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    Optical fiber

    (Optical fiber sensors)

    Institutionen frFysik ochMtteknik

    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

    (LANs), 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 m to 9 m. Multimode step-index fibers have diameters in

    the region from 50 m to 150 m or more.

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

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

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

    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,

    t, 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, crit, 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


    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 ( = 1.3 m) will

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

    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, A, 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, A,

    will contiue to the core-cladding

    interface at an incident angle roughly

    equal to the critical angle for total inter-

    nal reflexion, crit. Applying Snells 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.

    t 90o crit

    a2.405 2 NA( )------------------------=

    NA( ) nr12



    nro Asin NA( )=

    nro 1

    nro Asin nr1 tsin=


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    Further is and by applying Snells law for the critical angle,

    , we get

    and finally we get for the numerical aperture:


    The last expression can be modified by introducing the

    fractional refractive index difference, ,


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

    approximated as:


    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


    , 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.

    tsin critcos=

    nr1 critsin nr2=

    critcos 1 critsin( )2



    1nr1------- nr1


    2= = =

    NA( ) nro Asin nr12



    nr1 nr2


    NA( ) nr1 nr2+( ) nr1 nr2( ) nr1 nr2+( ) nr1 = =


    NA( ) nr1 2 =

    nr1 r( )[ ]2

    no( )2

    1 2 ra---

    2 =

    no nr2


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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


    where, = 2, and is the frequency of the light, is the propagation constant, is a phase

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

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

    axis, where the magnitude of th propagation vector is k = 2/, 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, , which is the z-componenet of the propagation vector,

    in order to avoid later confusion.

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

    solution will dep