1. Introduction

In the last decade, silicon-based multilayered structures have been intensively studied in the framework of all-silicon optoelectronics. It has been shown that microcavities (μCs) entirely based on hydrogenated amorphous Silicon Nitride (a-Si_1-xN_x:H) possess appealing optical properties such as resonant enhancement of photoluminescence [1].

In the present paper we investigate the second-order nonlinear properties of planar a-Si_1-xN_x:H Fabry-Pérot μCs. In such a centrosymmetric material, Second Harmonic Generation (SHG) can take place due to either surface nonlinearity or bulk quadrupolar contributions. As suggested in recently published works [2, 3, 4], the second-order nonlinearity in amorphous Silicon based homogeneous thin films and μCs could be mainly imputed to interfacial nonlin-earity. In this work we consider the dependance of SHG phenomena on the surrounding media geometry. In particular we compare two multilayer structures constituted by the same materials and differing only by the spatial ordering of layers. Although the two microcavities have similar linear response behavior, remarkable differences in the efficiency of the SH nonlinear conversion are observed. The explication of this effect is connected to the field distributions of both the pump and the SH radiation inside the structure. Such field distributions cannot be directly accessed by means of standard far-field optical characterization. Nevertheless, we can show direct Near-field measurements of the pump field distribution performed by means of an interferometric Scanning Near-field Optical Microscope (SNOM) raster scanning the cleaved facet of a planar μC. We finally interpret experimental far-field and near-field data on the basis of a theoretical model and propose a consistent explication of the observed effect.

2. Far-field characterization

Multilayered planar structures were grown by Plasma Enhanced Chemical Vapor Deposition (PECVD) on crystalline Silicon wafer. The composition of the a-Si_1-xN_x:H layers was controlled by operating on the ammonia fraction present in a SiH₄+NH₃ plasma. Refractive index was estimated by means of standard optical measurements [5], while their thickness was determined by taking into account the deposition rate previously fixed on homogeneous films. The total gas flux and pressure, substrate temperature and electrode distance were set at 75 sccm, 0.35 Torr, 200° and 20 mm respectively.

The Fabry-Pérot μCs considered here are composed of two specular 6-period Distributed Bragg Reflector (DBR) surrounding a half-wavelength spacer at each side. A pair of a-Si_1-xN_x:H quarter-wavelength layers having different refractive index constitutes each DBR elementary cell.

With the help of standard linear Transfer Matrix method, two μCs layouts were engineered in such a way that they present almost identical stop band, resonance frequency and Q-factor. In one configuration, the spacer (n_s=1.91) is surrounded by a couple of low-refractive index layers (Si₃N₄, n_L=1.76), while in the alternative configuration, the spacer is surrounded by a couple of high-refractive index layers (Si_rich, n_H=2.23). We refer to these two layouts as μC(A) and μC(B) respectively. In Fig. 1(a) and Fig. 1(b) the spatial profiles of refractive index in the two μCs are shown. Indeed, the first stratified configuration differs from the second one by a simple swap in the refractive index and thickness of DBRs elementary cells.

 

Fig. 1. One dimensional refractive index profile n(z) of the two cavity configurations: (a) μC(A), (b) μC(B).

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In Fig. 2 reflectance spectra measured at an incidence angle θ= 40° are shown. As expected, the two μCs present the same resonant behavior. They are both characterized by a stop band running from 1350 nm to 1650 nm and resonance wavelength λ_r = 1505 nm. The analysis of experimental spectra reveals a cavity quality factor Q ≃ 100.

Reflected SH signals were measured by tuning the optical frequency of the pump beam (ω) around the cavity resonance value, thus considering a SH (2ω) spectral interval from 680 nm to 780 nm.

The idler beam of a parametric oscillator pumped by a frequency tripled, Q-switched Nd:YAG laser provided sample excitation. The polarization of the pump beam was controlled via a double Fresnel rhomb. Detection was performed by means of a monochromator equipped with a photomultiplier. An analyzer placed in front of the monochromator allows the selection of the SH polarization state.

We first considered a P-P polarization scheme. In Fig. 3 the reflected SH signal is plotted for both μC(A) and μC(B). Incidence angle of the pump beam is θ= 40°.

 

Fig. 2. Measured reflectance of μC(A) (gray squares) and μC(B) (black circles). Illuminating light is P-polarized and incident at θ = 40°.

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Fig. 3. Measured reflected SH spectra referred to μC(A) (gray squares) and μC(B) (black circles). Calculated spectra: μC(A) (black solid line) and μC(B) (gray dashed line). Pump beam is P-polarized and incident at θ = 40°.

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The resonant behavior of both cavities can be recognized as the maximum of SHG corresponds to a pump wavelength λ = λ_r. In addition, we observe that μC(B) presents a sensibly higher SH signal as compared to μC(A). We quantify this difference by defining the following enhancement factor ${\eta }_{\mathrm{meas}}^P=\frac{I_{\mu }^P\left(B\right)\left(2\omega \right)}{I_{\mu }^P\left(A\right)\left(2\omega \right)}\simeq 9.$

The study of SHG in a-Si_1-xN_x:H microcavities illuminated with P-polarized radiation is rather complicated because it depends on three non-zero components of the nonlinear tensor χ⁽²⁾ [6]. Therefore, we simplify the problem by considering SHG in a S-P configuration. In this case, SHG is due only to a single non-zero χ⁽²⁾ tensor element, as later discussed.

 

Fig. 4. Measured reflected SH spectra referred to μC(A) (gray squares) and μC(B) (black circles). Calculated spectra: μC(A) (black solid line) and μC(B) (gray dashed line). Pump beam is S-polarized and incident at θ = 40°.

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Results on SH reflectance measurements are shown in Fig. 4. Although in S-P configuration SH signals are one order of magnitude weaker as compared to the P-P case [6], the same spectral features of both μCs are found, leading to the following enhancement factor:${\eta }_{\mathrm{meas}}^S=\frac{I_{\mu \left(B\right)}^S\left(2\omega \right)}{I_{\mu \left(A\right)}^S\left(2\omega \right)}\simeq 14.$ This experimental finding highlights a remarkable and unexpected difference in the nonlinear optical response of the two cavities. Since linear reflectance measurements cannot give an account of this effect, a direct knowledge of the field distribution inside the multilayer at ω and 2ω would be highly desirable. One of the most suitable tools for investigating (complex) optical field distribution on microstructured surfaces is represented by the Scanning Near-field Optical Microscope (SNOM) equipped with an heterodyne detection setup, as will be shown in the next section.

3. Near-field characterization

The experimental setup employed for the Near-field characterization is based on an apertureless heterodyne SNOM whose working scheme has been described in detail elsewhere [7, 8].

We focused our attention to μC(B). The sample was prepared by cleaving the planar μC along the cavity axis. The sample is vertically positioned on a 3-axis piezo table and raster scanned beneath an oscillating Atomic Force Microscopy (AFM) tip (Fig. 5). Sample illumination is provided by means of a properly designed optical fibre hosting a polymeric micro-lens directly grown at the very end of the fibre core [9]. Such a polymeric tip allows tight light focusing [10], thus minimizing unwanted scattering effects from the μC edge. The fibre is mounted on a 3-axis piezo-actuated linear stage and placed on the scanner table. Such a setup is particularly suited for Near-field measurements requiring a very localized illumination system solidary to the scanned sample.

Near-field intensity distribution have been collected while topographically scanning the cleaved section of the planar cavity with the AFM tip in intermittent contact mode. In order to reduce unwanted radiative contributions to the NF measurements, we tilted the optical fibre and managed to obtain a Total Internal Reflection condition on the scanned plane of the measured (b) and calculated (c) field intensity distribution in the z-direction. Measured intensity profile is obtained by averaging the collected field distribution over the dashed region. multilayer. In Fig. 6(a) we show the measured 2D intensity distribution referred to μC(B) illuminated with a pump beam at the cavity resonance wavelength. If we consider the z-profile of the intensity distribution (Fig. 6(b)) a good matching of measured data with the total electric energy distribution expected from calculations is found (Fig. 6(c)). As theoretically expected, the observed field localization at the cavity spacer and neighboring DBR interfaces becomes less evident as the illumination wavelength is shifted outside the resonance region (Fig. 7).

 

Fig. 5. Schematic of the apertureless SNOM setup employed for near-field mapping of μC(B).

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Fig. 6. (a). Measured intensity near-field distribution in μC(B) at resonance wavelength. 1D plot of the measured (b) and calculated (c) field intensity distribution in the z-direction. Measured intensity profile is obtained by averaging the collected field distribution over the dashed region.

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Fig. 7. (a). Measured off-resonance intensity near-field distribution in μC(B). 1D plot of the measured (b) and calculated (c) field intensity distribution in the z-direction. Measured intensity profile is obtained by averaging the collected field distribution over the dashed region.

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4. Discussion

The experimental results showed that μC(B) yields a SH signal ~ 10 times larger than μC(A) in both S-P and P-P configurations. In order to understand the physical origin of this effect, we calculated the nonlinear reflectance of the two microcavities by means of a nonlinear transfer matrix method suitable modified in order to describe surface effect [11, 12]. In fact, since we are dealing with amorphous materials, the second-order nonlinearity arises mainly from a surface contribution at layer interfaces, where the material symmetry is broken [14]. Satisfactory agreement with the experimental data is show in Fig. 3 and Fig. 4.

Beside the numerical calculation, we proposed a simple argument which directly links the pump and harmonic field distribution to the nonlinear response. Indeed, the large difference in SH signal intensity can be ascribed to the different distributions of electromagnetic field in the two structures.

We recall that in a one-dimensional system characterized by a second-order bulk nonlinearity χ⁽²⁾ _bulk(z) , an effective χ⁽²⁾ _eff tensor [15] can be defined:

where E² _ω(z) and E_2ω(z) represent complex electric fields associated to ω and 2ω radiation.

Here we intend to introduce an effective second-order tensor in the case of interface non-linearity χ⁽²⁾ _interf. For sake of simplicity, the entire model will refer to the S-P experimental configuration only. In this case, the term χ⁽²⁾ _∥∥⊥(z) represents the only contribution to SHG [6]. Therefore, if we neglect bulk quadrupolar contributions, we can set:

where the summation is extended over all the N interfaces placed at z_i, and characterized by χ⁽²⁾ _∥∥⊥,i. By using Eq. 2 in Eq. 1

which represents the effective nonlinear tensor. According to this simple model, the overall second order polarizability depends on the amplitude and phase distributions of electric fields E_ω(Z_i), E2_ω(Z_i) inside the multilayers and the value of χ⁽²⁾ _∥∥⊥,i at each layer interface [12, 13]. Since it has been demonstrated that SH intensity is ∝ |χ⁽²⁾ _eff|² [15],we can define the parameter η^S _calcas

and use it to compare SH intensities in the two cavities. A numerical evaluation of χ⁽²⁾ _eff for ,μC(A) and μC(B) can be performed by rigorously calculating the E_ω(z) and E2_ω,(z) distributions within the two multilayered structures. Values of χ⁽²⁾ _∥∥⊥,i are best-fit parameters used to match measured SH spectra with spectra calculated by means of a non-linear transfer matrix method (see Fig. 4). For an incident angle θ = 40°, calculations provide η^S _calc ≃ 11 which is in quite good agreement with the experimental η^S _meas. Thus the theoretical model proposed above suggests that the observed enhanced SHG emission is resulting from a collective complex response of the whole structure based on the phase matching together with a strong increase of the pump intensity at layer interfaces. The use of a χ⁽²⁾ _eff-based model provides a compact and unique description of the observed SHG enhancement as resulting from a non-trivial relationship among complex quantities.

 

Fig. 8. Calculated intensity distribution of the electric fields E_ω (a) and E_2ω (b) inside ,μC(A).

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In order to give an idea of the different field distribution in the two cavities, we report in Fig. 8 and Fig. 9 the light intensities |E_ω|² and |E_2ω|² for μC(A) and μC(B), respectively. Calculations have been performed by means of RCWA [16] method. As indicated by the arrows, the field E_ω is mostly localized at layer interfaces and in μC(B) it is almost twice as intense as in μC(A) because of the different refractive index contrast. On the other hand, the E_2ω field reveals only slight intensity differences in the two structures.

5. Conclusion

Experimental far-field measurements presented in this paper show that a-Si_1-xN_x:H Fabry-Pérot μCs can possess very different SHG properties even though characterized by almost identical linear reflectance spectra. We reported on a particular case in which two μCs exhibit SH intensity difference as large as one order of magnitude for both S-P and P-P configurations. This difference cannot be detected by means of linear reflectance measurements and depends on the intimate phase/amplitude relationship among the electric field distribution in the structure at ω and 2ω. Moreover, we experimentally demonstrate that optical near-field mapping can give useful insights on the spatial localization of the pump field E_ω in the multilayer.

 

Fig. 9. Calculated intensity distribution of the electric fields E_ω (a) and E_2ω (b) inside μC(B).

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With the help of a theoretical model we explain the experimental finding by accurately considering SHG as resulting from the whole linear and nonlinear response of the structure. In addition, we demonstrated that SHG phenomena due to surface nonlinearities in complex geometries can be described by the parameter χ⁽²⁾ _eff as defined in the text.

This work has been partially supported by the Italian National Project PRIN “Silicon-based photonic crystals“ (2004) and by the Piedmont Regional Project “Nanostructures for applied photonics“ (2004).