Virtual tunnels and green glass: The colors of common mirrors

M. V. Klein and T. E. Furtak, Optics, 2nd ed. (Wiley, New York, 1986), pp. 263–307.

R. S. Berns, Billmeyer and Saltzman’s Principles of Color Technology, 3rd ed. (Wiley, New York, 2000), pp. 44–66.

Parque de las Ciencias, Avda. del Mediterráneo s/n, 18006 Granada, Spain, 〈http://www.parqueciencias.com〉.

Thus in Fig. 2, the ascending “staircase” of floor images arises because the mirrors’ tops are closer than their bases. The mirror tunnel’s curvature is useful in itself for discussing the geometrical optics of plane mirrors and the behavior of their virtual images.

B. Schweig, Mirrors: A Guide to the Manufacture of Mirrors and Reflecting Surfaces (Pelham Books, London, 1973), pp. 3–4, 23–77.

H. G. Pfaender and H. Schroeder, Schott Guide to Glass (Van Nostrand Reinhold, New York, 1983), pp. 45, 66.

T.

Uchino

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K.

Nakaguchi

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Y.

Nagashima

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T.

Kondo

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M.

Rubin

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Handbook of Optical Constants of Solids, edited by E. D. Palik (Academic, Orlando, FL, 1985), Table IX, p. 356.

CM-2022 spectrophotometer from Minolta Corporation, 101 Williams Drive, Ramsey, NJ 07446;

HunterLab UltraScan spectrophotometer from HunterLab Associates, Inc., 11491 Sunset Hills Road, Reston, VA 20190-5280. Both instruments were calibrated shortly before we measured the spectra in Fig. 4.

Although the equal-energy illuminant is rarely encountered, we use its spectral simplicity to help us isolate the colorimetric effects of $Rλ$ in creating the mirror tunnel’s colors. For more realistic illuminants (for example, incandescent lighting), the mirror tunnel’s chromaticities depend on the wavelength-by-wavelength multiplication of $Rλ$ and a more complicated power spectrum.

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, New York, 1982), pp. 306–310. Here we follow convention and set the just-noticeable difference equal to the semimajor axis length of the MacAdam color-matching ellipse at the relevant chromaticity. For the equal-energy illuminant, this chromaticity is CIE 1931 $x=0.333 33,$ $y=0.333 33.$ We define chromaticity coordinates and other basic CIE metrics elsewhere in the main text.

Dominant wavelengths and purities are measured with respect to the equal-energy illuminant’s chromaticity. Here we define a white object as a diffuse reflector with constant $Rλ$ at visible wavelengths, say $Rλ=0.9;$ a photographer’s white card approximates this behavior.

Because each of the ∼12 reflected gray-card images visible in Fig. 2 occurs after one pair of reflections between the two mirrors, $N=24$ total reflections contribute to the color and brightness of its smallest gray card image. To measure the relative luminance (or radiance) of the $N=24$ mirror reflection, we restrict the spectroradiometer’s field of view to the 12th smallest image of some easily recognized object in the mirror tunnel (for example, the gray card in Fig. 2). Then we divide this reading by the luminance or radiance of the object seen directly (the $N=0$ image). Given the large luminance reductions involved, $N=50$ reflections is a generously large upper limit on the visibility of mirror tunnels formed by common mirrors.

On request, we will gladly provide readers with as many of our measured mirror spectra as desired, plus instructions on how to use this data in colorimetric calculations. We also can give readers smoothly interpolated complex refractive indices of glasses and metals in a form that is suitable for simulating mirrors’ chromaticities (see the model of Sec. IV).

The relationship between $Lν$ and N is not quite log-linear because luminance is calculated by a spectral integral that includes the changing shape of the mirrors’ $Rλ,N.$ If a mirror’s original spectral reflectances $Rλ,1$ were completely uniform, then both the reflected luminance and radiance would decrease exactly logarithmically with increasing N.

J.

Hernández-Andrés

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E. M.

Valero

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J. L.

Nieves

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J.

Romero

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For the complex refractive indices of pure silica, see Ref. 9, Table X, pp. 759–760.

We use data on pure silica’s very weak visible-wavelength absorption found in

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Keck

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R. D.

Maurer

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Schultz

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For the complex refractive indices of metallic aluminum, see Ref. 9, Table XII, pp. 398–399.

Cavity-Ringdown Spectroscopy: An Ultratrace-Absorption Measurement Technique, edited by K. W. Busch and M. A. Busch (American Chemical Society, Washington, DC, 1999).