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A wave is a wave is a wave

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					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.

				
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posted:11/10/2011
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