Page 303 - Optofluidics Fundamentals, Devices, and Applications

Optofluidic Resonators 277

I = T 2 = T 2
T + R − 2 cosϕ ⎛ ⎞
ϕ
2
1 R 2 2
1 ( − R) + 4 Rsin ⎜ ⎟
⎝ ⎠ 2
= T 2 2 ⎜ ⎛ 1 ⎞ ⎟
( (1 − R ) ⎡ 4R ⎤ ⎛ ⎞ϕ
⎜ 1 + ⎢ 2 ⎥ sin ⎜ ⎟ ⎟ ⎟
2
⎝ ⎣ (1 − R ) ⎦ ⎝ 2 ⎠ ⎠
2
ϕ
⎡ T ⎤ ⎡ ⎛ ⎞⎤ −1 ⎡ T ⎤ 2
ϕ
= ⎢ ⎥ ⎢ 1 + K sin 2 ⎜ ⎟⎥ = ⎢ ⎥ Α() (12-22)
⎣1 − ⎦ ⎣ ⎝ ⎠ 2 ⎦ 1 ⎣ ⎣ − R⎦
R
2
R 1
With K = 4 /( − R) , Α()ϕ is defined as the Airy function. The ampli-
tude of the resulting electric field vector of the back-reflected light
()
waves Em is obtained using again the principle of superposition
r
for m reflected waves:
1 1 2 {
iϕ
−+
+ −+
−+
+
E m = r + t t re iϕ 1 + rre + + rr ( 12 ) ) m −2 e im −2)ϕ }
(
()
r 1 12
t t re ( 1 − rre m )
− + i(m−1)ϕ
+− + iϕ
= r + + 11 2 1 2 (12-23)
1 − −+ iϕ
1 rre
12
For an infinite number of reflections m → ∞
+− +
tt r e iϕ
+
11 2
E → E ∞ = r + (12-24)
()
−+
r r 1 1 − rre iϕ
12
Considering two identical dielectric surfaces, using Eqs. (12-21) and
(12-24):
+− +
iϕ
tt r e iϕ 1 ( − Re iϕ − Te ) 1− e iϕ
+
E = r − 11 1 = R = R (12-25)
r 1 − + 2 iϕ iϕ iϕ
1 re 1 1− Re 1− Re
1
which leads to
⎛ ⎞
ϕ
4 Rsin 2 ⎜ ⎟
(22 ϕ) ⎝ 2 ⎠ ⎠
− cos
∗
I = E E = R =
R r r + 2 ϕ ⎛ ⎞
1 R − 2 Rcos ϕ
− )
(1 R 2 + 4R sin 2 ⎜ ⎟
⎝ ⎠ 2
⎛ ⎞ ⎡ ⎛ ϕ⎞⎤ − 1
ϕ
+
⎢
= K sin 2 ⎜ ⎟ 1 K sin 2 ⎜ ⎟⎥ (12-26)
n
⎝ ⎠ 2 ⎝ 2⎠
⎣ ⎦