This work is organized as follows. Section 2 presents a continuous-wave (cw) pumped Nd:YAG laser passively Q-switched by Cr4+:YAG SA with emission at 1.06 m. Although the laser pulse energy (Ep) was low, of 270 J at the repetition rate of ~9 KHz, and the pulse peak power was of only 16 kW, this device was the first passively Q-switched laser realized in our laboratory. Furthermore, it was used to demonstrate the first passively Q-switched Nd:YAG-Cr4+YAG laser with generation into green visible spectrum at 532 nm by intracavity frequency doubling with LiB3O5 (LBO) nonlinear crystal. In these experiments, both active Nd:YAG gain medium and Cr4+:YAG SA were of single-crystal nature. Section 3 is dedicated to repetitively-pumped, passively Q-switched Nd:YAG-Cr4+:YAG lasers with high pulse energy and few-MW level peak power. Results obtained with single-crystals, Nd:YAG and Cr4+:YAG discrete elements are given in Section 3.1. A detailed investigation of laser emission obtained with ceramics Nd:YAG and Cr4+:YAG was performed (Section 3.2): This was a step toward establishing ceramics materials as solution for a microchip laser used in laser ignition of an engine. Lastly, composite, all-ceramics Nd:YAG/Cr4+:YAG monolithic laser with two- and three-beam output were realized. Various characteristics of these devices are given and discussed in Section 3.3. The paper conclusions are presented in Section 4. The lasers described in this work will enable studies on the performances of internal combustion engines with multi-point ignition.

#### **2. Infrared at 1.06 m and green at 532 nm passively Q-switched Nd:YAG lasers**

Passive Q-switching technique is attractive particularly for scientific, medical, or industrial applications that do not require temporal accuracy better than microseconds range. This method yields lower output compared to electro-optic or acousto-optic Q-switched lasers, but has the advantages of a simple design, with good efficiency, reliability and compactness. The first cw, diode end-pumped Nd:YAG laser passively Q-switched by Cr4+:YAG SA

included optics for pumping, an 11-mm long resonator, as well as optics that collimated and focused the beam to dimension required for fuel ignition, was assembled in a device that

Various papers have also reported that multi-point ignition increases significantly the combustion pressure and shortens the combustion time compared to single-point ignition (Weinrotter et al., 2005a; Phuoc, 2000; Morsy et al., 2001). The experiments employed combustion chambers in which two laser beams were inserted through different windows, and thus distance between the ignition points was adjusted easily. However, the use of a single laser beam that was focused in three points with a diffractive lens failed to demonstrate improved combustion (Weinrotter et al., 2005a), opposite to the two-point ignition experiments. The result was attributed to the short distance between the ignition points. Therefore, study of the influence of multi-point ignition on the performances of a real car engine would require realization of passively Q-switched Nd:YAG/Cr4+:YAG lasers

In this work we report passively Q-switched Nd:YAG/Cr4+:YAG micro-lasers with multiple (two and three)-beam output, each beam inducing air-breakdown in points at adjustable distance. Opposite to the previous realized lasers that used discrete Nd:YAG and Cr4+:YAG single-crystals components (Koefler et al., 2007; Kroupa et al., 2009; Tsunekane et al., 2008; Tsunekane at al., 2010; Sakai et al., 2008), these lasers consist of composite, all-ceramics Nd:YAG/Cr4+:YAG monolithic media that were pumped by similar, independent lines. This work is organized as follows. Section 2 presents a continuous-wave (cw) pumped Nd:YAG laser passively Q-switched by Cr4+:YAG SA with emission at 1.06 m. Although the laser pulse energy (Ep) was low, of 270 J at the repetition rate of ~9 KHz, and the pulse peak power was of only 16 kW, this device was the first passively Q-switched laser realized in our laboratory. Furthermore, it was used to demonstrate the first passively Q-switched Nd:YAG-Cr4+YAG laser with generation into green visible spectrum at 532 nm by intracavity frequency doubling with LiB3O5 (LBO) nonlinear crystal. In these experiments, both active Nd:YAG gain medium and Cr4+:YAG SA were of single-crystal nature. Section 3 is dedicated to repetitively-pumped, passively Q-switched Nd:YAG-Cr4+:YAG lasers with high pulse energy and few-MW level peak power. Results obtained with single-crystals, Nd:YAG and Cr4+:YAG discrete elements are given in Section 3.1. A detailed investigation of laser emission obtained with ceramics Nd:YAG and Cr4+:YAG was performed (Section 3.2): This was a step toward establishing ceramics materials as solution for a microchip laser used in laser ignition of an engine. Lastly, composite, all-ceramics Nd:YAG/Cr4+:YAG monolithic laser with two- and three-beam output were realized. Various characteristics of these devices are given and discussed in Section 3.3. The paper conclusions are presented in Section 4. The lasers described in this work will enable studies on the performances of

matched the dimensions of an electrical spark plug (Tsunekane at al., 2010).

with multiple-beam output and with size close to that of an electrical spark plug.

internal combustion engines with multi-point ignition.

**lasers**

**2. Infrared at 1.06 m and green at 532 nm passively Q-switched Nd:YAG**

Passive Q-switching technique is attractive particularly for scientific, medical, or industrial applications that do not require temporal accuracy better than microseconds range. This method yields lower output compared to electro-optic or acousto-optic Q-switched lasers, but has the advantages of a simple design, with good efficiency, reliability and compactness. The first cw, diode end-pumped Nd:YAG laser passively Q-switched by Cr4+:YAG SA crystal delivered pulses with 11-μJ energy and 337-ps duration, at 6-kHz repetition rate (Zayhowsky and Dill III, 1994). The laser included a composite structure that was made of a thin piece of Nd3+:YAG single-crystal gain medium bonded to a short Cr4+:YAG SA single crystal. Later, laser pulses with increased energy of 100 μJ and duration of 36 ns at 15-kHz repetition rate were obtained from a passively Q-switched Nd:YAG-Cr4+:YAG laser, with medium average output power (Agnesi et al., 1997). Furthermore, laser pulses with high energy Ep= 3.4 mJ, but long duration of 99 ns, were achieved by employing side-pumping geometry of a Nd:YAG-Cr4+:YAG laser (Song et al., 2000).

A sketch of the cw-pumped Nd:YAG-Cr4+:YAG laser developed in our laboratory is shown in Fig. 1 (Pavel et al., 2001a). The gain medium was a composite Nd:YAG rod that was fabricated by diffusion bonding of a Nd:YAG single crystal (thickness of 5.0 mm, 1.1-at.% Nd doping) to an undoped, 1.0-mm thick YAG. The concept of combining doped and undoped components was used, in the beginning, to modify the configuration of the thermal field induced by pumping in solid-state laser rods, and successfully employed to improve the output performances of Nd:YAG (Hanson, 1995), Nd:YVO4 (Tsunekane et al., 1997), or Yb:YAG (Bibeau et al., 1998) lasers. This method has also found applications in the passive Q-switching technique. Thus, microchip structures that consisted of undoped YAG caps, Nd:YAG and Cr4+:YAG SA bonded together to form a monolithic resonator were demonstrated to produce linearly polarized, single-longitudinal mode output pulses in quasi-cw or pulsed pumping regimes (Zayhowski et al., 2000; Aniolek et al., 2000).

Fig. 1. A passively Q-switched Nd:YAG-Cr4+:YAG laser, pumped by cw diode laser is shown. Composite YAG/Nd:YAG was used for thermal management. The gain Nd:YAG medium as well as Cr4+:YAG SA were single crystals, discrete elements.

A fiber-bundled diode (OPC-B030-mmm-FC, OptoPower Co.; 1.55-mm diameter, 0.11 NA) was used for the pump at 807 nm (p). The YAG/Nd:YAG surfaces (S1 and S2) were antireflection (AR) coated at both the laser wavelength of 1.064 m (em) and p. A planeplane resonator with the pump-mirror M1 coated for high reflectivity (HR) at em and high transmission (HT) at p, and that was placed very close to Nd:YAG, was used. A collimating lens (L1) and a focusing lens (L2) were used to image the fiber bundle end into Nd:YAG, to a diameter of 800 m. The focusing point was 2.0 mm below surface S1 of Nd:YAG. Cr4+:YAG SA (CASIX Inc., China) with initial transmission T0 of 0.89, 0.85, and 0.80 and AR coated at em on both surfaces (F1 and F2) were used for Q-switching. Each Cr4+:YAG SA crystal was placed close to the out-coupling mirror (OCM) M2.

Figure 2 presents characteristics of Q-switched laser emission obtained from a 40-mm long resonator and an OCM with T= 0.10. A maximum average power of 3.8 W resulted for the Cr4+:YAG with T0= 0.89 (Fig. 2a) with a beam factor M2 of 1.4. The laser ran at a frequency as

All-Poly-Crystalline Ceramics Nd:YAG/Cr4+:YAG

whereas lines represent modeling.

experimental results and the calculated values was obtained.

Monolithic Micro-Lasers with Multiple-Beam Output 63

Figure 3 compares experimental (symbols) and calculated (continuous lines) pulse energy for two pump rates. In simulation, spectroscopic parameters of Nd:YAG and Cr4+:YAG were σg= 2.310-19 cm2 , σSA= 4.310-18 cm2, and σESA= 8.210-19 cm2 (Shimony, et al., 1995). The laser beam sizes in the gain crystal and in Cr4+:YAG were evaluated by the PARAXIA software package (Sciopt Enterprises, San Jose, California), in which the active medium was described as a thin lens. For the absorbed power of 9.2 W (near to the threshold) the calculated active element focal length was 33.8 cm and the ratio Ag/ASA amounted to 1.1, while for the absorbed pump power of 17.5 W (close to the maximum pump power) the focal length decreases to 13.2 cm and Ag/ASA was 1.3. Good agreement between the

Fig. 3. Laser pulse energy versus transmission T0 of Cr4+:YAG. Symbols are experimental data (open and filled signs for absorbed pump power of 9.2 W and 17.5 W, respectively),

In order to realize a passively Q-switched Nd:YAG-Cr4+:YAG laser with generation into green visible spectrum at 532 nm (), the set-up of Fig. 1 was modified such to include a nonlinear crystal. We used a V-type resonator, as shown in Fig. 4 (Pavel et al., 2001b). The YAG/Nd:YAG crystal, the Cr4+:YAG SA and a glass plate (BP) positioned at Brewster angle for polarization were placed in the resonator arm (of 80-mm length) made between mirrors M1 and M2. The nonlinear crystal was a 10-mm long LBO (type I, = 900, = 11.40; operation at 25oC) that was placed between mirrors M2 and M3 (the arm length was 90 mm). The LBO surfaces were AR coated at both em and wavelengths. The concave mirror M3 has a radius of 50 mm. This arrangement makes use of the high peak power available inside the

cavity, and enables high conversion efficiency of the fundamental wavelength em.

Figure 5 presents characteristics of the green laser pulses. For a Cr4+:YAG with T0= 0.90, the maximum average power at 532 nm was 0.95 W (Fig. 5a) at the absorbed pump power of 13.1 W; the laser beam quality was characterized by an M2 factor of 1.8. The green pulse energy was 226 J (Fig. 5b), and the laser runs with a 4.2-kHz rate of repetition and pulse duration of 86 ns. A slightly higher average power of 1.0 W was obtained with the Cr4+:YAG of T0= 0.85. However, the pulse energy reduced at 131 J and the pulse duration increased at 96 ns. Green pulse peak power reached 2.6 kW for the Cr4+:YAG with T0= 0.90 (Fig. 5b). The characteristics of Q-switched laser pulses at 532 nm were described with a model of rate equation for photon density inside the resonator (, for the inversion of population in

high as 24.8 kHz. The pulse energy and the pulse duration (tp, FWHM definition) were Ep= 152 J and tp= 25.4 ns (Fig. 2b), respectively; the pulse peak power was ~6.0 kW. When a SA with a lower transmission T0 was used, pulses with higher peak power were generated with a reduced average output power. For a Cr4+:YAG SA with T0= 0.85, pulses with tp= 18.0 ns at a repetition rate of 16.1 kHz resulted for the maximum absorbed power of 21.0 W. Energy Ep and pulse peak power was 213 J and 11.8 kW, respectively. For a Cr4+:YAG with T0= 0.80 an average power of 2.6 W at the absorbed pump power of 18.6 W resulted. The laser ran at 9.1 kHz repetition rate with pulses of Ep= 272 J energy and 16.2 kW peak power.

Fig. 2. Cw pumped, passively Q-switched Nd:YAG-Cr4+:YAG laser: (a) Average output power and the laser-beam M2 factor; (b) Laser pulse energy and laser pulse duration.

The Q-switched laser performances were evaluated with a rate equation model (Pavel et al., 2001a; Degnan, 1995; Zhang et al., 1997). The laser pulse energy is given by general relation:

$$E\_p = \frac{\hbar \nu^\prime \cdot A\_{\frac{\psi}{\xi}}}{2 \,\gamma\_{\beta} \sigma\_{\beta}} \cdot \ln \mathbb{R} \cdot \ln \left(\frac{n\_{g\ell}}{n\_{g^{\pm}}}\right) \tag{1}$$

where h is the photon energy at 1.06 m, g represents Nd:YAG stimulated emission, g is the inversion reduction factor, Ag is the effective area of the laser beam in Nd:YAG, and the OCM reflectivity is R= (1-T). The initial population inversion density, ngi is:

$$m\_{\mathcal{g}^i} = \frac{-\ln R + L - \ln T\_0^2}{2\sigma\_{\mathcal{g}}\ell\_{\mathcal{g}}} \tag{2}$$

where L represents the resonator round-trip residual loss and *g* is the Nd:YAG length. The final population inversion density, ngf and ngi are related by equation:

$$\left\{ \left( 1 - \frac{n\_{\rm gf}}{n\_{\rm gi}} \right) + \left[ 1 + \frac{\left( 1 - \delta \right) \cdot \ln T\_0^2}{\left( - \ln R + L - \ln T\_0^2 \right)} \right] \cdot \ln \left( \frac{n\_{\rm gf}}{n\_{\rm gi}} \right) + \frac{1}{a} \cdot \frac{\left( 1 - \delta \right) \cdot \ln T\_0^2}{\left( - \ln R + L - \ln T\_0^2 \right)} \cdot \left| 1 - \left( \frac{n\_{\rm gf}}{n\_{\rm gi}} \right)^a \right| = 0 \tag{3}$$

= ESA/SA, with SA and ESA the absorption cross section and exited-state absorption cross section of Cr4+:YAG, respectively. Parameter is = (SASA)/(gg)(Ag/ASA); SA is the inversion reduction factor for Cr4+:YAG and ASA is the laser beam effective area in Cr4+:YAG.

high as 24.8 kHz. The pulse energy and the pulse duration (tp, FWHM definition) were Ep= 152 J and tp= 25.4 ns (Fig. 2b), respectively; the pulse peak power was ~6.0 kW. When a SA with a lower transmission T0 was used, pulses with higher peak power were generated with a reduced average output power. For a Cr4+:YAG SA with T0= 0.85, pulses with tp= 18.0 ns at a repetition rate of 16.1 kHz resulted for the maximum absorbed power of 21.0 W. Energy Ep and pulse peak power was 213 J and 11.8 kW, respectively. For a Cr4+:YAG with T0= 0.80 an average power of 2.6 W at the absorbed pump power of 18.6 W resulted. The laser ran at

9.1 kHz repetition rate with pulses of Ep= 272 J energy and 16.2 kW peak power.

Fig. 2. Cw pumped, passively Q-switched Nd:YAG-Cr4+:YAG laser: (a) Average output power and the laser-beam M2 factor; (b) Laser pulse energy and laser pulse duration.

The Q-switched laser performances were evaluated with a rate equation model (Pavel et al., 2001a; Degnan, 1995; Zhang et al., 1997). The laser pulse energy is given by general relation:

> ln ln 2 *g gf*

where h is the photon energy at 1.06 m, g represents Nd:YAG stimulated emission, g is the inversion reduction factor, Ag is the effective area of the laser beam in Nd:YAG, and the

*g g gi hA n E R n*

0 ln ln

where L represents the resonator round-trip residual loss and *g* is the Nd:YAG length. The

(1 ) ln 1 (1 ) ln 1 1 ln 1 0 ( ln ln ) ( ln ln ) *gf gf gf gi gi gi nnn T T nnn RL T RL T*

= ESA/SA, with SA and ESA the absorption cross section and exited-state absorption cross section of Cr4+:YAG, respectively. Parameter is = (SASA)/(gg)(Ag/ASA); SA is the inversion reduction factor for Cr4+:YAG and ASA is the laser beam effective area in Cr4+:YAG.

*g g RL T*

2 2 0 0 2 2 0 0

2

(2)

(1)

(3)

*p*

OCM reflectivity is R= (1-T). The initial population inversion density, ngi is:

*n*

final population inversion density, ngf and ngi are related by equation:

2 *gi*

Figure 3 compares experimental (symbols) and calculated (continuous lines) pulse energy for two pump rates. In simulation, spectroscopic parameters of Nd:YAG and Cr4+:YAG were σg= 2.310-19 cm2 , σSA= 4.310-18 cm2, and σESA= 8.210-19 cm2 (Shimony, et al., 1995). The laser beam sizes in the gain crystal and in Cr4+:YAG were evaluated by the PARAXIA software package (Sciopt Enterprises, San Jose, California), in which the active medium was described as a thin lens. For the absorbed power of 9.2 W (near to the threshold) the calculated active element focal length was 33.8 cm and the ratio Ag/ASA amounted to 1.1, while for the absorbed pump power of 17.5 W (close to the maximum pump power) the focal length decreases to 13.2 cm and Ag/ASA was 1.3. Good agreement between the experimental results and the calculated values was obtained.

Fig. 3. Laser pulse energy versus transmission T0 of Cr4+:YAG. Symbols are experimental data (open and filled signs for absorbed pump power of 9.2 W and 17.5 W, respectively), whereas lines represent modeling.

In order to realize a passively Q-switched Nd:YAG-Cr4+:YAG laser with generation into green visible spectrum at 532 nm (), the set-up of Fig. 1 was modified such to include a nonlinear crystal. We used a V-type resonator, as shown in Fig. 4 (Pavel et al., 2001b). The YAG/Nd:YAG crystal, the Cr4+:YAG SA and a glass plate (BP) positioned at Brewster angle for polarization were placed in the resonator arm (of 80-mm length) made between mirrors M1 and M2. The nonlinear crystal was a 10-mm long LBO (type I, = 900, = 11.40; operation at 25oC) that was placed between mirrors M2 and M3 (the arm length was 90 mm). The LBO surfaces were AR coated at both em and wavelengths. The concave mirror M3 has a radius of 50 mm. This arrangement makes use of the high peak power available inside the cavity, and enables high conversion efficiency of the fundamental wavelength em.

Figure 5 presents characteristics of the green laser pulses. For a Cr4+:YAG with T0= 0.90, the maximum average power at 532 nm was 0.95 W (Fig. 5a) at the absorbed pump power of 13.1 W; the laser beam quality was characterized by an M2 factor of 1.8. The green pulse energy was 226 J (Fig. 5b), and the laser runs with a 4.2-kHz rate of repetition and pulse duration of 86 ns. A slightly higher average power of 1.0 W was obtained with the Cr4+:YAG of T0= 0.85. However, the pulse energy reduced at 131 J and the pulse duration increased at 96 ns. Green pulse peak power reached 2.6 kW for the Cr4+:YAG with T0= 0.90 (Fig. 5b).

The characteristics of Q-switched laser pulses at 532 nm were described with a model of rate equation for photon density inside the resonator (, for the inversion of population in

All-Poly-Crystalline Ceramics Nd:YAG/Cr4+:YAG

>

the optical resonator are considered.

with those delivered by a single SA crystal.

2 0 2 2 20 2

2 1 ( ln )

*h T n n d E ALT*

Monolithic Micro-Lasers with Multiple-Beam Output 65

2 2 0 0 2 2 0 0

*L T nd n*

*gf gf*

*gf gf gi g*

  *d*

(6)

*i*

(1 )ln 1 (1 )ln 1 1

*d T T n n d L T n dd dd L T n*

Figure 6 shows modeling of the green pulse energy, for the Cr4+:YAG with T0= 0.90, versus the absorbed pump power, at various values of the losses L. Agreement with experimental results is good, especially if uncertainties in evaluation of L, or of laser beam variation inside

2

0

Fig. 6. The green pulse energy versus absorbed pump power for the Cr4+:YAG SA with T0= 0.90. Signs represent experiments and modeling is given by the continuous lines.

This was the first passively Q-switched Nd:YAG-Cr4+:YAG laser intra-cavity frequency doubled with LBO nonlinear crystal. The laser performances (green pulse of 226 J energy and 2.6 kW peak power, with ~1 W average power) were much higher than previously developed systems. For example, green laser pulses with 2.5-J energy (190-mW average power) were obtained from a Nd:LSB gain medium passively Q-switched by Cr4+:YAG and intra-cavity frequency doubled by KTiOPO4 (KTP) in a linear resonator (Ostroumov et al., 1997). Furthermore, a Nd:YAG laser that was passively Q-switched by GaAs semiconductor and intra-cavity frequency doubled by KTP, in a V-type laser resonator, yielded green laser pulses with 20.5-J energy and ~250-mW average power (Kajava and Gaeta, 1997). Later, a Nd:GdVO4-Cr4+:YAG-KTP laser with 21-J energy per pulse (average power of ~400 mW) was realized (Liu et al., 2004). More recently, passively Q-switched Nd:YAG (An et al., 2006) or Nd:LuVO4 lasers (Cheng et al., 2011) intra-cavity frequency doubled with KTP were reported. Novelty of these last two devices is the use of two SA crystals, Cr4+:YAG and GaAs, for the purpose of obtaining shorter and more symmetrical pulses, in comparison

(1 )ln ( ln ) 1 ln 1

*g g gi gi*

( ) ( ln ) ( 1) ( ) ( ln )

Fig. 4. A passively Q-switched Nd:YAG-Cr4+:YAG laser, intra-cavity frequency doubled by LBO nonlinear crystal.

Fig. 5. Characteristics of the laser emission obtained from the Nd:YAG-Cr4+:YAG-LBO laser: (a) Average output power and laser beam M2 factor; (b) Laser pulse energy and peak power.

Nd:YAG (ng) and for the population density in Cr4+:YAG (nSA), and in which out-coupling of the cavity field by frequency conversion was considered (Pavel et al., 2001b). The initial population inversion density, ngi is given by relation:

$$m\_{\mathcal{S}^i} = \frac{L - \ln T\_0^2}{2\sigma\_{\mathcal{S}}\ell\_{\mathcal{S}}} \tag{4}$$

The final inversion density ngf and ngi are related by the transcendental equation:

$$\left[1+\frac{\left(\mathbf{1}-\mathcal{S}\right)\cdot\ln T\_{0}^{2}}{L-\ln T\_{0}^{2}}\right]\cdot\left[1-\left(\frac{n\_{\mathcal{g}f}}{n\_{\mathcal{g}^{\circ}}}\right)^{d}\right]-\frac{d}{d-1}\cdot\frac{n\_{\mathcal{g}f}}{n\_{\mathcal{g}^{\circ}}}\left[1-\left(\frac{n\_{\mathcal{g}f}}{n\_{\mathcal{g}^{\circ}}}\right)^{d-1}\right] = \frac{\left(1-\mathcal{S}\right)\cdot\ln T\_{0}^{2}}{L-\ln T\_{0}^{2}}\cdot\frac{d}{d-a}\cdot\left(\frac{n\_{\mathcal{g}f}}{n\_{\mathcal{g}^{\circ}}}\right)^{a}\cdot\left[1-\left(\frac{n\_{\mathcal{g}f}}{n\_{\mathcal{g}^{\circ}}}\right)^{d-a}\right] \tag{5}$$

where d= k/(cgg), and the coefficient k is given by the second-harmonic generation theory (Eimerl, 1987; Honea et al., 1998). The green pulse energy is 2( ) 2 2 *E h A k t dt c* with A the green beam effective area, h the photon energy at , and c the optical length of the resonator. Finally, the analytical expression deduced for the pulse energy E was:

Fig. 4. A passively Q-switched Nd:YAG-Cr4+:YAG laser, intra-cavity frequency doubled by

Fig. 5. Characteristics of the laser emission obtained from the Nd:YAG-Cr4+:YAG-LBO laser: (a) Average output power and laser beam M2 factor; (b) Laser pulse energy and peak power. Nd:YAG (ng) and for the population density in Cr4+:YAG (nSA), and in which out-coupling of the cavity field by frequency conversion was considered (Pavel et al., 2001b). The initial

2 *gi*

*n*

The final inversion density ngf and ngi are related by the transcendental equation:

(1 ) ln (1 ) ln 11 1 1

1 2 2 0 0 2 2 0 0

ln 1 ln

2 0 ln

*d d d gf gf gf gf gf gi gi gi gi gi*

(4)

  *c*

with

(5)

*g g L T*

*T T n nn d d n n L T n dn n L T dn n*

where d= k/(cgg), and the coefficient k is given by the second-harmonic generation theory (Eimerl, 1987; Honea et al., 1998). The green pulse energy is 2( ) 2 2 *E h A k t dt*

A the green beam effective area, h the photon energy at , and c the optical length of the resonator. Finally, the analytical expression deduced for the pulse energy E was:

population inversion density, ngi is given by relation:

LBO nonlinear crystal.

$$\begin{split} E\_{2o} &= \frac{\hbar \nu\_{2o}}{2\gamma\_{\mathcal{S}}\sigma\_{\mathcal{S}}} \cdot A\_{2o} \cdot (L - \ln T\_{0}^{2}) \cdot \left\{ \left[1 + \frac{(1-\delta)\ln T\_{0}^{2}}{(L-\ln T\_{0}^{2})}\right] \times \ln \frac{n\_{\mathcal{Y}}}{n\_{\mathcal{g}^{\circ}}} + \frac{d}{d-1} \left(1 - \frac{n\_{\mathcal{Y}}}{n\_{\mathcal{g}^{\circ}}}\right) \\ &+ \frac{d}{a(d-a)} \cdot \frac{(1-\delta)\ln T\_{0}^{2}}{(L-\ln T\_{0}^{2})} \cdot \left[1 - \left(\frac{n\_{\mathcal{Y}}}{n\_{\mathcal{g}^{\circ}}}\right)^{a}\right] - \left[\frac{1}{d(d-1)} + \frac{a}{d(d-a)} \cdot \frac{(1-\delta)\ln T\_{0}^{2}}{(L-\ln T\_{0}^{2})}\right] \times \left[1 - \left(\frac{n\_{\mathcal{Y}}}{n\_{\mathcal{g}^{\circ}}}\right)^{d}\right] \end{split} \tag{6}$$

Figure 6 shows modeling of the green pulse energy, for the Cr4+:YAG with T0= 0.90, versus the absorbed pump power, at various values of the losses L. Agreement with experimental results is good, especially if uncertainties in evaluation of L, or of laser beam variation inside the optical resonator are considered.

Fig. 6. The green pulse energy versus absorbed pump power for the Cr4+:YAG SA with T0= 0.90. Signs represent experiments and modeling is given by the continuous lines.

This was the first passively Q-switched Nd:YAG-Cr4+:YAG laser intra-cavity frequency doubled with LBO nonlinear crystal. The laser performances (green pulse of 226 J energy and 2.6 kW peak power, with ~1 W average power) were much higher than previously developed systems. For example, green laser pulses with 2.5-J energy (190-mW average power) were obtained from a Nd:LSB gain medium passively Q-switched by Cr4+:YAG and intra-cavity frequency doubled by KTiOPO4 (KTP) in a linear resonator (Ostroumov et al., 1997). Furthermore, a Nd:YAG laser that was passively Q-switched by GaAs semiconductor and intra-cavity frequency doubled by KTP, in a V-type laser resonator, yielded green laser pulses with 20.5-J energy and ~250-mW average power (Kajava and Gaeta, 1997). Later, a Nd:GdVO4-Cr4+:YAG-KTP laser with 21-J energy per pulse (average power of ~400 mW) was realized (Liu et al., 2004). More recently, passively Q-switched Nd:YAG (An et al., 2006) or Nd:LuVO4 lasers (Cheng et al., 2011) intra-cavity frequency doubled with KTP were reported. Novelty of these last two devices is the use of two SA crystals, Cr4+:YAG and GaAs, for the purpose of obtaining shorter and more symmetrical pulses, in comparison with those delivered by a single SA crystal.

All-Poly-Crystalline Ceramics Nd:YAG/Cr4+:YAG

Monolithic Micro-Lasers with Multiple-Beam Output 67

 Fig. 8. Characteristics of Q-switched pulses yielded by the Nd:YAG-Cr4+:YAG laser shown in Fig. 7, versus initial transmission T0 of Cr4+:YAG: a) Energy and duration; b) Peak power. From the experimental observations, stable breakdown in air was observed for laser pulse energy Ep larger than 1.5 mJ and pulse duration tp below 1 ns using an aspheric focus lens of 10-mm focal length. Based on these results, a Cr4+:YAG SA single crystal with initial transmission T0= 0.30 was selected for the laser igniter. The first prototype micro-laser module that was built in our laboratory and that has the same dimensions as a spark plug is shown in Fig. 9. The device includes not only the pumping optics from fiber to the Nd:YAG gain material, but also a beam expanding and focusing optics for ignition. The laser igniter has the same optical design and similar performances as the experimental module shown in Fig. 7, and it is physically possible to ignite a real engine by installing it instead of an electrical spark plug to a plug hole (Tsunekane et al., 2010). For real operation on an engine, however, the mechanical design inside the module has be improved in order to sustain the

high temperatures (up to 150°C) and vibrations of a real engine.

Fig. 9. Nd:YAG-Cr4+:YAG laser with one-beam output. Nd:YAG gain medium and

Cr4+:YAG SA were single crystals; all optical components (including the output mirror) were discrete elements. Air breakdown is shown and size comparison is made with a spark plug.

#### **3. High-peak power passively Q-switched Nd:YAG/Cr4+:YAG lasers**

#### **3.1 Nd:YAG-Cr4+:YAG micro-lasers based on single-crystal components**

The cw pumped, passively Q-switched lasers have large pulse-to-pulse energy fluctuations and large timing jitters (Huang et al., 1999; Tang et al., 2003) due to thermal and mechanical instabilities. The purpose of our next research was to realize a Nd:YAG that is passively Q-switched by Cr4+:YAG SA and that can be used for ignition of an automobile engine. The operation frequency of igniters in internal combustion engines is less than 60 Hz, corresponding to an engine speed of 7200 rpm; the duty cycle is less than 5% for automobiles. In such a low frequency range, passively Q-switched lasers that are quasi-cw pumped with a low duty cycle are expected to operate stably due to initialization of the thermal and mechanical conditions during pulses.

Figure 7 is a drawing of a passively Q-switched laser module developed in our laboratory for preliminary experiments (Tsunekane et al., 2008). The active medium was a 1.1-at.% Nd:YAG single crystal (Metal Mining Co., Ltd., Japan) with a length of 4 mm. AR (R<0.2%) and HR (R>99.8%) coatings at p and em, respectively, were deposited on the pumped surface S1 of Nd:YAG. HR (R>90%) and AR (R<0.2%) coatings atp and em, respectively, were deposited on the intra-cavity surface S2 of Nd:YAG. AR coatings at em were deposited on both surfaces of a Cr4+:YAG SA (4-mm thick single crystal; Scientific Materials Corp., USA). The output coupler was flat with transmission T= 0.50 at em. Cavity length was 10 mm. The Nd:YAG was end pumped by a fiber coupled, conductive cooled, 120-W peak power laser diode (JOLD-120-QPXF-2P, Jenoptik, Germany) with emission at p= 807 nm; fiber core diameter was 600 m and numerical aperture NA was 0.22. The fiber end was imaged into Nd:YAG to a spot size of 1.1-mm diameter. Pump energy was controlled by changing the pump pulse duration, whereas the peak pump power was maintained constant at 120 W. The maximum pump duration was 500 μs and the repetition rate was 10 Hz.

Fig. 7. Schematic drawing of the passively Q-switched Nd:YAG-Cr4+:YAG laser that was build of discrete, Nd:YAG and Cr4+:YAG single crystals.

Figure 8a shows energy of the laser pulse delivered by the Nd:YAG-Cr4+:YAG laser versus initial transmission T0 of a Cr4+:YAG SA. Ep increases from 0.45 mJ for a Cr4+:YAG with T0= 0.80 to 4.3 mJ for a Cr4+:YAG with T0= 0.15. Corresponding pump pulse energy (Epump) was 53 and 5.2 mJ, respectively. The pulse duration was measured with a 10 GHz, InGaAs detector (ET-3500, Electro-Optics Technology, Inc.) and with a 12 GHz oscilloscope (DSO81204B, Agilent Technology). The shortest pulse width of 300 ps was obtained with a Cr4+:YAG of T0= 0.15 (Fig. 8a). Pulse peak power was 0.16 MW for a Cr4+:YAG with T0= 0.80 and a record of 14.5 MW for a Cr4+:YAG with T0= 0.15 (as shown in Fig. 8b).

The cw pumped, passively Q-switched lasers have large pulse-to-pulse energy fluctuations and large timing jitters (Huang et al., 1999; Tang et al., 2003) due to thermal and mechanical instabilities. The purpose of our next research was to realize a Nd:YAG that is passively Q-switched by Cr4+:YAG SA and that can be used for ignition of an automobile engine. The operation frequency of igniters in internal combustion engines is less than 60 Hz, corresponding to an engine speed of 7200 rpm; the duty cycle is less than 5% for automobiles. In such a low frequency range, passively Q-switched lasers that are quasi-cw pumped with a low duty cycle are expected to operate stably due to initialization of the

Figure 7 is a drawing of a passively Q-switched laser module developed in our laboratory for preliminary experiments (Tsunekane et al., 2008). The active medium was a 1.1-at.% Nd:YAG single crystal (Metal Mining Co., Ltd., Japan) with a length of 4 mm. AR (R<0.2%) and HR (R>99.8%) coatings at p and em, respectively, were deposited on the pumped surface S1 of Nd:YAG. HR (R>90%) and AR (R<0.2%) coatings atp and em, respectively, were deposited on the intra-cavity surface S2 of Nd:YAG. AR coatings at em were deposited on both surfaces of a Cr4+:YAG SA (4-mm thick single crystal; Scientific Materials Corp., USA). The output coupler was flat with transmission T= 0.50 at em. Cavity length was 10 mm. The Nd:YAG was end pumped by a fiber coupled, conductive cooled, 120-W peak power laser diode (JOLD-120-QPXF-2P, Jenoptik, Germany) with emission at p= 807 nm; fiber core diameter was 600 m and numerical aperture NA was 0.22. The fiber end was imaged into Nd:YAG to a spot size of 1.1-mm diameter. Pump energy was controlled by changing the pump pulse duration, whereas the peak pump power was maintained constant at 120 W. The

**3. High-peak power passively Q-switched Nd:YAG/Cr4+:YAG lasers 3.1 Nd:YAG-Cr4+:YAG micro-lasers based on single-crystal components**

thermal and mechanical conditions during pulses.

maximum pump duration was 500 μs and the repetition rate was 10 Hz.

build of discrete, Nd:YAG and Cr4+:YAG single crystals.

Fig. 7. Schematic drawing of the passively Q-switched Nd:YAG-Cr4+:YAG laser that was

and a record of 14.5 MW for a Cr4+:YAG with T0= 0.15 (as shown in Fig. 8b).

Figure 8a shows energy of the laser pulse delivered by the Nd:YAG-Cr4+:YAG laser versus initial transmission T0 of a Cr4+:YAG SA. Ep increases from 0.45 mJ for a Cr4+:YAG with T0= 0.80 to 4.3 mJ for a Cr4+:YAG with T0= 0.15. Corresponding pump pulse energy (Epump) was 53 and 5.2 mJ, respectively. The pulse duration was measured with a 10 GHz, InGaAs detector (ET-3500, Electro-Optics Technology, Inc.) and with a 12 GHz oscilloscope (DSO81204B, Agilent Technology). The shortest pulse width of 300 ps was obtained with a Cr4+:YAG of T0= 0.15 (Fig. 8a). Pulse peak power was 0.16 MW for a Cr4+:YAG with T0= 0.80

Fig. 8. Characteristics of Q-switched pulses yielded by the Nd:YAG-Cr4+:YAG laser shown in Fig. 7, versus initial transmission T0 of Cr4+:YAG: a) Energy and duration; b) Peak power.

From the experimental observations, stable breakdown in air was observed for laser pulse energy Ep larger than 1.5 mJ and pulse duration tp below 1 ns using an aspheric focus lens of 10-mm focal length. Based on these results, a Cr4+:YAG SA single crystal with initial transmission T0= 0.30 was selected for the laser igniter. The first prototype micro-laser module that was built in our laboratory and that has the same dimensions as a spark plug is shown in Fig. 9. The device includes not only the pumping optics from fiber to the Nd:YAG gain material, but also a beam expanding and focusing optics for ignition. The laser igniter has the same optical design and similar performances as the experimental module shown in Fig. 7, and it is physically possible to ignite a real engine by installing it instead of an electrical spark plug to a plug hole (Tsunekane et al., 2010). For real operation on an engine, however, the mechanical design inside the module has be improved in order to sustain the high temperatures (up to 150°C) and vibrations of a real engine.

Fig. 9. Nd:YAG-Cr4+:YAG laser with one-beam output. Nd:YAG gain medium and Cr4+:YAG SA were single crystals; all optical components (including the output mirror) were discrete elements. Air breakdown is shown and size comparison is made with a spark plug.

All-Poly-Crystalline Ceramics Nd:YAG/Cr4+:YAG

Monolithic Micro-Lasers with Multiple-Beam Output 69

little below those measured with Nd:YAG (single crystals or poly-crystalline ceramics) of low concentrations. On the other hand, each Nd:YAG poly-crystalline ceramics showed a slight decrease of these efficiencies, compared with its counterpart Nd:YAG single crystal.

Fig. 11. Characteristics of laser emission in free-generation regime (i.e. without Cr4+:YAG): (a) Laser pulse energy versus pump pulse energy; (b) Overall optical efficiency (0) and

Figure 12 presents characteristics of Q-switched laser pulses obtained with a Cr4+:YAG SA single crystal (SA1) of initial transmission T0= 0.40, and various OCM transmission T. Laser pulses with energy Ep~1.7 mJ (Fig. 12a) and duration tp~1.5 ns were yielded by the 1.0-at.% Nd:YAG single crystal (OCM with T= 0.70). The highly-doped 2.0-at.% Nd:YAG single crystal yielded laser pulses with energy Ep= 1.22 mJ and duration tp= 1.45 ns. The corresponding pulse peak power was 1.1 MW for the 1.0-at.% Nd:YAG and 0.84 MW for the 2.0-at.% Nd:YAG single crystal (Fig. 12c). The Cr4+:YAG single crystal was then replaced with a Cr4+:YAG (SA2) ceramics of the same initial transmission T0= 0.40. The Q-switched laser pulse energy was Ep= 1.0 mJ for the 1.1-at.% Nd:YAG ceramics (sample A\*), and Ep ~1.1 mJ for the 2.0-at.% Nd:YAG ceramics (sample B\*) (Fig. 12b). Corresponding pulse peak power was 0.53 and 0.73 MW, respectively. Generally, pulse energy Ep was lower than that measured with the Cr4+:YAG single crystal (SA1), whereas the pulse duration was longer.

Fig. 12. Q-switched pulse energy versus OCM transmission obtained with Nd:YAG gain media and a Cr4+:YAG SA of T0= 0.40: (a) The single-crystal Cr4+:YAG (SA1); (b) The poly-

crystalline Cr4+:YAG (SA3) ceramics; (c) Laser pulse peak power is shown.

slope efficiency (s) for the available pump pulse energy of 36.1 mJ.

#### **3.2 Ceramics versus single-crystals Nd:YAG-Cr4+:YAG micro-lasers**

The Nd:YAG as well as the Cr4+:YAG SA media used in the previous reports were single crystals. The advancement in ceramic techniques has reached a maturity stage, especially in obtaining poly-crystalline cubic laser media of very good optical quality. It is recognized that laser ceramics has become a serious challenge to crystalline optics, especially due to an easier manufacturability and a lower price. The use of poly-crystalline ceramics could decrease the price of the Nd:YAG-Cr4+:YAG laser, which is a critical condition for realizing and using of a laser spark plug for engine ignition. We have therefore conducted an investigation of laser output characteristics obtained from a passively Q-switched Nd:YAG-Cr4+:YAG laser that employs single crystals and poli-crystalline ceramics as Nd:YAG active media as well as Cr4+:YAG SA elements.

Fig. 10. A sketch of the experimental set-up used for comparative investigation of laser emission with Nd:YAG and Cr4+:YAG single crystals and poly-crystalline ceramics.

A sketch of the experimental set-up is shown in Fig. 10. The laser media were Nd:YAG single crystals with doping level of 1.0-at.% Nd (sample A; Japan) and 2.0-at.% Nd (sample B; Germany), and poly-crystalline Nd:YAG ceramics with 1.1-at.% Nd (sample A\*; Baikowski Japan Co., Ltd.) and 2.0-at.% Nd (sample B\*; Baikowski Japan Co., Ltd.) doping level. The thickness of sample B\* was 3 mm, whereas the other Nd:YAG media had 4 mm in thickness. Side S1 of each Nd:YAG was coated as HR (R> 99.9%) at em, and as HT (T> 97%) at p. The other side (S2) was AR coated (T> 99.9%) at em, and as HR (R> 95%) at p. The Cr4+:YAG SA had initial transmission, T0 between 0.80 and 0.20, and were single crystals provided by two different venders (SA1 and SA2, China), as well as poly-crystalline ceramics (SA3; Baikowski Japan Co., Ltd.). Both sides of a Cr4+:YAG SA were AR coated at em. Q-switched emission was reported previously in all-ceramics Nd:YAG-Cr4+:YAG (Feng et al., 2004) or Yb:YAG-Cr4+:YAG (Dong et al., 2006; Dong et al, 2007) compact lasers, the pumping being made with diode lasers of low power (few watts) in cw mode. In our experiments, the optical pumping was made with a fiber-coupled diode laser (JOLD-120- QPXF-2P, Jenoptik, Germany) in quasi-cw regime. The pump repetition rate was 10 Hz and the pump pulse duration was fixed at 250 s; pump energy was controlled by changing the diode current. An optical system made of two L1 and L2 lenses was used to image the fiber end (600-m in diameter, NA= 0.22) into Nd:YAG to a spot size of 1.1 mm in diameter. A linear resonator made between side S1 of Nd:YAG and a plane OCM was employed.

Figure 11 presents output performances measured in free-generation regime, using a 35-mm long resonator equipped with an OCM of transmission T= 0.20. The slope efficiency, s was in the range of 0.65 to 0.61, the highest value being recorded with the 1.0-at.% Nd:YAG single crystal (sample A), as shown in Fig. 11a. The overall optical-to-optical efficiency, 0 for the maximum pump level of 36.1 mJ per pulse is given in Fig. 11b. Laser pulses with 22.7 mJ energy (optical to optical efficiency 0 of ~0.63) were measured from the 1.0-at.% Nd:YAG (sample A). Efficiencies s and 0 recorded with the highly-doped Nd:YAG were a

The Nd:YAG as well as the Cr4+:YAG SA media used in the previous reports were single crystals. The advancement in ceramic techniques has reached a maturity stage, especially in obtaining poly-crystalline cubic laser media of very good optical quality. It is recognized that laser ceramics has become a serious challenge to crystalline optics, especially due to an easier manufacturability and a lower price. The use of poly-crystalline ceramics could decrease the price of the Nd:YAG-Cr4+:YAG laser, which is a critical condition for realizing and using of a laser spark plug for engine ignition. We have therefore conducted an investigation of laser output characteristics obtained from a passively Q-switched Nd:YAG-Cr4+:YAG laser that employs single crystals and poli-crystalline ceramics as Nd:YAG active

Fig. 10. A sketch of the experimental set-up used for comparative investigation of laser emission with Nd:YAG and Cr4+:YAG single crystals and poly-crystalline ceramics.

linear resonator made between side S1 of Nd:YAG and a plane OCM was employed.

Figure 11 presents output performances measured in free-generation regime, using a 35-mm long resonator equipped with an OCM of transmission T= 0.20. The slope efficiency, s was in the range of 0.65 to 0.61, the highest value being recorded with the 1.0-at.% Nd:YAG single crystal (sample A), as shown in Fig. 11a. The overall optical-to-optical efficiency, 0 for the maximum pump level of 36.1 mJ per pulse is given in Fig. 11b. Laser pulses with 22.7 mJ energy (optical to optical efficiency 0 of ~0.63) were measured from the 1.0-at.% Nd:YAG (sample A). Efficiencies s and 0 recorded with the highly-doped Nd:YAG were a

A sketch of the experimental set-up is shown in Fig. 10. The laser media were Nd:YAG single crystals with doping level of 1.0-at.% Nd (sample A; Japan) and 2.0-at.% Nd (sample B; Germany), and poly-crystalline Nd:YAG ceramics with 1.1-at.% Nd (sample A\*; Baikowski Japan Co., Ltd.) and 2.0-at.% Nd (sample B\*; Baikowski Japan Co., Ltd.) doping level. The thickness of sample B\* was 3 mm, whereas the other Nd:YAG media had 4 mm in thickness. Side S1 of each Nd:YAG was coated as HR (R> 99.9%) at em, and as HT (T> 97%) at p. The other side (S2) was AR coated (T> 99.9%) at em, and as HR (R> 95%) at p. The Cr4+:YAG SA had initial transmission, T0 between 0.80 and 0.20, and were single crystals provided by two different venders (SA1 and SA2, China), as well as poly-crystalline ceramics (SA3; Baikowski Japan Co., Ltd.). Both sides of a Cr4+:YAG SA were AR coated at em. Q-switched emission was reported previously in all-ceramics Nd:YAG-Cr4+:YAG (Feng et al., 2004) or Yb:YAG-Cr4+:YAG (Dong et al., 2006; Dong et al, 2007) compact lasers, the pumping being made with diode lasers of low power (few watts) in cw mode. In our experiments, the optical pumping was made with a fiber-coupled diode laser (JOLD-120- QPXF-2P, Jenoptik, Germany) in quasi-cw regime. The pump repetition rate was 10 Hz and the pump pulse duration was fixed at 250 s; pump energy was controlled by changing the diode current. An optical system made of two L1 and L2 lenses was used to image the fiber end (600-m in diameter, NA= 0.22) into Nd:YAG to a spot size of 1.1 mm in diameter. A

**3.2 Ceramics versus single-crystals Nd:YAG-Cr4+:YAG micro-lasers**

media as well as Cr4+:YAG SA elements.

little below those measured with Nd:YAG (single crystals or poly-crystalline ceramics) of low concentrations. On the other hand, each Nd:YAG poly-crystalline ceramics showed a slight decrease of these efficiencies, compared with its counterpart Nd:YAG single crystal.

Fig. 11. Characteristics of laser emission in free-generation regime (i.e. without Cr4+:YAG): (a) Laser pulse energy versus pump pulse energy; (b) Overall optical efficiency (0) and slope efficiency (s) for the available pump pulse energy of 36.1 mJ.

Figure 12 presents characteristics of Q-switched laser pulses obtained with a Cr4+:YAG SA single crystal (SA1) of initial transmission T0= 0.40, and various OCM transmission T. Laser pulses with energy Ep~1.7 mJ (Fig. 12a) and duration tp~1.5 ns were yielded by the 1.0-at.% Nd:YAG single crystal (OCM with T= 0.70). The highly-doped 2.0-at.% Nd:YAG single crystal yielded laser pulses with energy Ep= 1.22 mJ and duration tp= 1.45 ns. The corresponding pulse peak power was 1.1 MW for the 1.0-at.% Nd:YAG and 0.84 MW for the 2.0-at.% Nd:YAG single crystal (Fig. 12c). The Cr4+:YAG single crystal was then replaced with a Cr4+:YAG (SA2) ceramics of the same initial transmission T0= 0.40. The Q-switched laser pulse energy was Ep= 1.0 mJ for the 1.1-at.% Nd:YAG ceramics (sample A\*), and Ep ~1.1 mJ for the 2.0-at.% Nd:YAG ceramics (sample B\*) (Fig. 12b). Corresponding pulse peak power was 0.53 and 0.73 MW, respectively. Generally, pulse energy Ep was lower than that measured with the Cr4+:YAG single crystal (SA1), whereas the pulse duration was longer.

Fig. 12. Q-switched pulse energy versus OCM transmission obtained with Nd:YAG gain media and a Cr4+:YAG SA of T0= 0.40: (a) The single-crystal Cr4+:YAG (SA1); (b) The polycrystalline Cr4+:YAG (SA3) ceramics; (c) Laser pulse peak power is shown.

All-Poly-Crystalline Ceramics Nd:YAG/Cr4+:YAG

was 11 mm. (b) Energy of the pump pulse is shown.

intensity of the pulse incident on the focusing lens.

Monolithic Micro-Lasers with Multiple-Beam Output 71

experiments (Pavel et al., 2010a) and with the available Nd:YAG gain media and Cr4+:YAG SA components, the best laser performances were obtained with single-crystals elements. The laser pulse peak power (shown in Fig. 14c) was also better when we used Nd:YAG single crystal with Cr4+:YAG single crystal (compared with its counterparts ceramics).

 Fig. 15. (a) Q-switched laser pulse energy and peak power versus OCM transmission for the 1.1-at.% Nd:YAG ceramics and the Cr4+:YAG ceramics with T0= 0.40. The resonator length

In the final experiment the resonator length was reduced to 11 mm. Figure 15 summarizes results obtained with a combination of all-poly-crystalline ceramics, 1.1-at.% Nd:YAG gain medium and Cr4+:YAG SA with T0= 0.40. Laser pulse energy was around 1.4 mJ when OCM's transmission was higher than T= 0.50, whereas the pulse duration was tp~ 550 ps. Therefore, corresponding pulse peak power overcomes 2.5 MW (Fig. 15a). Air breakdown was realized with a focusing lens of 11-mm focal length. The pump pulse energy varied between 9.6 mJ when OCM transmission was T= 0.20 (low pulse energy Ep= 0.95 mJ, and tp= 650 ps) and Epump= 14.7 mJ when OCM transmission was increased at T= 0.70 (Fig. 15b). It is known that the intensity required for optical breakdown depends on pulse duration. There are not many reports on this subject: According to (Paschotta, 2008), an optical intensity of ~2×1013 W/cm2 is required for air breakdown with laser pulses of 1-ps duration. Therefore, experiments were performed in order to evaluate the optical intensity of nsduration laser pulses that realizes air breakdown. A Nd:YAG-Cr4+:YAG laser, as shown in Fig. 10, was used. The laser pulse duration was varied by changing the OCM transmission T, and with the help of two Cr4+:YAG that had initial transmission T0 of 0.39 and 0.29. The resonator length was fixed at 15 mm: The laser beam M2 factor was ~1.50, as determined by knife-edge method. The air breakdown was observed after a convergent lens of 7.5 mm focal length. A half waveplate and a polarizer were placed after the laser, in order to vary the

Figure 16 presents optical intensity that induced air breakdown (in laboratory conditions). In experiments, the pulse duration tp could be varied between 0.4 and 1.1 ns; corresponding laser pulse optical intensities that induced air breakdown were ~0.651013 W/cm2 and ~0.401013 W/cm2, respectively. We therefore concluded that optical intensity of a laser

pulse with 1-ns duration that induced air breakdown is ~0.51013 W/cm2.

Fig. 13. Pump pulse energy necessary for laser emission with: (a) The single-crystal Cr4+:YAG (SA1); (b) The poly-crystalline Cr4+:YAG (SA3) ceramics.

The pump pulse energy was Epump= 14.7 mJ for 1.0-at.% Nd:YAG (sample A) (Fig. 13a), lower than 15.4 mJ energy of the pump pulse required for Q-switched emission of the 2.0 at.% Nd:YAG sample B (OCM with T= 0.70). On the other hand, Epump necessary for the 1.1 at.% Nd:YAG ceramics (sample A\*) was only 13 mJ (Fig. 13b), while Epump required for laser operation of the 2.0-at.% Nd:YAG ceramics was the highest of 22.3 mJ. Generally, higher pump pulse energy was necessary for laser operation of a Nd:YAG gain medium that was Q-switched by Cr4+:YAG ceramics (Fig. 13b), compared with emission of the same laser medium that was Q-switched with a Cr4+:YAG single crystal (Fig. 13a).

Fig. 14. The influence of Cr4+:YAG initial transmission T0 on Q-switched laser pulse energy obtained from: (a) The 1.0-at.% Nd:YAG single crystal; (b) The 1.1-at.% Nd:YAG ceramics. (c) Laser pulse peak power is shown.

Figure 14 presents performances of the Q-switched laser pulses obtained with the 1.0-at.% Nd:YAG single crystal (Fig. 14a), the 1.1-at.% Nd:YAG ceramics (Fig. 14b), and using the available Cr4+:YAG SA. The OCM has transmission T= 0.50 and the resonator length was fixed at 35 mm. Differences between pulses obtained with the SA1 and SA2 Cr4+:YAG single crystals were observed at the same initial transmission T0. Most probably, the final transmissions of the Cr4+:YAG single crystals were a little different, depending of the growth process used by the companies that delivered the SA. Therefore, at this stage of the

The pump pulse energy was Epump= 14.7 mJ for 1.0-at.% Nd:YAG (sample A) (Fig. 13a), lower than 15.4 mJ energy of the pump pulse required for Q-switched emission of the 2.0 at.% Nd:YAG sample B (OCM with T= 0.70). On the other hand, Epump necessary for the 1.1 at.% Nd:YAG ceramics (sample A\*) was only 13 mJ (Fig. 13b), while Epump required for laser operation of the 2.0-at.% Nd:YAG ceramics was the highest of 22.3 mJ. Generally, higher pump pulse energy was necessary for laser operation of a Nd:YAG gain medium that was Q-switched by Cr4+:YAG ceramics (Fig. 13b), compared with emission of the same laser

Fig. 14. The influence of Cr4+:YAG initial transmission T0 on Q-switched laser pulse energy obtained from: (a) The 1.0-at.% Nd:YAG single crystal; (b) The 1.1-at.% Nd:YAG ceramics.

Figure 14 presents performances of the Q-switched laser pulses obtained with the 1.0-at.% Nd:YAG single crystal (Fig. 14a), the 1.1-at.% Nd:YAG ceramics (Fig. 14b), and using the available Cr4+:YAG SA. The OCM has transmission T= 0.50 and the resonator length was fixed at 35 mm. Differences between pulses obtained with the SA1 and SA2 Cr4+:YAG single crystals were observed at the same initial transmission T0. Most probably, the final transmissions of the Cr4+:YAG single crystals were a little different, depending of the growth process used by the companies that delivered the SA. Therefore, at this stage of the

Fig. 13. Pump pulse energy necessary for laser emission with: (a) The single-crystal

Cr4+:YAG (SA1); (b) The poly-crystalline Cr4+:YAG (SA3) ceramics.

medium that was Q-switched with a Cr4+:YAG single crystal (Fig. 13a).

(c) Laser pulse peak power is shown.

experiments (Pavel et al., 2010a) and with the available Nd:YAG gain media and Cr4+:YAG SA components, the best laser performances were obtained with single-crystals elements. The laser pulse peak power (shown in Fig. 14c) was also better when we used Nd:YAG single crystal with Cr4+:YAG single crystal (compared with its counterparts ceramics).

Fig. 15. (a) Q-switched laser pulse energy and peak power versus OCM transmission for the 1.1-at.% Nd:YAG ceramics and the Cr4+:YAG ceramics with T0= 0.40. The resonator length was 11 mm. (b) Energy of the pump pulse is shown.

In the final experiment the resonator length was reduced to 11 mm. Figure 15 summarizes results obtained with a combination of all-poly-crystalline ceramics, 1.1-at.% Nd:YAG gain medium and Cr4+:YAG SA with T0= 0.40. Laser pulse energy was around 1.4 mJ when OCM's transmission was higher than T= 0.50, whereas the pulse duration was tp~ 550 ps. Therefore, corresponding pulse peak power overcomes 2.5 MW (Fig. 15a). Air breakdown was realized with a focusing lens of 11-mm focal length. The pump pulse energy varied between 9.6 mJ when OCM transmission was T= 0.20 (low pulse energy Ep= 0.95 mJ, and tp= 650 ps) and Epump= 14.7 mJ when OCM transmission was increased at T= 0.70 (Fig. 15b).

It is known that the intensity required for optical breakdown depends on pulse duration. There are not many reports on this subject: According to (Paschotta, 2008), an optical intensity of ~2×1013 W/cm2 is required for air breakdown with laser pulses of 1-ps duration. Therefore, experiments were performed in order to evaluate the optical intensity of nsduration laser pulses that realizes air breakdown. A Nd:YAG-Cr4+:YAG laser, as shown in Fig. 10, was used. The laser pulse duration was varied by changing the OCM transmission T, and with the help of two Cr4+:YAG that had initial transmission T0 of 0.39 and 0.29. The resonator length was fixed at 15 mm: The laser beam M2 factor was ~1.50, as determined by knife-edge method. The air breakdown was observed after a convergent lens of 7.5 mm focal length. A half waveplate and a polarizer were placed after the laser, in order to vary the intensity of the pulse incident on the focusing lens.

Figure 16 presents optical intensity that induced air breakdown (in laboratory conditions). In experiments, the pulse duration tp could be varied between 0.4 and 1.1 ns; corresponding laser pulse optical intensities that induced air breakdown were ~0.651013 W/cm2 and ~0.401013 W/cm2, respectively. We therefore concluded that optical intensity of a laser pulse with 1-ns duration that induced air breakdown is ~0.51013 W/cm2.

All-Poly-Crystalline Ceramics Nd:YAG/Cr4+:YAG

L3 that has an 11-mm long focal length.

0.8 0.8 0.8

Ep (mJ) tp (ns) Epump

4.1 2.5 2.4

0.30 and all the OCM employed in the experiments.

9.0 9.5 10.5

poly-crystalline ceramics laser that was build of discrete components.

OCM with transmission T= 0.50 was coated on surface S2 of Cr4+:YAG.

was bended with a prism (patent pending), and finally focussed.

OCM, T

> 0.40 0.50 0.70

Monolithic Micro-Lasers with Multiple-Beam Output 73

The laser medium was a 1.1-at.% Nd:YAG ceramics (Baikowski Japan Co., Ltd.) with thickness of 5 mm and a 10-mm diameter. Surface used for pumping was coated HR at the lasing wavelength of 1.06 m (em) and HT at p. The second surface was coated AR at both em and p. Around 90% of the pump radiation was absorbed in Nd:YAG. Cr4+:YAG ceramics with initial transmission T0 of 0.70, 0.50, and 0.30 were employed for Q-switching, both surfaces of a SA ceramics being coated AR at em. The resonator length was 12 mm, and a plane output mirror (OCM) was used for out-coupling. The laser beams path was bent with a prism placed after the resonator, whereas air breakdown was observed behind a lens

T0= 0.70 T0= 0.50 T0= 0.30

2.3 1.5 1.4 12.0 13.0 15.0

(mJ) Ep (mJ) tp (ns) Epump

1.0 0.6 0.7

1.7 2.1 2.3 (mJ)

18.5 20.0 21.5

(mJ) Ep (mJ) tp (ns) Epump

Table 1. Characteristics of Q-switched laser pulses obtained from the Nd:YAG-Cr4+:YAG all-

Characteristics of the Q-switched laser pulses obtained from the Nd:YAG-Cr4+:YAG laser are summarized in Table 1, at various OCM transmission T. Laser pulses of few-ns duration and energy Ep of 0.8 mJ were measured for the Cr4+:YAG with T0= 0.70. Energy Ep overcame 2 mJ and tp shortened below 1 ns when combination of Cr4+:YAG with T0= 0.30 and OCM with T of 0.50 or 0.70 was used. Air breakdown was successfully for the Cr4+:YAG with T0=

The next step of our investigations constituted realization of a compact Nd:YAG/Cr4+:YAG laser with two-beam output and dimensions close to an electrical spark plug. Figure 18a shows the experimental set-up. The laser medium was a composite Nd:YAG/Cr4+:YAG ceramics: Research experience of Baikowski Japan Co., as well as available optics on market (for example one could visit: www.thorlabs.com), allowed realization of this medium as a parallelepiped with 1015 mm2 surface area, as presented in Fig. 18b. The Nd:YAG doping level was 1.1-at.% Nd, and its length was increased at 8 mm: In this way, the gain medium absorption efficiency at p was better than 0.95, which avoided bleaching effects of Cr4+:YAG by the pump beam (Jaspan et al., 2004). The 3-mm thick Cr4+:YAG SA ceramics had initial transmission T0= 0.30. Surface S1 of Nd:YAG was coated HR at em and HT at p, and the

The compact pumping line imaged the fiber end to a spot size of 1.0-mm into Nd:YAG. Each laser beam was expanded and then collimated in the "expander" section. Next, the beam

The composite, all-ceramics, passively Q-switched Nd:YAG/Cr4+:YAG monolithic laser with two-beam output is shown in Fig. 19a. Figure 19b presents the air breakdown realized with this laser, whereas an electrical spark plug used in industrial gas engine is given for comparison. Each beam delivered Q-switched laser pulses with ~2.5 mJ energy and ~800 ps duration, which corresponds to a peak power of 3.1 MW. Minimal pump pulse energy was ~27 mJ. The laser pulse jitter, which was estimated from 500 consecutive pulses, improved

1.0 1.3 1.3

Fig. 16. Optical intensity of a laser pulse with ns duration that induced air breakdown.

#### **3.3 Composite all-ceramics Nd:YAG/Cr4+:YAG monolithic micro-lasers with two and three-beam output for engine ignition**

Figure 17 is a sketch of the set-up used to demonstrate, in preliminary experiments, a passively Q-switched Nd:YAG-Cr4+:YAG laser with two-beam output. Generally, one could choose to use one line for pumping, and to divide the high-energy output laser beam into two (ore more) fascicles, which has to be directed at necessary angle and then focused. This solution increases probability of damaging the laser media (due to the high intensity of the laser beam, or due to thermal effects), and could complicate the guiding line. Therefore, our choice was to employ similar, independent, multi-pumping lines, and then to change the optical path of a laser beam before focusing it.

The pump was made at 807 nm (p) with two fiber-coupled (600-m diameter and numerical aperture NA=0.22) diode lasers (JOLD-120-QPXF-2P, Jenoptik, Germany). Pump repetition rate and pump pulse duration were 5 Hz and 250 s, respectively. The fiber end was imaged into Nd:YAG to a spot size of 1.1-mm diameter. The two-pump beams were inserted into Nd:YAG with a metal-coated prism. Furthermore, distance between the pumping positions on Nd:YAG input surface was changed by forward and backward translation of this prism.

Fig. 17. Schematic of the experimental set-up used for "on table" demonstration of a passively Q-switched Nd:YAG-Cr4+:YAG all-ceramics laser with two-beam output.

Fig. 16. Optical intensity of a laser pulse with ns duration that induced air breakdown.

**three-beam output for engine ignition**

optical path of a laser beam before focusing it.

backward translation of this prism.

**3.3 Composite all-ceramics Nd:YAG/Cr4+:YAG monolithic micro-lasers with two and**

Figure 17 is a sketch of the set-up used to demonstrate, in preliminary experiments, a passively Q-switched Nd:YAG-Cr4+:YAG laser with two-beam output. Generally, one could choose to use one line for pumping, and to divide the high-energy output laser beam into two (ore more) fascicles, which has to be directed at necessary angle and then focused. This solution increases probability of damaging the laser media (due to the high intensity of the laser beam, or due to thermal effects), and could complicate the guiding line. Therefore, our choice was to employ similar, independent, multi-pumping lines, and then to change the

The pump was made at 807 nm (p) with two fiber-coupled (600-m diameter and numerical aperture NA=0.22) diode lasers (JOLD-120-QPXF-2P, Jenoptik, Germany). Pump repetition rate and pump pulse duration were 5 Hz and 250 s, respectively. The fiber end was imaged into Nd:YAG to a spot size of 1.1-mm diameter. The two-pump beams were inserted into Nd:YAG with a metal-coated prism. Furthermore, distance between the pumping positions on Nd:YAG input surface was changed by forward and

Fig. 17. Schematic of the experimental set-up used for "on table" demonstration of a passively Q-switched Nd:YAG-Cr4+:YAG all-ceramics laser with two-beam output.

The laser medium was a 1.1-at.% Nd:YAG ceramics (Baikowski Japan Co., Ltd.) with thickness of 5 mm and a 10-mm diameter. Surface used for pumping was coated HR at the lasing wavelength of 1.06 m (em) and HT at p. The second surface was coated AR at both em and p. Around 90% of the pump radiation was absorbed in Nd:YAG. Cr4+:YAG ceramics with initial transmission T0 of 0.70, 0.50, and 0.30 were employed for Q-switching, both surfaces of a SA ceramics being coated AR at em. The resonator length was 12 mm, and a plane output mirror (OCM) was used for out-coupling. The laser beams path was bent with a prism placed after the resonator, whereas air breakdown was observed behind a lens L3 that has an 11-mm long focal length.

Table 1. Characteristics of Q-switched laser pulses obtained from the Nd:YAG-Cr4+:YAG allpoly-crystalline ceramics laser that was build of discrete components.

Characteristics of the Q-switched laser pulses obtained from the Nd:YAG-Cr4+:YAG laser are summarized in Table 1, at various OCM transmission T. Laser pulses of few-ns duration and energy Ep of 0.8 mJ were measured for the Cr4+:YAG with T0= 0.70. Energy Ep overcame 2 mJ and tp shortened below 1 ns when combination of Cr4+:YAG with T0= 0.30 and OCM with T of 0.50 or 0.70 was used. Air breakdown was successfully for the Cr4+:YAG with T0= 0.30 and all the OCM employed in the experiments.

The next step of our investigations constituted realization of a compact Nd:YAG/Cr4+:YAG laser with two-beam output and dimensions close to an electrical spark plug. Figure 18a shows the experimental set-up. The laser medium was a composite Nd:YAG/Cr4+:YAG ceramics: Research experience of Baikowski Japan Co., as well as available optics on market (for example one could visit: www.thorlabs.com), allowed realization of this medium as a parallelepiped with 1015 mm2 surface area, as presented in Fig. 18b. The Nd:YAG doping level was 1.1-at.% Nd, and its length was increased at 8 mm: In this way, the gain medium absorption efficiency at p was better than 0.95, which avoided bleaching effects of Cr4+:YAG by the pump beam (Jaspan et al., 2004). The 3-mm thick Cr4+:YAG SA ceramics had initial transmission T0= 0.30. Surface S1 of Nd:YAG was coated HR at em and HT at p, and the OCM with transmission T= 0.50 was coated on surface S2 of Cr4+:YAG.

The compact pumping line imaged the fiber end to a spot size of 1.0-mm into Nd:YAG. Each laser beam was expanded and then collimated in the "expander" section. Next, the beam was bended with a prism (patent pending), and finally focussed.

The composite, all-ceramics, passively Q-switched Nd:YAG/Cr4+:YAG monolithic laser with two-beam output is shown in Fig. 19a. Figure 19b presents the air breakdown realized with this laser, whereas an electrical spark plug used in industrial gas engine is given for comparison. Each beam delivered Q-switched laser pulses with ~2.5 mJ energy and ~800 ps duration, which corresponds to a peak power of 3.1 MW. Minimal pump pulse energy was ~27 mJ. The laser pulse jitter, which was estimated from 500 consecutive pulses, improved

All-Poly-Crystalline Ceramics Nd:YAG/Cr4+:YAG

was measured by the knife-edge method, was 3.7.

Pulse duration (ps)

> 850 650 660

Peak power (MW)

> 2.79 3.12 2.08

>

2 0 2

Table 2. Characteristics of Q-switched laser pulses obtained from the composite, all-polly-

In order to explain the influence of pump-beam spot size on Q-switched laser performances, we used a rate equation model (Zhang et al., 2000; Li et al., 2007) in which the pump beam was assumed to have a top-hat distribution of radius *wp* and the laser beam was taken as Gaussian with a spot size of radius *wg.* Both *wp* and *wg* were considered constant along the Nd:YAG/Cr4+:YAG medium. The laser pulse energy is given by Eq. (1), while the initial

> ln ln 2 1 exp 2 *gi g g*

 *a*

with *a= wp/wg*. The final population inversion density, ngf and ngi are related by relation:

Figure 21 presents ngf/ngi versus ratio *a= wp/wg*. In simulation losses were L= 0.06 (0.01 for Nd:YAG and 0.05 for Cr4+:YAG final transmission), while spectroscopic parameter of

*RL T*

2 2 0 0 (1 ) ln (1 ) ln 1 1 ln 1 0 *gf gf gf gi gi gi n nn T T n nn*

Pump pulse

27 32.8 32

energy (mJ) M2 factor

(7)

3.7 4.0 4.0

(8)

Pulse energy (mJ)

> 2.37 2.03 1.37

crystalline Nd:YAG/Cr4+:YAG ceramics.

population inversion density, ngi was written as:

with parameter : = (-ln*R*+*L*-ln*T*02)/[1-exp(-2*a*2)].

*n*

Nd (at.%)

> 1.1 1.5 2.0

Monolithic Micro-Lasers with Multiple-Beam Output 75

electrical spark plug used in automobile industry. The experimental set-up is shown in Fig. 20a. Three composite, all-ceramics Nd:YAG/Cr4+:YAG media (Baikowski Japan Co., Ltd.), each with a 9-mm diameter were prepared for the experiments. While the Cr4+:YAG SA has initial transmission T0= 0.30 and a thickness of 2.5 mm, the influence of Nd-doping level on Qswitched laser characteristics was investigated by using Nd:YAG with 1.1-at.% Nd (7.5-mm thick), as well as highly-doped 1.5-at.% Nd (thickness of 5 mm) and 2.0-at.% Nd (thickness of 3.5 mm). Again, surface S1 of Nd:YAG was coated HR at em and HT at p. The OCM with T= 0.50 at em was coated on surface S2 of Cr4+:YAG SA. A photo of two composite Nd:YAG/Cr4+:YAG ceramics is shown in Fig. 20b. The optical pumping was realized through three independent, similar, and compact pumping lines (marked by 1 to 3 in Fig. 20a), each line containing a pair of an aspheric collimating lens and an aspheric focusing lens with short focal length and high NA. We mention that in order to fulfill dimensions of an automobile spark plug, diameter of all lenses was reduced and a new design of the fiber end was made. The characteristics of the Q-switched laser pulses measured from the Nd:YAG/Cr4+:YAG ceramics are given in Table 2. The energy of the laser pulse yielded by the 1.1-at.% Nd:YAG ceramics was 2.37 mJ, with a pulse peak power of 2.8 MW. The laser beam M2 factor, which

from 3 s to 1 s when the pump pulse energy was increased from 27 mJ to 34 mJ respectively, while the pulse standard deviation decreases from 0.53 s to 0.18 s. The prism angle was chose such as the distance between the ignition points (c) was 13 mm, whereas the depth of the ignition (bc) was 9 mm (Pavel et al., 2011a).

Fig. 18. (a) A sketch of the composite, all-ceramics Nd:YAG/Cr4+:YAG monolithic laser with two-beam output is presented. (b) The rectangular-shaped laser medium is shown.

Fig. 19. (a) The Nd:YAG/Cr4+:YAG laser with two-beam output is presented. (b) An electrical spark plug is shown for comparison and air breakdown in two points is illustrated.

Fig. 20. (a) Schematic of a passively Q-switched, composite, all-ceramics Nd:YAG/Cr4+:YAG monolithic laser with three-beam output. (b) Photo of two composite media is shown.

Once the Nd:YAG/Cr4+:YAG monolithic laser with two-beam output was build, the final goal of our work was realization of a laser device with three-beam output and a size that fits an

from 3 s to 1 s when the pump pulse energy was increased from 27 mJ to 34 mJ respectively, while the pulse standard deviation decreases from 0.53 s to 0.18 s. The prism angle was chose such as the distance between the ignition points (c) was 13 mm, whereas

Fig. 18. (a) A sketch of the composite, all-ceramics Nd:YAG/Cr4+:YAG monolithic laser with

two-beam output is presented. (b) The rectangular-shaped laser medium is shown.

 Fig. 19. (a) The Nd:YAG/Cr4+:YAG laser with two-beam output is presented. (b) An

electrical spark plug is shown for comparison and air breakdown in two points is illustrated.

Fig. 20. (a) Schematic of a passively Q-switched, composite, all-ceramics Nd:YAG/Cr4+:YAG monolithic laser with three-beam output. (b) Photo of two composite media is shown.

Once the Nd:YAG/Cr4+:YAG monolithic laser with two-beam output was build, the final goal of our work was realization of a laser device with three-beam output and a size that fits an

the depth of the ignition (bc) was 9 mm (Pavel et al., 2011a).

electrical spark plug used in automobile industry. The experimental set-up is shown in Fig. 20a. Three composite, all-ceramics Nd:YAG/Cr4+:YAG media (Baikowski Japan Co., Ltd.), each with a 9-mm diameter were prepared for the experiments. While the Cr4+:YAG SA has initial transmission T0= 0.30 and a thickness of 2.5 mm, the influence of Nd-doping level on Qswitched laser characteristics was investigated by using Nd:YAG with 1.1-at.% Nd (7.5-mm thick), as well as highly-doped 1.5-at.% Nd (thickness of 5 mm) and 2.0-at.% Nd (thickness of 3.5 mm). Again, surface S1 of Nd:YAG was coated HR at em and HT at p. The OCM with T= 0.50 at em was coated on surface S2 of Cr4+:YAG SA. A photo of two composite Nd:YAG/Cr4+:YAG ceramics is shown in Fig. 20b. The optical pumping was realized through three independent, similar, and compact pumping lines (marked by 1 to 3 in Fig. 20a), each line containing a pair of an aspheric collimating lens and an aspheric focusing lens with short focal length and high NA. We mention that in order to fulfill dimensions of an automobile spark plug, diameter of all lenses was reduced and a new design of the fiber end was made. The characteristics of the Q-switched laser pulses measured from the Nd:YAG/Cr4+:YAG ceramics are given in Table 2. The energy of the laser pulse yielded by the 1.1-at.% Nd:YAG ceramics was 2.37 mJ, with a pulse peak power of 2.8 MW. The laser beam M2 factor, which was measured by the knife-edge method, was 3.7.

Table 2. Characteristics of Q-switched laser pulses obtained from the composite, all-pollycrystalline Nd:YAG/Cr4+:YAG ceramics.

In order to explain the influence of pump-beam spot size on Q-switched laser performances, we used a rate equation model (Zhang et al., 2000; Li et al., 2007) in which the pump beam was assumed to have a top-hat distribution of radius *wp* and the laser beam was taken as Gaussian with a spot size of radius *wg.* Both *wp* and *wg* were considered constant along the Nd:YAG/Cr4+:YAG medium. The laser pulse energy is given by Eq. (1), while the initial population inversion density, ngi was written as:

$$m\_{g^i} = \frac{-\ln R + L - \ln T\_0^2}{2\sigma\_{g^i}\ell\_{g^i}\left[1 - \exp\left(-2a^2\right)\right]}\tag{7}$$

with *a= wp/wg*. The final population inversion density, ngf and ngi are related by relation:

$$
\left[1 - \frac{n\_{\mathcal{g}f}}{n\_{\mathcal{g}i}}\right] + \left[1 + \frac{(1-\delta)\cdot\ln T\_0^2}{\beta}\right] \cdot \ln\left(\frac{n\_{\mathcal{g}f}}{n\_{\mathcal{g}i}}\right) \cdot \frac{(1-\delta)\cdot\ln T\_0^2}{\beta} \cdot \left[1 - \left(\frac{n\_{\mathcal{g}f}}{n\_{\mathcal{g}i}}\right)^\alpha\right] = 0\tag{8}
$$

with parameter : = (-ln*R*+*L*-ln*T*02)/[1-exp(-2*a*2)].

Figure 21 presents ngf/ngi versus ratio *a= wp/wg*. In simulation losses were L= 0.06 (0.01 for Nd:YAG and 0.05 for Cr4+:YAG final transmission), while spectroscopic parameter of

All-Poly-Crystalline Ceramics Nd:YAG/Cr4+:YAG

Nd (at.%)

1.1 1.5 2.0

of the Nd:YAG/Cr4+:YAG ceramics media.

Nd:YAG/Cr4+:YAG medium is given in Fig. 22.

Monolithic Micro-Lasers with Multiple-Beam Output 77

2.34 2.03 1.34

Table 3. Average laser pulse energy and standard deviation measured along Ox and Oz axis

Very important for performances of the monolithic laser is the uniformity of the ceramic Nd:YAG/Cr4+:YAG material. Table 3 presents average values of the laser pulse energy determined along Ox and Oz axes, estimated by scanning the medium with a 0.5-mm step. The laser pulse energies were very close to those measured at the media center. Moreover, standard deviation was small, below 3% for the 1.1-at.% Nd:YAG, and less than 4% for the highly-doped Nd:YAG. The results indicate a very good homogeneity as well as quality of the composite all-ceramics Nd:YAG/Cr4+:YAG media, in spite of the high, 9-mm diameter. As an example, laser pulse energy measured from the highly-doped, 2.0-at.%

Fig. 22. Laser pulse energy measured along horizontal and vertical axis of the 2.0-at.%

The three-beam output micro-laser (Fig. 23a) was realized with the composite, all-ceramics 1.1-at.% Nd:YAG/Cr4+:YAG medium. The guiding line (Pavel et al., 2011b) was designed to assure a distance between a focusing point and the laser axis, c of 4.5 mm and a depth of the focusing point inside the combustion chamber, bc of 9 mm. Air breakdown is illustrated

Nd:YAG/Cr4+:YAG ceramics (inset shows the composite medium).

in Fig. 23b, and an automobile electrical spark plug is shown for comparison.

(mJ) Standard deviation (mJ)

0.02 0.03 0.05 0.06 0.03 0.02

Ox Oz Ox Oz

Average pulse energy

2.36 2.03 1.33

Nd:YAG was σg= 2.6310-19 cm2 (Taira, 2007). If *wp*<*wg* the overlap between pump and laser beam is good, but ngf/ngi increases when *wp/wg* decreases. Although the initial inversion of population ngi is high, a small fraction of it used for lasing and therefore Q-switched laser pulse energy is low. If *wp/wg* has a large value, the central part of the inversion of population interacts with laser mode, whereas some outside part could be depleted by spontaneous emission. Increasing *wp/wg* decreases ngf/ngi: The final inversion of population is low and a pulse laser with high energy is obtained. The expected values of the Q-switched laser pulse at various sizes *wg* of the laser mode were also shown in Fig. 21.

Fig. 21. Ratio ngf/ngi versus *wp/wg* and Q-switched laser pulse energy for various laser beam radii *wg*. The pulse energy obtained from the 1.1-at.% Nd:YAG is given by the sign (■).

A plane-plane resonator operates due to thermal effects induced by optical pumping in the laser medium. The focal length *f* of Nd:YAG thermal lens can be evaluated by relation: f= (Kc*w*p 2)/[Ph(dn/dT)] (Innocenzi et al., 1990). Nd:YAG has thermal conductivity Kc= 10.1 Wm-1K-1 (Sato & Taira, 2006), thermal coefficient of the refraction index is dn/dT= 0.7310-5 K-1, while ~0.24 of the absorbed pump power is transformed into heat (Ph) under efficient laser emission at 1.06 m. The average thermal lens of the 1.1-at.% Nd:YAG was evaluated as ~6.5 m, whereas value of *wg* was determinate by an ABCD description of the resonator as ~420 m. Sign of Fig. 21 is the experimental value of the laser pulse energy obtained from the 1.1-at.% Nd:YAG. Agreement with theoretical modeling is good, especially if uncertainties in evaluation of thermal focal lens or of other system parameters (such as losses L) are considered. The model can be improved by taking into account variation of pump beam radius *wp* and of laser beam spot size *wg* along the resonator length.

Energies of 2.03 mJ and 1.37 mJ were measured from the 1.5-at.% and 2.0-at.% Nd ceramics, respectively. The pump pulse energy increased from 27 mJ for the 1.1-at.% Nd:YAG to 33 mJ for the 1.5-at.% Nd:YAG, and to 32 mJ for the 2.0-at.% Nd:YAG. The decrease of the 4F3/2 upper-level lifetime with Nd-doping level could be a reason for lower laser performances recorded with the highly-doped Nd:YAG compared with the 1.1-at.% Nd:YAG. The OCM transmission has also to be optimized for the highly-doped Nd:YAG ceramics.

Nd:YAG was σg= 2.6310-19 cm2 (Taira, 2007). If *wp*<*wg* the overlap between pump and laser beam is good, but ngf/ngi increases when *wp/wg* decreases. Although the initial inversion of population ngi is high, a small fraction of it used for lasing and therefore Q-switched laser pulse energy is low. If *wp/wg* has a large value, the central part of the inversion of population interacts with laser mode, whereas some outside part could be depleted by spontaneous emission. Increasing *wp/wg* decreases ngf/ngi: The final inversion of population is low and a pulse laser with high energy is obtained. The expected values of the Q-switched laser pulse

Fig. 21. Ratio ngf/ngi versus *wp/wg* and Q-switched laser pulse energy for various laser beam radii *wg*. The pulse energy obtained from the 1.1-at.% Nd:YAG is given by the sign (■).

A plane-plane resonator operates due to thermal effects induced by optical pumping in the laser medium. The focal length *f* of Nd:YAG thermal lens can be evaluated by relation:

10.1 Wm-1K-1 (Sato & Taira, 2006), thermal coefficient of the refraction index is dn/dT= 0.7310-5 K-1, while ~0.24 of the absorbed pump power is transformed into heat (Ph) under efficient laser emission at 1.06 m. The average thermal lens of the 1.1-at.% Nd:YAG was evaluated as ~6.5 m, whereas value of *wg* was determinate by an ABCD description of the resonator as ~420 m. Sign of Fig. 21 is the experimental value of the laser pulse energy obtained from the 1.1-at.% Nd:YAG. Agreement with theoretical modeling is good, especially if uncertainties in evaluation of thermal focal lens or of other system parameters (such as losses L) are considered. The model can be improved by taking into account variation of pump beam radius *wp* and of laser beam spot size *wg* along the

Energies of 2.03 mJ and 1.37 mJ were measured from the 1.5-at.% and 2.0-at.% Nd ceramics, respectively. The pump pulse energy increased from 27 mJ for the 1.1-at.% Nd:YAG to 33 mJ for the 1.5-at.% Nd:YAG, and to 32 mJ for the 2.0-at.% Nd:YAG. The decrease of the 4F3/2 upper-level lifetime with Nd-doping level could be a reason for lower laser performances recorded with the highly-doped Nd:YAG compared with the 1.1-at.% Nd:YAG. The OCM

transmission has also to be optimized for the highly-doped Nd:YAG ceramics.

2)/[Ph(dn/dT)] (Innocenzi et al., 1990). Nd:YAG has thermal conductivity Kc=

at various sizes *wg* of the laser mode were also shown in Fig. 21.

f= (Kc*w*p

resonator length.

Table 3. Average laser pulse energy and standard deviation measured along Ox and Oz axis of the Nd:YAG/Cr4+:YAG ceramics media.

Very important for performances of the monolithic laser is the uniformity of the ceramic Nd:YAG/Cr4+:YAG material. Table 3 presents average values of the laser pulse energy determined along Ox and Oz axes, estimated by scanning the medium with a 0.5-mm step. The laser pulse energies were very close to those measured at the media center. Moreover, standard deviation was small, below 3% for the 1.1-at.% Nd:YAG, and less than 4% for the highly-doped Nd:YAG. The results indicate a very good homogeneity as well as quality of the composite all-ceramics Nd:YAG/Cr4+:YAG media, in spite of the high, 9-mm diameter. As an example, laser pulse energy measured from the highly-doped, 2.0-at.% Nd:YAG/Cr4+:YAG medium is given in Fig. 22.

Fig. 22. Laser pulse energy measured along horizontal and vertical axis of the 2.0-at.% Nd:YAG/Cr4+:YAG ceramics (inset shows the composite medium).

The three-beam output micro-laser (Fig. 23a) was realized with the composite, all-ceramics 1.1-at.% Nd:YAG/Cr4+:YAG medium. The guiding line (Pavel et al., 2011b) was designed to assure a distance between a focusing point and the laser axis, c of 4.5 mm and a depth of the focusing point inside the combustion chamber, bc of 9 mm. Air breakdown is illustrated in Fig. 23b, and an automobile electrical spark plug is shown for comparison.

All-Poly-Crystalline Ceramics Nd:YAG/Cr4+:YAG

**4. Conclusion**

Monolithic Micro-Lasers with Multiple-Beam Output 79

Fig. 25. Q-switched laser pulse energy and pump pulse energy versus pump repetition rate. Previous experiments were performed at 5-Hz pump repetition, and temperature of the composite Nd:YAG/Cr4+:YAG ceramics was not controlled. However, higher repetition rates are necessary for operation of a car engine, usually up to 60 Hz. Therefore, variation of laser pulse characteristics was investigated versus the pump repetition rate. Experiments concluded that an increase from 5 to 100 Hz of the pump repetition rate improved the laser pulse energy from 2.37 mJ to 2.51 mJ, i.e. by only a 6% fraction, as shown in Fig. 25. This change required a small (5 to 6%) increase of the minimal pump energy, from ~27 mJ at 5 Hz pump repetition rate to 28.4 mJ at the pump repetition rate of 100 Hz. Variation with temperature of the laser performances was behind this work purpose. Nevertheless, in prior papers (Tsunekane and Taira, 2009; Dascalu and Pavel, 2009; Pavel et al., 2010b) we have measured only a slight increase of the laser pulse energy when temperature of a Nd:YAG laser passively Q-switched by Cr4+:YAG SA (build of discrete, single-crystals components) was increased up to 150oC. Future experiments would also consider testing of the laser to

shock and vibration conditions that are similar to those experienced in a car engine.

performances of internal combustion engines with multi-point ignition.

A passively Q-switched Nd:YAG/Cr4+:YAG giant-pulse emitting micro-laser with up to three-beam output has been realized. The device incorporates a composite, all-ceramics Nd:YAG/Cr4+:YAG monolithic structure that was pumped by similar, independent lines. The laser size is comparable to that of an electrical spark plug, being the first demonstration of this kind of device to the best of our knowledge. Laser pulses with energy of ~2.4 mJ and 2.8-MW peak power at 5-Hz repetition rate were obtained from a 10-mm thick Nd:YAG/Cr4+:YAG ceramics, just such as "giant micro-photonics". Increasing pump repetition rate up to 100 Hz improved the laser pulse energy by 6% and required only a 6% increase of the pump pulse energy compared with operation at 5 Hz. Pulse timing of the laser beams can by controlled by changing the pump energy of each individual line. On the other hand, simultaneous multi-point ignition is possible by less than 5% tuning of the pump energy of each individual pump line. These lasers will enable studies on the

Fig. 23. (a) The passively Q-switched Nd:YAG/Cr4+:YAG micro-laser with three-beam output is shown. (b) Air breakdown in three points is realized.

Various other characteristics of a Q-switched laser pulse, such us the delay time (i.e. the time between the moment when the pump pulse begins and the moment when the laser pulse develops), or the pulse jitter and standard deviation were determined. Figure 24 presents these parameters function of the pump pulse energy. As expected, delay time decreases with the increase of the pump pulse power. Therefore, the use of independent pumping lines allows control of the air breakdown timing by changing the pump energy of each line. Furthermore, real simultaneous ignition in all three points can be obtained by a small (less than 5%) tuning of the pump energy of each individual line. Time jitter is low (2.1 s at the pump energy of 26.7 mJ and 1.0 s at 32-mJ available energy of the pump pulse) and thus it would not have a negative impact on an automobile engine that is ignited by the laser. A second laser pulse was not observed. Nevertheless, increasing the pump pulse duration would enable obtaining of multiple laser pulses (Weinrotter et al., 2005; Tsunekane et al., 2010), which are useful for engine ignition especially if lean fuel-air mixtures are used.

Fig. 24. Time delay of the Q-switched laser pulse, and time jitter and standard deviation versus pump pulse energy, for the repetition rate of 5 Hz (250-s pump pulse duration).

Various other characteristics of a Q-switched laser pulse, such us the delay time (i.e. the time between the moment when the pump pulse begins and the moment when the laser pulse develops), or the pulse jitter and standard deviation were determined. Figure 24 presents these parameters function of the pump pulse energy. As expected, delay time decreases with the increase of the pump pulse power. Therefore, the use of independent pumping lines allows control of the air breakdown timing by changing the pump energy of each line. Furthermore, real simultaneous ignition in all three points can be obtained by a small (less than 5%) tuning of the pump energy of each individual line. Time jitter is low (2.1 s at the pump energy of 26.7 mJ and 1.0 s at 32-mJ available energy of the pump pulse) and thus it would not have a negative impact on an automobile engine that is ignited by the laser. A second laser pulse was not observed. Nevertheless, increasing the pump pulse duration would enable obtaining of multiple laser pulses (Weinrotter et al., 2005; Tsunekane et al., 2010), which are useful for engine ignition especially if lean fuel-air

Fig. 23. (a) The passively Q-switched Nd:YAG/Cr4+:YAG micro-laser with three-beam

Fig. 24. Time delay of the Q-switched laser pulse, and time jitter and standard deviation versus pump pulse energy, for the repetition rate of 5 Hz (250-s pump pulse duration).

output is shown. (b) Air breakdown in three points is realized.

mixtures are used.

Fig. 25. Q-switched laser pulse energy and pump pulse energy versus pump repetition rate.

Previous experiments were performed at 5-Hz pump repetition, and temperature of the composite Nd:YAG/Cr4+:YAG ceramics was not controlled. However, higher repetition rates are necessary for operation of a car engine, usually up to 60 Hz. Therefore, variation of laser pulse characteristics was investigated versus the pump repetition rate. Experiments concluded that an increase from 5 to 100 Hz of the pump repetition rate improved the laser pulse energy from 2.37 mJ to 2.51 mJ, i.e. by only a 6% fraction, as shown in Fig. 25. This change required a small (5 to 6%) increase of the minimal pump energy, from ~27 mJ at 5 Hz pump repetition rate to 28.4 mJ at the pump repetition rate of 100 Hz. Variation with temperature of the laser performances was behind this work purpose. Nevertheless, in prior papers (Tsunekane and Taira, 2009; Dascalu and Pavel, 2009; Pavel et al., 2010b) we have measured only a slight increase of the laser pulse energy when temperature of a Nd:YAG laser passively Q-switched by Cr4+:YAG SA (build of discrete, single-crystals components) was increased up to 150oC. Future experiments would also consider testing of the laser to shock and vibration conditions that are similar to those experienced in a car engine.

#### **4. Conclusion**

A passively Q-switched Nd:YAG/Cr4+:YAG giant-pulse emitting micro-laser with up to three-beam output has been realized. The device incorporates a composite, all-ceramics Nd:YAG/Cr4+:YAG monolithic structure that was pumped by similar, independent lines. The laser size is comparable to that of an electrical spark plug, being the first demonstration of this kind of device to the best of our knowledge. Laser pulses with energy of ~2.4 mJ and 2.8-MW peak power at 5-Hz repetition rate were obtained from a 10-mm thick Nd:YAG/Cr4+:YAG ceramics, just such as "giant micro-photonics". Increasing pump repetition rate up to 100 Hz improved the laser pulse energy by 6% and required only a 6% increase of the pump pulse energy compared with operation at 5 Hz. Pulse timing of the laser beams can by controlled by changing the pump energy of each individual line. On the other hand, simultaneous multi-point ignition is possible by less than 5% tuning of the pump energy of each individual pump line. These lasers will enable studies on the performances of internal combustion engines with multi-point ignition.

All-Poly-Crystalline Ceramics Nd:YAG/Cr4+:YAG

*Appl. Opt.,* Vol. 43, No. 12, pp. 2555-2560.

pp. 1831-1833.

322-327.

014202 (5 pages).

4, pp. 233-240.

2011, paper CA7.1.

Vol. 88, No. 2, pp. 221-226.

*Phys. B,* Vol. 64, No. 3, pp. 301-3.5.

Verlag GmbH & Co. KGaA, Weinheim.

Monolithic Micro-Lasers with Multiple-Beam Output 81

Innocenzi, M. E.; Yura, H. T.; Fincher, C. L. & Fields, R. A. (1990). Thermal modeling of

Jaspan, M. A.; Welford, D. & Russell, J. A. (2004). Passively Q-switched microlaser

Kajava, T. T & Gaeta, A. L. (1997). Intra-cavity frequency-doubling of a Nd:YAG laser passively *Q*-switched with GaAs . *Opt. Commun.,* Vol. 137, No. 1-2, pp. 93-97. Kofler, H.; Tauer, J.; Tartar, G.; Iskra, K.; Klausner, J.; Herdin, G. & Wintner, E. (2007). An

Kroupa, G.; Franz, G. & Winkelhofer, E. (2009). Novel miniaturized high-energy Nd:YAG

Li, S. T.; Zhang, X. Y.; Wang, Q. P.; Li, P.; Chang, J.; Zhang, X. L. & Cong, Z. H. (2007).

Liu, J.; Yang, J. & He, J. (2004). Diode-pumped passively Q-switched intracavity frequency doubled Nd:GdVO4/KTP green laser. *Opt. & Laser Techn.,* Vol. 36, No. 1, pp. 31-33. Ma, J. X.; Alexander, D. R. & Poulain, D. E. (1998). Laser spark ignition and combustion characteristics of methane-air mixtures. *Comb. Flame,* Vol. 112, No. (4), pp. 492-506. Morsy, M. H.; Ko, Y. S.; Chung, S. H. & Cho, P. (2001). Laser-induced two-point ignition of premixture with a single-shot laser. *Comb. Flame,* Vol. 125, No. 4, pp. 724-727. Ostroumov, V. G.; Heine, F.; Kück, S; Huber, G; Mikhailov, V. A. & Shcherbakov, I. A.

Paschotta, R. (2008). Encyclopedia of Laser Physics and Technology. Handbook/Reference

Pavel, N.; Saikawa, J. & Taira, T. (2001b). Diode end-pumped passively Q-switched Nd:YAG

Pavel, N.; Saikawa, J.; Kurimura, S. & Taira, T. (2001a). High average power diode end-

Pavel, N.; Tsunekane, M. & Taira, T. (2010a). High peak-power passively Q-switched all-ceramics

Pavel, N.; Tsunekane, M.; Kanehara, K. & Taira, T (2011a). Composite all-ceramics, passively

Pavel, N; Tsunekane, M. & Taira, T. (2011b). Passively Q-switched Nd:YAG/Cr4+:YAG all-

absorber. *Jap. J. Appl. Phys.,* Vol. 40, Part. 1, No. 3A, pp. 1253-1259.

*Conference*, Baltimore, Maryland, USA, 1-6 May, 2011, paper CMP1. Pavel, N; Tsunekane, M. & Taira, T. (2010b). Enhancing performances of a passively Q-

coupler. *Opt. Lett.* Vol. 35, No. 10, pp. 1617-1619.

continuous-wave end-pumped solid-state lasers. *Appl. Phys. Lett.* Vol. 56, No. 19,

performance in the presence of pump-induced bleaching of the saturable absorber.

innovative solid-state laser for engine ignition. *Laser Phys. Lett.,* Vol. 4, No. 4, pp.

laser for spark ignition in internal combustion engines. *Opt. Eng.,* Vol. 48, No. 1, art.

Modeling of Q-switched lasers with top-hat pump beam distribution. *Appl. Phys. B,*

(1997). Intracavity frequency-doubled diode-pumped Nd: LaSc3(BO3)4 lasers. *Appl.*

Book. ISBN-10: 3-527-40828-2, ISBN-13: 978-3-527-40828-3; 2008 WILEY-VCH

laser intra-cavity frequency doubled by LBO crystal. *Opt. Commun.,* Vol. 195, No. 1-

pumped composite Nd:YAG laser passively Q-switched by Cr4+:YAG saturable

Nd:YAG/Cr4+:YAG lasers. Proceedings SPIE, Vol. 7469, Micro- to Nano-Photonics II - ROMOPTO 2009 Conference, 31 August – 03. Sept., 2009, Sibiu, Romania; paper 746903.

Q-switched Nd:YAG/Cr4+:YAG monolithic micro-laser with two-beam output for multi-point ignition. In: *CLEO 2011, Laser Science to Photonics Applications*

switched Nd:YAGCr4+:YAG microlaser with a volume Bragg grating output

ceramics, composite, monolithic micro-lasers with multi-beam output for laser ignition. In: *CLEO Europe - EQEC 2011 Conference*, Münich, Germany, 22-26 May,

#### **5. Acknowledgment**

This work was financed by Japan Science and Technology Agency, and partially supported by DENSO Company, Japan. Permanent support from and fruitful discussions with Mr. Kenji Kanehara of Nippon Soken. Inc. is acknowledged.The authors thank Mr. Nobuo Mizutani of IMS Equipment Development Division for the help with the laser module design. N. Pavel acknowledges partial support of the Romanian Ministry of Education and Research, project 12106/01.10.2008.

#### **6. References**

This work was financed by Japan Science and Technology Agency, and partially supported by DENSO Company, Japan. Permanent support from and fruitful discussions with Mr. Kenji Kanehara of Nippon Soken. Inc. is acknowledged.The authors thank Mr. Nobuo Mizutani of IMS Equipment Development Division for the help with the laser module design. N. Pavel acknowledges partial support of the Romanian Ministry of Education and

Agnesi, A.; Dell'Acqua, S. & Reali, G. C. (1997). 1.5 Watt passively Q-switched diodepumped cw Nd:YAG laser. *Opt. Commun.,* Vol. 133, No. 1-6, pp. 211-215. An, J; Zhao, S. Z.; Li, G. Q.; Yang, K. J.; Sun, Y. M.; Li, D. C.; Wang, J. & Li, M. (2006). Doubly

generator-optical parametric amplifier. *Opt. Lett*. Vol. 25, No. 8, pp. 557-559. Bibeau, C.; Beach, R. J.; Mitchell, S. C.; Emanuel, M. A.; Skidmore, J.; Ebbers, C. A.; Sutton, S.

Cheng, K.; Zhao, S.; Li, Y.; Li, G.; Li, D., Yang, K.; Zhang, G.; Ge, H. & Yu, Z. (2011). Diode-

and GaAs saturable absorbers. *Opt. Commun.,* Vol. 284, No. 1, pp. 344-349. Dascalu, T. & Pavel, N. (2009). High-temperature operation of a diode-pumped passively Qswitched Nd:YAG/Cr4+:YAG laser. *Laser Phys.* Vol. 19, No. 11, pp. 2090-2095. Degnan, J. (1995). Optimization of passively Q-switched lasers. *IEEE J. Quantum Electron*.,

Dong, J.; Shirakawa, A.; Takaichi, K.; Ueda, K.; Yagi, H.; Yanagitani, T. & Kaminskii, A.A.

Dong, J.; Ueda, K.; Shirakawa, A.; Yagi, H.; Yanagitani, T. & Kaminskii, A.A. (2007).

Eimerl, D. (1987). High average power harmonic generation. *IEEE J. Quantum Electron.,* Vol.

Feng, Y.; Lu, J.; Takaichi, K.; Ueda, K.; Yagi, H.; Yanagitani, T. & Kaminskii, A.A. (2004).

Hanson, F. (1995). Improved laser performance at 946 and 473 nm from a composite

Honea, E. C.; Ebbers, C. A.; Beach, R. J.; Speth, J. A.; Skidmore, J. A.; Emanuel, M. A. &

Huang, S.; L.; Tsui; T.; Y., Wang, C. H. & Kao, F. J. (1999). Timing jitter reduction of a passively Q-switched laser, *Jpn. J. Appl. Phys.*, Vol. 38, Part 2, No. 3A, pp. L239-241.

Nd:Y3Al5O12 rod. *Appl. Phys. Lett.*, Vol. 66, No. 26, pp. 3579-3551.

passively Q-switched intracavity-frequency-doubling Nd3+:YAG/KTP green laser with GaAs and Cr4+: YAG saturable absorbers. *Opt. Eng*., Vol. 45, No.12, Art. No. 124202. Aniolek, K. W.; Schmitt, R. L.; Kulp, T. J.; Richman, B. A.; Bisson, S.E. & Powers, P. E. (2000).

Microlaser-pumped periodically poled lithium niobate optical parametric

B. & Jancaitis, K. S. (1998). High-average-power 1-m performance and frequency conversion of a diode-end-pumped Yb:YAG laser. *IEEE J. Quantum Electron.,* Vol.

pumped doubly passively Q-switched Nd:LuVO4/KTP green laser with Cr4+:YAG

(2006). All-ceramic passively Q-switched Yb:YAG/Cr4+:YAG microchip laser.

Composite Yb:YAG/Cr4+:YAG ceramics picosecond microchip lasers. *Opt. Express*,

Passively *Q*-switched ceramic Nd3+:YAG/Cr4+:YAG lasers. *Appl. Opt.,* Vol. 43, No.

Payne, S. A. (1998). Analysis of an intracavity-doubled diode-pumped *Q*-switched Nd:YAG laser producing more than 100 W of power at 0.532 mm. *Opt. Lett.*, Vol. 23,

**5. Acknowledgment**

**6. References**

Research, project 12106/01.10.2008.

34, No. 10, pp. 2010-2019.

Vol. 31, No. 11, pp. 1890-1901.

Vol. 15, No. 22, pp. 14516-14523.

23, No. 5, pp. 575-592.

No. 15, pp. 1203-1205.

14, pp. 2944-2947.

*Electron. Lett.,* Vol. 42, No. 20, pp. 1154-1156.

**1. Introduction**

width of these lines is on the order of 6 *E E*/ ~ 10 .

absorption cross section near the resonant energy is2:

respectively, and

**5**

*2AOSense, Inc.*

(2)

*USA*

**Laser Pulses for Compton**

Sheldon S. Q. Wu1, Miroslav Y. Shverdin2, Felicie Albert1 and Frederic V. Hartemann1

(1)

Nuclear Resonance Fluorescence (NRF) is an isotope specific process in which a nucleus, excited by gamma-rays, radiates high energy photons at a specific energy. This process has been well known for several decades, and has potential high impact applications in homeland security, nuclear waste assay, medical imaging and stockpile surveillance, among other areas of interest. Although several successful experiments have demonstrated NRF detection with broadband bremsstrahlung gamma-ray sources1, NRF lines are more efficiently detected when excited by narrowband gamma-ray sources. Indeed, the effective

NRF lines are characterized by a very narrow line width and a strong absorption cross section. For actinides such as uranium, the NRF line width is due to the intrinsic line width

> *R kT*

where *ER* is the resonant energy (usually MeV), *M* the mass of the nucleus, *k* the Boltzmann constant, *c* the speed of light and *Teff* the effective temperature of the material. This model is valid as long as 0 2*kTD* where *TD* is the Debye temperature. Because in most cases 0 1 eV, the total width is just determined by the thermal motion of the atoms. Debye temperatures for actinides are usually in the range 100-200 K. Within this context, the NRF

*E*

3/2 0 2 1 ( ) exp 2 21

are on the order of tens of meV wide with an absorption cross section around 10 barns.

*i J E E E J*

where *i J* and *j J* are the total angular momentum for the ground and excited states

2 *eff*

2 2

*j R*

the radiation wavelength. Typically, strong M1 resonances at MeV energies

*Mc*

0 connecting the excited state to the ground state and to the Doppler width,

**Scattering Light Sources**

*1Lawrence Livermore National Laboratory*

### **Laser Pulses for Compton Scattering Light Sources**

Sheldon S. Q. Wu1, Miroslav Y. Shverdin2, Felicie Albert1 and Frederic V. Hartemann1 *1Lawrence Livermore National Laboratory 2AOSense, Inc. USA*

#### **1. Introduction**

82 Laser Systems for Applications

Phuoc, T. X. & White, F. P. (1999). Laser-induced spark ignition of CH4/Air mixtures. *Comb.*

Phuoc, T. X. (2000). Single-point versus multi-point laser ignition: Experimental measurements of combustion times and pressures. *Comb. Flame,* Vol. 122, No. 4, pp. 508-510. Sakai, H.; Kan, H. & Taira T., (2008). >1 MW peak power single-mode high-brightness passively Q-switched Nd3+:YAG microchip laser. *Opt. Express,* Vol. 16, No. 24, pp. 19891-19899. Sato, Y. & Taira, T. (2006). The studies of thermal conductivity in GdVO4, YVO4, and

Shimony, Y.; Burshtein, Z. & Kalisky, Y. (1995). Cr4+:YAG as passive Q-switch and Brewster

Song, J.; Li, C.; Kim, N. S. & Ueda, K. I. (2000). Passively Q-switched diode-pumped

Taira, T. (2007). RE3+-ion-deped YAG ceramic lasers. *IEEE J. Sel. Top. Quantum Electron.*, Vol.

Tang, D. Y.; Ng, S. P.; Qin, L. J. & Meng, X. L. (2003). Deterministic chaos in a diode-pumped

Tsunekane, M. & Taira, T. (2009). Temperature and polarization dependences of Cr:YAG

Tsunekane, M.; Inohara, T.; Ando, A.; Kanehara, K. & Taira, T. (2008). High peak power,

Tsunekane, M.; Inohara, T.; Ando, A.; Kido, N.; Kanehara, K. & Taira, T. (2010). High peak

Tsunekane, M.; Taguchi, N.; Kasamatsu, T. & Inaba, H. (1997). Analytical and experimental

Weinrotter, M.; Kopecek, H.; Tesch, M.; Wintner, E; Lackner, M. & Winter, F. (2005a). Laser

Zayhowski, J.; Cook, C. C.; Wormhoudt, J. & J. H. Shorter, J. H. (2000). Passively Q-switched

Zayhowsky, J. J. & Dill III, C. (1994). Diode-pumped passively Q-switched picosecond

Zhang, X.; Zhao, S., Wang, Q., Zhang, Q., Sun, L. & Zhang, S. (1997). Optimization of Cr4+-

Zhang, X.; Zhao, S.; Wang, Q.; Ozygus, B. & Weber, H. (2000). Modeling of passively Q-

switched lasers. *J. Opt. Soc. Am. B*., Vol. 17, No. 7, pp. 1166-1175.

America, 2008); 27-30 January, 2008, Nara, Japan, paper MB4.

pressure," *Exp. Therm. Fluid Science,* Vol. 29, No. 8, pp. 569-577.

America, 2000), Davos, Switzerland, Feb. 2000, paper WB4.

microchip lasers. *Opt. Lett*., Vol. 19, No. 18, pp. 1427-1429.

Y3Al5O12 measured by quasi-one-dimensional flash method. *Opt. Express*, Vol. 14,

plate in a pulsed Nd-laser. *IEEE J. Quantum Electron.,* Vol. 31, No. 10, pp. 1738-1741.

continuous-wave Nd:YAG–Cr4+:YAG laser with high peak power and high pulse

Nd:YAG laser passively Q switched by a Cr4+:YAG crystal, *Opt. Lett*., Vol. 28, No. 5,

transmission for passive Q-switching. In: *Conference on Lasers and Electro-Optics/International Quantum Electronics Conference*, Baltimore, Maryland, USA, May

passively Q-switched Cr:YAG/Nd:YAG micro-laser for ignition of engines. In: *Advanced Solid-State Photonics*, OSA Technical Digest Series (CD) (Optical Society of

power, passively Q-switched microlaser for ignition of engines. *IEEE J. Quantum*

studies on the characteristics of composite solid-state laser rods in diode-endpumped geometry. *IEEE J. Sel. Top. Quantum Electron*., Vol. 3, No. 1, pp. 9-18. Weinrotter, M.; Kopecek, H. & Wintner, E. (2005b). Laser ignition of engines. *Laser Phys.,*

ignition of ultra-lean methane/hydrogen/air mixtures at high temperature and

214.8-nm Nd:YAG/Cr4+:YAG microchip-laser system for the detection of NO. In *Advanced Solid State Lasers*, OSA Technical Digest Series (Optical Society of

doped saturable-absorber Q-switched lasers, *IEEE J. Quantum Electron*., Vol. 33, No.

*Flame,* Vol. 119, No. 3, pp. 203-216.

energy. *Appl. Opt*., Vol. 39, No. 27, pp. 4954-4958.

No. 22, pp. 10528-10536.

13, No. 3, pp.798-809.

31 - June 5, 2009, paper JTuD8.

*Electron.,* Vol. 46, No. 2, pp. 277-284.

Vol. 15, No. 7, pp. 947-953.

12, pp.2286-2294.

pp. 325-327.

Nuclear Resonance Fluorescence (NRF) is an isotope specific process in which a nucleus, excited by gamma-rays, radiates high energy photons at a specific energy. This process has been well known for several decades, and has potential high impact applications in homeland security, nuclear waste assay, medical imaging and stockpile surveillance, among other areas of interest. Although several successful experiments have demonstrated NRF detection with broadband bremsstrahlung gamma-ray sources1, NRF lines are more efficiently detected when excited by narrowband gamma-ray sources. Indeed, the effective width of these lines is on the order of 6 *E E*/ ~ 10 .

NRF lines are characterized by a very narrow line width and a strong absorption cross section. For actinides such as uranium, the NRF line width is due to the intrinsic line width 0 connecting the excited state to the ground state and to the Doppler width,

$$
\Delta = E\_R \sqrt{\frac{kT\_{eff}}{Mc^2}}\tag{1}
$$

where *ER* is the resonant energy (usually MeV), *M* the mass of the nucleus, *k* the Boltzmann constant, *c* the speed of light and *Teff* the effective temperature of the material. This model is valid as long as 0 2*kTD* where *TD* is the Debye temperature. Because in most cases 0 1 eV, the total width is just determined by the thermal motion of the atoms. Debye temperatures for actinides are usually in the range 100-200 K. Within this context, the NRF absorption cross section near the resonant energy is2:

$$
\sigma(E) = \pi^{3/2} \left(\frac{\lambda}{2\pi}\right)^2 \left(\frac{2J\_j + 1}{2J\_i + 1}\right) \frac{\Gamma\_0}{\Lambda} \exp\left[-\left(\frac{E - E\_R}{\Lambda}\right)^2\right] \tag{2}
$$

where *i J* and *j J* are the total angular momentum for the ground and excited states respectively, and the radiation wavelength. Typically, strong M1 resonances at MeV energies are on the order of tens of meV wide with an absorption cross section around 10 barns.

Laser Pulses for Compton Scattering Light Sources 85

radiate energy at the same frequency as that of the incident wave. The power radiated per unit solid angle in some observation direction **q**ˆ can be expressed in the cgs system of units as

> 2 2 2 2 0 2 ˆ ˆ . 4

From a scattering perspective, it is convenient to introduce a scattering cross section,

Energy radiated/unit solid angle/unit time Incident energy flux/unit area/unit time

> 0 1 ˆ ˆ *d dP r*

2 25 2

where 2 2 *r e mc* 0 / is the classical electron radius. The result in Eq. (5) is valid for radiation polarized along a specific direction **e**ˆ . For unpolarized incident radiation, the scattering

> 2 0 2 1 cos

2

0 8 6.65 10 3 *T r cm*

The Thomson cross section is only valid at low energies where the momentum transfer of the radiation to the particle can be ignored and the scattered waves has the same frequency as the incident waves. In 1923, Arthur Compton at Washington University in St. Louis found that the scattered radiation actually had a lower frequency than the incident waves. He further showed that if one adopts Einstein's ideas about the light quanta, then conservation laws of energy and momentum in fact led quantitatively to the observed frequency of the

energy 2 *mc* , part of the initial photon momentum is transferred to the particle. This would become known as the Compton effect. For this effect to be neglected, we must have

 *mc* in the frame where the charged particle is initially at rest. If this condition is violated, important modifications to Eq. (6) must be made due to the Compton effect. Quantum electrodynamical calculations give the modified scattering cross section for

*dP e cE*

*d Sd*

> *d r d*

total scattering cross section obtained by integration over all solid angles is

is the scattering angle between the incident ˆ

scattered waves. In other words, when the photon energy

unpolarized radiation, in the initial rest frame of the charged particle,

or better known as the Thomson cross section.

 *mc*

*d*

(4)

**e q** (5)

(6)

(7)

**k** and scattered **q**ˆ directions. The

approaches the particle's rest

. Thus the scattering cross section is

2 2

*m* **x E** . Such an accelerated motion will

**e q** (3)

charge is nonrelativistic, the particle acceleration is *e*

*d d*

where the incident flux is 2 *S cE* 0 / 4

defined by

cross section is

where

2

Currently, Compton scattering is among the only physical processes capable of producing a narrow bandwidth radiation (below 1%) at gamma-ray energies, using state-of-the art accelerator and laser technologies. In Compton scattering light sources, a short laser pulse and a relativistic electron beam collide to yield tunable, monochromatic, polarized gammaray photons. Several projects have recently utilized Compton scattering to conduct NRF experiments: Duke University2, Japan3 and Lawrence Livermore National Laboratory (LLNL)4. In particular, LLNL's Thomson-Radiated Extreme X-rays (T-REX) project demonstrated isotope specific detection of low-Z and low density materials (7Li) shielded behind high-Z and high-density materials (Pb, Al).

The T-REX device and the MEGa-ray machine currently in progress at LLNL are laser-based Compton sources that produce quasi-monochromatic, tunable, polarized gamma-rays via Compton scattering of energetic short duration laser pulses from high-brightness relativistic electron bunches. The detection of low-Z and low-density objects shielded by a high-Z dense material is a long-standing problem that has important applications ranging from homeland security and nonproliferation5 to advanced biomedical imaging and paleontology. X-rays are sensitive to electron density, and x-ray radiography yields poor contrast in these situations. Within this context, NRF offers a unique approach to the socalled inverse density radiography problem. Since NRF is a process in which nuclei are excited by discrete high-energy (typically MeV) photons and subsequently re-emit gammarays at discrete energies determined by the structure of the nucleus, and the resonance structure is determined by the number of neutrons and protons present in the nucleus, NRF provides isotope-specific detection and imaging capability6. Thus Compton sources have a lot of potential high profile applications, since NRF can be used for nuclear waste assay and management, homeland security and stockpile stewardship, among others. Other processes and physics can be investigated with narrowband gamma-rays, such as near-threshold photofission, giant dipole resonances, and detailed studies of nuclear structure.

This chapter presents, primarily within the context of NRF-based applications, the theoretical and experimental aspects of a mono-energetic gamma-ray Compton source. In Section 2, we will introduce the fundamental concepts and basic physics involved in Compton scattering. In Section 3, we will discuss concepts necessary for understanding laser technology underlying high brightness Compton sources. Section 4 will cover amplification applications and review of some laser pulse measurement techniques. Nonlinear conversion will be discussed in Section 5. Finally, Section 6 contains descriptions of some simulation tools.

#### **2. Mechanics of compton scattering**

An electromagnetic wave that is incident upon a free charged particle will cause it to oscillate and so emit radiation. The accelerated motion of a charged particle will generate radiation in directions other than that of the incident wave. This process may be described as Compton scattering of the incident radiation. In this section, we will review the elementary physics of this interaction.

#### **2.1 Overview**

Consider an electromagnetic wave with the electric field 0 **E e kx** ˆ*E t* cos( ) is incident upon a charged particle of charge *e* and mass *m*. Assuming that the resultant motion of the

Currently, Compton scattering is among the only physical processes capable of producing a narrow bandwidth radiation (below 1%) at gamma-ray energies, using state-of-the art accelerator and laser technologies. In Compton scattering light sources, a short laser pulse and a relativistic electron beam collide to yield tunable, monochromatic, polarized gammaray photons. Several projects have recently utilized Compton scattering to conduct NRF experiments: Duke University2, Japan3 and Lawrence Livermore National Laboratory (LLNL)4. In particular, LLNL's Thomson-Radiated Extreme X-rays (T-REX) project demonstrated isotope specific detection of low-Z and low density materials (7Li) shielded

The T-REX device and the MEGa-ray machine currently in progress at LLNL are laser-based Compton sources that produce quasi-monochromatic, tunable, polarized gamma-rays via Compton scattering of energetic short duration laser pulses from high-brightness relativistic electron bunches. The detection of low-Z and low-density objects shielded by a high-Z dense material is a long-standing problem that has important applications ranging from homeland security and nonproliferation5 to advanced biomedical imaging and paleontology. X-rays are sensitive to electron density, and x-ray radiography yields poor contrast in these situations. Within this context, NRF offers a unique approach to the socalled inverse density radiography problem. Since NRF is a process in which nuclei are excited by discrete high-energy (typically MeV) photons and subsequently re-emit gammarays at discrete energies determined by the structure of the nucleus, and the resonance structure is determined by the number of neutrons and protons present in the nucleus, NRF provides isotope-specific detection and imaging capability6. Thus Compton sources have a lot of potential high profile applications, since NRF can be used for nuclear waste assay and management, homeland security and stockpile stewardship, among others. Other processes and physics can be investigated with narrowband gamma-rays, such as near-threshold

photofission, giant dipole resonances, and detailed studies of nuclear structure.

This chapter presents, primarily within the context of NRF-based applications, the theoretical and experimental aspects of a mono-energetic gamma-ray Compton source. In Section 2, we will introduce the fundamental concepts and basic physics involved in Compton scattering. In Section 3, we will discuss concepts necessary for understanding laser technology underlying high brightness Compton sources. Section 4 will cover amplification applications and review of some laser pulse measurement techniques. Nonlinear conversion will be discussed in Section 5. Finally, Section 6 contains

An electromagnetic wave that is incident upon a free charged particle will cause it to oscillate and so emit radiation. The accelerated motion of a charged particle will generate radiation in directions other than that of the incident wave. This process may be described as Compton scattering of the incident radiation. In this section, we will review the

upon a charged particle of charge *e* and mass *m*. Assuming that the resultant motion of the

is incident

Consider an electromagnetic wave with the electric field 0 **E e kx** ˆ*E t* cos( )

behind high-Z and high-density materials (Pb, Al).

descriptions of some simulation tools.

elementary physics of this interaction.

**2.1 Overview**

**2. Mechanics of compton scattering**

charge is nonrelativistic, the particle acceleration is *e m* **x E** . Such an accelerated motion will radiate energy at the same frequency as that of the incident wave. The power radiated per unit solid angle in some observation direction **q**ˆ can be expressed in the cgs system of units as

$$\frac{dP}{d\Omega} = \frac{cE\_0}{4\pi} \left(\frac{e^2}{mc^2}\right)^2 \left(\hat{\mathbf{e}} \times \hat{\mathbf{q}}\right)^2. \tag{3}$$

From a scattering perspective, it is convenient to introduce a scattering cross section, defined by

$$\frac{d\sigma}{d\Omega} = \frac{\text{Energy radiated/unit solid angle/unit time}}{\text{Incident energy flux/unit area/unit time}}\tag{4}$$

where the incident flux is 2 *S cE* 0 / 4. Thus the scattering cross section is

$$\frac{d\sigma}{d\Omega} = \frac{1}{S} \frac{dP}{d\Omega} = r\_0^2 \left(\hat{\mathbf{e}} \times \hat{\mathbf{q}}\right)^2\tag{5}$$

where 2 2 *r e mc* 0 / is the classical electron radius. The result in Eq. (5) is valid for radiation polarized along a specific direction **e**ˆ . For unpolarized incident radiation, the scattering cross section is

$$\frac{d\sigma}{d\Omega} = \frac{r\_0^2}{2} \left( 1 + \cos^2 \theta \right) \tag{6}$$

where is the scattering angle between the incident ˆ **k** and scattered **q**ˆ directions. The total scattering cross section obtained by integration over all solid angles is

$$
\sigma\_T = \frac{8\pi}{3} r\_0^{-2} = 6.65 \times 10^{-25} cm^2 \tag{7}
$$

or better known as the Thomson cross section.

The Thomson cross section is only valid at low energies where the momentum transfer of the radiation to the particle can be ignored and the scattered waves has the same frequency as the incident waves. In 1923, Arthur Compton at Washington University in St. Louis found that the scattered radiation actually had a lower frequency than the incident waves. He further showed that if one adopts Einstein's ideas about the light quanta, then conservation laws of energy and momentum in fact led quantitatively to the observed frequency of the scattered waves. In other words, when the photon energy approaches the particle's rest energy 2 *mc* , part of the initial photon momentum is transferred to the particle. This would become known as the Compton effect. For this effect to be neglected, we must have 2 *mc* in the frame where the charged particle is initially at rest. If this condition is violated, important modifications to Eq. (6) must be made due to the Compton effect. Quantum electrodynamical calculations give the modified scattering cross section for unpolarized radiation, in the initial rest frame of the charged particle,

Laser Pulses for Compton Scattering Light Sources 87

fields are experimentally measured. To facilitate this, we will need generalizations of Eqs. (8) and (9) valid in an arbitrary inertial frame. The expressions are well known but their

ˆ 1 ˆ 1 1 ˆ ˆ

 **β k β q kq**

*pp p p X*

are products of the 4-vectors *k c* / ,

 is the 4-momentum of the charged particle after the scattering event. When the photon polarizations are not observed, this expression must be summed over the final

2 22 22 22 22 1 2

 (14)

*mc mc mc mc X X*

3 0 ( ) *e d d p k q q nn d d dqd d xd d m k*

simply to preserve energy-momentum conservation relation expressed in Eq. (12).

The above describes the interaction of a single electron with an incident electromagnetic wave. For realistic laser-electron interactions, one needs to take into account the appropriate photon and electron phase space. The most useful expression to describe the source is typically the local differential brightness, which may be written in terms of the number of scattered photons, *N*, per unit volume per unit time per solid angle per energy interval8:

**β k**

0 2 2 ˆ 2 1

*d r X*

2 2

1 1

 (11)

(12)

. The spin-averaged relativistic

 **k** , *q c* / , **q**

2

(13)

represents the relative velocity between

. The delta function in 0 *q q* is required

, it is

3 3

**p k** (15)

**k** represent the electron beam phase space density and laser photon

*m k*

Let us briefly apply some of theory discussed above to a realistic Compton source. Consider a Compton scattering geometry in Figure 2. Here, **p u** *m c*0 . Let us assume 1

are the incident and scattered 4-polarizations, respectively;

derivation is slightly involved, so we will simply quote the results7:

*d*

where **β** is the initial particle velocity and 2

2

and

and averaged over the initial polarizations, giving

2 *pol*

1 1 2

>

6

*N*

*t*

invariant *X* can be expressed as

 **β** ;

, *q p*

where *k p*

*p p k q*

and *p mc* ,

where 3 *n de* **p** and 3 *n d*

phase space density, respectively. The factor *p k*

the electron and photon and is equal to *c* for 1

$$\frac{d\sigma}{d\Omega} = \frac{r\_0}{2} \left(\frac{\alpha'}{\alpha}\right)^2 \left(\frac{\alpha}{\alpha'} + \frac{\alpha'}{\alpha} - \sin^2\theta\right) \tag{8}$$

where and are the incident and scattered frequencies, respectively. They are related by the Compton formula:

$$\frac{\alpha'}{\alpha o} = \frac{1}{1 + \frac{\hbar \alpha o}{mc^2} (1 - \cos \theta)}\tag{9}$$

The total cross section for scattering is then modified to

$$\sigma\_{\rm KN} = 2\pi \eta\_0^2 \left\{ \frac{1+\varepsilon}{\varepsilon^3} \left[ \frac{2\varepsilon \left(1+\varepsilon\right)}{1+2\varepsilon} - \ln\left(1+2\varepsilon\right) \right] + \frac{\ln\left(1+2\varepsilon\right)}{2\varepsilon} - \frac{1+3\varepsilon}{\left(1+2\varepsilon\right)^2} \right\} \tag{10}$$

where 2 /*mc* is the normalized energy of the incident light. This is called the Klein-Nishina cross section, first derived by Klein and Nishina in 1928. For 1 , Eq. (8) reduces to the Thomson cross section of Eq. (6). For scattering by high energy photons, the total cross section decreases from the Thomson value, as shown in Figure 1.

Fig. 1. Plot of the Klein-Nishina expression for the Compton scattering cross section. For small values of , the Compton cross section approaches the Thomson limit. At large values of , the Compton cross section approaches the asymptotic solution 3 1 8 2 ~ (ln 2 ) *T KN* .

While the scattering expressions are simple in the frame where the particle is initially at rest, it is often desirable to do calculations in the laboratory frame where the electromagnetic

0 2 sin

1 1 1 cos *mc*

(9)

(10)

, Eq. (8) reduces

8 2 ~ (ln 2 ) *T*

 .

*KN*

are the incident and scattered frequencies, respectively. They are related

2

/*mc* is the normalized energy of the incident light. This is called the Klein-

1 2 2 1 2

0 3 2 1 2 1 ln 1 2 1 3 2 ln 1 2

to the Thomson cross section of Eq. (6). For scattering by high energy photons, the total cross

Fig. 1. Plot of the Klein-Nishina expression for the Compton scattering cross section. For

, the Compton cross section approaches the asymptotic solution 3 1

While the scattering expressions are simple in the frame where the particle is initially at rest, it is often desirable to do calculations in the laboratory frame where the electromagnetic

, the Compton cross section approaches the Thomson limit. At large values

(8)

2 2

2

The total cross section for scattering is then modified to

 section decreases from the Thomson value, as shown in Figure 1.

Nishina cross section, first derived by Klein and Nishina in 1928. For 1

2

*KN r*

where 2

small values of

of  *d r d*

where

 and

by the Compton formula:

fields are experimentally measured. To facilitate this, we will need generalizations of Eqs. (8) and (9) valid in an arbitrary inertial frame. The expressions are well known but their derivation is slightly involved, so we will simply quote the results7:

$$\frac{d\sigma}{d\Omega} = \frac{r\_0^{\prime 2} X}{2\gamma^2 \left(1 - \oint \cdot \hat{\mathbf{k}}\right)^2} \left(\frac{\rho \prime}{\rho \nu}\right)^2 \tag{11}$$

$$\frac{\alpha^{\prime}}{\alpha^{\prime \alpha}} = \frac{1 - \mathfrak{P} \cdot \hat{\mathbf{k}}}{1 - \mathfrak{P} \cdot \hat{\mathbf{q}} + \frac{\mathcal{E}}{\mathcal{T}} \left(1 - \hat{\mathbf{k}} \cdot \hat{\mathbf{q}}\right)}\tag{12}$$

where **β** is the initial particle velocity and 2 1 1 . The spin-averaged relativistic

invariant *X* can be expressed as

$$X = \frac{1}{2} \left( \frac{\kappa}{\kappa'} + \frac{\kappa'}{\kappa} \right) - 1 + 2 \left( \epsilon\_{\mu} \pi^{\mu} - \frac{\epsilon\_{\mu} p^{\mu} \pi\_{\nu} p^{\nu \prime}}{\kappa} + \frac{\epsilon\_{\mu} p^{\prime \mu} \pi\_{\nu} p^{\nu}}{\kappa'} \right)^{2} \tag{13}$$

where *k p* , *q p* are products of the 4-vectors *k c* / , **k** , *q c* / , **q** and *p mc* , **β** ; and are the incident and scattered 4-polarizations, respectively; *p p k q* is the 4-momentum of the charged particle after the scattering event. When the photon polarizations are not observed, this expression must be summed over the final and averaged over the initial polarizations, giving

$$\overline{X} = \frac{1}{2} \sum\_{pol} X = \frac{\kappa}{\kappa'} + \frac{\kappa'}{\kappa} + 2 \left( \frac{m^2 c^2}{\kappa} - \frac{m^2 c^2}{\kappa'} \right) + \left( \frac{m^2 c^2}{\kappa} - \frac{m^2 c^2}{\kappa'} \right)^2 \tag{14}$$

The above describes the interaction of a single electron with an incident electromagnetic wave. For realistic laser-electron interactions, one needs to take into account the appropriate photon and electron phase space. The most useful expression to describe the source is typically the local differential brightness, which may be written in terms of the number of scattered photons, *N*, per unit volume per unit time per solid angle per energy interval8:

$$\frac{d^6N}{dq d\Omega d^3x dt} = \frac{d\sigma}{d\Omega} \delta(q - q\_0) n\_e n\_\lambda \frac{p\_\mu k^\mu}{\gamma m \hbar k} d^3 \mathbf{p} d^3 \mathbf{k} \tag{15}$$

where 3 *n de* **p** and 3 *n d* **k** represent the electron beam phase space density and laser photon phase space density, respectively. The factor *p k m k* represents the relative velocity between the electron and photon and is equal to *c* for 1 . The delta function in 0 *q q* is required simply to preserve energy-momentum conservation relation expressed in Eq. (12). Let us briefly apply some of theory discussed above to a realistic Compton source. Consider

a Compton scattering geometry in Figure 2. Here, **p u** *m c*0 . Let us assume 1 , it is

Laser Pulses for Compton Scattering Light Sources 89

*mc e* / . For brevity, QED units are adopted for the remainder of this section: charge is measured in units of the electron charge *e*, mass in units of the electron rest mass *m* , length

By using the derived nonlinear 4-velocity in conjunction with the energy-momentum

() , *u k q kq c*

0

*AA uA uAk k q k q u k*

0

*k uk u AA q kq u k*

In the geometry described in Figure 3, Eq. (12) can be rewritten using Eq. (19) as:

*k A A*

2 ( ) 2 , *c*

0 . 2 *c*

2

*A*

Here the small angle is different for each electron and represents the emittance of the

variation of *q* as a function of all the different parameters in Eq. (20), one finds for on-axis

/ /,

*qq kk*

*q q q q q q*

1 / , 4 / 2 /, / ,

/ . 1

*q q A*

*u k*

This new relation may be understood as a modified form of the Compton formula, now including the nonlinear ponderomotive force of the laser field. It is important to note that the general form of the laser potential is, for a linearly polarized plane wave, 0 ( ( )cos( ),0,0) *A g*

is the laser 4-potential measured in units of

, and time in units of

the laser pulse envelope.  By looking at the

2 cos *c*

(17)

, and the vacuum dispersion

(18)

(19)

(21)

 ,

(20)

*c* /*c* .

where 0 *u*

conservation relation, rewritten as:

one immediately obtains:

relation, *k k* 0

where  is the initial 4-velocity and *A*

0

, simplifies to:

is the phase and *g*( )

electron beam. Note also that *A A*

observation, 0

:

which after applying the Lorentz gauge condition, *k A* 0

0 0

*q u*

consequence of the inhomogeneous nature of the laser electrical field.

cos 1 cos

in units of the reduced electron Compton wavelength *c mc*

evident from Eq. (12) that the scattered photon energy maximizes for on-axis scattering events, with , 0 . For a relativistic electron with 1 , we may use the approximation, 2 1 1 2 , and find that Eq. (12) simplifies to 2 4 . In the case of highly relativistic electrons, the scattered radiation can be many orders of magnitude more energetic than the incident radiation! One can intuitively understand why this occurs. The counter-propagating incident radiation in the electron rest frame is up-shifted by a factor 2 due to the relativistic Doppler effect. The scattered radiation is emitted at approximately the same energy in the electron frame. The Lorentz transformation back to the laboratory frame gives another factor of 2 . Using typical experimental values for an interaction laser operating at 532 nm, 2.33 eV, colliding head-on with a relativistic electron with 227 , we find that / 0.002 1 which justifies neglecting the Compton effect to first

order. More importantly, the scattered radiation will have energy 2 5 4 4.8 10 eV which is well within the gamma-ray regime. The electron energy is decreased by a corresponding amount.