Page 235 - Fiber Optic Communications Fund

216 Fiber Optic Communications

2
Here F is called the “finesse” of the resonator and F is the ratio of the “free spectral range” FSR = ∕(2nL),
0
which is the spacing between RC enhancement maxima and the spectral FWHM, which is the opti-
cal “bandwidth.” The finesse in resonators corresponds to the quality factor Q = 2Δf ∕f in LC
LC 3dB o
(inductor–capacitor) electrical resonators at frequency f , with 2Δf being the 3-dB bandwidth around f ,
o 3dB o
which is the same as FWHM. The principal difference between an LC resonator and a Fabry–Perot cavity
is that the LC resonator is a lumped-element resonator with only one resonance frequency f , whereas the
o
Fabry–Perot cavity is a delay resonator with periodic resonance frequencies and spacing FSR between the
wavelengths corresponding to these frequencies. The last expression in Eq. (5.51) suggests that a high R
1
is favorable for RC enhancement, e.g., R = 0.9 would provide an RC enhancement ≈ 750. However, the
1
narrowing of the bandwidth of RCE at high R sets tight requirements for precision of distances in the
1
Fabry–Perot cavity, so that the inaccuracy =(ΔL∕L) < (0.5∕F) of the resonator length is a fraction of the
L
RCE bandwidth.
Example 5.5
Givenahigh R = 0.9 in the last approximate expression in Eq. (5.54), the RCE relative bandwidth is 1∕F =
1
√ √ √
(1 − 0.9)∕ = 1.6%. Using a conservative value for the refractive index of Si, n = ( ∕ )= (11.9)≈
Si
o
3.5, and choosing = 850 nm, then for integer = 4, the resonator length is L = integer × ∕(2n)= 4 ×
o
o
850 nm∕(2 × 3.5)≈ 486 nm. Consequently, to be within the resonance, the resonator should be fabricated
with inaccuracy ΔL < L × 0.5∕F = 486 nm × 0.5 × 1.6%≈ 4nm.
Comments: The fabrication of a stack of heterogeneous materials for two mirrors and silicon in-between with
accuracy of 4 nm is not simple. The materials might not be perfect, or the calculation might not be accurate,
in order to guarantee 0.8% accuracy; e.g., the refractive index of Si is not 3.5, but 3.65 at = 850 nm, which
o
gives a much larger error of 4% in the calculation. Thus, we cannot really exploit RCE with large values. In
real structures, RCE is in the range of 10, partially because of inaccuracy and additionally because R > 0.9
2
for the back mirror is also difficult to achieve (metals have reflections of about this value and DBR requires
more than four undulations of Si–SiO for higher reflections from the Bragg mirror).
2
Another limitation for RCE is that the external quantum efficiency is not a monotonic function of reflections
and absorption in the Fabry–Perot cavity. The terms in Eq. (5.51) that determine the maximum RCE external
quantum efficiency as a ratio of the material quantum efficiency at the resonant condition
max
(2L +Ψ +Ψ )=(4nL∕ +Ψ +Ψ )= 2 × integer (5.55)
1
o
2
1
2
are arranged in the following equation:
I ∕P opt
ph
max (4nL∕ = 2 × integer)=
0
hf ∕q
0
[1 − exp (−W)][1 + R exp (−W)]
2
= (1 − R ), (5.56)
√ S
[1 − R R exp (−W)] 2
1 2
with R ≈ R . Eq. (5.58) is for the minimum RCE external quantum efficiency as a ratio of the material
S 1 min
quantum efficiency at the antiresonant condition
(2L +Ψ +Ψ )=(4nL∕ +Ψ +Ψ )= + 2 × integer, (5.57)
2
2
o
1
1