Abstract

We estimate the optical fields of a phase-conjugating Bessel–Gauss resonator (PCBGR) by use of the matrix eigenvalue algorithm. According to the theoretical expression of Bessel–Gauss beams, we discuss the desired phase of the phase-conjugating mirror (PCM) for the production of Bessel–Gauss beams. On the basis of the given phase distribution of the PCM and the Huygens–Fresnel diffraction integral formula, we conduct the eigenmode matrix equation of the PCBGR and evaluate the field profiles of the PCBGR. We show from the numerical calculation that the PCBGR can produce the desired Bessel–Gauss mode provided that the parameters of the PCBGR are accurately chosen.

© 2006 Optical Society of America

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  1. J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, "Diffraction free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
    [CrossRef] [PubMed]
  2. J. Durnin, "Exact solutions for nondiffracting beams.I. The scalar theory," J. Opt. Soc. Am. A 4, 651-654 (1987).
    [CrossRef]
  3. G. Indebetouw, "Nondiffracting optical fields: some remarks on their analysis and synthesis," J. Opt. Soc. Am. A 6, 150-152 (1989).
    [CrossRef]
  4. J. Turunen, A. Vasara, and A. T. Friberg, "Holographic generation of diffraction-free beams," Appl. Phys. Lett. 27, 3959-3962 (1988).
  5. A. Vasara, J. Turunen, and A. T. Friberg, "Realization of general nondiffracting beams with computer-generated holograms," J. Opt. Soc. Am. A 6, 1748-1754 (1989).
    [CrossRef] [PubMed]
  6. G. Scott and N. McArdle, "Efficient generation of nearly diffraction-free beams using an axicon," Opt. Eng. 31, 2640-2643 (1992).
    [CrossRef]
  7. F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
    [CrossRef]
  8. A. N. Khilo, E. G. Katranji, and A. A. Ryzhevich, "Axiconbased Bessel resonator: analytical description and experiment," J. Opt. Soc. Am. A 18, 1986-1992 (2001).
    [CrossRef]
  9. J. Rogel-Salazar, G. H. C. New, and S. Chávez-Cerda, "Bessel-Gauss beam optical resonator," Opt. Commun. 190, 117-122 (2001).
    [CrossRef]
  10. J. C. Gutiérrez-Vega, R. Rodriguez-Masegosa, and S. Chávez-Cerda, "Bessel-Gauss resonator with spherical output mirror: geometrical- and wave-optics analysis," J. Opt. Soc. Am. A 20, 2113-2122 (2003).
    [CrossRef]
  11. C. L. Tsangaris, G. H. C. New, and J. Rogel-Salazar, "Unstable Bessel beam resonator," Opt. Commun. 223, 233-238 (2003).
    [CrossRef]
  12. R. I. Hernández-Aranda, S. Chávez-Cerda, and J. C. Gutiérrez-Vega, "Theory of the unstable Bessel resonator," J. Opt. Soc. Am. A 22, 1909-1917 (2005).
    [CrossRef]
  13. P. Paakkonen and J. Turunen, "Resonators with Bessel-Gauss modes," Opt. Commun. 156, 359-366 (1998).
    [CrossRef]
  14. A. Hakola, S. C. Buchter, T. Kajava, H. Elfstrom, J. Simonen, P. Paakkonen, and J. Turunen, "Bessel-Gauss output beam from adiode-pumped Nd:YAG laser," Opt. Commun. 238, 335-340 (2004).
    [CrossRef]
  15. A. G. Fox and T. Li, "Resonant modes in a maser interferometer," Bell Syst. Tech. J. 40, 453-458 (1961).
  16. A. G. Fox and T. Li, "Resonant modes in an optical maser," Proc. IRE 48, 1904-1905 (1960).
  17. D. Ling, Y. Fu, D. Xu, and Y. Guan, "Finite-sum matrix analysis of eigen-mode fields of the Gaussian-reflectivity plano-concave resonator," in High-Power Lasers and Applications II, D.Fan, K.A.Truesdell, and K.Yasui, eds., Proc. SPIE 4914, 371-381 (2002).
  18. Z. Wei, R. Wang, and Z. Wang, "Numerical analysis of mode-fields of unstable ring resonators 90° beam rotation," Acta Opt. Sin. 15, 696-702 (1995).
  19. S. A. Collins, Jr., "Lens-system diffraction integral written in terms of matrix optics," J. Opt. Soc. Am. 60, 1168-1177 (1970).
    [CrossRef]

2005

2004

A. Hakola, S. C. Buchter, T. Kajava, H. Elfstrom, J. Simonen, P. Paakkonen, and J. Turunen, "Bessel-Gauss output beam from adiode-pumped Nd:YAG laser," Opt. Commun. 238, 335-340 (2004).
[CrossRef]

2003

2001

A. N. Khilo, E. G. Katranji, and A. A. Ryzhevich, "Axiconbased Bessel resonator: analytical description and experiment," J. Opt. Soc. Am. A 18, 1986-1992 (2001).
[CrossRef]

J. Rogel-Salazar, G. H. C. New, and S. Chávez-Cerda, "Bessel-Gauss beam optical resonator," Opt. Commun. 190, 117-122 (2001).
[CrossRef]

1998

P. Paakkonen and J. Turunen, "Resonators with Bessel-Gauss modes," Opt. Commun. 156, 359-366 (1998).
[CrossRef]

1995

Z. Wei, R. Wang, and Z. Wang, "Numerical analysis of mode-fields of unstable ring resonators 90° beam rotation," Acta Opt. Sin. 15, 696-702 (1995).

1992

G. Scott and N. McArdle, "Efficient generation of nearly diffraction-free beams using an axicon," Opt. Eng. 31, 2640-2643 (1992).
[CrossRef]

1989

1988

J. Turunen, A. Vasara, and A. T. Friberg, "Holographic generation of diffraction-free beams," Appl. Phys. Lett. 27, 3959-3962 (1988).

1987

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, "Diffraction free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

J. Durnin, "Exact solutions for nondiffracting beams.I. The scalar theory," J. Opt. Soc. Am. A 4, 651-654 (1987).
[CrossRef]

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

1970

1961

A. G. Fox and T. Li, "Resonant modes in a maser interferometer," Bell Syst. Tech. J. 40, 453-458 (1961).

1960

A. G. Fox and T. Li, "Resonant modes in an optical maser," Proc. IRE 48, 1904-1905 (1960).

Buchter, S. C.

A. Hakola, S. C. Buchter, T. Kajava, H. Elfstrom, J. Simonen, P. Paakkonen, and J. Turunen, "Bessel-Gauss output beam from adiode-pumped Nd:YAG laser," Opt. Commun. 238, 335-340 (2004).
[CrossRef]

Chávez-Cerda, S.

Collins, S. A.

Durnin, J.

J. Durnin, "Exact solutions for nondiffracting beams.I. The scalar theory," J. Opt. Soc. Am. A 4, 651-654 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, "Diffraction free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, "Diffraction free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Elfstrom, H.

A. Hakola, S. C. Buchter, T. Kajava, H. Elfstrom, J. Simonen, P. Paakkonen, and J. Turunen, "Bessel-Gauss output beam from adiode-pumped Nd:YAG laser," Opt. Commun. 238, 335-340 (2004).
[CrossRef]

Fox, A. G.

A. G. Fox and T. Li, "Resonant modes in a maser interferometer," Bell Syst. Tech. J. 40, 453-458 (1961).

A. G. Fox and T. Li, "Resonant modes in an optical maser," Proc. IRE 48, 1904-1905 (1960).

Friberg, A. T.

A. Vasara, J. Turunen, and A. T. Friberg, "Realization of general nondiffracting beams with computer-generated holograms," J. Opt. Soc. Am. A 6, 1748-1754 (1989).
[CrossRef] [PubMed]

J. Turunen, A. Vasara, and A. T. Friberg, "Holographic generation of diffraction-free beams," Appl. Phys. Lett. 27, 3959-3962 (1988).

Fu, Y.

D. Ling, Y. Fu, D. Xu, and Y. Guan, "Finite-sum matrix analysis of eigen-mode fields of the Gaussian-reflectivity plano-concave resonator," in High-Power Lasers and Applications II, D.Fan, K.A.Truesdell, and K.Yasui, eds., Proc. SPIE 4914, 371-381 (2002).

Gori, F.

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Guan, Y.

D. Ling, Y. Fu, D. Xu, and Y. Guan, "Finite-sum matrix analysis of eigen-mode fields of the Gaussian-reflectivity plano-concave resonator," in High-Power Lasers and Applications II, D.Fan, K.A.Truesdell, and K.Yasui, eds., Proc. SPIE 4914, 371-381 (2002).

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Gutiérrez-Vega, J. C.

Hakola, A.

A. Hakola, S. C. Buchter, T. Kajava, H. Elfstrom, J. Simonen, P. Paakkonen, and J. Turunen, "Bessel-Gauss output beam from adiode-pumped Nd:YAG laser," Opt. Commun. 238, 335-340 (2004).
[CrossRef]

Hernández-Aranda, R. I.

Indebetouw, G.

Kajava, T.

A. Hakola, S. C. Buchter, T. Kajava, H. Elfstrom, J. Simonen, P. Paakkonen, and J. Turunen, "Bessel-Gauss output beam from adiode-pumped Nd:YAG laser," Opt. Commun. 238, 335-340 (2004).
[CrossRef]

Katranji, E. G.

Khilo, A. N.

Li, T.

A. G. Fox and T. Li, "Resonant modes in a maser interferometer," Bell Syst. Tech. J. 40, 453-458 (1961).

A. G. Fox and T. Li, "Resonant modes in an optical maser," Proc. IRE 48, 1904-1905 (1960).

Ling, D.

D. Ling, Y. Fu, D. Xu, and Y. Guan, "Finite-sum matrix analysis of eigen-mode fields of the Gaussian-reflectivity plano-concave resonator," in High-Power Lasers and Applications II, D.Fan, K.A.Truesdell, and K.Yasui, eds., Proc. SPIE 4914, 371-381 (2002).

McArdle, N.

G. Scott and N. McArdle, "Efficient generation of nearly diffraction-free beams using an axicon," Opt. Eng. 31, 2640-2643 (1992).
[CrossRef]

Miceli, J. J.

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, "Diffraction free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

New, G. H. C.

C. L. Tsangaris, G. H. C. New, and J. Rogel-Salazar, "Unstable Bessel beam resonator," Opt. Commun. 223, 233-238 (2003).
[CrossRef]

J. Rogel-Salazar, G. H. C. New, and S. Chávez-Cerda, "Bessel-Gauss beam optical resonator," Opt. Commun. 190, 117-122 (2001).
[CrossRef]

Paakkonen, P.

A. Hakola, S. C. Buchter, T. Kajava, H. Elfstrom, J. Simonen, P. Paakkonen, and J. Turunen, "Bessel-Gauss output beam from adiode-pumped Nd:YAG laser," Opt. Commun. 238, 335-340 (2004).
[CrossRef]

P. Paakkonen and J. Turunen, "Resonators with Bessel-Gauss modes," Opt. Commun. 156, 359-366 (1998).
[CrossRef]

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Rodriguez-Masegosa, R.

Rogel-Salazar, J.

C. L. Tsangaris, G. H. C. New, and J. Rogel-Salazar, "Unstable Bessel beam resonator," Opt. Commun. 223, 233-238 (2003).
[CrossRef]

J. Rogel-Salazar, G. H. C. New, and S. Chávez-Cerda, "Bessel-Gauss beam optical resonator," Opt. Commun. 190, 117-122 (2001).
[CrossRef]

Ryzhevich, A. A.

Scott, G.

G. Scott and N. McArdle, "Efficient generation of nearly diffraction-free beams using an axicon," Opt. Eng. 31, 2640-2643 (1992).
[CrossRef]

Simonen, J.

A. Hakola, S. C. Buchter, T. Kajava, H. Elfstrom, J. Simonen, P. Paakkonen, and J. Turunen, "Bessel-Gauss output beam from adiode-pumped Nd:YAG laser," Opt. Commun. 238, 335-340 (2004).
[CrossRef]

Tsangaris, C. L.

C. L. Tsangaris, G. H. C. New, and J. Rogel-Salazar, "Unstable Bessel beam resonator," Opt. Commun. 223, 233-238 (2003).
[CrossRef]

Turunen, J.

A. Hakola, S. C. Buchter, T. Kajava, H. Elfstrom, J. Simonen, P. Paakkonen, and J. Turunen, "Bessel-Gauss output beam from adiode-pumped Nd:YAG laser," Opt. Commun. 238, 335-340 (2004).
[CrossRef]

P. Paakkonen and J. Turunen, "Resonators with Bessel-Gauss modes," Opt. Commun. 156, 359-366 (1998).
[CrossRef]

A. Vasara, J. Turunen, and A. T. Friberg, "Realization of general nondiffracting beams with computer-generated holograms," J. Opt. Soc. Am. A 6, 1748-1754 (1989).
[CrossRef] [PubMed]

J. Turunen, A. Vasara, and A. T. Friberg, "Holographic generation of diffraction-free beams," Appl. Phys. Lett. 27, 3959-3962 (1988).

Vasara, A.

A. Vasara, J. Turunen, and A. T. Friberg, "Realization of general nondiffracting beams with computer-generated holograms," J. Opt. Soc. Am. A 6, 1748-1754 (1989).
[CrossRef] [PubMed]

J. Turunen, A. Vasara, and A. T. Friberg, "Holographic generation of diffraction-free beams," Appl. Phys. Lett. 27, 3959-3962 (1988).

Wang, R.

Z. Wei, R. Wang, and Z. Wang, "Numerical analysis of mode-fields of unstable ring resonators 90° beam rotation," Acta Opt. Sin. 15, 696-702 (1995).

Wang, Z.

Z. Wei, R. Wang, and Z. Wang, "Numerical analysis of mode-fields of unstable ring resonators 90° beam rotation," Acta Opt. Sin. 15, 696-702 (1995).

Wei, Z.

Z. Wei, R. Wang, and Z. Wang, "Numerical analysis of mode-fields of unstable ring resonators 90° beam rotation," Acta Opt. Sin. 15, 696-702 (1995).

Xu, D.

D. Ling, Y. Fu, D. Xu, and Y. Guan, "Finite-sum matrix analysis of eigen-mode fields of the Gaussian-reflectivity plano-concave resonator," in High-Power Lasers and Applications II, D.Fan, K.A.Truesdell, and K.Yasui, eds., Proc. SPIE 4914, 371-381 (2002).

Acta Opt. Sin.

Z. Wei, R. Wang, and Z. Wang, "Numerical analysis of mode-fields of unstable ring resonators 90° beam rotation," Acta Opt. Sin. 15, 696-702 (1995).

Appl. Phys. Lett.

J. Turunen, A. Vasara, and A. T. Friberg, "Holographic generation of diffraction-free beams," Appl. Phys. Lett. 27, 3959-3962 (1988).

Bell Syst. Tech. J.

A. G. Fox and T. Li, "Resonant modes in a maser interferometer," Bell Syst. Tech. J. 40, 453-458 (1961).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

J. Rogel-Salazar, G. H. C. New, and S. Chávez-Cerda, "Bessel-Gauss beam optical resonator," Opt. Commun. 190, 117-122 (2001).
[CrossRef]

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

P. Paakkonen and J. Turunen, "Resonators with Bessel-Gauss modes," Opt. Commun. 156, 359-366 (1998).
[CrossRef]

A. Hakola, S. C. Buchter, T. Kajava, H. Elfstrom, J. Simonen, P. Paakkonen, and J. Turunen, "Bessel-Gauss output beam from adiode-pumped Nd:YAG laser," Opt. Commun. 238, 335-340 (2004).
[CrossRef]

C. L. Tsangaris, G. H. C. New, and J. Rogel-Salazar, "Unstable Bessel beam resonator," Opt. Commun. 223, 233-238 (2003).
[CrossRef]

Opt. Eng.

G. Scott and N. McArdle, "Efficient generation of nearly diffraction-free beams using an axicon," Opt. Eng. 31, 2640-2643 (1992).
[CrossRef]

Phys. Rev. Lett.

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, "Diffraction free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Proc. IRE

A. G. Fox and T. Li, "Resonant modes in an optical maser," Proc. IRE 48, 1904-1905 (1960).

Other

D. Ling, Y. Fu, D. Xu, and Y. Guan, "Finite-sum matrix analysis of eigen-mode fields of the Gaussian-reflectivity plano-concave resonator," in High-Power Lasers and Applications II, D.Fan, K.A.Truesdell, and K.Yasui, eds., Proc. SPIE 4914, 371-381 (2002).

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Figures (8)

Fig. 1
Fig. 1

Geometrical configuration of the PCBGR.

Fig. 2
Fig. 2

Radial phase distributions of the PCM for m = 0 , λ = 1.064 μ m , and L = 50 cm . (a) The first term of Eq. (12) is used, and (b) both the first term and the second term are used.

Fig. 3
Fig. 3

Radial phase distributions of the PCM for m = 2 , λ = 1.064 μ m , and L = 50 cm . (a) The first term of Eq. (12) is used, and (b) both the first term and the second term are used.

Fig. 4
Fig. 4

Computing model of the PCBGR. Diffracting interfaces 1 and 3 are placed just before the phase-conjugating mirror, and diffracting interface 2 is located just before the plane mirror.

Fig. 5
Fig. 5

Radial amplitudes of the fundamental mode (00) across the plane just before the phase-conjugating mirror. (a) The first term of Eq. (12) is used, and (b) both the first term and the second term are used.

Fig. 6
Fig. 6

Radial amplitudes of the fundamental mode (00) across the plane just before the output mirror (a). The first term of Eq. (12) is used, (b) both the first term and the second term are used.

Fig. 7
Fig. 7

Field distributions of the fundamental mode (00) (a) across the phase-conjugating mirror and (b) across the output mirror.

Fig. 8
Fig. 8

Field distributions of mode 20 (a) across the phase-conjugating mirror and (b) across the output mirror.

Tables (1)

Tables Icon

Table 1 Eigenvalues and Losses for the Low-Loss Modes of the PCBGR

Equations (25)

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U m ( r , φ , 0 ) = C J m ( α r ) exp ( r 2 w 0 2 ) exp ( j m φ ) ,
U m ( r , φ , z ) = C w 0 w ( z ) J m ( α r 1 + j z z R ) exp { [ 1 w ( z ) 2 j k 2 R ( z ) ] ( r 2 + θ 2 z 2 ) } exp [ j Φ ( z ) ] exp ( j m φ ) ,
z R = 1 2 k w 0 2 .
w ( z ) = w 0 ( z ) [ 1 + ( z z R ) 2 ] 1 2 ,
R ( z ) = z + z R 2 z ,
Φ ( z ) = β z arctan ( z z R ) ,
β = k α 2 2 k .
θ = α k .
t m ( r ) = U m * ( r , φ , L ) U m ( r , φ , L ) = exp [ j k R ( L ) ( r 2 + θ 2 L 2 ) ] exp { j 2 arg [ J m ( α r 1 + j L Z R ) ] } exp [ j 2 Φ ( L ) ] .
U m ( r , φ , L ) = A exp [ j φ m ( r ) ] ,
t m ( r ) = exp [ j 2 φ m ( r ) ] .
Φ m ( r ) = 2 φ m ( r ) = 2 arg [ J m ( α r 1 + j L z R ) ] + k r 2 R ( L ) + k θ 2 L 2 R ( L ) ,
E 2 ( r 2 , φ 2 ) = j k exp ( j k L ) 2 π B 1 S 1 t m ( r 1 ) E 1 ( r 1 , φ 1 ) exp { j k 2 B 1 [ A 1 r 1 2 + D 1 r 2 2 2 r 1 r 2 cos ( φ 1 φ 2 ) ] } r 1 d r 1 d φ 1 ,
E 1 ( r 1 , φ 1 ) = j k exp ( j k L ) 2 π B 2 S 2 E 2 ( r 2 , φ 2 ) exp { j k 2 B 2 [ A 2 r 2 2 + D 2 r 1 2 2 r 1 r 2 cos ( φ 1 φ 2 ) ] } r 2 d r 2 d φ 2 ,
E 1 ( r 1 , φ 1 ) = E 1 ( r 1 ) exp ( j m φ 1 ) ,
E 2 ( r 2 , φ 2 ) = E 2 ( r 2 ) exp ( j m φ 2 ) ,
E 1 ( r 1 , φ 1 ) = E 1 ( r 1 ) exp ( j m φ 1 ) ,
E 2 ( r 2 ) = ( j ) m + 1 k exp ( j k L ) B 1 0 a 2 t m ( r 1 ) E 1 ( r 1 ) J m ( k r 1 r 2 B 1 ) exp [ j k 2 B 1 ( A 1 r 1 2 + D 1 r 2 2 ) ] r 1 d r 1 ,
E 1 ( r 1 ) = ( j ) m + 1 k exp ( j k L ) B 2 0 a 1 E 2 ( r 2 ) J m ( k r 1 r 2 B 2 ) exp [ j k 2 B 2 ( A 2 r 2 2 + D 2 r 1 2 ) ] r 2 d r 2 ,
E 2 ( r 2 ) l = p = 1 P G l p E 1 ( r 1 ) p ,
E 1 ( r 1 ) l = p = 1 P H l p E 2 ( r 2 ) p ,
G l p = ( j ) m + 1 k p a 1 2 exp ( j k L ) exp [ J Φ m ( a 1 p P ) ] L P 2 J m ( k l p a 1 a 2 L P 2 ) exp [ i π λ L P 2 ( a 1 2 p 2 + a 2 2 l 2 ) ] ,
H l p = ( j ) m + 1 k p a 2 2 exp ( j k L ) L P 2 J m ( k l p a 1 a 2 L P 2 ) exp [ i π λ L P 2 ( a 2 2 p 2 + a 1 2 l 2 ) ] ,
γ E 1 = ( H G ) E 1 ,
γ E 2 = ( G H ) E 2 .

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