Pixel non-linearity

This tool produces the pixel non-linearity coefficients that ExoSim needs as input, starting either from physical assumptions or from the non-linearity correction that is measured in the lab.

The detector non-linearity is usually written as a polynomial,

\[Q_{det} = Q \cdot (1 + \sum_i a_i \cdot Q^i)\]

where \(Q_{det}\) is the charge read by the detector and \(Q\) is the ideal count, \(Q = \phi t\), with \(\phi\) the number of electrons generated per unit time and \(t\) the elapsed time. The tool retrieves the \(a_i\) coefficients.

From physical assumptions

The PixelsNonLinearity tool derives the \(a_i\) coefficients from a simple physical model of the pixel.

Treating the pixel as a capacitor, the collected charge is

\[Q_{det} = \phi \tau \cdot \left(1 - e^{-Q/\phi \tau}\right)\]

where \(\tau\) is the capacitor time constant, so the product \(\phi \tau\) is constant, and \(Q = \phi t\) is the response of an ideal linear detector.

The pixel is taken to be saturated when the charge at the well depth, \(Q_{det, \, wd}\), falls 5% short of the ideal well depth \(Q_{wd}\):

\[Q_{det} = (1-5\%)Q_{wd}\]

so that

\[\phi \tau \cdot \left(1 - e^{-Q_{wd}/\phi \tau}\right) = (1-5\%)Q_{wd}\]

Solving this numerically gives

\[\frac{Q_{wd}}{\phi \tau} \sim 0.103479\]

and therefore

\[Q_{det} = \frac{Q_{wd}}{0.103479} \cdot \left(1 - e^\frac{- 0.103479 \, Q}{Q_{wd}}\right)\]

which a 4th-order Taylor expansion approximates as

\[ \begin{align}\begin{aligned}Q_{det} = Q\left[ 1- \frac{1}{2!}\frac{0.103479}{Q_{wd}} Q\\+ \frac{1}{3!}\left(\frac{0.103479}{Q_{wd}}\right)^2 Q^2\\- \frac{1}{4!}\left(\frac{0.103479}{Q_{wd}}\right)^3 Q^3\\+ \frac{1}{5!}\left(\frac{0.103479}{Q_{wd}}\right)^4 Q^4 \right]\end{aligned}\end{align} \]

The result is the set of coefficients for a 4th-order polynomial:

\[Q_{det} = Q \cdot (a_1 + a_2 \cdot Q + a_3 \cdot Q^2 + a_4 \cdot Q^3 + a_5 \cdot Q^4)\]

The only input needed is the saturation level, well_depth:

<channel> channel_name
    <detector>
        <well_depth> 25000 </well_depth>
    </detector>
</channel>

Then run the tool:

import exosim.tools as tools

tools.PixelsNonLinearity(options_file='tools_input_example.xml',
                         output='pnl_map.h5')

With this example, the expected non-linearity shape is:

No two pixels are identical, so the tool also produces a map with a set of coefficients per pixel. Each coefficient is drawn from a normal distribution around its mean value, with the standard deviation given in the configuration; if no standard deviation is given, the coefficients are held constant.

<channel> channel_name
    <detector>
        <spatial_pix> 200 </spatial_pix>
        <spectral_pix> 200 </spectral_pix>
        <pnl_coeff_std> 0.005 </pnl_coeff_std>
    </detector>
</channel>

Here the detector sizes and the coefficient spread have been added to the configuration, giving:

The output is a map of \(a_i\) coefficients per pixel, which feeds ApplyPixelsNonLinearity.

From measured correction coefficients

Write the non-linearity model as

\[Q_{det} = Q \bigtriangleup (1 + \sum_i a_i \cdot Q^i)\]

where \(\bigtriangleup\) is the operator that relates \(Q_{det}\) to \(Q\), and its meaning depends on how the coefficients \(a_i\) are defined.

In practice it is the inverse relation that is measured, since its coefficients can be found empirically:

\[Q ={Q_{det}}\bigtriangledown ( b_1 + \sum_{i=2} b_i \cdot Q_{det}^i)\]

where \(\bigtriangledown\) is the inverse of \(\bigtriangleup\). Depending on how the non-linearity was estimated, this operator is either a division (\(\div\)) or a multiplication (\(\times\)); if it is not specified, a division is assumed.

The PixelsNonLinearityFromCorrection tool converts the measured correction coefficients \(b_i\) into the \(a_i\) coefficients that ExoSim uses.

List the \(b_i\) coefficients in the configuration with the pnl_coeff keyword, in alphabetical order: pnl_coeff_a for \(b_1\), pnl_coeff_b for \(b_2\), pnl_coeff_c for \(b_3\), and so on. Any number of coefficients can be listed and they are parsed automatically. Note that with this notation \(b_1\) is not forced to be unity.

<channel> channel_name
    <detector>
        <well_depth> 25000 </well_depth>
        <pnl_coeff_a>  1.00117667e+00 </pnl_coeff_a>
        <pnl_coeff_b> -5.41836850e-07 </pnl_coeff_b>
        <pnl_coeff_c> 4.57790820e-11 </pnl_coeff_c>
        <pnl_coeff_d> 7.66734616e-16 </pnl_coeff_d>
        <pnl_coeff_e> -2.32026578e-19 </pnl_coeff_e>
        <pnl_correction_operator> / </pnl_correction_operator>

        <pnl_coeff_std> 0.005 </pnl_coeff_std>
    </detector>
</channel>

The example coefficients above are taken from Hilbert 2009, “WFC3 TV3 Testing: IR Channel Nonlinearity Correction” (link).

The tool retrieves the \(a_i\) coefficients for a 4th-order polynomial,

\[Q_{det} = Q \cdot (a_1 + a_2 \cdot Q + a_3 \cdot Q^2 + a_4 \cdot Q^3 + a_5 \cdot Q^4)\]

giving the expected non-linearity shape:

As before, it also produces a per-pixel map of the coefficients: