Deck 11: Integrated Optics and Photonics

Question 1

Assume the gain profile of an optical amplifier is
where 0 is the peak-gain wavelength and is the spectral width of the amplifier gain. If = 25 nm, find the FWHM (the 3-dB gain) of the amplifier gain if the peak gain at 0 is 30 dB.

Answer 1

Since we have The FWHM is given by 2( - 0), so that from the above equation, with the 3-dB gain G = 27 dB being 3 dB below the peak gain, we have
  which is the expected result for a gaussian gain profile.

Question 2

Compare the maximum theoretical power conversion efficiency (PCE) for 980-nm and 1475-nm pumping in an EDFA for a 1545-nm signal. Contrast this with actual measured results of PCE = 50.0 percent and 75.6 percent for 980-nm and 1475-nm pumping, respectively.
( b ) Using the actual results for PCE given in ( a ), plot the maximum signal output power as a function of pump power for 0 P p ,in 200 mW for pump wavelengths of 980 nm and 1475 nm.

Answer 2

From Eq. (11.17), the maximum PCE is given by  for 980-nm pumping, and by   for 1475-nm pumping

Question 3

Assume we have an EDFA power amplifier that produces P s , out = 27 dBm for an input level of 2 dBm at 1542 nm.
( a ) Find the amplifier gain.
( b ) What is the minimum pump power required

Answer 3

(a) 27 dBm = 501 mW and 2 dBm = 1.6 mW.
Thus the gain is (b) From Eq. (11.19), .With a 1.6-mW input signal, the pump power needed is

Question 4

To see the relative contributions of the various noise mechanisms in an optical amplifier, calculate the values of the five noise terms in Eq. (11.30) for operational gains of G = 20 dB and 30 dB. Assume the optical bandwidth is equal to the spontaneous emission bandwidth (30-nm spectral width) and use the following parameter values: ( b ) To see the effect of using a narrowband optical filter at the receiver, let B o = 1.25 × 10 11 Hz (125 GHz at 1550 nm) and find the same five noise terms for G = 20 dB and 30 dB.

Question 5

Plot the penalty factor F ( G ) givenby Eq. (11.43) as a function of amplifier gain for gains ranging from 0 to 30 dB. Assuming the fiber attenuation is 0.2 dB/km, draw a distance axis parallel to the gain axis to show the transmission distances corresponding to the gain values.

Question 6

Consider a cascaded chain of k fiber-plus-EDFA combinations, as shown in Fig. 11.24.
Fig. 11.24 Cascaded EDFA chain
( a ) Show that the path-averaged signal power is
( b ) Derive the path-averaged ASE power given by Eq. (11.42).

Question 7

Consider a long-distance transmission system containing a cascaded chain of EDFAs. Assume each EDFA is operated in the saturation region and that the slope of the gain-versus-input power curve in this region is -0.5; that is, the gain changes by ±3 dB for a 6-dB variation in input power. Let the link have the following operational parameters: ( a ) Suppose there is a sudden 6-dB drop in signal level at some point in the link. Find the power output levels after the degraded signal has passed through 1, 2, 3, and 4 succeeding amplifier stages.
( b ) Repeat part ( a ) for a signal-level drop of 12 dB.

Question 8

Let the electric field of an optical signal at a carrier frequency v i be
where P i is the signal power at the carrier frequency v i and i is the carrier phase. If N optical signals each at a different frequency v i are traveling along a fiber, show that the signal power is
where jk = 2( v j - v k ) represents the beat frequency at which the carrier population oscillates.

Question 9

Consider an EDFA with a gain of 26 dB and a maximum output power of 0 dBm.
( a ) Compare the output signal levels per channel for 1, 2, 4, and 8 wavelength channels, where the input power is 1 W for each signal.
( b ) What are the output signal levels per channel in each case if the pump power is doubled

Question 10

Recall from Eq. (7.13) that the bit-error rate (BER) can be given in terms of a Q factor, where, from Eq. (7.14),
When thermal noise is dominant, we have . However, for an EDFA the existence of signal-ASE beat noise produces the condition . In this case, from Eq. (11.30),
and
Noting that the receiver sensitivity P R is half the signal power of a transmitted 1 bit for a uniform distribution of ones and zeros (i.e., P s = 2 P R ), show that in terms of the Q factor the receiver sensitivity is
where F is the noise figure given by Eq. (11.35).

Question 11

Consider an InGaAsP semiconductor optical amplifier that has the following parameter values: If a 100-mA bias current is applied, find ( a ) the pumping rate R p ( b ) the maximum (zero- signal) gain, ( c ) the saturation photon density, ( d ) the photon density if a l- W signal at 1310 nm enters the amplifier. Compare the results of ( c ) and ( d ).

Question 12

Verify that the gain expression in Eq. (11.13) follows from the integral relationship Eq. (11.12 b ).

Question 13

Solving Eq. (11.13) numerically, make plots of the normalized amplifier gain ( G / G 0 ) versus the normalized output power ( P s , out / P amp , sat ) for unsaturated amplifier gain values of G 0 = 30 dB, 15 dB, and 10 dB.

Question 14

The output saturation power P out sat is defined as the amplifier output power for which the amplifier gain G is reduced by 3 dB (a factor of 2) from its unsaturated value G 0. Assuming G 0 » 1, show the amplifier output saturation power is

Question 15

Since the gain constant a depends on the frequency, the amplifier gain is also frequency- dependent. The 3-dB bandwidth (full-width at half-maximum, FWHM) is defined as the frequency for which the power gain G ( v ) is reduced by a factor of 2. Assume the gain parameter g has a gaussian profile
where v is the optical bandwidth (the spectral width of the gain profile) and v 0 is the maximum-gain frequency. Starting with Eq. (11.9), show that the ratio of the 3-dB bandwidth 2( v - v 0 ) to the optical bandwidth v is
where log 2 X is the base-2 logarithm of X. What does this equation show concerning the relationship between the amplifier gain and the optical bandwidth