# Fig. 2-1: Spherical and plane wave fronts 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 m. 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 2 = 90, 1 = c is the Snells Law: n1 Sin 1 = n2 Sin 2 Critical Angle c=Sin-1(n2/n1 ) Step Index Multimode Fiber

Fractional n 12 n 22 n2 1 refractive-index 2 2 n1 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 n1 Core: d = 8 - 10 m n2 The of a typical single-mode with a tight buffer Onlycross one section propagation mode is allowed in fiber a given wavelength.

tube. = diameter) This (d is achieved by very small core diameter (8-10 m) SMF offers bit rate, most 1999 S.O. Kasap,highest Optoelectronics (Prentice Hall) widely 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. 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. n2 O O'

O'' 3 2 1 2 3 n1 n2 1999 S.O. Kasap, Optoelectronics (Prentice Hall) Total Internal Reflection in Graded Index Fiber (a) TIR

n decreases step by step from one layer to next upper layer; very thin layers. (b) TIR Continuous decrease inn gives a ray path changing continuously. (a) A ray in thinly stratifed medium becomes refracted as it passes from one layer to the next upper layer with lower n and eventually its angle satisfies TIR. (b) In a medium where n decreases 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

A B'' nb na Ray 1 M 1999 S.O. Kasap, Optoelectronics (Prentice 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 fromO

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 the 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 Ray path along the fiber 4 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 (Prentice Hall) Maxwells Equations In a linear isotropic dielectric material with no currents and free of charges, ..(1) (Faradays Law) E: Electric Field .(2) (Maxwells Faraday equation) H: Magnetic Field .(3) (Gauss Law) .(4) (Gauss Law for magnetism) Taking the curl of (1) and using

and .(5) The parameter is permittivity and is permeability. Maxwells Equations But from the vector identity (6) Using (5) and (3), Note v 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 2 E 2 r 2 2 r r r r z Maxwells Equation Electrical and magnetic vectors in cylindrical coordinates are give by, ...(9)

(10) Substituting (9) and (10) in Maxwells curl equations .(11) .(12) .(13) Maxwells 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 , E, Hr , H, can be determined from (17) to (20). Maxwells 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 is increased by 2. Thus (25) Where 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 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 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 k 22 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,

2 2a 2 2 V 2 (u 2 w2 )a 2 k12 k 22 a 2 n1 n2 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, 2a( NA) V 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 constantb vs. V-number for a step index fiber for various LP modes. 1999 S.O. Kasap, Optoelectronics (Prentice 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,z,t ) = E(y)cos(t 0z) E(y) Light

m=0 n1 n2 The electric field pattern of the lowest mode traveling wave along the guide. This mode hasm = 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 vg1 1 < cut-off E(y) vg2 > vg1 Core 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 01 (c) The intensity in LP 11 (d) The intensity in LP 21 Core

E Cladding E 01 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 01 , LP 11 and LP 21 modes. 1999 S.O. Kasap, Optoelectronics (Prentice Hall) Fiber Key Parameters Fiber Key Parameters

Effects of Dispersion and Attenuation Dispersion for Digital Signals Fiber Information Digital signal Emitter t Photodetector Information Input Output Input Intensity

Output Intensity Very short light pulses 0 T t t 0 ~2 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 (Prentice Hall) Modal Dispersion High order mode Light pulse Low order mode 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 vg( 1 ) Emitter vg( 2 ) Very short light pulse Intensity Intensity Core Output

Intensity Spectrum, Spread, 1 o 2 0 t t

All excitation sources are inherently non-monochromatic and emit within a spectrum, , 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 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 30 -1 nm -1 ) Dm 20 10 Dm + D w

0 -10 Dw 0 -20 -30 1.1 1.2 1.3 ( m) 1.4

1.5 1.6 M a t e ri a l d i sp e rsi o n c o e ffi c i e n t (D m ) fo r t h e c o re m a t e ri a l (t a k e n a s S i O 2 ), w a v e g u i d e d i sp e rsi o n c o e ffi c i e n t (D w ) (a = 4 .2 m ) a n d t h e t o t a l o r c h ro m a t i c d i sp e rsi o n c o e ffi c i e n t D ch (= D m + D w ) a s a fu n c t i o n o f fre e sp a c e w a v e l e n g t h , 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 SiO2-13.5%GeO2 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 = D m + Dw -10 Dw -20 Thin 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 TTGVD ~= TTmat For Single Mode Fibers: But Group Velocity Disp. Hence, (TTpol 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 to Zero bits) : B TTotal 0.35 For half width pulses (Return to Zero bits) : B TTotal Disp. & Attenuation Summary Electrical signal (photocurrent) 1 0.707 Fiber Sinusoidal signal Emitter

t f = Modulation frequency Pi = Input light power 0 Photodetector Optical Input Optical Output Po = Output light power t 0 1 kHz

1 MHz 1 GHz 1 MHz 1 GHz f fel Sinusoidal electrical signal Po / Pi 0.1 0.05 t

1 kHz f op f An optical fiber link for transmitting analog signals and the effect of dispersion 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 distribution 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 Preform feed

Furnace 2000C Thickness monitoring gauge Polymer coater Ultraviolet light or furnace for curing Take-up drum Capstan The Fiber Cable Schematic illustration of a fiber drawing tower. 1999 S.O. Kasap, Optoelectronics (Prentice Hall)