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Chapter 2
Purpose of this chapter: understanding of the physical structure and
waveguiding properties.
2.1 The Nature of Light
2.2 Basic Optical Laws and Definitions
2.3 Optical Fiber Modes and Configurations
2.4 Mode Theory for Circular Waveguides
2.5 Single‐Mode Fibers 2.6 Graded‐Index Fiber Structure
2.8 Photonic Crystal Fibers**
1
2.1 The nature of light
2.1.1 Linear Polarization
2.1.2 Elliptical and Circular Polarization
¾ Light is an electromagnetic wave; based on Maxwell’
Maxwell’ equations, light is
transverse wave, i.e., both E and H are perpendicular to the direction
direction of
propagation
¾ What is polarization of the wave? The polarization of the wave is the
description of the behavior of the vector E in the plane x, y, perpendicular
to the direction of propagation z.
¾ Linearly polarized: If the vector E does remain in a fixed direction
¾ Circular / elliptical polarized: If E can rotate uniformly in the plane x, y
¾ Randomly polarized: If the vector E changes randomly with time
http://www.ee.buffalo.edu/faculty/cartwright/java_applets/polarization/simulations/rightpol.html
2
2.1 The nature of light
• The electric or magnetic field of plane linearly polarized waves traveling in a direction k, where the wave vector k = 2π/λ is the wave propagation JG G
G
G G
constant:
A( x, t ) = e x A exp[ j (ωt − k ⋅ z )]
0
ω = 2πν
ω: Angular frequency
ν : frequency of the light
k = 2π / λ
k : wave propagation constant
3
2.1 The nature of light
¾ A general state of Polarization :
JG
G
E x ( z , t ) = e x E0 x cos(ωt − kz )
JG
G
E y ( z , t ) = e y E0 y cos(ωt − kz + δ )
δ : relative phase difference between the two waves
¾ Circular / elliptical polarized: If δ is not zero.
¾ General equations :
2
⎛ Ex ⎞ ⎛ E y ⎞
⎛ E
⎟⎟ − 2 ⎜ x
⎜
⎟ + ⎜⎜
⎝ E0 x ⎠ ⎝ E0 y ⎠
⎝ E0 x
2
⎞ ⎛ Ey
⎟ ⎜⎜
⎠ ⎝ E0 y
⎞
2
⎟⎟ cos δ = sin δ
⎠
4
2.1 The nature of light
⎛ Ex ⎞ ⎛ E y
⎜
⎟ + ⎜⎜
⎝ E0 x ⎠ ⎝ E0 y
2
2
⎞
⎛ E ⎞ ⎛ Ey
⎟⎟ − 2 ⎜ x ⎟ ⎜⎜
⎝ E0 x ⎠ ⎝ E0 y
⎠
⎞
2
⎟⎟ cos δ = sin δ
⎠
¾ Discussion: The value of δ determines the state of polarizations
δ = 0, ±2π , ±4π ....
Linear polarization
π 3π 5π
δ = ± , ± , ± .... Elliptical polarization
2
2
2
Circular polarization :
Circular polarization
Ey
Ex
=±
E0 y
E0 x
2
E0 x = E0 y
π 3π 5π
δ = + , + , + ....
Right circular polarized
π 3π 5π
δ = − , − , − ....
Left circular polarized
2
2
2
2
2
2
⎛ Ex ⎞ ⎛ E y ⎞
⎟⎟ = 1
⎜
⎟ + ⎜⎜
⎝ E0 x ⎠ ⎝ E0 y ⎠
2
5
2.1 The nature of light
6
2.1 The nature of light
2.1.3 The Quantum Nature of Light
¾ Wave theory can explain all phenomena involving transmission of light
¾ However, when dealing with the interaction of light and matter, such as,
emission and absorption of light, neither wave theory and particle
particle theory is not
appropriate.
¾ Instead, we have to turn to Quantum Theory,
Theory, which indicates that optical
radiation has particle as well as wave properties.
Wave:
λ, ν, c, s = c / n
Particle: Wp = hν
¾ Light energy is always emitted or absorbed in discrete
units, called Quanta or Photons.
Photons.
¾ Light energy of a Photon is given by :
h : Planck’
Planck’s constant
ν : frequency of the light
E1
E = hν
h = 6.625 ×10−34 J ⋅ s
E2
hν
¾ Energy level in threethree-level laser :
E0
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2.2 Basic Optical Laws and Definitions
The ratio of the speed of light c in a vacuum to the speed v in matter is the index of refraction (refractive index) n of the material: n = c/v ≥ 1.0
8
2.2 Basic Optical Laws and Definitions
¾ Fundamental parameters:
- Refractive index : n
s=c/n
- Frequency : ν
λ=v/ν
- Wavelength :λ
:
- Speed of light in materials : s
- Speed of light in free space (vacuum) :
c =3*108 m/s
¾ Snell’ law of reflection:
¾ Snell’ law of refraction:
θ1 = θ 2
n1 sin φ1 = n2 sin φ2
Note: definition of θ , φ in this book
¾ Total Internal Reflection
- Critical angle:
sin φc = n2 / n1
9
2.2 Basic Optical Laws and Definitions
Applications of Snell’s Law
10
2.2 Basic Optical Laws and Definitions
¾ A few concepts:
- Plane of incidence (plane including incident ray and normal line)
line)
- p polarization (E parallel to plane of incidence)
- s polarization (E perpendicular to plane of incidence)
- TE (s polarization, E z = 0 ) Æ Transverse electric wave
- TM (p polarization, Hz = 0 ) Æ Transverse magnetic wave
¾ Note : on page 36, book uses N, p for s and p polarization
11
2.2 Basic Optical Laws and Definitions
¾ Fresnel equations for the reflected and transmitted E field in p
polarization (TM) and s polarization (TE):
Please see Pedrotti Optics page 495 – 495, Eq. 23 -27 and Eq. 23 – 28.
rp =
rs =
− n 2 cos θ + n 2 − sin 2 θ
+ n 2 cos θ + n 2 − sin 2 θ
cos θ − n 2 − sin 2 θ
n = n2 / n1
cos θ + n 2 − sin 2 θ
Note:
Due to the definition difference of θ in this book and Pedrotti book, we shall
replace cos (sin ) by sin (cos) function
12
2.2 Basic Optical Laws and Definitions
Question: what happen if incident angle φ1 is greater than critical angle ?
¾ Phase shifts: When light is totally internal
reflection, a phase shift δ occurs:
- Fresnel’
Fresnel’ Eqs.
Eqs. gives reflective coefficient
rp, rs for s and p polarization:
Note: θ in the following eqs. represents θ1 in the figure, which is relative to boundary
rp =
rs =
− n22 sin θ + n1 n22 − n12 cos 2 θ
n22 sin θ + n1 n22 − n12 cos 2 θ
When
n1 sin θ − n22 − n12 cos 2 θ
φ1 > φc
Complex number
n1 sin θ + n − n cos θ
2
2
2
1
2
n1 > n2
With
sin φc =
rp =
rs =
n2
n1
−n22 sin θ + jn1 n12 cos 2 θ − n22
n22 sin θ + jn1 n12 cos 2 θ − n22
n1 sin θ − j n12 cos 2 θ − n22
n1 sin θ + j n12 cos 2 θ − n22
r= re
iδ s , p
(Note:
(Note: φ is the incident angle and θ = π/2 – φ )
13
2.2 Basic Optical Laws and Definitions
¾ Phase Shifts (or call phase change) :
r= re
δ p , δs :
¾
iδ s , p
The phase shifts between Er and Ei
for s and p polarizations
eiθ = cos θ + i sin θ
Euler’ identity:
¾ Expression of Phase Change for reflection coefficients :
tan
tan
δp
2
δs
2
=−
n n 2 cos 2 θ − 1
sin θ
(2.19a)
=−
n 2 cos 2 θ − 1
n sin θ
(2.19b)
With
n = n1 / n2
Correct: Eq 2-19 on p36 needs a negative sign
14
2.3 Optical Fiber Modes and Configurations
Optical waveguides
Slab (or Planar) waveguide
Cylindrical optical fiber
15
2.3 Optical Fiber Modes and Configurations
Slab Waveguide Types
Electrical field mode distribution
16
2.3 Optical Fiber Modes and Configurations
2.3.1 Fiber Types
¾ An optical fiber is a dielectric waveguide (WG) that operates at
at optical frequency
¾ It confines electromagnetic energy in the form of light to within
within its surfaces and
guides the light in a direction parallel to its axis
¾ A set of guided EM waves called the modes of the WG are used to describe the
propagation of light along WG
¾ Only a certain discrete number of modes are capable of propagating
propagating along fiber.
These modes are EM waves that satisfy Maxwell’
Maxwell’ homogeneous equation in fiber
and boundary condition at the WG surface.
¾ Schematic of singlesingle-fiber structure:
17
2.3 Optical Fiber Modes and Configurations
2.3.1 Fiber Types
¾ Fiber type according to index profile:
stepstep-index; gradedgraded-index fiber (GRIN)
¾ Fiber type according to modes:
singlesingle-mode fiber ; multimulti-mode fiber
¾ Advantage / disadvantage of multimulti-mode
fiber :
18
2.3 Optical Fiber Modes and Configurations
2.3.1 Fiber Types
• The indices are uniform in a
step‐index fiber
Cladding
Cladding
2a
2a
Core
Core
• The index varies with the core radius in a graded‐index fiber
Typical diameters
SM core: 8‐
SM core: 8‐10 μ
10 μm
SM cladding: 125 μ
SM cladding: 125 μm
MM core: 50 or 62.5 μ
MM core: 50 or 62.5 μm
MM cladding: 125 μ
MM cladding: 125 μm
(SM = single mode)
(MM = multimode)
n1
n1
n2
Step-index fiber
n1
n1
n2
Graded-index fiber
(a) Basic fiber types
(b) Sample tailored profiles
19
2.3 Optical Fiber Modes and Configurations
2.3.3 StepStep-Index Fiber Structure
• In step‐index fibers the core of radius a has a refractive index n1, which is typically equal to 1.48. This is surrounded by a cladding of slightly lower index n2.
• Δ is the core‐cladding index difference or the index difference. Typical Δ
values range from 1‐3 % for multimode fibers and from 0.2‐1.0 % for single‐mode fibers.
n − n2
Δ= 1
n1
20
2.3 Optical Fiber Modes and Configurations
2.3.4 Ray Optics Representation
Propagation mechanism in an ideal step‐index waveguide
• Light enters the core at an angle θ0 from a medium of index n • Propagating rays are totally internally reflected at core/clad interface
• Meridional rays follow a zig‐zag path along the fiber core
• Snell’s law determines the minimum angle or n
sin φ c = 2
critical angle for total internal reflection:
n1
• Numerical aperture is related to the Maximum acceptance angle of a fiber : NA = n sin θ 0 ,max =
n12 − n 22 ≈ n1 2 Δ
21
2.3 Optical Fiber Modes and Configurations
Numerical Aperture Example
22
2.3 Optical Fiber Modes and Configurations
2.3.2 Rays and Modes
¾ What does SingleSingle-mode fiber mean?
¾ What are modes ?
‰ A set of guided EM waves called the
modes of the WG are used to describe the
propagation of light along WG
‰ The stable field distribution in the x
direction with only periodic z dependence
is known as a mode
LP01 HE11
TE01
LP11 TM01
HE21
LP21
EH11
HE31
23
2.3 Optical Fiber Modes and Configurations
2.3.2 Rays and Modes
¾ EM light field guided along fiber can be
represented by a superposition of bound or
trapped modes.
¾ Guided modes consists of a set of simple
EM field configurations.
Intensity plots for
different LP modes
LP01
LP11
LP21
24
Optical fiber (single
Optical fiber (single‐‐mode):
mode):
d = 8 μm; ncore= 1.444; ncladding= 1.437; λ = 1550 nm
The following results are obtained from commercial software COMSOL.
The effective index n = 1.4406
25
Optical fiber (single‐
Optical fiber (single‐mode):
d = 8 μm; ncore= 1.444; ncladding= 1.437; λ = 1550 nm
LP01: HE11x2 (Two polarizations)
26
Optical fiber (multimode
):
Optical fiber (multimode):
d = 10 μm; ncore= 1.444; ncladding= 1.437; λ = 1550 nm
LP01: HE11x2 (Two polarizations)
Fundamental mode LP01
Effective index neff = 1.4414
27
Optical fiber (multimode
):
Optical fiber (multimode):
d = 10 μm; ncore= 1.444; ncladding= 1.437; λ = 1550 nm
LP11: TE01, TM01, HE21x2 (Two polarizations)
TE01
TM01
Effective index neff = 1.4381
Effective index neff = 1.4381
HE21
HE21
Effective index neff = 1.4381
Effective index neff = 1.4381
28
Optical fiber (multimode
):
Optical fiber (multimode):
d = 10 μm; ncore= 1.444; ncladding= 1.437; λ = 1550 nm
LP11: TE01, TM01, HE21x2 (Two polarizations)
29
2.3 Optical Fiber Modes and Configurations
2.3.2 Rays and Modes
¾ Two methods to theoretically study the propagation characteristics
characteristics of light in an
optical fiber : Ray-tracing approach; Mode Theory (Maxwell’s equations)
¾ RayRay-tracing approach provides a good approximation to light acceptance
acceptance and guiding
properties of optical fiber when the ratio of fiber radius to the
the wavelength is large,
which is known as smallsmall-wavelength limit.
¾ RayRay-tracing approach could give a more direct physical interpretation
interpretation of light
propagation characteristics in an optical fiber, and provide an intuitive picture of
propagation mechanism in optical fiber
¾ Mode theory could provide accurate solution, give field distribution,
distribution, and analyze
coupling efficiency.
30
2.3 Optical Fiber Modes and Configurations
2.3.5 Wave representation in a dielectric slab waveguide
Ray-tracing approach
¾ Wavefront : The surfaces joining all points of equal phase are known
known as wavefronts.
¾ Phase change between the two different tracings with same phase front must be an
integer multiple of 2π
2π
¾ Phase shift : When light wave travels through materials, it undergoes a phase
shift given by: k ⋅ s = 2π n ⋅ s
s: physical distance
λ
31
2.3.5 Dielectric slab waveguide
Goal: 1) Mode condition equation (Eigenvalue equation); 2) Effective refractive indices of the modes
¾ Transverse Resonance condition : Phase changes along complete round trip (along
(
vertical direction) is 2mπ
n1
n2
β
θ
n2
d
Core index: n1
Cladding index: n2
4π n1 d sin θ
λ
k
h
+ 2δ = 2 π m
Phase shifts due to TIR
p polarization : δ
4π n1 d
s polarization : δ
sin θ
Δ = k sin θ ⋅ 2 d =
tan
Physical phase:
λ
p
s
tan
δp
2
δs
2
=−
n n 2 cos 2 θ − 1
sin θ
=−
n 2 cos 2 θ − 1
n sin θ
Propagation constant β and effective refractive index neff :
β = k cos θ =
2π n1
λ
cos θ =
2π n eff
λ
neff = n1 cos θ
If we can find θ, then we can obtain effective32index!
2.3.5 Dielectric slab waveguide
Mode condition equation (or Eigenvalue equations) for TE and TM modes:
TM mode:
TE mode:
2
2
⎛ π n d sin θ π m ⎞ n n cos θ − 1
tan ⎜ 1
−
⎟=
λ
2 ⎠
sin θ
⎝
(2.26b)
n 2 cos 2 θ − 1
⎛ π n d sin θ π m ⎞
tan ⎜ 1
−
⎟=
λ
2 ⎠
n sin θ
⎝
Transcendental equations
If we can find θ, then we can obtain effective index!
m = 0, 1, 2, 3,……
3,…… Corresponding to different guide modes
Modes: TE0 , TE1 , TE2 ……
TM0 , TM1 , TM2……
33
2.3.5 Dielectric slab waveguide
Mode condition equation in Normalized form: n2
n1
Symmetric WG:
Normalized parameters:
V variable: V = kd n12 − n22
θ
d
Normalized guide index: b =
n2
Transverse resonance condition:
4π n f d sin θ
λ
2
neff
− n22
n12 − n22
+ 2 δ = 2π m
Mode condition equation (Eigenvalue Eq.) in normalized form:
For TE modes: V 1 − b = mπ + 2 tan − 1
b
1− b
⎛ n12
For TM modes: V 1 − b = mπ + 2 tan − 1 ⎜⎜
2
⎝ n2
b
1− b
Or
(
)
tan V 1 − b =
2 b (1 − b )
1 − 2b
⎞
⎟⎟
⎠
34
2.3.5 Dielectric slab waveguide
b
Mode condition equation in Normalized form: V 1 − b = mπ + 2 tan − 1
1− b
(TE modes)
Normalized b-V diagram of a planar WG – TE modes
m: the order of modes
V=
2π d
λ
m=0
0.9
n12 − n22
0.8
m=1
0.7
2
neff
− n22
0.6
n12 − n22
m=2
bb
b=
1
b range: 0 to 1
V: related to d, n1, n2 and λ
0.5
m=3
0.4
0.3
m=4
Cutoff condition: 0.2
b= 0
V = mπ
0.1
0
0
2
4
6
8
10
12
14
16
18
V
V
V=
2.3.5 Dielectric slab waveguide
2π d
V 1 − b = mπ + 2 tan − 1
No propagation for guiding mode:
b=
20
n12 − n22
λ
Cutoff condition: no propagation for guiding mode
35
2
neff
− n22
n12 − n22
b
1− b
V = mπ
b= 0
1
Cutoff wavelength: The minimum wavelength at
m=0
0.9
which the slab waveguide will support only one
propagating mode is referred to as the cutoff wavelength
0.8
m=1
0.7
0.6
b
m=2
λc =
2d
m
n12 − n22
0.5
m=3
0.4
0.3
m=4
0.2
0.1
0
0
2
4
6
8
10
V
12
14
16
18
36
20
2.3.5 Dielectric slab waveguide
Mode condition equation for asymmetrical WG (for TE modes): nc
nf
cover
Transverse resonance condition:
4π n f d sin θ
+ δ c + δ s = 2π m
θ
d
ns
substrate
TIR phase shifts:
(1)
λ
tan
δ sTE,c
2
=−
n 2f cos 2 θ − ns2,c
n f sin θ
Subscript: s for substrate,
(2) c for cover
Propagation constant β and effective index neff: neff = n f cos θ
β =
2π neff
λ
= k 0 neff
Also set: h 2 = k02 n 2f − β 2 ; qc2 = β 2 − k02 nc2 ; qs2 = β 2 − k02 ns2
Eq. (2) becomes:
δ TE
q
tan s ,c = − s ,c
h
2
⎛q ⎞
⎛q ⎞
Eq. (2) becomes: 2hd − 2arctan ⎜ c ⎟ − 2arctan ⎜ s ⎟ = 2π m
⎝h⎠
⎝h⎠
tan( hd ) =
Mode condition equation for TE modes: h ( qc + q s )
h 2 − qc q s
37
2.3.5 Dielectric slab waveguide
Mode condition equation for asymmetrical WG (TE modes): 2π d n 2f − ns2
Cutoff Wavelength (nc > ns ≥ nc): λc =
arctan
tan( hd ) =
h ( qc + q s )
h 2 − qc q s
ns2 − nc2
+ mπ
n 2f − ns2
Special cases for symmetric WG with nc = ns: Mode condition equation splits into 2 equations, for Even and Odd modes
Even mode Odd mode hd
q
⎛
⎞
h
⎛ hd ⎞
qc = q s = q
tan ⎜
tan ⎜
⎟=
⎟=−
2
h
⎝
⎠
q
⎝ 2 ⎠
Two solutions:
Even
Even
Odd
38
2.3.5 Dielectric slab waveguide
Electric field distributions of lower‐order guided modes in a slab WG:
Evanescent tails extend into the cladding
Zeroth
order mode
First
order mode
Second
order mode
Zeroth-order mode = Fundamental mode
‰ For lowlow-order modes the fields are tightly concentrated near the center of slab (WG,
optical fiber), with little penetration into cladding region
‰ For highhigh-order modes, the fields are distributed more toward the edges of the guide
and penetrate further into cladding region
39
2.3.5 Dielectric slab waveguide
Numerical calculations to find effective index, etc:
using Matlab
tan( hd ) =
h ( qc + q s )
h 2 − qc q s
Find initial values first by graphs:
clear
close all
nf = 1.56; nc = 1.2; ns = 1; d = 3e-6; lambda = 1.55e-6; k0= 2*pi/lambda;
%Set as hd = x, where hd = sqrt(nf^2-beta^2)*k0*d;
Graphical solution of the eigenvalue
8
Vs=(nf^2-ns^2)*(k0*d)^2;
Vc=(nf^2-nc^2)*(k0*d)^2;
6
%qc = sqrt((nf^2-nc^2)*(k0*d)^2-x^2)=sqrt(Vs-x*x);
4
%qs = sqrt((nf^2-ns^2)*(k0*d)^2-x^2)=sqrt(Vc-x*x;
2
% F is function handle version
0
F = @ (x) (x*(sqrt(Vs-x^2) + sqrt(Vc-x^2))./...
(x^2 - sqrt(Vs-x^2)*sqrt(Vc-x^2)));
ezplot(@tan,[0,4*pi,-5,5])
-4
hold on
-6
h=ezplot(F,[0,4*pi]); set(h, 'Color', 'm');
title('Graphical solution of the eigenvalue')
xlabel('hd')
m=0
-2
m=1
m=2
-8
0
2
4
6
hd
8
10
40
12
2.3.5 Dielectric slab waveguide
Numerical calculations to find effective index, etc: tan( hd ) =
h ( qc + q s )
h 2 − qc q s
Solve the equation:
clear
close all
n1 = 1.56; n2 = 1.2; n3 = 1; d = 0.3e-5; lambda = 1.55e-6;
% n1,n2,n3 are indices for core, cover and substrate respectively
% here x is set as hd.
k = 2*pi/lambda;
V12 = sqrt(n1^2-n2^2)*k*d;
V13 = sqrt(n1^2-n3^2)*k*d;
F = @(x)(x*(sqrt(V12^2 - x.^2)+sqrt(V13^2 - x.^2))./...
(x.^2 - sqrt(V12^2 - x.^2).*sqrt(V13^2 - x.^2) ));
ModeEq = @(x)(F(x)-tan(x));
x(1) = fzero(ModeEq,3);
x(2) = fzero(ModeEq,5.5);
x(3) = fzero(ModeEq,8.5);
neff=sqrt(n1^2-(x./(k*d)).^2);
theta=acosd(neff/(n1));
format short g
[x' neff' theta']
Numerical results
hd
neff
2.7264
5.4356
8.1026
1.5438
1.4946
1.4106
θ (º)
8.2627
16.65
25.284
41
2.3.5 Dielectric slab waveguide
Maxwell’s equations
JG
JG
∂B
∇× E = −
∂t
JG
JJG JG ∂ D
∇× H = J +
∂t
JG
∇ ⋅ D = ρv
JG
∇⋅B = 0
Time-harmonic fields
JG
JJG
∇ × E = − jωμ H
JJG JG
JG
∇ × H = J + jωε E
JG
∇ ⋅ E = ρv / ε
JG
∇⋅B = 0
Wave Equations:
→
∂2
∇ 2 E − εμ E
= 0,
∂t 2
→
→
∇2 H
→
∂2
− εμ H
=0
∂t 2
E x , y , z ( x, y , z ) = E xo, y , z ( x, y )e j β z
o
o
Six components: Ex , y , z , H x , y , z
For simplicity, we use
E x , y , z for E o
x, y, z
JG
JJG
∇ × E = − jωμ H
JJG
JG
∇ × H = jωε E
JG
Source free
∇⋅E = 0
JG
∇⋅B = 0
JG
JJG
∇ × E = − jωμ H
∂Ez ∂E y
−
= − jωμ H x
∂y
∂z
∂Ex ∂Ez
−
= − jωμ H y
∂z
∂x
∂E y ∂Ex
−
= − jωμ H z
∂x
∂y
JJG
JG
∇ × H = jωε E
∂H z ∂H y
−
= jωμ Ex
∂y
∂z
∂H x ∂H z
−
= jωμ E y
∂z
∂x
∂H y ∂H x
Ez
−
= jωμ
42
∂x
∂y