Optical Fiber Concept

Optical Fiber Concept Optical fibers are light pipes Communications signals can be transmitted over these hair-thin strands of glass or plastic Concept is a century old But only used commercially for the last ~30 years when technology had matured 5 5

Why Optical Fiber Systems? Optical fibers have more capacity than other means (a single fiber can carry more information than a giant copper cable!) Price Speed Distance Weight/size Immune from interference Electrical isolation Security 6 6

Optical Fiber Applications Optical fibers are used in many areas > 90% of all long distance telephony > 50% of all local telephony Most CATV (cable television) networks Most LAN (local area network) backbones Many video surveillance links Military 7 7

Optical Fiber Technology An optical fiber consists of two different types of solid glass Core Cladding Mechanical protection layer 970: first fiber with attenuation (loss) <0 db/km 979: attenuation reduced to 0. db/km commercial systems! 8 8

Optical Fiber Communication Optical fiber systems transmit modulated infrared light Fiber Transmitter Components Receiver Information can be transmitted over very long distances due to the low attenuation of optical fibers 9 9

00 km 0 km km 00 m 0 m m 0 cm cm Frequencies in Communications wire pairs coaxial cable waveguide frequency 3 khz Submarine cable Telephone 30 khz Telegraph 300 khz TV Radio Satellite Radar 3 MHz 30 Mhz 300 MHz 3GHz 30 GHz µm optical fiber wavelength ee Telephone Data Video 300 THz 0 0

Frequencies in Communications Data rate Optical Fiber: > Gb/s Micro-wave ~0 Mb/s Short-wave radio ~00 kb/s Long-wave radio ~4 Kb/s Increase of communication capacity and rates requires higher carrier frequencies Optical Fiber Communications!

Optical Fiber Optical fibers are cylindrical dielectric waveguides Dielectric: material which does not conduct electricity but can sustain an electric field Cladding diameter 5 µm n Core diameter from 9 to 6.5 µm n Cladding (pure silica) Core silica doped with Ge, Al Typical values of refractive indices Cladding: n.460 (silica: SiO ) Core: n.46 (dopants increase ref. index compared to cladding) A useful parameter: fractional refractive index difference δ (n -n ) /n <<

Fiber Manufacturing Optical fiber manufacturing is performed in 3 steps Preform (soot) fabrication deposition of core and cladding materials onto a rod using vapors of SiCCL 4 and GeCCL 4 mixed in a flame burned Consolidation of the preform preform is placed in a high temperature furnace to remove the water vapor and obtain a solid and dense rod Drawing in a tower solid preform is placed in a drawing tower and drawn into a thin continuous strand of glass fiber 3 3

Fiber Manufacturing Step Steps &3 4 4

Light Propagation in Optical Fibers Guiding principle: Total Internal Reflection Critical angle Numerical aperture Modes Optical Fiber types Multimode fibers Single mode fibers Attenuation Dispersion Inter-modal Intra-modal 5 5

Total Internal Reflection Light is partially reflected and refracted at the interface of two media with different refractive indices: Reflected ray with angle identical to angle of incidence Refracted ray with angle given by Snell s law! Angles θ & θ defined with respect to normal! Snell s law: n sin θ n sin θ θ θ n > n θ n n Refracted ray with angle: sin θ n / n sin θ Solution only if n / n sin θ 6 6

Total Internal Reflection Snell s law: If θ >θ sin θ c n / n c No ray n sin θ n sin θ is refracted! θ n θ c n n θ n > n n θ n n n n θ For angle θ such that θ >θ C, light is fully reflected at the core-cladding interface: optical fiber principle! 7 7

Numerical Aperture For angle θ such that θ <θ max, light propagates inside the fiber For angle θ such that θ >θ max, light does not propagate inside the fiber n θ max n θ Example: n.47 c n.46 NA 0.7 NA n n n sinθmax n n n δ with δ << n n Numerical aperture NA describes the acceptance angle θ max for light to be guided 8 8

Theory of Light Propagation in Optical Fiber Geometrical optics can t describe rigorously light propagation in fibers Must be handled by electromagnetic theory (wave propagation) Starting point: Maxwell s equations E B T () H J + D T () D ρ f (3) B 0 (4) with B µ 0 H + M : Magnetic flux density D ε 0 E + P : Electric flux density J 0 : Current density ρ f 0 : Charge density 9 9

Theory of Light Propagation in Optical Fiber Pr,t ( ) P L ( r,t)+ P NL ( r,t) P L P NL + ( r,t) ε 0 χ () ( t t )Er,t ( ) dt : Linear Polarization χ () : linear susceptibility ( r,t): Nonlinear Polarization We consider only linear propagation: P NL (r,t) negligible 0 0

Theory of Light Propagation in Optical Fiber Ert (, ) PL (,) r t Ert (,) + µ 0 c t t + i t We now introduce the Fourier transform: E% ω ( r, ω) E(r,t)e dt k Ert (,) k ( i ω ) E% ( r, ω ) k t ω c () And we get: Er %(, ω) Er %(, ω ) + µ 0ε0χ ( ω) ω Er %(, ω) which can be rewritten as ω () Er % (, ω ) + cµ 0ε0χ ( ω) Er (, ω) 0 % c ω i.e. Er % (, ω) ε( ω) Er % (, ω) 0 c -

Theory of Propagation in Optical Fiber αc ε(ω) n + i + () ω with n R χ (ω) and α [ ] f ti i d ω cn(ω) I χ () (ω) [ ] n: refractive index α: absorption E (r,ω) ( E (r,ω)) E (r,ω) E (r,ω) E (r,ω) D (r,ω) 0 ( ) E (r,ω) + n ω c E (r,ω) 0 : Helmoltz Equation!

Theory of Light Propagation in Optical Fiber Each components of E(x,y,z,t)U(x,y,z)e jωt must satisfy the Helmoltz equation n n for r a U + n k 0 U 0 with n n for r > a k 0 π /λ Assumption: the cladding radius is infinite In cylindrical coodinates the Helmoltz equation becomes n n U U U for r a U r + r r + r ϕ + z + n k 0 U 0 with n n for r > a k 0 π /λ 0 Note: λω /c y x E r φ E z z r E φ 3 3

Theory of Light Propagation in Optical Fiber U U(r,φ,z) U(r)U(φ) U(z) Consider waves travelling in the z-direction U(φ) must be π periodic U(φ) e -j lφ, l0,±,± integer U ( r, ϕ, z) d F + dr r F( r) e df + n dr jlϕ k 0 e jβz β with l 0, ±, ±... Plugging into the Helmoltz Eq. one gets : l r F U(z) e -jβz n n for r a 0 with n n for r > a k0 π / λ0 One can define an effective index of refraction n eff β k 0 n eff is the such that β ω c n, n < n propagation eff eff < n constant 4 4

Theory of Propagation in Optical Fiber A light wave is guided only if nk0 β nk 0 We introduce κ n k 0 ( ) β γ β ( n k 0 ) κ + γ k ) 0 ( n n ) k 0 NA : constant! Note : κ,γ 0 κ,γ :real We then get : d F dr + df r dr + κ l F 0 for r a r d F dr + df r dr γ + l F 0 for r > a r 5 5

Theory of Propagation in Optical Fiber The solutions of J ( κr ) ( γr ) the equations are of F ( r) J for ρ a l l l : Bessel function of F ( r ) K for ρ > a l K l l st the form : kind with order l : Modified Bessel function of with κ ( n k 0 ) β ( ) γ β n k 0 κ + γ k 0 n ( n ) k 0 NA : constant! st kind with order l 6 6

Examples l0 l3 K 0 (γr) J 0 (κr) a K 0 (γr) r K 3 (γr) a J 3 (κr) a K 3 (γr) r F(r) J 0 (κr) for r a K 0 (γr) for r < a F(r) J 3 (κr) for r a K 3 (γr) for r < a 7 7

8 Characteristic Equation Boundary conditions at the core-cladding interface give a condition on the propagation constant β (characteristics equation) β solving can be found by The propagation constant g p p g β ( q ) β β l K J n K J lm l l l l + Γ + Κ Γ + Κ ) ( ) ( ) ( ) ( the characteristic equation : can be found by solving The propagation constant ' ' ' ' aκ aγ k n K J n K J l l l l Γ Κ Γ + Κ Γ Γ + Κ Κ Γ Γ + Κ Κ and with ) ( ) ( ) ( ) ( 0 For each l value there are m solutions for β Each value β lm corresponds to a particualr fiber mode 8

Number of Modes Supported by an Optical Fiber Solution of the characteristics equation U(r,φ,z)F(r)e -jlϕ e -jβ lm z is called a mode, each mode corresponds to a particular electromagnetic field pattern of radiation The modes are labeled LP lm Number of modes M supported by an optical fiber is related to the V parameter defined as M is an increasing function of V! V ak 0 NA π a λ n n If V <.405, M and only the mode LP 0 propagates: the fiber is said to be Single-Mode 9 9

Number of Modes Supported by an Optical Fiber π a λ Number of modes well approximated by: M V /, where V ( n n) n eff n n n.0 0.8 0.6 0.4 core LP 0 LP 0 3 4 3 6 5 3 03 3 4 0. 7 04 5 8 33 0 4 6 8 0 V Example: a 50 µm n.46 V7.6 δ0.005 M55 λ.3 3 µm If V <.405, M and only the mode LP 0 propagates: Single-Mode fiber! cladding 30 30

Examples of Modes in an Optical Fiber λ 0.638 µm a 8.335 µm n.4640 δ 0.034 3 3

Examples of Modes in an Optical Fiber λ 0.638 µm a 8.335 µm n.4640 δ 0.034 3 3

Cut-Off Wavelength The propagation constant of a given mode depends on wavelength [β (λ)] The cut-off condition of a mode is defined as β (λ)-k 0 n β (λ)-4π n /λ 0 There exists a wavelength λ c above which only the fundamental mode LP 0 can propagate 0 p p g π V <.405 λc na δ. 84an δ.405.405 λc λc or equivalently a 0.54 n δ n δ π Example: a 9. µm n.4690 δ0.004 λ c ~. µm 33 33

Single-Mode Guidance In a single-mode fiber, for wavelengths λ >λ c ~.6 µm only the LP 0 mode can propagate 34 34

Mode Field Diameter The fundamental mode of a single-mode fiber is well approximated by a Gaussian function F ( ρ ) Ce ρ w 0 where C is a constant and w the mode size A good approximation for the mode size is obtained from w w 0 0.69.879 a 0.65 + + for. < V 3/ 6 V V a for V >. 4 ln( V ) 0 <. 4 Fiber Optics Communication Technology-Mynbaev & Scheiner 35 35

Types of Optical Fibers Step-index single-mode n Cladding diameter 5 µm Core diameter from 8 to 0 µm n Refractive index profile n n n δ 0.00 r 36 36

Types of Optical Fibers Step-index multimode n Cladding diameter from 5 to 400 µm Core diameter from 50 to 00 µm n Refractive index profile n n n δ 0.0 r 37 37

Types of Optical Fibers Graded-index multimode n Cladding diameter from 5 to 40 µm Core diameter from 50 to 00 µm n n Refractive index profile n n r 38 38

Attenuation Signal attenuation in optical fibers results form 3 phenomena: Absorption Scattering Bending Loss coefficient: α α depends on wavelength P Out P e in αl P Out 0 0log 0 α L 4.343 α L P in ln(0) α is usually expressed in units of db/km : α db 4.343α For a single-mode fiber, α db 0. db/km @ 550 nm 39 39

Scattering and Absorption Short wavelength: Rayleigh scattering induced by inhomogeneity of the refractive index and proportional to /λ 4 Absorption Infrared band Ultraviolet band 4 st window nd 3rd 80 nm.3 µm.55 µm IR absorption.0 0.8 Rayleigh scattering α /λ 4 Water peaks 0.4 UV absorption 0. 3 Transmission windows 80 nm 300 nm 550 nm 0. 0.8.0..4.6.8 Wavelength (µm) 40 40

Macrobending Losses Macrodending losses are caused by the bending of fiber Bending of fiber affects the condition θ < θ C For single-mode fiber, bending losses are important for curvature radii < cm 4 4

Microbending Losses Microdending losses are caused by the rugosity of fiber Micro-deformation along the fiber axis results in scattering and power loss 4 4

Attenuation: Single-mode vs. Multimode Fiber 4 Fundamental mode Higher order mode 0.4 SMF 0. MMF 0. 0.8.0..4.6.8 Wavelength (µm) Light in higher-order modes travels longer optical paths Multimode fiber attenuates more than single-mode fiber 43 43

Dispersion What is dispersion? Power of a pulse travelling though a fiber is dispersed in time Different spectral components of signal travel at different speeds Results from different phenomena Consequences of dispersion: pulses spread in time t t 3 Types of dispersion: Inter-modal dispersion (in multimode fibers) Intra-modal dispersion (in multimode and single-mode fibers) Polarization mode dispersion (in single-mode fibers) 44 44

Dispersion in Multimode Fibers (inter-modal) Input pulse Input pulse Output pulse t t In a multimode fiber, different modes travel at different speed temporal spreading (inter-modal dispersion) Inter-modal dispersion limits the transmission capacity The maximum temporal spreading tolerated is / bit period The limit is usually expressed in terms of bit rate x distance 45 45

Dispersion in Multimode Fibers (Inter-modal) Fastest ray guided along the core center Slowest ray is incident at the critical angle n n θ c Slow ray θ L T Fast ray n n n L L c n c n L c T T with T v L L FAST FAST SLOW n n v SLOW FAST L n n T L vfast c n SLOW L cosθ L c δ FAST FAST and T SLOW L π sin cos θ L v SLOW SLOW L sin θ n n C θ L 46 46

Dispersion in Multimode Fibers If bit rate B b L n i.e. δ < c n B cn or L B < nδ s We must have T < B Example: n.5and δ 0.0 B L<0Mb s - p Capacity of multimode-step index index fibers B L 0 Mb/s km 47 47

Dispersion in graded-index Multimode Fibers Input pulse Input pulse Output pulse t t Fast mode travels a longer physical path Slow mode travels a shorter physical path Temporal spreading is small Capacity of multimode-graded index fibers B L Gb/s km 48 48

Intra-modal Dispersion In a medium of index n, a signal pulse travels at the group velocity ν defined as: g dωω λ d β v g dβ πc d λ Intra-modal dispersion results from phenomena Material dispersion (also called chromatic dispersion) Waveguide dispersion Different spectral components of signal travel at different speeds The dispersion parameter D characterizes the temporal pulse broadening T per unit length per unit of spectral bandwidth λ: T D λ L d λ d β DIntra modal in units of ps/nm km dλ vg πc dλ 49 49

Material Dispersion Refractive index n depends on the frequency/wavelength of light Speed of light in material is therefore dependent on frequency/wavelength Input pulse, λ t Input pulse, λ t t 50 50

Material Dispersion Refractive index of silica as a function of wavelength is given by the Sellmeier Equation A λ Aλ A3λ n( λ) + + + λ λ λ λ λ λ with A A A 3 0.8974794, λ 3 9896.6 nm 0.696663, λ 68.4043 nm 0.407946, λ 6.44 nm 3 5 5

5 Material Dispersion λ λ λ β π λ d dn n c d d c v g / λ λ λ Input pulse, λ λ λ T p p t t L Input pulse, λ t λ λ λ λ λ λ d n d c L v d d L D L T g 5

Material Dispersion ps/nm/km m) Material (p 0-00 -400-600 -800 Material 0@ µ 0.4 0.6 0.8.0..4.6.8 Wavelength (µm) D M -000 DMaterial0@.7 µm λ d n DMaterial (units : ps/nm km) c dλλ 53 53

Waveguide Dispersion The size w 0 of a mode depends on the ratio a/λ : λ λ >λ w 0.69.879 a 0.65 + + 3/ V V 6 Consequence: the relative fraction of power in the core and cladding varies This implies that the group-velocity ν g also depends on a/λ D Waveguide d dλ vg λ π nc d λ d w λ 0 where w 0 is the mode size 54 54

Total Dispersion D Intra modal D Material + D Waveguide D Intra-modal <0: normal dispersion region D Intra-modal >0: anomalous dispersion region Waveguide dispersion shifts the wavelength of zero-dispersion to.3 µm 55 55

Tuning Dispersion Dispersion can be changed by changing the refractive index Change in index profile affects the waveguide dispersion Total dispersion is changed 0 n n 0 Single-mode Fiber n Single-mode Fiber n Dispersion shifted Fiber 0 Single-mode fiber: D0 @ 30 nm Dispersion shifted Fiber: D0 @ 550 nm -0 Dispersion shifted Fiber.3.4.5 Wavelength (µm) 56 56

57 Dispersion Related Parameters s/km of :group delay in units β β ω β d n c eff y g p modal Intra λ π β λ ω ω β λ β λ β ω c d d d d d d v d d D d v g g /km s group velocity dispersion parameter in units of : β λ λ ω λ λ d d d v d g 57

Polarization Mode Dispersion Optical fibers are not perfectly circular y x x In general, a mode has polarizations (degenerescence): x and y Causes broadening of signal pulse T L v gx v gy D Polarization L 58 58

Effects of Dispersion: Pulse Spreading Total pulse spreading is determined as the geometric sum of pulse spreading resulting from intra-modal and inter-modal dispersion T T Intermodal + T Intra-modal + T Polarization Multimode Fiber : T Single - Mode Fiber : T ( D L) + ( D λ L) Inter modal ( D λ L) + ( D L ) Intra modal Intra modal Polarization Examples: Consider a LED operating @.85 µm λ50 nm after L km, T5.6 ns D Inter-modal.5 ns/km D Intra-modal 00 ps/nm km Consider a DFB laser operating @.5 µm λ.nmafterl00km, T0.34 ns! D Intra-modal 7 ps/nm km D Polarization 0.5 ps/ km 59 59

Effects of Dispersion: Capacity Limitation Capacity limitation: maximum broadening</f a bit period T < B For Single - Md Mode Fiber, T LB < D Intra modal λ LD (neglecting polarization effects) Intra modal λ Example: Consider a DFB laser operating @.55 µm λ 0. nm D 7 ps/nm km LB<50 Gb/s km If L00 km, B Max.5 Gb/s 60 60

Advantage of Single-Mode Fibers No intermodal dispersion Lower attenuation No interferences between multiple modes Easier Input/output coupling Single-mode fibers are used in long transmission systems 6 6

Summary Attractive characteristics of optical fibers: Low transmission loss Enormous bandwidth Immune to electromagnetic noise Low cost Light weight and small dimensions Strong, flexible material 6 6

Summary Important parameters: NA: numerical aperture (angle of acceptance) V: normalized frequency parameter (number of modes) λ c : cut-off wavelength (single-mode guidance) D: dispersion (pulse broadening) Multimode fiber Used in local area networks (LANs) / metropolitan area networks (MANs) Capacity limited by inter-modal dispersion: typically 0 Mb/s x km for step index and Gb/s x km for graded index Single-mode fiber Used for short/long distances Capacity limited by dispersion: typically 50 Gb/s x km 63 63