1. Introduction

Creation of compact inexpensive sources of radiation operating efficiently in visible, UV, or soft X-ray domains is one of the most important directions in the development and investigation of Free-Electron Lasers (FEL). A short-wavelength radiation can be generated by a FEL using either a high-energy (multy-GeV) electron beam or a short-period undulator. One of the ideas often mentioned and discussed in the latter context is that of a FEL using Media with Periodically Modulated Refractive Index (MPMRI). MPMRI can be considered as a kind of a volume diffraction grating. The following two types of MPMRI have been proposed: (1) a gas-plasma medium with periodically varied density or degree of ionization [1]–[8] and (2) a spatially periodical solid-state superlattice-like (SLL) structure, which can be composed, e.g., of a series of layers of different materials with different refractive indices [9]–[19]. The experimental feasibility of a transition radiation FEL was shown recently [20]. In this report we restrict our consideration to the case of SLL MPMRI. Modulation is characterized by its depth 2α, period λ₀, and wave vector q(to be parallel to the z-axis), q = 2π/λ₀. For a given frequency ω, eigenmodes of MPMRI consist of a superposition of partial plane waves (PPW) with wave-vectors: k _n = k ₀ + n q, where n = 0, ±1,±2,… and k ₀(ω) is the wave-vector of the zero-order PPW. The phase velocities of PPW are equal to ${\upsilon }_n^{\left(\mathit{\text{ph}}\right)}$ = ω/k_n (ω). If an electron beam moves in such a medium with a velocity v _e, the best conditions for efficient transfer of energy from the beam electron to the n-th PPW are provided if the projection of v _e upon k _n, (v _e; k _n)/|k _n)|, is close to ${\upsilon }_n^{\left(\mathit{\text{ph}}\right)}$:

In this paper, the maximum gain achievable in MPMRI FEL, the corresponding saturation field and efficiency are found and analyzed. The case of a large modulation is considered. The conditions of a large modulation regime are found and are shown to differ from a “naive” requirement of 2α ∼ 〈ε〉, where 〈ε〉 is the mean permeability of the medium. Advantages of a large-modulation MPMRI FEL as compared to the earlier considered FELs based on weakly modulated media [2], [7]–[12], [14] are discussed. For any given light frequency ω the gain is maximized with respect to a choice of PPW to be on resonance and a direction of the electron motion. In this report, we discuss requirements on the quality of the electron beam and the quality of the modulated medium. We do not consider many other technical problems, which are related to creation of MPMRI FEL in practice. The relative energy and angular spread of the electron beam, as well as error tolerance on the medium periodicity and mean permeability were assumed to be small enough to exclude inhomogeneous broadening of the gain spectrum G(ω): Δε_el /ε_el ∼ γ²/k_nL ∼ 1/nN ∼ 10^-3 - 10^-4, Δθ/θ ∼ 1/k_nLθ ², Δθ ∼ 1/k_nLθ ∼ 2 · 10^-4 - 10^-5 rad, Δ〈ε〉 ∼ 1/(ωL/c) ∼ 10^-4 - 10^-5, Δλ₀/λ₀ ≤ 1/nqL ∼ 1/nN = λ₀/nL ∼ 10^-3 - 10^-4 (N = L/λ₀ ∼ 10³ is the number of modulation periods and n is the index of the resonant PPW).

For the transverse size of the medium d ∼ 1 cm, γ = 10, and θ ∼ 1/γ = 0.1, and Δθ ∼ 10^-3 θ = 10^-4, the emittance of the beam ε could be estimated to be about d · Δθ∼1 - 2 mm·mrad or less. The required emittance can be increased if the electron beam propagates at the angle θ, that is smaller than its optimum value maximizing the gain. The required emittance can be increased by an order of magnitude or less because the decrease of θ leads to the decrease of the FEL gain G ∼ θ ². The acceptable emittance ε equals to ∼d/k_nLθ ∼ 20 mm·mrad in this case. In particular, we do not discuss methods for creation of such media, scattering of the electron beam, absorption of amplified radiation in the medium, stability of the medium to the impact of the electron beam. It was shown (see, e.g., the discussions in [4], [5], [9], [10], [15]–[19]) that these problems can be solved and that their solution impose certain limitations on the FEL operation. We concentrate our attention only on the most fundamental features of MPMRI FEL following from the nonlinearity of equations of motion. Such an analysis is also absolutely necessary for a prognosis what can be expected from MPMRI FEL provided that all the technical problems are solved.

2. Field eigenmodes of MPMRI

Let us consider an electromagnetic wave in MPMRI. It is assumed to be plane-polarized and propagating along the z-axis, k ₀∥q∥(0z) and, hence, k _n∥q. Let the x and y-axes be parallel to the electric and magnetic fields of the wave, E(z,t) and H(z,t), respectively. The above-mentioned expansion of E(z,t) and H(z,t) in PPW is characterized by the equation:

where, as long as we are interested in eigenmodes of MPMRI, E_n and H_n are some constants. They are given by the Maxwell equations for the fields E(z,t) and H(z,t) in the modulated medium. In the simplest model of MPMRI, its permeability ε(z) is taken in the form leading to the Mathieu equation for the field eigenmode:

In the case of SLL MPMRI, 〈ε〉 = (ε ₁ + ε ₂)/2, and α = |ε ₁-ε ₂| /4, where ε ₁ and ε ₂ are the permeabilities of two kinds of layers composing a superlattice, and usually 〈ε〉 > 1 in the visible or UV domains. Being substituted into the Maxwell equations, Eqs. (2) and (3) lead to the following set of algebraic equations for E_n and H_n [21], [22]:

This set of equations can be considered as an explicit formulation of the eigenfunction - eigenvalue problem with eigenvalues given by k ₀/q and the corresponding eigenfunc-tions {E_n,H_n } determining the eigenmodes of MPMRI. Eqs. (4) can be used to estimate qualitatively the amount of relatively large-amplitude PPW in the field eigenmode {E(z, t),H(z, t)}. By assuming that at maximal and minimal numbers of such PPW, n ₊ and n _-, the amplitudes E _-1+n±, E _n±, and E _1+n± are of the same order of magnitude, by eliminating H_n from Eq. (4), and by assuming that α ≪ 1, k ₀ ∼ 〈ε〉^1/2 ω/c≫nq, one can find n ₊ and n _- determining the range of indexes n of relatively large-amplitude PPW:

A large-modulation regime is achieved if

and if the frequency of the amplified electromagnetic wave lies within the transparency domain of the modulated medium.

3. Electron - Light Interaction

The electron beam dynamics can be described by the classical single-particle approach under assumption that the plasma frequency of the beam is not large, ω_{p beam} L/cγ ≪ 1, where L is the length of the MPMRI (in the z-direction) [24]. The electron motion in an electromagnetic eigenmode of MPMRI is found using the relativistic Lorentz equation:

where the fields E and H are defined in Section 2, p and v are the electron momentum and velocity, p = ε v/c ², and ε is the electron energy.

Under the resonance condition (1) Eq. (7) can be averaged over fast oscillations to keep only the electron interaction with the resonant (n - th) PPW. The averaged equation can be reduced to the form of the pendulum equation

Here φ = (k ₀ + nq)z(t) - ωt is the phase of the resonant PPW at the electron trajectory z(t), and Ω is given by [25]:

where θ is a small angle between v and k ₀. Eq. (7) can be solved by the method of perturbation theory with respect to E and H. Then, the change of the electron energy Δε per pass through the MPMRI layer can be found from the equation

The gain per pass is determined by the energy conservation rule: the energy lost by electrons is gained by the whole field eigenmode,

where n_e is the electron number density, angular brackets denote averaging over the initial phase of the resonant PPW φ(t = 0), and E_mode = [∑_ń $E_{\stackrel{\mathit{´}}{n}}^2$]^1/2. Not dwelling upon any details of calculations, let us write down the final expressions for the linear gain G_lin , saturation fields of the resonant PPW E_{n sat} and that of the amplified electromagnetic eigenmode E_{mode sat} , and efficiency η. The gain G_lin in the linear regime is given by

where L is the length of the MPMRI (in the z-direction) and u_n = (k_n υ _{0 z} - ω)L/2c is the dimensionless detuning from resonance (1). The saturation field can be found from the condition ΩL/c ∼ 1 to give

where the factor E_mode/E_n in Eq. (12) accounts for the difference between the amplitude of the resonant PPW and the amplitude of eigenmode as a whole. This factor has to be found numerically. At last, the efficiency is determined as

4. Operation parameters of the FEL

We optimized operation parameters of the MPMRI FEL to determine its maximum possible gain G_lin of Eq. (12). For any given electron velocity υ_z along the z-axis, the optimum transverse velocity υ_x is given by

where the optimum electron relativistic factor γ_opt is related to the velocity υ_z as follows:

Under the optimum resonance detuning u_n corresponding to d/du_n (sin u_n/u_n )² = 0.54 (and υ_z ≈ ${\upsilon }_n^{\left(\mathit{\text{ph}}\right)}$ according to the resonance condition of Eq. (1)) the maximized linear gain G_lin is given by:

where r_e = e ²/mc ² is the classical electron radius.

The calculations were performed for the SLL-structure composed of KCl and amorphous quartz layers. Its transparency window is rather wide, 2.8 · 10¹⁵ s ^-1 < ω < 10¹⁶ s ^-1. The parameters of such a superlattice used for calculations were: L = 0.5 cm and λ₀ = 3.3 · 10^-3 cm. Both the mean permeability 〈ε〉 and the modulation amplitude 2α of such a structure grow monotonously with the increasing frequency ω. In the calculations, the electric current density of a beam was taken to be j = 5 A/cm ².

The relative energy and angular spread of the electron beam, as well as error tolerance on the medium periodicity and mean permeability were assumed to be small enough to exclude inhomogeneous broadening of the gain spectrum G(ω): Δε_el /ε_el ∼ γ²/k_nL ∼ 1/nN ∼ 10^-3 - 10^-4, Δθ/θ ∼ 1/k_nLθ ², Δθ ∼ 1/k_nLθ ∼ 2 · 10^-4 - 10^-5 rad, Δ〈ε〉 ∼ 1/(ωL/c) ∼ 10^-4 - 10^-5, Δλ₀/λ₀ ≤ 1/nqL ∼ 1/nN = λ₀/nL ∼ 10^-3 - 10^-4 (N = L/λ₀ ∼ 10³ is the number of modulation periods and n is the index of the resonant PPW).

For the transverse size of the medium d ∼ 1 cm, γ = 10, and θ ∼ 1/γ = 0.1, and Δθ ∼ 10^-3 θ = 10^-4, the emittance of the beam ∊ could be estimated to be about d · Δθ∼1 - 2 mm·mrad or less. The required emittance can be increased if the electron beam propagates at the angle θ, that is smaller than its optimum value maximizing the gain. The required emittance can be increased by an order of magnitude or less because the decrease of θ leads to the decrease of the FEL gain G ∼ θ ². The acceptable emittance ε equals to ∼d/k_nLθ ∼ 20 mm·mrad in this case.

Data for the maximized gain G_lin shown in Fig. 1a indicate that the optimum PPW index n maximizing the gain grows with the increasing frequency ω. This agrees with the above-estimated amount of excited PPW Eq. (5) for modulation amplitude α(ω) being a growing function of ω.

The gain achievable at each particular PPW is an oscillating function of the frequency ω. This is related to the oscillatory dependence of the PPW amplitudes E_n (2)-(4) on ω.

 

Fig. 1a. The maximized gain G _max and the corresponding optimum PPW index.

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Fig. 1b. The electron relativistic factor γ maximizing the gain G.

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Fig. 1c. The saturation field of the amplified electromagnetic mode

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Fig. 1d. The efficiency of a FEL η

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Our calculations indicate that the gain achievable in FEL based on SLL-struc-tures can be rather high. Qualitatively, this result is explained by a large mean permeability of a SLL structure, 〈ε〉 > 1, which makes phase velocities of many low-order PPW smaller than the light speed c. It follows that the resonance electron velocity is low relativistic too, that makes it easier for electrons to respond to the amplified electromagnetic wave. In the small-modulation regime, ω·α(ω)≪qc〈ε〉^1/2, it is the large-amplitude zero-order PPW {E ₀,H ₀} that maximizes the gain and makes it pretty high. In this case, the FEL under consideration is similar to the stimulated Cherenkov FEL [23]. In the large-modulation regime, ω·α(ω)∼≫qc〈ε〉^1/2, the gain can be maximized at higher-order PPW. In this case, the MPMRI FEL can be considered conventionally as the Cherenkov FEL at higher-order PPW of the field eigenmode. The MPMRI FEL operating in the large-modulation regime is also similar to a Resonant-Transition-Radiation (RTR) FEL [9], [10], [13]. The large-modulation MPMRI FEL can provide larger gain, as compared to a RTR FEL. Indeed, the gain is proportional to the squared field amplitude E_n of the resonant PPW. The PPW amplitudes E_n of a MPMRI eigenmode (21), (22) do not fall while n _- < n < n ₊ and can exceed PPW amplitudes of a RTR FEL E_nRTR ∼ 1/n.

In Fig. 1b we present the optimum electron relativistic factor as a function of the frequency ω. In accordance with Eqs. (13), (14) and (17), the maximized linear gain G_lin and the corresponding efficiency η at saturation conditions are related so that η ⁵·$G_{\mathit{\text{lin}}}^2$ < 2.5 · 10^-4 Lλ³ $r_e^2$ $n_e^2$, η ³·G_lin < 3 · 10^-2γλ² r_en_e , and η·G_lin < 2.3L ² r_en_e /γ³, where the factor $E_n^2$/$E_{\mathit{\text{mode}}}^2$ < 1 of Eq. (17) was assumed to be of the order of 1. It follows that the saturation field and efficiency shown in Figs. 1c, 1d corresponding to the high gain of Fig. 1a are rather low. Indeed, the estimates for L = 0.5 cm, n_e = 10⁹ cm ^-3, υ_z /c ≈ 1, and for the gain G fixed at the level 0.1 indicate that the maximum efficiency of the FEL is limited by

Hence, the efficiency of the FEL operating in the visible or higher frequency domains, ω > 2.8 · 10¹⁵ s ^-1, is limited by η ≤ 6 · 10^-6. However, both the gain and efficiency can be large in the centimeter or millimeter wavelength domains.

The maximized gain of the MPMRI FEL was also found as a function of the modulation period λ₀. The data shown in Fig. 2a were obtained for the frequency ω = 2.8 · 10¹⁵ s ^-1. The SLL structure and the current density were the same as for Fig. 1.

 

Fig. 2a. The maximized gain G _max and the corresponding optimum PPW index.

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Fig. 2b. The electron relativistic factor γ maximizing the gain G.

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Fig. 2c. The saturation field of the amplified electromagnetic mode

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Fig. 2d. The efficiency of a FEL η

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At short modulation period the highest gain is achieved at the zero-order PPW. At large modulation period there are many large-amplitude PPW formed inside the medium and the maximum gain is achieved at high-order PPW, n ≫ 1. This result agrees with the criterion for the large-modulation regime of Eq. (6).

It was found that in dependence on λ₀, the largest gain is achieved at short modulation periods corresponding to the small-modulation regime. We attribute this large gain to the larger PPW amplitudes E_n , that occur at the small-modulation regime. Indeed, in small-modulation regime the energy of the electromagnetic eigenmode is distributed among a smaller amount of relatively larger-amplitude PPWs. It follows from the results of Fig. 2 that the large-modulation regime makes it possible to use long-period SLL structures for the FEL operation. Although the small-period structures are preferable for the large gain, long period SLL MPMRI can have certain advantages, such as a larger lifetime under the action of the electron beam.

5. Conclusions

As a resume, it is shown that the large-modulation regime offers an opportunity to construct a compact low-electron-energy free-electron laser operating in various frequency domains. The large-modulation regime occurs if ω·α(ω)≫cq, or |ε ₁ - ε ₂| ≫ λ/λ₀, where λ = 2πc/ω. In this case, the largest gain is shown to be due to the resonance interaction of the electron beam with the higher-order PPW composing the field eigenmode of the MPMRI. The corresponding maximized gain is calculated and shown to be rather large up to ω ∼ 10¹⁶ s ^-1. This is an evident extension of the small-modulation MPMRI FEL, that can use only the first-order PPW, n = 1. Such a case was considered earlier for a gas-plasma FEL [7], [8] and a periodic dielectric FEL [9]–[14]. Other types of MPMRI and peculiarities of the corresponding FEL schemes are considered elsewhere [9], [15]–[19], [25].

If operated at the IR or far-IR frequency domains, the proposed FEL can compete in power and saturation field with other low-cost compact infrared FELs, e.g., a CIRFEL [26] that uses a picosecond pulsed 7 - 14 Meυ electron beam and produces a pulsed radiation of a peak power about a megawatt. As compared to CIRFEL, the FEL considered is able to switch its operation frequency by changing a resonant PPW without a change of the electron beam energy. It is also able to generate multicolor radiation by using different PPWs simultaneously, that is important for spectroscopy. These features make the FEL considered rather promising candidate for a low-cost low-power medical applications oriented commercial FEL.