Internship Report – 2nd year

Design of photonic crystals applied to two different structures of laser

Laser, Photonic Crystal Simulations

– Confidential – Student : Besnier Alexandre

Mentor : Mink Jan De Run 4403 5503 LS Veldhoven The Nertherlands

June–September 2009 – Version 1

Aknowlegement I would like to thank Jan Mink and Gillian Mimnagh for the opportunity of this internship and for their help to integrate the company. I also thank Jan for his help along all my internship about my work, and Gillian who introduce us Eindhoven. I thank Oleg Guziy from university of TU Deflt who help me to validate some models. My regards go also to Raimond Dumoulin, Tom Vrancken and all other people working in 2M. I thank Niall Drievers, Brian Magner, Seán Nyhan and Martin Kennelly for their welcome and all unforgettable moments spent with them. Lastly I thank Lucas Bétend, Érik Lucas and Niall Drievers for their friendship and their good mood every days.

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Résumé Mon stage s’est déroulé au sein de 2M Engineering Ltd (Pays-Bas), spécialisée dans les capteurs, la métrologie, les lasers et le conditionnement. J’ai travaillé sur deux projets concernant les lasers à cristaux photoniques : Procryla et Dynamic Blue, où j’ai eu à réduire le spectre d’émission de deux différents types de lasers. Je présente dans ce rapport une introduction aux cristaux photoniques, en développant leur physique, l’interprétation du diagramme de bande, les différentes structures et les outils de modélisation. Je développe aussi la structure utilisée pour le projet Procryla, et étudie l’influence de chaque paramètre (rayon, périodes, permittivités), ainsi que les conséquences des défauts de fabrication en introduisant des défauts de forme elliptique. Finalement, je propose une solution où j’ai adapté l’ensemble des paramètres à l’application désirée. Enfin j’aborde, avec le projet Dynamic Blue, les lasers à émission verticale. J’en développe un modèle, le valide en utilisant la littérature, et explore l’influence des défauts elliptiques.

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Abstract My internship has took place in 2M Engineering Ltd (Netherlands), specialized in sensors, measurement, lasers and packaging. I have worked on two projects concerning photonic crystal lasers : Procryla and Dynamic Blue, where I have had to reduce the emission spectrum of two different kinds of lasers. I present in this report an introduction on photonic crystals, developing their physics, the band diagram interpretation, the different structures existing and modeling tools. I develop also the structure used for Procryla project, and study influence of each parameter (radius, periods, permittivities), and consequence of manufacturing fault introducing defects with ellipse shape. Finally I propose a solution where I have accorded all parameters to the wanted application. Lastly I tackle, with Dynamic Blue project, the vertical emitting lasers. I develop a model for them, validate it comparing with literature, and explore elliptic defects influence.

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Contents Aknowlegement

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1 Intership presentation 1.1 2M Engineering Ltd . . . . . . 1.2 Phocryla project . . . . . . . 1.2.1 Presentation . . . . . . . 1.2.2 Organization . . . . . . 1.3 Dynamic Blue project . . . . . 1.3.1 Presentation . . . . . . . 1.3.2 Organization . . . . . . 1.4 The objectives of my internship

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2 Photonic crystals 2.1 Presentation . . . . . . . . . . . . . . . . . . . 2.1.1 Periodic structure . . . . . . . . . . . . 2.1.2 Natural examples . . . . . . . . . . . . 2.1.3 Photonic band gap . . . . . . . . . . . 2.2 PhC physics . . . . . . . . . . . . . . . . . . . 2.2.1 Eigenvalue problem . . . . . . . . . . . 2.2.1.1 Floquet-Bloch solutions . 2.2.1.2 Plan wave method . . . . . . 2.2.2 Finite Difference Time Domain (Fdtd) 2.3 Dispersion diagram . . . . . . . . . . . . . . . 2.3.1 1D PhC diagram . . . . . . . . . . . . 2.3.2 2D PhC diagram . . . . . . . . . . . . 2.3.3 Slab structure . . . . . . . . . . . . . . 2.3.4 Diffracted wave-vectors . . . . . . . . .

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3 Phocryla project 21 3.1 Forchel’s lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 Mpb model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2009

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3.2.1 Structure . . . . . . . . . . . . 3.2.2 Validation . . . . . . . . . . . . Two-dimensional PhC simulations . . . 3.3.1 Radius variation . . . . . . . . 3.3.2 Periods influence . . . . . . . . 3.3.3 Permittivity role . . . . . . . . Influence of PhC manufacturing faults 3.4.1 Defects with ellipse shape . . . 3.4.2 Angle variation . . . . . . . . . Optimization of the structure . . . . .

4 Dynamic Blue project 4.1 Description . . . . . . . . . . 4.1.1 Laser’s structure . . . 4.1.2 Model : Slab PhC . . . 4.2 Three-dimensional simulations 4.2.1 Validation . . . . . . . 4.2.2 More results . . . . . .

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A Implementation of ellipses in Mpb B More results presentation B.1 Periods influence . . . . . . . . B.2 Permittivity role . . . . . . . . B.2.1 Basic structure : εh > εl B.2.2 Other structure : εh < εl B.3 Defects with ellipse shape . . . B.3.1 TM mode . . . . . . . . B.3.2 TE mode . . . . . . . .

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List of Figures 1.1 1.2 1.3 1.4

Principle of Phocryla project. . . . . . . . . . . . . . . . . . . . . Shift in ray spectrum for Raman scattering. The x-axis is center on laser wavelength. . . . . . . . . . . . . . . . . . . . . . . . . . . Skin cancer detection propose by Dynamic Blue poject. . . . . . . . Spectrum emission of different kind of lasers. (a) Laser without PhC, (b) PhC laser, (c) monofrequency laser. . . . . . . . . . . . .

Example of the three kind of photonic crystals. Each color represent different dielectric constants.[10] . . . . . . . . . . . . . . . . . . . . 2.2 Opals : natural example of photonic crystal. Multicolour effect even in the absence of pigment. . . . . . . . . . . . . . . . . . . . . . . . 2.3 Example of band diagram for a two-dimensional crystal. Some frequencies are forbidden into the PhC. . . . . . . . . . . . . . . . . . 2.4 Position of E and H components on the cubic grid. [17] . . . . . . . 2.5 1D photonic crystal. On left, structure with ε = 1 and d = 0.8a for the blue layer, and ε = 11 and d = 0.2a for the green layer [10]. On right, the dispersion diagram. . . . . . . . . . . . . . . . . . . . . . 2.6 Kinds of structures with a background’s epsilon higher than defect’s epsilon. Rectangular (a), cylinder-shaped (b), elliptic (c) defect. Rectangular (d) and hexagonal (e) grid. . . . . . . . . . . . . . . . . 2.7 Wavevector evolution used for 2D PhC band diagram. . . . . . . . . 2.8 Band diagram example for a 2D PhC. . . . . . . . . . . . . . . . . . 2.9 Gap map for different structures. (a) Square lattice with dielectric cylinder-shaped defects, (b) square lattice with air cylinder-shaped defects, (c) hexagonal lattice with dielectric cylinder-shaped defects, (d) hexagonal lattice with air cylinder-shaped defects [10]. . . . . . 2.10 Example of slab structure. . . . . . . . . . . . . . . . . . . . . . . . 2.11 Example of slab band diagram. Guided modes are below black curve. 2.12 (a) Schematic of the 2D-PhC. Arrows in the right side indicate particular direction Γ-X and Γ-J. (b) Reciprocal lattice of this PhC. K1 and K2 indicate the grating vectors of this lattice. (c) The PhC band diagram. I-IV points indicate particular points developed in the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.13 Wave-vectors diagram at point I (a) and point II (b). ki and Kd are the incident and diffracted wave-vectors, and K1 and K2 are lattice vectors, respectively. [9] . . . . . . . . . . . . . . . . . . . . . . . . 2.14 Wave-vectors diagram at point III (a) and point IV (b). [9] . . . . . 2.15 Out-plane wavevectors for the point III (a) and the point IV (b-c). For (a) and (b), wave-vectors are orthogonal to the substrate, but for (c) there is an angle compared with the norm. [9] . . . . . . . . 3.1 3.2

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(a) Laser structure used for Phocryla project. (b) PhC top view. [8] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PhC structure used in Forchel lasers. Λh and Λv represent horizontal and vertical periods, εl and εh the low an high permittivity, ans r the hole’s radius. The rectangle plot at the center is the unit cell use by Mpb for calculation. . . . . . . . . . . . . . . . . . . . . Map of the band gap between the first and the second band for a square lattice when radius change. (a) Theoretical result (TM mode) [10]. (b) My model result (TM mode). . . . . . . . . . . . . Epsilon map for each radius. The radius range is from 10nm to 110nm with a step of 20nm. . . . . . . . . . . . . . . . . . . . . . . Band diagram when radius change. (left) TE mode, (right) TM mode. Gap evolution when radius change. (left) gap value, (right) gap center. Gap evolution when periods change for TM mode and in M direction (surface). Others curves (1.4-0.2) show contours with a step of 0.2 c/a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gap center evolution when periods change for TM mode and in M direction (surface). Others curves (3-0.5) show contours with a step of 0.5 c/a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Band diagram when vertical period change (lv). Horizontal period is fixed and equal to 0.1µm. (left) TE mode, (right) TM mode. . . Band diagram when vertical period change (lv). Horizontal period is fixed and equal to 0.7µm. (left) TE mode, (right) TM mode. . . Gap evolution when permittivities changed. This graph is for M direction and TM mode. Others curves show contours with a step of 2.10−4 c/a. For εl ≥ εh , values are equal to 0, but it’s an artefact for a better visualization. . . . . . . . . . . . . . . . . . . . . . . . . Gap center evolution when permittivities changed. This graph is for M direction and TM mode. Others curves show contours with a step of 0.5c/a. For εl ≥ εh , values are equal to 0, but it’s an artefact for a better visualization. . . . . . . . . . . . . . . . . . . . . . . . . Gap evolution when permittivities changed. This time there isn’t any artefact for visualization, This graph is for M direction and TM mode. Others curves show contours with a step of 0.1c/a. . . . . . Internship report - Version 1

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3.14 Difference between the first band of the simulation with ellipses and the first band of simulation with cylinders where the diameter is equal to ellipses short-axis. It’s for TM mode in M direction. Others curves show contours with a step of 0.2c/a. . . . . . . . . . . . . . . 3.15 Epsilon map when ellipse angle change (0, 15, 30 and 45 degrees). . 3.16 Band diagram when ellipse angle change. (left) TE mode, (right) TM mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.17 Gap evolution when ellipse angle change. (left) gap value, (right) gap center. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.18 Band diagram for optimized parameters. (left) TE mode, (right) TM mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 4.2 4.3

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(a) Laser structure used for Dynamic Blue project. (b) PhC top view. (c) PhC cross section view. [12] . . . . . . . . . . . . . . . . . Example of slab structure used for simulations. . . . . . . . . . . . xy-plane structure used for Mpb. l is the period, l and h are the low and high permittivity, and r the hole’s radius. The rectangle plot at the center is the unit cell used by Mpb for calculation. . . . Band diagram in Γ direction with a zoom on point III. Points B and D are degenerated. . . . . . . . . . . . . . . . . . . . . . . . . . . . Point III modes presented in Imada’s et al. article [9]. Yellow circles indicate lattice points, blue and red areas the orthogonal magnetic field. (a) Point A, hexapole; (b-c) point B, quadrupoles; (d) point C, monopole and (e-f) point D, dipoles. . . . . . . . . . . . . . . . . Point III modes obtained with my model. White circles indicate lattice points, blue and red areas the orthogonal magnetic field. (a) Point A, hexapole; (b-c) point B, quadrupoles; (d) point C, monopole and (e-f) point D, dipoles. . . . . . . . . . . . . . . . . . Band diagram in Γ direction with a zoom on point III when ellipses are used in place of cylinders. Points B and D are now not degenerated. The first mode (point A) is now linear. . . . . . . . . . . . . . . . .

A.1 Block distortion with a rotation. . . . . . . . . . . . . . . . . . . . . A.2 Diagram of the problem for the first vector e~1 . . . . . . . . . . . . . A.3 No distortion with the new equation system. . . . . . . . . . . . . . B.1 Gap evolution when periods change for TE mode and in M direction (surface). Others curves (0.6-0.2) show contours with a step of 0.2 c/a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Gap center evolution when periods change for TE mode and in M direction (surface). Others curves (4-1) show contours with a step of 1 c/a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2009

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B.3 Gap evolution when permittivities changed. This graph is for M direction and TE mode. Others curves show contours with a step of 2.10−3 c/a. For εl ≥ εh , values are equal to 0, but it’s an artefact for a better visualization. . . . . . . . . . . . . . . . . . . . . . . . . B.4 Gap center evolution when permittivities changed. This graph is for M direction and TE mode. Others curves show contours with a step of 0.5c/a. For εl ≥ εh , values are equal to 0, but it’s an artefact for a better visualization. . . . . . . . . . . . . . . . . . . . . . . . . . . B.5 Gap center evolution when permittivities changed. This graph is for M direction and TM mode. Others curves show contours with a step of 0.5c/a. For εl ≥ εh , values are equal to 0, but it’s an artefact for a better visualization. . . . . . . . . . . . . . . . . . . . . . . . . B.6 Gap evolution when permittivities changed. This time there isn’t any artefact for visualization, This graph is for M direction and TE mode. Others curves show contours with a step of 0.05c/a. . . . . . B.7 Gap center evolution when permittivities changed. This graph is for M direction and TE mode. Others curves show contours with a step of 0.5c/a. For εh ≥ εl , values are equal to 0, but it’s an artefact for a better visualization. . . . . . . . . . . . . . . . . . . . . . . . . . . B.8 Gap evolution when ellipse axis change for TM mode and in M direction. Others curves show contours with a step of 0.05c/a. . . . B.9 Gap center evolution when ellipse axis change for TM mode and in M direction. Others curves show contours with a step of 0.05c/a. . B.10 Difference between the first band of the simulation with ellipses and the first band of simulation with cylinders where the diameter is equal to ellipses short-axis. It’s for TE mode in M direction. Others curves show contours with a step of 0.2c/a. . . . . . . . . . . . . . . B.11 Gap evolution when ellipse axis change for TE mode and in M direction. Others curves show contours with a step of 0.05c/a. . . . . B.12 Gap center evolution when ellipse axis change for TE mode and in M direction. Others curves show contours with a step of 0.05c/a. .

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Bibliography [1] Mpb software website : http://ab-initio.mit.edu/wiki/index.php/MIT_ Photonic_Bands. [2] 2m engineering website : www.2mel.nl. [3] Alexandre Besnier. Developement of tools for the simulation of photonic crystals. Master’s thesis, Ensps, 2009. [4] W.W. Bewley, C.L. Felix, I. Vurgafman, R.E. Bartolo, J.R. Lindle, J.R. Meyer, H. Lee, and R.U. Martinelli. Mid-Infrared photonic-crystal distributedfeedback lasers. Solid-State Electronics, 46:1557–1566, 2002. [5] Gérald Courpasson. Optical coupling of photonic crystal laser. Master’s thesis, National School of Physic of Strasbourg (ENSPS), 2008. [6] Frédéric Fantoni. Design and test setup for photonic crystal laser. Master’s thesis, National School of Physic of Strasbourg (ENSPS), 2008. [7] D.J. Garnier, P.R. Graves, H.J. Bowley, et al. Practical Raman Spectroscopy. Springer-Verlag, 1989. [8] H. Hofmann, H. Scherer, S. Deubert, M. Kamp, and A. Forchel. Spectral and spatial single mode emission from a photonic crystal distributed feedback laser. Applied Physics Letters, 90, March 2007. [9] Masahiro Imada, Alongkarn Chutinan, Susumu Noda, and Masamitsu Mochizuki. Multidirectionnaly distributed feedback photonic crystal lasers. Pysical Review B, 65(195306), 2002. [10] John Joannopoulos, Steven Johnson, Joshua Winn, and Robert Meade. Photonic crystals, Modeling the flow of light. Princeton University Press, second edition, 2008. [11] Jean-Michel Lourtioz. Cristaux photoniques et gaps de photons, aspects fondamentaux. Technique de l’ingénieur, July 2004. [12] Hideki Matsubara, Susumu Noda, Susumu Yoshimoto, Hirohisa Saito, Yue Jianglin, and Yoshinori Tanaka. Gan photonic crystal surface emitting laser at blue-violet wavelenghts. Science, 319(445), 2008. 2009

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Bibliography

[13] Susumu Noda, Mitsuru Yokoyama, Masahiro Imada, Alongkarn Chutinan, and Masamitsu Mochizuki. Polarization Mode Control of Two-dimensional Photonic Crystal Laser by Unit Cell Structure Design. Science, 293:1123– 1125, 2001. [14] Cédric Perrotton. Modeling of photonic crystal for laser structures. Master’s thesis, National School of Physic of Strasbourg (ENSPS), 2008. [15] Daniel Sjoberg, Christian Engstrom, Gherhard Kristensson, David Wall, and Niklas Wellander. A floquet-block decomposition of maxwell’s equation, applied to homogenization. Lund Institute of Technologie, Sweden, 2003. [16] I. Vurgaftmann and J.R. Meyer. Photonic crystal distributed feedback lasers. Applied Physics Letters, 78(11), January 2001. [17] Kane Yee. Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media. Ieee Transactions on Antennas and Propagation, 14(3):302–307, May 1966.

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