geometrical optics: some applications of the law of intensitydls/publications/spie6289/... · 2013....

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’slaw, ray tracing, the optical path length and phase. The geometrical optics law of intensity and the opticalwavefront are also well known, but are perhaps less used as a basis for optical design than ray tracing. Thispaper will first review how the geometrical optics law of intensity leads to the invariance of the product of theintensity of light times an element of area along a bundle of rays. Then, we will discuss how the geometricaloptics law of intensity provides a good foundation for understanding not only nonimaging optics through directways to design optical systems for prescribed illumination requirements but also imaging applications through acomplete 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 associatedwith 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 phenomenaand represents an approximate solution of Maxwell’s equations [5, pp. 375 - 392] for propagation of radiationin optical media when the wavelength of the light is small compared to the working dimensions and when thephase and amplitude of the optical field are smooth functions of the spatial coordinates over regions of the size ofmultiple wavelengths.3 The intensity law of geometrical optics,1 which is also known as the transport equationof 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 ofoptical technologies, such as, nonimaging optics, illumination engineering, and beam shaping, as well as a basisfor a geometrical understanding of the structure the optical image via caustic theory. The tools of differentialgeometry6,7 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 developtechnologies 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., Frénet equations,as a foundation for analysis of the structure of a wavefront propagating in homogeneous media, based on thegeneral integral of the eikonal equation developed by Stavroudis.4,8–13 Explicitly, we show that a linear Taylorexpansion of a ray propagation vector and application of the Frénet equations [6, Eq. (11.5)] lead to an importantresult 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 anda 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 completedescription of how the wavefront normal curvatures and torsion propagate through an optical system, which areknown as the generalize ray trace equations. The intensity law of geometrical optics provides a foundation fornonimaging 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 nonimagingoptical design. First, we illustrate some properties of caustic surfaces for lens systems of increasing complexityand correction of aberrations. Then, we describe some nomimaging optical systems whose designs are based onshaping 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.

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2. GEOMETRICAL OPTICS2.1. Principles of Geometrical OpticsThe concepts of rays, wavefronts, and energy propagation are fundamental in using geometrical optics for designof imagining or nonimagining optical systems. A brief overview of these concepts are presented in this sectionwhile illustrating how the geometrical optics field is evaluated by first determining the rays, which are usedto determine the phase of the optical field independently from its amplitude, and then, by determining theamplitude of the field by evaluating the variation of the amplitude along a ray path from the intensity lawwithout considering the field contributions from other ray paths. This approach of evaluating the optical fieldis in contrast to exact approach of wave optics where Maxwell’s equations5 are solved for the complex field.

pb

t

Figure 1. Three-dimensional plot of a helix toillustrate the unit tangent vector t, the unit prin-cipal normal vector p, and the unit bi-normal vec-tor b where b = t× p.

2.1.1. Concepts of Rays

Fermat’s principle can be used to determine the ray paths in aregion where the index of refraction n is a function of positionr. A ray path from a point r0 to point r is a curve C throughthese points which makes the line integral of the optical pathlength stationary. When n (r) is a smooth function, it can beshown from the calculus of variations that the trajectory of theray paths r (s) , expressed in terms of the curvilinear abscissa salong a ray path, satisfies the vector ray equation

d

ds

[n (r)

drds

]= ∇n (r) , (1)

where the derivative of the position vector r of a point on a raypath with respect to the arc-length s along the ray path is theunit tangent vector of the ray path

t̂ =drds

. (2)

The rate of change of the unit tangent vector to a ray path givesthe curvature of C

dt̂ds

=p̂ρ

, (3)

where p̂ is the unit principal normal vector to ray path and isperpendicular to t̂, and ρ specifies the radius of curvature of theray path. We know further from the Frénet equations [6, pp. 20

- 54] that we can define a third unit vector b̂(s) = t̂(s)× p̂(s) shown in Fig. 1. The three unit vectors (̂t, p̂, b̂)form a right-handed moving triplet of orthogonal unit vectors along the space curve (ray path in this case). Theoscillating plane contains t̂ and p̂. For any parametric curve, the Frénet equations are given by Eq. (3) and thefollowing two equations

dp̂ds

= − t̂ρ

+ τ b̂, (4)

db̂ds

= −τ p̂, (5)where τ represent the torsion of C, which measures the tendency of the curve to leave the local oscillating plane.The Frénet equations will be used in sect. 2.1.2 when showing how the wavefront curvatures propagate throughan optical system. From Eqs. (1) and (3), it follows

∇n (r) = dds

[n (r) t̂

]=

dn (r)ds

t̂ + n (r)p̂ρ

, (6)

so that the rays lie locally in the osculating plane containing t̂ and ∇n. The magnitude of the curvature of aray path is obtained by dividing Eq. (6) by n (r) and taking the scalar produce of Eq. (6) with p̂


1ρ

= p̂ · ∇ (lnn) , (7)

which says a ray tends to bend so that ∇n points into the direction where the center of curvature of the raypath is located. The torsion [7, pp. 15 - 18] of a ray path can be found by vector multiplication of t̂ and the, p̂with Eq. (7).

The rays originating from a point r0 will propagate into some region of space, and for each point r in theregion 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 knownas the eikonal. Stavroudis [4, pp. 29-43] applies the Hilbert Integral to geometrical optics and shows that

∇×[n (r)

dr(s)ds

]= 0, (8)

which implies that

n (r)dr(s)ds

= ∇φ (r) . (9)

The surface defined by φ (r) = constant is perpendicular to the unit tangent vector of a ray path, represents asurface of constant optical path distance from a reference point, and is called a wavefront. The phase of theoptical field at r is equal to the product of the free space wave number k0 [= ω/c = 2π/λ0 at frequency ω] timesφ (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 vectoris 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 wavewith wave vector k, or as k·dr. Solimeno et al.3 provide discussion of the eikonal and ray equations as theyfollow 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 ofthe wavelength.

2.1.2. Eikonal Equation

From Eq. (10), the eikonal equation in cartesian coordinates is given by

(∂φ

∂x

)2+

(∂φ

∂y

)2+

(∂φ

∂z

)2= n2 (r) , (12)

and is also known as the Hamilton-Jacobi equation. For homogeneous media, Stavroudis4,9, 10 has obtained ageneral 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 indicationpartial 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)


where ns is optical path length from the source to the wavefront surface that satisfies Eq. (15), s is the arc-lengthof the ray path from source (or reference point) to wavefront, and n the index of refraction of medium, whichhas been assumed to be a constant as part of developing this general integral solution of the eikonal equation.Solving Eqs. (13a) - (13c) for the Cartesian coordinates (x, y, z) of a point of the wavefront surface as a functionof (u, v) , when using Eq. (14) to eliminate w, leads to the following results

W =p

n2S−H, (16)

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)

S (u, v)=uex+vey +√

n2 − u2 + v2ez, (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 cur-vatures of the general wavefront defined by Eq. (16).4,10 Now, the Frénet equations are used to express theLaplacian of the eikonal in terms of the principal curvatures of a wavefront. In the next section, we will expressthe 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. Forexample, 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 wavefrontas it propagates some distance in terms of the principal radii of curvature of the wavefront; and Solimeno etal. [3, pp. 69 - 73] make a Taylor expansion to second order of the eikonal to express ∇2φ in terms of theprincipal curvatures of the wavefront. Here, we will expand the unit propagation vector of a ray path t̂ with aTaylor expansion to first order in displacements along the principal coordinate axes of the wavefront and use thesecond Frénet equation (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) oftwo curves formed on the wavefront surface when two planes containing the surface normal t̂ (r) and each planealigned along one principal direction esi of the wavefront at the point r on the wavefront. We know fromdifferential geometry [6, pp. 89 - 95] that a general surfaces has two principal curvatures (κ1, κ2) , or the radii ofthe principal curvatures of a surface are (ρ1, ρ2) where κi = 1/ρi. The principal axes are perpendicular to eachother. Then, we can write a displacement as

δr =hs1es1 + hs2es2 , (18)

and expanding t̂ in a Taylor series in the vicinity of a point W (r0) on a wavefront, we obtain

t̂ (r0 + δr) = t̂ (r0) + hs1∣∣∣∣∂t̂ (r)∂s1

∣∣∣∣r0

+ hs2∣∣∣∣∂t̂ (r)∂s2

∣∣∣∣r0

+ .... (19)

Equation (19) can be simplified and expressed in terns of the radii of the principal curvatures (ρ1, ρ2) of thewavefront at the point r0 by using the second Frénet equation. Now, we consider the two curves on the wavefrontformed by two perpendicular planes, where the unit tangent vector to one curve on the wavefront is es1and theunit principal normal vector is

(−t̂) , where es2 = es1 ×(−t̂) . Therefore, the second Frénet equation (4) for the

parallel curve characterized by the triplet vectors[es1 ,

(−t̂) , es2]

can be written explicitly as

∂t̂ (r)∂s1

= +es1ρ1

− τes2 . (20)

Similarly, for the perpendicularly curve∂t̂ (r)∂s2

= +es2ρ2

− τes1 . (21)


Equation (19) with Eqs. (20) - (21) becomes

t̂ (r0 + δr) = t̂ (r0) + hs1[+

es1ρ1

− τes2]

+ hs1[es2ρ2

− τes1]

, (22)

where we have truncated the Taylor series with the linear terms and omitted labeling the evaluation at r0 tosimplify notation. Since t̂ (r) = ∇φ (r) /n, we obtain the desired expression for ∇2φ (r) by taking the divergenceof t̂ (r0 + δr) given by Eq. (22) where hs1 and hs2 are the linear projections of δr on the coordinate unit vectors(es1 , es2) as follows

∇ · t̂ (r0 + δr) = + 1ρ1

+1ρ2

. (23)

Thus, we have shown for any point on the wavefront that the Laplacian of the eikonal φ (r) is given by

∇2φ (r)n

=1

ρ1 (0) + s+

1ρ2 (0) + s

. (24)

where ρ1 (0) and ρ2 (0) are the principal radii of curvature of the wavefront passing through the origin, wheres = 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 thepropagation 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 opticalfield within the region of space for which φ (r, r0) is defined. Next, we describe how the amplitude A (r) of theoptical field is evaluated in this region of space. We assume that the energy of the field propagates along the rayswith the speed v = c/n. From introductory physics, we know that the energy density of a field is proportionalto the square of the amplitude of the field. Physically, we know that the intensity I represents the energy perunit 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 energyalong 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 perpendicularto the ray paths. Equation (26) can be extended to apply to cases when either (or both) element of surfacearea for the input (or output) beam is inclined at some angle α with respect to a cross-sectional plane which isperpendicular 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 originatesfrom a point source or passes through a focus where the element of area on the wavefront goes to zero, theintensity law would be applied by considering the energy that leaves source per unit time per unit solid angle asthe 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 waveoptical field. Maxwell’s equations5 provide a complete description of the wave optics field where the phase andamplitudes are not independently determined as done when evaluating the ray optics field. For monochromaticfields oscillating with an angular frequency of ω, Maxwell’s equations for propagation in an inhomogeneous


medium, for which n (r) is a slowly varying function of r and/or E is perpendicular to ∇n, reduce to a vectorwave equation for E and H. As an approximation, we analyze the implications of the ansatz that the rayoptics field satisfies a scalar wave equation. By substituting the ray optics field u (r) = A (r) exp [−ik0φ (r)] at afrequency ω into the scalar wave equation,

∇2u (r) + k20n2 (r)u (r) = 0, (29)we obtain the following equation

k20

{[∇φ (r)]2 − n2 (r)

}+ (−ik0)

[∇2φ (r) + 2∇φ (r) · ∇A (r)

A (r)

]+∇2A (r)A (r)

= 0. (30)

For the optical field u (r) to satisfy the scalar wave equation (29), the coefficients of the real and imaginary termsin Eq. (30) must be small or set equal to zero. As we shall see below, ray optics follows from setting the firsttwo groups of terms in Eq. (30) equal to zero. Ray optics requires that the last term in Eq. (30) is very smalland can be neglected for small wavelengthes:

∇2A (r)A (r)

v 0 for small wavelengths, (31)

which is equivalent to saying that the amplitude of the ray optics field must be a smooth or slowly varyingfunction of position. Equation (31) is known to not be valid near the focal region or caustic of an opticalsystem where the cross-sectional area of a bundle of rays goes to zero. Appendix A contains a summary ofthe Luneburg and Kline asymptotic solution3 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) requiresevaluation of the Laplacian of the ray optics field amplitude, which has been neglected based on the assumptionthat both eikonal and ray optics amplitude are slowly varying functions of position over dimensions of the orderof wavelengths. Now, we show how the eikonal equation and the intensity law follows from Eqs. (30) and (31).

Setting the coefficient of the k20-term equal to zero gives the eikonal equation, and setting the coefficient ofthe ik0-term equal to zero is equivalent to the intensity law, as can be seen by multiplying this term by A2 (r)and noting that

A2 (r)∇2φ (r) + 2A (r)∇φ (r) · ∇A (r) = ∇ · [A2 (r)∇φ (r)] (32)= ∇ · [A2 (r)n (r) t̂] = 0, (33)

where the unit vector t̂ along a ray path (or normal to a wavefront) has been defined by Eq. (2), but can alsowritten as

t̂ =∇φ (r)|∇φ (r)| =

∇φ (r)n (r)

, (34)

when Eqs. (11) and (10) were used to obtain Eq. (34), and then Eq. (33). The quantity A2 (r)n (r) t̂ isanalogous to the the Poynting vector for the scalar field A (r) exp [−ik0φ (r)] , and Eq. (33) can be interpreted asthe conservation of the power flux. Once it is written in integral form by applying Gauss’s theorem to a smallvolume made up by the rays contiguous to the trajectory r (s) , it determines the law of variation of A (s) alonga ray path to the relation

A (s) = A (s0)[n (s0) dS (s0)n (s) dS (s)

]1/2(35)

where dS (s0) and dS (s) represent elements of cross-section area of the ray tube relative to s and s0. Thetransport equation, Eq. (32), can also be written with Eq. (34) as

∇2φ + 2n (r) dds

(ln A) = 0, (36)

which has the integral

A (s) = A (s0) exp{−1

2

∫ ss0

∇2φn (s′)

ds′}

. (37)


We can therefore write Eq. (33) as

n (s0) dS (s0)n (s) dS (s)

=(

A (s)A (s0)

)2= exp

[−

∫ ss0

∇2φn (s′)

ds′]

. (38)

One consequence of Eqs. (32) - (38) is that when dS → 0, then ∇2φ → −∞, which is connected with the conceptthat a caustic surface is the loci of points where the Laplacian of the eikonal is a divergent quantity as wellas 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 amplitudeof 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

exp[−

∫ ss0

∇2φn (s′)

ds′]

= exp{−

∫ ss0=0

[1

ρ1 (0) + s′+

1ρ2 (0) + s′

]ds′

}(39a)

= exp{− [ln (ρ1 (0) + s′) + ln (ρ2 (0) + s′)|s0

}(39b)

= exp {− [ln (ρ1 (0) + s)]− ln (ρ1 (0)) + ln (ρ2 (0) + s)− ln (ρ2 (0))} (39c)

= exp{− ln

[(ρ1 (0) + s) (ρ2 (0) + s)

(ρ1 (0)) (ρ2 (0))

]}, (39d)

which gives

A (s) = A (0)[

(ρ1 (0))(ρ1 (0) + s)

]1/2 [ (ρ2 (0))(ρ2 (0) + s)

]1/2(40)

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 ratioof the element of areas on the geometrical optics wavefront after propagation by a distance s from the origin by:

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

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 anoutput 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 bygeneralization of Eq. (41), has been studied by many during past fifty years.2,14–20 We can write J (s) as aquadratic function of the ray propagation distance between the two elements of area. By expanding out termsin Eqs. (40)-(41) to obtain the following expression for the intensity of the optical field at a distance s from theorigin

I (s) =I (0) ρ1ρ2

|ρ1ρ2 + (ρ1 + ρ2) s + s2| , (42a)

=I (0)

|1 + (κ1 + κ2) s + κ1κ2 s2| , (42b)

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 Itshould be noted that for propagation of a wavefront in homogeneous media, one is free to fix the coordinate axesalong the principal directions, as was done in deriving Eq. (24). However, when the wavefront is incident uponan optical system, the coordinate axes are more appropriately determined by symmetry of the optical system


or plane of incidence. Therefore, it is more useful to express Eq. (42a) in terms of the normal curvatures andtorsion of the wavefront than the principal curvatures. Stavroudis has obtained useful relationships between theprincipal curvatures and the normal curvatures and torsion of a surface, as expressed in terms of two differentorthogonal coordinates axes on a surface. [4, pp. 57 - 59] From these results, one can show

κ1 + κ2 = κ‖ + κ⊥, (43a)

κ1κ2 = κ‖κ⊥ − τ2, (43b)

where(κ‖, 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 t̂ and will be considered in sect. 3to be the plane of incidence. Using Eqs. (43a) - (43b), we can write Eq. (42b) in terms of the normal curvatures(κ‖, κ⊥

)and torsion τ :

I (s) =I (0)∣∣1 + (κ‖ + κ⊥

)s +

(κ‖κ⊥ − τ2

)s2

∣∣ . (44)

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 tothe equation J (s) = 0 which are the centers of curvature of the wavefront, which are located at s = −ρ1and 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 specificparameters used to generate the wavefront under analysis.

3. When a wavefront encounters a discrete optical interface, such as a mirror or lens, the boundary conditionsfrom the Electromagnetic Theory, i.e., Maxwell’s equations, which apply to the parallel and perpendicularcomponents of E (r) and H (r) at the interface between two optical regions, must be used to determinehow the wavefront propagates beyond an optical system.

4. For design of illumination systems that will transform a specific input intensity distribution into a specifiedoutput intensity distribution over a given surface, Eqs. (42a) - (44) can be solved for the shape of theoptical 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 radiiof curvature of the wavefront, and in sect. 2.2, we have shown how the optical field amplitude depends on theradii of the principal curvatures and propagation distance of the wavefront. Stavroudis4,10 describes how tocompute the principal curvatures of any known wavefront. Now, assume that we know the principal radii ofcurvature of a wavefront incident upon an optical system. For example, some important shapes of wavefrontsincident upon optical systems include planar, spherical, and general astigmatic elliptical. We assume that weknow 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 oftwo normal section of the wavefront formed by the intersection of the plane of incidence (parallel plane) and theplace perpendicular to the plane of incidence and the surface normal of the lens or mirror surface. The generalizeray trace equations4,8, 9, 15,19,20 are used to compute refracted wavefront curvatures when the incident wavefrontand refracting surface curvatures are known. A brief discussion of the derivation and use of the generalized raytrace equations are presented in this section.

The generalized ray trace equation can be derived by taking the total differential of the vector refractionequation and using the second Frénet equation (4) to replace the partial derivatives of the incident ray unitvector A (i) ; the unit surface normal vector to refracting surface N; and the refracted ray unit vector A (r) withrespect to the arc-length the curves formed by intersection of the plane of incidence and a plane perpendicularto the plane of incidence. Finally, we use boundary conditions to relate elements of arc-length on these surfaces


at their interface. We know from application of Snell’s law and the coplanarity requirement of the incident,surface normal of refracting surface, and refracted ray vectors

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 oneof the surfaces: the incident wavefront, refracting surface, and refracting wavefront, where the coordinate axes inthe tangent planes of these surfaces are expressed as

(e‖, e⊥

)with different arguments as indicated. The total

differential of Eq. (45) gives

∂A (r)∂sr⊥

dsr⊥ +∂A (r)

∂sr‖dsr‖ = γ

[d∂A (i)∂si⊥

dsi⊥ +∂A (i)∂si‖

dsi‖

]+ Ω

[∂N∂s⊥

ds⊥ +∂N∂s‖

ds‖

]+ NdΩ (46)

where the second Frénet equation (4) is used to express all the partial derivatives in Eq. (46) in terms of theirprojections on the corresponding surface coordinate axes, curvature, and torsion, such as,

∂A (i)∂si‖

= −e‖ (i)ρ‖ (i)

+ τe⊥ (i) . (47)

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

e⊥ (i) = e⊥ = e⊥ (r) and (48)

dsi⊥ = ds⊥ = dsr⊥, (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 refractingsurface coordinate axes to obtain

ds⊥{e‖ [κ⊥ (r)− γκ (i)− Ωκ⊥]− e⊥ [τ (r) cos r − γτ (i) cos i− Ωτ ]

}

+ ds‖{e‖ [τ (r) cos r − γτ (i) cos i− Ωτ ]− e⊥

[−κ‖ (r) cos2 r + γκ‖ (i) cos2 i− Ωκ‖]}

= 0. (51)

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

κ⊥ (r) = γκ⊥ (i) + Ωκ⊥, (52a)

κ‖ (r) cos2 r = γκ‖ (i) cos2 i + Ωκ‖, (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 andperpendicular normal sections of the wavefront before and after refraction. The parallel and perpendicularnormal curvatures and torsion of the output wavefront are combined with Eq. (44) for evaluating the intensityas wavefront leaves the optical surface. The principal radii of curvature of the wavefront can be evaluated fromEqs. (43a) - (43b) from the parallel and perpendicular normal curvatures and torsion of refracted wavefront fromEqs. (52a) - (52c). The principal radii of curvature of a refracted wavefront will be used in sect. 4 to computethe caustic surfaces.


4. IMAGING APPLICATIONS

The caustic surfaces have been defined to be the loci of focal points (centers of principal curvature) of a wave-front.4,9, 17 The caustic surfaces are generally a two-sheeted surface, since there are two principal radii ofcurvature for each point on a wavefront. Rays are also tangent to the caustic surfaces, and each ray passesthrough two focal points, which define points on each sheet of caustic. There is a phase change of π/2 at eachfocal point along a ray. [3, pp. 73, 287] The intensity of the ray optics field becomes infinity over the caus-tic surface,17,18,21 while the density of rays tangent to the caustic surfaces varies over the caustics.22 In thissection, we shall illustrate some properties of caustics by first outline how to compute coordinates of pointson the 3-dimensional caustic surfaces from results of applying the generalize ray trace questions to an opticalsystem. Then, we shall describe some properties of the meridional cross-section of a caustic surface for severallens systems. The classic lens aberrations of spherical aberration, coma, astigmatism, and field curvature havebeen identified with spatial features of the caustic surfaces.21 Minimizing the size of a caustic does conceptuallyimply that an optical system will produce a sharper image. However, since the density of rays tangent to acaustic surface varies over the caustic, one may conclude that a stronger imaging criteria would require thoseparts of a caustic with the greatest density of ray tangents to have their spatial extent minimized.

For a given optical system, the generalize ray trace equations can be applied to all rays to evaluated theparallel and perpendicular normal curvatures and torsion in addition to direction cosine, tout, and coordinatesof Xout of all rays leaving the optical system. The distance along a ray to the caustic surfaces is equal to theprincipal radii of curvature of ray at the last optics surface. The principal curvatures can be computed fromthe normal curvatures and torsion of each ray from Eqs. (43a) and (43b) to obtain:

κ1 (r) =12

[κ⊥ (r) + κ‖ (r)

]+

12

√[κ‖ (r)− κ⊥ (r)

]2 + τ2 (r), (53a)

κ2 (r) =12

[κ⊥ (r) + κ‖ (r)

]− 12

√[κ‖ (r)− κ⊥ (r)

]2 + τ2 (r), (53b)

Figure 2. Meridional intersection of the tangen-tial and sagittal sheets of the caustic surfaces of asinglet lens. (From Ref. [23, Fig. 1])

where the radii of the principal curvatures are compute fromρi = 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 passesthe tangential sheet of the caustic, which is physically associatedwith focusing of adjacent rays in the meridional (parallel) plane,and then passes the sagittal sheet of caustic, which is associatedadjacent rays in the sagittal (perpendicular) plane. A rim raycrosses the optical axis and is diverging in both the tangentialand sagittal directions as it passes the tangential caustic for thesecond time to define the size of the circle of least confusion.For the rotationally symmetric case shown in Figure 2, the 3-dimensional caustic surface is obtained by rotating the tangentialcaustic around the symmetry axes. The sagittal caustic in thiscase degenerates to a spike or line. Figure 3 shows plots of theray 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 highestnear the paraxial focus where the tangential and sagittal caustics meet.22 Figure 4 presents the meridionalcross-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 threeray paths for each field angle. The caustic surfaces are shown as solid lines to the right of lens. For off-axislight, the caustic sheets separate as a result of astigmatism and move away from the image plane as a result offield curvature. Figure 5 shows the meridional cross-section of caustic surfaces of a triplet lens, which does not


suffer from the large amounts of off-axis aberrations of astigmatism and field curvature present in the singletlens. Shealy [21, Figs. 5-15] present caustic cross-sections for additional classic lens configurations with axes foreach plot to allow comparisons. Collectively, it is clear that better designed lenses do have smaller spread oftheir caustic surfaces.

Figure 3. Refraction of a collimated beam by a plano-convex singlet lens is shown with (a) shows the density of tangentrays over the tangential caustic; (b) shows the tangential and sagittal caustics formed by singlet lens; and (c) shows theray 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 improveperformance of some classic designs. However, decisive or compelling results have not been obtained to justifymore wide spread use of a caustic-based merit function in optical design. Caustic surfaces do provide visual andimportant information about the imaging properties of optical systems. The generalized ray trace equations arecomplex, 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 opticalsurface is zero. [19, Appendix A] If the incident wavefront also does not have torsion, then the refracted wavefrontwill not have any torsion, and Eq. (52c) does not contribute to evaluation of the principal curvatures of thewavefront. 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)

κ‖ (r) cos2 r = γκ‖ (i) cos2 i + Ωκ‖, (56)

where the parallel (perpendicular) normal curvatures contribute to the tangential (sagittal) image point. Toreduce astigmatism of an optical system, one needs only to minimize

∣∣κ⊥ (r)− κ‖ (r)∣∣ , or to set

κ⊥ (r) = κ‖ (r) . (57)

Then, one can solve for the sag of one surface such that the optical system will have no astigmatism. Thisapproach 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 andbrightness rather than with image quality, which are important goals of both illumination engineering and laserbeam shaping. During the past fifteen years, the SPIE has held ten conferences with primary focus on nonimagingoptics, such as, the Nonimaging Optics and Efficient Illumination Systems.27 There has been a similar level ofinterest in laser beam shaping as reflected with SPIE hosting seven conferences on laser beam shaping duringthe past ten years, such as the Laser Beam Shaping VI Conference.28 The intensity law of geometrical opticsis one of the important principles used in both illumination engineering and laser beam shaping. We shall shalldiscuss 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 tocompute the intensity or irradiance along a bundle of rays propagating through an optical system. The intensity


Figure 4. Meridional cross-section of the caustic surfaces for a singlet lens with shape factor of 0.9. (Ref. [21, Fig. 1])

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


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 achievemaximum power transfer between sources, fibers, and detectors in opto-electronics, in radiative heat transfer, andin illumination engineering.30 A typical application of the intensity law in nonimaging optics involves determiningthe configuration of optical components which will illuminate a detector or substrate surface with a specifiedirradiance distribution. The intensity law may then be considered as a constraint applied to the entire systemas well as to each ray while the optical components and source are varied in search of a solution for systemconfiguration that provides the desired irradiance distribution over the output surface. Ray tracing methodswith a merit function based on the intensity law have become an efficient approach for design of nonimagingoptical system. For example, Evans and Shealy31–33 describe several merit functions used with genetic algorithmsfor design of laser-based illumination optics, where the number of optical components in the system has beenconsidered as a parameter during the global optimization.

r

z(r)

z

Mirror

Source Detector

ρ

θ

I(θ) E(R)

Figure 6. Configuration of nonimage systemwith source, mirror and detector.

For illumination systems with rotational symmetry, intensity lawcan be written as a differential equation for the shape of one opticalsurface. Figure 6 illustrates one configuration of a nonimaging sys-tem with a source, mirror, and detector. Assume that we want todetermine the shape of a rotationally symmetric mirror that will uni-formly illuminate the detector. The intensity distribution of sourceis described by I(θ) and the desired intensity distribution on the de-tector is described by E(R). The intensity law for the configurationshown in Figure 6 is given by

2πE (R)RdR = I (θ) 2πr2 sin θdθ. (58)

Equation (58) expresses conservation of energy. If we seek to uni-formly illuminate the detector (E = E0) with an isotropic pointsource

[I (θ) = I0/r2

], then integrating Eq. (58) over the acceptance

angles (θ0, θm) and maximum radius (Rm) of detector, we obtain thefollowing relationship

E0 = 2I0(cos θ0 − cos θm)

R2m, (59)

which fixes the relationship between total output power of source which reaches detector. To determine the sagof mirror, we need to use the ray trace equation for light form source and reflected from mirror to detector

R− rZ (R)− z (r) =

ArAz

=

(ρ′2 sin θ + 2ρρ′ cos θ − ρ2 sin θ)

(ρ′2 cos θ + 2ρρ′ sin θ − ρ2 cos θ) , (60)

where ρ′ = dρ/dθ, z = ρ cos θ, r = ρ sin θ as shown in Fig. 6. Equation (60) can be solved with suitableboundary conditions for the sag of the mirror. Reference [30, pp. 333-338] provides more details and results forthis example.

For many laser beam shaping applications, which seek to illuminate a surface in a prescribed manner, it issufficient 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 producea collimating output beam, then one must use the intensity law and the constant optical path length conditionwhen designing the beam shaping optics. Kreuzer35,36 described how to design a two-plano aspheric beam shaperwhich redistributes a Gaussian input laser beam into a more uniform, collimated output beam. Hoffnagle andJefferson37,38 reduced to practice a two-plano aspheric beam shaper using Kreuzer’s method. Many practicalbeam shaping applications well described in the literature.28

6. CONCLUSIONS

First, we reviewed the concepts of rays, wavefronts, and energy propagation using the Frénet equations fromthe mathematical theory of curves to build a foundation for an analysis of the structure and properties of the


wavefront. We have shown that making a linear Taylor expansion of the ray propagation vector in homogeneousmedia and applying the Frénet equations leads to an important expression for the Laplacian of the eikonal interms of the principal curvatures of the wavefront. We used the intensity law of geometrical optics to obtain thefunctional dependence of the amplitude of the ray optics field on its principal curvatures and propagation distance.Thus, we have shown that the intensity of ray optics field has an inverse quadratic functional dependence onthe propagation distance and the principal curvatures of the wavefront. The generalized ray trace equationsprovide a formalism for propagating the wavefront curvatures through an optical system. The generalized raytrace equations were derived by taking the total differential of the vector refraction equation and using theFrénet equations to establish this iterative relationships between the incident and refractive wavefront normalcurvatures and torsion, or the principal curvatures. Knowing the principal curvatures of a refracted wavefrontenables evaluation of its caustic surfaces. The meridional intersection of the caustic surfaces for several lenssystems were presented, and it was shown that the spatial properties of the caustics can be associated with classicaberrations. In particular, for a rotationally symmetric optical system the generalized ray trace equations reducea generalized Coddington equations for the sagittal and tangential focal points. The intensity law has beenused in beam shaping and illumination systems for design of the optics required to provide specified irradiancedistributions over a surface. Other beam shaping applications require a collimated output beam with a moreflattened profile, which is achieved by using the intensity law and the constant optical path length condition fordesign of the optical components.

APPENDIX A. ASYMPTOTIC SOLUTION OF SCALAR WAVE EQUATION

Assume that the Luneburg and Kline asymptotic series39,40

u (r) = e−ik0S(r)N∑

m=0

Am (r)(−ik0)m + o

(k−N0

), (A1)

is a solution of the scalar wave equation, Eq. (29). The Landau symbol o(k−N0

)indicates a function that

vanishes more rapidly than k−N0 , 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 electromagneticfield. Equation (A1) provides a physical model of electromagnetic propagation that is more complete than theone 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]

N∑m=0

Qm (r)(−ik0)m = o

(k−N0

), (A3)

where

Q0 (r) = (∇φ)2 − n2 (r) , (A4a)Q1 (r) =

(∇2φ + 2∇φ · ∇)A0, (A4b)Qm (r) =

(∇2φ + 2∇φ · ∇)Am−1 +∇2Am−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) = 0leads to the eikonal equation.


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