1. Introduction

The ability to localise light within a photonic crystal defect and achieve a strong field enhancement in a small volume has intrigued researchers for many years and more sophisticated and optimized designs have emerged leading to higher Q-factors and lower mode volumes.

Until recently, the major emphasis was on microresonator modes in thin 2D photonic crystals. Here the defects consist of one or several “missing” holes in an otherwise periodically perforated high index slab [1–4]. Fine adjustment of the holes surrounding the cavities lead to Q-factors far above 10⁴ [5]. Recently it was discovered that similarly high Q-factors can also be obtained in cavities of nanobeam waveguides [6,7]. In these structures a line of pores forms a Bragg-mirror confining the defect mode along the propagation direction, while the lateral confinement is achieved by total internal reflection.

The high Q-factors and corresponding field enhancements of these cavities represent an ideal integrated optical test bed for nonlinear optical effects like wave mixing, harmonic generation, phase modulation, and opto-optical switching etc. For this, either the waveguide material itself has to exhibit nonlinear optical properties or other nonlinear optical materials (e.g. organic molecules, chalcogenide glasses etc.) have to be brought into close proximity to the cavity, so that they “see” the high cavity field. This latter very flexible approach results in hybrid cavities which rely on a high index material for the light confinement and waveguiding, while the nonlinear optical functionality is added by another medium. To achieve higher interaction between the light and the nonlinear material slotted waveguides can be used [8]. Here, a slot is cut into a ridge waveguide which then is guiding the light is mainly in the slot. When creating nonlinear devices this helps to avoid losses inside the silicon, e.g. from two-photon-absorption [9]. The light-matter-interaction is further enhanced by the fact that due to the discontinuity of the TE electric field, the electric field inside the slot is strongly enhanced by a factor of $\left(n_{Wg}^2/n_{\mathit{Slot}}^2\right)$ [10]. In combination with photonic heterostructure microcavities the field enhancement in the slot, and thus the enhanced sensitivity to refractive index changes, has been exploited to create photonic crystal based chemical sensors [11]. Other silicon waveguide geometries with permanent slot are investigated by Koos et al. [12–15]. Here, the slot is filled and the waveguide is surrounded with nonlinear material achieving high nonlinear coefficients. Infiltrated slot waveguides in photonic crystals are proposed in Refs. [16–18]. In this paper we present a new type of hybrid slot microcavity, where a slot is cut into the microcavity region of the nanobeam (Fig. 1). This slot is locally filled with an optical nonlinear material to create an optically active device [19]. The Q-factor of the infiltrated microcavity is investigated by means of 3D finite element method (FEM) simulations and rules for designing hybrid slot microcavities are denoted.

 

Fig. 1 Schematic of a photonic wire slot microcavity. The slot microcavity is surrounded by tapers consisting of four pores with linearly altered diameter followed by dielectric mirorrs with five pores.

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2. Influence of the slot width and refractive index on the effective mode index

The infiltrated nanobeam slot cavity can be interpreted as a section of an infiltrated slot-waveguide between two Bragg mirrors. To obtain a high Q-cavity it is first necessary that the effective waveguide index n_eff of the TE-polarized mode in the infiltrated slot section is larger than the cladding index n_SiO2 = 1.445 to allow bound modes (Fig. 2). With an infiltrated low index material (e.g. polymer, n_slot = 1.5) the waveguide index surpasses the cladding index only for very narrow slots with widths below 130 nm. Confining the mode further into a cavity by adding Bragg-mirrors along the propagation direction would then easily push the mode to higher frequencies and correspondingly low indices. This would result in leaky behaviour and low Q-values. In the following study we therefore concentrate on nonlinear materials with the higher refractive index of 2.3 (e.g. chalcogenide glasses like AsSe), which are only locally infiltrated into the slot. In this case the resulting n_eff is large enough that even further confinement will result in a bound cavity mode. For our further studies we fix the slot width at 100 nm. This represents a good balance between the tendency for better mode confinement and higher field enhancement (which come with more narrow slots) and the current nanofabrication limits governed by standard reactive ion etching and e-beam writing.

 

Fig. 2 Waveguide index n_eff at λ₀ = 1.55 μm for hybrid slot waveguides depending on the slot width. The dependency for two locally infiltrated materials is shown - one with n = 1.5 (e.g. polymer) and one with n = 2.3 (e.g. chalcogenide glass). Insets: Electric field distribution of the single TE-mode for different slot widths (100 nm and 175 nm). The material of the two rails of the waveguide is silicon. The upper halfspace is air with n_air = 1 while the lower halfspace is silicon dioxide with n_SiO2 = 1.445. The calculations have been performed with 2D FEM eigenfrequency analysis.

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In a first step an untapered slot microcavity was investigated. The structure consists of a 500 nm wide silicon (n = 3.479) beam with a height of 220 nm on a silicon oxide substrate (n = 1.445). The Bragg mirrors adjacent to the slot contain nine pores with a diameter of 200 nm. Their center-to-center distance is 385 nm. This waveguide and Bragg mirror geometry for TE-polarized light was derived from the design published by Zain et al. [20], which has led to high Q nanobeam cavities with cavity wavelengths around 1.55 μm. The 100 nm wide slot has circular terminations (Fig. 1) whose radius of curvature is 50 nm. These round endings can be interpreted as “inscribed” pores at the end of the slot. The center-to-center distance between the first adjacent mirror pore and this “inscribed” pore is also 385 nm.

A systematic theoretical investigation of the Q-factors of the TE-polarized third order cavity mode was carried out for different slot lengths based on finite element calculations with the commercial program COMSOL Multiphysics. The quality factors shown in this paper were calculated with the 3D mode solver. The convergence of the Q values was ensured by testing by using higher mesh resolutions in the waveguide and the slot. To alter the wavelength of the cavity mode, the length of the slot has been varied and a linear dependency between resonance wavelength and slot length [Fig. 4] was observed. Tuning the slot length to 1120 nm results in a third order resonance at a wavelength of 1.548 μm close to the design wavelength of 1.55 μm. The third order mode, which shows three intensity maxima in the slot (Fig. 3 a) presents a good compromise between a high Q factor of Q ≈ 6,400 and a small mode volume V = 0.017λ³ (V = 0.062 μm³). The volume of the cavity mode was calculated according to Ref. [21]. To check the Q-values of the resonances at 1.55 μm we independently extracted them from the transmission spectra calculated for harmonic propagating waves (see inset in Fig. 4 – 6).

 

Fig. 3 Distribution of the electric field amplitude confined in a slot microcavity, without taper (a) and with a combination of tapered pore distance and pore diameter (b).

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Fig. 4 Dependence of the cavity Q-factor on the slot length. The mode wavelength is shifted by varying the length of the slot (the red line indicates the design wavelength at 1.55 μm). Here, the cavity is surrounded by a dielectric mirror consisting of 9 pores.

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Fig. 5 a) Dependence of the cavity Q-factor on the slot length. The mode wavelength is shifted by varying the length of the slot (the red line indicates the design wavelength at 1.55 μm). The cavity is surrounded by a dielectric mirror consisting of 5 pores and an inner taper consisting of 4 pores with adapted center-to-center distance to match effective cavity mode index and the Bloch index of the mirror mode. b) Cross section of the electric field intensity distribution of the resonant mode for a slot length of 1080 nm at the middle of the slot (top) and at the middle of the adjacent taper pore (bottom).

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Fig. 6 a) Dependence of the cavity Q-factor on the slot length. The mode wavelength is shifted by varying the length of the slot (the red line indicates the design wavelength at 1.55 μm). The cavity is surrounded by a dielectric mirror consisting of 5 pores and an inner taper consisting of 4 pores with adapted center-to-center distance and linearly varied pore diameter to achieve mode matching between cavity mode and mirror Bloch mode. b) Cross section of the electric field intensity distribution of the resonant mode for a slot length of 960 nm at the middle of the slot (top) and at the middle of the adjacent taper pore (bottom).

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Figure 4 shows that the Q-factor peaks at a resonance wavelength of 1.52 μm for a slot length of 1060 nm. For longer wavelengths the cavity modes approach the lower band edge of the Bragg mirror and the Bragg reflectivity drops leading to losses due to transmission through the Bragg mirrors. At shorter wavelength the losses due to lateral scattering into the substrate and the surrounding air increase. Overall this leads to a maximum Q-factor at a wavelength which lies between the photonic bandgap center of the Bragg mirror and its lower photonic band edge. This fact agrees well with the study of Bragg mirror reflectivites and scattering losses in nanobeams performed by Lalanne et al. [22].

3. Tapering of pore diameter and pore distance for Q factor enhancement

A possible explanation for the reduced quality factor at the design wavelength is the modal mismatch of the electric fields at the interface between slot cavity and Bragg mirrors. As shown in Ref. [23], scattering losses at a defect-mirror-interface can be reduced by gradually matching the electric field distribution of the cavity mode and the Bloch mode of the Bragg mirror. Normally, the proper parameters are obtained by successive variation of all taper parameters resulting in extensive calculations.

However we follow a simple design rule here, which was given earlier by Lalanne et al. [24]. It says that the effective refractive indices of the cavity waveguide mode (n_eff) and the Bloch mode of the Bragg mirror (n_B = λ₀/2a) should match (n_eff = n_B). This corresponds to the demand k_eff = Re(k_B), where k_eff is the wave vector of the cavity waveguide mode and Re(k_B) = π/a is the real part of the complex Bloch wave vector for frequencies within the photonic bandgap. This has been considered and a tapered geometry was assumed, where the distance between the four inner mirror pores has been varied to create a linear transition of the effective index from n_B = 2.013 to n_eff = 1.929 (Table 1). The resulting Q-values are shown in Fig. 5. The maximum Q-value appears now for a resonance at the design wavelength so that a modest enhancement of its Q-factor from 6,400 to Q ≈ 8,000 is observed, while the mode volume is unchanged. This relatively moderate increase can be explained by the fact that the difference between n_eff of the slot and n_B of the mirror is rather small, hence introducing a pore distance taper will not have a great impact. To discover the cause of further losses limiting the Q-factor, the electric field distribution at different xy-cross sections of the waveguide is plotted (Fig. 5). While the field is tightly concentrated within the center of the slot in the cavity, it is more spread out in the center of the first pore adjacent to the cavity. This means that the field of the cavity mode experiences an abrupt change in shape and not a smooth adiabatic transition. Several scattered waves are therefore excited at the cavity/Bragg mirror interface. To achieve an adiabatic mode transition and minimize scattering we create a smooth linear transition of the pore diameter from the 200 nm wide Bragg mirror pores to the 100 nm diameter of the slot. This “diameter taper” is performed in addition to the already existing “period taper” over the 4 inner pores on each side of the cavity. This combined “period-diameter taper” has indeed a profound impact on the Q-factor reaching a maximum of 35,000 at λ₀ = 1.55 μm for a slot with a length of 960 nm. The mode volume V = 0.023λ³ (V = 0.084 μm³) is slightly enhanced, which can be explained by the deeper penetration of the electric field into the mirror because of the smoothed tapering. The transition of the mode profile from the slot to the first adjacent pore is now indeed very gradual (see Fig. 6) due to the reduction of the diameter of the inner pores so that an adiabatic transition can be assumed. In general the Q factor of the cavity is limited by scattering losses and losses to the waveguide. However the losses to the waveguide seem to be dominant in this case. Adding another pore to the Bragg mirror clearly increases the Q-factor by improving the reflectivity of the Bragg mirror and reducing the losses towards the feeding waveguide. On the other hand extending the taper and keeping the total number of pores constant does not lead to a significant enhancement of the Q. This indicates that the scattering losses are already minimized by the described four-pore taper. In the light of these results a revisiting of the different causes for scattering losses helps to deepen the understanding of the behaviour of the Q-factor. For this the picture of a Fabry-Perot-cavity that consists of a waveguide with two adjacent Bragg-mirrors is applied [23]. As long as the light travels in a usual propagating waveguide mode within the center of the cavity, scattering does not appear due to the bound nature of the mode. However when the light enters the Bragg mirror a transition from a propagating waveguide mode to an exponentially damped oscillating mode can lead to several mode mismatches, which result in scattered waves.

Table 1. Pore distances a for tapering n_eff = 1.929 to n_B = 2.013

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If the wavelength of oscillation is different between waveguide and mirror region then its abrupt change at the waveguide/mirror interface leads to a disturbance of the usual sinusoidal spatial behaviour of the main “carrier wave” causing interface scattering. This is prevented by following the rule k_eff = k_B and a period tapering. However, in addition, the exponential decay of the wave within the Bragg mirror also causes scattering. This exponential decaying oscillating wave is represented by a Lorentz function in the wave vector spectrum which is concentrated at k_B = π/a (the carrier wavelength) [25]. If the decay is fast, the Lorentz function has appreciable wave vector components within the light cone leading to strong lateral scattering losses. As a result a small exponential decay of the field at least in the immediate vicinity of the cavity has to be chosen to minimize these losses. Finally the field cross section of the mode has to be adjusted gradually to match the field pattern of cavity waveguide mode and Bragg mirror mode to avoid scattering at the interface. This is accomplished by tapering the pore sizes.

4. Conclusion

In conclusion a novel concept of hybrid nanobeam cavities consisting of a locally infiltrated slot cavity and surrounding waveguide Bragg mirrors has been proposed. The Q-factor of the third order resonance was increased to 35,000 by tapering the period and diameter of the 4 pores, which are closest to the slot. The different causes for scattering losses were discussed and the importance of the gradual transition of the mode profile emphasized.