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Figure 13.1 shows the basic construction of an optical fibre.. In the figure a light ray is incident at the end of the fibre at an angle θ0 to the fibre axis.. Lost ray Cone of acceptanc

Fibre Optics in Metrology

13.1 INTRODUCTION

With a carrier frequency of some 1014 Hz, light has the potential of being modulated at much higher frequencies than radio waves Since the mid-1960s the idea of communication through optical fibres has developed into a vital branch of electro-optics Great progress has been made and this is now an established technique in many communication systems From the viewpoint of optical metrology, optical fibres are an attractive alternative for the guiding of light An even more important reason for studying optical fibres is their potential for making new types of sensors

13.2 LIGHT PROPAGATION THROUGH

OPTICAL FIBRES

More extensive treatments on optical fibres can be found in Senior (1985), Palais (1998), Keiser (1991) and Yu and Khoo (1990)

Figure 13.1 shows the basic construction of an optical fibre It consists of a central

cylindrical core with refractive index n1, surrounded by a layer of material called the

cladding with a lower refractive index n2 In the figure a light ray is incident at the end of

the fibre at an angle θ0 to the fibre axis This ray is refracted at an angle θ1 and incident

at the interface between the core and the cladding at an angle θ2 From Snell’s law of refraction we have

where n0 is the refractive index of the surrounding medium From the figure, we see that

θ1= π

If θ2 is equal to the critical angle of incidence (cf Section 9.5), we have

sin θ2 = n2

n1

( 13.3)

Optical Metrology Kjell J G˚asvik

ISBN: 0-470-84300-4

Lost ray

Cone of

acceptance

Figure 13.1 Basic construction of an optical fibre

which combined with Equations (13.1) and (13.2) gives

θ0 ≡ θa= sin−1







n21− n2 2

n0



For θ0< θathe light will undergo total internal reflection at the interface between the core and the cladding and propagate along the fibre by multiple reflections at the interface,

ideally with no loss For θ0> θa some of the light will transmit into the cladding and after a few reflections, most of the light will be lost

This is the principle of light transmission through an optical fibre The angle θa is an important parameter when coupling of the light into a fibre, usually given by its numerical aperture NA:

N A = n0sin θa=n2

1− n2

In practice, coupling of the light into the fiber can be accomplished with the help of a lens, see Figure 13.2(a) or by putting the fibre in close proximity to the light source and

Cladding

Core

(a)

(b)

Core

Index matching liquid

Figure 13.2 Coupling of light into a fibre by means of (a) a lens and (b) index-matching liquid

LIGHT PROPAGATION THROUGH OPTICAL FIBRES 309

linking them with an index-matching liquid to reduce reflection losses, Figure 13.2(b) When using the method in Figure 13.2(a), it is important to have the angle of the incident

cone less than θa to get maximum coupling efficiency

The above description of light propagation through an optical fibre is not fully complete

To gain better understanding, the fibre must be treated as a waveguide and the electro-magnetic nature of the light must be taken into account If a waveguide consisting of a transparent layer between two conducting walls is considered, the electric field across the waveguide will consist of interference patterns between the incident and reflected fields,

or equivalently, between the incident field and its mirror image, see Figure 13.3 The

path-length difference l between these fields is seen from the figure to be

where d is the waveguide diameter and θ is the angle of the incident beam From

the boundary conditions for such a waveguide we must have destructive interference

at the walls, i.e the path-length difference must be equal to an integral number of half the wavelength:

l = m λ

which gives

sin θ = mλ

where m is an integer Thus we see that only certain values of the angle of incidence are

allowed Each of the allowed beam directions are said to correspond to different modes

of wave propagation in the waveguide The field distribution across the waveguide for the lowest-order guided modes in a planar dielectric slab waveguide are shown in Figure 13.4 This guide is composed of a dielectric core (or slab) sandwiched between dielectric claddings of lower refractive index As can be seen, the field is non-zero inside the

Incident beam

d sin q

Reflected beam Conducting wall

Conducting wall

Mirror image of reflected beam

Mirror image of

incident beam

q

d

Figure 13.3 A conducting slab waveguide

Cladding

Core

Cladding

Figure 13.4 Electric field distribution of the lowest-order guided transversal modes in a dielectric slab waveguide

cladding This is not in contradiction with the theory of total internal reflection (see Section 9.5) which predicts an evanescent wave decaying very rapidly in the cladding material

The lowest number of modes propagating through the waveguide occurs when the

angle of incidence is equal to θa Then (assuming n0= 1 for air)

sin θa= mλ

2d =n2

1− n2

or

d

2



n21− n2 2

( 13.10)

To have only the lowest-order mode (m = 0) propagating through the waveguide, we

therefore must have

d

λ <

1 2



n21− n2 2

( 13.11)

An exact waveguide theory applied to an optical fibre is quite complicated, but the results are quite similar The condition for propagating only the lowest-order mode in an optical fibre then becomes

d

λ <

2.405 2π



n2

1− n2 2

= 2.405

2π(N A) = 0.383

A fibre allowing only the lowest-order mode to propagate is called a single-mode fibre,

in contrast to a multimode fibre which allows several propagating modes

13.3 ATTENUATION AND DISPERSION

That light will propagate through a fibre by multiple total internal reflections without loss is an idealization In reality the light will be attenuated The main contributions to

attenuation is scattering (proportional to λ−4) in the ultraviolet end of the spectrum and absorption in the infra-red end of the spectrum Therefore it is only a limited part of

ATTENUATION AND DISPERSION 311

First window

Total loss

Rayleigh scattering

Second

peak

Third window

0 0.5 1.0 1.5 2.0 2.5 3.0

Wavelength (nm)

Figure 13.5 Attenuation in a silica glass fibre versus wavelength showing the three major wave-length regions at which fibre systems are most practical (From Palais, J C (1998) Fiber Optic Communications (4th edn), Prentice Hall, Englewood Cliffs, N.J.) Reproduced by permission of Prentice Hall Inc.)

the electromagnetic spectrum where fibre systems are practical Figure 13.5 shows the attenuation as a function of wavelength for silica glass fibres Here are also shown the three major wavelength regions at which fibre systems are practical These regions are dictated by the attenuation, but also by the light sources available

Another source of loss in fibre communication systems is dispersion Dispersion is due

to the fact that the refractive index is not constant, but depends on the wavelength, i.e

n = n(λ) In fibre systems one talks about material dispersion and waveguide dispersion.

Here we will briefly mention material dispersion That the refractive index varies with wavelength means that a light pulse from a source of finite spectral width will broaden as

it propagates through the fibre due to the different velocities for the different wavelengths This effect has significant influence on the information capacity of the fibre The parameter

describing this effect is the pulse spread per unit length denoted τ/L where τ is the

difference in travel time for two extreme wavelengths of the source’s spectral distribution

through the length L This gives

τ

L = 



1

νg

( 13.13)

In dispersive media a light pulse propagates at the group velocity (Senior 1985) defined by

νg = dω

With the relations

ω = kc = 2π c

β = kn = 2π n

we get

1

νg = dβ

dω = dβ

dλ

dλ

dω =



−λ2

2π c

2π



1

λ

dn

dλ − n

λ2

= 1

c



n − λ dn dλ

( 13.16)

This gives

τ

L = 



1

νg

= 



n − λdn/dλ c

( 13.17)

The pulse spread per unit length per wavelength interval λ becomes

τ Lλ = d

dλ



1

νg

= d

dλ



n

c− λ

c

dn dλ

= −λ

c

d2n

1.45 n

(a)

0

(b)

2 / d

Wavelength Wavelength

Figure 13.6 (a) Refractive index versus wavelength for SiO2glass and (b) The second derivative

of the curve in (a)

DIFFERENT TYPES OF FIBRES 313

The material dispersion is defined as M = (λ/c)(d2n/ dλ2) The pulse spread per unit length then can be written as

τ

The refractive index for pure silicon dioxide (SiO2) glass used in optic fibres has the

wavelength dependence shown in Figure 13.6(a) At a particular wavelength λ0, there is

an inflection point on the curve Because of this, d2n/ dλ2= 0 at λ0 as seen from the curve of the second derivative in Figure 13.6(b) For pure silica, the refractive index is

close to 1.45 and the inflection point is near λ0= 1.3 µm Therefore this wavelength is

very suitable for long distance optical fibre communication

13.4 DIFFERENT TYPES OF FIBRES

Another construction than the step-index (SI) fibre sketched in Figure 13.1 is the so-called graded-index (GRIN) fibre It has a core material whose refractive index varies with distance from the fibre axis This structure is illustrated in Figure 13.7 As should be easily realized, the light rays will bend gradually and travel through a GRIN fibre in the oscillatory fashion sketched in Figure 13.7(d) As opposed to an SI fibre, the numerical aperture of a GRIN fibre decrease with radial distance from the axis For this reason, the coupling efficiency is generally higher for SI fibres than for GRIN fibres, when each has the same core size and the same fractional refractive index change

Conventionally, the size of a fibre is denoted by writing its core diameter and then its cladding diameter (both in micrometers) with a slash between them Typical dimensions

(d)

2

r

n

a 0

r

2

Figure 13.7 Graded index fibre: (a) refractive index profile; (b) end view; (c) cross-sectional view; and (d) ray paths along a GRIN fibre

of SI fibres are 50/125, 100/140 and 200/230 and typical dimensions of multimode GRIN fibres are 50/125, 62.5/125 and 85/125

SI fibres have three common forms: (1) a glass core cladded with glass, (2) a silica glass core cladded with plastic (termed plastic-cladded silica (PCS) fibres), and (3) a plastic core cladded with another plastic All-glass fibres have the lowest losses and the smallest pulse spreading, but also the smallest numerical aperture PCS fibres have higher losses and larger pulse spreads and are suitable for shorter links, normally less than a few hundred metres Their higher NA increase the coupling efficiency All-plastic fibres are used for path lengths less than a few tens of meters Their high NA gives high coupling efficiency Single-mode fibres have the highest information capacity GRIN fibres can transmit at higher information rates than SI fibres Table 13.1 shows representative numerical values

of important properties for the various fibres Somewhat different characteristics may be found when searching the manufacturers’ literature

Table 13.1 (From Palais, J C (1998) Fiber Optic Communication (4th edn), Prentice Hall,

Engle-wood Cliffs, New Jersey) Reproduced by permission

Description Core

Diameter (µm) NA (dB/km)Loss

(τ/L)

(ns/km)

Source Wavelength

(nm) Multimode

Glass

PCS

Plastic

Single mode

Polyurethane, 3.8 mm Kevlar, 2 mm Hytrel secondary buffer, 1 mm Silastic primary buffer, 0.4 mm Fibre, 0.23 mm

Figure 13.8 Light-duty, tight-buffer fibre cable (Siecor Corporation) The dimensions given are

the diameters (From Palais, J C (1998) Fiber Optic Communications (4th edn), Prentice Hall,

Englewood Cliffs, N.J.) Reproduced by permission of Prentice Hall

FIBRE-OPTIC SENSORS 315

The amount of protection against the environment of a fibre varies from one application

to another Various cable designs have been implemented A representative light-duty cable

is sketched in Figure 13.8 This cable weighs 12.5 kg/km and can withstand a tensile load

of 400 N during installation and can be loaded up to 50 N in operation

Fibre-optic communications developed very quickly after the first low-loss fibres were produced in 1970 Today, over 10 million km of fibre have been installed worldwide, numerous submarine fibre cables covering the Atlantic and Pacific oceans and many other smaller seas are operational In addition, installation of fibre-optic local area networks (LANs) is increasing

13.5 FIBRE-OPTIC SENSORS

Over the past few years, a significant number of sensors using optical fibres have been

developed (Kyuma et al 1982; Culshaw 1986; Udd (1991, 1993)) They have the potential

for sensing a variety of physical variables, such as acoustic pressure, magnetic fields, temperature, acceleration and rate of rotation Also sensors for measuring current and voltage based on polarization rotation induced by the magnetic field around conductors due to the Faraday effect in optical fibres have been developed It should also be mentioned that a lot of standard optical equipment has been redesigned using optical fibres The Laser Doppler velocimeter is an example where optical fibres have been incorporated to increase the versatility of the instrument

Figure 13.9 shows some typical examples of fibre-optic sensors In Figure 13.9(a) a thin semiconductor chip is sandwiched between two ends of fibres inside a steel pipe The light is coming through the fibre from the left and is partly absorbed by the semi-conductor This absorption is temperature-dependent and the amount of light detected

at the end of the fibre to the right is therefore proportional to the temperature and

Optical fibre

Stainless holder

Semiconductor absorber

Optical fibre (a)

Pressure plate

Fibre

To detector

Input light (b)

Figure 13.9 (a) Fibre-optic temperature sensor and (b) Fibre-optic pressure sensor

we have a fibre-optic temperature sensor Figure 13.9(b) shows a simplified sketch of

a pressure-sensing system The optical fibre is placed between two corrugated plates When pressure is applied to the plates, the light intensity transmitted by the fibre changes, owing to microbending loss Such systems have also been applied as hydrophones and accelerometers

Figure 13.10 shows the principle of a class of fibre-optic sensors based on interferome-try The fibres A and B can be regarded as either arm in a Mach–Zehnder interferometer The detector will record an intensity which is dependent on the optical path-length differ-ence through A and B When, for example, fibre A is exposed to loads such as tension, pressure, temperature, acoustical waves, etc., the optical path length of A will change and one gets a signal from the detector varying as the external load

Figure 13.11 shows a special application of optical fibres In Figure 13.11(a) two fibre bundles, A and B, are mixed together in a bundle C in such a way that every second fibre in the cross-section of C comes from, say, bundle A Figure 13.11(b) shows two neighbouring fibres, A and B Fibre A emits a conical light beam Fibre B will receive

light inside a cone of the same magnitude If a plane surface is placed a distance l in front

A

B

Light source

Detector

Figure 13.10 Interferometric fibre-optic sensor

(a)

(b)

(c)

C

IB

l

l

Figure 13.11