Page 275 - Phase Space Optics Fundamentals and Applications

256 Chapter Eight

8.6.2 Asymptotic Treatment and Ray
Equations
Approaches 1 and 2 mentioned in Sec. 8.2 can be used to separate
Eq. (8.55). For approach 2, it is assumed that both B and are real,
and Eq. (8.55) is separated into real and imaginary parts. This leads to
a momentum-space flux-line formalism where no two trajectories ever
have the same optical momentum at a given z, although they can cross
freely in position. (For quantum mechanics, the equivalent procedure
would lead to a momentum-space Bohmian formalism.) However,
except for very simple refractive index distributions, the two resulting
equations would be very complicated and would involve terms of
many orders in k. This approach is therefore not considered further.
Ontheotherhand,approach1,whichcorrespondstotheasymptotic
treatment, is tractable. We start by expanding B as a Debye series:

∞
,
B j
B(p,z) = (8.56)
(ik) j
j=0
The substitution of this series into Eq. (8.55) gives analogs of the
eikonal and transport equations
2
∂ ∂
2 2
|p| + = n − ,z (8.57a)
∂z ∂p

∂ 2 ∂ 1 ∂ 2 ∂n 2 ∂
B 0 + · B 0 − ,z = 0 (8.57b)
∂z ∂z 2 ∂p ∂x ∂p
¯
Equation (8.57a) can be solved by parameterizing p = P(z, ) and
defining
∂
¯ ¯
X(z, ) =− (P,z) (8.58)
∂p
∂
¯
¯ H(z, ) = (P,z) (8.59)
∂z
Notice that the derivative with respect to z of both sides of Eq. (8.58)
gives

2
∂ ∂ ∂
˙ ¯ ˙ ¯
X =− P · − (8.60)
∂p ∂p ∂p ∂z
To eliminate the cross-derivative term, let us consider the vector
derivative with respect to the transverse momentum of both sides