Report

The Optical Fiber and Light Wave Propagation Xavier Fernando Ryerson Comm. Lab The Optical Fiber • Fiber optic cable functions as a ”light guide,” guiding the light from one end to the other end. • Fiber categories based on propagation: – Single Mode Fiber (SMF) – Multimode Fiber (MMF) • Categories based on refractive index profile – Step Index Fiber (SIF) – Graded Index Fiber (GIF) Step Index Fiber n1 n2 n1>n2 • Uniform ref. index of n1 (1.44 < n1 < 1.46) within the core and a lower ref. index n2 in the cladding. • The core and cladding radii are a and b. Typically 2a/2b are 8/125, 50/125, 62.5/125, 85/125, or 100/140 µm. • SIF is generally made by doping high-purity fused silica glass (SiO2) with different concentrations of materials like titanium, germanium, or boron. Different Light Wave Theories • Different theories explain light behaviour • We will first use ray theory to understand light propagation in multimode fibres • Then use electromagnetic wave theory to understand propagation in single mode fibres • Quantum theory is useful to learn photo detection and emission phenomena Refraction and Reflection When Φ2 = 90, Φ1 = Φc is the Critical Angle Snell’s Law: n1 Sin Φ1 = n2 Sin Φ2 Φc=Sin-1(n2/n1 ) Step Index Multimode Fiber Fractional n12 n22 n2 1 refractive-index 2 2n1 n1 profile Ray description of different fibers Single Mode Step Index Fiber r Buffer tube: d = 1mm Protective polymerinc coating Cladding: d = 125 - 150 m n Core: d = 8 - 10 m n1 n2 The cross of a typical with a wavelength. tight buffer Only onesection propagation modesingle-mode is allowed fiber in a given tube. = diameter) This (d is achieved by very small core diameter (8-10 µm) SMF offers highest bit rate, most widely © 1999 S.O. Kasap, Optoelectronics (Prentice Hall) used in telecom Step Index Multimode Fiber • Guided light propagation can be explained by ray optics • When the incident angle is smaller the acceptance angle, light will propagate via TIR • Large number of modes possible • Each mode travels at a different velocity Modal Dispersion • Used in short links, mostly with LED sources Graded Index Multimode Fiber • Core refractive index gradually changes towards the cladding • The light ray gradually bends and the TIR happens at different points • The rays that travel longer distance also travel faster • Offer less modal dispersion compared to Step Index MMF Refractive Index Profile of Step and Graded Index Fibers n1 n2 a n= Graded Step n1 b n2 a n= b Step and Graded Index Fibers n2 n1 3 2 1 O n (a) Multimode step index fiber. Ray paths are different so that rays arrive at different times. n2 O O' O'' 3 2 1 2 3 n1 n2 © 1999 S.O. Kasap,Optoelectronics(P rentice Hall) n (b) Graded index fiber. Ray paths are different but so are the velocities along the paths so that all the rays arrive at the same time. Total Internal Reflection in Graded Index Fiber (a) TIR (b) TIR n decreases step by step from one layer Continuous decreas e inn gives a ray path changing continuously. to next upper layer; very thin layers . (a) A ray in thinly stratifed medium bec omes refracted as it pass es from one layer to the next upper layer with lowern and eventually its angle satisfies TIR. (b) In a medium wheren decreas es continuously the path of the ray bends continuously. © 1999 S.O. Kasap, Optoelectronics (Prentice Hall) O Total Internal Reflection in Graded Index Fiber - II nc B' B 2 B 1 B' c/nb c/na B' Ray 2 A B'' nb na A Ray 1 M © 1999 S.O. Kasap,Optoelectronics(P rentice Hall) We can visualize a graded index fiber by imagining a stratified medium with the layers of refractive b indices na > nb > nc ... Consider two close rays 1 and 2 launched from O at the same time but with slightly a different launching angles. Ray 1 just suffers total internal reflection. O' Ray 2 becomes refracted at B and reflected at B'. c Skew Rays Along t he fiber 1 1, 3 3 (a) A meridional ray always crosses the fiber axis. Meridional ray Fiber axis 2 2 1 2 1 Skew ray Fiber axis 5 3 5 4 4 Ray path along the fiber 2 3 (b) A skew ray does not have to cross the fiber axis. It zigzags around the fiber axis. Ray path projected on to a plane normal to fiber axis Illustration of the difference between a meridional ray and a skew ray. Numbers represent reflections of the ray. © 1999 S.O. Kasap,Optoelectronics(Prent ice Hall) Maxwell’s Equations In a linear isotropic dielectric material with no currents and free of charges, ……..(1) (Faraday’s Law) E: Electric Field ……….(2) (Maxwell’s Faraday equation) H: Magnetic Field ……….(3) (Gauss Law) ……….(4) (Gauss Law for magnetism) Taking the curl of (1) and using and …….(5) The parameter ε is permittivity and μ is permeability. Maxwell’s Equations But from the vector identity ……(6) Using (5) and (3), Notev p 1 …….(7) Similarly taking the curl of (2), it can be shown ………(8) (7) and (8) are standard wave equations. Note the Laplacian operation is, 1 E 1 2 E 2 E E 2 r 2 2 r r r r z 2 Maxwell’s Equation • Electrical and magnetic vectors in cylindrical coordinates are give by, .…..(9) ……(10) • Substituting (9) and (10) in Maxwell’s curl equations ….(11) ….(12) ….(13) Maxwell’s Equation • Also ----------(14) ----------(15) ----------(16) • By eliminating variables, above can be rewritten such that when Ez and Hz are known, the remaining transverse components Er , Eφ, Hr , Hφ, can be determined from (17) to (20). Maxwell’s Equation …………..(17) …………..(18) …..........(19) .………… (20) Substituting (19) and (20) into (16) results in ….…(21) …….(22) Electric and Magnetic Modes Note (21) and (22) each contain either Ez or Hz only. This may imply Ez and Hz are uncoupled. However. Coupling between Ez and Hz is required by the boundary conditions. If boundary conditions do not lead to coupling between field components, mode solution will imply either Ez =0 or Hz =0. This is what happens in metallic waveguides. When Ez =0, modes are called transverse electric or TE modes When Hz =0, modes are called transverse magnetic or TM modes However, in optical fiber hybrid modes also will exist (both Ez and Hz are nonzero). These modes are designated as HE or EH modes, depending on either H or E component is larger. Wave Equations for Step Index Fibers • Using separation of variables ………..(23) • The time and z-dependent factors are given by ………..(24) • Circular symmetry requires, each field component must not change when Ø is increased by 2п. Thus …………(25) • Where υ is an integer. • Therefore, (21) becomes ….(26) Wave Equations for Step Index Fibers • Solving (26). For the fiber core region, the solution must remain finite as r0, whereas in cladding, the solution must decay to zero as r∞ • Hence, the solutions are – In the core, (r < a), Where, Jv is the Bessel function of first kind of order v – In the cladding, (r > a), Where, Kv is the modified Bessel functions of second kind Bessel Functions First Kind Modified Bessel first kind Bessel Functions Second kind Modified Bessel Second kind Propagation Constant β • From definition of modified Bessel function • Since Kv(wr) must go to zero as r∞, w>0. This implies that • A second condition can be deduced from behavior of Jv(ur). Inside core u is real for F1 to be real, thus, • Hence, permissible range of β for bound solutions is Meaning of u and w Inside the core, we can write, Outside the core, we can write, q 2 u 2 k12 2 w2 2 k22 • Both u and w describes guided wave variation in radial direction – u is known as guided wave radial direction phase constant (Jn resembles sine function) – w is known as guided wave radial direction decay constant (recall Kn resemble exponential function) V-Number (Normalized Frequency) Define the V-Number (Normalized Frequency) as, 2a 2 2 V 2 (u 2 w2 )a 2 k12 k22 a 2 n1 n2 2 Define the normalized propagation const b as, w2 a 2 ( / k ) 2 n22 b 2 V n12 n22 All but HE11 mode will cut off when b = 0. Hence, for single mode condition, V 2a ( NA) Vc 2.405 b 1 LP 01 0.8 LP 11 0.6 LP 21 0.4 LP 02 0.2 0 V 0 1 2 3 2.405 4 5 6 Normalized propagation constant b vs. V-number for a step index fiber for various LP modes. © 1999 S.O. Kasap,Optoelectronics(P rentice Hall) 1.5 1 V[d2(Vb)/dV2] 0.5 0 0 1 2 3 V - number [d2(Vb)/dV2] vs. V-number for a step index fiber (after W.A. Gambling et al., The Radio and Electronics Engineer, 51, 313, 1981) © 1999 S.O. Kasap,Optoelectronics(Prentice Hall) Field Distribution in the SMF Field of evanescent wave (exponential decay) y n2 Field of guided wave E(y) E(y,z, t) = E(y)cos( t – 0z) Light m= 0 n1 n2 The electric field pattern of the lowest mode traveling wave along the guide. This mode has m = 0 and the lowest . It is often referred to as the glazing incidence ray. It has the highest phase velocity along the guide. © 1999 S.O. Kasap,Optoelectronics(Prentice Hall) Mode-field Diameter (2W0) In a Single Mode Fiber, E(r ) E0 exp(r 2 / w02 ) At r = wo, E(Wo)=Eo/e Typically Wo > a Cladding Power Vs Normalized Frequency Modes Vc = 2.4 Power in the cladding Lower order modes have higher power in the cladding larger MFD Higher the Wavelength More the Evanescent Field y y Cladding 2 > 1 1 > c v g1 1 < cut-off E(y) Core v g2 > v g1 2 < 1 Cladding The electric field of TE 0 mode extends more into the cladding as the wavelength increases. As more of the field is carried by the cladding, the group velocity increases. © 1999 S.O. Kasap,Optoelectronics(Prentice Hall) Light Intensity (a) The electric field of the fundamental mode (b) The intensity in the fundamental mode LP 0 1 (c) The intensity (d) The intensity in LP 1 1 in LP 2 1 Core Cladding E E0 1 r The electric field distribution of the fundamental mode in the transverse plane to the fiber axis z. The light intensity is greatest at the center of the fiber. Intensity patterns in LP 0 1, LP1 1 and LP 2 1 modes. © 1999 S.O. Kasap,Optoelectronics(Prent ice Hall) Fiber Key Parameters Fiber Key Parameters Effects of Dispersion and Attenuation Dispersion for Digital Signals Fiber Digital signal Informat ion Emitt er t Photodet ect or Informat ion Input Output Input Int ensit y Output Int ensit y ² Very short light pulses 0 T t t 0 ~2² An optical fiber link for transmitting digital information and the effect of dispersion in the fiber on the output pulses. © 1999 S.O. Kasap,Optoelectronics(Prent ice Hall) Modal Dispersion Hi gh ord er mo d e Lo w o rder mo d e Light pulse Broadened light pulse Cladding Core Intensity Intensity Axial 0 Spread, t Schematic illustration of light propagation in a slab dielectric waveguide. Light pulse entering the waveguide breaks up into various modes which then propagate at different group velocities down the guide. At the end of the guide, the modes combine to constitute the output light pulse which is broader than the input light pulse. © 1999 S.O. Kasap, Optoelectronics (Prentice Hall) t Major Dispersions in Fiber • Modal Dispersion: Different modes travel at different velocities, exist only in multimode fibers • This was the major problem in first generation systems • Modal dispersion was alleviated with single mode fiber – Still the problem was not fully solved Dispersion in SMF • Material Dispersion: Since n is a function of wavelength, different wavelengths travel at slightly different velocities. This exists in both multimode and single mode fibers. • Waveguide Dispersion: Signal in the cladding travels with a different velocity than the signal in the core. This phenomenon is significant in single mode conditions. Group Velocity (Chromatic) Dispersion = Material Disp. + Waveguide Disp. Cladding Input v g ( 1 ) Emitt er v g ( 2 ) Very short light pulse Int ensit y Int ensit y Core Output Int ensit y Spectrum, ² Spread, ² 1 o 2 0 t All excitation sources are inherently non-monochromatic and emit within a spectrum, ² , of wavelengths. Waves in the guide with different free space wavelengths travel at different group velocities due to the wavelength dependence of n1. The waves arrive at the end of the fiber at different times and hence result in a broadened output pulse. © 1999 S.O. Kasap,Optoelectronics(Prentice Hall) Group Velocity Dispersion t Modifying Chromatic Dispersion GVD = Material Disp. + Waveguide dispersion • Material dispersion depends on the material properties and difficult to alter • Waveguide dispersion depends on fiber dimensions and refractive index profile. These can be altered to get: – 1300 nm optimized fiber – Dispersion Shifted Fiber (DSF) – Dispersion Flattened Fiber (DFF) Material and Waveguide Dispersions Dispersion coefficient (ps km -1 nm-1 ) 30 Dm 20 10 Dm + Dw 0 0 -10 Dw -20 -30 1.1 1.2 1.3 1.4 (m) 1.5 1.6 Material dispersion coefficient (Dm) for the core material (taken as SiO2 ), waveguide dispersion coefficient (Dw ) (a = 4.2 m) and the total or chromatic dispersion coefficient Dch (= Dm + Dw ) as a function of free space wavelength, Dispersion coefficient (ps km-1 nm-1) 20 Dm 10 SiO2-13.5%GeO2 0 a (m) Dw 4.0 3.5 3.0 –10 2.5 –20 1.2 1.3 1.4 1.5 1.6 (m) Material and waveguide dispersion coefficients in an optical fiber with a core SiO 2-13.5%GeO 2 for a = 2.5 to 4 m. Different WG Dispersion Profiles WGD is changed by adjusting fiber profile Dispersion Shifting/Flattening (Standard) (Low Dispersion throughout) (Zero Disp. At 1550 nm) Dispersion coefficient (ps km -1 nm-1) n 30 20 r Dm 10 0 2 1 Dch = Dm + Dw -10 Dw -20 T hin layer of cladding with a depressed index -30 1.1 1.2 1.3 1.4 (m) 1.5 1.6 1.7 Dispersion flattened fiber example. The material dispersion coefficient ( Dm) for the core material and waveguide dispersion coefficient ( Dw) for the doubly clad fiber result in a flattened small chromatic dispersion between 1 and 2. Specialty Fibers with Different Index Profiles 1300 nm optimized Dispersion Shifted Specialty Fibers with Different Index Profiles Dispersion Flattened Large area dispersion shifted Large area dispersion flattened Polarization Mode Dispersion • Since optical fiber has a single axis of anisotropy, differently polarized light travels at slightly different velocity • This results in Polarization Mode Dispersion • PMD is usually small, compared to GVD or Modal dispersion • May become significant if all other dispersion mechanisms are small X and Y Polarizations A Linear Polarized wave will always have two orthogonal components. These can be called x and y polarization components Each component can be individually handled if polarization sensitive components are used Polarization Mode Dispersion (PMD) Each polarization state has a different velocity PMD Birefringence • Birefringence is the decomposition of a ray of light into two rays types of (anisotropic) material • In optical fibers, birefringence can be understood by assigning two different refractive indices nx and ny to the material for different polarizations. • In optical fiber, birefringence happens due to the asymmetry in the fiber core and due to external stresses • There are Hi-Bi, Low-Bi and polarization maintaining fibers. Total Dispersion For Multi Mode Fibers: (Note for MMF ΔTGVD ~= ΔTmat For Single Mode Fibers: But Group Velocity Disp. Hence, (ΔTpol is usually negligible ) Permissible Bit Rate • As a rule of thumb the permissible total dispersion can be up to 70% of the bit period. Therefore, 0.7 • For full width pulses (Non Return toZerobits) : B TTotal 0.35 • For half width pulses (Return toZerobits) : B TTotal Disp. & Attenuation Summary Electrical signal (photocurrent) 1 0.707 Fiber Sinuso idal signal Emitt er t Optical Input f = Modulation frequency Pi = Input light power 0 Ph oto detect or Optical Output Po = Output light power t 0 1 kHz 1 MHz 1 GHz 1 MHz 1 GHz f f el Sinuso idal elect rical sign al Po / Pi 0.1 0.05 t 1 kHz fop f An optical fiber link for transmitting analog signals and the effect of disp ersion in the fiber on the bandwidth, fop. © 1999 S.O. Kasap,Optoelectronics (Prentice Hall) Fiber Optic Link is a Low Pass Filter for Analog Signals Attenuation Vs Frequency Fiber attenuation does not depend on modulation frequency Attenuation in Fiber Attenuation Coefficient P (0)dB P ( z )dB dB/km z • Silica has lowest attenuation at 1550 nm • Water molecules resonate and give high attenuation around 1400 nm in standard fibers • Attenuation happens because: – Absorption (extrinsic and intrinsic) – Scattering losses (Rayleigh, Raman and Brillouin…) – Bending losses (macro and micro bending) All Wave Fiber for DWDM Lowest attenuation occurs at 1550 nm for Silica Attenuation characteristics Bending Loss Field dist ribution Microbending Escaping wave Cladding Core Note: Higher MFD Higher Bending Loss c R Sharp bends change the local waveguide geometry that can lead to waves escaping. The zigzagging ray suddenly finds itself with an incidence angle that gives rise to either a transmitted wave, or to a greater cladding penetration; the field reaches the outside medium and some light energy is lost. Micro-bending losses Fiber Production The Fiber Cable