Page 302 - Optofluidics Fundamentals, Devices, and Applications

276 Cha pte r T w e l v e

waves, and those traveling from right to left are defined as negative
waves. The coefficients are complex numbers. The back-and-forth
traveling waves in the resonator generate a phase difference that is
calculated to be
π(
ϕ = 22nd cos( θ)) (12-16)
λ
0
Using the principle of superposition, the amplitude for a wave traveling
from left to right through the resonator after m passes is given by

E m) = { rr ( ) m−1 }
−+
+ +
iϕ
(
)
( t t 1 + rre + + −+ e im−1)ϕ
t 12 1 2 1 2
⎡
m
− +
++
tt 1 −(r r ) e im ϕ⎤ ⎦
⎣
12
1 2
= (12-17)
−+
1 − rre ϕ i
12
−+
For an infinite number of reflections m → ∞ and rr < :
1
12
++
tt
E → E ∞ = 12 (12-18)
()
t t − −+ iϕ
1 rre
12
The resulting transmitted intensity is given by
++
tt 2
12
∗
I = E E = (12-19)
t t t −+ 2 −+
1 + rr − 2 rr cos ψ
12 12
with Ψ = ϕ ε; ε is a correction factor for the phase difference occur-
+
ring during the reflection.
ε= argr − + argr + (12-20)
1 2
If the surfaces of the Fabry-Perot resonator are made out of the same
dielectric layers, the coefficients r and t can be considered as being
real numbers. Then for a single reflecting surface:
+ −
− 2
+ 2
R
tt = ; r + = −r − ; () = () = ; R + T = 1 (12-21)
T
r
r
R and T are coefficents for the intensity of the reflection and the trans-
mission of the surface. Using Eq. (12-19) in Eq. (12-21) and ε = 0 and
−
+
t = t r =; + r :
−
2 1 2 1