supernumerary ice-crystal halos?

Supernumerary ice-crystal halos?

Michael V. Berry

Geometric-optics singularities in the intensity profiles of refraction halos formed by randomly oriented icecrystals are softened by diffraction and decorated with fine supernumerary fringes. If the crystals have afixed symmetry axis (as in parhelia), the geometric singularity is a square-root divergence, as in therainbow. However, the universal curve that describes diffraction is different from the rainbow's Airyfunction, with weak maxima (supernumerary fringes) on the geometrically dark region inside the halo(and even fainter fringes outside); these are much smaller than their counterparts on the light side ofrainbows. If the crystals have no preferred orientation (as in the 220 halo), the geometric singularity is astep. In this case the universal diffraction function has no maxima, and its supernumeraries areshoulders rather than maxima. The low contrast of the fringes is probably the main reason whysupernumerary halos are rarely if ever seen.

1. Introduction

It is well knownl 2 that both rainbows and ice-crystalhalos are consequences of the minimum deviation ofsunlight. The origins of the minima are, however,different in the two cases. In the rainbow, the raysthrough an individual raindrop have a minimumdeviation at a directional caustic (corresponding tothe Descartes ray), whereas in halos the rays througheach crystal form a collimated beam, and the mini-mum is over beams from differently oriented crystals(this is the same phenomenon as the minimumdeviation of a beam by a rotated prism, e.g., on aspectrometer turntable). In spite of this difference,at the level of geometric optics rainbows and halosshare the property that their differential scatteringcross sections, as functions of the deviation angle,possess singularities. For rainbows, this is an in-verse square root (arising from the Jacobian of themapping between angles of incidence and deviation,that is, from x to x2). For halos, the form of thesingularity (see Section 2) can be an inverse squareroot or a step, depending on the nature of thedistribution of crystal orientations.

Geometric optics fails on fine scales, and in the caseof the rainbow the intensity singularity is softened bydiffraction according to Ai2{ - p4, where Ai is the Airy

The author is with the H. H. Wills Physics Laboratory, Univer-sity of Bristol, Royal Fort, Tyndall Avenue, Bristol, Avon BS8 1TL,UK

Received 12 July 1993; revised manuscript received 18 October1993.

0003-6935/94/214563-06$06.00/0.© 1994 Optical Society of America.

function 3 and p is a scaled rainbow-crossing coordi-nate. The oscillations of Ai2{ - pl are the interferencefringes that are visible as supernumerary rainbowson the bright side of the geometric maximum (p > 0).It is natural to ask what are the analogous diffractionmodifications of the geometric singularities for halosand in particular whether there are supernumeraryhalos. These are the questions addressed here.

Minnaert4 claimed that supernumerary halos "havebeen seen at times" but gave no details. Visser56

employed diffraction theory to calculate their direc-tions and claimed that diffraction is responsible forthe observed wide variations in halo colors. I knowof no recent evaluation of these claims, but thecalculations that follow make it seem unlikely thathalo fringes have been seen in nature. Knnen 7

suggests that observations previously interpreted assupernumerary halos might instead be of halos withunusual radii produced (geometrically) by pyramidalcrystals; a particularly striking example was analyzedby Neiman. 8

Given the different mechanisms for the minimumdeviation, it is not surprising that the forms of thediffraction cross sections turn out to be different forhalos than for rainbows, but it is a little surprisingthat the halo cross sections share the rainbow prop-erty of depending on functions of a single variable,which can be evaluated in closed form. The univer-sal halo functions are calculated for two idealizedcrystal orientation distributions, called case I andcase II. In case I (Section 3) plate crystals share acommon axis perpendicular to the incident beam, andtheir orientation depends on a single angle describingrotation about this axis; this distribution corresponds

20 July 1994 / Vol. 33, No. 21 / APPLIED OPTICS 4563

to parhelia (sun dogs). In case II (Section 4), theaxes too are randomly oriented; for pencil crystalsthis distribution corresponds to the common 22° halo.Both halo diffraction functions are compared withtheir rainbow counterpart, and their properties areused to show why supernumerary halos are notcommon.

2. Geometric Singularities

Attention is restricted to an idealized model involvingrefraction by (truncated) 60° prisms in randomlyoriented hexagonal crystal columns or plates. Acollimated beam of monochromatic light strikes aprism face oriented near minimum deviation. Onlydirections close to minimum deviation are considered,so the variation of the Fresnel reflection coefficientcan be ignored. Polarization effects9"l0 are also ig-nored.

In case I, all crystals share a common (vertical)symmetry axis that is parallel to the edges of theeffective 600 prisms. The prisms are randomly ro-tated about their edges. Assume for simplicity thatthe incident sunlight is perpendicular to the edges,i.e., horizontal or almost so, so there are no skew rays.(The theory is almost the same for higher Sunelevations when there are skew rays; the essentialpoint is the common axis, which ensures that theskewness is constant.) Let the angle of incidence ona crystal be

i = imin + id (1)

where imin ( 40.920 for the refractive index 1.31corresponding to yellow light) corresponds to mini-mum deviation, and tr, which describes the orienta-tion, is small. The deviation angle of the emergentbeam is

Dgeom = Dmin + A1 2 + higher terms, (2)

where Dmin is the geometric halo angle (i.e., theminimum deviation, equal to 21.84°) and A is aconstant.

Each crystal contributes a beam described by adelta function in the direction Dgeom, so the intensitygenerated by the whole collection is the average overorientations iP. This is

(Igeom, I(D)) = dO S(D - Dmin -A 2)

1

=[A(D - Dmin)]'/ 2 EO(D - Dmin), (3)

where 0 denotes the unit step (the integration rangehas been extended to infinity because the only signifi-cant orientations are those near minimum deviation,where it is essential that the orientation distributionis smooth). The singularity here, at each of theminimum deviation spots on either side of the Sun(i.e., the parhelia), is exactly the same as that at thecaustic line of the rainbow, in spite of the two

different mechanisms for the formation of this singu-larity.

In case II, the axes of the prisms are random aswell, so that it is necessary to consider skew rays.An important observation, first made clearly by K6n-nen,9 is that this alters the nature of the singularity.Let the prism axis make an angle X + r/2 with theincident direction; thus X measures the skewness.The deviation angle now depends on X as well as on 4i,but not on the third angle that specifies the orienta-tion of the prism, say, s4, which describes rotations ofthe edge about the incident direction. It is known'that

Dgeom = Dmin + AqI2 + BX2 + higher terms. (4)

Instead of Eq. (3), we have

(Igeom, I(D)) = dt* f dX 8(D - Dmin - Aql2 - BX 2 )

Tr

=(A)/2 O(D - Dmn). (5)

Because of the randomness in A, this singularityforms a circular arc around the Sun, which is, ofcourse, the 22° halo. The fact that the singularity isweaker-a finite step rather than a divergence-is areason why these halos are less prominent thanrainbows (another reason is that halos, unlike rain-bows, are on the bright side of the sky).

3. Diffraction for Case I (Parhelia)

Now we assume that the wavelength X of the light ismuch smaller than the crystal diameter d (assumedthe same for all crystals). In the simplest approxima-tion, diffraction spreads the beams emerging fromeach crystal, the spreading in D being that from a slitwhose width is approximately w = d cos imjn/2 0.38d. Thus the intensity from a given crystal isproportional to

I(D) W sin[7rw(D - Dgeom)/X] 2) lrrw(D - Dgeom)/X J (6)

[The normalization is chosen to make I(D) ->8(D - Dgeom) in the geometric-optics limit x -> 0.]

For case I we have, instead of the geometric averageof Eq. (3), the diffraction intensity

(I (Do) = I dI sin[Tw(D - Dmin - A12

)/X]

X 7~irw(D - Dmi q2

(7)

Scaling now gives

(II(D)) = " / (8)

4564 APPLIED OPTICS / Vol. 33, No. 21 / 20 July 1994

singularity of Eq. (3) in the limit X - 0 [after thescalings of Eqs. (8) and (9)] are

1(-9 > 0),

- 0 H < 0)- (14)

Note that - > 0 corresponds to the geometricallybright sides of the parhelia. This gives the horizon-tal spreading across the geometric edge. [There willalso be diffraction in the vertical, described by thesame function as Eq. (6), with w replaced by thethickness of the crystal plates; this is not consideredfurther.]

The function p(-q) is the parhelion analog of theAiry function, and we now examine its properties.First note that it depends on a single scaling variableiq that does not involve the parameter A describingthe sharpness of the minimum deviation. Second,note that a change of the integration variable gives

1 At dv (sin 2

PH 'IT (V + q)1 /2 V I (10)

showing that the cross section is the geometric singu-larity convolved with the slit diffraction function.Third, this observation enablesp(-q) to be evaluated inclosed form because the Fourier transforms of bothcomponents are known. A calculation gives

Cos (2i +P -)= '(i~ + - 2( --)C[ 2 (y) /

+ sign(-q 2 + S[2( ) l

The sine terms in expressions (13) describe supernu-merary parhelia; note that these occur on both sidesof the geometric edge q = 0.

Fifth, the graph of p(Ti) (Fig. 1) indicates that thesupernumerary parhelia are of very low contrast andbarely visible on the bright side ( > 0) of the halo.Even on the dark side, the maximum contrast is only(Pma - Pnin)/(Pm. + Pmin) = 0.178 (these values re-fer to the first minimum at X = -2.949 and the firstsubsidiary maximum at ii = -4.020).

There are several striking differences between theparhelion functionp(q) and its rainbow analog (insetsin Fig. 1), namely, Ai2(-p), where," for a drop ofrefractive index n and diameter d,

p = (D - in) (n2 - 1)1/2 (2rd )2/(4 - n2)1/6 3X (15)

First, supernumerary rainbows occur only on the

1

(11)

where C and S denote the Fresnel integrals2 definedby -10 -5 0 77 5 10

(a)

C(z) + iS(z) - dt exp i t2) . (12)

-1

Fourth, the following limiting forms can easily befound from Eq. (11):

- sin(2T + )

pus) - q 42 1/2

4

Lrl3/ - sin(2,1 + )

4 1q I'/'3 4Ti(2,rm) 2

The terms that give rise to the

(v >> 1),

(i = 0),

10 -5 y

0.5

-1

-1.5

-2

11

5 10

lOglqp

(b)(vi << -1) (13) Fig. 1. (a) Thick curve, universal parhelion diffraction function

p('q) [Eq. (11)]; thin line; geometric-optics approximation [approxi-mation (14)]; inset, the corresponding rainbow function Ai

2 {-p}.geometric-optics (b) Logarithm ofp(ri): inset, logarithm of Ai2f-p}.


where

T" -1.187(D - Dmin)d

X

1 sin( - U2)2p() -_ - d u (9)

bright side of the geometric caustic, whereas thereare supernumerary parhelia on both sides of the edge,with higher contrast on the dark side (Fig. 1).

Second, the separation of supernumerary rainbowsscales as X2/3, whereas that of supernumerary parhe-lia scales as , so parhelion fringes are much closerthan rainbow fringes for a crystal and a drop with thesame diameter d (provided d >> ). The separationof the first two Airy intensity maxima is

2/3AD = 1.7641 d (16)

whereas parhelion fringe maxima are separated by[cf. Eq. (9) with An = r)

XAD = 2.647

a(17)

An important value is AD = 0.5°, the diameter of theSun, which happens to be similar to AD 0.60 for thegeometric color broadening arising from refractivedispersion. Then, for = 600 nm, Eq. (16) showsthat supernumerary rainbows are quenched for dropslarger than d 1.7 mm, whereas Eq. (17) shows thatsupernumerary parhelia are eliminated for crystalslarger than d 0.18 mm. However, it is known12

that halos (and, presumably, parhelia as well) can beformed by crystals several times smaller than this, sotheir supernumeraries would not be quenched in thisway.

Third, the contrast of rainbow fringes is muchgreater than that of parhelia. The minima of Ai2( - p)are zeros, so supernumerary rainbows have contrastunity. On the other hand, we have seen that thecontrast of parhelion fringes is at most 0.178. Thelow value means that supernumerary parhelia aremore vulnerable to quenching by variations in thesize of ice crystals. I think this is the main reasonwhy supernumeraries are not normally seen in parhe-lia, whereas supernumerary rainbows are frequentlyseen (especially near the rainbow top13). However,supernumerary parhelia ought to be observable in thelaboratory, for example in monochromatic collimatedlight scattered by a small rapidly spinning prism orcrystal.

Fraser' 4 shows simulations of the intensity distribu-tion across the 220 halo for different sizes of crystal,using the same physical theory as that presented herefor parhelia (and in particular neglecting skew rays),with the additional feature that the variation of theFresnel coefficients is included. Measurements onhis Fig. 2 show that his logarithmic curves can bescaled with the variables of Eqs. (9) to fit the univer-sal formula of Eq. (11) and so have the universalshape of Fig. 1(b).

4. Diffraction for Case 11(220 Halos)

To include skew rays, we simply replace the deltafunction in the geometric average of Eq. (5) by the slit

diffraction function of Eq. (6).

(III(D)) = A do | dx sin[7w(D -Dgo.)/ 2X -r I rw(D -Dgeom)/X12(1t

After scaling [cf. Eq. (8)], we obtain

(III(D)) = (AB)'/ 2 h(q), (1'

where X is as given in Eqs. (9), and h(vi) is

h(-i) = i du dvsin(J - v[ - 2 )2 (20)

This is the universal halo function that describesdiffraction softening of the geometric step of Eq. (5),whose properties are now examined. First note thath(-i) depends on a single scaling variable q, whichdoes not involve the parameters A and B describingthe sharpness of the minimum deviation. Second,the function is easily evaluated by transforming topolar coordinates in the u, v plane and integrating byparts. The result is

1 sin2 n Si(2,i)h(e) = d- sin + ie) (21)where Si denotes the sine integral 2 defined by

Si(z) mdz sin zz

(22)

Third, the limiting forms are

1 sin(2-q)h~n) 279 27rvq2

= 1/2

1 sin(2i)

2rrv 27rrv2

( >> 1),

(N = 0),

(vi << -1). (23)

These limits incorporate the geometric step, as, fromEqs. (9), small X implies large 1j9 1. The sine terms inEqs. (23) describe the supernumerary halos. Aswith parhelia, these occur on both sides of the step, atX = nr (n •0 ).

Fourth, the supernumeraries are shoulders, that iszero-slope inflections, rather than maxima. This isbecause Eq. (21) implies

d sin2 1

- h(q) = sn 2 Iq TI (24)

which is never negative. These faint supernumer-ary fringes can be seen in the graphs of h(-i) and logh(vi) (Fig. 2).


This gives

1h

0.8

0.6

(a)

10 -5

-0.25

-0,

-5-1

1.25

-1.5

-1.75

5 0

logloh

[Eq. (21)], (b)

(b)Fig. 2. (a) Universal halo diffraction function h()logarithm of h().

The remarks in the penultimate three paragraphsof Section 3, which contrast supernumerary parheliawith their rainbow counterparts, also apply to thesupernumerary halos described by h(-q). Indeed, theargument that supernumerary parhelia are unlikelyto be observed because of their low contrast appliesmore strongly to supernumerary halos, because, ashas just been explained, these have zero intensitycontrast.

5. Concluding Remarks

Using idealized models for the simplest cases, I havecalculated halo difraction functions in the asymptoticregime of directions close to the geometric singulari-ties for crystals that are large in comparison with thewavelength of light, and I have argued that thesupernumerary fringes predicted by these functionsare too weak to be observed. The fringes will be evenweaker in circumstances in which these idealizationsbreak down, such as crystals larger than 0.18 mm,large variations in crystal size, or7 variations in theangles between crystal faces that occur while crystalsare growing.

But supernumeraries are not the only effect ofdiffraction. Another is broadening of the main maxi-mum onto the geometrically dark side. From Figs. 1

and 2 this is of the same order of magnitude as that ofEq. (17) and so is probably visible, as Visser argued,5 6

in spite of broadening from the Sun and dispersion,for crystals smaller than d 0.18 mm, even when thesupernumerary parhelia and halos are obscured byvariations in crystal size.

More generally, it is interesting to consider thegeometry of halos and related phenomena from thestandpoint of singularity theory. I think that thissubject is not completely understood, although consid-erable progress has been made by Tape. 5"16 In thecase of caustics, that is, focal singularities of smoothfamilies of rays, of which the rainbow is an example,the singularities are those of catastrophe theory.' 7

This provides a classification that enables compli-cated caustics to be understood. We can ask whetherthe same mathematics can be employed to classifymore complicated halos. It cannot. Although bothcaustics and halos are singularities of smooth maps,the map for halos (linking crystal orientation to beamdeflection) is not of the gradient type assumed incatastrophe theory. In two dimensions (that is, onthe sky) all smooth maps share line and cusp singulari-ties, and some halos do indeed show cusps. 5"16 Butwith additional parameters (such as the elevation ofthe Sun), the singularities of gradient and nongradi-ent mappings are different.'8 Therefore we do notexpect to see halos involving some of the highercatastrophes, such as umbilics, or sections close totheir singularities. However, swallowtail singulari-ties are stable in both gradient and nongradient mapsfrom R3 - R3, and it is surprising that neither thissingularity nor its characteristic two-cusped, self-crossing section appears in any of the numeroussimulations of halos.

A related problem concerns the circumstances inwhich the geometric intensity singularities are stepsor inverse-square-root divergences or singularities ofanother sort. I agree with Kbnnen and Tinbergen'sconjecture that the step is the generic case. Forexample, the inverse square root for parhelia isunstable in the sense that it is replaced by a (large)step if there is any randomness in the distribution ofcrystal axis directions (physically, this instability isimportant only if the variation in direction exceedsthe diffraction spreading of the singularity).

Finally, it is worth noting that caustics and the halomechanisms elaborated here do not exhaust thepossibilities for producing geometric singularities.For example, it appears that the inverse-square-rootsingularity shared by rainbows and parhelia can alsoarise directly from a singularity in the distribution ofcrystal orientations, as in the rare Bottlinger's rings,formed by reflection, around the subsun.' 9

I thank R. G. Greenler for encouraging me to carryout the calculation reported here, G. P. Kdnnen forpointing out a serious error in an early version of thispaper and for some helpful references, and J. F. Nyefor a thorough reading of the manuscript and severalsuggestions.


-

References1. R. A. R. Tricker, Introduction to Meteorological Optics (Elsevier,

New York, 1970), Chap. 4.2. R. Greenler, Rainbows, Halos and Glories (Cambridge U.

Press, Cambridge, 1980).3. M. Abramowitz and I. A. Stegun, Handbook of Mathematical

Functions (National Bureau of Standards, Washington, D.C.,1964), Sec. 10.4.

4. M. G. J. Minnaert, Light and Color in the Outdoors (Springer,New York, 1993), p. 2 1 2 .

5. S. W. Visser, "Over de Buiging van het Licht bij de Vormingvan Halo's," K. Ned. Akad. Wet. Versl. Gewone Vergad. Afd.Natuurkd. 25, 1328-1351 (1918).

6. S. W. Visser, "Die Halo-Erscheinungen," in Handbuch derGeophysik, F. Linke and F. Mdller, eds. (Springer-Verlag,Berlin, 1961), Vol. 8, Chap. 15, pp. 1027-1081.

7. G. P. Kbnnen, Royal Netherlands Meteorological Society, DeBilt, The Netherlands (personal communication, 1993).

8. P. J. Neiman, "The Boulder, Colorado, concentric halo displayof 21 July 1986," Bull. Am. Meteorol. Soc. 70, 258-264 (1989).

9. G. P. K6nnen, "Polarisation and intensity distributions ofrefraction halos," J. Opt. Soc. Am. 73, 1629-1640 (1983).

10. G. P. Kbnnen and J. Tinbergen, "Polarimetry of a 22 halo,"Appl. Opt. 30, 3382-3400 (1991).

11. H. M. Nusssenzveig, Diffraction Effects in Semiclassical Scat-tering (Cambridge U. Press, Cambridge, 1992), p. 108.

12. W. Tape, "Some ice crystals that made halos," J. Opt. Soc. Am.73, 1641-1645(1983).

13. A. B. Fraser, "Why can the supernumerary bows be seen in arain shower?" J. Opt. Soc. Am. 73, 1626-1628 (1983).

14. A. B. Fraser, "What size of ice crystals causes the halos?" J.Opt. Soc. Am. 69, 1112-1118 (1979).

15. W. Tape, "Geometry of halo formation," J. Opt. Soc. Am. 69,1122-1132 (1979).

16. W. Tape, "Analytic foundations of halo theory," J. Opt. Soc.Am. 70, 1175-1192 (1980).

17. M. V. Berry and C. Upstill, "Catastrophe optics: morpholo-gies of caustics and their diffraction patterns," Prog. Opt. 18,258-346 (1980).

18. A. Thorndike, C. R. Cooley, and J. F. Nye, "The structure andevolution of flow fields and other vector fields," J. Phys. A 11,1455-1490 (1978).

19. D. K. Lynch and F. C. Mertz, "Bottlinger's rings," in Lightand Color in the Open Air, Vol. 13 of 1993 OSA TechnicalDigest Series (Optical Society of America, Washington, D.C.,1993), pp. 73-75.


supernumerary ice-crystal halos?

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