Editing Fermat's principle (section)

== Modern version ==

=== Formulation in terms of refractive index ===

Let a path {{math|Γ}} extend from point {{mvar|A}} to point {{mvar|B}}. Let {{mvar|s}} be the arc length measured along the path from {{mvar|A}}, and let {{mvar|t}} be the time taken to traverse that arc length at the ray speed <math>v_{\mathrm{r}}</math> (that is, at the radial speed of the local secondary wavefront, for each location and direction on the path). Then the traversal time of the entire path {{math|Γ}} is
{{NumBlk||<math display="block">T = \int_A^B\!dt = \int_A^B\frac{ds}{\,v_{\mathrm{r}}\,}</math>|1}}
(where {{mvar|A}} and {{mvar|B}} simply denote the endpoints and are not to be construed as values of {{mvar|t}} or {{mvar|s}}). The condition for {{math|Γ}} to be a ''ray'' path is that the first-order change in {{mvar|T}} due to a change in {{math|Γ}} is zero; that is,
<math display="block">\delta T =\, \delta\int_A^B\frac{ds}{\,v_{\mathrm{r}}\,} \,=\, 0\,.</math>

Now let us define the ''optical length'' of a given path (''[[optical path length]]'', ''OPL'') as the distance traversed by a ray in a homogeneous isotropic reference medium (e.g., a vacuum) in the same time that it takes to traverse the given path at the local ray velocity.<ref>[[#BW|Born & Wolf, 2002]], p.{{nnbsp}}122.</ref> Then, if {{mvar|c}} denotes the propagation speed in the reference medium (e.g., the speed of light in vacuum), the optical length of a path traversed in time {{mvar|dt}} is {{math|1=''dS'' = ''c'' ''dt''}}, and the optical length of a path traversed in time {{mvar|T}} is {{math|1=''S'' = ''cT''}}.{{tsp}} So, multiplying equation&nbsp;'''(1)''' through by {{mvar|c}}, we obtain
<math display="block">S \,= \int_A^B\!dS \,= \int_A^B\frac{\,c\,}{v_{\mathrm{r}}}\,ds = \int_A^B\!n_{\mathrm{r}}\,ds\,,</math>
where <math>\,n_{\mathrm{r}}=c/v_{\mathrm{r}}\,</math> is the ''ray index'' &ndash; that is, the [[refractive index]] calculated on the ''ray'' velocity instead of the usual [[phase velocity]] (wave-normal velocity).<ref>[[#BW|Born & Wolf, 2002]], p.{{nnbsp}}795, eq.{{nnbsp}}(13).</ref> For an infinitesimal path, we have{{tsp}} <math>dS=n_{\mathrm{r}\,}ds\,,\,</math> indicating that the optical length is the physical length multiplied by the ray index: the OPL is a notional ''geometric'' quantity, from which time has been factored out. In terms of OPL, the condition for {{math|Γ}} to be a ray path (Fermat's principle) becomes
{{NumBlk||<math display="block">\delta S = \delta\int_A^B\!n_{\mathrm{r}}\,ds = 0.</math>|2}}
This has the form of [[Maupertuis's principle]] in [[classical mechanics]] (for a single particle), with the ray index in optics taking the role of momentum or velocity in mechanics.<ref>Cf. [[#Chaves16|Chaves, 2016]], p.{{nnbsp}}673.</ref>

In an isotropic medium, for which the ray velocity is also the phase velocity,<ref group=Note>The ray direction is the direction of constructive interference, which is the direction of the [[group velocity]]. However, the "ray velocity" is defined not as the group velocity, but as the phase velocity measured in that direction, so that "the phase velocity is the projection of the ray velocity on to the direction of the wave normal" (the quote is from [[#BW|Born & Wolf, 2002]], p.{{nnbsp}}794). In an isotropic medium, by symmetry, the directions of the ray and phase velocities are the same, so that the "projection" reduces to an identity. To put it another way: in an isotropic medium, since the ray and phase velocities have the same direction (by symmetry), and since both velocities follow the phase (by definition), they must also have the same magnitude.</ref> we may substitute the usual refractive index {{mvar|n}} for&nbsp;{{math|''n''<sub>r</sub>}}.{{nnbsp}}<ref>Cf. [[#BW|Born & Wolf, 2002]], p.{{nnbsp}}876, eq.{{nnbsp}}(10a).</ref>{{r|veselago-2002-1099}}

=== Relation to Hamilton's principle ===

If {{math|''x''}}, {{math|''y''}}, {{math|''z''}} are Cartesian coordinates and an overdot denotes differentiation with respect to {{mvar|s}}, Fermat's principle '''(2)''' may be written<ref>Cf. [[#Chaves16|Chaves, 2016]], pp.{{nnbsp}}568–9.</ref>
<math display="block">\begin{align}
 \delta S
 &=\,\delta\int_A^B\!n_{\mathrm{r}}\,\sqrt{dx^2+dy^2+dz^2}\\
 &=\,\delta\int_A^B\!n_{\mathrm{r}}\,\sqrt{\dot{x}^2+\dot{y}^2+\dot{z}^2}~ds
 \,=\,0\,.
 \end{align}</math>
In the case of an isotropic medium, we may replace {{math|''n''<sub>r</sub>}} with the normal refractive index{{hsp}} {{math|''n''(''x'', ''y'', ''z'')}}, which is simply a [[scalar field]]. If we then define the ''optical [[Lagrangian mechanics|Lagrangian]]''<ref>[[#Chaves16|Chaves, 2016]], p.{{nnbsp}}581.</ref> as
<math display="block">L(x,y,z,\dot{x},\dot{y},\dot{z}) \,=\, n(x,y,z)\,\sqrt{\dot{x}^2+\dot{y}^2+\dot{z}^2}\,,</math>
Fermat's principle becomes<ref>[[#Chaves16|Chaves, 2016]], p.{{nnbsp}}569.</ref>
<math display="block">\delta S =\, \delta\int_A^B\!L(x,y,z,\dot{x},\dot{y},\dot{z})\,ds\,=\,0\,.</math>
If the direction of propagation is always such that we can use {{mvar|z}} instead of {{mvar|s}} as the parameter of the path (and the overdot to denote differentiation w.r.t.&nbsp;{{mvar|z}} instead of {{mvar|s}}), the optical Lagrangian can instead be written<ref>Cf. [[#Chaves16|Chaves, 2016]], p.{{nnbsp}}577.</ref>
<math display="block">L\big(x(z),y(z),\dot{x}(z),\dot{y}(z),z\big) = n(x,y,z)\,\sqrt{1+\dot{x}^2+\dot{y}^2}\,,</math>
so that Fermat's principle becomes
<math display="block">\delta S =\, \delta\int_A^B\!  L\big(x(z),y(z),\dot{x}(z),\dot{y}(z),z\big)\,dz\,=\,0.</math>
This has the form of [[Hamilton's principle]] in classical mechanics, except that the time dimension is missing: the third spatial coordinate in optics takes the role of time in mechanics.<ref>Cf.{{tsp}} [[#BW|Born & Wolf, 2002]], pp.{{nnbsp}}853–4,{{px2}}868; [[#Chaves16|Chaves, 2016]], p.{{nnbsp}}669.</ref> The optical Lagrangian is the function which, when integrated w.r.t.&nbsp;the parameter of the path, yields the OPL; it is the foundation of [[Hamiltonian optics|Lagrangian and Hamiltonian optics]].<ref>[[#Chaves16|Chaves, 2016]], ch.{{nnbsp}}14.</ref>