opt comm chap 12

8/6/2019 Opt Comm Chap 12

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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 eld, thereby recovering a time-varying electrical current which, under certaincircumstances, is an accurate replica of the initially transmitted eld.

A systems 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 diodes electrodes, as is depicted inFigure 12.2. Simplicity is also the rule in the receiver end. Here the optical power can be simply detectedand preamplied for launch into a load, as is depicted in Figure 12.3. Fidelity in such a system is oftensimply dened in terms of how well the signal supplied to the load compares with that supplied to thesources 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 was

discussed all the way back in the rst 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. If one doesnt 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 systems block diagram.

1


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CHAPTER 12. ANALOG COMMUNICATIONS 2

Figure 12.2: Direct detect transmitter end where the electrical signal is simply added to the optical sourcesdc 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 superuous 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 dene 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 ltering 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 ber 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 amplier. Some discussion of rare earth

elements and their properties was given in section 7.2.

12.1 Signal-to-Noise Analysis

A clear problem with nding probability distributions for signals is that signals need to vary with time inorder to carry information. Yet, when we nd count distributions pk (k)sinevitably 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 dene 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


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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 nd that count distributions are usefulin the digital case. In the analog case, we will nd 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 nd signal-to-noise

ratios for some transmission formats. In the next chapter, in the rst section on information theory, wellsee that the signal-to-noise ratio can be interpreted as an effective number of bits of information, but wellhave no need to go further with this interpretation. A point to note here is that the signal-to-noise ratios(SNR) as dened in this chapter are electrical SNRs which are the square of the optical SNRs calculatedfrom count distributions.

Here well reprint Figure 3.8 as gure 12.4, which illustrates what we mean by an ensembles of stochasticprocesses and, by example then, an ergodic process. The point of the gure is that an ensemble should bean innite set of statistically identical realizations of the given process. If we average down an ensemble atsome given time t, this should dene a complete set of moments, etc. If the statistics of a realization wereconstant in time, then it shouldnt 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 in the expression

f (t) =1

t

t f (t ) dt (12.1)as long enough, then the statistics we get should be as good as averaging down enough members of the

ensemble as correspond to the number of temporal modes contained in . If the statistics are constant forall time, then the process is called ergodic.

A gure 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


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

E (x(t)) independent of tE (x(t1)x(t2)) dependent only on = t2 t1

. (12.2)

Wide-sense stationary processes are those for which there is but one time increment of innite 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 denea nite 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 , . . . , x k ; t1 , t2 , . . . , t k ) = px (x1 , x2 , . . . , x k ; t1 T, t2 T, . . . , t k 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 R i ( ). A possible denition for the correlation function of a wide-sensestationary signal could be

R i ( ) = limT

E i(t + )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 nd the spectral density S i () inthe form

S i () = limT

E iT ()

2

2T , (12.5)

where

iT (t) =i(t), T/ 2 < t < T/ 20, otherwise (12.6)

and where

iT () =T

T

i(t )e jt 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 hereone that employs ensemble averages.


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

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

R12 (t1 , t 2) = E i(t1)i(t2) . (12.8)

The point here is that the current is generally dened by

i(t) =k ( t d ,t )

0h(t t i ), (12.9)

and the expectation is dened as an ensemble average. What (12.8) then indicates is an average of manyindependent, identically distributed (iid) samples taken over a xed interval. This is essentially the sameprinciple of operation as is used in spectrum analyzers and sampling oscilloscopes. To nd the correlation,we can use the two-point characteristic function. That is, we know that the two-point characteristic functioni 1 i 2 (1 , 2) should have an expansion of the form

i 1 i 2 (1 , 2) = 1 + j1m10 + j2m01 + j1 j2m11 +( j1)2

2m20 +

( j2)2

2m02 + . (12.10)

Clearly, m11 is the expectation

m11 = E i(t1)i(t2) , (12.11)

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


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

The two-point characteristic function must be of the form

i 1 i 2 (1 , 2) = E [ej 1 i 1 ej 2 i 2 ]. (12.12)

Writing out the currents explicitly, we have

i 1 i 2 (1 , 2) = E exp j1k

j =1h(t1 t j ) + j2

k

j =1h(t2 t j ) . (12.13)

Then, taking expectations over the Markov process yields

i 1 i 2 (1 , 2) = E n E k

t e

t b exp j1h(t1 t ) + j2h(t2 t )n(t )

mdt

k

, (12.14)

where

te = max( t1 , t2)tb = min( t1 , t2) d , (12.15)

as it is tacitly assumed that h(t) = 0 for t < 0 and t > d . 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 E k . 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 todene E k is a good approximationif an approximation at all. Using the denition of the E k operator, wenote that

i 1 i 2 (1 , 2) = E nmk

k!e m

t e

t b exp j1h(t1 t ) + j2h(t2 t )n(t )

mdt

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 denition of the innite sum for an exponential, we obtain

i 1 i 2 (1 , 2) = E n expt e

t b ej 1 h ( t 1 t ) ej 2 h ( t 2 t ) 1 n(t ) dt . (12.17)

We can now make an expansion of the quantity in the brackets before taking the E n expectation to nd

i 1 i 2 (1 , 2) = E n 1 +t e

t b (ej 1 h ( t 1 t ) ej 2 h ( t 2 t ) 1)n(t ) dt

+t e

t b dtt e

t b dt (ej 1 h ( t 1 t ) ej 2 h ( t 2 t ) 1)(ej 1 h ( t 1

t ) ej 2 h ( t 2 t ) 1)n(t )n(t ) + , (12.18)

which can then have the n expectation operator applied to yield


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i 1 i 2 (1 , 2) = 1 +t e

t b (ej 1 h ( t 1 t ) ej 2 h ( t 2 t ) 1)E n n(t ) dt

+ 12t e

t b

dtt e

t b

dt (ej 1 h ( t 1 t ) ej 2 h ( t 2 t ) 1)(ej 1 h ( t 1 t ) ej 2 h ( t 2 t ) 1)E n(t )n(t )

+ . (12.19)The last step is to expand the exponentials to nd

i 1 i 2 (1 , 2) = 1 + j1t e

t b h(t1 t )E n n(t ) dt + j2t e

t b h(t2 t )E n n(t ) dt+ ( j 1 )

2

2

t e

t b h2(t1 t )E n n(t ) dt +

( j 2 ) 2

2

t e

t b h2(t2 t )E n n(t ) dt

+ ( j 1 )2

2

t e

t b dtt e

t b dt h(t1 t )h(t2 t ) +( j 2 ) 2

2

t e

t b dtt e

t b dt h(t2 t )h(t2 t )+ j1 j2

t e

tb

h(t1

t )h(t2

t )E n n(t ) dt

+ j 1 j 22t e

t b dtt e

t b dt h(t1 t )h(t2 t ) + h(t2 t )h(t1 t ) E n n(t )n(t )+ (12.20)From the above, we can read off our correlation function, which is of the form

R i d (t1 , t2) =t e

t b h(t1 t )h(t2 t )E n n(t ) dt+

t e

t b dtt e

t b dt h(t1 t )h(t2 t ) + h(t1 t )h(t2 t ) E n n(t )n(t ) . (12.21)Noting that we can always write

t2 = tt1 = t2 + , (12.22)

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

R i d (t, ) =t e

t b h(t + t ) h(t t ) E n n(t ) dt+ 12

t e

t b

dtt e

t b

dt h(t + t ) h(t t ) E n n(t ) n(t )+ 12

t e

t b dtt e

t b dt h(t t ) h(t + t ) E n n(t ) n(t ) , (12.23)which now appears to depend explicitly on the time t as well as on the delay period . But the assumptionthat the information stream is random and stationary says that the t-dependence is not real and will averageout to leave an R i d ( ). What we need to do now is to identify terms that we consider noise and those thatwe consider signal. The rst 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


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noise is buried in the pn (n) of E n . 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 rst 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

t e

t b h(t + t )h(t t )E n n(t ) dt = nt e

t b h(t + t )h(t t ) dt . (12.24)If we now invoke the fact that h(t) is only nonzero between 0 and d , then we would want to pick limits tband te such that

t e

t b h(t + t )h(t t ) dt =t

t d h(t + 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 d h(t + t )h(t t ) dt = d

0 h(t + )h(t ) dt . (12.26)At this point, it does not matter whether we extend the limits to innity, however, as there are no contribu-tions to the integral from outside these limits. We can, therefore, write that

d

0 h(t + )h(t ) dt =

h(t + )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

d ej

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

where

H () =

ejt h(t ) dt

h(t) =

e j t H ( ) d (12.29)

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

i2SN = n

|H ()|2 d. (12.30)


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Without detailed knowledge of the h(t) (the important quantity, as well see, is really T (), the overalltransfer function of the receiver), we really cant go much farther. In the limit where h(t) is somewhatsquare and of height roughly e/ d , we can use the Parseval relation

h2(t) dt =

|H ()

|2 d (12.31)

to show that

i2SN MAX=

ne 2

d, (12.32)

where the bandwidth B would be 1/ 2 d (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 dont 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 and R i ( ) must be symmetric in in order that S i () be real. Applying similartechniques of limit changing and variable substitution such as we did to the shot noise term above, we canwrite

t e

t b dtt e

t b dt h(t + t )h(t t )Rn (t , t ) =

dt

dt h(t + )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 + )h(t )Rn (t , t ) =

dt h(t + ) 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 rst is a cross-correlation. Symbolically, we could then write that

dt h(t + )

dt Rn (t t )h(t ) = h( ) (Rn ( )h( )) , (12.36)where

f ( ) g( ) =

dt f (t + )g(t) (12.37)

and


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f ( )g( ) =

dt f ( t)g(t). (12.38)

We further should recall the Fourier transform theorems

F [f ( ) g( )] = F ()G() (12.39)

and

F [f ( )g( )] = F ()G(). (12.40)We can therefore take the spectra to nd

F [h( ) (Rn ( )h( ))] = H ()H ()S n (). (12.41)

We see that our spectral density of the current is

S i d () = [ n + S n ()] |H ()|2 . (12.42)

To nd an SNR, well 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 ltering, there is adirect way to nd the power from the correlation function. In terms of the correlation function, the electricalpower P e is given by

P e =

d

d ej R i ( ). (12.43)

By changing the order of integration and noting

d ej = ( ), (12.44)

we see that P e is given by

P e = R i (0) . (12.45)

With our general expression for R i ( ), we see then that

P e = 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 ber amplier, or EDFA)but probably no signal-dependent noise, so our process probably appears as


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n(t) = n s(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) = Rn s (t1 , t 2) + Rn n (t1 , t2) + 2 E [n s]E [nn ]. (12.49)

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

Rn n (t , t ) = nn 2(t t ). (12.50)We also know (it will be demonstrated in the rst 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 B s satisfying

B s > kT and the factor approaches zero. However, the current that is beinginjected into the junction will linearly increase N 2 and linearly decrease N 1 such that

N 2 = CI N 1 = ( N 1 + N 2) CI. (12.163)

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

N 2N 1

= I = e h/kT eff . (12.164)

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

n =1

eh/kT eff 1. (12.165)

Assuming h > kT eff then, one can write

n = e h/kT eff = I, (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 d

= 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.


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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.


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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 eld 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 diode

is 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 chanceto stimulate emission. This positive feedback causes a eld buildup. This eld 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 locks the b and the are pulledto the b. With this simplication (see Chapter 6), the b can be shown to satisfy a potential equation of the 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 of the 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 , and really the potential surfacecan only qualitatively describe the behavior at or below threshold. In general, this regime exhibits ercemode competition, and operation is very nonlinear in drive current, modulation, etc., which is completelyunlike the behavior of thermal sources or LEDs which are congured such that there is no locking mechanismwhatsoever at any drive current.) That is to say that the eld has a zero mean but a nite standard deviation.For a higher injection level, there is a threshold, and the eld 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 qualied 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 largemany GHz/mA. This does not cause the lasersmodulation characteristic to go nonlinear but does cause the channel dispersion to exhibit effects that arehard to track in a linear model.) For example, lets say that we were to write equations for n = bb and dkabout an operating point d0 . Then we would nd (see, for example, Haken 1980, Haken 1985, or Mickelson1993)


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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 d

=d0 d 2bn. (12.171)

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

d + c1 d + c2d = t d0 =

ti(t), (12.172)

which has a form analogous to the equations describing the circuit of Figure 12.30. Note that the circuit of 12.30 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. Lets say that there is a disturbance propagating througha medium of index n. The phase of the wave is given by


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= kz t (12.173)for a forward-propagating wave. The instantaneous angular frequency of a wave is given by

i = ddt . (12.174)If the index of refraction is now allowed to be slowly varying in time, the phase can be written in the form

= k0n(t)z t, (12.175)and, therefore, the instantaneous frequency as

i = k0zdndt

. (12.176)

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

= 1 + c2 m , (12.177)where is the total broadening and m is the modulation angular frequency, can be as large as 7. Itshould be noted that the chirp factor is essentially the same as , 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 ( > 1 Gbs) 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 electroabsorption modulator (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 eld 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 eld ( 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 smalleron 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 of the 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 elds 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


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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 dened byvoltages corresponding to the distance between turn-on and turn-off of this device. However, for small signaloperation around either of the operation points OP 1 or OP 2 , the power out of the device could be quitelinear in the incident voltage. What might a circuit representation of the P versus V in 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 nite substrate capacitance. If the electrodes are either lossy enough orlong enough (as they would be if operated in traveling wave conguration), 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. Lets say that the modulation frequency we

wish 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 LiNbO 3 , they are so different that the optical signal (which we dene 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 =1 d

t

t d e i m t dt = sinc( f m d ), (12.178)

where the d is given by

d =(ne no)

c, (12.179)

with ne the electrical index and no the optical index. Clearly, for either low-frequency ( f m


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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.


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

k = , (12.180)

where the k for an electrooptic material is

k = k0 n = k0rn 3r2d

(12.181)

where

k0 = 2 /. (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 =d

rn 3, (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 atten 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 conguration. The eld at theoutput of the directional coupler can be expressed in the form

(z = zout , t ) = assei ( s z out t ) + aa a ei ( s z out

t ) ei ( t ) , (12.184)

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

(z = zout , t ) = a(s + ei a )ei ( s z t ) . (12.185)


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Using that at the output,

s =1 + 2

2

a =1 2

2, (12.186)

we nd

(z = zout , t ) = a 11 + ei

2+

1 ei2

ei ( s z t ) . (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 conguration. These modes are interfered atthe output. The Mach-Zhender modulator has a single input which is split into two channels. In push-pullcongurations, 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 eld at the output, which is something like

(z = zout , t ) = a (ei + e i )eiz e it , (12.188)

can written as

(z = zout , t ) = a cos eiz e it , (12.189)

and the 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 opticalber. By the mid 1980s, the program had taken on a life of its own, and various agencies and companies werefunding research. Present aircraft contain signicant quantities of ber along with LEDs and detectors. Butthere is an area of the exchange of wire for ber that remains an active research topicthe 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 xing 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 ber 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, well discuss some of the properties of the use


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

of ber 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 modulationthat 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

P 0(t) = P 0(1 + cos c t). (12.190)Assuming the ber is single-mode, we would expect a minimum dispersion of roughly

D 1 psec

nm km(12.191)

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

=

f

f , (12.192)

we see that, for 1 .3 m operation, the linewidth corresponds to a frequency spread f of

f =0.05nm1300 nm

2.5 104 Hz = 10 GHz . (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,D , will be given by


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D = 0 .5psec/ km. (12.194)

Even at 100 GHz, where the period is 10 psec, dispersion should be small for links less than 2 km. This of course 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 ber at 1.3- m wavelength will have a 500 MHz/kmdispersion, saying that a 10-GHz signal could only be propagated less than 20 m. A single-mode ber at1.3 m 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- W 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) = en 0 cos c t, (12.195)

giving that

R i s ( ) = e2n202

2cos c, (12.196)

where

n0 = P 0 . (12.197)

One can therefore evaluate the squared current by

S i s () = e2

d

d e i R s ( ) (12.198)

to be equal to

S i s () =e2n202

2. (12.199)

The shot noise term is given by

S sn () d = e2

n02 d (12.200)

and the thermal noise term by

i2n d =2kT d RL

(12.201)

to give


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CNR =e2n202 / 2

e 2 n 02 d +

4kT d R L

. (12.202)

Lets evaluate this CNR for some characteristic numbers. For 10- W optical power at 1 .3 m, the n0 willbe given by

n0 = 10 14 / sec. (12.203)

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

CNR SN = m 0.005, (12.204)where m = n0 d . At 10 GHz, d 10 10 , giving

CNR SN = 50 = 17dB . (12.205)

This is not especially good. The signal out of the antenna will have both phase and amplitude uctuations.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 d of the CNR) at RF than atmicrowave frequencies. Shortening the link wont 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 amplication at the element. The microwave power will be something like the signal currentsquared times the load, presumed to be something like 50 . However, effective microwave current will beroughly 1 W, which corresponds to a microwave power of 50 pW. Generally, one wants to drive an antennawith mWs but doesnt want to have more than 20 or so dB gain at the element, as one would like to have a

low-noise amplier at the element. Again, we see that high-speed antenna drive turns into a problem whenusing high optical power links.Using a ber link to remote a receiving antenna is also problematic. Received signals can be as low as

pWs or even smaller at the minimum. Modulator drivers often for full modulation require 5V into 50 ora half watt of power. Again, the amplication 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 this

alternative 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


52/59


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 nd 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

i () = em (eji 0 1) e

2 i 2n2 (12.206)

with the attendent moments

m0 = 1m1 = mi o

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

and therefore

2 = mi 20 + i2n , (12.208)and the electrical current signal-to-noise ratio is

m1

= m2m + i 2ni 20 . (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 aboveorexpressible as

SNR = m2

m + i2n

i 20

. (12.210)

The m for a stationary increment would be given by

m = n d , (12.211)

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

n = P S , (12.212)

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

d =1

2B. (12.213)

Also denoting

n i n =i2n d

e2(12.214)


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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 = 2P 2S / 2BP S + n i n

, (12.215)

which is a reasonably standard expression but is really only valid when there is no information present. The 2P 2S in the numerator is generally referred to as the signal power, the P S in the denominator as the shot noise, and the i2n /e 2 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

= 2P 2S / 2BP S + n i n

(12.216)

or in the form

S N

=P S / 2B1 + n iP S

. (12.217)

The shot noise limit corresponds to the limit where

P S >> n i n (12.218)

and therefore that


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S N

P S2B

. (12.219)

Generally, though, we have the opposite situation in direct detect. Lets 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 a

constant-amplitude local oscillator signal which is coherent with the incident signal and of much larger am-plitude. For present, we wont 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 a 0 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 i j as

i je

=2 |a 0 aS |

2 . (12.221)

Writing that

P 0 = |a 0 |2

2(12.222)

and doing the same for P S , we see that

2i1e

= P 0 + P 0 P S + P S2

i2

e

= P 0

P 0 P S + P S . (12.223)

The result of the coherent subtraction will then yield

ie

= P 0 P S . (12.224)Although we have differenced out the P 0 from the current, one can assume that the P S is much smallerthan the P 0 term, and there will therefore remain the shot noise caused by this P 0 term which will notdifference as it is not coherent. We can then note that the signal current will be of the form

i2Se2

= P S P 0 , (12.225)

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

i2ne2

= P 0 + n i n , (12.226)

which gives us a signal-to-noise ratio of

S N

=P 0 P S / 2B

P 0 + n i n, (12.227)


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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 =

P S / 2B

1 + n i nP 0 . (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 dened 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 signicant 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 eld. For high enough local elds, 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

P x (x) =1

21

1 + x3/ 2 exp

x2

2 a + x, (12.229)

where

x =n s n s

nsn s = n pG

= n pF eF e 12ns = n pG

2F e

F e = keG + 2 1G

(1 ke), (12.230)


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where n s is the output distribution given a Poisson input with mean n p 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

G

2

=G

2

+ var (G). (12.231)The signal-to-noise ratio of the APD current, then, will appear as

SNR =G2 2P 2S / 2B

P S G2F + n i n, (12.232)

where the noise factor F is given by

F = 1 +varG

G2. (12.233)

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

SNR =P S / 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 Amplication

There has been a good amount of previous discussion in section 10.2 of Chapter 10 on optical amplication.The point of putting an optical amplier directly in front of the detector would be to try to boost the signalto achieve shot noise-limited amplication. We have seen that thermal noise currents are on the order of tens 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 amplication is that thereis a beat noise term of the form

i2B = 2 enG h/ d (12.235)if the input is Poisson. It is this noise current that needs to exceed the thermal noise.

In a chain of ampliers, it is the rst one with the assumed Poisson input which suffers a 1.5-dB SNR 0(3.0 dB SNR e) penalty. The second amplier 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 ampliers will remain Laguerre-distributed but with ever growing noisebut ever better

noise gures due to the growing excess noise relative to that generated in the amplier. Practically, however,there is probably a technical limit on F .

Problems

1. Consider a signal

E s (t) = a s (t)ei s ( t ) e i s t ,

which is to be direct detected. What is the resulting spectral density S n () in terms of S m () andS d () if


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(a) a s (tP s 1 + m(t) , s (t) = d(t);(b) a s (t) = a s (0), s (t) = d(t);(c) a s (t) = P s 1 + m(t) , s (t) = 0.

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

q(t) =m

=1

(t t i ),

where the t i are Poisson distributed. Find the spectral density S q() 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 S q() assuming a reasonable rate of arrival n(t). Use = 2 /T as the

unit on the horizontal axis.

3. If one wishes to design an output lter that maximizes the SNR t , 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 D t = 2 B0T time modes, each having width 1 / 2B0 sec. The count kicorresponds to the i th mode interval, and |S i |2 is the integrated signal intensity in this interval. Theproblem can now be stated as determining the weighting coefficients i so that the modied count

y =D t

i=1

i ki

has a maximum

SNR t =E [y] 2

var( y).

Maximize i

for the

|S

i |2 and apply the Schwarz inequality to nd

i.

4. It is convenient to dene a photomultiplier noise gure that describes the reduction in the shot-limited SNR p 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 gure in terms of the parameters i .


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