On the analysis of rotational symmetric microstructured surfaces

Pablo Benítez; Juan C. Miñano; Asunción Santamaría; Maikel Hernández

doi:10.1364/OE.15.002219

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=y3=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 P₂ 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+ϕ₀, where ϕ₀ is either 0 or π. Thus this mapping is given by:

x_{2} = ρ (β) \cos (ϕ + ϕ_{0})

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

Fig. 1. Coordinate system for tolerance study of rotational symmetric optical systems

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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 P₁ = (x ₁,y ₁,z) of the input space will, in the neighborhood of the ray R_N that passes through P₂ (in the output space), point towards the direction v₂ = (p ₂, q ₂, +(1- p ² ₂- q ² ₂)^1/2) given by:

(\begin{matrix} p_{2}^{*} \\ q_{2}^{*} \end{matrix}) = (\begin{matrix} a - b \cos (2 ϕ) & b \sin (2 ϕ) & c \cos ϕ \\ b \sin (2 ϕ) & a + b \cos (2 ϕ) & - c \sin ϕ \end{matrix}) (\begin{matrix} x_{1} \\ y_{1} \\ z \end{matrix})

where P ^* ₂ = p ₂ cos(ϕ₀), q ^* ₂ = q ₂ cos(ϕ₀) and

a = - \frac{n_{1}}{2} (\frac{\cos β}{ρ' (β)} + \frac{\sin β}{ρ (β)})

Note that since ϕ₀ is either 0 or π, P ^* ₂ = p ₂, q ^* ₂ = q ₂ for ϕ₀ = 0 and P ^* ₂ = -p ₂, q ^* ₂ = -q ₂ for ϕ₀ = 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 y₁cos(ϕ)=x ₁sin(ϕ) (with ∣x₁∣ and ∣y₁∣ 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:

\frac{\sin β}{ρ (β)} \geq 0 \Leftrightarrow a + b \leq 0

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:

Point - spot type \equiv \frac{\cos β}{ρ' (β)} > 0 Ring - spot type \equiv \frac{\cos β}{ρ' (β)} < 0

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

Point - spot type \Leftrightarrow b > a Ring - spot type \Leftrightarrow b < a

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

Point - spot type ⇛ - a \geq ∣ b ∣ \geq 0, a \neq 0 Ring - spot type ⇛ - b \geq ∣ a ∣ \geq 0, b \neq 0

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 P₁ (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 P₂ is defined by:

p_{2}^{*} = ε (a - b \cos (2 ϕ))

Let us analyze the curve obtained in the P ^* ₂ - q ^* ₂ plane when P₂ 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 P₂). The center (p ^* _C, q ^* _C) and angular radius α of the circumference (8) are:

p_{c}^{*} = aε q_{c}^{*} = 0 α = ∣ b ∣ ε

Therefore, according to (7):

Point - spot type ⇛ ∣ p_{c}^{*} ∣ \geq α \geq 0, p_{c}^{*} \neq 0 Ring - spot type \Rightarrow α \geq ∣ p_{c}^{*} ∣ \geq 0, α \neq 0

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).

Fig. 2. Locus of the ray directions emitted by P₁ (x ₁=ε>0, y ₁=0, z=0) passing through points P₂ on circumference x ² ₂+y ² ₂=ρ²(β) (a) for point-spot optical systems and (b) for ring-spot optical systems. The difference between both is summarized saying that ring-spot type systems always enclose the origin, while point-spot type never does it.

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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 P₁ (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).

Fig. 3. (a) Point-spot type (at bottom) and ring-spot type (at top) slices of a parabolic reflector. (b) Ray traces obtained from an off-axis point source, shown as a yellow sphere in (a).

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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:

α = 0 \Leftrightarrow = \frac{\sin β}{ρ (β)} = \frac{\cos β}{ρ' (β)} \Leftrightarrow ρ (β) = f \sin β

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:

p_{c}^{*} = - \frac{ε}{f}

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:

p_{c}^{*} = 0 \Leftrightarrow \frac{\sin β}{ρ (β)} = - \frac{\cos β}{ρ' (β)} \Leftrightarrow ρ (β) = \frac{f}{\sin β}

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

α = \frac{ε}{f} \sin^{2} β

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.

Fig. 4. In the First Order Optics framework, (a) point-spot system when α=0 (aplanatic), in which the spot is point-like and its center independent of β (so the whole system produce sharp imaging) (b) ring-spot system when p_C = 0, in which the spot is a ring centered on the optical axis and its radius α varies with β.

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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_d varies monotonically with the incident angle v_i (both shown in Fig. 5), so that the sign of dθ_d/dθ_i does not change.

Fig. 5. Regular and anomalously deflecting microstructures in two dimensions.

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In two dimensions the microstructures were classified in two groups, the regular type for which dθ_d/dθ_i ≥ 0 and the anomalous type for which dθ_d/dθ_I ≤ 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.

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)^mρ ′(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)):

p_{c} = 0 \Leftrightarrow \frac{\sin β}{ρ (β)} = \frac{\cos β}{ρ′ (β)} \Leftrightarrow ρ (β) = f \sin β

and then

α = \frac{ε}{f}

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).

Fig. 6. The TIR lens is an example of anomalous structured surface. When considered with finite facet size (i.e., out of the infinitesimal limit) such anomalism translates in the function ρ (β) being discontinuous, with its envelope having positive derivative but ρ’(β) < 0 in each facet.

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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)).

Fig. 7. (a) The use of anomalous microstructures allows the conception of devices that produce a well collimated ring pattern for chips placed off-axis, and thus color mixing collimators. (b) The introduction of a small angle holographic diffuser eliminates the central dark region, if desired.

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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.

Fig. 8. (a) Far field pattern produced by an off-axis chip in a specific color mixing device, which performs close to an anomalous aplanatic system. (b) Horizontal and vertical cross sections of that pattern produced by this device for a single off-axis light source showing that FWHM = 5° (chip is modelled as a Lambertian square with side = 0.25 mm, chip center being set at a distance of 0.5 mm to the optical axis; the optics diameter is 50 mm).

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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 P₁=(x ₁, y ₁, z) with direction v₁=(p ₁,q ₁,r ₁) (fulfilling ∣v₁∣ = n ₁).that passes through P₂ 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:

p_{2} = p_{2} (x_{1}, y_{1}, z; β, ϕ)

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:

p_{2} (x_{1}, y_{1}, z; β, ϕ;) = p_{2 x 0} (β, ϕ;) x_{1} + p_{2 y 0} (β, ϕ) y_{1} + p_{2 z 0} (β, ϕ) z + \dots

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

p_{2 x 0} (β, ϕ) = {[\frac{\partial p_{2}}{\partial x_{1}}]}_{0} p_{2 y 0} (β, ϕ) = {[\frac{\partial p_{2}}{{\partial y}_{1}}]}_{0} p_{2 z 0} (β, ϕ) = {[\frac{{\partial p}_{2}}{\partial z}]}_{0}

In matrix form:

(\binom{p_{2}}{q_{2}}) = M_{2,0} (β, ϕ) (\begin{matrix} x_{1} \\ y_{1} \\ z \end{matrix}) + \dots M_{2,0} (β, ϕ) = (\begin{matrix} p_{2 x 0} (β, ϕ) & p_{2 y 0} (β, ϕ) & p_{2 z 0} (β, ϕ) \\ q_{2 x 0} (β, ϕ) & q_{2 y 0} (β, ϕ) & q_{2 z 0} (β, ϕ) \end{matrix})

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:

{p'}_{2} = p_{2} ({x'}_{1}, {y'}_{1}, z; β, ϕ + φ)

where:

(\begin{matrix} {x'}_{1} \\ {x'}_{2} \end{matrix}) = R (φ) (\begin{matrix} x_{1} \\ y_{1} \end{matrix}) (\begin{matrix} {p'}_{2} \\ {q'}_{2} \end{matrix}) R (φ) = (\begin{matrix} p_{2} \\ q_{2} \end{matrix}) R (φ) = (\begin{matrix} \cos φ & \sin φ \\ - \sin φ & \cos φ \end{matrix})

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

(\begin{matrix} p_{2} \\ q_{2} \end{matrix}) = R^{t} (φ) M_{2,0} (β, ϕ + φ) R_{E} (φ) (\begin{matrix} x_{1} \\ y_{1} \\ z \end{matrix}) + \dots

where the superscript t indicates transposition and:

R_{E} = (\begin{matrix} \cos φ & \sin φ & 0 \\ - \sin φ & \cos φ & 0 \\ 0 & 0 & 1 \end{matrix})

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

R^{t} (φ) M_{2,0} (β, ϕ + φ) R_{E} (φ) \equiv M_{2,0} (β, ϕ)

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:

M_{2,0} (β, ϕ) \equiv R (ϕ) M_{2,0} (β, 0) R_{E}^{t} (ϕ)

Eq. (26) shows the ϕ dependence of matrix M_2,0(β,ϕ). Next, we need to find the coefficients in the matrix M_2,0(β, 0), which we will get by application of the skew invariant and the etendue conservation.

Since ϕ=0 in M_2,0(β, 0), point P₂ in Eq. (1) is given by:

x_{2} = ρ (β) \cos (ϕ_{0})

where ϕ₀ 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:

y_{1} p_{1} (x_{1}, y_{1}, z; β, 0) - x_{1} q_{1} (x_{1}, y_{1}, z; β, 0) = - \cos (ϕ_{0}) ρ (β) q_{2} (x_{1}, y_{1}, z; β, 0)

Taking into account that

p_{1} (00, 0; β, ϕ) = n_{1} \sin β \cos ϕ

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

n_{1} y_{1} \sin β + \dots = - \cos (ϕ_{0}) ρ (β) (q_{2 x 0} (β, 0) x_{1} + q_{2 y 0} (β, 0) y_{1} + q_{2 z 0} (β, 0) z) + \dots

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

q_{2 x 0} (β, 0) = q_{2 z 0} (β, 0) = 0 q_{2 y 0} (β, 0) = - \cos (ϕ_{0}) \frac{n_{1} \sin β}{ρ (β)}

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 P₁ = w₁ t, where w₁ = (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 P₂ 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 P₁ = w₁ t with direction v₁ = (p ₁, q ₁, r ₁) (where ∣v₁∣ = n ₁) has the coordinates:

x_{1} = a_{1} t p_{1} = p_{1} (t, β)

Taking into account Eq. (27), this bundle is transformed at z ₂=z_ap into:

x_{2} = ρ (β) \cos (ϕ_{0}) p_{2} = p_{2} (t, β)

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

d x_{1} d p_{1} + d y_{1} d q_{1} + d z_{1} d r_{1} \equiv d x_{1} d p_{2} + d y_{2} d q_{2} + d z_{2} d r_{2}

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

|\begin{matrix} \frac{{\partial x}_{1}}{\partial t} & \frac{{\partial x}_{1}}{\partial β} \\ \frac{{\partial p}_{1}}{\partial t} & \frac{{\partial p}_{1}}{\partial β} \end{matrix}| + |\begin{matrix} \frac{{\partial y}_{1}}{\partial t} & \frac{{\partial y}_{1}}{\partial β} \\ \frac{{\partial q}_{1}}{\partial t} & \frac{{\partial q}_{1}}{\partial β} \end{matrix}| + |\begin{matrix} \frac{{\partial z}_{1}}{\partial t} & \frac{{\partial z}_{1}}{\partial β} \\ \frac{{\partial r}_{1}}{\partial t} & \frac{{\partial r}_{1}}{\partial β} \end{matrix}| = |\begin{matrix} \frac{{\partial x}_{2}}{\partial t} & \frac{{\partial x}_{2}}{\partial β} \\ \frac{{\partial p}_{2}}{\partial t} & \frac{{\partial p}_{2}}{\partial β} \end{matrix}| + |\begin{matrix} \frac{{\partial y}_{2}}{\partial t} & \frac{{\partial y}_{2}}{\partial β} \\ \frac{{\partial q}_{2}}{\partial t} & \frac{{\partial q}_{2}}{\partial β} \end{matrix}| + |\begin{matrix} \frac{{\partial z}_{2}}{\partial t} & \frac{{\partial z}_{2}}{\partial β} \\ \frac{{\partial r}_{2}}{\partial t} & \frac{{\partial r}_{2}}{\partial β} \end{matrix}|

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

|\begin{matrix} a_{1} & 0 \\ \frac{{\partial p}_{1}}{\partial t} & \frac{{\partial p}_{1}}{\partial β} \end{matrix}| + |\begin{matrix} b_{1} & 0 \\ \frac{{\partial q}_{1}}{\partial t} & \frac{{\partial q}_{1}}{\partial β} \end{matrix}| + |\begin{matrix} c_{1} & 0 \\ \frac{{\partial r}_{1}}{\partial t} & \frac{{\partial r}_{1}}{\partial β} \end{matrix}| = |\begin{matrix} 0 & ρ' (β) \cos (ϕ_{0}) \\ \frac{{\partial p}_{2}}{\partial t} & \frac{{\partial p}_{2}}{\partial β} \end{matrix}|

that is:

\frac{{\partial p}_{1}}{\partial β} a_{1} + \frac{{\partial q}_{1}}{\partial β} b_{1} + \frac{{\partial r}_{1}}{\partial β} c_{1} = - ρ' (β) \cos (ϕ_{0}) \frac{{\partial p}_{2}}{\partial t}

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:

p_{1} (t, β) = n_{1} \sin β + [p_{1 x 0} (β, 0) a_{1} + p_{1 y 0} (β, 0) b_{1} + p_{1 z 0} (β, 0) c_{1}] t + \dots

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

(n_{1} \cos β) a_{1} + (- n_{1} \sin β) c_{1} + . . . = ρ' (β) \cos (ϕ_{0}) [p_{2 x 0} (β, 0) a_{1} + p_{2 y 0} (β, 0) b_{1} + p_{2 z 0} (β, 0) c_{1}] + . . .

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

[n_{1} \cos β + ρ ´ (β) \cos (ϕ_{0}) p_{2 x 0} (β, 0)] a_{1} + [ρ ´ (β) \cos (ϕ_{0}) P_{2 y 0} (β, 0)] b_{1} + [{- n}_{1} \sin β + ρ ´ (β) \cos (ϕ_{0}) P_{2 z 0} (β, 0)] c_{1} = 0

Since this equation must be fulfilled for any arbitrary w₁ = (a ₁, b ₁, c ₁), the coefficients of a ₁, b ₁ and c ₁ must vanish, and thus:

p_{2 x 0} (β, 0) = - \frac{n_{1} \cos β}{ρ' (β)} \cos (ϕ_{0})

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

M_{2,0} (β, ϕ) = R (ϕ) (\begin{matrix} - n_{1} \frac{\cos β}{ρ' (β)} \cos (ϕ_{0}) & 0 & n_{1} \frac{\sin β}{ρ' (β)} \cos (ϕ_{0}) \\ 0 & - n_{1} \frac{\sin β}{ρ (β)} \cos (ϕ_{0}) & 0 \end{matrix}) R_{E}^{t} (ϕ)

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

n_{ap} (ρ, z) = \sqrt{n^{2} (ρ, z) - \frac{h^{2}}{ρ^{2}}}

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:

v_{mer}^{2} = g^{2} + r^{2} = n^{2} (ρ, z) - \frac{h^{2}}{ρ^{2}} = n_{ap}^{2} (ρ, z)

Let v_i denote the three-dimensional incident ray vector on a point X=(ρ, ϕ, z) rotational microstructured surface, v_d the deflected ray vector and 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_i and v_d as the sum of their projection on the meridian plane and their projection along the azimuthal unit vector u_ϕ, that is:

v_{i} = (v_{i} \cdot u_{ϕ}) u_{ϕ} + v_{i, mer} v_{d} = (v_{d} \cdot u_{ϕ}) u_{ϕ} + v_{d, mer}

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

v_{d} \cdot u_{ϕ} = v_{i} \cdot u_{ϕ}

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_i and ∣v_d,2D∣ = n_d, where n_i and n_d 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 n_micro (which is not visible at the limit since its volume will vanish). Such a dependence can be made explicit by writing v_d,2D = F_2D(v _i,2D,N _2D;n_micro). Following Luneburg’s approach, taking into account that in rotational symmetry N3=N, the projected meridian vectors v_i,mer and v_d,mer will be deflected as:

v_{d, mer} = F_{2 D} ({v_{i, mer}, N;}_{} \sqrt{n_{micro}^{2} - (\frac{h^{2}}{ρ^{2}})})

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

\sqrt{n_{micro}^{2} - (\frac{h^{2}}{ρ^{2}})} = n_{micro} - \frac{1}{{2 n}_{micro}} (\frac{h^{2}}{ρ^{2}}) + . . .

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:

v_{d, mer} \approx F_{2 D} (v_{i, mer}, N)

Combining Eq. (45), (46) and (49) we get the deflection law in three dimensions, i.e. v_d as a function of v_i:

v_{d} = F_{3 D} (v_{i}, N) \approx (v_{i} \cdot u_{ϕ}) u_{ϕ} + F_{2 D} (v_{i} - (v_{i} \cdot u_{ϕ}) u_{ϕ}, N)

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:

d E_{i} = dρd g_{i} + dzd r_{i} + dθd h_{i} \equiv d l_{mer} \cdot d v_{i, mer} + dθd h_{i}

where d I_mer = (dρ,dz) and d v_i,mer = (dg _i, dr_i). Analogously, the etendue of the deflected bundle is:

d E_{d} = dρd g_{d} + dzd r_{d} + dθd h_{d} \equiv d l_{mer} \cdot d v_{d, mer} + dθd h_{d}

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_i,mer, 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:

d ρ d g_{i} + dzd r_{i} + dθdh \equiv \pm (d ρd g_{d} + dzd r_{d}) + dθdh

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:

d ρ_{i} d g_{i} + d z_{i} d r_{i} + d θ_{i} dh \equiv {(- 1)}^{m} (d ρ_{d} d g_{d} + d z_{d} d r_{d}) + d θ_{d} dh

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

d x_{1} d p_{1} + d z_{1} d r_{1} + d y_{1} d q_{1} \equiv - (d x_{2} d p_{2} + d z_{2} d r_{2}) + d y_{2} d q_{2}

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

\frac{\partial p_{1}}{\partial β} a_{1} + \frac{\partial q_{1}}{\partial β} b_{1} + \frac{\partial r_{1}}{\partial β} c_{1} = - η ˈ (β) \cos (ϕ_{0}) \frac{\partial p_{2}}{\partial t}

where η′(β) = (-1)^mρ′(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)

On the analysis of rotational symmetric microstructured surfaces

Abstract

1. Introduction

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

3. Extension to rotational microstructured surfaces

4. Application: LED color mixing devices

4. Conclusions

Appendix I: Derivation of Eq. (2)

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

Acknowledgments

References and links

Cited By

Figures (8)

Equations (75)

Optics Express