Pixels Non-Linearity

This tools helps the user to find the pixel non-linearity coefficients to as inputs for ExoSim, starting from the measurable pixel non-linearity correction.

In fact, the detector non linearity model, is usually written as polynomial such as

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

where \(Q_{det}\) is the charge read by the detector, and \(Q\) is the ideal count, as \(Q = \phi_t\), with \(\phi\) being the number of electrons generated and \(t\) being the elapsed time.

Pixels Non-Linearity from scratch

The PixelsNonLinearity tool retrieves the \(a_i\) coefficients, starting from physical assumptions.

The detector non linearity model, is written as polynomial such as

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

where \(Q_{det}\) is the charge read by the detector, and \(Q\) is the ideal count, as \(Q = \phi_t\), with \(\phi\) being the number of electrons generated and \(t\) being the elapsed time.

Considering the detector as a capacitor, the charge \(Q_{det}\) is given by

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

where \(\phi\) is the charge generated in the detector pixel, and \(\tau\) is the capacitor time constant. In fact the product \(\phi \tau\) is constant \(Q\) is the response of a linear detectror is given by \(Q = \phi t\)

The detector is considered saturated when the charge \(Q_{det}\) at the well depth \(Q_{det, \, wd}\) differs from the ideal well depth \(Q_{wd}\) by 5%.

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

Then

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

This equation can be solved numerically and gives

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

Therefore the detector collected charge is given by

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

Which can be approximated by a polynomial of order 4 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 results are the coefficients for a 4-th 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 user should indicate the expected non-linearity shape, by setting the saturation parameter, called well_depth:

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

Then the tool can be run as

import exosim.tools as tools

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

With the given example, we obtain the following expected non-linearity shape:

However, each pixel is different, and therefore, this class also produces a map of the coefficient for each pixel. Each coefficient is normally distributed around the mean value, with a standard deviation indicated in the configuration. If no standard deviation is indicated, the coefficients are assumed to be 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>

To obtain the map we added to the configuration the detector sizes and the standard deviation of the coefficients.

The code output is a map of \(a_i\) coefficients for each pixel, which can be injected into ApplyPixelsNonLinearity.

Pixels Non-Linearity from correcting coefficients

Let’s write again the detector non linearity model as

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

where \(Q_{det}\) is the charge read by the detector, and \(Q\) is the ideal count, as \(Q = \phi_t\), with \(\phi\) being the number of electrons generated and \(t\) being the elapsed time. In the equation above, \(\bigtriangleup\) is the operator used to defined the relation between \(Q_{det}\) and \(Q\), which depends on the definition of the coefficients \(a_i\) (see also equation below).

However, it is usually the inverse operation that is known, as it’s coefficients are measurable empirically:

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

Where \(\bigtriangledown\) is the inverse operator of \(\bigtriangleup\). Depending on the way the non linearity is estimated, the operator can either be a division (\(\div\)) or a multiplication (\(\times\)). If not specified, a division is assumed.

The PixelsNonLinearityFromCorrection can determine the coefficients \(b_i\) from the measured non linearity correction.

The \(b_i\) correction coefficients should be listed in the configuration file using the pnl_coeff keyword in increasing alphabetical order: pnl_coeff_a for \(b_1\), pnl_coeff_b for \(b_2\), pnl_coeff_c for \(b_3\), pnl_coeff_d for \(b_4\), pnl_coeff_e for \(b_5\) and so on. The user can list any number of correction coefficients, and they will be automatically parsed. Please, note that using this notation, \(b_1\) is not forced to be the 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>

as example of non linearity, we used the parameters from Hilbert 2009: “WFC3 TV3 Testing: IR Channel Nonlinearity Correction” (link).

This class will restrieve the \(a_i\) coefficients, starting from the the indicated \(b_i\). The results are the coefficients for a 4-th order polynomial:

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

With the given example, we obtain the following expected non-linearity shape:

However, each pixel is different, and therefore, this class also produces a map of the coefficient for each pixel as before.