# Interpixel Capacitance

The nodal capacitance C0 derived by the shot noise method appears to be too large. Since Eq. (1) shows that C0 is inversely proportional to the variance, the measured variance of the detector signal (v should be larger to obtain a smaller but more plausible nodal capacitance. The discrepancy cannot be explained by excess noise of the data acquisition chain as more noise would only make the variance larger not smaller. On the contrary, a mechanism has to be introduced which does not store signal charge but reduces the photon shot noise of a single pixel. The larger capacitance seen by the shot noise may be explained by coupling capacitance between neighboring pixels. As a consequence of interpixel capacitance the signal of a pixel is spread by capacitive coupling to adjacent pixels, which reduces the apparent photon shot noise. The photon shot noise method does not yield the pixel capacitance C0, but the provides a measure of the sum of C0 and all the coupling capacitors in series with the nodal capacitors of the neighboring pixels as shown in Fig. 7.

Figure 8. A surface plot of numerical simulation illustrating the effect of interpixel capacitance. In this example, the interpixel capacitance is made relatively large, x = 0.3, to dramatically demonstrate the effect. Upper half: short time exposure showing signal of individual, Poisson distributed photons. Lower half: long time integration with reduced intensity scale. Right half: no interpixel capacitance. Left half: interpixel capacitance, signal of single photon is spread to closest neighbors. Long exposures show a smoother surface.

Using the coupling capacitance Cc and the ratio of coupling capacitance and nodal capacitance x = Cc/C0, a simple model of the apparent capacitance seen by the shot noise is C = C0 (5x +1) /(x +1) . For simplicity, only coupling to the 4 closest neighbors is considered here. By applying Kirchhoffs law it can also be shown that the total signal with coupling V0+4Vi is equal to the total signal V without coupling, i.e., V0+4Vi =V with V being the signal for Cc=0. This implies that the interpixel capacitive coupling conserves photometry. For uniform illumination of the array no signal charge is stored on the coupling capacitors. The coupling capacitors reduce the photon shot noise because the voltage response of a photongenerated electron is not confined to a single pixel, but spread over all neighboring pixels. This nominal reduction of noise is accompanied by reduced image contrast and degraded detector MTF.

The effect of interpixel capacitance has been simulated numerically by generating random Poisson distributed short time exposures as shown in the upper halves of Fig. 8 as surface plots, and in the upper halves of Fig. 9 as images. The right sides of Figures 8 and 9 represent a detector without interpixel coupling whereas the left sides introduce an interpixel capacitive coupling of x = 0.3. The crosstalk can be clearly seen in the upper left quarter of both the surface plot of Fig. 8 and the image of Fig. 9. The lower half of Fig. 8 and Fig. 9 show the images with and without interpixel coupling for a much longer integration time with a larger intensity scale.

Figure 9. Image of numerical simulation illustrating the effect of interpixel capacitance.

Noise reduction versus coupling capacity

Figure 9. Image of numerical simulation illustrating the effect of interpixel capacitance.

Noise reduction versus coupling capacity

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Figure 10. Ratio of standard deviation with interpixel capacitive coupling Gc and without interpixel capacitive coupling c0 as function of coupling Cc/C0. Diamonds: simulation of random Poisson distributed short time exposures. Solid line: ratio of total capacitance C and nodal capacitance C0 taking into account the four closest neighbors C/C0=(5x+1)/(x+1).

The pixel to pixel variance without coupling on the right side is equal to the average number of integrated photons in agreement with Poisson statistics. The variance with capacitive interpixel coupling is smaller as shown by the smoother surface and image in the lower left quadrants of Figures 8 and 9. The ratio of the standard deviation with capacitive interpixel coupling cc and the standard deviation without coupling c0 as derived from the numerical simulation using Poison statistics is shown by diamonds in Fig. 9. For small coupling ratios Cc/C0 , the ratio cc/c0 follows the ratio of the total capacitance C and the nodal capacitance C0 which is represented by the solid line in Fig. 9. This ratio is given by the expression C/C0=(5x+1)/(x+1) only taking into account the coupling to the four closest neighbors.