**4. Time-frequency decomposition**

#### **4.1 Wavelet theory**

The wavelets are very particular elementary functions, these are the shortest vibrations and most elementary that one can consider. One can say that the wavelet east carries out a zooming on any interesting phenomenon of the signal which place on a small scale in the vicinity of the point considered [23].

*θ*ð Þ¼ Ω, *z*

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

**Figure 19.**

**23**

*E*0 2*:* ffiffiffi *π* p *γ*

*Femtosecond Laser Pulses: Generation, Measurement and Propagation*

in (**Figure 19**). Under these circumstances, we have

*E w*ð Þ*:exp* � ð Þ *<sup>w</sup>* � <sup>Ω</sup> <sup>2</sup>

As already mentioned, *τwavelet* is large enough to ensure that analyzing function has only non negligible values over a spectral range lying in the neighborhood of Ω

*(a) Initial pulse, (c) pulse after propagation of the 10 cm in glass SF56 (e) contour of the wavelet, (g) the wavelet representation, (b) initial pulse, (d) pulse after propagation of the 10 cm in the silica medium,*

*(f) contour of the wavelet, (h) the wavelet representation [22].*

4*γ* " #

*:exp j* ½ � ∅ð Þ *w :* (52)

#### **4.2 Wavelet techniques**

Starting with a signal *e t*ð Þ, in plane *z* ¼ 0, we define wavelet centered at Ω by (**Figure 18**):

$$\theta(\Omega) = E(w).exp\left[-\frac{\left(w - \Omega\right)^2}{4\gamma}\right], \text{with } E(w) = \frac{E\_0}{2\pi} \sqrt{\frac{\pi}{\Gamma}} \exp\left[\frac{\left(w - w\_0\right)^2}{4\cdot\Gamma}\right], \tag{49}$$

$$\theta(\mathbf{t}, z = \mathbf{0}) = T F \{\theta(\Omega, z = \mathbf{0})\}$$

TF it is the Fourier Transform equation z is the distance of propagation

$$\theta(t, \mathbf{z} = \mathbf{0}) = E\_0 \sqrt{\frac{\chi}{\chi + \Gamma}} \exp\left[\frac{-\left(w\_0 - \Omega\right)^2}{4\left(\chi + \Gamma\right)}\right] \cdot \exp\left[-\frac{\chi\Gamma}{\chi + \Gamma} t^2\right] \cdot \exp\left[j\frac{\chi w\_0 + \Gamma\Omega}{\chi + \Gamma} t\right] \tag{51}$$


#### **Figure 18.**

*Gaussian envelope decomposed on a number of waveletwecalculates the electric field associated with the wavelet θ Ω*ð Þ , *z* ¼ 0 *:*

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

**4. Time-frequency decomposition**

*Recent Advances in Numerical Simulations*

The wavelets are very particular elementary functions, these are the shortest vibrations and most elementary that one can consider. One can say that the wavelet east carries out a zooming on any interesting phenomenon of the signal which place

Starting with a signal *e t*ð Þ, in plane *z* ¼ 0, we define wavelet centered at Ω by

, with *E w*ð Þ¼ *<sup>E</sup>*<sup>0</sup>

2*:π*

*:exp* � *γΓ*

ffiffiffi *π Γ* r

*θ*ð Þ¼ *t*, *z* ¼ 0 *TF*f g *θ*ð Þ Ω, *z* ¼ 0 (50)

*γ* þ *Γ t* 2 � �

*<sup>γ</sup>*þ*<sup>Γ</sup> :*

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

4*:Γ* " #

2

*:exp j <sup>γ</sup>w*<sup>0</sup> <sup>þ</sup> *<sup>Γ</sup>*<sup>Ω</sup>

*γ* þ *Γ*

� �

*t*

(51)

, (49)

on a small scale in the vicinity of the point considered [23].

4*γ* " #

• In time, the pulse is also Gaussian, of parameter *γΓ*

frequency of analysis on Gaussian of parameter *γ* þ *Γ*.

*:exp* �ð Þ *<sup>w</sup>*<sup>0</sup> � <sup>Ω</sup> <sup>2</sup> 4ð Þ *γ* þ *Γ* " #

• The maximum of amplitude of the wavelet *θ*ð Þ *t*, *z* ¼ 0 vary with Ω, center

• The signal propagates in the positive *z* direction in a linear dispersive and transparent medium, which fills the half space *z*>0 and whose refractive index is *n w*ð Þ. After propagation, the wavelet *θ Ω*ð Þ , *x* may be written as

*Gaussian envelope decomposed on a number of waveletwecalculates the electric field associated with the wavelet*

**4.1 Wavelet theory**

**4.2 Wavelet techniques**

*<sup>θ</sup>*ð Þ¼ <sup>Ω</sup> *E w*ð Þ*:exp* � ð Þ *<sup>w</sup>* � <sup>Ω</sup> <sup>2</sup>

z is the distance of propagation

r

TF it is the Fourier Transform equation

ffiffiffiffiffiffiffiffiffiffiffi *γ γ* þ *Γ*

(**Figure 18**):

*θ*ð Þ¼ *t*, *z* ¼ 0 *E*<sup>0</sup>

**Figure 18.**

*θ Ω*ð Þ , *z* ¼ 0 *:*

**22**

$$\theta(\Omega, z) = \frac{E\_0}{2\sqrt{\pi \chi}} E(w) \exp\left[-\frac{\left(w - \Omega\right)^2}{4\chi}\right] \exp\left[j\mathcal{Q}(w)\right].\tag{52}$$

As already mentioned, *τwavelet* is large enough to ensure that analyzing function has only non negligible values over a spectral range lying in the neighborhood of Ω in (**Figure 19**). Under these circumstances, we have

#### **Figure 19.**

*(a) Initial pulse, (c) pulse after propagation of the 10 cm in glass SF56 (e) contour of the wavelet, (g) the wavelet representation, (b) initial pulse, (d) pulse after propagation of the 10 cm in the silica medium, (f) contour of the wavelet, (h) the wavelet representation [22].*

$$\begin{split} \mathcal{Q}(w) &= \mathcal{Q}(\Omega) + (w - \Omega) \frac{d\mathcal{Q}}{dw} \bigg|\_{} + \frac{1}{2!} (w - \Omega)^2 \frac{d^2 \mathcal{Q}}{dw^2} \bigg|\_{w = \Omega} + \dots + \frac{1}{n!} (w - \Omega)^n \frac{d^n \mathcal{Q}}{dw^n} \bigg|\_{w = \Omega} \\ &+ \theta(w) \cdot\_{w = \Omega} \end{split} \tag{53}$$

Pulse of the wavelet: Δ*τwavelet* ¼ 1000 *fs* Longer of the medium: *z* ¼ 10 *cm*

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

silica fiber are the same modification.

**5. Conclusion**

**Author details**

Mounir Khelladi

**25**

To describe the propagation of the pulse, we only consider the propagation of

**Figure 19(c)** shows that when pulse propagate inside SF56 glass present minor

**Figure 19(e)** shows that the SF56 glass resist in temporal domain than silica fiber as shown in **Figure 19(f)**. but, in frequency domain the both SF 56 glass and

Generating ultrashort light pulses requires a laser to operate in a particular regime, called mode-locking which many be illustrated either in the frequency or the time domain. Depending on the particular case, one description is much more intuitive than the other and we have chosen to present the simpler approach. The generation of femtosecond laser pulses via mode locking is described in simple physical terms. As femtosecond laser pulses can be generated directly from a wide variety of lasers with wavelengths ranging from the ultraviolet to the infrared no

The ability to accurately measure ultrashort laser pulses is essential to creating, using, and improving them, but the technology for their measurement has consistently lagged behind that for their generation. The result has been a long and sometimes quite painful history of attempts—and failures—to measure these exotic and ephemeral events. The reason is that many pulse-measurement techniques have suffered from, and continue to suffer from, a wide range of complications, including the presence of ambiguities, insufficient temporal and/or spectral resolution and/or range, an inherent inability to measure the complete pulse intensity and/or phase, an inability to measure complex pulses, and misleading results due to the loss of information due to idiosyn-

Finally, we have demonstrated here the possible decomposition of an ultrashort pulse into an infinite number of longer Fourier transform limited wavelets which propagate without any deformation through a dispersive medium. After propagation through the medium, the pulse may be visualized in a three-dimensional

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

the maximum of each wavelet in a three dimensional representation:

*Femtosecond Laser Pulses: Generation, Measurement and Propagation*

dispersion and distortion compared to the silica fiber in **Figure 19d**.

**Figure 19(g)** and **(h)** shown the amplitude of the wavelet

attempt is made to cover different technical approaches.

crasies of the technique or multishot averages over different pulses.

representation by the locus of the wavelet maxima.

Abou-bekrBelkaid University of the Tlemcen, Algeria

provided the original work is properly cited.

\*Address all correspondence to: mo.khelladi@gmail.com

Neglecting the higher terms in Eq. (49):

$$\mathcal{Q}(w) = \mathcal{Q}(\Omega) + (w - \Omega) \frac{d\mathcal{Q}}{dw}\bigg|\_{w=\Omega} + \frac{1}{2!}(w - \Omega)^2 \frac{d^2\mathcal{Q}}{dw^2}\bigg|\_{w=\Omega} + \theta(w). \tag{54}$$

$$\begin{split} \theta(\Omega, \mathbf{z}) &= \frac{E\_0}{2\sqrt{\pi}\eta} \sqrt{\frac{\pi}{\Gamma}} \exp\left[-\frac{(w - w\_0)^2}{4\Gamma}\right] \\ \propto \exp\left[-\frac{(w - \Omega)^2}{4\gamma}\right] \cdot \exp\left[j\mathcal{Q}^{(0)} + j(w - \Omega)\mathcal{Q}^{(1)} + \frac{1}{2}j(w - \Omega)^2 \mathcal{Q}^{(2)}\right] \end{split} \tag{55}$$
 
$$\theta(\mathbf{t}, \mathbf{z}) = \frac{1}{2\pi} \int\_{-\infty}^{+\infty} \theta(\Omega, \mathbf{z}) \cdot \exp\left(jwt\right) dw \tag{56}$$

We calculates the temporal electric field associated with the wavelet *θ Ω*ð Þ , *z :*

$$\theta(t,\boldsymbol{z}) = \frac{1}{2\pi} \frac{E\_0}{2\sqrt{\pi}\gamma} \sqrt{\frac{\pi}{\Gamma}} \mathcal{E} \left[ \frac{-\frac{(\boldsymbol{\Delta} - \boldsymbol{w}\_0)^2}{4\gamma}}{\Gamma} \right] \mathcal{E} \left( \boldsymbol{j} \mathcal{O}^{(0)} \right) \times \boldsymbol{\varepsilon} \left[ \frac{1}{4\gamma} + \frac{1}{\gamma} \frac{1}{2} \mathcal{O}^{(2)} \right] \mathcal{E}^2 \left[ \frac{(\boldsymbol{\Delta} - \boldsymbol{w}\_0)}{2\gamma} \boldsymbol{j} \mathcal{O}^{(1)} \right] \boldsymbol{\Omega}$$

$$\times \left( \int\_{-\infty}^{+\infty} \boldsymbol{\varepsilon}^{-\left[ \frac{1}{4\gamma} + \frac{1}{4\gamma} \frac{1}{2} \mathcal{O}^{(2)} \right] \boldsymbol{w}^2} \cdot \boldsymbol{\mathcal{E}}^{\left[ \frac{1}{4\gamma} + \frac{1}{4\gamma} \frac{1}{2} \mathcal{O}^{(2)} \right] \boldsymbol{\Omega} \boldsymbol{w}} \times \boldsymbol{\varepsilon} \left[ \frac{-\frac{(\boldsymbol{\Delta} - \boldsymbol{w}\_0)^2}{2\gamma} \boldsymbol{j} \mathcal{O}^{(1)}}{2\gamma} \boldsymbol{j} \mathcal{O}^{(1)} \right] \boldsymbol{w} \right) d\boldsymbol{w} \tag{57}$$

The amplitude of the incident Ω wavelet is given from Eq. (58) by

$$\theta(t,x) = \frac{E\_0}{2\sqrt{\pi}\eta} \sqrt{\frac{\Gamma(z)}{\Gamma}} \exp\left(j\mathcal{Q}^{(0)}\right) \exp\left(-\Gamma(z)\left[t + \frac{z}{V\_\xi(\Omega)}\right]^2\right)$$

$$\lambda \times \exp\left(-\frac{\left(\Omega - w\_0\right)^2}{4\Gamma} \left[1 - \frac{\Gamma(z)}{\Gamma}\right]\right) \times \exp\left[j\left(1 - \frac{\Gamma(z)}{\Gamma}\right)\Omega + \frac{\Gamma(z)}{\Gamma} w\_0\right] \left(t + \frac{z}{V\_\xi(\Omega)}\right). \tag{58}$$

This wavelet is characterized by a Gaussian envelope. This decomposition is valid only for the values of Δ*w* much larger than *δw* (Δ*w* ≫ *δw*).

The delay of group of the wavelet *<sup>t</sup>* <sup>þ</sup> *<sup>z</sup> Vg* ð Þ *Ω* h i is characterized by a Gaussian envelope which is the temporal width.

The delay of group of the wavelet is inversely proportional to the velocity of group its envelope propagates without deformation [22].

#### **4.3 Simulations**

#### *4.3.1 Parameters of the simulations*

Pulse initial: Δ*τ*<sup>0</sup> ¼ 20 *fs* (Wavelength)*λ* ¼ 800 *nm* *Femtosecond Laser Pulses: Generation, Measurement and Propagation DOI: http://dx.doi.org/10.5772/intechopen.95978*

Pulse of the wavelet: Δ*τwavelet* ¼ 1000 *fs*

Longer of the medium: *z* ¼ 10 *cm*

To describe the propagation of the pulse, we only consider the propagation of the maximum of each wavelet in a three dimensional representation:

**Figure 19(c)** shows that when pulse propagate inside SF56 glass present minor dispersion and distortion compared to the silica fiber in **Figure 19d**.

**Figure 19(e)** shows that the SF56 glass resist in temporal domain than silica fiber as shown in **Figure 19(f)**. but, in frequency domain the both SF 56 glass and silica fiber are the same modification.

**Figure 19(g)** and **(h)** shown the amplitude of the wavelet
