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Department of Electrical and Computer Engineering hotonics research aboratory Nano-Photonics (1) W. P. Huang Department of Electrical and Computer Engineering McMaster University Hamilton, Canada 1 Agenda Introduction: Nature of light: • Light as electromagnetic waves • Light as a quantum photons Confinement and guidance of light at nano-scale • Conventional waveguides • Advanced waveguides • Surface plasma polariton waveguides Confinement and resonance of light at nano-scale – Scattering of Light by Metal Particles – Surface plasma polariton resonators Light-matter interaction in nano-crystals – Optical properties of nano-crystals 2 Introduction to Nano-Photonics 3 What is Nano-Photonics? Science and technologies that investigate and utilize phenomena of light confinement, guidance, resonance, and interaction with nano-structures Topics: – Near Field Optics – Surface Plasma Polariton (SPP) Waveguides and Resonators – Dipole Energy Transfer 4 Why Nano-Photonics? Optical wavelength (100nm -- 1000nm) is much larger than typical dimension of nano-structures and hence interaction between light and nanostructures is usually very weak, less understood, and not sufficient utilized We need to understand how we may enhance, engineering and utilize the interaction between light and nano-structures to develop new optical functionalities on the smallest spatial dimension, at the lowest energy level, and within shortest temporal scale. 5 Methods for Construction of Nano-Structures 6 Nature and Properties of Light 7 Waves Theory Nature of Light Huygens (1690) Light is a wave traveling through ether (invisible medium) Newton (1704): Light is a stream of corpuscles Planck (1900): Radiation theory included “quantization” of energy Young(1801): Interference experiments supporting the wave theory Maxwell (1865): Light is electromagnetic wave. Particle Theory Quantum Theory of Light Bohr 1913 Bose (1924) de Broglie (1924) Einstein (1905): Photoelectric effect light is a stream of particles (photons) 8 Maxwell’s Equations Faraday Law of Induction: Er, t Generalized Amplere’s Law : Hr, t t Br, t t Dr, t J r, t Gauss’s Law for magnetic field : Br, t 0 Gauss’s Law for electric field : Dr, t ρr, t Er, t : volts/m Br, t : webers/m 2 Dr, t : coulombs/m 2 Hr, t : amperes/m J r, t : amperes/m 2 ρr, t : coulombs/m Conservation Law for electric charge and current: J r, t ρr, t 0 t 9 3 James Clerk Maxwell (1831 - 1879) A Scottish mathematical physicist who is widely regarded as the nineteenth century scientist who had the greatest influence on twentieth century physics. Maxwell demonstrated that electrical and magnetic forces are two complementary aspects of electromagnetism. He showed that electromagnetic fields travel through space, in the form of waves, at a constant velocity of 3.0 × 108 m/s. He also proposed that light was a form of electromagnetic radiation. Heinrich Hertz (1857 - 1894) A German physicist who was the first to broadcast and receive radio waves. Between 1885 and 1889, he produced electromagnetic waves in the laboratory and measured their wavelength and velocity. He showed that the nature of their reflection and refraction was the same as those of light, confirming that light waves are electromagnetic radiation obeying the Maxwell equations. 10 Maxwell’s Equations and EM Theory The study of Maxwell’s equations, devised in 1863, to represent the relationships between electric and magnetic fields in the presence of electric charges and currents, whether steady or rapidly fluctuating, in a vacuum or in matter. The equations represent one of the most elegant and concise ways to describe the fundamentals of electricity and magnetism. They pull together earlier results known from the work of Gauss, Faraday, Ampère, Biot, Savart and others, in a consistent and unified manner. Remarkably, Maxwell’s equations are perfectly consistent with the transformations of special relativity as well as the quantum field theory of electrodynamics 11 About Maxwell’s Equations Boltzmann: Was it a God who wrote these lines… Pierce: To anyone who is motivated by anything beyond the most narrowly practical, it is worthwhile to understand Maxwell’s equations simply for the good of his soul… Crease: When we discover that two or more heterogeneous empirical laws of nature can be unified under one principle that comprises them both, the discovery does give rise to a noticeable pleasure...even an admiration that does not cease when we have become fairly familiar with its object. This delight is more than having our expectations fulfilled or surprised, more than about the domination and control of nature, more than a biological product. The pleasure is a feature of the exercise of the human intellect. 12 Maxwell’s Equations in Linear, Isotropic and Stationary Media Constitutive Relations Dr, t εr Er, t Br, t μr Hr, t Jr, t σr Er, t Curl Equations Er, t μ Hr, t t Hr, t ε Er, t σ Er, t t The fields can be expressed in terms of E and H and described by the curl equations Divergence Equations ε Er, t ρr, t σ Er, t ρr, t 0 t μ Hr, t 0 The electric charge can be obtained as an left-over after E and H are solved. 13 Wave Equation in Free Space 2E E E E B μ o H μ o ε o 2 t t t 2 1 2E E 2 2 0 c t 2 c Similarly, 1 μ oεo 1 2H H 2 0 2 c t 2 299,792,458 m/s The sun is about 1.5x1011 m from the earth. How long does it take light to get here? 1.5 1011 m t 500 s 8.8 min 3 108 m/s Maxwell’s Predictions: 1. Electromagnetic fields may exist in the form of transverse waves 2. Light is electromagnetic in nature 14 Historical Accuracy for speed of light Date 1600 1676 1729 1849 1879 1950 1958 1972 1974 1976 1983 Experimentor Country Galileo Roemer Bradley Fizeau Michelson Michelson Essen Froome Evenson et al. Blaney et. al Woods et al. Italy France England France United States United States England England United States England England International Experimental Method Speed (10^8m/s) Lanterns and shutters "Fast" Moons of Jupiter 2.14 Aberration of Light 3.08 Cog Wheel 3.14 Rotating mirror 2.9991 Rotating mirror 2.99798 Microwave cavity 2.997925 Interferometer 2.997925 Laser Method 2.99792457 Laser Method 2.99792459 Laser Method 2.99792459 2.99792458 Relative Uncertainty (m/s) Error ? ? ? ? 75000.0 22000.0 1000.0 100.0 1.1 0.6 0.2 0.0 28% 2.70% 4.70% 400 in 10^6 18 in 10^6 0.1 in 10^6 0.1 in 10^6 2 in 10^9 3 in 10^9 3 in 10^9 Exact Maxwell: The velocity of transverse undulations in our hypothetical medium, calculated from the electromagnetic experiments of MM Kohlrausch and Weber, agrees so exactly with the velocity of light calculated from the optical experiments of M Fizeau, that we can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the source of the electric and magnetic phenomena. 15 A Special Solution: Sinusoidal Traveling Plane Wave E x z, t E cosωt kz H y z, t H cosωt kz General solution for forward wave Dispersion Relation Phase Velocity vp k ω μ oεo Group Velocity ω c k vg Characteristic Impedance Zo μo E 377 H εo Field Expressions z E x z, t E cos ω t c H y z, t E z cos ω t Zo c ω c k Wavelength vs Frequency λf c 16 Relationship between E and H Fields Er, t μ Hr, t ε Hr, t t E x z, t μ H y z, t z t Er, t t H y z, t ε E x z, t z t x z z H y z, t H t H t c c Ex Ex Hy z z E and H are perpendicular to each other and also to the direction of wave propagation. The three orientations obey the right hand rule. 1 E H c 1 H E c E Z H Zo 0 377 0 y Hy E and H are proportional to each other and the ratio is the characteristic wave impedance of the media 17 Power Flow and Stored EM Energies z Ex z, t E cos t c Stored Electric Energy Density 2 z 1 1 2 we Ex z, t E cos2 t 2 2 c H y z, t z E cos t Zo c Stored Magnetic Energy Density 2 z 1 1 2 wm H y z, t H cos 2 t 2 2 c Power Flow Density z S z Ex H y E H cos2 t c Relationship between Stored Energies and Power Flow Sz v p we wo Power Flow = Velocity Total Stored EM Energy 18 Time Average Power Flow and Stored Energies Stored Electric Energy Density 2 z 1 1 2 we Ex z, t E cos2 t 2 2 c Stored Magnetic Energy Density 2 z 1 1 2 wm H y z, t H cos2 t 2 2 c T 2 1 1 2 wm H y z, t dt H 2 4 0 T 2 1 1 2 we Ex z, t dt E 2 4 0 Power Flow Density z S z Ex H y E H cos t c 2 T 1 S z Ex H y dt E H 2 0 Relationship between Stored Energies and Power Flow Sz vp w w we wo Power Flow = Velocity Total Stored EM Energy 19 Momentum and Pressure z E x z, t E cos ω t c H y z, t Maxwell’s Stress Tensor Momentum Density pz Sz 1 1 z 2 E H E H cos ω t x y c2 c2 c2 c pz ε o 2 2 z E cos ω t c c T ε 1 p z 2 E x H y dt o E 2 c 0 2c pz w c E z cos ω t Zo c ε o E 2x μ o H 2y 1 T 0 2 0 μ0 Ex Zo Hy εo 0 ε o E 2x μ o H 2y 0 ε o E 2x μ o H 2y 0 0 z T ε o E 2 cos 2 ω t zˆ zˆ c T 1 T ε o zˆ zˆ E 2x dt ε o E 2 zˆ zˆ 2 0 T w zˆ z 20 Application of Radiation Pressure: Optical Tweezers A low power, continuous wave laser that is focused through a high N.A. objective can trap particles of diameter 10 m. Can move the trapped particle by moving the laser or stage, hence the laser acts as a “tweezer” by picking up and moving an individual particle. 21 Working Principle of Optical Tweezers Two regimes of operation – Rayleigh regime (diameter of particle << ) – Mie regime (diameter of particle >> ) Two main forces – Scattering force caused by reflection of incident beam – Gradient force caused by the deflection (transmission gradient force) of incident beam Gradient force dominates scattering force 22 Particle Nature of Light Quantum theory describes light (or more generally EM fields) as a particle called a photon According to quantum theory, a photon has an energy and momentum given by E f c λ p f λ c h= 6.6x10-34 [Js] Planck’s constant The energy and momentum of the light are proportional to the frequency (inversely proportional to the wavelength) ! The higher the frequency (lower wavelength) the higher the energy and momentum of the photon. One photon of visible light contains about 10^-19 Joules! Quantum theory describes experiments to astonishing precision, whereas the classical wave description cannot. 23 Illustration of Particle Characteristics of Light Photographs taken in dimmer light look grainier. Very very dim Bright Very dim Very bright Dim Very very bright When detecting very weak light, we find that it is made up of particles. is the "photon flux," i.e., number of photons/sec in a beam of power P = P / ħf 24 Wave-Particle Duality of EM Fields Wave E f p c λ Particle f λ c Characterized by: Characterized by: Amplitude (A) Frequency (f) Wavelength () Amplitude (A) Energy (E) Momentum (p) Energy of wave: A2 Probability of particle: A2 25 The Electromagnetic Spectrum Shortest wavelengths (Most energetic photons) More particle-like E =hf = hc/ h = 6.6x10-34 [J*sec] (Planck’s constant) More wave-like Longest wavelengths (Least energetic photons) 26 Notes on EM Spectrum Radio Waves Used in radio and television communication systems Microwaves Wavelengths from about 1 mm to 30 cm Well suited for radar systems Microwave ovens are an application Infrared waves Incorrectly called “heat waves” Produced by hot objects and molecules Readily absorbed by most materials Visible light Part of the spectrum detected by the human eye Most sensitive at about 560 nm (yellow-green) Gamma rays Emitted by radioactive nuclei Highly penetrating and cause serious damage when absorbed by living tissue 27 Human Visual Response of Colors Red + Blue = Magenta Red + Green = Yellow Blue + Green = Cyan The way electronic display (e.g., computer screen, TV, etc.) makes colors. Magenta + Green = White Cyan + Red = White Yellow + Blue = White 28 Gamut of Display Technologies Conventional Technologies Laser Technologies CIE 1976 UCS Chromaticity Diagram. Source: Joe Kane Productions, California 29 Laws of Electrons and Photons Laws of Electron Laws of Photon – Primarily particle like – Primarily wave like – Localization and – Globalization and concentration – Properties easy to change and control – Rich in information processing power – Lack in information transmission distribution – Properties hard to change and control – Rich in information transmission power – Lack in information processing It is much more difficult to confine and control light than current! 30 Light Propagation in Free Space 31 Classical Optical Waveguides Key Idea: To Use Objective Lens to Re-Focus the Diffracted Light so as to maintain guidance Device Feature Size >> Optical Wavelength 32 Focus of Parallel Light by Lens: Geometric Optics A lens followed by propagation by one focal length: x out 1 f 1 0 x in 0 f x in 0 θ 0 1 1 f 1 0 1 f 1 0 x in out f Parallel rays Focused rays f f For all rays xout = 0! Assume all input rays have qin = 0 At the focal plane, all rays converge to the z axis (xout = 0) independent of input position. Parallel rays at a different angle focus at a different xout. 33 The limitation of Geometric Optics The focused EM waves or beams will diffract as they propagate in free space and therefore can only focus with a minimum width (i.e., the diffraction limit) which is NOT zero as predicted by the ray theory under the geometrical optics approximation. The expression for a beam electric field is given by: E ( x, y, z ) exp ikz i ( z ) w( z ) x2 y 2 x2 y 2 exp 2 i w ( z ) R ( z ) w(z): the spot size vs. distance from the waist R(z) : the beam radius of curvature, and Ψ(z) : a phase shift. This expression is the solution to the wave equation when we require that the beam be well localized at some point (i.e., its waist). 34 Focusing a Gaussian beam winput f wfocus f A lens will focus a collimated Gaussian beam to a new spot size: wfocus f / winput which is limited by wavelength, focus length and the spot size of the input beam 35 Gaussian beam spot size, radius, and phase The expressions for the spot size, radius of curvature, and phase shift: w( z ) w0 1 z / zR Rz z z / z 2 R where zR is the Rayleigh Range (the distance over which the beam remains about the same diameter), and given by: 2 zR w0 / 2 ( z ) arctan( z / z R ) 36 Light Diffraction and Spatial resolution – If S1, S2 are too close together the Airy patterns will overlap and become 4 0 2 -2 0 -4 -2 -6 -4 -8 S2 37 -6-10 S1 0.5 6 2 8 4 10 6 g 8 10 g indistinguishable 0.5 1.0 Suppose two point sources or objects are far away (e.g. two stars) Imaged with some optical system Two Airy patterns 1.0 Maximum Resolution due to Diffraction Limit Rayleigh criterion – Resolving two equal intensity airy spots – Separation ~ 0.61 l0 / NA (circular aperture) – Practical limitations ~ 0.5um @ visible 38 Light Propagation in Waveguides 39 Conventional Dielectric Optical Waveguides John Tyndall Jean-Daniel Colladon 40 Glass Optical Fibers Kao, K.C. and Hockham, G.A., "Dielectric-fibre Surface Waveguides for Optical Frequencies", Proc. I.E.E. Vol. 113, No. 7, July 1966, pp. 1151-1158. 41 Integrated Optics S. E. Miller, “Integrated optics: an introduction,” Bell Syst. Tech. J., 48, 2059-2069, 1969. E. A. J. Marcatili, “Bends in optical dielectric guides,” Bell Syst. Tech. J., 48, 2103-2132, 1969. P. K. Tien, Light waves in thin films and integrated optics, Appl. Opt. 10, 1971. ``Research in integrated optics has two goals: One is to apply thin-film technology to the formation of optical devices and circuits. The other is the integration of a large number of optical devices on a small substrate, so forming an optical circuit reminiscent of the integrated circuit in microelectronics‘’, P. K. Tien, 1978. ``Integrated Optics has a long history. Yet, practical applications of integrated optics are still only few. Optical components in current use are large compared with a wavelength. This puts a fundamental limit on the density of the integrated components. By using structures with a large index contrast one may, at best, reduce the structure size to the order of one wavelength. In this limit, the structures resemble microwave components that are of the order of a single wavelength in size.’’ H.A.Haus,2002 42 Waveguide Models Assumptions – CW steady state – Longitudinally invariant structures – Arbitrary transverse geometries and index profiles Concepts – Propagation constants (modal indices) – Field patterns (modal fields) Forward Propagating Modes z y Backward Propagating Modes General Solutions for Waveguides A general solution to Maxwell’s equations under waveguide assumption ErT , z E rT e jβ z HrT , z H rT e Modal Field Patterns jβ z rT x xˆ y y The transverse position vector Modal Propagation Constants (Eigen Values) (Eigen Functions) The transverse fields E rT E t E z The longitudinal fields H rT H t H z +: Forward Traveling Wave _: Backward Traveling Wave 1D Waveguide Structures 0 x y TEM Fields Hx TEM Fields Maxwell’s Equations are reduced to two identical sub-sets as 1 Ey jω z Hz 0 1 Ey Hx j z Ex 1 Hy j z Ez 0 Ex E y Et H x H y Ht Hy Et dH t j dz 1 Ht Et H t 1 Ex j z 1 dEt j dz 45 Governing Equations and Modal Solutions Et 1 dH t j dz d 2 Et 2 Et 0 2 dz Et z E e j z E e j z Forward Propagating Wave Ht 1 dEt j dz d 2 Ht 2 H t 0 2 dz Ht z H e j z H e j z Backward Propagating Wave Propagation Constant TEM Modes 46 2D Waveguide Structures z x z x n n2 y Parallel-Plate y n1 n2 Dielectric Slab The y-dimension variation is much slower than that of x so that only onedimensional structure needs to be considered over the waveguide cross section 0 TM Fields TE Fields y 1 Hx Ey j z 1 Hz Ey j x 1 Ey j H H z x x z Maxwell’s Equations are reduced to two decoupled equations such that Ex 1 Hy j z Ez 1 Hy j x Hy 1 E Ex z j x z 47 Guided Waves in Parallel Metal Plate Structures x +a –a R+ A A' B' R– z B The plane wave is reflected at the upper and lower interfaces and propagates along a zig-zeg path (e.g., A'ABB‘). Phase-shifts due to transverse round-trip Conditions for Guidance 1. Total reflection at the upper and lower interfaces 2. Round-trip transverse phase-shift leads to constructive interference The transverse resonance condition R R 1 k x (4a) 2m R R exp jkx 4a 1 Phase-shifts due to reflection m=0,1,2,… 48 TE Modes on Parallel Plate Waveguide R R 1 R R 1 kx π m 1 2a k x2 2 n 2 k 2 2 k x (4a) 2m m 1 2 2 n k 2 a 2 m=0,1,2,… The transverse wave vector kx and the propagation constant β becomes discrete! For a given k, only certain discrete transverse wave number and therefore propagation constants are allowed! β nk For a given k2a, there β0 β1 are only limited number of real β that exist k π/2a π/a 49 Field Patterns of the TE Modes in Parallel Plate Waveguide z x –a +a m 1 E y Eo sin 2a x exp j m z Standing wave along x Traveling wave along z m 1 Hz j cos x exp j m z 2a Eo k x Only specific standing-wave field patterns are permitted due to the transverse resonance condition Due to the constructive interference between the forward and the back propagating waves along x, the guided waves are "quantized " into modes with discrete propagation constants and specific field patterns 50 Different modes correspond to plane waves at different angles q3 q2 q1 π β1 n 2 k 2 2 a m=0 2π β2 n k 2a m=1 2 Ey(x) 2 –a +a Ey(x) –a +a 2 2 Ey(x) m=2 3π β3 n k a 2 2 2 –a +a m+1=number of peaks in the field patterns The mode index m increases Decrease in propagation constants Increase in field oscillations 51 Mode Cut-Off Conditions Guided Modes Above Cut-Off: λ λ Cm If m f fC 2 fC m nk 1 m nk 1 f C then The highest order guided modes is therefore Below Cut-Off: If C m f fC m real 2nka M max 1 m 1 n k 0 2 a 2 Cut-off Condition: The cutoff m 1 kC wave number 2na m 2 2 The cutoff wavelength C 4na m 1 2 C The cutoff frequency fC m 1 c 4a n Evanescent Modes 2 then m j m jnk m 1 jnk C 2 fC m 1 imaginary f 52 Field Profiles of Evanescent Modes 4 3 n2 k 2 2a Ey(x) 2 m=3 –a +a z x Ey(x) m=4 5 4 n2 k 2 2a 2 –a +a z x m+1=number of nulls in the field patterns The mode index m increases Increase in field oscillations Increase in decay constants 53 Mode spectrum for parallel plate waveguide α – nk –β0 – β1– β2 – α3 – α4 Backward Guide Modes – α5 Backward Evanescent Modes –α6 α6 α5 α4 α3 Forward Evanescent Modes Forward Guide Modes β2 β1 β0 nk β Guided Modes + Evanescent Modes = Complete Orthogonal Modal Set 54 How Small A Metallic Waveguide Can Be? Fundamental Guided Mode If λ a 4n If λ a 4n then 1 nk 1 4na 0 jnk 1 4na 2 then 2 real imaginary The total width of the waveguide must be greater than half the wavelength in the media filling the waveguide to support the lowestorder guided mode. For waveguide of width smaller than λ/4n, no propagating mode exists. 55 Guided Waves in Dielectric Slab Structures Bound Rays→Guided Modes Conditions for Guidance Refracting Rays→Leaky Modes Conditions for Leakage n core n cladding n core > n cladding and q qc Limited guided modes with discrete mode indices and specific standing-wave field patterns are allowed. Mode Equation R 2exp jk x 4d 1 Transverse Resonance Conditions or q qc For anti-guides or guides below cut-off, leaky modes with complex propagation constants exist (as an approximate representation of the radiation modes). 56 TE Modes on Slab Waveguide R R exp j 2TE R R 1 n22 2 sin q1 2 n1 1 4 tan cos q1 2 TE n 22 2 sin θ1 2 n1 2 tan 1 cosθ1 n1k cos q1 (4a) 2m m=0,1,2,… kx q1 n k 1 β Only certain discrete angles are allowed! The dispersion Relations kx n1 k cos q1 k x2 2 n12 k 2 k x2 x2 n12 n22 k 2 x2 2 n22 k 2 x n12 k 2 sin 2 q1 n22 k 2 tan k x a x Even Modes kx 1 x 4 tan 4k x a 2m kx k tan k a x Odd Modes x x 57 Solutions for Propagation Constants x a kx a tan kx a Even Modes n 2 1 x a kx a cot kx a k x2 x2 n12 n22 k 2 αxa Odd Modes n 22 ka kxa The transverse wave vector kx and the propagation constant β becomes discrete! For a given kd, there are only limited number of real β that exist 58 Mode Indices for Slab Waveguide n2 n q 1 q n2 1 Ray Cycle 3.60 90º 88º 87º 86º 85º Effective Refractive Index TEo, TMo TE 1, TM1 3.59 TE 2, TM2 TE 3, TM3 3.58 84º 83º TE4, TM4 3.57 82º 3.56 81º TE 5, TM5 3.55 80.4º 0 1 2 d/ 3 4 5 59 Mode Fields for Slab Waveguides The different guided modes correspond to the plane waves at different angles q3 q2 q1 TE0 TE TE1 0 d TE 1 TE2 TE TE3 TE 2 3 m+1=number of peaks in the field patterns The mode index m increases Decrease in propagation constants Increase in field oscillations 60 Cut-off Condition For Slab Waveguides αx 0 x a kx a tan kx a k x2 x2 n12 n22 k 2 tan kx a 0 k x2 n12 n22 k C2 k x a m VC m Below Cut-Off: qC q cot kx a 0 m 1 kx a 2 Define V ka n12 n22 q>qC x a kx a cot kx a VC m 1 2 The total internal reflection condition is no longer satisfied n2 qC cos n1 1 Radiation Modes 61 Waveguide Parameters Single Mode Condition 2d 1 λ 2 2n For symmetric step-index slab Field Confinement Factor Eˆ Hˆ ẑda ˆ H ˆ ẑda E GuidingRegion n1 n 2 n n n1 n 2 2 Mode Attenuation Factor Entire Region Total Loss=Absorption + Leakage 62 Single Mode Condition For single-mode dielectric waveguide, the full width of waveguide depends on index contrast and average index of the waveguide. 63 Bending Radius Vs Index Contrast Slab Dielectric Waveguide TE Mode TM Mode The minimum bending radius is defined for bending loss of 0.5dB/cm and waveguide width adjusted to meet single-mode condition at λ=1.550μm 64 Field Confinement Vs Index Contrast Symmetric Slab Dielectric Waveguide TE Mode TM Mode The calculations are performed at the minimum bending radius for the single-mode slab waveguide at λ=1.550μm 65 Polarization Dependence Vs Index Contrast 66 Confinement Factors ˆ H ˆ ẑda E ˆ H ˆ ẑda E GuidingRegion Entire Region Symmetrical Step-Index Slab Waveguide e TE 2k x cos 2 k x w 2 1 k w sin k w x x x 1 o TE 2k x sin k x w 2 1 k w sin k w x x x 1 2 What if the slab is not symmetric? e TM n 2k cos k w 2 x 1 1 x n2 x k x w sin k x w 1 1 o TM n 2 2k sin 2 k w 2 x 1 1 x n2 x k x w sin k x w 2 2 Confinement Factors for Guided Modes in Symmetric Step-Index Slab Note that if d/λ→0, thenΓ →0 for the fundamental mode. Therefore, confinement of light by a waveguide of sub-wavelength dimension is difficult S. -L. Chuang, Phys. Optoeletron. Dev., John Wiley, pp. 253-257, 1995 Advanced Waveguides 69 Advanced Waveguides Bragg Waveguides (or Photonic Crystal Waveguides) χ Slot Waveguides Surface Plasma Polariton (SPP) Waveguides Features of the Advanced Optical Waveguides: 1) Confining and guiding light in low-index media 2) Confinement and guidance of light in sub-wavelength scale 70 Coupled Slab Waveguide TE Mode TE Mode n 1 Normalized Intensity nc=1.44, nh=1.99 Normalized Intensity 1 0.8 0.6 0.4 0.2 0 -2 -1 0 X ( m) 1 0.8 0.6 0.4 0.2 0 -2 2 -1 TM Mode hy 0 X ( m) 1 2 1 2 TM Mode W 1 Normalized Intensity X Normalized Intensity Y 1 0.8 0.6 0.4 0.2 0 -2 -1 0 X ( m) 1 W=1 μm 2 0.8 0.6 0.4 0.2 0 -2 -1 0 X ( m) W=50nm 71 Guiding Mechanism for Slot Waveguides nh2 E xh 0 nl2 E xl 0 E xc e Exl 0 nh2 2 1 h Ex 0 nl E xl E xh g c x d 2 d 2 o e g l x x ws ws ws ws 1 g l ws 1 g l 72 Slot Waveguides V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, Guiding and confining light in void nanostructure, Opt. Lett., vol.29, 1209-1211, 2004 73 Confinement Factor vs. Slot Layer Thickness and Index 0.5 tl=80nm tl=60nm tl=40nm tl=20nm tl=10nm tl=5nm Total Confinement in Nano-layer Single Slot tl increases 0.4 0.3 0.2 0.1 0.0 1.0 1.5 2.0 2.5 3.0 3.5 Refractive Index of the Slot Layer 74 Bending Loss and Confinement Factor For Different Slot Indices I SiO2 n w1 w II III Silicon w1 IV V n n SiO2 Silicon SiO2 y z x 10 10 10 0 0.4 Air SiO2 -5 -10 -15 0 20 40 60 Radius (um) 80 100 Confinement Loss (dB/um) 10 0.35 Air SiO2 0.3 0.25 0.2 0 20 40 60 80 100 Radius (um) 75 Bending Characteristics of 2D Slot WG 10 4 10 3 10 2 10 1 10 0 0.46 0.44 10 -1 10 -2 Horizontal Slots Vertical Slots 0.42 0.40 0.38 0.36 2 4 6 Bending Radius (m) 8 y (m) 5 x (m) Ex component y (m) 10 Ey component Total Confinement in Slot-layers Bending Loss (dB/cm) 5 times better 10 x (m) Bending radius: R=3m 76 Surface Plasma Polariton 77 Why metals for optical waveguides? Higher Frequencies at smaller dimensions 78 Dielectric Constants of Bulk Metal -20 Model Drachev Model Vial Data J&C Data Palik -40 Drude Model Real Part of Permittivity p2 ig : plasma frequency p g : damping coefficient Gold: g 2 18.36 1012 Aluminium: p -120 900 1000 1100 1200 1300 1400 Wavelength/nm 1500 1600 1700 16 14 100nm Imaginary Part of Permittivity h p 14eV -100 -160 800 9.0685 p 2 2.1556 10 2 c -80 -140 15 p -60 12 10 8 6 4 Model Drachev Model Vial Data J&C Data Palik 2 Vial et al. Phys. Rev. B 71, 085416 (2005) 0 800 900 1000 1100 1200 1300 1400 Wavelength/nm 1500 1600 79 1700 Propagation in bulk metal E x, y, z E exp i t z exp i z Plane wave along z-axis: H x, y, z H exp i t z exp i z r ii , i 0 The propagation length in optical wavelengths 1 2i Propagation in bulk gold 26 24 Power Propagation Distance (nm) B 850nm 22 EM Wave Can Not Propagate in Bulk Metal at Optical Wavelengths 20 18 1310nm 16 14 1550nm 12 10 800 900 1000 1100 1200 1300 1400 Wavelength (nm) 1500 1600 1700 80 Surface Electromagnetic Waves Zenneck Modes Radio frequency surface electromagnetic waves that occur at the surface of absorbent medium Brewster Modes Damping brings ‘Brewster case’ rays into two exponentially decaying away from the interface waves Fano Modes The only surface normal modes that exist at the surface in absence of damping 81 Concepts of Surface Polariton A polariton is an electromagnetic wave that is linearly coupled to an electric or magnetic dipole active elementary excitation in a condensed medium. A surface polariton is a polariton whose associated electromagnetic field is localized at the surface of the medium. 82 Guided Waves by Interface between Media ε o ε̂ 2 , μ o x z ε o ε̂ 1 , μ o TM Fields TE Fields Hx 1 Ey jω μo z Ex 1 Ey jω o x Ez Hz Ey 1 H H z x jω x z Hy 1 Hy jω z 1 Hy jω x 1 jω o E x z z E x TE Guided Mode Solution d 2 Ey dx 2 ˆ x N 2 k 2 E y 0 Region 1 Evanescent Wave Ey Eo exp 1 x jNkz 12 N 2 ˆ1 k 2 N: Mode Effective Index Region 2 Interface ˆ dE y Eˆ y , continuous dx 1 2 Evanescent Wave Ey Eo exp 2 x jNkz 22 N 2 ˆ2 k 2 N 2 ˆ1 N 2 ˆ2 Impossible! No TE Modes Can Be Supported by the Interface TM Guided-Mode Solution d 1 dHˆ y ˆ ˆ x N 2 k 2 Hˆ y 0 N: Mode Effective Index dx ˆ dx Region 1 Interface Evanescent Wave H y Ho exp 1x jNkz 1 dHˆ y ˆ Hy, continuous ˆ dx 12 N 2 ˆ1 k 2 1 0 Region 2 Decay along –x Evanescent Wave Ey Eo exp 2 x jNkz 1 2 ˆ 1 ˆ2 N 2 ˆ1 ˆ1 22 N 2 ˆ2 k 2 N 2 ˆ2 ˆ2 2 0 ˆ1 ˆ2 N ˆ1 ˆ2 Decay along +x 2 ˆ12 k2 ˆ1 ˆ2 2 1 2 2 ˆ22 ˆ1 ˆ2 k2 85 Conditions for Surface Plasma Polariton (SPP) Surface Wave with Field Decay Away from the Interface 1 ˆ 1 0 2 ˆ2 ˆ1 0 ˆ 2 Decay along x ˆ1 ˆ2 N 0 ˆ1 ˆ2 Guided Wave Propagating along the interface or 2 ˆ1 ˆ2 Propagate along z ε̂ 1 ε̂ 2 ˆ12 k2 0 ˆ1 ˆ2 2 1 ˆ1 ˆ2 0 ε̂ 0 2 2 ˆ22 ˆ1 ˆ2 k2 0 If ε2>0, then ε1<0, so the condition for lossless guidance with transverse confinement is that the dielectric function of the metal must be sufficiently negative! 86 Negative Dielectric Functions Conductors The Drude model for simple metals p2 2 0 if p2 4 ne2 me the plasma frequency p Insulators The Lorentz model for insulators 0 02 0 0 0 if 02 2 0 the natural frequency 2 0 0 02 87 Guided Modes Effective Mode Index ε̂1ε̂ 2 N R ε̂1 ε̂ 2 ε̂ 22 α2 k ε̂ 1 ε̂ 2 ε̂ 12 α1 k ε̂ 1 ε̂ 2 Mode Field Distribution Hy exp α1 x x 0 H y H o exp jNkz exp α 2 x x 0 ε2 z x ε1 x Hy ε1 ε2 z z 88 Guided Mode on Dielectric-Metal Interface: Ideal Metal ε̂ d ε do constant Dielectric: d P2 ˆm mo 2 x z If the frequency is smaller than the plasma frequency, the dielectric constant of metal becomes negative Metal: m = m' + m" ˆm 0 Further, if ˆmˆd Nk k k ˆm ˆd P mo mo m k mo mo do 2 P 2 P 2 2 2 0 do mo P2 2 mo do 2 P 2 0 do2 d k 0 2 2 mo P do 89 Drude Models and Experimental Data for Ag Drude model: p2 p2 m 1 2 , m 3 50 " Modified Drude model: 0 -50 -100 Measured data: ' " Drude model: ' " Modified Drude model: ' p2 m mo 2 p2 m 3 ' -150 200 400 600 800 1000 1200 Wavelength (nm) 1400 1600 1800 Contribution of bound electrons Ag: mo 3.4 90 Existence of the Bound SPP modes: m < − d Drude model 50 ε m 1 " -d -100 Measured data: ' " ' Drude model: ' " bound SP mode: m < -d Modified Drude model: ' 400 600 800 1000 1200 , ε m 1400 1600 p2 m mo 2 p2 m 3 Contribution of bound electrons -150 200 ω 2 ω 2p τω 3 Modified Drude model 0 -50 ω 2p 1800 Ag: ε mo 3.4 Wavelength (nm) 91 Surface Plasmon Polariton Dispersion Relation Radiative modes 'm > 0) Volume Plasma real β imaginary α p c sp p do Quasi-bound modes d < 'm < 0) imaginary β imaginary α 1 do mo 1 Surface Plasma 1/ 2 ˆmˆd k ˆ ˆ m d Bound modes ('m < d) Re β real β real α 92 Critical Points for SPP Dispersion Relation Frequency ω0 ω ω sp ω ωp ω Metal Ɛ SPP Mode Index ε̂ m ω N 1 ε̂ m ω sp ε̂ d ε̂ m ω p 0 ε̂ m 1 N N Decay in Metal α m ωP c ω2ε d αd 2 ωP c αm αd ε do ε mo 1 ε mo 12 α m jk ε mo 1 ε do ε do ε mo 1 N ωsp ε do ε mo ε mo ε do Decay in Dielectric ε 2mo α m jk ε mo ε do 2 ε do α d jk ε mo 1 ε do ε do2 α d jk ε mo ε do ωP ε mo ε do 93 SPP Mode Dispersion Relation (Ag/SO2) β<ω/c β=ω/c β>ω/c ωs ω0 1 ε mo ne 2 ωp moεo β A.M.Gadou, et.al., Epypt.J.Sol.,Vol.23,No.1,pp.13-26,2000 94 SPP Mode Fields exp d x H y H o exp jNkz exp m x x0 x0 x d d ,m 1 m D d m 1 ˆ exp d x x 0 d Ex o NH o exp jNkz 1 exp x x 0 d ˆ d d x0 ˆ exp d x 1 d Ez H o exp jNkz j o m exp x x 0 m ˆ m x 1 Ex x 95 Propagation Losses of SP Modes ˆmˆd k ˆm ˆd R j I Let 12 ˆ d R k 2 ˆ ˆ ˆ 2 d I R ˆd I k ˆ ˆ d R ˆm ˆR jˆI ˆ 2 ˆ 2ˆ 2 ˆ 4 d I e e 2 12 12 2 ˆI2 ˆd ˆI 12 2 ˆ 2 ˆ 2ˆ 2 ˆ 4 e d I e ˆe2 ˆR2 ˆI2 ˆd ˆR Surface Plasmon Propagation Length SP 1 ˆI1 2 I 96 Case A: Poor Conductor If ˆR ˆI ˆR ˆd 12 ˆRˆd R k ˆ ˆ R d 1 ˆ I k I2 2 ˆR ˆd ˆR ˆ ˆ R d 32 1 ˆI R3 2 k 2ˆR2 R k ˆd Case B: Good Conductor If ˆR ˆI ˆR 0 or ˆI ˆI2 ˆd2 R k ˆd ˆI 2 ˆd2 ˆI2 12 R k ˆd ˆd3ˆI I k 2 2 2 2 ˆ ˆ ˆ ˆ ˆ 2 d I I I d 12 97 SPP Dispersion Relations F. Yang, et.al., Physical Review B, Vol.44, No.11, pp.5855-5872,1991 98 Transverse and Longitudinal SPP Scales Transverse Length Longitudinal Length Au/air interface Au/air interface 1.8 260 1.6 240 Propagation Length (um) Transverse Length (um) 1.4 1.2 1 0.8 0.6 0.4 0.2 220 200 180 160 140 0.8 1 1.2 1.4 Wavelength (um) 1.6 1.8 2 120 0.8 1 1.2 1.4 Wavelength (um) 1.6 1.8 2 The transverse confinement length is in sub-wavelength, whereas the longitudinal propagation length is more than 100 times wavelength. 99 Surface Plasmon Polariton Length Scales Transverse Length Scale Longitudinal Length Scale Ideal Scenario: SP Waveguide Design Criteria δ m , δ d λ δ SP Small transverse δt Large longitudinal δl Barnes, et.al., Nature, Vol. 424, pp. 824-840, 2003 100 Propagation Distance of SPP at Different Wavelengths 100 m High loss in region of small SP 101 Transverse Skin Depths of SPP Confinement in Dielectric Reduces for Longer Wavelength Confinement in Metal Does not Change for Longer Wavelength 102 Intensity Profiles of SPPs x (nm) 103 Average Power Flow Density of SPPs 1 1 S E H Ex xˆ Ez zˆ H y yˆ 2 2 Ex Hy Ez 1 Hy j x 2 1 1 1 j S E H H y zˆ H y H y xˆ 2 2 2 x S 1 1 2 zˆ j xˆ E H H o exp 2 z 2 2 104 Gain-Assisted Lossless Propagation Gain Medium: 1 k ˆ1 jˆ1ˆ2 jˆ2 ˆ1 ˆ2 j ˆ1 ˆ2 n k ˆn jˆn ˆ1 ˆ2 j ˆ1 ˆ2 x z 2 Metal:ε2 12 n k k ˆ1 ˆ2 ˆ1 ˆ2 2 2 2 2 ˆ ˆ ˆ ˆ 2 2 2 2 1 2 2 1 ˆ2 jˆ2 ˆ1 ˆ1 ˆ ˆ2 ˆ1 ˆ2 ˆ1 ˆ2 ˆ1 ˆ2 2 ˆ1 ˆ2 2 ˆ1 ˆ2 ˆ ˆ n j n 1 j ˆ ˆ 2 1 2 Gain Required Lossless Transmission 2 ˆ2 ˆ1 2 2 ˆ2 ˆ1 ˆ1 0 ˆ2 ˆ ˆ 1 1 M.P.Neshad,et.al.,Opt.Express,Vol.12,No.17,pp.4072-4079,2004 ˆ2 2 2 ˆ2 105 Various SPP Propagation Regimes ε̂ 2 Gain Required for Lossless SPP Propagation 32 ε̂1 γk ε̂ 2 2 2 ε̂2 ε̂1 M.P.Neshad,et.al.,Opt.Express,Vol.12,No.17,pp.4072-4079,2004 106 Longitudinal and Transverse Characteristics Lossless Propagation δ SP β α 2 Loss of Transverse Confinement? M.P.Neshad,et.al.,Opt.Express,Vol.12,No.17,pp.4072-4079,2004 107 Summary Conventional metallic and dielectric waveguide structures can only confine and guide light at transverse dimension greater than quarter of the wavelength due to the wave nature and can achieve low-loss propagation for distance of many wavelength in length Slot waveguides can confine and guide light at sub-wavelength nano dimension and through distance of many wavelength in length, but limited to only the TE mode SPP waveguide can confine light within sub-wavelength nano dimension near the metal-dielectric interface by plasmonic resonance, but limited in propagation distance due to attenuation of the metal Long-reach sub-wavelength optical waveguides are important for miniaturized, high-density photonic ICs, but yet to be developed for practical applications 108