Topology-optimized Dual-Polarization Dirac Cones

TL;DR

This paper uses topology optimization to design photonic crystal structures with dual-polarization and dual-wavelength Dirac cones, enabling advanced control over light propagation in periodic media.

Contribution

It introduces a large-scale computational approach for inverse design of photonic Dirac cones in various crystal geometries with different symmetries.

Findings

Designs of photonic crystals with Dirac cones at different Brillouin zone points.

Demonstration of dual-polarization and dual-wavelength Dirac cones.

Potential applications in topological photonics and zero-index media.

Abstract

We apply a large-scale computational technique, known as topology optimization, to the inverse design of photonic Dirac cones. In particular, we report on a variety of photonic crystal geometries, realizable in simple isotropic dielectric materials, which exhibit dual-polarization and dual-wavelength Dirac cones. We demonstrate the flexibility of this technique by designing photonic crystals of different symmetry types, such as ones with four-fold and six-fold rotational symmetry, which possess Dirac cones at different points within the Brillouin zone. The demonstrated and related optimization techniques could open new avenues to band-structure engineering and manipulating the propagation of light in periodic media, with possible applications in exotic optical phenomena such as effective zero-index media and topological photonics.

Full text

Dirac cones (DC), or conical dispersions, have received broad attention due to their special properties affecting light transport in photonic systems, such as effective zero-index behavior Chan2011 ; moitra2013realization ; li2015chip ; Hajian2016 ; He2016 ; Engheta2017 ; Kita2017 , exceptional points zhen2015 ; lin2016inverse , photonic Zitterbewegung zhang2008observing , and topologically protected states lu2014topological ; khanikaev2013photonic . So far, DCs have been primarily studied in simple geometries based on circular pillars or air holes on a periodic lattice Chan2011 ; moitra2013realization ; li2015chip ; Vulis2016chip ; He2016 ; Kita2017 . One exception is our previous work lin2016inverse , which exploited topology optimization (TO) techniques to demonstrate higher-order DCs (precursors to exceptional points  zhen2015 ; lin2016inverse ) in complex structures. TO, which was first proposed more than a decade ago sigmund2003 , employs gradient-based algorithms to efficiently handle a very large design space, considering every pixel or voxel as a degree of freedom (DOF) in an extensive 2D or 3D computational domain Jensen11 . Recently, inverse-designed materials based on TO have been utilized to improve the performance of optical devices such as mode splitters, de-multiplexers, and wavelength converters Jensen11 ; liang2013formulation ; piggott2015inverse ; lu2013nanophotonic ; shen2015integrated ; lin2016cavity . Here, we apply TO toward the design of unprecedented dispersion features in photonic crystals (PhC), namely, two overlapping DCs with dual polarizations (DPDC): one having transverse magnetic polarization ($\mathbf{H}\cdot\mathbf{\hat{z}}=0$) and the other having transverse electric polarization ($\mathbf{E}\cdot\mathbf{\hat{z}}=0$). We show that if designed at the $\mathbf{\Gamma}$ point of the Brillouin zone, such PhCs exhibit effective zero-index behavior. Furthermore, we demonstrate DPDCs at the $\mathbf{K}$ point of a hexagonal PhC, which has implications for all-dielectric topological photonics khanikaev2013photonic ; pti3d16 .

Recent years have witnessed an exciting quest for exotic composite materials along with unusual states of matter involving enhanced optical, mechanical, and quantum properties wang2016lightweight ; drozdov2015conventional ; qin2017mechanics ; wong2016lasing . However, there has been comparatively less effort devoted to discovering unconventional structures that can enhance the functionality of ordinary materials, such as ubiquitous low-loss isotropic dielectrics. Our work represents an effort to leverage the capabilities of established but under-utilized inverse design tools to uncover increased functionalities for simple dielectrics.

Dual-polarization Dirac Cones.— Power emitted by a time-harmonic current source $\mathbf{J}\sim e^{i\omega t}$, proportional to the local density of states (LDOS), offers a convenient optimization framework for designing eigenmodes at a given frequency $\omega$. This follows the well-known principle that emitted power, f(\mathbf{E},\mathbf{J},\omega;\epsilon)=-\mathrm{Re}\Big{[}\int\mathbf{J}^{*}\cdot\mathbf{E}~{}d\mathbf{r}\Big{]}, is maximized when the source couples to a long-lived resonance Liang13 . Here, the electric field $\mathbf{E}$ is simply the solution of the steady-state Maxwell equation, $\nabla\times{1\over\mu}\nabla\times\mathbf{E}-\omega^{2}\epsilon(\mathbf{r})\mathbf{E}=i\omega\mathbf{J}$. The goal of TO is to discover the dielectric profile $\epsilon(\mathbf{r})$ that maximizes $f$ for any given $\mathbf{J}$ and $\omega$. In what follows, we judiciously choose $\mathbf{J(\mathbf{r})}$ and the symmetries of the unit cell to construct PhCs with a variety of intriguing spectral features (see supplemental material for details). In particular, we apply TO to design six accidentally degenerate modes of monopolar (M), dipolar (D) and quadrupolar (Q) profiles (inset of Fig. 1) that transform according to the $A$ and $E$ irreducible representations of the $C_{4v}$ point group, and which, in turn, give rise to conical dispersions in the vicinity of their degeneracy Sakoda12 ; mei2012first . We emphasize that designing such a six-fold degeneracy poses a significant challenge for conventional design and even for sophisticated, heuristic optimization algorithms such as particle swarms, simulated annealing, or genetic algorithms anneal ; genetic ; swarm , but can be susceptible to efficient gradient-based TO techniques, in combination with a proper problem formulation.

Figure 1 shows a topology-optimized PhC unit cell and its associated band structure, which exhibits two overlapping DCs at the $\Gamma$ point, one with transverse electric (TE) and the other with transverse magnetic (TM) polarization. Note that the DC in our design is the so-called generalized DC typically characterized by the presence of an extra flat band Dóra2011 . Within the DPDC, TM Dirac bands are formed by the degeneracy of one monopolar (M) and two dipolar (D) modes, whereas TE Dirac bands are formed by the degeneracy of two dipolar (D) and one quadrupolar (Q) modes. The optimized structure consists of high dielectric regions ($\epsilon_{\rm r}=5.5$), typical of common materials such as silicon nitride or titania, in a background of air ($\epsilon_{\rm r}=1$). Intermediate permittivities, $\epsilon_{\rm r}\in(1,5.5)$, are also seen as a result of fine-tuning the necessary modal frequencies to ensure a perfect overlap. The resulting gray-scale PhC has altogether six DPDC modes whose frequencies are degenerate to within $0.1\%$, an accuracy limited only by numerical resolution. Here, we note that in a few initial rounds of optimization, we deliberately optimize for a TM quadrupole at $f_{\rm TMQ}\sim 0.75f_{\rm D}$ and a TE monopole at $f_{\rm TEM}\sim 1.25f_{\rm D}$ in addition to the six degenerate modes forming the DC. We find that such a procedure for “mode separation” helps ensure well-isolated conical dispersions.

DCs at the center of the Brillouin zone correspond to zero-index behavior when the appropriate homogenization criteria are met Chan2011 ; li2015chip . We perform full-wave FDTD analysis on our DPDC structures and show that they indeed exhibit various zero-index characteristics. One characteristic of a zero-index medium (ZIM) is observed in the so-called “prism” test li2015chip , where plane waves normally incident on a facet of a zero-index prism emerge at right angle from another facet. Alternatively, one can also simulate the complex transmission and reflection coefficients of the zero-index medium, from which effective constitutive parameters can be extracted Chan2011 ; li2015chip . As shown in Figure 1(c), we perform a prism test by illuminating one side of a 45-45-90∘ triangular region made up of DPDC unit cells and then measuring the far-field patterns emerging out of the diagonal (hypotenuse) facet. Note that $\theta$ is the refraction angle between the direction of the emerging beam and the facet normal. Figure 1(d) shows smooth Gaussian beam profiles in the far field with the refraction angles crossing zero around the Dirac-cone wavelengths for both TE and TM polarizations. Index retrievals (Fig. 1e) also confirm the zero-index behavior with the effective index crossing $n=0$ between $1.6$–$1.7~{}\lambda/a$. It must be noted that in our structures, zero-index behavior is only observed for normal incidence; illumination at oblique incident angles excite modes which do not exhibit zero-index behavior, as is the case for most Dirac-cone zero-index media oblique15 .

While Fig. 1 demonstrates a perfect DPDC, the gray-scale dielectric profile poses a significant fabrication challenge (though it may be implemented in the radio frequency regime). Here, we examine a binary regularized version better suited for experimental realization. The modified structure (Fig. 2a) is obtained via two stages: first, we apply threshold projection filters wang2011 during the optimization process to produce a binary design; then, we feed the intermediate binary design into a post-optimization pixel-averaging routine to weed out the fine features and regularize the structure (see also the supplemental material). The associated band structure (Fig. 2b) shows a DPDC albeit with a spoiled overlap due to a small frequency gap of $\sim 1\%$ between the TE and TM DCs. Figure 2(c,d) show the corresponding prism tests and index retrieval analyses. Arguably, the optimally fine-tuned gray-scale structure shows better performance than the completely binary version. This is due to following reasons: for the gray-scale version, the zero-index crossing is perfectly linear and virtually degenerate for both polarizations; for the binary version, the crossings are separated by about $1\%$ and real part of the effective index shows a constant zero value while the imaginary part depicts a bump around the zero crossing, which corresponds to a small bandgap near the Dirac-point frequency li2015chip . Nevertheless, the modified structures clearly feature a range of wavelengths where near-zero-index behavior is observed for both polarizations, which make them realistic candidates for practical applications. We note that an approach to realize DPDCs and polarization-independent zero-index behavior was recently proposed wang2016full , which necessitates the use of complex meta-crystals based on patterning an anisotropic elliptic metamaterial. In contrast, we identify DPDCs by virtue of unconventional geometries that can be imprinted on simple ordinary isotropic dielectrics.

The appearance of complicated features in the DPDC geometry (Fig. 1) can be attributed to numerous stringent conditions imposed upon the optimization process. As noted above, one such condition is the mode separation constraint which pushes certain extraneous modes away from the Dirac degeneracy. We find that relaxing this constraint leads to a simple DPDC structure with regular geometric features (Fig. 3a) although the proximity of an unintentional TM quadrupole mode in the band structure engenders an anti-crossing (aka mode mixing joannopoulos2011photonic ) off the $\Gamma$ point near the Dirac frequency (Fig. 3b). Nevertheless, FDTD analyses of Fig. 3(a) clearly show the near-zero-index behavior (Fig. 3c,d) for both polarizations. Note that the simple “four-hole” structure (Fig. 3a) has a relative permittivity $\epsilon=3.3$, making it suitable for fabrication with common polymer materials.

Dirac Cones at the $\mathbf{K}$ point.— To demonstrate the versatility of our approach, we proceed to design DPDCs based on a hexagonal lattice with symmetry properties distinct from those found on a square lattice. In particular, we focus on the $\mathbf{K}$ point of the Brillouin zone, where two dipolar eigenmodes that transform according to the $E$ irreducible representations of $C_{3v}$ point group form a deterministic DC, i.e, a DC that arises as a consequence of the symmetry of the lattice Sakoda12 ; mei2012first . We show that we can overlap two such DCs, one with TM polarization and the other with TE polarization, thus restoring electromagnetic duality khanikaev2013photonic in the vicinity of the four-fold degenerate Dirac point. Specifically, we employ the LDOS TO formulation to design degenerate TM and TE dipolar modes while imposing $C_{3v}$ symmetry via suitable transformations which ensure the concurrence of the corresponding degenerate partner for each polarization, leading to DPDCs.

Figure 4 shows complex geometries discovered by TO and the corresponding band structure with overlaid TE and TM DCs at $\mathbf{K}$ point. The gap between the two Dirac points is as small as $<0.1\%$, only limited by numerical discretization errors. To our knowledge, this structure is the first proof-of-principle 2D design, based on ordinary isotropic dielectric materials, that hosts overlaid TE/TM DCs at a non-$\mathbf{\Gamma}$ point of a PhC. Moreover, this structure stands in contrast to more sophisticated recent designs using 2D metacrystals khanikaev2013photonic or 3D hexagonal PhC pti3d16 . Since DPDCs at the $\mathbf{K}$ point of a hexagonal lattice are important precursors to non-trivial topological states lu2014topological ; khanikaev2013photonic ; pti3d16 , our method suggests an alternative precursor from which one may realize a so-called photonic topological insulator (PTI). Since our focus here is realizing DPDCs, we will not pursue making a PTI here. However, it is worth mentioning that there are well-known techniques to introduce non-trivial topological bandgaps into DPDCs based on suitable bi-anisotropic perturbations, such as by introducing off-axis propagation $(k_{z}\neq 0)$, by systematic reduction of mirror symmetry, or by modifications that mix TE and TM polarizations while preserving the pseudospin distinction khanikaev2013photonic . Although the TO-discovered geometry might be quite challenging to fabricate due to the existence of pixel-thin hairy features, we note that these features do not indicate a fundamental limitation of our technique but are an artifact of underlying image-transformation steps which impose undue constraints on the optimization process. In the supplement, we discuss such drawbacks as well as possible ways to mitigate them.

Conclusion and remarks.— While some of the optimized designs we have presented might prove challenging, though not impossible, to fabricate at visible or near-infrared frequencies, they can be readily realized at mid- to far-IR as well as microwave frequencies via existing technologies such as computerized machining, 3D printing, laser cutting, additive manufacturing, or two-photon lithography, self-assembly of DNA molecules Borisov98 ; Lewis15 ; Vukusic16 ; Rothemund2006 . Furthermore, thin isolated features which typically beset topology-optimized designs can be removed by a variety of advanced regularization and averaging techniques Jensen11 . The appearance of such features indicates an optimization process that is severely constrained by the design requirements of realizing TE and TM modes with the same modal profile at the same frequency. The fundamental issue underlying such a design is that in a generic structured isotropic 2D medium, TM bands tend to be at lower frequencies than TE bands, breaking the so-called electromagnetic duality. While we have shown that our TO formulation is capable of restoring this duality and finding DPDCs on a 2D lattice, this comes at the expense of a highly irregular structure which needs to be fine-tuned with thin sensitive features. In contrast, we surmise that three-dimensional platforms will offer even greater flexibility. For example, it is known that TM modes tend to experience effectively different index of refraction relative to TE modes in 3D PhC slabs, e.g. depending on whether the PhC geometry consists of holes or pillars joannopoulos2011photonic . In future work, we will consider optimization in full 3D multi-layered geometries, which we expect will open up even more exciting opportunities for new structural designs in the fields of metasurfaces, metamaterials and topological photonics.

Acknowledgements.— The authors thank Philip Camayd-Muñoz and Orad Reshef for discussions. This work was partially supported by the Air Force Office of Scientific Research under Contract No. FA9550-14-1-0389, by the National Science Foundation under Grants No. DMR-1454836 and DMR-1360889, and by the Princeton Center for Complex Materials, a MRSEC supported by NSF Grant No. DMR-1420541. Z. Lin is supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE1144152.