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Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 9 — May. 2, 2005
  • pp: 3310–3322
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Coupling analysis of heterogeneous integrated InP based photonic crystal triangular lattice band-edge lasers and silicon waveguides

Haroldo T. Hattori, Christian Seassal, Xavier Letartre, Pedro Rojo-Romeo, Jean L. Leclercq, Pierre Viktorovitch, Marc Zussy, Lea di Cioccio, Loubna El Melhaoui, and Jean-Marc Fedeli  »View Author Affiliations


Optics Express, Vol. 13, Issue 9, pp. 3310-3322 (2005)
http://dx.doi.org/10.1364/OPEX.13.003310


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Abstract

In recent years, many groups have envisioned the possibility of integrating optical and electronic devices in a single chip. In this paper, we study the integration of a photonic crystal laser fabricated in InP with a silicon passive waveguide. The coupling of energy between a 2D photonic crystal (PhC) triangular lattice band-edge laser and waveguide positioned underneath is analyzed in this paper. We show that a 40% coupling could be achieved provided the distance between the laser and the waveguide is carefully adjusted. A general description of the fabrication process used to realize these devices is also included in this paper.

© 2005 Optical Society of America

1. Introduction

In this paper, we will focus on the sub-system constituted by the optical source and the passive waveguide. As device integration on the silicon chip should exhibit a small footprint, we will only consider photonic structures that enable a very high confinement of photons. In our approach, the light source is a photonic crystal laser. Indeed, photonic crystals (periodic structures with high index contrast) allow the confinement of light in small volumes, and so lead to the fabrication of ultra-compact and relatively low threshold lasers. Moreover, fabrication of bi-dimensional (2D) PhCs is now well controlled and uses fabrication procedures very similar to those used to realize silicon integrated circuits. A multitude of devices has already been produced, such as filters [3

3. S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Channel drop filters in photonic crystals,” Opt. Express 3, 4–11 (1998), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-3-1-4. [CrossRef] [PubMed]

5

5. T. Matsumoto and T. Baba, “Photonic crystal k-vector superprism,” IEEE/OSA J. Lightwave Technol. 22, 917–922 (2004). [CrossRef]

], waveguide bends and branches [6

6. S. Fan, S. G. Johnson, J. D. Joannopoulos, C. Manolatou, and H. A. Haus, “Waveguide branches in photonic crystals,” J. Opt. Soc. Am. B 18, 162–165 (2001). [CrossRef]

8

8. J. Smajic, C. Hafner, and D. Erni, “Design and optimization of an achromatic photonic crystal bend,” Opt. Express 11, 1378–1384 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-12-1378 [CrossRef] [PubMed]

], lasers [9

9. 0. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’ Brien, P. D. Dapkus, and I. Kim, “Two- dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999). [CrossRef] [PubMed]

22

22. J. Topol’ancik, S. Pradhan, P-C Yu, S. Gosh, and P. Bhattacharya, “Electrically injected photonic crystal edge-emitting quantum-dot laser source,” IEEE Photon. Technol. Lett. 16, 960–962 (2004). [CrossRef]

], etc.

Fig. 1. A general scheme of a guided optical interconnect

Many laser configurations have already been analyzed in the literature, such as defect-mode [9

9. 0. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’ Brien, P. D. Dapkus, and I. Kim, “Two- dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999). [CrossRef] [PubMed]

13

13. K. Srinivasan, P. E. Barclay, O. Painter, J. Chen, A. Y. Cho, and C. Gmachl, “Experimental demonstration of a high quality factor photonic crystal microcavity,” Appl. Phys. Lett. 83, 1915–1917 (2003). [CrossRef]

], surface-emitting [14

14. D. S. Song, S. H. Kim, H. G. Park, C. K. Kim, and Y. H. Lee, “Single-fundamental-mode photonic-crystal vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 80, 3901–3903 (2002). [CrossRef]

17

17. D. Ohnishi, T. Okano, M. Imada, and S. Noda, “Room temperature continuous wave operation of a surface-emitting two-dimensional photonic crystal diode laser,” Opt. Express 12, 1562–1568 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1562 [CrossRef] [PubMed]

] and band-edge [18

18. C. Monat, C. Seassal, X. Letartre, P. Viktorovitch, P. Regreny, M. Gendry, P. Rojo-Romeo, G. Hollinger, E. Jalaguier, S. Pocas, and B. Aspar, “InP two-dimensional photonic crystal on silicon: In-plane Bloch mode laser,” Appl. Phys. Lett. 81, 5102–5104, (2002). [CrossRef]

21

21. L. Florescu, K. Busch, and S. John, “Semiclassical theory of lasing in photonic crystals,” J. Opt. Soc. Am. B 19, 2215–2223 (2002). [CrossRef]

] lasers. The introduction of defects in a PhC lattice allows the design of high quality factor (Q) cavities [13

13. K. Srinivasan, P. E. Barclay, O. Painter, J. Chen, A. Y. Cho, and C. Gmachl, “Experimental demonstration of a high quality factor photonic crystal microcavity,” Appl. Phys. Lett. 83, 1915–1917 (2003). [CrossRef]

], fact that may lead to the fabrication of low threshold laser devices. However, as mentioned by Ohnishi et al. [17

17. D. Ohnishi, T. Okano, M. Imada, and S. Noda, “Room temperature continuous wave operation of a surface-emitting two-dimensional photonic crystal diode laser,” Opt. Express 12, 1562–1568 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1562 [CrossRef] [PubMed]

], broad-area lasers such as band-edge lasers, have the advantage of providing larger output power, and better heat dissipation. Moreover, localized defects have a tendency to exhibit a radiation pattern with a weak directivity, what can make the coupling of the emitted light to waveguides more difficult.

Band-edge lasers, that operate at the edges of the first Brillouin zone (for example, the K or M point in a triangular lattice of air holes), can be considered as the photonic crystal equivalents of DFB lasers. In many cases, in these high symmetry points, a given mode exhibits extrema in its band diagram. At these extreme points, the group velocity can be made very small, meaning that the average lifetime of photons in the active area can be made large, significantly reducing the size of the device [18

18. C. Monat, C. Seassal, X. Letartre, P. Viktorovitch, P. Regreny, M. Gendry, P. Rojo-Romeo, G. Hollinger, E. Jalaguier, S. Pocas, and B. Aspar, “InP two-dimensional photonic crystal on silicon: In-plane Bloch mode laser,” Appl. Phys. Lett. 81, 5102–5104, (2002). [CrossRef]

]. Moreover, as was argued by Florescu et al. [21

21. L. Florescu, K. Busch, and S. John, “Semiclassical theory of lasing in photonic crystals,” J. Opt. Soc. Am. B 19, 2215–2223 (2002). [CrossRef]

], the fact that the group velocity of light is reduced near the band edges may provide an optical gain enhancement. If this high-symmetry point of operation is below the light line, we may expect that the chosen resonant mode in the PhC structure will be well confined in the vertical direction. Band-edge lasers in triangular [18

18. C. Monat, C. Seassal, X. Letartre, P. Viktorovitch, P. Regreny, M. Gendry, P. Rojo-Romeo, G. Hollinger, E. Jalaguier, S. Pocas, and B. Aspar, “InP two-dimensional photonic crystal on silicon: In-plane Bloch mode laser,” Appl. Phys. Lett. 81, 5102–5104, (2002). [CrossRef]

] and square [19

19. S. H. Kwon, H. Y. Ryu, G. H. Kim, and Y. H. Lee, “Photonic bandedge lasers in two-dimensional square lattice photonic crystal slab,” Appl. Phys. Lett. 83, 3870–3872 (2002). [CrossRef]

] lattices of air holes were analyzed recently. However, in those analysis, light was radiated in air. The goal of this paper is to analyze the coupling of light from the laser into a waveguide and, in the long-term, to use this sub-system in a complete optical link. At this stage we present simulation results and the fabrication scheme of this heterogeneous structure.

In this paper we restrict our attention to band-edge devices that consist of triangular lattices of air holes in a slab structure. The resonant mode in the PhC is assumed to be Transverse Electric (TE) (using the same convention as in [6

6. S. Fan, S. G. Johnson, J. D. Joannopoulos, C. Manolatou, and H. A. Haus, “Waveguide branches in photonic crystals,” J. Opt. Soc. Am. B 18, 162–165 (2001). [CrossRef]

]) and we consider the first mode in the valence band. The waveguide is single-mode both in the vertical and lateral directions (width of about 300 nm). No gain is introduced in the materials, since we are mainly concerned with the analysis of light coupling into waveguides, as shall be described in the text.

2. General analysis

The first structure to be analyzed is the one shown in Fig. 2(a) and 2(b). The PhC is a triangular lattice of air holes fabricated in the InP layer. Triangular lattice of air holes provide a larger bandgap than square lattice of air holes and that is the main reason for this choice. The InP layer has a thickness (h1) of 250nm, while the thickness of the silicon waveguide (h3) is chosen as 220 nm. With these two values of h1 and h3, the structures in the PhC and waveguide layers are single-mode in the vertical direction. Moreover, as h1 is small, the evanescent field outside the InP and Si layers are more intense and they overlap quite efficiently. This allows a stronger coupling of light generated in the PhC laser into the waveguide. Here, it is assumed that the silica layer between the silicon wafer and the silicon waveguide is thick enough so that light leakage into the wafer is inhibited. The thickness of the intermediate layer (h2) controls the coupling in the waveguide: if h2 is too high, the coupling into the waveguide is very weak, if it is low, the coupling may be strong, but the optical losses induced in the PhC structure will be too high for lasing operation. More details about the choices of h2, the Si waveguide width and the PhC parameters will be given in the text.

Fig. 2. (a). Z-cut view of the structure, showing a triangular lattice PhC and the waveguide situated in the bottom of the PhC. The waveguide is oriented in the ΓK direction (b) Vertical structure of the device, showing different layers. The silicon waveguide is surrounded (laterally) by a silica layer

The band diagram for the PhC structure, calculated by MIT’s photonic bands software [23

23. S. G. Johnson and J. Joannopoulos, “Bloch-iterative frequency domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173 [CrossRef] [PubMed]

], is shown in Fig. 3 in the case of TE polarization and for a 47.5% filling factor (FF, ratio of the area occupied by the air holes in a primitive cell divided by the area of the primitive cell). The normalized frequency is expressed in units of Λ/λ (Λ is the lattice parameter of the PhC and λ the free-space wavelength). In order to avoid diffractive coupling of light into silica and air, band edges that stand under the light line should be used. Namely, our devices based on a triangular PhC structure are assumed to operate at the K or M point of the first Brillouin zone. The lightline with respect to the silica layer is introduced in this band diagram as a dotted curve.

Fig. 3. Band diagram of the triangular lattice PhC structure

In order to perform an efficient optical coupling between the PhC resonant mode and the waveguide, it is necessary to have a good phase matching between the waveguide’s propagation constant (β) and the wave vector along the considered direction of symmetry. The magnitude of the wave vectors along the ΓK and ΓM directions (at the first Brillouin zone, |K⃗ ΓK| and |K⃗ ΓM|) can be expressed as,

KΓK=4π3Λ
(1)
KΓM=2πΛ3
(2)

Having these parameters, we have to choose the operation point either at the K or M point, and the width (W) of the waveguide. If the K point is used, a 470 nm lattice constant should be chosen in order to operate close to λ=1550 nm. For this lattice constant, the magnitude of the wave vectors along the ΓK directions is |K⃗ ΓK|=8.91×106| rad/m. A simple effective index estimate shows that the propagation constant of the silicon waveguide is about 8.5×106 rad/m, for W=300 nm. In that case, the phase matching to the PhC resonant mode along the ΓK direction is reasonable. If the M point is used, since |K⃗ ΓM| is lower, phase matching can be achieved along the ΓM direction, but provided the waveguide width is reduced. This could be at the expense of additional propagation losses in the silicon waveguide. We finally chose to exploit the K-point of the first Brillouin zone (indicated with an arrow in Fig. 3).

1Qt=1Qi+1Qc
(3)

ηc , the estimated amount of power coupled into the waveguide with respect to the incident power, is then given by:

ηc=1Qc1Qi+1Qc
(4)

Therefore, we may consider this value of ηc as an upper bound of the actual coupling efficiency. Figures 4(b) and 4(c) respectively show the variation of Qt and Qc as function of h 2. One could note that the resonant wavelength changes only slightly with h 2. Figure 4(d) shows ηc as a function of h 2. On the basis of former experimental results [18

18. C. Monat, C. Seassal, X. Letartre, P. Viktorovitch, P. Regreny, M. Gendry, P. Rojo-Romeo, G. Hollinger, E. Jalaguier, S. Pocas, and B. Aspar, “InP two-dimensional photonic crystal on silicon: In-plane Bloch mode laser,” Appl. Phys. Lett. 81, 5102–5104, (2002). [CrossRef]

, 24

24. C. Monat, C. Seassal, X. Letartre, P. Regreny, P. Rojo-Romeo, P. Viktorovitch, M. Le Vassor d’Yerville, D. Cassagne, J.P. Albert, E. Jalaguier, S. Pocas, and B. Aspar, “Modal analysis and engineering of InP based two-dimensional photonic crystal microlasers on a silicon wafer,” IEEE J. Quantum Electron. 39, 419–425 (2003) [CrossRef]

], with our setup and with our investigated materials, we expect the quality factor should be over 600 to observe laser emission. This yields h 2>700 nm and a maximum coupling efficiency of around 30%. Q factors well over 600 could be obtained by increasing h 2, in order to reduce the threshold of a laser based on such a PhC structure, but at the expense of a reduced coupling coefficient.

Fig. 4. (a). 3D FDTD spectrum of |Hz2| as a function of the wavelength for the basic structure with no waveguide (b) Total quality factor (measured in the waveguide) and (c) quality factor related to coupling losses as a function of h2. (d) Coupling efficiency as a function of h2. Results for the basic structure.

The field distributions for Hz in the structure are shown in Fig. 5(a) to 5(c), in the case h 2=600 nm. The PhC area is 6.58 µm (x-direction) by 5.66 µm (y-direction) and is centered in the computation area. Figure 5(a) shows a z-cut of Hz in the PhC region, at the main resonant peak, clearly showing the mapping of the fundamental mode. Part of the light escapes through the sides of the PhC structure (in the InP layer). Figure 5(b) shows that light is coupled into both directions of the waveguide underneath and that the mode is symmetric with respect to the x- axis. Figure 5(c) shows a y-cut of the structure and we can observe that light is significantly coupled into the waveguide.

Fig. 5. (a). Field distribution (Hz component) in the PhC region (x-y plane). (b) Hz field distribution in the waveguide (x-y plane). (c) y-cut of the device, showing the field distribution of Hz. All these Fig. are for h2=600 nm

From the results presented in Fig. 5, it appears that radiation losses in the vertical direction are low. However, a significant part of the light is lost (i) by coupling to the InP slab, beside the PhC area and (ii) because light is coupled to the waveguide into both directions, which is generally not relevant to the application. Then, one could imagine a more efficient design that select mainly one channel of optical losses. Following this view, we propose a more “directive” structure, schematized in Fig. 6(a). We have inserted a 1D grating in one side of the silicon waveguide, so that the coupled light can only travel in one direction. The grating has a lattice constant Λ1=490 nm (constituted of alternating 270 nm long silica regions and 220 nm long silicon regions) and a width of 300 nm. Moreover, the main PhC, with a filling factor (FF) of 47.5 %, is surrounded, on three of its sides, by another PhC with FF=30% and same lattice constant, which acts as a shielding zone. This shielding PhC operates at its bandgap and the “shielding effect” improves with the number of rows.

The 3D FDTD spectrum for the situation where there is no bottom waveguide is shown in Fig. 6(b). Its main peak appears at λ=1531.24 nm with a quality factor of 2000 (indicated with an arrow in Fig. 6(b)). Based upon the increase of the Q, it looks that the shielding has effectively reduced the lateral losses along three sides of the main PhC. On the other hand, more peaks appear in this spectrum, but the main peak is the one which exhibits the highest Q and also corresponds to the fundamental mode of the structure. The variation of Qt , Qc and ηc as a function of h 2 are shown respectively in Fig. 7(a), 7(b) and 7(c). In this second design, the shielding increases the PhC resonant mode localization, and therefore, its k-vector distribution is larger. The consequence is that phase matching between the PhC resonator and the waveguide is worse ; finally, Qc is higher than in the case of the first design. On the other hand, the shielding decreases the optical losses of the PhC resonator (Qi ). Finally, both Qc and Qt are increased, and ηc remains unchanged. In particular, to achieve laser emission in the PhC structure, we should have h 2>300 nm. This leads to a coupling efficiency over 60%.

Fig. 6. (a). Z-cut view of the structure, showing the main PhC shielded by another triangular lattice PhC with lower FF (operating at its bandgap). The waveguide situated at the bottom of the structure has a 1D grating in one of his sides, also operating at the bandgap. (b) 3D FDTD spectrum of |Hz2| as a function of the wavelength for this structure with no waveguide.
Fig. 7. (a). Total quality factor (measured in the waveguide) as a function of h2. (b) Quality factor corresponding to coupling losses into the waveguide. (c) Coupling efficiency as a function of h2. Results for the shielded structure.

Figure 8(a) shows the field distribution of |Hz |, at the main resonant peak, in the PhC region. It can be observed that the field is well confined in the PhC region and can only escape in the +x direction. Figure 8(b) shows the field distribution of |Hz | in the waveguide. It can be clearly observed that the resonant mode is coupled to an even mode in the waveguide and that the coupling is quite directional. One could note that evanescent coupling into the waveguide is not highly localized ; therefore, the properties of the structure are not very sensitive to the position of the 1D grating. Figure 8(c) shows the field amplitudes of |Hz | into the PhC and the waveguide, for h 2=500 nm, showing that the coupling into the waveguide is efficient.

Fig. 8. (a). |Hz| field distribution in the PhC region. (b) |Hz | field distribution in the waveguide. (c) |Hz| vertical field distribution in the x-z plane. All these Fig. are for h2=500 nm.

3. Fabrication of the device

A specific process was developed to fabricate these heterogeneous III–V/silicon structures. The process flow is given in Fig. 9. The optical waveguide circuit is fabricated in a 200mm SOI (Silicon On Insulator) wafer that includes a 320 nm thick silicon layer on top of a 1µm thick buried oxide layer. Using this wafer, 0.3 to 1µm wide silicon wire waveguides are fabricated. For this study, electron beam lithography, hard mask, and reactive ion etching with HBr/Cl were used. Then, the passive photonic circuit is embedded in silica and planarised by CMP so as to obtain the proper thickness of silica above the guides. The active heterostructure has been grown by molecular beam epitaxy (MBE) on a 50mm InP wafer. This epilayer may contain an InAsP/InP quantum wells, designed to emit photons around 1.5µm. The total epilayer thickness is λ/2n, i.e. 240 nm in our case. A 10nm thin silica layer is then deposited by plasma-enhanced chemical vapor deposition assisted by electron cyclotron resonance (PECVD-ECR) on top of the active layer and the III–V wafer is then molecular bonded at the centre of the silicon wafer containing the photonic circuits. The final silica thickness between the passive silicon waveguides and the active layer (h2) is the key parameter that controls the coupling coefficient between the laser mode and the guided mode.

After bonding, the InP wafer is chemically etched from the backside in HCl solution until the InGaAs etch-stop layer is reached, then this layer is selectively removed in a FeCl3 solution. Electron beam lithography is used to pattern the PhC holes, with alignment accuracy better than ± 200nm. Finally, the PhC holes are transferred into the III–V heterostructure by reactive ion etching, using a CH4:H2 plasma. Figure 10 is a top view of a typical device including a PhC cavity and a Si wire waveguide underneath as a preliminary demonstration of the technological feasibility of these relatively complex and heterogeneous devices. This configuration might not be relevant with respect to optical coupling efficiency, but this example shows that the alignment between the Si waveguide and the PhC can be obtained with a good accuracy.

Fig. 9. Schematic description of the process steps used to fabricate PhC III–V microlaser on top of Si waveguides.
Fig. 10. SEM view of a III–V PhC micro-resonator (an hexagonal cavity) processed on top of a Si waveguide

4. Conclusions

We have analyzed III–V based photonic crystal band-edge lasers coupled to silicon waveguides positioned underneath. The initial basic structure could couple up to 30% of the incident power without sensibly degrading the lasing effect in the PhC region. By shielding three sides of the PhC, much higher coupling efficiency and directivity could be achieved, without degrading the properties of the PhC laser. For h 2=300nm, the coupling efficiency is over 60%. Achievable fabrication tolerances on the thickness of the intermediate silica thickness of ±100 nm, result in a coupling efficiency kept between 35% and 50 %, while the quality factor remains compatible with laser emission; this design is therefore robust to fabrication tolerances.

Acknowledgments

This work was supported by the European Union in the context of the European project FP6-2002-IST-1-002131-PICMOS.

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13.

K. Srinivasan, P. E. Barclay, O. Painter, J. Chen, A. Y. Cho, and C. Gmachl, “Experimental demonstration of a high quality factor photonic crystal microcavity,” Appl. Phys. Lett. 83, 1915–1917 (2003). [CrossRef]

14.

D. S. Song, S. H. Kim, H. G. Park, C. K. Kim, and Y. H. Lee, “Single-fundamental-mode photonic-crystal vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 80, 3901–3903 (2002). [CrossRef]

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18.

C. Monat, C. Seassal, X. Letartre, P. Viktorovitch, P. Regreny, M. Gendry, P. Rojo-Romeo, G. Hollinger, E. Jalaguier, S. Pocas, and B. Aspar, “InP two-dimensional photonic crystal on silicon: In-plane Bloch mode laser,” Appl. Phys. Lett. 81, 5102–5104, (2002). [CrossRef]

19.

S. H. Kwon, H. Y. Ryu, G. H. Kim, and Y. H. Lee, “Photonic bandedge lasers in two-dimensional square lattice photonic crystal slab,” Appl. Phys. Lett. 83, 3870–3872 (2002). [CrossRef]

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L. Florescu, K. Busch, and S. John, “Semiclassical theory of lasing in photonic crystals,” J. Opt. Soc. Am. B 19, 2215–2223 (2002). [CrossRef]

22.

J. Topol’ancik, S. Pradhan, P-C Yu, S. Gosh, and P. Bhattacharya, “Electrically injected photonic crystal edge-emitting quantum-dot laser source,” IEEE Photon. Technol. Lett. 16, 960–962 (2004). [CrossRef]

23.

S. G. Johnson and J. Joannopoulos, “Bloch-iterative frequency domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173 [CrossRef] [PubMed]

24.

C. Monat, C. Seassal, X. Letartre, P. Regreny, P. Rojo-Romeo, P. Viktorovitch, M. Le Vassor d’Yerville, D. Cassagne, J.P. Albert, E. Jalaguier, S. Pocas, and B. Aspar, “Modal analysis and engineering of InP based two-dimensional photonic crystal microlasers on a silicon wafer,” IEEE J. Quantum Electron. 39, 419–425 (2003) [CrossRef]

OCIS Codes
(130.0130) Integrated optics : Integrated optics
(140.2020) Lasers and laser optics : Diode lasers
(220.4610) Optical design and fabrication : Optical fabrication

ToC Category:
Research Papers

History
Original Manuscript: March 18, 2005
Revised Manuscript: April 15, 2005
Published: May 2, 2005

Citation
Haroldo Hattori, Christian Seassal, Xavier Letartre, Pedro Rojo-Romeo, Jean Leclercq, Pierre Viktorovitch, Marc Zussy, Lea di Cioccio, Loubna El Melhaoui, and Jean-Marc Fedeli, "Coupling analysis of heterogeneous integrated InP based photonic crystal triangular lattice band-edge lasers and silicon waveguides," Opt. Express 13, 3310-3322 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-9-3310


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