Temporal and frequency response of avalanche photodiodesfrom noise measurements
Torbjbrn Andersson, Alan R. Johnston, and Hans Eklund
This paper describes a method of obtaining the temporal and frequency response of avalanche photodiodes(APD) by performing simple noise measurements. From the measured noise spectrum and by using the Hil-bert transformation technique, the complex transfer function of the detector is determined. The temporalresponse can then easily be calculated by means of fast Fourier transforming. The method has been appliedon a high speed APD, with a bandwidth of 2 GHz, and on a relatively slow APD, with a bandwidth of 0.2GHz, to calculate the pulse response from a short optical pulse. The calculated pulse width for the fast APDwas 215 psec, and the corresponding measured width was 210 psec, while for the slow APD the calculatedand the measured widths both were 3.1 nsec. Also the shapes of the pulse responses showed excellent agree-ment. The method depends on the essentially identical frequency response of an APD and associated cir-cuits for noise due to steady-state illumination and for a signal.
1. Introduction
In high speed optical fiber communication systemsavalanche photodiodes (APD) are commonly used asdetectors because of their low noise, high internal gain,and high speed response.1 Many commercially avail-able APDs are today specified to have a gain-bandwidthproduct exceeding 200 GHz. As one reaches for higherand higher bit rates, >1 Gbit/sec, the bandwidth limi-tation becomes a problem. It is therefore of great im-portance to determine the time and frequency charac-teristics of APDs.
This paper describes a method of determining thetemporal and frequency response of a high speed APDwithout requiring a picosecond pulse laser system,which is not likely to be found in a fiber-optic systemslaboratory. The method utilizes the close correspon-dence between the shot-noise spectrum and the fre-quency response of the photodetector and associatedpreamplifier, if present. Using the suggested approach,
Torbj6rn Andersson is with Chalmers University of Technology,Department of Electrical Measurements, S-412 96 Gteborg 5, Swe-den; Alan Johnston is with California Institute of Technology, JetPropulsion Laboratory, Pasadena, California 91103; and Hans Eklundis with LM Ericsson Telephone Company, S-126 25 Stockholm,Sweden.
the detector response can easily be obtained withcommonly used spectrum analyzers.
For high frequency APDs the bandwidth is limitedby three factors: the RC-time constant correspondingto the equivalent circuit; the carrier transit time; andthe time for multiplication in the avalanche region (theavalanche buildup time).23 At low multiplicationfactors the first two effects are usually dominant whilewith increasing gain the avalanche buildup time be-comes more and more important.4 - 6
Noise spectrum measurements to obtain the fre-quency characteristics of shot-noise limited IR detectorshave been performed by Peyton et al. 7 8 and for P-I-Ndiodes by Dumant et al.9 Several investigators haveused the method to estimate the intrinsic response timeof an APD,4 10-1 2 but the use of a noise spectrum mea-surement directly to obtain the over-all detector re-sponse time (including temporal as well as frequencyperformance of an APD) has not been described.
The proposed method, which utilizes the concept ofKaneda et al.,4 uses noise spectrum measurements toobtain the frequency characteristics of the amplitudeof the complex detector transfer function. Using theHilbert transformation technique to obtain the phase,we are able to calculate the temporal response by fastFourier transforming (FFT) techniques. Furthermore,it is easy to distinguish which limitation dominates ina device. The suggested method has been applied ona commercial APD of the silicon reach-through typewith a gain-bandwidth product exceeding 200 GHz and,for comparison, on a similar APD with a gain-band-width product of 20 GHz.
In this section we will present the detector model,which is used when calculating the frequency and thetemporal response of the detector from the shot-noisespectrum.
We define the detector transfer function accordingto Fig. 1 as
G(jw) = [V3 (j)]/[P,(,c)], (1)
where V8(jo,) is the output signal voltage, and P, (j ,)is the frequency spectrum of the incident optical signal.The signal current of an APD can be given by an ex-pression of the form3
I, = RM(j)D(jc)PsU(kt),
Fig. 1. Assumed equivalent circuit of the APD, where i and in are
the signal and noise currents, R, and C are the parallel resistance and
capacitance, R. is the series resistanceL is the lead inductance, andRo is the load resistance (50 Q).
(2)
where A, the responsivity, is assumed frequency inde-pendent, and D(jw) is a delay factor introduced becauseof the finite carrier transit time of the drift region. Thefactor M(jw) is the frequency response of the internalgain. Using Eqs. (1) and (2) we obtain the detectortransfer function
Gjw) = RM(jw)D(jw)Z(jw), (3)
where Z(jc) is the transfer function, with load Ro, forthe equivalent circuit given in Fig. 1. Alternatively,with appropriate modification, Z(jcv) could include theresponse of an associated preamplifier. Provided thatthe APD is shot noise limited the noise spectrum at thedetector terminals can be obtained as3
V'(w) = 2e IM(co)I2FPo]?BID(c)1 2 1 Z(jw)12 , (4)
where we have assumed that the shot-noise spectrumis subject to the same delay factor as the signal.13 -4
Here Po is the light intensity from an unmodulated lightsource, e is the electron charge, F is the excess noisefactor, and B is the noise bandwidth of the detectionsystem. Comparing Eqs. (3) and (4) we see that thefrequency characteristics of the amplitude spectrumG(jw) I and of \/V (co) are the same, since we have as-
sumed that the excess noise factor is frequency inde-pendent. 10 The multiplication factor is given by theexpression- 6
M(jw) = Mo/(l + joriMo). (5)
The low frequency gain MO and the intrinsic responsetime rl can be determined experimentally. The fre-quency response due to the finite carrier transit time ofthe drift region can be expressed as
where -y is the product of the absorption constant andthe width of the drift region, and -rp and rn are thetransit times for holes and electrons, respec-tively. 13 -1 5
To calculate the phase 0(co) we use the Hilberttransform theory according to which the phase can becalculated from the magnitude provided that the
1 0
C-:3
3:0
a
0£
4-I
(n
-o
-10
-20
-30 -. 5 5.
Frequency (GHz)
Fig. 2. Measured shot-noise spectra V~n for different gain factors.
Curves are normalized to the noise level corresponding to M = 100for low frequencies. Low and high frequency slopes are assumed to
be 0 and 24 dB/oct, respectively.
transfer function is minimum phase, i.e., has no polesor zeros in the right half-plane.' 6 From Eqs. (5) and (6)and Fig. 1 we find that this condition is satisfied.
We can now determine both the magnitude and thephase of the transfer function from the noise spectrumand obtain the complex transfer function as
(7)
It is then possible to calculate the temporal response ofan APD for a known input pulse by FFT techniques.
III. Experiments
Measurements were performed on two differentAPDs, one with a gain-bandwidth product exceeding200 GHz (Telefunken BPW 28) and one with a gain-bandwidth product of 20 GHz (RCA 30817). The noisespectrum measurements were performed using a highfrequency spectrum analyzer, a broadband low noiseamplifier, and a dc light source. The amplifier, whichhad a flat frequency response up to 3.15 GHz, was onlynecessary to use at low shot-noise levels, i.e., for low gainfactors, to increase the sensitivity of the spectrum an-alyzer. As a dc light source we used an incandescentlamp or an unmodulated GaAlAs diode laser biasedbelow threshold with similar results.
The APDs were operated under such conditions thatthe shot noise was the predominant noise source. Thiswas tested and found to be valid by measuring andcomparing the total noise from the unilluminated APDwith that from the illuminated APD. For the highspeed APD the measured noise level at low frequencieswith the APD unilluminated, i.e., Po = 0, was found tobe equal to a shot-noise level corresponding to a gainfactor of -10 at a dc light intensity Po of 5 mW. Theshot-noise spectra for different gain factors were thendetermined by calculating the difference between thenoise levels obtained with the APD illuminated andunilluminated.
The shot-noise spectra for the high speed APD atdifferent gain factors are shown in Fig. 2. They all havethe same typical shape, and the bandwidth reductiondue to increasing gain is very weak, corresponding to anintrinsic response time -r < 0.2 psec in agreement withKaneda et al. 4 However, before accepting this valueof rl we must confirm that the bandwidth of theequivalent circuit does not increase when the reversebias voltage is increased to obtain higher gain. Ourdetectors are of the reach-through type indicating thatat relatively low reverse bias voltage the structure isfully depleted, so the diode capacitance should not de-crease further when the bias voltage is increasing. Timedomain reflectometry (TDR) measurements verifiedthat the diode capacitance remained constant for a re-verse bias voltage exceeding that corresponding to a gainfactor of -5.
In Fig. 3 we show the magnitude of the transferfunction I G (o) derived for the high speed APD andthe corresponding phase 0(U). Here we have assumeda constant slope of 24 dB/oct for high frequencies, whilefor low frequencies we have assumed a constant shot-noise level to reduce the measuring time.
To separate the equivalent circuit parameters, TDRmeasurements were performed. They resulted in aninductance of about L = 3 nH and a capacitance C = 0.8pF. The series resistance was 50 , while R wasgreater than 1 k. From this we conclude that in thehigh speed APD both the circuit and the intrinsic re-sponse time are limiting factors, in agreement withBerchtold et al. 17
The same measurement approach was carried out onthe slow APD with a bandwidth of 0.2 GHz. The re-sultant shot-noise spectrum is shown in Fig. 4. For thisAPD the bandwidth of the equivalent circuit was de-termined from TDR measurements to be 0.8 GHz.Moreover, we see that the shot-noise spectrum is of theform expressed by Eq. (6) indicating that this diode istransit time limited.
Knowing the transfer functions of the APDs it is easyto obtain the response to any optical input signal usingthe FFT technique. To determine the pulse responseof the high speed APD, we used a semiconductor in-jection laser modulated to obtain very short opticalpulses. According to computer calculations using thelinear single-mode rate equations, the resultant outputpulse had a 65-psec width. 8- 9 This laser pulse wasthen used to determine the pulse response of the APDs
0
m
-SI0
- 10
-20
-30
-40
-50
. S 5. 0Frequency (GHz)
. 00
-3. 1
_U
(D
-a
-6. 28
Fig. 3. Frequency characteristics of the fast APD for a gain factorof 100: (a) magnitude obtained from the noise spectrum and TDR
measurements; (b) phase calculated using the Hilbert transform.
0
m_0
3C-S00L
0
0C
c.o0n
-10
-20
-30
Frequency (GHz) 1. 0
Fig. 4. Measured shot-noise spectrum for the slow APD.
both experimentally and by calculation using thetransfer functions determined above. The result isshown in Fig. 5, where we have plotted both the mea-sured and calculated output from the fast APD. Thepulse widths are 210 and 215 psec, respectively, with adifference that can be fully explained by the measure-ment errors. No diffusion tail is observed or is it ex-pected since the diode is fully depleted (reach-throughstructure). However, it was not possible to confirmindependently the calculated pulse width of 65 psecfrom the laser, since our detector is not fast enough.The result for the slow APD is shown in Fig. 6, where weused a wider input pulse with a 1.4-nsec width to im-prove the SNR at the output of the APD. The mea-sured and calculated output pulse widths are both 3.1nsec.
To calculate the phase we have assumed a constantslope for high frequencies. From Eqs. (4)-(6), and theequivalent circuit in Fig. 1, we find that assumingy a, i.e., absorption in a very narrow surface region, thefrequency behavior of the magnitude I G (X) I at highfrequencies can be approximated by
I G () - sin 2P / W4, (8)
which indicates a slope of 24 dB/oct for high frequen-cies. To test the influence of various slopes, we calcu-lated, for the high speed APD, the pulse response to an
Fig. 5. Temporal response of the fast APD: (a) calculated detector
signal for a 65-psec wide laser pulse; (b) measured detector signal.
Optical input pulse 65 psec wide, when the slope variedfrom 12 dB/oct to 36 dB/oct. The only difference be-tween the output pulses was that the delay time be-tween the input and the output pulses increased forsteeper slopes, while the width and shape of the outputremained essentially unchanged. We also calculatedthe pulse response using different amounts of smooth-ing to test the influence of stocastic errors in the mea-surement values. For moderate averaging no changescould be observed in the pulse response.
IV. Conclusions
We have described a method of obtaining the tem-poral and frequency response of APDs from simplenoise measurements. The usefulness of the method hasbeen shown for a fast APD (2 GHz) as well as for a rel-atively slow one (0.2 GHz). The agreement betweencalculated pulse responses, obtained using this method,and independently measured pulse responses is excel-lent, e.g., the calculated and measured pulse widths forthe high speed APD are 215 and 210 psec, respectively.The method, which can be applied both on APDs andP-I-N diodes, is especially beneficial when one wantsto determine the temporal and frequency response ofan unknown high speed detector (>2 GHz) and a shortoptical test pulse is not readily available.
Limitations to this method of measuring the fre-quency characteristics of -an APD are mainly due to thesensitivity of the spectrum analyzer or, if used, thebandwidth and noise figure of the amplifier. Fur-thermore, it can be necessary to insert an attenuatorbetween the detector and the spectrum analyzer toimprove the standing wave ratio. These problemsprevented us from measuring the noise spectrum tofrequencies well beyond cutoff. However, since thecalculated pulse response is relatively unaffected fordifferent slope assumptions, this is not a serious draw-back.
The authors wish to thank S. T. Eng, S. Lundqvist,P. Torphammar, and R. Tell for helpful discussions ofthe work described in this paper. We also acknowledge
LO/
L0
0 5 lo0 1 5 20T i me (s)
Fig. 6. Temporal response of the slow APD for a 1.4-nsec wide laser
the support from J. Johansson for developing thecomputer programs.
This work was supported by the Swedish Board forTechnical Development.
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