Exact wavefronts and caustic surfaces produced by planar ripple lenses

Maximino Avendaño-Alejo; Luis Castañeda; Iván Moreno

doi:10.1364/OE.23.021637

1. Introduction

Smooth arbitrary planar curved lenses can potentially deliver higher performing, more compact, and lighter systems in a wide range with potential applications such as microscopy, corrector plates, solar concentrators, waveform shaping or in illumination systems.

The caustic can be defined as the locus of the principal centers of curvature of a wavefront, also, as the envelope for either refracted or reflected rays crossing an optical system. In other words, if equal optical paths are measured along each ray from the source, the surface constructed by the end points will be normal to all the rays. These surfaces are the phase fronts of the wave system, for which the rays are defined within the geometrical optics approximation [1]. Although the caustics and wavefronts either by reflection or refraction are part of a well known subject, the contribution in this work is to provide simple formulas of the caustic and wavefront surfaces caused by refraction on plano-curved lenses, whose shape is represented by a smooth surface lying in a meridional plane, in other words, it lies on a normal section of the refracting surface, coinciding with the tangential plane according to [2]. As is well known for rotationally symmetric optical system, the sagittal caustic for an axial object point degenerates into a segment of a straight line lying on the axis of symmetry and therefore the sagittal caustic is not considered in this manuscript. Finally, the condition for rays that undergo Total Internal Reflection (TIR), and an additional condition imposed on the refracting surface to overcome the TIR increasing the amount of refracted rays outside of the lens are provided, in such a way that we get an optimized shape for a smooth surface to refract all the light outside of the lens.

The treatment in this work to obtain the refracted wavefront is in agreement with the results presented by Shealy and Hoffnagle [3], where they have provided a simple expression for the k-function associated with the general solution of Stavroudis to the eikonal equation for refraction of a plane wave through an arbitrary surface. In our analysis we consider exact ray tracing in order to obtain the envelope for all refracted rays through smooth arbitrary planar curved lenses, and considering a proper transformation gives a formula for the caustic by reflection, which is also called catacaustic. Finally regarding the Huygens’s Principle we provide a simple expression for refracted wavefronts through planar ripple lenses. In other words, the results presented here illustrate the equivalence of the Huygens’ principle and the eikonal treatment. On a practical level, this Huygens’ approach has advantages: since the treatment here involves performing a sum of wavelets whose envelope will form the wavefront and does not imply solving a differential equation, we do not need to construct initial boundary conditions if we know a priori the parametric formulas which represent the refracting surface. Furthermore if we demand a particular shape for the refracted wavefront are able to obtain the surface through which is possible to produce this wavefront, the final result is a relatively simple solution that can be used to directly compute the smooth surface by solving numerically at first glance a coupled set of first order non-linear differential equations. From an optical design point of view, this is applicable to design corrector plates and generally speaking to design arbitrary lenses for wavefront shaping or collimator lenses for solar concentrators.

2. Exact caustic produced by a smooth arbitrary surface

Throughout this manuscript lower cases for (z, y) will be used to designate either formulas for the caustic surfaces or refracted rays through lenses and upper cases for (Z, Y) will be used to designate the geometrical wavefront. We define that the Z axis is parallel to the optical axis, we assume that Y − Z-plane is the plane of incidence, which is a cross section of an arbitrary surface whose shape is represented through a smooth arbitrary curve, with an origin of the system $O$ placed at the vertex of the lens. Without loss of generality we assume that light rays enter from the left and a plane wave is incident on the lens propagating along the optical axis, crossing the plane face of the lens without being deflected, and the rays are extended until they are refracted by the smooth surface as is shown in Fig. 1(a). A parametrized curve $S = (ℱ (τ), G (τ))$ in the real Euclidean space $ℛ^{2}$ where each function of the entries for S has derivatives of all orders, with S′(τ) ≠ 0, it moves along S as τ varies, and it never stops or turns round, since S′(τ) is never zero, the curves which satisfy this condition are called smooth curves, in other words, a curve is smooth at a point if its partial derivatives of its representative function exist, are continuous and are single valued.

Fig. 1 (a) A smooth surface and their parameters involved in the process of refraction. (b). Exact ray tracing through a smooth lens and their associated caustics.

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Let ±H be the entrance aperture, n_i the index of refraction of the lens for a predefined wavelength which is immersed in a medium with index of refraction n_a, (n_i > n_a), and where we have assumed that S represents the equation for the refracting surface in a meridional plane which is composed by two differentiable arbitrary functions provided as

S = (ℱ (τ), G (τ)),

considering that

G (τ)

is related to the heights for all arbitrary incident rays

G \in [- H, H]

. We assume that there is rotational symmetry about the Z axis, where the maximum thickness of the lens t_max lies on the optical axis and for simplicity we have called t. Let P₁ be an image point formed by intersection of the refracted rays A and B, and P₂ is the image point formed by the refracted rays B and C as is shown in Fig. 1(a). In this way, every refracted ray cuts the one next to it, the locus of their points of intersection is the required caustic by refraction. In order to obtain the caustic we consider the refracted ray P_iP₂ as representative for all refracted rays, assuming that P_i is a point on refracting surface whose coordinates are given by

P_{i} = (z_{i}, y_{i}) = (t + ℱ [τ], G [τ])

. Therefore, P_iP₂ can be written simply as

y \cos (θ_{a} - θ_{i}) + z \sin (θ_{a} - θ_{i}) = G \cos (θ_{a} - θ_{i}) + [t + ℱ] \sin (θ_{i} - θ_{i}),

where θ_i and θ_a are the incident and refractive angles respectively. This is a parametric family of refracted rays as is shown in Fig. 1(b). The caustic is the envelope of this family of rays. It is very easy to show that, θ_a is related through Snell’s law to θ_i, and they can be written as a function of an arbitrary parameter τ where the ray impinges on the smooth surface according to

θ_{i} = \arctan [- \frac{ℱ_{τ}}{G_{τ}}], θ_{a} = \arcsin [\frac{- n_{i} ℱ_{τ}}{n_{a} \sqrt{ℱ_{τ}^{2} + G_{τ}^{2}}}],

for θ_a means that an incident ray travels from a medium with index of refraction n_i (commonly an isotropic medium), to a different medium with index of refraction n_a (which is usually the index of refraction of air), where

ℱ_{τ}

and

G_{τ}

are the first derivatives with respect to τ. In order to obtain the envelope of the family of refracted rays [4], we differentiate Eq. (2) with respect to τ and reducing similar factors we obtain

- y \sin (θ_{a} - θ_{i}) + z \cos (θ_{a} - θ_{i}) = - G \sin (θ_{a} - θ_{i}) + [t + ℱ] \cos (θ_{i} - θ_{i}) + ℋ,

where we have defined

ℋ = \frac{G_{τ} \cos (θ_{a} - θ_{i}) + ℱ_{τ} \sin (θ_{a} - θ_{i})}{\partial θ_{a} / \partial τ - \partial θ_{i} / \partial τ} .

Finally, by solving Eqs. (4) and (2) for (z, y) and reducing further we get

\begin{matrix} z d (τ) = t + ℱ + \frac{[n_{a}^{2} G_{τ}^{2} + (n_{a}^{2} - n_{i}^{2}) ℱ_{τ}^{2}] [n_{a}^{2} G_{τ} + n_{i} \sqrt{n_{a}^{2} G_{τ}^{2} + (n_{a}^{2} - n_{i}^{2}) ℱ_{τ}^{2}}]}{n_{a}^{2} (n_{i}^{2} - n_{a}^{2}) [ℱ_{τ} G_{τ τ} - G_{τ} ℱ_{τ τ}]}, \\ y d (τ) = G + \frac{[n_{a}^{2} G_{τ}^{2} + (n_{a}^{2} - n_{i}^{2}) ℱ_{τ}^{2}] ℱ_{τ}}{n_{a}^{2} [ℱ_{τ} G_{τ τ} - G_{τ} ℱ_{τ τ}]}, \end{matrix}

where the subscript d means diacaustic (caustic by refraction), and

ℱ_{τ τ}

,

G_{τ τ}

are the second derivatives with respect to τ. It is important to remark that Eq. (5) provides the coordinates of the locus of points that parametrically represent the dicaustic produced by a plano-curved lens in a meridional plane as a function of τ when the point source is placed at infinity. If we consider that

ℱ \to S_{h}

, demanding that τ = h, hence

ℱ_{τ} \to {S^{'}}_{h}

,

ℱ_{τ τ} \to {S^{″}}_{h}

,

G \to h

,

G_{τ} \to 1

,

G_{τ τ} \to 0

and substituting these values into Eq. (5) it is reduced to Eq. (7) according to [5] for an aspheric lens. On the other hand, if we consider that

ℱ \to z

, where z = z(r) is the sag of aspheric lens surface, demanding that τ = r, hence

ℱ_{τ} \to z^{'}

,

ℱ_{τ τ} \to z^{″}

,

G \to r

,

G_{τ} \to 1

,

G_{τ τ} \to 0

, t → 0, n_i/n_a → n, and substituting these values into Eq. (5) it is reduced to Eq. (25) provided in [2] for an aspheric lens. To sum up,

S = (ℱ, G)

is a one-parameter smooth curve, expressed in terms of an auxiliary variable τ, and as is well known a change of parameterization does not change the geometry of the curve in any way, such that Eq. (5) could be considered as a general re-parameterization of the tangential caustic surface, without the necessity of having recourse to the explicit expression of either S_h(h) or z(r) respectively as were regarded in the examples explained above. Furthermore, making a proper transformation with the following assumptions:

z \to ℱ

,

r \to G

,

z^{'} \to ℱ_{τ} / G_{τ}

,

z^{″} \to (ℱ_{τ τ} G_{τ} - ℱ_{τ} G_{τ τ}) / {(G_{τ})}^{3}

, n → n_i/n_a and substituting into Eq. (25) of [2] we obtain Eq. (5) as we expected. Additionally, it is important to state that properly using Eq. (5) with the assumptions t = 0, n_i → −n_a and reducing further yields

(z_{c} (τ), y_{c} (τ) (ℱ + \frac{G_{τ} [ℱ_{τ}^{2} - G_{τ}^{2}]}{2 (ℱ_{τ} G_{τ τ} - ℱ_{τ τ} G_{τ})}, G + \frac{ℱ_{τ} G_{τ}^{2}}{ℱ_{τ} G_{τ τ} - ℱ_{τ τ} G_{τ}}),

where the subscript c means catacaustic (also known as caustic by reflection) which provide the coordinates of the locus of points that parametrically represent the tangential catacaustic produced by a smooth curved mirror as a function of τ, considering that a plane wavefront propagating along the optical axis is impinging on the smooth mirror. Alternatively, it is straightforward to show that assuming the following transformations:

ℱ \to z

,

ℱ_{τ} \to z^{'}

,

ℱ_{τ τ} \to z^{″}

,

G_{τ} \to 1

,

G_{τ τ} \to 0

, and substituting into Eq. (6) yields

(z_{c} (r), y_{c} (r)) = (z + \frac{1 - {z^{'}}^{2}}{2 z^{″}}, r - \frac{z^{'}}{z^{″}}),

where the subscript c means catacaustic, regarding that z = z(r) represent the sag of aspheric mirrors having rotational symmetry.

From Eq. (5) if the radical $n_{a}^{2} G_{τ}^{2} + (n_{a}^{2} - n_{i}^{2}) ℱ_{τ}^{2} \leq 0$ , then physically there are rays that undergo Total Internal Reflection (TIR), additionally, $(ℱ_{τ} / G_{τ})$ is related to the angle of incidence according to Eq. (3). Thus, for an arbitrary lens the condition for TIR imposes that

\pm \frac{n_{a}}{\sqrt{n_{i}^{2} - n_{a}^{2}}} = [\frac{ℱ_{τ}}{G_{τ}}] .

By solving Eq. (8) for τ whose real values we define as τ_ca, where the subscript ca means critical angles, providing the following considerations: If $G |_{τ_{c a}} \leq H$ , being H is the entrance aperture of the lens, and demanding that for τ → τ_ca then $n_{a}^{2} G_{τ_{c a}}^{2} + (n_{a}^{2} - n_{i}^{2}) ℱ_{τ_{c a}}^{2} = 0$ , and substituting these values into Eq. (5) then the caustic surface touches the smooth surface at $[t + ℱ (τ_{c a}), G (τ_{c a})]$ as is shown in Fig. 1(b), we can clearly see that there are regions in this configuration where the transmission of light is limited by TIR, on the other hand if we consider that the refracting surface is formed by parts of concave and convex surfaces, then will be formed both virtual and real caustic surfaces respectively, and from Eq. (8) we could get several solutions for τ_ca yielding the gaps for TIR. Finally, if $G |_{τ_{c a}} \leq H$ , then the caustic does not touch the lens and therefore there is no TIR when considering a plane wave passing through the refracting surface, it was explained briefly in [6]. It is worth to say that indices of refraction play a very important role for designing properly plano-curved lens to overcome loss due to TIR. The main idea is to reduce the gap produced by TIR between concave and convex regions and additionally to obtain a refraction angle θ_max at the borders of the lens as greater as it could be possible with potential applications such as diffusers of light or even more like collimators for solar cells.

3. Example: caustic produced by an arbitrary cycloid

A famous plane curve, called the cycloid, has been considered as an example in [7] providing a formula for the catacaustic, which can also be obtained by suitably using Eq. (6), as follows: Considering the cycloid $(G, ℱ) = (1 - \cos τ, τ - \sin τ)$ , and substituting into Eq. (6) we obtain (y_cc, z_cc) = (1/2)(1 − cos Γ, Γ − sin Γ), where the subscript cc means catacaustic of cycloid and we define Γ = 2τ, such that for one arch of a cycloid the catacaustic becomes two arches cycloid with identical beginning and endpoints and twice the period. On the other hand, we consider a slight modification of the cycloid as follows: We define as the locus of points traced out by a point on a circle of radius ”b” when a concentric circle of radius ”a” rolls without slipping along a straight line being it a cycloid of two parameters, mathematically this is represented by

(G, ℱ) = (\frac{- b}{2 π} (1 + \cos τ), \frac{a}{10 π} (τ - π - \frac{b}{a} \sin τ)), τ \in [- π, 3 π] .

The cycloid is called prolate cycloid if b > a and curate cycloid if a > b [8], in this work we have consider curate cycloids exclusively. The shapes of the cycloids are shown in Fig. 2(a). For a ≈ b the cycloid is going to present a singularity. For a ≫ b the cycloid is going to be a flat surface. If we extend the range for τ and considering rotational symmetry a possible diffusor of light takes the shape as is shown in Fig. 2(b). The main reason to consider a modified cycloid is its versatility, we can see from Eq. (9) that by expanding for $ℱ$ and $G$ in Taylor’s series as a function of τ, yields an even-order and an odd-order polynomial as a function of τ for each entry respectively, this opens the doors to designing analytically corrector plates. By taking the first and second derivatives from Eq. (9) we get

\begin{array}{l} (ℱ_{τ}, G_{τ}) = (\frac{b \sin τ}{2 π}, \frac{a - b \cos τ}{10 π}), \\ (ℱ_{τ τ}, G_{τ τ}) = (\frac{b \cos τ}{2 π}, \frac{a - b \sin τ}{10 π}) . \end{array}

Fig. 2 (a) Different curate cycloids for τ ∈ [−π, 3π] with arbitrary dimensions. (b) A smooth lens considering a cycloid with a = 5.0, b = 0.88, for τ ∈ [−12π, 20π].

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Where $(ℱ_{τ}, G_{τ}) \neq (0, 0)$ in the range for τ ∈ [−π, 3π], which also satisfies the condition imposed to be a smooth surface. By substituting Eqs. (9) and (10) into Eq. (5) and reducing further it is written as

(z_{c c}, y_{c c}) = (t + ℱ + \frac{Δ [n_{a}^{2} (a - b \cos τ) + n_{i} \sqrt{Δ}]}{50 π b n_{a}^{2} (n_{i}^{2} - n_{a}^{2}) [b - a \cos τ]}, G + \frac{Δ \sin τ}{10 π n_{a}^{2} [b - a \cos τ]}),

where we have defined

Δ = n_{a}^{2} {(a - b \cos τ)}^{2} + 25 b^{2} (n_{a}^{2} - n_{i}^{2}) \sin^{2} τ .

In other words, (z_cc,y_cc) from Eq. (11) gives the coordinates of the locus of points that parametrically represent the diacaustic produced by an arbitrary curate cycloid as are shown in Fig. 3, where we have chosen the following parameters: a = 5.0cm, t = 0.8cm, n_a = 1 and n_i = 1.5 for τ ∈ [−π,3π], varying slightly the values for b in all cases. It is very easy to show that if Δ → 0 from Eq. (11) is reduced to $(z_{c c}, y_{c c}) |_{Δ \to 0} = (t + ℱ, G)$ , which is a point lying on the refracting surface as was explained above. In Fig. 3(a) we can see that there are four gaps which undergo TIR, reducing the transmission of light through the lens, the values for the angles where the refraction is forbidden are also provided in Fig. 3(a), for instance θ_max = 48.189°, is provided for either τ = 1.0359rad or τ = 1.7516rad, whose points on the surface are (0.5741,−0.3608) and (0.6773, −0.2506) respectively in the range for τ ∈ [0, π], in other words for two rays impinging on the surface at these two positions they will be refracted outside of the lens in parallel lines with θ_max. Alternatively, as we can see in Fig. 3(b) there is no region having TIR, furthermore the caustic surface is a curve which changes from real (caustic outside of the lens) to virtual (caustic inside of the lens) without discontinuities, the principal idea is to get the right shape for a smooth surface avoiding gaps by TIR, increasing the amount of light transmitted through the planar ripple lens, as will be explained later. Thus, for instance the maximum angle of transmission considering this case yields θ_max = 48.189° for τ = 1.3937rad whose point on the refracting surface is given by (0.6357, −0.3051) in the range for τ ∈ [0, π]. Finally, in Fig. 3(c) there are no gaps due to TIR, and the caustic is a discontinuous curve, whose shape of the cycloid is going to be a flat surface, and therefore the maximum slope due to refraction is reduced considerably being θ_max = 33.773° for τ = 1.4060rad having a point on the lens given by (0.6480, −0.3019) in the range for τ ∈ [0, π].

Fig. 3 (a) a = 5cm, b = 0.94cm, with θ_max = 48.189°. (b) a = 5cm, b = 0.88045cm, with θ_max = 48.189°. (c) a = 5cm, b = 0.82cm, with θ_max = 33.7731°, for τ ∈ [−π, 3π].

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The angle for each of the refracted rays outside of the lens is given by θ_m(τ) = θ_a − θ_i, as is shown in Fig. 1(a), therefore either maximum or minimum angles which leave the lens according to our reference system leads to solve for τ the following expression

\frac{d [θ_{a} - θ_{i}]}{d τ} = [\frac{(n_{i}^{2} - n_{a}^{2})}{n_{i} G_{τ} + \sqrt{n_{a}^{2} G_{τ}^{2} + (n_{i}^{2} - n_{a}^{2}) ℱ_{τ}^{2}}}] [\frac{G_{τ} ℱ_{τ τ} - ℱ_{τ} G_{τ τ}}{\sqrt{n_{a}^{2} G_{τ}^{2} + (n_{i}^{2} - n_{a}^{2}) ℱ_{τ}^{2}}}] = 0,

where we have considered the values for θ_a and θ_i from Eq. (3). It is very easy to show that the denominator in the first term of Eq. (12) is never zero because it is reduced to

ℱ_{τ}^{2} + G_{τ}^{2} \neq 0

. On the other hand, the radical in the denominator in the second term from Eq. (12) imposes the critical angle as was explained above, and it also yields a removable singularity by using properly its numerator. On the other hand, this numerator is related directly to the second derivative

d^{2} G / d ℱ^{2}

and therefore provides information about the inflection points, furthermore, it is well known that at a point of inflection the radius of curvature is infinite. In summary, the main idea is to consider the radical equal to zero, which also provides the critical angle and simultaneously demands the numerator to be zero providing the coordinates for inflection points, in such a way that the gap produced by TIR can be reduced as much as it is possible, until it finally it coincides between a ray having the coordinates for TIR and an inflection point where the radius of curvature is infinite. Thus, from Eq. (12) we get

\begin{matrix} G_{τ} ℱ_{τ τ} - ℱ_{τ} G_{τ τ} = 0, \\ n_{a}^{2} G_{τ}^{2} + (n_{a}^{2} - n_{i}^{2}) ℱ_{τ}^{2} = 0. \end{matrix}

To sum up, by solving adequately the set of coupled differential equations from Eq. (13), we get an optimized smooth refracting surface, which does not have TIR and simultaneously it refracts all the light outside of lens having the maximum slopes for refracted rays through this surface. For example, let $G$ and $ℱ$ be arbitrary functions given by Eq. (9), and substituting the values from Eq. (10) into the first formula from Eq. (13) and reducing similar terms it can be written as follows

b = a \cos τ .

Alternatively, reducing the second formula from Eq. (13) yields: $ℱ_{τ} / G_{τ} = \pm n_{a} / {[n_{i}^{2} - n_{a}^{2}]}^{1 / 2}$ , and substituting into the first formula of Eq. (13), and demanding that $G_{τ} G_{τ τ} \neq 0$ , we have

\frac{ℱ_{τ τ}}{G_{τ τ}} = \pm \frac{n_{a}}{\sqrt{n_{i}^{2} - n_{a}^{2}}},

introducing the values for

ℱ_{τ τ}

and

G_{τ τ}

from Eq. (10) into Eq. (15), and solving for τ we obtain

τ = \pm \arctan [\frac{5 \sqrt{n_{i}^{2} - n_{a}^{2}}}{n_{a}}],

for example, considering the values given above for n_a = 1 and n_i = 1.5, thus from Eq. (16) yields τ = ±1.3937. Finally, substituting Eq. (16) into Eq.(14) we get

b = a [\frac{n_{a}}{\sqrt{25 n_{i}^{2} - n_{a}^{2}}}],

again as was considered before, we also use the values for n_a and n_i provided above and we hold on the value for a = 5 to be a constant, obtaining for b = 0.88045, which leads to the right shape for a smooth surface without undergoes TIR, which we have called an optimized surface, as is shown if Fig. 4, where we can clearly see that the slopes for refracted rays are greater for a surface having TIR than for an optimized surface up to the critical angle. Furthermore, the final contribution to refract all the light outside of the lens is greater for an optimized surface than for other surfaces.

Fig. 4 Behavior of the slopes for refracted rays outside of the smooth surfaces showing either continuity or discontinuity through the surface.

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4. Exact wavefront produced by a smooth arbitrary surface: Huygens’ principle

We assume that a plane wave is incident on the lens parallel to the optical axis, crossing the plane face of the lens without being deflected, and this is propagated through the lens until it reaches the smooth surface at P₀ as is shown in Fig. 5(a). In this way the Huygens’ principle assumes that a wavefront is the envelope of an aggregate of wavelets centered on a previous wave front in the wave-front train. The idea here is to find the family of wavelets that are centered on the smooth refracting surface and whose envelope yields the refracted wavefront, which is also called the phase front, zero-distance phase front or alternatively the archetype wavefront [9]. In order to obtain the refracted wavefront outside of the lens, we consider that the Optical Path Length, $O P L = n_{i} \bar{P_{0} P_{1}} + n_{a} \bar{P_{1} P_{3}}$ , is null. Thus, we have that $\bar{P_{0} P_{4}} = - (n_{i} / n_{a}) \bar{P_{0} P_{2}}$ , therefore the wavelets for the refracted wavefront outside of the lens are represented simply by

{(Z - [t + ℱ])}^{2} + {(Y - G)}^{2} = {(\frac{n_{i}}{n_{a}} ℱ)}^{2}, G \in [- H, H] .

Fig. 5 (a) Process of refraction produced by a smooth lens, and its associated parameters. (b) Wavelets and the archetype wavefront produced by a smooth surface, considering a plane wavefront incident on the lens.

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In order to obtain the wavefront, which is the envelope of the wavelets represented by Eq. (18), we differentiate Eq. (18) with respect to τ and reducing similar factors we have

Y - G = - {Z - [t + ℱ] + {(\frac{n_{i}}{n_{a}})}^{2} ℱ} [\frac{ℱ_{τ}}{G_{τ}}],

Finally, solving for Z and Y by using Eqs. (18) and (19) and reducing further we obtain

\begin{matrix} Z \pm (τ) = t + \frac{ℱ \sqrt{n_{a}^{2} G_{τ}^{2} + (n_{a}^{2} - n_{i}^{2}) ℱ_{τ}^{2}}}{n_{a}^{2} [ℱ_{τ}^{2} + G_{τ}^{2}]} [\sqrt{n_{a}^{2} G_{τ}^{2} + (n_{a}^{2} - n_{i}^{2}) ℱ_{τ}^{2} \pm n_{i} G_{τ}}], \\ Y \pm (τ) = G - \frac{n_{i} ℱ ℱ_{τ}}{n_{a}^{2} [ℱ_{τ}^{2} + G_{τ}^{2}]} [n_{i} G_{τ} \pm \sqrt{n_{a}^{2} G_{τ}^{2} + (n_{a}^{2} - n_{i}^{2}) ℱ_{τ}^{2}}], \end{matrix}

where the subscript (+) sign provides the equation of a wavefront progressing in a retrograde direction and is ignored, and for (−) sign it yields a wavefront progressing in the forward direction as shown in Fig. 5(b), both wavefronts are tangent to each of the wavelets centered along the smooth arbitrary surface passing through the vertex V of the lens. Note that the blue continuous circles in the figures indicate the envelope of the forward propagating wavefront, also called the archetype wavefront, on the other hand the blue dotted circles form the envelope of a backward wavefront which is a mathematical solution but it does not have any physical sense, therefore it is not taken into account in the subsequent analysis. Furthermore, the red dotted circles have their centers inside the region of TIR, they do not contribute to the formation of any refracted wavefront [9]. It is worth saying that both backward and forward wavefronts are described by discontinuous curves, although due to TIR these curves are broken with singularities. Even more, the wavelets with centers lying inside the gaps of TIR are not bounded, in other words, these wavelets are not tangent to neither backward nor forward wavefronts as we expected, but they are limited between these two zero-distance phase front. We consider exclusively from Eq. (20) a wavefront progressing in the forward direction and reducing further it becomes

\begin{array}{l} Z_{w} (τ) = t + \frac{(n_{a}^{2} - n_{i}^{2}) ℱ \sqrt{n_{a}^{2} G_{τ}^{2} + [n_{a}^{2} - n_{i}^{2}] ℱ_{τ}^{2}}}{n_{a}^{2} [n_{i} G_{τ} + \sqrt{n_{a}^{2} G_{τ}^{2} + [n_{a}^{2} - n_{i}^{2}] ℱ_{τ}^{2}}]}, \\ Y_{w} (τ) = G + \frac{n_{i} (n_{a}^{2} - n_{i}^{2}) ℱ ℱ_{τ}}{n_{a}^{2} [n_{i} G_{τ} + \sqrt{n_{a}^{2} G_{τ}^{2} + [n_{a}^{2} - n_{i}^{2}] ℱ_{τ}^{2}}]}, \end{array}

where the subscript w means wavefront. It is important to state that Eq. (21) provides the coordinates of the locus of points that parametrically represent the zero-distance phase front or the archetype wavefront, which is produced by a smooth arbitrary surface in a meridional plane as a function of τ for a plane wavefront incident on the lens. From Eq. (21) assuming that the radical is less than zero, either the rays or wavelets undergo Total Internal Reflection (TIR), see for example [9].

5. Propagation of the archetype wavefront produced by a smooth arbitrary surface

Since either the zero-distance phase front or the archetype wavefront can also be considered as a source of such a congruence, all ensuing phase fronts are parallel in the geometrical sense [10]. That is, points on each phase surface are equidistant where the distances are measured along the common normal between points on each of the phase front surfaces, according to Huygens’ principle as are shown in Fig. 5(b). Let $W_{0}$ be a point on a phase front whose profile is given parametrically by $W_{0} = (F (τ), G (τ))$ , then $W$ will be on a parallel phase front at an arbitrary distance ℒ according to the following equation

W = (F \pm \frac{[\partial G / \partial τ] ℒ}{\sqrt{{(\partial F / \partial τ)}^{2} + {(\partial G / \partial τ)}^{2}}}, G \mp \frac{[\partial F / \partial τ] ℒ}{\sqrt{{(\partial F / \partial τ)}^{2} + {(\partial G / \partial τ)}^{2}}}) .

In this way, Eq. (22) for $W_{(-, +)}$ provides a retrograde wavefront and for $W_{(+, -)}$ provides the forward wavefront. Substituting Eq. (21) into Eq. (22) for F → Z_w and G → Y_w and reducing further we get

\begin{matrix} Z_{‖_{w}} (τ) = Z_{w} + [\frac{n_{a}^{2} G_{τ} + n_{i} \sqrt{n_{a}^{2} G_{τ}^{2} + [n_{a}^{2} - n_{i}^{2}]}}{[n_{i} G_{τ} + \sqrt{n_{a}^{2} G_{τ}^{2} + [n_{a}^{2} - n_{i}^{2}] ℱ_{τ}^{2}}]}] \frac{ℒ}{n_{a}}, \\ Y_{‖_{w}} (τ) = Y_{w} - [\frac{(n_{a}^{2} - n_{i}^{2}) ℱ_{τ}}{[n_{i} G_{τ} + \sqrt{n_{a}^{2} G_{τ}^{2} + [n_{a}^{2} - n_{i}^{2}] ℱ_{τ}^{2}}]}] \frac{ℒ}{n_{a}}, \end{matrix}

where the subscript ║_w means wavefronts propagating in parallel form produced by an arbitrary plano-curved lenses, where (Z_w, Y_w) are defined in Eq. (21). We can see that for ℒ = 0, Eq. (23) is reduced to Eq. (21) providing the zero-distance phase front or archetype wavefront as we expect.

For example, by substituting the values from Eqs. (9) and (10) into Eqs. (21) and (23) and reducing further we have

\begin{matrix} Z_{‖_{w z}} (τ) = t + \frac{n_{a}^{3} ℒ + {2 π n_{i} n_{a} ℒ - (n_{a}^{2} - n_{i}^{2}) [1 + \cos τ] b} \sqrt{Δ}}{2 π n_{a}^{2} [n_{i} [a - b \cos τ] + \sqrt{Δ}]}, \\ Z_{‖_{w z}} (τ) = \frac{a}{10 π} [τ - π - \frac{b}{a} \sin τ] - \frac{5 b (n_{a}^{2} - n_{i}^{2}) {n_{i} [1 + \cos τ] b + 2 π n_{a} ℒ} \sin τ}{2 π n_{a}^{2} [n_{i} [a - b \cos τ] + \sqrt{Δ}]}, \end{matrix}

where the subscript ║_wc means wavefront parallels for a cycloid, and Δ has been defined above. The propagation of several wavefronts are shown in Fig. 6, considering different lengths for ℒ, assuming two different surfaces which do not present TIR. It is worth remarking that all the rays always stay perpendicular to the wavefronts as it should be, at specific positions on the refracted wavefront. Additionally, as they propagates large distances along the optical axis, the wavefront can be approximated satisfactorily by a spherical wavefront with radius ℒ and its center at vertex of the lens. To finalize, Eq. (5) has great potential applications in the field of illumination, for example to design the shape of a LED lens to produce a diffusor with a high efficiency of emission of light [11 – 13]. Additionally Eq. (23) has great potential applications to design corrector plates to correct aberrated wavefronts which are propagated through optical systems. Furthermore this kind of surface is described by a quartic surface [14, 15]. Using this formalism further work will be done in this field of optics.

Fig. 6 Propagation of refracted wavefronts through two cycloids with the following parameters: (a) a = 5cm, b = 0.82cm. (b) a = 5cm, b = 0.88045cm, for τ ∈ [−π, 3π]

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Conclusions

We have obtained simple formulas for either wavefronts and caustics produced by plano-curved smooth lenses having two arbitrary parametric functions, considering a plane wavefront propagating along the optical axis. The shape of the wavefronts can be modified by changing the parameters of the lens and the distance from which they are observed. When the condition of total internal reflexion is satisfied the wavelets do not contribute to the formation of the refracted wavefront, since these wavelets are unlimited, therefore they are not perpendicular to the refracted rays. Upon inspection one can see that our caustics can reduce the amount of TIR if we demand that the condition for the inflection points coincides simultaneously with a point where TIR occurs, with potential applications such as diffusers of light or collimators for solar cells. We do not treat the sagittal plane, but we could by using differential geometry to obtain the wavefront for the sagittal plane. The net result is a simplification in the formalism that gives us the simple set of Eqs. (5) and (23) in this manuscript that allowed a direct calculation to obtain the caustic and wavefronts refracted by planar ripple lenses. We believe that this method for obtaining either the caustics and wavefronts reported here are straightforward, giving a relationship between caustics and wavefronts propagating outside of the lens.

Acknowledgments

This work has been partially supported by a Programa de Apoyo a Proyectos de Investigación e Innovación Tecnológica – Universidad Nacional Autónoma de México (PAPIIT-UNAM) under project # IN 114414, Consejo Nacional de Ciencia y Tecnología (CONACyT) under project # 168570, and Escuela Superior de Ingeniería Mecánica y Eléctrica Unidad Ticomán, Instituto Politécnico Nacional, through Project no. SIP-20150465. The corresponding author is grateful to I. Goméz-García for their valuable assistance and comments.

References and links

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Exact wavefronts and caustic surfaces produced by planar ripple lenses

Abstract

1. Introduction

2. Exact caustic produced by a smooth arbitrary surface

3. Example: caustic produced by an arbitrary cycloid

4. Exact wavefront produced by a smooth arbitrary surface: Huygens’ principle

5. Propagation of the archetype wavefront produced by a smooth arbitrary surface

Conclusions

Acknowledgments

References and links

Cited By

Figures (6)

Equations (26)

Optics Express