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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 10 — May. 14, 2007
  • pp: 6102–6112
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Compact and efficient injection of light into band-edge slow-modes

P. Velha, J. P. Hugonin, and P. Lalanne  »View Author Affiliations


Optics Express, Vol. 15, Issue 10, pp. 6102-6112 (2007)
http://dx.doi.org/10.1364/OE.15.006102


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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/vg). 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

In recent years, faster optical telecommunication and data processing have motivated research towards solutions to try to minimize the involvement of electronics in signal manipulation and to keep signals in the optical domain as long as possible. For true all-optical signal processing, one has to use optical non-linearities. Unfortunately, these non-linearities are extremely weak, thus requiring large interaction lengths or huge operational powers. Different approaches may be used to reinforce light-matter interactions including the elaboration of new materials [1

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

] or of new structural geometries like coupled-resonator optical waveguides [2–3

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

] or photonic-crystal (PC) microcavities [4–5

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

]. In the quest for ultimate miniaturization, slow waves obtained by introducing a periodic corrugation along the z-axis of a waveguide appears as a promising approach [6–7

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]

]. Clearly, a crucial issue for integrated circuits using slow-wave waveguides is the realization of efficient light injectors between uniform z-invariant waveguides and slow-wave z-periodic waveguides.

In contrast, we hereafter consider slow-wave injectors that operate as interference filters (both backward and forward Bloch modes participate in the tapering process) and that can provide efficient injection in an ultra-compact way. Typically, we report on the design of efficient injectors with characteristics lengths of a few wavelengths. In Section 2, we start with the Bloch modes of periodic layers and analytically study the group-velocity impedance-mismatch problem arising at an interface between a uniform medium and a z-periodic layer stack. In particular we derive closed-form expression for the injection efficiency as a function of the group-velocity of injected light. This preliminary study is motivated by the fact that the injection problem is in essence one-dimensional and that simple intuitive analysis have not yet been presented for the back-reflection resulting at the interface between z-invariant and z-periodic media, despite its importance for our problem. Section 3 is devoted to the study of light injection at a single frequency. We show that injectors can be designed as simple Bragg mirrors that allows 100% coupling efficiencies for arbitrary small group velocities and that the injector length scales as log(c/vg). 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]

] and of optimisation techniques, we design efficient and wavelength-long injectors that couple light into slow waves (vg/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

We start with the scattering problem defined in Fig. 1(a). An incident plane wave in a uniform medium (refractive index n) is normally incident onto a semi-infinite periodic thin-film stack composed of lossless alternate layers with refractive indices nH and nL. 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 nH.

2.1 Injection efficiency

This scattering problem can be solved by considering the modes of the two semi-infinite media at a given frequency ω. We denote by k = ω/c the modulus of the free-space wave vector. The modes in the uniform medium are the forward- and backward- propagating plane waves, denoted ∣P+> exp(jknz) and ∣P-> exp(-jknz), with a unitary Poynting vector ∣P+> = [Ex, Hy] = [(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(jkneff z) and ∣B-(z)> exp(-jkneff z), with ∣B-> and ∣B+> two periodic functions of the z-coordinate and with neff 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

14. P. Yeh, Optical waves in layered media, (J. Wiley and Sons, New York1988).

].

B+(z=z0+pa)>=AHP+>+BHP>,
(1)

The ratio u=BH/AH 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 vg=0, ∣u∣=1. Outside the photonic gap, u is real for lossless materials and using the analytical expressions obtained in [14–15

14. P. Yeh, Optical waves in layered media, (J. Wiley and Sons, New York1988).

], it can be further shown that

0<u<1,
(2a)
1<u<0,
(2b)

for the valence and conduction bands, respectively. Similarly, the backward-propagating Bloch mode can be denoted by

B(z=z0+pa)>=A'HP+>+B'HP>,
(3)

Note that in general, there is no relation between ∣B-> 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]

]. In this case, B'H=AH and A'H=BH showing that the u-factor of ∣B-> is simply the inverse of that of ∣B+> for the bi-layer stack.

r=[(nnH)+u(n+nH)exp()][(n+nH)+u(nnH)exp()],
(4)

Fig. 1. Impedance mismatch problem. (a): Scattering at an interface between a uniform medium (refractive index n) and a semi-infinite bi-layer periodic stack (refractive indices nH and nL). (b) and (c): Injection efficiency T as a function of the group-velocity of the periodic-stack Bloch mode for different fill factors (0.1<f<0.9). Colormaps and white curves are exact numerical results obtained with the 2×2 transfer-matrix method in [14], and the superimposed solid-red curves are obtained using the approximate closed-form expressions. (b): data obtained in the vicinity of the valence band edge of a weak-modulation (nH=1.5 and nL=1.4) stack for n=nH. (c): data obtained in the vicinity of the conduction band edge of a strong-modulation stack (nH=3.495 and nL=1) for n=nL.

2.2 Approximate closed-form expression for the injection efficiency

(vgc)[α(1+u2)+2βu]=2nH(1u2),
(5)

with α=(1+f)nH 2 +(1-f)nL 2 and β=(1-f) cos(ϕ) (nH 2 - nL 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 vg and the stationary ratio u. To express ϕ as a function of vg, one may expand it in a power series of vg/c, ϕ=ϕ(0)+O(vg/c), and by retaining only the first term ϕ(0), Eq. (5) can be solved for u(vg)

uVB=1(α+βVB2nH)vgc+O(vgc)2,and
(6a)
uCB=1+(αβCB2nH)vgc+O(vgc)2,
(6b)

where βVB and βCB correspond to β(ϕ = ϕ(0)) at the valence and conduction band edges, respectively. For quarter-wave periodic stacks, f = (1+nH/nL)-1, a closed-form expression for the phase delay ϕ(0) exists [14

14. P. Yeh, Optical waves in layered media, (J. Wiley and Sons, New York1988).

], ϕ(0) = π/2 ± asin (nHnLnH+nL), the plus and minus signs holding for the conduction and valence bands. However for arbitrary f values, the calculation of ϕ(0) requires to solve for a transcendental equation, see Section 6.2 in Ref. [14

14. P. Yeh, Optical waves in layered media, (J. Wiley and Sons, New York1988).

]. To retrieve full analyticity, one may use the classical coupled-wave method [16

16. H. A. Haus, Waves and fields in optoelectronics (Prentice-Hall International, London, 1984).

], 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, ε1=(nH 2-nL 2) f sinc(πf). Within this approach that is all the more accurate as nH-nL is small, we have

ϕ(0)πfnH[1±ε1(2ε0)]ε012,
(7)

where ε0= fnH 2+(1-f) nL 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 nH, nL and n and of the fill factor f in the limit of small group velocities. As noted before, for n=nH, the expressions largely simplifies and one obtains

TVB=(α+βVBnH)vgc,TCB=(αβCBnH)vgc.
(8)

The predictions of the coupled-wave model are shown by the superimposed solid-red curves in Figs. 1(b) and 1(c) for a low- and high-index contrasts, respectively. Despite the large refractive-index modulation used in Fig. 1(c), which is likely not to be accurately described by the two-first Fourier coefficients of the relative permittivity, the agreement with the transfer-matrix results is quantitative. As we checked with computational results obtained by solving the transcendental equation for the phase delay ϕ(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).

For small-index contrasts, β is much smaller than α, and the injection efficiencies in Eq. (8) are almost identical, TVB≈TCB. Additionally, since α≈ nH 2 + nL 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≠nH, 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 nH/nL=3.495 and for n≠nH. Finally, note that the expressions in Eq. (8) largely differ from the usual ansatz [17–18

17. K. Sakoda, Optical properties of photonic crystals, (Springer-Verlag, Berlin2001) Chap. 11.

], r=(vg1-vg2) / (vg1+vg2), where vg1 and vg2 are the group-velocities of the incident and transmitted waves. For vg2<<vg1, the ansatz, which leads to T=4vg2/vg1, 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

The group-velocity impedance mismatch problem illustrated in Fig. 1 evidences the necessity of designing injectors for efficiently coupling light into slow modes. Indeed injectors are crucial for successful implementation or characterisation of systems relying on slow-mode field enhancements. In this Section, we show that, by engineering the interface between the uniform medium and the periodic stack, injectors with very-short length can be designed even for small group velocities. This result is established for 1D thin-film stacks and 2D periodic waveguides composed of slits in a slab waveguide.

3.1 Perfect injection in 1D thin-film stacks

Fig. 2. A slow-mode injector at a single frequency is basically a mirror. (a) Definition of the injector parameters : length L , interface S1 between the coupler and the uniform medium and interface S2 between the coupler and the periodic stack supporting a slow Bloch mode. (b) Perfect injector at a single frequency for a plane wave incident from the uniform medium. (c) Reciprocal problem. The incident illumination is the reciprocal Bloch mode propagating towards the negative z-direction. (d) General solution of the synthesis problem : the injector is composed of a mirror and of a phase plate.

L=log(cvg),
(9)

with g a constant parameter that depends on n, nL, nH 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

14. P. Yeh, Optical waves in layered media, (J. Wiley and Sons, New York1988).

], for a bi-layer periodic stack with nH=3.495, nL=1 and f=0.5. The injectors are simply quarter-wave Bragg mirrors composed of alternated layers with the same refractive indices nH and nL, 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 vg 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 (vg)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.

Fig. 3. Coupling efficiency into the slow-mode of a periodic stack (nH, nL) as a function of the group velocity (log scale) of the slow mode. Different injectors are considered. They are all composed of Bragg mirrors (nH, nL) with an increasing number of repeated pairs, m=1, …5. The results are obtained in the vicinity of the valence band edge of the periodic stack. The horizontal arrows indicate the bandwidths in GHz of the different injectors for T=0.8 (-1dB).

3.2 Broadband injection in 2D periodic waveguides

Fig. 4. Broadband injection from a planar waveguide to a periodic waveguide near the valence-band edge (λ=1.65 μm). (a) Injector geometry optimized for coupling at vg/c≈0.01. From left to right, the injector slit- and ridge-widths are 84, 143,127, 158, 174 nm and 239, 213, 173, 166, 166 nm, respectively. The superimposed red curve represents the squared modulus of the transverse electric field at optimal coupling. (b) Performance of the injector predicted by fully vectorial computational results for the radiation loss L (dashed curve), the modal reflection R (blue circles) and the transmission T (actually 1-T is shown with a solid black curve). All quantities are displayed in a log scale. Inset: Injection efficiency T as a function of vg with (red) and without (blue) injector. The maximum injection efficiency is as large as 0.999. In the absence of injector, this efficiency is only 0.25%.

As initial guesses for the optimisation, we have considered various Bragg mirrors with different groove and ridge lengths and different target group velocities. The optimisation has revealed that many different injector geometries can be used to inject light efficiently and that the most difficult criteria to fulfil is the broadband injection. Although the parameter hyperspace is likely not to be fully explored by the optimisation procedure, we believe that the set of solutions we obtained by repeating the optimisation with various initial guesses is likely to provide a good picture of the possible geometries that lead to efficient injection. Figure 4(a) shows a typical example of such a solution. This geometry has been obtained for a target group velocity of 0.01c in the first valence band. The superimposed red curve represents ∣E∣2 as a function of the z-coordinate. The field intensity in the periodic waveguide is roughly 40 times larger than that in the z-invariant waveguide. The optimised injector parameters are given in the figure caption. We note that the five slit widths progressively increase, while the five ridge widths progressively decrease along the injector. This progressive variation is well traditional for tapered mirrors engineered for ultrahigh Q microcavities, and is understood as a progressive tranverse-mode-profile matching of the various Bloch modes involved in the tapered geometry [21

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]

]. This guaranties small scattering losses, and therefore a high injection efficiency in the present context.

Figure 4(b) shows the injector performance as a function of the frequency of the incident guided mode, or equivalently as a function of the group velocity of the periodic-waveguide Bloch mode. The maximal injection efficiency T (solid black curve) is obtained for vg=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.

Similar performances have been obtained for smaller group velocities but with smaller bandwidths. For instance, for vg=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

In periodic media, Bloch modes with small group velocities are interference patterns and in their simplest form are created by the superposition of a forward- and a backward-propagating mode that together form a standing or a slowly-moving mode pattern. When illuminated from a z-invariant medium, light is only weakly coupled into the slow Bloch mode. For bi-layer periodic structures, we have derived closed-form expressions for the coupling efficiencies in the vicinity of the valence and conduction bands. The expressions evidence that the impedance mismatch essentially arises from the standing-wave character of the slow Bloch mode and that the injection efficiency is proportional to the group velocity of the Bloch mode, the proportionality factor being weakly dependant on the actual geometric parameters.

Acknowledgments

This research is partly supported under the European contract SPLASH of the 6th PCRD and by the Agence Nationale de la Recherche under contract MIRAMAN of the French ANR Nano2006. The authors thank C. Sauvan and G. Lecamp for fruitful discussions. P. Velha acknowledges a BDI CNRS-CEA fellowship. He is also at the Laboratoire Silicium Nanoélectronique Photonique et Structure of the CEA/DRFMC and at the Laboratoire des Technologies de la Microélectronique in Grenoble.

References and links

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]

3.

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]

4.

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

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 13, 2678–2687 (2005). [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]

7.

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]

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. 35, 365–379 (2003). [CrossRef]

10.

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]

11.

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]

12.

M. Povinelli, S. Johnson, and J. Joannopoulos, “Slow-light, band-edge waveguides for tuneable time delays,” Opt. Express 13, 7145–7159 (2005). [CrossRef] [PubMed]

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]

14.

P. Yeh, Optical waves in layered media, (J. Wiley and Sons, New York1988).

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]

16.

H. A. Haus, Waves and fields in optoelectronics (Prentice-Hall International, London, 1984).

17.

K. Sakoda, Optical properties of photonic crystals, (Springer-Verlag, Berlin2001) Chap. 11.

18.

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]

19.

L. A. Coldren and S. W. Corzine, Diode lasers and photonic integrated circuits, (J. Wiley and Sons, New York, 1995).

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]

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]

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]

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 72, 161318 (2005). [CrossRef]

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 15, 219–226 (2007). [CrossRef] [PubMed]

26.

H. A. Macleod, Thin-film optical filters, (Adam Hilger LTD, London1969).

27.

R. E. Collin, Field theory of guided waves, (London, Section 9 1960).

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

  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]
  3. 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]
  4. M. F. Yanik and S. Fan, "Stopping light all optically," Phys. Rev. Lett. 92, 083901 (2004). [CrossRef] [PubMed]
  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 13, 2678-2687 (2005). [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]
  7. 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]
  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. 35, 365-379 (2003). [CrossRef]
  10. 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]
  11. 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]
  12. M. Povinelli, S. Johnson, and J. Joannopoulos, "Slow-light, band-edge waveguides for tuneable time delays," Opt. Express 13, 7145-7159 (2005). [CrossRef] [PubMed]
  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]
  14. P. Yeh, Optical waves in layered media, (J. Wiley and Sons, New York 1988).
  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]
  16. H. A. Haus, Waves and fields in optoelectronics (Prentice-Hall International, London, 1984).
  17. K. Sakoda, Optical properties of photonic crystals, (Springer-Verlag, Berlin 2001) Chap. 11.
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