Download 2.2 Basic Optical Laws and Definitions

2.2 Basic Optical Laws and Definitions
¾ Fundamental parameters:
- Refractive index : n
v=c/n
- Frequency : f
λ=v/f
- Wavelength :λ
- Speed of light in materials : v
- Speed of light in free space (vacuum) :
c =3*108 m/s
θ 1= θ 2 , φ 2= φ 2,
¾ Snell’ law of reflection:
¾ Snell’ law of refraction:
n1 sinφ 1 = n2 sinφ 2
Note: definition of θ , φ in this book
¾ Total Internal Reflection
- Critical angle:
sin φc = n2 / n1
2.2 Basic Optical Laws and Definitions
¾ A few concepts:
- Plane of incidence (plane including incident ray and normal 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 : in page 35, book uses N, p for s and p polarization
2.2 Basic Optical Laws and Definitions
Question: what happen if incident angle is greater than critical angle ?
¾ Phase shifts: When light is totally internal reflection, a phase shift δ occurs:
- Fresnel’ Equation gives reflective coefficient rp, rs for s and p polarization:
rp =
rs =
n22 sin θ − n1 n22 − n12 cos 2 θ
n sin θ + n1 n − n cos θ
2
2
2
2
2
1
2
n1 sin θ − n22 − n12 cos 2 θ
n1 sin θ + n − n cos θ
2
2
With
2
1
n1 > n2
2
When
φ > φc
Complex number
rp =
rs =
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
n2
sin φc =
n1
(Note: φ is the incident angle and θ = π/2 – φ )
r= re
iδ s , p
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
Euler’ identity:
eiθ = cos θ + i sin θ
¾ Expression of Phase Change for reflection coefficients :
δp
n n 2 cos 2 θ − 1
=−
tan
2
sin θ
δs
n 2 cos 2 θ − 1
tan = −
2
n sin θ
With n = n1 / n2
(Eq.2-19a need a negative sign)
2.3 Optical Fiber Modes and Configurations
2.3.1 Fiber Types
¾ An optical fiber is a dielectric waveguide (WG) that operates at optical
frequency
¾ It confines electromagnetic energy in the form of light to 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 along
fiber. These modes are EM waves that satisfy Maxwell’ homogeneous
equation in fiber and boundary condition at the WG surface.
¾ Schematic of single-fiber structure:
2.3 Optical Fiber Modes and Configurations
2.3.1 Fiber Types
¾ Fiber type according to index profile:
step-index; graded-index fiber (GRIN)
¾ Fiber type according to modes:
single-mode fiber ; multi-mode fiber
¾ Advantage / disadvantage of multimode fiber :
2.3 Optical Fiber Modes and Configurations
2.3.2 Rays and Modes
¾ What does Single-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
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 Each of these guided modes consists of a set of simple EM field
configurations.
¾ EM light field:
E0 ei (ωt − β z )
β : z component of wave propagation
constant k = 2π /λ.
β is main parameter of interest in
describing fiber modes
2.3 Optical Fiber Modes and Configurations
2.3.2 Rays and Modes
¾ Two methods to theoretically study the propagation characteristics of light in
an optical fiber : Ray-tracing approach; Mode Theory (Maxwell’ equations)
¾ Ray-tracing approach provides a good approximation to light acceptance and
guiding properties of optical fiber when the ratio of fiber radius to the
wavelength is large, which is known as small-wavelength limit.
¾ Ray-tracing approach could give a more direct physical 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, and
analyze coupling efficiency.
2.3 Optical Fiber Modes and Configurations
2.3.3 Step-Index Fiber Structure
2.3.4 Ray Optics Representation
¾ Core-cladding index difference Δ :
also call fractional refractive index change
Δ=
n1 − n 2
n1
¾ Meridional rays; skew rays
¾ Using Meridional rays to examine the propagation will suffice to obtain a
general picture of rays propagating in a fiber
¾ Minimum angle that support total internal reflection : sin φ min =
n2
n1
¾ Numerical aperture is related to the Maximum acceptance angle of a fiber
NA = n sin θ 0 ,max =
n12 − n 22 ≈ n1 2 Δ
2.3 Optical Fiber Modes and Configurations
2.3.5 Wave representation in a dielectric slab waveguide
¾ Condition of wave propagation : All points on the same phase front
(wave front) of a plane wave must be in phase
¾ That means: Phase change between the two different tracings with
same phase front must be an integer multiple of 2π
¾ Wavefront : The surfaces joining all points of
equal phase are known as wavefronts.
¾ Huygens’s Principle : every point on a
propagating wavefront serves as the
source of spherical secondary wavelets,
such that the wavefront at some later time
is the envelope of these wavelets.
2.3 Optical Fiber Modes and Configurations
2.3.5 Wave representation in a dielectric slab waveguide
¾ Condition of wave propagation : All points on the same phase front of a
plane wave must be in phase
¾ That means: Phase change between the two different tracings with
same phase front must be an integer multiple of 2π
¾ Phase shift : When light wave travels through materials, it undergoes a
phase shift given by: Δ = k ⋅ s
-- propagation constant k :
k = 2π n / λ = k 0 n
2π n1
λ
( s1 − s 2 ) + 2δ = 2π m
-- δ : phase shift due to total internal reflection
4π n1 d sin θ
λ
+ 2 δ = 2π m
E
F
2.3 Optical Fiber Modes and Configurations
2.3.5 Wave representation in a dielectric slab waveguide
¾ Another method to find mode condition is from Resonance condition :
Phase change along complete round trip is integer multiple of 2π
β
θ
k
h
Δ = k sin θ ⋅ 2 d =
4π n1 d sin θ
λ
4π n1 d
λ
sin θ
+ 2 δ = 2π m
p polarization : δ
s polarization : δ
p
s
δp
n n 2 cos 2 θ − 1
=−
tan
2
sin θ
δs
n 2 cos 2 θ − 1
tan = −
n sin θ
2
2.3 Optical Fiber Modes and Configurations
2.3.5 Wave representation in a dielectric slab waveguide
p polarization :
s polarization :
m = 0, 1, 2, 3,……
Corresponding to different
guide modes
Modes:
TE0 , TE1 , TE2 ……
TM0 , TM1 , TM2……
2
2
⎛ π n1d sin θ π m ⎞ n n cos θ − 1
tan ⎜
−
⎟=
λ
2
sin θ
⎝
⎠
n 2 cos 2 θ − 1
⎛ π n1d sin θ π m ⎞
tan ⎜
−
⎟=
λ
2 ⎠
n sin θ
⎝