STUDIES OF A TERAWATT X-RAY FREE-ELECTRON LASER

STUDIES OF A TERAWATT X-RAY FREE-ELECTRON LASER H.P. Freund, 1,2,3 1 Department of Electrical and Computer Engineering, University of New Mexico, Albuquerque, New Mexico USA 2 Department of Electrical and Computer Engineering, Colorado State University, Fort Collins, Colorado, USA 3 NoVa Physical Science and Simulations, Vienna, Virginia, USA The possibility of constructing terawatt (TW) x-ray free-electron lasers (FELs) has been discussed using novel superconducting helical undulators [5]. In this paper, we consider the conditions necessary for achieving powers in excess of 1 TW in a 1.5 Å FEL using simulations with the MINERVA simulation code [7]. Steady-state simulations have been conducted using a variety of undulator and focusing configurations. In particular, strong focusing using FODO lattices is compared with the natural, weak focusing inherent in helical undulators. It is found that the most important requirement to reach TW powers is extreme transverse compression of the electron beam in a strong FODO lattice. The importance of extreme focusing of the electron beam in the production of TW power levels means that the undulator is not the prime driver for a TW FEL, and simulations are also described using planar undulators that reach near-tw power levels. In addition, TW power levels can be reached using pure self-amplified spontaneous emission (SASE) or with novel selfseeding configurations when such extreme focusing of the electron beam is applied. PACS numbers: 41.6.Cr, 52.59.Rz I. INTRODUCTION The number of x-ray Free-Electron Lasers (FELs) is increasing around the world [1-4] and the user community for these light sources is growing as well. Along with this growth of the user community, we can confidently expect that novel and important new applications will be found. The first operational x-ray FEL, the Linac Coherent Light Source (LCLS) at the Stanford Linear Accelerator Center [1], produces approximately 2 GW pulses of 1.5 Å photons at a repetition rate of 12 Hz, and the other x-ray FELs produce similar power levels. In order to be proactive and produce higher photon fluxes for the rapidly developing user community, research is under way around the world in techniques to produce still higher peak powers. Recent simulations [5] based on the GENESIS simulation code [6] and using a novel super-conducting helical undulator design and a quadrupole FODO lattice indicate that a terawatt (TW) x-ray FEL is possible. The configuration that was studied was based upon a selfseeding scheme whereby the optical field at an early stage of the SASE interaction is passed through a monochromator and then resynchronized with a fresh section of the electron bunch. This seed is assumed to be at power levels far in excess of the noise power; hence, the subsequent interaction is equivalent to that of a seeded x- ray FEL. The seed power is assumed to be at MW power levels, and TW output power levels are found after an additional m of a tapered undulator line. Both steadystate and time-dependent simulations with GENESIS were described where it was found, as expected, that the power levels were reduced when time-dependence was included in the simulations. This result represents an enhancement of the output power in the tapered undulator configuration of nearly two orders of magnitude over the saturated power in a uniform undulator. It has long been recognized that efficiency enhancements are possible in FELs using a tapered undulator [7,8]. Historically, however, experiments have shown efficiency enhancements using a tapered undulator of factors of 3 5 [9-11]. It is important, therefore, to understand what gives rise to such extreme efficiency enhancements with tapered undulators. In this paper, we consider various configurations necessary to achieve TW power levels in an x-ray FEL based upon both helical and planar undulators in conjunction with a strong-focusing FODO lattice. Simulations are described using the MINERVA simulation code [12]. This represents a preliminary analysis using steady-state simulations. Time-dependent simulations are in progress and will be reported in a future publication. The principal result we find is that near-tw or TW power levels can be obtained using extreme focusing of the electron beam by the FODO lattice with either a helical or planar tapered undulator. Hence, it is the extremely tight focusing imposed by the FODO lattice that gives rise to such extreme efficiency enhancements. For the cases under consideration, the rms electron beam radius can be as small as 7 8 m and the peak current densities reach 25 3 GA/cm 2. In contrast to expectations, we find that optical guiding occurs even for a tapered undulators in the limit of such extreme focusing. Simulations are performed using both long, continuous undulators and segmented undulators, and the taper is optimized with respect to both the start-taper point and the taper slope. This technique for efficiency enhancement can be used for pure SASE FELs as well as schemes where a monochromator is used to selectively narrow the SASE linewidth. The organization of the paper is as follows. A brief description of the MINERVA simulation code is given in Sec. II. The numerical simulations are discussed in Sec. III. This includes simulations of a long, single-section helical undulator. Here we compare the performance of a natural, weak-focusing helical undulator with that for a helical undulator with a strong-focusing FODO lattice. While it is not practical to construct a single, m long undulator, an analysis of this configuration serves to illustrate the effects of varying focusing strengths on the performance of a tapered undulator. We also compare the performance of the single, long undulator with that of a segmented

undulator with identical period and field strength. Further simulations are discussed for a planar undulator system with strong-focusing showing similar increases in the output power when extreme focusing of the electron beam is applied. A summary and discussion is given in Sec. IV. II. THE MINERVA SIMULATION CODE The MINERVA simulation code [12] is based on a threedimensional, time-dependent nonlinear formulation of the interaction that is capable of modeling a large variety of FELs including amplifier, oscillator, and self-amplified spontaneous emission (SASE) configurations. MINERVA employs the Slowly-Varying Envelope Analysis (SVEA) in which the optical field is represented by a slowly-varying amplitude and phase in addition to a rapid sinusoidal oscillation. The optical field is described by a superposition of Gaussian modes. The field equations are then averaged over the rapid sinusoidal time scale and, thereby, reduced to equations describing the evolution of the slowly-varying amplitude and phase. Time-dependence is treated using a breakdown of the electron bunch and the optical pulse into temporal slices each of which is one wave period in duration. The optical slices slip ahead of the electron slices at the rate of one wavelength per undulator period. MINERVA integrates each electron and optical slice from z z + z and the appropriate amount of slippage can be applied after each step or after an arbitrary number of steps by interpolation. Particle dynamics are treated using the full Newton- Lorentz force equations to track the particles through the optical and magnetostatic fields. The formulation tracks the particles and fields as they propagate along the undulator line from the start-up through the (linear) exponential growth regime and into the nonlinear postsaturation state. MINERVA includes three-dimensional descriptions of linearly polarized, helically polarized, and elliptically polarized undulators including the fringing fields associated with the entry/exit transition regions. This includes an analytical model of an APPLE-II undulator. Additional magnetostatic field models for quadrupoles and dipoles are also included. These magnetic field elements can be placed in arbitrary sequences to specify a variety of different transport lines. As such, we can set up field configurations for single or multiple wiggler segments with quadrupoles either placed between the undulators or superimposed upon the undulators to create a FODO lattice. Dipole chicanes can also be placed between the undulators to model various optical klystron and/or highgain harmonic generation (HGHG) configurations. The fields can also be imported from a field map. It is important to remark that the use of the full Newton-Lorentz orbit analysis allows MINERVA to treat self-consistently both the entry/exit taper regions of undulators, and the generation of harmonics of the fundamental resonance. In order to apply the formulation to the simulation of FEL oscillators, an interface has been written between MINERVA and the optical propagation code OPC [13,14]. Oscillator simulations proceed by tracking the output optical pulse from the undulator as simulated by MINERVA, through the resonator and back to the undulator entrance using OPC, after which the optical field is then imported into MINERVA for another pass through the undulator. This process is repeated for as many passes through the undulator and resonator as required for the oscillator to achieve a steady-state. III. NUMERICAL SIMULATIONS The simulations that we discuss herein are in the steadystate limit. This permits relatively rapid computational scans over a large variety of configurations while also capturing the essential physics underlying the interactions. In particular, many simulation runs are needed to optimize a tapered undulator configuration with respect to the starttaper point and the taper slope. However, we expect that the slippage of the optical field relative to the electronswill result in some degradation of the interaction and timedependent simulations are required to accurately describe an actual experimental configuration. Such simulations are in progress and will be reported in a future publication. A. The Case of a Helical Undulator The configuration that we consider is based upon a selfseeding scenario in which the interaction in a SASE FEL is halted at an early stage by passing the optical field through a monochromator to extract a narrow band of the SASE radiation after which this narrow bandwidth light is then re-injected in synchronism with a portion of the electron beam that acts as a fresh bunch. Hence, the light acts as the seed in a Master Oscillator Power Amplifier (MOPA). The electron beam is assumed to be characterized by an energy of 13.22 GeV, a peak current of 4 A, an rms energy spread of.1% and normalized emittances of.3 mm-mrad in both the x- and y-directions. The transverse profile of the electron beam is assumed to be characterized by a Gaussian distribution, and the beam is matched into either the natural focusing of a helical undulator or the FODO lattice/helical undulator system. We initially study the case of a single, long undulator with a period of 2. cm and a peak on-axis amplitude of 16.1 kg. This is equivalent to an undulator parameter K = 3.1, and yields a resonance at a wavelength of 1.5 Å. The simulations do not include the initial SASE region prior to the monochromator, and the undulator in the MOPA region is assumed to be m in length. While it is not practical to construct an undulator that is m in length, this model is useful to study the essential physics of the interaction in a tapered undulator with strong focusing applied. Since optical guiding ceases in the gaps between undulator segments, it is expected that the interaction will be less efficient in a segmented undulator, and an equivalent simulation of a segmented undulator will be described for comparison. We first consider the case of weak, natural focusing. In this case, the -function is 38.7 m and the matched rms beam radius is 29.9 m. Assuming that the seed power is 5 MW, saturation in the uniform undulator is found after about 12.5 m at a power level of about 4 GW. Two taper 2

Pierce Parameter Power (W) profiles have been considered: linear and quadratic. The growth of the power for each profile is shown in Fig. 1, where the optimal values of the start-taper point and the total decrease in the undulator field are indicated. As shown in the figure, there is little difference in the output powers for the two taper profiles after a length of m. This is typically what we found in the strong-focusing cases as well, and we will limit subsequent discussions to the case of a linear down-taper. Observe that the maximum power achieved after m is about.3 TW in either case which represents an enhancement of the efficiency over the saturated power in the uniform undulator by about a factor of about 7.5. Note, however, that the quadratic taper would result in increasingly higher powers relative to the linear taper for substantially longer undulators. 3.5 1 11 3 1 11 2.5 1 11 2 1 11 1.5 1 11 1 1 11 5 1 1 Linear Taper z = 12.3 m =.33 Quadratic Taper z = 13. m =.22 2 4 6 8 Fig. 1 The growth of the power for optimized linear and quadratic taper profiles for a weak-focusing helical undulator. (m) Gradient (kg/cm) 2 ( x + 2 y ) 1/2 (m) Radius ( m) x rms/y rms ( m) 2.2 26.4 6.49 1.3 8.3/6.22 3.3 18.69 9.61 12.2 9.68/7.47 4.4 14.2 12.32 14. 11.31/8.72 6.6 9.35 19.26 16.9 13.9/1.7 8.8 7.1 25.69 19.2 16./12.3 11.1 5.56 32.41 21. 18./13.8 Table 1 FODO lattice parameters. In order to study the effect of increasingly strong focusing, we consider six different FODO lattices as shown in Table 1. In each case the quadrupole length is assumed to be.74 m. The leftmost column in the table represents the length of the FODO cell while the second column is the field gradient. The third column is a measure of the average-function, while the two rightmost columns describe the rms beam radius and the initial beam sizes in the x- and y-directions. The Twiss- parameters are not shown but are x 1.3 and y.77. A summary plot showing the optimized output power for a linear taper profile after m for the six different FODO lattices and the weak-focusing undulator is shown in Fig. 2 for an assumed seed power of 5 MW. As expected, strongfocusing enhances the interaction with respect to the weakfocusing undulator. However, there appear to be two regimes associated with strong-focusing. At the longer FODO cell lengths ( = 6.6 m, 8.8 m, and 11.1 m), MINERVA predicts output powers within the range of approximately.7.8 TW. However, there is a more dramatic increase in the output powers as the FODO cell length decreases below 4.4 m. In this regime, we observe an approximately linear increase in the output power with decreasing -functions, and where the maximum output power exceeds 1.7 TW for the cases under consideration. 1.8 1.6 1.4 1.2 1..8.6.4.2 5 1 15 2 25 3 35 4 ( x 2 + y 2 ) 1/2 Weak Focusing Fig. 2 Optimized output power after m for the six FODO lattices and the weak-focusing case. The current density increases as the FODO cell length decreases so that the Pierce parameter (Fig. 3) increases with decreasing -functions, and this leads to increasing interaction strengths. The reason for the dramatic increase in the output power when the FODO cell length decreases below 4.4 m is twofold. In the first place, the stronger interaction strength yields optical guiding even in the tapered regime. In the second place, the smaller beam size results in a more coherent interaction with the optical field and this results in a higher trapping fraction. 1.3 1-3 1.2 1-3 1.1 1-3 1. 1-3 9. 1-4 8. 1-4 7. 1-4 Weak Focusing 6. 1-4 5 1 15 2 25 3 35 4 ( 2 + 2 ) 1/2 x y Fig. 3 The variation in the Pierce parameter with the - function. 3

.6.5.4.3.2.1 (a) = 4.4 m 8 6 4 2.7.6.5.4.3.2.1 (b) = 11.1 m 3 25 2 15 5. 2 4 6 8 1.2 1. (c) L = 3.3 m FODO 8.8 6.6 4.4.2 2. 2 4 6 8 2. (e) 1.5 L FODO 8 6 1. 4.5 2. 2 4 6 8. 2 4 6 8.8 35.7 (d) L = 8.8 m FODO 3.6 25.5 2.4.3 15.2.1 5. 2 4 6 8.8 35.7 (f) L = 6.6 m FODO 3.6 25.5 2.4.3 15.2.1 5. 2 4 6 8 Fig. 4 Plots of the power (left axis, blue) and spot size (right axis, red) versus distance along the undulator for different levels of focusing. The green line represents the rms electron beam radius. Observe that the optical field is guided when the length of the FODO cell is less than about 4.4 m. The evolution of the output power (left axis, blue) and the spot size of the optical field (right axis, red) versus position in the undulator is shown in Fig. 4 for each of the six choices of the FODO lattice. The green line in the figures represents the rms electron beam radius. In all of these cases, saturation in the uniform undulator occurs after between 1 15 m, and the optimal start-taper points range from about 9 12 m. The figures on the right (Figs. 4b, 4d, and 4f) correspond to the longest FODO cells with the lower power levels (.7.8 TW). It is evident from these figures that substantial diffraction occurs after the start of the taper where the optical field expands from about 1 2 m at the start-taper point to between 25 35 m after m. This is in stark contrast with what is found when the FODO cell is 4.4 m or shorter in length (shown in Figs. 4a, 4c, and 4e). As shown in the figure, the optical mode experiences substantial guiding in the tapered regime and the maximum expansion found after m of undulator is to a mode radius of between 4 6 m. This optical guiding in the case of extreme transverse compression of the electron beam is an important factor in reaching TW power levels. Comparable behavior in the case of a weak-focusing helical undulator is shown in Fig. 5 where we plot the power (left axis, blue) and spot size (right axis, red) versus distance along the undulator, and where the green line represents the rms electron beam radius. Observe that there is very little optical guiding if the field in the tapered region after about 15 m, but that the spot size of the optical mode 4

Trapping Fraction (%) P(E) Trapping Fraction (%) remains below about m due to the relatively large transverse size of the electron beam..3.25.2.15.1.5 Weak Focusing. 2 4 6 8 Fig. 5 Plot of the evolution of the optimized power and the spot size versus distance for the case of a weakfocusing helical undulator. Optical guiding occurs when the characteristic growth length of the optical field is shorter than the Rayleigh range; hence, the central power grows faster than diffraction. This is usually the case in the exponential growth regime, but, it is not typically found in a tapered undulator where the power grows more slowly than exponential. However, the extreme focusing that we find when 4.4 m leads to sufficiently rapid growth that optical guiding can occur. This is seen by noting that the power grows in the linear tapered region as P(z) = P [1 + (z z )/L G], where P is the power at the start of the tapered region, z is the start-taper point, and L G is the characteristic growth length. The Rayleigh range is about 15 m at the start of the tapered region. When = 4.4 m, the characteristic growth length is L G 13 m which is shorter than the Rayleigh range indicating that optical guiding is possible. The characteristic growth length decreases as decreases further. In contrast, L G is approximately 21 m or greater when 6.6 m and optical guiding cannot occur. 4 35 3 25 2 15 1 5 5 1 15 2 25 3 35 4 ( 2 + 2 ) 1/2 x y 8 6 4 2 Weak Focusing Fig. 6 Variation in the trapping fraction after m with the -function. The second major factor in achieving such extreme enhancements of the efficiency is how strong-focusing affects the trapping fraction. In this regard, we find that the trapping fraction is highest at about 38% of the beam for the shortest FODO cell length ( ) and decreases as the FODO cell length increases. This is shown in Fig. 6 where we plot variation in the trapping fraction after m of undulator versus the -function. This is related to the optical guiding where the mode size remains closest to that of the transverse extent of the electron beam and this enhances the coupling of the field to the electrons. Observe that the trapping fraction displays a local maximum for the weak-focusing helical undulator. An example of the spent beam distribution for the highest power case ( ) is shown in Fig. 7..12.1.8.6.4.2 z = m Trapping Fraction = 38%. 12 122 124 126 128 13 132 134 E (MeV) Fig. 7 The spent beam distribution for LFODO. In view of the fluctuations associated with SASE, the power that the monochromator delivers to the MOPA stage of the undulator can vary by large amounts from shot-toshot. As a result, it is important to understand how variations in the seed power affect the output from the tapered undulator. In order to study this issue, we have optimized the taper assuming a range of seed powers from.25 MW 25 MW. This represents a variation of two orders of magnitude in the seed power. The variation in the optimized output power and trapping fraction as the seed power is varied over this range is shown in Fig. 8 for the shortest FODO cell length. It is clear that the output power varies relatively little over this range of seed powers. 2.5 2. 1.5 1..5..1 1 1 P (MW) seed Fig. 8 The variation in the optimized output power (left axis, blue) and in the trapping fraction (right axis, red) as a function of the seed power for LFODO. 6 5 4 3 2 1 5

Power (W) Power (W) In general, the optimal start-taper point increases with decreasing seed power, but the optimal start-taper point varies relatively slowly between 12.2 m 8.7 m as the seed power increasers over this range. The optimal slope of the taper varies by about 2% over this range. In each case, however, the peaks in performance with respect to these parameters is relatively broad. As a result, we expect that the output power will exhibit relatively small variations due to the SASE-induced fluctuations in the seed power produced in the monochromator. We now consider the case of a segmented helical undulator with the strong-focusing FODO lattice with a cell length of 2.2 m. In order to configure this undulator/fodo lattice properly, the undulators are.96 m in length with 46 periods (for a 2. cm period) over with one period each in the entry and exit transition, and the gaps between the undulators are.16 m in length. Because MINERVA does not automatically select the optimum phase shift between the undulator sections, we have adjusted the wiggler field strength slightly to 16.135 kg in order to optimize the phase shift between undulator segments in the uniform undulator section for a resonance at 1.5 Å. It should be remarked that since the undulator field strength decreases in the tapered section, the factors controlling the optimal phase shift will also vary. The optimal phase shift can be selected by inserting phase shifters between the undulators or by varying the gap lengths. However, the gaps are quite short given the short length of the FODO cell and this would make it difficult to insert phase shifters in the configuration under consideration here. In addition, changing the gap lengths could, in principle, affect the lengths of the FODO cells. Of course, it may be that the interaction can be further optimized by changing the focus of the electron beam along the undulator which implies varying the parameters of the FODO lattice. As a result, it is clear that a more complete optimization of the segmented undulator configuration is a complicated procedure which is beyond the scope of the present study. As such, therefore, we shall restrict the present analysis to an optimization over the start-taper segment and the (linear) taper profile, but it should be recognized that this is not a complete optimization. The evolution of the power for an optimized taper profile and the corresponding spot size are shown in Fig. 9. The start-taper point is found to be the 16 th undulator for a seed power of 5 MW, and the optimal (linear) down taper is 6%. It is expected that the effect of the segmented undulator is to degrade the interaction because the optical field is not guided in the gaps and the phase shift may not be optimized either, and this is found to be the case where the output power reaches approximately.73 TW which is reduced relative to the 1.7 TW found for the single, long undulator. Nevertheless, this still represents an enhancement by a factor of more 15 over the saturated power in a uniform undulator. 8 1 11 7 1 11 6 1 11 5 1 11 4 1 11 3 1 11 2 1 11 1 1 11 1 st Tapered Wiggler = 16, =.6 2 4 6 8 5 4 3 2 Fig. 9 Evolution of the power and spot size for an optimized tapered undulator line. B. The Case of a Planar Undulator We now consider a flat-pole-face planar undulator with a 3. cm period and an on-axis field magnitude of 12.49 kg. These parameters correspond to the period and magnitude of the undulator in the LCLS; however, we consider the case of a single, long undulator for the present study. The electron beam has an energy of 13.64 GeV, a peak current of 3 A, a normalized emittance of.4 mm-mrad, and an rms energy spread of.1%. This configuration is resonant at a wavelength of 1.5 Å. The performance of the selfseeded MOPA configuration is studied using two FODO lattices. The FODO lattice used in the LCLS had a FODO cell length of approximately 7.3 m and a field gradient of 4.5 kg/cm. Note that the quadrupole length that we have been using (.74 m) also corresponds to the quadrupoles used in the LCLS. We compare the performance of the selfseeded MOPA based upon this FODO lattice with the extreme-focusing lattice shown in Table 1 with a FODO cell length of 2.2 cm. A seed power of 5 MW is used to determine the performance of the self-seeded MOPA for both FODO lattices. 1.4 1 11 1.2 1 11 1 1 11 8 1 1 6 1 1 4 1 1 2 1 1 Linear Taper z = 2. m =.19 = 7.3 m B Q = 4.5 kg/cm 2 4 6 8 2 15 Fig. 1 The evolution of the power (left axis, blue) and spot size (right axis, red) for the optimized taper using the lattice with the 7.3 m long FODO cell. The green line represents the rms electron beam radius. The Twiss parameters used to match the electron beam into the 7.3 m long FODO cell correspond to initial rms sizes of 22 m in the x-direction and 19 m in the y- 5 6

Power (W) Average Average Power (W) Average direction with Twiss- parameters of x = 1.1 and y =.82. Saturation for this lattice is found after about 25 m at a power level of close to 1 GW. The optimized linear down-taper is found to correspond to a start-taper point of 2 m with a total down-taper of 1.9% over the additional 8 m of tapered undulator. The evolution of the power and spot size for this optimized taper is shown in Fig. 1. The output power reaches.12 TW after m of undulator, which represents an enhancement over the saturation power for the uniform undulator by a factor of 12. Electron beam propagation is determined largely by the FODO lattice rather than the undulator, and the initial Twiss parameters specified in Table 1 for the 2.2 m FODO cell length are applicable for this planar undulator as well. Saturation in the uniform undulator for this FODO lattice is found after about 23.5 m at a power level of 15 GW. The optimized linear taper profile is characterized by a starttaper point of 2.2 m and a total down-taper of 4.8% over the length of the taper region. The evolution of the power and spot size for the optimized taper profile is shown in Fig. 11, where the output power reaches.73 TW. This corresponds to an enhancement by a factor of about 5 over the saturated power in the uniform undulator, and a factor of six greater than the output power in the FODO lattice with the 7.3 m cell length. 8 1 11 7 1 11 6 1 11 5 1 11 4 1 11 3 1 11 2 1 11 1 1 11 Linear Taper z =.48 B Q = 26.4 kg/cm 2 4 6 8 Fig. 11 The evolution of the power (left axis, blue) and spot size (right axis, red) for the optimized taper using the lattice with the 2.2 m long FODO cell. The green line represents the rms electron beam radius. It is clear, therefore, that and extreme-focusing FODO lattice will also bring the performance using a planar undulator to near-tw power levels in m long undulators. C. The Case of Pure SASE The utility of extreme focusing by means of a FODO lattice to achieve near-tw or TW output powers for the self-seeded MOPA configuration suggests that such extreme focusing will also improve the performance of pure SASE FELs. MINERVA [12] is capable of simulating the start-up from shot noise on the electron beam within the context of steady-state simulations, and we now consider SASE using the long, single, helical undulator described previously in conjunction with the extreme FODO lattice with a cell length of 2.2 m. As in the previous discussion, 2 15 5 7 the resonance at 1.5 Å is found using the electron beam energy of 13.22 GeV, a peak current of 4 A, an rms energy spread of.1% and normalized emittances of.3 mm-mrad in both the x- and y-directions. The transverse profile of the electron beam is characterized by a Gaussian distribution, and the beam is matched into the FODO lattice as described in Table 1. The evolution of the average power along a uniform undulator/fodo lattice is shown in Fig. 12. The average is over an ensemble of 3 different initial random phase distributions. As shown in the figure, saturation is found after about 16 m at a power of about 3 GW. 1 1 1 8 1 6 1 4 1 2 5 1 15 2 25 Fig. 12 Evolution of the average power along the undulator for SASE simulations using 3 initial random phase distributions. A (linearly) tapered undulator has been optimized based upon the saturation of the average power shown in Fig. 12 for the extreme-focusing FODO lattice, and the result is compared with that found for the self-seeded MOPA configuration (see Fig. 4e). The self-seeded MOPA simulation is based upon a seed power of 5 MW. As indicated in Fig. 12, the average SASE performance indicates that the 5 MW power level is reached after about 7 m of undulator. As a result, it is appropriate to consider a 17 m undulator/fodo lattice for purposes of the comparison with the self-seeded MOPA performance. 1.6 1.4 1.2 1..8.6.4.2. 2 4 6 8 Fig. 13 The evolution of the average power (left axis, blue) and spot size (right axis, red) for the optimized taper using the lattice with the 2.2 m long FODO cell. The green line represents the rms electron beam radius. 6 5 4 3 2 1

The optimal performance for the tapered undulator is found for a start-taper point of 14. m and a total downtaper of about 9.1% over the length of the tapered region. The optimization was performed over an ensemble of 3 different noise realizations for the initial phase distribution, and the evolution of the average power and spot size is shown in Fig. 13. The average output power at the end of the undulator is 1.5 TW with a standard deviation over the 3 different noise seeds of about 3.5%. It is expected that the tapered undulator interaction will be less effective for SASE than for a seeded MOPA [15]; however, this represents only a small penalty in comparison with the 1.7 TW found for the self-seeded MOPA. We also point out that, as in the case of the self-seeded MOPA, that there is substantial optical guiding is also found here. We conclude that extreme focusing in an undulator/fodo lattice can be used to reach TW power levels in pure SASE x-ray FELs. IV. SUMMARY AND DISCUSSION The physics underlying efficiency enhancement in tapered undulators has been understood for decades [7,8]; however, tapered undulator experiments have historically shown enhancements in the efficiency over the saturated power in a uniform undulator of less than five [9-11]. In this paper, we have examined the effect of extreme focusing on the performance of tapered-undulator x-ray FELs using the MINERVA simulation code and found that enhancements over the saturated power in a uniform undulator by a factor of 5 are possible. We considered resonant interactions at 1.5 Å with electron beams with energies and currents in the excess of 13 GeV and 3 A respectively. The emittances we assumed were.3.4 mm-mrad with an rms energy spread of.1%. These parameters are consistent with what is achieved in the current range of x-ray FELs. Our simulations indicate that the most important factor in achieving near-tw or TW power levels is the extreme focusing in a strong FODO lattice in conjunction with a long tapered undulator. This level of performance was found using either helical or planar undulators where the transverse focusing of the electron beam reached current densities in excess of 2 GA/cm 2. These extreme current densities are associated with large values for the Pierce parameter which give rise to extremely strong interactions that result in substantial optical guiding even in a tapered undulator configuration. It is important to remark that the advantages accruing from such extreme focusing are found for both self-seeded MOPAs and pure SASE FELs. Finally, it is important to extend the steady-state simulations discussed herein to time-dependent simulations. It is expected that slippage effects between the optical field and the electron bunch will result in some degradation in performance, and this should be quantified to determine limitations on the extreme-focusing model. Such simulations are in progress and will be reported in future publications. REFERENCES 1. P. Emma et al., Nat. Photon. 4, 641 (21). 2. H. Tanaka et al., Nat. Photon. 6, 529 (212). 3. http://www.xfel.eu/news/217/european xfel generates its first laser light/. 4. http://pal.postech.ac.kr/paleng/. 5. C. Emma, K. Fang, J. Wu, and C. Pellegrini, Phys. Rev. Accel. Beams, 19, 275 (216). 6. S. Reiche, Nucl. Instrum. Methods Phys. Res. A429, 243 (1999). 7. P. Sprangle, C.-M. Tang, and W.M. Manheimer, Phys. Rev. Lett. 43, 1932 (1979). 8. N.M. Kroll, P.L. Morton, and M. Rosenbluth, IEEE J. Quantum Electron. 17, 1436 (1981). 9. T.J. Orzechowski et al., Phys. Rev. Lett. 57, 2172 (1986). 1. X.J. Wang, H.P. Freund, W.H. Miner, Jr., J.B. Murphy, H. Qian, Y. Shen, and X. Yang, Phys. Rev. Lett. 13, 15481 (29). 11. D. Ratner et al., SLAC-PUB-14194 (21). 12. H.P. Freund, P.J.M. van der Slot, D.L.A.G. Grimminck, I.D. Setya, and P. Falgari, New J. Phys. 19, 232 (217). 13. J. Karssenberg et al., J. Appl. Phys., 9316 (26). 14. http://lpno.tnw.utwente.nl/opc.html. 15. H.P. Freund and W.H. Miner, Jr., J. Appl. Phys. 15, 11316 (29). 8