4.4 Reflection of a Gaussian Beam from a Layer with a Slowly Varying Thickness
79
× exp
−i z
1 −
0.5ε
2 y
2 k
2∧
1y
n
2
1 k 2
−
0.5ε
2 x
2 k
2∧
1x
n
2
1 k 2
−
ε
2 x
2
ε
2 y
2 k
2∧
1x k
2∧
1y
4n
4
1 k 4
+ O(ε
4
)
×
× exp[ik
∧
1x εx
+ ik
∧
1y εy
]
∞
−∞
∞
−∞
dξ
∧
1 dξ
∧
2 exp[−ik
∧
1x ξ
∧
1 − ik
∧
1y ξ
∧
2 ]Φ(ξ
∧
1 , ξ
∧
2 ).
If |k
∧
1x | | k, |k
∧
1y | | k, the square root in the exponential
k 2 n
2
1 − ε 2 x 2 k
2∧
1x − ε 2 y 2 k
2∧
1y
can be expanded into a series in which only terms quadratic in k 1x and k 1y would be
retained. Then,
E inc =
1
(2π) 2
∞
−∞
∞
−∞
dk
∧
1x dk
∧
1y ×
× exp
−i z
1 −
0.5ε
2 y
2 k
2∧
1y
n
2
1 k 2
−
0.5ε
2 x
2 k
2∧
1x
n
2
1 k 2
+ O(ε
4
)
+ ik
∧
1x εx
+ ik
∧
1y εy
×
×
∞
−∞
∞
−∞
dξ
∧
1 dξ
∧
2 exp[−ik
∧
1x ξ
∧
1 − ik
∧
1y ξ
∧
2 ]Φ(ξ
∧
1 , ξ
∧
2 ).
Let us relate the coordinate systems (x
, y
, z
) and (x, y, z)
kn 1 x
= k 11 x + k 12 y + k 13 z,
kn 1 y
= k 21 x + k 22 y + k 23 z,
kn 1 z
= k 31 x + k 32 y + k 33 z,
k 11 = kn 1 a 11 , k 12 = kn 1 a 12 , k 13 = kn 1 a 13 , k 21 = kn 1 a 21 ,
(4.52)
k 22 = kn 1 a 22 , k 23 = kn 1 a 23 , k 31 = kn 1 a 31 , k 32 = kn 1 a 32 , k 33 = kn 1 a 33 , (4.53)
a 11 = cos(ϕ) cos(ψ) − sin(ϕ) cos(θ ) sin(ψ),
(4.54)
a 12 = − sin(ϕ) cos(ψ) − cos(ϕ) cos(θ ) sin(ψ),
(4.55)
a 13 = sin(θ ) sin(ψ),
(4.56)
a 21 = cos(ϕ) sin(ψ) + sin(ϕ) cos(θ ) cos(ψ),
(4.57)
a 22 = − sin(ϕ) sin(ψ) + cos(ϕ) cos(θ ) cos(ψ),
(4.58)