ls1_unit_2 - ON CLASSICAL ELECTROMAGNETIC FIELDS II. RAYS :...

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ON CLASSICAL ELECTROMAGNETIC FIELDS 6 II. RAYS: THE EIKONAL TREATMENT OF GEOMETRIC OPTICS 6 Since ancient times, the notion of ray or beam propagation has been one of the most enduring and fundamental concepts in optical physics. As a zeroth order approximation we might consider a plane wave to be a model of a beam and its propagation vector to be a model of a ray . This is a reasonable start, but it is a much too restricted view and we can do much better. What we need is a solution to Maxwell's equations which is like a plane wave, but limited in spatial extent. One approach, the simplest, is called variously ray, Gaussian or geometric optics. A M AXWELLIAN D ERIVATION OF THE E IKONAL E QUATION: To fully understand geometric optics in the context of Maxwell's equations, we start by writing the electric and magnetic fields as pseudo-simple waves -- viz. r E r r , ϖ ( 29 = r e r r , ϖ ( 29 exp - i k 0 S r r , ϖ ( 29 [ ] [ II-1a ] r H r r , ϖ ( 29 = r h r r , ϖ ( 29 exp - i k 0 S r r , ϖ ( 29 [ ] [ II-1b ] where k 0 = ϖ μ 0 ε 0 c It is assumed that r e r r , ϖ ( 29 and r h r r , ϖ ( 29 are weak functions of position . The scalar phase function S r r , ϖ ( 29 is the spatially varying phase of the pseudo-simple wave. For the cases of pseudo-plane waves and pseudo-spherical waves the phase function is given, respectively, by k 0 S r r , ϖ ( 29 = x k x + y k y + z k z [ II-2a ] 6 See, for example, Max Born and Emil Wolf, Principle of Optics , Pergamon Press (1986), Chapter 3.
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7 and k 0 S r r , ϖ ( 29 = k 0 x 2 + y 2 + z 2 [ II-2b ] We now substitute these pseudo-simple wave expressions ( i.e. Equations [ II-1]) into Maxwell's equations to obtain exp - i k 0 S r r , ϖ ( 29 [ ] r ∇× r e r r , ϖ ( 29 - i k 0 r S r r , ϖ ( 29 × r e r r , ϖ ( 29 { } =- i μ 0 ck 0 r h r r , ϖ ( 29 exp - i k 0 S r r , ϖ ( 29 [ ] [ exp - i k 0 S r r , ϖ ( 29 [ ] r ∇× r h r r , ϖ ( 29 - i k 0 r S r r , ϖ ( 29 × r h r r , ϖ ( 29 { } = i ε r r , ϖ ( 29 ck 0 r e r r , ϖ ( 29 exp - i k 0 S r r , ϖ ( 29 [ ] [ exp - i k 0 S r r , ϖ ( 29 [ ] r ∇⋅ε r r , ϖ ( 29 r e r r , ϖ ( 29 [ ] - i k 0 ε r r , ϖ ( 29 r S r r , ϖ ( 29 r e r r , ϖ ( 29 { } = 0 [ exp - i k 0 S r r , ϖ ( 29 [ ] r ∇⋅ r h r r , ϖ ( 29 - i k 0 r S r r , ϖ ( 29 r h r r , ϖ ( 29 { } = 0 [ Rearranging, we obtain r S r r , ϖ ( 29 × r e r r , ϖ ( 29 0 c r h r r , ϖ ( 29 = i k 0 [ ] - 1 r ∇× r e r r , ϖ ( 29 [ II-4a ] r S r r , ϖ ( 29 × r h r r , ϖ ( 29 r r , ϖ ( 29 c r e r r , ϖ ( 29 = i k 0 [ ] - 1 r ∇× r h r r , ϖ ( 29 [ II-4b ] r S r r , ϖ ( 29 ⋅ε r r , ϖ ( 29 r e r r , ϖ ( 29 [ ] = i k 0 [ ] - 1 r ∇⋅ε r r , ϖ ( 29 r e r r , ϖ ( 29 [ ] [ II-4c ] r S r r , ϖ ( 29 r h r r , ϖ ( 29 = i k 0 [ ] - 1 r ∇⋅ r h r r , ϖ ( 29 [ II-4d ] In the ray , Gaussian or geometric approximation we assume that we may neglect the RHS's of these equations. To get something useful we multiply through the first equation ( i.e. Equation [ II-4a ]) as follows:
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ls1_unit_2 - ON CLASSICAL ELECTROMAGNETIC FIELDS II. RAYS :...

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