The success of the wave theory was remarkable, and it led to a number of important devices, some of which are described in this chapter. He did this by expressing the then-known laws of electromagnetism in such a way as to allow him to derive from them a wave equation see Appendix A. This wave equation permitted free-space solutions that corresponded to electro1 2 Polarization in Optical Fibers magnetic waves with a velocity equal to the known experimental value of the velocity of light. The consequent recognition of light as an electromagnetic phenomenon was probably the single most important advance in the progression of its understanding.
The relation 1. These materials belong to the class known as dielectrics and many are electrical insulators. Thus we may write 1. This polarization depends on the mobility of the electrons, within the atom or molecule, in the face of resistance by molecular forces. Thus, 1. The total energy flowing across unit area in unit time in the direction Oz will be that contained within a volume c m3, where c is the wave velocity.
Clearly I is proportional to the square of the electric field amplitude and also, from 1. The quantity I has SI units of watts. As is to be expected, in some more exotic cases e. This result should be noted for reference to the case of evanescent waves, which will be considered later. Note that I is proportional 2 to E 0 : this is an important relationship which will be re-emphasized shortly.
First, it relates a quantity that is directly measurable I with one which is not E 0. Second, it provides the actual numerical relationship between I and E 0 , and this is valuable when designing devices and systems, as we shall discover later. This topic will be dealt with much more comprehensively in Chapter 3. Thus, we are able to describe this solution completely by means of two waves: one in which the electric field lies entirely in the xz plane, and the other in which it lies entirely in the yz plane Figure 1.
E x is said to be linearly polarized in the direction Ox, and E y linearly polarized in the direction Oy. This corresponds to the condition that E x and E y are either in phase or in antiphase. The polarization properties of light waves are especially important for propagation within anisotropic media, in which the physical properties vary with direction. The polarization state of the light will now become dependent upon the propagation distance, and on the state of the medium. This, also, will be covered in detail in Chapter 3. These represent plane waves traveling in the Oz direction.
We shall now investigate the behavior of such waves, with particular regard to the effects that occur at the boundaries between different optical media. The Wave Theory of Light 9 Of course, other types of solution are also possible. Remember that intensity is proportional to the square of the amplitude. It is interesting and valuable to note that the propagation of a plane wave such as in Figure 1. On a given wavefront the waves at each point begin in phase this is the definition of a wavefront , so that they remain strictly in phase only in a direction at right angles to the front Figure 1.
Hence the plane wave appears to propagate in that direction. This principle of equivalence, first enunciated by Huygens and later shown by Kirchhoff to be mathematically sound , is very useful in the study of wave propagation phenomena generally. The laws of reflection and refraction were first formulated in terms of rays of light. It had been noticed around that, when dealing with point sources, the light passed through apertures consistently with the view that it was composed of rays traveling in straight lines from the point.
The practical concept was legitimized by allowing such light to pass through a small hole so as to Figure 1. Such rays were produced, and their behavior in respect of reflection and refraction at material boundaries was formulated, thus: 1. On reflection at a boundary between two media, the reflected ray lies in the same plane as that of the incident ray and the normal to the boundary at the point of incidence the plane of incidence ; the angle of reflection equals the angle of incidence.
These two laws form the basis of what is known as geometrical optics, or, ray optics. The majority of bulk optics e. However, it has severe limitations. For example, it cannot predict the intensities of the refracted and reflected rays. If, in the attempt to isolate a ray of light of increasing fineness, the aperture is made too small, the ray divergence appears to increase, rather than diminish. This occurs when the aperture size becomes comparable with the wavelength of the light, and it is under this condition that the geometrical theory breaks down.
Diffraction has occurred and this is, quintessentially, a wave phenomenon. The wave theory provides a more complete, but necessarily more complex, view of light propagation. We shall now deal with the phenomena of reflection and refraction using the wave theory, but we should remember that, under certain conditions apertures much larger than the wavelength , the ray theory is useful for its simplicity: a wave can be replaced by a set of rays in the direction of propagation, normal to surfaces of constant phase, and obeying simple geometrical rules.
After striking the boundary there will, in general, be a reflected and a refracted transmitted, t wave. This fact is a direct consequence of the boundary conditions that must be satisfied at the interface between the two media. Tangential components of E and H are continuous across the boundary. Normal components of B and D are continuous across the boundary. The above conditions must be true at all times and at all places on the boundary plane.
They can only be true at all times at a given point if the 12 Polarization in Optical Fibers frequencies of all the waves i. Furthermore, since the phase and amplitude of the incident wave must be constant on the boundary plane along any line for which x is constant see Figure 1. To go further it is necessary to give proper mathematical expression to the waves. Any given wave is, of course, a sinusoid, whose amplitude, frequency and phase define the wave completely, and the most convenient representation of such waves is via their complex exponential form.
Suppose Figure 1. To do this we match the components of E, H, D, B, separately. A further complication is that the values of these quantities at the boundary will depend on the direction of vibration of the E, H, fields of the incident wave, relative to the plane of the wave. Therefore, we need to consider two linear, orthogonal polarization components separately, one in the xz plane, the other normal to it. Any other polarization state can be resolved into these two linear components, so that our solution will be complete.
Let us consider the two stated linear components in turn. E in the plane of incidence; H normal to the plane of incidence The incident wave can now be written in the form see Figure 1. Having done this we can impose the boundary conditions to obtain the required relationships between wave amplitudes. We shall now derive these relationships—that is, that between the reflected and incident electric field amplitudes, and that between the refracted and incident electric field amplitudes for this case. We know that the exponential factors are all identical at the boundary if we are going to be able to satisfy the boundary conditions at all; let us, therefore, write the universal exponential factor as F.
For the incident i wave, from 1. The equations contain several points worthy of emphasis. First, we note that there is a possibility of eliminating the reflected wave. For E in the plane of incidence, we find from 1. It is instructive to understand the physical reason for the disappearance of the reflected ray at this angle when the electric field lies in the plane of incidence.
Referring to Figure 1. Hence these oscillations cannot generate any transverse waves in the required direction of reflection. Since light waves are, by their very nature, transverse, the reflected ray must be absent. If we ask the same question of the polarization that has E normal to the plane of incidence, we find from 1. If, then, a wave of arbitrary polarization is incident on the boundary at the Brewster angle, only the polarization with E normal to the plane of incidence is reflected.
This is a useful way of linearly polarizing a wave. The second point worthy of emphasis is the condition at normal incidence. Equations 1. This figure can be reduced by antireflection coatings. We shall now look at a rather different type of reflection where this is not the case. We now have, from 1. This represents a wave traveling in the Ox direction in the second medium i. Hence the amplitude of the wave in the second medium is reduced by a factor of 5. The wave is called an evanescent wave. Even though the evanescent wave is propagating in the second medium, it transports no light energy in a direction normal to the boundary.
All the light is totally internally reflected at the boundary. The fields which exist in the second medium give a Poynting vector which averages to zero in this direction, over one oscillation period of the light wave. All the energy in the evanescent wave is transported parallel to the boundary between the two media. The totally internally reflected wave now suffers a phase change which depends both on the angle of incidence and on the polarization.
Taking 1. This is consistent with the incident ray being reflected from a parallel plane that lies a short distance within the second boundary Figure 1. This view is also consistent with the observed phase shift, which now is regarded as being due to the extra optical path traveled by the ray. The displacement is known as the Goos-Hanchen effect and provides an entirely consistent alternative explanation of TIR. This provides food for further interesting thoughts, which we shall not pursue since they are somewhat beyond the scope of this book.
We know that these fields are vector fields since they represent forces on unit charge and unit magnetic pole, respectively.
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The fields will thus add vectorially. Consequently, when two light waves are superimposed on each other, we obtain the resultant by constructing their vector sum at each point in time and space, and this fact has already been used in consideration of the polarization of light Section 1. If two sinusoids are added, the result is another sinusoid. Consider now the arrangement shown in Figure 1. The portions of the wave that pass through the slits will interfere on the screen S, a distance d away.
Now each of the slits will act as a source of cylindrical s Incident plane wave p d Screen S Figure 1. Moreover, since they originate from the same plane wave, they will start in phase. These variations will be viewed as fringes i. Such interference is an essential feature of any wave motion, and light interference effects and phenomena pervade the whole of optical physics, phenomena which can be used in a wide variety of complex and, sometimes, quite subtle ways. In Section 1. Consequently wavefronts can interfere with themselves and with other, separate, wavefronts. To the former usually is attached the name diffraction, and to the latter interference, but the distinction is somewhat arbitrary and, in several cases, far from clear-cut.
Diffraction of light may be regarded as the limiting case of multiple interference as the source spacings become infinitesimally small. Consider the slit aperture in Figure 1. This slit is illuminated with a uniform plane wave and the light which passes through the slit is observed on a screen which is sufficiently distant from the slit for the light which falls upon it to be effectively, again, a plane wave.
These are the conditions for Fraunhofer diffraction. If source and screen are close enough to the slit for the waves not to be plane, we have a more complex situation, known as Fresnel diffraction. Fraunhofer diffraction is by far the more important of the two, and is the only form of diffraction we shall deal with here. Fresnel diffraction usually can be transformed into Fraunhofer diffraction, in any case, by the use of lenses which render the waves effectively plane, even over short distances.
Suppose that in Figure 1. This is an important result. Let us see how this works for some simple cases. Take first a uniformly illuminated slit of width s. This variation is shown in Figure 1. This form of variation occurs frequently in physics across a broad range of applications and it is instructive to understand why. In this case each infinitesimal element of the slit provides a wave amplitude adx and at the center of the screen all of these elements are in phase, producing a total amplitude, as.
Hence it is possible to represent all these elementary vectors as a straight line since they are all in phase of length as [Figure 1. The vector addition of all the vectors thus leads to a resultant that is the chord across the arc in Figure 1. Now the total length of the arc is the same as that of the straight line when all vectors were in phase i. The reason for the ubiquity of this variation in physics can now be seen to be due to the fact that one very often encounters situations where there is a systematically increasing phase difference among a large number of infinitesimal vector quantities: optical interference, electron interference, mass spectrometer energies, particle scattering, and so on.
The principles which lead to the sinc function are all exactly the same, and are those which have just been described. Let us return now to the intensity diffraction pattern for a slit. This is an important determinant of general behavior in optical systems. As a second example, consider a sinusoidal variation of amplitude over the aperture. Thus, the diffraction pattern consists of just two lines of intensity equally spaced about the center position of the observing screen Figure 1. Consequently, it will not comprise the original aperture, which must have positive and negative amplitude in order to yield just two lines in its diffraction pattern.
Thus, while this example illustrates well the strong relationship that exists between the two functions, it also serves to emphasize that the relationship is between the amplitude functions, while the observed diffraction pattern is in the absence of special arrangements the intensity function. Finally, we consider one of the most important examples of all: a rectangular-wave aperture amplitude function. The function is shown in Figure 1.
This is equivalent to a set of narrow slits i. If the aperture function extended to infinity in each direction then the Diffracted intensity f x Aperture function Figure 1. To fix these ideas, consider a grating of N slits, each of width d, and separated by distance s. Clearly each wavelength present in the light incident on a diffraction grating will produce its own separate diffraction 32 Polarization in Optical Fibers pattern.
This fact is used to analyze the spectrum of incident light, and also to select and measure specific component wavelengths. It has already been noted that refractive index is dependent upon the optical frequency. These oscillations then radiate their own, secondary, radiation. The extent of this interaction depends upon the relationship between the frequency of the primary, driving, wave and the fixed frequencies of the atomic resonances.
The atomic oscillators scatter some of the power in the primary wave away from the forward-propagating direction. They also absorb some of it, this component being redistributed as a heating of the material. Both scattering and absorption processes thus lead to attenuation of the primary wave. Another component of the secondary radiation propagates in the forward direction, combining with the primary wave to produce a resultant forwardpropagating wave.
This phase change is equivalent to a velocity difference, and this defines the refractive index. The strength of all of these effects is greatest when the frequency of the driving wave coincides with that of an atomic resonance. Hence this also will be the point at which the refractive index is changing most rapidly with optical frequency. All real sources of light provide their radiation over a range of frequencies. This range is large for an incandescent radiator such as a light bulb, and very small for a gas laser; but it can never be zero. Consequently, in the cases of a medium whose refractive index varies with frequency, different portions of the source spectrum will travel at different velocities and thus will experience different refractive indices.
This causes dispersion of the light energy, and the medium is thus said to be optically dispersive. The phenomenon has a number of manifestations and practical consequences. In the modern idiom of present-day optoelectronics we are rather more concerned with the effect that dispersion has on the information carried by a light beam, especially a guided one; so it is useful to quantify the dispersion effect with this in mind.
The Wave Theory of Light 35 which is the mean velocity of the two waves. This generalizes to be true for a continuous spread of frequencies, over a small range, such as would be emitted by any practical light source. These ideas may readily be generalized to include the complete spectrum of a practical source. In classical i. These corresponded to the electromagnetic wave frequencies that the atom was able to emit when excited into oscillation.
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Conversely, when light radiation at any of these frequencies fell upon the atom, the atom was able to absorb energy from the radiation in the way of all classical resonant system driving-force interactions. However, these ideas are incapable of explaining why, in a gas discharge, some frequencies which are emitted by the gas are not also absorbed by it under quiescent conditions; neither can it explain why, in the photoelectric effect where electrons are ejected from atoms by the interaction with light radiation , the energy with which the electrons are ejected depends not on the intensity of the light, but only on its frequency.
We know that the explanation of those facts is that atoms and molecules can exist only at discrete energy levels. These energy levels may be listed in order of ascending magnitude, E 0 , E 1 , E 2 ,. In this context we must think of the light radiation as a stream of photons. Similarly, any other quantity defined within the wave context also has its counterpart in the particulate context.
In attempting to reconcile the two views, the electromagnetic wave should be regarded as a probability function whose intensity at any point in space defines the probability of finding a photon there. But only in the specialized study of quantum optics are such concepts of real practical significance. For almost all other purposes including the present one either the wave representation or the particle representation is appropriate in any given practical situation, without any mutual contradiction.
We analyze polarization phenomena via the effects that polarization elements have on the optical powers. The processes which enable light powers to be measured accurately depend directly upon the effects which occur when photons strike matter. In most cases of quantitative measurement the processes rely on the photon to raise an electron to a state where it can be observed directly as an electric current.
We shall not deal in detail with the solid-state physics of photodetection; this is covered in many excellent specialized texts see, for example, . However, the essentials can be readily appreciated, as follows. An optical wave arrives at a photodetector as a random stream of particles photons , obeying usually Poisson statistics. The physical contact between these two types of semiconductor i.
The result is to establish an electric field across the junction as a consequence of the charge polarization. Suppose now that a photon is incident upon the region of the semiconductor exposed to this field. If this photon has sufficient energy to create an electron-hole pair, these two new charge carriers will be swept quickly in opposite directions across the junction to give rise to an electric current which can then be measured. This simple picture of the process enables us to establish two important relationships appropriate to such photodetection devices which are called photodiodes.
A typical spectrum for a silicon photodiode is shown in Figure 1. Metal contacts The Wave Theory of Light 39 0. Thus the current is proportional to the optical power. This means that the electrical power is proportional to the square of the optical power. It is important, therefore, when specifying the signal-to-noise ratio for a detection process, to be sure about whether the ratio is stated in terms of electrical or optical power.
This is a fairly common source of confusion in the specification of detector noise performance. Hence optical power is measured via a measurement of the electric current to which it gives rise in a photodiode. Using the wave description, we have seen in this chapter how it is possible to explain satisfactorily the phenomena of reflection, refraction, interference, and diffraction. We noted that the light wave is comprised of field vibrations that take place transversely to the propagation direction; we have touched only briefly, however, on the effects that depend upon the particular transverse direction in which this takes place—that is, upon the polarization state of the light.
In order to measure light powers, we need to invoke its corpuscular properties. A flow of corpuscles photons gives rise, in a photodetector, to a stream of electrons that is then measured as an electric current. References  Bleaney, B. Bleaney, Electricity and Magnetism, Oxford, U.
Wolf, Principles of Optics, 5th ed. Selected Bibliography Born, M. Guenther, R. Hecht, E. Lipson, S. Lipson, Optical Physics, Cambridge, U. The natural tendency for light from a localized source to spread in space, via the phenomenon of diffraction, implies an intrinsic lack of control over the final destination of the optical disturbance. In order to effectively manipulate light at any level of sophistication, such control is clearly required, and this is obtained by means of waveguiding.
Waveguides are physical channels that restrict and configure the physical paths which the light can take, and allow a defined passage between given points in space. The principles of waveguiding rely on just those wave properties of refraction, interference, polarization and total internal reflection that were established in Chapter 1.
In order to make use of these principles for the design and application of practical optical waveguides, it is necessary as always to develop an appropriate mathematical description of waveguiding action. This is the task on which we now embark. Here we have an infinite in width and length dielectric slab of refractive index n 1 , sandwiched between two other infinite slabs each of refractive index n 2.
This is the easiest arrangement to analyze mathematically, yet it illustrates all the important principles. Since the wave is thus confined to the first medium, it is said to be guided by the structure, which is consequently called a waveguide. Let us, firstly, consider guided light which is linearly polarized normal to the plane of incidence. The electric field of the wave represented by ray i see Figure 2. These two waves will be superimposed on each other and will thus interfere.
In order to progress down the guide indefinitely, the waves from the successive boundaries must interfere constructively, forming what is essentially a continuous, stable, interference pattern down the guiding channel. Equation 2. The allowed interference patterns are called the modes of the waveguide, for they are determined by the properties geometrical and physical of the guide. Secondly, however, if we choose to interpret 2. In fact, 2. To obtain this, let us start with 2. This we might have expected from the physics, since the propagation lies partly in the guide and partly in the outer medium evanescent wave.
We shall be returning to this point later. For a given value of m i. Hence, in order to proceed further, this dependence must be considered. The expressions for the phase changes which occur under TIR at a given angle were derived in Section 1. The terms refer, of course, to the direction of the stated fields with respect to the plane of incidence of the ray.
We can use 2. The solutions of the equations can be separated into odd and even types according to whether m is odd or even. If, on the same axes, we also plot the function aq tan aq, then 2. For obvious reasons these are called dispersion curves, and are important determinants of waveguide behavior. The three lowest order modes for a typical slab waveguide are shown in Figure 2.
A final point of great importance should be made. This is also apparent in Figure 2. Clearly the number of possible modes depends upon the waveguide parameters a, n 1 , and n 2. However, it is also clear that there will always be at least one mode, since the circle will always intersect the tan curve at one point, even for a vanishingly small circle radius. If there is only one solution, for any value of m, then Figure 2. It is given the symbol V and is called the normalized frequency, or, quite often, simply the V number.
It represents an important case, since the existence of just one mode in a waveguide simplifies considerably the behavior of radiation within it, and thus facilitates its use in, for example, the transmission of information along it. Physically, 2. Clearly, a very similar analysis can be performed for the TM modes, using 2. Look again now at Figure 2. It is shown that there are waves traveling in the outer media with amplitudes falling off the farther we go from the central channel, for the intensity of the interference pattern is nonzero in the cladding. This matter was dealt with in Section 1.
We know from 2. The answer to this question was given in Section 1. It is a fact of any physical analysis that all parameters of mathematical importance will always have a simple physical meaning. So the evanescent waves are waves that propagate in the outer media parallel with the boundary but with amplitude falling off exponentially with distance from the boundary. These evanescent waves are very important. Secondly, since energy is traveling in the Oz direction! We shall consider these aspects in more detail in Section 2. In this, waves are guided by planar channels and are processed in a variety of ways.
An example is shown in Figure 2. This is an electro-optic modulator, a device that modifies the power of light passing through it proportionally to an applied electric field. However, the electric field is acting on a waveguide which, in this case, is a channel such as we have just been considering surrounded by outer slabs called here a substrate.
The electric field is imposed by means of the two substrate electrodes, and the interaction path is under close control, as a result of the waveguiding. The material of which both the substrate and the waveguide are made should, in this case, clearly be an electro-optic material, such as lithium tantalate LiTaO3. Many other functions are possible using suitable materials, geometries, and field influences. One of the advantages of this integrated optics technology is that the structures can be produced to high manufactured tolerance by mass-production d Electrodes L Substrate Guide Figure 2.
Optical Waveguiding 53 methods, allowing them to be produced cheaply if required numbers are large, as is likely to be the case in optical telecommunications, for example. A potentially very powerful development is that of the Optoelectronic Integrated Circuit OEIC which combines optical waveguide functions with electronic ones such as optical source control, photodetection and signal processing, again on a single, planar, readily-manufacturable chip.
Note, finally, that in Figure 2. This is thus an example of an asymmetrical planar waveguide, the analysis of which is more complex than the symmetrical one that we have considered. However, the basic principles are the same; the mathematics is just more cumbersome, and is covered in many other texts see, for example, .
This is the geometry of the optical fiber, the central region being known as the core and the outer region as the cladding. In this case the same basic principles apply as for the dielectric slab, but the circular, rather than planar, symmetry complicates the mathematics. The mathematical manipulations are tedious, but are somewhat eased by using the so-called weakly guiding approximation. The ray must bounce down the core almost at grazing incidence. This means that the wave is very nearly a transverse wave, with very small z components of electric and magnetic fields.
By neglecting the longitudinal components H z , E z , a considerable simplification of the mathematics results [Figure 2. Since the wave is, to a first approximation, transverse, it can be resolved conveniently into two linearly polarized components, just as for free space propagation. Optical Waveguiding 57 2. Some of the low-order LP modes of intensity distribution are shown in Figure 2. There are, then, two possible linearly polarized optical fiber modes.
For the cylindrical geometry the single-mode condition is [analogously to 2. Some important practical features of optical-fiber design can be appreciated from the above two equations. First, the material of which an optical fiber is made must be transparent to optical wavelengths. Glass, which consists largely of fused silica SiO2 usually is chosen, for reasons which will be elaborated upon in Section 2. Silica has a refractive index of 1. Further important features can best be appreciated by reversion to geometrical ray optics.
As an example, let us consider, first, the problem of launching light into the fiber. Referring to Figure 2. The discrete values of reflection angle which are allowed by the transverse resonance condition within the TIR condition can be represented by the ray propagations shown in Figure 2. This makes clear that for a large number of allowable rays i. It is also clear, however, geometrically, that the rays will progress down the guide at velocities which depend on their angles of reflection: the smaller the angle, the smaller the velocity.
This leads to large modal dispersion at large NA since, if the launched light energy is distributed among many modes, the differing velocities will lead to varying times of arrival of the energy components at the far end of the fiber. This is undesirable in, for example, communications applications, since it will lead to a limitation on the communications bandwidth. In a digital system, a pulse cannot be allowed to spread into the pulses before or after it. For greatest bandwidth only one mode should be allowed, and this requires a small NA.
Thus a balance must be struck between good signal level large NA and large signal bandwidth small NA. A fiber design which attempts to attain a better-balanced position between these is shown in Figure 2. This fiber is known as graded-index GI fiber and it possesses a core refractive index profile that falls off parabolically approximately from its peak value on the axis. This profile constitutes, effectively, a continuous convex lens, which allows large acceptance angle while limiting the number of allowable modes to a relatively small value. GI fiber is used widely in short and medium distance communications systems.
For trunk systems single-mode fiber is invariably used, however. This ensures that the modal dispersion is entirely absent, thus removing this limitation on bandwidth. Single-mode fiber possesses a communications bandwidth that is an order of magnitude greater than that of multimode fiber. However, it is not without its problems. It is time now to deal with the communications application for optical fiber in a more coherent fashion. The basic arrangement for an optical-fiber communications system is shown in Figure 2.
A laser source provides light that is modulated by the information required to be transmitted, the information being in the form of Optical Waveguiding 61 Optical fiber Output signal Launch optics Optical source Optical modulator Photodetector Signal Figure 2. This light is then launched into an optical fiber that guides it to its destination. At the destination the light emerges from the fiber and falls onto a photodetector that converts it into an electrical signal. This electrical signal will be a close reproduction of that which was used to modulate the laser source: the closer it is, the better is the communications link.
The primary advantage of such an optical arrangement is the enormous communications bandwidth that it offers, for bandwidth is equivalent to information-carrying capacity.
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The higher the frequency of the carrier, the smaller the relative effect of a given modulation bandwidth, for any modulation signal will spread the carrier signal over a band at least equal to the modulation bandwidth. The smaller perturbation of the carrier frequency means that the properties of all the components in the communications system are substantially constant over the transmission bandwidth, and this applies especially to the transmission medium. For, if the medium acts differently for different frequencies over the modulation bandwidth, the information becomes distorted, and the communications link performance degrades.
Hence, at optical frequencies, using opticalfiber waveguides, very high-bandwidth systems can be expected. Digital systems are thus very robust in terms of signal level, the only requirement being that the level should be above a certain threshold, but they do require more bandwidth than analog systems. Opticalfiber communications systems readily provide this. However, even optical fibers both attenuate and distort the transmitted signals to some extent. It is necessary to understand the processes that lead to attenuation and distortion in fibers in order to get the best from them for communications purposes.
These are the subjects for the next two sections. Having removed these, the problem of the O-H resonance remained: this was the result of residual water in the structure, and it proved very difficult to remove. Clearly, under these conditions, the larger the optical wavelength the smaller the frequency , the smaller will be the attenuation, and the better will be the communications link. Thus 1. Losses as low as 0. However, there are considerations other than just attenuation to consider in the choice of the working communications wavelength.
One of these is the availability of suitable sources. Another problem is that of the dispersion. We shall now take a closer look at this last feature. Clearly, any optical energy propagating in a material medium will comprise a range of wavelengths. It is not possible to devise a source of radiation that has zero spectral width. Consequently, in the face of optical dispersion in the medium, different parts of the propagating energy will travel at different velocities; and if that energy is carrying information i. The further it travels, the greater will be the distortion; the greater the wavelength spread, the greater will be the distortion; the greater the dispersion power of the medium, again, the greater will be the distortion.
For good communications we need, therefore, to choose our sources, wavelengths, and materials very carefully, and in order to make these choices we must understand the processes involved. The effect of dispersion in a waveguide is to limit its communicationscarrying capacity i. This is seen most readily by considering a digital communications system—that is, one which transmits information by means of a stream of pulses Figure 2.
A stream of clear, distinct pulses is launched into the fiber for example by modulating a laser source. As the pulses propagate down the fiber, the spread of optical wavelengths of which Optical Waveguiding 65 Input pulses Distinguishable pulses After a distance L 2L Scarcely distinguishable pulses Pulse spreading Interference 3L Distance along fiber Figure 2. When the broadening has become so great that it is no longer possible to distinguish between two successive pulses, the communications link fails.
Clearly, for a given dispersive power, the broadening will increase linearly with distance. Let us look now at the particular causes of dispersion in optical fibers. This dispersion exists only in multimode fibers, since it results from the differing velocities of the range of modes supported by the fiber. Optical energy is launched into the fiber and will be launched into many, perhaps all, of the modes supported by the fiber. The effect of modal dispersion clearly will depend on how the propagating energy is distributed among the possible modes, and this will vary along the fiber as the energy redistributes itself according to local conditions e.
In order to get a feel for its order of magnitude, however, we can very easily calculate the difference in time of flight, over a given distance, between the fastest and slowest modes supported by the fiber. The fastest mode will be that which travels almost straight down the fiber, along the axis Figure 2.
Clearly Figure 2. Multimode fiber clearly is seriously limited in its bandwidth capability. It is instructive also to relate B to the amount of optical power which can be launched into the fiber from a given source. From Section 2. We can see from Figure 2. Hence the greater the value of NA 2, the greater will be the launched power. If we also assume that the noise on the received signal is independent of fiber length a fair assumption since almost all the noise will be shot and thermal noise generated in the receiver , then it follows that the detection signal-to-noise ratio SNR is proportional to the launched power for a given fiber length and thus to NA 2.
Increasing the NA for example may increase the power into the fiber, but this is at the expense of a reduced bandwidth, owing to the increased modal dispersion which results from the greater NA. Optical Waveguiding 69 for multimode optical-fiber links. These relationships allow a glimpse of the kinds of compromise that must be faced by optical-fiber communications systems designers.
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In order to minimize multimode dispersion, and thus maximize the bandwidth for a given fiber length, it is clear that the number of modes must be minimized. The absolute minimum number that we can have is one: a monomode fiber. It is for this reason that monomode or single mode fibers are the preferred medium for optical-fiber communications: only quite short distance Material dispersion is due to the fact that the refractive index of any optical material will vary with wavelength, owing to the structure of the atomic resonances. From Section 1. If it is not, then different portions of the source spectrum travel at different velocities and this will result in, among other effects, the broadening of an optical pulse.
It follows that this is the preferred wavelength for maximum bandwidth in a silica fiber. It is extremely fortuitous that this wavelength also corresponds to a minimum in the absorption spectrum for silica see Figure 2. It was the combination of these two factors that led to the rapid progress of monomode optical-fiber communications technology in the s. Let us insert some typical numbers into our equations to get a feel for practicalities. From Figure 2. Therefore, for a link of length km, for example, we shall have a bandwidth of 3. This is a respectable bandwidth but it should be possible to do better over this distance.
Before settling for this, however, there is yet another source of dispersion to worry about in regard to monomode fibers. This is known as waveguide dispersion and will now be considered. This lowest order mode is, of course, the only mode propagating in a monomode fiber. The physical origins of this form of dispersion lie in two factors. Second, as the frequency varies and the angle at the boundary varies, the penetration of the evanescent wave into the second medium cladding varies, in accordance with 2.
Hence the relative amounts of power in the guiding channel e. Clearly, all these arguments are reversed as the wavelength decreases. In particular, a decreasing wavelength causes the refractive index to tend towards that of the core; indeed, for very small wavelengths—very much less than the core diameter—the wave is, to first order, unaware of the boundary between the media and propagates as if it were doing so in an unrestricted core medium, of refractive index n 1.
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This waveguide dispersion can, in fact, be very useful, for it can be arranged to oppose the material dispersion. The waveguide dispersion depends upon the fiber geometry 2. Unrepeatered trunk telecommunications systems of several hundred kilometer lengths can be installed using such fiber. With its aid, it is possible to confine light and to direct it to where it is needed, over short, medium, and long distances. Furthermore, with the advantage of confinement, it is possible to control the interaction of light with other influences, such as electric, magnetic, or acoustic fields, which may be needed to impress information upon it.
Control also can be exerted over its intensity distribution, its polarization state and its nonlinear behavior. In short, optical waveguiding is crucial to the control of light. For the designers of devices and systems this control is essential. References  Syms, R. IEE, Vol. Selected Bibliography Marz, R.
Midwinter, J. Najafi, S. In the present chapter we shall expand and develop the ideas in regard to their general position in optical physics, before going on, in the following chapters, to look at the particular relevance and application of the ideas to optical fibers. We know that the electric and magnetic fields, for a freely propagating light wave, lie transversely to the propagation direction and orthogonally to each other. Normally, when discussing polarization phenomena, we fix our attention on the electric field, since it is this which has the most direct effect when the wave interacts with matter.
In saying that an optical wave is polarized we are implying that the direction of the optical field is either constant or is changing in an ordered, prescribable manner. In general, the tip of the electric vector circumscribes an ellipse, performing a complete circuit in a time equal to the period of the wave, or in a distance of one wavelength.
Clearly, the two descriptions are equivalent in this respect. It is also clear that the optical field can only change in any given, ordered fashion as long as the wave remains coherent, for the coherence determines the length, or time, over which the phase relationships, between orthogonal field components, remain constant. This implies that, for any other than the linear polarization state, any given polarization state can only be retained for the coherence length, or coherence time, of the light.
Electrons can move more easily along the chains than transversely to them, and thus the optical wave transmits easily only when its electric field lies along this acceptance direction. The material is cheap and allows the use of large optical apertures. It thus provides a convenient means whereby, for example, a specific linear polarization state can be defined; this state then provides a ready polarization reference that can be used as a starting point for other manipulations.
In order to study these manipulations and other aspects of polarization optics, we shall begin by looking more closely at the polarization ellipse. In other words, we say that the two waves must have a large mutual coherence. If this were not so, then relative phases and hence resultant field vectors would vary randomly within the detector response time, giving no ordered pattern to the behavior of the resultant field and thus presenting to the detector what would be, essentially, unpolarized light.
Assuming that the mutual coherence is good, we may investigate further the properties of the polarization ellipse. Note, first, that the ellipse always lies in the rectangle shown in Figure 3. Figure 3. Linear and circular states of polarization may be regarded as special cases where the polarization ellipse degenerates into a straight line or a circle, respectively. A linear state is obtained with the components in 3. The waves will be right-hand circularly polarized when m is even and left-hand circularly polarized when m is odd.
Light can become polarized as a result of the intrinsic directional properties of matter: either the matter that is the original source of the light, or the matter Elements of Polarization Optics 79 through which the light passes. These intrinsic material directional properties are the result of directionality in the bonding that holds together the atoms of which the material is made. This directionality leads to variations in the response of the material according to the direction of an imposed force, be it electric, magnetic, or mechanical. The best known manifestation of directionality in solid materials is the crystal, with the large variety of crystallographic forms, some symmetrical, some asymmetrical.
The characteristic shapes that we associate with certain crystals result from the fact that they tend to break preferentially along certain planes, known as cleavage planes, which are those planes between which atomic forces are weakest. It is not surprising, then, to find that directionality in a crystalline material is also evident in the light that it produces, or is impressed upon the light that passes through it.
In order to understand the ways in which we may produce polarized light, control it and use it, we must make a gentle incursion into the subject of crystal optics. In our previous discussions the forced oscillation was assumed to take place in the direction of the driving electric field, but in the case of a medium whose physical properties vary with direction, an anisotropic medium, this is not necessarily the case.
If an electron in an atom or molecule can move more easily in one direction than another, then an electric field at some arbitrary angle to the preferred direction will move the electron in a direction which is not parallel with the field direction Figure 3. The consequences, for the optics of anisotropic media, of this simple piece of physics are complex.
Immediately we can see that the already-discussed relationship between the electric displacement D and the electric field E, for an isotropic i. A tensor is a physical quantity which characterizes a particular physical property of an anisotropic medium, and takes the form of a matrix. Clearly D is not now in general parallel with E, and the angle between the two also will depend upon the direction of E in the material. Also, symmetrical tensors can be cast into their diagonal form by referring them to a special set of axes the principal axes which are determined by the crystal structure .
Hence the medium is offering two refractive indices to the wave traveling in this direction: we have the phenomenon known as double refraction or birefringence. Hence the phase difference between the two components will steadily increase and the composite polarization state of the wave will vary progressively from linear to circular and back to linear again. This behavior is, of course, a direct consequence of the basic physics which was discussed earlier: it is easier, in the anisotropic crystal, for the electric field to move the atomic electrons in one direction than in another.
Hence, for the direction of easy movement, the light polarized in this direction can travel faster than when it is polarized in the direction for which the movement is more sluggish. Harry Potter. Popular Features. New Releases. Notify me. Description Optical fibers are central to today's telecommunication and sensor technologies, and polarization behavior within these fibers has a profound impact on their performance.
This work provides the first definitive treatment of polarization phenomena, and delivers a wealth of analytical and practical knowledge that will be invaluable to engineers seeking to optimize and advance optical fiber performance in these industries. This exhaustively detailed resource explores the various polarization effects, their impact on communications and sensing systems, and the latest techniques to mitigate them.
It provides full details on polarization mode dispersion PMD and other effects that influence communications. Written by the field's foremost expert, this book will be welcomed by all engineers involved in the design and optimization of telecommunications and measurement-sensing systems. Product details Format Hardback pages Dimensions Other books in this series. Add to basket. Table of contents Principles of Optical-Fiber Propagation.
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