optical waveguide theory - georgia institute of...
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OPTICAL WAVEGUIDE THEORYWhat we need to know about fibers …
- Types of optical fibers
- Modes in an optical fiber
- How fibers guide light
- Numerical aperture
- Polarization in Fibers
- Attenuation and Dispersion in fibers
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Important to understand …
- Optical fibers have been developed for telecommunications and not for analytical chemistry or sensing!
- Approx. 80% of commercially available fibers serve telecommunications markets; 15% light delivery (e.g. lasers, LEDs); 5% waveguide based sensing and others.
- Advantage: demands on the properties of optical fibers from telecommunications usually much higher and stringent compared to analytical applications!
- Consequently: if we can use telecommunications technology for optical sensing we get more than we need (e.g. low-OH silica fibers, diode lasers, etc.)!
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Telecom example and analytical demands:
I need a fiber that will conduct NIR light.
Ideally, the fiber/fibers will conduct light in the UV/Vis, MIR and FIR as well!
I have to keep a tight pulse pattern.
For ultrafast laser spectroscopy (e.g. fs lasers) – YES; for optical sensing (e.g. evanescent field) – I DON’T CARE!
It must couple into from an LED.
Yes, but sometimes also from a broadband light source!
What do I do?
Hope that telecom is interested in the entire analytically relevant spectral range, because for analytics/opt. sensing alone new fibers will not be
developed!
HENCE: multimode AND single mode fibers of interest in sensing!
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There is always a history …
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Optical fiber materials
0 5
SiO2 (0.35-1.40)
Low OH SiO2 (0.4-2.5)
Sapphire (0.5-3)Fluoride ZBLAN (0.4-5)
Chalcogenide (1-11)
Tellurium halides (TeX) (3-13)
Silver halides (AgX) (4-18)
Hollow waveguides (3-17)
Transmission [µm]
10 2015
Material
500-1500 @ 10.6 µm
500 @ 11 µm
1000 @ 9 µm
400 @ 6 µm
15 @ 2.5 µm
5000 @ 2.9 µm
12 @ 0.82 µm
0.4 @ 1.32 µm
Attenuation
[dB/km]
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Optical fiber materials
0 5
SiO2 (0.35-1.40)
Low OH SiO2 (0.4-2.5)
Sapphire (0.5-3)Fluoride ZBLAN (0.4-5)
Chalcogenide (1-11)
Tellurium halides (TeX) (3-13)
Silver halides (AgX) (4-18)
Hollow waveguides (3-17)
Transmission [µm]
10 2015
Material
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What is an Optical Fiber?
An optical fiber is a waveguide for light
Generally consists of :
core inner part where wave propagates
cladding outer part used to keep wave in core
buffer protective coating
jacket outer protective shield
can have a connector
too
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Types of Fibersst
ep-in
dex
mul
timod
e n2
n2
n1
n2
n2
n1
n2
n2
n1
step
-inde
xsi
ngle
mod
eG
RIN
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The V-number of a fiber
NAaoλ
π= 2V
•Also known as the ‘V-parameter’ or ‘fiber parameter’
•The V-number governs the number of modes transported in the fiber
•In single-mode fiber: V<2.405Hence: no modal dispersion, only a single light mode transported
•Ideal for long-range optical networks (telcom) or if combined with a single-mode light source in optical sensing (e.g. laser diode)
a = fiber core diameterλo = incident wavelength
NA = numerical aperture (Note: we will come back to that later!)
)( 22
21 nnNA −=
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Fibers carry modes of light
2V~M modesofNumber
BUT:
What is a light mode?
More specifically:
Step-index fiber: Graded-index fiber:(GRIN)2
2VM =4
2VM ≈
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Modes of light (1)Let’s use the wave model and a simple example:- A plane wave moving in x-direction can be written as
y(x,t) = Asin(2πx/λ - 2πt/T),
with wavelength λ and period T (the inverse of the frequency).- We can also write
y(x,t) = Asin(kx - ωt),
where k = 2π/λ is the wavevector, and ω = 2π/T is the angularfrequency.- A wave, which is the sum of two waves, may be written as
y = y1 + y2.
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Modes of light (2)- If two waves oscillate at the same frequency they are ‘in phase’; at a point theamplitudes will add (constructive interference).- It they are ‘out of phase’ the amplitudes will subtract (destructive interference).- Now we consider a wave reflected at two end points where the vibration is set to zero.- EXAMPLE: a violin string attached at both ends to the instrument.- The total wave at any point of the string is the sum of two counter-propagating waves y = y1 + y2.- NOW:
y1 = Asin(kx - ωt), a wave moving to the right andy2 = Asin(kx + ωt), a wave moving to the left, resulting in the sum
y = A{sin(kx - ωt) + sin(kx + ωt)} = 2Asin(kx) cos(ωt)
- This wave has nodes at which y = 0 (resulting from destructive interference) and anti-nodes at which the amplitude is a maximum (2A) resulting from constructive interference.
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Modes of light (3)
Now we can answer the question ‘What is a light mode?’
• A solution to the wave equation!
- All vibrations of the string can be expressed as a sum of allowed modes.- Mathematically, the modes are solutions of a wave equation and satisfy specified boundary conditions.- In this case the boundary conditions are that y = 0 at the two ends ofthe string.- For the string, one end is at x = 0 and the other at x = a.-The condition y = 0 at x = 0 is automatically satisfied in the previous expression for y, since sin 0 = 0.- BUT: the condition y = 0 at x = a is only satisfied for certain ‘allowed values’ of k, i.e. values of k such that sin(ka) = 0.-Fulfilled only when ka = nπ, or equivalently when k = nπ/a, with n being a positive integer.CONCLUSION: Only certain modes of vibration are allowed with values determined by the boundary conditions.
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Controlling the # of Modes
From the V-number, we see that we can reduce the number of modes in a fiber by reducing:
(1) NA(difficult, because material properties need to be tuned)
(2) diameter(much easier!)
This is exactly the case in single mode fibers.
NAaoλ
π= 2V
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Example of # of Modes @ 850nm
Silica step-index fiber has nf = 1.452, nc = 1.442 (NA = 0.205)
2.4 E692 E322 E31.4 E32# step-index modes
1.2 E646 E311 E37161# GRIN modes
2.5 20050 1000400diameter (microns)
SELFOC graded index fiber with same NA
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Wait a minute …
• If all you want to do is conduct light, then a simple glass rod would be sufficient …
• So … why use fiber optics ?????
If you want to preserve characteristics of the light then you need fiber optics or other optical wave guides.
Examples: temporal profile, polarization, …
Fiber is more flexible and generally more adaptable to a given problem.
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The General Principle
The classical understanding of waveguiding in fiber optics comes from our longtime friend, Snell’s Law!
• Step index fibers: Total Internal Reflection
• GRIN fibers: layered changes in n
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Total Internal Reflection
n i sin θ i. n t sin θ t.n i sin θ i. n t θ t
n i sin θ c. n t sin 90( ).n i sin θ c. n t
n i sin θ c. n t 1( ).n i sin θ c. n t
sin θ cn tn i
n tn i
nt
ni
θi
really weakevanescent wave
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Light guiding by total internal reflection
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Step Index Fiber
( ) tii
tc n
nn ≡=θsin
nt
ni
θiθi
θi
cladding
core
escapes core (freedom!)
stuck in core(did not graduate)
escapes core (freedom!)
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Evanescent field
E
IR beam
IR beam
x
dp
Waveguide
Polymer layern1
n2
n1
n > n1 2
Θ
n2
d
nnn
p =
−
λ
π θ2 12 2
1
2
sin
PENETRATION DEPTH
E E ez
dp= ⋅−
0
EVANESCENT FIELDINTENSITY
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Graded Index Fibernc
nc
nf
nva
ries
quad
ratic
ally
like a “restoring” force !
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NA of a Fiber
maxθ
( )maxsin θ= outsidenNA
The NA defines a cone of acceptance for light that will be guided by the fiber
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NA of a Step Index Fiber
90-θtθt
θmax
nf
nc
must be > critical angle
( )maxsin θ= outsidenNA
22cfstep nnNA −=
ni
i
cfstep n
nnNA
22 −=
Since for air ni = 1
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NA of a GRIN Fiber
2DnNA f α=
Condition below assures a ray will have enough fiber to bend back towards the center axis:
α is a parameter describing how n changes in the GRIN fiber; D = dispersion coefficient
NA in air
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NA and # of Modes
smallNA
largeNA
propagated ray
killed ray
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Types of fiber ends
beam patterns can be:
spherical
cylindrical
bundles
90 degree
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Angle Preservation (1)
In an ideal fiber, the angle of incidence will equal the exit angle.
Rough surfaces, bending, and other real-world imperfections will case a change in the exit cone.
example: critical bend radius
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Angle Preservation 2)
θ2θ
θ
2θα
βα
−≈β 1
f
i
nn
θ
2θ
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Critical Bend radius
source: RPI website
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Fiber Tapers
θ1
θ2
( ) ( )2211 sinsin θ=θ dd
d1 d2
way to change the acceptance angles (NA) of a fiber
sometimes used to collimate light
does not necessarily “get you more light” since NA changes
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Polarization and Fibers
cladding
core
meridional ray: stays in the same plane
skewed ray: rotates in many planes about center
entrance exit
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Polarization
If the ray rotates during propagation, then the polarization state will change
linear polarized beam translates into elliptical beam
Sr
Sr
Er
Er
Br
Br
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What to do?
•In some applications, polarization is not needed:
example: spectroscopy, sensing
•In others it is critical:
example: polarization sensitive optical communication
•Solution: must remove the circular symmetry of the fiber
change n profile so that polarizations are not coupledeffect comparable to birefringence
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Attenuation & Dispersion
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Attenuation
−=
in
out
PPA log10
Fibers are made of “glass”
- commonly high-quality fused silica (SiO2)- some trace impurities (usually controlled)
Losses due to:
- Rayleigh scattering (~ λ-4)- absorption (“low-OH” in UV versus “high-OH” in IR)- mechanical stress- coatings
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Attenuation Profiles
absorption and scattering in fiber
in the IR: “low-OH” fiberts
page 297RayleighScattering
IR absorption
89% transmission
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Attenuation Profiles
The window at 1.5 µm is the reason for telcom happening at that frequency!
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Dispersion: The Basics
Light propagates at a finite speed
fastest ray
slowest ray
slowest ray: one entering at highest angle (“high order” mode)fastest ray: one traveling down middle (“axial mode”)
there will be a difference in time for these two rays
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Step index fiber
usually the biggest dispersion problem in step index multi-mode fibers
• Abrupt index change from core to cladding• Fractional reflective index change
• Modal dispersion: Input impulse spreadsto pulse width σ τ
where L=fiber length, c1=speed of light in core
02.0001.01
21 −=−=∆n
nn
∆≈12c
Lτσ
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Step index fiber
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GRIN fiber
• Gradual index decrease from n1 (on axis) to n2(core/cladding boundary)
• Optimal index profile approximated by parabola• Fractional reflective index change:• Modal dispersion: (Factor of ∆/2 smaller
than step-index fibre, butusually not met because profile not perfect)
1<<∆
2
14∆≈
cL
τσ
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Modal dispersion in GRIN Fibers
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Types of Dispersion in Fibers
modal - time delay from path length differences- usually the biggest culprit in step-index
material - n(λ) : different times to cross fiber-(note: smallest effect ~ 1.3 µm)
waveguide - changes in field distribution -(important for SM)
non-linear - n can become intensity-dependent
NOTE: GRIN fibers tend to have less modal dispersion
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Effect of Modal Dispersion
time time time
modal example: step index ~ 24 ns km -1GRIN ~ 122 ps km-1
initial pulse farther down farther still
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Effect of Modal Dispersion
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Review
1. Optical fibers carry modes of light
2. Step-index, GRIN, single mode & multimode
3. NA is related to acceptance cone and n’s.
4. How Step-index and GRIN fibers propagate light.
5. Factors that change light propagation in fibers:
a. mechanical aspects (bending, tapers, etc)
b. attenuation
c. dispersion
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Other ResourcesOriel (www.oriel.com)
Melles Griot (www.mellesgriot.com)
Polymicro Technologies
Useful Textbooks
“Optics,” by Hecht