Abstract

This study relates to a refringent sphere illuminated by a point source placed at a distance h from its center; for h the light beam becomes parallel. A selection of variables, principally angular with the center of the sphere as a common point, allows a global, straightforward, and geometrically transparent way to the rays, caustics, and wavefronts, internal as well as external, for every k order, k being the number of internal reflections. One obtains compact formulas for generating the rays and the wavefronts.

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References

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  1. J. A. Lock and T. A. McCollum, “Further thoughts on Newton’s zero-order rainbow,” Am. J. Phys. 62, 1082–1089 (1994).
    [Crossref]
  2. J. A. Lock and E. A. Hovenac, “Internal caustic structure of illuminated liquid droplets,” J. Opt. Soc. Am. A 8, 1541–1553 (1991).
    [Crossref]
  3. M. Avendano-Alejo, L. Castaneda, and I. Moreno, “Caustics and wavefronts by multiple reflections in a circular surface,” Am. J. Phys. 78, 1195–1198 (2010).
    [Crossref]
  4. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).
  5. S. Cornbleet, Microwave and Optical Ray Geometry (Wiley, 1984).
  6. J.-C. A. Chastang and R. T. Farouki, “The mathematical evolution of wavefronts,” Opt. Photon. News 3(1), 20–23 (1992).
    [Crossref]
  7. J. Bernouilli, “Lineae Cycloidales, Evolutae, Anti-Evolutae, Causticae, Anti-Causticae, Peri-Causticae. Earum usus & simplex relatio ad se invicem. Spira mirabilis,” Band 5, 62–74 (Birkhäuser, 1999), Die Werke von Jakob Bernoulli.
  8. A. Quetelet, “Démonstration et développements des principes fondamentaux de la théorie des caustiques secondaires,” Nouv. Mem., Acad. R. Sci. B.-L., Bruxelles 5, 5–52 (1829).
  9. A. Cayley, “A memoir upon caustics,” Philos. Trans. R. Soc. London 147, 273–312 (1857).
    [Crossref]
  10. R. A. Herman, Treatise on Geometrical Optics (Cambridge University, 1900).
  11. J. E. Eaton, “The zero phase-front in microwave optics,” IRE Trans. Antennas Propag. 1, 38–41 (1952).
    [Crossref]
  12. R. Damien, Théorème sur les Surfaces d’Onde en Optique Géométrique (Gauthier-Villars, 1955).
  13. O. N. Stavroudis, “Refraction of wavefronts: a special case,” J. Opt. Soc. Am. 59, 114–115 (1969).
    [Crossref]
  14. R. T. Farouki and J.-C. A. Chastang, “Curves and surfaces in geometrical optics,” in Mathematical Methods in Computer Aided Geometric Design II (Academic, 1992), pp. 239–260.
  15. M. Avendano-Alejo, L. Castaneda, and I. Moreno, “Exact wavefronts and caustic surfaces produced by planar ripple lenses,” Opt. Express 23, 21637–21649 (2015).
    [Crossref]
  16. A. Gitin, “Zero-distance phase front of an isoplanar optical system,” Opt. Commun. 367, 50–58 (2016).
    [Crossref]

2016 (1)

A. Gitin, “Zero-distance phase front of an isoplanar optical system,” Opt. Commun. 367, 50–58 (2016).
[Crossref]

2015 (1)

2010 (1)

M. Avendano-Alejo, L. Castaneda, and I. Moreno, “Caustics and wavefronts by multiple reflections in a circular surface,” Am. J. Phys. 78, 1195–1198 (2010).
[Crossref]

1994 (1)

J. A. Lock and T. A. McCollum, “Further thoughts on Newton’s zero-order rainbow,” Am. J. Phys. 62, 1082–1089 (1994).
[Crossref]

1992 (1)

J.-C. A. Chastang and R. T. Farouki, “The mathematical evolution of wavefronts,” Opt. Photon. News 3(1), 20–23 (1992).
[Crossref]

1991 (1)

1969 (1)

1952 (1)

J. E. Eaton, “The zero phase-front in microwave optics,” IRE Trans. Antennas Propag. 1, 38–41 (1952).
[Crossref]

1857 (1)

A. Cayley, “A memoir upon caustics,” Philos. Trans. R. Soc. London 147, 273–312 (1857).
[Crossref]

1829 (1)

A. Quetelet, “Démonstration et développements des principes fondamentaux de la théorie des caustiques secondaires,” Nouv. Mem., Acad. R. Sci. B.-L., Bruxelles 5, 5–52 (1829).

Avendano-Alejo, M.

M. Avendano-Alejo, L. Castaneda, and I. Moreno, “Exact wavefronts and caustic surfaces produced by planar ripple lenses,” Opt. Express 23, 21637–21649 (2015).
[Crossref]

M. Avendano-Alejo, L. Castaneda, and I. Moreno, “Caustics and wavefronts by multiple reflections in a circular surface,” Am. J. Phys. 78, 1195–1198 (2010).
[Crossref]

Bernouilli, J.

J. Bernouilli, “Lineae Cycloidales, Evolutae, Anti-Evolutae, Causticae, Anti-Causticae, Peri-Causticae. Earum usus & simplex relatio ad se invicem. Spira mirabilis,” Band 5, 62–74 (Birkhäuser, 1999), Die Werke von Jakob Bernoulli.

Castaneda, L.

M. Avendano-Alejo, L. Castaneda, and I. Moreno, “Exact wavefronts and caustic surfaces produced by planar ripple lenses,” Opt. Express 23, 21637–21649 (2015).
[Crossref]

M. Avendano-Alejo, L. Castaneda, and I. Moreno, “Caustics and wavefronts by multiple reflections in a circular surface,” Am. J. Phys. 78, 1195–1198 (2010).
[Crossref]

Cayley, A.

A. Cayley, “A memoir upon caustics,” Philos. Trans. R. Soc. London 147, 273–312 (1857).
[Crossref]

Chastang, J.-C. A.

J.-C. A. Chastang and R. T. Farouki, “The mathematical evolution of wavefronts,” Opt. Photon. News 3(1), 20–23 (1992).
[Crossref]

R. T. Farouki and J.-C. A. Chastang, “Curves and surfaces in geometrical optics,” in Mathematical Methods in Computer Aided Geometric Design II (Academic, 1992), pp. 239–260.

Cornbleet, S.

S. Cornbleet, Microwave and Optical Ray Geometry (Wiley, 1984).

Damien, R.

R. Damien, Théorème sur les Surfaces d’Onde en Optique Géométrique (Gauthier-Villars, 1955).

Eaton, J. E.

J. E. Eaton, “The zero phase-front in microwave optics,” IRE Trans. Antennas Propag. 1, 38–41 (1952).
[Crossref]

Farouki, R. T.

J.-C. A. Chastang and R. T. Farouki, “The mathematical evolution of wavefronts,” Opt. Photon. News 3(1), 20–23 (1992).
[Crossref]

R. T. Farouki and J.-C. A. Chastang, “Curves and surfaces in geometrical optics,” in Mathematical Methods in Computer Aided Geometric Design II (Academic, 1992), pp. 239–260.

Gitin, A.

A. Gitin, “Zero-distance phase front of an isoplanar optical system,” Opt. Commun. 367, 50–58 (2016).
[Crossref]

Herman, R. A.

R. A. Herman, Treatise on Geometrical Optics (Cambridge University, 1900).

Hovenac, E. A.

Lock, J. A.

J. A. Lock and T. A. McCollum, “Further thoughts on Newton’s zero-order rainbow,” Am. J. Phys. 62, 1082–1089 (1994).
[Crossref]

J. A. Lock and E. A. Hovenac, “Internal caustic structure of illuminated liquid droplets,” J. Opt. Soc. Am. A 8, 1541–1553 (1991).
[Crossref]

McCollum, T. A.

J. A. Lock and T. A. McCollum, “Further thoughts on Newton’s zero-order rainbow,” Am. J. Phys. 62, 1082–1089 (1994).
[Crossref]

Moreno, I.

M. Avendano-Alejo, L. Castaneda, and I. Moreno, “Exact wavefronts and caustic surfaces produced by planar ripple lenses,” Opt. Express 23, 21637–21649 (2015).
[Crossref]

M. Avendano-Alejo, L. Castaneda, and I. Moreno, “Caustics and wavefronts by multiple reflections in a circular surface,” Am. J. Phys. 78, 1195–1198 (2010).
[Crossref]

Quetelet, A.

A. Quetelet, “Démonstration et développements des principes fondamentaux de la théorie des caustiques secondaires,” Nouv. Mem., Acad. R. Sci. B.-L., Bruxelles 5, 5–52 (1829).

Stavroudis, O. N.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

Am. J. Phys. (2)

J. A. Lock and T. A. McCollum, “Further thoughts on Newton’s zero-order rainbow,” Am. J. Phys. 62, 1082–1089 (1994).
[Crossref]

M. Avendano-Alejo, L. Castaneda, and I. Moreno, “Caustics and wavefronts by multiple reflections in a circular surface,” Am. J. Phys. 78, 1195–1198 (2010).
[Crossref]

IRE Trans. Antennas Propag. (1)

J. E. Eaton, “The zero phase-front in microwave optics,” IRE Trans. Antennas Propag. 1, 38–41 (1952).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Nouv. Mem., Acad. R. Sci. B.-L., Bruxelles (1)

A. Quetelet, “Démonstration et développements des principes fondamentaux de la théorie des caustiques secondaires,” Nouv. Mem., Acad. R. Sci. B.-L., Bruxelles 5, 5–52 (1829).

Opt. Commun. (1)

A. Gitin, “Zero-distance phase front of an isoplanar optical system,” Opt. Commun. 367, 50–58 (2016).
[Crossref]

Opt. Express (1)

Opt. Photon. News (1)

J.-C. A. Chastang and R. T. Farouki, “The mathematical evolution of wavefronts,” Opt. Photon. News 3(1), 20–23 (1992).
[Crossref]

Philos. Trans. R. Soc. London (1)

A. Cayley, “A memoir upon caustics,” Philos. Trans. R. Soc. London 147, 273–312 (1857).
[Crossref]

Other (6)

R. A. Herman, Treatise on Geometrical Optics (Cambridge University, 1900).

J. Bernouilli, “Lineae Cycloidales, Evolutae, Anti-Evolutae, Causticae, Anti-Causticae, Peri-Causticae. Earum usus & simplex relatio ad se invicem. Spira mirabilis,” Band 5, 62–74 (Birkhäuser, 1999), Die Werke von Jakob Bernoulli.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

S. Cornbleet, Microwave and Optical Ray Geometry (Wiley, 1984).

R. T. Farouki and J.-C. A. Chastang, “Curves and surfaces in geometrical optics,” in Mathematical Methods in Computer Aided Geometric Design II (Academic, 1992), pp. 239–260.

R. Damien, Théorème sur les Surfaces d’Onde en Optique Géométrique (Gauthier-Villars, 1955).

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Figures (13)

Fig. 1.
Fig. 1. Glass sphere illuminated from the bottom up by a quasi-point source placed very close to its surface. The number 0 marks the order 0, and so on. The corresponding rainbows’ theoretical directions are given in Fig. 13.
Fig. 2.
Fig. 2. Meridional cross section of a refringent sphere illuminated by a point light source S. The ray is drawn for β>0.
Fig. 3.
Fig. 3. h/a=1.057, n=1.53, k=0, and the vertical scale is compressed vertically. The top of the figure shows the progression of the rays and their virtual extension. On the graph, the dotted line, which is a sort of passage through the infinite between the virtual and the real parts of the tangential caustic, gives the direction of the k=0 rainbow. The sagittal caustic, passing through the infinite limits of the vertical axis, extends from Ba=3.9, the cusp point of the virtual part of the tangential causitic, to Ba=2.2, the other extremity.
Fig. 4.
Fig. 4. Refringent sphere pictured in gray, the spherical screen of radius Bk, and the exit zero phase-front.
Fig. 5.
Fig. 5. h/a=1.057, n=1.53, k=0; exit wavefronts corresponding to a sequence of optical path lengths Qc.
Fig. 6.
Fig. 6. Zero phase-front is unique; it does not come from any actualization of the exit wavefront equation which, moreover, is nonvalid for Qc=0.
Fig. 7.
Fig. 7. h/a=1.057, n=1.53, k=0: the exit equivalent rays progress from the exit zero phase-front and cross exit wavefronts. Branches β<0 and β>0 of the tangential caustic and of the exit zero phase-front are shown.
Fig. 8.
Fig. 8. Dotted circle represents the meridional section of the imaginary spherical screen of radius Dk. The drawing is not to scale. The internal zero phase-front belongs to the “n” optical space, which is inside the sphere.
Fig. 9.
Fig. 9. h/a=1.057, n=1.53, k=0, k=1: internal wavefronts and internal zero-phase fronts.
Fig. 10.
Fig. 10. Ray generators. The exit rays at the bottom of the figure were traced by means of the polar ray Eq. (70). The two exit ray generators present different shapes: an S shape for h/a=1.057 with the presence of a rainbow, and a C shape for h/a=2.0 with the presence of a point image instead of a rainbow as noted at the end of Section 11.
Fig. 11.
Fig. 11. Exit wavefront generator, n=1.53, h/a=1.057, and k=1. The equivalent wavefronts are generated from the exit zero phase-front for which Q/a=0 to the wavefront for which Q/a=7.2.
Fig. 12.
Fig. 12. Rays’ big picture for the orders k=0 to k=3. The grey circle of radius z=1/n=1/1.53 corresponds to the optical space inside the sphere; it contains the four internal ray generators k=0, k=1, k=2, and k=3. The whole circle of radius z=1 corresponds to the optical space outside the sphere; it contains the incident and the exit ray generators K=0, K=1, K=2, and K=3. The story of the z=0.8 ray begins with the intersection point A between the incident ray generator and the dotted circle z=0.8. The segment of line between A and the point source allows the determination of the point Fs, where the incident ray enters the sphere and changes its direction. Points B, C, D, and E are the points where the exit generators are cut by this circle z=0.8. The straight line BC cuts the internal generator k1 at the point c, which belongs to the circle of radius z/n=0.8/1.53=0.523 and so on. This correspondence between for example C and c has been proved for the story of any z ray. Consequently, by means of the radii and the tangents, which are equivalent to the arms of the generators’ squares, one easily traces the internal and exit parts of the progressing z=0.8 ray, the exit points as F0 and F1, and that for any z ray. Moreover, as shown for the order k=1, the rainbows directions are easily determined by the exit generators’ points of minimum angular value, as required by Eq. (66). The rainbow point, defined in Fig. 10, at the “summit” of the K=1 S-shaped generator has to be noted. The internal ray generator always presents an S shape and consequently should also refer to a sort of internal rainbow. For a sphere of a very large diameter, a spherical screen placed inside it near the surface would intercept a circle of angular extent nearly equal to the one of this rainbow.
Fig. 13.
Fig. 13. Exit ray generator and the theoretical rainbows’ directions for k=0 to k=8; n=1.53, h/a=1.057, and β>0 as it is in Figs. 1, 3, 5, 6, 7, 9, 10, 11, and 12.

Equations (73)

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sini=c/a=(h/a)·sinβ=n·sinrz,
α0=iβ+(π2r)π/2,
αk=iβ+(k+1)(π2r)π/2,
δk=αk+i,
μk=αk+r.
(sinδkhasinβ)(xex)ka(cosδkhasinβ)(yex)ka=1.
(xex)ka=(ha)[cosδkcosβδkβ+sinδksinβ],
(yex)ka=(ha)[sinδkcosβδkβcosδksinβ].
(xin)ka=(1n)(ha)[cosμkcosβμkβ+sinμksinβ],
(yin)ka=(1n)(ha)[sinμkcosβμkβcosμksinβ].
(xex)ka=cosδk21z22(k+1)n2z2+zsinδk,
(yex)ka=sinδk21z22(k+1)n2z2zcosδk,
(xin)ka=(1n)cosμk11z2(2k+1)n2z2+zsinμkn,
(yin)ka=(1n)sinμk11z2(2k+1)n2z2zcosμkn.
(yex)ka=(ha)[(1)k2(ha)12(ha)(k+1n)],
(yin)ka=(ha)[(1)kn(ha1)(ha)(2k+1)];
(yex)ka=nsin(kπ+π/2)2n2(k+1)=n(1)k2(nk1),
(yin)ka=sin(kπ+π/2)n(2k+1)=(1)kn(2k+1).
(zca)=(1)p2p1nand(ρca)=0,
(yex)ka=(ha)(sinβcosδk)=(sinicosδk),
(yin)ka=(1n)(ha)(sinβcosμk)=(sinrcosμk).
ρk=arcsin(siniBk/a),
σk=αk+iarcsin(siniBk/a),
σk=δkarcsin(siniBk/a).
(Bk=a)(σk=αk),
(Bk)(σkδk).
L1=hcosβacosi,
n(k+1)L2=n(k+1)2acosr.
(L3)2+2(acosi)L3+(a2Bk2)=0;
L3=acosi+a2(cosi)2+(Bk2a2).
L1/a=vt,
n(k+1)L2/a=2(k+1)u,
L3/a=((Bk/a)2z2)t.
Qk/a=(2)t+2(k+1)u+v+(Bk/a)2z2,
Bk/a=z2+[Qk/a+2t2(k+1)uv]2.
Qk(z)=Qc.
a(sini)=(Bk)0sinϕk=Bksinρk,
(σk)0=δk+π+ϕk,
(σk)0=δk+π+arcsin(sini(Bk)0a),
(Bk/a)Qccosρk=[(Bk/a)Qc]2z2,
(Bk)0/a=[(Bk/a)Qc]2+(Qca)22BkaQcacosρk,=[(Bk/a)Qc]2+(Qca)22Qca(Bka)2z2,=z2+(Qca)2+[2t2(k+1)uv]2(Qca)2,(Bk)0/a=z2+[2t2(k+1)uv]2.
(σk)0=σk|Bk=(Bk)0+π+2arcsin(sini(Bk)0/a).
sinrDk=sinγka=sin(νk+r)a,
νk=arcsin(sinrDka)r.
(σk)1=μkarcsin(sinrDka).
L2=2acosr,
(σk)2=(σk)1+2νk+2rπ,
(σk)2=μk+arcsin(sinrDka)π.
Dk2=a2+(L2L2)22a(L2L2)cosr,
L2L2=acosr±(acosr)2(a2Dk2),
L2L2=a[1z2n2+(Dka)2z2n2],
(Qk)1=L1+nkL2+n(L2L2),
(Qk)2=L1+n(k+1)L2n(L2L2).
(Qk)1/a=vt+(2k+1)u+n2(Dk/a)2z2,
(Qk)2/a=vt+(2k+1)un2(Dk/a)2z2.
Dka=1nz2+[Qca+t(2k+1)uv]2.
(Dka)12=(zn),
(σk)12=αk+rπ/2=μkπ/2.
(Dk)0=(Dk)Qc2+(Qc/n)22cosγk(Dk)Qc(Qc/n),
2cosγk[Dka]Qc=2n[Dka]Qc2z2,
((Dk)0a)=[Dka]Qc2+[Qcn·a]22cosγk[Dka]Qc[Qcn·a],=z2n2+[t(2k+1)uv]2n2+2[Qcn·a]22[Qcn·a]2,(Dk)0a=1nz2+[t(2k+1)uv]2.
a(sinr)=(Dk)0sinψk,
(σk)0=μk+π+ψk,
(σk)0=μk+π+arcsin(sinr(Dk)0a).
(Bka)Gext=z,
(σk)Gext=δkπ/2.
(Bka)GS=z,
(σk)GS=2πarcsin(zha)arcsin(zha).
pinternal=zsin(μkθ),
pexit=zsin(δkθ),
pincident=zcos(β+θ),
xka=Qacosδk+(Bk)0acos(σk)0,
yka=Qasinδk+(Bk)0asin(σk)0.

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