A Theory of Broadband Matching for Optical Heterodyne Receivers

A Theory of Broadband Matching forOptical Heterodyne Receivers

V. K. Prabhu

A theory of broadband matching for an optical heterodyne receiver consisting of a semiconductor photo-diode followed by an IF broadband amplifier is presented in this paper. It is assumed that a finite linearlumped lossless interstage network is used for the broadband matching of the optical receiver, and the re-strictions thereby imposed by the diode on the gain of the optical receiver are obtained both in integraland nonintegral forms. Several types of rational function approximations to an ideal flat gain character-istic of the optical receiver are then considered, and synthesis procedures for thelossless interstage networksto realize these approximations are presented. Explicit expressions are given for the amount of toleranceof broadband performance obtained with these networks.

1. Introduction

Semiconductor photodiodes like Schottky barrierdiodes or conventional p-n or p-i-n diodes are increas-ingly being used'-' 2 for detection purposes in opticalheterodyne (double detection) receivers. They arenormally fast and efficient, converting up to 70%of the photons of the light beam into photoelectriccurrent." Because of the level of intensity of light,almost all photodiodes used in optical detection givean output proportional to the intensity of light.'However, this output is normally so small that furtheramplification is required, and so any practical receiverconsist-s of a photodiode followed by a high gain, lownoise IF amplifier.*

For a constant input optical signal power, it isknown" that the available output power of the diodeis a function of frequency. If the signal bandwidthis nonzero, even if a flat gain IF amplifier is used in thesystem, the available output power of the opticalreceiver will change with frequency of the signalleading to its linear distortion. It is this behavior ofthe photodiode that makes it essential to use a linearlossless interstage network between the diode and theamplifier to obtain a broadband matching of the opticalreceiver. t

The author is with Bell Telephone Laboratories, Incorporated,Crawford Hill Laboratory, Holmdel, New Jersey 07733.

Received 9 October 1967.* The noise performance of optical heterodyne receivers will be

considered in a future publication.t In the literature, the linear lossless interstage network used for

broadband matching of a system is also referred to as an equal-izer network or a matching etwork.13-'8

The characteristics of the solid-state photodiodeare briefly considered in Sec. II of this paper. Ex-pressions have been obtained for the available outputpower of the diode for an optical signal of power P8at an angular frequency o.

In Sec. III of this paper we show that the availablegain of the two-port amplifier is a function of its sourceadmittance' 8 and attains its optimum value Gamax for acertain source admittance Y = Go, + jBo,. TheIF amplifier is assumed to have an ideal broadbandflat gain characteristic so that G, is a constant at allfrequencies of interest and Bo, is zero.

We then consider in Sec. IV the role played by thelinear lossless interstage network in matching theadmittance of the photodiode to the optimum amplifieradmittance Yo,,. Results obtained in the theory ofbroadband matching of linear systems'3-' 7 have beenutilized in getting integral and nonintegral expressionsthat relate the available output power of the opticalreceiver to other parameters of the optical system.The integral relation shows that it is impossible toobtain a perfect match over any finite nonzero band offrequencies.

In the case of a double detection optical receiver, it isshown in Sec. V that Chebyshev and Butterworthapproximations to a flat gain characteristic of the IFamplifier are realizable, and the value E

2 of tolerance ofmatch for these kinds of approximations is obtained.For the photodiodes normally encountered in practice,it is shown that the tolerance e2 for a Butterworthapproximation is a monotonically increasing function

t The consideration of the case in which Go, and B, are bothfunctions of frequency is very complicated and is not discussedin this paper.

April 1968 / Vol. 7, No. 4 / APPLIED OPTICS 657

OPTICALLOSSLESS -

PHOTODIODE INTERSTAGE IF ZNETWORK - L

AMPLIFIERLOCALOSCILLATOR

OPTICALSIGNAL

Fig. 1. A double detection optical receiver. The input to thephotodetector is the sum of the local oscillator beam and the

incoming signal beam.

R

a- Ytp)

average current Io is caused both by the time average il-lumination and by the electrons and holes that are gener-ated at or near the junction. The signal current I iscaused by that portion of the total illumination at ornear the signal frequency of interest. In general,C(Vo) and R(Vo) are functions of the bias voltage.Assuming that in practical cases the excursions aroundthe bias point are small, we shall henceforth assume thatC(Vo) is a constant capacitance C and the series re-sistance R(Vo) is a constant resistance R. An equiva-lent circuit of the photodiode is shown in Fig. 2.

The peak photoelectric current I for a doubledetection optical receiver can be shown 9 20 to be givenby

Fig. 2. Equivalent circuit of photodiode. The physical sourcesof noise that may be present in the diode are not shown. Also theconductance G, that appears in parallel with C is usually sosmall (of the order of 10-' mho) that it can be neglected for all

practical purposes if c # 0.

of n, the order of complexity of the lossless interstagenetwork. For a Chebyshev approximation the toler-ance e2 is shown to be a monotonically decreasing func-tion of n, but that no great improvement can beobtained by using very high values of n. The case ofn = 1,2 is considered in great detail for the Chebyshevapproximations, and general methods have been pre-sented to synthesize any lossless interstage networkthat will meet the sufficiency requirements imposedby the diode.

I. Photodiode and Its Available Output PowerThe optical heterodyne receiver shown in Fig. 1

consists of a photodiode followed by a lossless inter-stage network, and an IF amplifier of center frequencyQo and semibandwidth W. The geometrical centerfrequency c0 of the IF amplifier is defined as

coo = [(o - W)(Qo + W)]i. (1)

The available gain Ga of the IF amplifier is assumedto be flat for 20o - W < w < Qo + W. We assume thatthe large amplitude local oscillator used in the doubledetection optical receiver does not have any effect onthe frequency behavior of the optical receiver followingthe photodiode. Any effects of beam misalignment,any effects of nonuniformity of the surface of thediode, etc. on the frequency behavior of the opticalreceiver are not taken into account in our analysis.

The diode is normally so arranged that the junction,or portions of it close to the function, is illuminatedby the sum of the local oscillator beam and the incom-ing signal beam. The electron-hole pairs thus createdby the incoming photons give rise to a small signalcurrent.' In operation, a reverse bias Vo is put on thediode, and the characteristics of the device' 2 for smallexcursions around Vo are that of a signal current genera-tor 1 and a direct current generator lo in parallel witha capacitance C(Vo), and this combination in serieswith a parasitic series resistance R(Vo). The time

1 = 2,,1q/hv(PbPs)2, (2)

where -q is the quantum efficiency of the photodiode, *q is the charge on an electron, h is Planck's constant,v is the optical frequency, P8 is the signal power, andPO is the local oscillator power. t t

It will be assumed in this paper that 1 in Eq. (2)is independent of frequency. Since the broadbandperformance of the optical receiver is not normallyaffected by the dc current o, we do not consider oany more in this paper.

From Fig. 2 the available power of the diode can berepresented as**

Sad = I,1l/8a'C'R. (3)

111. Available Power Gain of the IF Amplifier

The IF amplifier used in the optical receiver isassumed to be an ideal broadband amplifier, and itsterminal characteristics can be represented as (seeFig. 3)

(4)[II] = 2y2 y 2 YV

I2 Y2 Y22 V2

Let such an amplifier be driven by a current source I,with an internal admittance Y, (see Fig. 4).

The available power gain Ga of the two-port shownin Fig. 4 is defined as the ratio of the available power

* A quantum efficiency of greater than 70% has been obtained"for Schottky barrier photodiodes.

t In practice, a fraction k < 1 of incident photons is absorbedin the active region of the diode. To account for this effect, 1Iis usually multiplied by a factor k. Also for some diodes, I, ismultiplied' by a transit time reduction factor If(wT)I < 1, r beingsome effective transit time. We assume in this paper that f(T)

can be considered independent of X in i2o - W < co < 2o + W, and

f(wr) I = 1. We also assume that k = 1.t To account for any mismatch between local oscillator beam

and signal beam, I, is also usually multiplied'0 by a beam match-ing factor , where 3 < 1. We assume that A = 1.

§ The availability of ideal components is assumed in this paper.** From Eq. (3) it appears that if the series resistance R were

zero, this combination could give infinite available power. Ofcourse, the actual device cannot do this; the stated equivalentcircuit is only valid for small signal voltages.

658 APPLIED OPTICS / Vol. 7, No. 4 / April 1968

Tot I't I C

T

Using Eqs. (8)-(11), we can rewrite Eq. (7) as 8

Fig. 3. IF amplifier characterized by its admittance parameters.

TSt X S VI V2

Fig. 4. IF amplifier driven by a current source with an internaladmittance Y,.

Soa at the output of the network to the available powerSia from the source. We can, therefore, write

G. = Soa/Sia (5)

(1/Ga) = (1/Gamox) + (Rf/G.)IY.- Yogi ,

where

Rf = Re Y22/Y212.

It is also possible to write Eq. (14) in the form8

(G - Gh)2 + (B - Bh)' = G,

(14)

(15)

(16)

where G, B, and G can be found from Eq. (14).Equation (16) represents a family of circles.

If no matching network were used between the diodeand the IF amplifier, it is clear from Eqs. (3) and (14)that the available output power of the optical receiverwill be different at different frequencies for the sameinput power P. This will result in severe distortionof the signal. Also from Eqs. (2), (3), and (14) themaximum available output power Pomax of the doubledetection optical receiver at any frequency can berepresented as

Pomar = Gamax(?7q/hv)1PoPa/w'C'R,and t

Si. = I'/2(Y + Y*). (6)

(17)

or

It can be shown'8 from Eqs. (4)-(6) thatt

G, I~~Y21 12( Y + .*) 7(y22 + Y22*)Y11 + Y.1 - Re [Y12Y21(Yll* + Y*)] (7)

We can now show from Eq. (7) that G attains itsmaximum value Gamaxr for a source admittance Y,

Go,-+jBo,, where

Gamax = Y2/Y12I 1/[X + (X2 - 1)1], (8)

G09 = [IY12Y21I/(Y22 + Y22*)](X2 - 1)1 (9)

Bog = -Im y11 + Im (2y21)/(Y11 + y11*), (10)

and

- 2 Re (yui) Re (22) - Re (2y21) (11)

1Y12Y211

Since the amplifier is assumed to be an ideal broad-band IF amplifier, it follows that Gam, Go,, Bo,and X are independent of frequency. § We furtherassume that the amplifier under consideration isstable2 '-2 3 for all linear passive input and outputterminations. In such a case, it can be shown2123that

X > 1. (12)

From Eq. (9), we have

Pomax =4,oR(wc/X)', (18)

where

w, = 1/RC, (19)

and

,Do = Gamax 7 q/hv)2PoP. (20)

Let us design the lossless two port equalizer* shown inFig. 1 so that the actual available output power Poa isas close to Poma. as possible over the band width of in-terest.

Without any loss of generality, we normalize all ad-mittances with respect to Go,. Also because of ourassumption about the ideal characteristic of the IFamplifier, X0, = . t

IV. Theory of Broadband Matching

Refer to Fig. 5. Let

L S 812

S8=_S21 822_

denote the scattering matrix of the reactive two-portequalizer normalized to Y(p) on the left and 12 on theright.2 4 Let us define two reflection coefficients Pi(P)and P2(P), where

(21)

P1(p) = [Y,(p) - Y(-p)]/[Y(p) + Y(p)l,

P2(P) = [Y2(P) - 1]/[Y2(P) + 11,

t A* denotes the complex conjugate of A.I Re z and Im z denote respectively the real and imaginary parts

of the complex number z.§ The study of the case in which Garma, Go8 , Bog, and X may be

functions of frequency is very complicated, and is beyond thescope of this paper.

* Equation (18) shows that there is 1/w2 dependence in Pomax.If this is undesirable, a suitable transmission shaping networkcan be placed at the output of the receiver.

t For any realizable admittance, Bo,,(w) should be an odd func-tion of w.


Go, > 0. (13) (22)

(23)

1 p2 W2 E f( P - Po )"

7r P + o k= o

F(p) =g(P) = _ Z Ok(P -POkg~)-21(p) -. = o

Fig. 5. Losslessoptical receiver.

Ye Y2

equalizer used in broadband matching of theY(p) is the admittance of the photodiode as

shown in Fig. 2.

Also, let eq(p) be a regular all-pass network such that

'7(p) = II P -1=1 P + Ill*

and

and p = - + jw is the complex frequency variable.Since the network N is lossless, it can be shown24 that

IP,(jW)I = IP2(jW)!. (24)

Also from Eqs. (14), (23), and (24):

1 1 1-+± 4RfG, 0 . (25)

!ps(jW)j' IP2(jW)l2 (1 Ga) - (1/Gamax)

Let us denote the poles of Y(-p) in Re p > 0 by a,,ca, . . . ,am and set

3(p) = H (p - a)/(p + ar*).r= 1

(26)

From Eqs. (22) and (26), we can also write

(p){1 - Pi(p)} = O(p)[Y(p) + Y(-p)]/[Yl(p) + Y(p)].(27)

Equation (27) shows that, regardless of the losslessinterstage network used in the receiver, every zero ofY(P) = 2 [I + Y(-p)/Y(p) I in Rep > 0 must also be

a zero of

. 1P) = 3(p)[1 - pi(p)].

In 77(p) = E 71k(p - po)5.k=O

(37)

uIs are a set of points in the right half of the com-plex plane.

Let

1 4R 1Go,=1± (38)

and let so(p) be such that all its zeros and poles are inthe left-half plane.* It is now clear that if t

O(p)pt(p) = (p) = 7/(p)SO(p),

1 1 1

Is(j ' - PIC(jw)I' - sO(jU )i'

= 1 + - 4Rf0Go(1/G) - (Ganax)

(39)

(40)

We may now show"11-'7 that Yj(p) is a positive-realadmittance if and only if:

I. At every Class I transmission zero p0 of order k,

(28)

A zero p of -y(p) in Re p 2 0 of multiplicity k is said tobe a zero of transmission of the admittance Y(p) oforder k. Youla distinguishes between four kinds ofsuch zeros.'5 Class I contains all those in the strictright-half plane. Class II contains all those on the realfrequency axis which are simultaneously zeros of Y(p).Class III contains all those on the real frequency axisfor which 0 < Y(po) I < a, and Class IV contains allthose for which Y(po)l = . The restrictions im-posed on p(p) through Eq. (27) are formulated"-' 7

most compactly in terms of coefficients of the powerseries expansions of the following quantities:

Sr = Br, 0 < r < k - 1; (41)

or

bo = eorj + 0 - folu I + (i/O)- (l/Gaa dc,

=0,if d [1 + 4RfGoaa] 0,

= 1 if i I + 4RfGog ] 0doI (1/OnG ) - (1/Gara,.)1 = 0

(42)

and

S(p) = (p)P'(p) = E Sk(p - PO)',k=0

In s(p) = E sip - po),k=O

$(p) = E Bk(p - po)',k=O

InI 3(p) = E b 5(p - p)k.k =o

F(p) = (p)[Y(p) + Y(-p)] = E Fk(p - pO)',k=o

(29)

= r - f r + (i/Ga) - /Gun] dw,

1 < r < k- 1. (43)

(30)

(31) * It will be recognized that so(p) is a minimum phase function."

t Multiplication of p,(p) by 3(p) is necessary to make it analytic(32) in the right-half plane. This multiplication makes s(p) a bounded

real scattering coefficient.' 5 :Multiplication of so(p) by '1(p) intro-duces right-half plane zeros. This is sometimes necessary'4

(33) and is done so that s(p) can satisfy all the constraints imposed byY(P).


II 12

in(34)

(35)

(36)

II. At every Class II transmission zero jwo of orderk,

S = Br, O < r < k- 1, (44)

and

(Sk - B)/(Fk+ 1) < 0; (45)

or

b = 7r1 - + / (l/Gamax],

0 <r < k - 1.

and*

(bk - S)/(0k4 1) > 0-

If Iwo! = 0, or a, Eq. (47) may be replaced by

bk - 7 + fkln 1 +- 4IGlo, d, 0 1/Ga - (1/Gamax) dw

0ic+1

(46)

where a-, is the residue of Y(p) at p0 = jwo; or

br = 'r -X frill [I + (1 GO - IG .... )] dw,

O < r < k-1, (56)

and

(2°k-l)/(bk - k) 2 a-l, (57)

with equality if and only if the matching network isnondegenerate. If lwoo = 0 or , Eq. (57) may bereplaced by

20k- _ 4 a-1. (58)

(47) bk- nTh + fkin + (1/G) -4JiGGma dw

If Eqs. (41)-(58) are satisfied, Y(p) is a positive-real admittance. If Yj(p) is positive-real, the Darling-

(48) ton method can be used to obtain the lossless two-portinterstage network needed in the receiver.

III. At every Class III transmission zero of orderk^,

Sr = B., 0 < r < k - 2, (49)

and

(Sk-, - Bk-,)/Fk < 0; (50)

or

b, = - f [1 + (1/G) RjG, ( ] dw,

< r < k-1, (51)

and

(bk-1 - Sk-,)/0k < 0, (52)

with equality if and only if the matching network is non-degenerate. If lwol = 0 or , Eq. (52) may be re-placed by

bk-, - ,- + f, [ 1 + 4RGf G ] dw

ok> 0.

(53)

IV. At every Class IV zero of order k,

Sr = Br, 0 r < k - 1,

and

(Fk-*)I(S - Bk) a-I,

(54)

(55)

IV. Derivation of the Integral Constraint onthe Gain of the Optical Receiver

The theory of broadband matching presented inSec. III of this paper is now utilized to derive the con-straints imposed by the photodiode on the gain of theoptical receiver.26 The admittance Y(p) of the photo-diode shown in Fig. 3 can be written as*

Y(p) = (1/RGog)(p/p + we). (59)

From Eq. (27), it can be shown that the only transmis-sion zero of Y(p) lies at pa = 0, and is of order 1. Alsofrom Eq. (26),

$(p) = (p - cO)/(p + c)

= -1 + (2p/lwr) - (2p/co,2 ) + . . .

From Eq. (33), we can write

F(p) 1 2p1C

Go0, wc,(1 + p/wc)'

= 1 2CPS 4Cp'Go L c 2o . I'

Since the transmission zero is ofwrite from Eqs. (44)-(45) that

So = -1,

and

S < 2/ w,

where

s(p) = i-(p)sO(p),

(60)

(61)

(62)

(63)

Class II, we can

(64)

(65)

* If the matching network is nondegeneratel4"15 Eq. (47) be-comes an equality. If Y(wCo)I , the network is said to benondegenerate if and only if Y(jwo) + Y(jwo) 0. If |Y(jwo)|= X, the network is said to be nondegenerate if and only ifI Y. (ico)1 5d X,

(39)

* It must be noted that the equivalent circuit shown in Fig. 2for the photodiode is valid for frequencies co >> 0.

April 1968 / Vol. 7, NQ. 4 / APPLIED OPTICS 661

I5

I0

Po. FOR I

P0FOR300 MHz n-2 \

2~~~~~~~~~~~'I0 0 100 200 300 400 100 600 700

FREQUENCY f * -S IN MHZ-

Fig. 6. Normalized available output power P 0 /10 lR as a functionof for Butterworth approximations of order n = 1,2. It is

assumed thatf0 = co,/27r = 31.83 GHz, and = 1.

and

so(p),so(-p) = [1 + (1 4R,00 0 ]. (38)

Also from Sec. III, Eqs. (46)-(48), it follows that

1-f -2In 1 + ( -/GG) (G ]2 V A I )

(66)

where /A 's are a set of points in the right-half plane.Since Re (/,ul) 0 for all 1, we put 1(p) = 1. We,therefore, have

1 in 1I + (l/G ) - (/Gnox)] . (67)

Denoting the actual available output power of theoptical receiver at any frequency co by P, and usingEqs. (17)-(20), Eq. (67) may be rewritten as

1In 1 ± (hv)2 8R1 0 (w/co)2 ]Gdo <2W ^2 77q RPoP (/Poa,) -(/o_.]x dw W,

(68)

It must be noted that Rr and Go, are completely deter-mined by the IF amplifier used in the system, and if weassume that the signal power P, remains constant at allfrequencies of interest, Eq. (68) shows that Poa cannotbe made equal to Pomax over any nonzero band of fre-quencies in spite of the fact that any arbitrary linearlossless interstage network may be used in the receiver.This is one of the important results of this paper. Itmust be noted that the equivalent circuit shown in Fig.2 has been assumed in deriving Eq. (68). This equiv-

alent circuit very well describes the behavior of thediode 2 provided the lowest frequency of the signal oc-curring in the system is very far from zero. Also wemust observe from Eq. (68) that Poa can be made equalto P0o. at a finite number of discrete frequencies.' 4

V. Rational Function Approximations to theAvailable Output Power of the Optical Receiver

We have shown that Poa cannot be equal to Pomaxover any nonzero interval Qo - W < X < Qo + W of thefrequency spectrum.* We shall, therefore, make somerational function approximations to a flat gain charac-teristic of the optical receiver. If these rational func-tion approximations satisfy all the constraints of Sec.IV, a finite linear lumped lossless network can be found2 5

which realizes this kind of gain for the optical receiver.A complete treatment of this problem is beyond thescope of this paper, and certain kinds of approximationswidely used in network theory are considered.

A. Butterworth ApproximationsThe problem at hand is to approximate Po, as close to

Pomax as possible over the range Qo -W < co < Qo+ W.A set of Butterworth polynomials can be used for thispurpose. 2,27 Let

1 + 6'[(C02 - co2)/2WV]"'

(69)

where n is the order of complexity of the interstage net-work to be used in the broadband matching of theoptical receiver, and n is the order of the Butterworthpolynomial.2 7 It may be verified that Poa approxi-mates POmax in a maximally flat manner,2 7 and the be-havior of Poa as a function of w is shown in Fig. 6.Since it can be shown that Po, in Eq. (69) can be madeto satisfy Eqs. (64)-(68) by properly choosing E2 for allvalues of n, the approximation of Eq. (69) is realizable.

From Eq. (38),

so(p)so(p) =1

4RGoGamax

12[(W2 wo1)/2wW]2,(70)

It can be shown2 7 that

sO(p) =

t4- (p 2 + 102)4(p2 + cool)" + a-i(2pW)(p2 + &cO2)-1 + ... + a(2pW),

(71)

where

(4RfGooGamax/62)"12,-= sin(7r/2n) (72)

* Since the gain Ga is a real and even function of co, we onlyconsider the behavior of Poa for co > 0.


Since no useful purpose is served by using highervalues of n, we shall only consider the case n = 1. Forn = 1,

2 4RiOGGamax,Em- -

[(wO/Wc,)(0o/W)] 2

(W) 2 1P~a = '1 co Iu + 62[(12 - wos2)/2coIV]2

no *

Fig. 7. A typical plot of e2_,in/(4RfGogGamax) as a function of nwhen Butterworth approximations to a flat gain characteristic areused. Even though n is a discrete variable, the plot is given for

all n > 1.

and

S(p) = so(p) = -p 2 + 002

p2 + 2p(wo2 /w,,) + 002

C(fto+W)l (no- W)

AMPLIFIER

Fig. 8. Lossless interstage network for a Butterworth approxi-mation of order n = 1. The ideal transformer ratio t is given by

t = (RGo)-.

We can now expand Eq. (71) into a Taylor series*about p = 0. We have

so(p) = -1 + a- 1(2W/CU02)p - .... (73)

Now

S7 (p) = T A + (36)i1P + *

= (-1)'[1 - pz~l l + * + (74)

From Eqs. (39), (65), (73), and (74), we have

2W 'j, (1 1 )2an-W2 + a < - (75)

Since [(ls,) + (l/,Z)*] 2 0 for all 1, let us put 2(p) =1. We can then write from Eqs. (72) and (75) thatt

2 > 4 RfGoG7amax-eo(w/)(wo/W) sin(r/2n) (76)

A typical value of f = c/2 7r for a photodiode is about31.83 GHz. From this value of f, fo = o/27r = 300MHz, and (2W/Qo) = 100%, we have plotted, in Fig. 7,Emin2 /(4RfGyoGamax) as a function of n. It may be seenfrom the plot that min2 is a monotonically increasingfunction of n. This behavior of Emi,2 canbe explained bythe fact that Butterworth polynomials approximate theideal flat gain characteristic of the optical receiver in amaximally flat fashion.2

* We choose negative sign for s(p) so as to satisfy Eq. (64).This does not entail any loss in generality."5

t Equation (76) can also be obtained by using Eq. (68).$ Typical values of R and C for a photodiode are28 C = 1 F,

andR = 5 2.

Now from Eqs. (22), (39), and (79), it can be shownthat

Y 1(p) = 1 wo2/co,

RGog p + co/cl (so)

The circuit to realize Y(p) is shown in Fig. S. It mustbe remembered that 0o

2 is the geometric mean of theband of frequencies of interest, and

L = 1C(Q- W)(Qo + W)

n = (GoR)1.

(81)

(82)

This circuit agrees very well with our physical intuition.

B. Chebyshev ApproximationsOf the various means of approximating a given func-

tion, the Chebyshev method is one of the most interest-ing and important. It can be shown2 9 that given a setof n parameters, a function f(cw2) approximates g(co2) inthe Chebyshev sense if the parameters are determinedin such a way that the largest value of g(co2) -f(co2)Iin a given interval is minimum.** Since for a givencomplexity of the structure the maximum amount oftolerance for a Chebyshev approximation is the samethroughout the band, this type of approximation seemsto be the most desirable in the broadband performanceof optical receivers.

Let us try to approximate Poa by:

PO = PO-aX 1 I1 + 62T 2[(CL2 - Wo2)12wiV] (83)

where Tn(x) is an nth degree Chebyshev polyilomialgiven by 25'27' 29

T,,(x) = cos(n cos-'x). (84)

The behavior of Po, as a function of for n = 1,2 isshown in Fig. 9. The equiripple behavior of P isevident from Eq. (84). We can also show that the ap-

§ Since we are interested in mininum value of e, we have usedthe equality sign in Eq. (77).

** The Chebyshev approximating function has the equirippleproperty. 2 5,27,29


(77)

(78)

(79)

a.

200 300 400 500FREQUENCY f =w/2r IN MHZ

Fig. 9. Normalized available output power Poa/'4oR as a func-tion of for Chebyshev approximations of order n = 1,2. It is

assumed thatfc = co,/27r = 31.83 GHz, and e2 = 1.

proximation of the type given in Eq. (83) can be madeto satisfy Eqs. (64)-(68). It can also be shown that:

(p2 + W02), + b 2(2pV)'(p2 + W02)n-2 +(p2 + ,o2)n + a,,-1(2pW)(p2 + W02)-1 + .

where

a,,_ =sinh{ (1/n) silh-1[2(RfGOoGamax)/e] }

sinwr/2n

(85)

and

[fEmi.'In-2 2 1.25. (91)

[6min'] 7l-° 8 9

From Eqs. (90)-(91) and Fig. 10, we conclude that6

min2 is a monotonically decreasing function of n, butthat no great improvement in the value of Emm.2 is ob-tained by using very high values of n. Since the com-plexity of the network increases with n, we shall onlyconsider the cases n = 1,2. For n =2, Emm.2 attains1.25 times the minimum possible value. For n = 1the Butterworth and Chebyshev approximations arethe same. For n = 2, from Eq. (89)

Emi.2 = 2RfGogGamax(W.:c/0o)2(W/1o)2

and from Eqs. (85), (86), and (92), we can write

So(P) =

(pl + Cwo2)2 + (2pW)2(p2 + &Wo2)2 + (wo/w)(o/W)(2pW)(p2 + Wo2) + .(2pW)'

(92)

(93)

Equations (22), (39), and (93) show that

1ip W02 P2 + (,0

p) Go, Xc p3 + p2(wo02/c) + p(wo' + 2W 2) + (W0c 4/W)

(94)

The lossless interstage network realizing Y1(p) in Eq.(94) is shown in Fig. 11.

Similar methods can be used to determine the losslessinterstage networks when n > 2. We have shown,however, that no great improvement can be obtainedby using very high values of n.

(86)

If we expand so(p) about p = 0, we can show that

so(p) = -1 + pa,-1(2W/WO2) + .... (87)

We again put qj(p) = 1 to obtain minimum C2. FromEqs. (65) and (87)

fmi.2 __ sinh2 4 RfGoGa ... (88)6mm,,' = in sinh'[(wo/1W,)(co/W) siin(7r/2n)] I' 88

In practical cases, assuming the wo/coc < 1/(15)1,(co/wc)(wo/W) sin(7r/2n) < 1, for o/IW < ( 1 5 )1.*In this case, we can write

16lIGoGamac

r2(co/Wc)(Wo/W) [(sin7r/2n)/(r/2n)] 2

t

'51

E 6,,N O

cra:

(89)

A normalized plot of e,6nmn for fo = 0o/ 2 7r = 300 MHz,f = 31.83 GHz, and 2W/1 = 1007 is given in Fig. 10.We note that

[emin2] _1 7,r2 25m ,.__= - m2.a

[mmin21 n-a7 4(90)

* It can be shown from Eq. (1) that wa/W < (15)" for 2W/2o> ;-.

n -_

Fig. 10. A typical plot of emin_/(4RfGogGamax) as a function of nwhen Chebyshev approximations to a flat gain characteristic areused. Even though n is a discrete variable the plot is given for all

n > 1.


Emin2 =

2 ( W 2

CWo2 W

R C2

C ( 0 ) X gOPLIFIER

Fig. 11. Lossless interstage network for a Chebyshev approxima-tion of order n = 2. The ideal transformer ratio t is given by

t = (RG.,) 1.

C. Butterworth and ChebyshevApproximations with Less than Maximum Gainin the Passband of the IF Amplifier

In the preceding parts of this section Butterworthand Chebyshev polynomials are used to approximatethe ideal flat gain characteristic of the IF amplifier insuch a way that the maximum passband gain is Gamax.They can also be used"" 6 in a manner in which themaximum passband gain is KGamax, where

O<K< 1. (95)

Such approximations are given by

1 + [(W2 - Wo2)/2wW'21 (96)

and

Po = P--- K (71 + 2Tn2 [(w2 - C0o2)/(2jWT) (

where T3 (x) is an nth degree Chebyshev polynomial.It can be seen from Eqs. (96) and (97) that

[Po-]ma = KPomax < PomaX. (98)

If Eqs. (64)-(68) are to be satisfied, it can be shownfrom Eq. (96) that(1 -K + 4KRiGoGamax)"/2n - (1 -K)/2n

< (o/c)(wo/W) sin(7r/2n). (99)

Also, if P 3 in Eq. (97) is to be realizable, it can beshown that the following constraint must be satisfied:

sinh sinh (1 - K + 4KRfGo,,Gamax)1]In E

-sinh[(l/n) sinh-'(1 - K)1 /e]

< (o/wc)(wj/W) sin7r/2n. (100)

In general, for arbitrary n, Eqs. (99) and (100) can onlybe solved numerically. The numerical solution ofEqs. (99) and (100) requires that the value of RfGo,Gamax, wolwc, and w0 /W be known. For any specific IFamplifier, the values of K and 2 can be determinedfrom Eqs. (99), (100), and the matching network canthen be synthesized. Since we do not propose to gointo the characteristics of the IF amplifier, we do notconsider Eqs. (99) and (100) any more in this paper.

Other kinds of approximations like maximum averagegain approximation and least-squares approximationcan also be used in the theory of broadband matching ofthe optical receiver. If these approximations satisfythe restrictions that are imposed by the photodiode andthat are given in Eqs. (64)-(68), the methods given inSec. IV can be used to obtain a positive-real Y(p).This Yj(p) enables us to determine the lossless matchingnetwork required in the broadband matching of theoptical receiver.

VI. Conclusions

A theory of broadband matching for optical hetero-dyne receivers is presented in this paper. It is shownthat the following constraint must be satisfied by anylossless matching network used for the broadbandmatching of the optical receiver:

- Z- [1 + (hp) 2 Rf Go, (W/C) ]I X W la2 +G RPoP (/Poa) - ( I/Pomax)

X d < 2/w,. (68)

Equation (68) shows that it is impossible to obtain aflat gain from the optical receiver for any nonzero bandof frequencies and for any realizable lossless matchingnetwork.

Certain rational function approximations to an idealflat gain characteristic of the IF amplifier are then con-sidered. It is shown that Butterworth approximationsto an ideal flat gain characteristic are realizable, but thatthe tolerance of matching is a monotonically increasingfunction of n, the order of complexity of the lossless in-terstage network. However, by approximating anideal flat gain characteristic by Chebyshev polynomials,it can be shown that 2 is a monotonically decreasingfunction of n. For the photodiodes usually encounteredin practice, no great improvement in 62 can be obtainedby using very high values of n. We have shown that[62 Jn=2/ [2 In=m = 1.25, and realizations of networks forn = 1,2 are given.

Design methods and equations for any kind of ap-proximations to the ideal flat gain characteristic aregiven, and the constraints to be satisfied by these ap-proximations so that the gain can be realized by arealizable lossless matching network are explicitlystated.

I would like to express my very sincere thanks to C.L. Ruthroff for his help and advice in the preparationof this paper.

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A Theory of Broadband Matching for Optical Heterodyne Receivers