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.

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The distance must be much larger than the focal distance of the curved cylinder interface, f=Rn/(n1), where R is the cylinder's radius. Practically speaking, a distance 10R is fine.

17.

Huygens (Ref. 18) hypothesized horizontal cylinders of arbitrary in-plane orientation to describe what was at that time thought to be a tangent arc to the 46° circular halo (Refs. 9 and 24).

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Usually, the effective index of refraction n is discussed for rays entering a dense material from air, as given in Eq. (7) (Refs. 12–14 and 19). For the inverse situation, note that refraction from n = n0 to n = 1 is mathematically equivalent to refraction from n = 1 into n=1/n0. Equation (1) follows by inverting the corresponding effective inverse index of refraction, [(1/n0)]1.

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23.

The ratio r=D/w=0.347, where D is the CZA diameter and w is the image width, may be used to extract the angular distance to the zenith, ec=arctan(rX/2f)=16°. This value is in agreement with Eq. (2), e=27° (Ref. 22) and n = 1.31. The focal length of the rectilinear projection lens was f=14mm and X=23.6mm the APS-C sensor's x-dimension.

24.

For the natural halo, the exit angle's altitude e is close to tangential (Ref. 12) to the circular 46° halo, i.e., e+Dm, where Dm=2arcsin[n0sin(A/2)]A is the minimum deviation angle through a A=90° prism. However, it is not precisely equal (Refs. 13 and 14) except for e=arccos(n0/2)=22°.

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The factor cote only appears in case of the natural flat hexagonal plate crystals, see Ref. 11. We also neglect statistical misalignments of the ice crystals, cf. Refs. 9 and 18.

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29.

To see this, consider the dot product cosφ=î·n̂ of the incidence ray's unit vector î=(0,sinec,cosec) and the cylinder surface normal n̂=(sinϕ,cosϕ,0) at the point of refraction, as shown in Figs. 3(a)–3(d) 

30.

Figures 3(e)–3(h) show that tane=z/ρ,sinϕ=x/ρ, and cosϕ=l/ρ, where ρ2=l2+x2.

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36.

To include the effect of the reduced projection intensity on the wall due to the inclination of the rays, one may further include a factor cosϕ. We choose to ignore it here to be in line with the natural CHA intensity.

37.
See, for example, the international commission on illumination, <http://www.cie.co.at> and Wikipedia, <https://en.wikipedia.org/wiki/SRGB>.
38.
L.
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and
M.
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39.

The deflection through a cylinder of radius R2 with an inner (centered) opaque cylindrical obstacle of radius R1 is Δ=2(αβ), where α and β are the incidence and refracted angles, respectively, related by sinα=nsinβ. The symmetrical geometry of the corresponding least-deflected ray touching the inner cylinder tangentially shows that sinβ=R1/R2. Demanding that its deflection equals the actual prismatic angle of minimum deviation (the parhelion azimuth) (Ref. 24), Dm(n)=Δ, one finds after lengthy algebra, sin2α=n2sin2(A/2) (here, A=60°). Using this solution yields R1/R2=1/2. Illuminating (through the side walls) a filled glass with half its interior blocked produces thus a parhelion-like projection and represents the (false) mechanism Huygens conceived of (Ref. 7 and 16). It is thus not an analog of the actual parhelion mechanism (Refs. 6–10 and 12–15).

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