We present a theoretical analysis of the influence of structural imperfections on the device performance of optomechanical crystal nanobeam cavity. The quality factor, resonant frequency and optomechanical coupling properties are investigated statistically according to the various defects of the positions, radii, alignments, and surface roughness. Our results reveal the predominant influence and suggest an approach of manipulation towards these parameters, which provide important information for engineering a desired optomechanical crystal nanobeam cavity.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Microscopic cavity optomechanics structures, with optical and acoustic resonances co-localized in two or three spatial dimensions, have emerged as a vibrant field pursued by precision sensing and quantum information processing [1–6]. Among the currently types of optomechanical systems, the architecture based on thin-film optomechanical crystals (OMC) is a promising candidate for achieving scalable, cost-effective and low energy consumption devices, with the ability to create appropriate photonic and phononic bandgap [7–11]. Such structures are typically fabricated with bonded wafers, consisting of a slab, a sacrificing layer and a handling layer. Generally, after forming a periodic structure on the slab, the sacrificing layer is removed to achieve high-index contrast. Nevertheless, due to the intricate fabrication procedures, extrinsic imperfections may occur in the slab layer and contribute to degraded device performance. Up to now, lots of studies have been focused on the effect of imperfections for controlling and optimizing the fabrication procedures [12–18], while much effort has been devoted merely to optical or mechanical performance, without analysis of optomechanical coupling characteristics, which is also important in quantum information application. In this work, we comprehensively and quantitatively illuminate the influence of the nanometer scale imperfections on device performance of an OMC nanobeam cavity. The finite element methods are performed for the defects introduced during the fabrication processes, such as fluctuation of air hole radii and position, shift of air hole line, surface roughness and so on. Our results reveal the dominant imperfections for certain deteriorative characteristic and provide potential clues for engineering the desired OMC nanobeam device.
The schematic diagram of silicon OMC nanobeam cavity characterized in our analysis is shown in Fig. 1(a). The unit cell parameters in the mirror region, given in Fig. 1(b), are w = 529 nm, hx = 165 nm, hy = 366 nm, t = 220 nm and lattice constant a = 436 nm. A defect is generated by locally and smoothly changing the lattice spacing and ellipse dimensions, with the variations as a function of the hole number plotted in Fig. 1(c). According to the Nelder-Mead method , the device is designed to have a fundamental TE optical mode around 193.55 THz, and a mechanical breathing mode near 5.18 GHz, with mode profiles shown in Fig. 1(d) and Fig. 1(e), respectively. For the ideal situation, the variation of unit cells preserves the symmetry of the device for reflections about mirror planes perpendicular to y and z axes. Consequently, phonons and photons are confined in the defect region, without leaking into propagating waves in the mirror regions. However, the structural imperfections may break the symmetry of device and introduce the leakage of energy. Here, considering the processes needed to obtain the OMC structure from a wafer, we take into account six types of imperfections, such as (a) unitary variation of air hole radii, (b) shift of air hole line, (c) horizontal fluctuation of air hole positions, (d) perpendicular fluctuation of air hole positions, (e) fluctuation of air hole radii, and (f) surface roughness of air holes and nanobeam. Among them, similar with the conventional photonic crystal cavities [15,20], the imperfections of (a), (c), (d), (e) and (f) could be inevitable introduced in the processes of E-beam lithography and etching. The imperfection of (b) is due to the misalignment.
Finite element simulation methods are utilized to analyze the imperfection influence on synthetical performance of our OMC device. For eliminating optical crosstalk in the finite-size nature of the system, the scattering boundary and perfectly matched layer conditions are implemented in our method. Thus, the leakage rate and resonance frequency can be acquired through calculating the optical and mechanical eigenmodes, as shown in Fig. 1(d)-(e). Subsequently, we can analyze the dynamical interactions of electromagnetics and elastodynamics by multi-physics approaches for achieving optomechanical coupling parameter go. Here, the imperfections of (a) and (b) belong to the unitary variation of structural patterns, hence line shift and size change of the air holes can be defined as d and Δr, respectively. Otherwise, the random imperfections of (c)-(f) are applied to the simulation model using the Box Muller method, for which fluctuations and roughness obey a Gaussian distribution with a standard deviation of σ, i.e., random variation of structure parameters in the range between -σ and σ. Here, six sets of random scenarios with the same σ are calculated and all the simulation parameters are averaged to avoid the influence of structural randomness on the calculation reliability. Correlation within and between the imperfections can be ignored due to the good randomness quality of the random number sequences created in this method [15,16,21]. Furthermore, we can independently change the magnitudes of variations for the same structural imperfection by varying σ. It is remarkable that, compared with the fluctuations, an additional constructive method of synthesizing surface data is employed to characterize surface roughness by utilizing a sum of trigonometric functions similar to a Fourier series expansion,
3. Results and discussion
The numerically calculated results of optomechanical characteristics due to imperfections of (a) and (b) are given in Fig. 2(a)–2(h). For the variation of Δr from −12 nm to 12 nm with a step of 2 nm, with the increase of the air holes radii, the optical resonance frequency fo and mechanical resonance frequency fm emerge as an approximate linear trend and slopes are estimated to be 0.36 THz/nm and −0.03 GHz/nm, respectively. Optical quality factor Qo exponentially increases from 8.01×104 to 1.96×106, and Qm vibrates in the range from 2.1×106 to 1.9×107. This phenomenon could be explained through the optical and mechanical mode field distribution shown in Fig. 2(a). Distinctly the optical energy leaks from the defect region while mechanical energy preserved in the cavity. Moreover, the optomechanical coupling parameter go is proportional to the amplification of Δr with a slope 0.01 MHz/nm. In the case of air hole line shift, it is unambiguous that fo and fm keep stable relatively, thanks to the maintenance of photonic and phononic bandgap. The similar phenomenon can also be observed with respect to go. Furthermore, qualitatively analogy to the results in Ref. 18, phonon leakage is sensitive to the line shift d, namely a two-order decrease of Qm from 7.42×106 to 1.84×104 with the departure from center line, while Qo declines merely with a 32.5% drop. Opposite to the case of unitary variation of air holes radii, the mechanical energy leaks obviously from the defect region, as shown in Fig. 2(e).
The influence of perpendicular fluctuation of air hole positions is shown in Fig. 3(a)–3(d). With the increase of standard deviation σdp_rad from 0.005a to 0.03a, the parameters, including fo, fm and go, are insensitive to the perpendicular fluctuation, similar to the case of air hole line shift shown in Fig. 2. Then, Qm and Qo monotonically decrease by one order of magnitude. Figure 3(e)–3(l) give the influence due to fluctuation of air hole horizontal positions and radii. With the growth of standard deviation σdh_rad and σr_rad, the variation of Qo is quantitatively similar to the case of perpendicular fluctuation, while all the other parameters vary randomly. As mentioned above, the modeling of surface roughness is relatively complex. Here, the applied standard deviation σrou of surface amplitudes g (m, n) are set from 2 nm to 5 nm with a step of 0.5 nm, for which a bigger value indicates a greater surface fluctuation. Then, a uniform random function of phase angle f (m, n) is set in the interval between –π/2 and π/2 and the cutoff spatial frequencies in the x and y direction are both fixed at 1/20 (M = N = 20). The calculation results are given in Fig. 3(m)–3(p). For a larger σrou, Qo and Qm appear a decline trend by one order of magnitude, due to a larger leakage of photon and phonon. Obviously, roughness is the primary imperfection for the deterioration of optical and mechanical properties, considering that the equivalent variation of Qo and Qm is applied with smaller interval of σ. Besides, it can be seen that fo, fm and go vary in a small range. In particular, due to the effect of randomness, great difference of device performance (maximal value up to 88%) may exist among the six scenarios with the same σ. Statistics of simulation parameters is implemented to neutralize the effect and acquire the average characteristics of OMC nanobeam device with some structural imperfections. The device performance parameters with a larger error bar shown in Fig. 3 are more sensitive to the corresponding structural imperfections.
As shown in our results, the mechanical quality factor Qm can both decrease or increase due to fabrication imperfections, compared to the general monotonous variation of Qo from the ideal situation. This is because unlike photons, leaking into free space through many channels [16,18], phonons confined in one-dimensional geometry can only scatter into a single channel. As a result, the leakage channel of phonons due to fabrication imperfections may introduce only positively or negatively influence to the mechanical resonance. Analogously, the fabrication imperfections change the distribution of the intensity of mechanical and optical mode, which may both constructively and destructively affect the moving dielectric boundary and photoelastic effect, contributing to the random variation of the optomechanical coupling parameter go. The influences on resonance frequencies are neglectable, which are induced by imperfections of (b) or (d), corresponding to position variations in perpendicular direction. Oppositely, they are determined by the unitary variation of air hole radii. The similar phenomenon can be found on optomechanical coupling parameters. Considering all the mentioned imperfections, it is remarkable that enlargement of air hole radii is an exclusive approach to improve optical and optomechanical coupling properties of our OMC nanobeam device. Moreover, both of rough surface and air hole line deviation can be the main reason leading to the drastic deterioration of mechanical quality factors Qm by at least one order of magnitude. Nevertheless, in the latter case, it is possible to keep the optical quality factors Qo while opening up the phonon cavity.
In summary, we have numerically calculated the effects of six-type structural imperfections on the optomechanical characteristics of OMC nanobeam device. The variation in quality factor, shift of resonant frequency and fluctuation of optomechanical coupling properties are characterized with nanometer scale fabrication imperfection. Our analysis reveals the influence on quality factor could be decreased when the E-beam lithography and etching processes are improved to reduce surface roughness and asymmetry of air holes. Meanwhile, uniformly adjusting the radii of air holes opens a way to the manipulation of resonant frequency and optomechanical coupling parameter. The phonon leakage can be engineered through inducing cavity perturbation of air hole line shift, without affecting the lifetime of photon, which could also be applied to explore the possibility of on-chip phonon networks . Our analysis provides potential methods for improving the fabrication process and enabling applications of OMC nanobeam cavity.
National Natural Science Foundation of China (61308041, 61405030, 61704164, 61705033, 61775025, 91836102, U19A2076); National Key Research and Development Program of China (2018YFA0306102, 2018YFA0307400); Sichuan Province Science and Technology Support Program (2020YFG0289); Open-Foundation of Key Laboratory of Laser Device Technology, China North Industries Group Corporation Limited (KLLDT202008).
The authors declare no conflicts of interest.
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
1. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014). [CrossRef]
2. D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics 4(4), 211–217 (2010). [CrossRef]
3. R. Van Laer, B. Kuyken, D. Van Thourhout, and R. Baets, “Interaction between light and highly confined hypersound in a silicon photonic nanowire,” Nat. Photonics 9(3), 199–203 (2015). [CrossRef]
4. A. H. Safavi-Naeini, J. Chan, J. Hill, T. Alegre, A. Krause, and O. Painter, “Observation of Quantum Motion of a Nanomechanical Resonator,” Phys. Rev. Lett. 108(3), 033602 (2012). [CrossRef]
5. F. G. S. Santos, Y. A. V. Espinel, G. O. Luiz, R. S. Benevides, G. S. Wiederhecker, and T. P. Mayer Alegre, “Hybrid con- finement of optical and mechanical modes in a bullseye optomechanical resonator,” Opt. Express 25(2), 508 (2017). [CrossRef]
6. E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. Kip- penberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature 482(7383), 63–67 (2012). [CrossRef]
7. M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (2009). [CrossRef]
8. A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, J. Chan, S. Gröblacher, and O. Painter, “Two-Dimensional Phononic- Photonic Band Gap Optomechanical Crystal Cavity,” Phys. Rev. Lett. 112(15), 153603 (2014). [CrossRef]
9. K. Fang, M. H. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity- optomechanical circuits,” Nat. Photonics 10(7), 489–496 (2016). [CrossRef]
10. H. Liang, L. Rui, H. Yang, H. Jiang, and L. Qiang, “High-quality lithium niobate photonic crystal nanocavities,” Optica 4(10), 1251 (2017). [CrossRef]
11. D. Navarro-Urrios, E. Kang, P. Xiao, M. F. Colombano, and G. Fytas, “Optomechanical crystals for spatial sensing of submicron sized particles,” Sci. Rep. 11(1), 7829 (2021). [CrossRef]
12. S. Lan, K. Kanamoto, T. Yang, S. Nishikawa, Y. Sugimoto, N. Ikeda, H. Nakamura, K. Asakawa, and H. Ishikawa, “Similar role of waveguide bends in photonic crystal circuits and disordered defects in coupled cavity waveguides: An intrinsic problem in realizing photonic crystal circuits,” Phys. Rev. B 67(11), 115208 (2003). [CrossRef]
13. L. C. Andreani and M. Agio, “Intrinsic diffraction losses in photonic crystal waveguides with line defects,” Appl. Phys. Lett. 82(13), 2011–2013 (2003). [CrossRef]
14. T. Asano, B. S. Song, and S. Noda, “Analysis of the experimental Q factors (∼1 million) of photonic crystal nanocavities,” Opt. Express 14(5), 1996–2002 (2006). [CrossRef]
15. X. Mao, Y. Huang, K. Cui, C. Zhang, W. Zhang, and J. Peng, “Effects of structure parameters and structural deviations on the characteristics of photonic crystal directional couplers,” J. Lightwave Technol. 27(18), 4049–4054 (2009). [CrossRef]
16. H. Hagino, Y. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Effects of fluctuation in air hole radii and positions on optical characteristics in photonic crystal heterostructure nanocavities,” Phys. Rev. B 79(8), 085112 (2009). [CrossRef]
17. Y. Yamaguchi, S. W. Jeon, B. S. Song, Y. Tanaka, T. Asano, and S. Noda, “Analysis of Q-factors of fabrication imperfections in triangular cross-section nanobeam photonic crystal cavities,” J. Opt. Soc. Am. B 32(9), 1792–1796 (2015). [CrossRef]
18. R. N. Patel, C. J. Sarabalis, W. Jiang, J. T. Hill, and N. A. H. Safavi, “Engineering phonon leakage in nanomechanical resonators,” Phys. Rev. Appl. 8(4), 041001 (2017). [CrossRef]
19. J. Chan, T. M. Alegre, N. A. H. Safavi, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011). [CrossRef]
20. Q. Quan, P. B. Deotare, and M. Lončar, “Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide,” Appl. Phys. Lett. 96(20), 203102 (2010). [CrossRef]
21. G. Box and M. E. Muller, “A Note on the Generation of Random Normal Deviates,” Ann. Math. Statist. 29(2), 610–611 (1958). [CrossRef]