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Theory of information transfer in optical fibers: A tutorial review
Optical Fiber Technology 10 (2004) 150–170 www.elsevier.com/locate/yofte
Tutorial
Theory of information transfer in optical fibers: A tutorial review Akira Hasegawa Soliton Communications, Kyoto, Japan Received 24 June 2003; revised 11 October 2003
Abstract A tutorial and comprehensive review of theoretical development in information transfer in optical fibers is presented with particular emphasis on nonlinear effects and application of solitons. During the decade of 1990th, a remarkable progress has been achieved in the amount and distance of information transfer using optical fibers, thanks to progress in fiber manufacturing, ultra-high speed optical devices, and theoretical understanding of optical pulse propagation in fibers. The review covers basics of lightwave propagation in fibers, derivation of the master equation that describes information transfer in fibers, introduction of various practical problems, and solutions based on optical solitons. 2003 Published by Elsevier Inc.
1. Introduction A dielectric fiber can guide light wave since it has index of refraction larger than the air. In particular, fibers made of glass have achieved the minimum loss of about 0.15 dB/km at the wavelength around 1550 nm, the level of theoretical limit given by resonant absorption and Rayleigh scattering. Lightwaves in fibers face dispersion both in phase velocity and in group velocity. In particular, the dispersion in group velocity distorts information transmitted in fibers, since the lightwave information propagates at the group velocity and the group velocity dispersion (often called group dispersion) results in different arrival time of different components of information spectra. The dispersion property of a fiber is designated by the group delay, D, having a unit of ps/nm/km which indicates a delay of arrival time in ps of information E-mail address: [email protected]. URL: http://www.solitoncomm.com. 1068-5200/$ – see front matter 2003 Published by Elsevier Inc. doi:10.1016/j.yofte.2003.11.002
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carried by a wavelength separated by 1 nm from another wavelength over a distance of 1 km. Normally convention of D > 0 (D < 0) is used when the information at a shorter wavelength propagates faster (slower), a case often called anomalous (normal) dispersion. The group dispersion is controllable by designing proper profile of index of refraction of a fiber in the fiber cross section and fibers having a range of D of ±20 ps/nm/km have been made at various wavelengths. In particular, the dispersion-shifted fiber (DSF) that has zero group dispersion at the wavelength (∼1550 nm) of minimum loss has been produced in order to achieve distortion-less transmission of information. However, when a DSF is used, it has soon been discovered that the fiber nonlinearity due to the Kerr effect distorts information and induces cross talks among different channels in wavelength division multiplex (WDM) transmissions. As a result, attentions to the optical soliton, a stable pulse in the presence of both Kerr nonlinearity and (anomalous) dispersion, have emerged. The optical soliton in fibers was discovered earlier in 1973 and was introduced as an information carrier in fibers having group dispersion. It is somewhat ironical that the value of optical soliton has been recognized after the dispersion problem has been eliminated. This manuscript is a comprehensive tutorial review of information transfer in optical fibers. It is written in the manner appropriate for those who have elementary knowledge in optics and are interested in this subject but without much background knowledge. The manuscript is designed as self-contained so that the readers do not have to bother searching references. As the result, the manuscript has no citations. Those who are interested in more detailed knowledge on optical transmissions and optical solitons, the author recommend a reference for optical solitons, A. Hasegawa, M. Matsumoto, Solitons in Optical Fibers, Springer, Heidelberg, 2002, and for fiber nonlinear optics in general, G.P. Agrawal, Nonlinear Fiber Optics, Academic Press, San Diego, 1989.
2. Dielectric fibers as light waveguides In this section, we describe lightwave propagation in dielectric fibers. 2.1. Electric properties of dielectric materials In order to understand the lightwave propagation in optical fibers, we need some basic knowledge of electric properties of a dielectric material. When an electric field is applied to a dielectric, average position of electrons are shifted and the material polarizes and polarization current is induced. This current, like the current in a condenser is proportional to the time variation of the electric field. The effect of polarization is expressed by the use of electric displacement vector, D, as D = ε0 E + P.
(2.1)
(= 8.864 × 10−12
F/m) is the dielectric constant of vacuum and P represents the Here, ε0 polarization and is given by P = −ene ξ(E).
(2.2)
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In this expression, ne is the density of electrons that participate to the polarization, ξ represents the displacement of electron position in a dielectric molecule induced by the electric field E. One can obtain ξ as a function of E phenomenologically by means of equation of motion of electrons trapped in ionic potential of the dielectric. If we can assume that the potential is parabolic in ξ , the response becomes linear and the polarization becomes a linear function of electric field. The linear relation is conveniently expressed in Fourier amplitudes as P(ω) = χ(ω) E(ω).
(2.3)
Here ω is the angular frequency, the bar indicates the Fourier amplitude and χ(ω) is called the susceptibility. Substituting Eq. (2.3) into (2.1) gives the linear relation between D and E, D(ω) = ε0 ε∗ (ω) E(ω).
(2.4)
Here, ε∗ (= 1 + χ ) is the relative permittivity constant of the material. The light wave in dielectric material propagates at a speed lower than that in vacuum. The ratio of the phase velocity of the light wave in vacuum and that in a material is called the index of refraction often expressed by n0 (ω) and is defined by n0 (ω) =
ck √ ∗ = ε . ω
(2.5)
Here, k is the wave number. As is obvious from this derivation, if χ > 0, ε∗ is larger than unity and so is the index of refraction. In addition, it is clear that the group velocity, vg ≡ ∂ω/∂k = 1/k and the group dispersion, ∂vg /∂ω = −k vg2 are in general a function of frequency. We also note that these dispersion parameters in dielectric are given by k ≈
n0 (ω) c
and k ≈
2 ∂n0 . c ∂ω
(2.6)
For a glass, it is known that k (ω) becomes zero at the wavelength of 1300 nm and is negative (positive) for a longer (shorter) wavelength. The ionic potentials, however, are not exactly parabolic primarily due to the influence of neighboring ions. In many dielectric materials where the potential is symmetric, the lowest order modification to the parabolic potential appears as the reduction of the parabolic potential in proportion to ξ 4 . Since ξ is proportional to the electric field, this reduction 2 and the resultant index of induces increase of index of refraction in proportional to |E| refraction may be given by 2 = n0 (ω) + n2 (ω)|E| 2. n ω, |E| (2.7) Here n2 represents the coefficient of the nonlinear response and is caked the Kerr coefficient. For the fiber material, the value of n2 is very small and typically is about 10−22 (m/V)2 . Since the electric field of a lightwave in fibers with 1 mW of power has a value around 105 V/m, the nonlinear correction of the index of refraction is about 1 part 10−12 . It will be shown, however, that this tiny correction provides significant effects on information transfer in fibers.
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2.2. Fibers as light waveguides Let us now study the dispersion and linear effects in dielectric fiber guides. The dielectric wave guide may be characterized by an index of refraction n0 (ω) that depends on the radial coordinate. By the use of the electric displacement vector D and the magnetic flux density vector B, the Maxwell equations can be expressed as ∂B , (2.8) ∂t ∂D ∇ × B = µ0 . (2.9) ∂t If we take curl of Eq. (2.8) and substitute Eq. (2.9) and use the Fourier amplitude expression in time for the electric field, E, ∇ ×E=−
ω2 ∇ 2 E − ∇(∇ · E) + 2 E= 0. cD
(2.10)
Here c c (2.11) cD = √ = ∗ n0 ε is the speed of light in the dielectric material, which is a function of transverse coordinates. Because of the inhomogeneity of n0 , D ∇ ·E=∇ · = 0. (2.12) ε0 ε ∗ Thus we note that unlike the case of a metallic waveguides, the field cannot be separated to either TM or TE mode. The exact field in a fiber has all the three components both in magnetic and in electric fields and can be described by the wave equation (2.10). However, if the variation of n0 is small so that λ∂ ln(n0 )/∂r 1, where λ is the wavelength, the field may be divided to either TE or TM mode. Let us take a case of TE mode. The electric field may be expressed by a scalar function φ in the form E = ∇ × φz = ∇⊥ φ × z and the wave equation becomes 2 ω 2 n (r) − k ∇⊥ φ = 0. ∇ 2 ∇⊥ φ + 0 c2
(2.13)
(2.14)
Here, ∇⊥ is the gradient operator in the transverse directions and z is the unit vector in the axial (z) direction. The dispersion relation k(ω) may be formally obtained from (2.14) in terms of the eigenfunction for ∇⊥ φ, 2 2 ω2 /c2 |∇⊥ φ|2 n20 (r) dS − |∇⊥ φ| dS 2 . (2.15) k = 2 |∇⊥ φ| dS This expression shows clearly that the dispersion relation is determined by the combination of dielectric property of the fiber material and the waveguide property of the fiber. With a
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Fig. 1. Dispersion relation of a single mode fiber (solid line).
proper design of the profile of the index of refraction in the radial direction, the wavelength at which the group dispersion vanishes can be shifted to a longer wavelength. In particular, the dispersion-shifted fiber (DSF) has zero group dispersion at the wavelength of 1550 nm, where the fiber loss is minimum. The property of the eigenfunction depends on the two-dimensional mode numbers in radial and azimuthal directions. Therefore, the dispersion property also depends on the mode number. However, the dispersion relation is uniquely determined for a lowest mode number that exists at wavelength longer than the cutoff wavelength for higher modes as seen in Fig. 1. A fiber that allows a single mode for certain range of wavelength is called a single mode fiber (SMF). Normally SMF has a negative (positive) k for wavelengths longer (shorter) than 1300 nm. Most fibers, albeit small, have asymmetric dielectric properties in the cross sectional plane. Then the eigenfunction depends on the polarization of the mode. In this case the group dispersion also depends on the polarizations of the mode. The group dispersion that originates from the polarization is called the polarization mode dispersion (PMD). PMD leads to the breakdown of the single mode property. Since such asymmetry normally varies randomly along the axial direction of the fiber, PMD induces random walk of each polarization mode around the average group velocity.
3. Deformation of information In this section we discuss how the information is transferred in fibers that have group dispersion and nonlinearity. To study how the information is transmitted it is essential to study how the information, or modulation, propagates in a fiber. Information may be lost not only by the fiber loss but also by its deformation that arises during the propagation in fibers due to various properties of the fiber. As is well known, the information propagates
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propagates at the group veat the group velocity, ∂ω/∂k. So the modulation amplitude E locity. Here, the modulation amplitude is related to the full electric field amplitude E(z, t) through 1 t)ei(k0 z−ω0 t) + c.c. . (3.1) E(z, t) = E(z, 2 In this expression, k0 and ω0 are the wave number and frequency of the carrier wave. We t) is a slowly varying function of coordinate also note that the modulation amplitude E(z, and time as compared with E(z, t). t); i.e., if The loss of information originates from a deformation of the modulation E(z, E(z, t) does not change in z, there is no loss of information. t) Information can be put on the lightwave at z = 0 by choosing a proper shape of E(0, in time. The amount of information depends on how rapidly E(0, t) varies, which can be designated by the width of the Fourier spectrum, E(Ω), ∞ E(Ω) =
t)eiΩt dt. E(0,
(3.2)
−∞
The amount of information (bit/s) that the lightwave carries is approximately given by the spectral width Ω. Therefore the more the amount in information, the wider the spectral width becomes. Consequently to achieve an ultra-high speed of information transfer in a t) having a wide spectral width. fiber, we should study the behavior of E(z, Deformed information may be reconstructed by a regeneration scheme. The scheme generally requires detecting of information, which may be done by rectification of the R , t) at the repeater located at z = zR , by regenerating the original lightwave to extract E(z t) from E(z R , t) and by modulating a fresh lightwave by E(0, t). In an information E(0, early stage of optical communication, this scheme has commonly been adapted. However, it has soon become clear that the repeater cost is the major obstacle for the increase of transmission rate because it requires electronic circuits having increasingly higher speed of operation. Consequently, most high-speed transmission systems at present uses an alloptical scheme with loss compensated for by periodic optical amplifications. When the fiber loss is effectively eliminated the major limitations come from the group velocity dispersion and nonlinearity of the fiber. Let us first discuss the effect of the group dispersion. The group dispersion is the effect in which the group velocity varies as a function of the lightwave frequency. The group dispersion originates from the combination of the wave guide property and the material property of the fiber. In the presence of the group dispersion, information carried by differ t) propagate at different speed thus arrives at different ent frequency components of E(0, time. The relative delay t of arrival time of information at frequencies ω1 , and ω2 at distance z is given by tD =
∂vg /∂ω(ω2 − ω1 )z z z − = . vg (ω1 ) vg (ω2 ) vg2
(3.3)
∂vg = −k vg2 , ∂ω
(3.4)
If we use vg =
1 ∂ω = , ∂k k
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Eq. (3.3) becomes tD = −k (ω2 − ω1 )z.
(3.5)
Equation (3.4) shows that the difference of arrival time of information is proportional to the group dispersion, k , the difference of the frequency components ω2 − ω1 , and the distance of probation z. We note that if k < 0 (called anomalous dispersion regime), higher frequency component of information arrives earlier and for normal dispersion, k > 0, vise versa. If the information at different frequency components arrives at different time, the information may be lost. The problem becomes more series if the amount of information is large so that ω2 − ω1 is large. For a practical purpose, the group dispersion is designated by a group delay parameter, D, as defined by delay of arrival time in ps unit for two wavelength components separated by 1 nm over a distance of 1 km. From Eq. (3.5), the group delay may be described by 2π 2π 2π − z = −k c 2 λz. D = k c λ1 λ2 λ Thus, by taking λ = 1 nm and z = 1 km, D in the unit of ps/nm/km is related to k through 2πck . (3.6) λ2 For a single mode fiber, D becomes 0 (16 ps/nm/km) at λ ≈ 1300 nm (1550 nm) while for a dispersion shifted fiber, D becomes 0 around at λ = 1550 nm. This indicates that two frequency components of a pulse with the wavelength separation of 1 nm arrive with a time delay on the order of a few pc when they propagates over a distance of 1 km in a typical fiber. An important quantity that designates the fiber dispersion effect is the dispersion distance z0 defined by the distance over which a pulse width becomes doubled due to the group delay. Since the pulse width τ is given by the inverse of the spectral width ω, the dispersion distance is given from Eq. (3.5), as D=−
z0 =
τ 2 . |k |
(3.7)
We now discuss the effect of fiber nonlinearity on the information transfer. For an ordinary fiber, the lowest order nonlinearity originates from the Kerr effect where the index 2 . In the presence of refraction, n, changes in proportion to the electric field intensity |E| of the Kerr effect, the index of refraction n is given by Eq. (2.7). Thus the wave number becomes ω 2 . n0 (ω) + n2 (ω)|E| (3.8) k= c Equation (3.8) indicates that the Kerr effect induces a nonlinear phase shift ΦN through the nonlinear part of the wave number kN given by ΦN = kN z =
2z 2πn2 |E| . λ
(3.9)
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becomes about 105 V/m in a typical fiber. For a lightwave with a peak power of 1 mW, |E| 2 −12 2 = 10−12 , since z/λ = 1012 Thus n2 |E| has a value of 10 . Therefore, even if n2 |E| for z ≈ 1500 km and λ ≈ 1500 nm, ΦN between the high intensity portion and the low intensity portion can become 2π over this distance of propagation. This indicates that the phase information is lost over a distance of 1000 km if the light wave power is as low as 1 mW. In addition it creates a mixture of information in amplitude and phase. This means that information transfer by means of coherent modulation is not appropriate for a light wave in fibers for a propagation distance beyond 1000 km.
4. Master equation of information transfer in fibers with dispersion and nonlinearity The information carried by the lightwave in fibers is expressed by the modulation t) shown in Section 3. We now derive the equation that describes evolution of E(z, t) E(z, along the direction z of the propagation of information. The most convenient way to derive 2 ), around the carrier the envelope equation is to Taylor expand the wave number k(ω, |E| 2 , frequency ω0 and the electric field intensity |E| 1 ∂k 2 k − k0 = k (ω0 )(ω − ω0 ) + k (ω0 )(ω − ω0 )2 + |E| , 2 2 ∂|E|
(4.1)
and to replace k − k0 by the operator i∂/∂z and ω − ω0 by −i∂/∂t, and to operate on the t). The resulting equation reads electric field envelope, E(z, 1 ∂ 2E ∂E ∂E ∂k 2 − k 2 + + k i |E| E = 0. (4.2) 2 ∂z ∂t 2 ∂t ∂|E| We note that to obtain k in this expression, we should go back to (2.15) and take the second derivative of k with respect to ω. It is often convenient to study the evolution of in the coordinate moving at the group velocity τ = t − k z. Then the envelope equation E becomes 1 ∂ 2 E ∂k 2 ∂E − k + |E| E = 0. i (4.3) 2 2 ∂z 2 ∂τ ∂|E| For a lightwave envelope in a fiber, the coefficients of this equation depends on the fiber geometry and modal structure of the guided lightwave. In particular, for a single mode fiber (SMF) k = 0 occurs at λ = 1300 nm, which is determined primarily by the glass property itself, while, k becomes zero at λ = 1550 nm for a dispersion-shifted fiber (DSF) because of the wave-guide property. A linear wave packet deforms due to the group velocity dispersion k . For a lightwave pulse with a scale size of τ , the deformation takes place at the dispersion distance z0 = τ 2 /|k |. Thus it is convenient to introduce the distance Z normalized by z0 and time T normalized by τ . For a fiber having anomalous dispersion, Eq. (4.3) reduces to i ∂ 2q ∂q = + i|q|2q ∂Z 2 ∂T 2
(4.4)
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and for a fiber having normal dispersion, the coefficient of the first term on the right hand side become negative. In this expression, q is the normalized amplitude given by ω0 n2 gz0 1/2 q= E. (4.5) c Here, g is a reduction factor (∼1/2) of the electric field intensity due to the wave-guide effect. If we compare this expression with the nonlinear phase shift, Eq. (3.9), we can identify that q is nothing but the nonlinear phase shift at the dispersion distance. This indicates that if the amplitude is such that the nonlinear phase shift becomes order unit at the dispersion distance, the normalized amplitude, q becomes order unity. In other words, the nonlinear term and dispersion term in Eq. (4.4) becomes comparable for such amplitude. Equation (4.4) is the master equation that describes the evolution of information propagation in fibers with dispersion and nonlinearity for an electric field having intensity such that the nonlinear phase shift at the dispersion distance is order unit. The equation is often called the nonlinear Schrödinger equation. In a practical system, the fiber dispersion k often varies in Z. In addition, fiber may have amplifiers with gain G(Z) and the loss with loss rate Γ per dispersion distance. Then Eq. (4.4) should be modified to i ∂ 2q ∂q = d(Z) 2 + i|q|2q + G(Z) − Γ q. ∂Z 2 ∂T
(4.6)
Here d(z) is the group dispersion normalized by its average value.
5. Linear and nonlinear responses of fibers In this section, we discuss evolution of wave packets in fibers in the presence of dispersion and nonlinearity. To understand the problem, we treat these effects separately. As the nonlinear effects, we discuss self and cross phase modulation, Raman effect and four wave mixing. 5.1. Linear response of a fiber with group dispersion In the absence of the nonlinear term, Eq. (4.6) can be easily integrated by means of the Fourier transformation in time, ∞ q(Z, ¯ Ω) =
q(Z, T )eiΩT dT .
(5.1)
−∞
Applying the Fourier transformation on Eq. (4.6) and solving the result for q(Z, ¯ Ω) we obtain
Z Z Ω2 (5.2) G(Z ) − Γ dZ . q(Z, ¯ Ω) = q(0, ¯ Ω) exp −i d(Z ) dZ + 2 0
0
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q(Z, T ) is then obtained from the inverse transformation, 1 q(Z, T ) = 2π
∞
q(Ω, T )e−iΩT dΩ.
(5.3)
−∞
In particular we note that if the average (or integrated) dispersion is zero and the average (or integrated) gain balances with the loss q(Z, T ) can recover the initial modulation q(0, T ) exactly. That is if the fiber response is completely linear and if the amplifier has no noise one can design a fiber transmission line such that the initial information can be transmitted without loss by making d(Z) = 0 and G(z) = Γ . As an example of linear transmission, let us consider a case of a loss-less fiber in which the initial pulse shape is given by a Gaussian, q0 2 2 e−T /(2T0 ) . (5.4) q(0, T ) = √ 2πT0 The Fourier transform of q is obtained by the formula ∞ a −y 2 /(4a) −ax 2 −ixy e e e dx = , π
(5.5)
−∞
to give q0 q(0, ¯ Ω) = √ 2πT0
∞
e−T
2 /(2T 2 )+iΩT 0
dT = q0 e−Ω
2 T 2 /2 0
.
(5.6)
−∞
If we substitute Eq. (5.6) into (5.2) and assume d(Z) to be a constant, D0 , we have q(Z, ¯ Ω) = q0 e−(1/2)(T0 +iD0 Z)Ω . 2
2
(5.7)
If we further substitute this result into Eq. (5.3), we can obtain the wave packet q at a given distance Z, q0 q(Z, T ) = 2π =
∞
e−(1/2)(T0 +iD0 Z)Ω e−iΩT dΩ 2
2
−∞
−T02 T 2 + iD0 ZT 2 exp . 2(T04 + D02 Z 2 ) 2π(T02 + iD0 Z) q0
(5.8)
Inspecting this result, we can observe various interesting points. First, we note that the pulse width T02 + D02 Z 2 /T02 increases with Z independent of the sign of the dispersion. That is the fiber dispersion induces the increase of the pulse width approximately in proportion to the distance of propagation. Next we note that the anomalous dispersion, D0 > 0, induces the frequency to decrease in time (called a negative chirping), while for a normal dispersion, a positive chirping is induced. In a practical system, amplifiers always add noise, which is proportional to the gain and the gain bandwidth. As a result, one needs to provide a sufficiently large initial amplitude
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in order to maintain a proper signal to noise ratio. The finite level of intensity induces the nonlinear self-phase shift due to the Kerr effect as shown in Eq. (3.9) and also in 2 in Eq. (3.9) is properly Section 5.2. We note, however, that if the time dependence of |E| chosen, the self-phase shift may counter balance the phase shift induced by the (anomalous) dispersion shown in Eq. (5.8). This is the essential feature of the soliton formation. We will discuss ways to overcome the frequency chirp and pulse spread by means of nonlinearity in the next section. 5.2. Self phase modulation We now study the evolution of the wave packet due to the nonlinearity. If we retain only the nonlinear term in Eq. (4.6), we have ∂q (5.9) = i|q|2q. ∂Z If we assume |q|2 is constant in distance, Eq. (5.9) can be formally integrated to give Z |q|2 dZ .
q(Z, T ) = q(0, T ) exp i
(5.10)
0
This result indicates that the phase of q varies along the direction of propagation. This is the reconstruction of the nonlinear phase shift introduced in (3.9). This phenomenon is called the self phase modulation. For example, if we expand the pulse shape in time and approximate it by 1 − T 2 , we can see from Eq. (5.10) that the phase varies in proportion to −T 2 . Thus we can see that the nonlinearity also induces frequency chirp. We further note that in an anomalous dispersion regime, the direction of the chirp produced by the nonlinearity is opposite of that produced by the dispersion. This indicates that nonlinearity induced chirp can cancel the dispersion induced chirp in anomalous dispersion regime. This process leads to the formation of a soliton in the wave envelope. 5.3. Cross phase modulation In the presence of other waves or modulations, the other waves induce phase modulation through the Kerr effect in a manner similar to the self phase modulation. Let us assume that the modulation q consists of two components, q = q1 + q2 ≡ q¯1 e−iΩ1 T + q¯2 e−iΩ2 T .
(5.11)
Here Ω1 and Ω2 are the modulation frequencies of the two waves. The nonlinear response for the q1 wave then becomes ∂ q¯1 = i |q¯1 |2 + 2|q¯2|2 q¯1 + i q¯22 q1∗ e−i(2Ω2 −Ω1 )T . (5.12) ∂Z We can see here that in addition to the self phase modulation, the phase modulation on q1 is induced also by q2 wave as shown by the second term on the right hand side. This effect is called the cross phase modulation and plays a detrimental role in information transfer in wavelength division multiplex (WDM) transmission systems. The third term on the right hand side plays no significant role since the effect is averaged out in time.
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5.4. Four wave mixing In the presence of many other waves, the Kerr nonlinearity induces additional important effect called the four wave mixing. We consider a case in which there exist four waves given by q≡
4
q¯j e−iΩj T .
(5.13)
j =1
In this case, in addition to the self and cross phase modulations, the evolution of q1 wave is influenced by the phase mixing term that generated the third term in Eq. (5.12). That is, ∂ q¯1 = 2i q¯2 q¯3 q¯4∗ e−i(Ω2 +Ω3 −Ω4 −Ω1 )T + 2i q¯ 2q¯3∗ q¯4 e−i(Ω2 −Ω3 +Ω4 −Ω1 )T ∂Z + 2i q¯2∗ q¯3 q¯4 e−i(−Ω2 +Ω3 +Ω4 −Ω1 )T .
(5.14)
From this expression it is clear that when the resonant condition, Ω1 + Ω4 = Ω2 + Ω3 , Ω1 + Ω3 = Ω2 + Ω4 , or Ω1 + Ω2 = Ω3 + Ω4 , the phase of q1 is systematically modified without phase mixing. This effect is called the four wave mixing (FWM) and also plays a crucial role in WDM systems. In the presence of group dispersion, however, the four wave mixing is suppressed due to the phase mixing in the wave number space. Let us show how this occurs. In the presence of modulation, Ω, the wave number, K is induced, where K is given by K = Ω/vg = k Ω. In the absence of group dispersion, k is independent of Ω, thus when the resonant condition in Ω is met, the wavenumber resonance also takes place. However, in the presence of group dispersion, k depends on Ω and the wavenumber resonance ceases. This results in the phase variation in the products of three amplitudes in the right hand side of Eq. (5.14) that is proportional to k Ω 2 Z and the systematic phase modulation disappears. Thus the effect of FWM may be avoided in the presence of fiber having significant group dispersion. However, as shown in Section 5.1 the group dispersion distorts information in linear transmission systems. The modern technology in WDM systems focuses on means to suppression information distortion due to the group dispersion and to the FWM simultaneously. 5.5. Raman effect In addition to the Kerr nonlinearity fiber possesses an important nonlinear effect called Raman effect. The Raman effect is a process of photon scattered off by material resonances at a frequency, ωp . In fibers the optical phonon plays the role of the material resonance. As a result of the scattering, the photon at frequency ω generates a new photon having a frequency ω −ωp . It is known that the resonant frequency ωp is around at the wavelength of 100 nm. Phenomenologically the Raman effect may be expressed in terms of the imaginary part of the Kerr coefficient, say, α. If we take this effect in the cross phase modulation term, we can construct a coupled equation between q1 and q2 given by ∂ q¯1 (5.15) = i |q¯1 |2 + 2|q¯2|2 q¯1 − 2α|q¯2 |2 q¯1 ∂Z
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and ∂ q¯2 = i |q¯2 |2 + 2|q¯1|2 q¯2 + 2α|q¯1 |2 q¯2 . (5.16) ∂Z Here, q1 and q2 waves have frequencies given respectively by ω and ω − ωp . The Raman effect, as can be seen from these equations, transfers energy from the higher frequency wave to the lower frequency mode. The quantity α, the Raman coefficient normalized by the Kerr coefficient, has a value around at 0.1 when the two wave has a wavelength separation of around 100 nm. The Raman effect also plays important role in WDM in cascading energy from shorter wavelength modes to lower wavelength modes. The Raman effect can also be used as a mean of amplifying optical signal. If the pump wave is introduced in a fiber, the Raman gain is distributed along the fiber, Raman amplifiers effectively provides amplifications having amplifier spacing much shorter than discrete amplifiers such erbium doped fiber amplifiers (EDFA). As a result, the integrated noise figure of a Raman system can be made significantly lower than an EDFA system. In advanced WDM transmission systems, the Raman amplifications are more favorably chosen.
6. Optical soliton solution In the previous section, we learned that fibers have group dispersion and Kerr nonlinearity and these effects distort information as it transferred through a fiber. What is interesting and fortunate is the fact that Eq. (4.4) can be integrated for an arbitrary initial condition, q(0, T ) and in particular it has an exact and stable solution called a soliton given by sech(T ). The solution can be characterized by the amplitude η, the time central position T0 , and the phase σ , 2 η q(T , Z) = η sech η(T − T0 ) exp i Z + iσ . (6.1) 2 It is straightforward to check that sech(T ) does satisfy the nonlinear Schrödinger equation (4.4). In addition by adding the amplitude η, as well as the phase and position, one can identify (6.1) can be seen to be a solution of (4.4). This solution is called the one-soliton solution. The solution is obtained by the balance between the group dispersion and Kerr nonlinearity. In addition, the stability of the solution against various perturbation has been proven and numerically checked. In other words, the soliton solution is free from either the dispersive distortion or from self-phase modulation. Thus it is quite natural to use solitons as information carrier in fibers since all other format will face distortion either from dispersion or nonlinearity. We note that the solution (6.1) is obtained in the coordinate moving at the group velocity in the rest frame. Therefore in the rest frame, soliton moves at the group velocity. If there exists a frequency shift, the group velocity is modified due to the group dispersion. The amount of the group velocity shift is proportional to the frequency shift. In the presence of a frequency shift, the solution (6.1) is thus further modified to
i (6.2) q(T , Z) = η sech η(T + κZ − T0 ) exp −iκZ + η2 − κ 2 Z + iσ . 2
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Here, κ is the frequency shift normalized by 1/t0 . This solution is the Galilean transformation of (6.1). Thus the one-soliton solution of the nonlinear Schrödinger equation (1.7) is given by a sech(T ) function, which is characterized by four parameters, the amplitude η (also the pulse width), the frequency κ (also the pulse speed), time position T0 , and the phase σ . In particular, we note that the soliton speed κ is a parameter independent of the amplitude unlike soliton solution of other equations. This fact is very important for the use of an optical soliton as a digital signal since if pulses of different amplitudes propagate at different speeds, information carried by such pulses are all mixed up and lost after some distance of propagation. When one soliton is used to represent one digit, the major cause that leads to the loss of information is the timing jitter which originates from the variation of the soliton position T0 since the wave form of the soliton is robust. This situation is quite different from a non-soliton pulse where the major cause of information loss is due to the deformation of the waveform itself. The variation of T0 originates from the frequency shift κ and the finite dispersion (since Z is normalized by the dispersion distance). This is because when the soliton frequency shifts, it modifies the group velocity through the group dispersion, and results in the change in soliton position. In real transmission systems, various effects induce the frequency shift of solitons. For example, the amplifier noise randomly modulates the soliton frequency and induces random variation of the soliton position (random walk). Interactions between adjacent soliton also produce timing jitter caused by the frequency shift. In order to avoid the frequency shift due to the interaction, two adjacent solitons should be separated enough ( six times pulse width in time). This requirement is a severe drawback for soliton based communications because it requires bandwidth at least three times wider than linear systems where the pulse separation can be twice or less than twice than the pulse width. When solitons are used in a wavelength division multiplex (WDM) system, they are free from distortions caused by either the cross phase modulation or from the four wave mixing since solitons are robust against collisions with other solitons. Therefore solitons are better choice than linear pulses also in WDM systems. However, imperfect (or asymmetric) collision between solitons at different wavelength channels can induce frequency shift since the robustness is broken in non-ideal situation where the nonlinear Schrödinger equation breaks down. Imperfect collisions occur at the input, output, or at amplifiers. Although solitons are clearly better choice as the information carrier in optical fibers because of their robust nature, the timing jitter caused by various effect as described above can lead to the loss of information even though the wave shape is intact through out the transmission. These problems can be solved to some extent by various means of soliton control that either reduce or restore the frequency shifts.
7. Dispersion managements and dispersion managed solitons As was shown in Section 5.1, an optical pulse can propagate free of distortion if the fiber response is completely linear and integrated dispersion and the loss compensated by gain over the entire span is zero even if the local value of dispersion is finite and fiber has a finite loss. This fact has been extensively used in linear transmission systems to avoid
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Fig. 2. A canonical dispersion map for dispersion managed soliton systems.
nonlinear degradation of the information. Various dispersion profiles (or dispersion maps) have been tried to minimize the nonlinear effects. Basically the idea is to introduce locally relatively large dispersion to make nonlinear effects relatively less and to compensate the accumulated dispersion at the end of the line so that the integrated dispersion becomes zero. A method of programming dispersion is called the dispersion management. Dispersion management was also found to be effective in soliton transmissions since the timing jitter, which is the major cause of bit error in soliton systems, also depends on the local dispersion. 7.1. Dispersion managements To illustrate effects of programmed dispersion, let us consider the propagation of an optical pulse in a lossless fiber in which the group velocity dispersion varies periodically as shown in Fig. 2. If the fiber is linear, the pulse that starts at point a can recover the original shape completely at point e if the dispersion in anomalous region d0 at a Z < b and d Z < e is opposite from that in normal dispersion region, −d0 , at b Z < d, provided ab + de = bd. However, if the dispersion map is not symmetric, the pulse at Z = e cannot recover the original shape. Let us consider what happens if we take into account of the fiber nonlinearity. As was discussed in Section 5, the major nonlinearity of a fiber originates from the Kerr effect. This produces self-induced phase shift and induces chirp in the pulse. The fiber dispersion induces also a chirp. In an anomalous (normal) dispersion region, the direction of the chirp due to nonlinearity is opposite from (same as) that due to the dispersion. A soliton is produced if the self-induced chirp due to nonlinearity counteracts that due to the anomalous dispersion. Let us now consider what happens if the local dispersion is much larger than that which allows a soliton solution for the given pulse intensity. Then the pulse acquires certain amount of chirp at the end of the anomalous dispersion section Z = b in Fig. 2. However, this amount of chirp is reversed as the pulse propagates through the normal dispersion section and may be completely reversed at point Z = d. Then the amount of the chirp can again become zero at the end of periodicity at Z = e. In fact, it can be shown that for a proper choice of the initial pulse width and pulse intensity, one can construct a nonlinear pulse that can recover the initial pulse shape at each end of the periodic dispersion map for a wide range of the value of the average dispersion d. This includes d > 0, d = 0, and even d is slightly negative (average normal dispersion). Furthermore it can be shown that even if the pulse does not recover its original shape at
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the end of one period, it recovers the original shape after several spans of the periodic map with a proper choice of the initial pulse width and the intensity. Such nonlinear stationary pulses are produced by a balance between properly averaged dispersion, weighted by the spectral change, and nonlinearity and behave quite similar to the ideal soliton in terms of their stability and dynamic range of nonlinearity and dispersion. Consequently, they are often called a dispersion-managed soliton (DMS). Dispersion managed solitons are attractive because, by taking a map having zero or close to zero average dispersion, their timing jitter induced by frequency modulation can be made to zero or close to zero, since the rate of change of the time position T0 is proportional to the amount of the dispersion. As is clear from the above discussion, dispersion managed solitons have much larger dynamic range as compared with linear pulses that require exact cancellation of dispersion. 7.2. Dispersion managed soliton In this section we show how a dispersion-managed soliton is created using the Lagrangian method. The envelope equation for properly normalized electric field of optical field in a fiber with the group velocity dispersion variation, d(Z) in the direction of propagation Z is given by Eq. (4.6). This equation may be reduced to a Hamiltonian structure by introducing a new amplitude u, u = q/a,
(7.1)
where da = −Γ + G(Z) a. dZ u then satisfies
(7.2)
∂u d(Z) ∂ 2 u =i + ia 2 (Z)|u|2 u. (7.3) ∂Z 2 ∂T 2 Equation (7.3) is not integrable because of inhomogeneous coefficients a(Z) and d(Z). The dispersion managed soliton normally requires the average d(Z), d(Z), much smaller than the local |d(Z)|,
1 d(Z) ≡ Lp
Lp d(Z ) dZ d(Z),
(7.4)
0
where Lp is a periodic length of the dispersion map. As was shown in Section 5.1, if
d(Z) = 1 and in the absence of nonlinearity, Eq. (7.3) has an exact periodic solution given by a Gaussian with frequency chirp. Thus, let us apply an additional transformation of u to a new amplitude f (T , Z) by eliminating the chirp C and frequency shift κ through u(T , Z) = p(Z)f p(Z) T − T0 (Z) , Z
2 C(Z) (7.5) × exp i T − T0 (Z) − iκ(Z)T + iθ0 (Z) , 2
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where p, C, and κ represent the inverse of the pulse width, chirp coefficient and frequency shift all real functions of Z. T0 and θ0 represent the soliton position and the phase. When Eq. (7.5) is substituted into Eq. (7.3), the magnitude of the dispersion term gives 1 ∂2 1 √ ∂2 d(Z) 2 = d(Z)p2 p 2 , 2 2 ∂T ∂η
η = p(T − T0 ),
and the nonlinear term √ a 2 (Z)|u|2 u = p pa 2 (Z)f 3 .
(7.6)
(7.7)
We note here that the dispersion coefficient now becomes p2 d. This is because if the pulse width changes in Z, spectrum width changes in proportion to p and the dispersive effect becomes proportional to p2 d. This implies that a nonlinear stationary solution in the case of inhomogeneous dispersion is obtained by a balance between the reduced dispersion
p2 d(Z) and the nonlinearity pa 2 (Z) rather than the dispersion and nonlinearity like in the case of an ideal soliton in a constant dispersion. We now apply Lagrangian method to construct the solution of Eq. (7.3). We first note that Eq. (7.3) can be constructed by the variation of the Lagrangian density L given by 1 1 i uZ u∗ − u∗Z u + a 2 (Z)|u|4 − d(Z)|uT |2 . (7.8) 2 2 2 Here, uZ and uT are respectively partial derivatives with respect to Z and T . It is straightforward to see that Eq. (7.3) can be obtained by taking the functional derivative of L with respect to u∗ ,
∞ n δL ∂ ∂ ∂ ∂ n ∂ L = 0. (7.9) = + (−1) − δu∗ ∂u∗ ∂T n ∂u∗nT ∂Z ∂u∗Z L(T , Z) =
n=1
The merit of using Lagrangian method is that once the Lagrangian density is known, one can construct evolution equations for parameters that characterize the solution (7.4) from the variational principle. If we substitute the ansatz (7.5) into (7.8) and assume explicit variation of f in Z is negligible, and integrate the result over T , one can construct the Lagrangian L for parameters p, C, κ, T0 , and θ0 , ∞ L(p, C, κ, T0 , θ0 ) ≡
L(T , Z) dT
−∞
=
I2 dC a 2 pI4 dp2 I3 dC 2 I2 dκ 2 I1 − − − 2 − 2 2 2 2p 2 2p dZ dθ0 dκ − , + I1 T0 dZ dZ
where ∞ I1 = −∞
∞ 2
f (T ) dT ,
I2 = −∞
T 2 f 2 (T ) dT ,
(7.10)
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∞ I3 = −∞
df dT
2
167
∞ dT ,
I4 =
f 4 (T ) dT .
(7.11)
−∞
Now, variation of the Lagrangian gives the Lagrangian equation of motion ∂ ∂L ∂L = 0, − ∂Z ∂ r˙ ∂r
(7.12)
for rj = p, C, κ, T0 , and θ0 . Here, r-dot is the derivative with respect to Z. By taking a specific pulse shape for f (T ), one can evaluate the various moments in expression (7.11). In particular, if we take a Gaussian pulse of the type (5.4), Eq. (7.12) can be expressed by a coupled set of evolution equations for these parameters, dp = −pCd, dZ dC a 2 p3 = −C 2 d − √ + 2dp4 , dZ 2 dκ = 0, dZ dT0 = −κd, dZ dθ0 κ 2 d 5a 2p = − dp3 + √ . dZ 2 4 2
(7.13) (7.14) (7.15) (7.16) (7.17)
By solving these coupled equations, for a given map profile, d(Z), one can obtain the evolution of the pulse properties as a function of Z. Let us study the evolution of the pulse property by taking a canonical dispersion map that has piece-wise constant dispersion d = ±d0 as shown in Fig. 2. If we eliminate Z from Eqs. (7.13) and (7.14), we can obtain the phase space relationship between the pulse width and the chirp parameters, 2 C2 a 2 p2 d C2 = + √ − p3 . (7.18) dp 2 p 2 2 2d If d is piece-wise constant, Eq. (7.18) can be easily integrated for a lose-less fiber a(Z) = a0 (= constant), C2 p4 = C0 p2 − − Ap3 , 2 2
(7.19)
a2 A = √0 2 2d0
(7.20)
where
represents the strength of nonlinearity. One can obtain a periodic solution with the periodicity Lp by a proper choice of initial conditions p(0) and C(0) so that p(Lp ) = p(0) and C(Lp ) = C(0). For a linear pulse with A = 0, the periodic solution for f is a Gaussian with a periodically varying chirp parameter C(Z) and the trajectories in p–C plane at
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Fig. 3. Phase space trajectories of p and C for the dispersion map of Fig. 2.
d = d0 and d = −d0 completely overlays themselves. However, the nonlinearity produces a gap in these trajectories as shown in Fig. 3, because of the frequency chirp produced by the self-induced phase shift. The peak value of p at point c (normal dispersion regime) is less than that at anomalous dispersion regime because of the nonlinearity induced self phase shift. This makes
p2 d > 0 even if d = 0 and the balance between the weighted average p2 d and the nonlinearity produces the stationary solution. This situation is analogous to an ideal soliton solution that is constructed by a balance of (constant) dispersion and nonlinearity. However, in the dispersion-managed case, the above argument indicates that the nonlinear stationary solution is possible even for d < 0 or d = 0, provided that p2 d > 0 and proper nonlinearity exists. An ideal soliton solution for a fiber with d = d0 = constant, can be constructed for an arbitrary value of d0 by a proper choice of the amplitude, while the linear stationary solution exists only for d0 = 0. Similarly for a dispersion managed case, a nonlinear stationary periodic pulse can be constructed for an arbitrary value of d by a choice of initial amplitude and chirp, while the linear stationary pulse is possible only for d = 0. This allows DM solitons to have much larger tolerance in the fiber dispersion. We here note again that, if the system is linear, the trajectory in p–C plane shown in Fig. 3 returns at the original point d only when the average dispersion d is exactly zero. We further note that in a nonlinear system, if p(0) and/or C(0) is not chosen so that after one period p(L) and/or C(L) does not return to the original value, it was numerically confirmed p(nL) and/or C(nL) returns to a limit area in p–C plane, where n = 1, 2, . . . . In other words Eqs. (7.13) and (7.14) in general have doubly periodic solution.
8. Review of experimental results Experiments on optical solitons in fibers have a long history now. The first demonstration of existence of soliton and soliton phenomena was made in 1980 [1] some seven years after the first theoretical prediction. Here 7 ps pulse having a peak power over 1 W was sent to a single mode fiber of 700 m long. A color center laser is used as the light source.
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The first long distance transmission of a soliton pulse was achieved in 1988 [2]. Raman amplifications are used to compensate for the fiber loss. The experiment was done on 42 km recirculating loop of a single mode fiber using 55 ps pulse. The soliton pulse could maintain its width over a distance of 4000 km much beyond the distance of dispersive spread of a linear pulse. This is the first experimental result of an all-optical transmission system. This experiment also has demonstrated the limitation of the transmission distance by the timing jitter of the soliton pulse due to the amplifier noise. After the emergence of the concept of dispersion management, transmission experiments have been focused on the dispersion-managed soliton. In addition experiments of actual information transfer by means of coded pulse trains have become popular. The record single channel error free transmission of DMS has been achieved in 1998 using 40 Gb/s pulse over a distance of 10,000 by means of periodic erbium doped fiber amplifications in a recirculating loop [3]. Experiments of applying DMS for wavelength division multiplexed transmissions have also become the center of transmission experiments toward the end of 1990th to the beginning of 2000th. Achievement of error free transmission of over 1 Tb/s WDM transmission of DMS having 40 Gb/s per channel now became possible over a distance of several thousand kilometers [4]. Detail of some of these experimental results can been seen in the recently published book by Hasegawa and Matsumoto introduced earlier. It is remarkable that close agreements between theoretical predictions and experimental demonstrations have always been possible in soliton related phenomena in fibers. These facts indicate validity of the nonlinear Schrödinger equation model, (4.4) and (4.6) of description of the evolution of wave packet in fibers.
9. Concluding remarks A comprehensive review of theory of information transfer in fibers is presented in a self-contained manner. It is shown that for a typical high bit-rate optical signals having a pulse width of a few to a few tens of picosecond in a long haul fiber transmission system, effects of fiber dispersion and nonlinearity become comparable and equally important. As a result, design of optical transmission systems utilizing optical fibers requires careful considerations of these effects. There are basically two alternative approaches in solving the problems arising from dispersion and nonlinearity. One is to minimize these effects. For example, dispersion effect may be reduced by using a fiber having very small dispersion, while the nonlinear effect may be reduced by using low noise amplifiers, thereby the signal power level may be reduced proportionally. However, these attempts have limitations in ultra-high bit-rate systems since because both of these effects are enhanced in short optical pulses. The alternative approach is to take advantage of the combined effect of nonlinearity and dispersion to construct a stationary and stable signal by balancing these effects. This is the approach to use the optical soliton. In particular, the dispersion-managed soliton is a powerful solution that is capable to reducing the timing jitter caused by amplifier noise.
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By multiplexing the wavelengths of carriers, several terabit per second of ultra-high speed optical transmissions have been achieved. In this manner, the rate of information transfer has been increased more than three orders of magnitudes in the recent decade. As a matter of fact we did too well. At the end of 1980th, 85% of optical transmission systems have been occupied, while ten years later, most optical transmission systems are empty, because of the enormous capacity of the newly developed fiber systems. The author is convinced, however, that the glut of the optical transmission capacity will soon be solved as the access speed is increased proportionally and services that require broadband access prevail. At that point, our effort will be truly rewarded as a contributor to revolutionize the infrastructure of the modern society.
References [1] L.F. Mollenauer, R.H. Stolen, J.P. Gordon, Experimental observation of picosecond pulse narrowing and solitons in optical fibers, Phys. Rev. Lett. 45 (1980) 1095. [2] L.F. Mollenauer, K. Smith, Demonstration of soliton transmission over more than 4000 km in fiber with loss periodically compensated by Raman gain, Opt. Lett. 13 (1988) 675. [3] I. Morita, K. Tanaka, N. Edagawa, M. Suzuki, 40 Gbit/s single-channel soliton transmission over 10200 km without active Inline transmission control, in: 1998 European Conference on Optical Communication (ECOC 98), Vol. 3, Madrid, Spain, 1998, pp. 47–52. [4] K. Fukuchi, M. Kakui, A. Sasaki, T. Ito, Y. Inada, T. Tsuzaki, T. Shitomi, K. Fujii, S. Shikii, H. Sugahara, A. Hasegawa, 1.1-Tb/s (55 × 20-Gb/s) dense WDM soliton transmission over 3020-km widely dispersionmanaged transmission line employing 1.55/1.58-µm hybrid repeaters, in: Technical Digest of ECOC 99, Nice, France, September, 1999, PD2-10.
Tutorial
Theory of information transfer in optical fibers: A tutorial review Akira Hasegawa Soliton Communications, Kyoto, Japan Received 24 June 2003; revised 11 October 2003
Abstract A tutorial and comprehensive review of theoretical development in information transfer in optical fibers is presented with particular emphasis on nonlinear effects and application of solitons. During the decade of 1990th, a remarkable progress has been achieved in the amount and distance of information transfer using optical fibers, thanks to progress in fiber manufacturing, ultra-high speed optical devices, and theoretical understanding of optical pulse propagation in fibers. The review covers basics of lightwave propagation in fibers, derivation of the master equation that describes information transfer in fibers, introduction of various practical problems, and solutions based on optical solitons. 2003 Published by Elsevier Inc.
1. Introduction A dielectric fiber can guide light wave since it has index of refraction larger than the air. In particular, fibers made of glass have achieved the minimum loss of about 0.15 dB/km at the wavelength around 1550 nm, the level of theoretical limit given by resonant absorption and Rayleigh scattering. Lightwaves in fibers face dispersion both in phase velocity and in group velocity. In particular, the dispersion in group velocity distorts information transmitted in fibers, since the lightwave information propagates at the group velocity and the group velocity dispersion (often called group dispersion) results in different arrival time of different components of information spectra. The dispersion property of a fiber is designated by the group delay, D, having a unit of ps/nm/km which indicates a delay of arrival time in ps of information E-mail address: [email protected]. URL: http://www.solitoncomm.com. 1068-5200/$ – see front matter 2003 Published by Elsevier Inc. doi:10.1016/j.yofte.2003.11.002
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carried by a wavelength separated by 1 nm from another wavelength over a distance of 1 km. Normally convention of D > 0 (D < 0) is used when the information at a shorter wavelength propagates faster (slower), a case often called anomalous (normal) dispersion. The group dispersion is controllable by designing proper profile of index of refraction of a fiber in the fiber cross section and fibers having a range of D of ±20 ps/nm/km have been made at various wavelengths. In particular, the dispersion-shifted fiber (DSF) that has zero group dispersion at the wavelength (∼1550 nm) of minimum loss has been produced in order to achieve distortion-less transmission of information. However, when a DSF is used, it has soon been discovered that the fiber nonlinearity due to the Kerr effect distorts information and induces cross talks among different channels in wavelength division multiplex (WDM) transmissions. As a result, attentions to the optical soliton, a stable pulse in the presence of both Kerr nonlinearity and (anomalous) dispersion, have emerged. The optical soliton in fibers was discovered earlier in 1973 and was introduced as an information carrier in fibers having group dispersion. It is somewhat ironical that the value of optical soliton has been recognized after the dispersion problem has been eliminated. This manuscript is a comprehensive tutorial review of information transfer in optical fibers. It is written in the manner appropriate for those who have elementary knowledge in optics and are interested in this subject but without much background knowledge. The manuscript is designed as self-contained so that the readers do not have to bother searching references. As the result, the manuscript has no citations. Those who are interested in more detailed knowledge on optical transmissions and optical solitons, the author recommend a reference for optical solitons, A. Hasegawa, M. Matsumoto, Solitons in Optical Fibers, Springer, Heidelberg, 2002, and for fiber nonlinear optics in general, G.P. Agrawal, Nonlinear Fiber Optics, Academic Press, San Diego, 1989.
2. Dielectric fibers as light waveguides In this section, we describe lightwave propagation in dielectric fibers. 2.1. Electric properties of dielectric materials In order to understand the lightwave propagation in optical fibers, we need some basic knowledge of electric properties of a dielectric material. When an electric field is applied to a dielectric, average position of electrons are shifted and the material polarizes and polarization current is induced. This current, like the current in a condenser is proportional to the time variation of the electric field. The effect of polarization is expressed by the use of electric displacement vector, D, as D = ε0 E + P.
(2.1)
(= 8.864 × 10−12
F/m) is the dielectric constant of vacuum and P represents the Here, ε0 polarization and is given by P = −ene ξ(E).
(2.2)
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In this expression, ne is the density of electrons that participate to the polarization, ξ represents the displacement of electron position in a dielectric molecule induced by the electric field E. One can obtain ξ as a function of E phenomenologically by means of equation of motion of electrons trapped in ionic potential of the dielectric. If we can assume that the potential is parabolic in ξ , the response becomes linear and the polarization becomes a linear function of electric field. The linear relation is conveniently expressed in Fourier amplitudes as P(ω) = χ(ω) E(ω).
(2.3)
Here ω is the angular frequency, the bar indicates the Fourier amplitude and χ(ω) is called the susceptibility. Substituting Eq. (2.3) into (2.1) gives the linear relation between D and E, D(ω) = ε0 ε∗ (ω) E(ω).
(2.4)
Here, ε∗ (= 1 + χ ) is the relative permittivity constant of the material. The light wave in dielectric material propagates at a speed lower than that in vacuum. The ratio of the phase velocity of the light wave in vacuum and that in a material is called the index of refraction often expressed by n0 (ω) and is defined by n0 (ω) =
ck √ ∗ = ε . ω
(2.5)
Here, k is the wave number. As is obvious from this derivation, if χ > 0, ε∗ is larger than unity and so is the index of refraction. In addition, it is clear that the group velocity, vg ≡ ∂ω/∂k = 1/k and the group dispersion, ∂vg /∂ω = −k vg2 are in general a function of frequency. We also note that these dispersion parameters in dielectric are given by k ≈
n0 (ω) c
and k ≈
2 ∂n0 . c ∂ω
(2.6)
For a glass, it is known that k (ω) becomes zero at the wavelength of 1300 nm and is negative (positive) for a longer (shorter) wavelength. The ionic potentials, however, are not exactly parabolic primarily due to the influence of neighboring ions. In many dielectric materials where the potential is symmetric, the lowest order modification to the parabolic potential appears as the reduction of the parabolic potential in proportion to ξ 4 . Since ξ is proportional to the electric field, this reduction 2 and the resultant index of induces increase of index of refraction in proportional to |E| refraction may be given by 2 = n0 (ω) + n2 (ω)|E| 2. n ω, |E| (2.7) Here n2 represents the coefficient of the nonlinear response and is caked the Kerr coefficient. For the fiber material, the value of n2 is very small and typically is about 10−22 (m/V)2 . Since the electric field of a lightwave in fibers with 1 mW of power has a value around 105 V/m, the nonlinear correction of the index of refraction is about 1 part 10−12 . It will be shown, however, that this tiny correction provides significant effects on information transfer in fibers.
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2.2. Fibers as light waveguides Let us now study the dispersion and linear effects in dielectric fiber guides. The dielectric wave guide may be characterized by an index of refraction n0 (ω) that depends on the radial coordinate. By the use of the electric displacement vector D and the magnetic flux density vector B, the Maxwell equations can be expressed as ∂B , (2.8) ∂t ∂D ∇ × B = µ0 . (2.9) ∂t If we take curl of Eq. (2.8) and substitute Eq. (2.9) and use the Fourier amplitude expression in time for the electric field, E, ∇ ×E=−
ω2 ∇ 2 E − ∇(∇ · E) + 2 E= 0. cD
(2.10)
Here c c (2.11) cD = √ = ∗ n0 ε is the speed of light in the dielectric material, which is a function of transverse coordinates. Because of the inhomogeneity of n0 , D ∇ ·E=∇ · = 0. (2.12) ε0 ε ∗ Thus we note that unlike the case of a metallic waveguides, the field cannot be separated to either TM or TE mode. The exact field in a fiber has all the three components both in magnetic and in electric fields and can be described by the wave equation (2.10). However, if the variation of n0 is small so that λ∂ ln(n0 )/∂r 1, where λ is the wavelength, the field may be divided to either TE or TM mode. Let us take a case of TE mode. The electric field may be expressed by a scalar function φ in the form E = ∇ × φz = ∇⊥ φ × z and the wave equation becomes 2 ω 2 n (r) − k ∇⊥ φ = 0. ∇ 2 ∇⊥ φ + 0 c2
(2.13)
(2.14)
Here, ∇⊥ is the gradient operator in the transverse directions and z is the unit vector in the axial (z) direction. The dispersion relation k(ω) may be formally obtained from (2.14) in terms of the eigenfunction for ∇⊥ φ, 2 2 ω2 /c2 |∇⊥ φ|2 n20 (r) dS − |∇⊥ φ| dS 2 . (2.15) k = 2 |∇⊥ φ| dS This expression shows clearly that the dispersion relation is determined by the combination of dielectric property of the fiber material and the waveguide property of the fiber. With a
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Fig. 1. Dispersion relation of a single mode fiber (solid line).
proper design of the profile of the index of refraction in the radial direction, the wavelength at which the group dispersion vanishes can be shifted to a longer wavelength. In particular, the dispersion-shifted fiber (DSF) has zero group dispersion at the wavelength of 1550 nm, where the fiber loss is minimum. The property of the eigenfunction depends on the two-dimensional mode numbers in radial and azimuthal directions. Therefore, the dispersion property also depends on the mode number. However, the dispersion relation is uniquely determined for a lowest mode number that exists at wavelength longer than the cutoff wavelength for higher modes as seen in Fig. 1. A fiber that allows a single mode for certain range of wavelength is called a single mode fiber (SMF). Normally SMF has a negative (positive) k for wavelengths longer (shorter) than 1300 nm. Most fibers, albeit small, have asymmetric dielectric properties in the cross sectional plane. Then the eigenfunction depends on the polarization of the mode. In this case the group dispersion also depends on the polarizations of the mode. The group dispersion that originates from the polarization is called the polarization mode dispersion (PMD). PMD leads to the breakdown of the single mode property. Since such asymmetry normally varies randomly along the axial direction of the fiber, PMD induces random walk of each polarization mode around the average group velocity.
3. Deformation of information In this section we discuss how the information is transferred in fibers that have group dispersion and nonlinearity. To study how the information is transmitted it is essential to study how the information, or modulation, propagates in a fiber. Information may be lost not only by the fiber loss but also by its deformation that arises during the propagation in fibers due to various properties of the fiber. As is well known, the information propagates
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propagates at the group veat the group velocity, ∂ω/∂k. So the modulation amplitude E locity. Here, the modulation amplitude is related to the full electric field amplitude E(z, t) through 1 t)ei(k0 z−ω0 t) + c.c. . (3.1) E(z, t) = E(z, 2 In this expression, k0 and ω0 are the wave number and frequency of the carrier wave. We t) is a slowly varying function of coordinate also note that the modulation amplitude E(z, and time as compared with E(z, t). t); i.e., if The loss of information originates from a deformation of the modulation E(z, E(z, t) does not change in z, there is no loss of information. t) Information can be put on the lightwave at z = 0 by choosing a proper shape of E(0, in time. The amount of information depends on how rapidly E(0, t) varies, which can be designated by the width of the Fourier spectrum, E(Ω), ∞ E(Ω) =
t)eiΩt dt. E(0,
(3.2)
−∞
The amount of information (bit/s) that the lightwave carries is approximately given by the spectral width Ω. Therefore the more the amount in information, the wider the spectral width becomes. Consequently to achieve an ultra-high speed of information transfer in a t) having a wide spectral width. fiber, we should study the behavior of E(z, Deformed information may be reconstructed by a regeneration scheme. The scheme generally requires detecting of information, which may be done by rectification of the R , t) at the repeater located at z = zR , by regenerating the original lightwave to extract E(z t) from E(z R , t) and by modulating a fresh lightwave by E(0, t). In an information E(0, early stage of optical communication, this scheme has commonly been adapted. However, it has soon become clear that the repeater cost is the major obstacle for the increase of transmission rate because it requires electronic circuits having increasingly higher speed of operation. Consequently, most high-speed transmission systems at present uses an alloptical scheme with loss compensated for by periodic optical amplifications. When the fiber loss is effectively eliminated the major limitations come from the group velocity dispersion and nonlinearity of the fiber. Let us first discuss the effect of the group dispersion. The group dispersion is the effect in which the group velocity varies as a function of the lightwave frequency. The group dispersion originates from the combination of the wave guide property and the material property of the fiber. In the presence of the group dispersion, information carried by differ t) propagate at different speed thus arrives at different ent frequency components of E(0, time. The relative delay t of arrival time of information at frequencies ω1 , and ω2 at distance z is given by tD =
∂vg /∂ω(ω2 − ω1 )z z z − = . vg (ω1 ) vg (ω2 ) vg2
(3.3)
∂vg = −k vg2 , ∂ω
(3.4)
If we use vg =
1 ∂ω = , ∂k k
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Eq. (3.3) becomes tD = −k (ω2 − ω1 )z.
(3.5)
Equation (3.4) shows that the difference of arrival time of information is proportional to the group dispersion, k , the difference of the frequency components ω2 − ω1 , and the distance of probation z. We note that if k < 0 (called anomalous dispersion regime), higher frequency component of information arrives earlier and for normal dispersion, k > 0, vise versa. If the information at different frequency components arrives at different time, the information may be lost. The problem becomes more series if the amount of information is large so that ω2 − ω1 is large. For a practical purpose, the group dispersion is designated by a group delay parameter, D, as defined by delay of arrival time in ps unit for two wavelength components separated by 1 nm over a distance of 1 km. From Eq. (3.5), the group delay may be described by 2π 2π 2π − z = −k c 2 λz. D = k c λ1 λ2 λ Thus, by taking λ = 1 nm and z = 1 km, D in the unit of ps/nm/km is related to k through 2πck . (3.6) λ2 For a single mode fiber, D becomes 0 (16 ps/nm/km) at λ ≈ 1300 nm (1550 nm) while for a dispersion shifted fiber, D becomes 0 around at λ = 1550 nm. This indicates that two frequency components of a pulse with the wavelength separation of 1 nm arrive with a time delay on the order of a few pc when they propagates over a distance of 1 km in a typical fiber. An important quantity that designates the fiber dispersion effect is the dispersion distance z0 defined by the distance over which a pulse width becomes doubled due to the group delay. Since the pulse width τ is given by the inverse of the spectral width ω, the dispersion distance is given from Eq. (3.5), as D=−
z0 =
τ 2 . |k |
(3.7)
We now discuss the effect of fiber nonlinearity on the information transfer. For an ordinary fiber, the lowest order nonlinearity originates from the Kerr effect where the index 2 . In the presence of refraction, n, changes in proportion to the electric field intensity |E| of the Kerr effect, the index of refraction n is given by Eq. (2.7). Thus the wave number becomes ω 2 . n0 (ω) + n2 (ω)|E| (3.8) k= c Equation (3.8) indicates that the Kerr effect induces a nonlinear phase shift ΦN through the nonlinear part of the wave number kN given by ΦN = kN z =
2z 2πn2 |E| . λ
(3.9)
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becomes about 105 V/m in a typical fiber. For a lightwave with a peak power of 1 mW, |E| 2 −12 2 = 10−12 , since z/λ = 1012 Thus n2 |E| has a value of 10 . Therefore, even if n2 |E| for z ≈ 1500 km and λ ≈ 1500 nm, ΦN between the high intensity portion and the low intensity portion can become 2π over this distance of propagation. This indicates that the phase information is lost over a distance of 1000 km if the light wave power is as low as 1 mW. In addition it creates a mixture of information in amplitude and phase. This means that information transfer by means of coherent modulation is not appropriate for a light wave in fibers for a propagation distance beyond 1000 km.
4. Master equation of information transfer in fibers with dispersion and nonlinearity The information carried by the lightwave in fibers is expressed by the modulation t) shown in Section 3. We now derive the equation that describes evolution of E(z, t) E(z, along the direction z of the propagation of information. The most convenient way to derive 2 ), around the carrier the envelope equation is to Taylor expand the wave number k(ω, |E| 2 , frequency ω0 and the electric field intensity |E| 1 ∂k 2 k − k0 = k (ω0 )(ω − ω0 ) + k (ω0 )(ω − ω0 )2 + |E| , 2 2 ∂|E|
(4.1)
and to replace k − k0 by the operator i∂/∂z and ω − ω0 by −i∂/∂t, and to operate on the t). The resulting equation reads electric field envelope, E(z, 1 ∂ 2E ∂E ∂E ∂k 2 − k 2 + + k i |E| E = 0. (4.2) 2 ∂z ∂t 2 ∂t ∂|E| We note that to obtain k in this expression, we should go back to (2.15) and take the second derivative of k with respect to ω. It is often convenient to study the evolution of in the coordinate moving at the group velocity τ = t − k z. Then the envelope equation E becomes 1 ∂ 2 E ∂k 2 ∂E − k + |E| E = 0. i (4.3) 2 2 ∂z 2 ∂τ ∂|E| For a lightwave envelope in a fiber, the coefficients of this equation depends on the fiber geometry and modal structure of the guided lightwave. In particular, for a single mode fiber (SMF) k = 0 occurs at λ = 1300 nm, which is determined primarily by the glass property itself, while, k becomes zero at λ = 1550 nm for a dispersion-shifted fiber (DSF) because of the wave-guide property. A linear wave packet deforms due to the group velocity dispersion k . For a lightwave pulse with a scale size of τ , the deformation takes place at the dispersion distance z0 = τ 2 /|k |. Thus it is convenient to introduce the distance Z normalized by z0 and time T normalized by τ . For a fiber having anomalous dispersion, Eq. (4.3) reduces to i ∂ 2q ∂q = + i|q|2q ∂Z 2 ∂T 2
(4.4)
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and for a fiber having normal dispersion, the coefficient of the first term on the right hand side become negative. In this expression, q is the normalized amplitude given by ω0 n2 gz0 1/2 q= E. (4.5) c Here, g is a reduction factor (∼1/2) of the electric field intensity due to the wave-guide effect. If we compare this expression with the nonlinear phase shift, Eq. (3.9), we can identify that q is nothing but the nonlinear phase shift at the dispersion distance. This indicates that if the amplitude is such that the nonlinear phase shift becomes order unit at the dispersion distance, the normalized amplitude, q becomes order unity. In other words, the nonlinear term and dispersion term in Eq. (4.4) becomes comparable for such amplitude. Equation (4.4) is the master equation that describes the evolution of information propagation in fibers with dispersion and nonlinearity for an electric field having intensity such that the nonlinear phase shift at the dispersion distance is order unit. The equation is often called the nonlinear Schrödinger equation. In a practical system, the fiber dispersion k often varies in Z. In addition, fiber may have amplifiers with gain G(Z) and the loss with loss rate Γ per dispersion distance. Then Eq. (4.4) should be modified to i ∂ 2q ∂q = d(Z) 2 + i|q|2q + G(Z) − Γ q. ∂Z 2 ∂T
(4.6)
Here d(z) is the group dispersion normalized by its average value.
5. Linear and nonlinear responses of fibers In this section, we discuss evolution of wave packets in fibers in the presence of dispersion and nonlinearity. To understand the problem, we treat these effects separately. As the nonlinear effects, we discuss self and cross phase modulation, Raman effect and four wave mixing. 5.1. Linear response of a fiber with group dispersion In the absence of the nonlinear term, Eq. (4.6) can be easily integrated by means of the Fourier transformation in time, ∞ q(Z, ¯ Ω) =
q(Z, T )eiΩT dT .
(5.1)
−∞
Applying the Fourier transformation on Eq. (4.6) and solving the result for q(Z, ¯ Ω) we obtain
Z Z Ω2 (5.2) G(Z ) − Γ dZ . q(Z, ¯ Ω) = q(0, ¯ Ω) exp −i d(Z ) dZ + 2 0
0
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q(Z, T ) is then obtained from the inverse transformation, 1 q(Z, T ) = 2π
∞
q(Ω, T )e−iΩT dΩ.
(5.3)
−∞
In particular we note that if the average (or integrated) dispersion is zero and the average (or integrated) gain balances with the loss q(Z, T ) can recover the initial modulation q(0, T ) exactly. That is if the fiber response is completely linear and if the amplifier has no noise one can design a fiber transmission line such that the initial information can be transmitted without loss by making d(Z) = 0 and G(z) = Γ . As an example of linear transmission, let us consider a case of a loss-less fiber in which the initial pulse shape is given by a Gaussian, q0 2 2 e−T /(2T0 ) . (5.4) q(0, T ) = √ 2πT0 The Fourier transform of q is obtained by the formula ∞ a −y 2 /(4a) −ax 2 −ixy e e e dx = , π
(5.5)
−∞
to give q0 q(0, ¯ Ω) = √ 2πT0
∞
e−T
2 /(2T 2 )+iΩT 0
dT = q0 e−Ω
2 T 2 /2 0
.
(5.6)
−∞
If we substitute Eq. (5.6) into (5.2) and assume d(Z) to be a constant, D0 , we have q(Z, ¯ Ω) = q0 e−(1/2)(T0 +iD0 Z)Ω . 2
2
(5.7)
If we further substitute this result into Eq. (5.3), we can obtain the wave packet q at a given distance Z, q0 q(Z, T ) = 2π =
∞
e−(1/2)(T0 +iD0 Z)Ω e−iΩT dΩ 2
2
−∞
−T02 T 2 + iD0 ZT 2 exp . 2(T04 + D02 Z 2 ) 2π(T02 + iD0 Z) q0
(5.8)
Inspecting this result, we can observe various interesting points. First, we note that the pulse width T02 + D02 Z 2 /T02 increases with Z independent of the sign of the dispersion. That is the fiber dispersion induces the increase of the pulse width approximately in proportion to the distance of propagation. Next we note that the anomalous dispersion, D0 > 0, induces the frequency to decrease in time (called a negative chirping), while for a normal dispersion, a positive chirping is induced. In a practical system, amplifiers always add noise, which is proportional to the gain and the gain bandwidth. As a result, one needs to provide a sufficiently large initial amplitude
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in order to maintain a proper signal to noise ratio. The finite level of intensity induces the nonlinear self-phase shift due to the Kerr effect as shown in Eq. (3.9) and also in 2 in Eq. (3.9) is properly Section 5.2. We note, however, that if the time dependence of |E| chosen, the self-phase shift may counter balance the phase shift induced by the (anomalous) dispersion shown in Eq. (5.8). This is the essential feature of the soliton formation. We will discuss ways to overcome the frequency chirp and pulse spread by means of nonlinearity in the next section. 5.2. Self phase modulation We now study the evolution of the wave packet due to the nonlinearity. If we retain only the nonlinear term in Eq. (4.6), we have ∂q (5.9) = i|q|2q. ∂Z If we assume |q|2 is constant in distance, Eq. (5.9) can be formally integrated to give Z |q|2 dZ .
q(Z, T ) = q(0, T ) exp i
(5.10)
0
This result indicates that the phase of q varies along the direction of propagation. This is the reconstruction of the nonlinear phase shift introduced in (3.9). This phenomenon is called the self phase modulation. For example, if we expand the pulse shape in time and approximate it by 1 − T 2 , we can see from Eq. (5.10) that the phase varies in proportion to −T 2 . Thus we can see that the nonlinearity also induces frequency chirp. We further note that in an anomalous dispersion regime, the direction of the chirp produced by the nonlinearity is opposite of that produced by the dispersion. This indicates that nonlinearity induced chirp can cancel the dispersion induced chirp in anomalous dispersion regime. This process leads to the formation of a soliton in the wave envelope. 5.3. Cross phase modulation In the presence of other waves or modulations, the other waves induce phase modulation through the Kerr effect in a manner similar to the self phase modulation. Let us assume that the modulation q consists of two components, q = q1 + q2 ≡ q¯1 e−iΩ1 T + q¯2 e−iΩ2 T .
(5.11)
Here Ω1 and Ω2 are the modulation frequencies of the two waves. The nonlinear response for the q1 wave then becomes ∂ q¯1 = i |q¯1 |2 + 2|q¯2|2 q¯1 + i q¯22 q1∗ e−i(2Ω2 −Ω1 )T . (5.12) ∂Z We can see here that in addition to the self phase modulation, the phase modulation on q1 is induced also by q2 wave as shown by the second term on the right hand side. This effect is called the cross phase modulation and plays a detrimental role in information transfer in wavelength division multiplex (WDM) transmission systems. The third term on the right hand side plays no significant role since the effect is averaged out in time.
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5.4. Four wave mixing In the presence of many other waves, the Kerr nonlinearity induces additional important effect called the four wave mixing. We consider a case in which there exist four waves given by q≡
4
q¯j e−iΩj T .
(5.13)
j =1
In this case, in addition to the self and cross phase modulations, the evolution of q1 wave is influenced by the phase mixing term that generated the third term in Eq. (5.12). That is, ∂ q¯1 = 2i q¯2 q¯3 q¯4∗ e−i(Ω2 +Ω3 −Ω4 −Ω1 )T + 2i q¯ 2q¯3∗ q¯4 e−i(Ω2 −Ω3 +Ω4 −Ω1 )T ∂Z + 2i q¯2∗ q¯3 q¯4 e−i(−Ω2 +Ω3 +Ω4 −Ω1 )T .
(5.14)
From this expression it is clear that when the resonant condition, Ω1 + Ω4 = Ω2 + Ω3 , Ω1 + Ω3 = Ω2 + Ω4 , or Ω1 + Ω2 = Ω3 + Ω4 , the phase of q1 is systematically modified without phase mixing. This effect is called the four wave mixing (FWM) and also plays a crucial role in WDM systems. In the presence of group dispersion, however, the four wave mixing is suppressed due to the phase mixing in the wave number space. Let us show how this occurs. In the presence of modulation, Ω, the wave number, K is induced, where K is given by K = Ω/vg = k Ω. In the absence of group dispersion, k is independent of Ω, thus when the resonant condition in Ω is met, the wavenumber resonance also takes place. However, in the presence of group dispersion, k depends on Ω and the wavenumber resonance ceases. This results in the phase variation in the products of three amplitudes in the right hand side of Eq. (5.14) that is proportional to k Ω 2 Z and the systematic phase modulation disappears. Thus the effect of FWM may be avoided in the presence of fiber having significant group dispersion. However, as shown in Section 5.1 the group dispersion distorts information in linear transmission systems. The modern technology in WDM systems focuses on means to suppression information distortion due to the group dispersion and to the FWM simultaneously. 5.5. Raman effect In addition to the Kerr nonlinearity fiber possesses an important nonlinear effect called Raman effect. The Raman effect is a process of photon scattered off by material resonances at a frequency, ωp . In fibers the optical phonon plays the role of the material resonance. As a result of the scattering, the photon at frequency ω generates a new photon having a frequency ω −ωp . It is known that the resonant frequency ωp is around at the wavelength of 100 nm. Phenomenologically the Raman effect may be expressed in terms of the imaginary part of the Kerr coefficient, say, α. If we take this effect in the cross phase modulation term, we can construct a coupled equation between q1 and q2 given by ∂ q¯1 (5.15) = i |q¯1 |2 + 2|q¯2|2 q¯1 − 2α|q¯2 |2 q¯1 ∂Z
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and ∂ q¯2 = i |q¯2 |2 + 2|q¯1|2 q¯2 + 2α|q¯1 |2 q¯2 . (5.16) ∂Z Here, q1 and q2 waves have frequencies given respectively by ω and ω − ωp . The Raman effect, as can be seen from these equations, transfers energy from the higher frequency wave to the lower frequency mode. The quantity α, the Raman coefficient normalized by the Kerr coefficient, has a value around at 0.1 when the two wave has a wavelength separation of around 100 nm. The Raman effect also plays important role in WDM in cascading energy from shorter wavelength modes to lower wavelength modes. The Raman effect can also be used as a mean of amplifying optical signal. If the pump wave is introduced in a fiber, the Raman gain is distributed along the fiber, Raman amplifiers effectively provides amplifications having amplifier spacing much shorter than discrete amplifiers such erbium doped fiber amplifiers (EDFA). As a result, the integrated noise figure of a Raman system can be made significantly lower than an EDFA system. In advanced WDM transmission systems, the Raman amplifications are more favorably chosen.
6. Optical soliton solution In the previous section, we learned that fibers have group dispersion and Kerr nonlinearity and these effects distort information as it transferred through a fiber. What is interesting and fortunate is the fact that Eq. (4.4) can be integrated for an arbitrary initial condition, q(0, T ) and in particular it has an exact and stable solution called a soliton given by sech(T ). The solution can be characterized by the amplitude η, the time central position T0 , and the phase σ , 2 η q(T , Z) = η sech η(T − T0 ) exp i Z + iσ . (6.1) 2 It is straightforward to check that sech(T ) does satisfy the nonlinear Schrödinger equation (4.4). In addition by adding the amplitude η, as well as the phase and position, one can identify (6.1) can be seen to be a solution of (4.4). This solution is called the one-soliton solution. The solution is obtained by the balance between the group dispersion and Kerr nonlinearity. In addition, the stability of the solution against various perturbation has been proven and numerically checked. In other words, the soliton solution is free from either the dispersive distortion or from self-phase modulation. Thus it is quite natural to use solitons as information carrier in fibers since all other format will face distortion either from dispersion or nonlinearity. We note that the solution (6.1) is obtained in the coordinate moving at the group velocity in the rest frame. Therefore in the rest frame, soliton moves at the group velocity. If there exists a frequency shift, the group velocity is modified due to the group dispersion. The amount of the group velocity shift is proportional to the frequency shift. In the presence of a frequency shift, the solution (6.1) is thus further modified to
i (6.2) q(T , Z) = η sech η(T + κZ − T0 ) exp −iκZ + η2 − κ 2 Z + iσ . 2
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Here, κ is the frequency shift normalized by 1/t0 . This solution is the Galilean transformation of (6.1). Thus the one-soliton solution of the nonlinear Schrödinger equation (1.7) is given by a sech(T ) function, which is characterized by four parameters, the amplitude η (also the pulse width), the frequency κ (also the pulse speed), time position T0 , and the phase σ . In particular, we note that the soliton speed κ is a parameter independent of the amplitude unlike soliton solution of other equations. This fact is very important for the use of an optical soliton as a digital signal since if pulses of different amplitudes propagate at different speeds, information carried by such pulses are all mixed up and lost after some distance of propagation. When one soliton is used to represent one digit, the major cause that leads to the loss of information is the timing jitter which originates from the variation of the soliton position T0 since the wave form of the soliton is robust. This situation is quite different from a non-soliton pulse where the major cause of information loss is due to the deformation of the waveform itself. The variation of T0 originates from the frequency shift κ and the finite dispersion (since Z is normalized by the dispersion distance). This is because when the soliton frequency shifts, it modifies the group velocity through the group dispersion, and results in the change in soliton position. In real transmission systems, various effects induce the frequency shift of solitons. For example, the amplifier noise randomly modulates the soliton frequency and induces random variation of the soliton position (random walk). Interactions between adjacent soliton also produce timing jitter caused by the frequency shift. In order to avoid the frequency shift due to the interaction, two adjacent solitons should be separated enough ( six times pulse width in time). This requirement is a severe drawback for soliton based communications because it requires bandwidth at least three times wider than linear systems where the pulse separation can be twice or less than twice than the pulse width. When solitons are used in a wavelength division multiplex (WDM) system, they are free from distortions caused by either the cross phase modulation or from the four wave mixing since solitons are robust against collisions with other solitons. Therefore solitons are better choice than linear pulses also in WDM systems. However, imperfect (or asymmetric) collision between solitons at different wavelength channels can induce frequency shift since the robustness is broken in non-ideal situation where the nonlinear Schrödinger equation breaks down. Imperfect collisions occur at the input, output, or at amplifiers. Although solitons are clearly better choice as the information carrier in optical fibers because of their robust nature, the timing jitter caused by various effect as described above can lead to the loss of information even though the wave shape is intact through out the transmission. These problems can be solved to some extent by various means of soliton control that either reduce or restore the frequency shifts.
7. Dispersion managements and dispersion managed solitons As was shown in Section 5.1, an optical pulse can propagate free of distortion if the fiber response is completely linear and integrated dispersion and the loss compensated by gain over the entire span is zero even if the local value of dispersion is finite and fiber has a finite loss. This fact has been extensively used in linear transmission systems to avoid
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Fig. 2. A canonical dispersion map for dispersion managed soliton systems.
nonlinear degradation of the information. Various dispersion profiles (or dispersion maps) have been tried to minimize the nonlinear effects. Basically the idea is to introduce locally relatively large dispersion to make nonlinear effects relatively less and to compensate the accumulated dispersion at the end of the line so that the integrated dispersion becomes zero. A method of programming dispersion is called the dispersion management. Dispersion management was also found to be effective in soliton transmissions since the timing jitter, which is the major cause of bit error in soliton systems, also depends on the local dispersion. 7.1. Dispersion managements To illustrate effects of programmed dispersion, let us consider the propagation of an optical pulse in a lossless fiber in which the group velocity dispersion varies periodically as shown in Fig. 2. If the fiber is linear, the pulse that starts at point a can recover the original shape completely at point e if the dispersion in anomalous region d0 at a Z < b and d Z < e is opposite from that in normal dispersion region, −d0 , at b Z < d, provided ab + de = bd. However, if the dispersion map is not symmetric, the pulse at Z = e cannot recover the original shape. Let us consider what happens if we take into account of the fiber nonlinearity. As was discussed in Section 5, the major nonlinearity of a fiber originates from the Kerr effect. This produces self-induced phase shift and induces chirp in the pulse. The fiber dispersion induces also a chirp. In an anomalous (normal) dispersion region, the direction of the chirp due to nonlinearity is opposite from (same as) that due to the dispersion. A soliton is produced if the self-induced chirp due to nonlinearity counteracts that due to the anomalous dispersion. Let us now consider what happens if the local dispersion is much larger than that which allows a soliton solution for the given pulse intensity. Then the pulse acquires certain amount of chirp at the end of the anomalous dispersion section Z = b in Fig. 2. However, this amount of chirp is reversed as the pulse propagates through the normal dispersion section and may be completely reversed at point Z = d. Then the amount of the chirp can again become zero at the end of periodicity at Z = e. In fact, it can be shown that for a proper choice of the initial pulse width and pulse intensity, one can construct a nonlinear pulse that can recover the initial pulse shape at each end of the periodic dispersion map for a wide range of the value of the average dispersion d. This includes d > 0, d = 0, and even d is slightly negative (average normal dispersion). Furthermore it can be shown that even if the pulse does not recover its original shape at
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the end of one period, it recovers the original shape after several spans of the periodic map with a proper choice of the initial pulse width and the intensity. Such nonlinear stationary pulses are produced by a balance between properly averaged dispersion, weighted by the spectral change, and nonlinearity and behave quite similar to the ideal soliton in terms of their stability and dynamic range of nonlinearity and dispersion. Consequently, they are often called a dispersion-managed soliton (DMS). Dispersion managed solitons are attractive because, by taking a map having zero or close to zero average dispersion, their timing jitter induced by frequency modulation can be made to zero or close to zero, since the rate of change of the time position T0 is proportional to the amount of the dispersion. As is clear from the above discussion, dispersion managed solitons have much larger dynamic range as compared with linear pulses that require exact cancellation of dispersion. 7.2. Dispersion managed soliton In this section we show how a dispersion-managed soliton is created using the Lagrangian method. The envelope equation for properly normalized electric field of optical field in a fiber with the group velocity dispersion variation, d(Z) in the direction of propagation Z is given by Eq. (4.6). This equation may be reduced to a Hamiltonian structure by introducing a new amplitude u, u = q/a,
(7.1)
where da = −Γ + G(Z) a. dZ u then satisfies
(7.2)
∂u d(Z) ∂ 2 u =i + ia 2 (Z)|u|2 u. (7.3) ∂Z 2 ∂T 2 Equation (7.3) is not integrable because of inhomogeneous coefficients a(Z) and d(Z). The dispersion managed soliton normally requires the average d(Z), d(Z), much smaller than the local |d(Z)|,
1 d(Z) ≡ Lp
Lp d(Z ) dZ d(Z),
(7.4)
0
where Lp is a periodic length of the dispersion map. As was shown in Section 5.1, if
d(Z) = 1 and in the absence of nonlinearity, Eq. (7.3) has an exact periodic solution given by a Gaussian with frequency chirp. Thus, let us apply an additional transformation of u to a new amplitude f (T , Z) by eliminating the chirp C and frequency shift κ through u(T , Z) = p(Z)f p(Z) T − T0 (Z) , Z
2 C(Z) (7.5) × exp i T − T0 (Z) − iκ(Z)T + iθ0 (Z) , 2
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where p, C, and κ represent the inverse of the pulse width, chirp coefficient and frequency shift all real functions of Z. T0 and θ0 represent the soliton position and the phase. When Eq. (7.5) is substituted into Eq. (7.3), the magnitude of the dispersion term gives 1 ∂2 1 √ ∂2 d(Z) 2 = d(Z)p2 p 2 , 2 2 ∂T ∂η
η = p(T − T0 ),
and the nonlinear term √ a 2 (Z)|u|2 u = p pa 2 (Z)f 3 .
(7.6)
(7.7)
We note here that the dispersion coefficient now becomes p2 d. This is because if the pulse width changes in Z, spectrum width changes in proportion to p and the dispersive effect becomes proportional to p2 d. This implies that a nonlinear stationary solution in the case of inhomogeneous dispersion is obtained by a balance between the reduced dispersion
p2 d(Z) and the nonlinearity pa 2 (Z) rather than the dispersion and nonlinearity like in the case of an ideal soliton in a constant dispersion. We now apply Lagrangian method to construct the solution of Eq. (7.3). We first note that Eq. (7.3) can be constructed by the variation of the Lagrangian density L given by 1 1 i uZ u∗ − u∗Z u + a 2 (Z)|u|4 − d(Z)|uT |2 . (7.8) 2 2 2 Here, uZ and uT are respectively partial derivatives with respect to Z and T . It is straightforward to see that Eq. (7.3) can be obtained by taking the functional derivative of L with respect to u∗ ,
∞ n δL ∂ ∂ ∂ ∂ n ∂ L = 0. (7.9) = + (−1) − δu∗ ∂u∗ ∂T n ∂u∗nT ∂Z ∂u∗Z L(T , Z) =
n=1
The merit of using Lagrangian method is that once the Lagrangian density is known, one can construct evolution equations for parameters that characterize the solution (7.4) from the variational principle. If we substitute the ansatz (7.5) into (7.8) and assume explicit variation of f in Z is negligible, and integrate the result over T , one can construct the Lagrangian L for parameters p, C, κ, T0 , and θ0 , ∞ L(p, C, κ, T0 , θ0 ) ≡
L(T , Z) dT
−∞
=
I2 dC a 2 pI4 dp2 I3 dC 2 I2 dκ 2 I1 − − − 2 − 2 2 2 2p 2 2p dZ dθ0 dκ − , + I1 T0 dZ dZ
where ∞ I1 = −∞
∞ 2
f (T ) dT ,
I2 = −∞
T 2 f 2 (T ) dT ,
(7.10)
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∞ I3 = −∞
df dT
2
167
∞ dT ,
I4 =
f 4 (T ) dT .
(7.11)
−∞
Now, variation of the Lagrangian gives the Lagrangian equation of motion ∂ ∂L ∂L = 0, − ∂Z ∂ r˙ ∂r
(7.12)
for rj = p, C, κ, T0 , and θ0 . Here, r-dot is the derivative with respect to Z. By taking a specific pulse shape for f (T ), one can evaluate the various moments in expression (7.11). In particular, if we take a Gaussian pulse of the type (5.4), Eq. (7.12) can be expressed by a coupled set of evolution equations for these parameters, dp = −pCd, dZ dC a 2 p3 = −C 2 d − √ + 2dp4 , dZ 2 dκ = 0, dZ dT0 = −κd, dZ dθ0 κ 2 d 5a 2p = − dp3 + √ . dZ 2 4 2
(7.13) (7.14) (7.15) (7.16) (7.17)
By solving these coupled equations, for a given map profile, d(Z), one can obtain the evolution of the pulse properties as a function of Z. Let us study the evolution of the pulse property by taking a canonical dispersion map that has piece-wise constant dispersion d = ±d0 as shown in Fig. 2. If we eliminate Z from Eqs. (7.13) and (7.14), we can obtain the phase space relationship between the pulse width and the chirp parameters, 2 C2 a 2 p2 d C2 = + √ − p3 . (7.18) dp 2 p 2 2 2d If d is piece-wise constant, Eq. (7.18) can be easily integrated for a lose-less fiber a(Z) = a0 (= constant), C2 p4 = C0 p2 − − Ap3 , 2 2
(7.19)
a2 A = √0 2 2d0
(7.20)
where
represents the strength of nonlinearity. One can obtain a periodic solution with the periodicity Lp by a proper choice of initial conditions p(0) and C(0) so that p(Lp ) = p(0) and C(Lp ) = C(0). For a linear pulse with A = 0, the periodic solution for f is a Gaussian with a periodically varying chirp parameter C(Z) and the trajectories in p–C plane at
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Fig. 3. Phase space trajectories of p and C for the dispersion map of Fig. 2.
d = d0 and d = −d0 completely overlays themselves. However, the nonlinearity produces a gap in these trajectories as shown in Fig. 3, because of the frequency chirp produced by the self-induced phase shift. The peak value of p at point c (normal dispersion regime) is less than that at anomalous dispersion regime because of the nonlinearity induced self phase shift. This makes
p2 d > 0 even if d = 0 and the balance between the weighted average p2 d and the nonlinearity produces the stationary solution. This situation is analogous to an ideal soliton solution that is constructed by a balance of (constant) dispersion and nonlinearity. However, in the dispersion-managed case, the above argument indicates that the nonlinear stationary solution is possible even for d < 0 or d = 0, provided that p2 d > 0 and proper nonlinearity exists. An ideal soliton solution for a fiber with d = d0 = constant, can be constructed for an arbitrary value of d0 by a proper choice of the amplitude, while the linear stationary solution exists only for d0 = 0. Similarly for a dispersion managed case, a nonlinear stationary periodic pulse can be constructed for an arbitrary value of d by a choice of initial amplitude and chirp, while the linear stationary pulse is possible only for d = 0. This allows DM solitons to have much larger tolerance in the fiber dispersion. We here note again that, if the system is linear, the trajectory in p–C plane shown in Fig. 3 returns at the original point d only when the average dispersion d is exactly zero. We further note that in a nonlinear system, if p(0) and/or C(0) is not chosen so that after one period p(L) and/or C(L) does not return to the original value, it was numerically confirmed p(nL) and/or C(nL) returns to a limit area in p–C plane, where n = 1, 2, . . . . In other words Eqs. (7.13) and (7.14) in general have doubly periodic solution.
8. Review of experimental results Experiments on optical solitons in fibers have a long history now. The first demonstration of existence of soliton and soliton phenomena was made in 1980 [1] some seven years after the first theoretical prediction. Here 7 ps pulse having a peak power over 1 W was sent to a single mode fiber of 700 m long. A color center laser is used as the light source.
A. Hasegawa / Optical Fiber Technology 10 (2004) 150–170
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The first long distance transmission of a soliton pulse was achieved in 1988 [2]. Raman amplifications are used to compensate for the fiber loss. The experiment was done on 42 km recirculating loop of a single mode fiber using 55 ps pulse. The soliton pulse could maintain its width over a distance of 4000 km much beyond the distance of dispersive spread of a linear pulse. This is the first experimental result of an all-optical transmission system. This experiment also has demonstrated the limitation of the transmission distance by the timing jitter of the soliton pulse due to the amplifier noise. After the emergence of the concept of dispersion management, transmission experiments have been focused on the dispersion-managed soliton. In addition experiments of actual information transfer by means of coded pulse trains have become popular. The record single channel error free transmission of DMS has been achieved in 1998 using 40 Gb/s pulse over a distance of 10,000 by means of periodic erbium doped fiber amplifications in a recirculating loop [3]. Experiments of applying DMS for wavelength division multiplexed transmissions have also become the center of transmission experiments toward the end of 1990th to the beginning of 2000th. Achievement of error free transmission of over 1 Tb/s WDM transmission of DMS having 40 Gb/s per channel now became possible over a distance of several thousand kilometers [4]. Detail of some of these experimental results can been seen in the recently published book by Hasegawa and Matsumoto introduced earlier. It is remarkable that close agreements between theoretical predictions and experimental demonstrations have always been possible in soliton related phenomena in fibers. These facts indicate validity of the nonlinear Schrödinger equation model, (4.4) and (4.6) of description of the evolution of wave packet in fibers.
9. Concluding remarks A comprehensive review of theory of information transfer in fibers is presented in a self-contained manner. It is shown that for a typical high bit-rate optical signals having a pulse width of a few to a few tens of picosecond in a long haul fiber transmission system, effects of fiber dispersion and nonlinearity become comparable and equally important. As a result, design of optical transmission systems utilizing optical fibers requires careful considerations of these effects. There are basically two alternative approaches in solving the problems arising from dispersion and nonlinearity. One is to minimize these effects. For example, dispersion effect may be reduced by using a fiber having very small dispersion, while the nonlinear effect may be reduced by using low noise amplifiers, thereby the signal power level may be reduced proportionally. However, these attempts have limitations in ultra-high bit-rate systems since because both of these effects are enhanced in short optical pulses. The alternative approach is to take advantage of the combined effect of nonlinearity and dispersion to construct a stationary and stable signal by balancing these effects. This is the approach to use the optical soliton. In particular, the dispersion-managed soliton is a powerful solution that is capable to reducing the timing jitter caused by amplifier noise.
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By multiplexing the wavelengths of carriers, several terabit per second of ultra-high speed optical transmissions have been achieved. In this manner, the rate of information transfer has been increased more than three orders of magnitudes in the recent decade. As a matter of fact we did too well. At the end of 1980th, 85% of optical transmission systems have been occupied, while ten years later, most optical transmission systems are empty, because of the enormous capacity of the newly developed fiber systems. The author is convinced, however, that the glut of the optical transmission capacity will soon be solved as the access speed is increased proportionally and services that require broadband access prevail. At that point, our effort will be truly rewarded as a contributor to revolutionize the infrastructure of the modern society.
References [1] L.F. Mollenauer, R.H. Stolen, J.P. Gordon, Experimental observation of picosecond pulse narrowing and solitons in optical fibers, Phys. Rev. Lett. 45 (1980) 1095. [2] L.F. Mollenauer, K. Smith, Demonstration of soliton transmission over more than 4000 km in fiber with loss periodically compensated by Raman gain, Opt. Lett. 13 (1988) 675. [3] I. Morita, K. Tanaka, N. Edagawa, M. Suzuki, 40 Gbit/s single-channel soliton transmission over 10200 km without active Inline transmission control, in: 1998 European Conference on Optical Communication (ECOC 98), Vol. 3, Madrid, Spain, 1998, pp. 47–52. [4] K. Fukuchi, M. Kakui, A. Sasaki, T. Ito, Y. Inada, T. Tsuzaki, T. Shitomi, K. Fujii, S. Shikii, H. Sugahara, A. Hasegawa, 1.1-Tb/s (55 × 20-Gb/s) dense WDM soliton transmission over 3020-km widely dispersionmanaged transmission line employing 1.55/1.58-µm hybrid repeaters, in: Technical Digest of ECOC 99, Nice, France, September, 1999, PD2-10.