4.7. Spectrum Classes

4.7.1. LightSpectrum

A LightSpectrum defines wavelength-dependent emittance of light, with all spectral values being \(\geq 0\). LightSpectrum objects are needed when creating a RaySource or to render spectral distribution of detectors.

4.7.1.1. Creating the Spectrum

Units

For line spectra (modes "Monochromatic" and "Lines") the spectral unit are Watts, while for all other modes the spectral power density is given as unit W/nm. Distribution and shape parameters (val, line_vals, ...) are given in the same units. For spectra used in raytracing, the absolute scaling is unimportant, as the values will be rescaled by the parent ray source, which specifies the overall power.

Constant

This mode defines a constant spectrum with value val:

spec = ot.LightSpectrum("Constant", val=12.3)

Monochromatic

This mode implements a monochromatic source with wavelength wl.

spec = ot.LightSpectrum("Monochromatic", wl=423.56, val=3)

Lines

A line spectrum consist of multiple monochromatic sources. The argument lines is a list of wavelengths, while line_vals is a list with the same number of elements describing the height/power of each wavelength.

spec = ot.LightSpectrum("Lines", lines=[458, 523, 729.6], line_vals=[0.5, 0.2, 0.1])

Rectangle

The following equations defines a spectrum with a rectangular function with bounds wl0, wl1 and a scaling factor val:

spec = ot.LightSpectrum("Rectangle", wl0=520, wl1=689, val=0.15)

Gaussian

A Gaussian spectrum is modelled mathematically with a scaling factor \(S_0\), a center wavelength \(\lambda_0\) and a standard deviation \(\sigma_\lambda\):

(4.18)\[S(\lambda) = S_0 \exp \left( -\frac{\left(\lambda - \lambda_0\right)^2}{2 \sigma_\lambda^2}\right)\]

The spectrum object is created with mode "Gaussian", a mean value mu and standard deviation sig, all given in nanometers. Note that the Gaussian function will be truncated to the visible range [380nm, 780nm].

spec = ot.LightSpectrum("Gaussian", mu=478, sig=23.5, val=0.89)

Blackbody Radiator

The spectral radiance of a blackbody according to Planck’s Law is given as: [1]

(4.19)\[B_\lambda (\lambda, ~T) = \frac{2 h c^2}{\lambda^5} \frac{1}{\exp\left(\frac{h c } {\lambda k_\text{B} T}\right) - 1}\]

The equation contains the speed of light \(c\), the Planck constant \(h\) and the Boltzmann constant \(k_\text{B}\):

(4.20)\[\begin{split}c =&~ 299792458 ~\text{m/s}\\ h =&~ 6.62607015\cdot 10^{-34} ~\text{J s}\\ k_\text{B} =&~ 1.380649 \cdot 10^{-23} ~\text{J/K}\\\end{split}\]

Note that \(\lambda\) must be specified in meters in the above equation.

Note

The spectral radiance \(B_\lambda\) (Power per solid angle, source area and wavelength) is given in units \(\text{W}/(\text{m}^3~\text{sr})\), whereas the units in this class should be \(\text{W/nm}\) (Power per wavelength). Since \(B_\lambda\) is constant over the source area and angle independent, converting it corresponds to a simple rescaling.

There is an option to normalize the spectrum, so the peak value equals one. This can prove useful for plotting the spectrum. If the peak wavelength is inside the visible range, then the Stefan–Boltzmann law can be applied to calculate the normalization factor. Otherwise the maximum value will lie at one of the edges of the visible range.

A blackbody radiator, following Planck’s law, with a specific temperature of T in Kelvin, is initialized as:

spec = ot.LightSpectrum("Blackbody", T=3890, val=2)

The val parameter defines the peak value in W/nm.

User Function/Data

With the Data/Function mode, the spectrum is modelled by a user function/ data set. With a dataset, the data will be linearly interpolated.

This function requires a wavelength array in nm as input and returns a numpy array of the same shape.

spec = ot.LightSpectrum("Function", func=lambda wl: np.arctan(wl - 520)**2)

If a function with multiple parameters is utilized, additional arguments can be provided in the func_args parameter dictionary.

spec = ot.LightSpectrum("Function", func=lambda wl, c: np.arctan(wl - c)**2, func_args=dict(c=489))

For discrete datasets, the "Data" mode proves useful. In this case the LightSpectrum constructor takes a wavelength array wls and a value array vals as arguments, where both must be of the exact same one-dimensional shape.

wls = np.linspace(450, 600, 100)
vals = np.cos(wls/500)

spec = ot.LightSpectrum("Data", wls=wls, vals=vals)

Note that wls needs to be monotonically increasing with the same step size and needs to be inside the visible range [380nm, 780nm].

Histogram

This spectrum type is not user created, but is rendered on a detector or source. It consists of a list of bins and bin values.

4.7.1.2. Calculating Spectral Values

The LightSpectrum object can be called with a wavelength array to calculate the spectral values:

>>> wl = np.linspace(400, 500, 5)
>>> spec(wl)
array([0.        , 0.        , 0.62160997, 0.58168242, 0.54030231])

4.7.1.3. Wavelength Characteristics

Table 4.4 Wavelength characteristics functions

Function	Unit	Meaning
peak_wavelength	nm	wavelength for the spectral peak
centroid_wavelength	nm	power-weighted average wavelength, see Centroid Wavelength
fwhm	nm	full-width-at-half-maximum wavelength range
dominant_wavelength	nm	same hue wavelength, see Dominant Wavelength np.nan if not existent
complementary_wavelength	nm	opposite hue wavelength, see Dominant Wavelength np.nan if non-existent

For instance, we can calculate the peak wavelength of the LED B1 standard illuminant by doing:

>>> spec = ot.presets.light_spectrum.led_b1
>>> spec.peak_wavelength()
605.00225...

Note that with multiple same height peaks or a broad constant peak region the first peak value will be returned.

The centroid wavelength for the spectrum is:

>>> spec.centroid_wavelength()
592.39585...

The dominant wavelength is calculated using:

>>> spec.dominant_wavelength()
584.75088...

When dominant or complementary are not existent (e.g. magenta can’t be described by a wavelength), the values are set to NaN (not a number). You can find a visualization on both dominant and complementary wavelengths on this Wiki page.

The FWHM (full width at half maximum) is calculated usin:

>>> spec.fwhm()
129.18529...

The method calculates the smallest FWHM around the highest peak. While it is possible to calculate this value for all spectral shapes, it is only meaningful as width characterization for functions with a distinctive peak and an outward fall-off. For instance this metric does not make sense for a spectrum consisting of multiple separated bell-shaped curves.

4.7.1.4. Power

The spectral power can be calculated with:

>>> spec.power()
3206.9749...

And the luminous power in lumen units with:

>>> spec.luminous_power()
999886.86...

4.7.1.5. Rendering a LightSpectrum

Read section Rendering a RenderImage for details on rendering images, rendering spectra is done in a similar way. Analogously to rendering a source image, we can render a spectrum with source_spectrum and by providing a source_index parameter (default to zero). With a raytracer object RT, a source spectrum from source 1 is rendered with:

spec = RT.source_spectrum(source_index=1)

For a detector spectrum the detector_spectrum function is applied. It takes a detector_index argument, that defaults to zero.

spec = RT.detector_spectrum(detector_index=0)

Additionally we can render only a specific source by providing a source_index or limit the detector area by providing the extent parameter, as we did for the detector_image method.

spec = RT.detector_spectrum(detector_index=0, source_index=1, extent=[0, 1, 0, 1])

The above methods return a LightSpectrum object with type spectrum_type="Histogram".

4.7.2. TransmissionSpectrum

The Filter class requires a TransmissionSpectrum. All relative transmission values need to be inside the [0, 1] range. The TransmissionSpectrum provides less modes than the LightSpectrum class. Compared to the latter, the scaling factor vall now becomes important. This class defines a new inverse parameter, that subtracts the defined function from a value of one. This has the effect of turning the transmittance behavior into absorptance. A Gaussian bandpass becomes a notch filter, a rectangular bandpass a rectangular blocking filter.

Constant

A neutral density filter is defined with mode "Constant" and the linear transmittance value.

spec = ot.TransmissionSpectrum("Constant", val=0.5)

Gaussian

Colored filters (most commonly bandpass filters) can be created with a Gaussian function.

spec = ot.TransmissionSpectrum("Gaussian", mu=550, sig=30, val=1)

A Gaussian notch filter is easily defined with parameter inverse=True.

spec = ot.TransmissionSpectrum("Gaussian", mu=550, sig=30, val=1, inverse=True)

Rectangle

A rectangular pass filter is modelled by a rectangular function.

spec = ot.TransmissionSpectrum("Rectangle", wl0=500, wl1=650, val=0.1)

A rectangular blocking filter can be defined with inverse=True.

spec = ot.TransmissionSpectrum("Rectangle", wl0=500, wl1=650, inverse=True)

Creating an edgepass filter becomes easy by setting the bound to the edge of the visible range.

spec = ot.TransmissionSpectrum("Rectangle", wl0=500, wl1=780)

User Data/Function

Creating a TransmissionSpectrum with discrete data is equivalent to a LightSpectrum. However, all function/data values need to be inside the range [0, 1].

Getting Spectral Values

As for the LightSpectrum object, we can get the spectral values with:

>>> wl = np.linspace(400, 550, 5)
>>> spec(wl)
array([0., 0., 0., 1., 1.])

4.7.3. Spectrum

Spectrum is the parent class of both LightSpectrum and TransmissionSpectrum. It defines the following modes: "Monochromatic", "Rectangle", "List", "Function", "Data", "Gaussian", "Constant". Compared to LightSpectrum, only modes "Histogram" and "Blackbody" are missing. Generally the Spectrum class is not exposed to the user. But, for instance, the color matching functions ot.presets.spectrum.x, ot.presets.spectrum.y, ot.presets.spectrum.z are objects of this type.

4.7.4. Plotting

See Spectrum Plotting.

4.7.5. Spectral Lines Presets

optrace provides some spectral wavelength lines in its presets.

Table 4.5 Fraunhofer lines commonly used for Abbe number determination [2]

Name	Wavelength in nm	Element	Color
h	404.6561	Hg	violet
g	435.8343	Hg	blue
F’	479.9914	Cd	blue
F	486.1327	H	blue
e	546.0740	Hg	green
d	587.5618	He	yellow
D	589.2938	Na	yellow
C’	643.8469	Cd	red
C	656.272	H	red
r	706.5188	He	red
A’	768.2	K	IR-A

Due to limitations in python variable names, presets with a trailing apostrophe are named with an trailing underscore. For instance, F’ is named F_.

>>> ot.presets.spectral_lines.F_
479.9914

The most common wavelength combinations for Abbe numbers are FdC, FDC, FeC and F’eC’.

>>> ot.presets.spectral_lines.F_eC_
[479.9914, 546.074, 643.8469]

The following table provides the dominant wavelengths of the sRGB primaries (ITU-R BT.709). Dimensioning the scaling factors in the provided way produces D65 sRGB-white for equal R, G, B mixing ratios.

Table 4.6 Dominant wavelengths of sRGB primaries. Derived by optimization.

Name	Wavelength in nm	Scaling Factor
R	611.2826	0.5745000
G	549.1321	0.5985758
B	464.3118	0.3895581

These wavelengths are useful for simulating color mixing.

>>> ot.presets.spectral_lines.rgb
[464.3118, 549.1321, 611.2826]

4.7.6. Spectrum Presets

The following figures demonstrate the predefined presets for Spectrum and LightSpectrum.

Fig. 4.27 CIE standard illuminants. Available as ot.presets.light_spectrum.<name> with a, d50, ... as <name>	Fig. 4.28 CIE standard illuminants LED series. Available as ot.presets.light_spectrum.<name> with led_b1, led_b2, ... as <name>
Fig. 4.29 CIE standard illuminants Fluorescent series. Available as ot.presets.light_spectrum.<name> with fl2, fl7, ... as <name>	Fig. 4.30 Possible sRGB primary spectra. Available as ot.presets.light_spectrum.<name> with srgb_r, srgb_g, ... as <name>
Fig. 4.31 CIE color matching functions. Available as ot.presets.spectrum.<name> with x, y, z as <name>

Other presets include spectra from spectral lines combination in Section 4.7.5. Namely ot.presets.light_spectrum.<name> with FdC, FDC, FeC, F_eC_, rgb as <name>.

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References

[1]

Wikipedia. Planck’s law. 2022. URL: https://en.wikipedia.org/wiki/Planck%27s_law (visited on 2022-12-01).

[2]

Wikipedia. Abbe-Zahl. 2022. URL: https://de.wikipedia.org/wiki/Abbe-Zahl (visited on 2022-12-16).