## Coherent-form energy conservation relation for the elastic scattering of a guided mode in a symmetric scattering system |

Optics Express, Vol. 21, Issue 20, pp. 24093-24098 (2013)

http://dx.doi.org/10.1364/OE.21.024093

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### Abstract

We propose a coherent-form energy conservation relation (ECR) that is generally valid for the elastic transmission and reflection of a guided mode in a symmetric scattering system. In contrast with the classical incoherent-form ECR, |*τ*|^{2} + |*ρ*|^{2}≤1 with *τ* and *ρ* denoting the elastic transmission and reflection coefficients of a guided mode, the coherent-form ECR is expressed as |*τ* + *ρ*|≤1, which imposes a constraint on a coherent superposition of the transmitted and reflected modes. The coherent-form ECR is rigorously demonstrated and is numerically tested by considering different types of modes in various scattering systems. Further discussions with the scattering matrix formalism indicate that two coherent-form ECRs, |*τ* + *ρ*|≤1 and |*τ*−*ρ*|≤1, along with the classical ECR |*τ*|^{2} + |*ρ*|^{2}≤1 constitute a complete description of the energy conservation for the elastic scattering of a guided mode in a symmetric scattering system. The coherent-form ECR provides a common tool in terms of energy transfer for understanding and analyzing the scattering dynamics in currently interested scattering systems.

© 2013 OSA

## 1. Introduction

*τ*and a reflection coefficient

*ρ*. Classical ECR [2] has imposed a constraint of |

*τ*|

^{2}+ |

*ρ*|

^{2}≤1 on an incoherent superposition of

*τ*and

*ρ*. In this work, we propose that besides the classical ECR,

*τ*and

*ρ*also satisfy an ECR expressed as |

*τ*+

*ρ*|≤1, in which

*τ*and

*ρ*are superposed coherently. Rigorous demonstration of the coherent-form ECR is provided and its general validity is tested for different types of modes in various scattering systems. Besides imposing impact on classical theories of optical waveguides [2], the coherent-form ECR provides a common tool for analyzing the energy transfer via guided modes and the resultant properties of currently interested scattering systems, for instance, for deriving resonance conditions driven by the multiple scattering of some guided modes [13].

## 2. Numerical test of the coherent-form energy conservation relation

*τ*+

*ρ*|≤1 generally holds for various symmetric scattering systems and for different types of modes. Figures 1(a) and 1(b) present for different wavelengths the values of |

*τ*+

*ρ*| for the scattering of the SPP at two line scatterers, a periodic chain of infinite-depth holes [Fig. 1(a)] and an infinite-depth slit [Fig. 1(b)] in gold substrate, which are the elementary scatterers constituting the periodic metallic apertures supporting the extraordinary optical transmission [3

3. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature **391**(6668), 667–669 (1998). [CrossRef]

7. H. T. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature **452**(7188), 728–731 (2008). [CrossRef] [PubMed]

14. 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). [CrossRef]

*τ*and a reflected SPP with a coefficient

*ρ*(see the insets). The incident, transmitted and reflected SPP modes are assumed obeying the same normalization of fields. The elastic SPP scattering coefficients

*τ*and

*ρ*can be obtained as the scattering matrix elements with a fully-vectorial aperiodic Fourier modal method (a-FMM) [15

15. J. P. Hugonin and P. Lalanne, “Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A **22**(9), 1844–1849 (2005). [CrossRef] [PubMed]

7. H. T. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature **452**(7188), 728–731 (2008). [CrossRef] [PubMed]

*τ*+

*ρ*|≈1) as the wavelength is larger than the chain period (940nm), since now the transmitted and reflected higher-order surface waves (propagating nonperpendicularly to the chain that acts as a grating) are evanescent and do not carry energy. Also note that all the modes in the subwavelength holes are evanescent and do not carry energy [3

3. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature **391**(6668), 667–669 (1998). [CrossRef]

7. H. T. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature **452**(7188), 728–731 (2008). [CrossRef] [PubMed]

*τ*+

*ρ*| is distinctly smaller than one due to the appearance of higher-order propagative surface waves.

14. 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). [CrossRef]

*τ*+

*ρ*| could be distinctly smaller than one due to the lost energy carried away by the propagative SPP mode in the slit, as confirmed by the results in Fig. 1(b).

*τ*+

*ρ*| for the fundamental SPP mode on a gold nano-wire in air scattered by an air nano-gap. This represents an elementary scattering process in resonant nano-antennas for achieving a strong enhancement of field in the nano-gap [5

5. P. Mühlschlegel, H. J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science **308**(5728), 1607–1609 (2005). [CrossRef] [PubMed]

*τ*+

*ρ*| for the gap width of 100nm [Fig. 1(d)] is distinctly smaller than that for the gap width of 30nm [Fig. 1(c)], which is due to the higher energy loss scattered into free space at wider air gaps.

17. Z. H. Han and S. I. Bozhevolnyi, “Plasmon-induced transparency with detuned ultracompact Fabry-Perot resonators in integrated plasmonic devices,” Opt. Express **19**(4), 3251–3257 (2011). [CrossRef] [PubMed]

*τ*+

*ρ*| for the scattering of the SPP mode at an infinite-depth slit, a finite-depth groove and an orthogonal corner, respectively. Note that for Fig. 1(g), for which the transmission waveguide is not collinear with the incidence waveguide, the coefficient

*τ*of the transmitted SPP is extracted from the total field with the use of the mode orthogonality (see Eq. (1.36) in chapter 1.2.6 of [2]). Similar method has been used to extract the coefficient of the SPP mode at a single dielectric-metal interface [18

18. P. Lalanne, J. P. Hugonin, and J. C. Rodier, “Theory of surface plasmon generation at nanoslit apertures,” Phys. Rev. Lett. **95**(26), 263902 (2005). [CrossRef] [PubMed]

*τ*is expressed as an overlap integral between the SPP mode and the total field (with the same expression as the

*α*

^{+}in Eq. (4) of [18

18. P. Lalanne, J. P. Hugonin, and J. C. Rodier, “Theory of surface plasmon generation at nanoslit apertures,” Phys. Rev. Lett. **95**(26), 263902 (2005). [CrossRef] [PubMed]

*E*

_{SP}and

*H*

_{SP}should be replaced by the fundamental SPP mode). For the SPP scattering at the slit [Fig. 1(e)], |

*τ*+

*ρ*| is shown distinctly smaller than one for all wavelengths, which is due to the lost energy carried away by the propagative SPP mode in the slit. In comparison, |

*τ*+

*ρ*| for the SPP scattering at the groove [Fig. 1(f)] is distinctly smaller than one only at some specific wavelengths, and at other wavelengths, |

*τ*+

*ρ*| is quite close to unity. Further calculations show that the low values of |

*τ*+

*ρ*| are due to the Fabry-Perot resonance of the propagative SPP mode in the groove, which occurs at some specific wavelengths and which causes a considerable absorbance of energy by the lossy metal. For the results in Fig. 1(g), as expected, |

*τ*+

*ρ*| is quite close to one for all wavelengths since almost no energy is carried away or absorbed by channels other than the transmitted and the reflected SPP modes.

6. L. M. Tong, F. Zi, X. Guo, and J. Y. Lou, “Optical microfibers and nanofibers:A tutorial,” Opt. Commun. **285**(23), 4641–4647 (2012). [CrossRef]

*y*polarized) in [9

9. K. J. Huang, S. Y. Yang, and L. M. Tong, “Modeling of evanescent coupling between two parallel optical nanowires,” Appl. Opt. **46**(9), 1429–1434 (2007). [CrossRef] [PubMed]

*τ*+

*ρ*| is shown quite close to one when a strong coupling occurs (|

*τ*|≈1) for a coupling length

*L*~2.5μm, which results in a weak scattering loss of energy into free space, while |

*τ*+

*ρ*| is distinctly smaller than one when the coupling is weak.

## 3. Theoretical demonstration of the coherent-form energy conservation relation and further discussions with the scattering matrix formalism

*t*and

*r*, respectively. Classical grating theories provide |

*t*|

^{2}+ |

*r*|

^{2}≤1. The coherent-form ECR predicts that

*t*and

*r*should also satisfy |

*t*+

*r*|≤1. This prediction is confirmed by the numerical results shown in Fig. 3, which are obtained with a fully-vectorial rigorous coupled wave analysis (RCWA) [19

19. L. F. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A **14**(10), 2758–2767 (1997). [CrossRef]

*t*+

*r*|≤1 is satisfied over the whole considered range of wavelengths. There appears a sharp dip of |

*t*+

*r*| at a wavelength slightly larger than the array period (940nm), near which the extraordinary optical transmission occurs [3

3. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature **391**(6668), 667–669 (1998). [CrossRef]

**452**(7188), 728–731 (2008). [CrossRef] [PubMed]

**452**(7188), 728–731 (2008). [CrossRef] [PubMed]

*i*(at

*λ*= 1μm) to be 6.84

*i*. The result is shown in Fig. 3(b). Now the dip of |

*t*+

*r*| disappears, and as the wavelength is larger than the array period of 940nm, |

*t*+

*r*| is exactly equal to 1 since now only the zeroth-order plane wave is propagative and all other higher-order plane waves are evanescent and do not carry energy. As the wavelength is smaller than the array period, |

*t*+

*r*| becomes smaller than 1 due to the appearance of higher-order propagative plane waves.

## 4. Conclusion

*τ*+

*ρ*|≤1 for the elastic transmission (with a coefficient

*τ*) and reflection (coefficient

*ρ*) of a guided mode in a symmetric scattering system. Rigorous demonstration of the ECR is provided showing that the ECR is generally valid for approximately lossless waveguides. Its general validity is numerically confirmed by considering different types of modes such as those in plasmonic waveguides, dielectric waveguides or free space and by considering different scattering systems. The proposed ECR is shown to reach the equality as the considered transmitted and reflected mode is the sole channel that carries away the energy. Further discussions with the scattering matrix formalism show that for the elastic scattering of a guided mode in a symmetric scattering system, what we can obtain at most from the energy conservation is no more than two coherent-form ECRs, |

*τ*+

*ρ*|≤1 and |

*τ*−

*ρ*|≤1, along with the classical incoherent-form ECR |

*τ*|

^{2}+ |

*ρ*|

^{2}≤1. The present results may be extended to the more general or complex cases, such as the asymmetric scattering systems, the inelastic scattering between modes of different types, or the scattering of Bloch modes in periodic waveguides [20

20. G. Lecamp, J. P. Hugonin, and P. Lalanne, “Theoretical and computational concepts for periodic optical waveguides,” Opt. Express **15**(18), 11042–11060 (2007). [CrossRef] [PubMed]

## Acknowledgments

## References and notes

1. | M. Born and E. Wolf, |

2. | C. Vassallo, |

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

4. | J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature |

5. | P. Mühlschlegel, H. J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science |

6. | L. M. Tong, F. Zi, X. Guo, and J. Y. Lou, “Optical microfibers and nanofibers:A tutorial,” Opt. Commun. |

7. | H. T. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature |

8. | J. Yang, C. Sauvan, H. T. Liu, and P. Lalanne, “Theory of fishnet negative-index optical metamaterials,” Phys. Rev. Lett. |

9. | K. J. Huang, S. Y. Yang, and L. M. Tong, “Modeling of evanescent coupling between two parallel optical nanowires,” Appl. Opt. |

10. | X. P. Huang and M. L. Brongersma, “Rapid computation of light scattering from aperiodic plasmonic structures,” Phys. Rev. B |

11. | G. Y. Li, F. Xiao, L. Cai, K. Alameh, and A. S. Xu, “Theory of the scattering of light and surface plasmon polaritons by finite-size subwavelength metallic defects via field decomposition,” New J. Phys. |

12. | Here the guided mode is defined as a propagative waveguide mode, which obeys an exponential propagation rule exp( |

13. | For instance, the derivation of the resonance condition in [7] and [8] requires | |

14. | J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. |

15. | J. P. Hugonin and P. Lalanne, “Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A |

16. | E. D. Palik, |

17. | Z. H. Han and S. I. Bozhevolnyi, “Plasmon-induced transparency with detuned ultracompact Fabry-Perot resonators in integrated plasmonic devices,” Opt. Express |

18. | P. Lalanne, J. P. Hugonin, and J. C. Rodier, “Theory of surface plasmon generation at nanoslit apertures,” Phys. Rev. Lett. |

19. | L. F. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A |

20. | G. Lecamp, J. P. Hugonin, and P. Lalanne, “Theoretical and computational concepts for periodic optical waveguides,” Opt. Express |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(260.2160) Physical optics : Energy transfer

(290.5825) Scattering : Scattering theory

**ToC Category:**

Scattering

**History**

Original Manuscript: August 5, 2013

Revised Manuscript: September 13, 2013

Manuscript Accepted: September 16, 2013

Published: October 1, 2013

**Citation**

Haitao Liu, "Coherent-form energy conservation relation for the elastic scattering of a guided mode in a symmetric scattering system," Opt. Express **21**, 24093-24098 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-20-24093

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### References

- M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).
- C. Vassallo, Optical Waveguide Concepts (Elsevier, 1991).
- T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature391(6668), 667–669 (1998). [CrossRef]
- J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature455(7211), 376–379 (2008). [CrossRef] [PubMed]
- P. Mühlschlegel, H. J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science308(5728), 1607–1609 (2005). [CrossRef] [PubMed]
- L. M. Tong, F. Zi, X. Guo, and J. Y. Lou, “Optical microfibers and nanofibers:A tutorial,” Opt. Commun.285(23), 4641–4647 (2012). [CrossRef]
- H. T. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature452(7188), 728–731 (2008). [CrossRef] [PubMed]
- J. Yang, C. Sauvan, H. T. Liu, and P. Lalanne, “Theory of fishnet negative-index optical metamaterials,” Phys. Rev. Lett.107(4), 043903 (2011). [CrossRef] [PubMed]
- K. J. Huang, S. Y. Yang, and L. M. Tong, “Modeling of evanescent coupling between two parallel optical nanowires,” Appl. Opt.46(9), 1429–1434 (2007). [CrossRef] [PubMed]
- X. P. Huang and M. L. Brongersma, “Rapid computation of light scattering from aperiodic plasmonic structures,” Phys. Rev. B84(24), 245120 (2011). [CrossRef]
- G. Y. Li, F. Xiao, L. Cai, K. Alameh, and A. S. Xu, “Theory of the scattering of light and surface plasmon polaritons by finite-size subwavelength metallic defects via field decomposition,” New J. Phys.13(7), 073045 (2011). [CrossRef]
- Here the guided mode is defined as a propagative waveguide mode, which obeys an exponential propagation rule exp(ikz) with the propagation constant k being real or approximately real.
- For instance, the derivation of the resonance condition in [7] and [8] requires |τ + ρ|≈1 for the elastic scattering of SPPs, which is just the coherent-form ECR under the energy conservative condition.
- 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). [CrossRef]
- J. P. Hugonin and P. Lalanne, “Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A22(9), 1844–1849 (2005). [CrossRef] [PubMed]
- E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).
- Z. H. Han and S. I. Bozhevolnyi, “Plasmon-induced transparency with detuned ultracompact Fabry-Perot resonators in integrated plasmonic devices,” Opt. Express19(4), 3251–3257 (2011). [CrossRef] [PubMed]
- P. Lalanne, J. P. Hugonin, and J. C. Rodier, “Theory of surface plasmon generation at nanoslit apertures,” Phys. Rev. Lett.95(26), 263902 (2005). [CrossRef] [PubMed]
- L. F. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A14(10), 2758–2767 (1997). [CrossRef]
- G. Lecamp, J. P. Hugonin, and P. Lalanne, “Theoretical and computational concepts for periodic optical waveguides,” Opt. Express15(18), 11042–11060 (2007). [CrossRef] [PubMed]

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