Nonlinear Theory Of A Free-electron Laser Exploiting ... - IEEE Xplore

24

IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 34, NO. 1, JANUARY 1998

Nonlinear Theory of a Free-Electron Laser Exploiting Media with a Periodically Modulated Refractive Index Alexander I. Artemyev, Mikhail V. Fedorov, John K. McIver, and Evgenii A. Shapiro

Abstract—The nonlinear theory of a free-electron laser (FEL) exploiting media with a periodically modulated refractive index is presented. The gain and the saturation parameters are found for different operating regimes of such a FEL. The system discussed could be used for the amplification of light in the optical and ultraviolet regions of the spectrum. Index Terms—Compact free-electron lasers, nonlinear effects, periodically modulated media.

I. INTRODUCTION.

A

N IMPORTANT direction in the development of freeelectron lasers (FEL’s) is a search for methods that will permit construction of a small-size FEL operating at high frequencies. One of the ways to achieve this goal consists of using small-period periodic structures and, in particular, media with periodically modulated refractive indexes [1]–[8]. Such media can be constructed by using a solid-state superlattice [1], [2], a structure working in the construction material absorption edge frequency domain [3], [4], periodically spaced foil strips [5], [6], an ion-acoustic wave in a plasma [7], a gas-plasma medium with a periodically modulated degree of ionization [8], etc. To be more specific, let us discuss briefly the problem of preparation of a gas-plasma medium. Such a medium can consist of a series of breakdowns in a gas produced, e.g., by a train of pulses of an external laser. Such a train can be focused by a mirror that is turned around an axis perpendicular to the laser beam direction of propagaion. Subsequent laser pulses are focused into spots equally separated from each other. As a result, a gas-plasma medium with periodically modulated degree of ionization is formed [8]. Some other possibilities to create a series of breakdowns in a gas consist of using a sinusoidal transparent diffraction grating as a focusing system, Manuscript received February 6, 1997; revised September 12, 1997. The work of A. I. Artemyev was supported by the Russian Foundation for Basic Research under Grant 96-02-18241, in part by the International Science Foundation under Grant M1I300, and by a fellowship of INTAS Grants 93-2492 and 93-2492-ext provided by the research program of International Center of Fundamental Physics in Moscow. The work of E. A. Shapiro was supported by the Russian Foundation for Basic Research under Grant 96-0218241 and by a grant for graduate students awarded by the Soros Foundation. A. I. Artemyev, M. V. Fedorov, and E. A. Shapiro are with the General Physics Institute, Russian Academy of Sciences, 117942 Moscow, Russia. J. K. McIver is with the Center for Advanced Studies, Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 87031 USA. Publisher Item Identifier S 0018-9197(98)00356-X.

or using a standing microwave in a waveguide filled with a gas, or employing a gas discharge formed by a net-shaped electrode. These and other methods to construct a medium with a periodically modulated refractive index will be compared elsewhere [17]. Both spontaneous emission and linear amplification of light by an electron beam propagating in media with periodically modulated refractive index have been thoroughly described earlier [1]–[11]. In this paper the nonlinear theory of a FEL of the above-described type is presented. The saturation field is found, and the conditions at which the nonlinear gain becomes a falling function of the field strength of the amplified wave are determined. is a very important parameter of any FEL which has to be known before a specific device is designed. In particular, in accordance with one of the results derived below, the saturation field of the most effective operating regime of the FEL under consideration appears to be proportional to the squared electron relativistic factor and appears to be very small at small . This means that though formally the gain of the FEL under consideration can be rather large at nonrelativistic electron energies, construction of such a device is hardly reasonable because of a very low saturation field. Only if is as large as 10 or higher, the saturation field appears to be high enough to make the corresponding device interesting practically. The two specific cases considered below correspond to the electron beam propagating either along the direction of propagation of the light beam or at some angle that is larger than the critical angle , with respect to it. In the first of these two cases ) the gain appears to be much lower than in the second case ( ), and for this reason, practically, the case can hardly compete with the case of an oblique propagation of electrons. However, theoretically, the main equations describing the FEL under consideration in the case are very interesting. They cannot be reduced to the usual pendulum equation. Moreover, as it is shown below, these equations have a very simple analytical solution that is valid, at proper detunings from resonance, at any fields, weak, strong, or intermediate. Theoretically, this result is very interesting, and it is in contrast with the usual theory of FEL’s, where approximate analytical solutions can be found only in the cases of weak and strong fields [16]. In the case of an oblique propagation ( ), the nonlinear theory of the FEL under consideration appears to be much more similar for the usual nonlinear theory of a FEL with

0018–9197/98$10.00  1998 IEEE

ARTEMYEV et al.: NONLINEAR THEORY OF A FREE-ELECTRON LASER EXPLOITING MEDIA

undulator. However, even in this case, the definitions of the saturation field found below are different from those of the undulator-FEL, because of a difference in the physical origin of periodocity in a magnetic undulator and in a medium with periodically modulated refractive index. The main physical equations solved and derived below are presented in the form that is general enough to be valid for both cases of small and large modulation of a medium. Then, at the final stage, these equations are specified to describe the small-modulation case (for both linear and nonlinear theory). However, the above-described general form of consideration is very important. The general results derived will be used in our further investigation of the FEL exploiting a medium with a large modulation of its refractive index. The results of such an investigation will be publishd separately but, preliminarily, it can be mentioned here that the large-modulation regime can provide unproportionally higher gain and efficiency of FEL than the usually considered case of a small modulation. It should be mentioned, finally, that while trying to construct a FEL, exploiting a medium with periodically modulated refractive index, one can meet many technical problems. Some of them concern creation of the most appropriate medium, propagation of electron and light beams in the medium, stability of both electron beam and the medium itself, absorbtion and scattering of light in the medium, etc. Though very important and serious, these problems seem to have solutions, as it is shown, e.g., in the review paper by Kaplan et al. [3]. These practical problems are not addressed in this paper. Our main goal consists in answering the question: how high is the field up to which amplification in a FEL under consideration can occur? This is just the saturation field . Derivation of expressions for and discussion of the saturation effects under various conditions are the main contents of this paper. II. FIELD CONFIGURATION Let us assume that by some means a medium is prepared in which the refractive index or the permeability is periodically modulated along the axis (1) is the modulation depth, is the periodicity conwhere stant, , and is the period of modulation. The field eigenmodes in such a medium are the well-known Floquet–Bloch waves [12]–[15]. The electric ( ) and magnetic ( ) fields of the wave that propagates along the axis and has a frequency can be presented in the form (2)

25

The first of these equations has the form of the well-known Mathieu equation [12]. The Floquet–Bloch solutions of (3) are given by superpositions (4) can be either real or complex, depending on the where relations between , , and . For such a form of the field and , (3) are reduced to the infinite set amplitudes of algebraic equations for the constants and

(5) In accordance with the general Floquet–Bloch expansion of and [(4)], the electric and the field amplitudes magnetic field strengths of the wave and [(2)] have the form of superpositions of plane waves propagating along the axis with phase velocities (6) If modulation is weak, the constant is real [12], and both the expansion (4) and the set of equations (5) can be truncated so that only three lowest order terms with and are retained. In this case, (5) yields

(7) In this form, the truncation procedure is equivalent to the use of perturbation theory with respect to the modulation depth . Perturbation theory is correct only if , or , where (8) In the most typical case, the period of modulation is much longer than the wavelength of light , that is equivalent to or . This means that even a very small modulation (such that ) can result in a large difference between the plain wave in the vacuum and the eigenmodes in the media with the modulated refractive index. In the weak-modulation case ( ) the phase velocities of the three plane waves ( and ) of the superposition (4) are equal to

and are the complex ampliwhere the functions tudes of the fields obeying equations (9) (3)

The weak-modulation approximation described above, and (7) and (9) were used earlier [8] in the linear theory of FEL under

26

IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 34, NO. 1, JANUARY 1998

consideration. Here, we will also use this approximation in the final expressions for the nonlinear gain, resonance frequency, and saturation parameter. However, most of the intermediate equations will be written in a form general enough to be applicable to any modulation. In particular, it should be noted that, in the general case of an arbitrarily strong modulation, the wavevector can deviate from the linear function of (7), can deviate from the simple and the phase velocities expressions given by (9). III. EQUATIONS DESCRIBING ELECTRON–LIGHT INTERACTION In this paper, we will assume that electrons interact with the periodic medium only through the modulated refractive index. The effects associated with collisions of electrons with the grating will not be considered here. We assume that, at relativistic electron energies and at small length of a device under consideration, these effects only slightly perturb an electron beam. In a single-particle classical theory, it is assumed that the electron–light interaction is described by the relativistic equations of the electron motion in the presence of electromagnetic fields and defined in Section II (2)–(7) (10) where

and are the electron momentum and velocity, , and is the electron energy obeying the equation (11)

The electron–light interaction is most efficient if one of the plane waves in the superposition given by (4) has a phase close to the projection of the electron velocity velocity on the -axis,

. For this specific plane wave, its phase (12)

, appears to be calculated at the electron trajectory, a slow function of time . For example, by substituting the unperturbed free-electron coordinate into (12), we find

(13) The constant in the equation for characterizes an arbitrary instant of time when the given electron enters the interaction region at . Equation (13) shows explicitly when , and, hence, is a slow that . As for all other plane waves function of time if in the superposition of (4), under the same conditions, their phases at the electron trajectory are not slow functions. For example, in the same approximation of the unperturbed electron motion, for any , (14) By assuming that , we find the following criterion determining the condition under which the phase of

the resonant ( th) plane wave phases of all other plane waves

varies much slower than : (15)

In the weak-modulation approximation considered in the previous section, only one of the three plane waves in the truncated superposition of (4) can satisfy the condition of (15). This is for which , whereas for other two the wave with and , . For the wave with waves with , in the weak-modulation approximation, the condition yields (16) where is the electron relativistic factor that corresponds to which the unperturbed electron energy ; is a small angle, at is assumed to be large, . As which electrons move with respect to the axis, is the resonance frequency in the vicinity of which usual, electron–light interaction is expected to be most efficient. is determined by In the case of arbitrary modulation, or by the solution of the equation the condition (17) Equation (13) can be used to estimate the degree of applicability of (15). In the weak-field theory, the change in plays the role resonance phase is the length of the of the dimensionless detuning, where interaction region. The weak-field gain [8] is not small only . This yields where if . Hence, typically the ratio of the right- to where the left-hand sides of (15) is of the order of is the number of periods in the interaction region. As , the condition of (15) is satisfied and the long as above-described difference between slow (resonance) and fast (off-resonance) motions is well pronounced. By differentiating twice both sides of (12) and by using (10) and (11), we can find that, in the general case, the phases obey equations (18) where and are the velocity projections on the and axis, respectively. In accordance with the idea of separation of slow and fast is a slow function, and all motions, we can assume that with are fast functions of . By averaging (18) on over fast oscillations, we retain only one term with the right-hand side of this equation that, finally, takes the form (19)

ARTEMYEV et al.: NONLINEAR THEORY OF A FREE-ELECTRON LASER EXPLOITING MEDIA

By using the definition of the phase [(12)], we can simplify the expression in the right-hand side of (19):

27

Vice versa, if reduced to

is large enough (

), (24) is (27)

with

given by

(20) where, owing to the resonance condition of (15), approximated by

. The term

is estimated here

as being less or of order of times less than gives a contribution of about As a result, (19) is reduced to

. This term .

The initial conditions for these equations are

(29) (21)

Under the assumption that the total change of the electron energy is small, the electron energy and relativistic factor on the right-hand side of (21) can be replaced by their initial values. By recalling that and rot , where is the vector potential of the amplified wave, we can find that the component of the right-hand side of (10) is equal to the total time-derivative of . This equation is integrated to give (22) where, in terms of phases

(28)

is

where is the detuning. Solutions of (25)–(29) will be presented in the next sections. By combining (11) and (21), one can obtain the expression for the change of the electron energy

(30) As it is well known, the gain per pass is determined by the energy conservation rule: the energy lost by electrons is gained by the field

, (31) (23)

In the resonance approximation, only the one term correspondcan be retained on the ing to the resonance (slow) phase right-hand side of (23). With the help of (22), in the approximation of a small change of the electron energy, our general equation for the slow phase [(21)] can be rewritten in the form

(24) where is the resonance phase. Equation (24) is simplified in two cases when the first or second terms in the square brackets on the right-hand side are small. In the case of very small angle , such as is much less than the critical angle , , (24) takes the form (25) where (26) Here and in many other expressions below mated by .

is approxi-

where is the electron number density in the electron beam, and angular brackets denote averaging over . IV. ELECTRON MOTION ALONG DIRECTION OF MODULATION

THE

In the case , the slow (resonance) phase is determined by (25) and its solution. This equation is rather unusual. In contrast to the pendulum equation that is well known in the theory of FEL with undulators [16], (25) contains explicit dependence on the initial phase in addition to the dependence of on arising from the initial conditions[(29)]. On the other hand, similarly to the pendulum equation, (25) has the first integral of the form (32) where is given by (29). As in the case of the pendulum equation, the first and second terms on the left-hand side of (32) can be interpreted as the doubled kinetic and potential energy of a mechanical system that now can be referred to as “modified pendulum.” However, the potential energy of this “modified pendulum” has a structure different from that of the usual pendulum: it has minima in the points and , depending on the initial phase . In accordance with the initial conditions [(29)], the

28

IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 34, NO. 1, JANUARY 1998

the modified pendulum equation [(25)] and derivation of the strong-field gain appear to be very simple, even much simpler than in the usual case of the FEL with magnetic undulators [16]. In accordance with our assumption , the phase is always close to its initial value . By expanding and in powers of and retaining only the lowest order terms, we find that (25) is reduced to the equation of a harmonic oscillator with the oscillation frequency equal to : (38)

Fig. 1. Potential energey U of the “modified pendulum.” The pendulum starts 2= . its motion from the point ' '0 with the initial kinetic energy '=

_ 2=1 2

=

“modified pendulum” starts its motion from the point with the initial kinetic energy (Fig. 1). If , the motion of the modified pendulum is infinite and is only slightly affected by the field. If, in addition, , (32) can be solved by using perturbation theory with respect to the potential energy. By substituting the zerothorder solution of (25) and (29), , into the potential of (32), we get

After averaging over (33) yields

(39) and substituting it into (30) By averaging this result over we can find the strong-field change of the electron energy

(33)

(40)

and substitution into (30) and (31),

where is the zeroth-order Bessel function. The dependence of on the saturation parameter is shown in Fig. 2. The dashed line shows the dependence on calculated by using (40), and the solid line shows the same dependence as calculated by numerical solution of (25). The cases shown are 0.05, 0.2, and 1. It can be seen from Fig. 2 that (40) describes the electron–light and completely wrong if interaction correctly if . The strong-field gain can be obtained from (40) with the help of (31):

(34) where radius, and

Equation (38) describes electron motion correctly everyexcept two small zones where over the region and . The width of around the points each of these zones is of the order of , and hence, in our approximation, the contribution of these zones to the solution of (25) averaged over the initial phase can be neglected. The solution of (38) that obeys the initial conditions is given by

,

is the classical electron (35)

This result looks rather unusual. First, the spectral dependence is determined by the factor which is different from the usual one [16]. Second, at fixed , the gain of (34) is proportional to the interaction length to the first power, in contrast to the usual dependence . ), In the case of weak modulation ( , , and . Under these conditions, (34) yields

(41) The maximal value of the gain is achieved when . This occurs at the boundary of the region of applicability of the approximation used above. At this point, is estimated as (42)

(36)

If, in addition, the weak-modulation approximation is valid ( ), (42) yields

The weak-field approximation becomes invalid when the saturation parameter (37) becomes larger then one, detuning of (29) is small,

. If, in addition, the resonance (Fig. 1), then solution of

(43) If

, and

, we get (44)

ARTEMYEV et al.: NONLINEAR THEORY OF A FREE-ELECTRON LASER EXPLOITING MEDIA

29

given by (48) In terms of the incident wave, (48) yields (49) , i.e., By inverting (48), we can find the saturation length the length providing for any given , , and , (50) The linear gain (in the case is given by

1E

Fig. 2. The averaged change in the electron energy h i in dependence on i on the saturation parameter . The dashed line shows the dependence h  calculated by using (39), the solid line shows the result of the numerical 0.05, (b) = 0.2, and (c) calculation. The cases shown are (a) = = 1.

1 =

1 =

The saturation field

1 =

corresponding to

) found from (27)–(31)

(51)

1E

is given

where, as previously, . The spectral width of the linear gain is determined by the condition . In the weak-modulation approximation, the expression for the linear gain coincides with that of [8]:

by (52)

(45) In the weak-modulation approximation, in terms of the incident wave, (45) yields

The optimal linear gain is achieved at , is estimated as

. If

(46) It should be noted that, in the case of a strong modulation , the net saturation field of the light mode as a whole can be several orders larger than because a light eigenmode can consist of a very large number of partial plane waves.

(53) When

is substituted by the saturation length takes the form

[(50)],

(54) V. OBLIQUE PROPAGATION

OF

ELECTRONS

If the angle is large enough (though ), the second term in the square brackets of (24) can be dropped, and the equation for the phase is given by (27), i.e., by the usual pendulum equation. Solutions of this equation and both linear and nonlinear gain are well known [16]. The only difference with the usual undulator FEL consists of a difference in definitions of [in the case under consideration, is determined by (28)]. The saturation parameter is now given by (47) As usual, the saturation field of the resonant th partial is determined by the condition . In plane wave ), is the weak-modulation approximation (

and can berealThe strong-field region corresponds to ized either by increasing above , or above . The strong-field gain falls as [16]. The strong-field spectral [16]. width of the gain is determined by the condition VI. ESTIMATES

AND

DISCUSSION

In this paper, we described two regimes that can occur in a FEL exploiting a medium with a modulated refractive index. These regimes correspond to propagation of electrons at small angles or under a larger angle with respect to it. To compare these regimes, let us consider (34) and (45) of Section IV with (48) and (51) of Section V. An example of the gain dependencee gain on the field strength in both cases is presented in Fig. 3. Equations (34) and (51) show that at

30

IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 34, NO. 1, JANUARY 1998

Fig. 3. The gain Gopt in the two cases of oblique and longitudinal electron propagation at 0 0.1 mm, L 20 cm, 25, and ne 1010 cm03 .

=

=

=

=

small values of , , and . For the , the gain starts to saturate at , case of the gain starts to saturate at much whereas for , where . The stronger fields saturation parameters (37) and (47) are related to each other by the equation . At , the nonlinear gain (54) is of the order of . At (or ) and, hence, . , the gain also saturates and, in In the region accordance with (43), . By comparing the derived and , one can find that these two dependencies , at . At functions match each other, , , i.e., the gain even stronger fields, becomes larger than the gain (see Fig. 3). However, such strong fields ( ) correspond to a rather extreme situation which can hardly be of a practical interest. and , i.e., the oblique-incidence Most typically, gain is much larger than the gain achievable at . To estimate possible gains, let us assume that 10 cm , 0.1 mm, 20 cm, and 25. This estimate assumes the use of the scheme with periodical gas–plasma medium [8] and a moderate-current electron beam. at the resonant For these parameters, (53) gives wavelength 0.3 m. The saturation field is equal to 4000 V/cm. These results show that, in principle, the scheme under consideration can operate efficiently as an amplifier in the regime of oblique electron motion for the amplification at high frequencies. The important consequence of (48) is that the saturation field is very small at small values of . For example, the same gas-plasma scheme with an electron beam with would saturate at 6.5 V/cm. This illustrates the fact that all laser schemes that use small-period periodical media and nonrelativistic electron beams have very small saturation fields. To escape such an extremely low saturation field, one would have to miniaturize the grating and to decrease strongly the amount of grids that would decrease unproportionally the gain and make the amplification line wider. The estimate given above corresponds to the smallmodulation conditions. In the framework of this approximation, one can hardly obtain much better results: a larger gain or a shorter wavelength . However, as it was mentioned in

the Section I, these parameters can be significantly improved in the large-modulation regime. The conditions under which such a regime can occur as well as the main results of the corresponding analysis will be reported separately [17]. Briefly, in the large-modulation regime, one gets a chance to optimize the gain significantly by choosing the most appropriate partial plane waves of the field eigenmode to be in the resonance with electrons. This optimization can increase the gain by up to two orders of magnitude. As one does not need such an enormous gain, it can be reduced to a quite reasonable value (e.g., 10%) by choosing a smaller period of modulation and a faster electron beam than in the estimate given above. This would result in a shorter wavelength of the amplified wave. By using the results of this paper, one can make even more complicated optimization to get reasonably high gain and saturation field and reasonably short wavelength . We do believe that such a consideration will make an important step in solving the problem formulated in Section I, a search of methods to construct a compact FEL operating effectively in the ultraviolet and X-ray regions of wavelength . ACKNOWLEDGMENT The authors thank Prof. V. V. Apollonov for stimulating discussions. . REFERENCES [1] A. E. Kaplan and S. Datta, “Extreme-ultraviolet and x-ray emission and amplification by nonrelativistic electron beams traversing a superlattice,” Appl. Phys. Lett., vol. 44, pp. 661–663, 1984. [2] A. P. Apanasevich and V. A. Yarmolkevich, “Resonance transition radiation and its observation in multilayer interference structures,” Zh. Tech. Fiz., vol. 62, no. 4, pp. 120–125, 1992; also Sov. Phys. - Tech. Phys., vol. 37, no. 4, pp. 423–428, 1992. [3] A. E. Kaplan, C. T. Law, and P. L. Shkolnokov, “X-ray narrowline transition radiation source based on low-energy electron beams traversing a multilayer nanostructure,” Phys. Rev. E, vol. 52, pp. 6795– 6808, 1995 [4] C. J. Pincus, M. A. Piestrup, D. G. Boyers, Q. Li, J. L. Harris, X. K. Maruyama, D. M. Skopik, R. M. Silzer, H. S. Caplan, and G. B. Rothbart, “Measurements of X-ray emission from photoabsorbtion-edge transition radiation,” J. Appl. Phys., vol. 72, pp. 4300–4307, 1992. [5] M. A. Piestrup and P. F. Finman, “The prospects of an X-ray freeelectron laser using simulated resonance transition radiation,” IEEE J. Quantum Electron., vol. QE-19, pp. 357–364, 1983. [6] M. A. Piestrup, M. J. Moran, D. J. Boyers, C. I. Pincus, J. O. Kephart, R. A. Gearhart, and X. K. Maruyama, “Generation of hard x-rays from transititon radiation using high-density foils and moderate-energy electrons,” Phys. Rev. A, vol. 43, pp. 2387–2396, 1991. [7] K. R. Chen and J. M. Dawson, “Amplification mechanism of ion-ripple lasers and its possible applications,” IEEE Trans. Plasma Sci., vol. 21, pp. 151–155, 1993. [8] M. V. Fedorov and E. A. Shapiro, “Free-electron lasers based on media with periodically modulated refractive index,” Laser Phys., vol. 5, pp. 735–739, 1995. [9] K. F. Casey and C. Yech, “Transition radiation in a periodically stratified plasma,” Phys. Rev. A, vol. 2, pp. 810–818, 1970. [10] K. F. Casey, C. Yech, and Z. A. Kaprielian, “Cerenkov radiation in inhomogeneous periodic media,” Phys. Rev. B, vol. 140, pp. 768–775, 1965. [11] C. Elachi, “Cerenkov and transition radiation in space-time periodic media,” J. Appl. Phys., vol. 43, pp. 3719–3723, 1972. [12] N. W. McLachlan, Theory and Application of Mathieu Functions. New York: Oxford, 1947. [13] K. F. Casey and C. Yech, “Wave propagation in sinusoidally stratified plasma media,” J. Math. Phys., vol. 10, pp. 891–897, 1969. [14] P. St. J. Russel, “Optics of Floquet-Bloch waves in dielectric media,” Appl. Phys. B, no. 39, pp. 231–246, 1986.

ARTEMYEV et al.: NONLINEAR THEORY OF A FREE-ELECTRON LASER EXPLOITING MEDIA

[15] C. Elachi, “Waves in active and passive media,” Proc. IEEE, vol. 64, pp. 1666–1698, 1976. [16] M. V. Fedorov, “Interaction of intense laser light with free electrons,” in Laser Science and Technology, an International Handbook, V. S. Letokhov, C. V. Shank, Y. R. Shen, and H. Walther, Eds. Chur, Switzerland: Harwood, 1991. [17] V. V. Apollonov, A. I. Artemyev, M. V. Fedorov, J. K. McIver, and E. A. Shapiro, “Large-modulation regime of the free-electron laser exploiting a medium with periodically modulated refractive index,” Phys. Rev. A, submitted for publication.

Alexander I. Artemyev was born in Moscow, Russia, on December 25, 1966. He graduated from the Moscow Institute of Physics and Technology in 1989. He works in the General Physics Institute, Moscow, as a Junior Scientist. His area of scientific interests includes free-electron lasers, laser electron accelerators, and atomic physics.

Mikhail V. Fedorov was born in Moscow, Russia, on December 15, 1940. He graduated from the Moscow State University in 1964. He received the Candidate of of Sciences and Doctor of Sciences degrees from Lebedev Physical Institute, Academy of Sciences of the USSR, in 1967 and 1981, respectively. His current position is the head of the laboratory in the General Physics Institute, Moscow. His areas of expertise are laser physics, atomic physics, and field theory.

31

John K. McIver received the Ph.D. degree from the University of Rochester, Rochester, NY, in 1978. He then joined the University of New Mexico, Albuquerque, and currently holds a joint position with the European Office of Aerospace Research and Development of U.S. Air Forces. His research interests include multiphoton processes in atoms and solids, the influence of the transient and the bandwidth on the phase conjugation and frequency conversion, optical processes in photorefractive materials, nonlinear absorption in solids, and optical properties of nonlinear coupled elements in an optical cavity.

Evgenii A. Shapiro was born in Moscow, Russia, on November 1, 1972. He graduated from the Moscow institute of Physics and Technology in 1995. He is with the General Physics Institute, Moscow, as a Junior Scientist. His areas of scientific interests are free-electron lasers and atomic physics.