How To Say What You Want Describing Signals 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 = n0 sin qm = (n12 - n22)1/2. 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. How do we send signals? • Radio antenna (AM frequencies around 1000 kHz, FM frequencies around 100 MHz) • TV antenna (VHF frequencies are around 100 MHz, on either side of FM frequencies, UHF frequencies around 500 MHz) These are public transmissions, and so the carrier frequencies are set and regulated • Coaxial cable These are private • Optical waveguides transmissions, and sent over range of frequencies • ISDN Optical waveguides pros and cons • Message remains private • Flexibility • Low Loss • Insensitive to EM interference • Very high bandwidth BUT • Expensive to connect to every house • Require electricity-to-light converters • Either multi-modal, or less efficient (better coupling makes this less relevant) Chromatic Dispersion • Index of refraction is dependent on wavelength. • Typical materials exhibit higher indices of refraction for lower wavelengths (higher energies) • Thus violet light bends the most through a prism or water and appears on the outside of a rainbow. Optical Fiber Dispersion/Attenuation • 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 Modes in Optical Fibers • Modes in a fiber are specific field distributions that are independent of the length traveled down the fiber. However, these modes travel at slightly different speeds which matters more with longer fibers. • Fields of modes look like harmonics of standing waves Modes Combine to Give Path of Light • 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 • (Figures adapted from Photonics – not to scale) Modes in Optical Fibers (cont.) • Modes are like sines and cosines in Fourier series: each path that light takes down a fiber can be expressed as a sum of modes. • 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 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 Some ways to modulate signals • Amplitude modulation: – A signal with a constant carrier frequency is sent – The original signal becomes the amplitude of the transmitted signal – Since the transmitted signal is not a simple sine wave, it has a bandwidth of Fourier components • Frequency modulation: – A signal with a constant carrier frequency is sent – The original signal becomes the change in frequency of the transmitted signal – Since the transmitted signal is not a simple sine wave, it has a bandwidth of Fourier components – FM is easier to amplify, since only the frequency determines the signal. Schemes for Encoding • ASCII - The American Standard Code for Information Interchange is a standard seven-bit code that was proposed in 1963, and finalized in 1968. ASCII was established to achieve compatibility between various types of data processing equipment. • ASCII is the common code for microcomputer equipment. The standard ASCII character set consists of 128 decimal numbers ranging from zero through 127 assigned to letters, numbers, punctuation marks, and the most common special characters. The Extended ASCII Character Set also consists of 128 decimal numbers and ranges from 128 through 255. Pulses and Data • Can represent binary data with pulses in a variety of ways • 10110 could look like . . . Notice that the NRZ takes half the time of the others for the same pulse widths Non-return-to-zero (NRZ) Manchester Coding Return-to-zero (RZ) Bipolar Coding Distortion • No physical change is instantaneous • If change is too slow, won’t have time to rise before needs to fall • Results in data loss Sharp edges Sizeable rise time Really Distorted • Since rise is generally exponential, we define “rise time” to be time from 10% of max value to 90% of max; “fall time” is time from 90% to 10% • To be able to resolve data, the rise time and fall time must be less than 70% of the bit width What exactly is a decibel? • A ratio, often of power • BUT, in logarithmic form: • dB = 10 log (P2/P1) • e.g., if my received signal is 1/10 as big as my transmitted signal, my “gain” would be gain dB = 10 log (1/10) = -10 • The minus sign denotes loss, or a second power less than the initial power Why care about decibels? • Signal-to-noise ratios are often given in decibels • You want the signal to be larger than the noise, so the ratio (in dB) should be positive • For digital data, we use bit error rate, not signal-to-noise • Bit error rate is ratio of wrong bits to total bits - it should be small, whereas SNR should be large • Bit error rate can be expressed as a plain number, or in decibels Before the next class, . . . • Read On-Line Reading on Interference and Diffraction • Do Reading Quiz • Finish Homework 7 (start 8) • Do Activity 06 Evaluation by Midnight tonight.