A Photon Collider at Relativistic Intensity

Transcription

1 A Photon Collider at Relativistic Intensity Dissertation zur Erlangung des akademischen Grades Doctor rerum naturalium (Dr. rer. nat.) vorgelegt dem Rat der Physikalisch-Astronomischen Fakultät der Friedrich-Schiller-Universität Jena von M. Sc. Ben Liesfeld, geboren am in Frankfurt am Main

2 Gutachter 1. Prof. Dr. Roland Sauerbrey Institut für Optik und Quantenelektronik Friedrich-Schiller-Universität, Jena, Germany 2. Prof. Dr. Francois Amiranoff Laboratoire pour l Utilisation des Lasers Intenses Ecole Polytechnique, Palaiseau, France 3. Prof. Dr. Dino A. Jaroszynski Terahertz to Optical Pulse Source University of Strathclyde, Glasgow, United Kingdom Tag der letzten Rigorosumsprüfung: Tag der öffentlichen Verteidigung:

3 Abstract In this work a novel, powerful - but compact and versatile - experimental scheme, coined photon collider, is presented. It provides counter-propagating, focused, ultra-short laser pulses and the diagnostic means to accurately achieve spatial and temporal overlap of these pulses. The functionality and usability of the photon collider was demonstrated experimentally. For the first time an autocorrelation measurement was carried out at full relativistic intensity exactly in the laser focus and therefore under realistic experimental conditions. In a second experiment soft x-rays were generated through Thomson backscattering from laser-accelerated electrons. Our results represent the first observation of Thomson backscattered photons in an all-optical setup. The backscattered radiation was used to obtain time-resolved spectra of the electrons in the plasma during the acceleration process. To our knowledge these are the first time-resolved spectra of laser-accelerated electrons ever recorded. The photon collider, as a novel diagnostics tool, will shed light on such revolutionary plasma acceleration concepts like bubble acceleration. Its potential applications, however, have not yet been exploited with the presented experiments. Work is under way which will transform the photon collider to a true collider - of electrons. The creation of a laser-based positron source is envisioned considering the new high-intensity lasers currently under construction. Future experiments may also explore the realm of non-linear quantum electrodynamics in the optical regime.

4 Contents 1 Introduction 4 2 Laser-matter interaction Non-linear Thomson scattering Perturbative approximation of the scattering process Rigorous treatment of the scattering process Light propagation in a plasma Linear propagation Nonlinear propagation Electron acceleration Laser wakefield acceleration Self-modulated laser wakefield acceleration Direct laser acceleration Bubble acceleration Thomson backscattering in the linear regime Scattering from a single electron Scattering from a laser-accelerated electron bunch Single-shot autocorrelation at relativistic intensity The Jena Ti:Sa laser system (JETI) Protection of the laser system Experimental setup Alignment of the photon collider Exemplary scenarios Experimental results

5 4 Thomson backscattering from laser-accelerated electrons Experimental setup X-ray diagnostics Background and competing processes Temporal change of total backscattered radiation Electron spectra Future prospects of the photon collider Thomson backscattering as x-ray source Electron collider and positron production Non-linear QED Bibliography 90

6 1 Introduction A light intensity of I W/cm 2 represents a relativistic threshold in the nearinfrared wavelength regime. Electrons that are subjected to light intensities exceeding this value oscillate with relativistic velocities and a number of fascinating phenomena occur which are due to the relativistic nature of the electron motion. In fact, the term relativistic optics is used to describe this intensity regime [1]. On the basis of relativistic phenomena ultra-high laser intensities have opened up new exciting prospects in areas such as laser particle acceleration [2 6], laser-generated high energy photons [7 9] or laser induced nuclear reactions [8, 10 14]. All these past experiments were based on the use of a single, ultra-short and ultraintense laser pulse interacting with matter. In the course of this work experiments will be presented introducing a new concept into relativistic optics: the use of counterpropagating laser pulses. The analogy to conventional facilities is so prominent that we coined this concept the photon collider. The photon collider was developed by the high-intensity laser group at the Institut für Optik und Quantenelektronik, Jena, to be used with the Jena titanium:sapphire terawatt laser system (JETI). It represents one of the compact terawatt lasers operating at near-infrared wavelengths which have become popular in the past decade and which are capable of producing relativistic intensities. They are also referred to as table-top laser systems which means that they can still be installed on industry-standard optical tables and that they are small enough to be operated by medium-sized academic institutions. In 2002 intensities exceeding W/cm 2 were demonstrated with JETI [15]. To have some kind of reference to the real world considering this huge number: One would obtain such ultra-high intensity focusing all the sun light incident on earth onto a spot of 0.2 mm 2. A simplified diagram of the photon collider is shown in Fig In this scheme an ultra-short laser pulse is divided into two laser pulses by a beam-splitter. The two laser pulses are focused each by an off-axis parabolic mirror in a symmetric 4

7 1 Introduction Figure 1.1: The photon collider setup: A high-intensity laser pulse is divided into two laser pulses by a beam-splitter. Both pulses are focused by 45 off-axis parabolic mirrors onto the same point in space so that they are exactly counter-propagating. The time delay τ between the pulses may be adjusted by moving the beam-splitter. The counter-propagating pulses constitute a standing wave in the focus (electric field indicated in blue, magnetic field indicated in green). configuration where the plane of the beam-splitter is the symmetry plane. Why do counter-propagating laser pulses offer the means for novel laser plasma experiments? What are the powerful new aspects of the photon collider? It is, actually, quite obvious: Instead of a single ultra-intense laser pulse the scientist now has control over two. Since ultra-short laser pulses incident on matter are such a rich source of many kinds of radiation and accelerated particles (see e. g. [16]), the photon collider may combine any two of these powerful sources or a particular source and a laser pulse or just two laser pulses (and nothing in between - this will briefly be discussed in Ch. 5). A necessary prerequisite to make this work practically is that the location of the two laser pulses in space and in time can be controlled very accurately. The spatial accuracy must be as good as the focal radius ( 2 µm) and the temporal accuracy better than the pulse duration ( 80 fs). The first experiment presented here is dedicated to the task to demonstrate that the required accuracy can be achieved in the photon collider setup. To this aim a single-shot autocorrelation at relativistic intensity was carried out. Beyond the proof of principle of accurate control this represents the first pulse duration measurement at relativistic intensity in the laser focus [17]. The pulse duration, in turn, is important 5

8 1 Introduction to determine the laser intensity. The autocorrelation measurement is presented in detail in Ch. 3. The second, even more challenging experiment which was carried out with the photon collider is Thomson backscattering from relativistic laser-accelerated electrons - a combination of laser-based particle accelerator and counter-propagating ultra-intense laser pulse. This is, again, the first measurement of its kind in an all-optical setup [18]. The photon collider in the Thomson backscattering configuration is, however, not only a source of backscattered radiation - which may be quite powerful by itself - but also an on-line diagnostics tool for the energy distribution of laser-accelerated electrons. The first in situ and time-resolved electron spectra are presented in Ch. 4. During the short period of time that the photon collider has been operational (only little more than about a year) only a small fraction (exactly these two previously mentioned experiments) of the possible applications could be carried out. Ch. 5 will list three more examples and describe what lies ahead of us. Finally the now following chapter covers, subdivided into several short sections, some theoretical background necessary for the analysis of the experiments. 6

9 2 Laser-matter interaction The mechanisms that are involved in the interaction of an ultrashort laser pulse with matter, in our particular case with a gaseous target, are numerous and complex. In the context of this work we will focus on topics which are most important for the analysis of the experiments described later. Still, this section may not at all be called a complete description of the underlying physical principles. It will rather be a summary of well-established facts or elaborate on examples which are closely related to the experiments. First, the theory of non-linear Thomson scattering will be discussed, which is the basis for the single-shot autocorrelation experiment (Sec. 3), followed by short overviews over the propagation of a laser pulse in a plasma and electron acceleration mechanisms. At the end of this chapter I will return to a specific case of Thomson scattering: Thomson backscattering in the linear regime in an all-optical setup. This section will be the basis for the analysis of the backscattering experiment in Sec. 4 and makes use of almost all phenomena described up to then. 2.1 Non-linear Thomson scattering Thomson scattering is a classical description of scattering of photons from electrons in the regime where the scattered photon energy is small with respect to the electron energy. For electrons at rest this corresponds to photon energies of less than 511 kev. If photon energies above this threshold occur, one usually refers to this process as Compton scattering. In very intense laser fields the classical description of Thomson scattering is no longer valid since the electron motion becomes relativistic and the influence of the magnetic field on the electron motion can no longer be neglected. The motion of electrons in intense fields is non-linear and therefore radiation at frequencies different from the fundamental laser frequency is emitted. This is the non-linear Thomson scattering regime. 7

10 2 Laser-matter interaction A parameter useful to determine the nature of the electron motion is the dimensionless laser field strength parameter a 0 = ee m e ω 0 c, where e is the electron charge, E the amplitude of the electric field, m the electron rest mass, ω 0 the laser frequency and c the speed of light. In practical units a 0 is given by a 0 = 0.85 λ I 1 µm W/cm, 2 where λ is the laser wavelength and I the laser intensity. Physically, a 0 represents the electron quiver momentum in an oscillating field in units of m e c. For a 0 1 the electron motion can be considered classical. An extensive relativistic theory of non-linear Thomson scattering has been made available by Esarey et al. [19] and Lau et al. [20]. In the moderately relativistic regime (a 0 < 1), however, the equations of motion of an electron in an electromagnetic wave may be solved by perturbation theory. It can then be shown that the non-linear Thomson scattered light at the second harmonic frequency is a non-linear effect of second order. This allows to revert to well-known autocorrelation techniques for the interpretation of our experimental results. I will therefore start with the rather intuitive perturbative treatment of the scattering process and subsequently discuss the rigorous scattering theory. Since this section only provides the background for the analysis of the experiment in Sec. 3 and 4 it seems appropriate to provide some reference values from the experiment for the following theoretical considerations. Simplifications arising from experimental conditions will be indicated as well. The laser light used in the experiment has a central wavelength of λ 0 = 2πc/ω 0 = 795 nm. This is located in the near-infrared, invisible for the human eye. At this wavelength a laser pulse of 85 fs duration contains about 30 oscillations of the electric field. Here, the pulse duration is typically determined as the full width at half maximum (FWHM) of a Gaussian envelope. One photon of this wavelength has the energy of 1.56 ev. The laser pulses may be described with reasonable accuracy with Gaussian beams. Please note that when focusing Gaussian beams the curvature of the wavefronts becomes negligible in the focal region and the wave may be approximated by a plane wave for many 8

11 2 Laser-matter interaction Figure 2.1: Observation of the Thomson scattered light: The electromagnetic wave propagates along the z-axis, the electric field is aligned with the x-axis and the electron oscillates in the x-z-plane. The electron orbit is shown in the reference frame of the electron (co-moving observer). The second-harmonic Thomson scattered light is observed along the x-axis. applications. The second harmonic of the fundamental laser frequency ω 0 has a wavelength of λ 2ω = 398 nm which is visible blue light. The laser light is linearly polarized and we will from now on assume that the electric field vector E of any electromagnetic wave discussed in this section is aligned with the x-axis. The propagation of the laser light is set to be along the z-axis which lets the magnetic field vector B be aligned with the y-axis. In our experiments we chose to observe the interaction region along the x-axis, the detector being placed in the x-z-plane. Fig. 2.1 shows the orientation of the axes and the position of the observer. The trajectory of an electron undergoing oscillations is indicated by a bold line (here, shown in the moving frame of the observer). This electron orbit, the characteristic figure 8 movement, will be described in more detail in Sec Perturbative approximation of the scattering process The perturbative treatment starts with the formulation of the equations of motion of the electron. The scaling of the radiation emitted by the electron will be evaluated without specifying if the exciting electromagnetic wave is a standing or traveling wave. The validity of the approximation will be verified computing higher order terms and an example of two counter-propagating Gaussian pulses will be discussed which will be directly applicable to the experiment in Sec. 3. 9

12 2 Laser-matter interaction Equations of motion The relativistic Lorentz equation for a single electron in an electromagnetic field reads: d (γmṙ) = e(e + ṙ B), (2.1) dt where γ is the Lorentz factor, m the electron rest mass, ṙ the electron velocity and e the electron charge. In the weakly relativistic regime the influence of the magnetic field on the motion of the electron will be small. Therefore its velocity along the z-axis will be small compared to its velocity along the x-axis. One may therefore write the Lorentz factor of the electron as γ = (ṙ/c) 2 1 ( ẋ2 /c 2 ) = const., (2.2) where the brackets in ẋ 2 indicate the average over one laser period. Since the electric field is aligned with the x-axis, E = (E x, 0, 0), it follows that ṙ B = ( żb y, 0, ẋb y ) = ( że x /c, 0, ẋe x /c)). The equations of motion can now be simplified to γmẍ = ee x + eż c E x, (2.3) γmÿ = 0, (2.4) γm z = eẋ c E x. (2.5) Neglecting the second term in Eq. 2.3 and using the initial conditions ẋ( ) = 0 and ż( ) = 0 and the electron being born in a maximum of the electric field, one can integrate Eq. 2.3: ẋ (0) = e γm t E(t ) dt. (2.6) Please note that the Lorentz factor γ was set to be constant in Eq Inserting Eq. 2.6 into Eq. 2.5 one obtains the acceleration of the electron along the z-axis z (1) = t e2 γ 2 m 2 c E(t) E(t ) dt. (2.7) 10

13 2 Laser-matter interaction The explicit form of E(t) has not yet been specified in any way which means that this formula is valid for traveling as well as for standing waves. Emitted radiation To calculate the radiation emitted by an electron at the second-harmonic (2ω 0 ) frequency one may use the classic relation (see Jackson [21, p. 665]) E T (ρ, θ, t) = e z(t ρ/c) sin θ, (2.8) 4πε 0 c 2 ρ where ρ is the distance of the observer to the source, θ the angle of observation with respect to the z-axis and ε 0 the vacuum permittivity. The experimental conditions for the detection of the Thomson scattered light as described in Sec. 3.3 are as follows: 1. The radiation is detected in the x-z-plane under an angle of θ = 90 with respect to the z-axis while the distance ρ to the source is constant (see Fig. 2.1). 2. The envelope of the electric wave E(t) is varying slowly compared to the carrier frequency and goes to zero for t. 3. The exposure of the detector (a CCD camera) is much longer than the timescale of the interaction. Taking condition (1) into account and inserting Eq. 2.7 into Eq. 2.8 the emitted intensity at 2ω 0 may be written as I 2 = 1 2 ε 0c E T 2 = Condition (2) allows to simplify e 6 (4π) 2 ε 0 c 5 γ 4 m 4 eρ 2 E(t) t E(t ) dt 2. (2.9) t E(t ) dt 1 iω 0 E(t). (2.10) 11

14 2 Laser-matter interaction Condition (3) means that one has to integrate the emitted radiation over time and then arrives at the following relation for the detected Signal S 2 : S 2 E 2 (t) 2 dt. (2.11) Eq states that in the weakly relativistic regime the intensity of the non-linear Thomson signal, S 2, scales with the square of the laser intensity, I 2. The 2ω-emission of the laser plasma can therefore be used as a second-order autocorrelation signal. Validity of the approximation An estimate of the next higher order terms in the expansion of z (Eq. 2.5) will now be made. In order to carry out the time integrals it will be assumed that the electric field has the form E(t) = E 0 cos ω 0 t, which is a valid assumption for traveling as well as for standing waves. The integration will be carried out from t = 0 using the initial conditions ẋ(0) = 0, x(0) = 0, ż(0) = 0, z(0) = 0. Starting again with Eq. 2.3 ẍ (0) = e γm E 0 cos ω 0 t = ωc γ a 0 cos ω 0 t, where the dimensionless parameter a 0 was inserted, integration yields ẋ (0) = c γ a 0 sin ω 0 t. Please note that it is still assumed that Eq. 2.2 is a valid approximation. This result in turn is inserted into Eq. 2.5 which yields z (1) = ω 0c 2γ 2 a2 0 sin 2ω 0 t (2.12) and thus ż (1) = c 4γ 2 a2 0(1 cos 2ω 0 t). Returning to Eq. 2.3 and using the last result one obtains ẍ (2) = ω 0c 8γ a [ 3 0 (a 2 0 8γ 2 ) cos ω 0 t a 2 0 cos 3ω 0 t ] 12

15 2 Laser-matter interaction which leads through integration to ẋ (2) = One can then calculate following Eq. 2.5 c 8γ a [ 3 0 (a 2 0 8γ 2 ) sin ω 0 t a 2 0 sin 3ω 0 t ]. z (3) = ω 0c [ 48γ 4 a2 0 (24γ 2 2a 2 0) sin 2ω 0 t a 2 0 sin 4ω 0 t ]. Comparing this result with Eq one sees that the corrections to the amplitude of the second-harmonic signal arising from higher-order expansion terms are of the order of a 2 0/12γ 2. Corrections to the observed power of the second-harmonic signal will be of the order of (a 2 0/12γ 2 ) 2. Please note that this estimate of accuracy reflects the accuracy of the perturbative approximation for which the basic assumption γ = const was used. In Sec the perturbative approximation will be compared to the exact solution. Example: focused counter-propagating Gaussian laser pulses The non-linear Thomson signal generated by two focused and counter-propagating Gaussian laser pulses will be examined. The notation that is used here is the most commonly used definition of Gaussian pulses (see [22, 23]). The fields of the laser pulses incident from the left and from the right will be called E L and E R, respectively. The laser pulse incident from the right reaches its focus at t = τ and its focus is not necessarily located at z = 0 but at some arbitrary position z = z 1 : ] ] w 0 E L (r, z, t) = E 0L [ w(z) exp r2 exp [ i kr2 w 2 (z) 2R(z) [ ( ) ] 2 t z/c exp exp [i(ω 0 t kz + ϕ(z)], τ L [ ] [ ] w 0 E R (r, z z 1, t τ) = E 0R w(z z 1 ) exp r 2 kr 2 exp i w 2 (z z 1 ) 2R(z z 1 ) [ ( ) ] 2 t τ + (z z1 )/c exp τ L exp [i(ω 0 (t τ) + k(z z 1 ) + ϕ(z z 1 )]. 13

16 2 Laser-matter interaction Here, E 0L and E 0R are the field amplitudes, w 0 is the waist of the Gaussian beams, w(z) = w (z/z0 ) 2 the beam diameter, z 0 the Rayleigh length, R(z) = z(1 + (z 0 /z) 2 ) the wavefront curvature, τ L the laser pulse length and ϕ(z) = arctan(z/z 0 ). τ L is related to the laser intensity pulse duration at FWHM by τ = τ L 2 ln 2. The integrand of the autocorrelation signal according to Eq is now calculated: E 2 (t) 2 = E 2 R + E 2 L + 2E R E L 2 = E 2 R 2 + E 2 L ER 2 E L 2 +E 2 RE 2 L +2E R E L E 2 R + E 2 R E 2 L + 2E 2 RE RE L + 2E RE LE 2 L + 2E R E L E 2 L. (2.13) This result can be simplified taking into account the limited spatial resolution of the applied optical imaging system (see Sec. 3.3). The autocorrelation obtained in the experiment is not interferometric, i. e. spatially fast oscillating terms in Eq can be neglected. Spatially fast oscillating terms are those in which terms like exp[ ikz] and exp[ikz] do not cancel out. Therefore the only remaining terms are: E 2 (t) 2 = E 2 R 2 + E 2 L ER 2 E L 2. In the experiment the signal is integrated over r which does not change the z-dependence of the expression. Carrying out the integration over time one finds S 2ω I 2 0L ( ) 2 + z 1 + z I 2 0R ( ) 2 z z 1 z 0 I 0L I 0R ( ) 2 z z1 /2 z z z 0 exp [ 4 τ 2 L ( z z1 /2 c τ 2 ) 2 ]. (2.14) This function and its application and some special cases will be discussed in Sec Rigorous treatment of the scattering process The non-linear Thomson scattering theory [19, 20, 24, 25] allows to compute the radiation spectrum of a single electron (with known trajectory r(t)) in an electromagnetic wave decomposed into harmonic frequencies. In the following, a short summary of 14

17 2 Laser-matter interaction this theory will be given. The energy radiated by a moving charge per unit solid angle Ω and per unit frequency ω is given by Jackson [21, p. 676]: d 2 dωdω W = e 2 ω 2 16π 3 ε 0 c n (n F(ω)) 2, (2.15) F(ω) = β(t) exp [iω(t n r(t)/c] dt, (2.16) where n the unit vector pointing in the direction of observation and β(t) the velocity of the electron normalized to the speed of light. With the help of Eq one may calculate the radiation spectrum of any given electron motion, if the trajectory and then β(t) is known. If the motion of the electron is a periodic function of time with period T and if a net displacement r 0 of the electron is associated with each period of motion, one may write β(t + mt ) = β(t), (2.17) r(t + mt ) = mr 0 + r(t), (2.18) m being an integer number. Then, F(ω) can be written using the periodicity of the motion [20]: F(ω) = m= F m δ(ω mω 1 ) (2.19) ω 1 = 2π T n r 0 /c, (2.20) F m = ω T 1 β(t) exp [imω 1 (t n r(t)/c] dt. 2π (2.21) 0 Here, ω 1 is the fundamental frequency of the emitted radiation. From Eq one can see that this frequency is not necessarily equal to the laser frequency. The latter is a valid approximation only in the limit of weakly relativistic interaction, as will be shown for the case of the traveling wave. The movement of a single electron in a strong plane electromagnetic wave has 15

18 2 Laser-matter interaction been discussed in the literature at great length [19, 20, 24, 26 28]. I will give a short summary of the results since this is one of the few cases where a closed form solution can be obtained. In the following, convenient normalizations are introduced: time is normalized by 1/ω 0 and distance by c/ω 0. Normalized coordinates are indicated by a hat. For a linearly polarized plane wave E = E 0 e x cos(ˆt ẑ), with E 0 the electric field amplitude, Eq. 2.1 can be written in the following form: d dˆt (γβ x) = a 0 (1 β z ) cos(ˆt ẑ), (2.22) d dˆt (γβ y) = 0, (2.23) d dˆt (γβ z) = a 0 β x cos(ˆt ẑ). (2.24) The electric field strength is expressed in terms of the dimensionless parameter a 0 = ee 0 /mω 0 c. The electron trajectory has a closed form solution for the initial conditions: ˆx(ˆt = 0) = 0, ŷ(ˆt = 0) = 0, ẑ(ˆt = 0) = ẑ in β x (ˆt = 0) = β x0, β y (ˆt = 0) = β y0, β z (ˆt = 0) = β z0, if the trajectory is expressed as a function of the phase parameter θ = ˆt ẑ. The problem can be further simplified assuming the initial conditions: β x0 = β y0 = 0, since then β y = 0 for all times. The initial phase under which the electron is born in the electromagnetic field is θ in = ẑ in. The trajectory of the electron is given by [20, 26] γ = γ 0 + (a2 0 sin θ sin θ in ) 2, 2γ 0 (1 β z0 ) γβ z = γ γ 0 (1 β z0 ), γβ x = 1 γ 0 (1 β z0 ) a 0(sin θ sin θ in ), 16

19 2 Laser-matter interaction (cos θ in cos θ) (θ θ in ) sin θ in ˆx = a 0, γ 0 (1 β z0 ) ˆt = (θ θ [ ( in) 1 + a2 0(1 + β z0 ) 1 1 β z sin2 θ in + a2 0(1 + β z0 ) 2(1 β z0 ) [ sin 2θ 4 )] + 2 cos θ sin θ in 3 sin 2θ in 4 The velocity components β x and β z are periodic functions of θ with a period of 2π. The time period of the electron motion ˆT can be calculated by subtracting ˆt(θ + 2π) from ˆt(θ): ˆT = 2π 1 β z0 ]. [ ( )] 1 + a2 0(1 + β z0 ) sin2 θ in. (2.25) Fig. 2.2 shows sample trajectories of an electron in a plane electromagnetic wave for various θ in and a 0. In Fig. 2.2a) and c) the trajectory in the laboratory frame is shown while in Fig. 2.2b) a reference frame is introduced (primed coordinates) which moves with the electron along the z-axis. The moving frame representation makes the figure-8 movement of the electron visible. (β z0 = 0, γ 0 = 1) and θ in = 0 the trajectory simplifies to ˆx = a 0 (1 cos θ), ẑ = a2 0 4 (θ 1 sin 2θ). 2 For an initially resting electron The fact that the amplitude of the motion scales with a 0 along the x-axis but scales with a 2 0 along the z-axis is illustrated by Fig. 2.2a) and b). For increasing a 0 the acceleration along the z-axis, the component induced by the magnetic field, becomes dominant. The comparison of the exact solution of the equations of motion to the perturbative solution from the previous section is shown in Fig For a 0 = 0.1 the perturbative and the exact solution for β z are almost identical (Fig. 2.3a)). This is no longer the case when a 0 approaches unity. Inserting the Lorentz factor from the exact solution averaged over one period γ into Eq. 2.12, the amplitude of the oscillating acceleration are approximately equal but the frequencies contained in the oscillation are not (Fig. 2.3b)). The total radiated power may therefore be estimated correctly by the perturbative solution but the spectral distribution will differ. The spectral properties of the emitted radiation may be derived from the exact 17

20 2 Laser-matter interaction Figure 2.2: a) Trajectories of a free electron in a traveling electromagnetic wave for different laser field strengths a 0 = 1 and a 0 = 2 calculated for the initial conditions θ in = 0 and β z0 = 0 in the laboratory frame. b) The trajectories in the moving frame of the electron (indicated by the primed coordinate) exhibit the characteristic figure-8 movement. Laser field strength and initial conditions corresponding to a) were used. c) Electron trajectories in the laboratory frame for a 0 = 1, β z0 = 0 and θ in = 0.1, 0.2 and

21 2 Laser-matter interaction Figure 2.3: Comparison of perturbative (solid line) and exact solution (dashed line) of β z (t) for θ in = 0. a) a 0 = 0.1. Perturbative and exact solution are almost identical. b) a 0 = 0.5. Perturbative and exact solution yield approximately the same amplitude of β z which means that the calculated total radiated power will be similar. The frequencies contained in the oscillation differ, however. solution. Using the relation dθ/dˆt = γ 0 (1 β z0 )/γ one can transform Eq into F m = ω 1 ω 0 θ in +2π θ in γβ(θ) exp [ ] n ˆr(θ) 2πimˆt(θ) dθ, (2.26) ˆT n r 0 which is more suitable for calculations using the parametrized equations. For a given initial phase θ in of the electron one may now calculate the power of each of the harmonics emitted into a certain direction. This can generally be done solving the equations of motion numerically and applying Eq to the numerical solution, but since the exact solution is known, the radiated power may be calculated with much less effort through Eq For simplicity, I will confine myself to the configuration of the experiment described in the beginning of Sec. 2.1, i. e. observing under an angle of 90 with respect to the z-axis and along the axis of the electric field (x-axis). The dependency of the frequency of the emitted radiation on the initial phase and on the laser field strength a 0 according to Eq is shown in Fig. 2.4a). Here, the second-harmonic frequency is given in units of the laser frequency ω 0. The frequency of the second harmonic drops with increasing laser field strength - a circumstance which is not quite intuitively comprehensible. It is due to the fact that the period of the electron motion ˆT becomes elongated for large a 0 (cf. Eq. 2.25). The frequency of the emitted radiation also strongly depends on the initial phase θ in of the electron. 19

22 2 Laser-matter interaction Figure 2.4: Dependency of the frequency ω = mω 1 /ω 0 of the emitted radiation on the laser field strength a 0. a) Magnitude of the second-harmonic frequency (m = 2) emitted by the electron over laser field strength a 0 and initial phase θ in. b) Magnitude of the fundamental (red) and the first two harmonic frequencies (green and blue). The limiting lines of the areas correspond to an initial phase θ in = π/2 (lower line) and θ in = 3π/2 (upper line). The grey bar represents the frequency band of the interference filter used in the experiment. In Fig. 2.4b) the frequency bands of the fundamental and the first two harmonic frequencies over a 0 are shown. A single electron born at a certain phase in the electromagnetic wave still emits radiation at discrete frequencies but a number of electrons born each at different initial conditions lead to a broader emission band structure. At higher intensities the frequency bands of the harmonics overlap. In the limit of large a 0 this finally leads to an almost continuous emission. In our experiment the non-linear emission of the laser plasma is observed through a narrowbandwidth interference filter with a fixed central wavelength of 2ω 0. This narrow band of detection is indicated with a grey bar in Fig. 2.4b). In order to estimate the dependency of the detected radiation on a 0 the initial phase of the electrons emitting at the fixed frequency 2ω 0 and the power of the emitted radiation were determined. The result is displayed in Fig The contributions of the first three harmonics were calculated (dotted lines) as well as the sum of all contributions (solid line). For small a 0 the total radiated power at 2ω 0 is proportional to a 4 0 as was expected from the perturbative solution. For a 0 approaching unity the radiated power increases faster, proportional to a 5 0, but beyond a 0 = 1 a maximum is reached and the radiated power then decreases. This may have an impact on the 20

23 2 Laser-matter interaction Radiated power / arb. u ~a 4 0 ~a 5 0 m = 2 m = m = 4 a 0 Figure 2.5: Dependency of the power emitted per unit solid angle at the constant frequency 2ω 0 on the laser field strength a 0. The contributions from the harmonic frequencies mω 1 for m = 2, 3, 4 are displayed as dotted lines. The solid line represents the sum of these contributions. For a 0 < 0.5 the radiated power is proportional to a 4 0, for a 0 approaching unity this changes to a 5 0. Above a 0 = 1 a maximum is reached and the radiated power decreases. measured autocorrelation signal: The autocorrelation signal of laser pulses with a peak field strength of about a 0 = 1 may lead to shorter pulse lengths than they are in reality. Laser pulses exceeding a 0 = 1.3 on the other hand will appear broader. Until now, the movement of an electron in an intense traveling electromagnetic wave was discussed. However, counter-propagating laser pulses form a standing wave in the focal region. The description of the electron motion in a standing wave turns out to be more complicated, and appears to be erratic [29]. The solution cannot be given in closed form except for very particular initial conditions. The equations of motion for a standing wave are: d dˆt (γβ x) = 2a 0 (cos ˆt cos ẑ β z sin ˆt sin ẑ), d dˆt (γβ y) = 0, d dˆt (γβ z) = 2a 0 β x sin ˆt sin ẑ. These equations can no longer be parametrized introducing a universal phase θ 21

24 2 Laser-matter interaction Figure 2.6: a) Traveling wave: Electrons (indicated by circles) born at different moments in time (denoted by A, B, C ) and at different positions along the z-axis experience the same initial conditions in the electromagnetic field since only the phase relative to the wave is relevant. b) Standing wave: The initial conditions of electrons born in a standing wave cannot be reduced to a single parameter. and a systematic analytical analysis is impossible. Fig. 2.6 illustrates the qualitative difference between the traveling and the standing wave problem: Fig. 2.6a) shows electrons (circles) which are born in the electromagnetic wave at different positions z 0. One may find an appropriate time t 0 for each of the electrons such that they experience the same initial conditions and therefore one may attribute to these electrons the universal initial phase θ in. This is no longer possible for a standing wave (see Fig. 2.6b)) because the maximum amplitude of the field depends on z and on z only. Summary The rigorous non-linear Thomson scattering theory yields some important results which cannot be obtained from perturbation theory: 1. The phase at which the electron is born in the electromagnetic wave strongly determines the electron trajectory and therefore the radiation spectrum. 2. The fundamental frequency of the emitted radiation depends on the laser field strength a 0 and on the initial conditions. 3. The radiated power is shifted to higher harmonics for larger a 0. In the limit a 0 1 an almost continuous spectrum is obtained. 22

25 2 Laser-matter interaction Especially result (1) makes the use of particle-in-cell simulations (PIC) necessary. The temporal step size used in these simulations must be small enough to calculate the higher harmonics emission. This clearly means a large increase in required computation time and memory size. The effect of an infinitely extended plane and intense electromagnetic wave on a single electron has been discussed. The results permit to describe the radiation spectrum of a single electron and to estimate the emitted radiation of a population of plasma electrons. I will now turn to the more complex interaction of a focused laser pulse with a gas-jet. 2.2 Light propagation in a plasma A laser plasma can be treated with the so-called two-fluid theory, one fluid being the light plasma electrons, the other fluid being the heavy ions. The light electrons will exhibit rather fast oscillations compared to the slower (acoustic) oscillations supported by the ions. In our case, where the time scale of laser-matter interaction is determined by the laser pulse length which is much smaller than 1 ps, one may neglect the slow movement of the ions and consider the ion density as constant [30]. The oscillations which an electron plasma can support are characterized by the plasma frequency ω p = (e 2 n e /ε 0 m) 1/2, where n e is the electron density in the plasma, and ε 0 the vacuum permittivity. The dispersion relation of the linear wave equation in a uniform plasma is given by ω 2 = ω 2 p + k 2 c 2 (2.27) The wave number k becomes imaginary for ω < ω p which means that the wave is absorbed since the electrons are responding to the light wave in a characteristic time ωp 1. This condition defines a critical electron density above which propagation is impossible for a light wave of given frequency. The critical density for laser light of λ = 795 nm, as was used in the experiments described below, is n c = /λ 2 [µm 2 ] cm 3 = cm 3. Target material is commonly called underdense (overdense) if its density is below (above) the critical density. When an ultra-short laser pulse of an intensity I > W/cm 2 impinges on a He gas-jet those parts of the pulse exceeding W/cm 2 ionize the gas through 23

26 2 Laser-matter interaction multi-photon and tunnel ionization. This means the rising edge of the laser pulse will already have fully ionized the He gas before the main part of the pulse interacts with the gas-jet. The plasma and variations of its electron density in turn change the propagation of the laser pulse. One can attribute a refractive index η to the plasma and using Eq one can write η = ck/ω = 1 ω 2 p/ω 2 = 1 n e /n c. (2.28) Linear propagation If the plasma density is sufficiently low, the propagation of an intense laser pulse through a gaseous medium may be considered linear under certain conditions. The criterion for linear propagation will be the phase shift φ that the laser pulse undergoes due to ionization. This regime will become important for experiments where effects of propagation must be excluded and will therefore be discussed in the following. In the focal region of a Gaussian beam one may approximate the beam with a plane wave E(z, t) = E 0 exp iφ, where φ = ω 0 t k 0 η(t)z is the phase of the wave, k 0 the wavenumber in vacuum and η(t) the time-dependent index of refraction of the plasma. One has to assume that the refractive index depends on time since it depends on the ionization state of the gas. For ω ω p and therefore n e n c one may approximate η(t) 1 1 n e (t). 2 n c The rate of change of the electron density may be written as dn e /dt = R(t)(n i n e /Z), where R is a function of the pulse envelope, n i the neutral gas density and Z the ionization state of the gas atoms (Z = 2 for He gas). Since the early rising edge of a relativistic laser pulse fully ionizes the He gas one may assume n e = 2n i and thus dn e /dt = 0 for the main interaction. The phase difference between the part of the wave propagating through vacuum and the part propagating through the plasma is then φ = k 0 n i n c z, (2.29) where z is the extension of the plasma along pulse propagation. While this is valid 24

27 2 Laser-matter interaction Figure 2.7: a) Ponderomotive potential of a Gaussian laser pulse perpendicular to the axis of propagation (z-axis). b) Corresponding ponderomotive force vector field. The axes are normalized to the radius (half width at half maximum) of the laser pulse. In b) the extension of the laser pulse along the z-axis was set equal to its extension perpendicular to the z-axis which is not true for the laser parameters used in the experiments (focal radius of the order of few microns, pulse length of the order of 100 fs corresponding to 30 µm). for a homogeneous plasma, in the case of a gas-jet with Gaussian profile as used in the experiment n i z must be replaced by the integration n i (z)dz for an accurate estimate. For the propagation to be considered unaltered the phase shift must hold the condition φ < 1. For a homogeneous plasma of 1 cm length and laser parameters as used in our experiment (n c = cm 3 ), the gas density must be n i < cm 3 to ensure that the refractive index of the plasma does not alter the propagation of the laser pulse significantly Nonlinear propagation At high intensities a laser pulse may have a large impact on the laser plasma density distribution. The plasma index of refraction is then dependent on the laser intensity which gives rise to non-linear effects. 25

28 2 Laser-matter interaction The Ponderomotive force The laser pulses generating relativistic intensities are tightly focused - quite in contrast to the conditions in the earlier simplified discussions of electrons in plane waves. Averaging the equation of motion of an electron in such an inhomogeneous field over the fast laser oscillations one arrives in the case of low laser field strength a 0 1 at the description of a force acting on the electron [30] e2 F p = 4mω 2 E 2. (2.30) This force is called ponderomotive force and is directed along the gradient of the laser intensity. For a Gaussian beam profile this means that electrons are expelled from the optical axis. The ponderomotive vector field and the corresponding potential φ, which can be obtained from F p = φ, are visualized in Fig The concept of the ponderomotive force remains valid even for higher intensities as was shown in [29, 31]. In the fully relativistic description an additional factor of 1/ γ, where γ is averaged over the fast laser oscillations, is introduced: e 2 F p = 4 γ mω 2 E 2. In a quasi-static picture the electrons are driven away from the beam axis by the ponderomotive force until the resulting electric field between electrons and (immobile) ions counteracts this effect. At high laser intensities the ponderomotive force can lead to complete cavitation [32]. In the light of these findings one has to reformulate Eq since the electron density now depends on the distance from the optical axis r: η = 1 n e (r)/n c. A radial gradient of the electron density n e / r < 0 as generated by the ponderomotive force implies a gradient η/ r < 0 and this in turn means that the phase velocity v ph = cη on axis is smaller than farther outside. The wavefronts are bent and the plasma acts like a positive lens: The laser pulse is focused. This effect is called ponderomotive self-focusing. 26

29 2 Laser-matter interaction Relativistic self-focusing The electrons in a tightly focused laser pulse are oscillating with relativistic velocities (as shown for intense plane waves). Their velocity will depend on the local laser field strength and therefore the Lorentz factor will exhibit a spacial dependency γ = γ(r). The properties of the electrons oscillating in the field are changed and therefore the plasma frequency ω p is altered. Replacing ω p with ω p /γ(r) the refractive index of the plasma now reads η = 1 ω 2 p/(γ(r)ω 2 ). For moderate intensities (a 0 < 1) the electrons are mainly oscillating along the electric field vector (see Sec ) and γ = 1 + (p/mc) a 2 /2. Therefore a gradient in intensity a 2 / r < 0 again leads to a refractive index profile η/ r < 0 which leads to self-focusing. Summarizing all these contributions to the refractive index in the limit ω 2 p ω 2 one may write η(r) 1 ω2 p (1 a2 2ω δn ), n 0 where the a 2 /2 term represents relativistic self-focusing and the δn/n 0 term ponderomotive self-focusing. Here, δn represents a perturbation of the homogeneous plasma density n 0 induced by the ponderomotive force such that the resulting plasma density n e may be written as n e = n 0 + δn. Relativistic self-focusing sets in as soon as the laser power exceeds a certain critical power P c. frequency only [33]: This critical power depends on the ratio of laser frequency to plasma P c 16.2 ω2 [GW] = 16.2 n c [GW]. (2.31) ωp 2 n e This critical power is not very high compared to the power delivered by today s multiterawatt table-top lasers. For a given laser power increasing the plasma density results in lowering the threshold for relativistic self-focusing. This process has, of course, its limit in the critical density. For a laser pulse with a pulse energy of 600 mj and a pulse duration of 80 fs focused into a He gas-jet with a particle density of cm 3 (being fully ionized by the rising edge of the pulse) the critical power P c 0.6 TW is exceeded by a factor of

30 2 Laser-matter interaction a) n e,i b) I z- ct z- ct Figure 2.8: a) LWFA: A short laser pulse (cτ λ p ) shown as a dashed line drives a plasma wave. b) SM-LWFA: An initially long laser pulse (dashed line) breaks up into a train of shorter pulses which match the condition of LWFA and resonantly drives a plasma wave (after [41]). 2.3 Electron acceleration Since many years ago laser plasmas have been seen as an ideal medium for high-field, but compact-sized (millimeter range) accelerators [34, 35]. The plasma as an ionized medium may sustain much higher fields than is possible to generate with conventional accelerator technology where material breakdown imposes a limit at about 55 MV/m [36]. Electric fields in the GV/m range were inferred to be present in laser plasmas observing the acceleration of injected electrons [37 40]. In many experiments during the past two decades a variety of acceleration mechanisms was identified. A selection of these mechanisms will briefly be described in the following with the focus on schemes where only a single laser pulse is needed to accelerate initially resting electrons to relativistic energies. All these schemes have in common that a laser pulse is focused into a gas-jet generating an underdense plasma Laser wakefield acceleration An ultra-short laser pulse impinging on a plasma (the previously mentioned gas-jet is quickly ionized) drives a plasma wave with frequency ω p. The plasma wave follows the driving laser pulse with a phase velocity determined by the laser pulse group velocity v plasma ph = v g = cη with η as defined in Eq The electric fields associated with the plasma wave are now longitudinal. An electron can ride on the plasma wave and be accelerated to relativistic energies in the direction of laser propagation. 28

31 2 Laser-matter interaction This process is most efficient when the laser pulse length cτ, where τ is the pulse duration, is shorter than the plasma wavelength λ p = 2πc/ω p. This condition is depicted in Fig. 2.8a). When a large number of electrons acquire a velocity close to the phase velocity of the plasma wave, wave-breaking occurs [41]. The fast electrons are surfing on the wake of the plasma wave with velocity v plasma ph. The name for this process coined by Tajima and Dawson [34] is laser wakefield acceleration (LWFA) Self-modulated laser wakefield acceleration If the laser pulse length is longer than the plasma wavelength cτ > λ p, the laser pulse undergoes a self-modulation instability. The leading edge of the laser pulse drives a plasma wave. The electron density modulation of the plasma wave in turn represents a periodic modulation of the refractive index. It acts on the long laser pulse such that the pulse is self-modulated and breaks up into a train of short pulses (see Fig. 2.8b)). These shorter pulses now match the conditions for LWFA and can resonantly drive a plasma wave. The self-modulated laser wakefield acceleration (SM-LWFA) is not as efficient as the pure LWFA, but still high-energy, even quasi-monoenergetic electron beams can be generated Direct laser acceleration Another acceleration process which is quite different in nature to the wakefield acceleration is direct laser acceleration (DLA). It was first proposed by Gahn et al. [42] and is closely related to the formation of a relativistic channel. The ponderomotive force expels electrons from the laser beam axis and generates a radial quasi-static electric field. Electrons which are accelerated along the laser propagation generate - being a current of charges - an azimuthal magnetic field. Both these fields combined result in an effective potential well for relativistic electrons. Electrons trapped in this well will oscillate at the frequency ω β = ω p /(2 γ), the betatron frequency [43]. If the trapped electron is moving fast enough along the laser propagation, the laser oscillations may be in phase with the betatron oscillations in the frame of the electron. In this case, an efficient energy coupling is possible. The energy gained by the electron in this process directly results from the laser field and therefore the name direct laser acceleration is appropriate. 29

32 2 Laser-matter interaction a) 40 b) dn/de / 10 5 msr/mev T = 7 MeV Energy / MeV dn/de / 10 5 msr/mev Energy / MeV Figure 2.9: a) Typical exponential electron spectrum measured with a high-resolution magnet spectrometer. b) Electron spectrum recorded under similar experimental conditions as a) exhibiting prominent quasi-monoenergetic peaks. The laser pulse parameters in this experiment were: intensity I = W/cm 2, pulse energy E = 660 mj, pulse duration τ = 80 fs, peak plasma density n e = cm 3 [44]. SM-LWFA and direct laser acceleration cannot be discriminated experimentally. Both lead to similar electron energy distributions. Only in PIC simulations the various contributions may be separated. A typical exponential electron spectrum generated by SM-LWFA and direct laser acceleration is shown in Fig. 2.9a) [44]. The laser parameters in this experiment match the conditions of SM-LWFA. Fig. 2.9b) shows an electron spectrum with a significant monoenergetic peak. Monoenergetic features appear at a fraction of the laser shots and are not stable in terms of energy or intensity. Both spectra shown here were recorded under the same experimental conditions with a high-resolution dipole magnet spectrometer. The energy resolution of this kind of spectrometer relies on a small aperture which implies a small acceptance angle thus cutting out a small fraction of the total accelerated electron bunch. Small directional fluctuations of the electron beam create large changes in the recorded spectra. Conventional electron spectrometers as the one used here are basically offline diagnostics. After recording the electron spectra of a sequence of ten shots onto an imaging plate the target chamber must be vented and the information on the image plates must be processed further. 30

33 2 Laser-matter interaction Bubble acceleration A novel regime of laser wakefield acceleration, which was coined bubble acceleration, was proposed by Pukhov and Meyer-ter-Vehn [45] in 2002 on the basis of PIC simulations. Short (τ < 7 fs) and intense (a 0 > 1) laser pulses were predicted to produce quasi-monoenergetic electrons at energies exceeding 50 MeV. Since then, experiments in a transition regime between LWFA and bubble acceleration (also called forced laser wakefield regime, FLWF) demonstrated electron spectra with a significant, non-exponential shape [36], and, recently, quasi-monoenergetic electron beams in the range of MeV with an energy spread of a few percent, limited by the resolution of the detectors [3, 5, 46, 47]. Extensive numerical studies of the bubble regime were carried out which led to rule of thumb limits for the bubble regime as well as to scaling laws for the produced electron beams [48]. According to these studies the optimum configuration for a laser pulse in the bubble regime is defined by k p R a 0, τ R c, where k p = ω p /c is the plasma wave number and R the laser focus radius. The energy of the monoenergetic electrons E e is scalable according to the relation E e 0.65mc 2 P P rel cτ λ 0, where P is the laser pulse power, P rel 8.5 GW the relativistic power unit and λ 0 the laser wavelength. The number of electrons contained in the monoenergetic peak may also be determined by N e 1.8 P, k 0 r e P rel where k 0 = 2π/λ 0 is the laser wave number and r e = e 2 /4πε 0 mc 2 is the classical electron radius. Remarkably, the conversion efficiency of the laser energy into electrons is constant and about 20%. A conversion efficiency of 10% has already been demonstrated in the transition regime [46]. Putting numbers into the rule of thumb scaling laws one obtains very promising results: With a pulse duration of about 5 fs (which is the projected pulse duration 31

34 2 Laser-matter interaction x, y Pump pulse and electrons Probe pulse z Scattered x-rays Figure 2.10: Thomson backscattering of a weak, non-relativistic laser pulse (incident from the right) from laser-accelerated electrons (incident from the left). The angle ϑ is the scattering angle of the photons with respect to the z-axis which will also be called angle of observation. dω is the differential solid angle covered by the detector. e. g. for the new laser facility at the Max-Planck Institute of Quantum Optics, Garching, Germany) the expected electron energies are: 40 MeV for a pulse energy of 50 mj, 150 MeV for 500 mj and 250 MeV for 5 J. The prospects and applications of these ultra-short electron bunches in conjunction with the photon collider are discussed in Sec Thomson backscattering in the linear regime A rather complex interaction scheme will be discussed in the following which makes use of several phenomena presented above. A relativistic electron bunch that was accelerated by one or several of the various acceleration mechanisms is subjected to a weak, non-relativistic counter-propagating electromagnetic wave. This process is commonly referred to as relativistic Thomson backscattering or inverse Compton scattering. With the photon collider in mind we imagine having one strong laser pulse accelerating a bunch of electrons and another, weak and non-relativistic laser pulse counter-propagating and scattering from the electrons (see Fig. 2.10). This situation is actually already included in the thorough treatment of the nonlinear Thomson scattering (Sec ) in the limit a 0 1, but there I confined myself to initially resting electrons and radiation scattered under an angle of 90. It is very useful for the interpretation of the experiment in Sec. 4, however, to look at the problem of backscattered radiation in more detail. 32

35 2 Laser-matter interaction There is an analogous experimental scenario from accelerator physics which has been worked on extensively in the past and which was introduced into text books long ago: undulator radiation. It is generated in synchrotrons by sending relativistic electrons through static but spatially alternating magnetic fields. The differences to the laser-accelerated electrons interacting with counter-propagating laser photons are: The energy of the electrons is higher (synchrotrons are operating with GeV electrons) and the wavelength of the undulator field that the electrons experience is larger (of the order of cm in the laboratory frame). A fully laser-based scheme may offer certain advantages over large synchrotron facilities and are discussed in Sec. 5. It seems useful to start with reviewing a few well known facts from undulator theory [49, pp. 135ff.]. First, the emission of a single electron will be discussed and then the emission of a population of electrons of a given energy distribution Scattering from a single electron The steps to the calculation of the emitted radiation will be to determine the fields that the electron experiences in its frame of reference, to calculate the radiation pattern it emits in its frame of reference and then to transform this back to the laboratory frame. Many properties of undulator radiation can probably most easily be understood considering the classical Doppler effect: An electron which is at rest or moving slowly compared to the speed of light and undergoing an oscillation (induced by external fields) will emit radiation in a dipole pattern and the frequency of the radiation will not strongly depend on the position of the observer. If the electron is moving, however, the wavefronts are compressed along the propagation of the electrons. The Doppler effect is strongly angle dependent. In the relativistic case the observer will detect short wavelength x-ray emission along the propagation of the electrons. Towards larger angles the emission will have larger wavelengths up to infrared and radio waves depending on the velocity of the electron. For a static, weak undulator field the electron traveling at relativistic speed will experience a Lorentz-contracted field of wavelength λ = λ u /γ, where λ u is the wavelength of the undulating magnetic field and γ the Lorentz factor of the electron. The wavelength λ of the observed radiation is given by the undulator equation in the 33

36 2 Laser-matter interaction relativistic limit (β 1): λ = λ u (1 β cos ϑ) λ u 2γ 2 (1 + γ2 ϑ 2 ), (2.32) where ϑ is the angle of observation with respect to the axis along which the electrons propagate (see Fig. 2.10). The approximation in Eq is valid for ϑ 1. Eq represents a simplified version of the complete undulator equation (see [49]) and is shown in Fig. 2.11a) for a value of γ = 10. The undulator equation can easily be adapted to the situation where a counterpropagating electromagnetic wave (a laser pulse) substitutes the static undulator field. The frequency of the electromagnetic wave in the electron frame is then given by ω = ω β 1 β = (1 + β)γω 0 2γω 0 or λ λ 0 2γ, where ω 0 and λ 0 are the frequency and the wavelength of the electromagnetic wave, respectively. Eq will therefore only be changed by a factor of two. Considering only radiation emitted along the z-axis (ϑ = 0) one arrives at the equation for the energy of the emitted photons ω x = 2γ 2 ω u or, in the case of an electromagnetic wave, ω x = 4γ 2 ω 0. (2.33) Since as far as our experiment is concerned the emitted photons have energies in the x-ray range, all quantities related to them will be labeled with a subscript x. From Eq one can easily understand the great advantage of a Thomson scattering scheme over conventional undulators: The laser wavelength is typically a factor of 10 4 shorter than the period of commonly used undulators. This allows for the use of 100 times less energetic electrons to generate x-rays of a given energy. In the frame of the electron (the primed coordinate system) the emitted radiation will have a natural bandwidth λ /λ 1/N 0, where N 0 is the number of oscillations 34

37 2 Laser-matter interaction that the electron undergoes, corresponding to the number of periods of the undulating field. In the laboratory frame this bandwidth will be emitted in a central radiation cone of half angle: ϑ cen 1 γ N 0. (2.34) This central cone will become narrower the more energy the electron has. For a 5 MeV electron (γ 10) submitted to 30 laser oscillations, corresponding to an 80 fs probe pulse, the expected x-ray photon energy in forward direction is 620 ev and the opening half angle of the central emission cone is ϑ cen = 0.02 rad. The energy radiated by the electron per unit solid angle Ω and per unit frequency ω can be calculated integrating the Liénard-Wiechert potentials and is given e. g. by [16, 19]. For our purposes it is sufficient to consider the limit of weak electromagnetic waves (a 0 1), highly relativistic electrons (γ 1) and small observation angle (ϑ 1): ( ) d 2 2 I ω dω x dω = r emcγ 2 N0 2 a 2 0 R(ω 4γ 2 x, ω 0 ), ω 0 where r e is the classical electron radius. The resonance function R(ω, ω 0 ) has the form ( ) sin(k 2 L/2) R(ω x, ω 0 ) =, (2.35) k L/2 where k = k x (1 + γ 2 ϑ 2 )/(4γ 2 ) k 0, k x = ω x /c, k 0 = ω 0 /c and L = N 0 λ 0 the interaction length. The resonance function is strongly peaked at or written in terms of a resonant electron energy ω r = 4γ 2 ω γ 2 ϑ 2, (2.36) γ 2 r = ω x 4ω 0 /(1 ω xϑ 2 4ω 0 ). (2.37) The relation between γ r and ω x is displayed in Fig. 2.11b). Clearly, the angle of observation has a large impact on this relation. This will become important when the dependency of radiated spectra on ϑ is discussed later in this section or in the analysis of the experiment in Sec

38 2 Laser-matter interaction a) x / ë b) ã z / ë - 1 Energy / ev Figure 2.11: a) Relativistic angular dependent Doppler shift: This graph displays the wavelength of the radiation emitted by an electron moving relativistically (γ = 10) along the z-axis according to Eq The radiation is observed in the laboratory frame. b) The Lorentz factor γ of the electron as a function of the observed backscattered photon energy for various observation angles ϑ. Lower curve: ϑ = 0, middle curve: ϑ = 0.04 rad, upper curve: ϑ = 0.06 rad Scattering from a laser-accelerated electron bunch The total spectra radiated by a bunch of electrons of a given energy distribution f(γ) may be obtained by integrating the spectrum of a single electron multiplied by f(γ) over γ: d 2 I T dω x dω = d 2 I f(γ) dγ. (2.38) dω x dω Here the electron distribution function is given as the normalized function f(γ) = 1 dn e (γ), N b dγ where N b is the number of electrons in the bunch. For a large number of periods N 0 of the undulating field the resonance function (Eq. 2.35) may be approximated by a delta function: R ω r δ(ω x ω r ) = γ r δ(γ γ r ), where ω r = ω r /N 0 and γ r = 2γ 3 r ω 0 /(N 0 ω r ). This approximation is valid for wide 36

39 2 Laser-matter interaction energy distributions like exponential distributions which were measured in the SM- LWFA regime (see Fig. 2.9). The line broadening inherent to the single-electron spectrum is narrow compared to the contribution of the wide energy distribution. Carrying out the integration in Eq using the delta function approximation one arrives at γ 3 d 2 I T dω x dω = 1 2 r em e cn 0 a 2 0 f(γ). (2.39) 1 + γ 2 ϑ2 The intensity distribution of the Thomson backscattered radiation can be assumed flat over the solid angle Ω 1. The number of photons N x per frequency interval ω x backscattered into this collection angle can then be calculated by dividing Eq by the photon energy E x = ω x and multiplying by the number of electrons in the bunch N b : N x ω x Ω = N b ω x d 2 I T dω x dω = α 8ω L N b N 0 a 2 0γf(γ), (2.40) where the fine-structure constant α = e 2 /(4πε 0 c) was inserted. photons observed in the corresponding energy interval reads The number of N x E x Ω = where E 0 = ω 0 is the energy of a laser photon. α 8E L N b N 0 a 2 0γf(γ), (2.41) The Thomson backscattered photon spectrum is typically recorded with a constant detector solid angle Ω with a constant energy resolution E x. From Eq the electron spectrum may therefore be calculated by N b f(γ) = dn e(γ) dγ = N x E x 8E 0 αn 0 a 2 0γ Ω, or in terms of electron energy expressed in MeV using E e = mc 2 (γ 1) dn e (E e ) de e = N x 8E 0 E x αn 0 a 2 0γ Ω dγ (2.42) de e Please note that γ as shown in Eq is dependent on the observation angle ϑ (see also Fig. 2.11). In order to accurately calculate the electron spectrum according to Eq the angle ϑ has to be determined experimentally. 37

40 2 Laser-matter interaction Assuming an exponential energy distribution f(γ) of N b electrons with f(γ) = f 0 exp[ γ/t e ], where T e is the electron temperature given in units of γ and f 0 a normalization factor, Eq may be rewritten in terms of the scattered photon energy E x : dn x (E x ) = α [ ] Ωa 2 E x de x 8E 0N b N 0 f 0 0 4E 0 (1 Exϑ2 4E 0 ) exp E x 4E 0 (1 Exϑ2 4E 0 ) /T e. The maximum of the photon spectra is located at (2.43) E max = 4 T 2 e E T 2 e ϑ 2. The backscattered photon spectrum for a given electron distribution function of T e = 10 was calculated and is shown in Fig. 2.12a). The shape of the spectrum is strongly dependent on the observation angle ϑ. For larger ϑ the spectrum falls off steeper towards higher energies, the maximum of the spectrum shifts to lower energies. The maximum of the spectrum for ϑ = 0 is located at 624 ev. In Fig. 2.12b) backscattered spectra are displayed for various electron temperatures of T e = An increase in temperature shifts the maximum of the photon emission towards higher photon energies. Please note that the spectra shown in Figs. 2.12a) and b) were calculated using an absolute spectral bandwidth de x and not a relative bandwidth of 0.1% as common in the accelerator and synchrotron community (see e. g. [50]). Eq leads to several important conclusions: 1. The scenario in which a weak (non-relativistic) laser pulse and a laser-accelerated relativistic electron bunch are counter-propagating may be used to generate ultra-short and well collimated x-ray pulses. The spectra of these x-ray pulses will be broad if the energy distribution function of the electrons is broad. In turn, quasi mono-energetic electron bunches will produce peaked x-ray spectra. 2. Eq shows that the divergence angle of the emitted x-rays decreases with increasing energy. The higher the energy of the electrons, the more power is 38

41 2 Laser-matter interaction Figure 2.12: a) The backscattered Thomson spectra strongly depend on the observation angle ϑ (cf. Eq. 2.39). This graph shows backscattered photon spectra generated from electrons with an exponential energy distribution (T e = 10 in units of γ) for observation angles in the range ϑ = rad. b) Backscattered x-ray spectra calculated from Eq for ϑ = 0 assuming an exponential electron energy distribution with electron temperatures of T e =

42 2 Laser-matter interaction radiated in forward direction. 3. The measurement of the Thomson backscattered radiation immediately reveals the energy distribution function of the laser-accelerated electrons using Eq Thomson backscattering can therefore be used for in-situ timeresolved diagnostics of the electron acceleration process [51]. I would like to point out that no assumptions about the electron spectrum must be made which is, however, the case for other methods using e. g. nuclear reactions [14]. 40

43 3 Single-shot autocorrelation at relativistic intensity The accurate measurement of relativistic laser intensities is a difficult task. The commonly used method to determine laser intensities is the simultaneous, careful measurement of pulse energy E, pulse duration τ and focus area A, and the calculation of laser intensity according to I E/(τ A). The experimental determination of each of the parameters entering this equation is problematic at high intensities. For example classical autocorrelation techniques or phase resolved methods such as Spider or Frog for the determination of the pulse duration require the use of nonlinear optical materials. Even at intensities many orders of magnitude below the relativistic regime these materials are destroyed by the laser pulse. Therefore the pulse duration is usually measured at low intensities, frequently using only part of the laser beam, and then extrapolated to the experimentally relevant intensity range. Another method to directly measure maximum intensities employs field ionization of dilute gases by the laser pulse. The maximum charge state measured in a mass spectrometer is related to the maximum intensity by the Ammosov-Delone-Krainov (ADK) scaling law, which is well-established for non-relativistic intensities [52]. The experimental difficulty of this approach lies in the clear distinction of highly charged states of atoms generated by field ionization as opposed to other processes such as collisional ionization or recombination. Ions generated in the high-intensity focal region may also be shielded by ions generated in lower fields and never reach the detector [53]. It is under investigation whether the classical scaling law for field ionization used for the intensity measurement remains valid in the relativistic intensity regime [54]. Relativistic intensities have also been estimated through the ponderomotive scaling of electron temperature [55 60] or laser-induced nuclear reactions [14, 61]. While these latter methods yield reasonable lower bounds for the laser intensity they are, however, indirect and typically average over many thousand shots. It is 41

44 3 Single-shot autocorrelation at relativistic intensity Figure 3.1: A schematic representation of the Jena laser system in which key components introduced to prevent damage from back-reflected laser pulses during the photon collider experiments are highlighted in bold print: (1) Faraday isolator between regenerative amplifier and 4-pass amplifier, (2) diagnostics unit consisting of beam-profiler and energy meter for monitoring of the back-reflected pulse which is coupled out at the isolator, (3) spatial filter. therefore of interest to utilize a genuinely relativistic effect that is well understood theoretically for the determination of laser pulse parameters at relativistic intensities. Such an effect is non-linear Thomson scattering of light from free electrons (see Sec ). With the help of the photon collider and a He gas-jet as target we carried out the first single-shot autocorrelation measurement at relativistic intensity [17], which will be described below. I will start the discussion of the experiment with the description of the Jena laser system and its adaption to this experiment and continue with the experimental setup and the alignment procedure in detail. Various autocorrelation signals will be discussed both in theory and on the basis of experimental data. 3.1 The Jena Ti:Sa laser system (JETI) The Jena Ti:Sa laser system (JETI) is based on the principle of chirped pulse amplification (CPA) [62, 63]. A schematic setup is shown in Fig A commercial oscillator generates pulses of 10 nj, 45 fs at a central wavelength of 795 µm with a 42

45 3 Single-shot autocorrelation at relativistic intensity repetition rate of 80 MHz. A pulse picker reduces the repetition rate to 10 Hz. The pulses are lengthened to a duration of 150 ps in a grating stretcher and amplified in three stages to a pulse energy of J corresponding to an amplification of In all these three stages frequency-doubled Nd:YAG lasers are used as pump lasers. The first stage consists of a regenerative amplifier yielding an output energy of 2 mj. The second and third stage are multi-pass amplifiers. The 4-pass amplifier, pumped by mj, yields an output energy of mj, the 3-pass amplifier, pumped by J, finally provides up to 1.6 J pulse energy. After amplification the beam profile is enlarged to a diameter of 50 mm and the pulses are shortened again in a 4-grating vacuum compressor down to 80 fs. Starting from the compressor the beam must be guided in vacuum to the target chamber. Several diagnostics are installed for regular use to characterize the laser pulses (naming the ones relevant for this experiment): A third harmonic generation (THG) autocorrelator measuring the pulse length with a dynamic range of up to 10 5 in a time interval of a few picoseconds around the pulse maximum and an interferometric pulse front tilt diagnostic system. The pulse front tilt detection is necessary to achieve high intensity since a slight tilt of a laser pulse of 50 mm diameter and 30 µm length induces a substantial increase of the effective pulse duration in the laser focus. 3.2 Protection of the laser system The symmetric design of the photon collider on the one hand facilitates the alignment tremendously but on the other hand creates a great risk: Assuming there is no matter in the focal region partially absorbing the laser pulses, theoretically, if a 50/50 beamsplitter is used, 50% of the original pulse energy are reflected back from the vacuum chamber towards the laser system. 60% of that amount will pass the compressor so that 30% of the original energy on target will reach the last amplifier. Two measures are in place to protect the laser system: 1. A spatial filter was introduced between the 3-pass amplifier and the 4-pass amplifier (see Fig. 3.1). The filter consists of a 1:1.5 mirror telescope with a quartz pinhole (diameter µm). At this point returning pulses that are not perfectly aligned with the beam path will be suppressed. 43

46 3 Single-shot autocorrelation at relativistic intensity 2. A Faraday isolator was installed between 4-pass and regenerative amplifier (see Fig. 3.1). This is a crucial point in the laser setup since here a backreflected pulse will be scaled down in diameter by the 9:1 lens telescope which corresponds to an increase in intensity of a factor 9 2. The Faraday isolator consists of a Faraday rotator, two polarizing beam-splitters and a λ/2-waveplate. Looking from the regenerative amplifier along laser pulse propagation: The first ( input ) polarizer is aligned with the laser polarization. The Faraday rotator then turns the polarization of the pulse by -45. The waveplate then turns the polarization back to 0 (rotating +45 ). The second ( output ) polarizer is oriented along this polarization such that the forward propagating pulse is transmitted unimpeded. A pulse, back-reflected from the target chamber, will be polarized along the orientation of the output polarizer. It will be rotated by the λ/2-waveplate by -45 and rotated another -45 by the Faraday rotator since the rotation is insensitive to direction of propagation. The polarization of the back-reflected pulse will now be vertically aligned to the input polarizer. According to the contrast of the input polarizer ( ), most of the back-reflected light will be coupled out of the beam path onto a CCD camera and a power meter for monitoring. With these measures in place there are still some risks remaining: The 3-pass amplifier is unprotected from back-reflected pulses. Back-reflected pulses are amplified in the 3-pass amplifier as well as - if not eliminated by the spatial filter - in the 4-pass amplifier. The input polarizer of the Faraday isolator plays the most important role for the laser protection and will have to stand very high strain. In the experiment it turned out that this optical element will be damaged first. 3.3 Experimental setup The setup of the photon collider for the autocorrelation measurement is as follows (see Fig. 3.2): The main laser pulse, the polarization of which is oriented along the x-axis, is divided into two pulses of equal energy by a 50/50 beam-splitter of 2.5 mm 44

47 3 Single-shot autocorrelation at relativistic intensity a) Laser pulse 2ù probe beam Beamsplitter x Off-axis parabolic mirror z Äô Off-axis parabolic mirror z 1 He-gas jet 2ù Interference filter CCD b) Figure 3.2: a) Schematic of the experimental setup for two colliding laser pulses: The main laser pulse is divided into two pulses of equal energy by a beam-splitter which may be moved along the z-direction. Each pulse is focused into a He-gas jet by a 45 off-axis parabolic mirror. The 2ω self-emission of the plasma and the shadowgrams generated by the 2ω probe pulse are observed perpendicular to the z-axis. b) Color image of the He gas-jet during the experiment: (1) gas nozzle, (2) target mount, (3) parabolic mirror with quartz wafer debris shielding (removed in autocorrelation measurement). The bright overexposed region in the center of the image is due to plasma emissions. 45

48 3 Single-shot autocorrelation at relativistic intensity thickness. These are focused by two F/ off-axis parabolic mirrors into a pulsed He gas-jet and their foci are overlapped in space. The time delay between the pulses is adjusted so that they arrive at the focus at the same time by moving the beamsplitter. The electrons generated early in the laser pulse by ionization of the He gas are driven by both the electric and the magnetic field of the electromagnetic wave in such a way that Thomson scattered light close to the second harmonic of the laser frequency is generated. The self-emission of the laser plasma is imaged by a quartz lens onto a CCD camera under an angle of 84. In the weakly relativistic regime the second-harmonic Thomson signal observed approximately perpendicular to the z-axis is spectrally close to the second harmonic of the laser frequency and will be called second harmonic (axes are indicated in Fig. 3.2a)). This can be seen from Eq in Sec An interference filter of 10 nm bandwidth is introduced into the imaging setup to insure that only secondharmonic plasma emission is recorded. It was verified that the polarization direction of the second harmonic was in z-direction which is expected for second harmonic light generated by non-linear Thomson scattering. The signal on the CCD camera is integrated along the y-axis (columns) and is spatially resolved along the laser propagation (z-axis, rows) and it represents an autocorrelation trace of the laser pulse of second order at full intensity as was derived in Sec With the same imaging setup shadowgrams of the laser plasma can be acquired using a frequency-doubled ultra-short probe pulse of variable delay which is backlighting the interaction region. For the autocorrelation measurement the gas-jet must be so dilute that propagation effects through ionization can be neglected (see Sec ). The particle density of the gas-jet which exhibits a Gaussian profile along laser propagation and an exponential profile along the nozzle axis was determined by time-resolved interferometry with nitrogen gas (nitrogen having a much larger index of refraction than helium). The results are shown in Fig In our autocorrelation measurements the peak density of the gas-jet did not exceed cm 3. We estimate the upper limit of the integral phase shift to φ < 0.5 (cf. Eq. 2.29) [64]. The laser was operated at 100 mj pulse energy on target, i. e. 50 mj in each of the counter-propagating pulses. The intensity of the laser pulses was determined to W/cm 2. This corresponds to a laser field strength parameter a

49 3 Single-shot autocorrelation at relativistic intensity Figure 3.3: The gas-density n in front of the orifice of the subsonic, cylindrical nozzle (nozzle center at y = z = 0 µm) at two different timings: a) opening time of the valve of 1 ms, b) opening time 170 µs. Sharp peaks in the low gas-density measurement are artifacts due to noise from the interferometric measurements. The gas-density at the focal position in the autocorrelation measurement was determined through extrapolation. 3.4 Alignment of the photon collider The experimental setup was designed in such a way that the foci of the laser pulses are located in the center of the octagonal vacuum chamber. The parabolic mirrors can be tilted along the x- and y-axes and translated in all three dimensions but they cannot be rotated around the incident beam axis. This degree of freedom has to be adjusted in the very beginning of the experiment when the mirrors are fixed in their mounts. With the help of a HeNe laser the foci are adjusted to be in the center of the chamber. The later fine-adjustment of the foci through tilting the mirrors changes the position of the foci which can be corrected translating the parabolic mirrors in x, y, z but in principle this first fixation defines the angle under which the two laser pulses counter-propagate. This may be most easily understood if one remembers that the parabolic mirror is a circular cut-out of a (virtual) larger, axially symmetric paraboloid. The position of the focus with respect to the surface is fixed. The rotation of our parabolic mirror is therefore a degree of freedom necessary for the alignment. 47

50 3 Single-shot autocorrelation at relativistic intensity Figure 3.4: Intensity distribution in the laser focus a) of the laser pulse propagating through the beam-splitter (6.5 µm 2 FWHM) and b) of the laser pulse reflected from the beamsplitter (5.9 µm 2 FWHM). Focus optimization The foci are optimized in vacuum by tilting the mirrors along x and y while imaging the focal spot onto a CCD camera with a 40 microscope objective. The optimization is carried out at reduced laser power introducing an attenuator into the beam path between the last amplifier and the compressor, achieving a variable attenuation of up to Multiple images of the focal spots are taken using different sets of neutral density filters and are recomposed with the use of a computer program. This allows for a focus analysis with a dynamic range of 2000 using a common (and affordable) 8bit CCD camera. The intensity distribution in the laser foci measured before the experiment is shown in Fig From the resulting images the FWHM-area A of the focal spot and the ratio q of pulse energy inside the peak over energy outside the peak are determined. The intensity I is then calculated with I = E q A τ, where E is the pulse energy, and τ is the pulse duration (both parameters measured separately). Due to the limited field of view of the CCD camera the value of q measured in such a way represents an upper limit of the real value (see Fig. 3.4). 48

51 3 Single-shot autocorrelation at relativistic intensity Overlap in space In the next step the laser foci in low-density He gas are examined. The plasma self-emission of each of the beams is imaged under 90 onto an untriggered CMOS camera located in the vacuum chamber through two 8 microscope objectives (see Figs. 3.5a) and b)). These microscope objectives are aligned each with the x- and the y-axis. A filter for 800 nm prevents the camera from saturation by stray light. With this technique the laser foci can be overlapped within an accuracy of 10 µm given by the resolution of the imaging setup. Since the CMOS camera cannot be operated in triggered mode the resulting standard video signal has to be de-interlaced to obtain meaningful images. The images shown in Fig. 3.5c)-f) are averaged over 10 laser shots. The overall brightness of the images varies depending on the observation axis (the emission patterns of the non-linear harmonics are non-homogeneous) and experimental conditions (stray light). Overlap in time Now that the laser foci are overlapped within a few microns the time delay between the ultra-short laser pulses has to be adjusted. To this aim the interaction region of the laser pulse with the He gas-jet is back-lighted with a frequency-doubled probe pulse. Shadowgrams are recorded with the same imaging setup as used for the detection of plasma self-emission. The probe pulse is approximately as short as the main pulse and its delay with respect to the main pulse can be adjusted with a resolution of about 100 fs, i. e. changes of the ionization front propagating through the plasma can be consistently traced with 100 fs resolution. The delay adjustment takes place in two steps: First, the delay of the probe pulse is adjusted to the particular moment in time when the ionization front of the laser pulse incident from the right (propagating through the beam-splitter) reaches the position of the vacuum focus. We have now marked this certain moment in time with the arrival of the probe pulse and can use it as a reference. Secondly, the time delay of the laser pulse incident from the left is adjusted by moving the beam-splitter such that it reaches the focus at the same time delay of the probe pulse. The fringes that are visible in each picture of Fig. 3.6 are due to diffraction of the probe pulse on a plasma cavity generated by each of the laser pulses. This phenomenon has been thoroughly examined under a wide range of experimental conditions. In fact, it enables us to monitor the evolution of relativistic plasma 49

52 3 Single-shot autocorrelation at relativistic intensity a) b) Adapter to x-y-z translation stage CMOS Camera y x Mirror Beamsplitter Mirror Microscope objective (y-axis) Focal point Microscope objective (x-axis) c) d) 200 µm e) f) Figure 3.5: a) Picture of the diagnostics unit with four microscope objectives. Two 40 objectives are mounted along the z-axis (perpendicular to the plane of the image) to monitor the focus size. Two smaller 8 objectives are mounted along the x- and y- axes to image the side-scattered light from the focus onto a CMOS camera. The four objectives are highlighted by a white circle. The camera is mounted on top of the unit and can be operated in vacuum. b) Drawing of the diagnostics unit (the z-axis objectives were omitted for clarity). c)-f) Images of the self-emission of a laser plasma generated by one laser pulse and observed through the x- and y-objectives. c) Laser focus side view (along x-axis) of the pulse incident from the left, d) side view of the pulse incident from right, e) top view (along y-axis) of the pulse incident from left, f) top view of the pulse incident from right. 50

53 3 Single-shot autocorrelation at relativistic intensity a) b) 200 µm c) d) Figure 3.6: Shadowgrams of the interaction region: In a first step the delay of the backlighting probe pulse was adjusted such that it passed the interaction region when the pulse incident from the right reached its focus. a) The laser pulse incident from the left is still delayed by τ = 360 fs to the pulse incident from the right. Its ionization front is visible about 100 µm left of the focus. b)-d) The delay of the pulse incident from the left is gradually adjusted by 120 fs moving the beam-splitter such that both pulses pass the focus at the same moment in time. channels with 100 fs resolution and will be discussed elsewhere [65]. 3.5 Exemplary scenarios If the foci of the two parabolic mirrors are on the same optical axis, different situations may occur. Let us re-examine Eq. 2.14: S(z, z 1, τ) 1 ( ) 2 + z 1 + z R 2 ( ) 2 z z 1 z 0 4R ( ) 2 z z1 /2 z z z0 2 exp [ 4 τ 2 L ( z z1 /2 c τ 2 ) 2 ],(3.1) where we divided by I 0L and inserted the ratio of the intensities R = I 0R /I 0L. τ L is the laser pulse duration, τ the time delay between the pulses and z 0 the distance between the foci. τ L is related to the laser intensity pulse duration FWHM τ by τ = τ L 2 ln 2. The first two summands are Lorentzian peaks at the position of the foci (z = 0 and z = z 1 ). The third summand is a product of a Lorentzian and a Gaussian, the 51

54 3 Single-shot autocorrelation at relativistic intensity a) b) c) x d) z S z Figure 3.7: Illustrations of exemplary scenarios: The upper graph in each of the illustrations shows a snapshot of the two counter-propagating Gaussian laser pulses indicating the position of the foci and the time delay between the pulses. In each of the lower graphs the expected autocorrelation signal is plotted (grey: interferometric autocorrelation, black: spatially averaged signal). a) perfect overlap in space and time (z 1 = 0 and τ = z 1 /(2c)), b) overlap in space but a time delay remaining (z 1 = 0 and τ > 0), c) foci separated in space and no time delay (z 1 > 0 and τ = 0), d) foci separated in space and time delay (z 1 > 0 and τ = 0). In all these examples the ratio of the intensities R was set to one. 52

55 3 Single-shot autocorrelation at relativistic intensity center of which is located in the middle between the foci at z = z 1 /2. Far away from this center the Gaussian dominates the behaviour of the term since it decays exponentially. Close to the center, where the Gaussian does not vary much, the Lorentzian peak shape dominates. Please note that the information about the time duration of the pulse is contained in the Gaussian. For plane waves (z 0 ), a common focus of the two beams (z 1 = 0) and equal intensities from the left and the right (R = 1) Eq. 3.1 reduces to the well known intensity autocorrelation signal of second order [66]. The resulting signal may exhibit a variety of shapes. A single peak structure, two or even three distinct peaks are possible. We will discuss four exemplary situations: Overlapping foci. If the foci are at the same position z = 0, the two pulses propagating in opposite direction will cross each other at a position determined by the position of the beam-splitter. For zero delay τ between the two pulses they will cross exactly in the focus generating the maximum intensity in the standing wave (Fig. 3.7a). For τ 0 the overlap will occur outside the focus leading to a reduced signal with a slightly asymmetric shape (Fig. 3.7b). Separate foci. If the foci are at different positions z = 0 and z = z 1 on the optical axis, two approximately equal signals are expected due to each individual laser pulse if the delay τ between the pulses satisfies the condition τ z 1 /c. If both pulses overlap at or between the two foci, a more complex spatial signal appears that may consist of one (Fig. 3.7c), two (Fig. 3.7d) or even three peaks (not shown). The height of the observed peaks (in the case of a two-peak structure) is not necessarily equal, even if the laser pulses are of equal intensity. 3.6 Experimental results Fig. 3.8b) shows experimental data obtained for z 1 = 0 compared with calculated curves based on Eq. (3.1). The data is integrated over 10 laser shots to increase the accuracy of the non-linear curve-fit. Excellent agreement between the experimentally determined signals and the fitted curves of Eq. 3.1 are obtained for both the signal with zero delay ( τ = 0) as well as the two signals with a delay of τ = ±66 fs. The data taken at non-zero delay between the pulses exhibits an asymmetric shape 53

56 3 Single-shot autocorrelation at relativistic intensity a) c) b) d) Figure 3.8: a) Counter-propagating laser pulses with identical foci. b) Non-linear Thomson signal obtained for two counter-propagating laser pulses with identical foci. The experimental data represented by filled squares was measured for zero delay ( τ = 0), while the data represented by open squares and circles correspond to a delay of τ = ±66 fs. The solid lines were obtained by carrying out a non-linear curve-fit to the experimental data according to Eq For better visibility the data at non-zero delay (open symbols) was scaled by a factor of two (left scale) with respect to the data at zero delay (right scale). c) Counter-propagating laser pulses with foci separated by a distance z 1. d) Non-linear Thomson signal obtained for z 1 = 45 µm with the pulses intersecting each other in the left focus (filled squares) and for z 1 = 22 µm (open squares). 54

57 3 Single-shot autocorrelation at relativistic intensity Figure 3.9: a) Autocorrelation obtained from a THG autocorrelator using a small part of the beam cross-section. The dashed line indicates a Gaussian curve-fit with τ = 85 fs FWHM. b) Impact of the pulse front tilt on the effective pulse length in the laser focus. The intensity cross-section of two Gaussian pulses along the x-z-axis is shown. The pulse front of the left pulse is tilted with respect to the x-axis by an angle of 45 which is greatly exaggerated. This leads to an increase of the effective pulse length τ eff in the laser focus as is indicated by the line graphs of the temporal pulse shapes. as expected from the exemplary scenarios (Fig. 3.7b)). The curves are not perfectly symmetric to z = 0 due to the delicate optical setup: Small movements of the camera or any other optical element in the imaging setup had a large impact on the position of the image on the CCD chip. In Fig. 3.8d) autocorrelation results are shown for the case of non-identical foci of the two parabolic mirrors. The black curve shows an experimental non-linear Thomson scattering signal obtained for a focus separation of z 1 = 45 µm when the delay τ was adjusted such that the counter-propagating pulses intersected at the left focus at z = 0, as shown in Fig. 3.7d). When the focus position is then moved to z = 22 µm a symmetric signal (red curve) is obtained. The data in Fig. 3.8d) does not contain enough information about the Gaussian component of the autocorrelation function in Eq. 3.1 to carry out a non-linear curve-fit. Please note that only the Gaussian component is determined by the laser pulse length. The pulse duration τ and the Rayleigh length z 0 were obtained for the fitted curves in Fig. 3.8b). For the pulse duration of the full energy laser pulse at the focus position we obtain τ = (112 ± 11) fs and for z 0 = (9 ± 1) µm. This value of the pulse 55

58 3 Single-shot autocorrelation at relativistic intensity duration is larger than the value of τ = (84 ± 5) fs measured with the classic THG autocorrelator (see Fig. 3.9a)). The THG operates at low intensity and takes only a (2 2) mm 2 central cut-out of the full beam (Ø = 50 mm). The difference between the two measurements may be explained by a slight tilt of the pulsefront. At the entry of the compressor a single laser pulse resembles a disk which is very large in diameter (50 mm), but which has a thickness of only 30 µm. A very small tilt in the pulse front thus has a large impact on the effective pulse length (see Fig. 3.9b)). Our wavefront interferometer has a detection limit of 1 of arc. The relative increase of the pulse duration τ by a wavefront tilt is given by [67] τ eff τ = 1 + d FWHM sin α. (3.2) cτ The increase that is not detectable with the interferometer can therefore be as large as 13%. Another possible reason for the larger pulse duration determined by the photon collider setup is related to the dynamics of the electrons in the laser focus. Sec it was assumed for simplicity that the electrons are oscillating locally. In an experiment at relativistic intensities, however, the electrons may be accelerated over a large distance and for instance be scattered out of the focal region. This may have a blurring effect on the autocorrelation signal and lead to a seemingly larger pulse length. PIC simulations are under way to quantitatively assess this effect. The pulse duration measurement with the photon collider presented in this chapter represents the first single-shot autocorrelation measurement that has ever been carried out at relativistic intensity. Since it takes place in the laser focus where later the actual experiment is carried out (e. g. laser-acceleration) it is a method which is much closer to the true experimental laser parameters than conventional methods. In 56

59 4 Thomson backscattering from laser-accelerated electrons The process of laser pulses scattered from accelerated electrons is closely related to the generation of x-rays in undulator assemblies of synchrotrons. The close analogy was used in Sec to deduce some relations useful for the analysis and interpretation of the laser-generated radiation. So far, in Thomson scattering experiments the relativistic electron beams have been generated separately by conventional high energy accelerators [68 72]. Apart from serving as a bright x-ray source, the backscattered radiation was also used for electron beam characterization [73, 74] and fundamental studies of non-linear Compton scattering [75] and positron production from photon-photon interactions [7]. Recently, much progress has been made in laser-based accelerators producing relativistic electrons with low divergence and a small energy spread [3 5]. These advances have greatly increased the interest in the observation of Thomson scattering with an all-optical interaction scheme, where a laser-based accelerator is used in conjunction with a synchronized scattering laser pulse [50, 51, 76, 77]. The improvement in laser-acceleration of electrons apparently involved complex electron acceleration mechanisms. To date, they can only be understood through PIC simulations which clearly manifests the need for in situ and time-resolved diagnostics. However, conventional diagnostic methods like electron magnetic spectrometers only allow for the spectral analysis of a small cone of electrons at a distance from the interaction region outside the plasma after the acceleration process is complete. Catravas et al. [50] and Tomassini et al. [51] pointed out that - similarly to diagnosing the accelerator electron beam - the backscattered radiation in a laser-based setup can be used to infer energy- and time-resolved information of the electron acceleration process in the laser plasma. In the photon collider setup we can generate a strong and a weak (non-relativistic) 57

60 4 Thomson backscattering from laser-accelerated electrons laser pulse using an appropriate 90/10 beam-splitter. The strong laser pulse is focused into a He gas-jet and accelerates electrons in the laser plasma to relativistic energies. The counter-propagating weak pulse is then scattered from those electrons. It was shown that each measured backscattered photon spectrum is directly related to the electron energy distribution at the specific moment in time when the electron bunch and the counter-propagating laser pulse interact (see Sec. 2.10). Since the photon collider offers accurate control over both spacial and temporal overlap of pump and probe pulse it can be used to directly monitor the acceleration of the electrons. This direct access to the processes in the laser plasma has not been available to date. Our results represent the first observation of Thomson scattering from laser-accelerated electrons. 4.1 Experimental setup The schematics of the experimental setup are shown in Fig The main laser pulse is divided by a 90/10 beam-splitter into a pump pulse and a probe pulse, the stronger pump pulse being reflected from the beam-splitter. Each of the pulses is focused by an F/ off-axis parabolic mirror into a He gas-jet. The time delay between the laser pulses may be adjusted by moving the beam-splitter as indicated in Fig To achieve an optimum focal spot the parabolic mirrors may be tilted around two axes and to achieve spatial overlap both mirrors may be moved in three dimensions. The setup is located in a vacuum chamber and is entirely computer controlled. The adjustment of the exact spatial and temporal overlap of the two laser pulses was described earlier (see Sec. 3.4 and [17]). JETI was operated at the following parameters: The pulse length of 85 fs of the main laser pulse was determined by a THG autocorrelation. The energy of the undivided laser pulse was 370 mj focused to an intensity of W/cm 2. The probe pulse was focused to an intensity of W/cm 2. These nominal intensities correspond to laser field strengths of a 0 = 3 and a 0 = 0.8, respectively. Non-linear effects occurring in the beam-splitter substrate are negligible since at that point the beam diameter is about 50 mm and the energy of the probe pulse passing through the beam-splitter is only 10% of the main pulse energy. This was confirmed by calculations carried out with Lab2 [78]. 58

61 4 Thomson backscattering from laser-accelerated electrons Figure 4.1: a) Experimental setup of the Thomson backscattering experiment: The main laser pulse is divided into two pulses by a 90/10 beam-splitter both of which are focused by off-axis parabolic mirrors into a He gas-jet. The backscattered radiation is observed with an x-ray CCD camera through a hole in the mirror of the probe beam. The timedelay between pump and probe may be varied adjusting the position of the beam-splitter. A frequency-doubled laser pulse is used for shadowgraphy of the interaction area (angle of incidence exaggerated for better visibility). b) Setup of the x-ray diagnostics. The dimensions of the respective element and its distance to the source are given in mm. A: laser focus, B: pinhole in probe beam parabolic mirror, C: lead pinhole and metallic filter, D: CCD chip. 59

62 4 Thomson backscattering from laser-accelerated electrons Figure 4.2: a) Position of the relativistic channel generated by the pump pulse in the gasjet (indicated by a red line). The pump pulse is incident from the left. The position of the vacuum focus (z = 0) was determined from second-harmonic emission at low gas density. b) Side image of the relativistic channel at high gas density in false colors. The line graphs (white) were obtained binning the image along the y- and z-axis, respectively. A white arrow indicates the position of the vacuum focus (x = z = 0). A cylindrical subsonic gas nozzle was used which created a pulsed He gas-jet with a density profile of Gaussian shape along the laser axis and with a peak gas density of cm 3 (see Fig. 4.2). The gas density was determined using time-resolved interferometry with an error of 20%. When focused into the rising edge of the He gas-jet the pump pulse undergoes relativistic self-focusing and generates a plasma channel. In this configuration electrons are accelerated efficiently in forward direction to relativistic energies by the pump pulse. The spectrum of the high-energy electrons is dominated by an exponential shape and was determined in earlier experiments (see Fig. 2.9 in Sec and [14, 44]). The temperature of the electron spectrum usually ranges from 3 to 7 MeV. The probe pulse is scattered from these high-energy electrons and the backscattered x-ray photons are observed. The position of the laser focus and of the relativistic channel with respect to the gas-jet as well as the dimensions of the channel were determined from side images. At low gas density, similar to the condition in the autocorrelation experiment, the position of the vacuum focus is indicated by faint but highly peaked second-harmonic emission. At higher gas densities a bright and more than 100 µm long relativistic channel is generated by the pump pulse. The gas density profile along the optical axis and the extension of the relativistic channel are shown in Fig. 4.2a). A typical 60

63 4 Thomson backscattering from laser-accelerated electrons side view image of second-harmonic light emitted from the relativistic channel is displayed in Fig. 4.2b). A white arrow indicates the position of the pump pulse vacuum focus. In all side view images the pump pulse is incident from the left. In Fig. 4.2b) it is clearly visible that channeling of the pump pulse (and thus the non-linear emission) sets in before the pump pulse reaches its vacuum focus. On the other hand, in the case of overlapping foci, the probe pulse (incident from the right) must propagate through most of the gas-jet to reach the nominally highest intensity in its focus. Under these conditions the probe pulse does not undergo relativistic selffocusing and is not forming a single, stable plasma channel. Contrarily, filamentation reduces the effective intensity of the pulse which has to be taken into account in the analysis of backscattered radiation. This was confirmed by side-imaging of the interaction region: The probe pulse did not generate second-harmonic emissions detectable with our imaging setup. An ultra-short, frequency-doubled probe pulse with variable delay propagating at an angle of approx. 90 through the interaction region was deployed to record shadow images of the laser plasma with a time resolution of 100 fs. Monitoring the ionization fronts of the laser pulses in the shadow images the spatial and temporal point of interaction of the laser pulses was reliably established. Fig. 4.3a)-d) shows a typical set of shadowgrams of the interaction region where the strong pump pulse is incident from the left generating the relativistic channel and the probe pulse is incident from the right. The structures that are visible on these images are plasma regions ionized by the propagating laser pulses. The bright area at the very center of the images is due to the non-linear self-emission (cf. Fig. 4.2b)) of the laser plasma and indicates the position of the relativistic plasma channel generated by the pump pulse. The self-emission is visible here since the exposure time of the camera is long compared to the duration of the interaction. In the experimental situation shown in Fig. 4.3 the probe pulse has already undergone filamentation which can be seen from the irregular filamented structure of the probe pulse in contrast to the clear ionization front of the pump pulse. An upper limit of 45 µm for the probe pulse focal diameter may be estimated from the shadow images (nominal diameter 3 µm). A hole of 3 mm diameter was drilled into the parabolic mirror of the probe beam in such a way that it is aligned with the axis of the focused beams (z-axis in Fig. 4.1a)). An x-ray CCD camera (Andor DO420 BN) was placed on this axis for energy-resolved 61

64 4 Thomson backscattering from laser-accelerated electrons Figure 4.3: a)-d) Sequence of shadow images of two counter-propagating laser pulses with identical foci and both pulses interacting in the focus. The pump pulse is incident from the left, the probe pulse from the right. The white region in the center of the images is due to the self-emission of the relativistic channel of the pump pulse. e) Inverted image of the scintillating screen (Kodak KF x-ray intensifier screen) which was used for monitoring the energy of the electrons passing through the hole in the parabolic mirror (see Fig. 4.1). The dotted lines indicate the limits of the gap between the dipole magnets. 62

65 4 Thomson backscattering from laser-accelerated electrons detection of Thomson backscattered photons. A pair of dipole magnets was used to prevent electrons from reaching the CCD chip while the camera was additionally shielded with a lead pinhole from background bremsstrahlung. A filter set was introduced between CCD and laser focus to block irradiation by laser light. In conjunction with an x-ray intensifier screen (Konica KF) the dipole magnets were also used to infer the energy of the electrons passing through the hole in the probe beam parabolic mirror in a range of MeV. Electron energies in this range were expected from earlier measurements with high-resolution electron spectrometers (see Fig. 2.9). A typical image taken from the fluorescent backside of the screen is shown in Fig. 4.3e). The plane of the magnets was not perfectly aligned to the axis of the electron beam which led to helical electron trajectories. Towards lower electron energies the electron trace is limited by the edge of the upper dipole magnet. The spectrum of the electrons passing through the hole reflects the energy distribution of only a small fraction of accelerated plasma electrons. At large distances the energy distribution of the electrons has been found strongly anisotropic and angle dependent [36, 42, 60, 79]. The information from the scintillating screen was therefore primarily used as an on-line diagnostic to optimize the laser-acceleration process. 4.2 X-ray diagnostics The x-ray CCD camera was operated in single-photon counting mode. A single x-ray photon incident on the CCD chip creates electron-hole pairs in an amount which is proportional to the photon energy, on average 29 ev/count in the analyzed spectral range. The charge cloud created by a single photon may diffuse over several pixels (blooming). For an accurate, energy-resolved measurement it is therefore necessary to keep the number of photons incident on the chip so low that the charge clouds do not overlap in order to distinguish single events. For the analysis of the x-ray images an algorithm must be applied that recognizes the patterns of pixels which are known to be created by a single photon. The rejection of all pseudo-events which do not match any of the known patterns ensures the accuracy of the spectral information. Fig. 4.4a) shows a typical image ( pixels) recorded by the camera which 63

66 4 Thomson backscattering from laser-accelerated electrons a) b) Figure 4.4: Analysis of the x-ray CCD images: a) CCD image of the x-rays generated by one laser-shot. b) Enlargement of the image region indicated in a). Single photon events are distinguishable. was exposed to a single laser shot. The electronic dark current was subtracted. The magnified part of the image shown in Fig. 4.4b) clarifies that single photon events are discriminable. The position of the CCD camera relative to the laser focus is displayed in Fig. 4.1b). Two pinholes ensure that only radiation originating from the laser focus is detected by the CCD chip. The first pinhole is constituted by the hole in the probe pulse parabolic mirror of 3 mm diameter which is located at a distance of 150 mm from the focus. A second lead pinhole of 10 mm diameter and 50 mm thickness combined with a thin metallic filter at a distance of 320 mm ensures that bremsstrahlung radiation and stray light are blocked. The CCD chip of 26.6 mm 6.7 mm is positioned at a distance of 1500 mm from the laser focus. The general functionality of the analysis algorithm is described in Fig. 4.5a)-d). The algorithm recognizes the single- and multi-pixel events listed in Fig. 4.5d). The squares represent pixels on the x-ray CCD chip. Two different thresholds were applied: In order to be recognized as a valid pattern, the center pixel (marked red) must contain more counts than threshold No. 1 (T 1 ). Concurrently, adjacent pixels (marked blue) must be higher than background (threshold No. 2, T 2, where T 1 > T 2 ) and smaller than the center pixel. All neighboring pixels not belonging to the pattern (empty squares) are required to be on background level ( T 2 ), otherwise the event is rejected. We found that the optimum thresholds for our application were T 1 = 7 64

67 4 Thomson backscattering from laser-accelerated electrons Figure 4.5: Details of the analysis algorithm: a) Histograms of the various pattern classes of a typical x-ray image as indicated in the graph and the sum of all contributions (dashed line). b) Event counts broken down into pattern classes. For this particular image, most of the events are contributed by class 2 patterns. c) Comparison of a raw image histogram (grey bars) and the resulting analyzed data (identical to the dashed line in a)). d) List of patterns recognized by the algorithm. Two different thresholds (T 1 and T 2, where T 1 > T 2 ) are used: The red center pixel is required to be higher than T 1. Blue pixels must be higher than background (T 2 ) but lower than the center pixel. White pixels must be on background level ( T 2 ). Parts of the x-ray image which do not match these patterns are disregarded. 65

68 4 Thomson backscattering from laser-accelerated electrons a) 1 b) 1 Transmission, Efficiency 0,1 Transmission, Efficiency 0,1 0, Energy / ev 0, Energy / ev Figure 4.6: Quantum efficiency of the x-ray CCD camera and transmission of the filters used in the experiment: The Quantum efficiency of the x-ray CCD camera given by the manufacturer is shown as a dotted line in both graphs. At 1740 ev the K α -edge of Si is visible. Both graphs also display the filter transmission (dashed line) and the resulting efficiency curve according to Eq. 4.1 (solid line). a) Mylar-aluminum filter consisting of 6 µm Mylar foil vaporized with 300 nm aluminum. b) 300 nm Ni filter consisting of nm Ni foil and 3 Ni support grids of 5 µm thickness. Filter transmission data taken from Henke et al. [80]. and T 2 = 3. Each class of patterns contributes to a different part of the spectrum which is illustrated by Fig. 4.5a). This is due to the fact that the probability of generating many-pixel events rises with increasing photon energy. Additionally, the energy threshold for the detection of many-pixel events is higher than for few-pixel events. We therefore take only into account the part of the histogram where all pattern classes are considered, i. e. in particular we disregard all channels below channel 16. For highly exposed images this algorithm may only recognize and analyze a fraction of all events registered by the CCD chip (down to 30%). This is compensated by scaling the resulting spectrum with the total number of counts in the image. The accuracy of this procedure was verified in experiments with characteristic x-ray line emissions of variable intensity using a conventional x-ray tube. In order to obtain a photon spectrum from the image data the histogram of the analyzed data (like the one shown in Fig. 4.5c)) must be divided by the energy dependent quantum efficiency of the camera and by the transmission of the filter set. The efficiency of the CCD chip and the transmission of filters used in this experiment are displayed in Fig. 4.6a) and b). The quantum efficiency of the chip is shown in 66

69 4 Thomson backscattering from laser-accelerated electrons both graphs as a dotted line and the transmission curves of the filters as dashed lines. The electronic noise of the camera as well as the analysis algorithm led to a decreased energy resolution of the obtained spectra. In a separate experiment with a conventional x-ray tube line spectra at energies of 452 ev (Ti L α ), 930 ev (Cu L α ) and 1486 ev (Al K α ) were recorded. The energy broadening was estimated to be 10 channels ( 290 ev) from the width of the emission lines and the energy calibration was determined to an average of 29 ev/count. It is known from [81] that at lower energies the energy resolution of an x-ray CCD is no longer linear. This was confirmed in our experiments, but in the observed energy range the deviation of the energy calibration was < 2 ev. If the photon spectrum R(E) generated in the laser-plasma interaction is slowly varying with energy E compared to the broadening function g(e), one may write the photon distribution S(E) measured with the CCD as S(E) = T (E ) q(e ) R(E ) g(e E ) de R(E) T (E ) q(e ) g(e E ) de = R(E) T (E), (4.1) where T (E) is the transmission of the filter and q(e) the efficiency of the camera. The broadening function was assumed to be of Gaussian shape with a width of 290 ev. T (E) represents the blurred response function of the filter-detector assembly and is shown as solid black line in Figs. 4.6a) and b). The CCD is sensitive to photons of up to 10 kev of energy. The number of photons incident on the chip may be adjusted by decreasing the space angle covered by the CCD (increasing the distance to the source) or by the use of filters of various materials and thickness. The advantages of a CCD in single-photon counting mode compared to imaging soft x-ray spectrometers are the easy alignment of the camera with a collimated cw-laser running along the z-axis, through the laser focus and through the hole in the parabolic mirror and that optimization of the signal strength is easily achieved. The x-ray CCD was therefore the detector of choice in the Thomson backscattering experiment. 67

70 4 Thomson backscattering from laser-accelerated electrons 4.3 Background and competing processes In order to obtain the Thomson backscattered signal from the recorded spectra, it had to be separated from background. Background is caused e. g. by accelerated plasma electrons generating bremsstrahlung radiation in the vacuum vessel or characteristic x-ray emission from material of the nozzle body (steel). From [82, 83] it is also known that a single relativistic laser pulse incident on a gas-jet may produce x-rays in the range of a few kev through betatron oscillations of the electrons in the relativistic channel. This phenomenon is also called plasma wiggler and was first observed with high energy electrons (GeV range) propagating through a preformed plasma [84, 85]. X-rays are generated when relativistic electrons propagate through an ion channel and undergo betatron oscillations with the betatron fundamental frequency ω b = ck b = ω p / 2γ (see also Sec ). The properties of the plasma wiggler radiation are very similar to the radiation generated in a conventional synchrotron wiggler (see [49]).The plasma wiggler strength is given by K = γk b r 0, where r 0 is the excursion of the electrons in the ion channel. For K 1 the radiated spectrum becomes continuous. The critical (maximum) frequency ω c is determined by ω c [MeV] = γ 2 n e [cm 3 ] r 0 [µm], and the maximum of the radiated spectrum is located at about 0.29 ω c. The half angle of the central emission cone is θ cen = K/γ [48, 86]. The betatron radiation may therefore be highly collimated to an emission angle as low as 50 mrad [82] which is of the same order as the expected half angle of emission of Thomson backscattered radiation. Since important quantities like the excursion r 0 of the electrons is not accessible in an experiment, the betatron spectrum cannot be predicted from the laser parameters alone unless PIC simulations are carried out. Comparing our laser parameters to those of Rousse et al. [82] it is apparent that photons of much lower energy and of larger angular spread are to be expected. Its contribution to the background should therefore be small. Background spectra were recorded with only the pump pulse incident on the He 68

71 4 Thomson backscattering from laser-accelerated electrons Figure 4.7: a) Signal and background detected by the x-ray CCD camera. Enlarged region of ev where significant Thomson signal was measured. b) X-ray spectrum up to 7 kev showing characteristic lines from the nozzle material. The data shown here was averaged over 10 laser shots and corresponds to the data point #13 at the maximum in Fig. 4.10b). Please note that a) and b) are shown with a logarithmic scale. c) Single shot spectra (linear scale) obtained by subtracting background. Blue and red curves: ten shots recorded sequentially under constant experimental conditions. The hindmost spectrum clearly stands out from the others and represents a remarkably bright shot. Black curve: average over these ten shots which was used for further analysis. 69

72 4 Thomson backscattering from laser-accelerated electrons gas-jet. The Thomson signal was then identified as the difference between spectra recorded with both pulses incident on the target and these background spectra. Earlier experiments showed that gradual destruction of the nozzle body by accelerated ions leads to a continuously decreasing number of accelerated electrons. Due to the deformation of the nozzle orifice constant optimum conditions for laser-acceleration cannot be provided over several hundred laser shots. It was therefore necessary to record background spectra in regular intervals. Two consecutive background spectra were averaged and subtracted from spectra containing both background and Thomson signal. Fig. 4.7a)-b) shows a sample x-ray signal (including the background) and the corresponding background spectrum. The x-ray spectra typically extend beyond 7 kev but significant Thomson backscattering signal was measured in the range up to 2 kev only with our experimental configuration. In the range of kev the K α -lines from the nozzle material (mostly Fe, Cr) are visible. The Thomson backscattering signal as determined by subtraction amounts to about 30% background. The pure Thomson backscattering signal obtained from this measurement by subtraction is shown in Fig. 4.7c). The red and blue curves represent 10 single-shot spectra which were recorded under constant experimental conditions. The black curve is the average spectrum calculated from these ten shots. From the shot-to-shot fluctuations in these spectra a standard error for each channel may be calculated. The resulting error bars are displayed e. g. in Figs and An important property of Thomson backscattered x-rays in an all-optical setup is the small source size which is determined by the laser-generated electron beam profile. In our experiment the Thomson backscattering source size was not directly accessible since we were not able to discriminate between Thomson scattered and background photons. However, the overall x-ray source size was determined which represents an upper limit of the Thomson backscattering source size. The shadow of the filter support grids becomes visible when adding all recorded low-exposure images (see Fig. 4.8a)). In a simple geometric optics model the source size S 1 may be estimated from the softness of the shadow edges: S 1 = L 1 L 2 S 2, where L 1 and L 2 are the distance of the source and of the detector from the grid, 70

73 4 Thomson backscattering from laser-accelerated electrons a) b) Figure 4.8: Determining the source size from the filter grid. a) Adding all x-ray images recorded during the experiment yields an image of the shadow of the filter support grids. b) Cross-section around a grid shadow. The shadow edge extends over a distance of about 3 pixels corresponding to 78 µm on the CCD chip. respectively, and S 2 the extension of the shadow edge. The setup of the x-ray CCD is shown in Fig. 4.1b) and S 2 was determined from an image cross-section as shown in Fig. 4.8b). The source size S 1 was calculated to µm 2 which is in accordance with results published in [82, 83] and with the diameter of the plasma channel (20 µm FWHM) as shown in Fig Temporal change of total backscattered radiation Another independent indication that the recorded signal in fact arises from Thomson backscattering is the circumstance that the total number of backscattered photons is dependent on the delay between pump and probe pulse. Let us first consider a configuration where the pump and probe pulse vacuum foci are identical. The delay between pump and probe may be adjusted by moving the 45 beam-splitter by a distance s, where s is measured normal to the beam-splitter surface. The change of the pump pulse optical path amounts to σ 1 = 2 s (cf. Fig. 4.9b)). We now define the delay between pump and probe pulse as the difference between the moment in time when the pump pulse passes its vacuum focus and the moment in time when pump and probe pass each other. The choice of the pump pulse passing its focus as reference point in time is arbitrary. This choice turns out to be reasonable, however. We will apply this frame of reference even to situations where the pump 71

74 4 Thomson backscattering from laser-accelerated electrons Figure 4.9: Enlarged sections of the experimental setup shown in Fig. 4.1a) illustrating the impact of moving the parabolic mirror of the probe beam a) and of moving the beamsplitter on the time delay between pump and probe pulse b). and probe vacuum foci are not identical. The change of the optical path σ 1 of the pump pulse shifts the point of interaction by σ 1 /2 or - in other words - changes the delay between pump and probe by τ = σ 1 /2c. The situation is slightly more complicated, if one also considers configurations where the probe pulse focus is moved away from the pump pulse focus by a distance z by moving the probe parabolic mirror. The change of the probe pulse optical path can be estimated using geometric optics (see Fig. 4.9a)) to be σ 2 = (1 + 1/ 2) z. The corresponding time delay τ = σ 2 /2c inflicted on the moment of interaction must be considered when comparing experiments with different focal configurations. Two different experiments were carried out: In a first experiment A an Al-Mylar filter was used in front of the CCD (transmission shown in Fig. 4.6). However, for this experiment the observation angle ϑ is unknown (see Fig. 2.10). Its results may therefore serve for qualitative analysis of the total backscattered photon yield only and will not be considered later in Sec. 4.5 for spectral analysis. In a second experiment B a 300 nm Ni filter was deployed and the number of photons detected in the range of ev could be increased fourfold. The observation angle ϑ was determined experimentally. The signal to background ratio remained the same in both experiments (up to 0.3). In Fig the number of photons integrated over the interval ev and 72

75 4 Thomson backscattering from laser-accelerated electrons a) b) Number of photons / Number of photons / Delay / fs Delay / fs Figure 4.10: Integral number of photons in the range of ev. a) Data recorded in experiment A: The data indicated by squares was recorded with pump and probe focus being identical. Subsequently the focus of the probe pulse was moved away from the pump focus by 200 µm (data indicated by circles). A delay of 0 fs corresponds to the laser pulses interacting in the pump pulse focus. b) Data recorded in experiment B. Filled squares indicate a first sequence of measurements, empty squares a second. Both delay curves a) and b) show good agreement. One error bar in each graph indicates the standard error of the measurement deduced from shot-to-shot fluctuations of ten laser shots. averaged over 10 shots is shown with respect to the pump-probe delay. Fig. 4.10a) shows data acquired in experiment A. The photon yield in the configuration of overlapping vacuum foci (indicated by squares) shows a strong dependence on the pumpprobe delay. It is close to zero at a delay of 200 fs and rises to a maximum at 300 fs, then slowly decreases again. In order to determine if the dependence on the delay is due to the spacial convergence of the probe pulse only (opening angle 30 ) the probe pulse vacuum focus was moved by z = 200 µm from the pump pulse vacuum focus towards the incident probe pulse. If the dependence obtained from the earlier measurement was due only to the properties of the probe pulse, the maximum of the delay curve should have shifted by τ = z/c 700 fs towards positive delay. The measured data indicated by circles in Fig. 4.10a), however, fits nicely into the previously determined curve. Please note that the timescale of this additional data was adjusted according to the considerations at the beginning of this section. The increase in backscattered photons must therefore be related to an increase in high-energy electrons in the laser plasma. This seems reasonable considering that the probe pulse undergoes filamentation in the gas-jet and is consequently not strongly focused. 73

76 4 Thomson backscattering from laser-accelerated electrons Figure 4.11: a) The normalized photon yield in the interval ev averaged over 10 laser shots is plotted over the delay between the laser pulses. In this way data from different experiments (indicated by different symbols) can be compared. At delays where several data points are available an average is carried out and indicated as a solid line. b) Sketches of three scenarios of different pump-probe delay where the pump pulse is incident from the left generating a relativistic channel in which electrons are accelerated. The region where pump and probe pulse overlap is indicated with a hatched box. The photon yield obtained from experiment B exhibits the same dependency, but with an improved signal level (see Fig. 4.10b)). The maximum of the delay curve is more pronounced. After recording a first data set (filled squares) the measurement was repeated (empty squares) to ensure that the assumed Thomson signal did not decrease continuously with an increasing number of shots. Due to the deformation of the nozzle body the Thomson signal drops overall, as does the background (not shown). The results of the time-resolved photon yield measurements are summarized in Fig All data acquired so far is displayed in Fig. 4.11a). The number of photons detected in the energy interval ev was normalized for each experiment to facilitate comparison. At those delays, where more than one data point was available, an average was calculated and is shown as solid line. The underlying processes may be visualized schematically as shown in Fig. 4.11b). Three different scenarios are selected and indicated accordingly in Fig. 4.11a): 1. A delay of 200 fs: The probe pulse passes the pump pulse vacuum focus before the pump pulse arrives. Both pulses meet before the pump pulse has 74

77 4 Thomson backscattering from laser-accelerated electrons undergone self-focusing. At this moment in time effectively no electrons have been accelerated yet. 2. A delay of 200 fs: The probe pulse interacts with the laser-accelerated electrons towards the end of the relativistic channel. At this moment in time the number of accelerated electrons reaches its maximum. 3. A delay of 600 fs: The probe pulse interacts with the electrons outside the relativistic channel and the electron beam is now diverging. At later delays the divergence of the probe pulse may become relevant as well. The timescale of the changes in the photon yield agrees well with our observations of the non-linear Thomson scattering (Fig. 4.2). Typical side view images of the interaction region at the second-harmonic frequency show bright emission over a distance of µm along the z-axis. A laser pulse would need fs to propagate through this region corresponding to the rising edge of the graph in Fig. 4.11a). 4.5 Electron spectra In Sec. 2.4 the principles of Thomson backscattering and their application to the alloptical photon collider setup were discussed. It was shown that the detected x-ray spectra contain direct information about the electron energy at the location of interaction and at the certain moment in time of interaction. Thomson backscattering may therefore serve for on-line time-resolved electron spectroscopy. We will briefly recall the most important interrelations from Sec The x-ray photon energy E x = ω x may be calculated from the electron energy using E x = 4γ2 E γ 2 ϑ 2, where γ is the Lorentz factor of the electron, E 0 = ω 0 the laser photon energy and ϑ the angle of observation with respect to the direction of electron propagation (see Fig for the definition of ϑ). Conversely, one may calculate the energy of the 75

78 4 Thomson backscattering from laser-accelerated electrons electron (in terms of γ), which scattered the x-ray photon, by (cf. Eq and 2.37) γ = E x 4E 0 (1 Exϑ2 4E 0 ). (4.2) Having measured the backscattered photon spectrum N x / E x and having determined experimentally the observation angle ϑ we may now calculate the underlying electron spectrum N e / E e using N e = N x 8E 0 E e E x αn 0 a 2 0mc 2 Ω 1 γ, (4.3) where everything except γ, which may be calculated from Eq. 4.2, is experimental data or known constants. In experiment B, where the Ni filter was deployed, the constants in Eq were determined as follows: The number of oscillations N 0 contained in an 85 fs pulse of λ 0 = 795 nm is about 30. The solid angle of detection Ω = 80 µsr in our experiment was limited solely by the size of the CCD chip and not by pinholes. The angle of observation was experimentally determined to ϑ = 60 mrad by introducing a crosshair into the beam path of the collimated laser marking the center of the beam profile. Behind the laser focus the shadow of the cross-hair indicated the laser axis and therefore the axis of electron propagation and was used to directly measure the angle of observation ϑ on the surface of the parabolic mirror. It can easily be seen from Eq. 4.2 that a ϑ as large as 60 mrad will have a sizeable impact on the calculated electron spectrum. In fact, this ϑ is larger than the opening half angle of the central radiation cone for electrons of γ > 3. Therefore the detected backscattered Thomson signal is much weaker than the backscattered radiation directly in forward direction where most of the radiated power is concentrated. On the other hand, observation under a non-zero angle ϑ may offer certain advantages due to the γ 2 dependency of the x-ray energy E x : If one observes the scattered radiation exactly in forward direction and analyzes an x-ray spectrum covering two octaves, the obtained electron spectrum extends over only 2 octaves which may be a quite limited part of the electron spectrum. Observation under a non-zero angle ϑ, however, increases the covered electron energy range. After discussing the dependency of the total backscattering photon yield on the 76

79 4 Thomson backscattering from laser-accelerated electrons Data set #11: 0 fs 1000 dn x /de x / (29 ev) dn e /de e / (10 4 MeV -1 ) MeV 3.3 MeV Energy / ev Data set #12: 100 fs Energy / MeV 1000 dn x /de x / (29 ev) dn e /de e / (10 4 MeV -1 ) MeV 6.5 MeV Energy / ev Energy / MeV dn x /de x / (29 ev) Data set #13: 200 fs dn e /de e / (10 4 MeV -1 ) T e = 1.3 MeV T e = 6.1 MeV Energy / ev Energy / MeV Figure 4.12: Thomson backscattering spectra recorded with the x-ray CCD camera and averaged over 10 shots (graphs on the left) and the electron spectra calculated from these averaged photon spectra (graphs on the right). The data set numbers and delays correspond to those given in Fig

80 4 Thomson backscattering from laser-accelerated electrons pump-probe delay in the previous section it is of special interest to examine the change in the electron spectra with respect to the delay. Electron spectra were calculated from selected backscattered x-ray spectra obtained from experiment B. The corresponding total photon yields are indicated by data set numbers in Fig The x-ray spectrum and the calculated electron spectrum are shown in Figs and 4.13 arranged in rows. The error bars indicated in the x-ray spectra are standard errors derived from shot-to-shot fluctuations of ten laser shots. At the timing condition when the highest photon yield was generated the statistically most reliable spectra were obtained (here: set #13). The electron spectra are composed of two distinct exponential energy distributions with temperatures of about 1 MeV and 6 MeV, respectively. The temperatures are indicated by dashed lines in the electron spectra. The hotter temperature agrees well with our previous measurements using a conventional spectrometer. The surprising result is that the electron temperatures do not change significantly in the displayed delay range of 600 fs. From early delays on the two populations exist. At delay times far from optimum delay, however, little data is available to reliably carry out non-linear curve fits. The total number of electrons N b contained in the spectrum may be calculated from Eq It was pointed out earlier that the laser field strength a 0 of the probe pulse was reduced due to filamentation and the effective value was estimated from shadow images to be in the range of (0.8 being the nominal field strength). Inserting the parameters from the non-linear curve fits from set #13 yields an electron bunch charge of 0.07 nc nc. Other groups reported total bunch charges of nc [36, 42, 79, 87, 88]. This indicates that our effective probe pulse laser field strength is about 0.1 and thus significantly smaller than the nominal value. Future experiments will improve this value using specifically tailored, supersonic gas-jets. We interpret our results from the time-resolved electron spectroscopy as follows: The laser acceleration occurs on a very short timescale of the order of the laser pulse duration. During this short time two electron populations (one of low temperature, about 1 MeV, one of higher temperature, about 6 MeV) are formed. The idea that the acceleration of a particular set of electrons would take place over the whole length of the relativistic channel is therefore wrong. The acceleration length is much shorter. While the pump laser pulse propagates through the channel, the total number of 78

81 4 Thomson backscattering from laser-accelerated electrons Data set #16: 400 fs 1000 dn x /de x / (29 ev) dn e /de e / (10 4 MeV -1 ) MeV 5.5 MeV Energy / ev Data set #18: 500 fs Energy / MeV 1000 dn x /de x / (29 ev) dn e /de e / (10 4 MeV -1 ) MeV 6.8 MeV Energy / ev Data set #19: 600 fs Energy / MeV 1000 dn x /de x / (29 ev) dn e /de e / (10 4 MeV -1 ) MeV 7.4 MeV Energy / ev Energy / MeV Figure 4.13: Thomson backscattering spectra recorded with the x-ray CCD camera and averaged over 10 shots (graphs on the left) and the electron spectra calculated from these averaged photon spectra (graphs on the right). The data set numbers and delays correspond to those given in Fig

82 4 Thomson backscattering from laser-accelerated electrons relativistic electrons changes (increases), but not their energy distribution. These measurements represent the first energy- and time-resolved diagnostics of laser-accelerated electrons. So far, electrons accelerated in the SM-LWFA regime which exhibited an exponential energy distribution were examined. It will be most exciting to apply this technique to the regime where quasi-monoenergetic electrons are generated which has been shown to be possible also at the Jena laser facility [44]. It is believed that in the transition regime towards bubble acceleration the process of acceleration takes place over a range in the order of millimeters [46]. The time-dependent evolution of this process may be monitored with the photon collider. 80

83 5 Future prospects of the photon collider In the previous section the first experimental observation of Thomson backscattered photons from laser-accelerated electrons in an all-optical setup was presented. There, the focus was set on the identification of the Thomson backscattered signal, its separation from background and its application as an electron diagnostics tool. The prospects of the photon collider setup, however, reach far beyond that. It may serve as a short-pulsed x-ray source with unique properties or it may actually provide colliding particle beams - measuring up to its name. The photon collider may also be the basis of a visionary experiment demonstrating non-linear quantum electrodynamics (QED). These applications will be discussed in the following. 5.1 Thomson backscattering as x-ray source In order to assess the unique x-ray source properties of the photon collider appropriately, I will first briefly review the requirements which are made on short-pulsed x-ray sources in modern experiments and then list sources which are available today or which will be made available in a few years time. Finally, the capabilities of the photon collider will be presented and compared to those of other sources. One important application of short x-ray pulses are pump-probe experiments. The goal of these experiments is to achieve a better understanding of processes governed by atomic motion studying excited systems prior to vibrational relaxation, far from equilibrium. In such an experiment a pump pulse (typically realized by an ultra-short optical laser pulse) induces e. g. a conformational change in the sample which is then probed by an other pulse after a variable delay. If the probe pulse is an x-ray pulse, direct observation of real-time changes in the sample by means of x-ray diffraction 81

84 5 Future prospects of the photon collider Source Principle / Method Energy range / kev Source size / µm ALS 5.0 Synchrotron APS Type A Synchrotron Bessy II U49 Synchrotron PLEIADES Dense laser plasmas Thomson backscattering Characteristic x-ray emission B pk up to Plasma wiggler / LOA Betatron radiation XFEL FEL Photon collider Photon collider (projected) Thomson backscattering Thomson backscattering Table 5.1: Characteristic parameters of selected x-ray sources: Undulators operating at the fundamental frequency of Advanced Light Source (ALS, Berkeley), Advanced Photon Source (APS, Chicago), Berliner Elektronenspeicherring-Gesellschaft für Synchrotronstrahlung (Bessy, Berlin). Further sources are Picosecond Laser-Electron InterAction for the Dynamic Evaluation of Structures (PLEIADES, Livermore), the plasma wiggler at the Laboratoire de l optique appliquée (LOA, Palaiseau) and the European X- ray Laser project (XFEL, Hamburg). The peak spectral brightness B pk is given in photons/s/mm 2 /mrad 2 /0.1%BW, values are taken from literature given in the text. is possible (in reciprocal space). The timescale of interest for these experiments is in the range of a few 100 fs which is the order of vibrational periods. The x-ray probe pulse must therefore be ultrashort and very intense since the diffraction efficiency of the sample may be very low. A small x-ray source size is advantageous since small sources can be focused onto small areas (yielding higher photon flux) using x-ray focusing optics and allow for better imaging quality (if the application involves imaging). Additionally, the delay between pump and probe must be well-known - a priori or a posteriori. Tab. 5.1 lists examples of brilliant x-ray sources which are available for optical pump x-ray probe experiments. As characteristic parameters the energy range, the source size and the peak spectral brightness were selected which do not represent all important source properties but which may be a guideline for comparison. 82

85 5 Future prospects of the photon collider Third-generation synchrotron facilities Modern existing synchrotron facilities typically consist of a linear accelerator providing electron bunches in the GeV range which are then injected into storage rings. They provide short-pulsed x-rays in the energy range of a few hundreds of ev up to tens of kev sending the GeV electrons through bending magnets, undulator or wiggler structures in the storage ring. The electrons are kept circulating for as long as possible to maximize the usable beam-time. Synchrotrons are therefore inflexible concerning the timing of the electron bunches [89]. The x-ray pulse duration is determined by the electron bunch length which is typically 100 ps. In order to achieve better time-resolution, time-resolving detectors, streak cameras, are deployed which are limited in resolution to 1 ps [90, 91]. The use of a streak camera in turn lowers the effective number of photons available for the measurement since the long x-ray pulse is distributed over many slices registered by the streak camera. Novel electron bunch slicing techniques and ultrafast x-ray mirrors have been developed to generate truly femtosecond synchrotron radiation [92, 93]. However, these techniques reduce the x-ray intensity to the same degree as slicing by a streak camera. Generally, control over the pump-probe delay must be achieved by external synchronization which is very difficult at these time-scales. A jitter of 3 ps RMS between the pump laser pulse and the x-ray source may typically not be overcome [94]. Laser-plasma based x-ray sources When an ultra-short and ultra-intense laser pulse impinges onto a thin foil a microplasma is produced which is a source of many kinds of radiation and accelerated particles. In particular, a large number of accelerated, supra-thermal electrons generate bremsstrahlung and characteristic x-ray line emission within the foil target. The line emission may be considered monochromatic (0.01% BW) and is isotropic. The source size is small, i. e. of the order of the laser focus (a few tens of microns). The generated x-ray pulse duration is of the order of the laser pulse duration [95 98]. A fraction of the isotropic emission may be collected and refocused using x-ray optics photons from a source size of µm 2 within 100 fs emitted into a solid angle of 4π are reported in [99]. Recently, two novel laser-plasma x-ray sources have been demonstrated: Betatron 83

86 5 Future prospects of the photon collider radiation from relativistic electrons inside a plasma channel (the plasma wiggler, [82]) which was briefly discussed in Sec. 4.3 and bremsstrahlung from relativistic, laser-accelerated electrons [9, 14, 100]. The plasma wiggler represents a very bright, collimated and ultra-short (laser pulse-length limited) soft x-ray source with a size of down to 20 µm. Its peak spectral brilliance amounts to B pk = (this is different from the figure given in [82] which was miscalculated). The bremsstrahlung source produces hard x-rays and gamma-rays with an opening angle of 3, a source size of 400 µm and a pulse duration of a multiple of the laser pulse length depending on the converter target. Hybrids: Thomson scattering with linear accelerators (LINACs) This scheme, where a laser pulse is scattered from accelerated electrons, was proposed in the early 60s [101, 102] but did not achieve competitive brightness until the development of high-intensity lasers. It has been demonstrated in many facilities [68 71, 75, 103, 104] and maximum photon energies up to the GeV range have been observed [7]. The achieved intensities were, however, only moderate. The x-ray pulse duration is either determined by the laser pulse duration in a 90 scattering geometry (short-pulsed but low brightness x-rays) or by the electron bunch length in collinear geometry (long-pulsed and high brightness). Expensive and highly complex mechanisms must be installed to obtain control over the time jitter between laser and electron bunch which is 1.5 ps at best [69, 71, 72]. The only way to overcome this shortcoming is to carry out sophisticated a posteriori delay measurements with limited time-resolution [105] leading to a Monte-Carlo-like experimental strategy [94]. X-ray free electron lasers (X-FELs) They are the novel tool which will revolutionize the field - whereas tool is a short word for an installation of more than 3 km length. An X-FEL consists of a linear accelerator followed by an undulator of exceptional length (number of periods 10 3 ). The radiation is produced based on the same principles as in a synchrotron undulator with a few additional effects which are caused by a phenomenon widely known as the FEL collective instability [106]. Electrons propagating through the undulator interact with the electromagnetic Spectral brightness is given in units of photons/mm 2 /mrad 2 /s/0.1%bw which are commonly used in the synchrotron community. The units are omitted in the text for brevity. 84

87 5 Future prospects of the photon collider field generated by other electrons. The interaction changes their energy and the change is modulated at the x-ray wavelength. Additionally, due to the electron energy spread, electrons will bunch together within a wavelength. Spatially coherent radiation is generated, the single electron fields superimpose and the bunching mechanism grows even stronger. The gain is then proportional to the number of electrons squared which is huge. The laser-like properties of the radiation and the exceptional requirements on the electrons necessary for the FEL process lead to a bright x-ray source surpassing today s existing sources by at least eight orders of magnitude. FELs generating long-wavelength radiation may be compact-sized [107], but in order to reach the hard x-ray regime electrons in the GeV range must be used which calls for large accelerator facilities. Currently, X-FELs at two locations are under construction: The Linac Coherent Light Source (LCLS) at Stanford, USA [108], and the European X-ray Laser Project (XFEL) at Hamburg, Germany [109]. The former will be operational in 2009 at up to 8 kev, the latter in 2012 at up to 12 kev. The peak spectral brightness B pk of XFEL is projected to be But even though X-FEL facilities will provide radiation of unprecedented quality it will be difficult to conduct pump-probe experiments since - as it is with all accelerator-based sources - sub-picosecond synchronization is hard to achieve. A posteriori delay measurements will have to be developed to fully take advantage of the ultra-short X-FEL radiation [94, 105]. The photon collider How does the photon collider compare to the aforementioned sources? Since it is an all-optical scheme, it has intrinsical, built-in, synchronization at highest level as do have all other purely laser-based sources. It is, in this sense, predestined for pump-probe experiments. In our experiment we detected a maximum of Thomson backscattered photons at optimum delay. This corresponds to an average spectral brightness of B av = and a peak spectral brightness B pk = assuming a pulse duration of 85 fs. However, the observation angle was not optimized for maximum photon yield but rather large, larger than the central emission cone of the detected radiation. In forward direction the peak spectral brightness is calculated to be B pk = This is still not the best that the photon collider at our laser parameters could do: The probe beam was filamented before it interacted with the electron 85

88 5 Future prospects of the photon collider bunch thus decreasing the effective probe pulse intensity. Calculations have been carried out by Catravas et al. [50] assuming perfect spatial overlap between the laser-accelerated electron bunch and the probe laser focus. Catravas et al. obtained B av = and B pk for a typical, LWFA generated electron bunch of 5 nc and exponential energy distribution. The spectral brightness using narrower, quasi-monoenergetic laser-accelerated electron bunches may be even higher. The photon collider has therefore the potential to advance to a regime which was until today reserved for large synchrotron facilities. Rousse et al. used only a single laser pulse to obtain a brightness of B pk why would one bother to set up colliding laser pulses? There are important properties of Thomson backscattered photons which make the photon collider superior to the betatron source. Considering e. g. the scalability: How may higher photon energies be achieved in these schemes? Generally, in both cases the photon energy scales with γ 2. However, in the case of the plasma wiggler, the process of laser-acceleration is closely linked to the wiggler motion and cannot be separated. The conditions for optimum laser-acceleration may not coincide with optimum wiggler conditions. The plasma density, e. g., must be low (< cm 3 ) to obtain quasi-monoenergetic electron bunches [46], whereas the maximum photon yield from the plasma wiggler is obtained at maximum plasma density: The gain in radiated power is proportional to the plasma density squared [48]. Rousse et al. carried out the plasma wiggler experiments producing electron beams of 20 MeV temperature. If laser photons would be scattered from these electrons in a photon collider geometry, the maximum of the resulting photon spectrum would be located at 10 kev (see Sec. 2.4) compared to an exponential photon spectrum from the plasma wiggler in the range of kev. For 40 MeV which are reported in [9] the maximum would be shifted to 40 kev. Another advantage of the photon collider is that it conserves polarization, i. e. the emitted radiation has exactly the polarization of the scattering laser pulse. Polarized intense x-rays are of great interest for probing magnetic materials, helical structures and other samples with polarization dependent properties [49, Ch. 5 and references therein]. With the photon collider the parameters for laser-acceleration can be optimized independently from the x-ray generation. In consideration of the recent advances in laser electron acceleration [3 5, 44] the photon collider thus offers the means to generate unparalleled bright, polarized and short-pulsed x-rays in energy ranges 86

89 5 Future prospects of the photon collider Figure 5.1: The mini-orange is part of a positron detector specially designed for experiments with the photon collider. This image was created by K. Haupt in a ray-tracing simulation. Green, dark blue, cyan and red lines indicate positron trajectories of different energy generated in the laser focus. The positron trajectories are bent towards the top by the magnetic field where they can be guided to a detector by quadrupole magnets. exceeding 100 kev. 5.2 Electron collider and positron production Ultrashort laser pulses have proven to efficiently accelerate electrons and even generate quasi-monoenergetic electron beams. The next generation of high-power laser systems aims at generating shorter pulses at higher intensities. These relativistic few-cycle pulses are predicted to produce monoenergetic electron pulses of up to GeV energy. The scaling laws governing the underlying processes have been studied thoroughly in PIC simulations and are thought to be reliable since the simulations also explain the recent results in the transition regime between LWFA and bubble acceleration (see Sec. 2.3). Having these ultra-short pulsed particle sources at hand it seems logical to conduct electron collider experiments. When relativistic electrons collide, electron-positron pairs may be generated. The photon collider may therefore serve as a positron 87