**3. Focusing ultrashort laser pulses**

linearly increases as the energy becomes comparable to the saturation energy of the amplifier medium. Finally, the output energy is saturated at a certain energy level which is close to the

**Figure 9.** Fluence of the laser pulse with respect to the number of round trip. The input fluence was 1 mJ and the small

**Figure 10.** Diagram explaining the gain narrowing effect. The origin of the gain narrowing effect is an un-equal gain at a different wavelengths. The wavelength component at a higher gain grows faster than that at a lower gain. The gain

A series of amplifier system including a regenerative amplifier and multiple-stage amplifiers are used for energy amplification. The final output energy ranges from a couple of J to ~100 J, depending on the peak power level. The pulse energy should be amplified by a factor of ~1012 while keeping the pulse characteristics the same. This is not easy because of the gain narrowing effect induced by the different gains at different wavelengths. The gain narrowing phenomenon happens because a wavelength component located at a higher gain becomes stronger than a wavelength component at a lower gain as shown in **Figure 10**. The gain narrowing broadens the pulse duration of a compressed pulse. Several techniques, such as input intensity modulation, wavelength mismatch between the input and gain spectrum, gain

saturation, and so on, have been developed to minimize the gain narrowing effect.

narrowing effect broadens the pulse duration of the compressed pulse.

saturation fluence.

18 High Energy and Short Pulse Lasers

signal gain was assumed to be 3.5.

An amplified and compressed laser pulse is focused on solid or gas target for laser-matter interaction studies. Concave mirrors are generally used and the intensity reaches at a relativ‐ istic level, >1018 W/cm2 . The size of a focal spot is proportional to the focal length of a mirror and a shorter focal length is preferable to reach a higher intensity. Thus, one particular research interest is to tightly focus a laser pulse to reach an unprecedented intensity level. The paraxial approximation, which is commonly used in calculating focal spots under high *f*-number conditions, becomes invalid under tight focusing (low *f*-number) conditions. Intensities of a focal spot that have other polarization components different from an incident polarization are assumed to be negligible in the paraxial approximation. However, under tight focusing conditions, intensities at different polarizations increase and modify the overall intensity distribution of a focused laser pulse.

The intensity distributions of all polarization components of a focal spot formed under a tight focusing condition can be calculated by vector diffraction integrals developed by Stratton and Chu [16]. Recently, the intensity distributions of a focused fs high-power laser pulse under a tight focusing condition were intensively examined [17]. In this section, the intensity distri‐ butions of a tightly focused laser spot are described. The accurate assessment of the peak power and information on the intensity distribution are beneficial in simulating and predicting the motion of charged particles under a super-strong laser pulse that is provided by a tight focusing scheme.

#### **3.1. Modeling of focusing scheme with low** *f***-number parabolic mirror**

The parabolic mirror is used as a focusing mirror because of its quadratic surface profile. A linearly polarized (x-polarized) laser pulse having an electric field distribution, *E*inc(*θS*, *ϕS*), is incident on a parabolic mirror along the negative *z*-direction (**Figure 12**). By using Sttraton and Chu's vector diffraction integrals, the electric fields at all polarization components can be expressed as follows:

$$E\_s\left(x\_p, y\_p, z\_p\right) \sim \int\_{\rho\_\Theta}^{z} \int\_0^{2\pi} d\theta\_s d\phi\_s E\_{\omega\_\Theta}\left(\theta\_s, \phi\_s\right) \exp\left\{-ik\rho\left(x\_p, y\_p, z\_p, \theta\_s, \phi\_s\right)\right\} \frac{2f\sin\theta\_s}{\left(1-\cos\theta\_s\right)}\tag{31-1}$$

$$\times \left\{1 - \frac{\sin\theta\_s \cos\phi\_s}{1-\cos\theta\_s} \left(1 - \frac{1-\cos\theta\_s}{i2f}\right) \frac{2f\sin\theta\_s\cos\phi\_s - x\_r\left(1-\cos\theta\_s\right)}{2f}\right\}\tag{31-1}$$

$$\begin{split} E\_{\boldsymbol{r}}\left(\boldsymbol{x}\_{\boldsymbol{r}},\boldsymbol{y}\_{\boldsymbol{r}},\boldsymbol{z}\_{\boldsymbol{r}}\right) &\sim \int\_{\rho\alpha}^{\pi} \int\_{0}^{2\pi} d\theta\_{s} d\phi\_{s} E\_{\boldsymbol{\omega}\boldsymbol{w}}\left(\theta\_{s},\phi\_{s}\right) \exp\left\{-i\boldsymbol{\rho}\left(\boldsymbol{x}\_{\boldsymbol{r}},\boldsymbol{y}\_{\boldsymbol{r}},\boldsymbol{z}\_{\boldsymbol{r}},\theta\_{s},\phi\_{s}\right)\right\} \frac{2f\sin\theta\_{s}}{\left(1-\cos\theta\_{s}\right)^{3}} \\ &\times \left\{\sin\theta\_{s}\cos\phi\_{s}\left(1-\frac{1-\cos\theta\_{s}}{i2kf}\right)\frac{2f\sin\theta\_{s}\sin\phi\_{s}-\boldsymbol{y}\_{\boldsymbol{r}}\left(1-\cos\theta\_{s}\right)}{2f}\right\} \end{split} \tag{31-2}$$

$$\begin{split} \operatorname{E}\_{r}\left(\boldsymbol{x}\_{\rho},\boldsymbol{y}\_{\rho},\boldsymbol{z}\_{\rho}\right) &\sim \int\_{\rho\alpha}^{\pi} \int\_{0}^{2\pi} d\theta\_{s} d\phi\_{s} E\_{i\omega} \left(\theta\_{s},\phi\_{s}\right) \exp\left\{-i\rho\left(\boldsymbol{x}\_{\rho},\boldsymbol{y}\_{\rho},\boldsymbol{z}\_{\rho},\theta\_{s},\phi\_{s}\right)\right\} \frac{2f\sin\theta\_{s}}{\left(1-\cos\theta\_{s}\right)^{\frac{\gamma}{\gamma}}} \\ &\times \sin\theta\_{s}\cos\phi\_{s} \left\{1 - \left(1 - \frac{1-\cos\theta\_{s}}{i2kf}\right) \frac{2f\cos\theta\_{s} - \boldsymbol{z}\_{\rho}\left(1-\cos\theta\_{s}\right)}{2f}\right\} \end{split} \tag{31-3}$$

and

$$\varphi\left(\mathbf{x}\_{\rho},\mathbf{y}\_{\rho},\mathbf{z}\_{\rho},\theta\_{\mathbf{s}},\phi\_{\mathbf{s}}\right) = k\left(z\_{\rho}\cos\theta\_{\mathbf{s}} + \mathbf{x}\_{\rho}\sin\theta\_{\mathbf{s}}\cos\phi\_{\mathbf{s}} + \mathbf{y}\_{\rho}\sin\theta\_{\mathbf{s}}\sin\phi\_{\mathbf{s}}\right). \tag{31-4}$$

Here, *f* is the focal length of a mirror. *xP*, *yP*, and *zP* represent positions at vicinities of the focal point. *θS* is the polar angle measured from the positive *z*-axis and *ϕ<sup>S</sup>* is the rotational angle measured from the positive *x*-axis. The minimum angle, *θ*min, determines the *f*-number of the mirror. The distance between the source (s) and observation (p) points is expressed as 2*f*/(1 − cos *θS*) for the intensity of a laser spot and as 2*f*/(1 − cos *θS*) − *ρP*{cos *θS* cos *θ<sup>P</sup>* + sin *θS* sin *θ<sup>P</sup>* cos(*ϕS* − *ϕP*)} for the phase of the spot.

conditions, becomes invalid under tight focusing (low *f*-number) conditions. Intensities of a focal spot that have other polarization components different from an incident polarization are assumed to be negligible in the paraxial approximation. However, under tight focusing conditions, intensities at different polarizations increase and modify the overall intensity

The intensity distributions of all polarization components of a focal spot formed under a tight focusing condition can be calculated by vector diffraction integrals developed by Stratton and Chu [16]. Recently, the intensity distributions of a focused fs high-power laser pulse under a tight focusing condition were intensively examined [17]. In this section, the intensity distri‐ butions of a tightly focused laser spot are described. The accurate assessment of the peak power and information on the intensity distribution are beneficial in simulating and predicting the motion of charged particles under a super-strong laser pulse that is provided by a tight

The parabolic mirror is used as a focusing mirror because of its quadratic surface profile. A linearly polarized (x-polarized) laser pulse having an electric field distribution, *E*inc(*θS*, *ϕS*), is incident on a parabolic mirror along the negative *z*-direction (**Figure 12**). By using Sttraton and Chu's vector diffraction integrals, the electric fields at all polarization components can be

(31-1)

(31-2)

(31-3)

(31-4)

**3.1. Modeling of focusing scheme with low** *f***-number parabolic mirror**

distribution of a focused laser pulse.

20 High Energy and Short Pulse Lasers

focusing scheme.

expressed as follows:

and

**Figure 12.** On-axis focusing scheme for an aberrated laser beam with a low *f*-number parabolic mirror.

The wavefront aberration of a laser pulse is one of the factors that determines the intensity distribution of a focal spot. The wavefront aberration is the phase delay function across the laser beam and included in the incident electromagnetic field of a laser pulse as follows:

$$E\_{\rm av} \left( \theta\_{\rm s}, \phi\_{\rm s} \right) = E\_{\rm o} \left( \theta\_{\rm s}, \phi\_{\rm s} \right) \exp \left\{ ikW\_{\rm av} \left( \theta\_{\rm s}, \phi\_{\rm s} \right) \right\}. \tag{32}$$

Here, *θn* can be interpreted as a normalized radius defined by (*π* − *θS*)/(*π* − *θ*min). The wavefront aberration is expressed by the Zernike polynomials as *Winc* (*θ*, *<sup>ϕ</sup>*)=∑*<sup>u</sup>*,*<sup>v</sup> cu v Zu v* (*θn*, *ϕ<sup>S</sup>* ). In this case, *cu v* means the Zernike coefficient, and *Zu v* (*θn*, *ϕ<sup>S</sup>* ) is the Zernike polynomial for the *u*th radial and the *v*th azimuthal orders, respectively. The entire phase function on the mirror surface is modified as *k*(*zP* cos *θS* + *xP* sin *θS* cos *ϕS* + *yP* sin *θS* sin *ϕS*) + *kWinc*(*θn*, *ϕS*). For a high *f*-number case, *θS* and *θn* are almost the same, and the wavefront, *WS*(*θn*, *ϕS*), on the mirror surface is almost equivalent to *W*inc(*θS*, *ϕS*). However, as the *f*-number of a parabolic mirror decreases, the wavefront on the mirror surface becomes different from the wavefront of an incident wave. In this case, the normalized radius, *θn*, on the mirror surface is modified and given by

$$\theta\_s = \frac{\cos \theta\_s}{1 - \cos \theta\_s} \frac{1 - \cos \theta\_{\rm min}}{\cos \theta\_{\rm min}} \frac{\tan \left(\pi - \theta\_s\right)}{\tan \left(\pi - \theta\_{\rm min}\right)}.\tag{33}$$

The change in wavefront aberration due to the polarization rotation after reflection from a mirror surface should be considered for the effect of polarization. Thus, after reflection from a parabolic mirror, the normal vector to the wavefront surface is expressed by 2*n* ^ (*S* <sup>→</sup> ⋅*n* ^ )−*S* <sup>→</sup> as shown below:

$$2\hat{n}\left(\vec{S}\cdot\hat{n}\right) - \vec{S} = \left(\hat{x}\sin\theta\_s\cos\phi\_s + \hat{y}\sin\theta\_s\sin\phi\_s - \hat{z}\cos\theta\_s\right)W\_s\left(\theta\_v, \phi\_s\right) \tag{34}$$

Expressions for normal vectors on a parabolic mirror surface which are given by

$$n\_s = \frac{\sin \theta\_s \cos \phi\_s}{\left[2\left(1 - \cos \theta\_s\right)\right]^{1/2}}, \ n\_r = \left(\frac{1 - \cos \theta\_s}{2}\right)^{1/2}, \text{and} \ n\_r = \frac{\sin \theta\_s \sin \phi\_s}{\left[2\left(1 - \cos \theta\_s\right)\right]^{1/2}}\tag{35}$$

are used in the calculation of Eq. (34). Finally, the wavefront component that propagates to the *ρ*-direction contributes to the formation of an intensity distribution near the focal plane and is given by

$$\left[\Box\hat{n}\left(\vec{S}\cdot\hat{n}\right)-\vec{S}\right]\cdot\vec{k}\_{\rho}=\frac{2\pi}{\lambda}W\_{s}\left(\theta\_{v},\phi\_{s}\right)\left\{\sin^{2}\theta\_{s}\cos2\phi\_{s}+\cos^{2}\theta\_{s}\right\}\tag{36}$$

with *ρ* ^ = *x* ^ sin*θ<sup>S</sup>* cos*ϕ<sup>S</sup>* <sup>−</sup> *<sup>y</sup>* ^ sin*θ<sup>S</sup>* sin*ϕ<sup>S</sup>* <sup>−</sup> *<sup>z</sup>* ^ cos*θ<sup>S</sup>* . But, as expected in Eq. (36), the contribution by the {⋅} term is not significant when sin<sup>2</sup> *θS* cos 2*ϕS* << cos2 *θS*.

#### **3.2. Coherent superposition of monochromatic fields for femtosecond focal spot**

A femtosecond laser pulse typically has a broad spectrum of several tens of nm, thus the effect of broad spectrum of a femtosecond laser pulse on the focal spot should be considered in order to accurately describe the focal spot of a femtosecond laser pulse. The spatial and temporal profiles of a femtosecond focal spot can be calculated by the superposition of monochromatic electric fields near the focal point. The resultant electric fields for a femtosecond focal spot are expressed with spectral amplitude and phase as below (see **Figure 13**):

$$\begin{split} E\_{\boldsymbol{s},\boldsymbol{\chi},\boldsymbol{\chi}}\left(\boldsymbol{x}\_{\boldsymbol{\rho}},\boldsymbol{\chi}\_{\boldsymbol{\rho}},\boldsymbol{z}\_{\boldsymbol{\rho}}\right) &= \boldsymbol{R}\_{\boldsymbol{\lambda}}\exp\left(i\boldsymbol{\alpha}\_{\boldsymbol{\omega}\boldsymbol{\upbeta}}\right)E\_{\boldsymbol{s},\boldsymbol{\chi},\boldsymbol{\upbeta}}\left(\boldsymbol{\lambda}\_{\boldsymbol{\gamma}};\boldsymbol{x}\_{\boldsymbol{\rho}},\boldsymbol{\upgamma}\_{\boldsymbol{\rho}},\boldsymbol{z}\_{\boldsymbol{\rho}}\right) + \boldsymbol{R}\_{\boldsymbol{\lambda}\boldsymbol{\Omega}}\exp\left(i\boldsymbol{\alpha}\_{\boldsymbol{\omega}\boldsymbol{\upbeta}}\right)E\_{\boldsymbol{s},\boldsymbol{\upgamma},\boldsymbol{\upbeta}}\left(\boldsymbol{\lambda}\_{\boldsymbol{\gamma}};\boldsymbol{x}\_{\boldsymbol{\rho}},\boldsymbol{\upgamma}\_{\boldsymbol{\rho}},\boldsymbol{z}\_{\boldsymbol{\rho}}\right) + \\ &\cdots + \boldsymbol{R}\_{\boldsymbol{\lambda}\boldsymbol{\sigma}}\exp\left(i\boldsymbol{\alpha}\_{\boldsymbol{\omega}\boldsymbol{\upalpha}}\right)E\_{\boldsymbol{s},\boldsymbol{\upbeta}}\left(\boldsymbol{\lambda}\_{\boldsymbol{\omega}};\boldsymbol{x}\_{\boldsymbol{\rho}},\boldsymbol{y}\_{\boldsymbol{\uprho}},\boldsymbol{z}\_{\boldsymbol{\rho}}\right) \end{split} \tag{37}$$

Here, *Rλ* defined by *I<sup>λ</sup>* / *Iλ*,max and *αλ* are the relative amplitude and the spectral phase at a given wavelength, respectively. The subscripts (*x*, *y*, *z*) represent the polarization directions and *Ex*,*y*,*<sup>z</sup>*(*λn* : *xP*, *yP*, *zP*) induces the monochromatic electric field. Contrary to the monochromatic case, a different field oscillation period at a different wavelength induces a phase mismatch among waves at different wavelengths and reduces the intensity quickly as the observation position moves away from the origin of the focal plane. The intensity distribution along the propagation direction can be interpreted as the temporal profile of a femtosecond focal spot. Thus, the resultant electric fields, *Ex*,*y*,*<sup>z</sup>*(*xP*, *yP*, *zP*), provide the spatial and temporal (spatiotem‐ poral) intensity distributions of a laser focal spot. The resultant electric fields are numerically calculated. In the calculation, the spectrum is sliced into *n* components. The monochromatic electric field distributions at all polarization components are obtained from Eqs. (31-1)–(31-4). The relative amplitude ratio and the spectral phase are obtained by the measurement of a laser pulse. After calculating the resultant electric fields, the final intensity distributions at all polarization components become

(33)

(34)

(35)

(36)

(37)

^ (*S* <sup>→</sup> ⋅*n* ^ )−*S* <sup>→</sup> as

The change in wavefront aberration due to the polarization rotation after reflection from a mirror surface should be considered for the effect of polarization. Thus, after reflection from

are used in the calculation of Eq. (34). Finally, the wavefront component that propagates to the *ρ*-direction contributes to the formation of an intensity distribution near the focal plane and is

*θS* cos 2*ϕS* << cos2

A femtosecond laser pulse typically has a broad spectrum of several tens of nm, thus the effect of broad spectrum of a femtosecond laser pulse on the focal spot should be considered in order to accurately describe the focal spot of a femtosecond laser pulse. The spatial and temporal profiles of a femtosecond focal spot can be calculated by the superposition of monochromatic electric fields near the focal point. The resultant electric fields for a femtosecond focal spot are

Here, *Rλ* defined by *I<sup>λ</sup>* / *Iλ*,max and *αλ* are the relative amplitude and the spectral phase at a given wavelength, respectively. The subscripts (*x*, *y*, *z*) represent the polarization directions and *Ex*,*y*,*<sup>z</sup>*(*λn* : *xP*, *yP*, *zP*) induces the monochromatic electric field. Contrary to the monochromatic case, a different field oscillation period at a different wavelength induces a phase mismatch

**3.2. Coherent superposition of monochromatic fields for femtosecond focal spot**

expressed with spectral amplitude and phase as below (see **Figure 13**):

^ cos*θ<sup>S</sup>* . But, as expected in Eq. (36), the contribution by

*θS*.

a parabolic mirror, the normal vector to the wavefront surface is expressed by 2*n*

Expressions for normal vectors on a parabolic mirror surface which are given by

shown below:

22 High Energy and Short Pulse Lasers

given by

with *ρ* ^ = *x*

^ sin*θ<sup>S</sup>* cos*ϕ<sup>S</sup>* <sup>−</sup> *<sup>y</sup>*

the {⋅} term is not significant when sin<sup>2</sup>

^ sin*θ<sup>S</sup>* sin*ϕ<sup>S</sup>* <sup>−</sup> *<sup>z</sup>*

$$I\_{\boldsymbol{x}\_{\boldsymbol{x},\boldsymbol{y},\boldsymbol{z}}}\left(\boldsymbol{x}\_{\boldsymbol{\rho}},\boldsymbol{y}\_{\boldsymbol{\rho}},\boldsymbol{z}\_{\boldsymbol{\rho}}\right) \propto \left|E\_{\boldsymbol{x}\_{\boldsymbol{\rho}},\boldsymbol{z}}\left(\boldsymbol{x}\_{\boldsymbol{\rho}},\boldsymbol{y}\_{\boldsymbol{\rho}},\boldsymbol{z}\_{\boldsymbol{\rho}}\right)\right|^{\boldsymbol{\epsilon}}.\tag{38}$$

This approach provides information on intensity distribution at all polarization components both in temporal and spatial domains and it is also valid under high *f*-number focusing conditions as well. The sum of all polarization components given by *Ix*(*x*, *y*, *z*) + *Iy*(*x*, *y*, *z*) + *Iz*(*x*, *y*, *z*) is the intensity distribution measured by an image-sensing device.

**Figure 13.** Spectrum and spectral phase for calculating the femtosecond focal spot.

**Figure 14.** Three-dimensional intensity distribution of the continuous wave and spatially uniform laser beam under loose focusing condition. The x-polarized beam is assumed and the laser beam propagates along the −*z* direction. Un‐ der the far-field approximation, the x-polarization component is only considered to calculate the intensity distribution.

#### **3.3. Intensity distribution in the focal plane and its vicinity**

Under the loose focusing condition (*f*/# >> 1), the intensity distribution having the same polarization as an incoming laser pulse is only considered, and other polarization components (*Iy*(*x*, *y*, *z*) and *Iz*(*x*, *y*, *z*)) are ignored. In this case, the far-field approximation is applied and the Fourier transform of an incoming electric field, which is derived from the scalar diffraction integral, is widely used to obtain the intensity distribution of a focal spot. **Figure 14** shows the typical intensity distributions in the *x*-*y* plane and the *x*-*z* plane.

**Figure 15.** The change in intensity distributions as the *f*-number decreases. The intensity distributions for a continuous wave and uniform laser beam are calculated in the focal plane. (a) The ideal uniform laser beam profile without wave‐ front aberration is assumed as an input. (b) The uniform beam profile with wavefront aberration is assumed as an in‐ put.

Intensities having other polarizations different from an incoming laser pulse increase under the tight focusing condition. Typical aspects under tight focusing conditions are the increase in the intensity of a longitudinal polarization component and the elongation of a focal spot along the polarization direction. **Figure 15(a)** shows the change in the intensity distribution when the *f*-number of a parabolic mirror decreases from 5 to 0.25. The peak intensity of a longitudinal component, *Iz*, increases up to 41% of that of *Ix* under *f*/0.25 condition. But, compared to the x-polarization component, the peak intensity of the y-polarized component, *Iy*, is still negligible. Because of the increase of intensity in the longitudinal component and the deformation of x-polarized intensity, the resultant intensity is elongated in the polarization direction as shown in **Figure 15(a)**.

**3.3. Intensity distribution in the focal plane and its vicinity**

24 High Energy and Short Pulse Lasers

typical intensity distributions in the *x*-*y* plane and the *x*-*z* plane.

Under the loose focusing condition (*f*/# >> 1), the intensity distribution having the same polarization as an incoming laser pulse is only considered, and other polarization components (*Iy*(*x*, *y*, *z*) and *Iz*(*x*, *y*, *z*)) are ignored. In this case, the far-field approximation is applied and the Fourier transform of an incoming electric field, which is derived from the scalar diffraction integral, is widely used to obtain the intensity distribution of a focal spot. **Figure 14** shows the

**Figure 15.** The change in intensity distributions as the *f*-number decreases. The intensity distributions for a continuous wave and uniform laser beam are calculated in the focal plane. (a) The ideal uniform laser beam profile without wave‐ front aberration is assumed as an input. (b) The uniform beam profile with wavefront aberration is assumed as an in‐

put.

**Figure 15(b)** shows the change of a focal spot for an aberrated laser pulse as the *f*-number decreases. A small amount of wavefront aberration (*c*<sup>2</sup> −2 = 0.07 μm, *c*<sup>3</sup> −3 = 0.05 μm, *c*<sup>3</sup> −1 = 0.04 μm, and *c*<sup>3</sup> 1 = 0.02 μm) was introduced to the laser pulse to investigate the effect of wavefront aberration on the focal spot. The figure shows how the focal spot of an aberrated laser pulse is influenced by the focusing condition. Under a high *f*-number condition (*f*/5), the focal spot of an aberrated laser pulse is determined by the spatial characteristics of the laser pulse, such as wavefront aberration and spatial profile. The shape of the focal spot was almost same as that obtained with the Fourier transform method because the focusing condition and the amount of wavefront aberration did not violate the far-field and thin-lens approximations. Instead, under lower *f*-number conditions, focal spots are also influenced by the vectorial properties, resulting in the elongation along the polarization direction. With a given amount of wavefront aberration, the peak intensity of a longitudinal component, *Iz*, increases up to 40% of that of *Ix* under *f*/0.25 condition. Further calculation with a higher amount of wavefront aberration (*c*<sup>2</sup> 2 = *c*<sup>2</sup> −2 = *c*<sup>3</sup> −1 = 0.15 μm) shows that intensity distribution under *f*/0.5 focusing condition was still different from the intensity distribution obtained with the Fourier transform method.

**Figure 16** shows spatiotemporal intensity distributions of femtosecond focal spots for an aberrated laser pulse under two different focusing conditions (*f*/3 and *f*/0.5). The Zernike coefficient that are used again include *c*<sup>2</sup> −2 = 0.07 μm, *c*<sup>3</sup> −3 = 0.05 μm, *c*<sup>3</sup> −1 = 0.04 μm, and *c*<sup>3</sup> 1 = 0.02 μm. In the figure, the intensity distributions in the *x*-*y* plane provide information on spatial profiles of a femtosecond focal spot, and the intensity distributions in the *x*-*z* plane provide information on temporal profiles. By assuming a 12 fs, 10 PW, uniformly circular, and aberrated laser pulse as an input, peak intensities for x-polarized component increases up to ~8.8 × 1022 W/cm2 for *f*/3 and ~ 2.5 × 1024 W/cm2 for *f*/0.5, respectively. Under same conditions, peak intensities for longitudinal component rapidly increase to ~3.1 × 1020 W/cm2 and ~2.4 × 1023 W/cm2 . These intensities along the *z*-direction should be taken into account to better describe the motion of charged particles under an extremely strong EM field that is formed by tightly focusing a femtosecond high-power laser pulse.

**Figure 16.** The spatiotemporal intensity distributions of a focal spot with (a) an *f*/3 parabolic mirror and (b) an *f*/0.5 parabolic mirror. The peak intensities of *I*x reach ~8.8 × 1022 W/cm2 and ~2.5 × 1024 W/cm2 for *f*/3 and *f*/0.5 focusing con‐ ditions, respectively. The transverse intensity distribution is expressed in the *x*-*y* plane and the longitudinal intensity distribution is expressed in the *x*-*z* plane.

#### **4. Interaction of an intense laser pulse with plasma**

Under a strong electromagnetic field, the motion of an electron is governed by the Lorentz force as follows:

$$\frac{d\left(\chi m\_0 \tilde{\mathbf{v}}\right)}{dt} = -e\tilde{E} - e\left(\frac{\tilde{\mathbf{v}}}{c} \times \tilde{B}\right). \tag{39}$$

Here, *m*<sup>0</sup> is the electron rest mass, *c* is the speed of light, and *γ* is the Lorentz factor. When the electromagnetic field is not strong enough, the *<sup>v</sup>* → *<sup>c</sup>* × *B* <sup>→</sup> term on the right-hand side is much less than the first term on the right-hand side and negligible. In this case, the Lorentz force is reduced to *d*(*mv* <sup>→</sup> ) / *dt* = −*eE* <sup>→</sup> . By assuming the sine wave for the electric field and replacing the time derivative by − *iω*, then the speed of an electron is

Generation of High-Intensity Laser Pulses and their Applications http://dx.doi.org/10.5772/64526 27

$$
\nu = \frac{e}{m\phi} E\left(t\right). \tag{40}
$$

The maximum speed of an electron is given by *v*max = *eE*0/*mω*. By comparing the maximum speed of an electron and the speed of light, we define *β* = *v*/*c*. In the nonrelativisitic approach, we can consider *β* = 1 as a reference. Then, the intensity required for an electron to have the speed of light *c* is given by

$$I = \frac{\mathcal{S}\_o c}{2} \left(\frac{2\pi m c^2}{\lambda e}\right)^2. \tag{41}$$

The intensity for the speed of light is ~ 2.14 × 1018 W/cm2 for the 0.8 μm wavelength. In the nonrelativistic approach, the intensity of 1018 W/cm2 is roughly estimated for electrons to have a quiver motion in which the speed is close to the speed of light. The intensity of 1018 W/cm2 is known as the relativistic intensity for the electromagnetic field.

As shown in the previous section, the relativistic intensity is easily obtained by focusing a femtosecond high-power laser pulse. The femtosecond focal spot has a finite extent in the temporal and spatial domains. Let us expand the electric field of a high-power laser pulse in the Taylor series at a position of *x*0, then we obtain

$$E\_r\left(r\right) \simeq E\_r\left(r\right)\Big|\_{r=0} \cos\left(kz - \alpha t\right) + x \frac{\partial E\_r\left(r\right)}{\partial x}\Big|\_{r=0} \cos\left(kz - \alpha t\right) + \cdots \tag{42}$$

By inserting the first term on the right-hand side in Eq. (42) into the first term in Eq. (39) and solving the equation, the velocity and the position of electron are given by

$$\mathbf{v}\_{\times} = -\frac{eE\_{\times}\left(r\right)\Big|\_{\times \times 0}}{m\alpha}\sin\left(kz - \alpha t\right) \text{ and } \left.\mathbf{x} - \frac{eE\_{\times}\left(r\right)\Big|\_{\times \times 0}}{m\alpha^{\dagger}}\cos\left(kz - \alpha t\right)\right. \tag{43}$$

In order to see the effect of the intensity (or field) gradient of a focused intensity, let us put the expression of *x* in Eq. (43) into Eq. (42) and consider the Lorentz force again. Then, we obtain

$$m\frac{d}{dt}\nu\_{\,\,\,\epsilon} = -\frac{e^{\circ}}{2m\alpha^{\circ}}\frac{\partial E\_{\,\,\,\epsilon}^{\,\,\,}(r)}{\partial x}\bigg|\_{r=0} \cos^{\circ}\left(kx - \alpha t\right). \tag{44}$$

By taking the cycle average of the force, Eq. (44) becomes

**Figure 16.** The spatiotemporal intensity distributions of a focal spot with (a) an *f*/3 parabolic mirror and (b) an *f*/0.5

ditions, respectively. The transverse intensity distribution is expressed in the *x*-*y* plane and the longitudinal intensity

Under a strong electromagnetic field, the motion of an electron is governed by the Lorentz

Here, *m*<sup>0</sup> is the electron rest mass, *c* is the speed of light, and *γ* is the Lorentz factor. When the

than the first term on the right-hand side and negligible. In this case, the Lorentz force is

→ *<sup>c</sup>* × *B*

and ~2.5 × 1024 W/cm2

<sup>→</sup> . By assuming the sine wave for the electric field and replacing the

for *f*/3 and *f*/0.5 focusing con‐

<sup>→</sup> term on the right-hand side is much less

(39)

parabolic mirror. The peak intensities of *I*x reach ~8.8 × 1022 W/cm2

electromagnetic field is not strong enough, the *<sup>v</sup>*

time derivative by − *iω*, then the speed of an electron is

<sup>→</sup> ) / *dt* = −*eE*

**4. Interaction of an intense laser pulse with plasma**

distribution is expressed in the *x*-*z* plane.

26 High Energy and Short Pulse Lasers

force as follows:

reduced to *d*(*mv*

$$m\frac{d}{dt}\nu\_{\cdot} = -\frac{e^{\mathbb{1}}}{4m\alpha^{\mathbb{1}}}\frac{\partial E\_{\cdot}^{\mathbb{1}}\left(r\right)}{\partial \mathbf{x}}\Bigg|\_{r=0}.\tag{45}$$

Eq. (45) means that an electron can be pushed by the intensity or the field gradient. The force due to the intensity gradient is known as the ponderomotive force.

When a high-power laser pulse is focused in a gas target, the target immediately turns into the plasma medium. Electrons in the plasma medium feel the ponderomotive force by the laser pulse in temporal and spatial domains, and those are pushed by a focused laser field and separated from the background ions. The separation of electrons from background ions induces a strong electric field by the space charge effect. The periodic motion of oscillation for electrons occurs around heavy ions as the laser pulse propagates. The resultant pattern of alternating positive and negative charges is known as the plasma wave or laser wake. The laser wake field supports a very strong longitudinal electric field of 1 GeV/cm. Some of returning electrons can be captured into the laser wake and accelerated by the laser wake field up to GeV level. This is a short description of the laser wake field acceleration [18] (**Figure 17(a)**). Recent experiments using the laser wake field acceleration showed the quasimonoenergetic multi-GeV electron beams by focusing petawatt laser pulses [19–21]. The acceleration of electrons to 10 GeV or even 100 GeV level is now being pursued for the development of a compact electron accelerator.

Protons are also accelerated by a high-power laser pulse. In this case, a high-power laser pulse is focused onto a solid target. When a high-power laser pulse is focused on a thin metal target, the target immediately turns into plasma, and electrons in the plasma are accelerated toward the laser beam propagation direction by the ponderomotive force. Then there exists an electric field between accelerated electrons and background ions. The electric field can be used to accelerate protons existing in the metal as impurities [22, 23] (**Figure 17(b)**). At a lower laser intensity, the energy distribution for electrons is broad and the resultant proton energy distribution is also broad. As the laser intensity increases, proton energy distribution can be reduced by a narrow electron energy distribution by the radiation pressure. This is an indirect proton acceleration using electron acceleration. Protons can be directly accelerated to the light speed by an electromagnetic field as shown in Eq. (45). However, because of the proton mass, reaching to the speed of light by directly accelerating proton with an electromagnetic field requires a higher laser intensity up to ~1024 W/cm2 , which is sometimes called the ultrarela‐ tivistic laser intensity. One of the ways for efficiently reaching at the ultrarelativistic laser intensity is to employ a tight focusing scheme. Based on the recent progress in the high-power laser, the demonstration of ultrarelativistic laser intensity will be possible soon.

Energetic charged particles driven by high-power laser pulses are directly used for medical applications including radiation therapy and imaging. For example, energetic proton beams having an energy range of 100–200 MeV can be used for the radiation tumor therapy [24]. When proton beams is irradiated to tumors in human body, protons dramatically lose their energy and produce x-rays in the tumor. The produced x-ray destroys DNA chains in a tumor cell and eventually kills the tumor cell. Electron beams with an energy range of 6–20 MeV can also be used for treating cancers locating at skin and lip, chest-wall and neck, respiratory and digestive-track lesions, or lymph nodes [25]. Research on stable and reliable production of energetic particles is of great interest for developing a compact particle accelerator for medical applications.

**Figure 17.** (a) Electron acceleration though the laser wake field. The laser pulse is focused onto the gas target. The elec‐ trons are captured in the plasma cavity and accelerated by the cavity. (b). Proton acceleration. The accelerated elec‐ trons by the laser pulse pull proton on the metal surface.

High-brightness and high-energy photons (x-ray and γ-ray) can be produced through the laser-plasma accelerator as well. By comparing to the large-scale facilities, such as synchrotron and XFEL, the laser-plasma accelerator produces high-energy photon providing an attosecond temporal resolution and subatomic spatial resolution in a small size and reasonable cost. Highenergy photon can be used for research pertaining to ultrafast electron dynamics in atoms, molecules, plasmas, and solids. In the laser-plasma accelerator, many processes producing energetic photon sources, such as high harmonic generation [26], undulator radiation [27], betatron radiation [28], and Compton scattering [29], were proposed and some of them have been experimentally demonstrated. So far, basic applications for high-intensity laser pulses were described. Other interesting research topics related to fundamental physical processes are well described elsewhere [30].

### **5. Conclusion**

Eq. (45) means that an electron can be pushed by the intensity or the field gradient. The force

When a high-power laser pulse is focused in a gas target, the target immediately turns into the plasma medium. Electrons in the plasma medium feel the ponderomotive force by the laser pulse in temporal and spatial domains, and those are pushed by a focused laser field and separated from the background ions. The separation of electrons from background ions induces a strong electric field by the space charge effect. The periodic motion of oscillation for electrons occurs around heavy ions as the laser pulse propagates. The resultant pattern of alternating positive and negative charges is known as the plasma wave or laser wake. The laser wake field supports a very strong longitudinal electric field of 1 GeV/cm. Some of returning electrons can be captured into the laser wake and accelerated by the laser wake field up to GeV level. This is a short description of the laser wake field acceleration [18] (**Figure 17(a)**). Recent experiments using the laser wake field acceleration showed the quasimonoenergetic multi-GeV electron beams by focusing petawatt laser pulses [19–21]. The acceleration of electrons to 10 GeV or even 100 GeV level is now being pursued for the development of a compact electron

Protons are also accelerated by a high-power laser pulse. In this case, a high-power laser pulse is focused onto a solid target. When a high-power laser pulse is focused on a thin metal target, the target immediately turns into plasma, and electrons in the plasma are accelerated toward the laser beam propagation direction by the ponderomotive force. Then there exists an electric field between accelerated electrons and background ions. The electric field can be used to accelerate protons existing in the metal as impurities [22, 23] (**Figure 17(b)**). At a lower laser intensity, the energy distribution for electrons is broad and the resultant proton energy distribution is also broad. As the laser intensity increases, proton energy distribution can be reduced by a narrow electron energy distribution by the radiation pressure. This is an indirect proton acceleration using electron acceleration. Protons can be directly accelerated to the light speed by an electromagnetic field as shown in Eq. (45). However, because of the proton mass, reaching to the speed of light by directly accelerating proton with an electromagnetic field

tivistic laser intensity. One of the ways for efficiently reaching at the ultrarelativistic laser intensity is to employ a tight focusing scheme. Based on the recent progress in the high-power

Energetic charged particles driven by high-power laser pulses are directly used for medical applications including radiation therapy and imaging. For example, energetic proton beams having an energy range of 100–200 MeV can be used for the radiation tumor therapy [24]. When proton beams is irradiated to tumors in human body, protons dramatically lose their energy and produce x-rays in the tumor. The produced x-ray destroys DNA chains in a tumor cell and eventually kills the tumor cell. Electron beams with an energy range of 6–20 MeV can also be used for treating cancers locating at skin and lip, chest-wall and neck, respiratory and digestive-track lesions, or lymph nodes [25]. Research on stable and reliable production of energetic particles is of great interest for developing a compact particle accelerator for medical

laser, the demonstration of ultrarelativistic laser intensity will be possible soon.

, which is sometimes called the ultrarela‐

due to the intensity gradient is known as the ponderomotive force.

requires a higher laser intensity up to ~1024 W/cm2

accelerator.

28 High Energy and Short Pulse Lasers

applications.

The high-power laser facility is being developed for performing research on the laser-matter interaction in the relativistic and ultrarelativistic intensity regimes. The high-power laser pulse immediately ionizes solid and gas targets and makes the target medium plasma. The intensity can make use of the laser pulse as a small-scale and versatile particle accelerator. This is a primary purpose for developing high-intensity laser facilities. The interaction between an intense laser pulse and energetic charged particles produces high-energy photon as well. Many interesting schemes, such as undulator radiation, betatron radiation, and inverse Compton scattering, have been studied for producing high-energy photons. The high-energy photons can be used in many disciplines including industrial application, medical imaging, nuclear engineering, national security, and so on. As the intensity obtainable with the high-power laser pulse increases over 1024 W/cm2 , some of the fundamental physical processes can be investi‐ gated by light pulses with an ultrashort time scale. Since the invention of laser, the application field of laser has been dramatically expanded as the laser intensity increases. Now, the acceleration of charged particle by intense coherent light field became possible in the relativ‐ istic laser intensity regime, and new era for studying the laser-plasma interaction in the ultrarelativistic laser intensity regime will be open soon.
