Dichroic directional excitation of surface plasmon based on an integer-programming model Quansheng Chen, Yuanchao Sun, and Yueke Wang* Optical Information Science and Technology Department, Jiangnan University, Wuxi, Jiangsu 214122, China *Corresponding author: [email protected] Received 6 January 2015; revised 17 February 2015; accepted 19 February 2015; posted 19 February 2015 (Doc. ID 231982); published 23 March 2015
A silver film perforated with two subwavelength uniform slits is proposed for dichroic directional excitation of surface plasmon polaritons (SPPs). Under backside oblique illumination, the SPPs for two work wavelengths can propagate along the two opposite directions or in the same direction. Based on SPP interference, an integer-programming model is established for dichroic directional excitation of SPPs. The branch and bound method is introduced to find the optimal solutions for the integer-programming model, and therefore, the parameters of the structure and illumination angles can be obtained. The field distribution of the structure is investigated by using the finite-difference time-domain method (FDTD) to verify our design. Our theoretical model can achieve dichroic directional excitation of SPPs, simultaneously. © 2015 Optical Society of America OCIS codes: (240.6680) Surface plasmons; (240.6700) Surfaces; (250.5403) Plasmonics. http://dx.doi.org/10.1364/AO.54.002625
1. Introduction
Modern information processing demands ultracompact photonic devices at the chip scale. Surface plasmon polaritons (SPPs) are a mixed mode of an electromagnetic field coupled to collective electronic excitations propagating along the metal-dielectric interface [1]. These SPPs are considered as a promising candidate in future highly integrated photonic devices because they can break the diffraction limit [2]. Also, the subwavelength locality and near-field enhancement of the SPPs suggest other applications such as nanoscale optical waveguides [3], nanolithography [4], and biosensing [5]. Thus, the directional control of SPP excitation and propagation is an important issue in designing plasmonic devices. By adding asymmetrical gratings on both sides of the nanoslit, Gan et al. [6] have proposed a device capable of guiding SPPs of two different wavelengths, which correspond to the grating periods, propagating in opposite directions. Based on the 1559-128X/15/102625-05$15.00/0 © 2015 Optical Society of America
same principle, bidirectional or multidirectional splitters for SPPs in the visible, terahertz, and microwave frequencies have been experimentally realized [7–9]. Furthermore, SPP splitting structures can be used to measure the pulse width [10] and split the polarization [11]. Another principle, which is used to design unidirectional plasmonic excitation devices, is based on SPP interference. The SPPs can be excited unidirectionally along the direction in which SPPs interfere constructively. Several subwavelength metallic structures have been proposed to achieve the unidirectional excitation of SPPs, such as twoslits [12], one slit and one groove [13], two cavities [14], one stepped slit [15], and two grooves [16]. In addition, research progress for graphene plasmonics has provided a new way to achieve tunable directional excitation of SPPs [17,18]. In this paper, a silver film with two subwavelength uniform slits is proposed. Under backside oblique illumination, SPPs can be excited and they propagate along the silver–air interface unidirectionally based on SPP interference. To achieve dichroic directional excitation of surface plasmon, 2 × 2 equations based on SPP interference are listed, and an 1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS
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integer-programming model (IPM) [19] is established to find the optimal solution to the 2 × 2 equations. By using the branch and bound method (B-BM) [20], the geometrical parameters and illumination angle can be obtained. We take 632.8 and 532.8 nm as the two work wavelengths. Under the optimized parameters, the SPPs for the two wavelengths can propagate along opposite directions or in the same direction, and a finite-difference timedomain method (FDTD) simulation is employed to verify our design.
φ2
2π λspp1
2π φ1 − φ2 d sin θ: λ
(1)
Under oblique illumination, the left and right SPPs can be controlled to interfere constructively to one side and destructively to the other side, thereby realizing unidirectional excitation of SPPs. For example, if we want to excite SPPs, in relation to the wavelength that is λspp1 , only to the right side, the following phase matching conditions should be satisfied: φ1
2π λspp1
d φ2 2m1 π;
(2)
φ1
2π λspp2
φ2
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d φ2 2m3 1π;
2π d φ1 2m4 π; λspp2
(4)
(5)
where m3 and m4 are arbitrary integers, respectively. When Eqs. (2)–(5) are satisfied, the SPP mode of λspp1 can only propagate along the right side and the SPP mode of λspp2 can only propagate along the left side. Based on the above Eqs. (1)–(5), there are two undetermined geometrical parameters and the 2 × 2 undetermined integer, but it only has five equations. Therefore, it is impossible to find the exact solution. To obtain the optimal solution to this problem, an IPM based on Eqs. (1)–(5) is established as follows: 1 2π 2π d d sin θ − 2m1 π ; min λ λ π
(6.1)
2π 2π ≤ σπ; d − d sin θ − 2m 1π 2 λ λ
(6.2)
2π 2π d d sin θ − 2m3 1π ≤ σπ; λ λ
(6.3)
1
spp1
1
spp1
2
spp2
2π 2π d − d sin θ − 2m4 π ≤ σπ: λ λ
(6.4)
2
spp2
Here, Eq. (6.1) serves as an objective function in the IPM. Equations (6.2)–(6.4) serve as constraint conditions. When the objective function Eq. (6.1) reaches a minimal value and other constraint conditions in Eqs. (6.2)–(6.4) are satisfied, an optimal solution, which is very close to the exact solution, to Eqs. (1)–(5) has been found. In order to search for the optimal solution quickly, other constraint conditions can be satisfied according to the actual situations:
Fig. 1. Schematic of the structure composed of silver film with two slits.
(3)
where m1 and m2 are arbitrary integers, respectively. Moreover, if we want to excite SPPs, in relation to the wavelength that is λspp2, only to the left side, the following phase matching conditions should be satisfied:
2. Model and Method
The proposed structure is depicted in Fig. 1, in which two subwavelength uniform slits spaced by d are perforated into a silver film with thickness t. Other areas are air. When a TM polarized light whose magnetic field vector is parallel with the slits impinges on the structure from the backside, the scattered light from each slit may excite SPPs along the silver–air interface. At oblique incidence, asymmetrical excitation of SPPs can be achieved by modulating the phase difference (and thereby the interference) of SPPs launched separately from the two slits. The initial phases of SPPs generated by the left and right slits are denoted as φ1 and φ2 , respectively. With the illumination angle being illustrated by θ and the distance between the two slits being d, the phase difference, which is solely induced by the oblique incidence, is shown as follows:
d φ1 2m2 1π;
Lmin ≤ d ≤ Lmax ; θ ≤ π∕2
7
Here, Lmin and Lmax are the lower and upper limit of the distance between the two subwavelength slits, respectively. This IPM [Eqs. (6) and (7)] can be solved by B-BM.
The optimization procedure using B-BM can be described as follows: Step 1: The IPM can be changed to a linear programming model (LPM). The optimal parameters d, θ, m0 m1 ; m2 ; m3 ; m4 and the value of objective function min can be obtained by solving the LPM. Any element of m0 ranges from mimin to mimax . At first, all initial elements of m0 range from −∞ to ∞. If the solution U 0 d0 ; θ0 ; m0 cannot be found, STOP; the IPM has no solutions. Step 2: If all elements of m0 are all integers, then skip to Step 9. Otherwise, we choose the first element mi of m0, which is not an integer. The value range of mi can be divided into two branches (mimin , [mi ] and mi 1, mimax ). ([x] represents the maximum integer that is less than x.) Step 3: The solution U 0 and the optimal value min can be obtained by solving the LPM in the two branches of Step 2, respectively. U 1 , min1 and U 2 , min2 are the solutions and the values of objective functions in branches one and two, respectively. Step 4: If there is no solution in the two branches, STOP; the IPM has no solutions. Step 5: If the U 1 , min1 cannot be found, skip to Step 8. If the U 2 , min2 cannot be found, skip to Step 7. Step 6: If the M 1 is closer to min than M 2 . Skip to Step 7. Otherwise, skip to Step 8. Step 7: mimin mi 1 and U 0 U 1 , min min1 . Skip to Step 2. Step 8: mimax mi and U 0 U 2 , min min2 . Skip to Step 2. Step 9: STOP: the optimal parameters of the IPM are d0 , D0 , and m0 , and the optimal value is min. 3. Simulation Results
We take the two work wavelengths λ1 532.8 nm and λ2 632.8 nm for examples. The thickness of silver is chosen as t 400 nm, and the width of the two uniform slits is 50 nm. The permittivity of silver is given by the Drude model, εm ω ε∞ − ω2p ∕ ωω iγ, where ε∞ 4.2 is the relative permittivity at an infinite frequency, ωp 1.346 × 1016 rad∕s is the bulk plasma frequency, γ 9.13 × 1013 rad∕s is the collision frequency, and ω is the frequency of the incident light [21]. Here, two SPP modes of λ1 532.8 nm and λ2 632.8 nm propagate along the right side and left side, respectively. Additionally, the two SPP mode wavelengths are λspp1 509 nm and λspp2 614 nm. When the error value is chosen as σ 0.03, Lmin and Lmax are chosen as 1 and 6 μm, respectively. The optimal geometrical parameters are obtained as follows: d 2.92 μm and θ 36.4° by B-BM. The objective function min is 1e−25 , and m1 9, m2 2, m3 7, and m4 2. To verify our design method, FDTD is conducted, and the calculated time-average intensity jHj distribution is shown in Fig. 2. Figure 2(a) shows that the incident light illuminates on the upper surface of the silver film, and a strong density for jHj is found on the lower right
Fig. 2. Calculated time-average intensity jHj distribution: the indent light (a) λ1 532.8 nm and (b) λ2 632.8 nm.
surface of the film. Figure 2(b) shows that the incident light illuminates on the upper surface, and a strong density for jHj is found on the left lower interface. The intensity distribution of jHj determines the excitation directionality of the SPPs. To illustrate the SPPs intensity, we calculate the spatial frequency spectrum along the x-direction by fast Fourier transform (FFT), as shown in Fig. 3. Figure 3 shows the spatial frequency spectra for Hx corresponding to Figs. 2(a) and 2(b). The spatial frequencies of the SPPs along the silver–air interface are kspp ≈ 1.05k0 and kspp ≈ 1.03k0 for the incident wavelengths λ 532.8 and 632.8 nm, respectively. Thus, the normalized intensities jHkx 1.05k0 j2 and jHkx 1.03k0 j2 represent the relative intensities of the SPPs for the incident wavelengths λ 532.8 and 632.8 nm, propagating rightward and leftward, respectively. The solid line, which is obtained on the basis of the Hx, 20 nm away from the lower surface of the silver film in Fig. 2(a) shows that jHkx1 1.05k0 j2 ∕jHkx1 −1.05k0 j2 8.6, and the SPPs are directionally propagating along the lower right. The dotted line, which is obtained on the basis of the Hx, 20 nm away from the lower surface of silver film in Fig. 2(b) shows that jHkx2 1.03k0 j2 ∕jHkx2 −1.03k0 j2 1∕10.5, and the SPPs are directionally propagating along the lower left. In the same way, if we want to excite the two SPPs, the wavelengths of which are λspp1 and λspp2 and both propagate along the left side unidirectionally, the following phase matching conditions should be satisfied: φ1
2π d φ2 2m1 1π; λspp1
(8)
Fig. 3. Spatial frequency spectrum along the x-direction; the solid and dotted lines correspond to Figs. 2(a) and 2(b), respectively. 1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS
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φ2
φ1
2π λspp1
d φ1 2m2 π;
(9)
2π d φ2 2m3 1π; λspp2
(10)
2π d φ1 2m4 π; λspp2
(11)
φ2
where m1 ; m2 ; m3 , and m4 are arbitrary integers, respectively. The above equations also can establish an IPM model. We still use the two work wavelengths λ1 532.8 nm and λ2 632.8 nm for examples. The thickness of silver is chosen as t 400 nm, and the width of the slit is 50 nm. When the error value is chosen as σ 0.1; Lmin and Lmax are chosen as 1 and 6 μm, respectively. The optimal geometrical parameters are obtained as follows: d 5.7 μm and θ 35.5° by B-BM. The objective function min is 1e−4, and m1 17, m2 5, m3 14, and m4 4. To verify our design method, FDTD is conducted, and the calculated time-average intensity jHj distribution is shown in Fig. 4. Figure 4(a) shows that the incident light illuminates on the upper surface of the silver film, and a strong density for jHj is found on the lower left surface of the film. Figure 4(b) shows that the incident light illuminates on the upper surface, and a strong density for jHj is found on the left lower interface. The intensity distribution of jHj determines the excitation directionality of the SPPs. To illustrate the SPP intensity, we calculate the spatial frequency spectrum along the x-direction by FFT, as shown in Fig. 5. Figure 5 shows the spatial frequency spectra for Hx corresponding to Figs. 4(a) and 4(b). The
Fig. 4. Calculated time-average intensity jHj distribution: the indent light (a) λ1 532.8 nm and (b) λ2 632.8 nm.
Fig. 5. Spatial frequency spectrum along the x-direction; the solid and dotted lines correspond to Figs. 4(a) and 4(b), respectively. 2628
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solid line, which is obtained on the basis of the Hx, 20 nm away from the lower surface of the silver film in Fig. 4(a) shows that jHkx1 −1.05k0 j2 ∕ jHkx1 1.05k0 j2 9.9, and the SPPs are directionally propagating along the lower left. The dotted line, which is obtained on the basis of the Hx, 20 nm away from the lower surface of silver film in Fig. 4(b) shows that jHkx2 −1.03k0 j2 ∕jHkx2 1.03k0 j2 5.7, and the SPPs are directionally propagating along the lower left. 4. Conclusion
In this article, a silver film with two subwavelength uniform slits was proposed. Under backside illumination, SPPs were excited and they propagated along the metal–air interface unidirectionally based on SPP interference. To achieve dichroic directional excitation of surface plasmon, 2 × 2 equations based on SPP interference were listed, and an IPM was established to find the optimal solution to the 2 × 2 equations. By using B-BM, the geometrical parameters and illumination angle were obtained. We took 532.8 and 632.8 nm as the two work wavelengths. Under the specific parameters, the SPPs for the two wavelengths could propagate along opposite directions or in the same direction, and FDTD simulation was employed to verify our design. This project was supported by the Fundamental Research Funds for the Central Universities of China under Grant No. JUSRP1037 and the National Natural Science Foundation of China under Grant Nos. 11404143 and 11347214. References 1. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, 1988). 2. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature 391, 667–669 (1998). 3. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440, 508–511 (2006). 4. W. Srituravanich, N. Fang, C. Sun, Q. Luo, and X. Zhang, “Plasmonic nanolithography,” Nano Lett. 4, 1085–1088 (2004). 5. M. Skorobogatiy and A. V. Kabashin, “Photon crystal waveguide-based surface plasmon resonance biosensor,” Appl. Phys. Lett. 89, 143518 (2006). 6. Q. Q. Gan, B. S. Guo, G. F. Song, L. H. Chen, Z. Fu, Y. J. Ding, and F. J. Bartoli, “Plasmonic surface-wave splitter,” Appl. Phys. Lett. 90, 161130 (2007). 7. Q. Q. Gan and F. J. Bartoli, “Bidirectional surface wave splitter at visible frequencies,” Opt. Lett. 35, 4181–4183 (2010). 8. H. Caglayan and E. Ozbay, “Surface wave splitter based on metallic gratings with sub-wavelength aperture,” Opt. Express 16, 19091–19096 (2008). 9. Y. J. Zhou and T. J. Cui, “Multidirectional surface-wave splitters,” Appl. Phys. Lett. 98, 221901 (2011). 10. S. B. Choi, D. J. Park, Y. K. Jeong, Y. C. Yun, M. S. Jeong, C. C. Byeon, J. H. Kang, Q. H. Par, and D. S. Kim, “Directional control of surface plasmon polariton waves propagating through an asymmetric Bragg resonator,” Appl. Phys. Lett. 94, 263115 (2009). 11. F. Lu, G. Y. Li, F. Xiao, and S. Xu, “Compact bidirectional polarization splitting antenna,” IEEE Photon. J. 4, 1744– 1751 (2012).
12. T. Xu, Y. H. Zhao, D. C. Gan, C. T. Wang, C. L. Du, and X. G. Luo, “Directional excitation of surface plasmons with subwavelength slits,” Appl. Phys. Lett. 92, 101501 (2008). 13. F. López-Tejeira, S. G. Rodrigo, L. Martin-Moreno, F. J. Garcia-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, M. U. Gonzalez, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3, 324–328 (2007). 14. G. Lerosey, D. F. P. Pile, P. Matheu, G. Bartal, and X. Zhang, “Controlling the phase and amplitude of plasmon sources at a subwavelength scale,” Nano Lett. 9, 327–331 (2009). 15. J. J. Chen, Z. Li, S. Yue, and Q. H. Gong, “Efficient unidirectional generation of surface plasmon polaritons with asymmetric single-nanoslit,” Appl. Phys. Lett. 97, 041113 (2010).
16. J. S. Q. Liu, R. A. Pala, F. Afshinmanes, W. S. Cai, and M. L. Brongersma, “A submicron plasmonic dichroic splitter,” Nat. Commun. 2, 525 (2011). 17. C. Zeng, X. M. Liu, and G. X. Wang, “Electrically tunable graphene plasmonic quasicrystal metasurfaces for transformation optics,” Sci. Rep. 4, 5673 (2014). 18. C. Zeng, J. Guo, and X. M. Liu, “High-contrast electro-optic modulation of spatial light induced by graphene-integrated Fabry-Pérot microcavity,” Appl. Phys. Lett. 105, 121103 (2014). 19. H. P. Williams, Model Building in Mathematical Programming, 3rd ed. (Wiley, 1990). 20. A. H. Land and A. G. Doig, “An automatic method of solving discrete programming problems,” Econometrica 28, 497–520 (1960). 21. E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).
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A silver film perforated with two subwavelength uniform slits is proposed for dichroic directional excitation of surface plasmon polaritons (SPPs). Under backside oblique illumination, the SPPs for two work wavelengths can propagate along the two opposite directions or in the same direction. Based on SPP interference, an integer-programming model is established for dichroic directional excitation of SPPs. The branch and bound method is introduced to find the optimal solutions for the integer-programming model, and therefore, the parameters of the structure and illumination angles can be obtained. The field distribution of the structure is investigated by using the finite-difference time-domain method (FDTD) to verify our design. Our theoretical model can achieve dichroic directional excitation of SPPs, simultaneously. © 2015 Optical Society of America OCIS codes: (240.6680) Surface plasmons; (240.6700) Surfaces; (250.5403) Plasmonics. http://dx.doi.org/10.1364/AO.54.002625
1. Introduction
Modern information processing demands ultracompact photonic devices at the chip scale. Surface plasmon polaritons (SPPs) are a mixed mode of an electromagnetic field coupled to collective electronic excitations propagating along the metal-dielectric interface [1]. These SPPs are considered as a promising candidate in future highly integrated photonic devices because they can break the diffraction limit [2]. Also, the subwavelength locality and near-field enhancement of the SPPs suggest other applications such as nanoscale optical waveguides [3], nanolithography [4], and biosensing [5]. Thus, the directional control of SPP excitation and propagation is an important issue in designing plasmonic devices. By adding asymmetrical gratings on both sides of the nanoslit, Gan et al. [6] have proposed a device capable of guiding SPPs of two different wavelengths, which correspond to the grating periods, propagating in opposite directions. Based on the 1559-128X/15/102625-05$15.00/0 © 2015 Optical Society of America
same principle, bidirectional or multidirectional splitters for SPPs in the visible, terahertz, and microwave frequencies have been experimentally realized [7–9]. Furthermore, SPP splitting structures can be used to measure the pulse width [10] and split the polarization [11]. Another principle, which is used to design unidirectional plasmonic excitation devices, is based on SPP interference. The SPPs can be excited unidirectionally along the direction in which SPPs interfere constructively. Several subwavelength metallic structures have been proposed to achieve the unidirectional excitation of SPPs, such as twoslits [12], one slit and one groove [13], two cavities [14], one stepped slit [15], and two grooves [16]. In addition, research progress for graphene plasmonics has provided a new way to achieve tunable directional excitation of SPPs [17,18]. In this paper, a silver film with two subwavelength uniform slits is proposed. Under backside oblique illumination, SPPs can be excited and they propagate along the silver–air interface unidirectionally based on SPP interference. To achieve dichroic directional excitation of surface plasmon, 2 × 2 equations based on SPP interference are listed, and an 1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS
2625
integer-programming model (IPM) [19] is established to find the optimal solution to the 2 × 2 equations. By using the branch and bound method (B-BM) [20], the geometrical parameters and illumination angle can be obtained. We take 632.8 and 532.8 nm as the two work wavelengths. Under the optimized parameters, the SPPs for the two wavelengths can propagate along opposite directions or in the same direction, and a finite-difference timedomain method (FDTD) simulation is employed to verify our design.
φ2
2π λspp1
2π φ1 − φ2 d sin θ: λ
(1)
Under oblique illumination, the left and right SPPs can be controlled to interfere constructively to one side and destructively to the other side, thereby realizing unidirectional excitation of SPPs. For example, if we want to excite SPPs, in relation to the wavelength that is λspp1 , only to the right side, the following phase matching conditions should be satisfied: φ1
2π λspp1
d φ2 2m1 π;
(2)
φ1
2π λspp2
φ2
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APPLIED OPTICS / Vol. 54, No. 10 / 1 April 2015
d φ2 2m3 1π;
2π d φ1 2m4 π; λspp2
(4)
(5)
where m3 and m4 are arbitrary integers, respectively. When Eqs. (2)–(5) are satisfied, the SPP mode of λspp1 can only propagate along the right side and the SPP mode of λspp2 can only propagate along the left side. Based on the above Eqs. (1)–(5), there are two undetermined geometrical parameters and the 2 × 2 undetermined integer, but it only has five equations. Therefore, it is impossible to find the exact solution. To obtain the optimal solution to this problem, an IPM based on Eqs. (1)–(5) is established as follows: 1 2π 2π d d sin θ − 2m1 π ; min λ λ π
(6.1)
2π 2π ≤ σπ; d − d sin θ − 2m 1π 2 λ λ
(6.2)
2π 2π d d sin θ − 2m3 1π ≤ σπ; λ λ
(6.3)
1
spp1
1
spp1
2
spp2
2π 2π d − d sin θ − 2m4 π ≤ σπ: λ λ
(6.4)
2
spp2
Here, Eq. (6.1) serves as an objective function in the IPM. Equations (6.2)–(6.4) serve as constraint conditions. When the objective function Eq. (6.1) reaches a minimal value and other constraint conditions in Eqs. (6.2)–(6.4) are satisfied, an optimal solution, which is very close to the exact solution, to Eqs. (1)–(5) has been found. In order to search for the optimal solution quickly, other constraint conditions can be satisfied according to the actual situations:
Fig. 1. Schematic of the structure composed of silver film with two slits.
(3)
where m1 and m2 are arbitrary integers, respectively. Moreover, if we want to excite SPPs, in relation to the wavelength that is λspp2, only to the left side, the following phase matching conditions should be satisfied:
2. Model and Method
The proposed structure is depicted in Fig. 1, in which two subwavelength uniform slits spaced by d are perforated into a silver film with thickness t. Other areas are air. When a TM polarized light whose magnetic field vector is parallel with the slits impinges on the structure from the backside, the scattered light from each slit may excite SPPs along the silver–air interface. At oblique incidence, asymmetrical excitation of SPPs can be achieved by modulating the phase difference (and thereby the interference) of SPPs launched separately from the two slits. The initial phases of SPPs generated by the left and right slits are denoted as φ1 and φ2 , respectively. With the illumination angle being illustrated by θ and the distance between the two slits being d, the phase difference, which is solely induced by the oblique incidence, is shown as follows:
d φ1 2m2 1π;
Lmin ≤ d ≤ Lmax ; θ ≤ π∕2
7
Here, Lmin and Lmax are the lower and upper limit of the distance between the two subwavelength slits, respectively. This IPM [Eqs. (6) and (7)] can be solved by B-BM.
The optimization procedure using B-BM can be described as follows: Step 1: The IPM can be changed to a linear programming model (LPM). The optimal parameters d, θ, m0 m1 ; m2 ; m3 ; m4 and the value of objective function min can be obtained by solving the LPM. Any element of m0 ranges from mimin to mimax . At first, all initial elements of m0 range from −∞ to ∞. If the solution U 0 d0 ; θ0 ; m0 cannot be found, STOP; the IPM has no solutions. Step 2: If all elements of m0 are all integers, then skip to Step 9. Otherwise, we choose the first element mi of m0, which is not an integer. The value range of mi can be divided into two branches (mimin , [mi ] and mi 1, mimax ). ([x] represents the maximum integer that is less than x.) Step 3: The solution U 0 and the optimal value min can be obtained by solving the LPM in the two branches of Step 2, respectively. U 1 , min1 and U 2 , min2 are the solutions and the values of objective functions in branches one and two, respectively. Step 4: If there is no solution in the two branches, STOP; the IPM has no solutions. Step 5: If the U 1 , min1 cannot be found, skip to Step 8. If the U 2 , min2 cannot be found, skip to Step 7. Step 6: If the M 1 is closer to min than M 2 . Skip to Step 7. Otherwise, skip to Step 8. Step 7: mimin mi 1 and U 0 U 1 , min min1 . Skip to Step 2. Step 8: mimax mi and U 0 U 2 , min min2 . Skip to Step 2. Step 9: STOP: the optimal parameters of the IPM are d0 , D0 , and m0 , and the optimal value is min. 3. Simulation Results
We take the two work wavelengths λ1 532.8 nm and λ2 632.8 nm for examples. The thickness of silver is chosen as t 400 nm, and the width of the two uniform slits is 50 nm. The permittivity of silver is given by the Drude model, εm ω ε∞ − ω2p ∕ ωω iγ, where ε∞ 4.2 is the relative permittivity at an infinite frequency, ωp 1.346 × 1016 rad∕s is the bulk plasma frequency, γ 9.13 × 1013 rad∕s is the collision frequency, and ω is the frequency of the incident light [21]. Here, two SPP modes of λ1 532.8 nm and λ2 632.8 nm propagate along the right side and left side, respectively. Additionally, the two SPP mode wavelengths are λspp1 509 nm and λspp2 614 nm. When the error value is chosen as σ 0.03, Lmin and Lmax are chosen as 1 and 6 μm, respectively. The optimal geometrical parameters are obtained as follows: d 2.92 μm and θ 36.4° by B-BM. The objective function min is 1e−25 , and m1 9, m2 2, m3 7, and m4 2. To verify our design method, FDTD is conducted, and the calculated time-average intensity jHj distribution is shown in Fig. 2. Figure 2(a) shows that the incident light illuminates on the upper surface of the silver film, and a strong density for jHj is found on the lower right
Fig. 2. Calculated time-average intensity jHj distribution: the indent light (a) λ1 532.8 nm and (b) λ2 632.8 nm.
surface of the film. Figure 2(b) shows that the incident light illuminates on the upper surface, and a strong density for jHj is found on the left lower interface. The intensity distribution of jHj determines the excitation directionality of the SPPs. To illustrate the SPPs intensity, we calculate the spatial frequency spectrum along the x-direction by fast Fourier transform (FFT), as shown in Fig. 3. Figure 3 shows the spatial frequency spectra for Hx corresponding to Figs. 2(a) and 2(b). The spatial frequencies of the SPPs along the silver–air interface are kspp ≈ 1.05k0 and kspp ≈ 1.03k0 for the incident wavelengths λ 532.8 and 632.8 nm, respectively. Thus, the normalized intensities jHkx 1.05k0 j2 and jHkx 1.03k0 j2 represent the relative intensities of the SPPs for the incident wavelengths λ 532.8 and 632.8 nm, propagating rightward and leftward, respectively. The solid line, which is obtained on the basis of the Hx, 20 nm away from the lower surface of the silver film in Fig. 2(a) shows that jHkx1 1.05k0 j2 ∕jHkx1 −1.05k0 j2 8.6, and the SPPs are directionally propagating along the lower right. The dotted line, which is obtained on the basis of the Hx, 20 nm away from the lower surface of silver film in Fig. 2(b) shows that jHkx2 1.03k0 j2 ∕jHkx2 −1.03k0 j2 1∕10.5, and the SPPs are directionally propagating along the lower left. In the same way, if we want to excite the two SPPs, the wavelengths of which are λspp1 and λspp2 and both propagate along the left side unidirectionally, the following phase matching conditions should be satisfied: φ1
2π d φ2 2m1 1π; λspp1
(8)
Fig. 3. Spatial frequency spectrum along the x-direction; the solid and dotted lines correspond to Figs. 2(a) and 2(b), respectively. 1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS
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φ2
φ1
2π λspp1
d φ1 2m2 π;
(9)
2π d φ2 2m3 1π; λspp2
(10)
2π d φ1 2m4 π; λspp2
(11)
φ2
where m1 ; m2 ; m3 , and m4 are arbitrary integers, respectively. The above equations also can establish an IPM model. We still use the two work wavelengths λ1 532.8 nm and λ2 632.8 nm for examples. The thickness of silver is chosen as t 400 nm, and the width of the slit is 50 nm. When the error value is chosen as σ 0.1; Lmin and Lmax are chosen as 1 and 6 μm, respectively. The optimal geometrical parameters are obtained as follows: d 5.7 μm and θ 35.5° by B-BM. The objective function min is 1e−4, and m1 17, m2 5, m3 14, and m4 4. To verify our design method, FDTD is conducted, and the calculated time-average intensity jHj distribution is shown in Fig. 4. Figure 4(a) shows that the incident light illuminates on the upper surface of the silver film, and a strong density for jHj is found on the lower left surface of the film. Figure 4(b) shows that the incident light illuminates on the upper surface, and a strong density for jHj is found on the left lower interface. The intensity distribution of jHj determines the excitation directionality of the SPPs. To illustrate the SPP intensity, we calculate the spatial frequency spectrum along the x-direction by FFT, as shown in Fig. 5. Figure 5 shows the spatial frequency spectra for Hx corresponding to Figs. 4(a) and 4(b). The
Fig. 4. Calculated time-average intensity jHj distribution: the indent light (a) λ1 532.8 nm and (b) λ2 632.8 nm.
Fig. 5. Spatial frequency spectrum along the x-direction; the solid and dotted lines correspond to Figs. 4(a) and 4(b), respectively. 2628
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solid line, which is obtained on the basis of the Hx, 20 nm away from the lower surface of the silver film in Fig. 4(a) shows that jHkx1 −1.05k0 j2 ∕ jHkx1 1.05k0 j2 9.9, and the SPPs are directionally propagating along the lower left. The dotted line, which is obtained on the basis of the Hx, 20 nm away from the lower surface of silver film in Fig. 4(b) shows that jHkx2 −1.03k0 j2 ∕jHkx2 1.03k0 j2 5.7, and the SPPs are directionally propagating along the lower left. 4. Conclusion
In this article, a silver film with two subwavelength uniform slits was proposed. Under backside illumination, SPPs were excited and they propagated along the metal–air interface unidirectionally based on SPP interference. To achieve dichroic directional excitation of surface plasmon, 2 × 2 equations based on SPP interference were listed, and an IPM was established to find the optimal solution to the 2 × 2 equations. By using B-BM, the geometrical parameters and illumination angle were obtained. We took 532.8 and 632.8 nm as the two work wavelengths. Under the specific parameters, the SPPs for the two wavelengths could propagate along opposite directions or in the same direction, and FDTD simulation was employed to verify our design. This project was supported by the Fundamental Research Funds for the Central Universities of China under Grant No. JUSRP1037 and the National Natural Science Foundation of China under Grant Nos. 11404143 and 11347214. References 1. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, 1988). 2. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature 391, 667–669 (1998). 3. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440, 508–511 (2006). 4. W. Srituravanich, N. Fang, C. Sun, Q. Luo, and X. Zhang, “Plasmonic nanolithography,” Nano Lett. 4, 1085–1088 (2004). 5. M. Skorobogatiy and A. V. Kabashin, “Photon crystal waveguide-based surface plasmon resonance biosensor,” Appl. Phys. Lett. 89, 143518 (2006). 6. Q. Q. Gan, B. S. Guo, G. F. Song, L. H. Chen, Z. Fu, Y. J. Ding, and F. J. Bartoli, “Plasmonic surface-wave splitter,” Appl. Phys. Lett. 90, 161130 (2007). 7. Q. Q. Gan and F. J. Bartoli, “Bidirectional surface wave splitter at visible frequencies,” Opt. Lett. 35, 4181–4183 (2010). 8. H. Caglayan and E. Ozbay, “Surface wave splitter based on metallic gratings with sub-wavelength aperture,” Opt. Express 16, 19091–19096 (2008). 9. Y. J. Zhou and T. J. Cui, “Multidirectional surface-wave splitters,” Appl. Phys. Lett. 98, 221901 (2011). 10. S. B. Choi, D. J. Park, Y. K. Jeong, Y. C. Yun, M. S. Jeong, C. C. Byeon, J. H. Kang, Q. H. Par, and D. S. Kim, “Directional control of surface plasmon polariton waves propagating through an asymmetric Bragg resonator,” Appl. Phys. Lett. 94, 263115 (2009). 11. F. Lu, G. Y. Li, F. Xiao, and S. Xu, “Compact bidirectional polarization splitting antenna,” IEEE Photon. J. 4, 1744– 1751 (2012).
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