Outline. Propagation of Signals in Optical Fiber. Outline. Geometric Approach. Refraction. How do we use this?

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1 Outline Propagation of Signals in Optial Fiber Geometri approah Wave theory approah Loss and Bandwidth Galen Sasaki University of Hawaii Galen Sasaki University of Hawaii Outline Geometri approah Wave theory approah Loss and Bandwidth Geometri Approah Ray theory or geometrial optis: Light behaves like rays Index of refration: speed of light Snell s Law: how light bends Modal dispersion: effet on bit-rate x distane Fiber types Step index Graded index Single mode Galen Sasaki University of Hawaii 3 Galen Sasaki University of Hawaii 4 speed of light = /n n n Refration θ angle of refration angle of refletion θ θ Snell s Law: n x sin θ = n x sin θ Critial angle = sin - n/n then θ = π/ total internal refletion Galen Sasaki University of Hawaii 5 Light soure How do we use this? Fiber Contain the light down the fiber Make sure light pulses keep their shape Galen Sasaki University of Hawaii 6

2 θ0 Air n0 Step Index Fiber θ Cladding n Core n Definitions Numerial Aperature (NA) = n0 sin θ max 0 A measure of light gathering = (n-n)/n (reall n = ore, n = ladding) is typially small, e.g., 0.0 For small, NA n air max θ0 < θ0 = sin n n Galen Sasaki University of Hawaii 7 n 0 Example: n=.5 for silia and =0.0 NA is about 0.. Thus θ 0 max = degrees Galen Sasaki University of Hawaii 8 Modal Dispersion L Modal Dispersion T 0 T L /B L Intersymbol Interferene (ISI) Center: T f = L n /. Critial angle: T s = L n /n Galen Sasaki University of Hawaii 9 following ritial angle down the ore Galen Sasaki University of Hawaii 0 Modal Dispersion L Center: T f = L n /. Critial angle: T s = L n /n L n δt = Ts Tf = < n B Bit rate Galen Sasaki University of Hawaii Modal Dispersion Bit-rate distane produt BL (Mb/s)-km, B Mbps over L km. (Loss is the main limitation) Produt is usually onstrained by a onstant Modal dispersion onstraint on the produt n BL < n Step index fiber Galen Sasaki University of Hawaii

3 Graded-Index Fiber Index hanges more smoothly from ore to ladding in a quadrati way Different δt: =0.0, n=.5 4 Graded index BL < n n Step index fiber BL < n =8(Gb/s)-km =0(Mb/s)-km Galen Sasaki University of Hawaii 3 Geometri Optis Approah Okay for multi-mode beause ore radius a is µm, larger than wavelengths Single-mode fiber has smaller ore Diffration effet Refous spreading light beam Explained by wave theory Galen Sasaki University of Hawaii 4 Outline Geometri approah Wave theory approah Loss and Bandwidth Galen Sasaki University of Hawaii 5 Wave Theory Review of eletromagneti basis Eletri and magneti fields Five harateristis of the medium Wave equations Fiber modes Polariation Light propagation in wave guides Galen Sasaki University of Hawaii 6 Review of EM Review of EM Eletri Field Magneti Field qe = F qq0 F = 4πε 0 r F q E = = q0 4πε 0 r q F = qv B F = qvbsinθ v θ q B B Galen Sasaki University of Hawaii 7 Galen Sasaki University of Hawaii 8 3

4 E(t): eletri field H(t): magneti field Fourier Transforms: Wave Theory E ( = E( t)exp( iωt) dt Galen Sasaki University of Hawaii 9 Wave Theory P: Indued eletri polariation (or polariation) M: Magneti polariation D: Eletri flux density B: Magneti flux density εo: permittivity of vauum µo: permeability of vauum D = ε 0 E + P B = µ 0( H + M ) funtion of E =0 in silia Galen Sasaki University of Hawaii 0 Relationship between P and E Relationship between P and E affets dispersion and nonlinearities Five affets and their affet on P and E Loality of response Isotropy Linearity Homogeneity Losslessness Galen Sasaki University of Hawaii Loality of Response For any P(r) at r is only dependent on E(r) Approximately true for silia nm Example of nonloal response: P(r) is dependent on V ( where V(r) is some volume around r r) E( x, y, ) dxdyd Galen Sasaki University of Hawaii Isotropy Refrative index is the same in all diretions This means E and P have the same orientation Silia is isotropi Perfetly ylindrial fiber is isotropi Imperfet fiber is not isotropi, i.e., birefringent Galen Sasaki University of Hawaii 3 P( w) = ε Linearity P( t) = ε 0 χ( t u) E( u) du 0 t χ ( E( P may have a delayed response to E Chromati dispersion Linear suseptibility Linearity is true for silia at moderate powers and bit rates Galen Sasaki University of Hawaii 4 4

5 χ(t) = χ(t), Homogeneity i.e., eletromagneti properties are independent of position Losslessness Silia has ero loss, kind of Silia is homogeneous, but fiber is not n ( ω ) : = + χ ( Fiber has ore with different index of refr. than ladding Galen Sasaki University of Hawaii 5 For the time being, we ll assume all five properties. Galen Sasaki University of Hawaii 6 From Maxwell s Equations This ditates the EM waves that go through the medium Wave Equations ω n ( E + E = 0 ω n ( H + H = 0 = + + x y Galen Sasaki University of Hawaii 7 What do the Wave Equations do for us? The wave equations tell us how signals travel down a medium, like a fiber ore step index fiber ladding Fiber mode: solutions for wave equations for the ore wave equations for the ladding boundary onditions between ladding and ore Galen Sasaki University of Hawaii 8 Fiber Modes Suppose = diretion of eletromagneti wave fiber properties are independent of step index fiber Then eletri and magneti fields of a fiber mode are dependent on as exp(iβ), a sinusoid propagation onstant Galen Sasaki University of Hawaii 9 Propagation Constant β Propagation onstant β is ωn/ = πn/λ Propagation onstant is kn, where k = π/λ the wave number kn < β < kn ladding neff = β/k ore In general β is a funtion of ω. Galen Sasaki University of Hawaii 30 5

6 Normalied Propagation Constant k n b : = β k n k n n = n eff n n b( V ) ( / V ) Single Mode π Conditions for single mode: V : = a n n λ a n n < Typial values: a=4000nm, = (n-n)/n=0.003, V Galen Sasaki University of Hawaii 3 Galen Sasaki University of Hawaii 3 Multimode For large V (multimode), # modes is approximately V / Multimode --> several hundred modes Polariation Let s take a look at the signal on a single mode fiber ω n ( E + E = 0 linearly indep solutions Solution ero nonero Solution ero nonero y x n n Longitudinal omponent Nonero E ( t) = E eˆ + E eˆ + E eˆ x x y y Galen Sasaki University of Hawaii 33 Galen Sasaki University of Hawaii 34 Polariation Properties of the eletri field on a single mode fiber Linearly polaried: its diretion is onstant with time Solution = α Solution + α Solution. It is transverse sine it has no omponent along the diretion of propagation (atually very small) Polariation Mode Dispersion (PMD) If the fiber is not perfetly irula the x and y omponents have different speeds Galen Sasaki University of Hawaii 35 E Polariation = πj ( x, y)exp( iβ) E ( E x y J ( x, y) = l ) = πj ( x, y)exp( iβ) t J l (x,y) and J t (x,y) are dependent on (x,y) only through ρ = x + y E( = πj ( x, y)exp( iβ( )ˆ( e x, y) J l ( x, y) + J ( x, y) Galen Sasaki University of Hawaii 36 t 6

7 Light Propagation in Dieletri Waveguides Slab rather than a ylinder Outline Geometri approah Wave theory approah Loss and Bandwidth Galen Sasaki University of Hawaii 37 Galen Sasaki University of Hawaii 38 Loss and Bandwidth Loss Formula Attenuation formula Loss mehanisms Bandwidth Pout = Pin exp(-αl) Length of medium Customary to express the loss in db/km αdb = 0 log0 (Pout/Pin) db/km Different from the α above αdb =(0 log 0 e)α approx α Galen Sasaki University of Hawaii 39 Galen Sasaki University of Hawaii 40 Example fiber 0.5dB/km How far an a signal go? Typially, a signal an suffer 0-30 db loss before being amplified or attenuated The signal an go 0/0.5 to 30/0.5 km Galen Sasaki University of Hawaii 4 Loss Mehanisms Material Absorption due to impurities in silia negligible for nm Rayleigh Sattering dominant loss mehanism due to flutuations in the density of the medium at the mirosopi level loss oeffiient α R = A/λ 4 Galen Sasaki University of Hawaii 4 7

8 Loss (db/km) Fiber Attenuation 800nm nm 550nm 0.5 f = /λ f approx ( λ)/λ Measuring bandwidth x Bandwidth Bandwidth Wavelength λ (µm) 35 TH of bandwidth in single mode fiber Galen Sasaki University of Hawaii 43 Galen Sasaki University of Hawaii 44 8