IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-30, NO. 1, MARCH 1981 51

the measurement of complex permittivity," IEEE Trans. Instrum. Meas., vol. IM-23, pp. 434-438, Dec. 1974.

[3] M. C. Decreton and M. S. Ramachandraiah, "Nondestructive mea­surement of complex permittivity for dielectric slabs," IEEE Trans. Microwave Theory Tech., vol. MTT-23, pp. 1077-1080, Dec. 1975.

[4] M. S. Ramachandraiah and M. C. Decreton, "A resonant cavity ap­proach for the nondestructive determination of complex permittivity at microwave frequencies," IEEE Trans. Instrum. Meas., vol. IM-24, pp. 287-291, Dec. 1975.

[5] M. Gex-Fabry, J: R. Mosig, and F. E. Gardiol, "Reflection and radiation of an open-ended circular waveguide; Application to nondestructive measurement of materials," Arch. E lek. Übertragung, vol. 33, pp. 473-478, 1979.

[6] R. G. Bosisio, M. Giroux, and D. Couderc, "Paper sheet moisture measurements by microwave phase perturbation techniques," J. Mi­crowave Power, vol. 5, no. 1, pp. 25-34, Mar. 1970.

[7] S. Stuchly, private communication, 1973.

Abstract—This paper describes a computer system, which has been used to study high capacity components in optical-fiber links. The system includes an HP 21 MX minicomputer with a 32K main memory and a disk, making it possible to handle easily 4096 data points in both the time and the frequency domains. The modular organization of the software and the command file control of the processing have created a flexible system that has proved to be useful in evaluating models of semiconductor lasers and avalanche photodiodes. Furthermore, the system has been used to develop modulation methods for semiconductor lasers in the gigabit per second range.

I. INTRODUCTION

IN THE NEAR FUTURE an increasing demand of high capacity communication systems can be expected. Ex­

periments with optical-fiber communication in the gigabit per second range have already started [1], [2]. However, much work has still to be done, especially in optimizing the compo­nents and the modulation schemes [3], [4]. Computerized models of the involved components are, therefore, important tools in system design.

Most of the earlier works concerning computer simulation in the field of fiber optics have dealt with the fiber or with the laser separately. Dannwolf et al. have described a method to

Manuscript received December 8, 1979. This work was supported by the Swedish Board for Technical Development (STU).

The authors are with the Department of Electrical Measurements, Chal­mers University of Technology, S-412 96 Gothenburg, Sweden.

[8] B. Forssell, "Nondestructive measurements of the glass fiber content in reinforced plastics by means of microwaves," in Proc. 4th European Microwave Conf. (Montreux, Switzerland), pp. 132-136, 1974.

[9] E. Tanabe and W. T. Joines, "A nondestructive method for measuring the complex permittivity of dielectric materials at microwave frequencies using an open transmission line resonator," IEEE Trans. Instrum. Meas., vol. IM-25, pp. 222-226, Sept. 1976.

[10] N. Marcuvitz, Waveguide Handbook. New York: McGraw-Hill, 1950.

[11] J. Galejs, Antennas in Homogeneous Media. Oxford, England: Per-gamon, 1969.

[12] R. F. Harrington, Time Harmonic Electromagnetic Fields. New York: McGraw-Hill, 1961.

[13] L. Lewin, Advanced Theory of Waveguides. London, England: Iliffe, 1951.

[14] N. Bleistein and R. A. Handelsman, Asymptotic Evaluation of Integrals. New York, Holt, Rinehart and Winston, 1975.

measure the impulse response of multimode fibers and to transform the time-domain data into the frequency domain using a computer [5]. A baseband frequency response mea­surement system for optical components has been reported by Maeda et al. [6]. Models describing the output light from modulated semiconductor lasers have been reported by several authors [7]-[12].

In this paper we describe a computer system that is used to simulate component performance in high data-rate optical-fiber links. Our main interest is on the laser and the detector and how to separate the laser response from the detector signal. However, this system can also be used to study other compo­nents like fibers, amplifiers, and repeaters. An input pulse can be traced from the input terminals of the laser to the output terminals of the detector amplifier. For linear systems, e.g., photodetectors and amplifiers, we use models consisting of complex transfer functions, which also can be expressed as impulse responses in the time domain. For a nonlinear system, like a semiconductor injection laser, the model may consist of the well-known rate equations [4].

II. MEASUREMENT SYSTEM

A setup for measurements in the time domain on an opti­cal-fiber link is shown in Fig. 1. A signal transmitted through the link can be measured at three different points: after the pulse generator, the detector, and the amplifier. Furthermore,

A Computer System for Analysis of High Capacity Components in Optical-Fiber Links

JAN E. JOHANSSON, MEMBER, IEEE, TORBJÖRN ANDERSSON, PER TORPHAMMAR, HANS EKLUND, AND SVERRE T. ENG, MEMBER, IEEE

0018-9456/81/0300-0051 $00.75 ©1981 IEEE

52 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-30, NO. 1, MARCH 1981

PULSE (GENERATORI

D/A

LASER OPTICAL

FIBER DETECTOR

? ?

SAMPLING

OSCILLO­

SCOPE

A/b

l _ f TIME-DOMAIN MEASUREMENT DATA

MINI­COMPUTER) HP2116

MINI­COMPUTER HP 21 MX

DATA

Fig. 1. Block diagram showing the measurement setup.

it is possible to estimate the influence of the fiber by comparing measurements with and without it.

The data acquisition is made by means of an 18-GHz sampling oscilloscope controlled by an HP 2116 minicomputer. When a command is given to the program, the computer sweeps the sampling oscilloscope using a 12-bit D/A converter. The output is monitored by means of a 12-bit A/D converter and the digitized values are stored in the computer memory. Noise reduction is made by averaging a specified number of measurement values at each point during a sweep and also by averaging a number of sweeps. The measurement data are then transferred to a larger HP 21 MX minicomputer with a disk memory and stored on a data file for future processing. It is not necessary to use two computers, but it is very convenient to control the data acquisition with the small HP 2116 and use the larger HP 21 MX for other purposes simultaneously.

A typical measurement takes about 40 s when averaging 10 values at each of 1024 points in the time domain. The pro­cessing in the HP 21 MX computer is made by means of a general-purpose program package that uses the disk for data storage. Some special-purpose modules have been added for the application described in this paper. The details are given in Section IV.

III. MODEL DESCRIPTIONS

In this section we present the computer models for a GaAlAs double heterostructure (DH) semiconductor injection laser and for a silicon avalanche photodiode (APD) as well as methods for determination of the involved model parame­ters.

Laser Model

The dynamic behavior of a semiconductor laser can be de­scribed with different models of varying complexity. We have found that the two simplified single-mode rate equations

and

dN ^ TV n — = GN + 7 — at τρ re

dn _m n — = -GN + P dì τΡ

(1)

(2)

very well fulfill our requirements. The quantity TV is the photon density; n is the density of excited electrons; rp and re are the lifetimes of photons and excited electrons, respectively. The term y(n/re) represents the rate of spontaneous light injected

into the lasing mode, and P is the. rate of carrier injection into the laser cavity. We have assumed that the laser cavity is ideal with homogeneous population inversion, and that the gain is related to the excited electron density as

G = rpn0

(3)

The quantity no is approximately equal to the excited electron density at threshold [12].

To solve (1) and (2) numerically we need the parameters re, τρ, and 7. The lifetime of excited electrons re is calculated from the relation

Te = U In /., h + U --/, th,

- 1 (4)

where tj is the measurable turn-on delay when the laser is modulated by a step current Is and biased with a current /& not exceeding the threshold current /th [13]. The photon lifetime τρ can be obtained from the damped relaxation oscillation frequency ω^ according to

1 HH+ I2

4r?/?h

- 1 (5)

where / is the total current value after a step [14]. Equation (5) can readily be found from a linearization of the rate equations (1) and (2). The constant 7 is obtained from [15].

Qualitatively, this model shows good agreement with the real laser and can be used to find the optimum modulation condition in the gigabit per second range. The optimization is performed by finding the proper pulse current that after each pulse makes the density of excited electrons return as fast as possible to steady state, thereby, obtaining minimum pulse interaction [8], [12].

Detector Model

The most commonly used detector in optical-fiber links is the APD mainly because of its high sensitivity and its high gain-bandwidth product. The bandwidth of an APD is limited by three factors: the time constant corresponding to the equivalent circuit, the carrier transit time, and the multipli­cation time in the avalanche region [ 16]- [ 18]. We define the transfer function for the diode as

G(jœ) = ÏÏM(jw)D(j<iû)Z(jœ) (6)

where the responsivity ui is assumed frequency independent. The quantity D(jœ) is a delay factor corresponding to the finite carrier transit time. The factorM(/o>) is the frequency response of the internal gain, and Ζ(/ω) is the transfer function for the equivalent circuit.

The straightforward method of determining the frequency behavior of G(Jœ), by modulating a laser with a sinusoidal signal and study the detector output, is not applicable in this case since semiconductor lasers do not have a linear response at high enough frequencies. Furthermore, a pulse from a semiconductor laser cannot be used for impulse response measurements on the diode since it is difficult to experimen­tally confirm that the laser pulse is short enough without knowing the detector transfer function. Instead we have taken

JOHANSSON et al: HIGH CAPACITY COMPONENTS IN OPTICAL-FIBER LINKS 53

LIGHT SOURCE APD

BIAS VOLTAGE

AMPLIFIER

D/A SPECTRUM ANALYZER A/D

ν (̂ω) Μ(ω) MINI­

COMPUTER HP 2116

1 k

BIAS VOL ΓΑ6Ε

SAMPLING HEAD

1 '

SAMPLING OS >C.

TUNNEL DIODE

t k

Jp|(inCD

i~3 D/A A/D

Ζ(ω) W '

Σ_ϊ MINI­

COMPUTER HP 2116

Fig. 2. The experimental setup for measurement of the detector parameters. (a) Setup for measurement of the noise spectrum and the gain factor, (b) Setup for the TDR measurement performed to determine the equivalent circuit parameters.

advantage of the similarities in the frequency behavior of the transfer function G(jw) and the shot-noise power spectrum νΙ(ω). The frequency behavior of the noise spectrum can be expressed as [19]

ν2η(ω) ~ |Λ/(/ω)|2|Ζ)(/·ω)|2|ΖΟ·ω)|2 (7)

A typical setup for measurement of the shot-noise spectrum and the frequency characteristics of the gain factor for an avalanche photodiode is shown in Fig. 2(a). The shot-noise spectrum is easily obtained as the difference between the noise levels from the illuminated and the unilluminated APD, while the frequency characteristics of the gain factor are evaluated from comparative studies of shot-noise spectra for different dc gain factors. With a time-domain reflectometry (TDR) measurement, shown in Fig. 2(b), we can determine the equivalent circuit parameters by fitting a calculated step re­flection to the step reflection measured with the sampling os­cilloscope [19]. This will enable us to determine whether the APD is limited by the RC time, the transit time, or the mul­tiplication time.

To calculate the phase </>(ω), we use the Hilbert transform theory according to which the phase can be calculated from the magnitude as [20]

7Γ Jo "2 — '^2 ωΑ ■ dr\ (8)

resulting in

<7(/'ω) = \G(jœ)\eJ*^ (9)

provided that the transfer function is minimum phase, i.e., has no poles or zeros in the right half plane. Minimum phase be­havior is expected since the gain factor, the delay factor, and

the transfer function of the equivalent circuit all show mini­mum phase behavior [16], [17], [19]. Knowing both magni­tude and phase of the transfer function, one can calculate the electrical output arising from any optical input using FFT techniques.

IV. COMPUTER P R O G R A M S

The data processing needed for the evaluation of the com­ponents of an optical-fiber communication link is made using a general-purpose signal-processing package. The package has the capability to store several data buffers, each containing a number of (*, y) data points. The variable x is a real number representing the time or the frequency in a linear scale, and y is a complex number stored either as a real and an imaginary part or as amplitude and phase. The data processing is per­formed by means of several different program modules that can be either run interactively or controlled by special com­mand files.

Data Flow

The logical data flow for simulation of an optical-fiber link is shown in Fig. 3. The structure is related to the measurement setup shown in Fig. 1. The input signal to the laser model can either be read from a data file containing measurement data from the measurement setup or produced by a program mod­ule. Many different signals can be created, e.g., square waves and Gaussian pulses. The laser model described above has been implemented in a program module. The two rate equations (1) and (2) are numerically integrated to produce a laser output signal in the time domain. This signal is transformed into the frequency domain using a fast Fourier transform (FFT) pro­gram module. The influence of a fiber can be taken into ac­count by means of a linear transfer function that is multiplied by the laser output in the frequency domain. The output from the fiber is then multiplied by the transfer function of the de­tector. From a TDR measurement we calculate Z(/co) and from noise spectrum measurements, performed at different dc gain factors, we obtain M(jœ) and V2

n(œ). The phase is then calculated by means of a Hilbert transform module. The output of the detector in the frequency domain can then be transformed to the time domain using the FFT module and compared with measured data obtained from the measurement setup. The amplifier output can also be calculated by multi­plying the input by the transfer function in the frequency do­main and transforming the result to the time domain.

Program Organization

The use of the program package is not restricted to the data flow shown in Fig. 3, as can be seen from the actual program organization described in Fig. 4. All calculation activities are concentrated to the main data buffer, which can be processed by different program modules. The logical data flow shown in Fig. 3 is carried out by a sequence of commands given to a main program that activates the appropriate program modules. These commands can be entered to the main program inter­actively one at a time, or sequences of commands can be read from special command files. The command file handling in­cludes conditional execution of commands, DO loops, and subroutine nesting of command files to more than 50 levels.

54 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-30, NO. 1, MARCH 1981

TIME DOMAIN

FREQUENCY DOMAIN

PROGRAM MODULE

D DATA BUFFER NOISE DATA

TOR DATA

COMPLEX MULT

COMPLEX MULT

FIBER TRANSFER FUNCTION

AMPLITUDE CALCULATION

DETECTOR TRANSFER FUNCTION

HILBERT TRANSFORM

l O i DATA COMPARE

=ΓΤ-

COMPLEX MULT

AMPLIFIER TRANSFER FUNCTION

Fig. 3. The logical data flow of the computer calculations.

OTHER PROGRAM MODULES

. À

FFT

11 M il r MAIN YRE DATA Y,M BUFFER X

♦ PLOTTING u_

LASER 1 MODEL

1 HILBERT

TRANSFORM

DATA TRANSFER

see

.1. D D D D

DATA BUFFERS

Fig. 4. The structure of the program package, where other program modules include, e.g., test signal generation, pseudo noise generation, smoothing, statistics, and scaling.

The command files are used for convenient handling of fre­quently used sequences of commands. They also eliminate the need of an operator during time-consuming processing which often is necessary when the models are evaluated with several different parameter values.

All the program modules interact with the main data buffer, which is a file on the disk memory. It is possible to transfer data from external data files to the main data buffer. The additional data buffers have the same format as the main one and they are used for data storage and mathematical operations. The laser model uses the main data buffer as input and the output electron and photon densities are stored on two different ad­ditional data buffers. The FFT module converts the >>-data of the main data buffer to the corresponding transform and changes the jc-axis from time to frequency or vice versa. The Hilbert transform module uses one half of the ̂ -buffer as the input amplitude data and stores the calculated phase in the other half. Different data output modules are included in the package, e.g., plotting on a TV monitor and on a graphics plotter.

Other program modules include various standard features such as test signal generation, pseudo noise generation, fil­tering, smoothing, statistics, search for minimum and maxi­mum values, and scaling. Since all parameters and data buffers

are stored on the disk memory, it is very easy to add new pro­gram modules to the program package. The only interaction between the existing modules and the new ones is that they operate on the same main data buffer and that they are acti­vated by the same main program. If common areas in the computer memory were used instead of the disk for data storage, it would have resulted in a strong dependence between all the program modules. As a consequence, most of the modules would have had to be rewritten when new ones were added. Furthermore, the storage of parameter values on the disk reduces the amount of numerical input since all previous values are saved between different executions of the pro­grams.

V. PRACTICAL SYSTEM CONSIDERATIONS

The need of high resolution both in the time and the fre­quency domains requires large data buffers. The number of data points needed is

N = -1

(10) AtAf

where At and Δ/are the resolutions in the time and the fre­quency domains, respectively. Equation (10) is derived from the properties of the discrete Fourier transform. The number TV should be chosen as a power of two since the FFT module, which uses the Cooley-Tukey algorithm, will work with highest efficiency under this condition [21].

The HP 21 MX is a 16-bit computer that uses two words to represent a floating-point number. Since the data buffers consist of complex numbers, every data point will require four 16-bit computer words for storage. The computer has a 32K memory and if both the program and all the data have to be in the memory at the same time, a practical limit for TV is 4096. It is possible to exceed this limit but it will result in an increased need for disk accesses and, therefore, much longer execution time of the programs. This is especially significant for the FFT module because of the nature of the algorithm. For the appli­cation described in this paper we have chosen to work with TV = 4096, At = 10 ps, and Af= 24.4 MHz. The total range of

JOHANSSON et al: HIGH CAPACITY COMPONENTS IN OPTICAL-FIBER LINKS 55

Fig. 5. The normalized electron density calculated using the laser model with a 300-ps-wide Gaussian input current pulse of varying height. The central curve corresponds to the optimum modulation condition.

the data buffer in the time domain is then ( i V - l ) X A / = 40.95 ns and in the frequency domain Af X N/2 = 50 GHz.

The data acquisition in the time domain is made with a sampling interval of 10 ps resulting in a Nyquist folding fre­quency of 50 GHz as shown above. This is well above the 18-GHz range of the sampling oscilloscope making aliasing errors in the measurement procedure insignificant. Before being used in the data processing, the measurement data must be preprocessed. A special correction is made to remove trends in the data, defined as any frequency component whose period is longer than the total sample length. Such a trend, e.g., may be the result of a small drift in the dc level in the output of the sampling oscilloscope.

Most of the calculations in the data processing are made using single-precision floating-point numbers with 24-bit mantissa and 8-bit exponent. This corresponds to a resolution of about 6 decimal digits and a range of about 77 decades. Some of the program modules have the capability to work with 40-bit double-precision mantissa values. However, this will increase the execution time by at least a factor of 10 and, therefore, it cannot be used as a standard. We have tested the double-precision option for all critical calculations, e.g., the numerical integration of the laser rate equations (1) and (2). A comparison shows that it is unnecessary to use the double precision in most cases. However, it should be checked for each calculation situation if an increased resolution could give a better result.

The best way of testing the influence of measurement errors in the data processing is to check the sensitivity of each pro­gram module for disturbances. This should be done by adding both a constant, corresponding to a systematic error, and pseudo noise, corresponding to imprecision in the measure­ment, to the input data.

VI. EXAMPLES

In this section we will show some examples of data pro­cessing performed with the computer system.

Typical results from calculations using the laser model are shown in Fig. 5. The pulse current height was varied in search for the optimum modulation condition under which the density of excited electrons, as fast as possible after the modulating pulse, returns to steady state [12]. The calculations were controlled by a command file essentially built up by a DO loop

T 0

F requency (GHz)

Fig. 6. The frequency characteristics of the APD. {a) Amplitude data ob­tained from noise and TDR measurements, {b) Phase calculated using the Hilbert transform module.

in which a 300-ps-wide Gaussian current pulse of varying height is processed using the laser model. The output electron density was derived from numerical integration of ( 1 ) and (2), using a time step of 1 ps. This short time step was used to obtain a sufficient accuracy of the calculation. For each different pulse height every tenth value of the output was stored in consecutive data buffers for later plotting, shown in Fig. 5. The execution time for each repetition of the DO loop was in this case about 25 s.

An example of a complex detector transfer function ob­tained from noise spectrum and TDR-data measurements is shown in Fig. 6. After smoothing and assuming a constant slope of 24 dB per octave for frequencies above 3 GHz [19], the phase was calculated in 2048 points, 24.4 MHz apart, by means of the Hilbert transform module. Due to the large number of data points, the execution time was about 1.5 min.

An example of simulation of the laser and the detector is shown in Fig. 7. The input signal to the laser is a measured pulse, with a halfwidth time of 300 ps, from a step recovery diode [22]. The laser output, calculated using the rate equa­tions (1) and (2), is a 65-ps-wide pulse. The calculated output from the APD is derived using its transfer function obtained from noise measurements in the frequency domain. The result is a 215-ps-wide pulse to be compared with the measured APD output pulse that is 210 ps in width. The difference is fully explained by the inaccuracy of the measurements and by the limitations of the models.

VII. CONCLUSIONS

The work described in this paper shows the usefulness of a flexible computer system with a versatile program package for evaluation and measurements on components in high capacity optical-fiber links. The modular organization of the software makes it easy to add new program modules or change old ones. Furthermore, the command file control of the calculations has proved to very useful, e.g., when the component models are tested with many different parameter values. The system is built up around a minicomputer with a 32K main memory and a disk. This makes it possible to easily handle 4096 data points in both the time and the frequency domains. However, it is possible to increase the number of data points and, thereby,

56 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-30, NO. 1, MARCH 1981

(a)

(b)

(c)

c D X L 0 L •P _û L 0

0 c en

•H 0)

c •H 0 E 0

TJ φ E

(d)

. o 2 . O

Cns)

Fig. 7. A typical example of data processing with the computer system, (a) Measured input current pulse to the laser, (b) Calculated light output of the laser, (c) and (d) Calculated and measured output signals, respectively, from the APD.

achieve a higher resolution, but this will result in a much longer execution time due to the large amount of disk accesses nec­essary. Improvements are currently being done to shorten the execution time of the program modules by assembler coding and also by microprogramming of some time-critical disk-handling routines.

ACKNOWLEDGMENT

The authors wish to thank Dr. A. R. Johnston of the Jet Propulsion Laboratory, Pasadena, California, and R. Tell for helpful discussions of the work described in this paper. They also acknowledge the support from B. Olsson and C.-O. Fait during the preparation of the computer programs.

REFERENCES

[ 1 ] J. Gruber, M. Holz, P. Marten, R. Petschacher, P. Russer, and E. We-idel, "A 1 Gbit/s fiber optic communications link," Fourth European Conf. Optical Communication, Genova, Sept. 12-15, 1978.

[2] C. Baack, G. Eire, B. Enning, G. Heydt, H. Knupke, R. Löffler, and G. Waif, "1.12 Gbit/s regenerator experiment for an optical transmission system," Frequenz, vol. 32, pp. 151-153, May 1978.

[3] W. Harth and D. Siemsen, "Modulation characteristics of injection lasers including spontaneous emission—1. theory," AEG, vol. 30, no. 9, pp. 343-348, Sept. 1976.

[4] G. Arnold and P. Russer, "Modulation behaviour of semiconductor injection lasers," Appi. Phys. vol. 14, pp. 255-268, June 1977.

[5] J. W. Dannwolf, S. Gottfried, G. A. Sargent, and R. C. Strum, "Opti­cal-fiber impulse-response measurement system," IEEE Trans. Instrum. Meas., vol. IM -25, pp. 401-406, Dec. 1976.

[6] M. Maeda, K. Nagano, Y. Minai, and M. Tanaka, "Baseband frequency response measurement system for optical components," Appi. Optics, vol. 17, pp. 651-654, Feb. 1978.

[7] R. Lang and K. Kobayashi, "Suppression of the relaxation oscillation in the modulated output of semiconductor lasers," IEEE J. Quant. Electron., vol. QE-12, pp. 194-199, Mar. 1976.

[8] M. Danielsen, "A theoretical analysis for gigabit/second pulse code modulation of semiconductor lasers," IEEE J. Quant. Electron., vol. QE-12, pp. 657-660, Nov. 1976.

[9] P. Russer, G. Arnold, and K. Petermann, "High-speed modulation of DHS lasers in the case of coherent light injection," Third European Conf. Optical Communication, München, pp. 139-141, Sept. 1977.

[10] T. Ozeki and T. Ito, "A new method for reducing pattern effect in PCM current modulation of DH-GaAlAs lasers," IEEE J. Quant. Electron., vol. QE-9, pp. 1098-1101, Nov. 1973.

[11] M. Danielsen, J. Buus, F. Mengel, K. Mortensen, and K. Stubkjaer, "Characteristics of semiconductor stripe lasers as a fibre optic source," Third European Conf. Optical Communication, München, pp. 142-144, Sept. 1977.

[12] P. Torphammar, R. Tell, H. Eklund, and A. R. Johnston, "Minimizing pattern effects in semiconductor lasers at high rate pulse modulation," IEEEJ. Quant. Electron., vol. QE-15, pp. 1271-1276, Nov. 1979.

[13] K. Konnerth and C. Lanza, "Delay between current pulse and light emission of a GaAs injection laser," Appi. Phys. Lett., vol. 4, pp. 120-121, Apr. 1964.

[14] H. Kressel and J. K. Butler, Semiconductor Lasers and Heterojunction LEDs. New York: Academic, 1977.

[15] M. Ito and S. Machida, "Fractional spontaneous emission coupled into AlGaAs laser mode," Electron. Lett., vol. 14, pp. 693-695, Oct. 1978.

[16] T. Kaneda and T. Takanashi,."Avalanche buildup time of silicon ava­lanche photodiodes," Appi. Phys. Lett., vol. 26, pp. 642-644, June 1975.

[17] T. Kaneda, H. Takanashi, H. Matsumoto, and T. Yamaoka, "Avalanche buildup time of silicon reach-through photodiodes," J. Appi. Phys., vol. 47, pp. 4960-4963, Nov. 1976.

[18] P. P. Webb, R. J. Mclntyre, and J. Conradi, "Properties of avalanche photodiodes," RCA Rev., vol. 35, pp. 235-278, June 1974.

[19] T. Andersson, H. Eklund, and A. R. Johnston, "Temporal and frequency response of avalanche photodiodes from noise measurements," J. Appi. Opt., vol. 19, pp. 3496-3499, Oct. 1980.

[20] E. A. Guillemin, Theory of Linear Physical Systems. New York: Wiley, 1963.

[21] J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measure­ment Procedures. New York: Wiley, 1971.

[22] R. Tell, P. Torphammar, and S. T. Eng, "A 5 Gbit/s multiplexer for fiber-optical communication systems," Electron. Lett., vol. 13, pp. 765-766, Dec. 1977.