April 15, 2015 / Vol. 40, No. 8 / OPTICS LETTERS
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Discrete plasmonic Talbot effect in finite metal waveguide arrays Xiaoyan Shi,1,4 Wu Yang,2,5 Huaizhong Xing,1,* and Xiaoshuang Chen3 1
Department of Applied Physics and State Key Laboratory for Modification of Chemical Fibers and Polymer Materials, Donghua University, Ren Min Road 2999, Songjiang District, Shanghai 201620, China 2 3
College of Science, Henan University of Technology, Zhengzhou 450001, China
National Laboratory for Infrared Physics Shanghai Institute of Technical Physics, Chinese Academy of Sciences, 500 Yutian Road, Shanghai 200083, China 4
College of Science, Information Engineering University of PLA, Zhengzhou 450001, China 5 e-mail:
[email protected] *Corresponding author:
[email protected] Received December 11, 2014; revised March 11, 2015; accepted March 11, 2015; posted March 13, 2015 (Doc. ID 229002); published April 3, 2015
We introduce supermode theory into the propagation of surface plasmon polaritons (SPPs) in nanoscale metal waveguide arrays (MWGAs). The SPP supermodes in finite MWGAs are analyzed and the coefficient of excited supermodes can be determined quantitatively. The field intensity distributions in finite MWGAs can be explained by the superposition of the excited SPP supermodes. The discrete plasmonic Talbot effect in a finite MWGA is achieved successfully by adjusting different intensity for each input field. We also find that the period condition of the input fields in MWGAs is not the same with conventional dielectric waveguides. The theory is verified by the finite difference time-domain (FDTD) method. © 2015 Optical Society of America OCIS codes: (110.6760) Talbot and self-imaging effects; (240.6680) Surface plasmons; (130.2790) Guided waves; (160.3900) Metals. http://dx.doi.org/10.1364/OL.40.001635
The Talbot effect is a traditional optics phenomenon which has been studied in the past years [1–3]. Discrete Talbot effect has been demonstrated by Iwanow et al., in waveguide arrays, in which the Talbot effect occurs only as the spatial period of the input fields are P 1, 2, 3, 4, and 6 [4]. The plasmonic Talbot effect [5–8] and subwavelength metal waveguide arrays (MWGAs) [9–13] have also attracted more attention. Recently, discrete plasmonic Talbot effect was found in nanoscale metal waveguide arrays [14] and in monolayer graphene sheet arrays [15]. Usually, the waveguide array structures for discrete Talbot effect are quite large, while the number of waveguides is infinite for numerical simulation and hundreds for experiments. However, the Talbot effect in a finite MWGA has not been heavily researched. In this Letter, we attempt to achieve the discrete Talbot effect in finite MWGAs. We introduce supermode theory into the propagation of surface plasmon polaritons in nanoscale MWGAs. The SPP supermodes in finite MWGAs are analyzed. The coefficient of excited supermodes can be determined quantitatively by the input fields. Then the field intensity distributions in finite MWGAs can be explained by the superposition of the SPP supermodes. According to the SPP supermode theory, we succeeded in achieving the discrete Talbot effect in a finite MWGA by adjusting different intensity for each input field. The Talbot effect in an infinite array can also be explained by the theory of SPP supermodes. We also validate that the period condition of the input fields in infinite MWGAs is not the same with conventional dielectric waveguides. Our theoretical results are verified by the finite difference time-domain (FDTD) method [16]. Supermodes are the modes of propagation of a periodic array of waveguides, which consist of N identical 0146-9592/15/081635-04$15.00/0
waveguides [17]. The wave functions of the supermodes are written as linear combinations of the unperturbed wave functions of the individual waveguides: Hx; ye−iβz
X N
C n ψ n x; y e−iβz ;
(1)
n1
where C n is a constant, β is the propagation constant to be determined, and ψ n x; y is the unperturbed wave function of the mode supported by the nth waveguide when the separation between the waveguides is infinite. We assume that only the nearest neighbor coupling is present because the two waveguides are not too close to each other. The perturbation equation is J − Iβ2 − β20 C n K − β2 − β20 C n1 J − Iβ2 − β20 C n2 0;
(2)
where −β0 is the propagation constant of the mode of a single individual unperturbed waveguide. I is the overlap integral of the two individual wave functions that are not orthogonal to each other, K is the dielectric perturbation to one of the waveguides because of the presence of the other waveguide, and J represents the exchange coupling between the two waveguides [17]. According to the boundary condition C N1 C 0 0, the solution of Eq. (2) can be written C s;n sin
nsπ N 1
n 1; 2; 3; ; N;
where s is a supermode index given by © 2015 Optical Society of America
(3)
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s 1; 2; 3; ; N:
(4)
There are N independent solutions, each corresponding to an integer s. As the overlap integral I is much less than 1, the solution of Eq. (2) for the propagation constants of the sth supermode is β2s β20 K 2J cos
sπ : N 1
(5)
Above is the supermode theory of conventional dielectric waveguide arrays. Similarly, the supermode theory is still valid for SPPs in metal waveguide arrays. Next, we will study the supermode propagation of SPPs in an MWGA. We consider a system containing only two adjacent metal waveguides (N 2), where SPP modes are composed of an SPP symmetric mode and an SPP asymmetric mode, with propagation constants of βs and βa , respectively [9]. We think βs and βa are essentially the same as the propagation constants of the first and second supermodes. βs and βa are calculated by the rigorous field analysis approach in Ref. [9]. The structure is shown in Fig. 1(a); h is the guide width and d is the thickness of Ag film. Figure 1(b) shows the dependence of the propagation constant β1 and β2 on d, as h 30 nm, εair 1 and εAg −15.7 0.94j (the measured value at λ 632.8 nm [18]). When d 24 nm,
βs 1.523×107 m−1 : βa 1.985×107 m−1
Thus, the values of β0 K and J in Eq. (5) are
β20 K 3.130 J −0.810
×1014 m−2 : ×1014 m−2
10
According to Eqs. (3), (5), and (10), we can conclude that the propagation constants and the wave functions of SPP supermodes for more adjacent metal waveguides. Figures 2(a) and 2(b) are the simulation results of the field intensity distributions with the input period P 2 in an infinite array and a finite array of 13 metal waveguides, respectively. Figure 2(a) is the Talbot effect in an infinite array and familiar to us. Figure 2(b) is markedly different from Fig. 2(a), which the Talbot effect only occurs at the left center region and diminishes by degrees with the propagation distance. We will analyze the SPP supermodes of Fig. 2(b) in the following sections. In Fig. 2(b), the discrete system contains 13 adjacent two-dimensional metal waveguides (N 13) composed of Ag and air layers, in which the widths are d 24 nm and h 30 nm, respectively. The total field is formed by the superposition of all these supermodes, which can be written as φx; z; t
13 X
as H s xeiωt−βs z ;
(11)
s1
where as is the coefficient of each excited supermode.
6
H s xe−iβ1 z
X N
C s;n ψ n x; y e−iβs z ;
(12)
n1
Applying the supermode theory to the two metal waveguides, according to Eq. (5), the propagation constants of supermodes are β21 β20 K J;
(7)
β22 β20 K − J:
(8)
The SPP symmetric mode and asymmetric mode correspond to the first and second supermodes, i.e.,
βs β1 : βa β2
9
Fig. 1. (a) Scheme of two adjacent two-dimensional metal waveguides. The widths are d 24 nm and h 30 nm. (b) Dependence of the propagation constant of SPPs on the thickness of metal film d in the MWGA. βs is the SPP symmetric mode and βa is the SPP asymmetric mode. βs and βa correspond to the supermodes β1 and β2 , respectively.
C s;n is the coefficient of wave function in the nth waveguide of the s-order supermode, which can be calculated by Eq. (3). Since the input fields, which have identical intensity, are in the first, third, fifth, …, and thirteenth waveguides in Fig. 2(b), as can be calculated by 0 B B B @
C 1;1 C 1;2 .. .
C 2;1 C 2;2 .. .
C 1;13
C 2;13
C 13;1 C 13;2 .. . C 13;13
10 CB CB CB A@
a1 a2 .. . a13
0 1 1 0C C B C C B 1 C: CB B A @ .. C .A 1
(13)
1
Fig. 2. Metal waveguide arrays are composed of Ag and air layers. The simulation results of the Talbot effect with the input period P 2 in (a) an infinite metal waveguide arrays and (b) a finite array of thirteen metal waveguides. The maximum values indicated by arrows in the second (third) waveguide occur in 0.38, 1.14, and 1.84 (0, 0.72, and 1.46).
April 15, 2015 / Vol. 40, No. 8 / OPTICS LETTERS
Fig. 3. Intensity variations by the theoretical analysis of the SPP supermodes for Fig. 2(b) (without considering the transmission loss). The maximum values in the second waveguide occur in 0.39, 1.15, and 1.85. The maximum values in the third waveguide occur in 0, 0.74, and 1.50.
The right-hand side of the above equation is the input field’s intensity in each waveguide. The solution of Eq. (13) is as f0.74; 0; 0.37; 0; 0.18; 0; 0.10; 0; 0.18; 0; 0.37; 0; and 0.74g. As a2 a4 a12 0, only the symmetric supermodes are excited. The field distribution in Fig. 2(b) is the superposition of the excited symmetric supermodes. The intensity distributions in the second and third waveguides by the theoretical analysis of the SPP supermodes are shown in Fig. 3 (without considering the transmission loss). The solid line (the dashed line) indicates the field intensity in the second (third) waveguide. In Fig. 3, the theoretical maximum values in the second (third) waveguide occur in 0.39, 1.15, and 1.85 (0, 0.74, and 1.50). In Fig. 2(b), the simulational maximum values in the second (third) waveguide occur in 0.38, 1.14, and 1.84 (0, 0.72, and 1.46). Obviously, the theoretical results are consistent with the simulation results. The results of other waveguides come to similar conclusions. The theory of SPP supermodes is confirmed. Next, we attempt to achieve the discrete Talbot effect in the array of Fig. 2(b). According to Eq. (3), C 1;n , and C 13;n are equal in the first, third, fifth, … and thirteenth waveguide. We adjust the intensity of each input field in Fig. 2(b) as C 1;n ; the coefficient of the excited supermode can be calculated by 0
C 1;1 B C 1;2 B B .. @ . C 1;13
C 2;1 C 2;2 .. . C 2;13
0 1 10 1 C 1;1 C 13;1 a1 B C C B 0 C B C 13;2 C CB a2 C B C 3;1 C C; .. CB .. C B .. C . A@ . A B @ . A a13 C 13;13 C 13;1
(14)
where the right-hand side is the input field’s intensity in each waveguide. The solution of Eq. (14) is as f0.5; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; and 0.5g. Only the first-order supermode H 1 x and the thirteenth-order supermode H 13 x are excited. The total field is formed by the superposition of the two symmetric supermodes, in which the interference picture is the same with Talbot effect. To demonstrate the analytical results, we numerically simulate the propagation behavior of SPPs in the MWGA. The boundaries are set with perfectly matched layer boundary conditions. The relative permittivity of Ag is taken to be the measured value of εAg −15.7 0.94j at λ 632.8 nm [18]. The intensity of each input field is adjusted as C 1;n . The simulative intensity distribution is shown in
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Fig. 4. Field intensity distribution with the input period P 2 in the array of Fig. 2(b). Here the intensity of each input field is adjusted as C 1;n . The Talbot effect occurs clearly.
Fig. 4. We can see a periodic repetition of planar field distributions. The Talbot effect clearly occurs in the finite MWGA. The simulation value of the Talbot distance is 0.688 μm in Fig. 4. The theoretical value of the Talbot distance is 2π∕β13 − β1 . The values of β1 and β13 are β21 β20 K 2J cos
π 1.55×1014 m−2 ; 14
(15)
β213 β20 K 2J cos
13π 4.71×1014 m−2 : 14
(16)
Thus, the theoretical value of the Talbot distance is 0.680 μm. The simulation value shows a good agreement with the theoretical value. The discrete Talbot effect is achieved successfully in a finite array. The Talbot effect in an infinite array can also be explained by the theory of SPP supermodes. When N ∞, as can be calculated by 0
C 1;1 B C 1;2 B B .. @ . C 1;N
C 2;1 C 2;2 .. . C 2;N
0 1 10 1 1 C N;1 a1 B0C B C B C C N;2 C CB a 2 C B C .. CB .. C B 1 C: C . A@ . A B @0A . aN C N;N ..
(17)
The solution of Eq. (14) is as ≈ f0.64; 0; 0.21; 0; 0.1; 3; 0 ; 0; 0.13; 0; 0.21; 0; and 0.64g. Among these excited supermodes, the first-order supermode H 1 x and the N-order supermode H N x are dominant, with a coefficient of 0.64. The total field is mainly formed by the superposition of the two supermodes, of which the interference picture is the Talbot effect. By the SPP supermode theory, we can also explain the distribution with the input field pattern given with f1; 0; −1; 0; …g, which is mentioned in Refs. [4] and [15]. Since the input field pattern is f1; 0; −1; 0; …g, as can be calculated by
Fig. 5. Input field pattern is given with f1; 0; −1; 0; …g in a finite array of 13 metal waveguides. Left is the field intensity distribution. Right is the field amplitude distribution. Only the seventh-order supermode is excited, in which the field coefficients in the waveguides are C 7;n f1; 0; −1; 0; 1; 0; −1; 0; 1; 0; −1; 0; 1g.
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Fig. 6. Intensity distributions of (a) P 5 and (b) P 6. The discrete Talbot effect does not occur at P 6, but it occurs at P 5 in an infinite MWGA.
0
C 1;1 B C 1;2 B B .. @ . C 1;13
C 2;1 C 2;2 .. . C 2;13
10 1 0 1 1 C 13;1 a1 C B 0 C B C 13;2 C C CB a2 C B −1 C: .. CB .. C B B C . A@ . A @ 0 A .. a13 C 13;13 .
(18)
Thus, as f0; 0; 0; 0; 0; 0; 1; 0; 0; 0; 0; 0; 0g. Only the seven-order supermode is excited of which the field coefficients in waveguides are C 7;n f1; 0; −1; 0; 1; 0; −1; 0; 1; 0; −1; 0; and 1g. Thus, the field intensity in the second, fourth, …, and twelfth waveguide is zero. The intensity distribution and the amplitude distribution are shown in Fig. 5. The number of waveguides in Fig. 5 is 13, which can be expanded to infinity. Finally, we will check whether the period conditions P 1, 2, 3, 4, and 6 of conventional dielectric waveguide arrays are suitable for MWGAs, of which the number of waveguide is infinite. In Ref. [14], the discrete plasmonic Talbot effect in an infinite array was shown only with the input period P 2. The other periods have not been discussed and validated. We perform numerical simulations for these conditions in the infinite MWGAs. We find that the discrete Talbot effect does not exist at P 6, but it occurs at P 5 in the infinite MWGAs. The intensity distributions of P 5 and P 6 are shown in Fig. 6, so the period conditions of infinite MWGAs are P 1, 2, 3, 4, and 5. In conclusion, we introduce supermode theory into the propagation of surface plasmon polaritons in nanoscale MWGAs. The SPP supermodes in finite MWGAs are analyzed. The coefficient of excited supermodes can be determined quantitatively. Then the field intensity distributions in finite MWGAs can be explained by the superposition of the SPP supermodes. According to the SPP
supermode theory, we achieve the discrete Talbot effect in an array with finite metal waveguides by adjusting different intensities for each input field. We also supply a reasonable explanation with the SPP supermode theory for the Talbot effect in an infinite array. In addition, we verify that the period condition of the input fields in infinite MWGAs is not the same with conventional dielectric waveguides, but P 1, 2, 3, 4, and 5. This work is supported by the Plan of Nature Science Fundamental Research in Henan University of Technology (2012JCYJ15), and the National Natural Science Foundation of China (61376102). References 1. H. F. Talbot, Philos. Mag. 9(56), 401 (1836). 2. M. Wen, Y. Zhang, and M. Xiao, Adv. Opt. Photon. 5, 83 (2013). 3. S. Longhi, Phys. Rev. A 90, 043827 (2014). 4. R. Iwanow, D. A. May-Arrioja, D. N. Christodoulides, and G. I. Stegeman, Phys. Rev. Lett. 95, 053902 (2005). 5. M. R. Dennis, N. I. Zheludev, and F. J. Garc’ıa de Abajo, Opt. Express 15, 9692 (2007). 6. A. A. Maradudin and T. A. Leskova, New J. Phys. 11, 033004 (2009). 7. W. W. Zhang, C. L. Zhao, J. Y. Wang, and J. S. Zhang, Opt. Express 17, 19757 (2009). 8. Y. Yu, D. Chassaing, T. Scherer, B. Landenberger, and H. Zappe, Plasmonics 8, 723 (2013). 9. X. Fan, G. Wang, J. Lee, and C. T. Chan, Phys. Rev. Lett. 97, 073901 (2006). 10. G. Bartal, D. A. Genov, and X. Zhang, Phys. Rev. B 79, 201103 (2009). 11. L. Verslegers, P. B. Catrysse, Z. F. Yu, and S. H. Fan, Phys. Rev. Lett. 103, 033902 (2009). 12. M. Conforti, M. Guasoni, and C. D. Angelis, Opt. Lett. 33, 2662 (2008). 13. G. D. Valle and S. Longhi, Opt. Lett. 35, 673 (2010). 14. Y. Wang, K. Zhou, X. Zhang, K. Yang, Y. Wang, Y. Song, and S. Liu, Opt. Lett. 35, 685 (2010). 15. Y. Fan, B. Wang, K. Wang, H. Long, and P. X. Lu, Opt. Lett. 39, 3371 (2014). 16. A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000). 17. A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications (Oxford University, 2007). 18. E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).
