Chromatic dispersion. Optical fiber (fiber optics) Chromatic dispersion in optical fibers

An important parameter of an optical fiber is dispersion, which determines its information throughput.

An optical fiber transmits not just light energy, but also a useful information signal. Pulses of light, the sequence of which determines the information flow, blur during the propagation process. With a sufficiently large broadening, the pulses begin to overlap, so that it becomes impossible to separate them during reception (Figure 3).

Figure 3 - Effect of dispersion

Dispersion is the dispersion in time of the spectral or mode components of an optical signal, which leads to an increase in the duration of the optical radiation pulse as it propagates through the optical fiber and is determined by the difference in the squares of the pulse durations at the output and input 0V:

The smaller the dispersion value, the greater the flow of information that can be transmitted along the fiber. Dispersion not only limits the frequency range of the OF, but significantly reduces the signal transmission range, since the longer the line, the greater the increase in pulse duration.

Dispersion is generally determined by three main factors:

The difference in the propagation speeds of the guided modes (inter-mode dispersion),

Guiding properties of optical fiber (waveguide dispersion),

Parameters of the material from which it is made (material dispersion).

Figure 4 - Types of dispersion

The main reasons for the occurrence of dispersion are, on the one hand, a large number of modes in the optical fiber (intermode dispersion), and on the other hand, the incoherence of radiation sources actually operating in the wavelength spectrum (chromatic dispersion).

Intermode dispersion

It predominates in multimode OFFs and is caused by the difference in the time it takes for modes to travel through the OFF from its input to its output. For an optical fiber with a stepped refractive index profile, the speed of propagation of electromagnetic waves with a wavelength is the same for all modes. The difference in the propagation paths of guided modes at a fixed frequency (wavelength) of radiation from an optical source leads to the fact that the travel time of these modes through the optical fiber is different. As a result, the pulse they generate at the output of the OF is broadened. The magnitude of the pulse broadening is equal to the difference in the propagation time of the slowest and fastest modes. This phenomenon is called intermode dispersion.

The formula for calculating intermode dispersion can be obtained by considering the geometric model of the propagation of guided modes in the OF. Any guided mode in a stepped optical fiber can be represented by a light beam, which, when moving along the fiber, repeatedly experiences total internal reflection from the core-cladding interface. The exception is the main fashion HE 11 , which is described by a light beam moving without reflection along the axis of the fiber.

With a length of OB equal to L , the length of the zigzag path traversed by a light beam propagating at an angle and z to the fiber axis is L/cos and z (Figure 5).

Figure 5 - Paths of propagation of light rays in a two-layer optical fiber

The speed of propagation of electromagnetic waves with wavelength l is the same in the fiber under consideration and is equal to:

Where With - speed of light, km/s.

Usually in OV n 1 ? n 2, so it takes the form:

where is the relative value of the core-cladding refractive indices.

It is clear from the formula that the pulse broadening due to intermode dispersion is smaller, the smaller the difference in the refractive indices of the core and cladding. This is one of the reasons why in real stepwise OFs they try to make this difference as small as possible.

In practice, due to the presence of inhomogeneities (mainly microbends), individual modes, when passing through the optical fiber, influence each other and exchange energy.

Intermodal dispersion in stepwise OFs can be completely eliminated if the structural parameters of the OF are selected appropriately. So, if we make the dimensions of the core and? so small, then only one mode will propagate along the fiber at the carrier wavelength, i.e., there will be no mode dispersion. Such fibers are called single-mode. They have the highest throughput. With their help, large bundles of channels can be organized on communication highways.

Pulse dispersion can also be significantly reduced by appropriately selecting the refractive profile across the cross section of the OF core. Thus, the dispersion decreases when moving to gradient OBs. The intermode dispersion of gradient optical fibers is, as a rule, lower by an order of magnitude and more than that of stepped fibers.

In such gradient optical fibers, in contrast to optical fibers with a stepwise propagation profile, light rays no longer propagate in a zigzag manner, but along wave- or helical spiral trajectories.

2.1.Causes and types of dispersion

The main reason for the occurrence of dispersion in the fiber is the incoherence of the radiation source (laser). An ideal source emits all power at a given wavelength λ 0, but in reality the radiation occurs in the spectrum λ 0 ± Δλ (Fig. 2.1), since not all excited electrons return to the same state from which they were removed during pumping.

Fig.2.1. Real laser radiation

The refractive index is a frequency-dependent quantity, that is, n is a function of λ: n = f (λ), see Fig. 2.2.

Fig.2.2. Dependence of refractive index on wavelength

Consequently, when propagating a signal consisting of a mixture of wavelengths λ 0 ± Δλ, parts of the signal travel at different speeds, and dispersion occurs:

λ ± Δλ → n ± Δn → c /(n ± Δn) → v ± Δv → Δτ.

This type of dispersion is called material dispersion.

The transverse wave propagation constant (along the fiber radius) also depends on the wavelength, that is, the mode area and the area of ​​that part of the cladding that is captured by the mode area extending beyond the boundaries of the core depend on the wavelength. Light propagates along the part of the shell bordering the core at a higher speed than along the core, which contributes to a change in dispersion. This dispersion is called waveguide dispersion. Both of these dispersions, material and waveguide, are collectively called chromatic dispersion. They add up arithmetically. Figure 2.3 shows the dependences of the material and waveguide dispersion and their sum on the wavelength. For standard single-mode fiber at λ = 1300 nm, these dispersions are equal and opposite in sign, and the total dispersion is zero.

Fig.2.3. Wavelength dependence of material and waveguide dispersion in standard single-mode fiber (nm)

In multimode fiber, in addition to chromatic dispersion, there is also intermode dispersion. If there are several modes, then each propagates along the fiber at its own speed, which can differ significantly from each other. Figure 2.4 shows graphs of phase velocities of some modes.

Rice. 2.4. Graph of phase velocities of some modes as a function of frequency.

If the fiber parameters change, for example, the core diameter changes randomly, mode tuning occurs and the modes exchange energy. Intermode dispersion is an order of magnitude greater than chromatic dispersion, which was the reason for the development of single-mode cables in which there is no intermode dispersion. Table 2.1 shows the approximate ratio of the values ​​of dispersion types for different types of fibers.

Table 2.1. Relationship between different types of variance

The total dispersion is defined as the square root of the sum of the squares of the chromatic and mode dispersion:

(2.1)

Material and waveguide dispersions are calculated using the formulas

τ mat = ∆λ∙ М(λ)∙ L (2.2),

τвв = ∆λ∙ В(λ)∙ L (2.3),

where ∆λ is the laser radiation bandwidth, nm;

М(λ) and В(λ) – specific material and waveguide dispersion, ps/(nm km);

L – line length, km.

The values ​​of M(λ) and B(λ) are given in reference books.

τ Σ = [τ mm 2 +(τ mat + τ vv) 2 ] 1/2

Option table 2.1. Approximate dispersion values ​​for various fiber types

2.2. Polarization mode dispersion (PMD)

Light represents vibrations transverse to the direction of light propagation (Fig. 2.5). If the end of the field vector describes a straight line, then such polarization is called linear; if it is a circle or an ellipse, then it is called circular or elliptical. Most people, with rare exceptions, do not feel the polarization of light; only a few (such as Leo Tolstoy) clearly distinguish between polarized and non-polarized light. A conventional integrated light detector (diode) also reacts only to the intensity of the wave, and not to its polarization. However, some optical devices, such as certain types of amplifiers, have polarization-dependent gain.

Rice. 2.5. Types of linear polarization

In addition, the polarization of the vector is of great importance in the processes of reflection and refraction, since the Fresnel coefficients, which characterize the amplitudes of the reflected and refracted wave, generally depend on the direction of the polarization vector (Fig. 2.6). Figure 2.6 shows how a mixture of rays of parallel (dash) and perpendicular (dot) polarizations is reflected with respect to the propagation plane when passing through the horizontal interface plane. It can be seen from the figure that at a certain angle (Brewster angle) all reflected waves have perpendicular polarization, and refracted ones have parallel polarization.

Rice. 2.6. Reflection of waves of different polarization.

In a classic single-mode fiber, the only mode is the HE wave 11. However, if polarization is taken into account, then the fiber contains two mutually orthogonal modes corresponding to the horizontal and vertical axes x and y. In a real situation, the fiber is not always a perfect circle in cross-section, but often, due to certain features of the technology, is a small ellipse. In addition, when winding the cable and when laying it, asymmetrical mechanical stresses and deformations of the fiber occur, which leads to birefringence. The refractive index will change due to the additional stress, and the propagation speeds of orthogonal modes in different areas will differ from each other, which will introduce different time delays in the propagation of orthogonal modes. The pulse as a whole will experience a statistical broadening over time, which is called polarization mode dispersion (PMD). Since the PMD in different sections of the line is different and obeys statistical laws, root mean square summation is usually used, and the PMD is calculated using the formula

Along with the attenuation coefficient of the optical fiber, the most important parameter is dispersion, which determines its capacity for transmitting information.

Variance – This is the scattering in time of the spectral and mode components of the optical signal, which lead to an increase in the duration of the optical radiation pulse as it propagates through the optical fiber.

Pulse broadening is defined as the quadratic difference in the pulse duration at the output and input of the optical fiber according to the formula:

and the values ​​of i are taken at the level of half the pulse amplitude (Figure 2.8).

Figure 2.8

Figure 2.8 - Pulse broadening due to dispersion

Dispersion occurs for two reasons: the incoherence of radiation sources and the existence of a large number of modes. The dispersion caused by the first cause is called chromatic (frequency) , it consists of two components - material and waveguide (intra-mode) dispersions. Material dispersion is due to the dependence of the refractive index on the wavelength, waveguide dispersion is associated with the dependence of the propagation coefficient on the wavelength.

The dispersion caused by the second reason is called modal (intermode).

Mode dispersion is characteristic only of multimode fibers and is due to the difference in the travel time of modes along the optical fiber from its input to its output. IN OF with a stepped refractive index profile the speed of propagation of electromagnetic waves with wavelength is the same and equal to: , where C is the speed of light. In this case, all rays incident on the end of the optical fiber at an angle to the axis within the aperture angle propagate in the fiber core along their zigzag lines and, at the same speed of propagation, reach the receiving end at different times, which leads to an increase in the duration of the received pulse. Since the minimum propagation time of an optical beam occurs when the incident beam is , and the maximum is when , we can write:

where L is the length of the light guide;

Refractive index of the fiber core;

C is the speed of light in vacuum.

Then the value of the intermode dispersion is equal to:

Mode dispersion of gradient optical fibers an order of magnitude or more lower than that of stepped fibers. This is due to the fact that due to a decrease in the refractive index from the axis of the optical fiber to the shell, the speed of propagation of the rays along their trajectory changes. So, on trajectories close to the axis it is less, and on trajectories remote it is greater. Rays propagating along the shortest trajectories (closer to the axis) have a lower speed, and rays propagating along longer trajectories have a higher speed. As a result, the propagation time of the rays is leveled out, and the increase in pulse duration becomes smaller. With a parabolic refractive index profile, when the profile exponent q=2, the mode dispersion is determined by the expression:

The mode dispersion of the gradient OB is several times less than that of the step OB at the same values. And since it is usual, the mode dispersion of the indicated OFs can differ by two orders of magnitude.

In calculations when determining mode dispersion, it should be borne in mind that up to a certain line length, called the mode coupling length, there is no intermodal coupling, and then at a process of mutual conversion of modes occurs and a steady state occurs. Therefore, when the dispersion increases according to a linear law, and then, when - according to a quadratic law.

Thus, the above formulas are valid only for length. For line lengths, use the following formulas:

- for stepped light guide

- for gradient light guide,

where is the length of the line;

Mode coupling length (steady state), equal to km for stepped fiber and km for gradient fiber (established empirically).

Material dispersion depends on the frequency (or wavelength) and the OF material, which is usually quartz glass. Dispersion is determined by the electromagnetic interaction of the wave with bound electrons of the medium material, which, as a rule, is nonlinear (resonant) in nature.

The occurrence of dispersion in the light guide material, even for single-mode fibers, is due to the fact that the optical source exciting the fiber (light-emitting diode - LED or semiconductor laser PPL) generates light radiation having a continuous wave spectrum of a certain width (for LEDs this is approximately nm, for multimode PPLs - nm , for single-mode nm laser diodes). Different spectral components of light radiation propagate at different speeds and arrive at a certain point at different times, leading to broadening of the pulse at the receiving end and, under certain conditions, to distortion of its shape. The refractive index varies with wavelength (frequency), with the level of dispersion depending on the range of wavelengths of light introduced into the fiber (usually the source emits multiple wavelengths), as well as the central operating wavelength of the source. In region I, the transparency window is where longer wavelengths (850nm) move faster compared to shorter wavelengths (845nm). In region III of the transparency window, the situation changes: shorter ones (1550 nm) move faster compared to longer ones (1560 nm). Figure 2.9

Figure 2.9 – Wavelength propagation speeds

The length of the arrows corresponds to the speed of the wavelengths, with a longer arrow corresponding to faster movement.

At some point in the spectrum, the speeds coincide. This coincidence for pure quartz glass occurs at a wavelength of nm, called the zero-dispersion wavelength of the material, since . When the wavelength is below the zero dispersion wavelength, the parameter has a positive value; otherwise, it has a negative value. Figure 2.10

Material dispersion can be determined through specific dispersion using the expression:

.

The quantity - specific dispersion, , is determined experimentally. With different compositions of alloying impurities in the OM, it has different values ​​depending on (Table 2.3).

Table 2.3 – Typical values ​​of specific material dispersion

Waveguide (intra-mode) dispersion – This term denotes the dependence of the delay of a light pulse on the wavelength, associated with a change in the speed of its propagation in the fiber due to the waveguide nature of propagation. Pulse broadening due to waveguide dispersion is similarly proportional to the width of the source radiation spectrum and is defined as:

,

where is the specific waveguide dispersion, the values ​​of which are presented in Table 2.4:

Table 2.4

– is due to the differential group delay between beams with the main polarization states. The distribution of signal energy over different polarization states changes slowly over time, for example due to changes in ambient temperature, refractive index anisotropy caused by mechanical forces.

In a single-mode fiber, not one mode propagates, as is commonly believed, but two perpendicular polarizations (modes) of the original signal. In an ideal fiber, these modes would propagate at the same speed, but real fibers do not have an ideal geometry. The main cause of polarization mode dispersion is the non-concentricity of the fiber core profile, which occurs during the manufacturing process of the fiber and cable. As a result, two perpendicular polarization components have different propagation speeds, which leads to dispersion (Figure 2.11)

Figure 2.11

The coefficient of specific polarization-mode dispersion is normalized per 1 km and has the dimension . The polarization-mode dispersion value is calculated using the formula:

Due to its small value, it must be taken into account exclusively in single-mode fiber, and when high-speed signal transmission (2.5 Gbit/s and higher) with a very narrow spectral band of radiation of 0.1 nm or less is used. In this case, chromatic dispersion becomes comparable to polarization mode dispersion.

The specific PMD coefficient of a typical fiber is usually .

Chromatic dispersion consists of material and waveguide components and occurs during propagation in both single-mode and multimode fiber. However, it manifests itself most clearly in single-mode fiber, due to the absence of intermode dispersion.

Material dispersion is due to the dependence of the refractive index of the fiber on the wavelength. The expression for the dispersion of a single-mode fiber includes the differential dependence of the refractive index on the wavelength.

Waveguide dispersion is due to the dependence of the mode propagation coefficient on the wavelength

where the coefficients M(l) and N(l) are introduced - specific material and waveguide dispersions, respectively, and Dl (nm) - wavelength broadening due to the incoherence of the radiation source. The resulting value of the coefficient of specific chromatic dispersion is determined as D(l) = M(l) + N(l). The specific dispersion has the dimension ps/(nm*km). If the waveguide dispersion coefficient is always greater than zero, then the material dispersion coefficient can be either positive or negative. And here it is important that at a certain wavelength (approximately 1310 ± 10 nm for a stepped single-mode fiber), mutual compensation of M(l) and N(l) occurs, and the resulting dispersion D(l) becomes zero. The wavelength at which this occurs is called the zero-dispersion wavelength l 0 . Usually a certain range of wavelengths is indicated within which l 0 can vary for a given specific fiber.

Corning uses the following method to determine specific chromatic dispersion. Time delays are measured during the propagation of short light pulses in a fiber no less than 1 km long. After obtaining data samples for several wavelengths from the interpolation range (800-1600 nm for MMF, 1200-1600 nm for SF and DSF), delay measurements are resampled at the same wavelengths, but only on a short reference fiber (2 m length ). The delay times obtained on it are subtracted from the corresponding times obtained on the long fiber to eliminate the systematic error component.

For single-mode stepped and multimode graded fiber, the empirical Sellmeier formula is used: t (l) = A + Bl 2 + Cl -2. Coefficients A, B, C are adjustable, and are chosen so that the experimental points fit better on the t (l) curve. Then the specific chromatic dispersion is calculated by the formula:

where l 0 = (C/B) 1/4 is the zero dispersion wavelength, the new parameter S 0 = 8B is the zero dispersion slope, its dimension is ps/(nm 2 * km)), and l is the operating wavelength for which the specific chromatic dispersion is determined.

a) multimode gradient fiber (62.5/125)

b) single-mode stepped fiber (SF)

c) single-mode dispersion-shifted fiber (DSF)

Rice. 1.2

For a dispersion-shifted fiber, the empirical time delay formula is written as t(l) = A + Bl + Cl lnl, and the corresponding specific dispersion is given by

with parameter values ​​l 0 = e -(1+B/C) and S 0 = C/l 0, where l is the operating wavelength, l 0 is the zero dispersion wavelength, and S 0 is the zero dispersion slope.

Chromatic dispersion is related to specific chromatic dispersion by the simple relation t chr (l) = D(l)·Dl, where Dl is the width of the source radiation spectrum. The use of more coherent radiation sources, for example laser transmitters (Dl ~ 2 nm), and the use of an operating wavelength closer to the zero dispersion wavelength leads to a reduction in chromatic dispersion.

Specifications are taken from fibers produced by Corning.

3.3 OPTICAL FIBER

There are four main phenomena in optical fiber that limit the performance of WDM systems: chromatic dispersion, first- and second-order polarization mode dispersion, and nonlinear optical effects.

3.3.1 Chromatic dispersion

An important optical characteristic of the glass used in the manufacture of fiber is the dispersion of the refractive index, which manifests itself as the dependence of the speed of signal propagation on the wavelength - material dispersion. In addition, during the production of single-mode fiber, when a quartz filament is drawn from a glass preform, deviations in the geometry of the fiber and in the radial profile of the refractive index occur to varying degrees. The fiber geometry itself, together with deviations from the ideal profile, also makes a significant contribution to the dependence of the signal propagation speed on the wavelength; this is waveguide dispersion.

The combined influence of material and waveguide dispersions is called chromatic dispersion of the fiber, Fig. 3.16.

Fig. 3.16 Dependence of chromatic dispersion on wavelength

The phenomenon of chromatic dispersion weakens as the spectral width of the laser radiation decreases. Even if it were possible to use an ideal source of monochromatic radiation with zero lasing linewidth, then after modulation by an information signal, a spectral broadening of the signal would occur, and the greater the broadening, the higher the modulation speed. There are other factors that lead to spectral broadening of radiation, of which chirping of the radiation source can be distinguished.

Thus, the original channel is represented not by a single wavelength, but by a group of wavelengths in a narrow spectral range - a wave packet. Since different wavelengths propagate at different speeds (or more precisely, with different group velocities), an optical pulse that has a strictly rectangular shape at the input of the communication line will become wider and wider as it passes through the fiber. If the propagation time in the fiber is long, this pulse can mix with neighboring pulses, making it difficult to accurately reconstruct them. As the transmission speed and link length increase, the influence of chromatic dispersion increases.

Chromatic dispersion, as already mentioned, depends on the material and waveguide components. At a certain wavelength λ o chromatic dispersion becomes zero - this wavelength is called the zero dispersion wavelength.

Single-mode step-index silica fiber exhibits zero dispersion at 1310 nm. This fiber is often referred to as undispersion-biased fiber.

Waveguide dispersion is primarily determined by the refractive index profile of the fiber core and inner cladding. In a fiber with a complex refractive index profile, by changing the relationship between the dispersion of the medium and the dispersion of the waveguide, it is possible not only to shift the zero-dispersion wavelength, but also to select the desired shape of the dispersion characteristic, i.e. the form of the dependence of dispersion on wavelength.

The shape of the dispersion characteristic is key for WDM systems, particularly over dispersion-shifted fiber (ITU-T Rec. G.653).

In addition to the parameter λ o, the parameter S o is used, which describes the slope of the dispersion characteristic at wavelength λ o, Fig. 3.17. In general, the slope at other wavelengths is different from the slope at wavelength λo. The current value of the slope S o determines the linear component of the dispersion in the vicinity of λ o .

Rice. 3.17 Basic parameters of the dependence of chromatic dispersion on wavelength: λ o - wavelength of zero dispersion and S o - slope of the dispersion characteristic at the point of zero dispersion

Chromatic dispersion τ chr(usually measured in ps) can be calculated using the formula

τ chr = D(λ) Δτ L,

Where D(λ)- chromatic dispersion coefficient (ps/(nm*km)), and L- length of communication line (km). Note that this formula is not accurate in the case of ultra-narrowband radiation sources.

In Fig. Figure 3.18 shows separately the dependences of waveguide dispersion for fiber with unbiased (1) and biased (2) dispersion and material dispersion on wavelength.

Rice. 3.18 Dependence of dispersion on wavelength (chromatic dispersion is defined as the sum of material and waveguide dispersions.)

The chromatic dispersion of the transmission system is sensitive to:
increasing the length and number of communication line sections;
increasing the transmission speed (since the effective width of the source generation line increases).

It is less affected by:
reducing the frequency interval between channels;
increasing the number of channels.

Chromatic dispersion decreases when:
reducing the absolute value of the chromatic dispersion of the fiber;
dispersion compensation.

In WDM systems with conventional standard fiber (ITU-T Rec. G.652), chromatic dispersion should be given special attention as it is large in the 1550 nm wavelength region.