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Abstract
Generating a prescribed irradiance distribution given a source distribution is an inverse problem that sits at the heart of illumination design. The growing prevalence of freeform optics has inspired several design methods for obtaining a prescribed irradiance distribution possessing no symmetry. Up to now, these methods have relied exclusively on freeform optical surfaces for generating freeform irradiances. This paper presents a design method that, for the first time, applies gradient-index (GRIN) optics to solving this inverse problem. Using a piecewise-continuous freeform gradient-index (F-GRIN) profile, a single optic with two planar surfaces can be designed to produce a far-field prescribed irradiance distribution from a point source. The design process is herein presented along with two design examples which demonstrate some of the unique properties of F-GRIN illumination optics.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The fundamental objective of illumination design is finding an optical system that transforms a given source spatial and angular distribution into a desired irradiance and intensity distribution while abiding by étendue conservation. Historically, this inverse problem of finding the requisite optical system was solved using the principles of geometrical optics in order to obtain relatively tame, symmetric distributions. Advances in fabricating non-rotationally symmetric optics have opened the floodgates to a new era of illumination design, targeting far more complex, asymmetric distributions. These advancements working in concert with compact, energy efficient LED and laser sources have recently spurred far more creative and flexible illumination solutions than previously thought possible. This illumination revolution has been felt across many applications including automotive, street, architectural, and structured lighting.
Freeform optical surfaces have been key in recent developments. Various design methods have been proposed that use freeform surfaces to solve this illumination inverse problem [1]. It has been demonstrated that a reflective or refractive freeform surface capable of generating a prescribed irradiance distribution can be obtained by deriving and solving a highly non-linear, second-order partial differential equation of Monge-Ampère (MA) type [2,3]. The vast complexity of solving this equation only allows for numerical solutions, which has been done in several ways [4–12]. Some have instead considered the problem as a ray mapping problem and likened it to the Monge-Kantorovich mass transportation problem [13–18]. Others have obtained the sought freeform surface by the supporting quadratic method (SQM), which applies sections of conic surfaces to discretely create the prescribed irradiance [19–22]. Although well adopted in freeform illumination design, there are aspects of these different methods that pose limitations.
First, all of these designs rely on freeform surfaces, which present a challenge in fabrication and alignment, although less so compared to the tight tolerances of freeform surfaces used in imaging [23]. Typically, fabrication requires a five-axis surface generator for either the optical surface or a mold, making single optics expensive and mass customization difficult. These difficulties are amplified for designs requiring freeform surfaces that are not smooth. Note that this is not considering the case of general faceted illumination optics for which fillets at slope discontinuities are tolerable. Generally, as the complexity of the irradiance distribution increases, manufacturing error becomes a bottleneck with only nanometer surface error being tolerable [10].
Furthermore, a shortcoming of current freeform methods is limitations in achievable irradiance distributions. A smooth freeform surface is incapable of creating an irradiance distribution containing a not simply connected domain; in other words, there are unconnected regions and/or holes in the distribution [5,16,18]. This poses a significant problem for many desirable irradiance distributions such as letters or a logo. Consequently, MA-based methods which rely exclusively on smooth surfaces are incapable of creating such irradiance distributions, including those with perfectly null backgrounds [7]. These obstacles can be overcome, however, by introducing illumination optics that impart phase discontinuities in a wavefront. For example, irradiance distributions containing holes and unconnected regions have been obtained with the use of piecewise-continuous freeform surfaces possessing slope discontinuities [16–18]. Alternatively, Desnijder et al. proposed a design using a Fresnel-like array of freeform lenses with surface discontinuities [24]. In both of these cases, however, the difficulty of fabricating freeform surfaces is compounded with the challenge of fabricating surfaces that are discontinuous in either slope or surface. Although offering greater irradiance possibilities, fabricating a not smooth freeform surface is a significant limitation when it comes to implementation.
Freeform illumination design methods can be classified as either forming the prescribed irradiance distribution on a near-field or far-field surface, where the far-field region is defined by illumination distances significantly larger than the spatial extent of the optic. This notion of prescribed irradiance (optical power per unit area) in the far-field can also be considered as a prescribed intensity (optical power per unit solid angle), evaluated on a surface at some distance in the far-field. For irradiances containing unconnected regions or holes, the far-field problem is readily posed as a Monge-Kantorovich mass transportation problem with a solution in the form of a piecewise-continuous freeform surface [17,18]. On the other hand, the near-field problem must be constructed as a generalized mass transportation problem [16,19], making it a more challenging computational problem.
Lastly, creating a prescribed irradiance from an extended source poses a different problem. The aforementioned freeform design methods often rely on zero-étendue source approximations, considering either an ideal point or collimated source. For cases where the source extent is small compared to the source-optic distance, there is slight but acceptable blurring at the edges of the irradiance distribution [25]. This blurring can be corrected somewhat by adapting the target irradiance for the source extent [26]. For large source extents, however, zero-étendue methods are inadequate. More advanced methods can account for extended sources [25,27,28] but at the cost that multiple freeform surfaces are now required, adding a toll on fabrication and alignment.
In addition to freeform surfaces, gradient-index (GRIN) optics also offer a means of obtaining exotic irradiance distributions. Specifically, freeform gradient-index (F-GRIN) media, which possess rotational and/or axial variations in their refractive index profile, have been recently presented [29] and present many new possibilities. The fabrication of F-GRIN media has been demonstrated using additive manufacturing techniques with optical plastics [29,30]. This is done by employing inkjet printing technology to deposit ink droplets of varying refractive index [31] with a resolution on the order of tens of microns. Due to the nature of additive manufacturing, discontinuities in both the refractive index and the gradient can be incorporated. Not a mature technology, additive manufacturing of GRIN media also has drawbacks including scattering, a limited refractive index range, and limited aperture and thickness dimensions. Finally, applying GRIN to challenging illumination problems has been done previously. For example, Kunkel et al. demonstrated that a GRIN profile is capable of performing mode conversion for a diffracting beam [32].
In this paper, a novel method is presented for designing an illumination optic to achieve a prescribed freeform irradiance distribution without the need for freeform surfaces. For the first time to the authors’ knowledge, a GRIN optic is applied to solving the arbitrary illumination inverse problem. It is shown here that a single F-GRIN optic with only planar surfaces can produce an arbitrary far-field irradiance distribution from a zero-étendue point source (see Fig. 1). The design method for such an F-GRIN optic is outlined, and it is seen that several of the applied techniques also appear in freeform surface design methods. It will also be shown that applying F-GRIN to the inverse illumination problem offers advantages over freeform surface designs by addressing some of the aforementioned problems. Most significantly, fabricating an F-GRIN optic using additive manufacturing readily allows for incorporating discontinuities in the refractive index gradient. Similar to slope discontinuities in freeform surfaces, gradient discontinuities in an F-GRIN medium impart phase discontinuities in the wavefront. Consequently, F-GRIN illumination optics can use discontinuities in the refractive index gradient to create complex irradiance distributions containing holes, unconnected regions, and entirely null backgrounds. Finally, the plane parallel surfaces of the F-GRIN optic may also offer advantages in mounting, alignment and cleaning.
Fig. 1. Freeform illumination using a freeform gradient-index (F-GRIN) optic. The design objective is to solve the inverse problem of determining the refractive index profile $n\left (x, y, z\right )$ that yields the prescribed irradiance. The color map shows differences in refractive index.
Prior to examining the F-GRIN design process, the way by which rays traverse a GRIN medium must first be understood.
2. Linear GRIN ray tracing
Within all isotropic media, homogeneous and inhomogeneous alike, ray paths abide by Fermat’s principle. Formally, Fermat’s principle states that in a medium of refractive index $n$, the optical path integral
is stationary with respect to variations in path between any start and end points within the medium $s_{0}$ and $s_{1}$, respectively, and where $s$ is the path length along the ray [33].For homogeneous media, the refractive index $n$ is constant, and the stationary optical path solution is the familiar linear ray path. For inhomogeneous refractive index profiles, however, the ray path must be solved using the Euler-Lagrange equation. For Cartesian coordinates, the Lagrangian of Eq. (1) can be written as
where $'$ indicates $d/ds$. Applying this Lagrangian, the Euler-Lagrange equation becomes where $\boldsymbol {r}$ is the position vector [33].Analytically solving Eq. (3) proves to be challenging except in special cases of refractive index profiles possessing some degree of symmetry. In order to ray trace through any arbitrary refractive index profile including those lacking symmetry, a numerical Runge-Kutta based method from Sharma et al. [34] is often used, which relies on a series expansion and iteration. Meanwhile, freeform illuminator design methods typically rely on known analytical forms of the ray paths. As a result, the design of an F-GRIN illumination optic poses a challenge due to the unavailability of an analytical ray path. For this reason, the presented design method relies initially on linear GRIN, for which the analytical ray path is known. As discussed in Sec. 3, an array of discrete linear GRIN media can then be used to reconstruct a final piecewise-continuous F-GRIN design. With the illuminator initially relying on a linear GRIN array for a stepping stone to the final design, it is required to understand how rays propagate in a linear GRIN medium.
A linear GRIN with a gradient in the $\hat {\rho }$-direction has a refractive index profile defined by
where $n_{0}$ is the base refractive index at the origin, $\alpha$ is the magnitude of the refractive index gradient (i.e. the slope), and $\rho$ is the spatial dimension in the direction of the gradient (see Fig. 2(a)). Specifically, the F-GRIN illuminator design relies on linear GRIN media with gradients all lying in the $x$-$y$ plane. This can be formalized with the gradient unit vector defined as $\hat {\rho }=\langle \sin \theta _{G},\cos \theta _{G},0\rangle$ where $\theta _{G}$ is the angle of the gradient in the $x$-$y$ plane with respect to the $y$-axis. Using this unit vector, the gradient spatial dimension $\rho$ can be written as a linear combination in terms of Cartesian coordinates as $\rho =\sin \left (\theta _{G}\right )x+\cos \left (\theta _{G}\right )y$. Moreover, the definition of $\hat {\rho }$ entails that there is no refractive index change in the $z$-dimension, as is also the case for the final F-GRIN design.Fig. 2. Linear GRIN refractive index profile and ray path. (a) The refractive index gradient direction $\hat {\rho }$ lies in the $x$-$y$ plane and is oriented at an angle $\theta _{G}$ with respect to the $y$-axis. (b) The ray path $\rho \left (z\right )$ through a linear GRIN has the form of a hyperbolic cosine and lies in the $\rho$-$z$ plane.
As mentioned, the Euler-Lagrange equation for the ray path in Eq. (3) can be solved analytically with the help of some assumed degree of symmetry. As discussed in Appendix A, for GRIN profiles possessing symmetry about the $z$-axis with no refractive index change in $z$, the Euler-Lagrange equation can be simplified to
where $\rho \left (z\right )$ is the ray path and a dot indicates a derivative with respect to $z$ [33,35]. Equation (5) implies that the ray path within a linear GRIN is confined to a two-dimensional plane formed by the gradient and the initial direction of propagation, which in this case is the $z$-direction.The solution to this equation for the ray path in a linear GRIN medium is shown in Appendix A to have the form of a hyperbolic cosine function:
Another important feature of a linear GRIN is how ray bundles behave in transmission. For a linear GRIN in air with plane-parallel surfaces, a beam is angularly deflected, yet its original convergence is approximately conserved. In other words, the linear GRIN changes the beam direction without imparting optical power or significant aberration. For example, an input collimated beam exits approximately collimated but redirected in angle (see Fig. 3(a)). The amount of angular deflection experienced by the beam is directly proportional to the gradient magnitude $\alpha$ while the direction of deflection is determined by the gradient direction $\theta _{G}$. Together, these two parameters can be used to steer a beam to an arbitrary point in the far field (see Fig. 6). It is interesting to note that for a collimated beam, although the angle of refraction is approximately equal after the second surface, the angle of incidence within the GRIN is noticeably different. The constant angle of refraction is due to the refractive index gradient increasing along the second surface while the angle of incidence is decreasing, maintaining the Snell’s law product $n\sin \theta$ to be approximately constant along the second surface. Similarly, for a diverging or converging ray bundle from a point source, the beam is angularly deflected, yet its original numerical aperture (NA) is approximately conserved (see Fig. 3(b), c). This feature of linear GRIN to preserve beam convergence is an approximation with error on the order of arc-seconds across a beam for reasonable refractive index values and, therefore, is taken to be valid for the presented design process.
Fig. 3. Beam angular characteristics are approximately conserved when transmitting through a linear GRIN medium, such as for (a) collimated, (b) diverging, or (c) converging bundles of rays. The color map shows differences in refractive index.
The ability to redirect a beam in a specified direction using a linear GRIN is the foundation for the design method. With a linear GRIN array, sections of a diverging ray bundle can be redirected to specified points in space to build up a prescribed irradiance distribution. Also, since linear GRIN has no optical power, the irradiance spot from each redirected beamlet scales axially in space based on the NA of the original diverging beam. For this reason, irradiance distributions containing holes and a null background can be easily obtained using a linear GRIN array by only redirecting light where it is prescribed. After using a linear GRIN array to reconstruct the final F-GRIN design, this feature is maintained.
Stepping back, one might notice that the effect a linear GRIN has on a beam is very similar to that of a planar mirror in that the beam direction is changed while no optical power is contributed. Going a step further, an array of small linear GRIN sections, therefore, functions very similarly to that of a micromirror array. Operating similarly to how a digital micromirror device (DMD) functions as a spatial light modulator (SLM) in a digital projector, a linear GRIN array can function as a form of static SLM, which when properly designed, can project a prescribed irradiance distribution.
Next, the design method is examined for how to obtain the requisite linear GRIN array parameters and reconstruct a piecewise-continuous F-GRIN design to produce some prescribed irradiance.
3. Design process
The objective of the design process is to obtain the two-dimensional F-GRIN refractive index profile $n\left (x,y\right )$ that produces a specified irradiance distribution. Several constraints are established to guide the design and bound the solution space. First, the design process relies on a zero-étendue point source approximation, although extended sources that are small relative to the source distance can be accommodated as well. Next, for ease of fabrication, the F-GRIN illumination optic has simply two parallel planar surfaces. Third, the desired irradiance target is defined by a binary raster image file as well as the target throw ratio, the target size per axial distance from the optic. Lastly, there are several self-imposed constraints on refractive index values to match limitations of current GRIN additive manufacturing techniques. First, refractive index values should remain within the bounds of $1.45 \leq n\leq 1.55$. Second, the center thickness of the optic should be $\leq 5$ mm. The maximum attainable clear aperture, however, is large enough that the constraint does not become active in the design process.
The F-GRIN illuminator design process consists of four steps (see Fig. 4). First, mapping is performed from discrete points in the irradiance target to discrete elements in a linear GRIN array. Second, the linear GRIN array is formed such that each array element possesses the GRIN parameters required to redirect a beamlet from the source to its mapped point in the target. Third, the linear GRIN array is used to reconstruct a piecewise-continuous F-GRIN illuminator design that produces the specified irradiance distribution. Fourth, the design is evaluated via ray tracing.
3.1 Target mapping
Given an irradiance target, the design process begins by finding a mapping between all the discretized points in the target and an equal number of elements in a uniform rectangular array at the optic. The mapping of which array elements map to which target points is desired to produce largely continuous regions in the final piecewise-continuous F-GRIN design. This is done by solving a linear assignment problem (LAP) with an appropriately chosen cost function. Note that this concept of going from discrete points on the optic to points in the target is very similar to SQM used for freeform surface illumination design.
The cost function for the LAP was chosen as the three-dimension Euclidean distance between all array points $\left (x_{a},y_{a}, z_{a}\right )$ and all target points $\left (x_{t},y_{t}, z_{t}\right )$:
Upon examination of Eq. (9), it can be seen that such a cost function promotes spatially contiguous points in the target being formed by corresponding contiguous regions in the optic. This allows for large regions of continuous refractive index to be integrated, as described in Sec. 3.3. This choice of cost function is empirically conceived and other useful cost functions surely exist. For example, a different cost function exists that minimizes the total refractive index change of the linear GRIN array but sacrifices large, integrable regions in the final piecewise-continuous F-GRIN design.The LAP is solved by finding the mapping such that the Euclidean distance cost described in Eq. (9) is minimized across all lenslet-target pairs (see Fig. 5). An LAP can be solved numerically in several ways, but in this case it was solved using a variant of the Hungarian algorithm [36,37]. It is worth noting that LAPs appear in freeform surface design as well. Some use the Monge-Kantorovich mass transportation problem, which in discrete form can be posed as an LAP, to find optimal ray mappings [16–18].
When determining the mapping, all target points are located on a plane positioned at some arbitrary distance in the far-field, $z_{t}$. As a result, the obtained F-GRIN design will produce the desired irradiance at this plane as well as any other plane in the far-field but not in the near-field. Although not demonstrated here, the presented design method can be adapted to the near-field freeform illumination problem by considering a plane position $z_{t}$ in the near-field.
There are two complicating factors in the mapping that are worth noting. First, the linear GRIN array consists of a rectangular grid, so the product of the two grid dimensions must equal the total number of samples in the target. This is an issue when the target image file is discretized, for example, into a prime number of samples, which would only allow for a one-dimensional grid solution. Consequently, the irradiance distribution must be scaled prior to mapping such that the number of samples in the target can form a grid with an aspect ratio close to square. Second, a beneficial feature of this design method is that small extended sources can also be accommodated. This is done when performing target discretization using an irradiance adaptation technique similar to the one discussed by Wester et al. [26] for freeform surfaces.
Fig. 5. Target mapping by solving the linear assignment problem. The chosen cost function is to minimize the total three-dimensional Euclidean distance between node pairs. Demonstrated in two-dimensions, black nodes represent array points, and colored nodes represent target points.
3.2 Linear GRIN array
The second step in the design process is to obtain the requisite discrete linear GRIN array. This is done by determining for each array element the required linear GRIN parameters: the base refractive index $n_{0}$, gradient magnitude $\alpha$, and gradient direction in the $x$-$y$ plane $\theta _{G}$. Together, these three values must result in the beam from each array element being redirected to its corresponding target pixel based on the determined mapping. As can be seen in Fig. 6, different combinations of $\alpha$ and $\theta _{G}$ can steer the beam to different positions within a relatively large region.
Fig. 6. The effect of different linear GRIN parameters on the ray trajectory. Radial contours depict ray positions for different gradient magnitudes $\alpha$. Azimuthal contours depict ray positions for different gradient directions $\theta _{G}$. The center point is the ray position for a homogeneous medium. The ray trajectories are evaluated at $z=1$ m for a $15.8^{\circ }$ angle of incidence at the GRIN.
For all array elements, the value chosen for the base refractive index is $n_{0}=1.5$ in order to be centered within the manufacturable range. This lost degree of freedom is later regained when performing F-GRIN reconstruction. Given $n_{0}$ and the source, optic, and target positions, the gradient values $\alpha$ and $\theta _{G}$ are obtained for each array element by a ray tracing based iterative optimization routine. An example linear GRIN array design is depicted in Fig. 7.
Fig. 7. Example linear GRIN array midpoint design. Each array element has a unique gradient magnitude $\alpha$ and direction $\theta _{G}$ in order to illuminate its corresponding mapped point in the target. The refractive index profile is depicted in the $x$-$y$ plane and possesses no axial dependence. This array was used in reconstructing the final piecewise-continuous F-GRIN profile for the flower design in Fig. 9.
At this point, the designed linear GRIN array successfully produces the prescribed irradiance distribution. Several practical limitations exist, however, which require the final step of transforming the array into a piecewise-continuous F-GRIN optic. For one, crosstalk between adjacent array elements, where rays entering one element refract into a neighbor before exiting, produces erroneous rays that contribute to image artifacts and stray light. Another limitation of the array design is diffraction effects in the far-field due its the periodicity. Finally, fabrication limitations also restrict forming arrays of elements smaller than $\sim 0.5$ mm. The final design step of F-GRIN reconstruction solves all three of these problems.
3.3 Piecewise-continuous F-GRIN
The final step in the design process is going from the obtained linear GRIN array to a piecewise-continuous F-GRIN design. Each linear GRIN element in the array can be defined by its gradient via the parameters $\alpha$ and $\theta _{G}$. In order to create a continuous F-GRIN profile from this discrete linear GRIN array, the array elements must first be unwrapped by staggering the values of the base refractive index $n_{0}$. The gradients can then be integrated to form a continuous reconstructed refractive index profile that maintains approximately the same gradient as from the individual array elements (see Fig. 8). As with the linear GRIN array, the reconstructed F-GRIN profile has no axial change in refractive index.
Fig. 8. F-GRIN reconstruction and interpolation. (a) A linear GRIN array is created as a midpoint in the design process. (b) Reconstruction is performed with a modified version of the Southwell algorithm. (c) The reconstructed result then undergoes bicubic interpolation to obtain a continuous refractive index profile. The color maps show differences in refractive index and are not set to the same scale.
One may note that this process is very similar to zonal reconstruction for a Shack-Hartmann wavefront sensor, where a grid of two-dimensional local wavefront slopes can be reconstructed into an estimated continuous wavefront. Accordingly, F-GRIN reconstruction is conveniently performed with the Southwell algorithm [38], which was developed and is frequently used for zonal wavefront reconstruction. The Southwell algorithm provides a wavefront, or in this case refractive index, estimation point at each provided slope measurement. This means that for the F-GRIN reconstruction, the number of estimated refractive index points is equal to the number of elements in the linear GRIN array and the number of samples in the target (see Fig. 8(b)). For irradiance evaluation, ray tracing with the Sharma algorithm [34] requires the refractive index value and gradient at all points, so due to the sparseness of the refractive index sampling after reconstruction, interpolation must also be performed. The presented method employs bicubic interpolation to define the reconstructed refractive index profile continuously (see Fig. 8(c)).
Just as a smooth freeform surface cannot form a discontinuous irradiance, an entirely continuous F-GRIN profile cannot create a discontinuous irradiance distribution. For this reason, it is necessary that some gradient discontinuities from the linear GRIN array remain in the F-GRIN profile in order to obtain a discontinuous irradiance distribution. The refractive index and gradient discontinuities that remain in the final F-GRIN design are identified based on an adjustable threshold for neighbor differences in linear GRIN parameters $\alpha$ and $\theta _{G}$. An example of such F-GRIN discontinuities are depicted in Fig. 10. A consequence of these discontinuities is that the Southwell algorithm, which only considers continuous rectangular grids of points, needed to be modified to accommodate reconstructing around discontinuities. The process was similar to that reported by Li et al. [39] which handled non-rectangular boundaries, except now internal discontinuities are accounted for as well.
Recall, there are self-imposed design constraints based on what gradient refractive index values are achievable with additive manufacturing. Based on current technology, the refractive index should remain within the bounds of $1.45 \leq n\leq 1.55$. This corresponds to a base refractive index $n_{0}=1.5$ and a total refractive index change $\Delta n\leq 0.1$. The base refractive index is a relatively weak variable, so the refractive index profile can be shifted in the design process with minimal effect. On the other hand, the $\Delta n$ constraint significantly drives the design. The design process initially relies on an array of linear GRIN profiles, which possess no optical power. Due to this lack of power, the target throw ratio depends entirely on the divergence of the used source NA for creating the appropriately sized irradiance. This fact allows the piecewise-continuous F-GRIN designs to be scaled to meet manufacturing constraints with little to no loss in irradiance fidelity. For a fixed throw ratio, the $\Delta n$ and F-GRIN dimensions (clear aperture and center thickness) can be scaled as desired while adjusting the source distance to maintain the required source NA. This allows for very flexible illumination solutions ranging from very small illumination optics with compact source distances to larger optics and longer source distances all with manufacturable values for $\Delta n$. For demanding applications that require a high source NA (large optics with short source distances), the $\Delta n$ constraint can also be overcome by wrapping the refractive index profile and, consequently, introducing additional phase discontinuities, similar to Fresnel-like optics.
4. Design examples
Based on the presented method, two different design examples are demonstrated to produce two very different prescribed irradiance distributions (see Fig. 9). Both irradiance targets possess holes, sharp edges, and entirely null backgrounds, which could only be attainable with an illumination optic capable of imparting phase discontinuities (see Fig. 10). Both targets were discretized into $\sim 10^{5}$ points in order to highly sample all target features. Each design consists of a piecewise-continuous F-GRIN optic with no axial change in refractive index and two planar surfaces. Both designs create the target irradiance with a throw ratio of $0.5$, meaning the largest dimension of the target would be $1$ m at a distance of $2$ m, for example. Both designs were wrapped and scaled to have a total refractive index change of $\Delta n=0.1$ in order to be attainable with current additive manufacturing techniques. Similarly, both designs were scaled to reasonably small clear apertures for compact source distances. For example, the flower design has a clear aperture of $7\times 7$ mm$^{2}$ and a center thickness of $2$ mm for a source distance of $25$ mm. The computation time for each design was approximately 35 minutes (MacBook Pro, Dual-Core i5 @ 2.9 GHz, 16 GB RAM) with about 10 minutes being spent on solving the linear assignment problem for $\sim 10^{5}$ points.
Fig. 9. Two piecewise-continuous F-GRIN designs that produce a prescribed irradiance distribution, evaluated at $z=1$ m. (a) Two different targets were considered, both of which have holes and sharp discontinuities. A target throw ratio of $0.5$ was considered for both cases. (b) The designed F-GRIN illumination optics have both refractive index and gradient discontinuities with a total refractive index change $\Delta n=0.1$. The refractive index profile is depicted in the $x$-$y$ plane and possesses no axial dependence. The F-GRIN optics have planar surfaces. (c) The produced relative irradiance distribution for each design was evaluated using a Monte Carlo ray trace with $10^{6}$ rays.
Fig. 10. Refractive index and gradient discontinuities in the flower F-GRIN design in Fig. 9. Gradient discontinuities in the F-GRIN profile impart phase discontinuities which create holes and sharp discontinuities in the irradiance. Discontinuities are depicted by black lines in the expanded view. The stair-step nature of the discontinuities is a lasting effect from the linear GRIN array midpoint. The refractive index profile is depicted in the $x$-$y$ plane and possesses no axial dependence.
The designs are evaluated for produced irradiance using Monte Carlo ray tracing. The Sharma algorithm [34] was employed to trace $10^{6}$ rays from a point source with uniform angular emittance. On a surface in the far-field, relative irradiance is considered as a ray-based metric of ray hits per unit area. For both designs, the target distribution is recreated to a high degree of fidelity (see Fig. 9). Minor image artifacts can be seen, particularly in the “U of R” example, due to crosstalk when rays cross refractive index discontinuities. For this reason it is best to minimize the number of discontinuities. Although not implemented here, these artifacts can instead be eliminated by using an absorbing layer along discontinuities, a technique that is attainable with additive manufacturing. Finally, there is also minor clipping at the aperture edges that can be seen in the left side of the “U” and the right side of the “R” for example. This artifact can also be mitigated by extending the optic aperture boundary to avoid clipping.
Designed for the far-field, the presented ray-based irradiance is maintained over a very large range of axial distances over which the prescribed irradiance is preserved and scales according to the target throw ratio. For the “U of R” example, the relative irradiance profile remains virtually unchanged from the mapping distance $z_{t}=1$ m all the way to $z=100$ m (see Fig. 11). Considering the size of the geometrical range depicted in Fig. 11, diffraction effects including those from refractive index discontinuities must be analyzed to determine at what distance irradiance fidelity is lost. Finally, in all cases the absolute irradiance in the far-field will decrease with distance from the optic according to the inverse square law.
Fig. 11. Relative irradiance distribution at different evaluation distances $z$ produced by the “U of R” F-GRIN design. The different distributions scale in size with $z$ according to the target throw ratio of $0.5$. With design mapping performed at distance $z_{t}=1$ m, the ray-based irradiance is maintained into the far-field.
5. Conclusion
In this paper, for the first time gradient-index media are applied to solving the prescribed irradiance inverse problem. The target mapping was found by solving a linear assignment problem, and the result was used to obtain an intermediate solution in the form of a linear GRIN array. The final design was created by using the GRIN array to reconstruct a piecewise-continuous F-GRIN profile capable of creating the prescribed irradiance distribution. Two design examples were demonstrated and evaluated to highlight some of the unique aspects of F-GRIN illuminator designs as compared with freeform surface designs.
Future work remains for F-GRIN methods to reach the level of maturity of the several well established freeform surface based methods. First, an F-GRIN design can soon be fabricated and evaluated for produced irradiance distribution. A process can also be developed to tolerance a design’s sensitivity to refractive index error. Higher order effects including diffraction and chromatic dispersion must also be taken into account. Finally, more advanced designs with axial change in refractive index can be investigated to accommodate, for example, near-field irradiances, grayscale targets, or large extended sources.
Appendix A: Derivation of the linear GRIN ray path
The ray path in a linear GRIN medium with a gradient oriented in the $x$-$y$ plane is here derived starting from the Euler-Lagrange equation.
The Lagrangian for the refractive index can be written as
where the optical axis is taken to be in the $z$-direction and a dot denotes $d/dz$. Valid ray paths must satisfy the Euler-Lagrange equation, which written in Cartesian coordinates isConsider a linear GRIN refractive index profile $n\left (\rho \right )=n_{0}+\alpha \rho$ with a gradient lying along the spatial axis $\rho$ in the $x$-$y$ plane (see Fig. 2). The gradient spatial axis $\rho$ can be written as a linear combination in terms of Cartesian coordinates as $\rho =\sin \left (\theta _{G}\right )x+\cos \left (\theta _{G}\right )y$ where $\theta _{G}$ is the angle of the gradient in the $x$-$y$ plane with respect to the $y$-axis. Due to the symmetry of the refractive index profile about the $\rho$-axis as well as the lack of axial dependence, the coupled differential equations in Eq. (12) can be simplified to
meaning the ray can be considered exclusively in the $\rho$-$z$ plane formed by the gradient and the optical axis [35].To solve this boundary value problem, the ray’s initial conditions are chosen to be
Substituting into Eq. (13) for $n\left (\rho \right )=n_{0}+\alpha \rho$ and solving for $\ddot {\rho }$ yields where $\kappa =n_{0}/\alpha$.Letting $v=\dot {\rho }$, $\ddot {\rho }$ can now be written in terms of $v$ as $\ddot {\rho }=v dv/d\rho$. Substituting into Eq. (15):
Performing separation of variables and integrating gives which can be solved by change of variables as where $c$ is a constant of integration. Reverting to $\dot {\rho }$ from $v$: Applying the initial conditions at $z=0$ solves for $c=(1+\beta ^{2})/\kappa ^{2}$. Inserting this expression for $c$ and applying the quadratic formula:Since $\kappa$, $\gamma$ are constants, $\rho$ can now be integrated:
Although this derivation was for a ray initially incident at $\rho =0$, any other intersection point can also be accommodated by shifting the base refractive index according to Eq. (4).
Acknowledgments
We thank the City of Rochester, New York for allowing us to use the “Flower City” logo for a design example.
Disclosures
The authors declare no conflicts of interest.
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