Abstract

The equivalent Fresnel numbers of two unstable resonators of current interest for application in free-electron lasers are negative. For a resonator of this class, the equivalent collimated round-trip propagation of Siegman’s canonical formulation is a time-reversed propagation, and the modes and eigenvalues are the complex conjugates of those corresponding to a resonator having a positive equivalent Fresnel number with the same magnitude.

© 1986 Optical Society of America

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References

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  1. A. E. Siegman, IEEE J. Quantum Electron. QE-12, 35 (1976).
    [CrossRef]
  2. A. H. Paxton, T. C. Salvi, Opt. Commun. 26, 305 (1978); T. C. Salvi, A. H. Paxton, Appl. Opt. 18, 2098 (1979).
    [CrossRef]
  3. M. Piche, P. Lavigne, F. Martin, P.A. Belanger, Appl. Opt. 22, 1999 (1983).
    [CrossRef] [PubMed]
  4. K. E. Oughstun, P. A. Slaymaker, K. A. Bush, IEEE J. Quantum Electron. QE-19, 1558 (1983).
    [CrossRef]
  5. P. G. Gobbi, G. C. Reali, Opt. Commun. 52, 195 (1984); P. G. Gobbi, S. Morosi, G. C. Reali, A. S. Zarkasi, Appl. Opt. 24, 26 (1985).
    [CrossRef] [PubMed]
  6. A. H. Paxton, L. M. Gutheinz, Electro-Opt. Syst. Design 14, 31 (1982).
  7. G. T. Moore, Proc. Soc. Photo-Opt. Instrum. Eng. 453, 255 (1983).
  8. S. A. Mani, J. H. Hammond, in Proceedings of the International Conference on Lasers 81, C. B. Collins, ed. (STS, McLean, Va., 1981), p. 586.
  9. P. R. Akkapeddi, P. Glenn, A. Fuschetto, Q. Appert, V. K. Viswanathan, presented at Southwest Conference on Optics, March 4–8, 1985, Albuquerque, N.M.
  10. A. Sarnik, P. Glenn, presented at Southwest Conference on Optics, March 4–8, 1985, Albuquerque, N.M.
  11. If, however, the resonator mode has a focus between the image location and the outcoupling aperture, an image location before the outcoupling aperture corresponds to a negative equivalent Fresnel number. A similarity transformation of the round-trip propagation matrix will move the image location past the outcoupling aperture, with no intervening focus. The appropriate similarity transformation corresponds to the placement of a lens with some focal length f1 immediately in front of the outcoupling aperture and a lens with focal length −f1 immediately behind the outcoupling aperture.
  12. E. A. Sziklas, A. E. Siegman, Appl. Opt. 14, 1874 (1975).
    [CrossRef] [PubMed]

1984 (1)

P. G. Gobbi, G. C. Reali, Opt. Commun. 52, 195 (1984); P. G. Gobbi, S. Morosi, G. C. Reali, A. S. Zarkasi, Appl. Opt. 24, 26 (1985).
[CrossRef] [PubMed]

1983 (3)

G. T. Moore, Proc. Soc. Photo-Opt. Instrum. Eng. 453, 255 (1983).

K. E. Oughstun, P. A. Slaymaker, K. A. Bush, IEEE J. Quantum Electron. QE-19, 1558 (1983).
[CrossRef]

M. Piche, P. Lavigne, F. Martin, P.A. Belanger, Appl. Opt. 22, 1999 (1983).
[CrossRef] [PubMed]

1982 (1)

A. H. Paxton, L. M. Gutheinz, Electro-Opt. Syst. Design 14, 31 (1982).

1978 (1)

A. H. Paxton, T. C. Salvi, Opt. Commun. 26, 305 (1978); T. C. Salvi, A. H. Paxton, Appl. Opt. 18, 2098 (1979).
[CrossRef]

1976 (1)

A. E. Siegman, IEEE J. Quantum Electron. QE-12, 35 (1976).
[CrossRef]

1975 (1)

Akkapeddi, P. R.

P. R. Akkapeddi, P. Glenn, A. Fuschetto, Q. Appert, V. K. Viswanathan, presented at Southwest Conference on Optics, March 4–8, 1985, Albuquerque, N.M.

Appert, Q.

P. R. Akkapeddi, P. Glenn, A. Fuschetto, Q. Appert, V. K. Viswanathan, presented at Southwest Conference on Optics, March 4–8, 1985, Albuquerque, N.M.

Belanger, P.A.

Bush, K. A.

K. E. Oughstun, P. A. Slaymaker, K. A. Bush, IEEE J. Quantum Electron. QE-19, 1558 (1983).
[CrossRef]

Fuschetto, A.

P. R. Akkapeddi, P. Glenn, A. Fuschetto, Q. Appert, V. K. Viswanathan, presented at Southwest Conference on Optics, March 4–8, 1985, Albuquerque, N.M.

Glenn, P.

P. R. Akkapeddi, P. Glenn, A. Fuschetto, Q. Appert, V. K. Viswanathan, presented at Southwest Conference on Optics, March 4–8, 1985, Albuquerque, N.M.

A. Sarnik, P. Glenn, presented at Southwest Conference on Optics, March 4–8, 1985, Albuquerque, N.M.

Gobbi, P. G.

P. G. Gobbi, G. C. Reali, Opt. Commun. 52, 195 (1984); P. G. Gobbi, S. Morosi, G. C. Reali, A. S. Zarkasi, Appl. Opt. 24, 26 (1985).
[CrossRef] [PubMed]

Gutheinz, L. M.

A. H. Paxton, L. M. Gutheinz, Electro-Opt. Syst. Design 14, 31 (1982).

Hammond, J. H.

S. A. Mani, J. H. Hammond, in Proceedings of the International Conference on Lasers 81, C. B. Collins, ed. (STS, McLean, Va., 1981), p. 586.

Lavigne, P.

Mani, S. A.

S. A. Mani, J. H. Hammond, in Proceedings of the International Conference on Lasers 81, C. B. Collins, ed. (STS, McLean, Va., 1981), p. 586.

Martin, F.

Moore, G. T.

G. T. Moore, Proc. Soc. Photo-Opt. Instrum. Eng. 453, 255 (1983).

Oughstun, K. E.

K. E. Oughstun, P. A. Slaymaker, K. A. Bush, IEEE J. Quantum Electron. QE-19, 1558 (1983).
[CrossRef]

Paxton, A. H.

A. H. Paxton, L. M. Gutheinz, Electro-Opt. Syst. Design 14, 31 (1982).

A. H. Paxton, T. C. Salvi, Opt. Commun. 26, 305 (1978); T. C. Salvi, A. H. Paxton, Appl. Opt. 18, 2098 (1979).
[CrossRef]

Piche, M.

Reali, G. C.

P. G. Gobbi, G. C. Reali, Opt. Commun. 52, 195 (1984); P. G. Gobbi, S. Morosi, G. C. Reali, A. S. Zarkasi, Appl. Opt. 24, 26 (1985).
[CrossRef] [PubMed]

Salvi, T. C.

A. H. Paxton, T. C. Salvi, Opt. Commun. 26, 305 (1978); T. C. Salvi, A. H. Paxton, Appl. Opt. 18, 2098 (1979).
[CrossRef]

Sarnik, A.

A. Sarnik, P. Glenn, presented at Southwest Conference on Optics, March 4–8, 1985, Albuquerque, N.M.

Siegman, A. E.

A. E. Siegman, IEEE J. Quantum Electron. QE-12, 35 (1976).
[CrossRef]

E. A. Sziklas, A. E. Siegman, Appl. Opt. 14, 1874 (1975).
[CrossRef] [PubMed]

Slaymaker, P. A.

K. E. Oughstun, P. A. Slaymaker, K. A. Bush, IEEE J. Quantum Electron. QE-19, 1558 (1983).
[CrossRef]

Sziklas, E. A.

Viswanathan, V. K.

P. R. Akkapeddi, P. Glenn, A. Fuschetto, Q. Appert, V. K. Viswanathan, presented at Southwest Conference on Optics, March 4–8, 1985, Albuquerque, N.M.

Appl. Opt. (2)

Electro-Opt. Syst. Design (1)

A. H. Paxton, L. M. Gutheinz, Electro-Opt. Syst. Design 14, 31 (1982).

IEEE J. Quantum Electron. (2)

K. E. Oughstun, P. A. Slaymaker, K. A. Bush, IEEE J. Quantum Electron. QE-19, 1558 (1983).
[CrossRef]

A. E. Siegman, IEEE J. Quantum Electron. QE-12, 35 (1976).
[CrossRef]

Opt. Commun. (2)

A. H. Paxton, T. C. Salvi, Opt. Commun. 26, 305 (1978); T. C. Salvi, A. H. Paxton, Appl. Opt. 18, 2098 (1979).
[CrossRef]

P. G. Gobbi, G. C. Reali, Opt. Commun. 52, 195 (1984); P. G. Gobbi, S. Morosi, G. C. Reali, A. S. Zarkasi, Appl. Opt. 24, 26 (1985).
[CrossRef] [PubMed]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

G. T. Moore, Proc. Soc. Photo-Opt. Instrum. Eng. 453, 255 (1983).

Other (4)

S. A. Mani, J. H. Hammond, in Proceedings of the International Conference on Lasers 81, C. B. Collins, ed. (STS, McLean, Va., 1981), p. 586.

P. R. Akkapeddi, P. Glenn, A. Fuschetto, Q. Appert, V. K. Viswanathan, presented at Southwest Conference on Optics, March 4–8, 1985, Albuquerque, N.M.

A. Sarnik, P. Glenn, presented at Southwest Conference on Optics, March 4–8, 1985, Albuquerque, N.M.

If, however, the resonator mode has a focus between the image location and the outcoupling aperture, an image location before the outcoupling aperture corresponds to a negative equivalent Fresnel number. A similarity transformation of the round-trip propagation matrix will move the image location past the outcoupling aperture, with no intervening focus. The appropriate similarity transformation corresponds to the placement of a lens with some focal length f1 immediately in front of the outcoupling aperture and a lens with focal length −f1 immediately behind the outcoupling aperture.

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Figures (5)

Fig. 1
Fig. 1

The canonical model for one complete round trip in an arbitrary unstable resonator (taken from Ref. 1).

Fig. 2
Fig. 2

Section of periodic lens train representing resonator with negative equivalent Fresnel number.

Fig. 3
Fig. 3

The canonical method for one complete round trip in an unstable resonator with negative equivalent Fresnel number.

Fig. 4
Fig. 4

Geometric edges of the mode of the double-negative-branch unstable resonator (after Ref. 7).

Fig. 5
Fig. 5

Unstable ring resonator with grazing-incidence beam expanders.

Equations (9)

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N C = M a 2 / B λ
N eq = M 2 - 1 2 M 2 N C ,
U ( y ) = i N C - 1 1 M - 1 / 2 U 0 ( y 0 / M ) × exp [ - i π N C ( y - y 0 ) 2 ] d y 0 .
U ( y ) = γ M M - 1 / 2 i N C - 1 / M 1 / M U ( y ) × exp [ - i π N C ( y - M y ) 2 ] d y ,
T = [ 1 - d / f 2 d - d 2 / f d / f 2 - 2 / f 1 + d 2 / f 2 - 3 d / f ] = [ A B C D ]
N eq = a 2 ( M 2 - 1 ) 2 M B λ = a 2 ( M 2 - 1 ) 2 M λ ( 2 d - d 2 / f ) ,
M = [ ( g 2 - 1 ) 1 / 2 - g ] 2 ,             g = 1 - d / 2 f .
N C = ( i 1 N i ) - 1 ,
N C = [ λ ( L 1 a 2 + L 2 a b - L 3 b 2 + L 4 M a b + L 5 M 2 a 2 ) ] - 1 ,

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