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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 16 — Aug. 9, 2004
  • pp: 3652–3663
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Analysis of the Bloch mode spectra of surface polaritonic crystals in the weak and strong coupling regimes: grating-enhanced transmission at oblique incidence and suppression of SPP radiative losses

D. Gérard, L. Salomon, F. de Fornel, and A. V. Zayats  »View Author Affiliations


Optics Express, Vol. 12, Issue 16, pp. 3652-3663 (2004)
http://dx.doi.org/10.1364/OPEX.12.003652


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Abstract

The Bloch mode spectrum of surface plasmon polaritons (SPPs) on a finite thickness metal film has been analyzed in the regimes of weak and strong coupling between SPP modes on the opposite film interfaces. The SPP mode dispersion and associated field distributions have been studied. The results have been applied to the description of the light transmission through thick and thin periodically structured metal films at oblique incidence. In contrast to normal incidence, all SPP Bloch modes on a grating structure participate in the resonant photon tunnelling leading to the transmission enhancement. However, at the angle of incidence corresponding to the crossing of different symmetry film SPP Bloch modes, the far-field transmission is suppressed despite the enhanced near-field transmission. The combined SPP mode consisting of the two film SPPs having different symmetries that is achieved at the crossing frequency exhibits no radiative losses on a structured surface.

© 2004 Optical Society of America

1. Introduction

Fig. 1. Schematic of a nanostructured film.

In this paper we present numerical studies of the SPP Bloch mode spectrum of a finite thickness metal film in both weak and strong coupling regimes between SPP modes on the opposite film interfaces. The SPP mode dispersion and associated field distributions are discussed. This allows us to generally define conditions of the strong and weak coupling of the SPP modes on the interfaces of the finite-thickness structured films. The role of the various SPP Bloch modes in the optical transmission through a continuous (without apertures), optically thick metal film with a ridge-grating structure are considered at oblique incidence. In contrast to normal incidence, all SPP Bloch modes on a grating structure participate in the resonant photon tunnelling leading to richer spectrum of the resonant transmission enhancement. At the angle of incidence corresponding to the crossing of the different symmetry film SPP Bloch modes the transmission is suppressed. The combined SPP mode consisting of the two film SPPs having different symmetries that is achieved at the crossing frequency exhibits no radiative losses on a structured surface.

2. Numerical model

Numerical modelling was performed using differential method with the S-algorithm taking into account the Li remarks [21

21. M. Nevière and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design, Marcel Dekker, New-York, 2003.

, 22

22. E. Popov and M. Nevière, “Differential theory for diffraction gratings: a new formulation for TM polarization with rapid convergence,” Opt. Lett. 25, 598–600 (2000). [CrossRef]

]. The latter two modifications allow modelling of deep gratings with arbitrary grove shape and significantly improve the convergence rate of the differential method. Moreover, the use of the S-matrix propagation algorithm permits to calculate the electromagnetic field inside the structure. The structure under consideration is considered as three zones: two modulated zones (the ridges) and one homogenous layer (the continuous metal film). The electromagnetic field above and below the structure is presented as an expansion over the Rayleigh modes. The complex amplitudes of the Rayleigh waves are obtained by solving the propagation equations with the boundary conditions at the interfaces. This numerical procedure allows to recover the electromagnetic field distribution in and around the structure and to calculate spectral dependencies of its optical properties.

3. Results and discussion

The spectra of the SPP resonances entirely determine the spectra of absorption, reflection, and transmission of apertureless nanostructured metal films [6

6. S.A. Darmanyan and A.V. Zayats, “Light tunneling via resonant surface plasmon polariton states and the enhanced transmission of periodically nanostructured metal films: an analytical study,” Phys. Rev. B 67, 035424 (2003). [CrossRef]

]. The angular behaviour of the spectra is different in the weak and strong coupling regime determined by the interaction between the SPP modes on the opposite film interfaces. This behaviour depends on the thickness of a metal film, and more precisely on the mutual positions of the band-gaps in the spectrum of the SPP Bloch modes which are either weakly interacting SPPs on the opposite film interfaces or film SPP modes if the coupling is strong.

The SPP modes in the vicinity of the even SPP band-gaps are important at the angles of incidence close to the normal one [6

6. S.A. Darmanyan and A.V. Zayats, “Light tunneling via resonant surface plasmon polariton states and the enhanced transmission of periodically nanostructured metal films: an analytical study,” Phys. Rev. B 67, 035424 (2003). [CrossRef]

]. In the general case of a finite thickness metal film, these modes can be distinguished by their origin as film SPP modes: the lower energy f + mode with the symmetric electric field Ex distribution in a film and the higher energy antisymmetric mode (f -). Both these modes will be split into the set of the Bloch modes with lower (g +) and higher (g -) frequencies at the edges of the Brillouin zones. Thus, near the second band-gap, four SPP Bloch modes should be considered which can be arranged in two different types of spectra with overlapping or not-overlapping band-gaps (Fig. 2). The realization of one or another type of the spectrum depends on the film thickness and grating parameters that determine the band-gap width for each SPP mode.

Fig. 2. Schematics of the band-gap structure near the second SPP band-gap in the case of weak (a) and strong (b) coupling regimes: (f +, f -) film SPP Bloch modes, (g +,g -) SPP modes in the different Brillouin zones.
Fig. 3. (Colour) Reflection (a), absorption (b), and transmission (c) spectra of the nanostructured silver film (H=100 nm) at different angles of incidence: (black) θ=0°, (blue) θ=2°, (red) θ=4°. The structure consists of the silver ridges (h=20 nm, d=250 nm and D=500 nm) on both film interfaces.

3.1. Weak-coupling regime

Fig. 4. Dispersion of the SPP Bloch modes in the vicinity of the second band-gap on a periodic structure in a weak coupling regime. The parametres of the structure are the same as in Fig. 3

3.2. Strong-coupling regime

In the case of a strong-coupling between the SPPs on the film interfaces, the consideration of weakly-perturbed SPP states on the interfaces are no longer viable, and film SPP modes should be introduced. The shift of the frequencies of the two modes is significant so that they are clearly distinguishable in the spectra at normal incidence (Fig. 6). The lower frequency mode (f +) corresponds to the symmetric electric field field Ex distribution in the film and experiences strong Ohmic losses, while the high frequency mode (f -) has the antisymmetric field distribution and low losses [7

7. H. Raether, Surface Plasmons, Springer-Verlag, Berlin, 1988.

].

Fig. 5. (Colour) The magnetic field Hz distribution in the near-field region of the metallic structure at the wavelengths corresponding to (a) lower (λ=564 nm) and (b) upper (λ=506 nm) branches of the SPP Bloch modes around the second band-gap in a weak-coupling regime. Angle of incidence is θ=4°. The parametres of the film are the same as in Fig. 3. Geometry of the film is also shown.

In spite of the significant near-field transmission (the field in the vicinity of the interface opposite to the illuminated one) (Fig. 8(a)) since the SPP modes are efficiently exited on both film interfaces, there is no far-field radiation at the crossing frequency: the field intensity above the structure rapidly decays from the surface at the distance shorter than the wavelength (Fig. 8(b)). Physically, this can be understood considering opposite phases of the two SPP modes on the interface: each of them is coupled to the outgoing far-field wave, however, due to the difference in the phase, the outgoing fields cancel each other in the far-field region. The intensity distribution across the surface has 2-fold symmetry over the period of the structure confirming that the two competing SPP Bloch modes are completely “identical” (in the case of non-zero far-field transmittance, the intensity distribution is periodic only with the period of the structure). In the absence of radiative losses, these SPP mode propagate on the surface until converted into heat due to Ohmic losses, thus leading to the increased absorption observed at the crossing frequency.

Fig. 6. (Colour) Reflection (a), absorption (b), and transmission (c) spectra of the nanostructured silver film (H=40 nm) at different angles of incidence: (black) θ=0°, (blue) θ=1°, (green) θ=2°, (red) θ=4°. The structure consists of the silver ridges (h=20 nm, d=250 nm and D=500 nm) on both film interfaces.

The similar SPP mode crossing can occur at kSP=0 at some film thickness [20

20. S.A. Darmanyan, M. Nevière, and A.V. Zayats, “Analytical theory of optical transmission through periodically structured metal films via tunnel-coupled surface polariton modes,” Phys. Rev. B70, 15AUG (2004). [CrossRef]

]. This condition can be used to define the film thickness of a nanostructured film at which transition between weak and strong coupling regimes takes place. If at kSP=0 the g + Bloch mode of the antisymmetric SPP film mode is situated at lower frequencies than the g - Bloch mode of the symmetric film SPP mode, the SPP coupling can be considered as weak (weaker than periodic structure effects) and no mode crossing occurs. For smaller film thicknesses (strong coupling), the film SPP Bloch modes will be mixed due to their opposite dispersion at some kSP and there will be a mode crossing at some angle of incidence (Fig. 2).

Using the symmetry properties of the fields, the origin of the SPP Bloch modes on a finite thickness nanostructured films can be identified. This can be illustrated by the field distributions at normal incidence when only one branch of the SPP Bloch mode near the second band-gap is optically active and which is split into the two film SPP modes in a strong-coupling regime with symmetric and antisymmetric field distributions in the film but with the same phase across the surface (Fig. 9). The field distributions on the illuminating side of the film are about the same for both film SPP modes. However, the field distributions on another film interface have the opposite phase for different modes, resulting in the different frequency position of resonances. It should be noted that the Ex and Hz field distributions have opposite behaviour with the Ex field being antisymmetric (f -) and symmetric (f +) with respect to the film plane and the Hz field having opposite symmetry.

Fig. 7. Dispersion of the SPP Bloch modes in the vicinity of the second band-gap on a periodic structure in a strong coupling regime. The parametres of the structure are the same as in Fig. 6.
Fig. 8. (Colour) (a) The magnetic field Hz distribution in the near-field region of the metallic structure and (b) the intensity distribution of the transmitted field over the structure at the wavelength corresponding to the crossing of the SPP modes of different symmetries in a strong-coupling regime (λ=539 nm, θ=2°). The parametres of the structure are the same as in Fig. 6.
Fig. 9. (Colour) The magnetic field Hz (a,b) and electric field Ex (c,d) distributions in the near-field of the metallic structure at the wavelengths corresponding to (a,c) f - g + (λ=527 nm) and (b,d) f + g + (λ=552 nm) SPP Bloch modes at around the second-band gap in a strong-coupling regime. Angle of incidence is θ=0°. The parametres of the structure are the same as in Fig. 6. Geometry of the film is also shown.

Fig. 10. (Colour) The magnetic field Hz distribution in the near-field region of the metallic structure at the wavelengths corresponding to (a) f - g - (λ=493 nm), (b) f - g + (λ=555 nm), (c) f + g - (λ=527 nm), and (d) f + g + (λ=579 nm) film SPP Bloch modes around the respective second-band gaps in a strong-coupling regime. Angle of incidence is θ=4°. The parametres of the structure are the same as in Fig. 6. Geometry of the film is also shown.

4. Conclusion

SPP Bloch modes on finite-thickness periodically structured metal films have been investigated in the case of weak and strong coupling. It has been shown that the combination of the two film SPP modes having different symmetries that can be achieved at some crossing frequency results in the suppression of the radiative losses associated with these SPP modes on a periodically structured surface. This effect may be of significant importance for the development of the applications that involve SPP guiding on structured surfaces. Understanding of SPP modes properties on a finite thickness film is also needed to tailor optical properties of metal films governed by surface plasmon polaritons, in particular light transmission, absorption, and reflection. Combined with nonlinear optical materials, SPP crystals can be used to control optical properties, such as transmission, reflection and absorption of light in metallic nanostructures [5

5. I.I. Smolyaninov, A.V. Zayats, A. Stanishevsky, and C.C. Davis, “Optical control of photon tunneling through an array of nanometer scale cylindrical channels,” Phys. Rev. B 66, 205414 (2002). [CrossRef]

] or SPP propagation on a structured metal surface [23

23. A.V. Krasavin, N.I. Zheludev, and A.V. Zayats, “High-contrast modulation of light with light by control of surface plasmon-polariton wave coupling,” to be published.

]. Both passive and active optical elements of SPP optics based on SPP crystals can find numerous applications in all-optical photonics.

Acknowledgments

This work was supported in part by the UK Engineering and Physical Sciences Research Council. LS gratefully acknowledges the support from International Research Centre for Experimental Physics, The Queen’s University of Belfast, under the Distinguished Visiting Fellowship scheme.

References and links

1.

A.V. Zayats and I.I. Smolyaninov, “Near-field photonics: surface plasmons polaritons and localized surface plasmons,” J. Opt. A: Pure Appl. Opt. 5, S16–S50 (2003). [CrossRef]

2.

S.I. Bozhevolnyi, J. Erland, K. Leosson, P.M.W. Skovgaard, and J.M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86, 3008–3011 (2001). [CrossRef] [PubMed]

3.

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B 54, 6227–6244 (1996). [CrossRef]

4.

M. Kretschmann and A.A. Maradudin, “Band structures of two-dimensional surface-plasmon polaritonic crystals,” Phys. Rev. B 66, 245408 (2002). [CrossRef]

5.

I.I. Smolyaninov, A.V. Zayats, A. Stanishevsky, and C.C. Davis, “Optical control of photon tunneling through an array of nanometer scale cylindrical channels,” Phys. Rev. B 66, 205414 (2002). [CrossRef]

6.

S.A. Darmanyan and A.V. Zayats, “Light tunneling via resonant surface plasmon polariton states and the enhanced transmission of periodically nanostructured metal films: an analytical study,” Phys. Rev. B 67, 035424 (2003). [CrossRef]

7.

H. Raether, Surface Plasmons, Springer-Verlag, Berlin, 1988.

8.

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

9.

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through sub-wavelength holes,” Phys. Rev. B 58, 6779–6782 (1998). [CrossRef]

10.

T. J. Kim, T. Thio, T. W. Ebbesen, D. E. Grupp, and H. J. Lezec, “Control of optical transmission through metals perforated with subwavelenth holes arrays,” Opt. Lett. 24, 256–258 (1999). [CrossRef]

11.

L. Salomon, F. Grillot, A. V. Zayats, and F. de Fornel, “Near-field distribution of optical transmission of periodic subwavelength holes in a metal film,” Phys. Rev. Lett. 86, 1110–1113 (2001). [CrossRef] [PubMed]

12.

L. Martin-Moreno, F.J. García-Vidal, H.J. Lezec, K.M. Pellerin, T. Thio, J.B. Pendry, and T.W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001). [CrossRef] [PubMed]

13.

U. Schröter and D. Heitmann, “Surface-plasmon-enhanced transmission through metallic gratings,” Phys. Rev. B 58, 15419–15422 (1998). [CrossRef]

14.

J. A. Porto, F. J. García-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845–2848 (1999). [CrossRef]

15.

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62, 16100–16108 (2000). [CrossRef]

16.

A.V. Zayats, L. Salomon, and F. de Fornel, “How light gets through periodically nanostructured metal films: a role of surface polaritonic crystals,” J. Microsc. 210, 344–349 (2003). [CrossRef] [PubMed]

17.

N. Bonod, S. Enoch, P.F. Li, E. Popov, and M. Nevière, “Resonant optical transmission through thin metallic films with and without holes,” Opt. Express 11, 482–490 (2003). [CrossRef] [PubMed]

18.

A.M. Dykhne, A.K. Sarychev, and V.M. Shalaev, “Resonant transmittance through metal films with fabricated and light-induced modulation,” Phys. Rev. B 67, 195402 (2003). [CrossRef]

19.

D. Gérard, L. Salomon, F. de Fornel, and A.V. Zayats, “Ridge-enhanced optical transmission through a continuous metal film,” Phys. Rev. B 69, 113405 (2004). [CrossRef]

20.

S.A. Darmanyan, M. Nevière, and A.V. Zayats, “Analytical theory of optical transmission through periodically structured metal films via tunnel-coupled surface polariton modes,” Phys. Rev. B70, 15AUG (2004). [CrossRef]

21.

M. Nevière and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design, Marcel Dekker, New-York, 2003.

22.

E. Popov and M. Nevière, “Differential theory for diffraction gratings: a new formulation for TM polarization with rapid convergence,” Opt. Lett. 25, 598–600 (2000). [CrossRef]

23.

A.V. Krasavin, N.I. Zheludev, and A.V. Zayats, “High-contrast modulation of light with light by control of surface plasmon-polariton wave coupling,” to be published.

OCIS Codes
(050.0050) Diffraction and gratings : Diffraction and gratings
(240.6680) Optics at surfaces : Surface plasmons
(240.7040) Optics at surfaces : Tunneling

ToC Category:
Focus Issue: Extraordinary light transmission through sub-wavelength structured surfaces

History
Original Manuscript: May 20, 2004
Revised Manuscript: July 20, 2004
Published: August 9, 2004

Citation
D. Gérard, L. Salomon, F. de Fornel, and A. Zayats, "Analysis of the Bloch mode spectra of surface polaritonic crystals in the weak and strong coupling regimes: grating-enhanced transmission at oblique incidence and suppression of SPP radiative losses," Opt. Express 12, 3652-3663 (2004)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-16-3652


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References

  1. A.V. Zayats and I.I. Smolyaninov, ???Near-field photonics: surface plasmons polaritons and localized surface plasmons,??? J. Opt. A: Pure Appl. Opt. 5, S16???S50 (2003). [CrossRef]
  2. S.I. Bozhevolnyi, J. Erland, K. Leosson, P.M.W. Skovgaard, and J.M. Hvam, ???Waveguiding in surface Plasmon polariton band gap structures,??? Phys. Rev. Lett. 86, 3008???3011 (2001). [CrossRef] [PubMed]
  3. W.L. Barnes, T.W. Preist, S.C. Kitson, and J.R. Sambles, ???Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,??? Phys. Rev. B 54, 6227???6244 (1996). [CrossRef]
  4. M. Kretschmann and A.A. Maradudin, ???Band structures of two-dimensional surface-plasmon polaritonic crystals,??? Phys. Rev. B 66, 245408 (2002). [CrossRef]
  5. I.I. Smolyaninov, A.V. Zayats, A. Stanishevsky, and C.C. Davis, ???Optical control of photon tunneling through an array of nanometer scale cylindrical channels,??? Phys. Rev. B 66, 205414 (2002). [CrossRef]
  6. S.A. Darmanyan and A.V. Zayats, ???Light tunneling via resonant surface plasmon polariton states and the enhanced transmission of periodically nanostructured metal films: an analytical study,??? Phys. Rev. B 67, 035424 (2003). [CrossRef]
  7. H. Raether, Surface Plasmons, Springer-Verlag, Berlin, 1988.
  8. T.W. Ebbesen, J. Lezec, H.F. Ghaemi, T. Thio, and P.A. Wolff, ???Extraordinary optical transmission through subwavelength hole arrays,??? Nature (London) 391, 667???669 (1998). [CrossRef]
  9. H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, ???Surface plasmons enhance optical transmission through sub-wavelength holes,??? Phys. Rev. B 58, 6779???6782 (1998). [CrossRef]
  10. T.J. Kim, T. Thio, T.W. Ebbesen, D.E. Grupp, and H.J. Lezec, ???Control of optical transmission through metals perforated with subwavelenth holes arrays,??? Opt. Lett. 24, 256???258 (1999). [CrossRef]
  11. L. Salomon, F. Grillot, A.V. Zayats, F. de Fornel, ???Near-field distribution of optical transmission of periodic subwavelength holes in a metal film,??? Phys. Rev. Lett. 86, 1110???1113 (2001). [CrossRef] [PubMed]
  12. L. Martin-Moreno, F.J. Gar??ýa-Vidal, H.J. Lezec, K.M. Pellerin, T. Thio, J.B. Pendry, and T.W. Ebbesen, ???Theory of extraordinary optical transmission through subwavelength hole arrays,??? Phys. Rev. Lett. 86, 1114???1117 (2001). [CrossRef] [PubMed]
  13. U. Schröter and D. Heitmann, ???Surface-plasmon-enhanced transmission through metallic gratings,??? Phys. Rev. B 58, 15419???15422 (1998). [CrossRef]
  14. J.A. Porto, F.J. Gar??ýa-Vidal, and J.B. Pendry, ???Transmission resonances on metallic gratings with very narrow slits,??? Phys. Rev. Lett. 83, 2845???2848 (1999). [CrossRef]
  15. E. Popov, M. Nevière, S. Enoch, and R. Reinisch, ???Theory of light transmission through subwavelength periodic hole arrays,??? Phys. Rev. B 62, 16100???16108 (2000). [CrossRef]
  16. A.V. Zayats, L. Salomon, and F. de Fornel, ???How light gets through periodically nanostructured metal films: a role of surface polaritonic crystals,??? J. Microsc. 210, 344???349 (2003). [CrossRef] [PubMed]
  17. N. Bonod, S. Enoch, P.F. Li, E. Popov, and M. Nevière, ???Resonant optical transmission through thin metallic films with and without holes,??? Opt. Express 11, 482???490 (2003) <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-5-482">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-5-482</a>. [CrossRef] [PubMed]
  18. A.M. Dykhne, A.K. Sarychev, and V.M. Shalaev, ???Resonant transmittance through metal films with fabricated and light-induced modulation,??? Phys. Rev. B 67, 195402 (2003). [CrossRef]
  19. D. Gérard, L. Salomon, F. de Fornel, and A.V. Zayats, ???Ridge-enhanced optical transmission through a continuous metal film,??? Phys. Rev. B 69, 113405 (2004). [CrossRef]
  20. S.A. Darmanyan, M. Nevière, and A.V. Zayats, ???Analytical theory of optical transmission through periodically structured metal films via tunnel-coupled surface polariton modes,??? Phys. Rev. B 70, 15AUG (2004). [CrossRef]
  21. M. Nevière and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design, Marcel Dekker, New-York, 2003.
  22. E. Popov, M. Nevière, ???Differential theory for diffraction gratings: a new formulation for TM polarization with rapid convergence,??? Opt. Lett. 25, 598???600 (2000). [CrossRef]
  23. A.V. Krasavin, N.I. Zheludev, and A.V. Zayats, ???High-contrast modulation of light with light by control of surface plasmon-polariton wave coupling,??? to be published.

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