Detectors, pixels and response

A detector converts a physical interaction into data

A telescope delivers radiation to a detector. In an optical semiconductor example, absorbed photons can produce charge carriers that are collected and read out. A CCD accumulates charge and transfers it through a readout process; CMOS architectures use different arrangements of sensing and readout electronics. Their records require an instrument-specific response model.

A pixel is a sensing or sampling element associated with a portion of the delivered image. It is not automatically a separate astronomical object or a perfectly isolated sky direction. The optical point response can spread one compact source over several pixels, and multiple sources can contribute to a pixel.

The response depends on wavelength, collection efficiency, detector state and operating conditions. Raw values therefore describe the measurement system as well as the sky. Calibration relates those values to a useful signal quantity under specified conditions.

Keep the conversion stages separate: photons incident at a declared location, collected photoelectrons, numerical readout and the final calibrated estimate. Calling all of them “counts” without a definition can conceal an incorrect unit conversion.

Calculate an expectation in one optical model

Consider an invented unsaturated linear optical response at one stated wavelength. An expected 1000 incident photons produce, on average, 0.5 collected source electrons per incident photon. The expected source charge is therefore 500 electrons. This is a stipulated optical yield, not a universal detector constant.

Define the electronic gain here as 2 electrons per data number, abbreviated DN. A DN is a numerical readout unit; it is not inherently a photon or an electron. Also stipulate an electronic bias offset of 100 DN, with no sky or dark contribution in this conversion example.

Ignoring digitisation and random fluctuations, the expected raw value is bias + source electrons/gain = 100 + 500/2 = 350 DN. Subtracting the bias and multiplying by the stated gain gives 2 × (350 − 100) = 500 electrons.

The gain convention must be explicit: some descriptions use the reciprocal scale. Changing a numerical representation does not create incoming light. Gain settings can also affect an actual instrument’s readout behaviour and dynamic range, so the arithmetic is not a complete performance model.

This simple one-electron optical picture must not be extended to every wavelength or sensor. In some detector regimes, one incident photon can generate more than one collected electron. Interpret a quoted quantum efficiency or electron yield using that instrument’s definitions.

Subtracting a mean does not subtract its fluctuations

Bias is an electronic offset. Dark signal comes from detector processes accumulated without the target light. Sky or foreground signal is radiation added to the source measurement. Calibration can estimate and remove appropriate mean contributions, with uncertainty in those estimates.

Even if the background mean were known exactly, random background arrivals would still fluctuate in each observation. Source arrivals, dark counts and readout can also contribute uncertainty. A calibrated estimate is therefore not a noiseless copy of the sky.

A restricted independent-noise model gives SNR = S/√(S + B + D + npix Nread σread²). Here S, B and D are expected source, sky and dark electron totals over the same extraction and integration; npix is the number of extracted pixels, Nread the number of reads, and σread the per-pixel per-read noise standard deviation in electrons.

For invented S = 100, B = 80, D = 20, npix = 4, Nread = 1 and σread = 5 electrons, the variance is 100 + 80 + 20 + 4 × 25 = 300 in electron-count variance units. The ratio is about 100/√300 = 5.77. This model assumes independent Poisson counts, independent read errors and negligible uncertainty in the estimated background mean.

Correlated pixels, calibration error, source variability and other effects require additional treatment. The numerical ratio is not a universal detection criterion.

A detector has a usable range and a spatial response

A detector or readout can leave its linear range when a signal becomes too large. Charge capacity and digitiser limits are different constraints. A clipped digital value does not retain the missing amplitude simply because a calibration multiplier is later applied.

Some detector-specific charge-spill measurements can be recovered with validated methods under appropriate conditions. That is different from assuming every saturated or digitally clipped sample is still an ordinary linear measurement. Flag the regime and use the instrument’s documented treatment.

Smaller pixels can sample a delivered pattern more finely, but they do not by themselves shrink the optical or atmospheric blur. Extracting the same source across more pixels can also change the read-noise contribution. Pixel scale, resolving power and sensitivity need separate descriptions.

The next lesson considers what filters and spectrographs put in front of the detector. Their wavelength-dependent responses determine which arriving photons contribute to each image or spectral bin before the numerical record is interpreted.

Worked examples

Try the reasoning

1. With an expected 2000 incident optical photons and stipulated yield 0.25 electron/photon, what source charge is expected?

   Show the answer and reasoning

   500 electrons.

   Multiply the input expectation by the declared yield. This is not a claim that every exposure gives exactly 500.

2. May a gain expressed in electrons/DN be used as if it were DN/electron?

   Show the answer and reasoning

   No.

   The conversion direction and units must be retained.

3. Does halving a pixel’s angular scale guarantee twice the physical angular resolution?

   Show the answer and reasoning

   No.

   The delivered point response and sampling conditions also matter.

4. Does multiplying a digitally clipped value by a calibration factor automatically reconstruct its missing amplitude?

   Show the answer and reasoning

   No.

   The clipped record lacks the ordinary linear amplitude information. Any recovery requires additional justified information and an applicable method.

What to carry forward

• Photons, electrons, DN and calibrated quantities are different stages with declared conversion units.
• Mean calibration corrections do not erase arrival fluctuations or read noise.
• Linearity, saturation, sampling and instrument-specific response limit what the numerical record preserves.