From: pcdhSpamMeSenseless@electrooptical.net   
      
   On 7/26/2015 10:14 PM, RichD wrote:   
   > On July 24, Phil Hobbs wrote:   
   >>>>> Recently, I attended a seminar of Hamilton's cone, or   
   >>>>> whatever it's called, inside crystals.   
   >>>>> At one point, the speaker said, "the direction of energy flow   
   >>>>> is different than the wave direction."   
   >>>>> ??   
   >>>>> If I recall correctly, the wave direction is given   
   >>>>> by the Poynting vector:  E x H.  Am I now to believe   
   >>>>> the power doesn't flow the same way?   
   >>   
   >>>> You have to be careful about your terms.  Phase propagates   
   >>>> as usual, exp(i k x - omega t) for a plane wave.  However,   
   >>>> in an anisotropic crystal, the main part of the optical power   
   >>>> doesn't necessarily propagate along k.   
   >>>   
   >>> No doubt you're right, but I don't get it.  It's partly   
   >>> a matter of semantics - I don't grok the difference   
   >>> between 'phase propagation', 'wave propagation', and   
   >>> 'energy propagation'   
   >>>   
   >>> My memory of the physics, is that at a particular point   
   >>> and moment,  the field is represented by a 6-vector   
   >>> (3 x E, 3 x H), and the energy is given by E x H.   
   >>> That vector then 'travels', loosely speaking, to the   
   >>> neighboring point; hence velocity.  I don't understand   
   >>> how that could differ from energy flow (or optical power flow).   
   >>   
   >> In an anisotropic medium, there are still plane wave solutions   
   >> [i.e. something times exp(i k dot x - omega t) ].  However,   
   >> due to the anisotropic electric and magnetic susceptibility,   
   >> E and/or H aren't orthogonal to _k_.  That means that E cross   
   >> H doesn't lie along _k_, so the Poynting vector (energy   
   >> propagation) is in a different direction from _k_ (phase   
   >> propagation).   
   >   
   > OK, I think I get the idea: the wave propagates as   
   > usual, but Poynting points in a different direction.   
    >   
    > As I suspected, it's a matter of semantics -   
    > 'energy flow' doesn't imply flow, as a river, but   
    > indicates the direction of E x H.   
      
   No.  Phase propagates as usual, i.e. the plane wave is still a plane   
   wave.  However, energy propagates in a different direction in general,   
   leading to a phenomenon called _beam_walkoff_.   
      
   If you shine a laser beam at normal incidence into an isotropic medium,   
   it continues on undeviated.  Phase matching at the boundary dictates   
   that the transverse component of _k_ is zero on both sides, so apart   
   from the front surface reflection, it just continues straight on.   
      
   If the crystal is anisotropic, however, the beam will split into two   
   beams.  The phase matching condition is the same for both, so the   
   transverse _k_ is zero for both, and the phase of both propagates   
   straight on into the crystal, though at slightly different speeds.  But   
   in one beam (the e-ray), E is not orthogonal to _k_, so the Poynting   
   vector is not parallel to _k_, and the beam goes off along the Poynting   
   vector.  It's really very striking to see--not subtle at all.   
      
   Anisotropic crystals also exhibit birefringence (double refraction)   
   which is a different phenomenon that occurs at off-normal incidence.   
   This effect makes the walkoff less visually obvious, though it still occurs.   
      
   There are lots of polarization gizmos that rely on walkoff, of which the   
   most important is the walkoff plate polarizer used in large quantities   
   in telecoms, and the most interesting (I think) is the Nomarski prism, a   
   variation on the Wollaston beam splitting prism in (which due to   
   walkoff) the beams actually intersect outside the prism.  That is, the   
   direction of walkoff is opposite to the direction of beam deviation in   
   the two beams, so one zigs left inside the crystal and then zigs right   
   afterwards, and the other one does the opposite.   
      
   Conical refraction of the sort I think you're describing only occurs in   
   biaxial materials (ones with 3 different values of n rather than two as   
   in the more common uniaxial materials like calcite), and only at special   
   angles.   
      
    >   
    > Thanks. Can you recommend a text which explores this further?   
    >   
      
   The two best books I have on crystal optics are Born & Wolf's   
   "Principles of Optics" and Landau & Lifshitz's "Electrodynamics of   
   continuous media".   
      
      
   Cheers   
      
   Phil Hobbs   
   --   
   Dr Philip C D Hobbs   
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