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
Surface plasmon polaritonic (SPP) crystals formed on metal film interfaces have recently attracted much attention [1, 2, 3, 4]. Periodically nanostructured metal surfaces exhibit the optical properties similar to those of two-dimensional photonic crystals and can be used to guide and direct SPP waves in linear or bent defects of the structure . Moreover, due to intrinsically two-dimensional nature of SPPs and the electromagnetic field enhancement associated with them, the nonlinear optical effects can be achieved at such structures at much smaller light intensities than in conventional photonic crystals [1, 5]. In most cases the behaviour of SPPs in a periodic structure and the SPP Bloch modes were studied on the surface of a semi-infinite metal [3, 4, 6]. In the case of a thin metal film, the interaction between the SPP modes on the opposite interfaces leads to additional effects related to the film SPP modes formation that results in the modification of the SPP spectra . If, in addition, one of or both film interfaces periodically structured, the film SPP modes and SPP Bloch mode formation must be simultaneously taken into account.
The understanding of the SPP states on a finite thickness periodically structured metal film is also extremely important from the point of view of the conventional optical properties of metallic structures such as reflection, absorption, and transmission. The optical properties of periodic arrays of apertures (holes or slits) in a metal film have been intensively studied during last years [8, 9, 10, 11, 12]. It was shown that a periodical arrangement of subwavelength apertures enables suitably polarised light to be efficiently transmitted through the film. In addition to the aperture-related transmission which is important under the certain conditions [13, 14, 15], the resonant tunnelling between surface plasmon polaritons states on the film interfaces has been dubbed a general mechanism of optical transmission through a periodic metallic structure [1, 16]. Thus, continuous (without apertures) metal films can also exhibit the enhanced optical transmission [6, 17, 18, 19, 20]. The enhancement is observed providing that some kind of periodicity exists on a surface and/or in a metal film. Since in this case there are no apertures in the film, the light transmission takes place only due to the resonant tunnelling of photons through the metal film via the SPP Bloch modes which can be excited on one of or both film interfaces. At normal incidence, however, only the states of the lower branches of SPP Bloch modes at the edges of even band-gaps in the SPP spectra on a grating can be optically addressed and contribute to the transmission [6, 19]. This completely hides the effects associated with other SPP states which are inactive at normal incidence of light. The consideration of the oblique incidence is needed to investigate a role of all possible SPP states in the optical properties of a periodically structured 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
We have studied a structure (Fig. 1) consisting of a silver film of the thickness H with a periodic set of rectangular ridges (D is the grating period, d is the ridge width, and h is the ridge height). Since there are no apertures in a film, all effects related to one or another mechanism of the light transmission directly through the slits are absent, and the resonant tunnelling through a metal film is the only mechanism responsible for the enhanced transmission.
Numerical modelling was performed using differential method with the S-algorithm taking into account the Li remarks [21, 22]. 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 . 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 . 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.
In the case of a periodically structured film, the weak coupling regime can be considered if the film thickness is so large that the interaction between the SPP modes on the opposite film interfaces results in the smaller frequency shift than the band-gap width which is determined by the periodic structure (Fig. 2(a)). In the strong coupling regime (Fig. 2(b)), the SPP frequency shift due to the interaction through a thin film is stronger than the band-gap effects leading to the crossing of the Bloch mode branches which are related to the different film SPP modes. There is no anti-crossing in this case since the symmetries of the film Bloch modes are different.
3.1. Weak-coupling regime
In the weak-coupling regime, only one reflection, absorption, and transmission peak is observed at normal incidence in the visible spectral range for a given parameters of the periodic structure (Fig. 3) [6, 19]. This is related to the fact that this spectral range only covers the wavelengths around the second SPP band-gap where two SPP Bloch modes exist but the upper branch of the SPP Bloch mode near the even band-gaps is nonradiative and cannot interact with photons. Therefore, only lower SPP Bloch mode can be excited optically and participates in the light tunnelling through the film that leads to the observed enhanced transmission. At the same time, the splitting of the SPP resonances due to the interaction of SPPs on different interfaces is small and cannot be observed due to the finite width of the SPP resonances. Under oblique illumination, the photons can efficiently interact with both branches of the SPP Bloch waves near the band-gap, and two peaks are observed in the spectra on both sides from the peak that was seen at normal incidence. This behaviour is in accordance with the dispersion of the SPP Bloch modes (Fig. 4, cf. Fig. 2(a)).
The same as in the case of normal incidence, at the wavelengths of resonant transmission, the reflection has minimum while the absorption has maximum. The difference in the transmission amplitudes and the peak width at the frequencies of lower and upper SPP Bloch modes at oblique incidence is mainly related to different densities of the SPP states and Ohmic losses at the respective wavelengths. This also explains why the difference in the transmission peak amplitudes associated with different Bloch modes increases with the angle of incidence (the spectra for large incidence angles are not shown).
The field distributions above the structure show that the strongest near-field is related to the edges of the ridges (Fig. 5). The near-field of the g +-mode is much stronger close to the surface and has a longer extension range in the dielectric compared to the g --mode. This g + SPP mode provides stronger far-field transmission (Fig. 3). For both modes the near-field transmission is significant above ridges as well as groves of the grating. The fields of both g + and g - SPP Bloch modes have the same symmetry with respect to the film plane. The phase-shift between different SPP Bloch mode branches across the interface is also seen that corresponds to the different Brillouin zones in the extended zone model. Please note that in contrast to normal incidence , when only the standing SPP Bloch modes exists, the field distributions presented here correspond to the propagating SPP modes on the structured interfaces.
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 .
For oblique incidence, the same as in a weak coupling regime, two additional peaks appear in the spectra due to the contribution of the upper branches of the SPP Bloch modes. However, due to mutual position of the band-gap edges associated with the two film modes (Fig. 7)-the top edge of the symmetric film SPP Bloch mode (f + g -) is positioned below the bottom edge of the antisymmetric film mode (f - g +)-the spectral features associated with the different film SPP modes are mixed. With the increase of the angle of incidence, f - g +-mode shifts to the longer-wavelength spectral range, while f + g --mode shifts to higher frequencies. So that at some angle of incidence the both frequencies coincide and crossing of the dispersion curves occurs (Fig. 7, cf. Fig. 2(b)). The angle of incidence at which the crossing takes place depends on the film thickness and the grating parameters. The crossing leads to the interplay between the two modes of different symmetries and opposite phases so that the outgoing fields are tend to cancel each other, and null far-field transmittance accompanied by the increased absorption is observed at the crossing wavelength . With the further increase of the angle of incidence, the states of the SPP Bloch modes become again separated and the related transmission resonances are observed.
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.
The crossing does not result in the additional band-gap formation (anti-crossing) since the symmetries of the involved SPP modes are different. In the case of a strong coupling regime, the film SPP Bloch modes have different symmetries associated with (i) the branches of the g + and g - SPP Bloch modes having different field distribution across the surface and (ii) the film SPP modes (f + and f -) with symmetric and antisymmetric field distribution in the film. Generally, g + and g - modes cannot co-exist at the same frequency and this is the origin of the band-gap in the SPP spectrum . However, different film SPP Bloch modes characterised by symmetries f - g + and f + g - can exist at the same frequency because the two Bloch modes are related to different film SPP modes. It should be noted that the lower frequency SPP Bloch mode branch (g +) has a negative refraction index (dω/dkSP<0) with the opposite group and phase velocities of SPP waves.
The similar SPP mode crossing can occur at kSP=0 at some film thickness . 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.
For the film SPP Bloch modes in different Brillouin zones, the film mode symmetry is super-imposed on the respective Bloch mode symmetries. This can be clearly seen from the near-field distributions in the case of a strong coupling when all 4 SPP modes are significantly separated (Fig. 10). The comparison of the near-field intensities at the resonant wavelengths with the far-field transmission shows the difference in the near-field and far-field transmittance related to the different strength of SPP coupling to photons. For example, the f + g + mode provides highest far-field transmission but the strongest near-field is associated with the f - g + mode probably due to smaller losses associated with this antisymmetric (in electric field) SPP mode. The symmetry of the near-field distributions corresponds to the symmetries of the film SPP Bloch modes in different Brillouin zones (Fig. 10): the magnetic field is symmetric with respect to the film plane for f --modes (antisymmetric Ex-field) and antisymmetric for f +-modes as should be expected, at the same time for both film SPP modes there is a phase difference between the g + and g - Bloch modes corresponding to different Brillouin zones similar to the case of a weak-coupling regime (Fig. 5).
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  or SPP propagation on a structured metal surface . Both passive and active optical elements of SPP optics based on SPP crystals can find numerous applications in all-optical photonics.
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.
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