OPTICAL COMMUNICATIONUNIT-1

DISADVANTAGES

PROPAGATION WITHIN THE FIBER Principle of operation TOTAL INTERNAL REFLECTION (TIR).

Diffraction refers to various phenomena which occur when a wave encounters an obstacle. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1665.[2]

HYPERLINK "http://en.wikipedia.org/wiki/Diffraction" \l "cite_note-2"[3] In classical physics, the diffraction phenomenon is described as the apparent bending of waves around small obstacles and the spreading out of waves past small openings. Similar effects occur when light waves travel through a medium with a varying refractive index or a sound wave through one with varying acoustic impedance. Diffraction occurs with all waves, including sound waves, water waves, and electromagnetic waves such as visible light, x-rays and radio waves. As physical objects have wave-like properties (at the atomic level), diffraction also occurs with matter and can be studied according to the principles of quantum mechanics.

Richard Feynman [4] says that

"no-one has ever been able to define the difference between interference and diffraction satisfactorily. It is just a question of usage, and there is no specific, important physical difference between them."

He suggests that when there are only a few sources, say two, we call it interference, as in Young's slits, but with a large number of sources, the process is labelled diffraction.

While diffraction occurs whenever propagating waves encounter such changes, its effects are generally most pronounced for waves where the wavelength is roughly similar to the dimensions of the diffracting objects. If the obstructing object provides multiple, closely-spaced openings, a complex pattern of varying intensity can result. This is due to the superposition, or interference, of different parts of a wave that traveled to the observer by different paths (see diffraction grating).

The formalism of diffraction can also describe the way in which waves of finite extent propagate in free space. For example, the expanding profile of a laser beam, the beam shape of a radar antenna and the field of view of an ultrasonic transducer can all be analysed using diffraction equations.

Solar glory at the steam from hot springs. A glory is an optical phenomenon produced by light backscattered (a combination of diffraction, reflection and refraction) towards its source by a cloud of uniformly-sized water droplets.

The effects of diffraction are often seen in everyday life. The most striking examples of diffraction are those involving light; for example, the closely spaced tracks on a CD or DVD act as a diffraction grating to form the familiar rainbow pattern seen when looking at a disk. This principle can be extended to engineer a grating with a structure such that it will produce any diffraction pattern desired; the hologram on a credit card is an example. Diffraction in the atmosphere by small particles can cause a bright ring to be visible around a bright light source like the sun or the moon. A shadow of a solid object, using light from a compact source, shows small fringes near its edges. The speckle pattern which is observed when laser light falls on an optically rough surface is also a diffraction phenomenon. All these effects are a consequence of the fact that light propagates as a wave.

Diffraction can occur with any kind of wave. Ocean waves diffract around jetties and other obstacles. Sound waves can diffract around objects, which is why one can still hear someone calling even when hiding behind a tree.[5] Diffraction can also be a concern in some technical applications; it sets a fundamental limit to the resolution of a camera, telescope, or microscope.Thomas Young's sketch of two-slit diffraction, which he presented to the Royal Society in 1803

The effects of diffraction of light were first carefully observed and characterized by Francesco Maria Grimaldi, who also coined the term diffraction, from the Latin diffringere, 'to break into pieces', referring to light breaking up into different directions. The results of Grimaldi's observations were published posthumously in 1665.[6]

HYPERLINK "http://en.wikipedia.org/wiki/Diffraction" \l "cite_note-6"[7]

HYPERLINK "http://en.wikipedia.org/wiki/Diffraction" \l "cite_note-7"[8] Isaac Newton studied these effects and attributed them to inflexion of light rays. James Gregory (16381675) observed the diffraction patterns caused by a bird feather, which was effectively the first diffraction grating to be discovered.[9] Thomas Young performed a celebrated experiment in 1803 demonstrating interference from two closely spaced slits.[10] Explaining his results by interference of the waves emanating from the two different slits, he deduced that light must propagate as waves. Augustin-Jean Fresnel did more definitive studies and calculations of diffraction, made public in 1815[11] and 1818,[12] and thereby gave great support to the wave theory of light that had been advanced by Christiaan Huygens

HYPERLINK "http://en.wikipedia.org/wiki/Diffraction" \l "cite_note-12"[13] and reinvigorated by Young, against Newton's particle theory.

[edit] Mechanism

Photograph of single-slit diffraction in a circular ripple tankDiffraction arises because of the way in which waves propagate; this is described by the HuygensFresnel principle and the principle of superposition of waves. The propagation of a wave can be visualized by considering every point on a wavefront as a point source for a secondary spherical wave. The wave displacement at any subsequent point is the sum of these secondary waves. When waves are added together, their sum is determined by the relative phases as well as the amplitudes of the individual waves so that the summed amplitude of the waves can have any value between zero and the sum of the individual amplitudes. Hence, diffraction patterns usually have a series of maxima and minima.

There are various analytical models which allow the diffracted field to be calculated, including the Kirchhoff-Fresnel diffraction equation which is derived from wave equation, the Fraunhofer diffraction approximation of the Kirchhoff equation which applies to the far field and the Fresnel diffraction approximation which applies to the near field. Most configurations cannot be solved analytically, but can yield numerical solutions through finite element and boundary element methods.

[edit] Systems

It is possible to obtain a qualitative understanding of many diffraction phenomena by considering how the relative phases of the individual secondary wave sources vary, and in particular, the conditions in which the phase difference equals half a cycle in which case waves will cancel one another out.

The simplest descriptions of diffraction are those in which the situation can be reduced to a two-dimensional problem. For water waves, this is already the case; water waves propagate only on the surface of the water. For light, we can often neglect one direction if the diffracting object extends in that direction over a distance far greater than the wavelength. In the case of light shining through small circular holes we will have to take into account the full three dimensional nature of the problem.

Some of the simpler cases of diffraction are considered below.

[edit] Single-slit diffraction

Main article: Diffraction formalismDiffraction of red laser beam on the hole

Numerical approximation of diffraction pattern from a slit of width four wavelengths with an incident plane wave. The main central beam, nulls, and phase reversals are apparent.

Graph and image of single-slit diffraction

A long slit of infinitesimal width which is illuminated by light diffracts the light into a series of circular waves and the wavefront which emerges from the slit is a cylindrical wave of uniform intensity.

A slit which is wider than a wavelength produces interference effects in the space downstream of the slit. These can be explained by assuming that the slit behaves as though it has a large number of point sources spaced evenly across the width of the slit. The analysis of this system is simplified if we consider light of a single wavelength. If the incident light is monochromatic, these sources all have the same phase. Light incident at a given point in the space downstream of the slit is made up of contributions from each of these point sources and if the relative phases of these contributions vary by 2 or more, we may expect to find minima and maxima in the diffracted light. Such phase differences are caused by differences in the path lengths over which contributing rays reach the point from the slit.

We can find the angle at which a first minimum is obtained in the diffracted light by the following reasoning. The light from a source located at the top edge of the slit interferes destructively with a source located at the middle of the slit, when the path difference between them is equal to /2. Similarly, the source just below the top of the slit will interfere destructively with the source located just below the middle of the slit at the same angle. We can continue this reasoning along the entire height of the slit to conclude that the condition for destructive interference for the entire slit is the same as the condition for destructive interference between two narrow slits a distance apart that is half the width of the slit. The path difference is given by so that the minimum intensity occurs at an angle min given bywhere d is the width of the slit,

min is the angle of incidence at which the minimum intensity occurs, and is the wavelength of the light

A similar argument can be used to show that if we imagine the slit to be divided into four, six, eight parts, etc., minima are obtained at angles n given by

Where n is an integer other than zero.

There is no such simple argument to enable us to find the maxima of the diffraction pattern. The intensity profile can be calculated using the Fraunhofer diffraction equation as

Where I() is the intensity at a given angle,I0 is the original intensity, and

1. the sinc function is given by sinc(x) = sin(x)/(x) if x 0, and sinc(0) = 1

This analysis applies only to the far field, that is, at a distance much larger than the width of the slit.

2-slit (top) and 5-slit diffraction of red laser light

Diffraction of a red laser using a diffraction grating

A diffraction pattern of a 633 nm laser through a grid of 150 slits

[edit] Diffraction grating

Main article: Diffraction gratingA diffraction grating is an optical component with a regular pattern. The form of the light diffracted by a grating depends on the structure of the elements and the number of elements present, but all gratings have intensity maxima at angles m which are given by the grating equation

where

i is the angle at which the light is incident,

d is the separation of grating elements, and

m is an integer which can be positive or negative.

The light diffracted by a grating is found by summing the light diffracted from each of the elements, and is essentially a convolution of diffraction and interference patterns.

The figure shows the light diffracted by 2-element and 5-element gratings where the grating spacings are the same; it can be seen that the maxima are in the same position, but the detailed structures of the intensities are different.

A computer-generated image of an Airy diskComputer generated light diffraction pattern from a circular aperture of diameter 0.5micrometre at a wavelength of 0.6micrometre (red-light) at distances of 0.1cm 1cm in steps of 0.1cm. One can see the image moving from the Fresnel region into the Fraunhofer region where the Airy pattern is seen.

[edit] Circular aperture

Main article: Airy diskThe far-field diffraction of a plane wave incident on a circular aperture is often referred to as the Airy Disk. The variation in intensity with angle is given by

where a is the radius of the circular aperture, k is equal to 2/ and J1 is a Bessel function. The smaller the aperture, the larger the spot size at a given distance, and the greater the divergence of the diffracted beams.[edit] General aperture

The wave that emerges from a point source has amplitude at location r that is given by the solution of the frequency domain wave equation for a point source (The Helmholtz Equation),

where is the 3-dimensional delta function. The delta function has only radial dependence, so the Laplace operator (aka scalar Laplacian) in the spherical coordinate system simplifies to (see del in cylindrical and spherical coordinates)

By direct substitution, the solution to this equation can be readily shown to be the scalar Green's function, which in the spherical coordinate system (and using the physics time convention e it) is:

This solution assumes that the delta function source is located at the origin. If the source is located at an arbitrary source point, denoted by the vector and the field point is located at the point , then we may represent the scalar Green's function (for arbitrary source location) as:

Therefore, if an electric field, Einc(x,y) is incident on the aperture, the field produced by this aperture distribution is given by the surface integral:

On the calculation of Fraunhofer region fields where the source point in the aperture is given by the vector In the far field, wherein the parallel rays approximation can be employed, the Green's function, simplifies to

as can be seen in the figure to the right (click to enlarge). The expression for the far-zone (Fraunhofer region) field becomes

Now, since

and

the expression for the Fraunhofer region field from a planar aperture now becomes,

Letting,

and

the Fraunhofer region field of the planar aperture assumes the form of a Fourier transform

In the far-field / Fraunhofer region, this becomes the spatial Fourier transform of the aperture distribution. Huygens' principle when applied to an aperture simply says that the far-field diffraction pattern is the spatial Fourier transform of the aperture shape, and this is a direct by-product of using the parallel-rays approximation, which is identical to doing a plane wave decomposition of the aperture plane fields (see Fourier optics).

[edit] Propagation of a laser beam

The way in which the profile of a laser beam changes as it propagates is determined by diffraction. The output mirror of the laser is an aperture, and the subsequent beam shape is determined by that aperture. Hence, the smaller the output beam, the quicker it diverges.Diode lasers have much greater divergence than HeNe lasers for this reason.

Paradoxically, it is possible to reduce the divergence of a laser beam by first expanding it with one convex lens, and then collimating it with a second convex lens whose focal point is coincident with that of the first lens. The resulting beam has a larger aperture, and hence a lower divergence.

[edit] Diffraction-limited imaging

Main article: Diffraction-limited system

The Airy disk around each of the stars from the 2.56 m telescope aperture can be seen in this lucky image of the binary star zeta Botis.

The ability of an imaging system to resolve detail is ultimately limited by diffraction. This is because a plane wave incident on a circular lens or mirror is diffracted as described above. The light is not focused to a point but forms an Airy disk having a central spot in the focal plane with radius to first null of

where is the wavelength of the light and N is the f-number (focal length divided by diameter) of the imaging optics. In object space, the corresponding angular resolution is

where D is the diameter of the entrance pupil of the imaging lens (e.g., of a telescope's main mirror).

Two point sources will each produce an Airy pattern see the photo of a binary star. As the point sources move closer together, the patterns will start to overlap, and ultimately they will merge to form a single pattern, in which case the two point sources cannot be resolved in the image.ThRayleighcriterion specifies that two point sources can be considered to be resolvable if the separation of the two images is at least the radius of the Airy disk, i.e. if the first minimum of one coincides with the maximum of the other.

Thus, the larger the aperture of the lens, and the smaller the wavelength, the finer the resolution of an imaging system. This is why telescopes have very large lenses or mirrors, and why optical microscopes are limited in the detail which they can see.

[edit] Speckle patterns

Main article: speckle patternThe speckle pattern which is seen when using a laser pointer is another diffraction phenomenon. It is a result of the superpostion of many waves with different phases, which are produced when a laser beam illuminates a rough surface. They add together to give a resultant wave whose amplitude, and therefore intensity varies randomly.

[edit] Patterns

The upper half of this image shows a diffraction pattern of He-Ne laser beam on an elliptic aperture. The lower half is its 2D Fourier transform approximately reconstructing the shape of the aperture.Several qualitative observations can be made of diffraction in general:

The angular spacing of the features in the diffraction pattern is inversely proportional to the dimensions of the object causing the diffraction. In other words: The smaller the diffracting object, the 'wider' the resulting diffraction pattern, and vice versa. (More precisely, this is true of the sines of the angles.)The diffraction anglesare invariant under scaling; that is, they depend only on the ratio of the wavelength to the size of the diffracting object.When the diffracting object has a periodic structure, for example in a diffraction grating, the features generally become sharper. The third figure, for example, shows a comparison of a double-slit pattern with a pattern formed by five slits, both sets of slits having the same spacing, between the center of one slit and the next.

[edit] Particle diffraction

See also: neutron diffractionand electron diffractionQuantum theory tells us that every particle exhibits wave properties. In particular, massive particles caninterfere and therefore diffract. Diffraction of electrons and neutrons stood as one of thepowerful arguments in favor of quantum mechanics. The wavelength associated with a particle is the de Broglie wavelength

where h is Planck's constant and p is the momentum of the particle (mass velocity for slow-moving particles). For most macroscopic objects, this wavelength is so short that it is not meaningful to assign a wavelength to them. A sodium atom traveling at about 30,000m/s would have a De Broglie wavelength of about 50 pico meters.Because the wavelength for even the smallest of macroscopic objects is extremely small, diffraction of matter waves is only visible for small particles, like electrons, neutrons, atoms and small molecules. The short wavelength of these matter waves makes them ideally suited to study the atomic crystal structure of solids and large molecules like proteins.

Relatively larger molecules like buckyballs were also shown to diffract.[14][edit] Bragg diffraction

Following Bragg's law, each dot (or reflection), in this diffraction pattern forms from the constructive interference of X-rays passing through a crystal. The data can be used to determine the crystal's atomic structure.

For more details on this topic, see Bragg diffraction.

Diffraction from a three dimensional periodic structure such as atoms in a crystal is called Bragg diffraction. It is similar to what occurs when waves are scattered from a diffraction grating. Bragg diffraction is a consequence of interference between waves reflecting from different crystal planes. The condition of constructive interference is given by Bragg's law:

where is the wavelength,d is the distance between crystal planes, is the angle of the diffracted wave.and m is an integer known as the order of the diffracted beam.Bragg diffraction may be carried out using either light of very short wavelength like x-rays or matter waves like neutrons (and electrons) whose wavelength is on the order of (or much smaller than) the atomic spacing.[15] The pattern produced gives information of the separations of crystallographic planes allowing one to deduce the crystal structure. Diffraction contrast, in electron microscopes and x-topography devices in particular, is also a powerful tool for examining individual defects and local strain fields in crystals.[edit] CoherenceMain article: Coherence (physics)The description of diffraction relies on the interference of waves emanating from the same source taking different paths to the same point on a screen. In this description, the difference in phase between waves that took different paths is only dependent on the effective path length. This does not take into account the fact that waves that arrive at the screen at the same time were emitted by the source at different times. The initial phase with which the source emits waves can change over time in an unpredictable way. This means that waves emitted by the source at times that are too far apart can no longer form a constant interference pattern since the relation between their phases is no longer time independent.The length over which the phase in a beam of light is correlated, is called the coherence length. In order for interference to occur, the path length difference must be smaller than the coherence length. This is sometimes referred to as spectral coherence, as it is related to the presence of different frequency components in the wave. In the case of light emitted by an atomic transition, the coherence length is related to the lifetime of the excited state from which the atom made its transition.If waves are emitted from an extended source, this can lead to incoherence in the transversal direction. When looking at a cross section of a beam of light, the length over which the phase is correlated is called the transverse coherence length. In the case of Young's double slit experiment, this would mean that if the transverse coherence length is smaller than the spacing between the two slits, the resulting pattern on a screen would look like two single slit diffraction patterns.In the case of particles like electrons, neutrons and atoms, the coherence length is related to the spatial extent of the wave function that describes the particle.[edit] See alsoAtmospheric diffractionPulse dispersion in a graded index optical fiber is given by

where

is the difference in refractive indices of core and cladding,

is the refractive index of the cladding,

is the length of the fiber taken for observing the pulse dispersion,

is the speed of light, and

is the constant of graded index profile.

Splices n connectors

Numerical of first unit

Unit 2

Losses in optical fibers

Attenuation with graphs

ATTENUATION IN OPTICAL FIBERS

Attenuation and pulse dispersion represent the two most important characteristics of an optical fiber that determine the information-carrying capacity of a fiber optic communication system. Obviously, the lower the attenuation (and similarly, the lower the dispersion) the greater can be the required repeater spacing and therefore the lower will be the cost of the communication system. Pulse dispersion will be discussed in the next section, while in this section we will discuss briefly the various attenuation mechanisms in an optical fiber. The attenuation of an optical beam is usually measured in decibels (dB). If an input power P1 results in an output power P2, the power loss in decibels is given by (dB) = 10 log10 (P1/P2) (7-12) Thus, if the output power is only half the input power, the loss is 10 log 2 3 dB. Similarly, if the power reduction is by a factor of 100 or 10, the power loss is 20 dB or 10 dB respectively. If 96% of the light is transmitted through the fiber, the loss is about 0.18 dB. On the other hand, in a typical fiber amplifier, a power amplification of about 1000 represents a power gain of 30 dB. Figure 7-10 shows the spectral dependence of fiber attenuation (i.e., loss coefficient per unit length) as a function of wavelength of a typical silica optical fiber. The losses are caused by various mechanisms such as Rayleigh scattering, absorption due to metallic impurities and water in the fiber, and intrinsic absorption by the silica molecule itself. The Rayleigh scattering loss varies as 1/0 4, i.e., shorter wavelengths scatter more than longer wavelengths. Here 0 represents the free space wavelength. This is why the loss coefficient decreases up to about 1550 nm. The two absorption peaks around 1240 nm and 1380 nm are primarily due to traces of OH ions and traces of metallic ions. For example, even 1 part per million (ppm) of iron can cause a loss of about 0.68 dB/km at 1100 nm. Similarly, a concentration of 1 ppm of OH ion can cause a loss of 4 dB/km at 1380 nm. This shows the level of purity that is required to achieve low-loss optical fibers. If these impurities are removed, the two absorption peaks will disappear. For 0 > 1600 nm the increase in the loss coefficient is due to the absorption of infrared light by silica molecules. This is an intrinsic property of silica, and no amount of purification can remove this infrared absorption tail.

Figure 7-10 Typical wavelength dependence of attenuation for a silica fiber. Notice that the lowest

attenuation occurs at 1550 nm [adapted from Miya, Hasaka, and Miyashita].

As you see, there are two windows at which loss attains its minimum value. The first window is around 1300 nm (with a typical loss coefficient of less than 1 dB/km) where, fortunately (as we will see later), the material dispersion is negligible. However, the loss attains its absolute minimum value of about 0.2 dB/km around 1550 nm. The latter window has become extremely important in view of the availability of erbium-doped fiber amplifiers.

Calculation of losses using the dB scale become easy. For example, if we have a 40-km fiber link (with

a loss of 0.4 dB/km) having 3 connectors in its path and if each connector has a loss of 1.8 dB, the total

loss will be the sum of all the losses in dB; or 0.4 dB/km 40 km + 3 1.8 dB = 21.4 dB.

Let us assume that the input power of a 5-mW laser decreases to 30 W after traversing through

40 km of an optical fiber. Using Equation 7-12, attenuation of the fiber in dB/km is therefore

[10 log (166.7)]/40 0.56 dB/km.Mechanisms of attenuationLight attenuation by ZBLAN and silica fibers

Main article: Transparent materialsAttenuation in fiber optics, also known as transmission loss, is the reduction in intensity of the light beam (or signal) with respect to distance traveled through a transmission medium. Attenuation coefficients in fiber optics usually use units of dB/km through the medium due to the relatively high quality of transparency of modern optical transmission media. The medium is usually a fiber of silica glass that confines the incident light beam to the inside. Attenuation is an important factor limiting the transmission of a digital signal across large distances. Thus, much research has gone into both limiting the attenuation and maximizing the amplification of the optical signal. Empirical research has shown that attenuation in optical fiber is caused primarily by both scattering and absorption.

[edit] Light scattering

Specular reflection

Diffuse reflection

The propagation of light through the core of an optical fiber is based on total internal reflection of the lightwave. Rough and irregular surfaces, even at the molecular level, can cause light rays to be reflected in random directions. This is called diffuse reflection or scattering, and it is typically characterized by wide variety of reflection angles.

Light scattering depends on the wavelength of the light being scattered. Thus, limits to spatial scales of visibility arise, depending on the frequency of the incident light-wave and the physical dimension (or spatial scale) of the scattering center, which is typically in the form of some specific micro-structural feature. Since visible light has a wavelength of the order of one micrometre (one millionth of a meter) scattering centers will have dimensions on a similar spatial scale.

Thus, attenuation results from the incoherent scattering of light at internal surfaces and interfaces. In (poly)crystalline materials such as metals and ceramics, in addition to pores, most of the internal surfaces or interfaces are in the form of grain boundaries that separate tiny regions of crystalline order. It has recently been shown that when the size of the scattering center (or grain boundary) is reduced below the size of the wavelength of the light being scattered, the scattering no longer occurs to any significant extent. This phenomenon has given rise to the production of transparent ceramic materials.

Similarly, the scattering of light in optical quality glass fiber is caused by molecular level irregularities (compositional fluctuations) in the glass structure. Indeed, one emerging school of thought is that a glass is simply the limiting case of a polycrystalline solid. Within this framework, "domains" exhibiting various degrees of short-range order become the building blocks of both metals and alloys, as well as glasses and ceramics. Distributed both between and within these domains are micro-structural defects that provide the most ideal locations for light scattering. This same phenomenon is seen as one of the limiting factors in the transparency of IR missile domes.[30]At high optical powers, scattering can also be caused by nonlinear optical processes in the fiber.[31]

HYPERLINK "http://en.wikipedia.org/wiki/Optical_fiber" \l "cite_note-31"[32][edit] UV-Vis-IR absorption

In addition to light scattering, attenuation or signal loss can also occur due to selective absorption of specific wavelengths, in a manner similar to that responsible for the appearance of color. Primary material considerations include both electrons and molecules as follows:

1) At the electronic level, it depends on whether the electron orbitals are spaced (or "quantized") such that they can absorb a quantum of light (or photon) of a specific wavelength or frequency in the ultraviolet (UV) or visible ranges. This is what gives rise to color.

2) At the atomic or molecular level, it depends on the frequencies of atomic or molecular vibrations or chemical bonds, how close-packed its atoms or molecules are, and whether or not the atoms or molecules exhibit long-range order. These factors will determine the capacity of the material transmitting longer wavelengths in the infrared (IR), far IR, radio and microwave ranges.

The design of any optically transparent device requires the selection of materials based upon knowledge of its properties and limitations. The Lattice absorption characteristics observed at the lower frequency regions (mid IR to far-infrared wavelength range) define the long-wavelength transparency limit of the material. They are the result of the interactive coupling between the motions of thermally induced vibrations of the constituent atoms and molecules of the solid lattice and the incident light wave radiation. Hence, all materials are bounded by limiting regions of absorption caused by atomic and molecular vibrations (bond-stretching)in the far-infrared (>10m).

Thus, multi-phonon absorption occurs when two or more phonons simultaneously interact to produce electric dipole moments with which the incident radiation may couple. These dipoles can absorb energy from the incident radiation, reaching a maximum coupling with the radiation when the frequency is equal to the fundamental vibrational mode of the molecular dipole (e.g. Si-O bond) in the far-infrared, or one of its harmonics.

The selective absorption of infrared (IR) light by a particular material occurs because the selected frequency of the light wave matches the frequency (or an integer multiple of the frequency) at which the particles of that material vibrate. Since different atoms and molecules have different natural frequencies of vibration, they will selectively absorb different frequencies (or portions of the spectrum) of infrared (IR) light.

Reflection and transmission of light waves occur because the frequencies of the light waves do not match the natural resonant frequencies of vibration of the objects. When IR light of these frequencies strikes an object, the energy is either reflected or transmitted.

[[edit] Practical issues

[edit] Optical fiber cables

The propagation constant of an electromagnetic wave is a measure of the change undergone by the amplitude of the wave as it propagates in a given direction. The quantity being measured can be the voltage or current in a circuit or a field vector such as electric field strength or flux density. The propagation constant itself measures change per metre but is otherwise dimensionless.

The propagation constant is expressed logarithmically, almost universally to the base e, rather than the more usual base 10 used in telecommunications in other situations. The quantity measured, such as voltage, is expressed as a sinusoidal phasor. The phase of the sinusoid varies with distance which results in the propagation constant being a complex number, the imaginary part being caused by the phase change. [edit] Alternative names

The term propagation constant is somewhat of a misnomer as it usually varies strongly with . It is probably the most widely used term but there are a large variety of alternative names used by various authors for this quantity. These include, transmission parameter, transmission function, propagation parameter, propagation coefficient and transmission constant. In plural, it is usually implied that and are being referenced separately but collectively as in transmission parameters, propagation parameters, propagation coefficients, transmission constants and secondary coefficients. This last occurs in transmission line theory, the term secondary being used to contrast to the primary line coefficients. The primary coefficients being the physical properties of the line; R,C,L and G, from which the secondary coefficients may be derived using the telegrapher's equation. Note that, at least in the field of transmission lines, the term transmission coefficient has a different meaning despite the similarity of name. Here it is the corollary of reflection coefficient.

[edit] Definition

[edit] Attenuation constant

In telecommunications, the term attenuation constant, also called attenuation parameter or coefficient, is the attenuation of an electromagnetic wave propagating through a medium per unit distance from the source. It is the real part of the propagation constant and is measured in nepers per metre. A neper is approximately 8.7dB. Attenuation constant can be defined by the amplitude ratio;

The propagation constant per unit length is defined as the natural logarithmic of ratio of the sending end current or voltage to the receiving end current or voltage.

[edit] Copper lines

The attenuation constant for copper lines (or ones made of any other conductor) can be calculated from the primary line coefficients as shown above. For a line meeting the distortionless condition, with a conductance G in the insulator, the attenuation constant is given by;

however, a real line is unlikely to meet this condition without the addition of loading coils and, furthermore, there are some frequency dependant effects operating on the primary "constants" which cause a frequency dependence of the loss. There are two main components to these losses, the metal loss and the dielectric loss.

The loss of most transmission lines are dominated by the metal loss, which causes a frequency dependency due to finite conductivity of metals, and the skin effect inside a conductor. The skin effect causes R along the conductor to be approximately dependent on frequency according to;

Losses in the dielectric depend on the loss tangent (tan) of the material, which depends inversely on the wavelength of the signal and is directly proportional to the frequency.

[edit] Optical fibre

The attenuation constant for a particular propagation mode in an optical fiber, the real part of the axial propagation constant.

[edit] Phase constant

In electromagnetic theory, the phase constant, also called phase change constant, parameter or coefficient is the imaginary component of the propagation constant for a plane wave. It represents the change in phase per metre along the path travelled by the wave at any instant and is equal to real part of the angular wavenumber of the wave. It is represented by the symbol and is measured in units of radians per metre.

From the definition of (angular) wavenumber;

For a transmission line, the Heaviside condition of the telegrapher's equation tells us that the wavenumber must be proportional to frequency for the transmission of the wave to be undistorted in the time domain. This includes, but is not limited to, the ideal case of a lossless line. The reason for this condition can be seen by considering that a useful signal is composed of many different wavelengths in the frequency domain. For there to be no distortion of the waveform, all these waves must travel at the same velocity so that they arrive at the far end of the line at the same time as a group. Since wave phase velocity is given by;

it is proved that is required to be proportional to . In terms of primary coefficients of the line, this yields from the telegrapher's equation for a distortionless line the condition;

However, practical lines can only be expected to approximately meet this condition over a limited frequency band.

[edit] Filters

The term propagation constant or propagation function is applied to filters and other two-port networks used for signal processing. In these cases, however, the attenuation and phase coefficients are expressed in terms of nepers and radians per network section rather than per metre. Some authors[1] make a distinction between per metre measures (for which "constant" is used) and per section measures (for which "function" is used).

The propagation constant is a useful concept in filter design which invariably uses a cascaded section topology. In a cascaded topology, the propagation constant, attenuation constant and phase constant of individual sections may be simply added to find the total propagation constant etc.

[edit] Cascaded networks

Three networks with arbitrary propagation constants and impedances connected in cascade. The Zi terms represent image impedance and it is assumed that connections are between matching image impedances.

The ratio of output to input voltage for each network is given by,[2]The terms are impedance scaling terms[3] and their use is explained in the image impedance article.

The overall voltage ratio is given by,

Thus for n cascaded sections all having matching impedances facing each other, the overall propagation constant is given by,

Attenuation

Light power propagating in a fiber decays exponentially with length due to absorption and scattering losses.

Attenuation is the single most important factor determining the cost of fiber optic telecommunication

systems as it determines spacing of repeaters needed to maintain acceptable signal levels. In the near infrared and visible regions, the small absorption losses of pure silica are due to tails of absorption bands in the far infrared and ultraviolet. Impuritiesnotably water in the form of hydroxyl ions are much more dominant causes of absorption in commercial fibers. Recent improvements in fiber purity have reduced attenuation losses. State-of-the-art systems can have attenuation on the order of 0.1 dB/km.

Scattering can couple energy from guided to radiation modes, causing loss of energy from the fiber. There are unavoidable Rayleigh scattering losses from small scale index fluctuations frozen into the fiber when it solidifies. This produces attenuation proportional to l/4. Irregularities in core diameter and geometry or changes in fiber axis direction also cause scattering. Any process that

Typical Spectral Attenuation in Silica

Dispersion

As a pulse travels down a fiber, dispersion causes pulse spreading. This limits the distance and the bit rate of data on an optical fiber. Symbols become unrecognizable 1 0 1

Figure 2Dispersion

6

5

4

3

2

1

0

Attenuation (dB/km)Wavelength (m)

0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6

Figure 3Typical Spectral Attenuation in Silica

Polarization Mode Dispersion.

Polarization Mode Dispersion (PMD) is actually another form of material dispersion. Single-mode fiber supports one mode, which consists of two orthogonal polarization modes. Ideally, the core of an optical fiber has an index of refraction that is uniform over the entire cross section, unless the fiber is graded index. However, mechanical stresses, i.e. bending, can cause slight changes in the index of refraction in one dimension. This can cause one of the orthogonal polarization modes to travel faster than the other, hence causin dispersion of the optical pulse. imposes dimensional irregularities such as microbendingincreases scattering and hence attenuation.

Absorption losses

VI. Attenuation in Optical Fibers

Attenuation and pulse dispersion represent the two most important characteristics of an optical fiber that determine the information-carrying capacity of a fiber optic communication system. Obviously, the lower the attenuation (and similarly, the lower the dispersion) the greater can be the required repeater spacing and therefore the lower will be the cost of the communication system. Pulse dispersion will be discussed in the next section, while in this section we will discuss briefly the various attenuation mechanisms in an optical fiber.

The attenuation of an optical beam is usually measured in decibels (dB). If an input power P1 results in an output power P2, the power loss in decibels is given by

(dB) = 10 log10 (P1/P2)(7-12)

Thus, if the output power is only half the input power, the loss is 10log23dB. Similarly, if the power reduction is by a factor of 100 or 10, the power loss is 20dB or 10dB respectively. If 96% of the light is transmitted through the fiber, the loss is about 0.18dB. On the other hand, in a typical fiber amplifier, a power amplification of about 1000 represents a power gain of 30dB.

Figure7-10 shows the spectral dependence of fiber attenuation (i.e., loss coefficient per unit length) as a function of wavelength of a typical silica optical fiber. The losses are caused by various mechanisms such as Rayleigh scattering, absorption due to metallic impurities and water in the fiber, and intrinsic absorption by the silica molecule itself. The Rayleigh scattering loss varies as 1/04, i.e., shorter wavelengths scatter more than longer wavelengths. Here 0 represents the free space wavelength. This is why the loss coefficient decreases up to about 1550nm. The two absorption peaks around 1240 nm and 1380 nm are primarily due to traces of OH ions and traces of metallic ions. For example, even 1part per million (ppm) of iron can cause a loss of about 0.68 dB/km at 1100 nm. Similarly, a concentration of 1ppm of OHion can cause a loss of 4 dB/km at 1380 nm. This shows the level of purity that is required to achieve low-loss optical fibers. If these impurities are removed, the two absorption peaks will disappear. For 0 > 1600 nm the increase in the loss coefficient is due to the absorption of infrared light by silica molecules. This is an intrinsic property of silica, and no amount of purification can remove this infrared absorption tail.

As you see, there are two windows at which loss attains its minimum value. The first window is around 1300 nm (with a typical loss coefficient of less than 1dB/km) where, fortunately (as we will see later), the material dispersion is negligible. However, the loss attains its absolute minimum value of about 0.2 dB/km around 1550 nm. The latter window has become extremely important in view of the availability of erbium-doped fiber amplifiers.

Example 7-3

Calculation of losses using the dB scale become easy. For example, if we have a 40-km fiber link (with a loss of 0.4dB/km) having 3 connectors in its path and if each connector has a loss of 1.8dB, the total loss will be the sum of all the losses in dB; or 0.4dB/km40km+31.8dB=21.4dB.

Example 7-4

Let us assume that the input power of a 5-mW laser decreases to 30W after traversing through 40km of an optical fiber. Using Equation7-12, attenuation of the fiber in dB/km is therefore [10log(166.7)]/400.56dB/km.

Scattering

Bending losses

Leaky modesMode coupling losses combined losses in the fiber

Numerical sbased ojn these topics

Dispersion

Effect of dispersion on pulse transmissionPULSE DISPERSION IN STEP-INDEX FIBERS (SIF)

In digital communication systems, information to be sent is first coded in the form of pulses and these pulses of light are then transmitted from the transmitter to the receiver, where the information is decoded. The larger the number of pulses that can be sent per unit time and still be resolvable at the receiver end, the larger will be the transmission capacity of the system. A pulse of light sent into a fiber broadens in time as it propagates through the fiber. This phenomenon is known as pulse dispersion, and it occurs primarily because of the following mechanisms:

1. Different rays take different times to propagate through a given length of the fiber. We will discuss this for a step-index multimode fiber and for a parabolic-index fiber in this and the following sections. In the language of wave optics, this is known as intermodal dispersion because it arises due to different modes traveling with different speeds.4 2. Any given light source emits over a range of wavelengths, and, because of the intrinsic property of the material of the fiber, different wavelengths take different amounts of time to propagate along the same path. This is known as material dispersion and will be discussed in Section IX.

3. Apart from intermodal and material dispersions, there is yet another mechanismreferred to as waveguide dispersion and important only in single-mode fibers. We will briefly discuss this in Section XI. In the fiber shown in Figure 7-7, the rays making larger angles with the axis (those shown as dotted rays) have to traverse a longer optical path length and therefore take a longer time to reach the output end. Consequently, the pulse broadens as it propagates through the fiber (see

4 We will have a very brief discussion about modes in Section XI. We may mention here that the number of modes in a step index fiber is about V2/2 where the parameter V will be defined in Section XI. When V < 4, the fiber supports only a few modes and it is necessary to use wave theory. On the other hand, if V > 8, the fiber supports many modes and the fiber is referred to as a multimoded fiber. When this happens, ray optics gives accurate results and one need not use wave theory.

Figure 7-11). Even though two pulses may be well resolved at the input end, because of the broadening of the pulses they may not be so at the output end. Where the output pulses are not resolvable, no information can be retrieved. Thus, the smaller the pulse dispersion, the greater will be the information-carrying capacity of the system.

Figure 7-11 Pulses separated by 100 ns at the input end would be resolvable at the output end of 1 km

of the fiber. The same pulses would not be resolvable at the output end of 2 km of the same fiber.

We will now derive an expression for the intermodal dispersion for a step-index fiber. Referring back to Figure 7-7b, for a ray making an angle with the axis, the distance AB is traversed in time.where c/n1 represents the speed of light in a medium of refractive index n1, c being the speed of light in free space. Since the ray path will repeat itself, the time taken by a ray to traverse a length L of the fiber would be

t n L

L c cos = 1

The above expression shows that the time taken by a ray is a function of the angle made bythe ray with the z-axis (fiber axis), which leads to pulse dispersion. If we assume that all rays

lying between = 0 and = c = cos1(n2/n1) (see Equation 7-8) are present, the time taken by

The following extreme rays for a fiber of length L would be given by

t n L

min = c1 corresponding to rays at = 0 (7-16)

t n L

max cn = 1

corresponding to rays at =c = cos1(n2/n1) (7-17)

Hence, if all the input rays were excited simultaneously, the rays would occupy a time interval

at the output end of duration

i c

= = LN M

OQ P

t t n L n

max min n 1 1

2

1

or, finally, the intermodal dispersion in a multimode SIF is

i n L

where has been defined earlier [see Equations 7-5 and 7-6] and we have used Equation 7-11. The quantity i represents the pulse dispersion due to different rays taking different times in propagating through the fiber, which, in wave optics, is nothing but the intermodal dispersion and hence the subscript i. Note that the pulse dispersion is proportional to the square of NA. Thus, to have a smaller dispersion, one must have a smaller NA, which of course reduces the acceptance angle im and hence the light-gathering power. If, at the input end of the fiber, we have a pulse of width 1, after propagating through a length L of the fiber the pulse will have a width 2 given approximately by

That is, a pulse traversing through the fiber of length 1 km will be broadened by 50 ns. Thus, two pulses separated by, say, 500 ns at the input end will be quite resolvable at the end of 1 km of the fiber. However, if consecutive pulses were separated by, say, 10 ns at the input end, they would be absolutely unresolvable at the output end. Hence, in a 1-Mbit/s fiber optic system, where we have one pulse every 106 s, a 50-ns/km dispersion would require repeaters to be placed every 3 to 4 km. On the other hand, in a 1-Gbit/s fiber optic communication system, which requires the transmission of one pulse every 109s, a dispersion of 50 ns/km would result in intolerable broadening even within 50 meters or so. This would be highly inefficient and uneconomical from a system point of view.

From the discussion in the above example it follows that, for a very-high-information-carrying system, it is necessary to reduce the pulse dispersion. Two alternative solutions existone involves the use of near-parabolic-index fibers and the other involves single-mode fibers.

Types of dispersion- intermodal and intramodels Intermodal Dispersion As its name implies, intermodal

dispersion is a phenomenon between different modes in an optical fiber. Therefore this category of dispersion only applies to mulitmode fiber. Since all the different propagating modes have different group velocities, the time it takes each mode to travel a fixed distance is also different. Therefore as an optical pulse travels down a multim ode fiber, th e pulses begin to spread, until they eventually spread into one another. This effect limits both the bandwidth of multimode fiber as well as the distance it can transport data.

Intramodal Dispersion

Intramodal dispersion, sometimes called material dispersion, is a category of dispersion that occurs

within a single-mode. This dispersion mechanism is a result of material properties of optical fiber and applies to both single-mode and multimode fibers. There are two distinct types of intramodal dispersion: chromatic dispersion and polarization mode dispersion.

Chromatic Dispersion. In silica, the index of refraction is dependent upon wavelength. Therefore different wavelengths will travel down an optical fiber at different velocities. This implies that a pulse with a wider FWHM will spread more than a pulse with a narrower FWHM. This dispersion limits both the bandwidth and the distance that information can be supported. This is why for long communications links, it is desirable to use a laser with a very narrow linewidth. Distributed Feedback (DFB) lasers are popular for communications because they have a single longitudinal mode with a very narrow linewidth.

Material and waveguide dispersion

MATERIAL DISPERSION

We first define the group index. To do this we return to Equation 7-1 where we noted that the

velocity of light in a medium is given by v = c/n (7-24)

Here n is the refractive index of the medium, which, in general, depends on the wavelength. The

dependence of the refractive index on wavelength leads to what is known as dispersion,

discussed in Module 1.3, Basic Geometrical Optics. In Figure 7-13 we have shown a narrow

pencil of a white light beam incident on a prism. Since the refractive index of glass depends on

the wavelength, the angle of refraction will be different for different colors. For example, for

crown glass the refractive indices at 656.3 nm (orange), 589.0 nm (yellow), and 486.1 nm

(green) are respectively given by 1.5244, 1.5270, and 1.5330. Thus, if the angle of incidence

i = 45 the angle of refraction, r, will be r = 27.64, 27.58, and 27.47 for the orange, yellow,

and blue colors respectively. The incident white light will therefore disperse into its constituent

colorsthe dispersion will become more evident at the second surface of the prism as seen in

Figure 7-13.

Figure 7-13 Dispersion of white light as it passes through a prism

Now, the quantity v defined by Equation 7-24 is usually referred to as the phase velocity.

However, a pulse travels with what is known as the group velocity, which is given by

vg = c/ng (7-25)

where ng is known as the group index and, in most cases its value is slightly larger than n. In

Table 7.1 we have tabulated n and ng for pure silica for different values of wavelength lying

between 700 nm and 1600 nm. The corresponding spectral variation of n and ng for pure silica is

given in Figure 7-14.

Figure 7-14 Variation of n and ng with wavelength for pure silica. Notice that ng has

a minimum value around 1270 nm.

In Sections VIII and IX we considered the broadening of an optical pulse due to different rays

taking different amounts of time to propagate through a certain length of the fiber. However,

every source of light has a certain wavelength spread, which is often referred to as the spectral

width of the source. Thus, a white light source (like the sun) has a spectral width of about

300 nm. On the other hand, an LED has a spectral width of about 25 nm and a typical laser

diode (LD) operating at 1300 nm has a spectral width of about 2 nm or less. Each wavelength

component will travel with a slightly different group velocity through the fiber. This results in

broadening of a pulse. This broadening of the pulse is proportional to the length of the fiber and

to the spectral width of the source. We define the material dispersion coefficient Dm, which is

measured in ps/km-nm. Dm represents the material dispersion in picoseconds per kilometer

length of the fiber per nanometer spectral width of the source. At a particular wavelength, the

value of Dm is a characteristic of the material and is (almost) the same for all silica fibers. The

values of Dm for different wavelengths are tabulated in Table 7.1. A negative Dm implies that

the longer wavelengths travel faster; similarly, a positive value of Dm implies that shorter

wavelengths travel faster. However, in calculating the pulse broadening, only the magnitude

should be considered.

Example 7-6

The LEDs used in early optical communication systems had a spectral width 0 of about 20 nm

around 0 = 825 nm. Using Dm in Table 7.1 (at 850 nm), such a pulse will broaden by

m = Dm L = 84.2 (ps/km-nm) 1 (km) 20 (nm) ~ 1700 ps = 1.7 ns

in traversing a 1-km length of the fiber. It is very interesting to note that, if we carry out a similar

calculation around 0 1300 nm, we will obtain a much smaller value of m; thus

m = Dm L = 2.4 (ps/km-nm) 1 (km) 20 (nm) ~ 0.05 ns

OP T I C A L WA V E G U I D E S A N D F I B E R S

2000 University of Connecticut 271

in traversing 1-km length of the fiber. The very small value of m is due to the fact that ng is

approximately constant around 0 = 1300 nm, as shown in Figure 7-14. Indeed, the wavelength

0 1270 nm is usually referred to as the zero material dispersion wavelength, and it is because of

such low material dispersion that the optical communication systems shifted their operation to

around 0 1300 nm.

Example 7-7

In the optical communication systems that are in operation today, one uses laser diodes (LD) with

0 1550 nm having a spectral width of about 2 nm. Thus, for a 1-km length of the fiber, the

material dispersion m becomes

m = Dm L = 21.5 (ps/km-nm) 1 (km) 2 (nm) ~ 43 ps

the positive sign indicating that higher wavelengths travel more slowly than lower wavelengths.

[Notice from Table 7.1 that, for 0 1300 nm, ng increases with 0.]

Waveguide dispersion

In Section IX we discussed material dispersion that results from the dependence of the

refractive index of the fiber on wavelength. Even if we assume the refractive indices n1 and n2 to

be independent of 0, the group velocity of each mode does depend on the wavelength. This

leads to what is known as waveguide dispersion. The detailed theory is rather involved [see,

e.g., Chapter 10, Ghatak and Thyagarajan]; we may mention here two important points:

1. The waveguide dispersion is usually negative for a given single-mode fiber. The

magnitude increases with an increase in wavelength.

2. If the core radius a (of a single-mode fiber) is made smaller and the value of is made

larger, the magnitude of the waveguide dispersion increases. Thus we can tailor the

waveguide dispersion by changing the refractive index profile.

The following two examples demonstrate how one can tailor the zero-dispersion wavelength by

changing the fiber parameters.

Example 7-17

We consider the fiber discussed in Example 7-12 for which n2 = 1.447, = 0.003, and a = 4.2 m.

The variations of the waveguide dispersion (w), material dispersion (m), and total dispersion

OP T I C A L WA V E G U I D E S A N D F I B E R S

2000 University of Connecticut 277

(tot = w + m) with 0 are shown in Figure 7-15. From the figure it can be seen that the total

dispersion passes through zero around 0 1300 nm. This is known as zero total-dispersion

wavelength and represents an extremely important parameter.

Figure 7-15 The variations of m,, w, and tot with 0 for a typical conventional single-mode fiber

(CSF) with parameters given in Example 7-17. The total dispersion passes through zero at around

0 1300 nm, known as zero total dispersion wavelength.

Example 7-18

We next consider the fiber discussed in Example 7-13 for which n2 = 1.444, = 0.0075, and

a = 2.3 m. For this fiber, at 0 1550 nm,

w = 20 ps/km-nm, as seen in Figure 7-16.

On the other hand, the material dispersion at this wavelengthper km and per unit wavelength

interval in nmis given by Table 7.1 as

Dm = m = +21 ps/km-nm

We therefore see that the two expressions are of opposite sign and almost cancel each other.

Physically, because of waveguide dispersion, longer wavelengths travel more slowly than shorter

wavelengths. And, because of material dispersion, longer wavelengths travel faster than shorter

wavelengths. So the two effects compensate for each other, resulting in a zero total dispersion

around 1550 nm. Thus we have been able to shift the zero-dispersion wavelength by changing the

fiber parameters. These are known as the dispersion-shifted fibers, the importance of which will be

discussed in the next section. The variations of m, w, and tot with 0 are plotted in Figure 7-16,

showing clearly that tot is near zero at o = 1550 nm.

F U N D AM E N T AL S O F P H O T O N I C S

278 2000 University of Connecticut

Figure 7-16 The variations of m, w, and tot with 0 for a typical dispersion-shifted single-mode

fiber (DSF) with parameters given in Example 7-18. The total dispersion passes through zero at

around 0 1550 nm.X. DISPERSION AND MAXIMUM BIT RATE

We may mention here briefly that, in a digital communication system employing light pulses,

pulse broadening would result in an overlap of pulses, resulting in loss of resolution and leading

to errors in detection. Thus pulse broadening is one of the mechanisms (other than attenuation)

that limits the distance between two repeaters in a fiber optic link. It is obvious that, the larger

the pulse broadening, the smaller will be the number of pulses per second that can be sent down

a link. Different criteria based on slightly different considerations are used to estimate the

maximum permissible bit rate (Bmax) for a given pulse dispersion. However, it is always of the

order of 1/, where is the pulse dispersion. In one type of extensively used coding (known as

NRZ) we have

Bmax 0.7/ (7-26)

This formula takes into account (approximately) only the limitation imposed by the pulse

dispersion in the fiber. In an actual link the source and detector characteristics would also be

taken into account while estimating the maximum bit rate (see Module 1-8, Fiber Optic

Telecommunication). It should also be pointed out that, in a fiber, the pulse dispersion is caused,

in general, by intermodal dispersion, material dispersion, and waveguide dispersion. However,

waveguide dispersion is important only in single-mode fibers and may be neglected in carrying

out analysis for multimode fibers. Thus (considering multimode fibers), if i and m are the

dispersion due to intermodal and material dispersions respectively, the total dispersion is given

by

= + i m

2 2 (7-27)

F U N D AM E N T AL S O F P H O T O N I C S

272 2000 University of Connecticut

Example 7-8

We consider a step-index multimode fiber with n1 = 1.46, = 0.01, operating at 850 nm. For such a

fiber, the intermodal dispersion (for a 1-km length) is

which is usually written as

1 49 ns/km

If the source is an LED with = 20 nm, using Table 7.1, the material dispersion m is 1.7 ns/km

[see Example 7-6]. Thus, in step-index multimode fibers, the dominant pulse-broadening

mechanism is intermodal dispersion and the total dispersion is given by

= i + m =

2 2 49 ns/km

Using Equation 7-26, this gives a maximum bit rate of about

Bmax 0.7/ 14 Mbit-km/s

Thus a 10-km link can at most support 1.4 Mbit/s, since increases by a factor of 10, causing Bmax

to decrease by the same factor.

Example 7-9

Let us now consider a parabolic-index multimode fiber with n1 = 1.46, = 0.01, operating at

850 nm with an LED of spectral width 20 nm. For such a fiber, the intermodal dispersion, using

Equation 7-22, is

im = L n

c

1 2

2

0.24 ns/km

The material dispersion is again 1.7 ns/km. Thus, in this case the dominant mechanism is material

dispersion rather than intermodal dispersion. The total dispersion is

= 0.242 +1.72 = 1.72 ns/km

This gives a maximum bit rate of about

Bmax 0.7/ 400 Mbit-km/s

giving a maximum permissible bit rate of 20 Mbit/s for a 20-km link.

Example 7-10

If we now shift the wavelength of operation to 1300 nm and use the parabolic-index fiber of the

previous example, we see that the intermodal dispersion remains the same at 0.24 ns/km while the

material dispersion (for an LED of 0 = 20 nm) becomes 0.05 ns/km (see Example 7-6). The

OP T I C A L WA V E G U I D E S A N D F I B E R S

2000 University of Connecticut 273

material dispersion is now negligible in comparison to intermodal dispersion. Thus the total

dispersion and maximum bit rate are respectively given by

0.242 + 0.052 0.25 ns/km; Bmax = 2.8 Gbit-km/s

Example 7-11

If, in Example 7-10, we replace the LED with a laser diode of spectral width 2 nm, the material

dispersion becomes 0.17 ns/km, which is now smaller than the intermodal dispersion. The total

dispersion is

= 0.242 + 0.172 = 0.29 ns/km

giving a maximum bit rate of

Bmax = 0.7/ 2.4 Gbit-km/s

We should reiterate that, in the examples discussed above, the maximum bit rate has been estimated

by considering the fiber only. In an actual link, the temporal response of the source and detector

must also be taken into account.

Chromatic dispersion

Group delay etc. derivation

Total dispersionTransmission rate

Dispersion-shifted fibers

In Section VI we learned that the attenuation of a silica fiber attains its minimum value of about

0.2 dB/km at around 0 1550 nm. The second- and third-generation optical communication

systems operated around 0 1300 nm, where the dispersion was extremely small but the loss

was about 1 dB/km, and therefore the repeater spacing was limited by the loss in the fiber. Since

the lowest loss lies at around 0 1550 nm, if the zero-dispersion wavelength could be shifted

to the 0 1550-nm region, one could have both minimum loss and very low dispersion. This

would lead to very-high-bandwidth systems with very long (~ 100 km) repeater spacings. Apart

from this, extremely efficient optical fiber amplifiers capable of amplifying optical signals in the

1550-nm band have also been developed. Thus, shifting the operating wavelength from 1310 nm

to 1550 nm would be very advantageous. As discussed in Example 7-18, by reducing the core

size and increasing the value of , we can shift the zero-dispersion wavelength to 1550 nm,

which represents the low-loss window. Indeed, the current fourth-generation optical

communication systems operate at 1550 nm, using dispersion-shifted single-mode fibers with

repeater spacing of about 100 km, carrying about 10 Gbit/s of information (equivalent to about

150,000 telephone channels) through one hair-thin single-mode fiber.

We may mention here that, if one is interested in carrying out accurate calculations for total

dispersion, one may use the software described in Ghatak, Goyal, and Varshney.DISPERSION Since optical fiber is a waveguide, light can propagate in a number of modes

If a fiber is of large diameter, light entering at different angles will excite different modes while narrow fiber may only excite one mode

Multimode propagation will cause dispersion, which results in the spreading of pulses and limits the usable bandwidth

Single-mode fiber has much less dispersion but is more expensive to produce. Its small size, together with the fact that its numerical aperture is smaller than that of multimode fiber, makes it more difficult to couple to light sources

Types of fibers

Both types of fiber described earlier are known as step-index fibers because the index of refraction changes radically between the core and the cladding

Graded-index fiber is a compromise multimode fiber, but the index of refraction gradually decreases away from the center of the core

Graded-index fiber has less dispersion than a multimode step-index fiber

Dispersion in fiber optics results from the fact that in multimode propagation, the signal travels faster in some modes than it would in others

Single-mode fibers are relatively free from dispersion except for intramodal dispersion

Graded-index fibers reduce dispersion by taking advantage of higher-order modes

One form of intramodal dispersion is called material dispersion because it depends upon the material of the core

Another form of dispersion is called waveguide dispersion

Dispersion increases with the bandwidth of the light source

Losses

Losses in optical fiber result from attenuation in the material itself and from scattering, which causes some light to strike the cladding at less than the critical angle

Bending the optical fiber too sharply can also cause losses by causing some of the light to meet the cladding at less than the critical angle

Losses vary greatly depending upon the type of fiber

Plastic fiber may have losses of several hundred dB per kilometer

Graded-index multimode glass fiber has a loss of about 24 dB per kilometer

Single-mode fiber has a loss of 0.4 dB/km or less

Fiber-Optic Cables

There are two basic types of fiber-optic cable

The difference is whether the fiber is free to move inside a tube with a diameter much larger than the fiber or is inside a relatively tight-fitting jacket

They are referred to as loose-tube and tight-buffer cables

Both methods of construction have advantages

Loose-tube cablesall the stress of cable pulling is taken up by the cables strength members and the fiber is free to expand and contract with temperature

Tight-buffer cables are cheaper and generally easier to use

Group and phase velocity

Another consequence of dispersion manifests itself as a temporal effect. The formula above, v = c / n calculates the phase velocity of a wave; this is the velocity at which the phase of any one frequency component of the wave will propagate. This is not the same as the group velocity of the wave, which is the rate that changes in amplitude (known as the envelope of the wave) will propagate. The group velocity vg is related to the phase velocity by, for a homogeneous medium (here is the wavelength in vacuum, not in the medium): The group velocity vg is often thought of as the velocity at which energy or information is conveyed along the wave. In most cases this is true, and the group velocity can be thought of as the signal velocity of the waveform. In some unusual circumstances, where the wavelength of the light is close to an absorption resonance of the medium, it is possible for the group velocity to exceed the speed of light (vg > c), leading to the conclusion that superluminal (faster than light) communication is possible. In practice, in such situations the distortion and absorption of the wave is such that the value of the group velocity essentially becomes meaningless, and does not represent the true signal velocity of the wave, which stays less than c. The group velocity itself is usually a function of the waves frequency. This results in group velocity dispersion (GVD), which causes a short pulse of light to spread in time as a result of different frequency components of the pulse travelling at different velocities. GVD is often quantified as the group delay dispersion parameter (again, this formula is for a uniform medium only):

If D is less than zero, the medium is said to have positive dispersion. If D is greater than zero, the medium has negative dispersion.If a light pulse is propagated through a normally dispersive medium, the result is the higher frequency components travel slower than the lower frequency components. The pulse therefore becomes positively chirped, or up-chirped, increasing in frequency with time. Conversely, if a pulse travels through an anomalously dispersive medium, high frequency components travel faster than the lower ones, and the pulse becomes negatively chirped, or down-chirped, decreasing in frequency with time.

For ordering assistance, please contact Customer Service at 877-529-9114 (sales@fiberoptic.com) or visit our website at: www.fiberoptic.com

Chromatic Dispersion

Chromatic dispersion (optics)

Knowledgebase at FiberOptic.com

The result of GVD, whether negative or positive, is ultimately temporal spreading of the pulse. This makes dispersion management extremely important in optical communications systems based on optical fiber, since if dispersion is too high, a group of pulses representing a bit-stream will spread in time and merge together, rendering the bit-stream unintelligible. This limits the length of fiber that a signal can be sent down without regeneration. One possible answer to this problem is to send signals down the optical fibre at a wavelength where the GVD is zero (e.g. around ~1.3-1.5 m in silica fibres), so pulses at this wavelength suffer minimal spreading from dispersionin practice, however, this approach causes more problems than it solves because zero GVD unacceptably amplifies other nonlinear effects (such as four wave mixing). Another possible option is to use soliton pulses in the regime of anomalous dispersion, a form of optical pulse which uses a nonlinear optical effect to self-maintain its shape

solitons have the practical problem, however, that they require a certain power level to be maintained in the pulse for the nonlinear effect to be of the correct strength. Instead, the solution that is currently used in practice is to perform dispersion compensation, typically by matching the fiber with another fiber of opposite-sign dispersion so that the dispersion effects cancel; such compensation is ultimately limited by nonlinear effects such as self-phase modulation, which interact with dispersion to make it very difficult to undo. Dispersion control is also important in lasers that produce short pulses. The overall dispersion of the optical resonator is a major factor in determining the duration of the pulses emitted by the laser. A pair of prisms can be arranged to produce net negative dispersion, which can be used to balance the usually positive dispersion of the laser medium. Diffraction gratings can also be used to produce dispersive effects; these are often used in highpower laser amplifier systems. Recently, an alternative to prisms and gratings has been developed: chirped mirrors.

These dielectric mirrors are coated so that different wavelengths have different penetration lengths, and therefore different group delays. The coating layers can be tailored to achieve a net negative dispersion.

Dispersion in waveguides

Optical fibers, which are used in telecommunications, are among the most abundant types of waveguides.

Dispersion in these fibers is one of the limiting factors that determine how much data can be transported on a single fiber.

The transverse modes for waves confined laterally within a waveguide generally have different speeds (and field patterns) depending upon their frequency (that is, on the relative size of the wave, the wavelength) compared to the size of the waveguide.

In general, for a waveguide mode with an angular frequency () at a propagation constant (so that the

electromagnetic fields in the propagation direction z oscillate proportional to ei(z t)), the group-velocity dispersion parameter D is defined as:

where = 2c / is the vacuum wavelength and vg = d / d is the group velocity. This formula generalizes the one in the previous section for homogeneous media, and includes both waveguide dispersion and material dispersion. The reason for defining the dispersion in this way is that |D| is the (asymptotic) temporal pulse spreading t per unit bandwidth per unit distance travelled, commonly reported in ps / nm km for optical fibers.

A similar effect due to a somewhat different phenomenon is modal dispersion, caused by a waveguide having multiple modes at a given frequency, each with a different speed. A special case of this is polarization mode dispersion (PMD), which comes from a superposition of two modes that travel at different speeds due to random imperfections that break the symmetry of the waveguide.

For ordering assistance, please contact Customer Service at 877-529-9114 (sales@fiberoptic.com) or visit our website at: www.fiberoptic.com

Chromatic Dispersion

Chromatic dispersion (optics)

Knowledgebase at FiberOptic.com

Dispersion in gemology

In the technical terminology of gemology, dispersion is the difference in the refractive index of a material at the B

and G Fraunhofer wavelengths of 686.7 nm and 430.8 nm and is meant to express the degree to which a prism

cut from the gemstone shows fire, or color. Dispersion is a material property. Fire depends on the dispersion,

the cut angles, the lighting environment, the refractive index, and the viewer

Dispersion in imaging

In photographic and microscopic lenses, dispersion causes chromatic aberration, distorting the image, and various techniques have been developed to counteract it

Dispersion in pulsar timing

Pulsars are spinning neutron stars that emit pulses at very regular intervals ranging from milliseconds to seconds.

It is believed that the pulses are emitted simultaneously over a wide range of frequencies. However, as observed on Earth, the components of each pulse emitted at higher radio frequencies arrive before those emitted at lower frequencies. This dispersion occurs because of the ionised component of the interstellar medium, which makes the group velocity frequency dependent. The extra delay added at frequency is

where the dispersion measure DM is is the integrated free electron column density ne out to the pulsar at a distance d[4].

Of course, this delay cannot be measured directly, since the emission time is unknown. What can be measured is the difference in arrival times at two different frequencies. The delay T between a high frequency hi and a low frequency lo component of a pulse will be

and so DM is normally computed from measurements at two different frequencies. This allows computation of the absolute delay at any frequency, which is used when combining many different pulsar observations into an integrated timing solution.

Introduction

Telecommunications service providers have to face continuously growing bandwidth demands in all

networks areas, from long-haul to access. Because installing new communication links would require huge investments, telecommunications carriers prefer to increase the capacity of their existing fiber links by using dense wavelength-division multiplexing (DWDM) systems and/or higher bit rates systems. However, most of the installed optical fibers are old and exhibit physical characteristics that may limit their ability to transmit high-speed signals.

The broadening of light pulses, called dispersion, is a critical factor limiting the quality of signal transmission over optical links. Dispersion is a consequence of the physical properties of the transmission medium. Single-mode fibers, used in high-speed optical networks, are subject to Chromatic Dispersion (CD) that causes pulse broadening depending on wavelength, and to Polarization Mode Dispersion (PMD) that causes pulse broadening depending on polarization. Excessive spreading will cause bits to overflow their intended time slots and overlap adjacent bits. The receiver may then have difficulty discerning and properly interpreting adjacent bits, increasing the Bit Error Rate. To preserve the transmission quality, the maximum amount of time dispersion must be limited to a small proportion of the signal bit rate, typically 10% of the bit time.

With optical networks moving from 2.5 Gbps to 10 Gbps and onto 40 Gbps, the acceptable tolerance of

dispersion is drastically reduced. For instance, the amount of acceptable chromatic dispersion decreases by a factor of 16 when moving from 2.5 to 10 Gbps, and by an additional factor of 16 moving from 10 to 40 Gbps. These tight tolerances of high-speed networks mean that every possible source of pulse spreading should be addressed. Operating companies need to measure the dispersion of their networks to assess the possibility of upgrading them to higher transmission speeds, or to evaluate the need for compensation. This paper presents the causes and effects of dispersion and describes the different ways to measure it.

Chromatic Dispersion (CD)

CD Definition and Origin

Light within a medium travels at a slower speed than in vacuum. The speed at which light travels is

determined by the mediums refractive index. In an ideal situation, the refractive index would not depend on the wavelength of the light. Since this is not the case, different wavelengths travel at different speeds within an optical fiber.

Figure 1: CD in single-mode fiber

Laser sources are spectrally thin, but not monochromatic. This means that the input pulse contains several

wavelength components, traveling at different speeds, causing the pulse to spread. The detrimental effects

of chromatic dispersion result in the slower wavelengths of one pulse intermixing with the faster

wavelengths of an adjacent pulse, causing intersymbol interference.

The Chromatic Dispersion of a fiber is expressed in ps/(nm*km), representing the differential delay, or time spreading (in ps), for a source with a spectral width of 1 nm traveling on 1 km of the fiber. It depends on the fiber type, and it limits the bit rate or the transmission distance for a good quality of service.

CD of current network spans

As a consequence of its optical characteristics, the Chromatic Dispersion of a fiber can be changed by

acting on the physical properties of the material. To reduce fiber dispersion, new types of fiber were

invented, including dispersion-shifted fibers (ITU G.653) and non-zero dispersion-shifted fiber (ITU G.655).

The most commonly deployed fiber in networks (ITU G.652), called dispersion-unshifted singlemode fiber, has a small chromatic dispersion in the optical window around 1310 nm, but exhibits a higher CD in the 1550 nm region. This dispersion limits the possible transmission length without compensation on OC- 768/STM-256 DWDM networks.

ITU G.653 is a dispersion-shifted fiber (DSF), designed to minimize chromatic dispersion in the 1550 nm

window with zero dispersion between 1525 nm and 1575 nm. But this type of fiber has several drawbacks, such as higher polarization mode dispersion than ITU G.652, and a high Four Wave Mixing risk, rendering DWDM practically impossible. For these reasons, another singlemode fiber was developed: the Non-Zero Dispersion-Shifted Fiber (NZDSF). NZDSF fibers have now replaced DSF fibers, which are not used anymore.

The ITU G.655 Non-Zero Dispersion-Shifted Fibers were developed to eliminate non-linear effects

experienced on DSF fibers. They were developed especially for DWDM applications in the 1550 nm window. They have a cut-off wavelength around 1310 nm, limiting their operation around this wavelength.

Poly-chromatic

incident light

1

2

Single-Mode Fiber

Refractive index: n()

Figure 2: Chromatic dispersion profiles of different fiber types

CD Limit and Compensation

The chromatic dispersion in fiber causes a pulse broadening and degrades the transmission quality, limiting

the distance a digital signal can travel before needing regeneration or compensation. For DWDM systems

using DFB lasers, the maximum length of a link before being affected by chromatic dispersion is commonly

calculated with the following equation:

L is the link distance in km, CD is the chromatic dispersion in ps/(nm * km),

and B is the bit rate in Gbps.

As an example, consider a typical network transmitting data at 10 Gbps on a channel at 1550 nm over a

standard ITU G.652 fiber, having a CD coefficient of 17 ps/(nm*km). In this case, the theoretical maximum

distance of the link, before adding CD compensators, will be around 61 km, which is close to the typical

node distance in Europe. This length will be divided by 16 when upgrading the network to a 40 Gbps bit rate,

thus falling under 4 km. This distance is around 20 km for a 40 Gbps transmission through NZDSF. We can

thus see that CD is a major limiting factor in high-speed transmission.

Fortunately, CD is quite stable, predictable, and controllable. Dispersion Compensation Fiber (DCF), with its

large negative CD coefficient, can be inserted into the link at regular intervals to minimize its global

chromatic dispersion.

Tx Rx

DC modules

fiber span

delay [ps]

0

L

Figure 3: Chromatic dispersion compensation scheme

While each spool of DCF adequately solves chromatic dispersion for one channel, this is not usually the

case for all channels on a DWDM link. At the extreme wavelengths of a band, dispersion still accumulates

and can be a significant problem. In this case, a tunable compensation module may be necessary at the

receiver.

Dispersion [ps/nm-km]

1525

1530

1535

1540

1545

1550

1555

1560

SMF

NZDSF

Dispersion

Shifted Fiber

5

0

-5

Wavelength [nm]

10

1550nm DWDM window

1565

Fiber Type 1310 nm 1550 nm

ITU G.652 conventional 0 17

ITU G.653 DSF -15 0

ITU G.655 NZDSF -12 3

Chromatic Dispersion

ps / (nm * km)

delay [ps]

0

1

2

3

Figure 4: Residual chromatic dispersion on a DWDM-compensated link

Therefore, chromatic dispersion measurement is essential in the field to verify the types of installed fibers.

Such measurements assess if and how the fibers can be upgraded to transmit higher bit rates, verify fiber

zero point and slope for new installations, and carefully evaluate compensation plans.

CD Measurement Methods

In the field, there are three main methods for determining the chromatic dispersion of an optical fiber. These

are described by three TIA/EIA industry standards: the pulse-delay method (FOTP-168 standard), the

modulated phase-shift method (FOTP-169 standard), and the differential phase shift method (FOTP-175

standard).

These methods all measure first the time delay, in ps, as a function of the wavelength. They then deduce

the chromatic dispersion coefficient, in ps/(nm*km), from the slope of this delay curve and from the length of

the link. The delay curve plotted by CD analyzers is a fitted trace based on the acquired time delays at

discrete wavelength points. Therefore, the accuracy of the computation of the CD coefficient depends on

the amount of the measured data.

Figure 5: Curve fit to calculate CD coefficient

Phase-shift and differential phase-shift methods are quite similar. In both methods, a modulated source is

injected at the input of the fiber under test. The phase of the sinusoidal modulating signal is analyzed at the

output of the fiber and compared to the phase of a reference signal, modulated with the same frequency. In

the phase-shift method, the reference signal has a fixed wavelength, while the other modulated signal is

tuned in wavelengths. In the differential phase-shift method, both signals are tuned in wavelengths, with a

fixed wavelength interval. The analyzed modulated signal, tuned in wavelengths, is compared to a close

reference signal, also tuned in wavelength, but the wavelength gap is constant. The time delay of the link is

deduced from the phase-shift measurement, using the relationship between the delay (t), the phase (), and

the modulation frequency (f):

f

t

2

=

The phase-shift methods assume there is access to both ends of the fiber under test, with a transmitter unit

connected at the input, and a receiver unit at the output of the fiber. The wavelength selection can be

achieved either at the transmitter unit level, using a tunable laser, or at the receiver unit level, with a

broadband