Mirror tunnel photographed at the Science Museum in Granada, Spain. The large white circle framing the mirror tunnel is one of Fig. 1’s eyeholes, which also appear as small dark circles within the tunnel itself. 

Contexts in source publication

Context 1

... the centuries, mirrors have given us windows onto unusual worlds. Some of these worlds are as familiar as they are fundamentally odd: the one that lets us see our own face or the one that shows us a transposed twin of the room in which we stand. Yet the oddest mirror world may be the one found between two plane mirrors that face each other. If these mirrors are parallel ͑ or nearly so ͒ , then looking into either of them brings us to the vertiginous edge of a visual tunnel that seems to recede endlessly into the virtual distance. We call this phenomenon of repeated mirror reflections a mirror tunnel , and here we are interested in how one formed by two common mirrors transforms the colors of reflected objects. ͑ We define common mirrors as inexpensive second- surface plane mirrors such as those found in homes. ͒ Under- standing these transformations can teach students not only about geometrical optics and spectral transfer functions, but also how knowing the complex refractive indices of the metal backing and glass in such mirrors lets us predict their reflectance spectra via the Fresnel equations. 1 In a student experiment, two especially instructive ͑ and surprising ͒ results are that common mirrors are not spectrally neutral reflectors, and their reflectance spectra depend on the absorbing and reflecting properties of both the glass substrate and its metal backing. Because mirror tunnels are so visually compelling, we use them as a pedagogical tool in this paper. In Sec. II we de- scribe how to set up a mirror tunnel and observe its color shifts, and in Sec. III we discuss the measured reflectance spectra of some common mirrors and show how we use such spectra to calculate the chromaticity trends seen in mirror tunnels. In Sec. IV we use existing data on the complex refractive indices of silver and soda–lime silica glasses in a simple model that calculates the reflectance spectra and chromaticities for these mirrors. At a minimum, students should already be familiar with the optical significance of the mate- rials’ absorption, reflection, and transmission spectra. For deeper insights, they also should understand how such purely physical properties are translated into the psychophysical metric of colorimetry. A very readable introduction to the colorimetric system of the Commission Internationale de l’Eclairage ͑ CIE ͒ is given in Ref. 2. One well-designed mirror tunnel is seen in Granada, Spain’s Science Museum. 3 This virtual tunnel is produced simply by placing two tall mirrors parallel to and facing each other ͑ see Fig. 1 ͒ . Viewers look at one mirror through two small holes in the back of the other mirror; the holes are spaced to match the average distance between human eyes. This viewing geometry avoids the problem that can occur if the viewer’s head is between the two mirrors, where it can block the angularly smallest, most interesting regions of the mirror tunnel. On looking through the eyeholes, one sees a long progres- sion of ever-smaller reflected images of the two mirrors that plunges into the virtual distance, thus giving the illusion of an infinite tunnel ͑ Fig. 2 ͒ . If the mirrors are not exactly parallel ͑ as is true in Fig. 2 ͒ , then the tunnel curves in the same direction that the mirrors tilt toward each other. 4 After relish- ing the vertiginous thrill of this virtual tunnel, we find an- other, subtler feature: the farther we look into the tunnel, the greener and darker its reflected objects appear. To track this color and brightness shift in Fig. 2, we put a photographer’s gray card at the base of one mirror. Although the card’s image becomes progressively darker and more yellow–green with successive reflections, no such shifts occur for distant real objects seen by a single reflection in either mirror. This simple observation lies at the heart of our paper, and it begs the question that students must answer: why do repeated reflections change the mirror images’ color and brightness? Geometrical optics is silent on this point, because it predicts only the images’ location and size. Thus, to explain the brightness and color changes, we must consider the physical optics of common mirrors. Because color depends on spectral variability, we start by examining the spectral reflectances and transmissivities of the metal and glass used to make these mirrors. Common household mirrors have long been made by de- positing a thin film of crystalline silver on the rear surface of float-process flat glass. 5 This silver film is optically thick ͑ minimum thickness ϳ 10 ␮ m ͒ , and its rear surface is pro- tected from oxidation and abrasion by successive films of copper and lacquer. The polished glass itself is usually clear soda–lime silica whose constituents ͑ by weight ͒ are about 72% SiO 2 , 14% Na 2 O ( ϩ K 2 O), 9% CaO, 3% MgO, 1% Al 2 O 3 , and variable, smaller amounts of Fe 2 O 3 ͑ Ͻ 1% ͒ . 6,7 Other backing metals ͑ for example, aluminum ͒ and special ‘‘colorless’’ ͑ that is, lower iron oxide content ͒ or tinted glass can be used to change a mirror’s reflectance spectrum appre- ciably. Nonetheless, a soda–lime silica glass substrate with silver backing forms the optical core of most common mirrors. Figure 3 shows the reflectance and transmission spectra of these two materials in air at visible wavelengths ␭ (380 р␭ р 780 nm). 8,9 Although the glass transmits most at 510 nm ͑ a green ͒ , its broad transmission maximum only imparts a pastel greenish cast to white light, even after it has been transmitted through several centimeters of the glass. This greenish cast is best seen by looking obliquely through a plate of common glass, bearing in mind that refractive dis- persion can complicate the colors that students see. In other words, students should learn to distinguish between the uniform green or blue–green that results from glass absorption and the many prismatic colors that result from glass refrac- tion. In contrast, although Fig. 3’s silver spectrum shows a fairly steep decrease in spectral reflectance R ␭ below 460 nm, at longer wavelengths the metal reflects much more spectrally uniformly. When we form a composite common mirror from these two materials, what reflectance spectra result? Figure 4 shows some representative spectra both from Granada’s Science Museum mirrors and from common mirrors in our own laboratories and homes (400 р␭р 700 nm, measured in 10 or 5 nm steps ͒ . We measured these spectra at normal incidence using either of two portable spectrophotometers: a Minolta CM-2022 or a HunterLab UltraScan. 10 In Fig. 4, all spectral reflectances except those labeled ‘‘ R ␭ ͑ lab 1 ͒ ’’ were measured with the Minolta CM-2022. Given the instruments’ designs, these spectra necessarily include both external specular reflections from the mirror glass and multiple reflections from within the mirror. One of Fig. 4’s most striking features is how much the overall reflectance R varies from mirror to mirror ͑ and, as our measurements show, even within a given mirror ͒ ; mean R values range from 80.5% for the ‘‘museum 1’’ mirror to 95.6% for the ‘‘home 2’’ mirror. Although some of this variability may be attributed to spectrophotometer errors, more variability likely comes from differences in the mirrors’ original fabrication. Nevertheless, Fig. 4’s spectra all have similar shapes: they peak on average at 545 nm, a yellowish green. To restate these spectral differences in colorimetric terms, after a single reflection of a spectrally flat illuminant ͑ the equal-energy spectrum ͒ , 11 the most dissimilar mirrors in Fig. 4 ͑ the museum 1 and home 2 mirrors ͒ slightly change that illuminant’s color into two new colors. However, these two colors differ by only 14% more than the local MacAdam just-noticeable difference. In other words, after a single reflection, although many observers might be able to distinguish between colors reflected by Fig. 4’s most dissimilar mirrors when viewed side by side, this is by no means certain. Thus, the color shifts caused by Fig. 4’s very different-looking reflectance spectra will appear nearly identical. Yet each mirror’s color shift is by itself only marginally perceptible. Using the same equal-energy illuminant as before, we find that after one reflection, even the most spectrally selective mirror in Fig. 4 ͑ the museum 1 mirror ͒ increases that light’s colorimetric purity by Ͻ 3% or ϳ 2.7 just- noticeable differences. More significantly for the mirror tunnel, the dominant wavelength of this low-purity reflected ...

Context 2

... The 50th reflection has a much weaker and spectrally narrower reflectance spectrum than the first reflection. Obviously, this makes colors in the mirror tunnel’s most ‘‘distant’’ images darker and more saturated. In particular, 50 reflections by the museum 1 mirror give a white object a dominant wavelength ϳ 552 nm and colorimetric purity ϳ 71%, 13 compared with the 3% purity resulting from one reflection. Equally dramatic is the fact that the white object’s reflected luminance is reduced by a factor of 5780 after 50 reflections. 14 Between these extremes, the mirror tunnel’s luminances and chromaticities follow easily predicted, although dis- tinctly nonlinear, paths. To provide a context for these chromaticities, Fig. 6 shows the entire CIE 1931 x, y chromaticity diagram, 2 to which we have added a dashed chromaticity locus for blackbody or Planckian radiators. The dimension- less CIE 1931 chromaticity coordinates x, y are derived from psychophysical experiments in which subjects matched the colors of test lights by mixing red, green, and blue reference lights in varying intensities. To a first approximation, the x coordinate represents the relative amounts of green and red in a color, and the y coordinate indicates the relative amounts of green and blue. The chromaticity diagram’s curved border in Fig. 6 con- sists of monochromatic spectrum colors that increase clock- wise from short wavelengths at the diagram’s left vertex ͑ near x ϭ 0.175, y ϭ 0.0) to long wavelengths at its right vertex ͑ near x ϭ 0.735, y ϭ 0.265). This curved part of the border is called the spectrum locus . A straight line connects these two spectral extremes, and mixtures of monochromatic red and blue along it generate purples. Thus, purples are not spectrum colors proper ͑ they are not monochromatic ͒ , but they are the purest possible colors that link the ends of the visible spectrum. One point in Fig. 6’s interior is a white or achromatic stimulus , and for the equal-energy illuminant its chromaticity coordinates are x ϭ 0.333 33, y ϭ 0.333 33. We can form any other color by additively mixing a specific amount of this achromatic stimulus with a spectrum color ͑ or a purple ͒ . An arbitrary color’s dominant wavelength is found by extending a straight line from the achromatic stimulus through the arbitrary color and then on to intersect the spectrum locus. The monochromatic spectrum color at this inter- section defines the arbitrary color’s dominant wavelength. Furthermore, the arbitrary color’s fractional distance between white and its spectrum color defines its colorimetric purity . Figure 7 expands the boxed area in Fig. 6, and in it we plot chromaticity curves for a white object seen after N mirror- to-mirror reflections by the mirrors in Fig. 4. Each curve in Fig. 7 is constructed by repeatedly drawing from the chromaticity coordinates x N , y N for the R N reflection to x N ϩ 1 , y N ϩ 1 for the R N ϩ 1 reflection (1 р N р 50). We calculate each x N , y N by multiplying the equal-energy illuminant’s power spectrum by a white object’s constant R ␭ and the mirrors’ changing R ␭ , N . To show the maximum color range for mirror tunnels resulting from our measured spectra, we substi- tute in Fig. 7 a slightly more greenish common mirror ͑ ‘‘home 4’’ ͒ for the ‘‘lab 1’’ mirror in Fig. 4. Note that all colorimetric calculations given here can be done easily in an electronic spreadsheet that contains mirror spectra and the human color-matching functions. 15 As expected, Fig. 7’s chromaticities grow steadily purer with increasing N . However, for the largest N , the most distant colors in the tunnel are difficult ͑ or impossible ͒ to see because they are so dark. In fact, Fig. 8 shows that the reflected luminance L ␯ decreases nearly logarithmically with N , as one would expect from Eq. ͑ 1 ͒ . 16 Figure 8 does not include the ‘‘home 4’’ L ␯ shown in Fig. 7 because they would overlie the L ␯ of museum 1. Note that after 50 reflections the latter mirror not only yields the darkest colors ͑ Fig. 8 ͒ , but also the purest ones ͑ Fig. 7 ͒ . This decreased brightness is no coincidence, because increased spectral purity necessarily comes at the price of reduced luminance for nonfluo- rescing reflectors ͑ see Fig. 4 ͒ . One of the most instructive and satisfying aspects of the mirror–tunnel exercise for students is that they can simulate a fairly complex optical system with only modest mathemati- cal effort. We start with the facts that a second-surface mirror’s spectral reflectance R ␭ depends on the mirror surface’s preparation, the illuminant’s incidence angle and polarization state, the glass thickness and composition, and the optical properties of the metal itself. We then base our model on the following assumptions. ͑ 1 ͒ The mirror is a polished flat substrate of soda–lime silica (SiO 2 ) glass that is 5 mm thick and is backed by an optically thick silver ͑ Ag ͒ film. This glass matches the thickness of the mirror glass shown in Fig. 2. ͑ 2 ͒ We assume normal incidence for all reflections. Strictly speaking, the eyeholes in an actual mirror tunnel ͑ see Figs. 1 and 2 ͒ keep us from seeing normal-incidence reflections between the two mirrors, but this fact only negligibly affects the relevance of the color and luminance trends simulated here. ͑ 3 ͒ We include the effects on R ␭ of 5 reflections within each mirror ͑ see Fig. 9 ͒ ; higher-order internal reflections contribute almost nothing to R ␭ . However, we exclude in- terference among the internally reflected rays because the mirror glass is sufficiently thick ͑ϳ 8620 times the mean vis- ible so that its optical pathlengths are far greater than the illuminant’s coherence length. 17 Thus we add the intensities, rather than the amplitudes, of successive internal reflections. ͑ 4 ͒ As was true in Sec. III, we assume an unpolarized, equal-energy illuminant. During the light’s traversal of the mirror, it variously undergoes absorption within a medium or reflection and transmission at an interface between two media. Note that all the following equations are implicit functions of wavelength. At the air–glass interface, external specular reflection r 0 is given by the normal-incidence Fresnel equation: ...

Context 3

... the centuries, mirrors have given us windows onto unusual worlds. Some of these worlds are as familiar as they are fundamentally odd: the one that lets us see our own face or the one that shows us a transposed twin of the room in which we stand. Yet the oddest mirror world may be the one found between two plane mirrors that face each other. If these mirrors are parallel ͑ or nearly so ͒ , then looking into either of them brings us to the vertiginous edge of a visual tunnel that seems to recede endlessly into the virtual distance. We call this phenomenon of repeated mirror reflections a mirror tunnel , and here we are interested in how one formed by two common mirrors transforms the colors of reflected objects. ͑ We define common mirrors as inexpensive second- surface plane mirrors such as those found in homes. ͒ Under- standing these transformations can teach students not only about geometrical optics and spectral transfer functions, but also how knowing the complex refractive indices of the metal backing and glass in such mirrors lets us predict their reflectance spectra via the Fresnel equations. 1 In a student experiment, two especially instructive ͑ and surprising ͒ results are that common mirrors are not spectrally neutral reflectors, and their reflectance spectra depend on the absorbing and reflecting properties of both the glass substrate and its metal backing. Because mirror tunnels are so visually compelling, we use them as a pedagogical tool in this paper. In Sec. II we de- scribe how to set up a mirror tunnel and observe its color shifts, and in Sec. III we discuss the measured reflectance spectra of some common mirrors and show how we use such spectra to calculate the chromaticity trends seen in mirror tunnels. In Sec. IV we use existing data on the complex refractive indices of silver and soda–lime silica glasses in a simple model that calculates the reflectance spectra and chromaticities for these mirrors. At a minimum, students should already be familiar with the optical significance of the mate- rials’ absorption, reflection, and transmission spectra. For deeper insights, they also should understand how such purely physical properties are translated into the psychophysical metric of colorimetry. A very readable introduction to the colorimetric system of the Commission Internationale de l’Eclairage ͑ CIE ͒ is given in Ref. 2. One well-designed mirror tunnel is seen in Granada, Spain’s Science Museum. 3 This virtual tunnel is produced simply by placing two tall mirrors parallel to and facing each other ͑ see Fig. 1 ͒ . Viewers look at one mirror through two small holes in the back of the other mirror; the holes are spaced to match the average distance between human eyes. This viewing geometry avoids the problem that can occur if the viewer’s head is between the two mirrors, where it can block the angularly smallest, most interesting regions of the mirror tunnel. On looking through the eyeholes, one sees a long progres- sion of ever-smaller reflected images of the two mirrors that plunges into the virtual distance, thus giving the illusion of an infinite tunnel ͑ Fig. 2 ͒ . If the mirrors are not exactly parallel ͑ as is true in Fig. 2 ͒ , then the tunnel curves in the same direction that the mirrors tilt toward each other. 4 After relish- ing the vertiginous thrill of this virtual tunnel, we find an- other, subtler feature: the farther we look into the tunnel, the greener and darker its reflected objects appear. To track this color and brightness shift in Fig. 2, we put a photographer’s gray card at the base of one mirror. Although the card’s image becomes progressively darker and more yellow–green with successive reflections, no such shifts occur for distant real objects seen by a single reflection in either mirror. This simple observation lies at the heart of our paper, and it begs the question that students must answer: why do repeated reflections change the mirror images’ color and brightness? Geometrical optics is silent on this point, because it predicts only the images’ location and size. Thus, to explain the brightness and color changes, we must consider the physical optics of common mirrors. Because color depends on spectral variability, we start by examining the spectral reflectances and transmissivities of the metal and glass used to make these mirrors. Common household mirrors have long been made by de- positing a thin film of crystalline silver on the rear surface of float-process flat glass. 5 This silver film is optically thick ͑ minimum thickness ϳ 10 ␮ m ͒ , and its rear surface is pro- tected from oxidation and abrasion by successive films of copper and lacquer. The polished glass itself is usually clear soda–lime silica whose constituents ͑ by weight ͒ are about 72% SiO 2 , 14% Na 2 O ( ϩ K 2 O), 9% CaO, 3% MgO, 1% Al 2 O 3 , and variable, smaller amounts of Fe 2 O 3 ͑ Ͻ 1% ͒ . 6,7 Other backing metals ͑ for example, aluminum ͒ and special ‘‘colorless’’ ͑ that is, lower iron oxide content ͒ or tinted glass can be used to change a mirror’s reflectance spectrum appre- ciably. Nonetheless, a soda–lime silica glass substrate with silver backing forms the optical core of most common mirrors. Figure 3 shows the reflectance and transmission spectra of these two materials in air at visible wavelengths ␭ (380 р␭ р 780 nm). 8,9 Although the glass transmits most at 510 nm ͑ a green ͒ , its broad transmission maximum only imparts a pastel greenish cast to white light, even after it has been transmitted through several centimeters of the glass. This greenish cast is best seen by looking obliquely through a plate of common glass, bearing in mind that refractive dis- persion can complicate the colors that students see. In other words, students should learn to distinguish between the uniform green or blue–green that results from glass absorption and the many prismatic colors that result from glass refrac- tion. In contrast, although Fig. 3’s silver spectrum shows a fairly steep decrease in spectral reflectance R ␭ below 460 nm, at longer wavelengths the metal reflects much more spectrally uniformly. When we form a composite common mirror from these two materials, what reflectance spectra result? Figure 4 shows some representative spectra both from Granada’s Science Museum mirrors and from common mirrors in our own laboratories and homes (400 р␭р 700 nm, measured in 10 or 5 nm steps ͒ . We measured these spectra at normal incidence using either of two portable spectrophotometers: a Minolta CM-2022 or a HunterLab UltraScan. 10 In Fig. 4, all spectral reflectances except those labeled ‘‘ R ␭ ͑ lab 1 ͒ ’’ were measured with the Minolta CM-2022. Given the instruments’ designs, these spectra necessarily include both external specular reflections from the mirror glass and multiple reflections from within the mirror. One of Fig. 4’s most striking features is how much the overall reflectance R varies from mirror to mirror ͑ and, as our measurements show, even within a given mirror ͒ ; mean R values range from 80.5% for the ‘‘museum 1’’ mirror to 95.6% for the ‘‘home 2’’ mirror. Although some of this variability may be attributed to spectrophotometer errors, more variability likely comes from differences in the mirrors’ original fabrication. Nevertheless, Fig. 4’s spectra all have similar shapes: they peak on average at 545 nm, a yellowish green. To restate these spectral differences in colorimetric terms, after a single reflection of a spectrally flat illuminant ͑ the equal-energy spectrum ͒ , 11 the most dissimilar mirrors in Fig. 4 ͑ the museum 1 and home 2 mirrors ͒ slightly change that illuminant’s color into two new colors. However, these two colors differ by only 14% more than the local MacAdam just-noticeable difference. In other words, after a single reflection, although many observers might be able to distinguish between colors reflected by Fig. 4’s most dissimilar mirrors when viewed side by side, this is by no means certain. Thus, the color shifts caused by Fig. 4’s very different-looking reflectance spectra will appear nearly identical. Yet each ...

Context 4

... the centuries, mirrors have given us windows onto unusual worlds. Some of these worlds are as familiar as they are fundamentally odd: the one that lets us see our own face or the one that shows us a transposed twin of the room in which we stand. Yet the oddest mirror world may be the one found between two plane mirrors that face each other. If these mirrors are parallel ͑ or nearly so ͒ , then looking into either of them brings us to the vertiginous edge of a visual tunnel that seems to recede endlessly into the virtual distance. We call this phenomenon of repeated mirror reflections a mirror tunnel , and here we are interested in how one formed by two common mirrors transforms the colors of reflected objects. ͑ We define common mirrors as inexpensive second- surface plane mirrors such as those found in homes. ͒ Under- standing these transformations can teach students not only about geometrical optics and spectral transfer functions, but also how knowing the complex refractive indices of the metal backing and glass in such mirrors lets us predict their reflectance spectra via the Fresnel equations. 1 In a student experiment, two especially instructive ͑ and surprising ͒ results are that common mirrors are not spectrally neutral reflectors, and their reflectance spectra depend on the absorbing and reflecting properties of both the glass substrate and its metal backing. Because mirror tunnels are so visually compelling, we use them as a pedagogical tool in this paper. In Sec. II we de- scribe how to set up a mirror tunnel and observe its color shifts, and in Sec. III we discuss the measured reflectance spectra of some common mirrors and show how we use such spectra to calculate the chromaticity trends seen in mirror tunnels. In Sec. IV we use existing data on the complex refractive indices of silver and soda–lime silica glasses in a simple model that calculates the reflectance spectra and chromaticities for these mirrors. At a minimum, students should already be familiar with the optical significance of the mate- rials’ absorption, reflection, and transmission spectra. For deeper insights, they also should understand how such purely physical properties are translated into the psychophysical metric of colorimetry. A very readable introduction to the colorimetric system of the Commission Internationale de l’Eclairage ͑ CIE ͒ is given in Ref. 2. One well-designed mirror tunnel is seen in Granada, Spain’s Science Museum. 3 This virtual tunnel is produced simply by placing two tall mirrors parallel to and facing each other ͑ see Fig. 1 ͒ . Viewers look at one mirror through two small holes in the back of the other mirror; the holes are spaced to match the average distance between human eyes. This viewing geometry avoids the problem that can occur if the viewer’s head is between the two mirrors, where it can block the angularly smallest, most interesting regions of the mirror tunnel. On looking through the eyeholes, one sees a long progres- sion of ever-smaller reflected images of the two mirrors that plunges into the virtual distance, thus giving the illusion of an infinite tunnel ͑ Fig. 2 ͒ . If the mirrors are not exactly parallel ͑ as is true in Fig. 2 ͒ , then the tunnel curves in the same direction that the mirrors tilt toward each other. 4 After relish- ing the vertiginous thrill of this virtual tunnel, we find an- other, subtler feature: the farther we look into the tunnel, the greener and darker its reflected objects appear. To track this color and brightness shift in Fig. 2, we put a photographer’s gray card at the base of one mirror. Although the card’s image becomes progressively darker and more yellow–green with successive reflections, no such shifts occur for distant real objects seen by a single reflection in either mirror. This simple observation lies at the heart of our paper, and it begs the question that students must answer: why do repeated reflections change the mirror images’ color and brightness? Geometrical optics is silent on this point, because it predicts only the images’ location and size. Thus, to explain the brightness and color changes, we must consider the physical optics of common mirrors. Because color depends on spectral variability, we start by examining the spectral reflectances and transmissivities of the metal and glass used to make these mirrors. Common household mirrors have long been made by de- positing a thin film of crystalline silver on the rear surface of float-process flat glass. 5 This silver film is optically thick ͑ minimum thickness ϳ 10 ␮ m ͒ , and its rear surface is pro- tected from oxidation and abrasion by successive films of copper and lacquer. The polished glass itself is usually clear soda–lime silica whose constituents ͑ by weight ͒ are about 72% SiO 2 , 14% Na 2 O ( ϩ K 2 O), 9% CaO, 3% MgO, 1% Al 2 O 3 , and variable, smaller amounts of Fe 2 O 3 ͑ Ͻ 1% ͒ . 6,7 Other backing metals ͑ for example, aluminum ͒ and special ‘‘colorless’’ ͑ that is, lower iron oxide content ͒ or tinted glass can be used to change a mirror’s reflectance spectrum appre- ciably. Nonetheless, a soda–lime silica glass substrate with silver backing forms the optical core of most common mirrors. Figure 3 shows the reflectance and transmission spectra of these two materials in air at visible wavelengths ␭ (380 р␭ р 780 nm). 8,9 Although the glass transmits most at 510 nm ͑ a green ͒ , its broad transmission maximum only imparts a pastel greenish cast to white light, even after it has been transmitted through several centimeters of the glass. This greenish cast is best seen by looking obliquely through a plate of common glass, bearing in mind that refractive dis- persion can complicate the colors that students see. In other words, students should learn to distinguish between the uniform green or blue–green that results from glass absorption and the many prismatic colors that result from glass refrac- tion. In contrast, although Fig. 3’s silver spectrum shows a fairly steep decrease in spectral reflectance R ␭ below 460 nm, at longer wavelengths the metal reflects much more spectrally uniformly. When we form a composite common mirror from these two materials, what reflectance spectra result? Figure 4 shows some representative spectra both from Granada’s Science Museum mirrors and from common mirrors in our own laboratories and homes (400 р␭р 700 nm, measured in 10 or 5 nm steps ͒ . We measured these spectra at normal incidence using either of two portable spectrophotometers: a Minolta CM-2022 or a HunterLab UltraScan. 10 In Fig. 4, all spectral reflectances except those labeled ‘‘ R ␭ ͑ lab 1 ͒ ’’ were measured with the Minolta CM-2022. Given the instruments’ designs, these spectra necessarily include both external specular reflections from the mirror glass and multiple reflections from within the mirror. One of Fig. 4’s most striking features is how much the overall reflectance R varies from mirror to mirror ͑ and, as our measurements show, even within a given mirror ͒ ; mean R values range from 80.5% for the ‘‘museum 1’’ mirror to 95.6% for the ‘‘home 2’’ mirror. Although some of this variability may be attributed to spectrophotometer errors, more variability likely comes from differences in the mirrors’ original fabrication. Nevertheless, Fig. 4’s spectra all have similar shapes: they peak on average at 545 nm, a yellowish green. To restate these spectral differences in colorimetric terms, after a single reflection of a spectrally flat illuminant ͑ the equal-energy spectrum ͒ , 11 the most dissimilar mirrors in Fig. 4 ͑ the museum 1 and home 2 mirrors ͒ slightly change that illuminant’s color into two new colors. However, these two colors differ by only 14% more than the local MacAdam just-noticeable difference. In other words, after a single reflection, although many observers might be able to distinguish between colors reflected by Fig. 4’s most dissimilar mirrors when viewed side by side, this is by no means certain. Thus, the color shifts caused by Fig. 4’s very different-looking reflectance spectra will appear nearly identical. Yet each mirror’s color shift is by itself only marginally perceptible. Using the ...

Context 5

... 1. Schematic diagram of the mirrors used to produce the virtual mirror tunnel photographed in Fig. 2. The circles on the right mirror are eyeholes through which observers view the mirror tunnel.  ...