The Noise Measured by Back-illuminated Silicon Sensors

Roy T. Smart1, Charles C. Kankelborg1, and Jacob D. Parker2

1Montana State University

2NASA/GSFC

246th Meeting of the American Astronomical Society

June 9th, 2025

1 / 27

IRIS Noise Mystery

IRIS measures less noise than predicted (Wulser et al. 2018).

Photon transfer curve is too shallow

Expected slope ratio > 2

Measured slope ratio = 1.5

SNR is ~25% better than expected

2 / 27

HST/WFC3 Noise

Wide Field Camera 3 (WFC3) on Hubble also has this issue (Marinelli & Green 2024)

“Quantum yield” is lower

Expected ~1.7 @ 200 nm

Measured 1.09 @ 200 nm

SNR is ~25% better than expected

3 / 27

Outline

Signal

Quantum yield

Charge-collection efficiency

Absorption

Noise

Simple noise model

Shot noise

Fano noise

Partial-charge collection noise

Results

4 / 27

Signal

5 / 27

Quantum Yield

Quantum yield (QY) is the number of electron-hole pairs generated per photon

QY increases as photon energy increases

QY = E / (3.65 eV) where E is the energy of the photon

Ramanathan & Kurinsky (2020) QY model

6 / 27

Back-illuminated Sensor Model

Thin layer of silicon dioxide (~1 nm)

Light-sensitive silicon (16 μm)

Partial-charge collection (PCC) region (~100 nm)

Field-free region

Depletion region (8 μm)

7 / 27

Partial-Charge Collection

Photons are absorbed according to the Beer-Lambert law (red)

Photons absorbed in PCC region lose some electrons

Charge-collection efficiency (CCE) is the fraction of charge measured.

Piecewise-linear differential CCE introduced by Stern et al. (1994)

Denoted $𝜂 (𝑧)$

8 / 27

Penetration depth

Some ultraviolet wavelengths have a penetration depth smaller than the PCC region

~40 – 100 A

~500 – 4500 A

Sensor is much less efficient in these regions

9 / 27

Average Charge-Collection Efficiency

Integrate Beer-Lambert law against differential CCE

Average CCE is needed for correct photometry

10 / 27

Absorbance

Energy absorbed by light-sensitive layer

Reflections from each interface

Oxide absorption

Transmittance through entire sensor

Computed using standard optical constants (Henke 1993, Palik 1997)

11 / 27

Quantum Efficiency

Quantum Efficiency (QE) is the factor used to convert from incident photons to measured electrons

Allowed to be larger than one!

$QE \equiv \frac{𝑁_{𝑒}}{𝑁_{𝛾}}$

$= QY \times CCE \times Absorbance$

12 / 27

Effective Quantum Efficiency (EQE)

Measured quantity

Compare sensor to NIST photodiode

Between 0 and 1

Often used to compute QE

$EQE = CCE \times Absorbance$

13 / 27

Noise

14 / 27

Variance-to-mean ratio

Convenient measure of noise

If $𝑋$ is Poisson-distributed, $VMR (𝑋) = 1$

Constant vs. signal, unlike SNR

Equal to the slope of the photon-transfer curve

Same units as $𝑋$

$VMR (𝑋) = \frac{Var (𝑋)}{⟨ 𝑋 ⟩}$

15 / 27

Simple Noise Model

Shot noise only

All-or-nothing charge collection

Used by IRIS and WFC3

$𝜎_{𝑁_{𝑒}} = \sqrt{EQE \times ⟨ 𝑁_{𝛾} ⟩}$

$\Rightarrow VMR (𝑁_{𝑒}) = QY$

$\Rightarrow VMR (𝑁_{𝛾}) = \frac{QY}{QE} = \frac{1}{EQE}$

16 / 27

A Slightly More Complicated Noise Model

Shot noise

Fano noise

Inherent randomness in the charge-generation process

PCC noise

Uncertain CCE due to random absorption depth

17 / 27

Shot Noise

Absorbed photons are Poisson-distributed

$𝑁_{𝛾}^{'} ~ Poisson (𝐴 \times ⟨ 𝑁_{𝛾} ⟩)$

$VMR (𝑁_{𝛾}) = \frac{1}{𝐴}$

18 / 27

Fano Noise

Quantum yield uncertainty

Described by Fano factor

$ℱ = VMR (QY) ≃ 0.1$

Ramanathan & Kurinksy (2020) QY model

Small compared to other noise sources

19 / 27

PCC Noise

Since each photon is absorbed at a different depth, each experiences a different CCE.

$𝑛_{𝑖} \leftarrow Binomial ({QY}_{𝑖}, 𝜂 (𝑧_{𝑖}))$

$𝑁_{𝑒} = \sum_{𝑖 = 0}^{𝑁_{𝛾}^{'}} 𝑛_{𝑖}$

20 / 27

Total Noise

Sum of shot, Fano, and PCC noise

VMR is smaller than the simple model in the UV

The partial events improve the noise performance

21 / 27

SNR improvement

SNR improves by up to $~ 30 %$ compared to the simple model

$= \sqrt{\frac{simple VMR}{total VMR}}$

22 / 27

Charge spreading?

Charge spreading is when electrons diffuse into adjacent pixels.

Proportional to size of field-free region

Could explain the discrepancy since this increases the correlations between pixels

Stern et al. (2003)

23 / 27

Mean Charge Capture (MCC)

Average fraction of charge captured by central pixel

Janesick (2001) charge spreading model

Stern et al. (2004) measurements

Fit size of field-free region

Predicts a similar amount of charge spreading for both UV and visible light

24 / 27

Compare to Measurements

A table with numbers and a number AI-generated content may be incorrect.

25 / 27

Affected Instruments

AIA

94 Å

1600 Å

1700 Å

IRIS

1330 Å

1400 Å

MUSE

108 Å

26 / 27

Conclusions

Partial-charge collection is mostly responsible for IRIS noise mystery

Charge spreading is less important

Need to use a more complicated expression to compute noise

The model predicts better SNR in some wavelengths than designers expected

Results affect AIA, IRIS, WFC3, ESIS, FURST, and MUSE.

Code provided in the optika.sensors Python module.

Published on PyPI

27 / 27