Caustic of colors in Newton’s prism

M. V. Berry

doi:10.1364/JOSAA.474473

1. INTRODUCTION

Emil Wolf’s first research papers [1–3], written while he was a student at Bristol University in the 1940s, concerned geometrical optics. Therefore, I have chosen to mark the centenary of his birth with a contribution that is also about geometrical optics. It was stimulated by a paper [4] pointing out that many online illustrations of Isaac Newton’s prism experiments, in which white light is split into colors that are then recombined into white, contain errors: rays traveling in impossible directions. This led to a study of the optics of a single prism, revealing a feature apparently previously unrecognized (and certainly not widely known).

Consider the familiar geometrical-optics idealization of a single ray of white light incident on a glass prism. The light refracts in and refracts out, emerging as a diverging beam of colored light, as illustrated in Fig. 1(a). Now consider the emerging light, including the virtual rays inside the prism as well as the real rays outside. Figure 1(b) illustrates this beam for the same dispersion as Fig. 1(a), and Fig. 1(c) is a magnification of Fig. 1(b). The striking feature is that the rays emerge as though not from a virtual focal point, but from a virtual caustic: the lower boundary of a region reached by two rays. My purpose here is to study this caustic in detail.

Fig. 1. White light dispersed by a prism, with dispersion, exaggerated for clarity, represented by ${\mu _{\rm{max}}} = 0.1$ [notation in Eq. (1) in Section 2]. (a) Incident white ray, and colored rays in the prism and emerging from it; (b) only the emerging light, including virtual rays; (c) magnification of (b) (stretched in $y$), with the prism omitted and the caustic of colors shown in red.

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This unusual type of caustic is the envelope of the virtual emergent rays inside the prism, parameterized by their colors; I will call it a “caustic of colors.” This terminology emphasizes the difference from the more familiar colored caustics, which are polychromatic families of monochromatic families of rays, i.e., families of families, each of which has its own caustic. The differently colored monochromatic caustics, usually decorated by diffraction fringes [5], are assembled, incoherently on relevant time scales, to form the colored caustic. The most familiar colored caustics are rainbows [6], in which the caustics of the individual wavelengths, decorated by diffraction [7], are slightly shifted by the dispersion of water [8]; rainbow colors result from the combination of dispersion (Newton’s colors) and diffraction (Airy’s fringes) [9]. By contrast, in the situation considered here, the incident white light is a family of coincident colored rays, each of which propagates as an individual ray, so there are no monochromatic caustics: the caustic of colors is the envelope of the single family of differently colored rays, and incoherence means that there is no diffraction decoration.

Section 2 describes the exact geometry of the caustic of colors. In practice, dispersion is weak, and an approximate description is possible, as described in Section 3, with the advantage that the caustic can be described analytically. In regions where rays cross [particularly evident in Fig. 1(c)], the colors are determined by additive mixing, as explained in Section 4, leading to rendering of the colors, particularly in the interesting region where two rays overlap, as they would be perceived by (non-color-blind) humans if the virtual rays were focused to create a real image of the caustic. In practice, the incident white light is more accurately represented by a beam of finite width than a single ray. For the colors to be observable, this beam must be narrower than the caustic and also should not spread significantly during its passage along it. An estimate in Section 5 leads to the conclusion, implied by the extreme narrowness of the caustic of colors and the two-ray overlap region, that this would require an unrealistically large prism; however, if the incident “white” light is extended beyond the visual spectrum, the caustic might be detectable (though of course not its colors).

In one of Newton’s schemes for recombining the colors (Prop XI, prob VI, pp. 186-191 and Fig. 16 in Book One of [8]), the light emerging from a prism is focused by a lens (assumed achromatic) onto a second prism, from which the light should emerge as the original white ray. This assumes reversibility: the image of the caustic of colors in the second prism should be the reflection of that inside the first one. But it is not: as illustrated in Fig. 2, the image caustic is the longitudinal translation, not the reflection, of the original (it is reflected transversely but that is irrelevant here). This has an intriguing consequence: in strict geometrical optics, with the idealization of a single incident white ray, the recombination would not work. In practice, the extreme smallness of the caustic of colors and the finite width of the incident beam combine to obscure the colors, which by an argument analogous to that in Section 5 explains why Newton was able to observe the recombination as a white beam.

Fig. 2. Caustic of colors imaged by an achromatic lens of focal length $F$ [size of caustic greatly exaggerated, and prism omitted as in Fig. 1(c)].

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In another of Newton’s schemes (Exper. 2 on p. 116 and Fig. 3 of Book One of [8]), there is no second prism, and the image of the original virtual caustic, tacitly assumed to be a focal point, appears as a white spot. Again this is strictly impossible as illustrated in Fig. 2, and again the finite width of practical incident white beams restores whiteness in the image focused by the lens.

Fig. 3. Schematic of prism and notation for a colored ray.

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2. EXACT THEORY

Referring to Fig. 3, let the prism have apex angle $\alpha$, color-dependent refractive index $n$, and size specified by a length $L$, denoting the distance from the apex to the point of incidence; until Section 5, we will use units such that $L = 1$. Dispersion, relative to a representative value ${n_g}$ (“green”) in the middle of the visible spectrum, is specified by the color parameter $\mu$:

(1)$$n = {n_g}(1 +\mu).$$

In Fig. 1 and the visualizations to follow, we will take a “standard prism,” with $\alpha = \pi /3$, ${n_g} = 1.5$. The limits ${n_{\rm{blue}}}$ and ${n_{\rm{red}}}$ of the visible spectrum are denoted by ${\mu _{\rm{max}}}$:

(2)$${-}{\mu _{\rm{max}}} \le \mu \le {\mu _{\rm{max}}},\quad {\rm i.e.},\quad {\mu _{\rm{max}}} = \frac{{{n_{\rm{blue}}} - {n_{\rm{red}}}}}{{2{n_g}}}.$$

Usually, ${\mu _{\rm{max}}}$ is very small: for borosilicate crown glass, ${\mu _{\rm{max}}} \approx 0.004$. This is too small for effective visualization, so in pictures, we will use the larger value ${\mu _{\rm{max}}} = 0.1$, as in Figs. 1(a), 1(b), and 1(d).

For the incident white light, we choose the direction corresponding to minimum deviation for ${n_g}$, given by

(3)$$\sin {\theta _0} = {n_g}\sin \left({\frac{1}{2}\alpha} \right).$$

(For the standard prism, ${\theta _0}={ 48.60^ \circ}$.) The angle of refraction inside the prism of the ray with color $\mu$ is

(4)$$\sin {\theta _1}(\mu) = \frac{{\sin \left({\frac{1}{2}\alpha} \right)}}{{1 + \mu}}.$$

For this ray, the direction of emergence from the prism is

(5)$$\sin {\theta _3}(\mu) = {n_g}(1 +\mu)\sin \left({\alpha - {\theta _1}(\mu)} \right).$$

Its point of emergence (distance from the apex) is

(6)$${L_1}(\mu) = \frac{{\cos {\theta _1}(\mu)}}{{\cos \left({\alpha - {\theta _1}(\mu)} \right)}}.$$

It follows that the formula for this ray, referred to as coordinates in Fig. 3, is

(7)$$\begin{split}{y(x,\mu)}&={ \cos \left({\frac{1}{2}\alpha} \right)\left({1 - {L_1}(\mu)} \right)}\\ &\quad-{ \left({x - {L_1}(\mu)\sin \left({\frac{1}{2}\alpha} \right)} \right)\tan \left({{\theta _3}(\mu) - \frac{1}{2}\alpha} \right).}\end{split}$$

For positions $\{x,y\}$, there are zero, one, or two rays in the family with ${-}{\mu _{\rm{max}}} \le \mu \le {\mu _{\rm{max}}}$, determined by inversion:

(8)$$y(x,\mu) = y \Rightarrow \mu = {\mu _ \pm}(x,y).$$

The geometric-optics intensity (ray density) at $\{x,y\}$ is the inverse Jacobian of the map from color $\mu$ to $\{x,y\}$, namely,

(9)$$I(x,y) = \sum\limits_ \pm {\left({{{\left| {\frac{{\partial y(x,\mu)}}{{\partial\mu}}} \right|}_{{\mu _ \pm}(x,y)}}} \right)^{- 1}}.$$

The caustic of colors, $y = {y_c}(x)$, is the singularity of the map, that is, the envelope of the family of colored rays, where the Jacobian vanishes; $\partial y$ vanishes to a higher order than $\partial \mu$ (focusing corresponds to “a lot goes into a little”). Thus

(10)$${\partial _\mu}y(x,\mu) = 0 \Rightarrow \mu = {\mu _c}(x),$$

and the caustic of colors is

(11)$${y_c}(x) = y({x,{\mu _c}(x)} ).$$

3. SMALL DISPERSION

It is convenient to work with explicit approximate formulas, obtained by exploiting the fact that in practice, dispersion is weak: ${\mu _{\rm{max}}} \ll 1$. A long though straightforward calculation, not given here, gives the rays [Eq. (7)] to lowest relevant orders of smallness in $\mu$:

(12)$$y(x,\mu) \approx A - Bx + \mu (C - Dx) + {\mu ^2}(E - Fx),$$

where the coefficients $A,B,C,D,E,F$ are functions of ${n_g}$ and $\alpha$. Explicit expressions are lengthy; it suffices to give the numerical values for the standard prism ${n_g} = 1.5$, $\alpha = \pi /3$:

(13)$$\begin{split}A &= 0.168175,\quad B = 0.33635,\quad C = 0.796939,\\D &= 2.52434,\quad E = 3.60432,\quad F = 4.75036.\end{split}$$

For the illustrations to follow, dispersion is small enough for approximate formulas to be visually almost indistinguishable from the exact, and (with some abuse of notation) we will henceforth write expressions using $=$ rather than $\approx$.

The color parameters [Eq. (8)] at each point, for the ray family with dispersion ${\mu _{\rm{max}}}$, are obtained by inversion:

(14)$$\begin{split}{{\mu _ \pm}(x,y)}&={ \frac{{Dx - C \pm \sqrt {{{(C - Dx)}^2} - 4(E - Fx)(A - Bx - y)}}}{{2(E - Fx)}}}\\ &\quad {|{\mu _ \pm}(x,y)| \le {\mu _{\rm{max}}}.}\end{split}$$

On the caustic, where the two overlapping rays coincide, the square root vanishes, and the color parameter is

(15)$${\mu _c}(x) = - \frac{{(C - Dx)}}{{2(E - Fx)}},\quad |{\mu _c}(x)| \le {\mu _{\rm{max}}}.$$

This gives the position $x$ where the caustic has color $\mu$:

(16)$${x_c}(\mu) = \frac{{2E\mu + C}}{{D + 2F\mu}},\quad (|\mu | \le {\mu _{\rm{max}}}).$$

Thus the explicit formula for the caustic of colors is

(17)$$\begin{split}{{y_c}(x,{\mu _{\rm{max}}})}&={ A - Bx - \frac{{{{(C - Dx)}^2}}}{{4(E - Fx)}}}\\ &{\left({{x_c}(- {\mu _{\rm{max}}}) \le x \le {x_c}({\mu _{\rm{max}}})} \right).}\end{split}$$

These explicit formulas enable calculation of the regions occupied by different numbers of rays, illustrated in Fig. 4(a). The upper boundary of the region occupied by the ray family is

Fig. 4. (a) One-ray regions 1 and 2 (yellow) and the two-wave region 3 (red), for ${\mu _{\rm{max}}} = 0.1$; (b) intensity Eqs. (22) and (23) across the vertical lines in (a), with left, middle, and right corresponding to ${l_1}$, ${l_3}$, ${l_2}$ (for ${l_1}$, the spike is a slope discontinuity, as for ${l_3}$, not a divergence as across the caustic in ${l_3}$).

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(18)$${y_{\rm{max}}}(x,{\mu _{\rm{max}}}) = {\rm max}(y(x, - {\mu _{\rm{max}}}),y(x,{\mu _{\rm{max}}})).$$

The lower boundary is

(19)$${y_{\rm{min}}}(x,{\mu _{\rm{max}}}) = {\rm min}(y(x, - {\mu _{\rm{max}}}),y(x,{\mu _{\rm{max}}}),{y_c}(x,{\mu _{\rm{max}}})).$$

Outside the region between these two boundaries, there are no rays. Within it, there can be one or two rays. The sub-region where two rays overlap [region 3 in Fig. 4(a)] involves

(20)$${y_{\min 0}}(x,{\mu _{\rm{max}}}) = {\rm min}(y(x, - {\mu _{\rm{max}}}),y(x,{\mu _{\rm{max}}})),$$

and is

(21)$${y_{\rm{min}}}(x,{\mu _{\rm{max}}}) \le y \le {y_{\min 0}}(x,{\mu _{\rm{max}}}).$$

The complementary one-ray region is divided into two parts: region 1, where ${-}{\mu _{\rm{max}}} \le {\mu _ -}(x,y) \le {\mu _{\rm{max}}}$, and region 2, where ${-}{\mu _{\rm{max}}} \le {\mu _ +}(x,y) \le {\mu _{\rm{max}}}$.

The intensity Eq. (9) is

(22)$$I(x,y) = s(x,y){\rm Re}\frac{1}{{\sqrt {{{(C - Dx)}^2} - 4(E - Fx)(A - Bx - y)}}},$$

(the real part conveniently excludes unphysical values automatically), where

(23)$$s(x,y) = \left\{{\begin{array}{ll}1&{{\rm in \;\; regions\; 1\; or\; 2}}\\2&{{\rm in \;\; region\; 3}}\\0&{{\rm elsewhere}}\end{array}} \right..$$

The intensity has step discontinuities at the upper boundary Eq. (18), at the boundaries between regions 1 and 3 and 2 and 3, and at the lower boundary Eq. (19), excluding the caustic singularity where there is a square-root divergence. Figure 4(a) shows the upper and lower boundaries of the regions reached by real and virtual rays, and Fig. 4(b) shows sections of the geometrical ray intensity.

4. RENDERING THE COLORS

To simulate the caustic of colors by rendering the ray pattern on a computer monitor, we use the CIE 1931 color space, and transform to the RGB coordinates at each point in the beam. This procedure will be outlined here; it is described in detail elsewhere, e.g., [10,11]—see also Section 4 of [12]. For simplicity, I will calculate the appearance of the caustic of colors for incident white light with uniform intensity over the visual spectrum; for other illuminations (for example, thermal light), the colors look slightly different, but this is a small effect.

The CIE color space is based on wavelengths $\lambda$. To convert from the color parameter $\mu$ defined in Eq. (1), we use the simplest dispersion formula for the refractive index, $n = {A_0} + {B_0}/{\lambda ^2}$, leading to

(24)$$\!\!\!\lambda (\mu ,{\mu _{\rm{max}}}) = \frac{1}{2}\left({{\lambda _{\rm{red}}} + {\lambda _{\rm{blue}}} + ({\lambda _{\rm{red}}} - {\lambda _{\rm{blue}}})\frac{\mu}{{{\mu _{{\max}}}}}}\! \right),\!$$

where ${\lambda _{\rm{red}}}$ and ${\lambda _{\rm{blue}}}$ approximate the limits of the human visual spectrum. The next step involves the color matching functions,

(25)$${\boldsymbol {\bar r}}(\lambda) = \{\bar x(\lambda),\bar y(\lambda),\bar z(\lambda)\} ,$$

of the three cones in the eye of a standard observer, accurately and conveniently represented by the analytical formula in Section 2.2 of [13]. These are converted to the three CIE tristimulus values

(26)$$\begin{split}{\textbf{R}(x,y)}&={ \{X(x,y),Y(x,y),Z(x,y)\}}\\ &={ \sum\limits_ \pm {\boldsymbol {\bar r}}\left({\lambda ({\mu _ \pm}(x,y),{\mu _{\rm{max}}})} \right),}\end{split}$$

where the sum includes the zero, one, or two contributing rays at $\{x,y\}$ (there are no intensity factors because where two rays overlap, their intensities [Eq. (22)] are equal). The sum incorporates the linearity corresponding to additive color mixing in the CIE space.

Next, the tristimulus values must be converted to the computer’s RGB color coordinates. This depends on the red, green, and blue of the monitor’s pixels, but a convenient representative conversion [14] involves the CIEtoRGB matrix

(27)$$\textbf{M} = \left({\begin{array}{*{20}{c}}{2.36462}&{- 0.89654}&{- 0.46807}\\{- 0.51517}&{1.42641}&{0.08876}\\{0.00520}&{- 0.01441}&{1.0092}\end{array}} \right),$$

and the matrix product

(28)$$\{{\rm red}(x,y),{\rm green}(x,y),{\rm blue}(x,y)\} = {\boldsymbol {MR}}(x,y).$$

After a subtraction to eliminate negative values (out-of-gamut colors) and incorporating nonlinearity of our perception (gamma correction), the final RGB values are

(29)$$\begin{split}{\{{\rm R}(x,y),}&{{\rm G}(x,y),{\rm B}(x,y)\}}\\ &={ {{\left[{\frac{{\{{\rm red}(x,y),{\rm green}(x,y),{\rm blue}(x,y)\}}}{{{\rm max}\{{\rm red}(x,y),{\rm green}(x,y),{\rm blue}(x,y)\}}}} \right]}^{5/9}}.}\end{split}$$

Figure 5 shows the rendered real and virtual emergent beam, corresponding to Fig. 1(b) and the regions in Fig. 4(a). In one-ray regions 1 and 2, the light is monochromatic. The overlap region 3 [red in Fig. 4(a)], where colors add, is shown in Fig. 6. Figures 6(a) and 6(b) show the separate (monochromatic) colors corresponding to ${\mu _ +}(x,y)$ and ${\mu _ -}(x,y)$. Figure 6(c) shows their additive mixture; note the tiny regions near the top and right boundaries, where the mixture is white.

Fig. 5. Colors in the real and virtual emergent beam, comprising the three regions shown in Fig. 4(a).

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Fig. 6. Colors in the overlap region [red in Fig. 4(a)] for ${\mu _{\rm{max}}} = 0.1$, magnified and stretched in $y$; (a) monochromatic colors corresponding to ${\mu _ +}(x,y)$; (b) monochromatic colors corresponding to ${\mu _ -}(x,y)$; (c) their additive mixture.

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5. CAN THE CAUSTIC OF COLORS BE OBSERVED?

The feasibility of observing the colors in the two-ray region [Fig. 6(c)] depends on how big this is, so we start by calculating its $x$ length $\Delta x$ and $y$ width $\Delta y$. The $x$ length is, to lowest order in ${\mu _{\rm{max}}}$, using Eq. (16), and reinstating the prism size $L$ (Fig. 3):

(30)$$\begin{split}{\Delta x({\mu _{\rm{max}}},L)}&={ L\left({{x_c}({\mu _{\rm{max}}}) - {x_c}(- {\mu _{\rm{max}}})} \right)}\\ &={ \frac{{4L{\mu _{\rm{max}}}}}{{{D^2}}}(DE - FC)}\\ &={ 3.3343L{\mu _{\rm{max}}},}\end{split}$$

where the last member refers to the standard prism. The $y$ width is the separation between the crossing point, where [cf. Eq. (18)] $y(x, - {\mu _{\rm{max}}}) = y(x,{\mu _{\rm{max}}})$, and the corresponding point on the caustic ${y_c}(x)$. From Eq. (12), the crossing occurs at $x = C/D$, so, also using Eq. (17),

(31)$$\begin{split}{\Delta y({\mu _{\rm{max}}},L)}&={ y\left({\frac{C}{D},{\mu _{\rm{max}}}} \right) - {y_c}\left({\frac{C}{D},{\mu _{\rm{max}}}} \right)}\\ &={ L\mu _{\rm{max}}^2\left({E - \frac{{FC}}{D}} \right)}\\ &={ 2.1046L\mu _{\rm{max}}^2}.\end{split}$$

These sizes are very small. For a crown glass prism of size $L = 10\;{\rm cm}$ and dispersion ${\mu _{\rm{max}}} = 0.004$, $\Delta x = 1.35\;{\rm mm}$, $\Delta y = 3.37\,\,\unicode{x00B5}{\rm m}$. The highest-dispersion optical glass I could find is Schott F2, for which (2) and Table 3 of [15], with the visible range $380\;{\rm nm}\le \lambda \le 700\;{\rm nm}$, gives ${\mu _{\rm{max}}} = 0.0143$, leading to $\Delta x = 4.8$ mm, $\Delta y = 43\,\,\unicode{x00B5}{\rm m}$.

The dimensions of the overlap region increase linearly with the prism size $L$. To estimate how large $L$ must be to observe the caustic of colors, we replace the fictional idealization of a single white ray by a beam, conveniently taken to be Gaussian, with waist width $W$, centered on the caustic region. Resolving the colors requires $W \ll \Delta y$, so we take $W = \sigma \Delta y$, with $\sigma \ll 1$. In addition, the beam should not spread significantly over the caustic length $\Delta x$. The spreading distance in the prism, conveniently represented by light near the middle of the visual spectrum, is ${D_s} = \pi {W^2}{n_g}/{\lambda _g}$. Therefore the non-spreading condition is

(32)$${D_s} \gt \Delta x \Rightarrow \frac{{\pi {\sigma ^2}\Delta {y^2}{n_g}}}{{{\lambda _g}}} \gt \Delta x,$$

which with Eqs. (30) and (31) gives

(33)$$L \gt \frac{{0.240{\lambda _g}}}{{\mu _{\rm{max}}^3{\sigma ^2}{n_g}}}.$$

The caustic of colors is a virtual object, so to observe it, the emergent rays must be focused to create a real image. The emerging beam will spread by diffraction, and it is necessary to check that this does not obscure the dispersion spreading of interest here. The dispersion spreading angle, from Eqs. (12) and (13), is

(34)$$\begin{split}{{\theta _{\rm{disp}}}}&={ \mathop {\lim}\limits_{x \to \infty} \left| {\frac{{y(x,{\mu _{\rm{max}}}) - y(x, - {\mu _{\rm{max}}})}}{x}} \right| = 2D{\mu _{\rm{max}}}}\\ &={ 5.049{\mu _{\rm{max}}},}\end{split}$$

and for a Gaussian beam, the diffraction spreading angle, again estimated by the middle of the visual spectrum, is

(35)$${\theta _{\rm{diff}}} = \frac{{{\lambda _g}}}{{\pi W{n_g}}} = \frac{{{\lambda _g}}}{{\pi \sigma \Delta y{n_g}}}.$$

With Eqs. (31) and (33), this gives the ratio

(36)$$\frac{{{\theta _{\rm{diff}}}}}{{{\theta _{\rm{disp}}}}} \lt 0.125\sigma ,$$

which is comfortably small, confirming that diffraction spreading would not obscure the dispersion.

For a crown glass prism (${\mu _{\rm{max}}} = 0.004$), and choosing $\sigma = 0.1$ to resolve the caustic of colors, the estimate Eq. (33) gives $L \gt 130\;{\rm m}$. This unrealistically enormous prism size is a consequence of the small dispersion ${\mu _{\rm{max}}}$ and sensitive dependence $1/\mu _{\rm{max}}^3$. With Schott F2 glass, $L \gt 2.7\;{\rm m}$, much smaller but still probably impractically large.

The foregoing estimates, based on the simplest observation scheme, suggest that the caustic of colors would be difficult to observe. This is an extreme example of a known phenomenon: although caustics are robust in the sense of being mathematically stable under perturbation, they can often be so delicate that their fine details are obscured. A familiar example [16] is fine structure in the caustic on the bottom of a swimming pool, generated by sunlight refracted by waves on the water, blurred by the finite width of the sun.

The estimate Eq. (33) shows that the required prism size would decrease dramatically if the dispersion ${\mu _{\rm{max}}}$ could be increased. One way to accomplish this is to artifically extend the incident “white” illumination beyond the visible, for example, into the infrared. Then the dispersions for glasses [17,18], especially chalcogenide glasses [19] can exceed that for crown glass in the visible by a factor of 10. And for water, in the range of $0.2\,\,\unicode{x00B5}{\rm m} \lt \lambda \lt 2.65\,\,\unicode{x00B5}{\rm m}$ where it is transparent [20], ${\mu _{\rm{max}}} \approx 0.068$, approximately 17 times greater than for crown glass in the visible, reducing the estimate Eq. (33) to 8.5 cm. Water prisms are relatively easy to make. But of course, with this strategem, the colors of the caustic of colors could not be simulated or seen, though the geometrical intensity and its caustic singularity could be detected.

6. CONCLUDING REMARKS

It is disappointing that observation of the caustic of colors, a phenomenon hidden inside Newton’s prism but well-defined within geometrical optics, would require such large prisms, at least according to the simple schemes just described. This does not exclude the possibility of more ingenious schemes that would require smaller prisms.

I would not be surprised if the geometrical-optics calculation leading to the structure of the caustic of colors has been published already during the 350 years since Newton described his experiments with prisms. Newton himself described his experiments in great detail [8], but never mentioned caustics, nor does a detailed analysis of his experiments [21]. And I have not found any other publication reporting the caustic in the prism. The closest is in a 19th century textbook [22] containing calculations of the failure of achromaticity of Newton’s two-prism experiments, without identifying the virtual caustic.

Funding

Leverhulme Trust (Emeritus Fellowship).

Acknowledgment

I thank John Hannay, Pragya Shukla, and Jon Tarrant for helpful suggestions.

Disclosures

The author declares no conflicts of interest.

Data availability

No data were generated or analyzed in this research.

REFERENCES

1. E. Wolf and W. S. Preddy, “On the determination of aspheric profiles,” Proc. Phys. Soc. 59, 704–711 (1947).

2. E. Wolf, “On the designing of aspheric profiles,” Proc. Phys. Soc. 61, 494–503 (1948).

3. E. Wolf, A Contribution to the Theory of Aspheric Optical Systems (Bristol University, 1948).

4. J. Tarrant, “Web of confusion,” Phys. World 35, 52 (2022). [CrossRef]

5. M. V. Berry and S. Klein, “Colored diffraction catastrophes,” Proc. Natl. Acad. Sci. USA 93, 2614–2619 (1996). [CrossRef]

6. R. Lee and A. Fraser, The Rainbow Bridge: Rainbows in Art, Myth and Science (Pennsylvania State University/SPIE, 2001).

7. G. B. Airy, “On the intensity of light in the neighbourhood of a caustic,” Trans. Cambridge Philos. Soc. 6, 379–403 (1838).

8. I. Newton, Opticks: or a Treatise of the Reflections, Inflections and Colours of Light, 4th ed. (Dover, 1730).

9. R. L. Lee, “What are ‘all the colors of the rainbow’?” Appl. Opt. 30, 3401–3407 (1991). [CrossRef]

10. K. T. Travis, Effective Color Displays: Theory and Practice (Academic, 1991).

11. J. Walker, “Colour rendering of spectra,” 1996, http://www.fourmilab.ch/documents/specrend/.

12. M. V. Berry, “Coloured phase singularities,” New J. Phys. 4, 61–66 (2002). [CrossRef]

13. C. Wyman, P.-P. Sloan, and P. Shirley, “Simple analytic approximations to the CIE XYZ color matching functions,” J. Comput. Graph. Tech. 2, 1–11 (2013).

14. CIE, “CIE 1931 color space,” 2019, https://en.wikipedia.org/wiki/CIE_1931_color_space.

15. P. Hartmann, “Optical glass: deviation of relative partial dispersion from the normal line—need for a common definition,” Opt. Eng. 54, 105112 (2015). [CrossRef]

16. M. V. Berry and J. F. Nye, “Fine structure in caustic junctions,” Nature 267, 34–36 (1977). [CrossRef]

17. G. F. Brewster, J. F. Kunz, and J. L. Rood, “Dispersion of some optical glasses in the visible and infrared,” J. Opt. Soc. Am. 48, 534–536 (1958). [CrossRef]

18. I. H. Malitson, G. W. Cleek, O. N. Stavroudis, and L. E. Sutton, “Infrared dispersion of some oxide glasses,” Appl. Opt. 2, 741–747 (1963). [CrossRef]

19. H. G. Dantanarayana, N. Abdel-Moneim, Z. Tang, L. Sojka, S. Sujecki, D. Furniss, A. B. Seddon, I. Kubat, O. Bang, and T. M. Benson, “Refractive index dispersion of chalcogenide glasses for ultra-high numerical-aperture fiber for mid-infrared supercontinuum generation,” Opt. Mater. Express 4, 1444–1455 (2014). [CrossRef]

20. J. Turner, “Optical properties of water and ice,” 2021, https://en.wikipedia.org/wiki/Optical_properties_of_water_and_ice.

21. J. A. Lohne, “Experimentum crucis,” Notes Rec. R. Soc. London 23, 169–199 (1968). [CrossRef]

22. R. S. Heath, A Treatise on Geometrical Optics (Cambridge University, 1887).

Caustic of colors in Newton’s prism

Abstract

1. INTRODUCTION

2. EXACT THEORY

3. SMALL DISPERSION

4. RENDERING THE COLORS

5. CAN THE CAUSTIC OF COLORS BE OBSERVED?

6. CONCLUDING REMARKS

Funding

Acknowledgment

Disclosures

Data availability

REFERENCES

Data availability

Cited By

Figures (6)

Equations (36)

Journal of the Optical Society of America A