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Optoelectronics

Optical fiber

(Optical fiber sensors)

Institutionen förFysik ochMätteknik

Arno Platau11-02-2001Version 1

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Table of content

1. Introduction. 3

2. Some propertis of optical fibers. 4

2.1. Graded index lenses. 7

3. Wave optics and modes in optical fibers. 8

Project #1: Measuring the numerical aperture. 11

Project #2: Observing fiber modes. 12

Project #3: Proximity sensor. 12

Project #4: Pressure sensor. 14

Project #5: Interferometric temperature sensor. 16

References. 19

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1. Introduction.

In recent times optical fiber have been developed for application in communication systems.

The achievement of low-loss transmission, along with the additional advantages of large infor-

mation carrying capacity, immunity from electromagnetic interference, and small size and

weight, has created a new technology. Optical fiber has become the medium of choice for

communications applications. For example, the TAT-8 (Trans-AtlanticTelephon #8) system

(1988), is a 6500-km all-fiber link which has trans-Atlantic telephone capacity to the equivalent

of 20 000 voice channels. Compare this with TAT-1, completed in 1955, which carried 50 voice

channels over coaxial cable. Optical fiber is also being used extensively in Local Area Networks

(LAN’s), which are used for voice or data communications within or between buildings. Many

new buildings are now being built with fiber installed in their framework for future LAN use.

Optical fiber is also used in sensor applications, where the high sensitivity, low loss,

and electromagnetic interference immunity of the fibers can be expoited. Optical fibers are

versatile and sensors can be designed to detect many physical parameters, such as temperature,

pressure, strain, and electrical and magnetic fields, using either the power transmission

properties of multimode fibers or the phase sensitive properties of single-mode fibers.

In the laboratory You will have the opportunity to study five projects i fiber optics.

1. Measuring the numerical aperture .

2. Observing fiber modes.

3. Proximity sensor.

4. Pressure sensor.

5. Interferometric temperature sensor.

In the first two projects some basic properties of the optical fibers are examinated.

The other projects show how optical fiber can be used as sensors.

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2. Some propertis of optical fibers.

The optical fiber is a cylindrical waveguide. The basic structure consists of a central

light-carrying portion, called the core, which is surrounded by a cylindrical region, called the

cladding. The cladding is then covered with a protective plastic jacket.

The three major fiber configurations used in communication today are illustrated in fig. 1.

a) the single-mode step-index fiber, b) the multimode step-index fiber, and

c) the multimode graded index fiber.

The refractive index as a function of the radius is shown in fig. 1 above the corresponding fiber.

(nr1 is the refractive index of the core, nr2 ist the refractive index of the cladding.)

a) b) c)

Fig. 1. a) Single-mode step-index fiber, b) multimode step-index fiber, and

c) multimode graded index fiber.

The only difference between the single-mode step-index fiber, a), and the mulitmode multimode

step-index fiber, b), is the size of the core. Typical core diameters for single-mode step-index

fibers range in the region from 2 µm to 9 µm. Multimode step-index fibers have diameters in

the region from 50 µm to 150 µm or more.

The outer diameters of the cladding (which are of course larger than the core diameters) are

roughly 50 µm to 250 µm for both kinds of optical fibers.

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Typically, a multimode graded index fiber has a core diameter of about 20 µm to 90 µm.

From theoretical treatment of light propagation along the step-index fiber it can be shown

that only certain electromagnetic waves can propagate along the fiber as ‘guided’ modes.

Different modes correspond to light rays at different angles relative to the fiber axis.

The lowest mode corresponds to a ray which propagates into the axial direction. The highest

allowed mode corresponds to a ray which has the largest allowed angle relative to the fiber axis,

θt, but in order to be a guided mode the ray should arrive at the core-cladding interface with an

incident angle just larger than the critical angle for total internal reflexion, θcrit, thus

. (See fig. 2.)

In a single-mode step-index fiber only the ray propagating into the axial direction is guided.

The maximum core radius allowed for a single-mode optical fiber is according to theory

(1)

λ is the wavelenght of the light, (NA) is the numerical aperture, .

Notice: A step-index fiber which is a single-mode fiber using infra-red light (λ = 1.3 µm) will

show some higher modes when HeNe-laser light is used (λ = 0.633 µm).

On the other hand, if the radius of the core is much larger than the expression in equation (1),

many modes can propagate along the optical (multimode) fiber.

The numerical aperture is connected to the acceptance angle, θA, in the following manner:

nro is the index of refraction of the surrounding medium (air, ).

As seen in fig. 2, the ray which has the

maximal allowed incident angle, θA,

will contiue to the core-cladding

interface at an incident angle roughly

equal to the critical angle for total inter-

nal reflexion, θcrit. Applying Snell’s law

for the refraction air-core we get:

Fig. 2. Step-index fiber. The refractive index

nro is the refractive index of air ( ). profile is shown at the right.

θt 90o θcrit–≤

a2.405 λ⋅2π NA( )⋅------------------------=

NA( ) nr12

nr22

–=

nro θAsin⋅ NA( )=

nro 1≈

nro θAsin⋅ nr1 θtsin⋅=

1≈

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Further is and by applying Snell’s law for the critical angle,

, we get

and finally we get for the numerical aperture:

(2)

The last expression can be modified by introducing the

fractional refractive index difference, ∆,

(3)

and if , called as weakly-guiding approximation, the numerical aperture can be

approximated as:

(4)

In step-index fibers the wave-guiding property is due to total internal reflexion at the

core-cladding interface.

In graded index fibers the wave-guiding property is due to a bending of the rays

towards the fiber axis because of the radial variation of the refractive index.

The most usefull refractive index profile, nr1(r), is given by

(5)

, the fractional refractive index is for this case , no is the refractive index at the

fiber axis, and nr2 is the refractive index of the cladding. (a is the radius of the core.)

The path of a ray follows a sine function in space.

θtsin θcritcos=

nr1 θcritsin⋅ nr2=

θcritcos 1 θcritsin( )2– 1

nr2

nr1-------

� �� �

2–

1nr1------- nr1

2nr2

2–⋅= = =

NA( ) nro θAsin⋅≡ nr12

nr22

–=

∆nr1 nr2–

nr1---------------------=

NA( ) nr1 nr2+( ) nr1 nr2–( )⋅ nr1 nr2+( ) nr1 ∆⋅ ⋅= =

∆ 1«

NA( ) nr1 2 ∆⋅⋅=

nr1 r( )[ ] 2no( )2

1 2 ∆ ra---

� �� �

2⋅ ⋅–⋅=

∆ ∆no nr2–

no

------------------=

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A fan of rays injected at a point in a

graded index fiber spreads out and then

recrosses the axis at a common point just

as rays from a small object are reimaged

by a lens.

The distance it takes for a ray to traverse

one full sine path is called the pitch of

the fiber. The length of the pitch is

determined by ∆ , Fig. 3. Graded index fiber. The refractive

the fractional index difference. . index profile is shown at the right.

Diverging rays are refocussed at a point further down the fiber.

1.1 Graded index lenses.

If a graded index fiber is cut to a length of one quarter of the pitch of the fiber it can serve as

extremely compact lens, somtimes called GRIN lens. Light focussed on an axial point will

leave the GRIN lens parallell. Increasing the lenght of the GRIN lens to 0.29 of a pitch will give

the possibility to refocusse divergent light. Both 0.25 and 0.29 pitch GRIN lenses are usefull for

coupling light sources to optical fibers and fibers to detectors.

The HeNe-laser in our project is mainly coupled using objective lenses (20X and 40X).

.Fig. 4 Graded index (GRIN) lens.

a) 0.25 pitch lens.

, b) 0.29 pitch lens.

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3. Wave optics and modes in optical fibers

The description for the modes that propagate in a fiber is found by solving the wave equation in

cylindrical coordinates for the electric field of the light in the fiber. The solutions, which are

found to be harmonic in space and time, are of the form

(6)

where, ω = 2πν, and ν is the frequency of the light, β is the propagation constant, γ is a phase

constant, and q is an integer. The parameter, β, is important for specifying how light propagates

in a fiber. In the ray optics description, β is the projection of the propagation vector on the z

axis, where the magnitude of th propagation vector is k = 2π/λ, λ being the wavelength of light

in vacuum. It is important to make the distinction between the magnitude of the propagation

vector, k, and the propagation constant, β, which is the z-componenet of the propagation vector,

in order to avoid later confusion.

Solutions for β, f(r), and q are obtained by substituting (6) into the wave equation. The

solution will depend on the particular fiber geometry and index profile, including both the core

and the cladding, under consideration. The step-index profile is one of the few refractive index

profiles for which exact solutions may be obtained. For this case the solutions for f(r) are Bessel

functions.

An important quantity in determining which modes of an electromagnetic field will be

supported by a fiber is a parameter called the characteristic waveguide parameter or the

normalized wavenumber, or, simply the V-number of the fiber. It is written as

(7)

where k is the free space wavenumber, , is the radius of the core, and (NA) is the numerical

aperture of the fiber.

E r ϕ z, ,( ) f r( ) ωt βz– γ+( ) qϕ( )cos⋅cos⋅=

V k a NA( )⋅ ⋅=2πλ

------ a

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When the propagation constants,

(β-s) of the fiber modes are plotted as a

function of the V-number, it is easy to

determine the number of modes that can

propagate in a particular fiber. In fig. 5,

such a plot is given for some of the

lowest order modes. The number of

propagating modes is determined by the

number of curves that cross a vertical

line drawn at the V-number of the fiber.

Not that for fibers with V<2.405, only a

single mode will propagate in the fiber. Fig. 5. Low order modes of an optical fiber

This is the single-mode region.

The wavelength at which V = 2.40 is called the cut-off wavelength, denodet by λc, because, for

a particular product of core diameter and (NA), as the wavelength of radiation is increased, this

is the wavelength at which all higher-order modes are cut off and only a singele mode will prop-

agate in the fiber. A fiber which propagates onely th HE11 mode is said to be a

single-mode fiber.

In the weakly-guiding approximation ( ),

the exact solutions of waveguide theory, HEmm, can

be replaced by a set of modes which are linearly po-

larized, called the LP modes.. These LP modes are

combinations of the modes found from the exact

theory of the waveguide. These linearly-polarized

modes may be characterized by two subscripts, m

and n. The first subscript, m, gives the number of az-

imuthal, or angular, nodes (zeroes) that occur in the

electric field distriburtion of the mode; the second

subscript, n, gives the number of radial nodes that

occur. They can be identified by pattern in the output

of the fiber as it illuminates a screen. Fig. 6. Irradiance pattern of some low

order lineraly polarized modes.

∆ 1«

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The patterns are symmetric about the center of the beam and show bright regions separated by

dark regions (the nodes that determine the order numbers m and n). Some of these are schown

in fig. 6. It ist assumed that the zero field at the outer edge of the field distribution is counted as

a node, so . For the azimuthal nodes, . The lowest order HE11 mode consists of two

P01 modes with polarization at right angles to one another. Fig. 7 shows the propagation

constants of these modes as a function of V-number.

(Compare this figure with the exact solutions in fig. 6.)

When the V-number is greater

than 2.405 (the value at which the

first zero of the zero-order Bessel

function occurs), the next linearly-

polarized mode, LP11, can be sup-

ported by the fiber, so that both the

LP01 and LP11 modes will propagate.

For a fiber with a V-number of 3.832

(corresponding to th first zero of the

first-order Bessel function), two

more linearly-polarized modes can

propagate: the LP21 and the LP02 Fig. 7. Low order linearly polarized modes of an

modes. By changing the position optical fiber. Compare with fig. 5.

and angle of the input beam incident

on a low-V-number multimode fiber, individual linaearly-polarized modes can be lauched in

the fiber and observed at the output. The propagation of individual modes in such a fiber will

be observed in project #2. This will help overcome one of the difficulties of the concept of

modes in optical fiber, which is understanding what they are and how they differ from one an-

other.

n 1≥ m 0≥

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Project #1: Measuring the numerical aperture.

Values of the numerical aperture range from about 0.1 for single-mode fibers to 0.2-0.3

for multimode communications fibers up to about 0.5 for large-core fibers.

The way in which light is lauched into the fiber in

the method used here to measure the fiber (NA) is

shown in fig. 8. The light from the laser represents a

wave front propagating in the z-direction. The width of

the laser beam is much larger than the diameter of the

fiber core. In the neighborhood of the fiber core, the

wave front of the laser light takes on the same value at

all points having the same z, so we say that we have a

plane wave propagating parallel to the z-axis. Fig. 8. Geometry of a plane-wave launch

When a plane wave is incident on the end face of of a laser beam into an optical fiber.

a fiber, then we can be sure that all of the light

launched into the fiber has the sam incident angle, θc, in fig. 8.

If the fiber end face is then rotated obout

the point O in fig. 8, we can then measure the

amount of light accepted by the fiber as a function

of the incident angle.

The Electronic Industries Association uses

the angle at which the accepted power has fallen

to 5% of the peak accepted power as the

definition of the experimentally determinde (NA). Fig. 9. Laboratory set-up for

determination of fiber (NA).

The 5% intensity points are chosen as a

compromise to reduce requirements of the power level which has to be distinguished from back-

ground noise.

Rotate the fiber and measure the accepted power as a function of the angle. Make a plot and

determine the acceptance angle and (NA).

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Compare Your result with a rough measure as shown in

fig. 10. Launche laser light into one end face of the fiber by a

objetive lense and measures the diameter of the light cone, w,

which the other end face will project onto a screen at a dis-

tance L. The acceptance angle, θA, is calculated from:

(8) Fig. 10. Approximate measure of

the (NA) of a fiber.

Project #2: Observing fiber modes.

Lauche light of the HeNe-laser by an objective lens into a single-mode fiber (of a length

of 1-2 m). (Single-mode, when using infra-red light.)

Observe on a screen the light which is emerging from the other end face of the fiber.

How many modes do You get. What kind of modes ?

Project #3: Proximity sensor.

‘Fotonic sensor’.

The proximity sensor is one of the simplest

extrinsic fiber-based sensors for measurement

of position or movement. It was one of the first

commercially available displacement sensors

named ‘Fotonic sensor’.

It uses a bundle of fibers, half of which are

connected to a source of radiation, the other

half to a detector. ( See fig. 11.) Fig. 11 Illustration of the ‘Fotonic sensor.

If the bundle is placed in close proximity to a reflecting surface then the light will be reflected

back from the illuminating fibers into the detecting fibers. The ammount detected will depend

on the distance from the fiber ends to the surface.

θAtan12--- w

L----⋅=

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The form of the relationship between displacement

and light output can be estimated from a calculation

on a model consisting of two fibers as shown in fig. 12.

Taking a core radius, a = 100 µm, a core separation,

s = 100 µm, and a numerical aperture, NA = 0.4, results

in a coupling efficiency as function of the distance d as

shown in fig. 13.

Fig. 12 Two-fiber model

Fig. 13 Coupling efficiency as function of the distance calculated for a two-fiber model.

For the sensor shown in fig. 11 the form of the relationship between light output and displace-

ment is similar to that for the two-fiber model.

The ‘Fotonic sensor’ was originally developed for non-contact vibration analysis.

Measure the power of the output as a function of the displacement.

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Project #4: Pressure sensor.

In fig. 14 a pressure sensor is illustrated.

Unpolarized light is linearly polarized by the

polaroid, P, at an angle of 45o with respect to

the stress axis, S. Stress applied to the plexi-

glass piece, G, causes stress-induced

birefringence. Within the plexi-glass the

incident light splits into two linearly polarized

rays, one polarized parallell to the stress-axis,

the other perpendicular to the first.

Fig. 14. Pressure sensor using the photoelastic effect.

The difference in phase velocity of the two linearly polarized rays depends on the amount of

stress applied (and on the material properities). After passing the plexi-glass piece the rays will

have a phase difference and their superposition will result in elliptically polarized light.

A second polarizer, A, which acts as an analyzer selects the component of the transmitted beam

perpendicular to the first polarizer.

For the configuration shown, the optical power transmitted by the sensor is

(9)

where t ist the thickness of the plexi-glass cube, S is the applied stress, and f = 2tSo, where So

is the stress required to make the transmitted power through the transducer go from a maximum

to a minimum. It is called the material fringe value. Typical values for f are in the range of

0.2-0.3 MPa-m. This type of sensor has been used to measure pressures of up to 21 MPa with a

resolution of better than .

I I0πtS

f--------

� �� �sin

2⋅=

P∆( )P

------------ 104–

=

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In fig. 15 the pressure sensor assembly is

illustrated. The polarizers, P and A, are

glued to the plexi-glass surfaces.

Multimode graded index fibers are used as

light pipe-lines.

Two 1/4-pitch GRIN-rod lenses

(GRIN = graded index) are uses in order to

maximize the fiber-to-fiber coupling

through the assembly.

Fig. 15. Pressure sensor assembly

In fig. 16 the labortory set-up of the pressure sensor is shown.

Fig. 16. Laboratory set-up of the pressure sensor.

Measure the power of the detected light as a function of the applied pressure.

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Project #5: Interferometric temperature sensor.

Interferometric sensors are sophisticated sensors

which detect the influence of physical perturba-

tions on the phase of the light propagating in a

single-mode fiber. These sensors offer the

potential of extremely high sensititvity, but also

require the best single-mode fiber technology

available for their construction.

The most common phase sensors use the

Mach-Zehnder interferometer. Fig. 17. Mach-Zehnder fiber optic

A fiber optics version is illustrated in fig. 17. interferometer configuration.

As first beamsplitter a bidirectional

coupler is used and the light is lauched into two single-mode fibers of equal lengths. One of the

fibers serves as the reference arm, which is kept isolated from external perturbations, while the

other fiber serves as the sensor arm of the interferometer, which is exposed to the perturbation

to be measured. The external perturbation induces the phase shift in the sensing arm by means

of a change in the optical path length, which is caused by a change in the index of refraction of

the glass of the fiber and/or a change in the length of the fiber. The phase shift is detected when

the two beams are recombined at the receiver end of the sensor. This results in fringes which

can be detected and counted. Sensors using the Mach-Zehnder

configuration can be constructed to measure a wide variety of physical parameters, including

stress, strain, acoustic waves, magnetic field, and temperature. Single- ode optical fibers are em-

ployed in sensors using other types of interferometers, also.

When the Mach-Zehnder interferometer has been constructed, the light from the two

fibers interferes to form a series of bright and dark fringes. A change in the phase of the light in

the sensor fiber with respect to the phase of the light in the reference fiber appears as a

displacement of the fringe pattern; a phase change of 2πradians causes a displacement of one

fringe. The magnitude of the change in the physical parameter to be measured can be

determined directly by counting the fringe displacement.

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The phase of a light wave which travels a distance, L, in an optica fiber is given by

φ= βL, where β is the propagation constant of the light in the fiber. Changing any physical

parameter of the fiber’s environment causes a phase change given by

(10)

The first term on the right hand side of the equation is due to a change in the lenght of the

fiber, while the second term is due to a change in the propagation constant. The length, L, now

represents the length over which the physical change affects the fiber. The quantity which we

actually wish to determine is the phase change per fiber length per physical stimulus (δφ)/δSL,

where S is the stimulus. The magnitude of the stimulus can then be measured by counting the

shift of the fringes for a fiber of known interaction length.

As an example, consider the effect of a temperature change, δT, which affects a length, L,

of the fiber in the sensor arm of the interferometer. There are two effects which occur: the

change of length due to thermal expansion or contraction, and the change of the propagation

constant due to the temperature dependence of the index of refraction.

Thus equation (10) becomes

(11)

where and

For the case of a fused silica fiber and a HeNe laser sourec, the following values for pure

silica might be used:

n = 1.456 and

This gives

(12)

this corresponds to 17 fringes per oC per meter of fiber.

δφ βδL Lδβ+=

δφδTL---------- 2π

λ------ n

L---δL

δT------ δn

δT------+=

δLδT------

L T1( ) L T2( )–[ ]T1 T2–

---------------------------------------=δnδT------

n T1( ) n T2( )–[ ]T1 T2–

--------------------------------------=

1L--- δL

δT------⋅ 5

7–×101oC

--------=δnδT------ 10

6–×101oC

--------=

λ 632.89–×10 m=

δφδLT---------- 107

radiansoC m⋅

--------------------=

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Polarization problems.

The polarization of the light traveling in the interferometric sensor is important because

if the polarization of the two output beams are no plane parallel, sharp fringes will not be seen.

In the worst case, when the two polarizations are orthogonal to each other, no interference will

occur at all.

When long lengths of fiber are used in the interferometer, polarization-preserving fiber is

used. When short lenths of fiber, such as the approximately two meters required for this project,

are used in the interferometer, standard single-mode fiber may be used. If plane-polarized light

is launched in the fiber, the light will stay pretty well in its original plane-polarized mode as long

as the perturbations which would cause mode coupling are kept to a minimum. The output ends

of the fibers can then be manipulated to cause the polarizations of the output beams to be

parallalel.

This last statement may sound highly qualitative, but this approach works well for shorter

fiber lenghts, and avoids the need for handling an aligning the polarization-preserving fibers.

Doing the experiment:

1. In the discussion above, it was found that the fringe displacement of the interferometer

is highly temperature dependent. You can get a good feeling for this by trying a simple

qualitative demonstration. After the interferometer has been set up, place your hand around one

of the fibers withour touching it. The heat from your hand will cause a rapid displacement of the

fringes, even though you are not in direct contact with the fiber. As you perform this and the

following exercise, be aware of the environmental conditions of your interferometer, especially

if there is an air-conditioner outlet near your apparatus.

2. A simple quantitative experiment to calibrate the temperature sensitivity of the

interferometer can be done using ice. Before doing the experiment, use the data from equation

.... to calculate the expected fringe displacement between a typical room temperature and 0oC.

You should ge a value of about 9 fringes for an interaction lenght of about 2 cm.

3. Get some ice cubes and allow them to stand until they have become wet to the touch.

If they feel dry, they may be at a temperature which is significantly less than 0oC. If they feel

wet then their surfaface should be right at 0oC.

4. Lay one of the ice cubes on a straight section of one of the fibers on the breadboard.

Because of the large thermal mass of the breadboard, it may take 30-60 seconds for the fiber to

come into thermal equilibrium with the ice. This will ensure that you have plenty of time to

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count the fringes as they change.

5. Compare the number of fringes which you counted with the value that you got when

you did the calculation in step 2. How good is the “order of magnitude” calculation leading to

equation (12)?

References.

Newport Corporation: Projects in fiber optics.

Hecht: Optics.

Wilson/Hawkes: Optoelectronics.

Singh: Optoelectronics