2. Review First Exam

3. What have we learned? Any traveling sinusoidal wave may be described by y = y m sin(kx   t +  ) Light always reflects with an angle of reflection equal to the angle of incidence (angles are measured to the normal). When light travels into a denser medium from a rarer medium, it slows down and bends toward the normal. The Fourier spectrum of a wider pulse will be narrower than that of a narrow pulse, so it has a smaller bandwidth. Your bandwidth B must be as large as the rate N at which you transfer different amplitudes. The rise time of each pulse must be no more than 70% of the duration of the pulse

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 Else Have We Learned? Can represent binary data with pulses in a variety of ways 10110 could look like... Non-return-to-zero (NRZ) Return-to-zero (RZ) Bipolar Coding Notice that the NRZ takes half the time of the others for the same pulse widths Other schemes use tricks to reduce errors and BW requirements.

6. Optical Waveguides Summary Dispersion means spreading Signals in a fiber will have several sources of dispersion: –Chromatic: Material: index of refraction depends on wavelength (prism) Waveguide: some of wave travels through cladding with different index of refraction (primarily single-mode) – leads to wavelength-dependent effects –Modal: different modes travel different paths and so require different amounts of time to travel down fiber (CUPS) Also have attenuation/loss due to scattering/absorption by fiber material, which depends on wavelength/frequency

7. Optical Waveguide Summary (cont.) Modes in a fiber are specific field distributions that are independent of “z”, or length traveled down the fiber Fields of modes look like harmonics of standing waves Can make a single-mode fiber by: –reducing diameter of fiber so smaller cone of light enters –reducing NA of fiber so smaller cone of light is trapped

8. Interference of Waves  If crests match crests, then waves interfere constructively  Crests will match if waves are one wavelength, two wavelengths, … apart: path difference = m A max 2A max wave 1 wave 2 sum A max

9. Destructive Interference  If crests match troughs (180° out of phase), then waves interfere destructively  Crests will match troughs if waves are one/half wavelength, three/half wavelengths, … apart: path difference = (m+½) wave 1 wave 2 sum A max

10. What This Means for Light  Light is electromagnetic radiation  A light wave is oscillating electric and magnetic fields  The amplitude of the oscillation represents the maximum electric (or magnetic) field and determines the intensity of light  Intensity depends on the square of the maximum electric field: I = E max 2 /(2c  0 )  Constructive interference produces brighter light; destructive interference produces dimmer light.

11. Comparing Interference E max 2E max Medium amplitude of electric field yields medium intensity light Double amplitude of electric field yields quadruple intensity (very bright) light Zero amplitude of electric field yields zero intensity (no) light

12. Coherent vs. Incoherent Light “Everyday light” is incoherent Laser light is an example of coherent light Simple wave equation describes coherent waves y = y m sin(kx   t +  )

13. Diffraction Math  The locations of successive minima are given by  tan  = y/D  for small angles, sin  ~  ~ tan  = y/D

14. Diffraction by a circular aperture  A circular aperture of diameter d  Single slit of width a

15. Resolvability  Two objects are just resolved when the central diffraction maximum of one object is at the first minimum of the other. (Rayleigh’s criterion)  As before,  approximately y/L

16. Comments on Resolvability  If want to resolve objects closer to each other (smaller y), need smaller wavelength of light or larger aperature  This is called the diffraction limit

17. Why Do We Care? CD-ROMS and other optical storage devicesCD-ROMS

18. Before the next class,... Prepare for the First Exam! –Exam on Thursday, Feb. 14.