Expansion Method for Photonic Crystal Slabs

Transcription

1 Pavia University Phys. Dept. A. Volta The Guided-Mode Expansion Method for Photonic Crystal Slabs Lucio Claudio Andreani Dipartimento di Fisica Alessandro Volta, Università degli Studi di Pavia, via Bassi 6, 700 Pavia, Italy COST P Training School, Univ. of Nottingham, June 9-, 006 Collaboration: Acknowledgements Mario Agio (now at Physical Chemistry Laboratory, ETH-Zurich) Dario Gerace (now at Institute of Quantum Electronics, ETH Zurich) Support: MIUR (Italian Ministry for Education, University and Research) under Cofin program Silicon-based photonic crystals and FIRB project Miniaturized electronic and photonic systems INFM (National Institute for the Physics of Matte under PRA (Advanced Research Project) PHOTONIC

2 Outline. Required notions Basic properties of photonic band structure Plane-wave expansion Dielectric slab waveguide Waveguide-embedded photonic crystals. Guided-mode expansion method Photonic mode dispersion Intrinsic diffraction losses Convergence tests, comparison with other methods Examples of applications Photonic crystals are materials in which the dielectric constant is periodic in D, D or D

3 leading to the formation of photonic bands and gaps. A full photonic band gap in D may allow to achieve suppression of spontaneous emission (E. Yablonovitch, 987) localization of light by disorder (S. John, 987) Books: Joannopoulos JD, Meade RD and Winn JN, Photonic Crystals Molding the Flow of Light (Princeton University Press, Princeton, 995). Johnson SG and Joannopoulos JD, Photonic Crystals: the Road from Theory to Practice (Kluwer Academic Publishers, Boston, 00). Sakoda K: Optical Properties of Photonic Crystals, Springer Series in Optical Sciences, Vol. 80. (Springer, Berlin, 00). Busch K, Lölkes S, Wehrspohn RB, Föll H: Photonic Crystals Advances in Design, Fabrication and Characterization (Wiley-VCH, Weinheim, 004). Lourtioz JM, Benisty H, Berger V, Gérard JM, Maystre D, Tchelnokov A: Photonic Crystals: Towards Nanoscale Photonic Devices (Springer, Berlin, 005).

4 Maxwell equations in matter no sources, unit magnetic permebility First-order, time-dependent: B E = c t D H = c t D B = 0 = 0 D( = ε ( E( H ( = B( Second-order, harmonic time dependence ψ ( t) = ψe iωt : ω E( = ε ( E( c [ ε ( E( ] = 0 c H( = i E( ω ω H( H( ( = ε c H( = 0 c E( = i H( ω ε( Translational invariance If the photonic lattice is invariant under translations by the vectors R of a Bravais lattice, ε ( r + R) = ε (, then Bloch-Floquet theorem holds for any component of the fields: ψ ik r nk ( = e unk (, unk ( r + R) = unk ( The frequencies are grouped into photonic bands: ω = ω n (k) N.b. the system is assumed to be infinitely extended along the directions of periodicity the Bloch vector k is real. It is usually chosen to lie in the first Brillouin zone (band folding).

5 Eigenvalue problem & scale invariance ω H( = H( ε ( c ˆ ω ΘH( = λh, λ = c ˆ Θ = ε ( Hermitian eigenvalue problem (like for electron states in quantum mechanism) Hermitian operator r r' = sr, ε '( r') = ε ( ωn( k) ω n'( k') = s Scale invariance: photonic band structure is unique when expressed in dimensionless units ωa/(πc) versus ka (a=lattice constant) ε ( ε '( r'). Required notions Basic properties of photonic band structure Plane-wave expansion Dielectric slab waveguide Waveguide-embedded photonic crystals. Guided-mode expansion method Photonic mode dispersion Intrinsic diffraction losses Convergence tests, comparison with other methods Examples of applications

6 Plane-wave expansion in D Expansion of the magnetic field in plane waves with reciprocal lattice vectors G: ) H ( ) = + + +, λ (, λ) ε (, λ)e i( k G) r nk r G c n k G k G where ˆ ε( k + G, λ) ˆ ελ, λ =, are two orthogonal unit vectors perpendicular to k+g. The second-order equation for H becomes: ω G' λ' Hk + G, λ; k' + G', λ' cn( k + G', λ') = c ( k + G, λ) n c ˆ ε ˆ ε' ˆ ε ˆ ε' H k + G, λ; k+ G', λ' = k + G k + G' ε ( G, G') ˆ ε ˆ ε' ˆ ε ˆ ε' with the inverse dielectric matrix being defined as (v=unit cell volume) i( G' G) r ε ( G, G') = ε ( e dr, v Choice of plane waves The infinite sum over G is truncated to a finite number N of reciprocal lattice vectors, usually chosen by introducing a wavevector cutoff: G < Λ The dimension of the linear eigenvalue problem is (N)x(N). The matrix H is * real symmetric if ε( has a center of inversion (taken to be the origin) * complex hermitian if ε( has no center of inversion N.b. The cut-off condition restricts the numbers N to be used: e.g., for the fcc lattice, N=,9,5, If other values of N are chosen, unphysical splittings at k=0 may arise

7 Direct and inverse dielectric matrix i( G' G) r ε( G, G') = ε( e dr v i( G' G) r ε ( G, G') = ε ( e dr v () () In the limit N, it is equivalent to evaluate ε ( G, G') from () or as a numerical inversion of the direct dieletric matrix (): ε ( G, G') = [ ε( G, G')] Ho, Chan and Soukoulis, PRL 65, 5 (990) However, the two procedures have different convergence properties for finite N. The procedure by Ho, Chan and Soukoulis generally yields much faster convergence as a function of N. Fourier factorization rules Consider the Fourier transform A of a product of functions B and C: + AG = BG G' CG' G' = When the sum is truncated to a finite number N of harmonics, the above expression does not converge uniformly if both B and C are discontinuous at the same point while A=BC remains continuous. In this case the inverse rule yields uniform convergence and should be used: = + M AG G' = M CG' B G G' In the presence of a sharp discontinuity of ε(, expressions like εe should be treated according to the direct (inverse) rule when the continuous (discontinuous) component E (E ) of the field is involved. L.F. Li, JOSA A, 870 (996); P. Lalanne, PRB 58, 980 (998); S.G. Johnson and J.D. Joannopoulos, Opt. Expr. 8, 7 (00); A. David et al., PRB 7, (006)

8 Complete photonic band gap in D: the diamond lattice of dielectric spheres K.M. Ho, C.T. Chan and C.M. Soukoulis, Phys. Rev. Lett. 65, 5 (990) D photonic crystals: parity separation Specular reflection in the xy plane is a symmetry operation of the system, which we denote by σˆ xy. Thus for in-plane propagation we can distinguish y x Even, σ xy =+ modes (TE, H modes): components E x, E y, H z Odd, σ xy = modes (TM, E modes): components H x, H y, E z y E H y H E x x

9 Plane-wave expansion in D The expansion of the field contains D vectors k and G: ) H ( ) = + + +, λ (, λ) ε (, λ)e i( k G) r nk r G c n k G k G The polarization vectors can be chosen as: ˆ ε ( k + G,) zˆ, ˆ( ε k + G,) zˆ kˆ Thus the (N)x(N) matrix of the eigenvalue problem decouples as: = ( k G) ( k G') 0 Hk G, λ; k G', λ' ε ( G, G') 0 k + G k + G' where the upper (lowe block corresponds to H (E) modes. N.b. off-plane propagation, k z 0: no parity separation, D PWE Polarization-dependent gap in D: the square lattice TE or H modes: H z, E x, E y TM or E modes: E z,h x,h y dielectric pillars dielectric veins Gap for H (TE) modes favoured by connected dielectric lattice (veins) Gap for E (TM) modes favoured by dielectric columns (pillars)

10 Complete photonic gap in D Triangular lattice of holes, r/a=0.45 Graphite lattice of pillars, r/a=0.8 Gap maps of triangular and graphite lattices triangular (holes) graphite (pillars) Gap for H (TE) modes favoured by connected dielectric lattice (holes) Gap for E (TM) modes favoured by dielectric columns

11 . Required notions Basic properties of photonic band structure Plane-wave expansion Dielectric slab waveguide Waveguide-embedded photonic crystals. Guided-mode expansion method Photonic mode dispersion Intrinsic diffraction losses Convergence tests, comparison with other methods Examples of applications Books Yariv A., Quantum Electronics (Wiley, New York, 989). Yariv A. and Yeh P., Optical Waves in Crystals (Wiley, New York, 984). Marcuse D., Light Transmission Optics (Van Nostrand Reinhold, New York, 98) Marcuse D., Theory of Dielectric Optical Waveguides, nd ed. (Academic Press, New York, 99)

12 Dielectric slab waveguide All field components satisfy ( ) = 0 xy + ε z ψ z t i( k ) Assuming (, t) e xx ωt ψ r = ψ ( z), we get / 0, ( ) + q = q = ω ψ ε z k ω x z c z ε clad ε core ε clad ck x /n qcore,q clad clad real leaky modes qcore real, q clad imag. d ω>ck x /n q >0, propagating field ω<ck x /n q <0, exponentially damped guided modes ck x /n core Guided modes exist when ε core >ε clad (confinement by total internal reflection) q core,q clad imaginary no solutions k x Leaky vs. guided modes k ck z θ = arctan( x ) = arcsin( x ) kz ncoreω x θ Limiting angle: θ lim n = arcsin( clad ) ncore ω θ < θlim k x < nclad c No total internal reflection, leaky mode. e.m. energy flux at z=± ω θ > θlim k x > nclad c Total internal reflection, guided mode. e.m. energy flux only in xy plane

13 Dielectric slab waveguide: polarizations Taking the in-plane wavevector k = k x xˆ, the xz plane is a mirror plane and specular reflection σˆ xz is a symmetry operation (even if the slab is asymmetric in the vertical direction). Thus all modes (leaky and guided) separate into z ε ε d ε TE modes: E y, H x, H z TM modes: H y, E x, E z (odd under σˆ xz ) (even under σˆ xz ) E z y H H x E TE guided modes The transverse field component E y, assuming i( k ) E (, t) e xx ωt y r = Ey ( z), is found as z d/ -d/ ε ε ε d χ ( z d / ) Ae, E ( z) = Ae iqz + Be iqz y, χ( z+ d / ) Be, = ( ω χ ) / kx ε c ω q = ( ε ) / k x c ω / χ = ( kx ε ) c d z > d z < d z < Continuity of E y, H x, H z leads to the implicit equation q( χ + χ)cos( qd) + ( χχ q )sin( qd) = 0 Relation between coefficients: transfer-matrix method

14 TM guided modes d ε ε ε z The transverse field component H y, assuming, is found as d/ -d/ ) ( e ), ( ) ( z H t H y t x k i y x ω = r Continuity of H y, E x, D z =εe z leads to the implicit equation 0 ) )sin( ( ) )cos( ( = + + qd q qd q ε ε χ ε χ ε χ ε χ ε < = < = + > = = + ) (, ) (, ) (, ) ( / ) / ( / / ) / ( d z c k D e d z k c q e D e C d z c k e C z H x d z x iqz iqz x d z y ω ε χ ω ε ω ε χ χ χ Mode dispersion and effective index TM TE ωd/c ε = ε =ε = TM TE Effective Index ck/ω kd TM TE kd TM TE ε = ε =., ε =

15 General features of waveguide dispersion TM modes have higher frequencies than TE modes of the same order Symmetric waveguide: no cutoff for lowest-order TE and TM modes ( there is at least one confined photonic state at each frequency and for each polarization) Asymmetric waveguide: finite cutoffs even for lowest-order TE and TM modes, all cutoffs are polarization-dependent Effective mode index increases smoothly from cladding refractive index (close to cutoff, guided mode extends into claddings) to core refractive index (at large wavevectors, guided mode is well confined). Required notions Basic properties of photonic band structure Plane-wave expansion Dielectric slab waveguide Waveguide-embedded photonic crystals. Guided-mode expansion method Photonic mode dispersion Intrinsic diffraction losses Convergence tests, comparison with other methods Examples of applications

16 D photonic crystals embedded in planar waveguides (photonic crystal slabs) GaAs membrane (air bridge) Chow et al., Nature 407, 98 (000) GaAs/AlGaAs PhC slab with internal source H. Benisty et al, IEEE J. Lightwave Technol. 7, 06 (999) W waveguide in Si membrane S.J. McNab et al, Opt. Expr., 9 (00) Photonic crystal slabs: structures Air bridge GaAs/AlGaAs Silicon-on-Insulator (SOI)

17 Photonic crystal slabs: the light-line issue Only modes that lie below the light line of the cladding material are guided with no intrinsic losses truly guided modes. Modes that lie above the light line have an intrinsic loss mechanism due to out-ofplane diffraction quasi-guided modes. The issue of losses (intrinsic vs. extrinsic) is a crucial one for prospective applications of PhC slabs to integrated optics. Energy (ev) leaky 0. guided ka/π. Required notions Basic properties of photonic band structure Plane-wave expansion Dielectric slab waveguide Waveguide-embedded photonic crystals. Guided-mode expansion method Photonic mode dispersion Intrinsic diffraction losses Convergence tests, comparison with other methods Examples of applications

18 Basic idea of guided-mode expansion Photonic crystal slabs combine the features of D photonic crystals: control of light propagation in the xy plane by Bragg diffraction y Slab waveguides: control of light propagation in the vertical (z) direction by total internal reflection z x represent the electromagnetic field as a combination of D plane waves in xy and guided modes along z General eigenvalue problem Equation for the magnetic field: with the transversality condition ω H( = H( ε ( c H = 0. Expand magnetic field in a set of basis states as H ( = µ cµ Hµ (, orthonormal according to * Hµ ( Hν ( dr = δµν. We obtain a linear eigenvalue problem with a Hamiltonian matrix, ω ν Hµνcν = c µ c H ( ( H * µν = ε µ ( ) ( Hν ( ) dr whose eigenvalues and eigenvectors give the photonic band dispersion and magnetic field, respectively.

19 Electric field and normalization Once the magnetic field is obtained, the electric field is calculated as c E ( = i H(. ω ε ( The eigenmodes E n (, H n ( of the electromagnetic field are orthonormal according to * Hn( Hm( dr = δnm, * En( Em( ε ( dr = δnm. Physical meaning: the e.m. field energy of an eigenmode is equally shared between the electric and magnetic fields. The same relations can be used for second quantization of the fields: see C.K. Carniglia and L. Mandel, Phys. Rev. D, 80 (97). Effective waveguide We consider a PhC slab with semi-infinite claddings. For the basis states H µ (, we choose the guided modes of an effective homogeneous waveguide with an average dielectric constant ε jin each layer j=,,. z d/ d/ ε ε ε We usually take ε j to be the spatial average of ε j (xy) ε j (ρ) in each layer: ε j = cellε j( ρ) dρ, A=unit cell area A The average dielectric constants must fulfill ε > ε, ε.

20 Expansion in the basis of guided modes guided Hk ( = G, α c( k + G, α ) H k+ G, α ( k G α guided H k+ G, α ( = Bloch vector, chosen to be in the first Brillouin zone = reciprocal lattice vectors of the D Bravais lattice = index of guided mode = guided modes of effective waveguide at k+g Dimension of linear eigenvalue problem is (N PW N α ) (N PW N α ). Since the guided modes are simple trigonometric functions, the matrix elements can be calculated analytically. Both TE and TM guided modes appear in the expansion of a PhC slab mode. The index α includes mode order and polarization. Matrix elements: the dielectric tensor The matrix elements between guided modes depend on the inverse dielectric matrices in the three layers, i( G' G) ρ η j ( G, G') = ε j ( ρ) e dρ A Like in D plane-wave expansion, they are evaluated from numerical inversion of the direct dielectric matrices: ( G, G') = [ j ( G, G')] η j ε i( G' G) ρ ε j ( G, G') = ε j ( ρ) e dρ A Convergence properties, possible optimization beyond the Ho-Chan- Soukoulis procedure are similar to D plane-wave expansion. The off-diagonal components of η j (G,G ) are responsible for the folding of photonic bands in the first BZ, gap formation, splittings etc. mat. el.

21 Discussion of GME method (dispersion) Main drawback: the basis set of guided modes of the effective waveguide is not complete, since leaky modes are not included the GME method is an APPROXIMATE one Main advantage: folded photonic modes in the first Brillouin zone may fall above the light line guided and quasi-guided photonic modes are obtained, without the need of introducing any artificial layers (PML ) in the vertical direction. When a few guided modes are sufficient, as it often happens, the numerical effort is comparable to that of a D plane-wave calculation. Very suited for parameter optimization, design The dispersion of photonic bands in a PhC slab can be compared with that of the ideal D system and with the dispersion of free waveguide modes. Multimode slabs are naturally treated. Symmetry properties, TE/TM mixing When the PhC slab is symmetric under reflection in the xy plane: separation of even (σ xy =+) and odd (σ xy =+) modes k = kˆ x Low-lying photonic modes are dominated by lowest-order modes of the effective waveguide: σ xy =+ modes are dominated by TE waveguide modes quasi-te σ xy = modes are dominated by TM waveguide modes quasi-tm However, in general, any PhC mode contains both TE and TM waveguide modes only the symmetry labels σ xy =± have a rigorous meaning.

22 . Required notions Basic properties of photonic band structure Plane-wave expansion Dielectric slab waveguide Waveguide-embedded photonic crystals. Guided-mode expansion method Photonic mode dispersion Intrinsic diffraction losses Convergence tests, comparison with other methods Examples of applications Diffraction losses: the photonic Golden Rule Modes above the light line are coupled to radiative PhC slab modes. Time-dependent perturbation theory, like in Fermi s golden rule for quantum mechanics, yields an imaginary part of the frequency ω k Im = H ω π +,rad ρ j ';. c k k G ',, j c G λ The matrix element between a quasi-guided and a radiative mode is H k ( H * k ( ) ( H rad, rad = r r r k G λ j ( )) d ε ( ) + ',, The sum is over G reciprocal lattice vectors (out-of-plane diffraction channels) λ=te, TM polarization of radiative PhC slab modes j=, radiative modes that are outgoing in medium, Main approximation: radiative PhC slab modes are replaced with those of the effective waveguide.

23 Photonic DOS and scattering states The D photonic density of states at fixed in-plane wavevector is c g / ( ) ( ) ; z z j c θ ω ω dk ω g k ε ε j ρ j g δ + =. = 0 4 c π c ε j π c g / ( ω ) ε j It depends only on the asymptotic form of the fields, for scattering states with a single outgoing component: z j= j= d/ d/ X W X W X W Coupling matrix element Using the guided-mode expansion for the magnetic field of a PhC slab mode, the matrix element with a radiative mode is where H * k, rad = c( k + G, α ) Hguided, rad G, α * rad H k, rad = ( Hk ( ) ( H r r r k G λ j ( )) d ε ( ) + ',, are matrix elements between guided and radiation modes of the effective waveguide, which are calculated analytically. Both guided and radiation modes are normalized according to H * µ ( Hν ( dr = δµν (within a large box for radiation modes). mat. el.

24 Quality factor and propagation loss The quality factor of a photonic mode is defined as ω Q =. Im( ω) Spatial attenuation is described by an imaginary part of the wavevector: Im( ω) Im( k) =, v g where dω v g = dk is the mode group velocity. Propagation losses in decibels are obtained as Loss = 4.4 Im( k). To treat line and point defects (linear waveguides and nanocavities): use a supercell Discussion of GME method (losses) Main drawback: radiative PhC slab modes are replaced with those of the effective homogeneous waveguide the calculation of losses is an APPROXIMATE one Main advantage: diffraction losses are calculated by perturbation theory the procedure is numerically very efficient (little additional effort beyond photonic mode dispersion). Calculated values are more accurate when diffraction losses are small good for line defects and for high-q nanocavities. The method can be extended to disorder-induced losses of trulyguided modes below the light line.

25 . Required notions Basic properties of photonic band structure Plane-wave expansion Dielectric slab waveguide Waveguide-embedded photonic crystals. Guided-mode expansion method Photonic mode dispersion Intrinsic diffraction losses Convergence tests, comparison with other methods Examples of applications Frequency ωa/(πc) Convergence tests Triangular lattice on membrane, ε=., d/a=0.5, r/a=0. Γ M K (a) Frequency ωa/(πc) 0.5 Γ M K (a) 0 - Im(ω)a/(πc) Im(ω)a/(πc) (b) 0-6 (b) Number of guided modes N α Core effective ε Few guided modes are sufficient for convergence Use of average ε is justified

26 Comparison with MIT code Triangular lattice on membrane, ε=, d/a=0.6, r/a= Frequency ωa/(πc) odd, σ xy = even, σ xy =+ 0.0 Γ M K Γ S.G. Johnson et al., PRB 60, 575 (999) Guided-mode expansion Comparison with FDTD* Triangular lattice on membrane, ε=.56, d/a=0.65 r/a=0.5 FDTD 0.5 GME 0.05 ωa/πc σ kz =+ σ kz =- Q - ( Im(ω)/ω) even, σ kz = + odd, σ kz = Γ K M Γ K *T. Ochiai and K. Sakoda, PRB 6, 507 (00)

27 Comparison with Fourier-modal method (scattering-matrix) Triangular lattice on membrane, ε=, d/a=0., r/a=0.4 σ kz =+ σ kz = IEEE-JQE 8, 89 (00) N.b. Along main symmetry directions: vertical parity symmetry σ kz Losses in W waveguide: a comparison. Membrane, ε=.56, h=0.6a, r=0.06a Energy (ev) σ kz = σ kz = Wavevector ka/π Loss (db/mm) FDTD & Fourier: M. Cryan et al., IEEE-PTL 7, 58 (005) GME Wavelength (nm)

28 . Required notions Basic properties of photonic band structure Plane-wave expansion Dielectric slab waveguide Waveguide-embedded photonic crystals. Guided-mode expansion method Photonic mode dispersion Intrinsic diffraction losses Convergence tests, comparison with other methods Examples of applications Applications of GME method Photonic band dispersion and gap maps: D and D lattices Linear waveguides: propagation losses Photonic crystal nanocavities: Q-factors Geometry optimization, design, comparison with expts Radiation-matter interaction: exciton-polaritons, strong coupling Extrinsic losses: effects of disorder Main reference on the method: L.C. Andreani and D. Gerace, Phys. Rev. B 7, 54 (5 June 006)

29 Triangular lattice with triangular holes 0.5 Frequency ωa/(πc) a L even (σ xy =+) gap 0.58 odd (σ xy =-) gap Γ M K Γ 0.48 Γ M K Γ Frequency ωa/(πc) σ 0.0 xy =+ σ xy = - Γ M K Γ M K Γ d/a=0.68, L/a=0.85: no complete photonic gap d/a=0.5, L/a=0.8: even gap at ω, odd gap at ω Phys. Rev. B 7, 54 (006) Dielectric mirror in a membrane: D PhC slab TE TM d/a=0.4 f air =0. DBR (ideal D) waveguide PC slab 0.8 TM,TE 0.6 ωa/(πc) TM TE ka/π TM TE ka/π ka/π

30 Gap maps in D PhC slabs d/a=0.4 d/a=0.4 Frequency ωa/πc 0,6 0,4 0, TE TM Both 0,0 0,0 0, 0, 0, 0,4 0,5 0,6 0,7 air fraction Frequency ωa/πc 0,6 0,4 0, TE 0,0 TM Both 0,0 0, 0, 0, 0,4 0,5 0,6 0,7 air fraction Frequency ωa/πc 0,6 0,4 0, Complete band gap 0,0 0,0 0, 0, 0, 0,4 0,5 air fraction 0,6 0,7 Gap maps for PhC slabs must consider both the modes below and above the light line. Important for the design of structures with defect modes Very different behavior as compared to the reference ideal D system: no complete band gap, TE/TM splitting D. Gerace and LCA, Phys. Rev. E 69, (004) SOI photonic crystal slabs: fabrication procedure (D. Peyrade, M. Belotti and Y. Chen, LPN Marcoussis) e - PMMA Patterning Resist development e-beam lithography system (50keV JEOL 5DU ) SILICON SILICON DIOXIDE METAL MASK Metal deposition Resist removal Lift-off (metal mask) Ni 0nm Si SiO Reactive-ion etching Sample L4

31 Reflectance and ATR on D SOI PhC slabs ΓΜ (y) (a) In-plane wavevector ΓΚ (x) k =n(ω/c)sinθ +G photonic bands..0 W 5 a θ ZnSe Energy (ev) θ=5 R=0.5 Reflectance 55 TE Si core θ=5 R= ATR SiO cladding M. Patrini et al, IEEE-JQE 8, 885 (00); M. Galli et al, IEEE-JSAC, 40 (005) Air layer 55 TE ka/π Photonic mode dispersion: air bridge Triangular lattice, hole radius r=0.4a, different core thicknesses d 0,7 d/a=0. 0,7 d/a=0.6 0,7 ideal D 0,6 0,6 0,6 0,5 0,5 0,5 0,4 0,4 0,4 ωa/πc 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,0 M Γ K M 0,0 M Γ K M 0,0 M Γ K M Red (blue) lines bands that are even (odd) with respect to reflection in the xy plane. Dotted lines dispersion of light in the effective core material and in air. Notice the vertical confinement effect, which increases with decreasing core thickness. Below the light line, good agreement with S.G. Johnson et al., PRB 60, 575 (999).

32 Gap maps: triangular lattice in air bridge 0,7 0,7 0,7 0,6 d/a=0. 0,6 d/a=0.6 0,6 ideal D 0,5 0,5 0,5 ωa/πc 0,4 0,4 0,4 0, 0, 0, 0, 0, even 0,0 0, 0, 0, 0,4 0,5 hole radius r/a 0, 0, 0,0 0, 0, 0, 0,4 0,5 hole radius r/a Notice the confinement effect of the even gap with respect to the ideal D case and the absence of a full photonic band gap in a waveguide. For d/a=0.6 the cutoff energy of a second-order mode is plotted. even L.C. Andreani and M. Agio, IEEE-JQE 8, 89 (00) 0, 0, even (H) odd (E) both 0,0 0, 0, 0, 0,4 0,5 hole radius r/a D triangular lattice: complex energies (D plot) Im(ω)a/πc 0,00 0,05 0,00 0,7 0,6 0,5 0,005 0,4 0, 0, 0, 0,000 0,0 M Γ K Re(ω)a/πc Air bridge, d=0.a, r/a=0.. The radiative width of quasiguided modes increases for some modes close to the light line, due to the D DOS. At the Γ point, only the dipoleallowed mode has nonzero radiative width. Phys. Status Solidi (b) 4, 9 (00)

33 0.7 Complex energies vs. reflectivity θ= Re(ω)a/πc Im(ω)a/πc Γ Wavevector TM pol 0. K Reflectance Γ Wavevector K Air bridge d=0.a, triangular lattice of holes r=0.a, ΓK, TM polarization. Positions and linewidths of spectral structures real and imaginary parts of energy. Synt. Met. 9, 695 (00) Trends: losses as a function of... waveguide dielectric contrast hole radius ε core =, d/a=0. triangular lattice r/a= ε core =, ε clad =, d/a=0. triangular lattice Im(ωa/πc) Im(ωa/πc) ε clad hole radius r/a Radiative width of dipole-allowed mode at the Γ point. The losses increase with the dielectric contrast between core and cladding and with the air fraction of the D lattice. Agreement with the model of H. Benisty et al, APL 76, 5 (000).

34 Linear waveguides in D PhC slabs w Single missing row of holes in triangular lattice, channel thickness (W waveguide): w= w0 a ε=, d/a=0.5, r/a=0.8, a=400 nm 0.5 a Propagation losses of truly guided modes in Si membranes (extrinsic): ~ 0.6 db/mm (Notomi et al., Vlasov et al.) Frequency ωa/πc even odd Wavevector ka/π ωa/(πc) Propagation losses in W waveguides 0,6 0,5 0,4 0, 0, 0, w 0,0 0,0 0, 0,4 0,6 0,8,0 Wavevector (π/a) r/a=0.6 LCA and M. Agio, Appl. Phys. Lett. 8, 0 (00) a Im(ω)a/(πc) v g /c Attenuation Length l/a (a) r/a=0.8 r/a=0. r/a=0.6 r/a=0.40 (b) 0 (c) ωa/(πc)

35 Mode dispersion on increasing channel width σ kz = 0. σ kz =+ 0. ωa/πc W.0 W. W ωa/πc W. W.4 W k x a/π k x a/π k x a/π M. Galli et al., PRB 7, 5 (005) Photonic crystal nanocavities Point defects in PC slabs behave as 0D cavities with full photonic confinement in very small volumes. The Q-factor can be increased by shifting the positions of nearby holes. Q= has been measured: T. Akahane, T. Asane, B.S. Song, and S. Noda, Nature 45, 944 (00). V= cm cm Q (0 )

36 Q-factor of photonic crystal cavities 0 5 L defect L defect L defect expt.(*) L 0 4 Q-factor L 0 L Displacement x/a Supercell in two directions + Golden Rule Im(ω) and Q=ω/Im(ω) *Expt: T. Akahane et al., Nature 45, 944 (00). LCA, D. Gerace and M. Agio, Photonics and Nanostruc., 0 (004) A model of disorder: size variations Random distribution of hole radii within a large supercell: ( r r ) P( = exp σ σ r= r.m.s. deviation Dielectric perturbation ε ( = εdis ( ε ( couples guided modes to leaky waveguide modes losses by perturbation theory: ω Im k c = π H ε dis( * leaky H guided dr ω ρ k; c Propagation loss: Im( ω ) Im( k) = k, v g v g dω = k dk

37 Dependence on disorder parameter 0.4 Frequency (ωa/πc) r/a=0.005 r/a=0.0 r/a=0.0 r/a=0.04 r/a=0.08 r/a=0.005 r/a=0.0 r/a=0.0 r/a=0.04 r/a= Wavevector (ka/π) Im(ωa/πc) Loss*a (db) The losses show a quadratic behavior as a function of the disorder parameter typical of Rayleigh scattering D. Gerace and L.C. Andreani, Opt. Lett. 9, 897 (004) Comparison with experimental results [] Transmission (db) Present theory Experiment of Ref. [] 0.09 mm 0.80 mm 0.00 mm -40 mm mm Frequency ωa/πc 0,0 0,5 0,0 0,05 0,00 0,95 0,95 0,00 0,05 0,0 Frequency ωa/πc 0,90 0, 0,4 0,6 0,8,0 Wavevector ka/π,7 db/mm Diffraction losses (db/mm) Theory with no fitting parameters air bridge W waveguide, d/a=0.494, r/a=0.7, r=5 nm, a=445 nm [] S.J. McNab, N. Moll, Yu. Vlasov, Optics Express, 9 (00)

38 Line-defects with increased channel width a/λ W.0 σ kz =+ σ kz = db/mm r=. nm a=400 nm W. 0.4 db/mm a/λ W>W a/λ W Wave vector (π/a) 0.07 db/mm 0. Losses (db/mm) Large single-mode frequency window with ultra-low losses D. Gerace and LCA, Opt. Expr., 499 (005); Photon. Nanostr., 0 (005) Conclusions The GME method is APPROXIMATE for both the photonic mode dispersion (as the basis set does not contain radiative modes of the effective waveguide) and for the losses (because radiative PhC modes are replaced with those of the effective waveguide). The results are close to those of exact methods in many common situation. The GME method is most reliable for high index-contrast PhC slabs (membranes, SOI) and for not too high air fractions. Main advantages: computational efficiency (especially in the case of low losses: high-q nanocavities ) and physical insight when comparing with ideal D and homogeneous waveguide systems. Possible extensions (some underway): Improve upon the approximations PhC slabs with more than three layers Radiation-matter interaction Disorder-induced losses

39 Appendix: matrix elements etc. Basis set for guided-mode expansion 0, ) )sin( ( ) )cos( ( = + + qd q qd q χ χ χ χ 0, ) )sin( ( ) )cos( ( = + + qd q qd q ε ε χ ε χ ε χ ε χ ε Let us denote by the D wavevector in the xy plane with modulus g a unit vector perpendicular to both and the frequency of a guided mode The guided modes frequencies of the effective waveguide are found from g g ˆ = g ẑ g z ˆ ˆ ˆ = g ε ĝ ω g / / /,, = = = c g g c q c g g g g ω ε χ ω ε ω ε χ TE modes TM modes

40 Guided mode profile: TE Guided mode profile: TM

41 Matrix elements between guided modes diel. tensor Radiative mode profiles ˆ ε g = zˆ gˆ TE modes. Normalization: W = 0, X = / ε or W = / ε, X = 0. TM modes. Normalization: Y = 0, Z = or Y =, Z = 0. N.b. when q j is imaginary, the mode does not contribute to losses.

42 Matrix elements guided radiative coupling