Soliton (optics)

An electric field is propagating in a medium showing optical Kerr effect, so the refractive index is given by:

${\displaystyle n(I)=n+n_{2}I}$

We recall that the relationship between irradiance and electric field is (in the complex representation)

${\displaystyle I={\frac {|E|^{2}}{2\eta }}}$

where ${\displaystyle \eta =\eta _{0}/n}$ and ${\displaystyle \eta _{0}}$ is the impedance of free space, given by

${\displaystyle \eta _{0}={\sqrt {\frac {\mu _{0}}{\varepsilon _{0}}}}\approx 377{\text{ }}\Omega .}$

The field is propagating in the ${\displaystyle z}$ direction with a phase constant ${\displaystyle k_{0}n}$. About now, we will ignore any dependence on the y axis, assuming that it is infinite in that direction. Then the field can be expressed as:

${\displaystyle E(x,z,t)=A_{m}a(x,z)e^{i(k_{0}nz-\omega t)}}$

where ${\displaystyle A_{m}}$ is the maximum amplitude of the field and ${\displaystyle a(x,z)}$ is a dimensionless normalized function (so that its maximum value is 1) that represents the shape of the electric field among the x axis. In general it depends on z because fields change their shape while propagating. Now we have to solve the Helmholtz equation:

${\displaystyle \nabla ^{2}E+k_{0}^{2}n^{2}(I)E=0}$

where it was pointed out clearly that the refractive index (thus the phase constant) depends on intensity. If we replace the expression of the electric field in the equation, assuming that the envelope ${\displaystyle a(x,z)}$ changes slowly while propagating, i.e.

${\displaystyle \left|{\frac {\partial ^{2}a(x,z)}{\partial z^{2}}}\right|\ll \left|k_{0}{\frac {\partial a(x,z)}{\partial z}}\right|}$

the equation becomes:

${\displaystyle {\frac {\partial ^{2}a}{\partial x^{2}}}+i2k_{0}n{\frac {\partial a}{\partial z}}+k_{0}^{2}\left[n^{2}(I)-n^{2}\right]a=0.}$

Let us introduce an approximation that is valid because the nonlinear effects are always much smaller than the linear ones:

${\displaystyle \left[n^{2}(I)-n^{2}\right]=[n(I)-n][n(I)+n]=n_{2}I(2n+n_{2}I)\approx 2nn_{2}I}$

now we express the intensity in terms of the electric field:

${\displaystyle \left[n^{2}(I)-n^{2}\right]\approx 2nn_{2}{\frac {|A_{m}|^{2}|a(x,z)|^{2}}{2\eta _{0}/n}}=n^{2}n_{2}{\frac {|A_{m}|^{2}|a(x,z)|^{2}}{\eta _{0}}}}$

the equation becomes:

${\displaystyle {\frac {1}{2k_{0}n}}{\frac {\partial ^{2}a}{\partial x^{2}}}+i{\frac {\partial a}{\partial z}}+{\frac {k_{0}nn_{2}|A_{m}|^{2}}{2\eta _{0}}}|a|^{2}a=0.}$

We will now assume ${\displaystyle n_{2}>0}$ so that the nonlinear effect will cause self focusing. In order to make this evident, we will write in the equation ${\displaystyle n_{2}=|n_{2}|}$ Let us now define some parameters and replace them in the equation:

• ${\displaystyle \xi ={\frac {x}{X_{0}}}}$, so we can express the dependence on the x axis with a dimensionless parameter; ${\displaystyle X_{0}}$ is a length, whose physical meaning will be clearer later.
• ${\displaystyle L_{d}=X_{0}^{2}k_{0}n}$, after the electric field has propagated across z for this length, the linear effects of diffraction can not be neglected anymore.
• ${\displaystyle \zeta ={\frac {z}{L_{d}}}}$, for studying the z-dependence with a dimensionless variable.
• ${\displaystyle L_{n\ell }={\frac {2\eta _{0}}{k_{0}n|n_{2}|\cdot |A_{m}|^{2}}}}$, after the electric field has propagated across z for this length, the nonlinear effects can not be neglected anymore. This parameter depends upon the intensity of the electric field, that's typical for nonlinear parameters.
• ${\displaystyle N^{2}={\frac {L_{d}}{L_{n\ell }}}}$

The equation becomes:

${\displaystyle {\frac {1}{2}}{\frac {\partial ^{2}a}{\partial \xi ^{2}}}+i{\frac {\partial a}{\partial \zeta }}+N^{2}|a|^{2}a=0}$

this is a common equation known as nonlinear Schrödinger equation. From this form, we can understand the physical meaning of the parameter N:

• if ${\displaystyle N\ll 1}$, then we can neglect the nonlinear part of the equation. It means ${\displaystyle L_{d}\ll L_{n\ell }}$, then the field will be affected by the linear effect (diffraction) much earlier than the nonlinear effect, it will just diffract without any nonlinear behavior.
• if ${\displaystyle N\gg 1}$, then the nonlinear effect will be more evident than diffraction and, because of self phase modulation, the field will tend to focus.
• if ${\displaystyle N\approx 1}$, then the two effects balance each other and we have to solve the equation.

For ${\displaystyle N=1}$ the solution of the equation is simple and it is the fundamental soliton:

${\displaystyle a(\xi ,\zeta )=\operatorname {sech} (\xi )e^{i\zeta /2}}$

where sech is the hyperbolic secant. It still depends on z, but only in phase, so the shape of the field will not change during propagation.

For ${\displaystyle N=2}$ it is still possible to express the solution in a closed form, but it has a more complicated form:[3]

${\displaystyle a(\xi ,\zeta )={\frac {4[\cosh(3\xi )+3e^{4i\zeta }\cosh(\xi )]e^{i\zeta /2}}{\cosh(4\xi )+4\cosh(2\xi )+3\cos(4\zeta )}}.}$

It does change its shape during propagation, but it is a periodic function of z with period ${\displaystyle \zeta =\pi /2}$.

Soliton's shape while propagating with N = 1, it does not change its shape

Soliton's shape while propagating with N = 2, it changes its shape periodically

For soliton solutions, N must be an integer and it is said to be the order or the soliton. For ${\displaystyle N=3}$ an exact closed form solution also exists;[4] it has an even more complicated form, but the same periodicity occurs. In fact, all solitons with ${\displaystyle N\geq 2}$ have the period ${\displaystyle \zeta =\pi /2}$.[5] Their shape can easily be expressed only immediately after generation:

${\displaystyle a(\xi ,\zeta =0)=N\operatorname {sech} (\xi )}$

on the right there is the plot of the second order soliton: at the beginning it has a shape of a sech, then the maximum amplitude increases and then comes back to the sech shape. Since high intensity is necessary to generate solitons, if the field increases its intensity even further the medium could be damaged.

The condition to be solved if we want to generate a fundamental soliton is obtained expressing N in terms of all the known parameters and then putting ${\displaystyle N=1}$:

${\displaystyle 1=N={\frac {L_{d}}{L_{n\ell }}}={\frac {X_{0}^{2}k_{0}^{2}n^{2}|n_{2}||A_{m}|^{2}}{2\eta _{0}}}}$

that, in terms of maximum irradiance value becomes:

${\displaystyle I_{\max }={\frac {|A_{m}|^{2}}{2\eta _{0}/n}}={\frac {1}{X_{0}^{2}k_{0}^{2}n|n_{2}|}}.}$

In most of the cases, the two variables that can be changed are the maximum intensity ${\displaystyle I_{\max }}$ and the pulse width ${\displaystyle X_{0}}$.

Propagation of various higher-order optical solitons (image series: low power (no soliton), then n1–n7)

Curiously, higher-order solitons can attain complicated shapes before returning exactly to their initial shape at the end of the soliton period. In the picture of various solitons, the spectrum (left) and time domain (right) are shown at varying distances of propagation (vertical axis) in an idealized nonlinear medium. This shows how a laser pulse might behave as it travels in a medium with the properties necessary to support fundamental solitons. In practice, in order to reach the very high peak intensity needed to achieve nonlinear effects, laser pulses may be coupled into optical fibers such as photonic-crystal fiber with highly confined propagating modes. Those fibers have more complicated dispersion and other characteristics which depart from the analytical soliton parameters.

The first experiment on spatial optical solitons was reported in 1974 by Ashkin and Bjorkholm[6] in a cell filled with sodium vapor. The field was then revisited in experiments at Limoges University[7] in liquid carbon disulphide and expanded in the early '90s with the first observation of solitons in photorefractive crystals,[8][9] glass, semiconductors[10] and polymers. During the last decades numerous findings have been reported in various materials, for solitons of different dimensionality, shape, spiralling, colliding, fusing, splitting, in homogeneous media, periodic systems, and waveguides.[11] Spatials solitons are also referred to as self-trapped optical beams and their formation is normally also accompanied by a self-written waveguide. In nematic liquid crystals,[12] spatial solitons are also referred to as nematicons.

Localized excitations in lasers may appear due to synchronization of transverse modes.

Confocal ${\displaystyle 2F}$ laser cavity with nonlinear gain and absorber slices in Fourier-conjugated planes

In confocal ${\displaystyle 2F}$ laser cavity the degenerate transverse modes with single longitudinal mode at wavelength ${\displaystyle \lambda }$ mixed in nonlinear gain disc ${\displaystyle G}$ (located at ${\displaystyle z=0}$) and saturable absorber disc ${\displaystyle \alpha }$ (located at ${\displaystyle z=2F}$) of diameter ${\displaystyle D}$ are capable to produce spatial solitons of hyperbolic ${\displaystyle \operatorname {sech} }$ form:[13]

${\displaystyle {\begin{aligned}E(x,z=0)&\sim \operatorname {sech} \left(\!{\frac {\pi xD}{2\lambda F}}{\sqrt {\frac {1-\alpha G}{G}}}\,\right)\\[3pt]E(x,z=2F)&\sim \operatorname {sech} \left(\!{\frac {2\pi x}{D}}{\sqrt {\frac {G}{1-\alpha G}}}\,\right)\end{aligned}}}$

in Fourier-conjugated planes ${\displaystyle z=0}$ and ${\displaystyle z=2F}$.[14]

In 1973, Akira Hasegawa and Fred Tappert of AT&T Bell Labs were the first to suggest that solitons could exist in optical fibres, due to a balance between self-phase modulation and anomalous dispersion.[15] [16] Also in 1973 Robin Bullough made the first mathematical report of the existence of optical solitons. He also proposed the idea of a soliton-based transmission system to increase performance of optical telecommunications.

Solitons in a fibre optic system are described by the Manakov equations.

In 1987, P. Emplit, J.P. Hamaide, F. Reynaud, C. Froehly and A. Barthelemy, from the Universities of Brussels and Limoges, made the first experimental observation of the propagation of a dark soliton, in an optical fiber.

In 1988, Linn Mollenauer and his team transmitted soliton pulses over 4,000 kilometres using a phenomenon called the Raman effect, named for the Indian scientist Sir C. V. Raman who first described it in the 1920s, to provide optical gain in the fibre.

In 1991, a Bell Labs research team transmitted solitons error-free at 2.5 gigabits over more than 14,000 kilometres, using erbium optical fibre amplifiers (spliced-in segments of optical fibre containing the rare earth element erbium). Pump lasers, coupled to the optical amplifiers, activate the erbium, which energizes the light pulses.

In 1998, Thierry Georges and his team at France Télécom R&D Centre, combining optical solitons of different wavelengths (wavelength division multiplexing), demonstrated a data transmission of 1 terabit per second (1,000,000,000,000 units of information per second).

In 2020, Optics Communications reported a Japanese team from MEXT, optical circuit switching with bandwidth of up to 90 Tbps (terabits per second), Optics Communications, Volume 466, 1 July 2020, 125677.

An electric field is propagating in a medium showing optical Kerr effect through a guiding structure (such as an optical fibre) that limits the power on the xy plane. If the field is propagating towards z with a phase constant ${\displaystyle \beta _{0}}$, then it can be expressed in the following form:

${\displaystyle E(\mathbf {r} ,t)=A_{m}a(t,z)f(x,y)e^{i(\beta _{0}z-\omega _{0}t)}}$

where ${\displaystyle A_{m}}$ is the maximum amplitude of the field, ${\displaystyle a(t,z)}$ is the envelope that shapes the impulse in the time domain; in general it depends on z because the impulse can change its shape while propagating; ${\displaystyle f(x,y)}$ represents the shape of the field on the xy plane, and it does not change during propagation because we have assumed the field is guided. Both a and f are normalized dimensionless functions whose maximum value is 1, so that ${\displaystyle A_{m}}$ really represents the field amplitude.

Since in the medium there is a dispersion we can not neglect, the relationship between the electric field and its polarization is given by a convolution integral. Anyway, using a representation in the Fourier domain, we can replace the convolution with a simple product, thus using standard relationships that are valid in simpler media. We Fourier-transform the electric field using the following definition:

${\displaystyle {\tilde {E}}(\mathbf {r} ,\omega -\omega _{0})=\int \limits _{-\infty }^{\infty }E(\mathbf {r} ,t)e^{-i(\omega -\omega _{0})t}\,dt}$

Using this definition, a derivative in the time domain corresponds to a product in the Fourier domain:

${\displaystyle {\frac {\partial }{\partial t}}E\Longleftrightarrow i(\omega -\omega _{0}){\tilde {E}}}$

the complete expression of the field in the frequency domain is:

${\displaystyle {\tilde {E}}(\mathbf {r} ,\omega -\omega _{0})=A_{m}{\tilde {a}}(\omega ,z)f(x,y)e^{i\beta _{0}z}}$

Now we can solve Helmholtz equation in the frequency domain:

${\displaystyle \nabla ^{2}{\tilde {E}}+n^{2}(\omega )k_{0}^{2}{\tilde {E}}=0}$

we decide to express the phase constant with the following notation:

${\displaystyle {\begin{aligned}n(\omega )k_{0}=\beta (\omega )&=\overbrace {\beta _{0}} ^{\text{linear non-dispersive}}+\overbrace {\beta _{\ell }(\omega )} ^{\text{linear dispersive}}+\overbrace {\beta _{n\ell }} ^{\text{non-linear}}\\[8pt]&=\beta _{0}+\Delta \beta (\omega )\end{aligned}}}$

where we assume that ${\displaystyle \Delta \beta }$ (the sum of the linear dispersive component and the non-linear part) is a small perturbation, i.e. ${\displaystyle |\beta _{0}|\gg |\Delta \beta (\omega )|}$. The phase constant can have any complicated behaviour, but we can represent it with a Taylor series centred on ${\displaystyle \omega _{0}}$:

${\displaystyle \beta (\omega )\approx \beta _{0}+(\omega -\omega _{0})\beta _{1}+{\frac {(\omega -\omega _{0})^{2}}{2}}\beta _{2}+\beta _{n\ell }}$

where, as known:

${\displaystyle \beta _{u}=\left.{\frac {d^{u}\beta (\omega )}{d\omega ^{u}}}\right|_{\omega =\omega _{0}}}$

we put the expression of the electric field in the equation and make some calculations. If we assume the slowly varying envelope approximation:

${\displaystyle \left|{\frac {\partial ^{2}{\tilde {a}}}{\partial z^{2}}}\right|\ll \left|\beta _{0}{\frac {\partial {\tilde {a}}}{\partial z}}\right|}$

we get:

${\displaystyle 2i\beta _{0}{\frac {\partial {\tilde {a}}}{\partial z}}+[\beta ^{2}(\omega )-\beta _{0}^{2}]{\tilde {a}}=0}$

we are ignoring the behavior in the xy plane, because it is already known and given by ${\displaystyle f(x,y)}$. We make a small approximation, as we did for the spatial soliton:

${\displaystyle {\begin{aligned}\beta ^{2}(\omega )-\beta _{0}^{2}&=[\beta (\omega )-\beta _{0}][\beta (\omega )+\beta _{0}]\\[6pt]&=[\beta _{0}+\Delta \beta (\omega )-\beta _{0}][2\beta _{0}+\Delta \beta (\omega )]\approx 2\beta _{0}\,\Delta \beta (\omega )\end{aligned}}}$

replacing this in the equation we get simply:

${\displaystyle i{\frac {\partial {\tilde {a}}}{\partial z}}+\Delta \beta (\omega ){\tilde {a}}=0}$.

Now we want to come back in the time domain. Expressing the products by derivatives we get the duality:

${\displaystyle \Delta \beta (\omega )\Longleftrightarrow i\beta _{1}{\frac {\partial }{\partial t}}-{\frac {\beta _{2}}{2}}{\frac {\partial ^{2}}{\partial t^{2}}}+\beta _{n\ell }}$

we can write the non-linear component in terms of the irradiance or amplitude of the field:

${\displaystyle \beta _{n\ell }=k_{0}n_{2}I=k_{0}n_{2}{\frac {|E|^{2}}{2\eta _{0}/n}}=k_{0}n_{2}n{\frac {|A_{m}|^{2}}{2\eta _{0}}}|a|^{2}}$

for duality with the spatial soliton, we define:

${\displaystyle L_{n\ell }={\frac {2\eta _{0}}{k_{0}nn_{2}|A_{m}|^{2}}}}$

and this symbol has the same meaning of the previous case, even if the context is different. The equation becomes:

${\displaystyle i{\frac {\partial a}{\partial z}}+i\beta _{1}{\frac {\partial a}{\partial t}}-{\frac {\beta _{2}}{2}}{\frac {\partial ^{2}a}{\partial t^{2}}}+{\frac {1}{L_{n\ell }}}|a|^{2}a=0}$

We know that the impulse is propagating along the z axis with a group velocity given by ${\displaystyle v_{g}=1/\beta _{1}}$, so we are not interested in it because we just want to know how the pulse changes its shape while propagating. We decide to study the impulse shape, i.e. the envelope function a(·) using a reference that is moving with the field at the same velocity. Thus we make the substitution

${\displaystyle T=t-\beta _{1}z}$

and the equation becomes:

${\displaystyle i{\frac {\partial a}{\partial z}}-{\frac {\beta _{2}}{2}}{\frac {\partial ^{2}a}{\partial T^{2}}}+{\frac {1}{L_{n\ell }}}|a|^{2}a=0}$

We now further assume that the medium where the field is propagating in shows anomalous dispersion, i.e. ${\displaystyle \beta _{2}<0}$ or in terms of the group delay dispersion parameter ${\displaystyle D={\frac {-2\pi c}{\lambda ^{2}}}\beta _{2}>0}$. We make this more evident replacing in the equation ${\displaystyle \beta _{2}=-|\beta _{2}|}$. Let us define now the following parameters (the duality with the previous case is evident):

${\displaystyle L_{d}={\frac {T_{0}^{2}}{|\beta _{2}|}};\qquad \tau ={\frac {T}{T_{0}}};\qquad \zeta ={\frac {z}{L_{d}}};\qquad N^{2}={\frac {L_{d}}{L_{n\ell }}}}$

replacing those in the equation we get:

${\displaystyle {\frac {1}{2}}{\frac {\partial ^{2}a}{\partial \tau ^{2}}}+i{\frac {\partial a}{\partial \zeta }}+N^{2}|a|^{2}a=0}$

that is exactly the same equation we have obtained in the previous case. The first order soliton is given by:

${\displaystyle a(\tau ,\zeta )=\operatorname {sech} (\tau )e^{i\zeta /2}}$

the same considerations we have made are valid in this case. The condition N = 1 becomes a condition on the amplitude of the electric field:

${\displaystyle |A_{m}|^{2}={\frac {2\eta _{0}|\beta _{2}|}{T_{0}^{2}n_{2}k_{0}n}}}$

or, in terms of irradiance:

${\displaystyle I_{\max }={\frac {|A_{m}|^{2}}{2\eta _{0}/n}}={\frac {|\beta _{2}|}{T_{0}^{2}n_{2}k_{0}}}}$

or we can express it in terms of power if we introduce an effective area ${\displaystyle A_{\text{eff}}}$ defined so that ${\displaystyle P=IA_{\text{eff}}}$:

${\displaystyle P={\frac {|\beta _{2}|A_{\text{eff}}}{T_{0}^{2}n_{2}k_{0}}}}$

Experiments have been carried out to analyse the effect of high frequency (20 MHz-1 GHz) external magnetic field induced nonlinear Kerr effect on Single mode optical fibre of considerable length (50–100 m) to compensate group velocity dispersion (GVD) and subsequent evolution of soliton pulse ( peak energy, narrow, secant hyperbolic pulse).[18] Generation of soliton pulse in fibre is an obvious conclusion as self phase modulation due to high energy of pulse offset GVD, whereas the evolution length is 2000 km. (the laser wavelength chosen greater than 1.3 micrometers). Moreover, peak soliton pulse is of period 1–3 ps so that it is safely accommodated in the optical bandwidth. Once soliton pulse is generated it is least dispersed over thousands of kilometres length of fibre limiting the number of repeater stations.