Abstract

We present a new and greatly simplified derivation of the tuning relation for a Čerenkov free-electron laser. The laser uses an electron beam to excite coherent light in a planar waveguide. The geometric (or zig-zag) theory of waveguides is used to model the set of guided modes, while properties of hte Čerenkov radiation dictate which modes are excited. The results obtained are identical to earlier research based on the formal application of Maxwell’s equations. However, the geometric approach adds physical insight into the Čerenkov free-electron laser design problem. The dependence of output wavelength on electron-beam voltage and resonator thickness is plotted. A sample laser is designed for operation from 150 to 825 μm.

© 1992 Optical Society of America

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References

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  1. E. Fisch, J. Walsh, Appl. Phys. Lett. 60, 1298 (1992).
    [CrossRef]
  2. J. E. Walsh, in Laser Handbook, W. B. Colson, C. Pelligrini, A. Renieri, eds. (Elsevier, Amsterdam, 1990), Vol. 6, p. 485and references therein.
  3. H. Kogelnik, in Integrated Optics, T. Tamir, ed. (Springer-Verlag, Berlin, 1975), pp. 15–79.
  4. The standard convention for mode labeling is employed: the first index refers to the number of half-wavelength variations in y (width), and the second index refers to the variations in x (height). Since the derivation imposed no restrictions on the width, this first index must be zero.
  5. We do not include the details of FEL gain and its impact on the resonator length and electron-beam parameters (current density, energy spread). Such considerations are beyond the scope of this Letter and have little impact on the wavelength tuning. They are, however, essential to the design of a feasible laser source.

1992 (1)

E. Fisch, J. Walsh, Appl. Phys. Lett. 60, 1298 (1992).
[CrossRef]

Fisch, E.

E. Fisch, J. Walsh, Appl. Phys. Lett. 60, 1298 (1992).
[CrossRef]

Kogelnik, H.

H. Kogelnik, in Integrated Optics, T. Tamir, ed. (Springer-Verlag, Berlin, 1975), pp. 15–79.

Walsh, J.

E. Fisch, J. Walsh, Appl. Phys. Lett. 60, 1298 (1992).
[CrossRef]

Walsh, J. E.

J. E. Walsh, in Laser Handbook, W. B. Colson, C. Pelligrini, A. Renieri, eds. (Elsevier, Amsterdam, 1990), Vol. 6, p. 485and references therein.

Appl. Phys. Lett. (1)

E. Fisch, J. Walsh, Appl. Phys. Lett. 60, 1298 (1992).
[CrossRef]

Other (4)

J. E. Walsh, in Laser Handbook, W. B. Colson, C. Pelligrini, A. Renieri, eds. (Elsevier, Amsterdam, 1990), Vol. 6, p. 485and references therein.

H. Kogelnik, in Integrated Optics, T. Tamir, ed. (Springer-Verlag, Berlin, 1975), pp. 15–79.

The standard convention for mode labeling is employed: the first index refers to the number of half-wavelength variations in y (width), and the second index refers to the variations in x (height). Since the derivation imposed no restrictions on the width, this first index must be zero.

We do not include the details of FEL gain and its impact on the resonator length and electron-beam parameters (current density, energy spread). Such considerations are beyond the scope of this Letter and have little impact on the wavelength tuning. They are, however, essential to the design of a feasible laser source.

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Figures (3)

Fig. 1
Fig. 1

Schematic view of the Čerenkov FEL. The presentation is intentionally vague because the design specifics (electron-beam voltage, resonator dimensions) are dictated by the chosen region of operation.

Fig. 2
Fig. 2

Geometric view of guided modes in the Čerenkov FEL. A planar resonator of index n1 and thickness d is suspended in vacuum. An electron beam propagates parallel to the surface of the resonator, along the z axis. A guided mode is represented by a ray that has total internal reflection at the dielectric–air interface. The angle of propagation is dictated by the angle of Čerenkov emission and is its complement. The detail illustrates the Goos–Hänchen shift. This is a geometric interpretation of the phase shift that occurs on reflection at an interface.

Fig. 3
Fig. 3

Normalized wavelength λ/d plotted versus the electron-beam voltage for quartz, sapphire, and germanium planar resonators (n1 = 2, 3.1, and 4, respectively) suspended in vacuum (n2 = 1). The output wavelength is shorter at lower energies, where the steepest part of the tuning curve is found. There is no Čerenkov emission at voltages that do not accelerate the electrons to superluminal velocities. Higher-order modes are similar but are shifted to shorter wavelengths.

Equations (9)

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δ TM = tan - 1 { ( n 1 n 2 ) 2 [ ( n 1 sin θ 1 ) 2 - n 2 2 ] 1 / 2 n 1 cos θ 1 } .
2 ω d c n 1 cos θ 1 - 4 δ TM = 2 M π ,
ω d c = 1 n 1 cos θ 1 × ( M π + 2 tan - 1 { n 1 cos θ 1 [ ( n 1 sin θ 1 ) 2 - 1 ] 1 / 2 } ) .
i q = ω c ( 1 - n 1 2 sin 2 θ 1 ) 1 / 2 .
θ c = cos - 1 ( 1 β n 1 )
ω d c = β ( n 1 2 β 2 - 1 ) 1 / 2 × ( M π + 2 tan - 1 { n 1 2 [ 1 - β 2 ( n 1 2 β 2 - 1 ) 1 / 2 ] 1 / 2 } ) .
σ b λ β γ 1.
V ( keV ) = [ ( 1 1 - β 2 ) 1 / 2 - 1 ] 511.
V T ( keV ) = [ ( n 1 2 n 1 2 - 1 ) 1 / 2 - 1 ] 511.

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