 And we're asked to look at case a where the outside, medium for the cable is air with indexed refraction of 1.

00:49 And a second case where it has an index of refraction of 1 .460.

00:53 And we're finding delta t in both cases.

00:55 So it starts up, we're going to be making use of snell's law.

00:58 In all cases, says that for a refraction, n1, sine theta 1 for the incoming light is equal to n2, sine theta 2 for the refracted light.

01:11 For a critical angle, for internal reflection, which is definitely what's occurring in a fiber optic cable, it's always the case of the critical angle or theta 2 is equal to 90 degrees.

01:28 So theta 2 is equal to 90 degrees.

01:31 So our sign of 90 degrees just becomes 1.

01:34 And we're left with n2 over n1 equals sign.

01:39 We'll call it theta c.

01:41 This is the same as theta 1.

01:42 It's the critical angle for internal reflection, total internal reflection.

01:49 So we're going to make use of this in a second.

01:51 Now to find the delta t, the difference in time between the two rays, the one traveling just down the axis, the one that's reflecting.

01:58 It's obviously defined as tb minus ta.

02:01 And these times are really just determined by the speed of light and the medium, and the length of the path difference.

02:07 Well, yeah, the length of each path...