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Chapter 12

Analog Communications

In this chapter we will consider the most straightforward implementation of optical communications, thatwhere an analog signal is directly impressed on the optical carrier and where the detection scheme is sim-ply to detect the optical field, thereby recovering a time-varying electrical current which, under certaincircumstances, is an accurate replica of the initially transmitted field.

A system’s block diagram for an analog direct detection system may well appear as that depicted inFigure 12.1. The encoder in such an analog transmission system is often quite simple in that it simply takesan input electrical signal and converts its voltage and current to those required by the modulator. Further,an analog modulator could be just the laser diode or light-emitting diode’s electrodes, as is depicted inFigure 12.2. Simplicity is also the rule in the receiver end. Here the optical power can be simply detectedand preamplified for launch into a load, as is depicted in Figure 12.3. Fidelity in such a system is oftensimply defined in terms of how well the signal supplied to the load compares with that supplied to thesource’s electrodes. This is the basic picture of the direct detect analog system.

A common use of analog optical systems has been to replace an antenna. That is, a radio frequency (RF)system generally takes an information stream and impresses it on a carrier or, in subcarrier modulation (oftencalled frequency-division multiplexing (FDM)), takes a number of information streams and places these onthe same number of different carriers. The composite signal is then transmitted via an antenna. As wasdiscussed all the way back in the first chapter, an antenna is a good instrument with which to broadcast overa limited area or, if assisted by a satellite, a good instrument to begin the process of broad-area broadcast. Ifone doesn’t want to broadcast but instead to send point-to-point, the high coherence and short wavelengthof optical sources and/or optical waveguides are going to be hard to beat. The RF techniques, though, aremature and standardized. If in Figure 12.2 the electrical signal s(t) is an antenna-ready, possibly subcarrier-multiplexed signal and the load in Figure 12.3 is the RF demultiplex and readout system generally used asan antenna load, then the information coding and decoding parts of the optical communication system can

Figure 12.1: A system’s block diagram.

1

CHAPTER 12. ANALOG COMMUNICATIONS 2

Figure 12.2: Direct detect transmitter end where the electrical signal is simply added to the optical source’sdc bias current.

Figure 12.3: A possible realization for a direct detect receiver.

be bought off-the-shelf in a standardized (and therefore inexpensive due to competition) form. One couldargue that the carrier is superfluous in a single-channel system, but this is not quite true. As was discussedin section 8.3, there is always 1/f present in any system. The carrier frequency causes the decoding to takeplace away from the 1/f noise peak and thereby in a low-noise frequency regime. So, subcarrier modulationcan even be worthwhile in a single-channel system.

This chapter is organized as follows. In section 12.1 we consider how to define spectra and, therefore,signal-to-noise ratios for analog optical communication systems. In the next section, 12.2, we discuss thetwo most ubiquitous transmission formats, amplitude modulation (AM) and frequency modulation (FM).Section 12.3 gives discussion to the problem of filtering to try to best separate the signal from the noise.In section 12.4, discussion turns to multiplexing schemes, including both subcarrier and wavelength-divisionmultiplexing techniques. Section 12.5 concerns techniques to optically drive and read out RF and microwaveantennas with analog optical fiber links. Section 12.6 gives discussion to the three most common techniquesused to increase receiver sensitivity through achieving shot noise-limited operation: optical heterodyning,use of a photomultiplier, and use of a rare earth-doped optical amplifier. Some discussion of rare earthelements and their properties was given in section 7.2.

12.1 Signal-to-Noise Analysis

A clear problem with finding probability distributions for signals is that signals need to vary with time inorder to carry information. Yet, when we find count distributions—pk(k)’s—inevitably we need to either takethe conditioning number m to be a constant or itself a time-independent density. In digital communications,information is carried in the “toggling” of the signal from one level to another. In this case, one could thinkthat under certain conditions we could define count distributions with different conditioning parameters foreach of the levels and proceed from there. With analog signals, it is hard to see how count statistics could be


Figure 12.4: Schematic depiction of a “locally” stationary process.

used at all and clear that something else needs to be used. In the following paragraph, the attempt will be tomake the possibilities and conditions a bit more clear. Indeed, we will find that count distributions are usefulin the digital case. In the analog case, we will find that is is still possible to identify average values of signaland noise, despite the signal dependence of the shot noise. In this chapter, we will then find signal-to-noiseratios for some transmission formats. In the next chapter, in the first section on information theory, we’llsee that the signal-to-noise ratio can be interpreted as an effective number of bits of information, but we’llhave no need to go further with this interpretation. A point to note here is that the signal-to-noise ratios(SNR) as defined in this chapter are electrical SNRs which are the square of the optical SNRs calculatedfrom count distributions.

Here we’ll reprint Figure 3.8 as figure 12.4, which illustrates what we mean by an ensembles of stochasticprocesses and, by example then, an ergodic process. The point of the figure is that an ensemble should bean infinite set of statistically identical realizations of the given process. If we average down an ensemble atsome given time t, this should define a complete set of moments, etc. If the statistics of a realization wereconstant in time, then it shouldn’t matter which instant we choose to average at. However, this is saying thesame thing, as we could do the average on any of the temporal realizations and, if we choose the averagingtime τ in the expression

〈f(t)〉 =1τ

t∫t−τ

f(t′) dt′ (12.1)

as long enough, then the statistics we get should be as good as averaging down enough members of theensemble as correspond to the number of temporal modes contained in τ . If the statistics are constant forall time, then the process is called ergodic.

A figure such as Figure 12.5 helps to illustrate the various classes of random processes that we comeacross in practice. There can be processes which have continually varying condition numbers, which are


the outer ring of the diagram. A process which has stationary increments is a process in which there areincrements Ti in which we can write that

E (x(t)) independent of t

E (x(t1)x(t2)) dependent only on τ = t2 − t1. (12.2)

Wide-sense stationary processes are those for which there is but one time increment of infinite length. Theseare the processes for which we say that the Wiener-Khintchine theorem allows us to calculate spectra fromcorrelations and vice versa. Of course, when we have a process with stationary increments, we can definea finite transform which extends only over the increment. We will again employ this picture in the nextchapter on digital signals. A strictly stationary process would be one for which

px(x1, x2, . . . , xk; t1, t2, . . . , tk) = px(x1, x2, . . . , xk; t1 − T, t2 − T, . . . , tk − T ) (12.3)

for all values of T . The ergodic processes are those for which the time and ensemble averages are completelyreversible. If we have a true binary code, then we could say that we have a process with stationary increments.If we do analog coding, we have strictly a random process. However, if one considers the analog informationstream to be a random process with stationary statistics, then we can consider the process as wide-sensestationary. This is the approach that will be taken here. We will see that this approach will allow us toseparate the shot noise (signal dependent) from the signal.

In order to speak about electrical power SNR, we will need to discuss quantities quadratic in current, aswe know that electrical power is proportional to the square of the current. An obvious quantity to consideris the current correlation function Ri(τ). A possible definition for the correlation function of a wide-sensestationary signal could be

Ri(τ) = limT→∞

E

[i(t + τ)i(t)

2T

]. (12.4)

There is, though, a problem with this expression for use with information-bearing signals. To demonstratethis problem, we apply the Wiener-Khintchine theorem to the above to find the spectral density Si(ω) inthe form

Si(ω) = limT→∞

E

[∣∣iT (ω)∣∣2

2T

], (12.5)

where

iT (t) =

i(t), −T/2 < t < T/20, otherwise

(12.6)

and where

iT (ω) =

T∫−T

i(t′)e−jωt′ dt′. (12.7)

Unfortunately, the expression of (12.5), which originally came from Schuster and Lees (1901) and whichis called the periodogram, is apparently problematic, as is pointed out by various authors (Davenport andRoot 1958, Papoulis 1965, Wiener 1964, Mullis and Roberts 1987). The expression for the spectral densityconverges in the mean to the spectrum of a stationary process, but the variances of higher-order momentsdiverge. We will employ another technique here—one that employs ensemble averages.


Figure 12.5: A Venn-type diagram illustrating the various classes of stationarity within the total class ofrandom processes.

We discussed another technique for finding a correlation function back in section 11.3.2 of Chapter 11.We know that the correlation function can be defined by

R12(t1, t2) = E[i(t1)i(t2)

]. (12.8)

The point here is that the current is generally defined by

i(t) =k(t−τd,t)∑

0

h(t − ti), (12.9)

and the expectation is defined as an ensemble average. What (12.8) then indicates is an average of manyindependent, identically distributed (iid) samples taken over a fixed interval. This is essentially the sameprinciple of operation as is used in spectrum analyzers and sampling oscilloscopes. To find the correlation,we can use the two-point characteristic function. That is, we know that the two-point characteristic functionψi1i2(ω1, ω2) should have an expansion of the form

ψi1i2(ω1, ω2) = 1 + jω1m10 + jω2m01 + jω1jω2m11 +(jω1)2

2m20 +

(jω2)2

2m02 + · · · . (12.10)

Clearly, m11 is the expectation

m11 = E[i(t1)i(t2)

], (12.11)

which is, by definition, the correlation function. Therefore, if we can find an explicit representation forthe two-point characteristic function, we can find the desired correlation function. We did this back in


section 11.3.2—but only for a coherent signal in a stationary increment, not for an information-bearingwaveform. That is what we’ll presently perform.

The two-point characteristic function must be of the form

ψi1i2(ω1, ω2) = E[ejω1i1ejω2i2 ]. (12.12)

Writing out the currents explicitly, we have

ψi1i2(ω1, ω2) = E

exp

jω1

k∑j=1

h(t1 − tj) + jω2

k∑j=1

h(t2 − tj′)

. (12.13)

Then, taking expectations over the Markov process yields

ψi1i2(ω1, ω2) = EnEk

te∫

tb

expjω1h(t1 − t′) + jω2h(t2 − t′)

n(t′)m

dt′

k

, (12.14)

where

te = max(t1, t2)tb = min(t1, t2) − τd, (12.15)

as it is tacitly assumed that h(t) = 0 for t < 0 and t > τd. It should be noted that there is some approximationin the averaging over the emission times. The process is, in general, conditional Poisson rather than strictlyPoisson. Clearly, a strictly Poisson process will generate a Markov sequence of counts. This property of thesequence being Markov was used in deriving the operator Ek. The conditioning, however, should not putany memory into the system, we would not think, so we would have to say that using a Markov sequence todefine Ek is a good approximation—if an approximation at all. Using the definition of the Ek operator, wenote that

ψi1i2(ω1, ω2) = En

∑ mk

k!e−m

te∫

tb

expjω1h(t1 − t′) + jω2h(t2 − t′)

n(t′)m

dt′

k

. (12.16)

Pulling the mk out of the integral to cancel the mk in front (m is already integrated from 0 to t) and usingthe definition of the infinite sum for an exponential, we obtain

ψi1i2(ω1, ω2) = En

exp

te∫tb

ejω1h(t1−t′)ejω2h(t2−t′) − 1

n(t′) dt′

. (12.17)

We can now make an expansion of the quantity in the brackets before taking the En expectation to find

ψi1i2(ω1, ω2) = En

[1 +

te∫tb

(ejω1h(t1−t′)ejω2h(t2−t′) − 1)n(t′) dt′

+te∫tb

dt′te∫tb

dt′′ (ejω1h(t1−t′)ejω2h(t2−t′) − 1)(ejω1h(t1−t′′)ejω2h(t2−t′′) − 1)n(t′′)n(t′) + · · ·]

, (12.18)

which can then have the n expectation operator applied to yield


ψi1i2(ω1, ω2) = 1 +te∫tb

(ejω1h(t1−t′)ejω2h(t2−t′) − 1)En

[n(t′)]dt′

+ 12

te∫tb

dt′te∫tb

dt′′ (ejω1h(t1−t′)ejω2h(t2−t′) − 1)(ejω1h(t1−t′′)ejω2h(t2−t′′) − 1)E[n(t′)n(t′′)

]+ · · · . (12.19)

The last step is to expand the exponentials to find

ψi1i2(ω1, ω2) = 1 + jω1

te∫tb

h(t1 − t′)En

[n(t′)]dt′ + jω2

te∫tb

h(t2 − t′)En

[n(t′)]dt′

+ (jω1)2

2

te∫tb

h2(t1 − t′)En

[n(t′)]dt′ + (jω2)

2

2

te∫tb

h2(t2 − t′)En

[n(t′)]dt′

+ (jω1)2

2

te∫tb

dt′te∫tb

dt′′h(t1 − t′)h(t2 − t′) + (jω2)2

2

te∫tb

dt′te∫tb

dt′′h(t2 − t′)h(t2 − t′′)

+ jω1jω2

te∫tb

h(t1 − t′)h(t2 − t′)En

[n(t′)]dt′

+ jω1jω22

te∫tb

dt′te∫tb

dt′′[h(t1 − t′)h(t2 − t′′) + h(t2 − t′)h(t1 − t′′)

]En

[n(t′)n(t′′)

]+ · · · (12.20)

From the above, we can read off our correlation function, which is of the form

Rid(t1, t2) =te∫tb

h(t1 − t′)h(t2 − t′)En

[n(t′)]dt′

+te∫tb

dt′te∫tb

dt′′[h(t1 − t′)h(t2 − t′′) + h(t1 − t′′)h(t2 − t′)

]En

[n(t′)n(t′′)

]. (12.21)

Noting that we can always write

t2 = t

t1 = t2 + τ, (12.22)

where τ can be positive or negative, we can recast (12.20) in the form

Rid(t, τ) =te∫tb

h(t + τ − t′)h(t − t′)En

[n(t′)]dt′

+ 12

te∫tb

dt′te∫tb

dt′′ h(t + τ − t′)h(t − t′′)En

[n(t′)n(t′′)

]+ 1

2

te∫tb

dt′te∫tb

dt′′ h(t − t′)h(t + τ − t′′)En

[n(t′)n(t′′)

], (12.23)

which now appears to depend explicitly on the time t as well as on the delay period τ . But the assumptionthat the information stream is random and stationary says that the t-dependence is not real and will averageout to leave an Rid(τ). What we need to do now is to identify terms that we consider noise and those thatwe consider signal. The first integral in (12.23) has an expectation that is linear in n(t). This term mustbe the shot noise term, as shot noise is inevitably linear in the optical power, as we have seen in Chapters 8and 11. The next two terms, then, must be the information-bearing signals. It should probably be pointedout at this point, though, that there may be noise imbedded in this information-bearing signal yet. That


noise is buried in the pn(n) of En. To decode an information stream, that stream must vary more slowlythan the h(t)—at least twice as slowly (sampling theorem) but probably even more slowly than that. Thiswe will be able to use.

The first term, as was mentioned above and as we will presently see, is what we previously called shotnoise. Using the wide-sense stationarity of the process n, we can write

te∫tb

h(t + τ − t′)h(t − t′)En

[n(t′)]dt′ = n

te∫tb

h(t + τ − t′)h(t − t′) dt′. (12.24)

If we now invoke the fact that h(t) is only nonzero between 0 and τd, then we would want to pick limits tband te such that

te∫tb

h(t + τ − t′)h(t − t′) dt′ =

t∫t−τd

h(t + τ − t′)h(t − t′) dt′. (12.25)

We can always now shift the origin of time for a wide-sense stationary process such that

t∫t−τd

h(t + τ − t′)h(t − t′) dt′ =

τd∫0

h(t′ + τ)h(t′) dt′. (12.26)

At this point, it does not matter whether we extend the limits to infinity, however, as there are no contribu-tions to the integral from outside these limits. We can, therefore, write that

τd∫0

h(t′ + τ)h(t′) dt′ =

∞∫−∞

h(t′ + τ)h(t′) dt′. (12.27)

The integral on the right-hand side is an auto-correlation of the h(t). This should indicate to us that a goodstrategy to simplify the expression would be to go to the spectral domain. Fourier transforming (12.27) gives

∞∫−∞

dτ ejωτ

∞∫−∞

h(t′ + τ)h(t′) dt′ = |H(ω)|2 , (12.28)

where

H(ω) =

∞∫−∞

ejωt′h(t′) dt′

h(t) =

∞∫−∞

e−jω′tH(ω′) dω′ (12.29)

is a Fourier transform pair. With this, we can find an expression for the squared shot noise current, i2SN bysimply integrating in frequency space to obtain

i2SN = n

∞∫−∞

|H(ω)|2 dω. (12.30)


Without detailed knowledge of the h(t) (the important quantity, as we’ll see, is really T (ω), the overalltransfer function of the receiver), we really can’t go much farther. In the limit where h(t) is somewhatsquare and of height roughly e/τd, we can use the Parseval relation

∞∫−∞

h2(t) dt =

∞∫−∞

|H(ω)|2 dω (12.31)

to show that

i2SN

∣∣∣MAX

=ne2

τd, (12.32)

where the bandwidth B would be 1/2τd (the noise powers are often expressed in terms of the bandwidth)and where the MAX indicates that, for anything other than a square response function, there will be lessshot noise. One should be able to convince oneself of this rather easily, although I don’t know how to proveit.

We only need consider one of the two remaining terms, as one can be obtained from the other by makingthe transformation τ → −τ and Ri(τ) must be symmetric in τ in order that Si(ω) be real. Applying similartechniques of limit changing and variable substitution such as we did to the shot noise term above, we canwrite

te∫tb

dt′te∫

tb

dt′′ h(t + τ − t′)h(t − t′′)Rn(t′, t′′) =

∞∫−∞

dt′∞∫

−∞dt′′ h(t′ + τ)h(t′′)Rn(t′, t′′). (12.33)

But we have already assumed that n(t) is a stationary process. We can then write

Rn(t′, t′′) = Rn(t′ − t′′), (12.34)

which says that we can write

∞∫−∞

dt′∞∫

−∞dt′′ h(t′ + τ)h(t′′)Rn(t′, t′′) =

∞∫−∞

dt′ h(t′ + τ)∫

dt′′ Rn(t′ − t′′)h(t′′). (12.35)

The second integral is a convolution as a function of t′. The form of the second integral taken together withthe first is a cross-correlation. Symbolically, we could then write that

∞∫−∞

dt′ h(t′ + τ)

∞∫−∞

dt′′ Rn(t′ − t′′)h(t′′) = h(τ) (Rn(τ) ∗ h(τ)) , (12.36)

where

f(τ) g(τ) =

∞∫−∞

dt f(t + τ)g(t) (12.37)

and


f(τ) ∗ g(τ) =

∞∫−∞

dt f(τ − t)g(t). (12.38)

We further should recall the Fourier transform theorems

F [f(τ) g(τ)] = F ∗(ω)G(ω) (12.39)

and

F [f(τ) ∗ g(τ)] = F (ω)G(ω). (12.40)

We can therefore take the spectra to find

F [h(τ) (Rn(τ) ∗ h(τ))] = H∗(ω)H(ω)Sn(ω). (12.41)

We see that our spectral density of the current is

Sid(ω) = [n + Sn(ω)] |H(ω)|2 . (12.42)

To find an SNR, we’ll need to integrate our process over frequency. We already approximated this forthe shot noise term. Although it is useful to know the spectral density for such things as filtering, there is adirect way to find the power from the correlation function. In terms of the correlation function, the electricalpower Pe is given by

Pe =

∞∫−∞

dω

∞∫−∞

dτ ejωτRi(τ). (12.43)

By changing the order of integration and noting

∞∫−∞

dω ejωτ = δ(τ), (12.44)

we see that Pe is given by

Pe = Ri(0). (12.45)

With our general expression for Ri(τ), we see then that

Pe = n

∞∫−∞

h2(t) dt +

∞∫−∞

dt′ h(t′)

∞∫−∞

dt′′Rn(t′ − t′′)h(t′′). (12.46)

We still need a form for Rn(t′ − t′′).Our channel could add some Gaussian noise to our signal (i.e. an erbium-doped fiber amplifier, or EDFA)

but probably no signal-dependent noise, so our process probably appears as


n(t) = ns(t) + nn(t). (12.47)

In general, the autocorrelation can be written as

Rn(t1, t2) = E [n(t1)n(t2)] , (12.48)

which here would give

Rn(t1, t2) = Rns(t1, t2) + Rnn(t1, t2) + 2E[ns]E[nn]. (12.49)

If the nn is coming from additive Gaussian noise, then we can write that

Rnn(t′, t′′) = nn2δ(t′ − t′′). (12.50)

We also know (it will be demonstrated in the first section of Chapter 13) that, in order to receive a signal,we must sample it twice per period. Therefore, the signal must be bandlimited to a bandwidth Bs satisfying

Bs <1

2τd. (12.51)

But this is to say that the signal really doesn’t have enough bandwidth to completely decorrelate over adetector period. A usual form to try for a decorrelation is an exponential, so we’ll try

Rns(t′, t′′) = ns

2e−t′−t′′

τc , (12.52)

where ns is the average value of the signal count ns during τd. With the above considerations, we can write

Rn(t′, t′′) = En [n(t′)n(t′′)] = ns2e−

t′−t′′τc + 2ns nn + n2

nτdδ(t′ − t′′). (12.53)

With this we can write

Pe = n

∞∫−∞

h2(t′) dt′ + ns2

∞∫−∞

dt′ h(t′) dt′′′ e−t′−t′′

τc h(t′′) + ns nn

∞∫−∞

h(t′) dt′

2

+ nn2

∞∫−∞

h(t′) dt′

2

.(12.54)

A couple of reasonable approximations can give us a simple form for the power. First let’s say, as before,that h(t) is flat and of e/τd. Then

∞∫−∞

h2(t′) dt′ ≈ e2

τd. (12.55)

Further, let’s say that τd << τc, which shouldn’t be too good, as it would imply that we spent way too muchmoney on our detector, but at least it will lead to a simple answer. Then our expression becomes

Pe =ne2

τd+ ns

2e2 + ns nne2 + nn2e2. (12.56)


An SNR can be defined, as there is but one signal term, whereas the noise is a sum of the shot noise term,the signal spontaneous beat term, and the Gaussian (spontaneous) noise term. With this,

SNR =ns

2

nτd

+ ns nn + nn2 + 2kT

RLτde2

. (12.57)

To get a spectral density, more care needs to be taken with the signal-bearing term. It probably does nothurt too badly to again take the h(t) to be flat, but the exponential (or some other finite cutoff) term mustbe included in the evaluation. Otherwise, the signal will appear all at dc. To find the sns(ω) then requiresevaluation of

Sns(ω) = ns2

∞∫−∞

dτ ejωτe−|τ|τc . (12.58)

The integral can be rewritten in two pieces,

∞∫−∞

dτ ejωτe−|τ|τc =

∞∫0

dτ ejωτe−ττc +

∞∫0

dτ e−jωτe−ττc , (12.59)

and then evaluated:

∞∫−∞

dτ ejωτe−|τ|τc =

11τc

− jω+

11τc

+ jω. (12.60)

Simplifying, one finds that

Sns(ω) =ns

2τc

1 + ω2τ2c

. (12.61)

With this, we can now write that

Sn(ω) = (ns2 + ns nn)

(τc

1 + ω2τ2c

)+ nn

2τd. (12.62)

The whole Sid(ω), then, including thermal noise in the front end of the receiver circuit, is given by

Sid(ω) =e2

τd

([n + (ns

2 + ns nn)(

ns2τc

1 + ω2τ2c

)+ n2

]+

2kT

RLτd

)sinc(fτd). (12.63)

Two more embellishments could be used to make the spectral density a bit more realistic. For example, if wethink about AM radio, for example, the information is indeed bandlimited, but it is also bandlimited fromzero as well as from infinity. That is, at each instant a tone at a (positive and negative) frequency bandedaway from zero and below some ωmax is sent out. The tones vary randomly but center about some ω. Amodel for this process could be used instead of a purely exponential decay Rns(τ), one of the form

Rns(τ) = ns2e−

|τ|τc cos ωτ, (12.64)

which would lead to an Sns(ω) of


Figure 12.6: Sketch of a possible spectral density of the current generated by an analog optical signal incidenton a detector with preamplifier indicating the contributions of shot noise, additive Gaussian noise, 1/f noise,signal Gaussian beat noise, and the signal, where the frequency space transfer function is T (ω).

Sns(ω) = ns2τc

[1

1 + (ω − ω)2τc+

11 + (ω + ω)2τc

]. (12.65)

With all of this, the real situation might appear as depicted in Figure 12.6. The signal envelope in the figureis |T (ω)|2, where it has been anticipated that either the RC time constant of the receiver front end or adeliberately designed noise-blocking filter will be much more restrictive in frequency space than the |H(ω)|2.Therefore, the important quantity is the overall frequency transfer function. The signal has been shiftedaway from the 1/f noise peak at the origin. It is quite common in analog communications (as we will see inthe section on subcarrier modulation) to transmit the signal on a carrier which is low-frequency compared tothe receiver passband but still sufficiently high to shift the signal out from the 1/f peak. The shot, thermal,and Gaussian noises would be flat if not for the filtering effect. As the Gaussian would be flat, the beat noisewill just take the spectral shape of the filtered signal.

12.2 Analog Transmission Formats

As we might recall from the introduction to Part IV, we mentioned that an archetypical realization ofa conversation may appear as depicted in Figure IV.3. The information there is actually coded on bothamplitude and frequency of the signal. In the telephone circuit, the fact that the modulation wasn’t toostrong was used to allow the stream to ride on a dc carrier for circuit transmission. Were we to want to senda conversation after dc biasing over an RF link, we would need to take the signal and multiply it in a mixerby an RF carrier in order to be able to transmit it. We might end up after this process with a signal asdepicted in Figure 12.7. Such a signal is called amplitude modulated. As was discussed in association withFigure IV.7, such a signal can be envelope detected to retrieve the original, which may appear as depictedin Figure 12.8 after a dc block is applied. An important point to make here is that, had we not added a dc


Figure 12.7: Schematic depiction of an amplitude-modulated signal.

Figure 12.8: Envelope-detected version of s(t) of Figure 12.7.

component to the voice signal, we would have obtained something such as that depicted in Figure 12.9(a),and an envelope detection of that signal would have appeared as in 12.9(b). The input to the mixer musthave no zero crossings if it is to be reconstructed by simple envelope detection or, for that matter, by anysimple heterodyne system. If it has a zero crossing, the carrier changes sign, and detection then requiressomething which is phase-sensitive at the carrier frequency, a requirement which destroys completely thesimplicity of the simple lowpass envelope detection circuitry or frequency-locked heterodyne techniques.Similar comments apply to frequency modulation, where the information is impressed on the carrier as aninstantaneous change in frequency, expressible mathematically as

s(t) = a cos

ωCt +

t∫0

f(t′) dt′

. (12.66)

Here, the ωCt must be greater than thet∫0

f(t′) dt′ for all t lest information be lost. FM signals are almost

as easily generated as AM (although in practice it took thirty years after the demonstration of AM in 1906for the demonstration of FM to be carried out in 1936), as a dc signal information stream injected into avoltage-controlled oscillator (VCO) will give a signal such as (12.66), and then decoding can take place bysuch techniques as zero counting or with the use of a phase-locked loop. But now, we’ll give some discussionto spectra of FM and AM signals.

In an amplitude-modulated signal, the s(t) can be expressed in the form

s(t) = s0

[1 + µf(t)

]cos ωct, (12.67)


where µ is a modulation index (strictly less than unity), f(t) varies from −1 to +1, and the ωc is the carrierangular frequency. In an FM signal, the information is coded on the frequency such that the signal can beexpressed as

s(t) = s0 cos θ(t), (12.68)

where

dθ

dt= ωi = ωc + m(t) (12.69)

such that

s(t) = s0 cos

ωct + µ

t∫0

f(t′) dt′

, (12.70)

where µ is again a modulation index, and m(t) is defined as

m(t) = µf(t), (12.71)

where f(t) is now scaled such that −1 ≤ f(t) ≤ 1. Now might be a good time to discuss something aboutthe spectral characteristics of these waveforms.

For the moment, let’s assume that the function f(t) can be taken to be a simple sinusoid such that

f(t) = cos ωmt, (12.72)

where the ωm must be less than the ωc of the carrier. Finding the spectrum of the composite carrier plusmodulation for an AM signal can be done by writing

s(t) = s0(1 + µ cos ωmt) cos ωct= s0 cos ωct + µ

2 s0 cos(ωc − ωm)t + µ2 s0 cos(ωc + ωm)t, (12.73)

where the spectrum (or spectral density for that matter, as we shall soon see) may well appear as thatdepicted in Figure 12.9. If the m(t) actually contained information which had an average transmission rateof 2π/ωm, then the sidebands would no longer be spikes but would have envelopes, as is also depicted inFigure 12.9. To discuss a random data stream, however, requires that we use the spectral density, as thespectrum has little meaning when it is random. We’ll soon see that they appear qualitatively the same. Wecan show the existence of the information envelopes in much the same way as in the previous section, butfirst we must take a bit of care about the δ functions in the transform of s(t), s(ω), which is given by

s(ω) =s0

2[δ(ω − ωc) + µ

2 (δ(ω − ωc − ωm) + δ(ω − ωc + ωm))

+δ(ω + ωc) + µ2 (δ + ωc + ωm) + δ(ω + ωc − ωm)

]. (12.74)

As was mentioned in the beginning of the chapter, it is not a good idea to do infinite time averaging. It ismuch better to do a finite time average and then an ensemble average over the selected time frame. If thetime frame is finite, period T , then the S(ω) goes to ST (ω) and takes the form

ST (ω) =s0T

2[sinc(f − fc)T + µ

2 (sinc(f − fc − fm)T + sinc(f − fc + fm)T )


Figure 12.9: (a) A directly mixed version of the signal of Figure 12.8 and (b) its envelope-detected versionafter a dc block, which is not at all identical with the original.


+sinc(f + fc)T + µ2 (sinc(f + fc + fm)T + sinc(f + fc − fm)T )

]. (12.75)

If the sinc functions have minimal overlap, then one can express the spectral density Ss(ω) and

Ss(ω) =s20T

2

4

[sinc2(f − fc)T + µ2

4

(sinc2(f − fc − fm)T + sinc2(f − fc + fm)T

)+sinc2(f + fc)T + µ2

4

(sinc2(f + fc + fm)T + sinc2(f + fc − fm)T

)]. (12.76)

If we are to use a µf(t), where f(t) is a stationary random process normalized to a maximum of unity, thenwe can write

ST (ω) =s0T

2

[sinc(f − fcT +

µ

2

[fT (ω − ωc)

]+ sinc(f + fc)T +

µ

2fT (ω + ωc)

], (12.77)

where it has been assumed that the f(t) has a spectrum fT (ω) over the interval T which is significantlybroader than that of the sinc, such that the sinc can be taken to be a delta function in that term. Again,assuming that there is little overlap between the terms, we can find the spectral density Ss(ω) in the form

Ss(ω) =s20T

4

[sinc2(f − fc)T +

µ2

4E

[∣∣∣∣fTω − ωc

T

∣∣∣∣2]

+ sinc2(f + fc)T +µ2

4E

[∣∣∣fT (ω + ωc)∣∣∣2]]

. (12.78)

The expectation of∣∣∣fT (ω)

∣∣∣2 is just the finite Fourier transform of the correlation function Rf (τ) by theWiener-Khinchine theorem (assuming some time window exists in the time domain). But we saw in the lastsection that a reasonable apprimation to the autocorrelation function of a stochastic signal with correlationlength τc with frequency content centered aboutt a frequency fc was

Rf (τ) = e−|τ|τc cos ωmτ. (12.79)

With this, we see that the spectral density can be written as

Ss(ω) = s20T4

[sinc2(f − fc)τ + τc

Tµ2

4

(1

1+(ω−ωc−ωm)2τ2c

+ 11+(ω−ωc+ωm)2τ2

c

)+sinc2(f + fc)τ + τc

Tµ2

4

(1

1+(ω+ωc+ωm)2τ2c

+ 11+(ω+ωc−ωm)2τ2

c

)]. (12.80)

Again, this is essentially what is depicted in Figure 12.10.When the m(t) takes on the form in equation (12.72), then one can write the FM signal of equation (12.68)

in the form

s(t) = cos(ωct + µ

ωmsinωmt

)= cos ωct cos(β sin ωmt) − sin ωct sin(β sin ωmt), (12.81)

where β is quite evidently defined by the equation to be µ/ωm. A standard Bessel function identity allowsus to write

s(t) = [J0(β) + 2∑∞

n=1 J2n(β) cos(2nωmt)] cos ωct− [2∑∞

n=1 J2n−1(β) sin(2n − 1)ωmt] sin ωct, (12.82)

which may well have a spectrum as depicted in Figure 12.11. The spectrum when the sidebands carryinformation are superimposed as they were in Figure 12.9. Each sideband will carry an identical informationpacket. Note that the magnitude of the spectrum has been taken and that in fact the sidebands at multiples


Figure 12.10: Fourier transform of the expression of equation (12.80), which represents an amplitude-modulated waveform.

of ωm from the carrier each have a phase. It is seen that the modulated power in this FM signal has been morewidely spread throughout frequency space. In the limit where β << 1, one can note that Jn+1(β) << Jn(β),and the expression (12.82) will reduce to

s(t) ≈ J0(β) cos ωct + J1(β)[cos(ωc + ωm)t − cos(ωc − ωm)t

], (12.83)

where comparison with (12.71) indicates that the spectral content is the same but the phases differ. Indeed,narrowband FM is quite similar to AM. It is the spreading effect of the wideband FM that yields decodingadvantages which lead to, among other things, improved fidelity of decoded music over what the AM formatallows. It will be left to the problems to find a spectral density for this format when the modulation is µf(t)with f(t) information-bearing.

To most simply implement AM on an optical carrier, one could simply take the output of the telephonemouthpiece and use it as a modulating signal directly on the optical carrier with either a direct lasermodulation scheme (Figure 12.12) or an external modulation scheme. (More discussion will be given tomodulation schemes and modulators in section 12.5.) The external modulation scheme has certain advantagessuch as the fact that the laser center frequency changes with current level, and therefore external modulationcan be “cleaner” than direct modulation. Unfortunately, it is also much more expensive as well as muchmore lossy in terms of optical power, as the modulator must both be coupled into and out of.

There is a problem with the simplest implementation scheme which again goes back to 1/f noise. Theinformation bandwidth of the current spectral density generated in the receiver front end will be centeredon zero frequency with some bandwidth, as is sketched out in Figure 12.13. However, if the signal is “low-frequency” information such as a telephone message or such, the whole information packet may well sit rightwithin the 1/f noise peak which is inevitably there. In considering SNR in section 12.1, we ignored the 1/fnoise until the end of the section, where we assumed that the signal was centered some distance away fromzero in frequency space. Even if there is a dip in the signal at zero frequency (human hearing is from above20 Hz, as are musical instruments), it doesn’t help much, as 1/f noise may be significant to tens of kilohertzor even megahertz in some cases.

An obvious solution to the problem of AM transmission is to first put the information stream on an RFcarrier before using either of the modulation schemes of Figure 12.12. This is basically the simplest schemeof subcarrier modulation, that of single-channel subcarrier modulation. Heterodyne RF/microwave systemsdown-convert the signal plus carrier to an intermediate frequency (IF) in order to retrieve the information.If a subcarrier is chosen equal to an IF, one can use a standard radio receiver behind the photodetector. Inthis subcarrier case the signal spectrum looks like that depicted in Figure 12.13. If we assume that the f(t)is a zero-mean process, then the shot noise in the detector, averaging over an information period or so, will


Figure 12.11: A sketch of the magnitude of the spectrum of the expression for the FM signal of Figure 12.10.

Figure 12.12: Two different ways of impressing an information stream on an optical carrier: (a) direct lasercurrent modulation and (b) use of an external modulator.


Figure 12.13: Sketch of the received spectrum of a varying signal f(t) on an optical carrier.

be given essentially by the average value of the carrier current squared. This is to say that the signal sentto modulate the laser is given by

s(t) = s0 [1 + µf(t)] cos ωct (12.84)

such that the intensity I(t) launched from the laser is

I(t) = I [1 + µo [1 + µef(t)] cos ωct] , (12.85)

where µo is the optical modulation index and µe is the electrical modulation index. Assuming that the I(t)incident on the detector is uniform in x and y across the detector face, we find

Po(t) =∫∫

detector

dx dy I(x, y, t) dx dy = AdI(t), (12.86)

where Po(t) is the incident optical power and Ad is the detector area. The rate of counts in the detector isthen given by

n(t) =ηe

hωAdI(t), (12.87)

where η is the detector efficiency, and one often writes that the responsivity R is given by

R =ηe

hω. (12.88)

We can now note that

n(t) = RAdIo [1 + µo [1 + µef(t)] cos ωct] , (12.89)

where Io is the attenuated version of I that reaches the detector and where we have assumed that thechannel has produced no signal distortion. We need the first moment of n in order to find the shot noisecurrent. Certainly, the first-order moment En(n) is given by


Figure 12.14: Sketch of the received current spectral density of an optical carrier carrying an RF-modulatedAM signal.

En [n(t)] =η

hωAdIo = n. (12.90)

With a slight generalization of a result in the last section, we can write that the signal spectrum is

N(ω) =nTsµo

2[(sinc(f − fc)Ts + sinc(f + fc)Ts) + µe [F ((f − fc)Ts) + F ((f + fc)Ts)]] , (12.91)

where Ts is the sampling period and F (ω) is the transform of f(t) and where it has been assumed that thefc is a much higher frequency than the highest information frequency, allowing one to write that the lasttwo terms are just shifted versions of the information packet. With this, the spectral density can be writtenas

Sid(ω) =n2Tsµ

2o

4[sinc2(f − fc)Ts + sinc2(f + fs)Ts + µ2

e (Sf (ω − ωc) + Sf (ω + ωc))], (12.92)

where, as before, we have assumed that the sincs are narrower than the spectrum of f(t). The result issketched in Figure 12.14.

To define an SNR, we really need to think about the subsequent electrical processing that will be discussedin more detail in the next section. The idea will be to filter around the carrier at fc and then to eitherenvelope detect (to be discussed in detail in the next section) or heterodyne detect (to be discussed in detailin Chapter 14) to remove the RF carrier and be left with only one of the information packets. It followsfrom earlier considerations that we can write the unfiltered version of the squared currents, assuming onlyfiltering by an assumed square h(t), as

is2

= n20

µ2oµ

2e

2e2


i2sn =n0e

2

τd

i2n =2kT

τdRL, (12.93)

which gives that

SNR =µoµe

4 n0τd

1 + 4kTRL

1n0e2

, (12.94)

where dark current and possible beat noise have been ignored, and where is2

is the squared current of thesignal, i2sn is the shot noise squared current, and i2n is the variance of the Johnson (thermal) noise current.

We could try to carry out direct FM modulation of the laser diode by either of the techniques in Fig-ure 12.11, where the f(t) would now represent the subcarrier-modulated signal, and in fact both have beendone. We will leave the derivation of current spectral density and electrical signal-to-noise ratio to theproblems. It should be pointed out here that another way to generate FM is to use the laser chirp and putthe FM directly on the optical carrier without need for a subcarrier. This can also be carried out by havingan external modulator cause chirp. Such modulation schemes are called frequency shift keying (FSK), anddiscussion of such techniques is given in Chapter 14. Generally, though, this is only done in digital codingschemes, whether they be binary FSK (BFSK) or higher M -ary such as quadrature FSK (QFSK). However,to demodulate such an optical FM signal requires optical heterodyne detection.

Perhaps one more point that should be mentioned here, which will be returned to in more detail insection 12.5, is that the shot noise does not necessarily leave the RF carrier unaffected. In the aboveargument on SNR, it was tacitly assumed that the RF electronics would perfectly remove the carrier fromthe information. This may be acceptable in the case considered here, but we’ll see that, in high dynamicrange microwave systems, carrier degradation can be a problem.

12.3 An Analog Receiver

A general block diagram for an optical receiver appears in Figure 12.15. It should be noted that the samediagram could apply to either an analog or digital receiver, the difference between the two being in theprocessor block. A properly biased detector looks pretty much like a current source with some parallelcapacitance. This detector output current is then input to a preamplifier, which in general operates as atransimpedance amplifier—that is, it has a low-input impedance but an output impedance which can bematched to a line impedance, for example. Placement of such an amplifier between the detector and the restof the circuit thereby effectively transforms a current source to a voltage source. An equalizer is sometimesused for moderately high-speed operation where the RC time constant of the detector capacitance togetherwith the preamplifier input impedance can damp higher frequencies of the modulated signal. A passiveequalizer simply damps all frequencies to the lowest signal level, whereas an active filter can do more in theway of shaping without necessarily only causing any dispersive attenuation. The linear amplifier afterwardsis simply to boost the signal. As the signal out of the detector is usually weak (high-speed detectors generallyare limited in power handling capabilities even when adequate signal is available), the thermal circuit noisecan be large enough to corrupt the signal. The linear amplifier will amplify this noise along with thesignal but boosts the signal to a sufficient level that sources of Johnson noise other than the resistance inthe transimpedance amplifier are negligible relative to the signal level. For this reason, the equalizer issometimes omitted when the power limitation is severe. Generally, the linear amplifier will add some noiseto the signal in that it may have a noise figure greater than 0 dB. (1–3 dB are typical.) The filter after thelinear amplifier is an important component of an analog receiver. In reality, this filter is actually built in tothe amplifier response that can actually be an active filter response. In frequency modulation/demodulation,often a phase-locked loop is employed and the filter blocks actually operate as a single block.


Figure 12.15: A block diagram for an archetypical analog receiver.

12.3.1 The Detector Transimpedance Amplifier Combination

As was discussed back in Chapter 4, a possible detector transimpedance amplifier realization can be thesimple one appearing in Figure 12.16. Figure 12.16(b) is a circuit equivalent for the simple PIN/preamplifiercircuit of Figure 12.16(a). Making the usual assumption that the operational amplifier has infinite inputimpedance and a gain A, one can write that

vti = −Avd (12.95)

and

id =[1 + A

RF+ jωeD

]vti. (12.96)

We then can write that the transimpedance zti in the form

zti =vti

id=

ARf

1 + A + jωRfCD. (12.97)

Taking A to be much greater than one but not limiting ω, one finds that

zti∼= Rf

1 + jωτti, (12.98)

where

τti =RfCD

A. (12.99)

We note that


Figure 12.16: (a) Actual circuit diagram for a detector and transimpedance amplifier and (b) the same as(a) but where the diode has been replaced by its equivalent circuit.

zti

Rf=

11 + jωτti

(12.100)

has the form of a one-pole lowpass filter transfer function. However, the operational amplifier has greatlyimproved the behavior that would have been realized by the circuit of Figure 12.17. One can see immediatelyhere that the transimpedance is given by

zti =R

1 + jωRCD(12.101)

and that the time constant will be A times larger than the time τti. One could argue that one could simplylower R to Rf/A, but there are two problems with this. One is that this would lower the output voltage byA and increase the Johnson noise, given by

i2n =2kT

τdRL, (12.102)

by A. Commercially available operational amplifiers are numerous and inexpensive for up to 1-MHz op-eration. More custom solutions can be found for frequencies up to 1 GHz or so. Some solutions can befound above this, but demonstration of operational amplifiers above roughly 10 GHz is still an area of activeresearch. (See, for example, Sano et al 1996.) The continuing development of heterojunction bipolar tran-sistors seems to be providing a possible route to high-speed transimpedance amplification. Such devices canhave fT’s of up to 200 GHz and very high voltage gain in their operating ranges.


Figure 12.17: A bad idea for a receiver front end.

12.3.2 Linear Amplifier

The main purpose of the linear amplifier is simply to boost the signal. Oftentimes this amplifier will useautomatic gain control (AGC) when fading may be a problem, although AGC can wreak havoc on dynamicrange. The AGC is more important in digital systems, where one knows apriori what level one wants goinginto the processor.

The excess noise of an amplifier is always an issue. It is quite generally characterized by a noise figureF , which is given by the ratio

F =SNR|out

SNR|in , (12.103)

which can also be written as

F =is

2(out)i2n(in)

is2(in)i2n(out)

. (12.104)

Of course, the way to take this into account in calculations is to simply transform our SNRin to SNRout.As far as statistics go, the amplifier noise can probably be considered as additive Gaussian noise. Whatis generally done to take the noise figure into account in SNR calculations is to use the concept of noisetemperature. The idea is that, if the noise at the amplifier were purely additive Gaussian circuit noise, thenoise energy at the input of the amplifier would be

Ni = kTrt, (12.105)

but that at the output would exceed the gain G times the noise energy at the input, which is

No = k(Trt + TA), (12.106)

where TA is defined as the amplifier noise temperature. Using the relation

P0 = GPi, (12.107)

then the noise figure is

F =Po

No

Ni

Pi= 1 +

TA

T(12.108)

or, written another way,


TA = (F − 1)Trt. (12.109)

Of course, if there are other sources of noise in the system, we could lump them into the input noisetemperature and call it Ti. The output noise temperature To would then be given by

To = Ti + TA, (12.110)

where

TA = (F − 1)Ti. (12.111)

That is, the noise temperatures for the various steps are additive. A problem with this in our case is that, ifthe input current were shot noise-limited, there is the extra 3-dB penalty that was discussed in Chapter 10 inregard to optical amplification. As with the optical amplifier, the noise figure and temperature are functionsof the statistics of the input distribution. All electrical amplifier noise characterization techniques are basedon having exceptionally thermally well-calibrated sources. Optical amplifiers are generally evaluated withquite Poisson-distributed laser sources.

12.3.3 Filter

As will be discussed in more detail in the next chapter, digital systems generally employ an all-pole filter(Butterworth, Bessel, Chebyshev, etc.) at the output of the linear amplifier before splitting the signal intotwo pieces, one for clock recovery and the other to be input into the decision circuitry. In an analog circuit,such filters are also a possibility, but here we will concentrate attention on optimal detection schemes forAM and FM decoding.

In general, a subcarrier-modulated signal will have a spectrum which appears as in Figure 12.18. Thepoint is that the IF keeps the signal away from the 1/f noise peak, but the flat spectrum of the shot noiseand any additive Gaussian noise permeates the spectrum. However, if a filter that were “matched” to thesignal spectrum could be found, one could obtain a spectrum such as that depicted in Figure 12.19. Thatis, if we had apriori knowledge of the spectrum of the signal, we could use that knowledge to construct theideal filter.

A candidate filter to use for optimal filtering when the spectrum of the signal to be received is knownapriori is any of the class of Bayes estimators. There is a general theory associated with such constructionwhich is called Bayesian estimation. To find the form of the filter, we need to have a criterion for whatit is we want to optimize. The Wiener filter requires us to minimize the least squares residual error. Thequantity to minimize then is

r(t) = E (i(t) − io(t))2, (12.112)

where the i(t) is the current we should have (one that corresponds to the spectral density Si(ω) that weknow apriori) that is expressible in terms of m(t) as

i(t) = m(t)i0, (12.113)

and io(t) is the current in the receiver after the filter—that is,

io(t) = hf(t) ∗ ii(t), (12.114)

where hf(t) is the filter response, given by the inverse Fourier transform of the filter frequency responsefunction Hf(ω), or


Figure 12.18: The spectrum of an AM signal including shot, Gaussian, and 1/f noise.

Figure 12.19: An ideally filtered AM signal including all the noises passed by the ideal filter.


hf(t) =∫

dω e−iωtHf(ω). (12.115)

The io(t) is therefore also expressible in the form

io(t) =k(t−τd,t)∑

i=1

hf(t) ∗ h(t), (12.116)

which we could rewrite as

io(t) =k(t−τd,t)∑

i=1

ht(t). (12.117)

With all of this, we can write that

r(t) = E

i2(t) − 2i(t)

k∑i=1

ht(t) +

(k∑

i=1

ht(t)

)2 , (12.118)

which can be rewritten as

r(t) = i2oE(m2) − 2ioEmEkET

[m(t)

k∑i=1

ht(t)

]+ EmEkET

[k∑

i=1

ht(t)

]2. (12.119)

Using the definition of ET, we then can show

r(t) = i2oEm(m2) − 2ioEmEk

[k∫

dt′ ht(t − t′)n(t′)]

+EmEk

[km

∫dt′h2

t (t − t′)n(t′) + k2−km2

∫dt′∫

dt′′ ht(t − t′)ht(t − t′′)n(t′)n(t′′)]. (12.120)

The definition of the Ek can then be used to yield

r(t) = i2o Em(m2) − 2ioτdEn

[n(t)∫

dt′ ht(t − t′)n(t′)]+ En

∫dt′ h2

t (t − t′)n(t′)+En

[∫dt′ ht(t − t′)n(t′)

∫dt′′ ht(t − t′′)n(t′′)

], (12.121)

where the relation m(t) = n(t)τd has been assumed. Using the definition of the autocorrelation function, wecan write

r(t) = i20 Em(m) − 2i0τd

∫dt′ ht(t − t′)Rn(t − t′) + n

∫h2

t (t − t′) dt′

+∫

dt′ ht(t − t′)∫

dt′′ ht(t − t′′)Rn(t′ − t′′). (12.122)

To optimize (12.122), we can use the calculus of variations. This essentially means that we take a functionalderivative of r(t) with respect to the transfer function ht(t) and set the result to zero, then solve for optimalfilter function h0(t). Symbolically, we can write

δr(t)δh(t)

∣∣∣∣h0(t)

= 0. (12.123)

In our case, however, we essentially just take a derivative of the expression with respect to h(t) to yield

n2ioτdRn(τ) =∫

dt′ h0(t − t′) (En [n(t′)] + Rn(τ)) . (12.124)


Figure 12.20: A simple circuit which could carry out envelope detection.

Taking a Fourier transform, we obtain

H0(ω) =Sn(ω)n2e2

ne2

τd+ Sn(ω)n2e2

. (12.125)

We see that the prescription that, in frequency regimes where the noise is small compared to the signal,the filter function is equal to unity, meaning it is fully transparent. As the signal weakens relative to thenoise, the filter needs to attenuate—progressively more with decreasing SNR(ω). The spectrum obtainedafter Wiener filtering is sketched in Figures 12.18 and 12.19. We note here that we really should include thethermal noise to obtain

H0(ω) =Sn(ω)n2e2

Sn(ω)n2e2 + ne2

τd+ 2kT

RLτd

, (12.126)

where the T can be an effective temperature that takes into account the excess noise of the linear amplifier.

12.3.4 The Processor

Amplitude-modulated (AM) signals are generally separated from their carriers by envelope detection. Suchdetection can be carried out by a circuit as simple as that depicted in Figure 12.20. The diode serves as ahalf-wave rectifier. With reference to Figure 12.8, this would have the effect of removing all of the signalbeam to the + axis. The combination of R and C1 then acts as a lowpass filter to remove the carrier. Thiswould correspond now to the fast oscillations of Figure 12.7 disappearing, leaving only a positive signal(envelope) above the axis. The C2 acts as a blocking capacitor to remove the dc component of the signal,causing the combination of C1, C2, and R to act as a bandpass filter. The result is a signal such as that inFigure 12.8. The output voltage vo(t) can then go to a decoder to be transformed to information. In thecase of an AM radio receiver, the decoder could be electroacoustic transducer such as a loudspeaker.

Detection of an FM-modulated signal requires removing the time-dependent phase from the carrier wave.The situation is depicted in Figures 12.21 and 12.22. The wave is first passed through a differentiator.A differentiator can be simply a bandpass filter operated in a linear regime on the leading edge of itscharacteristic. This is because the operation of differentiation corresponds to the multiplication by jω in thefrequency domain, and a bandpass filter with peak frequency at 2ωc should be quite linear in ω around ωc.After the differentiator, the signal is given by

∂

∂tcos

ωct +

t∫0

f(t′) dt′

= − (ωc + f(t)) sin

ωct +

t∫0

f(t′) dt′

. (12.127)

The signal is then lowpass filtered to remove the carrier and just give ωc+f(t). A dc block (series capacitance)can then be used to retrieve the desired signal f(t).


Figure 12.21: Block diagram of one form of FM decoder.

An alternate demodulation system uses a phase-locked loop to demodulate the f(t) from the carrier. Thephase-locked loop will be discussed in more detail in the next section. The phase lock was originally describedby de Ballescize in 1932 (de Bellescize 1932, Gardner 1979) for use in homodyne receivers. Homodynereceivers, even in the RF case, no less the optical case, have never found too much application for reasonsthat should become clear in Chapter 14 on coherent detection. However, the phase-locking techniques havebecome ubiquitous in electronics since their original push application of line synchronization in televisionreceivers (Wendt and Fredendall 1943 or Gardner 1979). Although the phase-locked loop is designed togenerate a signal locked to the phase of another, in the process of generating the locking signal, the loopmust generate an error voltage proportional to the derivative of the phase which is indeed the instantaneousfrequency of the input signal (Van Trees 1971 or Gardner 1979). As the phase-locked loop is a feedback controlsystem, the complexity is higher than for a passive envelope detection circuit, but phase lock techniques atpresent are quite common and advanced.

12.3.5 Phase-Locked Loops

A phase-locked loop can be depicted as in Figure 12.23. The basic idea is that a signal of the form vi cos θi(t)is input to the loop. This signal is beaten with a signal v0 sin θo(t) generated by the loop. The output ofthis phase comparison operation is a voltage vd(t) given by

vd(t) = Vd sin(θi + θo) + Vd sin(θo − θi). (12.128)

The signal is then lowpass filtered. Assuming that θi(t) is something that is ever increasing with t, thelowpass filter will pass only the lower sideband, and the control voltage vc(t) will be given by

vc(t) = Vc sin(θo − θi). (12.129)

If the loop is tracking, then θo − θi should be small and

vc(t) ≈ Vc(θo − θi). (12.130)

The voltage-controlled oscillator VCO then takes the input Vc(t) and converts it to a frequency. If θo = θi,the frequency doesn’t change. If θo is different from θi, then Vc(t) is nonzero and the VCO is tuned to a newfrequency until it reduces the beat note to zero.

Some further insight into the phase-locked loop can be gained by doing a Laplace transform analysis(Gardner 1979). First, let’s consider the VCO. The frequency of the VCO is controlled by vc(t) or, in theLaplace domain, Vc(s). As frequency is the derivative of phase, one can write that

dθ0

dt= κ0vc(t) (12.131)


Figure 12.22: The demodulation of an FM signal with (a) the initial signal, (b) the differentiated signal,(c) the differentiated signal after lowpass filtering, and (d) the f(t)—that is, the initial signal used asmodulation.


Figure 12.23: A block diagram of the phase-locked loop, where the demodulated output is included toemphasize the FM demodulation application of the last subsection.

or, in the Laplace domain,

θ0(s) =κ0Vc(s)

s, (12.132)

which tells us that the VCO acts like an ideal integrator. Let’s say also that the phase detector operatesas an ideal phase detector—that is, it lowpass filters as well as the loop is tracking, well enough thatsin ∆θ ≈ ∆θ—so that we can write that

Vd(s) = κd [θi(s) − θo(s)] . (12.133)

The last equation necessary is then

Vc(s) = F (s)Vd(s). (12.134)

Combining the equations, we find that the closed-loop transfer function H(s) takes the form

H(s) =θo(s)θi(s)

=κ0κdF (s)

s + κ0κdF (s). (12.135)

The loop filter serves the purpose of limiting the loop noise. Anytime there is a closed loop, questions ofstability and response time arise. Simplicity is generally the best thing for stability. Ideally, the simplest,most stable yet still responsive filter would be an ideal active integrator, which is not quite realizable inpractice. Were such possible, the loop response would be

H(s) =κ

s2 + κ. (12.136)

As to the application of the phase-locked loop to FM detection, we note that Vc(t) is just the derivativeof θo(t), which, when tracking, is just equal to θi(t). If the input signal were then given by

viN (t) = ViN cos

ωct +

t∫0

f(t′) dt′

, (12.137)


the loop output (vc(t)) would just be

vc(t) = cf(t). (12.138)

The phase-locked loop is therefore, among other things, an FM demodulator.

12.4 Subcarrier and Wavelength-Division Multiplexing

Subcarrier multiplexing, often called frequency-division multiplexing (FDM), and wavelength-division mul-tiplexing (WDM) are two ways of achieving another layer of multiplexing in our hierarchy of multiplexingschemes. The first level of multiplexing generally carried out is that of space-division multiplexing—that is,of simply using more channels (for example, more fibers). A technique we have previously mentioned andwill discuss again in the next chapter on digital communications is time-division multiplexing (TDM). Thismultiplexing technique is not applicable to analog communications, as it requires synchronized interleavingof signals. Bits can be interleaved synchronously, but in general analog signals can’t, as they don’t have acontinuum of frequencies in their spectra. It might be possible to “compress” an analog waveform by, forexample, doubling all the frequencies contained in the signal, but reconstruction could be difficult if relativephases are not perfectly preserved in the doubling and halving operations. Halving the temporal length ofa bit as well as restretching it is just a matter of timing. For this reason, the two most common techniquesfor multiplexing analog signals without increasing the number of channels are FDM and WDM.

The cable television industry uses subcarrier multiplexing. When the FCC cut up the frequency spectrumto allocate television, FM, and other radio bands in 1953 (AM having been allocated in 1934), the frequencyspace for television was divided into 6-MHz-wide strips extending from 40 or 50 MHz up to hundreds of MHz.A television signal, complete with AM picture and FM sound, takes up about 4 MHz of bandwidth. The6-MHz strips then allowed each station to have a 1-MHz “guard rail” on each side around its transmission.Televisions, of which there are plenty, are designed to be fed a signal from an antenna which contains thiswhole spectrum, and from it the receiver selects the channel it wants. The cable companies wanted to havea product to replace the antenna, so they needed to send as many channels as they could but still followthe FCC standard such that their product could essentially just be plugged into the antenna input of thetelevision. (The decoder box exists simply to unscramble a single channel that the superheterodyne in thereceiver has already picked out.) The subcarrier-multiplexed or FDM’ed signal would then take the form (ifall channels were AM for definiteness, which isn’t the case, as the audio is carried on a low-frequency FMchannel)

s(t) =N∑

i=1

si (1 + µifi(t)) cos ωcit, (12.139)

where the spectrum of s(t) might appear as in Figure 12.24. The idea, then, behind any subcarrier multi-plexing scheme is to electrically generate a signal such as that in Figure 12.24 and somehow transmit it to theuser. As the cable TV signal may take up hundreds of megahertz, one runs into a situation similar to thatwhich the telephone companies ran into in the trunk lines. Many repeaters become necessary to transmit ahigh-bandwidth signal over coaxial cable. Fibers will transmit such signals, however. The idea was to use amodulation scheme such as one of those illustrated in Figure 12.25. Direct modulation of the laser requiresa special equalization board to avoid any harmonics of the RF carriers showing up, as any harmonics orbeating between the various carriers causes serious channel crosstalk. Many cable systems have opted to useexternal modulators, but there are linearity and chirp problems there as well, as will be discussed in muchmore detail in the next section.

Wavelength-division multiplexing, or WDM, is analogous to FDM except that all operations are carriedout at the optical carrier frequencies or wavelengths. To present, WDM has primarily been applied to theproblem of digital transmission, although it could be applied to analog transmission as well. A reason forthis is that, as we will see in section 13.3, digital codes are already kicked up to some frequency away fromthe 1/f noise peak simply by the existence of a clock. Analog signals in general are not unless accompaniedby an RF carrier. The problem really comes down to power. In order to achieve a given SNR, one needs


Figure 12.24: A spectrum of a subcarrier-modulated signal.

a given power level. Power levels, as will be seen in section 13.2, can be quite low in digital systems inorder to achieve low bit error rates (BERs). One needs higher SNRs, and hence power levels, in analogsystems in order to achieve a comparable fidelity level. Although the smoothness of the passband of passivecircuit filters has been discussed, passband is relative. The center frequency of an optical wave is hundreds ofterahertz—not MHz or GHz. Even though, as we will soon see, the passband of a grating filter can be sharp,it is only an order or two of magnitude sharper than a passive electrical filter, which does not make up for thecenter frequency being six to nine orders of magnitude higher than in the electrical case. To illustrate, let’sreconsider Figure 12.23. In the case of cable television, spacings between the peaks correspond to 6 MHz.WDM spacings are on the order of 100 GHz. Presently, so-called dense wavelength-division multiplexing(DWDM) systems are pushing this limit down to roughly 50 GHz, which is 0.4 nm at a 1550-nm centerwavelength, in order to put more channels onto a single link. Laboratory tests of 25, 12.5 and 3v3n 6.25GHz spacings have either been performed or are underway. A problem being faced here is that power leadsto nonlinearity at some level. If each channel needs so much power for a given SNR, more channels lead toever more power in the guide. In DWDM systems, as will be discussed in more detail in the next chapter(Chapter 13), four-wave mixing in the channel has the same effect as modulator nonlinearity has a subcarriermultiplexing signal. A partial solution to this dilemma is to not use dispersion-shifted fiber but instead to letthe channels disperse from one another, destroying any phase matching, and then to dispersion-compensateperiodically. That is to say that DWDM systems at present operate right at the verge of disastrous nonlineareffects. In an analog system where one needs to have a subcarrier anyway, why not multiplex and demultiplexchannels in the RF domain where standardized solutions are available and, due to standardization, cheap?One could use FDM on each WDM channel if one wanted to transmit many channels, but the nonlinearityproblem would become still more severe.

To illustrate why an RF multiplexed/demultiplexed system may be a better solution for analog signals,let’s consider the pieces of a WDM system as illustrated in Figure 12.26. At the transmitter end, oneneeds to have a wavelength-stabilized laser per channel along with all the necessary drive circuitry. Externalmodulators can also be used, but this does not preclude the need for stabilization, which as we will discussin Chapter 14 is not generally inexpensive. One also needs precision fiber components such as combiners,which have a minimum of 3 dB penalty each due to the brightness theorem, as was discussed in section 6.1of Chapter 6. The receiver end of the link will also require a separate detector and receiver per channel. Buteven before this, the filtering must be done in the optical domain, meaning that each optical channel mustbe generated by a high-quality grating filter. Perhaps the time is ripe to consider some of the characteristicsof a grating filter, as is done in various references (i.e. Mickelson 1992). We will first consider the simple andunrealistic case of an amplitude transmission grating to illustrate the principle. In the next chapter we’lldiscuss realistic solutions which have become quite sophisticated.

Let’s consider a plane wave incident on a transparency (see Figure 12.26) with amplitude transmittancegiven by


Figure 12.25: Schematic depictions of (a) a WDM transmitter and (b) a WDM receiver.


t(x, y) = (1 + µ cos(2π)fxx) rect(

x

Lx

)rect(

y

Ly

), (12.140)

where fx is given by

fx =1d. (12.141)

When the lens is infinite in extent (which is not always a tenable approximation, as was discussed in chapter 5,but which is reasonable here), we can write that the scalar field on the screen is given by

ψ(xf , yf , 2f) = −i e2ikf

4λf

∞∫−∞

dx0

∞∫−∞

dy0t(x0, y0)e−ikf (x0xf +y0yf )

= −i e2ikf

4λf

Lx/2∫−Lx/2

dx0

Ly/2∫−Ly/2

dy0(1 + µ2 ei2πfxs0 + µ

2 e−i2πfxx0)(ei2πxfλf x0ei

2πyfλf y0), (12.142)

which can be integrated to yield

ψ(xf , yf , 2f) = −i 32ikf

4λf LxLy sinc(

yf Ly

λf

)·[sinc xf Lx

λf + µ2 sinc

[Lx

(xf

λf + 1d

)]+ µ

2 sinc[Lx

(xf

λf − 1d

)]], (12.143)

which is a pattern which, along the x-axis, appears much as that sketched on the screen in Figure 12.26.The idea is that two diffracted orders have appeared. Each radiates out at an angle to the normal to thegrating θp, given by

sin θp =xf max

f=

λ

d, (12.144)

and has a first zero at an angle θz with respect to the propagation direction θp given by

sin θz =xf zero

f=

λ

Lx. (12.145)

The width of the peak compared to its angle of propagation is often called the finesse F of the grating andis given by

F =Lx

d, (12.146)

which is the number of lines in the grating as well as the number of resolvable points in the output whenthe grating is used to separate different frequency components. That is, for a shift in wavelength of δλ suchthat sin θp increases by δλ/d, this shift will only be resolvable if

δλ

λ≥ d

Lx. (12.147)

If one could get all of the power into a single diffracted order, then one could separate some number ofdifferent wavelength carriers if they were all separated from each other by at least δλ.

A grating such as the one we just analyzed doesn’t seem to be very useful, as there are two equal-amplitude diffracted orders, and the lion’s share of the power will go into the zeroth order. One would liketo have all the power from all of the wavelengths go into a single diffracted order. The above analysis is very


Figure 12.26: A plane wave incident on a grating and focused onto a screen which contains a first-orderdetector.

low-order, however, in the sense that it was assumed that the grating was infinitesimally thin. The anglesand peak widths out of such an analysis may be reasonable, but an infinitesimally thin grating could notdiffract at all, as the wave could not interact with the material. The theory of thick gratings is complexand treated elsewhere (Mickelson 1992, Hutley 1982), but the results can be stated here. It is possible todesign a thick grating which will diffract close to 100% into a given diffracted order. Although it wasn’tdemonstrated in the above, the finesse increases linearly with the number of the order. If one can use thefifth order, for example, the δλ spacing can be reduced by a factor of five. In transmission gratings, 100%diffraction efficiency generally requires careful control of thickness and depth of modulation. In a reflectiongrating, high diffraction efficiency generally requires a surface relief pattern known as a blazing. In eithertype of grating, efficiency generally requires Bragg angle incidence—that is, one needs to illuminate thegrating at the negative of the λ/d angle that is to come off the grating. This Bragg incidence allows formaximum reflection off the grates in the proper direction and then leaves the correct phasing up to thegrating designer. That Bragg angle incidence is required is problematic, as one can only pick the Braggangle for a single wavelength, yet one wants to use the grating to separate wavelengths. The wavelengthsin a WDM system, then, must not be too far apart from each other, and the grating will require a largenumber of periods or grates. However, enough people are involved in the technology that progress is beingmade. Most of the present-day work in WDM and DWDM is aimed at increasing the throughput of highbit-rate digital transmission systems. For analog transmission systems, the subject of this chapter, it isgenerally cheaper and easier to use the subcarrier techniques previously described in this section, techniquesnot requiring multiple sources or any exotic optics, which can be expensive and tricky to set up. There isno reason for the status quo to last forever, however. It is always risky to predict the future. The majorproblem with subcarrier multiplexing is the linearity of the modulation. In the next section we will discussmodulators in some detail.

12.5 Modulators and Modulation

The challenge in any form of communication is to get a high-quality replica of a message to the outputof a receiver decoder. In optical communication, the first obstacle to be faced in achieving this goal is toplace an encoded message on an optical carrier. There are basically two problems with carrying out thismodulation. One is the intrinsic nonlinearity of modulators, and the second is the intrinsic chirp of themodulation process which causes the emitted modulated signal to have a linewidth greater than the Fouriertransform limit imposed by the modulation. This chirp effect leads to increased dispersion in the channelwhen the source is a narrow-line one such as a single-mode laser.

As was touched upon in the last section, nonlinearity in the modulation can really destroy, for example, asubcarrier-modulated signal, as nonlinearity will cause different modulation frequencies to mix and generate


all sorts of beat notes which, if there are many channels, may well end up interfering with other channels.For example, let’s take a five-frequency signal

ψ(t) =5∑

i=1

Ai(t) cos(ωi + tφi) (12.148)

and take

ωn = nω0. (12.149)

Let’s say that channels 2 and 3 beat due to a quadratic nonlinearity in the modulator such that

ψnL(t) = [A2 cos(ω2t + φ2) + A3 cos(ω3t + φ3)]2. (12.150)

The crossterm from the beating will be

2A2A3 cos(ω2t + φ2) cos(ω3t + φ3) = A2A3 cos [(ω3 − ω2)t + (φ3 − φ2)]+A2A3 cos [(ω3 + ω2)t + (φ3 + φ2)] . (12.151)

However,

ω3 − ω2 = ω1; ω3 + ω2 = ω5, (12.152)

and the interference will land right on top of two of the other channels. There are smarter ways of choosingthe frequencies, but they will use the available bandwidth in a less efficient way than the simple assignment of(12.149). (We saw in the last chapter that fiber χ3 nonlinearity can have the same effect on DWDM signals.)A good way to see the effects of nonlinearity as well as chirp in modulation is to use the semiclassicalequations of the interaction of radiation and matter that were introduced in Chapters 7, 8, and 9.

Recall that the equations describing the interaction of a field described by modes of time-varying ampli-tudes bλ and a set of atoms defined by number µ described by polarization αµ and each in an inversion statedµ can be expressed in the form

dbλ

dt+ iωλbλ + κλbλ = −i

∑µ

g∗µλαµ + fλ

dαµ

dt+ iωααµ + κµαµ = i

∑λ

gµλbλdµ + Γµ−

ddµ

dt+

dµ − d0

τd= 2i∑

λ

g∗µλb∗λαµ − gµλbλα∗µ + Γd, (12.153)

where ωλ(ωd) is the angular frequency of the mode λ (atom µ); κλ, κd, and 1/τd are the damping constantsof the mode λ, polarization of the µth atom, and inversion of the µth atom; d0 is the equilibrium inversion;gµλ is the overlap of the mode λ with the µth atom; and fλ(t), Γµ−(t), and Γd(t) are the Langevin sourceterms necessary to describe the interaction of the field and atoms with their external environment. For thepresent discussion, we will assume that spontaneous emission is small, the field is in a single mode (that is,either high above threshold or well below threshold), and that the medium is homogeneous. With this, wecan drop fλ, Γµ−, Γd, and the µ and λ subscripts to obtain


db

dt+ iωb + κb = −ig∗α

dα

dt+ iωαα + καα = igbd

dd

dt+

d − d0

τd= 2i(g∗b∗α − gbα∗). (12.154)

Let’s first consider the case of a light-emitting diode. In a light-emitting diode, the only pump is currentinjection. Thermal excitation is only incidental and, in fact, for lifetime purposes one would like to keep itminimal. The light that is spontaneously emitted is allowed to leave the cavity without reflection. This hasseveral consequences. One of these is that the term on the right-hand side of (12.154c) will be of the form

g∗b∗α − gbα∗ = g sin(ωb − ωα)t, (12.155)

which will for practically all ωα and ωb be rapidly varying. This means that these terms will not phasematch to the slowly varying terms on the other side of (12.154c). Similar arguments can be made for termson the right-hand sides of (12.154a) and (12.154b). In (a), the left-hand side will resonate about a line centerωb, yet the α drives the system at an ωα which is quite generally far from the resonance. In (12.154b), theconverse argument applies to the ωα on the left-hand side and ωb on the right. As we did earlier, we canformally solve (12.154b) and plug it into (12.154a) to obtain a delta-correlated fluctuating force F (t). Also,far enough below threshold, each line of the b becomes so broad in frequency space that there cease to beindividual lines per se. With these simplifications, the system of (12.154) can be rewritten in the form

b + iωλb + κλb = f(t)

d +d

τd=

d0

τd⟨f(t)f(t′)

⟩= 2nκδ(t − t′). (12.156)

At this point, it appears that the system is completely decoupled unless one recalls that the n in (12.156c)is determined from the steady-state values in the d equation. Clearly, (12.156a) can be integrated to yield

b(t) = e−iωλte−κλt

t∫f(t′)e+iωλt′eκλt′ dt′. (12.157)

Noting that the time-varying photon number n(t) is given by b∗(t)b(t), we can write

n(t) = b∗(t)b(t) = n(0)e−2κt + e−2κλt

∫f∗(t′)f(t′′)eiωλ(t′−t′′)eκλ(t′+t′′) dt′ dt′′. (12.158)

Using (12.154) in (12.158) and taking an average 〈〉 yields

⟨n(t)⟩

= n(0)e−2κλt + e−2κλt2κn

t∫e2κλt′ dt′, (12.159)

which can be integrated to yield

⟨n(t)⟩

= n, (12.160)

as indeed it should. It remains to find n.Recall that the dk is defined as


dk =N2 − N1

N2 + N1, (12.161)

where N2 is the upper-state population and N1 is the lower-state. Boltzman would tell us that, in thermalequilibrium,

N2

N1= e−hν/kT . (12.162)

Clearly, at room temperature, hν >> kT and the factor approaches zero. However, the current that is beinginjected into the junction will linearly increase N2 and linearly decrease N1 such that

N2 = CI

N1 = (N1 + N2) − CI. (12.163)

For a small enough current (lack of state depletion) and low enough temperature, therefore,

N2

N1= κI = e−hν/kTeff . (12.164)

The average number of thermal photons in the cavity, however, must be given by the Planck factor

n =1

ehν/kTeff − 1. (12.165)

Assuming hν > kTeff then, one can write

n = e−hν/kTeff = κI, (12.166)

which says that optical intensity radiated from the cavity will go linearly in the injection current. This limitwill break down for sufficiently high injection. In practice, though, the level of injection would be hard toachieve as thermal runaway can well precede it. Therefore we can conclude that, under reasonable limits,the LED is another linear circuit element. Compare also the result of (12.166) with our earlier discussionsof coherence. Typically, LEDs have temporal coherence times of tens of femtoseconds, whereas a laser couldhave a temporal coherence time of tens of nanoseconds. Tens of femtoseconds is not long enough to get morethan one photon in that temporal spatial mode. Many temporal and spatial modes with less than a photonoccupancy each is just the limit in which the negative binomial random variate statistics approach Poissonstatistics.

What does the LED look like as a circuit element? The above arguments could be extended to solvefor the transient behavior of the dk. Clearly, d0 in the above-described limit will be linear in the injectioncurrent such that equation (12.166b) can be written as

d +d

τd= κ′I(t). (12.167)

We immediately note that this equation has the form

C∂V

∂t+

V

R= i(t), (12.168)

which is the node equation for the circuit of Figure 12.27. Indeed, the LED is not only linear in modulationfrequency, but it also has a circuit model such that a complete circuit model of an LED-driven opticalcommunication system could appear as in Figure 12.28.


Figure 12.27: Circuit model for a light-emitting diode.

Figure 12.28: Circuit diagram analogous to that of Figure 12.26 but for an LED-driven optical communicationsystem.


Figure 12.29: An illustration of the potential surface for laser action which shows, as previously discussed(Chapters 8 and 10), that at some value of the inversion parameter d0 (or equivalently at a threshold valueof the injection current), the field begins to have a mean value that exceeds its standard deviation.

Attention now naturally turns to what happens in a directly modulated laser-driven system. A laser diodeis also governed by the equations of (12.154). However, laser operation has little or nothing in common withLED operation. The laser cavity has a positive feedback mechanism such that the light has a “second chance”to stimulate emission. This positive feedback causes a field buildup. This field buildup is sufficiently strongthat the driving term on the right-hand side of (12.154b), instead of being negligible, becomes all-important.The effect is one often referred to as slaving. The result of it is that the α locks the b and the ωα are pulledto the ωb. With this simplification (see Chapter 6), the b can be shown to satisfy a “potential” equation ofthe form

b = −∂V

∂b+ f(t), (12.169)

where the V (b) is given by a form

V (b) = −C(d0)|b|4 + G(d0)|b|24

, (12.170)

where the V (b) can be sketched as in Figure 12.29. The point is that, up to some threshold parameter ofthe inversion parameter, the laser light output appears as that of a thermal source or LED. (Please notethat the potential equation was derived for a locking condition of the α, and really the potential surfacecan only qualitatively describe the behavior at or below threshold. In general, this regime exhibits fiercemode competition, and operation is very nonlinear in drive current, modulation, etc., which is completelyunlike the behavior of thermal sources or LEDs which are configured such that there is no locking mechanismwhatsoever at any drive current.) That is to say that the field has a zero mean but a finite standard deviation.For a higher injection level, there is a threshold, and the field mean will exceed its standard deviation. Thisis what is meant by laser action.

Is there a limit in which the laser can exhibit at least a small-signal linear behavior? The answer isa qualified yes. If the laser is current-biased well above threshold and then AC-modulated such that itsoperating point is little affected, the behavior is somewhat linear. (A nastiness we will return to is frequencychirp. Any change in the bias point changes the number of carriers in the channel. This changes therefractive index of the material and thereby changes the resonant frequency of the Fabry-Perot cavity, whichin turn pulls the oscillation frequency. The pulling is large—many GHz/mA. This does not cause the laser’smodulation characteristic to go nonlinear but does cause the channel dispersion to exhibit effects that arehard to track in a linear model.) For example, let’s say that we were to write equations for n = b∗b and dk

about an operating point d0. Then we would find (see, for example, Haken 1980, Haken 1985, or Mickelson1993)


Figure 12.30: Circuit model for the small-signal modulation of a laser diode.

Figure 12.31: Comparison of modulation response of a laser diode with that of an LED where the scale isdecreasing due to the fact that laser diodes can be 100 times more efficient than LEDs.

n = an + bn2

d +d

τd=

d0

τd− 2bn. (12.171)

These equations could then be “linearized” about the operating point and converted to a single second-orderequation,

d + c1d + c2d = ∂td0 = κ∂

∂ti(t), (12.172)

which has a form analogous to the equations describing the circuit of Figure 12.30. Note that the circuit of12.30 looks like the LED circuit model except for the series inductor. Try not to draw too much meaningfrom this, however. This inductance is not a circuit inductance but one due to stimulated emission. One cancompare the two responses of an LED and a laser diode, and this is done in Figure 12.31. The peak in thelaser diode response is known as a relaxation peak. The inverse of this resonant frequency is a time whichcorresponds to the time that it takes the junction to return to an equilibrium state through a combinationof stimulated emission and spontaneous recombination due to an injected “event.”

The cause of chirp can be explained as follows. Let’s say that there is a disturbance propagating througha medium of index n. The phase of the wave is given by


φ = kz − ωt (12.173)

for a forward-propagating wave. The instantaneous angular frequency of a wave is given by

ωi = −dφ

dt. (12.174)

If the index of refraction is now allowed to be slowly varying in time, the phase can be written in the form

φ = k0n(t)z − ωt, (12.175)

and, therefore, the instantaneous frequency as

ωi = ω − k0zdn

dt. (12.176)

An increase in n therefore down-chirps the frequency just as we saw in Chapter 10 in the discussion of selfphase modulation. The coupled b(t), α(t), and d(t) equations can also give the chirp effect if one does notadiabatically eliminate the α. The point is that a first-order correction to adiabatic elimination leads to adb/dt term in the expansion of α, which when substituted in the db/dt leads to an effective shift in the ωb

oscillation frequency (Hjelme 1988, Hjelme and Mickelson 1989). The effect, however, is quite obvious fromthe physics. An increase in the number of carriers in a cavity will increase the optical density and thereforethe index. This change will increase the effective optical distance between the mirrors and thereby shifts theresonant frequency downward, as we saw above. The chirp effect in semiconductor lasers is large. The chirpfactor c, which is the factor in the expression

∆ω =√

1 + c2 ωm, (12.177)

where ∆ω is the total broadening and ωm is the modulation angular frequency, can be as large as 7. Itshould be noted that the chirp factor is essentially the same as α, the linewidth enhancement factor whichcauses the semiconductor laser linewidth to exceed that predicted by the Townes-Schalew formula (Vahalaet al 1983). The chirp can be a real problem as far as dispersion is concerned, especially when consideringhigh-speed (> 1Gbs) modulation. This the reason why external modulators are often employed.

Three of the most common external modulator types are depicted in Figure 12.31. The electroabsorptionmodulator (EAM) of Figure 11.31(a) is always semiconductor-based, as it operates by the Franz-Keldysheffect, the effect in which the application of an electric field shifts the valence band energy level, therebychanging the frequency of the absorption edge. The Franz-Keldysh effect is therefore the semiconductoranalog of the Stark effect in atomic media. The EAM can be analyzed much as can the LED or laser, exceptthat one needs to apply a field (bext(t)) as the source term and modulate at d0 < 0. As with the laser whenone exceeds the small modulation limit, there will be nonlinearity. For digital communications, clearly thesmall signal limit will be exceeded, but then for digital communications the nonlinearity is not so importantas it is with analog signals. As the nonlinearity can wreak havoc on subcarrier-modulated signals, communityaccess TV (CATV) really requires predistortion of the modulation signal, much as is done when the laser isdirectly modulated. Chirp is another consideration with the EAM. As carrier density is being modulated,there is also an index modulation. There are many fewer carriers being modulated, however, than in an anabove-threshold laser, so the c factor is much smaller—on the order of 1 (Koyama and Iga 1988).

The basic idea behind a directional coupler switch (Figure 12.32) is that, if light is launched into one ofthe input channels of the structure, it will exit the output channel corresponding to the other input channelif no voltage is applied but will be “switched” to stay in its input channel all the way to the output whensome value of applied voltage is used to set up opposing electric fields in the two channels. An opticalpower versus applied voltage curve for the device may appear as in Figure 12.33. Clearly, this device is not


Figure 12.32: The geometrical arrangements of (a) an electroabsorption modulator, (b) a directional couplermodulator, and (c) a Mach-Zhender modulator.

linear in the modulation frequency for high-voltage operation, where high-voltage operation is defined byvoltages corresponding to the distance between turn-on and turn-off of this device. However, for small signaloperation around either of the operation points OP1 or OP2, the power out of the device could be quitelinear in the incident voltage. What might a circuit representation of the P versus Vin to the device looklike in such a case?

In general, a device with separated electrodes such as the device in 12.33 will look basically like a capacitorto an exciting electrical signal. With reference to Figure 12.34, we further note that the power out will beproportional to the drop across the capacitor. It is important to include any resistance in the model, as thiswill be where the power supplied to the capacitor will be damped out. Clearly there can be series resistancein the electrodes, as there can be a finite substrate capacitance. If the electrodes are either lossy enough orlong enough (as they would be if operated in traveling wave configuration), then a series inductance shouldbe included in the model, as is depicted in Figure 12.35.

There can be one other nuisance to the external modulator. Let’s say that the modulation frequency wewish to use is sufficiently high to allow “walk-off.” The idea is illustrated in Figure 12.36. The idea is thatthe velocity of propagation of the electrical signal can be much lower than the propagation velocity of theoptical signal. For example, in LiNbO3, they are so different that the optical signal (which we define as apoint on the phase front of the optical wave) will appear to “run” right through the modulator, overtakingthe electrical signal as if it were standing still. If this were the case, then part of the modulation that occursat the beginning of the modulator will be undone by a part at the end. Mathematically, one could write thatthe modulation corresponds to a running average over the electrical waveform such that the actual efficiencyof the modulation becomes

Eff =

∣∣∣∣∣∣1τd

t∫t−τd

e−iωmt dt

∣∣∣∣∣∣ = sinc(fmτd), (12.178)

where the τd is given by

τd =(ne − no)

c, (12.179)

with ne the electrical index and no the optical index. Clearly, for either low-frequency (fm << 1/τd, whichis the limit in which the electrical wavelength is much longer than the electrode length) or when ne ≈ no

(the index-matched case in which the two waves travel together for the electrode length), the efficiency ofmodulation is good. Thus, walk-off can always be avoided through making the electrodes short. However,there is an associated cost, as the voltage required to switch is the voltage required that


Figure 12.33: Schematic depiction of a directional coupler waveguide modulator.

Figure 12.34: A P versus V curve for a directional coupler switch used as a modulator.

Figure 12.35: A circuit model for an optical modulator operated in the (linear) small signal regime.


Figure 12.36: Figure illustrating the concept of walk-off.

∆k = π, (12.180)

where the ∆k for an electrooptic material is

∆k = k0∆n = k0rn3r

2d(12.181)

where

k0 = 2π/λ. (12.182)

r is the electrooptic coefficient, n is the average substrate index, d is the spacing between the electrodes,and v is the applied voltage. The switching voltage will be given by

v =dλ

rn3, (12.183)

indicating that the switching voltage is inversely proportional to the electrode length. The switching poweris therefore inversely proportional to the second power of the length. Microwave modulation frequencies,where the line becomes truly greater than a wavelength, may well require that one supply an external resistorin series with the substrate conductance. This resistor on one hand is used to match the load to the lineimpedance as well as to flatten the electrode frequency response at the cost of lowering the modulationdepth.

Chirp is still an issue with a directional coupler, at least in its usual configuration. The field at theoutput of the directional coupler can be expressed in the form

ψ(z = zout, t) = asψsei(βszout−ωt) + aaψae

i(βszout−ωt)eiφ(t), (12.184)

which, for as = aa = a, can be rewritten as

ψ(z = zout, t) = a(ψs + eiφψa)ei(βsz−ωt). (12.185)


Using that at the output,

ψs =ψ1 + ψ2

2

ψa =ψ1 − ψ2

2, (12.186)

we find

ψ(z = zout, t) = a

[ψ1

1 + eiφ

2+ ψ

1 − eiφ

2

]ei(βsz−ωt). (12.187)

Clearly, when there is modulation there will be chirp in both of the output channels.The Mach-Zhender interferometric modulator has operation similar to that of the directional coupler

modulator. The directional coupler, when used a modulator, takes a single-channel input, splits it into asymmetric mode and an antisymmetric mode, and then delays or speeds up the antisymmetric mode withrespect to the symmetric mode by using a push-pull electrode configuration. These modes are interfered atthe output. The Mach-Zhender modulator has a single input which is split into two channels. In push-pullconfigurations, then, one channel is sped up and the other is slowed down, and then they are interfered at theoutput. The characteristic power-versus-voltage curve will be sinusoidal and thus only small-signal linear.An interesting point is that the field at the output, which is something like

ψ(z = zout, t) = aψ(eiφ + e−iφ)eiβze−iωt, (12.188)

can written as

ψ(z = zout, t) = aψ cos φeiβze−iωt, (12.189)

and the φ is no longer a phase in the exponent but part of the amplitude. The push-pull Mach-Zhendermodulator can be chirp-free. Of course, the directional coupler could be also, if the symmetric mode weredelayed by the amount that the antisymmetric mode is advanced. This probably requires either a doublingof the voltage or a doubling of the electrode length, so there seems to be a penalty to removing the chirp.

12.6 Radio Frequency Photonics

In a number of applications, it is the weight of wiring that is a serious problem. Airplanes, for example, havea tremendous amount of wiring for communicating with sensors, etc., located all over the planes. Militaryaircraft have still more wiring than do commercial aircraft. The Air Force began studies already in the 1970sand various programs in the 1980s to see about replacing as much wire as possible with lower-weight opticalfiber. By the mid 1980s, the program had taken on a life of its own, and various agencies and companies werefunding research. Present aircraft contain significant quantities of fiber along with LEDs and detectors. Butthere is an area of the exchange of wire for fiber that remains an active research topic—the use of optics todrive and read out antennas. This is really the area referred to as RF photonics. The skin of an aircraft couldbe covered by conformal antennas and, if one could connect them all to readouts, the data could probablybe used for any number of purposes from wind shear detection to terrain sensing to fixing optimal direction,etc. But optics actually holds more promise in this area than just as a replacement for wire, which is still thereason that research continues in the area despite the lack of success in early attempts. Even in ground-basedradars, one often wants to have the generator and processing equipment remote from the antenna. Fiber haslow enough dispersion that remoting is convenient. The large fiber bandwidth is also a reason to believe thatperhaps shorter pulses could be used with radars, thereby increasing resolution. There is also talk aboutusing optics to generate “true time delay” drive for phased arrays, thereby allowing for rapid high spatialresolution scans of all directions in space. In what follows, we’ll discuss some of the properties of the use


Figure 12.37: A possible realization of an optical drive scheme.

of fiber optics for the driving and readout of radar antennas. There will be more discussion of this topicin Chapter 14 on heterodyne techniques. Here, attention will be placed on intensity modulation and directdetection schemes.

Consider the optical drive scheme of Figure 12.37. In this scheme, we picture an external modulator forthe laser. The RF source could directly modulate the laser as well. The pulser is applied to the RF source,but this as well could be applied to the laser, an external modulator for the RF source, or the externalmodulator for the laser. Running the laser without external modulation—that is, continuous wave (CW)–though, leads to better laser performance in terms of chirp, center wavelength stability, and minimal excessnoise, but it does lead to some excess optical circuit complexity. If chirp is considered to be a problem,one probably wants to use a Mach-Zhender modulator (or chirp-corrected directional coupler) and use smallsignal modulation to minimize harmonics, which will also resonate in the antenna.

The optical power exiting the modulator can be expressed in the form

P0(t) = P0(1 + µ cos ωct). (12.190)

Assuming the fiber is single-mode, we would expect a minimum dispersion of roughly

D ≈ 1 psecnm · km

(12.191)

if we are operating at a fiber dispersion minimum. A single-mode laser can have a linewidth of 0.05 nm.Using the relation (Mickelson 1992) between linewidth δλ, center wavelength λ, frequency spread δf , andcenter frequency f ,

δλ

λ=

δf

f, (12.192)

we see that, for 1.3µm operation, the linewidth δλ corresponds to a frequency spread δf of

δf =0.05 nm1300 nm

2.5 × 104 Hz = 10GHz. (12.193)

Conceivably, we could be remoting a 94-GHz antenna, although the modulation and detection are problematicpast about 10 GHz, but even with this the spectral broadening with a chirp-free external modulator thetransform-limited spectral width would be roughly 0.5 nm. We thus see that pulse broadening per kilometer,Dδλ, will be given by


Dδλ = 0.5 psec/km. (12.194)

Even at 100 GHz, where the period is 10 psec, dispersion should be small for links less than 2 km. This ofcourse is a best case. Off the dispersion minimum, the dispersion could be 20 psec/nm·km. If the source weremodulated at 100 Ghz and chirped with a factor of 10, the linewidth could be 5 nm and the dispersion couldbe 100 psec/km, which would limit link length for 100-GHz operation to tens of meters. For a multimodelink, the numbers are even tighter. A good multimode fiber at 1.3-µm wavelength will have a 500 MHz/kmdispersion, saying that a 10-GHz signal could only be propagated less than 20 m. A single-mode fiber at1.3µm has a loss of roughly 0.5 dB/km, so the loss over 2 km is pretty negligible. The main sources of losswill be coupling into and out of the modulator and any connector loss near the detector. Lasers with greaterthan 100-µW output are available. Losses approaching 10 dB can be associated with external modulators.There is also the small signal modulation limit to deal with.

To evaluate a carrier-to-noise ratio, we need to evaluate the signal, the shot noise, and thermal noisecurrents in the receiver. The carrier term can be written as

is(t) = en0µ cos ωct, (12.195)

giving that

Ris(τ) = e2n2

0

µ2

2cos ωcτ, (12.196)

where

n0 = αP0. (12.197)

One can therefore evaluate the squared current by

Sis(ω) = e2

∞∫−∞

dω

∞∫−∞

dτ e−iωτRs(τ) (12.198)

to be equal to

Sis(ω) =

e2n20µ

2

2. (12.199)

The shot noise term is given by

∞∫−∞

Ssn(ω) dω =e2n0

2τd(12.200)

and the thermal noise term by

∞∫−∞

i2n dω =2kT

τdRL(12.201)

to give


CNR =e2n2

0µ2/2

e2n02τd

+ 4kTτdRL

. (12.202)

Let’s evaluate this CNR for some characteristic numbers. For 10-µW optical power at 1.3µm, the n0 willbe given by

n0 = 1014/sec. (12.203)

The thermal current at room temperature for a 50-Ω load will be circa 1µW, so we will consider only theshot noise limit. For µ ≈ 0.1, the shot noise-limited CNR will be

CNRSN = m × 0.005, (12.204)

where m = n0τd. At 10 GHz, τd ≈ 10−10, giving

CNRSN = 50 = 17 dB. (12.205)

This is not especially good. The signal out of the antenna will have both phase and amplitude fluctuations.The microwave linewidth at the microwave source could be from 1 Hz to 1 kHz. At the antenna, it is greaterthan 1% of the center frequency. The only way to improve on this is to increase the modulated optical powerinto the receiver. The problem would be less (by the linear dependence on τd of the CNR) at RF than atmicrowave frequencies. Shortening the link won’t really help, as the loss is all from coupling in order tokeep the link short enough to keep the link from being dispersion-limited. Another associated problem isthe need for amplification at the element. The microwave power will be something like the signal currentsquared times the load, presumed to be something like 50 Ω. However, effective microwave current will beroughly 1µW, which corresponds to a microwave power of 50 pW. Generally, one wants to drive an antennawith mWs but doesn’t want to have more than 20 or so dB gain at the element, as one would like to have alow-noise amplifier at the element. Again, we see that high-speed antenna drive turns into a problem whenusing high optical power links.

Using a fiber link to remote a receiving antenna is also problematic. Received signals can be as low aspWs or even smaller at the minimum. Modulator drivers often for full modulation require 5V into 50 Ω ora half watt of power. Again, the amplification at the element is a complicated problem.

12.7 Techniques to Achieve Shot Noise-Limited Operation

As was discussed in previous chapters, telecommunications was the prime mover behind optical communi-cations in the 1970s and 1980s, and the problems posed by telecommunications needs were a specialized set.The expense of rights of way and installation required one to try to design for maximal upgradeability nomatter how great the cost of this alternative and the other system problems that implementation of thisalternative created. The optical links designed to do this then needed to be loss-limited, not dispersion-limited. This way, one could simply turn up the frequency (bit rate) with the attendant linear-in-bit-ratepower increase to keep the signal-to-noise ratio constant when upgrading. This led to a situation in whichone would try to work with a receiver input power which would correspond to the minimum level detectableby the chosen detector type. As it turned out, at that point in time, this minimum detectable power levelwould correspond to a level which would exceed the thermal noise generated by the room-temperature re-ceiver circuitry but whose shot noise level was greatly below the thermal contribution. The receiver wouldtherefore operate in a thermal noise regime, despite the fact that achievable SNRs for a given received powerlevel were much greater in the shot noise limit regime. Operating in this shot noise-limited regime wouldtherefore allow one to drop the input power requirement still lower if it were possible to somehow boost the


signal to an effectively higher input level without affecting (or at least minimally affecting) the input noiselevel.

To demonstrate this, we noted back in Chapter 11 that we could find characteristic functions for thecurrent in some reasonably general cases. For stationary increment cases, we noted that the characteristicfunction for a Poisson optical signal converted in a “sharp-edged response function” detector and mixed withthermal noise in the front end of a receiver had a characteristic function of the form

ψi(ω) = em(ejωi0−1)e−ω2 i2n

2 (12.206)

with the attendent moments

m0 = 1m1 = mio

m2 = (m + m2)i20 + i2n, (12.207)

and therefore

σ2 = mi20 + i2n, (12.208)

and the electrical current signal-to-noise ratio is

m1

σ=

√√√√ m2

m + i2ni20

. (12.209)

Generally, we would refer to a signal-to-noise ratio in the receiver as being expressed as electrical power,not in terms of the optical power. The electrical power SNR would then be the square of the above—orexpressible as

SNR =m2

m + i2ni20

. (12.210)

The m for a stationary increment would be given by

m = nτd, (12.211)

where the n could be expressed in terms of the constant signal power PS by

n = αPS , (12.212)

where α is the conversion efficiency in units of counts per Watt incident. Oftentimes one defines an effectivereceiver front-end bandwidth B by writing

τd =1

2B. (12.213)

Also denoting

nin=

i2n τd

e2(12.214)


Figure 12.38: Sketch of the behavior of the signal-to-noise ratio (in dB) versus the signal power.

and using all of these in our SNR expression, we note that

SNR =α2P 2

S/2B

αPS + nin

, (12.215)

which is a reasonably standard expression but is really only valid when there is no information present. Theα2P 2

S in the numerator is generally referred to as the signal power, the αPS in the denominator as the shotnoise, and the i2n/e2 as the thermal noise. The situation is as depicted in Figure 12.38. A dark current termcould also be included in the denominator as an additive term.

12.7.1 Optical Heterodyning

In one archetypical direct detection system, we noted that our signal-to-noise ratio was given by equa-tion (12.215). Ignoring the dark current with respect to the thermal noise, one can rewrite that equation inthe form

S

N=

α2P 2S/2B

αPS + nin

(12.216)

or in the form

S

N=

αPS/2B

1 + ni

αPS

. (12.217)

The shot noise limit corresponds to the limit where

αPS >> nin(12.218)

and therefore that


S

N≈ αPS

2B. (12.219)

Generally, though, we have the opposite situation in direct detect. Let’s consider the arrangement of Fig-ure 12.39, wherein we choose to mix the signal, which comes from far away and may be weak, with aconstant-amplitude local oscillator signal which is coherent with the incident signal and of much larger am-plitude. For present, we won’t say how we will perform the mixing operation but will leave that to discussionsof heterodyne implementation in Chapter 14. Here we will just assume that detector 1 sees a signal whichis the sum of aS and a0 and the second photodetector sees a signal that is the difference of the two. Thetotal i which comes out of a differencer, then, will be of the form

i = i1 − i2, (12.220)

where we can express ij as

ije

=α

2η|a0 ± aS |2. (12.221)

Writing that

P0 =|a0 |22η

(12.222)

and doing the same for PS , we see that

2i1e

= αP0 + α√

P0PS + αPS

2i2e

= αP0 − α√

P0PS + αPS . (12.223)

The result of the coherent subtraction will then yield

i

e= α√

P0PS . (12.224)

Although we have differenced out the αP0 from the current, one can assume that the αPS is much smallerthan the αP0 term, and there will therefore remain the shot noise caused by this αP0 term which will notdifference as it is not coherent. We can then note that the signal current will be of the form

〈i2S〉e2

= αPSαP0 , (12.225)

and the noise current, which will be of the form of the sum of the shot noise and thermal noise, will be ofthe form

〈i2n〉e2

= αP0 + nin, (12.226)

which gives us a signal-to-noise ratio of

S

N=

αP0αPSχ/2B

αP0 + nin

, (12.227)


Figure 12.39: Schematic depiction of the mixing of a signal with a local oscillation at a receiver.

which can be rewritten as

S

N=

αPSχ/2B

1 + nin

αP0

. (12.228)

We can now see that if we turn the local oscillator to a high enough power level we can indeed reach theshot noise level, the fundamental quantum limit defined by the signal level incident on the detector, despitethe preponderance of the circuit noise. How to actually carry this out practically will be discussed furtherin Chapter 14.

12.7.2 Photomultiplication

There was much work done on the statistics of avalanche photodiodes (APDs) in the early 1970s (Personick1971a, Personick 1971b, McIntyre 1972, Presinidi 1973; Webb et al 1974) when it was thought that thisnew solid-state device might turn out to be the best technique to bootstrap oneself to the shot noise limit.Photomultiplier tubes had been around for a long time but were bulky and required very large bias voltages.APDs were solid-state and, although requiring significant voltage, were more compact and lower-voltage thantheir tube counterparts. There still seems to be some effort remaining to apply them in the very highest-endtelecommunications applications. The main idea behind their operation is that a photon incident on theactive p-n junction of the device will generate an electron-hole pair. This pair is then accelerated by thelocal electric field. For high enough local fields, then, secondary carriers are generated through collisions.As the process is therefore a random one, it can be described by a pdf. The one commonly used is the oneproposed by Webb, McIntire, and Conradi (Webb et al 1974). The distribution is given by

Px(x) =1√2π

1(1 + x

δ

)3/2exp

− x2

2(a + x

δ

)]

, (12.229)

where

x =ns − ns

σns

ns = npG

δ =√

npFe

Fe − 1σ2

ns = npG2Fe

Fe = keG +(

2 − 1G

)(1 − ke), (12.230)


where ns is the output distribution given a Poisson input with mean np and G is the gain. The rest should beself-explanatory. Samples of this distribution can be efficiently generated numerically for simulation purposesby a technique developed by Ascheid (1990). There is a mean gain G and a standard deviation G2 whichwill be given by

G2 = G2 + var(G). (12.231)

The signal-to-noise ratio of the APD current, then, will appear as

SNR =G2α2P 2

S/2B

αPSG2F + nin

, (12.232)

where the noise factor F is given by

F = 1 +varG

G2. (12.233)

As can be seen, for large enough G, the SNR can approach the shot noise limit expression

SNR =αPSχ/2B

F(12.234)

except for the multiplicative excess noise factor F . Unfortunately, this factor rises sharply with the averagegain G.

12.7.3 Optical Amplification

There has been a good amount of previous discussion in section 10.2 of Chapter 10 on optical amplification.The point of putting an optical amplifier directly in front of the detector would be to try to boost the signalto achieve shot noise-limited amplification. We have seen that thermal noise currents are on the order oftens to hundreds of nW. It is only necessary to amplify so that the effective noise exceeds this level in orderto (almost) achieve the shot noise limit. The penalty that we have seen before for amplification is that thereis a beat noise term of the form

√i2B = 2enG hω/τd (12.235)

if the input is Poisson. It is this noise current that needs to exceed the thermal noise.In a chain of amplifiers, it is the first one with the assumed Poisson input which suffers a 1.5-dB SNR0

(3.0 dB SNRe) penalty. The second amplifier should only incur half this penalty, as the input multiplyconvolved Laguerre distribution carries an excess noise equal to the noise added at the input. The outputof all subsequent amplifiers will remain Laguerre-distributed but with ever growing noise—but ever betternoise figures due to the growing excess noise relative to that generated in the amplifier. Practically, however,there is probably a technical limit on F .

Problems

1. Consider a signalEs(t) = as(t)eiδs(t)e−iωst,

which is to be direct detected. What is the resulting spectral density Sn(ω) in terms of Sm(ω) andSd(ω) if


(a) as(t√

Ps

[1 + m(t)

], δs(t) = d(t);

(b) as(t) = as(0), δs(t) = d(t);

(c) as(t) =√

Ps

[1 + m(t)

], δs(t) = 0.

2. Consider a noise process q(t) expressible in the form

q(t) =m∑

=1

δ(t − ti),

where the ti are Poisson distributed. Find the spectral density Sq(ω) by

(a) Fourier transforming the expression for q;

(b) squaring the transform and simplifying the resulting sums; and

(c) averaging the resulting expressions over T .

(d) Sketch the resulting Sq(ω) assuming a “reasonable” rate of arrival n(t). Use ω = 2π/T as theunit on the horizontal axis.

3. If one wishes to design an output filter that maximizes the SNRt, a shot noise correlator of linearweighting counts that depend on both the signal and noise intensities during each mode can be used.The interval (0, T ) is divided into Dt = 2B0T time modes, each having width 1/2B0 sec. The count ki

corresponds to the ith mode interval, and |Si|2 is the integrated signal intensity in this interval. Theproblem can now be stated as determining the weighting coefficients βi so that the modified count

y∆=

Dt∑i=1

βiki

has a maximum

SNRt =

(E[y])2

var(y).

Maximize βi for the |Si|2 and apply the Schwarz inequality to find βi.

4. It is convenient to define a photomultiplier “noise figure” that describes the reduction in the shot-limited SNRp when a multiplier is used. This is given by

F = 1 +var(G)(G)2

.

For a nonideal photomultiplier, the multiplier gain variance is expressed as a fraction of its mean gain.Consider the Poisson branching process in which an initial event is transformed into a Poisson processwith parameter m1, and then each of the possible counts is transformed to a Poisson process withparameter m2, and then the process repeats up through a last event with mn.

(a) Calculate the mean and variance of this process.

(b) Derive the effective noise figure in terms of the parameters δi.

Bibliography

[1] S. Andre and D. Leonard, “An Active Retrodirective Array for Satellite Communications,” IEEETrans Ant Prop AP-11, 181–186 (1963).

[2] G. Ascheid, “On the Generation of WMC-Distributed Random Numbers,” IEEE Trans Commun38:2117–2118 (1990).

[3] C. Cutler, R. Kompener, and L. Tillotson, “A Self-Steering Array Repeater,” Bell Sys Tech J42:2014–2032 (1963).

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