**Generation of High-Intensity Laser Pulses and their Applications**

Tae Moon Jeong and Jongmin Lee

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/64526

#### **Abstract**

The progress in the laser technology makes it possible to produce a laser pulse having a peak power of over PW. Focusing such high-power laser pulses enables ones to have unprecedentedly strong laser intensity. The laser intensity over 1019 W/cm2 , which is called the relativistic laser intensity, can accelerate electrons almost to the speed of light. The acceleration of charged particles using such a high-power laser pulse has been successfully demonstrated in many experiments. According to the recent calculation using the vector diffraction theory, it is possible, by employing a tight focusing geometry, to produce a femtosecond (fs) laser focal spot to have an intensity of over 1024 W/cm2 in the focal plane. Over this laser intensity, protons can be directly accelerated almost to the speed of light. Such ultrashort and ultrastrong laser intensities will bring ones many opportunities to experimentally study ultrafast physical phenomena we have never met before. This chapter describes how to generate a high-power laser pulse. And, then the focusing characteristics of a femtosecond high-power laser pulse are discussed in the scalar and the vector diffraction limits. Finally, the applications of ultrashort high-power laser are briefly introduced.

**Keywords:** ultrashort laser pulse, high-intensity laser pulse, chirped pulse amplifica‐ tion, charged particle acceleration, tight focusing

#### **1. Generation of ultrashort laser pulses**

Femtosecond (fs) high-power laser pulses having a peak power of PW or higher are being produced for the study of laser-matter interactions in the relativistic intensity regime. An ultrashort laser pulse is generated in a mode-locked laser oscillator in the front and its energy is amplified in the following amplifiers. The mode locking is a technique to produce laser

© 2016 The Author(s). Licensee InTech. 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, provided the original work is properly cited.

pulses having a pulse duration in the ultrashort time scale such as picosecond (ps) or fs [1, 2]. In the technique, a gain or a loss of an oscillator is modulated in an active or a passive manner. Saturable absorber is a typical optical element modulating a loss in an oscillator. Nonlinear effect dependent on the laser intensity is used to realize a saturable absorption instantane‐ ously responding to the intensity. Under the saturable absorption, a laser pulse experiences a lower loss at a higher intensity. As a result, a higher intensity part of a laser pulse grows much stronger and the temporal duration becomes shorter during the saturable absorption process.

As the pulse duration of a laser pulse decreases, the spectrum of the pulse becomes broader and the pulse encounters the dispersion effect in the medium. The dispersion effect frequently tends to broaden the pulse duration. Without any dispersion control device, the resultant pulse duration is determined by the balance between the pulse shortening due to the saturation absorption and the pulse broadening due to the dispersion. With a proper dispersion control device, the dispersion in a laser pulse is compensated and the pulse duration is mostly determined by the spectral bandwidth of the laser pulse. The minimum pulse duration obtainable with a spectral bandwidth is known as the transform-limited pulse duration. Up to date, sub-10 fs laser pulses from an oscillator are generated by compensating for the dispersion effect with prism pairs [3]. In this section, the basic principle of the mode-locking technique is explained for generating an ultrashort laser pulse and the formation of an ultrashort laser pulse is described.

#### **1.1. Short pulse generation by locking phase of longitudinal mode**

When a laser oscillator is formed with an optical length of *L*, the wavelength of standing waves inside the oscillator is determined by 2*L*/*m* (*m* is a positive integer), and alternatively, the frequency by *νm* = *m* × *c*/2*L* (or *ωm* = *m* × *πc*/*L*). The oscillating frequency in the oscillator is limited by a gain spectrum and it is called the longitudinal mode of the oscillator. A laser pulse can be decomposed into the summation of each electric field having different modes, and the resultant electric field of the pulse can be written as a superposition of oscillating modes:

(1)

In a free running laser, the phase relation among oscillating modes is random and this is the origin of a short-timescale random intensity fluctuation. The phase relation between modes can be constant (i.e., *νm* − *ν<sup>m</sup>* − 1 = constant) under a specific condition. The situation of having the constant phase relation between modes is mentioned as "mode-locked." In this case, the intensity of the resultant electric field is given by

$$I\left(z,t\right) = \left|E\left(z,t\right)\right|^{\mathrm{i}} = \left|E\_{\circ}\left(z,t\right)\right|^{\mathrm{i}} \frac{\sin^{\circ}\left\{\mathrm{mo}\_{\circ}\left(t-z\right/c\right)/2\right\}}{\sin^{\circ}\left\{\mathrm{o}\_{\circ}\left(t-z\right/c\right)/2\right\}}\,. \tag{2}$$

As can be seen in Eq. (2), a strong intensity peak can grow in the resonator when oscillating electric fields are added under the mode-locking condition. This is the basic principle for generating a mode-locked laser pulse (see **Figure 1**). As expected in Eq. (2), the pulse duration of a mode-locked laser pulse is determined by the number of oscillating modes. For example, a Ti:sapphire laser that typically produces 10-fs laser pulses contains several hundredthousand modes in the spectral bandwidth. Up to date, a number of mode-locking techniques have been introduced to generate ps and fs laser pulses, but the underlying physics is basically the same and the question is how to realize locking longitudinal modes.

**Figure 1.** Power at free running and mode-locked operations. When the phase relation is random among longitudinal modes, the intensity has fluctuation because of the beating among modes (left). On the other hand, a single high peak laser pulse is formed under the constant phase relation between modes (right).

#### **1.2. How to lock phases of longitudinal modes**

In the early history of mode-locking technique, an active loss element operating at an rffrequency was installed in an oscillator. The element periodically inducing intensity loss initiates an intensity modulation at a repetition rate corresponding to the round-trip time. A periodic loss at a round-trip time forces to form a laser pulse inside the oscillator. This is known as the active mode-locking technique. Another technique is to introduce a passive-type intensity modulation to the oscillator. Thus, in the passive mode-locking technique, an optical element that has an intensity-dependent loss is installed in the oscillator. The intensity peak in the temporal domain has higher transmittance and energy gain, but a lower intensity part has lower transmittance and energy gain. The lower intensity part is relatively suppressed by the intensity-dependent loss when an intensity fluctuation circulates in the oscillator. As the intensity peak grows, the number of oscillating modes becomes larger and larger in the spectral domain, and the phase relation between modes is automatically locked to form a laser pulse.

#### *1.2.1. Saturable absorption*

(1)

(2)

pulses having a pulse duration in the ultrashort time scale such as picosecond (ps) or fs [1, 2]. In the technique, a gain or a loss of an oscillator is modulated in an active or a passive manner. Saturable absorber is a typical optical element modulating a loss in an oscillator. Nonlinear effect dependent on the laser intensity is used to realize a saturable absorption instantane‐ ously responding to the intensity. Under the saturable absorption, a laser pulse experiences a lower loss at a higher intensity. As a result, a higher intensity part of a laser pulse grows much stronger and the temporal duration becomes shorter during the saturable absorption process.

As the pulse duration of a laser pulse decreases, the spectrum of the pulse becomes broader and the pulse encounters the dispersion effect in the medium. The dispersion effect frequently tends to broaden the pulse duration. Without any dispersion control device, the resultant pulse duration is determined by the balance between the pulse shortening due to the saturation absorption and the pulse broadening due to the dispersion. With a proper dispersion control device, the dispersion in a laser pulse is compensated and the pulse duration is mostly determined by the spectral bandwidth of the laser pulse. The minimum pulse duration obtainable with a spectral bandwidth is known as the transform-limited pulse duration. Up to date, sub-10 fs laser pulses from an oscillator are generated by compensating for the dispersion effect with prism pairs [3]. In this section, the basic principle of the mode-locking technique is explained for generating an ultrashort laser pulse and the formation of an

When a laser oscillator is formed with an optical length of *L*, the wavelength of standing waves inside the oscillator is determined by 2*L*/*m* (*m* is a positive integer), and alternatively, the frequency by *νm* = *m* × *c*/2*L* (or *ωm* = *m* × *πc*/*L*). The oscillating frequency in the oscillator is limited by a gain spectrum and it is called the longitudinal mode of the oscillator. A laser pulse can be decomposed into the summation of each electric field having different modes, and the resultant electric field of the pulse can be written as a superposition of oscillating modes:

In a free running laser, the phase relation among oscillating modes is random and this is the origin of a short-timescale random intensity fluctuation. The phase relation between modes can be constant (i.e., *νm* − *ν<sup>m</sup>* − 1 = constant) under a specific condition. The situation of having the constant phase relation between modes is mentioned as "mode-locked." In this case, the

ultrashort laser pulse is described.

4 High Energy and Short Pulse Lasers

intensity of the resultant electric field is given by

**1.1. Short pulse generation by locking phase of longitudinal mode**

Some materials have a property that the absorption of light decreases as increasing the light intensity. This kind of material is known as the saturable absorber. In the saturable absorber, the light propagating in the medium transfers its energy to electrons in the ground level and excites them to higher energy levels. The light intensity decreases as the light propagates in the medium. The light absorption becomes very weak when the number of electrons in the ground level becomes sufficiently low, and the rest of light energy almost transmits the medium. At a time later, the excited electrons spontaneously decay into the ground level and the number of ground electrons is recovered to be ready to absorb energy from light. This phenomenon is known as the saturable absorption. The saturable absorber can be divided into slow and fast saturable absorbers, depending on the recovery time *τr*. In the slow saturable absorber, the recovery time is slower than the pulse duration *τp* and it is assumed to be shorter than the round-trip time under the mode-locking condition. Most saturable absorbers used as the form of solid state and semiconductor have the slow recovery property. In a slow saturable absorber, the intensity-dependent loss is described as follows:

$$\frac{dL}{dt} = -\frac{L - L\_\circ}{\tau\_\circ} - \frac{I}{F\_\omega}L\_\circ \tag{3}$$

Here, *L*0 is the unsaturated loss and *F*sat is the saturation fluence. Since *τ<sup>r</sup>* ≫ *τp*, the second term on the right-hand side of Eq. (3), is dominant and the loss exponentially decreases with respect to the pulse fluence *∫* −∞ *t I*(*t*)*dt*. For the slow absorber, two mode-locking regimes are possible depending on the soliton effect. Without the soliton effect, a slow saturable absorber absorbs the leading part of a pulse while the trailing part is less absorbed. The pulse formation is mostly determined by balancing between the net gain and losses. As a result, a pulse profile becomes shortened, and the pulse duration obtainable in this case is estimated by [4]

$$
\pi\_{\nu} = \frac{1.07}{\Delta \nu\_{\text{g}}} \sqrt{\frac{\text{g}}{\Delta R}} \,. \tag{4}
$$

Here, *Δνg* is the FWHM gain bandwidth, assuming a Gaussian-shaped gain spectrum, and *ΔR* is the modulation depth. As will be discussed later, the laser pulse can be broadened by the dispersion. Under a proper condition, the shortening and broadening processes can be balanced. Thus, for a slow saturable absorber with the soliton effect, a short laser pulse is generated by the self-phase modulation (SPM) in combination with an appropriate amount of negative dispersion. In this case, the pulse duration can be estimated by [4]

$$
\pi\_{\rho} \approx 1.76 \times \frac{2 \, \text{ $D$ } \, \text{ ${}^{\circ}$ }}{\text{ ${}^{\circ}$ }\_{SPM} E\_{\rho}}. \tag{5}
$$

Here, *D* is the group delay dispersion (GDD) per cavity round trip, *γ*SPM is the SPM coefficient (in rad/W) per round trip, and *Ep* is the pulse energy. **Figure 2(a)** and **(b)** shows the pulse formation with the slow saturable absorber. The laser pulse is formed when the loss decreases below the gain. The gain can be either unsaturated or saturated during the saturation absorp‐ tion process. Under the unsaturated gain, the laser pulse gains energy quickly in the beginning of the saturation absorption process. When the gain is saturated during the saturable absorp‐ tion, the decrease in the gain is slightly delayed and thus the net gain exists for the pulse formation.

excites them to higher energy levels. The light intensity decreases as the light propagates in the medium. The light absorption becomes very weak when the number of electrons in the ground level becomes sufficiently low, and the rest of light energy almost transmits the medium. At a time later, the excited electrons spontaneously decay into the ground level and the number of ground electrons is recovered to be ready to absorb energy from light. This phenomenon is known as the saturable absorption. The saturable absorber can be divided into slow and fast saturable absorbers, depending on the recovery time *τr*. In the slow saturable absorber, the recovery time is slower than the pulse duration *τp* and it is assumed to be shorter than the round-trip time under the mode-locking condition. Most saturable absorbers used as the form of solid state and semiconductor have the slow recovery property. In a slow saturable

Here, *L*0 is the unsaturated loss and *F*sat is the saturation fluence. Since *τ<sup>r</sup>* ≫ *τp*, the second term on the right-hand side of Eq. (3), is dominant and the loss exponentially decreases with respect

depending on the soliton effect. Without the soliton effect, a slow saturable absorber absorbs the leading part of a pulse while the trailing part is less absorbed. The pulse formation is mostly determined by balancing between the net gain and losses. As a result, a pulse profile becomes

Here, *Δνg* is the FWHM gain bandwidth, assuming a Gaussian-shaped gain spectrum, and *ΔR* is the modulation depth. As will be discussed later, the laser pulse can be broadened by the dispersion. Under a proper condition, the shortening and broadening processes can be balanced. Thus, for a slow saturable absorber with the soliton effect, a short laser pulse is generated by the self-phase modulation (SPM) in combination with an appropriate amount of

Here, *D* is the group delay dispersion (GDD) per cavity round trip, *γ*SPM is the SPM coefficient (in rad/W) per round trip, and *Ep* is the pulse energy. **Figure 2(a)** and **(b)** shows the pulse formation with the slow saturable absorber. The laser pulse is formed when the loss decreases below the gain. The gain can be either unsaturated or saturated during the saturation absorp‐ tion process. Under the unsaturated gain, the laser pulse gains energy quickly in the beginning

shortened, and the pulse duration obtainable in this case is estimated by [4]

negative dispersion. In this case, the pulse duration can be estimated by [4]

*I*(*t*)*dt*. For the slow absorber, two mode-locking regimes are possible

(3)

(4)

(5)

absorber, the intensity-dependent loss is described as follows:

to the pulse fluence *∫*

6 High Energy and Short Pulse Lasers

−∞ *t*

**Figure 2.** Laser pulse formation with saturable absorbers. The gain is not saturated in (a), but the gain is saturated in (b) during the saturation absorption process. (c) The absorption is quickly recovered in the fast saturable absorber. The laser pulse is formed when the loss decreases below the gain.

The absorption by the material is assumed to be instantaneously recovered in the fast saturable absorber (see **Figure 2(c)**). Thus, a higher intensity in the pulse center experiences a higher transmittance and a lower intensity in the side is suppressed by the saturable absorption. When a fast saturable absorber is installed in an oscillator, the intensity of a transmitted laser pulse increases in a gain medium at a growing rate of

$$\frac{dl}{dz} = \frac{\mathbf{g}\_\* I}{1 + I \frac{l}{l} I\_{\*\*}}.\tag{6}$$

Here, *I*sat is the saturation intensity and *g*<sup>0</sup> is the unsaturated small signal gain. Thus, the pulse profile is controlled by the intensity, and a higher gain at a higher intensity leads to the pulse shortening. The pulse duration is given by [5]

$$
\pi\_{\rho} \cong \frac{0.79}{\Delta \nu} \left(\frac{\text{g}}{L}\right)^{1/2} \left(\frac{I\_{sat}}{I}\right)^{1/2},
\tag{7}
$$

with the assumption of a hyperbolic secant pulse profile. Here, *Δν* is the gain bandwidth, *g* is the gain defined by *g*0/(1 + *I*/*I*sat), and *L* is the saturated loss. In reality, the fast saturable absorbing material operating in the femtosecond regime does not exist. Instead, there are materials having a strong nonlinear effect. These materials can possess the property of ultrafast loss modulation that is induced by the nonlinear effect. The ultrashort pulse formation by these materials can be considered as the mode locking by the fast saturable absorber. In this section, self-phase modulation as a nonlinear effect which induces ultrafast change in reflection or transmission is discussed.

**Figure 3.** (a) Self-phase modulation in time induces the time-dependent phase variation. The lower angular frequency in the rising part and the higher angular frequency in the falling part are induced. (b) Self-phase modulation in space makes the wavefront quadratically curved.

When a light pulse passes through a medium, it experiences an intensity-dependent change in refractive index. This phenomenon is known as the Kerr effect. The Kerr effect can induce an instantaneous loss modulation and make a medium to act as a fast saturable absorber (see **Figure 3**). In order to derive how the Kerr effect is related with the instantaneous loss modu‐ lation, let us consider the refractive index depending on the laser intensity which is given by

$$m = n\_0 + n\_2 I. \tag{8}$$

Here, *n*<sup>0</sup> is the normal refractive index and *n*<sup>2</sup> is the nonlinear refractive index related with the Kerr effect. After a nonlinear medium, the phase of the laser pulse is modified by

$$
\phi = nkd = n\_0 kd + n\_2 \text{Ikd} \,. \tag{9}
$$

With a Gaussian pulse profile, *<sup>I</sup>* <sup>=</sup> *<sup>I</sup>*<sup>0</sup> exp( <sup>−</sup>2*<sup>t</sup>* <sup>2</sup> / *<sup>τ</sup><sup>p</sup>* 2 ), in time, the second term on the right-hand side in Eq. (8) induces time-dependent phase variation, and the angular frequency is calculated as

loss modulation that is induced by the nonlinear effect. The ultrashort pulse formation by these materials can be considered as the mode locking by the fast saturable absorber. In this section, self-phase modulation as a nonlinear effect which induces ultrafast change in reflection or

**Figure 3.** (a) Self-phase modulation in time induces the time-dependent phase variation. The lower angular frequency in the rising part and the higher angular frequency in the falling part are induced. (b) Self-phase modulation in space

When a light pulse passes through a medium, it experiences an intensity-dependent change in refractive index. This phenomenon is known as the Kerr effect. The Kerr effect can induce an instantaneous loss modulation and make a medium to act as a fast saturable absorber (see **Figure 3**). In order to derive how the Kerr effect is related with the instantaneous loss modu‐ lation, let us consider the refractive index depending on the laser intensity which is given by

Here, *n*<sup>0</sup> is the normal refractive index and *n*<sup>2</sup> is the nonlinear refractive index related with the

0 2

Kerr effect. After a nonlinear medium, the phase of the laser pulse is modified by

f

0 2 *n n nI* = + . (8)

== + *nkd n kd n Ikd*. (9)

transmission is discussed.

8 High Energy and Short Pulse Lasers

makes the wavefront quadratically curved.

$$\rho \phi = -\frac{d\,\phi}{dt} = -\frac{d}{dt}\left(nkd\,d\right) = -n\_z kd \,\frac{dI}{dt} = \frac{4n\_z kdI\_d t}{\tau\_p^2} \exp\left(-\mathfrak{D}^z \left\{\tau\_p^z\right\}\right). \tag{10}$$

Thus, after a nonlinear medium, a laser pulse has lower frequency components in the rising edge and higher frequency components in the falling edge. When a Gaussian pulse having these induced frequency components is coherently added to the original one, the constructive interference occurs at the pulse center, but the destructive interference occurs at the edge. The constructive and destructive interferences induce an instantaneous reflectance change in time. This leads to the pulse shortening effect in time. Nonlinear coupled-cavity mode-locking technique introduced as the additive-pulse mode locking (APM) uses the instantaneous reflectance change induced by the self-phase modulation [6].

A similar phenomenon happens in the spatial domain as well. With a Gaussian beam profile, *<sup>I</sup>* <sup>=</sup> *<sup>I</sup>*<sup>0</sup> exp( <sup>−</sup>2*<sup>r</sup>* <sup>2</sup> / *<sup>w</sup>*<sup>0</sup> 2 ), in space, the phase at a radial position, *r*, is given by

$$\phi\left(r\right) = n\_{\text{z}}kdl\left(r\right) = n\_{\text{z}}kdl\_{\text{o}}\exp\left(-2r^{\text{z}}\right/\text{w}\_{\text{o}}^{\text{i}}\right) \approx n\_{\text{z}}kdl\_{\text{o}}\left(1 - \frac{2r^{\text{z}}}{\text{w}\_{\text{o}}^{\text{i}}}\right) \,. \tag{11}$$

with an approximation of exp( <sup>−</sup>2*<sup>r</sup>* <sup>2</sup> / *<sup>w</sup>*<sup>0</sup> 2 ) <sup>≈</sup>(1−2*<sup>r</sup>* <sup>2</sup> / *<sup>w</sup>*<sup>0</sup> 2 ). The phase variation induced by the nonlinear effect makes the wavefront quadratically curved in the radial direction. This means that, after the nonlinear medium with a positive nonlinear refractive index, the phase at a higher intensity becomes retarded to the phase at a lower intensity. The focal length induced by the quadratic curvature is calculated as

$$f\_{\omega} = -\frac{dr}{d\phi} = \frac{\boldsymbol{w}\_{\circ}^{\circ}}{4n\_{\circ}\boldsymbol{dl}\_{\circ}}\,. \tag{12}$$

This phenomenon is known as the self-focusing. Kerr-lens mode-locking (KLM) technique employs the self-focusing to induce an instantaneous intensity-dependent transmittance [7]. In the KLM technique, a higher intensity part can be separated by the self-focusing in combi‐ nation with an aperture. A higher intensity part in time and space domain has a higher transmittance because of the self-focusing. As a result, a higher intensity grows as a laser pulse circulates in a oscillator. The KLM technique forms an ultrashort pulse using this pulse shortening process. In the technique, a gain medium in the resonator also acts as a nonlinear medium that induces the self-focusing.

#### **1.3. Dispersion**

When a laser pulse propagates in a material with a length of *d*, the phase is given by the refractive index, *n*(*ω*), of the medium as follows:

$$
\phi(\,\,\phi) = n(\,\,\phi) \cdot k \cdot d = \frac{n(\,\,\phi) \cdot \phi \cdot d}{c} \,. \tag{13}
$$

The refractive index of a medium is a function of the angular frequency and can be expressed as the Sellmeier's formula of wavelength as follows:

$$m^{\circ}\left(\mathcal{\mathcal{A}}\right) = 1 + \frac{B\_{\circ}\mathcal{\lambda}^{\circ}}{\mathcal{\lambda}^{\circ} - C\_{\circ}} + \frac{B\_{\circ}\mathcal{\lambda}^{\circ}}{\mathcal{\lambda}^{\circ} - C\_{\circ}} + \frac{B\_{\circ}\mathcal{\lambda}^{\circ}}{\mathcal{\lambda}^{\circ} - C\_{\circ}}.\tag{14}$$

with the help of definition, *λ* = 2*πc*/*ω*. Here, *B*1, *B*2, *B*3, *C*1, *C*2, and *C*3 are known as Sellmeier's coefficients for material. Because of the refractive index of material depending on the wave‐ length, the phase of an ultrashort laser pulse after material experiences a distortion known as the dispersion. The dispersion is responsible for the broadening of a pulse duration and the distortion of the pulse profile in time. In order to see the effect of dispersion, let us express the spectral phase depending on the angular frequency as the Taylor expansion,

$$\phi\left(o\right) = \sum\_{\nu=0}^{\circ} \left(o\nu - o\_{\circ}\right)^{\*} \frac{\left.\hat{\boldsymbol{\alpha}}^{\*}\phi\left(o\boldsymbol{\alpha}\right)\right|}{\left.\hat{\boldsymbol{\alpha}}o\boldsymbol{\alpha}^{\*}\right|\_{\boldsymbol{\alpha}=\boldsymbol{\alpha}\phi} = \frac{d}{c}\sum\_{\nu=0}^{\circ}\left(o\nu - o\_{\circ}\right)^{\*}\frac{\left.\hat{\boldsymbol{\alpha}}^{\*}\left\{o\boldsymbol{\alpha}\cdot\boldsymbol{n}\left(o\boldsymbol{\alpha}\right)\right\}\right|}{\left.\hat{\boldsymbol{\alpha}}o\boldsymbol{\alpha}^{\*}\right|\_{\boldsymbol{\alpha}=\boldsymbol{\alpha}}}\Big|\_{\boldsymbol{\alpha}=\boldsymbol{\alpha}}.\tag{15}$$

Now, we define derivatives as

$$\left. \frac{\partial^r \phi(\alpha)}{\partial \alpha^r} \right|\_{\alpha = \ast} = \left. \frac{\partial^r \left( \alpha \cdot n \left( \alpha \right) \right)}{\partial \alpha^r} \right|\_{\alpha = \ast} = D\_\ast \left( \alpha \nu = \alpha\_\ast \right). \tag{16}$$

Because the material has a refractive index depending on the frequency, Eqs. (15) and (16) show interesting properties when a laser pulse with a broad spectrum propagates in the material. The first term, *D*0 = *d* ⋅ *ω* ⋅ *n*/*c*, in the phase relation represents a phase propagation in the material. The second term defined by *D*1(*ω* = *ω*0) = ∂*ϕ*(*ω*)/∂*ω*|*<sup>ω</sup>* = *ω*<sup>0</sup> is known as the group delay (GD) that can be interpreted as *d*/*vg*. Here, *vg* is the group velocity and represents the pulse propagation in the material. The third term defined by *D*2(*ω* = *ω*0) = ∂<sup>2</sup> *ϕ*(*ω*)/∂*ω*<sup>2</sup> |*<sup>ω</sup>* = *ω*<sup>0</sup> is known as the group delay dispersion (GDD) that is responsible for the temporal broadening of a pulse. The temporal broadening by the group delay dispersion is sometimes known as the chirping which originally means the frequency change in time.

Two kinds of temporal broadenings are possible depending on the sign of *D*2. When the sign of *D*<sup>2</sup> is positive, a long (red-like) wavelength component travels faster than a blue-like one in the pulse. On the other hand, a short (blue-like) wavelength component travels faster than a red-like one with a negative sign of *D*2. A pulse is said to be positively chirped when a red-like wavelength component travels faster, or to be negatively chirped when a blue-like wavelength component travels faster (see **Figures 4** and **5**).

**1.3. Dispersion**

10 High Energy and Short Pulse Lasers

refractive index, *n*(*ω*), of the medium as follows:

as the Sellmeier's formula of wavelength as follows:

Now, we define derivatives as

When a laser pulse propagates in a material with a length of *d*, the phase is given by the

The refractive index of a medium is a function of the angular frequency and can be expressed

with the help of definition, *λ* = 2*πc*/*ω*. Here, *B*1, *B*2, *B*3, *C*1, *C*2, and *C*3 are known as Sellmeier's coefficients for material. Because of the refractive index of material depending on the wave‐ length, the phase of an ultrashort laser pulse after material experiences a distortion known as the dispersion. The dispersion is responsible for the broadening of a pulse duration and the distortion of the pulse profile in time. In order to see the effect of dispersion, let us express the

Because the material has a refractive index depending on the frequency, Eqs. (15) and (16) show interesting properties when a laser pulse with a broad spectrum propagates in the material. The first term, *D*0 = *d* ⋅ *ω* ⋅ *n*/*c*, in the phase relation represents a phase propagation in the material. The second term defined by *D*1(*ω* = *ω*0) = ∂*ϕ*(*ω*)/∂*ω*|*<sup>ω</sup>* = *ω*<sup>0</sup> is known as the group delay (GD) that can be interpreted as *d*/*vg*. Here, *vg* is the group velocity and represents the

known as the group delay dispersion (GDD) that is responsible for the temporal broadening of a pulse. The temporal broadening by the group delay dispersion is sometimes known as the

Two kinds of temporal broadenings are possible depending on the sign of *D*2. When the sign of *D*<sup>2</sup> is positive, a long (red-like) wavelength component travels faster than a blue-like one in the pulse. On the other hand, a short (blue-like) wavelength component travels faster than a red-like one with a negative sign of *D*2. A pulse is said to be positively chirped when a red-like

spectral phase depending on the angular frequency as the Taylor expansion,

pulse propagation in the material. The third term defined by *D*2(*ω* = *ω*0) = ∂<sup>2</sup>

chirping which originally means the frequency change in time.

(13)

(14)

(15)

(16)


*ϕ*(*ω*)/∂*ω*<sup>2</sup>

**Figure 4.** Refractive index depending on the wavelength induces the group delay dispersion (GDD). In the positive GDD, the long-wavelength electromagnetic field travels faster than the short-wavelength electromagnetic field in the medium.

**Figure 5.** Frequency chirping in the laser pulse. In the upper drawing, a short laser pulse experiences the positive chirping, thus the long-wavelength (red) component arrives faster than the short-wavelength (blue) component in the laser pulse. In the lower drawing, a short laser pulse experiences the negative chirping, thus the short-wavelength (blue) component arrives faster than the long-wavelength (red) component. The pulse duration is broadened by the positive or negative chirping.

Higher-order derivatives in the Taylor expansion affect the pulse profile in time as higherorder dispersions. Even-order dispersions are responsible for the symmetric distortion of a laser pulse in time and odd-order dispersions are responsible for the antisymmetric distortion in the laser pulse. The dispersion control and compensation are key techniques to have a transform-limited laser pulse with a given spectrum. Third-order dispersion (TOD) and fourth-order dispersion (FOD) should be considered to be compensated for the generation of transform-limited pulse.
