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New micro-cones targets can efficiently produce higher energy and lower divergence particle beams

Published online by Cambridge University Press:  07 September 2010

N. Renard-Le Galloudec  and
E. D'Humieres
N. Renard-Le Galloudec*
Affiliation:
Nevada Terawatt Facility, Department of Physics, University of Nevada, Reno, Nevada
E. D'Humieres
Affiliation:
CELIA, Université de Bordeaux - CNRS - CEA, Talence, France
*
Address correspondence and reprint requests to: N. Renard-Le Galloudec, Nevada Terawatt Facility, Department of Physics, University of Nevada, Reno, Nevada 89557. E-mail: nathalie@unr.edu

Abstract

Small conical targets have been used in high intensity laser target interaction mostly in the context of fast ignition. We demonstrate that when cone targets are shaped appropriately and used with specific interaction conditions, they can produce particle beams of higher maximum energy and number in a lower angular divergence than flat targets. This is relevant to fast ignition, small compact particle beams, medical applications, focused ion and/or electron beam microscopes. This fact carries the potential to produce particle beams that are no longer limited by the characteristics of the laser. Note that for fast ignition, reducing the divergence of the beam lowers the energy requirement and enhances the energy deposition into the compressed fuel.

Keywords

Cone targetsHigh-intensity laser interactionLaser-produced particle beams

Information

Type
Research Article
Information
Laser and Particle Beams , Volume 28 , Issue 3 , September 2010 , pp. 513 - 519
Copyright
Copyright © Cambridge University Press 2010

1. INTRODUCTION

Cone targets appeared in laser target interaction after a series of key steps in the pursuit of fusion. In 1963, applications of fusion were just starting to be studied (Basov & Krokhin, Reference Basov and Krokhin1963). Nuckols et al. (Reference Nuckols, Wood, Thiessen and Zimmerman1972) conceived the laser implosion concept to produce fusion and inertial confinement fusion research was born. Concepts of fusion through ion beams (Winterberg, Reference Winterberg1974) or laser generated ion beams and target design for ions were developed (Kindel & Lindman, Reference Kindel and Lindman1979, and references therein). Some decades later, Tabak et al. (Reference Tabak, Hammer, Glinsky, Kruer, Wilks, Woodworth, Campbell and Perry1994) introduced the concept of fast ignition and Kodama et al. (Reference Kodama, Norreys, Mima, Dangor, Evans, Fujita, Kitigawa, Krushelnick, Miyakoshi, Miyanaga, Norimatsu, Rose, Shozaki, Shigemori, Sunahara, Tampo, Tanaka, Toyama, Yamanaka and Zepf2001) introduced the idea of a cone target for fast ignition to allow the laser beam to get far enough into the compressed plasma to produce the fast electron beam that would deliver the ignition spark to the right place. Based on experimental results and simulations, Roth et al. (Reference Roth, Cowan, Key, Hatchett, Brown, Fountain, Johnson, Pennington, Snavely, Wilks, Yasuike, Ruhl, Pregoraro, Bulanov, Campbell, Perry and Powel2001) expanded this concept by adding a curved proton-producing interface that the laser hits first for the proton fast ignition concept. New target concepts along with new ideas to achieve ignition of fusion targets with laser and particle beams are presently of high interest and have a wide range of applications in the field of high energy density physics (Bieniosek et al., Reference Bieniosek, Henestroza and Ni2010; Holmlid et al., Reference Holmlid, Hora, Miley and Yang2009; Johzaki et al., Reference Johzaki, Sakagami, Nagatomo and Mima2007; Koresheva et al., Reference Koresheva, Aleksandrova, Koshelev, Nikitenko, Timasheva, Tolokonnikov, Belolipetskiy, Kapralov, Sergeev, Blazevic, Weyrich, Varentsov, Tahir, Udrea and Hoffmann2009; Nakamura et al., Reference Nakamura, Mima, Sakagami, Johzaki and Nagatomo2008, Reference Nakamura, Sakagami, Johzaki, Nagatomo and Mima2006a, 2006b; Sakagami et al., Reference Sakagami, Johzaki, Nagatomo and Mima2006; Tahir et al., Reference Tahir, Kim, Matvechev, Ostrik, Shutov, Lomonosov, Piriz, Cela and Hoffmann2008; Wu et al., Reference Wu, Zhou, He and Zhu2009; Winterberg, Reference Winterberg2004). We present here studies of the cone target along with other shapes that have paved the way to enable a better understanding of the cone physics, along with its relevance for an array of applications.

2. EFFICIENT USE OF CONES

For such a target to give its full potential, because of the physics occurring in a cone, some criteria need to be met. The cone target needs to be precisely aligned (Nakamura et al., Reference Nakamura, Chrisman, Tanimoto, Borghesi, Kondo, Nakatsutsumi, Norimatsu, Tampo, Tanaka, Yabuuchi, Sentoku and Kodama2009, 2010; Lasinski et al., Reference Lasinski, Langdon, Still, Tabak and Town2009; Yu, Reference Yu, Cao, Yu, Lei, Sheng, Cai, Mima and He2010). The laser enters the cone and starts hitting the faces when its diameter is about 3 to 4 times the inside tip size (Sentoku et al., Reference Sentoku, Mima, Ruhl, Toyama, Kodama and Cowan2004; Nakamura et al., Reference Nakamura, Sakagami, Johzaki, Nagamoto, Mima and Koga2007, Reference Nakamura, Mima, Sakagami, Johzaki and Nagatomo2008; Renard-Le Galloudec et al., Reference Renard-Le Galloudec, Cho, Osterholz and Ditmire2008). Under low pre-plasma conditions, so as to not destroy the conical shape the laser interacts with the cone micro-focuses the laser light into the tip (Sentoku et al., Reference Sentoku, Mima, Ruhl, Toyama, Kodama and Cowan2004; Renard-Le Galloudec et al., Reference Renard-Le Galloudec, Cho, Osterholz and Ditmire2008). At the same time, the laser interacts with the faces of the cone, creates electrons, and guides them along the faces to the tip where the electron beam exits (Renard-Le Galloudec et al., Reference Renard-Le Galloudec, Cho, Osterholz and Ditmire2008, 2009). This increases dramatically the electron density in the tip, enables a higher conversion efficiency of laser light into very energetic or hot electrons (Nakamura et al., Reference Nakamura, Kato, Nagatomo and Mima2004, Reference Nakamura, Sakagami, Johzaki, Nagatomo and Mima2006, Reference Nakamura, Sakagami, Johzaki, Nagamoto, Mima and Koga2007, Reference Nakamura, Chrisman, Tanimoto, Borghesi, Kondo, Nakatsutsumi, Norimatsu, Tampo, Tanaka, Yabuuchi, Sentoku and Kodama2009; Sentoku et al., Reference Sentoku, Mima, Ruhl, Toyama, Kodama and Cowan2004; Nakatsutsumi et al., Reference Nakatsutsumi, Kodama, Norreys, Awano, Nakamura, Norimatsu, Ooya, Tampo, Tanaka, Tanimoto, Tsutsumi and Yabuuchi2007) and thus enhances both electrons and protons characteristics (Chen et al., Reference Chen, Kodama, Nakatsutsumi, Nakamura, Tampo, Tanaka, Toyama, Tsutsumi and Yabuuchi2005). Note here that the cone, not the laser, defines the beam diameter (Renard-Le Galloudec et al., Reference Renard-Le Galloudec, d’Humieres, Cho, Osterholz, Sentoku and Ditmire2009). A smaller cone angle produces more energetic electrons compared to a more open cone (Noda et al., 2002; Chen et al., Reference Chen, Kodama, Nakatsutsumi, Nakamura, Tampo, Tanaka, Toyama, Tsutsumi and Yabuuchi2005; Nakamura et al., Reference Nakamura, Sakagami, Johzaki, Nagamoto, Mima and Koga2007). In addition, cones show an increased absorption of the laser light compared to flat targets (Nakamura et al., Reference Nakamura, Sakagami, Johzaki, Nagamoto, Mima and Koga2007; Lasinski et al., Reference Lasinski, Langdon, Still, Tabak and Town2009), which makes them more efficient. More complex cone-based geometries have also been studied (Flippo et al., Reference Flippo, d’Humières, Gaillard, Rassuchine, Gautier, Schollmeier, Nurnberg, Kline, Adams, Albright, Bakeman, Harres, Johnson, Korgan, Letzring, Malekos, Renard-Le Galloudec, Sentoku, Shimada, Roth, Cowan, Fernández and Hegelich2008) and also show an increased efficiency compared to flat targets. A similar increased efficiency and higher energy protons have been reported in simulations with a concave target (Bin et al., Reference Bin, Lei, Cao, Yang, Huang, Yu and Yu2009). In our case, both the inside and outside tip of the cone are slightly curved (Fig. 1). Figure 1a shows the concept of such a target shape and Figure 1b, the simulated target. Shaping the back of flat targets has been demonstrated to focus protons beams (Wilks et al., Reference Wilks, Langdon, Cowan, Roth, Singh, Hatchett, Key, Pennington, MacKinnon and Snavely2001; Ruhl et al., Reference Ruhl, Bulanov, Cowan, Lisefkina, Nickles, Pegorano, Roth and Sandner2001; Roth et al., Reference Roth, Allen, Audebert, Blazevic, Brambrink, Cowan, Fuchs, Gauthier, Geißel, Hegelich, Karsch, Meyer-ter-Vehn, Ruhl, Schlegel and Stephens2002a, Reference Roth, Blazevic, Geissel, Schlegel, Cowan, Allen, Gauthier, Audebert, Fuchs, Meyer-ter-Vehn, Hegelich, Karsch and Pukhov2002b; Patel et al., Reference Patel, MacKinnon, Key, Cowan, Foord, Allen, Price, Ruhl, Springer and Stephens2003; Snavely et al., Reference Snavely, Zhang, Akli, Chen, Freeman, Gu, Hatchett, Hey, Hill, Key, Izawa, King, Kitagawa, Kodama, Langdon, Lasinski, Lei, MacKinnon, Patel, Stephens, Tampo, Tanaka, Town, Toyama, Tsutsumi, Wilks, Yabuuchi and Zheng2007), it is however the first time that this concept is adapted to a cone geometry in order to use cones as an essential element of the particle beam production and reap the benefits of the increased efficiency of its shape. It does more than a standard flat or curved target. It adds three essential aspects. The first aspect is that making use of the cone faces by allowing the laser to spread on them greatly reduces the amount of pre-plasma filling the cone, thus enabling an efficient use of the cone shape (Renard-Le Galloudec et al., Reference Renard-Le Galloudec, Cho, Osterholz and Ditmire2008, 2009). It also uses the faces to create the electrons and guide them to the tip. Several articles have showed the imprint of the laser pattern on flat targets into the particle beam (Roth et al., Reference Roth, Allen, Audebert, Blazevic, Brambrink, Cowan, Fuchs, Gauthier, Geißel, Hegelich, Karsch, Meyer-ter-Vehn, Ruhl, Schlegel and Stephens2002a; Fuchs et al., Reference Fuchs, Cowan, Audebert, Ruhl, Gremillet, Kemp, Allen, Blazevic, Gauthier, Geissel, Hegelich, Karsch, Parks, Roth, Sentoku, Stephens and Campbell2003). As the laser bounces several times on the faces on its way to the tip, its imprint disappears. It creates, at the tip, a laser imprint free area of high energy density, enabling more uniform beams. Also, its best focus is positioned toward the entrance of the cone, then all of the laser light available gets in regardless of the f-number of the focusing optic compared to the cone angle. The second aspect is the fact that the cone, then, not the laser, defines the particle beam (Renard-Le Galloudec et al., Reference Renard-Le Galloudec, Cho, Osterholz and Ditmire2008). The particle beam diameter has the potential to be smaller or bigger than the laser best focus by defining the size of the inside tip of the cone (Renard-Le Galloudec et al., Reference Renard-Le Galloudec, d’Humieres, Cho, Osterholz, Sentoku and Ditmire2009). The laser is clearly not directly driving the characteristics of the beam produced. The third aspect is the ability to control the divergence of the output beam. The tip of the cone is slightly curved in our case. This results in a modification of the divergence of the output particle beam by effectively modifying the accelerating sheath shape, and can be adjusted by adjusting the amount of curvature. This efficiently produces a beam with extremely relevant characteristics to fast ignition (Nakamura et al., Reference Nakamura, Sakagami, Johzaki, Nagamoto, Mima and Koga2007), laser based accelerators (Dunne, Reference Dunne2006), proton beams for proton radiography of plasmas (Borghesi et al., Reference Borghesi, Campbell, Schiavi, Haines, Willi, MacKinnon, Patel, Gizzi, Galimberti, Clarke, Pegoraro, Ruhl and Bulanov2002, Reference Borghesi, Sarri, Cecchetti, Kourakis, Hoarty, Stevenson, James, Brown, Hobbs, Lockyear, Morton, Willi, Jung and Dieckmann2010; Kodama et al., Reference Kodama, Azechi, Fujita, Habara, Izawa, Jitsuno, Jozaki, Kitagawa, Krushelnick, Matsuoka, Mima, Miyanaga, Nagai, Nagatomo, Nakai, Nishimura, Norimatsu, Norreys, Shigemori, Shiraga, Sunahara, Tanaka, Tampo, Toyama, Tsubakimoto, Yamanaka and Zepf2004), isochoric heating (Patel et al., Reference Patel, MacKinnon, Key, Cowan, Foord, Allen, Price, Ruhl, Springer and Stephens2003) shocks (Koenig et al., Reference Koenig, Henry, Huser, Benuzzi-Mounaix, Faral, Martinolli, Lepape, Vinci, Batani, Tomasini, Telaro, Loubeyre, Hall, Celliers, Collins, DaSilva, Cauble, Hicks, Bradley, MacKinnon, Patel, Eggert, Pasley, Willi, Neely, Notley, Danson, Borghesi, Romagnani, Boehly and Lee2004), proton therapy (Bulanov & Khoroshkov, Reference Bulanov and Khoroshkov2002; Fourkal et al., Reference Fourkal, Shahine, Ding, Li, , , Tajima and Ma2002; Noda et al., 2002; Pegoraro et al., Reference Pegoraro, Atzeni, Borghesi, Bulanov, Esirkepov, Honrubia, Kato, Khoroshkov, Nishihara, Tajima, Temporal and Willi2004; Nishiushi et al., 2009, Yogo et al., Reference Yogo, Sato, Nishikino, Mori, Teshima, Numasaki, Murakami, Demizu, Akagi, Nagayama, Ogura, Sagisaka, Orimo, Nishiuchi, Pirozhkov, Ikegami, Tampo, Sakaki, Suzuki, Daito, Oishi, Sugiyama, Kiriyama, Okada, Kanazawa, Kondo, Shimomura, Nakai, Tanoue, Sasao, Wakai, Bolton and Daido2009), micro-beam radiation therapy (Slatkin et al., Reference Slatkin, Spanne, Dilmanian and Sandborg1992), positron emission tomography (Spencer et al., Reference Spencer, Ledingham, Singhal, McCanny, McKenna, Clark, Krushelnick, Zepf, Beg, Tatarakis, Dangor, Norreys, Clarke, Allott and Ross2001), focused ion beam milling machines (Reyntjens & Puers, Reference Reyntjens and Puers2002), ion beam microscopes (Li, Reference Li2007), and dual beam electron/ion microscopes (MoberlyChan, Reference MoberlyChan2009). For applications such as proton therapy, control of the characteristics of the beam are important (Toncian et al., Reference Toncian, Borghesi, Fuchs, d’Humieres, Antici, Audebert, Brambrink, Cecchetti, Pipahl, Romagnani and Willi2006) and a micro-magnetic device (Schollmeier et al., Reference Schollmeier, Becker, Geißel, Flippo, Blazevic, Gaillard, Gautier, Gruner, Harres, Kimmel, Nurnberg, Rambo, Schramm, Schreiber, Schutrumpf, Schwarz, Tahir, Atherton, Habs, Hegelich and Roth2008) separates the electron and or proton beam from the X-rays, or focus them (Nishiuchi et al., Reference Nishiuchi, Daito, Ikegami, Daido, Mori, Orimo, Ogura, Sagisaka, Yogo, Pirozhkov, Sugiyama, Kiriyama, Okada, Noda, Fadil, Iwashita, Morita, Nakamura, Shirai, Tongu, Yamazaki, Daido, Hayashi, Orimo, Yamakawa, Kato, Matsukado, Li, Noda, Yamada, Uesaka and Beutelpacher2002; Kanazawa et al., Reference Kanazawa, Kondo, Shimomura, Tanoue, Nakai, Sasao, Wakai, Sakaki, Bolton, Choi, Sung, Lee, Oishi, Fujii, Nemoto, Souda, Noda, Iseki, and Yoshiyuki2009). An ion-milling machine that includes electron microscopy capabilities to image the object, and non-destructive X-rays radiographs of the same object would be a possibility. With high repetition rate (Tümmler et al., Reference Tümmler, Jung, Stiel, Nickles and Sandner2009) and high-energy high-repetition rate lasers (Bayramian et al., Reference Bayramian, Armstrong, Ault, Beach, Bibeau, Caird, Campbell, Chai, Dawson, Ebbers, Erlandson, Fei, Freitas, Kent, Liao, Ladran, Menapace, Molander, Payne, Peterson, Randles, Schaffers, Sutton, Tassano, Telford and Utterback2007; Burns et al., Reference Burns, Sethian, Wolford, Myers, Giuliani, Hegeler, Friedman and Jaynes2009) as well as targets that are on the verge of cost effective mass production (Renard-Le Galloudec et al., Reference Renard-Le Galloudec, Adams, Korgan, Malekos, Cowan, Gaillard, Rassuchine, Sant and Sentoku2006; Alexander et al., Reference Alexander, Goodin and Stephens2007, Reference Alexander, Stephens, Goodin, Petzoldt, Lee, Sheliak, Tolley, Neely and Foster2009; Higginson et al., Reference Higginson, Stephens and Brocato2006), cost effective compact applications can be readily envisioned.

Fig. 1.

(Color online) Conceptual schematic of the curved cone target (a) and simulated target (b).

3. A NEW SHAPE FOR PARTICLE BEAM GENERATION

Because the new target shape proposed here has not been fabricated yet, we used the two-dimensional (2D) particle-in-cell (PIC) code PICLS (Sentoku et al., Reference Sentoku, Kemp, Presura, Bakeman and Cowan2007) to run collisionless simulations and assess the electro-magnetic fields structures and proton beam characteristics in comparison with flat targets. We ran several intensities to span the range available to short pulse lasers. Figure 2a shows the simulated cone target as a dotted line. The inner and outer tip diameters are respectively 10 and 30 µm. They are both curved. The target itself is 10 µm thick. The simulations box is 150 µm long to capture the emitted particles. The incident laser pulse (1 µm, 40 fs, 21 µm full width at half maximum transverse spot size at 3 × 1018 W/cm2) has a Gaussian temporal and transverse spatial profile. The pulse is injected to the left of a 120 × 150 µm box. The laser interacts with the target at normal incidence, with its polarization in the simulation plane. The peak of the pulse enters the box 80 fs after the beginning of the calculation. The initial target density is 40n c and remains higher than the relativistic critical density a 0n c, where a 0 is the normalized laser amplitude and n c is the critical density (n c = 1.1 × 1021/λ2(μm)2 cm−3, λ is the laser wavelength). The plasma, composed of D ions, protons, and electrons, is initially fully ionized. The mesh size is Δx = Δy = 40 nm with 40 D ions or 40 protons and 40 electrons per cell. Two types of targets were investigated: cone D targets (shown in Fig. 1b) and flat D foils, both with a thin layer of protons at the back. An exponential pre-plasma consisting of protons and electrons is located inside the cone in the first case, or in front of the flat foil for the second case. It has a density 1% to n c over 50 µm with a characteristic length of 1 micron. The time step is equal to 0.132 fs.

Fig. 2.

(Color online) Proton energy density for the cone target at 3 × 1020 W/cm2 (a) and for the flat target for the same laser intensity (b).

Figure 2 represents the 2D proton energy density for a 10 µm thick curved-tip cone in a high intensity case at 3 × 1020 W/cm2. Figure 2b represents the same 2D proton energy density for a 10 µm flat target at the same intensity. We clearly see that the protons are a lot more confined in the cone than in the flat target where they tend to diffuse laterally. The protons emitted from the cone are much more collinear to the laser axis compared to the flat target where they expand perpendicularly to the sheath.

Figure 3 shows the proton divergence (py/px) as a function of the longitudinal position at 924fs for the cone (Fig. 3a) and the flat target (Fig. 3b) for 3 × 1020 W/cm2. In both cases, the average divergence is small, especially for the high-energy protons (those with a position from 120 to 140 µm). The cone target controls the divergence much better than the flat target over a wider range of energies. The curvature also allows to focus the most energetic protons in a specific location, and thus to deposit through the ions a higher energy in a smaller volume than in the case of a flat target, which is of special interest to isochoric heating.

Fig. 3.

(Color online) Proton divergence (py/px) as a function of the longitudinal position at 924fs for the cone (a) and the flat target (b) for 3 × 1020 W/cm2.

Figure 4 confirms that the cone is a much more efficient structure over a range of intensities. Figure 4a shows the maximum proton energy expected for both the cone target and the flat target over a range of intensities. We see that as we increase the intensity the maximum proton energy increases in general regardless of the target but the cone target clearly shows higher maximum proton energy than the flat target for all intensities. That difference increases with intensity. Especially evident at 3 × 1020 W/cm2 is, that both electrons (Fig. 4b) and protons (Fig. 4c) are accelerated to higher energies in a higher number for the cone target. Enhanced laser interaction results in much higher maximum proton energies at high intensities. Laser absorption is greatly increased in cone targets. It reaches 75.7% for the cone target and only 42.1% for the flat target in the high intensity case (3 × 1020 W/cm2). For the low intensity case (3 × 1018 W/cm2), the absorption reaches 83.4% for the cone target compared to 65.8% for the flat target. In the high intensity case, laser intensity reaches a maximum of 2.4 × 1021 W/cm2 in the tip of the cone (6 × 1020 W/cm2 for the flat target), highlighting the micro-focusing effect of the cone. The longitudinal electric field, the one accelerating the protons, reaches in this case 100 TV/m (17 TV/m for the flat target). In the low intensity case, laser intensity reaches a maximum of 4.3 × 1018 W/cm2 for the cone (7 × 1018 W/cm2 for the flat target) with a longitudinal electric field at 5 TV/m (3.6 TV/m for the flat target). In the low intensity case, the large preplasma present in both cases tends to give similar laser parameters evolutions, similar electric fields and a moderate increase of maximum proton energy. In the high intensity case, the laser intensity and the longitudinal electric field reach significantly larger values in the cone leading to an important increase in maximum proton energy. Note here that as the laser intensity increases, very often, so does the prepulse. It also becomes more difficult to contain the entire laser energy in the focal spot. In the case of having best focus toward the base of the cone, all the laser energy gets in and gets micro-focused by the cone shape into the tip. Additional studies need to be performed to further quantify the divergence, the effect of the pulse duration, laser intensity and prepulse.

Fig. 4.

(Color online) Maximum proton energies for three laser intensities from both the cone target and the flat target (a). Electron energy spectrum (b) for both cone and flat target at 660 fs, and proton energy spectrum (c) for both cone and flat target at 1.98 ps.

4. CONCLUSION

In conclusion, we show that a new conical target shape has the potential to produce proton beams of a higher maximum energy and a lower divergence. Because of the appropriate use of the cone structure itself, by using the faces leading to the tip, nor the laser imprint or the focal spot size have an impact on the particle beam characteristics. The contrast of the laser can also be mitigated and finally the f-number of the focusing optic is superseded by that of the cone target itself. All these parameters increase the potential for various groups to join in the research endeavor and pursue exciting new applications.

ACKNOWLEDGMENTS

The authors would like to thank Yasuhiko Sentoku for usage of the code. The first author would like to thank M. H. Key for fruitful discussions, R. B. Stephens, N. B. Alexander, J. Caird, and T. A. Melhorn for their contributions, J. S. Thompson, J. Kindel and T. Ditmire for their support. This work was supported by the National Nuclear Security Administration under cooperative agreements DE-FC52-03NA00156.

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Figure 0

Fig. 1. (Color online) Conceptual schematic of the curved cone target (a) and simulated target (b).

Figure 1

Fig. 2. (Color online) Proton energy density for the cone target at 3 × 1020 W/cm2 (a) and for the flat target for the same laser intensity (b).

Figure 2

Fig. 3. (Color online) Proton divergence (py/px) as a function of the longitudinal position at 924fs for the cone (a) and the flat target (b) for 3 × 1020 W/cm2.

Figure 3

Fig. 4. (Color online) Maximum proton energies for three laser intensities from both the cone target and the flat target (a). Electron energy spectrum (b) for both cone and flat target at 660 fs, and proton energy spectrum (c) for both cone and flat target at 1.98 ps.