LI ChangSheng, HUANG Dan, NIE JianJun, GUO JieRong

?

Dual photonic and phononic band gaps in Silicon dielectric waveguide

LI ChangSheng, HUANG Dan, NIE JianJun, GUO JieRong

(Department of Physics and Electronics Science, Hunan University of Arts and Science, Changde 415000, China)

Phononic and photonic band structures were theoretically investigated in a special silicon dielectric waveguides by plane wave extended method. Periodic stubs were designed along the usual waveguides in favor of creating phononic and photonic band gaps. At appropriate geometrical parameters, a complete phononic gap was obtained together with a photonic gap at given symmetries.

band structure; phononic crystal; photonic crystal; waveguide

Periodic structures that exhibit band gaps with a certain range of acoustic or optical waves inhibited to propagate are the so-called phononic or photonic crystals. They have attracted attentions since decades ago[1—4]. Phononic and photonic band gaps were found separately in highly periodic bulk materials, multilayer materials, and slabs[5—6]. With recent advances in nanoscale fabrication techniques, the control of phonons and photons in the same structure becomes possible. The simultaneous existence of photonic and phononic band gaps has been widely investigated in two dimensional structures like Silicon slabs drilled with periodic holes[7—10]. One dimensional structure like dielectric waveguide was usually used in photonics, recognized for its ability to manipulate light. Recent studies found that the elastic properties of such wire structures can greatly influence the optical behavior because of the so-called phonon-photon interaction or acousto-optical interaction. There has been an emerging research field of the so called opto-mechanical or nanomechanical materials[11—12]. On the other hand, the commonly used photonic dielectric waveguides usually don’t have a phononic band gap. Only those periodic stubbed structures could be in favor of creating a phononic band gap. The study of dielectric waveguide with periodic stubbed structures for dual band gaps becomes increasingly important. In this paper, we theoretically investigate the phononic and photonic band structures of stubbed dielectric waveguides, and search the optimal parameters to achieve dual band gap. The practical parameter to work at telecom range is also discussed.

1 Model and calculating details

The stubbed Silicon dielectric waveguide model is shown in Fig. 1. It is cut from the usual silicon PC plates in air. We choose silicon dielectric waveguides because of the fact that silicon dielectric waveguides are able to guide optical and acoustic waves and widely used in electronics and telecommunications. In simulation, Silicon is considered as a cubic material with elastic constants11=165.7 GPa,12=63.9 GPa,44=79.62 GPa, and mass density 2331 kg/m3. It is optically isotropic with a refractive index of 3.47.

The phononic or photonic band structures can be calculated by various methods, such as plane wave extended method (PWE), finite-difference time-domain (FDTD) method, or finite-element (FE) method. The PWE method is efficient in simulating band structures of highly periodic structures, while the FDTD method has advantage in simulating some structures with open boundaries. The FE method can be used in many structures and efficient in displacement analysis. Results calculated from PWE, FDTD and FE are coincided with each other in previous work[13-14].

Fig.1 Schematic view of the stubbed Silicon dielectric waveguide

In this work, we employed PWE method in phononic and photonic band structure calculations. For all the simulations, we have compared our PWE results with FE calculation results. For the photonic calculations, we have used up to 4 641 basis in the PWE calculation to achieve convergence.

2 Results and discussion

In the following we would like to show the band structure about the stubbed silicon waveguide. As seen in Fig.2, the simulation of the band structures has been performed in both phononic and photonic waveguides. The phononic band structure presents a large absolute band gap (with parameters/= 0.75;/= 0.25;w/= 0.75;w/= 1). In the same geometry, we have also calculated the photonic dispersion curves and observed clearly three absolute photonic band gaps for the odd modes and two for the even one. The lower and the upper gaps of the odd modes correspond to complete photonic band gaps of the structure (see black dashed lines in Fig.2).

Fig.2 Phononic and photonic dispersion curves of the stubbed nanowire structure (with parameters h/a = 0.75; w/a = 0.25; wi/a=0.75; We/a = 1).

Then we test the possibility to work experimentally in the first photonic band gap regime. For a wavelength located around 1 550 nm working with frequencies in the windows of optical communications, the following parameters can be estimated: For the photonic low frequency gap (at reduced frequency 0.258 1 in Fig. 2),=w= 400 nm,=w= 300 nm,= 100 nm; The corresponding phononic middle gap frequency is 6.4 GHz. For the photonic high frequency gap (at reduced frequency 0.40 in Fig. 2),=w= 617 nm,=w= 463 nm,= 154 nm; The corresponding phononic middle gap frequency is 4.1 GHz.

3 Summary

We have theoretically investigated the phononic and photonic band structures of stubbed silicon dielectric waveguides. Phononic and photonic band gaps were identified simultaneously within the same waveguide at appropriate geometrical ratios. The practical parameter to work experimentally at telecom range is also obtained.

[1] KushwahaMS, Halevi P, Dobrzynski L,et al. Acoustic band structure of periodic elasticcomposites [J]. Phys Rev Lett, 1993, 71: 2022—2025.

[2] Sigalas M, Economou E N. Band structure of elastic waves in two dimensional systems [J]. Solid State Commun, 1993, 86: 141—144.

[3] Yablonovitch E. Photonic band-gap structures [J]. J. Opt Soc Am B, 1993, 10: 283—295.

[4] Krauss T F, Delarue R M, Brandt S. Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths [J]. Nature, 1996, 383:699—702.

[5] Pennec Y, Vasseur J O, Djafari-Rouhani B, et al. Two-dimensional phononic crystals: Examples and applications [J]. Surface Science Reports, 2010, 65: 229—291.

[6] Charles C, Bonello B, Gannot F.Propagation of guided elastic waves in 2D phononic crystals [J]. Ultrasonics,2006, 44: 1209—1210.

[7] Akimov A, Tanaka Y, Pevtsov A, et al. Hypersonicmodulation of light in three-dimensional photonic and phononicband-gap materials [J]. Phys Rev Lett, 2008, 101: 033902-033905.

[8] Eichenfield M, Chan J, Camacho R M, et al. Optomechanical crystals [J]. Nature, 2009, 462: 78—82.

[9] Wu T T, Huang Z G, Tsai T C, et al. Evidence of complete band gap and resonances in a plate periodic stubbed surface [J]. Appl Phys Lett, 2008, 93: 111902-111905.

[10] Maldovan M, Thomas E L. Simultaneous complete elastic and electromagnetic band gaps in periodic structures [J]. Appl. Phys Lett,2006, 88: 251907-251910.

[11] Safavi-Naeini A H, Alegre TPM, Chan J, et al. Electromagnetically induced transparency and slow light with optome- chanics [J]. Nature, 2011, 472: 69—73.

[12] Safavi-Naeini A H, Chan J, Hill J T, et al. Observation of quantum motion of a nanomechanical resonator [J]. Phys Rev Lett, 2012, 108: 033602-033605.

[13] Pennec Y, Djafari-Rouhani B, Boudouti E, et al. Simultaneous existence of phononic and photonic band gaps in periodic crystalslabs [J]. Opt Express, 2010, 18: 14301-14306.

[14] Hassouani Y,Li C, Pennec Y, et al. Dual phononic and photonic band gaps in a periodic array ofpillars deposited on a thin plate [J]. Phys Rev B, 2010, 82: 155405-155410.

硅介電波導(dǎo)管的光子與聲子雙重帶隙

李長(zhǎng)生*, 黃 丹, 聶建軍, 郭杰榮

(湖南文理學(xué)院 物理與電子科學(xué)學(xué)院, 湖南 常德, 415000)

通過(guò)平面波展開法, 我們對(duì)一種特殊的硅介電波導(dǎo)管的聲子能帶結(jié)構(gòu)與光子能帶結(jié)構(gòu)進(jìn)行了理論研究. 為了同時(shí)產(chǎn)生光子帶隙和聲子帶隙, 在通常的波導(dǎo)管結(jié)構(gòu)中設(shè)計(jì)了一種周期性的樁型結(jié)構(gòu). 研究發(fā)現(xiàn): 在適當(dāng)?shù)慕Y(jié)構(gòu)參數(shù)條件下, 一個(gè)完整的聲子帶隙和特定極化對(duì)稱性的光子帶隙可以同時(shí)得到.

能帶結(jié)構(gòu); 聲子晶體; 光子晶體; 波導(dǎo)管

P 15

1672-6146(2014)02-0026-03

10.3969/j.issn.1672-6146.2014.02.006

通訊作者email: lcs135@163.com.

2014-05-06

國(guó)家自然科學(xué)基金(NSFC 11104069, NSFC 61204104); 湖南文理學(xué)院重點(diǎn)建設(shè)項(xiàng)目(光學(xué));光電信息集成與光學(xué)制造技術(shù)湖南省重點(diǎn)實(shí)驗(yàn)室項(xiàng)目.

(責(zé)任編校:劉剛毅)