This module estimates a monochromatic-equivalent intensity — not the real measured intensity —
that would produce the same total photon absorption in a material as the actual broadband light source.
It converts a measured total irradiance into an effective wavelength-specific value \( I_\text{target} \) (mW/cm²),
as if all absorbed photons came from a single wavelength \( \lambda_0 \).
Purpose
Light sources emit over broad spectral ranges, and both detectors and materials respond differently with wavelength.
The Spectral Weighted Intensity bridges that gap by estimating the equivalent monochromatic light intensity that
would deposit the same number of absorbed photons as the broadband source.
It accounts for:
1) the lamp’s emission spectrum \( I(\lambda) \),
2) the detector’s photon or energy response, and
3) the material’s wavelength-dependent absorptivity \( \varepsilon(\lambda) \).
Concept
The measured total irradiance \( mInt \) represents light distributed across wavelengths.
The model applies a spectral correction factor to convert this broadband value into an equivalent monochromatic intensity at \( \lambda_0 \):
\( I_\text{target} = mInt \times f_\text{intensity}(\lambda_0) \times f_\text{absorbance}(\lambda_0) \)
Here, \( I_\text{target} \) is
not the real irradiance at that wavelength,
but the
single-wavelength intensity that would yield the same total photon absorption rate in the sample.
1) Intensity Correction
The correction depends on whether the detector measures
photon flux or
energy flux:
(a) Photon-counting detector (e.g., photodiode):
Photodiodes respond to the
number of incident photons, not their energy.
Each absorbed photon ideally produces a single electron–hole pair, so the current is proportional to photon flux \( \Phi(\lambda) = I(\lambda) \, \lambda / hc \).
Because longer wavelengths carry less energy per photon, the same optical power produces
more photons at longer wavelengths.
The appropriate correction therefore weights by \( 1/\lambda \):
\( f_\text{intensity}(\lambda_0)
= \lambda_0
\frac{\displaystyle \int I(\lambda)/\lambda \, d\lambda}
{\displaystyle \int I(\lambda) \, d\lambda} \)
(b) Energy-measuring detector (thermal, thermopile, bolometer):
Thermal sensors respond to the
total absorbed energy as heat, regardless of photon energy.
They integrate power directly across wavelengths, giving an effectively flat spectral response (neglecting coating reflectivity).
Hence no photon weighting is required:
\( f_\text{intensity}(\lambda_0) = 1 \)
Justification for separate cases:
Photodiodes and thermal sensors fundamentally measure different physical quantities.
A photodiode converts
photon number to current (A ∝ photons/s), while a thermal detector converts
energy flux to temperature rise (W/cm²).
Because the photon energy \( E_\text{photon} = hc/\lambda \) decreases with wavelength,
a broadband lamp will appear relatively stronger to a photodiode than to a thermal sensor at longer wavelengths.
The model therefore applies distinct spectral corrections consistent with each detector’s underlying physics.
When intensity correction matters most:
For smooth, narrow spectra (e.g., a narrow LED), this correction has only a minor effect.
However, for
multi-peaked or discontinuous spectra — such as the
mercury arc lamp, which emits sharp lines at 405, 436, 546 nm, etc. —
the relative photon weighting can shift the effective intensity substantially.
In such cases, the correction is
critical for meaningful comparison between detectors or for matching experimental and simulated photon absorption.
Note: This neglects the
wavelength-dependent quantum yield (responsivity) of the detector,
which in practice can be significant.
However, such data are rarely accessible, so a flat responsivity is assumed for simplicity.
2) Absorbance Correction
Two spectra with the same total irradiance can drive different absorption rates depending on how light overlaps the
material’s absorbance.
This correction compares the absorbed photon flux for the actual spectrum versus a narrow monochromatic source:
\( f_\text{absorbance}(\lambda_0)
=
\frac{
\displaystyle \int I(\lambda)\,\varepsilon(\lambda)\,d\lambda
}{
\displaystyle \int G_{\lambda_0}(\lambda)\,\varepsilon(\lambda)\,d\lambda
} \)
where \( G_{\lambda_0}(\lambda) \) is a narrow Gaussian (FWHM ≈ 0.1 nm) centered at \( \lambda_0 \).
Result
The
Spectral Weighted Intensity is:
\( I_\text{target}(\lambda_0)
= mInt \times
\lambda_0
\frac{\displaystyle \int I(\lambda)/\lambda \, d\lambda}{\displaystyle \int I(\lambda)\,d\lambda}
\times
\frac{
\displaystyle \int I(\lambda)\,\varepsilon(\lambda)\,d\lambda
}{
\displaystyle \int G_{\lambda_0}(\lambda)\,\varepsilon(\lambda)\,d\lambda
} \)
This represents the
monochromatic-equivalent intensity that would produce the
same total photon absorption in the material as the real broadband source.
It should
never be interpreted as the actual irradiance at that wavelength,
but rather as a modeling construct useful for comparing spectral power distributions on an equivalent photon-absorption basis.
Adjustable parameters
Measured intensity (\( mInt \)) – total irradiance from experiment or datasheet (mW/cm²)
Target wavelength (\( \lambda_0 \)) – wavelength of interest (nm)
Detector type – photon-counting (photodiode) or energy-measuring (thermal)
Light source spectrum – user-defined emission shape \( I(\lambda) \)
Absorbance spectrum – material-specific molar absorptivity \( \varepsilon(\lambda) \)