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Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 17 — Aug. 23, 2004
  • pp: 3988–3995
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Small-volume waveguide-section high Q microcavities in 2D photonic crystal slabs

Ziyang Zhang and Min Qiu  »View Author Affiliations


Optics Express, Vol. 12, Issue 17, pp. 3988-3995 (2004)
http://dx.doi.org/10.1364/OPEX.12.003988


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Abstract

A series of microcavities in 2D hexagonal lattice photonic crystal slabs are studied in this paper. The microcavities are small sections of a photonic crystal waveguide. Finite difference time domain simulations show that these cavities preserve high Q modes with similar geometrical parameters and field profile. Effective modal volume is reduced gradually in this series of microcavity modes while maintaining high quality factor. Vertical Q value larger than 106 is obtained for one of these cavity modes with effective modal volume around 5.40 cubic half wavelengths [(λ/2nslab ) 3 ]. Another cavity mode provides even smaller modal volume around 2.30 cubic half wavelengths, with vertical Q value exceeding105.

© 2004 Optical Society of America

1. Introduction

The discovery of photonic crystals has opened up many new methods to manipulate light [1

1. E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 58, 2059 (1987) [CrossRef] [PubMed]

2

2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486 (1987) [CrossRef] [PubMed]

]. However, three dimensional photonic crystals usually involves complex connectivity and strict alignment, which make them rather challenging for fabrication [3

3. H. Benisty, “Modal analysis of optical guides with two-dimensional photonic band-gap boundaries,” J. Appl. Phys. 75, 4753 (1994)

4

4. A. Mekis, S. Fan, and J. D. Joannopoulos, “Bound states in photonic crystal waveguides and waveguide bends”, Phys. Rev. B 58, 4809 (1998) [CrossRef]

]. Alternatively, two dimensional photonic crystal slabs (2D PCS’s) have been proposed [5

5. T. F. Krauss, R. M. De La Rue, and S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths”, Nature 383, 699 (1996) [CrossRef]

6

6. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B , 60, 5751 (1999) [CrossRef]

], which promise easier fabrication using present techniques.

There have been many theoretical and experimental studies to improve the quality factor of microcavities in 2D PCS’s. It is known that total Q value (Qtot) can be separated into two parts, namely the in-plane Q value (Q ) and the vertical Q value (Q) by a simple relation 1Qtot=1Q+1Q. In a 2D PCS microcavity, light is confined within the defect region by two combined mechanisms, namely distributed Bragg reflection (in-plane) and total internal reflection (vertically). The in-plane confinement is determined by the number of periods of the host lattice surrounding the cavity, with the assumption that the resonant frequency of the defect mode lies within the in-plane guided mode bandgap. Thus, the more periods surrounding the defect, the stronger the in-plane light confinement and the larger the in-plane Q value (Q). The vertical confinement is due to the standard waveguiding by total internal reflection. In momentum (k) space analysis, vertical radiation losses occur when the cavity modes have in-plane momentum components (k) that lie within a region called ‘light cone’ [14

14. K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express 10, 670 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-670 [CrossRef] [PubMed]

15

15. J. Vuckovic, M. loncar, H. Mabuchi, and A. Scherer, “Optimization of the Q factor in photonic crystal microcavities,” IEEE J. of Quantum Electron. 38, 850 (2002) [CrossRef]

]. The larger the fraction of in-plane momentum components that lie within this light cone, the larger the vertical radiation loss and consequently the smaller the vertical Q value.

2. Calculation methods

We use the finite difference time domain method for our simulations [20

20. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas and Propagation,” 14, 302 (1966)

]. The structure size is 23a×23a for the entire cavity modes calculated in this paper, which is capable of providing sufficiently large Q so that Qtot is primarily limited by Q. The grid size is 0.05a in x, z direction and 0.0433a in y direction. Perfect matched layers are used as absorbing boundaries [21

21. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185 (1994) [CrossRef]

].

The Qtot value of the cavity modes is calculated using a combination of FDTD techniques and Padé approximation with Baker’s algorithm [22

22. M. Qiu, “High Q cavities in photonic crystal slabs: determining resonant frequency and quality factor accurately,” submitted for publication (2004)

23

23. W. H. Guo, W. J. Li, and Y. Z. Huang, “Computation of Resonant Frequencies and Quality Factors of Cavities by FDTD Technique and Padé Approximation,” IEEE Microwave Wireless Components Lett. 11, 223 (2001) [CrossRef]

], which is capable of achieving ultra accurate Q value using a much smaller number of time steps compared to other methods (typically <0.5% numerical errors and ~15,000 time steps for Q>105). For comparison reasons, some high Q values of certain cavity modes are also calculated using conventional method: Q=2π·f0UP, where f0 is the resonant frequency of the cavity mode, U is the time averaged energy stored in the cavity, and P is the power loss. Q and Q are calculated separately using this method.

V=ε(x,y,z)·E(x,y,z)2dxdydzmax[ε(x,y,z)·E(x,y,z)2],
(1)

where ε (x, y, z) is the spatial distribution of dielectric constant and

E(x,y,z)2=Ex(x,y,z)2+Ey(x,y,z)2+Ez(x,y,z)2.
(2)

3. Waveguide-section microcavities

3.1 Three missing hole waveguide-section cavity (M3)

The structure of our 2D PCS cavity is shown in Fig. 1, with three missing holes in the center. The refractive index of the slab is chosen to be 3.4, which corresponds to silicon at 1.55 µm. The thickness of the slab is 0.7a, where a is the lattice constant. The radius of the air holes is set to be R=0.29a. These two parameters are also the same for M2, M1 and M0 cavities.

For simplicity’s concern, we will use the same name (M3, M2 and etc.) for the cavity and its donor mode discussed in this paper, although all these cavities support multiple modes.

Fig. 1. Schematic diagram of 2D PCS microcavities with three central holes missing in a row. The refractive index of the slab is 3.4 and the thickness t is 0.7a. d and R1 are varied to find the largest Q of the M3 mode.

After careful scanning of d and R1, we find the highest Q value 8,000 can be achieved at d=0.24a and R1=0.23a. Figure 2(a) shows the H𝑧 field distribution of the M3 mode. The field is spread out into the surrounding lattice of the cavity and since the resonant frequency 0.3077 a/λ is near to the upper bandedge (~0.32 a/λ), the in-plane quality factor Q is limited to 35,000 even for a large size (23a×23a).

For the k space analysis, we take the 2D (kx, ky) Fourier transform of Ex, Ey and Hx, Hy field distribution in the air layer one FDTD grid cell above the slab surface, denoted by FT 2(Ex), FT 2(Ey), FT 2(Hx) and FT 2(Hy), respectively. The total radiated power P of the cavity mode can be estimated by the following equations [15

15. J. Vuckovic, M. loncar, H. Mabuchi, and A. Scherer, “Optimization of the Q factor in photonic crystal microcavities,” IEEE J. of Quantum Electron. 38, 850 (2002) [CrossRef]

].

P=η8λ2k2kkI·dkx·dky,
(3)

where η=μ0ε0,k=2πλ, and

I=FT2(Hy)+1ηFT2(Ex)2+FT2(Hx)1ηFT2(Ey)2
(4)

The M3 mode is not very exciting compared to the acceptor mode in Ref. [17

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

], which has Qtot=9.91×104 at resonant frequency 0.2618 a/λ when d=0.2a and slab thickness t=0.6a, according to our simulations. For d=0.2a and t=0.7a, Qtot equals 2.16×104 with resonant frequency 0.2549 a/λ. However, the donor mode becomes interesting when the waveguide-section is further shortened. Next, we study cavities with only two air holes removed but its surrounding geometrical structures retained.

Fig. 2. (a) H𝑧 field distribution of the M3 mode. (b) k space intensity profile I. The region inside the blue dashed circle (the light line) is the leaky region.

3.2 Two missing hole waveguide-section cavity (M2)

Fig. 3. (a) M2 cavity, with d=0.23a and R1=0.2a. (b) Hz field distribution of the M2 mode. (c) k space intensity profile I. Components inside the leaky region are reduced compared to the M3 mode in Fig. 2(b)

3.3 One missing hole waveguide-section cavity (M1)

It is interesting to note that M3, M2 and M1 modes all reach their highest Q value when d and R1 are tuned to around 0.22a. We can conclude that tuning d to around 0.22a allows light to penetrate deeper inside the surrounding lattice in the x direction and be reflected gently. Similarly, tuning R1 (and also R2 for M1 cavity) allows the reflections in the y direction to be smoother. The combined effect is that the spatial variation of the envelope function is tailored into a Gaussian profile so that the k components inside the light cone are reduced and thus Q improved. On the other hand, increasing d and decreasing R1 helps pull down the resonant frequency closer to the middle of the bandgap [14

14. K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express 10, 670 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-670 [CrossRef] [PubMed]

], which causes Q to increase.

Fig. 4. (a) M1 cavity, with d=0.21a, R1=0.22a and R2=0.25a (b) Hz field distribution of the M1 mode. (c) k space intensity profile I. Components inside the leaky region are greatly reduced compared to the M2 mode in Fig. 3(c)

3.4 Zero missing hole waveguide-section cavity (M0)

The Hz field distribution is shown in Fig. 5(b). The Hz field profile is now very simple and the number of the nodal points is much reduced. Figure 5(c) shows the k space electric intensity profile. There are very few k components inside the leaky region, which accounts for a large Q (≈1.9×105) of this mode.

Fig. 5. (a) M0 cavity, with d=0.14a and R1=0.27a. (b) Hz field distribution of the M0 mode. (c) k space intensity profile I.

We also calculated the Q value of the other modes supported by the M2, M1 and M0 cavity. The donor modes discussed in this paper have the highest Q value for each cavity respectively.

4. Comparisons

Fig. 6. Electric intensity distribution of (a) M3 mode, (b) M2 mode, (c) M1 mode, and (d) M0 mode.

Fig. 7. (a) Q value (the blue line with diamond marker) and modal volume (the green line with circle marker) comparison of the four modes. (b) Q (the blue line with diamond marker) and radiation factor RF (the green line with circle marker) comparison of the four modes. RF is normalized to the M1 mode.

5. Summary

Acknowledgments

This work was supported by the Swedish Foundation for Strategic Research (SSF) on Photonics and the Swedish Research Council (VR) under project 2003-5501.

References and links

1.

E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 58, 2059 (1987) [CrossRef] [PubMed]

2.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486 (1987) [CrossRef] [PubMed]

3.

H. Benisty, “Modal analysis of optical guides with two-dimensional photonic band-gap boundaries,” J. Appl. Phys. 75, 4753 (1994)

4.

A. Mekis, S. Fan, and J. D. Joannopoulos, “Bound states in photonic crystal waveguides and waveguide bends”, Phys. Rev. B 58, 4809 (1998) [CrossRef]

5.

T. F. Krauss, R. M. De La Rue, and S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths”, Nature 383, 699 (1996) [CrossRef]

6.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B , 60, 5751 (1999) [CrossRef]

7.

S. Fan, Pierre R. Villeneuve, and J. D. Joannopoulos, “Channel Drop Tunneling through Localized States,” Phys. Rev. Lett. 80, 960 (1998) [CrossRef]

8.

M. Qiu, “Ultra-compact optical filter in two-dimensional photonic crystal,” Electron. Lett. 40, 539 (2004) [CrossRef]

9.

M. Qiu and B. Jaskorzynska, “A design of a channel drop filter in a two-dimensional triangular photonic crystal”, Appl. Phys. Lett. 83, 1074 (2003). [CrossRef]

10.

S. Fan, Proceedings of the SPIE , v 3002, 1997, p 67–73 [CrossRef]

11.

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” S. M. , Nature 415, 621–623 (2002) [CrossRef]

12.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, Lidong Zhang, E. Hu, and A. Imamoglu, “Quantum Dot Single-Photon Turnstile Device,” Science 290, 2282 (2000) [CrossRef] [PubMed]

13.

C. Santori, D. Fattal, J. Vučković, G. S. Solomon, and Y. Yamamoto, “Indistinguishable photons from a single-photon device,” Nature 419, 594 (2002) [CrossRef] [PubMed]

14.

K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express 10, 670 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-670 [CrossRef] [PubMed]

15.

J. Vuckovic, M. loncar, H. Mabuchi, and A. Scherer, “Optimization of the Q factor in photonic crystal microcavities,” IEEE J. of Quantum Electron. 38, 850 (2002) [CrossRef]

16.

K. Srinivasan and O. Painter, “Fourier space design of high-Q cavities in standard and compressed hexagonal lattice photonic crystals,” Opt. Express 11, 579 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-6-579 [CrossRef] [PubMed]

17.

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

18.

H. Y. Ryu, M. Notomi, and Y. H. Lee, “High-quality-factor and small-mode-volume hexapole modes in photonic-crystal-slab nanocavities,” Appl. Phys. Lett. 83, 4294 (2003) [CrossRef]

19.

E. Yablonovitch and T. J. Gmitter, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380 (1991) [CrossRef] [PubMed]

20.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas and Propagation,” 14, 302 (1966)

21.

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185 (1994) [CrossRef]

22.

M. Qiu, “High Q cavities in photonic crystal slabs: determining resonant frequency and quality factor accurately,” submitted for publication (2004)

23.

W. H. Guo, W. J. Li, and Y. Z. Huang, “Computation of Resonant Frequencies and Quality Factors of Cavities by FDTD Technique and Padé Approximation,” IEEE Microwave Wireless Components Lett. 11, 223 (2001) [CrossRef]

24.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-Dimensional Photonic Band-Gap Defect Mode Laser,” Science 284, 1819(1999) [CrossRef] [PubMed]

25.

J. Vuckovic, M. loncar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E. 65, 016608 (2001) [CrossRef]

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

ToC Category:
Research Papers

History
Original Manuscript: July 12, 2004
Revised Manuscript: August 9, 2004
Published: August 23, 2004

Citation
Ziyang Zhang and Min Qiu, "Small-volume waveguide-section high Q microcavities in 2D photonic crystal slabs," Opt. Express 12, 3988-3995 (2004)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-17-3988


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References

  1. E. Yablonovitch, ཿInhibited Spontaneous Emission in Solid-State Physics and Electronics,ཿ Phys. Rev. Lett. 58, 2059 (1987) [CrossRef] [PubMed]
  2. S. John, ཿStrong localization of photons in certain disordered dielectric superlattices,ཿ Phys. Rev. Lett. 58, 2486 (1987) [CrossRef] [PubMed]
  3. H. Benisty, ཿModal analysis of optical guides with two-dimensional photonic band-gap boundaries,ཿ J. Appl. Phys. 75, 4753 (1994)
  4. A. Mekis, S. Fan, and J. D. Joannopoulos, ཿBound states in photonic crystal waveguides and waveguide bendsཿ, Phys. Rev. B 58, 4809 (1998) [CrossRef]
  5. T. F. Krauss, R. M. De La Rue, and S. Brand, ཿTwo-dimensional photonic-bandgap structures operating at near-infrared wavelengthsཿ, Nature 383, 699 (1996) [CrossRef]
  6. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, ཿGuided modes in photonic crystal slabs,ཿ Phys. Rev. B, 60, 5751 (1999) [CrossRef]
  7. S. Fan, Pierre R. Villeneuve, and J. D. Joannopoulos, ཿChannel Drop Tunneling through Localized States,ཿ Phys. Rev. Lett. 80, 960 (1998) [CrossRef]
  8. M. Qiu, ཿUltra-compact optical filter in two-dimensional photonic crystal,ཿ Electron. Lett. 40, 539 (2004) [CrossRef]
  9. M. Qiu and B. Jaskorzynska, ཿA design of a channel drop filter in a two-dimensional triangular photonic crystalཿ, Appl. Phys. Lett. 83, 1074 (2003). [CrossRef]
  10. S. Fan, Proceedings of the SPIE, v 3002, 1997, p 67-73 [CrossRef]
  11. Spillane, S. M., Kippenberg, T. J. and Vahala, K. J. ཿUltralow-threshold Raman laser using a spherical dielectric microcavity,ཿ S. M. , Nature 415, 621-623 (2002) [CrossRef]
  12. C. Santori, D. Fattal, J. Vu¿¡kovi¿ , G. S. Solomon, and Y. Yamamoto, ཿIndistinguishable photons from a single-photon device,ཿ Nature 419, 594 (2002) [CrossRef] [PubMed]
  13. K. Srinivasan and O. Painter, ཿMomentum space design of high-Q photonic crystal optical cavities,ཿ Opt. Express 10, 670 (2002), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-670">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-670</a> [CrossRef] [PubMed]
  14. J. Vu¿¡kovi¿ , M. lon¿ar, H. Mabuchi and A. Scherer, ཿOptimization of the Q factor in photonic crystal microcavities,ཿ IEEE J. of Quantum Electron. 38, 850 (2002) [CrossRef]
  15. K. Srinivasan and O. Painter, ཿFourier space design of high-Q cavities in standard and compressed hexagonal lattice photonic crystals,ཿ Opt. Express 11, 579 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-6-579">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-6-579</a> [CrossRef] [PubMed]
  16. V. Akahane, T. Asano, B. S. Song, and S. Noda, ཿHigh-Q photonic nanocavity in a two-dimensional photonic crystal,ཿ Nature 425, 944 (2003) [CrossRef] [PubMed]
  17. H. Y. Ryu, M. Notomi, and Y. H. Lee, ཿHigh-quality-factor and small-mode-volume hexapole modes in photonic-crystal-slab nanocavities,ཿ Appl. Phys. Lett. 83, 4294 (2003) [CrossRef]
  18. E.Yablonovitch and T. J. Gmitter, ཿDonor and acceptor modes in photonic band structure,ཿ Phys. Rev. Lett. 67, 3380 (1991) [CrossRef] [PubMed]
  19. K. S. Yee, ཿNumerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,ཿ IEEE Trans. Antennas and Propagation, 14, 302 (1966)
  20. J. P. Berenger, ཿA perfectly matched layer for the absorption of electromagnetic waves,ཿ J. Comput. Phys. 114, 185 (1994) [CrossRef]
  21. M. Qiu, ཿHigh Q cavities in photonic crystal slabs: determining resonant frequency and quality factor accurately,ཿ submitted for publication (2004)
  22. W. H. Guo, W. J. Li, and Y. Z. Huang, ཿComputation of Resonant Frequencies and Quality Factors of Cavities by FDTD Technique and Padé Approximation,ཿ IEEE Microwave Wireless Components Lett. 11, 223 (2001) [CrossRef]
  23. O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. OཿBrien, P. D. Dapkus, and I. Kim, ཿTwo-Dimensional Photonic Band-Gap Defect Mode Laser,ཿ Science 284, 1819 (1999) [CrossRef] [PubMed]
  24. J. Vu¿kovi¿, M. lon¿ar, H. Mabuchi and A. Scherer, ཿDesign of photonic crystal microcavities for cavity QED,ཿ Phys. Rev. E. 65, 016608 (2001) [CrossRef]
  25. P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, Lidong Zhang, E. Hu, and A. Imamoglu, ཿQuantum Dot Single-Photon Turnstile Device,ཿ Science 290, 2282 (2000)

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