Black-and-white fringes and the colors of caustics

Michael V. Berry and A. N. Wilson

Fine-scale interference fringes that decorate caustics produced with white light appear black and whitewith high contrast. This is surprising, as the elementary expectation, supported by theory and computersimulation, is that the fringes should be highly colored. The fringe separation is several arc minutes andtherefore close to the resolution limit of the eye. Under magnification (even of a photograph), the colorsare revealed. Therefore black-and-white fringes are an illusion, giving a dramatic naked-eye illustrationof the fact that the angular resolution of the visual system is better for luminance than for color.

1. Introduction

Interference fringes associated with focusing are easyto see with the unaided eye. For example theyappear as delicate lines decorating caustic curves inthe image of a distant light seen through raindropsthat have fallen on spectacle lenses (or a car wind-shield) on a rainy night. In the laboratory thefringes can be seen by viewing an illuminated pinholethrough a water drop or an irregular bathroom-window pane. Plate 49 shows an example. If theincident light is white, the fringes often appear vividlyblack and white (or only weakly colored), and manyare visible (up to 10).

These observations are surprising in several re-spects: first, because the conventional expectation isthat the visible fringes should be highly colored, asthe condition for constructive interference is satisfiedat different places for different colors. This wasrecognized by Simpson and MarstonI who remarkedon the black-and-white fringes they observed in whitelight scattered by oblate water-drops: ". . . the depen-dence on [wavelength] of the fine structure spacing... does not appear to influence the hue." Second,because differently colored rays should be refractedinto different directions (as in the rainbow). Andthird, because only a few fringes ought to be visible,as the rest should be quenched by overlapping colors.

When this work was performed the authors were with the H. H.Wills Physics Laboratory, Bristol University, Royal Fort, TyndallAvenue, Bristol BS8 1TL, UK. A. N. Wilson is now with theDepartment of Physics, The University, Chadwick Tower, P.O. Box47, Liverpool L69 3BX, UK.

Received 30 August 1993; revised manuscript received 18 Octo-ber 1993.

0003-6935/94/214714-05$06.00/0.3 1994 Optical Society of America.

The very term black-and-white interference fringesappears paradoxical.

The resolution of the paradox (Section 4) emergedfrom a theory (Section 3) leading to the descriptionand simulation of the expected colors in the fringepattern. This was based on diffraction theory (Sec-tion 2) for the wavelength and direction dependenceof light entering the eye.

2. Diffraction Theory

The theory we give here is one dimensional anddescribes the color pattern across a caustic curve (itwould be interesting to extend it to cover the colorsnear singularities of the caustics, such as cusps andhigher catastrophes2-5). Lamplight, whose spectralintensity as a function of wavelength is P(X), strikes atransparent screen (e.g., a hanging raindrop) withrefractive index n(X), whose (horizontal) thicknessas a function of a (vertical) coordinate is t(z).Elementary application of the Kirchhoff diffractiontheory shows that for wavelength X the far-field lightintensity in direction 0 is

I(x, 0) c X-1 dzexp(i(2rr/X){n(X)- 1]t(z) -Z}) P(X)

(1)

where the integration is over the height of the screen.In the observations of interest here, 0 never exceeds afew degrees.

Dominant contributions, corresponding to the raysof geometric optics, come from the stationary-phasepoints, where (n - 1)t'(z) = 0. Caustic directions 0,(local deflection maxima) are determined by the coales-cence of rays, that is, inflections of the screen profile,

4714 APPLIED OPTICS / Vol. 33, No. 21 / 20 July 1994

so that

Oc = (n - )t (zJ, [t ,(z) = 0]. (2)

Expanding the phase in expression (1) to the lowestsignificant order (cubic) in z - z, now gives theintensity near the caustic as

I(X, 0) C -1/3 Ai2 (x2[n(X) - 1t(z)} (0, - ))P().

(3)

Here Ai(e) is the Airy function6 whose oscillations for< 0 describe the interference fringes between the

two rays in each direction on the bright side of thegeometric caustic and whose decay for > 0 describesdiffraction on the dark side where there are no rays.

For the fringe spacing, defined as the angle Afbetween the first two maxima of Ai2, we find fromexpression (3) that

Avf = 1.11[ t2 | (4s)

It is surprising that for hanging water drops thisquantity is independent of the size of the drop. Thisfollows from Laplace's equation that relates surfacetension T and curvature to the difference betweenpressure inside the drop (increasing downwards be-cause of gravity) and outside the drop, which gives,for thin drops,

const. - pgz = Tt"(z), (5)

where g is the gravitational acceleration, and p is thedensity of water. Therefore

t" = const. = 1/L2 , (6)

where L = (T/pg)1/2 = 2.71 mm is the surface-tensionlength for water. From Eq. (4) it follows that foryellow light,

Af = 4.5 min. (7)

It is significant that this fringe size is near theresolution limit of the eye (- 1 min). The fringespacing is of the same order of magnitude for othercolors and for real water drops, which are not horizon-tal strips as in this simplified theory, and for ourexperiments with bathroom glass.

Across the visible spectrum, Af varies by a factor ofapproximately (700 nm/400 nm)2 /3

= 1.45 [Eq. (4)],and this gives a first crude indication of the influenceof diffraction on fringe colors. It is necessary tocompare the diffraction width Af with the shift associ-ated with refractive dispersion in the water (or glass),which is defined as Ar 6, 0, (400 nm) - 0, (700 nm).This effect would cause the caustics to have blueedges on their dark sides (the opposite of whathappens in the rainbow, which is a phenomenon ofminimum rather than maximum deflection). For

water, it follows from Eqs. (2) and (5) after a shortcalculation that O,, unlike the fringe size, does dependon the maximum thickness H of the drop, and isnegative:

0,(X) = - [n(X) - 1] - (8)

Dispersion (an/aA = 3.50 x 10-4 nm-l) implies7 thatAf A if H 0.3 mm, and Af dominates for thesmaller drops that we are interested in and that arecommon (this contrasts with rainbows, in which thecolors are usually dominated by refraction).

From formula (3), information about the colors ofcaustics is contained in the function

I(X, H, a) -1/3 Ai2 + PX x XD)() (9)

In this formula e is a scaled caustic-crossing coordi-nate, and a is a parameter that quantifies the relativeimportance of refractive dispersion and diffraction.For a hanging water drop, these quantities are givenby

= [0J(xD) - 0] x 120670 nm2 /3 ,

a = 6.5H2 nm-'/3 (H in mm), (10)

where XD is the wavelength of yellow light. Notethat e is independent of H, apart from a shift in thecaustic direction 0, [Eq. (8)], which does not affect thespacing of the fringes, but the refractive shift a doesdepend on H. Here our main interest is in diffraction-dominated caustics for which a is not large. For P(X)we chose the spectrum of a blackbody with a tempera-ture of 3063 K, modeling the quartz-halogen projec-tor lamp8 used in the experiments.

It is worth remarking that the same theory withnegative a describes the colors of rainbows, where, if ris the raindrop radius, k is related to angles in the skyby e = (Or - 0) X 20061r

2/3

nm2

/3 (r in millimeters),

and a = -1.779r2 /3 nm- 1/3 (r in millimeters).

3. Prediction of Diffraction Colors

For given a, the color at each point e in the pattern isdetermined by calculating the three CommissionInternational de L'Eclairage (CIE) tristimulus val-ues 7:

x a) = f dl(X, i, a)Yi(k),

where Xi _ {X, Y Z}, x-i _ {, y, z}. (11)

Here {x-(A)} are the spectral tristimulus values thatare derived from the mixture of primary colors thatmatch a monochromatic light of wavelength X and arerelated to the spectral responses of the three types ofcone in the eye of a standard observer. Y representsthe luminosity. It is convenient to transform X andY into the CIE (1931) chromaticity coordinates x and

20 July 1994 / Vol. 33, No. 21 / APPUED OPTICS 4715

y, which are defined by

x(t, a) X/(X + Y + Z),

y(E, a) _ Y/(X + Y + Z). (12)

The chromaticity locus {x(e), y(t)} as varies repre-sents the color (hue and saturation) across the causticspecified by the parameter a; together with Y(E), thislocus encodes the color of the pattern.

Figure 1(a) shows the luminosity Y, and Fig. 1(b)shows the chromaticity locus, for a = 0.2. (Similarcalculations of color loci have been given for interfer-ence colors, that is, cos2 fringes,9 and rainbows, thatis, a < 0.10-12) When a << 1, as in Fig. 1(a), severaloscillations are visible on the negative (lit side of thecaustic) side of the luminosity curve, and these persistbeyond the interval shown. This explains why somany fringes are visible. While a is increased to

-500 -300 -100 100

(a)

make refraction dominate diffraction, the oscillationsare quenched, so that the fringes disappear.

In Fig. 1(b), end D of the chromaticity locus corre-sponds to the far dark side of the caustic [ = 100 inFig. 1(a)]. Its large first few windings correspond tothe colors of the main Airy fringes. The smallwindings far on the bright side (large negative ) areasymptotic to the (rather reddish) white of the quartz-halogen lamp (x = 0.43, y = 0.40); this was evident inall our computations, and can be derived analyticallyfrom expression (9) and Eq. (11) by the use ofasymptotics of the Airy function.

If the fringes were really black and white, thechromaticity locus would either remain close to theunsaturated center of the diagram, or else its excur-sions towards the saturated boundary (the locus ofspectral colors) would be correlated with the minimaof luminosity (meaning that colors were confined tothe dark fringes). Neither is true. Therefore thefringes ought to be colored, appearances notwithstand-ing. It was not easy to get a clear idea of theexpected colors from Fig. 1, so we used the sameinformation to make computer simulations of thecolors (as has recently been done for the rainbow 2).

This required transforming {X, Y, ZI to the RGBinputs (between 0 and 1), driving the red, green, andblue phosphors on our (Apple high-resolution) moni-tor screen. The transformation is generated by aconstant matrix,13 which is determined by the tri-stimulus values of the screen's red, green, and blue,or alternatively by their chromaticity coordinates andthose of the screen's white. We measured thesequantities and obtained the results in Table 1.These imply the following transformation betweenRGB and tristimulus coordinates:

b0.8

y

0.6

0.4

0.2

00 0.2 0.4 0.6 X

(b)

Fig. 1. (a) Luminosity Y(t) across caustic, and (b) chromaticitylocus (thick curve), both calculated for a = 0.2. In (b), D marksthe dark end of the locus, the outer curve is the locus of purespectral colors and saturated purples, the triangle encloses thecolor gamut of the monitor screen (vortices R, G, B anticlockwisefrom right), and the filled circle represents the screen white.

=G)=MI a

B) \z

where

[ 3.78

M -1.20

0.03

-1.72 -0.57

-2.06 0.05

-0.19 0.76_

which we used with expression (9) and Eq. (11) to

Table 1. Measured Values of Luminosity and Chromaticity Coordinatesof Screen Phosphorsa and White of Apple Macintosh Color Monitor

Coordinates

Color Y x y

Red 22 0.60 0.34Green 75 0.29 0.58Blue 12 0.15 0.07White 102 0.27 0.29

aR, G, BI = 1, 0, 0}, {0, 1, O, and 0, 0, 11.b{R, G, B) - 1, 1,1). All values were measured with a Minolta

CS-100 chroma meter (the values should be accurate to ±5%).

4716 APPLIED OPTICS / Vol. 33, No. 21 / 20 July 1994

-

create the basis for simulations of the caustics withdifferent a.

The sets of raw RGB values thus computed con-tained values greater than unity. To remove these,each set was normalized by dividing by its largestmember. There were also some negative values,reflecting diffraction colors outside the gamut of themonitor screen [in Fig. 1(b) such colors correspond topoints where the chromaticity locus lies outside thetriangle]. To eliminate the negative values, we sim-ply replaced them by zero, a procedure that we foundto have little effect on the appearance of the simula-tions.

The final step in generating the simulations is toapply gamma corrections to correct for the nonlinear-ity of the screen. By measuring the brightness of thescreen for different RGB inputs and also white, wefound brightness c (R, G, or B)>, where y ranged from1.85 to 1.95. To correct for this, we replaced allcomputed (RGB) values by their 1/1.9th powers.It is essential to apply this correction because itmakes a big difference to the simulations, but theprecise value of y (within the measured range) is notimportant.

A simulation produced in this way and correspond-ing to a = 0.2 is shown in Plate 50; we produced manyothers, for negative as well as positive a. The colorsdepend on the refraction parameter a. For large a ,no fringes are visible, and the colors vary smoothlyacross the caustic; as mentioned above, this is thecase in which refraction dominates diffraction. Acurious feature for small negative values of a is thatthe effects of diffraction and refraction almost cancel,and several fringes are nearly monochromatic (areddish white); this is probably related to the whiterainbow' 5 observed with water drops of radius r <0.05 mm (although the simple Airy theory used hereapplies only within the main maximum). The appear-ance of the colors is also sensitive to the choice ofilluminant and the monitor matrix M.

4. Colors Revealed

The intensity and the variety of these predictedcolored fringes, together with their small angular size[Eq. (7)], suggested that the black-and-white fringesof Plate 49 are an illusion of perception and thatunder sufficient magnification the caustics would appearin their true colors. To test this, we produced severalsets of black-and-white fiinges (Plate 49 is an example)by refraction through irregular bathroom-window glass.The light source was a quartz-halogen lamp viewedthrough a 0.3-mm-diameter pinhole subtending 1.7 minat the glass, which was 600 mm away. The caustic wasgenerated from a 1 mm x 1 mm patch of the glass whoseirregularities were 2 mm (W) in lateral extent andraised 0.1 mm above the mean surface. From this weestimate by using t(z) (H/2)sin(27r/W) that themaximum deviation (angular size of the caustic patterns)is [from Eqs. (2)] 30, and the maximum fringe spacing[from Eq. (4)] is 10 min (i.e., rather larger than

those from hanging water drops). Both these valuesare in the range observed.

These caustics were magnified 15-20x by beingviewed through an eyepiece. This revealed abun-dant and subtle colors. The magnified images werefaint. This observation eliminates the possibilitythat the fringes in the unmagnified caustics are blackand white as a result of rod, rather than cone, vision,because then the magnified images, being fainter,would be more rod dominated and therefore lesscolored.

When the caustics and their colors were photo-graphed, no lenses were necessary: the diffractedlight was allowed to fall directly on the film, andmagnification was achieved simply by moving thecamera further from the screen. Because of thefaintness of the light, long exposures were required(up to 10 min). The colors are also visible in enlarge-ments of photographs taken without magnification ifthe grain size is sufficiently small to permit suchenlargements. For this we used Kodak Ektar 25,whose resolving power for high-contrast images (onthe negative) is stated as 5 plm, which corresponds to

0.4 min in angle and so is adequate.Plate 51 shows colors produced by a combination of

these techniques, that is, enlargement from a nega-tive photographed by lensless magnification (of acaustic in Plate 49). If Plate 51 is viewed from adistance in bright light (e.g., daylight) the colorsdisappear, and the fringes become black and whiteagain. The sequence of colors (black, white, yellow,red, black, blue, green, yellow, red, blue, green,red .. .) agrees quite well with that in Plate 50, whosea value was chosen to get the best match.

The agreement between theory and experiment isnot perfect. The widths of bands of color (particu-larly the white and the yellow in the main fringe) donot always match, and there are other subtle differ-ences (for example, traces of additional interference).A possible source of the mismatch is the difficulty ofgetting accurate photographic and printed color repro-ductions of the caustics and the monitor screen (thematch is better when the optically magnified fringesviewed by eye are compared directly with simulationson the screen). It is clear, however, that the theoryis correct in its essentials.

5. Discussion

It seems, then, that the illusion is the inability of theeye to resolve differences of color on fine angularscales where differences of intensity can be resolved.This is not a new observation. Loss of color visionfor fine detail explains why it is hard to match thecolors of fabrics with single threads, and the phenom-enon was exploited by television engineers16 17 toreduce the bandwidth of the channels that carrychrominance signals relative to the channel thatcarries luminance.

There have been several psychophysics experi-ments related to this phenomenon. Mullen'8 foundthat the bars of colored (e.g., red-green) isoluminant

20 July 1994 / Vol. 33, No. 21 / APPLIED OPTICS 4717

gratings were less easy to discriminate than those ofmonochromatic gratings with luminosity contrast.For high contrast, the smallest bar spacing that couldbe resolved in the colored gratings was 10 mincompared with 2 min for the monochromatic grat-ing. The spacings of our black-and-white fringesoften lie between these values in the range whereintensity variations, but not color variations, can beresolved. In Mullen's experiments the effects ofcolor were separated from those of luminosity, and sothey should not be compared directly with our obser-vations, in which strong and rapid variation of colorand lumunosity coexist.

Earlier, the perception of combined color and lumi-nance variations had been investigated by Hilz andCavonius19 and Hilz et al.20 They found (for severalgrating spacings covering the range of interest to us)that hue discrimination was improved in the presenceof a small luminance contrast C [which is defined as(Imax - Imin)/(Imax + m)]. However, this improve-ment did not persist as C was further increased.Color discrimination was best for C 0.05 (log-luminance increment = 0.045) and deterioratedrapidly for larger C (see Fig. 3 of Ref. 20). Thelargest luminance contrast they studied was 0.2(log-luminance increment = 0.18); this is muchsmaller than the contrast of our brightest fringes,which is close to unity (Fig. 1 and Plate 49).Therefore these experiments also should not be com-pared directly with our observations, although theeffects, and also those studied by Mullen, are probablyclosely related.

There may be few situations in nature for which itis important to resolve fine spatial color differences ina field that also varies in brightness on the same scale.If so, the poor spatial resolution of color differences,compared with luminance resolution, carries littlebiological disadvantage. On the contrary, the illu-sion reported here might serve a useful purpose, andwe offer the following speculation, based on a sugges-tion of Gregory.2' The illusion could be related tothe strategy of the visual system for compensatingthe severe chromatic aberration of the eye22 (whichcan be demonstrated simply23 24). The price of notseeing colors that are artifacts of chromatic aberra-tion (e.g., at the edges of objects) could be that somereal colors are not seen, as in the fine fringes studiedhere.

M. V. Berry thanks J. F. Nye for pointing outblack-and-white fringes in the early 1970's and Yu. N.Demkov for insisting on the need to understand thesefringes. We thank P. F. Heard and J. Wilson forassistance and advice in color rendering and T. Tro-

scianko for helpful comments. This research beganas A. N. Wilson's final-year undergraduate researchproject.

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