Over many millennia of human history, mankind has been interested in how events change in time, namely their dynamics. However, the time resolution of recording individual steps has been limited to direct sensory perception such as the eye’s ability (0.1 sec. or so) to recognize the motion, until 1800 AD when the technical revolution occurred following the industrial revolution. A famous motion picture of a galloping horse by E. Muybridge in 1878 is a good example of the technological development in time-resolved measurement. By this time, the nanosecond time resolution has been achieved; however, it took another century to break the nanosecond barrier as shown in Fig.1. The Advent of a laser has paved ways to ever shorter time resolution: in the 1980’s, the picosecond barrier was broken and the femtosecond science and technology has rapidly progressed in the 1990’s; at the turn of the 21st century, the femtosecond barrier has been broken (Hantschel et al., 2001), opening up the era of attosecond science and technology. The current shortest duration of a pulse achieved is 80 attoseconds around 100 eV of photon energy (Goulielmakis et al., 2008).
Femtosecond science and technology have allowed us to explore various ultrafast phenomena in physical (Siders et al., 1999), chemical (Zewail, 2000) and biological (Vos et al., 1999) systems. A great number of ultrafast atomic motions in biology, chemistry, and physics have been investigated with optical probes. In physics, the nature of atomic re-arrangements during phase transitions and the relation between amorphous, liquid and crystalline states has been interest (Afonso et al., 1996, Huang et al., 1998). Along with much interest in spintronics during the last decade, efforts have been made to understand spin dynamics in various pure and complex magnetic systems. In chemistry, the real time observation of atomic motions in chemical reactions has been long thought for. Femtosecond optical and IR technology has served this purpose in excellent ways. Femtosecond pulses have pumped molecules to create wavepackets. The observation of the motion of the wavepackets using femtosecond pulse probe or other methods has provided rich information on chemical reactions (Zewail, 2000). The various chemical bonds such as covalent, ionic, dative, metallic, hydrogen and van der Waals bonds have been studied in the varying complexity of molecular systems from diatomics to proteins and DNA. All of
these successful applications of femtosecond optical technology to follow reaction dynamics in chemistry have led to the award of Nobel prize to Prof. Zewail in 1999 who has initiated femtochemistry in the 1990’s. There are also many fundamental biological processes taking place in femtosecond time scale. Good examples are photo-induced isomerizations (Gai et al., 1998) and ligand dissociations (Perutz et al., 1998). Femtosecond optical pulses have been used for the investigation of these processes.
During the last several years, isolated attosecond pulses have been successfully exploited to control the localization of electron under chemical reaction (Kling et al., 2006), observe inner shell transitions in atoms (Drescher et al., 2002), and electron tunneling across Coulomb barrier (Uiberacker et al., 2007), electron transport in condensed matter (Cavalieri et al., 2007).
The demand on and the emphasis to the understanding of the ultrafast phenomena and the control of them is ever increasing. The Department of Energy of the United States of America asked the Basic Energy Science Advisory Committee (BESAC) to identify grand challenges in science to be pursued in the 21st century. The BESAC has identified 5 grand challenges in their report, the summary of which has been published in the July issue of Physics Today in 2008 (Fleming & Ratner, 2008). The first of the grand challenges is to control material properties at the level of electronic motion. The report recommended the development of attosecond and femtosecond metrology to measure and control electron dynamics.
Optical or ultraviolet light allows one to probe the dynamical changes of excited electronic states in a sample. This is the core of interest in many research topics such as electron-hole dynamics in solids, or excitation transfer in photosynthesis, bond breaking in chemical reactions. However, no or partial information on structure can be obtained if the process under investigation involves the structural changes.
X-ray imaging or diffraction has the potential to provide a global picture of structural changes in the fields of atomic and molecular physics, plasma physics, material science, chemistry and life science (Bloembergen, 1999; http://tesla.desy.de/new_pages /TDR_CD/PartV/fel.html). Since x-ray photons are scattered from all the electrons in a sample, the intensity of diffraction is directly related to the electronic density. Since the structural changes occur in the time scale of 100 fs or so, following the charge re-arrangement, the electronic density measured in this time scale closely reflects the atomic structure. On the other hand, attosecond x-ray pulse will enable us to follow even faster motions of electrons in intra-atomic or intra-molecular processes, which has not been achieved yet because of the lack of such sources.
Femtosecond time-resolved x-ray diffraction experiments have been used to study structural processes such as the observation of atomic structure and dynamics , the investigation of ultrafast phase transition in solid (Gaffney & Chapman, 2008) and time-resolved biomolecular imaging (Sokolowski-Tinten et al., 2003). These results are very impressive and of landmark: however, there are still many phenomena yet to be explored and a variety of attosecond and femtosecond X-ray sources yet to be further developed. They comprise a great challenge to the scientific community.
Figure 2 shows ultrafast sources available currently or in the immediate future. It is conspicuous that there is no source available in keV or higher energies faster than 10 fs. In fact, for wider exploration and manipulation of electron dynamics in a vast spectrum of natural phenomena, attosecond or a few fs keV pulses are demanded.
Several schemes have been proposed and/or demonstrated to generate an ultrashort keV x-ray pulse: the relativistic Doppler shift of a backscattered laser pulse by a relativistic electron beam (Sprangle et al., 1992, Hartemann, 1998, Esarey et al., 1993a, Chung et al., 2009), the harmonic frequency upshift of a laser pulse by relativistic nonlinear motion of electrons (Vachaspati, 1962, Brown & Kibble, 1964, Esarey et al,, 1992, Chen et al., 1998, 2000; Ueshima et al, 1999, Kaplan & Shkolnikov, 2002, Banerjee et al., 2002, Lee et al., 2003a, 2003b, 2005, 2008, Phuoc et al., 2003, Kim et al., 2009), high order harmonic generation in the interaction of intense laser pulse with solids (Linde et al., 1995, 1996, 1999, Norreys et al., 1996, Lichters et al., 1996, Tarasevitch et al., 2000) and x-ray laser using inner shell atomic transitions (Kim et al., 1999, 2001).
Ultrafast high-intensity X-rays can be generated from the interaction of high intensity femtosecond laser via Compton backscattering (Hartemann et al., 2005), relativistic nonlinear Thomson scattering (Ueshima et al., 1999, Kaplan & Shkolnikov 2002, Banerjee et al., 2002) and laser-produced betatron radiation (Phuoc et al., 2007). In synchrotron facilities, electron bunch slicing method has been adopted for experiments (Schoenlein, 2000, Beaud et al., 2007). Moreover, X-ray free electron lasers (Normile, 2006) were proposed and have been under construction. The pulse duration of these radiation sources are in the order of a few tens to hundred fs. There are growing demands for new shorter pulses than 10 fs.
The generation of intense attosecond or femtosecond keV lights via Thomson scattering (Lee et al., 2008, Kim et al., 2009) is attractive, because the radiation is intense and quasi-monochromatic. This radiation may be also utilized in medical (Girolami et al., 1996) and nuclear physics (Weller & Ahmed, 2003) area of science and technology.
When a low-intensity laser pulse is irradiated on an electron, the electron undergoes a harmonic oscillatory motion and generates a dipole radiation with the same frequency as the incident laser pulse, which is called Thomson scattering. As the laser intensity increases, the oscillatory motion of the electron becomes relativistically nonlinear, which leads to the generation of harmonic radiations. This is referred to as relativistic nonlinear Thomson scattered (RNTS) radiation. The RNTS radiation has been investigated in analytical ways (Esarey et al., 1993a, Chung et al., 2009, Vachaspati, 1962, Brown et al., 1964, Esarey & Sprangle, 1992, Chen et al., 1998, Ueshima et al., 1999, Chen et al., 2000, Kaplan & Shkolnikov, 2002, Banerjee et al., 2002). Recently, such a prediction has been experimentally verified by observing the angular patterns of the harmonics for a relatively low laser intensity of 4.4x1018 W/cm2 (Lee et al., 2003a, 2003b). Esarey et al. (Esarey et al., 1993a) has investigated the plasma effect on RNTS and presented a set of the parameters for generating a 9.4-ps x-ray pulse with a high peak flux of 6.5x1021 photons/s at 310 eV photon energy using a laser intensity of 1020 W/cm2. Ueshima et al. (Ueshima et al., 1999) has suggested several methods to enhance the radiation power, using particle-incell simulations for even a higher intensity. Kaplan and Shkolnikov et al. ( Kaplan & Shkolnikov, 2002) proposed a scheme for the generation of zeptosecond (10-21 sec) radiation using two counter-propagating circularly polarized lasers, named as lasertron.
Recently, indebted to the development of the intense laser pulse, experiments on RNTS radiation have been carried out by irradiating a laser pulse of 1018–1020 W/cm2 on gas jet targets (Kien et al., 1999, Paul et al., 2001, Hertz et al., 2001). A numerical study in the case of single electron has been attempted to characterize the RNTS radiation (Kawano et al., 1998) and a subsequent study has shown that it has a potential to generate a few attosecond x-ray pulse (Harris & Sokolov, 1998). Even a scheme for the generation of a zeptosecond x-ray pulse using two counter propagating circularly polarized laser pulses has been proposed (Kaplan & Shkolnikov, 1996).
In this chapter, we concern RNTS in terms of the generation of ultrafast X-ray pulses. The topics such as fundamental characteristics of RNTS radiations, coherent RNTS radiations, effects of the high-order fields (HOFs) under a tight-focusing condition, and generation of an intense attosecond x-ray pulse will be discussed in the following sections.
2. Fundamental characteristics of RNTS radiations
2.1. Electron dynamics under a laser pulse
The dynamics of an electron irradiated by a laser field is obtained from the relativistic Lorentz force equation:
The symbols used are: electron charge (e), electron mass (m e ), speed of light (c), electric field (
Eq. (1) can be analytically solved under a planewave approximation and a slowly-varying envelope approximation, which lead to the following solution (Esarey et al., 1993a):
intensity is low or
2.2. Harmonic spectrum by a relativistic nonlinear oscillation
Once the dynamics of an electron is obtained, the angular radiation power far away from the electron toward the direction,
where t’ is the retarded time and is related to t by
Then the angular spectrum is obtained by
Figure 5 shows how the spectrum is changed, as the laser intensity gets relativistic. The spectra were obtained by irradiating a linearly-polarized laser pulse on a counter-propagating relativistic electron with energy of 10 MeV, which is sometimes called as nonlinear Compton backscattering. One can see that higher order harmonics are generated as the laser intensity increase. It is also interesting that the spacing between harmonic lines gets narrower, which is caused by Doppler effect (See below). The cut-off harmonic number has been numerically estimated to be scaled on the laser intensity as
As shown in Fig. 6, the fundamental frequency,
In the case of an electron initially at rest (
Note that the amount of the red shift is different at different angles. The dependence on the laser intensity can be stated as follows. As the laser intensity increases, the electron’s speed approaches the speed of light more closely, which makes the frequency of the laser more red-shifted in the electron’s frame. No shift occurs in the direction of the laser propagation. The parasitic lines in the blue side of the harmonic lines are caused by the different amount of the red-shift due to rapid variation of laser intensity.
The angular distributions of the RNTS radiations show interesting patterns depending on harmonic orders [Fig. 7]. The distribution in the forward direction is rather simple, a dipole radiation pattern for the fundamental line and a two-lobe shape for higher order harmonics. There is no higher order harmonic radiation in the direction of the laser propagation. In the backward direction, the distributions show an oscillatory pattern on
For a laser intensity of 1020 W/cm2 (a o =6.4), the harmonic spectra from an electron initially at rest are plotted in Fig. 8 for different laser polarizations. In the case of a linearly polarized laser, the electron undergoes a zig-zag motion in a laser cycle. Thus the electron experiences severer instantaneous acceleration than in the case of a circularly polarized laser, in which case the electron undergoes a helical motion. This makes RNTS radiation stronger in intensity and higher in photon energy in the case of a linearly polarized laser. The most different characteristics are the appearance of a large-interval modulation in the case of a linear polarization denoted as ‘1’ in Fig. 8 (a). This is also related with the zig-zag motion of the electron during a single laser cycle. During a single cycle, the electron’s velocity becomes zero instantly, which does not happen in the case of the circular polarization. Thus a double peak radiation appears in a single laser cycle as shown in Fig. 9 (a). Such a double peak structure in the time domain makes the large-interval modulation in the energy spectrum. In both cases, there are modulations with small-interval denoted by ‘2’ in the Fig. 8 (a) and (b). This is caused by the variation of the laser intensity due to ultra-short laser pulse width. Such an intensity variation makes the drift velocity different for each cycle then the time interval between radiation peaks becomes different in time domain, which leads to a small-interval modulation in the energy spectrum.
The temporal structure or the angular power can be seen in Fig. 9. As commented above, in the case of the linear-polarization, it shows a double-peak structure. One can also see that the pulse width of each peak is in the range of attosecond. This ultra-short nature of the RNTS radiation makes RNTS deserve a candidate for as an ultra-short intense high-energy photon source. The pulse width is proportional to the inverse of the band width of the harmonic spectrum, and thus scales on the laser intensity as
The zig-zag motion of an electron under a linearly polarized laser pulse makes the radiation appears as a pin-like pattern in the forward direction as shown in Fig. 10 (a). However the radiation with a circularly polarized laser pulse shows a cone shape [Fig. 10 (b)] due to the helical motion of the electron. The direction of the peak radiation,
3. Coherent RNTS radiations
In the previous section, fundamental characteristics of the RNTS radiation are investigated in the case of single electron. It was also shown that the RNTS radiation can be an ultra-short radiation source in the range of attosecond. To maintain this ultra-short pulse width or wide harmonic spectrum even with a group of electrons, it is then required that the radiations from different electrons should be coherently added at a detector. In the case of RNTS radiation, which contains wide spectral width, such a requirement can be satisfied only if all the differences in the optical paths of the radiations from distributed electrons to a detector be almost the same. This condition can be practically restated: all the time intervals that scattered radiations from different electrons take to a detector,
3.1. Solid target
In the case of a solid target for distributed electrons (Lee et al, 2005), the time intervals that radiations take to a detector can be readily obtained with the following assumptions as the first order approximation: (1) plane wave of a laser field, (2) no Coulomb interaction between charged particles, thus neglecting ions, and (3) neglect of initial thermal velocity distribution of electrons during the laser pulse. With these assumptions, the radiation field
The condition for a coherent superposition in the z-x plane can now be formulated by setting Eq. (11) to be less than or equal to the pulse width of single electron radiation,
Equation (13) manifests that RNTS radiations are coherently added to the specular direction of an incident laser pulse off the target, if the target thickness, T hk is restricted to
Since the incident angle of the laser pulse can be set arbitrarily, one can set
The spectra in Fig. 13 (a) were obtained for a thick cylindrical target of 1 m in thickness and radius, and 1018 cm-3 in electron density under the normal incidence of a laser on its base. The spectrum in Fig. 13 (b) is for the case of oblique incidence on an ultra-thin target of 7 nm in thickness, 5 m in width, 20 m in length, 1016 cm-3 in electron density, and
The temporal shape at the specular direction for the case of coherent condition [Fig. 13 (b)] is plotted in Fig. 14 (a), which shows an attosecond pulse. The direction-matched coherent condition also leads to a very narrow angular divergence as shown in Fig. 14 (b). It should be mentioned that with a thick cylinder target, the radiation peak appears at
3.2. Electron beam
Exploiting a solid target for a coherent RNTS radiation may involve a complicated plasma dynamics due to an electrostatic field produced by a charge separation between electrons and ions. Instead, an idea using an electron beam has been proposed (Lee et al., 2006).
Following similar procedure in the previous section, the RNTS harmonic spectrum can be obtained with that from an electron at center and its integration over initial electron distributions with phase relationships as
where the following parameters are used: the number of electrons (N), radius (R), length (L), fractional energy spread (
In Eqs. (23) and (24), the subscript, ‘s’ represents either ‘g’ or ‘b’.
For the radiation scattered from an electron beam to be coherent up to a frequency
and the other for the restriction on the electron beam parameters:
Eq. (26) also shows why the direction of the beam velocity (
From the single electron calculation (the radiation from an electron of o = 20 under irradiation of a circularly polarized laser of a o = 5), it has been found that the peak radiation appears at
4. Effects of the high-order laser fields under tight-focusing condition
Paraxial approximation is usually used to describe a laser beam. However, when the focal spot size gets comparable to the laser wavelength, it cannot be applied any more. This is the situation where the RNTS actually takes place. A tightly-focused laser field and its effects on the electron dynamics and the RNTS radiation will be discussed in this section.
4.1. Tightly focused laser field
The laser fields propagating in a vacuum are described by a wave equation. The wave equation can be evaluated in a series expansion with a diffraction angle,
The laser fields are written up to the 5th order in
The zeroth order term in
Because ε is proportional to 1/w 0 , the high order fields (HOFs) become larger for smaller beam waist. Figure 17 shows that E y and E z get stronger as w o decreases. The peak field strengths of E y and E z amount to 2.6% and 15% of E x at w o = 1 μm, respectively. In the case of a counter-interaction between an electron and a laser pulse, HOFs much weaker than the zeroth-order field does not affect the electron dynamics. However, when the relativitic electron is driven by a co-propagating laser pulse, weak HOFs significantly affect the electron dynamics and consequently the RNTS radiation.
4.2. Dynamics of an electron electron with a tightly focused laser
The dynamics of a relativistic electron under a tightly-focused laser beam is investigated by the Lorentz force equation [Eq. (1)]. One can consider two extreme cases of interaction geometry as shown in Fig. 18. The counter-propagation scheme, or Compton back-scattering scheme is usually adopted to generate monochromatic x-rays. It has been shown in the previous section that the co-propagation scheme is more appropriate to generate the coherent RNTS radiation. For such schemes, the effect of HOFs will be investigated.
In the z-x plane,
However, when the electron co-propagates with the laser pulse (
The relativistic gamma factor is given by
Now the change of the gamma factor becomes significant and
4.3. Radiation from a co-propagating electron with a tightly focused laser
As shown in Sec. 2, nonlinear motion of an electron contributes to the harmonic spectra, or ultra-short pulse radiation. Thus it can be inferred that the enhancement of nonlinear dynamics with inclusion of HOFs, increase of gamma factor (or electron energy variation) by the first order field and the transverse acceleration by the second-order field, might enhance the RNTS radiations.
Figure 20 shows the effect on RNTS as HOFs are included. Figure 20 is the RNTS radiation obtained from the dynamics in Fig. 19 (b). The number on the plot is the number of order of HOF, up to which HOFs are included. Note that as the HOFs are included, the pulse duration gets shorter and the spectrum gets broader according. The radiation intensity is also greatly enhanced. It should be notices that it is the transverse acceleration that significantly enhances the RNTS radiations. Figure 20 shows the shorter pulse width or wider spectral width by a factor of more than 5 and the higher intensity by an order of magnitude.
In the tight-focusing scheme, the strong radiation can be assumed to be generated within the focal region. That is, the electron radiates when it passes through Rayleigh range approximately in length scale, or
The average photon energy can be approximated to the mean photon energy and then it can be estimated from the inverse of the pulse width according to the Fourier transform as
This shows that the average photon energy scales inversely with the square of the beam waist, as also shown later. The total radiation power, radiated power integrated over whole angle, from an accelerated electron is described as (Jackson, 1999)
The radiated powers in the electron’s own time (retarded time)
On the other hand, when
respectively. The effective fields for different harmonic orders can be approximated as
Then the total radiation energy can be calculated by the multiplication of the pulse width and the power. Using the estimated
With the effective fields [Eqs. (49)-(51)] and the radiation energy relations of Eqs. (47) and (48), the radiation energy for the field of certain order is evaluated as follows:
The effective strength of the zeroth-order field is much smaller than those of HOFs because
Since the transverse field is more effective than the longitudinal field for scattering radiation,
5. Generation of an intense attosecond x-ray pulse
In the case of the interaction of an electron bunch with a laser, there are three major interaction geometries: counter-propagation (Compton backscattering), 90 -scattering, and co-propagation (0 -scattering) geometry. To estimate the pulse width of the radiation in these interaction geometries, we need to specify the length (Lel) and diameter (LT) of the electron bunch, the pulse length (Llaser) of the driving laser and the interaction length (confocal parameter, Lconf). At current technology, the diameter of an electron bunch is typically 30 μm. The typical pulse widths of an relativistic electron bunch (v~c) and femtosecond high power laser are about 20 ps and 30 fs, respectively, which corresponds to Lel~ 6 mm and Llaser ~ 9 μm in length scale. For the beam waist of 5 μm at focus, the confocal parameter is Lconf ~ 180 μm for 800 nm laser wavelength. In other words, Le>> Lconf>> Llaser are rather easily satisfied. In the following, we confine our simulation to this case. In this situation, the radiation pulse width is then roughly estimated to be Δtcounter~2Lconf/c=600 fs, Δt90~(LT+Llaser)/c=130 fs, and Δtco~ Llaser/c=5 fs, for counter-propagation, 90 -scattering, and co-propagation geometry, respectively. When we consider the aspect of the x-ray pulse duration, the co-propagation geometry is considered to be adequate as an interaction geometry.
The co-propagating interactions between a femtosecond laser pulse and an electron bunch were demonstrated as series of simulations. The simulations are similar to those in Section 4. The difference is that the electron bunch and the pulsed laser, co-propagating along the +z direction, meet each other in the center of the tightly focused region (z = 0). The interaction between electrons is ignored because it is much weaker than the interaction between the laser field and electrons.
Figure 21 shows the temporal structure and the spectrum of the radiation from the interaction between an electron bunch and a co-propagating tightly focused fs laser. The pulse width and the wavelength of the laser is 5 fs FWHM and 800 nm (1.55 eV), respectively. The laser is focused to a beam waist of 5 μm at z=0 with an intensity of 2.1x1020 W/cm2 (
Figure 22 shows the total radiated energy and the averaged photon energy. The calculations have been done for various electron energies. To check the effect of the HOFs, the simulation have been carried out for various combinations of high order fields: (1) the zeroth-order only, up to (2) the first-, (3) the second-, and (4) the seventh-order fields. The electron bunch consists of
Figure 22 (a) also shows the
The study about an electron acceleration using a tightly-focused laser field by Salamin et al. showed that at least the fields up to the fifth-order are required to describe correctly the dynamics of an electron (Salamin & Keitel, 2002). In that calculation, a low energy (
As expected in Eq. (43), the average photon energy of the radiation is also proportional to
Because both of the photon energy
This represents a remarkable enhancement if we notice that when only the zeroth-order field is considered, the photon numbers in the co-propagation interaction are
The study also shows that the radiation efficiency of an electron increases as the focal spot gets smaller. The radiation efficiency can be defined by the radiated energy divided by both the number of the electron effectively participating in the radiation N eff and the energy of the driving laser I L :(I/N eff · I L ). Figure 23 shows the radiation efficiency of an electron with respect to the change of the beam waist. For fair comparison, the laser intensity is kept constant for different beam waists (a0 = 10 for all the data in Fig. 23). It is shown that the efficiency scales inversely to the 4th power of the beam waist. This scaling can be understood
as follows: The radius of the electron bunch is assumed to be much larger and the electrons are uniformly distributed so that the electron number participating in the radiation