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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 25 — Dec. 11, 2006
  • pp: 12451–12456
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Design of high-Q cavities in photonic crystal slab heterostructures by air-holes infiltration

Snjezana Tomljenovic-Hanic, C. Martijn de Sterke, and M. J. Steel  »View Author Affiliations


Optics Express, Vol. 14, Issue 25, pp. 12451-12456 (2006)
http://dx.doi.org/10.1364/OE.14.012451


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Abstract

We design novel photonic crystal slab heterostructures, substituting the air in the holes with materials of refractive index higher than n=1. This can be achieved by infiltrating the photonic crystal slab (PCS) with liquid crystal, polymer or nano-porous silica. We find that the heterostructures designed in this way can have quality factors up to Q=10 6. This high-Q result is comparable with the result of previously reported designs in which the lattice is elongated in one direction. Unlike conventional heterostructures, our design does not require nanometre-scale changes in the geometry. Additionally, infiltrated PCS can be constructed at any time after PCS fabrication.

© 2006 Optical Society of America

1. Introduction

A cavity is usually formed in either of two ways: forming a point cavity or forming a “hetero-structure”. A point defect may be formed by omitting one or more holes in the centre of the slab. In that case the optimization of the optical nanocavity design, i.e. a further increase of a quality factor, is possible by modifying the geometry of the lattice structure surrounding the cavity [11–13

11. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003). [CrossRef] [PubMed]

].

Another way to form a cavity is to use heterostructures. Song et al constructed double heterostructures, combining two PCSs with slightly different lattice constants [14

14. B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Mater. 4, 207–210 (2005). [CrossRef]

]. These lattices are combined similarly to the design illustrated in Fig. 1(b) and (c): in the outer regions (PC1) the lattice is hexagonal, whereas in the central region (PC2), the lattice is slightly elongated (by 10 nm) in the direction orthogonal to the heterostructure (parallel to the waveguide), but is otherwise identical. A waveguide introduced across these PCSs has different dispersion curves within the different parts of the PCS [15

15. B. S. Song, T. Asano, Y. Akahane, Y. Tanaka, and S. Noda, “Transmission and reflection characteristics of inplane hetero-photonic crystals,” Appl. Phys. Lett. 85, 4591–4593 (2004). [CrossRef]

]. Therefore within the same photonic band gap (PBG) there is a “mode-gap” between these curves. The mode “propagates” in the waveguide of the central structure and decays exponentially elsewhere. Consequently the light in one of the photonic crystal waveguides can be localized due to the mode-gap effect. With this design, Song et al experimentally achieved a quality factor of Q=6×105 [14

14. B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Mater. 4, 207–210 (2005). [CrossRef]

]. That is six times higher than they achieved with the point cavity design [11

11. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003). [CrossRef] [PubMed]

,12

12. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “Fine-tuned high-Q photonic-crystal nanocavity,” Opt. Express 13, 1202–1214 (2005). [CrossRef] [PubMed]

]. The effect of double heterostructures can be also achieved via lateral hole displacement [16

16. E. Kuramochi, M. Natomi, S. Mitsugi, A. Shinya, and T. Tanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88, 041112 (2006). [CrossRef]

]. In this way Kuramochi et al achieved experimentally a quality factor of Q=8×105.

2. Model and method

Our model is a PCS composed of a hexagonal array of cylindrical air holes in a silicon slab, as illustrated in Fig. 1. The structure has holes of radius R, a is the lattice constant and h is the thickness of the slab. Across the PCS there is a line defect, a W1 waveguide, in the Γ-K direction. We start with a homogeneous PCS, as illustrated in Fig. 1(a) and design the heterostructures by changing the holes’ refractive index in the central region of the PCS (indicated by the darker circles in Fig. 1(b) and (c)).

First we consider a bulk silicon-based (n=3.4) PCS that is infinite in the plane in order to obtain PBGs and associated eigenstates of a waveguide introduced in the Γ-K direction. As the second step in the design, a finite PCS, with 25a in the x-direction and 25a in the y-direction, is considered with the cavity in the centre.

Fig. 1 (a) Schematic of PCS with a W1 waveguide in the Γ-K direction and refractive index distribution in the plane of the structures considered (b) m=1 and (c) m=4.

In order to design a cavity, two numerical methods are used: the plane wave expansion (PWE) method for the PBG calculations and associated eigenstates of the photonic crystal waveguide, and the finite-difference time-domain (FDTD) method, combined with techniques of fast harmonic analysis [22

22. V. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals,” J. Chem. Phys. 107, 6756–6769 (1997). [CrossRef]

] for the quality factor calculations. This method exploits the knowledge that for a signal consisting of one or a few resonant modes, the electric field at an arbitrary point as a function of time can be represented as a sum of complex exponentials. By projecting the signal onto a Fourier basis in a narrow range around the resonant frequency, the complex frequencies can be found to very high accuracy, much greater than would be extracted from a standard Fourier transform. The error in the complex frequency is dominated entirely by the spatial grid resolution rather than the length of the simulation. Details on numerical parameters for the calculations can be found in Ref [5

5. S. Tomljenovic-Hanic, M. J. Steel, C. M. de Sterke, and J. Salzman, “Diamond based photonic crystal microcavities,” Opt. Express 14, 3556–3562 (2006). [CrossRef] [PubMed]

] with the satisfactory convergence obtained by using 32 points per period.

3. Results

The concept of the cavity design in heterostructures relies on the mode-gap effect [15

15. B. S. Song, T. Asano, Y. Akahane, Y. Tanaka, and S. Noda, “Transmission and reflection characteristics of inplane hetero-photonic crystals,” Appl. Phys. Lett. 85, 4591–4593 (2004). [CrossRef]

]. Therefore we first examine if there is a sufficient mode-gap between structures having materials other than air within the holes. In Fig. 2(a) we plot the dispersion curves for the regular structure (PC1) and modified structure (PC2).

Both structures have two guided modes below the light line in the lowest PBG, one in the middle of the bandgap and the other one in the lower part of the bandgap. The lower mode is the mode of interest [18

18. S. Tomljenovic-Hanic, M. J. Steel, C. M. de Sterke, and D. J. Moss, “High-Q cavities in photosensitive photonic crystals,” Opt. Lett. (to be published). [PubMed]

]. The dispersion curves of this mode for both regular structure, PC1, and the PC2 where air holes are filled with material having refractive index n=1.6, are plotted in Fig. 2(a). In the same figure the lower band edge is indicated by solid horizontal lines both for the regular and modified structure. Obviously, filling the holes with material of higher refractive index than air increases the refractive index of the structure in whole and consequently lowers the dispersion curve. The gap between these dispersion curves, measured at the edge of the Brillouin zone, is Δω˜=3.25×10-3, where ω˜=ωa/2πc. The size of the mode-gap is comparable with the mode-gap of the heterostructures formed of different lattice constants PCSs [10

10. B. Maune, M. Loncar, J. Wtzens, M. Hochberg, T. Baehr-Jones, and Y. Qiu, “Liquid-crystal electric tuning of a photonic crystal laser,” Appl. Phys. Lett. 85, 360–362 (2004). [CrossRef]

]. This indicates that the heterostructures formed of the PCSs that differ in the holes’ refractive index are also capable of the mode-gap operation.

Fig. 2 (a) Dispersion curves for W11 within the region of the lowest gap of the regular structure PC1 (empty triangles) and W12 of the structure PC2 (full circles), nholes=1.6; the dashed line represents the light line, the horizontal solid lines represent the lower band gap edge for PC1 and PC2, (b) quality factor Q (rectangles) and modal volume V (crosses) as a function of the refractive index of the central holes for m=1.

The resonant frequencies are also plotted in Fig. 3(a) and the mode-gap edges are indicated by the horizontal dotted lines. As the refractive index is fixed, the mode-gap that ranges from ω˜=0.2636 to ω˜=0.2607, does not change as m changes. The resonant frequency for m=1 occurs just below the upper mode-gap edge. As m increases the frequency crosses over the mode-gap almost linearly, passing the mid mode-gap closest to m=3. The resonant frequency that corresponds to the maximum, ω˜=0.2617, is in the lower half of the mode-gap. These results suggest that the relative position of the resonant frequency within the mode-gap is an important parameter in the design of high-Q heterostructures as well as the mode-gap position within the PBG as discussed above.

Fig. 3 Quality factor Q (rectangles) (a) and resonant frequencies (crosses) as a function of the number of periods within the cavity m, for fixed nholes=1.4, (b) as a function of the refractive index of the central holes for m=4 cavity.

We compare these results with the results presented in Fig. 2(b). The maximum occurs at different refractive index values, for m=1 at n=1.4 and for m=4 at n=1.25. However there is no contradiction as the resonant frequencies are very close, for m=1, ω˜=0.2628 and for m=4, ω˜=0.2625. As expected, increasing the number of layers lowers the resonant frequency.

Note that a second family of symmetric heterostructures, which are shifted by half a period in the horizontal direction from those in Figs 1(b) and (c), can be constructed. However, the Q values of these are approximately an order of magnitude smaller then those presented here. For this reason, we do not consider them here in more details.

4. Discussion and Conclusions

We designed high-Q cavities with quality factors that are comparable those obtained for heterostructures with lattice variation [14

14. B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Mater. 4, 207–210 (2005). [CrossRef]

]. The main advantage of our design is that it does not require changes in the geometry with nanometre precision [14

14. B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Mater. 4, 207–210 (2005). [CrossRef]

]. However there is no evidence that this design is less sensitive that the geometry based design. Susceptibility of the cavity properties to variations in the hole filling process will be addressed in future studies. The processing of air-hole infiltration can be done at any time after fabrication. If the structure is filled with LC, electro-optic or nonlinear polymer there is also the possibility of tuning these structures when voltage is applied. In future studies the effect of the top cladding layer produced during LC cell construction [10

10. B. Maune, M. Loncar, J. Wtzens, M. Hochberg, T. Baehr-Jones, and Y. Qiu, “Liquid-crystal electric tuning of a photonic crystal laser,” Appl. Phys. Lett. 85, 360–362 (2004). [CrossRef]

] should be addressed and quantified. Photonic crystal laser sources for chemical detection and LC tuned photonic crystal laser were experimentally demonstrated [9

9. M. Loncar and A. Scherer, “Photonic crystal laser sources for chemical detection,” Appl. Phys. Lett. 82, 4648–4650 (2003). [CrossRef]

,10

10. B. Maune, M. Loncar, J. Wtzens, M. Hochberg, T. Baehr-Jones, and Y. Qiu, “Liquid-crystal electric tuning of a photonic crystal laser,” Appl. Phys. Lett. 85, 360–362 (2004). [CrossRef]

] where a point cavity design was used for both applications.

In conclusion we have shown that ultrahigh-Q cavities can be designed in PCS heterostructures without change of the structure geometry. Quality factors of order Q~106 can be obtained by filling the holes in the central region of the homogenous PCS with nanoporous silica. The maximum values of this design achievable by using polymer materials or LC are higher than Q=7×105. This approach represents a novel technique for creating ultrahigh-Q cavities that furthermore opens the possibility of post-processing in PCS-based microcavities.

Acknowledgment

This work was produced with the assistance of the Australian Research Council (ARC) under the ARC Centres of Excellence Program. CUDOS (the Centre for Ultrahigh-bandwidth Devices for Optical Systems) is an ARC Centre of Excellence.

References and links

1.

B. S. Song, S. Noda, and T. Asano, “Photonic devices based on in-plane hetero photonic crystals,” Science 300, 1537 (2003). [CrossRef] [PubMed]

2.

A. Shinya, S. Mitsugi, E. Kuramochi, and M. Notomi, “Ultrasmall multi-channel resonant-tunneling filter using mode-gap of width-tuned photonic-crystal waveguide,” Opt. Express 13, 4202–4209 (2005). [CrossRef] [PubMed]

3.

H-G. Park, J-K. Hwang, J. Huh, H-Y Ryu, Y-h. Lee, and J-S. Kim, “Nondegenerate monopole-mode two-dimensional photonic band gap laser,” Appl. Phys. Lett. 79, 3032–3034 (2001). [CrossRef]

4.

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vuckovic, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95, 013904 (2005). [CrossRef] [PubMed]

5.

S. Tomljenovic-Hanic, M. J. Steel, C. M. de Sterke, and J. Salzman, “Diamond based photonic crystal microcavities,” Opt. Express 14, 3556–3562 (2006). [CrossRef] [PubMed]

6.

T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, “Fast bistable all-optical switch and memory on a silicon photonic crystal on-chip,” Opt. Lett. 30, 2575–2577 (2005). [CrossRef] [PubMed]

7.

C. Grillet, C. Smith, D. Freeman, S. Madden, B. Luther-Davies, E. C. Mägi, D. J. Moss, and B. J. Eggleton, “Efficient coupling o chalcogenide glass photonic crystal waveguide via silica optical fiber nanowires,” Opt. Express 14, 1070–1078 (2006). [CrossRef] [PubMed]

8.

M. Loncar and A. Scherer, “Microfabricated optical cavities and photonic crystals” in Optical microcavities, K. Vahala, ed. (World Scientific Publishing, 2004).

9.

M. Loncar and A. Scherer, “Photonic crystal laser sources for chemical detection,” Appl. Phys. Lett. 82, 4648–4650 (2003). [CrossRef]

10.

B. Maune, M. Loncar, J. Wtzens, M. Hochberg, T. Baehr-Jones, and Y. Qiu, “Liquid-crystal electric tuning of a photonic crystal laser,” Appl. Phys. Lett. 85, 360–362 (2004). [CrossRef]

11.

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003). [CrossRef] [PubMed]

12.

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “Fine-tuned high-Q photonic-crystal nanocavity,” Opt. Express 13, 1202–1214 (2005). [CrossRef] [PubMed]

13.

Z. Zhang and M. Qiu, “Small-volume waveguide-section high Q microcavities in 2D photonic crystal slabs,” Opt. Express 12, 3988–3995 (2004). [CrossRef] [PubMed]

14.

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Mater. 4, 207–210 (2005). [CrossRef]

15.

B. S. Song, T. Asano, Y. Akahane, Y. Tanaka, and S. Noda, “Transmission and reflection characteristics of inplane hetero-photonic crystals,” Appl. Phys. Lett. 85, 4591–4593 (2004). [CrossRef]

16.

E. Kuramochi, M. Natomi, S. Mitsugi, A. Shinya, and T. Tanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88, 041112 (2006). [CrossRef]

17.

T. Asano, B. S. Song, and S. Noda, “Analysis of the experimental Q factors (~1 million) of photonic crystal nanocavities,” Opt. Express 14, 1996–2002 92006). [PubMed]

18.

S. Tomljenovic-Hanic, M. J. Steel, C. M. de Sterke, and D. J. Moss, “High-Q cavities in photosensitive photonic crystals,” Opt. Lett. (to be published). [PubMed]

19.

E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interfaces,” Rev. Modern Phys. 78, 455–481 (2006). [CrossRef]

20.

G. P. Harmon, “Polymers for optical fibers and waveguides: An Overview,” in Optical polymers fibers and waveguides, J. P. Harmon and G. K. Noren, eds. (American Chemical Society, 2001) pp. 1–23.

21.

G. Wu, J. Wang, J. Shen, T. Yang, Q. Zhang, B. Zhou, Z. Deng, F. Bin, D. Zhou, and F. Zhang, “Properties of sol-gel derived scratch-resistant nano-porous silica films by a mixed atmosphere treatment,” J. Non-Cryst. Solids 275, 169–174 (2000). [CrossRef]

22.

V. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals,” J. Chem. Phys. 107, 6756–6769 (1997). [CrossRef]

OCIS Codes
(230.3990) Optical devices : Micro-optical devices
(230.5750) Optical devices : Resonators

ToC Category:
Photonic Crystals

History
Original Manuscript: October 25, 2006
Revised Manuscript: November 24, 2006
Manuscript Accepted: November 24, 2006
Published: December 11, 2006

Citation
Snjezana Tomljenovic-Hanic, C. Martijn de Sterke, and M. J. Steel, "Design of high-Q cavities in photonic crystal slab heterostructures by air-holes infiltration," Opt. Express 14, 12451-12456 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-25-12451


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References

  1. B. S. Song, and S. Noda, T. Asano, "Photonic devices based on in-plane hetero photonic crystals," Science 300,1537 (2003). [CrossRef] [PubMed]
  2. A. Shinya, S. Mitsugi, E. Kuramochi, and M. Notomi, "Ultrasmall multi-channel resonant-tunneling filter using mode-gap of width-tuned photonic-crystal waveguide," Opt. Express 13, 4202-4209 (2005). [CrossRef] [PubMed]
  3. H-G. Park, J-K. Hwang, J. Huh, H-Y Ryu, Y-h. Lee, J-S. Kim, "Nondegenerate monopole-mode two-dimensional photonic band gap laser," Appl. Phys. Lett. 79, 3032-3034 (2001). [CrossRef]
  4. D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vuckovic, "Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal," Phys. Rev. Lett. 95, 013904 (2005). [CrossRef] [PubMed]
  5. S. Tomljenovic-Hanic, M. J. Steel, C. M. de Sterke and J. Salzman, "Diamond based photonic crystal microcavities, " Opt. Express 14, 3556-3562 (2006). [CrossRef] [PubMed]
  6. T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, "Fast bistable all-optical switch and memory on a silicon photonic crystal on-chip," Opt. Lett. 30, 2575-2577 (2005). [CrossRef] [PubMed]
  7. C. Grillet, C. Smith, D. Freeman, S. Madden, B. Luther-Davies, E. C. Mägi, D. J. Moss, and B. J. Eggleton, "Efficient coupling o chalcogenide glass photonic crystal waveguide via silica optical fiber nanowires," Opt. Express 14, 1070-1078 (2006). [CrossRef] [PubMed]
  8. M. Loncar, and A. Scherer, "Microfabricated optical cavities and photonic crystals" in Optical microcavities, K. Vahala, ed. (World Scientific Publishing, 2004).
  9. M. Loncar and A. Scherer, "Photonic crystal laser sources for chemical detection," Appl. Phys. Lett. 82, 4648-4650 (2003). [CrossRef]
  10. B. Maune, M. Loncar, J. Wtzens, M. Hochberg, T. Baehr-Jones, and Y. Qiu, "Liquid-crystal electric tuning of a photonic crystal laser," Appl. Phys. Lett. 85, 360-362 (2004). [CrossRef]
  11. Y. Akahane, T. Asano, B. S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944-947 (2003). [CrossRef] [PubMed]
  12. Y. Akahane, T. Asano, B. S. Song, and S. Noda, "Fine-tuned high-Q photonic-crystal nanocavity," Opt. Express 13, 1202-1214 (2005). [CrossRef] [PubMed]
  13. Z. Zhang, and M. Qiu, "Small-volume waveguide-section high Q microcavities in 2D photonic crystal slabs," Opt. Express 12, 3988-3995 (2004). [CrossRef] [PubMed]
  14. B. S. Song, S. Noda, T. Asano and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Nature Mater. 4, 207-210 (2005). [CrossRef]
  15. B. S. Song, T. Asano, Y. Akahane, Y. Tanaka, and S. Noda, "Transmission and reflection characteristics of in-plane hetero-photonic crystals," Appl. Phys. Lett. 85, 4591-4593 (2004). [CrossRef]
  16. E. Kuramochi, M. Natomi, S. Mitsugi, A. Shinya, and T. Tanabe, "Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect," Appl. Phys. Lett. 88, 041112 (2006). [CrossRef]
  17. T. Asano, B. S. Song, and S. Noda, "Analysis of the experimental Q factors (~1 million) of photonic crystal nanocavities," Opt. Express 14, 1996-2002 (2006). [PubMed]
  18. S. Tomljenovic-Hanic, M. J. Steel, C. M. de Sterke and D. J. Moss, "High-Q cavities in photosensitive photonic crystals," Opt. Lett. (to be published). [PubMed]
  19. E. Istrate and E. H. Sargent, "Photonic crystal heterostructures and interfaces," Rev. Modern Phys. 78, 455-481 (2006). [CrossRef]
  20. G. P. Harmon, "Polymers for optical fibers and waveguides: An Overview," in Optical polymers fibers and waveguides, J. P. Harmon, and G. K. Noren, eds. (American Chemical Society, 2001) pp. 1-23.
  21. G. Wu, J. Wang, J. Shen, T. Yang, Q. Zhang, B. Zhou, Z. Deng, F. Bin, D. Zhou, and F. Zhang, "Properties of sol-gel derived scratch-resistant nano-porous silica films by a mixed atmosphere treatment," J. Non-Cryst. Solids 275, 169-174 (2000). [CrossRef]
  22. V. A. Mandelshtam and H. S. Taylor, "Harmonic inversion of time signals," J. Chem. Phys. 107, 6756-6769 (1997). [CrossRef]

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