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OPTICS LETTERS / Vol. 38, No. 20 / October 15, 2013
Photonic band gaps induced by submicron acoustic plate waves in dielectric slab waveguides Jin-Chen Hsu,1 Chiang-Hsin Lin,2 Yun-Cheng Ku,3 and Tzy-Rong Lin2,3,* 1
Department of Mechanical Engineering, National Yunlin University of Science and Technology, Douliou, Yunlin 64002, Taiwan 2 Institute of Optoelectronic Sciences, National Taiwan Ocean University, Keelung 20224, Taiwan 3
Department of Mechanical and Mechatronic Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan *Corresponding author:
[email protected] Received July 2, 2013; revised September 8, 2013; accepted September 10, 2013; posted September 11, 2013 (Doc. ID 193217); published October 7, 2013 We generate photonic bandgaps (PBGs) in dielectric slab waveguides by exciting their acoustic plate eigenmodes of submicron wavelength. We investigate the optical forbidden bands below the light line where the slab interfaces and index of refraction are periodically modulated by the acoustic fields. Results show that multiple scattering through the enhanced periodic acousto-optic (AO) interaction opens Bragg PBGs. A tunable bandgap width and transmittance are achieved. Transmitted optical waves are found to incur strong nonlinear modulation through AO interaction by a multiphonon exchange mechanism. The applications include tunable optomechanical and AO devices. © 2013 Optical Society of America OCIS codes: (230.1040) Acousto-optical devices; (230.5298) Photonic crystals; (230.7400) Waveguides, slab. http://dx.doi.org/10.1364/OL.38.004050
In recent years, the formation of allowed and forbidden frequency bands of classical waves in periodic media has received great interest [1,2]. Similar to crystalline solids, which exhibit electron gaps, periodic dielectric and elastic structures, which are known as the photonic and phononic crystals, can exhibit photonic and acoustic band gaps, respectively [3–7]. Recent developments in photonic crystals have demonstrated the possibility of controlling optical energy in compact and highly integrated micro and nanoscale devices, which provide promising applications for the next generation of photonics technology [8,9]. Conventional photonic crystals are constructed by two or more materials in which the periodical inclusions and the lattice constant are related to the photonic bandgap (PBG) frequency. To attain PBGs in the optical range of the spectrum, the lattice constant of the photonic crystals has to be brought down to the submicron scale. Very pure semiconductors (as insulators) are commonly used for this purpose because of their large dielectric constants and minimal light absorption [10]. Such structures with PBGs in the optical region are of interest in micro/nano lasers and optoelectronics [11,12]. In photonic crystals, most studies were based on passive structures in which the PBGs were immutable. Some studies have demonstrated tunable PBGs by external fields (e.g., electric, deformation, or strain fields) [13–15]. Recently, nonlinear modulation of light using acoustic phonons through a so-called acousto-optic (AO) interaction has been proposed [16–20]. Acoustic waves with frequencies in the range from hundreds of megahertz to several gigahertz can be generated by piezoelectric thin-film transducers, or electrostatic forces in microstructures [21,22]. Engineered optical modes in a simultaneous photonic and phononic bandgap structure have been used to achieve an enhanced modulation for active control of the photonic structures by acoustic wave energy. Promising applications include the AO modulator, optical switch, and the intensity and frequency modulation for micro/ nano lasers and sensors. 0146-9592/13/204050-04$15.00/0
In this Letter, we study the creation of PBGs in dielectric slab waveguides by AO interaction and the modulation of the refractive index through excitation of their acoustic eigenmodes. Specifically, we consider a silicon (Si) slab perturbed by its acoustic plate eigenmodes and monitor the modulation of the optical transmission over a broad range in the infrared region. The occurrence of the strong forbidden effect is due to the enhanced AO interaction between the guided optical and acoustic modes. Figure 1 shows the schematic of a Si slab where the acoustic plate waves of wavelength λa are generated. The acoustic plate waves generate a displacement field that causes the motion of the interfaces (interface effect) and a strain field that changes the refractive index (bulk effect) of the Si slab. These effects modulate the guided optical waves of reduced group velocity, and, in turn, enhance the AO interaction. With the periodic perturbation using the acoustic field, multiple scattering may generate PBGs. In the presented optical modulation, a longer acoustic wave train can enhance the optical scattering via an elongated interaction length, and PBGs can effectively form. Figure 2 shows the dispersion curves of the lowest three in-plane (y–z plane) acoustic plate waves in the Si slab of thickness d 500 nm. The slab completely Acoustic propagation direction
ka
Optical propagation direction
k
x
y
λa
z
d
Acoustic wavelength
Slab thickness
Fig. 1. Schematic of the acoustically perturbed dielectric slab optical waveguide, where λa denotes the wavelength of the acoustic plate waves and d is the unperturbed slab thickness. Both the guided optical waves (k) and acoustic plate waves (ka ) are assumed to propagate along the z direction. © 2013 Optical Society of America
October 15, 2013 / Vol. 38, No. 20 / OPTICS LETTERS
S0 mode A0 mode
2
0
0
0.2
0.4
0.6
0.8
Reduced Wavevector (ka d/2π)
Fig. 2. Dispersion curves of the lowest three in-plane acoustic plate modes: the A0, S0, and A1 modes. The dashed line denotes the chosen acoustic frequency f a 7.31 GHz excited in the Si slab waveguide of thickness d 500 nm. The corresponding acoustic wavelengths are λa 2π∕ka 640, 800, and 1905 nm.
confines the acoustic energy within the slab thickness, developing the fundamental antisymmetric and symmetric (A0 and S0) plate modes and their overtones (An and Sn, n 1; 2; 3; …). For an enhanced AO interaction, the optical and acoustic waves must have their wavelengths in the same order of magnitude. The mostly applied acoustic plate modes in microstructures are the A0, S0, A1, and S1 modes because of their simple displacement profiles which are easier to excite. Therefore, we consider the lowest three plate modes and a frequency corresponding to the submicron wavelengths. The acoustically induced strain field (partial derivative of the displacement field) changes the refractive indices via the photoelastic effect. The in-plane displacement fields of the acoustic eigenmodes, U y and U z , are governed by the following coupled stress-wave equations:
c11
∂2 U y ∂2 U y ∂2 U y ∂2 U z ρ 2 ; c44 c12 c44 2 2 ∂y∂z ∂y ∂z ∂t
c12 c44
∂2 U y ∂2 U y ∂2 U z ∂2 U z c44 c ρ ; 11 ∂y∂z ∂y2 ∂z2 ∂t2
(1)
∂U y 1 ∂U ny n0 − n30 p11 p12 z ; ∂y ∂z 2 ∂U y 1 3 ∂U z p11 ; nz n0 − n0 p12 ∂y ∂z 2
(2)
where p11 and p12 are strain-optic coefficients and n0 is the refractive index of unstrained Si. The calculations were implemented in a finite-element (FE) method based on COMSOL Multiphysics. The used material constants can be found in Refs. [20] and [23].
250
y
z
150 50 -50
Uz
-150 -250
(a)
Refractive Index
where ρ is the mass density, and c11 , c12 , and c44 are the three independent elastic stiffness constants of Si. We denote the acoustic wavevector and angular frequency as ka and ωa , respectively, and acoustic frequency f a ωa ∕2π. At the interfaces between Si and air (y d∕2), the null-stress conditions have to be satisfied for the acoustic plate modes. The perturbed indices are [23,24]
Position y (nm)
250
-5
-2.5
Uy 0
50
2.5
5
z
Uy
-50
Uz
-150
Displacement (nm)
250
y
150
Position y (nm)
4
-250
(b)
-5
-2.5
0
2.5
5
Displacement (nm)
-250
(c)
3.58
3.62
3.54
3.58
3.42 3.38
y ny
y nz
3.34
z
z
3.5 3.46 3.42 3.38
y ny
y nz z
3.34
z
3.3
600
(e)
0
200
400
600
Position z (nm)
800
2.5
5
y z nz
ny
y
3.38
3.3 400
0
Displacement (nm)
3.42
3.26
Position z (nm)
-2.5
3.46
3.26
200
-5
3.5
3.34
0
Uy
3.54
3.3
(d)
Uz
-50
3.54
3.46
z
50
3.58
3.5
y
150
-150
Refractive Index
A1 mode
Position y (nm)
6
Figures 3(a)–3(c) show the displacement fields of the three acoustic eigenmodes in the Si slab with an acoustic frequency 7.31 GHz. Their corresponding wavelengths are λa 640, 800, and 1905 nm, respectively. These acoustic modes cause symmetric (S0), or antisymmetric (A0 and A1) deformations of different periods along the z-direction with respect to the middle plane (y 0) on the top and bottom interfaces of the slab, while their associated strain fields generate different periodic patterns of refractive index variations in Si [Figs. 3(d)–3(f)]. The max. refractive index variations are about 1%–2% with the maximum acoustic displacement not exceeding 1% of the slab thickness (i.e., less than 5.0 nm), which is well below the material limit [19,20]. These acoustically induced spatial perturbations may contribute to the AO interaction via the interface and bulk effects to actively create a PBG. To show that the submicron wavelength acoustic plate waves can generate the optical forbidden frequency, the photonic band structures of unperturbed and acoustically perturbed Si slab waveguide are shown in Figs. 4(a) and 4(b), respectively. These photonic band structures can be built because the slab perturbed by single-mode acoustic waves becomes a periodic dielectric. In calculating the perturbed structures, the unit cell method with the Floquet boundary condition applied, in which the lengths of the unit cell are corresponding to the wavelengths of the acoustic plate modes. The acoustic fields are considered at the time instant (phase) of maximum displacement (amplitude). To obtain convergent results when using the FE method, we applied the moving mesh technique with a refined element size. The computation load was addressed using a commercialized computercluster system with parallel computing. Figure 4(b) shows that in the S0 acoustic plate mode, the AO interaction efficiently modulates the optical TE and TM modes near the first Brillouin zone (BZ) boundary, and PBGs from 70.76–71.09 THz (TE) and 95.59–95.8 THz (TM) are created, respectively, below the light line.
Refractive Index
Acoustic Frequency (GHz)
8
4051
(f)
z 0
400
800 1200 1600
Position z (nm)
Fig. 3. Total displacement fields (insets) of the three acoustic eigenmodes (a) A0 mode, (b) S0 mode, and (c) A1 mode, and their cross-sectional displacement components at the dashed lines in the insets. (d)–(f) Variations of the refractive indices ny (dashed curves) and nz (solid curves) caused by the three acoustic plate modes. (d) by the A0 mode at top surface y d∕2, (e) by the S0 mode at middle plane y 0, and (f) by the A1 mode at top surface y d∕2.
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on e
Optical Frequency (THz)
TM
60 640 nm
TE
40
800 nm
20 1905 nm
0 0
0.2
0.4
0.6
0.8
1
(a)
0.78
104 0.805 0.755 98
TE_S0
97
71
96
70
95
69 0.6
0.625
41 40
Wavevector (kd/2π)
105
82 0.755 73 72
1
TM_A0
106
83
TE
ht c
100
L ig
Optical Frequency (THz)
107
84
120
80
108 TE_A0
TE_A1
0.65
39
67
38
66
(b) 37
0.235
0.26
0.805
TM_S0
0.8 0.6 0.4
N=100 N=300 N=500
0.2 0 4.19
(a)
94 0.6 69 68
0.78
1
0.625
0.65
TM_A1
65 0.285 0.235
0.26
0.285
Wavevector ( kd 2π )
Fig. 4. (a) Photonic band structure of TE and TM modes below the light cone without acoustic perturbation. The vertical dashed lines denote the first BZ boundaries corresponding to the lattice constants set up by the three acoustic plate modes of different wavelengths 640, 800, and 1905 nm. (b) Close views of the acoustically perturbed TE and TM optical bands near the first BZ boundary by the A0, S0, and A1 acoustic plate modes of 7.31 GHz.
However, for the acoustic perturbation by the A0 and A1 modes at f a 7.31 GHz, there is no observable PBG for both the TE and TM modes. This suggests that the AO modulation of the optical waves in a dielectric slab waveguide is sensitive to the acoustic plate mode types. Also, Fig. 3 reveals interesting parity rules that allow for AO interaction in this structure. The symmetric interface variation and refractive indices with respect to the middle plane of the slab have the effect of opening the PBGs. On the other hand, the antisymmetric interface variation and refractive indices are useless to open any PBG. Unlike the S0 mode, the A0 and A1 modes deform the top and bottom interfaces of the Si slab antisymmetrically with respect to the middle plane, and this causes no variation in the slab thickness. The time variation of the edges and widths of the PBGs is shown in Fig. 5. Under the perturbation of the S0 mode, the PBGs have a switching frequency double the acoustic frequency. There is a symmetry of the displacement and perturbed index fields for the optical propagation as the acoustic phases in between 0–π and π–2π are exchanged. The scattering of the guided TE and TM waves in the perturbed Si slab can be intensified by increasing the AO interaction length and time, which can be achieved using a longer acoustic wave train. To show how the interaction length affects the AO modulation in generating the
Transmission
86
Transmission
4052
4.21
0.8 0.6 0.4
N=100 N=300 N=500
0.2 4.23
0 3.11
4.25
(b)
Wavelength (µm)
3.13
PBGs, transmissions are calculated in Fig. 6. We consider the scattering of the S0 mode at λa 800 nm in which the PBGs are generated most efficiently. In Fig. 6, spectra of optical transmission through the acoustic wave trains of 100, 300, and 500 waves are compared. Two transmission dips are observed: one at 70.9 THz for the TE mode and another at 95.7 THz for the TM mode. Increasing the number of the acoustic waves (N) lowers the transmission drastically. When N 500, the transmission coefficients drop to 0.152 (TE) and 0.219 (TM), and evident frequency forbidden bands are formed. This mechanism allows the submicron acoustic waves to tune the optical transmission to a switching frequency related to the acoustic frequency. Tuning the optical forbidden frequency is also anticipated by changing the acoustic frequency. Figure 7 shows the electric field distributions of the TE mode with and without the perturbation of the S0 mode in the slab. With the S0 mode (N 500), the AO interaction attenuates the intensity of the guided TE waves. Figure 7(b) shows that the electric field is exponentially decaying along the z direction in the acoustically induced PBG. The dynamic view of the structure is dependent on the propagating behavior of the acoustic waves. The periodic perturbation by the acoustic waves is a function of space and time. If the acoustic waves are traveling waves, so is Without AO
1
With AO
-1
(a)
71.1
70.9
70.7
(a)
0
π 2
π
3π 2 Acoustic Phase (ωa t )
2π
95.8
Lower edge upper edge
95.7
E x/Ei,x
Optical Frequency (THz)
Optical t Frequency (THz)
Lower edge Upper edge
95.9
3.17
Fig. 6. Optical transmission spectra of the (a) TE mode and (b) TM mode through the Si slab waveguide perturbed by the S0 mode acoustic wave train of N 100, 300, and 500. The acoustic perturbation results in transmission dips for the TE and TM modes around 70.9 and 95.7 THz, respectively. Shaded frequency ranges denote the PBGs in Fig. 4(b).
1 71.3
3.15
Wavelength (µm)
TE_S0
0.5 0 -0.5 -1
95.6
0 (b)
95.5 0
(b )
π π 2 3π 2 Acoustic Phase (ωa t )
2π
Fig. 5. Time varying PBG widths and gap-edge frequencies of the (a) TE and (b) TM modes as a function of phase of the S0 mode acoustic plate waves.
100
200
300
400
Position z (µm)
Fig. 7. (a) Electric field distributions of the TE waves at midgap frequency in the Si slab waveguide with and without acoustic perturbation (S0 mode). (b) Attenuation of the electric field of the S0 mode perturbed TE waves along the propagation direction.
October 15, 2013 / Vol. 38, No. 20 / OPTICS LETTERS -0.35
0.5
-0.37 0
π
2π
ωat
0.5
0
(a)
∆Ex/Ei,x
1
-0.36
∆Ex(n)/∆Ex(2)
∆Ex/Ei,x
∆Ex(n)/∆Ex(1)
1
0 -0.4 -0.8
π
0
2π
ωat
0 1
2
3
4
5
6
7
8
Order of Fourier Component
(b)
1
2
3
4
5
6
7
8
Order of Fourier Component
Fig. 8. Time variations of the acoustically perturbed transmitted optical field of TE mode (insets) and their associated spectra of Fourier components (a) at 65 THz, which is far from the optical bandgap, and (b) at the midgap frequency.
the periodic perturbation, which moves at their acousticwave velocity (a few times 103 m∕s). Since the velocity of the acoustic waves is five orders of magnitude smaller than that of light, the periodic perturbation is regarded as stationary. The problem reduces to light propagation in a periodic medium. However, if the acoustic plate waves are standing waves, the periodic perturbation in the slab will be in an oscillatory manner. The slab modulates light at an acoustic frequency, and the PBGs are time varying. In addition to the forbidden effect, the scattered optical waves experience the photon–phonon interaction. This interaction can cause elastic and inelastic light scattering that involves the absorption and/or emission of single or multiple phonons by photons. As a result, the reflected and transmitted optical waves have the form of a Fourier series that consists of infinite monochromatic waves with angular frequencies denoted by ω; ω ωa ; ω 2ωa ; … [16,24] E j y; z; t e−iωt
X
−inωa t ; E n j y; ze
(3)
n0;1;
is the nth order Fourier component of the where E n j scattered electric field E j . Figure 8 illustrates the time variations of the perturbed transmitted optical field (TE mode at the midgap and offgap frequencies) and their associated Fourier spectra [Eq. (3)]. Figure 8(a) shows that the Fourier spectrum of the transmitted optical field at 65 THz (far from the TE bandgap) perturbed by the acoustic waves is dominated by the first-order Fourier components with small contributions from the higher-order components. This means that the single-phonon processes dominate the photon–phonon exchange in the AO interaction. In Fig. 8(b), however, when the optical frequency is in the middle of its gap, the higher-order terms, especially the second-order term, contribute obviously to the Fourier spectrum. This implies that the multiphonon processes dominate. In the bandgaps, the optical waves have a very slow group velocity to pass through the acoustic wave trains. The transmitted light experiences stronger photon–phonon exchange near the acoustically generated PBGs, where the nonlinear AO interaction is enhanced by the prolonged interaction time. Correspondingly,
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the dynamic gap-width variations in Fig. 5 exhibit twoperiod variations during a time period of the S0 mode, showing a second harmonic modulation of the transmitted light with the time varying PBGs. In summary, we demonstrated the creation of PBGs in a Si slab by exciting its acoustic modes. We calculated the photonic band structures of TE and TM waves under the perturbation of the lowest three plate modes. The results show that the S0 mode can create PBGs of tunable width and transmission, which can be adjusted by changing the acoustic intensity and the length of the wave train. The authors thank the National Science Council of Taiwan for financial support (NSC 101-2221-E-224-015; NSC 101-2218-E-019-002). References 1. J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, Solid State Commun. 102, 165 (1997). 2. M. S. Kushwaha, P. Halevi, L. Dobrzynski, and B. Djafari-Rouhani, Phys. Rev. Lett. 71, 2022 (1993). 3. J. B. Pendry and A. MacKinnon, Phys. Rev. Lett. 69, 2772 (1992). 4. R. H. Olsson III and I. El-Kady, Meas. Sci. Technol. 20, 012002 (2009). 5. S.-C. S. Lin and T. J. Huang, Phys. Rev. B 83, 174303 (2011). 6. M. B. Assouar and M. Oudich, Appl. Phys. Lett. 99, 123505 (2011). 7. J.-C. Hsu and T.-T. Wu, Phys. Rev. B 74, 144303 (2006). 8. M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, Nature 404, 53 (2000). 9. E. Ozbay, E. Michel, G. Tuttle, R. Biswas, M. Sigalas, and K. M. Ho, Appl. Phys. Lett. 64, 2059 (1994). 10. P. Halevi and F. Ramos-Mendieta, Phys. Rev. Lett. 85, 1875 (2000). 11. J. Huang, S. H. Kim, J. Gardner, P. Regreny, C. Seassal, P. A. Postigo, and A. Scherer, Appl. Phys. Lett. 99, 091110 (2011). 12. J. Chan, A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, and O. Painter, Appl. Phys. Lett. 101, 081115 (2012). 13. X. Hu, Q. Zhang, Y. Liu, B. Cheng, and D. Zhang, Appl. Phys. Lett. 83, 2518 (2003). 14. W. Park and J. B. Lee, Appl. Phys. Lett. 85, 4845 (2004). 15. S. Kim and V. Gopalan, Appl. Phys. Lett. 78, 3015 (2001). 16. I. E. Psarobas, N. Papanikolaou, N. Stefanou, B. DjafariRouhani, B. Bonello, and V. Laude, Phys. Rev. B 82, 174303 (2010). 17. X.-S. Qian, J.-P. Li, M.-H. Lu, Y.-Q. Lu, and Y.-F. Chen, J. Appl. Phys. 106, 043107 (2009). 18. G. Gantzounis, N. Papanikolaou, and N. Stefanou, Phys. Rev. B 84, 104303 (2011). 19. Q. Rolland, M. Oudich, S. El-Jallal, S. Dupont, Y. Pennec, J. Gazalet, J. C. Kastelik, G. Lévêque, and B. Djafari-Rouhani, Appl. Phys. Lett. 101, 061109 (2012). 20. T.-R. Lin, C.-H. Lin, and J.-C. Hsu, J. Appl. Phys. 113, 053508 (2013). 21. C. M. Lin, W. C. Lien, V. V. Felmetsger, M. A. Hopcroft, D. G. Senesky, and A. P. Pisano, Appl. Phys. Lett. 97, 141907 (2010). 22. F.-C. Hsu, J.-C. Hsu, T.-C. Huang, C.-H. Wang, and P. Chang, Appl. Phys. Lett. 98, 143505 (2011). 23. A. Yariv and P. Yeh, Optical Wave in Crystals (Wiley, 2003). 24. F.-L. Hsiao, C.-Y. Hsieh, H.-Y. Hsieh, and C.-C. Chiu, Appl. Phys. Lett. 100, 171103 (2012).
