## Compact and efficient injection of light into band-edge slow-modes

Optics Express, Vol. 15, Issue 10, pp. 6102-6112 (2007)

http://dx.doi.org/10.1364/OE.15.006102

Acrobat PDF (206 KB)

### Abstract

We design compact (a few wavelength long) and efficient (>99%) injectors for coupling light into slow Bloch modes of periodic thin film stacks and of periodic slab waveguides. The study includes the derivation of closed-form expressions for the injection efficiency as a function of the group-velocity of injected light, and the proof that 100% coupling efficiencies for arbitrary small group velocities is possible with an injector length scaling as log(c/v_{g}). The trade-off between the injector bandwidth and the group velocity of the injected light is also considered.

© 2007 Optical Society of America

## 1. Introduction

1. S. E. Harris, “Electromagnetically induced transparency,” Phys. Today **50**, 36–42 (July 1997). [CrossRef]

2. J. Khurgin, “Expanding the bandwidth of slow-light photonic devices based on coupled resonators,” Opt. Lett. **30**, 2778–2780 (2005). [CrossRef] [PubMed]

4. M. F. Yanik and S. Fan, “Stopping light all optically,” Phys. Rev. Lett. **92**, 083901 (2004). [CrossRef] [PubMed]

6. A. Yu. Petrov and M. Eich, “Zero dispersion at small group velocities in photonic crystal waveguides,” Appl. Phys. Lett. **85**, 4866–4868 (2004). [CrossRef]

_{g}). In Section 4, we consider more realistic geometries composed of slits etched into a slab waveguide. Since radiation loss into the cladding is included into this problem, this geometry represents a test bed for even more realistic three-dimensional geometries, like PC waveguides, while preserving computational requirements at a moderate level. Using a combination of fully-vectorial methods [13

13. J. P. Hugonin and P. Lalanne, “Perfectly-matched-layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A. **22**, 1844–1849 (2005). [CrossRef]

_{g}/c=0.1-0.001) with bandwidths of the order of 250 GHz at -1 dB. Section 5 summarizes the results.

## 2. Group-velocity impedance mismatch problem

_{H}and n

_{L}. Hereafter, we denote by a the periodicity constant of the thin film stack and by

*f*(the fill factor) the fraction of material with refractive index n

_{H}.

### 2.1 Injection efficiency

^{+}> exp(jknz) and ∣P

^{-}> exp(-jknz), with a unitary Poynting vector ∣P

^{+}> = [E

_{x}, H

_{y}] = [(2/n)

^{1/2},(2n)

^{1/2}] and ∣P

^{-}> = [(2/n)

^{1/2},-(2n)

^{1/2}]. The modes of the periodic medium are the Bloch modes, ∣B

^{+}(z)> exp(jkn

_{eff}z) and ∣B

^{-}(z)> exp(-jkn

_{eff}z), with ∣B

^{-}> and ∣B

^{+}> two periodic functions of the z-coordinate and with n

_{eff}the Bloch-mode effective index. ∣B

^{+}> and ∣B

^{-}> are calculated as the eigenstates of the unit-cell transfer matrix of the periodic medium and neff as the associated eigenvalues [14].

_{H}, represented by the dashed vertical lines in Fig. 1(a) and located at planes z=z

_{0}+p

*a*, with p an integer. Similar planes exist for the layers of refractive index n

_{L}. In every layer of the periodic structure, the functions ∣B

^{+}> and ∣B

^{-}> can be expanded as a superposition of two counter-propagative plane waves. We will denote by A

_{H}and B

_{H}, the modal coefficients of this decomposition in the transversal plane z=z

_{0}, A

_{H}referring to the forward plane wave and B

_{H}to the backward plane wave. Thus, we have

*u*=B

_{H}/A

_{H}between the backward- and forward-modal coefficients is an important parameter which describes the stationary character of the Bloch mode. Inside the photonic gap and at the band edges where v

_{g}=0, ∣

*u*∣=1. Outside the photonic gap,

*u*is real for lossless materials and using the analytical expressions obtained in [14–15], it can be further shown that

^{-}> and ∣B

^{+}>, except if the periodic stack is composed of lossless materials or if it possesses a mirror-symmetry with respect to a transverse plane [15

15. J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structure,” Phys. Rev. E **53**, 4107–4121 (1996). [CrossRef]

_{H}=A

_{H}and A'

_{H}=B

_{H}showing that the

*u*-factor of ∣B

^{-}> is simply the inverse of that of ∣B

^{+}> for the bi-layer stack.

*z*< 0) can be expressed as ∣P

^{+}> exp(jknz) + r∣P

^{-}> exp(-jknz), where r is the modal reflectivity coefficient. Similarly, the field in the periodic medium is simply t∣B

^{+}>exp(jkn

_{eff}z), in the absence of illumination from the right side. By satisfying the boundary condition at z = 0 (continuity of E

_{y}and H

_{x}), one obtains

_{H}

*fa*is the phase delay associated to the propagation through the layers with a refractive index n

_{H}. The previous equation shows that the reflectance ∣r∣

^{2}and the injection efficiency, equal to T=1-∣r∣

^{2}in the absence of loss, depend on Fresnel-type back-reflections through the ratio (n-n

_{H})/(n+n

_{H}), on the phase delay and on the

*u*-factor of ∣B

^{+}>. Note that for n=n

_{H}, r is simply given by r=

*u*exp(jϕ) and the high reflectance ∣r∣

^{2}close to the band-gap edges is purely due to the stationary character of ∣B

^{+}> (∣u∣ ≈ 1). Using the 2×2 transfer-matrix formalism in Ref. [14], ϕ and

*u*can be implicitly calculated as a function of the group-velocity v

_{g}of ∣B

^{+}>. The injection efficiency T=1-∣r∣

^{2}(i.e. the coupling efficiency into ∣B

^{+}>), in the vicinity of the band-edges of the first valence and conduction bands, is shown in Fig. 1(b), for 0.1<

*f*<0.9.

### 2.2 Approximate closed-form expression for the injection efficiency

*f*)n

_{H}

^{2}+(1-

*f*)n

_{L}

^{2}and β=(1-

*f*) cos(ϕ) (n

_{H}

^{2}- n

_{L}

^{2}). Equation (5) is exact and provides a general relationship between the free-space wavevector k (β depends on k through ϕ), the group velocity v

_{g}and the stationary ratio

*u*. To express ϕ as a function of v

_{g}, one may expand it in a power series of v

_{g}/c, ϕ=ϕ

^{(0)}+O(v

_{g}/c), and by retaining only the first term ϕ

^{(0)}, Eq. (5) can be solved for

*u*(v

_{g})

_{VB}and β

_{CB}correspond to β(ϕ = ϕ

^{(0)}) at the valence and conduction band edges, respectively. For quarter-wave periodic stacks,

*f*= (1+n

_{H}/n

_{L})

^{-1}, a closed-form expression for the phase delay ϕ

^{(0)}exists [14], ϕ

^{(0)}= π/2 ± asin

*f*values, the calculation of ϕ

^{(0)}requires to solve for a transcendental equation, see Section 6.2 in Ref. [14]. To retrieve full analyticity, one may use the classical coupled-wave method [16], assuming that the Bloch modes of the periodic media is described by only two counter-propagative slowly-varying z-functions that are coupled at the band edges by the first-Fourier coefficient of the relative-permittivity modulation, ε

_{1}=(n

_{H}

^{2}-n

_{L}

^{2})

*f*sinc(π

*f*). Within this approach that is all the more accurate as n

_{H}-n

_{L}is small, we have

_{0}=

*f*n

_{H}

^{2}+(1-

*f*) n

_{L}

^{2}is the DC-component of the Fourier coefficients of the relative permittivity, the plus and minus signs holding for the conduction and valence bands. By substituting Eqs. (7) and (6) into Eq. (4), one easily obtains a closed-form expression for the injection efficiency T=1-∣r∣

^{2}as a function of the dielectric material properties n

_{H}, n

_{L}and n and of the fill factor

*f*in the limit of small group velocities. As noted before, for n=n

_{H}, the expressions largely simplifies and one obtains

^{(0)}at the band edges, the deviation in Figs. 1(b) and 1(c) mainly results from the coupled-wave-method approximation used to derive Eq. (7).

_{VB}≈T

_{CB}. Additionally, since α≈ n

_{H}

^{2}+ n

_{L}

^{2}, the injection efficiency weakly depends on

*f*, as illustrated in Fig. 1(b). For large index contrasts, β cannot be neglected and for n≠n

_{H}, Eq. (4) instead of Eq. (8) has to be used. Both situations result in a dependence of the injection efficiency with the fill factor through the phase delay ϕ in Eq. (4) or directly through the coefficients α and β. This is illustrated in Fig. 1(c), which holds for n

_{H}/n

_{L}=3.495 and for n≠n

_{H}. Finally, note that the expressions in Eq. (8) largely differ from the usual ansatz [17–18], r=(v

_{g1}-v

_{g2}) / (v

_{g1}+v

_{g2}), where v

_{g1}and v

_{g2}are the group-velocities of the incident and transmitted waves. For v

_{g2}<<v

_{g1}, the ansatz, which leads to T=4v

_{g2}/v

_{g1}, completely ignores the difference of injection efficiencies at the two band edges, which is predicted by the transfermatrix computational results or by the approximate formula of Eq. (8).

## 3. Slow-mode injectors

### 3.1 Perfect injection in 1D thin-film stacks

_{1}and all the incident power is coupled into the forward-propagating slow Bloch mode of the periodic stack. This slow Bloch mode ∣B

^{+}>exp(jkn

_{eff}z) is defined at plane S

_{2}by its forward and backward-modal coefficients, A

_{H}and B

_{H}, with ∣A

_{H}∣≈∣B

_{H}∣>>1 in the slow-mode regime and with ∣A

_{H}∣

^{2}-∣B

_{H}∣

^{2}=1 for the sake of normalization. Perfect injection is achieved if ∣t∣=1, t being the complex amplitude injection coefficient.

^{+}> in (c) and all waves are shown in (c). Since ∣A’

_{H}∣ ≈∣B’

_{H}∣>>1 with ∣B’

_{H}∣>∣A’

_{H}∣, Fig. 2(c) evidences that the injector basically acts as a mirror for the plane waves, which reflects most of the light with a modal reflectivity coefficient r

_{m}=A’

_{H}/B’

_{H}and which transmits light with a modal transmission ∣t/B’

_{H}∣

^{2}=1-∣r

_{m}∣

^{2}. Thus, designing an injector to couple light into a slow mode amounts to synthesize a mirror. In fact, the smaller the group velocity, the larger the mirror reflectance is. Figure 2(d) shows the general solution of the synthesis problem. The injector is composed of a mirror with reflectance ∣A’

_{H}/B’

_{H}∣

^{2}and eventually of a phase plate, the latter being used to guaranty that the argument of r

_{m}strictly matches the argument of the stationary-ratio (A’

_{H}/B’

_{H}) of the backward-propagating Bloch mode ∣B

^{-}>.

_{m}are available as a function of the number

*m*of alternate layer pairs. For a mirror form with the same materials as the periodic stack, it is shown [19] that r

_{m}=(1- (n

_{L}/n

_{H})

^{2m})/(1+(n

_{L}/n

_{H})

^{2m}), which is approximately equal to 1-2(n

_{L}/n

_{H})

^{2m}for large

*m*. By equally the latter expression with the stationary-ratio defined by Eqs. (6a) or (6b), we obtain that

_{VB}or β

_{CB}. Therefore, the injector length L scales as the logarithm of c/v

_{g}and we have

_{L}, n

_{H}and

*f*. The length in Eq. (9) largely contrasts with those obtained with an adiabatic approach. To check this, we have solved the scattering problem shown in Fig. 2(a) with the 2×2 transfer-matrix formalism [14], for a bi-layer periodic stack with n

_{H}=3.495, n

_{L}=1 and

*f*=0.5. The injectors are simply quarter-wave Bragg mirrors composed of alternated layers with the same refractive indices n

_{H}and n

_{L}, and designed at a Bragg wavelength corresponding to the band edge of the periodic stack. Figure 3 shows the transmission T into the slow Bloch mode of the periodic stack, as a function of the frequency of the incident plane wave (actually v

_{g}on the horizontal axis). The different curves are obtained for different

*m*values, ranging from

*m*=1 to

*m*=5. They have been calculated in the vicinity of the first valence band, but almost identical curves have been obtained for the conduction band. As expected, perfect injection (T=1) is achieved in all cases. Additionally we note that the group velocities corresponding to T=1 scale linearly with

*m*(in log scale), as predicted by Eq. (9). The horizontal arrows delimitate the 1dB (T=0.8) bandwidth of the injector in GHz. Indeed the bandwidth vanishes as (v

_{g})

^{2}, and for very small group velocities of ≈10

^{-4}, it is actually very small, ≈10

^{-2}GHz. Therefore, injectors designed as pure mirrors cannot be used for full-optical signal processing applications, but they may find applications for single frequency applications and may be incorporated into new architectures in DBR and DFB lasers for instance.

### 3.2 Broadband injection in 2D periodic waveguides

20. C. Sauvan, G. Lecamp, P. Lalanne, and J. P. Hugonin, “Modal-reflectivity enhancement by geometry tuning in photonic crystal microcavities,” Opt. Exp. **13**, 245–255 (2005). [CrossRef]

21. P. Lalanne and J. P. Hugonin, “Bloch-wave engineering for high Q’s, small V’s microcavities,” IEEE J. Quantum Electron. **39**, 1430–1438 (2003). [CrossRef]

20. C. Sauvan, G. Lecamp, P. Lalanne, and J. P. Hugonin, “Modal-reflectivity enhancement by geometry tuning in photonic crystal microcavities,” Opt. Exp. **13**, 245–255 (2005). [CrossRef]

*a*=350 nm and is composed of 185-nm large lamellar grooves etched down to the SiO

_{2}substrate. Because of the low cladding refractive indices, the periodic waveguide supports a single truly-guided slow-mode in the vicinity of the band edge (k≈π/

*a*) of the first valence and conduction bands. For broadband and lossless couplings, we have optimised injectors consisting of five slits and five ridges fully etched trough the waveguide core, the free parameters being the lengths of the ridges and grooves. An example of an optimised structure is shown in Fig. 4(a). For the optimisation we use the simplex search Nelder-Mead method. This direct method that does not use numerical or analytic gradients relies on an iterative simplex-minimization approach that progressively reduces the explored volume in the parameter hyper-space [22

22. J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead Simplex Method in Low Dimensions,” SIAM J. Optim. **9**,112–147 (1998). [CrossRef]

13. J. P. Hugonin and P. Lalanne, “Perfectly-matched-layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A. **22**, 1844–1849 (2005). [CrossRef]

*y*-axis) investigated in this work, very high accuracy (relative error below 0.1%) is achieved for the transmission, see details in Ref. [13

13. J. P. Hugonin and P. Lalanne, “Perfectly-matched-layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A. **22**, 1844–1849 (2005). [CrossRef]

_{g}=0.015 (λ≈1.65 μm) and is ≈99.9%. As the wavelength deviates from this value, the performance degrades. Actually, it is limited by the back-reflection R (blue circles) that rapidly increases, while the radiation losses L=1-T-R (magenta dashed curve) remain below 10

^{-3}over almost the entire spectral range. The 1dB (T>0.8) bandwidth is determined to be 275Ghz, a value approximately 10 times larger than that achieved with purely periodic injectors in Fig. 3. The broader bandwidth is a net effect of the large number of degrees of freedom that we have intentionally used for the 2D injector. The latter consists in 5 pairs of ridges-slits, while only m=2 layer pairs were used for the 1D injector in Fig. 3.

_{g}=0.002c, similar computations have shown that efficient injectors with a maximum injection efficiency of 99.4% can be designed with a 20 times smaller bandwidth. We believe that group velocities in the range of 0.01 that corresponds to a slow down factor of ≈40 may represent an interesting regime for on-chip optical processing with ≈250 GHz bandwidths. Smaller group velocities are likely to offer prohibitively small bandwidths and their associated propagation loss due to various disorders induced by fabrication inaccuracies may additionally be a critical issue [23–25

23. D. Gerace and L. C. Andreani, “Effects of disorder on propagation losses and cavity Q-factors in photonic crystal slabs,” Photonics and Nanostructures-fundamentals and applications **3**, 120–128 (2005). [CrossRef]

## 4. Conclusion

_{g}<c/100. There is a compromise. These conclusions have been reached for Bloch modes in thin-film stacks and in periodic slab waveguides, but are expected to remain quantitatively valid for other kinds of periodic ridge waveguides. Photonic crystal waveguides, like single-row-defect waveguides, deserves a specific study because the physical nature of the light confinement is different especially in the slow light regime. We expect that this prospective study will be helpful for further investigations in the field.

## Acknowledgments

## References and links

1. | S. E. Harris, “Electromagnetically induced transparency,” Phys. Today |

2. | J. Khurgin, “Expanding the bandwidth of slow-light photonic devices based on coupled resonators,” Opt. Lett. |

3. | J. Poon, L. Zhu, G. DeRose, and A. Yariv, “Transmission and group delay of microring coupled-resonator optical waveguides,” Opt. Lett. |

4. | M. F. Yanik and S. Fan, “Stopping light all optically,” Phys. Rev. Lett. |

5. | M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe “Optical bistable switching action of Si high-Q photonic-crystal nanocavities,” Opt. Express |

6. | A. Yu. Petrov and M. Eich, “Zero dispersion at small group velocities in photonic crystal waveguides,” Appl. Phys. Lett. |

7. | D. Mori and T. Baba, “Wideband and low dispersion slow light by chirped photonic crystal coupled waveguide,” Opt. Express |

8. | T. F. Krauss, “Photonic Crystals shine on,” Phys. World 32–36, (February 2006). |

9. | A. Melloni, F. Morichetti, and M. Martinelli, “Linear and Nonlinear propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. |

10. | G. Lenz, B. J. Eggleton, C. K. Madsen, and R .E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electronics |

11. | Y. A. Vlasov and S. J. McNab, “Coupling into the slow light mode in slab-type photonic crystal waveguides,” Opt. Lett. |

12. | M. Povinelli, S. Johnson, and J. Joannopoulos, “Slow-light, band-edge waveguides for tuneable time delays,” Opt. Express |

13. | J. P. Hugonin and P. Lalanne, “Perfectly-matched-layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A. |

14. | P. Yeh, |

15. | J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structure,” Phys. Rev. E |

16. | H. A. Haus, |

17. | K. Sakoda, |

18. | B. Momeni and A. Adibi, “Adiabatic stage for coupling of light to extended Bloch modes of photonic crystal,” Appl. Phys. Lett. |

19. | L. A. Coldren and S. W. Corzine, |

20. | C. Sauvan, G. Lecamp, P. Lalanne, and J. P. Hugonin, “Modal-reflectivity enhancement by geometry tuning in photonic crystal microcavities,” Opt. Exp. |

21. | P. Lalanne and J. P. Hugonin, “Bloch-wave engineering for high Q’s, small V’s microcavities,” IEEE J. Quantum Electron. |

22. | J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead Simplex Method in Low Dimensions,” SIAM J. Optim. |

23. | D. Gerace and L. C. Andreani, “Effects of disorder on propagation losses and cavity Q-factors in photonic crystal slabs,” Photonics and Nanostructures-fundamentals and applications |

24. | E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, “Disorder-induced scattering loss of line-defect waveguides in photonic crystal slabs,” Phys. Rev. B |

25. | M. D. Settle, R. J. P. Engelen, M. Salib, A. Michaeli, L. Kuipers, and T. F. Krauss, “Flatband slow light in photonic crystals featuring spatial pulse compression and terahertz bandwidth,” Opt. Express |

26. | H. A. Macleod, |

27. | R. E. Collin, |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(260.2110) Physical optics : Electromagnetic optics

(350.7420) Other areas of optics : Waves

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: March 16, 2007

Revised Manuscript: April 25, 2007

Manuscript Accepted: April 30, 2007

Published: May 2, 2007

**Citation**

P. Velha, J. P. Hugonin, and P. Lalanne, "Compact and efficient injection of light into band-edge slow-modes," Opt. Express **15**, 6102-6112 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-10-6102

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### References

- S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50, 36-42 (July 1997). [CrossRef]
- J. Khurgin, "Expanding the bandwidth of slow-light photonic devices based on coupled resonators," Opt. Lett. 30, 2778-2780 (2005). [CrossRef] [PubMed]
- J. Poon, L. Zhu, G. DeRose, and A. Yariv, "Transmission and group delay of microring coupled-resonator optical waveguides," Opt. Lett. 31, 456-458 (2006). [CrossRef] [PubMed]
- M. F. Yanik and S. Fan, "Stopping light all optically," Phys. Rev. Lett. 92, 083901 (2004). [CrossRef] [PubMed]
- M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi and T. Tanabe "Optical bistable switching action of Si high-Q photonic-crystal nanocavities," Opt. Express 13, 2678-2687 (2005). [CrossRef] [PubMed]
- A. Yu. Petrov and M. Eich, "Zero dispersion at small group velocities in photonic crystal waveguides," Appl. Phys. Lett. 85, 4866-4868 (2004). [CrossRef]
- D. Mori and T. Baba, "Wideband and low dispersion slow light by chirped photonic crystal coupled waveguide," Opt. Express 13, 9398-9408 (2005). [CrossRef] [PubMed]
- T. F. Krauss, "Photonic Crystals shine on," Phys. World 32-36, (February 2006).
- A. Melloni, F. Morichetti, and M. Martinelli, "Linear and Nonlinear propagation in coupled resonator slow-wave optical structures," Opt. Quantum Electron. 35, 365-379 (2003). [CrossRef]
- G. Lenz, B. J. Eggleton, C. K. Madsen, and R.E. Slusher, "Optical delay lines based on optical filters," IEEE J. Quantum Electronics 37, 525-532 (2001). [CrossRef]
- Y. A. Vlasov and S. J. McNab, "Coupling into the slow light mode in slab-type photonic crystal waveguides," Opt. Lett. 31, 50-52 (2006). [CrossRef] [PubMed]
- M. Povinelli, S. Johnson, and J. Joannopoulos, "Slow-light, band-edge waveguides for tuneable time delays," Opt. Express 13, 7145-7159 (2005). [CrossRef] [PubMed]
- J. P. Hugonin and P. Lalanne, "Perfectly-matched-layers as nonlinear coordinate transforms: a generalized formalization," J. Opt. Soc. Am. A. 22, 1844-1849 (2005). [CrossRef]
- P. Yeh, Optical waves in layered media, (J. Wiley and Sons, New York 1988).
- J. M. Bendickson, J. P. Dowling, and M. Scalora, "Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structure," Phys. Rev. E 53, 4107-4121 (1996). [CrossRef]
- H. A. Haus, Waves and fields in optoelectronics (Prentice-Hall International, London, 1984).
- K. Sakoda, Optical properties of photonic crystals, (Springer-Verlag, Berlin 2001) Chap. 11.
- B. Momeni and A. Adibi, "Adiabatic stage for coupling of light to extended Bloch modes of photonic crystal," Appl. Phys. Lett. 87, 171104 (2005). [CrossRef]
- L. A. Coldren and S. W. Corzine, Diode lasers and photonic integrated circuits, (J. Wiley and Sons, New York, 1995).
- C. Sauvan, G. Lecamp, P. Lalanne, and J. P. Hugonin, "Modal-reflectivity enhancement by geometry tuning in photonic crystal microcavities," Opt. Exp. 13, 245-255 (2005). [CrossRef]
- P. Lalanne and J. P. Hugonin, "Bloch-wave engineering for high Q’s, small V’s microcavities," IEEE J. Quantum Electron. 39, 1430-1438 (2003). [CrossRef]
- J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, "Convergence properties of the Nelder-Mead Simplex Method in Low Dimensions," SIAM J. Optim. 9,112-147 (1998). [CrossRef]
- D. Gerace and L. C. Andreani, "Effects of disorder on propagation losses and cavity Q-factors in photonic crystal slabs," Photonics and Nanostructures-fundamentals and applications 3, 120-128 (2005). [CrossRef]
- E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, "Disorder-induced scattering loss of line-defect waveguides in photonic crystal slabs," Phys. Rev. B 72, 161318 (2005). [CrossRef]
- M. D. Settle, R. J. P. Engelen, M. Salib, A. Michaeli, L. Kuipers, and T. F. Krauss, "Flatband slow light in photonic crystals featuring spatial pulse compression and terahertz bandwidth," Opt. Express 15, 219-226 (2007). [CrossRef] [PubMed]
- H. A. Macleod, Thin-film optical filters, (Adam Hilger LTD, London 1969).
- R. E. Collin, Field theory of guided waves, (London, Section 9 1960).

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