Fiber Optics Digital Transmission Fiber System
Advantages of Fiber Optics: Speed Distance Immunity to electromagnetic interference (EMI) Security
Optical Fiber Specifications
Fiber Specifications Optical Characteristics
Core Diameter (µm) 6-62.5 Attenuation @850nm (dB/km)
2.4 - 2.5
Clad Diameter (µm) 125 Attenuation @1300nm (dB/km)
0.3 - 0.7
Coating Diameter (µm) 245 Attenuation @1550nm (dB/km)
0.15 - 0.3
Numerical Aperture 0.17±.015
Zero Dispersion Wavelength (nm) 1297-1316
Group Refractive Index @1300nm 1.479
WDM/ DWDM: Upgrade fiber index profile to support more wavelength transmission
- Reducing / shifting / flattening dispersion - Reducing attenuation - Operating wavelength (1330 nm, 1550 nm) - EDFA
Basic configuration of EDFA
Developments in Photonics Photoelectric effect – Einstein E = hν - φ Laser – Townes and Schalow (1958) Laser types:
a) Solid state lasers: Ruby, Nd:YAG, Nd:glass b) Gas lasers: He-Ne, Argon ion, CO2 c) Liquid lasers: Dyes d) Semiconductor lasers
Semiconductor lasers: (injection laser or diode laser) Emitting wavelength – Material dependent
GaAs/AlGaAs -- 0.8 µm InGaAsP/InP -- 1.3 to 1.55 µm Quantum cascade long wavelength lasers -- 2 to 10 µm
Structure PN heterojuction Edge emitting Vertical cavity surface emitting
Threshold Output Power
Fiber Characteristics and Processing Structure of Fiber Core – higher index Cladding – lower index Coating Single mode Multimode Fabrication of optical waveguides: Deposition – MCVD, VAD Consolidation Drawing Coating Cable (Protection) Index control via doping SiO2
GeO2
P2O5
B2O3
Optical Fiber Spectral Attenuation Characteristics: Absorption UV – Electronic excitation IR – Vibration excitation Rayleigh Scattering λ-4 dependence Attenuation: dB/km
Fiber Bending Loss: Macrobending
Microbending
Fiber Drawing Tower:
Fiber Cable
Cross section of various types of fiber optic cable
Maxwell Equations: For charge free and perfect dielectric medium (i.e., with ρ=0 and σ=0)
where ED ε= and HB µ= Boundary conditions: Tangential component of electric field: Et is continuous Normal component of electric displacement: Dn is continuous Tangential component of magnetic field: Ht is continuous Normal component of magnetic induction: Bn is continuous
tBE∂∂−=×∆
tDH∂∂=×∆
0=×∆ D0=×∆ B
21 tt EE =
2211 nn EE εε =
21 tt HH =
2211 nn HH µµ =
Wave equation: (E-field)
Helmholtz equation: (E-field)
where nkk 0= Solution of Helmholtz equations yields the planar wave equations:
Using Maxwell equations, one obtains
The vectors E (electric field), H (magnetic field), and k (wave propagation direction) are at right angles to each other. Thus the field associated with a plane electromagnetic wave are transverse to the direction of propagation.
).(0
rktieEE −= ω
).(0
rktieHH −= ω
ωεHkE ×−=
ωµEkH ×=
012
2
22 =
∂∂−∇
tE
cE
tieE ω−~
0220
2 =+∇ EnkE
Polarization Linearly polarized light: Phase difference between Ex and Ey is nπ (n = 0, 1, ..) Circularly polarized light: Phase difference between Ex and Ey is (2n+1)π/2 (n = 0, 1, …) Elliptically polarized light: a) TE: Electric field lying perpendicular to the plane of incidence b) TM: Electric field lying in the plane of incidence Waves in metallic structures Two boundaries:
Modes: (ray model) Incidence angle determines phase of a wave propagation, giving rise to different modes. Rectangular waveguide: Simple boundary condition – metallic boundary: E=∝ E=0 -a 0 a x
Standing waves: Reflection by metallic mirror: y
02202
2
2
2
=+∂∂+
∂∂ Enk
yE
xE
Incoming wave: )( ykxk yxAe + Reflected wave: )( ykxk xxBe − x Applying boundary condition ay y = 0,
TE wave: E-field normal to the plane of incidence TM wave: H field normal to the field of incidence
02202
2
=+∂∂ Enk
xE
ikxikx BeAeE −+~kxkx sincos~ βα +
==
=,...2,sin,...,cos
0
23
20
ππ
ππ
kakxEkakxE
E
( ) BABAe xikx −=⇒=+ 0
Multi-dimensional problems Wave propagating between two parallel metallic plates: x a z -a
A different interpretation on wave propagating between two parallel metallic plates: x k0n z θ A wave bounces between two metallic boundaries. Effective component along z: βθ =cos0nk Effective component along x: θsin0 nk The wave must be periodic after bouncing one complete cycle, i.e.,
πθ nank =×2sin0 ank
m
02sin πθ =⇒
Hence ( )2222
0akm
eff nn π−= ⇒ Same result !
02202
2
2
2
=+∂∂+
∂∂ Enk
zE
xE
znikzi effeeE 0~ =β
( ) 022202
2
=−+∂∂ Ennk
xE
eff
( ) ,...3,2,22/1220 πππ=− annk eff
( )22
220ak
meff nn π−=⇒
Wave propagating along a rectangular metallic waveguide 2-D cross section of a rectangular metallic waveguide: y
02202
2
2
2
2
2
=+∂∂+
∂∂+
∂∂ Enk
zE
yE
xE
znikzi effeeE 0~ =β b -a a x -b
By separation of variables of the above partial differential equation,
with yx EEE = and ( )2220
22effyx nnkkk −=+
Normal incidence on one dimensional dielectric boundary: (ε = n2)
0)(2202
2
=+∂∂ Exnk
xE
n1 n2
x=0 x :0<x xnikxnik BeAeE 1010 −+= :0>x xnikxnik DeCeE 2020 −+=
Assume the wave comes from left. Therefore, .0=D Eǁ continuous CBA =+⇒ Hǁ continuous CnkBAnk 2010 )( =−⇒ C
nnBA
1
2=−⇒
⇒ Ann
nC21
12+
=
AnnnnB
21
21
+−=
022
2
=+∂∂
xx Ekx
E
022
2
=+∂∂
yy Eky
E
( ) 022202
2
2
2
=−∂∂+
∂∂ Ennk
yE
xE
eff
Poynting vector:
⋅
=
ncEA 0
2
2ε
Flux coming in: 201
2Acn
reflected: 2
21
21201
2
+−⋅
nnnnAcn
transmitted: 2
21
21201
)(4
2 nnn
Acn
+⋅
Energy conservation: Incoming flux = flux reflected + flux transmitted. http://www.cyber.rdg.ac.uk/ISP/infrared/technical_data/substrate_optical_properties/page01.htm Dielectric interface: y incident beam reflected A B θin n1 x n2 θr C refracted
Incident: )(sin1010 ysocxniknki ininAerAe ⋅−⋅=⋅ θθ
Reflected: )cos(sin10 yxnik ininBe θθ +
Refracted: )cos(sin20 yxnik rrCe θθ − TE wave: E ǁ Z Boundary condition 1: Eǁ continuous at y = 0
Snell’s law: rin nn θθ sinsin 21 = 2
1
sinsin
nn
in
r =⇒
θθ
( ) ee xnikxnik rin CBA θθ sinsin 2010 =+⇒
if n1 > n2, inr θθ <
Boundary condition 2: Hǁ continuous
( )E
nknkkji
Ek x
000cossin 1010 θθ −=× is continuous.
Using the above, and also noting that B⊥ is continuous, we have
Solving for B
Special Cases: 2
21
210
+−=⇒=
nnnnRinθ
190 =⇒= Rinθ
Consider innn θ2
22
21 sin1−
It can be imaginary if 1sin222
21 >inn
n θ
1
2sinnn
cr =θ or 1
21sinnn
cr−=θ critical angle
It happens when 21 nn > total internal reflection.
rin CnABnCBA
θθ coscos)( 21 −=−=+⇒
ABinr
inrnnnn
θθθθ
coscoscoscos
12
12+−=⇒
Ainin
n
n
ininn
n
nn
nn
θθ
θθ
cossin1
cossin1
12
22
21
2
12
22
21
2
+−
−−
=
If we define 220 cos irnk r =θ 110 cos knk in =θ
then AkirkirB
12
12
+−=
2
22
21
22
2122 A
rkrk
AB =++
⋅=⇒
and TEieAB φ2⋅= where
1
21tankr
TE−−=φ Goos-Hanchen phase shift
For TM polarization,
tEn
tDH
∂∂=
∂∂=×∇ 2
CBA =+
221
222
21 cos
cosnC
nn
nB
nA
in
r ⋅=−θθ
−=⇒ −
221
2121tan
nknr
TMφ
Planar waveguide: 2n d 1n θ
0n
20 nn = : symmetric waveguide 20 nn ≠ : asymmetric waveguide
( ) πφφθ mdnk 2222cos 101210 =++⋅
TE: 1
01
1
2110 tantancos
kr
krmdnk −− ++= πθ
Comparison between TE and TM: Snell’s law same
crθ same ( 90=crθ in Snell’s law) Reflection, crin θθ < different
For TE: AnnnnB
inr
inr
θθθθ
coscoscoscos
12
12
+−=
For TM: Annnn
Binr
inr
θθθθ
coscoscoscos
21
21
+−
=
inrnn θθ coscos21 >⇒< 0coscos 12 =− inr nn θθ impossible 0coscos 21 =− inr nn θθ possible using inr nn θθ sinsin 12 =
inr θπθ −=2
1
21tannn
in−=θ Brewster angle ( Bθ )
1
R 0 Bθ
2π inθ
Summary of equations for dielectric interface Reflection coefficient:
TE:
2
12
22
21
2
12
22
21
22
cossin1
cossin1
+−
−−
inin
inin
nnnn
nnnn
AB
θθ
θθ
TM: 21 nn ↔ Critical angle:
1
21sinnn
cr−=θ
Brewster angle:
1
21tannn
B−=θ
Goos-Hanchen phase shift:
TE: in
in
nnnn
θ
θφ
cos
1sintan
1
222
21
21
−−= −
TM: 21 nn ↔ Characteristic equation for modes:
πφφθ mdnk in =++ 101210 cos
Numerical examples:
5.1=glassn 1=airn
R @ 0=inθ : 04.05.25.0 22
12
12 =
=
+−=
nnnnR
crθ (glass-air): 8.415.1
1sin 1 == −crθ
Bθ (glass-air): 7.335.1
1tan 1 == −Bθ
(air-glass): 3.5615.1tan 1 == −
Bθ
TEφ @ 80 :
5.7680cos5.1
180sin15.11
tan
22
1 −=⋅
−
⋅−= −φ
TMφ @ 80 :
9.8380cos1
180sin15.15.1
tan
22
1 −=⋅
−
⋅−= −φ
Characteristic equation for modes can only be solved numerically by a computer program.
crin θθ ≥ Do loop 001.0+inθ Equation satisfied
Effective mode index ( effn ) 1n z θ
10nk = θcos10nk
θsin10nk ∴ Propagation along z has the following form: znike ⋅θsin10
010 sin knnk eff=≡ θβ where θsin1nneff =
90<<θθcr 12 nnn eff <<⇒
effn is, in general, a function of dk0 , 1n , 2n A typical “dispersion” curve: effn 1n
2n
0λd
to obtain: cut-off number of modes
Extension of the basic idea Multi-layered waveguide: 1=n 1.50 1.48 1.47 Two possibilities:
I) 3n
1n 1θ 1d
2n 2θ 2d
4n
πφφθθ mnkdnkd =+++ 241322021101 coscos where ijφ is the phase shift in reflected beam at the thij interface
II) πφφθ mnkd =++ 12131101 cos
Alternative solution to the planar waveguide x 0n
2d+ symmetric
z 1n waveguide
2d−
0n Helmholtz equation
0)(2202
2
2
2
=+∂∂+
∂∂ Exnk
zE
xE
zkinzi effeeE 0~ =β
( ) 022202
2
=−+∴ Ennkdx
Edeff
1nnn eff <<
⇒ sinusoidal solution for 2dx ≤
exponential solution for 2dx >
=
+
−−
xd
dx
Be
xkABe
E
2
1
2
2
2
cosγ
γ
or
−
+
−−
xd
dx
Be
xkABe
2
1
2
2
2
sinγ
γ
Boundary conditions: symmetric mode asymmetric mode TE: ||E BkA d =21cos BkA d =21sin
∂∂
xEH || BkAk d
2211 sin γ−=− BkAk d2211 cos γ−=
1
221tan
kk d γ
= 1
221cot
kk d γ
−=
πγ mk kd += −
1
2121 tan πγ
++= −
21
tan1
2121 mk
kd
⇒×2 πγ mdk k += −1
211 tan2 where innkk θcos101 ≡
πθ γ mdkn kin +=∴ −1
2101 tan2cos same as before!!
Boundary conditions (contd.): symmetric mode asymmetric mode
TM: ||H BkA d
2cos BkA d2sin
∂∂
xH
nE 2||
1 BnkkAn d 2
1222
1 sin γ−=− − BnkAkn d 2
022
2
1 cos γ−=
20
212
2tankn
nk d γ= 20
212
2cotkn
nk d γ−=
πγ mkn
nk d +
=⇒ −
20
2121
2 tan ( )πγ21
20
2121
2 tan ++
=⇒ − m
knnk d
What are the additional information ?
1. Field pattern ( )
22dxe −−γ
exp
kxcos kxsin
exp exp
2. Overlap dxEd
d∫−
2
2
2
3. Solution for non-uniform index distribution
)(xn varying in space gradually rather than abruptly 3LiNbO
iT
Acceptance angle Air Fiber
1 2n
crθ 1n maxθ φ
1
21sinnn
cr−=θ
crθφ −= 90
1sin
sin 1max n=φ
θ
crn θθ cossin 1max =⇒
Since 1
2sinnn
cr =θ
22
21
121
22 11cos nn
nnn
cr −=−=∴ θ
NAnn ≡−= 22
21maxsinθ
Example: ,48.11 =n 47.12 =n 17.0=⇒ NA 89.9max =θ
Cylindrical coordinate system r φ z φ
z Helmholtz equation will look like
2
2
2
2
22
22 11
zrrrr ∂∂+
∂∂+
∂∂+
∂∂=∇
φ
Maxwell equations:
⇒
∂∂−=×∇
tBE
( ) ze
zr
rz
HjEr
rErr
Hjr
EEj
HjEjEr
ωµφ
ωµβ
ωµβφ
φ
φ
φ
−=∂∂−
∂∂
=∂∂+
−=+∂∂
11
1
⇒
∂∂=×∇
tDH
( ) zr
zr
rz
EjHr
rHrr
Ejr
HHj
EjHjHr
εωφ
εωβ
εωβφ
φ
φ
φ
=∂∂−
∂∂
−=∂∂+
=+∂∂
11
1
nc=
εµ1
effnc=
βω
Only zE and zH are essential because
∂∂+
∂∂−=
∂∂−
∂∂−=
∂∂−
∂∂−=
∂∂+
∂∂−=
rEH
rqjH
Err
HqjH
rHE
rqjE
Hrr
EqjE
zz
zzr
zz
zzr
ωεφ
β
φωεβ
µωφ
β
φµωβ
φ
φ
2
2
2
2
where 2220
2 β−≡ nkq Helmholtz equations:
011 22
2
22
2
=+∂∂+
∂∂+
∂∂
zzzz EqE
rrE
rrE
φ
011 22
2
22
2
=+∂∂
+∂∂
+∂∂
zzzz Hq
Hrr
Hrr
Hφ
Solution by separation of variables: )()()()( 4321 tFzFFrAFEz φ= where )(
43 ~)()( ztjetFzF βω − Since the field is periodic, πφφ 2+= υφφ jeF =∴ )(2 , where υ is an integer
0112
221
21
2
=
−++ Fq
drdF
rdrFd
γυ : Bessel function
)()()( ztjjz eeurAJarE βωυφ
υ−=<
)()()( ztjjz eewrCKarE βωυφ
υ−=>
with 221
20
2 β−= nku 1n : core index 2
220
22 nkw −= β 2n : cladding index Likewise,
)()()( ztjjz eeurBJarH βωυφ
υ−=<
)()()( ztjjz eewrDKarH βωυφ
υ−=>
Boundary conditions at ar = :
0)()()()(
0)()(
'2
'221
21
=
−−
−−
=−
=−=−
wawKDwaKa
jCjuauJBuaJa
jAu
jEE
waCKuaAJEE zz
υυυυφφ
υυ
ωµυβω
ωµωβ
0)()()()(
0)()(
'22
'1221
21
=
−−
+−=−
=−=−
wawKCwaKa
jDjuauJAuaJa
jBu
jHH
waDKuaBJHH zz
υυυυφφ
υυ
ωευβω
ωευβ
To have non-zero A , B , C , D :
( )( )2
22
222
20
21
20
2'2
2'1
'2
'2
11
0
)()()()(
)(0)(0
)()()()(
0)(0)(
+
=ℵ+ℑℵ+ℑ⇒
=
−−−
−
wuanknk
waKaw
waKwj
uaJau
uaJuj
waKuaJ
waKw
jwaKaw
uaJu
juaJau
waKuaJ
βυ
βυωεβυωε
ωµβυωµβυ
υυυυ
υυυυ
υυ
υυυυ
υυ
where )()('
uauJuaJ
υ
υυ ≡ℑ and
)()('
wawKwaK
υ
υυ ≡ℵ
General behavior of effective mode index: (Figure 2-18, textbook) effn 1n 2n 1 2 3 4 5 6
( ) 212
221
2 nnaV −=λπ
The cut-off of higher-order modes: 405.2=V
Modes are in group: 11HE
01TE , 01TM , 21HE 11EH , 31HE , 12HE
o Modified index term: ( ) 2
2220 r
nnk effυ−
1n 0=υ 2n r 0≠υ
r
0)0(0==
≠υυ rJ : ⇒ Field is zero at the center of the guide. When 121 nnn <<− 22
22
1 β≈≈ kk
+±=ℵ+ℑ 22
11wua
υυυ
Categorized modes: 0
)()(
)()(
0
1
0
1 =+wawK
waKuauJ
uaJ TE, TM modes
0)()(
)()( 11 =+ ++
wawKwaK
uauJuaJ
υ
υ
υ
υ EH modes
0)()(
)()( 11 =− −−
wawKwaK
uauJuaJ
υ
υ
υ
υ HE modes
Order of Bessel function involved:
→−→+→
=HEEH
TMTEj
11
,1
υυ degenerate group
Modes in each degenerate group has similar effn versus V behavior.
j : jJ ⇒ the order of Bessel function TE , TM involve 1J mHE ,υ involve 1−υJ (special case: 1=υ )
mEH ,υ involve 1+υJ The characteristic equation has m solutions for each υ (depending on waveguide parameters, e.g., λ , a , NA .) Modes: mTE ,υ 0=zE
mTM ,υ 0=zH mEH ,υ zz HE > mHE ,υ zz EH >
Bessel function of the first kind:
Bessel function of the second kind:
Modified Bessel function of the first kind:
Modified Bessel function of the second kind:
Symmetry of fiber waveguide Helmholtz equation: 022
02 =+∇ EnkE :E zE
zH zji eerEE βυφ −⋅= )(
( ) 012
22202
2
=
−−+∂∂+
∂∂ E
rnnk
rE
rrE
effυ
Bessel functions: υJ sinusoidal for ar <
υK exponential for ar >
1, −⇒ υυ JHE m υφje m-th solution 1, +⇒ υυ JEH m υφje m-th solution
1,
, JTMTE
m
m⇒
υ
υ υφje m-th solution
Analysis of E-field pattern 011 JHE ⇒ a r rE 0 π2 φ
φE 0 π2 φ
Modes in the fiber
01LP mode distribution
MFD core diameter Mode field diameter of a single-mode fiber: Not all light travels through the core of the fiber, but is distributed through both the core and the cladding. The “mode field” is the distribution of light through the core and cladding of a particular fiber. Mode-Field Diameter (MFD) defines the size of the power distribution. MFD is important in determining the bending loss.
Degenerate modes: modes with similar β vs. V (dispersion curve)
11HE 0J 01LP 210101 ,, HETMTE 1J 11LP
3111, HEEH 2J 21LP 12HE 0J 02LP
4121, HEEH 3J 31LP Notes:
a) mlLP , corresponds to linearly polarized mode b) Subscript l corresponds to the order of Bessel function (e.g., 331 JLP → )
c) mm HELP 10 = d) mmm TMHELP 021 ,= e) )2(, ,1,1, ≥= −+ lEHHELP mlmlml
Fundamental mode of the fiber is 11HE mode. Depending on polarization of the laser, various groups of degenerate modes are launched into the fiber.
Graded index fiber
( )
>∆−
<<
∆−=
arn
ararnnr
21
21
21
021
1
1
α
21
22
21
2nnn −=∆
1n is the index at the fiber core axis. 2n is the cladding index α is approximately 2, therefore, the index profile within the core is parabolic.
Dispersion Pulse spreading and loss of peak power along an optical fiber:
Electric field of a wave propagating:
( )rtierEtrE ⋅−∧
= βω0),( where n
0
2λπβ =
Phase velocity: c
nf
nnfvp
11202
0
==== λπβω
λπ
Group velocity: 11 −−
=
==dkdc
dkd
ddk
ddvg
ββωβ
ω
Delay time: Ldd
cdkd
cL
vL
gg λ
βπλβτ
2
2
−===
Spreading in delay time: ( ) ( )δλλτ
δτdd g=
( )
+−= 2
222
2 λβλ
λβλδλ
π dd
dd
cL
Dispersion: λτ
dd
LD g1=
What makes β ω dependent ?
-- dispersion. Material dispersion:
)(2 λλπβ n=
−=− λλλτ
ddnn
cL
matg)(
2
2 )(1λλλ
dnd
cDmat −= [ps/(km.nm)]
Material dispersion characteristics:
Waveguide dispersion: Normalized parameters V , b
22
21
222
2
nnn
b k
−−
=β
11
21 <<−=∆n
nn
)1(2
21
2
+∆≅⇒
−−
≅⇒
bknnnn
b k
β
β
( ) ∆≅−= 22
2/122
21 kannnkaV
∆
=⇒
22anVk
∆+=− dkkbdnn
cL
wg)(
22τ
∆+=∴ − dVVbdnn
cL
wg)(
22τ
2
22 )(
dVVbdV
cnDw λ
−=
Dispersion characteristics of single-mode optical fiber:
Fiber Dispersion Characteristics: Zero dispersion region 3.1 mµ Chromatic dispersion in standard SMF (single mode fiber)
15 ps/km-nm @1.55 µm Chromatic dispersion in dispersion-shifted SMF <3 ps/km-nm @1.55 µm Polarization mode dispersion : 0.3 ps/km Polarization maintaining fiber Dispersion Control: DSF (Dispersion shifted fiber) – upgrade existing 1.3 µm fiber systems for 1.55 µm operation DFF (Dispersion flattened fiber) – by suitable adjustment of fiber
core/ cladding index profile DCF (Dispersion compensating fiber) – using fiber grating
(DFB/DBR)
Fiber index profiles Step index:
Depressed-clad index:
Triangular Index:
55.1 mµ DSF
Intermodal dispersion (in multimode fiber):
Generalized refractive index profile of fiber:
[ ]
≥∆−≤<∆−
=arn
ararnrn
q
),21(0,)/(21
)( 21
212
where q represents the exponent of the power law profile. ⇒∞=q a step index profile ⇒= 2q a parabolic index profile
Pulse dispersion in step index fiber:
1
maxnc
LT = , 2
minnc
LT =
( )21 nncLT −=∆⇒
)(121 nn
cLT −=∆
⇒
skm /103
01.05×
≅
30≅ ns/km very serious !! In parabolic index fiber )2( =q :
Lc
nLT 21
2∆=∆
Pulse dispersion is 25.0≈ ns/km In optimum profile ( 98.1=q ), pulse dispersion is 06.0≈ ns/km
Graded index multimode fiber:
Pulse Propagation and Broadening Dispersion in time: E pulse duration τ time
Optical cycle 0
21ωπ==
fT
power
2Eα time Optical cycle 1410−≈ sec Pulse duration 1210−≈ sec In frequency domain: (power spectral density) )(ωS ω∆ 0ω ω
Gaussian pulse (closed form representation): titeyxutyxE 0
2
),(),0,,( 0ωα +−=
Fourier transform of the time-dependent function:
Time domain: 2te α−
Frequency domain: α
πα4
2
41 Ω−e
Finite pulse width ⇔ additional spectral content The signal is no longer a single frequency signal at 0ω . There is a spectral
width at frequency Ω+0ω . Since the propagation constant, i.e., 0
2λπn is
frequency-dependent as: ( ) m+Ω+Ω+=Ω+ 22
2
00
0021)(
ωω ωβ
ωββωβ
dd
dd
We have ),( tzE propagating as a broadening pulse. Pulse broadening due to material dispersion:
−= 2
2
λλ
dnd
cD
Pulse broadening due to waveguide dispersion:
−=
2
2
λλ
dnd
cD eff
gvd
d 1
0
=ωω
β , gv : group velocity
ωω
β
ω ddv
va
dd g
g22
2
21
21
0
−== ⇒ group velocity dispersion
Pulse width: aLaLL0
2
20
02ln82ln81)(
ττττ ≅
+=
The shorter the initial pulses, the faster it broadens. Because, the broader is the spectral content. Equivalent situation: shorter focal length longer focal length Time – space Frequency – angle
“uncertainty principle” Fiber Nonlinearity Generation of new frequency components Intermixing of optical signals Raman, Stimulated Raman Scattering – molecular vibration Brillouin Scattering – Sound wave Four-Wave Mixing in DWDM Systems
Chirping: 2
2
λdnd
A systematic change in frequency during the pulse propagation – an effect of group velocity dispersion. (also refer to Fig. 3-12, text book) If chirping is systematic, we can use it to compress the pulse – optical pulse compression. We can also make giant pulses by amplifying chirped pulse and compress it to a very short duration. Techniques of pulse compression:
1) Fiber with opposite dispersion 2) Compression by grating
Polarization maintaining fiber: An optical fiber in which the polarization of lightwave launched into the fiber is maintained during propagation with little or no cross-coupling of optical power between the polarization modes. If a fiber is perfectly symmetric, it will not preserve the polarization, since Y
effXeff nn = .
Because of built-in stress, twist, etc., Y
effXeff nn ≠
( ) fXeff
Yeff Bknnk 00 ≡−≡β , where fB is called birefringence.
Cross-sections of polarization-maintaining optical fibers range from elliptical to rectangular. Such fibers possess internal birefringence.
βπ2=pL : beat length
810−≈fB for non-birefringent fiber 310−≈ for birefringent fiber.