Nanophotonics and Metamaterials*

Transcription

1 *)Krasnoyarsk Federal University, Russia lectures available on Nanophotonics and Metamaterials* Professor Vladimir M. Shalaev 5 nm

2 Overview of the Course Part I: a) Light interaction with matter and b) Photonic Crystals a) Maxwell s equations Dielectric properties of insulators, semiconductors and metals (bulk) Light interaction with nanostructures and microstructures (compared with λ) b) Electromagnetic effects in periodic media Media with periodicity in 1,, and 3-dimensions Applications: Omni-directional reflection, sharp waveguide bends, Light localization, Superprism effects, Photonic crystal fibers Part II: Metal optics (plasmonics) and nanophotonics Light interaction with 0, 1, and dimensional metallic nanostructures Guiding and focusing light to nanoscale (below the diffraction limit) Near-field optical microscopy Transmission through subwavelength apertures Part III: Metamaterials Metamaterials, optical magnetism, and negative refractive index Perfect lens How to make objects invisible: Cloaking

3 Overview in Images 5 nm K.S. Min et al. PhD Thesis K.V. Vahala et al, Phys. Rev. Lett, 85, p.74 (000) J. D. Joannopoulos, et al, Nature, vol.386, p (1997) T.Thio et al., Optics Letters 6, (001). S. Lin et al, Nature, vol. 394, p. 51-3, (1998) J.R. Krenn et al., Europhys.Lett. 60, (00)

4 Motivation Major breakthroughs are often materials related Stone Age, Iron Age, Si Age,.metamaterials People realized the utility of naturally occurring materials Scientists are now able to engineer new functional nanostructured materials Is it possible to engineer new materials with useful optical properties Yes! Wonderful things happen when structural dimensions are λ light and much less This course talks about what these things are and why they happen What are the smallest possible devices with optical functionality? Scientists have gone from big lenses, to optical fibers, to photonic crystals, to Does the diffraction set a fundamental limit? Possible solution: metal optics/plasmonics

5 A. Light Interaction with Matter: Maxwell Equations & Constitutive Relations Maxwell s Equations Divergence equations D = ρ f B =0 Curl equations B E = t D H = + t J D = Electric flux density E = Electric field vector ρ = charge density B = Magnetic flux density H = Magnetic field vector J = current density

6 Constitutive Relations Constitutive relations relate flux density to polarization of a medium Electric E ( ) D = ε E+ P E = ε E Electric polarization vector Material dependent!! ε 0 = Dielectric constant of vacuum = C N -1 m - [F/m] ε = Material dependent dielectric constant When P is proportional to E Total electric flux density = Flux from external E-field + flux due to material polarization Magnetic 0 0 ( ) B = μ H + μ M H Magnetic flux density Magnetic field vector Magnetic polarization vector μ 0 = permeability of free space = 4πx10-7 H/m Note: For now, we will focus on materials for which M = 0 B = μ0h

7 Speed of the EM wave: Speed of an EM Wave in Matter Compare E t E =μεε 0 0 r and E = 1 v E t v 1 1 c = = μ ε ε ε r r Where c 0 = 1/(ε 0 μ 0 ) = 1/((8.85x10-1 C /m 3 kg) (4π x 10-7 m kg/c )) = ( 3.0 x 10 8 m/s) Optical refractive index Refractive index is defined by: n c = = ε r = 1+ χ v Note: Including polarization results in same wave equation with a different ε r c becomes v

8 Dispersion relation: ω = ω(k) Derived from wave equation Substitute: E Dispersion Relation (, t) E r = (, t) { } ( zt, ) = Re E( z, ω) exp( ikr+ iωt) n c E r t Result: Check this! k n c ω = ω c ω = k n Group velocity: Phase velocity: v g r v g dω dk ω c c c v ph = = = = k n ε 1+ χ k

9 Solution to: Electromagnetic Waves (, t) E r = n c t (, t) E r { } Monochromatic waves: E( r, t) = Re E( k, ω) exp( ik r+ iωt) { } (, t) = Re (, ω) exp( i + iωt) H r H k k r Check these are solutions! TEM wave Optical intensity Symmetry Maxwell s Equations result in E H propagation direction Time average of Poynting vector: S ( r, t) = E( r, t) H( r, t)

10 EM waves in Dispersive Media Relation between P and E is dynamic EM wave: + ε0 (, t) = dt' x( t t' ) E( r, t' ) P r { } { ω ω } (, t) = Re (, ω) exp( i + iωt) E r E k k r (, t) = Re (, ) exp( i + i t) P r P k k r Relation between complex amplitudes (, ω) = εχω ( ) E( k, ω) P k 0 (Slow response of matter ω-dependent behavior) This follows by equation of the coefficients of exp(iωt)..check this! It also follows that: ( ) = + ( ) ε ω ε 1 χ ω 0

11 Absorption and Dispersion of EM Waves Transparent materials can be described by a purely real refractive index n { } EM wave: E( zt, ) = Re E( k, ω) exp( ikz+ iωt) c ω Dispersion relation ω = k k =± n n c Absorbing materials can be described by a complex n: n= n' + in'' ω ω ω α It follows that: k =± ( n' + in'' ) =± n' + i n'' ± β i c c c Investigate + sign: E( ) E( ) α zt, = Re k, ω exp iβz z+ iωt Traveling wave Decay ω Note: β = n' = k0n' n act as a regular refractive index c ω α = n'' = k0n'' α is the absorption coefficient c

12 Absorption and Dispersion of EM Waves n is derived quantity from χ (next lecture we determine χ for different materials) Complex n results from a complex χ: χ = χ' + iχ'' n = 1+ χ n= n' + in'' = 1+ χ = 1 + χ' + iχ'' α = kn'' 0 α n= n' i = 1 + χ ' + iχ '' k 0 Weakly absorbing media When χ <<1 and χ << 1: 1 + χ ' + iχ'' 1 + ( χ' + iχ'' ) Refractive index: 1 n' = 1 + χ ' α kn = '' = kχ '' Absorption coefficient: 0 0 1

13 Intermediate Summary (A) Maxwell s Equations Bold face letters are vectors! B D = ρ f B = 0 E = t Curl Equations lead to E P E = με μ 0 t t Linear, Homogeneous, and Isotropic Media P = ε χ E 0 (under certain conditions) D H = + J t Wave Equation n E ( r, t) E( r, t) = c t Solutions: EM waves α E( zt, ) = Re E( z, ω) exp iβz z+ iωt where β = kn 0 ' and α = kn 0 '' In real life: Response of matter (P) is not instantaneous + (, t) = ε dt' x( t t' ) E( r, t' ) P r 0 Phase propagation χ ' = χ'( ω) n' = n' ( ω) χ '' χ''( ω) = n'' = n'' ( ω) absorption

14 B. Optical Properties of Bulk and Nano 5 nm

15 Microscopic Origin ω-response of Matter Origin frequency dependence of χ in real materials Lorentz model (harmonic oscillator model) Insulators (Lattice absorption, color centers ) Semiconductors (Energy bands, Urbach tail, excitons ) Metals (AC conductivity, Plasma oscillations, interband transitions ) Real and imaginary part of χ are linked Kramers-Kronig But first.. When should I work with χ, ε, or n? They all seem to describe the optical properties of materials!

16 n and n vs χ and χ vs ε and ε All pairs (n and n, χ and χ, ε and ε ) describe the same physics For some problems one set is preferable for others another n and n used when discussing wave propagation α E( zt, ) = Re E( z, ω) exp iβz z+ iωt where β = kn 0 ' and α = kn 0 '' Phase propagation absorption χ and χ used when discussing microscopic origin of optical effects ε and ε As we will see today Inter relationships Example: n and ε From n = ε r n' + in'' = ε ' + iε '' r r ( ) ( ) ε r ' = n' n'' ε '' = nn ' '' r and n ' = n '' = ( ) ( ) ε ' + ε '' + ε ' r r r ( ) ( ) ε ' + ε '' ε ' r r r

17 Behavior of bound electrons in an electromagnetic field Optical properties of insulators are determined by bound electrons Lorentz model Charges in a material are treated as harmonic oscillators m a = F + F + F el E, Local Damping Spring d r dr m + mγ + Cr = ee exp L i t dt dt Guess a solution of the form: p Linear Dielectric Response of Matter ( ω ) = p0 exp i t ( ω ) The electric dipole moment of this system is: d p dp m + mγ + Cp = e E exp L i t dt dt dp ; = iωp0 exp dt mω p imγωp + Cp = e E L (one oscillator) p ( ω ) = e ( iωt) r d p dt E L C, γ r + ; ω p 0 exp = Solve for p 0 (E L ) Nucleus e -, m p = er ( iωt)

18 Atomic Polarizability Determination of atomic polarizability Last slide: mω p imγωp + Cp = e E L C e 0 iγω 0 0 L ω p p + p = E (Divide by m) m m Define as ω 0 (turns out to be the resonance ω) p = e 1 E ω ω iγω 0 m 0 L Atomic polarizability (in SI units) Define atomic polarizability: ( ) α ω p e 1 = ε ε m ω ω iγω 0 0E L 0 0 Resonance frequency Damping term

19 Characteristics of the Atomic Polarizability Response of matter (P) is not instantaneous ω-dependent response p0 e 1 Atomic polarizability: α ( ω) = Aexp iθ ( ω) ε = ε m ω ω iγω = 0E L 0 0 Amplitude A = e ε m 0 ( 0 ) 1 ω ω + γ ω 1/ Amplitude smaller γ Phase lag of α with E: 180 θ = tan 1 γω ω ω 0 Phase lag 90 smaller γ 0 ω 0 ω

20 Relation Atomic Polarizability (α) and χ: cases Case 1: Rarified media (.. gasses) Dipole moment of one atom, j : Polarization vector: Occurs in Maxwell s equation.. sum over all atoms p ( ) = εα ω E j 0 j L P 1 ε = = α = ε Nα V p E E j E-field photon Density 0 j j L 0 j L V j α j ( ω) = e 1 ε m ω ω iγω 0 0 P = Ne 1 L 0 m ω ω iγω E = E 0 Microscopic origin susceptibility: Plasma frequency defined as: ω p ( ε χ ) ( ) χ ω Ne ε m 0 = = χ ( ω ) L Ne 1 ε m ω ω iγω 0 0 = ω p ω ω iγω 0

21 Remember: ε and n follow directly from χ Frequency dependence ε Relation of ε to χ : ε = 1+ χ = 1+ ω p 0 ω ω iγω ε ' + iε '' = 1 + χ ' + iχ '' = 1+ ω p 0 ω ω iγω ε ( ω ) ( ) ε ' = 1 + χ ' ω = 1+ ( ) ε '' = χ '' ω = ω p ( ω0 ω ) ( 0 ) + ( 0 ) ω ω γ ω ωγω p ω ω + γ ω 1+ ω p ω 0 1 χ ' = 0 χ = ω p ω 0 ω 0 ω p ω

22 Propagation of EM-waves: Need n and n Relation between n and ε n = ε ε ( ω ) ( ) ( ) ε r ' = n' n'' ε '' = nn ' '' r 1+ ω p ω 0 1 n ( ω ) ω 0 ω ω << ω 0 : High n low v ph =c/n ω ω 0 : Strong ω dependence v ph n ' n '' n ' = 1 Large absorption (~ n ) ω >> ω 0 : n = 1 v ph =c ω 0 ω

23 χ = k Realistic Rarefied Media Realistic atoms have many resonances Resonances occur due to motion of the atoms (low ω) and electrons (high ω) Ne k 1 ε m ω ω iγω 0 k Where N k is the density of the electrons/atoms with a resonance at ω k Example of a realistic dependence of n and n α= k 0 n Molecule rotation Atomic vibration Electron excitation n n > 1 indicates presence high ω oscillators ω 1 ω ω 3 Log (ω)

24 Classification Matter: Insulators, Semiconductors, Metals Bonds and bands One atom, e.g. H. Schrödinger equation: H+ E Two atoms: bond formation H +? + H Every electron contributes one state Equilibrium distance d (after reaction)

25 Classification Matter ~ 1 ev Pauli principle: Only electrons in the same electronic state (one spin & one spin )

26 Atoms with many electrons Classification Matter Empty outer orbitals Partly filled valence orbitals Outermost electrons interact Form bands Energy Filled Inner shells Electrons in inner shells do not interact Do not form bands Distance between atoms

27 Classification Matter Insulators, semiconductors, and metals Classification based on bandstructure

28 Absorption Processes in Semiconductors Absorption spectrum of a typical semiconductor E E C Phonon Photon E V ω Phonon

29 Excitons: Electron and Hole Bound by Coulomb Analogy with H-atom Electron orbit around a hole is similar to the electron orbit around a H-core 1913 Niels Bohr: Electron restricted to well-defined orbits n = ev + n = -3.4 ev n = ev Binding energy electron: E B 4 me e 13.6 = = ev, n= 1,,3,... ( 4πε 0 n) n Where: m e = Electron mass, ε 0 = permittivity of vacuum, = Planck s constant n = energy quantum number/orbit identifier

30 Binding Energy of an Electron to Hole Electron orbit around a hole Electron orbit is expected to be qualitatively similar to a H-atom. Use reduced effective mass instead of m e : * 1/ = 1/ e + 1/ m m m h Correct for the relative dielectric constant of Si, ε r,si (screening). e - ε r,si h - Binding energy electron: E Typical value for semiconductors: B * m = ev, n= 1,,3,... m ε e E = 10meV 100meV B Note: Exciton Bohr radius ~ 5 nm (many lattice constants)

31 Example: Si nanocrystals Semiconductor Nanoparticles 4.0 E E C E V D: 1-5 nm Photoluminescence e h hν=e G Bandgap (ev) Diameter (nm) E B C.Delerue et al. Phys Rev. B 48, 1104 (1993)

32 Size and Material Dependent Optical Properties Fluorescence signal (a.u.) Wavelength (nm) Red series: InAs nanocrystals with diameters of.8, 3.6, 4.6, and 6.0 nm Green series: InP nanocrystals with diameters of 3.0, 3.5,and 4.6 nm. Blue series: CdSe nanocrystals with diameters of.1,.4,3.1, 3.6, and 4.6 nm M.Bruchez et al. (Alivisatos group), Science, 013, 81 (014)

33 Optical Properties of Metals (determine ε) Current induced by a time varying field Consider a time varying field: Equation of motion electron in a metal: E { } ( t) = Re E( ω) exp( iωt) d x dv v m = = m ee dt dt τ relaxation time ~ s { } Look for a steady state solution: v( t) = Re v( ω) exp( iωt) Substitution v into Eq. of motion: i mv( ) This can be manipulated into: The current density is defined as: J( ω ) = ne It thus follows: J( ω) ( ω) mv ω ω = ee( ω) τ e v( ω) = E( ω) m 1 τ iω ( ) v Electron density = ( ne m) ( 1 τ iω) E ( ω)

34 Determination conductivity From the last page: Optical Properties of Metals J ( ω) = ( ne m) ( 1 τ iω) E ( ω) = σ ( ω) E( ω) ( ω) ( ) σ ω ( ne m) = = σ ( 1τ iω) ( 1 iωτ) Definition conductivity: J ne τ where: σ 0 = m Both bound electrons and conduction electrons contribute to ε From the curl Eq.: For a time varying field: () t () D ε B E t H = + J = + J t t E { } ( t) = Re E( ω) exp( iωt) ε E σ ω H = + J = iωε ω E ω + σ ω E ω = ωε ε ω E ω ( ) ( ) ( ) ( ) i ( ) ( ) B B 0 B t iεω 0 ε EFF ( ω) Currents induced by ac-fields modeled by ε EFF σ For a time varying field: ε ε ε iε ω Bound electrons EFF = B = B + i σ ε ω 0 0 Conduction electrons 0 ( )

35 Optical Properties of Metals Dielectric constant at ω ω visible σ σ0 ( 1+ iωτ 0 ) σ0 σ0 Since ω vis τ >> 1: σ ( ω) = = + i 1 ωτ 1+ ω τ ωτ It follows that: ( i ) ( ωτ ) σ σ σ ε = ε + i = ε + i ε ω ε ω τ ε ω τ 0 0 EFF B B σ 0 ne p 10eV for metals ετ 0 ε0m Define: ω ( ) ε ω ω p p EFF = εb + i 3 ω ωτ Bound electrons Free electrons What does this look like for a real metal?

36 Excitation of a Metal Nanoparticle Energy (ev) Particle Volume = V 0 ε M = ε M + iε M E-field σ ext (nm ) Ag cluster D = 10 nm 1 n=1.5 n=3.3 R = 5 nm Host matrix ε H = ε H = n H ω σext ( ω) = 9 ε c ' 3/ H V λ (nm) ' εm ( ω) ( ) + + ( ) 0, ',, ε M ω εh εm ω

37 Applications Metallic Nanoparticles Engraved Czechoslovakian glass vase Ag nanoparticles cause yellow coloration Au nanoparticles cause red coloration Molten glass readily dissolves 0.1 % Au Slow cooling results in nucleation and growth of nanoparticles

38 C. Photonic crystal: An Introduction Photonic crystal: Periodic arrangement of dielectric (metallic, polaritonic ) objects. Lattice constants comparable to the wavelength of light in the material.

39 A worm ahead of its time Sea Mouse and its hair Normal incident light Off-Normal incident light 0cm

40 Fast forward to 1987 E. Yablonovitch Inhibited spontaneous emission in solid state physics and electronics Physical Review Letters, vol. 58, pp. 059, 1987 S. John Strong localization of photons in certain disordered dielectric superlattices Physical Review Letters, vol. 58, pp. 486, 1987 Face-centered cubic lattice Complete photonic band gap

41 The emphasis of recent breakthroughs The use of strong index contrast, and the developments of nanofabrication technologies, which leads to entirely new sets of phenomena. Conventional silica fiber, δn~0.01, photonic crystal structure, δn ~ 1 New conceptual framework in optics Band structure concepts. Coupled mode theory approach for photon transport. Photonic crystal: semiconductors for light.

42 Two-dimensional photonic crystal λ 1.5 μm a High-index dielectric material, e.g. Si or GaAs

43 Band structure of a two-dimensional crystal Displacement field parallel to the cylinder Frequency (c/a) air band dielectric band M Γ X TM modes Γ X M Γ Wavevector (π/a) QuickTime and a TIFF (Uncompressed) decompressor are needed to see this picture. Γ Wavevector determines the phase between nearest neighbor unit cells. X: (0.5*π/a, 0): Thus, nearest neighbor unit cell along the x-direction is 180 degree out-of-phase M: (0.5*π/a, 0.5*π/a): nearest neighbor unit cell along the diagonal direction is 180 degree out-of-phase

44 Bragg scattering Incident light e ikx a re -ikx Regardless of how small the reflectivity r is from an individual scatter, the total reflection R from a semi infinite structure: Diverges if R = re ikx + re ika e ikx + re 4ika e ikx +... = re ikx 1 1 e ika e ika = 1 k = π a Bragg condition Light can not propagate in a crystal, when the frequency of the incident light is such that the Bragg condition is satisfied Origin of the photonic band gap

45 A simple example of the band-structure: vacuum (1d) Vacuum: ε=1, μ=1, plane-wave solution to the Maxwell s equation: k He i(k r ωt ) with a transversality constraints: k H=0 A band structure, or dispersion relation defines the relation between the frequency ω, and the wavevector k. ω = c k ω For a one-dimensional system, the band structure can be simply depicted as: ω=ck k

46 Visualization of the vacuum band structure (d) For a two-dimensional system: ω ω = c k x + k y Light cone This function depicts a cone: light cone. A few ways to visualize this band structure : k y k x k y ω ω ω Continuum of plane wave Light line k x ω=ck x 0 k x (0,0) (k,0) (k,k) (0,0) k Constant frequency contour Projected band diagram Band diagram along several special directions

47 Maxwell s equation in the steady state Time-dependent Maxwell s equation in dielectric media: Hr,t ( )= 0 εe( r,t)= 0 Time harmonic mode (i.e. steady state): Maxwell equation for the steady state: ( ( )) Hr,t ( ) ε() r εer,t 0 t ( ( )) Er,t ( )+ μ 0Hr,t t Hr,t ( )= Hr ( )e iωt Er,t ( )= Er ( )e iωt Hr ( )+ iω( ε( r)ε 0 Er ( ))= 0 Er ( ) iω( μ 0 Hr ( ))= 0 = 0 = 0

48 Master s equation for steady state in dielectric Expressing the equation in magnetic field only: 1 ε() r Hr ()= ω c Hr ( ) c = 1 ε 0 μ 0 Thus, the Maxwell s equation for the steady state can be expressed in terms of an eigenvalue problem, in direct analogy to quantum mechanics that governs the properties of electrons. Field Eigen-value problem Operator Quantum mechanics Electromagnetism Ψ( r,t)= Ψ( r)e jωt Hr,t ( )= Hr ( )e iωt ˆ H Ψ r ( )= EΨ r ( ) H ˆ = m + V () r ΘH()= r ω Hr c Θ= 1 ε() r ()

49 Donor and Acceptor States 0.5 Air defect frequency (c/a) Dielectric defect Air defect radius of defect (c/a) Dielectric defect s-state p-state P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, Phys. Rev. B 54, 7837 (1996)

50 Line defect states: projected band diagram Electric field parallel wavevector Frequency (c/a) Guided Modes Wavevector (π/a)

51 Photonic crystal vs. conventional waveguide High-index region Low index region Conventional waveguide Photonic crystal waveguide

52 High transmission through sharp bends Polluck, Fundamentals of Optoelectronics, 1995 A. Mekis et al, PRL, 77, 3786 (1996)

53 Micro add/drop filter in photonic crystals Two resonant modes with even and odd symmetry. The modes must be degenerate. The modes must have the same decay rate. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, Physical Review Letters 80, 960 (1998).

54 Super-lens and Super-prism effects H. Kosaka et al, Phys. Rev. B. 58, 10096, 1998 H. Kosaka et al, Appl. Phys. Lett. 74, 1370, 1999

55 Snell s law in terms of a constant frequency circle Example: using constant frequency diagram to derive Snell s law and the condition for total internal reflection. Snell s law y n ω c = k x + k y n 1 = 1 θ 1 x n = 1.5 θ n 1 sinθ 1 = n sinθ

56 Constant frequency contour in a D crystal Constant frequency diagram for the first band At low frequencies, the constant frequency diagram approaches a circle, the photonic crystal behaves as a uniform dielectric as far as diffraction is concerned With increasing frequencies, the constant frequency contour becomes more complicated, leading to effects including superprism, superlens, negative refraction, and self-collimation. Luo et al, Phys. Rev. B 65, 01104, 00; M. Notomi, Phys. Rev. B 6, 1069, 000

57 Super-lens and constant frequency ω=0.165 πc/a P. C. M Γ X Vg = k ω(k) group velocity

58

59 V g = k ω(k) group velocity Photonic Band Engineering

60 3D photonic crystal with complete band gap 34 % when spheres are touching each other Complete band gap observed in both air spheres and dielectric spheres Refractive index ratio needs to exceed in order for band gap to open Optimal structure consists of connected dielectric and air networks. Ho, Chan, Soukoulis, PRL, 65, 315 (1990)

61 Variants of diamond structure, practical 3d structures Chan et al, Solid State Communication, 89, (1994) S. Lin et al, Nature, vol. 394, p. 51-3, (1998)

62 Self-assembled 3D photonic crystal structures Y. Vaslov et al, Nature, vol. 414, p. 89, (001)

63 Photonic crystal slab structures Low-index materials High-index materials In plane D photonic band gap provides complete in plane confinement. Out of plane confinement provided by high index guiding Ease of fabrication In complete confinement in the third dimension

64 Photonic band diagram for photonic crystal slabs Radiation modes above the light line. Losslessly guided modes below the light line. Incomplete band gap in the guided mode spectrum

65 Waveguides in dielectric slabs Oxide Magnetic field Si 0.40 Frequency (c/a) Radiation modes Slab modes gap Wavevector (π/a)

66 Photonic crystal fibers Photonic crystal fibers guide light by corralling it within a periodic array of microscopic air holes that run along the entire fiber length. P. Russell, Science, 99, 358, 003

67 Stack and Draw Technique Macroscopic preform with the required periodicity Furnace to soften the silica gas Photonic crystal fiber

68 A brief overview of conventional fiber structure Silica cladding: n 1 = 1.44 to 1.46 Silica core doped with Germanium, Boron or Titantium n n 1 n = ~ 0.0 Core diameter for single mode fiber about 8 μm.

69 Propagation loss in conventional optical fiber OH Peak Rayleigh scattering: from random localized variation of the molecular positions in glass which creates random inhomogeneities in index. Infrared absorption: from vibrational transitions. Absolute minimum at 1.55 micron, at 0.16dB/km, about 3.6% per km. Saleh and Teich, Fundamentals of Photonics, 1991

70 fiber optical amplifier for long distance communication Er, gain maximum close to 1.55 micron Usable bandwidth limited by the amplifier bandwidth to be approximately 30nm Improving bandwidth by removing amplifiers, guiding in air?

71 Band diagram for conventional fibers Guiding mode exists between the light line of the cladding and light line of the core. ω Light-line of cladding ω=ck/n 1 Radiation modes Guided modes Light-line of core ω=ck/n

72 Lower and higher order modes Fundamental mode ω Radiation modes Light-line of cladding ω=ck/n 1 Guided modes Light-line of core ω=ck/n V = πa λ Higher order mode The V-number determines the number of modes in the fiber ( n core n ) cladding

73 Conventional vs Photonic Crystal Fibers Conventional fiber Core diameter 9 micron Dielectric-core PCF Core diameter 5 micron Air-core PCF Core diameter 9 micron

74 Endless single mode photonic crystal fiber Solid core photonic crystal fiber. Solid core region nominally 4.6 μm wide The fiber supports a single mode over the range of at least nm Knight et al, Opt. Lett. 1, 1547, 1996

75 The cladding as a mode sieve The lower modes can not escape as the wire mesh are too narrow. The higher order modes can leak through the narrow strip. Increasing the relative size of the diameters of holes (d) with respect to the pitch (Λ) leads to the trapping of higher modes Single mode behavior occurs when d/λ < 0.4 Knight et al, Opt. Lett. 1, 1547, 1996

76 The band structure picture Much larger room for dispersion management. State-of-art loss figure at 0.58dB/km No complete band gap at β = 0 for silica/air type of index contrast. At finite β, band gap can appear. Band gaps arises from multiple reflection at the interfaces. At finite β, the reflectivity goes up, effectively increasing the inplane index contrast. In order to achieve guiding in air, the criteria is to find a band gap above the light line of air. Requires fairly large air holes (r ~ 0.47Λ) Possible region for air guiding

77 Air core photonic band gap fibers, experiments Transmission through a 100m fiber 13 db/km in propagation loss, comparable to early days of conventional optical fiber. Loss primarily due to the coupling of core modes to surface modes, and likely can be further reduecd significantly in newer design. Smith et al, Nature, 44, 657 (003)

78 The Bragg fiber Using multilayer-film reflection to replace metal and create a light pipe. The boundary condition for EM field at the boundary of core-film boundary can be designed to be rather similar to that at the metal boundary. P. Yeh, A. Yariv and E. Marom, J. Opt. Soc. Am. 68, 1196 (1978).

79 All dielectric co-axial waveguide Single polarization mode in dielectric waveguide, similar to the TEM mode

80 Hollow optical fiber, experiments Guiding of intense CO laser light at 10 micron wavelength range for high power applications Temelkuran et al, Nature, 40, 650 (00).