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Question on Two theorems for the group velocity in dispersive media

  1. Jul 17, 2009 #1
    Question on "Two theorems for the group velocity in dispersive media"

    In the paper (1), the authors show that the group velocity can be faster than light, slower than light, infinite, or even negative without contradicting the causality principle.

    I have no doubt about the validity of this result, simply by considering some examples.

    However, their derivation is based only on the causality principle.
    The causality principle, as I understand it, does not forbit faster than light transmission of a signal.
    The causality principle only assumes that effects follow their cause.

    In the derivation, the causality principle is formulated from the Kramers-Kronig relation.
    There is no further hypothesis about the refraction index.
    It is not even assumed that it must conform to classical relativistic physics.
    This where I have a problem.

    Indeed, the index of refraction is the result of the responses of the material to an excitation.
    These responses should satisfy the Maxwell's equations and the laws of motion from special relativity.
    Therefore, in these responses, the speed of light should play a particuliar role: never "a signal" should go faster than light.

    My impression is that the limitation by the speed of light is not taken into account by only using the Kramers-Kronig relation.
    Therefore, I cannot be totaly sure that the derivation in (1) is fully valid.
    I cannot exclude that, by taking the limitation by the speed of light into account, the result would not be modified.
    One way, maybe, to solve this problem could be the use a retarded potentials formulation where the limitations by the speed of light would be automatically taken into account.

    Could you help me on that question?

    (1) http://physics.princeton.edu/~mcdonald/examples/optics/bolda_pra_48_3890_93.pdf [Broken]
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Jul 17, 2009 #2


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    Re: Question on "Two theorems for the group velocity in dispersive media"

    I have not (and don't have time to) read the paper in detail, but they seem to use the equation v_g=dw/dk throughout. That equation is derived from the first order term in a Taylor expansion of w(k). It is only valid for slow variation of w(k), and not for the cases the consider. Several textbooks show that, for any variation of the index of refraction, the front of a wave pulse cannot exceed c. The top of the pulse may exceed c as the pulse distorts and breaks up, but not the front.
  4. Jul 17, 2009 #3
    Re: Question on "Two theorems for the group velocity in dispersive media"

    You are right Clem.
    However, I am interrested in the (possibly general) conditions under which the group velocity might be constrained below the speed of light, or in the conditions for being not contrained SLT.
    Therefore, my concern is more about understanding their assumptions and specially their use of causality.
    It seems to me that the hypothesis of causality is less restrictive than slower than light constraints. Therefore, I have some doubt about the reach and interrest of their theorem 1.
  5. Jul 17, 2009 #4


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    Re: Question on "Two theorems for the group velocity in dispersive media"

    My point is that anything based on dw/dk for anything is irrelevant in their circumstances.
  6. Jul 19, 2009 #5
    Re: Question on "Two theorems for the group velocity in dispersive media"


    I agreed totally with your remark: the relevance of the GV is limited.

    Nevertheless, the GV is always defined in an non-ambiguous way and can always be calculated, even though its meaning and relevance is limited.

    My question is about a property claimed in this referenced paper.
    I would like to know if their use of causality covers all the constraints that should be taken into account.
    I believe they did not take into account the delay related to the speed of light that arise from the Maxwell's equations.
    They did not constrain the refraction index in any way.
    Taking into account that the refractive index should be obtained from Maxwell's equation and relativistic mechanics is, I think, a stronger constraint than causality.
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