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Details of total internal reflection

  1. Nov 11, 2013 #1


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    Consider snell's law [itex] n_1 \sin{\theta_1}=n_2 \sin{\theta_2} [/itex]([itex] n_1 [/itex] and [itex] n_2[/itex] are real).
    We know that if [itex] n_2<n_1 [/itex], there exists an incident angle called critical angle that gives a refraction angle of ninety degrees i.e. [itex] \sin{\theta_c}=\frac{n_2}{n_1} [/itex].

    But if the incident angle is greater than the critical angle(i.e. [itex] \sin{\theta_1}>\frac{n_2}{n_1} [/itex]),Then:[itex] \sin{\theta_2}=\frac{n_1}{n_2}\sin{\theta_1}>1[/itex]

    But we know that [itex] \sin{\theta}>1 [/itex] can happen for no real [itex] \theta [/itex],so we say that [itex] \theta_2 [/itex] should be complex:
    [itex] \theta_2=\alpha+i \beta [/itex] and [itex] \sin{\theta_2}=\sin{(\alpha+i \beta)}=\sin{(\alpha)}\cos{(i \beta)}+\sin{(i \beta)}\cos{(\alpha)}=\sin{(\alpha)}\cosh{(\beta)}+i\cos{(\alpha)}\sinh{(\beta)} [/itex]

    But from snell'w law,we know that [itex] \sin{\theta_2}[/itex] should be real and so we should always have [itex] cos{\alpha}=0 \Rightarrow \alpha=\frac{\pi}{2} [/itex] and so [itex] \sin{\theta_2}=\cosh{\beta} [/itex].

    This means that the only variable which is capable of giving information about the Total reflected ray,is [itex] \beta [/itex]. But how?

  2. jcsd
  3. Nov 11, 2013 #2


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    Staff: Mentor

    You cannot use a law in a parameter range where it does not apply.

    What else do you need? The angle is just the same as the incident angle.
  4. Nov 11, 2013 #3
    No reason to use complex angles here. You simply defined [itex]\beta[/itex] such that [tex]\cosh \beta = \frac{n_1}{n_2}\sin \theta_1[/tex]
  5. Nov 11, 2013 #4
    Now, what you do get from the treatment of this problem with complex angles is an expression for the evanescent waves [itex]F = A e^{i\vec k_2\cdot \vec x},[/itex] where [itex]\vec k_2 = cos\theta_2\hat i + sin\theta_2\hat j[/itex], and [itex]\vec x = x\hat i + y\hat j[/itex]. Now if you plug in your complex parametrization [itex]\theta_2 = \frac{\pi}{2} + \beta[/itex], than you get [tex]F = A exp [iy\,cosh\beta - x sinh\beta][/tex]
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