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When I am studying the total reflection phenomenon, I calculated the Poynting vector of the transmitted wave, which can be written as $S_t=A(k_{x}\hat{x}+i\alpha\hat{z})$ A is some constant. I choosed $z=0$ as the interface, light incident from the region $z>0$, If total reflection occurs, the z-component become imaginary, for some reference the imaginary part is regarded as "reactive power" like in AC circuit.

In Hecht's text, stated that the energy is circulating across the interface. But how can I see it from mathematical expressions?

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2 Answers 2

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You probably use some unusual definition of the Poynting vector. The vector is always real: up to a constant factor, it's a vector product of the electric and magnetic fields, which are real. If you use complex expressions for the fields, you should modify the definition of the Poynting vector.

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  • $\begingroup$ I think I used proper definition from E cross H, usually S follows the directions of k, where k's z-component becomes imaginary when total reflection occurs. $\endgroup$
    – Andy Huang
    Jan 14, 2014 at 15:17
  • $\begingroup$ As I said, if you use complexified fields, this is not a proper definition (please see en.wikipedia.org/wiki/…) $\endgroup$
    – akhmeteli
    Jan 14, 2014 at 18:49
  • $\begingroup$ Well, I don't want a time-averaged quantity for this time being, I need a instantaneous S at first to observe its time varying behavior $\endgroup$
    – Andy Huang
    Jan 14, 2014 at 19:24
  • $\begingroup$ That does not matter. What matters is your definition of the Poynting vector cannot be used for complex fields. $\endgroup$
    – akhmeteli
    Jan 14, 2014 at 21:01
  • $\begingroup$ It is common practice to use complex amplitudes for power in the discussion of stationary sinusoidal oscillations in electrical networks. The imaginary part stands for the oscillation between capacitive and inductive energy. The natural generalization to complex Poynting vectors for sinusoidal electromagnetic waves is the oscillating energy density between the E- and the H-field (as energy density per time). If I remember right it is something like $j\omega$ times maximal stored energy density maybe times some factor $\frac12$ (field does not matter since energy oscillates between the fields). $\endgroup$
    – Tobias
    Jan 15, 2014 at 8:34
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The energy of the totally reflected wave has a non propagating component in the z direction in your coordinate system. the amplitude of that component decays exponentially in the z direction. It is no longer a transverse electromagnetic wave. It is called an evanescent wave and also an inhomogeneous wave. If you want to see it experimentally you can couple to this non propagating component by modifying the boundary conditions. Take an optical fiber, scratch the cladding and see light exit the fiber.

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