Citation for this paper:
Chen, S., Jin, S. & Gordon, R. (2014). Super-transmission from a finite
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Super-transmission from a finite subwavelength arrangement of slits in a metal film Shuwen Chen, Shilong Jin, and Reuven Gordon
May 2014
This article was originally published at:
Super-transmission from a finite subwavelength
arrangement of slits in a metal film
Shuwen Chen,1,2 Shilong Jin,1and Reuven Gordon 2,*
1Department of Optoelectronic Engineering, College of Optoelectronic Science and Engineering, National University of Defense Technology, Changsha, 410073, China
2Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC, V8W 3V6, Canada
*rgordon@uvic.ca
Abstract: A theory is presented for the transmission of transverse magnetic
waves through a finite number of subwavelength slits in metal film. While a single slit achieves the single channel limit on resonance, multiple slits show super-transmission exceeding the single channel limit. The phenomenon of super-transmission is revealed as a result of cross-coupling of modes and confirmed by simulations. The influence of finite permittivity in the IR and microwave regime is included by perturbative corrections to the theory. The theory agrees quantitatively with past experiments and finite-difference time-domain simulations. By considering two or more modes in the slit region, our theory provides an approach to the analysis of cross-coupling among slits, which allows for super-transmission and features of a Fano resonance.
©2014 Optical Society of America
OCIS codes: (050.6624) Subwavelength structures; (250.5403) Plasmonics; (230.4555) Coupled resonators.
References and links
1. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998).
2. F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82(1), 729–787 (2010).
3. Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, “Transmission of light through slit apertures in metallic films,” Opt. Express 12(25), 6106–6121 (2004).
4. R. Gordon, A. G. Brolo, D. Sinton, and K. L. Kavanagh, “Resonant optical transmission through hole-arrays in metal films: Physics and applications,” Laser Photonics Rev. 4(2), 311–335 (2010).
5. S.-H. Chang, S. K. Gray, and G. C. Schatz, “Surface plasmon generation and light transmission by isolated nanoholes and arrays of nanoholes in thin metal films,” Opt. Express 13(8), 3150–3165 (2005).
6. F. van Beijnum, C. Rétif, C. B. Smiet, H. T. Liu, P. Lalanne, and M. P. van Exter, “Quasi-cylindrical wave contribution in experiments on extraordinary optical transmission,” Nature 492(7429), 411–414 (2012). 7. J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow
slits,” Phys. Rev. Lett. 83(14), 2845–2848 (1999).
8. Y. Takakura, “Optical resonance in a narrow slit in a thick metallic screen,” Phys. Rev. Lett. 86(24), 5601–5603 (2001).
9. R. Gordon, “Angle-dependent optical transmission through a narrow slit in a thick metal film,” Phys. Rev. B 75(19), 193401 (2007).
10. S. A. Maier, “Plasmonic field enhancement and SERS in the effective mode volume picture,” Opt. Express 14(5), 1957–1964 (2006).
11. R. Gordon, “Near-field interference in a subwavelength double slit in a perfect conductor,” J. Opt. Soc. Amer. A 8(6), L1–L3 (2006).
12. H. F. Shi, X. G. Luo, and C. L. Du, “Young’s interference of double metallic nanoslit with different widths,” Opt. Express 15(18), 11321–11327 (2007).
13. P. Lalanne, J. P. Hugonin, and J. C. Rodier, “Theory of surface plasmon generation at nanoslit apertures,” Phys. Rev. Lett. 95(26), 263902 (2005).
14. L. Verslegers, Z. Yu, Z. Ruan, P. B. Catrysse, and S. Fan, “From electromagnetically induced transparency to superscattering with a single structure: a coupled-mode theory for doubly resonant structures,” Phys. Rev. Lett. 108(8), 083902 (2012).
15. G. D’Aguanno, D. de Ceglia, N. Mattiucci, and M. J. Bloemer, “All-optical switching at the Fano resonances in subwavelength gratings with very narrow slits,” Opt. Lett. 36(11), 1984–1986 (2011).
16. K. L. Lee, S. H. Wu, and P. K. Wei, “Intensity sensitivity of gold nanostructures and its application for high-throughput biosensing,” Opt. Express 17(25), 23104–23113 (2009).
17. H. Im, K. C. Bantz, N. C. Lindquist, C. L. Haynes, and S.-H. Oh, “Vertically oriented sub-10-nm plasmonic nanogap arrays,” Nano Lett. 10(6), 2231–2236 (2010).
18. Y. M. Bahk, J. W. Choi, J. Kyoung, H. R. Park, K. J. Ahn, and D. S. Kim, “Selective enhanced resonances of two asymmetric terahertz nano resonators,” Opt. Express 20(23), 25644–25653 (2012).
19. X. S. Chen, H. R. Park, M. Pelton, X. J. Piao, N. C. Lindquist, H. Im, Y. J. Kim, J. S. Ahn, K. J. Ahn, N. Park, D. S. Kim, and S. H. Oh, “Atomic layer lithography of wafer-scale nanogap arrays for extreme confinement of electromagnetic waves,” Nat. Commun. 4, 2361 (2013).
20. F. Yang and J. R. Sambles, “Determination of the microwave permittivities of nematic liquid crystals using a single-metallic slit technique,” Appl. Phys. Lett. 81(11), 2047–2049 (2002).
21. J. R. Suckling, A. P. Hibbins, M. J. Lockyear, T. W. Preist, J. R. Sambles, and C. R. Lawrence, “Finite conductance governs the resonance transmission of thin metal slits at microwave frequencies,” Phys. Rev. Lett. 92(14), 147401 (2004).
22. Z. Ruan and S. Fan, “Superscattering of light from subwavelength nanostructures,” Phys. Rev. Lett. 105(1), 013901 (2010).
23. D. C. Skigin and R. A. Depine, “Transmission resonances of metallic compound gratings with subwavelength slits,” Phys. Rev. Lett. 95(21), 217402 (2005).
24. A. P. Hibbins, I. R. Hooper, M. J. Lockyear, and J. R. Sambles, “Microwave transmission of a compound metal grating,” Phys. Rev. Lett. 96(25), 257402 (2006).
25. D. Xiang, L.-L. Wang, X.-F. Li, L. Wang, X. Zhai, Z.-H. Liu, and W.-W. Zhao, “Transmission resonances of compound metallic gratings with two subwavelength slits in each period,” Opt. Express 19(3), 2187–2192 (2011).
26. R. Gordon, “Light in a subwavelength slit in a metal: propagation and reflection,” Phys. Rev. B 73(15), 153405 (2006).
27. R. Gordon, “Vectorial method for calculating the Fresnel reflection of surface plasmon polaritons,” Phys. Rev. B 74(15), 153417 (2006).
28. R. Gordon and A. G. Brolo, “Increased cut-off wavelength for a subwavelength hole in a real metal,” Opt. Express 13(6), 1933–1938 (2005).
29. E. Palik, Handbook of Optical Constants of Solids (Academic Press, 1997).
30. A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82(3), 2257–2298 (2010).
1. Introduction
Since the discovery of extraordinary optical transmission (EOT) through subwavelength hole arrays in metal films [1], the study of the interaction between light and subwavelength apertures has become an active area of optics [2–4]. Compared with two-dimensional (2D) hole arrays [5,6], the system of slits is simpler and therefore allows for an analytic understanding of the underlying physics [7–9]. On the other hand, such slits exhibit no cut-off for the transverse magnetic (TM) mode [10], which is attractive for resonantly confining and squeezing light to the nanoscale. The slit system plays a fundamental role in understanding the mechanisms of the interaction of light with subwavelength apertures, including transmission resonances [7], near-field interference [11,12], surface plasmon generation [13], superscattering [14] and Fano resonances [15]. It also has practical significance for applications, such as biosensing [16], surface enhanced Raman scattering (SERS) [17] and terahertz (THz) field enhancement [18,19].
In 1999, transmission resonances of narrow slits in metallic films were demonstrated [7]. In 2001, a Fabry-Pérot-like model was introduced, providing an analytic expression of resonance wavelength for single slit in perfect conductor (PEC) [8]. That model was experimentally investigated at the microwave frequencies, and the influence of finite conductivity of metal was reported [20,21]. In 2007, a general angle-dependent analytic formulation for a single slit was developed [9]. Recently, the concept of single channel limit [22], that states the maximum scattering cross section of an atom cannot exceed 3λ π2 2 in
3D or 2λ πin 2D, has been explored in nano-optics systems. A deep subwavelength double-slit was carefully designed to surpass the single channel limit by the interaction of different channels and dielectric loading of one of the slits [14]. Resonant transmission and field enhancement by one-nanometer-scale slits in gold and silver films has been experimentally demonstrated in the near-infrared and THz regimes [19]. Although subwavelength slit systems have been intensely investigated [2], few works have departed from investigating only single slit or periodically arranged slits [23–25], and the rich physics of multi-slit system requires further investigation.
In this paper, we introduce a theory for multi-slit transmission. The theory is formulated for a perfect electric conductor (PEC), but modified perturbatively to include the effects of finite permittivity (in the visible-IR regime) and finite conductivity (in the microwave regime). Super-transmission through a subwavelength arrangement of multiple slits in a metal film is demonstrated. Not only does this multiple slit system surpasses the single channel limit, it also shows transmission that is considerably larger than one through single slit with the slit width equivalent to the total width of the multi-slit system. The Fano resonance invoked by the interference of different channels is observed. Comprehensive numerical simulations are carried out to verify the theory introduced in this work, and excellent agreement is found.
2. Mode matching theory for a multiple slit system
z = l z = 0 z x 0 E H k0 a (a) a z = l z = 0 d z x 0 E H k0 w (b)
Fig. 1. Schematic of single and multiple subwavelength slit systems under TM incidence.
Figure 1 shows an example of subwavelength multi-slit system. It includes three subwavelength slits with width a , separated by d in an infinitely wide metallic film of the thickness l . The dimensions are normalized by the incident wavelength. The even eigenmodes of this multiple slit can be written as
1 1 1
rect rect rect ,
2 2 2 x d x x d E a a a μ = + + + − (1)
1rect 1 rect 1rect .
2 2 2 x d x x d E a a a ν = + − + − (2)
When a TM plane wave impinge on the multi-slit system from the above region, the reflected and transmitted magnetic fields can be written as the expansion of plane wave modes. The following equations are found by applying the continuity of the tangential components of the electric and magnetic fields at boundary of z=0 and z l= , respectively:
( ) (
)
( ) (
)
( ) (
)
( ) (
)
0 0 1 exp d , 1 exp d , exp d , exp d , z x x x x x x z x x x x x x k k r k ik x k E E E E r k ik x k E E E E k k t k ik x k E E E E t k ik x k E E E E μ μ ν ν μ μ ν ν μ μ ν ν μ μ ν ν μ μ ν ν μ μ ν ν μ μ ν ν μ μ ν ν ∞ + − + − + − + − −∞ ∞ + − + − + − + − −∞ ∞ + − + − + − + − −∞ ∞ + − + − + − + − −∞ + = + + + − = − + − = + + + = − + −
(3)where r k
( )
x and t k( )
x are the field amplitudes of a plane waves with x component of thewave vector kx, Eμ+ and E
Eμ− and E
ν− the backward ones, and μ+, μ−, ν+, ν− are unknown coupling coefficients.
After some typical manipulations for mode matching using orthogonality and integrating over
x [26,27], the unknowns X are obtained as
1 , − = X A b (4) with
(
)
( ) ( ) ( ) ( )(
)
(
)
(
)
2 2 2 1 1 2 2 1 1 2 , , 0 0 v v v v v j l j l j l j l j l j l I a I a I I a I I I a I a a e I a I a e I e I e I e I e I a I a μ μ μ μ μ μμ μμ μν μν μν μν νν νν β β β β β μμ μμ μν μν β β β β β μν μν νν νν + − + − + − + + − − = − + = − + A b (5) where βμ and βν are the propagation constants of modes, and(
)
(
)
(
)
(
)
(
)
(
)
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 sin 1 1 cos 2 d , 2 1 sin 1 1 cos 2 d , 2 1 sin 1 cos 2 1 d . 2 1 au I du u u u au I du u u u au I du u u u μμ μν νν π π π π π π π π π ∞ −∞ ∞ −∞ ∞ −∞ = + − = − − = − −
(6)The transmitted power is calculated by the Poynting theorem, and the transmission cross section is found as
(
2 2 2 2)
T a .
σ = λ μ+ − μ− +ν+ −ν− (7)
This approach is readily generalized for any number slits. For N slits, the P orthogonal eigenmodes can be found as Ep =
Nn=1κnrect(
x x+ n)
an . Accordingly, the couplingcoefficients p+ and p− can be solved as above from matrix inversion.
3. Quantitative comparison with numerical simulations
0.5 1 1.5 2 2.5 3 0.5 1.0 1.5 λ ( μm) σT (μ m ) 1 slit theory 1 slit simulation 3 slits theory 3 slits simulation
Fig. 2. Transmission cross sections of single and 3 slits in PEC with thickness of 1µm and slit width of 0.15 µm. For the 3 slits, the separation d is 0.3µm
Figure 2 shows the transmission through PEC subwavelength slits calculated by the mode matching theory in Section 2. The transmission cross section σT is defined as the ratio between the amount of transmitted power and the intensity of the incident wave. To verify the theoretical calculations, a comprehensive numerical method of finite difference time domain (FDTD) is used as simulations. Excellent agreement is found for both the single slit and three slit systems. The single slit shows Fabry-Pérot resonances, which have been demonstrated from past works with the same geometry [8].
The three slits system shows similar Fabry-Pérot resoannces, but also a Fano-like behavior due to the interaction between multiple transmission channels. Around the Fano resonance, the transmission substantially increases then abruptly drops to the minimum as will be discussed in more detail later.
4. Influence of finite permittivity/conductance in a real metal
In the case of real metal, the propagation of TM electromagnetic wave in a narrow slit of metallic film may be governed by the gap mode [28]:
2 2 m 2 2 2 2 m 4 tan h 4 0, 2 4 a β π ε β π ε β π − − + = − (8)
where εm is the relative permittivity of metal, andβ is the normalized propagation constant. In the low frequency regime (far-IR, THz and microwave), the permittivity is typically dominated by the imaginary part and an accurate approximation can be found for Eq. (8), where the influence of finite conductance is equal to an increase of the slit length by
(
2 m)
. l l π ε a Δ = (9) 0.1 0.15 0.2 0.25 0.3 2.2 2.4 2.6 2.8 3.0 slit width (μm) λ (μ m ) PEC Aluminium simulation (b) 0 200 400 600 800 1000 65 66 67 68 69 slit width (μm) frequency (GHz) PEC Aluminium experiment [21] (a)Fig. 3. Resonant frequency of slits in Aluminium films with different slit widths. (a) Single slit of length 19.58 mm, the imaginary part of relative permittivity is −1.071 × 107, and experiment
data are taken from Ref. 21. (b) Three uniform slits of length 1μm,the permittivity used in the calculations and FDTD simulations is from experimental data [29].
Figure 3 shows the influence of finite conductance in real metal. In microwave regime, obvious resonant frequency shift emerges only in very deep subwavelength slits while in near-infrared regime substantially frequency shift happens in deep subwavelength slit. Comparing with experimental data (Fig. 3(a)) or simulation (Fig. 3(b)), it is clear that this perturbation can be applied to our theoretical model to capture the influence of the real metal’s response with high accuracy. It is clear from Fig. 3(a) that good accuracy for single mode-matching, within 1% for the resonance frequency, is obtained for slit widths up to 20% of the wavelength.
Our approach in this perturbation formulation is to assume that the mode shape is well-represented by the PEC case (e.g., Eqs. (1) and (2)), and that only the propagation constant within the slit changes, but otherwise the theory is identical to the PEC case. This
approximation is clearly not valid if the slit skin-depth becomes comparable to the slit-width. As we can see from Fig. 3(b), a small departure from the comprehensive simulations (~1%) is seen only for the 100 nm slit width, and here the skin depth is about 10 nm. So, as long as the mode-shape does not vary by more than 10% of the PEC case, good accuracy is expected from this perturbative approach.
5. The single channel limit and beyond
Figure 4 firstly shows that the Fabry-Pérot transmission resonances of a single slit achieve the single channel limit, which has not been widely noted in past works. It should be noted that the single channel limit for 2D transmission is /λ π, since there is also a reflected component with equal contribution.
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.5 1.0 1.5 2.0 size parameter normalized σ T ( λ / π ) channel limit single slit 3 slits
Fig. 4. Comparison of transmissions with the single channel limit. The size of these two slit systems is 1 µm in length and 0.2 µm in width. Size parameter refers to the ratio of the total width w and wavelength, i.e. 2πw/λ. The transmission cross sections are normalized by
/ .
λ π
Figure 4 also shows super-transmission exceeding the single channel limit through a three slit system in a PEC. There are two transmission resonances in one period in the three slit system with symmetric or asymmetric profiles. The symmetric transmission is basically restricted by the single channel limit. However, the asymmetric one exhibits larger transmission than that. The origin of the asymmetric resonance can be explained with Fano resonance [30]. The three slits support two different scattering channels that couple through the diffraction at the ends of the slits. It is well known that π phase jump occurs around the resonance, and this exhibits in-phase and out-phase interference to produce enhanced and reduced transmission.
An interesting result can be found in Fig. 4 by noting that the total width of the three slit structure is equal to the width of single slit. The smaller throughout area of three slits transmits more light than that transmitted by bigger single slit.
In microwave and THz regimes, metals are well-approximated by a PEC and super-transmission, exceeding the single channel limit, should be readily observed. In the visible and near infrared regime, metal loss will broaden the sharp Fano resonances and reduce the strength. For example, Fig. 5 shows an example of super-transmission for three 40 nm slits separated by 40 nm in a 140 nm thick silver film, and the associated single slit transmission for Ag in the visible regime.
500 550 600 650 700 750 800 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 λ(nm) normalized transmission 3 slits single slit
Fig. 5. Super-transmission at visible regime for three 40 nm slits separated by 40 nm in a 140 nm thick silver film. The single slit is the same width as the full width of the three slits. 6. Discussion
The theory introduced in this work is an application of mode matching; however, it is not the typical matching of a discrete set of modes on either side of the boundary, but mode-matching to a continuum, which considers the infinite modes in the semi-infinite region. This work moves forward from past works, by considering two or more modes in the slit region. This adds significantly complexity since each mode not only self-couples, but also cross-couples to the other modes. Interesting physics arises from this cross-coupling since it produces nearly degenerate resonances that exhibit Fano interference.
To clarify the effect of the cross-coupling, we plot the transmission of multi-slit again compared with that neglecting the coupling terms. As shown in Fig. 6, the cross-coupling plays the key role in super-transmission for deeply subwavelength multi-slit system. The rich physics of this system is that it allows for a Fano resonance in a subwavelength structure. This occurs precisely because it is one of the few physical systems that allows for degenerate resonances in a (laterally) subwavelength confined region. The constructive zone of the Fano resonance allows for superscattering beyond the single channel limit. Interestingly, it also allows for total suppression of radiation. Figure 7 shows the FDTD simulation results of the distributions of electric fields at points A and B indicated in Fig. 5. The intensity of the components in the direction normal to the slits is plotted in log scale. At point A, the fields of satellite slits are clearly enhanced and the collective effect is like two radiating dipoles. However, at point B, the fields of satellite slits are weaker than the central one, and this reduces the beam width and suppresses radiation.
Fig. 6. Transmitted cross section of three slits system compared with that of neglecting cross coupling terms. The dimensions are the same as that in Fig. 2.
Fig. 7. The intensity of x component of electric fields 2 0
/
x
E E of points A and B in Fig. 6, in log scale.
The approach introduced here for three slits can be extended to multiple slits in a straightforward way. For example, Fig. 8 shows the application to five slits, where there are three even modes cross-coupling and the transmission curve exhibits two Fano resonances in one period since the modes are not entirely degenerate. Extending to a greater number of slits can be obtained with the same approach. It should be noted that others have investigated the infinite slit case, but for that case, it is a standard mode-matching response [8].
For two slits, only one even mode is excited for normal incidence, so it cannot present super-transmission. For four slits, super-transmission is again possible. Therefore, by symmetry arguments, three slits is the minimum required to obtain super-transmission for normal incidence. In Ref. 14, a dielectric-loaded double-slit structure shows super-transmission at angled incidence, where symmetry breaking is present. In that work, a coupled-mode theory was used that treats the modes as generic oscillators, which requires extraction of the oscillator parameters ab extra, whereas our theory is self-contained. Also, the dielectric loading of the slits introduces experimental complexity.
Fig. 8. Transmission of 5 slits compared with FDTD simulation.
The dependence of Fano resonance on the thickness of the metal can be written as the usual Fabry-Pérot cavity condition βl+ =ϕ mπ [19], whereβ is the propagation constant in the slit cavity, l is the length of the slit, mis an integer and ϕ is the shifted phase due to the refection at the end of the slit. Figure 9 shows the example of the dependence for the three slit in a PEC film, where the propagation reduced as k0 =2π λ. The additional phase ϕ can be
found as 0.992π at Fano resonances by ϕ=mπ−2π λl resfrom Fig. 8 and 0.926π for the
extremely subwavelength, but also has a rapid transverse oscillation which makes the Iνν
integral of Eq. (6) predominantly imaginary. This difference comes from the distinction of the dominant modes and cross-coupling; that is, both modes play a role in the Fano resonance, while the dominant mode of Fabry-Pérot resonance is only the lowest order “bright” mode.
Fig. 9. The dependence of resonance on the thickness of metal film for three slits and single slit. The other dimensions are the same as in Fig. 2.
7. Conclusions
We presented a theory of transmission through multiple subwavelength slits in metal film. Taking the influence of finite conductance or permittivity into account, this theory shows excellent agreement with experiment and simulation for real metals over a broad range of frequencies, as well as for a PEC. We investigated the single channel limit in the single slit system and super-transmission from multiple slits, where the single-channel limit is exceeded through a Fano resonance through multiple channels. We provided an analytical approach to study the cross-coupling of slits cavity. The results presented are interesting to enhanced spectroscopy applications where squeezing light below the diffraction limit is required.
Acknowledgments
Shuwen Chen acknowledges the funding support from the Chinese Scholarship Council. This work is supported by the NSERC (Canada) Discovery Grant.
