In the 1990ies, when I worked as an astronomer at the Hoher List observatory (Germany) my boss was Prof. E. H. Geyer. He was an experienced astronomer and created a number of astronomical instruments and measuring devices. From the many fruitful discussions and private lessons I received from him, I remember one special fact, that gained my attraction: Once, he mentioned a specific number of photon events per second that will be recorded from a 12th magnitude star when observed with a 1-meter class telescope. Although I intuitively got, what he meant, it took me a while to unfold the full scope of this simple sentence he told me.
What will be the advantage of thinking in photon counts? First of all, as a physicist and astronomer, I really like the idea and concept behind it. At a time when photo-multiplier tubes were popular, it seemed also natural to think in photon count rates. Computing photon counts received from a star with given apparent magnitude has some advantages over unit-less digital numbers that rise from an astronomical picture. So the question is: What physical information can we extract from these unit-less digital numbers in an astronomical picture? Unfortunately, physics and astronomy won't work without mathematics, but computations aren't too hard. There are some puzzle pieces to collect from scientific literature.
Max Planck's law
Astronomers are interested in energy equivalents. Sometimes, when reading astronomical papers, even for astronomers, it is not even easy to understand measured fluxes from stars, gaseous nebulae and other astronomical targets, that are expressed in multiple different energy equivalents und thus different physical units, that are hard to compare. With the many diverging color bands and filter systems, it will even get harder to compare results (see for example Bessel, 1990). Each ground-based or satellite observatory has its own color filter system invented.
The deduction of the following mathematics is partly taken from one of my previous publications (Bauer, 2011).
Light is an electro-magnetic wave, that is sent from a distant star, for example. Light can also be understood as a particle stream wherever light interacts with material, like a photo-cathode or semiconductor device to detect light. Planck (1901) introduced a formula, that yields a relationship between the photo quantum (photon) energy E, the Planck constant h and the frequency of the light ν:
E = h ν
In a silicon imaging detector photons enter the semi-conductor crystal up to certain depth (transparent depth) where the photon energy will free electrons with certain probability (photo-effect). These free electrons can then be measured as a photo current, amplified and converted into digital numbers. With special photon counting devices photons can be counted individually. In a conventional CMOS device, however, the detection process will not count individual photons, but single electrons are amplified and a photo current is converted into digital numbers. Amplifier noise will add noise to the detected digital signal. So we cannot tell exact single photon count numbers. The light detection process, on the other side, includes a natural source of noise from the statistical detection process.
Figure – In the conversion process from photo quants (photons) emitted by a star quantum efficiency qν, camera gain gPC and noise from the statistical detection process define the digital numbers counted with each pixel of a modern CMOS or CCD imaging sensor.
A digital image shall be regarded as a set of numbers I(x), which are the measured pixel intensity values at the two-dimensional coordinate x. There are not only intensities stored with the pixel values, but also constraints from geometry and the physics of the device. The incoming flux of the light will cause electrons created within the silicon and collected over time. The pixel intensities I(x,t) therefore represent a flux from the source at time t. The electron current from the chip is then amplified and digitized by the electronics. This yields a certain conversion factor gPC between the freed photo electrons and the digital numbers obtained. The relation between the measured photo current NPE(x,t) and the actual number NQ(x,t) of photo events detected is given by a certain quantum efficiency qν. at frequency ν (Mullikin, 1994). Quantum efficiency is the probability to yield a photo electron freed by the energy of a photon, which is typically less than 1 (100%).
I(x,t) = gPC NPE(x,t) = gPC qν NQ(x,t)
With Planck's idea of the quantum nature of light this translates into:
∑n qν En = E qν NQ(x) ~ NPE(x)
This formula means: The photo current NPE(x) obtained from the number n of the individual photo electrons is proportional to the total energy E of the photo events detected. En is the energy of each individual photo quant collected with index n. In astronomy we typically use color filters, which cover certain wavelength range, like broad blue, green or red color filters or narrow band-pass nebula filters. Here, the probability to detect a photo event results from the spectral distribution of light emitted by the star or nebula multiplied by the spectral sensitivity of the detector plus filter. Each combination of filter and camera sensors results in a total the quantum efficiency qν, which depends on the wavelength portion observed. In simple words: Filter plus camera define the expected photo count within the spectral range of the light emitted by a colored star or nebula.
We found a conversion formula between digital numbers and the number of photon events, that physically arrived at the imaging detector. The individual photon events do not arrive at constant frequency, but rather follow an irregular process that causes noise to the detection of light. Taking multiple photos of the same star, the number of photons detected from the star will vary between images. The scattering of the digital number recorded between images follows a Poisson statistics. The Poisson distribution of values has an interesting property: Given an average number of photons detected its variance of the is the same as the mean value. The error of the digital numbers (intensities) recorded from multiple images taken can be computed by the square-root of the photon count variance including conversion factor gPC. This property of the noise process, the variance of the Poisson distribution, and also the physical law of energy conservation prohibit any ability to increase the signal-to-noise level of any astronomical image recorded in any way by post-processing the immutable digital image (Bauer, 2011).
Unity Gain
The factor gPC in the above fomula is the camera gain. Camera control allows to chose between arbitrary gain values. In case of scientific or dedicated astronomical cameras camera gain settings that the user may select often are arbitrary values. For conventional photo cameras camera gain is controlled by chosing ISO settings of the digital camera. Unity gain denotes the gain setting gPC≈1, where the digital unit of the pixel value is equal, or at least closest to the number of photo electrons freed by photons within the silicon device.
Unity gain condition: gPC≈1.
Certain manufacturers of astronomical imaging devices provide information about the gain value, that is closest to the condition. For conventional photo camera ISO settings that reflect unity gain are available from internet databases (see literature section). However, if precise measures are required the exact gain setting for a specific camera which is closest to the unity gain condition shall to be determined by conducting experiments with different gain settings. Variance is a mathematical term, that is defined as the average difference between random values and their mean value. A practical experiment to derive unity gain is to collect two or more flatfields in series with different gain settings. Then compute mean value and variance (average difference of the pixel intensities) for each set of flatfields that have the same gain setting. Unity gain condition is found for the gain settings, where the mean value of the digital numbers recorded is the same as the variance.
Unity gain enables single photon counting estimates, although this is accompanied by certain additional readout noise. However, modern CMOS cameras, such as those used in astronomy domain, have pretty low noise values and thus readout errors in the range of ±1 e- per pixel. The natural noise from the light detection process and resulting quantum photon count of the illuminated sky and nebula, for example, is larger compared to readout noise. Hence modern CMOS devices work in the region of a single photon counting device. At this point thinking in quantum counts provides certain advantage to directly estimate important physical properties from the numbers in a digital image.
My next article will shed light on the relationship between apparent magnitude and photon count rates...
[to be continued soon]
Literature
- Bessell, M. S., 1990. UBVRI Passbands. PASP, 102, 1181.
- Bauer, T., Efficient Pixel Binning of Photographs. IADIS International Journal on Computer Science and Information Systems, vol. VI, no. 1, p.1-13, P. Isaías and M. Paprzycki (eds.), 2011. ISSN: 1646-3692
- Planck, M., 1901. On the Law of Distribution of Energy in the Normal Spectrum. Annalen der Physik. vol.4, p.553
- Mullikin, J.C. et al, 1994. Methods for CCD Camera Characterization. Proc. SPIE, vol. 2173, 73
- Photons to photos internet database: https://www.photonstophotos.net/Charts/RN_e.htm