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Abstract
In externally seeded free-electron lasers (FELs) that rely on a frequency upconversion scheme to generate intense short-wavelength light pulses, the slippage effect in the radiator imposes a lower limit on the FEL pulse duration, which is typically on the order of a few tens of femtoseconds. Recently it was proposed that a combination of a chirped microbunch and a tapered undulator can be used to break this limit. Although the method has the potential to reduce the FEL pulse duration down to a level that cannot be achieved by current state-of-the-art technology, it requires a very short seed pulse ($\sim$ one optical cycle or less), making it challenging to put this concept into practical use. Here, we propose an alternative technique to relax the requirement on the seed pulse length. We show that the modified scheme allows generation of FEL pulses with durations much shorter than that determined by the seed pulse and the slippage effect. The performance of the method, which can easily be implemented at existing seeded FEL user facilities, is evaluated through a campaign of analytical calculations and simulations. For our set of typical seeded FEL parameters, we expect the generation of 1.6 fs long pulses at 26 nm with a peak power of 10 GW using a 20 fs long chirped seed pulse operating at 260 nm.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Shortening the laser pulse duration is one of the most important research areas in photon science. Even though a great deal of effort has been made to generate ultrashort pulses for probing fast dynamics, the quest is still on to further reduce the duration of a laser pulse. This also holds for free-electron lasers (FELs), i.e., light sources based on coherent emission of radiation from a high-energy electron beam. There is no theoretical lower limit on the lasing wavelength of an FEL, and a number of FEL facilities working in soft and hard x-ray regions are already fully operational [1–4].
Because the achievable pulse length scales linearly as the wavelength, an FEL has the potential to reach a much shorter pulse duration compared to an optical laser. To date, various schemes to shorten the FEL pulses have been proposed [5–11] and demonstrated [12–16]. It should be noted, however, that no schemes listed above can generate isolated FEL pulses whose pulse length is reduced down to the theoretical lower limit, i.e., monocycle duration. Apart from a number of technical limitations and difficulties, this is attributable to the so-called slippage effect; the radiation emitted by the electron beam overtakes the electrons as they travel under the influence of the sinusoidal magnetic field generated by an undulator. As a result, the coherent undulator radiation (CUR) emitted by the electron beam, which is the origin of the FEL pulse, is stretched as the number of undulator periods increases.
To overcome the above problem and generate monocycle FEL pulses, a new FEL concept has been proposed [17], in which a prebunched electron beam with a current profile given by a chirped sinusoid (chirped microbunch, CM) is injected into a tapered undulator; if the chirp rate of the CM and the gradient of the undulator taper satisfy a particular condition, the slippage-driven pulse stretch of CUR can be avoided. Let us call this concept “SC$^3$” throughout this paper, which stands for slippage controlled coherent radiation by chirped microbunching.
The SC$^3$ concept explained above can be applied to high-gain harmonic generation (HGHG) FELs [18], in which the high-harmonic radiation is generated by frequency upconversion of a seed pulse. Because the slippage-driven pulse stretch can be avoided, the high-harmonic radiation inherits the short-pulse property of the seed pulse. As a result, it is in principle possible to generate monocycle FEL pulses at much shorter wavelengths than what is currently available. The scheme is called “monocycle harmonic generation (MCHG),” and is attractive for future development of FEL facilities.
Two variations of the scheme have been proposed to realize the MCHG FELs [17,19]. In terms of the accelerator layout and requirement on the electron beam, the second scheme [19] is more feasible than the first one [17]. Although the second scheme was named differently (“subcycle harmonic generation”) to distinguish it from the first, both of them are collectively referred to as MCHG FELs in this paper.
The key point towards the realization of MCHG FELs is how to prepare the ultrashort seed pulse. Even with state-of-the-art laser technology, it is still challenging, albeit not impossible, to generate monocycle pulses, especially for application in seeded FELs in which the seed pulse needs to be highly stable to guarantee synchronization with the electron beam. It is thus preferable to relax the requirement on the seed pulse length. As explained in detail later, the performance attainable with the MCHG FEL is quite sensitive to the seed pulse duration, or more precisely, the seed bandwidth, and its effectiveness rapidly decreases for narrower bandwidths. This makes the realization of MCHG FELs technically challenging, and also makes the SC$^3$ concept less attractive for practical applications. The purpose of this paper is to propose a scheme to overcome the above difficulty, in which the SC$^3$ concept works even with seed pulse durations longer than a few cycles.
2. Principle of operation
2.1 Requirements on the seed pulse in a MCHG FEL
Let us first recall how the MCHG FEL works. As illustrated in Fig. 1(a), it is composed of two tapered undulators having the same specifications, which are referred to as the modulator and the radiator, and a magnetic chicane inserted between them. The electron beam is modulated in energy by interaction with the monocycle seed pulse while traveling along the modulator. After passing through the magnetic chicane with an optimized longitudinal dispersion, the energy modulation is converted to a density modulation whose profile is given by a chirped sinusoid; a CM reflecting the tapered field profile of the modulator is formed in the electron beam. If the energy modulation is strong enough, the CM contains high-frequency components as illustrated in the inset of Fig. 1(a), and the current peak interval is not constant but varies according to the taper gradient. To be specific, the $n$-th interval is equal to the slippage length corresponding to the $n$-th period of the modulator. Here, the slippage length, which is also referred to as the fundamental wavelength, is defined as the distance by which the emitted radiation overtakes the electron beam in one undulator period. The electron beam with the above CM profile emits an intense monocycle pulse when it passes through a radiator having the same specifications as the modulator. The mechanism of monocycle pulse formation was explained in detail in [17].
Fig. 1. Schematic layout of the (a) MCHG FEL and (b) proposed FEL scheme to take advantage of the SC$^3$ concept.
To investigate the impact of the seed pulse properties on the performance of the MCHG FEL, we discuss the temporal profile of the CM and the waveform of radiation emitted by the CM traveling along the radiator, using the relative coordinate $s=z-\bar {v}_z t$; here, $z$ denotes the longitudinal coordinate along the modulator and radiator axes, $t$ denotes time, and $\bar {v}_z$ denotes the average velocity of the electron beam along the $z$ axis. To facilitate the following discussion, it is convenient to introduce a normalized coordinate $\tau =s/\lambda _0$, where $\lambda _0$ is the central wavelength of the seed pulse and equals the average slippage length in the modulator, which is also the same as the fundamental wavelength of the radiator.
We assume that the temporal profile of the CM, $n(\tau )$, is given by a simple form,
with $N$ being the number of modulator periods. The above formula means that the CM is composed of $(N+1)$ local microbunches characterized by a function $f(\tau )$, and located at $\tau =h(k)$, which corresponds to the longitudinal position of the seed pulse when the electron beam passes through the $k$-th period of the modulator. Recalling that the modulator is linearly tapered, we have with $\alpha$ being a parameter defined by the taper gradient. To be specific, $\alpha$ denotes the total variation of the slippage length within the modulator, which is evident from $\alpha =h'(N/2)-h'(-N/2)$, with primes denoting differentiation with respect to $\tau$. It also defines the full bandwidth of the frequency spectrum of the CM, because the interval between local microbunches uniformly varies from $\lambda _0(1-\alpha /2)$ to $\lambda _0(1+\alpha /2)$.Note that the above discussion is valid only if the seed pulse is sufficiently short. To be more specific, the bandwidth of the seed pulse should be of the order of $\alpha$. If this is not satisfied, the slippage length at the $k$-th period given by $\lambda _0(1+\alpha k/N)$, goes beyond the bandwidth of the seed pulse in the upstream ($k<0$) and downstream ($k>0$) sides of the modulator, where the interaction between the electron beam and the seed pulse is lost and the local microbunches cannot be created. As a result, the bandwidth of the CM becomes narrower than $\alpha$.
The electric field of radiation emitted by the CM, whose temporal profile is represented by $n(\tau )$, is given by
where $\otimes$ is the convolution operation, and $E_1(\tau )$ is the electric field of radiation emitted by a single electron traveling along the radiator. Recalling that the radiator has the same specifications as the modulator, it is easy to understand that $E_1(\tau )$ has a simple form similar to $n(\tau )$, namely, where $g(\tau )$ characterizes the local radiation pulse generated in a single radiator period. Equation (4) means that the local pulse emitted at the $k$-th period is located at $\tau =-h(k)$, which is easily understood by recalling the slippage effect; the local pulse moves forward with respect to the electron, while it travels along the radiator.The radiation waveform emitted by the CM traveling along the radiator can be analytically investigated using Eqs. (1-4). As an example, let us consider an extreme case when the local microbunch and radiation pulse are infinitely short, and thus both $f(\tau )$ and $g(\tau )$ are described by a delta function. Omitting the unit-conversion coefficient for simplicity, we have $f(\tau )=g(\tau )=\delta (\tau )$. Then it is easy to show that
and thus we have where $\tilde {n}(\nu )$ is the temporal Fourier transform of $n(\tau )$ defined as and $\mathscr {F}^{-1}$ is the inverse Fourier transform with respect to the normalized frequency $\nu$. Note that the unit-conversion coefficient has been omitted again in the above formulation. The waveform $E(\tau )$, which is given by the inverse Fourier transform of a real function $|\tilde {n}(\nu )|^2$, has a typical temporal duration of $\alpha ^{-1}$, where $\alpha$ denotes the bandwidth of $|\tilde {n}(\nu )|^2$ as explained before. It is now obvious that the pulse length of radiation emitted by the CM can be controlled by tuning the parameter $\alpha$, and can ultimately go below the monocycle duration, if $\alpha$ can be large enough.To quantitatively investigate the impact of the parameter $\alpha$ on the performance of the MCHG FEL, let us calculate the radiation waveform $E(\tau )$ under a more realistic approximation,
Fig. 2. Current profiles $n(\tau )$ for different values of (a) $\alpha =1$, (b) 0.3, (c) 0.1 and (d) 0.03, with $N=$20.
If the taper gradient of the modulator is large, for example $\alpha =1$, the microbunch interval greatly varies along $\tau$ as found in Fig. 2(a). As a result, we can expect that an ultrashort pulse, which is completely isolated, is generated as found in Fig. 3(a). Although it is difficult to define the central wavelength, we may say that a monocycle pulse, or even a subcycle pulse, can be generated in this condition, which is obviously an advantage of the MCHG FEL. For smaller values of $\alpha$, however, this advantage is lost, and the waveform transforms from an isolated subcycle pulse to a train of subcycle pulses with alternating polarities. Although such a pulse structure may be employed in some special uses, this is not the case for most practical applications.
Fig. 3. Radiation waveforms $E(\tau )$ calculated for the current profiles shown in Figs. 2(a)–2(d).
Now let us discuss the requirement on the seed pulse length to achieve a given value of $\alpha$. As explained before, the bandwidth of the seed pulse should be broader than, or at least of the order of $\alpha$. In general, the root mean square (RMS) bandwidth of the pulse with the RMS transform-limited pulse length of $\sigma _{\tau }$ is given by $\sigma _{\lambda }/\lambda _0=(4\pi \sigma _{\tau })^{-1}$. Recalling that $\alpha$ denotes the full bandwidth of the frequency spectrum of the CM, its RMS bandwidth is given by $\alpha /2\sqrt {3}$. Now we find a condition,
or in the FWHM pulse length, This means that nearly monocycle or even shorter seed pulses ($\Delta \tau =0.65$) are required to satisfy the condition $\alpha =1$, corresponding to Figs. 2(a) and 3(a).2.2 Using a harmonic radiator to relax the condition on the seed pulse duration
Although generating an isolated subcycle pulse is an attractive goal of FELs based on the SC$^3$ concept, preparing the monocycle seed pulse is technically challenging. Instead of such an ultimate goal, it is preferable to have an alternative scheme to take advantage of the SC$^3$ concept to shorten the pulse length in a seeded FEL.
As explained in the previous section, a train of subcycle pulses is generated in the MCHG FEL, when the seed pulse is not sufficiently short, and thus $\alpha$ is not large enough. This comes from the fact that many high-harmonic components of radiation are synthesized in phase, which makes it difficult to define the central wavelength of radiation. Although such a pulse structure may potentially be useful for some special purposes, what is needed in most practical applications is an isolated short pulse, whose central wavelength is well defined.
The above discussion suggests an alternative layout as indicated in Fig. 1(b); instead of the tapered undulator with the same specifications as the modulator (hereafter referred to as a fundamental radiator) supposed in the MCHG FEL, a tapered undulator with an average slippage length of $\lambda _0/m$ is used as a radiator (harmonic radiator), where $m$ is an integer specifying the target harmonic order. In addition, a chirped pulse is used as a seed pulse together with a few-period modulator to form the CM in the electron beam, instead of the Fourier-transform limited pulse combined with an $N$-period modulator supposed in the original scheme. As a result, the pulse compression process of the seed pulse, which may bring about technical difficulties, especially in the case of monocycle pulses, can be eliminated. The layout after the above modifications is now quite similar to the HGHG scheme; we have just two differences: a chirped seed pulse and a tapered radiator. Even so, the output pulse is expected to have a much shorter pulse duration than the conventional HGHG FEL, with the central wavelength of $\lambda _0/m$.
Note that the parameter $N$ in this case should be defined as the number of the local microbunches instead of the number of periods of the modulator. To be specific, $N$ is roughly given by the pulse length of the chirped seed pulse divided by $\lambda _0$. In addition, the number of periods of the harmonic radiator should be $mN$ instead of $N$. This condition means that the total slippage length in the harmonic radiator should be equal to the length of the CM.
Although it is likely that the SC$^3$ concept works with the harmonic radiator as well, we need to prove that the condition (6) or equivalent is satisfied, and derive the optimum taper gradient of the harmonic radiator. For this purpose, the temporal profile of the CM is modified as follows (see Appendix)
Let us now consider a waveform of radiation given by
The waveform of radiation emitted by the CM traveling along the harmonic radiator, or the tapered undulator introduced above, is given as
If the bandwidth of $n_m$ is not too broad, harmonic components other than $m$ disappear. Then, neglecting the unit-conversion coefficient as before, we have which is equivalent to the condition (6) and thus the output pulse is expected to have a short duration, with the central wavelength of $\lambda _0/m$.As an example, Figs. 4(a)–4(d) plot the waveforms $E(\tau )$ with the target harmonic order of $m=10$, for the current profiles shown in Figs. 2(a)–2(d), respectively. The advantage of using the harmonic radiator is obvious in Figs. 4(c) and 4(d). In contrast to the train of subcycle pulses as found in Figs. 3(c) and 3(d), a short pulse that is almost isolated is generated, with the central wavelength of $\lambda _0/10$ as shown in the inset of Fig. 4(c). It should be noted, however, that the harmonic radiator is not advantageous for larger values of $\alpha$, as found in Figs. 4(a) and 4(b).
To investigate the reason, let us focus on the spectrum of $\tilde {n}_m$, i.e, the harmonic components of the CM. Figures 5(a) and 5(b) plot the amplitude and phase of $\tilde {n}_m$ as a function of $\nu$ for three harmonic orders of $m=9$, 10 and 11, with (a) $\alpha =0.1$ and (b) $\alpha =0.3$. In each case, the spectrum is characterized by a quadratic phase modulation reflecting the temporal profile of the CM. When $\alpha$ is not too large (a), the spectra of the neighboring harmonics do not overlap, i.e., the bandwidths are narrower than the intervals between the central frequencies of individual harmonics. Because the condition (15) is satisfied, the harmonic radiator works to generate an isolated pulse. This is not the case for $\alpha =0.3$ (b), where we find that the spectral profile of the 10th harmonic largely overlaps with the 9th and 11th harmonics, which have different phase modulations. As a result, two satellite pulses coming from these different-order harmonics are generated in addition to the main one. It is easy to understand that the condition $m\alpha \leq 1$ should be satisfied to avoid the generation of multiple pulses.
Fig. 5. Amplitude (solid lines) and phase (dashed lines) of $\tilde {n}_m(\nu )$ calculated with (a) $\alpha =0.1$ and (b) $\alpha =0.3$ for three different harmonic orders of $m=9$ (black), 10 (red) and 11 (blue).
Having verified that the harmonic radiator works in the SC$^3$ concept, let us discuss the optimum taper gradient of the harmonic radiator to give the normalized chirp rate of $2\pi m\alpha /N$. Recalling the definition of $\tau$, we have the condition
with $\omega (t)$ denoting the instantaneous frequency of radiation. Let $\beta$ be the taper gradient of the harmonic radiator, i.e., the deflection parameter $\kappa$ changes as with $K$ being the deflection parameter at the center of the harmonic radiator $(z=0)$. While the electron beam travels along the harmonic radiator, the fundamental frequency of undulator radiation changes as where $\gamma$ denotes the Lorentz factor of the electron beam and $\Lambda$ denotes the magnetic period. The chirp rate of this radiation is then given by Substituting Eqs. (18) and (19) into (16), and using the well-known relation for undulator radiation we have for the optimum taper of the harmonic radiator. Note that we have assumed that $\beta$ is much less than unity and replaced $\kappa (z)$ with $K$ where appropriate. In addition, we have applied the condition of the harmonic radiator, $2\pi c/\omega (0)=\lambda _0/m$.Now let us discuss the advantage of the proposed method against the MCHG FEL in terms of the spectral properties. For this purpose, we focus on the flux density $F(\nu )$ defined as
Let us consider the CM with the current profile $n(\tau )$ shown in Fig. 2(c) ($N=20$ and $\alpha =0.1$). Note that the radiation waveform $E_m(\tau )$ should be rigorously evaluated instead of Eq. (13), so that a quantitative comparison between the fundamental radiator (MCHG FEL) and the harmonic radiator (proposed method) is possible. Thus, we need to specify several parameters for each radiator. As an example, we assume $K=8.3$, $\Lambda =80$ mm and $\beta =0.021$ m$^{-1}$ for the fundamental radiator, while $K=4.3$, $\Lambda =30$ mm and $\beta =0.006$ m$^{-1}$ for the harmonic ($m=10$) radiator, respectively, where the taper gradient $\beta$ has been determined using Eq. (21). It is obvious that the number of periods of the fundamental radiator should be 20 ($=N$), while that of the harmonic radiator should be 200 ($=mN$).Figure 6 shows the comparison of $F(\nu )$ between the fundamental and harmonic radiators, where $E_m(\tau )$ is numerically evaluated using the above parameters with the synchrotron radiation code SPECTRA [20]. The spectrum obtained with the fundamental radiator (MCHG FEL) is composed of many odd-number harmonics, which reflects the spectral property of undulator radiation that even-number harmonics vanish in the forward direction [21]. Obviously, it is difficult to define the “central frequency” of radiation in this case. In contrast, the spectrum obtained with the harmonic radiator (proposed method) has a well-defined central frequency at $\nu =10$. What is more important is that the peak flux density in these high-harmonic regions is about 2 orders of magnitude higher.
Before closing this section, we explain the mechanism of the SC$^3$ concept using the harmonic radiator in the time domain, which helps the discussions in the next section. Let us consider the following “resonant condition”
which means that the microbunch interval (LHS) is equal to $m$ (=target harmonic order) times the “instantaneous wavelength” $2\pi c/\omega (z)$ at the position $z$ (RHS). It is obvious that the radiation is not amplified if this condition is not satisfied. Let $\tau _r$ be the coordinate at which the resonant condition is satisfied (hereafter referred to as the resonant coordinate). Under the assumption that the optimum taper gradient given by Eq. (21) is not too large, it follows from Eq. (23) that which means that the resonant coordinate moves forward along the electron beam, while it travels along the harmonic radiator. Recalling that the average slippage length of the harmonic radiator is $\lambda _0/m$, it is easy to show that the speed of this motion is the same as that of the radiation slippage. As a result, only a limited part of radiation, whose position coincides with the resonant coordinate in the electron beam, is amplified to form an isolated and intense pulse. This pulse, which is referred to as the resonant pulse, is originally generated in the tail of the electron beam, moves forward along the electron beam according to Eq. (24), and is continuously amplified by the local microbunches located at the resonant coordinate.3. Example
To demonstrate the potential of the proposed scheme, i.e., the FEL based on the SC$^3$ concept using a harmonic radiator, numerical simulations have been performed with the parameters summarized in Tables 1 and 2 , using the FEL simulation code SIMPLEX [22].
Table 1. Electron-beam and modulator parameters assumed in the numerical simulations.
Table 2. Seed-pulse and harmonic-radiator parameters assumed in the numerical simulations ($^{\dagger }$transform-limited pulse length, $^{\ddagger }$group delay dispersion, $^*$refer to the text for details).
For the electron beam, we use parameters that can typically be achieved at a seeded FEL user facility such as, e.g., the FERMI FEL [4], and the CM is supposed to be created using a two-period modulator. We consider three conditions for the seed pulse: (i) transform-limited (TL), (ii) positively chirped, and (iii) negatively chirped; note that all of them have the same pulse length (FWHM) of $\Delta t=$20 fs, and the chirped pulses have the TL pulse length of $\Delta t_{TL}=$ 5 fs with the group delay dispersion of $G=\pm$35 fs$^2$. The number of periods of the harmonic radiator (=200) has been determined so that the total slippage length is comparable to $c\Delta t$. Then, the parameter $N$, which roughly represents the number of local microbunches contributing to the formation of radiation, is given by $N\sim 200/10=20$. In addition, a helical undulator, instead of a conventional linear undulator, has been chosen as the harmonic radiator; this is to decrease the gain length and increase the peak power as much as possible.
Note that the condition (i) corresponds to the conventional HGHG FEL, and thus tapering the radiator is not necessarily required; in this example, however, the radiator is divided into two parts, and the 2nd half is tapered with the taper gradient optimized to maximize the pulse energy. The taper gradient for the other two conditions, (ii) and (iii), has been determined as follows.
Let us first recall the instantaneous frequency of the chirped seed pulse,
with $\omega _0=2\pi c/\lambda _0$ being the central frequency of the seed pulse, and $\sigma _{TL}=\Delta t_{TL}/2\sqrt {2 \ln 2}$ being the RMS pulse length of the TL pulse. The variation of the frequency (and thus the wavelength) of the chirped seed pulse relates directly to the positional variation of the local microbunches in the CM. Then it is easy to show that the parameter $\alpha$ is given byFigure 7 shows the current profiles of the electron beam at the entrance of the harmonic radiator, calculated for the conditions (i)-(iii) with optimized longitudinal dispersion of the chicane (1.8$\times 10^{-5}$m). For reference, the normalized coordinate $\tau$ is shown in the top axis with vertical grid lines. It is obvious that the local microbunches are regularly arranged with the interval of $\lambda _0=260$ nm in the condition (i), while the intervals linearly change along $s$ in the conditions (ii) and (iii), meaning that the CM is created as expected.
Fig. 7. Current profiles at the entrance of the harmonic radiator for the three different conditions (i)-(iii).
Figure 8(a) shows the temporal profiles of radiation for the conditions (i)-(iii), evaluated at the exit of the harmonic radiator. Note that the ordinate scale for the condition (ii) is different from that for the other two, because the peak power available with this condition is much lower than the other two, the reason for which is discussed later in detail. Besides this discrepancy, the pulse lengths in (ii) and (iii) are much shorter than that of the HGHG FEL (i) as expected. To be specific, the FWHM pulse lengths evaluated from the temporal profiles, as indicated by dashed lines, are (i) 22 fs, (ii) 2.1 fs and (iii) 3.0 fs.
Fig. 8. Temporal profiles of radiation (a) for the conditions (i)-(iii), and (b) for different collection angles in the case of the condition (iii). Note that the ordinate scale in (a) for the condition (ii) is different from the other two.
We stress here that the pulse duration in conditions (ii) and (iii) can be further shortened by collimation, or by intercepting the off-axis radiation. For example, Fig. 8(b) shows the variation of the temporal profile in the condition (iii), when the angular acceptance ($\Delta \theta _{x,y}$) is varied. Note that the horizontal and vertical angles are supposed to be the same. We find that the pulse length is reduced by decreasing the angular acceptance; although this is followed by the reduction of the peak power, the pulse length reaches the minimum value of 1.6 fs at $\Delta \theta _{x,y}=0.15$ mrad with the peak power of 10 GW, which is still comparable to that of the HGHG FEL (i).
The pulse shortening due to the reduced angular acceptance is related to the angular dispersion of undulator radiation, which is explained as follows.
It is well known that the fundamental frequency of undulator radiation propagating with a radial angle $\theta$ is given by
with which the resonant condition defined in Eq. (23) should be modified as follows The above equation means that the resonant coordinate $\tau _r$ is given as a function of $\theta$ as well as $z$. Assuming that $\alpha$ is much less than unity, we have with Because $\Delta \tau (\theta )$ is independent of $z$, the off-axis radiation propagating with an angle $\theta$ is continuously amplified by the local microbunches, which are located $|\Delta \tau |$ behind (or ahead of) those at $\tau =\tau _r(z,0)=z/m\Lambda$. As a result, the resonant pulse has a large angular divergence, with the angle of propagation ($\theta$) being correlated with the longitudinal position ($\tau$), whose relation is described by Eq. (30). In other words, the pulse length can be reduced by eliminating the off-axis radiation, as found in Fig. 8(b).Finally, let us discuss the discrepancy in the peak power available in the conditions (ii) and (iii), namely, with positively and negatively chirped seed pulses. To figure out the reason, we compare the variations of the CM while the electron beam travels along the harmonic radiator. As an example, let us focus on the local microbunch located near the center of the electron beam ($s=0$), whose current profiles evaluated at the entrance of the harmonic radiator ($z=-3$ m), and 2 m after the entrance ($z=-1$ m), are plotted in Figs. 9(a) and 9(b), respectively. Here, the origin of $z$ is defined as the center of the harmonic radiator.
Fig. 9. Current profiles (a,b) and macroparticle distributions (c,d) for the conditions (ii) and (iii); (a,c) are evaluated at the entrance, while (b,d) are at 2-m after the entrance of the harmonic radiator. Blue lines and dots correspond to the condition (ii), while red ones to (iii).
At the entrance of the harmonic radiator (a), the two current profiles are similar to each other. After the electron beam travels 2 m in the harmonic radiator (b), the profile in the condition (ii) significantly changes and becomes much broader than that in (iii). In other words, the microbunch in (ii) is spoiled through interaction with radiation generated in the harmonic radiator, while that in (iii) is rather enhanced. It should be noted that the resonant pulse is located at $s=\lambda _0z/m\Lambda =-0.87$ $\mu$m at $z=-1$ m, and has yet to interact with the local microbunch near $s=0$. Namely, the resonant pulse is not attributable to the variation of the current profiles explained above.
To understand the interaction between the CM and the radiation it generates, we plot the macroparticle distributions in the energy-time phase space in Figs. 9(c) and 9(d), which correspond to the conditions in Figs. 9(a) and 9(b), respectively. As found in (d), the phase of the energy modulation is completely different for the two conditions, (ii) and (iii). To be specific, the energy modulation in (ii) works to spoil the microbunch (debunching phase), while that in (iii) works to enhance the microbunch (bunching phase).
The above phase relation can be understood by focusing on the relative phase between a specific local microbunch, and radiation emitted by another microbunch located just behind. As an example, let us consider two local microbunches A and B, located at $\tau =h(0)$ and $\tau =h(-1)$. In addition, we consider a waveform of radiation, which is emitted by the microbunch B and later interacts with A. Figure 10 shows the schematic illustration of the phase relation between the radiation and microbunches for the conditions (i)-(iii), where we have assumed that $N$ is sufficiently large so that $h(-1)=-1$ regardless of the chirp rate $\alpha$. As a result, the microbunch B is located at the same coordinate $\tau =-1$ in all the conditions (i)-(iii).
Fig. 10. Illustration to explain the phase relation between the local microbunches (dots) and coherent radiation (solid lines) for the conditions (i)-(iii).
In the condition (i), the radiator is not tapered (at least not in the 1st half) and the wavelength of radiation is constant ($\lambda _0/m$). Thus, the interaction between the microbunch A and radiation emitted by B is done in phase. As a result, the microbunch A transfers its energy to radiation and amplifies it. In the condition (ii), where the harmonic radiator is tapered with a positive taper gradient, the wavelength of radiation is shorter than $\lambda _0/m$, at least before the resonant pulse arrives at A ($z<0$). Then the interaction is done out of phase, and thus the microbunch A does not exchange energy with the radiation on average. It should be noted, however, that an energy chirp will be given to the microbunch A in a debunching phase (let us recall that electrons lose energy by applying a positive radiation field). In the same manner, it is easy to show that an energy chirp with a bunching phase is applied to the microbunch A in the condition (iii).
Summarizing the above discussions, most of the local microbunches are more or less spoiled in the condition (ii) before the resonant pulse arrives. Thus, the amplification of the resonant pulse in (ii) is expected to be much lower than that in (iii), as found in the numerical example.
4. Summary
We have proposed a method to expand the applicability of the SC$^3$ concept for shortening the pulse length in seeded FELs. To overcome the stringent requirement on the seed pulse duration in the original MCHG FEL scheme, a harmonic radiator is used instead of the fundamental one. Although the output pulse length still depends on the seed pulse duration, an isolated pulse that is much shorter than the seed pulse/slippage can be generated in the proposed scheme. This is in contrast to the MCHG FEL, in which a train of subcycle pulses is generated if the seed pulse is not short enough. Analytical studies based on a simple model and detailed numerical simulations have been performed to reveal the effectiveness and capability of the proposed scheme, together with its limitations. It is obvious that this scheme helps to realize FELs based on the SC$^3$ concept.
Appendix: Expanding the temporal profile of the CM into harmonic components
To prove Eq. (11), we first modify the summation as follows
Funding
Japan Society for the Promotion of Science (KAKENHI Grant Number JP18H03691).
Disclosures
The authors declare no conflicts of interest.
References
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