## Theory of enhanced optical transmission through a metallic nano-slit surrounded with asymmetric grooves under oblique incidence |

Optics Express, Vol. 18, Issue 19, pp. 19495-19503 (2010)

http://dx.doi.org/10.1364/OE.18.019495

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### Abstract

A metallic nano-slit surrounded with asymmetric grooves is proposed as the plasmonic concentrator for oblique incident light. A theoretical model based on the surface plasmon polariton (SPP) coupled-mode method is derived for the extraordinary optical transmission (EOT) through such a structure under oblique incidence. The model is quantitatively validated with the finite element method. With the model, the physical insight of the EOT is then interpreted, i.e., the major contributions to the transmission include the vertical Fabry-Perot resonance of the slit, and the interference among slit modes excited by the incident light, by SPPs generated from groove arrays and their first-order reflections. This is quite different from the EOT through a nano-slit surrounded with symmetric grooves under normal incidence.

© 2010 Optical Society of America

## 1. Introduction

## 2. Theoretical model

12. P. Lalanne, J. P. Hugonin, H. T. Liu, and B. Wang, “A microscopic view of the electromagnetic properties of sub-*λ* metallic surfaces,” Surf. Sci. Rep. **64**, 453–469 (2009). [CrossRef]

12. P. Lalanne, J. P. Hugonin, H. T. Liu, and B. Wang, “A microscopic view of the electromagnetic properties of sub-*λ* metallic surfaces,” Surf. Sci. Rep. **64**, 453–469 (2009). [CrossRef]

*θ*, the period of a groove array should be properly chosen to compensate the missing momentum between SPPs and the incident TM-polarized free-space light [13

13. C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature **445**, 39–46 (2007). [CrossRef] [PubMed]

*n*is the effective refractive index of the SPP mode at a air-metal interface, and

_{sp}*m*is an integer. If the periods of grooves on the left and the right sides are set as

*p*

_{1}=

*mλ*/(Re(

*n*)− sin

_{sp}*θ*) and

*p*

_{2}=

*mλ*/(Re(

*n*) + sin

_{sp}*θ*) with “Re” referring to the real part, respectively, the generated SPPs by grooves on both sides will propagate toward the central slit under oblique incidence, as shown in Fig. 2(a). As a result, grooves surrounding the nano-slit in the proposed structure are asymmetric. Such structures are widely used in the optical off-axis beaming [14–16

14. D.-Z. Lin, T.-D. Cheng, C.-K. Chang, J.-T. Yeh, J.-M. Liu, C.-S. Yeh, and C.-K. Lee, “Directional light beaming control by a subwavelength asymmetric surface structure,” Opt. Express **15**, 2585–2591 (2007). [CrossRef] [PubMed]

9. O. T. A. Janssen, H. P. Urbach, and G.W. ’t Hooft, “Giant optical transmission of a subwavelength slit optimized using the magnetic field phase,” Phys. Rev. Lett. **99**, 043902 (2007). [CrossRef] [PubMed]

11. L. Cai, G. Li, Z. Wang, and A. Xu, “Interference and horizontal Fabry-Perot resonance on extraordinary transmission through a metallic nano-slit surrounded with grooves,” Opt. Lett. **35**, 127–129 (2010). [CrossRef] [PubMed]

*A*

_{1}(

*B*

_{2}) is the complex amplitude of the superposition of the left-going (right-going) SPP modes starting from the edge of the slit,

*A*

_{2}(

*B*

_{1}) is that of the left-going (right-going) SPP modes starting from the edge of the right (left) grooves,

*D*(

*U*) is that of the fundamental slit mode propagating downward (upward). Here we use the

*H*component instead of

_{z}*E*or

_{x}*E*to model the EOT, because the incident, the scattered, and the transmitted fields have strong

_{y}*H*components [9

_{z}9. O. T. A. Janssen, H. P. Urbach, and G.W. ’t Hooft, “Giant optical transmission of a subwavelength slit optimized using the magnetic field phase,” Phys. Rev. Lett. **99**, 043902 (2007). [CrossRef] [PubMed]

*r*

_{s1}and

*r*

_{s2}are the respective reflectance coefficients of the slit fundamental modes at the top and the bottom openings.

*ρ*and

*τ*are the reflectance coefficient and the transmittance coefficient of the SPP mode at the slit, respectively. The SPP mode at the air-metal interface and the slit fundamental mode can be converted to each other with a conversion coefficient

*α*, as shown in Figs. 2(b) and 2(d). The SPP generation coefficients and reflectance coefficients of the left and the right groove arrays are

*β*

_{1}(

*θ*),

*r*

_{g1}and

*β*

_{2}(

*θ*),

*r*

_{g2}, respectively.

*t*

_{0}(

*θ*),

*β*(−

_{s}*θ*), and

*β*(

_{s}*θ*) are the respective generation coefficients of the slit fundamental mode, the left-going and the right-going SPP modes by a single silt under the illumination of a plane wave with amplitude

*I*

_{0}and angle

*θ*. Since the slits considered are very small (usually

*w*<

_{sl}*λ*/2), we only take into account the slit fundamental mode with the effective complex refractive index

*n*. Finally, the coupled-mode equations lead to

_{sl}*w*

_{1}= exp(

*ik*

_{0}

*n*

_{sp}d_{1}),

*w*

_{2}= exp(

*ik*

_{0}

*n*

_{sp}d_{2}), and

*v*= exp(

*ik*

_{0}

*n*). Instead of the slit-groove center-to-center distances, we use the slit-groove edge-to-edge distances

_{sl}t*d*

_{1}and

*d*

_{2}, which are more convenient for the analysis as they are independent from the widths of the slit and grooves. Note that we set the phase of the incident magnetic field at the center of the slit’s upper opening (

*x*=

*y*= 0) to zero. As a result,

*β*

_{1}(

*θ*) includes a phase difference of the obliquely incident field exp[−

*ik*

_{0}sin

*θ*(

*d*

_{1}+

*w*/2)],

_{sl}*β*

_{2}(−

*θ*) includes exp[

*ik*

_{0}sin

*θ*(

*d*

_{2}+

*w*/2)], and

_{sl}*β*(±θ) includes exp(∓

_{s}*ik*

_{0}sin

*θw*/2). One should notice that

_{sl}*β*(±

_{s}*θ*) can be omitted in practice due to ∣

*β*(±

_{s}*θ*)∣ ≫ ∣

*β*

_{1}(

*θ*)∣ and ∣

*β*(±

_{s}*θ*)∣ ≫ ∣

*β*

_{2}(−

*θ*)∣ for any given

*θ*when the groove number

*N*is large enough (for example,

*N*> 5).

*B*

_{1},

*A*

_{1},

*B*

_{2},

*A*

_{2},

*D*,

*U*and the coupling among them indicate that multiple reflections and conversions are automatically incorporated in Eq. (2).

*r*

_{g1},

*r*

_{g2}and

*β*

_{1},

*β*

_{2}. Of course, the grooves can also be further modeled using the coupled-mode equations in terms of the contribution of every groove. However, in this case, the resulted equation system is so complex that there will be no closed-form solutions. A vivid example is on the SPP’s reflectance coefficient

*r*[18

_{g}18. G. Li, L. Cai, F. Xiao, Y. Pei, and A. Xu, “A quantitative theory and the generalized Bragg condition for surface plasmon Bragg reflectors,” Opt. Express **18**, 10487–10499 (2010). [CrossRef] [PubMed]

*α*

^{2}

*r*

_{g1}

*r*

_{s2}

*w*

^{2}

_{1}

*ν*and

*α*

^{2}

*r*

_{g2}

*r*

_{s2}

*w*

^{2}

_{2}

*ν*, then obtain

*δ*

_{1}=

*r*

_{g1}

*w*

^{2}

_{1}and

*δ*

_{2}=

*r*

_{g2}

*w*

^{2}

_{2}. Specially, for a slit surrounded with symmetric groove arrays under normal incidence, we have

*r*

_{g1}=

*r*

_{g2}=

*r*,

_{g}*w*

_{1}=

*w*

_{2}=

*w*,

*β*

_{1}=

*β*

_{2}=

*β*, and

*δ*

_{1}=

*δ*

_{2}=

*r*

_{g}w^{2}. Equation (3) is then reduced into

11. L. Cai, G. Li, Z. Wang, and A. Xu, “Interference and horizontal Fabry-Perot resonance on extraordinary transmission through a metallic nano-slit surrounded with grooves,” Opt. Lett. **35**, 127–129 (2010). [CrossRef] [PubMed]

*D*∣

^{2}instead of the transmission efficiency

*η*, which is defined as the total energy transmitted into the far field normalized to the energy incident on the slit opening [9

9. O. T. A. Janssen, H. P. Urbach, and G.W. ’t Hooft, “Giant optical transmission of a subwavelength slit optimized using the magnetic field phase,” Phys. Rev. Lett. **99**, 043902 (2007). [CrossRef] [PubMed]

*η*is proportional to ∣

*D*∣

^{2}for a nano-slit. The proportional coefficient denotes the conversion efficiency from the slit mode to the propagating light through the slit on the output side of the slit. Given the slit’s parameters, the coefficient is fixed, thus ∣

*D*∣

^{2}and

*η*result in identical conclusions, as will be validated in the next section.

## 3. Results and discussions

### 3.1. Model validations

*D*∣

^{2}predicted by the model with the transmission efficiency

*η*calculated by the finite element method (FEM) [19

19. G. Li and A. Xu, “Phase shift of plasmons excited by slits in a metal film illuminated by oblique incident TM plane wave,” Proc SPIE **7135**, 71350T (2008). [CrossRef]

*α*and

*t*

_{0}(

*θ*), are obtained analytically [20

20. P. Lalanne, J. P. Hugonin, and J. C. Rodier, “Approximate model for surface-plasmon generation at slit apertures,” J. Opt. Soc. Am. A **23**, 1608–1615 (2006). [CrossRef]

*r*

_{s1},

*r*

_{s2},

*ρ*,

*r*

_{g1},

*r*

_{g2}and the transmittance coefficient

*τ*are calculated by the a-FMM [17

17. E. Silberstein, P. Lalanne, J. P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A **18**, 2865–2875 (2001). [CrossRef]

*β*

_{1}(

*θ*) and

*β*

_{2}(

*θ*) can be calculated by either the method combining the a-FMM with the reciprocity theorem [21

21. H. Liu, P. Lalanne, X. Yang, and J. P. Hugonin, “Surface plasmon generation by subwavelength isolated objects,” IEEE J. Sel. Top. Quantum Electron. **14**, 1522–1529 (2008). [CrossRef]

*λ*= 800 nm, the substrate of silica with permittivity

*ε*= 1.46, and gold with relative permittivity

_{d}*ε*= −26.2 + 1.85

_{m}*i*[22] are used to illustrate our discussions throughout the paper.

*D*∣

^{2}as a function of slit-groove distances

*d*

_{1}and

*d*

_{2}. Obviously, the influences of

*d*

_{1}and

*d*

_{2}on the transmission under oblique incidence are not identical because of the asymmetric surrounding grooves, which have different reflectance coefficients for the SPP modes. This phenomenon will be further explained in the next section. From Fig. 3(a), the peak transmission position (point “Q”) and the dip one (point “P”) of

*d*

_{1}and

*d*

_{2}are determined. The corresponding strong transmission enhancement and suppression are shown in Figs. 3(d) and 3(e), respectively, validating the accuracy of the model predictions.

*D*∣

^{2}) and the FEM simulations (

*η*) on the proposed plasmonic concentrator for the incidence angle

*θ*= 20°. In this case, the periods for the left grooves and the right ones should be 1180 nm and 588 nm according to Eq. (1), respectively. As one can see, the general trends of the transmission efficiencies, especially the positions as well as the periods of transmission peaks, are well captured by the proposed model. Predictions on peak values may be inaccurate sometimes, as illustrated in Fig. 3(c). This is because our model is a “pure” SPP model which only takes into account the SPPs’ contributions, while the CWs’ are neglected. Due to the characteristic damping of CWs in the visible or near infrared regime [12

12. P. Lalanne, J. P. Hugonin, H. T. Liu, and B. Wang, “A microscopic view of the electromagnetic properties of sub-*λ* metallic surfaces,” Surf. Sci. Rep. **64**, 453–469 (2009). [CrossRef]

### 3.2. Physical interpretations

*ρ*, which is usually very small (∣

*ρ*∣ < 0.1) compared with

*τ*(∣

*τ*∣ > 0.85) for small slits (

*w*<

_{sl}*λ*/2) [23

23. O. T. A. Janssen, H. P. Urbach, and G. W.’t Hooft, “On the phase of plasmons excited by slits in a metal film,” Opt. Express **14**, 11823–11832 (2006). [CrossRef] [PubMed]

*I*

_{0}/(1 −

*r*

_{s1}

*r*

_{s2}

*ν*

^{2}), represents the vertical F-P resonance effect of the cavity formed by the top and the bottom openings of the slit, as mentioned in Ref. 24

24. A. Krishnan, T. Thio, T. J. Kima, H. J. Lezec, T. W. Ebbesen, P. A. Wolff, J. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Evanescently coupled resonance in surface plasmon enhanced transmission,” Opt. Commun. **200**, 1–7 (2001). [CrossRef]

*q*is an integer. Hereafter, we set

*w*= 100 nm to illustrate our discussions. Calculated by the a-FMM, the reflectance coefficients are

_{sl}*r*

_{s1}= −0.2389 − 0.4411

*i*and

*r*

_{s2}= 0.1081 − 0.4125

*i*, and the effective refractive index of the silt fundamental mode is

*n*= 1.2346 + 0.0082

_{sl}*i*. According to Eq. (6), the resonant film thickness should be

*t*= 174 nm (

*q*= 0).

*αw*

_{1}

*β*

_{1}(1 +

*τδ*

_{2}), and rewrite it as

*αw*

_{1}

*β*

_{1}+

*αw*

_{1}

*β*

_{1}

*τr*

_{g2}

*w*

^{2}

_{2}. This term means the interference between slit fundamental modes, which are generated by the part of SPPs propagating from the left grooves directly coupled into the slit, and the part crossing over the slit, reflected back by the right groove array, and then coupled into the slit from the right side, respectively, as illustrated in Fig. 4. The term

*αw*

_{2}

*β*

_{2}(1 +

*τδ*

_{1}) can be understood similarly. The corresponding constructive interference conditions for

*αw*

_{1}

*β*

_{1}(1 +

*τr*

_{g2}

*w*

^{2}

_{2}) and

*αw*

_{2}

*β*

_{2}(1+

*τr*

_{g1}

*w*

^{2}

_{1}) are given by

*m*

_{1}and

*m*

_{2}are integers. Obviously, the periods of these terms as functions of slit-groove distances are

*λ*/2, where

_{sp}*λ*=

_{sp}*λ*/Re(

*n*) is the wavelength of the SPP mode.

_{sp}*τ*

^{2}

*δ*

_{1}

*δ*

_{2}), i.e., 1/(1 −

*τ*

^{2}

*r*

_{g1}

*r*

_{g2}

*w*

^{2}

_{1}

*w*

^{2}

_{2}), results from the multiple reflection of SPPs in the cavity formed by the left and the right grooves. This effect was referred to as the horizontal F-P resonance in our previous work on normal incidence [11

11. L. Cai, G. Li, Z. Wang, and A. Xu, “Interference and horizontal Fabry-Perot resonance on extraordinary transmission through a metallic nano-slit surrounded with grooves,” Opt. Lett. **35**, 127–129 (2010). [CrossRef] [PubMed]

**35**, 127–129 (2010). [CrossRef] [PubMed]

*r*

_{g1}

*r*

_{g2}is usually very small for grooves which are designed for high SPP excitation efficiency [25

25. F. López-Tejeira, F. J. García-Vidal, and L. Martín-Moreno, “Scattering of surface plasmons by one-dimensional periodic nanoindented surfaces,” Phys. Rev. B **72**, 161405 (2005). [CrossRef]

*r*

_{g1}

*r*

_{g2}∣ < 0.1. As a result, we further assume 1 −

*r*

_{g1}

*r*

_{g2}

*τ*

^{2}

*w*

^{2}

_{1}

*w*

^{2}

_{2}≈ 1, and reduce Eq. (5) into

**35**, 127–129 (2010). [CrossRef] [PubMed]

*d*

_{1}and

*d*

_{2}in Figs. 3(a) to 3(c), where the peak separations are quite different. For grooves used in Fig. 3,

*r*

_{g1}= −0.3036 + 0.0377

*i*, and

*r*

_{g2}= −0.0362 − 0.1077

*i*. Then

*r*

_{g2}

*τw*

^{2}

_{2}can be further neglected because of small

*r*

_{g2}, indicating that the first-order reflection of SPPs propagating from grooves on the left side makes little contribution. As a result, we rewrite Eq. (8) as

*d*

_{1}(or

*w*

_{1}), the separation between peaks of ∣

*D*∣

^{2}as a function of

*d*

_{2}is approximate to

*λ*; however, given

_{sp}*d*

_{2}(or

*w*

_{2}), this peak separation, as a function of

*d*

_{1}, is approximate to

*λ*/2 because of the term 1 +

_{sp}*τδ*

_{1}.

*d*(

_{i}*i*= 1,2) paly key roles in the interference between SPPs generated from groove arrays on the left and the right sides, and their first-order reflections. As a result, one can use different

*d*to compensate the different phases of the SPPs launched from the left and the right to maximize the transmission.

_{i}## 4. Concluding remarks

*θ*, as illustrated in Fig. 5. Its transmission efficiency reaches the maximum at the designed angle, i.e.

*θ*= 0°, but decreases rapidly when

*θ*deviates from 0°, resulting a narrow FOV. The angular performance is similar for the proposed structure with asymmetric surrounding grooves, which is designed for the oblique incident light. It is clear that the angular pattern of the transmission efficiency for different structures are almost the same, except that transmission peaks are shifted to the designed angles.

## Acknowledgements

## References and links

1. | T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff. “Extraordinary optical transmission through subwavelength hole arrays,” Nature |

2. | Z. Yu, G. Veronis, S. Fan, and M. L. Brongersma, “Design of midinfrared photodetectors enhanced by surface plasmons on grating structures,” Appl. Phys. Lett. |

3. | R. D. Bhat, N. C. Panoiu, S. R. Brueck, and R. M. Osgood, “Enhancing the signal-to-noise ratio of an infrared photodetector with a circular metal grating,” Opt. Express |

4. | L. A. Dunbar, M. Guillaumée, F. de León-Pérez, C. Santschi, E. Grenet, R. Eckert, F. López-Tejeira, F. J. García-Vidal, L. Martín-Moreno, and R. P. Stanley, “Enhanced transmission from a single subwavelength slit aperture surrounded by grooves on a standard detector,” Appl. Phys. Lett. |

5. | T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, and T. W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. |

6. | F. J. Garcia-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martin-Moreno, “Multiple paths to enhance optical transmission through a single subwavelength slit,” Phys. Rev. Lett. |

7. | A. Degiron and T.W. Ebbesen, “Analysis of the transmission process through single apertures surrounded by periodic corrugations,” Opt. Express |

8. | B. Ung and Y. Sheng, “Interference of surface waves in a metallic nanoslit,” Opt. Express |

9. | O. T. A. Janssen, H. P. Urbach, and G.W. ’t Hooft, “Giant optical transmission of a subwavelength slit optimized using the magnetic field phase,” Phys. Rev. Lett. |

10. | Y. X. Cui and S. L. He, “A theoretical re-examination of giant transmission of light through a metallic nano-slit surrounded with periodic grooves,” Opt. Express |

11. | L. Cai, G. Li, Z. Wang, and A. Xu, “Interference and horizontal Fabry-Perot resonance on extraordinary transmission through a metallic nano-slit surrounded with grooves,” Opt. Lett. |

12. | P. Lalanne, J. P. Hugonin, H. T. Liu, and B. Wang, “A microscopic view of the electromagnetic properties of sub- |

13. | C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature |

14. | D.-Z. Lin, T.-D. Cheng, C.-K. Chang, J.-T. Yeh, J.-M. Liu, C.-S. Yeh, and C.-K. Lee, “Directional light beaming control by a subwavelength asymmetric surface structure,” Opt. Express |

15. | S. Kim, H. Kim, Y. Lim, and B. Lee, “Off-axis directional beaming of optical field diffracted by a single subwavelength metal slit with asymmetric dielectric surface gratings,” Appl. Phys. Lett. |

16. | Y. Lim, J. Hahn, S. Kim, J. Park, H. Kim, and B. Lee, “Plasmonic light beaming manipulation and its detection using holographic microscopy,” IEEE J. Quantum Electron. |

17. | E. Silberstein, P. Lalanne, J. P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A |

18. | G. Li, L. Cai, F. Xiao, Y. Pei, and A. Xu, “A quantitative theory and the generalized Bragg condition for surface plasmon Bragg reflectors,” Opt. Express |

19. | G. Li and A. Xu, “Phase shift of plasmons excited by slits in a metal film illuminated by oblique incident TM plane wave,” Proc SPIE |

20. | P. Lalanne, J. P. Hugonin, and J. C. Rodier, “Approximate model for surface-plasmon generation at slit apertures,” J. Opt. Soc. Am. A |

21. | H. Liu, P. Lalanne, X. Yang, and J. P. Hugonin, “Surface plasmon generation by subwavelength isolated objects,” IEEE J. Sel. Top. Quantum Electron. |

22. | E. D. Palik, |

23. | O. T. A. Janssen, H. P. Urbach, and G. W.’t Hooft, “On the phase of plasmons excited by slits in a metal film,” Opt. Express |

24. | A. Krishnan, T. Thio, T. J. Kima, H. J. Lezec, T. W. Ebbesen, P. A. Wolff, J. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Evanescently coupled resonance in surface plasmon enhanced transmission,” Opt. Commun. |

25. | F. López-Tejeira, F. J. García-Vidal, and L. Martín-Moreno, “Scattering of surface plasmons by one-dimensional periodic nanoindented surfaces,” Phys. Rev. B |

**OCIS Codes**

(050.1220) Diffraction and gratings : Apertures

(250.5403) Optoelectronics : Plasmonics

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: June 29, 2010

Revised Manuscript: August 19, 2010

Manuscript Accepted: August 19, 2010

Published: August 30, 2010

**Citation**

Lin Cai, Guangyuan Li, Feng Xiao, Zhonghua Wang, and Anshi Xu, "Theory of enhanced optical transmission through a metallic nano-slit surrounded
with asymmetric grooves under oblique incidence," Opt. Express **18**, 19495-19503 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-19-19495

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### References

- 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). [CrossRef]
- Z. Yu, G. Veronis, S. Fan, and M. L. Brongersma, “Design of midinfrared photodetectors enhanced by surface plasmons on grating structures,” Appl. Phys. Lett. 89,151116 (2006). [CrossRef]
- R. D. Bhat, N. C. Panoiu, S. R. Brueck, and R. M. Osgood, “Enhancing the signal-to-noise ratio of an infrared photodetector with a circular metal grating,” Opt. Express 16,4588–4596 (2008). [CrossRef] [PubMed]
- L. A. Dunbar, M. Guillaumée, F. de León-Pérez, C. Santschi, E. Grenet, R. Eckert, F. López-Tejeira, F. J. García-Vidal, L. Martín-Moreno, and R. P. Stanley, “Enhanced transmission from a single subwavelength slit aperture surrounded by grooves on a standard detector,” Appl. Phys. Lett. 95,011113 (2009). [CrossRef]
- T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, and T. W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. 26,1972–1974 (2001). [CrossRef]
- F. J. Garcia-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martin-Moreno, “Multiple paths to enhance optical transmission through a single subwavelength slit,” Phys. Rev. Lett. 90,213901 (2003). [CrossRef] [PubMed]
- A. Degiron and T.W. Ebbesen, “Analysis of the transmission process through single apertures surrounded by periodic corrugations,” Opt. Express 12,3694–3700 (2004). [CrossRef] [PubMed]
- B. Ung and Y. Sheng, “Interference of surface waves in a metallic nanoslit,” Opt. Express 15,1182–1190 (2007). [CrossRef] [PubMed]
- O. T. A. Janssen, H. P. Urbach, and G.W. ’t Hooft, “Giant optical transmission of a subwavelength slit optimized using the magnetic field phase,” Phys. Rev. Lett. 99,043902 (2007). [CrossRef] [PubMed]
- Y. X. Cui and S. L. He, “A theoretical re-examination of giant transmission of light through a metallic nano-slit surrounded with periodic grooves,” Opt. Express 17,13995–14000 (2009). [CrossRef] [PubMed]
- L. Cai, G. Li, Z. Wang, and A. Xu, “Interference and horizontal Fabry-Perot resonance on extraordinary transmission through a metallic nano-slit surrounded with grooves,” Opt. Lett. 35,127–129 (2010). [CrossRef] [PubMed]
- P. Lalanne, J. P. Hugonin, H. T. Liu, and B. Wang, “A microscopic view of the electromagnetic properties of sub-⌊ metallic surfaces,” Surf. Sci. Rep. 64,453–469 (2009). [CrossRef]
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