Optical couplers for coherent optical phase diversity systems

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Siuzdak, J. (1988). Optical couplers for coherent optical phase diversity systems. (EUT report. E, Fac. of Electrical Engineering; Vol. 88-E-190). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1988

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Optical Couplers for

Coherent Optical Phase

Diversity Systems

by

J. Siuzdak

EUT Report 88-E-190 ISBN 90-6144-190-0 March 1988

ISSN 0167- 9708

Faculty of Electrical Engineering Eindhoven The Netherlands

OPTICAL COUPLERS FOR COHERENT OPTICAL

PHASE DIVERSITY SYSTEMS

by

J. Siuzdak

EUT Report 88-E-190

ISBN 90-6144-190-0

Eindhoven

March 1988

CIP-GEGEVENS KONINKLIJKE BIBLIOTHEEK, DEN HAAG Siuzdak, J.

Optical couplers for coherent optical phase diversity systems / by J. Siuzdak. - Eindhoven: Eindhoven University of Technology, Faculty of Electrical Engineering. - Fig. - (EUT report,

ISSN 0167-9708; 88-E-190) Met I it. opg., reg.

ISBN 90-6144-190-0

SISO 668.8 UDC 621.372.83 NUGI 832 Trefw.: optische communicatie.

Abstract

Optical couplers, used in coherent optical phase diversity systems, are analysed. On the basis of the coupled modes propagation theory, expression for the output optical fields are obtained. Then, the signal to noise ratio (SNR) at the output of a receiver using these couplers is computed. The variation of SNR with changes of the coupling coefficient is examined for various couplers. The influence of coupling losses is also investigated.

Si uzdak, J.

OPTICAL COUPLERS FOR COHERENT OPTICAL PHASE DIVERSITY SYSTEMS

Faculty of Electrical Engineering, Eindhoven University of Technology, 1988

EUT Report 88-E-190

Adress of the author:

*

Dr. J. Si uzdak ,

TelecolIllIlunication Division, Faculty of Electrical Engineering, Eindhoven University of Technology, P.O. Box 513,

5600 MB Eindhoven, The Netherlands.

CONTENTS l.Introduction 2.Couplers

3.Photodetectors outputs 4.Noise

5.SNR at the output of the receiver

-2-6.SNR sensitivity over coupling coefficient changes 7.Losses 8.Conclusions 9.Acknowledgement 10. References 3 3 11 16 20 28 33 35 38 39

1. Introduction

The sensitivity of homodyne coherent detection is offset by the difficulty of optical phase locking of two independent coherent sources. One solution to this problem uses a multi port optical network which can give nearly optimum signal detection without phase locking. Fig.1 shows the block diagram of the multi port optical homodyne receiver [5]. It consists of a K-port optical network, K photodetectors, K low pass filters of bandwidth B, K envelope detectors and a summer. The optical network is usually realised by a 3 or 4 port directional coupler formed from 3 or 4 parallel optical fibers arranged to make a single compact passive stable junction. If the optical power launched into one input fiber is equally distributed at the outputs then the receiver is insensitive to the changes of the phase difference between the signal and the local oscillator i.e. the output does not depend on this phase difference. When the above assumption is not met, i.e. when coupling is not ideal, some degradation of the system performance is expected to occur. This problem is investigated in this report.

2. Couplers

In this section we shall analyse the wave propagation in the coupler. The following assumptions will be taken in the analySis:

signal

local

K port

optical

network

esc

i

llater

Fig. 1. Block diagram of the phase diversity receiver.

-4-.

- all the fibers are identic,

the fibers are symmetrically spaced, they are lossless,

- the coupling coefficients between equally spaced fibers are equal,

- if the fibers were not coupled then the modes of propagation would have the same propagation constants

/3

in all of them.

Let the direction of propagation be z. Then we have for coupled modes propagation [I]

1 = 1,2 ... N (1)

Here e. is the electromagnetic field in i-th fib«~r,

/3

is the propagation constant in each

1

fiber in the case there is no coupling between the fibers, kil is the eoupling coefficient between the i-th and the l-th fiber, N is the number of the fibers in the coupler. To make the solution of eqn. (1) easier we substitute

e·

=

a.exp(-j/3z)

1 1 (2)

This is an artificial substitution and it does not imply that the propagation constants in the coupler are the same as that of the ilingle fiber. Using the aforementioned assumptions, we have

for a 3 fibers coupler. Here k is the coupling coefficient between any two of the three fibers (see Fig.2a). For a 4 fibers coupler we have

da

~

+

_jka2

+

jaka₃

+

jka₄ = 0

da

dz

3

+

jaka₁

+

_jka2

+

jka₄ =

0

(4)

da

dz

4

+

_jka1

+

_jaka2

+

_jka4 = O.

The coupling coefficients are defined as in Fig.2b.

a)

b)

Fig. 2. Coupling between the fibers: a) [3x3] coupler,

2.1. 3 fibers coupler

Eqn.(3) has non....,zero solutions only when the following condition if, satisfied [2]

mkk kmk = 0 kkm

Then the general solution of eqn.(3) is given ill the form

3

a. = E b1·exp(jm1z)

I 1=1 I

i = 1,2,3

Here m₁ is the solution to eqn.(5). We have from eqn.(5)

m

l = -2k

m2

=

ID3

=

k

In eqn.(6) bli's should satisfy [2]

IDlb_ll

+

_bl2

+

_b13 = 0 b_ll

+

ID₁b₁₂

+

b₁₃ = 0 b_ll

+

_b12

+

_{ID1 b13} = 0

1 = IJ!~

FrOID eqns. (6),(7) we have the general solution in the form

(5)

(6)

a₁ = b

1exp(-2jkz)

+

b2exp(jkz) a2 = b I exp( -2jkz)

+

b:txp(jkz) a

3 = b1exp(-2jkz) - (b2

+

b3)exp(jkz)

When optical power is inserted to only one fiber, we have for z a

2 = a3 = O. In this case the solution is given by

AO 2AO a 1 =

3

exp(-2jkz)

+

-rexp(jkz) AO AO a2

=

a₃

=

"3

exp( -2jkz) -

"3

exp(jkz)

(8)

(9)

Here AO is the input optical field rms amplitude. (The optical power Po is then given by Po = A6)' We can rewrite eqn. (8) in the form

(10)

(11)

where "{ = kz and the values of 4>1 and 4>2 are given by

sin4>l = _{(2sin"{ - sin2,,{)/r1} (12)

cos4>l = (2cos"{

+

_cos2,,{)/r1

(13)

r 1 =.j5

+

4cos3"{

sin4>2 = -(sin"{

+

_sin2,,{)/r2 (14)

--8-Thus, the optical powers in the fibres are expressed by

2.2. 4 fibers coupler

Eqn. (4) has non-zero solutions only when the following condition is satisfied [2J

m k ak k k m k ak ak k m k =0 k ak k m This yields m! = (2 - a:)k ffi2 = -{2

+

a:)k

ffi3 = ffi4 = a:k

The general solution of eqn. (4) is given by

(15)

( 16)

(17)

3

a1 = 1~1 bliexp(jmlz) i = 1,2,3,4

The values of b_li should satisfy

mlb

_n

_{+ b12 + ab13 + b14 = 0} b

_n

+ m

l bl2 + bl3 + abl4 = 0 ab

_n

_{+ bl2 + mlbl3 + bl4 = 0} b_{ll + abl2 + bl3 + ml bl4 = 0}

The general solution follows from eqns. (19),(20)

1= 1,2,3

a₁ = b₁exp(jakz) + b₂exp(j(2-a)kz) + b

3exp(-j(2+a)kz) a

2 = b4exp(jakz) - b2exp(j(2-a)kz) + b3exp(-j(2+a)kz) a₃ = -b₁exp(jakz) + b₂exp(j(2-a)kz) + b₃exp(-j(2+a)kz) a₄ = -b₄exp(jakz) - b₂exp(j(2-a)kz) + b₃exp( -j(2+a)kz)

(19)

(20)

(21)

When the optical power is launched to one fiber only, we have the following initial conditions: a₁ = AO' a₂ = a₃ = a₄ = 0 for z = O. Then the solution is given by

A a₁ = ! (exp(jkaz) + cos2kz.exp(-jkaz)) A a 2 = a4 = -!jsin2kz.exp(-jakz) (22) A a

3 = ! (- exp(jkaz) + cos2kz.exp( -jkaz))

-10-0

A

a2 = _a4 =

.J-

sin2'Yoexp(jt2)

where 'Y

=

kz, 'YO

=

akz and +1' +2' +3 are given by

sin'Yo(1-coS2'Y)

sint 1 =

-;:::====:::;:==========

j

1 + cos22'Y +

2COS2'YoCOS2~~

cos 'YO (1 +cos2'Y)

cost 1 =

-;:::::==::::;-;:.===========

J

1 + cos22'Y + 2COS2'YoCOS2'Y(;

The optical powers in the fibers are

(23)

(24)

(25)

(27)

3. Photodetectors outputs

Two optical signals are fed into two input fibers: the local oscillator laser of amplitude AO and the received signal of amplitude BS and phase

(J

=

(J(t).

This phase term includes both the difference between the frequencies and the phase noises of both the lasers. All possible arrangements of inserting these signals into 3 and 4 fibers couplers are shown in Fig. 3. Eqns. (3), (4) are linear so the superposition holds and we can directly use the results of the previous paragraph. We also assume that AO

»

BS which will simplify many results.

a)

-12·-3.1. 3 fibers coupler

The outputs of the fibers are given by (10), (11)

A B

u₂ =

~~2(1-COS31)

exp(jt₂) + ;.ni+4cos3rexp(j(tl+ll) (28)

Photodetectors currents are

1= 1,2,3 (29)

Here R is the responsivity given by

R=*I

(30)

where J1. is the quantum efficiency, e is the electron charge, h is Planck's constant, and

The terms containing BS have been neglected. The first term of each of the above equations is a DC component which may be easily filtered out. Then we have from eqns. (12)-(15)

2RAOBS

9 [cosO(cos3-y-1) - sinO 3sin3'YJ

2RAOBS

12 = 9 [cosO(cos3-y-1) - sinO 3sin3'YJ (32)

Eqn. (32) gives us the output of the photodetectors.

:l.2. 1\ fibers coupler (Fig. 3b)

The outputs of the fibers are given by (23)

A

B

-14--A

B

u

2

= ;

sin2 ')expOt 2) +

..J-

J l+COs221+2COS21oCOS210 expO(

0++

1))

(33)

A

B

u

3 = ;

J

l+cos22')'-2COS21oCOS210 exp(jt3) + ..; sin21 exp(j(

0++ 2))

Photodetectors currents are

(34)

Neglecting the DC terms and using eqnso (24)-{26) we finally obtain the outputs of the photodetectors

(35)

3.3. 4 fibers coupler (Fil:. 3c)

The outputs of the fibers are given by (23)

A

B

u2

=

u4

= . ;

sin2, exp(jt 2) + . ; sin2, exp(j(

0++ 2))

(36)

Photodetectors currents are

RAOBS

-16-·

(37)

Neglecting the DC terms and using eqns. (24)-(26) we finally obl:ain the outputs of the photodetectors RAOBS 2 II = 2 [cosO(-sin 2'Y)+sinO(-2sin2'YOcos2'Y)J (38) RAOBS 2 13 = 2 [cosO(-sin 2'Y)+sin0(2sin2Yocos2'Y)J. 4. Noise

There are three main noise sources in the receiYer:

a. Thermal noise of the load resistance of the photodiode and the following amplifier. The power of this noise at the input of each squarer is given by [3J

(39)

Here kB is the Boltzmann's constant, T is the absolute temperature, B is the bandwidth, F(R

However, the power of the local oscillator laser is usually chosen so large that this noise may be neglected as compared with the other noise sources. Therefore we shall neglect the influence of this noise source.

b. Shot noise of the photodetector current. The power of this noise is given by [3]

(40)

Here Ip is the photocurrent, In is the dark current, and IB is the background radiation current. The photocurrent Ip dominates when the local oscillator power is large. Thus

( 41)

As AO

> >

BS the photocurrent Ip is determined by the local oscillator power terms in eqns. (31), (34), (37).

c. Relative intensity noise (RIN) of the local oscillator laser [4]. This noise is due to random changes of the local oscillator power. The RIN current is given by

(42)

where

(43)

Here B is the bandwidth and O'B is the factor depending on the laser itself. The value of II' is again determined by the local oscillator power terms in eqns. (31), (34), (37).

-18--We note that the signal power at the output; of the photodetector is proportional to (eqns. (32), (35), (38))

R2A~B~

where

A~, B~

are the typical powers of the local oscillator and the signal, respectively. As both these powers fluctuate due to the RlN, some noise source is contained even in the signal term. However, the RlN spectral density for the lasers driven high above threshold is of order -130 dB/Hz to -150 dB/Hz

[11].

Thus the power of this noise for

1

GHz bandwidth is 40 to 60 dB less than that of the signal and we neglect this noise as compared with other noise sources. At present we will compute the noises at the outputs of the photocletectors for all the schemes of Fig. 3. The thermal noises will be neglected.

4.1. 3 fibers coupler

The shot noise is from eqns. (31), (41)

2 .2 2eBRA_O lSI = 9 (5

+

cos3-y) (44) 2 .-2 .2 _4eBRAO lS2

=

IS3

=

9 (1 - cos3-y)

The RIN is from eqns. (31), (42)

(45) 2RA2

4.2. 4 fibers cOllpler (Fig. 3b)

We have from eqns. (34), (41) for the shot noise

(46)

2

r2 eBRAO 2

IS3 ""

2

(1+cos 2-y-2cos2'Ycos2'YO)

For the RIN we have from eqns. (34), (42)

(47)

4.3. 4 fibers cOllpler (Fig. 3c)

We have from eqns. (37), (41) for the shot noise the same expressions as eqn. (46). The same holds for the RIN and it is given by eqn. (47).

-20-·

5. SNR at the output of the receiver

The signals at the input of each squarer are

1= 1,2,3 (,4)

so the output is

The signal at the output of the summer is then

N

=

3,4 (48)

The noise is (N = 3,4)

(48a)

We have neglected the shot noise term because it is small if compared with the signal and the RlN. The signal power is

The noise power is from eqn. (48a)

N = 3,4 (50)

Here Ipl is the l-th photocurrent. In the derivation of eqn. (50) we have assumed that the shot noises in each photodetector are independent and they do not depend on the RIN. Furthermore we have assumed that the RIN has a Gaussian probability density function and used

<n~(t»

=

3«ni(t»)2

=

3(u_BB)2.

Apart from the detection scheme shown in Fig. 1 there is another slightly different receiver structure depicted in Fig. 4 [6]. In this receiver signals are subtracted before

3 4 2

signal

1

local

osc

i

llator

,---1

+

_>---1

squar .

+

_>---i

_{squar .}

Fig. 4. [4x4] receiver with full suppression of RIN.

-22--squaring, so only two squarers are needed. The signals at the inputs of the squarers are given by

and

After squaring and summing we have

(IcIl

+

_(I2-li

+

(iscisi

+

_(iS2-iSi

+

_(iR2-iR4)2

+

_(iR2-iR4)2

+

2(ICI3)(iSCiS3)

+

2(ICI3)(iRCiR3)

+

2(iSCiS3)(iRCiR3)

+

2(iS2-iS4)(I2-14)

+

2(I2-14)(iR2-iR4)

+

2(iS2-iS4)(iR2-iR4)

The signal power is given by

(.51 )

Neglecting the shot noise terms, the noise pow(~r is

(53)

5.1. 3 fibers coupler (Fir:. 3a)

The signal power may be readily computed from eqns. (32), (49). We have

(54)

If the optical powers are equally distributed we have from eqns. (16), (17) that cos3-y = - 0.5. In this case

(55)

that is, the signal does not depend on the phase difference between the transmitting laser and the local oscillator 0 =

0(

t). The noise power may be computed from eqns. (31), (32), (50). However, in the general case this computation is troublesome and it gives only little insight into SNR. Thus we will compute the noise power for equally distributed optical powers. We have then cos3-y = -1/2 and Ipl

=Ip2=Ip3=RA~/3.

Some cancellation of the RIN occurs as (eqn. (32))

3

E II = 0 1=1

and the third term in eqn. (50) turns to zero. The noise power is given by

-24--The signal to noise power ratio is from eqns. (55), (56)

If the RIN power may be neglected eqn. (57) yields

2 2

RBS /LBS SNR = 4e"B" = 4lll'B"

(57)

(58)

The last result shows that the sensitivity of this multiport receiver is 3 dB worse than that of the quantum limit [3,5,7].

5.2. 4 fibers coupler (Fig. 3b)

The signal power may be readily computed from eqns. (35), (49). We have

R2A2B2 . 22 [ 21l... 22 . 2:0( 22 22 )]

Vs

= 0 Ssm , cos ""m 'O+sm cos '0+ cos , (59)

If the optical powers are equally distributed we have from eqn. (27) that cos2, =

o.

In this case

(60)

and the phase noise is not yet cancelled. The (:ancellation of the 0

,=

O( t) terms occur when

One very important point must be stressed at this moment. Once we have chosen cos2'Y

=

0 the value of 'YO is also fixed as 'YO

=

k'Y. Thus the condition cos2'Y

=

0 need not imply that eqn. (61) is satisfied (implication holds only for k = 1/2). In other words, the suppression of the phase noise terms occurs in general for other values of 'Y than it is required for equal power distribution. The crucial parameter is k, which is determined by the fibers themselves and their spacing. However, one may say that it is possible to choose such a value of n ('YO = 7r/2

+

n7r) n - integer, that the condition (61) is satisfied at least approximately. In this case we have from eqn.

(60)

(62)

One may say that the condition (61) is not necessary as the mean value of the signal averaged over OCt) does not depend on 'YO and it is given by eqn. (62). In this case however, the signal fluctuates due to changes of

0,

and it will be shown later on that it leads to an increase of the bit error rate (BER).

The noise will be computed for equally distributed optical powers. In this case we obtain from eqn. (34)

4

Some cancellation of the RIN occurs as (eqn. (35)) E II = 0 and the third term in 1=1

eqn. (50) turns to zero. The noise power is then expressed by

(63)

-26--(1;1)

i.e. it is exactly the same as for the 3 fibers coupler.

5.3. 4 fibers coupler (Fig. 3c)

The signal power may be readily computed from eqns. (38), (49). We have

(65)

In this case the requirement of obtaining r~aximum average signal power is not

consistent with the need of cancelling the phse noise terms. Indeed, the maximum average power of the signal is reached for Sin22'1'O = 1 and cos22·Y = 1. It does not offer cancellation of

OCt)

terms. However, it will be shown later on that the BER depends rather on a minimum value of

Vs

tha:~ on its average. Thill minimum will be maximized if Sin22'1'O = 1 and sin 42'1' = 2Cos22.y. From here we get COS22'1' =

2-~

and (66)

It is necessary to stress that this does not correspond to the equally distributed powers. We have from eqn. (37)

and from eqns. (50), (66)

(68)

Comparing eqn. (68) with eqns. (57), (64) we readily see that the SNR given by eqn.

(68) is worse. As the arrangements for couplers from Fig. 3b and c are almost the same we shall not pay attention to the receiver from Fig. 3c any more.

5.4. 4 fibers coupler (Fig. 4)

The signal power may be readily computed from eqns. (35), (51). We have

2R2A2B2 [ . 22 2(} 22 . 2/ll . 22

v S = 0 S sm 1'0 cos

+

cos 1'0 sm VJ sm l' (69)

If the optical powers are equally distributed we have from eqn. (27) that cos21' =

o.

Then

(70)

The cancellation of the phase terms occurs when Sin221'0 = cos221'0 = 1/2 which is implied by the condition cos21'

=

0 only when k

=

1/2. Then

-28--We must stress again that the equal powE:r distribution is not coincident in the general case with the cancellation of

O(t)

terms. For equally distributed powers

In this case all the terms related to the RIN turn to zero in eqn, (5!!) so

and finally

RB2

S

SNR=4eB (72)

This scheme offers suppression of the RIN even if the condition

(li1)

is not met. For the cancellation of the RIN only the equal power distribution is needed. Comparing eqn. (72) with eqns. (57), (64) we readily SeE: that the last receiver has the greatest

SNR which may be very important if the RIN is substantial.

6. SNR sensitivity over coupling coefficient ch:mges

The most important parameter for a digital transmission link is the bit error rate (BER). In our case it is given by [8J (assuming that the gaussian approximation of the probability density functions holds)

BER =

lJ

exp ( -SNR/8) ~ 7r

SNR

If the optical powers are not equally distributed then the value of SNR depends on

O(t). In this case SNR = SNROf(O). Assuming the uniform probability density function of 0 we obtain

1

J21r

~

exp(-SNROf(O)j8)

BER = 21i' dO

'If

0

~

'lfSNR O

flO)

(74)

Using the steepest descent method [9] we may express the last integral as

(75)

for b'fSNR

_o

j8

> >

1, where Of is the variation of f( 0) over the 0 - 2'1f range. We can clearly see that, as we stated before, the value of BER depends mainly on the minimum value of the SNR as OM is the value of 0 for which SNR has the minimum. At present we will obtain the values of SNR = SNROf( 0) for slight deviations from the optimum and for different couplers. We assume that the RIN is negligible to make the comparison between different detectors possible. If this condition is not satisfied the receiver from Fig. 4 has always the superior performance. Then we have from eqn. (50) for the noise power of the detector of Fig. 3a

(76)

-30--(77)

Where 5"«< 1, cos 3"(0 = - 1/2, and Ip =

R.A~/3.

Using eqns. (31), (32) we obtain

(78)

(79)

Thus for 5"(

< <

1

In the same way the signal power is given by (.eqn. (54))

(81)

Finally from eqns. (80), (81)

SNR = SNRO [1 - [3" sin(20

+

1I"/6)5,),J (82)

where SNRO is given by eqn. (58). The noise power for the receiver from Fig. 3b is given by

(83)

Using eqns. (77) with the conditions sin22'}' = 1, cos2'}' = 0, sin2'}'0 = cos2,},0 = 1/./2, Ip =

RA~/4

and eqns. (34), (35) we obtain

(84)

The signal power is given by (eqn.

(59))

(85)

In the derivation of eqns. (84), (85) we used the fact that '}'O = k'}'. We have from eqns. (64), (84), (85)

SNR = SNRO (1+4cos20 o'}'O) (86)

-3~!-(87)

We have made use of eqns. (34),(35). The signal power is given by (eqn.(69))

(88)

Then we have finally from eqns.(87),(88)

SNR = SNRO (1

+

4cos20 010)

(89)

i.e. the sensitivity to 10 changes is the sam,~ for both the 4 fibers couplers. We can also express these sensitivities by means of the optical power differences between the fibers which can be easily measured. We ha.ve for the normalised power differences from eqns.(16), (17), (27)

(4 fibers coupler)

(3 fibers coupler)

Then the sensitivity coefficients of SNR chan@;es with respect to optical power changes are (~/2)

oP

4' (1/2)

oP

3' for the 4 and 3 fibers couplers, respectively. It follows that the 4 fibers coupler is more sensitive to the power changes. The BER for the 3 fibers coupler is given by (eqns. (74), (82))

2'11"

BER3= [2" exp( - SNR/S)

J

exp(f3" SNROsin(211

+

'11"/6)

07/S)

dll 2'11" ~ 'II" SNRO 0

Here 10 is the modified Bessel function of the zeroth order [10] and BERO = BER( 01 = 0). In the same way

(90a)

Since the function 10 increases when its argument increases we readily see that BER4

>

_{BER3 for the same values of 010 = 01, That is the 4 fibers couplers are more} sensitive to the coupling coefficient changes. For large values of SNROOI

»

1 we may use the asymptotic expansion of the Bessel function [10]. We have

BER4 _{exp(SNROol /2)}

~

2f3"'II" SNRO h/S

BER3

=

~

_{2'11" SNRO}

b

_{l /2 exp(f3" SNROol/S)} 0.66 exp(0.2S SNROol ) (91)

We must stress that it _{is an asymptotic formula and any substitution of SNROol} ~ 0 is misleading.

7. Losses

We shall examine the influence of losses for the 3 fibers coupler treating it as an example. The value of the propagation constant is now (30 - ja

_o

and from eqn. (9)

-34-(92)

where p

=

exp(-3a

_o

z). We assumed that AO/3

=

1 and neglected all the common factors. The powers are given by

The phase angles are expressed via

sin'l = 2sin1 - psin21

~

p2

+

4

+

4pcos3'Y cost l = 2cos 1

+

pcos21

~l

+

4

+

4pcos3'Y sint 2 = _ si n1

+

psin21

J

1

+

p2 - 2pcos3'Y cost 2 = - cos1

+

pcos21

~

1

+

p2 _ 2pcos3'Y

The signals at the outputs of the photodetecto:rs are given by

(93)

(94)

• 2

13 = 2R(1

+

p - 2pcos3,),) cosO (96)

Then the signal at the output of the receiver is

(97)

For a lossless medium p

=

1 and eqn. (97) reduces to eqn. (54) (with AO/3

=

BS/3

=

1). If we assume that the powers are equally distributed we have from eqn. (93) that

(98)

This condition may be satisfied only for p

>

1/2 i.e. for exp(-aOz)

>

1/3~

= 0.8. However, the condition (98) does not lead to the suppression of the

O(t)

terms in eqn. (97). It means that this cancellation occurs for a power distribution which differs from the equal power distribution mentioned before.

8. Conclusions

The analysis favours two schemes: the 3 fiber coupler of Fig. 3a and the 4 fibers coupler of Fig. 4. When the R1N is substantial the device from Fig. 4 has the best

-36--performance as it offers the complete suppression of the RIN in the ideal case. However, for correct operation the condition k = 1/2 is required as satisfying eqn. (61a) is rather doubtful. It means that the coupling coefficient between adjacent fibers must be twice that of pairs of other fibers. Manufacturing of such a fiber coupler may be rather difficult. On the other hand, the 3 libers coupler is less sensitive to changes of the coupling coefficient. It offers also some cancellation of the RIN and it is much easier to make, as the condition of equal power distribution is sufficient to the suppression of the phase noise terms. Therefore the three fibers cupler is the best choice when the RIN is negligible.

There is a need of a receiver that combines the advantages of both the above mentioned detectors and does not have their drawbacks. The proposed receiver is shown in Fig. 5. The signal from each photodetector is fed to two subtractors, the outputs of the subtractors are then squared. The signal photocurrents are given by eqn. (32). After subtracting and squaring we have for the output signal

:>

Isquarer

1----.

signal

_{_>>---i\squarer}

~---l

' - - - r - - '

LO 3x3

coupler

_>

Isquarer

~_---.J

(99)

wlllch gives for equal power distribution (cos31' = -1/2)

(100)

In tills case the RIN is completely suppressed. The shot noise current at the output is given by

The shot noise power is then (eqns. (31), (32), (41))

wlllch gives

RB2

S SNR =

4iiR

i.e. the same as for the other couplers in the absence of RIN.

(101)

(102)

Tills receiver offers also the possibility of suppression of the phase noise terms even if

the power is not equally distributed (non-ideal coupling). Indeed, it is necessary to insert only an adjustable attenuator (amplifier) at the output of each subtractor. (The attenuation of the photocurrents (1-3) and (2-3) must be the same in order to reduce sinO

cosO

terms). Then we have

-38-Here kl is the attenuation of the (ICI2) difference and k2 is the a.ttenuation of both (ICI3) and (12-13) differences. It _{is obvious that by correct choice of k1, k2 we are} able to cancel the phase noise terms.

It follows from the last paragraph that the los:!es must be kept as low as possible. It is necessary to stress that the given detection schemes prohibit the use of frequency and phase modulation.

9. Acknowledgement

The author would like to thank drjr. W. van Etten for many helpful discussions during preparation of this report. He is also grateful to Mrs. T. Pellegrino for retyping the text.

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MULTIPLE-BEAM GROUNDSTAT~FLECTOR ANTENNA EUT Report B7-E-171. 19B7. ISBN 90-6144-171-4

SYSTEM: A preliminary stu~y.

Bast;aans, M.J. and A.H.M. Akkermans

ERROR REDUCTION IN TWO-DIMENSioNAL PULSE-AREA MODULATION, TO COMPUTER-GENERATED TRANSPARENCIES.

EUT Report 87-E-172. 19B7. ISBN 90-6144-172-2

WITH APPLICATION

(173) Zhu Yu-Cai

DNA BOUND OF THE MODELLING ERRORS 'JF BLACK-BOX TRANSFER FUNCTION ESTIMATES. EUT Report 87-E-173. 1987. ISBN 90-5144-173-0

(174 ) Berkelaar, M.R.C.M. and J.F.M. TheeJwen

TECHNOLOGY MAPPING FROM BCOLEAN~~IONS TO STANDARD CELLS. EUT Report 87-E-174. 1987. ISBN 90-,5144-174-9

(175) Janssen, P.H.M.

FURTHER RESULTS ON THE MeM I LLAN DEGIlEE AND THE KRONECKER I ND I CES OF ARMA MODELS. EUT Report 87-E-175. 1987. ISBN 90-6144-175-7

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(177 )

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(179 )

Jan:>~en, P.H.M. and P. Stoica, T. Si5derstrom, P. E~khOff

MObEL STRUCTURE SELECT 10Nl'0R MULT1'IARiA8LE SYSTEM BY CROSS-VAll DAT ION MEHIODS. EUT Report 87-E-176. 1987. ISBN 90-6144-176-5

Stefanov, 8. and A. Veefki nd, l. Zai~kova

ARCS IN CESIUM SEEDED NOBLE GASES~,SULTING FROM A MAGNETICALLY INDUCED ELECTRIC FI ELD.

EUT Report 87-E-177. 19B7. ISBN 90-6144-177-3

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ON THE EXPECTATION OF THE PRODUCT OF FOUR MATRIX-VALUED GAUSSIAN RANDOM VARIABLES. EUT Report 87-E-17B. 1987. ISBN 90-6144-17B-l

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EUT Report 87-E-179. 1987. ISBN 90-6144-179-X

(180) Ginneken, L.P.P.P. van

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EUT Report 87-E-180. 1987. ISBN 90-6144-180-3

(181) Rolltm, M.II . .I. clnd P.T.H. Vac:...:;en

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EUT Report 87-E-181. 1987. ISBN 90-6144-181-1

(182) Zhu Yu-Cai

NACK-BOX IDENTIFICATION OF MIMO TR"NSFER FUNCTIONS: Asymptotic properlie; of

prediction error models.

EUT Report 87-E-182. 1987. ISBN 90-6144-182-X (183) Zhu Yu-Cai

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EUT Report 87-E-183. 1987. ISBN 90-6144-183-8

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ENHANCEMENT OF HEAT TRANSFER BY CORONA WIND. EUT Report 87-E-184. 1987. ISBN 90-6144-6

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Obit

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EUT Report 87-E-185. 1987. ISBN 90-6144-185-4

(186) Fu Yanhong

IRE INFLUENECE OF CONTACT SURFACE MICROSTRUCTURE ON VACUUM ARC STABILITY AND ARC VOLT AGE.

( 187)

rUT R!~p()rl A7-r-186. l~fj/. 1:;!iN 90-f.14 11-18G-l Kai ser, F. and L. Stok, R. van den Eiorn

DrSTCN AND IMPLEMENTATION OF A MODUlr-LIBRARY EUT Report 87-E-187. 1987. ISBN 90-6144-1B7-0

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EUT Report 88-E-188. 1988. ISBN 90-6144-188-9

(189) Pineda de Gyvez, J.

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EUT Report B8-E-189. 1988. ISBN 90-6144-189-7

(190) Siuzdak, J.

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(191) Bastiaans, M.J.

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(192) Worm, S.C.J~

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EUT Report B8-E-192. 19B8. ISBN 90-6144-192-7

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EUT Report 88-E-193. 1988. ISBN 90-6144-193-5

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MARTINU5 VAN MARUM: A Dutch scientist in a revolutionary time.

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