2. Those Interfering Signals Modes and Dispersion in Fibers

3. Review Light is trapped in an optical fiber if it strikes the sides of the fiber at angles greater than the critical angle for the core-cladding interface The core must have a higher index of refraction than the cladding for total internal reflection to occur. The numerical aperture (NA) of a fiber relates the maximum angle of incidence on the front of the fiber to the indices of refraction of the fiber: NA = n 0 sin  m = (n 1 2 - n 2 2 ) 1/2.

4. Review (cont.) Any periodic function of frequency f 0 can be expressed as a sum over frequency of sinusoidal waves having frequencies equal to nf 0, where n is an integer. The sum is called the Fourier series of the function, and a plot of amplitude (coefficient of each sin/cos term) vs. frequency is called the Fourier spectrum of the function. Any non-periodic function (so frequency f 0  0) can be expressed as an integral over frequency of sinusoidal waves having frequencies. The integral is called the Fourier transform of the function, and a plot of amplitude vs. frequency is called the Fourier spectrum of the function. The Fourier spectrum of a wider pulse will be narrower than that of a narrow pulse, so it has a smaller bandwidth.

5. What Exactly Is Bandwidth, and Why Do We Care? A range of frequencies Generally found by taking the frequencies with amplitudes more than half the maximum amplitude (e.g., on a Fourier spectrum) Bandwidth for a medium is the range of frequencies which can pass through that medium with a minimum of separation Sampling theory says that a signal transmitting N different amplitudes per second requires a bandwidth of at least N/2: B>N/2 Usually this ideal is not achieved, and the required bandwidth is larger –Grant says B approx N

6. Pulses and Data Can represent binary data with pulses in a variety of ways 10110 could look like... Non-return-to-zero (NRZ) Return-to-zero (RZ) Manchester Coding Bipolar Coding Notice that the NRZ takes half the time of the others for the same pulse widths

7. Phase differences and interference Light rays taking different paths will travel different distances and be reflected a different number of times Both distance and reflection affect the how rays combine Rays will combine in different ways, sometimes adding and sometimes canceling ii 00 n0n0 n1n1 n2n2

8. Modes Certain combinations of rays produce a field that is uniform in amplitude throughout the length of the fiber These combinations are called modes and are similar to standing wave on a string Every path can be expressed as a sum of modes (like Fourier series) ii 00 n0n0 n1n1 n2n2

9. Creating a Mode (Figures adapted from Photonics – not to scale) The resulting pattern is uniform throughout the length of the fiber – this is a mode of the fiber

10. Modes in a Fiber (Figures adapted from Photonics – not to scale) The field distributions of successive modes look like the harmonics of standing waves! – the phenomena are very similar Mode 1: Electric Field across the fiber Mode 1: Intensity across the fiber ~E 2 Mode 2: Intensity across the fiber ~E 2 Mode 3: Intensity across the fiber ~E 2 Mode 2: Electric Field across the fiber Mode 3: Electric Field across the fiber

11. Modes Combine to Give Path of Light (Figures adapted from Photonics – not to scale) To add Mode 1 and Mode 2, must add fields. BUT, modes travel at different speeds, so sum of fields changes as go down the fiber Result is one of the paths light will take Mode 1 Mode 2 2 Intensity Pattern of Sum

12. Modal Dispersion Since different modes travel different distances in the fiber, they will arrive at the end at different times. For graded-index fibers, not only do different modes travel different distances, they travel through different media!

13. Reducing the number of Modes Different modes interact differently with the fiber, so modes will spread out, or disperse If the fiber is narrow, only a small range of  0 will be able to enter, so the number of modes produced will decrease A small enough fiber can have only a single mode BUT, you will lose efficiency because not all the light from the source enters the fiber. ii 00 n0n0 n1n1 n2n2

14. Do the Activity Work as far as you can before Dr. Persans arrives

15. Before the next class,... Re-Read Chapter 3-4 of Grant, focusing on discussion of modes and of different types of dispersion. Start Homework 7, due next Thursday Do Activity 05 Evaluation by Midnight Friday.