A Simple Proof of Damien ’ s Theorem and Duality in Theory of the Zero-Distance Phase Front

Duality plays the main role in all mathematical theories. In wave optics based on the concept of “the zero-distance phase front,” duality takes the form of Damien’s theorem and the mirror symmetry between the conic refracting surfaces with the plane zerodistance phase front and the plane refracting surfaces with the conic zero-distance phase front. A systematic study of these dualities has been performed.


Introduction
In 1825 Joseph Gergonne [1] noticed the "symmetry" of roles played by points and lines in definitions and theorems of geometry: if we assume that all parallel lines intersect at infinity (the axiom of projective geometry), any theorem or definition one-to-one corresponds to another theorem or definition (the so-called dual statement), when we substitute "point" for "line," "lie on" for "pass through," "collinear" for "concurrent," "intersection" for "join," or vice versa.At present, various types of duality have been discovered in various branches of mathematics and theoretical physics (including optics [2,3]).
It is known that the application of the law of refraction together with the axioms of geometry gives rise to "optical geometry," implemented in design of aspheric lenses [4].In optical geometry based on the concept of "the zero-distance phase front" [5] duality takes the form of Damien's theorem and of the mirror symmetry between the conic refracting surfaces with the plane zero-distance phase front and the plane refracting surfaces with the conic zero-distance phase front [4,[6][7][8][9][10][11].
We propose a simple proof of Damien's theorem and describe dual pairs: "the refracting (reflecting) surface-the zero-distance phase front."

Refraction and Reflection
A monochromatic wave can be described by a locus of points of the wave with the same phase of oscillation, the so-called phase front [12].The wave propagation can be described by a family of the phase fronts.An alternative way of describing of wave propagation is given by orthogonal trajectories to the phase fronts, the so-called light rays.The shape of the phase fronts (and light rays) is usually determined by the geometry of the source and the properties of the medium.In a homogeneous medium a point source  generates a family of concentric spherical (circular) phase fronts with a common center at  (or a homocentric beam of light rays with a vertex at ) (Figure 1(a)), and a plane source generates plane phase fronts (or parallel rays) (Figure 1(b)).In an inhomogeneous medium phase fronts can be very complicated, however the principle of equal optical paths [12] states that, in an optical system consisting of refracting and reflecting elements, the optical path length (OPL) between any two phase fronts is identically the same for all rays (Figure 2).The OPL is defined by the line integral along any ray path : where (, , ) is the refraction index at each point along the ray path  and Let us consider the interaction of a spherical wave from a point source  with a single refracting (or reflecting) surface  separating two media with refraction indexes  1 and  2 at a distance  = 0 to the right of it (Figure 3(a)).For simplicity, but with no loss of generality, let us assume that two closely adjacent rays from a source  are focused by the refracting surface at a point , where the refracting surface is an axially symmetric surface, and that the axis of symmetry 0 passes through points  and  [6][7][8][9].A plane containing the axis 0 is a meridional plane.In this plane the curve  represents a refracting surface, closely adjacent rays are incident upon the plane  in a neighborhood of the point , and  and  are bipolar coordinates of  with respect to  and .
In case of the refracting surface  (Figure 3(a)) the principle of equal optical paths takes the form [6][7][8][9]  1 ⋅  =  2 ⋅ . ( According to the theorem of Malus and Dupin [6][7][8][9]12] the normal congruence between light rays and phase fronts will remain the normal congruence after refraction on .Thus,  and  are the opposites of the two right triangles with the common hypotenuse, so that the angles  and  are related by the equation where  and  are the angles between the incident and the refracted phase fronts with the surface  at .As rays are the normals to the phase fronts, the angles  and  can be interpreted as the angles between the incident and the refracted rays with the normal to the surface  at  (Figure 3  Let us design a refracting surface  which transforms a divergent beam of rays (spherical wave) radiated from the on-axis source  into a plane-parallel beam of rays (a plane wave).Let the axis  be perpendicular to the axis of symmetry (Figures 5(a) and 5(b)).From (2) we have where  =  ⋅ cos .Direct integration of (4) leads to The constant is being evaluated by the condition  =  when  = 0.So (5) gives the conic section: where  ≡  2 / 1 and  ≡ (1 − ).The source is located at the focus farthest from the conic surface.
In case of the reflecting surface  (Figure 3(b)), the principle of equal optical paths takes the form  = −  (7) or where  and  are the angles between the incident and the reflected phase fronts with the surface  at .As rays are the normals to the phase fronts, the angles  and  can be interpreted as the angles between the incident and the reflected rays with the normal to the surface  at  (Figure 4(b)).Equation ( 8) describes the law of reflection;  = −.Note, that ( 8) is (3) if we put  1 = 1 and  2 = −1.
Lemma 2. The incident angle  (and therefore the reflected angle ) is uniquely determined by the difference between the incident and the reflected angles: Δ ≡  − ( − ) (Figure 4(b)).Indeed, from the law of reflection (2) we easily obtain the incident angle  = (−Δ)/2 and the angle of reflection  = −.
Let us design a reflecting surface  which transforms a divergent beam of rays (a spherical wave) radiated from the source  into a plane-parallel beam of rays (a plane wave) (Figure 5(c)).In this case from (7) we have where  =  ⋅ cos .Direct integration of (9) leads to The constant is being evaluated by the condition  =  when  = 0.So (10) gives a parabola with a focal length :

Zero-Distance Phase Front
Phase fronts will be some changing surfaces: at one time they can be completely smooth but at the next moment they have cusps or other singular points [11].It is known that from a family of phase fronts can be extracted one in the simplest form which is a representative of the entire family nevertheless [11].This phase front is the so-called zero-distance phase front [4][5][6][7][8][9].Let us continue the refracted ray in its reverse direction to the point , where the negative sign indicating the reverse direction.As point  moves over the surface , point  moves over a surface ℎ.This surface ℎ is the locus of points {}, which is the desired zerodistance phase front (Figure 6(a)).The zero-distance phase front replaces the point source  and the refracting surface , so that this replacement does not affect the family of wavefronts in the second medium.
International Journal of Antennas and Propagation  For the case of a reflecting surface  we put  1 =  and  2 = − in (12) (Figure 6(b)).Therefore  is continued in the forward direction that is on the side of the reflector opposite to the source , and Note that the zero-distance phase front of a parallel beam of rays is a plane (Figures 5(a), 5(b), and 5(c)).Thus, we have the following.
If  2 >  1 (the refraction occurs from a less dense medium to a more dense medium), the single refracting surface giving the plane zero-distance phase front is a hyperbolic surface (Figure 5(a)).
If  2 <  1 (the refraction occurs from a more dense medium to a less dense medium), the single refracting surface giving the plane zero-distance phase front is an elliptical surface (Figure 5(b)).
If  1 = 1 and  2 = −1, the single reflecting surface giving the plane zero-distance phase front is a parabolic surface (Figure 5(c)).
Let us consider the case in which the refracting surface  is a plane surface [11].If  = || is the length of this ray from (−, 0) to (0, ) and  is the refracted angle, the point  of the zero-distance phase front (12) has coordinates The relationship between the angle of incidence  and the angle of refraction  is given by Snell's law (3).The height of  with respect to 0 and the length of the ray from  to  are given by Substituting the previous relations, we obtain Eliminating  between these two expressions and using the identity 1 + tan 2  = 1/cos 2 , we obtain The zero-distance phase front of a plane refracting surface is a conic section centered at the origin 0. In polar coordinates, (17) takes the form where The source is located at the focus of the conic surface.Thus, we have the following.
If  2 >  1 (the refraction occurs from a less dense medium to a more dense medium), the zero-distance phase front of the plane refracting surface is a hyperbolic surface (Figure 7(a)).
If  2 <  1 (the refraction occurs from a more dense medium to a less dense medium), the zero-distance phase front of the plane refracting surface is an elliptical surface (Figure 7(b)).It is known that if  1 = 1 and  2 = −1, the zero-distance phase front of the plane reflecting surface is a point (, 0).
Damien's theorem [4,[6][7][8][9][10] states the following: if a surface ℎ is a zero-distance phase front of the refracting surface  with respect to a point source , then the inversion in a circle centered on  leads to the transformation of  into a surface   and ℎ into a surface ℎ  , so that   is the zerodistance phase front for the refracting surface ℎ  with respect to the same point source  and for the same refraction indexes (Figure 8).A refracting surface inverts into a zero-distance phase front and vice versa.
Let us prove this theorem.The inversion in the circle centered at  leads to the transformation of  into   and ℎ into ℎ  .A straight line passing through points  and  intersects the surface   at the point   .A straight line passing through points  and  intersects the surface ℎ  at the point   .As points  and   and points  and   are inverse points with respect to the same inversion circle, they are connected by the following relation: where  is the radius of the inversion circle.Let us suppose that the surfaces  and ℎ  are refracting surfaces separating optically homogeneous media with refraction indexes  1 and  2 (the point source  within the first medium).Let us consider triangles Δ and Δ    , where ∠ = ∠    .According to (19a) and (19b), the lengths of the sides of the triangle Δ are proportional to the lengths of the sides of the triangle Δ.Thus, the triangles Δ and Δ    are similar and ∠ = ∠    = Δ; that is,  −  =   −   .By lemma for Snell's law, the angles of incidence at the refracting surfaces  and ℎ  are the same:  =   .From ( 4 refracting surface and the zero-distance phase front will be swapped.Similarly, if the circle of inversion has the radius  =  √ 2 and the center at the point source , the reflecting surface and the zero-distance phase front will be swapped too. In Figure 9 the inversion of the plane zero-distance phase front  = (1 −  1 / 2 )/ cos  with respect to a point source  and the radius  = √1 −  1 / 2 is a circle  = (1 −  1 / 2 ) cos  passing through the source , and the inversion of the conic refracting surface, (6), is the limacon of Pascal (, ) [4,[6][7][8][9], where Thus, according to Damien's inversion theorem, the pair "the conic refracting surface-the plane zero-distance phase front" (Figures 5(a The inversion of the plane zero-distance phase front  = 2/ cos  is a circle  =  ⋅ cos  passing through the source .The inversion of the parabolic reflecting surface, (11), with respect to a point source  and the radius  =  √ 2, is the cardioid (, ) [4,[6][7][8][9], where Thus, according to Damien's inversion theorem, the pair "the parabolic reflecting surface-the plane zero-distance phase front" (Figure 5(c)) corresponds to the dual pair "the spherical reflecting surface-the zero-distance phase front in the form of the cardioid" (Figure 9(c)).In Figure 10 the inversion of the plane refracting surface  = / cos  with respect to a point source  and the radius  = √1 −  1 / 2 is a circle  = √1 −  1 / 2 cos  passing through the source , and the inversion of the conic refracting surface (18) is the limacon of Pascal (, ) [4,[6][7][8][9], where International Journal of Antennas and Propagation Thus, according to Damien's inversion theorem, the pair "the plane refracting surface-the conic zero-distance phase front" (Figures 7(a According to Damien's inversion theorem, the pair "the plane reflecting surface-the point zero-distance phase front" (Figure 7(c)) corresponds to the dual pair "the point reflecting surface-the spherical zero-distance phase front" (Figure 10(c)).

Conclusion
(1) A simple proof of Damien's theorem has been proposed.
(2) A systematic study of the dual pairs "the optical surface-the zero-distance phase front" has been performed for both refractive and reflective optical surfaces.

Figure 3 :
Figure 3: (a) The law of refraction and (b) the law of reflection from the point of view of the principle of equal optical paths.

Lemma 1 .
(a)).Equation (3) describes Snell's law of refraction.The incident angle  (and therefore the refracted angle ) is uniquely determined by the difference between the incident and the refracted angles: Δ ≡  −  (Figure4(a)).

Figure 4 :
Figure 4: (a) The law of refraction and (b) the law of reflection from the point of view of ray optics.

Figure 6 :
Figure 6: The zero-distance phase front ℎ = {} with respect to vertex  of the homocentric beam of light rays.(a) The ray refracted at point  of the surface  goes in the opposite direction for the distance ( 2 / 1 ) || to the point .(b) The ray reflected at point  of the surface  goes in the opposite direction for the distance || to the point .

Figure 7 :Figure 8 :
Figure 7: Refraction of rays from a point  on a plane surface : (a)  <   ; (b)  >   .(c) Reflection of rays from a point  on a plane surface .

Figure 10 :
Figure 10: "Designing" the optical surfaces for the spherical zero-distance phase fronts passing through the source : (a) the refracting surface  1 <  2 ⇒ the limacon of Pascal with  > 0; (b) the refracting surface  1 >  2 ⇒ the limacon of Pascal with  < 0; (c) the reflecting surface ⇒ the point reflector (small mirror sphere).
) and 5(b)) corresponds to the dual pair "the spherical refracting surface-the zero-distance phase front in the form of the limacon of Pascal" (Figures9(a) and 9(b)).
) and 7(b)) corresponds to the dual pair "the refracting surface in the form of the limacon of Pascal-the spherical zero-distance phase front" (Figures10(a) and 10(b)).