## Abstract

A previous paper [2] presented an analysis of a class of microstructured optical surfaces in two dimensions, in which a classification of the microstructures was obtained (regular and anomalous) and a concept of 2D ideal microstructures was introduced. In this paper the study of those microstructured optical surfaces is extended to three dimensions with rotational symmetry. As a starting point, non-microstructured rotational optical systems in the First Order Approximation are also classified as point-spot type and ring-spot type, with remarkable perfect particular cases. This classification is also extended to the case in which ideal microstructured rotational surfaces are used, for both regular and anomalous type. The case of perfect ring-spot type system with an odd number of rotational, anomalous, ideal microstructures enables the definition of an anomalous aplanatic system that has direct application for mixing spatially and angularly the light emitted by several sources.

© 2007 Optical Society of America

## 1. Introduction

Structured optical surfaces, such as Fresnel lenses, TIR lenses, or Fresnel mirrors, are often used in nonimaging optical design [1]. When the structural element is small enough (microstructured surface), it can be approximated as infinitesimal for some calculations, and the macro-profile of the surface can be treated as a new type of optical surface with a certain deflection law, which will be either the reflection law or the Snell law. This asymptotic limit has been widely used previously (see [2] and references therein) to design both Fresnel and TIR lenses. In reference [2] we considered a certain class of microstructures in this limit, and we studied its properties in two-dimensions. In this paper, we study those microstructured surfaces in three dimensions with rotational symmetry. In section 2 we will introduce a classification of non-microstructured rotational optical systems, which we will extrapolate in section 3 to the class of ideal microstructures studied in [2]. Section 4 will present an example of application of rotational microstructure surfaces to color mixing.

## 2. Ring-spot type and point-spot type rotational optics

Consider the coordinate system shown in Fig. 1 and assume that the optical device has rotational
symmetry with respect to the *z* axis. The ray R_{N} is
emitted from the origin
*x*
_{1}=*y*3=*z*
_{1}=0
with direction given by the angles β and ϕ, and let us assume
that for any value of β and ϕ the optical system is such that
the ray exits the optical system parallel to the *z* axis (as a
parabolic mirror does). Without loss of generality we can consider the positive
*z* axis as the direction of those exit rays (situation shown in Fig. 1). Therefore, this optical system transforms a
spherical wavefront emitted from an on-axis point onto a plane wavefront normal to
the optical axis.

The results in this section can be generalized to other pair of wavefronts with
rotational symmetry with respect to the z axis. Two other common examples are the
case of two spherical wavefronts (*i.e*. elliptical or hyperbolical
mirror type systems) and two plane wavefronts (*i.e*. afocal
systems).

We will assume that the refractive index *n*
_{1} of the input
space (surrounding the origin) is homogeneous, in general
*n*
_{1}≠1, while there is air
(*n*
_{2}=1) at the output space (in which the plane
wavefront is immersed).

The exit point P_{2} of the ray RN shown in Fig. 1 at a plane
*z*=*z _{ap}* is a function of β and
ϕ. Due to the rotational symmetry,

*ϕ*

^{∗}=ϕ+ϕ

_{0}, where ϕ

_{0}is either 0 or π. Thus this mapping is given by:

$${y}_{2}=\rho \left(\beta \right)\mathrm{sin}\left(\varphi +{\varphi}_{0}\right)$$

where ρ(β) is a function that depends on the optical design.

We will assume in this section that the function ρ(β)
continuous and strictly monotonic (thus
ρ’(β)≠0). As proved in Appendix I, under
these assumptions on ρ(β) and in the First Order Optics
approximation [5], any ray R emitted at a point P_{1} =
(*x*
_{1},*y*
_{1},*z*)
of the input space will, in the neighborhood of the ray R_{N} that passes
through P_{2} (in the output space), point towards the direction
v_{2} = (*p*
_{2}, *q*
_{2},
+(1- *p*
^{2}
_{2}-
*q*
^{2}
_{2})^{1/2}) given by:

where *P*
^{*}
_{2} = *p*
_{2}
cos(ϕ_{0}), *q*
^{*}
_{2} =
*q*
_{2} cos(ϕ_{0}) and

$$b=\frac{{n}_{1}}{2}\left(\frac{\mathrm{cos}\beta}{\rho \text{'}\left(\beta \right)}-\frac{\mathrm{sin}\beta}{\rho \left(\beta \right)}\right)$$

$$c={n}_{1}\left(\frac{\mathrm{sin}\beta}{\rho \text{'}\left(\beta \right)}\right)$$

Note that since ϕ_{0} is either 0 or π,
*P*
^{*}
_{2} = *p*
_{2},
*q*
^{*}
_{2} = *q*
_{2} for
ϕ_{0} = 0 and *P*
^{*}
_{2} =
-*p*
_{2}, *q*
^{*}
_{2} =
-*q*
_{2} for ϕ_{0} = π. We
will also call **V**
^{*}
_{2} =
(*P*
^{*}
_{2},
*q*
^{*}
_{2}, + $\sqrt{1-{p}_{2}^{2}-{q}_{3}^{2}}$
).

The neighborhood of ray R_{N} in which Eq. (2) is accurate can be very small for certain functions
ρ (β) at certain values of β, for instance, if one
of the coefficients *a*, *b* or *c*
becomes infinite. As an example, if ρ’(β
=π/4) ≠ 0 but ρ (β = π/4)=0,
*a*, *b* become infinite, and thus the First Order
neighborhood around β = π/4 at any ϕ reduces to
rays R fulfilling
y_{1}cos(ϕ)=*x*
_{1}sin(ϕ) (with
∣x_{1}∣ and ∣y_{1}∣
small enough), which are just the meridian rays. This can be checked by direct
substitution in Eq. (2) (since under that condition
*P*
^{*}
_{2} and
*q*
^{*}
_{2} will not depend on the term that
becomes infinite, sin(β)/ρ(β)). We will exclude
such degenerate cases below.

By definition of the coordinate system, ρ ≥ 0 and 0 ≤β ≤ π, so:

For a given value of β ≠ π/2 (we will talk about
this excluded case later), let us define a rotational optical system as being of the
*point-spot type* at that β value if
ρ′(β) and cos(β) have the same sign, and
*ring-spot type* otherwise (the reason for this nomenclature will
become clear later). This can be written as:

It can be easily deduced from Eq. (3), taking into account the definitions (5), that:

Combining inequalities (4) and (6), is easy to show that:

From Eq. (7) we obtain that in point-spot type systems,
*a* is negative, while *b* is smaller in magnitude but
can be positive, negative or null. On the other hand, in ring-spot type systems,
*b* is negative, while *a* is smaller in magnitude
but can be positive, negative or null.

These Eq. (7) have significant implications about how point-spot and
ring-spot systems perform. To illustrate them, let us consider that
ρ(β) for all β values is such that the First Order
approximation is accurate for one point source located at P_{1}
(*x*
_{1}=ε>0,
*y*
_{1}=0, *z*=0), with ε small
enough. According to Eq. (2), the direction vector **v**
^{*}
_{2}
of the ray exiting from P_{2} is defined by:

$${q}_{2}^{*}=\epsilon \left(b\phantom{\rule{.2em}{0ex}}\mathrm{sin}\left(2\varphi \right)\right)$$

Let us analyze the curve obtained in the *P*
^{*}
_{2} -
*q*
^{*}
_{2} plane when P_{2} is revolved
along the *z* axis, i.e., when ϕ varies from 0 to
2π for a given angle β in Eq. (1). This curve is the circumference given by (8) in parametric
form, which has period π (i.e., a half of the period of revolution of
P_{2}). The center (*p*
^{*}
_{C},
*q*
^{*}
_{C}) and angular radius α of
the circumference (8) are:

Therefore, according to (7):

From Eq. (10) it can be deduced that in ring-spot systems, the
circumference (8) always encloses the optical axis (i.e., the direction
*p*=0, *q*=0), while in point-spot systems the
circumference (8) never encloses the optical axis, as illustrated in Fig. 2. This distinguishing feature means that, when the
light source center is displaced from the optical axis, in point-spot optical
systems the spot will be a simply-connected bright region, while in ring-spot
systems, necessarily the spot is not simply connected, showing a ring-shaped bright
region with a dark zone at the optical axis. This feature defines the nomenclature
introduced to describe the two types of behaviors (point-spot and ring-spot) of
optical systems near a certain value β≠π/2. The
excluded case β≠π/2 just corresponds to the case in
which the circumference (8) passes through the origin
(*p*
_{2}=0, *q*
_{2}=0).

Let us remark the implications of α being strictly different from zero in
the ring-spot systems. It means that under no circumstances the point source
P_{1} (*x*
_{1}=ε>0,
*y*
_{1}=0, *z*=0) can be perfectly imaged
in this First Order framework onto a point at infinity (which is obtained when
α = 0). However, point-spot systems allow such a perfect imaging
(although of course it is not guaranteed).

As an example, for the specific case of the parabolic reflectors, the region β>90° is point-spot type while the region β < 90° is ring-spot type. The fact that the ring-spot type region of these reflectors produces spots enclosing the optical axis was already known [6], and the two regions were called “backward” and “forward reflection regions” of the paraboloid,respectively. Figure 3 shows the result of a ray trace on two slices a parabolic reflector in both regions for off-axis point sources (the finite size of the ring in the pattern is due to the also finite thickness of the slices).

Let us present now two remarkable examples, which we will refer to *perfect
point-spot* and *ring-spot* optical systems. First, the
perfect point-spot system will be that producing sharp imaging in this First Order
framework, which is obtained when:

where *f* is the integration constant. This dependence
ρ(β) is the well-known Abbe sine condition and the optical
system is said to be aplanatic [2], and the parameter *f* is the focal length.
For these aplanatic systems, we deduce from (9) that:

which is independent of β, which means the well known result that the
whole optical system form a Dirac-delta intensity at the (*pc, qc*)
direction (in the First Order approximation).

We can also find the analogous example for ring-spot type, as that fulfilling:

where *f* is again an integration constant. From this dependence
ρ(β), we deduce from (9) that for these systems:

In this ideal ring-spot type case, the intensity at the exit is a centered ring
(*p _{c}*

^{*}= 0) for each angle β, but its angular radius α is not independent of β, with a maximum value of ε /

*f*. These two remarkable examples are shown in Fig. 4.

## 3. Extension to rotational microstructured surfaces

Following Ref. [2], we define a microstructured optical surface in two
dimensions as a line where the rays suffer a deflection, where deflection here means
a change of the ray vector direction. A class of microstructured surfaces can be
characterized by its “law of deflection”, which gives the new
direction of the deflected ray as a function of the direction of the incident ray
and the unit normal to the line. For instance, if **v**
_{i,2D} is
the incident ray vector in two dimensions, **v**
_{d,2D} is the
unitary deflected ray vector and **N**
_{2D} is the unitary normal
to the line, then a law of deflection is a function **v**
_{d,2D} =
**F**
_{2D}(**v**
_{i,2D},
**N**
_{2D}), where **F _{2D}** is vector function
that is defined by the specific microstructure. The microstructured surface is of
the transmissive or reflective type depending on incident and exiting rays laying at
the same or opposite side of the microstructured line, respectively (see Fig. 5). We restricted that study to the case in which
function F2D is continuous and where the exit angle in two dimensions

*v*varies monotonically with the incident angle

_{d}*v*(both shown in Fig. 5), so that the sign of

_{i}*dθ*/

_{d}*dθ*does not change.

_{i}In two dimensions the microstructures were classified in two groups, the
*regular* type for which
*dθ _{d}*/

*dθ*≥ 0 and the

_{i}*anomalous*type for which

*dθ*/

_{d}*dθ*≤ 0. Being regular or anomalous depends on the actual design of the microstructure and its corresponding deflection law. The same classification applies for reflecting or transmitting microstructures (see Fig. 5). According to this classification, non-microstructured surfaces (i.e., conventional refractive surfaces and mirrors) are of the regular type.

_{I}The analysis of the previous section cannot be directly applied to rotational
symmetric microstructured surfaces analyzed in Ref. [2] because in general the deflection law
**v**
_{d,2D}=**F _{2D}**(

**v**

_{i,2D},

**N**

_{2D}) does not guarantee that the etendue conservation applies and this conservation (given by Eq. (34)) has been used in the derivation of Eq. (2) (see Appendix I). However, for the particular case of an optical system containing only ideal microstructures as defined in Ref. [2], for which etendue is preserved in two dimensions, it can be proved (as it is done in Appendix II) that Eq. (3) and (5) are still valid if the function ρ’(β) is replaced by another function η′(β ) = (-1)

^{m}ρ ′(β), where

*m*is the number of anomalous ideal rotational microstructured surfaces that the rays impinge on. Then, if only regular microstructured surfaces are used or

*m*is even, η’(β)=ρ’(β) (so Eq. (3) and (5) are still valid as they are). On the other hand, if

*m*is an odd number, then η’(β) = -ρ’(β) (so Eq. (3) and (5) are modified). This change in sign also affects all the following equations in Section 2. Note that this change of sign affects deeply to the behavior of these rotational symmetric optics, because the nature of point-spot and ring-spot optical systems does depend on η′(β). For instance, a perfect ring-spot type rotational optical system with an odd number of ideal anomalous surfaces will have (on the contrary to Eq. (13) and (14)):

and then

Therefore, the spot will be a ring spot whose radius is constant (independent of
β). Such a system, which will have practical interest as shown in the
next section, will be called *anomalous aplanatic* system, in analogy
to the ideal point-spot type system.

In order to get a deeper insight in the change of sign in the equations in Section 2 due to the use of an odd number of anomalous rotational microstructures, we will consider one anomalous microstructure out of the infinitesimal limit, i.e., with finite structures sizes. For instance, this is the case of the TIR lens [4] shown in Fig. 6. Since it is not at the limit, its analysis can be done as slices of non-microstructured surfaces, and thus Section 2 applies for each separate slice (i.e., for each TIR facet). Each of these facets shows a decreasing function ρ =ρ(β), and since 0 ≤β < π/2 in this lens, cos(β)/ρ(β)<0, so each facet is ring-spot type according to the definition in Eq. (5). On the other hand, the envelope of function ρ =ρ (β) (shown as a dotted line in Fig. 6) is an increasing function. When the facet size is taken to the infinitesimal limit, the envelop becomes the function η(β) as defined at the beginning of this Section 3. Since a well design TIR lens is an ideal microstructure in the First Order approximation at the infinitesimal facet limit [2], the etendue conservation will guarantee the equality η′(β) = -ρ′(β) (see Appendix II).

## 4. Application: LED color mixing devices

The perfect ring-spot type systems have the interesting property of producing (in the
First Order approximation) a rotational symmetric intensity pattern (see Fig. 4(b)) from non-rotational symmetric source, for
instance, from an LED chip whose center is placed at a distance
ρ_{center} from the optical axis. That intensity pattern will
depend on the value ρ_{center} and not on the azimuthal position
of the chip. Therefore, if several chips are placed at the same distance
ρ_{center} to the optical axis, the intensity of the three
will coincide and they will simply add (the total relative intensity will remain
unchanged). This concept is presently being applied to conceive devices with LED
chips of different colors, providing spatial and angular color mixing.

On the contrary to perfect non-microstructured ring-spot systems, in which the radius
α of the ring cannot be kept constant (Eq. 14)) with β, anomalous microstructures can achieve
the anomalous aplanatic behaviour introduced in Section 2. This enables the design
of good color mixing collimators, since (in the First Order approximation) the far
field of any of the off-axis located chip in an anomalous aplanatic system will be a
collimated beam with the shape of a ring, whose inner radius is approximately given
by Eq. (16) for ε
=ρ_{center}-*R*
_{chip} and the outer
radius by the same equation with ε
=ρ_{center}+*R*
_{chip}, where
*R*
_{chip} is an average radius of the chip (see Fig. 7(a)).

Secondary optics can be added to those collimators so that the color mixing is preserved. For instance, a small angle holographic diffuser can be added to eliminate the dark zone at the center of the far field pattern, producing a zone of good uniformity, as indicated in Fig. 7(b). Ray trace results on a specific device are shown in Fig. 8.

## 4. Conclusions

For the study a class of three dimensional rotational symmetric microstructures we first classified rotational optical systems into point-spot and ring-spot types, with the remarkable perfect particular cases (Section 2). When applying this classification to the ideal microstructured rotational surfaces (Section 3), the novel concept of anomalous aplanatic system was shown to produce a sharp ring-type spot at the target for an off-axis point source. We have shown that this concept has application in color mixing from several light sources, solely by their location at the same radius from the optical axis.

## Appendix I: Derivation of Eq. (2)

Any ray R emitted at a point P_{1}=(*x*
_{1},
*y*
_{1}, *z*) with direction
**v _{1}**=(

*p*

_{1},

*q*

_{1},

*r*

_{1}) (fulfilling ∣

**v**∣ =

_{1}*n*

_{1}).that passes through P

_{2}at the exit, which is expressed as a function of β and ϕ according to Eq. (1), will point towards the direction

**v**

_{2}= (

*p*

_{2},

*q*

_{2}, (1-

*p*

^{2}

_{2}-

*q*

^{2}

_{2})

^{1/2}), which can be described by the following general equations:

$${q}_{2}={q}_{2}({x}_{1},{y}_{1},z;\beta ,\varphi )$$

The First Order Optics neighborhood of the ray R_{N} in Fig. 1 will correspond to the linear approximation of
Eq. (17) in variables
*x*
_{1}-*y*
_{1}-*z*
_{1}
around the origin O. To obtain it, consider the polynomial series expansion
of these functions around the origin O = (*x*
_{1}=0,
*y*
_{1}=0,
*z*
_{1}=0).Taking into account that
*p*
_{2}(0,0,0)=
*q*
_{2}(0,0,0)=0:

$${q}_{2}({x}_{1},{y}_{1},z;\beta ,\varphi )={q}_{2x0}(\beta ,\varphi ){x}_{1}+{q}_{2y0}(\beta ,\varphi ){y}_{1}+{q}_{2z0}(\beta ,\varphi )z+\dots $$

where the symbol “…” denotes the terms with order higher than 1, and:

$${q}_{2x0}(\beta ,\varphi )={\left[\frac{\partial {q}_{2}}{\partial {x}_{1}}\right]}_{0}{q}_{2y0}(\beta ,\varphi )={\left[\frac{{\partial q}_{2}}{{\partial y}_{1}}\right]}_{0}{q}_{2z0}(\beta ,\varphi )={\left[\frac{{\partial q}_{2}}{\partial z}\right]}_{0}$$

In matrix form:

Our goal in this Appendix is to derive the matrix
**M**
_{2,0}(β,ϕ)., The functions
*p*
_{2}, *q*
_{2} in Eq. (17) must fulfill the following equations because the
optics has rotational symmetry:

$${q\text{'}}_{2}\phantom{\rule{.2em}{0ex}}=\phantom{\rule{.2em}{0ex}}{q}_{2}\left({x\text{'}}_{1},{y\text{'}}_{1},z;\beta ,\varphi +\phi \right)$$

where:

Introducing this into the expansion in Eq. (18):

where the superscript *t* indicates transposition and:

Since Eq. (23) and Eq. (20) must be equal for all
(*x*
_{1}, *y*
_{1},
*z*), the coefficients must coincide, and thus:

Without loss of generality due to the rotational symmetry, we can set
ϕ=0, rename φ as ϕ and since
**R**
^{-1} = **R**
^{t} and
**R**
^{-1}
_{E} =
**R**
^{t}
_{E}, we obtain:

Eq. (26) shows the ϕ dependence of matrix
**M _{2,0}**(β,ϕ). Next, we
need to find the coefficients in the matrix

**M**(β, 0), which we will get by application of the skew invariant and the etendue conservation.

_{2,0}Since ϕ=0 in **M _{2,0}**(β, 0), point
P

_{2}in Eq. (1) is given by:

where ϕ_{0} is either 0 or π. From the skew
invariant, *h* =
*y*
_{1}
*p*
_{1} -
*x*
_{1}
*q*
_{1} =
*y*
_{2}
*p*
_{2} -
*x*
_{2}
*q*
_{2}, so that:

Taking into account that

$${q}_{1}\left(00,0;\beta ,\varphi \right)={n}_{1}\mathrm{sin}\phantom{\rule{.2em}{0ex}}\beta \phantom{\rule{.2em}{0ex}}\mathrm{sin}\phantom{\rule{.2em}{0ex}}\varphi $$

the polynomial expansion of both sides of Eq. (28) with respect to variables
(*x*
_{1}, *y*
_{1},
*z*) around the origin is:

Identifying the coefficients of the first order terms, we get:

In order to calculate the rest of coefficients of matrix
**M _{2,0}**(β, 0), consider now the
biparametric bundle formed by the rays emitted at the input by the points of
the line P

_{1}=

**w**

_{1}*t*, where

**w**= (

_{1}*a*

_{1},

*b*

_{1},

*c*

_{1}) is an arbitrary (but fixed) unit vector and

*t*is variable, and passing at the exit through the points P

_{2}of the line

*y*

_{2}=0, which are given by Eq. (27) in parametric form (with parameter β). Let us consider as parameters of the biparametric bundle the two parameters

*t*and β of the two lines. The ray emitted from P

_{1}=

**w**

_{1}*t*with direction

**v**= (

_{1}*p*

_{1},

*q*

_{1},

*r*

_{1}) (where ∣

**v**∣ =

_{1}*n*

_{1}) has the coordinates:

$${y}_{1}={b}_{1}t\hspace{1em}\hspace{1em}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}{q}_{1}={q}_{1}(t,\beta )$$

$${z}_{1}={c}_{1}t\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}{r}_{1}={r}_{1}(t,\beta )$$

Taking into account Eq. (27), this bundle is transformed at
*z*
_{2}=*z _{ap}* into:

$${y}_{2}=0\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}{q}_{2}={q}_{2}(t,\beta )$$

$${z}_{2}={z}_{\mathit{ap}}{\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}r}_{2}=\sqrt{1-{p}_{2}^{2}(t,\beta )-{q}_{2}^{2}(t,\beta )}$$

In general, the etendue conservation theorem for biparametric bundles [1] state that:

which in terms of the two parameters *t* and β, Eq. (34) can be written as:

Taking into account Eq. (32) and (33):

that is:

Consider now the polynomial series expansion of
*p*
_{1} =
*p*
_{1}(*t*,β),
*q*
_{1} =
*q*
_{1}(*t*,β),
*r*
_{1} =
*r*
_{1}(*t*,β) and
*p*
_{2} =
*p*
_{2}(*t*,β) around
*t* = 0:

$${q}_{1}\left(t,\beta \right)=\left[{q}_{1x0}(\beta ,0){a}_{1}+{q}_{1y0}(\beta ,0){b}_{1}+{q}_{1z0}(\beta ,0){c}_{1}\right]t+\dots $$

$${r}_{1}(t,\beta )={n}_{1}\mathrm{cos}\beta +\left[{r}_{1x0}(\beta ,0){a}_{1}+{r}_{1y0}(\beta ,0){b}_{1}+{r}_{1z0}(\beta ,0){c}_{1}\right]t+\dots $$

$${p}_{2}\left(t,\beta \right)=\left[{p}_{2x0}(\beta ,0){a}_{1}+{p}_{2y0}(\beta ,0){b}_{1}+{p}_{2z0}(\beta ,0){c}_{1}\right]t+\dots $$

where here the symbol “…” denotes the term
with order higher than 1 in *t*. Substituting Eq. (38) into Eq. (35) we obtain

where here the symbol “…” denotes the term
with order higher than 0 in *t*. Identifying the zero order
coefficients, we obtain the equation:

Since this equation must be fulfilled for any arbitrary
**w _{1}** = (

*a*

_{1},

*b*

_{1},

*c*

_{1}), the coefficients of

*a*

_{1},

*b*

_{1}and

*c*

_{1}must vanish, and thus:

$${p}_{2y0}\left(\beta ,0\right)=0$$

$${p}_{2z0}\left(\beta ,0\right)=\frac{{n}_{1}\mathrm{sin}\beta}{\rho \prime \left(\beta \right)}\mathrm{cos}\left({\varphi}_{0}\right)$$

Therefore, according to Eq. (26), (31) and (41), we get:

which coincide with Eq. (2), as can be easily checked by direct multiplication.

## Appendix II: Ideal microstructured rotational surfaces and Eq. (2)

Our objective in this Appendix is to find the analogous equations to Eq. (2)-(3) for ideal microstructured rotational surfaces. In order to achieve this, we will first deduce that when considering rotational symmetric microstructures, the deflection law in three dimensions is completely determined by the two-dimensional deflection law in the First Order Approximation. Then, the concept of ideal rotational microstructured surface will be defined and its implications will be deduced from their corresponding equations analogous to Eq. (2)-(3).

Let us consider a ray trajectory in cylindrical coordinates (ρ,
ϕ, *z*) in a rotationally symmetric optical system
(the optical axis of which is the z-axis of the coordinate system). The ray
trajectories are the solutions of the Hamilton system of differential
equations in parametric form, using the Hamitonian function H = ½
(*n*
^{2}(ρ,*z*) -
*g*
^{2}+
(*h*/ρ)^{2} +
*r*
_{2}), consistent with the condition H=0 [1]. Variables
*g*-*h*-*r* are the
conjugates variables of ρ, ϕ, *z* in
the Hamiltonian formulation (*h* is the skew invariant). The
direction vector of a ray in three dimensions is given by the vector
**v**=(*g*, *h*/ρ,
*r*) in the cylindrical coordinates, with magnitude such
that **v**
^{2} =
*g*
^{2}+(*h*/ρ)^{2}+*r*
_{2}
= *n*
^{2}(ρ). Note that *h*
= ρ (**v∙u**
_{ϕ}), where
**u**
_{ϕ} is the azimuthal unit vector
(perpendicular to the meridian plane).

As Luneburg proved [5], the ρ-*z* coordinates of
a three-dimensional ray trajectory describe also a ray trajectory in two
dimensional geometry (ρ,*z*) in which the apparent
refractive index distribution

which depends on the value of *h* of the ray (which remains
unchanged along the ray). The direction vector of that two-dimensional
trajectory is just **v _{mer}**= (

*g*,0,

*r*) (or =

**v**= (

_{mer}*g*,

*r*) for short), i.e.,

**vmer**the projection of

**v**on the meridian plane, whose magnitude fulfils:

Let **v _{i}** denote the three-dimensional incident ray
vector on a point

**X**=(ρ, ϕ,

*z*) rotational microstructured surface,

**v**the deflected ray vector and

_{d}**N**the unitary normal to the surface at

**X**. In order to calculate the deflection law in three dimensions,

**v**=

_{d}**F**(

_{3D}**v**,

_{i}**N**), let us express

**v**and

_{i}**v**as the sum of their projection on the meridian plane and their projection along the azimuthal unit vector

_{d}**u**, that is:

_{ϕ}Since the skew invariant *h* remains unchanged in the
deflection, and *h*=ρ
(**v∙uϕ**), we obtain:

As introduced in Section 3, the deflection law in two dimensions establishes
that
**v _{d,2D}**=

**F**(

_{2D}**v**

_{i,2D},

**N**

_{2D}), where the magnitudes ∣

**v**

_{i,2D}∣ =

*n*and ∣

_{i}**v**∣ =

_{d,2D}*n*, where

_{d}*n*and

_{i}*n*are the refractive indices before and after the deflection. Considering the microstructured surface out of the limit of infinisimal facet size, its design will in general may include local refractive index regions as

_{d}*n*(which is not visible at the limit since its volume will vanish). Such a dependence can be made explicit by writing

_{micro}**v**=

_{d,2D}**F**(

_{2D}**v**

_{i,2D},

**N**

_{2D};

*n*). Following Luneburg’s approach, taking into account that in rotational symmetry

_{micro}**N**3=

**N**, the projected meridian vectors

**v**and

_{i,mer}**v**will be deflected as:

_{d,mer}Eq. (47) is exact, but it has the practical problem that it
depends on *h* and *ρ*(and then the
knowledge of the internal design of the microstructure is required). Let us
see, however that in the First Order approximation, only the dependence of
**F3** upon *v*
_{i,2D},
**N**
_{2D} is needed. This is deduced because the rays
in the neighourhood of R_{N} are traced in a first-order
approximation, and the analogous expansion for Eq. (47) in cylindrical coordinates has the direction cosine
*h*/ρ as one the variables. Since
*h*=0 for R_{N}, and

we find that the square root can be approximated as the constant
*n*
_{micro} (i.e., the value of the refractive
index for *h*=0) up to first order. Therefore, in the First
Order approximation we can write Eq. (47) as:

Combining Eq. (45), (46) and (49) we get the deflection law in three dimensions, i.e.
**v _{d}** as a function of

**v**:

_{i}In general, not all the rays impinging on a real microstructured surface will
be deflected according to this deflection law. This can be due to the actual
design of the microstructure or, when dealing with extended ray bundles, due
to more fundamental reasons derived for the etendue conservation theorem. In
Ref. [2] an ideal microstructure was defined in two
dimensions as that in which all the impinging rays of the design extended
bundle are deflected according to the law, and conversely all the deflected
rays come from incident rays according to the law in two dimensions. Let us
show that those microstructures will also be ideal in three-dimensions with
rotational symmetry in the First Order Approximation. In cylindrical
coordinates, considering a differential element
(*dρ*,
*ρdθ*,*dz*) on the
microstructured surface, the etendue of a biparametric bundle incident on it
is given by:

where *d*
**I _{mer}** =
(

*dρ*,

*dz*) and

*d*

**v**= (

_{i,mer}*dg*

_{i},

*dr*). Analogously, the etendue of the deflected bundle is:

_{i}The skew invariant states that
*h _{i}*=

*h*=

_{d}*h*, while the ideality of the microstructure in two dimensions [2] states that

*d*

**Imer**∙

*d*

**v**= ±

_{i,mer}*d*

**I**∙

_{mer}*d*

**v**, where the + sign applies for regular microstructures and the - sign to the anomalous ones. Therefore, we find that for 2D ideal microstructures in the First Order approximation with rotational symmetry:

_{i,mer}Therefore, ideal regular microstructures (for which the + sign applies)
perform as non-microstructured ones in Eq. (53), because Eq. (54) with the + sign is just the cylindrical coordinates
version of Eq. (34). On the other hand, when the ray bundle crosses
*m* ideal anomalous microstructures, the successive
application Eq.(53) will lead to:

If *m* is even, Eq. (54) is again just the cylindrical coordinates version
of Eq. (34). If, however, *m* is odd, Eq. (54) implies that

should replace Eq. (34) in the derivation of Appendix I. Thus, Eq. (37) becomes

where η′(β) =
(-1)^{m}ρ′(β). Equation (56) implies that Eq. (3) and (5) remain valid when *m* is even but
ρ’(β) must be replaced by
η’(β) =
-ρ’(β) in those equations when
*m* is odd.

## Acknowledgments

This study was supported by Light Prescriptions Innovators and by the projects TEC2004-04316 and PCI2005-A9-0350 from the Spanish Ministerio de Educación y Ciencia. The authors also thank William A. Parkyn, Jr. for his help in editing the manuscript. Mr. Parkyn previously showed in ref [4] that the ring-spot pattern of a TIR lens means that it cannot be used for imaging, a particular case of the general concepts presented herein.

## References and links

**1. **R. Winston, J.C. Miñano, and P. Benítez, *Nonimaging Optics*,
(Elsevier, 2005)

**2. **P. Benítez, J. C. Miñano, and A. Santamaría, “Analysis of microstructured surfaces
in two dimensions,” Opt. Express **14**, 8561-8567
(2006).
http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-19-8561 [CrossRef] [PubMed]

**3. **M. Born and E. Wolf, *Principles of Optics*,
(Pergamon, Oxford,
1975).

**4. **W.A. Parkyn and D, Pelka, “Compact non-imaging lens with
totally internal reflecting facets”, in
*Nonimaging Optics: Maximum Efficiency Light Transfer*,
Roland Winston, ed., Proc. SPIE **1528**, 70-81,
(1991) [CrossRef]

**5. **R.K. Luneburg, *Mathematical theory of Optics*,
(U. California, Berkeley,
1964), chapter IV

**6. **D. Korsch, *Reflective Optics*, pg. 27,
(Academic, New York,
1999)