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Derivation from Maxwells equations
With the equations for a general time-harmonic field in a
nonconducting isotropic
medium,
it is found that the vectors and satisfy Maxwells equations when
independent of time.
In regions free of charge and current,
where, being the wavelength in a vacuum.Additionally, a
homogeneous plane wave in a medium with an index of refraction of
and
propagating in the direction of the unit vector s can be modeled
by the equations,
where eand h are vectors with constant and mostly complex
values. With this example,
along with findings of a monochromatic electric dipole field in
a vacuum,
where r is the distance from the dipole. In this example
however, e and h are not
constant vectors. But with distances very far away from the
dipole relative to the
wavelength andwith appropriate normalization of the dipole
moment, these vectors are
shown to be independent of .These two cases lend themselves to
examining regions many wavelengths away from the
sources with more general types of fields such that
where, described as the optical path, is a real scalar position
function and andare vector
functions of position.
Next, relating the functions , , and using various vector
identities,
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Problem 2.
Using ray matrices derive the overall round trip matrix for a
laser cavity comprising 2
spherical mirrors (curvatures R1 and R2) separated by a distance
d. Under what
conditions is this cavity stable in the Kogelnik and Li
sense?
Solution
The ABCD matrix transformation is as follows:
where
and
In order to find the eigenvalues, the determinant of the matrix
is calculated and solved in
the following manner:
The resulting quadratic formula is as follows:
