1. Introduction

In the tracking system for the spot sources, to sense the tilt angle of an incident wave-front dynamically, the quadrant detector is usually placed at the focal plane of the imaging lens to detect the centroid movements of the focal spot [1]. Thus the imaging lens and associated quadrant detector constitute a quadrant tracking sensor. When such a quadrant tracking sensor is applied in astronomical observation, or earth-satellite links, only a small fraction of the light energy received by the aperture of telescope can be used for tilt tracking. Most of the other incident energy is used for wave-front correction and imaging. Therefore, the tracking sensor is required to work in photon counting mode [2–4]. When a multianode PMT (photo-multiplier tube) is used as a quadrant detector, because of the crosstalk between anodes of the multianode, the photon counting accuracy is limited, and thus the spot centroid detection accuracy of the quadrant detector is influenced. As the de-crosstalk algorithm can be used to reduce the crosstalk, it is necessary to analyze if the de-crosstalk algorithm can effectively ameliorate the influences of crosstalk on tracking accuracy.

2. Principles of the quadrant tracking sensor

As shown in Fig. 1 , the principles of the quadrant tracking sensor are as follows [5]. The photosensitive surface of the quadrant detector is placed at the focal plane of the imaging lens. The incident wave-front forms a focal spot therein after collation through the imaging lens. The centroid of the focal plane spot would change with the tilt angle of the incident wave-front changes. This remodels the light energy distribution in each of the four quadrants (see Fig. 2 ). According to the photoelectron number output by each of the four quadrants, the offset direction of the spot centroid can be calculated, thus the tilt direction of the incident wave-front is found.

 

Fig. 1 Schematic of a quadrant tracking sensor (1): imaging lens, (2): quadrant detector, (3): focal plane spot

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Fig. 2 Detection of spot centroid

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Generally, the spot intensity variations on the photosensitive surface of the quadrant detector follow a Gaussian distribution. When the spot centroid is given in Cartesian coordinates as$(x_0,y_0)$, the single quadrant width of the quadrant detector is$\omega$, and the total photon number is$P_s$, the photoelectron number output by each quadrant should be [6]

Where, E is the quantum efficiency of the quadrant detector; $erf(\epsilon )$is the cumulative error function: $erf(\epsilon )=\frac{2}{\sqrt{\pi }}{\displaystyle {\int }_0^{\epsilon }{e}^{-t^2}}dt$; $G_s$is the Gaussian width of the spot.

The formula of the spot centroid $(x_c,y_c)$on the photosensitive surface of the quadrant detector is

Where, $N_i$ is the photoelectron number output by the i^th quadrant of the quadrant detector.

From Eq. (2), the spot centroid ($x_c$,$y_c$) is dimensionless, so a displacement coefficient $k_d$ should be defined. When $x_c$and $y_c$are in the linear region of the quadrant sensor, $k_d$is used to calculate the true spot centroid$\left(x_0,y_0\right)$ from $\left(x_c,y_c\right)$. In this situation, the calculation formulas of the tilt angles ${\alpha }_x$and${\alpha }_y$ of the incident wave-front in the x-direction and y-direction (both $x_0$ and $y_0$ are small) should be

Where, f is the effective focal length of the tracking sensor.

3. Crosstalk of the multianode PMT

The photoelectrons emitted from the photocathode of PMT can be multiplied rapidly by the multi-grade dynodes of PMT. Finally the number of electrons collected by the anode of PMT can be increased by between 10⁴ and 10⁸ times, which can permit the PMT to achieve the single-photon detection sensitivity. As a feature of its spatial design, several PMTs can be integrated into one package which is called the multianode PMT. The output signals from different pins correspond to the spatial position of the photocathode, thus the 2 × 2 multianode PMT can be regarded as a quadrant detector.

However, due to fabrication process limitations, there is no shielding at each of the anodes. When a multianode PMT is operating in photon counting mode, due to the coupling of inter-electrode capacitance, if one quadrant produces a photoelectron pulse, the rest quadrants will produce some smaller coupling pulses. These coupling pulses could be defined as photoelectron pulses by the photo-counting circuit, and then some pseudo photon counts are registered.

In terms of the principles of crosstalk production, when the i^th quadrant produces P_i true photoelectron numbers, because of the crosstalk, the j^th quadrant will produce P_j pseudo photon counts. The crosstalk ratio between the i^th and j^th quadrants can be defined as

To simplify the analysis, the crosstalk ratio of adjacent quadrants can be seen as $k_c$ in terms of the symmetry of the quadrants, while the crosstalk ratio of diagonally opposite quadrants is k_s, as shown in Fig. 3 .

 

Fig. 3 Crosstalk distribution diagram

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Therefore, when the spot moves, the relationship between the photoelectron vector $N_C=\left[N_1,N_2,N_3,N_4\right]\text{'}$ which is obtained by recording and the true photoelectron number $n_s=\left[n_1,n_2,n_3,n_4\right]\text{'}$ which is produced by the spot is

Where, C is the crosstalk matrix, and represents the bridge between the arriving photoelectron signals and those detected. So when there is crosstalk, the spot centroid obtained by calculation and the arriving photoelectron number is

Under different crosstalk conditions, the spot centroid detection curves of quadrant detector are as shown in Fig. 4 , where it shows that for larger crosstalk levels, the detection curve gradient becomes shallower.

 

Fig. 4 The influence of crosstalk on the calculated value of the spot centroid

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Since the optical paths of the tracking detector in both x-direction and y-direction are symmetrical, all the following discussion focuses on the x-direction, y-direction is similar.

4. The tracking error of the multianode-PMT-based quadrant tracking sensor

4.1 Spot displacement sensitivity

The spot displacement sensitivity ${\rho }_x$ is defined [7] as the partial derivative of the spot centroid x_c as obtained by calculation from the true centroid position x₀ of the spot. It denotes the changing trend of the calculated spot centroid when the true spot centroid changes elsewhere within a slightly different locus (see Eq. (7)).

From Eq. (1), Eq. (2), Eq. (6) and Eq. (7), Eq. (8) is derived thus

Where, $D_a=erf\left(\frac{\omega -x_0}{\sqrt{2}{G}_s}\right)+erf\left(\frac{x_0}{\sqrt{2}{G}_s}\right),D_{\text{b}}=erf\left(\frac{\omega +x_0}{\sqrt{2}{G}_s}\right)-erf\left(\frac{x_0}{\sqrt{2}{G}_s}\right)$, $D_a^{\text{'}}=\frac{\sqrt{2}}{\sqrt{\pi }{G}_s}\cdot \exp \left[-{\left(\frac{\omega -x_0}{\sqrt{2}{G}_s}\right)}^2\right]-\exp \left[-{\left(\frac{x_0}{\sqrt{2}{G}_s}\right)}^2\right],D_b^{\text{'}}=\frac{\sqrt{2}}{\sqrt{\pi }{G}_s}\cdot \exp \left[-{\left(\frac{x_0}{\sqrt{2}{G}_s}\right)}^2\right]-\exp \left[-{\left(\frac{\omega +x_0}{\sqrt{2}{G}_s}\right)}^2\right].$

In Eq. (7): $\frac{\left(D_a-D_b\right)\cdot \left(D_a^{\text{'}}+D_b^{\text{'}}\right)-\left(D_a+D_b\right)\cdot \left(D_a^{\text{'}}-D_b^{\text{'}}\right)}{{\left(D_a+D_b\right)}^2}$ is the displacement sensitivity of the spot with the Gaussian distribution, $\frac{1-k_s}{1+2k_c+k_s}$ is the influence of the crosstalk on spot displacement sensitivity. Obviously, the larger the crosstalk is, the lower the spot displacement sensitivity.

4.2 Spot centroid detecting error

According to the error theory [8], the variance formula of the centroid detecting error is given by Eq. (9).

Where, $U_N={\displaystyle \sum _{i=1}^4\left(e_i\cdot N_i\right)},V_N={\displaystyle \sum _{i=1}^4{N}_i},{\sigma }_{U_N}^2={\displaystyle \sum _{i=1}^4{\sigma }_i^2}+{\displaystyle \sum _{i=1,j=1,i\ne j}^4\left(e_i\cdot e_j\cdot {\sigma }_{ij}\right)},$ ${\sigma }_{V_N}^2={\displaystyle \sum _{i=1}^4{\sigma }_i^2+{\displaystyle \sum _{i=1,j=1,i\ne j}^4{\sigma }_{ij}}},{\sigma }_{U_N{V}_{{}_N}}={\displaystyle \sum _{i=1}^4\left(e_i\cdot {\sigma }_i^2\right)}+{\displaystyle \sum _{i=1,j=1,i\ne j}^4\left(e_i\cdot {\sigma }_{ij}\right)},$ $e_1=1,e_2=-1,e_3=-1,e_4=1,$${\sigma }_i^2$ is the fluctuation variance of the output signal in the i^th quadrant. ${\sigma }_{ij}$is the covariance of output signals between the i^th and j^th quadrants.

The photons, caused by random launch, follow a Poisson distribution. As the amplitude of each coupling pulse is random, the probability of pseudo photon counts produced by coupling pulse can be considered to follow a Poisson distribution too. Consequently, the variance of the output signal in the i^th quadrant is

The covariance of the output signal in the i^th and j^th quadrants is caused by crosstalk between them, so

Substituting Eq. (10) and Eq. (11) into Eq. (9), then the variance of centroid detection error for quadrant detector is given by Eq. (12).

Where, $V_N=\left(1+2k_c+k_s\right)\cdot E\cdot P_s,v_1=\frac{k_s^2-2k_s}{1+2k_c+k_s},v_2=\frac{4k_c^2+8k_c{k}_s+2k_s+3k_s^2}{1+2k_c+k_s}.$

4.3 Tracking error

It is known that the tracking error of a quadrant tracking sensor is

When the spot is at the centre of the quadrant detector, the displacement sensitivity and the variances of centroid detecting error are given by Eq. (14) and Eq. (15) respectively.

The variance of the tracking error of quadrant tracking sensor is

When seen from the perspective of Eq. (16) and Fig. 5 , the higher the crosstalk of quadrant detector is, the larger the tracking error of quadrant tracking sensor.

 

Fig. 5 Influence of crosstalk on the sensor’s tracking error

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5. Tracking error analysis of the de-crosstalk algorithm

When seen from the perspective of Eq. (5), in the presence of crosstalk, the recorded photoelectron vector $N_c$ and the true photoelectron number vector $n_s$ are linked by crosstalk matrix C. Theoretically, the reconstructed photoelectron number vector $n_{Rs}$ can be calculated by the recorded photoelectron vector $N_c$ and the matrix R which is the inverse matrix of crosstalk matrix C by

Where, $n_{Rs}=[n_{R_1},n_{R_2},n_{R_3},n_{R_4}]\text{'}$, because of $〈n_{R_i}〉=n_i$ ($〈\mathrm{...}〉$ means the average value of ...), it seems the crosstalk is decreased to 0, thus this algorithm is called as the de-crosstalk algorithm.

5.1 Spot displacement sensitivity of the de-crosstalk algorithm

After applying the de-crosstalk algorithm, the mean photoelectron count output by each of quadrants is equal to the mean $n_i$ of arriving photon numbers from each quadrant. So the spot centroid becomes

Therefore the spot displacement sensitivity ${\rho }_{Rx}$ of de-crosstalk algorithm is

5.2 Centroid detecting error of the de-crosstalk algorithm

In terms of the error theory, after applying the de-crosstalk algorithm, the centroid detecting error of the quadrant detector is

Where, $U_R={\displaystyle \sum _{i=1}^4\left(e_i\cdot n_{Ri}\right)},V_R={\displaystyle \sum _{i=1}^4{n}_{Ri}},{\sigma }_{U_R}^2={\displaystyle \sum _{i=1}^4{\sigma }_{R_i}^2}+{\displaystyle \sum _{i=1,j=1,i\ne j}^4\left(e_i\cdot e_j\cdot {\sigma }_{R_i{R}_j}\right)},{\sigma }_{V_R}^2={\displaystyle \sum _{i=1}^4{\sigma }_{R_i}^2+{\displaystyle \sum _{i=1,j=1,i\ne j}^4{\sigma }_{R_i{R}_j}}},$ ${\sigma }_{U_R{V}_{{}_R}}={\displaystyle \sum _{i=1}^4\left(e_i\cdot {\sigma }_{R_i}^2\right)}+{\displaystyle \sum _{i=1,j=1,i\ne j}^4\left(e_i\cdot {\sigma }_{R_i{R}_j}\right)}.(e_1=1,e_2=-1,e_3=-1,e_4=1).$

${\sigma }_{R_i}^2$ is the fluctuation variance of the output signal from the i^th quadrant after using the de-crosstalk algorithm. ${\sigma }_{R_i{R}_j}$ is the covariance between photoelectron numbers $n_{Ri}$ and $n_{Rj}$ as output by the i^th and j^th quadrants.

After using the de-crosstalk algorithm, the output photoelectron number of quadrant detector is

So,

Where, ${\sigma }_{N_l}^2$ is the fluctuation variance of $N_l$ which is determined by Eq. (10). ${\sigma }_{ij}$is the covariance between $N_i$ and $N_j$ which is determined by Eq. (11).

Statistical analysis according to covariance theory yields the following

In terms of Eqs. (22) to (23), after using the de-crosstalk algorithm, the centroid error variance formula of the de-crosstalk algorithm is

Where, $V_R=E\cdot P_s,v_{R_1}=\frac{k_s^2-k_s+2k_c+1}{{(1-k_s)}^2},v_{R_2}=\frac{3k_s^2+8k_c{k}_s+k_s-2k_c+4k_c^2-1}{{(1-k_s)}^2}.$

5.3 Tracking error of the de-crosstalk algorithm

For the case that the spot is located at the centre of the quadrant sensor, the displacement sensitivity of the de-crosstalk algorithm and the variances of centroid detecting error of the de-crosstalk algorithm respectively are shown as Eq. (25) and Eq. (26).

The tracking error of the de-crosstalk algorithm is

Comparing Eqs. (25)–(27) with Eqs. (14)–(16), after using the de-crosstalk algorithm, the spot displacement sensitivity improves by a multiple of $\frac{1+2k_c+k_s}{1-k_s}$ times, but the centroid detection error increases by a multiple of $\frac{1+2k_c+k_s}{1-k_s}$ times too. In summary, the de-crosstalk algorithm cannot improve the tracking accuracy of quadrant tracking sensor.

6. Experimental results

The quadrant detector used in the experiments was a 2 × 2 multianode PMT. The single quadrant width of the multianode PMT was 9 mm with a dead zoon width of 0.2 mm.

As shown in Fig. 6 , the photoelectron signal output by the quadrant detector was carried over a transmission line to a receiving computer. The intensity of the incident wave-front was adjusted by the use of attenuation plane to change the photon numbers arriving at the quadrant detector; the tilt angle of the incident wave-front was adjusted by the tilt mirror.

 

Fig. 6 Experimental block diagram (1). laser (2). attenuation plane (3). Pinhole (4). parallel light collimator (5). tilt mirror (6). reflect mirror (7). imaging lens (8). matched lens (9). pyramid (10). quadrant detector (11). computer (12). high-voltage amplifier (13). incident wave-front

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6.1 Measurement of crosstalk coefficient

The inter-quadrant crosstalk coefficient can be changed by changing the threshold level of the photon counting circuit at the multianode PMT in actual applications. The higher the threshold is, the smaller the inter-quadrant crosstalk coefficient. However, when the threshold was increased, the photon sensitivity of the quadrant detector was decreased. On the basis of crosstalk coefficient theory, when the spot was situated within one single quadrant, the photon number output by the other quadrants is produced solely by crosstalk. Crosstalk coefficient can be calculated by

Where, $N_{Ai}$ is the mean of the photon number output by the quadrant where the spot was located. $N_{Aj}$ is the mean of the pseudo photon number output by the quadrants where there was no spot present. Figure 7 shows the photon numbers output by the quadrant detector for different spot locations.

 

Fig. 7 Output photon numbers by quadrant for different spot locations

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According to Fig. 7, the crosstalk coefficients of the multianode PMT at the adopted threshold level were: $k_c=\text{0}\text{.4858}$ and$k_s=\text{0}\text{.3487}$. Obviously, the crosstalk was caused by inter-electrode capacitance in the multianode PMT, but the capacitance and distance were inversely proportional. So the coefficient $k_c\approx \sqrt{2}{k}_s$ can be found thus.

6.2 The calculated value of spot centroid

The aforementioned computer was used to correct static aberration in the system at a rate of 5 frames per second (fps), thereby making the spot move on a locus bounded by a rectangular-based pyramid. The influence curves of crosstalk on spot centroid were then evaluated. The photoelectron number output by the quadrant detector with different spot locations is measured and processed with the de-crosstalk algorithm. Thus the curves indicating the calculated value of spot centroid and the true spot centroid were shown in Fig. 8 .

 

Fig. 8 Spot centroid loci and true positions

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As seen in Fig. 8, the experimental data generally coincided with the theoretical values. The larger the crosstalk, the shallower the gradient of spot centroid curve, and the lower the spot displacement sensitivity.

6.3 Tracking error

Once more, the system’s static aberration was corrected by computer at a rate of 5 fps, thus keeping the spot in the centre of the pyramid, the incident intensity on the quadrant detector was adjusted by the attenuation plane, and then the influence of crosstalk on the vibration of the spot centroid was examined under different incident photon number conditions.

In the experimental system, the Gaussian width of the spot was 52 μm, and the effective focal length of imaging lens was 5.6 m. At this time, the jitter error applicable to the spot centroid when spot was located at the origin was converted into a tracking error, and its values shown in Fig. 9 .

 

Fig. 9 Tracking error

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Figure 9 demonstrated that the de-crosstalk algorithm could not improve the tracking accuracy effectively (partial offset was caused by incorrect measurement of each entry in the crosstalk matrix C). Acceptable correlation between experimental and theoretical values may also be noted.

7. Conclusion

The tracking error of the multianode-PMT-based quadrant tracking sensor was influenced by the error inherent in the detection of the spot centroid and the displacement sensitivity. The pseudo photon counts caused by inter-quadrant crosstalk were produced randomly, and they were found to follow a Poisson distribution. The analysis showed that the larger the crosstalk between quadrants, the larger the tracking error of the quadrant tracking sensor. The de-crosstalk algorithm can be used to reduce the crosstalk. Theory and experiment results showed that the de-crosstalk algorithm can improve the spot displacement sensitivity to the level associated with zero crosstalk, but at the expense of a negative impact on spot centroid detection accuracy. As a result of this, the de-crosstalk algorithm is incapable of any improvement to the tracking accuracy of the multianode-PMT-based quadrant tracking sensor.