Indian Joumal of Pure & Applied Physics Vol. 37, January 1999, pp. 66-72

Performance degradation of' an optical Costas loop due to non-ideal response of 90° hybrid

Be Sarkar and (Ms) R Hati

Physics Department, Burdwan University , Burdwan 713 104 (WB)

Received 27 July 1998; revised 23 November 1998

Effect of non-ideal response of a 90° hybrid in optical Costas loops has been examined. Even a very slight phase deviation '1', from the ideal hybrid phase shift of 90° leads to increased power penalties and receiver sensi tivities. Choice of the optimum power splitting ratio leading to a minimum BER, for a fixed laser linewidth, also increases. The greater the magnitude of the phase deviation, the more the system performance degradation. The nature of these variations does not depend significantly on the number of in-line optical amplifiers.

1 Introduction The homodyne receiver, in combination with PSK

modulation, provides the utmost sensitivity for coherent optical free space and fibre communications. Such receivers require phase synchronisation between the input signal and the local laser oscillator. One elegant method of achieving this goal uses the nonlinear CZostas phase locked loop (PLL)I-4. Its main advantages are that it does not rely on a physically present input carrier. It allows effective coupling in the front end and the low frequency modulation components are not tracked out as the loop bandwidth is increased. As a result, it overcomes the performance penalty associated with generating a pilot carrier, it rules out the problems associated with drift in dc­coupled loops and the choice of loop bandwidth is not limited by the low frequency content of the modulation spectrum. However, because the shot­noise sources are situated within the control loop, the performance becomes dependent on the signal splitting ratio used. A 90° optical hybrid5 is an important system component for an optical homodyne receiver with a Costas PLL. To implement such hybrids the following four techniques have been proposed6 : (1) the bulk optics implementations.7-

1O,

(2) the all-single-mode fiber implementation \{~ ' \(3) an integrated optics implemeniation lJ

. '4 and(4) an over­moded fiber implementation IS. In each of these methods a "nearly ideal" behaviour of the 90° hybrid has been acijieved9

. ' I.I3. '4 . A practical device will, however, suffer an inevitable power loss which depends on the hardware implementation I3

. In the

method outlined in Ref. 13, the value of the hybrid phase shift is controlled by a control voltage. Further, a phase fluctuation of ± 5° and a temperature induced phase drift of about 15°/ °C have been reported '4. These factors point to the fact that even a very small steady phase deviation from 90° may occur in these hybrids . This would inevitably affect the power penalty and laser linewidth requirements of the system. 'The present paper is concerned with the system performance of an optical Costas PLL receiver if a steady phase deviation (\jf) as small as -( 10-3)° occurs in the practical 90° optical hybrid used in it. The effects of laser phase noise, detector shot-noise and optical amplifier noise have been taken into consideration. The effect of \jf on the signal power penalties, corresponding receiver sensitivities and laser linewidth req'Uirements as well as their variations with it have been examined. The results have been compared with those of the ideal case l 6

, when \jf =0. The background theory for system performance analysis is given in Section 2. Section 3 presents the computational results showing the effect of \jf in the practical situation. Finally, Section 4 gI ves a qualitative discussion summansIng the results obtained.

2 Theory

Fig. I shows a Costas PLL receiver used for receiving PSK signal s passing through a number of in­line optical amplifiers . The incoming signal light and the local oscillator merge at the input end of a

..,

SARKAR & HA 11: OPTICAL COSTAS LOOP 67

"practical" 900 optical hybrid6-'\ characterised symbolically here by two beam wmbiners and a (90+ ",)0 phase shifter. Here", is a steady phase deviation( from the ideal 9Qo), of the order of as small as one thousandth of a degree and may be positive or negative. As shown in Fig .• , each of the four outputs of the 9Qo hybrid is followed by a photodetector. The electric field of the signal light going to theI-arm is

orntAL fIBER

lOCAL OSCILLATOII

1- orm

Fig. I--Block diagram of an optical Costas PLL receiver with a non-ideal 9ff optical hybrid [PD: photodetector, LPF: low-pass filter, OA: optical amplifier]

ESt = ~(l-a)Ps exp[j~s(t)]+ .J(l- a)a(t) ... (1)

+ j.J(l- a)b(t)

where Ps is the received signal power. a and (I-a) are the signal power splitting ratios into the Q-arm and 1-arm respectively. The in-phase component aCt) and the quadrature-phase component b(t) ' of the optical amplifier noise account for the impact of optical amplifiers. The optical phase <l>s(t) is given by

1t <t> s (t) = 2" d (t) + <t> SN (t ) ... (2)

where d(t) (= ±I) is the data being transmitted and <l>SN (t) is the laser phase noise. The electric field of the local oscillator light going to the I-arm, with a controlled phase <l>e(t) is

1t E u = ~aPL exp[J {2' + <I> LN (t) + <l>c(t)}] ... (3)

where PL is the local oscillator power, <l>LN (t) the random laser phase, a the local oscillator power splitting ratio into I-arm and <l>eCt) is given by

.. . (4)

with Gveo and ve(t) being the oscillator gain and input control voltage, respecti vel y. A constant phase rt/2 has been added in Eq.(3) to indicate a·rt/2 phase lead of the local oscillator light going to the I-arm with respect to that going into the Q-arm. Following the theoretical calculations explicitly shown in Ref. 16 and briefly outlined here, the electric fields of the lights at the inputs of photodetectors 1 and 2 are, respectively

... (5)

and

... (6)

After detection, the signals at the outputs of the respective photodetectors are

... (7)

and

... (8)

where R is the OlE conversion coefficient. The differential signal in the I-arm is therefore

.. . (9)

Also, the electric field of signal light into the Q­arm is given by

ESQ = ~aPs exp[J<I>s(t)]+.Jaa(t) + }.Jab(t)

... (10)

As seen from Fig. I, the local oscillator light into Q-arm is first phase shifted by (1tI2+ "'), taking the practical situation into consideratiotl . So the electric field of the local oscillator light for the Q-arm, denoted by ELQ, becomes

ELQ = ~(l-a)PL exp[J{<I>LN(t)+<\>c(t)-'V}]

... (11)

Similarly, the light at the inputs of photodetectors 3 and 4 are

68 INDIAN J PURE APPL PHYS. VOL 37, JANUARY 1999

... (12)

and

... ( 13)

respectively, while the signal at the corresponding outputs of the photodetecters, after detection, are

.... (14)

and

.. . (15)

The differential signal for the Q-arrn is .thus

... (16)

From Eqs (9) and (16), the output signal iM of the multiplier is found to be

i M = i lT X iQT

== 2R~a(1- a) Ps PL d (t)( <l>e (t) + 'V) X iQT

= 4R2a(l- a)PLPS (cj>,(t) + 'V)

- 4R 2a(1- a)PL ..jP;d(t)[a(t)cos(S(t) - '1')

+ b(t) sin(S(t) - 'V)] + 2R~a(l- a)PLPsd(t)nQ(t)

... (17)

where cj>e(t) [=cj>SN(t)-cj>LN(t)-cj>c(t)] is the phase error and Set) is equal to $LN(t)+$c(t) . The term no(t) [ = n4(t)­nlt)] arises due to the shot noise n)(t) and n4(t), associated with. photodetectors 3 and 4 respectively. Here, the authors have neglected the second, third and fourth terms on the right hand side of Eq.(9), (obtained on expanding it), since they are negligibly small. The authors have also assumed that the phase error $.(t) is small enough along with 'V, which is -(10-3)0, so that the following linear approximations can be made:

and

cos ($.(t) +'V)= I.

By substituting vc(t)=iM*j{t) in (4) wherej{t) is the impulse response of the loop filter and * denotes a convolution, one obtains

d$, d$N 2 Tt=Tt- Gvw 4R a(l-a)PLPS($,U)+'V)* J(t)

+ Gvcn 4R 2a(l- a)PL ..jP;d(t){a(t)c:os(S(t) - 'V)

+b(t)sin(8(t) - 'V)} * f(t) - Gvco2R~a(l-a)PLPsd(t)nQ(t) * f(t) ... ( 1'8)

Here, $N(t) [=$SN(t)-$LN(t)] represents the total laser phase noise. Considering the noises $N(t), a(t), b(t), net) and the random data d(t) all independent of each other, the power spectral density (PSD) of the phase error $eCt) becomes

S,(w) = w's ..... + 4R' a(l- a)PLP,G;oIF(W)I'[4R'a(l-a)PtI(a') + ~,'V' } + S-ol

liw +4R'a(l-a)PL~,G ... .F(w~'

... (19)

S.N and Sno being the PSD's of the random laser phase noise <!>N(t) and the shot noise no(t), respectively. The noise aCt) and b(t) actually being band-limited white noises, are however treated as white noises, since the noise bandwidth is much larger than the system bandwidth. Their PSD's are assumed to be identical and are given by

... (20)

where, N is the number of cascaded optical amplifiers, hv the photon energy, Ga the gain of each amplifier, n .. the spontaneous eIJ1ission factor and y the loss factor accounting for optical power attenuation in the fiber line.

.As in Ref. 16, the authors have used a first-order active filter, whose transfer function is

.. . (21)

with

G 't =---

1 (21tfJ2 ... (22)

and

-

SAlU<.AR & HATI: OPTICAL COSTAS LOOP 69

~ 't --2 - f

1t: II

... (23)

Here G [=4R2G"""a(l-a)PsPJ is the overall loop gain ; In and ~ are the loop natural frequency and <i<!mping ratio, respectively. Proceeding as in Ref. 16, by expressing the PSD of 4>0 in Eq.( 19) in terms of the closed loop transfer function H(,) ref. I , and then integrating it with respect to I (=oY21t),the phase-error variance is obtained as

... (24)

Here~vis the laser linewidth, q the electron charge and the PSD's S.N and SnQ are 2~v/1t1 and 2RqPL(1-a), respectively . The first term on the right­hand side of Eq.(24) represents the impact of laser phase noise while the second term accounts for the effects of optical amplifier noise and shot-noise" collectively. The optimum loop natural frequency is easily obtained from Eq.(24) as

n(l + 4~2 )[(a 2) + Ps'V 2 + 2!a]

... (25) IIpt

The phase error variance corresponding to this I for a fixed damping ratio then becomes

2 a 4le =

n

... (26)

From Eq.(26) it is evident that the phase error variance increases significantly due to the presence of even a very small phase deviation 'I' (from 900

). This is because of a marked increase in the order of the term Ps'l'2 from the other third bracketed terms in Eq.

(26) . Physically, it were as if the signal power coupled with '1'2 adds to the noise power owing to optical amplifier noise and shot noise, thereby increasing the value of (J,. . This results in a degradation of the system performance. Ideally, in the absence of 'I' (i.e.

'I' = 0) , (J .. would reduce to its expected value'6, and this effect would not occur. Also, since the phase error variance is a positive function in"" the sign of 'I' does not bear any consequence, i.e. it is immaterial whether the actual hybrid phase shift is slightly greater than or slightly less than 900

• It is only the amount of deviation from90° , i.e. 1'1' I which affects the· system performance. Using Eq.(26), the system performance can be evaluated in terms of the bit-error rate (BER) given by

BER=

where the function

erjc(x) = lc J;exp(-y')dy

and

an = [4R 2 PLa(l-a)(a 2 )B

+2qR(aPL +(1-a)Ps )B]"2

.. . (27)

... (28a)

... (28b)

B being the signal bandwidth. In Eq.(28b) , the term(l-a)ps being negligibly small , is ignored so that an is linear with ...JPL , making the BER given by Eq.(27) independent of the local oscillator power.

3 Computational Results

This section shows the degradation of system performance of long haul PSK communication systems with Costas PLL receivers employing practical 900 hybrids having a very small phase deviation 'I' [-( 10-3)0]. In all computations done here, the damping ratio ~ of the PLL is taken as 1/...J2, the OlE conversion factor R as 0.8, the spontaneous emission factor ns as 2.0, bit rate Rb as I G bit/sec . and a matched filter is used for the data detection port in the optical communication system with the optical wavelength A= 1.5J..lm. The signal bandwidth B is taken as 112 of Rb. All amplifier spacings are assumed to be in equal in length along the fiber line. The amplifier gain Ga is assumed to fully compensate for the optical power loss enabling us to write, <a2>=Nhvns_

70 INDIAN J PURE APPL PHYS. VOL 37, JANUARY 1999

3.1 Power penalties and laser Iinewidth requirements

In the absence of tracking error, the BER may be defined as

1 2R~PsPLa.(1-a) BERo = -eifc[ I ]

2 ""1/20 n

... (29)

which gives the shot-noise-plus-amplifier-noise limit for maintaining a fixed BER. Power penalty (in dB) due to incomplete phase tracking is defined as the extra signal power as compared to the shot-noise-plus amplifier-noise-limit required for maintaining the bit­error rate of 10.9 • Power penalties induced by incomplete phase tracking as a function of normalised laser linewidths for various cases of N are shown in Fig.2, both in absence and presence of", (=0.0005° ). Here the power penalty was computed by using the optimum value of (x, ( for a given .1v/Rb ) which leads to the minimum BER. This optimum value of (X is found to increase with increasing laser linewidth, the effect being more pronounced when N=O than when N increases '6. Fig. 2 shows an increase in power penalty for a given laser Iinewidth due to the presence of even such a small value of '" as compared to the ideal case. The increase in the number of optical amplifiers N, does not seem to reduce the rate of increase in the power penalty due to the occurrence of ",. In other words, increasing N does not reduce the effect of",.However, as in the case when .1v is zero, increase in N does seem to reduce the power penalty for a given ",.Fig. 3 shows the corresponding phase error standard deviations for the case of Fig.2. Fig. 4 shows the receiver sensitivity (defined as the signal power for BER= 10.9

) as a function of laser linewidth for ",=0 and",=0.OOO5°. Here also an optimum (X is used for each data point. The results for .1v=O, correspond to the shot-noise-plus-amplifier-noise limit. Here also, for a given laser linewidth, an increase in the receiver sensitivity occurs due to the presence of an additional phase shift ",. It is also clear from this figure that increasing N does not in anyway influence the effect that", has in the system degradation .

3.2 Optimum power splitting ratio

The choice of an optimum power splitting ratio (X, in order to obtain minimum BER for a base-line system (i.e. N=O), is well known l

? and the decrease in its significance with increasing N has also been

reported '6. However, for a given N, the effect of an additional phase shift", on the optimum value of (X is worth noting. Fig. 5 shows a plot of receiver sensitivity as a function of (X for various N in case of

.1v=O.29MHz both in absence and presence of · '" (=0.0015° ). Evidently, for low values of N (e.g. 0,2) the optimum value of (X «l.,P') yielding minimum BER, increases due to ",. For instance, when N=O, (x"p.=.0.13 for ",=0, while the same is 0.23 for ",==0.0015 0

• When · N=2, <lopt ~ 0.1 for ",=0, while it is 0.18 for ",=0.OO15°.However, it is clear that this increase in (Xop. due to '" becomes less distinct with increasing Nand ultimately loses significance.

---~=O-OOO5' 10 ~=O'

BER:10- 9

~g' ... - 6

~~ ,..-~ ~ 4 co ... .c 0.<>'

t~ ~ ~ 2 00. ILE

o u C

o

, /

/

/ /

/ /

I ! ,

2 3 4 ~V/Rb (.,0;3)

I

5 L 6

Fig. 2 - Laser-phase-noise-induced power penalty as a function of normalized laser Iinewidth for various N. both in the ideal (1j1=0)

and non-ideal (1jI=0.0005°) case

--- - 1/1=0-0005' 1 s·O 1/'=0'

BER :10-9

.. ~ .,. .. .., ?JoO !! > .. ..,

... t .. • '" i. 9.0

6

Fig. 3 - Phase error standard deviation at BER = 10.9 versus normalized laser Iinewidth for various N. both in the ideal (1jI=0) and non-ideal (1jI=0.0005°) case

,

SARKAR & HATI: OPTICAL COSTAS LOOP 711

- - - 'II = 0.0005' -2' -- '11=0'

81t rat.:, Gblts/s.c. -32

-1

-40

1-44 L.,;:::-;.;:-~:"--::/~4~-----"> ~-48 " .. .. .. .. > 'ii ¥ 0::

" N 2

--- ---

los.r Iin.width (x MHz)

Fig. 4 - (",=0.0005°) and absence(",=O) of",. taking Rb as I Gbls Receiver sensitivity as a function of laser linewidth for various N. both in presence

-40

, -_.-' ./ " -:A' J ::: ~,~~ ~ ?;.~ --~ ~ ~ ~ ~ ~ ------~ ~~ ~---~~ \ '

-46 \

-- -...

- -"l - ' - - - - - --N' 2

:t ·i -50

: \ ~ -52 \ '§ a:

-54

-5'~~7-~~~~~--~--~--~---L--L-~ o 0'1 0. 2 0') 0· 4 0·' 0.6 0·7 0-. 0" 1'0

lnpul power splilling ratio II

Fig. 5 - Receiver sensitivity as a function of a for various N. with Rb = IGb/sanMv=0.29 MHz. when", = 0 and '" = 0.0015° .

3.3 Effect of variation of additional phase shift

Fig. 6 shows the variation of receiver sensitivity with increasing value of '" for different N, taking ~v=0.29MHz. Evidently, for each case of N shown, the receiver sensitivity increase by about 7 dBm as '" goes from zero to as small as 0.0025°.The lines for

Bit rote:1Gbils/SfC ~v:0·29MHz

-35

-40 E N:IO

!8 N'6 ~ -451-..!!:.~---?:

:~ N=2 '~-50~""":"':~---.. 1/1

... .. ·~-55 N=O ~ ~....;.;..~---0:

.600'" --oJ... 5--'.J..0----::,"'-:. 5:----:2"'-:. O:---:''''":--~

Fig. 6 - Variation of receiver sensitivity with additional phase shilt ",. taking ~v=O.29 Mhz

10 Bit rate = 1 G bits/sec A\I: 0.29 MHz

8

en "C ~

>-(,

'0 c: .. a.. .

4 ... .. J 0 a.. 2

o

Fig. 7 - Plot of laser-phase-noise-induced power penalty versus ",. taking ~v=0.29 MHz

various N run fairly parallel to each other indicating that the nature of variation is identical in each case and does not depend on N. Fig. 7 shows a plot of the power penalty versus "', all other parameters remaining the same as in Fig. 6. The plots for N=2,6 and 10, overlap one another and are close to that of N=O. Here also, one finds that the power penalty increases considerably for", being as small as 0.0025°.

72 INDIAN J PURE APPL PHYS. VOL 37, JANUARY 1999

4 Discussion References

As explained earlier in the introduction, there is every possibility that a very small phase deviation 'If, from 9()<>, occurs in the practical implementation of a 900 hybrid used in Costas PLLs. This in tum leads to a number of consequences as regards system performance, depending on the magnitude of'll. The power penalties , phase error standard deviations and receiver sensitivities increase due to the presence of'lf. The increase in the number of optical amplifiers N, does not alter the degree to which the. system is affected for a given 'If. The laser linewidth and N 7

remaining fixed, the opti~um value of the power splitting ratio (Clop,), required for obtaining a minimum BER , increases due to the presence of 'If. This effect

2

3

4

5

6

8

9

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becomes less significant for larger values of N. The larger the value of 'If, the greH.ter the values of receiver sensitivity and power penalty for a given N and a fixed laser linewidth, indicating a greater perfomlance degradation. The nature of these variations with 'If does not depend significantly on N. Thus, the authors find that the system performance degrades considerably due to the non-ideal response of a 900

hybrid. These changes should be taken into consideration in all practical situations.

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Acknowledgement

The authors acknowledlge the University of Burdwan for the financial assistance provided to them for 'carrying out this work.

11 Kazovsky L G, Curtis L, Young W C & Cheung N K, Appl Opt, 26 (1987) 437.

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