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Michelson Interferometer: Optics

  1. Feb 18, 2014 #1
    1. The problem statement, all variables and given/known data

    A Michelson Interferometer has incident light in range 780-920 nm from a source. Intensity as a function of x (distance from central maxima) is given by:

    [tex]I_{(x)} = 3I_0 + 3I_0 cos(K_1x) cos (K_2x) - I_0 sin (K_1x) sin (K_2x)[/tex]

    where ##K_1 = 1.3 x 10^7 m^{-1}## and ##K_2 = 3.3 x 10^5 m^{-1}## and ##I_0## is intensity of central maxima.

    Part(a): Show that intensity can be written as sum of two monochromatic components.

    Part(b): Find their mean wavenumber ##\overline {v}##

    Part(c): Find their wavenumber separation ##\Delta \overline {v}##

    Part (d): Find their relative intensities

    Part (e): Over a larger range, it is found that periodic terms are in fact multiplied by:

    [tex]f_{(x)} \propto e^{(-K_3^2 x^2)} [/tex]

    Make a sketch of the interference pattern.

    2. Relevant equations



    3. The attempt at a solution

    Part(a)

    [tex]I_{(x)} = 3I_0 + 3I_0 cos(K_1x) cos (K_2x) - I_0 sin (K_1x) sin (K_2x)[/tex]
    [tex]= 3I_0 + 2I_0 cos\left ( (k_1 + k_2) x\right ) + I_0 cos \left ((k_1 - k_2)x \right )[/tex]
    [tex] = \frac{3}{2}I_0 + 2I_0 cos (k_3' x) + \frac{3}{2}I_0 + I_0 cos (k_4' x)[/tex]

    where ##k_3' = k_1 + k_2## and ##k_4' = k_1 - k_2##.

    Thus it is a sum of two monochromatic intensities with wavenumber ##k_3'## and ##k_4'##.

    Part (b)

    [tex]\overline {v} = \frac{k_3' + k_4'}{2} = k_1[/tex]

    Part (c)

    [tex] \Delta \overline {v} = \frac{k_3' - k_4'}{2} = k_2 [/tex]

    Part(d)

    [tex]\frac{I'_3}{I'_4} = \frac{\frac{3}{2} + 2 cos (k_3' x)}{\frac{3}{2} + cos (k_4' x)}[/tex]

    At x = 0,

    [tex]\frac{I'_3}{I'_4} = \frac{7}{4} [/tex]

    For relative intensities, do they want us to find it at a distance x away or at the center?

    Part (e)

    I can see that for large x, Intensity → 6I0

    [tex]I = \frac{3}{2}I_0 + 2I_0 cos (Ak_3' e^{x^2}) + \frac{3}{2}I_0 + I_0 cos (Ak_4' e^{x^2})[/tex]

    So the general idea is that it varies sinusoidally about 6I0 but finally settles along 6I0. Here's my sketch:

    2qlantz.png
     
  2. jcsd
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