Abstract

A diffraction formula for annular beam propagation is suggested. Significant computational savings are obtained without restriction to low azimuthal mode orders. Azimuthal mode discrimination is shown to exist in stable annular resonators. High-order azimuthal modes can suffer low diffraction losses with certain mirror parameters. These high-order modes are identified with azimuthal revolving rays that satisfy known geometric relations for multipass resonators.

© 1993 Optical Society of America

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  1. N. Hodgson, Q. Lü, S. Dong, B. Eppich, U. Wittrock, “High power solid state lasers in rod-, slab- and tube geometry,” Laser Optoelektron. 23, 82–92 (1991).
  2. P. Burlamacchi, R. Pratesi, “High-efficiency coaxial wave-guide dye laser with internal excitation,”; Appl. Phys. Lett. 23, 475–476 (1973).
    [CrossRef]
  3. U. Habich, A. Bauer, P. Loosen, H.-D. Plum, “Resonators for coaxial show-flow CO2 lasers,” in Eighth International Symposium on Gas Flow and Chemical Lasers, C. Domingo, J. M. Orza, eds., Proc. Soc. Photo-Opt. Instrum. Eng., 1397, 383–386 (1991).
  4. T. R. Ferguson, M. E. Smithers, “Toric unstable resonators,” Appl. Opt. 23, 2122–2126 (1984).
    [CrossRef] [PubMed]
  5. M. Morin, P.-A. Bélanger, “Diffractive analysis of annular resonators,” Appl. Opt. 31, 1942–1947 (1992).
    [CrossRef] [PubMed]
  6. J. W. Ogland, “Mirror systems for uniform beam transformation in high-power annular lasers,” Appl. Opt. 17, 2917–2924 (1978).
    [CrossRef] [PubMed]
  7. D. Ehrlichmann, U. Habich, H.-D. Plum, “Simple annular resonators with toric and helical mirrors,” in Laser Resonators and Coherent Optics: Modeling, Technology, and Applications, A. Bhowmik, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1868 (to be published).
  8. J. G. Xin, D. R. Hall, “Compact, multipass, single transverse mode CO2 laser,” Appl. Phys. Lett. 51, 469–471 (1987).
    [CrossRef]
  9. W. D. Murphy, M. L. Bernabe, “Numerical procedures for solving nonsymmetric eigenvalue problems associated with optical resonators,” Appl. Opt. 17, 2358–2365 (1978).
    [CrossRef] [PubMed]
  10. Y. Takada, H. Saito, T. Fujioka, “Eigenmode of an annular stable resonator,”; IEEE J. Quantum Electron. 24, 11–12 (1988).
    [CrossRef]
  11. L. W. Casperson, “Cylindrical laser resonators,” J. Opt. Soc. Am. 63, 25–29 (1973).
    [CrossRef]
  12. V. L. Gamiz, “Propagation of thin annular scalar field distributions,” in New Methods for Optical, Quasi-Optical, Acoustic, and Electromagnetic Synthesis, U. R. Stone, ed. Proc. Soc. Photo-Opt. Instrum. Eng.294, 13–18 (1981).
  13. U. Wittrock, B. Eppich, H. Weber, “Beam quality of the 1-kW inside-pumped Nd:YAG tube laser,” in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 94.
  14. A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell. Syst. Tech. J. 40, 453–488 (1961).
  15. H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
    [CrossRef]
  16. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1989), Chap. 12, pp. 381–453.
  17. Ref. 16, Chap. 6.4, pp. 170-176.
  18. M. Abromowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 9, pp. 355–389.
  19. C. M. Bender, S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978), Chaps. 3.7 and 3.8, pp. 107–136.
  20. U. Habich, “Diffusiongekühlte CO2-Laser für die industrielle Materialbearbeitung,” Ph.D. dissertation (Verlag der Augusti-nus-Buchhandlung, Aachen, Germany, 1992), pp. 69–70.

1992 (1)

1991 (1)

N. Hodgson, Q. Lü, S. Dong, B. Eppich, U. Wittrock, “High power solid state lasers in rod-, slab- and tube geometry,” Laser Optoelektron. 23, 82–92 (1991).

1988 (1)

Y. Takada, H. Saito, T. Fujioka, “Eigenmode of an annular stable resonator,”; IEEE J. Quantum Electron. 24, 11–12 (1988).
[CrossRef]

1987 (1)

J. G. Xin, D. R. Hall, “Compact, multipass, single transverse mode CO2 laser,” Appl. Phys. Lett. 51, 469–471 (1987).
[CrossRef]

1984 (1)

1978 (2)

1973 (2)

P. Burlamacchi, R. Pratesi, “High-efficiency coaxial wave-guide dye laser with internal excitation,”; Appl. Phys. Lett. 23, 475–476 (1973).
[CrossRef]

L. W. Casperson, “Cylindrical laser resonators,” J. Opt. Soc. Am. 63, 25–29 (1973).
[CrossRef]

1966 (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

1961 (1)

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell. Syst. Tech. J. 40, 453–488 (1961).

Abromowitz, M.

M. Abromowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 9, pp. 355–389.

Bauer, A.

U. Habich, A. Bauer, P. Loosen, H.-D. Plum, “Resonators for coaxial show-flow CO2 lasers,” in Eighth International Symposium on Gas Flow and Chemical Lasers, C. Domingo, J. M. Orza, eds., Proc. Soc. Photo-Opt. Instrum. Eng., 1397, 383–386 (1991).

Bélanger, P.-A.

Bender, C. M.

C. M. Bender, S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978), Chaps. 3.7 and 3.8, pp. 107–136.

Bernabe, M. L.

Burlamacchi, P.

P. Burlamacchi, R. Pratesi, “High-efficiency coaxial wave-guide dye laser with internal excitation,”; Appl. Phys. Lett. 23, 475–476 (1973).
[CrossRef]

Casperson, L. W.

Dong, S.

N. Hodgson, Q. Lü, S. Dong, B. Eppich, U. Wittrock, “High power solid state lasers in rod-, slab- and tube geometry,” Laser Optoelektron. 23, 82–92 (1991).

Ehrlichmann, D.

D. Ehrlichmann, U. Habich, H.-D. Plum, “Simple annular resonators with toric and helical mirrors,” in Laser Resonators and Coherent Optics: Modeling, Technology, and Applications, A. Bhowmik, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1868 (to be published).

Eppich, B.

N. Hodgson, Q. Lü, S. Dong, B. Eppich, U. Wittrock, “High power solid state lasers in rod-, slab- and tube geometry,” Laser Optoelektron. 23, 82–92 (1991).

U. Wittrock, B. Eppich, H. Weber, “Beam quality of the 1-kW inside-pumped Nd:YAG tube laser,” in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 94.

Ferguson, T. R.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1989), Chap. 12, pp. 381–453.

Fox, A. G.

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell. Syst. Tech. J. 40, 453–488 (1961).

Fujioka, T.

Y. Takada, H. Saito, T. Fujioka, “Eigenmode of an annular stable resonator,”; IEEE J. Quantum Electron. 24, 11–12 (1988).
[CrossRef]

Gamiz, V. L.

V. L. Gamiz, “Propagation of thin annular scalar field distributions,” in New Methods for Optical, Quasi-Optical, Acoustic, and Electromagnetic Synthesis, U. R. Stone, ed. Proc. Soc. Photo-Opt. Instrum. Eng.294, 13–18 (1981).

Habich, U.

U. Habich, “Diffusiongekühlte CO2-Laser für die industrielle Materialbearbeitung,” Ph.D. dissertation (Verlag der Augusti-nus-Buchhandlung, Aachen, Germany, 1992), pp. 69–70.

U. Habich, A. Bauer, P. Loosen, H.-D. Plum, “Resonators for coaxial show-flow CO2 lasers,” in Eighth International Symposium on Gas Flow and Chemical Lasers, C. Domingo, J. M. Orza, eds., Proc. Soc. Photo-Opt. Instrum. Eng., 1397, 383–386 (1991).

D. Ehrlichmann, U. Habich, H.-D. Plum, “Simple annular resonators with toric and helical mirrors,” in Laser Resonators and Coherent Optics: Modeling, Technology, and Applications, A. Bhowmik, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1868 (to be published).

Hall, D. R.

J. G. Xin, D. R. Hall, “Compact, multipass, single transverse mode CO2 laser,” Appl. Phys. Lett. 51, 469–471 (1987).
[CrossRef]

Hodgson, N.

N. Hodgson, Q. Lü, S. Dong, B. Eppich, U. Wittrock, “High power solid state lasers in rod-, slab- and tube geometry,” Laser Optoelektron. 23, 82–92 (1991).

Kogelnik, H.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Li, T.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell. Syst. Tech. J. 40, 453–488 (1961).

Loosen, P.

U. Habich, A. Bauer, P. Loosen, H.-D. Plum, “Resonators for coaxial show-flow CO2 lasers,” in Eighth International Symposium on Gas Flow and Chemical Lasers, C. Domingo, J. M. Orza, eds., Proc. Soc. Photo-Opt. Instrum. Eng., 1397, 383–386 (1991).

Lü, Q.

N. Hodgson, Q. Lü, S. Dong, B. Eppich, U. Wittrock, “High power solid state lasers in rod-, slab- and tube geometry,” Laser Optoelektron. 23, 82–92 (1991).

Morin, M.

Murphy, W. D.

Ogland, J. W.

Orszag, S. A.

C. M. Bender, S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978), Chaps. 3.7 and 3.8, pp. 107–136.

Plum, H.-D.

D. Ehrlichmann, U. Habich, H.-D. Plum, “Simple annular resonators with toric and helical mirrors,” in Laser Resonators and Coherent Optics: Modeling, Technology, and Applications, A. Bhowmik, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1868 (to be published).

U. Habich, A. Bauer, P. Loosen, H.-D. Plum, “Resonators for coaxial show-flow CO2 lasers,” in Eighth International Symposium on Gas Flow and Chemical Lasers, C. Domingo, J. M. Orza, eds., Proc. Soc. Photo-Opt. Instrum. Eng., 1397, 383–386 (1991).

Pratesi, R.

P. Burlamacchi, R. Pratesi, “High-efficiency coaxial wave-guide dye laser with internal excitation,”; Appl. Phys. Lett. 23, 475–476 (1973).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1989), Chap. 12, pp. 381–453.

Saito, H.

Y. Takada, H. Saito, T. Fujioka, “Eigenmode of an annular stable resonator,”; IEEE J. Quantum Electron. 24, 11–12 (1988).
[CrossRef]

Smithers, M. E.

Stegun, I. A.

M. Abromowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 9, pp. 355–389.

Takada, Y.

Y. Takada, H. Saito, T. Fujioka, “Eigenmode of an annular stable resonator,”; IEEE J. Quantum Electron. 24, 11–12 (1988).
[CrossRef]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1989), Chap. 12, pp. 381–453.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1989), Chap. 12, pp. 381–453.

Weber, H.

U. Wittrock, B. Eppich, H. Weber, “Beam quality of the 1-kW inside-pumped Nd:YAG tube laser,” in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 94.

Wittrock, U.

N. Hodgson, Q. Lü, S. Dong, B. Eppich, U. Wittrock, “High power solid state lasers in rod-, slab- and tube geometry,” Laser Optoelektron. 23, 82–92 (1991).

U. Wittrock, B. Eppich, H. Weber, “Beam quality of the 1-kW inside-pumped Nd:YAG tube laser,” in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 94.

Xin, J. G.

J. G. Xin, D. R. Hall, “Compact, multipass, single transverse mode CO2 laser,” Appl. Phys. Lett. 51, 469–471 (1987).
[CrossRef]

Appl. Opt. (4)

Appl. Phys. Lett. (2)

P. Burlamacchi, R. Pratesi, “High-efficiency coaxial wave-guide dye laser with internal excitation,”; Appl. Phys. Lett. 23, 475–476 (1973).
[CrossRef]

J. G. Xin, D. R. Hall, “Compact, multipass, single transverse mode CO2 laser,” Appl. Phys. Lett. 51, 469–471 (1987).
[CrossRef]

Bell. Syst. Tech. J. (1)

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell. Syst. Tech. J. 40, 453–488 (1961).

IEEE J. Quantum Electron. (1)

Y. Takada, H. Saito, T. Fujioka, “Eigenmode of an annular stable resonator,”; IEEE J. Quantum Electron. 24, 11–12 (1988).
[CrossRef]

J. Opt. Soc. Am. (1)

Laser Optoelektron (1)

N. Hodgson, Q. Lü, S. Dong, B. Eppich, U. Wittrock, “High power solid state lasers in rod-, slab- and tube geometry,” Laser Optoelektron. 23, 82–92 (1991).

Proc. IEEE (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Other (9)

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1989), Chap. 12, pp. 381–453.

Ref. 16, Chap. 6.4, pp. 170-176.

M. Abromowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 9, pp. 355–389.

C. M. Bender, S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978), Chaps. 3.7 and 3.8, pp. 107–136.

U. Habich, “Diffusiongekühlte CO2-Laser für die industrielle Materialbearbeitung,” Ph.D. dissertation (Verlag der Augusti-nus-Buchhandlung, Aachen, Germany, 1992), pp. 69–70.

V. L. Gamiz, “Propagation of thin annular scalar field distributions,” in New Methods for Optical, Quasi-Optical, Acoustic, and Electromagnetic Synthesis, U. R. Stone, ed. Proc. Soc. Photo-Opt. Instrum. Eng.294, 13–18 (1981).

U. Wittrock, B. Eppich, H. Weber, “Beam quality of the 1-kW inside-pumped Nd:YAG tube laser,” in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 94.

U. Habich, A. Bauer, P. Loosen, H.-D. Plum, “Resonators for coaxial show-flow CO2 lasers,” in Eighth International Symposium on Gas Flow and Chemical Lasers, C. Domingo, J. M. Orza, eds., Proc. Soc. Photo-Opt. Instrum. Eng., 1397, 383–386 (1991).

D. Ehrlichmann, U. Habich, H.-D. Plum, “Simple annular resonators with toric and helical mirrors,” in Laser Resonators and Coherent Optics: Modeling, Technology, and Applications, A. Bhowmik, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1868 (to be published).

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Figures (7)

Fig. 1
Fig. 1

Stable annular resonator. The center of the annular gap is r ¯ = ( b + a ) / 2.

Fig. 2
Fig. 2

Asymptotic form and exact values of Hl(Z), l = 500, imaginary part. Hl(Z) diverges for Z → 0 (a). Errors in frequency and amplitude increase with decreasing Z (a–c). The asymptotic form reveals phase errors even for Zl (c).

Fig. 3
Fig. 3

Diffraction losses versus azimuthal order l. The vertex of curvature is centered within the annular gap, ( r c = r ¯ ). The diffraction losses increase with l and are nearly degenerate for low azimuthal orders. High-order mode suppression improves for g = 1 − L/R → 1.

Fig. 4
Fig. 4

Intensity distributions for different azimuthal mode orders l. The center of intensity shifts outward with increasing azimuthal order and gives rise to increasing losses.

Fig. 5
Fig. 5

Diffraction losses for different vertices of curvature. Inward-shifted vertices of curvature lead to low-loss high-order azimuthal modes. The arrows represent the loss minimum positions according to geometrical considerations. The comparison for different curvatures g = 0, shown in (a), and g = 0.75, shown in (b), shows that the width of the minimum increases with curvature.

Fig. 6
Fig. 6

Revolving ray. The ray revolves with an azimuthal inclination θ, and the azimuthal advance per reflection is approximately ϕ ≈ Lθ/r.

Fig. 7
Fig. 7

Azimuthal mode order of centered (low-loss) beam versus shift of vertex of curvature. The mode order l is mostly affected by the vertex shift r ¯ r c for g = 0.

Equations (16)

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γ E = T E ,
E ( r , ϕ , z ) = l = + U l ( r , z ) exp ( il ϕ ) ,
U l ( r 2 , L ) = i l + 1 k L 1 exp ( ikL ) a b U l ( r 1 , 0 ) J l ( k r 1 r 2 / L ) × exp [ i k ( r 1 2 + r 2 2 ) / ( 2 L ) ] r 1 d r 1 ,
J l ( k b r 1 / L ) cos ( k b r 1 / L ) , exp [ i k r 1 2 / ( 2 L ) ] exp { i k [ b 2 + b ( r 1 b ) + ] / ( 2 L ) } exp [ ikb r 1 / ( 2 L ) ] .
J l ( Z ) [ 2 / ( π Z ) ] 1 / 2 cos ( Z π l / 2 π / 4 ) .
U l ( r 2 , L ) = [ i / λ L ] 1 / 2 exp ( ikL ) a b U l ( r 1 , 0 ) ( r 1 / r 2 ) 1 / 2 × exp [ i k ( r 1 r 2 ) 2 / ( 2 L ) ] d r 1 + i [ i / ( λ L ) ] 1 / 2 × exp [ i ( k L π l ) ] a b U l ( r 1 , 0 ) ( r 1 / r 2 ) 1 / 2 × exp [ i k ( r 1 + r 2 ) 2 / ( 2 L ) ] d r 1 .
U l ( r 2 , L ) [ i / ( λ L ) ] 1 / 2 exp ( ikL ) a b U l ( r 1 , 0 ) ( r 1 / r 2 ) 1 / 2 × exp [ i k ( r 1 r 2 ) 2 / ( 2 L ) ] d r 1 .
J l ( Z ) = [ H l ( 1 ) ( Z ) + H l ( 2 ) ( Z ) ] / 2
lim Z H l ( 1 ) ( Z ) = [ 2 / ( π Z ) ] 1 / 2 exp [ i ( Z l π / 2 π / 4 ) ] × k = 0 N = a l k Z k ,
a l 0 = 1 a l k = a l k 1 i 8 k | 4 l 2 ( 2 k 1 ) 2 | , k 0 .
U l ( r 2 , L ) i l + 1 k exp ( i k L ) / ( 2 L ) a b U l ( r 1 , 0 ) H l ( 1 ) ( k r 1 r 2 / L ) × exp [ i k ( r 1 2 + r 2 2 ) / ( 2 L ) ] r 1 d r 1 ,
H l ( 1 ) ( k r 1 r 2 / L ) exp [ i k ( r 1 2 + r 2 2 ) / ( 2 L ) ] exp [ i k r 1 r 2 / L ] exp [ i k ( r 1 2 + r 2 2 ) / ( 2 L ) ] , exp [ i k ( r 1 r 2 ) 2 / ( 2 L ) .
θ l / ( k r ) .
N = 2 π r / ( L θ ) ,
N π [ 2 / ( 1 g ) ] 1 / 2 [ r / ( r r c ) ] 1 / 2 ,
l = k r 2 L [ 2 ( 1 g ) ] 1 / 2 [ ( r r c ) / r ] 1 / 2 | r = r ¯ .

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