## Directional free-space coupling from photonic crystal waveguides |

Optics Express, Vol. 19, Issue 21, pp. 20586-20596 (2011)

http://dx.doi.org/10.1364/OE.19.020586

Acrobat PDF (1365 KB)

### Abstract

We present a general approach for coupling a specific mode in a planar photonic crystal (PC) waveguide to a desired free-space mode. We apply this approach to a W1 PC waveguide by introducing small index perturbations to selectively couple a particular transverse mode to an approximately Gaussian, slowly diverging free space mode. This “perturbative photonic crystal waveguide coupler” (PPCWC) enables efficient interconversion between selectable propagating photonic crystal and free space modes with minimal design perturbations.

© 2011 OSA

## 1. Introduction

2. T. Baba, “Slow light in photonic
crystals,” Nature Photonics **2**, 465–473
(2008). [CrossRef]

3. H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow
light in photonic crystal waveguides” Phys.
Rev. Lett. **94**, 073903 (2005). [CrossRef] [PubMed]

4. J. F. McMillan, M. Yu, D. L. Kwong, and C. W. Wong, “Observation of four-wave mixing in
slow-light silicon photonic crystal waveguides,”
Opt. Express **18**, 15484–15497
(2010). [CrossRef] [PubMed]

5. M. Soljacic, S. G. Johnson, S. Fan, M. Ibanescu, E. Ippen, and J. D. Joannopoulos, “Photonic-crystal slow-light enhancement
of nonlinear phase sensitivity,” J. Opt.
Soc. Am. B **19**, 2052–2058
(2002). [CrossRef]

6. H. Takano, B. S. Song, T. Asano, and S. Noda, “Highly efficient multi-channel drop
filter in a two-dimensional hetero photonic
crystal,” Opt. Express **14**, 3491–3496
(2006). [CrossRef] [PubMed]

8. S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by
single defect in a photonic bandgap structure,”
Nature (London) **407**, 608–610
(2000). [CrossRef]

9. M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H. Ryu, “Waveguides, resonators, and their
coupled elements in photonic crystal slabs,”
Opt. Express **12**, 1551–1561
(2004). [CrossRef] [PubMed]

10. P. Bienstman, S. Assefa, S. G. Johnson, J. D. Joannopoulos, G. S. Petrich, and L. A. Kolodziejski, “Taper structures for coupling into
photonic crystal slab waveguides,” J. Opt.
Soc. Am. B **20**, 1817–1821
(2003). [CrossRef]

11. P. E. Barclay, K. Srinivasan, and O. Painter, “Design of photonic crystal waveguides
for evanescent coupling to optical fiber tapers and integration with high-Q
cavities,” J. Opt. Soc. Am. B **20**, 2274–2284
(2003). [CrossRef]

12. B. Wang, J. Jiang, and G. P. Nordin, “Compact slanted grating
couplers,” Opt. Express **12**, 3313–3326
(2004). [CrossRef] [PubMed]

13. A. Mizutani, Naoki Ikeda, Y. Watanabe, N. Ozaki, Y. Takata, Y. Kitagawa, F. Laere, R. Baets, Y. Sugimoto, and K. Asakawa, “Planar focusing lens grating for vertical coupling on 2D photonic crystal slab waveguide,” in Lasers and Electro-Optics Society, 2006. LEOS 2006. 19th Annual Meeting of the IEEE, pp. 843–844 (2006).

14. A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vuckovic, “Dipole induced transparency in
waveguide coupled photonic crystal cavities,”
Opt. Express **16**, 12154–12162
(2008). [CrossRef] [PubMed]

15. M. Toishi, D. Englund, A. Faraon, and J. Vuckovic, “High-brightness single photon source
from a quantum dot in a directional-emission
nanocavity,” Opt. Express **17**, 14618–14626
(2009). [CrossRef] [PubMed]

17. Y. Tanaka, M. Tymczenko, T. Asano, and S. Noda, “Fabrication of two-dimensional photonic
crystal slab point-defect cavity employing local three-dimensional
structures,” Japanese J. Appl.
Phys. **45**, 6096–6102
(2006). [CrossRef]

*k*-space, and in Section 4 we analyze the transmittance and the modification of the shape and the directionality of the scattered beam.

## 2. Coupled mode theory

*ɛ*(

*r⃗*) is the relative dielectric constant and

*c*is the speed of light in vacuum. We define

*ɛ*as the relative dielectric constant for the waveguide, which is assumed to be periodic along the propagating direction. The solution of Eq. (1) with

_{w}*ɛ*=

*ɛ*is in form of

_{w}*B⃗*are Bloch states having periodicities as the multiples of the periodicity of

_{w}*ɛ*, where

_{w}*k⃗*represents the crystal momentum and

*x̂*is the propagating direction of the waveguide. We desire a scattered beam propagating in free space denoted by

*A⃗*(

*r⃗*).

*A⃗*is also a solution of Eq. (1) with the relative dielectric constant

*ɛ*=

*ɛ*. We assume the field to be a superposition of the waveguide and free space modes:

_{G}*a*(

*t*) is the slowly varying component of the Gaussian beam, and

*b*(

*k⃗*,

*t*) and

*c*(

*k⃗*,

*t*) are the slowly varying components of the forward and backward propagating Bloch modes for the waveguide, respectively. Substituting Eq. (3) into Eq. (1), we can obtain the full coupled mode equations describing the interaction of the coefficients

*a*(

*t*),

*b*(

*k⃗*,

*t*), and

*c*(

*k⃗*,

*t*) through Δ

*ɛ*. In particular, the coupling rate between

*a*(

*t*) and

*b*(

*k⃗*,

*t*) can be evaluated [7

7. E. Waks and J. Vuckovic, “Coupled mode theory for photonic
crystal cavity-waveguide interaction,” Opt.
Express **13**, 5064–5073
(2005). [CrossRef] [PubMed]

19. A. Yariv, “Coupled-mode theory for guided-wave
optics,” IEEE J. Quantum Electron. **9**, 919–933
(1973). [CrossRef]

*ɛ*=

_{w}*ɛ*–

_{t}*ɛ*, representing the index perturbations introduced into the already existing structures. The subscript letters

_{w}*w*and

*t*denote the existing waveguide and the target structure.

*A⃗*·

^{*}*B⃗*and the perturbation term Δ

_{k⃗}*ɛ*. Since the phase of A is nearly constant in the plane of the PC, whereas

_{w}*B⃗*varies rapidly along the direction of propagation, the spatial integral of the dot product

_{k⃗}*A⃗*·

^{*}*B⃗*will vanish. For a non-vanishing coupling constant, the index perturbation Δ

_{k⃗}*ɛ*should have the same periodicity as

_{w}*B⃗*, assuming an interaction region that is large compared to the period

_{k⃗}*a*.

*a*, refractive index of the slab material

*n*= 3.6, slab thickness

*h*= 0.7

*a*, and hole radius

*r*= 0.3

*a*. The full coupling system is as shown in Fig. 1(a). We used a 3D FDTD simulation to calculate the waveguide modes. Figure 1(b) shows the dispersion relation of the W1 photonic crystal waveguide. A fundamental and higher order mode inside the photonic bandgap (PBG), wedged by the areas shaded in gray, are shown in blue and green, respectively, and are confined below the light line, represented by the red line. Figures 2(a) and 2(b) show the

*x̂*and

*ŷ*electric field components of the fundamental mode, which we label as

*B⃗*, at wave-vector

_{k⃗}*kx*=

*π*/

*a*, and frequency

*ω*= 0.2436, in units of 2

*πc*/

*a*, where c is the speed of light in free space. As shown previously, the index perturbation must have a period of 2

*π*/

*k⃗*= 2

_{x}*ax̂*to scatter this

*k*-state to the point in

*k*-space, where the in-plane

*k*= 0. Note that because the

_{x}*x̂*and

*ŷ*components of the electric field of the waveguide mode have different symmetries along the

*ŷ*direction, an index perturbation with even symmetry across

*y*= 0 will scatter only the

*ŷ*-component, but not the

*x̂*-component of the field. Similarly, it is possible to scatter only the

*x̂*-component with a anti-symmetric perturbation distribution across the

*y*= 0 line. For the following analysis, we will choose to scatter the

*ŷ*-component. Furthermore, we suppose that we want to scatter into a vertical mode that has a Gaussian profile in the in-plane directions, as shown in Fig. 2(c). Figure 2(d) shows the dot product

*A⃗*·

^{*}*B⃗*, which appears in the integrand in Eq. (4). To find a perturbation that scatters light into a slowly diverging Gaussian beam, we calculate the coupling rate for beams with different divergence angles and compare the coupling constants among different index perturbation designs. We discuss in this paper, the perturbations are formed by enlarging three-hole clusters with periodicity 2

_{k⃗,y}*a*in the

*x̂*-direction. The blue and green lines in Fig. 2(e) represent the coupling constants

*κ*

_{1}and

*κ*

_{2}for Gaussian beams, focused in the plane of the photonic crystal, with waists of 2

*a*and

*a*, respectively, and the red line represents the ratio

*κ*

_{1}/

*κ*

_{2}. The horizontal axis denotes the perturbation size in units of the area of the unit cell of the triangular lattice, and the vertical axes denote the coupling constant (left) and the ratio

*κ*

_{1}/

*κ*

_{2}(right). We observe that a local maximum exists for

*κ*

_{1}/

*κ*

_{2}when the index perturbation size is 0.3244, in units of the fractional area of unit cell, that is, the light scattered in this case diverges less than a well-defined Gaussian beam. The perturbed structure for this case is shown in Fig. 2(f) and the corresponding coupling strength

*κ*

_{1}and the ratio

*κ*

_{1}/

*κ*

_{2}are 0.2228 and 1.815, respectively. As expected, the coupling rate into the Gaussian beam of smaller waist is lower than for the wider, less rapidly diverging beam. The brown shaded area in the right panel of Fig. 2(f) shows the index perturbation.

*ω*= 0.2436 is excited by a Gaussian beam source at one end of the PC waveguide. The

*ŷ*-component of the electric field in the plane of the PC is shown in Fig. 3(a), which contains five periods of the index perturbation shown in Fig. 2(f). The perturbation couples the incident waveguide mode upward, resulting in reduced waveguide transmission (< 0.5%) on the right side of the perturbation. In Fig. 3(b), we consider the effect of this perturbation on a higher-frequency mode at

*ω*= 0.3078, which also has even symmetry across the

*y*= 0 axis, but which has a periodicity of

*a*in the direction of propagation. Equation (4) predicts that if Δ

*ɛ*has a periodicity of 2

*a*, the coupling constant

*κ*will vanish in this case. This result is confirmed by a high in-plane transmission after the coupler (> 99%), as shown in Fig. 3(b). We will discuss this mode selective property of the coupler in greater detail in Section 3.

_{ba}*y⃗*component of the evanescent field and the upward component of the scattering field from the coupler. Figure 4(a) is a snapshot viewed in the vertical cross-section through a plane passing through the middle of waveguides, and Fig. 4(b) is the three dimensional view of the

*E*energy density isosurface. We observe a strong radiated field originating from the 5-period long perturbed region.

_{y}*r⃗*,

_{i}*u⃗*(

*r⃗*) denotes the spatial profile of the radiated field,

*k⃗*

_{0}is the

*k*-vector in free space,

*k⃗*= (

_{w}*π*/

*a*+Δ

*k*)

_{w,x}*x̂*is the crystal momentum of the perturbed PC waveguide where Δ

*k*indicates a possible shift in the

_{w,x}*k*-vector due to a slightly different dispersion in the perturbation region.

*r⃗*is th position of the

_{i}*i*-th scattering point, and

*r⃗*

_{1}is the position of the start of the coupler. The field dependence in Eq. (5) includes the decrease of the field amplitude along the coupler,

*α*, and the phase difference of the waveguide mode between perturbations. Because the perturbations are

*r⃗*have a periodicity of 2

_{i}*ax̂*, the phase difference is equivalent to 2

*a*Δ

*k*. Equation (5) therefore predicts a shifting of the diffraction angle with wavelength, as given by the waveguide’s dispersion relation. We observe this dependence in numerical simulations. In Fig. 5(a), the frequency of the incident waveguide mode is tuned from

_{w,x}*ω*= 0.2960 to

*ω*= 0.3120 as the scattering angle shifts from −4° to 20°.

## 3. Mode selectivity

*ω*= 0.3078. We now analyze this mode selectivity more closely in

*k*-space. Figure 6(a) compares the dispersion relations for the fundamental mode (the blue line) and a higher mode (the green line) inside the photonic band gap, having frequencies of

*ω*= 0.2436(2

_{A}*πc*/

*a*) and

*ω*= 0.3151 at

_{B}*k*=

_{x}*π*/

*a*. For the structure with index perturbations (the right panel), the dispersion is raised slightly because of the decrease in the effective refractive index. The periodicity of 2

*a*scatters modes near

*k*=

_{x}*π*/

*a*by a lattice vector of Δ

*k*=

_{x}*π*/

*a*to the point where

*k*= 0. Since these components are above the light line, they leak out of the PC plane and result in higher loss rate of the waveguide mode.

_{x}*γ*, normalized by the loss rate of the fundamental waveguide mode in the unperturbed structure, through different perturbation patterns. The green line shows

_{r}*γ*for the higher-order waveguide mode at

_{r}*ω*= 0.3151. As the perturbation size increases, the loss rate of the fundamental mode increases by over two orders of magnitude, while the loss rate of the higher-mode remains roughly constant. This result demonstrates a high degree of selectivity of the perturbation coupler between different kinds of waveguide modes. Figure 6(c) shows the frequency increase of the fundamental and higher order mode (the blue and green lines).

## 4. Transmittance and beam shape modification

*S*is the transmission out of the PC plane and the fraction quantifies the mode overlap of the upward(or downward)-traveling component of the radiated field with the target Gaussian beam profile. The scattering rate of the couplers with different index perturbations are shown in Fig. 7 (the red solid line). The perturbations not only scatter the incident waveguide mode into the Gaussian mode, but also induce backward reflection (the gray solid line) and other in-plane leakage (the gray dotted line). The maximal scattering rate is reached when

*r*

_{0}= 0 and the perturbation size in the sides is 0.4466, in units of the fractional area of unit cell. As shown in Fig. 7, the scattering rate doubles when the perturbation size increases about 40% (the region A), but slightly decreases as the radius of the central holes increases (the region B). If a perturbation is introduced only at the center of the waveguide (the region C in Fig. 7), then the incident light is mostly reflected and only weakly scattered out of plane.

*T*= 0.6989 × 89% = 62%.

_{c}*x̂*and

*ŷ*(transverse in-plane direction), we expand the coupling region and decrease the output beam divergence. This modified index perturbation is shown in Fig. 8(a), again optimized for the mode at

*ω*= 0.2436. Figures 8(b) and 8(c) show the intensity distributions of beams scattered from a small coupler (Fig. 3) and a broadened one (Fig. 8(a)), viewed in

*y*–

*z*plane through the centers of the couplers. Figures 8(d) and 8(e) compare the divergence of the radiated beams in both cases by fitting a Gaussian beam profile to the data. We assume 2

*θ*= 2

*W*

_{0}/

*z*

_{0}, where 2

*θ*is the divergence angle of the beam and

*W*

_{0}and

*z*

_{0}are the beam waist and depth of the focus, respectively. In the case of the broadened coupler, the divergence is decreased by 20% from that of the original smaller coupler.

*T*can be achieved by the methods described above, the overall efficiency of the PPCWC is so far limited to 50%. The upward scattering efficiency is defined as the directionality, where

_{c}*P*and

_{U}*P*are the powers radiated in the upward and downward directions, respectively. One method of increasing

_{D}*D*is to redirect the downward scattered field with a mirror. This may be achieved with a DBR structure, separated from the PC slab by some distance,

*d*. When

*d*= 0.4667

*h*, the downward radiated light reflected by the mirror interferes with the upward radiated light constructively and a local maximum of

*D*= 70% occurs. When we move the mirror further from the slab (

*d*> 0.6

*h*), there is minimal interference between two beams due to divergence of the beams. The upward scattering rate is close to the total scattering rate in this case, and we achieve

*D*= 95%. More sophisticated methods may be used to increase directional scattering, such as only partial etching of the PPCWC. This has been successfully employed for photonic crystal cavities [15

15. M. Toishi, D. Englund, A. Faraon, and J. Vuckovic, “High-brightness single photon source
from a quantum dot in a directional-emission
nanocavity,” Opt. Express **17**, 14618–14626
(2009). [CrossRef] [PubMed]

17. Y. Tanaka, M. Tymczenko, T. Asano, and S. Noda, “Fabrication of two-dimensional photonic
crystal slab point-defect cavity employing local three-dimensional
structures,” Japanese J. Appl.
Phys. **45**, 6096–6102
(2006). [CrossRef]

20. X. Chen, C. Li, C. K. Y. Fung, S. M. G. Lo, and H. K. Tsang, “Apodized waveguide grating couplers for
efficient coupling to optical fibers,” IEEE
Photon. Technol. Lett. **20**, 1156–1158
(2010). [CrossRef]

## 5. Conclusions

## Acknowledgments

## References and links

1. | J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, |

2. | T. Baba, “Slow light in photonic
crystals,” Nature Photonics |

3. | H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow
light in photonic crystal waveguides” Phys.
Rev. Lett. |

4. | J. F. McMillan, M. Yu, D. L. Kwong, and C. W. Wong, “Observation of four-wave mixing in
slow-light silicon photonic crystal waveguides,”
Opt. Express |

5. | M. Soljacic, S. G. Johnson, S. Fan, M. Ibanescu, E. Ippen, and J. D. Joannopoulos, “Photonic-crystal slow-light enhancement
of nonlinear phase sensitivity,” J. Opt.
Soc. Am. B |

6. | H. Takano, B. S. Song, T. Asano, and S. Noda, “Highly efficient multi-channel drop
filter in a two-dimensional hetero photonic
crystal,” Opt. Express |

7. | E. Waks and J. Vuckovic, “Coupled mode theory for photonic
crystal cavity-waveguide interaction,” Opt.
Express |

8. | S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by
single defect in a photonic bandgap structure,”
Nature (London) |

9. | M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H. Ryu, “Waveguides, resonators, and their
coupled elements in photonic crystal slabs,”
Opt. Express |

10. | P. Bienstman, S. Assefa, S. G. Johnson, J. D. Joannopoulos, G. S. Petrich, and L. A. Kolodziejski, “Taper structures for coupling into
photonic crystal slab waveguides,” J. Opt.
Soc. Am. B |

11. | P. E. Barclay, K. Srinivasan, and O. Painter, “Design of photonic crystal waveguides
for evanescent coupling to optical fiber tapers and integration with high-Q
cavities,” J. Opt. Soc. Am. B |

12. | B. Wang, J. Jiang, and G. P. Nordin, “Compact slanted grating
couplers,” Opt. Express |

13. | A. Mizutani, Naoki Ikeda, Y. Watanabe, N. Ozaki, Y. Takata, Y. Kitagawa, F. Laere, R. Baets, Y. Sugimoto, and K. Asakawa, “Planar focusing lens grating for vertical coupling on 2D photonic crystal slab waveguide,” in Lasers and Electro-Optics Society, 2006. LEOS 2006. 19th Annual Meeting of the IEEE, pp. 843–844 (2006). |

14. | A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vuckovic, “Dipole induced transparency in
waveguide coupled photonic crystal cavities,”
Opt. Express |

15. | M. Toishi, D. Englund, A. Faraon, and J. Vuckovic, “High-brightness single photon source
from a quantum dot in a directional-emission
nanocavity,” Opt. Express |

16. | N. Tran, S. Combrie, and A. De Rossi, “Directive emission from high-Q photonic
crystal cavities through band folding,”
Phys. Rev. B |

17. | Y. Tanaka, M. Tymczenko, T. Asano, and S. Noda, “Fabrication of two-dimensional photonic
crystal slab point-defect cavity employing local three-dimensional
structures,” Japanese J. Appl.
Phys. |

18. | A. Yariv, |

19. | A. Yariv, “Coupled-mode theory for guided-wave
optics,” IEEE J. Quantum Electron. |

20. | X. Chen, C. Li, C. K. Y. Fung, S. M. G. Lo, and H. K. Tsang, “Apodized waveguide grating couplers for
efficient coupling to optical fibers,” IEEE
Photon. Technol. Lett. |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(130.3120) Integrated optics : Integrated optics devices

(250.5300) Optoelectronics : Photonic integrated circuits

(130.5296) Integrated optics : Photonic crystal waveguides

(230.5298) Optical devices : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: August 4, 2011

Revised Manuscript: August 29, 2011

Manuscript Accepted: August 29, 2011

Published: October 3, 2011

**Citation**

Cheng-Chia Tsai, Jacob Mower, and Dirk Englund, "Directional free-space coupling from photonic crystal waveguides," Opt. Express **19**, 20586-20596 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-21-20586

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### References

- J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystal: Molding the Flow of Light, 2nd ed., (Princeton University Press, 2008).
- T. Baba, “Slow light in photonic crystals,” Nature Photonics2, 465–473 (2008). [CrossRef]
- H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides” Phys. Rev. Lett.94, 073903 (2005). [CrossRef] [PubMed]
- J. F. McMillan, M. Yu, D. L. Kwong, and C. W. Wong, “Observation of four-wave mixing in slow-light silicon photonic crystal waveguides,” Opt. Express18, 15484–15497 (2010). [CrossRef] [PubMed]
- M. Soljacic, S. G. Johnson, S. Fan, M. Ibanescu, E. Ippen, and J. D. Joannopoulos, “Photonic-crystal slow-light enhancement of nonlinear phase sensitivity,” J. Opt. Soc. Am. B19, 2052–2058 (2002). [CrossRef]
- H. Takano, B. S. Song, T. Asano, and S. Noda, “Highly efficient multi-channel drop filter in a two-dimensional hetero photonic crystal,” Opt. Express14, 3491–3496 (2006). [CrossRef] [PubMed]
- E. Waks and J. Vuckovic, “Coupled mode theory for photonic crystal cavity-waveguide interaction,” Opt. Express13, 5064–5073 (2005). [CrossRef] [PubMed]
- S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by single defect in a photonic bandgap structure,” Nature (London)407, 608–610 (2000). [CrossRef]
- M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H. Ryu, “Waveguides, resonators, and their coupled elements in photonic crystal slabs,” Opt. Express12, 1551–1561 (2004). [CrossRef] [PubMed]
- P. Bienstman, S. Assefa, S. G. Johnson, J. D. Joannopoulos, G. S. Petrich, and L. A. Kolodziejski, “Taper structures for coupling into photonic crystal slab waveguides,” J. Opt. Soc. Am. B20, 1817–1821 (2003). [CrossRef]
- P. E. Barclay, K. Srinivasan, and O. Painter, “Design of photonic crystal waveguides for evanescent coupling to optical fiber tapers and integration with high-Q cavities,” J. Opt. Soc. Am. B20, 2274–2284 (2003). [CrossRef]
- B. Wang, J. Jiang, and G. P. Nordin, “Compact slanted grating couplers,” Opt. Express12, 3313–3326 (2004). [CrossRef] [PubMed]
- A. Mizutani, Naoki Ikeda, Y. Watanabe, N. Ozaki, Y. Takata, Y. Kitagawa, F. Laere, R. Baets, Y. Sugimoto, and K. Asakawa, “Planar focusing lens grating for vertical coupling on 2D photonic crystal slab waveguide,” in Lasers and Electro-Optics Society, 2006. LEOS 2006. 19th Annual Meeting of the IEEE, pp. 843–844 (2006).
- A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vuckovic, “Dipole induced transparency in waveguide coupled photonic crystal cavities,” Opt. Express16, 12154–12162 (2008). [CrossRef] [PubMed]
- M. Toishi, D. Englund, A. Faraon, and J. Vuckovic, “High-brightness single photon source from a quantum dot in a directional-emission nanocavity,” Opt. Express17, 14618–14626 (2009). [CrossRef] [PubMed]
- N. Tran, S. Combrie, and A. De Rossi, “Directive emission from high-Q photonic crystal cavities through band folding,” Phys. Rev. B79, 041101 (2009). [CrossRef]
- Y. Tanaka, M. Tymczenko, T. Asano, and S. Noda, “Fabrication of two-dimensional photonic crystal slab point-defect cavity employing local three-dimensional structures,” Japanese J. Appl. Phys.45, 6096–6102 (2006). [CrossRef]
- A. Yariv, Optical Electronics in Modern Communications, 5th ed., (Oxford University Press, New York, 1997).
- A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron.9, 919–933 (1973). [CrossRef]
- X. Chen, C. Li, C. K. Y. Fung, S. M. G. Lo, and H. K. Tsang, “Apodized waveguide grating couplers for efficient coupling to optical fibers,” IEEE Photon. Technol. Lett.20, 1156–1158 (2010). [CrossRef]

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