general linear differential interferometers

346 J. Opt. Soc. Am. A/Vol. 12, No. 2 /February 1995 R. W. C. Vance and R. Barrow

General linear differential interferometers

R. W. C. Vance and R. Barrow

BHP Research, P.O. Box 188, Wallsend, New South Wales 2287, Australia

Received January 31, 1994; revised manuscript received September 13, 1994; accepted September 20, 1994

We investigate the conditions under which an interferometer’s output is a function only of the differencebetween two optical lengths. In certain applications, such differential behavior, characteristic of the classicalMichelson and Mach–Zehnder schemes [M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1980)],improves resilience to source frequency jitter. The question of whether systems of higher sensitivity thanthe Mach–Zehnder or Michelson schemes can have differential behavior is addressed, and it is proved that nolinear differential scheme can have a sensitivity greater than that of the classical Michelson interferometer.It is also proved that the scattering matrix elements for a linear N port cannot be differential functions of theoptical lengths within the network.

1. INTRODUCTION

Resonant interferometers such as the Fabry–Perot cav-ity or the fiber ring resonator1 can have outstandingsensitivities, and therefore great interest has recentlybeen shown in their use in attempts to detect theoret-ically foreseen but hitherto experimentally unobservedgravitational waves.2,3 However, in certain applicationsand with certain kinds of system-noise sources to bedescribed below, they can impose much harsher tempo-ral coherence requirements on their source lasers thando less sensitive two-beam interferometers such as theMichelson or Mach–Zehnder interferometers. In suchcases, the two-beam interferometer’s milder coherencerequirement arises because the output intensity is a func-tion only of the difference between two optical lengths.In this paper therefore we investigate the question ofwhether the two-beam interferometer is the only kindwith such differential behavior or whether a higher-sensitivity, resonant, differential scheme may be possible.

As an example, we are interested in the application ofinterferometers to magnetic field sensing for explorationof geophysical minerals.4 Hence the source laser must berugged, in field deployable, and highly portable; the centerfrequency of such lasers is not highly stable and wandersover an appreciable range, owing to environmental effectssuch as vibration and temperature drift. Such shifts inthe center frequency give rise to changes in the phasedelays imposed by the sensing optical lengths withinan interferometric system. These changes are indistin-guishable from those induced by the sensed magneticfield; thus serious, irreversible signal contamination canarise.

There are four methods of accommodating such jitter-noise:

• Shorten the sensing delays, thus rendering themless susceptible to source-frequency jitter.

• Frequency stabilize the source-laser.• Implement feedback control of the optical sensing

delay so as to counteract the effect of source frequencyjitter.

0740-3232/95/020346-08$06.00

• Use a differential interferometer with well-balancedoptical lengths, so that any source-frequency jitter inducesthe same phase change in both optical paths, thus notaffecting the interferometer’s output intensity.

In the current application, active control of the sensinglength is highly undesirable, since electrically active com-pensation systems can give rise to magnetic noise manyorders of magnitude greater than the sensed magneticfield. The magnetic-field-induced delay changes are pro-portional to the sensing length, so the first option is alsoundesirable. A frequency-stabilized source is difficult touse in the field; thus we are constrained to the last optionin the present application.

Therefore in certain applications differential interfer-ometer behavior can be highly advantageous. In manyapplications such as those in Refs. 2 and 3, however,the extreme sensitivity afforded by resonant behavioris essential, and since all contemporary resonant inter-ferometric systems are nondifferential, the engineeringproblems of source stabilization or closed-loop sensing-length control must be properly overcome for full benefitto be derived from these systems. Early attempts toachieve high sensitivity in fiber-ring resonators wereoften thwarted by inadequate solution of these engineer-ing problems; in Ref. 5 a resonant ring with a finesseof 100 was studied, yet the system was only twice assensitive as a corresponding Mach–Zehnder interfer-ometer, owing to high jitter-induced noise. Indeed, theextreme sensitivity of resonators to frequency shifts hasled to their often being used as laser cavities or sensi-tive frequency discriminators.6 The dearth of resonant-interferometer examples in a major 1989 interferome-try conference7 suggests that the engineering problemsof source stabilization and sensing-length control arenot trivial.

In summary, for differential interferometers the source-frequency instability acts on a length difference, whichcan be extremely small even though the sensing lengthsthemselves can be extremely long to enhance sensitivityto the sensed quantity. This is not the case for the ringresonator, wherein source jitter acts upon the whole sens-

1995 Optical Society of America

R. W. C. Vance and R. Barrow Vol. 12, No. 2 /February 1995 /J. Opt. Soc. Am. A 347

ing length. It is therefore apparent that an interferome-ter that would both (1) achieve a high sensitivity throughresonant behavior and (2) operate differentially would bea highly desirable interferometric system in some applica-tions. For example, the recycling Michelson interferome-ter in Ref. 2 has two active optical paths, and an exampleof a resonant optical fiber ring resonator with separatesensing and reference paths can be found in Ref. 8. It isnatural to ask whether such systems can be configuredso as to make their outputs wholly differential functionsof the sensing and reference lengths. In the followingsections it will be shown that if an interferometer is dif-ferential, then its output intensity as a function of thepath-length difference has the same form as the functionsobtained for the Michelson and Mach–Zehnder interfer-ometers (as functions of their path-length difference); i.e,the classical Mach–Zehnder and Michelson interferome-ters can attain the same phase sensitivities as any dif-ferential interferometer, so that it is futile to attempt toincrease sensitivity through resonant behavior if differen-tial interferometer operation is essential.

A further application of these results is in planar inte-grated optics, in which it is important to engender in-sensitivity to temperature drifts. A simple method ofachieving this goal is to require a system’s behaviorto be a differential function of pairs of optical lengthswithin it and to ensure, by appropriate fabrication, thatthese lengths undergo similar temperature-drift-inducedstrains. In the light of our results on interferometers,the most general functional dependence of such a circuit’sbehavior on its internal lengths can be inferred.

In Section 2 a general notation for an interferometricsystem is set up; it is illustrated in Section 3 by formula-tion of a description of a resonant Michelson interferome-ter with light recycling.2 In Section 4 our general resultsabout linear differential interferometers are derived.

2. FORMALISM FOR A GENERALLINEAR OPTICAL NETWORKIn what follows we shall study the general optical networkthat is schematically shown in Fig. 1. It is assumed that

• The source laser and all light output from or inputto any of the ports 1–5 is perfectly monochromatic.

• The ports 0–5, connecting the network to the out-side world, are single moded, and therefore the electro-magnetic field at each of them is fully defined by thecomplex amplitudes aj and bj of the modes incident uponand scattered from each port, respectively.

• The network is linear.• The network is stable, that is, if it has gain, the

system’s free oscillations decay exponentially with time,and so all quantities b0, a1, b1, . . . , a5, b5 respond to anybounded input a0 in a bounded and casual way.

• There is no feedback of the output b5 into any partof the network and, therefore, in particular, a5 0 in thisanalysis.

No assumption of reciprocity is needed, and the networkis not necessarily passive, only stable. It should be notedthat the form of Fig. 1 can cope with all feedback of theoutput into the network despite the assumption other-

wise; if there is feedback, the boundaries of the generallinear six port in Fig. 1 can be extended to include itseffect inside the black box (at the center of the figure),leaving a new output that is not fed back in any way(Fig. 2). Note that multimoded optical ports can also beaccommodated by the a simple modification of the abovenotation: for an N-moded port, N separate single-modedports are assigned in the general model of Fig. 1, the fieldat the nth port fully defined by two complex amplitudesdescribing the incident and the scattered nth modes.

In a scattering-matrix notation9 a 6 3 6 scattering ma-trix, S6, describes the general linear, homogeneous rela-tionship between the incident and the scattered waves:26666666664

b0

b1

b2

b3

b4

b5

37777777775 S6

26666666664

ao

a1

a2

a3

a4

a5

37777777775or

b0

b

b5

SD

GF

a0

a

a5

5 ,

(1)

where F is a scalar, D a 4 3 1 vector, G a 1 3 4 vector,and S a 4 3 4 matrix. a and b are the vectors of incidentand scattered waves, respectively, at ports 1, 2, 3, and 4.The partitions left blank in Eq. (1) are irrelevant to thefollowing analysis since, by assumption, a5 0 and b0

has no effect on the source laser. Thus the only relevantequations are

b5 Fa0 1 GT a ,

b Da0 1 Sa . (2)

Here s dT stands for transpose. The relevant scattering-matrix partitions have the following physical meaningsin the absence of the links in Fig. 1. F, the feed-forwardconstant, is the ratio of the system output b5 to inputa0 when there are no waves incident upon ports 1–4.The driving vector D describes the amplitudes b Da0

of the waves launched from ports 1–4 in response to

Fig. 1. Notation for a general linear interferometer.


(a)

(b)Fig. 2. System remodeling to remove output feedback: (a)system with output feedback, (b) equivalent system withoutfeedback.

the system input a0. The gathering vector G defines theoutput b5 GT a arising from the incidence of waves aonto ports 1–4. Finally, the reduced scattering matrixS describes the waves b Sa scattered from ports 1–4in response to waves a incident upon these ports.

The links in Fig. 1 are assumed to obey (here b is thelink propagation constant) the following:

a1 exps2ibL1db2 ,

a2 exps2ibL1db1 ,

a3 exps2ibL2db4 ,

a4 exps2ibL2db3 (3)

or

b zcCszdda ,

where

Cszdd

2666640 zd 0 0zd 0 0 00 0 0 zd

21

0 0 zd21 0

377775 , (4)

zc exp

"2i

b

2sL1 1 L2d

#,

zd exp

"2i

b

2sL1 2 L2d

#. (5)

Here the notation z is deliberately taken from thez-transform theory of discrete linear systems and signalprocessing; the z terms in the above equations correspondto pure delay lines, and therefore the present systemresembles a digital filter in many ways. Indeed, the sta-bility considerations below are very similar to those fordigital filters.10

In analogy to the theory of electronic differential ampli-fiers, we call zc and zd the common-mode and differential-mode phase delays. If the links are lossless, b is real andzc and zd lie on the unit circle. The factors in Eq. (5) areused instead of expfibsL1 1 L2dg and expfibsL1 2 L2dg; ifthe latter factors were used, the elements of the matrix inEq. (4) would be

pzczd and

pzcyzd rather than zc zd and

zcyzd. The reasoning in Section 4 below is based on theLaurent series with respect to zc and zd of the networktransfer function, and the square roots would invalidatethis reasoning by introducing branch points at the originof the complex plane.

The network can be analyzed by a matrix analogy of thederivation11 of the transfer function for the Fabry–Perotcavity or the fiber-ring resonator. In response to a unitinput, a wave of amplitude F is immediately output fromthe system, and a vector of waves D is launched intothe links. After these waves travel the links, the vec-tor zcCszddD is incident upon ports 1–4, generating acomponent zcGT CszddD at the output and relaunchingwaves of amplitudes zcSCszddD into the links. Thisprocess leads to an infinite number of light recircula-tions, the nth recirculation outputting a beam of ampli-tude zc

nGT fSCszddgnCszddD. The total system transferfunction, T b5ya0, is thus

T szc, zdd F 1 GT CszddDzc 1 GT CszddSCszddDzc2

1 GT fCszddSg2CszddDzc3 1 . . . . (6)

If the network is stable, this series converges; otherwisethe recirculations would grow exponentially with recircu-lation number and the output power would increase with-out bound. Practically, the output power would rise untilthe system were driven out of its linear operation regime,and lasing could follow. The condition for stability isseen to be that all the eigenvalues of SCszdd lie withinthe unit disk. Here we exclude, as being unattainable inpractical systems, the case of marginal stability, in whichthe system can be perfectly lossless, SCszdd has an eigen-value exactly on the unit circle, and thus light recirculatesprecisely with neither loss nor gain forever in the links.

Under the stated stability conditions the geometric se-ries can be summed; hence

T szc, zdd F 1 GT fI 2 zcCszddSg21zcCszddD , (7)

where I denotes the 4 3 4 identity matrix. Equation (7)can be derived by a direct solution of Eqs. (2) and (4);however, the geometric series in Eq. (6) gives insight intothe system stability and will be needed in Section 4 below.


(a)

(b)Fig. 3. (a) Resonant Michelson interferometer, (b) optical fiberequivalent.

3. RESONANT MICHELSONINTERFEROMETER EXAMPLETo illustrate the above general interferometer notation,we now apply it to the resonant Michelson interferome-ter considered in Ref. 2. The system, together with anoptical fiber equivalent, is shown in Fig. 3.

Fabry–Perot cavities are formed between reflectors m2and m4 and between m1 and m3. In the optical fibernetwork a symmetric 2 3 2 coupler is equivalent to thehalf-silvered mirror of the Michelson interferometer, andthe feedback links are shown as dashed lines. We firstderive the F, D, G, and S matrices by considering thenetwork’s behavior without the links. Let the scatteringmatrix description of the partially reflecting mirror m1 be"

A1

b1

#

"r1 t1

t1 r10

#"B1

a1

#. (8)

For a reciprocal mirror, the 2 3 2 matrix is symmetric9;hence the elements in positions (1, 2) and (2, 1) are thesame. The mirror will be assumed to appear the samewhatever side the waves are incident from (i.e., the mirrorappears the same to wave a1 as it does to wave B1) sothat r1 r1

0. A lossless mirror has a unitary scatteringmatrix,9 so that its scattering matrix must have the form

Sm1

24 r1 ip

1 2 r12

ip

1 2 r12 r1

35 , (9)

where r1 is the mirror’s reflection coefficient. Let

M 1

p2

"1 ii 1

#(10)

be the transfer matrix of the 2 3 2 symmetric coupler;this matrix is found by the same reasoning as that usedto derive Eq. (9). In the absence of the feedback links,the optical fiber network in Fig. 3 obeys the following:"

B1

B3

# M

"A0

0

#,

"B0

b5

# MT

"A1

A3

#,

"A1

A3

# r

"B1

B3

#1 i

p1 2 r2

"a1

a3

#,

"b1

b3

# i

p1 2 r2

"B1

B3

#1 r

"a1

a3

#,

A0 r0B0 1 iq

1 2 r02 a0 , (11)

where r1 r3 r and mirrors m1 and m3 are identical.We can readily solve these equations to eliminate A1, B1,A3, B3, A0, and B0 to obtain

"b1

b3

#

26664r 2r0s1 2 r2d

22i

r0s1 2 r2d2

2ir0s1 2 r2d

2r 1

r0s1 2 r2d2

37775"

a1

a3

#

2

p1 2 r2

p1 2 r0

2p

2

"1i

#a0 , (12)

b5

p1 2 r2p

2f2s1 1 rr0dis1 2 rr0dg

"a1

a3

#2 r

q1 2 r0

2 a0 .

(13)

In the absence of the feedback links, ports 2 and 4 arecompletely decoupled from the input, the output, and


ports 1 and 3. They obey a2 2b2 and a4 2b4. Thusfrom these two equations together with Eqs. (12) and (13)we find, for this network,

S

26666666664

r 2r0s1 2 r2d

20 2i

r0s1 2 r2d2

0

0 21 0 0

2ir0s1 2 r2d

20 r 1

r0s1 2 r2d2

0

0 0 0 21

37777777775,

(14)

D 2

p1 2 r2

p1 2 r0

2p

2

26666410i0

377775 , (15)

G 2

p1 2 r2p

2

2666642s1 1 rr0d

0is1 2 rr0d

0

377775 , (16)

F 2rq

1 2 r02 . (17)

Some aspects of this interferometer’s behavior, as cal-culated by Eq. (7) are shown in Fig. 4, where, for eachplot, zc is held constant, zd expsiud, and the intensityjT szc, zddj2 is plotted as a function of u. In (a), zc 1;(b) is a close-up of plot a for u ø py2, and (c) is the sameclose-up for zc exps0.01id, i.e., with the common-modephase varied by 0.01 rad. It is clear that T is a very sen-sitive function of zc in this case.

4. GENERAL DESCRIPTION OF LINEARDIFFERENTIAL INTERFEROMETERSFrom the general description of linear optical networksformulated in Section 2, we now derive complete charac-terizations of the class of interferometers whose outputsdepend only on the difference of two optical lengths.

A. Differential Intensity and Phase ResponseConsider first an element Sjk of the scattering matrix of ageneral linear optical network. Suppose that Sjk dependson two optical lengths. The physical interpretation of Sjk

is the complex amplitude of the wave output from port j inresponse to a unit-amplitude excitation of port k when allother ports are undriven. Hence Sjk can be interpretedas a network transfer function T szc, zdd, exactly as in theforegoing sections. Suppose we require Sjk T szc, zdd(both magnitude and phase) independent of the common-mode phase delay, zc. However, observe that Eq. (6) ex-presses T as its unique Taylor series with respect to zc

about the point zc 0; thus T is independent of zc ifand only if the terms GT CszddDzc, GT CszddSCszddDzc

2,. . . all vanish. However, this implies that T F and isindependent of zd [with Eq. (6) considered as the uniqueLaurent series in zd about zd 0] so that:

If an element of a linear optical network’s scatteringmatrix is a function of two optical lengths L1 and L2,then this scattering matrix element must have some de-pendence on the common-mode length, L1 1 L2. If not,

then it has no dependence on the differential mode lengthL1 2 L2, either, and is therefore completely independentof L1 and L2.

This result is relevant to integrated optical devices inparticular. For example, one may wish to implementan accurately known, absolute optical delay of a certainphase f between the two ports with constant attenuationA, so that Sjk A expsifd, A , 1. It is highly desir-able that both the magnitude and the phase of Sjk be dif-ferential functions of two optical lengths, thus achievinggood device temperature insensitivity if the lengths arewell matched and undergo similar thermal strains. Bythe above result, such an implementation is impossible.

(a)

(b)

(c)Fig. 4. Typical behavior of a resonant Michelson interferometer.


B. Differential Intensity Response OnlyIn interferometry a lesser requirement is sought. As wasdiscussed in Section 1, it may be desirable that the outputmagnitude only be a differential function of two lengths.The output phase is irrelevant since most detectors can-not detect phase, and, indeed, the whole purpose of in-terferometry is to convert phase variations to amplitudevariations. We now pose the following question:

Given the network of Fig. 1, when can two paths linkports 1–4 as shown such that the transfer function mag-nitude jT szc, zadj is a function of only zd for all zc, zd onthe unit circle?

The assumption that zc and zd must lie on the unit circleis equivalent to an assumption of lossless links [Eq. (5)].Lossy links can usually be readily accommodated by re-placement of zc with Azc, where A , 1 is an attenu-ation. The links’ attentuation varies far less than doestheir phase delay for the small length variations normallyencountered in interferometry.

It will now be proved that if an interferometer is differ-ential, its output intensity as a function of zd has the sameform as that of the classical Michelson or Mach–Zehnderinterferometers. The proof assumes the magnitude ofT szc, zdd to be independent of zc for unit magnitude zc andzd and then derives the most general possible T szc, zddfulfilling this condition.

Note that T szc, zdd is a rational function of zc and zd

[Eq. (7)], and a rational function vszd of z has a constantmagnitude on the unit circle if and only if its poles pk

and zeros qk are symmetrical with respect to the unitcircle10,12; i.e., qk 1ypk

p, and

vszd v0zn0

Yk

√z 2

1pk

p

!Y

k

sz 2 pkd, (18)

which is equivalent to (the ck are symmetric functions ofthe coefficients pk)

vszd v0zn0

3c0 1 c1z 1 c2z2 1 . . . 1 cN21zN21 1 cN zN

cNp 1 cN21

pz 1 cN22pz2 1 . . . 1 c1

pzN21 1 c0pzN

;

(19)

i.e., the denominator coefficients are the conjugates of thenumerator coefficients written in backward order. Heren0 is an integer. Now observe that T szc, zdd is a ratio ofpolynomials in zc whose coefficients are polynomial func-tions of zd and, by assumption, the magnitude of T szc, zddis independent of zc for jzcj jzdj 1. Therefore

T szc, zdd T0szddzcn0

3c0szdd 1 c1szddzc 1 . . . 1 cN szddzc

N

fcN szddgp 1 fcN21szddgpzc 1 . . . 1 fc0szddgpzcN

.

(20)

Here the functions T0szdd, c0szdd, . . . , cN szdd correspond tothe constants v0, c0, . . . , cN in Eq. (18); we must allow fortheir being functions of zd here. Note that zd

p zd21 for

jzdj 1; hence Eq. (20) can be written as

T szc, zdd T0szddzcn0

NXj0

√MX

k0

cj ,kzdk

!zc

j

NXj0

√MX

k0

cN2j ,kzdk

!p

zcj

T0szddzcn0zd

M zc2N

MXk0

√NX

j0

cj ,kzcj

!zd

k

MXk0

√NX

j0cj ,M2kzc

j

!p

zdk

T0szddP szc, zdd , (21)

where P szc, zdd has been defined as T szc, zddyT0szdd. Byconsidering P szc, zdd to be a ratio of polynomials in zd

whose coefficients are polynomials in zc, we can reversethe above reasoning to apply Eq. (19) to P szc, zdd; thisfunction is thus shown to have unit magnitude for allzc and zd, so that the only variation in transfer-functionmagnitude can arise from T0szdd. T0szdd is a rationalfunction of zd and will now be shown to have particularlysimple form.

T0szdd is expanded as the unique Laurent series aboutzd 0, and P szc, zdd is expanded as the unique Laurentseries with respect to zc about zc 0; thus T szc, zdd iswritten as an infinite series of terms of the form tj ,kzc

jzdk.

The term zcj zd

k represents a component of the outputlight that has traveled the first link s j 1 kdy2 times andthe second link s j 2 kdy2 times. Hence for real, causalinterferometers neither j 1 k nor j 2 k is negative, sothat j $ 0, jkj # j . Consider any term zc

uzdv in the

Laurent expansion for P szc, zdd. The powers of zd in theLaurent expansion for T0szdd cannot be greater than u 2 vnor less than 2su 1 vd, and the Laurent series for T0szddmust be finite:

T0szdd t0 1 t1zd 1 . . . 1 tmzd

m

zdn

, (22)

where m and n are integers. The number of zeros ofT0szdd, equal to m, is now calculated from Eq. (6). Thesezeros are zeros of T szc, zdd, which are independent of zc;from Eq. (6), at such a zero,

F 0; GT fCszddSgRCszddD 0

for R 0, 1, 2, 3, . . . . (23)

Only the terms for R 0, 1, 2, and 3 need to be consideredbecause the Cayley–Hamilton theorem13 shows that allhigher powers of CszddS are linear expressions in cubicor lower powers. Hence T0szdd has a zero if and only ifF 0, and

GT CszddD 1zd

st00 1 t02zd2d 0 ,

GT CszddSCszddD 1

zd2

st10 1 t12zd2 1 t14zd

4d 0 ,

GT fCszddSg2CszddD 1

zd2 st20 1 t22zd

2 1 t24zd4

1 t16zd6d 0 ,

GT fCszddSg3CszddD 1

zd2 st30 1 t32zd

2 1 t34zd4

1 t36zd6 1 t38zd

8d 0 (24)


Fig. 5. Generalized Michelson interferometer.

by direct computation of the terms in Eqs. (24). Thetjk are functions of the elements of D, G, and S. ThusT0szdd can have at most eight zeros in pairs of the form6z0. A straightforward direct computation of the termsin Eq. (23) shows that there can be only four zeros intwo pairs of the form 6z0. This computation is outlinedas follows. The number of zeros of T0szdd is bounded bythe lowest polynomial degree in Eqs. (24). Hence thereare at most two zeros 6

pt00yt02 i unless t00 t02

0, in which case a direct computation of the terms inEq. (23) shows that t20, t26, t30, and t38 are all divisible byeither t10 or t14. Then T0szdd can have at most four rootsdefined by the second equation in the set Eqs. (24) unlesst10 t12 t14 0. However, this case would imply thatt20 t26 t30 t38 0, and the remaining equationscan still have at most four roots in pairs of the form 6z0.Hence, for a wholly differential interferometer, the mostgeneral form of the system transfer-function magnitude is

jT0szddj jt0 1 t2zd2 1 t4zd

4j . (25)

This function can be realized by the generalized classi-cal Michelson interferometer shown in Fig. 5. The class-ical Mach–Zehnder interferometer corresponds to eithert0 0 or t4 0. Given the constraint that jT szddj # Gmax,a Lagrange-multiplier calculation shows that the interfer-ometer’s sensitivity jdT szddydzdj is greatest when d2 0and d0 d4. This corresponds to a classical two-armMichelson interferometer with a symmetric, equisplitting2 3 2 directional coupler as the beam splitter. If the sys-tem is passive, Gmax # 1; otherwise, Gmax . 1 correspondsto the presence of gain in the arms of a Michelson or aMach–Zehnder interferometer.

The above proof does not exclude the existence ofa resonant, differential interferometer. However, itstransfer-function magnitude has the same form as thatof the device in Fig. 5, and its sensitivity cannot exceedthat of the sensitivity-optimal Michelson interferometerdescribed above.

Note also that the proof does not exclude an interferom-eter that is both more sensitive than the Michelson andinsensitive to small common-mode path-length variationsat particular values of zd and zc. Equation (7) can havethe structure required by Eq. (19) at finitely many valuesof zd on the unit circle and still be a globally nonconstantfunction of zc.

C. Differential Intensity and Relative Phase ResponseUsing the above, we can now obtain one final result.Recall that a linear optical network’s scattering-matrixelements cannot be independent of both zc without beingindependent of zd also, where zc and zd refer to thecommon- and differential-mode delays imposed by any

two optical lengths. However, often one does not careabout absolute phases, and only the relative phases be-tween the scattering matrix elements are important.Let two scattering-matrix elements be T1szc, zdd andT2szc, zdd. We now investigate when the magnitudesjT1j, jT2j together with their relative phase argsT1yT2dare independent of zc. In this case the function

T1

T2

F1 1 G1

T C1szddD1zc 1 G1T C1szddS1C1szddD1zc

2 1 . . .F2 1 G2

T C2szddD2zc 1 G2T C2szddS2C2szddDzc

2 1 . . .

(26)

must be completely independent of zc. Here F1, F2, . . .are the quantities defined in Section 2 for each of thetwo functions T1 and T2. By examination of the Laurentseries, it is straightforward to show that exactly one powerof zc in both the numerator and the denominator Taylorseries in Eq. (26) must be nonzero; hence

T1

T2

G1T fC1szddS1gN21C1szddD1

G2T fC2szddS2gN21C2szddD2

. (27)

However, the magnitudes jT1j and jT2j are individuallyindependent of zc. Hence, by Eq. (25), Eq. (27) can besimplified so that


4j ,


4j ,

arg

"T1szddT2szdd

# arg

√t10 1 t12zd

2 1 t14zd4

t20 1 t22zd2 1 t24zd

4

!(28)

are the most general forms that the magnitudes and therelative phase can take if they are to be independent ofthe common-mode optical delay.

5. CONCLUSIONIn this paper we have discussed the potential advan-tages of differential operation in avoiding the detrimentaleffects of source laser frequency jitter and temperaturedrifts in integrated optical circuits.

The transfer function was derived for a general linearoptical network when the behavior of the transfer func-tion depends on two optical lengths. The derivation wasillustrated by application to a recently proposed resonantinterferometer. Using the general transfer function, wethen proved that no element of a general linear N port’sscattering matrix can be rendered a wholly differentialfunction of two lengths; i.e., the dependence on the sum ofthe lengths cannot be annulled without the dependence onthe difference between these lengths being annulled also.Next we showed that if the network’s transfer-functionmagnitude is a wholly differential function of two opticallengths, then this transfer function has the same form asthat of the Michelson or Mach–Zehnder interferometerand therefore that the Michelson interferometer attainsthe greatest sensitivity of all differential interferometers.Then the most general form of a linear network’s scat-tering parameters was derived for the care in which theparameter magnitudes and the relative phases must bewholly differential functions of two optical lengths.


ACKNOWLEDGMENTSThe authors thank the Broken Hill Proprietary CompanyLtd. for permission to publish this paper, Anis Inayat-Hussain and Glenn Bryant of BHP Research for theirreview of the mathematical content, and John Love ofthe Optical Sciences Centre, Australian National Univer-sity, for extremely valuable input on the basic theoryof linear optical fiber systems. Much of this researchwas done while R. W. C. Vance was a recipient of anAustralian Telecommunications and Electronics ResearchBoard postgraduate scholarship at the Optical SciencesCentre, Australian National University, Canberra.

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general linear differential interferometers

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