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I For the Fresnel Equations for TM light why is 1 + r not t?

  1. Jan 19, 2017 #1
    The complex amplitude ratios for light are defined as:

    rTM = ErTM / EiTM
    tTM = EtTM / EtTM

    I've done the derivation from Wikipedia and see that

    (n2/n1) * tTM = rTM + 1.

    But I don't understand what is going on physically. I understand that these values are not power or intensity so I can't really invoke conservation of energy (or can I?). Why is

    r + 1 = t

    for TE light, but not TM?

    Thanks.
     
  2. jcsd
  3. Jan 19, 2017 #2

    Charles Link

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    If I understand your question, you are using reflection coefficients for the magnetic component of the TEM wave. For the electric component, it has ## \nabla \times E =- \frac{\partial{ B}}{\partial{t}} ##. Using Stokes'theorem, you can make the rectangle around the area over which the line integral is performed arbitrarily thin, which makes ## E_1=E_2 ## so that ## \\ ## (1) ## E_i+E_r=E_t ## for the components of ## E ## parallel to the interface. ## \\ ## Also, energy (intensity) ## I=n E^2 ##. (assuming normal incidence and leaving out constants of proportionality.) With energy conservation ## \\ ## (2) ## I_i=n_1 E_i^2=n_1 E_r^2+n_2 E_t^2 ##. ## \\ ## The Fresnel coefficients for the electric field ## \rho=\frac{E_r}{E_i}=\frac{n_1-n_2}{n_1+n_2} ## and ## \tau=\frac{E_t}{E_i}=\frac{2n_1}{n_1+n_2} ## can be computed from the two equations above. ## \\ ## The magnetic components of these waves, other than proportionality constants, are basically ## B=\hat{n} \times E ##, where ## \hat{n} ## is a unit vector and points in the direction of propagation. I think this cross product puts a minus sign on the ## B_r ## term so that ## B_i-B_r =B_t ##. (The energy equation would be left unchanged by this cross product.) This would make any Fresnel relations different for the magnetic components. Hopefully this answers your question. ## \\ ## Note: For the electric field coefficients the result is ## 1-\rho=(\frac{n_2}{n_1}) \tau ## . For the magnetic field coefficients ## 1+\rho_m=(\frac{n_2}{n_1}) \tau_m ##.
     
    Last edited: Jan 19, 2017
  4. Jan 19, 2017 #3
    Thank you Charles, but I don't mean the reflection coefficients for the magnetic component of a TEM wave. I mean a TM polarized wave, that is to say, my wave doesn't have a magnetic field in the direction of propagation. It is also called a parallel polarized wave or a p polarized wave.

    I'm also looking for a more conceptual understanding.

    Thank for your help so far.
    Latempe
     
  5. Jan 19, 2017 #4

    Charles Link

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    I think what I did might be what you are looking for. The ## B ## is perpendicular to the direction of propagation. (along with the ## E ##). At normal incidence, the direction of polarization does not enter into the picture. The p polarization is polarized with the E field in the plane of incidence and reflection, and the perpendicular polarization has ## E ## perpendicular to this plane. At normal incidence, the Fresnel coefficients are independent of polarization. A google of the term TM polarized wave showed that it is just another name for what I have always known as parallel polarization of a transverse electromagnetic wave.
     
    Last edited: Jan 19, 2017
  6. Jan 19, 2017 #5
     
  7. Jan 19, 2017 #6

    Charles Link

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    Those are called the Kirchhoff relations, and they are presented in many E&M textbooks including J.D. Jackson's E&M textbook. They are a complicated function of the angle of incidence, (along with ## n_1 ## and ## n_2 ##), and are different for parallel and perpendicular polarization. ## \\ ## To describe them qualitatively, the energy reflection coefficient ## R=\rho^2 ## increases monotonically from ## \theta_i =0 ## to ## \theta_i=\pi/2 ##, for perpendicular polarization, from the normal incidence value (at ## \theta_i=0 ## ) to ## R= 1.0 ## at ## \theta_i=\pi/2 ##, (grazing incidence). The parallel polarization case takes a dip to ## R=0 ## at the Brewster angle before going to ## R=1.0 ## at the grazing angle of ## \theta_i=\pi/2 ##. ## \\ ## I think Halliday-Resnick's E&M textbook shows these two graphs. ## \\ ## Note: I edited this a couple of times, and now it hopefully reads reasonably well.
     
    Last edited: Jan 19, 2017
  8. Jan 19, 2017 #7

    The text I am referencing is Fundamentals of Photonics 2nd Ed. by Saleh and Tiech. They give the Fresnel equations as:

    $$ r_{TE} = \frac{ η_2*secθ_2 - η_1*secθ_1 } {η_2*secθ_2 + η_1*secθ_1 }, t_{TE} = 1 + r_{TE} $$

    $$ r_{TM} = \frac{ η_2*cosθ_2 - η_1*cosθ_1 } {η_2*cosθ_2 + η_1*cosθ_1 }, t_{TM} = (1+r_{TM}) \frac{cosθ_2} {cosθ_1} $$

    I have a fairly good understand of the derivation and know math mathematically why ## t_{TM} = (1+r_{TM}) \frac{cosθ_2} {cosθ_1} ##, but I don't grasp this conceptually.
     
  9. Jan 19, 2017 #8

    Charles Link

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    I believe the ## t_{TE}=1+r_{TE} ## equation is in error. At normal incidence it is incorrect, and thereby is incorrect in general. I would suggest you google another source on the subject. Once you understand the derivation for normal incidence, the derivations for other angles of incidence can be figured out with a little effort. Most textbooks also show these formulas with ## cos(\theta) ## instead of ## sec(\theta) ##. They could be converted to expressions with ## cos(\theta) ## for comparison to see if they are correct. ## \\ ## Editing: Unless they are doing something very different from what I think they are trying to do, the ## r_{TE} ## expression is completely wrong. In fact, other than getting a minus sign on the whole expression, it agrees with another google article for ## \rho_{||} ##. Meanwhile, the ## \rho_{TM} ## agrees with ##\rho_{perpendicular} ## ,other than again an overall minus sign. ## \\ ## The TM , according to the google,should be parallel polarization. I don't know that TM and TE are in widespread usage. They were always referred to as parallel and perpendicular in the courses that I took. ## \\ ## I highly recommend you google a couple of other sources and compare. I think Saleh and Tiech have it incorrect. ## \\ ## Additional comment: I see one source I googled uses the boundary conditions on ## B ## , instead of an energy conservation equation, as the second equation which it works with. In any case, the derivations are still similar.
     
    Last edited: Jan 19, 2017
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