3.1 NLSE including the resonant and nonresonant cubic susceptibility tensors

From Maxwell's equation, the field in fibers satisfies

$$\nabla^2 \overrightarrow{E} - \frac{1}{c^2} \frac{\partial^2 \overrightarrow{E}}{\partial t^2} = -u\_0 \frac{\partial^2 \overrightarrow{P\_L}}{\partial t^2} - u\_0 \frac{\partial^2 \overrightarrow{P\_{NL}}}{\partial t^2} \tag{15}$$

∂A ∂z þ i 2 β2 ∂2 A <sup>∂</sup>t<sup>2</sup> � <sup>1</sup> 6 β3 ∂3 A <sup>∂</sup>t<sup>3</sup> ¼ � <sup>a</sup> 2

Nonlinear Schrödinger Equation

DOI: http://dx.doi.org/10.5772/intechopen.81093

gain coefficients.

Then, there is

V t ^ ðÞ¼ �3k<sup>0</sup> 8nAeff χ ð Þ3

> n¼2 i n <sup>n</sup>! β<sup>n</sup> ∂nφ <sup>∂</sup>T<sup>n</sup> ¼ Eφ.

1 2 β2 ∂2 ϕ ∂t<sup>2</sup> þ i 6 β3 ∂<sup>3</sup>ϕ <sup>∂</sup>t<sup>3</sup> � <sup>3</sup>k<sup>0</sup> 8nAeff χ ð Þ3 NRj j <sup>ϕ</sup> <sup>2</sup>

Let

equation �∑<sup>k</sup>

21

3.2 The solution by Green function

The solution has the form

<sup>A</sup> <sup>þ</sup> <sup>i</sup> <sup>3</sup>k<sup>0</sup> 8nAeff χ ð Þ3 NRj j <sup>A</sup> <sup>2</sup>

<sup>R</sup> ð Þ <sup>t</sup> � <sup>τ</sup> j j <sup>A</sup>ð Þ<sup>τ</sup> <sup>2</sup>

k<sup>0</sup> ¼ ω0=c, where ω<sup>0</sup> is the center frequency. Aeff is the effective core area. n is the refractive index. The last term is responsible for the Raman scattering, selffrequency shift, and self-steepening originating from the delayed response:

�2ð Þ ω<sup>1</sup> þ ω<sup>2</sup> þ ω<sup>3</sup>

where gð Þ ω<sup>1</sup> þ ω<sup>2</sup> þ ω<sup>3</sup> is the Raman gain and fð Þ ω<sup>1</sup> þ ω<sup>2</sup> þ ω<sup>3</sup> is the Raman non-

<sup>f</sup>ðω<sup>1</sup> <sup>þ</sup> <sup>ω</sup><sup>2</sup> <sup>þ</sup> <sup>ω</sup>3Þ ¼ <sup>2</sup>ð Þ <sup>ω</sup><sup>1</sup> <sup>þ</sup> <sup>ω</sup><sup>2</sup> <sup>þ</sup> <sup>ω</sup><sup>3</sup> ð Þ <sup>1</sup> � j j <sup>Γ</sup>

A zð Þ¼ ; t φð Þt e

<sup>ϕ</sup> � <sup>k</sup>0gð Þ <sup>ω</sup><sup>s</sup> <sup>1</sup> � ifð Þ <sup>ω</sup><sup>s</sup> ½ � 2nAeff

<sup>2</sup>nAeff <sup>ð</sup>þ<sup>∞</sup>

and taking the operator V t ^ ð Þ as a perturbation item, we first solve the eigen

i 6 β3

Assuming E ¼ 1, we get the corresponding characteristic equation:

Its characteristic roots are r1, r2, r3. The solution can be represented as

� 1 2 β2r <sup>2</sup> <sup>þ</sup> <sup>β</sup><sup>3</sup> 6 r

∂<sup>3</sup>ϕ

�∞ χ ð Þ3

gðω<sup>1</sup> þ ω<sup>2</sup> þ ω3Þ¼ �2ð Þ ω<sup>1</sup> þ ω<sup>2</sup> þ ω<sup>3</sup>

<sup>H</sup>^ <sup>0</sup>ðÞ¼ <sup>t</sup>

NRj j <sup>ϕ</sup> � <sup>k</sup>0gð Þ <sup>ω</sup><sup>s</sup> <sup>1</sup> � ifð Þ <sup>ω</sup><sup>s</sup> ½ �

1 2 β2 ∂2 ϕ <sup>∂</sup>T<sup>2</sup> <sup>þ</sup>

1 2 β2 ∂2 ∂t<sup>2</sup> þ i 6 β3 ∂3

A Ð<sup>t</sup> �<sup>∞</sup> <sup>χ</sup> ð Þ3 A þ

<sup>2</sup> � <sup>2</sup>j j <sup>Γ</sup> <sup>þ</sup> j j <sup>Γ</sup> <sup>2</sup> h i (25)

ϕ ðþ<sup>∞</sup> �∞ χ ð Þ3

dτ

ik0gð Þ ω<sup>0</sup> ½ � 1 � ifð Þ ω<sup>0</sup> 2nAeff

<sup>2</sup> � <sup>2</sup>j j <sup>Γ</sup> <sup>þ</sup> j j <sup>Γ</sup> <sup>2</sup> (24)

�iEz (26)

<sup>R</sup> ð Þ <sup>t</sup> � <sup>τ</sup> j j ϕ τð Þ <sup>2</sup>

<sup>∂</sup>t<sup>3</sup> (28)

<sup>R</sup> ð Þ <sup>t</sup> � <sup>τ</sup> j j ϕ τð Þ <sup>2</sup>

<sup>∂</sup>T<sup>3</sup> <sup>¼</sup> <sup>E</sup><sup>ϕ</sup> (30)

<sup>3</sup> <sup>¼</sup> <sup>E</sup> (31)

ϕ ¼ c1ϕ<sup>1</sup> þ c2ϕ<sup>2</sup> þ c3ϕ<sup>3</sup> (32)

dτ ¼ Eϕ

(27)

dτ (29)

(23)

$$\begin{split} \overrightarrow{P}\_{L}\left(\overrightarrow{r},t\right) &= \varepsilon\_{0} \int\_{-\infty}^{+\infty} \chi^{(1)}(t-t') \overrightarrow{E}\left(\overrightarrow{r},t'\right) dt' \\ &= \varepsilon\_{0} \int\_{-\infty}^{+\infty} \chi^{(1)}(o) \overrightarrow{E}\left(\overrightarrow{r},o\right) \exp\left(iat\right) da \end{split} \tag{16}$$

$$\chi^{(1)}(\alpha) = \int\_{-\infty}^{+\infty} d\tau \chi^{(1)}(\tau) \exp\left(-j\alpha\tau\right) \tag{17}$$

where E ! is the vector field and χð Þ<sup>1</sup> is the linear susceptibility. PL ! and PNL ! represent the linear and nonlinear induced fields, respectively [30]. The cubic susceptibility tensor including the resonant and nonresonant terms is

$$
\chi^{(3)}(w) = \chi\_{NR}^{(3)} + \chi\_{R}^{(3)}(w) \tag{18}
$$

There are

$$
\begin{split}
\overrightarrow{P}\_{NL,NR}\left(\overrightarrow{r},t\right) &= \epsilon\_0 \iiint\_{\infty} dt\_1 dt\_2 dt\_3 \chi\_{NR}^{(3)}(t\_1,t\_2,t\_3) \vdots \,\overrightarrow{E}\left(\overrightarrow{r},t-t\_1\right) \cdot \overrightarrow{E}\left(\overrightarrow{r},t-t\_2\right) \cdot \overrightarrow{E}\left(\overrightarrow{r},t-t\_3\right) \\ &= \epsilon\_0 \iiint\_{\infty} d\alpha\_1 d\alpha\_2 d\alpha\_3 \chi\_{NR}^{(3)}(-\alpha\_1 - \alpha\_2 - \alpha\_3; \alpha\_1 + \alpha\_2 + \alpha\_3) \\ &\stackrel{\rightarrow}{E}\left(\overrightarrow{r},t\_1\right) \cdot \overrightarrow{E}\left(\overrightarrow{r},t\_2\right) \cdot \overrightarrow{E}\left(\overrightarrow{r},t\_3\right) \exp\left(jat\right) \delta(\alpha - \alpha\_1 - \alpha\_2 - \alpha\_3)
\end{split} \tag{19}
$$

$$\chi\_{\rm NR}^{(3)}(-a\_1 - a\_2 - a\_3; a\_1 + a\_2 + a\_3) = \iint\_{\infty} dt\_1 dt\_2 dt\_3 \chi\_{\rm NR}^{(3)}(t\_1, t\_2, t\_3) \tag{20}$$

$$\begin{aligned} \exp\left(-j\alpha\_1 t\_1 - j\alpha\_2 t\_2 - j\alpha\_3 t\_3\right) \\ \vec{P}\_{\text{NL},R}\left(\vec{r}, t\right) &= \varepsilon\_0 \iiint\_{\infty} dt\_1 dt\_2 dt\_3 \chi\_R^{(3)}(t, t\_1, t\_2, t\_3) \vdots \,\vec{E}\left(\vec{r}, t - t\_1\right) \cdot \vec{E}\left(\vec{r}, t - t\_2\right) \cdot \vec{E}\left(\vec{r}, t - t\_3\right) \\ &= \varepsilon\_0 \iiint\_{\infty} d\alpha\_1 d\alpha\_2 d\alpha\_3 \chi\_R^{(3)}(t, -\alpha\_1 - \alpha\_2 - \alpha\_3; \alpha\_1 + \alpha\_2 + \alpha\_3) \\ \vec{E}\left(\vec{r}, t\_1\right) &\cdot \vec{E}\left(\vec{r}, t\_2\right) \cdot \vec{E}\left(\vec{r}, t\_3\right) \exp\left(j\alpha t\right) \delta(\omega - \alpha\_1 - \alpha\_2 - \alpha\_3) \end{aligned} \tag{21}$$

$$\begin{split} \chi\_{R}^{(3)}(t) &= \frac{1}{\sqrt{2\pi}} \int\_{-\infty}^{+\infty} \frac{a}{a - (a\mathbf{1} + a\mathbf{2} + a\mathbf{3}) + i\Gamma} e^{-i\alpha t} d\alpha \\ &= -\sqrt{\frac{\pi}{2}} a \left( \mathbf{1} + \frac{\Gamma}{|\Gamma|} \right) e^{-|\Gamma|t + i(a\mathbf{1} + a\mathbf{2} + a\mathbf{3})t - i\frac{\pi}{2}} \end{split} \tag{22}$$

Γ and a are the attenuation and absorption coefficients, respectively [31]. Repeating the process of [3] E ¼ F xð Þ ; y A zð Þ ; t exp ð Þ iβz , there is

Nonlinear Schrödinger Equation DOI: http://dx.doi.org/10.5772/intechopen.81093

3. Green function method for the time domain solution of NLSE

∂<sup>2</sup> E ! <sup>∂</sup>t<sup>2</sup> ¼ �u<sup>0</sup>

ðþ<sup>∞</sup> �∞

ðþ<sup>∞</sup> �∞

> ðþ<sup>∞</sup> �∞

is the vector field and χð Þ<sup>1</sup> is the linear susceptibility. PL

ð Þ3 NR þ χ ð Þ3

> ! r !; <sup>t</sup> � <sup>t</sup><sup>1</sup> � �

NRð Þ �ω<sup>1</sup> � ω<sup>2</sup> � ω3;ω<sup>1</sup> þ ω<sup>2</sup> þ ω<sup>3</sup>

dt1dt2dt3χ

∞

<sup>R</sup> ð Þ t; �ω<sup>1</sup> � ω<sup>2</sup> � ω3; ω<sup>1</sup> þ ω<sup>2</sup> þ ω<sup>3</sup>

a ω � ðω<sup>1</sup> þ ω<sup>2</sup> þ ω3Þ þ iΓ

e�j j <sup>Γ</sup> <sup>t</sup>þið Þ <sup>ω</sup>1þω2þω<sup>3</sup> <sup>t</sup>�<sup>i</sup>

! r !; <sup>t</sup> � <sup>t</sup><sup>1</sup> � �

represent the linear and nonlinear induced fields, respectively [30]. The cubic

susceptibility tensor including the resonant and nonresonant terms is

<sup>χ</sup>ð Þ<sup>3</sup> ð Þ¼ <sup>ω</sup> <sup>χ</sup>

NRð Þ t1; t2; t<sup>3</sup> ⋮ E

� E ! r !; t<sup>3</sup> � �

<sup>R</sup> ð Þ t; t1; t2; t<sup>3</sup> ⋮ E

� E ! r !; t<sup>3</sup> � �

From Maxwell's equation, the field in fibers satisfies

¼ ε<sup>0</sup>

¼ ε<sup>0</sup>

<sup>χ</sup>ð Þ<sup>1</sup> ð Þ¼ <sup>ω</sup>

∇<sup>2</sup> E ! � 1 c2

Nonlinear Optics ‐ Novel Results in Theory and Applications

PL ! r !; t � �

¼ ε<sup>0</sup> ∭ ∞

¼ ε<sup>0</sup> ∭ ∞

> E ! r !; t<sup>1</sup> � �

¼ ε<sup>0</sup> ∭ ∞

¼ ε<sup>0</sup> ∭ ∞

> E ! r !; t<sup>1</sup> � �

> > χ ð Þ3 <sup>R</sup> ðÞ¼ t

Repeating the process of [3] E ¼ F xð Þ ; y A zð Þ ; t exp ð Þ iβz , there is

dt1dt2dt3χ

dω1dω2dω3χ

� E ! r !; t<sup>2</sup> � �

NRð�ω<sup>1</sup> � ω<sup>2</sup> � ω3;ω<sup>1</sup> þ ω<sup>2</sup> þ ω3Þ ¼ ∭

ð Þ3

ð Þ3

ðþ<sup>∞</sup> �∞

a 1 þ

Γ j j Γ � �

Γ and a are the attenuation and absorption coefficients, respectively [31].

ffiffiffi π 2 r

dt1dt2dt3χ

dω1dω2dω3χ

� E ! r !; t<sup>2</sup> � �

¼ �

1 ffiffiffiffiffi <sup>2</sup><sup>π</sup> <sup>p</sup>

ð Þ3

ð Þ3

where E !

There are

χ ð Þ3

NL,NR r !; t � �

P !

P ! NL,R r !; t � �

20

3.1 NLSE including the resonant and nonresonant cubic susceptibility tensors

∂2 PL ! <sup>∂</sup>t<sup>2</sup> � <sup>u</sup><sup>0</sup>

<sup>χ</sup>ð Þ<sup>1</sup> <sup>t</sup> � <sup>t</sup> <sup>0</sup> ð Þ E ! r !; t <sup>0</sup> � � dt0

<sup>χ</sup>ð Þ<sup>1</sup> ð Þ <sup>ω</sup> <sup>E</sup> ! r !;ω � �

∂2 PNL !

exp ð Þ iωt dω

<sup>d</sup>τχð Þ<sup>1</sup> ð Þ<sup>τ</sup> exp ð Þ �jωτ (17)

� E ! r !; <sup>t</sup> � <sup>t</sup><sup>2</sup> � �

exp ð Þ jωt δ ωð Þ � ω<sup>1</sup> � ω<sup>2</sup> � ω<sup>3</sup>

ð Þ3 NRð Þ t1; t2; t<sup>3</sup>

exp ð Þ �jω1t<sup>1</sup> � jω2t<sup>2</sup> � jω3t<sup>3</sup>

exp ð Þ jωt δ ωð Þ � ω<sup>1</sup> � ω<sup>2</sup> � ω<sup>3</sup>

e �iωt dω

> π 2

� E ! r !; <sup>t</sup> � <sup>t</sup><sup>2</sup> � �

<sup>∂</sup>t<sup>2</sup> (15)

!

<sup>R</sup> ð Þ ω (18)

and PNL !

> � E ! r !; <sup>t</sup> � <sup>t</sup><sup>3</sup> � �

� E ! r !; <sup>t</sup> � <sup>t</sup><sup>3</sup> � �

(19)

(20)

(21)

(22)

(16)

$$\frac{\partial A}{\partial \mathbf{z}} + \frac{i}{2} \beta\_2 \frac{\partial^2 A}{\partial t^2} - \frac{1}{6} \beta\_3 \frac{\partial^3 A}{\partial t^3} = -\frac{a}{2} A + i \frac{3k\_0}{8n A\_{\text{eff}}} \chi\_{NR}^{(3)} |A|^2 A + \frac{ik\_0 g(a\nu\_0)[1 - \dot{\mathbf{z}}f(a\nu\_0)]}{2n A\_{\text{eff}}}$$

$$A \int\_{-\infty}^t \chi\_R^{(3)}(t - \tau) |A(\tau)|^2 d\tau$$

k<sup>0</sup> ¼ ω0=c, where ω<sup>0</sup> is the center frequency. Aeff is the effective core area. n is the refractive index. The last term is responsible for the Raman scattering, selffrequency shift, and self-steepening originating from the delayed response:

$$f(a\_1 + a\_2 + a\_3) = \frac{2(a\_1 + a\_2 + a\_3)(1 - |\Gamma|)}{-2(a\_1 + a\_2 + a\_3)^2 - 2|\Gamma| + |\Gamma|^2} \tag{24}$$

$$\mathbf{g}(\alpha\_1 + \alpha\_2 + \alpha\_3) = \left[ -\mathbf{2}(\alpha\_1 + \alpha\_2 + \alpha\_3)^2 - \mathbf{2}|\Gamma| + |\Gamma|^2 \right] \tag{25}$$

where gð Þ ω<sup>1</sup> þ ω<sup>2</sup> þ ω<sup>3</sup> is the Raman gain and fð Þ ω<sup>1</sup> þ ω<sup>2</sup> þ ω<sup>3</sup> is the Raman nongain coefficients.
