Radiometric Calibration

 

Introduction
The colored filters which make up the Bayer array that covers a digital camera’s CCD let light of certain colors (wavelengths) pass while blocking others. These filters enable the camera to capture color images, but cause distortion in the spectrographs it records and the spectrograms derived from them. This distortion, along with other sources of distortion such as the CCD’s quantum efficiency, can be removed by deconvolving the spectrometer’s system function from its spectrograms. Once these effects have been removed the resulting spectrum will have linear relationship with the intensity of the light source. If a light source with a known irradiance is used as a reference all the other spectrums can be calibrated against.

 

Deconvolution Illustrated
Since the diffraction grating decomposes light by wavelength and the light source is held constant, the data recorded by the spectrometer is in the wavelength domain. This means that the spectrometer’s system function can be determined by capturing a spectrograph of a light source with a flat spectrum (i.e. truly white light) and that the distortion caused by the colored filters, quantum efficiency, and other effects can be removed by division. Unfortunately, a light source with a completely flat spectrum is hard to obtain; however, a fine alternative is a light source with a non-flat but mathematically well-defined spectrum. The illustrations below demonstrates this.

The 100 W tungsten incandescent bulb’s spectrum is denoted X and unknown. The spectrometer’s system function is denoted H and unknown. The spectrometer’s response to the 100 W bulb is denoted Y and is known. The spectrometer’s spectral response, H, is the desired spectrum.

 

 

Unfortunately, since X is unknown it cannot be deconvolved from Y by division to get H. However, X can be modeled as a black body using Planck’s law with corrections for tungsten’s emissivity. X’ is the model for X. Therefore, H’ = Y/X’. If X’ is a good model for X, then H’ should be a good substitution for H. For the remainder of this article no distinction is made between X and X’ as well as H and H’.

 

 

Modeling X and Determining H
The spectrum of the light produced by heating the tungsten filament of a 100 W light bulb to incandescence is very similar to an ideal black-body radiator. Therefore, the bulb’s spectrum can be mathematically modeled using Planck’s law with corrections for tungsten’s emissivity. Since a CCD counts photons, spectral photon irradiance is used. The equation for the spectral photon radiance of a tungsten emissivity corrected black body, represented by N(λ), is given below. The variable λ is in nm and the parameter T is in K. The emissivity function ε was interpolated from the table given in Absolute Optical Calibration using a Simple Tungsten Bulb

Irradiance is used in the calibration procedure because it allows the various light sources of interest to be different distances from the entrance slit. Therefore a weak source can be placed near the slit were its light is easily detected and a strong source can be placed further back so that it does not saturate the CCD. The photon spectral irradiance of an incandescent bulb is:

Where omega is the number of steradians covered, which is 4*pi since the light bulb is an isotropic source. The emitting area of the filament is represented by Af. The area of the sphere with a radius equal to the distance from filament to slit is denoted with Asphere.

 

The temperature of the filament determines the shape of the spectrum the incandescent lamp produces. This temperature can be calculated by first measuring the filament’s resistance while it is at room temperature and then when power is applied to it. The hot resistance is obtained indirectly by measuring the voltage across and the current through the filament.

Therefore to find the spectrometer’s system function:
1) take a spectrograph of a CFL and use it to derive a calibrated wavelength scale, λ.
2) take a spectrograph of a 100 W incandescent bulb and measure the resistance of the filament.
3) convert the incandescent spectrograph to a spectrogram, Y(λ).
4) derive a mathematical representation of the incandescent lamp’s irradiance spectrum undistorted by the spectrometer, X(λ)
5) find H(λ) by dividing Y(λ) by X(λ).

 

The diffraction grating spreads a light source’s spectrum across the pixels of the CCD in such a way that only a narrow band of wavelengths from the spectrum fall on any particular pixel and which band of wavelengths changes based on the pixels horizontal position. When a photon of one wavelength strikes the semiconductor material inside of a pixel it creates a free electron which is indistinguishable from a free electron created by a photon of a different wavelength or even from a free electron generated by heat. Because they are indistinguishable, electrons in a pixel can only be counted or summed up (i.e. integrated) like pebbles dropped into a cup. Therefore, if we assume the number of electrons created per wavelength in a pixel is constant over the narrow band of wavelengths that shine on it, then Y(λ) = g*Δλ*V(λ).

 

H(λ) represents the transformation of the photon spectral irradiance to a counts. Now if, for instance, we have an LED with an actual spectrum denoted by U(λ) while the spectrum which is captured by the spectrometer is called Z(λ), then Z(λ) = H(λ)*U(λ). Since U(λ) is desired and H(λ) and Z(λ) are known then U(λ) is simply equal Z(λ)/H(λ). Keep in mind that λ was translated from a discrete set of pixel numbers, therefore Y(λ) and V(λ) are point-series not continuous functions.

 

Code
Given the resistance values of the filament and lambda, the function generate_Wbb_spectrum() calculates the filament temperature and outputs the emissivity corrected spectrum of the incandescent tungsten filament bulb. Once you have the spectrometer’s system function you can remove it’s effects on your other spectrograms by dividing it out of them.

 

Links
Radiometry and Photometry FAQ

Radiometry and Photometry

Spectral Power Distribution Curves of Different GE Lamps

Spectral Response of Some Cameras

More Spectral Responses

Measuring System Gain (aka CCD Gain) and Noise

Measuring CCD Gain (counts/electron)

CCD Measurements

Quantum Efficiency