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A Optical Limit of Maxwell’s Equations
If we let ρ be the vector pointing from a position on a ray toward the center of
the osculating circle at that point on the ray’s path, and ρ be the osculating circle
radius, then geometry dictates that
d 2 r
ds 2 =
1
ρ
ρ .
From the above, the gradient of the index of refraction must lie in the plane of the
tangent vector to the ray and the normal to the ray pointing toward the center of the
osculating circle.
The radius of the osculating circle is given by
ρ =
1
ρ · ∇ ln n
.
A.3 Fermat’s Principle
Fermat’s principle states that light rays take that path between two points which
minimizes the time of travel: t =
dt = (1/c)
n ds. The principle is a
consequence of the behavior of optical rays. The time is minimized if
n ds =
n
dr
dτ
·
dr
dτ
dτ
reaches a minimum between two points. The Euler-Lagrange equations to minimize
this integral become
d
ds
n
dr
ds
= ∇n
which is the same as the ray equation.
The full set of laws for reflection and refraction can be derived from the ray
equation or from Fermat’s Principle.
A.4 Snell’s Law
Suppose a boundary exists between two regions of constant but distinct materials
with different indices of refraction. If the boundary is a plane, and coordinates are
selected with the z axis perpendicular to the plane, then the gradient of the index of
refraction is along the z axis. If a ray is in the x − z plane, then the projection of
the ray equation terms along the x axis reads (n sin θ ) = 0, where θ is the angle
between the ray tangent vector and the normal to the surface. This is Snell’s law:
n 1 sin θ 1 = n 2 sin θ 2 .
A Optical Limit of Maxwell’s Equations
If we let ρ be the vector pointing from a position on a ray toward the center of
the osculating circle at that point on the ray’s path, and ρ be the osculating circle
radius, then geometry dictates that
d 2 r
ds 2 =
1
ρ
ρ .
From the above, the gradient of the index of refraction must lie in the plane of the
tangent vector to the ray and the normal to the ray pointing toward the center of the
osculating circle.
The radius of the osculating circle is given by
ρ =
1
ρ · ∇ ln n
.
A.3 Fermat’s Principle
Fermat’s principle states that light rays take that path between two points which
minimizes the time of travel: t =
dt = (1/c)
n ds. The principle is a
consequence of the behavior of optical rays. The time is minimized if
n ds =
n
dr
dτ
·
dr
dτ
dτ
reaches a minimum between two points. The Euler-Lagrange equations to minimize
this integral become
d
ds
n
dr
ds
= ∇n
which is the same as the ray equation.
The full set of laws for reflection and refraction can be derived from the ray
equation or from Fermat’s Principle.
A.4 Snell’s Law
Suppose a boundary exists between two regions of constant but distinct materials
with different indices of refraction. If the boundary is a plane, and coordinates are
selected with the z axis perpendicular to the plane, then the gradient of the index of
refraction is along the z axis. If a ray is in the x − z plane, then the projection of
the ray equation terms along the x axis reads (n sin θ ) = 0, where θ is the angle
between the ray tangent vector and the normal to the surface. This is Snell’s law:
n 1 sin θ 1 = n 2 sin θ 2 .