Kerr effect
Spatial solitons: counters diffraction
Temporal solitons: counters dispersion
- Bright solitons
- Dark solitons
How to get squeezed light

i\partial _{t}\psi =-{1 \over 2}\partial _{x}^{2}\psi +N |\psi |^{2}\psi

i\partial _{t}\psi =-{1 \over 2}\partial _{x}^{2}\psi +N |\psi |^{2}\psi

\psi(x) = N \textrm{sech}(x)

\psi(x) = N \textrm{sech}(x)

NLSE

Fundamental solition $N = 1$

After this talk, you will understand:

Image: Wikipedia

A soliton is a solution of a non-linear partial differential equation, such that

1. Has a permanent form;
2. It is localized within a region;
3. It does not obey the superposition principle;
4. It does not disperse.

For the next few slides, please raise your hand if you think it is a soliton.

1. Has a permanent form;
2. It is localized within a region;
3. It does not obey the superposition principle;
4. It does not disperse.

(wavefunction of trans-polyacetylene doped by a counter ion)

kdV equation - Waves

Burgers equation

Nonlinear Schrodinger Equation - Optics

i\partial _{t}\psi =-{1 \over 2}\partial _{x}^{2}\psi +N |\psi |^{2}\psi

i\partial _{t}\psi =-{1 \over 2}\partial _{x}^{2}\psi +N |\psi |^{2}\psi

\partial _{t}\phi +\partial _{x}^{3}\phi -6\,\phi \,\partial _{x}\phi =0\,

{\displaystyle \partial _{t}\phi +\partial _{x}^{3}\phi -6\,\phi \,\partial _{x}\phi =0\,}

\varphi _{tt}-\varphi _{xx}+\sin \varphi =0,

{\displaystyle \varphi _{tt}-\varphi _{xx}+\sin \varphi =0,}

Sine-Gordon model - Josephson Junctions

...

Skyrmion model - BEC/Topological solitons

Abrikosov-Nielsen-Olesen model - Superconductivity

i\partial _{t}\psi =-{1 \over 2}\partial _{x}^{2}\psi +N |\psi |^{2}\psi

i\partial _{t}\psi =-{1 \over 2}\partial _{x}^{2}\psi +N |\psi |^{2}\psi

\partial _{t}\phi +\partial _{x}^{3}\phi -6\,\phi \,\partial _{x}\phi =0\,

{\displaystyle \partial _{t}\phi +\partial _{x}^{3}\phi -6\,\phi \,\partial _{x}\phi =0\,}

\varphi _{tt}-\varphi _{xx}+\sin \varphi =0,

{\displaystyle \varphi _{tt}-\varphi _{xx}+\sin \varphi =0,}

Dispersive terms in non-linear equation: energy is not conserved

Mathematical property common between all solition solutions: the PDEs are integrable.

Order

within

Chaos

Integrable systems

(solvable)

Fluid dynamics

(complete chaos)

Wave equation (complete order)

more dispersion

no dispersion

Image: Wikipedia

Solitons occur exactly at the regime where the system is non-linear but integrable.

Gaussian pulse in nonlinear media

Gaussian pulse in nonlinear media + third order dispersion

Full playlist: Christophe FINOT

N = 1 solition in nonlinear media

N = 3 soliton in nonlinear media

Full playlist: Christophe FINOT

N = 5 solition in nonlinear media

N = 3 soliton in nonlinear media, + third order dispersion

Full playlist: Christophe FINOT

\varphi (x)=k_{0}nL(x)

\varphi (x)=k_{0}nL(x)

Phase shift is a function of the geometry

Huygens Fresnel principle

Spatial solitons

Images: Wikipedia

\varphi (x)=k_{0}nL(x)

\varphi (x)=k_{0}nL(x)

Phase shift is a function of the geometry

\varphi (x)=k_{0}n(x)L=k_{0}L[n+n_{2}I(x)]

\varphi (x)=k_{0}n(x)L=k_{0}L[n+n_{2}I(x)]

If we had an intensity distribution that had the same amount of phase shift, we get a self-focusing effect while removing the $x$ dependence from $L$ .

Self-focusing: main mechanism to generate squeezed light

Image: http://jonsson.eu/research/lectures/lect10/lect10.pdf

Kerr effect

n = n_0 + n_2 I

n = n_0 + n_2 I

When the refractive index $n$ is proportional to the intensity of the wave.

Two types of solitons:

- Spatial solitons

- Temporal solitons

n = n_0 + n_2 I(x)

n = n_0 + n_2 I(x)

n = n_0 + n_2 I(t)

n = n_0 + n_2 I(t)

Proof sketch. Let $n(I)$ be the refractive index as a function of intensity.

n(I)=n+n_{2}I(x)

n(I)=n+n_{2}I(x)

Equation of non-linear media

\mathbf{E}(x,z,t)=a_x(z)e^{{i(k_{0}n(I)z-\omega t)}}

\mathbf{E}(x,z,t)=a_x(z)e^{{i(k_{0}n(I)z-\omega t)}}

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

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

The electric field inside

Helmholtz equation

Derivation of NLSE

\mathbf{E}(x,z,t)=a_x(z)e^{{i(k_{0}n(I)z-\omega t)}}

\mathbf{E}(x,z,t)=a_x(z)e^{{i(k_{0}n(I)z-\omega t)}}

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

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

The electric field inside

Helmholtz equation

\left|{\frac {\partial ^{2}a_x(z)}{\partial z^{2}}}\right|\ll \left|k_{0}{\frac {\partial a_x(z)}{\partial z}}\right|

{\displaystyle \left|{\frac {\partial ^{2}a_x(z)}{\partial z^{2}}}\right|\ll \left|k_{0}{\frac {\partial a_x(z)}{\partial z}}\right|}

Assuming the amplitude changes slowly respect to $z$

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

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

Obtain.

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

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

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

{\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}

Drop terms with $I^2$

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

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

Non linear schrodinger equation

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

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

Non Linear Schrodinger equation

(normalized)

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

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

$N \ll 1$ : Linear terms dominate

$N \gg 1$ : Non linear part dominates

For soliton solutions, $N$ must be an integer, called the order of the soliton.

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

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

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

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

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 )}}.

{\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 )}}.}

$N$ = 1

$N$ = 2

Solutions to NLSE

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

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

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 )}}.

{\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 )}}.}

$N$ = 1

$N$ = 2

Image: Wikipedia

Dispersion

Group velocity Dispersion

Non-linear dispersion relation implies $v_g \neq v_p$

D=\frac{d^2\omega}{dk^2} = \frac{dv_g}{dk}

{\displaystyle D=\frac{d^2\omega}{dk^2} = \frac{dv_g}{dk}}

Image: Wikipedia

$D > 0$ : anomalous dispersion. $k = \omega^2$

higher frequency arrives faster/[slower] than lower frequency

D < 0: normal dispersion

D > 0: anomalous dispersion

Image: http://jonsson.eu/research/lectures/lect10/lect10.pdf

Temporal solitons

\varphi (t)=\omega _{0}t-k_{0}z[n+n_{2}I(t)]

\varphi (t)=\omega _{0}t-k_{0}z[n+n_{2}I(t)]

\omega (t)={\frac {\partial \varphi (t)}{\partial t}}=\omega _{0}-k_{0}zn_{2}{\frac {\partial I(t)}{\partial t}}

\omega (t)={\frac {\partial \varphi (t)}{\partial t}}=\omega _{0}-k_{0}zn_{2}{\frac {\partial I(t)}{\partial t}}

Image: Wikipedia

Temporal solitons

D > 0

anomalous dispersion

Kerr effect: self focusing

Image: Wikipedia

PDE for temporal solitons still the NLSE: P roof on Wikipedia

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

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

The derivation is the exact same except now we need to derive the equation in frequency domain, Fourier transform back.

So far, we have assumed:

$D > 0$ (temporal solitons) or
$n_2 > 0$ (spatial solitons)

What if $D < 0$ or $n_2 < 0$ ?

Temporal solitons: $D > 0$

D > 0

anomalous dispersion

Kerr effect: self focusing

Kerr

GVD

What if $D < 0$ or $n_2 < 0$ ?

Temporal solitons: $D < 0$

D < 0

Normal dispersion

Kerr effect: self focusing

Kerr

GVD

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

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

$D > 0$ or $n_2 > 0$

Image: http://jonsson.eu/research/lectures/lect10/lect10.pdf

a(\xi, \zeta)=a_0 \tanh \left(a_0 \xi \right) \exp \left(i a_0^2 \zeta\right)

a(\xi, \zeta)=a_0 \tanh \left(a_0 \xi \right) \exp \left(i a_0^2 \zeta\right)

$D < 0$ or $n_2 < 0$

Image: http://jonsson.eu/research/lectures/lect10/lect10.pdf

Timeline of solitons

1973: Solitons in optical fibres proposed [1] [2]
1974: Spatial solitons first discovered [4]
1987: Experimental observation of dark solitons [11]
2020: 90TB per second with optical circuit switching [12]

Kerr effect
Spatial solitons: counters diffraction
Temporal solitons: counters dispersion
- Bright solitons
- Dark solitons
How to get squeezed light

i\partial _{t}\psi =-{1 \over 2}\partial _{x}^{2}\psi +N |\psi |^{2}\psi

i\partial _{t}\psi =-{1 \over 2}\partial _{x}^{2}\psi +N |\psi |^{2}\psi

\psi(x) = N \textrm{sech}(x)

\psi(x) = N \textrm{sech}(x)

NLSE

Fundamental solition $N = 1$

Image: Wikipedia

Main References

[1] A. Hasegawa and F. Tappert, Transmission of Stationary Nonlinear Optical Pulses in Dispersive Dielectric Fibers. I. Anomalous Dispersion, Appl. Phys. Lett. 23, 142 (1973).

[2] A. Hasegawa and F. Tappert, Transmission of Stationary Nonlinear Optical Pulses in Dispersive Dielectric Fibers. II. Normal Dispersion, Appl. Phys. Lett. 23, 171 (1973).

[3] Y. Song, X. Shi, C. Wu, D. Tang, and H. Zhang, Recent Progress of Study on Optical Solitons in Fiber Lasers, Applied Physics Reviews 6, 021313 (2019).

[4] J. E. Bjorkholm and A. A. Ashkin, Cw Self-Focusing and Self-Trapping of Light in Sodium Vapor, Phys. Rev. Lett. 32, 129 (1974).

[5] G. I. Stegeman and M. Segev, Optical Spatial Solitons and Their Interactions: Universality and Diversity, Science 286, 1518 (1999).

[6] F. Jonsson, Lecture Notes on Nonlinear Optics (2003). http://jonsson.eu/research/lectures/

[7] P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University Press, Cambridge, 1990).

References

[8] Y. S. Kivshar, Optical Solitons: From Fibers to Photonic Crystals / Yuri S. Kivshar, Govind P. Agrawal. (Academic Press, 2003).

[9] P. G. Drazin and R. S. Johnson, Solitons: An Introduction, 2nd ed. (Cambridge University Press, Cambridge, 1989).

[10] J. K. Shaw, Mathematical Principles of Optical Fiber Communications / J.K. Shaw. (Society for Industrial and Applied Mathematics, 2004).

[11] P. Emplit, J. P. Hamaide, F. Reynaud, C. Froehly, and A. Barthelemy, Picosecond Steps and Dark Pulses through Nonlinear Single Mode Fibers, Optics Communications 62, 374 (1987).

[12] K. Ikeda, K. Suzuki, R. Konoike, S. Namiki, and H. Kawashima, Large-Scale Silicon Photonics Switch Based on 45-Nm CMOS Technology, Optics Communications 466, 125677 (2020).

Optical Solitons

Optical Solitons

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