Abstract

A rigorous analysis of the unstable Bessel resonator with convex output coupler is presented. The Huygens–Fresnel self-consistency equation is solved to extract the first eigenmodes and eigenvalues of the cavity, taking into account the finite apertures of the mirrors. Attention was directed to the dependence of the output transverse profiles; the losses; and the modal-frequency changes on the curvature of the output coupler, the cavity length, and the angle of the axicon. Our analysis revealed that while the stable Bessel resonator retains a Gaussian radial modulation on the Bessel rings, the unstable configuration exhibits a more uniform amplitude modulation that produces output profiles more similar to ideal Bessel beams. The unstable cavity also possesses higher-mode discrimination in favor of the fundamental mode than does the stable configuration.

© 2005 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2004 (2)

A. Hakola, S. C. Buchter, T. Kajava, H. Elfström, J. Simonen, P. Pääkkönen, J. Turunen, “Bessel–Gauss output beam from a diode-pumped NdYAG laser,” Opt. Commun. 238, 335–340 (2004).
[CrossRef]

M. Guizar-Sicairos, J. C. Gutiérrez-Vega, “Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields,” J. Opt. Soc. Am. A 21, 53–58 (2004).
[CrossRef]

2003 (3)

J. C. Gutiérrez-Vega, R. Rodríguez-Masegosa, 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]

B. Duszcyk, M. P. Newell, S. J. Sugden, “Numerical methods for solving the eigenvalue problem for a positive branch confocal unstable resonator,” Appl. Math. Comput. 140, 427–443 (2003).
[CrossRef]

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

2001 (2)

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

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

1998 (2)

1997 (1)

1992 (1)

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

1990 (1)

1989 (3)

1988 (1)

1987 (2)

J. Durnin, J. J. Micely, 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]

1980 (1)

1978 (1)

1974 (1)

1961 (1)

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

Bernabe, M. L.

Bor, Zs.

Buchter, S. C.

A. Hakola, S. C. Buchter, T. Kajava, H. Elfström, J. Simonen, P. Pääkkönen, J. Turunen, “Bessel–Gauss output beam from a diode-pumped NdYAG laser,” Opt. Commun. 238, 335–340 (2004).
[CrossRef]

Cavallaro, J. R.

Chávez-Cerda, S.

Chen, N.-X.

Cong, W.-X.

Dente, G. C.

Durnin, J.

J. Durnin, J. J. Micely, 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]

J. Durnin, J. H. Eberly, “Diffraction free arrangement,” U.S. patent 4,887,885 (December 19, 1989).

Duszcyk, B.

B. Duszcyk, M. P. Newell, S. J. Sugden, “Numerical methods for solving the eigenvalue problem for a positive branch confocal unstable resonator,” Appl. Math. Comput. 140, 427–443 (2003).
[CrossRef]

Eberly, J. H.

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

J. Durnin, J. H. Eberly, “Diffraction free arrangement,” U.S. patent 4,887,885 (December 19, 1989).

Elfström, H.

A. Hakola, S. C. Buchter, T. Kajava, H. Elfström, J. Simonen, P. Pääkkönen, J. Turunen, “Bessel–Gauss output beam from a diode-pumped NdYAG laser,” Opt. Commun. 238, 335–340 (2004).
[CrossRef]

Erdélyi, M.

Fox, A. G.

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

Friberg, A. T.

Gu, B.-Y.

Guizar-Sicairos, M.

Gutiérrez-Vega, J. C.

Hakola, A.

A. Hakola, S. C. Buchter, T. Kajava, H. Elfström, J. Simonen, P. Pääkkönen, J. Turunen, “Bessel–Gauss output beam from a diode-pumped NdYAG laser,” Opt. Commun. 238, 335–340 (2004).
[CrossRef]

Horváth, Z. L.

Indebetouw, G.

Kajava, T.

A. Hakola, S. C. Buchter, T. Kajava, H. Elfström, J. Simonen, P. Pääkkönen, J. Turunen, “Bessel–Gauss output beam from a diode-pumped NdYAG laser,” Opt. Commun. 238, 335–340 (2004).
[CrossRef]

Katranji, E. G.

Khilo, A. N.

Kikuchi, H.

K. Uehara, H. Kikuchi, “Generation of nearly diffraction-free laser beams,” Appl. Phys. B 48, 125–129 (1989).
[CrossRef]

Latham, W. P.

Li, T.

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

Lissak, B.

McArdle, N.

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

Micely, J. J.

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

Murphy, W. D.

New, G. H. C.

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

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

Newell, M. P.

B. Duszcyk, M. P. Newell, S. J. Sugden, “Numerical methods for solving the eigenvalue problem for a positive branch confocal unstable resonator,” Appl. Math. Comput. 140, 427–443 (2003).
[CrossRef]

Pääkkönen, P.

A. Hakola, S. C. Buchter, T. Kajava, H. Elfström, J. Simonen, P. Pääkkönen, J. Turunen, “Bessel–Gauss output beam from a diode-pumped NdYAG laser,” Opt. Commun. 238, 335–340 (2004).
[CrossRef]

P. Pääkkönen, J. Turunen, “Resonators with Bessel-Gauss modes,” Opt. Commun. 156, 359–366 (1998).
[CrossRef]

Rodríguez-Masegosa, R.

Rogel-Salazar, J.

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

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

Ruschin, S.

Ryzhevich, A. A.

Scott, G.

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

Siegman, A. E.

Simonen, J.

A. Hakola, S. C. Buchter, T. Kajava, H. Elfström, J. Simonen, P. Pääkkönen, J. Turunen, “Bessel–Gauss output beam from a diode-pumped NdYAG laser,” Opt. Commun. 238, 335–340 (2004).
[CrossRef]

Sugden, S. J.

B. Duszcyk, M. P. Newell, S. J. Sugden, “Numerical methods for solving the eigenvalue problem for a positive branch confocal unstable resonator,” Appl. Math. Comput. 140, 427–443 (2003).
[CrossRef]

Szabó, G.

Tittel, F. K.

Tsangaris, C. L.

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

Turunen, J.

A. Hakola, S. C. Buchter, T. Kajava, H. Elfström, J. Simonen, P. Pääkkönen, J. Turunen, “Bessel–Gauss output beam from a diode-pumped NdYAG laser,” Opt. Commun. 238, 335–340 (2004).
[CrossRef]

P. Pääkkönen, J. Turunen, “Resonators with Bessel-Gauss modes,” Opt. Commun. 156, 359–366 (1998).
[CrossRef]

A. Vasara, J. Turunen, 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, A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
[CrossRef] [PubMed]

Uehara, K.

K. Uehara, H. Kikuchi, “Generation of nearly diffraction-free laser beams,” Appl. Phys. B 48, 125–129 (1989).
[CrossRef]

Vasara, A.

Appl. Math. Comput. (1)

B. Duszcyk, M. P. Newell, S. J. Sugden, “Numerical methods for solving the eigenvalue problem for a positive branch confocal unstable resonator,” Appl. Math. Comput. 140, 427–443 (2003).
[CrossRef]

Appl. Opt. (5)

Appl. Phys. B (1)

K. Uehara, H. Kikuchi, “Generation of nearly diffraction-free laser beams,” Appl. Phys. B 48, 125–129 (1989).
[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).
[CrossRef]

J. Opt. Soc. Am. A (8)

Opt. Commun. (4)

P. Pääkkönen, J. Turunen, “Resonators with Bessel-Gauss modes,” Opt. Commun. 156, 359–366 (1998).
[CrossRef]

A. Hakola, S. C. Buchter, T. Kajava, H. Elfström, J. Simonen, P. Pääkkönen, J. Turunen, “Bessel–Gauss output beam from a diode-pumped NdYAG laser,” Opt. Commun. 238, 335–340 (2004).
[CrossRef]

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

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

Opt. Eng. (Bellingham) (1)

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

Phys. Rev. Lett. (1)

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

Other (2)

J. Durnin, J. H. Eberly, “Diffraction free arrangement,” U.S. patent 4,887,885 (December 19, 1989).

A. E. Siegman, Lasers (University Science, 1986).

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Figures (12)

Fig. 1
Fig. 1

Design of the resonator with (a) reflective and (b) refractive axicon. The convex–spherical mirror is placed at a distance L from the axicon. (c) Lens-guide-equivalent resonator and self-reproducibility condition for ray trajectories after one and two round trips. RP stands for reference plane.

Fig. 2
Fig. 2

Transverse profiles of the magnitude and phase of the fundamental Bessel mode at (a), (b) the axicon plane and (c), (d) the output plane for a resonator with output flat mirror and a 2 = 5 mm .

Fig. 3
Fig. 3

Transverse profiles of the magnitude and phase of the fundamental Bessel mode at (a) (b) the axicon plane and (c) (d) the output plane for a UBR with R = 50 L and a 2 = 5 mm .

Fig. 4
Fig. 4

Transverse profiles of the magnitude and phase of the second-order Bessel mode at (a), (b) the axicon plane and (c), (d) the output plane for a UBR with R = 50 L and a 2 = 5 mm .

Fig. 5
Fig. 5

Passive three-dimensional intracavity field distribution in the UBR with R = 50 L . The eigenfield is first forward propagated from the axicon plane to the output plane and later returned backward to the axicon plane.

Fig. 6
Fig. 6

Transverse profiles of the magnitude of the fundamental Bessel mode at the axion and output planes of the UBR for several ratios R L .

Fig. 7
Fig. 7

Transverse profiles of the magnitude of the second-order Bessel mode at the axion and output planes of the UBR for several ratios R L .

Fig. 8
Fig. 8

(a) Diffractive losses Γ = 1 γ 2 in terms of the normalized radius R L for the fundamental mode with different wedge angles. Comparison of results with the matrix and Fox–Li methods. (b) Comparison of the results for different aperture sizes of the output mirror.

Fig. 9
Fig. 9

Transition between the stable and the unstable regions of the Bessel resonator. (a) Diffractive losses Γ = 1 γ 2 and (b) normalized phase shifts Δ β π are depicted as a function of the normalized radius R L . Mode crossings are present for higher-order modes. In (b) the phase curves for the considered modes in (a) are contained within the region between the phase curves for modes (0,1) and (0,4).

Fig. 10
Fig. 10

Comparison between the transverse modes for the SBR and the UBR.

Fig. 11
Fig. 11

Transverse field patterns at the output and axicon planes corresponding to the length factors μ = 0.8 , 1, and 1.2 for convex output mirror with R = 50 L and α = 0.5 .

Fig. 12
Fig. 12

Loss and resonant frequency shift behavior as a function of the length factor μ for the first two angular modes ( l , p ) = ( 0 , 1 ) and (1,1), with R = 50 L and α = 0.5 .

Tables (1)

Tables Icon

Table 1 Diffractive Losses for the First Ten Modes

Equations (23)

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L = a 1 2 tan θ 0 a 1 2 θ 0
u ( r ) = J l ( k t r ) exp ( r 2 w 2 ) exp [ i ( l ϕ + Φ ) ] ,
[ r 2 θ 2 ] = [ 1 0 2 θ 0 r 1 1 ] [ r 1 θ 1 ] ,
[ A B C D ] = [ 1 L 0 1 ] [ 1 0 2 R 1 ] [ 1 L 0 1 ] [ 1 0 2 θ 0 r 1 1 ]
= [ 1 + 2 L R + ( 2 θ 0 r 1 ) 2 L ( 1 + L R ) 2 L ( 1 + L R ) ( 2 θ 0 r 1 ) ( 1 + 2 L R ) + 2 R 1 + 2 L R ] .
[ A B C D ] ( r θ ) = ± ( r θ ) ,
det [ A 1 B C D 1 ] = 0 .
r one = ( L + R ) θ 0 , r two = L θ 0 = a 1 2 .
u l p ( r n ) = 0 a m K l ( r m , r n ) T m ( r m ) u l p ( r m ) d r m ,
K l ( r m , r n ) = ( i ) l + 1 ( k B ) r m J l ( k B r m r n ) exp [ i k 2 B ( A r m 2 + D r n 2 ) ] ,
T 1 ( r 1 ) = exp ( i 2 k θ 0 r 1 ) , r 1 a 1 ,
T 2 ( r 2 ) = exp ( i k r 2 2 R ) , r 2 a 2 ,
u l p ( r 2 ) = 0 a 1 K l ( r 1 , r 2 ) T 1 ( r 1 ) u l p ( r 1 ) d r 1 ,
u l p ( r 3 ) = 0 a 2 K l ( r 2 , r 3 ) T 2 ( r 2 ) u l p ( r 2 ) d r 2 .
u l p ( r 3 ) = 0 a 1 H l 13 ( r 1 , r 3 ) u l p ( r 1 ) d r 1 ,
H l 13 ( r 1 , r 3 ) = 0 a 2 H l 23 ( r 2 , r 3 ) H l 12 ( r 1 , r 2 ) d r 2 ,
γ l p u l p ( r 3 ) = 0 a 1 H l 13 ( r 1 , r 3 ) u l p ( r 1 ) d r 1 ,
γ l p = γ l p exp ( i β l p )
Γ l p = 1 γ l p 2
Φ = k L + β l p = q π ,
γ u l p = ( H W ) u l p ,
γ u l p = ( P 23 P 12 ) u l p ,
Δ ν = ν 0 ( Δ β π ) ,

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