optical waveguide theory - georgia institute of...

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

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.)!

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!

There is always a history …

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]

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

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

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

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 −=

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 ≈

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.

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.

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.

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

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

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.

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

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

Light guiding by total internal reflection

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!)

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

Graded Index Fibernc

nc

nf

nva

ries

quad

ratic

ally

like a “restoring” force !

NA of a Fiber

maxθ

( )maxsin θ= outsidenNA

The NA defines a cone of acceptance for light that will be guided by the fiber

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

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

NA and # of Modes

smallNA

largeNA

propagated ray

killed ray

Types of fiber ends

beam patterns can be:

spherical

cylindrical

bundles

90 degree

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

Angle Preservation 2)

θ2θ

θ

2θα

βα

−≈β 1

f

i

nn

θ

2θ

Critical Bend radius

source: RPI website

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

Polarization and Fibers

cladding

core

meridional ray: stays in the same plane

skewed ray: rotates in many planes about center

entrance exit

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

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

Attenuation & Dispersion

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

Attenuation Profiles

absorption and scattering in fiber

in the IR: “low-OH” fiberts

page 297RayleighScattering

IR absorption

89% transmission

Attenuation Profiles

The window at 1.5 µm is the reason for telcom happening at that frequency!

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

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

Step index fiber

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

Modal dispersion in GRIN Fibers

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

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

Effect of Modal Dispersion

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

Other ResourcesOriel (www.oriel.com)

Melles Griot (www.mellesgriot.com)

Polymicro Technologies

Useful Textbooks

“Optics,” by Hecht