How to model colored and Tristimulus Sources

This article describes the various models available in Non-Sequential Mode for defining broadband sources. These models fall into two categories: (1) source definitions based on measured spectra and (2) source definitions based on Tristimulus values. Knowledge of how to model different sources and analyze the photometric response is important for being able to create numerous optical systems including digital displays and projection systems. 

Authored By Sanjay Gangadhara

Introduction

A number of optical systems (e.g. digital displays and projection systems) require knowledge of the photometric (human eye) response of the system to a broadband source. In OpticStudio, a polychromatic source may be defined either by using multiple system wavelengths or by defining the Tristimulus (or associated) values for the source. The resultant color distribution of the output image may be visualized using the Detector Color object, as described in the article "How to measure and optimize color data." A brief overview of the various source models available in Non-Sequential mode is provided in this article.

Source modeling using measured spectra

There are a number of ways to model polychromatic sources in the Non-Sequential mode of OpticStudio. The default is to define multiple system wavelengths and weights in the Wavelength Data dialog box.

Wavelength_Data_dialog_box

Then set Source Color: System Wavelengths for the associated source object.

NSCE_Source_Color

This method allows up to 24 different wavelengths to be defined for the source, each with a (potentially) different value for the relative weight. There may be situations, however, in which you wish to model multiple sources in the system, each with a different spectral distribution. Or perhaps you wish to model a single source that is composed of more than 24 wavelengths, and you don’t want to use multiple configurations to do so.  These issues can be addressed by selecting for each source object something other than System Wavelengths for the Source Color option. The Source Color option is found in the Sources Tab of the Object Properties dialog box:

Object_Properties_dialog_box

A number of options are available for defining the spectral distribution of each source in a system, and a discussion of those options associated with Tristimulus source models will be provided in the next section. To define a source based on a measured spectrum, there are three options available (in addition to System Wavelengths):

  • Uniform Power Spectrum: The relative intensities are equal for all wavelengths defined
  • Black Body Spectrum: The relative intensities are determined from the blackbody emission curve at a particular temperature for all wavelengths defined
  • User Defined Spectrum: The relative intensities are given in a text file for all wavelengths defined

For the case of the Spectrum File, both the wavelength and corresponding relative intensity values are provided in a text file. You can find more information about it in the help file located as follow: “The Setup Tab...Editors Group (Setup Tab)...Non-sequential Component Editor...Object Properties (non-sequential component editor)...Sources...Defining a spectrum file." This allows users to explicitly provide measurement data for a source directly to OpticStudio. Up to 100 data points may be provided in the file.

For the Uniform Power Spectrum and Black Body Spectrum, the wavelengths that will be used to model the source are defined in the dialog box:

NSCE_Sources

The wavelength spectrum is uniformly spaced, going from the initial to the final wavelength values specified in the number of steps given by the “Spectrum” parameter. For example, in the above system the wavelengths used to model the Black Body source are 0.44, 0.54, and 0.64 microns (i.e. from 0.44 to 0.64 microns in 3 steps), and the relative weights for each of these three wavelengths are determined by the blackbody emission curve at a temperature of 10,000 Kelvin.

The manner in which wavelengths are used to define the source is the same for the Tristimulus source models as it is for the Uniform Power and Black Body source models. As we will see in the next section, information in the Prescription Data Report is available to determine the wavelength weights for all source models other than System Wavelengths.

The flexibility of the User Defined Spectrum allows definition of an arbitrary source composed of up to 100 wavelengths. However, in many cases the information provided for a source is not the measured spectra, but rather a Tristimulus characterization of the source. This information may also be used to model a source in Non-Sequential mode, as described next.

Source modeling using Tristimulus data

An overview of Tristimulus source modeling is provided here. As described in the article, the Tristimulus XYZ values characterize the color of a source as seen by the human eye.

CIE_standard_observer_color_matching_functions

Equation_1

Equation_2

Equation_3

The quantities x-bar(l), y-bar(l), and z-bar(l) represent the human eye response, while I(l) represents the spectral distribution of the source.

Since the Tristimulus values represent the integral of the spectral distribution over the human eye response, many different distributions can generate the same XYZ values. In other words, different spectra can generate the same color. Thus, in addition to specifying XYZ values for the source, in OpticStudio we must also specify the wavelengths associated with the source distribution. This is done in the same manner as it was done for the Uniform Power and Black Body source models, as described in the previous section.

To choose the Tristimulus XYZ model for a particular source object, select CIE 1931 Tristimulus XYZ under the Source Color setting in the Sources Tab of the Object Properties dialog box:

CIE_1931_Tristimulus_XYZ

Once this model has been selected, options to define the XYZ and the wavelength values for the source are shown. In the above example, the wavelengths used to model the source are 0.44, 0.54, and 0.64 microns (i.e. from 0.44 to 0.64 microns in 3 steps), and the Tristimulus values for the source are X = 1, Y = 0.5, and Z = 0.25.

Once the XYZ and the wavelength values are defined, OpticStudio performs a fitting to determine the relative intensity of light at each wavelength. The intensity values are chosen to generate the desired XYZ values, and in such a way as to keep the RMS of the intensity values at a minimum while ensuring that none of the wavelength weights are negative. The fit results are shown below the input data, along with a color bar showing the closest RGB representation for this fit.

In the above example, an exact match to the input XYZ values has been generated. When an exact match is not found, it is generally because the wavelength spectrum does not include enough wavelengths and/or the correct wavelengths to model the color of interest, or because the input XYZ values do not represent a color that can actually be seen by the human eye. More details are provided in the above linked article (e.g. see the chromaticity diagram), as well as in the section entitled “The spectrum fitting algorithm” located as follow in the OpticStudio Help File: “The Setup Tab...Editors Group (Setup Tab)...Non-sequential Component Editor...Object Properties (non-sequential component editor)...Sources...The spectrum fitting algorithm.”

Values for the intensity distribution may be observed in the Prescription Data Report.

Prescription_Data_Report

Prescription_Data_Report_2

In this report, we see the three wavelengths used to model the source listed in the first column, and the x-bar, y-bar, and z-bar values (i.e. the human eye response) for each wavelength listed in columns 3-5. The relative intensity (weight) for each wavelength is listed in column 2. The weights are normalized so that the total sum is unity, and the weights are always provided in terms of Watts. For example, if a source with the above chromatic characteristics launched 1 W of power, ~ 0.061 W of light would be radiated at 0.44 microns, ~ 0.064 W of light would be radiated at 0.54 microns, and ~ 0.875 W of light would be radiated at 0.64 microns.

The last column in the report shows how many Lumens would be seen at each wavelength for the given weightings. The number of Lumens is simply the product of the weight and the y-bar value, multiplied by 683. This final scale factor corresponds to the number of Lumens for 1 Watt of light at 0.555 microns, for which the y-bar value is 1.0. Thus, for the wavelengths shown above:

Equation_4

Equation_5

Equation_6

If this source launched 1 W of power, the human eye would see ~ 147.4 Lumens (= sum over all wavelengths). It is interesting to note that although the relative intensity in Watts for light at 0.54 microns is small for this source, the number of Lumens contributed at this wavelength is significant, because of the large y-bar value. In other words, the human eye is very responsive to the green portion of the wavelength spectrum.

As indicated above, the relative intensity listed in the Prescription Data Report is always in Watts, as rays are always traced in terms of Watts. If the source is defined in terms of Lumens (which can be done by changing the Source Units to Lumens under the Units tab of the General dialog box), OpticStudio will first convert the source to the equivalent number of Watts, using the Lumens/Watts conversion shown above. Once the source power in Watts is determined, the distribution at each wavelength is known from the relative weights. Once a ray of a particular wavelength hits a detector, its intensity may be converted back to Lumens, if the Detector Color object is used.

OpticStudio supports other source definitions which are equivalent to the Tristimulus XYZ definition. These are:

  • CIE 1931 Chromaticity xy: This source color model is essentially identical to the XYZ model above, except the normalized chromaticity coordinates are used instead.
  • CIE 1931 RGB: RBG values are converted to CIE XYZ coordiantes, and the method for computing the spectrum is then the same as for CIE 1931 Tristimulus values. 
  • D65 White: Defines X = 0.9505, Y = 1.0000, Z = 1.0890, which is the D65 white color on computer monitors.
  • Color Temperature: XYZ values that generate the same color as a blackbody at a particular temperature.
  • CIE 1976 Chromaticity u’v’: This source color model is essentially identical to the XYZ model above, except the normalized u’ and v’ chromaticity coordinates are used instead.

In all of the above cases, OpticStudio converts the input data into Tristimulus XYZ values. The same fitting algorithms are then used to determine the intensity distribution for the source over the inputted wavelength range. As described in the previous section, there are also sources in OpticStudio which require no fitting:

  • System Wavelengths: Relative intensities and wavelengths defined in Wavelength Data dialog box.
  • Uniform Power: Relative intensities are equal over all wavelengths.
  • Black Body: Rrelative intensities determined from true blackbody curve.
  • User Defined: Relative intensities and wavelengths given in a text file.

These sources may or may not generate an output that can be seen by the human eye, as a result. Nonetheless, in all these cases – except for System Wavelengths – the relative weightings and associated x-bar, y-bar, and z-bar values will be shown in the Prescription Data Report.

When System Wavelengths is chosen, then the wavelengths defined in the Wavelength Data dialog box are used for the source (default). In this case, the relative intensities are not provided in the Prescription Data Report, but they are of course listed in the Wavelength Data dialog box. If the source units are in Lumens, then the weights provided in the Wavelength Data dialog box are also in Lumens; this is the only time in which wavelength weights are not given in Watts.

KA-01431

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