*2.6.2.2.1 Examples*

### **3. Propagation of a light pulse in a transparent medium**

The frequency Fourier transform of a Gaussian pulse has already been given as

$$E(w) = \exp\left(\frac{-(w - w\_0)^2}{4\,\Gamma}\right). \tag{34}$$

consequence. The phase velocity *V*∅ð Þ *w*<sup>0</sup> measures the propagation speed of the plane wave components of the pulse in the medium. These plane waves do not carry

The second term in Eq. 40 shows that, after propagation over a distance z, the pulse keeps a Gaussian envelope. This envelope is delayed by an amount *z=Vg* , *Vg* being the group velocity. The second term in Eq. 40 also shows that the pulse envelope is distorted during its propagation because its form factor *Γ*ð Þ*z* , defined as

> *<sup>w</sup>*<sup>0</sup> <sup>¼</sup> *<sup>d</sup> dw*

This term is called the "Group Velocity Dispersion". The temporal width of the

In optical materials, the refractive index is frequency dependent. This dependence can be calculated for a given material using a Sellmeier equation, typically of the form

*i*¼1

where *wi* is the frequency of resonance and *Bi* is the amplitude of resonance. In the case of optical fibers, the parameters *wi* and *Bi* are obtained experimentally by fitting the measured dispersion curves to Eq. (44) with *m* ¼ 3 and depend on the

An ultrashort Fourier limited pulse has a broad spectrum and no chirp; whenit

propagates a distance through a transparent medium, the medium introduces dispersion to the pulse inducing an increase in the pulse duration. We consider dispersions of orders two. The pulse broadens on propagation because of group

In summary, the propagation of a short optical pulse through transparent medium results in a delay of the pulse, a duration broadening and a frequency chirp. All these phenomena are increase with distance of propagation. We shown in

Bi *1.81651732 0.428893631 1.07186278 λi*ð Þ *μm 0.0143704198 0.0592801172 121.419942*

*w*2

*Biw*<sup>2</sup> *i*

ð Þ¼ *<sup>w</sup>* <sup>1</sup> <sup>þ</sup>X*<sup>m</sup>*

The index of litharge SF56 is given by the following expression (43):

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 4*: Γ:k*<sup>00</sup> ð Þ*z*

1 *Vg* � �

*w*<sup>0</sup>

2 *:* *:* (41)

*<sup>i</sup>* � *<sup>w</sup>*<sup>2</sup> (43)

(42)

any information, because of their infinite duration.

*DOI: http://dx.doi.org/10.5772/intechopen.95978*

pulse at point z:

with *<sup>k</sup>*<sup>00</sup> <sup>¼</sup> *<sup>λ</sup>*<sup>3</sup>

core constituents [21].

**3.2 Parameter of dispersion**

velocity dispersion (GVD).

*Parameters for litharge SF56 glasses.*

**Table 1.**

**19**

2*:π:c*<sup>2</sup> *d*2 *n <sup>d</sup>λ*<sup>2</sup> *<sup>Γ</sup>* <sup>¼</sup> <sup>2</sup> *log* <sup>2</sup> Δ2 0 , Δ*τ*<sup>0</sup> initial pulse before propagation inside the medium

**3.1 Application in litharge index SF56 medium**

Bi: is the coefficients of glass see in **Table 1**

*n*2

Depends on the angular frequency *w* through *k*00ð Þ *w* ,

*Femtosecond Laser Pulses: Generation, Measurement and Propagation*

*k*<sup>00</sup> ¼ *<sup>d</sup>*<sup>2</sup> *k=dw*<sup>2</sup> � �

Δ*τ<sup>z</sup>* ¼ Δ*τ*<sup>0</sup>

*<sup>Γ</sup>* <sup>¼</sup> <sup>2</sup> *log* <sup>2</sup> Δ2 0 is coefficient of function Gaussian

An ultrashort Fourier limited pulse has a broad spectrum and no chirp; when it propagates a distance through a transparent medium, the medium introduces a dispersion to the pulse inducing an increase in the pulse duration. To investigate and determine the dispersion, we assume a Gaussian shape for the pulse. The electric field of the pulse is given as Eq. (35).

After the pulse has propagated a distance z, its spectrum is modified to

$$E(w, z) = E(w) \exp\left[\pm ik(w)z\right], \\ k(w) = \frac{n(w).w}{c}, \tag{35}$$

where k wð Þ is now a frequency-dependent propagation factor. In order to allow for a partial analytical calculation of the propagation effects, the propagation factor is rewritten using a Taylor expansion as a function of the angular frequency, assuming that Δ*w* ≪ *w*<sup>0</sup> (this condition is only weakly true for the shortest pulses). Applying the Taylor expansion to Eq. (37), the pulse spectrum becomes.

Δ*w* is bandwidth of the pulse and *w*<sup>0</sup> is central pulsation

$$k(w) = k(w\_0) + k'(w - w\_0) + \frac{1}{2}k''(w - w\_0)^2 + \dots,\tag{36}$$

$$\begin{array}{l} \text{where } k' = \left(\frac{dk(w)}{dw}\right)\_{w\_0} \text{ and } k'' = \left(\frac{d^2k(w)}{dw^2}\right)\_{w\_0}, \\\\ E(w, z) = \exp\left[-ik(w\_0)z - ik'z(w - w\_0) - \left(\frac{1}{4\Gamma} + \frac{i}{2}k''\right)(w - w\_0)^2\right]. \end{array} \tag{37}$$

The time evolution of the electric field in the pulse is then derived from the calculation of the inverse Fourier transform of Eq. (39),

$$e(t,z) = \int\_{-\infty}^{+\infty} E(w,z) \, e^{-iwt} dw \tag{38}$$

so that

$$e(t, z) = \sqrt{\frac{\Gamma(z)}{\pi}} \exp\left[iw\_0 \left(t - \frac{z}{V\_{\mathcal{Q}}(w\_0)}\right)\right] \times \exp\left[-\Gamma(z) \left(t - \frac{z}{V\_{\mathcal{G}}(w\_0)}\right)^2\right] \tag{39}$$

Where

$$\left(V\_{\mathcal{Q}}(w\_0) = \left(\frac{w}{k}\right)\_{w\_0}, V\_{\mathcal{g}}(w\_0) = \left(\frac{dw}{dk}\right)\_{w\_0}, \mathbf{1}/(\varGamma(\mathbf{z}) = \mathbf{1}/\varGamma + 2\mathbf{ik}\prime\prime\mathbf{z}.\tag{40}$$

In the first exponential term of Eq. 40, it can be observed that the phase of the central frequency *w*<sup>0</sup> is delayed by an amount *<sup>z</sup> <sup>V</sup>*<sup>∅</sup> after propagation over a distance z. Because the phase is not a measurable quantity, this effect has no observable

*Femtosecond Laser Pulses: Generation, Measurement and Propagation DOI: http://dx.doi.org/10.5772/intechopen.95978*

consequence. The phase velocity *V*∅ð Þ *w*<sup>0</sup> measures the propagation speed of the plane wave components of the pulse in the medium. These plane waves do not carry any information, because of their infinite duration.

The second term in Eq. 40 shows that, after propagation over a distance z, the pulse keeps a Gaussian envelope. This envelope is delayed by an amount *z=Vg* , *Vg* being the group velocity. The second term in Eq. 40 also shows that the pulse envelope is distorted during its propagation because its form factor *Γ*ð Þ*z* , defined as

Depends on the angular frequency *w* through *k*00ð Þ *w* ,

$$\left(k'' = \left(^{d^2k}\!\!/\_{dw}\right)\_{w\_0} = \frac{d}{dw}\left(\frac{1}{V\_{\mathfrak{g}}}\right)\_{w\_0}.\tag{41}$$

This term is called the "Group Velocity Dispersion". The temporal width of the pulse at point z:

$$
\Delta \tau\_x = \Delta \tau\_0 \sqrt{1 + 4.(\Gamma \, k^{\prime \prime} z)^2}. \tag{42}
$$

with *<sup>k</sup>*<sup>00</sup> <sup>¼</sup> *<sup>λ</sup>*<sup>3</sup> 2*:π:c*<sup>2</sup> *d*2 *n <sup>d</sup>λ*<sup>2</sup> *<sup>Γ</sup>* <sup>¼</sup> <sup>2</sup> *log* <sup>2</sup> Δ2 0 ,

**3. Propagation of a light pulse in a transparent medium**

is coefficient of function Gaussian

electric field of the pulse is given as Eq. (35).

*Recent Advances in Numerical Simulations*

*<sup>Γ</sup>* <sup>¼</sup> <sup>2</sup> *log* <sup>2</sup> Δ2 0

The frequency Fourier transform of a Gaussian pulse has already been given as

An ultrashort Fourier limited pulse has a broad spectrum and no chirp; when it propagates a distance through a transparent medium, the medium introduces a dispersion to the pulse inducing an increase in the pulse duration. To investigate and determine the dispersion, we assume a Gaussian shape for the pulse. The

2

*:* (34)

*<sup>c</sup>* , (35)

<sup>2</sup> <sup>þ</sup> … , (36)

2

*:* (37)

(39)

ð Þ *w* � *w*<sup>0</sup>

�*iwtdw* (38)

*Vg*ð Þ *w*<sup>0</sup>

� �<sup>2</sup> " #

, 1*= Γ*ð Þ¼ *z* 1*=Γ* þ 2*ik*<sup>00</sup> ð *z:* (40)

*<sup>V</sup>*<sup>∅</sup> after propagation over a distance z.

4*:Γ* !

*E w*ð Þ¼ *exp* �ð Þ *<sup>w</sup>* � *<sup>w</sup>*<sup>0</sup>

After the pulse has propagated a distance z, its spectrum is modified to

Applying the Taylor expansion to Eq. (37), the pulse spectrum becomes.

Δ*w* is bandwidth of the pulse and *w*<sup>0</sup> is central pulsation

*:* and *<sup>k</sup>*<sup>00</sup> <sup>¼</sup> *<sup>d</sup>*<sup>2</sup>

calculation of the inverse Fourier transform of Eq. (39),

*:exp iw*<sup>0</sup> *<sup>t</sup>* � *<sup>z</sup>*

,*Vg*ð Þ¼ *w*<sup>0</sup>

*e t*ð Þ¼ , *z*

� � � �

*k w*ð Þ¼ *k w*ð Þþ <sup>0</sup> *k*<sup>0</sup>

where *<sup>k</sup>*<sup>0</sup> <sup>¼</sup> *dk w*ð Þ

so that

*e t*ð Þ¼ , *z*

Where

**18**

*dw* � �

*E w*ð Þ¼ , *z exp* �*ik w*ð Þ<sup>0</sup> *z* � *ik*<sup>0</sup>

ffiffiffiffiffiffiffiffiffi *Γ*ð Þ*z π*

> *w k* � � *w*<sup>0</sup>

central frequency *w*<sup>0</sup> is delayed by an amount *<sup>z</sup>*

r

*V*∅ð Þ¼ *w*<sup>0</sup>

*w*<sup>0</sup>

*E w*ð Þ¼ , *<sup>z</sup> E w*ð Þ*exp* ½ � �*ik w*ð Þ*<sup>z</sup>* , *k w*ð Þ¼ *n w*ð Þ*:<sup>w</sup>*

ð Þþ *w* � *w*<sup>0</sup>

*k w*ð Þ *dw*<sup>2</sup> � �

*z w*ð Þ� � *w*<sup>0</sup>

The time evolution of the electric field in the pulse is then derived from the

ðþ<sup>∞</sup> �∞

*V*∅ð Þ *w*<sup>0</sup>

*dw dk* � �

Because the phase is not a measurable quantity, this effect has no observable

*w*<sup>0</sup>

In the first exponential term of Eq. 40, it can be observed that the phase of the

*w*<sup>0</sup> ,

� �

*E w*ð Þ , *z :e*

1 2 *k*00

> 1 4*Γ* þ *i* 2 *k*00 � �

ð Þ *w* � *w*<sup>0</sup>

� *exp* �*Γ*ð Þ*<sup>z</sup> <sup>t</sup>* � *<sup>z</sup>*

where k wð Þ is now a frequency-dependent propagation factor. In order to allow for a partial analytical calculation of the propagation effects, the propagation factor is rewritten using a Taylor expansion as a function of the angular frequency, assuming that Δ*w* ≪ *w*<sup>0</sup> (this condition is only weakly true for the shortest pulses).

Δ*τ*<sup>0</sup> initial pulse before propagation inside the medium

#### **3.1 Application in litharge index SF56 medium**

In optical materials, the refractive index is frequency dependent. This dependence can be calculated for a given material using a Sellmeier equation, typically of the form

$$m^2(w) = 1 + \sum\_{i=1}^{m} \frac{B\_i w\_i^2}{w\_i^2 - w^2} \tag{43}$$

Bi: is the coefficients of glass see in **Table 1**

The index of litharge SF56 is given by the following expression (43):

where *wi* is the frequency of resonance and *Bi* is the amplitude of resonance. In the case of optical fibers, the parameters *wi* and *Bi* are obtained experimentally by fitting the measured dispersion curves to Eq. (44) with *m* ¼ 3 and depend on the core constituents [21].

#### **3.2 Parameter of dispersion**

An ultrashort Fourier limited pulse has a broad spectrum and no chirp; whenit propagates a distance through a transparent medium, the medium introduces dispersion to the pulse inducing an increase in the pulse duration. We consider dispersions of orders two. The pulse broadens on propagation because of group velocity dispersion (GVD).

In summary, the propagation of a short optical pulse through transparent medium results in a delay of the pulse, a duration broadening and a frequency chirp. All these phenomena are increase with distance of propagation. We shown in


**Table 1.** *Parameters for litharge SF56 glasses.* **Figure 16** that the duration broadening is not linear for ultrashort pulse. Specially under 70 fs or less. The Eq. (43) is not applicable for pulse less than 70 fs. For minimize these parameters we introduce the nonlinear phenomena as Self phase modulation (SPM), Soliton pulse, dispersion compensate fiber.

#### **3.3 Group velocity dispersion**

The Group Velocity Dispersion (GVD) is defined as the propagation of different frequency components at different speeds through a dispersive medium. This is due to the wavelength-dependent index of refraction of the dispersive material [22].

*φ*ð Þ¼ *w φ*ð Þþ *w*<sup>0</sup> ð Þ *w* � *w*<sup>0</sup> *dφ dw* � � � � þ 1 2! ð Þ *<sup>w</sup>* � *<sup>w</sup>*<sup>0</sup> <sup>2</sup>*d*<sup>2</sup> *φ dw*<sup>2</sup> � � � � *w*¼*w*<sup>0</sup> þ … þ 1 *n*! ð Þ *<sup>w</sup>* � *<sup>w</sup>*<sup>0</sup> *<sup>n</sup> dn φ dw<sup>n</sup>* � � � � *w*¼Ω (44) *φ λ*ð Þ¼ <sup>2</sup>*<sup>π</sup> <sup>λ</sup> <sup>n</sup>*ð Þ*<sup>λ</sup> <sup>z</sup> dλ dw* ¼ � *<sup>λ</sup>*<sup>2</sup> 2*πc dφ dw* ¼ � *<sup>z</sup> c dn <sup>d</sup><sup>λ</sup>* � *<sup>n</sup>* � � *d*2 *φ dw*<sup>2</sup> ¼ þ *λ*3 4*π*<sup>3</sup>*c*<sup>2</sup> *d*2 *n <sup>d</sup>λ*<sup>2</sup> *<sup>z</sup> d*3 *φ dw*<sup>3</sup> ¼ � *<sup>λ</sup>*<sup>4</sup> <sup>4</sup>*π*<sup>2</sup>*c*<sup>3</sup> <sup>3</sup> *d*2 *n <sup>d</sup>λ*<sup>2</sup> <sup>þ</sup> *<sup>λ</sup> d*3 *n dλ*<sup>3</sup> " #*z d*4 *φ dw*<sup>4</sup> ¼ þ *λ*5 <sup>8</sup>*π*<sup>3</sup>*c*<sup>4</sup> <sup>12</sup> *<sup>d</sup>*<sup>2</sup> *n <sup>d</sup>λ*<sup>2</sup> <sup>þ</sup> <sup>8</sup>*<sup>λ</sup> d*3 *n <sup>d</sup>λ*<sup>3</sup> <sup>þ</sup> *<sup>λ</sup>*<sup>2</sup> *<sup>d</sup>*<sup>4</sup> *n dλ*<sup>4</sup> " #*z d*5 *φ dw*<sup>5</sup> ¼ � *<sup>λ</sup>*<sup>6</sup> <sup>16</sup>*π*<sup>4</sup>*c*<sup>5</sup> <sup>60</sup> *<sup>d</sup>*<sup>2</sup> *n <sup>d</sup>λ*<sup>2</sup> <sup>þ</sup> <sup>60</sup>*<sup>λ</sup> d*3 *n <sup>d</sup>λ*<sup>3</sup> <sup>þ</sup> <sup>15</sup>*λ*<sup>2</sup> *<sup>d</sup>*<sup>4</sup> *n <sup>d</sup>λ*<sup>4</sup> <sup>þ</sup> *<sup>λ</sup>*<sup>3</sup> *<sup>d</sup>*<sup>5</sup> *n dλ*<sup>5</sup> " #*z d*6 *φ dw*<sup>6</sup> ¼ þ *λ*7 <sup>32</sup>*π*<sup>5</sup>*c*<sup>6</sup> <sup>360</sup> *<sup>d</sup>*<sup>2</sup> *n <sup>d</sup>λ*<sup>2</sup> <sup>þ</sup> <sup>480</sup>*<sup>λ</sup> d*3 *n <sup>d</sup>λ*<sup>3</sup> <sup>þ</sup> <sup>180</sup>*λ*<sup>2</sup> *<sup>d</sup>*<sup>4</sup> *n <sup>d</sup>λ*<sup>4</sup> <sup>þ</sup> <sup>24</sup>*λ*<sup>3</sup> *<sup>d</sup>*<sup>5</sup> *n <sup>d</sup>λ*<sup>5</sup> <sup>þ</sup> *<sup>λ</sup>*<sup>4</sup> *<sup>d</sup>*<sup>6</sup> *n dλ*<sup>6</sup> " #*z* 8 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>< >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>: (45)

<sup>∅</sup>ð Þ *<sup>p</sup>* ¼ �ð Þ<sup>1</sup> *<sup>p</sup>*

**Figure 16.**

**Figure 17.**

**21**

calculation (**Figure 17**) [22].

*:*2*π:z*

*λ* 2*:π:c* � �*<sup>p</sup>*

*Femtosecond Laser Pulses: Generation, Measurement and Propagation*

*DOI: http://dx.doi.org/10.5772/intechopen.95978*

X *p*

*λ <sup>j</sup>*�<sup>1</sup>

*Temporal broadening of the transform- limited pulse for differentvalues of the propagation distance z.*

Analytically known and experimentally observed propagation affects such as spectral shift, pulse broadening and asymmetry in dispersive media can be easily brought out in the simulation using formalism presented here. In addition, such studies can be extended to pulses of arbitrary temporal shape without any further algorithmic complexity by numerical simulation. Higher order dispersion effects can be handled easily in the numerical simulation unlike in the case of analytical

*(a) The pulse broadens on propagation as a result of group velocity dispersion (GVD) (b) the pulse shape is no*

*longer Gaussian, and it becomes asymmetric due to higher order dispersions.*

*A p*ð Þ � 1, *<sup>j</sup>* � <sup>1</sup> *<sup>n</sup>*ð Þ*<sup>j</sup>* with *<sup>p</sup>*<sup>&</sup>gt; 2 (48)

*j*¼2

Itseemsto methatwe can write *<sup>ϕ</sup>*ð Þ*<sup>i</sup>* <sup>¼</sup> *<sup>d</sup><sup>i</sup> ϕ dw<sup>i</sup>* asarecurrence, giving *<sup>ϕ</sup>*ð Þ*<sup>i</sup>* basedonderivativesoforderi, the index of refraction. Matrix form, wecan writet

$$
\begin{bmatrix} \mathcal{Q}^{(2)} \\ \mathcal{Q}^{(3)} \\ \mathcal{Q}^{(4)} \\ \mathcal{Q}^{(5)} \\ \mathcal{Q}^{(6)} \end{bmatrix} = (-1)^{\pi} 2.\pi z \begin{bmatrix} \lambda \\ 3 & 1 & 0 & 0 & 0 \\ 12 & 8 & 1 & 0 & 0 \\ 60 & 60 & 15 & 1 & 0 \\ 360 & 480 & 180 & 24 & 1 \end{bmatrix} \tag{46}
$$

The various terms of the Taylor expansion to order n can be written in the shape of a matrix [A], which's we can express various terms A ij.

$$\mathcal{Q}(w) = \mathcal{Q}(w\_0) + (w - w\_0)\mathcal{Q}^{(1)} + \sum\_{i=2}^{p} \frac{1}{i!} (w - w\_0)^i \mathcal{Q}^{(i)}|\_{w = w\_0}. \tag{47}$$

<sup>∅</sup>ð Þ *<sup>w</sup>* is Taylor series of phase and <sup>∅</sup>ð Þ *<sup>p</sup>* is the orders of Taylor series

*Femtosecond Laser Pulses: Generation, Measurement and Propagation DOI: http://dx.doi.org/10.5772/intechopen.95978*

**Figure 16.**

**Figure 16** that the duration broadening is not linear for ultrashort pulse. Specially under 70 fs or less. The Eq. (43) is not applicable for pulse less than 70 fs. For minimize these parameters we introduce the nonlinear phenomena as Self phase

The Group Velocity Dispersion (GVD) is defined as the propagation of different frequency components at different speeds through a dispersive medium. This is due to the wavelength-dependent index of refraction of the dispersive material [22].

> *φ dw*<sup>2</sup> � � � � *w*¼*w*<sup>0</sup>

*<sup>λ</sup> <sup>n</sup>*ð Þ*<sup>λ</sup> <sup>z</sup>*

2*πc*

*d*2 *n <sup>d</sup>λ*<sup>2</sup> *<sup>z</sup>*

" #

*d*3 *n <sup>d</sup>λ*<sup>3</sup> <sup>þ</sup> *<sup>λ</sup>*<sup>2</sup> *<sup>d</sup>*<sup>4</sup>

" #

" #

" #

*d*3 *n dλ*<sup>3</sup>

*z*

*n dλ*<sup>4</sup>

*n <sup>d</sup>λ*<sup>4</sup> <sup>þ</sup> *<sup>λ</sup>*<sup>3</sup> *<sup>d</sup>*<sup>5</sup>

*n <sup>d</sup>λ*<sup>4</sup> <sup>þ</sup> <sup>24</sup>*λ*<sup>3</sup> *<sup>d</sup>*<sup>5</sup>

10 00 0 31 0 0 0 12 8 1 0 0 60 60 15 1 0 360 480 180 24 1

*z*

*n dλ*<sup>5</sup>

*n <sup>d</sup>λ*<sup>5</sup> <sup>þ</sup> *<sup>λ</sup>*<sup>4</sup> *<sup>d</sup>*<sup>6</sup>

*dw<sup>i</sup>* asarecurrence, giving *<sup>ϕ</sup>*ð Þ*<sup>i</sup>* basedonder-

*z*

*n dλ*<sup>6</sup>

*z*

(46)

*:* (47)

*λ*3 4*π*<sup>3</sup>*c*<sup>2</sup>

> *d*2 *n <sup>d</sup>λ*<sup>2</sup> <sup>þ</sup> *<sup>λ</sup>*

*d*3 *n <sup>d</sup>λ*<sup>3</sup> <sup>þ</sup> <sup>15</sup>*λ*<sup>2</sup> *<sup>d</sup>*<sup>4</sup>

*<sup>d</sup>λ*<sup>3</sup> <sup>þ</sup> <sup>180</sup>*λ*<sup>2</sup> *<sup>d</sup>*<sup>4</sup>

*ϕ*

The various terms of the Taylor expansion to order n can be written in the shape

*p*

1 *i*!

ð Þ *w* � *w*<sup>0</sup>

*i* ∅ð Þ*<sup>i</sup>* � � *w*¼*w*<sup>0</sup>

*i*¼2

<sup>∅</sup>ð Þ *<sup>w</sup>* is Taylor series of phase and <sup>∅</sup>ð Þ *<sup>p</sup>* is the orders of Taylor series

þ … þ

1 *n*! ð Þ *<sup>w</sup>* � *<sup>w</sup>*<sup>0</sup> *<sup>n</sup> dn*

*φ dw<sup>n</sup>* � � � � *w*¼Ω

(44)

(45)

ð Þ *<sup>w</sup>* � *<sup>w</sup>*<sup>0</sup> <sup>2</sup>*d*<sup>2</sup>

*φ λ*ð Þ¼ <sup>2</sup>*<sup>π</sup>*

<sup>4</sup>*π*<sup>2</sup>*c*<sup>3</sup> <sup>3</sup>

*n <sup>d</sup>λ*<sup>2</sup> <sup>þ</sup> <sup>60</sup>*<sup>λ</sup>*

ivativesoforderi, the index of refraction. Matrix form, wecan writet

*λ* 2*:π:c* � �*<sup>n</sup>*

*d*3 *n*

*n <sup>d</sup>λ*<sup>2</sup> <sup>þ</sup> <sup>8</sup>*<sup>λ</sup>*

*dλ dw* ¼ � *<sup>λ</sup>*<sup>2</sup>

*dφ dw* ¼ � *<sup>z</sup> c dn <sup>d</sup><sup>λ</sup>* � *<sup>n</sup>* � �

*d*2 *φ dw*<sup>2</sup> ¼ þ

*λ*5 <sup>8</sup>*π*<sup>3</sup>*c*<sup>4</sup> <sup>12</sup> *<sup>d</sup>*<sup>2</sup>

<sup>16</sup>*π*<sup>4</sup>*c*<sup>5</sup> <sup>60</sup> *<sup>d</sup>*<sup>2</sup>

*n <sup>d</sup>λ*<sup>2</sup> <sup>þ</sup> <sup>480</sup>*<sup>λ</sup>*

2*:π:z*

of a matrix [A], which's we can express various terms A ij.

<sup>∅</sup>ð Þ¼ *<sup>w</sup>* <sup>∅</sup>ð Þþ *<sup>w</sup>*<sup>0</sup> ð Þ *<sup>w</sup>* � *<sup>w</sup>*<sup>0</sup> <sup>∅</sup>ð Þ<sup>1</sup> <sup>þ</sup><sup>X</sup>

Itseemsto methatwe can write *<sup>ϕ</sup>*ð Þ*<sup>i</sup>* <sup>¼</sup> *<sup>d</sup><sup>i</sup>*

¼ �ð Þ<sup>1</sup> *<sup>n</sup>*

*d*3 *φ dw*<sup>3</sup> ¼ � *<sup>λ</sup>*<sup>4</sup>

*d*4 *φ dw*<sup>4</sup> ¼ þ

*d*5 *φ dw*<sup>5</sup> ¼ � *<sup>λ</sup>*<sup>6</sup>

∅ð Þ<sup>2</sup> ∅ð Þ<sup>3</sup> ∅ð Þ <sup>4</sup> ∅ð Þ<sup>5</sup> ∅ð Þ <sup>6</sup>

*λ*7 <sup>32</sup>*π*<sup>5</sup>*c*<sup>6</sup> <sup>360</sup> *<sup>d</sup>*<sup>2</sup>

*d*6 *φ dw*<sup>6</sup> ¼ þ

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

**20**

modulation (SPM), Soliton pulse, dispersion compensate fiber.

*dφ dw* � � � � þ 1 2!

**3.3 Group velocity dispersion**

*Recent Advances in Numerical Simulations*

*φ*ð Þ¼ *w φ*ð Þþ *w*<sup>0</sup> ð Þ *w* � *w*<sup>0</sup>

8

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><

*Temporal broadening of the transform- limited pulse for differentvalues of the propagation distance z.*

$$\mathcal{D}^{(p)} = (-1)^p.2\pi.z \left[\frac{\lambda}{2\pi.c}\right]^p \sum\_{j=2}^p \lambda^{j-1} A(p-1, j-1) n^{(j)} \text{ with } p > 2\tag{48}$$

Analytically known and experimentally observed propagation affects such as spectral shift, pulse broadening and asymmetry in dispersive media can be easily brought out in the simulation using formalism presented here. In addition, such studies can be extended to pulses of arbitrary temporal shape without any further algorithmic complexity by numerical simulation. Higher order dispersion effects can be handled easily in the numerical simulation unlike in the case of analytical calculation (**Figure 17**) [22].

**Figure 17.**

*(a) The pulse broadens on propagation as a result of group velocity dispersion (GVD) (b) the pulse shape is no longer Gaussian, and it becomes asymmetric due to higher order dispersions.*
