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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 20 — Oct. 7, 2013
  • pp: 24093–24098
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Coherent-form energy conservation relation for the elastic scattering of a guided mode in a symmetric scattering system

Haitao Liu  »View Author Affiliations


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 |τρ|≤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

As sketched by the insets in Fig. 1
Fig. 1 Numerical test of the coherent-form ECR |τ + ρ|≤1 for different types of modes in various symmetric scattering systems. (a)-(b) Scattering of a SPP at a periodic chain of infinite-depth square holes (a) or at an infinite-depth slit (b) in gold substrate with an air cladding (refractive index 1). The chain period is 940nm, and the hole side length and the slit width are 266nm. (c)-(d) For the fundamental SPP mode on a gold nano-wire in air scattered at an air gap with a width of 30nm (c) and 100nm (d). The nano-wire has a square cross section with a side length of 40nm. (e)-(g) For the fundamental SPP mode in a gold/air/gold waveguide (air-gap width 100nm), which is scattered at an infinite-depth slit (slit width 50nm) (e), at a finite-depth groove (groove width 50nm and depth 1μm) (f), and at an orthogonal corner (g). (h) Coupling of the fundamental mode of two tightly contacted silica nano-wires (diameter 400nm, refractive index 1.46 at wavelength λ = 633nm) in air. The considered fundamental mode is polarized in the plane determined by the two wire axes.
, the elastic scattering of a guided mode [12

12. 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.

] at a scatterer can be characterized by a transmission coefficient τ and a reflection coefficient ρ. Classical ECR [2

2. C. Vassallo, Optical Waveguide Concepts (Elsevier, 1991).

] 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

2. C. Vassallo, Optical Waveguide Concepts (Elsevier, 1991).

], 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

13. 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.

].

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

For the results shown in Fig. 1(b), the width of the subwavelength slit is chosen equal to the side length of the square holes in the chain. The intrinsic difference between the slit and the chain of holes is that the former always supports a fundamental SPP mode that is propagative [14

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]

]. This implies that for the slit, |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).

Figures 1(c) and 1(d) provide the values of |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]

]. It is seen that for all wavelengths, |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.

For the fundamental SPP mode in a metal-insulator-metal waveguide that may constitute compact plasmonic circuits due to its deep subwavelength confinement of the SPP mode [17

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]

], Figs. 1(e)-1(g) provide the values of |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

2. C. Vassallo, Optical Waveguide Concepts (Elsevier, 1991).

]). 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]

]. For the calculation the 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]

] where the single-interface SPP field ESP and HSP 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.

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

The theoretical demonstration proves that the coherent-form ECR is generally valid, provided that the scatterer is symmetric and that the waveguides on both sides of the scatterer are identical and approximately lossless.

To further check this general validity, Fig. 3
Fig. 3 Numerical test of the coherent-form ECR for the transmission (with a coefficient t) and reflection (coefficient r) of a normally incident plane wave at a metallic hole array drilled in a gold membrane in air (inset). The plane wave is polarized in one periodic direction of the grating. |t + r| for different wavelengths is shown. (a) For a real lossy metal of gold. (b) For an artificial lossless metal. The array period is 940nm, the side length of square holes is 266nm, and the membrane thickness is 200nm.
provides another example considering the transmission and reflection of a normally incident plane wave at a metallic hole array drilled in a gold membrane in air (see the inset). The transmission and reflection coefficients of the zeroth-order plane wave are denoted by 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]

]. As shown in Fig. 3(a) for a real lossy metal of gold (taking refractive indices from [16

16. E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

]), |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]

,7

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

]. This dip is due to the energy loss resulting from a resonant excitation of surface waves on the membrane surface which causes the extraordinary optical transmission [7

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

].

It may be interesting to examine the condition to achieve the equality of the coherent-form ECR. From the demonstration one can easily obtain the condition: for the waveguides on both sides of the scatterer, the transmitted and reflected guided mode is the sole mode that carries energy; within the region of the scatterer no energy is absorbed or carried away.

To confirm the analysis, we repeat the calculation of Fig. 3(a) but consider an artificial lossless metal. This is achieved by artificially removing the real part of the refractive index of the lossy gold, for instance, changing the gold refractive index of 0.25 + 6.84i (at λ = 1μm) to be 6.84i. 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

In summary, we have proposed a coherent-form ECR |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 |τρ|≤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

Financial supports from the 973 Program (2013CB328701), the Natural Science Foundation of Tianjin (11JCZDJC15400) and the Natural Science Foundation of China (61322508) are acknowledged. The author thanks Dr. Philippe Lalanne for providing helpful comments.

References and notes

1.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).

2.

C. Vassallo, Optical Waveguide Concepts (Elsevier, 1991).

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]

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 455(7211), 376–379 (2008). [CrossRef] [PubMed]

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]

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]

7.

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

8.

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]

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]

10.

X. P. Huang and M. L. Brongersma, “Rapid computation of light scattering from aperiodic plasmonic structures,” Phys. Rev. B 84(24), 245120 (2011). [CrossRef]

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. 13(7), 073045 (2011). [CrossRef]

12.

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.

13.

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.

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]

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]

16.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

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]

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]

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]

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]

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

  1. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).
  2. C. Vassallo, Optical Waveguide Concepts (Elsevier, 1991).
  3. 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]
  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,” Nature455(7211), 376–379 (2008). [CrossRef] [PubMed]
  5. 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]
  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]
  7. H. T. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature452(7188), 728–731 (2008). [CrossRef] [PubMed]
  8. 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]
  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]
  10. X. P. Huang and M. L. Brongersma, “Rapid computation of light scattering from aperiodic plasmonic structures,” Phys. Rev. B84(24), 245120 (2011). [CrossRef]
  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.13(7), 073045 (2011). [CrossRef]
  12. 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.
  13. 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.
  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]
  15. 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]
  16. E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).
  17. 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]
  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]
  19. 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]
  20. 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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