Abstract

Freeform optics enable irregular system geometries and high optical performance by leveraging rotational variance. To this point, for both imaging and illumination, freeform optics has largely been synonymous with freeform surfaces. Here a new frontier in freeform optics is surveyed in the form of freeform gradient-index (F-GRIN) media. F-GRIN leverages arbitrary three-dimensional refractive index distributions to impart unique optical influence. When transversely variant, F-GRIN behaves similarly to freeform surfaces. By introducing a longitudinal refractive index variation as well, F-GRIN optical behavior deviates from that of freeform surfaces due to the effect of volume propagation. F-GRIN is a useful design tool that offers vast degrees of freedom and serves as an important complement to freeform surfaces in the design of advanced optical systems for both imaging and illumination.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Optical designers have always pushed the envelope of what systems are possible using the tools available to them. Historically, the available toolkit has consisted of spherical refractive and reflective surfaces (of which planar is a special case) [1]. By introducing additional surfaces, the degrees of freedom (DFs) available to the designer increases, which in turn enables greater system specifications but at the cost of ballooning system size. More advanced options such as conic and aspheric surfaces are also available, offering more DFs per surface to achieve higher system specifications in smaller packages but now at the cost of more complex surface fabrication and metrology [2,3]. These conventional surface types all possess rotational symmetry, and their use in design is best limited to geometries maintaining a single shared optical axis. Otherwise, the tilting and/or decentering of surface axes introduces non-rotationally symmetric aberrations according to nodal aberration theory (NAT) [4]. This restriction on system symmetry includes designs such as the three-mirror anastigmat where off-axis sections of rotationally symmetric surfaces share a common optical axis [5]. The first century of modern optical design limited its scope to these rotationally symmetric design forms; however, there is no fundamental reason why optical design cannot leverage rotational variance in tilting and/or decentering elements.

With advancements in modeling, fabrication, and testing, a new realm of freeform optics has been introduced that deviates from traditional optical design. By straying from a single shared optical axis, optical elements may be tilted and decentered for more complex system geometries. In doing so, freeform aberrations are introduced which cannot be easily corrected with rotationally symmetric optics [6] but can be with the addition of freeform optics [7]. This freedom of breaking from a shared optical axis offers exciting new possibilities for system size, weight, and optical performance not previously attainable with conventional axially symmetric designs. Examples of such optical systems empowered by freeform surfaces include unobscured folded telescopes [8], compact spectrometers [9], augmented reality head-worn displays [10], and nonimaging optics for generating prescribed illumination distributions [11].

Freeform optics have matured in capability over the years [12]. The earliest freeform optics for imaging were anamorphic optical elements, which have a toroidal surface component that contributes optical power differently along orthogonal axes [13]. Early progressive addition lenses for ophthalmic applications were some of the first commercially available freeforms [14]. Alvarez introduced a cubic freeform surface that generates variable amounts of optical power when a pair are laterally sheared [15]. Two similar surfaces were also found in the viewfinder for the Polaroid SX-70 camera [16]. These early applications of freeform optics, although impactful, are much narrower in scope than current freeform capabilities. In general, freeform surfaces are defined by their sag as a function of two spatial dimensions, $z\left (x,y\right )$. This can be done using local representations such as non-uniform rational B-splines (NURBS) [17] or radial basis function (RBF) interpolation [18]. Global representations applying a polynomial basis are also common such as with Zernike polynomials [19] and Q-2D polynomials [20]. At times, freeform surfaces also possess discontinuities in either surface or slope for illumination applications [21].

The described progression of freeform surfaces has naturally led to some evolution in the definition of freeform optics. A widely accepted approach considers surface geometry. Rolland et al. defines a freeform surface as having no axis of rotational invariance either within or beyond the optic [12]. This is similar to the ISO standard 17450-1:2011 definition of a freeform surface shape as having no translational or rotational symmetry about axes normal to the mean plane and belonging to the class of complex invariance [22]. A different perspective considers the fabrication processes required for generating freeform surfaces. To create a surface with rotational variance, a third independent axis is required such as with the C-axis in generating and finishing equipment [12].

To this point, freeform optics have almost exclusively referred to freeform surfaces, but recent advancements in the manufacturing of gradient-index (GRIN) media [2330] have now enabled a new type of freeform optic. GRIN offers additional DFs in the form of spatially varying refractive index, but geometries have historically been limited by available fabrication processes [31]. Now, freeform GRIN (F-GRIN) media are available which have refractive index distributions that may vary arbitrarily in up to three spatial dimensions, $n\left (x,y,z\right )$. There are many similarities and differences between F-GRIN and freeform surfaces. Like freeform surfaces, recent work has shown that F-GRIN can influence freeform aberrations in imaging applications [32]. For nonimaging optical design, a prescribed illumination distribution can be generated using an F-GRIN optic as it can be with freeform surfaces [33,34]. On the other hand, F-GRIN provides DFs in three dimensions, $n\left (x,y,z\right )$, compared with freeform surfaces which are a function of two dimensions, $z\left (x,y\right )$, as depicted in Fig. 1. This added volume propagation component of F-GRIN results in different behavior than freeform surfaces and presents new opportunities for optical design. With most definitions of freeform optics being focused on surfaces, the question now arises of nomenclature for F-GRIN media with an emphasis on remaining as consistent as possible with current perspectives on freeform surfaces.

 figure: Fig. 1.

Fig. 1. Comparison of (a) freeform surface and (b) freeform GRIN (F-GRIN) with the same transverse dependence in sag and refractive index, respectively. The F-GRIN also has a third dimension of axial variation.

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This work will survey F-GRIN media as a new freeform optical design tool for both imaging and illumination. First, F-GRIN’s capabilities in optical design are analyzed, starting with its optical influence and novel impact on freeform aberrations. Next, available DFs along with mathematical representations are outlined. Then, the unique chromatic properties of GRIN are discussed, followed by the current status of fabrication and metrology as well as current and future applications of F-GRIN. Finally, with these factors in mind, a nomenclature is proposed for the different types of F-GRIN.

2. Freeform gradient-index (F-GRIN) media

2.1 Optical influence

In any given design, an optical element’s purpose is to impart a variation in phase spanning both the pupil and field-of-view. For a single field-of-view, this imparted phase difference $\Delta \phi \left (x,y\right )$ is directly proportional to the optical path difference $OPD\left (x,y\right )$ across the optical element,

$$\Delta\phi\left(x,y\right)=k\cdot OPD\left(x,y\right)$$
where $k=2\pi /\lambda$ is the wavenumber. $OPD\left (x,y\right )$ defines the spatial variation in optical path length, $OPL\left (x,y\right )$, relative to the value at some reference coordinate, $OPL\left (x_{0},y_{0}\right )$.

For a single ray, the $OPL$ is determined by the path integral within a refractive index field $n$,

$$OPL=\int_{s_{0}}^{s_{1}}n\hspace{0.5mm}ds$$
where $s$ is the ray path between points $s_{0}$ and $s_{1}$ [35]. According to Fermat’s principle, the very definition of a ray is when this path integral is stationary with respect to any variation in path $s$.

For homogeneous media, the refractive index $n$ is constant with $s$, and the stationary path integral in Eq. (2) simplifies to $n$ multiplied by the Euclidean distance between points $s_{0}$ and $s_{1}$. This dictates the optical influence of a surface within homogeneous media. For example, consider some freeform surface sag $z\left (x,y\right )$ on a base planar surface that forms an interface with homogeneous media of refractive indices $n$ and $n'$, as in Fig. 2(a). Neglecting refraction by use of the thin-element approximation (TEA) [36], the $OPD$ imparted upon an on-axis transmitted plane wave propagating along $z$ is

$$OPD\left(x,y\right)\approx\left(n-n'\right)z\left(x,y\right)$$
where at the reference coordinate $z\left (x_{0},y_{0}\right )=0$ and $OPD\left (x_{0},y_{0}\right )=0$. Similarly, for a reflective freeform surface with sag $z\left (x,y\right )$, the imparted $OPD$ is
$$OPD\left(x,y\right)\approx2\hspace{0.5mm}n\hspace{0.5mm}z\left(x,y\right)$$
since $n'=-n$ upon reflection. This shows freeform surface sag directly influences optical system behavior, which explains why surfaces with rotational variance can impact freeform aberrations caused by rotational asymmetry.

 figure: Fig. 2.

Fig. 2. Optical path difference illustrated for (a) freeform surface, (b) F-GRIN, and (c) freeform surface with F-GRIN. (a) The surface is immersed in homogeneous refractive indices $n$ and $n'$. (c) The surface is immersed in inhomogeneous refractive index fields $n\left (x,y,z\right )$ and $n'\left (x,y,z\right )$. Colormap represents arbitrary refractive index.

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Now consider an F-GRIN optic with a spatially varying refractive index distribution $n\left (x,y,z\right )$. Unlike for homogeneous media, $n$ may vary with ray path $s$ and must remain within the path integral in Eq. (2). Instead, the stationary ray path satisfying Eq. (2) can be solved for using the Euler-Lagrange equation with an appropriately chosen Lagrangian [35]. The result of doing so is a nonlinear differential equation governing ray paths within inhomogeneous media,

$$\frac{d}{ds}\left(n\left(\boldsymbol{r}\right)\frac{d\boldsymbol{r}}{ds}\right)=\nabla n\left(\boldsymbol{r}\right)$$
where $\boldsymbol {r}$ is the Cartesian position vector. As a result, rays traversing an inhomogeneous medium no longer follow straight paths. Instead, they follow curved paths where generally the curvature is proportional to the gradient of the refractive index field, $\nabla n$. These curved ray paths cannot in general be solved analytically, with known solutions limited to special cases of symmetry [35]. As a result, numerical techniques are often used for GRIN ray tracing [3739]. Without an analytical ray expression, it is not immediately clear if there is a general $OPD$ expression for F-GRIN like there is for freeform surfaces; however, a useful approximation can illustrate the optical influence of F-GRIN.

Consider an F-GRIN optic with only transverse refractive index variation, $n\left (x,y\right )$, and plane-parallel surfaces separated by distance $\Delta z$. By assuming the F-GRIN plate is thin ($\Delta z$ is small), ray paths are approximately linear. By TEA, this means the imparted $OPD$ for an on-axis plane wave propagating along $z$ is

$$OPD\left(x,y\right)\approx \left[n\left(x,y\right)-n_{0}\right]\Delta z$$
where the constant $n_{0}$ is some equivalent refractive index that provides a reference for the $OPD$. A convenient choice is $n_{0}=n\left (x_{0},y_{0}\right )$ to remain analogous with the freeform surface reference $z(x_{0},y_{0})=0$. For propagation along $z$, the expression in Eq. (6) is exact when $\Delta z$ is differential and becomes more approximate as the plate thickness increases and ray paths are no longer precisely linear. Also, note the resemblance of Eqs. (3)–(4) for a freeform surface and Eq. (6) for a transverse F-GRIN. For a freeform surface the $OPD$ is proportional to the surface sag $z\left (x,y\right )$ while for F-GRIN it is proportional to the transverse refractive index variation $n\left (x,y\right )$. This shows that both freeform surfaces and F-GRIN can influence freeform aberrations in tilted and decentered systems using rotational variance in either $z\left (x,y\right )$ or $n\left (x,y\right )$.

The behavior of F-GRIN does, however, deviate from that of freeform surfaces when there is longitudinal refractive index variation in addition to transverse variation, as depicted in Fig. 2(b). In this case, volume ray propagation presents additional optical influence not captured by Eq. (6). Instead, an F-GRIN volume $n\left (x,y,z\right )$ can be considered as a series of infinitesimal transverse F-GRIN plates $n_{i}\left (x,y\right )$ [32]. As a function of propagation direction, the full $OPD$ can be approximated as the linear sum of each plate’s $OPD$ contribution, although such an approximation neglects behavior due to ray curvature and induced aberrations. For example, for the on-axis field-of-view, the total $OPD$ imparted by an F-GRIN medium of thickness $t$ can be found approximately by integrating along the propagation direction $z$ [36]:

$$OPD\left(x,y\right)\approx \int_{0}^{t}\left[n\left(x,y,z\right)-n_{0}\right]dz={-}n_{0}t+ \int_{0}^{t}n\left(x,y,z\right)dz$$
where the equivalent refractive index $n_{0}=t^{-1}\int _{0}^{t}n\left (x_{0},y_{0},z\right )dz$ is now defined as the average index along $z$ at reference point $\left (x_{0},y_{0}\right )$. This on-axis $OPD$ for a longitudinally varying F-GRIN can also be imparted by a freeform surface [Eq. (3) or (4)] or an exclusively transverse F-GRIN [Eq. (6)]; however, the longitudinal variation does differ in behavior when evaluating the field dependence of the imparted $OPD$. The ramifications of this effect can be explained within the framework of NAT.

Fuerschbach et al. [7] demonstrated by NAT that a freeform surface at the aperture stop introduces some field constant aberration(s) while a freeform surface away from the stop introduces field constant as well as some proportion of field-dependent aberration(s). For example, a Zernike coma surface at the stop introduces field constant coma while away from the stop it introduces field constant coma; field asymmetric, field linear astigmatism; and field linear, medial field curvature. To correct field-dependent freeform aberrations introduced by tilted and/or decentered elements, a certain freeform surface away from the stop has no control over the type and proportion of field-dependent aberrations it imparts. Instead, multiple freeform surfaces are required to balance different orders of field-dependent aberrations [8].

Longitudinally varying F-GRIN differs in aberration field dependence compared with freeform surfaces due to the effect of volume propagation. Specifically, the order of F-GRIN longitudinal variation influences the order of field dependence in imparted aberrations [32]. This effect can be thought of as the series of transverse sections $n_{i}\left (x,y\right )$ in a longitudinally varying F-GRIN acting like a series of freeform surfaces. For example, away from the stop, an F-GRIN with transverse Zernike coma and no longitudinal variation introduces the same aberrations as the Zernike coma freeform surface [see Fig. 3(a)-(b)]. By changing the F-GRIN’s longitudinal variation from constant to linear, however, field constant coma is no longer introduced [Fig. 3(c)]. By again increasing the order of longitudinal variation from linear to quadratic, both field constant and field linear aberrations are removed [Fig. 3(e)]. What is particularly influential is specific field dependence can be isolated by applying a combination of longitudinal terms in a single F-GRIN. For example, the proper balance of constant and linear axial terms can yield only field constant aberrations but not field linear ones [Fig. 3(d)]. This means that for this Zernike coma F-GRIN, any combination of field constant and field linear aberrations can be introduced by applying the appropriate longitudinal refractive index change. The optical influence of combining transverse and longitudinal refractive index variation offers the designer valuable new DFs in a single freeform optic that can target specific field-dependent freeform aberrations introduced in a tilted and/or decentered system.

 figure: Fig. 3.

Fig. 3. Freeform aberration field dependence for Zernike coma (a) freeform surface and (b)-(e) F-GRIN of different longitudinal orders, $z^{j}$. (a) Freeform surface and (b) axially constant F-GRIN yield approximately the same result of field constant coma; field asymmetric, field linear astigmatism; and field linear, medial field curvature. (c) F-GRIN with linear axial variation removes the field constant aberration. (d) F-GRIN with a combination of constant and linear axial terms leaves the field constant aberration but removes field linear ones. (e) F-GRIN with quadratic axial variation removes both field constant and field linear aberrations. All deigns have the same field-of-view, the aperture stop positioned 15 mm away from the freeform, and a “perfect” unaberrated lens following the optic for image formation. The freeform surface is on a base plane, and F-GRIN designs have plane-parallel surfaces. Field curvature, astigmatism, and coma are shown across the field-of-view and on the same scale where icon size indicates the magnitude of present aberration.

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One final scenario of interest is the interplay of freeform surfaces combined with F-GRIN, as in Fig. 2(c). Consider a freeform surface of sag $z\left (x,y\right )$ immersed in plate of thickness $t$ with F-GRIN incident medium $n\left (x,y,z\right )$ and refracted medium $n'\left (x,y,z\right )$. For propagation along $z$, the imparted $OPD$ can again be obtained by TEA for surface and GRIN as

$$\begin{aligned} OPD\left(x,y\right)&\approx\int_{0}^{z\left(x,y\right)}\left[n\left(x,y,\zeta\right)-n\left(x_{0},y_{0},\zeta\right)\right]d\zeta \\ &+\int_{z\left(x,y\right)}^{t}\left[n'\left(x,y,\zeta\right)-n'\left(x_{0},y_{0},\zeta\right)\right]d\zeta \end{aligned}$$
with respect to reference point $\left (x_{0},y_{0}\right )$. This combined use of freeform surfaces and F-GRIN introduces vast DFs for even more advanced aberration control. For example, the influence of a complex surface immersed in F-GRIN has been leveraged for the design of a polychromatic annular folded lens [40].

2.2 Degrees of freedom

GRIN offers several valuable DFs that are helpful in improving optical performance. As described in Sec. 2.1, the optical effect of GRIN is closely tied to the dimensionality of the refractive index change. The coordinate system governing index variation is also very influential. Together, these factors determine what DFs are available to impart spatial refractive index change and enhance system performance.

GRIN profiles have historically been limited to those varying in only one coordinate due to limited fabrication techniques. As a result, three traditional GRIN geometries emerged that apply single coordinate change but in different coordinate systems. The first type is axial GRIN $n\left (z\right )$ which varies along one Cartesian coordinate. This axis $z$ is typically parallel to the optical axis due to its ability to correct spherical aberration at refractive surfaces [41], although this is not a requirement [35]. Radial GRIN $n\left (\rho \right )$ is a second common geometry defined in cylindrical coordinates and varies along the radial direction $\rho$ while the azimuthal $\theta$ and axial $z$ dependence is constant. This geometry gives rise to the Wood lens which can introduce optical power using planar refractive surfaces [42]. The third common geometry is spherical GRIN $n\left (r\right )$ which also varies along the radial coordinate $r$ but now in spherical coordinates where the azimuthal $\theta$ and polar $\phi$ dependence is constant. Spherical GRIN also introduces optical power and is able to correct field aberrations such as astigmatism and distortion as well as chromatic aberrations [43].

Recent developments in fabrication [2330] have now made possible GRIN with two or even three dimensions of arbitrary refractive index change to offer a wide variety of DFs. For two dimensions, GRIN is free to vary in the transverse plane, such as $n\left (x,y\right )$ used in [33]. In this case, GRIN behavior is analogous to that of freeform surfaces, $z\left (x,y\right )$, as discussed in Sec. 2.1. Also intriguing are designs applying change along transverse and longitudinal coordinates, such as $n\left (\rho,z\right )$ used in [40]. Finally, three-dimensional refractive index variation offers the greatest DFs, defined in either Cartesian coordinates $n\left (x,y,z\right )$, cylindrical coordinates $n\left (\rho,\theta,z\right )$, or spherical coordinates $n\left (r,\theta,\phi \right )$. As described in Sec. 2.1, this third axis of variation allows F-GRIN to offer behavior not achievable with a single freeform surface $z\left (x,y\right )$ (see Fig. 3). A summary of available DFs can be seen in Table 1 for all permutations across three coordinate systems.

Tables Icon

Table 1. Available GRIN degrees of freedom with different coordinate systems for one, two, and three coordinates of refractive index variation.

In addition to the spatial form of refractive index variation, additional DFs are present when designing with GRIN. A significant factor is the GRIN’s total refractive index change $\Delta n$. By Eqs. (6)–(8), the magnitude of $OPD$ imparted by a GRIN is directly proportional to $\Delta n$. This is similar to the sag departure of a freeform surface dictating the extent of optical influence, as in Eqs. (3)–(4). In either case, $\Delta n$ or sag, a larger value is typically desirable due to the increased power of available DFs, but similar to surface sag departure, GRIN $\Delta n$ is limited by fabrication and testing capabilities. The magnitude of the gradient $\nabla n$ captures the rate of $\Delta n$ per unit volume and also constrains possible fabrication and testing options.

An important characteristic that can be leveraged in design is the chromatic properties of GRIN. Unlike freeform surfaces which can be reflective, GRIN’s optical influence is dependent on transmission, and like all optical materials, GRIN suffers from dispersion. As such, GRIN presents refractive index variation not only spatially but also spectrally, $n\left (x,y,z,\lambda \right )$. Unlike for homogeneous media, the wavelength dependence of GRIN can be controlled by adjusting the homogeneous media constituting the GRIN which have their own dispersive properties. This offers very valuable DFs by allowing the GRIN dispersion to be engineered to correct present chromatic aberrations. The chromatic influence of GRIN will be discussed further in Sec. 2.4. Finally, other GRIN DFs that can be leveraged include thermal control (dn/dt, CTE) for athermalization [44,45], birefringence for polarization control [46], and even nonlinear effects [47].

Mathematical representations for some of these DFs are discussed next in Sec. 2.3. Later, a nomenclature is proposed in Sec. 3 that describes the different types of GRIN listed in Table 1.

2.3 Mathematical representations

An important factor in designing with freeform optical elements is the mathematical representation of the optic. For freeform surfaces, the various local and global mathematical representations strongly influence the resultant design [48]. Representation also has ramifications for ease of fabrication and testing such as by minimizing sag departure [49]. The mathematical representation of F-GRIN now requires specification throughout a volume, introducing an additional third dimension compared with freeform surfaces. Like freeform surfaces, GRIN can be described either locally or globally.

Local representations for GRIN varying in two dimensions can be defined using the same techniques as freeform surfaces [17,18] except now the dependent variable is refractive index rather than surface sag. For such definitions, the refractive index is taken to be constant along the remaining third dimension. For three-dimensionally varying GRIN, however, local representations must be extended to capture volumetric behavior. A simple yet powerful representation is a “point cloud” of discrete refractive index points throughout a volume. The refractive index field can then be defined continuously in three dimensions by interpolation. Basic schemes include nearest neighbor or trilinear interpolation; however, for most cases, a smooth representation is desirable, which can be achieved using tricubic interpolation [50]. Techniques like NURBS [51] and RBF interpolation [52] can also be generalized to three independent dimensions for local GRIN representations. Local representations are also convenient for describing GRIN with discontinuities in refractive index or gradient. For example, a piecewise-continuous F-GRIN optic for prescribed illumination has been demonstrated that is locally represented using bicubic interpolation around discontinuities [33].

Global representations for GRIN apply polynomial definitions in either one, two, or three coordinates. As with local representations, one or two coordinate global representations can be used where the GRIN is assumed to be constant in the remaining coordinate(s), such as with the traditional axial, radial, and spherical GRIN types. For variation in three coordinates, a natural choice of representation in Cartesian coordinates is the $xyz$ polynomial,

$$n\left(x,y,z,\lambda\right)=\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\sum_{k=0}^{\infty}c_{ijk}\left(\lambda\right)x^{i}y^{j}z^{k}$$
where $c_{ijk}\left (\lambda \right )$ are wavelength-dependent scalar coefficients and $c_{000}\left (\lambda \right )=n_{0}\left (\lambda \right )$ is a constant base refractive index. In cylindrical coordinates, a global representation can consist of the product of a two-dimensional polar polynomial basis $U_{ij}\left (\rho,\theta \right )$ and a one-dimensional polynomial basis $V_{k}\left (z\right )$,
$$n\left(\rho,\theta,z,\lambda\right)=\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\sum_{k=0}^{\infty}c_{ijk}\left(\lambda\right)U_{ij}\left(\rho,\theta\right)V_{k}\left(z\right)$$
where again $c_{ijk}$ are spectrally dependent scalar coefficients. A common choice in polar coordinates is the orthogonal Zernike polynomial basis [19] which is defined over the unit circle and frequently used in describing freeform surfaces. For F-GRIN, a representation applying Zernike polynomials and the orthogonal Legendre polynomials in $z$ presents a convenient choice. Yang et al. [32] show this powerful representation can be analyzed using NAT similar to freeform surfaces and is used in this work for the F-GRIN designs in Fig. 3. Finally, a global representation in spherical coordinates is also possible. For example, Zernike polynomials can be expanded to three dimensions, now defined over the unit sphere rather than the unit circle [53].

2.4 Chromatic properties

Freeform surfaces impart $OPD$ exclusively by surface contributions, and as such, both refractive and reflective freeform surfaces are possible. On the other hand, GRIN relies on both surface and transmission contributions for optical influence [54]. Like all optical materials, upon transmission GRIN presents chromatic dispersion due to a wavelength dependence of the refractive index, $n\left (x,y,z,\lambda \right )$. Dispersion introduces chromatic aberrations in optical designs; however, as discussed in Sec. 2.2, the unique chromatic properties of GRIN serve as valuable DFs in controlling present chromatic effects.

Often, two or more optical materials are used to constitute a GRIN. Each base material has its own dispersion curve that can be quantified by its Abbe number $\nu$ and partial dispersion $P_{F,d}$ at $d$, $F$, and $C$ wavelengths. The interplay of multiple base materials’ dispersion sets the dispersive properties of GRIN, and this allows the GRIN dispersion to be engineered by modifying the base materials used in composition. This dispersion engineering can produce desirable chromatic properties including ones unattainable with a single homogeneous material. For example, GRIN can itself be defined by an Abbe number $\nu _{GRIN}$ and partial dispersion $P_{F,d,GRIN}$ by

$$\nu_{GRIN}=\frac{\Delta n\left(\lambda_{d}\right)}{\Delta n\left(\lambda_{F}\right)-\Delta n\left(\lambda_{C}\right)},\hspace{5mm}P_{F,d,GRIN}=\frac{\Delta n\left(\lambda_{F}\right)-\Delta n\left(\lambda_{d}\right)}{\Delta n\left(\lambda_{F}\right)-\Delta n\left(\lambda_{C}\right)}$$
where $\Delta n\left (\lambda _{i}\right )$ is the difference in refractive index of base materials at wavelength $\lambda _{i}$ [55]. These definitions were originally proposed [56] for specific orders of index variation to be equivalent to the homogeneous Abbe number and partial dispersion for calculation of paraxial chromatic aberration. Nevertheless, the more general definitions in Eq. (12) prove useful in defining GRIN chromatic behavior even though only partially analogous to their conventional counterparts. For homogeneous media with Abbe numbers typically bound between 20 and 90, Eq. (12) shows that values for $\nu _{GRIN}$ can potentially range from $\left (-\infty, \infty \right )$. For a catalog of accessible GRIN base materials [26], the achievable values for $\nu _{GRIN}$ and $P_{F,d,GRIN}$ are depicted in Fig. 4. This highlights the unique dispersive properties of GRIN which prove valuable in controlling chromatic effects in optical design.

 figure: Fig. 4.

Fig. 4. GRIN chromatic properties from accessible base plastic materials [26]. (a) GRIN Abbe number $\nu _{GRIN}$ and (b) GRIN partial dispersion $P_{F,d,GRIN}$ are shown versus total material refractive index difference $\Delta n\left (\lambda _{d}\right )$. Both $\nu _{GRIN}$ and $P_{F,d,GRIN}$ present values unattainable with homogeneous optical materials. Quantities are calculated using the expressions in Eq. (12).

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The dependence of GRIN dispersion on chosen base materials raises the question of what spectrally dependent refractive index value is obtained when combining different materials. The simplest description is a binary linear composition model which assumes the total refractive index is the linear combination of the two base materials $n_{0}\left (\lambda \right )$, $n_{1}\left (\lambda \right )$, weighted by their respective concentrations $C_{0}$, $C_{1}$,

$$n\left(\lambda\right)=C_{0}n_{0}\left(\lambda\right)+C_{1}n_{1}\left(\lambda\right)$$
where the sum of the concentrations is unity [57]. This assumption that dispersion curves are linearly combined is not necessarily true and has been shown to be inadequate in modeling some material combinations [55]. To remain consistent with fabrication, the GRIN’s spectral behavior must be modeled with an accurate material model. Specifically, in design this constrains the possible wavelength dependence of coefficients such as $c_{ijk}$ in Eqs. (10)–(11). Finally, using more than two base materials increases the freedom of GRIN dispersion by accessing an area rather than a curve on the Abbe diagram [58].

2.5 Fabrication and metrology

Traditionally, GRIN fabrication techniques have been limited in what spatial refractive index variation could be imparted. Several recent advancements have introduced F-GRIN fabrication methods offering these greater DFs. One technique applies polymer nanolayers of different refractive indices that can be consolidated into axial GRIN sheets and then mechanically shaped into some alternate profile [23]. This process enables some F-GRIN distributions but is unable to obtain arbitrary profiles since it is limited by the capacity to shape the axial GRIN sheet. A second fabrication process with greater DFs is the additive manufacturing of GRIN. Using low- and high-index nanoparticles in a host polymer, arbitrary F-GRIN volumes can be constructed using drop on demand inkjet printing techniques [24,25]. Within the GRIN profile, the refractive index and dispersion can be tuned by varying the type and concentration of these nanoparticles (see Fig. 4 for the GRIN chromatic properties resulting from different printable material combinations [26]). A different GRIN additive manufacturing process has also been demonstrated with glass [27]. Finally, arbitrary gradients have been created in two dimensions using photopolymerization [28] and three dimensions using direct laser writing [29], including in materials for the infrared [30]. Nevertheless, all of these F-GRIN fabrication technologies were only recently introduced and have not reached maturity.

The metrological tools for F-GRIN also pose a challenge. One- and two-dimensional refractive index variation can be obtained by transmitted phase measurements such as with a Mach-Zehnder interferometer [59], although this is only a relative index measurement. For three-dimensional variation, metrological tools are lacking. One option is to cut an F-GRIN into sections of approximately constant axial refractive index, but this measurement process results in the destruction of the optic. Three-dimensional refractive index tomography using multi-directional data has been described but only in the refractionless regime and using measurements spanning a 180 degree field-of-view [60]. This is not a viable option for most F-GRIN parts. One promising technique that obtains longitudinal index information is laser beam deflectometry [61]. On the other hand, techniques such as optical coherence tomography [62] and optical diffraction tomography [63] cannot be used since (ideally) there are no scattering features in GRIN media. Overall, the state of F-GRIN metrology is largely in its infancy.

2.6 Applications

F-GRIN can be used to enhance designs across many of the same applications as freeform surfaces. Folded, unobscured telescopes commonly use freeform surfaces [8]. Meanwhile, a folded, unobscured Schmidt-like telescope has been demonstrated using an F-GRIN phase corrector plate at the stop [32]. Progressive addition lenses for ophthalmics are one of the earliest and most widespread use of freeform surfaces [14]. Similarly, Fischer showed that an F-GRIN profile can be used in place of a freeform surface for progressive lenses [64]. Another active field of interest is freeform surfaces in head-worn displays [10] where a prismatic F-GRIN eyepiece has also been demonstrated [65]. Finally, design methods have been developed to produce a prescribed radiance distribution using freeform surfaces [11]. It was recently showed that the same could be achieved using F-GRIN [33,34].

Future applications of F-GRIN are as numerous as those for freeform surfaces. For these cases, a designer could leverage some of the strengths of F-GRIN including its control of aberration field dependence (see Fig. 3), unique chromatic properties [58], athermalization capabilities [44,45], and powered planar surfaces [42].

3. Proposed F-GRIN definition and nomenclature

As shown in Sec. 2., the capabilities and practical considerations of F-GRIN are in many ways reminiscent of freeform surfaces, but the two fundamentally differ due to the added volume propagation component of F-GRIN. With this in mind, the discussion now turns to proposing a nomenclature for different F-GRIN distributions.

The definition of freeform surfaces has evolved through the years, but it has generally come to mean a surface with no axis of rotational symmetry within or beyond the surface [12]. As a result, most generally, freeform surfaces must be defined as a function of two coordinates, such as $z\left (x,y\right )$ or $z\left (\rho,\theta \right )$. Surfaces defined by a single coordinate such as $z\left (x\right )$ may possess rotational variance but remain translationally symmetric, an additional restriction often placed on freeform surfaces [22]. A different perspective considers the axes necessary to generate a surface in fabrication, but this can lead to ambiguity as fabrication methods evolve.

When deciding on a nomenclature for F-GRIN, it is difficult to strictly adhere to freeform surface definitions because a volume and a surface are fundamentally different entities, even though for certain cases their behavior is similar. There are several relevant questions to consider when attempting to classify F-GRIN. Should the definition be based on refractive index geometry or required fabrication dimensionality? Should it be independent of present surface interfaces with the GRIN? Should it be independent of reference frame or used field-of-view?

Upon consideration of these questions, the definition of freeform GRIN proposed here is:

Any GRIN medium that must be specified in two or more independent spatial coordinates is a freeform GRIN (F-GRIN).

This definition is based purely on the geometrical form of the refractive index variation and does not factor in present surfaces or reference frame, which could change GRIN classification in different circumstances. For example, an axial GRIN $n\left (z\right )$ with a coordinate system misaligned with that of its surfaces functions as a freeform optic; however, the refractive index distribution is not freeform and thus is not an example of F-GRIN. This is analogous to a tilted spherical surface introducing freeform aberrations yet not being classified as a freeform surface. Moreover, the presented viewpoint attempts to remain consistent with freeform surfaces by applying a geometry-based definition rather than considering more situation-dependent effects like imparted aberrations or fabrication methods [12]. Also, by placing an emphasis on two or more coordinates of variation, the DFs available to the designer are comparable when considering either a freeform surface or an F-GRIN.

Considering the different types of GRIN variation listed in Table 1, the bottom two sections are classified as F-GRIN based on their coordinate dependence and the proposed definition. For examples of these different types, see Fig. 5 for randomly generated F-GRIN refractive index profiles in Cartesian, cylindrical, and spherical coordinates. These various F-GRIN types differ in their optical influence, primarily due to differences in imparted aberration and associated field dependence (see Figs. 3 and 6). The applicability of these different profiles in design is dependent on the geometry and needed DFs of the total optical system. A design may also apply only a section of the F-GRIN volumes shown in Fig. 5. For example, a section of an F-GRIN in spherical coordinates may be useful in systems with some degree of concentricity.

 figure: Fig. 5.

Fig. 5. Different examples of freeform GRIN with two or three coordinates of refractive index variation based on the proposed definition and nomenclature. Colormap represents arbitrary refractive index.

Download Full Size | PPT Slide | PDF

 figure: Fig. 6.

Fig. 6. $\rho z$ F-GRIN profiles contain an axis of rotational symmetry in refractive index yet may impart freeform aberrations. (a) $\rho z$ F-GRIN of Zernike astigmatism possesses an axis of rotational symmetry along $z$ (dotted line). This $\rho z$ F-GRIN imparts freeform aberrations for fields propagating about $x$ including (b) field constant astigmatism and (c) field linear medial coma. Aberration calculations are performed for the $\rho z$ F-GRIN at the aperture stop and a “perfect” unaberrated lens following for image formation. The on-axis field-of-view is defined as propagation along $x$. Icon size indicates the magnitude of present aberration.

Download Full Size | PPT Slide | PDF

A key distinction of the proposed F-GRIN definition compared with that of freeform surfaces is that it does not exclude geometries with an axis of rotational symmetry in refractive index. In fact, F-GRIN with rotationally symmetric distributions are capable of imparting freeform aberrations whereas axially symmetric surfaces cannot. To understand why, consider the refractive index field in cylindrical coordinates $n\left (\rho,z\right )$, which has rotational symmetry about $z$. For rays generally propagating along $z$, ray behavior is unchanged with rigid rotation about $z$, meaning wavefront aberrations can be represented by a rotationally symmetric expansion [66]. Due to the volumetric characteristic of GRIN, however, the reference frame can be rotated 90 degrees, and light can now propagate orthogonal to the axis of rotational symmetry. By doing so, a freeform refractive index field $n\left (\rho,z\right )$ may impart freeform aberrations as it can for $n\left (x,y\right )$. For the case of Zernike astigmatism in $\rho,z$, see Fig. 6 where field constant astigmatism, a freeform aberration, is imparted for propagation along $x$ with a rotationally symmetric refractive index field.

In terms of proposed nomenclature, F-GRIN that varies in two coordinates rather than three will include a qualifier specifying which two axes possess refractive index change. For instance, $xy$ F-GRIN and $yz$ F-GRIN both vary along two coordinates but with different optical behavior. When changing in all three coordinates, no qualifier is needed. For examples of the proposed nomenclature see Fig. 5.

Supported by the recent work of many in this field [2330,3234,40,61,64,65], the proposed definition and nomenclature formalizes F-GRIN media as a type of freeform optic. Although F-GRIN and freeform surfaces are both types of freeform optics, one should not get the impression that they are merely substitutes for one another. Instead, the introduction of F-GRIN extends the concept of freeform optics further. F-GRIN and freeform surfaces possess different capabilities and limitations that exist in a continuum. As such, a holistic view of freeform optics is necessary to avoid stagnation and progressively expand the field.

4. Conclusion

The introduction of freeform gradient-index (F-GRIN) media offers a valuable new freeform design tool. Here, the optical influence of F-GRIN is evaluated in comparison with that of freeform surfaces and is shown to have different field-dependent behavior when varying longitudinally. The many available DFs of F-GRIN are discussed along with different local and global mathematical representations. Next, the unique chromatic properties of F-GRIN and the current state of fabrication and metrology are described. Then, current and future applications of F-GRIN are examined for both imaging and illumination design. Finally, an F-GRIN definition and nomenclature is proposed that focuses exclusively on the geometry and spatial dependence of refractive index variation. Although in its infancy, F-GRIN offers unique capabilities similar to that of freeform surfaces that will prove valuable for future optical design.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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References

  • View by:

  1. R. Kingslake, Lens design fundamentals (Academic Press, 1978).
  2. D. J. Schroeder, Astronomical optics (Academic Press, 2000), 2nd ed.
  3. P. P. Clark, “Mobile platform optical design,” in International Optical Design Conference 2014, vol. 9293 of Proc. SPIE (2014), p. 92931M.
  4. K. P. Thompson, “Aberration Fields in Tilted and Decentered Optical Systems,” Ph.D. thesis, University of Arizona (1980).
  5. L. G. Cook, “Three-Mirror Anastigmat Used Off-Axis In Aperture And Field,” in Space Optics II, vol. 0183 of Proc. SPIE (1979), pp. 207–211.
  6. S. Steven, J. Bentley, and A. Dubra, “Design of two spherical mirror unobscured relay telescopes using nodal aberration theory,” Opt. Express 27(8), 11205–11226 (2019).
    [Crossref]
  7. K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Theory of aberration fields for general optical systems with freeform surfaces,” Opt. Express 22(22), 26585–26606 (2014).
    [Crossref]
  8. A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9(1), 1756 (2018).
    [Crossref]
  9. J. Reimers, A. Bauer, K. P. Thompson, and J. P. Rolland, “Freeform spectrometer enabling increased compactness,” Light: Sci. Appl. 6(7), e17026 (2017).
    [Crossref]
  10. D. Cheng, Y. Wang, C. Xu, W. Song, and G. Jin, “Design of an ultra-thin near-eye display with geometrical waveguide and freeform optics,” Opt. Express 22(17), 20705–20719 (2014).
    [Crossref]
  11. R. Wu, Z. Feng, Z. Zheng, R. Liang, P. Benítez, J. C. Miñano, and F. Duerr, “Design of Freeform Illumination Optics,” Laser Photonics Rev. 12(7), 1700310 (2018).
    [Crossref]
  12. J. P. Rolland, M. A. Davies, T. J. Suleski, C. Evans, A. Bauer, J. C. Lambropoulos, and K. Falaggis, “Freeform optics for imaging,” Optica 8(2), 161–176 (2021).
    [Crossref]
  13. W. J. Smith, Modern optical engineering: the design of optical systems (McGraw Hill, 2008), 4th ed.
  14. H. J. Birchall, “Lens of variable focal power having surfaces of involute form,” U. S. patent2,475,275 (5July1949).
  15. L. W. Alvarez, “Development of variable-focus lenses and a new refractor,” J. Am. Optom. Assoc. 49(1), 24–29 (1978).
  16. W. T. Plummer, “Unusual optics of the Polaroid SX-70 Land camera,” Appl. Opt. 21(2), 196–202 (1982).
    [Crossref]
  17. M. P. Chrisp, B. Primeau, and M. A. Echter, “Imaging freeform optical systems designed with NURBS surfaces,” Opt. Eng. 55(7), 071208 (2016).
    [Crossref]
  18. O. Cakmakci, B. Moore, H. Foroosh, and J. P. Rolland, “Optimal local shape description for rotationally non-symmetric optical surface design and analysis,” Opt. Express 16(3), 1583–1589 (2008).
    [Crossref]
  19. A. B. Bhatia and E. Wolf, “On the circle polynomials of Zernike and related orthogonal sets,” Math. Proc. Cambridge Philos. Soc. 50(1), 40–48 (1954).
    [Crossref]
  20. G. W. Forbes, “Characterizing the shape of freeform optics,” Opt. Express 20(3), 2483–2499 (2012).
    [Crossref]
  21. D. A. Bykov, L. L. Doskolovich, A. A. Mingazov, E. A. Bezus, and N. L. Kazanskiy, “Linear assignment problem in the design of freeform refractive optical elements generating prescribed irradiance distributions,” Opt. Express 26(21), 27812–27825 (2018).
    [Crossref]
  22. “Geometrical product specifications (GPS) – General concepts – Part 1: Model for geometrical specification and verification,” ISO17450-1:2011 (2011).
  23. S. Ji, K. Yin, M. Mackey, A. Brister, M. Ponting, and E. Baer, “Polymeric nanolayered gradient refractive index lenses: technology review and introduction of spherical gradient refractive index ball lenses,” Opt. Eng. 52(11), 112105 (2013).
    [Crossref]
  24. S. D. Campbell, D. E. Brocker, D. H. Werner, C. Dupuy, S.-K. Park, and P. Harmon, “Three-dimensional gradient-index optics via injketaided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, (IEEE, 2015), pp. 605–606.
  25. D. M. Schut, C. G. Dupuy, and J. P. Harmon, “Inks for 3D printing gradient refractive index (GRIN) optical components,” U. S. patent9,447,299 (21April2016).
  26. E. Elliot, C. Dupuy, S. Sang, and N. Nguyen, “Nanovox LLC proprietary optical ink,” Voxtel (2021), https://voxtel-llc.com .
  27. R. Dylla-Spears, T. D. Yee, K. Sasan, D. T. Nguyen, N. A. Dudukovic, J. M. Ortega, M. A. Johnson, O. D. Herrera, F. J. Ryerson, and L. L. Wong, “3D printed gradient index glass optics,” Sci. Adv. 6(47), eabc7429 (2020).
    [Crossref]
  28. A. C. Urness, K. Anderson, C. Ye, W. L. Wilson, and R. R. McLeod, “Arbitrary GRIN component fabrication in optically driven diffusive photopolymers,” Opt. Express 23(1), 264–273 (2015).
    [Crossref]
  29. C. R. Ocier, C. A. Richards, D. A. Bacon-Brown, Q. Ding, R. Kumar, T. J. Garcia, J. van de Groep, J.-H. Song, A. J. Cyphersmith, A. Rhode, A. N. Perry, A. J. Littlefield, J. Zhu, D. Xie, H. Gao, J. F. Messinger, M. L. Brongersma, K. C. Toussaint, L. L. Goddard, and P. V. Braun, “Direct laser writing of volumetric gradient index lenses and waveguides,” Light: Sci. Appl. 9(1), 196 (2020).
    [Crossref]
  30. K. A. Richardson, M. Kang, L. Sisken, A. Yadav, S. Novak, A. Lepicard, I. Martin, H. Francois-Saint-Cyr, C. M. Schwarz, T. S. Mayer, C. Rivero-Baleine, A. J. Yee, and I. Mingareev, “Advances in infrared gradient refractive index (GRIN) materials: a review,” Opt. Eng. 59(11), 112602 (2020).
    [Crossref]
  31. D. T. Moore, “Gradient-index optics: a review,” Appl. Opt. 19(7), 1035–1038 (1980).
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2020 (8)

R. Dylla-Spears, T. D. Yee, K. Sasan, D. T. Nguyen, N. A. Dudukovic, J. M. Ortega, M. A. Johnson, O. D. Herrera, F. J. Ryerson, and L. L. Wong, “3D printed gradient index glass optics,” Sci. Adv. 6(47), eabc7429 (2020).
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D. H. Lippman and G. R. Schmidt, “Prescribed irradiance distributions with freeform gradient-index optics,” Opt. Express 28(20), 29132–29147 (2020).
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W. Minster Kunkel and J. R. Leger, “Numerical design of three-dimensional gradient refractive index structures for beam shaping,” Opt. Express 28(21), 32061–32076 (2020).
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C. R. Ocier, C. A. Richards, D. A. Bacon-Brown, N. A. Krueger, M. K. Clawson, J. A. N. T. Soares, and P. V. Braun, “Optically anisotropic porous silicon microlenses with tunable refractive indexes and birefringence profiles,” Opt. Mater. Express 10(4), 868–883 (2020).
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A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9(1), 1756 (2018).
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R. Wu, Z. Feng, Z. Zheng, R. Liang, P. Benítez, J. C. Miñano, and F. Duerr, “Design of Freeform Illumination Optics,” Laser Photonics Rev. 12(7), 1700310 (2018).
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A. M. Boyd, “Optical design of athermal, multispectral, radial gradient-index lenses,” Opt. Eng. 57(8), 085103 (2018).
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2017 (3)

J. Ye, L. Chen, X. Li, Q. Yuan, and Z. Gao, “Review of optical freeform surface representation technique and its application,” Opt. Eng. 56(11), 110901 (2017).
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J. Reimers, A. Bauer, K. P. Thompson, and J. P. Rolland, “Freeform spectrometer enabling increased compactness,” Light: Sci. Appl. 6(7), e17026 (2017).
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A. J. Yee and D. T. Moore, “Free-space infrared Mach–Zehnder interferometer for relative index of refraction measurement of gradient index optics,” Opt. Eng. 56(11), 111707 (2017).
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2015 (2)

2014 (2)

2013 (2)

S. Ji, K. Yin, M. Mackey, A. Brister, M. Ponting, and E. Baer, “Polymeric nanolayered gradient refractive index lenses: technology review and introduction of spherical gradient refractive index ball lenses,” Opt. Eng. 52(11), 112105 (2013).
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A. J. Visconti, J. A. Corsetti, K. Fang, P. McCarthy, G. R. Schmidt, and D. T. Moore, “Eyepiece designs with radial and spherical polymer gradient-index optical elements,” Opt. Eng. 52(11), 112102 (2013).
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2011 (1)

2008 (1)

2005 (1)

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2003 (1)

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (6)

Fig. 1.
Fig. 1. Comparison of (a) freeform surface and (b) freeform GRIN (F-GRIN) with the same transverse dependence in sag and refractive index, respectively. The F-GRIN also has a third dimension of axial variation.
Fig. 2.
Fig. 2. Optical path difference illustrated for (a) freeform surface, (b) F-GRIN, and (c) freeform surface with F-GRIN. (a) The surface is immersed in homogeneous refractive indices $n$ and $n'$. (c) The surface is immersed in inhomogeneous refractive index fields $n\left (x,y,z\right )$ and $n'\left (x,y,z\right )$. Colormap represents arbitrary refractive index.
Fig. 3.
Fig. 3. Freeform aberration field dependence for Zernike coma (a) freeform surface and (b)-(e) F-GRIN of different longitudinal orders, $z^{j}$. (a) Freeform surface and (b) axially constant F-GRIN yield approximately the same result of field constant coma; field asymmetric, field linear astigmatism; and field linear, medial field curvature. (c) F-GRIN with linear axial variation removes the field constant aberration. (d) F-GRIN with a combination of constant and linear axial terms leaves the field constant aberration but removes field linear ones. (e) F-GRIN with quadratic axial variation removes both field constant and field linear aberrations. All deigns have the same field-of-view, the aperture stop positioned 15 mm away from the freeform, and a “perfect” unaberrated lens following the optic for image formation. The freeform surface is on a base plane, and F-GRIN designs have plane-parallel surfaces. Field curvature, astigmatism, and coma are shown across the field-of-view and on the same scale where icon size indicates the magnitude of present aberration.
Fig. 4.
Fig. 4. GRIN chromatic properties from accessible base plastic materials [26]. (a) GRIN Abbe number $\nu _{GRIN}$ and (b) GRIN partial dispersion $P_{F,d,GRIN}$ are shown versus total material refractive index difference $\Delta n\left (\lambda _{d}\right )$. Both $\nu _{GRIN}$ and $P_{F,d,GRIN}$ present values unattainable with homogeneous optical materials. Quantities are calculated using the expressions in Eq. (12).
Fig. 5.
Fig. 5. Different examples of freeform GRIN with two or three coordinates of refractive index variation based on the proposed definition and nomenclature. Colormap represents arbitrary refractive index.
Fig. 6.
Fig. 6. $\rho z$ F-GRIN profiles contain an axis of rotational symmetry in refractive index yet may impart freeform aberrations. (a) $\rho z$ F-GRIN of Zernike astigmatism possesses an axis of rotational symmetry along $z$ (dotted line). This $\rho z$ F-GRIN imparts freeform aberrations for fields propagating about $x$ including (b) field constant astigmatism and (c) field linear medial coma. Aberration calculations are performed for the $\rho z$ F-GRIN at the aperture stop and a “perfect” unaberrated lens following for image formation. The on-axis field-of-view is defined as propagation along $x$. Icon size indicates the magnitude of present aberration.

Tables (1)

Tables Icon

Table 1. Available GRIN degrees of freedom with different coordinate systems for one, two, and three coordinates of refractive index variation.

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

Δ ϕ ( x , y ) = k O P D ( x , y )
O P L = s 0 s 1 n d s
O P D ( x , y ) ( n n ) z ( x , y )
O P D ( x , y ) 2 n z ( x , y )
d d s ( n ( r ) d r d s ) = n ( r )
O P D ( x , y ) [ n ( x , y ) n 0 ] Δ z
O P D ( x , y ) 0 t [ n ( x , y , z ) n 0 ] d z = n 0 t + 0 t n ( x , y , z ) d z
O P D ( x , y ) 0 z ( x , y ) [ n ( x , y , ζ ) n ( x 0 , y 0 , ζ ) ] d ζ + z ( x , y ) t [ n ( x , y , ζ ) n ( x 0 , y 0 , ζ ) ] d ζ
n ( x , y , z , λ ) = i = 0 j = 0 k = 0 c i j k ( λ ) x i y j z k
n ( ρ , θ , z , λ ) = i = 0 j = 0 k = 0 c i j k ( λ ) U i j ( ρ , θ ) V k ( z )
ν G R I N = Δ n ( λ d ) Δ n ( λ F ) Δ n ( λ C ) , P F , d , G R I N = Δ n ( λ F ) Δ n ( λ d ) Δ n ( λ F ) Δ n ( λ C )
n ( λ ) = C 0 n 0 ( λ ) + C 1 n 1 ( λ )

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