Fresnel analysis of Kretschmann geometry with a uniaxial crystal layer on a threelayered film Yu-Ju Hung, Yin-Rui Yen, and I-Sheng Lin Citation: AIP Advances 6, 045023 (2016); doi: 10.1063/1.4948509 View online: http://dx.doi.org/10.1063/1.4948509 View Table of Contents: http://aip.scitation.org/toc/adv/6/4 Published by the American Institute of Physics AIP ADVANCES 6, 045023 (2016) Fresnel analysis of Kretschmann geometry with a uniaxial crystal layer on a three-layered film Yu-Ju Hung,a Yin-Rui Yen, and I-Sheng Lin Department of Photonics, National Sun Yat-Sen University, Kaohsiung, Taiwan, 80424, R.O.C (Received 18 February 2016; accepted 20 April 2016; published online 27 April 2016) The use of total internal reflection within the prism coupling scheme is a simple approach to the generation of surface plasmon polariton waves on a metal/dielectric interface Unfortunately, an anisotropic layer on a metallic film complicates the derivation of resonance angle In this study, we present clear Fresnel analysis of a liquid crystal film on a metal surface Few current simulation packages enable the analysis of multiple layers with a single anisotropic layer The proposed formulation process is applicable to multi-layered structures C 2016 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/) [http://dx.doi.org/10.1063/1.4948509] Plasmonic devices have proven highly versatile implemented by advanced nano fabrication techniques The intrinsic characteristics of surface plasmon polaritons (SPP) - strong surface field, effectively short optical wavelength, and directional propagation have led to their application in research on imaging beyond the diffraction limit,1–3 the capabilities of nano-lithography,4,5 the implementation of strong surface fields in enhancing fluorescence,6 second harmonic generation,7 nano-waveguide routing for signal processing,8 the design of optical components,9,10 beam splitters,11 focusing lenses,12 metasurface phase manipulations,13 and holography involving metallic films.14 The optical tunability of SPP devices depends primarily on the dielectric layer on a structured metallic surface Following the completion of nano-fabrication on a metal layer, the structural design of the device cannot be altered This leaves the tuning of the dielectric layer above as the only means of producing active or reconfigurable devices Changes in temperature, variations in the refractive index induced by optical pumping, chemical modification, and electrical tuning are methods used to tune the criteria affecting resonance Soft liquid crystal materials that present a change in refractive index (∆n = 0.2 ∼ 0.5) provide a convenient solution to the problem of modulation.15,16 However, the optical analysis of uniaxial crystal films on a metal surface can be prohibitively complex, requiring highly detailed analysis to ensure that the boundary continuity conditions are matched At present, no existing simulation tools provide a module for this type of computation Numerous researchers have investigated the problem of surface plasmon polariton waves at the interface between an anisotropic layer and a metal film.17–20 An anisotropic medium on the surface of a metal covering an isotropic substrate provides a basic platform for the fabrication of active plasmonic devices Reference 18 provides detailed analysis pertaining to the derivation of the eigen propagation k-vectors for SPP modes in a three layered structure; however, this method cannot be used to determine the amount of light coupled with the surface plasmon mode This study developed the means to derive the optical reflectivity of such structures based on Kretschmann geometry This makes it possible to verify the resonance angle of SPP coupling with a high degree of precision, thereby enabling systematic improvements in overall resonance This also makes it possible to analyze the effect of resonance on layers with different structures, including an amplification layer Figure 1(a) illustrates the global coordinates XYZ pertaining to the geometry of the film exposed to TM-polarized incident light The wave vector of incident light is (k x , 0, k z ) with an a Electronic mail: yjhung@mail.nsysu.edu.tw 2158-3226/2016/6(4)/045023/6 6, 045023-1 © Author(s) 2016 045023-2 Hung, Yen, and Lin AIP Advances 6, 045023 (2016) FIG (a) XYZ coordinates for three-layered film structure; (b) Rotational angle definition between local coordinate X”Y”Z” and XYZ global coordinate (c) Optical axis (OA) of liquid crystal is aligned along X” incident angle θ1 at the first interface and an incident angle of α at the second interface The first layer is the dielectric prism with a dielectric constant of ε d The second and third layers are metal (ε m ) and LC (ε o & ε e ) respectively In this figure, “d” refers to the thickness of the metal layer with the center of the metal film positioned at Z = The definition of rotational angles between XYZ and X”Y”Z” is presented in Figure 1(b) The X”-axis is aligned with the optical axis (OA) of the liquid crystal, as shown in Fig 1(c) The two eigen fields, Eo” and Ee”, in the local coordinate system can be expressed as follows: k x′′ 2 2 E ′′ k − ne k0 x k y′′ E y′′ = k − no k02 Ez′′ k z′′ k − no k02 E ′′ x E y′′ = −k z′′ ′′ Ez′′ k y (1) (2) Following coordinate rotation, the two eigen fields associated with the global coordinates (metal surface) are expressed as follows: 045023-3 Hung, Yen, and Lin AIP Advances 6, 045023 (2016) k zo sin φ cos θ k0 kx k zo cos θ cos φ Eo : Eo sin θ − k0 k0 kx − sin ϕ cos θ k0 (3) k zo k zo k x sin θ cos θ cos ϕ − k k0 k0 Ho = E o ε o sin ϕ cos θ ω µ k x sin θ − k zo k x cos θ cos ϕ k 02 k 02 (4) k ze k x k zo sin θ − cos θ cos φ + k0 k0 −ε o sin ϕ cos θ Ee = A 2 k ze k x k ze − ε o k cos θ cos ϕ + sin θ k 02 k 02 k ze ε o sin ϕ cos θ k0 k ze k0 k x ε sin θ − ε cos θ cos ϕ He = A o o k0 ω µ k k x − ε o sin ϕ cos θ k0 Where k zo = no k02 − k x (5) (6) (7) kze = n o − n e cos θ sin θ cos φk x + n o 2n e n e 2sin2 θ + n o 2cos2θ k 02 − n 2o n 2e sin2θsin2ϕ + cos2ϕ + n o 4cos2θsin2ϕ k x2 n e 2sin2 θ + n o 2cos2 θ (8) A more detailed derivation can be found in a previous work.18 For TM incidence from z< − d/2, the application of continuity conditions at the boundaries makes it possible to obtain the reflection coefficients as follows: rT M → T E = −2ε 0k 02n 1(k z e − k z o ) cos α sin φ(k z o cos φ − k x tan θ) (9) {(k z o + k 0n cos α)[k z o (k z o n + ε 0k 0k z e cos α)cos2φ + k x 2(k z e n + ε 0k cos α)tan2 θ] + (k z o n + ε 0k cos α)[ε 0k 02(k z e + k 0n cos α)sin2φ − k x (k z e + k z o )(k z o + k n cos α) cos φ tan θ]} rT M → T M = −{k z o (−k z2 o n + ε 0k 0k z e cos α)(k z o + k n cos α)cos2φ + ε 0k 02(−k z o n + ε 0k cos α)(k z e + k n cos α)sin2 φ + k x (k z o + k n cos α) tan θ[(k z e + k z o )(k z o n − ε 0k cos α) cos φ + k x (−k z e n + ε 0k cos α) tan θ]} {ε 0k 02(k z o n + ε 0k cos α)(k z e + k n cos α)sin2φ + (k z o + k n cos α)[k z o (k z2 o n + ε 0k 0k z e cos α)cos2φ− k x (k z e + k z o )(k z o n + ε 0k cos α) cos φ tan θ + k x 2(k z e n + ε 0k cos α)tan2θ]} (10) 045023-4 Hung, Yen, and Lin AIP Advances 6, 045023 (2016) For TE incidence from z < −d/2, the reflection coefficient can be expressed as follows: r T E→ T M = −2ε 0k 02n 1(k z e − k z o ) cos α sin φ(k z o cos φ + k x tan θ) {(k z o + k 0n cos α)[k z o (k z o n + ε 0k 0k z e cos α)cos2 φ + k x 2(k z e n + ε 0k cos α)tan 2θ] (11) + (k z o n + ε 0k cos α)[ε 0k 02(k z e + k 0n cos α)sin2φ − k x (k z e + k z o )(k z o + k n cos α) cos φ tan θ]} r T E→ T E = −{k z o (k z2 o n + ε 0k 0k z e cos α)(k z o − k n cos α)cos2φ − ε 0k 02(k z o n + ε 0k cos α)(−k z e + k n cos α)sin2φ +k x (−k z o + k n cos α) tan θ[(k z e + k z o )(k z o n + ε 0k cos α) cos φ − k x (k z e n + ε 0k cos α) tan θ]} {ε 0k 02(k z o n + ε 0k cos α)(k z e + k 0n cos α)sin2φ + (k z o + k n cos α)[k z o (k z o n + ε 0k 0k z e cos α)cos2φ− k x (k z e + k z o )(k z o n + ε 0k cos α) cos φ tan θ + k x 2(k z e n + ε 0k cos α)tan2 θ]} (12) Beginning with from the interface between layer and the LC medium (medium 3), as shown in Fig (i i) (i i) Er = r 23 Ei E (i i) r T M→ T M M rT = (i i) ErT E rT M→ T E rT E→ T M rT E→ T E (13) E (i i) M iT (i i) EiT E (14) The reflection matrix element is described in Eq (14) The reflection between layer and layer can be expressed as follows: r m12 0 r 12 = (15) 0 t m12 0 t 12 = (16) 0 rm12 and tm12 represent the reflection and transmission coefficients from medium to medium 2, when the incident light is TM polarized The subscript “m” denotes TM polarization The reverse route is defined as follows: r m21 r 21 = (17) r e21 t m21 t 21 = (18) t e21 re21 and te21 represent the reflection and transmission coefficients from medium to medium 1, when the incident light is TE polarized The subscript “e” denotes TE polarization FIG Three-layered structure computed in this study 045023-5 Hung, Yen, and Lin AIP Advances 6, 045023 (2016) FIG TM reflectivity v.s incidence angle θ1 (a) The OA of the liquid crystal layer is rotated on the YZ plane The resonance angle is shifted approximately degrees (b) The OA of the layer is rotated on the XY plane The resonance angle is shifted degrees A change of ∆n = 0.23 is introduced in both cases The accumulated reflection ingredients in Fig are as follows: r 123 = r 12 + t 21r 23t 12ei 2ϕ + t 21r 23r 21r 23t 12ei 4ϕ +······ · · i 2ϕ [I + A + AA + AAA + · · · · · · · ·] t 12 2ϕ −1 = r 12 + t 21r 23ei X − λ1 X t 12 − λ2 = r 12 + t 21r 23e (19) where A = r 12r 23ei2ϕ and the eigen value of A is λ1 and λ2 X represents the composition of the corresponding eigen vectors X1 and X2 The phase term ϕ = n2 k o cos α ∗ d Two examples are given as follows: ne = 1.75, no = 1.52, λ = 632.8nm and ε d = 6.65 The optical axes of the LC lies along the Y-Z (φ = 90) and X-Y planes (θ = 0) Fig presents the TM reflection coefficients with respect to incident angle θ1 in the case of an Au film 50nm in thickness (ε = −11.1 + j ∗ 0.8) The resonance dip swept from the black curve to the red curve as the LC was gradually rotated on the corresponding planes The surface plasmon resonance dip FIG Maximum TE reflectivity with TM incidence v.s incidence angle θ1 The OA of the liquid crystal layer was rotated at the specific angles shown in the figure A very small proportion of the TM incident light was converted to TE polarized light due to the birefringence of the rotated liquid crystal layer situated on top of the metal layer 045023-6 Hung, Yen, and Lin AIP Advances 6, 045023 (2016) presents a shift of approximately degrees and 1.8 degrees on the two respective planes A small proportion of TM incident light is converted to TE polarized-reflection due to the birefringence effect of LC molecules, the principle axes of which are not oriented along the global coordinates Fig shows the maximum TE reflection and the corresponding OA angles in both cases of rotation If both dielectric layers are LC materials, then the ordinary incidence in medium is reflected as an ordinary component of reflection; however, in a three-layer system, the resonance angle of ordinary reflected light differs from that of extraordinary light The e-light and o-light transmitted into medium would be the superposition of TE and TM eigen fields A reflection matrix similar to Eq (14) could be used to depict multiple reflections within medium A similar derivation of the complex boundary conditions between medium (LC) and medium (metal) can be found in a previous paper.21 This paper reports Fresnel analysis on a three-layered structure with a single uniaxial crystal layer Due to the fact that the uniaxial crystal layer is at the top, TM incidence produces TM and TE reflections simultaneously The TE reflection component is very small but still exists This proposed procedure could be applied to an arbitrary number of multiple layers with a uniaxial medium inside, for which existing software solutions are not equipped We would like to thank the Ministry of Science and Technology, R.O.C for the funding of project “104-2112-M-110 -007 –” Z Jacob, L V Alekseyev, and E Narimanov, Opt Exp 14(7), 8247–8256 (2006) Z W Liu, H Lee, Y Xiong, C Sun, and X Zhang, Science 315, 1686–1686 (2007) Smolyaninov II, Y J Hung, and C C Davis, Science 315, 1699–1701 (2007) W Srituravanich, N Fang, C Sun, Qi Luo, and Xiang Zhang, Nano Lett 4(6), 1085–1088 (2004) Z.W Liu, Qi-Huo Wei, and Xiang Zhang, Nano Lett 5(5), 957–961 (2005) F Tam, G.P Goodrich, B.R Johnson, and N J Halas, Nano Lett 7(2), 496–501 (2007) H.J Simon, D.E Mitchell, and J.G Watson, Phys Rev Lett 33(26), 1531–1534 (1974) Y Fang and M Sun, Light: Science & Applications 4, e294, doi: 10.1038/lsa.2015.67 (2015) L Feng, Z.W Liu, V Lomakin, and Y Fainman, Appl Phys Lett 96, 041112 (2010) 10 Y.J Hung, M H Shih, J.H Liou, and J.Y Tai, Opt Exp 18(26), 27606–27612 (2010) 11 K.-W Chang and C.-C Huang, Sci Rep 6, 19609, doi: 10.1038/srep19609 (2016) 12 Z Liu, JM Steele, W Srituravanich, Y Pikus, C Sun, and X Zhang, Nano Lett 5(9), 1726–1729 (2005) 13 Dianmin Lin, Pengyu Fan, Erez Hasman, and Mark L Brongersma, Science 345, 298–302 (2014) 14 M Ozaki, J-I Kato, and S Kawata, Science 332, 218–220 (2011) 15 J Olson, A Manjavacas, T Basu, D Huang, A E Schlather, B Zheng, N J Halas, P Nordlander, and S Link, ACS Nano 10(1), 1108–1117 (2015) 16 D Franklin, Y Chen, A Vazquez-Guardado, S Modak, J Boroumand, D Xu, S.-T Wu, and D Chanda, Nature Comm 6, 7337 (2015) 17 K Zhelyazkova, M Petrov, B Katranchev, and G Dyankov, Journal of Physics: Conference Series 558, 012023 (2014) doi:10.1088/1742-6596/558/1/012023 18 Y-J Yen, T-H Li, Z-Y Wu, T-H Lin, and Y-J Hung, Opt Exp 23(25), 32377–32386 (2015) 19 R Luo, Y Gu, X K Li, L J Wang, I C Khoo, and Q H Gong, App Phys Lett 102(1), 011117 (2013) 20 X L Wang, P Wang, J X Chen, Y H Lu, H Ming, and Q W Zhan, App Phys Lett 98(2), 021113 (2011) 21 J Lekner, J Phys.:Condens Matter 3, 6121–6133 (1991) ...AIP ADVANCES 6, 045023 (2016) Fresnel analysis of Kretschmann geometry with a uniaxial crystal layer on a three- layered film Yu-Ju Hung ,a Yin-Rui Yen, and I-Sheng Lin Department of Photonics,... of resonance angle In this study, we present clear Fresnel analysis of a liquid crystal film on a metal surface Few current simulation packages enable the analysis of multiple layers with a single... (LC) and medium (metal) can be found in a previous paper.21 This paper reports Fresnel analysis on a three- layered structure with a single uniaxial crystal layer Due to the fact that the uniaxial