Lecture 3 Optical fibers as waveguides Maxwell s equations The wave equation Fiber modes Phase velocity, group velocity Dispersion Fiber Optical Communication Lecture 3, Slide 1
Maxwell s equations in an optical fiber (..1) H is the magnetic field [A/m] Ampère s circuital law, no current in fiber H D t E is the electric field [V/m] Faraday s law of induction E B t D is the electric flux density [C/m ] D No free electric charges in fiber B is the magnetic flux density [Vs/m ] B No free magnetic charges Fiber Optical Communication Lecture 3, Slide
The constitutive relations Relate the fields to the properties of the material P is the polarization field density The electric permittivity, ε 8.85 1 1 [As/Vm] D E P The magnetization is zero in a fiber The magnetic permeability μ 1.6 1 6 [Vs/Am] B H The permittivity and the permeability are related to the speed of light c 1/ Fiber Optical Communication Lecture 3, Slide 3
The text-book uses the definition Fourier transform definition ~ s ( ) s ( t)exp( i t) dt Many different definitions are used in different fields of science With this definition t i s( t) 1 ~ s ( )exp( it) d A wave traveling in the positive z-direction is described by exp( it) s( t) ~ s ( ) i E( z, t) A cos( z t ) Re[ Ae exp( iz i β is a propagation constant t )] Fiber Optical Communication Lecture 3, Slide 4
Derivation of the wave equation We derive the wave equation E t By using the Fourier transform of E we have ~ χ is the susceptibility The wave equation becomes H E P it it E Er, te dt P Pt e d E ~ ~ E 1 ~ E c c ε is the relative dielectric constant (dimensionless) This is often written as ε r (not in the course book, though) ~ t c 1 ~ ~ t E E ~ t P Fiber Optical Communication Lecture 3, Slide 5
Derivation of the wave equation The dielectric constant is related to the refractive index n and the loss α c n i The frequency dependence of n is referred to as material dispersion We use the vector rule E It is assumed that the medium is homogeneous Finally the wave equation reads (using k = ω/c and neglecting losses) ~ ~ ~ E E E ~ ~ E n ~ k E Fiber Optical Communication Lecture 3, Slide 6
Optical fiber modes (..) Modes are solutions to the wave equation Satisfy the boundary conditions Do not change spatial distribution as they propagate Assume that the cladding extends to infinity The wave equation for E z in cylindrical coordinates (ρ, φ, z) 1 Ez 1 Ez Ez n k E z z n = n 1 (ω) in the core (r < a) and n = n (ω) in the cladding (r > a) The same equation is obtained for the H z -component Maxwell s equations give the remaining four E- and H-field components The boundary conditions at z = determines whether E z or H z are excited E z = transverse electric (TE) mode H z = transverse magnetic (TM) mode E z, or H z hybrid mode, EH or HE Fiber Optical Communication Lecture 3, Slide 7
Solving the wave equation for the modes We use the method of separation of variables and obtain Z E z,, z F Zz z exp iz exp im where β is the propagation constant (to be determined) and m is an integer The equation for F is d F 1 df m n k F d d This is Bessel s equation and its solution are Bessel functions Fiber Optical Communication Lecture 3, Slide 8
In the core (ρ < a, n = n 1 ), Bessel s eq. has bounded solutions J m for β < k n 1 so that where F AJ p p n m k The Bessel functions 1 In the cladding (r > a, n = n ), Bessel s eq. has bounded solutions K m for β > k n so that where F CK q K m is exponentially decaying. m q n k Fiber Optical Communication Lecture 3, Slide 9
Initially given information: The propagation constant The fiber geometry, i.e., the core radius a, the core index n 1, and the cladding index n The operating wavelength/frequency, i.e. k = π/λ ω/c Current status: We have the solution for E z with two unknown constants (amplitude in the core and the cladding) The equation and solution for H z is analogous The integer m is chosen by us Then we get the solution! The boundary conditions give an equation for β mn (ω) A number of solutions for each m-value Maxwell s equations give the other field components Each mode has a specific, wavelength-dependent propagation constant Fiber Optical Communication Lecture 3, Slide 1
For m = either H z or E z are zero The mode family These are called TE n and TM n modes The other modes are hybrid modes; EH mn or HE mn The lowest order mode is HE 11, which exists for all wavelengths A Comsol simulation of HE 11 is seen to the right E x and E y are shown by the arrows E z is shown by the color The longitudinal (z) component is much smaller than the transverse Essentially linearly polarized Fiber Optical Communication Lecture 3, Slide 11
Further examples More calculated modes are shown below E z is not correct in third figure, top row Does not decrease properly with ρ Numerical artifact of an outer boundary Fiber Optical Communication Lecture 3, Slide 1
The effective index (mode index) The propagation constant must lie in the interval k n < β < k n 1 We define a mode index (or effective index) as n / k The value is between the core and cladding index n n n 1 The effective index gives a measure of the mode confinement HE 11 mode well confined mode n weakly confined mode a -a intensity n 1 n n 1 n n intensity Fiber Optical Communication Lecture 3, Slide 13
V-parameter and effective propagation constant The normalized frequency is defined V a n1 n an1 c The normalized propagation constant is n n b n n All modes except HE 11 are cutoff for V <.45 For large V, approximately V / modes are guided 1 Fiber Optical Communication Lecture 3, Slide 14
The single-mode condition: Examples (..3) A multimode fiber with a = 5 μm and Δ =.5 has a value of V = 18 at λ = 1.3 mm. It then supports 18/ = 16 modes Single-mode fibers are often designed to have a cut-off at λ = 1. μm. By taking n 1 = 1.45, Δ =.4, we find that the required core radius a = 4 μm Fiber Optical Communication Lecture 3, Slide 15
The HE 11 mode and the LP modes The z-components of the E/H fields for the HE 11 mode are quite small (for small Δ), and the E x or E y component is dominating This mode is essentially linearly polarized The polarization state depends on how the E-field vector is directed Thus, the single mode consists of two degenerate polarization modes Two linearly polarized modes, depending on whether E x or E y is excited The HE 11 mode is often referred to as the LP 1 mode LP means linearly polarized Fiber Optical Communication Lecture 3, Slide 16
Birefringence Real fibers have variations in the core shape/doping and experience nonuniform stress The LP-mode degeneracy is removed The fiber is birefringent with a slow and a fast axis The index difference is Δn = n x n y Even a single-mode fiber really has two orthogonally polarized modes The index difference between the two modes is called birefringence Leads to polarization rotation over a beat length L B = λ/δn (1 mm 1 m) Beat length = The length over which the polarization state changes one period in a birefringent material z x y L B Fiber Optical Communication Lecture 3, Slide 17
Propagating waves, phase velocity Each frequency component ω of the light propagates according to ~ ~ jz E( z, ) E(, ) e Shows that the power spectrum, which is proportional to E(ω), will not change during transmission The propagation constant, β, is a function of ω This is called the dispersion relation We Taylor expand β(ω) to get 1 1,, m d m d m A monochromatic wave propagates with the phase velocity v p Fiber Optical Communication Lecture 3, Slide 18
Phase velocity The propagating field is E( z, t) E cos( z t ) The phase velocity is v p Fiber Optical Communication Lecture 3, Slide 19
A pulse E(t) at z = will propagate as i.e. it moves with the group velocity Group velocity E( z, t) v g d d d E z d t = k n mode y = k n 1 Fiber Optical Communication Lecture 3, Slide
Group velocity E( z, t) v g d E z d t Fiber Optical Communication Lecture 3, Slide 1
Group velocity dispersion (GVD) (.3) If the group velocity is different for different frequency components, the medium is dispersive Single-mode fibers have two contributions to the GVD: The dependence of n 1, on ω This is called material dispersion The mode behavior, which makes β depend on ω This is called waveguide dispersion In multimode fibers, the different velocities of the different modes is the main source of dispersion This is called modal dispersion Fiber Optical Communication Lecture 3, Slide
Pulse broadening by chromatic dispersion Study a discrete spectrum! (Signal will be periodic) Fiber Optical Communication Lecture 3, Slide 3
Inter-symbol interference The pulse distortion from dispersion leads to intersymbol interference (ISI) Neighboring pulses will broaden and overlap Dispersion limits the bit-rate! The information capacity of an optical fiber is often quantified by the bit-rate distance product fiber type multi-mode, step-index multi-mode, graded-index single mode, step-index maximum BL Mbit/s km.5 Gbit/s km >.5 Gbit/s km Fiber Optical Communication Lecture 3, Slide 4