TECHNICAL NOTE: V803 Avalanche Photodiode Receiver Performance Metrics Introduction The following note overviews the calculations used to assess the noise equivalent power (NEP), noise equivalent input (NEI), and signal to noise ratio (STN) performance of avalanche photodiodes (APDs) and APD photoreceivers. Shown are the effects of the APD’s effective ionization constant, keff, which characterizes the noise of the avalanche process. Background APDs are photodetectors that can be regarded as the semiconductor analog of photomultiplier tubes (PMTs). One important difference is that APDs don’t have a photocathode that is physically separate from their current gain medium, and so they typically use primary photocarriers more efficiently than PMTs. For the same reason, the quantum efficiency of an APD does not degrade over the lifetime of the detector. A second difference is that the multiplication process in an APD is normally bi‐directional, so it has different statistics than a PMT in which the gain process is uni‐directional. Under reverse bias (typically less than 100 V for InGaAs APDs), APDs have internal current gain (M) due to impact ionization (avalanche multiplication), which results in a gain of approximately M=30 for most commercial InGaAs APDs, but up to M=8,000 for Voxtel’s latest APD models. Current gain is used in detector systems to boost the photocurrent from a weak optical signal above the noise floor of the amplifier. However, the gain provided by an APD contributes its own shot noise component that grows with increasing avalanche gain, so for any given level of amplifier noise, there is a limit to how much avalanche gain will be useful. All avalanche photodiodes generate excess noise due to the statistical nature of the avalanche process. The 'excess noise factor’ is generally denoted by the symbol 'F', and is the ratio of the mean square gain to the square of the mean gain – it is also the ratio by which the spectral intensity of shot noise on an APD’s current exceeds that which would be expected from a noiseless multiplier on the basis of Poisson statistics alone. Excess Noise The excess noise factor is a function of both the gain and the APD’s effective ionization coefficient ratio (k). Impact of keff on NEP, NEI, and STN As described above, the excess shot noise of an APD at a given gain depends upon its effective ionization coefficient ratio (k) according to equation (1). The values of k are significant because they correspond to different APD device technologies which are compatible with near infrared (NIR)‐sensitive InGaAs absorbers. The most common InGaAs APDs have bulk InP multipliers characterized by k=0.4. Voxtel sells InGaAs APDs with thin InAlAs multipliers that are characterized by k<0.2. Recently, Voxtel has developed InGaAs APDs with multiple gain stages which can operate with k~0.02. The excess noise factor is normally calculated using a formula derived by Mclntyre1 which is based upon the assumption of an avalanche medium with uniform characteristics and an impact‐ionization process that is independent of carrier history: 2

 M  1   . F ( M , k )  M 1  (1  k ) 
 
 M  

(1) Neglecting minor 1/f and thermal noise components, the noise spectral intensity of an APD (SI in A2/Hz) is approximately flat across its bandwidth, and it depends upon the excess noise factor, the gain, and the primary (i.e. unmultiplied) current Iprimary. Note that Iprimary includes both the primary dark current as well as the primary photocurrent. By application of the Burgess variance theorem (used to find the variance 1
Mclntyre, R.J., “Multiplication Noise in Uniform Avalanche Diodes,” IEEE Transactions on Electron Devices, ED13 (1966). Voxtel Inc. TECHNICAL NOTE ‐ Page 2 of a multiplied quantity) and an extension of Milatz’s theorem (used to relate a rate of occurrence to a spectral density),2 the commonly‐used formula for an APD’s noise spectral intensity is: S I  2 q M 2 F ( M , k ) I primary . (2) Equation (2) is appropriate for noise analysis when the signal is encoded in the instantaneous power level of an optical signal, such as in a telecommunications receiver where a resistive transimpedance amplifier (RTIA) is used in conjunction with the APD. In that case, one integrates SI over the bandwidth of the receiver to find the APD’s contribution to the variance of the current fluctuations. Alternatively, one may have some form of integrating receiver, as might be used to measure the number of photons in a pulse of light. In that case, a charge amplifier such as a capacitive transimpedance amplifier (CTIA) might be used, with a characteristic integration time . In that case, the noise contribution of the APD in RMS electrons is: I dp ( M ) 

  . noise  M 2 F ( M , k )    signal 
q


(3) Here, η is the unity‐gain quantum efficiency of the APD, Idp(M) is the primary dark current at gain M, and the signal level is in photons. Equation (3) may be recognized as the Burgess variance theorem, (or rather an RMS value found by taking the square root of the variance given by the theorem). Detector Noise‐Equivalent Power (NEP) To quantify detector noise, a common figure of merit is the noise‐equivalent power (NEP). NEP can be defined as either a power or a power spectral density. When defined as a power in units of W, it represents the incident optical power required to generate a photocurrent equal to the RMS noise current of the photodetector (in) at a specified frequency (f), and within a specific bandwidth (Δf). To find in one integrates SI over Δf and takes the square root. The NEP is found by setting the signal photocurrent equal to the RMS noise current, and solving for the optical power: 2
Van Der Ziel, A., Noise in Solid State Devices and Circuits (John Wiley & Sons, 1986), pp. 14‐18.
Voxtel Inc. TECHNICAL NOTE ‐ Page 3 NEPAPD  R( )  M 
S
I
df  f  2 q M 2 F ( M , k ) I primary . (4) f
Here, NEP is the (unknown) optical power and R(λ) is the unity‐gain responsivity. The difficulty in isolating NEP depends upon the accuracy of the approximation used for Iprimary. In the first place, Iprimary includes both the primary dark current and also the primary photocurrent; in other words, the product NEP×R(λ) which represents the signal photocurrent appears inside the radical as part of Iprimary. Moreover, if there is background illumination, then the resulting photocurrent must be added to Iprimary. Finally, if there is a significant portion of the dark current which bypasses the APD’s multiplying junction – for instance surface leakage – then it must have its own noise contribution split out and added in quadrature to the right‐hand side of equation (4), with M and F set to unity. Fortunately, in most practical cases, the shot noise contribution of an optical signal near the NEP of an APD is negligible and may be safely omitted from equation (4). Likewise, in most conditions the shot noise on the multiplied component of the dark current dominates any contribution from background illumination or unmultiplied dark current. In that case, the NEP is simply: NEPAPD 
f  2 q M 2 F ( M , k ) I dp
R ( )  M
W . (5) Note that some sources prefer to use the multiplied responsivity in place of the unity‐gain responsivity, in which case the avalanche gain does not appear explicitly separate in the denominator. We break it out to emphasize that the factor of M which appears in the numerator is canceled out by the factor of M in the denominator; avalanche gain improves the NEP of a receiver when the noise is dominated by amplifier noise, but it cannot improve the NEP of a stand‐alone APD. Voxtel Inc. TECHNICAL NOTE ‐ Page 4 When dealing with APDs as stand‐alone components, NEP is commonly quoted in units of W / Hz1/2, for which purpose the integration over a specific bandwidth is omitted, and the flat noise spectrum given by the spectral intensity theorem of equation (2) is used to estimate the spectral density of the noise current. In that case – and making the same approximations that omit the signal shot noise, background illumination, and unmultiplied dark current – the NEP is just the square root of equation (2) divided by the multiplied responsivity: NEPAPD
SI


R ( )  M
2 q M 2 F ( M , k ) I dp
R ( )  M
 W 
 Hz  . 

(6) Again, note that the avalanche gain does not appear explicitly in the denominator if the multiplied responsivity is used in equation (6). Figure 1 shows the spectral intensity form of NEP. It was calculated using equation (6) and assuming uniform primary dark current levels of 1 nA, and a responsivity of 0.875 A/W which corresponds to 70% quantum efficiency at 1550 nm. Receiver Noise‐Equivalent Power (NEP) To estimate the NEP of an RTIA‐based receiver, one must include the noise of the amplifier in the noise current. Very often, an RTIA’s specification gives the input‐referred RMS noise current of the amplifier Figure 1: Effect of k on NEP calculated for APDs using equation (6). Voxtel Inc. TECHNICAL NOTE ‐ Page 5 across its 3 dB bandwidth (ampRMS). If not, the RTIA’s input‐referred noise spectral intensity can be integrated over its 3 dB bandwidth in the same manner as the APD’s shot noise. One caveat is that if the APD’s bandwidth is wider than the RTIA’s bandwidth, then the RTIA will limit the noise bandwidth from the APD, and ΔfRTIA should be used instead of ΔfAPD when calculating the in‐band APD shot noise. Since the amplifier’s noise is uncorrelated with the dark current of the APD, the variance of the sum is equal to the sum of the variances, and equation (4) becomes: 2
. NEPRTIA receiver  R( )  M  f RTIA  2 q M 2 F ( M , k ) I primary  ampRMS
(7) Unlike the case of a stand‐alone APD, the NEP will improve as the APD is operated at higher gain, since the amplifier’s noise is independent of the APD’s gain. Ultimately, however, the APD’s shot noise will come to dominate the amplifier noise, and the receiver’s NEP will fall if the APD gain is increased further: NEPRTIA receiver 
2
f RTIA  2 q M 2 F ( M , k ) I dp  amp RMS
R ( )  M
W  . (8) If the optical signal is harmonic or can be efficiently detected without using the full bandwidth of the receiver, it may be advantageous to band‐limit the receiver. For that analysis, the spectral intensity version of NEP is superior. In that case, the average spectral intensity of the RTIA’s noise should be used instead of the RMS value; alternatively, if the spectral intensity is not published by the RTIA’s manufacturer, it can be approximated by squaring the RMS value and dividing by the RTIA’s bandwidth: NEPRTIA receiver 
2
2 q M 2 F ( M , k ) I dp  ampRMS
/ f RTIA
R ( )  M
 W  .  Hz 


(9) Figure 2 shows the NEP across a 2‐GHz bandwidth of a hypothetical receiver made from these parts and a hypothetical ultra‐low‐noise RTIA with 25 nA RMS input‐referred noise. Voxtel Inc. TECHNICAL NOTE ‐ Page 6 Receiver Noise‐Equivalent Input (NEI) The symbol NEI is used inconsistently in the radiometric community – variously for “noise‐equivalent irradiance” or “noise‐equivalent input.” Voxtel’s usage is the latter. NEI expresses the noise of an integrating receiver in terms of the number of input photons that would produce a charge signal equal to the RMS charge noise of the receiver. Some workers convert NEP to NEI by means of the photon energy, but the quantity which results must be used with some care. The problem is that power is an instantaneous rate whereas the number of photons is a quantity – if you integrate the noise‐equivalent photon rate over a sample time  then you must use Δf = 1/2 (the Nyquist frequency) in equations (5) and (8) instead of the APD or RTIA’s bandwidth. A second consideration is that a capacitive transimpedance amplifier (CTIA) is likely to be used for measurements of photon number, rather than an RTIA, and a CTIA’s read noise does not increase with sample integration time. For these reasons, to find the NEI of an APD or APD‐based receiver, it is preferable to start with equation (3) and write the analog of equation (4): I dp ( M ) 

  . NEI APD    M  M 2 F ( M , k )    NEI APD 
q


(10) Equation (10) assumes that noise contributions from background photocurrent and unmultiplied dark Figure 2: RTIA Receiver NEP (nW) vs APD Gain ‐ Effect of k on NEP calculated for a high‐bandwidth, ultra‐low‐noise RTIA receiver using equation (8). Voxtel Inc. TECHNICAL NOTE ‐ Page 7 Figure 3: CTIA Receiver NEI vs. APD Gain: Effect of k on NEI calculated for a CTIA receiver using equation (14). current leakage are negligible. If background illumination is significant, the resulting primary photocurrent can be found using the unity‐gain responsivity, and should be added to the primary dark current in equation (10). Inclusion of unmultiplied dark current would require addition of a separate term under the radical, with M and F set to unity. Under the assumption of negligible signal shot noise, the NEI term under the radical is omitted, and equation (10) reduces to: M 2 F (M , k ) 
NEI APD 
I dp ( M )
q

M
[photons] . (11) It is possible to arrive at equation (11) from the similar expression for NEP given by equation (5). First, the definition of unity‐gain responsivity must be used to replace R(λ) with η: R ( ) 
q
h
A
 W  . (12) Also, Δf must be set to 1/2, and the resulting expression for NEP must be divided by the photon energy (hν) to find a noise‐equivalent photon rate. Finally, NEI is found by multiplying by the sample time , which represents signal integration. However, this exercise is unhelpful if one wishes to analyze a CTIA‐based receiver. Instead, one begins with equation (3) and adds the CTIA’s RMS read noise (in electrons) in quadrature: Voxtel Inc. TECHNICAL NOTE ‐ Page 8 I dp ( M ) 

2
.    ampRMS
NEI CTIA receiver    M  M 2 F ( M , k )    NEI CTIA receiver 
q


(13) It is well to note that the CTIA read noise in equation (13) does not increase with integration time . Making the usual assumption and dropping the signal shot noise, equation (13) simplifies to: M 2 F (M , k ) 
I dp ( M )
q
M
NEI CTIA receiver 
2
  ampRMS
. (14) Figure 3 shows NEI for an integrating CTIA‐based receiver with 60 electrons RMS input‐referred read noise and a 50‐ns integration time. Signal to Noise Ratio (SNR) The signal to noise ratio (SNR) under illumination can be calculated by including the effects of the signal shot noise. Assuming that the signal and dark current can be characterized by the same excess noise factor, the noise variance is obtained by autocorrelation of the fluctuating photocurrent and dark current.  s2  is2 (t )  2 q M 2 F ( M , k ) ( I dark , primary  I signal )f (15) Where Isignal = R(λ)Pin (16) The SNR is found by taking the ratio of the signal after amplification (Pin(λ) × R ( ) × M) to the charge noise: SNR 
I p2
2
P R ( )M 
. F ( M , k )I ( M ) f
2

in
2qM 2
(17) dp
Depending on the operating conditions, the APD receiver’s SNR can be dominated by several different sources. Under low light conditions, where the amplifier noise dominates (limit σT >> σs) and neglecting dark current Voxtel Inc. TECHNICAL NOTE ‐ Page 9 SNRthermal
Pin2 M 2 RL R 2 ( )

4k BTL Fn f
(18) Under signal shot noise limited conditions: SNRshot 
Pin R ( )
Pin

2qF ( M , k )f 2hF ( M , k )f
(19) Alternatively, one may have some form of integrating receiver, as might be used to measure the number of photons in a pulse of light. In that case, a charge amplifier such as a capacitive transimpedance amplifier (CTIA) might be used, with a characteristic integration time . In that case, the noise contribution of the APD in RMS electrons is: noise 
I dp ( M ) 

M 2 F ( M , k )    signal 
  . q


(20) Here, η is the unity‐gain quantum efficiency of the APD, Idp(M) is the primary dark current at gain M, and the signal level is in photons. Equation (3) may be recognized as the Burgess variance theorem, (or rather, an RMS value found by taking the square root of the variance given by the theorem). Similarly, the noise of a CTIA‐based receiver can be found by adding the input‐referred charge noise of the amplifier (amp) in quadrature to (3): I dp ( M ) 

   amp 2 . noiseCTIA receiver  M 2 F ( M , k )    signal 
q


(21) The SNR is found by taking the ratio of the charge signal after amplification (signal × η × M) to the charge noise given by equation (4): Voxtel Inc. TECHNICAL NOTE ‐ Page 10 SNR 
signal    M
I dp ( M ) 

M 2 F ( M , k )    signal 
   amp 2
q


. (22) The SNR curves plotted above in Figure 4were calculated using (5), assuming η = 75%, amp = 40 electrons RMS, and τ = 1 μs, showing the impact of k on SNR for a fixed Idp. Figure 4: SNR vs APD gain calculated for an amplifier with a 40 e‐ RMS noise, and showing APDs characterized by various values of k. Conventional InGaAs APDs are characterized by k=0.4, whereas Voxtel’s baseline design has k=0.2. Our most advanced InGaAs APD operates with k=0.02. A primary dark current level of 3 pA was assumed. Voxtel Inc. TECHNICAL NOTE ‐ Page 11