OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 25 — Dec. 6, 2010
  • pp: 25693–25701
« Show journal navigation

Paired modes of heterostructure cavities in photonic crystal waveguides with split band edges

Sahand Mahmoodian, Andrey A. Sukhorukov, Sangwoo Ha, Andrei V. Lavrinenko, Christopher G. Poulton, Kokou B. Dossou, Lindsay C. Botten, Ross C. McPhedran, and C. Martijn de Sterke  »View Author Affiliations


Optics Express, Vol. 18, Issue 25, pp. 25693-25701 (2010)
http://dx.doi.org/10.1364/OE.18.025693


View Full Text Article

Acrobat PDF (2423 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We investigate the modes of double heterostructure cavities where the underlying photonic crystal waveguide has been dispersion engineered to have two band-edges inside the Brillouin zone. By deriving and using a perturbative method, we show that these structures possess two modes. For unapodized cavities, the relative detuning of the two modes can be controlled by changing the cavity length, and for particular lengths, a resonant-like effect makes the modes degenerate. For apodized cavities no such resonances exist and the modes are always non-degenerate.

© 2010 Optical Society of America

1. Introduction

Fig. 1 Schematic of DHCs and their dispersion curves. (a) DHC based on a W1 waveguide. The highlighted region indicates the underlying PCW being perturbed to create a cavity. (b) Dispersion curve of the W1 PCW. Broken cyan curve shows the dispersion curve of a PCW with the background index changed to nb = 3.005. Green highlighting shows the mode gap. Vertical dashed line indicates BZ edge. (c) A DHC based on a PCW where the holes adjacent to the waveguide have had their radius increased to engineer the dispersion of the band minimum. (d) Dispersion curve of dispersion engineered PCW. The two band minima are at kxd/π = ±0.8034 (a reciprocal lattice vector is added in figure).

On the other hand, it was found that the existence and characteristics of cavity modes are fundamentally modified if optical resonators are based on waveguides whose dispersion features ‘split band-edges’ (SBE), which appear inside the Brillouin zone (BZ). Examples of such resonators include sections of omniguide fibers [12

12. M. Ibanescu, S. G. Johnson, D. Roundy, Y. Fink, and J. D. Joannopoulos, “Microcavity confinement based on an anomalous zero group-velocity waveguide mode,” Opt. Lett. 30, 552–554 (2005). [CrossRef] [PubMed]

], periodic structures with anisotropic layers [13

13. A. Figotin and I. Vitebskiy, “Slow-wave resonance in periodic stacks of anisotropic layers,” Phys. Rev. A 76, 053839 (2007). [CrossRef]

15

15. K. Y. Jung and F. L. Teixeira, “Numerical study of photonic crystals with a split band edge: Polarization dependence and sensitivity analysis,” Phys. Rev. A 78, 043826 (2008). [CrossRef]

], and side-coupled periodic waveguides [16

16. S. Ha, A. A. Sukhorukov, A. V. Lavrinenko, and Yu. S. Kivshar, “Cavity mode control in side-coupled periodic waveguides: theory and experiment,” Photonics Nanostruct.: Fundam. Appl. 8, 310–317 (2010). [CrossRef]

]. In particular, it was shown that there appear two pairs of forward and backward propagating waves arbitrarily close to split band edges, and nontrivial coupling between all four waves can give rise to a pair of cavity modes with nontrivial tuning properties [16

16. S. Ha, A. A. Sukhorukov, A. V. Lavrinenko, and Yu. S. Kivshar, “Cavity mode control in side-coupled periodic waveguides: theory and experiment,” Photonics Nanostruct.: Fundam. Appl. 8, 310–317 (2010). [CrossRef]

].

2. Formulation of theory

We now need to make further approximations if we wish to proceed analytically. DHCs are typically created by weakly perturbing a PCW, and consequently their modes are weakly bound along the x-direction. Thus the typical width of a DHC mode is larger than a period of the PCW. C(kx) is then expected to be narrow in reciprocal space. Since there are two inequivalent band minima, C(kx) is then composed of two narrow peaks centred on each minimum. We can thus write C(kx)=F1(kxkL(1))+F2(kxkL(2)), where kL(1) and kL(2) are the Bloch wavevectors of the two band minima. We note that according to the general symmetry properties of Bloch modes in dielectric PCs [19

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

21

21. S. W. Ha, A. A. Sukhorukov, K. B. Dossou, L. C. Botten, A. V. Lavrinenko, D. N. Chigrin, and Yu. S. Kivshar, “Dispersionless tunneling of slow light in antisymmetric photonic crystal couplers,” Opt. Express 16, 1104–1114 (2008). [CrossRef] [PubMed]

],
kL(2)=kL(1),ωkx=ωkx,Ekx(r)=Ekx*(r).
(5)
Provided that the separation between kL(1) and kL(2) is large compared to their widths we can rewrite Eq. (4) as two equations, one where kx has values where the function F1(kxkL(1)) is dominant and another where F2(kxkL(2)) is dominant. The first of the two equations is
F1(kxkL(1))(ωkx2ω2)=ω2dkx[F1(kxkL(1))+F2(kxkL(2))]2drδɛ(r)Ekx*(r)Ekx(r).
(6)
We now multiply by exp[i(kxkL(1))x] and integrate over kx. On the left hand side this gives
M(ωiddx2ω2)f1(x)
(7)
where f1(x)=F1(kxkL(1))ei(kxkL(1))xdkx is an envelope function. The operator, ωid/dx2 denotes an expansion about the band minimum in powers of –id/dx and arises as a result of inverse Fourier transforming with respect to kx. It is given by ωid/dx2ωL2(ωL/DL)d2/dx2, where ωL is the frequency of the band-edge and DL is the curvature of the band-edge. We choose to keep only the leading order derivative as the band-edge is quadratic to first order and the envelope function along x is slowly varying compared to the period of the PCW. We now turn to the RHS of Eq. (6). Since F1 and F2 are narrow functions, the main Bloch mode contribution is from the Bloch mode corresponding to the Bloch wavevector upon which the minimum is centred, that is, modes corresponding to the Bloch wavevectors kL(1) and kL(2) respectively. We thus write Ekx(r)EkL(j)(r)ei(kxkL(j))x where j denotes either 1 or 2. After some manipulation, the final equation is
L^f1(x)=2πω2M[f1(x)δ11+f2(x)δ12].
(8)
where f1(x) and f2(x) are envelope functions and
L^=(ω2ωiddx2),δjm=dyδɛ(r)EkL(j)*(r)EkL(m)(r).
(9)
Due to the symmetry relations formulated in Eq. (5), δℰ11 = δℰ22 and δ12=δ21*. Carrying out the same process as above for the second equation we write the two equations in matrix form giving
[L^+ω2dδ11(x)ω2dδ12(x)ω2dδ12*(x)L^+ω2dδ11(x)][f1(x)f2(x)]=[00],
(10)
where we have recast the normalisation parameter M in terms of that over a unit cell, that is, M = (2π/d) with =celldxdyɛ(r)||E(r)||2. We note that the diagonal terms are equivalent and the off-diagonal terms are each other’s complex conjugates, as a consequence of symmetry relations in Eq. (5). Thus, when computing the frequency of the DHC modes the magnitude of the diagonal terms determine how deeply the frequency of the modes moves into the modegap, while the off-diagonal terms determine the relative frequency detuning of the two states. The states will then become degenerate if the contribution of the off-diagonal terms vanishes.

Finally, from Eq. (3) the cavity mode is given by
E(r)=f1(x)EkL(1)(r)+f2(x)EkL(2)(r),
(11)
which implies that the mode is composed of a superposition of band-edge Bloch modes each modulated by an envelope function given by solving Eq. (10). By using the symmetry relations in Eq. (5), we can rewrite Eq. (11) as
E(r)=fR(x)Re[EkL(1)(r)]+fI(x)Im[EkL(1)(r)],
(12)
where fR = f1 + f2 and fI = if1if2. Accordingly, the eigenmode of Eq. (10) can be reformulated for the real envelope functions fR(x) and fI(x),
[L^+ω2d[δ11(x)]+Re(δ12(x))]ω2dIm[δ12(x)]ω2dIm[δ12(x)]L^+ω2d[δ11(x)Re(δ12(x))]][fR(x)fI(x)]=[00].
(13)
We now can see from Eqs. (12) and (13) that the profiles of cavity modes are real-valued (up to a constant overall phase), being a superposition of real and imaginary parts of the complex Bloch waves with the corresponding envelope functions. Although the phase of a Bloch function can be chosen freely, leading to changes in its real and imaginary parts, the final mode profile does not depend on the overall phase of the Bloch wave.

3. Results and Discussion

We now use our theory to compute the DHC modes. We examine unapodized cavities where we create the DHC by altering the refractive index of the background material by Δn = 0.005, as well as apodized cavities where the index of the background is altered according to a Gaussian profile with a maximum index change of Δn = 0.005. Creating the cavity in this way is similar to the photosensitive cavities previously investigated numerically in [8

8. S. Tomljenovic Hanic, M. J. Steel, C. M. de Sterke, and D. J. Moss, “High-Q cavities in photosensitive photonic crystals,” Opt. Lett. 32, 542–544 (2007). [CrossRef] [PubMed]

] and experimentally in [22

22. M. W. Lee, C. Grillet, S. Tomljenovic Hanic, E. C. Magi, D. J. Moss, B. J. Eggleton, X. Gai, S. Madden, D. Y. Choi, D. A. P. Bulla, and B. Luther-Davies, “Photowritten high-Q cavities in two-dimensional chalcogenide glass photonic crystals,” Opt. Lett. 34, 3671–3673 (2009). [CrossRef] [PubMed]

]. Figures 2(a) and 2(d) show the frequency of the DHC modes as a function of the length of the cavity for unapodized and Gaussian cavities respectively. Here, the numerical calculations were carried out using the finite element package COMSOL. The existence of two modes is a result of the fact that there are two band minima. We highlight the intertwining of the two lines in Fig. 2(a). The difference in frequency of the two modes is shown in Figs. 2(b) and 2(e). This frequency splitting is associated with the off-diagonal terms in Eq. (10). There is a small discrepancy between the numerics and our theory for the splitting as result of using a single band-edge mode at each minimum and assuming weak coupling between the two minima. These assumptions lead to a slight difference between numerics and theory in the k-space distribution of the field, in turn leading to a slight shift in the positions of the zero-crossings in Fig. 2(b). To correct this discrepancy a higher order theory would be necessary. As shown in Figs. 2(c) and 2(f), the average frequency of the two modes is in excellent agreement with the numerics, which indicates that the diagonal terms in Eq. (10) accurately describe how deep the modes move into the bandgap. Importantly, our theory highlights the key difference between the apodized and unapodized cavities, that is, the unapodized cavities have modes whose detuning oscillates with width, while the apodized cavity’s modes have a detuning which decreases with cavity length.

Fig. 2 (a) Frequency of DHC modes versus cavity length for unapodized cavity in PCW with background index nb = 3.0, air hole radius 0.3d, where d is the period, and radius of holes adjacent to the waveguide are changed to 0.4d. The background index has been altered to nb = 3.005 to create the cavity. Broken red line corresponds to modes calculated using a fully numerical computation (COMSOL). The blue line corresponds to modes calculated using our perturbation theory. Top of figure corresponds to the band-edge frequency, d/λ = 0.26626, while the bottom is the frequency of the edge of the modegap, d/λ = 0.26584. (b) The detuning of the frequency of the two modes in (a). (c) Average of frequencies of the two modes in (a). (d)–(f) Same as (a)–(c) but for the Gaussian cavity with a maximum background index of nb = 3.005.

Figures 3(a) and 3(b) show the modes of the DHC with a length of 9d computed with our perturbation theory. Here, the odd mode is now the fundamental mode as, from Fig. 2(a), it is evident that the modes have crossed. Figure 3(c) shows a comparison between a full numerical calculation of the fields and our perturbation theory. The good agreement indicates that the theory correctly computes both the decay rate of the envelope functions as well as their phases. From these figures it is clear that the perturbation theory contains the underlying physics governing the behaviour of these modes. The theory thus has the potential to be used as a first-principles guide to designing such structures.

Fig. 3 Normalised amplitudes (arbitrary units) of |Ey| for the DHC modes where the DHC has a length of 9d and is centred on x/d = 0. (a) |Ey| for the fundamental mode with frequency d/λ = 0.26607 computed using perturbation theory. (b) |Ey| for the higher order mode with frequency d/λ = 0.26611 computed using perturbation theory. (c) Cross section of (a) taken along the centre of the PCW (blue curve), but with an added comparison to fully numerical calculations of the fields using a finite element method (COMSOL) (broken red curve). Broken vertical lines indicate the edges of the cavity.

We now analyse the modes of the unapodized cavity. As shown in Fig. 2(a) both the numerical solutions and the perturbation theory predict crossings of the two modes at periodic intervals, meaning that we expect the fundamental mode to alternate from being even to odd with respect to the cavity centre. From Eq. (10) it is apparent that this behaviour is associated with the contribution of the off-diagonal terms. When these terms are large, the difference in frequency of the two modes is large, and when their contribution vanishes, the equations decouple and the modes are degenerate. Here, the δℰij(x) terms depend on x and do not completely vanish unless the perturbation is zero. At the degenerate points however, they evidently have no net contribution to the frequency of the DHC modes. From the symmetry relations in Eq. (5), it is easy to see that the off-diagonal terms are oscillating complex valued functions. When these terms are such that they average to zero along the length of the cavity, the contributions of the off-diagonal terms vanish and the modes become degenerate.

Figure 2(b) shows the frequency of the modes for the apodized cavity. Here, the dependence of the frequency of the modes on the length of the cavity is different to that of the unapodized cavity. This difference between the apodized and unapodized cavities is best explained in terms of the envelope functions in reciprocal space. Recall that the functions F1(kxkL(1)) and F2(kxkL(2)) are the FTs of the envelope functions. As the envelope of the DHC mode is slowly varying, these functions are narrow in comparison to their separation in reciprocal space and thus couple weakly. In order to illustrate how the band extrema couple in reciprocal space, we solve Eq. (10) with the coupling terms artificially set to zero. This gives us the shape of F1(kxkL(1)) and F2(kxkL(2)) in the absence of coupling. This is shown for the unapodized cavity in Fig. 4(a). The sidelobes of the maximum centred on the origin are associated with the reflections at the abrupt sidewalls of the cavity [10

10. T. Asano, B. S. Song, Y. Akahane, and S. Noda, “Ultrahigh-Q nanocavities in two-dimensional photonic crystal slabs,” IEEE J. Sel. Top. Quantum Electron. 12, 1123–1134 (2006). [CrossRef]

]. The blue curve is for a cavity with length 10.5d and has a zero at kxdkL(1)d1.2. This zero coincides very closely to the centre of the equivalent peak for F2(kxkL(2)) and thus the frequencies of the two modes of a DHC with length 10.5d are almost degenerate [c.f. Fig. 2(a)]. The broken green line in Fig. 4(a) is for a cavity with length 8d. Its sidelobes do not have a zero near kx=jL(2) and thus its modes are strongly coupled. We thus deduce that when one of the sidelobes of F1(kxkL(1)) coincides with the point where F2(kxkL(2)) is centred, it decouples the contribution from the two band extrema and the modes become degenerate. These sidelobes lead to the intersections of the frequency of the DHC modes as a function of the cavity length shown in Fig. 2(a). The function F1(kxkL(1)) in the absence of coupling for a Gaussian cavity is shown in Fig. 4(b). Here, the reflections at the edges of the cavity are “gentle” and as a result such sidelobes do not exist in reciprocal space. The behaviour of the frequency of DHC modes for apodized cavities is then simple: as the cavity length increases the envelope becomes wider in space and more confined in reciprocal space and hence the coupling between the two band extrema decreases. This agrees with the behaviour shown in Fig. 2(b).

Fig. 4 Sidelobes of F1(kxkL(1)) (arbitrary units) in the absence of coupling. (a) Broken green curve is for an unapodized cavity of length 8d. The blue curve is for an unapodized cavity with length 10.5d, i.e. near a point where the modes become degenerate. Although not shown, F2(kxkL(1)) has the same shape as F1(kxkL(1)), but is centred on kx=kL(2) indicated by the vertical dashed line. (b) Same as (a) but for a Gaussian cavity. The broken green curve is for a cavity with a 1/e length of 4d, while the blue curve is for a cavity with a 1/e length of 5.25d.

Although we used a two-dimensional model to analyze the modes of split band DHCs, we expect to observe the same behavior for such modes in PC slab structures. In fact, our theoretical model can still be used in PC slab structures with minor modifications. A three-dimensional derivation ends up only differing via the integral in Eq. (9), which also contains an integration over the out-of-plane coordinate.

4. Conclusion

In conclusion, we have suggested and demonstrated new approaches for flexible control of cavity modes in double-heterostructure photonic-crystal cavities. We have shown that cavities created in dispersion engineered photonic crystal waveguides featuring split band edges inside the Brillouin zone generally support a pair of cavity modes. The mode detuning depends non-trivially on the length and shape (apodized or unapodized) of the cavity region, offering possibilities to either increase or fully cancel the mode detuning. We also anticipate that these results may lead to advances in realization of tailored nonlinear light-matter interactions based on the paired cavity modes.

Acknowledgments

The support of the Australian Research Council through its Centres of Excellence and Discovery Programs is acknowledged.

References and links

1.

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nat. Photonics 4, 477–483 (2010). [CrossRef]

2.

J. Vuĉković and Y. Yamamoto, “Photonic crystal microcavities for cavity quantum electrodynamics with a single quantum dot,” Appl. Phys. Lett. 82, 2374–2376 (2003). [CrossRef]

3.

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78–82 (2009). [CrossRef] [PubMed]

4.

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, 1819–1821 (1999). [CrossRef] [PubMed]

5.

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

6.

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

7.

Y. Tanaka, T. Asano, and S. Noda, “Design of photonic crystal nanocavity with Q-factor of ∼109,” J. Lightwave Technol. 26, 1532–1539 (2008). [CrossRef]

8.

S. Tomljenovic Hanic, M. J. Steel, C. M. de Sterke, and D. J. Moss, “High-Q cavities in photosensitive photonic crystals,” Opt. Lett. 32, 542–544 (2007). [CrossRef] [PubMed]

9.

S. Gardin, F. Bordas, X. Letartre, C. Seassal, A. Rahmani, R. Bozio, and P. Viktorovitch, “Microlasers based on effective index confined slow light modes in photonic crystal waveguides,” Opt. Express 16, 6331–6339 (2008). [CrossRef] [PubMed]

10.

T. Asano, B. S. Song, Y. Akahane, and S. Noda, “Ultrahigh-Q nanocavities in two-dimensional photonic crystal slabs,” IEEE J. Sel. Top. Quantum Electron. 12, 1123–1134 (2006). [CrossRef]

11.

D. Englund, I. Fushman, and J. Vuĉković, “General recipe for designing photonic crystal cavities,” Opt. Express 13, 5961–5975 (2005). [CrossRef] [PubMed]

12.

M. Ibanescu, S. G. Johnson, D. Roundy, Y. Fink, and J. D. Joannopoulos, “Microcavity confinement based on an anomalous zero group-velocity waveguide mode,” Opt. Lett. 30, 552–554 (2005). [CrossRef] [PubMed]

13.

A. Figotin and I. Vitebskiy, “Slow-wave resonance in periodic stacks of anisotropic layers,” Phys. Rev. A 76, 053839 (2007). [CrossRef]

14.

A. A. Chabanov, “Strongly resonant transmission of electromagnetic radiation in periodic anisotropic layered media,” Phys. Rev. A 77, 033811 (2008). [CrossRef]

15.

K. Y. Jung and F. L. Teixeira, “Numerical study of photonic crystals with a split band edge: Polarization dependence and sensitivity analysis,” Phys. Rev. A 78, 043826 (2008). [CrossRef]

16.

S. Ha, A. A. Sukhorukov, A. V. Lavrinenko, and Yu. S. Kivshar, “Cavity mode control in side-coupled periodic waveguides: theory and experiment,” Photonics Nanostruct.: Fundam. Appl. 8, 310–317 (2010). [CrossRef]

17.

S. Mahmoodian, C. G. Poulton, K. B. Dossou, R. C. McPhedran, L. C. Botten, and C. M. de Sterke, “Modes of Shallow Photonic Crystal Waveguides: Semi-Analytic Treatment,” Opt. Express 17, 19629–19643 (2009). [CrossRef] [PubMed]

18.

J. M. Luttinger and W. Kohn, “Motion of electrons and holes in perturbed periodic fields,” Phys. Rev. 97, 869–883 (1955). [CrossRef]

19.

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

20.

P. St. J. Russell, T. A. Birks, and F. D. Lloyd Lucas, “Photonic Bloch waves and photonic band gaps,” in Confined Electrons and Photons, E. Burstein and C. Weisbuch, eds., (1995), pp. 585–633.

21.

S. W. Ha, A. A. Sukhorukov, K. B. Dossou, L. C. Botten, A. V. Lavrinenko, D. N. Chigrin, and Yu. S. Kivshar, “Dispersionless tunneling of slow light in antisymmetric photonic crystal couplers,” Opt. Express 16, 1104–1114 (2008). [CrossRef] [PubMed]

22.

M. W. Lee, C. Grillet, S. Tomljenovic Hanic, E. C. Magi, D. J. Moss, B. J. Eggleton, X. Gai, S. Madden, D. Y. Choi, D. A. P. Bulla, and B. Luther-Davies, “Photowritten high-Q cavities in two-dimensional chalcogenide glass photonic crystals,” Opt. Lett. 34, 3671–3673 (2009). [CrossRef] [PubMed]

OCIS Codes
(130.2790) Integrated optics : Guided waves
(140.3948) Lasers and laser optics : Microcavity devices
(350.4238) Other areas of optics : Nanophotonics and photonic crystals
(130.5296) Integrated optics : Photonic crystal waveguides
(050.5298) Diffraction and gratings : Photonic crystals
(230.5298) Optical devices : Photonic crystals

ToC Category:
Integrated Optics

History
Original Manuscript: September 21, 2010
Revised Manuscript: November 18, 2010
Manuscript Accepted: November 20, 2010
Published: November 23, 2010

Citation
Sahand Mahmoodian, Andrey A. Sukhorukov, Sangwoo Ha, Andrei V. Lavrinenko, Christopher G. Poulton, Kokou B. Dossou, Lindsay C. Botten, Ross C. McPhedran, and C. M. de Sterke, "Paired modes of heterostructure cavities in photonic crystal waveguides with split band edges," Opt. Express 18, 25693-25701 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-25-25693


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, "Sub-femtojoule all-optical switching using a photonic-crystal nanocavity," Nat. Photonics 4, 477-483 (2010). [CrossRef]
  2. J. Vuckovic, and Y. Yamamoto, "Photonic crystal microcavities for cavity quantum electrodynamics with a single quantum dot," Appl. Phys. Lett. 82, 2374-2376 (2003). [CrossRef]
  3. M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, "Optomechanical crystals," Nature 462, 78-82 (2009). [CrossRef] [PubMed]
  4. 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, 1819-1821 (1999). [CrossRef] [PubMed]
  5. B. S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Nat. Mater. 4, 207-210 (2005). [CrossRef]
  6. E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, "Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect," Appl. Phys. Lett. 88, 041112 (2006). [CrossRef]
  7. Y. Tanaka, T. Asano, and S. Noda, "Design of photonic crystal nanocavity with Q-factor of ~109," J. Lightwave Technol. 26, 1532-1539 (2008). [CrossRef]
  8. S. Tomljenovic Hanic, M. J. Steel, C. M. de Sterke, and D. J. Moss, "High-Q cavities in photosensitive photonic crystals," Opt. Lett. 32, 542-544 (2007). [CrossRef] [PubMed]
  9. S. Gardin, F. Bordas, X. Letartre, C. Seassal, A. Rahmani, R. Bozio, and P. Viktorovitch, "Microlasers based on effective index confined slow light modes in photonic crystal waveguides," Opt. Express 16, 6331-6339 (2008). [CrossRef] [PubMed]
  10. T. Asano, B. S. Song, Y. Akahane, and S. Noda, "Ultrahigh-Q nanocavities in two-dimensional photonic crystal slabs," IEEE J. Sel. Top. Quantum Electron. 12, 1123-1134 (2006). [CrossRef]
  11. D. Englund, I. Fushman, and J. Vuckovic, "General recipe for designing photonic crystal cavities," Opt. Express 13, 5961-5975 (2005). [CrossRef] [PubMed]
  12. M. Ibanescu, S. G. Johnson, D. Roundy, Y. Fink, and J. D. Joannopoulos, "Microcavity confinement based on an anomalous zero group-velocity waveguide mode," Opt. Lett. 30, 552-554 (2005). [CrossRef] [PubMed]
  13. A. Figotin, and I. Vitebskiy, "Slow-wave resonance in periodic stacks of anisotropic layers," Phys. Rev. A 76, 053839 (2007). [CrossRef]
  14. A. A. Chabanov, "Strongly resonant transmission of electromagnetic radiation in periodic anisotropic layered media," Phys. Rev. A 77, 033811 (2008). [CrossRef]
  15. K. Y. Jung, and F. L. Teixeira, "Numerical study of photonic crystals with a split band edge: Polarization dependence and sensitivity analysis," Phys. Rev. A 78, 043826 (2008). [CrossRef]
  16. S. Ha, A. A. Sukhorukov, A. V. Lavrinenko, and Yu. S. Kivshar, "Cavity mode control in side-coupled periodic waveguides: theory and experiment," Photonics Nanostruct.: Fundam. Appl. 8, 310-317 (2010). [CrossRef]
  17. S. Mahmoodian, C. G. Poulton, K. B. Dossou, R. C. McPhedran, L. C. Botten, and C. M. de Sterke, "Modes of Shallow Photonic Crystal Waveguides: Semi-Analytic Treatment," Opt. Express 17, 19629-19643 (2009). [CrossRef] [PubMed]
  18. J. M. Luttinger, and W. Kohn, "Motion of electrons and holes in perturbed periodic fields," Phys. Rev. 97, 869-883 (1955). [CrossRef]
  19. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).
  20. P. St, J. Russell, T. A. Birks, and F. D. Lloyd Lucas, "Photonic Bloch waves and photonic band gaps," in Confined Electrons and Photons, E. Burstein and C. Weisbuch, eds., (1995), pp. 585-633.
  21. S. W. Ha, A. A. Sukhorukov, K. B. Dossou, L. C. Botten, A. V. Lavrinenko, D. N. Chigrin, and Yu. S. Kivshar, "Dispersionless tunneling of slow light in antisymmetric photonic crystal couplers," Opt. Express 16, 1104-1114 (2008). [CrossRef] [PubMed]
  22. M. W. Lee, C. Grillet, S. Tomljenovic Hanic, E. C. Magi, D. J. Moss, B. J. Eggleton, X. Gai, S. Madden, D. Y. Choi, D. A. P. Bulla, and B. Luther-Davies, "Photowritten high-Q cavities in two-dimensional chalcogenide glass photonic crystals," Opt. Lett. 34, 3671-3673 (2009). [CrossRef] [PubMed]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2 Fig. 3
 
Fig. 4
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited