Dielectric Waveguides and Optical Fibers. 高錕 Charles Kao

Dielectric Waveguides and Optical Fibers 高錕 Charles Kao 1

Planar Dielectric Slab Waveguide Symmetric Planar Slab Waveguide n 1 area : core, n 2 area : cladding a light ray can undergo TIR at the n 1 /n 2 boundary in realistic case, beam dia. >> 2a whole end of waveguide illuminated only at certain angles θ A and C in phase constructive interference only certain waves can exist in the guide ΔΦ(AC) = k 1 ( AB+BC ) 2φ = m( 2π ), m = 0, 1, 2 ( φ : phase change at TIR, k 1 : wavevector in n 1 ) AB + BC = BC cos(2θ) + BC = BC [ (2cos 2 θ 1 ) + 1 ] = 2d cosθ ( BC = d/cosθ, AB = BC cos2θ ) for wave to propagate k 1 2d cosθ 2φ = 2mπ ( Eletric field along x axis ) 2

Planar Dielectric Slab Waveguide Waveguide Condition for wave propagation along the waveguide k 1 2d cosθ 2φ = 2mπ for each m one allowed θ m and one corresponding φ m m θ m waveguide condition :, ( d = 2a ) ( a general condition, can be derived for the following situation also ) k can be resolved into 2 propagation const. ation const. along the guide) // guide axis z ransverse propagation const.) guide axis z const.) guide axis z const.) guide axis z onst.) guide axis z nst.) guide axis z 3

Planar Dielectric Slab Waveguide Field Distribution in Waveguide ray 1 and 2 meet at pt. C with the phase difference : ( w. ) ( y : a a ) E = E 1 + E 2 : a stationary standing wave E field pattern along y-direction travels along the guide axis z with a propagation constant β m : distribution E m (y) across the guide travels down the guide along z 4

Planar Dielectric Slab Waveguide Mode of Propagation m : mode number each m different θ m (κ m ), φ m E m (y) β m : distribution E m (y) across the guide travels down the guide along z different modes of propagation Light energy can be transported only along the guide via one or more of these possible modes of propagation. 5

Planar Dielectric Slab Waveguide Light Propagation in Slab Waveguide different m different E m (y) θ m β m different modes of propagation higher mode more reflection more penetration into cladding different modes travel down the guide at different group velocities a light pulse spreads as it travels along the guide 6

Planar Dielectric Slab Waveguide Single and Multimode Waveguides θ m must satisfy sinθ m > sinθ c ( i.e. θ m > θ c ) m ( 2V φ ) / π, V : V-number, V-parameter, normalized thickness, normalized freq. V-number depends on the waveguide geometry, 2a, and waveguide properties, n 1 and n 2 a characteristic parameter of the waveguide ( note : It can be shown that φ always 2V m 0 ) when V < π/2 2V/π < 1 m 2V / π < 1 m = 0 only fundamental mode (m=0) propagates along the dielectric slab waveguide single mode planar waveguide (and θ m 90 o, φ π ) @cut-off wavelength λ c V = π/2 when λ > λ c only fundamental mode will propagate 7

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Planar Dielectric Slab Waveguide TE and TM Modes Any electric field can be resolved into E and E // components E (plane of incidence & z) TE mode corresponding magnetic field of E //, B (corresponding magnetic field of E // ) (plane of incidence & z) TM mode E and E // experience different φ and φ // require different θ m to propagate different set of modes for E and E // modes associated with E : TE modes, TE m modes associated with E // : TM modes, TM m (n 1 -n 2 ) << 1 difference between φ at TIR for E and E // negligible small waveguide condition and cut-off condition same for TE and TM modes 9

Planar Dielectric Slab Waveguide Waveguide Modes Ex. A planar dielectric guide, 2a = 20 μm, n 1 = 1.455, n 2 = 1.440, λ = 900 nm find θ m for all the modes <Sol. > substitute waveguide condition : into θ c = arcsin(n 2 /n 1 ) = 81.77 o 10

Planar Dielectric Slab Waveguide V Number and Number of Modes Ex. A planar dielectric guide, 2a = 100 μm, n 1 = 1.490, n 2 = 1.470, λ = 1 μm estimate the number of modes <Sol. > since φ π ( for a multi-mode guide, V > 1 ) m 48.7 there are about 49 modes ( including m = 0 ) 11

Modal and Waveguide Dispersion Waveguide Dispersion Diagram v g along the guide = energy/information transportation velocity v g = dω/dβ θ m (λ) = θ m (ω) β m = k 1 sin θ m = β m (ω) v g of a given mode is a function of ω and the waveguide properties ( even if n 1 is independent of υ or λ ) with a, n 1, n 2, and m ω vs. β m ( dispersion diagram ) dω/dβ = v g all allowed propagation modes contained within the lines with slopes c/n 1 and c/n 2 ω cut-off corresponds to cut-off condition ( λ = λ c ) when V = π/2 v g changes from one mode to another (@ fixed freq.) v g changes with freq. (@ fixed mode) 12 λ c

Modal and Waveguide Dispersion Intermodal Dispersion at ω >> ω cut-off multimode operation lowest mode (m=0) has the slowest v g ( ~ c/n 1 ) highest mode has the highest v g different modes take different time to travel the guide modal dispersion, intermodal dispersion higher modes travel in cladding at higher speed ( n 2 <n 1 ) higher v g single mode waveguide has no intermodal dispersion modal dispersion : intermodal dispersion is usually smaller than the above due to an intermode coupling 13 λ c

Modal and Waveguide Dispersion Intramodal Dispersion when operated at a fixed mode : longer λ ( lower ω ) penetration in cladding propagate faster waveguide dispersion n 1 and n 2 dependence on λ broadening of a propagating light pulse material dispersion waveguide dispersion material dispersion intramodal dispersion 14 λ c

Step Index Fiber Step Index Fiber a cylindrically symmetric dielectric waveguide : often use coord. r, φ, z normalized index difference Δ = (n 1 n 2 ) / n 1 for practical fibers : Δ << 1 ( weakly guided fibers ) reflections and constructive interferences can occur in both x and y directions need two integers, l & m, to label possible traveling waves or guided modes 2 kinds of rays guided modes meridional ray skew ray 15

Step Index Fiber Guide Linear Polarized Wave meridional rays TE (no E z ) or TM (no B z ) type guided modes skew rays guided modes with both E z and B z ( or H z ) components hybrid modes ( EH or HE modes ) guided modes in weakly guided fibers ~ plane polarized ( E B z ) guided linear polarized ( LP ) mode along the fiber : 2l : # of max. around a circumference m : # of max. along r from core center l more skew ray contribution to the mode m associated with the reflection angle θ of the rays (as in the planar guide) 16

Step Index Fiber Intermodal Dispersion guided linear polarized ( LP ) mode along the fiber : different ( l,m ) modes different E lm (r,φ) β lm dispersion behavior (ω vs. β lm ) v g (l,m) different v g (l,m) intermodal dispersion broadening of input light pulse for fiber designed to allow only fundamental mode propagation ( LP 01 ) no intermodal dispersion 17

Step Index Fiber V- Number in Step Index Fiber V-number, normalized frequency, of a step index fiber :, n = (n 1 + n 2 ) / 2 w. proper choice of a and Δ can design V < 2.405 only fundamental mode LP 01 allowed single mode fiber if further reduce core size a LP 01 mode extends more into cladding, and may result in some power loss Typically single mode fibers have much smaller a and Δ than multimode fibers 18

Step Index Fiber V- Number in Step Index Fiber V-number, normalized frequency, of a step index fiber :, n = (n 1 + n 2 ) / 2 λ V, and when V > 2.405 multimode propagation at cut-off wavelength, λ c : V > 2.405 # of modes, M V 2 /2 a, n 1 M λ, n 2 M cladding diameter for a single-mode step index fiber is at least 10 times the core diameter to prevent field penetration of cladding and hence intensity loss 19

Step Index Fiber Normalized Propagation Constant propagation constant β lm depends on waveguide properties and the light source λ normalized propagation constant b, which depends only on V-number : ( β : β lm ) β = kn 2 (propagation in cladding material) b = 0 β = kn 1 (propagation in core material) b = 1 ( 0 < b < 1 ) given the V-number of the fiber b from the fig. β for allowed LP modes 20

Step Index Fiber Multimode and Single Mode Fiber Ex. A step index fiber, 2a = 100 μm, n 1 = 1.468, n 2 = 1.447, λ = 850nm the number of allowed modes =? <Sol. > since V > 2.405 there are numerous modes! - Ex. A step index fiber, n 1 = 1.468, n 2 = 1.447, λ = 1.3 μm a =? for single mode operation <Sol. > a 2.01 μm 1. very thin for easy coupling of the fiber to a source or a fiber 2. with a comparable to λ, simple geometric ray analysis may not be valid 21

Step Index Fiber Single Mode Cut Off Wavelength Ex. A step index fiber, 2a = 7 μm, n 1 = 1.458, n 2 = 1.452, λ = 1.3 μm λ c =? V-number =? mode field diameter (MFD) =? <Sol. > for single mode operation, for λ < 1.208 μm multimode propagation @ λ = 1.3 μm : 22

Step Index Fiber Group Velocity and Delay Ex. A single mode fiber, 2a = 3 μm, n 1 = 1.448, n 2 = 1.440, λ = 1.5 μm. Given that 1. β =? 2. change λ by 0.01% β =? 3. v g (1.5 μm) =?, and group delay, τ g, over 1 km =? <Sol. > = 1.910088 b = 0.3860859 β = 6.044796 x 10 6 m -1 τ g over 1 km = 4.83 μs 23

Numerical Aperture Numerical Aperture and Acceptance Angle only rays within a certain cone at the input of the fiber can propagate thru the fiber : α < α max sinθ c = n 2 /n 1 numerical aperture, NA : (: a characteristic parameter of a fiber) 2α max : total acceptance angle for meridional rays for multi-mode propagation, the majority are usually skew rays acceptance angle > 2α max 24

Numerical Aperture Multimode Fiber and Total Acceptance Angle Ex. A step index fiber, 2a = 100 μm, n 1 = 1.480, n 2 = 1.460, λ = 850 nm. NA =? acceptance angle from air =? # of modes =? <Sol. > sin α max = NA / n o = 0.2425 / 1 α max = 14 o, total acceptance angle = 28 o # of modes, M V 2 /2 = 4016 25

Numerical Aperture Single Mode Fiber Ex. A typical single mode fiber, 2a = 8 μm, n 1 = 1.46, normalized index difference is 0.3%, cladding diameter = 125 μm. NA =? acceptance angle =? single mode λ c =? <Sol. > substituting (n 1 -n 2 ) = n 1 Δ and (n 1 +n 2 ) 2n 1 for single mode propagation : V < 2.405 26

Dispersion in Single Mode Fibers Material Dispersion in single mode step-index fiber no intermodal dispersion for single mode fibers, still have 1/ material dispersion, 2/ waveguide dispersion, 3/ profile dispersion, and 4/ polarization dispersion no perfectly monochromatic light in practice and n 1 varies with λ wave propagation speed varies with λ waves w. different λ propagate at different v g material dispersion (depends on the material properties of the guide) v g = c/n g and N g for silica glass around 1.3 μm ~ const. no material dispersion around 1.3μm 27

Dispersion in Single Mode Fibers Material Dispersion Coefficient group delay ( signal delay per unit distance ) : τ g = τ / L = 1/v g = dβ 01 /dω ( L : fiber length, β 01 : fundamental mode propagation constant ) dispersion is also normally expressed as spread in arrival time per unit fiber length : Δτ / L = D m Δλ ( D m : material dispersion coefficient ) ( Δτ is the spread in τ g due to the dependence of β 01 on λ through N g, v g = c/n g ) For silica fiber core : D m 0 @ λ 1.27 μm, and when doped with germania (GeO 2 ) this D m vs. λ curve shifts slightly to higher λ output pulse Material dispersion coefficients for the core materia (taken as SiO 2 and a = 4.2 μm) as a function of free space wavelength λ. 28

Dispersion in Single Mode Fibers Waveguide Dispersion even when n 1 and n 2 are indep. of λ, v g still depends on λ by virtue of the guiding properties of the waveguide structure waveguide dispersion waveguide dispersion expressed as time spread per unit fiber length : Δτ / L = D ω Δλ ( D ω : waveguide dispersion coefficient ), for 1.5 < V < 2.4 ( N g2, n 2 : cladding refractive indices ) D ω 1/a, D m and D ω has opposite tendencies output pulse Material dispersion coefficients for the core materia (taken as SiO 2 and a = 4.2 μm) as a function of free space wavelength λ. 29

Dispersion in Single Mode Fibers Chromatic/Total Dispersion in single mode fibers, dispersion of a propagating pulse arises from the finite width Δλ of the source spectrum chromatic dispersion = (material dispersion) + (waveguide dispersion) first order approximation, chromatic dispersion : chromatic dispersion coef. : D ch = D m + D ω adjust a and hence D ω shift zero dispersion λ 0 (dispersion shifted fibers) e.g. a shift λ 0 to 1550 nm D ch = 0 does not mean there would be no dispersion at all. output pulse Material dispersion coefficients for the core materia (taken as SiO 2 and a = 4.2 μm) as a function of free space wavelength λ. 30

Dispersion in Single Mode Fibers Profile and Polarization Dispersion Effects v g (01) depends on (n 1 -n 2 ), i.e. Δ, and Δ = Δ(λ) profile dispersion : D p : profile dispersion coef., typically < 1 ps/(nm km) ( originates from material dispersion, also a part of chromatic dispersion because of dependence on Δλ, D ch = D m +D ω +D p ) fiber not perfectly symmetric and homogeneous refractive indices n 1 and n 2 not isotropic refractive index depends on the direction of the E field β of a given mode depends on its polarization polarization dispersion polarization dispersion usually < 1 ps/(nm km) and roughly L 2 31

Dispersion in Single Mode Fibers Dispersion Flattened Fibers waveguide dispersion D ω can be adjusted by changing waveguide geometry can alter the waveguide refractive index profile, e.g. use doubly clad adjust D ω dispersion flattened fiber, e.g. chromatic dispersion ~ 1 3 ps/(nm km) from 1.3 1.6 μm allows wavelength multiplexing, e.g. using a number of wavelengths, such as 1.3 μm and 1.55 μm, as communication channels 32

Dispersion in Single Mode Fibers Material Dispersion For LED and LD Ex. Usually use half-power width for Δλ and Δτ : Δλ 1/2 (called linewidth) and Δτ 1/2 which are widths between half intensity points. LED source : 1.55 μm, linewidth of 100nm material dispersion effect LD source : 1.55 μm, linewidth of 2 nm per km of the silica fiber =? < Sol. > at 1.55 μm D m = 22 ps/(km nm) LED : LD : 33

Dispersion in Single Mode Fibers Material, Wavegudie, and Chromatic Dispersion Ex. Consider a single mode optical fiber as in the figure. Excited by a laser : 1.5 μm, linewidth : Δλ 1/2 = 2 nm. (1) with 2a = 8 μm, dispersion per km =? and (2) 2a =? for zero chromatic dispersion? <Sol.> (1) (2) at λ = 1.5 μm D ω = D m when a = 3 μm D ch = 0 ( note : Although D ch = 0 at 1.5 μm, this is only at one λ, whereas the input radiation is over a range of λ so that in practice chromatic dispersion is never actually zero. ) 34

Bit Rate, Electrical and Optical Bandwidth Bit Rate In digital communications, signals are generally sent as light pulses. maximum rate at which the digital data can be transmitted along the fiber : bit rate capacity, B (bits per second) B is directly related to the dispersion characteristics dispersion is typically measured between half-power (or intensity) points and is called full width at half power (FWHP), or full width at half maximum (FWHM) : Δτ 1/2 avoid intersymbol interference peak to peak time separation > 2Δτ 1/2 ( intuitively ) T > 2 Δτ 1/2 35

Bit Rate, Electrical and Optical Bandwidth Gaussian Output Light Pulses for more rigorous analysis need to know output signal temporal shape for Gaussian output light pulses : peak separation > 4σ σ : rms dispersion ( or standard deviation from the mean ), σ = 0.425Δτ 1/2 B 1/T = 1/4σ = 0.25/σ = 0.59/ Δτ 1/2 (~ 18% higher than the 0.5/Δτ 1/2 guess) σ = L D ch σ λ, σ λ : light source rms spread of λ B 0.25 / (L D ch σ λ ) BL 0.25/ ( D ch σ λ ) : characteristic parameter of fiber ( indep. of L ) Ex. typical BL for a step-index single mode fiber operating @1300 nm with a laser diode : several Gb s -1 km σ 2 = σ 2 intermodal + σ 2 intramodal 36

Bit Rate, Electrical and Optical Bandwidth Optical and Electrical Bandwidth measure output signals with sinusoidal intensity inputs at different freq. fiber transfer characteristics optical bandwidth f op freq. range for modulated optical signal transfer intuitively, f op B if fiber dispersion characteristics are Gaussian : f op 0.75B 0.19/σ f op > f el f el is important to optical receiver system design NRZ bit rate = 2 x RZ bit rate ( NRZ : nonreturn-to -zero, RZ : return-to-zero ) 37

Bit Rate, Electrical and Optical Bandwidth Dispersion, Bit Rate, and Bandwidth Ex. 38

Graded Index Optical Fiber Graded Index (GRIN) Optical Fiber main drawback of single mode step index fiber : small NA multimode fiber : (1) greater NA and wider core 2a easier and more light coupling, but (2) suffers from intermodal dispersion graded index (GRIN) fiber : all rays arrive same point at the same time minimize intermodal dispersion 39

Graded Index Optical Fiber Fiber Comparisons Ex. 40

Light Absorption and Scattering Light Absorption some energy form the propagating wave is converted to other forms of energy, e.g. heat absorption lattice absorption : ionic polarization of the lattice atoms and vibration of the ions and ionic impurities along with the E field EM wave converted to lattice vibrational energy (heat) EM waver freq. ~ natural lattice vibrational freq. ( i.e. in IR region ) max. lattice absorption e.g. Si-O bond resonance λ ~ 9 μm Ge-O bond resonance λ ~ 11 μm 41

Light Absorption and Scattering Light Scattering light scattering : portion of the energy in a light beam is directed away from the original direction of propagation Rayleigh scattering : the size of a scattering region, whether an inhomogeneity, a small particle, or a molecule, is much smaller than λ of the incident light optical fiber drawn by freezing liquid-like flow random thermodynamic fluctuations in composition and structure frozen into fiber small inhomogeneous regions act as small dielectric particles Rayleigh scattering always there Rayleigh scattering peaks at UV frequencies where electronic polarization is max. Rayleigh scattering when λ Reyleigh scattering λ -4 42

Light Absorption and Scattering Attenuation in Optical Fibers attenuation coef. α : fractional decrease in optical power per unit distance α db, attenuation in decibles per unit length : lattice vibration due to Si-O and Ge-O bonds fundamental resonance of OH bond vibration 2.7 μm 1 st overtone ~ 1.4 μm 1310 nm : used for optical communication 1.55 μm : long-haul communication 43

Light Absorption and Scattering Microbending Loss sharp local bending of fiber changes in guide geometry and n profile loss of TIR or greater cladding penetration light energy radiating away from the guiding direction microbending loss higher mode propagate with incidence angle closer to θ c multimode fibers suffer more from bending loss than single mode fibers radius of bend, R microbending loss increase exponentially 44

Light Absorption and Scattering What s Next light sources : LED and LD photodetectors 45

Planar Dielectric Slab Waveguide Quiz #2 QUIZ #2 姓名學號 Please express β m in terms of n 1, λ, and θ m 46