Abstract

The power radiation pattern of Smith–Purcell radiation is measured at various latitudes and azimuth angles relative to the electron beam. The experimental data are used to evaluate the various models and the physical mechanisms previously suggested to describe Smith–Purcell radiation. Good agreement is observed between the experimental data and the theoretical curves derived from Van den Berg’s analysis [ J. Opt. Soc. Am. 63, 1588 ( 1973)]. The radiation mechanism proposed by Salisbury [ J. Opt. Soc. Am. 60, 1279 ( 1970)] was analyzed and shown to be too small to account for the measured radiation. The experiment and Van den Berg’s theory predict stronger emission at azimuthal angles off the plane perpendicular to the gratings. This observation leads to conclusions regarding the design of optical cavities for Smith–Purcell free-electron lasers and orotron millimeter-wavelength-radiation tube devices.

© 1984 Optical Society of America

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References

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  1. J. M. J. Madey, Nuovo Cimento B 50, 64 (1979).
    [Crossref]
  2. A. Gover, A. Friedman, S. Ruschin, G. Kuritzki, and A. Yariv. Faculty of Engineering, Quantum Electronics Laboratory Rep. 84/2 (Tel Aviv University, Tel Aviv, Israel, to be published). This paper derives the general relation between the spontaneous emission radiant intensity and the stimulated emission gain of any quasi-free-electron radiation scheme. The report is available on request.
  3. A. Gover and A. Yariv, Physics of Quantum Electronics, S. Jacobs, M. Sargent, and M. Scully, eds. (Addison-Wesley, Reading, Mass., 1977), Vol. 5, p. 197.
  4. A. Gover and Z. Livni, Opt. Commun. 26, 375 (1978).
    [Crossref]
  5. J. Wachtel, J. Appl. Phys. 50, 49 (1979).
    [Crossref]
  6. F. S. Rusin and G. D. Bogomolov, Proc. IEEE 57, 720 (1969).
    [Crossref]
  7. V. K. Korneyekov and V. P. Shestopolov, Sov. Radio Eng. Elect. Phys. 22, 148 (1977).
  8. K. Mizuno, S. Ono, and Y. Shibata, IEEE Trans. Electron Devices ED-20, 749 (1973).
    [Crossref]
  9. R. P. Levitt, D. E. Wortman, and H. Dropkin, IEEE J. Quantum Electron. QE-17, 1333, 1341 (1981).
    [Crossref]
  10. S. J. Smith and E. M. Purcell, Phys. Rev. 92, 1069 (1953).
    [Crossref]
  11. K. Ishiguro and T. Tako, Opt. Acta 8, 25 (1961).
    [Crossref]
  12. W. W. Salisbury, J. Opt. Soc. Am. 60, 1279 (1970).
    [Crossref]
  13. G. Toraldo di Francia, Nuovo Cimento 16, 61 (1960).
  14. P. M. van den Berg, J. Opt. Soc. Am. 63, 1588 (1973).
    [Crossref]
  15. J. P. Bachheimer, Phys. Rev. B 6, 2985 (1972).
    [Crossref]
  16. E. L. Burdette and G. Hughes, Phys. Rev. A 14, 1766 (1976).
    [Crossref]
  17. A. Gover, H. Freund, V. L. Granatstein, J. H. McAdoo, and C. M. Tang, Millimeter Components and Techniques, Part III of Infrared and Millimeter Waves, Vol. 12, K. J. Button, ed. (Academic, New York, to be published). Equation (16) can also be derived by integration over frequencies of the spectral radiant intensity of undulator radiation [e.g., see B. M. Kincaid, J. Appl. Phys. 48, 2684 (1977)].
    [Crossref]
  18. B. Adam and F. Kneubuhl, Appl. Phys. 8, 281 (1975).
    [Crossref]

1981 (1)

R. P. Levitt, D. E. Wortman, and H. Dropkin, IEEE J. Quantum Electron. QE-17, 1333, 1341 (1981).
[Crossref]

1979 (2)

J. M. J. Madey, Nuovo Cimento B 50, 64 (1979).
[Crossref]

J. Wachtel, J. Appl. Phys. 50, 49 (1979).
[Crossref]

1978 (1)

A. Gover and Z. Livni, Opt. Commun. 26, 375 (1978).
[Crossref]

1977 (1)

V. K. Korneyekov and V. P. Shestopolov, Sov. Radio Eng. Elect. Phys. 22, 148 (1977).

1976 (1)

E. L. Burdette and G. Hughes, Phys. Rev. A 14, 1766 (1976).
[Crossref]

1975 (1)

B. Adam and F. Kneubuhl, Appl. Phys. 8, 281 (1975).
[Crossref]

1973 (2)

P. M. van den Berg, J. Opt. Soc. Am. 63, 1588 (1973).
[Crossref]

K. Mizuno, S. Ono, and Y. Shibata, IEEE Trans. Electron Devices ED-20, 749 (1973).
[Crossref]

1972 (1)

J. P. Bachheimer, Phys. Rev. B 6, 2985 (1972).
[Crossref]

1970 (1)

1969 (1)

F. S. Rusin and G. D. Bogomolov, Proc. IEEE 57, 720 (1969).
[Crossref]

1961 (1)

K. Ishiguro and T. Tako, Opt. Acta 8, 25 (1961).
[Crossref]

1960 (1)

G. Toraldo di Francia, Nuovo Cimento 16, 61 (1960).

1953 (1)

S. J. Smith and E. M. Purcell, Phys. Rev. 92, 1069 (1953).
[Crossref]

Adam, B.

B. Adam and F. Kneubuhl, Appl. Phys. 8, 281 (1975).
[Crossref]

Bachheimer, J. P.

J. P. Bachheimer, Phys. Rev. B 6, 2985 (1972).
[Crossref]

Bogomolov, G. D.

F. S. Rusin and G. D. Bogomolov, Proc. IEEE 57, 720 (1969).
[Crossref]

Burdette, E. L.

E. L. Burdette and G. Hughes, Phys. Rev. A 14, 1766 (1976).
[Crossref]

Dropkin, H.

R. P. Levitt, D. E. Wortman, and H. Dropkin, IEEE J. Quantum Electron. QE-17, 1333, 1341 (1981).
[Crossref]

Freund, H.

A. Gover, H. Freund, V. L. Granatstein, J. H. McAdoo, and C. M. Tang, Millimeter Components and Techniques, Part III of Infrared and Millimeter Waves, Vol. 12, K. J. Button, ed. (Academic, New York, to be published). Equation (16) can also be derived by integration over frequencies of the spectral radiant intensity of undulator radiation [e.g., see B. M. Kincaid, J. Appl. Phys. 48, 2684 (1977)].
[Crossref]

Friedman, A.

A. Gover, A. Friedman, S. Ruschin, G. Kuritzki, and A. Yariv. Faculty of Engineering, Quantum Electronics Laboratory Rep. 84/2 (Tel Aviv University, Tel Aviv, Israel, to be published). This paper derives the general relation between the spontaneous emission radiant intensity and the stimulated emission gain of any quasi-free-electron radiation scheme. The report is available on request.

Gover, A.

A. Gover and Z. Livni, Opt. Commun. 26, 375 (1978).
[Crossref]

A. Gover, A. Friedman, S. Ruschin, G. Kuritzki, and A. Yariv. Faculty of Engineering, Quantum Electronics Laboratory Rep. 84/2 (Tel Aviv University, Tel Aviv, Israel, to be published). This paper derives the general relation between the spontaneous emission radiant intensity and the stimulated emission gain of any quasi-free-electron radiation scheme. The report is available on request.

A. Gover and A. Yariv, Physics of Quantum Electronics, S. Jacobs, M. Sargent, and M. Scully, eds. (Addison-Wesley, Reading, Mass., 1977), Vol. 5, p. 197.

A. Gover, H. Freund, V. L. Granatstein, J. H. McAdoo, and C. M. Tang, Millimeter Components and Techniques, Part III of Infrared and Millimeter Waves, Vol. 12, K. J. Button, ed. (Academic, New York, to be published). Equation (16) can also be derived by integration over frequencies of the spectral radiant intensity of undulator radiation [e.g., see B. M. Kincaid, J. Appl. Phys. 48, 2684 (1977)].
[Crossref]

Granatstein, V. L.

A. Gover, H. Freund, V. L. Granatstein, J. H. McAdoo, and C. M. Tang, Millimeter Components and Techniques, Part III of Infrared and Millimeter Waves, Vol. 12, K. J. Button, ed. (Academic, New York, to be published). Equation (16) can also be derived by integration over frequencies of the spectral radiant intensity of undulator radiation [e.g., see B. M. Kincaid, J. Appl. Phys. 48, 2684 (1977)].
[Crossref]

Hughes, G.

E. L. Burdette and G. Hughes, Phys. Rev. A 14, 1766 (1976).
[Crossref]

Ishiguro, K.

K. Ishiguro and T. Tako, Opt. Acta 8, 25 (1961).
[Crossref]

Kneubuhl, F.

B. Adam and F. Kneubuhl, Appl. Phys. 8, 281 (1975).
[Crossref]

Korneyekov, V. K.

V. K. Korneyekov and V. P. Shestopolov, Sov. Radio Eng. Elect. Phys. 22, 148 (1977).

Kuritzki, G.

A. Gover, A. Friedman, S. Ruschin, G. Kuritzki, and A. Yariv. Faculty of Engineering, Quantum Electronics Laboratory Rep. 84/2 (Tel Aviv University, Tel Aviv, Israel, to be published). This paper derives the general relation between the spontaneous emission radiant intensity and the stimulated emission gain of any quasi-free-electron radiation scheme. The report is available on request.

Levitt, R. P.

R. P. Levitt, D. E. Wortman, and H. Dropkin, IEEE J. Quantum Electron. QE-17, 1333, 1341 (1981).
[Crossref]

Livni, Z.

A. Gover and Z. Livni, Opt. Commun. 26, 375 (1978).
[Crossref]

Madey, J. M. J.

J. M. J. Madey, Nuovo Cimento B 50, 64 (1979).
[Crossref]

McAdoo, J. H.

A. Gover, H. Freund, V. L. Granatstein, J. H. McAdoo, and C. M. Tang, Millimeter Components and Techniques, Part III of Infrared and Millimeter Waves, Vol. 12, K. J. Button, ed. (Academic, New York, to be published). Equation (16) can also be derived by integration over frequencies of the spectral radiant intensity of undulator radiation [e.g., see B. M. Kincaid, J. Appl. Phys. 48, 2684 (1977)].
[Crossref]

Mizuno, K.

K. Mizuno, S. Ono, and Y. Shibata, IEEE Trans. Electron Devices ED-20, 749 (1973).
[Crossref]

Ono, S.

K. Mizuno, S. Ono, and Y. Shibata, IEEE Trans. Electron Devices ED-20, 749 (1973).
[Crossref]

Purcell, E. M.

S. J. Smith and E. M. Purcell, Phys. Rev. 92, 1069 (1953).
[Crossref]

Ruschin, S.

A. Gover, A. Friedman, S. Ruschin, G. Kuritzki, and A. Yariv. Faculty of Engineering, Quantum Electronics Laboratory Rep. 84/2 (Tel Aviv University, Tel Aviv, Israel, to be published). This paper derives the general relation between the spontaneous emission radiant intensity and the stimulated emission gain of any quasi-free-electron radiation scheme. The report is available on request.

Rusin, F. S.

F. S. Rusin and G. D. Bogomolov, Proc. IEEE 57, 720 (1969).
[Crossref]

Salisbury, W. W.

Shestopolov, V. P.

V. K. Korneyekov and V. P. Shestopolov, Sov. Radio Eng. Elect. Phys. 22, 148 (1977).

Shibata, Y.

K. Mizuno, S. Ono, and Y. Shibata, IEEE Trans. Electron Devices ED-20, 749 (1973).
[Crossref]

Smith, S. J.

S. J. Smith and E. M. Purcell, Phys. Rev. 92, 1069 (1953).
[Crossref]

Tako, T.

K. Ishiguro and T. Tako, Opt. Acta 8, 25 (1961).
[Crossref]

Tang, C. M.

A. Gover, H. Freund, V. L. Granatstein, J. H. McAdoo, and C. M. Tang, Millimeter Components and Techniques, Part III of Infrared and Millimeter Waves, Vol. 12, K. J. Button, ed. (Academic, New York, to be published). Equation (16) can also be derived by integration over frequencies of the spectral radiant intensity of undulator radiation [e.g., see B. M. Kincaid, J. Appl. Phys. 48, 2684 (1977)].
[Crossref]

Toraldo di Francia, G.

G. Toraldo di Francia, Nuovo Cimento 16, 61 (1960).

van den Berg, P. M.

Wachtel, J.

J. Wachtel, J. Appl. Phys. 50, 49 (1979).
[Crossref]

Wortman, D. E.

R. P. Levitt, D. E. Wortman, and H. Dropkin, IEEE J. Quantum Electron. QE-17, 1333, 1341 (1981).
[Crossref]

Yariv, A.

A. Gover, A. Friedman, S. Ruschin, G. Kuritzki, and A. Yariv. Faculty of Engineering, Quantum Electronics Laboratory Rep. 84/2 (Tel Aviv University, Tel Aviv, Israel, to be published). This paper derives the general relation between the spontaneous emission radiant intensity and the stimulated emission gain of any quasi-free-electron radiation scheme. The report is available on request.

A. Gover and A. Yariv, Physics of Quantum Electronics, S. Jacobs, M. Sargent, and M. Scully, eds. (Addison-Wesley, Reading, Mass., 1977), Vol. 5, p. 197.

Appl. Phys. (1)

B. Adam and F. Kneubuhl, Appl. Phys. 8, 281 (1975).
[Crossref]

IEEE J. Quantum Electron. (1)

R. P. Levitt, D. E. Wortman, and H. Dropkin, IEEE J. Quantum Electron. QE-17, 1333, 1341 (1981).
[Crossref]

IEEE Trans. Electron Devices (1)

K. Mizuno, S. Ono, and Y. Shibata, IEEE Trans. Electron Devices ED-20, 749 (1973).
[Crossref]

J. Appl. Phys. (1)

J. Wachtel, J. Appl. Phys. 50, 49 (1979).
[Crossref]

J. Opt. Soc. Am. (2)

Nuovo Cimento (1)

G. Toraldo di Francia, Nuovo Cimento 16, 61 (1960).

Nuovo Cimento B (1)

J. M. J. Madey, Nuovo Cimento B 50, 64 (1979).
[Crossref]

Opt. Acta (1)

K. Ishiguro and T. Tako, Opt. Acta 8, 25 (1961).
[Crossref]

Opt. Commun. (1)

A. Gover and Z. Livni, Opt. Commun. 26, 375 (1978).
[Crossref]

Phys. Rev. (1)

S. J. Smith and E. M. Purcell, Phys. Rev. 92, 1069 (1953).
[Crossref]

Phys. Rev. A (1)

E. L. Burdette and G. Hughes, Phys. Rev. A 14, 1766 (1976).
[Crossref]

Phys. Rev. B (1)

J. P. Bachheimer, Phys. Rev. B 6, 2985 (1972).
[Crossref]

Proc. IEEE (1)

F. S. Rusin and G. D. Bogomolov, Proc. IEEE 57, 720 (1969).
[Crossref]

Sov. Radio Eng. Elect. Phys. (1)

V. K. Korneyekov and V. P. Shestopolov, Sov. Radio Eng. Elect. Phys. 22, 148 (1977).

Other (3)

A. Gover, A. Friedman, S. Ruschin, G. Kuritzki, and A. Yariv. Faculty of Engineering, Quantum Electronics Laboratory Rep. 84/2 (Tel Aviv University, Tel Aviv, Israel, to be published). This paper derives the general relation between the spontaneous emission radiant intensity and the stimulated emission gain of any quasi-free-electron radiation scheme. The report is available on request.

A. Gover and A. Yariv, Physics of Quantum Electronics, S. Jacobs, M. Sargent, and M. Scully, eds. (Addison-Wesley, Reading, Mass., 1977), Vol. 5, p. 197.

A. Gover, H. Freund, V. L. Granatstein, J. H. McAdoo, and C. M. Tang, Millimeter Components and Techniques, Part III of Infrared and Millimeter Waves, Vol. 12, K. J. Button, ed. (Academic, New York, to be published). Equation (16) can also be derived by integration over frequencies of the spectral radiant intensity of undulator radiation [e.g., see B. M. Kincaid, J. Appl. Phys. 48, 2684 (1977)].
[Crossref]

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Figures (7)

Fig. 1
Fig. 1

Schematic of the Smith–Purcell radiation effect.

Fig. 2
Fig. 2

Definition of the latitude (η) and azimuth (ζ) angles of a Smith–Purcell diffraction order.

Fig. 3
Fig. 3

The Smith–Purcell radiation power detected by a detector as a function of η for various azimuth angles ζ as calculated from Van den Berg’s graphs for grating and beam parameters corresponding to the experiment.

Fig. 4
Fig. 4

Configuration of Salisbury’s radiation mechanism.

Fig. 5
Fig. 5

Measured −2-order Smith–Purcell radiation power as a function of the latitude angle ζ.

Fig. 6
Fig. 6

Measured −2-order Smith–Purcell radiation power as a function of the latitude angle ζ.

Fig. 7
Fig. 7

Proposed optimal resonator configurations for Smith–Purcell FEL’s. (a) Three-mirror open-cavity resonator. (b) Rectangular waveguide or closed-cavity resonator.

Equations (22)

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λ n = - D n ( c / v 0 - sin η ) ,
E ( γ , t ) = ( 2 π 2 ) - 1 Re 0 d ω - d k y E ( x , y , k y , ω ) × exp ( i k y y - i ω t ) ,
E ( x , y , k y , ω ) = n = - E n ( k y , ω ) exp ( i k x n x + i k z n z ) ,
k x n = k x 0 + n 2 π / D ,
k z n = ( k 2 - k y 2 - k x n 2 ) 1 / 2 ,
J ( x , z , ω ) = - e exp ( i ω v 0 z ) δ ( z - z 0 ) i ^ x
k x n = k sin η n ,
k y n = k cos η n sin ζ n ,
k z n = k cos η n cos ζ n .
E y n ϕ n ( k y , ω ) , E x n = - k y k x n k 2 - k y 2 ϕ n , E z n = - k y k z n k 2 - k y 2 ϕ n , H x n = - 0 μ 0 k k z n k 2 - k y 2 ϕ n , H y n = 0 , H z n = 0 μ 0 k k x n k 2 - k y 2 ϕ n .
H y n = ψ n ( k y , ω ) , H x n = - k y k x n k 2 - k y 2 ψ n , H z n = - k y k z n k 2 - k y 2 ψ n , E x n = μ 0 0 k k z n k 2 - k y 2 ψ n , E y n = 0 , E z n = - μ 0 0 k k x n k 2 - k y 2 ψ n .
½ E n × H n * = ½ ( k 2 - k y 2 ) - 1 ω ( 0 ϕ n ϕ n * + μ 0 ψ n ψ n * ) k n .
W = e 2 D 0 radiating waves n - π / 2 π / 2 π / 2 cos 2 η cos 2 ζ [ ( 1 / β ) - sin η ] 3 R n ( ζ , η ) 2 × exp [ - z 0 - z max h int , n ( ζ , η ) ] cos η d η d ζ ,
h int , n D ( β - 1 - sin η ) 4 π n [ β - 2 - 1 + cos 2 η sin 2 ζ ] 1 / 2 = λ n 4 π [ β - 2 - 1 + cos 2 η sin 2 ζ ] 1 / 2 .
I n ( ζ , η ) = e J 0 b L 4 π 0 D n × cos 2 η cos 2 ζ ( β - 1 - sin η ) 2 ( β - 2 - 1 + cos 2 η sin 2 ζ ) 1 / 2 × R n ( ζ , η ) 2 .
E x = 1 2 π μ 0 0 J 0 β 0 D sin ( θ i ) sin ( 2 θ i ) sin ( 4 π cos θ i D x ) ,
E z = 1 2 π μ 0 0 J 0 β 0 D sin ( θ i ) cos ( 2 θ i ) sin ( 4 π cos θ i D x ) ,
B y = 1 2 π c μ 0 0 J 0 D s i n ( θ i ) sin ( 4 π cos θ i D x ) .
I ( ζ , η ) = e I 0 a w 2 k w 2 L 32 π 2 0 γ 2 β 2 × cos 2 ζ ( β - 1 - sin η ) 2 + sin 2 ζ ( β - 1 sin η - 1 ) 2 ( β - 1 - sin η ) 5 ,
γ ( 1 - β 2 ) - 1 / 2 , a w e k w m c ( B w + 1 v 0 E w ) , = e D J 0 μ 0 / 0 sin θ i 2 π k w m c 2 ( 1 + 1 / β 0 2 )
α p q = 2 π 2 k 2 [ p 2 w y 3 Re ( 1 ν 2 - 1 ) + q 2 w z 3 Re ( ν 2 ν 2 - 1 ) ] ,
α p q = 2 cos 2 η [ sin 2 ζ w y Re ( 1 ν 2 - 1 ) + cos 2 ζ w z Re ( ν 2 ν 2 - 1 ) ] .

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