Abstract

We report on two resonator systems for producing Bessel–like beams with longitudinally dependent cone angles (LDBLBs). Such beams have extended propagation distances as compared to conventional Bessel–Gauss beams, with a far field pattern that is also Bessel–like in structure (i.e. not an annular ring). The first resonator system is based on a lens doublet with spherical aberration, while the second resonator system makes use of intra–cavity axicons and lens. In both cases we show that the LDBLB is the lowest loss fundamental mode of the cavity, and show theoretically the extended propagation distance expected from such beams.

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  1. J. Durnin, “Exact solutions for nondiffracting beams. The scalar theory,” J. Opt. Soc. Am. B 4(4), 651 (1987).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  7. J. Rogel-Salazar, G. H. C. New, and S. Chavez-Cerda, “Bessel-Gauss beam optical resonator,” Opt. Commun. 190(1-6), 117–122 (2001).
    [CrossRef]
  8. I. A. Litvin and A. Forbes, “Bessel–Gauss Resonator with Internal Amplitude Filter,” Opt. Commun. 281(9), 2385–2392 (2008).
    [CrossRef]
  9. V. N. Belyi, N. S. Kasak, and N. A. Khilo, “Properties of parametric frequency conversion with Bessel light beams,” Opt. Commun. 162(1-3), 169–176 (1999).
    [CrossRef]
  10. N. A. Khilo, E. S. Petrova, and A. A. Ryzhevich, “Transformation of the order of Bessel beams in uniaxial crystals,” Quantum Electron. 31(1), 85–89 (2001).
    [CrossRef]
  11. V. N. Belyi, N. S. Kazak, and N. A. Khilo, “Frequency conversion of Bessel light beams in nonlinear crystals,” Quantum Electron. 30(9), 753–766 (2000).
    [CrossRef]
  12. Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998).
    [CrossRef]
  13. I. A. Litvin, M. G. McLaren, and A. Forbes, “A conical wave approach to calculating Bessel-Gauss beam reconstruction after complex obstacles,” Opt. Commun. 282(6), 1078–1082 (2009).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  21. P. A. Bélanger, R. L. Lachance, and C. Paré, “Super-Gaussian output from a CO(2) laser by using a graded-phase mirror resonator,” Opt. Lett. 17(10), 739–741 (1992).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  24. T. Aruga, S. W. Li, S. Y. Yoshikado, M. Takabe, and R. Li, “Nondiffracting narrow light beam with small atmospheric turbulence-influenced propagation,” Appl. Opt. 38(15), 3152–3156 (1999).
    [CrossRef]
  25. V. Belyi, A. Forbes, N. Kazak, N. Khilo, and P. Ropot, “Bessel–like beams with z–dependent cone angles,” Opt. Express 18(3), 1966–1973 (2010).
    [CrossRef] [PubMed]

2010 (1)

2009 (2)

I. A. Litvin and A. Forbes, “Intra-cavity flat-top beam generation,” Opt. Express 17(18), 15891–15903 (2009).
[CrossRef] [PubMed]

I. A. Litvin, M. G. McLaren, and A. Forbes, “A conical wave approach to calculating Bessel-Gauss beam reconstruction after complex obstacles,” Opt. Commun. 282(6), 1078–1082 (2009).
[CrossRef]

2008 (1)

I. A. Litvin and A. Forbes, “Bessel–Gauss Resonator with Internal Amplitude Filter,” Opt. Commun. 281(9), 2385–2392 (2008).
[CrossRef]

2007 (1)

2001 (3)

A. N. Khilo, E. G. Katranji, and A. A. Ryzhevich, “Axicon-based. Bessel resonator: Analytical description and experiment,” J. Opt. Soc. Am. A 18(8), 1986 (2001).
[CrossRef]

N. A. Khilo, E. S. Petrova, and A. A. Ryzhevich, “Transformation of the order of Bessel beams in uniaxial crystals,” Quantum Electron. 31(1), 85–89 (2001).
[CrossRef]

J. Rogel-Salazar, G. H. C. New, and S. Chavez-Cerda, “Bessel-Gauss beam optical resonator,” Opt. Commun. 190(1-6), 117–122 (2001).
[CrossRef]

2000 (2)

V. N. Belyi, N. S. Kazak, and N. A. Khilo, “Frequency conversion of Bessel light beams in nonlinear crystals,” Quantum Electron. 30(9), 753–766 (2000).
[CrossRef]

T. Tanaka and S. Yamamoto, “Comparison of aberration between axicon and lens,” Opt. Commun. 184(1-4), 113–118 (2000).
[CrossRef]

1999 (2)

V. N. Belyi, N. S. Kasak, and N. A. Khilo, “Properties of parametric frequency conversion with Bessel light beams,” Opt. Commun. 162(1-3), 169–176 (1999).
[CrossRef]

T. Aruga, S. W. Li, S. Y. Yoshikado, M. Takabe, and R. Li, “Nondiffracting narrow light beam with small atmospheric turbulence-influenced propagation,” Appl. Opt. 38(15), 3152–3156 (1999).
[CrossRef]

1998 (2)

Z. Jaroszewicz and J. Morales, “Lens axicons: systems composed of a diverging aberrated lens and a perfect converging lens,” J. Opt. Soc. Am. A 15(9), 2383–2390 (1998).
[CrossRef]

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998).
[CrossRef]

1997 (2)

1992 (2)

C. Pare and P. A. Belanger, “Custom Laser Resonators Using Graded-Phase Mirror,” IEEE J. Quantum Electron. 28(1), 355–362 (1992).
[CrossRef]

P. A. Bélanger, R. L. Lachance, and C. Paré, “Super-Gaussian output from a CO(2) laser by using a graded-phase mirror resonator,” Opt. Lett. 17(10), 739–741 (1992).
[CrossRef] [PubMed]

1991 (3)

1989 (1)

1987 (2)

J. Durnin, “Exact solutions for nondiffracting beams. The scalar theory,” J. Opt. Soc. Am. B 4(4), 651 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

Aruga, T.

Belanger, P. A.

C. Pare and P. A. Belanger, “Custom Laser Resonators Using Graded-Phase Mirror,” IEEE J. Quantum Electron. 28(1), 355–362 (1992).
[CrossRef]

Bélanger, P. A.

Belyi, V.

Belyi, V. N.

V. N. Belyi, N. S. Kazak, and N. A. Khilo, “Frequency conversion of Bessel light beams in nonlinear crystals,” Quantum Electron. 30(9), 753–766 (2000).
[CrossRef]

V. N. Belyi, N. S. Kasak, and N. A. Khilo, “Properties of parametric frequency conversion with Bessel light beams,” Opt. Commun. 162(1-3), 169–176 (1999).
[CrossRef]

Bouchal, Z.

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998).
[CrossRef]

Burvall, A.

Chavez-Cerda, S.

J. Rogel-Salazar, G. H. C. New, and S. Chavez-Cerda, “Bessel-Gauss beam optical resonator,” Opt. Commun. 190(1-6), 117–122 (2001).
[CrossRef]

Chlup, M.

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998).
[CrossRef]

Dainty, C.

Davidson, N.

Durnin, J.

J. Durnin, “Exact solutions for nondiffracting beams. The scalar theory,” J. Opt. Soc. Am. B 4(4), 651 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

Forbes, A.

V. Belyi, A. Forbes, N. Kazak, N. Khilo, and P. Ropot, “Bessel–like beams with z–dependent cone angles,” Opt. Express 18(3), 1966–1973 (2010).
[CrossRef] [PubMed]

I. A. Litvin and A. Forbes, “Intra-cavity flat-top beam generation,” Opt. Express 17(18), 15891–15903 (2009).
[CrossRef] [PubMed]

I. A. Litvin, M. G. McLaren, and A. Forbes, “A conical wave approach to calculating Bessel-Gauss beam reconstruction after complex obstacles,” Opt. Commun. 282(6), 1078–1082 (2009).
[CrossRef]

I. A. Litvin and A. Forbes, “Bessel–Gauss Resonator with Internal Amplitude Filter,” Opt. Commun. 281(9), 2385–2392 (2008).
[CrossRef]

Friberg, A. T.

Friesem, A. A.

Goncharov, A. V.

Hasman, E.

Herman, R. M.

Jaroszewicz, Z.

Kasak, N. S.

V. N. Belyi, N. S. Kasak, and N. A. Khilo, “Properties of parametric frequency conversion with Bessel light beams,” Opt. Commun. 162(1-3), 169–176 (1999).
[CrossRef]

Katranji, E. G.

Kazak, N.

Kazak, N. S.

V. N. Belyi, N. S. Kazak, and N. A. Khilo, “Frequency conversion of Bessel light beams in nonlinear crystals,” Quantum Electron. 30(9), 753–766 (2000).
[CrossRef]

Khilo, A. N.

Khilo, N.

Khilo, N. A.

N. A. Khilo, E. S. Petrova, and A. A. Ryzhevich, “Transformation of the order of Bessel beams in uniaxial crystals,” Quantum Electron. 31(1), 85–89 (2001).
[CrossRef]

V. N. Belyi, N. S. Kazak, and N. A. Khilo, “Frequency conversion of Bessel light beams in nonlinear crystals,” Quantum Electron. 30(9), 753–766 (2000).
[CrossRef]

V. N. Belyi, N. S. Kasak, and N. A. Khilo, “Properties of parametric frequency conversion with Bessel light beams,” Opt. Commun. 162(1-3), 169–176 (1999).
[CrossRef]

Lachance, R. L.

Lewis, J. W.

Li, R.

Li, S. W.

Litvin, I. A.

I. A. Litvin and A. Forbes, “Intra-cavity flat-top beam generation,” Opt. Express 17(18), 15891–15903 (2009).
[CrossRef] [PubMed]

I. A. Litvin, M. G. McLaren, and A. Forbes, “A conical wave approach to calculating Bessel-Gauss beam reconstruction after complex obstacles,” Opt. Commun. 282(6), 1078–1082 (2009).
[CrossRef]

I. A. Litvin and A. Forbes, “Bessel–Gauss Resonator with Internal Amplitude Filter,” Opt. Commun. 281(9), 2385–2392 (2008).
[CrossRef]

McLaren, M. G.

I. A. Litvin, M. G. McLaren, and A. Forbes, “A conical wave approach to calculating Bessel-Gauss beam reconstruction after complex obstacles,” Opt. Commun. 282(6), 1078–1082 (2009).
[CrossRef]

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

Morales, J.

New, G. H. C.

J. Rogel-Salazar, G. H. C. New, and S. Chavez-Cerda, “Bessel-Gauss beam optical resonator,” Opt. Commun. 190(1-6), 117–122 (2001).
[CrossRef]

Pare, C.

C. Pare and P. A. Belanger, “Custom Laser Resonators Using Graded-Phase Mirror,” IEEE J. Quantum Electron. 28(1), 355–362 (1992).
[CrossRef]

Paré, C.

Parigger, C.

Petrova, E. S.

N. A. Khilo, E. S. Petrova, and A. A. Ryzhevich, “Transformation of the order of Bessel beams in uniaxial crystals,” Quantum Electron. 31(1), 85–89 (2001).
[CrossRef]

Plemmons, D. H.

Rogel-Salazar, J.

J. Rogel-Salazar, G. H. C. New, and S. Chavez-Cerda, “Bessel-Gauss beam optical resonator,” Opt. Commun. 190(1-6), 117–122 (2001).
[CrossRef]

Ropot, P.

Ryzhevich, A. A.

N. A. Khilo, E. S. Petrova, and A. A. Ryzhevich, “Transformation of the order of Bessel beams in uniaxial crystals,” Quantum Electron. 31(1), 85–89 (2001).
[CrossRef]

A. N. Khilo, E. G. Katranji, and A. A. Ryzhevich, “Axicon-based. Bessel resonator: Analytical description and experiment,” J. Opt. Soc. Am. A 18(8), 1986 (2001).
[CrossRef]

Takabe, M.

Tanaka, T.

T. Tanaka and S. Yamamoto, “Comparison of aberration between axicon and lens,” Opt. Commun. 184(1-4), 113–118 (2000).
[CrossRef]

Tang, Y.

Turunen, J.

Vasara, A.

Wagner, J.

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998).
[CrossRef]

Wiggins, T. A.

Yamamoto, S.

T. Tanaka and S. Yamamoto, “Comparison of aberration between axicon and lens,” Opt. Commun. 184(1-4), 113–118 (2000).
[CrossRef]

Yoshikado, S. Y.

Appl. Opt. (4)

IEEE J. Quantum Electron. (1)

C. Pare and P. A. Belanger, “Custom Laser Resonators Using Graded-Phase Mirror,” IEEE J. Quantum Electron. 28(1), 355–362 (1992).
[CrossRef]

J. Opt. Soc. Am. A (4)

J. Opt. Soc. Am. B (1)

J. Durnin, “Exact solutions for nondiffracting beams. The scalar theory,” J. Opt. Soc. Am. B 4(4), 651 (1987).
[CrossRef]

Opt. Commun. (6)

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998).
[CrossRef]

I. A. Litvin, M. G. McLaren, and A. Forbes, “A conical wave approach to calculating Bessel-Gauss beam reconstruction after complex obstacles,” Opt. Commun. 282(6), 1078–1082 (2009).
[CrossRef]

J. Rogel-Salazar, G. H. C. New, and S. Chavez-Cerda, “Bessel-Gauss beam optical resonator,” Opt. Commun. 190(1-6), 117–122 (2001).
[CrossRef]

I. A. Litvin and A. Forbes, “Bessel–Gauss Resonator with Internal Amplitude Filter,” Opt. Commun. 281(9), 2385–2392 (2008).
[CrossRef]

V. N. Belyi, N. S. Kasak, and N. A. Khilo, “Properties of parametric frequency conversion with Bessel light beams,” Opt. Commun. 162(1-3), 169–176 (1999).
[CrossRef]

T. Tanaka and S. Yamamoto, “Comparison of aberration between axicon and lens,” Opt. Commun. 184(1-4), 113–118 (2000).
[CrossRef]

Opt. Express (2)

Opt. Lett. (3)

Phys. Rev. Lett. (1)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

Quantum Electron. (2)

N. A. Khilo, E. S. Petrova, and A. A. Ryzhevich, “Transformation of the order of Bessel beams in uniaxial crystals,” Quantum Electron. 31(1), 85–89 (2001).
[CrossRef]

V. N. Belyi, N. S. Kazak, and N. A. Khilo, “Frequency conversion of Bessel light beams in nonlinear crystals,” Quantum Electron. 30(9), 753–766 (2000).
[CrossRef]

Other (1)

Z. Jaroszewicz, Axicons: Design and Propagation Properties, Vol. 5 of Research and Development Treatise (SPIE Polish Chapter, Warsaw, 1997).

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Figures (6)

Fig. 1
Fig. 1

(a) An extra–cavity design for producing LDBLBs by passing a Gaussian beam through a Galilean telescope system where the lenses L2 has spherical aberration; (b) the intra–cavity resonator equivalent where M1 has spherical aberration.

Fig. 2
Fig. 2

The phase of mirror M1 obtained by direct solution of Eq. (4) (a (blue)) and the intensity profiles on resonator mirror M1 and M2 after stabilization (a (red)). The fundamental mode stabilization as a function of the number of round trips (b).

Fig. 3
Fig. 3

The propagation properties of the output beam. (a) – longitudinal dependence of on-axial intensity, (b) – cross section of propagated beam on distance 0.7 m (1) and 1.3 m (2) correspondingly, (c) – spatial spectrum intensity view of obtained beam profiles, (d) – density plot of the propagated beam.

Fig. 4
Fig. 4

(a) An extra–cavity design for producing LDBLBs by passing a doublet of axicon-lens and axicon; (b) the intra–cavity resonator equivalent.

Fig. 5
Fig. 5

The phase of mirror M2 obtained by the direct solution of Eq. (7) (a (blue)) and the intensity profiles on resonator mirror M2 and M1 after stabilization (a (red)). The fundamental mode stabilization as a function of the number of round trips (b).

Fig. 6
Fig. 6

The propagation properties of the output beam of the intra–cavity axicons scheme. (a) – longitudinal dependence of the central peak, (b) – cross section of the propagated beam on distance 0.7 m (1) and 1.3 m (2) correspondingly, red dotes on (b1) – the Bessel function of 0 order with radial wave number equal to 2.05 × 104 m−1 (c) – spatial spectrum intensity view of the obtained beam profiles, (d) – density plot of the propagated beam.

Equations (7)

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a 1 ( ρ , z 1 ) = i k 0 2 z 1 exp ( i k 0 ρ 2 2 z 1 ) 0 R 0 exp ( ρ 1 2 w 0 2 + i k ρ 1 2 2 f 1 + i k ρ 1 2 2 z 1 ) J 0 ( k 0 ρ ρ 0 z 1 ) ρ 1 d ρ 1
a 1 ( ρ , z 1 ) = w 0 w ( z 1 ) exp [ ( ρ w ( z 1 ) ) 2 + i k ρ 2 2 R ( z 1 ) i α ( z 1 ) ]
a 2 = a 1 exp ( i k 0 ρ 2 / 2 f 2 + i k 0 β ρ 4 ) .
t M 1 = a 2 * / a 2 .
a 1 ( ρ , z 1 ) = i λ z 1 exp ( i k 0 ρ 2 2 z 1 ) 0 2 π 0 R a 2 exp ( ρ 1 2 ρ 0 2 i k 0 γ 2 ρ 1 i k 0 ρ ρ 1 z 1 cos ( ϕ ϕ 1 ) ) ρ 1 2 d ρ 1 d ϕ 1
a 1 ( ρ , z 1 ) = i ρ F 2 ( z 1 F ) 2 ( 1 γ 2 z 1 ρ ) 3 / 2 exp [ i k 0 2 z 1 ( ρ 2 + z 1 / F 1 + i z 1 / z 0 ( z 1 / F 1 ) 2 + ( z 1 / z 0 ) 2 ( ρ γ 2 z 1 ) 2 ) ]
a 2 = a 1 exp ( i k 0 γ 1 ρ ) .

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