2.3 Optical Fiber Configurations and Modes
57
first note that the distance from point A to point D is AD = AF − DF = (d/tan θ) −
d tan θ. Thus, the distance between points C and D is
s 2 = AD cos θ =
cos
2
θ − sin
2
θ
d/ sin θ
The requirement for wave propagation can then be written as
2π n 1
λ
(s 1 − s 2 ) + 2δ = 2π m
(2.24a)
where m = 0, 1, 2, 3, …. Substituting the expressions for s 1 and s 2 into Eq. (2.24a)
then yields
2π n 1
λ
d
sinθ
−
cos
2
θ − sin
2
θ
d
sinθ
+ 2δ = 2π m
(2.24b)
which can be reduced to
2π n 1 d sin θ
λ
+ δ = π m
(2.24c)
Considering only electric waves with components normal to the plane of
incidence, it follows from Eq. (2.19a) that the phase shift upon reflection is
δ = −2arctan
⎡
⎣
cos 2 θ −
n
2
2 /n
2
1
sinθ
⎤
⎦
(2.25)
The negative sign is needed here because the wave in the medium must be a
decaying and not a growing wave. Substituting this expression into Eq. (2.24c) yields
2π n 1 d sin θ
λ
− π m = 2arctan
⎡
⎣
cos 2 θ −
n
2
2 /n
2
1
sin θ
⎤
⎦
(2.26a)
or
tan
π n 1 d sin θ
λ
−
π m
2
=
⎡
⎣
n
2
1 cos 2 θ − n
2
2
n 1 sin θ
⎤
⎦
(2.26b)