A.

Excess noise factor and QEFR

During the characterization of a SWIR RAPID FPAs, we have observed some devices exhibiting a mean value of the measured excess noise factor (F_meas) value below 1. As an example a value 0,94 was measured for the 3.3μm cut off wavelength device. The F definition detailed later implies that a F value below 1 is not physic. A difference between the true F and the measured value has ever been observed by DRS [4]. The origin of this difference was explained by a variation of the pixel fill factor when the photodiode is biased. In our case an F_meas below 1 suggest that the fill factor slightly increases when the APD is biased. The pixel QE can be defined as the ratio between the number of incoming photons to the number of electrons collected by the photodiode before multiplication by the gain. From this simple definition, a fill factor increase will lead to a QE increase. This QE evolution has two main consequences that will be detailed in the following analysis.

The APD gain is usually measured by the ratio between high and low bias signals. If the QE slightly increases when the photodiode is biased to get the gain, the measured gain G_meas will contain both the true gain G and QE variation and gives an over-estimation of the gain. If Q₁ is the quantum efficiency low APD bias (at unity gain) and Q the QE at high bias, the only equation that can be written is:

The second consequence of QE variation with the bias is that the excess noise factor measurement can result in a value lower than the theoretical low limit of 1. F is defined as the ratio of SNR at the input of the APD to the SNR at the output. Here we are using the photonic F, the ratio between SNR limited by the photon noise before APD gain, SNR_{SN In} to the value of this SNR after gain, SNR_{SN Out}

For a given number of integrated photons N_ph, the shot noise limited SNR before APD gain is:

and so:

As Q is unknown, the SNR before the APD gain can’t be evaluated from (3).

The input shot noise limited SNR that can be measured is:

F_meas being defined as the value of F evaluated from SNR_{SN In Meas} and the measurement of SNR with the APD biased, SNR_{SN Out Meas}, then we get:

And making the ratio with (4):

It’s now clear that F_meas is underestimated when the QE increase with APD bias. Even if the true values of Q and F can’t be measured, the Q/F ratio or QEFR is accessible to the measure via the following relation simply deducted from the previous one:

B.

Noise equivalent number of photons and signal to noise ratio

The evaluation of the SNR that will provide an APD based FPA for a given number of incoming photons is crucial for all kind of applications. Another useful parameter is the number of photon producing a signal equal to the total noise of the FPA. For the following calculation, we will assume a negligible background flux. To avoid confusion with other formulation of the noise equivalent number photon taking into account the noise of the signal itself [5], we will name this one the dark noise equivalent number of photons and use the notation N_deph.

From a noise measurement in dark condition, we get an evaluation of the sum of the dark current and the read noise. If we name this total dark noise at the output of the APD σ_dark and from the previous the N_deph is:

Using (1), we get a formulation using parameters accessible to measurement:

Readout circuit noise of APD based FPA are often expressed in electrons back referred to the photodiode input (i.e. before APD gain). When the QE changes with bias, we don’t have access to the true gain and the back referred FPA noise expressed in electron is unknown. The FPA noise can only be expressed in Volt at the output of the FPA or in photons at the input of the photodiode.

The SNR at the output of the FPA is:

S_Out being the FPA output photonic signal expressed in electron and σ_{SN Out} the photonic shot noise in electron. By expressing the signal and noise of (4), it comes:

And then (11) gives:

The product FG²Q can be expressed from measurable FPA parameters. It’s the ratio .

This SNR formulation allows to evaluate the SNR at the output of the FPA for any number of incoming photons and from measureable figures of merit: the QEFR and N_deph. This is also valuable when the QE is stable with APD bias, but as we have seen before, if it’s not the case, some precautions have to be taken to evaluate QEFR and N_deph.

A.

Conversion capacitance evaluation

The conversion capacitance is calculated from the slope of the total current of the array as a function of the mean output voltage of all pixels. As for other photonic measurements, the FPA is in front of an extended black body and equipped with an f/4.1 cold screen. The current variation was obtain by changing the temperature of a black body in front of the FPA from 15°C to 30°C. A conversion capacitance C_int of 4.14 fF was determined from Fig. 1 for a low APD bias voltage (unity gain). For higher bias, the gain, QE and C_int evaluations can’t be distinguished.

Fig. 1 :

RAPID FPA integration capacitance evaluation at low APD bias voltage (-0.2 V).

B.

Electro optical FPA performances measurements and experimental SNR evaluation

The gain was measured from the FPA response to a black body temperature (T_bb) variation of 5°C around 17.5 C. The use of the response method avoids any confusion between the dark current and the gain as it can be when measuring the gain from a simple current ratio. For the 3.3 μm cut off FPA, a mean value gain of 31 is obtained at 8 Volt of reverse bias. The pixel to pixel or spatial standard deviation is quite low at 2.2. More important than the gain dispersion is the uniformity of an image after non uniformity correction (NUC). As an example, we have simply make a NUC from two images at different integration times (30 μs and 80 μs) at 8 Volt of APD reverse bias. Fig. 2 is the result of the NUC applied to an image at a third integration time of 60 μs. The 82 bad pixels deviates by more than 33 % of the mean value. The Y scale limits of the image are also fixed at more or less 33 % of the mean value. The residual fixed pattern noise (RFPN) in this case is less than 2.8 % of the mean value. This good uniformity after NUC is preserved when the photon flux on the FPA changes. Fig. 3 shows a flat field in front of a 20°C black body with the NUC recorded at 15°C used for Fig. 2. The spatial dispersion is very low at 1.3% and there is only 39 bad pixels.

Fig. 2 :

T_bb = 15°C flat field map and histogram for 8 Volt of APD reverse bias (measured gain of 31) after non uniformity correction.

Fig. 3 :

T_bb = 20°C flat field histogram for 8 Volt of reverse bias (measured Gain of 31) after non uniformity correction with a NUC performed at 15°C.

As it was done for the gain, the quantum efficiency is evaluated by changing the black body temperature and making the ratio between the calculated variation of the number of photons per pixel and the variation of integrated electrons. By working on difference we eliminate a possible background photonic signal. As we don’t use any optical filter, the calculation of the number of incoming photons is made from the measured spectral response of a photodiode coming from the same LPE-grown wafer than the APD pixels.

For F evaluation, we use an expression obtained from (7) and (12):

Using (1) e and the expression of the FPA output signal and noise voltages, and , we obatin a simple expression of F_meas:

Output voltage and noise values used for F calculation are the difference between short and long integration time measurements in order to eliminate the circuit offset and read noise. The mean excess noise measured at -8 Volt (gain 31) is 0.94. For this APD geometry configuration the quantum efficiency increases with photodiode bias.

The noise in the dark was measured at 600 μs integration time in a passively cooled cryostat designed for very low residual photon flux. This integration time allows a frame rate of 1.5 KHz representative of the typical value needed for AO. Given the weak dark current measured at 80 K, the noise in the dark is dominated by the ROIC noise.

The QEFR and N_deph maps evaluated from the previous gain, F, QE and noise in dark conditions measurements are presented Fig. 4. The QEFR shows a mean value of 0.59 with a weak spatial dispersion of 1.7 %. This dispersion only takes into account the QE dispersion at low APD bias as we use the mean value of F_meas for the QEFR evaluation. A mean N_deph of less than 3 photons is obtained. We can observe a tail for high values side of the N_deph histogram. The read out circuit is highly suspect to be the source of these noisy pixels. Indeed, most of them are common to low and high APD reverse bias voltage noise measurements. Furthermore, we have also observe this tail for a short integration time in all FPA tested. We have now identify the most probable source of excess noise in the ROIC. A new design and conception would clearly improve the ROIC noise performance.

Fig. 4 :

QEFR and N_deph histograms of the 3.3 μm cut off FPA at -8V (G_meas = 31).

For imaging applications at low signal level, it may be convenient to express SNR in amplitude rather than in power. The previous SNR_Out was expressed from the power definition in order to stay coherent with classical F definition. The SNR in amplitude (SNR_Amp) is given by the square root of (15). The evolution of SNR_Amp with the number of photons per pixel and per frame for the tested FPA is plotted Fig. 5. The shot noise limited SNR and the SNR that can be evaluated at unity gain from the dark noise and QE measurement are also plotted.

Fig. 5 :

SNR_Amp for the 3.3 μm cut off FPA calculated from the mean QEFR and N_deph measured values at -8V (blue curve). The red curve is evaluated for the same FPA without APD gain (with the measured dark noise unity gain).

An histogram of SNR_Amp for 20 photons per frame per pixel evaluated from QEFR and N_deph maps is also depicted in Fig. 6. The mean SNR_Amp of 3 is coherent with the value obtained from the curve of Fig. 5. From these curves we can also notice a FPA output SNR under 1.5 without gain and a shot noise limited value of 3.3. The noise histogram displays a noise tail coming from dark noise and then as we said before, probably from the ROIC. The noise histogram curve becomes much more symmetric when photonic noise overcomes this excessive read out noise (Fig. 7). This is a good indication that noisy pixels observed in dark measurements are not due to avalanche photodiodes with excessive F.

Fig. 6 :

example of calculated SNR_Amp histogram of the 3.3 μm cut off FPA from the QEFR and N_deph maps at -8 V and assuming a signal of 20 photons per pixel per frame.

Fig. 7 :

Back referred measured noise histogram of the 3.3 μm cut off FPA under illumination expressed in photons (divided by Q₁G_meas) at -8 V, 30 μs integration time and TCN = 15°C.

C.

Dark current of a 3 μm FPA

The RAPID FPAs have been used to measure the 3μm APD dark current evolution with temperature between 80 K and 200 K. The aim of these measurement was to get information about the limiting diode current process.

The mean dark current density of a 3μm FPA was measured at high and low APD reverse bias (Fig. 8). Currents values are normalized by the pixel area, (30 μm)², and by the measured gain for the high reverse bias. For both measurements we have also plotted a fit using the sum of a generation recombination (GR), a diffusion and a photonic current. Indeed we have observed a saturation in the decrease of the current with temperature at the same level for both APD bias. As the Si ROIC was not optimized to reduce the electroluminescence we believe that this limitation at 1×10^-12 A/cm² (or 56 e-/s/pixel) current comes from the circuit glow. This current value was used as photonic current to fit the experimental data. For this manual fit of our data, we have used the same value of diffusion current at low and high APD bias. At low bias, the GR current like contribution exceeds the diffusion like current under 150 K. For the same device biased at -6.3 V (gain of 7.3) the GR dominates above 180 K. If the current at high bias would follow the GR trend, the pixel dark current would reach 2.4×10^-15 A/cm² or 0.13 e-/s for our 30 μm pixel.

Fig. 8 :

3 μm cut off wavelength APD FPA current measurement in dark conditions at -0.2 V (left) and -6.3 V (right) reverse bias.