Artificial circumzenithal and circumhorizontal arcs Markus Selmkea) and Sarah Selmke Universitat€ Leipzig, 04103 Leipzig, Germany (Received 9 July 2016; accepted 18 May 2017) A glass of water, with white light incident upon it, is typically used to demonstrate a rainbow. On a closer look, this system turns out to be a rather close analogy of a different kind of atmospheric optics phenomenon altogether: circumzenithal and the circumhorizontal halos. The work we present here should provide a missing practical demonstration for these beautiful and common natural ice halo displays. VC 2017 American Association of Physics Teachers. [http://dx.doi.org/10.1119/1.4984802]
I. INTRODUCTION We will detail each experimental setup and show how to arrive at a quantitative description of several aspects of the Light that falls onto a transparent thin-walled cylinder 1,2 artificial halo analog, re-deriving well-known expressions (such as a drinking glass) filled with water gets refracted. from the natural atmospheric optics ice-halo phenomena. Several ray paths may be realized through what then effec- For ideal experimental results, one can use a round reflection tively represents a cylinder of water. Light can either illumi- cuvette. However, a beaker (e.g., borrowed from a French nate and enter through the side of the cylinder, or it can enter coffee press) or any other cylindrical glass and a focusable through the top or bottom interfaces, depending on its origin LED flashlight or a projector lamp (a bright source of paral- and direction of travel. Indeed, the former situation (illumina- lel white light) will work just fine. tion from the side and under a shallow inclination angle) reveals a rainbow in the backward direction. The reason for the backward rainbow being that the geometry mimics the II. ARTIFICIAL CIRCUMZENITHAL ARC incidence plane geometry of a light path though a spherical We begin with the artificial CZA, for which a ray is raindrop: refraction, internal reflection, and a second refrac- assumed to enter through the top air-water interface and tion upon exit, all occurring at the cylinder’s side wall, pro- to leave the cylinder through its side wall, as depicted in duce the familiar observable rainbow caustic in the backward 3–6 Figs. 3(a)–3(c). At the first interface, the ray changes its direction at around 42 from the incident light source. inclination angle e toward the horizontal plane according Now, returning to the initial claim, we consider illumina- to Snell’s law. We denote complementary angles by a sub- tion of the glass through the top water-air interface. If the script c, such that, for instance, e p=2 e [see Fig. 3(b)]. angle of incidence is shallow enough, light may exit through c ¼ À Thus, taking n 1 for air we have sin e n0 sin e0 , with an the cylinder’s side wall. Contrary to common belief,1,2 (cf. ¼ c ¼ c associated transmission coefficient T1 ec according to the also blogs, etc., found via an internet search for “glass water Fresnel equations. When later discussingð Þ intensities, we table rainbow”), this situation is not related to the rainbow. will consider polarization-averaged transmission coefficients Instead, this geometry equals the average geometry of light only, although this approach will not strictly be valid for the paths through an upright hexagonal ice prism, entering second refraction due to the partial polarization upon the first through the (horizontal) top face and leaving through either refraction. of its six (vertical) side faces, as shown in Fig. 1. The averag- The second (skew-ray) refraction now occurs under a geom- ing is meant to be over different prism orientations as indi- etry that may be decomposed into two parts.12–15,19,20 One is cated in the figure; this, in turn, is what typically causes the in the horizontal plane, shown from the top in Fig. 3(c),and natural atmospheric phenomenon known as the circumzeni- described by an effective index of refraction n (Bravais’ index thal arc (CZA) halo,6–15 an example of which is shown in 0 of refraction for inclined rays) given by21 Fig. 2(a). In the experiment, an analogous curved spectrum is observed when the refracted light is projected on the floor 16 cos2e (the horizontal plane) some distance from the cylinder, as 0 (1) n0 2 2 : shown in Fig. 2(b). ¼ s1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi=n0 sin e0 Similarly, illuminating the glass at a very steep angle at its À side, the light may enter through the side wall and leave through the top surface. Now, apart from top and bottom being reversed, this geometry equals the average geometry of light entering a rectangular (vertical) side face of a hexag- onal plate crystal and leaving through its bottom (horizontal) hexagonal face. This is the situation corresponding to the natural halo phenomenon known as the circumhorizontal arc (CHA).6–10,12–15 Anecdotally, it is puzzling why Huygens, who was the first to establish an extensive quantitative framework for halos Fig. 1. First refraction: Rays entering through the top face (faintly ruled) of both a cylinder (left) and a hexagonal prism (right) experience an equal based on the (false) assumption of refracting cylinders, did inclination refraction. Second refraction of the skew rays by the side faces in not conceive of this CZA mechanism and instead invoked a both cases are equivalent when the effect of rotational averaging of the 16,17 more complicated one. prism is considered. The same holds true for the reverse ray paths.
575 Am. J. Phys. 85 (8), August 2017 http://aapt.org/ajp VC 2017 American Association of Physics Teachers 575 inclination angle e the deflected rays to lie on a curve x / ; y / . We will neglect the finite size of the glass as anð ð approximationÞ ð ÞÞ and thereby assume its diameter 1 to be much smaller than the vertical distance l to the surface onto which the ray is projected (1 l). The CZA curve may then be parametrized by the angle / p=2; p=2 , see Fig. 3(c), as 2½À
x sin /00 l tan ec00 ; (3) y ¼ cos /00
wherein /00 /00 / , or sin /00 / n0 sin / . Equation ¼ ð Þ ð þ Þ¼ ð Þ (3) describes a circle of radius l tan ec00, shown as a dashed line in Figs. 2(b) and 2(c).However,itturnsoutthatonlya segment of the circle is attainable by the exiting rays due to the occurrence of total internal reflection. The solid black line in Fig. 2(c) shows this limit. The critical internal angle of inci- dence may be found from / arcsin 1=n0 ,suchthat/0 TIR ¼ ð Þ ¼ p=2 marks the onset of total internal reflection. Herein n0 is a 12,15 function of e0,whichisafunctionofe. One finds
n2 1 / arccos 0 À ; (4) TIR cos ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffie
which translates into a corresponding azimuthal limit /TIR00 p=2 /TIR of the (projected artificial) CZA. ¼ A similarÀ reasoning leads to the existence of a critical elevation angle eTIR above which the internal second refrac- tion becomes a total internal reflection, e00 0 in Eq. (2). c ¼ Equivalently, one may set /TIR 0 and solve Eq. (4) for e to obtain11–13,15 !
2 eTIR arccos n 1 : (5) ¼ ð 0 À Þ qffiffiffiffiffiffiffiffiffiffiffiffiffi Equation (5) shows that at around eTIR 28 even the ¼ last glimpse of the red, with index n0 red 1:332, is less ð Þ¼ refracted than blue, with index n0 blue 1:341, so part of ð Þ¼ the artificial (water) CZA disappears. For ice, taking n0 1:31, the corresponding critical solar elevation above ¼ 11–15 which this halo can no longer be observed is 32. Equation (5) also shows that any material with n0 > p2 19,33–35 Fig. 2. (a) Natural circumzenithal arc (CZA) halo display. The solar elevation (such as glass) will not produce a CZA (nor a CHA). For this reason alone, and in order to not have to constructffiffiffi a was e 27 (Ref. 22), and the angular distance to the zenith was ec00 16 as determined¼ with the help of the vanishing lines of vertical and parallel¼ fea- water-filled hexagonal prism, it is nice to have a simple ana- tures of the building (dashed lines) (Ref. 23). (b) Artificial CZA produced by log demonstration experiment to overcome this practical illuminating the top surface of a water-filled acrylic cylinder under a shallow limitation. The full azimuthal width of the CZA is D/00 angle (outer diameter: 50 mm, inner diameter: 46 mm, length: 50 mm, height: CZA 2/TIR00 and is an increasing function of the elevation, l 1 m). (c) Artificial CZA curves according to Eq. (3) for e 5; 15; 25 ¼ ¼ ¼f g starting from 125 and approaching a half-circle, or 180, for and for red and blue color each. Shading according to the intensity I /00 , given in Eq. (6). Full color-spectrum versions are described in Appendixð A. Þ e eTIR. In this limit, Eqs. (4) and (5) show that light emerges! only from a small section around / 0, where the ¼ There is also a second inclination refraction described by the effective index of refraction diverges n0 eTIR whereby ð Þ!1 actual material’s index of refraction n0. The exiting ray, which the exiting refraction deviates rays at a right angle /0 p=2 hits the cylinder’s side wall under an xy-projected incidence towards the left and the right. angle of / (to the normal), is thus deflected in the horizontal The complementary angle ec00 of the final exit ray’s inclina- plane by /00 /0 /, where Snell’s law connects the latter tion, Eq. (2), corresponds to the angular distance to the azi- ¼ À 22–24 two angles via n0 sin / sin /0.Theinclinationangletothe muth of the natural CZA halo phenomenon. This ¼ plane changes according to n0 sin e0 sin e00,suchthatoverall angular distance is independent of the azimuth /00 (or /), the exit angle to the vertical becomes¼11–13,15 such that the natural CZA appears as a true circle around the zenith, as shown in Fig. 2(a),12,13,15 just as the artificial CZA 2 2 e00 arccos n cos e : (2) c ¼ ð 0 À Þ is a circle in the xy-plane, as shown in Fig. 2(b). One may qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi also observe (both in natural displays of the phenomenon as Referring to the experiment’s setup and coordinates as well as in the experiment) that the angular width of the visi- defined in Fig. 3(a), one finds for each light source ble spectrum, or the chromatic angular dispersion Dec00
576 Am. J. Phys., Vol. 85, No. 8, August 2017 M. Selmke and S. Selmke 576 Fig. 3. Geometry and setup for the CZA experiment [panels (a)–(d)] and the CHA experiment [panels (e)–(h)]. See text for details.
e00 red e00 blue , remains roughly constant at 1:6 refraction is then captured by this factor peaking around ¼ c ð ÞÀ c ð Þ (which may be compared to the dispersion 1:2 and 2:2 of /00 0. However, in this case, no divergence of the intensity the primary and secondary rainbows, respectively). Only at appears (meaning this factor remains finite at all times). The very small inclination angles is a broadening observed11,15 chromatic dispersion factor accounts for the changing width before the shrinking CZA eventually disappears as it con- De00 de00=dn0 n0 blue n0 red of the arc, such that c ð Þ½ ð ÞÀ ð Þ verges towards the zenith (or the point below the cylinder), its apparent brightness is 1=Dec00. Overall then, cf. Figs. 2(a) and 2(c). / 1 1 All of the above characteristics of the CZA can be readily I /00; e T1T2A d/00=db À de00=dn0 À ; (6) observed with a focusable flashlight in hand and a glass of ð Þ/ ð Þ ð Þ water resting at the edge of a table. For instance, by slowly which is plotted in Fig. 4. This expression quantifies the lowering the incidence light direction on the water surface observed azimuthal decay in intensity away from the forward (while keeping aim by observing the shadow), the CZA can direction and towards zero for /00 /TIR00 due to the second interface’s transmission going smoothly! to zero as the limit of be seen to only emerge when the angle gets below 28, start- ing with a wide and weak arc of small radius and good spec- total internal reflection is reached. The transmission coefficient tral separation below the glass, to then transition to a brighter T2 u [blue dashed line in Fig. 4(a)], upon the second refrac- tion,ð Þ depends on the actual angle u to the normal, as seen in but forwardly more confined arc, to then eventually disappear 29 1 as a faint arc of largest radius, as shown in Fig. 2(c). Fig. 3(d),whichisgivenby u arccos cos / cos en0À ). The parametric curves in Fig. 2(c) have¼ beenð shaded according III. ARC INTENSITY to this overall intensity function and match the appearance of experimentally observed CZAs projections and natural halo Without treating the situation in full detail, a description of the displays quite well. Together with the theoretical CZA azi- approximate intensity along the azimuthal coordinate /00 requires muthal width D/CZA00 , the above considerations quantify and 11,18 several key factors to be considered including: T1,thetrans- reiterate the often-stated rule of thumb that a natural CZA mission at the first interface; T2,thetransmissionatthesecond rarely appears to exceed a quarter of a full circle. interface; the fact that A cos / sin e cot e0,ageometricfac- Plotting the intensity in the forward direction at the position 25 1 / tor; d/00=db À ,araybundlingfactorwithb sin / (see of maximum CZA intensity (at /00 / 0) as a function of ð Þ 1 ¼ ¼ ¼ below); de00=dn0 À ,achromaticdispersionfactor;andIAM, the elevation (inclination), I e , one observes a wide peak, as atmosphericð attenuationÞ (for natural halos only). seen by the thick black lineð inÞ Fig. 4(b). This means that for The cross-sectional factor A takes the projected surface of some solar elevations (or light source inclinations) the (artifi- the top face into account, which admits refraction along the cial) CZA is brighter as compared to others. For water and ice ray path described by the internal angle /, see dashed area in (including the atmospheric attenuation), this peak occurs at 25,26 Fig. 3(c). The ray bundling factor is a caustic intensity around 17 and 20, respectively. For ice then, this value corre- factor for the cylinder experiment, and a fake caustic inten- sponds to the solar elevation at which the CZA is best observ- sity factor for the natural counterpart.20 Its concept is similar able in nature (see, for instance, Refs. 7, 11, 13,and15). For to that of the rainbow caustic,27 and accounts for the predom- water, this is the light source inclination for which the experi- inance of certain deflection directions as outcomes of the ment produces the brightest CZA projection. Both T1 (thin gray refraction of an incident parallel bundle of rays. The fact dashed line) and the geometric factor (thin gray solid line) that more rays experience a small azimuthal in-plane describe the decay towards e 0. For the natural halo, the !
577 Am. J. Phys., Vol. 85, No. 8, August 2017 M. Selmke and S. Selmke 577 Fig. 4. (a) The (artificial) CZA intensity distribution along the azimuth coor- dinate /00 at e 17. Only for /00 /00 ; /00 is I /00 > 0. (b) The ¼ 2½À TIR þ TIR ð Þ CZA intensity at /00 0 for different elevation (inclination) angles e. Only ¼ for e 0; eTIR is I e > 0 and the CZA observable. The refractive index of water2 (1.33)½ was usedð Þ here; the picture for ice is very similar. Fig. 5. (a) Artificial circumhorizontal arc (CHA) projection. The circle at atmospheric attenuation also affects the decay when the sun is the ceiling, crossing the shadow of the water-filled cylinder, corresponds to 28 low and dim. At the other end of the curve, T2 (thick gray the natural parhelic circle halo (the external reflection contribution). (b) Plot dashed line) along with the ray bundling factor (thick gray solid of the CHA curve, Eq. (8), for elevations e 65; 75; 85 and for both red and blue (l 1 m). (c) The acrylic cylinder¼f as placed in theg center of a line) describe the decay towards e eTIR.Thedecreaseofthe ¼ ! spherical projection screen (Ref. 31). (d) Two views showing the CHA pro- ray bundling factor may be understood as follows. jection onto the spherical screen. The inclination angle e0 increases for increasing elevation angles e such that the effective index of refraction n0 of Eq. (1) increases as well (eventually diverging as we have seen). The second refraction at the top water-air interface only This in turn causes the exit rays, which are deflected by /00, changes the inclination towards the horizontal, n sin e to sweep more rapidly across the forward (/00 0) direction as 0 c0 ¼ sin ec00, and no in-plane refraction takes place since the the impact parameter b crosses the symmetry axis. Accordingly ¼ 2 refracting interfaces’ normal is vertical. One finds cos e00 then, the bundling of rays in the forward direction is reduced as 2 2 e increases. n0 sin e, which is the analog of Eq. (2). Referring to ¼the experiment’sÀ setup and coordinates as defined in Figs. 3(e)–3(g), the curve (x, z) described by the artificial IV. ARTIFICIAL CIRCUMHORIZONTAL ARC CHA on the vertical wall is parametrized by the angle / as We now turn to the artificial CHA, as shown in Fig. 5(a), x l sin /00 for which the first refraction is at the side wall of the cylinder ; (8) z ¼ cos /00 tan /00 in the experiment, as seen in Figs. 3(e)–3(g). Thus, it is this refraction that must be treated according to the inclined 2 2 2 2 2 skew-ray theory of Bravais. Again, we decompose the prob- where /00 / /0. Since z x l tan e00, and tan e00 is a constant¼ (independentÀ of¼ðx),30þtheÞ artificial projected lem into two parts. The material’s index of refraction n0 directly determines the change in the inclination angle, CHA for each inclination e is a hyperbola in the xz-plane, as seen in Fig. 5(b). If the projection were onto a sphere, the sin e n0 sin e0, and the in-plane azimuthal refraction (i.e., ¼ natural CHA halo’s geometry of a circle segment parallel to as seen from above) follows sin / n0 sin /0, with the effec- tive index of refraction being12–15,19¼,20 the horizon (i.e., at constant elevation) were to be recovered, as discussed in Ref. 31 and Figs. 5(c) and 5(d). 2 2 The angle to the surface normal, which determines the n0 sin e n0 À : (7) transmission coefficient T1 u , is u arccos cos / cos e , as cos2e ð Þ ¼ ð Þ ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi seen in Fig. 3(h). The transmission coefficient T2 e0 ð cÞ
578 Am. J. Phys., Vol. 85, No. 8, August 2017 M. Selmke and S. Selmke 578 corresponds to the second refraction changing the elevation reflections from the stem of the glass as well as artificial hel- only. The intensity may then be analyzed along the same iac arcs8 due to external reflections by the conical surface. lines as for the CZA, but with the cross-sectional factor Also, placing an opaque obstacle inside the glass of water of being cos / cos e.36 Again, this approach was used to set the the original experiment, a parhelion-like appearance can be transparency of the curves in Fig. 5(b). We find that the obtained.39 Likely, many more halo counterparts may be CHA is brightest at around e 69 in the experiment and realized along the lines presented here. ¼ 65 for the natural halo, whereas azimuthally it decays to zero for / p=2 (grazing incidence) where accordingly ! 12,15 APPENDIX A: COLORS /00 p=2 arcsin 1=n0 . The corresponding full azi- ¼ À ð Þ muthal width D/CHA00 of twice that value therefore ranges Throughout the article the computations focused mainly on from 125 180 in the experiment and 116 180 for ! ! the shape of the artificial halos. This appendix shall briefly the natural ice halo phenomenon. The half-circle limit is outline how the visual color perception of either artificial or reached in the reverse situation as compared to the CZA natural halos may be described. Perceived colors can be com- (here for grazing incidence with e p=2, as compared to ¼ puted as follows. First, the tristimulus values (trichromatic grazing exit e00 p=2 for the CZA). The natural CHA only c ¼ color space coordinates) X; Y; Z are computed via the CIE forms when the inclination is larger than p=2 eTIR 58 f g 15 À ¼ standard colorimetric observer color matching functions (ice ) and 62 (water), where the critical inclination as x"; y"; "z and from the spectral radiance L, via, e.g., determined for the CZA, in Eq. (5), can be reused. The halo f g X 01 L k x" k dk. We assumed a linear relation between rises in altitude (elevation) as the sun approaches the zenith this¼ quantityð Þ andð Þ the spectral power density of a given light (90 angle of incidence onto the side face), where it will be Ð 12,14 source. For a single wavelength, a narrow (FWHM 1 nm) e00 32. The experimental CHA reproduces these ¼ Gaussian line spectrum was chosen. Then, a linear transfor- behaviors closely. mation (a matrix product) converts these to linear RGB values, RL; GL; BL M X; Y; Z , which subsequently are V. CONCLUSION AND OUTLOOK convertedð to the commonÞ¼ Áð sRGB colorÞ space values through a We have argued that a very simple experiment, namely, a non-linear transformation. Finally, the values are normalized glass filled with water illuminated under various directions to give a valid sRGB color. The required data can be found online.37 The hereby obtained sRGB color values R ; G ; B of incidence, provides a rich phenomenology. We have f s s sg shown that the emerging projections of light closely corre- for each wavelength were multiplied by an intensity factor spond to two natural atmospheric ice halos: the circumzeni- I=I0 to recreate the azimuthal intensity decay of the corre- spondingly colored CZH segments corresponding to n0 k . thal and the circumhorizontal arcs. The general angular ð Þ characteristics derived and validated by the experiment also Corresponding computations assuming a solar spectrum, a apply to their natural counterparts. This demonstration LED light source spectrum, and an incandescent light bulb spectrum along with the dispersion n0 k of water are shown experiment may complement more complex ones based on ð Þ spinning glass crystals.19,31–35 It also produces a purer spec- in Fig. 7 for different elevations. These images can be com- pared to experiments or displays computed with dedicated trum as compared to a rainbow demonstration experiment 38 since, similar to the action of a prism, ideally no color- halo simulation software. Note that in the experiment col- overlap occurs. The experiment may be used as a halo alter- ors may superimpose (and green disappear) if the projection native to popular rainbow demonstrations,5 as an illustrative distance l is too small. example of skew-ray refraction, or simply to show dispersion. APPENDIX B: ARTIFICIAL SUNCAVE PARRY ARC We end with an outlook on similar experiments. One may use a water-filled martini cocktail glass as a refracting cone, We keep the notation of Fig. 3(a), where we imagine the as shown in Fig. 6. By the same idea that led us to the CZA cylinder being replaced by the cone of Fig. 6, and acknowl- analogy, one may confirm (see Appendix B) that the average edge that for the second refraction (the first being analogous geometry of light entering through the top air-water interface to the CZA) the normal vector is now n^ cos a=2 sin /; and leaving through the lateral cone surface results in an cos a=2 cos /; sin a=2 for the cone’s¼ ðÀ lateralð surface.Þ analogy to Parry’s halo.6,8,15 Using a cocktail glass, one may À ð Þ ð ÞÞ ^ The incidence direction vector i 0; sin ec0 ; cos ec0 makes find in addition an artificial parhelic circle due to external an angle u arccos ^i n^ to¼ð this normal.À The refractedÞ ¼ ðÀ Á Þ angle is then determined by n0 sin u sin u0. To get the refracted ray, one may rotate the external¼ unit vector n^ À according to the right hand rule by u0 about an axis perpendic- ular to the plane of incidence, defined by the unit vector k^ ^i n^= sin u. This can be done via Rodrigues’ rotation formula,¼  yielding the exiting ray’s unit direction vector r^ ¼ r^x; r^y; r^z n^ n0 cos u cos u0 ^i n0. The line described byð this unitÞ¼ directionð vectorÀ andÞþ starting at a height l above the origin (at the position of the glass) intersects the horizontal xy-(projection-)plane at the point x; y; 0 l e^z l ^r=^ez r^. Fig. 6. Analogy (in the sense of Fig. 1) of the parallel light refraction by a This is the artificial projected suncaveð ParryÞ¼ arc6,8À,15 parame-Á cone (e.g., a filled martini glass) and the ray path responsible for the upper trized by . suncave Parry arc (Refs. 6, 8, and 15). While the martini glass typically has / The complementary angle to the elevation is e00 an apex angle of a 70 instead of the 60 required for a perfect analogy, c ¼ the resulting pattern is very similar. Best results are obtained when a slit- arccos e^ ^r when r^ > 0, and negative that value when ðÀ z Á Þ y aperture is used to constrain illumination to a band across the diameter only. r^y < 0, signifying a Parry arc beyond the zenith with e00
579 Am. J. Phys., Vol. 85, No. 8, August 2017 M. Selmke and S. Selmke 579 Fig. 7. Light source spectral power densities for (a) sun light, (b) LED light, and (c) incandescent light bulb. The colors of the sources are shown in the spec- trum’s insets. Panels (d)–(f) give the corresponding perceived colors of the artificial CZAs in the xy-plane.
> p=2 (in front of the cocktail glass when viewed from the 12A. Bravais, “Memoire sur les halos et les phenome`nes optiques qui les accompagnent,” J. de l’ Ecole Royale Polytech. 31(18), 1–270 direction of incident light). The azimuths are /00 6arccos ¼ (1847). 0; 1; 0 r^ ; r^ ; 0 = r^ ; r^ ; 0 , where the plus sign 13 ðð ÞÁð x y Þ jð x y ÞjÞ W. J. Humphreys, Physics of the Air, 2nd ed. (McGraw-Hill Book applies to r^x < 0 and the negative sign to r^x > 0. The arc Company, Inc., New York and London, 1929). appears concave towards the shadow of the glass (corre- 14R. A. R. Tricker, Introduction to Meteorological Optics (Mills & Boon, sponding to the sun position and being suncave as its London, 1970). natural counterpart), and the angular distance to this 15A. Wegener, Theorie der Haupthalos (L. Friederichsen & Co, Hamburg, shadow, D arccos 0; cos e ; sin e ^r , corresponds 1926). S 16The distance must be much larger than the focal distance of the curved to the distance¼ to theðð sun forð theÞ À naturalð ÞÞ halo. Á Þ The hereby cylinder interface, f Rn0= n0 1 , where R is the cylinder’s radius. obtained angular coordinates agree with those in Ref. 15. Practically speaking, a¼ distanceð տÀ 10ÞR is fine. Alternatively then, the projected artificial Parry arc curve on 17Huygens (Ref. 18) hypothesized horizontal cylinders of arbitrary in-plane the floor may be computed via Eq. (3) if the following orientation to describe what was at that time thought to be a tangent arc to 15 replacements via the variables of Wegener’s chapter 12 are the 46 circular halo (Refs. 9 and 24). 18C. Huygens, Oeuvres Comple`tes Tome XVII: L’horloge a pendule de made: e00 p=2 h and /00 d and Wegener’s u being c ! À r ! 1651 a 1666. Travaux divers de physique, de mecanique et de technique the parametrization. Using glasses with smaller apex angles de 1650 a 1666. Traite des couronnes et des parhelies (1662 ou 1663) (ed. a (walls becoming increasingly vertical and more cylinder- J.A. Vollgraff. Martinus Nijhoff, Den Haag, 1932); see arc THS in Fig. 22, like), the arc inverts from suncave to sunvex and approaches §37, and its discussion. 19 the CZA in the limit a 0. M. Selmke, “Artificial Halos,” Am. J. Phys. 83(9), 751–760 (2015). ! 20G. P. K€onnen, “Polarization and intensity distributions of refraction hal- os,” J. Opt. Soc. Am. 73(12), 1629–1640 (1983). a)Electronic mail: [email protected]; URL: http:// 21Usually, the effective index of refraction n is discussed for rays entering a photonicsdesign.jimdo.com 0 dense material from air, as given in Eq. (7) (Refs. 12–14 and 19). For the 1C. J. Lynde, Gilbert Light Experiments for Boys (The A. C. Gilbert Co., inverse situation, note that refraction from n n to n 1 is mathemati- New Haven, Conn., 1920), Experiment No. 94. 0 cally equivalent to refraction from n 1 into ¼n 1=n .¼ Equation (1) fol- 2M. Edgeworth and R. L. Edgeworth, Practical Education 1, 55-56 (Printed 0 lows by inverting the corresponding effective¼ inverse¼ index of refraction, for J. Johnson, St. Paul’s Church-Yard, London, 1798). 1 1=n 0 À . 3D. Ivanov and S. Nikolov, “Optics demonstrations using cylindrical 0 22Wolfram½ð Þ Alpha, Wolfram Alpha LLC,
580 Am. J. Phys., Vol. 85, No. 8, August 2017 M. Selmke and S. Selmke 580 28Air mass (solar energy), from Wikipedia, the free encyclopedia,
581 Am. J. Phys., Vol. 85, No. 8, August 2017 M. Selmke and S. Selmke 581