Two integrated devices based on the vertical coupling between a photonic crystal microcavity and a silicon (Si) ridge waveguide are presented in this paper. When the resonator is coupled to a single waveguide, light can be spectrally extracted from the waveguide to free space through the far field emission of the resonator. When the resonator is vertically coupled to two waveguides, a vertical add-drop filter can be realized. The dropping efficiency of these devices relies on a careful design of the resonator. In this paper, we use a Fabry-Perot (FP) microcavity composed of two photonic crystal (PhC) slab mirrors. Thanks to the unique dispersion properties of slow Bloch modes (SBM) at the flat extreme of the dispersion curve, it is possible to design a FP cavity exhibiting two quasi-degenerate modes. This specific configuration allows for a coupling efficiency that can theoretically achieve 100%. Using 3D FDTD calculations, we discuss the design of such devices and show that high dropping efficiency can be achieved between the Si waveguides and the PhC microcavity.
© 2010 OSA
It is now well known that due to their dispersion properties, photonic crystals (PhC) offer one of the best platforms to realize a wide range of photonic devices. 3D integration of such devices is the next step towards a future multi level photonic integrated circuit. One of the main key functions to forward integration of photonic crystal based devices is the coupling between such PhC structures to Si waveguides. Indeed, Si/SiO2 ridge waveguides are easy to fabricate, exhibit very low losses around 1.55µm, and operate on a wide wavelength scale. In addition to these advantages, PhC can control light in the three directions of space making this association very attractive for 3D photonic integration.
In this paper, we focus on the coupling between a Si ridge waveguide and photonic crystal based resonators, in particular a vertical Fabry-Perot cavity composed of two 1D photonic crystal slab mirrors. This microcavity has already been extensively studied in [1, 2]. As depicted in Fig. 1 , the Si waveguide is coupled vertically to the PhC microcavity. Usually, the reported studies on the coupling between PhC devices and waveguides operate as “in-plane” devices [3–8]. Some groups study and use the coupling between PhC devices and optical fibres [9, 10]. To our knowledge, only one vertical design has been reported leading to a reflection-type filter . Very recently, an experimental demonstration of a W1 PhC laser coupled to a Si waveguide has been published . In this last case, the coupling between the Si ridge waveguide and the W1 PhC waveguide involves a Bloch mode located below the light cone, preventing a fine control of the PhC far field pattern.
Despite of the fact that more technological steps are needed to realize such devices, the study of this vertical configuration is motivated by several advantages. First, the gap between PhCs and waveguides can be accurately controlled using material growth or deposition. Moreover, the use of different materials for waveguides and PhC slabs is made easier than in the case of an in-plane device , in the frame of heterogeneous integration (using molecular bonding for instance) . The fabrication of this device requiring successive deposits of different materials, such as amorphous Si and Si dioxide (SiO2), is completely CMOS compatible. Additionally, using suspended PhC membrane, these devices can also be tuned by electrostatic actuation of the PhC slab (MOEMS approach ). Finally, a device consisting in several levels is an exciting way to achieve a new 3D photonic integrated circuit.
The paper is organized as follows. After the introduction, in the first part we describe the coupling mechanism between a resonator that supports stationary modes and a ridge waveguide. The design of the PhC microcavity is addressed in a second part. The microcavity is constituted by two 1D PhC slab mirrors spaced by a λ SiO2 gap. The unique dispersion properties resulting from the use of PhC slab mirrors allow the presence of two quasi-degenerate modes in the PhC cavity, the essential property needed to reach 100% of dropping efficiency [16, 17]. In the last two parts, we discuss the design of new devices based on the vertical coupling between such PhC microcavity and Si ridge waveguides. The first device will be used to extract light, spatially and spectrally from the Si waveguide to free space. Indeed, by using slow Bloch modes of the PhC slab mirror located at the Γ-point of the dispersion curve, one can control accurately the far field emission of the microcavity. This device can then be coupled either to an optical fibre or to another level of an optical 3D circuit. Lastly, we demonstrate that a vertical add-drop filter consisting in the coupling between the PhC microcavity and two silicon waveguides can be realized. These two devices, given in Fig. 1, allowing a selective and directive communication between several levels can be seen as new components for multi-level photonic integrated circuits.
2. Coupling conditions
2.1. Phase matching
Evanescent coupling between the microcavity and the waveguide is made possible by the hybrid nature of the slow Bloch mode (SBM) inside the PhC mirror which is located at the Γ-point of the dispersion curve. Indeed, such a SBM is partly radiated in the vertical direction in the cavity, and partly guided inside the PhC membrane. The guided component of the SBM can be used to transfer energy between the cavity mode and the propagating mode of the waveguide. The in-plane wave-vector k// of the SBM in the mirror is obviously in a direction perpendicular to the PhC slits, thus the waveguide is logically positioned as shown in Fig. 1.
An efficient coupling between the PhC mirror SBM and the waveguide mode thus requires the phase matching between the guided component of the PhC mirror SBM (the in-plane wave-vector, k//) and the propagation constant β of the guided mode. The lattice parameter of the 1D PhC mirror is denoted a. The spatial distribution of the field inside the PhC slab is periodic with period a, and the amplitude of the SBM component having k//=2π/a is the most dominant in the Bloch function (Γ-point). The phase matching condition between the two wave-vectors, k// and β, then requires that the effective index neff of the guided mode equals λ/a. For a given microcavity design, this condition will set the effective index of the guided mode and thus the dimensions (width and thickness) and material (refractive index) of the refractive waveguide. We work with silicon (n=3.5 at 1.5µm) for the design of the PhC microcavity and waveguides and the device is entirely embedded in silica (n=1.45 at 1.5µm). The phase matching condition can be achieved using such materials (waveguide and PhC cavity dimensions are detailed in section 3).
2.2. Basics of coupling
In this part, using the coupled mode theory in time , we highlight the parameters that govern the evanescent coupling between the PhC microcavity and the waveguide.
Theoretically, for a resonator that supports stationary modes, dropping efficiency of 100% can be reached if two stationary degenerate modes with opposite symmetry are used for resonant transfer . A careful design of the resonator is thus needed to achieve a high dropping efficiency.
The studied system is depicted in Fig. 2 . Light propagating in the Si waveguide can be partially coupled to the resonator and then returned to the waveguide in the same propagation direction (transmission) or in the opposite direction (reflection). The light transferred to the resonator can be coupled to the continuum (light extraction). The amplitude of the incoming wave in the waveguide is denoted S+1, the reflected part S-1 and the transmitted part S-2 after coupling with the resonator. The PhC SBM will interact with the waveguide mode in the forward and backward directions along the interaction length L. The quality factor of the uncoupled cavity Q0 can be defined as where ω is the mode frequency and τ0 the lifetime of photons inside the resonator before their escape out of the PhC microcavity. We can also express the coupling time, by where 1/τc is the rate at which photons leak from the microcavity to the waveguide. In Fig. 2, the subscript s (a) is used to denote the symmetric (anti-symmetric) mode. In this perturbative approach, additional losses (diffraction losses) in free space due to the coupling are neglected.
Using the coupled mode theory in time, the reflected and transmitted amplitudes can be expressed as follows :
Let us now extend the coupling scheme described above (between a resonator and a single waveguide) to the case where the same resonator is vertically coupled to two waveguides: the light propagating in the first waveguide, which called the bus waveguide, can be transferred in a direction defined by the specific symmetries of the two degenerate modes in the second (receiver) waveguide (see Fig. 3 ). The device is called an add-drop filter. In the configuration described in Fig. 2, it is expected  that at resonance the light in the bus will be transferred in the forward direction in the receiver waveguide.
Similarly to the coupling with a single waveguide, the coupled mode theory in time allows us to calculate the reflected and transmitted amplitudes in the bus, and the light transferred in the backward and forward directions in the receiver waveguide. These quantities can be expressed as follows:
Assuming that the two modes are perfectly degenerate (ωa=ωs and τ0a=τ0s) and the coupling rates are the same for the symmetric and anti-symmetric modes (τca=τcs), the reflected and transmitted powers in the bus waveguide as well as the powers in the receiver waveguide in the backward and forward directions can be written (at resonance) as:
3. 3D FDTD calculation method
In order to characterize the dropping efficiency of the coupled devices, we use a 3D FDTD calculation method with perfectly matched layers (PML) developed in our laboratory . The total simulation domain is 22.6*28.6*6.95µm along the x, y and z directions and the spatial step of the simulation is 0.05µm. A gaussian dipole source located at one end of the waveguide emits between 1.2 and 1.8μm (see Fig. 3). In the case of simple light extraction from a single Si waveguide, two detector planes are placed at each end of the waveguide to calculate the reflectance, which is the part of the field which is coupled from the PhC SBM to the propagating mode of the waveguide in the backward direction, and the transmittance which is the same quantity calculated in the forward direction. In the case of an add-drop device, four detector planes are placed at each end of both waveguides. The calculated reflected and transmitted powers in each waveguide are normalized with respect to the reflected and transmitted powers calculated without the resonator. Quality factors and frequencies of modes are obtained using “harminv” , a free MIT program, on the temporal files resulting from the 3D FDTD simulation.
4. Microcavity design
In the first part of this paper, we explained the necessity for the resonator to present two degenerate modes of opposite symmetries to achieve a dropping efficiency of 100%. By using a FP microcavity composed of two PhC mirrors, we can exploit the dispersion characteristics of this PhC based FP microcavity to obtain two quasi-degenerate modes of opposite symmetries inside the cavity . By quasi-degenerate, we mean that the spectral distance between both modes is much lower than their linewidth. The modes are never strictly degenerate, thus we will use in the following the term “quasi-degenerate”. Indeed, the FP microcavity has finite lateral sizes, thus enabling the presence of higher order cavity modes. The first two lateral cavity modes in the y-direction (along the PhC periodicity) are spectrally close and the spectral distance between these two modes depends not only on the lateral size of the cavity, but also on the band curvature of the cavity mode.
A flatter band curvature around the Γ-point leads to spectrally closer lateral FP modes. Contrary to a classical FP microcavity based on two Bragg mirrors, the band curvature for a FP cavity based on two PhC slab mirrors can be made extremely flat (this property is due to the use of slow Bloch modes at a Γ-point extreme of the dispersion characteristics with negative curvature: for a rigorous demonstration of this point, see ). Thus, in the PhC based microcavity, one can obtain more easily quasi-degeneracy between the two first lateral modes.
The studied microcavity consists in two identical photonic crystal slab mirrors spaced by a λ SiO2 gap. Each mirror is a 1D Si PhC slab where the lattice parameter is 950nm, the Si filling factor 50% and the slab thickness 300nm. This 1D PhC mirror presents a broadband reflection in the range 1.45-1.65μm for light polarized parallel to the slits (TE polarization) . Using 3D FDTD, we then design a FP microcavity composed of these two PhC mirrors spaced by a silica gap of 1.25µm. The microcavity has a lateral size of 20µm. In this microcavity, we obtain two modes spectrally separated by 0.5nm (λ1 = 1555.6nm and λ2 = 1556.1nm), their respective quality factors being 12000 and 3000. These modes are not perfectly degenerate and the small differences between their wavelengths and quality factors will slightly decrease the dropping efficiency. Nevertheless, high light extraction can be obtained as it will be shown in next section.
The working wavelength is 1.55µm and the 1D PhC lattice parameter is 0.95µm. Using the phase matching condition given in section 2.1, we set the width of the Si waveguide to 0.24µm and its thickness to 0.2µm.
Concerning the sensitivity of the microcavity to technological imperfections, it can be noticed that the alignment between the cavity mirrors (lateral positioning) is not a critical factor for coupling efficiency. Indeed, the microcavity resonance wavelength is shifted by 0.1nm when the lateral misalignment between the two mirrors is 200nm and 0.3nm when the misalignment is 475nm (which is half the lattice parameter of the PhC mirror) and the quality factor remains of a few thousands (>10000). However, the flatness of band curvature is sensitive to the filling factor of the PhC slab (slit width) . Thus, a careful control of the filling factor should be done in order to obtain two nearly degenerate modes in the microcavity.
5. Light extraction
First, when the microcavity is coupled to a single Si waveguide (Fig. 1 a) the device acts as a light extractor. Indeed, light propagating in the waveguide is efficiently emitted vertically in free space thanks to the coupling with the resonator.
We first calculate using 3D FDTD, the quality factors of the two quasi degenerate modes of the PhC based microcavity when coupled to the waveguide as a function of the coupling distance. Results are given in Fig. 4 . These quality factors and the associated frequencies of the two modes were inserted in the coupled mode Eq. (1) given in section 2.2 in order to calculate reflected, transmitted and extracted powers (see Fig. 5 ).
The coupled mode theory described in section 2.2 shows that the maximum dropping efficiency is reached when the quality factors of the two modes are half the quality factors of the unloaded microcavity. This is obtained for a coupling distance around 900nm (Fig. 4). Furthermore, numerical results given in Fig. 5 show that the extracted power reaches 96% at this coupling distance.
The coupled mode theory in time does not take into account the additional diffracted losses induced by the coupling between the silicon waveguide and the PhC mirrors. In order to estimate these losses and thus obtain the quality factor of the device at resonance, we use 3D FDTD calculations. Figure 6 shows the reflected, transmitted and extracted power spectra for a coupling distance of 900nm. The spectra exhibit different peaks corresponding to different lateral modes of the microcavity. At 1.555µm (lowest order mode), the extracted power in the waveguide reaches 90%. The slight difference between theoretical results obtained from the coupled mode theory in time (Fig. 5) and the 3D FDTD simulations comes from the spectral resolution of the 3D FDTD calculation (Δλ≈0.3nm) on the one hand, and from the generation of losses due to the coupling, which are accounted for in the 3D FDTD calculation on the other hand.
Electric field maps of the device are given in Fig. 7 , for a coupling distance of 900nm. One can see that the phase matching between the travelling waveguide mode and the PhC Bloch mode is satisfied. Additionally, these figures show the simultaneous presence of the two quasi-degenerate modes in the microcavity, responsible for the high extraction efficiency. We also report in Fig. 7 the far field patterns of the coupled device and the bare microcavity. The former exhibit only one lobe located at an off-normal angle of 6°, showing the strong directionality of the coupling.
In the fabrication of this device, the distance between the waveguide and microcavity is likely to be a critical factor but it can be very accurately controlled through material deposition, silicon dioxide for example (to a few nm).
6. Vertical add-drop filter
In the previous case, the light propagating in the waveguide was selectively transferred to free space with emission directions defined by the far field pattern of the microcavity. As the microcavity emission is localized near the vertical direction, this device can be easily coupled to an optical fibre or another optical level of a 3D integrated circuit.
The vertical add-drop configuration allows for a spectrally and spatially resolved light transfer to a waveguide located on another vertical level.
The reflected and transmitted powers in each waveguides have been calculated using the coupled mode equations for an add-drop filter (see Eq. (3)) and the quality factors and frequencies of the quasi-degenerate modes (obtained using 3D FDTD as a function of the coupling distance between the microcavity and the two waveguides). Results are given in Fig. 8 .
Figure 8 shows that the reflected power (R=Dbackward) is almost zero in accordance to the coupled mode theory in time presented in section 2.2. Using 3D FDTD, we also calculated the transmitted and reflected powers in the bus and receiver waveguides for four coupling distances. Results are given in Fig. 9 .
The comparison between the dropping efficiency obtained using the coupled mode theory and the numerical 3D FDTD calculations allows to distinguish two regimes:
- - For coupling distances >700nm, dropping efficiency values obtained using 3D FDTD are very similar to those obtained using the coupled mode theory. For high coupling distance, the dropping efficiency remains low because the coupling strength does not permit to compensate the losses. However, as it can be seen in Fig. 9 (d), diffracted losses due to the coupling are small (the off resonances transmitted power is close to 100%).
- - For coupling distances <700nm, diffracted losses due to the coupling significantly increase. Dropping efficiencies obtained using 3D FDTD are not well reproduced by those obtained using the coupled mode theory. Two main effects are responsible: first, for small coupling distances diffracted losses due to the coupling highly increase. Second, as the coupling strength is different for the symmetric and anti-symmetric modes with the waveguide (due to the different spatial repartition of each mode), the spectral distance of both modes increases (see e.g. Figure 9a where a two components peak can be clearly identified) leading to a reduction of the dropping efficiency.
The maximum power transfer to the receiver in the forward direction reaches 65% for a coupling distance of 700nm. The quality factor is a few hundreds (>500) at resonance and the signal to noise ratio (which is the dropping power over all powers travelling in the two waveguides) has been estimated to be around 20 (or 13 dB). The transmitted power in the bus waveguide remain very small at resonance, even for high coupling distances (<10% at 900nm) showing the good dropping efficiency between the waveguides through the microcavity.
However, in our 3D calculations the dropping efficiency is limited by two main effects:
- - Losses due to the coupling are estimated around 15% for a coupling distance of 700nm (only 5% for a coupling distance of 900nm). This point can be improved by using PhC mirrors with gradual modification of the filling factor on the edges (impedance matching between the propagating waveguide mode and the PhC slow Bloch mode). Furthermore, this design allows a significant increase of the intrinsic quality factors of modes in the microcavity .
- - The studied microcavity modes are not perfectly degenerate (ωa≠ωs and τ0a≠τ0s) and the quality factors of both modes are about a few thousands. The increase of the PhC lateral sizes is a good way to improve the degeneracy of both modes, their quality factors and thus the dropping efficiency. Indeed, by using a 25*25µm microcavity, quality factors of a few ten of thousands have been obtained. 3D FDTD calculations have not been performed with these microcavity sizes since it leads to very long calculation times and very high memory resources. However, in an experimental design, various PhC microcavity sizes can be realized.
In the add-drop scheme, the main issue in the fabrication of the device is the alignment between the two waveguides which has to be carefully controlled in order to ensure even coupling rates to the microcavity. However, as in the previous device (light extraction), the coupling distance and silica cavity spacer between the mirrors can be accurately controlled through material deposition and the lateral misalignment between the patterns is still not a critical factor for coupling efficiency.
In this paper, coupling schemes between a Si ridge waveguide and a PhC based device are analysed. Using 3D FDTD, we showed that an extracting efficiency as high as 90% can be achieved from refractive Si ridge waveguide through a PhC microcavity. As the microcavity emission is radiated around the vertical direction, this device can be easily coupled to an optical fibre or other optical levels of a 3D integrated circuit. We also studied a vertical add-drop filter. It consists in the coupling of two Si waveguides with the PhC microcavity. Dropping efficiency of 65% has been obtained in such a configuration. Although this efficiency is limited, increasing the lateral size of the microcavity or modifying the PhC mirror edges should significantly increase the light transfer. This device allows for a spectrally and spatially resolved light transfer to a waveguide located on other vertical levels.
We presented two passive devices that are completely compatible with CMOS processing (successive deposits of amorphous silicon, silicon dioxide…). Heterogeneous integration of III-V semiconductors, through molecular bonding for instance, will also allow for the realization of a PhC microcavity laser coupled to a Si ridge waveguides.
Lastly, these two vertical operating devices open the way to future integration of photonic crystal based devices in a 3D photonic integrated circuit.
The authors would like to thank Fabien Mandorlo for fruitful discussions, Esther Wertz and Frederic Bordas for their careful reading of the manuscript and Laurent Carrel for his technical support on 3D FDTD. This work was partly developed into the frame of the FP6 “ePiXnet” european network of excellence. It benefits from the support of the Rhône-Alpes region and of the European FP7 HELIOS IP project.
References and links
1. S. Boutami, B. Ben Bakir, X. Letartre, J.-L. Leclercq, and P. Viktorovitch, “Photonic crystal slab mirrors for an ultimate vertical and lateral confinement of light in vertical Fabry Perot cavities,” Proc. SPIE 6989, 69890V (2008). [CrossRef]
2. S. Boutami, B. Benbakir, X. Letartre, J. L. Leclercq, P. Regreny, and P. Viktorovitch, “Ultimate vertical Fabry-Perot cavity based on single-layer photonic crystal mirrors,” Opt. Express 15(19), 12443–12449 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-15-19-12443. [CrossRef]
3. C. Smith, R. De La Rue, M. Rattier, S. Olivier, H. Benisty, C. Weisbuch, T. Krauss, R. Houdré, and U. Oesterle, “Coupled guide and cavity in a two-dimensional photonic crystal,” Appl. Phys. Lett. 78(11), 1487 (2001). [CrossRef]
4. M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H. Ryu, “Waveguides, resonators and their coupled elements in photonic crystal slabs,” Opt. Express 12(8), 1551–1561 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?&id=79588. [CrossRef]
5. Z. Zhang and M. Qiu, “Compact in-plane channel drop filter design using a single cavity with two degenerate modes in 2D photonic crystal slabs,” Opt. Express 13(7), 2596–2604 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-7-2596. [CrossRef]
6. C. Seassal, Y. Desieres, X. Letartre, C. Grillet, P. Rojo-Romeo, P. Viktorovitch, and T. Benyattou, “Optical coupling between a two-dimensional photonic crystal based microcavity and single-line defect waveguide on InP membranes,” IEEE J. Quantum Electron. 38(7), 811–815 (2002). [CrossRef]
7. H. Takano, B. Song, T. Asano, and S. Noda, “Highly efficient in-plane channel drop filter in a two-dimensional heterophotonic crystal,” Appl. Phys. Lett. 86(24), 241101 (2005). [CrossRef]
8. H. Takano, Y. Akahane, T. Asano, and S. Noda, “In-plane-type channel drop filter in a two-dimensional photonic crystal slab,” Appl. Phys. Lett. 84(13), 2226 (2004). [CrossRef]
9. C. Grillet, C. Monat, C. L. Smith, B. J. Eggleton, D. J. Moss, S. Frédérick, D. Dalacu, P. J. Poole, J. Lapointe, G. Aers, and R. L. Williams, “Nanowire coupling to photonic crystal nanocavities for single photon sources,” Opt. Express 15(3), 1267–1276 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?id=125713. [CrossRef]
12. Y. Halioua, T. J. Karle, F. Raineri, P. Monnier, I. Sagnes, G. Roelkens, D. Van Thourhout, and R. Raj, “Hybrid InP-based photonic crystal lasers on silicon on insulator wires,” Appl. Phys. Lett. 95(20), 201119 (2009). [CrossRef]
13. K. Nozaki, H. Watanabe, and T. Baba, “Photonic crystal nanolaser monolithically integrated with passive waveguide for effective light extraction,” Appl. Phys. Lett. 92(2), 021108 (2008). [CrossRef]
14. P. Rojo Romeo, J. Van Campenhout, P. Regreny, A. Kazmierczak, C. Seassal, X. Letartre, G. Hollinger, D. Van Thourhout, R. Baets, J. M. Fedeli, and L. Di Cioccio, “Heterogeneous integration of electrically driven microdisk based laser sources for optical interconnects and photonic ICs,” Opt. Express 14(9), 3864–3871 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=OE-14-9-3864. [CrossRef]
15. S. Boutami, B. Ben Bakir, J.-L. Leclercq, X. Letartre, P. Rojo-Romeo, M. Garrigues, P. Viktorovitch, I. Sagnes, L. Legratiet, and M. Strassner, “Highly selective and compact tunable MOEMS photonic crystal Fabry-Perot filter,” Opt. Express 14(8), 3129–3137 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-8-3129. [CrossRef]
16. C. Manolatou, M. Khan, S. Fan, P. Villeneuve, H. Haus, and J. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35(9), 1322–1331 (1999). [CrossRef]
20. E. Drouard, H. Hattori, C. Grillet, A. Kazmierczak, X. Letartre, P. Rojo-Romeo, and P. Viktorovitch, “Directional channel-drop filter based on a slow Bloch mode photonic crystal waveguide section,” Opt. Express 13(8), 3037–3048 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-8-3037. [CrossRef]
21. L. Ferrier, S. Boutami, F. Mandorlo, X. Letartre, P. Rojo-Romeo, P. Viktorovitch, P. Gilet, B. B. Bakir, P. Grosse, J. M. Fedeli, and A. Chelnokov, “Vertical microcavities based on photonic crystal mirrors for III-V/Si integrated microlasers”, ”, Proceedings of SPIE, Photonics Europe, Proc. SPIE, Vol. 6989, 69890W, Photonic Crystal Materials and Devices VIII (2008).
22. L. Ferrier, P. Rojo-Romeo, E. Drouard, X. Letatre, and P. Viktorovitch, “Slow Bloch mode confinement in 2D photonic crystals for surface operating devices,” Opt. Express 16(5), 3136–3145 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-5-3136. [CrossRef]