Photonic crstals: from Bloch modes to T-matrices B. Gralak, S. Enoch, G. Taeb Institut Fresnel, Marseille, France Objectives : Links between Bloch modes and grating theories (transfer matri) Stud of anomalous refraction at the band edges and applications i k r Bloch wave: ψk ( r) = e uk ( r ) Dispersion relation of the Bloch modes But: the crstal is infinite and fills the whole space no incident field onl real Bloch vectors are considered Studied structure: T 2D photonic crstal made of dielectric rods of radius.475, with optical inde n = 3 arranged on a square lattice with period d = 1.27 The eigenvalues of the transfer matri T give us more information than the Bloch modes.
T-matri and Bloch modes d Bloch mode: solution with onl phase shifts for the elementar translations d and. A crstal made of 3 gratings. Infinite along the -direction. Finite etent with respect to the -direction. Plent of place for an incident field! = = - d T T-matri point of view: d translation gives phase shift (pseudo-periodicit). translation gives phase shift for eigenvectors of the T operator if the eigenvalue has a modulus equal to 1. Harmonic problem ω given. Pseudo-periodic component of the field k given: u( + d, ) = ep(i k d) u(, ). Eigenvector of the T-matri, with eigenvalue µ : T u = µ u T u = ep(i arg( µ )) u, if µ = 1 Each eigenvalue µ of T with µ = 1 is associated with a propagating Bloch mode. The -component of the Bloch vector is: k arg( µ ) k = ω given grating problem k k given ()* Dispersion curve of Bloch modes
Dispersion relation of the Bloch modes in a 2D photonic crstal In the infinite structure the field is can be represented as a sum of Bloch i k modes: ψ ( r) = e u ( r) k r k.8.6 Dispersion diagram for E// polarization. The first reduced Brillouin zone is represented below. ω d / (2 πc).4 k M (π/d,π/d).2. M Γ X M Γ X k.6.55.5 ω d / (2πc).45.4.35.3.25.2.15.1 3D view of the dispersion diagram of the Bloch modes..5...5 1. k 1.5 2...5 2. 1.5 1. k
ω d / (2πc).54.53.52.51.5.49.48.47.46..5 1. k 1.5 2. For a fied wavelength λ the dispersion relation is the intersection between a horizontal plane and the sheets...5 1. 1.5 k 2. Enlargement of the previous figure at the upper limit of the second band gap. k 2. 1. -2. -1. 1. 2. Dispersion curves -1. k The energ flow is given b ω k -2. e ω + k and points toward the ascending side of the sheet. e : it is perpendicular to the curve
Ultra-refraction k Circle: vacuum dispersion curve Crstal dispersion curves k 5 4 3 2 1-1 -2-3 -4-5 -5-4 -3-2 -1 1 2 3 4 5 1..9.85.8.75.7.65.6.55.5.45.4.35.3.25.2.15.1.5 Bounded 2D photonic crstal Modulus of the electric field. λ = 2.54, θ = 6.4
Negative refraction k Crstal dispersion curves k 5 4 3 2 1-1 -2-3 -4-5 -5-4 -3-2 -1 1 2 3 4 5 Modulus of the electric field. λ = 2.54, θ = 4
Application: ultrarefractive optical components 1 - lens.4 k.3.2.1 λ 1 = 2.56 k neff λ 2 = 2.55 With λ = 2.56, the curve is almost a circle and simulates a homogeneous material with optical inde =. 86 n eff...1.2.3.4 1 k -1-2 -3-4 -5-6 -7 53 R: radius of curvature of the concave side. Focal length: R f = 1 n eff 5 = = 55 1.86-8 -9-5 -4-3 -2-1 1 2 3 4 5 "Ultrarefractive" microlens: a convergent lens with focal length = 21 λ. The width of the lens is about 25λ. Ver high inde contrast ( 1/.86 12)
Application: ultrarefractive optical components 2 - microprism 5 4 3 θ 2 1-1 Width of the prism = 14 λ -2-3 -4-3 -2-1 1 2 3 4 6 λ = 2.48 5 SER 4 3 2 1 λ = 2.5 λ = 2.52 Band edge rapid variations with the wavelength. The photonic crstal prism is more dispersive than usual prisms or gratings. 34 36 38 4 42 44 Angle de diffraction θ (degrés) Ver dispersive material.
3 - Directive sources 5 4 3 2 1 The heagonal lattice has been epanded along the direction ( 1.126) in the upper region of the crstal. 161 dielectric rods with radius r =.6, optical inde n = 2.9. A wire source is located at =, = 34. -3-2 -1 1 2 3.17 1/λ.16.15.14.13..2.4 k.6.8 1...2.4 k.6.8 Dispersion relation of the Bloch modes for the heagonal (left) and the epanded lattice (right). The bottom plane of the figure is for λ = 7.93..16 1/λ.14..2.4 k.6.8..2.4.6 k.8
Allowed propagation directions outside the crstal.8 k.6.4 circle: dispersion curve of the Crstal dispersion vacuum curve. Gives the allowed propagation constants in the crstal.2...2.4.6.8 k 9 4 12 6 3 2 15 3 1 18 Radiation pattern of the photonic crstal source λ = 7.93.
7 6 5 4 3 2 1 2. 1.8 1.6 1.4 1.2 1..8.6.4.2-1 -4-3 -2-1 1 2 3 4 Modulus of the electric field. Wire source is located at =, =34. λ = 7.93. The computation time (161 rods) is about 5 seconds to compute the coefficients of the field and 2 seconds to compute the field map (88 points) on a desktop computer.
Conclusion Classical Bloch stud Grating, T-matri More realistic problem Infinite crstal filling the Slice of grating. Finite structure entire space Finite in one direction (microlens, microprism, No incident field Simple link with the incident field (diffraction case, emitting situation) Onl propagating modes Also evanescent modes. information on the transmission inside a bandgap Dispersion relations of Eigenvalues of the the Bloch modes T-matri k, k ω ω, k k Includes all the features of a classical Bloch stud, plus more More accurate, faster. Snthetic approach (visual support given b the 3D dispersion diagrams of the Bloch waves). Helpful in the understanding of the photonic crstal properties. Permits to obtain easil the parameters giving rise to: ultrarefraction (the photonic crstal simulates a homogeneous material with a ver low optical inde), negative refraction, and more generall to get the allowed directions for the mean energ flu in the photonic crstal. antenna, ) Incident field (plane wave, limited beam, source in the crstal) Also evanescent modes Rigorous numerical computations Confirm the epected behaviors Quantitative results Check the effects of the boundaries of the photonic crstal
References B. Gralak, S. Enoch and G. Taeb. "Anomalous refractive properties of photonic crstals." J. Opt. Soc. Am. A 17, 112-12 (2). S. Enoch, G. Taeb and D. Mastre, "Numerical evidence of ultrarefractive optics in photonic crstals", Optics Communications 161, 171-176 (1999) G. Taeb, D. Mastre. "Rigorous theoretical stud of finite size twodimensional photonic crstals doped b microcavities". J. Opt. Soc. Am. A 14, 3323-3332 (1997).