1. INTRODUCTION
Recent progress in laser technology opened new research fields in high energy density physics, such as X-ray generation with high brightness (Hartemann et al., Reference Hartemann, Tremaine, Anderson, Barty, Betts, Booth, Brown, Crane, Cross, Gibson, Fittinghoff, Kuba, Le Sage, Slaughter, Wootton, Hartouni, Springer, Rosenzweig and Kerman2004), charged particle acceleration with high quality (Hatchett et al., Reference Hatchett, Brown, Cowan, Henry, Johnson, Key, Koch, Langdon, Lasinski, Lee, Mackinnon, Pennington, Perry, Phillipps, Roth, Sangste, Singh, Snavely, Stoyer, Wilks and Yasuike2000; Katsouleas, Reference Katsouleas2004; Fuchs et al., Reference Fuchs, Sentoku, Karsch, Cobble, Audebert, Kemp, Nikroo, Antici, Brambrink, Blazevic, Campbell, Fernandez, Gauthier, Geissel, Hegelich, Pepen, Poescu, Renard-Legalloudec, Roth, Schreiber, Stephens and Cowan2005; Flippo et al., Reference Flippo, Hegelich, Albright, Yin, Gautier, Letzring, Schollmeier, Schreiber, Schulze and Fernandez2007; Karmakar & Pukhov, Reference Karmakar and Pukhov2007; Koyama et al., Reference Koyama, Adachi, Miura, Kato, Masuda, Watanabe, Ogata and Tanimoto2006, Lifschitz et al., Reference Lifschitz, Faure, Glinec, Malka and Mora2006; Nickles et al., Reference Nickles, Ter-Avetisyan, Schnuerer, Sokollik, Sandner, Schreiber, Hilscher, Jahnke, Andreev and Tikhonchuk2007; Yin et al., Reference Yin, Albright, Hegelich and Fernandez2006; Strangio et al., Reference Strangio, Caruso, Neely, Andreoli, Anzalone, Clarke, Cristofari, Del Prete, Di Giorgio, Murphy, Ricci, Stevens and Tolley2007), fast ignition research in inertial fusion (Tabak et al., Reference Tabak, Hammer, Glinsky, Kruer, Wils, Woodworth, Campbell, Perry and Mason1994; Johzaki et al., Reference Johzaki, Sakagami, Nagatomo and Mima2007; Sakagami et al., Reference Johzaki, Nagatomo, Sakagami, Nakamura, Mima, Nakao and Yokota2006; Zvorykin et al., Reference Zvorykin, Didenko, Ionin, Kholin, Konyashchenko, Krokhin, Levchenko, Mavritskii, Mesyats, Molchanov, Rogulev, Seleznev, Sinitsyn, Tenyakov, Ustinovskii and Zayarnyi2007), and so on. In fast ignition scheme, cone targets are used to guide the heating laser pulse close to the core plasma which is surrounded by large-scale corona plasma, and to generate high energy charged particles which heat up the core. Advantages in using cone targets were confirmed in experiments, which show huge increase of neutron yield (Kodama et al., Reference Kodama, Shiraga, Shigemori, Toyama, Fujioka, Azechi, Fujita, Habara, Hall, Izawa, Jitsuno, Kitagawa, Krushelnick, Lancaster, Mima, Nagai, Nishimura, Norimatsu, Norreys, Sakabe, Tanaka, Youssef, Zepf and Yamanaka2002). But many physical issues are not clearly understood yet, which are now intensively studied experimentally and numerically (Stephens et al., Reference Stephens, Snavely, Aglintskiy, Amiranoff, Andersen, Batani, Baton, Cowan, Freeman, Hall, Hachett, Hill, Key, King, Koch, Koenig, Mackinnon, Lancaster, Martinolli, Norreys, Perelli-Cippo, Rabec-Legloahec, Rousseaux, Santos and Scianitti2004; Chen et al., Reference Chen, Kodama, Nakatsutsumi, Nakamura, Tampo, Tanaka, Toyama, Tsutsumi and Yabuuchi2005; Campbell et al., Reference Campbell, Kodama, Melhorn, Tanaka and Welch2005). One of the key issues among them is the generation of high energy particles from the cone target. It is shown in three-dimensional (3D) particle-in-cell (PIC) simulation that cone targets focus laser energy and high energy electrons at the cone tip which result in higher coupling efficiency from laser to electrons (Sentoku et al., Reference Sentoku, Ruhl, Mima, Kodama, Tanaka and Kishimoto1999). In this paper, we investigate the interaction between intense laser pulses and cone targets by using two-dimensional (2D) PIC simulation in order to study the cone geometry dependence on laser-cone interaction. We introduce two important parameters which characterize the laser-cone inetarction, and show the optimum value of them for fast ignition. To understand the dependence of laser-cone interaction on target geometry is crucial for designing the cone targets for fast ignition, and becoming basis for using them as the devices of generating high energy charged particles (Kodama et al., Reference Kodama, Sentoku, Chen, Kumar, Hatchett, Toyama, Cowan, Freeman, Fuchs, Izawa, Key, Kitagawa, Kondo, Matsuoka, Nakamura, Nakatsutsumi, Norreys, Norimatsu, Snavely, Stephens, Tampo, Tanaka and Yabuuchi2004).
The paper is composed as follows. In Section 2, electron acceleration processes taking place in laser-cone interaction are studied and two parameters which characterize laser-cone interaction are introduced. In Section 3, the cone angle dependence on laser intensification and electron acceleration are studied. Electron confinement around cone tip is discussed. In Section 4, effect of another important parameter which is a ratio laser spot size and cone tip size is discussed. The conclusions are given in Section 5.
2. ELECTRON ACCELERATION PROCESSES IN LASER-CONE INTERACTION
First, we compare the electron characteristics generated from a cone target and a plane target which are irradiated by ultra-intense laser pulses. In Figure 1, electron spectra from the cone target and the plane target are plotted. Parameters and conditions of the cone target simulation are as follows. The target density is 100 times the critical density which is defined as n c = mɛ0ω0/e 2 where m and e are the electron mass and the charge, ɛ0 and ω are the dielectric constant in vacuum and the laser frequency, respectively. The preplasma exists inner side on the cone target which has exponential profile whose scale lengths are 1.0 µm and 0.27 µm at the cone tip and cone side wall, respectively. At the rear side of the cone target, overdense plasma which models the corona plasma is located whose density is 2n c. The initial electron temperature is 10 keV, and ions are kept immobile. The laser pulse irradiate the target from the left boundary whose intensity is 5.0 × 1019 W/cm2 with 1.0 µm wavelength, which leads to the normalized vector potential which is defined by 
, where I and λL are the laser intensity in units of W/cm2 and the laser wavelength in unit of µm. The laser field is linearly polarized in the y-direction with a Gaussian profile whose spot size is 10.0 µm (full width at half maximum: FWHM). The laser pulse rises up in five laser cycles and sustains its peak intensity for 150 fs. A plane target is used for comparison, which is modeled to have the same conditions as the cone target, and is made by flattening the cone geometry into flat one, i.e., the target maximum density is 100n c with preplasma of 1.0 µm scale length, and it is surrounded by 2n c plasma at the rear side. The laser conditions are exactly the same as those used in the cone simulation.
Fig. 1. Comparison of electron energy spectrums generated from (a) cone target and (b) plane target. Spectra are observed at target rear side, which are irradiated by intense laser pulses of 1.5 × 1019 W/cm2 and 150 fs.
In Figures 1a and 1b, time-integrated electron energy spectrums observed at 2 µm behind the targets are plotted. In plane target case, electron spectrum is fitted by Maxwell distribution with two temperatures. Lower temperature is 0.4 MeV for electrons with energy <2.0 MeV. Higher temperature is 2.5 MeV, which is well approximated by ponderomotive energy as 
(Wilks et al., Reference Wilks, Kruer, Tabak and Langdon1992). Electron spectrum from cone target shown in Figure 1b is fitted by Maxwell distribution with three temperatures. The lowest temperature is T low ~ 0.35 MeV which is almost the same as in the plane case. But other two temperatures differ from the ponderomotive energy of incident laser pulse. The electrons whose energy is 
is fitted with T mid ~ 1.9 MeV which is lower than the ponderomotive energy, and electrons with higher energy is fitted with T high ~ 5.0 MeV, which is much higher than the ponderomotive energy of incident lasers. These two components account for electrons generated at cone wing and cone tip (Nakamura et al., Reference Nakamura, Sakagami, Johzaki, Nagatomo and Mima2006, Reference Nakamura, Sakagami, Johzaki, Nagatomo, Mima and Koga2007). Since the laser irradiates the target obliquely, its intensity decreases on the surface, which results in electrons whose temperature is lower than initial ponderomotive energy. And at the cone tip, laser field is intensified due to multiple reflections, which results in electrons with temperature higher than ponderomotive energy.
These results indicate that there are two dominant acceleration processes taking place in laser-cone interaction; acceleration at wing and tip. The acceleration at cone wing depends on laser intensity and more importantly, cone angle, since the interaction strongly depends on incident angle for oblique irradiation. The acceleration at the cone tip depends on cone-focused laser fields, which is characterized by the ratio of laser spot size and cone tip as well as cone angle. Therefore two parameters which are cone angle and laser spot/cone tip ratio are characterizing the laser-cone interaction. In the following sections, we show how these parameters characterize the interaction and optimize them for the fast ignition.
3. ELECTRON CONFINMENT AROUND THE CONE TIP
The laser-cone interaction depends on the cone angle because the irradiation angle of laser field at cone sidewall and the propagation path of laser light inside cone change. The laser light irradiates cone sidewall with relatively large angle, e.g., 75° for 30° cone target. The energy absorption rate drastically changes around irradiation angle of 75° (Ruhl & Cairns, Reference Ruhl and Cairns1997), which is the case of laser-cone interaction. In addiction to the irradiation angle, the laser light propagation inside the cone target changes due to the cone angle.
A sample ray of laser light propagating in 30° cone target is drawn in Figure 2, where the specular reflection at wall is assumed. The laser light reflects four times before the light propagates backward, and the light located within 2R at the cone entrance is focused down to 0.53R spot diameter, indicating that laser light is intensified at cone tip about four times. The reflection angle is bigger for cone targets with larger cone angle, which results in less reflection time for larger angle cone targets, such as three times reflection in 45° cone and two times reflection in 60° cone. This affects the energy absorption rate.
Fig. 2. A sample ray of light propagation inside 30° cone target. The rays located in 2R at cone entrance are focused to the diameter of 0.53R.
In Table 1, the energy absorption rate evaluated by PIC simulation is shown for different cone angle. The simulation conditions are following. The laser intensity is 1.0 × 1020 W/cm2 with duration of 150 fs whose spot size is 10 µm. The target density is 100n c, and diameter is 30 µm at cone entrance and 3 µm at the cone tip. As the cone angle becomes larger, the reflection time decreases to reduce absorption rate. When the high energy coupling from laser to electron is desired such as the fast ignition, cone targets with smaller angle are beneficial. Since the absorption rate sharply drops as the irradiation angle goes over 75°, 30° is small enough for achieving the high absorption rate.
Table 1. Energy flux ratio of electrons propagating out from cone to input laser in left column and absorption rate in right column. The ratio of stored energy to laser energy is also written in side the bracket
In Table 1, along with the absorption rate, the ratio of the energy which is stored inside cone at 300 fs after terminating the laser pulse to the input laser energy is written. Even 300 fs later the termination, roughly half of the absorbed energy is still left inside the cone target. The temporal evolution of energy stored inside target for 30° cone is shown in Figure 3. These results show that the energy absorbed by the cone target is not released smoothly from the target. For high energy electrons flow out, the return current to sustain current-neutrality is necessary, which is achieved by the compensation of the bulk electrons. The electric field is induced to accelerate and heat up the bulk electrons for the compensation.
Fig. 3. Temporal evolution of electron energy inside the cone target. The energy is gradually released from the cone target after the laser irradiation.
The distributions of electron kinetic energy and electro-magnetic field energy at 150 fs after terminating the laser pulse are shown in Figure 4. It is clearly shown that high energy electrons are confined around the cone tip and prevented from flowing out. Electric and magnetic field energy is also localized around the cone tip. The electric field is disturbed and turbulent, which confine electrons at the tip. This phenomenon depends on target conductivity, therefore on target material, temperature, density, and so on. In the above simulation, the current density flowing in the forward direction toward the cone tip reaches up to 25× (−encc). When there exist abundant free and relatively high energy bulk electrons, which ensures current neutrality, this disturbed electric field is not induced.
Fig. 4. (Color online) (top) Electron kinetic energy density distribution which is normalized by ncmc2. (bottom) Electric and magnetic field energy (E 2 + B 2)/2 which is normalized by ncmc2.
This confinement is not negligible in fast ignition since the moderate energy electrons (~ 1 MeV) are kept releasing from the cone target toward core plasma for relatively long time. This long time effect is studied in integrated simulation of 1D-PIC and 2D Fokker-Planck code and shown to be important in the core heating (Sakagami et al., Reference Sakagami, Johzaki, Nagatomo and Mima2006).
4. DEPENDENCE OF RATIO BETWEEN LASER SPOT AND CONE TIP ON LASER-CONE INTERACTION
In addition to the cone angle, the ratio between the laser spot and cone tip determines how much the laser field is intensified at the cone tip by the cone guiding. The cone targets with different cone tip size by keeping the cone angle and the entrance diameter are compared. The diameter at entrance is 30 µm and diameter at tip is varied as 0, 3, 5, and 10 µm. The cone angle is 30° and the target density is 100n c. The electron temperature is 1 keV with immobile ions. The laser spot size is 10 µm in diameter and intensity is 1.0 × 1021 W/cm2 with duration of 150 fs. In Table 2, the energy absorption rate and energy flux through the cone tip within 3 µm diameter are summarized. As the tip size becomes smaller, the interaction region at cone wing increases and the laser field is intensified more at the cone tip, which results in higher coupling. But the energy flux passing through the tip is maximum for 3 µm case, since the high energy electrons highly concentrated at the cone tip, which leads to diverging of the beam due to the space charge. Numerical simulations also show that the electron energy flux at rear side of the cone target is localized within the diameter comparable to the tip size. As a result, the tip diameter is chosen to be small enough to focus laser light on the tip and to increase absorption rate, and large enough to prevent the excess concentration of high energy electrons. For 30° cone targets, the spot size of about one-fourth of laser spot diameter is maximizing the wall interaction and focus all the laser beam to the tip. For a smaller tip, the laser light irradiating at the cone entrance outside the circle whose diameter is about four times the tip diameter does not reach to the tip. For a larger tip, laser intensification is smaller and wall interaction length is also shorter, which results in smaller absorption. In reality, the reflection at wall is not specular and laser profile at the tip is modulated by pre-plasmas. So the number one-fourth is not the strict number, but gives an insight in designing the cone target.
Table 2. Energy flux ratio of electrons propagating out from cone to input laser (left) and absorption rate (right) for cone targets with different cone tip size. The laser focus spot and diameter of cone entrance are same for all simulations
CONCLUSIONS
The laser-cone interaction is studied by 2D PIC simulations. Two important parameters which characterize the laser-cone interaction are introduced, which are cone angle and ratio of laser spot and tip diamter. These paramters control the laser propagation inside the target, laser intensification at the tip, and wall interaction. For the angle, 30° cone target is suitable for fast ignition since the high energy absorption and high energy flux. For the ratio of laser spot and tip diameter, about one-fourth of tip size is maximizing the lsaer fucusing effect and wall interaction. The confinement of high energy electrons at cone tip is observed, which is due to the disturbed electric field at the cone tip. The electric field is generated in order to satisfy the current neutrality. Therefore, the phenomena depends on the target conditions, such as material, density, temperature and so on. Further study of this effect on electron transport in fast ignition is considered to be necessary.
