## Strong coupling of in-plane propagating plasmon modes and its control |

Optics Express, Vol. 21, Issue 11, pp. 13187-13192 (2013)

http://dx.doi.org/10.1364/OE.21.013187

Acrobat PDF (1151 KB)

### Abstract

We show anti-crossings due to strong in-plane coupling of grating excited propagating plasmon modes in dielectric-metal-dielectric structure with 2D dielectric pattern on top. Grating coupled propagating plasmon modes along with their complete dispersion in the measurement range and all different sample orientations are calculated first. Further a coupled mode theory is presented for the specific geometry presented here. Experimentally measured anti-crossing widths are compared with those calculated by coupled mode theory. It is shown that the coupling strength of the plasmon modes and thus the anti-crossing width can be controlled by the orientation of the sample.

© 2013 OSA

## 1. Introduction

1. L. Novotny, “Strong coupling, energy splitting, and level crossings: A classical perspective,” Am. J. Phys. **78**(11), 1199–1202 (2010). [CrossRef]

3. E. A. Stinaff, M. Scheibner, A. S. Bracker, I. V. Ponomarev, V. L. Korenev, M. E. Ware, M. F. Doty, T. L. Reinecke, and D. Gammon, “Optical signatures of coupled quantum dots,” Science **311**(5761), 636–639 (2006). [CrossRef] [PubMed]

4. D. Sarid, R. T. Deck, and J. J. Fasano, “Enhanced nonlinearity of the propagation constant of a long-range surface-plasma wave,” J. Opt. Soc. Am. **72**(10), 1345–1347 (1982). [CrossRef]

8. Y. Chu and K. B. Crozier, “Experimental study of the interaction between localized and propagating surface plasmons,” Opt. Lett. **34**(3), 244–246 (2009). [CrossRef] [PubMed]

9. G. S. Agarwal and S. Dutta Gupta, “Interaction between surface plasmons and localized plasmons,” Phys. Rev. B Condens. Matter **32**(6), 3607–3611 (1985). [CrossRef] [PubMed]

16. A. Christ, S. G. Tikhodeev, N. A. Gippius, J. Kuhl, and H. Giessen, “Waveguide-plasmon polaritons: Strong coupling of photonic and electronic resonances in a metallic photonic crystal slab,” Phys. Rev. Lett. **91**(18), 183901 (2003). [CrossRef] [PubMed]

18. S. D. Gupta, “Theoretical study of plasma resonance absorption in conical diffraction,” J. Opt. Soc. Am. B **4**(11), 1893–1898 (1987). [CrossRef]

19. S. Kasture, P. Mandal, A. Singh, A. Ramsay, A. S. Vengurlekar, S. Dutta Gupta, V. Belotelov, and A. Venu Gopal, “Near dispersion-less surface plasmon polariton resonances at a metal-dielectric interface with patterned dielectric on top,” Appl. Phys. Lett. **101**(9), 091602 (2012). [CrossRef]

## 2. Sample fabrication and measurement geometry

_{x}) and 625 nm (a

_{y}) in the two orthogonal directions. The fill factors (air to dielectric ratio) are 0.65 and 0.7 in x and y directions, respectively. Figure 1 shows the schematic of the sample along with the measurement geometry and the SEM image. Further details about the structure are in Ref.19

19. S. Kasture, P. Mandal, A. Singh, A. Ramsay, A. S. Vengurlekar, S. Dutta Gupta, V. Belotelov, and A. Venu Gopal, “Near dispersion-less surface plasmon polariton resonances at a metal-dielectric interface with patterned dielectric on top,” Appl. Phys. Lett. **101**(9), 091602 (2012). [CrossRef]

_{x}which excites the SPPs. Similarly, for φ = 90° the finite electric field component along a

_{y}excites the SPPs. Angle θ is the launch angle in the plane perpendicular to the XY-plane.

## 3. Experimental results

20. S. G. Romanov, N. Vogel, K. Bley, K. Landfester, C. K. Weiss, S. Orlov, A. V. Korovin, G. P. Chuiko, A. Regensburger, A. S. Romanova, A. Kriesch, and U. Peschel, “Probing guided in a monolayer colloidal crystal on a flat metal film,” Phys. Rev. B **86**(19), 195145 (2012). [CrossRef]

21. M. López-García, J. F. Galisteo-López, A. Blanco, J. Sánchez-Marcos, C. López, and A. García-Martín, “Enhancement and directionality of spontaneous emission in hybrid self-assembled photonic-plasmonic crystals,” Small **6**(16), 1757–1761 (2010). [CrossRef] [PubMed]

19. S. Kasture, P. Mandal, A. Singh, A. Ramsay, A. S. Vengurlekar, S. Dutta Gupta, V. Belotelov, and A. Venu Gopal, “Near dispersion-less surface plasmon polariton resonances at a metal-dielectric interface with patterned dielectric on top,” Appl. Phys. Lett. **101**(9), 091602 (2012). [CrossRef]

**101**(9), 091602 (2012). [CrossRef]

_{x}. Use of thin metal layer results in coupling of SPP modes at the top and bottom interfaces resulting in symmetric and antisymmetric modes. Thus, modes with subscript “A” are anti-symmetric and those with “S” are symmetric modes. Modes with subscript “d” are Rayleigh anomalies [19

**101**(9), 091602 (2012). [CrossRef]

## 4. Coupled mode theory for calculating anti-crossing width

22. H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. **43**(5), 2327–2335 (1972). [CrossRef]

23. R. Daendliker, “Coupled waves: A powerful concept in modern optics,” SPIE Proc. Fifth International Topical Meeting on Education and Training in Optics **3190,** 279–288 (1997). [CrossRef]

24. A. Kolomenskii, S. Peng, J. Hembd, A. Kolomenski, J. Noel, J. Strohaber, W. Teizer, and H. Schuessler, “Interaction and spectral gaps of surface plasmon modes in gold nano-structures,” Opt. Express **19**(7), 6587–6598 (2011). [CrossRef] [PubMed]

^{x}, H

^{y}, and E

^{z}) and calculate the coupling constants. For a three layer system with a thin metal film sandwiched between dielectric materials, solutions to Maxwell’s equations with appropriate boundary conditions at the interfaces for TM polarized wave are well known [25]. In the three regions, (in the top dielectric, in the metal and in the bottom dielectric), expressions for the three relevant field components (H

^{y}, E

^{z}and E

^{x}) are in 3 unknown constants. We simplify these equations and write them in terms of a single constant which is solved for by using the power normalization relation,where the integration extends into all the three layers along z-axis and φ is the azimuthal angle with respect to the x-axis. In this case, when

_{β}governs the evolution of the field. When two of the TM modes are coupled, the in-plane and normal to plane coupling constants are given similar to the overlap integrals [26].

*K*= 2π/a

_{x}_{x}and

*K*= 2π/a

_{y}_{y}with a

_{x}and a

_{y}being the periodicities in the x and y directions, respectively. On substituting this in the field propagation equation, it reduces to,where the transverse and the in-plane coupling constants for two interacting modes, α and β, are given by,where φ

_{α}and φ

_{β}are the angles made by the propagation vector of each of the waves with the x-axis and E

^{x}is the electric field component along the propagation direction of each mode. The total coupling constant is the sum of the transverse and in-plane coupling constants. Next we calculate the specific k values allowed by the sample and excitation geometry to find the coupling between coplanar SPP modes. For the three layer system with appropriate boundary conditions when we invoke continuity of the field and its derivative at each interface, we get the SPP dispersion relation given by [25],where

_{x}component and the second term is the k

_{y}component. For given structure (that is, a

_{x}and a

_{y}) and the measurement geometry (θ, φ), we find the k

_{in-plane}values satisfying both Eq. (5) and Eq. (6) simultaneously. We calculate the coupling constant between two such resonant modes A1(x) and A2(x) given by,

_{α}= k

_{β}+ mK

_{x}, splitting comes out to be Δ = 2K

_{αβ}in the k

_{x}component. It may be seen from Eq. (4) that the azimuthal angle φ gives control on the coupling strength.

## 5. Conclusion

## References and links

1. | L. Novotny, “Strong coupling, energy splitting, and level crossings: A classical perspective,” Am. J. Phys. |

2. | R. M. Stevenson, V. N. Astratov, M. S. Skolnick, D. M. Whittaker, M. Emam-Ismail, A. I. Tartakovskii, P. G. Savvidis, J. J. Baumberg, and J. S. Roberts, “Continuous wave observation of massive polariton redistribution by stimulated scattering in semiconductor microcavities,” Phys. Rev. Lett. |

3. | E. A. Stinaff, M. Scheibner, A. S. Bracker, I. V. Ponomarev, V. L. Korenev, M. E. Ware, M. F. Doty, T. L. Reinecke, and D. Gammon, “Optical signatures of coupled quantum dots,” Science |

4. | D. Sarid, R. T. Deck, and J. J. Fasano, “Enhanced nonlinearity of the propagation constant of a long-range surface-plasma wave,” J. Opt. Soc. Am. |

5. | W. R. Holland and D. G. Hall, “Surface-plasmon dispersion relation: Shifts induced by the interaction with localized plasma resonances,” Phys. Rev. B |

6. | S. D. Gupta, G. V. Varada, and G. S. Agarwal, “Surface plasmons in two-sided corrugated thin films,” Phys. Rev. B Condens. Matter |

7. | K. F. MacDonald, Z. L. Samson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photonics |

8. | Y. Chu and K. B. Crozier, “Experimental study of the interaction between localized and propagating surface plasmons,” Opt. Lett. |

9. | G. S. Agarwal and S. Dutta Gupta, “Interaction between surface plasmons and localized plasmons,” Phys. Rev. B Condens. Matter |

10. | W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, “Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film,” Phys. Rev. Lett. |

11. | A. Christ, T. Zentgraf, S. G. Tikhodeev, N. A. Gippius, O. J. F. Martin, J. Kuhl, and H. Giessen, “Interaction between localized and delocalized surface plasmon polariton modes in a metallic photonic crystal,” Phys. Status Solidi |

12. | Z. Chen, I. R. Hooper, and J. R. Sambles, “Grating-coupled surface plasmon polaritons and waveguide modes in a silver-dielectric-silver structure,” J. Opt. Soc. Am. A |

13. | H. Gao, J. Henzie, M. H. Lee, and T. W. Odom, “Screening plasmonic materials using pyramidal gratings,” Proc. Natl. Acad. Sci. U.S.A. |

14. | A. Ghoshal, I. Divliansky, and P. G. Kik, “Experimental observation of mode-selective anticrossing in surface-plasmon-coupled metal nanoparticle arrays,” Appl. Phys. Lett. |

15. | J. Li, H. Lu, J. T. K. Wan, and H. C. Ong, “The plasmonic properties of elliptical metallic hole arrays,” Appl. Phys. Lett. |

16. | A. Christ, S. G. Tikhodeev, N. A. Gippius, J. Kuhl, and H. Giessen, “Waveguide-plasmon polaritons: Strong coupling of photonic and electronic resonances in a metallic photonic crystal slab,” Phys. Rev. Lett. |

17. | H. Raether, |

18. | S. D. Gupta, “Theoretical study of plasma resonance absorption in conical diffraction,” J. Opt. Soc. Am. B |

19. | S. Kasture, P. Mandal, A. Singh, A. Ramsay, A. S. Vengurlekar, S. Dutta Gupta, V. Belotelov, and A. Venu Gopal, “Near dispersion-less surface plasmon polariton resonances at a metal-dielectric interface with patterned dielectric on top,” Appl. Phys. Lett. |

20. | S. G. Romanov, N. Vogel, K. Bley, K. Landfester, C. K. Weiss, S. Orlov, A. V. Korovin, G. P. Chuiko, A. Regensburger, A. S. Romanova, A. Kriesch, and U. Peschel, “Probing guided in a monolayer colloidal crystal on a flat metal film,” Phys. Rev. B |

21. | M. López-García, J. F. Galisteo-López, A. Blanco, J. Sánchez-Marcos, C. López, and A. García-Martín, “Enhancement and directionality of spontaneous emission in hybrid self-assembled photonic-plasmonic crystals,” Small |

22. | H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. |

23. | R. Daendliker, “Coupled waves: A powerful concept in modern optics,” SPIE Proc. Fifth International Topical Meeting on Education and Training in Optics |

24. | A. Kolomenskii, S. Peng, J. Hembd, A. Kolomenski, J. Noel, J. Strohaber, W. Teizer, and H. Schuessler, “Interaction and spectral gaps of surface plasmon modes in gold nano-structures,” Opt. Express |

25. | S. A. Maier, |

26. | J. M. Liu, |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(240.1485) Optics at surfaces : Buried interfaces

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: March 7, 2013

Revised Manuscript: May 16, 2013

Manuscript Accepted: May 16, 2013

Published: May 23, 2013

**Citation**

Sachin Kasture, Prasanta Mandal, S. Dutta Gupta, and Achanta Venu Gopal, "Strong coupling of in-plane propagating plasmon modes and its control," Opt. Express **21**, 13187-13192 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-11-13187

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### References

- L. Novotny, “Strong coupling, energy splitting, and level crossings: A classical perspective,” Am. J. Phys.78(11), 1199–1202 (2010). [CrossRef]
- R. M. Stevenson, V. N. Astratov, M. S. Skolnick, D. M. Whittaker, M. Emam-Ismail, A. I. Tartakovskii, P. G. Savvidis, J. J. Baumberg, and J. S. Roberts, “Continuous wave observation of massive polariton redistribution by stimulated scattering in semiconductor microcavities,” Phys. Rev. Lett.85(17), 3680–3683 (2000). [CrossRef] [PubMed]
- E. A. Stinaff, M. Scheibner, A. S. Bracker, I. V. Ponomarev, V. L. Korenev, M. E. Ware, M. F. Doty, T. L. Reinecke, and D. Gammon, “Optical signatures of coupled quantum dots,” Science311(5761), 636–639 (2006). [CrossRef] [PubMed]
- D. Sarid, R. T. Deck, and J. J. Fasano, “Enhanced nonlinearity of the propagation constant of a long-range surface-plasma wave,” J. Opt. Soc. Am.72(10), 1345–1347 (1982). [CrossRef]
- W. R. Holland and D. G. Hall, “Surface-plasmon dispersion relation: Shifts induced by the interaction with localized plasma resonances,” Phys. Rev. B27(12), 7765–7768 (1983). [CrossRef]
- S. D. Gupta, G. V. Varada, and G. S. Agarwal, “Surface plasmons in two-sided corrugated thin films,” Phys. Rev. B Condens. Matter36(12), 6331–6335 (1987). [CrossRef] [PubMed]
- K. F. MacDonald, Z. L. Samson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photonics3(1), 55–58 (2009). [CrossRef]
- Y. Chu and K. B. Crozier, “Experimental study of the interaction between localized and propagating surface plasmons,” Opt. Lett.34(3), 244–246 (2009). [CrossRef] [PubMed]
- G. S. Agarwal and S. Dutta Gupta, “Interaction between surface plasmons and localized plasmons,” Phys. Rev. B Condens. Matter32(6), 3607–3611 (1985). [CrossRef] [PubMed]
- W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, “Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film,” Phys. Rev. Lett.92(10), 107401 (2004). [CrossRef] [PubMed]
- A. Christ, T. Zentgraf, S. G. Tikhodeev, N. A. Gippius, O. J. F. Martin, J. Kuhl, and H. Giessen, “Interaction between localized and delocalized surface plasmon polariton modes in a metallic photonic crystal,” Phys. Status Solidi243(10), 2344–2348 (2006) (b). [CrossRef]
- Z. Chen, I. R. Hooper, and J. R. Sambles, “Grating-coupled surface plasmon polaritons and waveguide modes in a silver-dielectric-silver structure,” J. Opt. Soc. Am. A24(11), 3547–3553 (2007). [CrossRef] [PubMed]
- H. Gao, J. Henzie, M. H. Lee, and T. W. Odom, “Screening plasmonic materials using pyramidal gratings,” Proc. Natl. Acad. Sci. U.S.A.105(51), 20146–20151 (2008). [CrossRef] [PubMed]
- A. Ghoshal, I. Divliansky, and P. G. Kik, “Experimental observation of mode-selective anticrossing in surface-plasmon-coupled metal nanoparticle arrays,” Appl. Phys. Lett.94(17), 171108 (2009). [CrossRef]
- J. Li, H. Lu, J. T. K. Wan, and H. C. Ong, “The plasmonic properties of elliptical metallic hole arrays,” Appl. Phys. Lett.94(3), 033101 (2009). [CrossRef]
- A. Christ, S. G. Tikhodeev, N. A. Gippius, J. Kuhl, and H. Giessen, “Waveguide-plasmon polaritons: Strong coupling of photonic and electronic resonances in a metallic photonic crystal slab,” Phys. Rev. Lett.91(18), 183901 (2003). [CrossRef] [PubMed]
- H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1986).
- S. D. Gupta, “Theoretical study of plasma resonance absorption in conical diffraction,” J. Opt. Soc. Am. B4(11), 1893–1898 (1987). [CrossRef]
- S. Kasture, P. Mandal, A. Singh, A. Ramsay, A. S. Vengurlekar, S. Dutta Gupta, V. Belotelov, and A. Venu Gopal, “Near dispersion-less surface plasmon polariton resonances at a metal-dielectric interface with patterned dielectric on top,” Appl. Phys. Lett.101(9), 091602 (2012). [CrossRef]
- S. G. Romanov, N. Vogel, K. Bley, K. Landfester, C. K. Weiss, S. Orlov, A. V. Korovin, G. P. Chuiko, A. Regensburger, A. S. Romanova, A. Kriesch, and U. Peschel, “Probing guided in a monolayer colloidal crystal on a flat metal film,” Phys. Rev. B86(19), 195145 (2012). [CrossRef]
- M. López-García, J. F. Galisteo-López, A. Blanco, J. Sánchez-Marcos, C. López, and A. García-Martín, “Enhancement and directionality of spontaneous emission in hybrid self-assembled photonic-plasmonic crystals,” Small6(16), 1757–1761 (2010). [CrossRef] [PubMed]
- H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys.43(5), 2327–2335 (1972). [CrossRef]
- R. Daendliker, “Coupled waves: A powerful concept in modern optics,” SPIE Proc. Fifth International Topical Meeting on Education and Training in Optics 3190, 279–288 (1997). [CrossRef]
- A. Kolomenskii, S. Peng, J. Hembd, A. Kolomenski, J. Noel, J. Strohaber, W. Teizer, and H. Schuessler, “Interaction and spectral gaps of surface plasmon modes in gold nano-structures,” Opt. Express19(7), 6587–6598 (2011). [CrossRef] [PubMed]
- S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).
- J. M. Liu, Photonic Devices (Cambridge University, 2005).

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