1. INTRODUCTION

The experimental and theoretical research on the light passing through and coming out of a glass sphere illuminated by a point source placed at a distance $h$ from its center is the origin of the study presented here. Figure 1 gives an idea of the experimental setup. A more or less symmetrically curved white cardboard intercepting the light coming out of the sphere, about 145 mm in diameter, and illuminated by a quasi-point source placed 8 mm below its surface shows the relative positions of the caustics for the 0th, 1st, 2nd, 3rd, 4th, 5th, 7th, and 8th orders. The caustic for the order 0 (no internal reflection), which shines almost vertically, should be noted; the corresponding rainbow was considered by Lock and McCollum [1]. Flat and cylindrical screens were used for the measurements, and a system of coordinates and variables elaborated, with which the derivation of the caustics is made.

 

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.

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However, the idea of a spherical screen led to a sort of synthesis variable that allows formulas for the wavefronts, internal as well as exit ones, for every $k$ order, $k$ being the number of internal reflections. Moreover, the so-called zero internal and exit phase-fronts valid for each $k$ order were derived. From them, ray and wavefront “generators” were devised.

2. POINT SOURCE AND REFRINGENT SPHERE

Figure 2 shows the meridional section of a refringent sphere with radius $a$ and index of refraction $n$ illuminated by a point source $S$ placed at a distance $h$ below the center $C$. The refractive index of the surrounding is 1. The incident ray making the angle $\beta$ with the axis $SC$ meets the boundary at $A_{k-1}$ with the angle of incidence $i$. Refracted partially with the angle $r$, it crosses the sphere and then splits into two subrays at $A_k$: one is refracted outside in the direction ${\delta }_k$ with the angle $i$, and the other is reflected with the angle $r$. This process is repeated for each $A_k$ on the boundary, with $k=\mathrm{0,1},\mathrm{2,3},\dots$. The angle ${\alpha }_k$ denotes the angular position of $A_k$, ${\mu }_k$, the direction of the ray going from $A_{k-1}$ to $A_k$, and ${\delta }_k$ the direction of the ray exiting at $A_k$. Using the sine, refraction, and reflection laws, one obtains

 

Fig. 2. Meridional cross section of a refringent sphere illuminated by a point light source $S$. The ray is drawn for $\beta >0$.

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These equations are valid for both $\beta \ge 0$ and $\beta \le 0$.

3. SPHEREBOWS, SCREENBOWS, AND RAINBOWS

The variable $\beta$ refers to the angular position of the rays as they leave the source. This corresponds to the original deployment of the rays: it is the basic independent variable. $h/a$ being given, one can use $i$ or $z$ as independent variable.

${\alpha }_k$ corresponds to the deployment of the rays on the surface of the sphere with a concentration in the vicinity of ${({\alpha }_k)}_{\mathrm{crit}}$, the “spherebow” with ${({\alpha }_k)}_{\mathrm{crit}}={({\alpha }_k)|}_{\frac{d{\alpha }_k}{d\beta }=0}$. In Fig. 1, one sees easily the spherebows for ${({\alpha }_0)}_{\mathrm{crit}}$ and ${({\alpha }_1)}_{\mathrm{crit}}$; they are denoted “caustic rings” by Lock and Hovenac [2] in a paper related to the internal caustic structure of illuminated liquid droplets by parallel incident rays. ${\delta }_k$ corresponds to the directional deployment with a concentration in the vicinity of ${({\delta }_k)}_{\mathrm{crit}}$, the rainbow with ${({\delta }_k)}_{\mathrm{crit}}={({\delta }_k)|}_{\frac{d{\delta }_k}{d\beta }=0}$. $R_k$, which refers to the coordinate $x$ of the intersection point of the exit ray with the plane screen, corresponds to a planar deployment with a concentration in the vicinity of ${(R_k)}_{\mathrm{crit}}$, the “screenbow” with ${(R_k)}_{\mathrm{crit}}=(R_k){|}_{\frac{d{R}_k}{d\beta }=0}$. In Fig. 1, all the colored lines except the ones on the sphere are continuous sequences of screenbows, the intersection curves of the caustics by a curved white cardboard. The rainbows are screenbows at the “ends” of the caustics, which extend to infinity.

4. EXIT AND INTERNAL TANGENTIAL CAUSTICS

The rays constitute a family of lines in the $x–y$ plane. One obtains from the geometry in Fig. 2 the $x$ and $y$ intercepts of the ray and subsequently its equation. With $\beta$ as independent variable and the subscripts “ex” for “exit” and “in” for “internal,”

This equation constitutes the family of the exit rays of which the parameter is $\beta$. The exit tangential caustic is obtained by the solution of the system formed by Eq. (6) and its derivative with respect to $\beta$,

In the same way, for the internal tangential caustics,

Equations (9) and (10) correspond with Eq. (5) in the Avendano-Alejo et al. [3] paper related to caustics resulting from multiple reflections in a circular surface.

For parallel incident rays, with $z$ as independent variable,

As any ray, the one for $i=0$ passes through the caustic. The nearby ones for which $i<0$ and $i>0$ are by symmetry mutually divergent or convergent. Consequently, there exists a cusp point for $i=0$ on the $y$ axis. Then, for this point, $x=0$ and

With $p\equiv k+1$, ${\rho }_c\equiv {(x_{\text{in}})}_k$, and $z_c\equiv {(y_{\text{in}})}_k$,

 

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 $\frac{B}{a}=-3.9$, the cusp point of the virtual part of the tangential causitic, to $\frac{B}{a}=2.2$, the other extremity.

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5. EXIT AND INTERNAL SAGITTAL CAUSTICS

As evoked by van de Hulst [4] in his book Light Scattering by Small Particles on page 202, some rays cross the $SC$ symmetry axis and cause concentration points on the sagittal planes between two definite limits on the axis, one of which is a cusp point of the tangential caustic. With $\frac{{(x_{\text{ex}})}_k}{a}=0$ and $\frac{{(x_{\text{in}})}_k}{a}=0$, one obtains the resultant axial segments, i.e., the exit and the internal sagittal caustics. From Eqs. (7)–(10), they are given by

With $r\equiv {\theta }_r$ and ${\mu }_k\equiv \gamma -\pi /2$, Eq. (21) corresponds with Eq. (8) in the Lock and Hovenac [2] paper quoted earlier.

6. SPHERICAL SCREEN AND SYNTHESIS VARIABLE

Figure 4 illustrates the refringent sphere of radius $a$ pictured in gray and surrounded by a spherical screen of variable radius $B_k$. The exit ray meets this screen at the point $E_k$, of which the angular coordinate ${\sigma }_k$ constitutes a sort of synthesis in that the variables ${\alpha }_k$ and ${\delta }_k$ are part of it. In $\mathrm{\Delta }$ [$E_k/C/A_k$],

 

Fig. 4. Refringent sphere pictured in gray, the spherical screen of radius $B_k$, and the exit zero phase-front.

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The last two equations show how ${\alpha }_k$ and ${\delta }_k$ are related to ${\sigma }_k$. And, regardless of the value of $h/a$,

7. EXIT WAVEFRONT

In Fig. 4, $L_1$, $n{L}_2$, and $L_3$ are the sections of the optical path length ${\mathbf{Q}}_0$ between $S$ and $E_0$; for $k=1$, $L_2$ is doubled and so on; then

In $\mathrm{\Delta }$ [$E_k/C/A_k$], from the law of cosines,

From Eq. (1) with $z\equiv \sin i$, $t\equiv \sqrt{1-z^2}$, $u\equiv \sqrt{n^2-z^2}$, and $v\equiv \frac{h\cos \beta }{a}=\sqrt{{(\frac{h}{a})}^2-z^2}$,

Then, ${\mathbf{Q}}_k$ being the optical path length between $S$ and $E_k$,

When $\beta \equiv 0$ (parallel incident rays), $v=\frac{h\cos \beta }{a}=\frac{h}{a}$; then $v$ represents the relative distance to the center $C$ of an incident plane wavefront.

Let ${\mathbf{Q}}_c$ be the optical distance between the source and the wavefront; then

The minimum value of ${\mathbf{Q}}_c$ is the minimum one of ${\mathbf{Q}}_k(z)$ with $B_k/a=1$.

Equations (24) and (35), in which ${\mathbf{Q}}_k(z)={\mathbf{Q}}_c$, constitute parametric equations in $z$ for the ${\mathbf{Q}}_c$ wavefront. The rectangular coordinates for a point on its curve are $[(B_k/a)\cos {\sigma }_k,(B_k/a)\sin {\sigma }_k]$.

An example of a sequence of exit wavefronts is shown in Fig. 5.

 

Fig. 5. $h/a=1.057$, $n=1.53$, $k=0$; exit wavefronts corresponding to a sequence of optical path lengths ${\mathbf{Q}}_c$.

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8. EXIT ZERO PHASE-FRONT

In Fig. 4, the point $E_k$ belongs to a wavefront for which ${\mathbf{Q}}_k(z)={\mathbf{Q}}_c$. The point ${(E_k)}_0$ is such that its distance from $E_k$ equals ${\mathbf{Q}}_c$. Its polar coordinates are $[{(B_k)}_0,{({\sigma }_k)}_0]$ with the subscript 0 for “zero phase-front.” The totality of the points ${(E_k)}_0$ corresponding to the totality of the points $E_k$ that belong to the ${\mathbf{Q}}_c$ wavefront constitutes the zero phase-front. Is this front independent of ${\mathbf{Q}}_c$? It must be so in order to be meaningful.

In $\mathrm{\Delta }$ [${(E_k)}_0/C/E_k$] and from Eqs. (22) and (35),

Equations (39) and (41) define the exit zero phase-front and show that ${(B_k)}_0/a$ and ${({\sigma }_k)}_0$ are independent of ${\mathbf{Q}}_c/a$; they depend only on $n$, $h/a$, $k$, and naturally on $i$ or on $\beta$. From Eq. (35), it is to be observed that $B_k$ where ${\mathbf{Q}}_k/a=0$ and ${(B_k)}_0$ are identical. But, ${({\sigma }_k)}_0$ and ${\sigma }_k$ where $B_k={(B_k)}_0$ are different. Indeed, from Eqs. (24) and (39),

Consequently, the exit zero phase-front is unique. It is not, among the sequence of the exit wavefronts, the one for which ${\mathbf{Q}}_k/a=0$, as shown in Fig. 6. The minimum value of ${\mathbf{Q}}_k/a$ for the exit wavefronts is given by $L_1+L_2$. The zero phase-front defined through ${(B_k)}_0$ and ${({\sigma }_k)}_0$ is independent of $Q_k/a$, as it is for the source of the rays. The equation of any $\beta$ ray crossing a given wavefront is the one of the line joining the intersection point [$B_k/a\cos {\sigma }_k$, $B_k/a\sin {\sigma }_k$] to the corresponding “zero” point [${(B_k)}_0/a\cos {({\sigma }_k)}_0$, ${(B_k)}_0/a\sin {({\sigma }_k)}_0$] as shown in Fig. 7.

 

Fig. 6. Zero phase-front is unique; it does not come from any actualization of the exit wavefront equation which, moreover, is nonvalid for $Q_c=0$.

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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 $\beta <0$ and $\beta >0$ of the tangential caustic and of the exit zero phase-front are shown.

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9. INTERNAL WAVEFRONT

In Fig. 8, the dotted circle represents an imaginary spherical screen of variable radius $D_k$. In $\mathrm{\Delta }$ [$A\text{out}/C/{(F_k)}_1$], with ${\nu }_k\le \pi /2$,

 

Fig. 8. Dotted circle represents the meridional section of the imaginary spherical screen of radius $D_k$. The drawing is not to scale. The internal zero phase-front belongs to the “$n$” optical space, which is inside the sphere.

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The internal ray crosses the imaginary screen at the points ${(F_k)}_1$ and ${(F_k)}_2$, of which the angular coordinates are ${({\sigma }_k)}_1$ and ${({\sigma }_k)}_2$, which is negative in the drawing, with ${({\sigma }_k)}_1\ge {({\sigma }_k)}_2$. As shown in Fig. 8, ${\nu }_k={\alpha }_k-{({\sigma }_k)}_1$. Then,

In $\mathrm{\Delta }$ [($A_{\mathrm{in}}/C/A_{\mathrm{out}}$)] and from Eqs. (44) and (45),

In $\mathrm{\Delta }$ [$A_{\mathrm{in}}/C/{(F_k)}_1$], from the law of cosines,

In Fig. 8, ${{(\mathbf{Q}}_k)}_1$ denotes the optical path length of the ray between $S$ and ${(F_k)}_1$, and ${{(\mathbf{Q}}_k)}_2$ between $S$ and ${(F_k)}_2$ with

From Eqs. (31), (32), and (51), and $t$, $u$, and $v$ defined in Section 7,

From Eqs. (54) and (55), for ${{(\mathbf{Q}}_k)}_1$ and also for ${{(\mathbf{Q}}_k)}_2$, for a wavefront with the optical path length ${\mathbf{Q}}_c$,

The internal ray meets the imaginary screen (of variable radius) at two (variable) points, ${(F_k)}_1$ and ${(F_k)}_2$; their coincidence point ${(P_{\text{in}})}_k$ determines the boundary ${(G_{\text{int}})}_k$ between the $Q-1$ and $Q-2$ domains. This coincidence implies that $(L_2-L_{2-})=L_2/2=a\cos r=a\sqrt{1-\frac{z^2}{n^2}}$. Then, from Eqs. (45) and (48) and Eq. (51), with the subscript 12 for the boundary,

The last two equations determine the polar coordinates of the boundary curve ${(G_{\text{int}})}_k$ between the $Q-1$ and $Q-2$ domains. Figure 9 shows the combination $Q-1+Q-2$. Equations (45) and (56) define the wavefronts in $Q-1$, and Eqs. (48) and (56) the ones in $Q-2$.

 

Fig. 9. $h/a=1.057$, $n=1.53$, $k=0$, $k=1$: internal wavefronts and internal zero-phase fronts.

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10. INTERNAL ZERO PHASE-FRONT

The same procedure as the one in Section 8 is followed.

In Fig. 8, ${(F_k)}_0$ belongs to the “$n$”-optical space; its distance to ${(F_k)}_1$ is $Q_c/n$. In $\mathrm{\Delta }$ [${(F_k)}_0/C/{(F_k)}_1$], from Eqs. (43) and (56),

In Fig. 8, ${({\sigma }_k)}_0$ being the angular coordinate of ${(F_k)}_0$,

The same result is obtained with ${(F_k)}_2$.

Equations (61) and (64) define the internal zero phase-front. The ones for $h/a=1.057$, $n=1.53$, $k=0$, and $k=1$ are shown in Fig. 9.

The equation of any $\beta$ ray crossing a given wavefront is the one of the line joining the intersection point [$D_k/a\cos {({\sigma }_k)}_1$, $D_k/a\sin {({\sigma }_k)}_1$] (or [$D_k/a\cos {({\sigma }_k)}_2$, $D_k/a\sin {({\sigma }_k)}_2$]) to the corresponding “zero” point [${(D_k)}_0/a\cos {({\sigma }_k)}_0$, ${(D_k)}_0/a\sin {({\sigma }_k)}_0$] as shown at the bottom of Fig. 9.

11. RAY GENERATOR

A. Internal Ray Generator

Each point of the boundary curve ${(G_{\text{int}})}_k$ defined at the end of Section 9 is the middle point of the corresponding internal ray, which constitutes a chord of the circle of radius $a$. From the property of the circle, this ray is perpendicular to the line joining this middle point and the circle center $C$. The two form a sort of adjustable carpenter square. This fact allows the tracing of any internal ray by means of the curve ${(G_{\text{int}})}_k$, hereafter called the “internal ray generator.”

This property of the boundary curve ${(G_{\text{int}})}_k$ becomes evident from another point of view. In Fig. 8, $c_{\text{in}}/a=\sin r=z/n$, and the line segment $c_{\text{in}}$ intersects the middle point ${(P_{\text{int}})}_k$ of the interior $k$ ray, which is a chord of the circle of radius $a$. From that, Eqs. (57) and (58) follow. In the second drawing from the top of Fig. 10, one sees the internal and the exit ray generators for $n=1.53$, $k=0$, and $h/a=2.0$.

 

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.

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B. Exit Ray Generator

In Fig. 4, in the same way as for $c_{\text{in}}$ and the internal $k$ ray, the line segment $c_{\text{ex}}$ perpendicular to the exit $k$ ray is such that $c_{\text{ex}}/a=\sin i=z$. This segment $c_{\text{ex}}$ intercepts the exit $k$ ray at the middle point ${(P_{\text{ext}})}_k$ of the chord $\overline{A_k^{\prime }{A}_k}$ formed by its intercepts points with the circle of radius $a$. In an analog way, the next equations follow for the exit ray generator ${(G_{\text{ext}})}_k$:

Examples of exit ray generators are given in Fig. 10.

C. Incident Ray Generator

It also gives a ray generator ${(G_S)}_k$ for the incident part of the ray. Of course, one does not need it in order to trace an incident ray. However, it is useful, as shown in Section 12. In an analog way, one obtains the polar coordinates of the point ${(P_{\text{inc}})}_S$, which is the middle point of the chord formed by the intersection of the extended incident ray with the circle of radius $a$,

D. Polar Ray Equations

The ray generator uses the polar coordinates of the middle point of the chord supporting the ray as it is for the equation in polar coordinates of the line supporting the ray. Equations (57), (58), (65), (66), (67), and (68) being taken into account, the normal forms of the straight lines supporting the $k$ internal ray, the $k$ exit ray, and the $k$ incident ray are respectively given by

E. Generator Shape

The S shape of the branches of the exit ray generator at the top of Fig. 10 indicates the presence of a rainbow; the direction is given by the “rainbow point” for which the angular coordinate is an extremum. In the middle, the exit and the incident ray generators present a C shape with no rainbow; the exit one refers to a point image, and the incident one to a point object.

12. k RAYS’ BIG PICTURE

Figure 12 gives the big picture for the rays of orders $k=0$ to $k=3$. The graduated circular grid takes the place of the generator adjustable square. A circle radius constitutes the variable arm of the square; the tangent to this circle, which is the other arm, gives the ray direction; and the z-number of the ray is given by the circle radius scale. The circumference scale allows the determination of the angular ${\sigma }_k$ position of any point of a generator curve, the determination of the ${\alpha }_k$ position of the ray, and the determination of its ${\delta }_k$ or ${\mu }_k$ direction by means of its translation in such a way that it passes through the axes origin. Moreover, one determines easily the direction of the $k$ rainbow, the approximate position of the cusp point of the tangential caustic, and those of the extremities of the sagittal caustic. The precision of such operations depends on the size of the circular grid. But, regardless of its size, it gives an overall picture of the situation.

 

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 $Q/a=7.2$.

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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 $F_s$, 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 $k_1$ 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 $F_0$ and $F_1$, 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.

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13. ZERO PHASE-FRONT AS WAVEFRONT GENERATOR AND RAY GENERATOR

The zero phase-front is a virtual wavefront, and as such the equivalent rays traced from it are orthogonal to any wavefront as a consequence of the theorem of Malus and Dupin (Cornbleet [5], p. 23). Let $Q/a$ be the optical relative distance from the source at the intersection point between an exit ray and a generated wavefront, $[x_k,y_k]$ being the $x$ and $y$ coordinates of the intersection point, ${\delta }_k$ the ray direction, and $[\frac{{(B_k)}_0}{a}\cos {({\sigma }_k)}_0,\frac{{(B_k)}_0}{a}\sin {({\sigma }_k)}_0]$ the coordinates of the corresponding point of the $k$ exit zero phase-front. Then, any ray being a straight line,

The two last equations define parametrically the $Q/a$ exit wavefront generated by the $k$ exit zero phase-front acting as a wavefront generator. An example is given in Fig. 11. With ${\mu }_k$ taking the place of ${\delta }_k$ and ${({\sigma }_k)}_0$ being changed accordingly [Eq. (64) instead of Eq. (39)], these equations define parametrically the $Q/a$ internal wavefront generated by the $k$ internal wavefront generator.

The line joining the points $[{(B_k)}_0\cos {({\sigma }_k)}_0,{(B_k)}_0\sin {({\sigma }_k)}_0]$ and $[x_k,y_k]$ defines the $k$-generated exit ray, and so correspondingly for the internal ray, as shown at the bottom of Fig. 9.

By the way, the zero phase-front has a long history. Chastang and Farouki [6] wrote about it. In short, in regard to its name, one can present an indicative list of dates where it appears with different designations:

The appellation “zero phase-front” arose within the context of the microwave optics. In 1952, Eaton [11] published the short article “The zero phase-front in microwave optics,” where he defines the zero phase-front as “the virtual phase-front in image space that has the same phase as some reference phase-front.” In an example where a point source was considered, he writes, “[...] the virtual phase-front with the same phase as the illuminating point source [...]”

14. CONCLUSION

A solution to the problem of a point source of light placed at an arbitrary distance from a refringent sphere, valid also for parallel incident rays, has been elaborated. However, the intensity of light and the states of polarization have not been considered.

Figures 1 (see also Fig. 13) and 3 show experimentally and theoretically the existence of the rainbow of order $k=0$ for a sufficiently near point source.

The caustics have been deduced by means of the fact that the meridional rays constitute a family of lines in the $x–y$ plane.

The principal novelty of this approach, as far as I know, lies in the idea of the spherical screen from which were defined the key variables $B_k$ and ${\sigma }_k$ for the exit rays and the corresponding variables for the internal rays. As shown in Fig. 4 for the exit rays and in Fig. 8 for the internal rays, the optical path length can be geometrically and algebraically related to them. This connection leads to the wavefront equation and to the zero-phase-front one. The deduction of the internal wavefront equation requires a more complicated apparatus. But, as a by-product, it leads to the ray generator, which finds its best utility in the rays’ big picture, as shown in Fig. 12 for the orders $k=0$ to $k=3$.

In Fig. 7, the rays are traced by means of Eqs. (39) and (41) and Eqs. (24) and (35), in which ${\mathbf{Q}}_k(z)$ can only take values corresponding to real exit wavefronts.

Quite the contrary, through the procedure explained in Section 13, the zero phase-front generates independently not only equivalent rays as in Fig. 9 but also equivalent wavefronts, virtual and real, as in Fig. 11.

More is to be done regarding this particularity.

 

Fig. 13. Exit ray generator and the theoretical rainbows’ directions for $k=0$ to $k=8$; $n=1.53$, $h/a=1.057$, and $\beta >0$ as it is in Figs. 1, 3, 5, 6, 7, 9, 10, 11, and 12.

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