Introduction to Optical Networks Chapter 2 Propagation of Signals in Optical Fiber 1 2.Propagation of Signals in Optical Fiber Advantages • Low loss ~0.2dB/km at 1550nm • Enormous bandwidth at least 25THz • Light weight • Flexible • Immunity to interferences • Low cost Disadvantages and Impairments • Difficult to handle • Chromatic dispersion • Nonlinear Effects 2 Cladding SiO2 , refractive index ≈1.45 SiO2 core 8~10μm, 50μm, 62.5μm doped 2.1 Light Propagation in Optical Fiber 3 4 2.1.1Geometrical Optical Approach (Ray Theory) This approach is only applicable to multimode fibers. 1 : incident angle (入射角) 2 : refraction angle (折射角) 1r : reflection angle (反射角) 1r 1 Snell’s Law n1 sin1 n2 sin2 n1, n2 : refractive indices 5 2f n1 n2 and when 2 / 2 n2 , n1 =>Critical angle c Sin When 1 c , total internal reflection occurs. 1 let 0 = air refractive index 0max= acceptance angle (total reflection will 6 occur at core/cladding interface) n0 sin0max n1 sin1max n2 sinc , n1 c sin( 2 2 1max max 1 n2 ) n1 n1 cos 1max n2 sin 1max n22 1 2 n1 n0 sin 0max 0max sin1 n12 n22 n12 n22 n0 (2.2) 7 n1 n2 Denote n1 n12 n22 (n1 n2 )(n1 n2 ) (n1 n2 )n1 If Δ is small (less than 0.01) n12 n22 n1 max 0 sin For n1 1.5 1 n1 2 2 n0 (multimode) 0.01 n0 1 0max 12 max n sin n1 2 Numerical Aperture NA= 0 0 Because different modes have different lengths of paths, intermodal dispersion occurs. 8 Infermode dispersion will cause digital pulse spreading Let L be the length of the fiber The ray travels along the center of the core T f Ln1 / C The ray is incident at c (slow ray) Ln1 Ts c cos1max Ln12 n2 max cos1 cn2 n1 T Ts T f Ln12 Ln1 cn2 c Ln1( n1 n2 ) cn2 Ln12 cn2 9 Assume that the bit rate = Bb/s 1 Bit duration T B T 1 T 2 2B Ln12 1 cn2 2B The capacity is measured by BL (ignore loss) n2c c BL 2 2n1 2n1 Foe example, if 0.01, n1 1.5 BL (10mb / s ) km 10 For optimum graded-index fibers, δT is shorter than that in the step-index fibers, because the ray travels along the center slows down (n is larger) and the ray traveling longer paths travels faster (n is small) 11 The time difference is given by (For Optical graded-index profile) and If Ln1 2 T 8c 1 2B 4c BL n1 2 T (single mode BL 0.01, n1 1.5 BL 8 (Gb / s ) km c 2n1 ) Long haul systems use single-mode fibers 12 2.1.2 Wave Theory Approach Maxwell’s equations D B 0 B t D H J t D.1 D.2 D.3 D.4 ρ : the charge density, J: the current density D : the electric flux density, B: the magnetic flux density : the electric field, H : the magnetic field 13 Because the field are function of time and location in the space, we denote them by E ( r, t ) and H( r, t ) , where r and t are position vector and time. Assume the space is linear and time-invariant the Fourier transform of E( r, t ) is 2.4 E( r, w)= E( r, t )exp(iwt )dt - w 2 f let P be the induced electric polarization D 0 E P 0 : the permittivity of vacuum B 0 ( H M ) M : the magnetic polarization : the permeability of vacuum 注意有些書Fourier transform定義為 E ( r, w)= E( r, t ) exp( jwt )dt - 2.5 2.6 0 - E( r, t ) exp( j 2 ft )dt E ( r, t )= E( r, w) exp( j 2 ft )dt - 14 Locality of Response: P and E related to dispersion and nonlinearities If the response to the applied electric field is local P(r1 ) depends only on E(r1 ) not on other values of E(r1 ), r r1 This property holds in the 0.5~2μm wavelength Isotropy: The electromagnetic properties are the same for all directions in the medium Birefringence: The refraction indexes along two different directions are different (lithium niobate, LiNbO , modulator, isolator, tunable filter) 3 15 Linearity: P(r, t ) 0 x(r, t t ' )E(r,t ' )dt ' (Convolution Integral) 2.7 x( r, t ) : linear susceptibility The Fourier transform of P( r, t ) is P(r, w) 0 x(r, w)E(r, w) 2.8 Where x( r, w) is the Fourier transform of E( r, t ) ( x( r, t ) is similar to the impulse response) x( r, w) is function of frequency => Chromatic dispersion 16 Homogeneity: A homogeneous medium has the same electromagnetic properties at all points x( r, t ) x(t ) The core of a graded-index fiber is inhomogeneous Losslessness: No loss in the medium At first we will only consider the core and cladding regions of the fiber are locally responsive, isotropic, linear, homogeneous, and lossless. The refractive index is defined as def 2 2.9 n ( w) 1 x( w) n 1.5 For silica fibers x 1.25 17 From Appendix D D B 0 B t D H J t For 0 (zero charge) 0 (zero conductivity, dielectric material) J E 0 D 0 E P B 0 ( H M ) For nonmagnetic material M 0 18 E B t ( 0 H ) t 2 D ( 0 ) 2 t t 2 0 (0 E P ) 2 t 2 2 0 0 E 0 P 2 2 t t Assume linear and homogenence E ( r, w ) E ( r, t ) exp(iwt )dt 1 E ( r, t ) E ( r, t ) exp( iwt )dw 2 iw t 19 2 2 E( r, t ) 0 0 2 E( r, t ) 0 2 P( r, t ) t t Take Fourier transform ( t iw) 2 2 E(r, w) 0 0 w E(r, w) 0 w P(r, w) Recall P(r, w) 0 x(r, w)E(r, w) 2.8 E(r, w) 0 0 w2 E(r, w) 0 0 w2 x(r, w) E(r, w) Denote c 1 0 0 c: speed of light n( w) 1 x( w) (Locally response, isotropic, linear, homogeneous, lossless) E(r, w) 0 0 w2 (1 x(r, w))E(r, w) w2 n2 2 E( r, w) c 2.9 20 E ( r, w) ( E ( r, w)) E ( r, w) 2 2 2 w n E ( r, w) 2 E ( r, w) ( E ( r, w)) c 2 palacian operation 2 E ( r, t ) 0, 2 2 w n ( w) E ( r, w ) E ( r, w ) 0 2 c 2 2 2 or E ( r, w) n ( w)K 0 E ( r, w) 0 2 where K 0 w c 2 2.10 (free space wave number) 21 For Cartesian coordinates 2 2 2 2 x2 y 2 z 2 For Cylindrical coordinatesρ. φ and z 2 Ez 1 Ez 1 2 Ez 2 Ez 2 2 n2 k02 Ez 0 2 2 z n1 a n:{ n a 2 a: radius of the core 2 2 w n ( w) 2 H ( r , w ) H (r, w) 0 2 Similarly c Boundary conditions 0 E is finite , E 0, and continuity of field at ρ=a 2.11 References: G.P. Agrawal “Fiber-Optical Communication System” Chapter 2 John Senior “Optical Fiber Communications, Principles and practice” John Gowar “Optical Communication Systems” 注意有些書在 time domain運算 有些書在frequency domain運算 22 Fiber Modes cladding core x z y 23 E core , E cladding , H core , and H cladding must satisfy 2.10, 2.11 and the boundary conditions. let E( r, w) Ex e x E y e y Ez e z Where e x , e y , and e z are unit vectors For the fundamental mode, the longitudinal component is Ez 2 J e ( x, y ) exp(i z ) wn 2 fn 2 n c c : the propagation constant 24 J ( x, y ) : Bessel functions The transverse components ( Ex and E y ) Ex 2 J t ( x, y )exp(i z ) For cylindrical symmetry of the fiber J ( x, y ) and J t ( x, y ) depend only on x y 2 2 In general, we can write E( r, w) 2 J ( x, y )exp(i ( w) z )e( x, y ) (Appendix E) 25 Where J ( x, y) J ( x, y) J ( x, y) 2 2 t The multimode fiber can support many modes. A single mode fiber only supports the fundamental mode. Different modes have different β, such that they propagate at different speeds.=>mode dispersion (We can think of a “mode” as one possible path that a guided ray can take) 26 For a fiber with core n1and cladding n2, if a wave propagating purely in the core, then the propagation constant is wn1 2 n1 1 c kn1 λ: free space wavelength The wave number k 2 Similarly if the wave propagating purely in the cladding, then 2 kn2 The fiber modes propagate partly in the cladding and partly in the core, so kn2 kn1 Define the effective index neff k n n n 2 eff 1 The speed of the wave in the fiber= c n eff 27 For a fiber with core radius a , the cutoff condition is def 2 V a n12 n22 2.405 V : normalized wave number n1 n2 Recall n 1 V↓ when a↓ and △ ↓ For a single mode fiber, the typical values are a=4μm and △=0.003 V 2 at 1550nm 28 The light energy is distributed in the core and the cladding. 29 30 Since Δ is small, a significant portion of the light energy can propagate in the cladding, the modes are weakly guided. The energy distribution of the core and the cladding depends on wavelength. kn1 kn2 n k n2 neff n1 eff It causes waveguide dispersion (different from material dispersion) ( Appendix E ) For longer wave, it has more energy in the cladding and vice versa. 31 A multimode fiber has a large value of V 2 V The number of modes 2 For example a=25μm, Δ=0.005 V=28 at 0.8μm Define the normalized propagation constant (or normalized effective index) def b k n 2 2 2 2 k n k n 2 2 1 2 2 2 2 eff 2 1 n n 2 2 2 2 n n b(V ) (1.1428 0.9960 / V ) 2 HE11 mode ( H z Ez ) b(V ) is used to investigate the wave propagation in fibers 32 Polarization Two fundamental modes exist for all λ. Others only exist for λ< λcutoff, E( r, t ) E x e x E y e y E z e z E z : longitudinal component E x , E y : transverse components Linearly polarized field : Its direction is constant. For the fundamental mode in a single-mode fiber E x , E y E z 33 34 35 Fibers are not perfectly circularly symmetric. The two orthogonally polarized fundamental modes have different β =>Polarization-mode dispersion (PMD) Differential group delay (DGD) ps km Δτ=Δβ/w ~typical value Δτ=0.5 100 km => 50 ps Practically PMD varies randomly along the fiber and may be cancelled from an segment to another segment. ps km Empirically, Δτ ~0.1-1 Some elements such as isolators, circulators, filters may have polarization-dependent loss (PDL). 36 2.2 Loss and Bandwidth Pout Pin e L L : length of fiber : fiber loss in dB km Pout 10 log10 dB Pin dB (10 log10 e) 4.343 Two main loss mechanisms : material absorption and Rayleigh scattering The material absorption is negligible in 0.8 m ~ 1.6 m 37 38 c Recall f 2 Take the bandwidth over which the loss in dB/km is with a factor of 2 of its minimum. 80nm at 1.3μm, 180nm at 1.55μm =>BW=35 THz 39 Erbium-Doped Fiber Amplifiers (EDFA) operate in the c and L bands, Fiber Raman Amplifiers (FRA) operate in the S band. All Wave fiber eliminates the absorption peaks due to water. 40 41 2.2.1 Bending loss A bend with r = 4cm, loss < 0.01dB r↓ loss↑ 2.2.3 Chromatic Dispersion Different spectral components travel at different velocities. a. Material dispersion n(w) b. Waveguide dispersion, different wavelengths have different energy distributions in core and cladding =>different β, kn2< β < kn1 1 dB dw , 1 : group velocity 1 2 2 d B : group velocity dispersion (GVD ) parameter dw2 If 2 0 zero dispersion 2 0, the dispersion is normal 2 0, the dispersion is anomalous 42 2.3.1 Chirped Gaussian Pulses Chirped: frequency of the pulse changes with time. Cause of chirp: direct modulation, nonlinear effects, generated on purpose. (soliton) 43 Appendix E, or Govind P. Agrawal “ Fiber- Optic Communication Systems” 2nd Edition, John Wiley & Sons. Inc. PP47~51 A chirped Gaussian pulse at z=0 is given by 1 ik t 2 T0 G(t ) R A0 e A0 e A0 e 1 t 2 T0 2 1 t 2 T0 2 2 e i0 t 2 k t cos 0 t 2 T0 cos (t ) kt 2 (t ) 0 t 2T0 The instantaneous angular frequency d (t ) k 0 t dt T0 T0 Pulse width 44 k = The chirp factor Define: The linearly chirped pulse: the instantaneous angular frequency increases or decreases with time, (k=constant) i0t G ( t ) R A o , t e Note A A i A2 Solve 1 2 2 o with 2the initial z t 2 t 1 ik t condition A(o, t ) A0 e A0T0 We get A( z, t ) 2 T0 (E.7) (1 ik )(t 1z )2 exp 2 2 2 T0 i 2 z(1 ik ) T0 i 2 z(1 ik ) 2 t 1z 1 ik Az exp 2 T02 i 2 z(1 ik ) (E.8) A(z,t) is also Gaussian pulse 45 Tz T0 2 kz i 2 z Tz T0 T0 2 kz 2 z 2 2 2 2 z 2 kz 1 2 T T 0 0 2 Broadening of chirped Gaussian pulses They have the same of broadening length. Note a b distance 2LD , c d distance 0.4 LD 46 In Fig 2.9, 2 0, it is true for standard fibers at 1.55μm LD Let T02 2 If z LD , dispersion can be neglected T02 z z 2 If be the dispersion length 2 T (2.13) 2 0 T02 Tz T0 2 z 2 1 Tz T0 1 and k 0 (unchirped pulse ) z LD For 2.5 Gb / s systems at 1.55 m ( return to zero pulse) T 0.2ns ( half pulse duration) 2 For 10 Gb / s, T0 0.05ns LD 115km let T0 LD 1800km For NRZ ( return to zero ) LD 600km for 2.5 Gb / s 47 For kβ2 < 0 Tz T0 2 2 z k 2 z 1 2 T02 T0 ↓decreases 2 increases ↑ for certain z For β2 > 0, high frequency travels faster => the tail travels fasters => compression (Fig 2.10) => make βk < 0, LD increases 48 Note β2> 0 , k< 0 β2k<0 49 2.3.2 Controlling the Dispersion Profile Def: Chromatic dispersion parameter D = 2 c 2 2 in ps / nm km D = DM + D w The standard single mode fiber has small chromatic dispersion at 1.3 μm but large at 1.55 μm 50 At 1.55μm loss is low, and EDFA is well developed. Dispersion becomes an issue We have not much control over DM, but Dw can be controlled by carefully designed refractive index profile. Dispersion shifted fibers, which have zero dispersion in 1.55μm band 51 52 2.4 Nonlinear Effects For bit rate ≦2.5 Gb/s, power a few mw Linear Assumption is valid Nonlinear effect appears for high power or high bit rate ≧ 10 Gb/s and WDM systems The first category relates to the interaction of lightwave with phonons (molecular vibrations) - Rayleigh Scattering - Stimulated Brillouin Scattering (SBS) - Stimulated Raman Scattering (SRS) 53 The second category is due to the dependence of the refractive index on the intensity - self-phase modulation (SPM) - four-wave mixing (FWM) SBS and SRS transfer energy from short λ (pump) to long λ(stokes wave) Scattering gain coefficient, g, is measured in meter/watt and Δf. SPM induces chirping In a WDM system, variation of n depending on the intensity of all channels. =>Yields Cross-phase modulation (CPM) =>interchannel crosstalk 54 ● FWM, f1, f 2 , ... f n fi , f j , f k ( fi f j f k ), e.g 2 fi f j , fi f j f k 55 2.4.1 Effective Length and Area The nonlinear effect depends on fiber length and cross-section. P( z ) P0 e P0 Le L z 0 z P( z )dz when Le : effective length Le 1 e L Typically 0.22 dB km at 1.55 m L 1 (for long link) Le 20 km 56 In addition nonlinear effect intensity Ae effective cross sectional area 2 F ( r , ) rdrd r 4 F ( r, ) rdrd r 2 F ( r, ) : Fundamental mode intensity Ie P effective intensity Ae SMF Ae ~ 85 m 2 , DSF , Ae ~ 50 m 2 由 power point 50, DSF n1 大 n2 dispersion compensating fiber ( DCF ) n1 及 n2 差最多 Ae 更小, nonlinear effect 更嚴重 57 2.4.1 Stimulated Brillouin Scattering (SBS) The scattering interaction occurs with acoustic phonons over Δf =15 MHz, at 1.55μm, stokes and pump waves propagate in opposite directions. If spacing > 20 MHz => no effects on different channels Ps(0) Pp(L) SBS pumping 58 g B 4 10 11 m w independent of dI s gB I p Is Is dz dI p gB I p Is I p dz I s : Intensity of stokes, Ps I s Ae (2.14) (2.15) I p : Intensity of pump, Pp I p Ae Ae : effective area Assuming I s is small, g B I p I s dI p Ip I p I p ( z ) I p (0)e z dz Pp ( L ) Pp (0)e L L : length g B Pp (0 ) Le Ps (0) Ps ( L )e L e Le = 1-e L (2.16) (P.78) Pp ( L ) Pp (0)e Ae L 59 2.4.3 Stimulated Roman Scattering (SRS) SRS will deplete short wave power and amplifier long wave. 60 2.4.4 Propagation in a Nonlinear Medium In a nonlinear medium, Fourier Transfer is not applicable. When the electrical field has only one component, we can write E ( r, t ) and P( r, t ) as the scalar functions E ( r, t ) , and P( r, t ) . Appendix F, P( r, t ) contains higher order terms P( r, t ) 0 0 0 t x(1) ( r, t t1 )E( r, t1 )dt1 t t t t x(2) (t t1, t t2 )E( r, t1 )E( r, t2 )dt1dt2 t x(3) (t t1, t t2 , t t3 )E ( r, t1 )E ( r, t2 )E (r, t3 )dt1dt2dt3 ( F .1) x(1) ( r, t ) : the linear susceptibility x( i ) ( r, t ) : higher order nonlinear susceptibilities P( r, t ) PL ( r, t ) PNL ( r, t ) linear polarization nonlinear polarization 61 Because of symmetry x(2) ( r, t ) 0 , and x(i ) 0 i 4. 5... t t t PNL (r, t ) 0 x(3) (t t1, t t2, t t3 )E(r,t1)E(r,t2 )E(r,t3 )dt1dt2dt3 ( F.2) The nonlinear response occurs less than 100x10-15sec. If the bit rate is less than 100 Gb/s, then x(3) (t t1, t t2 , t t3 ) x(3) (t t1 ) ( t t2 ) (t t3 ) PNL ( r, t ) 0 x(3) E 3 ( r, t ) E 3 (2.19) x(3): the third-order nonlinear susceptibility independent of time For simplicity, assume that the signals are monochromatic plane waves E(r,t ) E( z,t ) E cos( w0t i z ) E is constant in the plane perpendicular to the dispersion of propagation In WDM systems with n wavelengths at the angular frequencies w1, w2 ... wn .,( 1, 2 ... n ) 0 E ( r, t ) E ( z, t ) n E cos( w t z ) i 1 i i i i 62 2.4.5 Self-phase Modulation (SPM) Because n is intensity – dependent =>induces phase shift proportional to the intensity =>creates chirping => pulse broadening It is significant for high power systems. Consider a single channel case E( z, t ) E cos( w0t 0 z ) PNL ( r, t ) 0 x(3) E 3 cos3 ( w0t 0 z ) 1 3 0 x(3) E 3 cos( w0t 0 z ) cos(3 w0t 3 0 z ) 4 4 3 w0 0 3 (2.20) shorter wavelength, the last term is negligible 3 PNL ( r, t )=( 0 x(3) E 2 )E cos( w0t 0 z ) 4 E( z, t ) (2.21) 63 (1) Recall n ( w) 1 x for linear medium 2 n Now, we have to modify ( w) as 2 3 (3) 2 n ( w) 1 x x E 4 (1) 2 We get w 0 0 c let 1 x n 1 x 2 (1) 3 (3) 2 x E 4 (1) w0 n 3 (3) 2 0 1 x E 2 c 4n x(3) is very small w 3 (3) 2 0 0 ( n x E ) c 8n (2.22) propagation constant changes with E 2 => Phase changes with E 2 intensity 64 E( z, t ) E cos( w0t 0 z ) ,whose phase changes as Ez2 , this phenomenon is referal as self- phase modulation (SPM) The intensity of the electrical field 1 I 0 cnE 2 in w 2 m 2 The intensity-dependent refractive index is n( E ) n nI (2.23) The nonlinear index coefficient n 2 3 x( 3 ) 0 cn 8 n 2 2.2 3.4 10 8 m in silica fiber We take n 3.2 10 8 m w for example Because a pulse has its finite temporal extent =>The phase shift is different in different parts of the pulse The leading edges have positive frequency shift The tailing edges have negative frequency shift => SPM causes positive chirping n w 2 65 2.4.6 SPM-induced chirp for Gaussian Pulses Consider an unchirped pulse with envelope U (0, ) e 2 which has unit peak amplitude and 2 -width T0=1, and the peak power P0=1 Define the nonlinear length as e 1 e LNL A 2 nP0 If link length ≧ LNL => nonlinear effect is severe 66 From Appendix E, (E.18) U ( z, ) U (0, z )e U (0, z )e iz U (0, ) 2 LNL iz E.18 LNL 2 e After propagation L distance, The SPM-induced phase change is ' L LNL e 2 The instantaneous frequency is given by w( ) w0 2 L 2 e , w0 : central freg. LNL and the chirp factor is k SPM ( ) 2 L 2 e (1 2 2 ) LNL References: Appendix E, and (Arg97) 67 k SPM ( ) 2 L 2 (12 2 ) e LNL Recall Le < 1 e L increases with L effective length (2.25) 1 At the center of the pulse 0 Ae 2 k SPM , L NL LNL 2 nP0 At 1.55 m, 0.22 dB For P0 1mw P0 10mw LNL km 384km LNL 38km negligible significant 68 2.4.7 Cross-Phase Modulation In WDM systems, the intensity-dependent nonlinear effects (phase shift) are enhanced by other signals, this effect is referred to as cross-phase modulation (CPM) Consider two channels E(r, t ) E1 cos( w1t 1z ) E2 cos( w2t 2 z ) (3) 3 P ( r , t ) x E ( r, t ) Recall NL 0 0 x (3) (2.19) E1 cos( w1t 1z ) E2 cos( w2t 2 z ) 3 69 2w1+w2, 2w2+w1, 3w1and 3w2 can be neglected 2w1-w2, 2w2-w1, are part of FWM. Consider the w1 channel, the CPM term is 3 0 x(3) ( E12 2 E22 ) E1 cos( w1t 1z ) 4 (2.27) CPM If E1=E2 SPM Apparently CPM effect is twice of SPM. In practice, β1 and β2 are different => The pulses corresponding to individual channel walk away from each other. => can not interact further => CPM is negligible for standard fibers Note for DSF, they travel at same velocity, CPM is significant 70 2.4.8 Four-Wave Mixing (FWM) 71 wi, wj ,wk (three waves) generate wi ± wj ± wk (fourth wave) For example, channel spacing Δw w2 = w1 + Δw, w3 = w1 + 2Δw w1- w2+ w3 = w2, 2w2-w1 = w3 72 Define wijk wi w j wk , i, j k The degeneracy factor d i. j .k i j i j 3 6 ( eq. 2.30) ( eq. 2.33) The normalized Pijk(z,t) is given by 0 x(3) Pijk ( z, t ) dijk Ei E j Ek cos ( wi w j wk ) t ( i j k )z 4 (2.36) If we assume that the optical signals propagate as plane waves over Ae and distance L, then the power is wijk dijk x Pijk 8 Ae neff c (3) 2 2 P L PP i j k (using Fig 2.15 and 2.36) Pi . Pj and Pk are powers at wi w j wk 73 For example Pi Pj Pk 1mw , Ae 50 m 2 wi w j , d ijk 6 n 3.0 10 8 m2 w L 20 km Pijk 9.5 w about 20dB below Pi 1mw If another channel at wijk Then FWM will interfere the wijk channel. Practical FWM lacks of phase matching => No significant influence (in normal fibers) 74 2.4.9 New Optical Fiber Types A. DSF is not suitable for WDM due to nonlinear effect. To reduce nonlinear effect (different group velocities lack phase matching) =>to develop nonzero-dispersion fibers (NZ-DSF) a chromatic dispersion 1~6 ps/nm-km or -1 ~ -6 ps/nm-km NZ-DSF has most advantage of DSF (in c-band) 75 76 Large Effective Area Fiber (LEAF) 1 nonlinear effect A for fix power e 77 78 Positive and Negative Dispersion Fibers For Chromatic dispersion compensation 79 80