Abstract
In recent years, inverse spectra were investigated with imaging optics and a quantitative description with radiometric units was suggested (Rang in Phänomenologie komplementärer Spektren, Logos, Berlin, 2015). It could be shown that inverse spectra complement each other additively to a constant intensity level. Since optical intensity in radiometric units is a power area density, it can be expected that energy densities of inverse spectra also fulfill an inversion equation and complement each other. In this contribution we report findings on a measurement of the power area density of inverse spectra for the near ultraviolet, visible and the infrared spectral range. They show the existence of corresponding spectral regions ultra-yellow and infra-cyan in the inverted spectrum and thereby present additional experimental evidence for equivalence of inverse spectra beyond the visible range.
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Notes
Optical contrasts, which merge into one another by reversing their luminance, are called inverse. Then \( L\left( {x, \vartheta , \varphi } \right) + \bar{L}\left( {x, \vartheta , \varphi } \right) = {\text{const}} \) holds true for the luminance L and the reversed luminance \( \bar{L} \) for all positions x with the same constant (Rang 2015). The inversion of spectra can be equivalently defined. To specify this definition for irradiance instead of luminance has the advantage of being consistent with the units of the measured values. Inverse spectra add up to a wavelength-independent, but not necessarily white irradiance standard. In the case of a broadband illumination, which includes at least the entire VIS region, the irradiance standard becomes white and one speaks of complementary spectra. Therefore, complementary spectra are a subclass of inverse spectra.
The conservation of radiation energy plays a key role also in the proof of Babinet's theorem. According to it the inversion of diffraction structures (as for example slit apertures) leads to identical diffraction images. The theorem applies only to diffractive structures as imaging elements. In contrast, slit apertures are not used as imaging elements but as imaged elements in spectroscopical applications. In the latter case inverse structures result in inverse images, irrespective of whether the aperture is imaged by a prism, a grating or an inverted grating.
After the present paper had been accepted for publication Jan Henrik Wold brought up an unpublished manuscript from André Bjerke's estate at the National Library in Oslo. This dated on the 15th of November 1957 and tells the results of qualitative measurements of the “temperature” in the “Ultra-Yellow” region as well as in the “Infra-Blue” region of the inverse spectrum. Although Bjerke's experiment and findings can not be reconstructed from the manuscript, it should be mentioned, that his idea to extend optical complementarity to more general radiation properties of the inverse spectrum beyond the optical region is rather close to the task of the present paper.
If the mirror slit aperture (Fig. 1)—and thus the effective contrast element—is replaced by a perfect mirror, the projection screen is at any point homogeneously illuminated and shows no wavelength dependency over the full measuring range. Besides the mirror slit aperture the mountings of the optical elements are also contrasts. However, these do not result in a spectral decomposition on the screen, if the positioning is correct in the concatenated beam path. Thus through the prism, the screen is illuminated just as it is with removed prism, except for the reflection losses at the glass surfaces (Rang 2015).
The wavelength allocation between detector position and spectral component was calculated by means of the Sellmeier-equations with coefficients of BK7 and the deviation angles at the prism and then calibrated by a known xenon peak at approximately 840 nm (Osram datasheet) and tested by measured positions of five known mercury emission lines.
The ultra-violet range of the spectrum is suppressed due to the lower reflection of the aluminum coatings in the ultra-violet. This is particularly noticeable here due to the fivefold reflection on the optical components (Fig. 1).
The extent of the visible spectrum can be estimated by means of the light sensitivity curve of the eye, which has its maximum for colour and daylight in the green spectral range at approximately 555 nm and decreases in the red and violet region of the spectrum. At 430 nm and at 690 nm it is only about 1% of its maximum value.
Due to the higher background and the increased irradiance of the inverse spectrum, a slightly higher typical standard deviation of the mean value of 1.04 nW (0.04 nW/mm2) is calculated here.
In both the ordinary and the inverse spectrum the background was not quite flat, so that the subtraction in both cases led to a small change of the spectral profiles. This is due to an inhomogeneous angular emission of the light source. The emission of the light source is affected at moderate opening angles by a decrease of the luminous area caused by perspective and minimal shadowing of the arc caused by the electrodes of the xenon lamp. Additionally the angular dependency of reflection loss at the prism and inhomogeneous properties of optical coatings may have an effect on the background. In the inverse spectrum, however, the illumination inhomogeneity was significantly higher in the measuring range.
Mathematically, an element b is called a (two-sided) inverse element to a when a · b = b · a = e, where e is the neutral element of the binary operation. In the present case of an additive operation \( \bar{E}^{\text{r}} \) is designated as an additive inverse element to \( E^{r} \) with the neutral element 0. From the commutativity of the additive operation it follows that \( E^{\text{r}} \) is also the additive inverse element to \( \bar{E}^{\text{r}} \). In addition to this mathematical analogy of inverse elements in sets and inverse spectra, the above inversion equation can also be derived with a radiometric formalism from optical considerations (Rang 2015).
References
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Acknowledgements
We thank Olaf Müller for encouraging us in the above investigation, Johannes Kühl and Oliver Passon for constructive discussions, David Auerbach and Laura Liska for language proofreading and the DAMUS-DONATA e.V. for the financial support that enabled us to realize the project.
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Rang, M., Grebe-Ellis, J. Power Area Density in Inverse Spectra. J Gen Philos Sci 49, 515–523 (2018). https://doi.org/10.1007/s10838-017-9394-8
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DOI: https://doi.org/10.1007/s10838-017-9394-8