Subscriber access provided by University of Florida | Smathers Libraries
Article
Limitations and Opportunities for Optical Metafluids to Achieve Unnatural Refractive Index Kwangjin Kim, SeokJae Yoo, Ji-Hyeok Huh, Q-Han Park, and Seungwoo Lee ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.7b00546 • Publication Date (Web): 08 Aug 2017 Downloaded from http://pubs.acs.org on August 9, 2017
Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.
ACS Photonics is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.
Page 1 of 27
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
ACS Photonics
Limitations and opportunities for optical metafluids to achieve unnatural refractive index Kwangjin Kim1,+, SeokJae Yoo2,+*, Ji-Hyeok Huh1, Q-Han Park2, and Seungwoo Lee1,3* 1
SKKU Advanced Institute of Nanotechnology (SAINT), Sungkyunkwan University (SKKU), Suwon 16419, Republic of Korea 2
Department of Physics, Korea University, Seoul 02841, Republic of Korea
3
Department of Nano Engineering and School of Chemical Engineering, Sungkyunkwan University (SKKU), Suwon 16419, Republic of Korea *Email:
[email protected];
[email protected] +
Equally contributed to this work
Keywords: Metafluids, Meta-Atoms, Metamolecules, Refractive Index, Soft fluidity
Abstract: Optical metafluids have held a special position among the platforms of metamaterials, because other than the lithography-based hard approaches, the soft fluidity-based solution process not only enables their immediate practical utility, but also allows for reconfigurable and adaptable nanophotonic systems. However, the fundamental limits of the available effective parameters of optical metafluids are not yet clearly defined. Of particular interest is the accessible range of the refractive index under a practically available volume fraction ϕ and the structural motifs of building blocks. In addition, previously reported theoretical works are based on an effective medium theory which excludes dipolar coupling between building blocks. Using these initial approaches, the interaction between the building blocks at a relatively higher ϕ was not accurately rationalized. In this work, we advance an effective medium theory by using the 3D dressed polarizability. Then, we successfully rationalize the dipolar coupling between each of the building blocks and systematically exploit the fundamental limits of optical metafluids in terms of accessible effective parameters. Also, for the first time, we discuss both the phase transition of metafluids and uniaxial characteristics of fluidic crystals in terms of engineering effective parameters. Thereby, the practically available range of effective parameters from the concept of an optical metafluid is realistically defined. It is revealed that an unnaturally near-zero-refractive index and an ultrahigh refractive index can be attainable through optical metafluids. Given the fundamental limits defined by 3D dressed polarizability, a comprehensive perspective of the limits and merits of optical metafluids is provided. In 2006, A. Alù and N. Engheta first suggested that the assembly of metallic nanoparticles (NPs) into a subwavelength ring geometry (i.e., plasmonic metamolecules or lumped magnetic molecules) offers versatile yet highly efficient control over artificial magnetism at optical frequencies.1 Since then, the colloidal suspensions of plasmonic metamolecules exhibiting magnetic responses have been collectively referred to as “metafluids.”2 Other than lithographically developed planar metamaterials, metafluids can provide versatile solution-processability, which makes them more likely to have immediate practical utility. In line with these advances, much theoretical groundwork has been explored over the last decade; various cluster motifs including tetrahedral, octahedral, icosahedral, and raspberry-typed clusters made of Au or Ag nanospheres (NSs) have been ACS Paragon Plus Environment
ACS Photonics
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
intensively exploited as potential building blocks of metafluids.2−8 Indeed, several seminal theoretical works proved that a negative refractive index can be achieved, in particular when the filling fraction ϕ of plasmonic metamolecules is 0.74 (a maximum value for close-packed facecentered-cubic (FCC) or body-centered-cubic (BCC) crystals). Also, along with theoretical predictions, the assembly of plasmonic metamolecules, which have recently been benefitting from advances in soft matter engineering, has seen considerable progress, from an initial demonstration of raspberry clusters to the recent feedback-controlled assembly of an asymmetric gold nanorod (AuNR) dimer.9−19 Yet, experimentally achieving such extreme effective parameters of metafluids has proven to be significantly difficult.10−12,14,15 The existence of experimental hurdles is unsurprising because the quest for negative effective permeability µeff, permittivity εeff, and refractive index n with metafluids platform demands addressing an even more challenging task: achieving an unrealistically high ϕ of plasmonic metamolecules, for example, 0.74.1,5 Of course, through an increase in entropy, colloidal NPs and clusters can be spontaneously packed into ordered superlattices (e.g., FCC) even in the presence of liquids, where ϕ is larger than 0.5 (i.e., a phase transition from a pure fluid to an intermediate state between fluids and crystals, such as liquid crystals).20−22 However, it has been experimentally proven that beyond a ϕ of ~ 0.58, the phase of the colloidal suspension becomes glasslike rather than crystalline because it would be difficult to make an internal energy to be perfectly zero in a realistic experiment.22 Therefore, a ϕ of 0.74, the theoretically suggested requirement for a negative refractive index,1,5,8 can be achieved only by solvent evaporationenabled assembly of metamolecules into a solid-state superlattice.22 The concept of metafluids has been thereby limited to a theoretical peculiarity far from a readily realizable technology.1−19 If our technological goal is still limited to attaining a negative refractive index, then the significant constraints of the metafluids platform discourages their practical use (e.g., just use metafluids as precursors for self-assembled optical metamaterials). In this work, we will argue that the scope of metafluids needs to be expanded to conceive ever more realistic and powerful strategies for the versatile manipulation of light. First, we reimagine metafluids to explore the possibility of expanding the space of accessible effective parameters, including a near-zero-refractive index and an unnaturally high refractive index. In essence, a single plasmonic NP as well as plasmonic clusters were added in the library of possible building blocks of metafluids; each of those building blocks were designed to be dispersed in fluid with a realistically achievable ϕ. Because the previous research has mostly dealt with the effective parameters of metafluids, mainly for a ϕ of 0.74,1,3,5,8 the practically available effective parameters with a realistic ϕ of NPs and plasmonic clusters are not yet well defined. Also, the effect of a metafluid phase which is changed from a pure liquid (herein referred to as a “fluid” phase) to a crystalline liquid (herein referred to as “fluid+crystal” phase), according to ϕ should be accounted into the engineering of effective parameters of realistic metafluids. Especially, in the “fluid+crystal” phase, the coupling of the dipolar resonance between each individual building block should be included when engineering effective parameters for metafluids; however, the previous works have not reflected this.1,2,5,8 Herein, we advance an effective medium model via the rational implementation of the 3D dressed polarizability effect into the Clausius-Mossotti relation. Thus, the effective parameters of metafluids are more realistically engineered by adjusting the building block motifs and ϕ. It is revealed that even at a relatively low ϕ (less than 0.74), metafluids empower a fine control over the effective permittivity εeff, permeability µeff, and refractive index n beyond naturally ACS Paragon Plus Environment
Page 2 of 27
Page 3 of 27
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
ACS Photonics
available regimes. Thereby, the other degrees of freedom, which are not available with natural fluids, could come to the fore. Advancing effective medium theory for a realistic engineering of the effective parameters of metafluids Building blocks and phases of optical metafluids. In this work, three different NP motifs, including single NP, tetrahedral cluster, and octahedral cluster, were used as the building blocks of the optical metafluids (Figure 1a). Spherical gold (Au) NPs of 20 nm, 30 nm, and 60 nm diameter were used as the representative meta-atoms, which are all deep-subwavelength scales to justify the use of effective medium theory at the wavelength of interest (i.e., 500 nm ~ 2000 nm). Generally, increasing the size of metallic NPs induces the surface modes, which in turn can enhance the radiative loss. However, the polarizability of the NS whose radius is R scales with R3. Thus, we selected such size range of Au NPs to address this trade-off relationship between the polarizability and radiative loss. In addition, the gap between NPs of clusters (i.e., metamolecules) was set to be 1 nm. Herein, we include the results of Au NPs rather than silver (Ag) NPs for the following two reasons. First, highly uniform Au NPs with perfectly spherical shapes have become available with recent advances in chemical synthesis, whereas it is still difficult to obtain perfectly spherical Ag NPs.23−26 Second, it is well known that Au NPs are more robust against chemical oxidation compared with Ag NPs. With respect to ϕ, the phase of metafluids can be divided into two categories (see Figure 1b): (i) a “fluid” phase, in which individual building blocks are randomly dispersed within the liquid (ϕ less than 0.50) and (ii) a “fluid+crystal” phase in which individual building blocks are assembled into a FCC crystal within the liquid (ϕ larger than 0.5). The 3D crystallization of spherical NPs should choose FCC rather than other geometrical lattices because of the packing argument (i.e., FCC is the densest lattice for spherical colloids). Here, the gap between the crystallized building blocks was varied according to ϕ. From a practical point of view, we restricted the upper limit of ϕ to 0.1 for the “fluid” phase. Especially, concentrating plasmonic colloids beyond ϕ of 0.1 would be difficult, in stark contrast to dielectric colloids. As the van der Waals force between constituents of metallic colloids is much stronger than that for dielectric colloids, meta-atoms and metamolecules are likely aggregated or undergo sedimentations at a relatively high ϕ (no longer a colloidal suspension). However, as reported recently, coating metallic NPs with a polymer brush can transform the NPs to quasi-hard spheres; the controlled sedimentation within the microfluidic channel allows the polymer-coated metallic colloids to be packed into crystals, even in the presence of liquid.27 Thereby, it is reasonable to assume that the phase of the metafluid can be directly transited from the “fluid” (a ϕ less than 0.1) to the “fluid+crystal phase” (a ϕ higher than 0.5). Additionally, we assumed that despite its non-spherical overall shape, the metamolecules can be packed into a FCC crystal. This can be justified because metamolecules could be encapsulated with a spherical shell by using a controlled chemical reaction (see inset in the left panel of Figure 1b).13 Of course, the spherical encapsulation could further increase the refractive index around metamolecules from 1.33 to 1.40 and thus enhance both electric and magnetic dipole polarizabilities, as the example is shown in Figure S1, Supporting Information. However, for the simplicity of the numerical simulation, we excluded a spherical shell. This also can be justified by a ACS Paragon Plus Environment
ACS Photonics
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 4 of 27
small difference of refractive index between water and a spherical shell (i.e., organic polymer). Consequently, the sedimentation together with entropic-packing enables the formation of an FCC crystal composed of metamolecules (see right panels of Figure 1b). Water was used as a representative liquid because Au NPs can be thoroughly dispersed without aggregation.9−12 Effective medium theory for “fluid” phase-metafluids. First, we obtained effective parameters of “fluid” phase-metafluids. The effective parameters of “fluid” phase-metafluids cannot be obtained by the S-parameter retrieval method, which has been widely used in the field of metamaterials.28−30 This is because the S-parameter retrieval method requires periodicity of the unit structure. Individual NPs or metamolecules in the “fluid” phase cannot be periodic; thus, the interaction between building blocks should be characterized by the ensemble average.
t
t
First, we numerically retrieved the electric and magnetic polarizability tensors ( αe and αm ) of the building blocks using the method proposed in Ref. [31] (see more details in Method part). Herein, numerical calculations were supported by the commercially available COMSOL Multiphysics package. The electromagnetic fields, scattered by subwavelength meta-atoms or metamolecules, can
t
be described by the fields radiated by the point electric dipole p = ε0εhαe E0 and the magnetic
t
dipole m = µh µ0αmH0 . ε0 and µ0 denote the vacuum permittivity and permeability, respectively, while εh and µh indicate those respective values for the host medium (i.e., water). The electric and
t
t
magnetic polarizability tensors αe and αm , respectively, have 9-components to be determined, and we can obtain the 18 total components by varying the polarization and the propagation direction of the incident plane wave along three axes.32 In the second step, we applied the rotational average to
t
t
the polarizability tensors αe and αm because meta-atoms or metamolecules in the “fluid” phase have random orientations. The rotationally averaged polarizability tensors are given by t α e .m = tr ( Λ e , m −1 α e , m Λ e , m ) / 3 with the corresponding eigenmatrices Λe.m . Finally, we obtained the
effective permittivity εeff and permeability µeff from the Clausius-Mossotti relation that correlates the microscopic polarizability with the macroscopic effective parameters.33 The effective permittivity εeff and permeability µeff are given by
εeff =
µeff =
3+ 2N α e 3− N α e
ε hε0 ,
3 + 2N αm µ h µ0 , 3 − N αm
(1)
(2)
where N is the number density of NPs or metamolecules. Then, the refractive index is determined by neff = ε eff µeff . A more detailed extension of the equations is described in the Method part.
The 3D dressed polarizability for “fluid+crystal” phase-metafluids. When φ is higher than 0.5, meta-atoms and metamolecules are crystallized into the FCC lattice. Consequently, the dipolar coupling between each building block must be reflected in the determination of the effective parameters.34 To rationalize the dipolar coupling, we implemented the notion of dressed ACS Paragon Plus Environment
Page 5 of 27
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
ACS Photonics
polarizability into the Clausius-Mossotti relation.35 In particular, we newly extended the notion of dressed polarizability from 2D to 3D to realistically predict the effective parameters of the “fluid+crystal” phase-metafluids. Given the size of Au NSs and cluster, the coupling of the higher order mode (e.g., quadrupole) can be excluded. When dipoles are periodically arranged in the crystal lattice, fields radiated by neighboring dipoles can affect dipole moments p and m . Considering the dipolar coupling between neighboring building blocks, the local fields Eloc and H loc are given by33,36 t t Eloc = E0 + ω 2 µ ∑ G i p + iωµ ∑ ∇ × G i m i
H loc
i
t t , = H 0 + ω εµ ∑ G i m − iω ∑ ∇ × G i p 2
i
(3)
i
with the angular frequency ω , the electric permittivity ε , the magnetic permeability µ , the t incident electric field E 0 , and the incident magnetic field H0 . Green’s dyadic function G i and its curl are determined by the location of dipoles at r , and are defined by36
t eikr 1 t rr 2 2 G= − 1 + ikr + k r I + ( 3 − 3ikr − k 2 r 2 ) 2 , ( ) 2 2 4π r k r r
(4)
t eikr 1 t ∇× G = r × I ( −1 + ikr ) , 2 4π r r
(5)
(
)
Once the Green’s dyadic functions are determined according to the crystal lattice, we can obtain the dipole moments, modified by the relations of p = εαeEloc and m = µαmHloc . Here, we interpreted the effect of the dipolar coupling by using the dressed polarizability, in which the dipole moments t t t t become p = ε α e, d E0 and m = µα m ,d H 0 with the dressed polarizability tensors α e,d ≡ γ eα e and t t α m,d ≡ γ mα m . For the periodic lattice with primitive axis vectors ai (i=1, 2, 3), the local-field coupling between two nearest neighbors at r = ±ai can be canceled out due to the sign flipping term t ∇ × G according to r = ±ai . Therefore, the local fields (Eq. (3)) become t Eloc = E 0 + ω 2 µ ∑ G i p i (6) t . 2 H loc = H 0 + ω εµ ∑ G i m i
Using the periodicity of the crystal lattice, the expressions for the dipole moments (Eq. (3)) are simplified to t p = εα e Eloc = εα e E0 + ω 2 µ ∑ G i p i . (7) t 2 m = µα m H loc = µα m H 0 + ω εµ ∑ G i m i
t
t
By solving for p and m , we can obtain the dressing tensors γe and γ m as follows: t −1 t t 2 γ e = I − εα e ω µ ∑ G i , i ACS Paragon Plus Environment
(8)
ACS Photonics
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
t −1 t t γ m = I − ω 2 εµ 2α m ∑ G i . i
Page 6 of 27
(9)
When the dressed polarizabilities are written as tensors because of anisotropic local fields inside the crystal lattice, we obtain the effective parameters t t 3 + 2 Nα e ε= (10) t ε hε 0 , 3 − Nα e
t t 3 + 2 Nαm µ= t µh µ0 . 3 − Nα m
(11)
In the Supporting Information, we provide the detailed descriptions for deriving the effective parameters of the FCC, the simple cubic (SC), and the BCC lattices. Basically, the unit cell of the FCC lattice consists of 14 individual spherical Au NPs, with a coordination number of 6 (Figure 2a). In other words, one Au NP is surrounded by the 6 nearest neighboring Au NPs and coupled with each other via dipolar local fields (hereafter, referred to as “nearest neighbor” interaction, marked by light blue-colored lines). Of course, the “next nearest” neighbor”, highlighted by orange-colored lines in Figure 2a, can be possible. However, it turned out interaction is negligible compared with its “nearest neighbor” counterpart. Thus, in the following results, we excluded “next nearest” neighbor” for simplicity. We explicitly validated our effective medium theory by comparing it with the numerical retrieval method.28−30 To do this, we used the FCC crystal made of single 30 nm Au NPs as an exemplary model (see Figure 2a-b). Herein, air was used as a fluid for simplicity and the gap between Au NPs was set to be 2 nm (ϕ of 0.56), as shown in Figure 2b. The incident electric and magnetic fields together with the wavevector are presented in Figure 2a. S-parameters were obtained by the finitedifference time-domain (FDTD) method.37,38 Figures 2b-c summarize the refractive indices of air-hosted 30 nm AuNP FCC crystals, which were respectively obtained by (i) the original Clausius-Mossotti relation (red line), (ii) the 3D dressed polarizability-implemented effective medium model (blue line), and (iii) the numerical retrieval method (green line). The individual plasmonic NPs induce a broad electric dipole (ED) resonance, whereas magnetic dipole (MD) resonance is negligible. Consequently, the random dispersing of AuNPs in the fluid leads to the enhancement of both the effective permittivity εeff and refractive index n at both the ED resonance and the off-resonant long-wavelength regime (i.e., electric metafluid).39 Once individual AuNPs are ordered into 2D or 3D crystals with a deep-subwavelength scale gap, the induced EDs can be strongly coupled with each other (i.e., capacitive coupling), so as to enhance the electric polarizability. Thereby, the effective permittivity εeff and the refractive index n can be further increased.35,37,38 It is clearly visible that the 3D dressed polarizability-based model successfully reflected the capacitive coupling between Au NPs, as evidenced by both the red-shifted ED resonance and the enhanced refractive index compared with the Clausius-Mossotti relation. The results from the rigorous retrieval method match well with those from the 3D dressed polarizabilityimplemented effective medium model.
ACS Paragon Plus Environment
Page 7 of 27
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
ACS Photonics
Unnatural effective parameters of optical metafluids Electric metafluids. Previously, the scope of functions that can be achieved using optical metafluids was mainly limited to artificial magnetism, as a negative refractive index at optical frequencies was an important goal in the initial stages of the metamaterial field.1−3,6,8 By contrast, achieving an unnaturally high-refractive-index should be another important goal, as paving the way for advancing light trapping in solar cells and enhancing the resolution of imaging and lithography.40,41 To exploit this possibility, 20 nm, 30 nm and 60 nm Au NPs were assumed to be dispersed in water as representative example of electric metafluids. Figure 3a summarizes the ED polarizabilities of 20 nm, 30 nm, and 60 nm Au NPs-dispersed water (i.e., the “fluid” phase with a ϕ less than 0.1). These can all be viewed as bare polarizabilities.35 The peaks of these bare polarizabilities are the hallmarks of ED resonances; the increase in Au NP size leads to enhancement and red-shift of the ED polarizability. Thus, as shown in Figures 3b-d, the “fluid” phase of electric metafluids exhibits the maximum refractive indices at the ED resonance (up to 1.86 for 60 nm Au NPs); in the regime of off-resonant long-wavelengths, the available refractive indices (i.e., 1.60 – 1.65) can be still higher than those of natural fluids (e.g., 1.27 – 1.35), while the optical loss is relatively low. The corresponding figure-of-merit (FOM, defined by Re(n)/Im(n)) at a ϕ of 0.1 is summarized in Figure S2, Supporting Information.
Next, we investigate the electric metafluids with a “fluid-crystal” phase. First, it is noteworthy that the dressed polarizability is varied according to the primitive axes of the FCC lattice (marked by light blue, green, and orange lines in Figure 4a). This is because the FCC lattice is a uniaxial crystal, indicating an optical material whose effective parameters are different with respect to the primitive axes. In the Supporting Information, we mathematically proved the uniaxial characteristics of the FCC lattice in greater detail. As demonstrated in our previous work, the dressed ED polarizability can be enhanced or quenched according to the angular deviations between the induced ED and the “nearest neighbor” interaction.36 For example, when EDs are induced in two Au NPs that are horizontally aligned along the direction of the “nearest neighbor” interaction, the capacitive coupling becomes maximized. As such, the dressed ED polarizability is fully enhanced (bottom left panel of Figure 4b). In contrast, the induced EDs can be perpendicularly aligned with respect to the direction of the “nearest neighbor” interaction, and the dressed ED polarizability consequently becomes fully quenched (top left panel of Figure 4b). These two cases are the only accessible possibilities for a 2D tetragonal lattice.36 However, in the FCC crystal, the possible dressed ED polarizability can be more diversified than these two extreme cases due to the intrinsic 3D complexity of Au NP spatial arrangements. For instance, the directions of the induced ED and the “nearest neighbor” interaction cannot be horizontally or perpendicularly aligned with each other. Therefore, the dressed ED polarizability can be partially enhanced and quenched as well (right panels of Figure 4b). To quantify the 3D dressed ED polarizability of the FCC with respect to the primitive axes, first we visualized the near field coupling (capacitive coupling) of three “nearest neighbor” interactions at ED resonances via FDTD-enabled spatial mapping of the electric field intensity (see Figures 4c-e): three “nearest neighbor” interactions including Au NP 12-1, 12-2, and 12-6 were mapped. Herein the FCC lattice of 30 nm Au NPs dispersed in water (ϕ of 0.56) was employed as a representative model. This analysis reveals that the capacitive coupling can be strongly induced for axis 1, whereas for the axes 2 and 3, capacitive coupling is partially enhanced and quenched, respectively. ACS Paragon Plus Environment
ACS Photonics
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Then, 3D dressed ED polarizabilities for each axis of 30 nm Au NP-based “fluid+crystal” phase metafluid (Figures 5a-b) were calculated as shown in Figure 5c. Indeed, compared with the bare polarizability (see Figure 3a), the dressed ED polarizability for axis 1 is strongly enhanced together with the significant red-shift. The partially enhanced and quenched ED polarizabilities for axes 3 and 2 are also visible. Figures 5d-f show the corresponding refractive indices of this electric metafluid as function of ϕ (0.5~0.74): the figure of merit (FOM) for each axis is included in Figure S3, Supporting Information. For axis 1, the peak of the refractive index is observed at a wavelength outside of the range of interest due to the strongly enhanced and red-shifted ED polarizabilities. Thus, for axis 1, the optical frequencies of interest are included in the short wavelength regime of the plasmonic resonance;2 the achievable refractive indices in turn are reduced to unnaturally near zero values. Meanwhile, for axes 2 and 3, an enhancement of the refractive index is visible at the wavelength of interest; this mainly originates from the increased ϕ of the “fluid-crystal” phase. However, it is still clear that for axis 3, the partially enhanced ED polarizability is responsible for the respectively increased and red-shifted refractive index compared with axis 2. Particularly, the refractive indices can be unnaturally increased to 9 and 4.5 respectively at the ED resonance and the off-resonant long-wavelength regimes. In addition, according to the gap between Au NPs (controllable via adjusting ϕ of the FCC-crystallized Au NPs), the strength of the capacitive coupling and the resultant refractive index can be precisely tuned. In line with this, a wide gamut of high-refractive-index regions with low loss can be achieved through the concept of electric metafluids. In addition to ϕ, the size of the Au NPs allows for further tuning of the effective parameters (Figure 6). Figures 6a-d and 6e-h present the dressed ED polarizabilities (for a ϕ of 0.74) and refractive indices for 20 nm and 60 nm Au NP metafluids in the “fluid+crystal” phase. Interestingly, the increase in Au NP size results in both reduced ED polarizabilities per volume and refractive indices: the FCC consisting of a 20 nm Au NP can increase refractive indices up to 10 at the ED resonance and to 5.6 in the off-resonant long-wavelength regime, when illuminated by light polarized along axis 3. This achievable refractive index is much higher than upper limit of the representative solid state dielectrics at the wavelength of interest (e.g., ~ 3.5 for silicon, ~ 4.7 for germanium, ~ 1.7 for sapphire or alumina). Also, this behavior found in the “fluid + crystal” phase is contrasted with that of the “fluid” phase, indicating that radiative loss becomes more elucidated when Au NPs are assembled into a crystal. Thus, 20 nm Au NPs can be advantageous over larger Au NPs in terms of enhancing the refractive index.
Magnetic metafluids. When Au NPs are assembled into a cluster, a metallic ring motif with a small gap can be formed.9 Thereby, the MD resonance can be boosted via the circulating displacement current along Au NP ring motifs.1−8 In particular, the 3D clustering of metallic NPs, which can be achieved by the droplet-enabled confined self-assembly of colloids,42 allows for nearly isotropic responses of the ED and MD.13 First, we exploited ED and MD responses of tetrahedral and octahedral clusters consisting of 60 nm Au NPs (Figures 7a-b): the clusters made of 20 nm and 30 nm Au NPs exhibit a negligible MD at optical frequencies. Both clusters of 60 nm Au NPs show much stronger MD resonances compared to a single Au NP. As the MD of octahedral clusters are found to be more efficient than that of their tetrahedral counterparts, we used them as representative magnetoplasmonic building blocks of metafluids. ACS Paragon Plus Environment
Page 8 of 27
Page 9 of 27
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
ACS Photonics
In the “fluid” phase (see schematic in Figures 7c-d), both the effective permittivity εeff and permeability µeff are varied according to the ED (at 675 nm) and MD (825 nm) resonances of individual octahedral clusters as shown in Figures 7e-f. As a result, the refractive index can be smaller or higher than that of water respectively at MD and ED resonances (Figures 7g-h). However, the gamut of refractive index available with the “fluid” phase is comparable to that of naturally occurring fluids (1.33 – 1.55). This is because the permeability µeff is negligibly modulated (almost a constant value of 1.0). Also, the radiative loss is quite significant; consequently, the modulation range of the effective permittivity εeff is relatively narrow (Figure 7e). Especially, the total size of the octahedral cluster is about 145 nm, and its radiative loss is much stronger than that of the 20 ~ 60 nm Au NPs. Once the octahedral clusters are crystallized into the FCC lattice (see schematic in Figure 8a), the dressed polarizability for both the ED and MD can be partially enhanced or quenched according to the geometrical relation between clusters and dipoles, as shown in Figure 8b. Figures 8c-d compare the dressed polarizabilities for each axis with the bare polarizabilities (for a ϕ of 0.74). The individual EDs in the octahedral cluster can interact with each other via capacitive coupling, as evidenced by the red-shifted peak of the dressed ED polarizabilities compared with that of bare polarizability (Figure 8c). Nevertheless, for ED, the dressed polarizability was weaker than bare polarizability owing to the strong radiative loss: the strength of the dressed ED polarizability is in the increasing order of axes 1, 3, and 2. The significant radiative loss is also confirmed from the broadened peaks of the dressed ED polarizabilities. Interestingly, the peak position of the MD polarizability remains almost unchanged (Figure 8d) after crystallization. This is because the induced MDs resulting from the circulating displacement current cannot interact with each other through capacitive coupling. This non-capacitive coupling of each induced MD is further evidenced by the non-broaden peaks of the dressed MD polarizabilities. However, the strength of the dressed MD polarizabilities is found to be smaller than that of the bare MD polarizability mainly due to the enhancement of radiative loss, which is caused by MD coupling: the strength of dressed MD polarizabilities is in the increasing order of axes 2, 3, and 1. As a result, the position of the MD resonance is significantly detached from that of the ED resonance, meaning that it would be difficult to overlap the positions of the electric and magnetic resonances. Given such resonant behaviors of the ED and MD, the effective permittivity εeff and permeability µeff were calculated at optical frequencies of interest (Figures 8e-f). Due to a significant radiative loss, the effective permittivity εeff is broadly tuned from 0.2 to 1.5 (for axis 1) and from 1.5 to 3.5 (for axes 2 and 3). In contrast, the effective permeability µeff can be sharply reduced to 0.7 at the MD resonance (e.g., for axis 2). Thereby, MD resonance acts as a determining factor for the modulation of the refractive index n of the “fluid+crystal” phase-magnetic metafluid (see schematic in Figures 9a-b). As shown in Figures 9c-h, at the MD resonance, the refractive index n of this magnetic metafluid can be reduced to 1.1 (e.g., for axis 1), but in stark contrast to previous researches,1,5,8 it would be difficult to meet the following condition for a negative refractive index even with ϕ of 0.74.
Re ( ε ) Im ( µ ) + Re ( µ ) Im ( ε ) < 0 .
ACS Paragon Plus Environment
(12)
ACS Photonics
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
A tetrahedral cluster made of 60 nm Au NPs is also unable to meet such condition (see Figure S4, Supporting Information). Of course, the further engineering of the cluster motifs, for example, (i) using Ag rather than Au and (ii) symmetry-breaking,14,15 could increase the strength of both the ED and MD resonances; thereby, the effective permittivity εeff and permeability µeff in turn could be reduced below 0 at a specific wavelength. However, the ED and MD resonances in the FCC lattice of cluster are spectrally detached as shown above. Consequently, in considering the dipolar coupling between plasmonic metamolecules, the ED and MD resonance-enabled modulation window of the effective permittivity εeff and the permeability µeff cannot sufficiently overlap. Therefore, achieving a negative refractive index at optical frequencies using magnetic metafluid is a more challenging proposition than suggested by a less comprehensive analysis. Our results indicate that a new design paradigm regarding metafluid is required to attain the unnatural negative index. One possible route toward this may be using magnetodielectric building blocks.43−46 This perspective will be addressed in a separate paper.
Conclusion We systematically exploit the practically accessible space of the metafluid’s effective parameters by using an advanced effective medium model (i.e., the 3D dressed polarizability-based ClausiusMossotti relation). The colloidal suspensions, which could be readily materialized with currently available pallets of building block motifs and values of ϕ, are found to be already efficient at greatly expanding accessible effective parameters into an unnatural regime (i.e., a near-zero-refractive index or unnaturally high refractive index). Despite the difficulty in achieving an unnaturally negative refractive index, optical metafluids can still be practically useful, for example, as a pivotal component of optofluidic devices. Throughout the past decade, microfluidic technology has been expertly implemented in the development of various optical components fully made out of fluids (i.e., the whispering gallery mode (WGM) cavity, waveguide, and microlens). These technologies are jointly addressed by the overarching concept of optofluidics.47−49 In particular, soft fluidity offers an otherwise impossible potential to dynamically adapt and reconfigure both the morphology and basic optical properties of photonic device components; thus, enabling flexible control over light flow within materials and devices. Moreover, the atomically smooth interfaces between fluidic components provide another advantage when trying to attain a high performance for optical devices (e.g., a high Q-factor of cavity). However, the fluid pallets used in optofluidics so far have been limited to naturally occurring water (n of 1.33) or oil such as Fluorinert FC-770 (n of 1.27) and toluene (n of 1.45); consequently, the refractive index contrast between two immiscible fluids forming optical components was relatively low (less than 0.2). Therefore, by the implementation of optical metafluids into common optofluidic device architectures, on-chip manipulation of light flow with an unprecedented degree of freedom can be achieved. Herein we coin the term “metafluidics” to describe this research direction.
Acknowledgement This work was supported by Samsung Research Funding Center for Samsung Electronics under Project Number SRFC-MA1402-09. ACS Paragon Plus Environment
Page 10 of 27
Page 11 of 27
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
ACS Photonics
Methods Determining effective parameters of metafluids (Clausius-Mossotti theory). Conventional parameter retrieval method assumes the periodic boundary condition (PBC).28-30 Although metamolecules are isotropic, the relative arrangements of metamolecules in metafluids are not isotropic. Therefore, the parameter retrieval, based on S-parameter, cannot be applied to optical metafluids. In this method, we derive the effective index determination method for optical metafluids by using the ClausiusMossotti relation. Polarizability retrieval technique for building blocks. We derive the retrieval method for an isotropic molecule (dielectric) which can be modeled by the relation p and m with scalar polarizaiblities αe and αm . We numerically retrieve the polarizability tensor of bi-anisotrpoic molecules using a commercial finite element method (FEM) software (COMSOL Multiphysics). Note that the retrieval method for the bi-anisotropic metamolecules can be found in elsewhere.50 The scattered far fields are k 2 eikr 1 E sc = n × m , (13) ( n × p ) × n − 4πε hε 0 r cµ0
H sc =
1
η
n × E sc ,
(14)
where n is the unit vector in the direction of observation, and ε h is the relative permittivity of the host medium. η = µ / ε is the medium impedance. We choose the observation direction to be +z. The incident field is x-polarized. This implies p = ( px , 0 ) , m = ( 0, m y ) , and n = zˆ . Therefore, Eq. (13) becomes E sc
+z
=
e ikr 1 1 − m y , m x ) = γ ( r ) p x + m y , 0 , ( ( px , p y ) − 4πε h ε 0 r cµ0 η0 k2
E sc
−z
1 = γ (r ) px − my , 0 , η0
(15)
(16)
eikr for convenience. Wavenumber is given by k = nk0 . Solving Eq. (15) and 4πε hε 0 r (16) for px, and my, we have where γ ( r ) ≡
k2
px =
my =
η0 Esc , x 2γ
(
(
+z
+z
− Esc , x
1 E sc , x 2γ
+ E sc , x
−z
−z
)=ε ε α E , h
)=µ µ α h
0
m
0
e
0
H 0 = α m E0 / η .
(17)
(18)
From the relations px = ε hε0αe E0 and my = µh µ0αm H0 = αmE0 / η , the numerically retrieved electric and magnetic polarizability are given by 1 (19) αe = Esc , x + z + Esc , x − z , 2γε hε 0 E0
(
)
ACS Paragon Plus Environment
ACS Photonics
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
αm =
Page 12 of 27
η0 2 Esc , x + z − Esc, x − z . 2γε hε 0 E0
(
)
(20)
Clausius-Mossotti relation for an electromagnetic dipole. First, we briefly introduce the derivation of the Clausius-Mossotti relation for an electric dipole. Local field due to electric dipoles is given by
E = E0 +
P 3εhε0
.
(21)
The electric dipole moment of a molecule is p = ε hε 0αeE , and thus the polarization is P P = N p = N ε h ε 0α e E = N ε h ε 0α e E 0 + 3 ε hε 0
.
(22)
Solving for P, we have
P=
Nαe ε hε 0E0 . 1 − Nαe / 3
(23)
From the constitutive relation, D = ε E0 = ε0E0 + P , we have P = ( ε − ε hε 0 ) E0 . Plugging this into Eq. (23) and solving for the polarizability α , we can arrive the Clausius-Mossotti relation,
Nαe ε − ε hε 0 = . 3 ε + 2ε hε 0
(24)
This can be rewritten for permittivity as follows:
ε=
3 + 2Nαe ε hε 0 . 3 − Nαe
By the duality of electromagnetism, exchanges
µ=
(25)
αe ↔ αm and ε ↔ µ leads to
3 + 2Nαm µh µ0 . 3 − Nαm
(26)
For anisotropic dipoles whose polarizabilities are written in diagonal tensors αt e and αt m , the effective permittivity and permeability tensor are straightforwardly obtained by51
t t 3 + 2Nαe ε= t ε hε 0 , 3 − Nαe
(27)
t t 3 + 2Nαm µ= t µh µ0 . 3 − Nαm
(28)
Numerical simulation. We calculate optical responses of plasmonic metamolecules using a commercial FEM software (COMSOL Multiphysics) and FDTD software (CST Microwave Studio 2014). We took tabulated data for the electric permittivity of gold in Jonhson and Christy.52 Incident plane wave with the intensity of the unity is applied to the entire calculation domain enclosed by the perfect matched layer (PML). The maximum size of meshes in the simulation is limited to a quarter of the wavelength. The distance between the detection point of the scattered fields and the center of the metamolecule was chosen to be larger than the longest wavelength in the ACS Paragon Plus Environment
Page 13 of 27
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
ACS Photonics
spectral range of interest. We also confirmed the convergence error is lower than the order of 104 ~10-3.
Supporting Information Details about effective medium theory, FOM of optical metafluids, and effective parameter analysis on tetrahedral cluster-based optical metafluids are available in Supporting Information. This material is available free of charge via the internet at http://pub.acs.org.
References 1. Alù, A.; Salandrino, A.; Engheta, N. Negative Effective Permeability and Left-Handed Materials at Optical Frequencies. Opt. Express 2006, 14, 1557−1567. 2. Urzhumov, Y. A.; Shvets, G.; Fan, J.; Capasso, F.; Brandl, D.; Nordlander, P. Plasmonic Nanoclusters: a Path towards Negative-Index Metafluids. Opt. Express 2007, 15, 14129−14145. 3. Alù, A.; Engheta, N. Dynamical Theory of Artificial Optical Magnetism Produced by Rings of Plasmonic Nanoparticles. Phys. Rev. B 2008, 78, 085112. 4. Merlin, R. Metamaterials and the Landau-Lifshiftz Permeability Argument: Large Permittivity Begets High-Frequency Magnetism. Proc. Natl. Acad. Sci. USA 2009, 106, 1693−1698. 5. Simovski, C. R.; Tretyakov, S. A. Model of Isotropic Resonant Magnetism in the Visible Range Based on Core-Shell Clusters. Phys. Rev. B 2009, 79, 045111. 6. Alù, A.; Engheta, N. The Quest for Magnetic Plasmons at Optical Frequencies. Opt. Express 2009, 17, 5723−5730. 7. Sheikholeslami, S. N.; Carcía-Etxarri, A.; Dionne, J. A. Controlling the Interplay of Electric and Magnetic Modes via Fano-like Plasmon Resonances. Nano Lett. 2011, 11, 3927−3934. 8. Vallecchi, A.; Albani, M.; Capolino, F. Collective Electric and Magnetic Plasmonic Resonances in Spherical Nanoclusters. Opt. Express 2011, 19, 2754−2772. 9. Fan, J. A.; Wu, C.; Bao, K.; Bao, J.; Bardhan, R.; Halas, N. J.; Manoharan, V. N.; Nordlander, P.; Shvets, G.; Capasso, F. Self-Assembled Plasmonic Nanoparticle Clusters. Science 2010, 328, 1135−1138. 10. Mühlig, S.; Rockstuhl, C.; Yannopapas, V.; Bürgi, T.; Shalkevich, N.; Lederer, F. Optical Properties of a Fabricated Self-Assembled Bottom-Up Bulk Metamaterial. Opt. Express 2011, 19, 9607−9616. 11. Mühlig, S.; Cunningham, A.; Scheeler, S.; Pacholski, C.; Bürgi, T.; Rockstuhl, C.; Lederer, F. Self-Assembled Plasmonic Core-Shell Clusters with an Isotropic Magnetic Dipole Response in the Visible Range. ACS Nano 2011, 5, 6586−6592. ACS Paragon Plus Environment
ACS Photonics
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
12. Dintinger, J.; Mühlig, S.; Rockstuhl, C.; Scharf, T. A Bottom-Up Approach to Fabricate Optical Metamaterials by Self-Assembled Metallic Nanoparticles. Opt. Mater. Express 2012, 2, 269−278. 13. Urban, A. S.; Shen, X.; Wang, Y.; Large, N.; Wang, H.; Knight, M. W.; Nordlander, P.; Chen, H.; Halas, N. J. Three-Dimensional Plasmonic Nanoclusters. Nano Lett. 2013, 13, 4399−4403. 14. Sheikholeslami, S. N.; Alaeian, H.; Koh, A. L.; Dionne, J. A. A Metafluid Exhibiting Strong Optical Magnetism. Nano Lett. 2013, 13, 4137−4141. 15. Yang, S.; Ni, X.; Yin, X.; Kante, B.; Zhang, P.; Zhu, J.; Wang, Y.; Zhang, X. FeedbackDriven Self-Assembly of Symmetry-Breaking Optical Metamaterials in Solution. Nat. Nanotechnol. 2014, 9, 1002−1006. 16. Qian, Z.; Hastings, S. P.; Li, C.; Edward, B.; McGinn, C. K.; Engheta, N.; Fakhraai, Z.; Park, S.-J. Raspberry-like Metamolecules Exhibiting Strong Magnetic Resonances. ACS Nano 2015, 9, 1263−1270. 17. Yang S.; Wang, Y.; Ni, X.; Zhang, X. Optical Modulation of Aqueous Metamaterial Properties at Large Scale. Opt. Express 2016, 23, 28736−28741. 18. Gomez-Graña, S.; Le Beulze, A.; Treguer-Delapierre, M.; Mornet, S.; Duguet, E.; Grana, E.; Cloutet, E.; Hadziioannou, G.; Leng, J.; Salmon, J.-B.; Kravets, V. G.; Grigorenko, A. N.; Peyyety, N. A.; Ponsinet, V.; Richetti, P.; Baron, A.; Torrent, D.; Barois, P. Hierarchical Self-Assembly of a Bulk Metamaterial Enables Isotropic Magnetic Permeability at Optical Frequencies. Mater. Horiz. 2016, 3, 596−601. 19. Höller, R. P. M.; Dulle, M.; Thomä, S.; Mayer, M.; Steiner, A. M.; Förster, S.; Fery, A.; Kuttner, C.; Chanana, M. Protein-Assisted Assembly of Modular 3D Plasmonic Raspberrylike Core/Satellite Nanoclusters: Correlation of Structure and Optical Properties. ACS Nano 2016, 10, 5740−5750. 20. Wood, W. W.; Jacobson, J. D. Preliminary Results from a Recalculation of the Monte Carlo Equation of State of Hard Spheres. J. Chem. Phys. 1957, 27, 1207−1208. 21. Alder, B. J.; Wainwright, T. E. Phase Transition for a Hard Sphere System. J. Chem. Phys. 1957, 27, 1208−1209. 22. Pusey, P. N.; van Megen, W. Phase Behaviour of Concentrated Suspensions of Nearly Hard Colloidal Spheres. Nature 1986, 320, 340−342. 23. Lee, Y.-J.; Schade, N. B.; Sun, L.; Fan, J. A.; Bae, D. R.; Mariscal, M. M.; Lee, G.; Capasso, F.; Sacanna, S.; Manoharan, V. N.; Yi, G.-R. Ultrasmooth, Highly Spherical Monocrystalline Gold Particles for Precision Plasmonics. ACS Nano 2013, 7, 11064−11070. 24. O’Brien, M. N.; Jones, M. R.; Brown, K. A.; Mirkin, C. A. Universal Noble Metal Nanoparticle Seeds Realized Through Iterative Reductive Growth and Oxidative Dissolution Reactions. J. Am. Chem. Soc. 2014, 136, 7603−7606.
ACS Paragon Plus Environment
Page 14 of 27
Page 15 of 27
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
ACS Photonics
25. Kim, D.-K.; Hwang, Y. J.; Yoon, C.; Yoon, H.-O.; Chang, K. S.; Lee, G.; Lee, S.; Yi, G.-R. Experimental Approach to the Fundamental Limit of the Extinction Coefficients of UltraSmooth and Highly Spherical Gold Nanoparticles. Phys. Chem. Chem. Phys. 2015, 17, 20786−20794. 26. Kim, M.; Lee, S.; Lee, J.; Kim, D. K.; Hwang, Y. J.; Lee, G.; Yi, G.-R.; Song, Y. J. Deterministic Assembly of Metamolecules by Atomic Force Microscope-Enabled Manipulation of Ultra-Smooth, Super-Spherical Gold Nanoparticles. Opt. Express 2015, 23, 12766−12776. 27. Henzie, J.; Grünwald, M.; Widmer-Cooper, A.; Geissler, P. L.; Yang, P. Self-Assembly of Uniform Polyhedral Silver Nanocrystals into Densest Packings and Exotic Superlattices. Nat. Mater. 2012, 11, 131−137. 28. Chen, X.; Grzegorczyk, T. M.; Wu, B.-I.; Pacheco, J.; Kong J. Robust Method to Retrieve the Constitutive Effective Parameters of Metamaterials. Phys. Rev. E 2004, 70, 016608. 29. Smith, D. R.; Vier, D. C.; Koschny, Th,; Soukoulis, C. M. Electromagnetic Parameter Retrieval from Inhomogeneous Metamaterials. Phys. Rev. E 2005, 71, 036617. 30. Liu, R.; Cui, T. J.; Huang, D.; Zhao, B.; Smith, D. R. Description and Explanation of Electromagnetic Behaviors in Artificial Metamaterials Based on Effective Medium Theory. Phys. Rev. E 2007, 76, 026606. 31. Asadchy, V. S; Faniayeu, I. A.; Younes R.; Tretyakov, S. A. Determining Polarizability Tensors for an Arbitrary Small Electromagnetic Scatterer. Phot. Nano. Fund. Appl. 2014, 12, 4, 298. 32. Arango, F. B.; Koenderink A. F. Polarizability tensor retrieval for magnetic and plasmonic antenna design, New J. Phys 2013, 15, 073023. 33. Jackson, J. D. Classical Electrodynamics, 3rd Eds; John Wiley & Sons, INC., 1999. 34. Ross, M. B.; Blaber, M. G.; Schatz, G. C. Using Nanoscale Anisotropic to Engineer the Optical Response of Thee-Dimensional Plasmonic Metamaterials. Nat. Commun. 2014, 5, 4090. 35. Yoo, S.; Park, Q-H. Effective Permittivity for Resonant Plasmonic Nanoparticle Systems via Dressed Polarizability. Opt. Express 2012, 20, 16480−16489. 36. Novotny, L.; Hecht, B. Principles of Nano-Optics, 2nd Eds; Cambridge University Press, 2012. 37. Lee, S. Colloidal Superlattices for Unnaturally High-Index Metamaterials at Broadband Optical Frequencies. Opt. Express 2015, 23, 28170−28181. 38. Chung, K.; Kim, R.; Chang, T.; Shin, J. Optical Effective Media with Independent Control of Permittivity and Permeability based on Conductive Particles. Appl. Phys. Lett. 2016, 109, 021114.
ACS Paragon Plus Environment
ACS Photonics
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
39. Kubo, S.; Diaz, A.; Tang, Y.; Mayer, T. S.; Khoo, I. C.; Mallouk, T. E. Tunability of the Refractive Index of Gold Nanoparticle Dispersions. Nano Lett. 2007, 7, 3418−3423. 40. Yablonovitch, E. Statistical Ray Optics. J. Opt. Soc. A 1982, 72, 899−907. 41. Abbe, E. Beitra ̈ge zur theorie des mikroskops und der mikroskopischen wahrnehmung. Archiv F. Mikrosk. Anatom. 1973, 9, 413−418. 42. Manoharan, V. N.; Elsesser, M. T.; Pine, D. J. Dense Packing and Symmetry in Small Clusters of Microspheres. Science 2003, 301, 483−487. 43. Fu, Y. S.; Kuznetsov, A. I.; Miroshnichenko, A. E.; Yu, Y. F.; Zhang, J.; Luk’yanchuk, B. Directional Visible Light Scattering by Silicon Nanoparticles. Nat. Commun. 2013, 4, 1527. 44. Shi, L.; Harris, J. T.; Fenollosa, R.; Rodriguez, I.; Lu, X.; Korgel, B. A.; Meseguer, F. Monodisperse Silicon Nanocavities and Photonic Crystals with Magnetic Response in the Optical Region. Nat. Commun. 2013, 4, 1904. 45. Person, S.; Jain, M.; Lapin, Z.; Sáenz, J. J.; Wicks, G.; Novotny, L. Demonstration of Zero Optical Backscattering from Single Nanoparticles. Nano Lett. 2013, 13, 1806−1809. 46. Cho, Y.; Huh, J.-H.; Park, K. J.; Kim, K.; Lee, J.; Lee, S. Using Highly Uniform and Smooth Selenium Colloids as Low-Loss Magnetodielectric Building Blocks of Optical Metafluids. Opt. Express 2017, 8, 395−401. 47. Tang, S. K. Y.; Stan, C. A.; Whitesides, G. M. Dynamically Reconfigurable Liquid-Core Liquid-Cladding Lens in a Microfluidic Channel. Lab Chip. 2008, 8, 395−401. 48. Tang, S. K. Y.; Li, Z.; Abate, A. R.; Agresti, J. J.; Weitz, D. A.; Psaltis, D.; Whitesides, G. M. A Multi-Color Fast Switching Microfluidic Droplet Dye Laser. Lab Chip. 2009, 9, 2767−2771. 49. Tang, S. K. Y.; Mayers, B. T.; Vezenov, D. V.; Whitesides, G. M. Optical Waveguiding Using Thermal Gradients Across Homogeneous Liquids in Microfluidic Channels. Appl. Phys. Lett. 2006, 88, 061112. 50. Asadchy, V. S.; Faniayeu, I. A.; Ra’Di, Y.; Tretyakov, S. A. Determining polarizability tensors for an arbitrary small electromagnetic scatterer. Photonics Nanostructures Fundam. Appl. 2014, 12, 298–304. 51. Palffy-Muhoray, P.; Balzarini, D. A. The Clausius-Mossotti relation for anisotropic molecular fluids. Can. J. Phys. 1981, 59, 375–377. 52. Johnson, P. B.; Christy, R. W. Optical Constants of the Noble Metals. Phys. Rev. B 1972, 6, 4370-4379.
ACS Paragon Plus Environment
Page 16 of 27
Page 17 of 27
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
ACS Photonics
Figure 1. Schematic for (a) building blocks and (b) phase transition of optical metafluids. Individual gold (Au) nanoparticles (NPs) and cluster made of Au NPs can act as meta-atoms and metamolecules (i.e., building blocks), respectively. According to the volume fraction of the building blocks (ϕ), the phase of optical metafluid is changed from the “fluid” phase to the “fluid+crystal” phase. The effective parameters of optical metafluids should be rationalized with respect to the phase of optical metafluids.
ACS Paragon Plus Environment
ACS Photonics
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Figure 2. (a) Schematic for face-centered-cubic (FCC) crystal made of individual Au NPs. (b) Schematic for 30 nm Au NP-based FCC crystal, designed to be dispersed in air (ϕ of 0.56). This model was used for the validation of 3D dressed polarizability-based effective medium by comparison it with numerical retrieval method. (c) Real and (d) imaginary parts of the effective refractive index of the 30 nm Au NP-based FCC crystal in air (ϕ of 0.56), calculated by (i) original Clausius-Mossotti relation (red line), (ii) 3D dressed polarizability-based Clausius-Mossotti relation (blue line), and (iii) numerical retrieval method (green line).
ACS Paragon Plus Environment
Page 18 of 27
Page 19 of 27
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
ACS Photonics
Figure 3. (a) The electric dipole (ED) polarizabilities for 20 nm, 30 nm, and 60 nm Au NPdispersed water (i.e., the “fluid” phase-electric metafluid). ED polarizabilities are normalized with respect to volume. (b-d) Effective refractive indices of the “fluid” phase-electric metafluid with respect to ϕ. 20 nm (b), 30 nm (c), and 60 nm (d) Au NPs act as the electric meta-atoms.
ACS Paragon Plus Environment
ACS Photonics
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Figure 4. (a) Schematic for three primitive axes of FCC crystal. (b) Schematic of the possibilities for enhanced or quenched ED and magnetic dipole (MD) dressed polarizabilities. (c) Spatial distribution of electric field intensity when FCC is illuminated by three different polarizations along
ACS Paragon Plus Environment
Page 20 of 27
Page 21 of 27
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
ACS Photonics
primitive axes 1, 2, and 3. Three kinds of “nearest neighbor” neighboring interactions are characterized.
Figure 5. (a-b) Schematic for the “fluid+crystal” phase-electric metafluids with respect to ϕ. Individual Au NP should be organized into FCC lattice than other lattices due to the packing argument; the gap between Au NP can be changed according to ϕ. (c) ED polarizabilities of the “fluid+crystal” phase-electric metafluids (ϕ of 0.74) with respect to the different incident electric fields along the primitive axes 1, 2, and 3. Herein, 30 nm Au NPs are used as electric meta-atoms. (d-f) The effective refractive indices (real part) of the 30 nm AuNP-based “fluid+crystal” phaseelectric metafluids with respect to ϕ, when the structure is illuminated by the different incident electric fields along the primitive axes 1 (d), 2 (e), and 3 (f).
ACS Paragon Plus Environment
ACS Photonics
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 ACS Paragon Plus Environment
Page 22 of 27
Page 23 of 27
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
ACS Photonics
Figure 6. (a) ED polarizabilities of the 20 nm Au NP “fluid+crystal” phase-electric metafluids (ϕ of 0.74) with respect to different incident electric fields along the primitive axes 1, 2, and 3. (b-d) The effective refractive indices (real part) of the 20 nm AuNP “fluid+crystal” phase-electric metafluids with respect to ϕ, when the structure is illuminated by the different incident electric fields along the primitive axes 1 (b), 2 (c), and 3 (d). (e) ED polarizabilities of the 60 nm Au NP “fluid+crystal” phase-electric metafluids (ϕ of 0.74) with respect to different incident electric fields along the primitive axes 1, 2, and 3. (f-h) The effective refractive indices (real part) of the 60 nm AuNP “fluid+crystal” phase-electric metafluids with respect to ϕ, when the structure is illuminated by the different incident electric fields along the primitive axes 1 (f), 2 (g), and 3 (h).
ACS Paragon Plus Environment
ACS Photonics
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Figure 7. (a) Spatial mapping of electric field (small blue arrows) and magnetic field intensities for the tetrahedral and octahedral clusters made of 60 nm Au NPs. The gap between Au NPs are designed to be 1 nm. (b) ED and MD polarizabilities of the tetrahedral and octahedral clusters made of 60 nm Au NPs (ϕ of 0.1). (c-d) Schematic for the “fluid” phase-magnetic metafluids, in which octahedral clusters are dispersed in water. (e-h) The collective set of the effective parameters of the “fluid” phase-magnetic metafluids with respect to ϕ. ACS Paragon Plus Environment
Page 24 of 27
Page 25 of 27
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
ACS Photonics
Figure 8. (a) Schematic for the “fluid+crystal” phase-magnetic metafluids. The magnetic building blocks, i.e., octahedral clusters made of 60 nm Au NPs, should be organized into FCC crystal rather than other lattices. Herein, the octahedral clusters can be encapsulated by a spherical shell.13 (b) Schematic of the possibilities for the partially enhanced and quenched ED/MD polarizabilities between the FCC-crystalized octahedral cluster. (c) The comparison between bare and 3D dressed polarizabilities for the ED of the FCC-crystalized octahedral cluster (ϕ of 0.74). (d) The comparison between the bare and 3D dressed polarizabilities for the MD of the FCC-crystalized octahedral cluster (ϕ of 0.74). (e) The effective permittivity εeff for each different primitive axis of the “fluid+crystal” phase-magnetic metafluids (ϕ of 0.74). (f) The effective permeability µeff for each different primitive axis of the “fluid+crystal” phase-magnetic metafluids (ϕ of 0.74). ACS Paragon Plus Environment
ACS Photonics
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Figure 9. (a-b) Schematic for the “fluid+crystal” phase-magnetic metafluids with respect to ϕ. (c-h) The effective refractive indices (both real and imaginary parts) of the “fluid+crystal” phasemagnetic metafluids with respect to ϕ, when the structure is illuminated by the different incident electric fields along the primitive axis 1 (c-d), 2 (e-f), and 3 (g-h). ACS Paragon Plus Environment
Page 26 of 27
Page 27 of 27
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
ACS Photonics
Table of Contents
ACS Paragon Plus Environment
