FOT_MEng2004_FEUP

Fibre Optic Technology
Mestrado Eng. Electrotécnica e de Computadores 
(2003/2005) 
António Lobo
Prof. Associado da UFP
Prof. Convidado da FEUP

© A.Lobo (2004) 2
Course Topics
1. Optical Fibre: Basic Characteristics
2. Light Propagation in Fibres and Related Optical
Effects
3. Types of Optical Fibre
4. Passive Fibre Optic Devices
5. Active Fibre Optic Devices
6. Other Fibre Optic (Hybrid) Devices
7. Fibre Device Fabrication Techniques

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Reading Material
Optical Fiber Communications
▪ G. Keiser, Optical Fiber Communications, 3rd Ed., McGraw-Hill (2000).
▪ J. M. Senior, Optical Fiber communications, Prentice Hall (1985).
▪ G. P. Agrawal, Non-Linear Fiber Optics, 2nd Ed., Academic Press (1995).
▪ H. Kolimbiris, Fiber Optics Communications, Pearson Prentice Hall (2004).
Optical Fiber Technology
▪ K.T.V. Grattan and B.T. Meggitt, Optical Fiber Sensor Technology, Vol. 1 and 2, Chapman & Hall
(1995).
▪ B. Culshaw and J. Dakin, Optical Fiber Sensors: Principles and Components, vol.II, Artech House
(1988).
▪ B. Culshaw and J. Dakin, Optical Fiber Sensors: Components and Subsystems, vol.III, Artech House
(1997).
▪ R. Kashyap, Fiber Bragg Gratings, Academic Press (1999).
▪ C. Hentschel, Fiber Optic Handbook, HP Electronics Instruments Dept, ISBN:3-9801677-0-4.
▪ D. Derickson, Fiber Optic Test and Measurement, Hewlett-Packard Co., Prentice-Hall Publs. (1998).
▪ Jeff Hecht, Understanding Fiber Optics, 4th Ed., Pearson Prentice Hall (2002).
General Optics
▪ E. Hecht, Optics, 4th Ed., Addison Wesley (2002)
▪ M. Born and E. Wolf, Principles of Optics, 7th Ed. Cambridge Univ. Press (1999).
Other material
▪ Class notes and handouts with each of the class topics provided in this syllabus.
▪ Optical Fiber Sensors (OFS) Conference Proceedings.
▪ Optical Fiber Communications (OFC) Conference Proceedings.

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Assessment to Final Grade
The grade is based on the following:
◗ Final examination (50%)
◗ Home Assignments (20%)
◗ Term Paper Project (30%)
Examination is based on class lectures, reading
assignments, homework problems and class handouts.
A Term Paper is required.

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Term Paper Project
LISTING OF POTENTIAL SUBJECTS FOR TERM PAPER
(Paper is due May 29, 2004)
1. Optical fibre propagation in fibres
2. Non-linear optical fibre effect
3. High strength fibres
4. Fibre optic sensors (many choices here)
5. Environmental effects in fibres
6. Fibre optic cable designs
7. Wavelength division multiplexing devices
8. Fibre optic filters.
9. Fibre Bragg gratings
10. Microstructured Fiber Devices
11. Fibre optic modulators
12. Dispersion compensation devices
13. Plastic optical fibre for communications
14. Aging effects in fibre optics
15. Photonic crystal fibres
16. Active fibre optic devices in communications
17. Optical power propagation in fibres
18. Fibre Lasers
19. Long wavelength fibre applications
20. All-Fibre optic signal processing
21. Optical fibre technology for automotive applications
22. Optical fibre technology for biomedical applications
23. Optical fibre technology for environmental applications

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Acknowledgements
▪ Prof. David Jackson – Head of Applied Optics Group, School Physical Sciences, Univ. Kent (UK).
▪ Prof. Philip Russell – Head of Optoelectronics Group, Univ. Bath & CTO of Blaze Photonics Ltd. (UK)
▪ Prof. Govind Agrawal – Optical Communications Group, Univ. Rochester (USA).
▪ Prof. Xiaoyi Bao – Fiber Optics Group, Univ. Ottawa (Canada).
▪ Prof. Jose L. Santos – Head of UOSE, INESC Porto & FCUP, Univ. Porto (Portugal).
▪ Dr. Adrian Podoleanu - Applied Optics Group, School Physical Sciences, Univ. Kent (UK).
▪ Dr. David Webb – Photonics Research Group, Aston Univ. (UK).
▪ Dr. Hector Guerrero – Lab. Optoelectronica, Dept. Ciencias Espacio, INTA (Spain).
▪ Dr. Ralf Pechstedt, Bookham Technology Plc. (UK).
▪ Prof. Y. J. Rao – Dept. optoelectronics Eng., Chongquing Univ. & General Manager of Chongqing Bao
Tong Optical Fiber Technology Ltd. (China)
▪ Dr. Christian Hentschel – Boeblingen Instruments Div., Hewlett-Packard GmbH (Germany).

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▪ Roman Times (Italy) Glass is drawn into fibers
▪ ?? (Venice) Decorative flowers made of glass fibers
▪ 1626 Snell (Holland) Snell’s Law
▪ 1713 Reaumur (France) Makes spun glass fibers
▪ 1841 Colladon (Swiss) Light guiding in jet of water of Geneva
▪ 1854 Tyndall (UK) Light guiding in a thin water jet
▪ 1880 Wheeler (USA) System of light pipes to illuminate homes
▪ 1888 Roth & Reuss (Austria) Bent glass to illuminate body cavities
▪ 1897 Rayleigh (UK) Analysis of waveguide
▪ 1926 Hansell (USA) Principles of fiber optic imaging bundle
▪ 1930 Lamm (Germany) Assembles first bundle of transparent fibers
▪ 1953-54 Heel (Holland) Simple bundles of clad fiber
▪ 1954 Hopkins, Kapany (UK) Image transmission w. unclad fiber bundles
▪ 1956 Curtiss (USA) Makes the first glass-clad fiber
▪ 1957 Hirschowitz & Curtiss (USA) First test fiber-optic endoscope in a patient
▪ 1960 Goubau & Christian (USA) Hollow optical guides w. periodic lenses
1.1 Historical Introduction

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▪ 1960 (May) Maiman et.al. (USA) First laser (ruby laser)
▪ 1960 (Dec.) Javan et.al. (USA) Operation of He-Ne laser
▪ 1961 Kapany and Snitzer (UK) Mode analysis of optical fiber – SingleMode Fiber
▪ 1962 Hall’s et. Al. (USA) Operation of semiconductor laser
▪ 1964 Goubau and Christian (USA) Light guide with periodic lenses
▪ 1966 Kao and Hockham (UK) Suggest that fiber loss could be <20 dB/km
▪ 1967 Kawakami (Japan) Proposes graded-index optical fiber
▪ 1970 Maurer, Keck, Schultz (USA) Fiber transmission loss 17dB/km @ 633 nm
▪ 1070-71 Dyot & Kapron (USA) Find pulse spreading lower at 1.2 to 1.3 µm
▪ 1972 Gambling et.al. (UK) Gigahertz bandwidth over 1km
▪ 1974 Kaiser & Astle (USA) Air-silica microstructured optical fiber
▪ 1975 Payne and Gambling (UK) Calculate zero material dispersion at 1.3 µm
▪ 1976 Horiguchi & Osanai (Japan) First fiber w. 0.47dB/km @ 1.2 µm & Open 3rd
Window @ 1.55 µm.

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▪ 1978 NTT (Japan) Makes SMF w. 0.2dB/km @ 1.55 µm
▪ 1981 British Telecom (UK) Transmits 140Mb/s in 49 km of SMF @ 1.3 µm
▪ 1987 Payne et.al.(UK) Develops EDFA @ 1.55 µm
▪ 1988 Mollenauer (USA) Soliton transmission on 4000 km of SMF
▪ 1991 Russell (UK) Proposes the Photonic Crystal optical fiber
▪ 1993 Nakazawa (Japan) Soliton signals over 180 million kms
▪ 1993 Mollenauer et.al.(USA) 10 Gbits through 11000 km of SMF using soliton system
▪ 1995-96 Russell et.al. (UK) Demonstrates the first Photonic Crystal optical fiber
▪ 1999 Cregan, Russell et.al. (UK) Demonstrates the Photonic Bandgap optical fiber

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1.2 Fibre Optic Physical Structure
Core
(8-12 µm)
Cladding
(125 µm)
Buffer Coating
(250 ou 900 µm)
R. Maurer, P. Schultz, D.Keck
(Corning Corp., 1970)
Human Hair ~ 70 µm

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1.2 Fibre Optic Physical Structure

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1.3 Index of Refraction and Index Difference
r r
o o
c
n
v
εµ
ε µ
ε µ
= = =
c = speed of light in vacuum (≈
3×108 m/s)
v = speed of light in medium
1
o o
c
ε µ
=
Permeability in
vacuum = 4π×10-7
N s2/C2
Permitivity in
vacuum = 8.854
×10-12 C2/N m2
For most materials in the optical region:
1r rnµ ε≈ ⇒ ≈
Substance n
Fused silica 1.458
Diamond 2.419
@ λ = 589.29 nm (Sodium D light)

© A.Lobo (2004) 13
2
2
2 2
1
j
j j
A
n
λ
λ λ
= +
−
∑
Sellmeier’s Equation
Absorption Bands
Visible

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2
2 2
2
1 withj
j j
j j
A
n b
b
λ
λ
λ
= + =
−
∑
A1=0.6961663 b1=0.004629148
A2=0.4079426 b2=0.01351206
A3=0.8974994 b3=97.934062
Refractive index variation of fused silica

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1 2
1
for 1
n n
n
−
Δ ≈ Δ ≪
2 2
1 2
2
12
n n
n
−
Δ =
and 1 2n n>
Refractive Index Difference, also called Refractive Index Contrast

© A.Lobo (2004) 16
1.4 Total Internal Reflection
1 1 2 2sin sinn nφ φ⋅ = ⋅
Snell’s Law
Increasing θ1 progressively until θ2 = π/
2 :
2
1
sin c
n
n
φ =
Critical angle
“Critical angle is the minimum angle necessary to obtain total internal reflection”

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1.5 Numerical Aperture and Acceptance Angle
2 2
1 2NA sino An n nθ= = −
Numerical Aperture (NA)
“Numerical Aperture express de ability of the fibre to collect light”
or
“Numerical Aperture measures spreading of light from the end of the fibre”

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Relation between Numerical Aperture and Index Difference
1NA 2n≅ Δ
Fibre type NA Δ
Step-Index SM 0.1 2×10-3
Graded-index MM 0.2 1×10-2
Step-index MM 0.3 2×10-2

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1. Using the Snell’s relation, how can you estimate the
amplitudes of the optical rays?
2. “An optical ray at any angle θ less than the critical
angle (θc) propagate along the fiber.” Is this true?
3. Definition of NA described before for singlemode fibre
is frequently wrong! Why?
4. Why the index of refraction is frequency dependent?
Questions to think:

Light Propagation in Fibres and
Related Optical Effects
FIBRE OPTIC TECHNOLOGY COURSE
António Lobo
Prof. Associado (UFP)

© A.Lobo (2004) 2
2. Light Propagation in Fibres and Related Optical Effects
2.1 The Wave nature of light.
2.2 Rays and Modes representation.
2.3 Mode Theory for fibres.
2.4 The Cut-off wavelength and V-number.
2.5 Singlemode and multimode fibre propagation.
2.6 Mode Field Diameter (MFD)
2.7 Phase velocity and group velocity.
2.8 Absorption and Scattering losses
2.9 Dispersion: Group delay and Material dispersion.
2.10 Chromatic Dispersion (CD).
2.11 Polarization effects and Birefringence.
2.12 Polarization Dependence Loss (PDL)
2.13 Polarization Mode Dispersion (PMD).
2.14 Non-linear Optical Effects
◗ Stimulated Raman Scattering
◗ Stimulated Brillouin Scattering
◗ Four-Wave Mixing
◗ Self-Phase and Cross-Phase Modulation

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2.1 Wave Nature of the Light
❧ Waves
◗ Electromagnetic radiation consisting of propagating electric and
magnetic fields (interference & diffraction)
❧ Photons
◗ Quanta of energy (photoelectric effect)
The two views are related: the energy in a photon is proportional to the
frequency of the wave.
DUAL NATURE

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Waves
[ ]( , ) cos ( v )
A
r t k r t
r
ψ
" #
= ⋅ ±% &
' (
[ ]( , ) cos )x t A kx tψ ω= ⋅ ±
2
2
2
1
v t
ψ
ψ
∂
∇ = ⋅
∂

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A Single Photon
(“short” packet wave)
Photon
❧ Quanta of Energy: Photon
(photoelectric effect)
E hν=
E → Energy of 1 photon in Joules (J)
h → Planck’s constant: 6.626×10-34 J-s
ν→frequency in Hz

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c → Speed of light = 2.9979 ×108 m/s; λ → Wavelength in meters; ν→Frequency in Hz
n→Refractive index (vaccum=1.0000; standard air= 1.0003; silica fibre: 1.44 to 1.48)
c nλν=

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21
o
o
S E B c E Bε
µ
= × = ×
! ! ! ! !y zE cB=
Poynting Vector

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About Reflection (Fresnel Equations): E ⊥Plane-of-Incidence
v
0
B k E
k E
= ×
=
!! !
! !
i
cos( )
cos( )
i oi r r
t ot t t
E E k r t
E E k r t
ω
ω
= −
= −
!! ! !
i
!! ! !
i
ot oi orE E E= +
! ! !
cos cos
cos cos
2 cos
cos cos
or i i t t
oi i i t t
ot i i
oi i i t t
E n n
r
E n n
E n
t
E n n
θ θ
θ θ
θ
θ θ
⊥
⊥
⊥
⊥
# $ −
= =& '
+( )
# $
= =& '
+( )

© A.Lobo (2004) 9
About Reflection (Fresnel Equations): E ||Plane-of-Incidence
cos cos
cos cos
2 cos
cos cos
or t i i t
oi i t t i
ot i i
oi i t t i
E n n
r
E n n
E n
t
E n n
θ θ
θ θ
θ
θ θ
" # −
= =% &
+' (
" #
= =% &
+' (
!
!
!
!

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Total Internal Reflection (case: ni > nt):θi ≥ θc
sin t
c
i
n
n
θ =
Critical Angle:
2 2 2
2 2 2
2 2 2 2
2 2 2 2
cos sin
cos sin
cos sin
cos sin
t
t
i i i i t
i i i i t
i i i i t
i i i i t
n i n n
r
n i n n
n in n n
r
n in n n
θ θ
θ θ
θ θ
θ θ
⊥
− −
=
+ −
− −
=
+ −
!
* *
1
e 0
1
r i t
r r r r
I I I
R
⊥ ⊥
"= =
⇒ = =$
= %
! !
sint t i ii k xn n t
t otE E e e
θ ωβγ % &−( )
= ∓
" "
Evanescent wave:
Goos-Hänchen depth
1
0
4 2
x
n
λ
< Δ <
Δ

© A.Lobo (2004) 11
Total Internal Reflection (case: ni > nt):θi ≥ θc
*
2 2
1
i i
r rr
r r e e− ΔΦ − ΔΦ
= =
= =
2 2 2
1
2 2 2
1
2
sin
tan
cos
sin
tan
cost
i i t
i i
i i i t
i
n n
n
n n n
n
θ
θ
θ
θ
−
⊥
−
$ %−
& 'ΔΦ =
& '* +
$ %−
& 'ΔΦ =
& '* +
!
⊥ΔΦ
ΔΦ!
2
1
i
t
n
n
=
=
Polarization TE →ΔΦ⊥
Polarization TM →ΔΦ||

© A.Lobo (2004) 12
About Reflection (case: ni > nt): Phase-shifts
tan t
p
i
n
n
θ =
Brewster Angle:
The component of electric field normal to plane-
of-incidence undergoes a phase shift of π
radians upon reflection when the incident
medium has a lower index than the transmitting
medium.

© A.Lobo (2004) 13
1
1 2
2
sin tan 1 ; is an integer
sin 2
kn a m m
π
φ
φ
−
$ %Δ
− − =' (
) *
Propagation Condition:
The optical field distribution that satisfies this phase matching condition is called MODE
Goos-Hänchen shift (ΔΦ⊥)
n1
n2
n2

© A.Lobo (2004) 14
Formation of modes

© A.Lobo (2004) 15
At angles for which the condition of self-consistently (i.e., as a wave reflects twice it
duplicates itself) is satisfied, the two waves interfere and create a pattern that does not
change with z direction.
z

© A.Lobo (2004) 16
V
ξ
1 2V kn a= Δ
sin
2
φ
ξ =
Δ
Single-mode condition
1.571
2
cV v
π
< = =
Normalized frequency:
Propagation constant:
Dispersion Equation
1
cos
2
m
V
π
ξ
ξ
−
+
=
(Slab Waveguide)

© A.Lobo (2004) 17
Coordinate System
Notação
2
grad
div
rot
lap
f f
A A
A A
f f
= ∇
= ∇
= ∇×
= ∇
! !
i
! !
Cylindrical Coordinates (r,φ,z)

© A.Lobo (2004) 18
Maxwell´s Equations
0
B
E
t
E
B E
t
E
B
µε µσ
ρ
ε
∂
∇× = −
∂
∂
∇× = +
∂
∇ =
∇ =
!
!
!
! !
!
i
!
i
1
0
B
E dl dS
t
E
B dl E dS
t
E dS dV
B dS
µε µσ
ρ
ε
∂
⋅ = − ⋅
∂
' (∂
⋅ = + ⋅) *
∂+ ,
⋅ = ⋅
⋅ =
∫ ∫∫
∫ ∫∫
∫∫ ∫∫∫
∫∫
!
! !!
!
! !! !
!!
!!
"
"
#
#

© A.Lobo (2004) 19
Maxwell´s Equations (2)
0
B
E
t
D
H J
t
D
B
ρ
∂
∇× = −
∂
∂
∇× = +
∂
∇ =
∇ =
!
!
!
! !
!
i
!
i
2 1
2 1
2 1
2 1
0
0
0
t t
t t
n n
n n
E E
H H
D D
B B
σ
− =
− =
− =
− =
! !
! !
Boundary Conditions
(for conductors)
o
o
D E E P
B H H M
J E
ε ε
µ µ
σ
= = +
= = +
=
! ! ! !
! ! ! !
! !
Constitutive Equations

© A.Lobo (2004) 20
2
2
2
2
2
2
0
0
E
E
t
H
H
t
µε
µε
∂
∇ − =
∂
∂
∇ − =
∂
!
!
!
!
In dielectric, non-conducting media, σ = 0
Wave Equations (1)
2
2
2
2
2
2
0
0
E E
E
t t
H H
H
t t
µε µσ
µε µσ
∂ ∂
∇ − − =
∂ ∂
∂ ∂
∇ − − =
∂ ∂
! !
!
! !
!

© A.Lobo (2004) 21
2 2 2
2 2 2
( ) ( ) 0
( ) ( ) 0
E r n k E r
H r n k H r
∇ + =
∇ + =
! !! !
! !! !
Wave Equations (2)
Helmholtz Equations
2
2 2 2
2
n k
c
ω
µεω= =
Waves in photonics are often monochromatic, with a frequency that stays the same
across the material boundaries, that is:
( , ) ( ) j t
E r t E r e ω
=
! !! !

© A.Lobo (2004) 22
Coordinate System
Notação
2
grad
div
rot
lap
f f
A A
A A
f f
= ∇
= ∇
= ∇×
= ∇
! !
i
! !
Cylindrical Coordinates (r,φ,z)

© A.Lobo (2004) 23
( )
( )
( , ) ( , )
( , ) ( , )
j t z
j t z
E r t E r e
H r t H r e
ω β
ω β
φ
φ
−
−
=
=
! !!
! !!
2 2 2
2 2 2
( ) ( ) 0
( ) ( ) 0
E r n k E r
H r n k H r
∇ + =
∇ + =
! !! !
! !! !
2 2 2
2 2
2 2 2 2
2 2 2
2 2
2 2 2 2
1 1
( , ) 0
1 1
( , ) 0
z z z z
z
z z z z
z
E E E E
n r k E
r r r r z
H H H H
n r k H
r r r r z
φ
φ
φ
φ
∂ ∂ ∂ ∂
+ ⋅ + ⋅ + + =
∂ ∂ ∂ ∂
∂ ∂ ∂ ∂
+ ⋅ + ⋅ + + =
∂ ∂ ∂ ∂
The longitudinal component of the electric field
does not change though either propagation or
reflection at the cylindrical surface

© A.Lobo (2004) 24
Since the equations for Er and Eθ are coupled, we first solve for Ez (Hz is a solution of the same
Helmholtz equation and its solutions have the same form ). We find all other field components
form Ez and Hz using Mawell’s equations.
We look for solution of the form:
( , , ) ( ) ( ) ( )zE r z R r Z zφ φ= ⋅Φ ⋅
2 2 2
2 2 2
2 2 2
( ) 0
R R Z
r r r knr
r r zφ
∂ ∂ ∂ Φ ∂
+ + + + =
∂ ∂ ∂ ∂
In the core we find: ( )
( )
( ) ( ) ( )
j z j z
j j
Z z ae be
ce de
R r gJ r hN r
β β
νφ νφ
ν ν
φ
κ κ
−
−
= +
Φ = +
= + 2 2 2
( ) and 0,1,2,...onkκ β ν= − =

© A.Lobo (2004) 25
We can simplify these solutions noting that:
• Often we have only forward going waves, thus a =0.
• The Neumann function Nν(κr) goes to minus infinity at r =0, so it is unphysical (h =0).
Therefore Jν(κr) is the proper solution in the core.

© A.Lobo (2004) 26
2
2
( ) j z Z
Z z be
z
β
β− ∂
= ⇒ = −
∂
2 2
2 2 2
2 2
0
R R
r r r R
r r r
ν
κ
# $∂ ∂
+ + − =' (
∂ ∂ ) *
• Due to predominant propagation of the field along the z axis an oscillatory characteristic is
assumed for the z dependence
• We need both the clockwise and counter-clockwise circulating exponentials that describe
the φdependence of the eigenmodes
2
2
2
( ) j j
ce deνφ νφ
φ ν φ
φ
− ∂ Φ
Φ = + ⇒ = −
∂
Bessel Equation
2 2 2
( ) and 0,1,2,...onkκ β ν= − =

© A.Lobo (2004) 27
2
2
2
0
r
ν
κ
# $
− >& '
( )
CASE 1: CORE
CASE 2: CLADDING
Modified Bessel functions
Bessel functions
2
2
2
0
r
ν
κ
# $
− <& '
( )
( )
( )
j z
z
j z
z
E AJ r e
H BJ r e
νφ β
ν
νφ β
ν
κ
κ
−
−
=
=
( )
( )
j z
z
j z
z
E CK r e
H DK r e
νφ β
ν
νφ β
ν
γ
γ
−
−
=
=
A,B,C and D are constants determined by the boundary conditions
2 2
γ κ= −

© A.Lobo (2004) 28
After some maths….
We get this “small” CHARACTERISTIC EQUATION for an optical fibre
( ) ( )
2' ' ' '
2
1
2
2 2
1
( ) ( ) ( ) ( )1 1 1 1
( ) ( ) ( ) ( )
1 1
o
J a K a J a K an
a J a a K a a J a n a K a
n k a a
ν ν ν ν
ν ν ν ν
κ γ κ γ
κ κ γ γ κ κ γ γ
βν
κ γ
% &% & % &
' (⋅ + ⋅ ⋅ ⋅ + ⋅ ⋅ =' ( ' (' (* +* + * +
, -% &
. /= ⋅ +' (
' (. /* +0 1
2 2 2
1
2 2 2
2
( )
( )
o
o
n k
n k
κ β
γ β
= −
= −
( ) ( )
2 22
V a aκ γ= +
2 2
1 2where oV k a n n= −
The characteristic equation is used with:
to find values for κ, γ, β and neff

© A.Lobo (2004) 29
About the Effective Index (neff)
A plane wave propagates with a phase term ejnkz, where k=2π/λo is the free-space wave-vector.
We can define an effective index for a guided wave that has a phase factor ejβz with:
2 eff
o
nπ
β
λ
≡
Then;
2 12 2
o o
n nπ π
β
λ λ
< < 2 1effn n n< <
The effective index is an “average” index seen by the guided mode.

© A.Lobo (2004) 30
Meridional Modes (ν = 0)
For modes that correspond to bouncing
meridional rays, there is no φ dependence.
The characteristic equation simplifies
greatly. Modes are of two types – TE0µ
(Ez=0) and TM0µ (Hz=0) with µ=1,2, ….
The values κ, γ can be found graphically
Curves of the characteristic equation of the
TE0µ and TM0µ modes

© A.Lobo (2004) 31
Skew Modes (ν ≠ 0)
Modes have both Ez≠0 and Hz≠0 and thus are
called “hybrid” modes. The hybrid modes are
labeled EHνµ and HEνµ depending on whether Ez
or Hz is dominant.
The values κ, γ can be found graphically
Curves of the characteristic equation of the
HE1µ modes

© A.Lobo (2004) 32
Field Distributions in Optical Fibers (1)
1 0
1 0
0
( )
( )
0
r
r
E
E J r
H J r
H
φ µ
µ
φ
κ
κ−
∼
∼
∼
∼
1 0
1 0
( )
0
0
( )
r
r
E J r
E
H
H J r
µ
φ
φ µ
κ
κ
−∼
∼
∼
∼
TE Modes TM Modes
This Bessel function (J1) has a zero at the origin and one maximum in the core

© A.Lobo (2004) 33
Field Distributions in Optical Fibers (2)
1 21 1 21
1 21 1 21
( )cos(2 ) ( )sin(2 )
( )sin(2 ) ( )cos(2 )
r rE J r H J r
E J r H J rφ φ
κ φ κ φ
κ φ κ φ
−
− −
∼ ∼
∼ ∼
HE21 Mode
This Bessel function (J1) has a zero at the origin and one maximum in the core

© A.Lobo (2004) 34
Linearly Polarized (LP) Modes
When the refractive index of the core n1 ≈ n2 , the characteristic equation (CE) can be
simplified. This is called the “weakly guiding approximation”. The CE can be written
in the unified form as:
1 1
( ) ( )
( ) ( )
m m
m m
J a K a
J a K a
κ γ
κ κ γ γ− −
=
1 for TM and TE modes
1 for EH modes
1 for HE modes
m ν
ν
→#
$
= + →%
$ − →'
LP modes can be constructed from sums of EH and HE modes that have the same propagation constant.

© A.Lobo (2004) 35
Linearly Polarized (LP) Modes

© A.Lobo (2004) 36
( ) ( )
2 22
V a aκ γ= +
2 2 2
1 2 1 2o oV k a n n k an= − = Δ
2 2 2
1
2 2 2
2
o
o
u a a k n
w a a k n
κ β
γ β
= = −
= = −
This is a dimensionless number that determines how many modes a fibre can support
2 22
2
2 2 2
1 2
( / )ok nw
b
V n n
β −
= =
−
Normalized Propagation Constant*
2 2
1 2
2
cutoff
a
n n
V
π
λ = −
Cut-off Wavelength
* D. Gloge, “Weakly guiding fibers”, Applied Optics 10, 2252-2258 (1971)

© A.Lobo (2004) 37
min
1
2.405
3.7cutoff
V V
anλ λ
< =
> = Δ
Singlemode Condition
( )
( ) 10log
( )
straight
loop
P
R
P
λ
λ
λ
" #
= $ %$ %
& '
See recommendation ITU-T G.650

© A.Lobo (2004) 38
2.405V ≤
Single-mode Condition
Single-modeRegion
LP
Step-index singlemode fibre

© A.Lobo (2004) 39
2
2
V
M =
2 2
1 2NA n n= − 2 2 2 2
1 2sin n nπ θ πθ πΩ = ≈ = −
Numerical Aperture Solid Angle
Total number of Modes in the fibre
(Step-index Fibre)

© A.Lobo (2004) 40
In the weakly guiding approximation, the big steps in this figure become perfectly vertival
(eg. TE01, TM01 and HE21 have the same V at cutoff. Groups of modes with the same cutoff
also have the same propagation constant.
Singlemode
Condition

© A.Lobo (2004) 42
HE11 mode
Solid lines – E field
Dashed lines – H field
With permission of C.D. Cantrell and D.M. Hollenbeck, “Fiberoptic Mode Functions: A Tutorial”, Erick
Jonsson School of Eng. and Computer Science, Univ. Texas at Dallas, Course EE6314.

© A.Lobo (2004) 43
HE21 mode

© A.Lobo (2004) 44
TM01 mode TE01 mode

© A.Lobo (2004) 45
Optical Power in the Mode
( ) ( )* * *1 1
2 2
z z r rS E H u E H E Hθ θ= × ⋅ = −
! ! !
( )
2 2
* *
0 0 0 0
1
2
a a
core z r rP S r dr d r E H E H dr d
π π
θ θφ φ= ⋅ ⋅ = − ⋅ ⋅∫ ∫ ∫ ∫
( )
2 2
* *
0 0
1
2
cladding z r r
a a
P S r dr d r E H E H dr d
π π
θ θφ φ
∞ ∞
= ⋅ ⋅ = − ⋅ ⋅∫ ∫ ∫ ∫
22
2 2 2
1 1
( )( )
1 1
( ) ( )
1
core m
total m m
cladding core
total total
P K aa
P V K a K a
P P
P P
γκ
γ γ− +
$ %
= − −& '
( )
= −

© A.Lobo (2004) 46
Optical Power in the Mode LPml

© A.Lobo (2004) 47
2.6 Mode Field Diameter (MFD).
Gaussian Beam (1)
2
( )
r
w
oE r E e
! "
−$ %
& '
=
2
2
( ) o
r
w
oI r I e
! "
− $ %
& '
=

© A.Lobo (2004) 48
Gaussian Beam (2)
1/22
2
( ) 1
( ) 1
o
R
R
z
w z w
z
z
R z z
z
! "# $
% &= + ' (
% &) *+ ,
! "# $
= +% &' (
) *% &+ ,
2
o
R
w
z
π
λ
=
2
ow
λ
π
Θ =Divergence Angle:
Rayleigh Range:
for large ( )
o
z
z w z
w
λ
π
⇒ ≈
H. Kolgelnik and T. Li, “Laser beams and resonators”, Applied Optics 5, 1550-1566 (1966)

© A.Lobo (2004) 49
Petermann I Integral (Near-Field):
M. Artiglia et al, “Mode Field Diameter Measurements in Single-Mode Optical Fibers”, Journal of Lightwave Technology, Vol. 7, No. 8.
(1989)
1/2
2 3
0
2
0
( )
2 2
( )
I
E r r dr
MFD
E r r dr
∞
∞
" #
⋅% &
% &= ⋅
% &
⋅% &
% &' (
∫
∫
Petermann II Integral* (Far-Field):
1/2
3
2 ( )sin cos
( )sin cos
II
I d
MFD
I d
θ
θ
θ
θ
θ θ θ θ
λ
π
θ θ θ θ
−
−
% &
⋅( )
( )= ⋅
( )
⋅( )
( )* +
∫
∫*TIA/EIA FOTP-191, ITU-T G.650E

© A.Lobo (2004) 50
Near-Field Profiles
100X
40X
J. L. Guttman, “Mode-Field Diameter and “Spot Size” Measurements of Lensed and Tapered Specialty Fibers”, NIST Symposium on
Optical Fiber Measurements, September 24-26, 2002

© A.Lobo (2004) 51
Far-Field Profiles
J. L. Guttman, “Mode-Field Diameter and “Spot Size” Measurements of Lensed and Tapered Specialty Fibers”, NIST Symposium on
Optical Fiber Measurements, September 24-26, 2002
a.) Fiber #1 b.) Fiber #2

© A.Lobo (2004) 52
Petermann II Integral* (Far-Field):
1/2
3
2 ( )sin cos
( )sin cos
II
I d
MFD
I d
θ
θ
θ
θ
θ θ θ θ
λ
π
θ θ θ θ
−
−
% &
⋅( )
( )= ⋅
( )
⋅( )
( )* +
∫
∫(*TIA/EIA FOTP-191, ITU-T G.650E)
• Errors [1,2]
• Obliquity Factor and Aperture Field
•Elliptical Fiber [3]
• Radial Symmetry for Hankel Transform
• Field Within Fiber vs Field at Focus
[1] M. Young, “Mode-field Diameter of single-mode optical fiber by far-field scanning”, Applied Optics, Vol. 37, No. 24, August 1998
[2] R. C. Wittmann and M. Young, “Are the Formulas for Mode-Field Diameter Correct?”, NIST SOFM 1998
[3] M. Artiglia et al, “Mode Field Diameter Measurements in Single-Mode Optical Fibers”, Journal of Lightwave Technology, Vol. 7, No. 8.
August 1989

© A.Lobo (2004) 53
1.5 6
1.619 2.879
0.65w a
V V
! "
= + +# $% &
D. Marcuse, Bell Systems Tech. Journal 56, 703-718 (1977)
Marcuse Model:

© A.Lobo (2004) 54
1.5 6
1.619 2.879
0.65w a
V V
! "
= + +# $% &
For the best fit between a Gaussian function and the Bessel function in the core we can use
the Marcuse Model:
Satisfying this condition gives about 96% overlap between the Gaussian and the Bessel
function mode profiles. At the cut-off condition (V ≈ 2.405) we obtain:
1.1w a≈

© A.Lobo (2004) 55
1.5 6
1.237 1.429
0.761w a
V V
! "
= + +# $% &
Other MFD models
1.5 6
1.289 1.041
0.759w a
V V
! "
= + +# $% &
1.5 6
1.66 0.987
0.616w a
V V
! "
= + +# $% &
( )
1.5
1
for V>1
ln
w a
V
=Snyder and Sammut Model
Myslinski Model
Desurvire Model
Whitley Model
Several models have been developed to obtain better agreement with experimentally
observed data (particularly gain factors →Erbium doped fibres)

© A.Lobo (2004) 56
2.7 Absorption and Scattering losses.
3
8 2
4
8
3
R T B Fn p k T
π
γ β
λ
=
SCATTERING
NON-LINEAR
LINEAR
• Rayleigh (diameter physical anomalies < λ/10)
• Mie (diameter physical anomalies > λ/10)
• Brillouin (acoustic phonon →SBS)
• Raman (optical phonon →SRS)
n – refractive index of the material, βT – isothermal compressibility, p – photoelastic coefficient,
TF – Solidification temperature, kB – Boltzman constant, L - the fibre length.
Rayleigh Scattering Coefficient
RL
RayleighF e γ−
=
Transmission Loss factor

© A.Lobo (2004) 57
ABSORPTION
INTRINSIC
INDUCED Curvature (micro and macrobending)
EXTRINSIC
Interaction of free electrons and the λ within
the fibre material.
Impurities atoms in glass material (metal ions)
4.582
3 [ ]
[ / ] 1.108 10
UV
m
UV dB km Ce e
λ
λ µλ
α −
= = ×
48.48
11 [ ]
[ / ] 4 10 m
IR dB km eλ µ
α −
= ×
Urbach’s rule (empirical relationship) Macrobending (critical radius)
2
1
2 2
1 2
3
4 ( )
MMF
n
R
n n
λ
π
=
−
3
2 2
1 2
20 0.996
2.748SMF
cutoff
R
n n
λ λ
λ
−
# $
= −% &% &− ' (

© A.Lobo (2004) 58
ATTENUATION
z
dz
P P+dP
1 (0)
( ) (0) ln [1/km or 1/m]
( )
LdP P
L P L P e
dz L P L
α
α α− # $
= − ⇒ = ⇒ = & '
( )
(0) (0)
10log 10log 10 log 4.343
( ) ( )
L LP P
e e L e
P L P L
α α
α α
" #
" #= ⇔ = = =% & ' (
' (
10 (0)
[ / ] log
( )
[ / ] 4.343 [1/ ]
P
dB km
L P L
dB km km
α
α α
" #
= $ %
& '
=
scattering absorption bendingα α α α= + +

© A.Lobo (2004) 60
Theoretically expected minimum attenuation
Governed by:
1. Rayleigh scattering at short λ
2. Multi-phonon absorption at long λ

© A.Lobo (2004) 61
Fibre λ (nm) α (dB/km)
MMF-GI 850 2.5
SMF 1300 0.5
SMF 1550 ≤0.25
F-POF (Fluorinated) 800 to 1340 60
ZBLAN 1550 0.02
SMFF
(Fluoride glass)
2300 0.005
Typical minimum attenuation values for several fibres

© A.Lobo (2004) 62
2.8 Phase velocity and group velocity.
phv
ω
β
=g
d
v
d
ω
β
=
1
1
g
c
v
dn
n
d
λ
λ
=
" #
−% &
' (
Group Velocity Phase Velocity
Guide Group Index (ng)
1
ph
c
v
n
=

© A.Lobo (2004) 63
1
g
L d L d
L L
v d c dk
β β
τ β
ω
= = = ⋅ =
Group Delay
1
o
d
D
L d
τ
λ
= ⋅
Dispersion Parameter
Origin of the Dispersion: Frequency dependence of the mode index n(ω)
2
0 1 0 2 0
1
( ) ( ) ( ) ( ) ...
2
n
c
ω
β ω ω β β ω ω β ω ω= = + − + − +
1
2 2 3 2
2 2 2 2 2
1 1
1
2
2
g
g
ndn
n
c d c v
dn d n d n d n
c d d c d c d
β ω
ω
ω λ
β ω
ω ω ω π λ
% &
= + = =' (
) *
% &
= + ⋅ ⋅' (
) *
! !

© A.Lobo (2004) 64
2
1
22 2
2d c d n
D
d c d
β π λ
β
λ λ λ
= = − − ⋅!Dispersion Parameter:
Group Velocity Dispersion (GVD): 1
2 2
1 1 g
g g
dvd d
d d v v d
β
β
ω ω ω
# $
= = = − ⋅' (' (
) *
(Contains the information about the variation of the
group velocity with wavelength)
2
g g
d L d L
t L LD
d v d v
δ δω β δω δλ δλ
ω λ
% & % &
= = ⋅ = = ⋅( ) ( )( ) ( )
* + * +
If a pulse with spectral width (Δδ) input to a fibre with length L, the output pulse
broadening is:
Limitation on the bit rate: 2
1
or 1t B t BL BLD
B
δ δ β δω δλ< = ⋅ ⋅ <

© A.Lobo (2004) 65
2
1
2
1 g
mat
dn d n
D
c d c d
λ
λ λ
= ⋅ = − ⋅
Material Dispersion
0gdn
dλ
=
Zero-dispersion wavelength = 1276 nm
Negative Dmat
Positive Dmat

© A.Lobo (2004) 66
2.10 Chromatic Dispersion (CD)
2 2
2
2 2
1 2
( / )ok n
b
n n
β −
=
−
Normalized propagation constant*:
2
1 2
( / )ok n
b
n n
β −
≅
−
for small Δ
2 (1 )o o effk n b k nβ = + Δ =
* See D. Gloge, “Weakly guiding fibers”, Applied Optics 10, 2252-2258 (1971)
D. Gloge, “Dispersion in weakly guiding fibers”, Applied Optics 10, 2442-2445 (1971).
2
1
2
1 g
mat
dn d n
D
c d c d
λ
λ λ
= ⋅ = − ⋅
2
2
2
( )
wg
n d Vb
D V
c dVλ
Δ
= −
mat wgD D D= +

© A.Lobo (2004) 67
2
2
2
( )
wg
n d Vb
D V
c dVλ
Δ
= −
Waveguide Dispersion

© A.Lobo (2004) 68
Anomalous dispersionNormal dispersion
λD ≈ 1320 nm

© A.Lobo (2004) 69
Anomalous Dispersion
(D > 0)
Normal Dispersion
(D < 0)

© A.Lobo (2004) 70
P.K. Bachman et.al., “Dispersion-flattened single-mode fibres prepared with PCVD: Performance, limitations, design and optimization”, J.
Lightwave Technology 4, 858-863 (1986)

© A.Lobo (2004) 71
0( ) ( )oD Sλ λ λ= −
4
0
( ) 1
4
oS
D
λ λ
λ
λ
" #$ %
= −' () *
+ ,' (- .
Typical values for S0 are 0.092 ps/(km·nm2)
for SMF, and between 0.06 and 0.08 ps/
(km·nm2) for DSF
1
( ) gd
D
L d
τ
λ
λ
= ⋅
Slope (ps/km·nm2)
(See recommendation ITU-T G.652)

© A.Lobo (2004) 73
k
h
v
k
h
v
0 cos( )x x x xE e E t zω β φ= − +
! !
0 cos( )y y y yE e E t zω β φ= − +
! !
TOTAL x yE E E= +
! ! !
nkn
λ
π
==β
2
x
y
POLARIZATION

© A.Lobo (2004) 74
time
h
v
time
h
v
-4 -2 2 4
h,(V/m)
-3
-2
-1
1
2
3
v,(V/m)
0 0cos( ) cos( )TOTAL x x y yE e E t z e E t zω β ω β π= − + − +
! ! !
LINEAR POLARIZATION

© A.Lobo (2004) 75
time
h
v
time
h
v
-1 -0.5 0.5 1
h,(V/m)
-1
-0.5
0.5
1
v,(V/m)
0 sin( ) cos( )TOTAL x x yE E e t z e t zω β ω β# $= − + −& '
! ! !
CIRCULAR POLARIZATION

© A.Lobo (2004) 76
time
h
v
time
h
v
-2 -1 1 2
h,(V/m)
-4
-2
2
4
v,(V/m)ELLIPTICAL POLARIZATION
0 1 0 2sin( ) cos( )TOTAL x x y yE e E t z e E t zω β φ ω β φ= − + + − +
! ! !

© A.Lobo (2004) 77
Birefringence effect of polarized light in the fibre
Birefringence
Beat Length
( )
or equivalently,
2
( )
x y
x y
B n n
n n
π
β
λ
= −
Δ = − 2
BL
B
π λ
β
= =
Δ
π/2 2π0 π/4 3π/4 π 5π/4 3π/2 7π/4

© A.Lobo (2004) 78
Jones Calculus* - applicable only to polarized waves
(*) E. Hecht, “Optics”, 4Ed. Addison Wesley, Chapter 8, 2002.
A1
0
0
x
y
i
x
in i
y
E e
E
E e
ϕ
ϕ
" #
= $ %
$ %& '
! xout
out
yout
E
E
E
! "
= # $
% &
!
11 12
21 22
xout xin
yout yin
E Ea a
E Ea a
! " ! "! "
=# $ # $# $
% &% & % &
0
0
0
1
1
x
x
x
i
ix
in xi
x
E e
E E e
E e
ϕ
ϕ
ϕ
" # " #
= =$ % $ %
& '& '
!
45º
11
12
E
! "
= # $
% &
!02 xi
xE eϕ
Dividing both terms by:

© A.Lobo (2004) 79
Jones Matrix
from E. Hecht, “Optics”, 4Ed. Addison Wesley, Chapter 8, 2002.

© A.Lobo (2004) 80
[from D.Derickson, “Fiber Optic Test and Measurement” Prentice Hall, Ch. 6 (1998)]
Measurement of the Jones Matrix of an optical element

© A.Lobo (2004) 81
Stokes Parameters – applicable to both totally or partially polarized light.
The elements describe the optical power in particular
reference polarization states.
0
1
2 45º 45º
3
total
LH LV
L L
RC LC
S I
S I I
S I I
S I I
+ −
" # " #
$ % $ %−$ % $ %=
$ % $ %−
$ % $ %
−$ % $ %& ' & '
2 2 2
1 2 3
0
S S S
DOP
S
+ +
=

© A.Lobo (2004) 82
Normalized Stokes Parameters
0
1
20
3
1
1 1
2
3
S
Ss
Ss S
Ss
! "! "
# $# $
# $# $ =
# $# $
# $# $
# $% & % &
2 2 2
1 2 3DOP s s s= + +

© A.Lobo (2004) 83
Poincaré Sphere – graphical tool in real, 3D space that allows convenient description of polarized
signals and of polarization transformations caused by propagation through
devices.

© A.Lobo (2004) 84
Measurement of retardance θ of a near λ/4-wave retarder
[from D.Derickson, “Fiber Optic Test and Measurement” Prentice Hall, Ch. 6 (1998)]

© A.Lobo (2004) 85
Twisting of a singlemode fibre [from Agilent HP8509C Application Note]

© A.Lobo (2004) 86
Muller Matrix – applicable to any degree of polarization.
00 01 02 030 0
10 11 12 131 1
20 21 22 232 2
30 31 32 333 3
out in
out in
out in
out in
m m m mS S
m m m mS S
m m m mS S
m m m mS S
! " ! "! "
# $ # $# $
# $ # $# $= ⋅
# $ # $# $
# $ # $# $
# $# $ # $& '& ' & '

© A.Lobo (2004) 87
Two polarization modes Ex , Ey of a singlemode fibre
© R. Ulrich, TU Hamburg-Harburg

© A.Lobo (2004) 88
From K.T.V.Grattan & B.T. Meggitt,
“Optical Fiber Sensor Technology”,
Chapman & Hall (1995)

© A.Lobo (2004) 89
From K.T.V.Grattan & B.T. Meggitt,
“Optical Fiber Sensor Technology”,
Chapman & Hall (1995)

© A.Lobo (2004) 90
2.12 Polarization Dependence Loss (PDL) .
Device Under Test
(DUT)
Constant Power
100% Polarized
Time
Pmáx
Pmin
max
min
10logdB
P
PDL
P
! "
= # $
% &
PDL measures the peak-to-peak difference in transmission for light with
various states of polarization.

© A.Lobo (2004) 91
PMD is a fundamental property of singlemode optical fibre and components in which signal
energy at a given wavelength is resolved into two orthogonal polarization modes of slightly
different propagation velocity. The resulting difference in propagation time between
polarization modes is called the differential group delay (Δτ).
g
L d
L
v d
n d n
L
c c d
β
τ
ω
ω
ω
Δ
Δ = =
Δ
Δ Δ% &
= + ⋅( )
* +
d
L L
d
φ β
φ β τ
ω ω
Δ
= Δ → = = Δ
Δ
Also, Frequency-domain manifestation

© A.Lobo (2004) 92
PMD is characterized by the PMD-vector, Ω(ω), in Stokes space, around which an output
state of polarization (SOP), s , rotates when the carrier frequency is changed.
'
dS
S S
dω
= = Ω×
!
! !!
τΔ = Ω
!
And the differential group delay (DGD) is:

© A.Lobo (2004) 93
CASE 1 - Ideal fibre section (βx = βy)
,
,
( )
x in
in
y in
E
E
E
ω
" #
= $ %
& '
! ( )out in
j L
in
E J E
e Eβ
ω
−
$ % $ %= & ' & '
=
! ! !
i
!
0
( )
0
j L
j L
e
J
e
β
β
ω
−
−
$ %
= & '
( )
!
0τΔ =

© A.Lobo (2004) 94
CASE 2 - linearly birefringent fibre section (βx ≠ βy)
,
,
( )
x in
in
y in
E
E
E
ω
" #
= $ %
& '
!
0
( )
0
x
y
j L
j L
e
J
e
β
β
ω
−
−
$ %
= & '
& '( )
!
2d
L n
d
L
n
c
π
τ
ω λ
% &
Δ = ⋅ Δ) *
+ ,
≅ Δ ⋅
,
,
( )
x
y
j L
x in
out j L
y in
e E
E
e E
β
β
ω
−
−
$ %
= & '
& '( )
!

© A.Lobo (2004) 95
CASE 3 - Birefringence axes aligned: PM fibre
0
( )
0
x
y
j L
total j L
e
J
e
β
β
ω
−
−
$ %
= & '
& '( )
!
out
out
dS
S
dω
= Ω×
!
!!
Ω
!
τΔ = Ω
!

© A.Lobo (2004) 96
CASE 4 - Random orientation of birefringence axes aligned: standard
long-length fibre
[ ] [ ] [ ] [ ] [ ] [ ]2 2 1 1total n nJ R J R J R J= ⋅ ⋅⋅⋅⋅ ⋅ ⋅ ⋅
!
( )out
out
dS
S
d
ω
ω
= Ω ×
!
!!

© A.Lobo (2004) 97
• Frequency domain scenario: PMD vector and principal states of polarization (PSP)
1,20 if outout
out
dS
S PSP
dω
= =
!
!
• Time domain scenario: Pulse splitting

© A.Lobo (2004) 98
• Frequency domain scenario : frequency dependence of PSP
0( ) ( ) ....ωω ω ωΩ = Ω + Ω Δ +
! ! !
0
d
d
ω
ω ω
ω =
Ω
Ω =
!
!
1st order approximation is valid
within “PSP-bandwidth”:
1
PSPω
τ
Δ ≅
Δ
• Time domain scenario: multi-path pulse transmission
[See J.L. Santos, Ph.D. Thesis, Chap.7,
Univ. Porto, (1992)]

© A.Lobo (2004) 99
2
2
2
2
3
2
e
τ
α
τ
π α
Δ
−Δ
⋅
with 8τ α πΔ =
Probability of finding DGD at value Δτ
is given by the Maxwellian density
distribution

© A.Lobo (2004) 100
Consequences of PMD
❧ DGD (Δτ) causes:
◗ Pulse broadening
◗ Reduction of eye openings / increase in BER
◗ Additional power penalty
◗ Increase of outage probability
❧ Instantaneous Δτ
◗ Is a random variable
◗ It varies due to environment (temnperature, strain)
◗ It can surpass its mean value by far
2
PMD Lτ τΔ = Δ ∝

© A.Lobo (2004) 101
2.14 Non-linear Optical Effects.
❧ Self-Phase Modulation (SPM) – pulses distiort as they propagate.
❧ Cross-Phase Modulation (XPM) – pulses interfere with one another
❧ Modulation Instability (MI) – CW beams break into pulses
❧ Solitons – a nonlinear way of transmitting pulses
❧ Four-Wave Mixing (FWM)
❧ Optical Kerr Effect – electric field imposed induces linear birefringence
❧ Stimulated Brillouin Scattering (SBS) – inelastic scattering from
acoustic phonons
❧ Stimulated Raman Scattering (SRS) – inelastic scattering from
molecular resonances
❧ Supercontinuum Generation (SG) – “white light” generation
Parametric
Process
(light induced modulation)
Non-Parametric
Process
Observed effect

© A.Lobo (2004) 102
Nonlinear optics is a result of anharmonic excitation of the medium. The induced
polarization is given by:
( )0 1 2 3 ...P E EE EEEε χ χ χ= ⋅ + ⋅ + ⋅ +
! ! ! ! ! ! !
where χ1, χ2, χ3 are 1st, 2nd, 3rd order susceptibilities. χ2 vanishes in centro-symmetric
materials like glass, so the lowest-oder nonlinear term is χ3.
One manifestation of this is the nonlinear refractive index:
where n2K is the nonlinear Kerr coefficient and is directly related to χ3. In most glasses, n2 is
positive, so the refractive index of the material increases at higher intensities. The value of
n2K for SiO2 is ~3.2×10-20 m2W-1.
2
0 2Kn n n E= +

© A.Lobo (2004) 103
❧ Nonlinear Coefficient (γ) – measure of the strength of the nonlinar response of
a particular fibre at frequency ω0 with effective area Aeff and n2k known.
2 0K
eff
n
cA
ω
γ =
❧ Effective Length (Leff) – effective nonlinear length of a fibre with physical length
L and loss given by α.
1 L
eff
e
L
α
α
−
−
=
❧ Nonlinear Length (LNL) – the fibre length required for nonlinear effects to
become important, for a given peak pump power. Explicity, the length for development
of a phase shift of unity. 0
1
NLL
Pγ
=
❧ Dispersion Length (Ld) – length over which the pulse length τ0 is significantly
dispersed in a fibre with β2.
2
0
2
dL
τ
β
=
DEFINITIONS

© A.Lobo (2004) 104
λo
Rayleigh radiation (λo) Brillouin radiation (λo±0.08 nm)
Raman radiation (λo
±80 nm )
Scattering spectra of an optical fibre
Wavelength
Energy
Incident radiation (λo=1550 nm)
2 s
B
o
nV
n
l
=Anti-Stokes Stokes
ElectrostrictionRaman shift ≈ -13 THz @ 1.55 µm
Brillouin shift ≈ -11 GHz @ 1.55 µm

© A.Lobo (2004) 105
2.14.1 Stimulated Raman Scattering (SRS)
(Spontaneous) Raman Scattering: a very
small amount of light in any molecular is
inelastic scattered. A lower-frequency
photon is produced, and the extra energy
goes into exciting a molecular vibrationsl
or rotational mode.
Stimulated Raman Scattering (SRS): the
lower-frequency radiation beats with the
pump beam to provide a field beating at
the Raman frequency. This drives the
Raman oscillations directly, so that the
shifted radiation experiences gain at the
expense of the pump beam.
0
P zS
SRS S S S
dI
g I I e I
dz
α
α−
= −
The growth of the Stokes wave along the
fibre in both spontaneous and stimulated
emission may be expressed in the form:
Raman gain

© A.Lobo (2004) 106
2.14.1 Stimulated Raman Scattering (SRS)
16
effSRS
th
SRS eff
kA
P
g L
≅
17 W km @ 1.55
µm
SRS
th effP L⋅ ≈ ⋅
Relative polarization factor
(1 ≤ k ≤ 2)

© A.Lobo (2004) 107
2.14.2 Stimulated Brillouin Scattering (SBS)
SBS Characteristics:
❧ Low power threshold
❧ Backward propagating Stokes wave
❧ Small Stokes shift (low phonon energy)
❧ Acoustic phonon lifetime is long (10ns),
so gain bandwidth is narrow.
❧ Need narrow linewidth pump source for
efficient excitation.
21 1 pump effSBS
th
B SBS eff
A
P
g L
ν
ν
Δ# $
≅ +& '
Δ( )
0.03 W km @ 1.55
µm
SBS
Electrostriction

© A.Lobo (2004) 108
2.14.3 Four-Wave Mixing (FWM)
2
6
3
4 2 2
31024
( ) eff L
ij FWM i j
eff
L
P L PP e
n c A
α
χπ
η
λ
−
' (' (
= ⋅* +* + * +, - , -
see A.R. Chraplyvy, Journal Lightwave Technology 8, 1548-1557 (1990).

© A.Lobo (2004) 109
2.14.4 Self-Phase (SPM) and Cross-Phase Modulation (XPM)
Kerr nonlinearity
Linear regime
2
effSPM
th
K eff
A
P
n L
λ
π
≅
1.5 W km @ 1.55
µm
SPM
( )2
2SPM K
d
n E
dt
νΔ ∝

© A.Lobo (2004) 110
2.14.4 Self-Phase (SPM) and Cross-Phase Modulation (XPM)
( )2
0 2
2
0 2
2
2
K
K
nL L
n n E
n n n E
π
φ π
φλ
λ
$
= %
⇒ = +'
%= + (
( )
0 0
2
0 2
' '
2
' K
d
dt
L d
n E
dt
φ
ω ω ω ω ω
π
ω ω
λ
= + ⇔ = + ⇒
⇒ = +
( )
( )
2
0
2
0
0 ' ( )
0 ' ( )
K
K
d
E t
dt
d
E t
dt
ω ω ω
ω ω ω
> ⇒ = −
< ⇒ = +
Pulse is
CHIRPED

© A.Lobo (2004) 111
Limitations over WDM
Limitations on the channel power
imposed by four nonlinear effects
(assumed λ=1.55 µm and fibre loss
of 0.2 dB/km)
see A.R. Chraplyvy, Journal Lightwave Technology 8, 1548-1557 (1990).

© A.Lobo (2004) 112
Soliton Propagation 2
0 2
3.11
FWHM
P
T
β
γ
!
Fundamental soliton
peak power

Types of Optical Fibre
António Lobo

© A.Lobo (2004) 2
3.1 Optical Fibre Fabrication.
3.2 Standard Singlemode Fibre.
3.3 Step-Index and Graded-Index Multimode Fibre.
3.4 Dispersion Shifted Fibre.
3.5 Dispersion Flattened Fibre.
3.6 Non-Zero Dispersion Shifted Fibre.
3.7 Dispersion Compensating Fibre.
3.8 E-Band Fibre.
3.9 Polarization Maintaining Fibres:
PANDA, Bow-Tie, Elliptical and Side-Hole types.
3.10 Rare-Earth Doped Fibres:
Erbium, Ytterbium, Neodymium, Praseodymium.
3.11 Long Wavelength Fibres (Fluoride and Chalcogenide).
3.12 Plastic Fibre.
3.13 Hollow Fibre.
3.14 Photonic Crystal Fibre.

© A.Lobo (2004) 3
3.1 Optical Fibre Fabrication Fibre Drawing
(Double-Crucible Method)
Cabelte S.A.

© A.Lobo (2004) 4
3.1 Optical Fibre Fabrication
Modified Chemical Vapour Deposition (MCVD)
Prof. J. Knight, Optoelectronics Group, Dept. Physics, University of Bath (UK) & Blaze Photonics Ltd. (UK)

© A.Lobo (2004) 5
3.2 Standard Singlemode Fibre (SMF)
❧ Generally standard singlemode fibre, have a step-index refractive profile.
❧ Main disadvantages: coupling difficulties with incoherent optical sources (like LEDs), small NA
❧ One of the most advanced commercial singlemode fibres: SMF-28 (by Corning)
Main Characteristic (Measurement methods comply with ITU-T G.650 and IEC 793-1)
❧ Attenuation coefficient: ≤0.40 dB/km @ 1310 nm , ≤0.3 dB/km @ 1550 nm
❧ NA = 0.13
❧ Cut-off wavelength: < 1260 nm
❧ Core diameter: 8.3 µm
❧ MFD: 9.30±0.50 µm @ 1310 nm, 10.50 ±1.00 µm @ 1550 nm
❧ Zero dispersion wavelength: 1301.5 nm ≤ λ0D ≤ 1321.5 nm
❧ Zero dispersion slope (So): 0.092 ps/nm2#km
❧ PMD (máx): ≤ 0.2 ps/√km
❧ Refractive index difference (Δ): 0.36%
❧ Effective group index refraction (Ng): 1.4675 @ 1310 nm, 1.4681 @ 1550 nm

© A.Lobo (2004) 6
SMF-28
From Plasma Optical Fiber B.V. (Eindhoven)

© A.Lobo (2004) 7
Depressed cladding 1300 nm Optimized SMF

© A.Lobo (2004) 8
3.3 Step-Index (SI-MMF) and Graded-Index Multimode Fibre (GI-MMF)
❧ SI-MMF are very similar to step-index SMF, but differ in their core diameter.
❧ SI-MMF have fibre core diameters around 50 to 62.5 µm, and cladding with 125 µm.
❧ GI-MMF are defined by a progressive decrease of refractive index from the centre of the core with
radius a toward the cladding, while still maintaining the fundamental relationship: n1 > n2.
❧ Different modes can interfere with each other generating modal noise.
❧ Refractive index profiles difficult to realize in practice → expensive and some flutctuations from
ideal profile are inevitable.
1( ) 1 2
y
r
n r n
a
! "
= − Δ% &
' (
Where y is the core characteristic
refractive index profile:
❧ y = ∞ for step-index profile
❧ y = 1 for triangular profile
❧y = 2 for parabolic profile

© A.Lobo (2004) 9
3.4 Dispersion Shifted Fibre (DSF)
❧ DSF is a singlemode fibre with very low dispersion at 1550 nm operating wavelength.
Main Characteristics (Measurement methods comply with ITU-T G.650 and IEC 793-1)
❧ Attenuation coefficient: ≤0.25 dB/km @ 1550 nm Cut-off wavelength: ≤ 1260 nm
❧ NA = 0.17 Core diameter: 8.3 µm MFD: 8.10 ±0.65 µm @ 1550 nm
❧ Zero dispersion wavelength: 1535 nm ≤ λ0D ≤ 1565 nm
❧ Zero dispersion slope (So): 0.085 ps/(nm2·km)
❧ Total dispersion coefficient: ≤ 2.7 ps/(nm·km)
❧ PMD (máx): ≤ 0.5 ps/√km

© A.Lobo (2004) 10
3.5 Dispersion Flattened Fibre (DF-SMF)
❧ DF-SMFF is a singlemode fibre with very low dispersion simultaneously at 1330 nm and 1550 nm
operating wavelengths.
Main Characteristics
❧ Attenuation coefficient: ≤0.5 dB/km @ 1310 nm , ≤0.3 dB/km @ 1550 nm
❧ Dispersion coefficient: ≤3.5 ps/(nm·km) @ 1310 nm , ≤ 3.5 ps/(nm·km) @ 1550 nm

© A.Lobo (2004) 11
3.6 Non-Zero Dispersion Shifted Fibre (NZ-DSF)
❧ NZ-DSF is a singlemode fibre specially designed for Dense Wavelength Division Multiplexing
(DWDM) technology. Non-linear effects such as Four-Wave Mixing (FWM) are minimized, by
moving the zero-dispersion wavelength outside the band used by EDFAs..
❧ Dispersion coefficient: -3.5 to -0.1 ps/(nm·km) over the range 1530 to 1560 nm
❧ PMD: ≤ 0.5 ps/√km
❧ Attenuation coefficient: ≤0.25 dB/km @ 1550 nm
From Corning Corp.

© A.Lobo (2004) 13
Large Area NZ-DSF
Fibre nonlinearities are inversely proportional to the core effective area, and so, increasing the effective
area will ultimately reduce nonlinearities through the reduction of light density propagating along the
core. The large effective area NZ-DSF is based on a triangular core and raised index ring.
Corning Corp. developed the LEAF NZ-DSF with a effective area of 72 µm2 as compared with to 55
µm2 standard NZ-DSF.
Shifts the zero dispersion wavelength to the
1550 nm range
Enhances the cross-sectional effective
area while simultaneously achieving a
relative low bending loss
Refractive Index Profile

© A.Lobo (2004) 14
Large Area NZ-DSF
2K input
eff
n LP
Nonlinearity
A
→
From Corning Corp.

© A.Lobo (2004) 15
Increasing the optical power on a SMF (G.652)
10,6 10,7 10,8 10,9 11,0 11,1 11,2
-45
-40
-35
-30
-25
-20
-15
-10
νTLS@25.7ºC
νBrillouin
OpticalPower(dBm)
Frequency Shift (GHz)
VOA=6dB
9,0 9,5 10,0 10,5 11,0 11,5 12,0
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
VOA=5dB
νTLS@25.5ºC
νBrillouin
OpticalPower(dBm)
Frequency Shift (GHz)
With permission of M. Melo and M.O. Berendt, Multiwave Networks Lda.. (Portugal)

© A.Lobo (2004) 16
3.7 Dispersion Compensating Fibre (DCF)
❧ DCF is a singlemode fibre with very high waveguide dispersion. The overall dispersion of this fibre
is opposite in sign and much larger in magnitude than that of standard fibre, so they can be used to
cancel out or compensate the dispersion in SMF.
❧ Dispersion coefficient (typ.): -80 to -150 ps/(nm·km) @ 1550 nm
❧ Dispersion slope: -0.40 ps/(nm2·km) @ 1550 nm
❧ PMD: 0.25~0.5 ps/√km
❧ Attenuation: ≤0.55 dB/km @ 1550 nm
a
b
From Fujikura Technical Review, 2001

© A.Lobo (2004) 17
3.8 E-Band Fibre (E-SMF)
E-SMF is a singlemode fibre specially designed for the Extend-Band (or E-Band) transmission window,
which is located at the wavelength range region from 1360 nm to 1460 nm, but with possibility to operate
also in the other bands, such as, C-band.
Main Characteristics:
❧ Attenuation (máx): 0.35 dB/km @ 1310 nm , 0.31 dB/km @ 1385 nm, 0.22 dB/km @ 1550 nm
❧ Zero dispersion wavelength: 1300 nm ≤ λ0D ≤ 1322 nm
❧ Zero dispersion slope (So): 0.092 ps/nm2#km
❧ PMD (typ.): ≤ 0.08 ps/√km
Commercial Examples:
❧ Corning SMF-28e® fibre → Corning Corp.
❧ AllWave® fibre → Lucent Technologies Inc.
❧ LightScope ZWP™ fibre → CommScope Inc.
❧ TeraSPEED ™ fibre → Avaya Inc.
❧ BBG ® -SMF-WF fibre → Hitachi Cable Ltd.
DEFINITION OF SIGNAL WAVELENGTH BAND (ITU-T)
Band Name
Wavelength (nm)
O-band (Original) 1260 - 1360
E-band (Extended) 1360 - 1460
S-band (Short Wavelength) 1460 - 1530
C-band (Conventinal) 1530 - 1565
L-band (Long Wavelength) 1565 - 1625
U-band (Ultralong Wavelength) 1625 - 1675

© A.Lobo (2004) 18
3.8 E-Band Fibre (E-SMF)
SMF
E-SMF (5th window)
“water peak” around 1385 nm

© A.Lobo (2004) 19
3.9 Polarization Maintaining Fibre (PMF)
PMF is a singlemode fibre specially designed to maintain the state of polarization (SOP) of guided
polarized light. In the most common optical fiber telecommunications applications, PM fiber is used to
guide light in a linearly polarized state from one place to another.
Commercial Types:
❧ PANDA fibre
❧ BowTie fibre
❧ Elliptical shape (core or cladding ) fibre
❧ D-shaped fibre
❧ Side-Hole fibre
❧ Side-Pit fibre
❧ Spun fibre
2
( )F SL n n
π
φ
λ
Δ = Δ −
Birefringence (typ.): 2×10-4 to 7 ×10-4

© A.Lobo (2004) 20
3.9 Polarization Maintaining Fibres (PMF) Fast axis
Slow axis
Bow Tie PANDA Side-Hole
Elliptical Stressed
Cladding
D-shaped
Elliptical Core
Elliptical Core

© A.Lobo (2004) 21
3.10 Rare-Earth Doped Fibres.
For more details see: www.webelements.com
❑ Erbium (Er)
❑ Ytterbium (Yb)
❑ Neodymium (Nd)
❑ Praseodymium (Pr)
❑ Thulium (Tm)
❑ Holmium (Ho)
Lanthanides are best characterized by observation that they
possess incomplete inner 4f levels and, to a large extent, they form
ions that exist solely in the 3+ state. This state is formed by the
removal of two outer 6s electrons and one inner 4f electron.
What are Rare Earth Ions?

© A.Lobo (2004) 22
Optical properties of the Rare Earth Ions
Fluorescence Up-Conversion

© A.Lobo (2004) 23
Erbium Doped Fibre (ErDF)
ABSORPTION BANDS:
❧ 520 nm
❧ 650 nm
❧ 800 nm
❧ 980 nm (free from excited-state absorption ESA)
❧1480 nm

© A.Lobo (2004) 24
Neodymium Doped Fibre (NdDF)
ABSORPTION BANDS:
❧ 520 nm
❧ 590 nm
❧ 820 nm

© A.Lobo (2004) 25
Thulium Doped Fibre (TmDF)
ABSORPTION BANDS:
❧ 520 nm
❧ 590 nm
❧ 820 nm

© A.Lobo (2004) 26
Holmium Doped Fibre (HoDF)
Pumped at 640 nm

© A.Lobo (2004) 27
Praseodymium Doped Fibre (PrDF)

© A.Lobo (2004) 28
Ytterbium Doped Fibre (YbDF)
ABSORPTION BANDS:
❧ 520 nm
❧ 590 nm
❧ 840 nm
This transition is significantly broadened

© A.Lobo (2004) 29
3.11 Long Wavelength Fibres (Fluoride and Chalcogenide).
❧ Fluorozirconate fibers transmit light between 0.4 and 5 µm.
❧ Silver Halide compounds fiber (AgBrCl) transmit light between 3 and 16 µm.
❧ Synthetic crystalline sapphire (Al2O3) can transmit light between 0.5 and 3.1 µm

© A.Lobo (2004) 30
3.12 Plastic Fibre (POF).
Fluorinated POF characteristics:
❧ Attenuation: <150 dB/km @ 650 nm 1.5 mm diameter
❧ NA: 0.4
❧ Index profile: Step-index and also GI-index
PMMA – Polymethyl Metacrylate

© A.Lobo (2004) 31
3.13 Hollow Fibre (HOF).
Hollow optical fiber with an inner core ring
Hollow silica tube filled with HCBD (hexachlorobutadiene)
D. N. Payne and W. A. Gambling, Electronics Letters, vol. 8, p374, 1972
K. Oh, FOR Lab., Dept. of Information and Communications
Kwangju Institute of Science and Technology (K-JIST)

© A.Lobo (2004) 32
3.13 Hollow Fibre (HOF).
K. Oh, FOR Lab., Dept. of Information and Communications Kwangju Institute of Science and Technology (K-JIST)
S. Choi, K. Oh, W. Shin, U. C. Ryu, Electron. Lett., vol. 37, no.13 , pp.823-825, Jun. 2001.
New type of Mode Coupler based on tapered HOF

© A.Lobo (2004) 33
3.14 Photonic Crystal Fibre (PCF).
Stack-and-Draw Fabrication Process

© A.Lobo (2004) 34
How can one make this….?
Differential Pressure !

© A.Lobo (2004) 35
Extrusion (Soft glass)

© A.Lobo (2004) 38
Prof. P.St Russell, Head of Optoelectronics Group, Dept. Physics, University of Bath (UK) & CSO of Blaze Photonics Ltd. (UK)
Endlessly SMF PM fibre
Multicore PCF
Highly Nonlinearity PCF
Hollow Core PCF Hollow Core Visible PCF

© A.Lobo (2004) 39
Random Hole Fibre (RHOF)
Virginia Polytechnic Institute
and State University , USA
High NA PCF
Crystal Fiber A/S (Denmark)
High NA Yb-Doped PCF
Crystal Fiber A/S (Denmark)

Passive Fibre Optic Devices
António Lobo

© A.Lobo (2004) 2
4.1 Directional Couplers: Splitters and Combiners.
4.2 WDM Couplers.
4.3 Tap Couplers
4.4 Biconical Fibre Filter.
4.5 Fibre Bragg Gratings.
4.6 Long Period Fibre Gratings.
4.7 Gain Equalizing Filters.
4.8 Mach-Zehnder and Michelson Filters (Interleavers).
4.9 Fibre Ring Filters.
4.10 Fibre Fabry-Perot Filters.
4.11 Fibre Polarizer.
4.12 Fibre Depolarizer.
4.13 Fibre Polarization Controller.
4.14 Fibre Optic Attenuator.

© A.Lobo (2004) 3
4.1 Directional Couplers: Splitters and Combiners
Directional Coupler. Theory of Coupled Modes
Between the waveguides 1 and 2 optical power is coupled continuously back and forth (in
both directions, 1→2 and 2→1). This results in a periodic spatial variation of the powers
Pi(z) in the waveguides (i=1,2) with a periodicity Lc (coupling length)

© A.Lobo (2004) 5
Directional Coupler: Coupled Mode Theory (*)
Coupled Wave Equations:
and
2 1
2 1
( )1
12 2
( )2
21 1
j z
j z
dA
j A e
dz
dA
j Ae
dz
β β
β β
κ
κ
− −
+ −
= −
= − 12 21κ κ κ= =
(*) for more detailed analysis see K. Okamoto, Fundamentals of Optical Waveguides, Academic Press (2001) – Chap.4
mode coupling coeff.
(coupler with geometric symmetry)
Solutions assumed in the form:
(for codirectional couplers: β1>0, β2>0)
1 1 1
2 2 2
( )
( )
jqz jqz j z
jqz jqz j z
A z a e a e e
A z a e a e e
δ
δ
+ − − −
+ − −
⎡ ⎤= +⎣ ⎦
⎡ ⎤= +⎣ ⎦
With initial conditions:
1 1 1
2 2 2
(0)
(0)
a a A
a a A
+ −
+ −
+ =
+ =2 1( )
2
β β
δ
−
=
and
2 2
q κ δ= +

© A.Lobo (2004) 6
Directional Coupler: Coupled Mode Theory(*)
Substituting the assumed solutions into the coupled wave equations and applying the initial
conditions, we obtain:
(*) for more detailed analysis see K. Okamoto, Fundamentals of Optical Waveguides, Academic Press (2001) – Chap.4
1 1 2
2 1 2
( ) cos( ) sin( ) (0) sin( ) (0)
( ) sin( ) (0) cos( ) sin( ) (0)
j z
j z
A z qz j qz A j qz A e
q q
A z j qz A qz j qz A e
q q
δ
δ
δ κ
κ δ
−⎡ ⎤⎛ ⎞
= + −⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦
⎡ ⎤⎛ ⎞
= − + −⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦

© A.Lobo (2004) 7
Directional Coupler: Coupled Mode Theory
For the most practical case, in which light is coupled into waveguide 1 only at z=0, we have
the conditions of A1(0)=Ao and A2(0)=0. Then the optical power flow along the z-direction is
given by:
2
1 2
1 2
2
2 2
2 2
( )
( ) 1 sin ( )
( )
( ) sin ( )
o
o
A z
P z F qz
A
A z
P z F qz
A
= = −
= =
2 1( )
2
β β
δ
−
=
2 2
q κ δ= +
( )
2
2
1
1
F
q
κ
δ
κ
⎛ ⎞
= =⎜ ⎟
⎝ ⎠ +
Power-coupling efficiency

© A.Lobo (2004) 8
Directional Coupler
0 1 2 3 4 5 6
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6
0
0.2
0.4
0.6
0.8
1
0 1Fδ = ⇒ = 0.5Fδ κ= ⇒ =
P1
P1
P2
P2
Normalized distance (qz) Normalized distance (qz)
For 50% coupling it is a necessary condition that δ ≤ κ
2 22 2
cL
q
π π
κ δ
= =
+
Coupling Length:
Lc
Lc/2
NormalizedPowers

© A.Lobo (2004) 9
Directional Fibre Coupler
Side-Polished coupler
(adjustable CR)
Fused coupler
(fixed CR, possibly λ-dependent)

© A.Lobo (2004) 12
1
3
(dB) 10log
P
CR
P
⎛ ⎞
= ⎜ ⎟
⎝ ⎠
Coupling Ratio (CR)
By varying the parameters δ and κ it is possible to adjust CR in a very wide range.

© A.Lobo (2004) 13
4.2 WDM Couplers.
Normal -3dB coupler at 1550 nm has interaction coupling length is typically 0.9 mm,
whereas for a WDM coupler that length can be from 100 to 500 mm.
Lc is depends on the optical wavelength λ

© A.Lobo (2004) 15
4.4 Biconical Fibre Filter.
n1
n2
n3
Refraction Index Profile
Core
Int. Cladding
Ext. Cladding
Biconical Filter
1487 1507 1527 1547 1567 1587
-68
-66
-64
-62
-60
TransmittedPower(dBm)
Wavelength (nm)
Depressed-cladding fibre
1 2
1 sin (
2
fT m
π
λ λ
⎡ ⎤⎛ ⎞
≈ + −⎜ ⎟⎢ ⎥Λ⎝ ⎠⎣ ⎦
A.B. Lobo Ribeiro, Ph.D. thesis, FCUP (1996)
• A.C. Boucouvalas, J. Lightwave Technol. 3, 1151-1158 (1985).
• A.C. Boucouvalas and G Georgiou, IEE Proc. 134, Pt.J, 191-195 (1987)

© A.Lobo (2004) 17
4
2 effn
λ
Λ =
λ1
λ2
λ3
λ4 λ1
λ2
λ3
λ4
2
( ) coseff zn z n n z
π
δ
⎛ ⎞
= + ⎜ ⎟
Λ⎝ ⎠
4 2 effnλ = Λ
2
tanh z
peak
B
L n
R
π δ
λ
⎛ ⎞⋅
= ⎜ ⎟
⎝ ⎠
2
2
2
2
B z
B
eff B
L n
n L
λ π δ
λ π
λ
⎛ ⎞
Δ = + ⎜ ⎟
⎝ ⎠
L
Peak Reflectivity
Stop-Band Width (FWHM)
Rpeak
ΔλB

© A.Lobo (2004) 18
Fibre Bragg Grating: Coupled Mode Theory(*)
(*) for more detailed analysis see R. Kashyap, Fiber Bragg Gratings, Academic Press (1999) – Chap.4
2 2
2
2 2 2
sinh ( )
( )
cosh ( )
L
R
L
α
λ ρ
α β
Ω
= =
Ω − Δ
2 2
2
2
z
eff
eff
n
n
n
π δ
α β
π π
β
λ
⋅
Ω =
Λ
= Ω − Δ
⎛ ⎞ ⎛ ⎞
Δ = −⎜ ⎟ ⎜ ⎟
Λ⎝ ⎠⎝ ⎠
2
(arg )
( )
2
d
c d
λ ρ
τ λ
π λ
= −
L = 20 mm
sinh( )
sinh( ) cosh( )
L
L i L
α
ρ
β α α α
−Ω
=
Δ −
Group delay:
Reflectivity:
ΩL = 4

© A.Lobo (2004) 19
Chirped Fibre Bragg Grating (CFBG)
2chirp eff chirpnλΔ = ΔΛ
0( ) 2
( ) G
chirp g
L
v
λ λ
τ λ
λ
−
≈ ⋅
Δ
λ1 λ2 λ3
Pulse compression
after CFBG
λ1
λ2
λ3
In most fibre at 1550 nm, short wavelength light tends to travel faster
Λ
L

© A.Lobo (2004) 20
Chirped Fibre Bragg Grating (CFBG)
Uniform Grating
Chirped Grating
Chirped Grating with Apodization
( )( ) cos ( )eff zn z n n z zδ κ= + ⋅
δnz
z

© A.Lobo (2004) 21
4.6 Long-Period Fibre Grating.
LPG promotes coupling between the propagating core mode and co-propagating cladding modes
ΛLPG
, ,( )LPG i eff clad i LPGn nλ = − Λ
The long-period grating (LPG) has a period
typically in the range 100 µm to 1 mm
L = 10mm, ΛLPG = 450 µm (SMF-28 fibre)

© A.Lobo (2004) 24
4.7 Gain Equalizing Filter.
© Teraxion Inc.
Gain Flattening Filter (GFF) with Concatenated Chirped-FBG (*)
(*) M. Rochette, M. Guy, S. LaRochelle, J. Lauzon and F. Trépanier, Photonics Technology Letters 11, 536-538 (1999).

© A.Lobo (2004) 27
0 1 2 3 4
0
1
0 4π3π2ππ
λ/2
Pontos de
quadratura
Iout
/Iin
φ (radianos)
high
sensitivity
points
Mach-Zehnder Michelson
1 2
1 2
1 2
2
( ) 1 ( ) cos
( )
out
I I
I I I
I I
γ τ φ
⎡ ⎤
= + ⋅ ± ⋅⎢ ⎥
+⎢ ⎥⎣ ⎦
2 n
L
π
φ
λ
= Δ
1 2L L LΔ = − 1 22( )L L LΔ = −

© A.Lobo (2004) 28
Optical interleavers
• Channel spacing management via optical interferometry (e.g., Michelson,
Mach-Zehnder) e.g. from 25 GHz to 50 GHz, or vice versa.
• Interleavers are needed in density populated WDM systems with 2.5 Gbit/s, 10
Gbit/s and even 40 Gbit/s transmitter/receivers

© A.Lobo (2004) 29
Cascaded Mach-Zehnders: 16-Channel Frequency Selector
0 1 2 3 4 5 6
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6
0
0.2
0.4
0.6
0.8
1
MZ1
ΔL 2ΔL 4ΔL 8ΔL
MZ2
MZ3
MZ4

© A.Lobo (2004) 30
Cascaded Mach-Zehnders: 1×4 Demultiplexer
0 1 2 3 4 5 6
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6
0
0.2
0.4
0.6
0.8
1
MZ1
ΔL
MZ2
MZ3
ΔL/2
ΔL/2
λ1, λ2, λ3, λ4
λ1
λ3
λ2
λ4

© A.Lobo (2004) 31
4.9 Fibre Ring Filter.
Ring (direct coupled)
0 180 360 540 720
0
1
4π3ππ 2π0
k = 0.5
k = 0.2
k = 0.1
Iout
/Iin
φ (radianos)
2
2 2
(1 ) 1
4(1 )sin ( / 2)
out in
k
I I
k k
δ
φ
⎡ ⎤
= − −⎢ ⎥− −⎣ ⎦
2
(1 ) L
k e α
δ −
= −
Ring: Direct coupled
Phase (!)
NormalizedTransmittedPower
k - power coupling coefficient of the coupler
(1-δ) – power coupling loss of the coupler
α – attenuation coeff. of the fibre ring
L – ring length
Iin Iout

© A.Lobo (2004) 32
0 180 360 540 720
0
1
(a) Acoplamento Cruzado
k = 0.5
k = 0.8
k = 0.9
I
out
/I
in
φ (radianos)
4π3π2ππ0
2
2 2
(1 )
(1 ) 1
(1 ) 4 sin ( / 2 / 4)
out in
k
I I
k k
δ
φ π
⎡ ⎤−
= − −⎢ ⎥− − −⎣ ⎦
2
(1 ) L
k e α
δ −
= −
Phase (!)
Ring (cross-coupled)
Ring: Cross-coupled
•  An advantage of this configuration is that
the entire ring can be made of one fibre, i.e.,
without cutting and splicing the fibre
Iin
Iout

© A.Lobo (2004) 33
Fibre Ring Filter: Sagnac Interferometer (*)
(*) X. Fang, H.Ji, C.T. Allen, K. Demarest and L. Pelz, IEEE Photonics Technology Letters 9, 458-460 (1997).
PMF
SMF
3 dB
0 10 20 30 40 50 60
0
0.2
0.4
0.6
0.8
1
2
( ) sin
2
out
in
I
I
φ
γ τ
⎛ ⎞
= ⋅ ⎜ ⎟
⎝ ⎠
Phase (!)
Iin
Iout

© A.Lobo (2004) 34
4.10 Fibre Fabry-Perot Filter.
L
M
M
0 180 360 540 720
0
1
4π3π2ππ0
R = 80%
R = 50%
R = 25%
R = 4%
Iout
/Iin
φ (radianos)
Phase (!)
2
2
2
1
1
2
1 sin
1 2
A
R
T
R
R
φ
⎛ ⎞
−⎜ ⎟
−⎝ ⎠=
⎛ ⎞ ⎛ ⎞
+⎜ ⎟ ⎜ ⎟
− ⎝ ⎠⎝ ⎠
T
2
2
4 1 sinnL
n
π θ
φ
λ
−
=
For normal incidence: θ = 0º
1
FSR R
Finesse
FWHM R
π
= =
−
A – Absorption loss of the mirror
R – Reflectivity of the mirror
For more detailed analysis see E. Hecht, Optics, 4ed. Chap.9.
FSR

© A.Lobo (2004) 36
Surface Plasmon Polarizer
A Surface plasmon is an electromagnetic wave that
propagates along the interface of two materials, one of
which has a negative dielectric constant. The plasmon
is a transverse magnetic effect and exhibits the
property of strongly attenuating one polarization
component from an optical beam while the other
polarization is unaffected. Thus only the TM
polarization is coupled to the plasmon and the TE
polarization is unaffected.
TMo Plasmonβ β=
Phase-Matching Condition

© A.Lobo (2004) 37
Surface Plasmon Polarizer
Thin-film metal layer
Fibre Core
Silica Block
Optical Fibre
Fibre Cladding
~50 nm
P. Perumalsamy, In-Line Fiber Polarizer, M.Sc. Thesis, Virginia Polythecnic Institute, Virgina, USA (1998).

© A.Lobo (2004) 38
4.12 Fibre Depolarizer.
#1 #2 #n
Depolarization occurs by averaging over many
different polarization states of the recirculating
beams.
Advantages are:
• All-fibre SMF device
• Low cost structure
• Insensitive to input polarization state
• Infinitesimal DOP is possible with increasing
numbers of cascaded fibre rings
(*) B. Paisheng and C. Lin, Electronics Letters 34, 1777-1778 (1998).
Passive Fibre Depolarizer
3 dB

© A.Lobo (2004) 39
Fibre Coil Polarization Controller (*)
(*) H.C. Lefevre, Electronics Letters 16, 778-780 (1980).
A.B. Lobo Ribeiro, M.Sc. Thesis, Univ. Kent, UK (1992).
2
0.836
( , )
r N m
R m N
λ
⋅ ⋅ ⋅
=
R – radius of the coil
r – nominal radius of the fibre (2r =125 µm, standard SMF)
N – number of turns of fibre on the coil
m –fractional-wave order (m=2, 4 or 8) to get a λ/m wave plate
λ – wavelength of the light
Free-space optics approach
Fibre optic approach

© A.Lobo (2004) 40
Fibre Squeezer Polarization Controller(*): The PolaRITE TM
(*) www.generalphotonics.com
Babinet Compensator: see E. Hecht, Optics, 4ed. Chap.8.
Free-space optics approach (Babinet-Soleil compensator)
Fibre optic approach (PolaRITE TM)
1 2
2
( ) o ed d n n
π
ϕ
λ
Δ = − ⋅ −

© A.Lobo (2004) 41
PM Fibre Twist Polarization Controller(*)
(*) www.fiberpro.com
PMF
SMF SMF
Free-space optics approach
(with λ/4-wave plates)
• No Fiber Squeezing
• No Damage on Fiber Jacket
TWIST

© A.Lobo (2004) 42
Pin Pout
( ) 10 log 10in
out
P
IL dB OD
P
⎛ ⎞
= ⋅ = ×⎜ ⎟
⎝ ⎠
OD : Optical Density
For example: An attenuator with optical density of 2, has a 20 dB ILoss
Standard SMF Standard SMF
Air Gap
λ = 1300 nm
Fresnel reflection 3.5% → return loss of 14.5 dB
( ) 10 log in
back
P
RL dB
P
⎛ ⎞
= ⋅ ⎜ ⎟
⎝ ⎠
Pback
More accurate analysis based on electric fields:
INTERFERENCE ANALYSIS

© A.Lobo (2004) 43
Absorbing SMF
(GeAl core+Rare Earth ions)
S. Chia, AMP Journal of Technology 5, 19-23 (1996) from AMP Inc.
Response of insertion loss to temperature cycling between -40ºC and
85ºC of five individual samples of a 5 dB OA measured at 1550 nm.
± 0.30 dB
5 dB attenuator.

Active Fibre Optic Devices
António Lobo

© A.Lobo (2004) 2
5.1 Thermally Tunable Fibre Filter and Modulator.
5.2 Twin-Core Fibre Switch.
5.3 Acousto-Optic Tunable Fibre Filter.
5.4 Piezoelectric Tunable Fibre Filter.
5.5 Acousto-Optic Fibre Modulator.
5.6 Piezoelectric Fibre Modulator.
5.7 Electric Poling Fibre Modulator.
5.8 Fibre Amplifier.

© A.Lobo (2004) 3
Temperature-induced optical phase shifts in fibres (*)
(*) N. Lagakos, J.A. Bucaro and J. Jarzynski, Applied Optics 20, 2305-2308 (1981).
1
z
T
L n n n
T
L n n T nρ
φ
ε
φ
Δ Δ Δ ∂ Δ⎛ ⎞ ⎛ ⎞
= + = + ⋅Δ +⎜ ⎟ ⎜ ⎟
∂⎝ ⎠ ⎝ ⎠
2 nLπ
φ
λ
=
εz , εr – Axial (z) and radial (r) strains in the core.
ρ – core density.
n – effective refractive index
p11 , p12 – are the Pockels coefficients of the core.
[ ]
2
11 12 12
1 1
( )
2
z r z
n n
p p p
T n T Tρ
φ
ε ε ε
φ
⎧ ⎫Δ ∂⎛ ⎞
= + − + +⎨ ⎬⎜ ⎟
Δ ∂ Δ⎝ ⎠ ⎩ ⎭
5 -1
0.7 10 C
bare
T
φ
φ
−Δ
= ×
Δ
o
5 -1
1.7 10 C
jacket
T
φ
φ
−Δ
= ×
Δ
o

© A.Lobo (2004) 4
Optical phase sensitivity of the fibre due to a physical parameter
2
X
n L n
S
L X L X X
φ π
λ
Δ ∂ ∂⎡ ⎤
= = ⋅ +⎢ ⎥Δ ∂ ∂⎣ ⎦
Physical Parameter (X) Sensitivity (SX)
Axial Deformation (ε) 107
rad m-1
strain-1
Temperature (T) 100 rad m-1
°C-1
Hydrostatic Pressure (P) 5*10-5
rad m-1
Pa-1
(For a wavelength of 850 nm)

© A.Lobo (2004) 5
Temperature-induced optical phase shifts in metal coated fibres (*)
(*) C.T. Shyu and L. Wang, J. Lightwave Technology 12, 2040-2048 (1994).
Metallic film
2
( )o nT T I R hA
t t V cρ
∂ − − Θ∂Θ
= =
∂ ∂ ( )
2
/
( ) 1 tnI R
t e
hA
τ−
Θ = −
( )
2
/
( ) 1 tnI R
t e
hA
τβ
φ −
Δ = −R – Electrical resistance of the metallic film.
ρ , c – density and specific heat of the fibre (respectively).
V , A – volume and surface area of the metallic film.
k – thermal conductivity of the film
h – heat transfer coefficient
In
cV
hA
ρ
τ =
T
φ
β
Δ
=
Δ
→ Fibre glass material

© A.Lobo (2004) 6
Temperature-induced optical phase shifts in metal coated fibres (*)
(*) C.T. Shyu and L. Wang, J. Lightwave Technology 12, 2040-2048 (1994).
gold coating
In
Coating length: 3 cm
Resistance: 60.6 Ω

© A.Lobo (2004) 7
Thermally fibre modulator (*)
(*) - C.T. Shyu and L. Wang, J. Lightwave Technology 12, 2040-2048 (1994).
- B.J. White et. al., J. Lightwave Technology 5, 1169-1174 (1987).
Metallic film
In cos (ωt)
Test current
Detected optical signal
Predicted signal
100 Hz

© A.Lobo (2004) 8
Thermally tunable fibre Bragg grating filter (*)
1 1B
X
B
n
S
X X X n X
λ
λ
Δ ∂Δ ∂
= = ⋅ + ⋅
Δ ∂ ∂
Physical Parameter (X) Sensitivity (SX)
Axial Deformation (ε) 0.78×10-6
µs t r a i n
-1
Temperature (T) 8×10-6
°C-1
Hydrostatic Pressure (P) 2.7×10-6
MPa-1
2B nλ = Λ
B
X

© A.Lobo (2004) 9
0.0 0.1 0.2 0.3
150
200
250
300
350
400
Time (min.)
Is
(mA)
Vaplicado
0.8
0.9
1.0
1.1
1.2
1.3
1.4
(a)
Vout
/Vref
(*) P.M. Cavaleiro, F.M. Araújo and A.B.Lobo Ribeiro, Electronics Letters 34, 1133-1135 (1998)
silver film
In
SMF fibre with FBG
0.0 0.5 1.0 1.5 2.0 2.5 3.0
835.2
835.4
835.6
835.8
836.0
836.2
836.4
836.6
BraggWavelength(nm)
Electrical current - squared (A
2
)
Resistance: 1.2 Ω

© A.Lobo (2004) 10
(*) L.Lin et. al., IEEE Photonics Technology Letters 15, 545-547 (2003)
Ti-Pt-Ni metal alloyed
coating (2 µm)
In
SMF fibre with FBG
Resistance: 7.5 Ω

© A.Lobo (2004) 11
11.2 nm
Wavelength (nm)
OpticalPower(dBm)
Thermally tunable fibre Bragg grating (*): Laser Application
(*) J.J. Pan et.al., Lightwaves 2020 Inc., US Pat.6018534
0.009 nm/ºCB
T
λΔ
Δ
;
0.16 nm/ºCB
DFBT
λΔ
Δ
;

© A.Lobo (2004) 12
5.2 Twin Core Fibre Switch.
Twin Core Fibre (*)
(*) - P.J. Severin, in Proc. SPIE vol.1314 – Fiber Optics 90, 348-364 (1990).
- R. Romaniuk, J. Dorosz, in Proc. SPIE vol.4887 (2001)
2 21
2 22
( )
1 sin
( )
sin
o
o
P z kz
F
P F
P z kz
F
P F
⎛ ⎞
= − ⎜ ⎟
⎝ ⎠
⎛ ⎞
= ⎜ ⎟
⎝ ⎠
2
2 1
1
1
2
F
k
β β
=
−⎛ ⎞
+⎜ ⎟
⎝ ⎠
Power-coupling efficiency:
Cladding
(125 µm)
4 µm
Cores
(10 µm)

© A.Lobo (2004) 13
Twin Core WDM coupler/filter (*)
1530 1535 1540 1545 1550 1555 1560
0
0.2
0.4
0.6
0.8
1
Wavelength [nm]
NormalizedPowers
P1
P2
(*) twin core fibre sample provided by Prof. P.J. Severin (Philips Eindhoven)
L = 20 cm
P2
P1 Po
Cores: 9 µm
Cores separation: 2 to 5 µm
Cladding: 100 µm

© A.Lobo (2004) 14
1530 1535 1540 1545 1550 1555 1560
0
0.2
0.4
0.6
0.8
1
(*) twin core fibre sample provided by
Prof. P.J. Severin (Philips Eindhoven)
P1 Po
Piezoelectric Transducer
V
Wavelength [nm]
NormalizedPower(P1/Po)
Length variation of 100 µm on 20 cm

© A.Lobo (2004) 16
FREQUENCY
SHIFTER
Optical Frequency
ωo
Optical Frequency
ωo ± ωac
Acoustic Frequency
ωac
( Acoustic frequency (ωa) can be generated
electrically or optically)
sin
2 4
o o ac
B B
ac acV
λ λ ω
π
Θ = ⇒ Θ ≅
Λ
Diffracted beam angle at:
For more information see: www.brimrose.com

© A.Lobo (2004) 17
(*) W. P. Risk, R. Youngquist, R.C. Kino, H.J. Shaw, Optics Letters 9, 309 (1984)
Surface Acoustic Wave (SAW) device (*)
(Used to produce surface acoustic waves)
cos SAW a
a B B
V
f L L
Λ
Θ = =Phase-matching condition:
Velocity of the SAW
Fibre beat length
SAW wavelength

© A.Lobo (2004) 18
Flexure-wave fibre device (*)
(*) B.Y. Kim, J.N. Blake, H.E. Engan, H.J. Shaw, Optics Letters 11, 389-391 (1986).
York HB600 fibre (HiBi fibre), with beat
length of 1.5 mm and an acoustic frequency
of ~0.2 MHz, producing a frequency shift of
~790 kHz

© A.Lobo (2004) 19
(*) T. E. Dimmick, G. Kakarantzas, T. A. Birks, and P. St. J.Russell, in Proc. OFC‘2000, FB4 pp.25-27, 2000.
11 21
2
ac
HE HE
π
β β
Λ =
−
Resonance condition:

© A.Lobo (2004) 20
(*) T. E. Dimmick, G. Kakarantzas, T. A. Birks, and P. St. J.Russell, in Proc. OFC‘2000, FB4 pp.25-27, 2000.
Filter bandwidth 0.8 ac
L
λ
λ
ΛΔ
=
2.9 nm bandwidth
(4 cm)
< 10 cm

© A.Lobo (2004) 21
1.53 1.54 1.55 1.56 1.57
0.2
0.4
0.6
0.8
1
L
M
M
PZTOptical fibre
A = 0
R = 95 %
L = 20 µm
ΔL = +1 µm2
2
2
1
1
2
1 sin
1 2
A
R
T
R
R
φ
⎛ ⎞
−⎜ ⎟
−⎝ ⎠=
⎛ ⎞ ⎛ ⎞
+ ⎜ ⎟ ⎜ ⎟
− ⎝ ⎠⎝ ⎠
FSR ≈ 60 nm
FWHM ≈ 2 nm
Finesse ≈ 30
Wavelength (µm)
Transmittance
Tunable Fibre FP Filter

© A.Lobo (2004) 23
Strain-induced optical phase shifts in fibres (*)
(*) A. Bertholds, R. Dändliker, J. Lightwave Technology 5, 895-900 (1987).
( )
2
T const
n L nL
π
φ
λ=
Δ = Δ + Δ
2 nLπ
φ
λ
=
εz , εr – Axial (z) and radial (r) strains in the core (εr= ν εz)
ν – Poisson ratio (=0.17 for silica).
n – effective refractive index (= 1.456 at λ=1550 nm)
p11 , p12 –Pockels coefficients of the core (p11=0.121 , p12=0.27 for silica)
[ ]
2
11 12 12
2
( )
2
z r z
n
nL p p p
π
φ ε ε ε
λ
⎧ ⎫
Δ − + +⎨ ⎬
⎩ ⎭
;
6 -1 -1
4 10 rad m strain
L
φ
ε
Δ
≈ × ⋅ ⋅
Δ
for λ = 1550 nm

© A.Lobo (2004) 24
PZT PZT
FBG
(*) A. Iocco, H.G. Limberger, R.P. Salathé, Electronics Letters 33, 2147-2148 (1997).
Tunable FBG Filter (*)

© A.Lobo (2004) 26
5.5 Acousto-optic Fibre Modulator.
All-fibre AO Frequency Shifter (*)
(*) D.O. Culverhouse et.al., IEEE Photonics Tech. Letters 8, 1636-1637 (1996).

© A.Lobo (2004) 27
5.5 Acousto-optic Fibre Modulator.
ZnO-Coated Optical Fibre AO Phase Modulator(*)
(*) A. Roeksabutr and P.L. Chu, J. Lightwave Technology 16, 1203-1211 (1998).
3
11 12( ) rn l p p
π
φ ε
λ
Δ ≈ +
Optical phase shift:
Max. phase shift (exp):~700 mrads @ 670 MHz

© A.Lobo (2004) 28
r
R
32
11 12
12
(1 ) ( )4
2 2
eff
eff
n p p rNC
n p
V R
νφ π
λ
⎧ ⎫+ +Δ ⎪ ⎪⎡ ⎤
= + −⎨ ⎬⎢ ⎥⎣ ⎦⎪ ⎪⎩ ⎭
(rad/volt)
for more details see: G. Martini, Optical & Quantum Electronics 19, 179 (1987).
N – number of fibre turns wrapped on the PZT
ν – Poisson ratio of silica (0.17)
p11 , p12 –Pockels coefficients of the fibre core
C – radial displacement of PZT per unit of voltage (p.ex. C≈0.61 nm/V for a PZT-5H)

© A.Lobo (2004) 30
(*) M.N. Zervas and I.P. Giles, Optics Letters 13, 404-406 (1988).
Fibre-Loop Phase Modulator(*)
2
phase
resonance
loop
V
f
L
=
Resonance Frequency
of the loop
As the advantage to remove the phase shift
dependence on the PZT characteristics

© A.Lobo (2004) 31
5.7 Electric Poling Fibre Modulator.
The fibre is 127 mm diameter with two holes of 45 mm
diameter. The optical fibres are thermally poled at around
250°C with an electric field in the order of 4kV. The
electrodes are inserted to lengths in excess of 20 cm.
(150 VAC at 4.5 kHz) EO coefficient can be calculated. EO
coefficients of ~0.2 pm/V have been measured in 1.55 µm
silica single mode fibres (for a LiNbO3: 7~30 pm/V)
See: Applied optics Group, Univ. Kent, Canterbury (U.K.) - www.kent.ac.uk/physical-sciences/aog
4.5 kHz
2
2
2
NL Kn L E
π
φ
λ
Δ =
n2K ≈ 3.2×10-20 m2/W
2
2eff o Kn n n E= + 2 11
2 for 10 V/mK on E n E≈ ≈

© A.Lobo (2004) 33
S-Ring Fibre Laser (*)
(*) O.G. Okhotnikov, A.B.Lobo Ribeiro, J.A.R. Salcedo, Applied Physics Letters 63, 2726-2728 (1993).

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