**************************************************************************** The Microwave Image Transimpedance Front-End Amplifier For Optical Receivers **************************************************************************** The image impedance network used in distributed amplifiers is employed in the design of a transimpedance optical receiver front-end at microwave frequencies. Compared with the conventional design, for the same transimpedance the current design has more than twice the bandwidth, or for the same bandwidth it has 3 dB better sensitivity. A front-end is designed for 10 Gb/s transmission with a sensitivity of -25 dBm for a p-i-n detector and -33 dBm for an APD. These results compare favorably with published high-impedance receiver designs. Transimpedance, noise and receiver sensitivity for both p-i-n and avalanche photodetectors, sensitivity degradation due to post amplifier noise figure, stability, and enhancement of bandwidth by compensation are discussed. "INTRODUCTION" We suggest here an image transimpedance (ITZ) front-end amplifier for use in optical receivers\*F. Presently, only analysis is available. We are building the receiver experimentally. Compared with the conventional transimpedance amplifier with the same transimpedance, the ITZ front-end has more than twice the bandwidth. Alternately, for the same bandwidth, it has 3 dB better sensitivity. Due to the difficulty in fabricating large inductors, the ITZ is best used in the microwave frequency range from 5 to 20 GHz. In GaAs FET distributed amplifiers, the gate-to-source (drain-to-source) capacitances of the FETs become the shunt capacitors of an artificial, lumped LC transmission line -- the gate (drain) line. The formation of the LC transmission line is based on the principle of image impedance. Such line has wide bandwidth. The use of the transmission line overcomes the bandwidth limitation due to the poles located at both the gate and drain nodes of the FETs because all capacitances become part of the transmission lines (the gate and drain lines). In theory a distributed amplifier has an infinite gain-bandwidth product. In practice, attenuation in the gate line limits the number of FETs to a few. As regards to bandwidth, in contrast to an ideal distributed line, the lumped nature of this LC transmission line does impose a bandwidth limit which is less than DELTA f sub I ~=~ 1 over {pi sqrt {LC sub T}}~, where L and $C sub T$ are the series inductance and shunt capacitance. Our simulated results show that the ITZ has a bandwidth larger than $DELTA f sub I$. For optical receivers, bandwidth of the transimpedance front-end amplifier is often limited by the pole at the input node to which the photodiode is connected because the total input capacitance at that node, comprised of the sum of the photodiode capacitance $C sub p$ and amplifier input capacitance $C sub a$, is usually larger than that at any other node. For a given $C sub p$, noise is minimized when $C sub a = C sub p$. In doing so, the input circuit can be made into a one-section image impedance network. In the following sections, first we will briefly summarize the relevant features of transmission lines formed by image impedance networks. The results are used to design an ITZ front-end for 10 Gb/s transmission using a commercial HEMT (High Electron Mobility Transistor). We will discuss the transimpedance, noise and sensitivity, and stability of this image transimpedance amplifier/receiver and compare them with that of the conventional transimpedance amplifier/receiver. The computed sensitivity of the ITZ is compared with that of published high-impedance designs. Finally, we describe how bandwidth can be further enhanced by parallel and series compensation of the amplifier output circuit. "IMAGE IMPEDANCE AND THE ITZ FRONT-END AMPLIFIER" For the infinitely cascaded chain of identical but alternately reversed 2-port networks shown in Fig. 1, the image impedances $Z sub {I1}$ and $Z sub {I2}$ are defined as the input impedances looking into ports 1 and 2, respectively. In Fig. 2, a voltage source $v sub s$ with source impedance $Z sub s ~=~ Z sub {I1}$ and a load impedance $Z sub L ~=~Z sub {I2}$ are connected to an image impedance network of one section. The ratio of the terminal voltages is\*(Rf G. L. Matthaei, L. Young and E. M. T. Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures, Artech House, Inc., Dedham, MA 1980. v sub 2 over v sub 1 ~=~ sqrt {Z sub {I2} over Z sub {I1}}~~e sup {-j beta}~. For |$v sub 2$| = |$v sub 1$|, it is required that $beta$ be real and $Z sub {I2} ~=~ Z sub {I1}$. Because the input node of the transimpedance amplifier can be cast into this form, we are particularly interested in the symmetrical $pi$-network of Fig. 3, in which a series inductor L is shunted on its sides by capacitors $C sub 1$ and $C sub 2$ ($C sub 1 = C sub 2 = C$). For this network, $Z sub {I1} = Z sub {I2} = Z sub I$, where $Z sub I$ is the image impedance. With $C sub T = 2C$ and $ omega =2 pi f$, the image impedance is Z sub I ( omega )~=~ Z sub {Io} over sqrt {1~-~{ omega sup 2 LC sub T /4}} where Z sub {Io} ~=~ sqrt {L/{C sub T}} and beta ~=~ 2 {sin} sup {-1} ( omega sqrt {LC sub T /4})~. For $f < DELTA f sub I ~==~ 1 over {pi sqrt {LC sub T }}$, $Z sub I ( omega is real and increases with frequency. It reaches infinity at $f~=~ DELTA f sub I $ and is imaginary thereafter. From (5), for $f << DELTA f sub I$, $1 over sqrt {LC sub T }$ is recognized to be the phase velocity and from (3) and (4) $Z sub {Io}$ the characteristic impedance of the transmission line. In theory, from (2), the symmetrical and matched $pi$-network has infinite bandwidth, even though the infinite impedance at $f = DELTA f sub I$ renders the network meaningless. In practice for distributed amplifiers, $Z sub s$ and $Z sub L$ are fixed resistors and the bandwidth becomes finite. .P Since the photodiode is a current source, we replace $v sub s$ and $Z sub s$ in Fig. 2 by a current source $i sub p$ and the 2-port network by one that has identical image impedances at each end ($Z sub {I1} = Z sub {I2} = Z sub I$), with port 2 now terminated by $Z sub L = Z sub I$. A terminal voltage $v sub 1 = i sub p Z sub I$ is developed which propagates to node 2. From (2) the magnitudes of $v sub 1$ and $v sub 2$ are equal; they differ only in phase by an amount $beta$. Hence, the transimpedance is $Z sub T = Z sub I e sup {-j beta}$ and as discussed previously it can exhibit infinite peaking. Similar to distributed amplifiers, if $Z sub L = R sub L = Z sub {Io}$ (a fixed resistance), the peaking becomes finite since the network is not matched over all frequencies ($Z sub L != Z sub I$, the latter increasing with frequency). The bandwidth of the current-to-voltage gain, $Z sub T (f)$ or transimpedance, is therefore finite. Given that $Z sub I$ increases with frequency for $f < DELTA f sub I$, a larger load resistance of $R sub L = sqrt 2 Z sub {Io}$ will provide a better match over that frequency range. In both cases, simulation results to be given below show that the bandwidth of $|Z sub T (f)|$ is slightly larger than $DELTA f sub I$. A conventional transimpedance amplifier is shown in Fig. 4. It consists of a feedback resistor $R sub f$ across a voltage amplifier of gain -A (measured with $R sub f$ in place). The input impedance of the amplifier is the parallel combination of $R sub {in} = R sub f over 1+A$ and $C sub a$. A photodiode with shunt capacitance $C sub p$ is connected to the input node. Minimum noise is attained when $C sub a$=$C sub p$\*(Rf R. G. Smith and S. D. Personick, "Receiver Design For Optical Fiber Communication Systems," in Semiconductor Devices for Optical Communication, 2nd Edition H. Knessel, Ed., Springer-Verlag, New York 1982. which will be assumed in the following discussion. For the input circuit, adding an inductor L between $C sub p$ and $C sub a$ of value (6A) L~=~{C sub T R sub {in} sup 2}~=~C sub T ({R sub f over {1+A}}) sup 2~~~~ ~~for~R sub {in}=Z sub {Io}~, or (6B) L~=~1 over 2 C sub T R sub {in} sup 2 ~=~ {{C sub T} over 2} ({R sub f over {1+}}) sup 2~~~~~~for~R sub {in}= sqrt 2 Z sub {Io}~, where $C sub T$=$C sub p$+$C sub a$, forms an approximation to an ideal image impedance network of one section to which the photocurrent source $i sub p$ is connected. We first study the frequency response of the magnitude and phase of the transimpedance $Z sub T (f)$ and the input impedance $Z sub {in} (f)$ of the circuit in Fig. 3 for $C sub 1$=$C sub 2$=0.2 pF (a value typical of GaAs FETs) and $R sub {in} ~=~ 50~OMEGA$ and $100~OMEGA$, as a function of L. The results for $R sub {in}=50~OMEGA$ are shown in Figs. 5A-6B. In Fig. 5A, for L=0, $|Z sub T (f)|$ has a bandwidth of 8 GHz. For other L's, $|Z sub T (f)|$ can have both dips at midband and peaks at the band edge, the amounts of which are tabulated in the insert. The band edge peaking is of lesser concern since the post amplifier gain roll-off will reduce the peaking as long as that peaking is not too large (<3 dB). Therefore, the bandwidth depends really on how much dip in $|Z sub T (f)|$ is allowed which in turn depends on the post amplifier frequency response. For the bandwidth to be useful, the post amplifier should not increase the dip to 3 dB. From (6A), L is 1 nH and there is no dip but a slightly excessive peak of 4 dB, with bandwidth of 17.2 GHz. From (6B), L=0.5 nH; the dip and peak are, respectively, 1.5 dB (at 12 GHz) and 2 dB (at 20 GHz), and bandwidth is increased to 24 GHz which is three times the input pole. L=0.75 nH appears to be a good compromise with only 0.3 dB dip and a bandwidth of about 20 GHz. That bandwidth is 2.5 times the input pole of $1 over {2 pi R sub {in} C sub T}$, the latter representing the bandwidth limit of the conventional transimpedance amplifier. From (1), for L=0.75 nH, $DELTA f sub I$ = 18.4 GHz . In this case, the 3-dB bandwidth of 20 GHz is larger than $DELTA f sub I$. Approximating the bandwidth of the image transimpedance amplifier by $DELTA f sub I ~=~ 1 over {pi R sub {in} C sub T}$, it is seen that the bandwidth of the image transimpedance amplifier is at least a factor of two greater than that of the conventional transimpedance amplifier. It is interesting to observe that the magnitude and phase of $Z sub T ( DELTA f sub I )$ are 1 and -180\(de. Phase linearity is desireable to minimize pulse distortion. From Fig. 5B, for L$<=$1 nH, $Z sub T (f)$ is linear in phase up to 0.75$DELTA f sub I$. The magnitude and phase of the input impedance $Z sub {in} (f)$ are plotted in Figs. 6A and 6B. |$Z sub {in} (f)$| has both midband dip and band edge peak which are correspondingly responsible for the dip and peak in $|Z sub T (f)|$. The peaking of |$Z sub {in} (f)$| at the band edge causes a larger voltage to be developed at node 1 which is consequently transmitted to node 2. The phase of $Z sub {in} (f)$ is initially close to 0\(de, i.e., $Z sub {in}$ is resistive, and is approximately flat with frequency up to 0.75$DELTA f sub I$ before it falls off from thereon. This deviation from constant phase in $Z sub {in} (f)$ causes the phase of $Z sub T (f)$ to depart from linearity. $|Z sub T (f)|$ for $R sub {in}=100~OMEGA$ is plotted in Fig. 7. L=3 nH yields a bandwidth of 10 GHz and a dip of 0.3 dB and may be considered optimal since the dip is very small and the peak is only 3 dB. Compared to the $R sub {in}=50~OMEGA$ case, $L sub {opt}~ alpha ~R sub {in} sup 2$ and bandwidth $alpha ~{1 over {R sub {in}}}$ which agree with theory. Note that from (6B) L is 2 nH which gives a dip and peak of 1.5 and 2 dB, respectively, and a bandwidth of 12 GHz which is three times larger than the input pole. These results are similar to the $R sub {in}$=50 $OMEGA$ case. For the image transimpedance amplifier, assuming the bandwidth to be $1 over {pi R sub {in} C sub T}$ and gain to be $R sub f$, the gain-bandwidth product (GB) is $1+A over {2 pi C sub p}$. For large GB, $C sub p$ should be as small as possible and A as large as possible. As we will show later, smaller $C sub p$ also results in lower circuit noise. At microwave frequencies, it is difficult to make L larger than 10 nH as parasitics will dominate. Therefore an L value less than 5 nH is preferable. This small value of L can be replaced by a section of transmission line. It can be shown that when $Z sub o sup 2 >> omega L R sub {in}$, then L$approx Z sub o {l over v }$, where $Z sub o$, l and v are, respectively, the characteristic impedance, length and phase velocity of the transmission line. Using $Z sub o = 200~OMEGA$, we obtained similar results to that of using an actual inductor. Finally, we wish to determine the sensitivity in frequency response to variation in element values of the circuit in Fig. 3 terminated by $R sub L$=100 $OMEGA$. For $C sub 2$=0.2 pF, we examine ${|Z sub T (f)|}$ with $C sub 1$ as a parameter for different values of L. In Fig. 8, we show the results of varying $C sub 1$ for L=4 nH. Overall, we conclude that 0.2 pF<$C sub 1$<0.3 pF and 3 nH<L<4 nH will give a bandwidth between 7 and 10 GHz while restricting the dip and peak to be less than 1 and 4 dBs, respectively. Therefore, the ITZ is not extremely sensitive to parameter variations. Up to $+-$20% variation in $C sub 1$ and $+-$15% variation in L is permitted. It is possible in practice to hold element values within this tolerance. The image impedance method is used in a transimpedance front-end amplifier design for 10 Gb/s transmission using a 0.5 $mu$m $times$ 300 $mu$m HEMT\*F Fujitsu FHR01X. with $C sub p$=0.2 pF, $R sub f = 500~ OMEGA$, and a load impedance of $R sub L = 200~OMEGA$ in parallel with $C sub L$=0.2 pF. $C sub L$ is the input capacitance of the next circuit stage. First we examine the input admittance of the feedback amplifier which is plotted in . Up to 8 GHz it corresponds approximately to a 92 $OMEGA$ resistor in parallel with a 0.2 pF capacitor, and therefore can be made into a one-section image impedance network. $C sub a$=0.2 pF, and for L=0, the original input pole is 3.4 GHz as shown in Fig. 10. To form the ITZ front-end, L=4 nH is added and the bandwidth increases to 7.6 GHz, more than doubling the original value. Due to parasitics in the FET and small mismatch between $C sub p$ and $C sub a$, this response is slight different from Fig. 7. The responses for other values of L are also shown in Fig. 10. "NOISE AND RECEIVER SENSITIVITY" We consider a digital system with binary transmission at 10 Gb/s supported by our front-end amplifier of 7.6 GHz bandwidth. (Because a raised cosine output pulse shape is most desirable, in practice a receiver bandwidth of about 70-80% of the bit rate is often sufficient. See the discussion on raised cosine pulse shape in the following section). We will use $DELTA f$ to represent bandwidth and B to represent bit rate. Simulation using SPICE show that the one-stage HEMT ITZ front-end amplifier produces less output noise than the conventional transimpedance (TZ) amplifier. Noise comparison is made between the ITZ amplifier of Fig. 10 with L=4 nH and $R sub f$ = 500 $OMEGA$ (giving $DELTA f$=7.6 GHz and $|Z sub T (0)|$=409 $OMEGA$) and the TZ amplifier of Fig. 10 with L=0 and $R sub f$ reduced to 195 $OMEGA$ (giving $DELTA f$=7.6 GHz and $|Z sub T (0)|$=149 $OMEGA$). Sensitivities of the ITZ and TZ receivers will be calculated following the method in [2] for the two cases of a full raised-cosine output pulse obtained by equalization and a non-raised-cosine output pulse shape. For the latter output pulse, there is intersymbol interference (ISI) during the decision process, causing sensitivity degradation. We treat this effect by finite extinction ratio of the optical signal. The shot noise due to the gate current, $I sub {gate}$, of the FET is assumed to be negligible. This assumption is not always valid in current GaAs HEMT technology as $I sub {gate}$ can be as high as 100 $mu$A, thus contributing significant shot noise. As the technology of HEMT improves, it is expected that this current can be made negligibly small. In the case of the Fujitsu HEMT used in our design, the sensitivity degradation due to $I sub {gate}$ is well under 0.1 dB, and can be neglected. We also assume that the detector dark current, even when it's multiplied as in the case of the APD detector, contributes such low shot noise that the corresponding degradation in receiver sensitivity is negligible. This approximation is valid even for relatively large values of multiplied dark current (up to 100 nA). "Equivalent Input Noise" The total rms output noise voltage computed by using SPICE for the TZ and ITZ front-end amplifiers are, respectively, 0.194 mV and 0.224 mV. Because the two amplifiers have different transimpedance, it is appropriate instead to express noise in terms of equivalent input noise. For the two front-ends under discussion, the ITZ is lower in equivalent input noise by more than a factor of two in rms value. The mean square output noise voltage, ${v sub o sup 2} bar$, and equivalent circuit noise current at the input, ${i sub c sup 2} bar$, of either the front-end amplifier or the complete receiver are given by [2] (7) {v sub o sup 2} bar ~=~ int from 0 to inf S ( omega ) {| Z sub T ( omega )|} sup 2 df~ and (8) {i sub c sup 2} bar ~=~ {{v sub o sup 2} bar } over {| Z sub T (0)|} sup 2 ~=~ {1 over {|Z sub T (0)|} sup 2} int from 0 to inf {S ( omega )} {|Z sub T ( omega )|} sup 2 df~, where $S( omega )$ is the total equivalent input noise-current spectral density and $Z sub T ( omega )$ the transfer function of the amplifier or the receiver, through which the output noise voltage is evaluated. From (8), ${i sub c sup 2} bar$ depends only on the shape of $|Z sub T ( omega )|$ but not its magnitude. (9A) {i sub c sup 2} bar ~=~ C sub 2 I sub 2 B~~~~~~for~S( omega )=C sub 2 ~~~~~ (white~noise), and (9B) {i sub c sup 2} bar ~=~ C sub 3 I sub 3 B sup 3~~~~~~for~S ( omega )=C sub 3 omega sup 2 , where (10A) I sub 2 ~=~ {1 over {{|Z sub T (0)|} sup 2 B}} int from 0 to inf {|Z sub T ( omega )|} sup 2 df and (10B) I sub 3 ~=~ {1 over {{|Z sub T (0)|} sup 2 B sup 3}} int from 0 to inf {|Z sub T ( omega )|} sup 2 f sup 2 df~. $I sub 2$ and $I sub 3$ for $Z sub T ( omega )$ that convert several different input waveforms to raised-cosine output waveforms are given in [2]. For both the 1-stage TZ and ITZ amplifiers, the two dominant sources of noise are the thermal noise associated with the feedback resistor $R sub f$ and the FET transconductance $g sub m$. For both amplifiers, the sum of the $R sub f ~(500 OMEGA )$ and $g sub m ~(50 mS)$ contributions relative to total noise is over 96% in rms values. The noise from the load resistor $R sub L (200 OMEGA )$ is very small relative to the above sources. We first discuss the TZ amplifier. The mean square output noise voltage due to $R sub f$, calculated by SPICE, is $1.65 times {10} sup {-8} ~v sup 2$. To see the amount of contribution from the various noise sources, their equivalent input noise-current density expressions are derived using the simplified circuit model for the FET as shown in Fig. 11 in which all extrinsic (parasitic) elements are neglected. The equivalent input noise-current density due to $R sub f$ is (11A) S sub {R sub f,TZ} ( omega ) ~=~ {4kT} over {R sub f} {left [ {{(G sub f + g sub m )} sup 2 ~+~ {( omega C sub T - {G sub f g sub m} over { omega C sub T})} sup 2} over {{(G sub f - g sub m )} sup 2 ~+~ {({G sub f sup 2 - G sub f g sub m} over {omega C sub T})} sup 2} right ] }~, .EN .DE .P where ${G sub f}={1 over {R sub f}}$ and $C sub T = C sub p + C sub a$. $C sub a$ is the sum of the gate-source and gate-drain capacitances of the FET, $C sub {gs}$ and $C sub {gd}$, plus any stray capacitance $C sub s$. For our circuit, the term within [ ] in (11A) increases monotonically from 1.24 to 1.74 from 0 to 10 GHz with a mean value of 1.42. In SPICE calculation, all parasitics are included and from SPICE, the above term is almost a constant, falling slightly with increasing frequency across the 10 GHz band with a mean of 1.22. The equivalent circuit derivation is seen to give good approximation to the exact result. Using the SPICE result for greater accuracy, (11B) S sub {R sub f,TZ} ( omega )~approx~{4.88kT} over R sub f and the output noise voltage or equivalent input noise current is proportional to B. ${v sub o sup 2} bar$ due to $g sub m$ calculated by SPICE is $1.7 times {10} sup {-8} ~v sup 2$, about the same as the $R sub f$ noise. Defining $S sub {g sub m,TZ} ( omega )$ to be the equivalent input noise-current density, using Fig. 11, we find the channel noise to be (12A) S sub {g sub m,TZ} ( omega ) ~=~ {4kT} over {g sub m} {left [ 1 over {R sub f sup 2} ~+~ {( omega C sub T )} sup 2 right ] } {left ( {g sub m R sub f} over {{g sub m R sub f}~-~1} right )} sup 2~. The last term in (12A) is a constant with a value of approximately 1.24. However, from SPICE, that term is frequency dependent, monotonically decreasing from 1.24 to 0.97 with a mean of 1.14 over the 10 GHz band. Therefore, from SPICE, (12B) S sub {g sub m,TZ} ( omega ) ~approx~ {4.56kT} over {g sub m} {left [ 1 over {R sub f sup 2} ~+~ {( omega C sub T )} sup 2 right ] }~. In the limit of large $R sub f$, the $R sub f$ term in (12B) can be neglected. In that case, from (9B), the $omega sup 2$ term in (12B) produces a circuit noise component proportional to $B sup 3$. This is the minimum circuit or FET noise since the $R sub f$ noise in (11B) would also be negligible. In practice, however, the $R sub f$ noise is not always negligible since the largest $R sub f$ value is limited by the required bandwidth. For the front-ends considered here, the $R sub f$ noise is almost equal to the $g sub m$ noise. We have used a noise bandwidth of 10 GHz in the calculation of equivalent input circuit noise because we assume that such bandwidth limitation exists, either in the post amplifier, equalizer, or the decision circuit, or through the cascade of these elements. Letting $S sub {R sub L,TZ} ( omega )$ to represent the equivalent input noise density due to $R sub L$ for the TZ amplifier, we find from Fig. 11 and (12B) that (13) S sub {R sub L,TZ} ( omega ) ~=~ {S sub {g sub m,TZ} ( omega )} over {g sub m R sub L} ~approx~ {4.56kT} over {g sub m sup 2 R sub L} {left [ 1 over {R sub f sup 2} ~+~ {( omega C sub T )} sup 2 right ] }~. This noise density is a factor of $g sub m R sub L approx 10$ smaller than that due to $g sub m$. From (11B), (12B), (13), (9A) and (9B) and neglecting the $R sub f$ term in (12B) and (13) since it is much smaller than the $C sub T$ term for f>5 GHz, the equivalent input noise current is (14) {i sub c sup 2} bar ~=~ 4kT left [ {{1.22 I sub 2 B} over {R sub f}} ~+~ {1.14 {{(2 pi C sub T )} sup 2} over {g sub m} I sub 3 B sup 3 {(1+{1 over {g sub m R sub L}})}} right ]~. (14) gives the relative contribution from $g sub m$, $R sub f$ and $R sub L$. The noise ratio between $g sub m$ and $R sub f$ is (15) {{i sub {c,g sub m} sup 2} bar} over {{i sub {c,R sub f} sup 2} bar} ~approx~ {{{(2 pi C sub T )} sup 2} I sub 3 R sub f B sup 2 } over {g sub m I sub 2}~. From (15), to maintain the same ratio, $R sub f$ should be varied as $B sup {-2}$. Thus, at high bit rates, $R sub f$ can be substantially reduced making the design of the transimpedance amplifier easier. As we shall show in a later section, the transimpedance amplifier compares favorably with the high-impedance design. We now discuss the ITZ amplifier. The mean square output noise voltages due to $g sub m$ and $R sub f$ are $2.1 times {10} sup {-8}~v sup 2$ and $2.8 times {10} sup {-8}~v sup 2$, respectively. The $R sub f$ noise is 33% larger than the $g sub m$ noise. From (8), the equivalent input noise current of the ITZ amplifier is actually less than that of the TZ amplifier because the ITZ amplifier has a much larger transimpedance. $S ( omega )$ for the ITZ amplifier varies with frequency in a complex manner and will be given below. For the ITZ amplifier, from Fig. 11, the expression for the equivalent input noise-current density due to $g sub m$ is (16) S sub {g sub m,ITZ} ( omega )~=~4kT {g sub m} {left {{{{[G sub f (1- {omega sup 2 C sub p L})]} sup 2}~+~{[ omega {(C sub T - omega sup 2 C sub a C sub p L)}]} sup 2} over {{(G sub f - g sub m)} sup 2} right } }~. Across the 10 GHz noise bandwidth, the term within { } in (16) ranges from 0.001 to 0.39 and closely approximates that calculated by SPICE. From Fig. 11, the equivalent input noise-current density for $R sub f$ is (17) S sub {R sub f,ITZ} ( omega ) ~=~ {4kT} over R sub f {left { {{G sub f sup 2 {(1 - omega sup 2 C sub p L)} sup 2} ~+~ {omega sup 2} {(C sub T - omega sup 2 C sub a C sub p L)} sup 2} over {{{(G sub f - g sub m )} sup 2} ~~ left [ {{G sub f sup 2 {(1 - omega sup 2 C sub p L)} sup 2} ~+~ omega sup 2 {(C sub T - omega sup 2 C sub a C sub p L)} sup 2} over {{g sub m sup 2 {(1 - omega sup 2 C sub p L)} sup 2} ~+~ {omega sup 2} {(C sub T - omega sup 2 C sub a C sub p L)} sup 2} right ] } right } }~. Across the noise bandwidth, the term within { } in (17) varies between 0.05 and 5.7 and turns out again to provide good approximation to SPICE calculations. .P Finally, the equivalent input noise-current density of $R sub L$ for the ITZ amplifier is (18) S sub {R sub L,ITZ} ( omega ) ~=~ S sub {g sub m,ITZ} over {g sub m R sub L} ~=~ {{4kT} over R sub L} {left {{{{[G sub f (1- {omega sup 2 C sub p L})]} sup 2}~+~{[ omega {(C sub T - omega sup 2 C sub a C sub p L)}]} sup 2} over {{(G sub f - g sub m)} sup 2} right } } which is again a factor of $g sub m R sub L approx 10$ smaller than that due to $g sub m$. The total equivalent input noise-current densities (approximately equal to the sum of the $R sub f$ and $g sub m$ contributions) for the TZ and ITZ amplifiers are plotted in Fig. 12. For the TZ amplifier, the $omega sup 2$ dependence is clearly seen. For the ITZ amplifier, the equivalent input noise-current density is smaller and almost independent of frequency up to 8 GHz; however, above 8 GHz, it increases rapidly with frequency. Up to 10 GHz, the ITZ amplifier is lower in equivalent input noise current than the TZ amplifier. As a result, the ITZ receiver has better sensitivity to be discuss next. "Receiver Sensitivity -- Noise-Free Post Amplifier" For sensitivity calculation, we follow the approach in [2], which assumes that the signal-dependent noise is negligible relative to the circuit noise and that the probability density function of the signal noise amplitude distribution is gaussian. The variance of this gaussian function is equal to ${i sub c sup 2} bar$. Assuming that the occurrence of a mark or space is equally probable, the average photocurrent needed to achieve a given bit-error-rate (BER) is [2] (19) {i sub p} bar ~=~ Q sqrt {{i sub c sup 2} bar } where ${i sub p} bar$ is the average photocurrent and Q is the argument of the complementary error function, erfc (x), and related to BER by (20) BER ~=~ erfc~(Q)~==~ 1 over sqrt {2 pi } int from Q to inf e sup {-({z sup 2} /2)} dz~. For BER in the range of ${10} sup {-9}$ to ${10} sup {-15}$, $Q ~approx~ 3~-~{1 over 3} {log} sub {10} (BER)$. For BER=${10} sup {-9}$, $Q approx 6$. In terms of optical measurement using a p-i-n detector, (19) becomes [2] (21) eta P bar ~=~ {({h nu } over q )}Q sqrt {{i sub c sup 2} bar }~, .EN .DE .P where $P bar$ is the required average optical power, $eta$ is the quantum efficiency, $h nu$ the photon energy and q the electronic charge. For an avalanche photodetector (APD), the magnitude of the signal-dependent noise in a time slot (assumed only due to the signal within that time slot) is a function of the average avalanche gain M and is no longer negligible in comparison to the signal-independent circuit noise. (21) becomes [2] (22) eta P bar ~=~ {({h nu } over q )}Q left [ { sqrt {{i sub c sup 2 } bar } over M } ~+~ qB I sub 1 Q F(M) right ] ~, where $I sub 1$ is given in [2] for raised-cosine output and various input pulse shapes. F(M) is the excess noise factor given by\*(Rf R. J. McIntyre, "Multiplication Noise in Uniform Avalanche Diodes," IEEE Trans. Electron Devices, vol. ED-13, pp.164-168, 1966. (23) F(M) ~=~ kM ~+~ (1-k)(2-{1 over M}) in which $k$ is the electron and hole ionization coefficient ratio. (22) is optimized when [2] (24) M~=~M sub {opt} ~=~ {1 over {k sup {half}}} { left [ {sqrt {{i sub c sup 2} bar } over {qB I sub 1 Q}} ~+~ k ~-~ 1 right ] } sup {half}~. For p-i-n detectors, the sensitivity is calculated for input and output pulse shapes of NRZ and full raised-cosine, respectively, and BER=${10} sup {-9}$ at 10 Gb/s, at $lambda$=1.3 $mu$m. From (21), (9A), (9B), (11B), (12B), (13), $I sub 2$=0.562 and $I sub 3$=0.087 for the assumed pulse shapes, we calculate for the TZ amplifier that $sqrt {{i sub c sup 2 } bar } ~=~ 1.0~mu A$, giving $eta P bar$ of -22.4 dBm. In comparison, for the ITZ amplifier, $sqrt {{i sub c sup 2 } bar } ~=~ 0,475 ~mu A$, giving $eta P bar$ of -25.6 dBm. $eta P bar$ against BER for both the TZ and ITZ receivers using the p-i-n detector are plotted in Fig. 13 as shown by the curves marked "NF=0 dB". It is seen that for B=10 Gb/s, in addition to providing over 2.7 times higher gain, the ITZ receiver gives 3.2 dB better sensitivity than the TZ receiver at ${10} sup {-9}$ BER. For APDs, one of the most advanced is the InGaAs SAGM (Separate Absorption and Multiplication regions) APD (k=0.35). Using this APD, at 10 Gb/s, from (24), $M sub {opt}$=23.3 for the TZ receiver and equals 16 for the ITZ receiver. Figure 14 is a plot of the calculated sensitivity of the TZ and ITZ receivers as a function of M. Especially for the ITZ receiver, the optimal range of M is broad, extending from about 11 to 23 with a $DELTA {eta P} bar$ of only 0.2 dB over that range. Therefore, in an actual ITZ receiver using the above APD, the avanlanche gain need not be set precisely in order to obtain near optimum sensitivity. With $I sub 1 approx$0.548 for NRZ input and full raised-cosine output pulse shapes, at B=10 Gb/s and BER=${10} sup {-9}$, $eta P bar$ equals -32.8 dBm for the TZ receiver and -34.2 dBm for the ITZ receiver. The sensitivity advantage of the ITZ over the TZ receiver is now reduced to 1.4 dB. $eta P bar$ versus BER for both TZ and ITZ receivers using this APD are plotted in Fig. 15 as shown by the curves marked "NF=0 dB". From Fig. 13 and 15, the improvement in sensitivity by using the InGaAs SAGM APD ranges from 8.6-10.4 dB over the p-i-n detector. In practice, however, the actual sensitivity improvement is less due to finite gain-bandwidth product of the APD. Currently, the best GB for an InGaAs SAGM APD is 70 GHz\*(Rf. B. L. Kasper and J. C. Campbell, "Multigigabit-per-Second Avalanche Photodiode Lightwave Receivers," Journal of Lightwave Technology, vol. LT-5, no. 10, Oct. 1987. Using this diode as an example, if a bandwidth of 10 GHz is required, the permissible gain of M=7 would be less than $M sub {opt}$, and from Fig. 14 would yield sensitivities of -30.3 dBm for the TZ receiver and -33 dBm for the ITZ receiver. In general, we find that practical receivers using the InGaAs SAGM APD should provide over 7 dB better sensitivity than using the p-i-n detector, for both types of front-ends. We now consider the degradation in sensitivity when the output pulse shape is not raised-cosine. For example, that situation would occur if the decision circuit were connected directly to the front-end output without equalization. The sensitivity degradation occurs as a result of two separate effects: intersymbol interference (ISI) from adjacent time slots and greater noise in the signal at the decision circuit input. The latter effect occurs because the equalizer not only optimizes output pulse shape to minimize ISI but also band-limits the output noise voltage. For the single amplifier stage of Fig. 10 we obtain from SPICE $sqrt {{i sub c sup 2 } bar }$=1.3 $mu$A for the TZ amplifier and $sqrt {{i sub c sup 2 } bar }$=0.55 $mu$A for the ITZ amplifier. Using (21), sensitivity is $eta P bar$ = -21.3 dBm for the TZ receiver and -25 dBm for the ITZ receiver, at ${10} sup {-9}$ BER, B=10 Gb/s and $lambda$=1.3 $mu$m, using a p-i-n detector. In comparison, the sensitivities for a raised-cosine output pulse are better by 1.1 and 0.6 dB, respectively, for the TZ and ITZ receivers. The reason is that for an NRZ input pulse, the transfer function which yields a raised-cosine output pulse is more band-limiting than our front-end amplifier response which is essentially flat up to B. This result and the effect of ISI to be discussed below indicate the importance in controlling the output pulse shape. Figure 16 shows the output pulse waveforms for a single 10 Gb/s NRZ input pulse, for both the TZ and ITZ amplifiers. We assume that these waveforms are input to the decision circuit and approximate the effect of ISI by that of finite extinction ratio, i.e., nonzero transmitted optical power corresponding to a space. Attributing the amplitude of the first positive side-lobe of the output pulse waveform to be due to the nonzero power detected during a space, we can use [2] to calculate the resulting sensitivity penalty. From Fig. 16, given that the ratio of the peak amplitudes of the first positive ripple to the main-lobe is approximately 0.075 for the ITZ receiver, we calculate that for a p-i-n detector, a 0.7 dB sensitivity penalty is incurred for the ITZ receiver; there is practically no ISI and therefore no sensitivity penalty for the TZ receiver. ISI is greater in the ITZ receiver since its transfer function deviates more from the ideal transfer function (which produces no ISI) than that of the TZ receiver, as seen from the insert of Fig. 16. Note that our calculation only considers the ISI from a single pulse from the previous time slot, and therefore is not the worst case. "Receiver Sensitivity - Including Post Amplifier Noise" From (12A), (13) and (9B), in the limit of very large $R sub f$ and $R sub L$, ${i sub c sup 2} bar ~alpha~ {C sub a sup 2} over {g sub m} ~alpha~ W sub g$, where $W sub g$ is the FET gate width. Noise is zero when $C sub a$=0. However, this would correspond to the absence of the front-end amplifier altogether. This apparent paradox arises because we have not considered post amplifier noise. When that noise is included, minimum noise is attained at non-zero value of $W sub g$. Therefore, for noise minimization, it is important to include the post amplifier noise which will be discussed next. So far we have assumed the post amplifier to be noise-free. If the equalizer is lumped together with the post amplifier, the functions of this circuit are to produce gain and provide pulse shaping. In reality, the post amplifier also produces noise which degrades receiver sensitivity. The noise property of an amplifier is often given in terms of noise figure (NF) which in turn is defined only for a resistive source impedance as follows: (25) NF ~==~ {input~(S/N)} over {output~(S/N)} or (26) NF ~==~ {Total~available~output~noise~power} over {Available~output~noise ~power~due~only~"to"~R sub s} where $(S/N)$ is the signal-to-noise power ratio of the amplifier and $R sub s$ is the source resistance. Strictly speaking, the output impedance of the front-end amplifier is not a pure resistance but also includes a reactance. However, the reactance is smaller than the resistance so that we can treat the output impedance as a resistor of value $R sub o$. For $R sub s$=$R sub o$ the noise of the post amplifier can be expressed in terms of an equivalent noise voltage at its input, ${v sub i sup 2} bar$, by (27) {v sub i sup 2} bar ~=~ 4kT {R sub o} {({NF} bar ~-~1)}B where ${NF} bar$ is the average of the spot noise figure of the post amplifier. In terms of ${i sub c sup 2} bar$ at the input of the front-end amplifier, (28) {i sub c sup 2} bar ~=~ {4kT {R sub o} {({NF} bar ~-~1)}B} over {{| Z sub {fe} (0) |} sup 2 } where $Z sub {fe} (0)$ is the transfer function of the front-end at f=0. Adding this to the original equivalent input noise current of the front-end amplifier, we can use (21) and (22) to compute the degradation in sensitivity due to noise in the post amplifier. The average $R sub o$ for both amplifiers is about 50 $OMEGA$. In Fig. 17 the calculated sensitivity is plotted against ${NF} bar$ for the ITZ and TZ receivers for BER=${10} sup {-9}$ at B=10 Gb/s and for both the p-i-n and APD detectors. As before the input and output pulse shapes are assumed to be NRZ and raised-cosine, respectively. Optimal avalanche gain at each ${NF} bar$ value is assumed for the APD. Relative to the APD, the sensitivity penalty variation with ${NF} bar$ in using a p-i-n detector is greater, for both the ITZ and TZ receivers, because the APD provides internal gain. In going from a noise-free post amplifier to one that has a 10 dB noise figure, for p-i-n detectors, the ITZ and TZ receivers suffer about 2.3 dB and 3.2 dB sensitivity penalty, respectively; for APDs, that degradation is 1 dB for the ITZ receiver and 1.5 dB for the TZ receiver. The ITZ receiver suffers less penalty than the TZ receiver because greater gain (transimpedance) is provided by the front-end amplifier in the former. In Figs. 13 and 15 we plotted sensitivity versus BER for the ITZ and TZ receivers over the range of BER from ${10} sup {-6}$ to ${10} sup {-15}$ as a function of post amplifier average noise figure (${NF} bar$). A family of curves each representing different ${NF} bar$s (0, 5 and 10 dB) is shown for each type of receiver. A practical number for ${NF} bar$ is 5 dB. Using that value, from Figs. 13 and 15, the degradation in receiver sensitivity for a p-i-n detector in going from ${NF} bar$=0 dB to ${NF} bar$=5 dB (over the entire BER range shown) is about 0.9 dB for the ITZ receiver and 1.3 dB for the TZ receiver. Therefore, for practical receivers using the p-i-n detector, at 10 Gb/s operation the ITZ receiver should provide over 3.5 dB better sensitivity than the conventional transimpedance receiver. Using the APD detector, for the same noise figure, the penalties are about 0.4 dB and 0.6 dB, respectively, for the ITZ and TZ receivers. These results assume large enough GB of the APD such that $M sub {opt}$ is always attained. $M sub {opt}$ increases with ${NF} bar$. For APD GB of 70 GHz and $M=7<M sub {opt}$, the sensitivity degradation going from ${NF} bar$=0 dB to ${NF} bar$=5 dB (over the ${10} sup {-6}$ to ${10} sup {-15}$ BER range) is 0.7 dB for the ITZ receiver and 1.1 dB for the TZ receiver. Therefore, for practical receivers using the above APD, at 10 Gb/s operation the ITZ receiver should be 3 dB better in sensitivity than the conventional transimpedance configuration. It is interesting to note that the ITZ/p-i-n receiver with a post amplifier of ${NF} bar$=10 dB still has about a 1 dB sensitivity superiority over the TZ/p-i-n receiver with a noise-free post amplifier, across the entire BER range; in comparison, that superiority for APDs is 0.4 dB. The calculated sensitivities at ${10} sup {-9}$ BER for the various receiver configurations, at average post amplifier noise figure values of 0, 5 and 10 dB, are summarized in Table 1. "Comparison With Published Results" Recently published results on high speed receivers include a complete APD/FET receiver operating at 8 Gb/s with 6.9 GHz bandwidth\*(Rf B. L. Kasper et al., "An APD/FET Optical Receiver Operating at 8 Gbit/s," Journal of Lightwave Technology, vol. LT-5, no.3, March 1987. and a p-i-n/FET front-end with 8 GHz bandwidth\*(Rf. J. L. Gimlett, "Low-Noise 8 GHz PIN/FET Optical Receiver," Electronics Letters, vol. 23, no.6, March 1987. Both receivers are of the high-impedance type. In the former, an InGaAs SAGM APD with GB=60 GHz was used in conjunction with a two-stage GaAs MESFET preamplifier, followed by an RC differentiator and a post amplifier comprised of cascaded commercial wide-band amplifiers plus a transversal equalizer (for bandwidth enhancement). Sensitivity of -25.8 dBm was measured. The latter receiver consists of an InGaAs p-i-n photodiode connected to a three-stage GaAs MESFET preamplifier, and reports a measured equivalent input noise-current density of <$1.44 times {10} sup {-22}~A sup 2 over Hz$. As discussed in Sec. 3.2, when using an InGaAs APD with GB=70 GHz, the ITZ receiver which incorporates a noise-free post amplifier and perfect equalization (to obtain raised-cosine output pulse shape) has an expected sensitivity of $eta P bar$ = -33 dBm. A 10 dB average post amplifier noise figure would cause the sensitivity to degrade by about 1 dB, and should ISI exist due to imperfect equalization, an additional dB of degradation may be incurred. In that case, the predicted sensitivity of $eta P bar$ = -31 dBm would still be about 5 dB better than the results of [6]. Of course, we must note that the comparison is between predicted and experimental results. In Fig. 18 we have plotted the square root of equivalent input noise-current densities of the TZ and ITZ front-ends as well as the front-end of [7]. For the front-end of [7] we show both the calculated and measured results. It is seen that the computed equivalent input noise current of the ITZ front-end and that of [7] are comparable, and we would expect that the two receivers have similar sensitivities. The above comparison lead to a very important and interesting discovery -- the ITZ receiver which is of the transimpedance configuration has equal or superior sensitivity in comparison with the best high-impedance receivers of comparable bandwidth. Up to this time, the high-impedance configuration has always boasted the best sensitivity [4]. However, in systems application, the transimpedance design is usually favored due to greater ease in manufacture as well as providing larger dynamic range. The ITZ receiver combines the advantages of both the high-impedance and transimpedance types. "STABILITY" Since the S-parameters of the HEMT are available, it is convenient to examine the stability of the front-end amplifier by the stability factors K and $B sub 1$\*(Rf, G. Gonzalez, Microwave Transistor Amplifiers Analysis and Design, Prentice-Hall, Inc., Englewood Cliffs, NJ 1984. and also the source and load stability circles, where (29) K ~=~ {1~-~{{|S sub {11}|} sup 2}~-~{{|S sub {22}|} sup 2}~+~{{| DELTA |} sup 2}} over {2 {|S sub {12} S sub {21}|}} and (30) B sub 1 ~=~ 1~+~{{|S sub {11}|} sup 2}~-~{{|S sub {22}|} sup 2}~-~{| DELTA |} sup 2 where $DELTA ~=~ S sub {11} S sub {22} ~-~ S sub {12} S sub {21}$ and $S sub {ij}$ is the ${ij} sup {th}$ component of the scattering matrix. A two-port network is unconditionally stable, i.e., independent of source and load impedances when $K$>1 and $B sub 1$>0. These conditions guarantee that the real parts of the input and output impedances of the two-port network are positive. When $K$<1 and/or $B sub 1$<0, the network is potentially unstable. In this case the source and load stability circles will give the range of source and load impedance values within which $K$>1 and $B sub 1$<1 for stable operation. The Fujitsu HEMT alone is potentially unstable due to internal feedback by the gate-drain capacitance $C sub {gd}$. However, when sufficient feedback by $R sub f$ is applied, the FET becomes unconditionally stable. For $K$ and $B sub 1$ calculation, we have included $C sub p$, $C sub L$ and $R sub L$ in the two-port network, i.e., the source and load impedances are infinite. $K$ and $B sub 1$ as a function of $R sub f$ (for $R sub f$ from 200 to 2 k$OMEGA$) for the TZ and ITZ amplifiers are plotted in Figs. 18A and 18B, respectively. In both cases, when $R sub f$<1 k$OMEGA$, both amplifiers are unconditionally stable. Since the $R sub f$ value used in our design is 500 $OMEGA$, there is substantial margin to guard against potential instability. "ENHANCEMENT OF BANDWIDTH BY PARALLEL AND SERIES INDUCTIVE COMPENSATION" In the above ITZ front-end using the Fujitsu HEMT, we found that the bandwidth can be further extended by compensating the output circuit. The output circuit can be parallel compensated by an inductor $L sub p$ in series with $R sub L$, and additionally, series compensated by an inductor $L sub s$ in series with $C sub L$. The output voltage is taken across $C sub L$ which represents the input capacitor of the next stage. $L sub p$ increases the impedance of $R sub L$ while $L sub s$ reduces the shunting effect of $C sub L$. The result for using $L sub p$ and $L sub s$ in conjunction with the image impedance front-end is shown in Fig. 19. The inductor at the input node (now referred to as $L sub {in}$) is decreased to 2 nH to provide a wider bandwidth but also greater midband dip and band edge peak. $L sub p$=8 nH compensates for the dip and $L sub s$=0.5 nH offsets the peak. The bandwidth increases from 7.6 GHz to over 12 GHz, and phase linearity is maintained up to 10 GHz. Therefore, using three inductors, the bandwidth is increased by almost a factor of four over the original TZ amplifier. Summary : We have suggested an image transimpedance front-end amplifier for an optical receiver at microwave frequencies. The photodiode capacitance and the input capacitance of the amplifier form the shunt capacitors of an artificial transmission line of one section. The bandwidth of the image transimpedance front-end is found to be two to three times larger than the conventional transimpedance front-end. .P Depending on the desired bandwidth, the image transimpedance may suffer minor midband dip and band edge peaking. For a 1.5 dB dip and 2 dB peak, the bandwidth is three times that of the conventional design. For 10 Gb/s transmission the image transimpedance amplifier is not sensitive to component variation up to $+-$15%. Using the image impedance method, the transimpedance and bandwidth of a $0.5~mu m~times~300~mu m$ Fujitsu HEMT amplifier are 409 $OMEGA$ and 7.6 GHz, respectively. In comparison, in the conventional transimpedance amplifier design for the same transimpedance, the bandwidth is only 3.2 GHz. Compared to the TZ amplifier redesigned for identical bandwidth as the ITZ amplifier, assuming noise-free post amplification and NRZ input and full raised-cosine output pulse shapes, in addition to providing 2.7 times greater transimpedance, the ITZ receiver has about 3.2 dB better sensitivity for p-i-n detectors and 1.4 dB better sensitivity for InGaAs SAGM APDs, over the BER range from ${10} sup {-6}$ to ${10} sup {-15}$. Using the same APD with a gain-bandwidth product of 70 GHz, the calculated sensitivity of either receiver is about 7 dB better than when using a p-i-n detector. A raised-cosine output pulse shape obtained with an equalizer is superior to a non-equalized output pulse shape in terms of both lower noise and zero ISI. Specifically, the lower noise translates to about 1 dB better sensitivity, for both receivers using p-i-n detectors at ${10} sup {-9}$ BER and B=10 Gb/s; the ISI caused by the non-raised-cosine output pulse shape causes about 0.7 dB sensitivity penalty. The noise contributed by the post amplifier can impose significant sensitivity penalty. For an average noise figure of 10 dB in the post amplifier with $R sub o$=50 $OMEGA$, the penalties range from 2.3-3.2 dB for a p-i-n detector and 1-1.5 dB for the InGaAs SAGM APD. For practical implementation of the ITZ receiver which includes post amplifier average noise figure of 5 dB and APD gain-bandwidth product of 70 GHz, it should provide 3 dB better sensitivity than the conventional transimpedance receiver for either the p-i-n or APD detector, at 10 Gb/s operation over the BER range from ${10} sup {-6}$ to ${10} sup {-15}$. The sensitivity of the ITZ receiver compares favorably with that of published high-impedance designs with the added advantages of large dynamic range and ease of fabrication. The bandwidth of the image transimpedance amplifier can be further enhanced by inductive compensation. For example, we have shown that when the load of our HEMT amplifier is both parallel and series compensated, the bandwidth is increased from 7.6 GHz to over 12 GHz. Figures 1-18 Table 1 Cover Sheet Only: DvMs 211, 213, 214 R. Gnanadesikan J. M. Rowell W. D. Warters Divisions 21360, 21470 DvMs/DsMs 213, 214 K. A. Bischoff M. M. Choy A. G. Chynoweth J. L. Gimlett Chinlon Lin E. Nussbaum Note: There are still some EMAC prompts in here, sorry about that..every time I try to edit them i get "Fatal Internal Error"..I wonder what that means <Smirk!> -DT.