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Beam bending through compound structure with two subwavelength metallic slits in each period
Optik 125 (2014) 1529–1532
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Beam bending through compound structure with two subwavelength metallic slits in each period Gaige Zheng a,b,∗ , Linhua Xu a,b , Yigen Wu a , Kun Cao a , Yunyun Chen a , Yuzhu Liu c a b c
School of Physics and Optoelectronic Engineering, Nanjing University of Information Science & Technology, Nanjing 210044, Jiangsu, China Optics and Photonic Technology Laboratory, Nanjing University of Information Science & Technology, Nanjing 210044, China Paul Scherrer Institute, WSLA-004, Switzerland
a r t i c l e
i n f o
Article history: Received 6 April 2013 Accepted 5 August 2013
Keywords: Metallic gratings Surface plasmon polaritons Subwavelength structures
a b s t r a c t To provide a more practical, easy-to-implement method to achieve directional modulation with a plasmonic lens, beam manipulation method via compound metallic gratings with two subwavelength slits filled with different dielectrics inside each period is proposed and numerically investigated by finitedifference time-domain (FDTD) method. Compared with conventional metal-grating based structures, phase retardation is tuned by the Fabry-Pérot (FP) resonant condition and light bending is achieved by constructing a carefully designed, curved phase front for the plasmonic lenses. Our designs have advantages in ease of fabrication and capability to perform in the far field. With these advantages, the designs are expected to be valuable in applications such as plasmonic circuits and photonic communication. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction Since the phenomena of extraordinary transmission [1] and directional beaming of light [2] were reported, much interest has been attracted in investigations about surface plasmon polaritons (SPPs) excited in subwavelength metallic structures, which opens up the potential applications in plasmonic elements. Metallic light beam manipulating devices such as splitters [3,4], deflectors [5], angle compensators [6] and lenses [7–18] have been demonstrated. Both theoretical predictions [7,8,10,13–18] and experimental demonstration [9,11,17] have been reported in the visible range. Such devices work by modulating the phase delay distribution at the metallic device’s surfaces according to the constructive light interference principle. In [18], plasmonic lenses cannot only focus, but can also bend electromagnetic (EM) waves have been designed and characterized. The bending effect is achieved by constructing an asymmetric phase front caused by varying phase retardations in EM waves and the control of the phase front profile is achieved through two mechanisms: phase retardation caused by the width and shape of the individual slits in the lens, and the position of these slits. For certain applications, it would be
∗ Corresponding author at: School of Physics and Optoelectronic Engineering, Nanjing University of Information Science & Technology, Nanjing 210044, Jiangsu, China. Tel.: +86 13851781730. E-mail addresses: [email protected], [email protected] (G. Zheng). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.08.041
highly desirable to realize positional modulation through a plasmonic lens without changing the period of array, or the length and width of slits. In this paper, we extend the previous studies and examine ways of controlling the phase front of the transmitted beam by introducing compound metallic gratings with two subwavelength slits filled with different dielectrics inside each period. When suitable dielectrics are chosen, the Fabry-Pérot (FP)-like modes with the different orders can be found inside the two adjacent slits at a certain resonant frequency, an asymmetric phase front caused by varying phase retardations in EM waves as they pass through the plasmonic lens can be constructed based on the FP resonant condition. The slits used in our designs are rectangular that guarantee the simplicity in fabrication using the focused ion beam (FIB) technique.
2. Structure and simulated model Fig. 1 illustrates the proposed compound structure under study, where each period of the grating comprises two slits, slit 1 and slit 2, with identical widths but filled with different dielectric media with relative permittivities εd1 and εd2 , respectively. The interspacing between the two slits (center to center) is denoted as d, the period and thickness of the grating are and L, respectively. It is worth noting that the distance between any two slits is much larger than the skin depth of surface plasmon inside the silver, hence, no surface plasmon cross-talk will occur between adjacent slits.
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Fig. 2. (a) The calculated transmission spectra of the simple metallic gratings. (b) Variation of the real part of neff with permittivity for coupld-SPPs inside the metallic slit. Fig. 1. Schematic diagram of the proposed structure: each period of the grating comprises two slits with identical widths but filled with different dielectric media. The input light is a TM-polarized plane wave from the left.
The frequency-dependent permittivity of the metal is expressed by the Drude model for the dielectric response of the metal which is mainly governed by its free electron plasmon ε(ω) = 1 −
ωp2 ω(ω + i)
(1)
where ωp is the plasma frequency, is the absorption coefficient, and ω stands for the frequency. The metal investigated is silver and the considered wavelength range is 600–1300 nm. The corresponding parameters are ωp = 1.37 × 1016 rad/s and = 3.21 × 1013 rad/s [19]. When ω < ωp , the real part of the permittivity is negative, the standard time iteration scheme of the FDTD method becomes unstable. However, this problem can be solved by introducing the current density into Maxwell’s equations [20]. We assume the monochromatic operation with the free space wave number k0 = ω0 /c, the time-dependency of exp(−jωt), and corresponding free space wave length 0 = 2/k0 . Here, c is the speed of the light in vacuum. To demonstrate the validity of designed structure, twodimensional FDTD simulation is performed with the perfectly matched layer (PML) absorbing boundary conditions in x and z directions of the simulation domain. Since the width of the bus waveguide is much smaller than the operating wavelength in the structure, only the fundamental waveguide mode is supported. The incident light for excitation of the SPP mode is a TM-polarized (the magnetic field is parallel to y axis) fundamental mode. In the following FDTD simulations, the grid size in the x and z directions are chosen to be x = z = 2 nm and t = x/2c, which are sufficient for the numerical convergence, here c is the velocity of light in vacuum. The normalized time-averaged magnetic field intensity |Hy |2 is employed to represent the field intensity. 3. Simulation results and discussion In calculation, the whole length of the structure in the xdirection (see Fig. 1) is 2 micrometers (m). In addition, two power monitors are placed at the input and output side of the grating for calculating the incident power (Pin ) and the transmitted power (Pout ). Subsequently, the transmission (Tr ) is defined to be Tr = Pout /Pin . Fig. 2(a) displays the calculated zero-transmission spectra of the simple gratings with subwavelength slits filled with different dielectric εd1 = εd2 = 1, εd1 = εd2 = 2.48 and εd1 = εd2 = 4.5, respectively. The parameters are chosen as: w = 50 nm, d = 200 nm, L = 700 nm, = 600 nm. For εd1 = εd2 = 1, the transmission peaks at wavelengths 730 nm and 1050 nm are associated with the thirdand second-order FP-like modes, respectively [21]. Similarly, the transmission peaks at wavelengths 670 nm, 800 nm and 1050 nm
for εd1 = εd2 = 2.48 attribute to the fifth-, fourth- and third-order FP-like modes, respectively, and the transmission peaks at wavelengths 650 nm, 740 nm, 860 nm and 1050 nm for εd1 = εd2 = 4.5 attribute to the seventh- sixth-, fifth- and fourth-order FP-like modes, respectively. For the bare metallic slit array, current density standing waves are established on both metal alls of each slit by incident oscillating light [22]. Owing to the existence of current flow, the distribution of charge density has been rearranged on the both surfaces of each slit. The length of FP cavity approximately equals to the distance of current flow on one wall, which is the extent of charge distribution in the y direction. The resonant wavelength of F–P mode in a bare slit array can be obtained by [23]: 2k0 Re(neff )L + ϕ = N · 2
(2)
where k0 = 2/ is the wave vector of light in vacuum, is the wavelength of the incident light in vacuum, neff is the effective refractive index of coupled-SPPs (waveguide mode) inside the slit, ϕ is an additional phase shift experienced by the fundamental mode when reflecting at the grating interfaces, and N is an arbitrary integer. For TM-polarized case, neff can be approximately solved from [24]: εd
εm
n2eff − εm
n2eff − εd
=
1 − exp(k0 w 1 + exp(k0 w
n2eff − εd )
n2eff − εd )
(3)
Fig. 2(b) plots the variation of the real parts of neff for the metallic slit filled with different dielectric media with four wavelengths. Fig. 3(a)–(d) further illustrate the corresponding field intensity distributions when the slits are filled with different material at FP resonant wavelengths. The results show that the energy emerging from the structure tends to focus within several micrometers in an extremely small region (i.e., small FWHM). As seen from Fig. 3(b) and (d), the focal length will be changed when the resonant wavelength is different. To deflect the beam, curved phase front of the transmitted beam should be achieved. The authors in Ref. [18] proposed a method to control the phase front profile by the width and shape of the individual slits and the position of these slits. Unlike previous studies [7–10,18], phase retardation in our structure is tuned by FP resonant condition. There is no need to change the period of array, or the length and width of slits. Thus the structure proposed in this paper is a much smaller, more compact device and convenient for integration. The position of focus and the bending angle are determined using the image processing software, ImageJ [25], to capture the area with the highest intensity within the main beam. Under normally incident radiation with different wavelengths, field distributions are shown in Fig. 4, where one can find the off-axis focusing phenomenon. Bam deflection can be observed from the simulation due to the constructive interference of the phase-delayed, transmitted waves from the rectangular slits. The positional and directional modulation functionalities of a plasmonic lens are determined by the relative phase difference
G. Zheng et al. / Optik 125 (2014) 1529–1532
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Fig. 3. Profiles of field intensity of the proposed structure when the slits are filled with different dielectric media. (a) εd1 = εd2 = 1 with a wavelength of 1050 nm, (b) εd1 = εd2 = 2.48 with a wavelength of 1050 nm, (c) εd1 = εd2 = 4.5 with a wavelength of 1050 nm and (d) εd1 = εd2 = 2.48 with a wavelength of 800 nm.
between neighboring slits. The phase differences between the two slits are determined by [26]:
ϕ = ϕ1 − ϕ2 = Re(ˇ1 L) + arg 1 −
− Re(ˇ2 L) − arg 1 −
1 − (ˇ1 /k0 ) 1 + (ˇ1 /k0 )
1 − (ˇ2 /k0 ) 1 + (ˇ2 /k0 )
2
exp(2jˇ1 L)
2
exp(2jˇ2 L)
(4)
By illuminating light on upper side of slits, SPPs can be excited in the subwavelength slits, the phase retardation is dominantly determined by the real part of propagation constant [13]. Therefore Eq. (4) can be approximated as Re(ˇ1 L) − Re(ˇ2 L). ˇ1 and ˇ2 are complex propagation constants in the two slits which can be calculated from [5,8,9]:
tanh
w 2
ˇi2
− εdi k02
=
−εd εAg
ˇi2 − εAg k02
ˇi2 − εdi k02
(i = 1, 2)
(5)
where w is slit width, εdi and εAg are relative dielectric constants for metal and the materials between slits, respectively. By calculating the propagation constants from Eq. (5), and by substituting their value in Eq. (4) we obtain the phase difference of the fields exiting from the slits. As a consequence of the phase difference, the beam exiting the nano-slit is deflected. By introducing compound metallic gratings with two subwavelength slits filled with different dielectrics inside each period, we can create a relative phase difference profile across the width of the lens. The position of the constructive interferences of the decoupled optical waves depends
strongly on the phase delay profile, and it is expressed as beam modulation effect. As seen from Fig. 4, the four lens designs vary not only in bending angle, but also in light distribution and intensity. Here we show beam deflection through appropriately choosing the refractive index of material filling in the metal slits. In order to deflect the beam, the phase front of the transmitted beam should be inclining to the exit plane of the slit array, thus the index distribution should be asymmetric instead of the symmetrical one as used in the focusing case. A main beam deflection of ∼11◦ can be observed from Fig. 4(a). When the wavelength is changed to = 1050 nm, a beam deflection of ∼20 ◦ can be observed from Fig. 4(b). When εd1 = 1, εd2 = 4.5, = 860 nm, the transmitted beam will bend to the negative direction, as can be seen from Fig. 4(c). When = 1050 nm, because εd = 1 and εd = 4.5 are associated with the second-order and fourth-order FP-like modes, respectively. Hence there is a focal spot on the horizontal direction, which is shown in Fig. 4(d). The simulated phase front of transmitted beam is consistent with our anticipated results. Fig. 5 shows the field intensities of the proposed structure for beam bending when εd1 = 2.48, εd2 = 4.5 with different wavelengths. Bending angle becomes larger with the increasing of wavelength. When the optical waves are incident at an angle to the optical axis will add an extra planar phase front that interferes with the asymmetric phase front of the plasmonic lens, leading to a larger bending angle off the axial direction. The slits used in our designs are rectangular that guarantee the simplicity in fabrication using the FIB technique. In addition, the proposed lenses may be further optimized by fine-tuning the geometrical parameters (such as width, length, and spacing of the slits).
Fig. 4. Profiles of field intensity of the proposed structure for beam bending. (a) εd1 = 1, εd2 = 2.48, = 790 nm; (b) εd1 = 1, εd2 = 2.48, = 1050 nm; (c) εd1 = 1, εd2 = 4.5, = 860 nm; (d) εd1 = 1, εd2 = 4.5, = 1050 nm. The other parameters are the same as in Fig. 3.
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Fig. 5. Profiles of field intensity of the proposed structure for beam bending. (a) εd1 = 2.48, εd2 = 4.5, = 650 nm; (b) εd1 = 2.48, εd2 = 4.5, = 670 nm; (c) εd1 = 2.48, εd2 = 4.5, = 860 nm; (d) εd1 = 2.48, εd2 = 4.5, = 1050 nm. The other parameters are the same as in Fig. 3.
4. Conclusion In conclusion, a method to achieve the directional tuning of light was proposed. The compound metallic gratings contain two subwavelength slits filled with different dielectrics inside each period. Numerical simulation results indicate that the beam bending effects are governed by the FP resonant conditions. There is no need to change the period of array, or the length and width of slits. Thus the structure proposed is a much smaller, more compact device and convenient for integration. Compared with existing plasmonic lenses [7–18] that can bend light, our lens designs have advantages in ease of fabrication and capability to perform in the far field. With these advantages, we expect our proposed structure to be helpful for designing subwavelength optical devices, as well as facilitating manufacturing processes. Acknowledgments This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 61203211 and 11204140), the Foundation for Outstanding Young Teachers of Nanjing University of Information Science & Technology (Grant No. 20110423), and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 13KJB140006). The authors would especially like to thank Dr. Xu Yi from Guangzhou University, Su Wei from Nanjing University of Science & Technology for their useful discussions. References [1] W.L. Barnes, A. Dereux, T.W. Ebbesen, Surface plasmon subwavelength optics, Nature 424 (2004) 824–830. [2] H.J. Lezec, A. Degiron, E. Devaux, R.A. Linke, L. Martin-Moreno, F.J. GarciaVidal, T.W. Ebbesen, Beaming light from a subwavelength aperture, Science 297 (2002) 820–822. [3] S.-Y. Lee, J. Park, I. Woo, N. Park, B. Lee, Surface plasmon beam splitting by the photon tunneling through the plasmonic nanogap, Appl. Phys. Lett. 97 (2010) 133113. [4] C.L. Zou, F.W. Sun, C.H. Dong, X.F. Ren, J.M. Cui, X.D. Chen, Z.F. Han, G.C. Guo, Broadband integrated polarization beam splitter with surface plasmon, Opt. Lett. 36 (2011) 3630–3632. [5] T. Xu, C.T. Wang, C.L. Du, X.G. Luo, Plasmonic beam deflector, Opt. Express 16 (2008) 4753–4759.
[6] L. Verslegers, P.B. Catrysse, Z. Yu, S. Fan, Planar metallic nanoscale slit lenses for angle compensation, Appl. Phys. Lett. 95 (2009) 071112. [7] Z.J. Sun, H. Kim, Refractive transmission of light and beam shaping with metallic nano-optic lenses, Appl. Phys. Lett. 85 (2004) 642–644. [8] G.G. Zheng, S.R. Xiao, C.Y. Zhang, W. Su, Y.Z. Liu, Plasmonic planar lens based on stack of metal-insulator-metal waveguides with height tuning, Optik (2013), http://dx.doi.org/10.1016/j.ijleo.2013.01.083. [9] S. Ishii, A.V. Kildishev, V.M. Shalaev, K.P. Chen, V.P. Drachev, Metal nanoslit lenses with polarization-selective design, Opt. Lett. 36 (2011) 451–453. [10] Y. Liu, Y.Q. Fu, X.L. Zhou, Z.W. Xu, G.S. Zeng, F.Z. Fang, Experimental study of indirect phase tuning-based plasmonic structures for finely focusing, Plasmonics 6 (2011) 227–233. [11] R.G. Mote, W. Zhou, Y.Q. Fu, Beaming of light through depth-tuned plasmonic nanostructures, Optik 121 (2010) 1962–1965. [12] S. Jia, L.H. Yan, J.H. Si, W.H. Yi, F. Chen, X. Hou, Subwavelength beam focusing by multiple-metal slits surrounded by chirped dielectric surface gratings, Opt. Commun. 285 (2012) 1519–1522. [13] Y.T. Yu, H. Zappe, Theory and implementation of focal shift of plasmonic lenses, Opt. Lett. 37 (2012) 1592–1594. [14] B. Lee, S. Kim, H. Kim, Y. Lim, The use of plasmonics in light beaming and focusing, Prog. Quant. Electron. 34 (2010) 47–87. [15] S.Y. Yang, W.B. Chen, R.L. Nelson, Q.W. Zhan, Miniature circular polarization analyzer with spiral plasmonic lens, Opt. Lett. 34 (2009) 3047–3049. [16] Y.Q. Fu, X.L. Zhou, S.L. Zhu, Ultra-enhanced lasing effect of plasmonic lens structured with elliptical nano-pinholes distributed in variant period, Plasmonics 5 (2010) 111–116. [17] Q. Chen, Effect of the number of zones in a one-dimensional plasmonic zone plate lens: simulation and experiment, Plasmonics 6 (2011) 75–82. [18] Y.H. Zhao, S.S. Lin, A.A. Nawaz, B. Kiraly, Q.Z. Hao, Y.J. Liu, T.J. Huang, Beam bending via plasmonic lenses, Opt. Express 18 (2010) 23458–23465. [19] F.I. Baida, D.V. Labeke, Three-dimensional structures for enhanced transmission through a metallic film: annular aperture arrays, Phys. Rev. B 67 (2003) 155314. [20] S.K. Gray, T. Kupka, Propagation of light in metallic nanowire arrays: finitedifference time-domain studies of silver cylinders, Phys. Rev. B 68 (2003) 045415. [21] D. Xiang, L.L. Wang, X.F. Li, L. Wang, X. Zhai, Z.H. Liu, W.W. Zhao, Transmission resonances of compound metallic gratings with two subwavelength slits in each period, Opt. Express 19 (2011) 2187–2192. [22] A.P. Hibbins, M.J. Lockyear, J.R. Sambles, The resonant electromagnetic fields of an array of metallic slits acting as Fabry-Pérot cavities, J. Appl. Phys. 99 (2006) 124903. [23] Y.H. Wang, Y.Q. Wang, Y. Zhang, S.T. Liu, Transmission through metallic array slits with perpendicular cuts, Opt. Express 17 (2009) 5014–5022. [24] J.A. Dionne, L.A. Sweatlock, H.A. Atwater, A. Polman, Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization, Phys. Rev. B 73 (2006) 035407. [25] Image Processing and Analysis in Java (ImageJ), http://rsbweb.nih.gov/ij/ [26] G.G. Zheng, Y.Y. Chen, C.Y. Zhang, M. Lai, W. Su, Y.Z. Liu, Beam focusing from double subwavelength metallic slits filled with nonlinear material surrounded by dielectric surface gratings, Photon. Nanostruct. 10 (2012) 560–567.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Beam bending through compound structure with two subwavelength metallic slits in each period Gaige Zheng a,b,∗ , Linhua Xu a,b , Yigen Wu a , Kun Cao a , Yunyun Chen a , Yuzhu Liu c a b c
School of Physics and Optoelectronic Engineering, Nanjing University of Information Science & Technology, Nanjing 210044, Jiangsu, China Optics and Photonic Technology Laboratory, Nanjing University of Information Science & Technology, Nanjing 210044, China Paul Scherrer Institute, WSLA-004, Switzerland
a r t i c l e
i n f o
Article history: Received 6 April 2013 Accepted 5 August 2013
Keywords: Metallic gratings Surface plasmon polaritons Subwavelength structures
a b s t r a c t To provide a more practical, easy-to-implement method to achieve directional modulation with a plasmonic lens, beam manipulation method via compound metallic gratings with two subwavelength slits filled with different dielectrics inside each period is proposed and numerically investigated by finitedifference time-domain (FDTD) method. Compared with conventional metal-grating based structures, phase retardation is tuned by the Fabry-Pérot (FP) resonant condition and light bending is achieved by constructing a carefully designed, curved phase front for the plasmonic lenses. Our designs have advantages in ease of fabrication and capability to perform in the far field. With these advantages, the designs are expected to be valuable in applications such as plasmonic circuits and photonic communication. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction Since the phenomena of extraordinary transmission [1] and directional beaming of light [2] were reported, much interest has been attracted in investigations about surface plasmon polaritons (SPPs) excited in subwavelength metallic structures, which opens up the potential applications in plasmonic elements. Metallic light beam manipulating devices such as splitters [3,4], deflectors [5], angle compensators [6] and lenses [7–18] have been demonstrated. Both theoretical predictions [7,8,10,13–18] and experimental demonstration [9,11,17] have been reported in the visible range. Such devices work by modulating the phase delay distribution at the metallic device’s surfaces according to the constructive light interference principle. In [18], plasmonic lenses cannot only focus, but can also bend electromagnetic (EM) waves have been designed and characterized. The bending effect is achieved by constructing an asymmetric phase front caused by varying phase retardations in EM waves and the control of the phase front profile is achieved through two mechanisms: phase retardation caused by the width and shape of the individual slits in the lens, and the position of these slits. For certain applications, it would be
∗ Corresponding author at: School of Physics and Optoelectronic Engineering, Nanjing University of Information Science & Technology, Nanjing 210044, Jiangsu, China. Tel.: +86 13851781730. E-mail addresses: [email protected], [email protected] (G. Zheng). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.08.041
highly desirable to realize positional modulation through a plasmonic lens without changing the period of array, or the length and width of slits. In this paper, we extend the previous studies and examine ways of controlling the phase front of the transmitted beam by introducing compound metallic gratings with two subwavelength slits filled with different dielectrics inside each period. When suitable dielectrics are chosen, the Fabry-Pérot (FP)-like modes with the different orders can be found inside the two adjacent slits at a certain resonant frequency, an asymmetric phase front caused by varying phase retardations in EM waves as they pass through the plasmonic lens can be constructed based on the FP resonant condition. The slits used in our designs are rectangular that guarantee the simplicity in fabrication using the focused ion beam (FIB) technique.
2. Structure and simulated model Fig. 1 illustrates the proposed compound structure under study, where each period of the grating comprises two slits, slit 1 and slit 2, with identical widths but filled with different dielectric media with relative permittivities εd1 and εd2 , respectively. The interspacing between the two slits (center to center) is denoted as d, the period and thickness of the grating are and L, respectively. It is worth noting that the distance between any two slits is much larger than the skin depth of surface plasmon inside the silver, hence, no surface plasmon cross-talk will occur between adjacent slits.
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Fig. 2. (a) The calculated transmission spectra of the simple metallic gratings. (b) Variation of the real part of neff with permittivity for coupld-SPPs inside the metallic slit. Fig. 1. Schematic diagram of the proposed structure: each period of the grating comprises two slits with identical widths but filled with different dielectric media. The input light is a TM-polarized plane wave from the left.
The frequency-dependent permittivity of the metal is expressed by the Drude model for the dielectric response of the metal which is mainly governed by its free electron plasmon ε(ω) = 1 −
ωp2 ω(ω + i)
(1)
where ωp is the plasma frequency, is the absorption coefficient, and ω stands for the frequency. The metal investigated is silver and the considered wavelength range is 600–1300 nm. The corresponding parameters are ωp = 1.37 × 1016 rad/s and = 3.21 × 1013 rad/s [19]. When ω < ωp , the real part of the permittivity is negative, the standard time iteration scheme of the FDTD method becomes unstable. However, this problem can be solved by introducing the current density into Maxwell’s equations [20]. We assume the monochromatic operation with the free space wave number k0 = ω0 /c, the time-dependency of exp(−jωt), and corresponding free space wave length 0 = 2/k0 . Here, c is the speed of the light in vacuum. To demonstrate the validity of designed structure, twodimensional FDTD simulation is performed with the perfectly matched layer (PML) absorbing boundary conditions in x and z directions of the simulation domain. Since the width of the bus waveguide is much smaller than the operating wavelength in the structure, only the fundamental waveguide mode is supported. The incident light for excitation of the SPP mode is a TM-polarized (the magnetic field is parallel to y axis) fundamental mode. In the following FDTD simulations, the grid size in the x and z directions are chosen to be x = z = 2 nm and t = x/2c, which are sufficient for the numerical convergence, here c is the velocity of light in vacuum. The normalized time-averaged magnetic field intensity |Hy |2 is employed to represent the field intensity. 3. Simulation results and discussion In calculation, the whole length of the structure in the xdirection (see Fig. 1) is 2 micrometers (m). In addition, two power monitors are placed at the input and output side of the grating for calculating the incident power (Pin ) and the transmitted power (Pout ). Subsequently, the transmission (Tr ) is defined to be Tr = Pout /Pin . Fig. 2(a) displays the calculated zero-transmission spectra of the simple gratings with subwavelength slits filled with different dielectric εd1 = εd2 = 1, εd1 = εd2 = 2.48 and εd1 = εd2 = 4.5, respectively. The parameters are chosen as: w = 50 nm, d = 200 nm, L = 700 nm, = 600 nm. For εd1 = εd2 = 1, the transmission peaks at wavelengths 730 nm and 1050 nm are associated with the thirdand second-order FP-like modes, respectively [21]. Similarly, the transmission peaks at wavelengths 670 nm, 800 nm and 1050 nm
for εd1 = εd2 = 2.48 attribute to the fifth-, fourth- and third-order FP-like modes, respectively, and the transmission peaks at wavelengths 650 nm, 740 nm, 860 nm and 1050 nm for εd1 = εd2 = 4.5 attribute to the seventh- sixth-, fifth- and fourth-order FP-like modes, respectively. For the bare metallic slit array, current density standing waves are established on both metal alls of each slit by incident oscillating light [22]. Owing to the existence of current flow, the distribution of charge density has been rearranged on the both surfaces of each slit. The length of FP cavity approximately equals to the distance of current flow on one wall, which is the extent of charge distribution in the y direction. The resonant wavelength of F–P mode in a bare slit array can be obtained by [23]: 2k0 Re(neff )L + ϕ = N · 2
(2)
where k0 = 2/ is the wave vector of light in vacuum, is the wavelength of the incident light in vacuum, neff is the effective refractive index of coupled-SPPs (waveguide mode) inside the slit, ϕ is an additional phase shift experienced by the fundamental mode when reflecting at the grating interfaces, and N is an arbitrary integer. For TM-polarized case, neff can be approximately solved from [24]: εd
εm
n2eff − εm
n2eff − εd
=
1 − exp(k0 w 1 + exp(k0 w
n2eff − εd )
n2eff − εd )
(3)
Fig. 2(b) plots the variation of the real parts of neff for the metallic slit filled with different dielectric media with four wavelengths. Fig. 3(a)–(d) further illustrate the corresponding field intensity distributions when the slits are filled with different material at FP resonant wavelengths. The results show that the energy emerging from the structure tends to focus within several micrometers in an extremely small region (i.e., small FWHM). As seen from Fig. 3(b) and (d), the focal length will be changed when the resonant wavelength is different. To deflect the beam, curved phase front of the transmitted beam should be achieved. The authors in Ref. [18] proposed a method to control the phase front profile by the width and shape of the individual slits and the position of these slits. Unlike previous studies [7–10,18], phase retardation in our structure is tuned by FP resonant condition. There is no need to change the period of array, or the length and width of slits. Thus the structure proposed in this paper is a much smaller, more compact device and convenient for integration. The position of focus and the bending angle are determined using the image processing software, ImageJ [25], to capture the area with the highest intensity within the main beam. Under normally incident radiation with different wavelengths, field distributions are shown in Fig. 4, where one can find the off-axis focusing phenomenon. Bam deflection can be observed from the simulation due to the constructive interference of the phase-delayed, transmitted waves from the rectangular slits. The positional and directional modulation functionalities of a plasmonic lens are determined by the relative phase difference
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Fig. 3. Profiles of field intensity of the proposed structure when the slits are filled with different dielectric media. (a) εd1 = εd2 = 1 with a wavelength of 1050 nm, (b) εd1 = εd2 = 2.48 with a wavelength of 1050 nm, (c) εd1 = εd2 = 4.5 with a wavelength of 1050 nm and (d) εd1 = εd2 = 2.48 with a wavelength of 800 nm.
between neighboring slits. The phase differences between the two slits are determined by [26]:
ϕ = ϕ1 − ϕ2 = Re(ˇ1 L) + arg 1 −
− Re(ˇ2 L) − arg 1 −
1 − (ˇ1 /k0 ) 1 + (ˇ1 /k0 )
1 − (ˇ2 /k0 ) 1 + (ˇ2 /k0 )
2
exp(2jˇ1 L)
2
exp(2jˇ2 L)
(4)
By illuminating light on upper side of slits, SPPs can be excited in the subwavelength slits, the phase retardation is dominantly determined by the real part of propagation constant [13]. Therefore Eq. (4) can be approximated as Re(ˇ1 L) − Re(ˇ2 L). ˇ1 and ˇ2 are complex propagation constants in the two slits which can be calculated from [5,8,9]:
tanh
w 2
ˇi2
− εdi k02
=
−εd εAg
ˇi2 − εAg k02
ˇi2 − εdi k02
(i = 1, 2)
(5)
where w is slit width, εdi and εAg are relative dielectric constants for metal and the materials between slits, respectively. By calculating the propagation constants from Eq. (5), and by substituting their value in Eq. (4) we obtain the phase difference of the fields exiting from the slits. As a consequence of the phase difference, the beam exiting the nano-slit is deflected. By introducing compound metallic gratings with two subwavelength slits filled with different dielectrics inside each period, we can create a relative phase difference profile across the width of the lens. The position of the constructive interferences of the decoupled optical waves depends
strongly on the phase delay profile, and it is expressed as beam modulation effect. As seen from Fig. 4, the four lens designs vary not only in bending angle, but also in light distribution and intensity. Here we show beam deflection through appropriately choosing the refractive index of material filling in the metal slits. In order to deflect the beam, the phase front of the transmitted beam should be inclining to the exit plane of the slit array, thus the index distribution should be asymmetric instead of the symmetrical one as used in the focusing case. A main beam deflection of ∼11◦ can be observed from Fig. 4(a). When the wavelength is changed to = 1050 nm, a beam deflection of ∼20 ◦ can be observed from Fig. 4(b). When εd1 = 1, εd2 = 4.5, = 860 nm, the transmitted beam will bend to the negative direction, as can be seen from Fig. 4(c). When = 1050 nm, because εd = 1 and εd = 4.5 are associated with the second-order and fourth-order FP-like modes, respectively. Hence there is a focal spot on the horizontal direction, which is shown in Fig. 4(d). The simulated phase front of transmitted beam is consistent with our anticipated results. Fig. 5 shows the field intensities of the proposed structure for beam bending when εd1 = 2.48, εd2 = 4.5 with different wavelengths. Bending angle becomes larger with the increasing of wavelength. When the optical waves are incident at an angle to the optical axis will add an extra planar phase front that interferes with the asymmetric phase front of the plasmonic lens, leading to a larger bending angle off the axial direction. The slits used in our designs are rectangular that guarantee the simplicity in fabrication using the FIB technique. In addition, the proposed lenses may be further optimized by fine-tuning the geometrical parameters (such as width, length, and spacing of the slits).
Fig. 4. Profiles of field intensity of the proposed structure for beam bending. (a) εd1 = 1, εd2 = 2.48, = 790 nm; (b) εd1 = 1, εd2 = 2.48, = 1050 nm; (c) εd1 = 1, εd2 = 4.5, = 860 nm; (d) εd1 = 1, εd2 = 4.5, = 1050 nm. The other parameters are the same as in Fig. 3.
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G. Zheng et al. / Optik 125 (2014) 1529–1532
Fig. 5. Profiles of field intensity of the proposed structure for beam bending. (a) εd1 = 2.48, εd2 = 4.5, = 650 nm; (b) εd1 = 2.48, εd2 = 4.5, = 670 nm; (c) εd1 = 2.48, εd2 = 4.5, = 860 nm; (d) εd1 = 2.48, εd2 = 4.5, = 1050 nm. The other parameters are the same as in Fig. 3.
4. Conclusion In conclusion, a method to achieve the directional tuning of light was proposed. The compound metallic gratings contain two subwavelength slits filled with different dielectrics inside each period. Numerical simulation results indicate that the beam bending effects are governed by the FP resonant conditions. There is no need to change the period of array, or the length and width of slits. Thus the structure proposed is a much smaller, more compact device and convenient for integration. Compared with existing plasmonic lenses [7–18] that can bend light, our lens designs have advantages in ease of fabrication and capability to perform in the far field. With these advantages, we expect our proposed structure to be helpful for designing subwavelength optical devices, as well as facilitating manufacturing processes. Acknowledgments This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 61203211 and 11204140), the Foundation for Outstanding Young Teachers of Nanjing University of Information Science & Technology (Grant No. 20110423), and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 13KJB140006). The authors would especially like to thank Dr. Xu Yi from Guangzhou University, Su Wei from Nanjing University of Science & Technology for their useful discussions. References [1] W.L. Barnes, A. Dereux, T.W. Ebbesen, Surface plasmon subwavelength optics, Nature 424 (2004) 824–830. [2] H.J. Lezec, A. Degiron, E. Devaux, R.A. Linke, L. Martin-Moreno, F.J. GarciaVidal, T.W. Ebbesen, Beaming light from a subwavelength aperture, Science 297 (2002) 820–822. [3] S.-Y. Lee, J. Park, I. Woo, N. Park, B. Lee, Surface plasmon beam splitting by the photon tunneling through the plasmonic nanogap, Appl. Phys. Lett. 97 (2010) 133113. [4] C.L. Zou, F.W. Sun, C.H. Dong, X.F. Ren, J.M. Cui, X.D. Chen, Z.F. Han, G.C. Guo, Broadband integrated polarization beam splitter with surface plasmon, Opt. Lett. 36 (2011) 3630–3632. [5] T. Xu, C.T. Wang, C.L. Du, X.G. Luo, Plasmonic beam deflector, Opt. Express 16 (2008) 4753–4759.
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