*2.1.1 Examples*

Nowadays Ti:sapphire lasers usually deliver several watts of average output power

field. The lowest order of this dependence can be written as follows:

*n r*ð Þ¼ *n*<sup>0</sup>

At high intensities, the refractive index depends nonlinearly on the propagating

1 2

�*gr*<sup>2</sup>

The refractive index changes with intensity along the optical path and it is larger in the center than at the side of the nonlinear crystal. This leads to the beam selffocusing phenomenon, which is known as the Kerr lens effect (see **Figure 2**). Consider now a seed beam with a Gaussian profile propagating through a nonlinear medium, e.g. a Ti:sapphire crystal, which is pumped by a cw radiation. For the stronger focused frequencies, the Kerr lens favors a higher amplification. Thus, the self-focusing of the seed beam can be used to suppress the cw operation, because the losses of the cw radiation are higher. Forcing all the modes to have equal phase (mode-locking) implies that all the waves of different frequencies will interfere (add) constructively at one point, resulting in a very intense, short light pulse.

where *n*<sup>2</sup> is the nonlinear index coefficient and describes the strength of the coupling between the electric field and the refractive index *n*. The intensity is:

*I r*ð Þ¼ *e*

*n*2*I r*ð Þ (1)

(2)

and produce pulses as short as 6.5 fs (**Figure 1**) [14].

n0: linear index refractive.

*Recent Advances in Numerical Simulations*

**Figure 1.**

**Figure 2.**

**4**

*The Kerr lens effect and self-focusing.*

*A Ti: Sapphire oscillator.*

Auto TPL Tripler for laser oscillator.

Sprite XT: Tunable ultrafast Ti: sapphire laser (**Figure 3**).

The modes are separated in frequency by *ν* ¼ *c=*2*L*, L being the resonator length, which also gives the repetition rate of the mode-locked lasers:

$$
\pi\_{\rm rep} = \frac{1}{T} = \frac{c}{2L} \tag{3}
$$

Where L is length of cavity and T is period.

#### **Figure 3.**

*The Kerr lens mode-locking (KLM) principle. (a) the net gain curve (gain minus losses). In this example, from all the longitudinal modes in the resonator (b), only six (c) are forced to have an equal phase.*

Moreover, the ratio of the resonator length to the pulse duration is a measure of the number of modes oscillating in phase. For example, if *L* ¼ 1 *m* and the emerging pulses have 100 *fs* time duration, there are 105 modes contributing to the pulse bandwidth. There are two ways of mode-locking a femtosecond laser: passive modelocking and active mode-locking. In a laser cavity, these modes are equally spaced (with spacing depending on the cavity length). The electric field distribution with N such modes in phase (considered to be zero, for convenience) can be written as:

$$E(t) = \sum\_{n}^{N-1} E\_n e^{i\nu\_0 + n\Delta\nu t} \propto \frac{e^{iN\Delta\nu t} - e^{i\nu\_0 t}}{e^{i\Delta\nu t} - 1} \tag{4}$$

Where *w*<sup>0</sup> is the central frequency and Δw is the mode spacing, this appears as a carried wave with frequency domain.

n: is integer from 1 to N.

The laser intensity is given by

$$\left[\mathbf{I(t)}\propto\left[\mathbf{E(t)}\right]^2 = \frac{\sin^2[(2\mathbf{n}+\mathbf{1})\Delta\mathbf{w}.\mathbf{t}/2]}{\sin^2(\Delta\mathbf{w}.\mathbf{t}/2)}\tag{5}$$

**2.3 Multipass and regenerative amplification**

*Femtosecond Laser Pulses: Generation, Measurement and Propagation*

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

Two of the most widely used techniques for amplification of femtosecond laser pulses are the multipass and the regenerative amplification. In the multipass ampli-

The regenerative amplification technique implies trapping of the pulse to be amplified in a laser cavity (see **Figure 2b**). Here the number of passes is not important. The pulse is kept in the resonator until all the energy stored in the amplification crystal is extracted. Trapping and dumping the pulse in and out of the resonator is done by using a Pockelcell (or pulse-picker) and a broad-band polarizer [15].

*Schematics of a chirped pulse amplification system, showing the duration and energy level of the signal at the*

Of all potential amplifier media, titanium- doped sapphire has been the most wide spread used. It has several desirable characteristics which make it ideal as a high-power amplifier medium such as a very high damage threshold (<sup>8</sup>*-*<sup>10</sup> *J/cm2*

By using a dispersive line (combination of gratings and/or lenses), the individual frequencies within a femtosecond pulse can be separated (stretched) from each

In normal materials, low frequency components travel faster than high frequency components; in other words, the velocities of large wavelength components are higher than that of shorter ones. These materials induce a positive group velocity dispersion on a propagating pulse. To compensate the positive GVD (Group Velocity Dispersion) and rephase the dephased components a setup which produces

Four identical gratings in a sequence as shown in **Figure 8** make up a grating compressor. A pulse impinges on the first gratings with an angle of *θ*. Then from the second grating the spectral components in the spectrum travel together in parallel

),

fication different passes are geometrically separated (see **Figure 6a**).

a high saturation fluence, and high thermal conductivity.

negative group velocity dispersion is needed [15].

**2.4 Stretcher-compressor**

**Figure 5.**

*different stages of the system.*

other in time (see **Figure 7a**).

*2.4.1 Grating compressor*

**7**

Where E(t): electrical field.

This is series of pulses with width inversely proportional to the number of modes that are locked in phase of the mode spacing. The concept of mode-locking is easier said than done.

**Figure 4**. shows how the time distribution of a laser output depends upon the phase relations between the modes. **Figure 4a** is the resultant intensity of two modes in phase **Figure 4b**, is the resultant intensity of five modes in phase and a period repetition of a wave packet from the resultant constructive interference can be seen.

#### **2.2 CPA laser system**

CPA is the abbreviation of chirped pulse amplification. Chirped pulse amplification is a technique to produce a strong and at the same time ultrashort pulse. The concept behind CPA is a scheme to increase the energy of an ultrashort pulse while avoiding very high peak power in the amplification process.

In the CPA technique, ultrashort pulses are generated typically at low energy �10�<sup>9</sup> J, with a duration around 10�<sup>12</sup> –10�<sup>14</sup> seconds and at a high repetition rate of about 10<sup>8</sup> 1/s in an oscillator (**Figure 5**).

#### **Figure 4.**

*The influence of the phase relation between oscillating modes on the output intensity of the oscillation. (a) Two modes in phase, and (b) five modes in phase.*

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

#### **Figure 5.**

Moreover, the ratio of the resonator length to the pulse duration is a measure of the number of modes oscillating in phase. For example, if *L* ¼ 1 *m* and the emerging pulses have 100 *fs* time duration, there are 105 modes contributing to the pulse bandwidth. There are two ways of mode-locking a femtosecond laser: passive modelocking and active mode-locking. In a laser cavity, these modes are equally spaced (with spacing depending on the cavity length). The electric field distribution with N such modes in phase (considered to be zero, for convenience) can be written as:

*iw*0þ*n*Δ*wt* <sup>∝</sup> *eiN*Δ*wt* � *eiw*0*<sup>t</sup>*

½ � ð Þ 2n þ 1 Δw*:*t*=*2

Where *w*<sup>0</sup> is the central frequency and Δw is the mode spacing, this appears as a

sin <sup>2</sup>

This is series of pulses with width inversely proportional to the number of modes that are locked in phase of the mode spacing. The concept of mode-locking is easier

**Figure 4**. shows how the time distribution of a laser output depends upon the phase relations between the modes. **Figure 4a** is the resultant intensity of two modes in phase **Figure 4b**, is the resultant intensity of five modes in phase and a period repetition of a wave packet from the resultant constructive interference can be seen.

CPA is the abbreviation of chirped pulse amplification. Chirped pulse amplification is a technique to produce a strong and at the same time ultrashort pulse. The concept behind CPA is a scheme to increase the energy of an ultrashort pulse while

In the CPA technique, ultrashort pulses are generated typically at low energy �10�<sup>9</sup> J, with a duration around 10�<sup>12</sup> –10�<sup>14</sup> seconds and at a high repetition rate

*The influence of the phase relation between oscillating modes on the output intensity of the oscillation. (a) Two*

*ei*Δ*wt* � <sup>1</sup> (4)

ð Þ <sup>Δ</sup>w*:*t*=*<sup>2</sup> (5)

*E t*ðÞ¼

carried wave with frequency domain. n: is integer from 1 to N. The laser intensity is given by

*Recent Advances in Numerical Simulations*

Where E(t): electrical field.

said than done.

**2.2 CPA laser system**

**Figure 4.**

**6**

*N* X�1 *n Ene*

I tð Þ∝½ � E tð Þ <sup>2</sup> <sup>¼</sup> sin <sup>2</sup>

avoiding very high peak power in the amplification process.

of about 10<sup>8</sup> 1/s in an oscillator (**Figure 5**).

*modes in phase, and (b) five modes in phase.*

*Schematics of a chirped pulse amplification system, showing the duration and energy level of the signal at the different stages of the system.*

#### **2.3 Multipass and regenerative amplification**

Two of the most widely used techniques for amplification of femtosecond laser pulses are the multipass and the regenerative amplification. In the multipass amplification different passes are geometrically separated (see **Figure 6a**).

The regenerative amplification technique implies trapping of the pulse to be amplified in a laser cavity (see **Figure 2b**). Here the number of passes is not important. The pulse is kept in the resonator until all the energy stored in the amplification crystal is extracted. Trapping and dumping the pulse in and out of the resonator is done by using a Pockelcell (or pulse-picker) and a broad-band polarizer [15].

Of all potential amplifier media, titanium- doped sapphire has been the most wide spread used. It has several desirable characteristics which make it ideal as a high-power amplifier medium such as a very high damage threshold (<sup>8</sup>*-*<sup>10</sup> *J/cm<sup>2</sup>* ), a high saturation fluence, and high thermal conductivity.

#### **2.4 Stretcher-compressor**

By using a dispersive line (combination of gratings and/or lenses), the individual frequencies within a femtosecond pulse can be separated (stretched) from each other in time (see **Figure 7a**).

In normal materials, low frequency components travel faster than high frequency components; in other words, the velocities of large wavelength components are higher than that of shorter ones. These materials induce a positive group velocity dispersion on a propagating pulse. To compensate the positive GVD (Group Velocity Dispersion) and rephase the dephased components a setup which produces negative group velocity dispersion is needed [15].

#### *2.4.1 Grating compressor*

Four identical gratings in a sequence as shown in **Figure 8** make up a grating compressor. A pulse impinges on the first gratings with an angle of *θ*. Then from the second grating the spectral components in the spectrum travel together in parallel

**Figure 6.**

*(a) The amplifier configuration uses two spherical mirrors in a multi-pass confocal configuration to make the signal pass eight times through the amplifying medium, (b) schematic principle of a regenerative amplifier.*

*2.4.2 Prism compressor*

*in order to remove the spatial dispersion shown.*

**Figure 8.**

frequencies.

will result

above

**Figure 9.** *Pulse compressor.*

**9**

*β*: is angle.

*GDD* <sup>¼</sup> <sup>4</sup>*Lλ*<sup>3</sup>

<sup>2</sup>*π*<sup>2</sup>*c*<sup>2</sup> *sin<sup>β</sup> <sup>d</sup>*<sup>2</sup>

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

*Femtosecond Laser Pulses: Generation, Measurement and Propagation*

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

A prism compressor is built of four sequentially arranged identical prisms used in a geometry similar to **Figure 9**; often at their minimum deviation (to decrease geometrical (spatial) distortion of prisms on the beam) and in their Brewster angle (to minimize power loss). Because of the symmetry in the arrangement, it is possible to place a mirror after the second prism (as we did in the grating compressor setup) perpendicular to the beam propagation direction. The first prism spreads the pulse spectral components out in space. In the second prism the red frequencies of

*Grating pairs used in the control of dispersion. R and b indicate the relative paths of arbitrary long- and shortwavelength rays. γ is the (Brewster) angle of incidence at the prism face. Light is reflected in the plane (p1–p2)*

The optical path of a ray propagating in the compressor is defined as [15]: GDD

The third order dispersion can also be evaluated in the same manner to that used

<sup>2</sup>*<sup>n</sup>* � <sup>1</sup> *n*3

� �<sup>2</sup> ( ) (9)

� 2 cos *<sup>β</sup> dn*

*dλ*

the spectrum must pass through a longer length in the glass than the blue

*dn dλ* � �<sup>2</sup>

" # � �

**Figure 7.**

*Principle of a stretcher (a) and a compressor (b). The stretcher setup extends the temporal duration of the femtosecond pulse, whereas the gratings' arrangement in the compressor will compress the time duration of the pulse. Both setups are used in femtosecond amplifiers.*

directions but with wavelength dependent position (spatially chirped). The gratings are set in such a way that their wavelength dispersions are reversed which implies that the exiting ray from the second grating is parallel to the incident ray to the first grating.

The group delay induced by the grating compressor is given

$$\mathbf{T\_g} = \frac{2\mathbf{L}}{\mathbf{c}\sqrt{\mathbf{1} - \left(\frac{\lambda}{\mathbf{d}} - \sin\eta\right)^2}} \left[\mathbf{1} + \left(\frac{\lambda}{\mathbf{d}} - \sin\eta\right)\sin\eta\right] \tag{6}$$

where, is the light wavelength, *L* is the distance between the gratings, *d* is the grating's constant and *γ* is the incidence angle of the beam to the first grating. Dispersion of the group delay is obtained as:

$$\text{GDD} = \frac{\text{d}^2 \mathcal{Q}}{\text{dw}^2} = -\frac{\lambda^3 \text{L}}{\pi \text{c}^2 \text{d}^2} \left[ 1 - \left( \frac{\lambda}{\text{d}} - \sin \eta \right)^2 \right]^{-3/2} \tag{7}$$

GDD is Group Delay dispersion.

The third order dispersion produced in the grating compressor will be:

$$\text{TOD} = \frac{1}{L} \frac{\text{d}^3 \mathcal{Q}}{\text{dw}^3} = -\frac{\text{d}^2 \mathcal{Q}}{\text{dw}^2} \frac{6 \pi \lambda}{\text{c}} \left[ \frac{1 + \frac{\lambda}{\text{d}} \sin \chi - \sin^2 \chi}{1 - \left(\frac{\lambda}{\text{d}} - \sin \chi\right)^2} \right] \tag{8}$$

TOD is Third Order Dispersion.

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

#### **Figure 8.**

directions but with wavelength dependent position (spatially chirped). The gratings are set in such a way that their wavelength dispersions are reversed which implies that the exiting ray from the second grating is parallel to the incident ray to the first

*Principle of a stretcher (a) and a compressor (b). The stretcher setup extends the temporal duration of the femtosecond pulse, whereas the gratings' arrangement in the compressor will compress the time duration of the*

*(a) The amplifier configuration uses two spherical mirrors in a multi-pass confocal configuration to make the signal pass eight times through the amplifying medium, (b) schematic principle of a regenerative amplifier.*

where, is the light wavelength, *L* is the distance between the gratings, *d* is the grating's constant and *γ* is the incidence angle of the beam to the first grating.

> L <sup>π</sup>c2d<sup>2</sup> <sup>1</sup> � <sup>λ</sup>

The third order dispersion produced in the grating compressor will be:

∅ dw2 6πλ c

1 þ <sup>λ</sup>

1 � <sup>λ</sup>

λ <sup>d</sup> � sin<sup>γ</sup> � �

� �

<sup>d</sup> � sin<sup>γ</sup> � �<sup>2</sup> " #�3*=*<sup>2</sup>

<sup>d</sup> sin<sup>γ</sup> � sin <sup>2</sup>

" #

<sup>d</sup> � sin<sup>γ</sup> � �<sup>2</sup>

γ

sinγ

(6)

(7)

(8)

The group delay induced by the grating compressor is given

1 � <sup>λ</sup>

∅ dw2 ¼ � <sup>λ</sup><sup>3</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>d</sup> � sin<sup>γ</sup> � �<sup>2</sup> <sup>q</sup> <sup>1</sup> <sup>þ</sup>

Tg <sup>¼</sup> 2L c*:*

Dispersion of the group delay is obtained as:

*pulse. Both setups are used in femtosecond amplifiers.*

*Recent Advances in Numerical Simulations*

GDD is Group Delay dispersion.

TOD is Third Order Dispersion.

TOD <sup>¼</sup> <sup>1</sup> L d3 ∅ dw3 ¼ � <sup>d</sup><sup>2</sup>

GDD <sup>¼</sup> <sup>d</sup><sup>2</sup>

grating.

**8**

**Figure 7.**

**Figure 6.**

*Grating pairs used in the control of dispersion. R and b indicate the relative paths of arbitrary long- and shortwavelength rays. γ is the (Brewster) angle of incidence at the prism face. Light is reflected in the plane (p1–p2) in order to remove the spatial dispersion shown.*

#### *2.4.2 Prism compressor*

A prism compressor is built of four sequentially arranged identical prisms used in a geometry similar to **Figure 9**; often at their minimum deviation (to decrease geometrical (spatial) distortion of prisms on the beam) and in their Brewster angle (to minimize power loss). Because of the symmetry in the arrangement, it is possible to place a mirror after the second prism (as we did in the grating compressor setup) perpendicular to the beam propagation direction. The first prism spreads the pulse spectral components out in space. In the second prism the red frequencies of the spectrum must pass through a longer length in the glass than the blue frequencies.

The optical path of a ray propagating in the compressor is defined as [15]: GDD will result

$$\text{GDD} = \frac{4L\lambda^3}{2\pi^2 c^2} \left\{ \sin\beta \left[ \frac{d^2 n}{d\lambda^2} + \left(\frac{dn}{d\lambda}\right)^2 \left(2n - \frac{1}{n^3}\right) \right] - 2\cos\beta \left(\frac{dn}{d\lambda}\right)^2 \right\} \tag{9}$$

*β*: is angle.

The third order dispersion can also be evaluated in the same manner to that used above

**Figure 9.** *Pulse compressor.*

$$TOD = -\frac{L^4}{4\pi^2 c^3 L} \left[ 3\frac{d^2 n}{d\lambda^2} - \lambda \frac{d^3 n}{d\lambda^3} \right] \tag{10}$$

The instantaneous frequency is given as

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

W0: central pulsation. *w t*ð Þ: instantaneous pulse. *φ*ð Þ*t* is instantaneous phase.

to as a *chirp* (**Figure 10**).

into:

**Figure 10.**

**11**

*2.5.2 Time domain description*

also has the Gaussian shape [16].

*w t*ðÞ¼ *w*<sup>0</sup> þ

*Femtosecond Laser Pulses: Generation, Measurement and Propagation*

*dφ*ð Þ*t*

The spectral intensity can be derived by taking the Fourier-transform of Eq.14, it

Since in this paper the main emphasis is on the temporal dependence, all spatial

�*iw*0*t*

dependence is neglected, i.e., *E x*ð Þ¼ , *y*, *z*, *t E t*ð Þ*:* the electric field *E t*ð Þ, is a real quantity and all measured quantities are real. However, the mathematical descrip-

*E t* <sup>~</sup>ðÞ¼ *A t* <sup>~</sup>ð Þ*:<sup>e</sup>*

where *A t* <sup>~</sup>ð Þ is the complex envelope, usually chosen such that the real physical field is twice the real part of the complex field, and *w*<sup>0</sup> is the carrier frequency, usually chosen to the center of the spectrum. In this way the rapidly varying is separated from the slowly varying envelope *A t* <sup>~</sup>ð Þ. *E t* <sup>~</sup>ð Þ can be further decomposed

> �*iφ*ð Þ*<sup>t</sup>* <sup>¼</sup> *E t* <sup>~</sup>ð Þ *:e <sup>i</sup>φ*<sup>0</sup> *:e*

*φ*ð Þ*t* is often to as the temporal phase of the pulse and *φ*<sup>0</sup> the absolute phase, which relates the position of the carrier wave to the temporal envelope of the pulse (see **Figure 11**). In ∅ð Þ*t* the strong linear term due to the carrier frequency, *wt*, is omitted.which means that a nonlinear temporal phase yields a time-dependent frequency modulation- the pulse is said to carry a chirp (illustrated in **Figure 11b**).

*Self-phase modulation. Variation in the instantaneous frequency w t*ð Þ *of the transmitted pulse after the*

*propagation through a nonlinear medium with a positive nonlinear index of refraction n2.*

tion is simplified if a complex representation is used:

*E t* <sup>~</sup>ðÞ¼ *E t* <sup>~</sup>ð Þ *:e <sup>i</sup>φ*<sup>0</sup> *:e*

The shortest possible pulse, for a given spectrum, is known as the *transformlimited pulse duration*. It should be noted that Eq. (13) is not equality, i.e. the product can very well exceed *K*. If the product exceeds *K* the pulse is no longer transform-limited and all frequency components that constitute the pulse do not coincide in time, i.e. the pulse exhibits frequency modulation is very often referred

*dt* <sup>¼</sup> *<sup>w</sup>*<sup>0</sup> � <sup>2</sup>*<sup>α</sup>*

*τ*2 *g*

*t* (15)

*:* (16)

�*i*ð Þ <sup>∅</sup>ð Þ�*<sup>t</sup> <sup>w</sup>*<sup>0</sup> *:* (17)

n: is refractive index.

#### **2.5 Mathematical description of laser pulses**

In order to understand the behavior of ultrashort light pulses in the temporal and spectral domain, it is necessary to formulate the relation between the two domains mathematically. It is important to introduce the concept of the amplitude and the phase of the electric field because the generation, measurement, and shaping of ultrashort laser pulses is based on measuring and influencing these properties. The electric field in the time-domain is invariably connected with its counterpart *E w*ð Þ in the frequency-domain via a Fourier transform:

#### *2.5.1 Pulse duration and spectral width*

The statistical definitions are usually used in theoretic calculations and given as

$$\langle t^2 \rangle = \frac{\int\_{-\infty}^{+\infty} t^2 |E(t)|^2 dt}{\int\_{-\infty}^{+\infty} |E(t)|^2 dt}; \langle w^2 \rangle = \frac{\int\_{-\infty}^{+\infty} w^2 |E(w)|^2 dw}{\int\_{-\infty}^{+\infty} |E(w)|^2 dw} \tag{11}$$

In case of Gaussian pulse is easy to determine pulse duration and spectral width by applying FWHM (Full Width Half at Maximum) of intensity. One can show that these quantities are related through the following universal inequality.

$$
\Delta w \Delta \tau \geq \mathbf{1}/2 \tag{12}
$$

Therefore, one defines the pulse duration Δ*τ* as the Full Width at Half Maximum (FWHM) of the intensity profile and the spectral width Δ*w* as the FWHM of the spectral intensity. The Fourier inequality is then usually given by

$$
\Delta w \Delta \tau \geq K \tag{13}
$$

where K is a numerical constant, depending on the assumed shape of the pulse.

#### *2.5.1.1 Gaussian pulse*

The Gaussian pulse, which is most commonly used in ultrashort laser pulse characteristics. The pulse is linearly chirped and represented by

$$A(t) = A\_0 \exp\left(\frac{-(1+ia)t^2}{\tau\_\text{g}^2}\right) \text{ with } \Delta\tau\_\text{p} = \sqrt{2\ln 2\tau\_\text{g}}\tag{14}$$

A0: amplitude, *α*: chirp,

*τ<sup>g</sup>* : pulse duration at FWHM.

Δ*τp*:pulse duration at FWHM after propagation.

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

The instantaneous frequency is given as

$$
\omega w(t) = w\_0 + \frac{d\rho(t)}{dt} = w\_0 - \frac{2a}{\tau\_\text{g}^2}t \tag{15}
$$

W0: central pulsation.

*TOD* ¼ � *<sup>L</sup>*<sup>4</sup>

n: is refractive index.

*Recent Advances in Numerical Simulations*

**2.5 Mathematical description of laser pulses**

in the frequency-domain via a Fourier transform:

þ Ð∞ �∞ *t* <sup>2</sup>j j *E t*ð Þ <sup>2</sup> *dt*

þ Ð∞ �∞

j j *E t*ð Þ <sup>2</sup> *dt*

these quantities are related through the following universal inequality.

Therefore, one defines the pulse duration Δ*τ* as the Full Width at Half Maximum (FWHM) of the intensity profile and the spectral width Δ*w* as the FWHM of the spectral intensity. The Fourier inequality is then usually

where K is a numerical constant, depending on the assumed shape

characteristics. The pulse is linearly chirped and represented by

*A t*ðÞ¼ *<sup>A</sup>*<sup>0</sup> *exp* �ð Þ <sup>1</sup> <sup>þ</sup> *<sup>i</sup><sup>α</sup> <sup>t</sup>*

Δ*τp*:pulse duration at FWHM after propagation.

The Gaussian pulse, which is most commonly used in ultrashort laser pulse

*τ*2 *g*

!

2

with Δ*τ<sup>p</sup>* ¼

*2.5.1 Pulse duration and spectral width*

*t* <sup>2</sup> � � <sup>¼</sup>

given by

of the pulse.

**10**

*2.5.1.1 Gaussian pulse*

A0: amplitude, *α*: chirp, *τ<sup>g</sup>* : pulse duration at FWHM. <sup>4</sup>*π*2*c*3*<sup>L</sup>* <sup>3</sup>

In order to understand the behavior of ultrashort light pulses in the temporal and spectral domain, it is necessary to formulate the relation between the two domains mathematically. It is important to introduce the concept of the amplitude and the phase of the electric field because the generation, measurement, and shaping of ultrashort laser pulses is based on measuring and influencing these properties. The electric field in the time-domain is invariably connected with its counterpart *E w*ð Þ

The statistical definitions are usually used in theoretic calculations and given as

; *<sup>w</sup>*<sup>2</sup> � � <sup>¼</sup>

In case of Gaussian pulse is easy to determine pulse duration and spectral width by applying FWHM (Full Width Half at Maximum) of intensity. One can show that

þ Ð∞ �∞

þ Ð∞ �∞

*<sup>w</sup>*<sup>2</sup>j j *E w*ð Þ <sup>2</sup>

j j *E w*ð Þ <sup>2</sup>

Δ*w*Δ*τ* ≥1*=*2 (12)

Δ*w*Δ*τ* ≥*K* (13)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 *ln* 2*τ<sup>g</sup>*

q

*dw*

*dw*

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

" #

*d*3 *n dλ*<sup>3</sup>

(10)

(11)

(14)

*w t*ð Þ: instantaneous pulse.

*φ*ð Þ*t* is instantaneous phase.

The spectral intensity can be derived by taking the Fourier-transform of Eq.14, it also has the Gaussian shape [16].

The shortest possible pulse, for a given spectrum, is known as the *transformlimited pulse duration*. It should be noted that Eq. (13) is not equality, i.e. the product can very well exceed *K*. If the product exceeds *K* the pulse is no longer transform-limited and all frequency components that constitute the pulse do not coincide in time, i.e. the pulse exhibits frequency modulation is very often referred to as a *chirp* (**Figure 10**).

### *2.5.2 Time domain description*

Since in this paper the main emphasis is on the temporal dependence, all spatial dependence is neglected, i.e., *E x*ð Þ¼ , *y*, *z*, *t E t*ð Þ*:* the electric field *E t*ð Þ, is a real quantity and all measured quantities are real. However, the mathematical description is simplified if a complex representation is used:

$$
\tilde{E}(t) = \tilde{A}(t).e^{-iw\_0t}.\tag{16}
$$

where *A t* <sup>~</sup>ð Þ is the complex envelope, usually chosen such that the real physical field is twice the real part of the complex field, and *w*<sup>0</sup> is the carrier frequency, usually chosen to the center of the spectrum. In this way the rapidly varying is separated from the slowly varying envelope *A t* <sup>~</sup>ð Þ. *E t* <sup>~</sup>ð Þ can be further decomposed into:

$$\tilde{E}(t) = \left| \tilde{E}(t) \right| \left| \mathcal{e}^{i\rho\_0} \mathcal{e}^{-i\rho(t)} = \left| \tilde{E}(t) \right| \mathcal{e}^{i\rho\_0} \mathcal{e}^{-i(\mathcal{Q}(t) - \mu\_0)}. \tag{17}$$

*φ*ð Þ*t* is often to as the temporal phase of the pulse and *φ*<sup>0</sup> the absolute phase, which relates the position of the carrier wave to the temporal envelope of the pulse (see **Figure 11**). In ∅ð Þ*t* the strong linear term due to the carrier frequency, *wt*, is omitted.which means that a nonlinear temporal phase yields a time-dependent frequency modulation- the pulse is said to carry a chirp (illustrated in **Figure 11b**).

**Figure 10.**

*Self-phase modulation. Variation in the instantaneous frequency w t*ð Þ *of the transmitted pulse after the propagation through a nonlinear medium with a positive nonlinear index of refraction n2.*

*E w*ð Þ¼ <sup>1</sup>

*E t* <sup>~</sup>ðÞ¼ <sup>1</sup>

matic waves. A common procedure is to employ Taylor expansion

1 *n*!

*n*¼1

terms will not change the temporal profile of the pulse.

components are also characterized gate pulses are used [17].

<sup>∅</sup>ð Þ¼ *<sup>w</sup>* <sup>∅</sup><sup>0</sup> <sup>þ</sup>X<sup>∞</sup>

**2.6 Ultrashort pulse measurement techniques**

*2.6.1 Non-interferometric techniques*

*A*ð Þ*τ* is quadrature detection

**13**

intensityautocorrelation function is defined by

I(t) is intensity and E(t) is electrical field

*2.6.1.1 Intensity autocorrelation*

Just as in the time domain, *E w* <sup>~</sup>ð Þ can be written as:

*Femtosecond Laser Pulses: Generation, Measurement and Propagation*

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

the time domain,

with pulsation.

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

*E w* <sup>~</sup>ð Þ¼ *E w* <sup>~</sup>ð Þ � � � �*e*

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

ðþ<sup>∞</sup> �∞

where *φ*ð Þ *w* now denotes the spectral phase. An inverse transform leads back to

*E t* <sup>~</sup>ð Þ*:<sup>e</sup>*

*an*ð Þ *<sup>w</sup>* � *<sup>w</sup>*<sup>0</sup> *<sup>n</sup>* with *an* <sup>¼</sup> *dn*

ðþ<sup>∞</sup> �∞

From Eq. (20) it is clear that *E t* <sup>~</sup>ð Þ can be seen as a superposition of monochro-

∅ð Þ *w* is spectral phase, w is pulsation, and *an* is variation of the spectral phase

It can be seen by inserting this Taylor expansion into Eq. (21) that the first, two

The most commonly used technique of ultrashort laser pulse width measurement is concerned with the study of the temporal intensity profile *I t*ð Þ through its second-order correlation function that is obtained by the second-harmonic generation. Ultrashort - pulse characterization techniques, such as the numerous variants of frequency resolved optical gating (FROG) and spectral phase interferometry for direct electric-field reconstruction (SPIDER), fail to fully determine the relative phases of wellseparated frequency components. If well-separated frequency

Complex electric field *E t*ð Þ corresponds to intensity *I t*ðÞ¼ j j *E t*ð Þ <sup>2</sup> and an

þ ð∞

�∞

Two parallel beams with a variable delay are generated, then focused into a second-harmonic-generation crystal to obtain a signal proportional to *E t*ðÞþ *E t*ð Þ þ *τ* . Only the beam propagating on the optical axis, proportional to the cross

*A*ð Þ¼ *τ*

*E t*ð Þ*:e*

�*iwt:dt:* (18)

*<sup>i</sup>φ*ð Þ *<sup>w</sup> :* (19)

*iwt:dw:* (20)

*:* (21)

∅ *dwn* � � � � *w*¼*w*<sup>0</sup>

*I t*ð Þ*I t*ð Þ þ *τ dt* (22)

#### **Figure 11.**

*(a) The electric field of an ultra-short lasers pulse, (b) the electric field of an ultrashortlasers pulse with a strong positive chirp.*

An ultrashort pulse of light will lengthen after it has passed through glass as the index of refraction, which dictates the speed of light in the material, depends nonlinearly on the wavelength of the light. The wavelength of an ultrashort pulse of light is formed from the distribution of wavelengths either side of the center wavelength with the width of this distribution inversely proportional to the pulse duration.

#### *2.5.2.1 Phase and chirp*

Instantaneous phase function of *E t*ð Þ can be described as the sum of temporal phase andproduct of carrier frequency with time by the relation *w t*ðÞ¼ *<sup>d</sup> dt* ∅ðÞþ*t w*0*t:*

Carrier frequency *w*<sup>0</sup> has been choosen by minimizing of temporal variationof phase ∅ð Þ*t* . The first deriviation of *w t*ð Þ is defined by temporally-dependentcarrier frequency as the result of applying the derivation we receive relation expended in series. Then carrier frequency time denotes quadratic chirp.

Positive chirp is when leading edge of pulse is red-shifted in relation to central wavelength and trailing edge is blue-shifted. Negative chirp happens in opposite case. Linear chirp, instantaneous frequency varies linearly with time. The presence of chirp results in significant different delays between the spectrally different components of laser pulse causing pulse broadening effect and leading to a duration-bandwidth.

Chirps always appear when ultrashort laser pulses propagate through a medium such as air or glass, where the spectral components of the pulse are subject to a different refractive index. This effect is called *Group Velocity Dispersion* (GVD).

#### *2.5.3 Lens frequency domain description*

The frequency representation is obtained from the time domain by a complex Fourier transform,

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

$$E(w) = \frac{1}{\sqrt{2\pi}} \int\_{-\infty}^{+\infty} E(t) \, e^{-iwt} \, dt. \tag{18}$$

Just as in the time domain, *E w* <sup>~</sup>ð Þ can be written as:

$$
\tilde{E}(w) = \left| \tilde{E}(w) \right| e^{i\rho(w)}.\tag{19}
$$

where *φ*ð Þ *w* now denotes the spectral phase. An inverse transform leads back to the time domain,

$$
\tilde{E}(t) = \frac{1}{\sqrt{2\pi}} \int\_{-\infty}^{+\infty} \tilde{E}(t) \, e^{iwt} \, dw. \tag{20}
$$

From Eq. (20) it is clear that *E t* <sup>~</sup>ð Þ can be seen as a superposition of monochromatic waves. A common procedure is to employ Taylor expansion

$$\mathcal{Q}(w) = \mathcal{Q}\_0 + \sum\_{n=1}^{\infty} \frac{1}{n!} a\_n (w - w\_0)^n \text{ with } a\_n = \frac{d^n \mathcal{Q}}{dw^n} \Big|\_{w = w\_0}. \tag{21}$$

∅ð Þ *w* is spectral phase, w is pulsation, and *an* is variation of the spectral phase with pulsation.

It can be seen by inserting this Taylor expansion into Eq. (21) that the first, two terms will not change the temporal profile of the pulse.

#### **2.6 Ultrashort pulse measurement techniques**

The most commonly used technique of ultrashort laser pulse width measurement is concerned with the study of the temporal intensity profile *I t*ð Þ through its second-order correlation function that is obtained by the second-harmonic generation. Ultrashort - pulse characterization techniques, such as the numerous variants of frequency resolved optical gating (FROG) and spectral phase interferometry for direct electric-field reconstruction (SPIDER), fail to fully determine the relative phases of wellseparated frequency components. If well-separated frequency components are also characterized gate pulses are used [17].

#### *2.6.1 Non-interferometric techniques*

#### *2.6.1.1 Intensity autocorrelation*

Complex electric field *E t*ð Þ corresponds to intensity *I t*ðÞ¼ j j *E t*ð Þ <sup>2</sup> and an intensityautocorrelation function is defined by

$$A(\tau) = \int\_{-\infty}^{+\infty} I(t)I(t+\tau)dt\tag{22}$$

*A*ð Þ*τ* is quadrature detection

I(t) is intensity and E(t) is electrical field

Two parallel beams with a variable delay are generated, then focused into a second-harmonic-generation crystal to obtain a signal proportional to *E t*ðÞþ *E t*ð Þ þ *τ* . Only the beam propagating on the optical axis, proportional to the cross

An ultrashort pulse of light will lengthen after it has passed through glass as the

*(a) The electric field of an ultra-short lasers pulse, (b) the electric field of an ultrashortlasers pulse with a strong*

Instantaneous phase function of *E t*ð Þ can be described as the sum of temporal

Carrier frequency *w*<sup>0</sup> has been choosen by minimizing of temporal variationof phase ∅ð Þ*t* . The first deriviation of *w t*ð Þ is defined by temporally-dependentcarrier frequency as the result of applying the derivation we receive relation expended in

Positive chirp is when leading edge of pulse is red-shifted in relation to central wavelength and trailing edge is blue-shifted. Negative chirp happens in opposite case. Linear chirp, instantaneous frequency varies linearly with time. The presence of chirp results in significant different delays between the spectrally different components of laser pulse causing pulse broadening effect and leading to a

Chirps always appear when ultrashort laser pulses propagate through a medium

The frequency representation is obtained from the time domain by a complex

such as air or glass, where the spectral components of the pulse are subject to a different refractive index. This effect is called *Group Velocity Dispersion* (GVD).

*dt* ∅ðÞþ*t w*0*t:*

phase andproduct of carrier frequency with time by the relation *w t*ðÞ¼ *<sup>d</sup>*

series. Then carrier frequency time denotes quadratic chirp.

index of refraction, which dictates the speed of light in the material, depends nonlinearly on the wavelength of the light. The wavelength of an ultrashort pulse of light is formed from the distribution of wavelengths either side of the center wavelength with the width of this distribution inversely proportional to the pulse duration.

*2.5.2.1 Phase and chirp*

*Recent Advances in Numerical Simulations*

**Figure 11.**

*positive chirp.*

duration-bandwidth.

Fourier transform,

**12**

*2.5.3 Lens frequency domain description*

product *E t*ð Þ*E t*ð Þ � *τ* is retained. This signal is then recorded by a slow detector, which measures

$$I(\tau) = \int\_{-\infty}^{+\infty} |E(t)E(t-\tau)|^2 dt = \int\_{-\infty}^{+\infty} I(t)I(t-\tau)dt\tag{23}$$

*2.6.1.2.1 Examples*

plate; SMF, single mode fiber.

*2.6.2 Interferometric techniques*

by a slow detector (**Figure 12**).

**Figure 12.**

*wavelength.*

**15**

*2.6.2.1 Interferometric autocorrelation*

Experimental setup. PARS microscopy with 532-nm excitation and 1310-nm integration beams. BC, beam combiner; GM, galvanometer mirror; L, lens; OL, objective lens; PBS, polarized beam splitter; PD, photodiode; QWP, quarter wave

Setup for an interferometric autocorrelator is similar to the field autocorrelator above, with the following optics added: L: converging lens, SHG: secondharmonic generation crystal, F: spectral filter to block the fundamental wavelength. A nonlinear crystal can be used to generate the second harmonic at the output of a Michelson interferometer in a collinear geometry. In this case, the signal recorded

j j *E t*ðÞþ *E t*ð Þ � *<sup>τ</sup>* <sup>2</sup>

*I*ð Þ*τ* is called the interferometric autocorrelation. It contains some information about the phase of the pulse: the fringes in the autocorrelation trace wash out as the

*Setup for an interferometric autocorrelator, similar to the field autocorrelator above, with the following optics added: L: Converging lens, SHG: Second-harmonicgeneration crystal, F: Spectral filter to block the fundamental*

*dt* (28)

*I*ð Þ¼ *τ*

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spectral phase becomes more complex (**Figure 13**).

þ ð∞

�∞

*I*ð Þ*τ* is exactly the intensity autocorrelation *A*ð Þ*τ* .

This numerical factor, which depends on the shape of the pulse, is sometimes called the deconvolution factor. If this factor is known, or assumed, the time duration (intensity width) of a pulse can be measured using an intensity autocorrelation. However, the phase cannot be measured [17, 18].

#### *2.6.1.2 FROG*

The technique of Frequency-Resolved Optical Gating (FROG) has been introduced by Trebino and coworkers. In FROG technique signal *E*<sup>1</sup> has been temporally shifted about τ through time-delay element in respect with signal *E*2. Then, two signals have been in nonlinear medium non-interferometrically overlapped. As the result of SFG or DFG process (at the efficient phase matching conversion) one receive the FROG signal.

In construction with process II also collinear geometry is possible.

$$E\_{FROG}(\Omega,\tau) \propto \int\_{-\infty}^{+\infty} E\_1(t-\tau)E\_2(t) \exp\left(i\Omega t\right)dt\tag{24}$$

$$\propto \int\_{-\infty}^{+\infty} \tilde{E}\_1(w)\tilde{E}\_2(\Omega - w) \exp\left(iw\tau\right) dw \tag{25}$$

$$\propto \int\_{-\infty}^{+\infty} dt \int\_{-\infty}^{+\infty} \tilde{E}\_1(w) E\_2(t) \exp\left(-iwt\right) \exp\left(i\Omega t + iw\tau\right) dw \tag{26}$$

The spectral intensity

$$I\_{FROG}(\mathfrak{Q}, \mathfrak{r}) \propto |E\_{FROG}(\mathfrak{Q}, \mathfrak{r})|^2 \tag{27}$$

Where:

*τ* is delay between two temporal electrical field E1(t) and E2(t)

*Ω* is delay between two spectral electrical field E1(w) and E2(w)

*IFROG*ð Þ *Ω*, *τ* is called FROG-trace. Relations (25), (26) are only the mathematical representation of Eq. (24). The task of receiving unknown complex signals *E*<sup>1</sup> and *E*<sup>2</sup> from measured FROG trace is known as the FROG reconstruction problem.

The resulting trace of intensity versus frequency and delay is related to the pulse's spectrogram, a visually intuitive transform containing both time and frequency information. Using phase retrieval concepts that the FROG trace yields the full intensity *I t*ð Þ and phase ∅ð Þ*t* of an arbitrary ultrashort pulse. As has been already mentioned, several schemes and methods exist for frequency-resolved optical gating as a technique for the full characterization of ultrashort optical signals as complex electric fields [14, 19].

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

### *2.6.1.2.1 Examples*

product *E t*ð Þ*E t*ð Þ � *τ* is retained. This signal is then recorded by a slow detector,

*dt* ¼

This numerical factor, which depends on the shape of the pulse, is sometimes

The technique of Frequency-Resolved Optical Gating (FROG) has been introduced by Trebino and coworkers. In FROG technique signal *E*<sup>1</sup> has been temporally shifted about τ through time-delay element in respect with signal *E*2. Then, two signals have been in nonlinear medium non-interferometrically overlapped. As the result of SFG or DFG process (at the efficient phase matching conversion) one

called the deconvolution factor. If this factor is known, or assumed, the time duration (intensity width) of a pulse can be measured using an intensity autocorre-

In construction with process II also collinear geometry is possible.

þ ð∞

�∞

*τ* is delay between two temporal electrical field E1(t) and E2(t) *Ω* is delay between two spectral electrical field E1(w) and E2(w)

and *E*<sup>2</sup> from measured FROG trace is known as the FROG reconstruction

*IFROG*ð Þ *Ω*, *τ* is called FROG-trace. Relations (25), (26) are only the mathematical representation of Eq. (24). The task of receiving unknown complex signals *E*<sup>1</sup>

The resulting trace of intensity versus frequency and delay is related to the pulse's spectrogram, a visually intuitive transform containing both time and frequency information. Using phase retrieval concepts that the FROG trace yields the full intensity *I t*ð Þ and phase ∅ð Þ*t* of an arbitrary ultrashort pulse. As has been already mentioned, several schemes and methods exist for frequency-resolved optical gating as a technique for the full characterization of ultrashort optical

þ ð∞

�∞

*I t*ð Þ*I t*ð Þ � *τ dt* (23)

*E*1ð Þ *t* � *τ E*2ð Þ*t* exp ð Þ *iΩt dt* (24)

*<sup>E</sup>*~1ð Þ *<sup>w</sup> <sup>E</sup>*~2ð Þ *<sup>Ω</sup>* � *<sup>w</sup>* exp ð Þ *iw<sup>τ</sup> dw* (25)

*<sup>E</sup>*~1ð Þ *<sup>w</sup> <sup>E</sup>*2ð Þ*<sup>t</sup>* exp ð Þ �*iwt* exp ð Þ *<sup>i</sup>Ω<sup>t</sup>* <sup>þ</sup> *iw<sup>τ</sup> dw* (26)

*IFROG*ð Þ *<sup>Ω</sup>*, *<sup>τ</sup>* <sup>∝</sup>j j *EFROG*ð Þ *<sup>Ω</sup>*, *<sup>τ</sup>* <sup>2</sup> (27)

j j *E t*ð Þ*E t*ð Þ � *<sup>τ</sup>* <sup>2</sup>

which measures

*2.6.1.2 FROG*

receive the FROG signal.

*I*ð Þ¼ *τ*

*Recent Advances in Numerical Simulations*

þ ð∞

�∞

*I*ð Þ*τ* is exactly the intensity autocorrelation *A*ð Þ*τ* .

lation. However, the phase cannot be measured [17, 18].

*EFROG*ð Þ *Ω*, *τ* ∝

∝ þ ð∞

�∞

∝ þ ð∞

The spectral intensity

Where:

problem.

**14**

�∞

signals as complex electric fields [14, 19].

*dt* þ ð∞ �∞

Experimental setup. PARS microscopy with 532-nm excitation and 1310-nm integration beams. BC, beam combiner; GM, galvanometer mirror; L, lens; OL, objective lens; PBS, polarized beam splitter; PD, photodiode; QWP, quarter wave plate; SMF, single mode fiber.
