## Deterministic design of wavelength scale, ultra-high Q photonic crystal nanobeam cavities |

Optics Express, Vol. 19, Issue 19, pp. 18529-18542 (2011)

http://dx.doi.org/10.1364/OE.19.018529

Acrobat PDF (1272 KB)

### Abstract

Photonic crystal nanobeam cavities are versatile platforms of interest for optical communications, optomechanics, optofluidics, cavity QED, etc. In a previous work [Appl. Phys. Lett. **96**, 203102 (2010)], we proposed a deterministic method to achieve ultrahigh *Q* cavities. This follow-up work provides systematic analysis and verifications of the deterministic design recipe and further extends the discussion to air-mode cavities. We demonstrate designs of dielectric-mode and air-mode cavities with *Q* > 10^{9}, as well as dielectric-mode nanobeam cavities with both ultrahigh-*Q* (> 10^{7}) and ultrahigh on-resonance transmissions (*T* > 95%).

© 2011 OSA

## 1. Introduction

*Q*), small mode volume (

*V*) [1] optical cavities provide powerful means for modifying the interactions between light and matter [2

2. K. J. Vahala, “Optical microcavities,” Nature **424**, 839–846 (2003). [CrossRef] [PubMed]

3. J. L. O’Brien, A. Furusawa, and J. Vuckovic, “Photonic quantum technologies,” Nat. Photonics **3**, 687–695 (2009). [CrossRef]

4. J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics **4**, 535–544 (2010). [CrossRef]

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

6. D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics **4**, 211–217 (2010). [CrossRef]

7. D. G. Grier, “A revolution in optical manipulationm,” Nature **424**, 21–27 (2003). [CrossRef]

8. D. Psaltis, S. R. Quake, and C. Yang, “Developing optofluidic technology through the fusion of microfluidics and optics,” Nature **442**, 381–386 (2006). [CrossRef] [PubMed]

9. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. **58**, 2059–2062 (1987). [CrossRef] [PubMed]

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

*Q*-factors [11

11. J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature **390**, 143–145 (1997). [CrossRef]

31. P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Loncar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. **94**, 121106 (2009). [CrossRef]

*Q*factors are typically obtained using extensive parameter search and optimization. In a previous work [32

32. Q. Quan, P. B. Deotare, and M. Loncar, “Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide,” Appl. Phys. Lett. **96**, 203102 (2010). [CrossRef]

*Q*PhC nanobeam cavity and verified our designs experimentally. The proposed method does not rely on any trial-and-error based parameter search and does not require any hole shifting, re-sizing and overall cavity re-scaling. The key design rules we proposed, that result in ultrahigh

*Q*cavities, are (i) zero cavity length (

*L*= 0), (ii) constant length of each mirror segment (‘period’=

*a*) and (iii) a Gaussian-like field attenuation profile, provided by linear increase in the mirror strength.

17. 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]

22. Y. Tanaka, T. Asano, and S. Noda, “Design of photonic crystal nanocavity with *Q*-factor of ∼10^{9},” J. Lightwave Technol. **26**, 1532 (2008). [CrossRef]

*Q*s on par with those found in slab-based geometries, but in much smaller footprints, and are the most natural geometries for integration with waveguides [23

23. M. Notomi, E. Kuramochi, and H. Taniyama, “Ultrahigh-Q nanocavity with 1D Photonic Gap,” Opt. Express , **16**, 11095 (2008). [CrossRef] [PubMed]

31. P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Loncar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. **94**, 121106 (2009). [CrossRef]

23. M. Notomi, E. Kuramochi, and H. Taniyama, “Ultrahigh-Q nanocavity with 1D Photonic Gap,” Opt. Express , **16**, 11095 (2008). [CrossRef] [PubMed]

33. E. Kuraamochi, H. Taniyama, T. Tanabe, K. Kawasaki, Y-G. Roh, and M. Notomi, “Ultrahigh-Q one-dimensional photonic crystal nanocavities with modulated mode-gap barriers on SiO_{2} claddings and on air claddings,” Opt. Express **18**, 15859–15869 (2010). [CrossRef]

*Q*cavities based on dielectric stacks that are of interest for realization of vertical-cavity surface emitting lasers (VCSELs) and sharp filters. Finally, it is important to emphasize that while our method is based on the framework of Fourier space analysis [35

35. J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Optimization of the Q factor in photonic crystal microcavities,” IEEE J. Quantum Electron. **38**, 850–856 (2002). [CrossRef]

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

*Q*-factors in our devices [38

38. M. Palamaru and P. Lalanne, “Photonic crystal waveguides: Out-of-plane losses and adiabatic modal conversion,” Appl. Phys. Lett. **78**, 1466–1468 (2001). [CrossRef]

39. P. Lalanne, S. Mias, and J. P. Hugonin, “Two physical mechanisms for boosting the quality factor to cavity volume ratio of photonic crystal microcavities,” Opt. Express **12**, 458–467 (2004). [CrossRef] [PubMed]

## 2. Numerical verification of the deterministic design approach

32. Q. Quan, P. B. Deotare, and M. Loncar, “Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide,” Appl. Phys. Lett. **96**, 203102 (2010). [CrossRef]

32. Q. Quan, P. B. Deotare, and M. Loncar, “Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide,” Appl. Phys. Lett. **96**, 203102 (2010). [CrossRef]

*L*=0), that is the hole-to-hole spacing between the two central holes is the same as the rest of the structure (

*a*). This minimizes the cavity loss and the mode volume simultaneously. The cavity loss is composed of the radiation loss into the free space (characterized by

*Q*

_{rad}) and the coupling loss to the feeding waveguide (

*Q*

_{wg}).

*Q*

_{wg}can be increased simply by adding more gratings along the waveguide.

*Q*

_{rad}can be increased by minimizing the spatial Fourier harmonics of the cavity mode inside the lightcone, achieved by creating a Gaussian-like attenuation profile [32

**96**, 203102 (2010). [CrossRef]

**96**, 203102 (2010). [CrossRef]

**r**

*=*

_{j}*j*·

*ax̂*,

*a*is the period, and

*j*= ±1, ±2... are integers.

*R*is the radius of the hole. Using plain wave expansion method [40] in the beam direction (

*x̂*), where

*G*= 2

*π*/

*a*. The zeroth (

*κ*

_{0}) and first (

*κ*

_{1}) order Fourier components can be expressed as [40]

*J*

_{1}is the first order Bessel function. Filling fraction

*f*=

*πR*

^{2}/

*ab*is the ratio of the area of the air-hole to the area of the unit cell. We note that the above expressions are calculated assuming nanobeam cavity has infinite thickness (i.e 2D equivalent case). Better estimation can be obtained by replacing

*ε*

_{air}and

*ε*

_{Si}with the effective permittivities.

*k*) for a given frequency (

*ω*) is a complex number, whose imaginary part denotes the mirror strength (

*γ*). For solutions near the band-edge, of interest for high-

*Q*cavity design [32

**96**, 203102 (2010). [CrossRef]

*δ*is the detuning from the mid-gap frequency) and the wavevector as

*k*= (1 +

*iγ*)

*π*/

*a*. Substituting this into the master equation, we obtain

*j*represents the

*j*

^{th}mirror segment counted from the center), at which point the mirror strength

*γ*

^{j}^{=1}= 0.

*γ*increases with

*j*. With

*ε*

_{air}= 1 and

*ε*

_{Si}= 3.46

^{2}, we calculate in Fig. 2(a) the

*γ*–

*j*relation for different tapering profiles. It can be seen that quadratic tapering profile results in linearly increasing mirror strengths, needed for Gaussian field attenuation [32

**96**, 203102 (2010). [CrossRef]

*γ*–

*f*relation (Fig. 2(c)). As shown in Fig. 2(d), linearly increasing mirror strength is indeed achieved after quadratic tapering.

*L*in between (Fig. 1(a)). Figure 2(e) shows the simulated

*Q*-factors for various

*L*s. Highest

*Q*

_{rad}is achieved at zero cavity length (

*L*=0), which supports the prediction in [32

**96**, 203102 (2010). [CrossRef]

*H*-field distribution in the plane right above the cavity, obtained from 3D FDTD simulation. As shown in Fig. 2(h), this field distribution can be ideally fitted with

_{z}*H*= sin(

_{z}*πx*/

*a*)exp(−

*σx*

^{2})exp(−

*ξy*

^{2}), with

*a*= 0.33,

*σ*= 0.14 and

*ξ*= 14. The fitted value

*a*agrees with the “period”, and

*σ*agrees with that extracted value from Fig. 2(d):

*H*distribution along the dashed line in Figs. 2(g) and 2(h). Therefore, we conclude that zero cavity length, fixed periodicity and a quadratic tapering of the filling fraction results in a Gaussian field profile, which leads to a high-

_{z}*Q*cavity [32

**96**, 203102 (2010). [CrossRef]

**96**, 203102 (2010). [CrossRef]

42. S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisbergs, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E **65**, 066611 (2002). [CrossRef]

**E**

_{||}is the component of

**E**that is parallel to the side wall surfaces of the holes and

**D**

_{⊥}is the component of

**D**that is perpendicular to the side wall surfaces of the holes. Under Gaussian distribution, the major field component

*D*= cos(

_{y}*π*/

*ax*)exp(−

*σx*

^{2})exp(−

*ξy*

^{2}),

*δε*perturbation occurs at

*r*= ±(

*j*− 1/2)

*a*+

*R*, where

_{j}*j*

^{th}hole (counted the center), with

*j*=2,3...

*N*,

*N*is the total number of mirror segments at each side. Since the cavity mode has a Gaussian profile,

*N*, with a nonzero intercept due to diffraction limit. For large

*N*, the intercept can be neglected, and thus

*f*= 0.2 to

*f*= 0.1 into Eq. (6), the frequency offset

*δλ*/

*λ*v.s

*N*can be obtained. Figure 2(f) shows the frequency offset for different total number of mirror pairs (

*N*), calculated from the perturbation theory, as well as using FDTD simulations. It can be seen that the deviation decreases as the number of modulated mirror segments increases, and is below 1% for

*N*> 15.

*Q*, dielectric-mode cavity resonant at a target frequency can be designed using the following algorithm:

- Determine a target frequency. For example in our case we want
*f*_{target}= 200THz. Since the cavity resonant frequency is typically 1% smaller than the dielectric band-edge of the central segment, estimated using the perturbation theory, we shift-up the target frequency by 1%, i.e.*f*_{adjusted}= 202THz. - Pick the thickness of the nanobeam - this is often pre-determined by the choice of the wafer. For example, in our case, the thickness of the nanobeam is 220nm, determined by the thickness of the device layer of our silicon-on-insulator (SOI) wafer.
- Choose periodicity according to
*a*=*λ*_{0}/2*n*_{eff}, where*n*_{eff}is effective mode index of the cavity and can be estimated by numerical modeling of a strip waveguide that nanobeam cavity is based on. However, we found that the absolute value of the periodicity is not crucial in our design, as long as there exists a bandgap. Therefore, we pick*n*_{eff}= 2.23, which is a median value of possible effective indices in the case of free standing silicon nanobeam (*n*_{eff}∈ (1, 3.46)). This results in*a*= 330nm. - Set nanobeam width. Large width increases the effective index of the cavity mode, pulls the mode away from the light line, thus reducing the in-plane radiation loss. On the other hand, a large beam width will allow for higher order modes with the same symmetry as the fundamental mode of interest. Using band diagram simulations, we found that the width of 700nm is good trade-off between these two conditions (Fig. 2(b)).
- Set the filling fraction of the first mirror section such that its dielectric band-edge is at the adjusted frequency: 202THz in our case. Band diagram calculations based on unit cells are sufficient for this analysis. We found that an optimal filling fraction in our case is
*f*_{start}= 0.2 (Fig. 2(b)). - Find the filling fraction that produces the maximum mirror strength for the target frequency. This involves calculating the mirror strength for several filling fractions (Fig. 2(c)), each of which takes one or two minutes on a laptop computer. In our case we found that
*f*_{end}= 0.1. - Pick the number of mirror segments (
*N*) to construct the Gaussian mirror: we found that*N*≥ 15 (on each side) are generally good to achieve high radiation-*Q*s. - Create the Gaussian mirror by tapering the filling fractions quadratically from
*f*_{start}(=0.2 in our case) to*f*_{end}(=0.1) over the period of*N*segments. From the above analysis, the mirror strengths can be linearized through quadric tapering (Fig. 2(d)). - Finally, the cavity is formed by putting two Gaussian mirrors back to back, with no additional cavity length in between (
*L*= 0). To achieve a radiation-limited cavity (*Q*_{wg}>>*Q*_{rad}), 10 additional mirrors with the maximum mirror strength are placed on both ends of the Gaussian mirror. We will show in the next section, no additional mirrors are needed to achieve a waveguide-coupled cavity (*Q*_{rad}>>*Q*_{wg}).

**96**, 203102 (2010). [CrossRef]

*H*field profile in the plane right above the cavity, obtained from FDTD simulation. Figure 3(d) shows the fitted field profile using the same parameters that are used in the original structure shown in Figs. 2(g)–2(i), but with sine function replaced by cosine function. Figure 3(e) shows the

_{z}*H*distribution along the dashed line in Figs. 3(c) and 3(d).

_{z}35. J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Optimization of the Q factor in photonic crystal microcavities,” IEEE J. Quantum Electron. **38**, 850–856 (2002). [CrossRef]

*σa*

^{2}<< 1, both distributions have their Fourier components strongly localized at

*k*= ±

*π*/

*a*, as is verified by FDTD simulations in Fig. 4(a) and 4(b). Since

*k*= 0. Therefore, dielectric-centered cavities (Fig. 1) should have higher

*Q*-factors. However, in high-Q cavity designs,

*σa*

^{2}<< 1 is satisfied and thus both dielectric-centered and air-centered cavities have comparable

*Q*-factors. FDTD simulation shows that the above

*Q*

_{tot}= 3.8 × 10

^{8}and

*Q*

_{tot}= 3.5 × 10

^{8}respectively. The mode volume of the

*λ*

_{res}/

*n*

_{Si})

^{3}, smaller than the

*V*= 0.76(

*λ*

_{res}/

*n*

_{Si})

^{3}).

*H*cavity has even less radiated power at small zenith angles, consistent with the above analysis. By integrating the zenith and azimuth angle dependent far field emission, we found that 32% and 63% of the power emitted to +

^{odd}*ẑ*direction can be collected by a NA=0.95 lens, respectively for

*H*cavity and

^{odd}*H*cavity.

^{even}## 3. Ultra-high *Q*, dielectric-mode photonic crystal nanobeam cavities

### 3.1. Radiation-Q limited and waveguide-coupled cavities

*V*than the

*Q*

_{wg}>>

*Q*

_{tot}). We find in Fig. 5 that

*Q*

_{tot}increases exponentially and

*V*increases linearly as the total number of mirror pairs in the Gaussian mirror (

*N*) increases. A record ultra-high

*Q*of 5.0 × 10

^{9}is achieved while maintaining the small mode volume of 0.9 × (

*λ*

_{res}/

*n*

_{Si})

^{3}at

*N*= 30.

17. 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]

31. P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Loncar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. **94**, 121106 (2009). [CrossRef]

*Q*and high transmissions (

*T*) cavities are possible with the above design steps (i)–(ix), with no additional “coupling sections” needed. We study

*T*and

*Q*

_{total}dependence on the total number of mirror pair segments in the Gaussian mirror (

*N*) in Fig. 5(b). Partial

*Q*-factors (

*Q*

_{rad},

*Q*

_{wg}) are obtained from FDTD simulations, and

*T*is obtained using

*Q*= 1.3 × 10

^{7},

*T*= 97% at

*N*= 25.

### 3.2. Higher order modes of the dielectric-mode cavity

*Q*mode that we deterministically designed is the fundamental mode of the cavity. Meanwhile, higher order cavity modes also exist. The number of higher order modes depends on the width of the photonic band gap and total number of mirror segments in the Gaussian mirror. To reduce the simulation time, we study the higher order modes of a waveguide-coupled cavity, that has a total number of 12 mirror pair segments, possessing a moderate

*Q*-factor. Figure 6(a) shows the transmission spectrum obtained from FDTD simulation, by exciting the input waveguide with a waveguide mode, and monitoring the transmission through the cavity at the output waveguide. The band-edge modes are observed at wavelengths longer than 1.6

*μ*m and shorter than 1.3

*μ*m. Figures 6(b)–6(d) shows the major field-component (

*E*) distribution of the three cavity modes. As expected, the eigenmodes alternate between symmetric and anti-symmetric modes. Symmetry plane is defined perpendicular to the beam direction, in the middle of the cavity (dashed line in Fig. 6). The total

_{y}*Q*-factors of modes I–III are 10,210, 1,077 and 286 respectively. Effective mode volumes of them are 0.55, 0.85 and 1.06 respectively. We note that transversely odd modes are well separated from the transversely symmetric cavity modes, hence were not considered in Fig. 6.

## 4. Ultra-high *Q*, air-mode photonic crystal nanobeam cavities

### 4.1. Radiation-Q limited cavity

4. J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics **4**, 535–544 (2010). [CrossRef]

7. D. G. Grier, “A revolution in optical manipulationm,” Nature **424**, 21–27 (2003). [CrossRef]

8. D. Psaltis, S. R. Quake, and C. Yang, “Developing optofluidic technology through the fusion of microfluidics and optics,” Nature **442**, 381–386 (2006). [CrossRef] [PubMed]

*Q*air-mode nanobeam cavity is realized by pulling the air-band mode of photonic crystal into its bandgap, which can also be designed using the same design principles that we developed for dielectric-mode cavities. In contrast to the dielectric-mode case, the resonant frequency of the air-mode cavity is determined by the air band-edge frequency of the central mirror segment. Then, to create the Gaussian confinement, the bandgaps of the mirror segments should shift to higher frequencies as their distances from the center of the cavity increase. This can be achieved by progressively increasing the filling fractions of the mirror segments away from the center of the structure (instead of decreasing in the dielectric-mode cavity case). One way to accomplish this is to increase the size of the holes away from the center of the cavity. While this may be suitable for non-waveguide coupled (radiation-

*Q*limited) cavities, it is not ideal for a waveguide-coupled cavity. For this reason, we employ the design that relies on tapering of the waveguide width instead of the hole size. Similar geometry was recently proposed by Ahn et. al. [44

44. B. H. Ahn, J. H. Kang, M. K. Kim, J. H. Song, B. Min, K. S. Kim, and Y. H. Lee, “One-dimensional parabolic-beam photonic crystal laser,” Opt. Express **18**, 5654–5660 (2010). [CrossRef] [PubMed]

*w*

_{start}= 1

*μ*m (Fig. 7(a)), with the hole radii kept constant at 100nm. Third, to create the Gaussian mirror, the beam widths are quadratically tapered from

*w*

_{start}= 1

*μ*m to

*w*

_{end}= 0.7

*μ*m, which produces the maximum mirror strength (band diagrams shown in Fig. 7(a)). This procedure involves calculating the mirror strength for several beam widths (Fig. 7(b)), each takes one or two minutes on a laptop computer. As shown in Fig. 7(c), the mirror strengths are linearized after the quadratic tapering. In order to achieve a radiation-

*Q*limited cavity, 10 additional mirror segments are placed at both ends of the Gaussian mirror that has beam width

*w*

_{end}= 0.7

*μ*m.

*Q*of nanobeam cavities that have different total number of mirror pair segments in the Gaussian mirrors. We have achieved a record ultra-high

*Q*of 1.4 × 10

^{9}, air-mode nanobeam cavity. As shown in Fig. 8(a), the effective mode volumes of the air-mode cavities are much larger than the dielectric-mode cavities.

### 4.2. Cavity strongly coupled to the feeding waveguide

*T*and

*Q*

_{total}dependence on the total number of mirror pair segments in the Gaussian mirror (

*N*) using FDTD simulations. As shown in Fig. 8(b), we achieve nanobeam cavity with

*Q*= 3.0 × 10

^{6},

*T*= 96% at

*N*= 25.

### 4.3. Higher order modes of the air-mode cavity

*Q*cavity that we were able to design is the fundamental mode of the cavity. Higher order modes coexist with the fundamental modes inside the band gap. Fig. 9(a) shows the transmission spectrum of a waveguide-coupled air-mode nanobeam cavity, that has 15 mirror pair segments in the Gaussian mirror. The band-edge modes are observed at wavelengths longer than 1.6

*μ*m. The modes in the range of 1.2

*μ*m to 1.35

*μ*m are formed by the higher order band modes in Fig. 7(a). Figures 9(b)–9(c) show the major field-component distribution (

*E*) of the two cavity modes inside the bandgap. The total

_{y}*Q*-factors of these two modes are 23,935 and 5,525 respectively. The effective mode volumes are 2.32 and 3.01 respectively.

## 5. Conclusion

*Q*photonic crystal nanobeam cavities. With this method,

*Q*> 10

^{9}radiation-limited cavity, and

*Q*> 10

^{7},

*T*> 95% waveguide-coupled cavity are deterministically designed. These

*Q*-factors are comparable with those found in whispering gallery mode (WGM) cavities [45

45. D. W. Vernooy, A. Furusawa, N. P. Georgiades, V. S. Ilchenko, and H. J. Kimble, “Cavity QED with high-Q whispering gallery modes,” Phys. Rev. A **57**, R2293–R2296 (1998). [CrossRef]

47. M. Soltani, S. Yegnanarayanan, and A. Adibi, “Ultra-high Q planar silicon microdisk resonators for chip-scale silicon photonics,” Opt. Express **15**, 4694–4704 (2007). [CrossRef] [PubMed]

*Q*nanobeam cavity design, and thus enable both fundamental studies in strong light and matter interactions, and practical applications in novel light sources, functional optical components (filters, delay lines, sensors) and densely integrated photonic circuits.

## Acknowledgments

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25. | S. Reitzenstein, C. Hofmann, A. Gorbunov, M Strauß, S. H. Kwon, C. Schneider, A. Loffler, S. Hofling, M. Kamp, and A. Forchel, “AlAs/GaAs micropillar cavities with quality factors exceeding 150000,” Appl. Phys. Lett. |

26. | A. R. Md Zain, N. P. Johnson, M. Sorel, and R. M. De La Rue, “Ultra high quality factor one dimensional photonic crystal/photonic wire microcavities in silicon-on-insulator (SOI),” Opt. Express |

27. | Y. Zhang and M. Loncar, “Ultra-high quality factor optical resonators based on semiconductor nanowires.” Opt. Express |

28. | M. W. McCutcheon and M. Loncar, “Design of a silicon nitride photonic crystal nanocavity with a Quality factor of one million for coupling to a diamond nanocrystal,” Opt. Express |

29. | L. D. Haret, T. Tanabe, E. Kuramochi, and M. Notomi, “Extremely low power optical bistability in silicon demonstrated using 1D photonic crystal nanocavity,” Opt. Express |

30. | J. Chan, M. Eichenfield, R. Camacho, and O. Painter, “Optical and mechanical design of a “zipper” photonic crystal optomechanical cavity”, Opt. Express |

31. | P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Loncar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. |

32. | Q. Quan, P. B. Deotare, and M. Loncar, “Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide,” Appl. Phys. Lett. |

33. | E. Kuraamochi, H. Taniyama, T. Tanabe, K. Kawasaki, Y-G. Roh, and M. Notomi, “Ultrahigh-Q one-dimensional photonic crystal nanocavities with modulated mode-gap barriers on SiO |

34. | Q. Quan, I. B. Burgess, S. K. Y. Tang, D. L. Floyd, and M. Loncar, “High-Q/V photonic crystal nanobeam cavities in an ultra-low index-contrast polymeric optofluidic platform,” arXiv:1108.2669 (2010). |

35. | J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Optimization of the Q factor in photonic crystal microcavities,” IEEE J. Quantum Electron. |

36. | K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express |

37. | D. Englund, I. Fushman, and J. Vuckovic, “General recipe for designing photonic crystal cavities,” Opt. Express |

38. | M. Palamaru and P. Lalanne, “Photonic crystal waveguides: Out-of-plane losses and adiabatic modal conversion,” Appl. Phys. Lett. |

39. | P. Lalanne, S. Mias, and J. P. Hugonin, “Two physical mechanisms for boosting the quality factor to cavity volume ratio of photonic crystal microcavities,” Opt. Express |

40. | K. Sakoda, |

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

42. | S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisbergs, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E |

43. | J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Optimization of three-dimensional micropost microcavities for cavity quantum electrodynamics,” Phys. Rev. E |

44. | B. H. Ahn, J. H. Kang, M. K. Kim, J. H. Song, B. Min, K. S. Kim, and Y. H. Lee, “One-dimensional parabolic-beam photonic crystal laser,” Opt. Express |

45. | D. W. Vernooy, A. Furusawa, N. P. Georgiades, V. S. Ilchenko, and H. J. Kimble, “Cavity QED with high-Q whispering gallery modes,” Phys. Rev. A |

46. | D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature |

47. | M. Soltani, S. Yegnanarayanan, and A. Adibi, “Ultra-high Q planar silicon microdisk resonators for chip-scale silicon photonics,” Opt. Express |

**OCIS Codes**

(140.4780) Lasers and laser optics : Optical resonators

(230.5298) Optical devices : Photonic crystals

(230.7408) Optical devices : Wavelength filtering devices

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: May 23, 2011

Revised Manuscript: August 8, 2011

Manuscript Accepted: August 18, 2011

Published: September 8, 2011

**Citation**

Qimin Quan and Marko Loncar, "Deterministic design of wavelength scale, ultra-high Q photonic crystal nanobeam cavities," Opt. Express **19**, 18529-18542 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-19-18529

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### References

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