## Free-electron laser exploiting a superlattice-like medium

Optics Express, Vol. 3, Issue 5, pp. 162-170 (1998)

http://dx.doi.org/10.1364/OE.3.000162

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### Abstract

Amplification in free-electron lasers exploiting media with periodically modulated refractive indices is studied in the regime of a large modulation. The conditions for realization of the large-modulation regime in a superlattice-like medium are established. The maximized gain, the corresponding saturation field and efficiency, as well as the optimal electron energy and propagation direction are determined. It is shown that the large-modulation regime makes it possible to extend significantly the operation frequency domain of the FEL employing a low-relativistic electron beam. Relationship with the Cherenkov and stimulated resonance-transition-radiation FELs is discussed. This research is partially supported by RFBR grant 97-02-17783.

© Optical Society of America

## 1. Introduction

1. K. R. Chen and J. M. Dawson, “Ion-ripple laser,” Phys. Rev. Lett. **68**, 29–32 (1992). [CrossRef] [PubMed]

9. M. A. Piestrup and P. F. Finman, “The prospects of an X-ray free-electron laser using stimulated resonance transition radiation,” IEEE J. Quantum Electron. **QE 19**, 357–364 (1983). [CrossRef]

20. H.C. Lihn, P. Kung., C. Settakron, H. Wiedemann, D. Bocek, and M. Hernandez, “Observation of stimulated transition radiation,” Phys. Rev. Lett. **76**, 4163–4166 (1996). [CrossRef] [PubMed]

_{0}, and wave vector

**q**(to be parallel to the

*z*-axis),

*q*= 2π/λ

_{0}. 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

*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

*d*∼ 1

*cm*, γ = 10, and

*θ*∼ 1/γ = 0.1, and Δ

*θ*∼ 10

^{-3}

*θ*= 10

^{-4}, the emittance of the beam

*ε*could be estimated to be about

*d*· Δ

*θ*∼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*

_{n}

*Lθ*∼ 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

4. V. A. Bazylev, T. J. Schep, and A. V. Tulupov, “Short-Wavelength Free-Electron Lasers with Periodic Plasma Structures,” J. of Phys. D - Appl. Phys. **27**, 2466–2469 (1994). [CrossRef]

5. V. A. Bazylev, V. Goloviznin, M. M. Pitatelev, A. V. Tulupov, and T. J. Schep, “On the possibility of construction of plasma undulators,” Nucl. Instr. & Meth. Phys. Res. A **358**, 433–436 (1995). [CrossRef]

9. M. A. Piestrup and P. F. Finman, “The prospects of an X-ray free-electron laser using stimulated resonance transition radiation,” IEEE J. Quantum Electron. **QE 19**, 357–364 (1983). [CrossRef]

10. M. B. Reid and M. A. Piestrup, “Resonance transition radiation X-Ray laser,” IEEE J. Quantum Electron. **QE 27**, 2440–2455 (1991). [CrossRef]

15. C. I. Pincus, M. A. Piestrup, D. G. Boyers, Q. Li, J. L. Harris, X. K. Maruyama, D. M. Skopik, R. M. Silzer, and H. S. Caplan, “Measurements of X-ray emission from photoabsorption-edge transition radiation,” Phys. Rev. A **43**, 2387–2396 (1991). [PubMed]

## 2. Field eigenmodes of MPMRI

*z*-axis,

**k**

_{0}∥

**q**∥(0

*z*) 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:

*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:

*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

21. K. F. Casey, C. Yech, and Z. A. Kaprielian, “Cherenkov radiation in inhomogeneous periodic media,” Phys. Rev. B **140**, 768–775 (1965). [CrossRef]

22. B. Pardo and J. M. Andre, “Transition radiation from periodic stratified structure,” Phys. Rev. A **40**, 1918–1925 (1989). [CrossRef] [PubMed]

*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:

## 3. Electron - Light Interaction

*ω*

_{p beam}

*L*/

*c*γ ≪ 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:

**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.

*n*-

*th*) PPW. The averaged equation can be reduced to the form of the pendulum equation

*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]:

*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 Δ

*ε*per pass through the MPMRI layer can be found from the equation

*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}

^{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

*L*is the length of the MPMRI (in the

*z*-direction) and

*u*

_{n}= (

*k*

_{n}

*υ*

_{0 z}-

*ω*)

*L*/2

*c*is the dimensionless detuning from resonance (1). The saturation field can be found from the condition Ω

*L/c*∼ 1 to give

*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

*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

*u*

_{n}corresponding to

*d/du*

_{n}(sin

*u*

_{n}

*/u*

_{n})

^{2}= 0.54 (and

*υ*

_{z}≈

*G*

_{lin}is given by:

*r*

_{e}=

*e*

^{2}/

*mc*

^{2}is the classical electron radius.

^{15}

*s*

^{-1}<

*ω*< 10

^{16}

*s*

^{-1}. The parameters of such a superlattice used for calculations were:

*L*= 0.5

*cm*and λ

_{0}= 3.3 · 10

^{-3}

*cm*. Both the mean permeability 〈

*ε*〉 and the modulation amplitude 2

*α*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}.

*G*(

*ω*): Δ

*ε*

_{el}/

*ε*

_{el}∼ γ

^{2}/

*k*

_{n}

*L*∼ 1/

*nN*∼ 10

^{-3}- 10

^{-4}, Δ

*θ/θ*∼ 1/

*k*

_{n}

*Lθ*

^{2}, Δ

*θ*∼ 1/

*k*

_{n}

*Lθ*∼ 2 · 10

^{-4}- 10

^{-5}

*rad*, Δ〈

*ε*〉 ∼ 1/(

*ωL/c*) ∼ 10

^{-4}- 10

^{-5}, Δλ

_{0}/λ

_{0}≤ 1/

*nqL*∼ 1/

*nN*= λ

_{0}/

*nL*∼ 10

^{-3}- 10

^{-4}(

*N*=

*L*/λ

_{0}∼ 10

^{3}is the number of modulation periods and

*n*is the index of the resonant PPW).

*d*∼ 1

*cm*, γ = 10, and

*θ*∼ 1/γ = 0.1, and Δ

*θ*∼ 10

^{-3}

*θ*= 10

^{-4}, the emittance of the beam

*∊*could be estimated to be about

*d*· Δ

*θ*∼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*

_{n}

*Lθ*∼ 20

*mm·mrad*in this case.

*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*.

*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

23. W. Becker and J. K. McIver, “Classical theory of stimulated Cerenkov radiation,” Phys. Rev. A **31**, 783–789 (1985). [CrossRef] [PubMed]

*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

9. M. A. Piestrup and P. F. Finman, “The prospects of an X-ray free-electron laser using stimulated resonance transition radiation,” IEEE J. Quantum Electron. **QE 19**, 357–364 (1983). [CrossRef]

10. M. B. Reid and M. A. Piestrup, “Resonance transition radiation X-Ray laser,” IEEE J. Quantum Electron. **QE 27**, 2440–2455 (1991). [CrossRef]

13. M. S. Dubovikov, “Transition radiation and Bragg resonances,” Phys. Rev. A **50**, 2068–2074 (1994). [CrossRef] [PubMed]

*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*.

*ω*> 2.8 · 10

^{15}

*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.

_{0}. The data shown in Fig. 2a were obtained for the frequency

*ω*= 2.8 · 10

^{15}

*s*

^{-1}. The SLL structure and the current density were the same as for Fig. 1.

*n*≫ 1. This result agrees with the criterion for the large-modulation regime of Eq. (6).

_{0}, 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

*ω·α*(

*ω*)≫

*cq*, or |

*ε*

_{1}-

*ε*

_{2}| ≫ λ/λ

_{0}, where λ = 2

*πc/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*∼ 10

^{16}

*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

8. A. I. Artemyev, M. V. Fedorov, J. K. McIver, and E. A. Shapiro, “Nonlinear theory of a free-electron laser exploiting media with periodically modulated refractive index,” IEEE J. Quantum Electron. **QE 34**, 24–31 (1998). [CrossRef]

**QE 19**, 357–364 (1983). [CrossRef]

**QE 19**, 357–364 (1983). [CrossRef]

15. C. I. Pincus, M. A. Piestrup, D. G. Boyers, Q. Li, J. L. Harris, X. K. Maruyama, D. M. Skopik, R. M. Silzer, and H. S. Caplan, “Measurements of X-ray emission from photoabsorption-edge transition radiation,” Phys. Rev. A **43**, 2387–2396 (1991). [PubMed]

26. A. Belliger, “Whats wrong with CIRFEL?”, http://viper.princeton.edu/~prebys/belliger.pdf, December 7, 1997.

*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.

## References and links

1. | K. R. Chen and J. M. Dawson, “Ion-ripple laser,” Phys. Rev. Lett. |

2. | R. N. Agrawal and V. K. Tripathi, “Ion-acoustic-wave pumped free-electron laser,” IEEE Trans. Plasma Science , |

3. | K. Nakajima, M. Kando, T. Kawakubo, T. Nakanishi, and A. Ogata, “A table-top X-ray FEL based on laser wakefield accelerator-undulator system,” Nucl. Instr. & Meth. Phys. Res. A |

4. | V. A. Bazylev, T. J. Schep, and A. V. Tulupov, “Short-Wavelength Free-Electron Lasers with Periodic Plasma Structures,” J. of Phys. D - Appl. Phys. |

5. | V. A. Bazylev, V. Goloviznin, M. M. Pitatelev, A. V. Tulupov, and T. J. Schep, “On the possibility of construction of plasma undulators,” Nucl. Instr. & Meth. Phys. Res. A |

6. | N. I. Karbushev, “Free-electron lasers with static and dynamic plasma wigglers,” Nucl. Instr. & Meth. Phys. Res. A |

7. | M.V. Fedorov and E.A. Shapiro, “Free-electron lasers based on media with periodically modulated refractive index,” Laser Physics |

8. | A. I. Artemyev, M. V. Fedorov, J. K. McIver, and E. A. Shapiro, “Nonlinear theory of a free-electron laser exploiting media with periodically modulated refractive index,” IEEE J. Quantum Electron. |

9. | M. A. Piestrup and P. F. Finman, “The prospects of an X-ray free-electron laser using stimulated resonance transition radiation,” IEEE J. Quantum Electron. |

10. | M. B. Reid and M. A. Piestrup, “Resonance transition radiation X-Ray laser,” IEEE J. Quantum Electron. |

11. | G. Bekefi, J. S. Wurtele, and I. H. Deutsch, “Free-electron laser radiation induced by a periodic dielectric medium,” Phys. Rev. A |

12. | A. E. Kaplan and S. Datta, “Extreme-ultraviolet and x-ray emission and amplification by nonrela-tivistic electron beams traversing a superlattice,” Appl. Phys. Lett. |

13. | M. S. Dubovikov, “Transition radiation and Bragg resonances,” Phys. Rev. A |

14. | C. S. Liu and V. K. Tripathi, “Short-wavelength free-electron laser operation in a periodic dielectric,”, IEEE Trans. Plasma Science |

15. | C. I. Pincus, M. A. Piestrup, D. G. Boyers, Q. Li, J. L. Harris, X. K. Maruyama, D. M. Skopik, R. M. Silzer, and H. S. Caplan, “Measurements of X-ray emission from photoabsorption-edge transition radiation,” Phys. Rev. A |

16. | M. A. Piestrup, D. G. Boyers, C. I. Pincus, J. L. Harris, X. K. Maruyama, J. C. Bergstrom, H. S. Caplan, R. M. Silzer, and D. M. Skopik, “Quasi-monochromatic X-ray source using photoabsorption-edge transition radiation,” Phys. Rev. A |

17. | C. K. Gary, R. H. Pantell, M. Ozcan, M. A. Piestrup, and D. G. Boyers, “Optimization of the channeling radiation source crystal to produce intense quasimonochromatic X rays,” J. Appl. Phys. |

18. | R. B. Fiorito, D. W. Rule, X. K. Maruyama, K. L. DiNova, S. J. Evertson, M. J. Osborne, S. Snyder, H. Rietdyk, M. A. Piestrup, and A. H. Ho, “Observation of Higher-order Parametric X-ray spectra in mosaic graphite and single silicon crystals,” Phys. Rev. Lett. |

19. | A. E. Kaplan, C. T. Law, and P. L. Shkolnikov, ”X-ray narrow-line transition radiation source based on low-energy electron beams traversing a multilayer nanostructure,” Phys. Rev. E |

20. | H.C. Lihn, P. Kung., C. Settakron, H. Wiedemann, D. Bocek, and M. Hernandez, “Observation of stimulated transition radiation,” Phys. Rev. Lett. |

21. | K. F. Casey, C. Yech, and Z. A. Kaprielian, “Cherenkov radiation in inhomogeneous periodic media,” Phys. Rev. B |

22. | B. Pardo and J. M. Andre, “Transition radiation from periodic stratified structure,” Phys. Rev. A |

23. | W. Becker and J. K. McIver, “Classical theory of stimulated Cerenkov radiation,” Phys. Rev. A |

24. | M. V. Fedorov, “Atomic and free electrons in a strong light field,” World Scientific Publishin, Singapore, New Jersey, London, Hong Kong (1997). |

25. | V. V. Apollonov, A. I. Artemyev, M. V. Fedorov, J. K. McIver, and E. A. Shapiro, “Gas-plasma and superlattice free-electron lasers exploiting a medium with periodically modulated refractive index,” Laser and Particle Beams, to appear in (1988). |

26. | A. Belliger, “Whats wrong with CIRFEL?”, http://viper.princeton.edu/~prebys/belliger.pdf, December 7, 1997. |

**OCIS Codes**

(140.2600) Lasers and laser optics : Free-electron lasers (FELs)

(140.3070) Lasers and laser optics : Infrared and far-infrared lasers

(140.3610) Lasers and laser optics : Lasers, ultraviolet

**ToC Category:**

Research Papers

**History**

Original Manuscript: June 11, 1998

Revised Manuscript: March 3, 1998

Published: August 31, 1997

**Citation**

V. V. Apollonov, Alexander Artemyev, Mikhail Fedorov, E. Shapiro, and John McIver, "Free-electron laser exploiting a superlattice-like medium," Opt. Express **3**, 162-170 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-3-5-162

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### References

- K. R. Chen and J. M. Dawson, "Ion-ripple laser," Phys. Rev. Lett. 68, 29-32 (1992). [CrossRef] [PubMed]
- R. N. Agrawal and V. K. Tripathi, "Ion-acoustic-wave pumped free-electron laser," IEEE Trans. Plasma Science 23, 788-791, (1995). [CrossRef]
- K. Nakajima, M. Kando, T. Kawakubo, T. Nakanishi, A. Ogata, "A table-top X-ray FEL based on laser wakefield accelerator-undulator system," Nucl. Instr. & Meth. Phys. Res. A 375, 593-596 (1996). [CrossRef]
- V. A. Bazylev, T. J. Schep, A. V. Tulupov, "Short-Wavelength Free-Electron Lasers with Periodic Plasma Structures," J. of Phys. D - Appl. Phys. 27, 2466-2469 (1994). [CrossRef]
- V. A. Bazylev, V. Goloviznin, M. M. Pitatelev, A. V. Tulupov, T. J. Schep, "On the possibility of construction of plasma undulators," Nucl. Instr. & Meth. Phys. Res. A 358, 433-436 (1995). [CrossRef]
- N. I. Karbushev, "Free-electron lasers with static and dynamic plasma wigglers," Nucl. Instr. & Meth. Phys. Res. A 358, 437-440 (1995). [CrossRef]
- M. V. Fedorov, E. A. Shapiro, "Free-electron lasers based on media with periodically modulated refractive index," Laser Physics 5, 735-739 (1995).
- A. I. Artemyev, M. V. Fedorov, J. K. McIver, E. A. Shapiro, "Nonlinear theory of a free- electron laser exploiting media with periodically modulated refractive index," IEEE J. Quantum Electron. QE 34, 24-31 (1998). [CrossRef]
- M. A. Piestrup, P. F. Finman, "The prospects of an X-ray free-electron laser using stimulated resonance transition radiation," IEEE J. Quantum Electron. QE 19, 357-364 (1983). [CrossRef]
- M. B. Reid, M. A. Piestrup, "Resonance transition radiation X-Ray laser," IEEE J. Quantum Electron. QE 27, 2440-2455 (1991). [CrossRef]
- G. Bekefi, J. S. Wurtele, I. H. Deutsch, "Free-electron laser radiation induced by a periodic dielectric medium," Phys. Rev. A 34, 1228-1236 (1986). [CrossRef] [PubMed]
- A. E. Kaplan, S. Datta, "Extreme-ultraviolet and x-ray emission and amplification by nonrelativistic electron beams traversing a superlattice," Appl. Phys. Lett. 44, 661-663 (1984). [CrossRef]
- M. S. Dubovikov, "Transition radiation and Bragg resonances," Phys. Rev. A 50, 2068-2074 (1994). [CrossRef] [PubMed]
- C. S. Liu, V. K. Tripathi, "Short-wavelength free-electron laser operation in a periodic dielectric,", IEEE Trans. Plasma Science 23, 459-464 (1995). [CrossRef]
- C. I. Pincus, M. A. Piestrup, D. G. Boyers, Q. Li, J. L. Harris, X. K. Maruyama, D. M.Skopik, R. M.Silzer, H. S. Caplan, "Measurements of X-ray emission from photoabsorption-edge transition radiation," Phys. Rev. A 43, 2387-2396 (1991). [PubMed]
- M. A. Piestrup, D. G. Boyers, C. I. Pincus, J. L. Harris, X. K. Maruyama, J. C. Bergstrom, H. S. Caplan, R. M. Silzer, D. M. Skopik, "Quasi-monochromatic X-ray source using photo- absorption-edge transition radiation," Phys. Rev. A 43, 3653-3661 (1991). [CrossRef] [PubMed]
- C. K. Gary, R. H. Pantell, M. Ozcan, M. A. Piestrup, D. G. Boyers, "Optimization of the channeling radiation source crystal to produce intense quasimonochromatic X rays," J. Appl. Phys. 70, 2995-3002 (1991). [CrossRef]
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