Reproduced from G.P. Agrawal (1997) Fiber Optic Communica-tions Systems,
2nd edn, John Wiley & Sons, New York by permission of John Wiley and Sons
Inc.
課程編號：941 課程編號：941 U0460 科目名稱：矽光子學 授課教師：黃鼎偉 時間地點：一678 時間地點：一678 明達館 303
Step-Index Optical Fiber 3.1 THE STRUCTURE OF OPTICAL FIBERS
– Typically the core of a singlesingle-mode fiber will be of the order of 2-10 μm.
3.2 MODES OF AN OPTICAL FIBER 3.3 NUMERICAL APERTURE AND ACCEPTANCE ANGLE 3.4 DISPERSION IN OPTICAL FIBERS 3.5 SINGLE-MODE FIBERS: MODE PROFILE, MODE-FIELD DIAMETER, AND SPOT SIZE 3.6 NORMALIZED FREQUENCY, NORMALIZED PROPAGATION CONSTANT, AND CUTOFF WAVELENGTH
Figure 3.1 The structure of the step-index optical fiber
Graded-Index Optical Fiber α r n r = n1 1 − 2Δ r ≤ a a n(r ) = n1 1 − 2Δ
TElm or TMlm modes Δ=
n1 − n2 n1
r≥a
– Communications fibers fall into this category with Δ usually being less than 3 % – For many applications, α = 2 is the optimum (i.e. a parabolic profile)
– The integer l represents the fact that there will be 2l field maxima around the circumference of the field distribution. – The integer m refers to the m field maxima along a radius.
Figure 3.2 Refractive index profile of the graded-index optical fiber core
Meridional rays The propagation angles of the skew rays are such that it is possible for components of both the E and H fields to be transverse to the fiber axis. HElm or EHlm depending on whether the E or H field dominates the transverse field. In weakly guiding fibers, the exact modal solutions are usually approximated by Linearly Polarized modes, designated LPlm modes
Consider a graded-index fiber with a parabolic refractive index profile (i.e. α = 2) α
Figure 3.3 Propagation constant of the exact fiber modes plotted against normalized frequency.
r n r = n1 1 − 2Δ a
r≤a
n(r ) = n1 1 − 2Δ
r≥a
Figure 3.4 Intensity profiles of the three lowest-order LP modes.
Reproduced from G.P. Agrawal (1997) Fiber Optic Communica-tions Systems, 2nd edn, John Wiley & Sons, New York by permission of John Wiley and Sons Inc.
Figure 3.5 Structure of the gradedindex optical fiber
Table 3.1 Relationship between approximate LP modes and exact modes. Source: Optical Fiber Communications, Principles and Practice, 2nd edn, J M Senior, Pearson Education Limited. Reproduced by permission of Pearson Education
Linearly polarized
Exact modes
LP01
HE11
LP11
HE21, TE01, TM01
LP21
HE31, EH11
LP02
HE12
LP31
HE41, EH21
LP12
HE22, TE02, TM02
LPlm
HE2m, TE0m, TM0m
LPlm(l ≠ 0 or 1)
HEl+1,m, EHl-1,m
Figure 3.6 Total internal reflection in a graded-index fiber. Source: Optical Fiber Communications, Principles and Practice, 2nd edn, J M Senior, Pearson Education Limited. Reproduced by permission of Pearson Education
NUMERICAL APERTURE AND ACCEPTANCE ANGLE
Critical angle θc, Acceptance angle θa. na sinθa = n1 sin 90 −θc = n1 cosθc = n1 1− sin2 θc sin θ c =
Figure 3.7 Mode trajectories in graded-index fiber. Source: Optical Fiber Commu-nications, Principles and Practice, 2nd edn, J M Senior, Pearson Education Limited. Reproduced by permission of Pearson Education
n2 n1
Figure 3.8 Acceptance angle of an optical fiber
NUMERICAL APERTURE AND ACCEPTANCE ANGLE
Numerical Aperture
na sinθa = n1 1− n22 / n12 = n12 − n22
NA = na sin θ a = n12 − n22
NA = n1 2Δ
Figure 3.8 Acceptance angle of an optical fiber
NUMERICAL APERTURE AND ACCEPTANCE ANGLE
For skew rays
NA = na sin θ a cos γ = n12 − n22
Dispersion in optical fibers means that parts of the signal propagate through the fiber at slightly different velocities, resulting in distortion of the signal. Fiber Digital signal Emitter
Photodetector
Information
Information
t
Input
Output
Input Intensity
Output Intensity
²δτ τ 1/2
Figure 3.9 The acceptance angle of skew rays. Source: Optical Fiber Communications, Principles and Practice, 2nd edn, J M Senior, Pearson Education Limited. Reproduced by permission of Pearson Education
Very short light pulses 0
t T
t
0 ~2 τ
1/2 ~2δτ
For a given pulse broadening, δτ . If pulses are not to overlap, then the maximum pulse broadening must be a maximum of half of the transmission period, T .
δτ ≤ 12 T The bit rate BT, is 1/T 1/T
BT ≤
1 2δτ
Some texts use a more conservative rule of thumb, making BT ≤ ¼ δτ For a gaussian pulse, with an r.m.s. r.m.s. width σ. A typical rule of thumb is BT ≤ 0.2/σ
The fastest and slowest modes possible in such a fiber will be the axial ray, and the ray propagating at the critical angle θc. ( sinθc = n2/n1 )
Figure 3.10 The origin of modal dispersion
Intermodal dispersion is a result of the different propagation times of different modes (different m) within a fiber. High order mode
Light pulse
In a multimode fiber, generally
Low order mode
Broadened light pulse
Cladding
t min =
distance L Ln = = 1 velocity c / n1 c
t max =
L / sin θ c L n1 = c / n1 c sin θ c
Core
Intensity
Intensity Axial
Spread, Δ τ
t 0
Schematic illustration of light propagation in a slab dielectric waveguide. Light pulse entering the waveguide breaks up into various modes which then propagate at different group velocities down the guide. At the end of the guide, the modes combine to constitute the output light pulse which is broader than the input light pulse.
t
or
t max
L / sin θ c L n12 = = c / n1 c n2
Δ is the relative refractive index (n1-n2)/n1
δ t si = t max − t min = For n1 ≈ n2
Ln Ln L n n1 − n2 L n = − 1= Δ c n2 c c n2 n1 c n2 2 1
δ t si =
2 1
2 1
Ln1Δ c
If we let Δ = 0.01 and n1 = 1.49. Hence δτ = 49.6 ns/km. The maximum bit rate becomes
BT =
1 2δτ
≈ 10 Mbit/s·km
n2 n1 3 2 1
O
n
(a) Multimode step index fiber. Ray paths are different so that rays arrive at different times.
n2
O
O'
O''
3 2 1 2 3
n1 n2
n
(b) Graded index fiber. Ray paths are different but so are the velocities along the paths so that all the rays arrive at the same time.
For gradedgraded-index multimode fiber, the velocity of the ray is inversely proportional to the local refractive index, it can be shown that
Ln1Δ2 δ t gi = 8c If we let Δ = 0.01 and n1 = 1.49. Hence δτ = 62 ps/km. ps/km. The maximum bit rate becomes
BT =
1 2δτ
≈ 8 Gbit/s· Gbit/s·km
y
Different spectral components of the light source may have different propagation delays, and hence pulse broadening may occur, even if transmitted by a single mode.
y
v g1
λ2 > λ1
ω1 < ωcut-off
Vp2 v g2 > v g1
Core
v g (λ 1 )
Emitter
λ1 & λ 2
v g (λ 2 )
Core
Output
λ1
Very short light pulse
Intensity
λ2
Intensity
Spectrum, Δλ λ
λ1 & λ 2
Spread, δτ τ
λ1
λo
λ
λ2
Vg
Cladding
Input
Intensity
λ2
ω2 < ω1 Cladding
E(y)
Vp1
λ1
Cladding
λ1 > λc
For λ1 Vp2 (Fast Light Packet Vg > Vp)
Delay time for a pulse with a specific frequency bandwidth Δω
∂τ ∂2β δτ ch = Δω = L Δω ∂ω ∂ω 2
Total chromatic dispersion D is the sum of the material dispersion DM and the waveguide dispersion Dw. There is zero dispersion close to 1.31 μm (λZD) in glass optical fiber.
Group velocity dispersion (GVD) parameter, D ps/(km·nm).
2πc ∂ 2 β D = − 2 2 λ ∂ω Thus
δτ ch = D LΔλ Figure 3.11 The variation in chromatic dispersion with wavelength.
Intramodal (Chromatic) Dispersion in DispersionDispersion-modified SingleSingle-mode Optical Fiber
There are two main contributions to chromatic dispersion: – Material dispersion. The refractive index of any medium is a function of wavelength, and hence different wavelengths that see different refractive indices will propagate with different velocities, resulting in intramodal dispersion. – Waveguide dispersion. Even if the refractive index is constant, and material dispersion eliminated, the propagation constant β would vary with wavelength for any waveguide structure, resulting in intramodal dispersion.
The contribution of waveguide dispersion is dependent on fiber parameters such as refractive indices and core diameter. Dispersion shifted fibers: fibers: shift λZD typically to 1.55 μm, where the optical loss of optical fiber is a minimum. Dispersion flattened fibers: fibers: the total chromatic dispersion is relatively small over the wavelength range 1.31.3-1.6 μm.
Figure 3.12 Dispersion-shifted and dispersion-flattened fibers.
Intramodal (Chromatic) Dispersion in Single-mode Optical Fiber For dispersion flattened single-mode optical fiber, operated near λZD , e.g. D = 1 ps/(nm·km)
BT =
δτch
≈ 500 Gbit/s·km
Comparison SI MM fiber BT ≈ 10 Mbit/s·km GI MM fiber BT ≈ 8 Gbit/s·km SI SM fiber BT ≈ 500 Gbit/s·km
SINGLE-MODE FIBERS: MODE PROFILE, MODE-FIELD DIAMETER, AND SPOT SIZE Cylindrical coordinates (r, φ and z) – exp jβz – exp jmφ – Jm(r), Km(r) E LP = Elm (r , ϕ ) exp j (ωt − β lm z )
Cartesian coordinates (x, y and z) – exp jβz – exp jmx or indep. indep. of x – cos kymy, exp -kymy Em = Em ( y ) exp j (ωt − β m z )
Figure 3.13 (a) Variation of the first four Bessel functions. (b) Variation of the first two modified Bessel functions.
MODE PROFILE – J0(r), K0(r)
Figure 3.14 Field shape of the fundamental mode of a step-index fiber, for normalized frequencies of V = 1.5 and V = 2.4.
Mode Field Diameter = 2w0 Spot size = w0
V +1 2w0 = 2a V
Normalized Frequency: V number
V=
2π
λ0
a n12 − n22 =
2π
λ0
an1 2Δ
Normalized propagation constant: b
b=
β / k0 2 − n22 n12 − n22
=
β / k0 2 − n22 2n12 Δ
a: fiber core radius Figure 3.15 Approximation to the fundamental mode, showing the mode-field diameter (MFD) and the spot size w0
Δ: relative refractive index difference λ0 operating wavelength
2.405
Figure 3.16 Propagation constant of the exact fiber modes plotted against normalized frequency.
Fitting curve for LP01
0.996 b ≈ 1.1428 − V
2
(1.5 < V < 2.5)
Cut-off Condition for Single Mode Fiber Vc = 2.405 for step-index fibers
λc =
2π an1 2Δ Vc
λc/λ0 = V/Vc.
λc =
Vλ0 2.405
