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Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 21 — Oct. 20, 2003
  • pp: 2769–2774
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Deformation-induced bandgap tuning of 2D silicon-based photonic crystals

Sukky Jun and Young-Sam Cho  »View Author Affiliations


Optics Express, Vol. 11, Issue 21, pp. 2769-2774 (2003)
http://dx.doi.org/10.1364/OE.11.002769


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Abstract

We address the issue of tuning the absolute bandgap in 2D silicon-based photonic crystals by mechanical deformation. The moving least-square (MLS) method, recently proposed by the authors for photonic bandgap materials, is employed for the real-space computation of band structures. The uniaxial tension mode is shown to be more effective for bandgap tuning than both pure and simple shear deformations. We verify that bandgap modifications are strongly influenced by the deformation-induced distortion of interfaces between inclusions and matrix. This result ensures the usefulness of real-space technique for the accurate calculation of strained photonic bandgap materials.

© 2003 Optical Society of America

1. Introduction

During last decade, photonic bandgap materials have attracted increasing interests of photophysics and optoelectronics communities due to the possibility of controlling light propagation through the materials [1

1. J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).

, 2

2. Y. Xia, “Photonic crystals,” Adv. Mater. 13, 369 (2001) and papers in this special issue. [CrossRef]

]. More recently, a growing number of research papers have specifically reported the tunability of bandgap in photonic crystals. For example, various techniques of bandgap tuning have been demonstrated by applying electric field [3

3. K. Busch and S. John, “Liquid-crystal photonic-band-gap materials: the tunable electromagnetic vacuum,” Phys. Rev. Lett. 83, 967–970 (1999). [CrossRef]

, 4

4. Y. Shimoda, M. Ozaki, and K. Yoshino, “Electric field tuning of a stop band in a reflection spectrum of synthetic opal infiltrated with nematic liquid crystal,” Appl. Phys. Lett. 79, 3627–3629 (2001). [CrossRef]

], temperature [5

5. K. Yoshino, Y. Shimoda, Y. Kawagishi, K. Nakayama, and M. Ozaki, “Temperature tuning of the stop band in transmission spectra of liquid-crystal infiltrated synthetic opal as tunable photonic crystal,” Appl. Phys. Lett. 75, 932–934 (1999). [CrossRef]

, 6

6. J. Zhou, C.Q. Sun, K. Pita, Y.L. Lam, Y. Zhou, S.L. Ng, C.H. Kam, L.T. Li, and Z.L. Gui, “Thermally tuning of the photonic band gap of SiO2 colloid-crystal infilled with ferroelectric BaTiO3,” Appl. Phys. Lett. 78, 661–663 (2001). [CrossRef]

], and magnetic field [7

7. Y. Saado, M. Golosovsky, D. Davidov, and A. Frenkel, “Tunable photonic band gap in self-assembled clusters of floating magnetic particles,” Phys. Rev. B 66, 195108–195113 (2002). [CrossRef]

], as well as by infiltrating liquid crystals [8

8. S.W. Leonard, J.P. Mondia, H.M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gsele, and V. Lehmann, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61, R2389–R2392 (2000). [CrossRef]

]. In addition, mechanical strain has also been examined as an alternative for band-structure modifications and, possibly, for tuning the bandgaps in photonic crystals as desired [9

9. H. Pier, E. Kapon, and M. Moser, “Strain effects and phase transitions in photonic resonator crystals,” Nature (London) 407, 880–883 (2000). [CrossRef] [PubMed]

13

13. V. Babin, P. Garstecki, and R. Holyst, “Photonic properties of an inverted face centered cubic opal under stretch and shear,” Appl. Phys. Lett. 82, 1553–1555 (2003). [CrossRef]

].

Our current analysis of the deformation (or strain) effect on bandgap modification, follows a distinguishing route from others that have previously addressed the strain-tunable photonic bandgap materials. First, using the MLS method, we numerically solve the linear elasticity problems of the unit cell of photonic crystal, in order to find the actual deformation in this highly inhomogeneous material subject to prescribed displacements at the outer boundaries of unit cell. Then, we perform the real-space band-structure calculation using the MLS method as well, which is very useful for modelling arbitrarily distorted interfaces due to the deformation.

Therefore, taking full advantage of the real-space technique, we focus on the realistic modelling of the deformation-induced change in the shape of interface between matrix and inclusion, and consequently on the effect of the distorted interface on bandgap modification. In this respect, our numerical study is different from some results in literatures where the emphases were put only on reconfiguring the lattice structure by mechanical strain without considering the detailed shape change along interface. The deformation studies and band-structure calculations are provided in section 2 and 3 respectively, followed by comprehensive discussions.

2. Deformations of 2D triangular photonic crystals

In mechanics of materials, strains or deformations are conventionally classified into fundamental modes such as pure shear, simple shear, and uniaxial tension, as shown in Fig. 1 where the undeformed square of dashed line is also given for clear illustration [15

15. J.M. Gere and S.P. Timoshenko, Mechanics of Materials (PWS Publishing Company, Boston, 1997).

]. Each deformation mode has its own characteristics. For example, the pure shear stress implies that only the shear stress component does not vanish at every point in the material. However, photonic crystal is the highly inhomogeneous material accompanying abrupt changes in material constants across the interface between matrix and inclusions. In mechanics’ point of view, the internal responses of matrix and inclusions to prescribed boundary conditions, are quite different from each other. For example, silicon matrix deforms in response to the applied load, while air columns do not transfer the load. It is therefore difficult to define the fundamental deformation modes of photonic crystals that are exactly equivalent to those of homogeneous materials illustrated in Fig. 1. Nonetheless, we borrow here the terminology of those three deformation modes, for convenience. We will use them to specify how we impose the corresponding displacement boundary conditions on the outer boundary of unit cell. The terminology can also be justified for inhomogeneous materials because those three modes provide the only means available to guide systematically how strains be applied in practice.

Fig. 1. Illustrations of fundamental deformation modes: (a) pure shear, (b) simple shear, and (c) uniaxial tension. Dashed square is the undeformed original shape.

We consider 2D triangular photonic crystals involving cylindrical air (ε=1.0) rods in silicon (ε=11.56) matrix. The inclusion’s volume fraction of the undeformed unit cell is given f=0.72. It is well known that this triangularly periodic structure of those dielectric materials has complete photonic bandgap. Three deformation modes mentioned above are separately implemented by imposing corresponding displacements (ux, uy) along the boundary of unit cell. That is, we first prepare the undeformed parallelogram unit cell as shown in Fig. 2, and then prescribe the displacement boundary conditions according to each deformation mode. In this paper, we restrict our study to relatively small deformation of 3% strain as in [11

11. S. Kim and V. Gopalan, “Strain-tunable photonic band gap crystals,” Appl. Phys. Lett. 78, 3015–3017 (2001). [CrossRef]

]. The strain is defined as εij=(∂ui/∂xj+∂uj/∂xi)/2 where u 1=ux, u 2=uy, x 1=x, and x 2=y in our notation. It is noted that the prescribed boundary condition is in fact imposed on the silicon’s part only (i.e., not on the air) along the outer boundary of unit cell.

Next, we solve numerically the typical governing equation of 2D plane strain linear elasticity to obtain the displacement field at all points in the silicon matrix. Refer to [16

16. T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, and P. Krysl, “Meshless methods: An overview and recent developments,” Comput. Methods Appl. Mech. Engrg. 139, 3–47 (1996). [CrossRef]

] on how to solve linear elasticity problem using the MLS method. Young’s modulus of 185 (GPa) and Poisson’s ratio of 0.26 are here used for silicon. The traction-free boundary condition is imposed on the circular interface between the silicon and the air, while the silicon’s part out of the outer boundary are fixed as prescribed above. In Fig. 2, the computed results under 3% elastic strains corresponding to each of the three deformation modes are given in comparison with the undeformed original unit cell. It is clearly shown that the circular interfaces are somewhat distorted due to deformation. In the following section, we explore how the distorted interfaces contribute to modifying the original band structures of photonic crystal.

3. Results on the band structures of deformed photonic crystals

Based on the deformed unit cells obtained from linear elasticity problem, we here compute the photonic band structures using the MLS method, and investigate the effect of each deformation mode on bandgap modification. Because the method is a real-space technique as stated before, it is suitable for accurately modelling the slightly distorted interfaces caused by deformation. The unit cell is discretized by 925 nodes and the associated 864 integration cells. 3×3 quadrature points for each cell, and [1, x, y] for polynomial basis vector are also employed in the MLS modelling. Detailed description regarding how the MLS method is used for the highly accurate computation of photonic band structures, can be found in our recent work [14

14. S. Jun, Y.-S. Cho, and S. Im, “Moving least-square method for the band-structure calculation of 2D photonic crystals,” Opt. Express 11, 541–551 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-6-541 [CrossRef] [PubMed]

].

Fig. 2. Undeformed (red) and deformed (blue) unit cells of 2D triangular photonic crystal with cylindrical air rods: (a) pure shear, (b) simple shear, and (c) uniaxial tension. In each mode, corresponding shear or tensile strain of 3% is applied.
Fig. 3. Schematic diagrams of symmetry points and zones in the reciprocal lattice of undeformed (left) and deformed (right) photonic crystals.

When computing the band structures of deformed photonic crystal, we do not take into account the stress-induced change of dielectric constant in silicon matrix, because the deformation considered in this paper is small enough to disregard it [17

17. M. Huang, “Stress effects on the performance of optical waveguides,” Int. J. Solids Struct. 40, 1615–1632 (2003). [CrossRef]

]. As well as the distorted interface, the lattice distortion deviated from the original perfect triangular lattice, is also brought about by deformation. Consequently, the perfect hexagonal symmetry in reciprocal lattice is broken. In computing band structures, we thus employ the quasi-hexagonal symmetry points of deformed photonic crystals, as illustrated in Fig. 3. For each deformation mode, we are able to obtain the exact geometry of each quasi-hexagon subjected to 3% strain. Three different symmetry zones, labelled by 1 to 3 in the figure, are considered in calculating the band structures [11

11. S. Kim and V. Gopalan, “Strain-tunable photonic band gap crystals,” Appl. Phys. Lett. 78, 3015–3017 (2001). [CrossRef]

].

Results on the band structures of deformation modes are given in Fig. 4 where the top, middle, and bottom rows are respectively for pure shear, simple shear, and uniaxial tension. In the figure, ω is the frequency of the monochromatic electromagnetic wave, c the speed of light, and a the radius of cylindrical air inclusions. In each row, the band structure of the undeformed photonic crystal is also given in the left column. The other three columns respectively correspond to the three different symmetry zones of deformed reciprocal lattice as depicted in Fig. 3. The two dashed horizontal lines in each row indicate the upper and lower bound of absolute bandgap of the undeformed original photonic crystal. This original bandgap is overlapped for the clear demonstration of bandgap changes due to each deformation mode.

In Fig. 4, it is notable that both pure and simple shear deformations cause little alteration of the absolute bandgap although band structures are slightly modified along the symmetry lines. In fact, the ‘widths’ of the absolute bandgaps are not significantly changed in all the three deformation cases, under this small amount of applied strain. This conclusion is contrary to a reference [11

11. S. Kim and V. Gopalan, “Strain-tunable photonic band gap crystals,” Appl. Phys. Lett. 78, 3015–3017 (2001). [CrossRef]

] which used the plane wave expansion method and thus did not consider the details of interface changes due to deformations. Our results are rather similar to those of more recent references although different types of materials were employed [13

13. V. Babin, P. Garstecki, and R. Holyst, “Photonic properties of an inverted face centered cubic opal under stretch and shear,” Appl. Phys. Lett. 82, 1553–1555 (2003). [CrossRef]

, 17

17. M. Huang, “Stress effects on the performance of optical waveguides,” Int. J. Solids Struct. 40, 1615–1632 (2003). [CrossRef]

].

Fig. 4. Photonic band structures under pure shear (top), simple shear (middle), and uniaxial tension (bottom). TM and TE modes are in blue and red, respectively. Dashed horizontal lines indicate the bandgap of undeformed original photonic crystal. Insets in top low illustrate the quasi-hexagonal symmetry zones of the deformed photonic crystal.

Table 1. Correlations between volume fraction and bandgap shift.

table-icon
View This Table

than in both shear cases. Consequently, the bandgap shift is evident in tensile mode only. This type of correlation between volume fraction and bandgap shift can also be found in the gap map for triangular lattice of air columns in medium of ε=11.4, as given in [1

1. J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).

] where the bandgap gets shifted down as the volume fraction decreases near f=0.72, without perceptible change of bandgap width. As already seen from Fig. 2, lattice distortions are somewhat greater in shear modes than in uniaxial tension mode. On the other hand the bandgap shift is prominent in the tension only as above. It is most probably because the effects of lattice distortions are compensated by the interface distortions in the shear modes, which is not the case of uniaxial tension. This strongly implies that, when computing band structures of strained photonic crystals, we must carefully consider not only the lattice distortion but also the shape changes of interfaces.

4. Conclusions

Acknowledgments

This work was supported by Korea Research Foundation Grant (KRF-2003-003-D00007).

References and links

1.

J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).

2.

Y. Xia, “Photonic crystals,” Adv. Mater. 13, 369 (2001) and papers in this special issue. [CrossRef]

3.

K. Busch and S. John, “Liquid-crystal photonic-band-gap materials: the tunable electromagnetic vacuum,” Phys. Rev. Lett. 83, 967–970 (1999). [CrossRef]

4.

Y. Shimoda, M. Ozaki, and K. Yoshino, “Electric field tuning of a stop band in a reflection spectrum of synthetic opal infiltrated with nematic liquid crystal,” Appl. Phys. Lett. 79, 3627–3629 (2001). [CrossRef]

5.

K. Yoshino, Y. Shimoda, Y. Kawagishi, K. Nakayama, and M. Ozaki, “Temperature tuning of the stop band in transmission spectra of liquid-crystal infiltrated synthetic opal as tunable photonic crystal,” Appl. Phys. Lett. 75, 932–934 (1999). [CrossRef]

6.

J. Zhou, C.Q. Sun, K. Pita, Y.L. Lam, Y. Zhou, S.L. Ng, C.H. Kam, L.T. Li, and Z.L. Gui, “Thermally tuning of the photonic band gap of SiO2 colloid-crystal infilled with ferroelectric BaTiO3,” Appl. Phys. Lett. 78, 661–663 (2001). [CrossRef]

7.

Y. Saado, M. Golosovsky, D. Davidov, and A. Frenkel, “Tunable photonic band gap in self-assembled clusters of floating magnetic particles,” Phys. Rev. B 66, 195108–195113 (2002). [CrossRef]

8.

S.W. Leonard, J.P. Mondia, H.M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gsele, and V. Lehmann, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61, R2389–R2392 (2000). [CrossRef]

9.

H. Pier, E. Kapon, and M. Moser, “Strain effects and phase transitions in photonic resonator crystals,” Nature (London) 407, 880–883 (2000). [CrossRef] [PubMed]

10.

S. Noda, M. Yokoyama, M. Imada, A. Chutian, and M. Mochizuki, “Polarization mode controll of two-dimensional photonic crystal laser by unit cell structure design,” Science 2931123–1125, (2001) [CrossRef] [PubMed]

11.

S. Kim and V. Gopalan, “Strain-tunable photonic band gap crystals,” Appl. Phys. Lett. 78, 3015–3017 (2001). [CrossRef]

12.

P.A. Bermel and M. Warner, “Photonic band structure of highly deformable self-assembling systems,” Phys. Rev. E 65, 10702–10705 (2002). [CrossRef]

13.

V. Babin, P. Garstecki, and R. Holyst, “Photonic properties of an inverted face centered cubic opal under stretch and shear,” Appl. Phys. Lett. 82, 1553–1555 (2003). [CrossRef]

14.

S. Jun, Y.-S. Cho, and S. Im, “Moving least-square method for the band-structure calculation of 2D photonic crystals,” Opt. Express 11, 541–551 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-6-541 [CrossRef] [PubMed]

15.

J.M. Gere and S.P. Timoshenko, Mechanics of Materials (PWS Publishing Company, Boston, 1997).

16.

T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, and P. Krysl, “Meshless methods: An overview and recent developments,” Comput. Methods Appl. Mech. Engrg. 139, 3–47 (1996). [CrossRef]

17.

M. Huang, “Stress effects on the performance of optical waveguides,” Int. J. Solids Struct. 40, 1615–1632 (2003). [CrossRef]

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(260.2110) Physical optics : Electromagnetic optics

ToC Category:
Research Papers

History
Original Manuscript: September 23, 2003
Revised Manuscript: October 15, 2003
Published: October 20, 2003

Citation
Sukky Jun and Young-Sam Cho, "Deformation-induced bandgap tuning of 2D silicon-based photonic crystals," Opt. Express 11, 2769-2774 (2003)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-21-2769


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References

  1. J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).
  2. Y. Xia, �??Photonic crystals,�?? Adv. Mater. 13, 369 (2001) and papers in this special issue. [CrossRef]
  3. K. Busch and S. John, �??Liquid-crystal photonic-band-gap materials: the tunable electromagnetic vacuum,�?? Phys. Rev. Lett. 83, 967-970 (1999). [CrossRef]
  4. Y. Shimoda, M. Ozaki, and K. Yoshino, �??Electric field tuning of a stop band in a reflection spectrum of synthetic opal infiltrated with nematic liquid crystal,�?? Appl. Phys. Lett. 79, 3627-3629 (2001). [CrossRef]
  5. K. Yoshino, Y. Shimoda, Y. Kawagishi, K. Nakayama, and M. Ozaki, �??Temperature tuning of the stop band in transmission spectra of liquid-crystal infiltrated synthetic opal as tunable photonic crystal,�?? Appl. Phys. Lett. 75, 932-934 (1999). [CrossRef]
  6. J. Zhou, C.Q. Sun, K. Pita, Y.L. Lam, Y. Zhou, S.L. Ng, C.H. Kam, L.T. Li, and Z.L. Gui, �??Thermally tuning of the photonic band gap of SiO2 colloid-crystal infilled with ferroelectric BaTiO3,�?? Appl. Phys. Lett. 78, 661-663 (2001). [CrossRef]
  7. Y. Saado, M. Golosovsky, D. Davidov, and A. Frenkel, �??Tunable photonic band gap in self-assembled clusters of floating magnetic particles,�?? Phys. Rev. B 66, 195108-195113 (2002). [CrossRef]
  8. S.W. Leonard, J.P. Mondia, H.M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. G¨ osele, and V. Lehmann, �??Tunable two-dimensional photonic crystals using liquid-crystal infiltration,�?? Phys. Rev. B 61, R2389-R2392 (2000). [CrossRef]
  9. H. Pier, E. Kapon, and M. Moser, �??Strain effects and phase transitions in photonic resonator crystals,�?? Nature (London) 407, 880-883 (2000). [CrossRef] [PubMed]
  10. S. Noda, M. Yokoyama, M. Imada, A. Chutian, and M. Mochizuki, �??Polarization mode controll of twodimensional photonic crystal laser by unit cell structure design,�?? Science 293 1123-1125, (2001) [CrossRef] [PubMed]
  11. S. Kim and V. Gopalan, �??Strain-tunable photonic band gap crystals,�?? Appl. Phys. Lett. 78, 3015-3017 (2001). [CrossRef]
  12. P.A. Bermel and M. Warner, �??Photonic band structure of highly deformable self-assembling systems,�?? Phys. Rev. E 65, 10702-10705 (2002). [CrossRef]
  13. V. Babin, P. Garstecki, and R. Holyst, �??Photonic properties of an inverted face centered cubic opal under stretch and shear,�?? Appl. Phys. Lett. 82, 1553-1555 (2003). [CrossRef]
  14. S. Jun, Y.-S. Cho, and S. Im, �??Moving least-square method for the band-structure calculation of 2D photonic crystals,�?? Opt. Express 11, 541-551 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-6-541">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-6-541</a> [CrossRef] [PubMed]
  15. J.M. Gere and S.P. Timoshenko, Mechanics of Materials (PWS Publishing Company, Boston, 1997).
  16. T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, P. Krysl, �??Meshless methods: An overview and recent developments,�?? Comput. Methods Appl. Mech. Eng. 139, 3-47 (1996). [CrossRef]
  17. M. Huang, �??Stress effects on the performance of optical waveguides,�?? Int. J. Solids Struct. 40, 1615-1632 (2003). [CrossRef]

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