20 IEEE JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. LT-3, NO. 1 , FEBRUARY 1985

Joel C. Masland received a B.S. in education from Temple University and an M S . in educa-

Park, PA. tion from Pennsylvania State University, College

He is a Member of the technical Staff in the Piezoelectric and Optical Components De- partment at AT&T Bell Laboratories, Allen- town, PA where he works with lightwave multiplexers.

* Stephanie A. Wartonick was born in Hunington, PA, on January 28, 1959. She received the B.S. degree in physics from Muhlenberg College, Allentown, PA and is currently working toward the M.S.E.E. a t Lehigh University, Bethlehem, PA.

In 1980, she joined AT&T Bell Laboratories, Allentown, PA, where she worked on liquid- crystal optical attenuators. From 1981 to 1983, she was involved in the development of wave- length division multiplexers. Currently she is

working on optical data links.

Theory of Laser Phase Noise in Recirculating Fiber-optic Delay Lines

MOSHE TUR, BEHZAD MOSLEHI, MEMBER, IEEE, AND JOSEPH W. GOODMAN, FELLOW, IEEE

Abstract-Optically driven recirculating delay lines, often used in fiber-optic signal processors, convert the source phase noise to spec- trally structured intensity noise at the loop output. This spectrum is characterized by deep notches at zero frequency as well as at other multiples of l/(loop delay). An expression is derived for the power spectral density of the output noise. The formulae obtained, are then used to investigate the dependence of the magnitude and shape of this noise on the source coherence time, the coupling ratio, the coupler and loop losses, and the fiber birefringence.

I. INTRODUCTION

an adjustable coupler [e.g., [2]). This simple structure has been used as: a) a transient buffer memory and data rate trans- former [ l ] , [3] ; b) a notch filter (over 18 GHz) [4], [ 5 ] ; c) a systolic processor [ 6 ] ; and d) a building block for single- mode fiber lattice filters [7].

In all of these applications, an electronic RF signal modulates the intensity of the light source, usually a semiconductor laser, and the processed guided waves are converted back to elec- tricity by a high-speed detector. Since the laser driving current and the detector output current are, respectively, proportional

R ECENTLY introduced single-mode fiber-optic signal pro- cessors make an extensive use of recirculating delay lines.

The basic recirculating loop [ l ] , shown in Fig. 1, is made by closing a continuous strand of single-mode fiber on itself using

to the injected and detected light intensities, overall linearity is ensured only if the optical summations, which take place at the various couplers are also effectively linear in the intensities of the interacting waves (rather than in their amplitudes). This requirement is easily met by using low coherence sources, e.g.,

Manuscript received May 18, 1984. This work was supported by multimode laser diodes, with a coherent time much shorter

Litton Systems, Inc. than any relevant loop delay. The use of low coherence sources M. Tur is with Stanford University, Stanford, CA 94305 on leave has the additional advantage of rendering these devices rela-

69978. from the School of Engineering, Tel-Aviv University, Tel-Aviv, Israel tively insensitive to environmental effects such as temperature

CA 94305. The dynamic range of these wide-band devices is limited B. Moslehi and J. W. Goodman are with Stanford University, Stanford, etc-

0733-8724/8S/0200-0020$01 .OO 0 1985 IEEE

T U R et al.: THEORY OF LASER PHASE NOISE 21

(b) Fig. 1. (a) The recirculating single-mode fiber delay line. (b) A manu-

ally adjustable directional coupler [ 21.

from above by: a) the available source output power; b) opti- cal nonlinearities in the fiber; and c) nonlinear responses of the source and detector. It is limited from below by the noise characteristics of the source radiation and eventually by the unavoidable detection shot noise. Since a short coherence time is associated with excessive phase noise [8], the output noise of these signal processing devices, when driven by a short coherence laser diode (see Fig. 2), is expected to be correlated with the laser phase noise. Indeed, it has been recently demon- strated [9], [ lo], that the power spectrum of the optical in- tensity at the output of a single-mode fiber recirculating delay line, driven by a multimode semiconductor laser, is not only much stronger than the laser intensity noise but it also exhibits a spectral structure with notches at zero frequency as well as at other integer multiples of l/(loop delay), as shown in Fig. 3. In addition, using the amplitude transfer characteristics of the coupler and a finite number of recirculations, it was possible [ 9 ] to relate the observed spectral structure to the laser phase noise.

The above mentioned characteristic structure of the power spectral density is not unique to fiber-optic recirculating delay lines. Indeed, Fig. 4 shows the intensity spectrum at the out- put of a bulk optics recirculating loop, driven by the same low coherence laser source.

Previous investigators of the implications of laser phase noise in fiber-optic devices, have dealt almost exclusively with non-

recirculating structures, e.g., Michelson or Mach-Zehnder sen- sors [l 1 ] and homodyne or heterodyne communication systems [12]. As predicted by these studies [13], [14], the phase- induced output power spectrum attains its maximum at zero frequency f = 0, and, for a source coherence time r, which is much shorter than the device delay r , the spectrum near f = 0 is relatively flat within several 117 units. It appears, therefore, that recirculating and nonrecirculating fiber structures convert the laser phase noise input into intensity noise at the output in very different ways.

In this paper we present a comprehensive analysis of the phase-induced intensity noise at the output of a single-mode fiber recirculating delay line driven by a laser source with a finite coherence time. Section I1 presents the basic model and its mathematical formulation. The expression for the intensity covariance function is constructed in Section I11 along with approximations which are valid when the source coherence time is much shorter than the loop delay. ?he expression for the power spectrum is presented in Section IV together with several useful figures. The limitations of the theory, as well as a few possible extensions, are considered in Section V.

11. MATHEMATICAL FORMULATION In this section we shall mathematically model the experi-

mental arrangement of Fig. 2, which includes a laser diode, a recirculating single-mode fiber-optic loop, a polarization

22 IEEE JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. LT-3, NO. 1, FEBRUARY 1985

A D J U S T A B L E D I R E C T I O N A L

M U L T I M O D E C O U P L E R J C W L A S E R DIODE

D E L A Y L I N E .

F I B E R

1 u S P E C T R U M A N A L Y Z E R

Fig. 2. Experimental setup for the measurement of the spectrum of the intensity noise at the output of a recirculating delay line, driven by a semiconductor laser. E,, Ea, and E3, E4 are, respectively, the input and output fields. The polarization controller is used to change the state of polarization of the propagating wave without modifying its intensity. The optical power at the detector was 130 mW, the loop length was varied from 27 cm to 200 m, and the amplifier gain =40 dB.

0 500 IOOOMHz

( 3 MHz RESOLUTION > Fig. 3. Experimental noise spectra for a loop length of 27 cm (7 =

1.35 ns): A the amplifier noise with the laser off;B the laser intensity noise (the coupler is disassembled, as in Fig. l(a)), and C phase- induced intensity noise with an assembled coupler (see Fig. l(b)).

D Same as C but with an intermediate setting of the coupler. Due to The coupling coefficient was adjusted to give the highest noise level.

the limited spectral range of the detection system, only to notches can be observed: one at f = 0 and another at f = 1 / ~ ( = 740 MHz). In this short loop, the spectrum was practically insensitive to adjust- ments of the polarization controller.

controller [15] , a high-speed p-i-n detector, and a spectrum analyzer.

A. The Laser Short coherence laser diodes are usually of the gain-guided

[16] multimode type. Their output amplitude can be modeled as the sum of a finite number of cavity modes, where each mode has the form

E(t) = Eo exp [i(oo t + cp(t)] . (1)

The complex vector amplitude Eo is assumed to be time- independent, thereby neglecting intensity noise [16] (this

* A RING RESONATOR

AVALANCHE ---- Tt . . . - . . . - , .

]ANALYZER

(3MHz RESOLUTION) Fig. 4. A bulk-optics recirculating delay line. Quite a few notches are

seen with a spacing of 150 MHz.

assumption, which is experimentally justified [9], is further discussed in Section 5). wo is the center optical frequency of the mode and q(t) represents the phase noise which can be also expressed in terms of the source frequency noise: q(t) = 1‘ dt’ [w(t’) - oO]. I t is common to assume that this fre- quency noise is a stationary normal (Gaussian) process with zero mean and flat spectrum [14]. Under this assumption, p( t ) is a Wiener-Levy random process [17] k it is a zero mean normal random process with a structure function, D ( t l , t z ) , of the form

D(tl - t z ) = ( [ c p ( t 1 ) - c p ( t 2 ) 1 2 ) = I t 1 - t a l

( 2 ) 7,

The angle brackets denote ensemble average and the propor- tionality factor 117, can be shown to be the mode coherence time [ & I . For gain-guided lasers, r , % 50-100 ps, which corre- sponds to a spectral width of several gigahertz. This form of the structure function, in (2), is also associated with a Lorentzian lineshape of the optical spectrum of the mode [8], [ 161. The finite bandwidth Aw, generated by cp(t) is assumed to be small enough so that the fiber dispersion can be neglected and the field at the output of a piece of fiber of IengthL is a delayed version of (I), namely, E(t - L/V), U is the phase velocity, aver- aged over Aw.

B. The Directional Coupler

A polarization-independent linear directional coupler [ 181 can be described by

El , E2 , E3, and E4 are, respectively, the input and output fields (see Fig. 2). The coupler power insertion loss is repre- sented by an amplitude transmission factor 6 o , and the complex

TUR e t a l . : THEORY OF LASER PHASE NOISE

numbers A , B , C, and D form a unitary (energy preserving) matrix [ 191

/AI2 + lBI2 = ICI2 -k [Dl2 = 1

CA* t DB" = 0 (4)

(The star denotes complex conjugation).

C. The Fiber A weakly guiding, ideal single-mode fiber can support two

degenerate, mutually orthogonal linear polarizations of the LPol type [20]. However, when a fiber is incorporated in a system, it is both bent and twisted, and the resulting linear and circular birefringences [15], [21], [22] distort the LPol modes to the extent that they are no longer degenerate eigenmodes. Nonetheless, there exist two orthogonal eigen- modes E, and Eb such that when propagating from port 1 to port 2 through the loop, their polarizations are, respectively, preserved.' Eu and Eb have different phase velocities and, therefore, experience different delays r, and 71, as they travel around the loop. Under normal bending and twisting condi- tions, and including the natural parasitic birefringence of single- mode fibers, we find [22] that the normalized difference between these two velocities is on the order of (which corresponds to a beat length L , = hv/Au of 1 m). In general, these two modes are also differently attenuated, but for sim- plicity, we shall assume the same amplitude attenuation coeffi- cient, for both E, and E,. The input field from the laser in (I), can be decomposed into

E(t) = g EU(t) t h Eb(t). (5 )

If E(t) , E,(t) and Eb(t) have equal intensities, then (using the orthogonality of E, and Eb)

1812 + lh/2 = 1 (6)

and the output intensity at port 4 is

= 1812 ' I h 1 2 Ib(t) (7 1 Zu and I , are, respectively, the output intensities of modes a and b.

D. The Polarization Controller An ideal polarization controller 1151 , [ 2 2 ] , can transform

the state of polarization of highly polarized light between any two given states, with no change in the light intensity. Its in- corporation in the loop allow us to modify E, and E,, and to adjust r, - rb.

E. The Output of the Spectrum Analyzer Since the spectrum analyzer input voltage is proportional to

Z(t)-the light intensity from port 4, the output of the ac- coupled spectrum analyzer gives the power spectrum of Z( t ) . According to the Wiener-Khinchine theorem [ 171 , this power spectrum is the Fourier transform o f the autocovariance func-

'These modes can be expressed as a linear combination of the LPol modes with coefficients which depend on the fiber-length coordinate 1221.

23

tion of Z ( t )

COVI(t1 , t2) = ( [ Z ( t l ) - ( I ) ] [ Z ( t 2 ) - (I)] )

= 1814 Cua(tl 3 t 2 ) + 21gI2 I h l 2 Cab(t1 9 t2)

+ I h 1 4 Cbb(tl 9 t2) (8)

Caa( t l , t 2 ) and Cbb(t1, t 2 ) are, respectively, the autocovari- ance functions of Z, ( t ) and Ib(t) and Cub(tl, t 2 ) is the cross- covariance of Z, and I D .

In the following sections we evaluate these three covariance functions.

111. EVALUATION OF COVI( t1 , t 2 )

A. The Output Modal Zntensity and Its Average Value

When dealing with either of the two eigenmodes E, or Eb, the output field at port 4 is the sum of contributions from an infinite number of recirculations

m

E4(t) = 6o CEl (t) + 6gDA n = l

. exp (- naoL) El ( t - n7) (9)

r is the loop delay (7 is T~ for E, and T b for E b ) , L is the loop length, and a. is the fiber (amplitude) attenuation per unit length which is on the order of several decibels per kilometer.

For a unit amplitude input field of the form

El ( t ) = exp ( i b o t + 9 0 ) ) )

we find (for N recirculations)

N E4(t) = Fn exp ( i [wo( t - n7) + cp(t - n r ) ] )

n=o

Fn = { ' O C n = O 60DB-1A(60B exp [ - a O L ] ) " , n > 1

The output intensityZ(t) = /E4(t)I2 is given by

Z ( t ) = FnFZ exp (i[cp(t- n r ) N N

n=o m=o

- cp(t - m7) + (m - n) 0~71).

The evaluation of the ensemble average (I(t)> involves the cal- culation of the ensemble average of the exponential term in (12). Since cp(t) is a normal process, cp(t - n7) and cp(t - m7) are jointly normal so that [ 171

(exp (i[cp(t - n7) - cp(t - mr) + (m - n) war])>

= exp [i(m - n) 007]

X exp [ - +D((m - n) T)]. (1 3)

D ( t ) is the phase structure function in (2), and the ensemble average of Z(t) is

N N ( I ( t ) ) = F,F; exp [i(m - n ) w07]

n = o m=o

. exp [ - + D((n - m) r)] .

Whenr>>r,,exp [ - i D ( ( m - n ) r ) ] I n g m x O a n d

( Z ( t ) ) = [Fnj2 . N

n=o

B. The General Expression for cab.

cab is the cross-covariance function of I, ( t ) and 1, ( t )

Cab(tl 3 t 2 = ( [ra (tl) - ( I a )I LZb(t2 - (1, ) I ) = ( r a ( t , ) I b ( f 2 ) ) - ( ra )<rb>. (1 6)

We shall first treat the correlation term (ZaIbb). Using (12) and asuming different coefficient series iFa,n), and <Fb,n] for the two modes, we find

where the exponent can be expressed in terms of the process structure function, in (2)

$ = D [(m - n) T,] - D [tl - t2 + nlrb - nr,]

t D [ t , - tz t m/rb - nr,] + D [ t , - t z + d T b - mT,]

- D [ t l - t2 +drb-?’?27a] + D [ ( m l - d ) T b ] . (19)

Equations (14), (1 7), and (1 8) can now be combined to give

m=o n = o ml=o nl=o m f n m l f nl

* exp [ i [ ( m - n) W o T a + (ml- nl) w o T b ] ]

. exp [ - i [ D [ ( m - n)r,] +D[(ml - n l ) r b ] ] ]

. {exp [++ [D[t l - t2 + nlrb - nr,]

- D [tl - t2 + mlrb - n ~ , ]

- D [ t l - t 2 + 121rb - mi-,]

+ D [tl - t 2 + mZTb - W I T , ] ] ] - 1). (20)

The conditions m f n and ml f nl were added to (20) to re- flect the fact that there is no contribution to cab from terms

L I b H I W A V E ICLHNULUbY, V U L . L I - 3 , NU. I , P‘OBKUAKY 1 9 8 5

with either m = n or ml = nl (the two terms in the curly brack- ets cancel each other).

c. An Approximate Evaluation of cab As mentioned in Section 11, the two orthogonally polarized

eigenmodes of a single-mode fiber propagate with slightly dif- ferent phase velocities. For a typical beat length of 1 m, as in Section 11-B

(7, - T b ( = 7 (2 1)

r is the average loop delay (=0.5(ra + rb)), This assumption al- lows us to replace both T, and T b in the arguments of the structure functions in (20) by r. This cannot be done in (20)’s phasor term: exp [i[(rn - n) mora t (ml - nl) W O T b ] ] since mo(rb - 7,) is on the order of 0.5 rad/m. Thus

N N N N Cab(tl > IZ?” Fa,nF:mFb,nlFz,rnl

m=o n = o ml=o nl=o m f n ml f nl

. exp [i [(m - n) O ~ T , + (ml- nl) W o r b ] ]

. {exp [- $K(na, n,ML,NL) (~ / rc>l

- exp [- 3 - 4 1 + J W - n9 j I ( d T c ) J l

(22)

where

M L = m l + A ; N L = n l + A (23)

A = ( t l - tz)/r (24)

K ( ~ , ~ , M L , N L ) = [ J m - H I + JML -NLJ - I N . - n J

+ 1~~ - nI t IN^ - m l - 1~~ - ml]

(25 1 and use was made of (2).

To ensure optical summation of intensities and to avoid in- terferometric sensitivity to environmental conditions, fiber- optic signal processors usually employ light sources with a coherence time much shorter than any loop delay

r/r, >> 1. (26)

This strong inequality can be used to substantially simplify (22). Since m # n and ml f nl, the exponent of the second term in the curly brackets in (22) is equaI or larger than r/rc and as a result of (26), this second term can be completely ignored. As for the first term in the curly brackets, it can be also neglected unless

K ( m , n , M L , N L ) x O . (27)

As can be concluded from the definition of K in (24), this condition, subject to m # n and ml # nl, can be met if and only if

m = NL = nl + ( t l - t2 )/r together with

n =ML = ml + (tl - tZ) / r .

TUR et al.: THEORY OF LASER PHASE NOISE 2 5

Thus Cab(t1, t z ) is significant only when (tl - t z ) is vety close - 1 < R e s < 1 (7, = T b )

to an integer multiple M of r, i.e., only when (3 8)

and

and (28) can be rewritten as

m=nl+M, n =ml + M .

When t1 and tz are given (subject to (29)), M is known, and in 1 4 4 8; u3

the quadruple summation of (22), only those terms for which (30) holds, will contribute to C a b ( t 1 , tz). At this point we combine all our previous observations to obtain (K(m, n, ML = G”= n - M , NL = m - M ) = 2t‘/r) IADI3 IC] . & ; u ( I M I + ~ )

+2---r *

(1 - U ) (1 - U2)’ P I M = O (39)

Cab( t l ’ t2 ) l t l - t 2=~?+t ’ ; l t ‘ l<o .5 - 2

IBI3 1 - u

. exp [-I tl - t2 - M T ~ / T ~ ] . (3 1)

where

4 = exp [i [WO (.a - r b ) ] 1 + (32)

Note that c a b approaches zero when I tl - tz - M T ~ >> r, (subject to It1 - t 2 - Mrl G 0 . 5 ~ ) .

The form of (31) suggests that cab can be expressed as a convolution

Thus AM(4) depends on q only through the multiplicative fac- tor Re S.

D. The General Approximate Expression for CovI(t, , t z ) C, and c b , may be derived from Cab by letting r, = r b , or

equivalently, by equating 4 to 1. Therefore, a general approxi- mate expression for Cov,(tl, t 2 ) is easily obtained from (8) and (33)

6 (t l - t 2 - Mr) is the Dirac 6 function, centered around Mr, and

where

and AM is givefi by

A“(4) (43)

N N E. Discussion of (41) = Fa,nF&nFb,(m-M)>oF$,(n-M)>Oq m - n .

m=o n=o < N Q N The approximations in (21) and (26), enabled us to obtain

manifests the different roles played by the various system

For simplicity, we shall evaluate A M ( 4 ) only for the case where a) The covariance function is affected by the state Of

Fa,n = F b , n , i.e., under the assumption that both the coupler polarization of the propagating wave, as determined by g7

and the attenuation coefficient in the fiber are polarization h, and Tu - r b , only through the frequency-independent mul- independent. substituting (1 1) in (35) N + M, we obtain two tiplicative factor T (which does not change the form of the nested infinite geometrical series which can be summed to give covariance function).

b j CovI(tl , t z ) is a function of the time difference t1 - t z , AM(4) = Re S * GM (36) and its form determined by the lineshape of the laser emission

where ( 1 tl/rc =D( t ) in (2)) convolved with an infinite sum of im- pulses, symmetrically located around t1 - tz = 0, at integer

1 - 6; exp [-2aoL] /BIZ multiples of r. The strengths of these impulses depend only

1 - 6; exp [ - ~ o I , z ] 1 ~ 1 ~ 4-l on the coupler’s characteristics and the fiber attenuation. As for the signs of these impulses, Go is obviously positive, and

m 2 n a relatively simple expression for CovI(tl , t z ) which clearly

(35) parameters.

s = q - l

= 4 - 1 1 - U the sign of G M , M # 0 , in (39), depends on the ratio W of its

1 - uq-1 (37) two summonds

26 IEEE JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. LT-3, NO. 1, FEBRUARY 1985

- 2 r - T i Fig. 5 . The autocovariance function of the output light intensity in

(41). Only the first four side-spikes are shown.

I Negative Term I [l - U ] [ l t U ]

I Positive Term1 U 2 W = = I CBIADI

(44)

Most practical systems involving fibers and directional couplers can be adequately modeled by a unitary coupling matrix in (3)-(4), and U < IBI2 < 1. These two conditions are suffi- cient (but in no way necessary) to ensure that W > 1 and G, is negative for all M f 0. The general form of Cov,(t, ~ t 2 ) ap- pears in Fig. 5 (only the first four negative spikes are shown). The paper by Tur and Moslehi [9] proposes a simplified ex- planation of this figure, showing the relationship between the first three terms in the expansion of (9) and the central and first two side-spikes of Fig. 5. More generally, it is evident from (35), that the spikes at +MT are generated by recircula- tions with order n greater or equal to M .

IV. THE POWER SPECTRUM OF THE OUTPUT INTENSITY A. Derivation

According to the Wiener-Khinchine theorem [17], the power spectrum of the output intensity is the Fourier transform of the covariance function. Within the approximations of (21), (26), and (41), it is given by the product

(45 1

where the last term in (45) is the Fourier transform of exp [ - ] t]/.r,] and S6 (f) i s the Fourier transform of C6 , (43)

The fiber birefringence affects the spectrum S( f) only through the multiplicative, frequency-independent factor T in (42). Plugging C6(tl - t2 = M T ) , in (43), into (46), we find the full

dependence of S6 (f) on the various system parameters

r

1 4 4 6 4 , ~ ~ exp [-2nifr] t 2 -- IBI4 (1 - U ) (1 - U 2 ) (1 - Uexp [-2n$r])

B. Results and Discussion

S6 (f) is shown in Fig. 6 for various values of B and 01~. The full spectrum, S(f) in (45), can then be obtained by multiply- ing Fig. 6 by the polarization factor T and by a decreasing en- velope (with a characteristic scale T ~ ) , which is determined by the actual lineshape of the mode. When the loop is lossless, Fig. 6(a), infinitely deep notches appear, but their depths de- crease as the loop loss increases Fig. 6(b).

Fig. 6 depicts the frequency dependence of the spectrum of the output intensity of the loop when driven by a low co- herence source. It is interesting to note that its form is very similar to the wavelength (or loop length) dependence of the output intensity of the same loop when performing as a reso- nator with a highly coherent HeNe laser input [23], [24]. This resemblance is probably a consequence of the nature of the low coherence source: its frequency changes very fast over many free spectral ranges [25] (T /T , >> 1). The similarity, though, is not complete. a) A lossless resonator is characterized by an infinitely deep but infinitesimally NUWOW notches. b) Even for a lossy resonator, there always exists a certain cou- pling ratio for which the resonator notches still have infinite depth. c) The location of the resonator notches is extremely sensitive to micrometer-size variations in the loop length, while with an incoherent source the output intensity spectrum is en- vironmentally stable.

Another interesting feature in Fig. 6 is the dependence of the form of the spectrum on the coupling ratio: the higher the cou- pling ratio, the flatter is the power spectral density within any given period. This dependence can be correlated with the fact that the number of (nonvanishing) recirculations increases with the coupling ratio (see Fig. 7). As noted in Section 111-E, the

TUR etal.: THEORY OF LASER PHASE NOISE 27

0

-10 - m Y

m

3 -20 c t- o W

m a

- 30

-03 l

0 1.00 2.00 3.00 FREQUENCY f r )

0

- 10

- 20

0 1.00 2.00 3.00 FREQUENCY f T )

Fig. 6. The power spectral density &(f), in (46). (a) A lossless loop (60 = 1,010 = 0) ; (b) 6 = 1 , a0 = 0.1: (1) B 2 = 0.1; IB I = IC1 = 0.32 and lAi=ID)=0.95;(2)B2=0.4;]Bl=lC(=0.63andIAI=)DI=0.77, (3) B2 = 0.7; ( B ( = IC( = 0.84 and (AI = /D I = 0.55 and (4) B2 = 0.9; /BI = IC1 = 0.95 and / A I = ID I = 0.32.

t 2 l B l =0.1

t z le1 =0.9

L I 2 3 4 5

(C) Fig. 7. The amplitudes of the first few output terms, in (9) , for several

coupling ratios.

pair of spikes at tl - t2 = kMr is generated only by recircula- tions with a high enough order: n >M. Upon performing the Fourier transformation that generates (4.9, this pair contrib- utes a spectral component at the frequency l /(Mr). There-

k22OMHz

-1lOd8

(a) (dl

-90dB

-4 -1lOdB

I n -90de- -IIOdBo L

-4 k-5OnS 100 200 MHz MHz

(c) (f)

Fig. 8. An experimental demonstration of the dependence of the shape of the spectrum on the coupling ratio for a 10-m loop. These figures wele obtained by modulating the CW laser output with a single 35-11s pulse whose polarity was such that the optical input into the loop was substantially decreased. (a)-(c) show the loop impulse response while (d)-(f) show the corresponding spectra. The loop loss and coupling ratios, as estimated from (a)-(c), roughly match the parameters of Fig. 6(b).

fore, a large number of recirculations increases the harmonic content of the spectrum and, consequently, flattens it. This dependence of the form of the spectrum on the coupling ratio is experimentally demonstrated in Fig. 8. For a weak coupling ratio, Fig. 8(a), only a very small amount of light is directly coupled from the input pulse to the output port (port 4 in Fig. 2). While most of the light leaves the loop after just one recirculation, a very small portion of the outgoing light is cou- pled back into the loop to form a weak third pulse, which is the last, hardly visible recirculation in Fig. 8(a). Fig. 8(d) shows the resulting spectrum, a fairly round one, as expected from the above discussion. The impulse response for a moder- ate coupling ratio, is shown in Fig. 8(b). The corresponding spectrum, Fig. 8(e), is an elevated version of Fig. 8(d), with flatter tops. Strong coupling is shown in Fig. 8(c) and 8(f). In spite of their low magnitude, several recirculations are visi- ble in Fig. 8(c). The depth of the notches is the same as in Fig. 8(d), but the tops are much flatter and lower. All these observations are in good agreement with Fig. 6. Also note, that since Fig. 8(a)-(c) were obtained by partially blanking the CW emission of the laser, the vertical widths of the oscil- loscope traces, in between the pulses and after the pulses die, are indicative of the magnitude of the .noise as it appears in the time domain. Indeed, the observed time-domain magni- tude completely correlates with the spectral magnitudes of Fig. 8(d)-(f).

When the coupling matrix is unitary and fr = n/2 (n an in- teger), the expression for s& (f) can be greatly simplified.

U[1 + (- 1)"U] [l - (- l), U/lB12]2

[l - ( - l ) ,U] [l - U ] 2 [1+ U ] .

. .

(48)

2s IEEE JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. LT-3, NO. 1 , FEBRUARY 1985

to give 2While Fig. 12 generally agrees with our theoretical results, it is seen r 2 - u~ [cos [uo(Ta - Tb)l - ul that the depth of the notches is slightly sensitive to the adjustment of

T =- the polarization controller. Two possible reasons may be suggested to (I t r)2 1 - 2ucos [ u o ( T , - T b ) ] t u2 explain this deviation from our theory (see (45)): a) the adjustment of

thc polarization controller also slightly modifies the loop loss and b) the (54) two modes E, and Eb have slightly different attenuation coefficients.

T U R e t al.: THEORY OF LASER PHASE NOISE 29

- 4

1 w 0 z W

COUPLING COEFFICIENT = 0.4

NO LOSS

L = nX

-12 0 2.00 4.00

8 = w , l r , - r b l ( R A D I A N S ) 0 I I I I

Fig. 11. The dependence of the birefringence factor T, in (54), on w 0 ( 7 ~ - T b ) for an input wave whose power is equally divided b e tween the Ea and Eb modes. The corresponding < above those shown here.

0 2.00 4.00 6.00 8.00 10.00 T/TC

lie Fig. 13 . The variance of the output intensity noise as a function of r / rC. [S[* = 0.4, the loop transmittance P is unity, and the loop length is assumed to be an integer multiple of the source wavelength.

much stronger than the intensity noise of the laser (see Fig. 3), thereby partially justifying the simple model of (1).

2) The Coupler: Equations (3) and (4) are only approxima- tions. In real couplers, such .as the mechanically polished ad- justable directional coupler [ 2 ] , a more accurate description [26] takes into account the normal modes of the coupler it-

MHz different loss factors. The deviations from our simple model

Fig. 12. The spectra for a 10-m recirculating loop on a condensed fre- are small and, in principle, can be incorporated in a more com-

quency scale, for two orthogonal adjustments of the polarization plicated treatment along the lines of this paper. controller. 3) The Approximation r/r, >> 1: The final results of this

work, in (41) and (49, are valid only for r/rc >> 1. However,

2 d 0 /

0 250 500 self, which usually have different propagation velocities and

(3MHr RESOLUTION)

V. LIMITATIONS AND POSSIBLE IMPROVEMENTS OF THE THEORY

the basic expression in (20) (with N = m), is valid for all values of TIT,. Fig. 13 shows the dependence of the variance of the output intensity, as obtained from (20) (with a large value of

The theory presented in this paper suffers from a few limita- tions that need improvement:

1) The Laser Model: In order to achieve a short coherence time, a multimode laser is usually used in fiber-optic signal processors, employing recirculating delay lines. This type of laser simultaneously emits many spectral lines, each with a fairly short coherence time which decreases with the line power [8]. All the results of this paper are derived from a single line representation, as in (1). If the amplitudes and phases of the various spectral lines were independent of one another, then our results would remain unchanged: Each line would supply its own contribution, as given by (45), and cross terms among different spectral lines (having an interline spacing of 50-100 GHz [SI) would not fall into our spectral range of interest. However, the lines are experimentally found to ex- change power among themselves, resulting in partition noise and fluctuating line amplitudes [16]. Moreover, the phase noise of the various lines is partially correlated at least for short delays [14]. In order to include these effects in the theory, a lot of information must be provided concerning the statistical interrelations among all the amplitudes and phases involved. This information is hardly available today, for multi- mode lasers. On the other hand, our relatively simple theory accounts for all the observed experimental evidence. Also, the phase-induced intensity noise is experimentally found to be

N and ra = rb), on the ratio r/rc. For small values of this ra- tio, the variance has an interferometric sensitivity to the loop length and, therefore, a loop length is assumed which is an in- teger multiple of the source wavelength. It is clearly seen that the curve saturates for r/rc FZ 4, thereby indicating that our final results, in (41) and (4.9, are valid even for fairly small values of r/rc.

VI. SUMMARY

This paper considered in detail the magnitude and structure of the phase-induced intensity noise produced by recirculating delay lines. Expressions were derived for the autocovariance function, as well as for the power spectral density of the out- put noise. The formulae obtained were used to investigate the dependence of the magnitude and shape of this noise on the source coherence time, the coupling ratio, the coupler and loop losses, and the fiber birefringence. This type of noise is important in assessing the dynamic range of fiber-optic signal processors. The results are also valid for bulk-optics devices.

ACKNOWLEDGMENT

The authors wish to thank Prof. C. C. Cutler and Prof. H. J. Shaw for helpful discussions. The couplers were fabricated by G . Kotler.

30 IEEE JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. LT-3, NO. 1 , FEBRUARY 1985

151

191

REFERENCES S. A. Newton, R. S. Howland, K. P. Jackson, and H. J. Shaw, “High-speed pulse-train generation using single-mode fiber recir- culating delay lines,” Electron. Lett., vd. 19, pp. 756-758, 1983. R. A. Bergh, G. Kotler, and H. J. Shaw, “Single-mode fibre optic directional coupler,” Electron. Lett., vol. 16, pp. 260-261, 1980. S. A. Newton, J. E. Bowers, and H. J. Shaw, “Single mode fiber recirculating delay line,”Proc. SPIE, vol. 326, pp. 108-115,1982. J. E. Bowers, S. A. Newton, W. V. Sorin, and H. J. Shaw, “Filter response of single mode fiber recirculating delay lines,” Electron. Lett.,vol. 18, pp. 110-111, 1982. S. A. Newton and P. S. Cross, “Micro-wave frequency response of an optical-fibre delay line filter,” Electron. Lett., vol. 19, pp.

M. Tur, J. W. Goodman, B. Moslehi, J. E. Bowers, and H. J. Shaw, “Fiber-optic signal processor with applications to matrix-vector multiplication and lattice filtering,” Opt. Lett., vol. 7, pp. 463- 465,1982. B. Moslehi, J. W. Goodman, M. Tur, and H. J. Shaw, “Fiber- optic lattice signal processing,” Proc. IEEE, July 1984. A. R. Reisinger, C. D. David Jr., K. L. Lawley, and A. Yariv, “Coherence of a room temperature CW GaAs/GaAlAs injection laser,” IEEE J. Quantum Electron., vol. QE-15, pp. 1382-1387, 1979. M. Tur and B. Moslehi, “Laser phase noise effects in fiber-optic signal processors with recirculating loops,” Opt. Lett., vol. 8, pp.

M. Tur, B. Moslehi, J. E. Bowers, S. A. Newton, K. P. Jackson, J. W. Goodman, C. C. Cutler, and H. J. Shaw, “Spectral structure of phase induced intensity noise in recirculating delay lines,” Proc. SPIE, vol. 412, Apr. 1983. A. Dandridge and A. B. Tveten, “Phase noise of single-mode diode lasers in interferometer systems,” Appl . Phys. Lett., vol. 39, pp.

Y. Yamamoto and T. Kimura, “Coherent optical fiber transmis- sion systems,” IEEE J. Quantum Electron., vol. QE-17, pp. 919-

480-481,1983.

229-231, 1983.

530-532,1981.

935,1981. - -~

r131 J. A. Armstrone. “Theorv of interferometric analysis of laser L 1

phase noise,”J. Idpt. SOC. Amer., vol. 56, pp. 1024-1031, 1966. K. Petermann and E. Weidel, “Semiconductor laser noise in an interferometer system,” IEEE J. Quantum Electron., vol. QE-17,

H. C. Lefevre, “Single-mode fibre fractional wave devices and polarization controllers,” Electron. Lett., vol. 16, pp. 778-780, 1980. K. Petermann and G. Arnold, “Noise and distortion characteris- tics of‘ semiconductor lasers in optical fiber communication sys- tems,” IEEE J. Quantum Electron., vol. QE-18, pp, 543-555, 1982. A. Papoulis, “Probability, random variables, and stochastic pro- cesses.” New York: McGraw-Hill, 1965. M.J.F. Digonnet, and H. J. Shaw, “Analysis of a tunable single mode fiber coupler,” IEEE J. Quantum Electron., vol. QE-18,

D. T. ‘‘inkbeiner, 11, “Introduction to matrices and linear trans- forma ons,” 2nd ed. San Francisco, CA: Freeman, 1966. H. G. Unger, Planar Optical Waveguides and Fibres. New York: Clarendon Press, 1977. R. Ulrich and A. Simon, “Polarization optics of twisted single mode fibers,”Appl. Opt., vol. 18, p. 2241, 1979. S. C. Rashleigh, “Origins and control of polarization effects in single fibers,” IEEE J. Lightwave Technology, vol. LT-I, pp.

L. F. Stokes, “Single-mode optical-fiber resonator and applica- tions to sensing,” Ph.D. dissertation, Stanford University, Stan-

pp. 1251-1256,1981.

pp. 746-754, 1982.

312-331, 198.:.

ford, CA, 1983: L. F. Stokes. M. Chodorow. and H. J. Shaw. “All-single-mode fiber resonator,” Opt. Lett., vol. 7, pp. 288-290, 1982. - A. Yariv, Introduction to Optical Engineering, 2nd ed. New York: Holt Rinehart and Winston, 1976. R. Youngqnist, and H. J. Shaw, “Asymmetric losses in direc- tional couplers: Effects on Sagnac and Mach-Zehnder fiber inter- ferometers,”Proc. SPIE, vol. 412, p. 36, 1983.

*

IEEE JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. LT-3, NO. 1, FEBRUARY 1985 31

listed in Who’s Who in the West, Who’s Who in America, and Interna- tional Who’s Who in Engineering.

Dr. Goodman chaired an I.E.E.E. ad hoc Committee on Optical anc’ Electrooptical Systems in 1969. and served on the Editorial Board of the PROC. IEEE for the years 1979 and 1980. He has also been a member of the Fellows Committee of the Santa Clara Valley Section of the IEEE for several years. For the Optical Society of America (OSA), he was elected a Director-at-Large for the years 1972-1974; he also served on the Board of Directors ex-officio through 1983 in his capacity as Editor of the JOURNAL OF THE OPTICAL SOCIETY OF AMERICA, a position he held for the years

1978-1983. For the International Optical Engineering Society (SPIE), he was elected to the Board of Governors for the years 1980-1982, and has served as a member and Chairman of the Awards Committee, as a member of the Nominating Committee, and as a member of the Publications Committee and the Technical Council. He is a Fellow of the OSA and the SPIE. In 1971, he was chosen recipient of the F.E. Terman award of the American Association for Engineering Education. He received the 1983 Max Born award of the Optical Society of America, for his contributions to physical optics, and in particular to holography, synthetic aperture optics. imaging processing, and speckle theory.

Radiation Losses from Couplers ALLAN W. SNYDER AND FRANK F. RUHL

Ahtract-The antisymmetric mode of couplers, composed of two parallel fibers, becomes cut off when V is sufficiently small. We show that the familiar perturbation modes of this coupler give an excellent approximation to the cutoff value of V previously obtained by an exact analysis. More importantly, by using the perturbation modes, we es- tablish that the rate of leakage below cutoff sets a genuine practical limitation to operation. To do this, ,we develop a scalar theory for radiation losses.

I. INTRODUCTION

A S SHOWN BY Love and Ankiewicz [ l ] , [2] couplers composed of two parallel, identical fibers fail when the

waveguide parameter V is sufficiently small. In particular, they find that the lowest order antisymmetric mode of the composite, two-fiber structure is “cut off” for a finite V value, while the symmetric mode propagates for all values of V.

It is well known that modes that are below their cutoff fre- quency continue to propagate but are leaky. Therefore, it is the rate of leakage and the fiber length that determine the V value for which the mode is sufficiently attenuated that it can be ignored. For example, some leaky modes of multimoded fibers can propagate with significant power for more than a kilometer even though they are well below their “cutoff” frequency [3 3 . Since couplers are comparatively short, say only centimeters in length, it is possible that the effective V for significant modal attenuation is considerably smaller than the “cutoff” frequency determined by Love and Ankiewicz [ l ] , [2]. Accordingly, it is of interest to calculate the modal leakage from optical couplers in order to determine how far below its cutoff a mode can be operated.

Manuscript received May 16,1984. The authors are with the Department of Applied Mathematics, R e

search School of Physical Sciences, Australian National University, Canberra, Australia.

This paper has two main objectives: i) to show how the fa- miliar “perturbation” modes [3] of the two-fiber coupler give an excellent approximation to the cutoff V obtained from the exact analysis of Love and Ankiewicz 111, [2] ; and ii) to de- velop a formalism for determining radiation losses from wave- guides such as the two-parallel-fiber coupler.

11. PERTURBATION MODES OF THE FIBER COUPLER

The modes of the composite two-fiber system shown in Fig. 1 can be derived directly from the scalar wave equation by elementary perturbation methods, provided the fibers are weakly guiding and sufficiently well separated. This procedure is described in detail by Snyder and Love [3, p. 3871, and we simply use their results here. Thus the electric field E?, of the 2 lowest order (+ or -) modes has the approximate form

E+ = **(x,y) e e (1)

where the unit vector e is either x- or y-directed and we have chosen the z-axis to coincide with the fiber axis. Perturbation analysis leads to the following approximation for the field

i (P+z-wt) - -

@* f @* - -

(2) where GI and 5, are the familiar weakly guiding fields of the fundamental mode on one fiber in isolation from the other; they are given in Appendix A-1 .

A. Propagation Constant

The propagation constants p+ for the two modes of the structure have the forms [ 3 ]

p+ =p*c (3)

where p is the propagation constant of the fundamental mode of a fiber in isolation (see Appendix A-1). For a circularly

0733-8724j85/0200-0031S01 .OO 0 1985 IEEE