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Optical Receivers 217

I ∕P opt
ph
(4nL∕ = 1 + 2 × integer)=
min 0
hf ∕q
0
[1 − exp (−W)][1 + R exp (−W)]
2
= √ (1 − R ), (5.58)
S
[1 + R R exp (−W)] 2
1 2
with R ≈ R . As follows from Eqs. (5.56) and (5.58), the undulation of RC enhancement between peak and
1
S
valley values, at resonance and antiresonance, is
[ √ ] 2
1 + R R exp (−W)
max 1 2
= √
min 1 − R R exp (−W)
1 2
( √ ) 2
1 + R R
1 2
≈ √ , when W << 1, since exp (−W)≈ 1;
1 − R R
1 2
( √ ) 2
1 + R 1
≈ √ , when W << 1, and also R ≈ 1;
2
1 − R 1
4 1 F
≤ ∼ ≈ , when W << 1, R ≈ 1, and R → 1, (5.59)
√ √ 2 1
(1 − R ) 2 1 − R 1
1
where the last line shows that the cavity finesse F and RCE undulation max ∕ min between peak and valley
values are related, recalling the approximate relations in Eq. (5.54).
The behavior of RCE is illustrated in Fig. 5.20 and compared with a non-resonant photodetector. The hori-
zontal axis is reversed, considering that a particular photodetector has fixed width W of the absorption layer,
and the absorption coefficient decreases for longer wavelengths, thus, the left-hand sides of the plots corre-
spond to shorter wavelengths, while the right-hand sides correspond to longer wavelengths. The behavior of
RCE is discussed further below. At shorter wavelengths in photodetectors with thick absorption layers, the
product × W is larger than 1. In this case, the light is absorbed before reaching the back mirror, and the
RCE structure behaves identically with the non-resonant photodetector–all lines overlap for × W > 3in
Fig. 5.20 and the back mirror, if any, is irrelevant. Of course, a portion of the incident light is reflected by the
front mirror (or the surface of the photodetector), and we desire R = R to be as low as possible.
S
1
At longer wavelengths in photodetectors with thin absorption layers, the product × W is smaller than 1,
and RCE becomes relevant. If the back mirror is ideal (R = 1, left-hand plot in Fig. 5.20), the resonance
2
in the Fabry–Perot resonator would help to increase the quantum efficiency (non-monotonic thin lines) and
√
even restore the ideal value max = 1 at condition exp (−W)= R . However, real mirrors have reflection
1
R < 1, and the decrease in back mirror reflection R to 0.9 and 0.8 (still high) degrades the ability of RCE to
2
2
restore the quantum efficiency, as seen in the middle and right-hand plots of Fig. 5.20, especially for high front
mirror reflection R (which is also the reflection from the photodetector surface, R ≈ R ). The condition for
S
1
1
maximum quantum efficiency becomes a complicated expression:
√ 1 + R [2exp (−W)− 1]
2
R R = , (5.60)
1 2
2 −(1 − R ) exp (−W)
1
but tends to (R × R )≈(1 − 2W) when (W) ≤ 0.1 and R ≥ 0.8. In addition, if the resonator is not tuned
1 2 2
at the wavelength, then RCE will suppress the quantum efficiency, as shown with symbols on dashed lines in
Fig. 5.20. The suppression is less than 3 dB, which is not a dramatic decrease in responsivity, but we realize
that the RCE is not favorable if the cavity is not precisely tuned at the wavelength of interest, e.g., in cases of