One of the main directions in modern optics is the development of quantum technologies, such as quantum communications, imaging, and sensing, where single-photon [1–4] or few-photon states of light are generated, manipulated, and detected [5–8]. Development of dedicated methods for characterizing detectors in this context is necessary, as widely recognized inside the radiometric community [9]. The single-photon regime is far from the one where traditional radiometry operates and where the best accuracy is available [10]. Some specific activities in this context are already ongoing [11], in particular, related to the calibration of single-photon detectors exploiting the Klyshko’s twin-photon coincidence technique [11–18] and its developments [19–26].

Besides the commonly used on–off and single-photon avalanche photodiodes without spatial resolution, new types of detectors are considered for specific applications to overcome their limitations, namely, photon number resolving and spatially resolving detectors. Among them, electron-multiplying charge-coupled device (EMCCD) cameras represent a commercial and diffuse approach for single- and few-photon imaging when high spatial resolution is required. EMCCDs and intensified CCDs have been used also for top-level experiments in quantum optics and quantum information technology when many spatial modes have to be detected, from sub-shot noise imaging [27], quantum illumination [28], twin-beam correlations investigations [29], and ghost imaging [30] to the detection of the EPR state and entanglement of orbital angular momentum [31–35].

In this Letter, we present the absolute calibration of an EMCCD camera, operated in the photon counting regime as a threshold detector (“on–off”) pixel by pixel, by the measurement of spatially multimode quantum correlation in a squeezed vacuum. The “on–off” behavior is achieved by applying a discriminating threshold $T$ on the electron counts $n_e$ at each pixel: a photon is detected if $n_e>T$. In this regime, the camera works as a nonlinear photon number resolving detector, counting the number of incident photons in a region of many pixels (spatial multiplexing) and acquiring many frames (time multiplexing). Hereinafter, we assume to work under the condition of low illumination (negligible probability to have more than one photon per pixel per frame); in such a condition, the device can be approximated as a linear photon counting detector.

The main difference between the Klyshko’s twin-photon coincidence technique and the method presented here stems from the fact that we compared the number of detected photons in correlated areas in a large integration time, exploiting a technique that was developed for a CCD in the analog regime [36–41]. In particular, we measure the noise reduction factor $\zeta =〈{[\delta (N_1-\alpha N_2)]}^2〉/〈N_1-\alpha N_2)〉$ and the correlation $C=〈N_1{N}_2〉-〈N_1〉〈N_2〉$, where $\alpha =〈N_1〉/〈N_2〉$, and $N_1$ and $N_2$ are the numbers of photons in the two correlated areas. These quantities are related to the mean detection efficiency $\eta$ by $\zeta \simeq \frac{1+\alpha }{2}-\eta A$ and $C\simeq \eta A〈N_1〉$, both in the analog and photon counting regimes for low photon numbers per mode, where $A$ is a geometrical parameter [41].

Even if in principle, also in the case of an EMCCD operating in photon counting, it would be possible to exploit the Klyshko’s twin-photon technique, there are a certain number of practical reasons preventing its use: Klyshko’s technique needs an illumination regime providing not more than one coincidence per frame, but in this illumination range the noise becomes dominant. Moreover, the read time of an EMCCD is higher with respect to single-photon detectors, yielding Klysko’s technique as very slow.

It is important to note that the detection efficiency includes both the transmission efficiency of the optical detection system and the CCD camera quantum efficiency. The efficiency of the CCD could be measured by performing a conventional calibration of the losses in the optical channel through a transmissivity measurement (as in the Klyshko’s twin-photon technique).

In this work, we exploit the same procedure for the absolute calibration of the detection efficiency both for the EMCCD operating in the proportional (analog) regime (i.e., without electron multiplication) and when it is operating in the photon-counting regime under the assumption of low illumination. Then, we compare the results obtained in the two regimes for the same device and the same experimental setup (only the power of the source of the twin-photon has to be tuned, as well as the mode of operating of the camera). We have to consider that in the photon counting regime, the detection efficiency is a function of the threshold. Therefore, it is not possible to perform a direct comparison between the efficiencies for the two regimes. However, it is possible to calculate the dependence $\eta (T)$ by the measured value of the analog quantum efficiency ${\eta }_0$ and to compare this with the measured efficiency in photon counting.

The quantum efficiency measurements, obtained for the same detector operated in proportional and photon-counting regimes, are the most significant results of this Letter that take advantage of the unique versatility in terms of the regimes of operation of our detector.

In order to obtain the theoretical behavior for the noise and for the quantum efficiency $\eta (T)$ of our camera, we analyze the typical model of an EMCCD [41,42], and then we estimate the parameters involved in this model by means of a set of measurements that are independent of the absolute calibration.

For $n$ photoelectrons at its input, the multiplication stage of each pixel provides a random number of electron counts $x$ following the distribution [43]:

The most relevant noise contributions typically affecting an EMCCD are read noise, dark current, and spurious charges.

The read noise, generated by the on-chip output amplifier, follows a Gaussian distribution $P_{rn}(x;\mu ,\sigma )$, where the mean value $\mu$ is the bias level of the read-out distribution, and the standard deviation $\sigma$ characterizes the fluctuation of read noise.

The dark current is due to thermally generated charges and strongly depends on temperature and acquisition time. Generally, it can be independently measured and subtracted from data. In our case, the camera parameters can be set to have a negligible contribution of dark current.

Spurious charges, also called clock-induced charges (CICs), are created during the fast clock variations required for shifting the photoelectrons to the readout register. CICs generate an electron count distribution $P_{\mathrm{sc}}(x|n)$ that has the same behavior as Eqs. (1) and (2) but with gain $g_{\mathrm{sc}}$ lower than $g$.

In a reasonable operating configuration, the probability to have more than one spurious charge is negligible. Therefore, the random probability distribution of the electron counts, in the absence of illumination (i.e., electrons generated by photon absorption), is

The definition of detection efficiency, for a camera in single-photon-counting regime, is $\eta (T)=P_{\text{true}}(T)/p_{\mathrm{ph}}$, where $P_{\text{true}}(T)$ is the probability that a pixel clicks ($x\ge T$) due to an incident photon, and $p_{\mathrm{ph}}$ is the probability to have an incident photon. When operating with a sufficiently low-light level, it is possible to assume that at most one photon is detected per pixel. Under this condition, the probability that a pixel clicks is

The function $P_1(x)$ represents the electron count distribution under the assumption that there is at most one photon per pixel, that there are no spurious counts, and take into account the contribution of the read noise

For what concerns our camera, we have estimated independently all the parameters involved in the models. Therefore, we are able to predict the behavior of detection efficiency and noise as a function of an arbitrary threshold $T$ and as a function of the analog detection efficiency ${\eta }_0$.

Figure 1(a) presents a histogram of the electron count distribution for a frame acquired with a closed shutter (no input light), showing excellent agreement between the experimental data and the theoretical prediction in Eq. (4). Two different behaviors are clearly visible: at low-count levels, the Gaussian contribution of read noise is dominant; at high-count levels, only the exponential contribution of spurious charges is observable. By fitting the theoretical distributions of read noise, on the first part of the histogram we estimate $\mu =507.9(0.3)\text{counts}/\text{pixel}$ and $\sigma =24.88(0.03)\text{counts}/\text{pixel}$. By fitting the distribution of spurious charge on the second part of the histogram, we estimate the parameters $p_{\mathrm{sc}}=0.0044(0.0006)$ and $g_{\mathrm{sc}}=141(2)\text{counts}/\text{pixel}$.

 

Fig. 1. Logarithmic scale histogram of the electron counts distribution for a frame acquired without incident light (a) and with incident light (b). Gaussian curve fit (red line). Exponential curve fit (orange line).

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Figure 1(b) shows the histogram of the electron count distribution for a frame acquired with input light. The light intensity was selected to avoid double events and to guarantee that spurious charges are negligible with respect to photoelectrons. By fitting the theoretical distributions ${\mathcal{P}}_1(x)$ on the part of the histogram in which the read noise is negligible, we estimate the mean multiplication gain of the camera $g=147(2)\text{counts}/\text{pixel}$.

All the uncertainties are evaluated on a set of 35 frames acquired under the same illumination conditions.

Our setup (Fig. 2) is composed of a diode laser, operating at $\lambda =406\mathrm{nm}$ in pulsed mode and synchronized with the exposure time of an EMCCD camera. The laser is coupled in a single-mode fiber, providing a well-Gaussian-shaped beam. The beam is then collimated, and a polarizing beam splitter (PBS) selects the vertical component of the polarization. The laser beam pumps a $5\mathrm{mm}\times 5\mathrm{mm}\times 15\mathrm{mm}$ BBO nonlinear crystal of Type II, where two correlated beams are produced through spontaneous parametric downconversion. The generated twin beams are sent to the EMCCD by two plane mirrors. A far-field lens with a focal length of $f=10\mathrm{cm}$ is located in a $f-f$ configuration between the output surface of the crystal and the detection plane. An interference filter at 800 nm with 40 nm bandwidth and a central transmissivity of 99% is put in front of the camera. The phase matching is set to maximizing the emission around the degenerate wavelength ($\lambda =812\mathrm{nm}$).

 

Fig. 2. Schematic representation of the experimental apparatus.

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Figure 3 shows the behavior of $\eta (T)$ derived by the model of EMCCD [Eq. (8)] and compares it with the experimental results, showing perfect agreement with the theoretical prediction. The analog quantum efficiency, derived using the same technique based on the twin beam in the analog regime [40,41], is ${\eta }_0=0.54(0.02)$. It is worth recalling that ${\eta }_0$ and $\eta$ represent the detection efficiency of the beam at 812 nm, including both the transmission efficiency of the optical detection system and the CCD camera quantum efficiency. It is important to note that the model and the measurement technique, based on twin beams, are not valid for low-threshold levels when the noise contribution becomes dominant. In our system, this happened for approximately $T<560$ counts. Figure 4 shows the level of noise in the photon counting regime and the corresponding theoretical level provided by Eq. (4).

 

Fig. 3. Quantum efficiency, as a function of the threshold $T$, estimated exploiting spatial correlations of quantum twin beams. The continuous line shows the theoretical prediction calculated by Eqs. (8) and (9) and by mean of the preliminary estimation of ${\eta }_0$.

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Fig. 4. Measured probability to have a click per pixel per frame due to a noise event as a function of the threshold (uncertainty bars are smaller than dots). The continuous line represents the prediction obtained by our model: Eqs. (4) and (5).

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The results shown in Fig. 3 represent the first absolute calibration of the quantum efficiency of an EMCCD operating as a threshold detector.

The comparison shows that the behavior of the detection efficiency of the EMCCD, in the on–off regime after threshold application, is completely predictable by a simple model that is a function of the measured value of the gain of the electro-multiplication register, of the efficiency measured in the analog regime, and of the chosen threshold $T$. Thus, we achieve a complete characterization and understanding of the behavior of a device acquiring more and more importance in the field of single-photon measurements. On the other hand, we establish a bridge between the light-intensity level typical of classical radiometric measurements, the analog regime, and quantum radiometry operating at the single-photon level [9,11].

The possibility to use the same absolute calibration technique, for the same device, both for the photon counting regime and the analog regime provides a radiometric link between low the illumination regime (few photons) to the mesoscopic and to the classical macroscopic ones. This represents an important step in the development of quantum radiometry, providing traceability of measurements at the few-photon level, the relevant-illumination level for most of the emerging quantum technologies.