Comprehensive three-dimensional analysis of surface plasmon polariton modes at uniaxial liquid crystal-metal interface

Yin-Ray Yen; Tsun-Hsiun Lee; Zheng-Yu Wu; Tsung-Hsien Lin; Yu-Ju Hung

doi:10.1364/OE.23.032377

1. Introduction

The analysis of surface plasmon waves at the interface between an anisotropic dielectric layer and a noble metal is an intriguing and complex phenomenon in the field of metamaterial research [1–10]. Anisotropy on a metal surface contributes to distinct functions, such as scattering-free two-dimensional wave optics [11], superresolution [1,12,13], extended surface plasmon polariton (SPP) propagation length [14,15], nonlinear optical phenomena [16], sensitive field probing ability [3] and tunable reflective spectrums [17]. From a practical perspective, an anisotropic layer can be fabricated using e-beam lithography, bulk crystal growth, or through the application of spin-coated materials, such as liquid crystals (LC) or anisotropic organic layers. LC materials are an obvious choice with regard to the optical and electrical tailoring of the anisotropic properties. No physical alteration in the structure is required to modulate the optical properties of the resulting devices [18]. Many papers on such scheme emphasize on the tunability of the optical spectrum [17,19–22]. In [23], the researchers presented a thorough discussion on optical behavior at the interface between a substrate and a layer of anisotropic crystal undergoing three-dimensional (3D) rotation. Until recently, two-dimensional (2D) rotations of LCs on a metal surface have specifically been derived and the associated eigen SPP waves were discussed [24–28]. Different layer combinations (two interfaces) were discussed in [15,29]. [15,23–29] investigated a single interface (with 2D and 3D rotation of the anisotropic layer) and two interfaces (with 2D rotation of the material undergoing analysis). A lack of analysis on the 3D rotation of LC molecules on the thin metal film (two interfaces) restricts our understanding of this category phenomenon. Comprehensive wave analysis associated with the 3D rotation of LC optical axis (OA) has yet to be achieved. In this paper, we extend the scope of [15,25] by performing 3D analysis for the LC on a thin metal film deposited on a glass susbtrate, specifically for E7 material, which has n_o = 1.52 and n_e = 1.75 at 532nm. In our paper, we observed strong agreement between experiment and simulation results, thereby demonstrating the efficacy of the derived formulation. Most metallic structures that are incorporated within a liquid crystal layer are fabricated on a thin metal film and supported by a substrate. This fact underlines the importance of elucidating the two-interface problem with 3D LC OA tuning.

2. Rotation of elliptical k-space

Figure 1(a) defines the SPP propagation direction using XYZ global coordinates and the LC rotation angles with X”Y”Z” local coordinates. The OA of the LC is along the X” direction, as shown in Fig. 1(b). The interface lies on the XY plane and the k-vector indicates the SPP eigen mode direction of propagation.

Fig. 1 (a) XYZ -global coordinate, and X”Y”Z”-LC coordinate; (b) k-vector indicating the eigen mode (SPP mode) propagation direction. The optical axis of the LC lies along the X” axis.

Download Full Size | PDF

The rotation matrix between the two coordinates is defined as follows:

[\begin{matrix} X^{″} \\ Y^{″} \\ Z^{″} \end{matrix}] = [\begin{matrix} \cos θ \cos ϕ & \cos θ \sin ϕ & \sin θ \\ - \sin ϕ & \cos ϕ & 0 \\ - \sin θ \cos ϕ & - \sin θ \sin ϕ & \cos θ \end{matrix}] [\begin{matrix} X \\ Y \\ Z \end{matrix}] = R(θ, ϕ) [\begin{matrix} X \\ Y \\ Z \end{matrix}]

In [28], Yeh described how the dispersion relationship of uniaxial crystals could be categorized using two equations:

o-light:

\frac{k_{x} {^{' ​'}}^{2}}{k_{0}^{2}} + \frac{k_{y} {^{' ​'}}^{2}}{k_{0}^{2}} + \frac{k_{z} {^{' ​'}}^{2}}{k_{0}^{2}} = n_{o}^{2}

in which the characteristic E-field is expressed as

{E^{″}}_{o} = [\begin{matrix} {E^{″}}_{x} \\ {E^{″}}_{y} \\ {E^{″}}_{z} \end{matrix}] = [\begin{matrix} 0 \\ - {k^{″}}_{z} \\ {k^{″}}_{y} \end{matrix}]

e-light:

\frac{{k^{″}}_{x}^{^{2}}}{n_{o}^{2} k_{0}^{2}} + \frac{{k^{″}}_{y}^{^{2}}}{n_{e}^{2} k_{0}^{2}} + \frac{{k^{″}}_{z}^{^{2}}}{n_{e}^{2} k_{0}^{2}} = 1

in which the characteristic E-field is expressed as

{E^{″}}_{e} = [\begin{matrix} {E^{″}}_{x} \\ {E^{″}}_{y} \\ {E^{″}}_{z} \end{matrix}] = \frac{1}{{k^{″}}_{x} (k^{2} - n_{o}^{2} k_{0}^{2})} [\begin{matrix} {k^{″}}_{x}^{^{2}} - n_{o}^{2} k_{0}^{2} \\ {k^{″}}_{x} {k^{″}}_{y} \\ {k^{″}}_{x} {k^{″}}_{z} \end{matrix}]

To solve the eigen modes of surface plasmon polaritons at the interface between the metal and LC layer, we assumed that the SPP propagates along the x-direction (global coordinates), decaying exponentially into both mediums in the z-direction with

(k_{x}, 0, k_{z ​ o | z e})

, as shown in Fig. 1(b).

Application of the rotation matrix to the k-space dispersion relation results in the following:

For o-light:

k_{z o} = \sqrt{n_{o}^{2} k_{0}^{2} - k_{x}^{2}}

For e-light:

\begin{array}{l} k_{z e} = \\ \frac{(n_{o}^{2} - n_{e}^{2}) \cos θ \sin θ \cos ϕ k_{x} + \sqrt{n_{o}^{2} n_{e}^{2} (n_{e}^{2} \sin^{2} θ + n_{o}^{2} \cos^{2} θ) k_{0}^{2} - [n_{o}^{2} n_{e}^{2} (\sin^{2} θ \sin^{2} ϕ + \cos^{2} ϕ) + n_{o}^{4} \cos^{2} θ \sin^{2} ϕ] k_{x}^{2}}}{n_{e}^{2} \sin^{2} θ + n_{o}^{2} \cos^{2} θ} \end{array}

Any E fields within the global coordinates could be expressed using combinations of

{E^{″}}_{o} and {E^{″}}_{e}

. Applying the rotational matrix

R(θ, ϕ)

to

{E^{″}}_{o} and {E^{″}}_{e}

, and substituting k” for the global k-vector makes it possible to correlate these two eigen field expressions to global coordinates using the global k-vector.

In the three layer system in Fig. 2, z>d/2 is the LC layer, d/2>z>-d/2 is the metal layer, and z<-d/2 is the isotropic layer. A three layer derivation from uniform layers can be found in [30].

Fig. 2 Structure of LC-metal-isotropic medium

Download Full Size | PDF

The SPP k-vector is defined as

k = {\begin{cases} (β, 0, i q_{L C}) z>d/2 \\ (β, 0, \pm i q_{m}) -d/2<z<d/2 \\ (β, 0, - i q_{d}) z < - d / 2 \end{cases}

where β is the same as k_spp in later notation.

The E field can be expressed as

E = {\begin{matrix} E_{_{0}} e^{i (β x - ω t) - q_{_{L C}} z}, z > d / 2 \\ \begin{array}{l} E_{_{0}} e^{i (β x - ω t) \pm q_{_{m}} z}, - d / 2 < z < d / 2 \\ E_{_{0}} e^{i (β x - ω t) + q_{_{d}} z}, z < - d / 2 \end{array} \end{matrix}

In the LC layer,

q_{L C}

satisfies two different dispersion relations, resulting in two values:

q_{o}

and

q_{e}

same as Eq. (6) and Eq. (7). The E field in each layer is expressed as a constant M_1~8 such that the corresponding characteristic E field and H field can be derived using Maxwell’s equations.

E_{O} = M_{1} [\begin{matrix} i \frac{q_{o}}{k_{0}} \sin ϕ \cos θ \\ \frac{β}{k_{0}} \sin θ - i \frac{q_{o}}{k_{0}} \cos θ \cos ϕ \\ - \frac{β}{k_{0}} \sin ϕ \cos θ \end{matrix}] e^{i (β x - ω t) - q_{o} z}

\begin{array}{l} E_{e} = M_{2} [\begin{matrix} \frac{q_{o}^{2}}{k_{0}^{2}} \cos θ \cos ϕ + i \frac{q_{e} β}{k_{0}^{2}} \sin θ \\ - ε_{o} \sin ϕ \cos θ \\ i \frac{q_{e} β}{k_{0}^{2}} \cos θ \cos ϕ - \frac{q_{e}^{2} + ε_{o} k_{0}^{2}}{k_{0}^{2}} \sin θ \end{matrix}] e^{i (β x - ω t) - q_{e} z} \end{array}

E_{_{m}}^{T E \pm} = - M_{3 | 4} [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}] e^{i (β x - ω t) \mp q_{m} z}

E_{_{m}}^{T M \pm} = - M_{5 | 6} [\begin{matrix} \mp i \frac{q_{m}}{k_{0}} \\ 0 \\ \frac{β}{k_{0}} \end{matrix}] e^{i (β x - ω t) \mp q_{m} z}

E_{d}^{T E} = M_{7} [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}] e^{i (β x - ω t) + q_{d} z}

E_{d}^{T M} = M_{8} [\begin{matrix} i \frac{q_{d}}{k_{0}} \\ 0 \\ \frac{β}{k_{0}} \end{matrix}] e^{i (β x - ω t) + q_{d} z}

The fact that the upper and lower interfaces are asymmetrical means that the field associated with the metal is also asymmetrical. Therefore

M_{3} \neq M_{4} and M_{5} \neq M_{6}

. The application of field continuity conditions means that the following determinant must be 0, and

β

can be acquired numerically.

\begin{array}{l} olight elight {TE}_{m}^{+} {TE}_{m}^{-} {TM}_{m}^{+} {TM}_{m}^{-} {TE}_{d}^{-} {TM}_{d}^{-} \\ \begin{matrix} E_{x \frac{d}{2}} \\ E_{y \frac{d}{2}} \\ H_{x \frac{d}{2}} \\ H_{y \frac{d}{2}} \\ E_{x - \frac{d}{2}} \\ E_{y - \frac{d}{2}} \\ H_{x - \frac{d}{2}} \\ H_{y - \frac{d}{2}} \end{matrix} [\begin{matrix} \frac{i q_{o} \sin ϕ \cos θ}{k_{0}} e^{- \frac{q_{o} d}{2}} & \frac{q_{o}^{2} \cos θ \cos ϕ + i q_{e} β \sin θ}{k_{0}^{2}} e^{- \frac{q_{e} d}{2}} & 0 & 0 & \frac{- i q_{m}}{k_{0}} e^{- \frac{q_{m} d}{2}} & \frac{i q_{m}}{k_{0}} e^{\frac{q_{m} d}{2}} & 0 & 0 \\ \frac{β \sin θ - i q_{o} \cos θ \cos ϕ}{k_{0}} e^{- \frac{q_{o} d}{2}} & - ε_{o} \sin ϕ \cos θ e^{- \frac{q_{e} d}{2}} & e^{- \frac{q_{m} d}{2}} & e^{\frac{q_{m} d}{2}} & 0 & 0 & 0 & 0 \\ \frac{- q_{o}^{2} \cos θ \cos ϕ - i q_{o} β \sin θ}{k_{0}^{2}} e^{- \frac{q_{o} d}{2}} & \frac{i q_{e} ε_{o} \sin ϕ \cos θ}{k_{0}} e^{- \frac{q_{e} d}{2}} & \frac{- i q_{m}}{k_{0}} e^{- \frac{q_{m} d}{2}} & \frac{i q_{m}}{k_{0}} e^{\frac{q_{m} d}{2}} & 0 & 0 & 0 & 0 \\ ε_{o} \sin ϕ \cos θ e^{- \frac{q_{o} d}{2}} & \frac{β ε_{o} \sin θ - i q_{e} ε_{o} \cos θ \cos ϕ}{k_{0}} e^{- \frac{q_{e} d}{2}} & 0 & 0 & - ε_{m} e^{- \frac{q_{m} d}{2}} & - ε_{m} e^{\frac{q_{m} d}{2}} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{- i q_{m}}{k_{0}} e^{\frac{q_{m} d}{2}} & \frac{i q_{m}}{k_{0}} e^{- \frac{q_{m} d}{2}} & 0 & \frac{i q_{d}}{k_{0}} e^{- \frac{q_{d} d}{2}} \\ 0 & 0 & e^{\frac{q_{m} d}{2}} & e^{- \frac{q_{m} d}{2}} & 0 & 0 & e^{- \frac{q_{d} d}{2}} & 0 \\ 0 & 0 & \frac{- i q_{m}}{k_{0}} e^{\frac{q_{m} d}{2}} & \frac{i q_{m}}{k_{0}} e^{- \frac{q_{m} d}{2}} & 0 & 0 & \frac{i q_{d}}{k_{0}} e^{- \frac{q_{d} d}{2}} & 0 \\ 0 & 0 & 0 & 0 & - ε_{m} e^{\frac{q_{m} d}{2}} & - ε_{m} e^{- \frac{q_{m} d}{2}} & 0 & - ε_{d} e^{- \frac{q_{d} d}{2}} \end{matrix}] [\begin{matrix} M_{1} \\ M_{2} \\ M_{3} \\ M_{4} \\ M_{5} \\ M_{6} \\ M_{7} \\ M_{8} \end{matrix}] = 0 \end{array}

A simulation case is given:

λ = 532 n m, d = 40 n m, ε_{m} = - 4.37 + j * 1 . 63, n_{o} = 1.52, n_{e} = 1.75

and

(θ, ϕ {)=(0-90}^{o} {,80}^{o})

, the resulted dispersion curves are shown below in Fig. 3. The two-interface problem introduces symmetrical as well as antisymmetrical modes.

Fig. 3 Eigen values resolved using Eq. (16)

Download Full Size | PDF

3. Experimental Demonstration

PMMA gratings of various pitches with a step height of 170nm were fabricated using an E-beam lithography system (FEI Inspectf50). The pitch of the grating was a value in the range 350nm to 450nm with every 10nm spacing. A single 40nm layer of Au film was deposited on top of the gratings, whereupon an LC (E7) layer with a thickness of 6μm was sandwiched between the metallic corrugations and an ITO glass superstrate, as shown in Fig. 4. When LC is @ON, the OA lies perpendicular to the metallic surface, regardless of whether it lies over an edge or in a flat area of the grating. When LC is @OFF, OA lies along the nano-grooves parallel to the direction of rubbing on the upper polyimide layer on the ITO glass. According to Kretschamn geometry,the total-internal-reflection (TIR) at the first dielectric-1/metal interface provides a superluminal k-vector suitable for the coupling of light to the SPP mode at the (adjacent) second dielectric-2/metal interface where the refractive index of the dielectric-1 is higher than that of dielectric-2. In this study, we used PMMA gratings and liquid crystals (n = 1.52~1.75) as the first and second dielectric layers with little contrast in refractive index. Therefore, a grating structure is necessary in order to provide an additional k-vector to enable the incident light to match the k-vector of the SPP waves.

Fig. 4 LC cell with metallic grating structure. 532nm light is incident in TM polarization. When LC is in @ON state, the OA of the LC lies perpendicular to the metal surface, regardless of whether it is over the edge or the flat area of the grating.

Download Full Size | PDF

Figure 5 outlines the setup of the experiment. The sample was placed on top of a glass prism in accordance with Kretschamn geometry. An incident laser (TM polarized) with wavelength of 532nm was coupled to the grating region and the transmission was tuned by applying 10V to the LC layer. The stripes of the grating lay along the y-axis. The angle between the projection of the incidence on the x-y plane and the y-axis was ϕ_pr and the angle between the incidence and the z direction was θ_pr, as shown in Fig. 5(b). The analyzer in front of the imaging camera is perpendicular to the grating stripes. The corresponding SPP coupling determined by the momentum matching condition is listed in Eq. (17).

Fig. 5 (a) Prism coupling setup- grating stripe is along y-direction. (b) Definition of incident light direction, where “pr” refers to prism.

Download Full Size | PDF

k_{c o m p o s i t e}^{2} = {(n_{p r i s m} k_{0} \sin θ_{p r} \cos ϕ_{p r})}^{2} + {(n_{p r i s m} k_{0} \sin θ_{p r} \sin ϕ_{p r} \pm m \frac{2 π}{Λ})}^{2}

Figure 6 presents five groups of data with incident angles $θ_{p r}$ and $ϕ_{p r}$ ranging from 37° to 47° and 84° to 89°, respectively. Figure 6(a) shows the grating arranglement. Figure 6(b) presents a set of data collected using an objective lens with NA = 0.5, whereas (c)~(f) were collected using an objective lens with NA = 0.3. Figure 6(b) illustrates the pitches that were responsive to applied voltage (390nm/400nm/410nm); no other pitches responded to the applied voltage. As shown in Figs. 6(c)~(f), an increase in incident angle led to a reduction in the smallest transmissive pitch, which indicates that m in Eq. (17) must be negative.

Fig. 6 Large transmission through grating region: (a) pitches ranging from 350 to 450nm (an increase in $θ_{p r}$ resulted in smaller pitches in the transmissive region, which indicates that the momentum matching integer m must be negative); (b) collected under NA = 0.5; (c)~(f) collected under NA = 0.3

Download Full Size | PDF

Figure 7 presents transmission T_-1 with m = −1 order. The LC dielectric layer was maintained uniformly at n = 1.52 or n = 1.75. For the sake of simplicity, these values were computed using FEMLAB COMSOL with a 2D grating and no ϕ rotation (ϕ = 90°). 0th-order transmission was beyond the collection angle defined by an objective lens with NA = 0.3. Thus, only the m = −1 order could be collected and differences in the incidence angle resulted in different cut-off pitches, as indicated by the arrows in the figure. The transmission of light through pitches greater than the values indicated by the arrow results in smaller diffraction angles, thereby permitting collection using an objective lens with NA = 0.3. In the case of a fixed incidence angle, T_-1@ n_LC = 1.52 (similar to @OFF) was higher than T_-1@n_LC = 1.75 (similar to @ON), which is the opposite of what was observed in Fig. 6. These results are an indication of @OFF low transmission and @ON high transmission. From this we can deduce that evanescent T_-2 and T_-3 waves coupling with SPP modes must be involved in transimission.

Fig. 7 T_-1 transmission with m = −1 showing simulations at two incidence angles. In the case of a fixed incidence angle, T_-1@ n_LC = 1.52 is higher than T_-1@n_LC = 1.75. The fact that this differs from the results obtained in experiments means that SPP mode coupling must be involved. The arrow indicates the cutoff pitch when using an objective lens with NA = (0.3). No difficulties in the collection of light were encountered in cases of transmission through pitches exceeding the cutoff pitch.

Download Full Size | PDF

Figure 8 illustrates the value of $k_{c o m p o s i t e}$ in Eq. (17) in the cases where m = −2 and m = −3. Effective tuning pitches are highlighted. It should be noted that the effective working values for $k_{c o m p o s i t e}$ in corresponding pitches are centered around 1.67~1.80*k_o; i.e., $k_{s p p}$ for the antisymmetrical SPP mode (in E_x) in Fig. 8(a) and around 2.85~3.10*k_o for the symmetrical mode. In Fig. 8(b), the diffraction angles of m = −3 order reached the value of $k_{s p p}$ @ON and the longer grating periods adjacent to the highlighted regions match $k_{s p p}$ @OFF. This may explain why gratings with longer pitches (proximal to the highlighted region) also responded to applied voltage. In this study, we were concerned only with the tunability of transmission in pitches close to $k_{s p p}$ , rather than in regions with longer grating pitches (e.x. 450nm and above), which always remain transmissive. T_-1 transmission plays an important role in regions that are always bright. Figures 6(d) and (e) illustrate the inverse transmission behavior at a pitch of 450nm (intensity @OFF brighter than @ON), which indicates that T_-1 is dominant. A reference measurement was found in [31], in which broadband transmission was also observed at wavelengths above 450nm.

Fig. 8 (a) $k_{c o m p o s i t e}$ for gratings of each pitch when m = −2 order; (b) $k_{c o m p o s i t e}$ when m = −3 order. The two horizontal lines indicate $k_{s p p}$ values derived using the formulae in this study, in the case where LC is @ON and @OFF. The $k_{s p p}$ values at 1.67*k_o and 1.80*k_o are associated with an anti-symmetrical E_x field whereas 2.85*k_o and 3.10*k_o are associated with a symmetrical E_x field.

Download Full Size | PDF

It should be noted that with Δn = 0.23(for E7), the difference between k_spp@ON and OFF states was only 0.13k_o which is not particularly large. This indicates that the tuning function of SPP modes in the case of an LC layer on metallic structures is very sensitive. The Au-covered sidewalls of the PMMA grating stripes may have contributed to the effects of tuning; however, the thickness of the Au film was difficult to determine in this experiment. The issue of LC alignment on grating boundaries was also addressed in [17], in which curved surfaces were adopted for the tuning of LC molecules. The electrically modulated optical scattering of an LC layer on a metallic surface can usually be observed quite clearly, particularly when edge or surface defects are present. The issue of LC alignment at irregular boundaries is a complicated issue and worthy of further study. We believe that the formulae derived in this study to deal with the 3D rotation of LC on a thin metal film is the first step.

4. Conclusions

This paper derived a comprehensive formula with which to solve surface plasmon polariton modes between an anisotropic layer and metal interface within a three-layered system. The derived results enable the tuning of the large transmission of a grating coupling through the application of voltage into a liquid crystal layer. This is the first report to explicitly demonstrate and fit the sensitive tuning range of SPP modes in such a medium and provide a general guide for the electro-optical tuning of plasmonic devices with liquid crystal materials in a 3D environment.

5. Acknowledgements

We would like to thank Prof. Wei-Chih Lin for his support using FEMLAB COMSOL tools and Prof. S.-C. Jeng’s discussion on LC alignment. We would also like to thank the Ministry of Science and Technology, R.O.C. for the funding of project “104-2112-M-110 −007 –“.

References and links

1. I. I. Smolyaninov II, Y. J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315(5819), 1699–1701 (2007). [CrossRef] [PubMed]

2. L. Ferrari, C. H. Wu, D. Lepage, X. Zhang, and Z. W. Liu, “Hyperbolic metamaterials and their applications,” Prog. Quantum Electron. 40, 1–40 (2015). [CrossRef]

3. B. Stein, E. Devaux, C. Genet, and T. W. Ebbesen, “Plasmonic crystal enhanced refractive index sensing,” Appl. Phys. Lett. 104(25), 251111 (2014). [CrossRef]

4. M. A. Razumova and I. M. Dmitruk, “Splitting of Plasmon Frequency in Spherical Metal Nanoparticles in Anisotropic Medium,” Plasmonics 8(4), 1699–1706 (2013). [CrossRef]

5. A. Babu, C. Bhagyaraj, J. Jacob, G. Mathew, and V. Mathew, “Surface plasmon propagation in a metal strip waveguide with biaxial substrate,” Opt. Quantum Electron. 45(6), 481–490 (2013). [CrossRef]

6. K. V. Sreekanth and T. Yu, “Long range surface plasmons in a symmetric graphene system with anisotropic dielectrics,” J. Opt. 15(5), 055002 (2013). [CrossRef]

7. Y. Takeichi, Y. Kimoto, M. Fujii, and S. Hayashi, “Anisotropic propagation of surface plasmon polaritons induced by para-sexiphenyl nanowire films,” Phys. Rev. B 84(8), 085417 (2011). [CrossRef]

8. M. R. Shcherbakov, M. I. Dobynde, T. V. Dolgova, D. P. Tsai, and A. A. Fedyanin, “Full Poincare sphere coverage with plasmonic nanoslit metamaterials at Fano resonance,” Phys. Rev. B 82(19), 193402 (2010). [CrossRef]

9. L. Feng, Z. W. Liu, V. Lomakin, and Y. Fainman, “Form birefringence metal and its plasmonic anisotropy,” Appl. Phys. Lett. 96(4), 041112 (2010). [CrossRef]

10. Y. J. Hung, M. H. Shih, J. H. Liou, and J. Y. Tai, “Polarity-variable birefringence on hyperlens structure,” Opt. Express 18(26), 27606–27612 (2010). [CrossRef] [PubMed]

11. J. Elser and V. A. Podolskiy, “Scattering-free plasmonic optics with anisotropic metamaterials,” Phys. Rev. Lett. 100(6), 066402 (2008). [CrossRef] [PubMed]

12. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006). [CrossRef] [PubMed]

13. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007). [CrossRef] [PubMed]

14. A. A. Krokhin, A. Neogi, and D. McNeil, “Long-range propagation of surface plasmons in a thin metallic film deposited on an anisotropic photonic crystal,” Phys. Rev. B 75(23), 235420 (2007). [CrossRef]

15. R. Luo, Y. Gu, X. K. Li, L. J. Wang, I. C. Khoo, and Q. H. Gong, “Mode recombination and alternation of surface plasmons in anisotropic mediums,” Appl. Phys. Lett. 102(1), 011117 (2013). [CrossRef]

16. A. Piccardi, A. Alberucci, N. Kravets, O. Buchnev, and G. Assanto, “Power-controlled transition from standard to negative refraction in reorientational soft matter,” Nat. Commun. 5, 5533 (2014). [CrossRef] [PubMed]

17. D. Franklin, Y. Chen, A. Vazquez-Guardado, S. Modak, J. Boroumand, D. Xu, S.-T. Wu, and D. Chanda, “Polarization-independent actively tunable colour generation on imprinted plasmonic surfaces,” Nat. Commun. 6, 7337 (2015). [CrossRef] [PubMed]

18. G. Si, Y. Zhao, E. S. P. Leong, and Y. J. Liu, “Liquid-Crystal-Enabled Active Plasmonics: A Review,” Materials (Basel) 7(2), 1296–1317 (2014). [CrossRef]

19. Y. Zhao, Q. Hao, Y. Ma, M. Lu, B. Zhang, M. Lapsley, I.-C. Khoo, and T. J. Huang, “Light-driven tunable dual-band plasmonic absorber using liquid-crystal-coated asymmetric nanodisk array,” Appl. Phys. Lett. 100(5), 053119 (2012). [CrossRef]

20. Y. J. Liu, G. Y. Si, E. S. P. Leong, N. Xiang, A. J. Danner, and J. H. Teng, “Light-Driven Plasmonic Color Filters by Overlaying Photoresponsive Liquid Crystals on Gold Annular Aperture Arrays,” Adv. Mater. 24(23), OP131–OP135 (2012). [CrossRef] [PubMed]

21. M. Ojima, N. Numata, Y. Ogawa, K. Murata, H. Kubo, A. Fujii, and M. Ozaki, “Electric field tuning of plasmonic absorption of metallic grating with twisted nematic liquid crystal,” Appl. Phys. Express 2(8), 086001 (2009). [CrossRef]

22. W.-S. Chang, J. B. Lassiter, P. Swanglap, H. Sobhani, S. Khatua, P. Nordlander, N. J. Halas, and S. Link, “A Plasmonic Fano Switch,” Nano Lett. 12(9), 4977–4982 (2012). [CrossRef] [PubMed]

23. J. Lekner, “Reflection and refraction by uniaxial crystals,” J. Phys. Condens. Matter 3(32), 6121–6133 (1991). [CrossRef]

24. R. Li, C. Cheng, F. F. Ren, J. Chen, Y. X. Fan, J. P. Ding, and H. T. Wang, “Hybridized surface plasmon polaritons at an interface between a metal and a uniaxial crystal,” Appl. Phys. Lett. 92(14), 141115 (2008). [CrossRef]

25. X. L. Wang, P. Wang, J. X. Chen, Y. H. Lu, H. Ming, and Q. W. Zhan, “Theoretical and experimental studies of surface plasmons excited at metal-uniaxial dielectric interface,” Appl. Phys. Lett. 98(2), 021113 (2011). [CrossRef]

26. H.-H. Liu and H.-C. Chang, “Leaky surface plasmon polariton modes at an interface between metal and uniaxially anisotropic materials,” IEEE Photonics J. 5(6), 4800806 (2013). [CrossRef]

27. H. Zhou, X. Zhang, D. Kong, Y. Wang, and Y. Song, “Complete dispersion relations of surface plasmon polaritons at a metal and birefringent dielectric interface,” Appl. Phys. Express 8(6), 062003 (2015). [CrossRef]

28. P. Yeh and C. Gu, in Optics of Liquid Crystal Displays(John Wiley & Sons, Inc., New York, 1999), pp. 63–64.

29. X. Li, Y. Gu, R. Luo, L. Wang, and Q. Gong, “Effects of dielectric anisotropy on surface plasmon polaritons in three-layer plasmonic nanostructures,” Plasmonics 8(2), 1043–1049 (2013). [CrossRef]

30. J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B Condens. Matter 33(8), 5186–5201 (1986). [CrossRef] [PubMed]

31. B. K. Singh and A. C. Hillier, “Surface plasmon resonance enhanced transmission of light through gold-coated diffraction gratings,” Anal. Chem. 80(10), 3803–3810 (2008). [CrossRef] [PubMed]

Comprehensive three-dimensional analysis of surface plasmon polariton modes at uniaxial liquid crystal-metal interface

Abstract

1. Introduction

2. Rotation of elliptical k-space

3. Experimental Demonstration

4. Conclusions

5. Acknowledgements

References and links

Cited By

Figures (8)

Equations (17)

Optics Express