Abstract

Recent studies of optical reflectors as part of the vision apparatus in the eyes of decapod crustaceans revealed assemblies of nanoscale spherulites - spherical core-shell nanoparticles with radial birefringence. Simulations performed on the system highlighted the advantages of optical anisotropy in enhancing the functionality of these structures. So far, calculations of the nanoparticle optical properties have relied on refractive indices obtained using ab-initio calculations. Here we describe a direct measurement of the tangential refractive index of the spherulites, which corresponds to the in-plane refractive index of crystalline isoxanthopterin nanoplatelets. We utilize measurements of scattering spectra of individual spherulites and determine the refractive index by analyzing the spectral signatures of scattering resonances. Our measurements yield a median tangential refractive index of 1.88, which is in reasonable agreement with theoretical predictions. Furthermore, our results indicate that the optical properties of small spherulite assemblies are largely determined by the tangential index.

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

1. Introduction

Light scattering is a ubiquitous phenomenon, that is responsible for our visual perception of the objects around us. Particles much smaller than the wavelength of light, such as molecules, can be considered point dipole scatterers [1,2]. These particles exhibit a characteristic increase in scattering at shorter wavelengths [3,4]. Among other things, this is responsible for the blue colour of the sky. For particles whose size is comparable to the wavelength of light, scattering resonances can occur, which lead to enhanced scattering at certain wavelengths [2]. The spectral position of these resonances is determined by the shape, size and refractive index of the particles. When assembled into three dimensional ordered arrays, such particles can form photonic crystals, which can exhibit photonic stop gaps – bands of frequencies in the electromagnetic spectrum which are not allowed to propagate within the structure in a specific direction [5]. Resonant scatterers can be arranged in two-dimensional patterns to form metasurfaces which can be designed to perform a variety of tasks such as phase modulation and lensing [6,7]. In addition, partially disordered arrays that possess only short-range order, can exhibit phenomena such as non-iridescent structural colour [810].

Most of our ability to engineer the optical response of photonic structures was developed during the last few decades, as a result of developments in materials engineering, and nanofabrication. However, living organisms have used photonic structures to their advantage for millions of years, optimizing their properties in the course of evolution [11,12]. Examples are the light reflectors found in the eyes of aquatic animals, the structural colour developed by birds, insects and certain plants, and the almost ubiquitous iridescence of fish scales [1315]. These structures employ a diverse range of (nanostructured) materials to achieve the desired optical response, including crystals of small organic molecules such as guanine, proteins such as keratin and collagen, and polysaccharides such as cellulose and chitin [12]. To construct reflectors, aquatic organisms typically use crystalline guanine platelets, as their high in-plane refractive index ($n \approx 1.83$) enables better refractive index contrast with water ($n = 1.33$) and consequently, more efficient photonic structures [16].

Recently, studies on decapod crustaceans revealed an intriguing photonic structure. The tapetum layer behind the photosensitive elements in the eye is constructed from a partially disordered opal-like assembly of spherical isoxanthopterin nanoparticles [1719]. Isoxanthopterin belongs to a class of high refractive-index biogenic substances, called pterins, which typically occur as pigments in a variety of organisms [20,21]. In each nanoparticle, crystalline plates of isoxanthopterin, a few nanometers in thickness and a few tens of nanometers in lateral size, are arranged in a lamellar fashion to form hollow spherical shells whose diameter is approximately 400 nm, as illustrated in Fig. 1(a). Calculations based on density functional theory (DFT) revealed that the isoxanthopterin crystal structure gives rise to a very high optical anisotropy ($n_1 = 1.90$, $n_2 = 2.02$, and $n_3 = 1.4$) [22]. In the spherical shells, the crystalline plates are so arranged that the crystal direction corresponding to the lowest refractive index is always along the radial direction. The crystals are not oriented relative to each other in the lateral directions, therefore the two high indices are effectively averaged. As a consequence of the architecture of the particles and of the orientation of the crystals, the spherical structure is characterized by a low refractive index for light polarized in the radial direction and a large refractive index for light polarized tangential to its surface.

 figure: Fig. 1.

Fig. 1. (a) A schematic of the isoxanthopterin spherulites found in decapod crustacean eyes. The structure is characterized by a smaller refractive index $n_r$ along the radial direction (red arrow) and a larger index $n_t$ along the tangential direction (green arrows); (b) Scattering cross section of the spherulite. $Q_{\textrm {sca}}$ is the scattering cross section (all directions). The other curves show the contributions of various scattering multipoles. The prominent peaks result from the TE (magnetic) quadrupole and octupole modes.; (c) The variation of $Q_{\textrm {sca}}$ with the radius of the spherulite, with shell thickness $t = r/2$; (d) The variation of $Q_{\textrm {sca}}$ with the ratio of shell thickness to radius ($t/r$). ; (e) The variation of $Q_{\textrm {sca}}$ with the tangential index ($n_t$). Other parameters in (c),(d) and (e) are the same as in (b). The spectral position of the multipole resonances is strongly dependent on $n_t$ and $r$.

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Such a structure, characterized by a radial and tangential index, is termed a spherulite [23,24]. An arbitrary rotation about the centre of the spherulite does not affect the radial and tangential refractive indices; therefore, a spherulite possesses the same full rotational symmetry as an isotropic sphere. However, the strong optical anisotropy leads to an optical response that is quite different from optically isotropic structures of similar average refractive index. For example, isoxanthopterin spherulites in an aquatic environment have been shown to scatter more efficiently in the backward directions than their isotropic counterparts [17]. In addition, close packed assemblies of these spherulites exhibit gaps in their photonic band structure that are not present in similar assemblies of optically isotropic particles [17,18]. Recent theoretical work also suggests that radial optical anisotropy can enable highly directional scattering and optical cloaking [2527]. Radial anisotropy offers an additional parameter to tune scattering resonances of nanoparticles. These tailored resonances can be used to enhance nonlinear processes such as second harmonic generation [2830].

For the specific case of isoxanthopterin, the theoretical results summarized in the preceding paragraph were obtained using the computed refractive indices. However it is not straightforward to verify these values experimentally. When crystallized from solution, isoxanthopterin crystallizes in a structure that is different from its biogenic form [22]. Moreover, ensemble measurements of suspensions of spherulites are not possible due to the polydispersity of the particles and the presence of other organic matter in extracted samples. This necessitates measuring the refractive index directly from single biogenic particles. Such measurements require a deeper understanding of the scattering properties of spherulites and the ability to quantitatively model the measurements of scattered light, which is the goal of the present work.

Experimental measurements of refractive indices of other biogenic pterin pigments utilized optical absorbance measurements together with a Kramers-Kronig analysis of the spectra, and have been verified using rigorous computational electromagnetic modelling [20,21]. Isoxanthopterin, however is transparent to visible light [19]. This necessitates a more suitable technique to determine refractive index from nanoparticles. In this article, we demonstrate the utility of characterizing scattering resonances of the nanoparticles in determining their refractive index. By studying the spherulitic particles found in the eyes of the prawn Macrobrachium rosenbergii, which are typically about 400 nm in diameter, we demonstrate close agreement between the measured tangential refractive index of the spherulites and calculated values. In addition, our work sheds light on the advantages of using spherulites in self-assembled photonic structures by observing scattering from the smallest possible structure containing more than one spherulite - spherulite dimers.

2. Modelling the optical response of isoxanthopterin spherulites

Spherulites possess the same rotational symmetries as a sphere; therefore light scattered by spherulites exhibits symmetries similar to the case of a homogeneous isotropic sphere. Importantly, scattered fields can be decomposed into transverse electric (TE) and transverse magnetic (TM) multipole components, and these fields can be calculated following the approach of Mie theory [13,31]. The scattered electric field can be expressed in terms of a series of coefficients, $a_l$ and $b_l$, which correspond to the $l^{\textrm {th}}$ TM and TE multipole component of the field, respectively. At a point $P$ with coordinates $(r,\theta ,\phi )$ far away from the scatterer located at the origin, the scattered field is given by

$$\mathbf{E}_{s}\left(r,\theta,\phi\right) = \frac{e^{ik(r-z)}}{-ikr}\mathbf{X}\left(\theta,\phi\right)E_{i},$$
where $E_{i}$ is the incident plane wave field ($x$-polarized), $z = r\cos \theta$ and
$$\mathbf{X}\left(\theta,\phi\right) = S_{2}\left(\theta\right)\cos\phi \hat{\mathbf{e}}_{{\parallel}}\left(\theta,\phi\right) + S_{1}\left(\theta\right)\sin\phi \hat{\mathbf{e}}_{{\perp}}\left(\theta,\phi\right).$$
Here, $S_1$ and $S_2$ are the scattering coefficients along the directions perpendicular and parallel to the scattering plane (a plane containing the origin, $P$, and the propagation vector of the incident light), respectively. From the scattered amplitude, the total scattering cross section is obtained using the following integral over all possible angular coordinates [2].
$$Q_{\textrm{sca}} = \int_{0}^{2\pi}{\int_{0}^{\pi}{\dfrac{\left|\mathbf{X}\right|^{2}}{k^{2}}\sin\theta d\theta d\phi}}$$
In order to obtain the various cross sections, we computed the coefficients $a_l$ and $b_l$ using the method of Hertz-Debye potentials [2,31]. Figure 1(b) shows the typical scattering characteristics of an isoxanthopterin spherulite of outer radius $r = 200 \textrm { nm}$, shell thickness $t = 0.5r = 100 \textrm { nm}$, a hollow core (index $n_c = 1$), tangential index $n_t = 1.96$ and radial index $n_r = 1.4$ placed in vacuum (medium index $n_m = 1$). Upon inspection of the graph of $Q_{\textrm {sca}}$ vs $\lambda _{0}$, one can immediately identify two prominent peaks. These peaks correspond to the TE Quadrupole (TE2) and TE Octupole (TE3) scattering resonances of the particle. The natural frequencies of the particle corresponding to these multipoles, and consequently the position of peaks in the scattering spectrum, depend on the refractive indices of the materials that constitute the particle, and its geometry and dimensions [2]. This dependence can, in turn, be used to extract the refractive properties from the spectral positions of the resonances.

Figure 1(c) shows the calculated variation of the cross section $Q_{\textrm {sca}}$ with $r$. In this calculation, $t$ the shell thickness is kept at half the radius, and the remaining parameters are the same as for Fig. 1(b). The positions of the TE2 and TE3 resonances depend strongly on the radius, as the natural frequencies corresponding to a specific multipole shift to longer wavelengths as the particle size increases. Figure 1(e) illustrates the strong dependence of the TE2 and TE3 peaks in the scattering cross section on the tangential index $n_t$. As in the case of the radius, an increase in $n_t$ shifts each multipole to longer wavelengths. As the TE modes are not associated with any electric field component in the radial direction, their resonance frequencies are unaffected by a change in the radial index $n_r$. An increase in the thickness of the shell, for a fixed radius, does not affect the resonance frequencies for values of $t$ greater than approximately $r/2$. This is illustrated in Fig. 1(d), and is a result of the tendency of the modes to concentrate their energy in the high index shell. At $t = r/2$, the shell accounts for 87.5% of the volume of the spherulite, and increasing the thickness of the shell beyond this value only has a perturbative effect on the modes. For the same reasons, for similar values of $t/r$, deviations of the core index from $n_c = 1$ also do not have a significant effect on the spectral positions of these resonances. These results suggest that with a knowledge of $r$ and $t$, a measurement of the spectral position of the TE resonances can be used to determine $n_t$. Our measurements following this procedure on individual spherulites are described in the following section.

3. Determination of the refractive index

Spherulites extracted from the tapetum of a Macrobrachium rosenbergii specimen were dropped on a TEM finder grid (Formvar and carbon support 200 mesh copper grid, Micro to Nano). Multiple locations on this grid were then imaged at different magnifications using a TEM (Tecnai T12, operated at 120 kV), and the individual particles were numbered and their positions recorded.

Scattering spectra of individual particles were obtained by collecting scattered light in a dark field optical microscope (Zeiss Axio Observer) using a low numerical aperture objective (Zeiss Achroplan 20X, NA 0.45) and oblique illumination with white light [32,33]. The geometry of illumination and collection of scattered light in the dark field microscope is schematically shown in Fig. 2(a). The DF illumination angle is $\Theta_c = 25^{\circ }$. The scattered light was filtered at the Fourier plane by an iris to limit the collection angle to $13.5^{\circ }$ of the scattered light. Optical images were captured using a CMOS camera (Blackfly S USB3, FLIR) and an sCMOS camera (PCO Edge 5.5). The particles were located utilizing a camera placed in the image plane (Edge 5.5, PCO) and their scattering spectrum was then recorded with a confocal fiber-coupled grating spectrometer (Shamrock 303i, Andor).

 figure: Fig. 2.

Fig. 2. Dark field optical setup and TEM correlation.(a) Schematic of the dark-field scattering measurement geometry. Incident light is colored yellow and scattered light is colored green; (b),(c),(d): TEM images taken at different magnifications from a TEM finder grid; (e),(f),(g): DF optical microscope images taken from the same area on the finder grid as show in (b),(c),(d); Note the different lengths of the scale bars in panels (d) and (g).

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For each incident wavelength, the measured spectrum corresponds to the following quantity.

$$Q_{\textrm{DF}} = \iint_{C}{\dfrac{\left|\mathbf{X}\right|^{2}}{k^{2}}\sin\theta d\theta d\phi}$$
The integration over $\theta$ and $\phi$ is carried out over the cone of angles $C$ collected by the microscope. By placing an iris at an intermediate Fourier plane of the microscope, we precisely set the collection angle to $\Theta _c=13.5^{\circ }$. The angle of incidence of the illumination is $\Theta _i=25^{\circ }$.

As the dimensions of the particles are close to the wavelength of visible light, accurate measurements of $r$ and $t$ require electron microscopy. Therefore, spherulites extracted from the specimen were dispersed on a TEM grid and imaged at different magnifications. After TEM imaging, the same TEM grid was imaged in a dark-field optical microscope. To enable efficient correlative measurements, we used a finder grid, in which rows and columns are labelled. Figures 2(b)–(d) are TEM images of the sample dispersed on the grid. Figure 2(b) is a representative TEM image of an engraved section on the finder grid. An optical DF image of the same region is shown in Fig. 2(e). Upon zooming into a region of the grid shown in Figs. 2(c) and (f), individual particles can be resolved. The TEM images allow us to conveniently distinguish the spherical nanoparticles from other particulate matter, and identify particles of high quality (Section 2, Supplement 1). Figure 2(d) shows one of the spherulites shown in Fig. 2(c) at a higher magnification, which allowed us to precisely measure its diameter and shell thickness (Section 4, Supplement 1). Figure 2(g) shows an optical DF image of the same particle as in Fig. 2(d) (not at the same scale). As the size of the particle is close to the resolution limit of the collection objective (numerical aperture 0.45), hardly any detail can be resolved, and the image is similar to the point spread function. The size of this image is matched using collection optics to the diameter of the confocal fiber input of the spectrometer.

Experimental data and analysis for a representative particle are presented in Fig. 3. Figure 3(a) shows a TEM image of the particle, and Fig. 3(b) shows the measured DF spectrum, in which the peaks corresponding to the two multipole resonances can be clearly identified. The two peaks correspond to the TE2 and TE3 scattering resonances described in the previous section. Figure 3(c) shows the scattering cross section and DF scattering spectrum calculated using the theoretical procedure described in the preceding section for $n_t = 1.96$. The experimentally measured spectrum is in good qualitative agreement with the calculations.

 figure: Fig. 3.

Fig. 3. Determination of $n_t$. (a) A TEM image of a representative spherulite. The scalebar is equivalent to 200 nm; (b) Measured DF spectrum for the particle in (a); (c) The normalized cross section ($Q_{\textrm {sca}}$, blue curve) and the DF signal (red curve, scaled by a factor of 10) calculated for a particle of the same dimensions as in (a) using $n_t = 1.96$ and $n_r = 1.4$. The TE resonances are clearly identifiable in the DF spectrum; (d) The variation of the TE2 and TE3 peak positions with $n_t$. Peak positions from (b) and the corresponding values of $n_t$ are marked by dotted lines. We obtain values of 1.83 and 1.89 for $n_t$ from the TE2 and TE3 multipoles, respectively.

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Using the values of $r$ and $t$ obtained from TEM, we used Eq. (4) to calculate the variation of the DF spectrum with $n_t$. The resonant wavelength corresponding to these multipoles depends monotonically on the tangential index $n_t$. This dependence is shown in Fig. 3(d). Using the measured values of the peak positions, the value of $n_t$ can be read off the plot (as illustrated by dotted lines in Fig. 3(d)). Following this procedure, $n_t$ was determined for each particle from the dimensions and spectral positions of scattering resonances (Section 5, Supplement 1). We also performed finite-difference time-domain (FDTD) simulations using a commercial grade solver to ensure that the presence of the carbon and polymer films in the TEM grid do not significantly alter the positions of the Mie resonance peaks (Section 6, Supplement 1) [34]. From an analysis of the spectral position of TE3 resonances, we obtained a median refractive index $n_t \cong 1.89$. From a similar analysis of TE2 resonances we obtained $n_t \cong 1.86$. Since isoxanthopterin exhibits optical absorption peaks in the ultraviolet, we attribute the observed decrease in the estimated $n_t$ with increasing wavelength to material dispersion [19]. Similar trends in dispersion have been observed in other biogenic pterin pigments as well [20].

Our estimated values of $n_t$ are lower than the value predicted using DFT [17,22]. We note that the spectral position of a resonance depends on the distribution of the fields within the particle. Inhomogeneity of the material within the shell can therefore lead to deviations in resonant frequencies of different modes. Consequently this results in different estimated values of $n_t$ from different multipoles of the same particle. Also, the stacking of the isoxanthopterin nanoplatelets that constitute the shell may be imperfect. This can result in a lower effective index due to the presence of a lower refractive index material between the single crystal platelets. Therefore, it is fair to assume that our measurement sets a lower limit in-plane refractive index of a pure isoxanthopterin crystal. Our measured value thus corresponds reasonably well with the theoretically calculated value of 1.96. Notably, this value is indeed higher than other organic materials more commonly used in biogenic photonic structures, such as guanine. This highlights a potential reason for its use in the crustacean tapetum.

4. Spherulite dimers

The biogenic spherulites in the crustacean tapetum are not found isolated, but in dense aggregates. Both the internal resonances of each spherulite and near-field coupling contribute to the optical properties of such assemblies [10,35]. It is instructive to study the effects of aggregates of particles in order to understand emergent optical properties of self assembled photonic structures. The simplest such aggregate, the spherulite dimer, consists of two spherulites in close proximity. The dimer does not possess the rotational symmetry of the spherulite and therefore is expected to exhibit a scattering response that is strongly dependent on the orientation of the dimer relative to the direction of polarization of the incident light. This scattering formally depends on both the radial and tangential refractive indices, and offers a potential route to measuring the radial index $n_r$.

TEM imaging enabled us to identify particle dimers on the grid. One such dimer is illustrated in Fig. 4(a). We measured scattering spectra using the same DF microscope configuration as in the case of single particles with a polarizer introduced in the illumination path before the condenser. We measured the DF spectrum as a function of the polarization angle of the incident light relative to the dimer axis. A typical scattering spectrum measured from the dimer is shown in Fig. 4(b). As expected, the scattering spectrum exhibits a strong polarization dependence.

 figure: Fig. 4.

Fig. 4. Scattering from spherulite dimers. (a) A TEM image of a dimer of spherulites in close proximity. The length of the scalebar is 200 nm.; (b) Measured DF scattering spectra with varying angle between the dimer axis and the orientation of the illumination polarizer; (c, d) Simulated DF spectra for the dimer of the same dimensions as in (a) using $n_t = 1.96$, $n_r = 1.4$ (c) and $n_t = n_r = 1.96$ (d). Prominent qualitative features in the scattering spectrum do not depend significantly on the radial index $n_r$.

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The DF scattering spectrum simulated using FDTD for the same particle dimensions as the dimer in Fig. 4(a) are shown in Fig. 4(c), using $n_t = 1.9$ and $n_r = 1.4$. While we observe a scattering response that is strongly dependent on the polarization, we were unable to demonstrate a quantitative agreement with the results of simulations. This is primarily due to the lower structural quality of the particles constituting the dimer. Moreover, upon comparing the simulations of spectra for a spherulite with $n_r = 1.4$ (Fig. 4(c)) and an isotropic shell with $n_r = n_t = 1.9$ (Fig. 4(d)), we observe only relatively small differences between the simulated scattering response. This hinders an estimation of $n_r$ by directly comparing the simulations to the experimental results. However, the polarization dependence provides inspiration to study possibilities with efficient near-field interactions between spherulites in their assemblies.

Our results strongly suggest that the scattering properties of assemblies of spherulites are dominated by the effects of a high $n_t$, and are only weakly affected by $n_r$. This provides a further clue as to the evolutionary advantage of using isoxanthopterin spherulites in reflector assemblies since this enables to construct a structure with an effective index contrast of approximately 0.6 (between $n_t$ and cytoplasm). More generally, these findings support the hypothesis that the use of spherulites optimally utilizes the high in-plane refractive index of crystalline isoxanthopterin platelets in designing the tapetum reflector.

5. Conclusions

We used a combination of correlative transmission electron microscopy and darkfield optical scattering spectrometry to determine the refractive index of crystalline isoxanthopterin in biogenic nanoscale spherulites. Our results show that the optical response of these spherulites is largely determined by the transverse electric (TE) scattering multipole resonances, which depend only on the tangential refractive index $n_t$. By studying these resonances, we determined the value of $n_t$ to be $1.88$, which is in reasonable agreement with values predicted by DFT calculations. Since the material within the spherulite shell is not densely packed, our measurements likely represent a lower boundary on the value of $n_t$. This indicates that the refractive index of isoxanthopterin is indeed significantly higher than that of guanine, which is the most common material constituting photonic reflectors in aquatic animals. The high refractive index and optical anisotropy of crystalline isoxanthopterin, together with the ability to assemble the material into complex hierarchical photonic structures can potentially lead to several hitherto unexplored possibilities in nanophotonic design.

Funding

Minerva Foundation; Crown Photonics Center; Israel Science Foundation (583/17).

Acknowledgements

The authors thank Prof. Amir Sagi (Ben-Gurion University of the Negev, Dept. of Life Sciences) for providing specimens of M. rosenbergii. The authors also thank the anonymous reviewer for several suggestions that improved this article significantly. D.O. is the incumbent of the Harry Weinrebe Professorial Chair of Laser Physics.

Disclosures

The authors declare no conflicts of interest.

Data availability

Measured refractive indices are tabulated in Section 5 of Supplement 1. Raw data underlying the results presented in this paper may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

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32. C. Sännichsen and A. P. Alivisatos, “Gold Nanorods as Novel Nonbleaching Plasmon-Based Orientation Sensors for Polarized Single-Particle Microscopy,” Nano Lett. 5(2), 301–304 (2005). [CrossRef]  

33. J. Karst, N. H. Cho, H. Kim, H.-E. Lee, K. T. Nam, H. Giessen, and M. Hentschel, “Chiral Scatterometry on Chemically Synthesized Single Plasmonic Nanoparticles,” ACS Nano 13(8), 8659–8668 (2019). [CrossRef]  

34. FDTD Solutions, Lumerical Inc. https://www.lumerical.com/products/fdtd/.

35. V. Hwang, A. B. Stephenson, S. Magkiriadou, J.-G. Park, and V. N. Manoharan, “Effects of multiple scattering on angle-independent structural color in disordered colloidal materials,” Phys. Rev. E 101(1), 012614 (2020). [CrossRef]  

References

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  • |

  1. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999), 7th ed.
  2. C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  3. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330(3), 377–445 (1908).
    [Crossref]
  4. J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, 1941).
  5. E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987).
    [Crossref]
  6. A. I. Kuznetsov, A. E. Miroshnichenko, M. L. Brongersma, Y. S. Kivshar, and B. Luk’yanchuk, “Optically resonant dielectric nanostructures,” Science 354(6314), aag2472 (2016).
    [Crossref]
  7. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science 334(6054), 333–337 (2011).
    [Crossref]
  8. H. Noh, S. F. Liew, V. Saranathan, S. G. J. Mochrie, R. O. Prum, E. R. Dufresne, and H. Cao, “How Noniridescent Colors Are Generated by Quasi-ordered Structures of Bird Feathers,” Adv. Mater. 22(26-27), 2871–2880 (2010).
    [Crossref]
  9. J.-G. Park, S.-H. Kim, S. Magkiriadou, T. M. Choi, Y.-S. Kim, and V. N. Manoharan, “Full-Spectrum Photonic Pigments with Non-iridescent Structural Colors through Colloidal Assembly,” Angew. Chem. Int. Ed. 53(11), 2899–2903 (2014).
    [Crossref]
  10. L. Schertel, L. Siedentop, J. Meijer, P. Keim, C. M. Aegerter, G. J. Aubry, and G. Maret, “The Structural Colors of Photonic Glasses,” Adv. Opt. Mater. 7(15), 1900442 (2019).
    [Crossref]
  11. P. Vukusic and J. R. Sambles, “Photonic structures in biology,” Nature 424(6950), 852–855 (2003).
    [Crossref]
  12. S. Tadepalli, J. M. Slocik, M. K. Gupta, R. R. Naik, and S. Singamaneni, “Bio-Optics and Bio-Inspired Optical Materials,” Chem. Rev. 117(20), 12705–12763 (2017).
    [Crossref]
  13. B. A. Palmer, D. Gur, S. Weiner, L. Addadi, and D. Oron, “The Organic Crystalline Materials of Vision: Structure-Function Considerations from the Nanometer to the Millimeter Scale,” Adv. Mater. 30(41), 1800006 (2018).
    [Crossref]
  14. S. Kinoshita, S. Yoshioka, and J. Miyazaki, “Physics of structural colors,” Rep. Prog. Phys. 71(7), 076401 (2008).
    [Crossref]
  15. S. Kinoshita, Structural Colors in the Realm of Nature (World Scientific, 2008).
  16. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2011).
  17. B. A. Palmer, V. J. Yallapragada, N. Schiffmann, E. M. Wormser, N. Elad, E. D. Aflalo, A. Sagi, S. Weiner, L. Addadi, and D. Oron, “A highly reflective biogenic photonic material from core–shell birefringent nanoparticles,” Nat. Nanotechnol. 15(2), 138–144 (2020).
    [Crossref]
  18. V. J. Yallapragada and D. Oron, “Optical properties of spherulite opals,” Opt. Lett. 44(23), 5860–5863 (2019).
    [Crossref]
  19. B. A. Palmer, A. Hirsch, V. Brumfeld, E. D. Aflalo, I. Pinkas, A. Sagi, S. Rosenne, D. Oron, L. Leiserowitz, L. Kronik, S. Weiner, and L. Addadi, “Optically functional isoxanthopterin crystals in the mirrored eyes of decapod crustaceans,” Proc. Natl. Acad. Sci. 115(10), 2299–2304 (2018).
    [Crossref]
  20. B. D. Wilts, B. Wijnen, H. L. Leertouwer, U. Steiner, and D. G. Stavenga, “Extreme Refractive Index Wing Scale Beads Containing Dense Pterin Pigments Cause the Bright Colors of Pierid Butterflies,” Adv. Opt. Mater. 5(3), 1600879 (2017).
    [Crossref]
  21. M. J. Henze, O. Lind, B. D. Wilts, and A. Kelber, “Pterin-pigmented nanospheres create the colours of the polymorphic damselfly Ischnura elegans,” J. R. Soc. Interface. 16(153), 20180785 (2019).
    [Crossref]
  22. A. Hirsch, B. A. Palmer, A. Ramasubramaniam, P. A. Williams, K. D. M. Harris, B. Pokroy, S. Weiner, L. Addadi, L. Leiserowitz, and L. Kronik, “Structure and Morphology of Light-Reflecting Synthetic and Biogenic Polymorphs of Isoxanthopterin: A Comparison,” Chem. Mater. 31(12), 4479–4489 (2019).
    [Crossref]
  23. H. W. Morse and J. D. H. Donnay, “Spherulite optics,” Am. J. Sci. s5-23(137), 440–461 (1932).
    [Crossref]
  24. A. G. Shtukenberg, Y. O. Punin, E. Gunn, and B. Kahr, “Spherulites,” Chem. Rev. 112(3), 1805–1838 (2012).
    [Crossref]
  25. W. Liu, “Ultra-directional super-scattering of homogenous spherical particles with radial anisotropy,” Opt. Express 23(11), 14734–14743 (2015).
    [Crossref]
  26. H. Barhom, A. A. Machnev, R. E. Noskov, A. Goncharenko, E. A. Gurvitz, A. S. Timin, V. A. Shkoldin, S. V. Koniakhin, O. Y. Koval, M. V. Zyuzin, A. S. Shalin, I. I. Shishkin, and P. Ginzburg, “Biological Kerker Effect Boosts Light Collection Efficiency in Plants,” Nano Lett. 19(10), 7062–7071 (2019).
    [Crossref]
  27. L. Gao, T. H. Fung, K. W. Yu, and C. W. Qiu, “Electromagnetic transparency by coated spheres with radial anisotropy,” Phys. Rev. E 78(4), 046609 (2008).
    [Crossref]
  28. R. Savo, A. Morandi, J. S. Müller, F. Kaufmann, F. Timpu, M. Reig Escalé, M. Zanini, L. Isa, and R. Grange, “Broadband Mie driven random quasi-phase-matching,” Nat. Photonics 14(12), 740–747 (2020).
    [Crossref]
  29. J. H. Bahng, S. Jahani, D. G. Montjoy, T. Yao, N. Kotov, and A. Marandi, “Mie Resonance Engineering in Meta-Shell Supraparticles for Nanoscale Nonlinear Optics,” ACS Nano 14(12), 17203–17212 (2020).
    [Crossref]
  30. S. Jahani, J. H. Bahng, A. Roy, N. Kotov, and A. Marandi, “Confining light in all-dielectric anisotropic metamaterial particles for nano-scale nonlinear optics,” arXiv:2102.13087 [physics] (2021).
  31. J. Roth and M. J. Dignam, “Scattering and extinction cross sections for a spherical particle coated with an oriented molecular layer,” J. Opt. Soc. Am. 63(3), 308–311 (1973).
    [Crossref]
  32. C. Sännichsen and A. P. Alivisatos, “Gold Nanorods as Novel Nonbleaching Plasmon-Based Orientation Sensors for Polarized Single-Particle Microscopy,” Nano Lett. 5(2), 301–304 (2005).
    [Crossref]
  33. J. Karst, N. H. Cho, H. Kim, H.-E. Lee, K. T. Nam, H. Giessen, and M. Hentschel, “Chiral Scatterometry on Chemically Synthesized Single Plasmonic Nanoparticles,” ACS Nano 13(8), 8659–8668 (2019).
    [Crossref]
  34. FDTD Solutions, Lumerical Inc. https://www.lumerical.com/products/fdtd/ .
  35. V. Hwang, A. B. Stephenson, S. Magkiriadou, J.-G. Park, and V. N. Manoharan, “Effects of multiple scattering on angle-independent structural color in disordered colloidal materials,” Phys. Rev. E 101(1), 012614 (2020).
    [Crossref]

2020 (4)

B. A. Palmer, V. J. Yallapragada, N. Schiffmann, E. M. Wormser, N. Elad, E. D. Aflalo, A. Sagi, S. Weiner, L. Addadi, and D. Oron, “A highly reflective biogenic photonic material from core–shell birefringent nanoparticles,” Nat. Nanotechnol. 15(2), 138–144 (2020).
[Crossref]

R. Savo, A. Morandi, J. S. Müller, F. Kaufmann, F. Timpu, M. Reig Escalé, M. Zanini, L. Isa, and R. Grange, “Broadband Mie driven random quasi-phase-matching,” Nat. Photonics 14(12), 740–747 (2020).
[Crossref]

J. H. Bahng, S. Jahani, D. G. Montjoy, T. Yao, N. Kotov, and A. Marandi, “Mie Resonance Engineering in Meta-Shell Supraparticles for Nanoscale Nonlinear Optics,” ACS Nano 14(12), 17203–17212 (2020).
[Crossref]

V. Hwang, A. B. Stephenson, S. Magkiriadou, J.-G. Park, and V. N. Manoharan, “Effects of multiple scattering on angle-independent structural color in disordered colloidal materials,” Phys. Rev. E 101(1), 012614 (2020).
[Crossref]

2019 (6)

J. Karst, N. H. Cho, H. Kim, H.-E. Lee, K. T. Nam, H. Giessen, and M. Hentschel, “Chiral Scatterometry on Chemically Synthesized Single Plasmonic Nanoparticles,” ACS Nano 13(8), 8659–8668 (2019).
[Crossref]

H. Barhom, A. A. Machnev, R. E. Noskov, A. Goncharenko, E. A. Gurvitz, A. S. Timin, V. A. Shkoldin, S. V. Koniakhin, O. Y. Koval, M. V. Zyuzin, A. S. Shalin, I. I. Shishkin, and P. Ginzburg, “Biological Kerker Effect Boosts Light Collection Efficiency in Plants,” Nano Lett. 19(10), 7062–7071 (2019).
[Crossref]

V. J. Yallapragada and D. Oron, “Optical properties of spherulite opals,” Opt. Lett. 44(23), 5860–5863 (2019).
[Crossref]

M. J. Henze, O. Lind, B. D. Wilts, and A. Kelber, “Pterin-pigmented nanospheres create the colours of the polymorphic damselfly Ischnura elegans,” J. R. Soc. Interface. 16(153), 20180785 (2019).
[Crossref]

A. Hirsch, B. A. Palmer, A. Ramasubramaniam, P. A. Williams, K. D. M. Harris, B. Pokroy, S. Weiner, L. Addadi, L. Leiserowitz, and L. Kronik, “Structure and Morphology of Light-Reflecting Synthetic and Biogenic Polymorphs of Isoxanthopterin: A Comparison,” Chem. Mater. 31(12), 4479–4489 (2019).
[Crossref]

L. Schertel, L. Siedentop, J. Meijer, P. Keim, C. M. Aegerter, G. J. Aubry, and G. Maret, “The Structural Colors of Photonic Glasses,” Adv. Opt. Mater. 7(15), 1900442 (2019).
[Crossref]

2018 (2)

B. A. Palmer, D. Gur, S. Weiner, L. Addadi, and D. Oron, “The Organic Crystalline Materials of Vision: Structure-Function Considerations from the Nanometer to the Millimeter Scale,” Adv. Mater. 30(41), 1800006 (2018).
[Crossref]

B. A. Palmer, A. Hirsch, V. Brumfeld, E. D. Aflalo, I. Pinkas, A. Sagi, S. Rosenne, D. Oron, L. Leiserowitz, L. Kronik, S. Weiner, and L. Addadi, “Optically functional isoxanthopterin crystals in the mirrored eyes of decapod crustaceans,” Proc. Natl. Acad. Sci. 115(10), 2299–2304 (2018).
[Crossref]

2017 (2)

B. D. Wilts, B. Wijnen, H. L. Leertouwer, U. Steiner, and D. G. Stavenga, “Extreme Refractive Index Wing Scale Beads Containing Dense Pterin Pigments Cause the Bright Colors of Pierid Butterflies,” Adv. Opt. Mater. 5(3), 1600879 (2017).
[Crossref]

S. Tadepalli, J. M. Slocik, M. K. Gupta, R. R. Naik, and S. Singamaneni, “Bio-Optics and Bio-Inspired Optical Materials,” Chem. Rev. 117(20), 12705–12763 (2017).
[Crossref]

2016 (1)

A. I. Kuznetsov, A. E. Miroshnichenko, M. L. Brongersma, Y. S. Kivshar, and B. Luk’yanchuk, “Optically resonant dielectric nanostructures,” Science 354(6314), aag2472 (2016).
[Crossref]

2015 (1)

2014 (1)

J.-G. Park, S.-H. Kim, S. Magkiriadou, T. M. Choi, Y.-S. Kim, and V. N. Manoharan, “Full-Spectrum Photonic Pigments with Non-iridescent Structural Colors through Colloidal Assembly,” Angew. Chem. Int. Ed. 53(11), 2899–2903 (2014).
[Crossref]

2012 (1)

A. G. Shtukenberg, Y. O. Punin, E. Gunn, and B. Kahr, “Spherulites,” Chem. Rev. 112(3), 1805–1838 (2012).
[Crossref]

2011 (1)

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science 334(6054), 333–337 (2011).
[Crossref]

2010 (1)

H. Noh, S. F. Liew, V. Saranathan, S. G. J. Mochrie, R. O. Prum, E. R. Dufresne, and H. Cao, “How Noniridescent Colors Are Generated by Quasi-ordered Structures of Bird Feathers,” Adv. Mater. 22(26-27), 2871–2880 (2010).
[Crossref]

2008 (2)

S. Kinoshita, S. Yoshioka, and J. Miyazaki, “Physics of structural colors,” Rep. Prog. Phys. 71(7), 076401 (2008).
[Crossref]

L. Gao, T. H. Fung, K. W. Yu, and C. W. Qiu, “Electromagnetic transparency by coated spheres with radial anisotropy,” Phys. Rev. E 78(4), 046609 (2008).
[Crossref]

2005 (1)

C. Sännichsen and A. P. Alivisatos, “Gold Nanorods as Novel Nonbleaching Plasmon-Based Orientation Sensors for Polarized Single-Particle Microscopy,” Nano Lett. 5(2), 301–304 (2005).
[Crossref]

2003 (1)

P. Vukusic and J. R. Sambles, “Photonic structures in biology,” Nature 424(6950), 852–855 (2003).
[Crossref]

1987 (1)

E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987).
[Crossref]

1973 (1)

1932 (1)

H. W. Morse and J. D. H. Donnay, “Spherulite optics,” Am. J. Sci. s5-23(137), 440–461 (1932).
[Crossref]

1908 (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330(3), 377–445 (1908).
[Crossref]

Addadi, L.

B. A. Palmer, V. J. Yallapragada, N. Schiffmann, E. M. Wormser, N. Elad, E. D. Aflalo, A. Sagi, S. Weiner, L. Addadi, and D. Oron, “A highly reflective biogenic photonic material from core–shell birefringent nanoparticles,” Nat. Nanotechnol. 15(2), 138–144 (2020).
[Crossref]

A. Hirsch, B. A. Palmer, A. Ramasubramaniam, P. A. Williams, K. D. M. Harris, B. Pokroy, S. Weiner, L. Addadi, L. Leiserowitz, and L. Kronik, “Structure and Morphology of Light-Reflecting Synthetic and Biogenic Polymorphs of Isoxanthopterin: A Comparison,” Chem. Mater. 31(12), 4479–4489 (2019).
[Crossref]

B. A. Palmer, A. Hirsch, V. Brumfeld, E. D. Aflalo, I. Pinkas, A. Sagi, S. Rosenne, D. Oron, L. Leiserowitz, L. Kronik, S. Weiner, and L. Addadi, “Optically functional isoxanthopterin crystals in the mirrored eyes of decapod crustaceans,” Proc. Natl. Acad. Sci. 115(10), 2299–2304 (2018).
[Crossref]

B. A. Palmer, D. Gur, S. Weiner, L. Addadi, and D. Oron, “The Organic Crystalline Materials of Vision: Structure-Function Considerations from the Nanometer to the Millimeter Scale,” Adv. Mater. 30(41), 1800006 (2018).
[Crossref]

Aegerter, C. M.

L. Schertel, L. Siedentop, J. Meijer, P. Keim, C. M. Aegerter, G. J. Aubry, and G. Maret, “The Structural Colors of Photonic Glasses,” Adv. Opt. Mater. 7(15), 1900442 (2019).
[Crossref]

Aflalo, E. D.

B. A. Palmer, V. J. Yallapragada, N. Schiffmann, E. M. Wormser, N. Elad, E. D. Aflalo, A. Sagi, S. Weiner, L. Addadi, and D. Oron, “A highly reflective biogenic photonic material from core–shell birefringent nanoparticles,” Nat. Nanotechnol. 15(2), 138–144 (2020).
[Crossref]

B. A. Palmer, A. Hirsch, V. Brumfeld, E. D. Aflalo, I. Pinkas, A. Sagi, S. Rosenne, D. Oron, L. Leiserowitz, L. Kronik, S. Weiner, and L. Addadi, “Optically functional isoxanthopterin crystals in the mirrored eyes of decapod crustaceans,” Proc. Natl. Acad. Sci. 115(10), 2299–2304 (2018).
[Crossref]

Aieta, F.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science 334(6054), 333–337 (2011).
[Crossref]

Alivisatos, A. P.

C. Sännichsen and A. P. Alivisatos, “Gold Nanorods as Novel Nonbleaching Plasmon-Based Orientation Sensors for Polarized Single-Particle Microscopy,” Nano Lett. 5(2), 301–304 (2005).
[Crossref]

Aubry, G. J.

L. Schertel, L. Siedentop, J. Meijer, P. Keim, C. M. Aegerter, G. J. Aubry, and G. Maret, “The Structural Colors of Photonic Glasses,” Adv. Opt. Mater. 7(15), 1900442 (2019).
[Crossref]

Bahng, J. H.

J. H. Bahng, S. Jahani, D. G. Montjoy, T. Yao, N. Kotov, and A. Marandi, “Mie Resonance Engineering in Meta-Shell Supraparticles for Nanoscale Nonlinear Optics,” ACS Nano 14(12), 17203–17212 (2020).
[Crossref]

S. Jahani, J. H. Bahng, A. Roy, N. Kotov, and A. Marandi, “Confining light in all-dielectric anisotropic metamaterial particles for nano-scale nonlinear optics,” arXiv:2102.13087 [physics] (2021).

Barhom, H.

H. Barhom, A. A. Machnev, R. E. Noskov, A. Goncharenko, E. A. Gurvitz, A. S. Timin, V. A. Shkoldin, S. V. Koniakhin, O. Y. Koval, M. V. Zyuzin, A. S. Shalin, I. I. Shishkin, and P. Ginzburg, “Biological Kerker Effect Boosts Light Collection Efficiency in Plants,” Nano Lett. 19(10), 7062–7071 (2019).
[Crossref]

Bohren, C.

C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999), 7th ed.

Brongersma, M. L.

A. I. Kuznetsov, A. E. Miroshnichenko, M. L. Brongersma, Y. S. Kivshar, and B. Luk’yanchuk, “Optically resonant dielectric nanostructures,” Science 354(6314), aag2472 (2016).
[Crossref]

Brumfeld, V.

B. A. Palmer, A. Hirsch, V. Brumfeld, E. D. Aflalo, I. Pinkas, A. Sagi, S. Rosenne, D. Oron, L. Leiserowitz, L. Kronik, S. Weiner, and L. Addadi, “Optically functional isoxanthopterin crystals in the mirrored eyes of decapod crustaceans,” Proc. Natl. Acad. Sci. 115(10), 2299–2304 (2018).
[Crossref]

Cao, H.

H. Noh, S. F. Liew, V. Saranathan, S. G. J. Mochrie, R. O. Prum, E. R. Dufresne, and H. Cao, “How Noniridescent Colors Are Generated by Quasi-ordered Structures of Bird Feathers,” Adv. Mater. 22(26-27), 2871–2880 (2010).
[Crossref]

Capasso, F.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science 334(6054), 333–337 (2011).
[Crossref]

Cho, N. H.

J. Karst, N. H. Cho, H. Kim, H.-E. Lee, K. T. Nam, H. Giessen, and M. Hentschel, “Chiral Scatterometry on Chemically Synthesized Single Plasmonic Nanoparticles,” ACS Nano 13(8), 8659–8668 (2019).
[Crossref]

Choi, T. M.

J.-G. Park, S.-H. Kim, S. Magkiriadou, T. M. Choi, Y.-S. Kim, and V. N. Manoharan, “Full-Spectrum Photonic Pigments with Non-iridescent Structural Colors through Colloidal Assembly,” Angew. Chem. Int. Ed. 53(11), 2899–2903 (2014).
[Crossref]

Dignam, M. J.

Donnay, J. D. H.

H. W. Morse and J. D. H. Donnay, “Spherulite optics,” Am. J. Sci. s5-23(137), 440–461 (1932).
[Crossref]

Dufresne, E. R.

H. Noh, S. F. Liew, V. Saranathan, S. G. J. Mochrie, R. O. Prum, E. R. Dufresne, and H. Cao, “How Noniridescent Colors Are Generated by Quasi-ordered Structures of Bird Feathers,” Adv. Mater. 22(26-27), 2871–2880 (2010).
[Crossref]

Elad, N.

B. A. Palmer, V. J. Yallapragada, N. Schiffmann, E. M. Wormser, N. Elad, E. D. Aflalo, A. Sagi, S. Weiner, L. Addadi, and D. Oron, “A highly reflective biogenic photonic material from core–shell birefringent nanoparticles,” Nat. Nanotechnol. 15(2), 138–144 (2020).
[Crossref]

Fung, T. H.

L. Gao, T. H. Fung, K. W. Yu, and C. W. Qiu, “Electromagnetic transparency by coated spheres with radial anisotropy,” Phys. Rev. E 78(4), 046609 (2008).
[Crossref]

Gaburro, Z.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science 334(6054), 333–337 (2011).
[Crossref]

Gao, L.

L. Gao, T. H. Fung, K. W. Yu, and C. W. Qiu, “Electromagnetic transparency by coated spheres with radial anisotropy,” Phys. Rev. E 78(4), 046609 (2008).
[Crossref]

Genevet, P.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science 334(6054), 333–337 (2011).
[Crossref]

Giessen, H.

J. Karst, N. H. Cho, H. Kim, H.-E. Lee, K. T. Nam, H. Giessen, and M. Hentschel, “Chiral Scatterometry on Chemically Synthesized Single Plasmonic Nanoparticles,” ACS Nano 13(8), 8659–8668 (2019).
[Crossref]

Ginzburg, P.

H. Barhom, A. A. Machnev, R. E. Noskov, A. Goncharenko, E. A. Gurvitz, A. S. Timin, V. A. Shkoldin, S. V. Koniakhin, O. Y. Koval, M. V. Zyuzin, A. S. Shalin, I. I. Shishkin, and P. Ginzburg, “Biological Kerker Effect Boosts Light Collection Efficiency in Plants,” Nano Lett. 19(10), 7062–7071 (2019).
[Crossref]

Goncharenko, A.

H. Barhom, A. A. Machnev, R. E. Noskov, A. Goncharenko, E. A. Gurvitz, A. S. Timin, V. A. Shkoldin, S. V. Koniakhin, O. Y. Koval, M. V. Zyuzin, A. S. Shalin, I. I. Shishkin, and P. Ginzburg, “Biological Kerker Effect Boosts Light Collection Efficiency in Plants,” Nano Lett. 19(10), 7062–7071 (2019).
[Crossref]

Grange, R.

R. Savo, A. Morandi, J. S. Müller, F. Kaufmann, F. Timpu, M. Reig Escalé, M. Zanini, L. Isa, and R. Grange, “Broadband Mie driven random quasi-phase-matching,” Nat. Photonics 14(12), 740–747 (2020).
[Crossref]

Gunn, E.

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H. Noh, S. F. Liew, V. Saranathan, S. G. J. Mochrie, R. O. Prum, E. R. Dufresne, and H. Cao, “How Noniridescent Colors Are Generated by Quasi-ordered Structures of Bird Feathers,” Adv. Mater. 22(26-27), 2871–2880 (2010).
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R. Savo, A. Morandi, J. S. Müller, F. Kaufmann, F. Timpu, M. Reig Escalé, M. Zanini, L. Isa, and R. Grange, “Broadband Mie driven random quasi-phase-matching,” Nat. Photonics 14(12), 740–747 (2020).
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L. Schertel, L. Siedentop, J. Meijer, P. Keim, C. M. Aegerter, G. J. Aubry, and G. Maret, “The Structural Colors of Photonic Glasses,” Adv. Opt. Mater. 7(15), 1900442 (2019).
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B. A. Palmer, V. J. Yallapragada, N. Schiffmann, E. M. Wormser, N. Elad, E. D. Aflalo, A. Sagi, S. Weiner, L. Addadi, and D. Oron, “A highly reflective biogenic photonic material from core–shell birefringent nanoparticles,” Nat. Nanotechnol. 15(2), 138–144 (2020).
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S. Tadepalli, J. M. Slocik, M. K. Gupta, R. R. Naik, and S. Singamaneni, “Bio-Optics and Bio-Inspired Optical Materials,” Chem. Rev. 117(20), 12705–12763 (2017).
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B. D. Wilts, B. Wijnen, H. L. Leertouwer, U. Steiner, and D. G. Stavenga, “Extreme Refractive Index Wing Scale Beads Containing Dense Pterin Pigments Cause the Bright Colors of Pierid Butterflies,” Adv. Opt. Mater. 5(3), 1600879 (2017).
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B. D. Wilts, B. Wijnen, H. L. Leertouwer, U. Steiner, and D. G. Stavenga, “Extreme Refractive Index Wing Scale Beads Containing Dense Pterin Pigments Cause the Bright Colors of Pierid Butterflies,” Adv. Opt. Mater. 5(3), 1600879 (2017).
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V. Hwang, A. B. Stephenson, S. Magkiriadou, J.-G. Park, and V. N. Manoharan, “Effects of multiple scattering on angle-independent structural color in disordered colloidal materials,” Phys. Rev. E 101(1), 012614 (2020).
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N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science 334(6054), 333–337 (2011).
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H. Barhom, A. A. Machnev, R. E. Noskov, A. Goncharenko, E. A. Gurvitz, A. S. Timin, V. A. Shkoldin, S. V. Koniakhin, O. Y. Koval, M. V. Zyuzin, A. S. Shalin, I. I. Shishkin, and P. Ginzburg, “Biological Kerker Effect Boosts Light Collection Efficiency in Plants,” Nano Lett. 19(10), 7062–7071 (2019).
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R. Savo, A. Morandi, J. S. Müller, F. Kaufmann, F. Timpu, M. Reig Escalé, M. Zanini, L. Isa, and R. Grange, “Broadband Mie driven random quasi-phase-matching,” Nat. Photonics 14(12), 740–747 (2020).
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B. A. Palmer, V. J. Yallapragada, N. Schiffmann, E. M. Wormser, N. Elad, E. D. Aflalo, A. Sagi, S. Weiner, L. Addadi, and D. Oron, “A highly reflective biogenic photonic material from core–shell birefringent nanoparticles,” Nat. Nanotechnol. 15(2), 138–144 (2020).
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B. A. Palmer, D. Gur, S. Weiner, L. Addadi, and D. Oron, “The Organic Crystalline Materials of Vision: Structure-Function Considerations from the Nanometer to the Millimeter Scale,” Adv. Mater. 30(41), 1800006 (2018).
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B. D. Wilts, B. Wijnen, H. L. Leertouwer, U. Steiner, and D. G. Stavenga, “Extreme Refractive Index Wing Scale Beads Containing Dense Pterin Pigments Cause the Bright Colors of Pierid Butterflies,” Adv. Opt. Mater. 5(3), 1600879 (2017).
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N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science 334(6054), 333–337 (2011).
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R. Savo, A. Morandi, J. S. Müller, F. Kaufmann, F. Timpu, M. Reig Escalé, M. Zanini, L. Isa, and R. Grange, “Broadband Mie driven random quasi-phase-matching,” Nat. Photonics 14(12), 740–747 (2020).
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J. H. Bahng, S. Jahani, D. G. Montjoy, T. Yao, N. Kotov, and A. Marandi, “Mie Resonance Engineering in Meta-Shell Supraparticles for Nanoscale Nonlinear Optics,” ACS Nano 14(12), 17203–17212 (2020).
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S. Tadepalli, J. M. Slocik, M. K. Gupta, R. R. Naik, and S. Singamaneni, “Bio-Optics and Bio-Inspired Optical Materials,” Chem. Rev. 117(20), 12705–12763 (2017).
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M. J. Henze, O. Lind, B. D. Wilts, and A. Kelber, “Pterin-pigmented nanospheres create the colours of the polymorphic damselfly Ischnura elegans,” J. R. Soc. Interface. 16(153), 20180785 (2019).
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H. Barhom, A. A. Machnev, R. E. Noskov, A. Goncharenko, E. A. Gurvitz, A. S. Timin, V. A. Shkoldin, S. V. Koniakhin, O. Y. Koval, M. V. Zyuzin, A. S. Shalin, I. I. Shishkin, and P. Ginzburg, “Biological Kerker Effect Boosts Light Collection Efficiency in Plants,” Nano Lett. 19(10), 7062–7071 (2019).
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Nature (1)

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Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. E (2)

L. Gao, T. H. Fung, K. W. Yu, and C. W. Qiu, “Electromagnetic transparency by coated spheres with radial anisotropy,” Phys. Rev. E 78(4), 046609 (2008).
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V. Hwang, A. B. Stephenson, S. Magkiriadou, J.-G. Park, and V. N. Manoharan, “Effects of multiple scattering on angle-independent structural color in disordered colloidal materials,” Phys. Rev. E 101(1), 012614 (2020).
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Phys. Rev. Lett. (1)

E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987).
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Proc. Natl. Acad. Sci. (1)

B. A. Palmer, A. Hirsch, V. Brumfeld, E. D. Aflalo, I. Pinkas, A. Sagi, S. Rosenne, D. Oron, L. Leiserowitz, L. Kronik, S. Weiner, and L. Addadi, “Optically functional isoxanthopterin crystals in the mirrored eyes of decapod crustaceans,” Proc. Natl. Acad. Sci. 115(10), 2299–2304 (2018).
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S. Jahani, J. H. Bahng, A. Roy, N. Kotov, and A. Marandi, “Confining light in all-dielectric anisotropic metamaterial particles for nano-scale nonlinear optics,” arXiv:2102.13087 [physics] (2021).

Supplementary Material (1)

NameDescription
» Supplement 1       Supplement 1: Details of extraction of spherulites; Identifying intact particles; Details of the optical microscope setup; Determination of particle dimensions; Tabulated measured values of nt; Calculation of the effect of the TEM grid film.

Data availability

Measured refractive indices are tabulated in Section 5 of Supplement 1. Raw data underlying the results presented in this paper may be obtained from the authors upon reasonable request.

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Figures (4)

Fig. 1.
Fig. 1. (a) A schematic of the isoxanthopterin spherulites found in decapod crustacean eyes. The structure is characterized by a smaller refractive index $n_r$ along the radial direction (red arrow) and a larger index $n_t$ along the tangential direction (green arrows); (b) Scattering cross section of the spherulite. $Q_{\textrm {sca}}$ is the scattering cross section (all directions). The other curves show the contributions of various scattering multipoles. The prominent peaks result from the TE (magnetic) quadrupole and octupole modes.; (c) The variation of $Q_{\textrm {sca}}$ with the radius of the spherulite, with shell thickness $t = r/2$; (d) The variation of $Q_{\textrm {sca}}$ with the ratio of shell thickness to radius ($t/r$). ; (e) The variation of $Q_{\textrm {sca}}$ with the tangential index ($n_t$). Other parameters in (c),(d) and (e) are the same as in (b). The spectral position of the multipole resonances is strongly dependent on $n_t$ and $r$.
Fig. 2.
Fig. 2. Dark field optical setup and TEM correlation.(a) Schematic of the dark-field scattering measurement geometry. Incident light is colored yellow and scattered light is colored green; (b),(c),(d): TEM images taken at different magnifications from a TEM finder grid; (e),(f),(g): DF optical microscope images taken from the same area on the finder grid as show in (b),(c),(d); Note the different lengths of the scale bars in panels (d) and (g).
Fig. 3.
Fig. 3. Determination of $n_t$. (a) A TEM image of a representative spherulite. The scalebar is equivalent to 200 nm; (b) Measured DF spectrum for the particle in (a); (c) The normalized cross section ($Q_{\textrm {sca}}$, blue curve) and the DF signal (red curve, scaled by a factor of 10) calculated for a particle of the same dimensions as in (a) using $n_t = 1.96$ and $n_r = 1.4$. The TE resonances are clearly identifiable in the DF spectrum; (d) The variation of the TE2 and TE3 peak positions with $n_t$. Peak positions from (b) and the corresponding values of $n_t$ are marked by dotted lines. We obtain values of 1.83 and 1.89 for $n_t$ from the TE2 and TE3 multipoles, respectively.
Fig. 4.
Fig. 4. Scattering from spherulite dimers. (a) A TEM image of a dimer of spherulites in close proximity. The length of the scalebar is 200 nm.; (b) Measured DF scattering spectra with varying angle between the dimer axis and the orientation of the illumination polarizer; (c, d) Simulated DF spectra for the dimer of the same dimensions as in (a) using $n_t = 1.96$, $n_r = 1.4$ (c) and $n_t = n_r = 1.96$ (d). Prominent qualitative features in the scattering spectrum do not depend significantly on the radial index $n_r$.

Equations (4)

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

E s ( r , θ , ϕ ) = e i k ( r z ) i k r X ( θ , ϕ ) E i ,
X ( θ , ϕ ) = S 2 ( θ ) cos ϕ e ^ ( θ , ϕ ) + S 1 ( θ ) sin ϕ e ^ ( θ , ϕ ) .
Q sca = 0 2 π 0 π | X | 2 k 2 sin θ d θ d ϕ
Q DF = C | X | 2 k 2 sin θ d θ d ϕ

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