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The complex refractive index vs. permittivity

  1. Nov 4, 2014 #1
    I have a process of thought and would like to run past some other minds to point out if I am incorrect in my thinking.

    I am looking into conductivity in high frequencies and a lot of papers I am looking up list a complex refractive index. They list something as in nAg = 0.1453 + j11.3587. (Excuse the imaginary symbol, j, I come from an electrical engineering background.)

    Various sources inform that [itex]n = \sqrt {ε_r} [/itex]. [Link]

    Also that [itex] ε = ε' - jε'' = ε_1 - j \frac {σ}{ω}[/itex] [Electromagnetics for Engineers; Fawwaz T Ulaby]

    We can get from [Wikipedia.org] that:
    [itex] ε = ε_1 + jε_2 = (n + j κ)^2 = n^2 + j 2nκ - κ^2 [/itex]
    [itex] ε_1 = n - κ^2; ε_2 = 2nκ [/itex]
    *Possible contradiction in Wikipedia vs. Ulaby*
    Ulaby states - [itex] ε = ε' - jε''[/itex]
    Wikipedia states - [itex] ε = ε_1 + jε_2 [/itex]
    So, does:
    [itex] 2nκ = \frac {σ}{ω} [/itex]
    Where:
    n - real part refractive index
    κ - Complex part refractive index
    σ - conductivity
    ω - angular frequency
    I am kind of running this by so that someone can say "Yup" but also, I think, writing it down helps me to understand a little better. Also, this is the first time of me using LaTeX and wanted to keep trying it out.
     
    Last edited: Nov 4, 2014
  2. jcsd
  3. Nov 4, 2014 #2

    DrDu

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    I don't know why Ulaby uses a minus sign in the definition of epsilon''. Must be an engineering convention.
     
  4. Nov 4, 2014 #3

    dlgoff

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    Isn't that just an approximation?
    If you mean this equation, I believe it's a dash; not a negative sign.
     
  5. Nov 5, 2014 #4
    It was a proof from Maxwell's Equations
    [itex] \nabla \times H(t)= J(t) + jωεE(t) [/itex]
    [itex] J(t) = σE(t) [/itex]

    Substitute
    [itex] \nabla \times H(t) = (σ + jωε) E(t) [/itex]
    [itex] \nabla \times H(t) = jω (ε + \frac {σ}{jω}) E(t) [/itex]
    [itex] \nabla \times H(t) = jω (ε - j \frac {σ}{ω}) E(t) [/itex]
    [itex] \nabla \times H(t) = jω (ε' - j ε'') E(t) [/itex]
    Where:
    H(t) - Time Varying Magnetic Field with direction (Still getting used to the [itex] coding)
    E(t) - Time Varying Electric Field with direction
    J(t) - Current Density
    σ - Conductance
    ω - Angular Frequency (2π f)
    ε - Permittivity​

    Though I could well be wrong in my assumptions as this provides a relationship between Time Varying (TV) Magnetic fields and TV Electric fields...
     
  6. Nov 6, 2014 #5
    I suppose the question would then be, is the permittivity gained through complex refractive index similar to that of the permittivity relating Magnetic Fields and Electric fields?
     
  7. Nov 6, 2014 #6

    DrDu

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    It certainly is, but usually, we use it in other frequency regions. There are also different conventions. E.g. in optics it is usual to set B=H. All potential magnetic effects are included in a dependence of the dielectric function on the wavevector, i.e. ##\epsilon(\mathbf{k},\omega)##.
     
  8. Nov 13, 2014 #7
    Previously I had given the proof:
    [itex]\varepsilon - j \frac{\sigma}{\omega} = n^2 - K^2 + j2nK[/itex]
    Should it be:
    [itex]\varepsilon_r = n^2 - K^2 + j2nK[/itex]
    And so:
    [itex]\varepsilon_0 \varepsilon_r = \varepsilon_0 \varepsilon_r - j \frac{\sigma}{\omega} [/itex]
    [itex]\varepsilon_r = \varepsilon_r - j\frac{\sigma}{\omega \varepsilon_0}[/itex]
    Therefore:
    [itex]\varepsilon_r - j \frac{\sigma}{\omega \varepsilon_0} = n^2 - K^2 + j2nK[/itex]
    Does this sound correct?
     
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