Design of Silica Encapsulated High-Q Photonic ... - OSA Publishing

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JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 32, NO. 5, MARCH 1, 2014

Design of Silica Encapsulated High-Q Photonic Crystal Nanobeam Cavity Alexandre Bazin, Rama Raj, and Fabrice Raineri

Abstract—We report on the design of photonic crystal nanobeam cavity fully encapsulated in silica. The proposed design, based on the principle of gentle confinement of the electromagnetic field, is mostly analytical and emphasizes on the most realistic options for fabricating nanocavities, in particular in III–V semiconductor materials. After determining the field decay inside the photonic bandgap of a nanobeam photonic crystal, we engineer the envelope of the cavity mode into a Gaussian shape by shifting only progressively the lattice constant. We discuss the various implementations of such shifts and give a simple algorithm to position each hole. The resonant wavelengths are found to depend linearly on the central lattice constant and on the radius of the holes. High Q factors above 106 and modal volume V close to (λ/n)3 are obtained. In particular, Q factors remain high for a wide range of values of the central lattice constant and of holes radii, hence showing exceptional tunability properties as well as robustness with respect to common fabrication defects. Index Terms—Design, nanocavity, photonic crystals, Q factor.

I. INTRODUCTION HE planar photonic crystal (PhC) micro/nano cavity is one of the most studied structures and is emblematic in nanophotonics because of its outstanding ability to localise the electromagnetic (EM) field within a nearly diffraction limited volume (V ∼ (λ/n)3 ) of material or air. Because light-matter interaction is usually driven by the EM field intensity, it is clear that the design and the fabrication of high-Q small-V resonators constitute a major issue for the demonstration of low power consuming optical devices such as lasers [1], [2] or switches [3], [4], as well as for the study of quantum optics phenomena [5], [6] and opto-mechanical coupling [7], [8] within solid state systems. PhC nanocavities are always formed by the addition of a defect in a PhC periodic lattice. Their design relies strongly on the precise analysis of the optical losses arising

T

Manuscript received September 4, 2013; revised November 2, 2013 and November 30, 2013; accepted December 10, 2013. Date of publication December 18, 2013; date of current version January 17, 2014. This work was supported by the EU Project HISTORIC and Project COPERNICUS, and by ANR “Jeunes Chercheurs” PROWOC projects. A. Bazin was with the CNRS Laboratoire de Photonique et de Nanostructures, 91460 Marcoussis, France. He is now with the Institute of Industrial Science, the University of Tokyo, Tokyo 153—8505, Japan (e-mail: bazin@iis. u-tokyo.ac.jp). R. Raj is with the CNRS Laboratoire de Photonique et de Nanostructures, 91460 Marcoussis, France (e-mail: [email protected]). F. Raineri is with the CNRS Laboratoire de Photonique et de Nanostructures, 91460 Marcoussis, France and also with Universit´e Paris Diderot, Paris 75205, France (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JLT.2013.2295267

from the in-plane and out-of-plane coupling of the cavity to the outside world. While the in-plane losses (for instance in a PhC nanocavity formed by a missing hole in a lattice of etched holes) can usually be suppressed by increasing the number of holes around the defect, the decrease of the out-plane losses requires a deeper understanding of their origin. More than a decade ago, it was emphasized by Srinivasan et al. [9] that the out-of-plane losses of a planar PhC cavity mode were dependent on the EM field k-space distribution and particularly on the amount of k component that lay within the light cone of the surrounding material. In 2003, Akahane et al. [10] used a similar approach and proposed the concept of gentle confinement of the EM field. This concept aimed at giving a simple design rule for obtaining high-Q PhC cavity modes, with modal V comparable to (λ/n)3 . Their idea was to make the EM field decay at the cavity edges as “gentle” as possible in order to avoid the large increase of the momentum components within the light cone that usually occurs in “regular” defect cavities. To do so, they impose a Gaussian form to the EM field longitudinal envelope. Presently, many designs of planar PhC cavities are available which permit the attainment of Q factors higher than 106 [11]–[14] with V comparable to (λ/n)3 . Among them, a particular structure, the PhC nanobeam cavity, has lately emerged as a very interesting option because of its small footprint and its natural predisposition to be coupled to wire waveguides. These structures are composed of a step index wire waveguide drilled with an array of holes, which confine the light transversally by total internal reflexion and longitudinally by the photonic band gap (PBG) effect. In the PhC literature, several studies [13], [15]–[17] have shown that nanobeam cavities could theoretically exhibit highQ factors up to 1010 . Although not explicitly stated, those designs used the concept of gentle confinement of the field as they always showed some form of gradation in the geometry of the structure (hole radius, periodicity, width) to ensure that the EM field envelope profile is Gaussian. Ahn et al. [13] obtained numerically, Q factors up to 108 by increasing the width of an air-bridged III-V semiconductor nanobeam quadratically along the direction of propagation, keeping the modal V below (λ/n)3 . Quan et al. [14], [17] recently presented and discussed several designs of Gaussian mode profiles in Si nanobeam cavities. In the case of air-suspended nanobeams, they obtained Q factors greater than 109 with effective modal V around (λ/nSi )3 by varying quadratically the hole radius while keeping constant the hole-to-hole distance. This study focusses on the design of nanobeam cavities made of InP-based materials embedding InGaAsP/InGaAs quantum wells (n ∼ 3.35) for laser applications [13], [18]. We choose to

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BAZIN et al.: DESIGN OF SILICA ENCAPSULATED HIGH-Q PHOTONIC CRYSTAL NANOBEAM CAVITY

consider structures fully embedded in silica in place of the more usual air clad membranes. This is motivated by the fact that, for some applications such as lasers or switches, air-bridged structures might not fulfil the requirements in terms of mechanical stability and environmental sensitivity. Moreover, air-bridged nanobeams are strongly thermally isolated as air surrounds the structure in both transverse directions, which is a serious concern when active devices are at stake. A SiO2 encapsulation provides, indeed, a better heat sink than air reducing thereby the thermal resistivity of the devices by more than one order of magnitude [19]. The case of nanobeams embedded in a low-index dielectric environment, known to be less favourable to obtain high-Q factors than air-suspended structures, was addressed by both [16] and [17], where the cavities were made within a silicon on insulator (SOI) substrate. The two studies obtained numerically Qs higher than 107 and, experimentally, higher than 105 by implementing the Gaussian shaping of the EM field envelope by changing the holes radii along the nanobeam. Differently, our design uses a variation of the lattice constant along the structure to obtain a Gaussian envelope of the field. The reason for this choice hinges on the necessity for an extreme precision of the hole positioning during the fabrication process, as this precision is directly related to the e-beam lithographic system, precision which, today can be as small as a fraction of nm. On the other hand, designs using fine tuning of the hole radii, like in refs [14], [17], could be much more difficult to concretise as the dimensions of the etched holes depend on various process parameters such as the exposure dose and the proximity effects during the electron beam lithography, as well as the plasma etch-rate of the holes usually varying with their radius. The next section of this paper is dedicated to the description of the design method we followed to obtain a SiO2 -encapsulated nanobeam cavity exhibiting a Q factor greater than 106 and a modal V of the order of (λ/n)3 . II. DESIGN OF THE SILICA-ENCAPSULATED PHC NANOBEAM CAVITY The design of our SiO2 -encapsulated PhC nanobeam cavity is based on the achievement of a longitudinal Gaussian mode envelope for the EM field through a subtle variation of the lattice constant in the array of cylindrical holes drilled into the semiconductor wire waveguide. In order to calculate this variation, we used an analytical method adapted from the one developed in [20] and [21] where the authors designed air-clad and glass-clad 2-D PhC high-Q cavities made in a line defect waveguide formed in a triangular lattice of air holes. The method is established starting from the following consideration: when light is injected at a frequency within the PBG of a PhC, the wavevector of the EM field will have a non-zero imaginary part q meaning the EM field envelope will decay spatially as e−q x (here x denotes the distance in the direction along the nanobeam). To obtain a PhC nanobeam cavity mode with a Gaussian field envelope, it is then necessary to vary q linearly with the distance x from the cavity centre, so that the envelope

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Fig. 1. (a) Top view of the SiO2 -encapsulated PhC nanobeam. The lattice constant of the PhC nanobeam is here a = 350 nm, and the hole radius r = 120 nm. (b) Logarithm of the electric field intensity distribution along the PhC nanobeam and in a frequency range across the PBG. x = 0 indicates the centre of the first hole.

will decay as e−B x where 2

q = Bx.

(1)

To do so, the analytical procedure to design the structure relies on the retrieval of the dependence of q on the lattice constant a. Indeed, the value of q at a given frequency within the band gap depends directly on the distance of this frequency with respect to the band edges frequencies. As a change of the lattice constant induces modifications of the band structure, the value of q can thereby be tuned along the cavity. Our method consists in: 1) calculating q as a function of the frequency for different values of the lattice constant. 2) calculating the modification induced by a change of the lattice constant on the bandstructure. 3) finally calculating the variation of the lattice constant needed along the PhC to obtain a Gaussian field profile. A. Decay Factor q(f ) Inside the PBG of a PhC Nanobeam The calculation of q(f ) inside the PBG (outside the PBG q(f ) = 0) is done numerically with 3D FDTD by solving the propagation of a broadband pulse sent toward the perfectly periodic PhC nanobeam (see Fig. 1(a)). This method is preferred to that of proposed in [20], where q(f ) is deduced from the band-structure calculation by approximating its variation by extending, at the band edge, the dispersion curve of the allowed band using the analytic continuation method. At frequencies falling within the PBG, the field penetrates evanescently the periodic part of the PhC (of lattice constant a) with a penetration length proportional to 1/q. In order to monitor the decay at each frequency inside the PBG, the FDTD simulation window contains a linear monitor placed at the centre of the structure along the waveguide that performs a discrete temporal Fourier transform at each position x for each frequency. Fig. 1(b) shows the spatial distribution of the calculated electric field intensity in a 285 nm thick, 505 nm wide, InP-based (n ∼ 3.35) silica embedded PhC nanobeam of lattice constant a = 350 nm and with a hole radius r = 120 nm as a function of the frequency. It is plotted in log scale to perceive the effective variations of the field decay inside the PBG. The PBG

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Fig. 4. Parameter A a as a function of the lattice constant a of the PhC nanobeam. Error bars are represented by the segments. A linear fit of A a is plotted in a dashed line. Fig. 2. Calculated EM field decay q in the PBG of periodic PhC nanobeams (285 nm × 505 nm, n = 3.35, r = 120 nm) for lattice constants a = 350 nm (red), 380 nm (purple), and 410 nm (green).

Fig. 3. Cut-off frequency of the band-edge modes of a periodic PhC nanobeam (285nm×505nm, n = 3.35, r = 120nm) as a function of the inverse of a. The dashed line represents a linear fit in 1/a of the cut-off frequency of the dielectric band-edge.

is clearly noticeable as the field intensity is very low inside it (green zone). The values of q(f ) are obtained by fitting, for each frequency, the field amplitude (in the periodic part of the wire) with a function, which is a product of a cosine and a decaying exponential. The results are plotted on Fig. 2 for three different lattice constants, 350 nm, 380 nm and 410 nm. In the PBG, the field decay increases with the detuning from the band-edge modes and reaches its maximum around the mid-gap frequency. For a given value of the period a, the dispersion relation f (qa ) inside the gap can be approximated near the 2 band-edge frequencies by a parabolic function. From this point, we choose to follow on with the design of the cavity by considering the variation of q close to the dielectric (lower) band edge mode. This, as it will be shown further, will lead to cavity modes where the field is confined within the semiconductor materials. So, close to the dielectric band-edge, one can write: f − facut (2) q (f ) = Aa where facut is the cut-off frequency of the dielectric band-edge mode for a lattice constant a, and Aa a parameter also dependent on the lattice constant a. The calculated cut-off frequency of the 2 band-edge modes are plotted on Fig. 3 as a function of the inverse of a.

Fig. 5.

Schematics of the PhC nanobeam cavity.

As can be seen on the figure, facut is inversely proportional to a such as facut = K/a + f˜.

(3)

We find by fitting the curve on Fig. 3, K = 4.82 × 104 THz.nm and f˜ = 53.5 THz. The parameter Aa is finally obtained by fitting the curves of Fig. 2 using (1) near the dielectric band-edge mode. The fitting accuracy may vary with the accuracy of the qa values obtained and with the range of frequency that the fit explores. The latter dependence is the most critical in this procedure. Aa is plotted on Fig. 4 as a function of the lattice constant. We approximate Aa by a linear function such as Aa =A0 + α(a − a0 ). From Fig. 4, for the particular structure under study with a0 = 350 nm, we find, A0 = 5.8 ± 0.1 Hz.m2 and α = 24(±3)MHz.m. Now that we have determined the dependence of the decay factor of the EM field within the PBG of PhC nanobeams on the frequency and the lattice constant, we can calculate how the lattice constant a(x) should vary along the nanobeam cavity in order to obtain a resonant mode exhibiting a Gaussian field envelope. B. Calculation of the Spatially Varying Lattice Constant A schematic of the PhC nanobeam cavity is represented in Fig. 5. As indicated in the figure, the design of the cavity is based on the variation by parts of the lattice constant a(x) so that a


Fig. 6. (a) Lattice constant as a function of the distance from the centre of the cavity for α = 0 (blue line) and α = 24 MHz.m. (b) Spatial field envelope profile for α=0 (blue line) and α = 0.024 GHz.m. Here, B = 0.3 μm−2 .

linear increase of q with the distance x from the cavity centre is obtained to achieve the Gaussian profile. The choice of the lattice constant, a0 , at the centre of the cavity fixes the resonant frequency of the Gaussian mode close to the cut-off frequency facut . As pointed out by Quan et al. in [14], a perturbative calculation would show that the cavity resonance approaches the band-edge frequency determined by the lattice constant a0 as the number of holes used for the Gaussian shaping of the mode is increased. By applying (2) at f = fcav using (3), one can obtain the general expression of the spatially varying field decay: K 1 − 1 a0 a(x) q (fcav , x) = (4) Aa(x) where a(x) is the lattice constant at the x position and Aa (x) the associated value of Aa . Equalizing (1) and (4), one can write the equation giving the spatial evolution of the lattice constant to achieve a Gaussian field envelope, such as: K 2 2 2 2 2 ˜ αB x a (x) + A0 B x − a (x) + K = 0 (5) a0 with A˜0 = A0 − αa0 . This quadratic equation leads to a unique valid solution of a(x) which is: a(x) = − A˜0 B 2 x2 −

K a0

−

A˜0 B 2 x2 −

K a0

2

− 4αB 2 x2 K .

2αB 2 x2 √

(6)

Note this solution exists only when x ∈]0, X[ with ⎛ ⎞ ˜ ˜ A0 K ⎝ A0 X= + 2α + 2α + α ⎠. a0 a0 A˜0 B 2 The solution takes the following simpler form when α = 0: a0 a (x) = . (7) 2 1 − A 0 BK a 0 x2 We plot, on Fig. 6(a), the spatial evolution of a(x) for α = 24 MHz.m and α = 0 and, on Fig. 6(b), the corresponding spatial envelopes of the field obtained after calculating the variation of q(x) in both cases. B here is fixed at 0.3 μm−2 to obtain a

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full width at half maximum (FWHM) of the Gaussian envelope equal to 3 μm (arbitrary value). The lattice constant must increase with x to ensure a Gaussian field profile. As it is shown in this example, this progressive shift in a is in general very subtle close to the centre of the cavity. Indeed, here, a is augmented by only 2 nm for x = 1 μm. This observation has, of course, a direct impact on the way to implement this design for the fabrication, as the lattice constant cannot be varied below the lithographic system resolution. We can also see that the influence of α on the envelope profile seems rather limited. A slight discrepancy with the Gaussian profile is only observed when x becomes greater than 3 μm. Hence, in what follows, we will simply neglect the effect of α and use (7) to calculate a(x). Up to now, we have considered a(x) and q(x) as continuous functions even though a(x) varies by step in real structures. We will now describe the hole positioning method we implemented in this study to approach the continuous model. C. Construction of the PhC Nanobeam Cavity Similar to the construction method used in [19], the centre of our structure is situated between two holes. Because the structure is symmetric with respect to the plane x = 0, we will determine a(x) for x > 0. As indicated in Fig. 5, the cavity is formed by blocks of holes made in the nanobeam, in which the lattice constant is kept constant. The central “block” (lattice constant a0 ) is chosen to be three periods long so that the first two holes positions are p01 =

1 a0 2

and

p02 =

3 a0 2

The subsequent blocks are 2 periods long. Note that each block length could have been chosen to be 1 period long. However, in that case, the necessary lattice constant shifts may become much smaller than the lithographic system resolution. Instead, our choice of block lengths ensures lattice constant variations greater than 0.5 nm in most cases, which corresponds to an achievable resolution with an electron beam lithographic system. We note pik the position of kth hole of the ith block of the structure. The index k = 1 denotes the hole in the center of the block and k = 2 the subsequent one. One can write pik =

i−1 3 a0 + 2 aj + kai = C˜i + kai 2 j =1

(8)

where ai is the lattice constant in the ith block. In this study, we choose to fix ai to the lattice constant given by (7) at x = pi1 as indicated in Fig. 7. 2 2 ˜ Solving the equation by considering A 0 BK a 0 pi1 1, we find −(2C˜i − F ) − F (F − 4(a0 + C˜i )) (9) ai = 2 with F =

K A 0 B 2 a 20

.

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Fig. 7. Schematic representation for the holes positioning in the PhC nanobeam cavity.


Fig. 9. Electric field intensity distribution of the first 4 resonant modes of the InP-based nanobeam cavity encapsulated in SiO2 .

Fig. 8. (a) Spatially varying lattice constant in the cavity for the continuous model (blue line) and the discretized model (red line). (b) Spatial field envelope considering continuous values (blue line) and discretized values (red line) for the lattice constant.

The discretised values ai (x) of the lattice constant in the cavity are plotted, together with a(x) derived from the continuous model on Fig. 8(a). We calculate and plot on Fig. 8(b), the expected field envelope by using (4) and taking q(x) = q(ai (x)). One can see that the discretisation of a(x) results in a field envelope which can be fitted by a Gaussian function but with a broader width in comparison to the continuous case (FWHM = 4.3 μm instead of 3 μm). This correction should be taken into account when a particular FWHM is targeted. In this study, the number of blocks of varying lattice constant is chosen equal to 6 (12 holes starting from the centre of the cavity). A final block made of a perfectly periodic array of 11 holes is added to prevent the EM field from escaping the structure in the longitudinal direction. D. 3-D Finite Difference Time Domain (FDTD) Simulations Three-dimensional FDTD simulations were performed for cavities built using the procedure described previously, in order to obtain the different resonant frequencies, their respective Q factors and spatial distributions. The nanobeam cavities that we simulated are made of InPbased materials (n = 3.35) surrounded (including inside the holes) by SiO2 (n = 1.46). The beams are 505 nm wide and 285 nm high. The central period a0 is chosen to be 350 nm and the constant hole radius r is 120 nm. Fig. 9 shows the electric field intensity distribution and the wavelength of the first 4 resonant modes of a simulated cavity with B = 0.3 μm−2 , A0 = 5.8 Hz.m2 and K = 4.82 × 104 THz.nm.

Fig. 10. (a) Calculated E y profile along the nanobeam of the first resonant mode (red line) fitted by Gaussian-cosine function (black dotted line). (b) Fourier transform of the E y profile of the first resonant mode. The SiO2 light line is indicated by the dashed line.

For all the modes, the electric field intensity is confined in the high index material between the holes, which was expected as we built the cavity from the dielectric band-edge of a PhC nanobeam. It is, of course, important to achieve such a spatial distribution to achieve efficient laser emission, as a good overlap between the EM field and the active material is necessary. The lower wavelength mode (λ = 1572.1 nm) presents the expected Gaussian profile as shown in Fig. 10(a). The higher order modes exhibit several lobes in their profile and are spaced in wavelength from one another by approximately 28 nm. The Q factor of the first order mode is 955 000, as for the higher order modes it decreases to, respectively, 350 000, 85 000, and 6900. The Q factor of the first order mode is calculated for cavities of different FHWM of the Gaussian envelope. The cavities are 4In(2) . designed by setting the B parameter such as B = FW HM 2 The results are plotted on Fig. 11(a). As the FWHM is increased from 1.5 to 3.3 μm, Q rockets from 103 to more than 106 . When the FWHM is increased further to 4.8 μm, Q decreases slowly down to just below 105 . It is interesting to compare the evolution of Q with FWHM to the amount of the spatial Fourier Transform (FT) of the field that is contained in the leaky region. This region of the reciprocal space within the light cone of the surrounding medium is defined by values of k < 2πnSiO 2 /λ. In the example of the field distribution in the reciprocal space of the


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Fig. 11. (a) Q factors of cavities of different FHWM of the Gaussian profile (red diamonds). The black squares indicate the calculated integral ξ of the spatial Fourier Transform of the resonant mode along the direction k x within the SiO2 line cone. (b) Resonant wavelength (blue triangles) and modal volume (red diamonds) of cavities of different FHWM of the Gaussian profile.

first order mode depicted in Fig. 10(b), we can see clearly that the main contribution of the field is a Gaussian which peaks at a wavevector close to π/a0 (k = 9 μm−1 ). The amount of the field within the light cone (on the left side of the blue dashed line on Fig. 10(b)) is estimated for the different cavities by calculating the integral ξ of the FT of the field along the kx direction within the light cone. This is plotted together with the corresponding Q factor on Fig. 11(a). We observe that, in most cases, a reduction in ξ corresponds to an increase of Q of the same order of magnitude showing that the “smart confinement” of the field is at the origin of the boosting of the Q factor. Conversely, the fall of Q observed for the smallest and the largest FWHMs corresponds to an increase of ξ indicating it can be explained by a misadjusted Gaussian shaping of the cavity mode. Our option of fixing the number of blocks of varying lattice constants (equal to 6 here as in [20]) may lead to some unfavorable situations when the FWHM is changed. Indeed, the shaping zone of the cavity has to be longer than the targeted spatial extent of the EM field defined by FWHM. When this is not the case, as for cavities with FWHMs