**5. Conclusion**

<sup>∅</sup>ð Þ¼ *<sup>w</sup>* <sup>∅</sup>ð Þþ <sup>Ω</sup> ð Þ *<sup>w</sup>* � <sup>Ω</sup> *<sup>d</sup>*<sup>∅</sup>

*Recent Advances in Numerical Simulations*

þ *θ*ð Þ *w :w*¼<sup>Ω</sup>

*θ*ð Þ¼ Ω, *z*

*θ*ð Þ¼ *t*, *z*

�

ðþ<sup>∞</sup> �∞ *e* � <sup>1</sup> <sup>4</sup>*Γ*<sup>þ</sup> <sup>1</sup> 4*γ* –1 2 *<sup>j</sup>*∅ð Þ<sup>2</sup> ½ �*<sup>w</sup>*<sup>2</sup>

*θ*ð Þ¼ *t*, *z*

4*Γ*

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

**4.3 Simulations**

**24**

�

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

*dw* � � � � þ 1 2!

> *dw* � � � � *w*¼Ω þ 1 2!

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

1 2*π*

*<sup>e</sup>* �ð Þ *<sup>Ω</sup>*�*w*<sup>0</sup> <sup>2</sup> 4*Γ* h i

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

*:exp j*∅ð Þ <sup>0</sup> � �

*:exp j* <sup>1</sup> � *<sup>Γ</sup>*ð Þ*<sup>z</sup>*

*Vg* ð Þ *Ω* h i

This wavelet is characterized by a Gaussian envelope. This decomposition is

The delay of group of the wavelet is inversely proportional to the velocity of

*:e* <sup>1</sup> <sup>4</sup>*Γ*<sup>þ</sup> <sup>1</sup> 4*γ* �1 2 *<sup>j</sup>*∅ð Þ<sup>2</sup> ½ �<sup>2</sup>Ω*<sup>w</sup>*

ffiffiffiffiffiffiffiffiffi *Γ*ð Þ*z Γ*

valid only for the values of Δ*w* much larger than *δw* (Δ*w* ≫ *δw*).

r

<sup>1</sup> � *<sup>Γ</sup>*ð Þ*<sup>z</sup> Γ*

group its envelope propagates without deformation [22].

þ ð∞

�∞

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

*<sup>e</sup> <sup>j</sup>*∅ð Þ <sup>0</sup> ð Þ � *<sup>e</sup>*

4*Γ* " #

*:exp j*∅ð Þ <sup>0</sup> <sup>þ</sup> *j w*ð Þ � <sup>Ω</sup> <sup>∅</sup>ð Þ<sup>1</sup> <sup>þ</sup>

Neglecting the higher terms in Eq. (49):

ffiffiffi *π Γ* r

*θ*ð Þ¼ *t*, *z*

ffiffiffi *π Γ* r

<sup>∅</sup>ð Þ¼ *<sup>w</sup>* <sup>∅</sup>ð Þþ <sup>Ω</sup> ð Þ *<sup>w</sup>* � <sup>Ω</sup> *<sup>d</sup>*<sup>∅</sup>

*E*0 2*:* ffiffiffi *π* p *γ*

4*γ* " #

> 1 2*π*

*E*0 2*:* ffiffiffi *π* p *γ*

> *E*0 2*:* ffiffiffi *π* p *γ*

> > 2

The delay of group of the wavelet *<sup>t</sup>* <sup>þ</sup> *<sup>z</sup>*

envelope which is the temporal width.

*4.3.1 Parameters of the simulations*

Pulse initial: Δ*τ*<sup>0</sup> ¼ 20 *fs* (Wavelength)*λ* ¼ 800 *nm*

! � �

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

2

∅ *dw*<sup>2</sup> � � � � *w*¼Ω

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

� �

� <sup>1</sup> <sup>4</sup>*Γ*<sup>þ</sup> <sup>1</sup> 4*γ* –1 2 *<sup>j</sup>*∅ð Þ<sup>2</sup> ½ �*<sup>Ω</sup>*<sup>2</sup>

<sup>2</sup>*<sup>Γ</sup>* �*j*∅ð Þ<sup>1</sup> h i

*exp* �*Γ*ð Þ*z t* þ

� �

Ω þ

*Γ* � �

�*<sup>e</sup>* �ð Þ <sup>Ω</sup>�*w*<sup>0</sup> <sup>2</sup>

þ … þ

∅ *dw*<sup>2</sup> � � � � *w*¼Ω

> 1 2

*j w*ð Þ � <sup>Ω</sup> <sup>2</sup>

*θ*ð Þ Ω, *z :exp jwt* ð Þ*dw* (56)

*:e* ð Þ *<sup>Ω</sup>*�*w*<sup>0</sup> <sup>2</sup>*<sup>Γ</sup>* –*j*∅ð Þ<sup>1</sup> � �*<sup>Ω</sup>*

*z Vg*ð Þ *Ω*

*t* þ

*z Vg*ð Þ Ω � �

*:*

(58)

*dw* (57)

*:e jwt* !

� �2 !

*Γ*ð Þ*z Γ w*<sup>0</sup>

is characterized by a Gaussian

*:*∅ð Þ<sup>2</sup>

1 *n*! ð Þ *<sup>w</sup>* � <sup>Ω</sup> *nd<sup>n</sup>*

þ *θ*ð Þ *w :* (54)

∅ *dw<sup>n</sup>* � � � � *w*¼Ω

(53)

(55)

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 attempt is made to cover different technical approaches.

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 idiosyncrasies of the technique or multishot averages over different pulses.

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 representation by the locus of the wavelet maxima.
