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I have a theoretical question regarding the irradiance measurement of light sources. From the basics I know the following:

  1. Irradiance has the unit $\frac{w}{m^2}$, in the context of light that means: The sum of photon's energy over a unit time per unit area.
  2. Photon energy is $E = h \times v$.
  3. A photometer includes a photodiode, which has a material depending absorption probability influenced by temperature and photon's frequency $v$.
  4. If a photon gets absorbed and has sufficient energy ($≥$ band gap), an electron jumps from the valence band in the conduction energy band, which leads to a measurable photocurrent.

Consequently, photocurrent is only proportional to the number of absorbed photons per unit time which have sufficient energy and not necessarily proportional to the photon energy itself.

So I assume, there can be approximately 3 types of commercial photometers out there:

  1. The ones, which aren't calibrated at all. So just for relative photon flux measurements of the same light source.
  2. The ones, directly showing the irradiance in w/m^2. But those are only useful, if the measured light source has the exact same spectrum as the one, to which the photometer was calibrated. Because of the fact, that a photometers 1) can just measure the flux and apparently, 1000 photons with frequency 1Hz are lower in energy than 1000 photons with frequency 2Hz and the photometer just measures "1000" without the knowledge of the frequency and 2) see first (3).
  3. The ones, which are calibrated to show the photon flux density in $umol/cm^2/s$ and are provided with a sheet of relative absorption rates for different wavelengths + coefficients to convert the photon flux density in $\frac{w}{m^2}$ for different wavelength intervalls, so one must know the relative spectrum of the light source.

Personally, I find variant 3 the most useful one.

Then I asked myself, how are photon flux measurement devices calibrated in the first place? Does NIST use a cavity with a hole blackbody radiator, measuring the temperature and calculating the radiation irradiance theoretical with plank's law, than converting it to flux with the relation $n = \frac{E}{h \times v}$and fitting the measurable photocurrent to those values and also variably change the spectrum (so the temperature) to compensate the material specific wavelength dependent absorption rates of the photometer?

I also assume, that therefore the first photometers, which were capable of absolute flux or irradiance measurements, were only possible after plank's discovery because one obviously can't calibrate a photometer by the means of another photometer, so one must find another way to gain knowledge about the irradiance of a reference source (e.g. temperature measurements of a blackbody radiator)?

Please correct me, wherever you notice something wrong. I'm not very familiar with stuff like that.

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3 Answers 3

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so I suppose the current is proportional to the photons energy and the light flux (is that even correct?)

No, photocurrent in a photodiode is proportional to number of absorbed photons (neglecting reflection, recombination, or other losses and multiple carrier interactions for the time being). If a photon has energy above the bandgap it generates one electron-hole pair. the extra energy is lost to heat.

How are professional measurement devices calibrated, that they display mW/cm^2 of the light? I mean how they convert the given electrical current to the sum of the protons energy per second per area (correct interpretation?)?

They have to be calibrated against a known source power and spectrum. If you change the spectrum of what you are measuring, the calibration will not be accurate.

How do such devices calculate "sensor area", I mean the material itself have a given density so I can't just assume the light hits a continuous, uniformly and endless dense sensor area.

There is a specific active area of the detector that is part of the photodiode design. Otherwise a smaller aperture could be used, but this will typically hurt your sensitivity.

There are photodiodes with a range from 300-1200 nm wavelength, how can a single material absorb photons in that continuum if atoms only accept discrete energies???

The densely packed semiconductor atoms create energy "bands" which allow absorption of a wide range of energies.

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  • $\begingroup$ "The densely packed semiconductor atoms create energy "bands" which allow absorption of a wide range of energies." That makes sense, but the energy band is still discrete. I know that emittet photons must have a well defined energy as well because of the discrete energy levels of the emitting atoms but what about the bremsstrahlung? There could theoretical be photons falling somewhere in between two energy states of the band. I've already read that there is still the possibility of absorbtion due to different mechanisms but it's very unlikely. $\endgroup$
    – iwab
    Oct 8, 2023 at 1:29
  • $\begingroup$ "They have to be calibrated against a known source power and spectrum" but how is the power known in the first place? Just from theoretical calculations about how many photons of which frequency must be emitted every second given the properties of the material and so on..? $\endgroup$
    – iwab
    Oct 8, 2023 at 1:39
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    $\begingroup$ NIST-traceable power meters rely on black body absorbers and temperature measurements (or at least the ones I've used do). These have slow time response. The photodiode is then a fast indicator of the shape of the pulse. As for separation of energy states in the band, the energy difference is order $1 \over 10^{22}$ eV or so, so truly minute compared to thermal broadening. $\endgroup$
    – Jon Custer
    Oct 8, 2023 at 2:10
  • $\begingroup$ @JonCuster that sound very interesting, may I ask what material were used as a near perfect black body absorber? As far as I know a black body is only an ideal idea, which material is used by NIST? I suppose one with the densest energy band possible? $\endgroup$
    – iwab
    Oct 8, 2023 at 2:16
  • $\begingroup$ Look into bolometers and what they use (I recall carbon black but could be wrong). $\endgroup$
    – Jon Custer
    Oct 8, 2023 at 2:29
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If, for example I buy myself a luxmeter, I can't just measure any light source because I don't know the spectral properties, maybe my source radiate more wavelengths to which my device isn't very sensible - thus leading to a much lower value than it actually is.

So you measure the spectral properties. You buy a more expensive instrument, the illuminance spectrophotometer. Here, the light is passed through a diffraction grating and onto a linear array of photodetectors, essentially a 1-D camera. Because each detector pixel only measures a known, narrow wavelength range, the spectral response of the photodiodes can be calibrated out.

A cheaper method is to use a thermal sensor, these are typically thermopile (a stack of back-to-back thermocouples) behind a black absorber material. These tend to have very flat, broadband responses. However, they have a slow response (time constants around a second) and influenced by ambient temperature and airflow. These sensors often require frequent zeroing and are not as good at measuring low powers.

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I think that one specific consideration when talking about "fundamental calibration" is that many times the types of instruments that we use in day-to-day life are far more boring than we assume, in the sense that the quantum-like interactions with the materials are not specifically quantized, rather we take the macroscopic, measurable properties of materials and use those to characterize the devices.

Materials for optical detection

You can see above that several materials have a wide range of wavelengths to which they will respond electrically, more specifically, you will see it in the spectral sensitivity curves.

enter image description here

You can see that sensitivity is mA/W, therefore, the current (in mA) generated by a set input power (in Watts). As you correctly pointed out in your first point, the irradiance is measured in watts per square meter, which is a measurable amount and not tied directly to the photons impinging on the material, rather it is based on its macroscopic properties.

As to which sources you can measure:

https://www.oceaninsight.com/products/light-sources/calibration-sources/radiometric-calibrated/

There are manufacturers that can provide "radiometric calibrated" illumination sources.

https://nvlpubs.nist.gov/nistpubs/Legacy/SP/nistspecialpublication250-89.pdf

As you noted, these are generated by NIST by what is called a "cryogenic radiometer". The principle of operation of a cryogenic radiometer is very nicely and clearly discussed in the following 1985 paper by Martin, et Al.:

https://iopscience.iop.org/article/10.1088/0026-1394/21/3/007

The gist of the experiment is using a laser with known spectral components through a cryogenic chamber, in order to reduce thermal effects that can affect how a device behaves.

from https://iopscience.iop.org/article/10.1088/0026-1394/21/3/007/pdf

How you measure current here is through a thermistor, therefore you can see that in order to find how a specific variable affects a measurement, you need to control the environment, use a device that can accurately control its behavior based on one controlled parameter, and finally, use this as a standard to define your other device parameters.

Than converting it to flux with the relation $n=E/h×v$ and fitting the measurable photocurrent to those values and also variably change the spectrum (so the temperature) to compensate the material specific wavelength dependent absorption rates of the photometer?

Not necessarily. Note that in the cryogenic radiometer paper not once is quantization or Planck's law mentioned. All the measurements are purely classical and therefore traceable and measurable by existing laboratory methods.

In the paper a paper that measured the Stefan Boltzmann constant is mentioned: https://royalsocietypublishing.org/doi/epdf/10.1098/rsta.1985.0058 and you can see that the methods are increasingly built upon works that use thermally-stabilized devices which have been historically referenced as black body radiators, but for example that which was used in the Martin paper is a platinum box with a black interior.

I also assume, that therefore the first photometers, which were capable of absolute flux or irradiance measurements, were only possible after plank's discovery because one obviously can't calibrate a photometer by the means of another photometer, so one must find another way to gain knowledge about the irradiance of a reference source (e.g. temperature measurements of a blackbody radiator)?

It entirely depends on your setup and how the photometers are used in the setup. In the cryogenic radiometer paper, there are silicon detectors that are used to control the power of the krypton ion laser, but here what is important is not the absolute measurement, rather you care about the relative power difference between two temperatures, for example. Therefore, if you know what is the power ratio between two temperature measurements, you can calibrate your thermal source even with a "relative" photometer in play. What matters is which parameters you can control and which devices you use in your setup to corroborate your results.

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