3 The Fouks-Schubert model (FS model)

3.1 Origin of the model

The Si:Ga linear detector of the PHT-S channel of ISOPHOT has been extensively studied by Fouks and Schubert from experimental and theoretical points of view (Fouks 1992; Schubert et al. 1994; Fouks & Schubert 1995; Schubert et al. 1995; Schubert 1995). The LW ISOCAM detector is a Si:Ga array too. Its size is 32 $\times$ 32 pixels, 100 $\mu$m pitch and the thickness of 500 $\mu$m. PHT-S pixels size are 0.35 $\times$ 0.37 mm (for 64 pixels long) and the thickness is 2.0 mm. Wavelength ranges are 6.-12. $\mu$m for PHT-S and 4.-17. $\mu$m for LW ISOCAM.

In this paper, we will not describe the semi conductor physics equations but only indicate the more relevant references. Because of the intrinsic non linear nature of the equations and of the sensitivity to the boundary conditions, several steps were needed from the first linearized solutions (Suris & Fouks 1978; Suris & Fouks 1980) to the final non linear solutions covering a large range of detectors (Vinokurov & Fouks 1991; Fouks 1992). For the detectors of PHT-S and LW-ISOCAM, the most relevant paper to understand the physics is Fouks & Schubert (1995). It is based on the first order description of the physics inside the bulk of an extrinsic p-type photo-conductor. After several hypothesis about the dominant physical processes, a set of conditions and differential equations is obtained. Solving the set of differential equations assuming a constant electric field inside the bulk, a Riccati differential equation is found. Its solution which gives the temporal response is a non linear equation.

  
3.2 The equations

The first solution describes the response to a instantaneous step of flux at time t=0, from the constant level $J _{0} ^\infty$to the constant level $J _{1} ^\infty$:

$$J(t)= \beta J_{1} ^\infty + \frac{(1-\beta)J _{1} ^\infty J _... ...\infty +(J _{1} ^\infty -J _{0} ^\infty )~{\rm exp}(-t/\tau )}$$ (1)

$J _{1} ^\infty$ is the stabilized photo-current measured at time $t= +\infty$. It is also directly related to the observed flux, since a linear relationship is assumed between the flux and the photo-current after stabilization (Fouks & Schubert 1995). The parameter $\beta$ characterizes the instantaneous jump just after the flux change. The theory gives a simple relationship between the time constant $\tau$ and $J _{1} ^\infty$over several order of magnitude: $\tau = \lambda / J _{1} ^\infty$. Yet, the time constant is $\lambda$. This dependence has also been observed for the LW channel of ISOCAM (Sect. 3.4 and Abergel et al. 1999). Equation (1) does not allow us to compute J(t)if $J_{1} ^\infty= 0$, and for that case a first order development gives:

$$J(t)= \frac{(1-\beta)J _{0} ^\infty } {1 + \frac{t}{\lambda}J _{0} ^\infty }\cdot$$ (2)

We can see in Fig. 3 that the agreement of Eq. (2) with the data is perfect.

On data (ground based and in-flight data), starting from a level close to the dark level (lower than 1 ADU/G/s), an inflection point (see Sect. 2 and Fig. 2) is clearly visible for moderate upward steps (typically above 10 ADU/G/s and below 100 ADU/G/s). It is more difficult to observe it for strong upward steps since it is temporally very close to the jump. No inflection point is visible for small upward steps (ratio $J _{1} ^\infty/ J _{0} ^\infty$ between 1 and 2) and for downward steps. This inflection point is predicted and perfectly reproduced with Eq. (1). Its position $t^{{\rm ip}}$ can be easily calculated from Eq. (1). For $J _{0} ^\infty > 0$, we obtain:

$$t^{{\rm ip}}= \frac{\lambda}{J _{1} ^\infty} \times {\rm ln}... ...(\frac{J _{1} ^\infty -J _{0} ^\infty}{J _{0} ^\infty}\right).$$ (3)

Much information comes from this equation. An inflection point is not predicted for downward steps ( $(J _{1} ^\infty -J _{0} ^\infty) < 0$). The inflection point appears for upwards steps only when $J_{1}^\infty / J_{0}^\infty > 2$. For $J _{0} ^\infty = 0$, $t^{{\rm ip}}= + \infty$, so in that case the response is constant and, $\forall t>0, J(t)= \beta J_{1} ^\infty$. However the case $J _{0} ^\infty$ strictly equal to 0 (and obviously $J _{0} ^\infty < 0$) is not physical, since the detector always receives infrared photons due to thermal emissions of the different parts of the camera (even when it is switched off). For $J _{0} ^\infty > 0$, Fig. 4 illustrates the variations of the position of the inflection point with the value of the initial level J₀.

Figure 4: FS model response to upward and downward steps of flux, from Eq. (1) (see also Fig. 8 in Schubert & Fouks 1995). Upward steps of flux from a constant level J0to a constant level J1 occur at time t=100 s and downward steps from J1 to J0 at t=900 s. We have taken: J0 = 0.01, 0.1, 1.0, 2.0 and 5.0 ADU/G/s, and J1= 10 ADU/G/s. For all these simulations, the values of the parameters $\beta$ and $\lambda$ are constant. We see that Eq. (1) is very sensitive to the initial level (J0) for the upward steps: curves from 0.01, 0.1 and 1.0 ADU/G/s are very different. When the dark level is poorly estimated, such non linear effect can allow us to correct the value of J0

  
3.3 From two non stabilized levels

Equation (1) can only be used for transitions between stabilized levels. The following enhancement (Fouks & Schubert 1995) allows us to describe a not stabilized detector output J _n(t) from a recursive point of view (see Fig. 5 for notations). An observation is made of N integrations starting at time t_n. We assume that the input flux $J _{n} ^\infty$ is constant during each successive integration. During the integration number n, we have:

$$J _n(t)= \beta J_n ^\infty + \frac{(1-\beta)(J _{n} ^{{\rm in... ...ty -J _{n} ^{{\rm ini}} ) ~{\rm e}^{(-(t-t_{n})/\tau _{n} )}}.$$ (4)

Figure 5: Response to two successive upward steps of flux (at time tn-1 = 100 s and tn = 200 s respectively), corresponding to three blocks of constant input fluxes (number n-2, n-1 and n respectively). We have taken $\beta$ = 0.55, $\lambda$ = 600. ADU/G, $J _{n-2} ^{{\rm end}} = J _{n-2} ^\infty$ = 1. ADU/G/s, $J _{n-1} ^\infty$ = 10. ADU/G/s and $J _{n} ^\infty$ = 30. ADU/G/s. This figure is adapted from Fouks & Schubert (1995)

Basically we use the same notations as in Eq. (1). $J _{n} ^{{\rm ini}}$ is the response measured just after the jump at time t_n. J _n(t) is the response at any time t between t_n and t_n+1. In this equation, the state at the end of the previous integration n-1 is contained in $J _{n} ^{{\rm ini}}$, which can be related to the previous flux and the current stabilized flux values ( $J _{n-1} ^\infty$ and $J _{n} ^\infty$) and the final previous flux just before the jump $J _{n-1} ^{{\rm end}}$by:

$$J _{n} ^{{\rm ini}}=J _{n-1} ^{{\rm end}} +\beta (J _{n} ^\infty - J _{n-1} ^\infty ).$$ (5)

Finally, we obtain from Eq. (4) and Eq. (5):

 

$${J _n(t) = \beta J_n ^\infty + ...}$$

$$\frac{(1-\beta)(J _{n-1} ^{{\rm end}}-\beta J _{n-1} ^\infty ) J _{n... ...end}} -\beta J _{n-1} ^\infty)) {\rm e} ^{-(t-t_{n})J _{n} ^\infty /\lambda }}.$$ (6)

J _n(t) depends on $J _{n} ^\infty$, $J _{n-1} ^\infty$ and $J _{n-1} ^{{\rm end}}$ (Fig. 5).

If we take: $J _{n-1} ^\infty= J _{n-1} ^{{\rm end}}= 0$, we have $\forall t>0, J _n(t)=\beta J_{n} ^\infty$, and $\forall n' > n, J _{n'}(t)=\beta J_{n'} ^\infty$. Negative values for $J _{n-1} ^\infty$ are not physical and would give, after the instantaneous jump, a monotonous decrease with a vertical asymptote located after the jump (its position can be calculated from Eq. (1)).

  
3.4 Adjustment of the model

Figure 6: Block by block correction. These data are taken during the Performance Verification phase (Revolution number 16). The zodiacal background was continuously observed, and the input sky brightness modulated between two values by moving the filter wheel. Upper panel: Mean value of the 11 $\times$ 11 central square as a function of the time. The integration time per frame was 2.1 s. The unknown flux $J _{n} ^\infty$ is fitted for each block with constant ( $\beta ,\lambda$) parameters. The adjusted input fluxes are over plotted (dashed line), together with the response computed with the FS model (solid line). Lower panel: residue. Evidence of a not perfect fitting can be seen at the beginning and at the end of block 5

We have adjusted the model of the response on experimental data, both on the averaged response within the 11$\times$11 central square and for individual pixels. We have used data with limited flux steps, since the response for steps greater than $\sim$150 ADU/G/s is too fast to give accurate values (typically a few integrations).

We illustrate the results with data already presented in Sect. 2. The signal was stabilized before the flux changes. The values $J _{0} ^\infty$ and $J _{1} ^\infty$ are known. We have three free parameters to fit Eq. (1) with the data, the two parameters $\beta$ and $\lambda$, and an offset level. To add an offset is mandatory for some observations (especially for the ground base data) since the absolute dark level was poorly estimated while the response is very sensitive to the initial level before the flux step, therefore to the absolute value of the dark level (see Sect. 3.2 and Fig. 4). The three parameters are adjusted by least square fitting ($\chi ^2$ criterion with constant error). We see on the examples we present (Figs. 2 and  3) that the model remarkably reproduces both upward and downward steps. Other steps have been adjusted, with higher and lower amplitudes, with a similar accuracy. Especially, the position of the inflection point of upward steps is accurately reproduced.

We have also made adjustments when the signal was not stabilized before the changes of flux using Eq. (6) to estimate the value of $J^\infty$ for each block (one block corresponding to a constant configuration of the whole system, see Sect. 4). We assume that $\beta$ and $\lambda$ are constant over the whole observation. We see in Fig. 6 that the precision is $\sim 1\%$. We have the same kind of results for different datasets taken during the whole life of the satellite. We have also checked that the theoretical relationship $\tau=\lambda /J^\infty$predicted by Suris & Fouks (1980) is verified by the data, for uniform illumination of the detector and for upward and downward steps of flux in the range of a few ADU/G/s to $\sim$ 500 ADU/G/s.

  
3.5 Pixel to pixel variations of the parameters ( $\beta ,\lambda$)

The fitting method described in the previous section has been applied for the 32 $\times$ 32 pixels of the detector in order to derive for each pixel the values of $\beta$ and $\lambda$ for uniform illumination. Practically, we have used long-term observations of the zodiacal background taken successively with two filters. First the data have been dark corrected, then glitches have been removed using a temporal median filtering which detect all pixels deviating from the running median value by more than a threshold value. In order to remove glitches still contained in the data (especially those with significant memory effects, Désert 1999), two pass fitting is used. For the second pass, pixels deviating from the running averaged value of more than 3$\sigma$ are discarded. The correction of glitches with negative tail is much more difficult and manual removing is necessary for a limited number of case. In any case, this two pass fitting does not discard all glitches with memory effects.

Figure 7: Final ( $\beta ,\lambda$) maps fitted on upward steps from six independent data sets. Values with poor $\chi ^2$are discarded before computing the mean value. One column was disconnected during the whole mission. The lower map presents the spatial variations of the response ( flat field) to a uniform illumination

The FS model is not symmetrical. The adjustment of $\beta$ and $\lambda$is more accurate for upward steps than downward steps, because the $\chi ^2$ values present in the ( $\beta ,\lambda$) plane a steeper minimal value for upward steps than downward steps. On simulated noisy data, we have obtained a precision of $\Delta \beta/ \beta \sim 0.02$ and $\Delta \lambda / \lambda \sim 0.03$for upward steps, and only $\Delta \beta/ \beta \sim 0.2$and $\Delta \lambda / \lambda \sim 0.1$ for downward steps.

We have used upward steps of flux on quasi uniform illumination (essentially zodiacal emission) with $J _{0} ^\infty$ from 5 to 25 ADU/G/s and a ratio $J _{1} ^\infty/ J _{0} ^\infty$ in the range 2-5. This range is unfortunately limited for in-flight data by the zodiacal emission characteristics. Values of $J _{0} ^\infty$ lower than 10 ADU/G/s give poor adjustments, likely due to uncertainties in the dark level.

Actually, six independent data sets (from scientific observations of diffuse clouds taken during revolutions number 129, 321, 590, 658, 684 and 772) have been extensively analyzed. Because of (1) glitches with memory effects, (2) oscillations in the response curve and (3) in some cases low signals at the edge of the detector due to the optical vignetting, the values of $\beta$ and $\lambda$ cannot be accurately estimated for typically $\sim 15\%$of the pixels in an observation. About 5 to 10$\%$of the pixels in each data set present oscillations during the stabilization after the upward steps. Most of these pixels are close to the edges or the corners of the detector (Abergel et al. 1999). The FS model cannot describe such behavior, and only $\beta$ can be accurately estimated. These oscillations will be discussed in Sect. 5.6.

We have computed six independent maps of $\beta$ and $\lambda$from the six data sets we use. On these maps, we have pixel to pixel variations which are significant. They are reproducible on the six independent data sets. The final maps of $\beta$ and $\lambda$ are obtained by averaging the six maps taking into account for each pixel the $\chi ^2$ value for each data set and discarding values by $\sigma$ clipping. The final values of $\beta$ have an absolute accuracy better than $\pm$0.02 for 95$\%$ of the pixels. For the final values of $\lambda$, the accuracy is better than $\pm$20 ADU/G for $\sim$50$\%$ of the pixels (the lower left part of the map presented in Fig. 7), while for $\sim$10$\%$ of the pixels the accuracy is above 50 ADU/G. The final maps are presented in Fig. 7, compared to the map of the spatial variation of the response for uniform illumination (the flat field). Histograms are plotted in Fig. 8.

Figure 8: Histogram of $\beta$ (upper panel) and $\lambda$ (lower panel) for the 32 $\times$ 32 pixels and from data of Fig. 7

The accuracy on the final maps of $\beta$ and $\lambda$ is limited because (1) only six data sets have been used with a limited range of the $J _{1} ^\infty/ J _{0} ^\infty$ ratio, (2) at low $J _{1} ^\infty/ J _{0} ^\infty$ ratio the FS model is not very sensitive to $\lambda$, (3) several pixels are affected by glitches, (4) for some pixels only one or two independent values have been used and (5) uncertainties in the dark level.

The mean values of $\beta$ and $\lambda$ are equal to $\overline{\beta}$ = 0.51 and $\overline{\lambda}$ = 560 ADU/G. The data taken during the ground-based tests have also been extensively used to explore the transient behavior over a wide dynamic range. Different values ( $\overline{\beta} \sim$0.6 and $\overline{\lambda} \sim$400 ADU/G) are obtained due to a different setup of the detector.

3.6 Consequences for one of the model hypotheses

One important assumption of the FS model is (Vinokurov & Fouks 1991):

$${\rm ln}\left(\frac{J_1 ^{\infty}}{J_0 ^{\infty}}\right) << \frac{E_0}{E_j},$$ (7)

where E₀ is the electrical field between the contacts and E_j is a parameter which characterizes the quality of the contacts. We have no information about E_j but we can derive it from the parameters $\beta$ and $\lambda$, since:

$$E_j = \frac{\beta \lambda}{2 \epsilon \epsilon_0}.$$ (8)

We have found $E_j \sim 5.~10^3$ V/m and $E_0 \sim 4.~10^4$ V/m then the condition 7 is satisfied. For PHOT-S, Schubert (1995) indicates $E_j \sim 2.~10^3$ V/m and $E_0 \sim 1.~10^5$ V/m.