**2. Amplification of ultrashort laser pulses**

The energy of a mode-locked laser pulse with a pulse duration of 1 ps or below typically ranges from 10−12 to 10−10 J. The ultrashort laser pulse cannot be directly amplified in amplifiers because of damage issues in optical elements due to the nonlinear effect and the low-energy extraction efficiency. These hurdles were detoured by employing the chirped-pulse amplification (CPA) technique devised by Strickland and Mourou [8]. The key idea of the CPA technique is to temporarily stretch a laser pulse before amplification, to amplify the energy of the stretched pulse, and finally, after energy amplification, to compress the pulse duration to the original level. The CPA technique was well demonstrated in many systems around the world [9], and now it is used for producing the relativistic laser intensity (>1018 W/cm<sup>2</sup> ).

The control of pulse duration is usually performed by an optical setup which uses the GDD induced by the grating. The stretched pulse duration ranges from few hundreds of ps to nanosecond (ns). The stretched pulse is amplified in a series of amplifier chain including regenerative and/or multipass amplifiers. The output energy can be estimated from the Frantz-Nodvik equation. In this section, the basic principles for controlling the pulse duration and for amplifying the energy are explained.

#### **2.1. Stretching of an ultrashort laser pulse before amplification**

The control of pulse duration using the dispersion was first proposed by Treacy [10]. In the proposal, two gratings with a normal separation distance of *b* are placed in the parallel geometry to induce a negative GDD. The total amount of GDD can be controlled by the separation distance. According to the Treacy's proposal, when a laser pulse passes through an optical setup shown in **Figure 6**, the group delay dispersion (GDD) experienced by a laser pulse is given by

**Figure 6.** Parallel grating pulse stretching scheme. The parallel grating pulse stretcher introduces a negative GDD to the laser pulse.

Generation of High-Intensity Laser Pulses and their Applications http://dx.doi.org/10.5772/64526 13

$$\left. \frac{d^{\flat}\phi}{d\phi^{\flat}} \right|\_{\circ \circ} = -\frac{\lambda\_{\flat}}{2\pi c^{\flat}} \left( \frac{\lambda\_{\flat}}{d} \right)^{\flat} \frac{b}{\cos^{\flat}\theta'(\lambda\_{\flat})}.\tag{17}$$

Here, *d* is the groove spacing of grating and *θ* ' is the diffraction angle. The first-order diffraction is only considered in this case. The diffraction angle is calculated by the grating equation as follows:

**2. Amplification of ultrashort laser pulses**

12 High Energy and Short Pulse Lasers

amplifying the energy are explained.

pulse is given by

the laser pulse.

now it is used for producing the relativistic laser intensity (>1018 W/cm<sup>2</sup>

**2.1. Stretching of an ultrashort laser pulse before amplification**

The energy of a mode-locked laser pulse with a pulse duration of 1 ps or below typically ranges from 10−12 to 10−10 J. The ultrashort laser pulse cannot be directly amplified in amplifiers because of damage issues in optical elements due to the nonlinear effect and the low-energy extraction efficiency. These hurdles were detoured by employing the chirped-pulse amplification (CPA) technique devised by Strickland and Mourou [8]. The key idea of the CPA technique is to temporarily stretch a laser pulse before amplification, to amplify the energy of the stretched pulse, and finally, after energy amplification, to compress the pulse duration to the original level. The CPA technique was well demonstrated in many systems around the world [9], and

The control of pulse duration is usually performed by an optical setup which uses the GDD induced by the grating. The stretched pulse duration ranges from few hundreds of ps to nanosecond (ns). The stretched pulse is amplified in a series of amplifier chain including regenerative and/or multipass amplifiers. The output energy can be estimated from the Frantz-Nodvik equation. In this section, the basic principles for controlling the pulse duration and for

The control of pulse duration using the dispersion was first proposed by Treacy [10]. In the proposal, two gratings with a normal separation distance of *b* are placed in the parallel geometry to induce a negative GDD. The total amount of GDD can be controlled by the separation distance. According to the Treacy's proposal, when a laser pulse passes through an optical setup shown in **Figure 6**, the group delay dispersion (GDD) experienced by a laser

**Figure 6.** Parallel grating pulse stretching scheme. The parallel grating pulse stretcher introduces a negative GDD to

).

$$d\left(\sin\theta\,'\neg\sin\theta\right) = \mathbb{X}\,.\tag{18}$$

As shown in Eq. (17), the parallel grating geometry always introduces the negative GDD, and thus the blue-like wavelength component travels faster than the red-like one. The positive GDD can be either introduced by installing a telescope in the parallel grating geometry, which was proposed by Martinez [11]. A telescope is an optical device that induces an angular dispersion. The GDD induced by an angular dispersion is given by

$$\left. \frac{d^i \phi}{d\phi^i} \right|\_{\ast 0} \approx -\frac{L\_r \cdot \phi\_0}{c} \left( \left. \frac{d\alpha}{d\phi} \right|\_{\ast 0} \right)^\ast \tag{19}$$

with an approximation of cos *α*≫ sin *α*. In the equation, *α* is the deviation angle at the reference wavelength and *Lp* is the propagation distance after the surface of an angularly dispersive element. When a laser pulse propagates an optical setup shown in **Figure 7**, the angular dispersion is magnified by a factor of *M*, which is the magnification of a telescope. Then, the GDD induced by the angular dispersion after the propagation of *z* ' is

**Figure 7.** GDD control by the grating pair with a telescope inside. The grating pair with the telescope can induce the positive and negative GDD depending on the total length between gratings. The negative GDD is obtained by *L*/ 2 < (*f* + *f* '). The positive GDD is obtained by *L*/2 < (*f* + *f* ').

$$\left.\frac{d^2\phi}{d\phi^\circ}\right|\_{\ast 0} = -\frac{\alpha\_\circ}{c} \left(\left.\frac{d\alpha}{d\phi}\right|\_{\ast 0}\right)^\circ z^\ast M^\circ. \tag{20}$$

As shown in **Figure 7**, the propagation distance *z* ' is given by *L* − 2(*f* + *f* ') and the magnification by *f*/*f* '. The positive GDD can be obtained when *L* − 2(*f* + *f* ') < 0 or *L*/2 < (*f* + *f* '). This condition can be met by moving a second grating before the focal point *F* '. In general, the first lens can be placed at a position of *z* + *f*. Then, an additional GDD, *<sup>d</sup>* <sup>2</sup> *ϕ <sup>d</sup><sup>ω</sup>* <sup>2</sup> <sup>|</sup>*ω*<sup>0</sup> <sup>=</sup> <sup>−</sup> *<sup>ω</sup>*<sup>0</sup> *<sup>c</sup>* ( *<sup>d</sup><sup>θ</sup>* ' *<sup>d</sup><sup>ω</sup>* |*ω*<sup>0</sup> )2 *z*, by an angular dispersion after the propagation of *z* should be added to Eq. (20) to obtain

$$\left. \frac{d^\circ \phi}{d\phi^\circ} \right|\_{\phi^\circ} = -\frac{\alpha \flat\_\circ}{c} \left( \left. \frac{d\theta^\circ}{d\phi} \right|\_{\ast \ast} \right)^\circ \left( z^\circ M^\circ + z \right). \tag{21}$$

In many cases, a reflecting mirror can be put after the first lens to reduce the cost and space. The positive GDD induced by two grating geometry having a telescope can be compensated for with the parallel grating pair. This is important because the pulse duration stretched by the positive or negative GDD can be recompressed by the negative or positive GDD. This is the principle for stretching and compressing an utrashort laser pulse in the CPA technique. In a common CPA technique, a pulse stretcher introduces a positive GDD to the laser pulse and a pulse compressor introduces a negative GDD. The reason for this is that the material dispersion used in amplifier systems also produces a positive GDD. If a laser pulse has negative GDD by a stretcher, the pulse duration of a pulse is shortened as the pulse propagates in a medium having a positive GDD. This might induce damage on optical elements that the pulse propagates. The other combination that uses a pulse stretcher introducing negative GDD and a pulse compressor introducing positive GDD is also possible. This combination is known as the down-chirped pulse amplification (DCPA) technique and also demonstrated with a grating stretcher and bulk material compressor. Although the DCPA technique works for the energy amplification of an ultrashort laser pulse, the pulse duration of the pulse is somewhat broadened because higher-order dispersions, such as TOD and FOD, induced by media in the laser system remain uncompensated. As mentioned earlier, third-order dispersion (TOD), and fourth-order dispersion (FOD) should be corrected or optimized to obtain a nearly transformlimited pulse duration through the pulse compressor.

The misalignment in the parallelism of a grating induces an additional angular dispersion in the spatial domain. This is known as the spatial chirping. The spatial chirping can easily be examined by monitoring the intensity distribution of a focal spot. If there is the spatial chirping in the laser beam profile, a focal spot is elongated along the chirping direction. Sometimes, the elongation by the spatial chirping is confused with astigmatism in the beam. However, the spatial chirping can be discriminated by the through-the-focus image because the elongation by the spatial chirping is not rotated by 90 degrees while the elongation by astigmatism can be rotated.

#### **2.2. Rate equation**

When a laser pulse passes through an amplification medium, the pulse obtains energy gain from the medium. The energy gain comes from a stored energy in the medium which is provided by an external power source. Absorption by the transition between electronic energy levels is used to store an external energy. Electrons at a lower energy level are excited to a higher energy level through the pumping process. When an electromagnetic wave (photon) with a specific wavelength defined by the atomic energy transition is radiated to an excited atom, the atom emits the same electromagnetic wave (photon) as the incoming one. This means that an incoming electromagnetic wave is amplified in intensity. This dynamics can be described by the rate equation. In order to describe the situation mathematically, let us consider a four-energy-level system shown in **Figure 8**.

As shown in **Figure 7**, the propagation distance *z* ' is given by *L* − 2(*f* + *f* ') and the magnification by *f*/*f* '. The positive GDD can be obtained when *L* − 2(*f* + *f* ') < 0 or *L*/2 < (*f* + *f* '). This condition can be met by moving a second grating before the focal point *F* '. In general, the first lens can

In many cases, a reflecting mirror can be put after the first lens to reduce the cost and space. The positive GDD induced by two grating geometry having a telescope can be compensated for with the parallel grating pair. This is important because the pulse duration stretched by the positive or negative GDD can be recompressed by the negative or positive GDD. This is the principle for stretching and compressing an utrashort laser pulse in the CPA technique. In a common CPA technique, a pulse stretcher introduces a positive GDD to the laser pulse and a pulse compressor introduces a negative GDD. The reason for this is that the material dispersion used in amplifier systems also produces a positive GDD. If a laser pulse has negative GDD by a stretcher, the pulse duration of a pulse is shortened as the pulse propagates in a medium having a positive GDD. This might induce damage on optical elements that the pulse propagates. The other combination that uses a pulse stretcher introducing negative GDD and a pulse compressor introducing positive GDD is also possible. This combination is known as the down-chirped pulse amplification (DCPA) technique and also demonstrated with a grating stretcher and bulk material compressor. Although the DCPA technique works for the energy amplification of an ultrashort laser pulse, the pulse duration of the pulse is somewhat broadened because higher-order dispersions, such as TOD and FOD, induced by media in the laser system remain uncompensated. As mentioned earlier, third-order dispersion (TOD), and fourth-order dispersion (FOD) should be corrected or optimized to obtain a nearly transform-

The misalignment in the parallelism of a grating induces an additional angular dispersion in the spatial domain. This is known as the spatial chirping. The spatial chirping can easily be examined by monitoring the intensity distribution of a focal spot. If there is the spatial chirping in the laser beam profile, a focal spot is elongated along the chirping direction. Sometimes, the elongation by the spatial chirping is confused with astigmatism in the beam. However, the spatial chirping can be discriminated by the through-the-focus image because the elongation by the spatial chirping is not rotated by 90 degrees while the elongation by astigmatism can

When a laser pulse passes through an amplification medium, the pulse obtains energy gain from the medium. The energy gain comes from a stored energy in the medium which is provided by an external power source. Absorption by the transition between electronic energy

*ϕ <sup>d</sup><sup>ω</sup>* <sup>2</sup> <sup>|</sup>*ω*<sup>0</sup> <sup>=</sup> <sup>−</sup> *<sup>ω</sup>*<sup>0</sup>

*<sup>c</sup>* ( *<sup>d</sup><sup>θ</sup>* ' *<sup>d</sup><sup>ω</sup>* |*ω*<sup>0</sup> )2

*z*, by an angular

(21)

be placed at a position of *z* + *f*. Then, an additional GDD, *<sup>d</sup>* <sup>2</sup>

14 High Energy and Short Pulse Lasers

limited pulse duration through the pulse compressor.

be rotated.

**2.2. Rate equation**

dispersion after the propagation of *z* should be added to Eq. (20) to obtain

**Figure 8.** Diagram for energy levels, level transition rate, and the number of electrons at the energy level. In the fourlevel system, the storage of external energy is accomplished by the absorption due to the electronic transition from level 0 to level 3, and the lasing or energy gain is obtained by the electronic transition from level 2 to level 1.

In a four-level system shown in **Figure 8**, electrons at the lowest energy level 0 are excited to level 3 by the pumping process. The changing rate for the excited electron population increases by the electron population at level 0 and the pumping rate *Wp*. In a short time, electrons at level 3 lose their energy and decay into level 2 with a transition probability *W*32. Electrons at level 3 also decay into level 1 and 0 with probabilities *W*31 and *W*30. The changing rate for electron population at level 2 increases with the number of electrons at level 3 and the transition probability *W*32, and it decreases with the number of electrons at level 2 and transition probabilities *W*21 and *W*<sup>20</sup> to level 1 and level 0, respectively. The main lasing action or energy gain happens with the transition from level 2 to level 1. Under this circumstance, the rate equations for electrons at each level can be expressed as

$$\frac{dn\_{\circ}}{dt} = W\_{\circ}n\_{\circ} - W\_{\circ}n\_{\circ} - W\_{\circ}n\_{\circ} - W\_{\circ}n\_{\circ} \tag{22-1}$$

$$\frac{dn\_{\perp}}{dt} = W\_{\gg}n\_{\succ} - W\_{\ll}n\_{\succ} - W\_{\gg}n\_{\perp}.\tag{22-2}$$

$$\frac{dn\_{\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\text}}}}}}}}{dt} = W\_{\text{\tiny\text{\text{\tiny\text{\text{\tiny\text{\text{\tiny\text{\text}}}}}}} + W\_{\text{\text{\tiny\text{\text{\textgreater}}}}} n\_{\text{\text{\textgreater}}} - W\_{\text{\text{\textgreater}}} n\_{\text{\text{\textgreater}}}\tag{22-3}$$

$$\frac{dn\_s}{dt} = -W\_{\text{v}}n\_s + W\_{\text{v}s}n\_{\text{v}} + W\_{\text{v}s}n\_{\text{v}} + W\_{\text{v}s}n\_{\text{v}}.\tag{22-4}$$

Although rate equations for level 1 and level 0 are not explained here, those can be easily derived from **Figure 8**. In the four-level system, it is assumed that electron populations at levels 1 and 3 are very small because of the rapid transition to other levels, i.e., *n*2, *n*0 ≫ *n*3, *n*1. The total number, *n*, of electrons is determined by the sum of electron numbers at levels 0 and 2, i.e., *n* = *n*2 + *n*0. In the steady-state condition, the change of electron populations at levels 3 and 2 are very small as well; so, we assume *d n*<sup>3</sup> *dt* <sup>=</sup> *<sup>d</sup> <sup>n</sup>*<sup>2</sup> *dt* ≈0. From Eqs. (22-1) and (22-2), we obtain

$$\frac{m\_{\circ}}{m\_{\circ}} = \frac{W\_{\circ}}{\left(W\_{\circ\circ} + W\_{\circ\circ}\right)} \frac{W\_{\circ\circ}}{\left(W\_{\circ\circ} + W\_{\circ\circ} + W\_{\circ\circ}\right)}.\tag{23}$$

At level 2, the approximation of *W*<sup>21</sup> ≫ *W*20 is valid because the lasing action or gain is dominant. And, electron transition from level 3 to level 2 is most dominant to the other transition and thus *W*32 ≫ *W*31, *W*30. Under these conditions, Eq. (23) reduces to

$$\frac{m\_{\circ}}{m\_{\circ}} = \frac{W\_{\nu}}{W\_{\circ}}\tag{24}$$

According to Eq. (24), a laser pulse can have energy gain when *n*2 > *n*0. The population inversion happens when the difference, *Δn* = *n*<sup>2</sup> − *n*0, in electron numbers at levels 2 and 0 is positive. Under the heavily pumping condition, most electrons exist in level 2, and the number of electrons (*n*2) at level 2 approximately equals *n*0. The population inversion is given by

(25)

By using the relation of *W*21/*Wp* = *I*(*z*)/*I*sat, Eq. (25) becomes *Δn* ≈ *n*/(1 + *I*/*I*sat). When a low-intensity laser pulse propagates in the gain medium, the intensity growing rate is linear with the product of propagation distance and population inversion as shown below:

$$\frac{dI\left(z\right)}{dz} = I\left(z\right)\sigma\_{,\parallel}\Delta n\,,\tag{26}$$

Here, *σ*21 is the emission cross section. By inserting the relation of *Δn* ≈ *n*/(1 + *I*/*Isat*) into Eq. (26), the growing rate for the intensity becomes

$$\frac{dl\left(z\right)}{dz} = I\left(z\right) \frac{\sigma\_{,n}n}{1 + I\left(z\right)/I\_{ss}} \text{ or } \frac{dl\left(z\right)}{dz} = \text{g}\left(z\right)I\left(z\right). \tag{27}$$

In Eq. (27), the gain *g*(*z*) is defined by *g*0/(1 + *I*(*z*)/*I*sat) and *g*0 is defined by *σ*21*n*. When the intensity of a laser pulse is small enough, the intensity exponentially grows with *I*0 exp(∫*g*(*z*)*dz*). The gain, exp(∫*g*(*z*)*dz*), at a small input intensity is known as the small signal gain. As the intensity becomes stronger, the growing rate for the intensity starts to be lowered and the intensity linearly grows with *gL* in the saturation regime, where *L* is the medium length.

#### **2.3. Energy amplification**

(22-3)

(22-4)

(23)

(24)

(25)

(26)

Although rate equations for level 1 and level 0 are not explained here, those can be easily derived from **Figure 8**. In the four-level system, it is assumed that electron populations at levels 1 and 3 are very small because of the rapid transition to other levels, i.e., *n*2, *n*0 ≫ *n*3, *n*1. The total number, *n*, of electrons is determined by the sum of electron numbers at levels 0 and 2, i.e., *n* = *n*2 + *n*0. In the steady-state condition, the change of electron populations at levels 3 and

At level 2, the approximation of *W*<sup>21</sup> ≫ *W*20 is valid because the lasing action or gain is dominant. And, electron transition from level 3 to level 2 is most dominant to the other

According to Eq. (24), a laser pulse can have energy gain when *n*2 > *n*0. The population inversion happens when the difference, *Δn* = *n*<sup>2</sup> − *n*0, in electron numbers at levels 2 and 0 is positive. Under the heavily pumping condition, most electrons exist in level 2, and the number of

By using the relation of *W*21/*Wp* = *I*(*z*)/*I*sat, Eq. (25) becomes *Δn* ≈ *n*/(1 + *I*/*I*sat). When a low-intensity laser pulse propagates in the gain medium, the intensity growing rate is linear with the product

Here, *σ*21 is the emission cross section. By inserting the relation of *Δn* ≈ *n*/(1 + *I*/*Isat*) into Eq. (26),

electrons (*n*2) at level 2 approximately equals *n*0. The population inversion is given by

of propagation distance and population inversion as shown below:

the growing rate for the intensity becomes

*dt* ≈0. From Eqs. (22-1) and (22-2), we obtain

*d n*<sup>3</sup> *dt* <sup>=</sup> *<sup>d</sup> <sup>n</sup>*<sup>2</sup>

transition and thus *W*32 ≫ *W*31, *W*30. Under these conditions, Eq. (23) reduces to

2 are very small as well; so, we assume

16 High Energy and Short Pulse Lasers

The small signal gain describes how much intensity or energy can be achieved with a given small input intensity. The small intensity means an intensity level that does not affect the population inversion. In this subsection, we will describe the energy amplification in an amplifier system. A single-pass energy gain can be measured by putting a detector before and after the amplification medium during the energy measurement experiment. The small signal and single-pass gain, *G*0, at the first pass is given by exp(∫*g*(*z*)*dz*) or simply by

$$G\_\circ = \exp\left(\mathcal{g}\_\circ L\right). \tag{28}$$

Here, *g*0 is the measured gain coefficient and *L* is the medium length. Using the Frantz-Nodvik equation [12], the output energy of a laser pulse at the *i*th round trip in the amplifier can be expressed by

$$F\_i = F\_{\mu\nu} \ln\left[1 + G\_i \left\{ \exp\left(\frac{F\_{i-1}}{F\_{\mu\nu}}\right) - 1 \right\} \right]. \tag{29}$$

Here, *F* means the fluence of a laser pulse defined by *∫* −∞ ∞ *I*(*z*, *t*)*dt* and the subscripts (*i* and *i* − 1) mean the *i*th and (*i* − 1)th round trips. *F*sat is the saturation fluence. In a multipass amplifier system, an amplified laser pulse is reinjected into the amplifier medium. Thus, the gain decreases as the input intensity increases. The reduced gain at the *i*th round trip can be calculated from the gain and the fluence at the (*i* − 1)th round trip as follows:

$$G\_i = \left\{ 1 - \left( 1 - \frac{1}{G\_{\shortmid}} \right) \exp\left( -\frac{F\_{\shortmid}}{F\_{\shortmid}} \right) \right\}^{-1}.\tag{30}$$

**Figure 9** shows the amplified output energy as a function of the round trip in a multipass amplifier. In the calculation, the Ti:sapphire crystal is assumed as an amplifier medium. The saturation fluence of the Ti:sapphire crystal is 1.2 J/cm<sup>2</sup> and the small signal gain of 3.5 is assumed. The energy exponentially increases in the first few round trips, but the energy 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 saturation fluence.

**Figure 9.** Fluence of the laser pulse with respect to the number of round trip. The input fluence was 1 mJ and the small signal gain was assumed to be 3.5.

**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 narrowing effect broadens the pulse duration of the compressed pulse.

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.

The amplified spontaneous emission (ASE) occurred in a large size gain crystal reduces an overall gain and deteriorates the spatial profile of a laser pulse as shown in **Figure 11**. A spontaneous emission traveling in the transverse direction of the gain medium has energy gain before the laser pulse arrives. When the gain and the size of gain medium are small, the ASE is negligible during the amplification process. However, as the size of gain medium is large enough with a considerable gain, the ASE becomes significant. In order to reduce the ASE, the gain medium is enclosed by the light-absorption cooling liquid having a refractive index similar to the gain medium. With the cooling liquid, the spontaneous emission transmits the boundary between the gain medium and cooling liquid, and scattered in the mount. Thus, the ASE reflected from the boundary can be suppressed. Sometimes, the spontaneous emission has enough energy gain even in a single transverse pass. In this case, a delayed pumping scheme can be useful to reduce the ASE.

**Figure 11.** Laser beam profile with and without the amplified spontaneous emission (ASE). The ASE reduces energy gain and deteriorates beam profile.

Since the demonstration of laser in 1960, the laser technology has continuously advanced to build petawatt (PW) laser systems. In 1999, the first CPA PW laser has been demonstrated using a Ti:sapphire/Nd:glass hybrid system [13]. Almost a decade later, 30 fs 1 PW laser operating at 0.1 Hz repetition rate was developed [14] and more recently an amplifier for 5 PW laser system has been successfully demonstrated [15]. Now, fs and 10 PW laser systems are under construction through the European Extreme Light Infrastructure (ELI) program.
