Table of Contents:

Nonlinear Optics: 2

1. Introduction. 2

2. Second Harmonic Generation. 2

3. Kerr effect 7

3.1 Electrostatic Kerr Cell 8

3.2 Polarization and Phase relations. 9

3.3 Optical Kerr Shutter 10

3.4 Optical Kerr Effect: Theory. 11

3.5 Cross Phase Modulation (XPM) 14

3.6 Some applications. 17

4. Soliton Propagation in Optical Fibers. 18

4.1 Classes of Solitons. 20

4.2 Optical fiber and its non-linearity. 20

4.3 Spatial solitons. 23

4.4 Comparison between actual transmission system and soliton transmission 25

4.5 Applications: Light Guiding Light 27

5. Conclusions. 34

6. References. 35

Appendix 1: Solitons - Historical Background. 36

Nonlinear Optics:

1. Introduction

Some important advances in nonlinear optics:

Ø Townes (1960), invention of the laser.

Ø Franken (1961), First observation ever of nonlinear optical effects, second harmonic generation (SHG).

Ø Terhune (1962), First observation of third harmonic generation (THG).

Ø E. J. Woodbury and W. K. Ng (1962), first demonstration of stimulated Raman scattering.

Ø Armstrong (1962), formulation of general permutation symmetry relations in nonlinear optics.

Ø A. Hasegawa and F. Tappert (1973), first theoretical prediction of soliton generation in optical fibers.

Ø H. M. Gibbs (1976), first demonstration and explanation of optical bistability.

Ø L. F. Mollenauer (1980), first confirmation of soliton generation in optical fibers.

Recently, many advances in nonlinear optics has been made, with a lot of efforts with fields of, for example, Bose-Einstein condensation and laser cooling; these fields are, however, a bit out of focus from the subjects of this course, which can be said to be an introduction to the 1960s and 1970s advances in nonlinear optics. It should also be emphasized that many of the effects observed in nonlinear optics, such as the Raman scattering, were observed much earlier in the microwave range.

Some important applications in nonlinear optics:

Ø Optical parametric amplification (OPA) and oscillation (OPO), hw_p=>hw_s+hw_i.

Ø Second harmonic generation (SHG), hw+ hw => h(2w).

Ø Third harmonic generation (THG), hw+ hw+ hw => h(3w).

Ø Pockels effect, or the linear electro-optical effect (applications for optical switching).

Ø Optical bistability (optical logics).

Ø Optical solitons (ultra long-haul communication).

2. Second Harmonic Generation

In dielectric materials, when applying an electric field, we induce a small (compared to atomic dimensions) displacement of positive and negative charges. Every molecule of the material acquire an induced dipole moment and the material as a whole is said polarized. All these dipoles produce their own field, which adds to the external one. If the external field oscillates, with a frequency far from some resonant frequency of the material (i.e. there is no absorption) the polarization, defined as the sum of all the dipole moments, follows the external field. Thus, polarization can be related to the applied electrical field E through the so-called electrical susceptibility tensor X:

(2.1)

where we have separated the three spatial components of the polarization and highlighted the frequency dependence of the various terms. if the tensor X is not diagonal, the polarization, normally, is not parallel to the external field. If the electrical field become strong enough, the linear response regime, described by equation 2.1 doesn't hold anymore and we must consider non-linear terms of the polarization:

			(2.2)
			(2.3)

where with X⁽¹⁾ and X⁽²⁾ we indicate the linear and the second-order susceptibility; from now on we'll drop the tensor formalism. Notice that if we are in presence of two fields with different frequencies, the non-linear polarization allows the exchange of energy between a number of electromagnetic fields of different energies. Second harmonic generation (SHG) is one application of this phenomenon. In SHG, part of the energy of an optical wave of frequency ω propagating through a material is converted to that of a wave of frequency 2ω. Thus assume that electronic response to a driving electric field can be simulated by that of an electron in an anharmonic potential well. The equation of motion for the electron is then

(2.4)

where X is the deviation from the potential minimum, mDX² is the anharmonic restoring force (corresponding to the (m/3)DX³ in the potential), the driving electric field is E_ocos(wt) and γ the dumping term. Notice that in centrosymmetric crystals must be D=0, so that SHG can have place only in non-centrosymetric materials.

Figure 2.1: An applied sinusoidal electric field and the resulting polarization; (a) in a linear crystal and (b) in a crystal lacking inversion symmetry.

This process is illustrated in figures (2.1) and (2.2). In figure (2.1) is shown the difference between the linear response regime and the non-linear one: in the last, the polarization still have the same periodicity of the driving field, but it is not anymore a sinusoidal function. We can, thus, decompose the polarization function in its Fourier components, shown in figure (2.2).

Figure 2.2: Analysis of the nonlinear polarization wave: (a) polarization from figure (2.1), (b) first harmonic the P Fourier factorization, (c) second harmonic (2w) and (d) DC component.)

If the material is not invariant under inversion symmetry, will have a non-zero second-harmonic components and, thus, an oscillation at a frequency w2=2w. This means that an electromagnetic field with frequency 2w will be generated in our system. The amplitude of the second-harmonic field is always several orders of magnitude lower than the incoming (and transmitted) one. Nevertheless, it is possible greatly increase the conversion efficiency, fulfilling the so-called phase-matching condition. It can be demonstrated that for the conversion efficiency holds

(2.5)

where k^w is the wavevector of the incoming wave, k^2w is the wavevector secon-harmonic generated one and is the length of our dielectric material. Thus, to maximize the efficiency conversion, we must work in an experimental condition for which , or k^2w =k^w. If Δk≠0, the second-harmonic generated wave at some plane (e.g. X₁), having propagated to some other plane (X₂), is not in phase with the second-harmonic genearated wave at X₂. This results in the interference described by the factor (sin²(x)/x²)in equation (2.5). The common technique that is used to satisfy the phase-matching condition take advantage of the natural birefringence of anisotropic crystals, but it remains quite a stringent condition. A second technique for achieving phase matching in nonlinear optics is to modulate the nonlinearity periodically in one dimension.The resulting grating can be used to compensate for the wave vector mismatch in a way that is called quasi-phase matching(QPM). In 1998 Berger proposed extending the idea of QPM to multiple spatial dimensions in much the same way as conventional linear gratings have been extended to photonic crystals. The result is that the phase-matching condition is greatly relaxed. This happens because, if we consider a peridiocally varying nonlinear susceptibility X⁽³⁾=X⁽²⁾(r), we can write it as a Fourier expansion :

(2.6)

where the sum is extended over the whole 2D reciprocal lattice. Inserting the expansion in the calculations that brought to the phase-matching condition, the B vectors enter in the sinus argument leading to the quasi-phase matching condition

(2.7)

where the reciprocal lattice vector B has been labelled with the two integer coordinates (n,m), given in the (b₁,b₂) basis of the recirocal lattice. For each vector B_n,m and a prescribed k^w there is a unique angle of propagation for the second harmonic. The coupling strength of a phase matching process using B_n,m, results proportional to the corresponding Fourier coefficient of X⁽²⁾. If a particular Fourier coefficient is zero, then no SHG will be observed in that direction. The first experimental observation of SHG by a photonic crystal were made by Broderick in year 2000 and confirmed the Berger theory.

The figure below shows the setup for optical second harmonic generation in LiNbO3

Figure 2.3: The setup for optical second harmonic generation in LiNbO3.

One example on using SHG as shown in figure 2.4:

Figure 2.4: example: generation of fifth harmonic (5w)

3. Kerr effect

Kerr effect was, first discovered by John Kerr in 1875. Very

few materials are known to behave as doubly refracting at normal

condition and certain material become doubly refracting when placed

in strong electric field. This effect is generally termed as Kerr electro-

optic effect. Since it is observed at high electric field, it is not due to

only first order electric field. This nonlinear effect is considered due

to reorientation and rotation of molecular lattice of the medium,

called as Kerr medium.

P = ec_eE (For linear medium)

Where e is permitivity of medium and c_e is electric susceptibility

P = e(c¹ E + c² E ² + c³ E ³ +….) (For nonlinear medium)

Where c¹ is linear susceptibility tensor, c² is quadratic tensor

(3 x 3 x 3 matrix) and c³ is cubic tensor (cubic matrix with 3 x 3

matrix at each lattice point).

Since non-linear response time is of the order of 2- 4 femto seconds, by exploiting the Kerr effect, one can construct Kerr Shutter for many practical purposes. Electrostatic Kerr Shutter has been used for high-speed photography. All Optical Kerr Shutter (everything controlled by light) can be used in optical communication where the rate of pulse (data) transfer is of great importance.

High intensity pulsed laser beam can be used as pump together with low intensity probe beam in a fiber optic cable. Pump generates required electric field to change the polarization of low intensity probe beam. With Kerr Fiber, that preserves polarization for pump, and a polarizer, femto second Optical Kerr Shutter can be constructed.

3.1 Electrostatic Kerr Cell

If the Kerr medium is aligned such that the optical axis is parallel to the applied electric field, then the birefringence can be explained in terms of Dn, the difference in the refractive indices of extraordinary (E) and Ordinary (O) rays by empirical formula.

Dn = K l E² Where E is electric field. In terms of the phase change,

f = 2LK pV²/d²

Where K is Kerr coefficient, L is the length of electrode in “Kerr Cell”, V is potential difference applied across electrodes, d is interaction length of light in Kerr Cell.

By using a Polariscope one can measure the phase change (f) due to electric field and can measure Kerr Coefficient “K”.

Kerr Cell arrangement to measure Kerr coefficient

Fig 3.1. Without applying voltage

Fig 3.2. With voltage

3.2 Polarization and Phase relations

R is the plane polarized light vector incident on the field perpendicular to the xy plane and at an angle “i” with the field. The light is traveling in z direction.

Let, R = a sin wt

Then x = a cos i sin wt

y = a sin i sin wt

With the electric field applied,

x = a cos i sin(wt + f₁)

y = a sin i sin(wt + f₂)

Fig 3.3. Diagrammatic section of the Kerr cell system showing angular positions along optic axis

Let, Df = f₁ – f₂

ð (x²/cos²i)-(2xy cos Df)/(cos i sin i) +(y²/sin²i) = a² sin² Df

Expressing in polar coordinates by letting r as any component of the polarized vector R, emergent from the field at an angle q in the xy plane.

x = r cos(q +i)

y = r sin(q +i)

r² = (a² sin² Df/2) / (1-(1- tan²Df/2) cos²q) : (equation of ellipse)

With the angle i = 45⁰, the semi-axes of the ellipse are therefore,

A = a cosDf/2 for q = 0 B = a sinDf/2 for q = p/2

At Df = p/2, r² = a²/2, regardless of q and circular polarization exists.

For p/2 < Df < p, the polarization becomes elliptic again with major and minor axes interchanged, B > A.

For Df = p the polarization becomes plane once more and r = B = a.

If an analyzer is placed at – 45^o to the electric field (i.e. q = p/2)

Then transmitivity (T) of linearly polarized beam with its electric field vector at an angle

i = 45^o ,

T = sin² Df/2

This is the condition we are going to use for optical Kerr Shutter.

3.3 Optical Kerr Shutter

A high intensity pulsed laser beam (pump) can be sent through

a Kerr medium (optical fiber) along with a probe beam so that

the electric field generated by pump beam changes the

optical properties of the medium. If optical fiber is polarization

preserving for pump beam then with proper adjustment of

polarizer, one can make ultra fast optical shutter. When there

is pump beam, the probe beam passes through fiber, otherwise not.

Fig 3.4. Probe polarization is perpendicular to the direction of polarizer. Without pump beam, no light passes through polarizer.

Fig 3.5. Probe polarization is changed by pump beam. With proper choice of pump intensity the polarization of probe beam can be change by 90^o making shutter open for the duration of pump pulse.

3.4 Optical Kerr Effect: Theory

Starting from Maxwell’s equation

…………. (3.1)

Where, J_f and r_f are free current density and charge density. P is electric polarization and M is magnetic polarization. In case of optical fiber J_f = 0 and r_f = 0 and M = 0

In general to describe the relation between E and P, quantum mechanical approach is required, but phenomenological approach given by following equation can be used in the medium far from resonance.

P = c^(1).E + c⁽²⁾:EE + c⁽³⁾ : E.E.E +….. ………… (3.2)

For linear case, P can be expressed as,

………… (3.3)

Where c⁽¹⁾ is linear susceptibility. If E is a monochromatic plane wave with

………… (3.4)

Then Fourier transformation P(r, t) becomes

………… (3.5)

Then linear dielectric constant is given by,

e(k , w) = 1 + 4p c⁽¹⁾ (k , w) ………… (3.6)

In electric dipole approximation, c⁽¹⁾ (r, t) is independent of r. Hence c⁽¹⁾ (k, w) and e(k, w) are independent of k.

For nonlinear case, P can be expressed in power series of E.

…(3.7)

Within electric dipole approximation c⁽ⁿ⁾ (r,t) is independent of r and hence c⁽ⁿ⁾ (k, w) is independent of k. P can also be written as,

P(w) = P⁽¹⁾(w) + P⁽²⁾(w)+ P⁽³⁾(w)

Where superscripts denotes the order of nonlinearity in the polarization vector. It is easier to separate the linear part with nonlinear as,

P(r, t) = P_L(r, t) + P_NL(r, t).

If there is no free charge,

………… (3.8)

For the fiber cable like fused silica, molecules are symmetric hence c⁽²⁾ has no contribution. Hence P_NL (r, t) comes from c⁽³⁾ alone.

Approximations:

1. Electric dipole => medium response is local.
2. | P_NL | << | P_L |
3. =>
7. P_NL is small perturbation.

So that only linear contribution gives

8. dielectric constant

With nonlinear contribution

,

e’(w) = 1 + c⁽¹⁾(w) + e_NL(w) , ………… (3.9)

3. To be scalar approach valid polarization should be maintained along fiber.
4. Quasi monochromatic - optical field centered at w_o and (Dw/w_o) << 1. For w_o ~ 10¹⁵ s^-1, Dw ~ 0.1 ps, this condition is valid.
5. Slowly varying envelop approximation.

In slowly varying envelop approximation: The rapidly varying part of electric filed as well as polarization vectors can be separated

………… (3.10)

………… (3.11)

………… (3.12)

Where c.c. is complex conjugate. E’(r ,t), P_L’(r ,t), P_NL’(r ,t) are slowly varying parts, x is polarization unit vector. Putting equation (3.11) into (3.5)

………… (3.13)

Where, E’ (r, w) is Fourier transform of E’(r, t).

Similarly putting equation (3.12) into (3.7) and considering nonlinear response instantaneous and time dependence of c⁽³⁾ as three delta function of the form d(t – t₁)

P_NL(r, t) = e_o c⁽³⁾ :E(r, t) E(r, t)E(r, t) ………… (3.14)

Dispersion of c⁽³⁾ can be neglected by assuming instantaneous non-linear response. Then from equation (3.10), (3.12) and (3.14)

P’_NL(r, t) = e_o e_NLE’(r, t) ………… (3.15)

e_NL = ¾ c⁽³⁾_xxxx | E’ (r,t)| ² ………… (3.16)

e_NL non-linear contribution to dielectric constant

Now Fourier transform of E’(r,t) is given by,

………… (3.17)

When equation (3.17) is plugged in equation (3.8), we get the equation of the form,

………… (3.18)

Where,

k_o = w/c and e’(w) = 1 + c_xx⁽¹⁾(w) + e_NL ………… (3.9)

Defining,

n’(w) = n(w) + n₂|E’|² ………… (3.19)

Also using e= (n’ + ia/2k_o)² and from equations (3.15), (3.9) and (3.19), and neglecting (n₂)² terms we get,

n₂= (3/8n) c⁽³⁾_xxxx ………… (3.20)

3.5 Cross Phase Modulation (XPM)

3.5.1 Coupling between waves of different frequencies

Considering two waves having frequencies w₁ and w₂ coupled together, the x component of resulting electric field is given by,

………… (3.21)

Where E₁ and E₂ are the x components of the coupling waves. Substituting equation (3.21) in (3.14) and neglecting the dispersion of c_xxxx,

.… (3.22)

Where,

P_NL(w₁) = c_eff (|E₁|² + 2 |E₂|²) E₁

P_NL (w₂) = c_eff (|E₂|²+ 2 |E₁|²) E₂

P_NL (2 w₁- w₂) = c_eff E₁² E₂^* ………… (3.23)

P_NL (2 w₂- w₁) = c_eff E₂² E₁^*

c_eff = ¾ e_o c⁽³⁾_xxxx

Phase matching condition has to be satisfied to build up (2w₁- w₂) and (2w₂- w₁) frequency components significantly as in for four waves mixing. In general special arrangement is required to satisfy this condition. Hence we neglect those terms containing (2w₁- w₂) and (2w₂- w₁).

P_NL(w_j) = e_oe_j^NLE_j for (j = 1,2.)

P (w_j) = e_oe_jE_j ………… (3.24)

Where,

………… (3.25)

Here, n = n₁ = n₂ has been assumed. The phase difference is given by,

………… (3.26)

Here, first term is Self Phase Modulation (SPM) and second term is Cross Phase Modulation (XPM).

3.5.2 Coupling between Polarization component of same wave.

………… (3.27)

Where, E_x and E_y are complex amplitudes.

In isotropic medium

………… (3.28)

With P_x and P_y given by,

……… (3.29)

Where, i, j = x or y, and

c⁽³⁾_xxxx = c⁽³⁾_xxyy + c⁽³⁾_xyxy + c⁽³⁾_xyyx ………… (3.30)

Here, c⁽³⁾_{xxyy ,} c⁽³⁾_{xyxy and} c⁽³⁾_xyyx are three independent and non zero components of c⁽³⁾, which depend on physical mechanism that contribute to c⁽³⁾. In silica fiber, electronic contribution is dominant and all of these components are nearly equal in magnitude.
………… (3.31)

Again the last terms of equation (3.31) is analogous to the four waves mixing and contributes negligibly when the length of the fiber L >> L_{B , Where,}

is called beat length of the fiber.

Then, ………… (3.32)

The phase difference between x and y component of probe at the output of the fiber of length L is given by

………… (3.33)

Considering the case of pump linearly polarized along x-axis and probe field 45^o to pump field and also neglecting the probe contribution. From equation (3.25),

Dn_x

= 2n₂|E_p|^{2 ,} Where, |E_p|² is pump intensity.

Recalling equation (3.29) ……...(3.29)

There is no pump field component in y-direction. Only first term in the equation (3.29), hence the first term of equation (3.31), contribute to P_y and again neglecting self-contribution,

Dn_y = 2n₂ b |E_p|²

Where, b = c⁽³⁾_xxyy / c⁽³⁾_xxxx and b= 1/3 for silica fiber

_{From equation (33),}

………… (3.34)

n_2B is the Kerr Coefficient of the fiber.

The probe transmitivity T_p is now given by.

………… (3.34)

Thus, by changing the pump intensity (E_p), fiber length (L) or wavelength of probe (l), transmission of the probe signal through fiber can be controlled.

3.6 Some applications

3.6.1 Measuring the Speed of Light

Kerr Shutter can be used to measure the speed of light. A laser beam is chopped by a fast shutter (the Kerr cell, with the pulser). The oscilloscope traces a rising signal, starting at some definite time after the oscilloscope trace is triggered by the pulser. The delay in receiving the signal can be observed which is partly in the electronics, but partly also in the travel time necessary for the light pulse to reach the photodiode detector. By changing the light path the delay changes by an amount proportional to the change in distance. The constant of proportionality is just the speed of light.

1.

Figure 3.6: The delay is large for longer path length and small for shorter path length

In 1958 Froome got the value of c = 299,792.5 km/s using a microwave interferometer and a Kerr cell shutter. Now adopted value of c = 299,792.458 km/s (1983)

3.6.2 Kerr Shutter as fluorescence rejection system

Kerr Shutter can be used as fluorescence rejection system in spectroscopic measurement. If the spectroscopic signals are collected for longer time, fluorescence will pervade the real signal. If the spectroscopic excitation pulse and the Kerr Gating pulse (Pump) are synchronized, pure signal can be collected.

Raman spectra of acetonitrile and the laser dye DCM measured fig. (a) with gate and fig. (b) without gate for 1 mM DCM concentration. The probe wavelength was 588 nm (1 ps, 4.6 mJ) and the accumulation time 400 s.

4. Soliton Propagation in Optical Fibers

The term "soliton" was introduced in the 1960's, but the scientific research of solitons had started in the 19th century[1] when John Scott-Russell observed a large solitary wave in a canal near Edinburgh. In the days of Scott Russell, there was much debate concerning the very existence of this kind of solitary waves. Nowadays, many model equations of nonlinear phenomena are known to possess soliton solutions.

Solitons are very stable solitary waves in a solution of those equations. As the term "soliton" suggests, these solitary waves behave like "particles". When they are located mutually far apart, each of them is approximately a traveling wave with constant shape and velocity. As two such solitary waves get closer, they gradually deform and finally merge into a single wave packet; this wave packet, however, soon splits into two solitary waves with the same shape and velocity before "collision".

Figure 4.1: Solitons

The stability of solitons stems from the delicate balance of "nonlinearity" and "dispersion" in the model equations. Nonlinearity drives a solitary wave to concentrate further; dispersion is the effect to spread such a localized wave. If one of these two competing effects is lost, solitons become unstable and, eventually, cease to exist. In this respect, solitons are completely different from "linear waves" like sinusoidal waves. In fact, sinusoidal waves are rather unstable in some model equations of soliton phenomena.

So when both the dispersive and the nonlinear term are present in the equation the two effects can neutralize each other. If the water wave has a special shape the effects are exactly counterbalanced and the wave rolls along undistorted. The soliton shape can be expressed by equation (4.1) as follows:

u(x,t)  =   a sech²[b(x-vt)], (4.1)

with b=(a/12)^1/2 and v=3a. The constant a is the only free parameter in the solution. It defines the amplitude and the width in such a way that a large (tall) soliton will be narrow, while a low soliton will be broad. The constant v defines the velocity of the soliton. Since v=3a a tall soliton will move faster than a low one.

Figure 4.2: Sine wave

4.1 Classes of Solitons

Solitons can be classified as follows:

1. Bright temporal envelope solitons

Pulses of light with a certain shape and energy that can propagate unchanged over large distances.

2. Dark temporal envelope solitons

Pulses of "darkness" within a continuous wave, where the pulses are of a certain shape, and possess propagation properties similar to the bright solitons.

3. Spatial solitons

Continuous wave beams or pulses, with a transverse extent of the beam that via the refractive index change due to optical Kerr-effect can compensate for the diffraction of the beam. The optically induced change of refractive index works as an effective wave guide for the light.

Also Solitons can be classified into:

1. Fundamental Solitons:

Which represents the family of pulses that doesn’t change its shape as it propagates.

2. Higher-Order Solitons:

Represents those pulses that undergo periodic changes in its shape as it propagates.

4.2 Optical fiber and its non-linearity.

It is important to understand that an optical soliton cannot be propagated anywhere except in mater like optical fibers which possess two main properties such as:

Ø The Group Velocity Dispersion (GVD)

Ø Non-linearity term (the Kerr effect)

4.2.1 The effect of GVD

The group velocity dispersion is related to the introduced dispersion parameter as:

(4.2)

and hence the sign of the group velocity dispersion is the opposite of the sign of the dispersion parameter β. In order to get a qualitative picture of the effect of linear dispersion, let us consider the effect of the sign of β:

β> 0:

For this case, the group velocity dispersion is negative, since

(4.3)

This implies that the group velocity decreases with an increasing angular frequency. In other words, the “blue” frequency components of the pulse travel slower than the “red” components. Considering the effects on the pulse as it propagates, the leading edge of the pulse will after some distance contain a higher concentration of low (“red”) frequencies, while the trailing edge rather will contain a higher concentration of high (“blue”) frequencies. This effect is illustrated in Fig. 4.3.

Figure 4.3. Pulse propagation in a linearly dispersive medium with β > 0.

Whenever “red” frequency components travel faster than “blue” components, we usually associate this with so-called normal dispersion.

β < 0:

For this case, the group velocity dispersion is instead positive, since now

(4.4)

This implies that the group velocity increases with an increasing angular frequency. In other words, the “blue” frequency components of the pulse now travel faster than the “red”

components. Considering the effects on the pulse as it propagates, the leading edge of the pulse will after some distance hence contain a higher concentration of high (“blue”) frequencies, while the trailing edge rather will contain a higher concentration of low (“red”) frequencies.

This effect, being the inverse of the one described for a negative group velocity dispersion, is illustrated in Fig. 4.4.

Figure 4.4. Pulse propagation in a linearly dispersive medium with β < 0.

Whenever “blue” frequency components travel faster than “red” components, we usually associate this with so-called anomalous dispersion.

Notice that depending on the distribution of the frequency components of the pulse as it enters a dispersive medium, the pulse may for some propagation distance actually undergo pulse compression.

For β > 0, this occurs if the leading edge of the pulse contain a higher concentration of

“blue” frequencies, while for β < 0, this occurs if the leading edge of the pulse instead contain a higher concentration of “red” frequencies.

Where a Gaussian wave propagating in the presence of SPM only can be shown as in Fig.4.5 (Note that the envelope remains the same, but there is a chirping effect):

Figure 4.5: Gaussian wave propagating in the presence of SPM only.

4.2.2 Non-Linear Schrödinger (NLS) equation.

These two properties (GVD and Kerr effect) give us the non-linear Schrödinger equation:

(4.5)

Where q is the complex amplitude of he pulse, z represents the distance along the direction of propagation, t the time.

The second term is originated from the GVD and the third is due to the Kerr effect. Hasegawa in 1973 was the first to show that a pulse of light in a fiber suit exactly to the NLS, to solve this equation and to introduced the optical soliton as solution.

The soliton comes from the exact cancellation between the GVD and the non-linear effect. As a result, a soliton can travel without spreading its shape, through a fiber over thousands kilometers.

Figure 4.6: Solitary wave propagating through non-linear, dispersive

medium. The pulse shape does not change with time.

4.3 Spatial solitons

As a light beam with some limited spatial extent in the transverse direction enter an optical Kerr media, the intensity variation across the beam will via the intensity dependent refractive index n = n0 +n2I form a lensing through the medium. Depending on the sign of the coefficient n2 (the nonlinear refractive index"), the beam will either experience a defocusing lensing effect (if n₂ < 0) or a focusing lensing effect (if n2 > 0); in the latter case the beam itself will create a self-induced waveguide in the medium (see Fig. 4.7).

Figure 4.7. An illustration of the effect of self-focusing.

As being the most important case for beams with maximum intensity in the middle of the beam (as we usually encounter them in most situations), we will focus on the case n2 > 0. For this case, highly intense beams may cause such a strong focusing that the beam eventually break up again, due to strong diffraction effects for very narrow beams, or even due to material damage in the nonlinear crystal.

For some situations, however, there exist stationary solutions to the spatial light distribution that exactly balance between the self-focusing and the diffraction of the beam. We can picture this as a balance between two lensing effects, with the first one due to self-focusing, with an effective focal length L_foc (see Fig.4.8), and the second one due to diffraction, with an effective focal length of L_defoc (see Fig. 4.9).

Figure 4.8. Self-focusing seen as an effective lensing of the optical beam.

Figure 4.9. Diffraction seen as an effective defocusing of the optical beam.

Whenever these effects balance each other, we in this picture have the effective focal length L_foc + L_defoc = 0.

In the electromagnetic wave picture, the propagation of an optical continuous wave in optical Kerr-media is governed by the wave equation:

(4.6)

For simplicity we will from now on consider the spatial extent of the beam in only one transverse Cartesian coordinate x.

By introducing the spatial envelope A_w(x; z) according to

(4.7)

and using the slowly varying envelope approximation in the direction of propagation z, the wave equation takes the form

(4.8)

4.4 Comparison between actual transmission system and soliton transmission

4.4.1.Limits of actual system.

First, whatever is the bits’ format to send, it is important to precise that different wavelengths (colors) travel at the same time in the same fiber (with approximately the same speed), the multiplexing, more or less dense (DWDM), is made to send more data at the same time.

Actually most of transmission use NRZ (Non Return to Zero) and RZ (Return to Zero) method.(see figure 4.10)

Figure 4.10: Schematic illustration of NRZ and RZ format for a bit sequence

Both NRZ and RZ system are limited by the spreading of their shape, which requires a large pulses separation and fiber spans of limited length. This also imply to use electronic components at the end of each span (to combat the deformation of the bits), to receive the signal, to analyze it, then to rebuild it and to send it in the new span.

But nowadays, recent applications use more data such as videoconference, Internet etc…which imply an increasing request in higher rate system and what are the physical solutions?

First of all, a physical approach is to increase the frequency (data rate) in a channel.

For that we send pulses with less separation between them. A problem occurs: the broadening which gets worse as data rate increase (GVD).

Figure 4.11 What’s happens with higher frequency

Secondly, another way should be to increase the multiplexing in one band (add new colors). If we move away from 1550 nm, we are in the area where the losses of the fiber are more important so we have to add more power at the injection, and problems of NLS appear.

Finally, a conceivable solution should be to add another band (L band at longer wavelength) to increase the data rate, but a new problem occurs: the Raman effect which should distort the actual C band transmission (the L band pump the energy of the C band). As a conclusion of actual system, to go from 2.5 Gbit/s to 10 (or even to 40 Gbit/s we need more sensitive receivers and spans less long. But it is too expensive!

4.4.2 The soliton solution.

The soliton transmission system is a RZ transmission and the time of the pulse is in order to picosecond. This kind of transmission is characterized by two main advantages. First there is a slow rate of change of the pulse, so there is no need of transmitters and receivers but only optical amplifiers. Thus the cost of the installation is less expensive than a classical communication link. Moreover, the fact that there is not use of electronic components eliminates the problem relative to different kind of encoding between countries.

Secondly, we know that soliton travel in non-linear medium, this is important too because it allows higher power to be injected into the fiber. These two advantages induce in 8 dB of extra power margin (5 dB due to better signal over noise and 3 dB from higher power injection). Thus, the span can be longer and we can broaden the width of multiplexing (to move away from 1550 nm, in area where the losses of the fiber are a little bit more important).

4.4.3 The soliton transmission system.

With the first experiments, researchers conclude that solitons were suitable at higher bit rate only on submarine links but not for DWDM system (dense wavelength division multiplexing) on terrestrial link because of four-wave mixing (FWM).

The FWM results from interaction between different wavelength (multiplexing). That interaction generates additional optical carriers and is proportional to the spacing of the channel in DWDM system. To combat the FWM, we need a high dispersion when to avoid pulse broadening, a low average dispersion is required.

Technically we will have to use spans with more of 20 ps/nm.km of dispersion but with a total average of a few ps/nm/Km.

Recently, Mollenauer (in May 2000) succeeded to transmit solilton at a rate of 270 Gbit/s along 9000 km staying in the error free domain (less than 1 error for 10⁹ bits).

For that, he has used DSM (Dispersion Shift Management) fibers, filter sliding system, a channel spacing of 75 GHz and 27 channels (range from 1542.5 nm to 1558.1 nm). The amplification was realized with erbium doped fibers and Raman gain control system. Despite the fact that the optical amplification is still a problem, this test is a great success for solitons that have really broken the rules of the actual other techniques’ limits.

Furthermore, Mollenauer expects to send information at 1 THz with the same range but only 0.4 nm spacing, underlying that last experiment was only limited by technical problems.

4.5 Applications: Light Guiding Light

The present on-going revolution in photonics could see its summit with the ultimate all-optical device. Simply imagine a transparent cube like that of Fig.1 with a myriad of interconnections and components created by light alone. This is our dream. Circuits that lie on top of one another and are reconfigurable. Virtual circuitry. No physical "wires". Either light itself directs and manipulates light without any intervening fabricated components such as optical waveguides, or light is used to write ideal optical components and circuitry in photosensitive materials. Light controls its own destiny.

The story of guiding light with light and of creating virtual circuitry is not just about an emerging technology and the ingenious efforts on the part of many who are presently attempting to make it a reality. Research into this field has also revealed new conceptual and experimental approaches for understanding how curious light beams, known as optical spatial solitons, can be made to remain localized in space while performing some rather amazing acrobatics. Advances of late have been remarkably swift, coming from different groups across the globe: Spatial solitons, once the domain of high power lasers, can now be launched by an incandescent light bulb. Soliton dynamics, once the province of esoteric mathematics, is now accessible with undergraduate physics. Mere theoretical predictions of a few years ago, such as the possibility of one soliton being made to spiral about another, the fusion or the creation of solitons upon collision, and the transportation of a dim beam by a bright beam are now readily observable in the laboratory. Even popular science magazines have questioned whether fabricated optical components would eventually become obsolete for device applications.

Solitons from a Linear Perspective

The building blocks for light guiding light are free-standing beams, known as spatial solitons. Unlike linear waves which diffract, solitons create their own channel as they travel in a uniform nonlinear medium, remaining localized and preserving their shape. Beams in a linear medium do not influence each other. But solitons can attract, repel, spiral around each other and this interaction can even be described by the classical force laws treating the beams as particles with mass Whereas linear waves always pass through one another, solitons can be dramatically altered by collisions. They can annihilate one another, fuse (Fig. 2) or give birth to multiple solitons. These phenomena turn out to be of potential importance to the emerging technology of light guiding light and light written circuitry.

From this elementary perspective, we can appreciate that disparate types of solitons are actually the same animal. In the simplest case, a soliton is one mode of the waveguide it induces. A soliton can be two or more modes of the induced waveguide. This elaboration explains interesting soliton dynamics, incoherent solitons, multi-humped solitons and the coexistence of different classes of solitons.

Now, if a soliton can be composed of a number of modes, each travelling at a different speed, then it should be possible to decompose the soliton into its constituent modes in exact analogy to Newton's refraction of white light by a prism into its component colours. And this elementary physics foreshadows novelties such as symmetric soliton beams being transformed into asymmetric beams upon colliding with one another.

The fact that nonlinear propagation has a linear waveguide equivalent provides a powerful conceptual tool, one that guides us in a physical manner to the fundamental equations and to their solutions. It allows us to predict novel phenomena, motivate light written circuitry, and foreshadow the design of lossless waveguide components. Put simply, all soliton dynamics have a linear waveguide analogue, albeit some unusual shaped waveguide system. Vice versa, every linear waveguiding phenomenon has its soliton equivalent in some nonlinear medium. A self-consistency relation unites the linear and nonlinear equivalents.

A Simple Model of Soliton Dynamics

One major challenge is to find a simple analytical description of solitons and their interactions. To achieve this, we need only borrow from the literature of the linear harmonic oscillator.

Because every linear optical waveguide has a soliton equivalent, it is natural to first consider the simplest optical waveguide possible, one whose refractive index falls off parabolically. Light beams obey the linear harmonic oscillator in this medium. This reveals that Gaussian beams remain Gaussian shaped as they propagate. In general the beams undulate periodically, undergoing periodic trajectories.

Now, according to the above linear perspective, Gaussian shaped solitons must also exist in some homogeneous nonlinear medium with the same behaviour as beams in a parabolic index optical waveguide. The particular nonlinear medium is found by using the self-consistency relation. Several candidates exist. But the simplest medium is one whose nonlinear induced refractive index change depends on the beam total power only. This arises, for example, when the medium has a nonlocal response with a correlation length that is much larger than the beam diameter. In such a medium, Gaussian shaped soliton beams remain Gaussian and they are unaltered by colliding with one another.

For a special beam radius and power, a Gaussian beam will propagate without change. Such a beam is called a stationary soliton. It induces a graded index optical fiber which can guide a signal beam. All other beams "breathe" as they propagate with their radius oscillating periodically (Fig. 3).

What happens to two stationary solitons that are initially launched in parallel to each other? In a homogeneous linear medium they would diffract as they travel in a straight trajectory. In this nonlinear medium they can attract and undergo periodic collisions with one another or, if launched skew to each other, spiral about each other as shown in Fig. 4. Finally, a distant "dim" beam can remain localized and be guided and steered by a "bright" soliton beam.

Solitons as Bundles of Classical Particles

Most experiments to date have involved comparatively narrow solitons launched by a coherent source. At the other extreme it is possible to have comparatively large incoherent solitons. And, in a beautiful experiment, Mitchell and Segev have launched them from an ordinary incandescent light bulb!

Such "big" incoherent solitons can be very neatly viewed as being composed of an enormous number of modes of the multimoded waveguide they induce. But, recall that diffuse light propagation along multimoded waveguides can be described by classical geometric optics. So incoherent solitons can also be viewed as bundles of rays, each ray obeying the paraxial ray equation, or equivalently as a bundle of classical (non-interacting) point particles, each particle obeying Newton's laws of motion.

This leads to predictions that are unique to incoherent solitons. For example, they can have any shape in their two-dimensional cross-section, even travel in parallel without interacting, unlike coherent beams in the same intensity dependent medium.

Temporal Solitons for Telecommunications

It is insightful to contrast spatial with temporal optical solitons. Temporal solitons are pulses that propagate along optical glass fibersfor long distance telecommunications. Here the material nonlinearity is only weakly perturbed. This is a one-dimensional problem. Whereas, spatial solitons envisaged for device applications in bulk material are typically quasi-monochromatic beams that are localized in two-transverse dimensions and propagate only for millimeters. These beams sufficiently alter the refractive index of the bulk material to actually create their own waveguides. The extra dimension brings additional riches such as the possibility for beams to spiral around each other, but it also demands that the nonlinearity be saturating or nonlocalized if the beams are to be both stable and localized in space.

Device and Logic Applications: Switching Light with Light

The dream of photonics is to have a completely optical technology. Here the traditional carriers of information, electrons, are envisaged to be replaced by photons for devices based on switching and logic. Spatial solitons offer one potential way to achieve this dream. We have described how waveguides are induced by solitons. The challenge is to develop methods for controllable steering of these waveguides by light itself and to produce reconfigurable waveguides.

Because solitons can attract and guide beams, light can be used to switch light for various device and logic applications which are presently being performed by electronics. The solitons can be considered as the information flow itself or as inducing optical waveguides in which the information is carried. This information can take the form of a weak ("dim") probe beam at a different wavelength or different polarisation than the ("bright") soliton beam. In either scheme, intricate virtual circuitry can be written in bulk nonlinear media. Depending on the material, the circuitry has a life span which allows for the possibility of self-reconfigurable circuits. Such plasticity opens the door to adaptive circuits that can be designed to transform themselves to the desired application. We are led to the image of a transparent cube with thousands of dynamically interconnected "wires" all created, maintained and organised by light itself.

The speed required for switching depends on the application. It can be as slow as seconds for circuit reconfiguration in network application or as fast as picoseconds for optical computing. The steering of one soliton by another or of a weak signal beam by a soliton forms the simplest case of spatial switching. Alternatively, the coupling ratio of a soliton induced coupler can be adjusted by changing the pump signal in one arm. Structures can be created to tap a signal, make a copy or reroute it and these processes can be made dynamic. It is also possible for two colliding solitons to fuse or for a soliton to be split into two solitons by a weak probe beam, thus creating additional forms of spatial switching

Designer components and light written circuitry

We have shown how soliton dynamics can be approached from the perspective of linear waveguides. Curiously the reverse is also true. The phenomenon of soliton dynamics provides a method of actually fabricating waveguide components whose design had not even been foreshadowed from our knowledge of linear waveguides. The concept is elementary. Solitons are allowed to interact in the appropriate photosensitive material so that the desired induced waveguide configurations are then permanently written.

For example, consider the optical device known as the X-junction. This is one of many building block components used for processing optical signals. It can be used to mix or split two signals in any desired proportion. Figure 5 shows a signal propagating through a soliton induced X-junction. In this particular case, the junction is designed to be completely transparent.

Light written circuitry offers potential advantages over the more conventional fabrication processes such as ion diffusion, PECVD, and sol-gel. In some materials beams fuse upon colliding. But, unless beams collide at virtual grazing incidence, they always pass through one another without influencing each other. In this way it is possible to compress circuitry into a compact space with many circuits sharing the same physical location. Furthermore, certain photosensitive materials offer the potential for erasing one light written device and replacing it by another. Hence, we have the building blocks for dense reconfigurable virtual circuitry.

5. Conclusions

In Nonlinear Optics we are concerned about the study of interaction of light in matter, we can see that for a linear optics the material acts in a linear way regarding polarization, because in the linear region the electric field is much lower than Intra-Atomic field, and the refractive index “n” is independent from the light intensity “I”. Whereas in Nonlinear Region the Electric field is approximately equal to Intra-Atomic field resulting with modified electron distribution, so the material reacts nonlinearly with the light, and “n” depends on “I”.

Nonlinear optical phenomena (such as the Kerr effect and second-harmonic generation) could be not bad, if we think about all gains caused by nonlinear devices, so there are many useful applications that benefit from this nonlinear behavior. Applications of this effective nonlinearity is soliton generation (Solitons can exist when dispersion and nonlinearity counteract one another), mode-locking of lasers frequency-shifting, optical switching, and compression of pulses, etc...

Virtual circuitry could be another beautiful application, like a transparent cube with a myriad of interconnections created, maintained, and arranged by light itself. So we can control “n” by the light itself or manipulate one beam with the other, and this leads to a great variety of technical innovations.

6. References

v Applied Nonlinear Optics, Frits Zernike and John Midwinter, 1973, John Wiley.

v Optics and Lasers including fibers and integrated optics, M. Young, 2^nd revised edition, 1984, Springer-Verlag.

v Laser Age in Optics “translated from Russian”, L.V. Tarasov, 2^nd printing 1982, Mir publishers-Moscow.

v Fiber optic Communication, Joseph Palaris, 4^th edition 1998, Prentice Hall.

v Optical Fiber Communications, Gerd Keiser, 3^rd International Edition 2000,McGraw-Hill.

v Lecture Notes on Nonlinear Optics, Fredrik Jonsson, Department of Laser Physics and Quantum Optics, March 24, 2003,(http://www.lib.kth.se)

v Nonlinear Electro-Optic Effect and Kerr Shutter, Electrodynamics- II project work, (Advisor: Prof. C.D. Lin) By Jagat Shakya and Mim Lal Nakarmi, Department of Physics, Kansas State University, Manhattan, KS 66502, April 2001.

v Discrete Spatial Optical Solitons in Waveguide Arrays, H. S. Eisenberg and Y. Silberberg R. Morandotti, A. R. Boyd, and J. S. Aitchison, PHYSICAL REVIEW LETTERS,VOLUME 81, NUMBER 16, 19 OCTOBER 1998

v Optical Solitons, Mr. FOURRIER Jean-christophe, Mr. DUREL Cyrille, Quantum Electronics, Lecturer : Dr. Jean-paul MOSNIER, Applied Physics, Year 4, 2000

v Light Guiding Light. (Let light be the master of its own destiny)
by Allan W Snyder and François Ladouceur (http://people.deas.harvard.edu/~jones/ solitons/papers/lgl/lgl.html)

v Nonlinear optical interactions in optical fibers, R. Boyd and E. Buckland, Journal of nonlinear optical physics and Materials, Vol.7 No.1 1998, pp105-112.

v http://photonics.ece.umd.edu/pubs/misc-presentations/MP-1/solitons.pdf

v www.sccs.swarthmore.edu/users/02/lisal/ physics/presentations/soliton.pdf

v www.pma.caltech.edu/Courses/ph136/ yr2002/chap09/0209.1.pdf

v http://www.elettra.trieste.it

v http://www.fact-index.com/s/so/soliton.html

v http://www.ma.hw.ac.uk/solitons/

v homepages.tversu.ru/~s000154/collision/main.html

v www.ma.hw.ac.uk/solitons/press.html

v http://www.lib.kth.se

v http://photonics.ece.umd.edu/pubs/misc-presentations/ MP-1/solitons.pdf

v http://physics.usc.edu/~vongehr/solitons_html/solitons.pdf

Appendix 1: Solitons - Historical Background

Until very recently, physicists were quite content about their understanding of wave motion. They had theories that explained all of their observations. Every type of wave motion possible had been studied, including that resulting from vibrating strings, sound waves, water waves and electromagnetic waves. In fact, most of the information that we receive comes from wave motion.

In the mid sixties at Princeton University two mathematical physicists, Zabusky and Kruskal, had discovered the existence of special localized waves, which exhibited particle-like behavior. Namely, when any two of these waves interacted, they came out of the collision with their identities in tact.They named these stable waves solitons. As such behavior in waves was unusal, Zabusky, Kruskal and their colleagues sought to understand solitons. As a result of their efforts, they later found a method for solving the equation, which governed soliton propagation. It wasn't long before other researchers had found the presence of solitons in other systems and had applied similar techniques.

In 1973 Hasegawa and Tappert proposed that such solitons could be used in optical communications systems. Researchers had already been working on the possibility of using optical fibers as the conduit for long distance communications . Such systems have a greater transmission rate than conventional systems using cables. In the 1970's optical fibers were designed, which had relatively low losses. However, they still had to overcome the fact that wave pulses in optical fibers tended to broaden due to dispersion. This spreading was detrimental, since well separated pulses at the transmitter could begin to overlap by the time they had reached the receiver, causing the information being transmitted to become jumbled. However, solitons had the advantage that the information could be transmitted with very little loss or dispersion.

The Great Wave of Translation

The story begins in 1834 on the bank of the Edinburg-Glasgow canal in Scotland. A Scottish Engineer by the name of John Scott Russell made the following report to the British Association for the Advancement of Science about what he had seen that day.

"I believe that I shall best introduce this phenomenon by decribing the circumstances of my own first aquaintance with it. I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped - not so the mass of water in the channel which it had put into motion; it accumulated round the prow of the vessel in a state of agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original feature some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation, a name which it now very generally bears."

Russell was so impressed by the incident, that he went back to his lab and made a model of the canal, with which he could try to reproduce and study water waves of the same type.

In 1955 Fermi, Pasta, and Ulam were studying the finite heat conductivity of solids. Their model of a solid consisted of a set of masses connected by springs. In such a model the masses are the molecules of the solid and the springs represent the forces holding the molecules together. It was known that if the springs behaved according to Hooke's law, the same law governing the behavior of common springs, then heat can flow through the solid without there being a temperature difference between its ends. In this case the thermal conductivity is infinite. Fermi, Pasta and Ulam undertook a numerical study of a one-dimensional lattice at Los Alamos on a Manaic I computer. Starting out with the energy concentrated in the lowest modes, they had expected that the system would thermalize; i.e., they had expected that the initial energy would become distributed throughout all of the modes of the system.

What they had found instead was that the energy flowed back and forth throughout all the modes and eventually recollected near the initial state. This surprised everyone, as the system did not behave as expected. In 1967 Zabusky and Kruskal at Princeton University set out to understand this abnormal behavior. They approximated the discrete spring model by a continuous model, by letting the spacing between the masses approach zero. The resulting equation that they had obtained for wave propagation in the model was the KdV equation, the KdV equation is an equation which tells us how the spatial changes in the wave profile are related to its temporal change. Given the shape of the wave at one instant of time, one would like to determine from this equation what the profile will look like during the next instant. Using this new profile, one would then predict the profile at an even later instant.

The dispersive characteristics of a fiber can be minimized in the operating region of the
system by changing the core radius and the difference in the refractive indices of the core and the cladding. The main parameter that characterizes dispersion is the rate of change of the group velocity with respect to the frequency. This is known as group velocity dispersion, or GVD. The wavlength at which there is minimum dispersion is the zero-dispersion wavelength. For wavelengths below this point, the fiber exhibits positive, or normal, GVD; while, for larger wavelengths, the disperion is known as negative, or anomalous, GVD.

The relation between the polarization and the applied electric field has been assumed to be linear. However, with the advent of the laser, which produces large electric fields, nonlinear effects should also be included. One type of nonlinear effect, the Kerr effect, occurs when the medium becomes anisotropic due to the polarization of the molecules in the presence of an applied electric field. In particular, the index of refraction is found to depend on the square of the amplitude of the applied electric field. If the optical fibers are designed appropriately, the dispersive and nonlinear effects can be balanced, thereby allowing for the possibility of a localized pulse to propagate through the fiber without distortion due to either effect.

In 1973 Hasegawa and Tappert had proposed that soliton pulses could be used in optical communications through the balance of nonlinearity and dispersion. They showed that these solitons would propagate according to the nonlinear Schrodinger equation (NLS), which had been solved by the inverse scattering method a year earlier by Zakharov and Shabat. At that time there was no capability to produce the fibers with the proper characteristics for doing this and the dispersive properties of optical fibers were not known. Also, the system required a laser which could produce very small wavelengths, which also was unavailable. It wasn't until seven years later, when Mollenauer, Stollen and Gordon at AT&T Bell Laboratories had experimentally demonstrated the propagation of solitons in optical fibers.

The original communications systems employed pulse trains with widths of about one nanosecond. However, there was still some distortion due to fiber loss. This was corrected by placing repeaters every several of tens of kilometers. As the width of the available pulses was decreased, the spacing of the repeaters was increased. In the mid 1980's it was proposed that tby sending in an additional pump wave along the fiber, the dispersion of a soliton could be halted through a process known as Raman scattering. In 1988 Mollenauer and his group had shown that this could be done by propagating a soliton over 6000 km without the need for repeaters.

Research into the use of solitons as information carriers in optical systems is still being
heavily researched. Soliton pulses are not immune to fiber losses, leading to pulse broadening. As the pulses broaden, neighboring solitons will overlap and this overlap is not fully understood. Systems, which allow the propagation of envelope solitons, operate in the region of negative group velocity dispersion (GVD). However, for positive GVD it has been recently found that a different type of soliton, the so-called dark soliton, can propagate through optical fibers. The characteristics of these fibers is currently being investigated. These are just a few of the directions that theoretical and experimental investigations have taken over the last decade in this rapidly growing field. It is clear that solitons will play a major role in the next generation of optical communication systems.

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[1] See Appendix 1 for historical background on Solitons