From: "John T. Lowry" <jlowry@mcn.net>
Newsgroups: sci.math,sci.physics
Subject: Re: Snell's Law in differential format?
Date: Mon, 2 Nov 1998 16:49:08 -0700

I once wrote a (classified Confidential) paper, while in the submarine
force, on a method for depicting sound rays in the sea under the assumption
of (piecewise) constant velocity gradients. In a given layer, the rays
follow circular paths with radius of curvature R = v0/(k*cos(theta_0)),
where v0 is the speed coming into the layer, k is the velocity gradient
(index of refraction), and theta_0 is the incoming depression angle (angle
relative to the new surface). For depth z in the layer, v(z) = v0 + k*z.
With these geometrical defintions Snell's law is: cos(theta(z))/v(z) =
cos(theta_0)/v0. Hope this helps.
    John

John T. Lowry, PhD
Flight Physics; Box 20919; Billings MT 59104
Voice: 406-248-2606

Manuel Gamito wrote in message <87emrlhfhx.fsf@metacreations.com>...
>
>Snell's Law describes light refraction on a discontinuous boundary
>between two different media. But it is also known that light bends
>when crossing a medium of smoothly varying density (e.g. mirages).
>
>My question is: is there an equivalent to Snell's Law in terms of a
>differential equation that describes this bending of light through an
>inhomogeneous medium? Could you give me any pointers to this topic?
>
>Many thanks!
>Manuel Gamito
>
>--
>Your sword's blowing glue!!!
> ... let me say that again ...
> Your sword's glowing blue!!!, Dalbozz of Gurth
==============================================================================

From: "Peter Diehr" <diehr-peter@ic.net.NO!SPAM>
Newsgroups: sci.math,sci.physics
Subject: Re: Snell's Law in differential format?
Date: Tue, 3 Nov 1998 08:50:55 -0500

You do the natural thing ... let the index of refraction be a function of
postion, and then write Fermat's principle of least time as a variational
principle:

 (variation of) Integral{ n(position) * ds}  = 0, where ds is the
displacement.

After some tedium, you arrive at the "ray equation":

d/ds( n*dr/ds) = gradient n

where r is the postion vector, s is the displacement, and n the index of
refraction.  If you look up the derivation of the paraxial ray equation in
any advanced optics text (e.g., Born and Wolf), you should find this.

Another route is the eikonal equation, where S is a "potential" whose
equilevel surfaces S(r)=constant are everywhere normal to the rays.  Then
you can work through Fermat's principle again to come up with:

 (dS/dx)^2 + (dS/dy)^2 + (dS/dz)^2 = n^2,

where, of course, partial derivatives are meant, and n is a function of
position.

Best Regards, Peter

Manuel Gamito wrote in message <87emrlhfhx.fsf@metacreations.com>...
>
>Snell's Law describes light refraction on a discontinuous boundary
>between two different media. But it is also known that light bends
>when crossing a medium of smoothly varying density (e.g. mirages).
>
>My question is: is there an equivalent to Snell's Law in terms of a
>differential equation that describes this bending of light through an
>inhomogeneous medium? Could you give me any pointers to this topic?
==============================================================================

From: lrmead@orca.st.usm.edu (Larry Mead)
Newsgroups: sci.math,sci.physics
Subject: Re: Snell's Law in differential format?
Date: 3 Nov 1998 18:49:16 GMT

Manuel Gamito (mgamito@metacreations.com) wrote:

: Snell's Law describes light refraction on a discontinuous boundary
: between two different media. But it is also known that light bends
: when crossing a medium of smoothly varying density (e.g. mirages).

: My question is: is there an equivalent to Snell's Law in terms of a
: differential equation that describes this bending of light through an
: inhomogeneous medium? Could you give me any pointers to this topic?

: Many thanks!
: Manuel Gamito

Yes, The general differential equation is called the Eikonel equation,
discussed in advance optics books, and Goldstein's graduate text in
classical meehanics. Also, Feynman derives it in one of the three volumes
of his lectures.

--
Lawrence R. Mead Ph.D. (Lawrence.Mead@usm.edu)
Eschew Obfuscation! Espouse Elucidation!
www-dept.usm.edu/~physics/mead.html