Artificial circumzenithal and circumhorizontal arcs Markus Selmke and Sarah Selmke* *Universit¨atLeipzig, 04103 Leipzig, Germany∗ (Dated: September 2, 2018) We revisit a water glass experiment often used to demonstrate a rainbow. On a closer look, it also turns out to be a rather close analogy of a different kind of atmospheric optics phenomenon altogether: The geometry may be used to faithfully reproduce the circumzenithal and the circum- horizontal halos, providing a missing practical demonstration experiment for those beautiful and common natural ice halo displays. I. INTRODUCTION Light which falls onto a transparent thin-walled cylin- der (e.g. a drinking glass) filled with water gets refracted. Several ray paths may be realized through what then ef- fectively represents a cylinder made of water. Light may either illuminate and enter through the side of the cylin- der, or may enter through the top or bottom interfaces, depending on the angle and spot of illumination. Indeed, FIG. 1. Rays entering through the top face of both a cylinder in the former situation, i.e. illumination from the side and (left) and a hexagonal prism (right) experience an equivalent refraction. Refraction of the skew rays by the side faces are under a shallow inclination angle reveals a rainbow in the equivalent when the effect of rotational averaging of the prism backwards direction. The reason being that the geome- is considered. The same holds true for the reverse ray path. try mimics the incidence plane geometry of a light path though a spherical raindrop: Refraction, internal reflec- tion and a second refraction upon exit, all occurring at was the first to establish an extensive quantitative frame- the cylinder's side wall, produce the familiar observable work for halos based on the (false) assumption of refract- ◦ rainbow caustic in the backwards direction at around 42 ing cylinders, did not conceive of this CZA mechanism 2,3 towards the incidence light source. and instead invoked a more complicated one.13 Now, returning to the initial claim, we consider illumi- We will detail each experimental setup and show how nation of the glass through the top water-air interface. If to arrive at a quantitative description of several aspects of the angle of incidence is shallow enough, light may exit the artificial halo analoga, rederiving well-known expres- through the cylinder's side wall. Contrary to common sions from the natural atmospheric optics ice halo phe- belief,1 (cf. also blogs etc. found via an internet search for nomena. For ideal experimental results, one may use a \glass water table rainbow") this situation is not related round reflection cuvette. However, a beaker or any other to the rainbow. Instead, this geometry equals the aver- cylindrical glass and a focusable LED flashlight (source age geometry of light paths through an upright hexagonal of parallel white light) will work just fine. ice prism, entering through the (horizontal) top face and leaving through either of its six (vertical) side faces, cf. Fig.1. The averaging meant being over different prism II. ARTIFICIAL CIRCUMZENITHAL ARC orientations as indicated in the figure. This, in turn, is what causes the natural atmospheric phenomenon known 3{11 We begin with the artificial CZA, for which a ray is as- as the circumzenithal arc (CZA) halo, an example of sumed to enter through the top air-water interface and to which is shown in Fig.2(a). In the experiment, an anal- leave the cylinder through its side wall, cf. Fig.3(a)-(c). ogous curved spectrum is observed when the refracted At the first interface, the ray changes its inclination e to- light is projected on the floor (the horizontal plane) some wards the horizontal plane according to Snell's law. We arXiv:1608.08664v1 [physics.ao-ph] 30 Aug 2016 12 distance from the cylinder, see Fig.2(b). denote complementary angles by a subscript c, such that Similarly, illuminating the glass at a very steep angle for instance ec = π=2 − e, see Fig.3(b). Thus, we have at its side, the light may enter through the side wall and 0 sin (ec) = n0 sin (ec), with an associated transmission co- leave through the top surface. Now, apart from top and efficient T1 (ec) according to the Fresnel equations. When bottom being reversed, this geometry equals the average later discussing intensities, we will consider polarization- geometry of light entering a rectangular (vertical) side averaged transmission coefficients only, although this ap- face of a hexagonal plate crystal and leaving through its proach will not strictly be valid for the second refraction bottom (horizontal) hexagonal face. This is the situation due to the partial polarization upon the first refraction. corresponding to the natural halo phenomenon known as The second (skew-ray) refraction now occurs un- the circumhorizontal arc (CHA).3{6,8{11 der a geometry that may be decomposed into two Anecdotally, it appears puzzling why Huygens, who parts8{11,15,16: One in the horizontal plane (i.e. as seen 2 from the top, cf. Fig.3(c)) and described by an effec- tive index of refraction n0, Bravais' index of refraction (a) for inclined rays, and a second inclination refraction de- scribed by the actual material's index of refraction n0. d 17 re e The appropriate effective index of refraction reads: lu a) b s cos2 (e0) n0 = : (1) 2 2 0 1=n0 − sin (e ) zenith The exiting ray, which hit the cylinder's side wall under a xy-projected incidence angle of φ (to the normal), is thus deflected in the horizontal plane by φ00 = φ0 − φ, cirrus clouds where Snell's law connects the latter two angles via n0 sin (φ) = sin (φ0). The inclination angle to the plane (b) 0 00 changes according to n0 sin (e ) = sin (e ), such that over- all the exit angle to the vertical becomes7{9,11 q 00 2 2 ec = arccos n0 − cos (e) : (2) original image exposure corr. Then, referring to the experiment's setup and coordinates point as defined in Fig.3(a), one finds for each light source below glass inclination angle e the deflected rays to lie on a curve (x (φ) ; y (φ)). This CZA curve may be parametrized by the angle φ 2 [−π=2; π=2], see Fig.3(c), as stool x sin (φ00) = l tan (e00) ; (3) light incidence direction y c cos (φ00) (c) wherein φ00 = φ00 (φ), i.e. sin (φ00 + φ) = n0 sin (φ). Eq.3 describes a circle, see dashed line in Fig.2(b),(c). How- 0.4 ever, it turns out that only a segment of the circle is at- tainable by the exiting rays due to the occurrence of total 0.3 internal reflection. The solid black line in Fig.2(c) shows this limit. The critical internal angle of incidence may 0.2 c i 0 0 r be found from φ = arcsin (1=n ), such that φ = π=2 c TIR l 0 e marks the onset of total internal reflection. Herein n is 0.1 a function of e0 which is a function of e. One finds8,11 0.0 q -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 2 φTIR = arccos n0 − 1= cos (e) ; (4) which translates into a corresponding azimuthal limit 00 FIG. 2. (a) Natural circumzenithal arc (CZA) halo display. φ = π=2 − φTIR of the (projected artificial) CZA. TIR The solar elevation was e = 27◦,18 and the angular distance A similar reasoning leads to the existence of a critical 00 ◦ to the zenith was ec = 16 as determined with the help of the elevation angle eTIR above which the internal second re- 00 vanishing lines of vertical & parallel features of the building fraction becomes a total internal reflection, ec = 0, Eq. 19 0 (dashed lines). (b) Artificial CZA produced by illuminat- (2), even for φ = 0 where n is lowest. Equivalently, ing the top surface of a water-filled acrylic cylinder under a one may set φTIR ! 0 and solve Eq. (4) for e to arrive shallow angle (outer = 50 mm, inner = 46 mm, length: 7{9,11 ? ? at: 50 mm, height: l = 1 m). (c) Artificial CZA curves according ◦ ◦ ◦ q to Eq. (3) for e = f5 ; 15 ; 25 g and for red and blue color 2 each. Shading according to the intensity I (φ00), Eq. (6). eTIR = arccos n0 − 1 : (5) ◦ Eq. (5) shows that at around eTIR = 28 even the last p glimpse of the red (n0 (red) = 1:332, i.e. less refracted shows that any material with n0 > 2, i.e. glass, will 15,28{30 than blue n0 (blue) = 1:341) part of the artificial (wa- not produce a CZA (nor a CHA). For this reason ter) CZA disappears. For ice, taking n0 = 1:31, the alone, and in order to not have to construct a water-filled corresponding critical solar elevation above which this hexagonal prism, it is nice to have a simple analog demon- halo can no longer be observed is 32◦.7{11 Eq. (5) also stration experiment to overcome this practical limitation. 3 (a) (b) (c) (d) incidence xy-plane flat ice-crystals only: plane of incidence exit ray exit ray CZA exit ray (e) (f) exit ray (g) (h) CHA xy-plane plane of incidence incidence FIG. 3. Geometry and setup for (a)-(d) the CZA experiment and (e)-(h) the CHA experiment. Read text for details. 00 00 00 7 The full azimuthal width of the CZA is ∆φCZA = 2φTIR ordinate φ requires several key factors to be considered: and is an increasing function of the elevation, starting from 125◦ and approaching a half-circle, i.e.