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Need help finding origin of an equation

  1. Feb 8, 2012 #1
    I am doing a Michelson interferometer lab which instructs me to use the equation Fb=(Fl*v)/c, where Fb is the beat frequency, Fl is the frequency of the laser, and v is the velocity of a oscillating mirror. The interferometer has one stationary mirror and a mirror that is mounted on a speaker that oscillates from a signal provided by a function generator. I'm trying to find the origin of the Fb=... equation.
     
  2. jcsd
  3. Feb 8, 2012 #2
    It's derived using the relativistic Doppler effect formula.

    When the light is incident on the mirror, the frequency in the rest frame of the mirror is:
    [tex]
    f' = f_L \, \sqrt{\frac{1 + \beta}{1 - \beta} }, \ \beta = \frac{v}{c}
    [/tex]

    Then, as the mirror re-emits the light, the new frequency is:
    [tex]
    \tilde{f} = f' \, \sqrt{\frac{1 + \beta}{1 - \beta} } = f_L \, \frac{1 + \beta}{1 - \beta}
    [/tex]
    Then, use the fact that [itex]\beta \ll 1[/itex] and perform an expansion in powers of [itex]\beta[/itex]. To first order we have:
    [tex]
    \frac{1 + \beta}{1 - \beta} = (1 + \beta) ( 1 - \beta)^{-1} = (1 + \beta) \left[ 1 + (-1) (-\beta) + O(\beta^2) \right] = 1 + 2 \beta + O(\beta^2)
    [/tex]
    [tex]
    \tilde{f} \approx f_L + 2 \, f_L \, \beta
    [/tex]

    Then, consider the interference of the incoming light ray and the reflected ray with a slightly bigger frequency.

    [tex]
    u = A \, \cos \left[ 2 \pi f_L \left(t - \frac{x}{c} \right) \right] - A \, \cos \left[ 2 \pi \tilde{f} \left(t + \frac{x}{c} \right) \right]
    [/tex]

    You should use simple trigonometry to see that this represents a beating wave, with a beat frequency:
    [tex]
    f_B = \frac{\tilde{f} - f_L}{2}
    [/tex]
    which reproduces your formula.
     
  4. Feb 8, 2012 #3
    Thank you, I really appreciate it!
     
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