Bending and Bouncing Light
Standing Waves, Reflection, and
Refraction
What have we learned?
• Waves transmit information between two points without
individual particles moving between those points
• Transverse Waves oscillate perpendicularly to the direction
of motion
• Longitudinal Waves oscillate in the same direction as the
motion
• Any traveling sinusoidal wave may be described by
y = ym sin(kx wt + f)
• f is the phase constant that determines where the wave
starts.
What else have we learned?
• The time dependence of periodic waves can be described
by either the period T, the angular speed w, or the
frequency f, which are all related:
w = 2pf = 2p/T
• The spatial dependence of periodic waves can be described
by either the wavelength l or the wave number k, which
are related.
k = 2p/l
• The speed of a traveling wave depends on both spatial and
time dependence:
v = l/T = lf = w/k
Standing waves - graphically
v v v
v v v
v=0 v=0 v=0
Animation of Standing Wave Creation
Standing waves - mathematically
• Take two identical waves traveling in opposite directions
y1 = ym sin (kx - wt)
y2 = ym sin (kx + wt)
yT = y1 + y2 = 2ym cos wt sin kx
This uses the identity
sin a + sin b = 2cos½(a-b)sin½ (a+b)
• Positions for which kx = np will ALWAYS have zero field.
• If kx = np/2 (n odd), field strength will be maximum for
particular time
Standing waves - interpretation
y = 2ym cos wt sin kx
• Positions which always have zero field (kx = np) are
called nodes.
• Positions which always have maximum (or
minimum) field (kx = = np/2 (n odd)) are called
antinodes.
• The location of nodes and antinodes don’t travel in
time, but the amplitude at the antinodes changes
with time.
Standing waves - if ends are fixed
• If the amplitude must be zero at the ends of the
medium through which it travels, then standing
waves will only be created if nodes occur at the
endpoints.
– One example is a string with fixed ends, like a violin
string
• Then the wavelength will be some fraction of 2L,
where L is the length of the string/antenna/etc.
L=nl/2
Standing waves - if one end open
• If one end is open, the endpoint is an antinode
– This is similar to waves in a cavity with an open end, like
a wind instrument
– Think about shaking a rope to set up a wave. Your end is
free to move, and the wave amplitude cannot be greater
than the amplitude of your motion
• Then the wavelength will be some odd fraction of
4L, where L is the length of the string/antenna
L=nl/4, n odd
Why care about Standing Waves?
• Electromagnetic signals are produced by standing
waves on antenna, for example
• The length of the antenna can be no shorter than 1/4
the wavelength of the signal (since end of antenna is
not fixed)
• This puts practical constraints on what wavelengths
can be transmitted - need short wavelengths, or high
frequencies
• They are similar in concept to Fourier spectra and
modes in an optical fiber – both of which interest us
Summary of Reflection
• All angles determining the direction of light rays are
measured with respect to a normal to the surface.
• Light always reflects off a surface with an angle of reflection
equal to the angle of incidence.
• When light strikes a rough surface, each “ray” in the beam
has a different angle of incidence and so a different angle of
reflection – this is called Diffuse Reflection
Refraction
• When light travels into a denser medium from a
rarer medium, it slows down and decreases in
wavelength as the wave fronts pile up - animation
• The amount light slows down in a medium is
described by the index of refraction : n=c/v
• The wavelength in vacuum l0 is related to the
wavelength l in other media by the index of
refraction too: n = l0/l
• The frequency of the light, and so the energy,
remain unchanged.
Snell’s Law
• As light slows down and decreases in wavelength, it
bends - animation
• The relationship between angles of incidence and
refraction (measured from the normal!) is given by
Snell’s Law:
n1 sin q1 = n2 sin q2
Do the “Before You Start” Questions in
Today’s Activity
Total Internal Reflection
• Light traveling from a denser medium to a rarer
medium bends away from the normal, so the angle
in the rarer medium could become 90 degrees.
• When the angle of refraction is 90 degrees, the
angle of incidence is equal to the critical angle:
sin qc = n2/n1, where n1 is for the denser medium
• Any angles of incidence q1 qc result in Total
Internal Reflection, when the light cannot exit the
denser material.
Do the Rest of the Activity
What have we learned today?
• Identical sinusoidal waves traveling in opposite directions
combine to produce standing waves:
y = y1 + y2 = 2ym cos wt sin kx
• Nodes, or locations for which kx = np, will not move but
will always have zero displacement.
• If standing wave has both ends fixed (both nodes) a
distance L apart,
n l = 2L, n any integer
• If standing wave has one end fixed (node) and one end
open (antinode) a distance L apart,
n l = 4L, n odd integer
What else have we learned?
• The angle of incidence ALWAYS equals the
angle of reflection
• Light reflecting off a smooth surface undergoes
total reflection, while light reflecting off a rough
surface can undergo diffuse reflection
• Light entering a denser medium will
(a) slow down, v = c/n
(b) decrease in wavelength, l = l0/n
(c) and bend toward a normal to the interface of the
media,
n1 sinq1 = n2 sinq2
What else have we learned?
• Light entering a rarer medium can exhibit total
internal reflection (TIR) if the angle of incidence
is greater than the critical angle for the interface
sin qc = n2/n1
• TIR is the phenomenon underlying fiber optics;
the Numerical Aperture indicates the angles at
which light can enter a fiber and remain trapped
inside:
NA = n0 sin qm = (n12 - n22)1/2.
Before the next class, . . .
• Read the Assignment on Fourier Analysis
found on WebCT
• Read Chapter 3 from the handout from
Grant’s book on Lightwave Transmission
• Do Reading Quiz 4 which will be posted on
WebCT by Friday morning.
• Start Homework 5 (found on WebCT by
Friday AM), due next Thursday on material
from this and the previous class.