Geometrical Optics: Some Applications of the Law of Intensity

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Geometrical optics: some applications of the law of intensity

David L. Shealy University of Alabama at Birmingham, Department of Physics, 1530 3rd Ave. S, Birmingham, AL 35294-1170

ABSTRACT Geometrical optics is commonly associated with the ray-like properties of light, such as, law of reflection, Snell’s law, ray tracing, the optical path length and phase. The geometrical optics law of intensity and the optical wavefront are also well known, but are perhaps less used as a basis for optical design than ray tracing. This paper will first review how the geometrical optics law of intensity leads to the invariance of the product of the intensity of light times an element of area along a bundle of rays. Then, we will discuss how the geometrical optics law of intensity provides a good foundation for understanding not only nonimaging optics through direct ways to design optical systems for prescribed illumination requirements but also imaging applications through a complete understanding of the differential geometry of the optical wavefront and the caustic surfaces.

1. INTRODUCTION Geometrical optics1–4 provides a ray-based description for the propagation of light and is commonly associated with properties of light, such as, law of reflection, Snell’s law, ray tracing, the optical path length, aberrations, and optical design. Geometrical optics provides an approximate description of a wide-range of optical phenomena and represents an approximate solution of Maxwell’s equations [5, pp. 375 - 392] for propagation of radiation in optical media when the wavelength of the light is small compared to the working dimensions and when the phase and amplitude of the optical field are smooth functions of the spatial coordinates over regions of the size of multiple wavelengths.3 The intensity law of geometrical optics,1 which is also known as the transport equation of the ray-optics field, provides a systematic way to evaluate the structure and properties of the wavefront, phase, and amplitude of the ray-optics field. The intensity law has been the foundation for development of optical technologies, such as, nonimaging optics, illumination engineering, and beam shaping, as well as a basis for a geometrical understanding of the structure the optical image via caustic theory. The tools of differential geometry6, 7 are important in understanding the structure and properties of the optical wavefront, image (caustic) surfaces, and the irradiance distributions, while cutting-edge experimental techniques are required to develop technologies for fabrication, testing, and assembly of optical devises with nanometer or smaller tolerances. In this paper, we discuss concepts of rays using the mathematical theory of curves, i.e., Frénetequations, as a foundation for analysis of the structure of a wavefront propagating in homogeneous media, based on the general integral of the eikonal equation developed by Stavroudis.4, 8–13 Explicitly, we show that a linear Taylor expansion of a ray propagation vector and application of the Frénetequations [6, Eq. (11.5)] lead to an important result relating the Laplacian of the eikonal to the principal curvatures of the wavefront and propagation distance. Then, the intensity law is shown to provide a method for evaluating the intensity of a beam as it propagates and a relationship between optical field amplitude, principal curvatures of the wavefront, and propagation distance. Next, the differential properties of the vector refraction (reflection) equation are shown to provide a complete description of how the wavefront normal curvatures and torsion propagate through an optical system, which are known as the generalize ray trace equations. The intensity law of geometrical optics provides a foundation for nonimaging optical design and beam shaping, which have been established and developed into useful technologies. We close with discussion of applications of the generalized ray trace equations for both imaging and nonimaging optical design. First, we illustrate some properties of caustic surfaces for lens systems of increasing complexity and correction of aberrations. Then, we describe some nomimaging optical systems whose designs are based on shaping optics to produce desired illuminations for given input light distributions. Further author information: D.L.S.: E-mail: [email protected], Tel.: 205-934-8068

Copyright 2006 Society of Photo-Optical Instrumentation Engineers. This paper will be published in Proc. SPIE 5876, Sept. 2005, and is made available as an electronic preprint with permission of SPIE. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in the paper for a fee or for commercial purposes, or modification of the content of the paper are prohibited. 2. GEOMETRICAL OPTICS 2.1. Principles of Geometrical Optics The concepts of rays, wavefronts, and energy propagation are fundamental in using geometrical optics for design of imagining or nonimagining optical systems. A brief overview of these concepts are presented in this section while illustrating how the geometrical optics field is evaluated by first determining the rays, which are used to determine the phase of the optical field independently from its amplitude, and then, by determining the amplitude of the field by evaluating the variation of the amplitude along a ray path from the intensity law without considering the field contributions from other ray paths. This approach of evaluating the optical field is in contrast to exact approach of wave optics where Maxwell’s equations5 are solved for the complex field. 2.1.1. Concepts of Rays Fermat’s principle can be used to determine the ray paths in a region where the index of refraction n is a function of position r. A ray path from a point r0 to point r is a curve C through t these points which makes the line integral of the optical path length stationary. When n (r) is a smooth function, it can be b p shown from the calculus of variations that the trajectory of the ray paths r (s) , expressed in terms of the curvilinear abscissa s along a ray path, satisfies the vector ray equation · ¸ d dr n (r) = ∇n (r) , (1) ds ds where the derivative of the position vector r of a point on a ray path with respect to the arc-length s along the ray path is the unit tangent vector of the ray path dr ˆt = . (2) ds The rate of change of the unit tangent vector to a ray path gives the curvature of C Figure 1. Three-dimensional plot of a helix to dˆt ˆp = , (3) illustrate the unit tangent vector t, the unit prin- ds ρ cipal normal vector p, and the unit bi-normal vec- where ˆp is the unit principal normal vector to ray path and is tor b where b = t × p. perpendicular to ˆt, and ρ specifies the radius of curvature of the ray path. We know further from the Frénetequations [6, pp. 20 - 54] that we can define a third unit vector bˆ(s) = ˆt(s) × ˆp(s) shown in Fig. 1. The three unit vectors (ˆt, ˆp, bˆ) form a right-handed moving triplet of orthogonal unit vectors along the space curve (ray path in this case). The oscillating plane contains ˆt and ˆp. For any parametric curve, the Frénetequations are given by Eq. (3) and the following two equations dˆp ˆt = − + τ bˆ, (4) ds ρ dbˆ = −τ ˆp, (5) ds where τ represent the torsion of C, which measures the tendency of the curve to leave the local oscillating plane. The Frénetequations will be used in sect. 2.1.2 when showing how the wavefront curvatures propagate through an optical system. From Eqs. (1) and (3), it follows d £ ¤ dn (r) ˆp ∇n (r) = n (r)ˆt = ˆt + n (r) , (6) ds ds ρ so that the rays lie locally in the osculating plane containing ˆt and ∇n. The magnitude of the curvature of a ray path is obtained by dividing Eq. (6) by n (r) and taking the scalar produce of Eq. (6) with ˆp

The rays originating from a point r0 will propagate into some region of space, and for each point r in the region of space, a scalar function can be deﬁned as the optical path distance from r0 to r along a ray path. Hamilton introduced this function φ (r, r0) and named it the point characteristic function, which is also known as the eikonal. Stavroudis [4, pp. 29-43] applies the Hilbert Integral to geometrical optics and shows that · ¸ dr(s) ∇ × n (r) = 0, (8) ds

which implies that dr(s) n (r) = ∇φ (r) . (9) ds The surface defined by φ (r) = constant is perpendicular to the unit tangent vector of a ray path, represents a surface of constant optical path distance from a reference point, and is called a wavefront. The phase of the optical field at r is equal to the product of the free space wave number k0 [= ω/c = 2π/λ0 at frequency ω] times φ (r, r0) . Squaring Eq. (9) gives [∇φ (r)]2 = [n (r)]2 , (10) which is known as the eikonal equation and is used to solve for the eikonal function φ (r) . The propagation vector is defined by k (r) = k0∇φ (r, r0) , (11) and is also tangent to the ray path. In the vicinity of r on a wavefront, the phase varies locally as a plane wave with wave vector k, or as k·dr. Solimeno et al.3 provide discussion of the eikonal and ray equations as they follow from Maxwell’s equations when a small wavelength approximation is used as well as assuming that φ (r) and the amplitude of the optical field have a smooth spatial variation with respect to dimensions of the order of the wavelength.

2.1.2. Eikonal Equation From Eq. (10), the eikonal equation in cartesian coordinates is given by µ ¶ µ ¶ µ ¶ ∂φ 2 ∂φ 2 ∂φ 2 + + = n2 (r) , (12) ∂x ∂y ∂z

and is also known as the Hamilton-Jacobi equation. For homogeneous media, Stavroudis4, 9, 10 has obtained a general integral of the eikonal equation which is summarized below:

φ = ux + vy + wz + h (u, v) (13a)

uz = (x + hu) w (13b)

vz = (y + hv) w (13c)

where h (u, v) is an arbitrary function that is subject to appropriate boundary conditions, subscripts indication partial derivatives, such as, hu = ∂h/∂u and

u2 + v2 + w2 = n2. (14)

Stavroudis deﬁnes a speciﬁc wavefront from this general integral of the eikonal equation by setting

φ (x, y, z) = ns, (15)

with the following deﬁnitions, using the Cartesian unit vectors (ex, ey, ez) ,

W (u, v)=x (u, v) ex+y (u, v) ey + z (u, v) ez, (17a) p 2 2 2 S (u, v)=uex+vey + n − u + v ez, (17b)

H (u, v)=huex+hvey, (17c)

p = ns − h (u, v) + uhu + vhv. (17d)

Using standard techniques from differential geometry,6 one can evaluate the surface normal and principal curvatures of the general wavefront defined by Eq. (16).4, 10 Now, the Frénetequations are used to express the Laplacian of the eikonal in terms of the principal curvatures of a wavefront. In the next section, we will express the variation of the optical field amplitude in terms of the principal curvatures of the wavefront. There are a number of works in the literature which have evaluated the curvatures of wavefronts. For example, Stavroudis10 evaluates the principal curvatures of a wavefront in a direct, but very tedious manner; Born & Wolf [1, pp. 116 - 117] evaluate in a conceptual manner the ratio of an element of area on a wavefront as it propagates some distance in terms of the principal radii of curvature of the wavefront; and Solimeno et al. [3, pp. 69 - 73] make a Taylor expansion to second order of the eikonal to express ∇2φ in terms of the principal curvatures of the wavefront. Here, we will expand the unit propagation vector of a ray path ˆt with a Taylor expansion to first order in displacements along the principal coordinate axes of the wavefront and use the second Frénetequation (4) to obtain an expression for ∇2φ in terms of the principal curvatures of the wavefront. As the curvilinear coordinates (u, v) on the wavefront surface, it is convenient to use the arc-length (s1, s2) of two curves formed on the wavefront surface when two planes containing the surface normal ˆt (r) and each plane

aligned along one principal direction esi of the wavefront at the point r on the wavefront. We know from diﬀerential geometry [6, pp. 89 - 95] that a general surfaces has two principal curvatures (κ1, κ2) , or the radii of the principal curvatures of a surface are (ρ1, ρ2) where κi = 1/ρi. The principal axes are perpendicular to each other. Then, we can write a displacement as

s1 s2 δr =h es1 + h es2 , (18)

and expanding ˆt in a Taylor series in the vicinity of a point W (r0) on a wavefront, we obtain ¯ ¯ ¯ ¯ ¯∂ˆt (r)¯ ¯∂ˆt (r)¯ ˆt (r + δr) = ˆt (r ) + hs1 ¯ ¯ + hs2 ¯ ¯ + .... (19) 0 0 ¯ ∂s ¯ ¯ ∂s ¯ 1 r0 2 r0

Equation (19) can be simplified and expressed in terns of the radii of the principal curvatures (ρ1, ρ2) of the wavefront at the point r0 by using the second Frénetequation. Now, we consider the two curves on the wavefront formed by two perpendicular planes, where the unit tangent vector to one curve on the wavefront is e and the ¡ ¢ ¡ ¢ s1 unit principal normal vector is −ˆt , where e = e × −ˆt . Therefore, the second Frénetequation (4) for the s2 £ s1 ¡ ¢ ¤ ˆ parallel curve characterized by the triplet vectors es1 , −t , es2 can be written explicitly as ˆ ∂t (r) es1 = + − τes2 . (20) ∂s1 ρ1 Similarly, for the perpendicularly curve ˆ ∂t (r) es2 = + − τes1 . (21) ∂s2 ρ2

where we have truncated the Taylor series with the linear terms and omitted labeling the evaluation at r0 to simplify notation. Since ˆt (r) = ∇φ (r) /n, we obtain the desired expression for ∇2φ (r) by taking the divergence s1 s2 of ˆt (r0 + δr) given by Eq. (22) where h and h are the linear projections of δr on the coordinate unit vectors

(es1 , es2 ) as follows 1 1 ∇ · ˆt (r0 + δr) = + + . (23) ρ1 ρ2 Thus, we have shown for any point on the wavefront that the Laplacian of the eikonal φ (r) is given by

∇2φ (r) 1 1 = + . (24) n ρ1 (0) + s ρ2 (0) + s

where ρ1 (0) and ρ2 (0) are the principal radii of curvature of the wavefront passing through the origin, where s = 0. The corresponding centers of curvature (principal centers of curvature) are at s = −ρ1 (0) and s = −ρ2 (0) . Equation (24) is used in sect. 2.2 to evaluated the functional dependence of the ﬁeld amplitude on the propagation distance s.

2.1.3. Intensity Law

We can now evaluate a system of rays, the associated family of wavefronts, and the phase k0φ (r) of the optical field within the region of space for which φ (r, r0) is defined. Next, we describe how the amplitude A (r) of the optical field is evaluated in this region of space. We assume that the energy of the field propagates along the rays with the speed v = c/n. From introductory physics, we know that the energy density of a field is proportional to the square of the amplitude of the field. Physically, we know that the intensity I represents the energy per unit time crossing a unit element of surface area on the wavefront. Thus, we can write

I (r) = v |A (r)|2 . (25)

For a small bundle of rays passing through an element of area dS1 at r1 and dS2 at r2, conservation of energy along the bundle of rays requires I (r1) dS1 = I (r2) dS2, (26) where we are assuming in the ray optics approximation that no energy propagates in directions perpendicular to the ray paths. Equation (26) can be extended to apply to cases when either (or both) element of surface area for the input (or output) beam is inclined at some angle α with respect to a cross-sectional plane which is perpendicular to the direction of propagation of the beam

cos(α1) I1dS1 = cos(α2) I2dS2, (27)

where αi is the angle between the ray propagation direction and the surface normal. Similarly, if light originates from a point source or passes through a focus where the element of area on the wavefront goes to zero, the intensity law would be applied by considering the energy that leaves source per unit time per unit solid angle as the intensity IΩ. We can relate I1 to IΩ by the following

IΩdΩ = I1dS1. (28)

2.2. Ray Optics Field

The field A (r) exp [−ik0φ (r)] is known as the ray optics field, which is only an approximation of an exact wave optical field. Maxwell’s equations5 provide a complete description of the wave optics field where the phase and amplitudes are not independently determined as done when evaluating the ray optics field. For monochromatic fields oscillating with an angular frequency of ω, Maxwell’s equations for propagation in an inhomogeneous

2 2 2 ∇ u (r) + k0n (r) u (r) = 0, (29) we obtain the following equation · ¸ n o ∇A (r) ∇2A (r) k2 [∇φ (r)]2 − n2 (r) + (−ik ) ∇2φ (r) + 2∇φ (r) · + = 0. (30) 0 0 A (r) A (r) For the optical field u (r) to satisfy the scalar wave equation (29), the coefficients of the real and imaginary terms in Eq. (30) must be small or set equal to zero. As we shall see below, ray optics follows from setting the first two groups of terms in Eq. (30) equal to zero. Ray optics requires that the last term in Eq. (30) is very small and can be neglected for small wavelengthes: ∇2A (r) v 0 for small wavelengths, (31) A (r) which is equivalent to saying that the amplitude of the ray optics field must be a smooth or slowly varying function of position. Equation (31) is known to not be valid near the focal region or caustic of an optical system where the cross-sectional area of a bundle of rays goes to zero. Appendix A contains a summary of 3 the Luneburg and Kline asymptotic solution of the scalar wave equation for arbitrary k0, where Eq. (A4c) shows that evaluating the field amplitude coefficients beyond the ray optics approximation (m = 0) requires evaluation of the Laplacian of the ray optics field amplitude, which has been neglected based on the assumption that both eikonal and ray optics amplitude are slowly varying functions of position over dimensions of the order of wavelengths. Now, we show how the eikonal equation and the intensity law follows from Eqs. (30) and (31). 2 Setting the coefficient of the k0-term equal to zero gives the eikonal equation, and setting the coefficient of 2 the ik0-term equal to zero is equivalent to the intensity law, as can be seen by multiplying this term by A (r) and noting that £ ¤ A2 (r) ∇2φ (r) + 2A (r) ∇φ (r) · ∇A (r) = ∇ · A2 (r) ∇φ (r) (32) £ ¤ = ∇ · A2 (r) n (r)ˆt = 0, (33)

where the unit vector ˆt along a ray path (or normal to a wavefront) has been defined by Eq. (2), but can also written as ∇φ (r) ∇φ (r) ˆt = = , (34) |∇φ (r)| n (r) when Eqs. (11) and (10) were used to obtain Eq. (34), and then Eq. (33). The quantity A2 (r) n (r)ˆt is analogous to the the Poynting vector for the scalar field A (r) exp [−ik0φ (r)] , and Eq. (33) can be interpreted as the conservation of the power flux. Once it is written in integral form by applying Gauss’s theorem to a small volume made up by the rays contiguous to the trajectory r (s) , it determines the law of variation of A (s) along a ray path to the relation · ¸ n (s ) dS (s ) 1/2 A (s) = A (s ) 0 0 (35) 0 n (s) dS (s)

where dS (s0) and dS (s) represent elements of cross-section area of the ray tube relative to s and s0. The transport equation, Eq. (32), can also be written with Eq. (34) as d ∇2φ + 2n (r) (ln A) = 0, (36) ds which has the integral ½ Z s 2 ¾ 1 ∇ φ 0 A (s) = A (s0) exp − 0 ds . (37) 2 s0 n (s )

µ ¶2 · Z s 2 ¸ n (s0) dS (s0) A (s) ∇ φ 0 = = exp − 0 ds . (38) n (s) dS (s) A (s0) s0 n (s )

One consequence of Eqs. (32) - (38) is that when dS → 0, then ∇2φ → −∞, which is connected with the concept that a caustic surface is the loci of points where the Laplacian of the eikonal is a divergent quantity as well as the geometrical optics intensity. Also, the field amplitude along a ray depends on the way in which dS or, equivalently, ∇2φ varies. Thus, the knowledge of the trajectory of a ray is not sufficient to deduce the amplitude of the field. Equation (38) can be integrated for the case of homogenous media (assume n = 1) using results of Eqs. (??)-(24) to obtain functional variation of the amplitude variation with propation distance as follows

· Z s 2 ¸ ½ Z s · ¸ ¾ ∇ φ 0 1 1 0 exp − 0 ds = exp − 0 + 0 ds (39a) s0 n (s ) s0=0 ρ1 (0) + s ρ2 (0) + s © 0 0 sª = exp − [ln (ρ1 (0) + s ) + ln (ρ2 (0) + s )|0 (39b) = exp {− [ln (ρ1 (0) + s)] − ln (ρ1 (0)) + ln (ρ2 (0) + s) − ln (ρ2 (0))} (39c) ½ · ¸¾ (ρ (0) + s)(ρ (0) + s) = exp − ln 1 2 , (39d) (ρ1 (0)) (ρ2 (0)) which gives · ¸ · ¸ (ρ (0)) 1/2 (ρ (0)) 1/2 A (s) = A (0) 1 2 (40) (ρ1 (0) + s) (ρ2 (0) + s)

where the square root must be taken as a real positive or imaginary positive. For a spherical wave (ρ1 = ρ2) the phase along a ray undergoes a jump of π/2 in passing through each focal point, as Solimeno et al. [3, pp. 73, 287] have shown for a spherical wave with a ﬁnite aperture. From Eqs. (25), (26), (38) and (40), we can write an expression for the intensity along a ray path as a ratio of the element of areas on the geometrical optics wavefront after propagation by a distance s from the origin by:

I (0) I (s) = , (41) |J (s)|

where J (s) is the Jacobian relating the two elements of area dS (s) and dS (0) on the wavefront. Equation (41) can be generalized to complete optical systems there the Jacobian relates an input element of area dSin and an output element of area dSout or two any two elements of area along a ray path within a complex optical system. The general problem of analysis of the geometrical optical intensity or irradiance, which can be represented by generalization of Eq. (41), has been studied by many during past ﬁfty years.2, 14–20 We can write J (s) as a quadratic function of the ray propagation distance between the two elements of area. By expanding out terms in Eqs. (40)-(41) to obtain the following expression for the intensity of the optical ﬁeld at a distance s from the origin

I (0) ρ1ρ2 I (s) = 2 , (42a) |ρ1ρ2 + (ρ1 + ρ2) s + s | I (0) = 2 , (42b) |1 + (κ1 + κ2) s + κ1κ2 s |

where the intensity I (0) is for s = 0 and Eq. (42b) follows from Eq. (42a) by displaying κi instead of ρi. Equations (42a) - (42b) are special cases of the flux flow equation studied extensively by Shealy and Burkhard.17–20 It should be noted that for propagation of a wavefront in homogeneous media, one is free to fix the coordinate axes along the principal directions, as was done in deriving Eq. (24). However, when the wavefront is incident upon an optical system, the coordinate axes are more appropriately determined by symmetry of the optical system

κ1 + κ2 = κk + κ⊥, (43a) 2 κ1κ2 = κkκ⊥ − τ , (43b) ¡ ¢ where κk, κ⊥ are the normal curvatures of curves on the wavefront formed when two perpendicular planes intersect the wavefront. The parallel plane contains the normal to wavefront ˆt and will be considered in sect. 3 ¡to be the¢ plane of incidence. Using Eqs. (43a) - (43b), we can write Eq. (42b) in terms of the normal curvatures κk, κ⊥ and torsion τ : I (0) I (s) = ¯ ¡ ¢ ¡ ¢ ¯. (44) ¯ 2 2¯ 1 + κk + κ⊥ s + κkκ⊥ − τ s There are several interesting properties of Eqs. (42a) - (44) which merit further discussion and development:

1. The term J (s) is a quadratic function of the propagation distance s, and therefore, there are two roots to the equation J (s) = 0 which are the centers of curvature of the wavefront, which are located at s = −ρ1 and s = −ρ2 for the wavefront considered in Eq. (40). 2. For a general wavefront, the two roots of the equation J (s) = 0 locate focal points of the wavefront, where the loci of all focal points of a wavefront is the caustic surface of the optical system for the specific parameters used to generate the wavefront under analysis. 3. When a wavefront encounters a discrete optical interface, such as a mirror or lens, the boundary conditions from the Electromagnetic Theory, i.e., Maxwell’s equations, which apply to the parallel and perpendicular components of E (r) and H (r) at the interface between two optical regions, must be used to determine how the wavefront propagates beyond an optical system. 4. For design of illumination systems that will transform a specific input intensity distribution into a specified output intensity distribution over a given surface, Eqs. (42a) - (44) can be solved for the shape of the optical components required to produced the necessary intensity redistribution.

3. GENERALIZED RAY TRACE EQUATIONS In sect. 2.1.2, we have shown how the Laplacian of the eikonal can be expressed in terms of the principal radii of curvature of the wavefront, and in sect. 2.2, we have shown how the optical field amplitude depends on the radii of the principal curvatures and propagation distance of the wavefront. Stavroudis4, 10 describes how to compute the principal curvatures of any known wavefront. Now, assume that we know the principal radii of curvature of a wavefront incident upon an optical system. For example, some important shapes of wavefronts incident upon optical systems include planar, spherical, and general astigmatic elliptical. We assume that we know the geometry of the incident wavefront, such as, the principal directions and curvature of the wavefront. Then, techniques from differential geometry [4, pp. 57-59] allows one to evaluate the curvatures and torsion of two normal section of the wavefront formed by the intersection of the plane of incidence (parallel plane) and the place perpendicular to the plane of incidence and the surface normal of the lens or mirror surface. The generalize ray trace equations4, 8, 9, 15, 19, 20 are used to compute refracted wavefront curvatures when the incident wavefront and refracting surface curvatures are known. A brief discussion of the derivation and use of the generalized ray trace equations are presented in this section. The generalized ray trace equation can be derived by taking the total differential of the vector refraction equation and using the second Frénetequation (4) to replace the partial derivatives of the incident ray unit vector A (i) ; the unit surface normal vector to refracting surface N; and the refracted ray unit vector A (r) with respect to the arc-length the curves formed by intersection of the plane of incidence and a plane perpendicular to the plane of incidence. Finally, we use boundary conditions to relate elements of arc-length on these surfaces

A (r) = γA (i) + ΩN (45)

where γ = n (i) /n (r) and Ω = −γ cos i + cos r. Each of the vectors in Eq. (45) are surface normal vectors to one of the surfaces: the incident wavefront, refracting surface,¡ and¢ refracting wavefront, where the coordinate axes in the tangent planes of these surfaces are expressed as ek, e⊥ with diﬀerent arguments as indicated. The total diﬀerential of Eq. (45) gives " # · ¸ ∂A (r) r ∂A (r) r ∂A (i) i ∂A (i) i ∂N ∂N r ds⊥ + r dsk = γ d i ds⊥ + i dsk + Ω ds⊥ + dsk + NdΩ (46) ∂s⊥ ∂sk ∂s⊥ ∂sk ∂s⊥ ∂sk

where the second Fr´enetequation (4) is used to express all the partial derivatives in Eq. (46) in terms of their projections on the corresponding surface coordinate axes, curvature, and torsion, such as,

∂A (i) ek (i) i = − + τe⊥ (i) . (47) ∂sk ρk (i)

Also, take the cross product of N with the resulting equation to eliminate the dΩ term and impose the boundary conditions on the incident and refracted optical ﬁeld, which ensure continuity of tangential component of the ﬁeld at the interface as expressed by

e⊥ (i) = e⊥ = e⊥ (r) and (48) i r ds⊥ = ds⊥ = ds⊥, (49)

and conservation of ﬂux along the normal component of ﬁeld at the interface as expressed by

dS cos i = dW i and dS cos r = dW r, (50)

where dS in an element of area of refracting surface and dW i (dW r) is an element of area on the incident (refracted) wavefront. Finally, apply vector identiﬁes to express all terms as their components along the refracting surface coordinate axes to obtain © ª ds⊥ ek [κ⊥ (r) − γκ (i) − Ωκ⊥] − e⊥ [τ (r) cos r − γτ (i) cos i − Ωτ] © £ 2 2 ¤ª + dsk ek [τ (r) cos r − γτ (i) cos i − Ωτ] − e⊥ −κk (r) cos r + γκk (i) cos i − Ωκk = 0. (51)

Now, set the coeﬃcients of dsk and ds⊥ equal to zero, since the are independent, and set the coeﬃcients of ek and e⊥, since they are orthogonal. This give the generalized ray trace equations

κ⊥ (r) = γκ⊥ (i) + Ωκ⊥, (52a) 2 2 κk (r) cos r = γκk (i) cos i + Ωκk, (52b) τ (r) cos r = γτ (i) cos i + Ωτ, (52c)

which can be expressed in terms of the radii of curvatures and torsion by using ρi = 1/κi. Equations (52a) - (52c) provide relationships between the normal radii of curvature and torsion of the curves in the parallel and perpendicular normal sections of the wavefront before and after refraction. The parallel and perpendicular normal curvatures and torsion of the output wavefront are combined with Eq. (44) for evaluating the intensity as wavefront leaves the optical surface. The principal radii of curvature of the wavefront can be evaluated from Eqs. (43a) - (43b) from the parallel and perpendicular normal curvatures and torsion of refracted wavefront from Eqs. (52a) - (52c). The principal radii of curvature of a refracted wavefront will be used in sect. 4 to compute the caustic surfaces.

where the radii of the principal curvatures are compute from ρi = 1/κi. Thus, the caustic surfaces are deﬁne by

XC1 = Xout + ρ1 (r) tout, (54a)

XC2 = Xout + ρ2 (r) tout. (54b)

Figure 2 illustrates for the two rim rays how the ray first passes the tangential sheet of the caustic, which is physically associated with focusing of adjacent rays in the meridional (parallel) plane, and then passes the sagittal sheet of caustic, which is associated adjacent rays in the sagittal (perpendicular) plane. A rim ray crosses the optical axis and is diverging in both the tangential and sagittal directions as it passes the tangential caustic for the second time to define the size of the circle of least confusion. Figure 2. Meridional intersection of the tangen- For the rotationally symmetric case shown in Figure 2, the 3- tial and sagittal sheets of the caustic surfaces of a dimensional caustic surface is obtained by rotating the tangential singlet lens. (From Ref. [23, Fig. 1]) caustic around the symmetry axes. The sagittal caustic in this case degenerates to a spike or line. Figure 3 shows plots of the ray density over the tangential and sagittal caustic surfaces for refraction of collimated beam by a plano-convex singlet. These results suggest that the ray density is highest near the paraxial focus where the tangential and sagittal caustics meet.22 Figure 4 presents the meridional cross-section of the caustic surfaces for a singlet lens with a shape factor of 0.9 and index of refraction of 1.5168. ALPHA specifies the angle incident rays make with respect to the optical axis. The dotted lines represent three ray paths for each field angle. The caustic surfaces are shown as solid lines to the right of lens. For off-axis light, the caustic sheets separate as a result of astigmatism and move away from the image plane as a result of field curvature. Figure 5 shows the meridional cross-section of caustic surfaces of a triplet lens, which does not

Figure 3. Refraction of a collimated beam by a plano-convex singlet lens is shown with (a) shows the density of tangent rays over the tangential caustic; (b) shows the tangential and sagittal caustics formed by singlet lens; and (c) shows the ray density over the sagittal caustic.(From Ref. [22, Fig. 4])

Several authors23–25 have used the size of caustic surfaces as a merit function in optical design to improve performance of some classic designs. However, decisive or compelling results have not been obtained to justify more wide spread use of a caustic-based merit function in optical design. Caustic surfaces do provide visual and important information about the imaging properties of optical systems. The generalized ray trace equations are complex, but enable intensity calculations and deepen the understanding of imaging conditions. For example, for a rotationally symmetric optical system, the torsion of the curved form by the normal planes and each optical surface is zero. [19, Appendix A] If the incident wavefront also does not have torsion, then the refracted wavefront will not have any torsion, and Eq. (52c) does not contribute to evaluation of the principal curvatures of the wavefront. Rather, Eqs. (52a)-(52b) determine the sagittal (S) and tangential (T) focal points of the wavefront. Therefore, for rotationally symmetric systems, the generalized Coddington equations19 are given by

κ⊥ (r) = γκ (i) + Ωκ⊥, (55) 2 2 κk (r) cos r = γκk (i) cos i + Ωκk, (56)

where the parallel (perpendicular) normal curvatures contribute to the¯ tangential (sagittal)¯ image point. To ¯ ¯ reduce astigmatism of an optical system, one needs only to minimize κ⊥ (r) − κk (r) , or to set

κ⊥ (r) = κk (r) . (57) Then, one can solve for the sag of one surface such that the optical system will have no astigmatism. This approach has been used as part of design of multi-mirror anastigmat designs.26

5. NONIMAGING OPTICS APPLICATIONS Nonimaging optics applications cover a wide range of topics with a primary concern with power transfer and brightness rather than with image quality, which are important goals of both illumination engineering and laser beam shaping. During the past fifteen years, the SPIE has held ten conferences with primary focus on nonimaging optics, such as, the Nonimaging Optics and Efficient Illumination Systems.27 There has been a similar level of interest in laser beam shaping as reflected with SPIE hosting seven conferences on laser beam shaping during the past ten years, such as the Laser Beam Shaping VI Conference.28 The intensity law of geometrical optics is one of the important principles used in both illumination engineering and laser beam shaping. We shall shall discuss briefly in this section the use of the intensity law in illumination and beam shaping applications. It is clear from Eqs. (26)-(28) and (41)-(44) that the intensity law of geometrical optics enables one to compute the intensity or irradiance along a bundle of rays propagating through an optical system. The intensity

Figure 5. Meridional cross-section of the caustic surfaces for a Cooke’s triple for diﬀerent ﬁeld angles labeled by ALPHA in degrees of half-angle.

Copyright 2006 Society of Photo-Optical Instrumentation Engineers. This paper will be published in Proc. SPIE 5876, Sept. 2005, and is made available as an electronic preprint with permission of SPIE. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in the paper for a fee or for commercial purposes, or modification of the content of the paper are prohibited. law expresses conservation of energy and can be shown to follow from Maxwell’s equations. [1, pp. 115-117] The intensity law has been used for design of optical systems to concentrate solar radiation29 and to achieve maximum power transfer between sources, fibers, and detectors in opto-electronics, in radiative heat transfer, and in illumination engineering.30 A typical application of the intensity law in nonimaging optics involves determining the configuration of optical components which will illuminate a detector or substrate surface with a specified irradiance distribution. The intensity law may then be considered as a constraint applied to the entire system as well as to each ray while the optical components and source are varied in search of a solution for system configuration that provides the desired irradiance distribution over the output surface. Ray tracing methods with a merit function based on the intensity law have become an efficient approach for design of nonimaging optical system. For example, Evans and Shealy31–33 describe several merit functions used with genetic algorithms for design of laser-based illumination optics, where the number of optical components in the system has been considered as a parameter during the global optimization. For illumination systems with rotational symmetry, intensity law can be written as a differential equation for the shape of one optical Mirror surface. Figure 6 illustrates one configuration of a nonimaging system with a source, mirror, and detector. Assume that we want to r determine the shape of a rotationally symmetric mirror that will uni- formly illuminate the detector. The intensity distribution of source ρ is described by I(θ) and the desired intensity distribution on the detector is described by E(R). The intensity law for the configuration z(r) shown in Figure 6 is given by θ 2πE (R) RdR = I (θ) 2πr2 sin θdθ. (58) Equation (58) expresses conservation of energy. If we seek to uni- Source Detector z I(θ) E(R) formly£ illuminate the¤ detector (E = E0) with an isotropic point 2 source I (θ) = I0/r , then integrating Eq. (58) over the acceptance angles (θ0, θm) and maximum radius (Rm) of detector, we obtain the following relationship Figure 6. Configuration of nonimage system with source, mirror and detector. (cos θ0 − cos θm) E0 = 2I0 2 , (59) Rm which fixes the relationship between total output power of source which reaches detector. To determine the sag of mirror, we need to use the ray trace equation for light form source and reflected from mirror to detector ¡ ¢ 02 0 2 R − r Ar ρ sin θ + 2ρρ cos θ − ρ sin θ = = 02 0 2 , (60) Z (R) − z (r) Az (ρ cos θ + 2ρρ sin θ − ρ cos θ) where ρ0 = dρ/dθ, z = ρ cos θ, r = ρ sin θ as shown in Fig. 6. Equation (60) can be solved with suitable boundary conditions for the sag of the mirror. Reference [30, pp. 333-338] provides more details and results for this example. For many laser beam shaping applications, which seek to illuminate a surface in a prescribed manner, it is sufficient to use the intensity law for design of reflective, refractive, or diffractive optical elements.34 However, when the laser beam shaping application requires the optics to change both the intensity profile and to produce a collimating output beam, then one must use the intensity law and the constant optical path length condition when designing the beam shaping optics. Kreuzer35, 36 described how to design a two-plano aspheric beam shaper which redistributes a Gaussian input laser beam into a more uniform, collimated output beam. Hoffnagle and Jefferson37, 38 reduced to practice a two-plano aspheric beam shaper using Kreuzer’s method. Many practical beam shaping applications well described in the literature.28

6. CONCLUSIONS First, we reviewed the concepts of rays, wavefronts, and energy propagation using the Fr´enetequations from the mathematical theory of curves to build a foundation for an analysis of the structure and properties of the

APPENDIX A. ASYMPTOTIC SOLUTION OF SCALAR WAVE EQUATION Assume that the Luneburg and Kline asymptotic series39, 40

XN Am (r) ¡ ¢ u (r) = e−ik0S(r) + o k−N , (A1) (−ik )m 0 m=0 0

¡ −N ¢ is a solution of the scalar wave equation, Eq. (29). The Landau symbol o k0 indicates a function that −N vanishes more rapidly than k0 , while the propagation vector has been deﬁned by

k (r) = k0∇φ (r) . (A2)

The asymptotic series given in Eq. (A1) has also been labeled the ray optics representation of the electromagnetic ﬁeld. Equation (A1) provides a physical model of electromagnetic propagation that is more complete than the one traditional associated conventional geometrical optics.

To determine φ (r) and the Am’s, introduce Eq. (A1) into the scalar wave equation (29) to obtain [3, Ch. II]

XN Q (r) ¡ ¢ m = o k−N , (A3) (−ik )m 0 m=0 0 where

2 2 Q0 (r) = (∇φ) − n (r) , (A4a) ¡ 2 ¢ Q1 (r) = ∇ φ + 2∇φ · ∇ A0, (A4b) ¡ 2 ¢ 2 Qm (r) = ∇ φ + 2∇φ · ∇ Am−1 + ∇ Am−2 (m = 2, 3, 4...) . (A4c)

Felsen and Marcuvitz41 have shown that Eq. (A3) is satisﬁed for every N only if

Qm (r) = 0 (m = 0, 1, 2, ...) (A5)

which yields a set of equations which determine φ (r) and the Am’s. It is evident from Eq. (A4a) that Q0 (r) = 0 leads to the eikonal equation.

Copyright 2006 Society of Photo-Optical Instrumentation Engineers. This paper will be published in Proc. SPIE 5876, Sept. 2005, and is made available as an electronic preprint with permission of SPIE. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in the paper for a fee or for commercial purposes, or modification of the content of the paper are prohibited. REFERENCES 1. M. Born and E. Wolf, Principles of Optics, Cambridge University Press, Cambridge, UK, seventh ed., 1999. 2. G. A. Deschamps, “Ray techniques in electromagnetics,” Proceedings of the IEEE 60.9, pp. 1022–1035, 1972. 3. S. Solimeno, B. Crosignani, and P. DiPorto, Guiding, Diffraction, and Confinement of Opical Radiation, Academic Press, Inc., New York, 1986. 4. O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics, Wiley-VCH Verlag, Weinheim, Germany, 2006. 5. D. J. Griffiths, Introduction to Electrodynamics, Prentice Hall, Upper Saddle River, NJ 07458, third ed., 1999. 6. E. Kreyszig, Introduction to Differential Geometry and Riemannian Geometry, Mathematical Exposition No. 16, University of Toronto Press, 1968. 7. D. J. Struik, Lectures on Classical Differential Geometry, Addison-Wesley Publishing Company, Inc., Read- ing, MA, second ed., 1961. 8. S. C. Parker, “Properties and applications of generalized ray tracing,” technical report 71, Optical Sciences Center, University of Arizona, Tucson, AZ 85721, November 1971. 9. O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics, Academic Press, Inc., New York, 1972. 10. O. N. Stavroudis and R. C. Fronczek, “Caustic surfaces and the structure of the geometrical image,” J. Opt. Soc. Am. 66.8, pp. 795–800, 1976. 11. O. N. Stavroudis, R. C. Fronczek, and R.-S. Chang, “Geometry of the half-symmetric image,” J. Opt. Soc. Am. 68.6, pp. 739–742, 1978. 12. O. N. Stavroudis, “The k function in geometrical optics and its relationship to the archetypal wave front and the caustic surface,” J. Opt. Soc. Am. A 12.5, pp. 1010–1016, 1995. 13. O. N. Stavroudis and J. B. Hurtado-Ramos, “Maxwell equations and the k function,” J. Opt. Soc. Am. A 17.8, pp. 1469–1474, 2000. 14. J. B. Keller and H. B. Keller, “Determination of reflected and transmitted fields by geometrical optics,” J. Opt. Soc. Am. 40.1, pp. 48–52, 1950. 15. J. A. Kneisly, “Local curvature of wavefronts in an optical system,” J. Opt. Soc. Am. 54.2, pp. 229–235, 1964. 16. E. Bochove, “Geometrical optics field of general optical systems,” J. Opt. Soc. Am. 69.6, pp. 891–897, 1979. 17. D. L. Shealy and D. G. Burkhard, “Flux density for ray propagation in discrete index media expressed in terms of the intrinsic geometry of the deflecting surface,” Optica Acta 20.4, pp. 287–301, 1973. 18. D. L. Shealy and D. G. Burkhard, “Analytical illuminance calculation in a multi-interface optical system,” Optica Acta 22.6, pp. 485–501, 1975. 19. D. G. Burkhard and D. L. Shealy, “Simplified formula for the illuminance in an optical system,” Appl. Opt. 20.5, pp. 897–909, 1981. 20. D. G. Burkhard and D. L. Shealy, “A different approach to lighting and imaging: formuas for flux density, exact lens and mirror equations and caustic surfaces in terms of the differential geometry of surfaces,” in Materials and Optics for Solar Energy Conversion and Advanced Lighting Technology, C. M. Lampert and S. Holly, eds., Proc. SPIE 692, pp. 248–272, 1986. 21. D. L. Shealy, “Analytical illuminance and caustic surface calculations in geometrical optics,” Appl. Opt. 15.10, pp. 2588–2596, 1976. 22. D. G. Burkhard and D. L. Shealy, “Formula for the density of tangent rays over a caustic surface,” Appl. Opt. 21.18, pp. 3299–3306, 1982. 23. I. H. Al-Ahdali and D. L. Shealy, “Optimization of three- and four-element lens systems by minimizing the caustic surfaces,” Appl. Opt. 29.31, pp. 4551–4559, 1990. 24. R. S. Change and O. N. Stavroudis, “Generalized ray tracing, caustic surfaces, generalized bending, and the construction of a novel merit function for optical design,” 1980. 25. A. M. Kassim, D. L. Shealy, and D. G. Burkhard, “Caustic merit function for optical design,” Appl. Opt 28.3, pp. 601–606, 1989.

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