Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://vega.inp.nsk.su/~inest/halometer/Volpi_FiberOpticsRevAa.pdf Äàòà èçìåíåíèÿ: Sun Jun 17 15:09:49 2007 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 20:08:26 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: annular solar eclipse

Fundamentals of

Fiber Optics
An Introduction for Beginners
Reinhard Jenny, M.S. Physics

translated: Scott Kittelberger, M.S. Physics

Volpi AG Wiesenstrasse 33 CH 8952 Schlieren www.volpi.ch Volpi Manufacturing USA Co., Inc. 5 Commerce Way Auburn, NY 13210 www.volpiusa.com


Contents 1 2 2.1 2.2 2.3 2.4 Historical Overview Fundamentals of Light Propagation in Light Guides Total Internal Reflection Numerical Aperture of a Light Guide Guided Light Outside the Core Medium Phase Shift of Total Reflection

3 Light Guides - Modes 3.1 Mode Equation 3.2 Mode Number of a Light Guide 4 Light Intensity Distribution in a Light Guide

5 Fundamentals and Properties of Optical Fibers 5.1 Fiber Types 5.2 Loss Mechanisms in Fibers 5.2.1 Material Absorption 5.2.2 Material Scattering 5.2.3 Light Guide Specific Scattering Mechanisms 5.2.4 Radiation Losses due to Macrobending 5.2.5 Losses due to Microbending 5.2.6 Fiber Coupling Losses 6 Transit Time Behaviour of Light in Light Guide 6.1 Mode Dispersion 6.2.1 Material Chromatic Dispersion 6.2.2 Propagation Chromatic Dispersion 7 8 8.1 8.2 8.3 9 9.1 9.2 9.3 9.4 Emitted Mode Radiation of Fibers Properties of Fiber Bundles Emitted Radiation Characteristics of Bundles Bundle Losses Fiber Bundle Transmission Fiber Optic Illumination Transforming Light Distribution Fibers as Isolators Partitioning Luminous Flux with Fibers Fiber Optic Sensor Application

10 Overview of Common Fibers 10.1 Fiber for Producing Fiber Bundles 10.2 Single Fibers or Monofibers for Spectroscopy, Laser Light Propagation etc. 10.3 Fiber for Image Transmission

Fundamentals of Fiber Optics An Introduction for Beginners

Page - 1 -

© / Reinhard Jenny / 26.4.2000


1 Historical Overview The first attempts at guiding light on the basis of total internal reflection in a medium dates to 1841 by Daniel Colladon. He attempted to couple light from an arc lamp into a stream of water (Fig. 1). Several decades later, the medical men Roth and Reuss used glass rods to illuminate body cavities (1888). At the beginning of the 20 century light was successfully transmitted through thin glass fibers. In 1926 J.L.Baird received a patent for transmitting an image in glass rods and C.W.Hansell first began contemplating the idea of configuring an imaging bundle. In 1930 the medical student Heinrich Lamm of Munich produced the first image transmitting fiber bundle. In 1931 the first mass production of glass fibers was achieved by Owens ­ Illinois for Fiberglas. Arc Lamp Water
th

Light guided in water pipe

Fig. 1 : Historic attempt of D. Colladon to guide light in a stream of water (Geneva, 1841)

Attempts at patenting the idea of glass fibers with an enveloping clad glass was initiated by H.M.Moller in a patent by Hansell , however, refused. As a result the well-known scientists A.C.S. van Heel, Kapany and H.H.Hopkins produced the first fiber optic endoscope on the basis of fiber cladding in 1954. Curtiss developed an important requisite for the production of unclad glass fibers in 1956. He suggested that a glass rod be used as the core material with a glass tube of lower index of refraction melted to it on the outside. In 1961, E. Snitzer described the theoretical basis for very thin (several micron) fibers, which are the foundation for our current fiber optic communication network. The notion of launching light into thin films was suggested by von Karbowiak in 1963 In 1967, S. Kawakami proposed the concept of fiber whose index of refraction varied in a continuous, parabolic manner from the center to the edge (gradient index fiber). The main thrust of further activities in the development of fiber optics was in improving material quality of glass. High levels of purity were required of preform to address the enormous economic and technological potential of a worldwide communications network. 2 Fundamentals of Light Propagation in Light Guides

Fiber Optic light guides are media whose transverse dimension (diameter, thickness) can be very small, typically 10µm to 1 mm. They are very flexible and can be produced in virtually any
Fundamentals of Fiber Optics An Introduction for Beginners Page - 2 © / Reinhard Jenny / 26.4.2000


desired length. The material is usually glass, quartz or plastic. For special applications, other exotic materials such as liquid light guides, sapphire, fluoride or calcogenide may be used. There are some unavoidable requirements for good light transmission, such as pure glass materials for the core and cladding and high transparency for the spectrum of interest. Minimal optical dispersion is also desired. Process parameters such as glass transformation temperature, viscosity, inclusions and chemical affinity dictate the economics and quality of the fiber product. Light launched into a fiber will after a given length reach the core material boundary and pass to another medium (glass, air, etc.). Depending on the incident angle, some of the energy will be refracted outward (leaky modes) and some will reflect back into the core material (Figure 2). 2.1 Total Internal Reflection When the outer medium is less optically dense (lower index of refraction) than the core material, there is a distinct angle for which no light is refracted (Figure 3). Light is completely reflected back into the core material (Total Internal Reflection).

'

Refracted ray in Medium 2 ; n2 "Leaky Modes" Medium 1; n1 Interface reflected ray, guided by fiber



Incident ray Fig 2.: Light transmission in Medium with n1 > n2 Maximum light can only be transmitted through the light guide if total internal reflection occurs at the core-clad interface. In this cas e, > Min, where Min is the angle of incidence for which ' = 90°. n n2 Cladding n1
Min




Max

Core n1 > n2 n2

Fig. 3: Light transmission Medium n1 with total reflection of the transmitted ray; The light guide has a cladding material n2 ; n .. Index of refraction for the coupling medium (usually air; n = 1) 2.2 Numerical Aperture of a Light Guide Per the law of refraction, total refl ection at the core/clad interface obeys: sin(Min) = n2 / n1 (1) Max is the largest angle the fiber can accept. The Numerical Aperture, NA, of the light guide, describes this maximum angle:

Fundamentals of Fiber Optics An Introduction for Beginners

Page - 3 -

© / Reinhard Jenny / 26.4.2000


NA = n sin (Max ) = n1 - n2 (2) All angles, Max or Min, will be transmitted by the fiber with larger angles resulting as leaky modes (by refraction at the core/clad interface).
2 2

2.3 Guided Light Outside the Core Medium Clad fibers are an absolute necessity for transmitting light over long distance. If no cladding would be used, the environment (atmosphere, gases, dirt) would be the cladding material. Absorption would drastically reduce the transmitted luminous flux. One should note that for total internal reflection, a portion of the energy in the electric field penetrates medium 2 (evanescence field, Figure 4). Typically the penetration depth is 5 times the respective wavelength.

z

Cladding; Core;

n2 n1

n2 Fig. 4: The electromagnetic field penetrates the cladding glass at the point of total internal reflection. The penetration depth is z To simplify matters, subsequent mathematical descriptions of light transmission in waveguides are related to planar waveguide-configurations. Equation (3) gives the penetration depth of a electromagnetic wave (transverse) in medium n2 (for planar lightguides): n1 z = (3) 2 2 2 2 NA - n1 cos () Should reach the critical angle for total internal reflection (1), the penetration depth z becomes . If = 90°, then z = n1/(2 NA). The fact that a portion of the energy is transmitted in the cladding places certain demand on the cladding material. Further, it should be noted that the reflected wave experiences a phase shift dependent on . 2.4 Phase Shift of Total Reflection Fig. 5: Phase Shift, ,after Total Reflection

The phase shift, immediately after the reflection, causes the sine wave of the spreading ray to wander with the same periodicity (frequency) (Fig. 5). The phase shift, (), repeats every 2.
Fundamentals of Fiber Optics An Introduction for Beginners Page - 4 © / Reinhard Jenny / 26.4.2000


Figure 5 schematically shows the phase shift for transversal electric wave modes (TE -Modes): () = -2 arctan [ sin () ­ (n2/n1) cos ()
2 2

]

(4)

For the sake of completeness, it should be stated that transversal magnetic wave modes (TM) exist orthogonal to the electric wave modes (TE). Hence, for a mode number n there are two propagation wave modes (TE n and TM n). Equivalent relationship exists for (3) and (4) modified for TM wave modes. 3 Light Guide Modes 3.1 The Mode Equation A typical model for light transmission in the fiber core is a zigzag pattern. Every zigzag configuration has an angle pair, designated ( ; ), which is also called a mode. For the existence of such modes, electromagnetic wave theory requires waves to interfere constructively with each other (Light amplification by superposition). B l2 Ah



l1 C

dK

Wave Front

Fig. 6. Propagation within the light guide should produce the same amplitude at point A and point C (i.e. a maximum). The thickness of the propagating film is dK. In order to construct wave superposition, it is important that points A and C have the same wave amplitude (maximum or minimum). This is also means that over length l1+l2, the wave period and phase shift of the wave front over 2 reflections of produce the same amplitude as at point A. Expressed in phase space, this means that the phase difference between A and C be a whole multiple of 2 ( = 2m; m=0,1,2...). The phase delay over l1+l2 is therefore: dK n1cos() + 2 = 2m; for m=0,1,2.. (5) This equation is the fundamental condition for propagation of a wave in planar wave guides (thin film with cladding) of order m. This is also referred to as the Mode Equation. If definite values of dK, n1 and are inserted into (5), due to the integer values of m, the angle is not a free variable as in geometric optics. Rather a discreet series of angles m of order m results with a discret number of modes. In light guiding optics, the angle and the core index of refraction n1 are characteristic parameters for light propagation. Equation (6) formulates an associated effective index of refraction for a propagating mode: N = n1 sin ()
Fundamentals of Fiber Optics An Introduction for Beginners

=

4

where
Page - 5 -

Min < /2

(6)

© / Reinhard Jenny / 26.4.2000


Figure 7 is a graph of the first 4 TE modes in a planar wave guide as a function of core thickness. For a thickness of, for example, 2 µm only modes m = 0 to m = 3 are possible for the given parameters. As an aside, the associated TM modes are also present, but not shown. Figure 7 : Diagram of Mode existence (Dispersion-Relation) Graph of film thickness as a function of effective index of refraction (or as a function of angle where N = n1 sin() ) shows which TE modes can exist for given material and design parameters. Per (6) and (1) the effective index varies between n2 and n1. This means every N corresponds to a fixed angle . The smallest film thickness for each mode dF (= Cut-Off-Thickne ss) is also shown in the Diagram for N = n2.
Dispersion - Relations of Mode Orders 0 to 3 n1 = 1.52; n2 = 1.435; lamda= 500 nm 4 Film-Thickness (microns)

5

3

3rd order

2
2nd order

1

1st order

0 1.435

0-th order

1.455

1.475

1.495

1.515

n2

Effective Index of Refraction N

n1

3.2 Mode Number of a Light Guide If one attempts to define the angle increment, , from the mode equation (5) and (6), this describes the angle difference between modes. For N => n with => /2 and a defined film 1 thickness (note sin = n1cos ): = / (2dF ) (7)

The number of modes in a light guide can therefore be estimated based on the valid aperture angle, [ dimensions in radians; from NA = sin()], evenly distributed over the incremental angles. We therefore obtain: M 2( / ) = 4dF / (8) As shown in Fig. 7, for planar light guides and a film thickness of 2 µm, there are 8 modes (4 TE modes and 4 TM modes). For cylindrical light guides, the principle of superposition for mode propagation is practically the same as for planar light guides. The number of modes propagating in a fiber light guide is given by a configuration parameter called the V-Parameter. This calculated value is: DF ... Fiber diameter V = DF / n1 - n2 = DF NA /
2 2

(9)

For cylindrical fibers the number of modes is: 2 MFo V / 2

(10)

The same result as (10) is obtained in (12) if one takes the incremental angle difference between two neighboring modes with Fo = 2 / (DF ) (11) calculating the number of modes M as 2 MFo 2 (/Fo) (12)
Fundamentals of Fiber Optics An Introduction for Beginners Page - 6 © / Reinhard Jenny / 26.4.2000


Example: A fiber with a 50 µm core diameter and NA = 0.5 ( = 500 nm) has a V-Number V = 157.079. According to (9) - (12) there exists MFo 12337 Modes. 4 Light Intensity Distribution in a Light Guide Phenomena logically, one can interpret the intensity distribution of light for the simplest case of a th 0 order mode (m=0), ie.in a planar light guide with the superposition of two rays of the same mode. The interference of two rays produce interference stripes in the spatial superposition zone with a distance between stripes = / (2 n sin()) (13) n ... Index of refraction of propagating medium; ... Wavelength In the following picture is an example of a two ray interference pattern. S1 S2

Fig. 8: Two ray interference with interference pattern In a planar light guide, symmetrical rays of the same mode interfere, resulting in a periodic interference pattern. The intensity of the rays will be greatest in the areas of the light guide where the principle portion of wave energy in the propagating core is the greatest. In a light guide with a single propagating mode, the above described two ray interference results is a single intensity s tripe. The intensity falls off going from the core to the edge. One expects a Gaussian intensity destribution, I0 (see Fig. 9). The superposition of different modes results in an intensity distribution with one or more zero intensity nodes over the core cross section. I1 I0

Figure 9. Interference in a light guide; the superposition zone is concentrated in the light propagating volume. The exact solutions for the electromagnetic field components are t solutions of the Maxwell he equations from which the intensity distribution can be derived. For cylindrical fibers, the principle superpostition is the same as for planar light guides. Due to the cylindrical geometry, the solutions of Maxwell's equations for wave propagation are Bessel functions. Out of these sets of solutions, the field components can be determined. They are divided into 3 electric, and 3 magnetic components. They are designated as circularily symmetric modes, TE 0n and TM 0n, as well as non-circularily symmetric modes, EHlm and HE lm .

Fundamentals of Fiber Optics An Introduction for Beginners

Page - 7 -

© / Reinhard Jenny / 26.4.2000


The following table shows mode configurations and the possible number of modes. diameters and wave lengths are contained in the V number V-Number 0 - 2.4048 2.4048 - 3.8317 3.8317 ­ 5.1356 5.1356 ­ 5.5201 5.5201 ­ 6.3802 6.3802 ­ 7.0156 7.0156 ­ 7.5883 7.5883 ­ 8.4172 etc. Mode-Configuration He11 TE01, TM 01, He21 HE 12, EH11, HE 31 EH21, HE 41 TE02, TM 02, He22 EH31, HE 51 HE 13, EH12, HE 32 EH41, HE 61

Fiber

Total Number Modes 2 6 12 16 20 24 30 34

Field components and number of modes as a function of the V-Parameter From the table, one can see that for fibers with V < 2.4048, only one fundemental mode can be transmitted. The fundemental mode is comprised of two eigenmodes which differ only by their polarization. Such fibers are known as Single-Mode - or Mono-Mode -Fibers . 5 Fundamentals and Properties of Optical Fibers 5.1 Fiber Types Waveguides are classified, on the one hand by the index of refraction profile of the core material, and on the other hand by the mode propagating ability. As was previously suggested, there are therefore single mode and multimode fibers. In classifying the index of refraction profile, we differentiate between step index, gradient index and special profile fibers. Step index fibers have a constant index profile over the whole cross section. Gradient index fibers have a non-linear, rotationally symmetric index profile, which falls off from the center of the fiber outwards (Fig. 10). Fiber type Cross Section r Index n(r) Ray Propagation

Multimode ­ Step Index

Single mode Step Index

Multi mode Gradient Index n (r) = n1 ­ NA (2r/DF ) ; Figure 10: Overview of fundemental fiber types
2 2 2 g

0
Fundamentals of Fiber Optics An Introduction for Beginners

Page - 8 -

© / Reinhard Jenny / 26.4.2000


In the case of step index, multimode fibers the index of refraction is constant, therefore the profile parameter g = . For gradient index fibers, the index of refraction is reduced from the middle outwards. As opposed to travelling in a straight line, the rays travel in a spiral form around the optical axis. 5.2 Loss Mechanisms in Fibers The following effects can lead to losses in electromeagnetic energy propagating in fibers: material absorption, material scattering, waveguide scattering due to form-inhomogeneities, mode losses due to fiber bending and cladding losses. 5.2.1 Material-Absorption Absorption losses are largely due to impurities in glass material from residual foreign atomic substances and hydrogen/oxygen molecules. Lastly, there are attenuation maxima in small band wavelength regions. The fundamental attenuated wavelength (highest absorption) is due to (OH) ions. In quartz this is at = 2.7 µm. In the spectral region below this wavelength, there are other absorption bands at 1.38µm, 1.24µm, 950 nm and 720 nm. Between these wavelength bands there are "windows" of minimal attenuation. These spectral st nd rd regions are at 850nm (1 windows), at 1300 nm (2 window) and at 1550 nm (3 window). These spectral regions are used for data transmission (communication technology). Foreign substances include metal ions such as Cr , Fe and Cu . The associated absorption bands are between 500nm and 1000 nm. The bandwidth can be very different depending on the specific glass and metal ion being discussed. Attempting to transmit short wavelength light in quartz fibers (ie. UV light = 210 nm) can lead to a damage mechanism referred to as solarisation. In the quartz structure, there are absorption centers where anions (negativly charged ions) are replaced by an electron. These electrons can be excited, potenitially at resonance. These regions in the crystal are also called color centers, because the normally color neutral crystals (ie. NaCl) become characteristically discoloured. 5.2.2 Material Scattering One crucial scattering mechanism is Rayleigh Scattering. Spatially there are high density gradients (short compared to the wavelength) which alter the index of refraction and cause 4 scattering. The intensity of the scattered light is proportional to 1/ . The effect evidences itself in, among other things, strong reverse scattering. Another scattering mechanism is Mie Scattering, which mainly results in forward scattering. This mechanism comes from material inhomogeneities in larger wavelength spectrums. Stimulated Raman Scattering and Stimulated Brillouin Scattering are non-linear radiation induced effects, which exceed intensity thresholds. Transmitting laser light alone can exceed these threshold values. 5.2.3 Light Guide Specific Scattering Mechanisms So called intrinsic fiber characteristics can cause loss of energy. Some of these effects are: changes in core diameter, difference in refractive indices, index profile effects, mode coupling (double mechanisms) and scattered radiation in the cladding glass. Radiation losses can exist due to the conversion of core modes to non-porpagating modes (cladding modes). This results in a reduction in the carrying modes. Extrinsic causes for loss mechanisms come from such things as mechnicanical influences, such as micro and macrobending.
3+ 2+ 2+

Fundamentals of Fiber Optics An Introduction for Beginners

Page - 9 -

© / Reinhard Jenny / 26.4.2000


5.2.4 Radiation Losses due to Macrobending Fiber bending with a constant bend radius is referred to as macrobending. This produces at least 2 loss mechanisms: a) In multimode fibers, the number of propagating modes is reduced as a function of bend radius according to the following description: M(R) M0 (1 - DF n2 /(R NA )) M0 ... number of propagating modes without bending M(R) ... number of propagating modes with bending; n2 ... clad refractive index R ... bend radius; DF ... fiber diameter; NA ... numerical aperture The percent of light discoupled (mode leakage) is: M/M DF n2 /(R NA ) x 100 (%)
2 2 2 2

(14)

(15)

From (15) it can be seen that to minimalize mode losses, fibers with small diameters and high numerical apertures are best suited. b) An additional problem worth mentioning in bent fibers is electromagnetic radiation loss by differences in propagation (wave front) velocity The main portion of electromagnetic energy is concentrated in the fiber core, while other portions are transmitted in the cladding and a slight amount outside the cladding. In bending the fiber with bending radius R, the light will move with the mediums propagation velocity. In the fiber cross section , the area radially further from the radius center will need to move with a greater velocity than that of the fiber core to maintain the signal transport speed. At reaching a critical value, zkr, a barrier is reached. The speed of light in a medium can not exceed its natural value c = c0/nM (with c0 = 2.9979 108 m/s and nM .. index of refraction for Medium M). The transport velocity lies beyond this point. Seeing as the signal velocity no longer exists, light can no longer be transmitted in this configuration and relevant portion of the energy radiates into the surroundings. Figure 11: Bent fiber with bend radius R. z
kr

z

Field F(z)

The field on the far side of the center bend radius reaches the speed of light at distance zkr. As a result, light is radiated.

R M In this way, losses exist resulting in higher attenuation. For multimode fibers, this effect is relatively small when compared to the effect described in a). However, this type of attenuation does more seriously affect single mode fibers as bending is applied. For single mode fibers the reduction coefficient is calculated by: B = (c 1/R) exp (-c 2R) (16)

c 1, c 2 ... constants depending on fiber manufacture and wavelength. The stronger the electromagnetic field of transmitted modes out of the core, the more pronounced this effect.

Fundamentals of Fiber Optics An Introduction for Beginners

Page - 10 -

© / Reinhard Jenny / 26.4.2000


Modes of longer wavelength lead to larger field expansion, which should be taken into account in given cable configurations. 5.2.5 Losses due to microbending Along the length of the fiber, periodic or statistically distributed locations of curvatures occur, whose magnitude continously varies. The associated loss mechnism is mainly exhibited by a permanent transformation of the transmitted mode. 5.2.6 Fiber Coupling Losses Cleaved single fibers may be spliced. The splicing region can exhibit intrinsic (purely optical) and extrinsic (mechnical alignment) losses. The following diagram shows various configurations and transmission values for multimode fibers with cleaved terminations.

d1 differing fiber diameters NA1 differing fiber apertures g1

d2

T = (d2/d1)² d2 d1

NA2

T = (NA2 / NA1)² NA2 NA1

g2

T = (g2/g1){(g1+2)/(g2+2)}

differing index of refraction profiles d m d T = {(2/ )arccos(m/d) ­ 2 1/2 - (2m/d)(1-(m/d) ) }

transverse misalignment (m) d axial gap (v) d angular misalignment () n1 n1 v d d

T = {1 ­ (v/d) tan(c ) } sin(c ) = NA

T {1 ­ (16/3) (sin(/2)/sin(c ) }

n2 surface reflections

T = {1 ­ (n1 ­ n2) / (n1 + n2) }

2

2

Fundamentals of Fiber Optics An Introduction for Beginners

Page - 11 -

© / Reinhard Jenny / 26.4.2000


6 Transit Time Behaviour of Light in Light Guide When short pulses of light energy are coupled into a fiber, the time behaviour is strongly influenced by the fiber type, as well a the core and cladding materials. Transit time differences will s lead to a limitation of bandwidth, chiefly for telecommunication technology. In light transmission technology this effect is called dispersion. There are 2 basic types of dispersion: a) Mode dispersion b) Chromatic Dispersion : Material dispersion Light guide dispersion 6.1 Mode dispersion Mode dispersion comes from differing transit times for different modes due to differing optical paths (zig-zag patterns multiplied by index of refraction). This is obviously only for multimode fibers. Single mode fibers propagate only one mode, which has only one path and therefore no path difference or transit time difference. Gradient index fibers theoretically have the same optical path for all modes. Due to the decreasing index of refraction from the core to the cladding interface, light rays travel faster the closer they are to the interface. So different modes travel in different spiral paths. Lower modes have a shorter path, as they propagate nearer the optical axis, but are also in a larger index material. Higher mode modes have a longer path length, but travel in lower index material. The product of "path length and index of refraction" is constant. Transit time differences are therefore greatly compensated for in gradient index fibers. 6.2.1 Chromatic Dispersion ­ Material Dispersion The refractive indices of the core and cladding are wavelength dependent. This means the differing wavelengths travel in the same medium with differing refractive indices. As the velocity in the medium is given by v( ) = c/n() (c ... speed of light in vacuum, n ... refractive index of medium), it varies with varying wavelength. A light pulse with spectral bandwidth , leads to a transit time difference t. In digital communication, lasers generating short pulses are used. 6.2.2 Chromatic Dispersion - Light Guide Dispersion As a result of the differing refractive indices between core and cladding, and their associated wavelength dependence, light in light guides travels with differing velocities. Together with material effects, a light guide will have spectral transit time variations. Careful material selection can limit transit time differences for specific wavelength regions. 7 Emitted Mode Radiation of Fibe rs

In principle, the radiated energy of a fiber is enclosed in the fibers aperture angle 2 (Fig.12). Loss mechanism, which reduce the number of modes in the fiber core (ie. Macrobending), limit the presence of higher modes. The radiated angle is therefore smaller than the specified aperture angle. In bent cylindrical fibers, the first order approximation of the effective numerical aperture, NA*, is from (9), (10) and (14): 2 2 2 NA* = sin(*) = n1 ­ n2 (1+ DFo /2R) (18) Various scattering processes in the core (Rayleigh or Mie Scattering) can convert a portion of the light power from lower modes to higher modes and therefore "fill" the specified numerical aperture.Generally, the rule for the radiated angle characteristics of straight multimode fibers is: output light aperture, NA out , equal or less than that of the light coupled in the fiber, NA in, provided

Fundamentals of Fiber Optics An Introduction for Beginners

Page - 12 -

© / Reinhard Jenny / 26.4.2000


NA in does not exceed the aperture of the fiber NA radiating multimode fiber are shown. NA
a) Fiber

Fiber

. In Fig. 12, the essential charcteristics for

NA Entrance Exit 2

out

NA b)

NA out = NA in for NA in NA

Fiber

in



'

NA

out

NA '

in

c)

2

2

'

NA

out

NA '

in

d)



d1

Fiber ­ Taper (Cone) d2

'

NA

out

> NA

in

d1 sin () = d2 sin (') NA e) NA Bent Fiber R
out

2

2

2

2

out

< NA

in

NA

ein

Figrue 12. Relation of a fibers radiated angle to different input angle distributions. In case b) asymmetric input coupling leads to symmetric output coupling. In d) a fiber with varying cross section (taper) changes the numerical aperture 8 Characteristics of Fiber Bundles Many fibers gathered together into a bundle of diameter d , will maintain the essential properties B of single fibers. 8.1 Emitted Radiation Characteristics of Bundles A fiber bundle displays the same characteristics with regard to emitted radiation as a single fiber. Skewing the fiber arrangment off the bundle axis can influence radiated emission characteristics. Twisting fibers in the outer part of the bundle cross section can accept an inhomogenous intput distribution and create an even output distribution (Fig. 13b)

Fundamentals of Fiber Optics An Introduction for Beginners

Page - 13 -

© / Reinhard Jenny / 26.4.2000


Fiber bundle a) 2 2

b)

2 Skewed fibers Fibers arranged parallel

Figure 13: Various output characteristics with the same input. This is possible because the direction of skewed fibers is close to the direction of the coupled light at the input. As a consequence, after refraction and the bundle entrance, rays coupled into the fiber can propagate with the same orientation as the optical axis. The light in skewed fibers has a maximum intensity at 0°,ie on the fiber axis. Seeing as the fibers at the bundle exit are parallel to the bundle axis, the otherwis expected intensity minimum along the bundle axis is compensated for (Fig. 13a). 8.2 Losses in Fiber Bundles In principle, all previously discussed loss mechanisms in single fibers apply to fiber bundles. The fibers are closely packed and epoxied together. This leads to 2 additional losses, which are only relevant to bundles: interstitial spacing and cladding losses. In Fig. 14, both these spatial transmission losses (interstitial and cladding losses) are depicted Interstitial spaces (Epoxy); area => ( ca. 10 %)

Reflection losses At fiber entrance and exit (ca. 2 x 5%)

Fiber core DCore dFiber

Fiber Cladding Surface => (10 ­ 25 %) Figure 14. Cross section of a fiber bundle. Diagram of bundle specific sources of losses
Fundamentals of Fiber Optics An Introduction for Beginners Page - 14 © / Reinhard Jenny / 26.4.2000


A special process of ther