Transcript A wave is a wave is a wave

Those Interfering Signals
Modes and Dispersion in Fibers
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.
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
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
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
n0
q0
n2
qi
n1
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)
n0
q0
n2
qi
n1
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
Modes in a Fiber
Mode 1: Electric
Field across the fiber
Mode 1: Intensity
across the fiber ~E2
Mode 2: Electric
Field across the fiber
Mode 2: Intensity
across the fiber ~E2
Mode 3: Electric
Field across the fiber
Mode 3: Intensity
across the fiber ~E2
• The field distributions of successive modes look like the harmonics
of standing waves! – the phenomena are very similar
• (Figures adapted from Photonics – not to scale)
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)
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!
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 q0 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.
n0
q0
n2
qi
n1
Do the Activity
Work as far as you can before Dr.
Persans arrives
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.