June 1, 2015 / Vol. 40, No. 11 / OPTICS LETTERS
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Transformation-optics-inspired anti-reflective coating design for gradient index lenses Kenneth L. Morgan,* Donovan E. Brocker, Sawyer D. Campbell, Douglas H. Werner, and Pingjuan L. Werner Department of Electrical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802, USA *Corresponding author: [email protected] Received April 7, 2015; revised May 5, 2015; accepted May 6, 2015; posted May 8, 2015 (Doc. ID 237662); published May 21, 2015 Recent developments in transformation optics have led to burgeoning research on gradient index lenses for novel optical systems. Such lenses hold great potential for the advancement of complex optics for a wide range of applications. Despite the plethora of literature on gradient index lenses, previous works have not yet considered the application of anti-reflective coatings to these systems. Reducing system reflections is crucial to the development of this technology for highly sensitive optical applications. Here, we present effective anti-reflective-coating designs for gradient index lens systems. Conventional anti-reflective-design methodologies are leveraged in conjunction with transformation optics to develop coatings that significantly reduce reflections of a flat gradient index lens. Finally, the resulting gradient-index anti-reflective coatings are compared and contrasted with conventional homogeneous anti-reflective coatings. © 2015 Optical Society of America OCIS codes: (050.1755) Computational electromagnetic methods; (080.2710) Inhomogeneous optical media; (080.3620) Lens system design; (110.2760) Gradient-index lenses; (310.1210) Antireflection coatings. http://dx.doi.org/10.1364/OL.40.002521
The advent of transformation optics (TO) brought forth a renaissance of gradient-index (GRIN) designs [1–4]. However, the TO-based solutions generally require anisotropic and magnetic materials that make realizing such systems often infeasible. Quasi-conformal transformation optics (QCTO) relaxes the material requirements predicted by the more general TO approach, enabling the development of isotropic dielectric devices [5,6]. In particular, QCTO techniques have piqued interest in designing GRIN lens systems, where all-dielectric materials are desired. These GRIN systems hold great potential for reducing the size, weight, and power (SWaP) of optical devices by introducing additional design flexibility. However, anti-reflective coatings (ARCs) for such GRIN systems have not yet been investigated, leaving a crucial gap in the adaptation of the technology, since often highperformance optics necessitate ARCs to mitigate reflection losses. ARC design is well-established in literature for homogeneous lens systems [7,8], including a wide variety of methodologies extending beyond the narrowband quarter-wave coating [9], to encompass multi-layer [7], GRIN [8,10], and nature-inspired coatings [11,12] that are capable of achieving wideband solutions. Here, we demonstrate ARC designs to reduce reflection losses of GRIN lenses. In particular, conventional ARC design techniques are leveraged in conjunction with QCTO to develop coatings for narrowband and broadband systems. The resulting GRIN ARC designs are then compared with homogeneous ARCs, which are found to present a trade-off between design complexity and anti-reflective performance. This study considers a flattened converging lens based on a biconvex-to-flat mapping [13]. This system provides a suitable case study, where the behavior is welldocumented. The biconvex lens is assumed to be silicon, which at the design wavelength of λ0 3.5 μm corresponds to n 3.43. Specifically, we consider a f/1 microlens system with an aperture that is 15λ0 and a thickness of 2.14λ0 . Its performance was evaluated using a 2D fullwave simulation in COMSOL Multiphysics 4.3b. The lens 0146-9592/15/112521-04$15.00/0
is illuminated by a normally incident uniform 42-μm diameter beam at a wavelength of 3.5 μm, and the reflectance is calculated using S-parameter analysis. The resulting normalized electric field distribution is shown in Fig. 1(a). While mismatch between the lens material and free space results in a reflectance of 30% at the design wavelength, these reflections can be mitigated by incorporating ARCs. Quarter-wave coatings are applied to the lens surfaces, and the performance of the resulting system is shown in Fig. 1(b). The coatings reduce the reflectance at the design wavelength to approximately 0%. This single-layer coating provides a narrow bandwidth of 29% with respect to wavelength corresponding to a reflectance of less than 2%, as shown in Fig. 2. Using QCTO, the conventional bi-convex lens is mapped into a flat GRIN lens via a distortion of a virtual coordinate grid into a physical domain. The required material parameters to realize the equivalent distortion are then extracted, a process that is illustrated in Fig. 3. QCTO significantly relaxes the material requirements produced using traditional TO mappings, enabling
Fig. 1. Normalized electric field of a conventional bi-convex lens (a) without an ARC and (b) with an ARC. © 2015 Optical Society of America
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Fig. 2. Reflectance of a conventional bi-convex lens with single-layer quarter-wave ARCs.
Fig. 3. Bi-convex to flat transformation. (a) A virtual coordinate grid is mapped into a physical geometry. (b) The resulting equivalent index distribution of the flattened bi-convex lens.
designers to realize nearly isotropic, non-magnetic devices as shown by the refractive index distribution in Fig. 3(b). The flattened GRIN lens again features material discontinuities along its surfaces. Figure 4(a) shows that the GRIN lens has the same focusing functionality as the bi-convex lens in Fig. 1(a) and also suffers from significant reflections: specifically, a reflectance of 50%.
Fig. 4. Normalized electric field of a GRIN flattened bi-convex lens (a) without an ARC and (b) with a homogeneous ARC.
Fig. 5. Parametric study of reflectance of flattened lens system with a homogeneous ARC: reflectance versus ARC index (main), discretized lens interface index distribution (left inset), and histogram of the number of instances of each index value (right inset).
This lens features greater reflections than the bi-convex lens because the higher indices facilitate a greater mismatch between the lens region and free space. Nevertheless, as shown in Fig. 4(b), the reflections from the GRIN lens can also be mitigated by incorporating ARC layers to the lens surfaces. Possibly the simplest anti-reflective design is to coat the GRIN flat lens with homogeneous quarter-wave coatings. Since it is unclear which index to reference in the GRIN profile, a parametric study was performed to determine the optimal ARC index and corresponding quarter-wave thickness to achieve peak performance. Figure 5 shows that a homogeneous index of 2.06 provides the lowest reflectivity. To characterize this behavior in a more quantitative manner, consider the GRIN profile along the surface of the flat lens, shown in the left inset of Fig. 5. When observing this profile as a random variable, the optimal ARC index (n 2.06) corresponds to n 4.24, which is the mode of the discrete probability distribution function of the set, as depicted in the right inset of Fig. 5. The location of this reference index is slightly radially off-center in the lens. The performance of the resulting system is depicted in Fig. 4(b). These homogeneous ARCs reduce the system reflectance to approximately 0% at the design wavelength, but, as with the biconvex lens system, this coating provides a narrow operating bandwidth: here of 24% with reference to a reflectance below 2%, as shown in Fig. 6. This system does not perform as well as the ARC for the conventional lens system. This is because the homogeneous ARC does not provide optimal matching across the entire gradient distribution of the lens interface.
Fig. 6. Reflectance of a flattened GRIN lens with a homogeneous ARC, radial GRIN ARC, and an embedded GRIN ARC.
June 1, 2015 / Vol. 40, No. 11 / OPTICS LETTERS
Hence, incorporating a GRIN distribution into the ARC should enhance the performance of the system. Another design method considers implementing a radial GRIN distribution to the ARC layer. This GRIN ARC distribution corresponds to the geometric mean of free space index and the index distribution along the surface of the lens. The resulting coating realizes a radial GRIN, but remains homogeneous in the axial direction. The thickness of this radial GRIN ARC is uniform and the same thickness as the optimized homogeneous ARC, providing optimal performance and maintaining the flattened geometry. This system is also capable of achieving approximately 0% reflectance at the design wavelength, but does not feature significant performance improvement over the homogeneous ARC, as evident in Fig. 6. In fact, the performance is nearly identical, and the system realizes the same bandwidth as the purely homogeneous case. Alternatively, the ARC design can be executed simultaneously with the QCTO mapping. This method produces a single flattened GRIN region that performs the desired operation whilst mitigating reflections. As shown in Fig. 7(a), a homogeneous lens with homogeneous ARCs is concurrently flattened into a single inhomogeneous lens system with a GRIN lens profile and an embedded GRIN ARC. The resulting refractive index profiles are displayed in Fig. 7(b), where the GRIN distributions corresponding to the lens and the ARC layers are exploded to clearly discern the gradients. This system is also capable of achieving a reflectance of roughly 0% at the design wavelength. The primary performance distinction between this design and the previous implementations is its bandwidth. Here, the system again realizes a 24% bandwidth corresponding to a reflectance below 2%, as shown in Fig. 6. However, when considering a reflectance below 4%, this configuration outperforms the other designs. The extra bandwidth is acquired at the expense of additional complexity in the ARC profile. That is, the QCTO-derived index varies in both the radial and axial directions. However, probably most promising is the fact that this methodology enables optical designers to streamline the rich field of filter theory and ARC design directly into the QCTO process. That is, the system can be designed entirely using traditional methods and can later be incorporated as a single system, into the
Fig. 7. (a) QCTO mapping of a bi-convex lens with ARC coatings and (b) the resulting refractive index distribution of the lens system with embedded ARC layers.
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Fig. 8. Reflectance of a conventional bi-convex lens with a 3-layer binomial ARC.
QCTO mapping. Thus far, single-layer coatings have been shown to mitigate reflections over a relatively narrow bandwidth. However, the same design concepts are easily extended to multi-layer coatings for broadband operation. Next, the same conventional bi-convex lens is considered for broadband coatings. Although there are a plethora of techniques to realize broadband matching, here we consider a 3-layer binomial ARC design [9]. For a biconvex lens with an index of 3.43, the binomial approximation provides the corresponding indices for each layer of 2.64, 1.74, and 1.15. This multi-layer ARC design significantly enhances the bandwidth of the lens system. Fig. 8 depicts that the system achieves a bandwidth of 92% with a reflectance below 2%. Using the same process described for the single layer case, consider first homogeneous ARCs for a broadband GRIN system. Referencing the same index as in the single layer case, n 4.24, the binomial approximation is applied to the GRIN lens. The homogeneous indices of the resulting ARC layers are 3.00, 1.87, and 1.17, each with corresponding quarter-wave thicknesses. The multilayer homogeneous ARC greatly increases the operating bandwidth compared to the single-layer case. Here, the system achieves a bandwidth of 87%, as shown in Fig. 9. As with the narrowband design, the GRIN lens system with homogeneous ARCs does not perform as well as the conventional bi-convex lens system. The 3-layer ARC design can be incorporated with the QCTO mapping in the same manner as the single-layer case. This mapping starts with a homogeneous bi-convex lens with a 3-layer homogeneous ARC. The lens system is
Fig. 9. Reflectance of a flattened GRIN lens with a 3-layer homogeneous binomial ARC and an embedded 3-layer GRIN binomial ARC.
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predictable manner. However, QCTO-enabled embeddedGRIN ARCs yield the same functionality over a broader bandwidth. The choice of which solution to implement is largely a tradeoff between performance criteria and design/fabrication complexity. Most notably, it was also demonstrated that these design methodologies for narrowband solutions are also easily extended to multi-layer broadband solutions. This indicates that the rich fields of ARC design and filter theory can be leveraged in conjunction with QCTO to realize GRIN lenses, as well as more general QCTO systems with possible applications in integrated photonics components, with mitigated reflections. This work was supported in part by DARPA/AFRL under contract number FA8650-12-C-7225. Fig. 10. Refractive index profile of the GRIN lens with an embedded 3-layer GRIN binomial ARC.
then flattened, resulting in a single GRIN region depicted in Fig. 10. As with the single-layer design, the ARC layers possess gradients in both the radial and axial directions. Each ARC layer is displayed in a separate color map to exaggerate the index distribution. This system also significantly enhances the bandwidth compared to the single-layer implementation. From Fig. 9, we can see that this system achieves a bandwidth of 94% corresponding to a reflectance below 2%. As with the single-layer case, this implementation features a larger bandwidth than the homogeneous ARC design. However, this extra bandwidth is realized at the expense of increased material composition complexity. In this work, we have demonstrated that it is possible to develop ARCs for GRIN systems. Several ARC design paradigms were proposed and studied to eliminate the reflections of a flattened bi-convex lens. It was found that traditional homogeneous ARCs can still effectively mitigate surface reflections and can be designed in a
References 1. J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006). 2. U. Leonhardt, Science 312, 1777 (2006). 3. D.-H. Kwon and D. H. Werner, IEEE Antennas Propag. Mag. 52, 24 (2010). 4. D. H. Werner and D.-H. Kwon, Transformation Electromagnetics and Metamaterials: Fundamental Principles and Applications (Springer, 2014). 5. N. Kundtz and D. R. Smith, Nat. Mater. 9, 129 (2010). 6. H. F. Ma and T. J. Cui, Nat. Commun. 1, 124 (2010). 7. H. K. Raut, V. A. Ganesh, A. S. Nair, and S. Ramakrishna, Energy Environ. Sci. 4, 3779 (2011). 8. M. Chen, H. Chang, A. S. P. Chang, S.-Y. Lin, J.-Q. Xi, and E. F. Schubert, Appl. Opt. 46, 6533 (2007). 9. C. A. Balanis, Advanced Engineering Electromagnetics, 2nd ed. (John Wiley & Sons, 2012). 10. W. H. Southwell, Opt. Lett. 8, 584 (1983). 11. K. Forberich, G. Dennler, M. C. Scharber, K. Hingerl, T. Fromherz, and C. J. Brabec, Thin Solid Films 516, 7167 (2008). 12. Y. Li, J. Zhang, S. Zhu, H. Dong, Z. Wang, Z. Sun, J. Guo, and B. Yang, J. Mater. Chem. 19, 1806 (2009). 13. W. Tang, C. Argyropoulos, E. Kallos, W. Song, and Y. Hao, IEEE Trans. Antennas Propag. 58, 3795 (2010).
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Transformation-optics-inspired anti-reflective coating design for gradient index lenses Kenneth L. Morgan,* Donovan E. Brocker, Sawyer D. Campbell, Douglas H. Werner, and Pingjuan L. Werner Department of Electrical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802, USA *Corresponding author: [email protected] Received April 7, 2015; revised May 5, 2015; accepted May 6, 2015; posted May 8, 2015 (Doc. ID 237662); published May 21, 2015 Recent developments in transformation optics have led to burgeoning research on gradient index lenses for novel optical systems. Such lenses hold great potential for the advancement of complex optics for a wide range of applications. Despite the plethora of literature on gradient index lenses, previous works have not yet considered the application of anti-reflective coatings to these systems. Reducing system reflections is crucial to the development of this technology for highly sensitive optical applications. Here, we present effective anti-reflective-coating designs for gradient index lens systems. Conventional anti-reflective-design methodologies are leveraged in conjunction with transformation optics to develop coatings that significantly reduce reflections of a flat gradient index lens. Finally, the resulting gradient-index anti-reflective coatings are compared and contrasted with conventional homogeneous anti-reflective coatings. © 2015 Optical Society of America OCIS codes: (050.1755) Computational electromagnetic methods; (080.2710) Inhomogeneous optical media; (080.3620) Lens system design; (110.2760) Gradient-index lenses; (310.1210) Antireflection coatings. http://dx.doi.org/10.1364/OL.40.002521
The advent of transformation optics (TO) brought forth a renaissance of gradient-index (GRIN) designs [1–4]. However, the TO-based solutions generally require anisotropic and magnetic materials that make realizing such systems often infeasible. Quasi-conformal transformation optics (QCTO) relaxes the material requirements predicted by the more general TO approach, enabling the development of isotropic dielectric devices [5,6]. In particular, QCTO techniques have piqued interest in designing GRIN lens systems, where all-dielectric materials are desired. These GRIN systems hold great potential for reducing the size, weight, and power (SWaP) of optical devices by introducing additional design flexibility. However, anti-reflective coatings (ARCs) for such GRIN systems have not yet been investigated, leaving a crucial gap in the adaptation of the technology, since often highperformance optics necessitate ARCs to mitigate reflection losses. ARC design is well-established in literature for homogeneous lens systems [7,8], including a wide variety of methodologies extending beyond the narrowband quarter-wave coating [9], to encompass multi-layer [7], GRIN [8,10], and nature-inspired coatings [11,12] that are capable of achieving wideband solutions. Here, we demonstrate ARC designs to reduce reflection losses of GRIN lenses. In particular, conventional ARC design techniques are leveraged in conjunction with QCTO to develop coatings for narrowband and broadband systems. The resulting GRIN ARC designs are then compared with homogeneous ARCs, which are found to present a trade-off between design complexity and anti-reflective performance. This study considers a flattened converging lens based on a biconvex-to-flat mapping [13]. This system provides a suitable case study, where the behavior is welldocumented. The biconvex lens is assumed to be silicon, which at the design wavelength of λ0 3.5 μm corresponds to n 3.43. Specifically, we consider a f/1 microlens system with an aperture that is 15λ0 and a thickness of 2.14λ0 . Its performance was evaluated using a 2D fullwave simulation in COMSOL Multiphysics 4.3b. The lens 0146-9592/15/112521-04$15.00/0
is illuminated by a normally incident uniform 42-μm diameter beam at a wavelength of 3.5 μm, and the reflectance is calculated using S-parameter analysis. The resulting normalized electric field distribution is shown in Fig. 1(a). While mismatch between the lens material and free space results in a reflectance of 30% at the design wavelength, these reflections can be mitigated by incorporating ARCs. Quarter-wave coatings are applied to the lens surfaces, and the performance of the resulting system is shown in Fig. 1(b). The coatings reduce the reflectance at the design wavelength to approximately 0%. This single-layer coating provides a narrow bandwidth of 29% with respect to wavelength corresponding to a reflectance of less than 2%, as shown in Fig. 2. Using QCTO, the conventional bi-convex lens is mapped into a flat GRIN lens via a distortion of a virtual coordinate grid into a physical domain. The required material parameters to realize the equivalent distortion are then extracted, a process that is illustrated in Fig. 3. QCTO significantly relaxes the material requirements produced using traditional TO mappings, enabling
Fig. 1. Normalized electric field of a conventional bi-convex lens (a) without an ARC and (b) with an ARC. © 2015 Optical Society of America
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Fig. 2. Reflectance of a conventional bi-convex lens with single-layer quarter-wave ARCs.
Fig. 3. Bi-convex to flat transformation. (a) A virtual coordinate grid is mapped into a physical geometry. (b) The resulting equivalent index distribution of the flattened bi-convex lens.
designers to realize nearly isotropic, non-magnetic devices as shown by the refractive index distribution in Fig. 3(b). The flattened GRIN lens again features material discontinuities along its surfaces. Figure 4(a) shows that the GRIN lens has the same focusing functionality as the bi-convex lens in Fig. 1(a) and also suffers from significant reflections: specifically, a reflectance of 50%.
Fig. 4. Normalized electric field of a GRIN flattened bi-convex lens (a) without an ARC and (b) with a homogeneous ARC.
Fig. 5. Parametric study of reflectance of flattened lens system with a homogeneous ARC: reflectance versus ARC index (main), discretized lens interface index distribution (left inset), and histogram of the number of instances of each index value (right inset).
This lens features greater reflections than the bi-convex lens because the higher indices facilitate a greater mismatch between the lens region and free space. Nevertheless, as shown in Fig. 4(b), the reflections from the GRIN lens can also be mitigated by incorporating ARC layers to the lens surfaces. Possibly the simplest anti-reflective design is to coat the GRIN flat lens with homogeneous quarter-wave coatings. Since it is unclear which index to reference in the GRIN profile, a parametric study was performed to determine the optimal ARC index and corresponding quarter-wave thickness to achieve peak performance. Figure 5 shows that a homogeneous index of 2.06 provides the lowest reflectivity. To characterize this behavior in a more quantitative manner, consider the GRIN profile along the surface of the flat lens, shown in the left inset of Fig. 5. When observing this profile as a random variable, the optimal ARC index (n 2.06) corresponds to n 4.24, which is the mode of the discrete probability distribution function of the set, as depicted in the right inset of Fig. 5. The location of this reference index is slightly radially off-center in the lens. The performance of the resulting system is depicted in Fig. 4(b). These homogeneous ARCs reduce the system reflectance to approximately 0% at the design wavelength, but, as with the biconvex lens system, this coating provides a narrow operating bandwidth: here of 24% with reference to a reflectance below 2%, as shown in Fig. 6. This system does not perform as well as the ARC for the conventional lens system. This is because the homogeneous ARC does not provide optimal matching across the entire gradient distribution of the lens interface.
Fig. 6. Reflectance of a flattened GRIN lens with a homogeneous ARC, radial GRIN ARC, and an embedded GRIN ARC.
June 1, 2015 / Vol. 40, No. 11 / OPTICS LETTERS
Hence, incorporating a GRIN distribution into the ARC should enhance the performance of the system. Another design method considers implementing a radial GRIN distribution to the ARC layer. This GRIN ARC distribution corresponds to the geometric mean of free space index and the index distribution along the surface of the lens. The resulting coating realizes a radial GRIN, but remains homogeneous in the axial direction. The thickness of this radial GRIN ARC is uniform and the same thickness as the optimized homogeneous ARC, providing optimal performance and maintaining the flattened geometry. This system is also capable of achieving approximately 0% reflectance at the design wavelength, but does not feature significant performance improvement over the homogeneous ARC, as evident in Fig. 6. In fact, the performance is nearly identical, and the system realizes the same bandwidth as the purely homogeneous case. Alternatively, the ARC design can be executed simultaneously with the QCTO mapping. This method produces a single flattened GRIN region that performs the desired operation whilst mitigating reflections. As shown in Fig. 7(a), a homogeneous lens with homogeneous ARCs is concurrently flattened into a single inhomogeneous lens system with a GRIN lens profile and an embedded GRIN ARC. The resulting refractive index profiles are displayed in Fig. 7(b), where the GRIN distributions corresponding to the lens and the ARC layers are exploded to clearly discern the gradients. This system is also capable of achieving a reflectance of roughly 0% at the design wavelength. The primary performance distinction between this design and the previous implementations is its bandwidth. Here, the system again realizes a 24% bandwidth corresponding to a reflectance below 2%, as shown in Fig. 6. However, when considering a reflectance below 4%, this configuration outperforms the other designs. The extra bandwidth is acquired at the expense of additional complexity in the ARC profile. That is, the QCTO-derived index varies in both the radial and axial directions. However, probably most promising is the fact that this methodology enables optical designers to streamline the rich field of filter theory and ARC design directly into the QCTO process. That is, the system can be designed entirely using traditional methods and can later be incorporated as a single system, into the
Fig. 7. (a) QCTO mapping of a bi-convex lens with ARC coatings and (b) the resulting refractive index distribution of the lens system with embedded ARC layers.
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Fig. 8. Reflectance of a conventional bi-convex lens with a 3-layer binomial ARC.
QCTO mapping. Thus far, single-layer coatings have been shown to mitigate reflections over a relatively narrow bandwidth. However, the same design concepts are easily extended to multi-layer coatings for broadband operation. Next, the same conventional bi-convex lens is considered for broadband coatings. Although there are a plethora of techniques to realize broadband matching, here we consider a 3-layer binomial ARC design [9]. For a biconvex lens with an index of 3.43, the binomial approximation provides the corresponding indices for each layer of 2.64, 1.74, and 1.15. This multi-layer ARC design significantly enhances the bandwidth of the lens system. Fig. 8 depicts that the system achieves a bandwidth of 92% with a reflectance below 2%. Using the same process described for the single layer case, consider first homogeneous ARCs for a broadband GRIN system. Referencing the same index as in the single layer case, n 4.24, the binomial approximation is applied to the GRIN lens. The homogeneous indices of the resulting ARC layers are 3.00, 1.87, and 1.17, each with corresponding quarter-wave thicknesses. The multilayer homogeneous ARC greatly increases the operating bandwidth compared to the single-layer case. Here, the system achieves a bandwidth of 87%, as shown in Fig. 9. As with the narrowband design, the GRIN lens system with homogeneous ARCs does not perform as well as the conventional bi-convex lens system. The 3-layer ARC design can be incorporated with the QCTO mapping in the same manner as the single-layer case. This mapping starts with a homogeneous bi-convex lens with a 3-layer homogeneous ARC. The lens system is
Fig. 9. Reflectance of a flattened GRIN lens with a 3-layer homogeneous binomial ARC and an embedded 3-layer GRIN binomial ARC.
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predictable manner. However, QCTO-enabled embeddedGRIN ARCs yield the same functionality over a broader bandwidth. The choice of which solution to implement is largely a tradeoff between performance criteria and design/fabrication complexity. Most notably, it was also demonstrated that these design methodologies for narrowband solutions are also easily extended to multi-layer broadband solutions. This indicates that the rich fields of ARC design and filter theory can be leveraged in conjunction with QCTO to realize GRIN lenses, as well as more general QCTO systems with possible applications in integrated photonics components, with mitigated reflections. This work was supported in part by DARPA/AFRL under contract number FA8650-12-C-7225. Fig. 10. Refractive index profile of the GRIN lens with an embedded 3-layer GRIN binomial ARC.
then flattened, resulting in a single GRIN region depicted in Fig. 10. As with the single-layer design, the ARC layers possess gradients in both the radial and axial directions. Each ARC layer is displayed in a separate color map to exaggerate the index distribution. This system also significantly enhances the bandwidth compared to the single-layer implementation. From Fig. 9, we can see that this system achieves a bandwidth of 94% corresponding to a reflectance below 2%. As with the single-layer case, this implementation features a larger bandwidth than the homogeneous ARC design. However, this extra bandwidth is realized at the expense of increased material composition complexity. In this work, we have demonstrated that it is possible to develop ARCs for GRIN systems. Several ARC design paradigms were proposed and studied to eliminate the reflections of a flattened bi-convex lens. It was found that traditional homogeneous ARCs can still effectively mitigate surface reflections and can be designed in a
References 1. J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006). 2. U. Leonhardt, Science 312, 1777 (2006). 3. D.-H. Kwon and D. H. Werner, IEEE Antennas Propag. Mag. 52, 24 (2010). 4. D. H. Werner and D.-H. Kwon, Transformation Electromagnetics and Metamaterials: Fundamental Principles and Applications (Springer, 2014). 5. N. Kundtz and D. R. Smith, Nat. Mater. 9, 129 (2010). 6. H. F. Ma and T. J. Cui, Nat. Commun. 1, 124 (2010). 7. H. K. Raut, V. A. Ganesh, A. S. Nair, and S. Ramakrishna, Energy Environ. Sci. 4, 3779 (2011). 8. M. Chen, H. Chang, A. S. P. Chang, S.-Y. Lin, J.-Q. Xi, and E. F. Schubert, Appl. Opt. 46, 6533 (2007). 9. C. A. Balanis, Advanced Engineering Electromagnetics, 2nd ed. (John Wiley & Sons, 2012). 10. W. H. Southwell, Opt. Lett. 8, 584 (1983). 11. K. Forberich, G. Dennler, M. C. Scharber, K. Hingerl, T. Fromherz, and C. J. Brabec, Thin Solid Films 516, 7167 (2008). 12. Y. Li, J. Zhang, S. Zhu, H. Dong, Z. Wang, Z. Sun, J. Guo, and B. Yang, J. Mater. Chem. 19, 1806 (2009). 13. W. Tang, C. Argyropoulos, E. Kallos, W. Song, and Y. Hao, IEEE Trans. Antennas Propag. 58, 3795 (2010).
