Optical lens compression via transformation optics

D. A. Roberts; N. Kundtz; D. R. Smith

doi:10.1364/OE.17.016535

1. Introduction

Transformation optics is a newly developed technique for the design of complex electromagnetic media. While the transformation optical approach initially gained distinction as a means to design materials that function as “invisibility” cloaks [1–3], it has been rapidly adopted as a design strategy for a wide array of electromagnetic devices, including integrated optical components [4–7], focusing devices [8–14] and antenna components [15–19]. In developing transformation optical versions of conventional optical devices, we seek to leverage the flexibility available in varying the constitutive parameters of an inhomogeneous material throughout its volume, rather than confining our device engineering to the input and output interfaces to a homogeneous medium. The progression in recent years of increasingly complex, inhomogeneous metamaterials has revealed the precision with which carefully controlled gradients in either the magnetic or the electric constitutive parameters can be achieved in structured media to achieve focusing or guiding of electromagnetic waves [20–24].

While gradient index lenses and optics have proven to be extremely compatible with metamaterial implementations [21–23], conventional refractive optics based on unconventional material properties (such as negative refractive index) have been shown to have interesting advantages [25,26] and can also be implemented using metamaterials. In particular, metamaterials can be designed with both magnetic as well as electric response, so that impedance-matched, low reflectance lenses can be constructed. Likewise, negative index lenses can have substantially reduced spherical aberration and coma, with a generally lower overall aberration profile. A different figure-of-merit, the thickness or profile of the lens, can also be addressed by altering the composition of the underlying material. Although image processing schemes can be employed to achieve similar feature resolution with a lower numerical aperture (and hence thinner) lens [27], it is also possible to reduce the profile of or “flatten” a refractive lens by increasing the magnitude of the refractive index. In the absence of a magnetic component to the material, large index lenses require materials with large dielectric values. While the large dielectric value may lead to improved imaging performance, the increased reflection reduces the practicality of a lens flattened by this approach.

The transformation optical approach to flattening a standard refractive lens is to apply a coordinate transformation in which space is compressed along the optical axis. As is the usual process in transformation optical design, the coordinate transformation can then be applied to Maxwell’s equations, resulting in a renormalization of the permittivity and permeability tensors. Since space is compressed in only one dimension, the resulting constitutive parameters are inherently anisotropic, with both electric and magnetic response, and vary spatially. Since the form of the transformation that achieves the desired compression is not unique, it thus is possible to introduce various optimization schemes to increase the feasibility of the final design.

2. Lens compression

Lenses based on dielectric materials have polarization properties that depend on the polarization and the angle of incidence of the incoming waves. The difference in transmission and reflection behavior between, for example, transverse electric (TE) and transverse magnetic (TM) waves generally become much more severe when the lens material is made anisotropic, potentially allowing additional phenomena such as polarization rotation to complicate the imaging properties. If the correct anisotropic medium corresponding to the full transformation optical design is implemented, then all polarizations are maintained and the scattering behavior will be no worse for the flattened lens as compared with the original lens. However, achieving the permittivity and permeability distribution corresponding to the complete transformation remains a considerable challenge for metamaterials, and thus for a flattened lens that manages both polarizations.

The concept underlying the transformation optical approach is that a region of space can be described using a given set of (unprimed) coordinates, but have electromagnetic waves behave as if a second set of (primed) coordinates had been used. This behavior can be facilitated by introducing a set of spatially varying constitutive parameters, whose forms are determined according to

\begin{array}{l} ε^{i' j'} = \det {(A)}^{- 1} A_{i}^{i'} A_{j}^{j'} ε^{i j} \\ μ^{i' j'} = \det {(A)}^{- 1} A_{i}^{i'} A_{j}^{j'} μ^{i j}, \end{array}

where

A_{i}^{i'} = \frac{\partial x^{i'}}{\partial x^{i}} .

The indices (i, i’,j,j’) run from 1 to 3. In Cartesian coordinates these indices specify the x, y and z axes.

To achieve lens flattening, we consider transformations that compress space in the direction normal to the lens (along the optical axis). We identify three regions of space, defined by the differential ratios:

\begin{array}{l} I . \frac{\partial x^{i'}}{\partial x^{i}} = 1 \\ I I .0 < \frac{\partial x^{i'}}{\partial x^{i}} < 1 \\ I I I . \frac{\partial x^{i'}}{\partial x^{i}} = 1 \end{array}

It is clear that for a distance δx in the unprimed system, the corresponding distance δx’ in the primed system will be smaller, and the space will be compressed. A simple form for the element of the Jacobian matrix along the optical axis is as follows,

\frac{d x'}{d x} = {\begin{array}{c} a, & l_{1} < x < l_{2} \\ 1, & x \leq l_{1} \cup x \geq l_{2} \end{array},

where l₁ and l₂ are the bounds of the transformation in the unprimed space. This expression can be integrated to determine the coordinate map, giving

x' (x) = {\begin{array}{c} a x + c, & l_{1} < x < l_{2} \\ l_{1}' + (x - l_{1}), & x \leq l_{1} \\ l_{2}' + (x - l_{2}), & x \geq l_{2} \end{array} .

The integration constant is determined by a boundary condition that can be used to define certain optical positions of the lens, such as setting the position of

x' = x

. The bounds of the device in the primed system (which will correspond to the physical location of the device when implemented) are given by the maps

l_{1}' = x' (l_{1}) = a l_{1} + c

and

l_{2}' = x' (l_{2}) = a l_{2} + c

. The transformation extends to infinity, since the bounds of the transformation do not map to the same point in both systems (i.e.

l_{1}' \neq l_{1}

and

l_{2}' \neq l_{2}

). However, since

\partial x' / \partial x = 1

outside of this region, no “material” is required. A plot of the transformation and the transformed grid is shown in Fig. 1 .

Fig. 1 (a) Plot of the space compression transformation expressed by Eq. (5). The shaded region in the left plot is compressed by a factor of two in the right plot. In both plots, the lines are of constant x and constant y. The integration constant was chosen so that x = 0 maps to x’ = 0. (b) Plot of x’ versus x, illustrating the compression that occurs in the primed coordinate system.

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For the transformation defined by Eq. (5), Eqs. (1) can be used to find the forms for ε and μ as

\begin{array}{l} ε^{i j} = [\begin{array}{c} a & 0 & 0 \\ 0 & 1 / a & 0 \\ 0 & 0 & 1 / a \end{array}] n {(x, y)}^{2}, \\ μ^{i j} = [\begin{array}{c} a & 0 & 0 \\ 0 & 1 / a & 0 \\ 0 & 0 & 1 / a \end{array}] . \end{array}

We assume the index profile n(x,y) represents a conventional refractive optic, for which the index of refraction occurs via an electric response, with n = ε ^1/2. Equation (6) shows that the dielectric response of the lens must be transformed in addition to the free space surrounding it. Additionally, the position of lens is translated and must be defined through the integration constant.

To illustrate the lens-flattening procedure, we apply a transformation that compresses a planar-convex lens to half its original thickness. The uncompressed lens has a refractive index of 1.5, a thickness of 50 cm and a focal length of 200 cm.

To compress the lens to half its thickness requires that a = ½. The integration constant is chosen so that x = 0 is mapped to x’ = 0 (i.e. c = 0). This position corresponds to the planar side of the lens. The spherical surface of the lens extends to –d = −0.05 m. It is clear from Eq. (5) that this position is mapped to -d/2 = 0.025 m. The resulting transformation, which is plotted in Fig. 1c for these parameters, is

x' (x) = {\begin{array}{c} x + 0.025 \\ \frac{1}{2} x \\ x \end{array} \begin{array}{c} x \leq - 0.05 \\ - 0.05 < x < 0 \\ x \geq 0 \end{array} .

Equation (7) shows explicitly that for x < −0.05, the transformation is not unity, but has a uniform shift. Since this shift will not change the derivative in the Jacobian matrix, no change to the (free space) material is required. Following Eq. (6), the material parameters of the transformed lens are calculated. Figure 2d shows the non-zero material tensor components for the normal lens and the transformed lens.

Fig. 2 Plot of the z-component of the electric field for a lens with a focal length of 0.2 m: (a) Original planar-convex lens. (b) Transformation optically compressed lens. (c) Plot of intensity across the focal plane for the original planar-convex lens and the transformation optically compressed lens. (d) Material parameters: Non-zero permittivity and permeability components of the original lens and the transformed lens.

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A simulation was performed using the two dimensional finite-element solver of the COMSOL Multiphysics software package. In these simulations, a plane wave at 20 GHz was incident from the left and focused by the lens. The lens is apertured by an absorbing material. Both the original lens and the transformed lens were simulated. The results of the simulations are shown in Fig. 2a and Fig. 2b, respectively. As evidenced by the field plot and the ray trace of the power flow, both lenses focus the incident plane wave at the focal point. While the profile of the transformed lens has been reduced, it exhibits the same optical path length as the original lens. Moreover, all properties of the original lens—including the aberration profile, field-of-view and numerical aperture—are expected to be unchanged. Note that, in keeping with Eq. (7) the entire left hand region (Region I) has been shifted inwards by 0.025 m. Plots of the intensity along the focal plane were taken for each lens and are shown in Fig. 2c. The intensity distributions are identical, validating the success of this transformation.

3. Dielectric only implementation

The lens design simulated above is compelling because of the ability to re-shape the lens without the introduction of additional Fresnel reflections or otherwise changing the lens properties. However, the use of permeability components greater than unity introduces the need for resonant elements in a metamaterial implementation. The use of resonant elements implies bandwidth limitations and is often associated with larger absorption in the material. It is thus advantageous to search for implementations in which only dielectric response is necessitated and nonresonant metamaterial elements can be employed. Here we demonstrate that for waves restricted to the transverse magnetic (TM) polarization, a dielectric implementation may be found.

To find a reduced form for the lens material parameters, we consider the behavior of the lensing medium in the in the limit of ray optics. We can write the general wave equation for TM waves with magnetic field polarized along the z-direction:

\frac{1}{μ_{z} (x)} \frac{\partial}{\partial x} [\frac{1}{ε_{y} (x)} \frac{\partial H_{z}}{\partial x}] + \frac{1}{μ_{z} (x)} \frac{\partial}{\partial y} [\frac{1}{ε_{x} (x)} \frac{\partial H_{z}}{\partial y}] = \frac{\partial^{2} H_{z}}{\partial t^{2}},

where we have allowed for the constitutive parameters being dependent on the coordinate along the optical axis. The first term in (8) may be written as

(- \frac{1}{μ_{z}} \frac{1}{ε_{y}^{2}} \frac{\partial ε_{y}}{\partial x} \frac{\partial H_{z}}{\partial x} + \frac{1}{μ_{z} ε_{y}} \frac{\partial^{2} H_{z}}{\partial x^{2}}) .

Since the permittivity tensor does not vary spatially throughout either the compressed lens or the background medium, the first term is zero everywhere except at the lens interfaces. Thus, for most of the domain, a simplified wave equation applies:

\frac{1}{n_{x}^{2}} \frac{\partial^{2} H_{z}}{\partial x^{2}} + \frac{1}{n_{y}^{2}} \frac{\partial^{2} H_{z}}{\partial y^{2}} = \frac{\partial^{2} H_{z}}{\partial t^{2}}

in which

n_{x} = \sqrt{μ_{z} ε_{y}},

n_{y} = \sqrt{μ_{z} ε_{x}}

the index values parallel to and perpendicular to the optical axis, determine the wave propagation properties. Any combination of μ_z, ε_y and ε_x that leave n_x and n_y invariant will approximately reproduce the same focusing or imaging properties. Thus, if we define

ε_{y}' = μ_{z} ε_{y},

ε_{x}' = μ_{z} ε_{x},

μ_{z}' = 1,

both Eq. (11) and Eq. (12) remain invariant, but we arrive at an index only version of the compressed lens.

In the limit of ray optics the direction of the refraction at the lens boundary is solely dependent upon the index (Snell’s law). Therefore, for sufficiently short wavelengths, this approximation is justified. The effect of making this adjustment is that the Fresnel reflection coefficients will change, resulting in increased scattering at the boundaries of the medium. In the case of the material described here, the reduced material properties become

ε^{i j} = [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 / a^{2} & 0 \\ 0 & 0 & 1 \end{array}] n {(x, y)}^{2},

μ = 1.

All the permittivity components are greater than unity (since a is between 0 and 1). Figure 3a shows a COMSOL simulation of the resulting dielectric-only compressed lens. This transformation is no longer reflectionless, as evidenced by the power flow lines shown in the figure.

Fig. 3 (a) Plot of the z-component of the magnetic field for the dielectric-only compressed lens with a focal length of 0.2 m. (b) Plot of the z-component of the magnetic field for the dielectric-only gradient transformation compressed lens with a focal length of 0.2 m.

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One way to limit reflections is by adjusting the original transformation so that the final material profile is a smooth gradient from and to the free space regions. For a given range of permittivity, the implementation of a gradient will result in a smaller overall compression factor (a choice that will depend on the properties required for the lens). The following transformation yields a gradient index profile and compresses the lens via a parabolic transformation (as opposed to the jump transformation expressed in Eq. (4),

\frac{d x'}{d x} = {\begin{array}{c} (1 - a) {(\frac{2 x - l_{1} - l_{2}}{l_{1} - l_{2}})}^{2} + a, & l_{1} < x < l_{2} \\ 1, & x \leq l_{1} \cup x \geq l_{2} \end{array},

where again l₁ and l₂ are the bounds of the transformation in the unprimed space, and a is a real number (0,1). However, a is still a free parameter (related to the minimum permittivity value), but is no longer equal to the compression ratio. As before, this expression can be integrated to determine the coordinate map, giving

x' (x) = {\begin{array}{c} \frac{(1 - a)}{6 {(l_{1} - l_{2})}^{2}} {(2 x - l_{1} - l_{2})}^{3} + a x + c, & l_{1} < x < l_{2} \\ l_{1}' + (x - l_{1}), & x \leq l_{1} \\ l_{2}' + (x - l_{2}), & x \geq l_{2} \end{array} .

The discussion of the integration constant and boundary conditions remains the same as in Eq. (5). A plot of both Eq. (18) and Eq. (19) is shown in Fig. 4 . The compression ratio is given by

\frac{l_{2}' - l_{1}'}{l_{2} - l_{1}} = \frac{2}{3} a + \frac{1}{3} .

For a = ½ (which will enable the same range of material parameters as in the previous transformations), the compression ratio is reduced to 2/3.

Fig. 4 (a) Plot of the space compression transformation expressed by Eq. (19). The shaded region shows the compressed region. As opposed to the jump compression, the parabolic compression is graded in. The lines are of constant x and constant y. The integration constant was chosen so that x = 0 maps to x’ = 0. (b) Plot of x’ versus x for the parabolic transformation expressed by Eq. (19) illustrating the compression that occurs in the primed coordinate system. (c) Plot of dx’/dx versus x for the parabolic transformation expressed by Eq. (18).

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When the resultant anisotropic index profile is realized using dielectric-only materials Eq. (4) created a step barrier for the wave to scatter off. Equation (18) grades this impedance mismatch resulting in significantly lower scattering. A direct comparison of the terms in Eq. (9) demonstrates that the ray approximation is still appropriate. Figure 3b shows the resulting COMSOL simulation. It is clear from the power flow lines that the reflections are mitigated. The result is a dielectric-only spatially compressed lens.

4. Conclusions

We have successfully used a space compression transformation to reduce the profile of a lens. In this transformation we effectively transform all of space, while only requiring a transformation optical material in a small region, resulting in an overall reduction in the volume of a lens. The transformation introduces a simple form of anisotropy to the requisite material. The compression factor can be scaled as desired at the expense of a larger range of material parameters. For a given polarization, the lens can be engineered to be dielectric-only, making it broad band and essentially lossless. By introducing a gradient into the transformation, Fresnel reflections can be minimized. The transformation optical approach thus fuses the concepts of refractive and gradient index optics, providing a systematic means of designing improved optical devices starting with conventional paradigms.

References and links

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Optical lens compression via transformation optics

Abstract

1. Introduction

2. Lens compression

3. Dielectric only implementation

4. Conclusions

References and links

Cited By

Figures (4)

Equations (20)

Optics Express