Review

First Exam

What have we learned?• Any traveling sinusoidal wave may be described by

y = ym 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

Review (cont.)• Any periodic function of frequency f0 can be expressed as a

sum over frequency of sinusoidal waves having frequencies equal to nf0, 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 f0 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.

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.

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

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

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

Amax

2Amax

wave 1

wave 2

sum

Amax

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

Amax

Amax

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 = Emax

2/(2c0) Constructive interference produces brighter light;

destructive interference produces dimmer light.

Comparing Interference

Emax

2Emax

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

Coherent vs. Incoherent Light

• “Everyday light” is incoherent

• Laser light is an example of coherent light

• Simple wave equation describes coherent waves

y = ym sin(kx t + )

Diffraction Math

The locations of successive minima are given by

tan = y/D

for small angles, sin ~ ~ tan = y/D

.....)3 ,2 ,1(sin

,...)2 ,1 ,0( 2

1sin

2

nna

mma

Diffraction by a circular aperture A circular aperture of diameter d

Single slit of width a

minimum)(1st 22.1sind

minimum)(1st sina

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

ddR

22.122.1

sin 1

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

dD

yR

22.1

Why Do We Care?

• CD-ROMS and other optical storage devices

Before the next class, . . .

• Prepare for the First Exam!– Exam on Thursday, Feb. 14.