Propagation of signals in optical fibres

11 11 Fiber optic link OR Propagation of signals in optical fibres??? Simon-Pierre Gorza BEST Course - 213 http://opera.ulb.ac.be/

Outline: 2.1 Geometrical-optics description 2.2 Electromagnetic analysis Fields and modes 2.3 Linear properties of single mode optical fibers 2.4 Nonlinear properties of single mode optical fibers 2.5 Optical amplification 2

2.1 Geometrical-optics description Step-index (SI) waveguides Optical fibre generic structure: Structure of fibres for telecommunication applications: Core surrounded by a cladding The core and the cladding consist of ultrapure silica mixed with dopants The purpose of the dopants is to change the refractive index r Θ z 2b (r,θ,z) 2a Two-dimensional description (r,z) of the light confinement 3

refractive index profile r rz plan r +b n n 1 n 2 +a -a z -b core (also slab or film) n 1 >n 2 cladding Assumptions: -geometrical dimensions aand b >> wavelength λ - core and cladding: homogeneous dielectrics - translation symmetry in the z direction (i.e. no curvature) -no yor Θdependence - lossless medium -infinite extension of the cladding(b>> a) 4

Geometrical description = rays propagation Snell s law = refraction and (total) internal reflection cladding n 2 unguided ray air n = 1 θ i < θ ic θ r core n 1 > n 2 Φ Φ > Φ c Φ guided ray cladding n 2 If the corresponding ray is guided Φ c : criticalangle 5

Numerical Aperture NA NA is linked with the half angle of the acceptance cone of the waveguide Let: the normalized index difference and: the average refractive index weak guiding approximation (n 1 n 2 ): Typical order of magnitude: For silica-based optical fibers: n.1 & n 1 1.485 air n = 1 cladding n 2 = 1.47 θ ic = 12 θ i < 12 θ ic = 12 8 core n 1 = 1.485 82 82 NA.21 cladding n 2 = 1.47 6

Multipath dispersion (or Modal dispersion): Time of flight difference between the guided rays waveguide legth: L t I in I out t t cladding n 2 L v Φ c core n 1 L r cladding n 2 with n.1; n 1 1.485 & L = 1 mm t 5ps large NA large t( n) limitation of the bit rate 7

Bit rate x Length (BL) product: Limitation due to the largest time delay between the guided rays (modal dispersion) fibre length: L I in I out or 1? Inter-Symbol Interference(ISI) T B t T B t Oder of magnitude of the BLproduct limitation? (under the assumption that the size core >> λ) Typical order of magnitude (optical fibers): n,1 ; n 1 1,485 & L= 1 km BL < 2x1 1 bit s -1 m A signal at a bit rate of 2Mbits -1 can only be transmitted over a distance of 1 m (for multimode SI optical fiber)!! 8

2.2 Electromagnetic analysis Fields and modes Step-index (SI) fibres Electromagnetic description of wave-guiding = electromagnetic fields, solution of Maxwell s equations and satisfying the appropriate boundary conditions at the core-cladding interface Hypothesis: Media (core, cladding): Dielectric, nonmagnetic, linear, isotropic, lossless Homogeneous (step-index waveguide), z independent Core index = n 1 > n 2 = cladding index Infinite extension of the cladding (2b >> 2a) r Θ z 2b 2a! No hypothesis about the core size 9

Maxwell s equations Faraday s law Electric field Magnetic induction Electric displacement Magnetic field Ampère s law Gauss s law Dielectric Material properties: constitutive relations Nonmagnetic µ = magnetic permeability = 4 x 1-7 N A -2 ε = dielectric permittivity = 8,85 x 1-12 C 2 N -1 m -2? Polarization density field where 1

Polarization density field Microscopic origin: induced atomic dipole Dielectric medium: atomic density N +e -e +e -e Polarization density field Electric susceptibility linear, permanent isotropic, causal 11

Helmholtz equations in the frequency domain Resolution in the frequency domain (or equivalently for one frequency of the EM field) Fourier transform of the fields and dielectric constant: Helmhotz equation: negligible attenuation real κ Buck (1995) 12

6 coupled equations where λ= wavelength in the vacuum = 2 c/ω k = wavenumber in the vacuum = ω/c k = wavenumber = n k k 1 =n 1 k (in the core) k 2 = n 2 k = (in the cladding) Outline of the resolution: 1. take the symmetry properties of the structure (x,y,z) into account 2. separately solve the problem in each homogeneous medium (SI) of refractive index n 1 (i.e. in the core where x < a) or n 2 (i.e. in the cladding where x > a) 3. apply the boundary conditions at the core-cladding interfaces (x = ± a) i.e. continuity of the tangential components of E (E y and E z ) and H (H y and H z ), as well as of the normal component of D (D x = ε κ(x) E x ) and B (B x = µ H x ) 13

FromMaxellequations, with E z and/or H z E y = E y (E z, H z ) H y = H y (E z, H z ) 1 E x = E x (E z, H z ) H x = H x (E z, H z ) Solution: forward propagating wave along the z direction 2 Cylindrical coordinates r = x 2 + y 2 β: propagation constant symmetry of the problem 14

& 3 Separate the dependence on r and θ: solution of the form rotational symmetry (2πperiodicity in θ)? N(integer number): angular (or azimuthal) mode number? Bessel equation 15

Qualitative behavior of Bessel functions: J, J 1 Y, Y 1 Solution of! x 2 d 2 G N dx 2 +x dg N dx + (x2 N 2 )G N =! guided mode core (x < a) cladding (x a)! K, K 1! I, I 1 Solution of x 2 d 2 G N dx 2 +x dg N dx (x2 + N 2 )G N = 16

Bessel equation for R z The angular dependence of the field Θ z (θ) is an harmonic function: The radial dependence of the filed R z (r) is a Bessel function: at given N N=, ±1, ±2, ±3,... core (x < a) : cladding (x a): & & evanescent field (guided mode) where: σ 2 > & ξ 2 > guided mode n 1 k > β & β> n 2 k The wavenumber of a guided mode βreflects its spreading both in the core andthe cladding 17

Solution in the core: r a N Solution in the cladding: r a N where: where: For a given azimuthal number N, all the possible q solutions (radial number) have to be found (if the are any solutions ) How? by applying the field continuity conditions for E and H, D and B at the core-cladding interface (1) (2) E z core = E z cladding E θ core = E θ cladding ε 1 E r core = ε 2 E r cladding (3) (4) H z core = H z cladding H θ core = H θ cladding µ H r core = µ H r cladding at r = a homogenous system of 4 equations satisfied by A, B, Cand D 18

[M]. A B C D non trivial solution = det[m] = within the weakly guiding approximation n 1 n 2 ' J N (u) uj N (u) + K ' (w) N wk N (w) = ±N 1 u + 1 2 w 2 u J (u) m 1 J m (u) = w K (w) m 1 K m (w) Further simplification: properties of Bessel Functions Let V = ak n 1 2 n 2 2 is the normalized frequency Vis a dimensionless number that characterizes the waveguide for each mand for a given V: The qsolutionsu mq of this equation (and the corresponding β mq ) give the qguided LP mq modes 19

graphical resolution for m= 1 or (at V= 2): single-mode fiber V = 2 q=1 q=2 q=3 q=4 m= 1: LP 1q modes m= : LP q modes J 1 (u) J (u) = u w K 1 (w) K (w) J 1 (u) J (u) = +w u K 1 (w) K (w) J. Buck (24) q=1 q=2 zeros of the J Bessel function q=3 u ΗΕ11 = 1,5 only one HEsolution: mode LP 1 the LP 1 solution always exists!! For V< 2,45 only the LP 1 mode is guided: the waveguide is single-mode 2

V = 8 zeros of the J 1 Bessel function q=1 q=2 u=v=8 q=3 q=4 m= 1: LP 1q modes m= : LP q modes J 1 (u) J (u) = u w K 1 (w) K (w) J 1 (u) J (u) = +w u K 1 (w) K (w) q=1 q=2 q=3 J. Buck (24) zeros of the J Bessel function u ΗΕ11 = 2.15; u ΗΕ12 = 4.87; u ΗΕ13 = 7.4 u LP11 = 3.4; u LP12 = 6.1 5 solutions LP q or LP 1q ForV= 8, there are at least 5 LP mq modes that couldpropagate together: the waveguide is multimode 21

Transverse intensity distribution I(x,y) y LP1 I +a -a -a +a -a +a x LP11 LP21 y I J. Buck (1995) The only guided mode in a single-mode fiber (equiv. axial ray of the geometrical optics) y I +a +a -a -a +a x -a -a +a -a +a -a +a x 22

Transverse intensity distribution I(x,y) LP1 I y +a -a +a -a The only guided mode in a single-mode fiber (equiv. axial ray of the geometrical optics) The amplitude of the fieldof the fundamental modeis well approximated by a Gaussian distribution: 2w is known as the Mode field diameter (MFD x w > a -a +a at V = 2.4 LP 1 r/a [ G. Agrawal, "Fiber-optics communication systems", Wiley (1997)] J. Buck (1995) 23

Summary: In an optical fibre, a guided EM wave can be decomposed as a sum over the guided modes of the fibre Each mode has its own propagation constant (β mq ) and therefore propagates at is own phase and group velocity The propagation constant shows the spreading of the mode in the core and the cladding SI fibers are characterized by the dimensionless parameter V An optical fiber is single-mode if only the fundamental mode is guided, i.e. for SI fibres if: V = 2π λ a n 2 n 2 1 2 < 2,45 λ> λ cutoff small n small a (typ.: 3 x 1-3 ) (typ.: 5 µm) An optical fiber is single mode for wavelengths larger than its cutoff wavelength 24

Summary: Two types of fibers for telecommunication applications: Multimode fibers: core diameter 5 µm or 62.5 µm - cheap, easy to align - suffer from modal dispersion, suitable for applications with low B.L product Single mode fibers: core diameter around 9 µm - more expensive, require carful alignment - for applications with high B.L product, and typically for long-haul applications with more than 4 km link lengths 25

2.3 Linear properties of single mode optical fibers A. Signal attenuation in single-mode fibers 1 1??? L: length of the single-mode fiber When a signal is propagating in a fiber, its power is reduced due to fiber losses. Since the sensitivity of any detector is limited, a minimum power has to reach the detector. This sets a limitation to the maximum propagation distance (before detection or amplification), given the signal power at the fiber input. Decrease of the power (Beer s law) : dp(z) dz = α P P(L) = P()exp α m ( L m ) α m = Attenuation coefficient [m -1 ] P(z = L): Output power P(z = ): Power launched at the input end P z α[db/km] 1 1 km log 1 P(z = 1km) P(z = ) 26

Loss mechanisms and loss spectrum of single-mode fibers O-band (126nm 136nm) α[db km -1 ] OH - OH - OH - [ G. Agrawal, "Fiber-optics communication systems", Wiley (1997)] UV and IR absorption bysilica molecules Rayleigh scattering on fiber inhomogeneities(amorphous material) Material absorption by impurities in fibres:dopants (GeO 2, P 2 O 5,..) and OH ions C-band (153nm 1565nm) Typical absorption in SMF fibres:.2 db/km at 1.55 µm 27

B. Chromatic dispersion A single-mode fiber has a normalized frequency V that is sufficiently small to support only the fundamental LP 1 mode Single-mode waveguide no modal dispersion, the energy is carried by a single mode But: there is still a pulse broadening mechanism called group velocity dispersion (GVD) or intramodal dispersion or simply fiber dispersion This GVD comes from: 1. All signals carrying information and/or all optical sources have a finite spectral width (signal monochromatic wave) & 2. the refractive index n of silica is wavelength dependent, i.e. n is -slightly- different for each spectral components of the pulse = material dispersion 3. the propagation constant β of the guided mode HE 11 is wavelength dependent and is therefore -slightly- different for each spectral components of the pulse = waveguide dispersion 28

Length of the single-mode waveguide: L I in τ temporal intensity I out τ out τ + T t propagation T t f 1/τ spectral density f 1/τ f f f f where: & Group velocity v g : (n g = group index") (T= time of flight of the pulse ) The group velocity is not the same for all the pulse spectrum frequencies the pulse width changes when its propagates in the fibre 29

Temporal pulse spreading T: β 2 : GVD parameter D: Dispersion parameter [psnm -1 km -1 ] Spectral density (a.u.) Normalized intensity (a.u.) z = (km) 1.5 2 1-1 -15-1 -5 5 1 15-2 Time : t - z/v g (ps) 1 2 1.5-1 -2-1 1 2-2 Frequency : ω - ω (radthz) Phase (rad) Phase (rad) Spectral density (a.u.) Normalized intensity (a.u.) z = 1 (km) 1 2 1.5-1 -15-1 -5 5 1 15-2 Time : t - z/v g (ps) 1.5 2 1-1 -2-2 -1 1 2 Frequency : ω - ω (radthz) Phase (rad) Phase (rad) t= 1 ps over1 km 3

Different wavelengths propagate at a different groupvelocity First order estimate of the limitation of the BLproduct for a single-mode waveguide base on the spectral width λ: to avoid severe intersymbol interference but where λ is either the spectral width of the laser source or is related to the pulse duration Typical order of magnitude: D 1 psnm -1 km -1 at λ= 133 nm ; λ 1 nm (laser diode) BL < 1 +15 bit s -1 m BL < 1 Tbits -1 km Such lightwavesystems support -at the first order-a bit rate Bof 1 Gbits -1 over a distance of 1 km!! Importance of the spectral width of the source (coherence) for demanding applications where the BLproduct is large 31

Propagation of the E-field: description in the spectral domain (ω): The propagation constant β(ω) can be expanded in a Taylor series around the carrier frequency ω : where Slowly varying envelope (SVE) of the field: β : propagation constant at ω new variables: β = β-β and ω = ω - ω 32

ITF (β ) Up to second order: propagation quadratic spectral phase In the spectral domain, the propagation in a linear dispersive medium changes the spectral phase. The spectral density remains constant (lossless, linear). Spectral phase: Fourier transform properties: group delay of the pulse frequency dependent group delay temporal pulse broadening 33

Example: propagation of a Gaussian pulse in a single-mode fiber with GVD Parameters: -L = 1 km - D = 1ps/nm/km ( anomalous dispersion ) - λ= 133 nm T = 1ps 1 z = (km) 1 z = 1 (km) I (a.u.).5 1-1 1 Time (ps) 1-1 Phase (rad) I (a.u.).5 1-1 1 Time (ps) 1-1 Phase (rad) I (a.u.).5-2 -1 1 2 Frequency (radthz) 1-1 Phase (rad) I (a.u.).5-2 -1 1 2 Frequency (radthz) 1-1 Phase (rad) 34

T = 1ps 1 z = (km) 1 z = 1 (km) I (a.u.).5 1-1 1 Time (ps) 1-1 Phase (rad) I (a.u.).5 1-1 1 Time (ps) 1-1 Phase (rad) I (a.u.).5-2 -1 1 2 Frequency (radthz) 1-1 Phase (rad) I (a.u.).5-2 -1 1 2 Frequency (radthz) 1-1 Phase (rad) Shorter pulses broaden faster dispersion limitation of the bit rate Bfor a given propagation length 35

Physical origins and control of the dispersion D: 1. Material dispersion 2. Profile dispersion 3. Waveguide dispersion negligible D M D W D =D M +D W D < Normal dispersion D > Anomalous dispersion zero-dispersion wavelength ω D W <!! [ G. Agrawal, "Fiber-optics communication systems", Wiley (1997)] 36

The control of β(ω), through D W, allows to fix λ ZD in the wavelength range above 13 nm ( = λ ZD for pure silica) Different commercially available optical fibres: SMF DFF Reverse dispersion RDF DSF Dispersion compensating DCF [ G. Agrawal, "Fiber-optics communication systems", Wiley (1997)] ω SMF : Single mode Fibre DSF : Dispersion-Shifted Fibre NZDSF : Non-Zero Dispersion-Shifted Fibre DCF : Dispersion Compensating Fibre RDF : Reverse Dispersion Fibre DFF : Dispersion-Flattened Fibre 37

Dispersion management: Optical link= concatenation of pieces of the sametype of fibre temporal pulse spreading: L BL product limitation: quadratic spectral phase: Optical link= concatenation of Npieces of differenttypes of fibres I t β 2 L I t quadratic spectral phase: temporal pulse spreading: L 1 L 2 Thepulse spreading is minimum for I I I t t t -SSMF: Standard Single-Mode Fibre: D= 17 psnm -1 km -1 at 1,55 µm -DCF: Dispersion CompensatingFibre: D= -8ps.nm -1 km -1 at 1,55 µm. 38

C. Polarization mode dispersion: principal axis The fibre core of single mode fibres is not perfectly circular: polarization modes have slightly different propagation constants polarization modes have slightly different group velocities Mechanical and thermal stresses add also birefringence to the fibre The optical link is usually the result of splices between fibreswith different relative orientations of their principal polarization axes fast mode, lower β y slow mode, larger β x q 2 slow slow slow fast fast fast q n t Pulse spreading due to polarization mode dispersion The axis orientation is random statistical description of the pulse broadening (= average pulse broadening) through the parameter PMD C : 1 B L < 1 PMD c 39

Bit rate limitations 1 B L < 1 PMD c «Old» fibre: PMDc= 1-1 ps/ km Actualfibres: PMDc=.2 ps/ km Data rate (Gb/s) L max (km) (PMD c =.2 ps/ km) L max (km) (PMD c = 1 ps/ km) 1 2.5x1 5 1 4 16 5.25 When the signal wavelength is close to the zero dispersion wavelength of the fiber or when the chromatic dispersion is compensated to minimize its effect, PDMhas to be taken into account PMD becomes an important limiting parameter in high-bit rate systems 4

Summary: Fibre losses cause exponential attenuation of the signal power limit the maximum distance of the fiber link before detection or amplification Fibre attenuation is minimum around 1.5 µm in SMF made up of silica (~.2 db/km) In SMF, chromatic dispersion and polarization mode dispersion set a limit on the maximum BL product supported by a fiber link Chromatic dispersion in SMF can easily be compensated by propagating the signal in a piece of DCF fiber, providing that In actual systems, PMD becomes significant at ultra-high bit rate (1 Gbits/s and above) 41

2.4 Nonlinear properties of single modeoptical fibers Nonlinear interactions between the optical signal and the matter the optical fibers are made of (silica) begin to appear when the signal powers are increased to achieve longer propagation distances. When the input signal power is increased, various nonlinear effects take place in single- as well as in multi-channel systems. In WDM systems, nonlinear effects result in power transfer between channels and therefore degrade the signal-to-noise ratio and increase the crosstalk. Nonlinear effects - degrade the signal-to-noise ratio - increase the crosstalk - place an upper limit to the input power NL effects degrade the overall system performances Three of the most interesting phenomenon are Stimulated Brillouin Scattering, Stimulated Raman Scattering and Four-Wave Mixing 42

n Ι A. Stimulated Brillouin Scattering (SBS): interference pattern forward & backward waves z Brillouin scattering physical origin: Electrostriction (tendency of a material to be compressed in the presence of an electric field) Material compression refractive index rises due to the photoelastic effect Light is scattered by the refractive index variations in the backward direction @-. 1213!/ 8C12!G3Z; I J V HV V backward output power ` V H` HV J ` J V O6[1-13!/ 8C12!G3Z; I forward output power HB-OF_F7-14 There is a power thresholdabove which almost all the incident power is back scattered by SBS HV H` V ` A</ <!/ 8C12!G3Z; I! Optical Fiber, Cables and Systems, ITU-T manual (29) 43

B. (Spontaneous) Raman Scattering: Measured spectra in a SMF fiber virtual state Signal ω vibrational states Ω ground state (ω Ω) Raman scattering results in the excitation of the silica molecules input signal Spontaneously scattered light L. Thevenaz, Advacedfiber optics, chapter 9, EPFL PRESS (211) Raman scattering occurs in both forward and backward directions and decreases the signal output power The frequency shift between the incident and the scattered waves is of the order of Ω/2π 13 THz for silica fiber Because of the amorphous nature of silica glass, the bandwidth of Raman scattering is quite large ν 6 8 Thz 44

Stimulated Raman Scattering (SRS): virtual state Pump ω Signal: p ω s vibrational states Ω ground state Raman amplifiers take advantage of SRS to amplify the signal P p SMF In WDM systems, the signal at λ s behaves as a Raman pumpfor channels at lower wavelengths. λ SMF λ transfer of energy relative optical power signal depletion for short frequency channels & Raman-induced crosstalk for high frequency channels 45

C. Four-Wave mixing (FWM) Linear medium: Nonlinear medium: (local, isotropic, instantaneous) Far from electronic resonances, the optical nonlinear response can be describe by generalizing the expression of the polarization: 1 st order susceptibility: linear response 2 d order nonlinear susceptibility 3 d order nonlinear susceptibility In isotropic media as in silica fibers, the second order nonlinear susceptibility is zero. To the lowest order, the nonlinear polarization is thus: 46

Let E = Ee jω t + E * e jω t Two signals at ω 1 and ω 2 : E = E 1 e j(ω 1 ω )t + E 2 e j(ω 2 ω )t E 2 E = E 1 2 E 1 e j(ω 1 ω )t + E 1 2 E 2 e j(ω 2 ω )t + E 2 2 E 1 e j(ω 1 ω )t + E 2 2 E 2 e j(ω 2 ω )t +E 1 2 E 2 * e j(2ω 1 ω 2 ω )t + E 1 E 2 2 e j(ω 1 ω )t + E 2 E 1 2 e j(ω 2 ω )t + E 2 2 E 1 * e j(2ω 2 ω 1 ω )t source terms at new frequencies 2ω 1 -ω 2 and 2ω 2 -ω 1 E 1 c 2 E t 2 = µ P t 2 δω = ω 2 ω 1 2ω ω ω 1 ω 2ω ω 2 1 2 2 1 = ω 1 δω = ω 2 +δω ω E 1 = E 1 e jω 1 t + c.c E 2 = E 2 e jω 2 t + c.c 47

FWM is responsible for energy transfer from each channel to neighbouring channels Momentum conservation: = β(2ω 1 ω 2 )+ β(2ω 2 ω 1 ) β(ω 1 ) β(ω 2 ) = β(ω 1 )+ β 1.(ω 1 ω 2 )+ 1 2 β 2.(ω 1 ω 2 )2 + β(ω 2 )+ β 1.(ω 2 ω 1 )+ 1 2 β 2.(ω 2 ω 1 )2 β(ω 1 ) β(ω 2 ) = β 2.(ω 1 ω 2 ) 2 β 2 = (perfect phase matching) FWM even between distant frequencies FWM is reduced for distant channel and/or unequally spaced channels In WDM systems, to limit the impact of FWM responsible for interchannel crosstalk: β 2 is kept locally high, but zero (or near to zero) in average to allow a high BLproduct Non zero dispersion shifted fibre (NZDSF): D 2-4ps/nm/km The power per channel is limited to about 1 mwin WDM systems 48

2.5 Optical amplification Signals are attenuated when propagating in fibres Dectectorsneed a minimum signal level to allow for low Bit Error Rates Nonlinear effects set a limit to the power of the signal launched at the fibre input The solution is to amplify the signal (or many signals in WDM systems) thanks to erbium doped fibre amplifiers (EDFA) The optical amplification is based onstimulated emission by excited erbium ions of Er-doped fibres The erbium ions are optically excited by absorption of laser radiation at 148nm or 98nm The amplification is possible for signals in the range 153nm-1565nm, that is the C-band Typical small signal gain of EDFA is 2-3 db The propagation distance is limited by the noise added by the EDFA (spontaneous emission): L max 1 km for 4-km amplifier spacing pump laser signal in ω E 2 E 1 ω Er 3+ fiber E 2, N 2 E 1, N 1 signal out 49

Summary: Nonlinear effects limit the maximum power of the input signal Three of the most interesting nonlinear effect are SBS, SRS and FWM These NL effects decrease the performances both in single wavelength systems (decrease of the signal power) and in WDM systems (inter-channels crosstalk) Erbium-doped amplifiers (and Raman amplifiers) are used to compensate for the losses The maximum link length is limitted by the degradation of the signal (noise) in the amplifiers 5

References Fiber-optics communication systems, G.P. Agrawal, Wiley (21), 4 th ed. (ISBN 978--47-5511-3) Fundamentals of optical fibers John A. Buck, Wiley & Sons (24) 2nd ed. (ISBN 978--471-22191-3) Understanding fiber optics Jeff Hecht, Prentice Hall (22) 4th ed. (ISBN -13-27828-9) Optical Fibers, cables and systems, ITU-T Manual (29) 51