Phase-lock control considerations for coherently combined lasers

Phase-lock control considerations for coherently combinedlasers

James B. Armor, Jr., and Stanley R. Robinson

Fundamental performance limitations of a phase-lock control loop used to coherently combine the output oftwo lasers are presented. The phase-lock loop is designed to lock the differential phase (frequency andphase) between the two lasers to a specified reference phase. An optical heterodyne configuration is usedto determine the differential phase of the laser pair, which in turn is compared with the reference phase tocreate an error voltage. The error voltage is filtered and used to frequency modulate one of the lasers in anattempt to null the error. An integro-differential loop equation, valid for the linear operating range, is de-rived in terms of the reference phase, the heterodyne measurement noise, and the various laser phase insta-bilities. The solution of the equation results in an expression for the phase error variance in terms of theclosed-loop noise equivalent bandwidth WH. An expression for the value of WH which minimizes the phaseerror variance is developed. In addition to the noise effects, the steady-state and dynamic performance ofthe loop is examined for different loop filters and modulation formats. A design example pairing a CO 2waveguide and conventional laser is presented. Implications for coherent laser arrays are discussed.

1. Introduction

A well-known method of generating periodic, opticalpulse trains with high peak power, and extremely nar-row pulse widths is the mode-locked laser.' However,the modulation characteristics are largely dependenton the laser's cavity parameters. For example, thepulse repetition rate of a mode-locked laser is deter-mined primarily by the cavity length. Thus, the mod-ulation format is inflexible.

Conceptually, it is easy to think of each laser modeas being generated in individual laser cavities. So, ageneralized concept of mode-locking could be the co-herent combination of the outputs of an array of sin-gle-mode, but electronically phase-controlled, lasers.Such a configuration has a potential modulation formatthat is much more flexible. For the special case of eachindividual laser offset from the next by a constant fre-quency, such a configuration will result in the familiarmode-locked waveforms, as was recently experimentallydemonstrated.2 A discussion of the possible types ofwaveforms and their potential applications is presentedelsewhere. 3

Both authors are at Wright-Patterson Air Force Base, Ohio 45433.James Armor is with the Foreign Technology Division (AFSC);Stanley Robinson is with the Air Force Institute of Technology, De-partment of Electrical Engineering.

Received 16 January 1979.0003-6935/79/183165-11$00.50/0.© 1979 Optical Society of America.

The key requirement for such a system is the precisecontrol of the relative frequencies and phases of theindividual lasers of the array. A logical first step in theconstruction of such an array is the phase locking of asingle pair of lasers. This paper presents fundamentalperformance limitations and design considerations fora phase-lock feedback control loop designed to lock thefrequency/phase difference between the output beamsof two single-mode lasers to a desired controllable value.An integro-differential equation, valid for the linearizedoperating region of the loop, is derived in terms of thedesired frequency/phase control, the laser phase in-stabilities, and other loop component noises and in-stabilities. Expressions for the power spectra of thevarious noises and instabilities are used to solve theintegro-differential equation for the phase error vari-ance in terms of the closed-loop noise equivalentbandwidth of the system. The phase error varianceexpression can be used to determine the optimum (i.e.,minimum variance) closed-loop bandwidth. In addi-tion to noise effects, system performance parameters(such as loop acquisition time, frequency pull-in range,and steady-state error) are related to the loop filtercharacteristics and the closed-loop bandwidth for var-ious desired control schemes (such as step or rampchanges in frequency). A detailed design example forthe two-laser control loop is presented to illustrate theeffects of the various noise and control parameters onthe over-all phase-lock performance.

15 September 1979 / Vol. 18, No. 18 / APPLIED OPTICS 3165

II. Two-Laser Phase-Lock ProblemWhen one mentions the problem of phase-lock of two

sinusoidal sources, the first configuration that comesto mind is the well known phase-lock loop (PLL), shownin Fig. 1. Ideally, two lasers could be placed in phase-lock by letting the output field of one laser be the PLLinput signal and letting the other, its frequency beingvoltage adjustable, act as the voltage-controlled oscil-lator (VCO). Unfortunately, this scheme is not feasible,since there are no known methods for direct measure-ment of laser field e(Ft). All schemes detect the in-tensity or power density of the field | (Ft) 1 2, thus losingall phase information. Furthermore, the extremelyhigh laser frequencies, on the order of 1014 Hz, wouldalso present difficult electronic problems.

To circumvent these problems, a common procedureis to sum (or interfere) the output fields of the two lasersonto a single detector. The output of the detector willinclude a signal with a phase equal to the difference ofthe laser phases plus low-frequency terms. The phasedifference, or beat frequency, is more readily handledby available electronics. Typically, one laser has astable phase so that the phase instabilities of the othercan be examined. 4

Another reason this procedure is so attractive is thatexcellent noise performance can be obtained with simplepostdetector processing. (This well-known method,known as heterodyne detection, will be discussed later.)We are now ready to discuss the two-laser phase controlproblem (in relation to the well-known PLL) in de-tail.

A. Two-Laser PLL: Preliminary Discussion

The basic PLL of Fig. 1 can now be adapted to lockthe phase difference between two lasers to a referencephase, as shown in Fig. 2. The entire laser pair anddetector combination acts as the VCO. The filtered

error voltage modulates the frequency of one laser only,but this is translated to a modulation in the differentialphase seen at the detector. The important aspect of thetwo-laser loop is that the differential phase between thetwo lasers is locked to a reference phase; the absolutephase of the laser pair remains uncontrolled except bythe nature of the lasers themselves and thus is free tofluctuate.

The loop input reference signal has the form

vr(t) = V sinfr(t), (1)

where

,Ibr(t) = (27rfdt + ad) + Or(t)- (2)

The first term, (2 rfdt + Od), is the constant portion ofthe reference phase and represents the VCO quiescentfrequency. (All derivations will be made with respectto the VCO quiescent frequency fd.) The second term,Or(t), is the modulated term and consists of two por-tions:

Or(t) = (t) + yr(t), (3)

where (t) is the control modulation, and yr(t) is a zeromean random process representing the reference phaseinstabilities.

The voltage-controlled oscillator (VCO) consists ofthe two lasers, one of which is modulated by the filterederror voltage, and a heterodyne receiver (HR) thatutilizes samples of their output fields. The outputfields of the lasers are combined for use in the far field,as seen in Fig. 2. However, samples of the fields aretaken for use in the control loop. These sampled fieldsare assumed to be planar with constant amplitudes.Thus the sampled fields for the reference and modu-lated lasers are, respectively,

'o(Ft) = Ao cos4bo(t),

&,(Ft) = Al cosclhi(t).

The phase of the reference laser is

4'o(t) = (27rfot + ko) + yo(t),

(4)

(5)

(6)

where (27rfot + 00) is the operating phase, and yo(t) isa zero-mean, Gaussian random process describing themeasurable reference laser instabilities. Its statisticaldescription is discussed in detail in a later section.Since this laser is not modulated, fo is always thequiescent frequency fOq, that is,

Fig. 1. Basic phase-lock loop.

PHAS v (LOOP v W(tr It0 - DETECTOR'e FILTER

Kd A t } C

_ n|HETERODYNE RECEIVER |

COMB I N E R |SBEAeo (;, __ _EK _ _M_.D A

Fig. 2. Two-laser differential phase control loop.

fo - Oq.

The phase of the modulated laser is

41(t) = (27rfit + 01) + y(t),

(7)

(8)

where (27rfit + 01) is the operating phase, and yl(t) isagain a zero-mean, Gaussian random process repre-senting that laser's instabilities. More specifically, theoperating frequency f1 consists of a constant quiescentfrequency fiq portion and a portion modulated by thefiltered error. The modulated laser is frequencymodulated by applying the filtered error voltage vf (t)to an internal electrooptic or external piezoelectriccrystal, which changes the effective cavity length. Theinstantaneous frequency of the modulated laser is

3166 APPLIED OPTICS / Vol. 18, No. 18 / 15 September 1979

yr Wt)

vf (tI

d(t) = 27TKmvf(t) + d(t), (18)

where Km is the modulation gain due to the crystal inhertz per volt.

The two sampled laser fields are directed into a het-erodyne receiver (HR). The output of an HR is wellknown and will simply be stated in this paper. Theresults are conditioned on the following assumptions:

(1) The fields impinging on the detector are col-linear, planar, of constant magnitude, and normallyincident.

(2) The combined beam is incident on a detector ofactive area Ad.

(3) The fields, traveling in free space, have units ofV/mV'Q, and without loss of generality,

AO >> Al. (10)

(4) Finally, the bandpass filter (BPF) is ideal withimpulse response b(t) and transfer function B(j27rf).The filter IB(227rf) 12 has magnitude 1/C21 , bandwidthBo1 Hz, and center frequency

fd = flq - fOq.

where

d(t) = a1 (t) - yo(t). (19)

Using the quadrature component representation ofthe HR noise6 and assuming that the noise spectrum issymmetric about the VCO quiescent frequency fd, Eq.(12) can be rewritten as

Vd(t) = Vd cOSbd(t) + nc(t) cos(27rfdt + Od)

+ n,(t) sin(2rfdt + kd), (20)

where n, (t) and n, (t) are the (statistically independent)in-phase and quadrature components of the noise n(t),respectively. Equation (20) is an alternate form for theVCO output.

The reference and VCO signals, as seen in Fig. 2, meetin a phase detector (PD). The PD will be modeled asa multiplier with double frequency components re-moved.7 The output of the PD, called the error signal,is then

(11)

The filter characteristic is shown in Fig. 3.With these assumptions the output of the HR iS5

d(t) = Vd COS4 d(t) + n(t), (I

where

Vd = qqAdAoAlRhfoCo1

4'd(t) = (t) - o(t).

Ve(t) = KdLp[Vr(t)Vd(t) (21)

where Kd is the PD gain, and Lp [ I] denotes the low-passportion of the quantity in brackets. Substituting Eqs.

2) (1) and (2) into the above equation results in

(13)

(14)

In the above, q is the charge of an electron, -q is thequantum efficiency of the detector, h is Planck's con-stant, R is the effective load resistance of the detector,and n(t) is a zero-mean, Gaussian random process. TheHR noise arises from the fundamental quantum natureof the optical detection process and should not be con-fused with the noise arising from the laser phase insta-bilities. The statistical nature of n (t) will be examinedin a later section.

The phase of the HR output can be rewritten as

'Id(t) = 4 1(t) - 4'o(t) = 2fdt + ad + Od(t),

Ve(t) = [(KdVrVd)/2][sin4(t) + n'(t)],

where

n'(t) = [1/(Vd)I[nC(t) sinOr(t) + n8 (t) cosOr(t)],

4(t) = r(t) - 'd(t) = r(t) - Od(t).

(22)

(23)

(24)

The variable +(t) is called the loop phase error. As seenin the Eq. (23) the noise due to the HR is independentof the loop phase error, in contrast to classical PLLconfigurations.8

Referring again to Fig. 2, the error signal is nowpassed through a loop filter with impulse response f (t),which gives a filtered error voltage of

Vf(t) = [(KdVrVd)/2]sino(t) + n'(t) * f(t),

where * denotes convolution.

(25)

(15)

where

fd = flq - foq,

d = 01 - k0.

The phase-modulated portion is

Ct -

-;d ° {d

I B(j2 ,rf 2

I- Bo- -_

f(H,)

Fig. 3. Heterodyne receiver bandpass filter transfer function.

(16) B. Loop Equation

(17) in With a preliminary discussion of the signals presentin Fig. 2 now presented, we can complete the derivationof an equation for the total loop phase error. We notethat the filtered error voltage is next fed back into thefrequency modulator of the modulated laser changingits output frequency as described in Eq. (9). The in-stantaneous frequency of the VCO output is, from Eqs.(15) and (18),

Td(t) = 2 d + Kmvf(t)] + d(t ). (26)

Substituting into this expression the results derived forfiltered error signal, Eq. (25), gives

cd(t) = 2irfd+ K[sin4'(t) + n'(t)] * f(t) + Yd(t), (27)

where


l . . . . . .

(ij(t = 2rVi, + K,,,vf(t)] + �,(t), (9)

Finally, using Eq. (24), the nonlinear stochastic in-tegro-differential equation for the loop phase error is

+(t) = br(t) - K[sin4(t) + n'(t)] * f(t) - y(t). (29)

This equation will be linearized by assuming that +p(t)is small (less than 1 rad) so that sin+/(t) b(t). Addi-tionally, converting this equation into the frequencydomain by taking the Laplace transform yields8

(s) = or() - d() - KF(S) [(s) + N'(s)]. (30)

(It is assumed that the transform of the sample func-tions of the random processes exists.) This linearizedversion of the integro-differential equation is graphi-cally depicted in Fig. 4.

Of primary interest in the loop equation is the laserdifferential phase 4bd(t). Since d (t) = 2fdt + 'kd +0

d (t), it is sufficient to know only the modulated phaseOd(t). Substituting Or(s) - Od(s) for * (s) [from Eq.(24)] and solving for Od (s) give

Od(s) = H(s)[N'(s) + Or(s)] + [1-H(s)Ird(s), (31)

where

sKF(s)H s) + KF(s) (32)

H(s) is the closed-loop transfer function. From Eq. (3),the transform of the reference-modulated phase is

N' (e I d(s)

Fig. 4. Linear two-laser phase-lock loop model.

or(s) = z(s) + r(s), (33)

and from Eq. (19)

rd(S) = ri(s) - ro(s). (34)

Substituting these into the above gives

Od (s) = H(s)[N'(s) + Z(s) + r(s)I + [1 - H(s)][ri(s) - ro(s)].(35)

The laser differential phase error can thus be obtainedby passing the various noise, instability, and controlprocesses through a set of linear filters. Equation (35)is shown graphically in Fig. 5 for both Laplace and timedomains. Solving Eq. (30) for 'I(s) yields

'I(s) = [1 - H(s)][Or(s)- rd(s)] - H(s)N'(s). (36)

This is the loop phase error between the laser differ-ential phase and the desired reference phase. It willbecome important when considering the modulatedsteady-state response of the system in a later section.

With the loop response defined in terms of the loopequation, or equivalently a parallel set of filters, anevaluation of its performance can now be made. Tosimplify the problem, an equivalent model for H(s), theclosed-loop transfer function, will first be developed.

C. Noise Equivalent Bandwidth of the Closed-LoopTransfer Function

The closed-loop transfer function is from Eq. (32)

H(s) = KF(s)s + KF(s) (32)

For any loop filter F(s) considered here, we note that

IH(0)12= 1. (37)

The noise equivalent bandwidth (NEB) of H(s) is de-fined as an ideal low-pass filter (LPF) with magnitudeIH(0) 12 and one-sided bandwidth WH whose area underthe curve is the same as the area under IH(s) 12 acrossthe entire spectrum.8 That is,

NOISE I rI - Is)

N '(C R O -CONTROL ) Z4st

NOISE

CONTROL )

WH = 4-j SJ IH(s)I 2ds,

or in Fourier notation,

WH = f IH(j27rf)12 df.

(38)

(39)

This bandwidth is graphically depicted in Fig. 6.The exact size of WH will of course depend on the

exact nature of F(s). Although ideally F(s) is only theloop filter which follows the PD in the loop (Fig. 2), thisassumes that no filtering occurs elsewhere in the loop.

- edIt) , I H(j2rf}l2

/__\1

H.. .T-WH 0 WH f(Hz

Fig. 5. Equivalent linear filter model of the two-laser phase-lock loopfor (a) Laplace and (b) time domains.

Fig. 6. Noise equivalent bandwidth model of the closed-loop transferfunction.


C It)

K = 7KKdKrKd. (28)

However, realistically, F(s) includes contributions notonly from the designed loop filter, but from every othercomponent in the loop (e.g., the response time of thelaser phase modulator and the HR BPF); thus thecomponent with the narrowest bandpass response willset the maximum limit on the value of WH. In thismodel, the two dominant filtering sources are the de-signed filter and the HR BPF, the narrower of whichwill limit WH.

An important measure of the steady-state perfor-mance of the phase-lock control loop is a second mo-ment description of the laser differential phase Od (t).To find the mean and variance, the statistics of thenoises and instabilities and the nature of the controlmodulation must be known. A first and second mo-ment description of the noise and instabilities is thusdiscussed next.

I1. Statistical Description of Loop Noise Sources

There are three sources of noise preventing an idealsteady value of laser differential phase. They are theloop reference phase instabilities, the individual laserphase instabilities, and the heterodyne receiver noise.A second moment model for each will be presented inthis section.

A. Loop Input Reference Instabilities

As defined in Eq. (3) the phase modulation portion0 r(t) of the loop input reference phase consists of adeterministic control modulation component t(t) anda random instability component Yr (t). It will be as-sumed that the phase instability is stationary, zeromean, with a constant power spectral density. Thus

metric about fd, the spectrum of the quadrature com-ponents of n(t) is

Sn,(f) = Sn.(f) = hCAdR

= 0, elsewhere.

Il1 2

(44)

The loop phase noise due to the HR is given by Eq.(23). Since n,(t) and n,(t) are the quadrature com-ponents of the zero mean process n(t),

E[n(t)] = E[n,(t)] = E[n,(t)] = E[n'(t)] = 0. (45)

Using Eq. (23) plus the fact that the cross-correlationbetween n, (t) and n, (t) is zero [since Sn (f) is symmetricabout fd], we obtain the autocorrelation of n'(t),

(46)

(47)

Rn(t2, t)AE[n'(t2)n'(tj)]

= [1/(Vd)]Rn (O)E [cosAOr (-)I,

where

1r = t2- tl (48)

AO,(T) = Or(t2) - r(t)

= (t2) - (t1) + Yr(t2) - -Yr(tl). (49)

This result is strictly not stationary. However, it willbe assumed that the input reference is stable relativeto the rest of the system so that 'yr(t) 0. It is furtherassumed that the desired modulation (t) will notchange as fast as (will have a larger correlation timethan) the other system processes. So for time intervalsof interest it follows that8

AOr(T) 0 0 (50)

so that

E[Or(t)] = (t,

SO(f) = Sr(f) = Nr.

(40)

(41)

R.,(t 2,t1 ) = R.,(r) = [1/(Vd)]R.J(r)

or finally

B. Heterodyne Receiver Noise

The power spectral density of the output noise of anHR is well known and will simply be stated here. Theresults are conditioned on the previous assumptionsmade for the HR plus the following additional as-sumptions:

(1) For power or energy calculations only,

fo = fOq flq. (42)

(2) There is negligible excess noise. (An excessnoise term arises strictly from the random portion of thefields.5) With all the previous assumptions, the outputof an HR is given by Eq. (12), where n(t) is a zero-meanGaussian random process with spectrum,5

q2nfdA2R2

Sn (f) = 2hfoCo1

fd_-- If fd +B222

= 0, elsewhere. (43)

The variable Bo1 is the bandwidth of the HR bandpassfilter and 1/(C21) its magnitude. The other variablesare as defined earlier (see Fig. 3).

It is convenient to use a narrowband noise model forHR noise [see Eq. (20)]. Since this spectrum is sym-

Snf) [(V2)]S.c(f)J (52)

which from Eqs. (44) and (13} reduces to

S.'(f) = (hfo)I(AdA), If I - [(Bol)12].

= 0 elsewhere. (53)

The power spectrum of the loop noise due to the HR isthen proportional to the energy of a detected photonhfo.

C. Laser Phase Instabilities

The last type of noise limiting the performance of thetwo-laser phase-lock loop is the instability of the phase

SPECTRUMAALZER MEASUREDTEST LASER DETECTOR LASER

FIELDMODULATOR STANDARD Y III

LASER l I ~~~~~~~~~~~DISCRIMINATORI

STABILIZATION LOOP SPECTRUM INSTANTANEOUSSTABILIZATION LOOP I , F ANALYZER FREQUENCY

Fig. 7. Typical arrangement for measuring the spectra of a laser fieldand its instantaneous frequency.


(51)

of each laser. This section describes the power spec-trum of laser phase in terms of the laser linewidth(which is measurable from the laser output field).

The output field of a laser can be represented by'

6(t) = A cos[27rfot + Oo + y(t)], (54)

where fo and 00 are, respectively, the constant frequencyand phase of the field, and y(t) is a Gaussian randomprocess. Arbitrarily, y(t) will be given a mean of zeroby including any cor tant mean term in (2-7rfot + 4o).

An assumption about the stationarity of the phaseinstability y(t) is now made. Laser electric fields be-cause of their inherently high frequencies are typicallymeasured as a beat or relative field between two lasersas shown in Fig. 7.9-11 For that type of configuration,where the measurements are made with respect to astandard laser whose instabilities are negligible com-pared with those of the laser to be tested, it has beenshown that y(t) is a stationary Gaussian random pro-cess.9-'4 Such a measurement technique is assumed forall fields used in this paper. Thus, any statementsherein regarding laser fields or spectra will be inter-prete,' as the respective measured quantities. Thisinterpretation is consistent with the types of measure-ments required by the two-laser phase-lock loop we arediscussing.

Using the assumption that y(t) is a zero-mean, sta-tionary, Gaussian random process, a relationship be-tween the field spectrum and the phase instability hasbeen derived.9-' 5 The results will simply be statedhere.

The random phase instability y(t) of lasers is com-posed of two components.' 0 One is an external con-tribution Ye (t) due to acoustic noise, structural vibra-tions, plasma oscillations, thermal and pressure drifts,and other environmental disturbances. (It is also calledtechnical noise10 or extraneous modulation.12"13) Theother element is a quantum contribution -Yq (t), usuallymuch weaker. Quantum noise is the quantum-me-chanical or statistical limit of phase fluctuations andthus sets the minimum size of the phase spectrum.

Assuming that the form of the laser field spectrumdue to quantum noise alone is Lorentzian with linewidthAfq, the spectrum of the phase fluctuation due toquantum noise is

Syq(f) = (fq)/(2jrf2). (55)

The spectrum of the phase fluctuation due to externalnoise sources is

S (f) = Sq(f) + Se(f) R f1 [ 2C 2f, ISy~f)S~yqf)+Sye -2rf

2 I (f2 + f2)j (57)

Equation (57) is the phase spectrum of a single laser,but as defined in Eq. (19), the laser differential phaseinstability Yd (t) is due to two lasers, the reference laserand the modulated laser. Since it was previously as-sumed that the laser instabilities are zero mean,

E[Yd(t)] = 0. (58)

It will further be assumed that the instabilities in eachlaser are statistically independent, which is reasonableif they are in their own thermally and acoustically iso-lated cavities. Using this assumption, the spectrum ofthe difference is the sum of the individual spectra.Thus from Eq. (57) we obtained

SId(f) = (aoo+ l) of+0o IfOd1f2 j fo2) 2~ f~l + f2j (59)

where

a = (Afq)/27r

a =e 2 Af2

7r 8r n2

(60)

(61)

With the statistical nature of the reference instability,the lasers's instabilities, and the HR noise now specified,the performance of the laser differential phase from Eq.(35) can now be determined.

IV. Loop Performance

The performance of the control loop will be measuredin terms of the steady-state (SS) mean and variance ofthe laser differential phase (bd(t). This will be donefirst with no modulation, that is, (t) = 0. For the no-modulation case the noise equivalent bandwidth

WHmin,

which results in the minimum value of phase variance,will be determined. Next the SS mean will be deter-mined in the modulated case for different modulationschemes and filter types. The transient system re-sponse for the modulated case will also be examined.

A. No Modulation: Steady-State Mean and Variance

The laser differential phase error is given by Eq. (35).Using the expression for the expected value of laserphase noise [Eq. (58)], HR noise [Eq. (45)], loop inputreference phase modulation [Eq. (40)], and the fact thatt(t) = 0 (no modulation), the mean of the laser differ-ential phase error is

E[Od(t)] = E[Od(s)] = 0.

S-ye(f) = (fe2)/[7rf2(fc2+f2)], (56)

where fc is an arbitrarily small cutoff frequency. Thisresult assumes that the form of the laser field spectrumdue to external noise alone is Gaussian in shape with astandard deviation (rms linewidth) of o(e or equivalentlya linewidth of Afe,

The observed field spectrum is the sum of the quan-tum and external spectrum results. Adding Eqs. (55)and (56) gives

(62)

Thus the constant phase between the lasers from Eqs.(2) and (15) is

E[Fd(t)] = E [4r(t) = 2fdt + O/Id. (63)

That is, when the system is unmodulated, the phasedifference between the two lasers is equal to the loopreference phase. This, of course, is the desired re-sult.

The variance is a measure of fluctuations away fromthe desired mean. Equation (35) shows that the phase

3170 APPLIED OPTICS / Vol. 18, No. 18/ 15 September 1979

Yr

(a)

S f)

(b)

I Hhj2arf))12

WH EBo/2 f(Hz)

I\~ _-H(i2lTf 2

WH f(Hz)

Fig. 8. Graphical representation of contributions to the laser dif-ferential phase error variance due to (a) the heterodyne receiver and

input reference noise and (b) the laser's phase instabilities.

error is obtained by passing HR noise and referencephase noise through the filter H(s) and the laser insta-bilities through the filter [1 - H(s)]. The referencemodulation in this case is (t) = 0 [thus Z(s) = 0].Using the NEB of H(s) derived earlier and assumingthat all the noise processes are statistically independent,the variance of the phase error is

a2 = 2 f [S.,(f) + S,,(f)]df + 2 S,()df. (64)

The two terms of this integral are illustrated in Fig. 8to aid in showing the effect of WH on the resulting or

2.

The total cross-hatched area on the two graphs is thevariance. In Fig. 8(a), as previously noted, B0 1/2 cannever be less than WH. Thus from the point of view ofthe closed-loop transfer function, the HR noise n'(t) isnearly white.

Using the values for the spectral densities of Eqs. (41),(53), and (59) and integrating reduces Eq. (64) to

C2 = 2N W +2hfoWH 2 (ao + al) 2 fi _r

AdA1 WH fel IWH 2

+ tan1(H) +200 [ fo -+ tank (WH) .fc co WI 2 fco

C3 = [2(o + 01)]/3. (71)

The variable C1 is proportional to the energy of a de-tected photon hfo; C2 is proportional to /Afq, the sum ofthe quantum-limited laser field linewidths; and C3 isproportional to Afe, the sum of the laser field linewidthsdue to external effects. The conflicting effects of theclosed-loop bandwidth of the terms in the phase errorvariance indicate that some optimization of WH is de-sirable.

To optimize the loop SS performance, the varianceof the laser differential phase must be minimized. Theexpression for phase error variance is given by Eq. (68),and a plot showing the effect of the three different termsis given in Fig. 9. To find WH such that o-2 is minimal,we solve for the stationary point of Eq. (68) with respectto WH, yielding

WH min={C + Cs~ 2

+ C _~- JF(C32 C2 C 3 1V1/2] 1/3 C1 [(J (C3]1(72)

If C 2 C3, which is typically the case, since for mostlasers Afe >> \fq,

8 Eq. (72) simplifies to

WH C1) (73)

It is obvious from the examination of Fig. 9 that if theactual bandwidth is less than WH mn, the variance islimited by the laser phase instabilities (i.e., laser phasenoise dominates). If WH is greater than WHmin, thevariance is dominated by detector noise and input ref-erence instabilities. It will be shown that in manyconceivable systems,

WH < WHmin

because WHmin is very large, and thus the variance islimited by the phase instabilities of the lasers and isindependent of the HR measurement noise (see thenumerical example in a later section).

(65)

From the section on laser phase instabilities, fco and fiare arbitrarily small (experimentally less than 1 Hz9 ),so it is reasonable to assume that

WH/fCO >> 1; WH/fl >> 1- (66)

Using that result, the tan-' terms can be approximatedby the first three terms of their power series as

ta1,WH r fc ffc 2 WH 3 WH

,r2

c^W + W., - ,

(67)

Substituting this approximation back into Eq. (65) fi-nally yields an expression for the laser differential phaseerror variance of

a2 = C1WH + + ,

WH W~~2 (68)

where

(69)C1 = [(2hfo)/(nAdA)I) + 2Nr,

C2 = 2(ao + al), (70)Fig. 9. Closed-loop bandwidth vs laser differential phase error

variance.


- - l

SOO(f |

ID5

I"2

(r2

,. ...

w^. ,"..,.

Table 1. Basic Parameters for First-, Second-, and Third-Order Loop Filters

Loop filter Closed-loop Noisetransfer transfer equivalent

Loop function function bandwidthorder F(s) H(s) 1 - H(s) WH (Hz)

1 1 K s Ks+K s+K 4

2 1 +a K(s+a) s2K+a

s S2 +K(s+a) S2 +K(s+a) 43 1 + + b K(s2 + as + b) s3 K(aK + a2 -b)

s s2 s3 +K(s2+as+b) s3 +K(s2 +as+b) 4(aK-b)

Table II. Control Modulation Schemes In Temporal and LaplaceRepresentation

Modulation Time Laplacescheme domain representation

Step in phase A/ru(t) AO5

Step in frequency 27rAftu(t) 2rAf(ramp in phase) s2

Ramp in frequency 2U(t)(phase 2 S3acceleration)

B. Modulated Reference SignalsThe real power of a phase-lock control loop is its

ability to track a modulated reference input. The loopfilter F(s), the principal determinant of the loop'smodulated performance, is typically optimized for thetype of modulation desired. This section examines theperformance of the loop for some specified modulationschemes and filter types.

As stated earlier, F(s) in Fig. 2 actually includes allfiltering by any component in the loop. It will be as-sumed that all these components are controllable, andthus F(s) can be constructed as desired.

There are several loop filters that are commonly usedin phase-lock loops.1 6 These are the first-, second-, andthird-order filters and are listed in Table I. The ex-pressions for H(s) and WH were calculated using Eqs.(32) and (39), respectively.

The modulation schemes to be examined are listedin Table II in both Laplace and time domains. (Sincethe modulation schemes are causal, they are uniquetransform pairs.) The variable W(t) from Eq. (3) is thecontrol phase modulation of the loop reference phaseand is in units of radians.

Once the filter and modulation scheme are known,the mean steady-state response for the laser differentialphase error can be determined. For this determination,the loop phase error 4'(t) becomes important. From Eq.(24),

h/(t) = Or(t) - d(t), (24)

and using Eq. (40) [and the assumption that Or(t) nd

d (t) are statistically independent] and exchangingterms gives

E[Od(t)] = (t) - E[p(t)]. (74)

Thus the mean steady-state value for the laser differ-ential phase error can be found by knowing the meansteady-state value for the loop phase error and thecontrol modulation of the reference phase. For in-stance, if (t) is a ramp in frequency (Dt2)/2, and theloop is second order, using Eq. (36) and the final valuetheorem, the mean steady-state loop phase is

lim E[l(t)] = lim sE[4(s)] = lim s ( 2 K),S S ra+daS

Mq*()]hs = D/aK (rad).

(75)

(76)

Now utilizing Eq. (74) the mean steady-state laser dif-ferential phase error is

E[Od(t)]ss = [(Dt2)/2] - [D/(aK)]. (77)

That is, the loop will follow the frequency ramp (ac-celerating phase), but there will be a constant phaseerror offset of D/aK rad between the two lasers. Thisprocedure is followed for all combinations of filter or-ders and reference phase modulations schemes, and theresults are summarized in Table III. Only the third-order loop will follow accelerating phase with no con-stant phase offset. (The first-order loop has a phaseand frequency offset when trying to track acceleratingphase and is not calculated.)

C. Dynamic ResponseThe loop filter, as shown previously, is important in

determining the loop performance. First, the filterdetermines the closed-loop transfer function and re-sulting loop bandwidth (Table I). This in turn influ-ences the steady-state variance of the laser differentialphase error caused by the laser and input reference in-stabilities and detector noise. Second, the filter de-termines the steady-state response of the loop to variousmodulation patterns (Table III). In addition to thesesteady-state performances, the loop filter also deter-mines the transient or dynamic response of the loop.

The most versatile filter, as seen in Table III, is theone resulting in a third-order loop. The exact behaviorof a third-order loop is complex and difficult to obtain.16However, it can be roughly contrasted with the sec-ond-order loop whose behavior is relatively well known.


When a third-order loop is not initially locked to theloop reference phase, it is not as stable in its ability toacquire lock as the second-order loop. The second-order results could in this case act as an upper bound(best case) on the third-order loop performance. 1 7

Once locked, however, the third-order can outperformthe second-order loop, most notably in its ability tofollow accelerating phase (frequency ramp) with nosteady-state phase error. In this case, second-orderloop performance acts as a rough lower bound onthird-order loop performance. More specific resultsmust be obtained by computer simulation.

Some important performance equations for a sec-ond-order loop in a noiseless environment are listed inTable IV. Since they are valid for a noiseless environ-ment, they represent the best possible performance ina noisy environment.

Table Ill. Mean Steady-State Value of the Laser Differential Phase forDifferent Loop Orders and Modulation Schemes

E[Od(t)]s(rads) ¢(t) (rads)

Loop Dt 2

orders AO b(t) 2n-Aftu(t) 2 u(t)

1 AI 2Aft - 27r/f -aK

2 AO/ 2irAft Dt 2 D2 aK

3 A/I 2irAft Dt22

a The first-order loop has a phase and frequency offset when tryingto track accelerating phase and is not calculated.

These results are in terms of radian frequency w(rad/sec) instead of f (Hz), since w is typically used inthe literature when dealing with dynamic feedbacksystem behavior. (The general relationship is w = 27rf.)Some results utilize two loop-parameter dependentvariables called the natural frequency won and thedamping factor t. They are defined as follows17 :

(78)

(79)

co = (aK) 112,

= /2(K/a)112 = [KI(2w.~)].

The natural frequency is the system phase error oscil-lation frequency in response to a step in the input ref-erence frequency when there is no damping (Q = 0).The damping factor is a measure of how quickly theoscillations attenuate as the loop acquires lock. Manyof the expressions for the same parameter vary de-pending on the reference, so all sources are also listedin the table.

Two two main areas of transient behavior are trackingand acquisition. Tracking parameters assume that theloop is initially locked to the input reference phase.The hold-in range ACOH is the maximum amount thereference frequency can deviate from the VCO quies-cent frequency (HR output frequency) and have theloop remain locked. It is theoretically infinite for bothsecond- and third-order loops.17 Another importanttracking parameter is the frequency step limit Aw,,. Itis the maximum step in reference frequency belowwhich the loop will not skip cycles. Cycle skipping oc-curs when the difference between the reference andVCO frequencies is large enough to create a differentialfrequency (beat frequency) at the PD output. Finally,the maximum reference frequency ramp that the loopwill follow without dropping out of lock is called themaximum tracking sweep rate Dmax (rad/sec 2 ).

Table IV. Equations For Dynamic Performance of a Second-Order Phase-Lock Loop in the Absence of Noise

Tracking

Aw,, = 1.8w(Q + 1)Frequency step limit

Hold-in range

Maximum trackingsweep rate

= 2W2n (third-order loop)

Acquisition

AWL = 2WnLock-in frequency

Lock-in time

Pull-in frequency

Pull-in time

Maximum acquisitionsweep rate

= 2Wn ( + 0.6)TL = 15IWH

= r/n

Aw, = N

T, = (AW)2/2Mn

Rmax = n/2

= a(4WH - a)


AWH = -

Dmax = 2

rad

secradsecradsec2

Ref. 7

Ref. 17

Refs. 7, 16, 17

Refs. 16, 17

radsec

sec

radsec

sec

radsec2

Ref. 17

Ref. 17Ref. 17Ref. 7

Refs. 16, 17

Refs. 7, 17

Refs. 7, 17,

Ref. 16

Acquisition parameters assume that the loop is notinitially locked. The pull-in frequency Awp is themaximum amount the reference frequency can deviatefrom the VCO quiescent frequency and still permit theloop to converge to a locked state. This, like the hold-inrange, is also theoretically infinite. As the loop con-verges on the reference frequency, the VCO frequencyfinally gets close enough to the reference frequency sothat the loop stops skipping cycles and settles into thelocked state. The range within which the loop ceasesskipping cycles after pull-in is called the lock-in fre-quency AWL. The associated times for the loop topull-in and lock-in to a step change in reference fre-quency are, respectively, Tp and TL. The pull-in timeTp includes only the time that the loop cycle-skips.The lock-in time TL is the time from the last cycle-skipuntil lock is acquired. The tracking frequency steplimit Aws is often equated with the acquisition lock-infrequency AWL, since the basic criterion in the defini-tion of each is the maximum frequency difference belowwhich the loop does not skip cycles. It does not matterif this frequency difference occurs as a result of an inputreference frequency step with the loop initially lockedor as a result of pull-in with the loop attempting to ac-quire lock. This can readily be seen in the similarity ofthe expressions for AW, and AWL in Table IV. Thelock-in time TL thus applies to either case. Finally, themaximum frequency ramp that the loop can sweep andstill acquire lock is called the maximum acquisitionsweep rate Rmax (rad/sec 2 ).

It is important to note that these transient perfor-mance criteria may dominate simple noise consider-ations. If, for example, WH was chosen to minimize thephase error variance [Eq. (72)], it might not be largeenough to meet, say, the acquisition sweep rate de-manded of the system.

The steady-state mean and variance of 1?d (t) due tounwanted fluctuations (detector noise, laser instabili-ties, and reference instabilities) has been examinedwhen the loop input reference was not modulated. Thesteady-state and dynamic changes in Id (t) with variousmodulation schemes and filter designs were also ex-amined. The effect of any given individual parameter[e.g., F(s), Yd (t), n(t), (t)] on the over-all laser dif-ferential phase &Id(t) [or equivalently the modulatedportion Od(t)] can most readily be understood by ex-amination of Fig. 5 [Eq. (35)]. A design example for apotentially realizable two-laser control system will nowbe presented to illustrate the effects of the various pa-rameters on the performance of the loop.

V. Numerical Design ExampleThe specific system to be examined here pairs a

highly stable CO2 reference laser with a highly adjust-able CO2 waveguide laser. CO2 lasers were chosen be-cause of their widespread use. The waveguide laser waschosen because it has a very wide gain bandwidth andthus can operate over a wide range of frequenciesaround the centerline frequency.18

The transition of interest in this example is X = 10.6gim, which corresponds to

fo = 2.83 X 1013 Hz. (80)

The reference laser has the following characteristics:

Afqo = 6.28 X 10-4 Hz;

Afea = 7.77 X 103 Hz.

(81)

(82)

The waveguide laser is assumed to have the followingcharacteristics:

Afqi = 2.00 X 10-5 Hz;

Afel = 2.12 X 107 Hz.

(83)

(84)

Also, it has a usable gain linewidth (range over which theoutput frequency is adjustable and the laser power outdoes not drop below half its maximum value) of 7.00 X108 Hz. As is true for most lasers, the linewidths of bothlasers in this example are dominated by external dis-turbances, not quantum limitations. With the quan-tum and external linewidths for both lasers known, Eqs.(60), (61), (70), and (71) can be used to find

C2 = 2.06 X 10-4 sec',

C3 = 1.72 X 1013 sec-2 .

(85)

(86)

Clearly, Afei dominates in the term C3. The HR sam-ples 1 X 10-3 W (= AdA2) and has a quantum efficiencyof q = 0.5. The reference is assumed to be very stableso that Nr 0. Using Eq. (69) it is found that

C1 = 7.51 X 1i-17 sec. (87)

If the HR BPF is designed to allow the frequency torange over the entire usable gain bandwidth, Bo, mustbe greater than 7.00 X 108 Hz. Arbitrarily, Bo, is pickedto be equal to that amount:

Bo, = 7.00 X 108 Hz. (88)

If it is assumed that the loop filter does not limit thebandpass of the loop, WH is at most Bo1/2. The NEBWH of the closed-loop transfer function reasonablymust be somewhat less than that and is arbitrarily as-sumed to be

WH = 4.15 X 107 Hz. (89)

Using the results for C,, C2, C3, and WH the laser dif-ferential phase variance from Eq. (68) is

a2 = 9.99 X 10-3 rad2

or standard deviation of

= 0.100 rad = 5.75 deg.

(90)

(91)

This is fairly good performance. However, if WH isdecreased by only 1 order of magnitude to 4.15 X 106 Hz,the standard phase deviation will increase to = 1 rad(57.3 deg). The minimum variance bandwidth usingEq. (73) is

WHmin = 7.7 X 109 Hz. (92)

This is more than 2 orders of magnitude greater thanthe actual WH. Thus the system is dominated by laserphase noise. The variance at

WHmin

is

3174 APPLIED OPTICS / Vol. 18, No. 18/ 15 September 1979

a2 in = 8.68 X10-7 rad 2 (93)

or

K-min = 0.932 mrad (0.053 deg), (94)

which is the best noise variance this particular loop canattain.

A second-order loop will be used in this example. Acommonly employed compromise between stability andspeed of response for a second-order system is to let16

= 0.707. (95)

Using the expression for WH of a second-order loopfrom Table I and Eq. (79) to solve for K gives

K = 1.11 X 108 sec 1 . (96)

Again using Eq. (79) gives

a = 5.53 X 107 sec- 1 , (97)

and Eq. (78) results in

w = 7.83 X 107 rad/sec. (98)

The transient performance of the loop is found byusing the results of Table IV. The tracking frequencystep limit is

Aws = 2.41 X 108 rad/sec = 3.83 X 107 Hz. (99)

The maximum tracking sweep rate is

Dmax = 6.13 X 1015 rad/sec2 = 9.76 X 1014 sec-2, (100)

while the acquisition lock-in frequency is

AWL = 2.05 X 108 rad/sec = 3.26 X 107 Hz. (101)

As long as the changes in the reference input frequencyare less than AcoL (A 5co), the loop will have a lock-intime of no more than

TL = 40 nsec. (102)

If, however, the reference frequency changes by the fullamount of usable laser gain linewidth, 7.00 X 108 Hz, thepull-in time is

Tp = 722 nsec, (103)

which is in addition to the lock-in time, giving a maxi-mum time to lock of 762 nsec for the system. This isvery useful and could allow modulation on the order ofmilliseconds with the loop still in lock. The maximumacquisition sweep rate is

Rmax = 3.07 X 1015 rad/sec 2 = 4.88 X 1014 sec- 2.

There is, however, a steady-state phase error if thissecond-order loop tracks a frequency ramp. If, for ex-ample, the tracking sweep rate is at its maximum al-lowable value for the loop (Dmax = 6.13 X 1015 rad/sec 2 ),the steady-state phase error from Table III is 1 rad (57.3deg). A third-order loop must be used to eliminate thiserror.

This example has shown that even if one laser of thepair has a large field spectrum linewidth (-2.1 X 107

Hz), the loop can perform very well, with a standard

deviation between the two-laser phases of less than 6deg and a maximum locking time of about 762 nsec. Itis important to note that the system performance islimited by the laser phase instabilities (expecially thatof the waveguide laser) and not by detector noise. Thekey elements in improving performance are to make WHas large as possible and to make the lasers as stable aspossible.

VI. Conclusions

The basic result of this paper, given by Eq. (68), is thefundamental limitation in performance, in terms of thevariance of the differential phase, of a PLL designed toset the differential phase between the output fields oftwo lasers to a desired value. The fundamental noisesources are the HR noise and the laser phase instabili-ties, which are oppositely affected by the closed-loopbandwidth. As the example showed, the bandwidthmust be extremely large before HR noise becomes thedominant noise source. Thus, in most cases, it is likelythat the laser phase instabilities will ultimately limitloop performance. Other measures of performancewhen the loop is modulated were also discussed.

Portions of this work were presented at the 1978Annual Meeting of the Optical Society of America, 30October-3 November 1978, San Francisco.

References1. A. Yariv, Quantum Electronics (Wiley, New York, 1975).2. C. L. Hayes and L. M. Laughman, Appl. Opt. 16, 263 (1977).3. H. E. Hagemeier and S. R. Robinson, Appl. Opt. 18, 270

(1979).4. A. Yariv and M. M. Caton, IEEE J. Quantum Electron. QE-10,

509 (1974).5. W. K. Pratt, Laser Communication Systems (Wiley, New York,

1969).6. R. E. Ziemer and W. H. Tranter, Principles of Communications:

Systems, Modulation and Noise (Houghton Mifflin, Boston,1976).

7. F. M. Gardner, Phaselock Techniques (Wiley, New York,1966).

8. W. C. Lindsey, Synchronization Systems in Communication andControl (Prentice-Hall, Englewood Cliffs, N.J., 1972).

9. A. E. Siegman et al., IEEE J. Quantum Electron. QE-3, 180(1967).

10. A. E. Siegman and R. Arrathoon, Phys. Rev. Lett. 20, 901(1968).

11. K. R. Manes and A. E. Siegman, Phys. Rev. A 4, 373 (1971).12. E. D. Hinkley and C. Freed, Phys. Rev. Lett. 23, 277 (1969).13. T. S. Jaseja et al., Phys. Rev. Lett. 10, 165 (1963).14. B. Picinbono, Phys. Rev. A 4, 2398 (1971).15. J. B. Armor, Jr., "Phase-Lock Control Considerations for Mul-

tiple, Coherently Combined Lasers," Master's Thesis, Air ForceInstitute of Technology, Wright-Patterson Air Force Base, Ohio,GEO/EE/77D-2 (December 1977).

16. A. J. Viterbi, Principles of Coherent Communication(McGraw-Hill, New York, 1966).

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Phase-lock control considerations for coherently combined lasers