Photonic crystals can afford us complete control over light propagation. When the dielectric constant of the materials in the crystals are different enough and the absorption of light is minimal [1–3], this can provide very interesting phenomena. We can cite negative refraction [4], gaps [5], or the presence of flat bands in the dispersion diagram leading to very small values of group velocity [6]. Such slow light modes are usually used to enhance non-linear responses through the confinement of light induced by the energy storage. This phenomenon, also called Purcell effect (see ref. [7] and references therein), can be understood as an increase of the light-matter interaction delay. Several studies were based upon these slow light modes for 2D photonic crystals [8]. In most of them, the propagation direction is in the plane of the crystal i.e. perpendicular to the axis of the pattern (air or dielectric rod). In several other studies [9], off-plane propagation is considered and it corresponds to propagation along the axis of the holes. In that case, the 2D crystal is illuminated toward its thickness that becomes finite. Thus, no light propagation takes place along the x and y directions. According to this, the enhanced optical transmission (EOT) through metallic plates perforated with aperture arrays can be considered as a metallic photonic crystals used at the Γ point of the dispersion diagram. One can imagine to take advantage of EOT to conceive new functionalities.

In this paper, we consider annular aperture arrays (AAA) [10] embedded on a lithium niobate substrate to design a new device for the enhancement of the SHG by combining EOT with the propagation of a slow light mode. Recently, an important SHG enhancement in a GaAs photonic crystal was reported [11]; in that paper, there is a comparison between a nanostructured GaAs and a z-cut LiNbO ₃ bulk substrate. Thus, our configuration consists of filling a sub wavelength AAA, made in silver, by this dielectric material. This structure is embedded on a substrate made of the same dielectric (LiNbO ₃ substrate). Our choice of LiNbO ₃ is due to our aim to develop different integrated optical functions (i.e. electro-optical modulators, non-linear devices etc.) so we can conceive multifunctional micro-devices. Moreover lithium niobate is the most used material in telecommunication applications. The proposed device is schematically illustrated in Fig. 1.

When illuminated at normal incidence with a linearly polarized plane wave, the AAA structure exhibits EOT which is due to the excitation and the propagation of a guided mode inside the annular cavities [12]. This mode, named the TE₁₁-like mode, has a cut-off frequency for which the EOT occurs. At this cut-off wavelength, the real part of the effective index of the guided mode tends to zero so that the phase velocity becomes very large (see Fig. 4 in [14]). Consequently, for a finite-thick metallic layer of AAA this mode corresponds to a slow light mode (frozen mode) and thus a light confinement is obtained. In this work we propose this frozen mode to greatly enhance the SHG.

 

Fig. 1. Schematic view of the proposed structure. The period is defined by p. R_o and R_i are the outer and the inner radii respectively while h is the thickness of the metallic film. Note that the substrate material exists also inside the apertures.

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Light confinement associated with a high nonlinear coefficient χ ⁽²⁾ are necessary to generate an efficient SH signal. In order to be detected in the far-field, this SH field should be transmitted outside the cavities therefore it must have a second peak of transmission that should appear in the transmission spectrum of the AAA structure. Thus, the transmission spectrum must exhibit two peaks for both the fundamental and the SH wavelengths.

Therefore, our problem consists on finding a suitable design of an AAA structure with a specific transmission spectrum. In a recent work, Baida et al. [13] demonstrated that the position of the main transmission peak located at the cut-off wavelength of the TE 11-like mode almost depends on the geometrical parameters of each annular aperture. In a first approximation, we consider that it is independent of the metal thickness (λ TE11 c = f(Ro,Ri) where Ro and Ri are outer and inner radii of the aperture). Whereas its harmonics depends on the metal thickness (λ = f(h) where h is the thickness of the metal). In order to have an idea of these dependencies, we first calculate the cut-off wavelength of an infinite waveguide for different couples (Ro,Ri). Ri values are comprised between 50 nm and 85 nm nm and Ro varies between 120 nm and 145 nm. As already mentioned in ref. [14], Fig. 2(a) shows, contrarily to the case of perfect metal, that the cut-off wavelength of the TE 11-like mode for the silver coaxial waveguide is not linear versus the radii values. It rises up by increasing the outer radius and reaches its maximum when the radii are close to each other. The white line of Fig. 2(a) corresponds to the couples (Ro,Ri) in which the cut-off wavelength is equal to 1550 nm. In 2(b) we present the location dependence of the first peak transmission as a function of the inner and the outer radii for a finite structure. The thicknes metal is arbitrarily fixed to 90 nm. Ro and Ri were adjusted from Fig. 2(b) so that fabrication constraints are taken into account. Indeed, due to fabrication restrictions, we should maximize as much as possible the difference between the two radii Ro − Ri. Finally, the chosen values were Ro = 135 nm and Ri = 65 nm. In addition, the period of the AAA structure, p = 300 nm, has been chosen in order to eliminate the Surface Plasmon (SP) resonance near the transmission peak and thus avoiding its negative role on the transmission through the guided modes [15] for both the fundamental and the SH signals.

The numerical study is done with a 3D-FDTD homemade code, using a non uniform meshing. Metal dispersion is taken into account by introducing the Drude model in the FDTD algorithm ¹.

 

Fig. 2. (a) Trace of the cut-off wavelength of the guided mode as a function of inner and outer radii in nm. The white curve gives the different couples (Ro, Ri) where λ TE11 c = 1550 nm. (b) Dependence of the first peak location for a finite thickness as a function of the structure parameters. The metal thickness is arbitrarily fixed to 90 nm.

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Previously, we remarked that the position of the first harmonic of the TE ₁₁ guided mode depends on the metal thickness. Therefore we study the transmission spectra as a function of thickness. The R_o and R_i radii values selected before are respected. Figure 3 shows the transmission evolution for the MDPhC structure as a function of the incident wavelength for film thicknesses varying between 70 nm and 170 nm. As expected, the first harmonic peak (see Fig. 3) strongly depends on the metal thickness. For thicknesses values smaller than 100 nm, this later remains uneven. This is due to the fact that Fabry-Pérot phase matching cannot be obtained for small thicknesses values.

 

Fig. 3. Zero-order transmission spectra through AAA made in silver and filled by LiNbO ₃ deposited on a LiNbO ₃ substrate (n_subs = 2.143, p = 300 nm,R_o = 135 nm,R_i = 65 nm). The thickness of the film, denoted by h, varies between 70 nm and 170 nm.

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This resonance peak shifts to a greater wavelength (red shift) as thickness increases while the position of the cut-off wavelength remains almost constant. As a consequence, we opt to choose the metal thickness h = 120 nm inducing a first harmonic located at 775 nm. It should be noted that the transmission presented in Fig. 3 is normalized by the same transmission value calculated throughout the LiNbO ₃ bulk substrate where we replaced the metallic film by the dielectric.

The local field factor [16], that gives the field enhancement and the SHG efficiency, is directly linked to the group velocity associated to the light inside the structure :

It can also be expressed by the ratio of the average of the electric field over all the structure divided by the same quantity calculated without the metallic film :

Thus, as in [8], to evaluate this field factor, we will use these two expressions.

As previously stated, the period of the AAA structure is fixed so that no surface plasmon occurs in the spectral range between the two considered wavelengths (λ_SHG = 775 nm and λ_fund = 1550 nm). Consequently, we can consider that apertures act independently between them because there is no coupling via the two metallic interfaces. As a consequence, the effective index of the guided mode is equal to the one of a single coaxial waveguide. That is why a specific N-Order Body-Of-Revolution FDTD code (BOR-FDTD) [14] is used to determine the dispersion diagram for such a waveguide made in silver. The BOR-FDTD method is based on the discretization of the Maxwell equation when expressed in cylindrical coordinates. It is used to simulate axially symmetrical structure. It is considered as a 2.5 − D code as it allows the passage from a 3D to a 2D meshing. The azimuthal dependence is analytical and only axial and radial coordinates are discretized. Figure 4 illustrates the dispersion curves of the infinite coaxial waveguide.

In this manner, the group velocity can be deduced from the slope of the dispersion curve at a defined k ₀. The cut-off of the TE ₁₁-like mode corresponds to k_z = 0, we get ν_g = c/37.4074 from Fig. 4.

 

Fig. 4. Dispersion curves of the first modes in the case of a single infinite silver coaxial waveguide. The interspace media is LiNbO ₃ and the radius values are R_o = 135 nm,R_i = 65 nm.

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The group velocity in the bulk is given by ν_g = c/n_g, where n^g is the refractive index of the lithium niobate at λ = 1550 nm (n_g = 2.143) therefore we deduce that f ² = 37.4074/2.143 ≃ 17.45.

On the other hand, a BOR-FDTD homemade code is used to determine the light distribution inside one annular aperture when illuminated under normal incidence. The geometrical parameters are kept the same (R_o = 135 nm,R_i = 65 nm and h = 120 nm) and a non-uniform meshing is applied to faithfully describe the structure. The spatial steps vary from δ r = δ z = 2nm inside the coaxial apertures and increases continuously to attain Δ r = Δ z = 5nm up from the outside edge. A linearly polarized (along the x direction) pulsed plane wave impinges on the structure from the bottom of the substrate (see Fig. 1). This allows us to determine the response of the structure over a large spectral domain (λ ∈ [600; 2000] nm). The calculation is evaluated over 35000 times iterations and over 1800 × 500 spatial points (r = 16μm and z = 3μm). Since the calculations are made on one single aperture, the result should be multiplied by the filling factor $\left(F=\frac{\pi \left(R_o^2-R_i^2\right)}{p^2}=0.49\right)$.

 

Fig. 5. Zero-ordered transmission spectrum of the AAA silver film (blue solid line). The field factor for the same annular aperture arrays (green solid line) is compared to the one of the cylindrical hole arrays (green dashed line).

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Transmission (in blue) and field factor (in green) spectra through the AAA structure are presented in Fig. 5. One notices that the maximum of field factor is localized at the cut-off wavelength of the TE ₁₁ guided mode and is equal to τ = 17.7 which is in very good agreement with the value of the local field factor deduced from the dispersion diagram. As expected, we observe that the second peak is always at the half of the initial wavelength λ = 775 nm and it corresponds to the position of the Second Harmonic (SH) signal. Besides, the field factor spectra of a cylindrical aperture array made in silver and filled by LiNbO ₃ is also presented in the same Fig. 5. The geometrical parameters of this structure were also chosen in order to get a transmission peak at λ = 1550 nm (R = 225 nm,h = 200 nm and P = 550 nm). These parameters are obtained following the same chronological steps done for the annular aperture array. In this case, the field factor at this wavelength is τ ≃ 1. Note that in this paper we have chosen our parameters so that the structure can be realistically fabricated, and therefore we did not optimize them in order to have maximum of field factor at the TE ₁₁ cut-off wavelength.

In view of the fact that the maximum of field enhancement is located at the cut-off wavelength, we display in Fig. 6 the light distribution (the fifth root of the square modulus of the electric field) inside and nearby the aperture. The results are also attained using a BOR-FDTD code at a fundamental pump wavelength of 1550 nm, Fig. 6(a), and at the SH wavelength 775 nm Fig. 6(b). We can see that, at the fundamental wavelength, the intensity remains constant while one node appears in the middle of the cavity for the SH wavelength. It is clearly shown that light is essentially confined between the two metallic interfaces. This is certainly due to the excitation of a plasmonic mode inside the cavities.

 

Fig. 6. The fifth root of the calculated intensity distribution in cross section of an AAA in a 120 nm silver thick filled by lithium niobate for the first λ = 1550 nm (a) and its first harmonic λ = 775 nm(b). R_o = 135 nm, R_i = 65 nm and h = 120 nm.

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Fig. 7. Amplitude distributions in the xOy plane of the three cartesian components of the electric field inside a single annular aperture at the fundamental wavelength (λ= 1550 nm). The arrows indicate the direction of the incident electric field associated with the plane wave impinging the structure from the lithium niobate side.

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Figure 7 shows the spatial distributions of the three cartesian components of the electric field normalized with respect to the x components. The x polarized plane wave at 1550 nm wavelength is incident from the lithium niobate side. The horizontal sections are at the middle of the cavity. By examining the intensity of the three components, we can see that E_x dominates inside the cavity, E_y is two times smaller than E_x and E_z is almost zero. Moreover, by referring to lithium niobate second-order nonlinear tensor, we get that the greater non-linear coefficient is d ₃₃ which means that the optical electric field should be oriented to the crystal axis direction. In addition, it is important to notice that metal also possesses superficial non-linear susceptibility which can arise the second harmonic signal [17,18].

In summary, the properties of the AAA structures are used for SHG. For a given metal, they allow the design of a transmission spectrum with two peaks located at the pump and the SH frequencies by adjusting independently the aperture radii and the metal thickness. The EOT and the non-linear material, LiNbO ₃, successfully improve the enhancement factor of the SHG. The results show that the conversion of 1550 nm excitation light into 775 nm emission is theoretically enhanced by a factor 17 with respect to bulk LiNbO ₃. This value is independently estimated by two different numerical methods. The near field and the field distribution inside the cavities confirm this enhancement. An experimental realization of this structure is in progress in order to confirm our theoretical calculations, e-beam and ICP-RIE technologies will be used [19–21].

This work is partly supported by the project MAGNETO-PHOT n^o BLAN06-2-135594 funded by the ANR (Agence National de la Recherche).