Abstract

We evaluate the eigenfields of an unstable Bessel–Gauss resonator (UBGR) by use of the transfer-matrix method in which the transverse profiles and their corresponding losses of the UBGR are considered as the eigenvectors and eigenvalues of a transfer matrix so that the dominant mode fields and their losses of the UBGR can be readily extracted in terms of the matrix eigenvalue algorithm. Moreover, based on the eigenfields across two mirrors that resulted from the transfer-matrix method, we simulate the field distributions in the cavity and the propagation of output beams by means of the angular spectrum method. The computation results show that the UBGR easily produces a fundamental Bessel–Gauss mode of good quality, and the output beams retain the original Bessel–Gauss distribution during propagation.

© 2006 Optical Society of America

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References

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  1. F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
    [CrossRef]
  2. P. Pääkkönen and J. Turunen, "Resonators with Bessel-Gauss modes," Opt. Commun. 156, 359-366 (1998).
    [CrossRef]
  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, 1986-1992 (2001).
    [CrossRef]
  4. J. Rogel-Salazar, G. H. C. New, and S. Chávez-Cerda, "Bessel-Gauss beam optical resonator," Opt. Commun. 190, 117-122 (2001).
    [CrossRef]
  5. J. C. Gutiérrez-Vega, R. Rodríguez-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]
  6. C. L. Tsangaris, G. H. C. New, and J. Rogel-Salazar, "Unstable Bessel beam resonator," Opt. Commun. 223, 233-238 (2003).
    [CrossRef]
  7. 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]
  8. S. A. Collins, Jr., "Lens-system diffraction integral written in terms of matrix optics," J. Opt. Soc. Am. 60, 1168-1177 (1970).
    [CrossRef]
  9. W. Zaifu, W. Runwen, and W. Zhijiang, "Numerical analysis of mode-fields of unstable ring resonators 90° beam rotation," Acta Opt. Sin. 15, 696-702 (1995).
  10. 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).
  11. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  12. P. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, 1966).
  13. M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, 1997).

2005 (1)

2003 (2)

2001 (2)

A. N. Khilo, E. G. Katranji, and A. A. Ryzhevich, "Axicon-based 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 (1)

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

1995 (1)

W. Zaifu, W. Runwen, and W. Zhijiang, "Numerical analysis of mode-fields of unstable ring resonators 90° beam rotation," Acta Opt. Sin. 15, 696-702 (1995).

1987 (1)

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

1970 (1)

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, 1997).

Chávez-Cerda, S.

Clemmow, P.

P. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, 1966).

Collins, S. A.

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).

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

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.

Hernández-Aranda, R. I.

Katranji, E. G.

Khilo, A. N.

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).

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]

Pääkkönen, P.

P. Pääkkönen 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]

Rodríguez-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]

Runwen, W.

W. Zaifu, W. Runwen, and W. Zhijiang, "Numerical analysis of mode-fields of unstable ring resonators 90° beam rotation," Acta Opt. Sin. 15, 696-702 (1995).

Ryzhevich, A. A.

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.

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

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, 1997).

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).

Zaifu, W.

W. Zaifu, W. Runwen, and W. Zhijiang, "Numerical analysis of mode-fields of unstable ring resonators 90° beam rotation," Acta Opt. Sin. 15, 696-702 (1995).

Zhijiang, W.

W. Zaifu, W. Runwen, and W. Zhijiang, "Numerical analysis of mode-fields of unstable ring resonators 90° beam rotation," Acta Opt. Sin. 15, 696-702 (1995).

Acta Opt. Sin. (1)

W. Zaifu, W. Runwen, and W. Zhijiang, "Numerical analysis of mode-fields of unstable ring resonators 90° beam rotation," Acta Opt. Sin. 15, 696-702 (1995).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Commun. (4)

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

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

Other (4)

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).

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

P. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, 1966).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, 1997).

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Figures (9)

Fig. 1
Fig. 1

Geometric arrangement of an UBGR.

Fig. 2
Fig. 2

Computing model for the fields across two mirrors of the UBGR. Interfaces 1 and 3 are just before the axicon, and interface 2 is just before the output coupler.

Fig. 3
Fig. 3

Field distributions across the plane just before the axicon: (a) E 1(r 1, φ1)00 and (b) E 1(r 1, φ1)20.

Fig. 4
Fig. 4

Field distributions across the plane just before the output mirror: (a) E 2(r 2, φ2)00 and (b) E 2(r 2, φ2)20.

Fig. 5
Fig. 5

Radial amplitudes across the plane just before the output mirror: (a) E 2(r 2)00 and (b) E 2(r 2)20.

Fig. 6
Fig. 6

(Color online) Radial distributions of the intracavity fields from the plane just after the axicon to the convex-spherical mirror: (a) E 1(r, z)00 and (b) E 1(r, z)20.

Fig. 7
Fig. 7

Intracavity field profiles of E 1(x, y)00 across the (a) z = L∕3 and (b) z = 2L∕3 planes.

Fig. 8
Fig. 8

(Color online) Radial distributions of the output beams from the plane just after the convex-spherical mirror to the z = L plane: (a) E 2(r, z)00 and (b) E 2(r, z)20.

Fig. 9
Fig. 9

Output field profiles of E 2 (x, y)00 across the (a) z = L∕2 and (b) z = L planes.

Tables (1)

Tables Icon

Table 1 Eigenvalues and Losses of Dominant Modes

Equations (28)

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E 2 ( r 2 , φ 2 ) = i k exp ( i k L ) 2 π B 1 S 1 E 1 ( r 1 , φ 1 ) × exp { i 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 ) = i k exp ( i k L ) 2 π B 2 S 2 E 2 ( r 2 , φ 2 ) × exp { i 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 ( i n φ 1 ) ,
E 2 ( r 2 , φ 2 ) = E 2 ( r 2 ) exp ( i n φ 2 ) ,
E 1 ( r 1 , φ 1 ) = E 1 ( r 1 ) exp ( i n φ 1 ) ,
E 2 ( r 2 ) = ( i ) n + 1 k exp ( i k L ) B 1 0 a 2 E 1 ( r 1 ) J n ( k r 1 r 2 B 1 ) × exp [ i k 2 B 1 ( A 1 r 1     2 + D 1 r 2     2 ) ] r 1 d r 1 ,
E 1 ( r 1 ) = ( i ) n + 1 k exp ( i k L ) B 2 0 a 1 E 2 ( r 2 ) J n ( k r 1 r 2 B 2 ) × exp [ i k 2 B 2 ( A 2 r 2     2 + D 2 r 1     2 ) ] r 2 d r 2 ,
T 1 = [ A 2 B 2 C 2 D 2 ] = [ 1 2 θ 0 L / r 1 L 2 θ 0 / r 1 1 ] ,
T 2 = [ A 1 B 1 C 1 D 1 ] = [ 1 2 L / R L 2 / R 1 ] ,
θ 0 = arcsin ( n sin α ) α ( n 1 ) α .
E 2 ( r 2 ) m = n = 1 M U m n E 1 ( r 1 ) n ,
E 1 ( r 1 ) m = n = 1 M V m n E 2 ( r 2 ) n ,
U m n = ( i ) n n + 1 k n a 1     2 exp ( i k L ) B 1 M 2 J n n ( k m n a 1 a 2 B 1 M 2 ) × exp [ i π B 1 λ M 2 ( A 1 a 1     2 n 2 + D 1 a 2     2 m 2 ) ] ,
V m n = ( i ) n n + 1 k n a 2     2 exp ( i k L ) B 2 M 2 J n n ( k m n a 1 a 2 B 2 M 2 ) × exp [ i π B 2 λ M 2 ( A 2 a 2     2 n 2 + D 2 a 1     2 m 2 ) ] ,
γ E 1 = W E 1 = ( V U ) E 1 ,
γ E 2 = W E 2 = ( U V ) E 2 .
U ( x , y ) = F 1 { A ( f X , f Y ) } = A ( f X , f Y ) exp [ j 2 π ( f X x + f Y y ) ] d f X d f Y .
A ( f X , f Y ) = F { U ( x , y ) } = U ( x , y ) exp [ j 2 π ( f X x + f Y y ) ] d x d y ,
A ( cos α λ , cos β λ ) = U ( x , y ) × exp [ j 2 π ( cos α λ x + cos β λ y ) ] d x d y ,
A ( f X , f Y ) = A 0 ( f X , f Y ) H ( f X , f Y ) ,
H ( f X , f Y ) =
{ exp [ j 2 z π λ 1 - ( λ f X ) 2 ( λ f Y ) 2 ] f X       2 + f Y       2 < 1 / λ 2 0 f X      2 + f Y      2 1 / λ 2 ,
U ( x , y ) = F 1 { A 0 ( f X , f Y ) × exp [ j 2 z π λ 1 ( λ f X ) 2 ( λ f Y ) 2 ] } ,
U ( x , y ) = F 1 { F { U 0 ( x , y ) } × exp [ j 2 z π λ 1 ( λ f X ) 2 ( λ f Y ) 2 ] } .
E in ( x , y ) = F 1 { F { E 1 ( x 1 , y 1 ) T 1 ( x 1 , y 1 ) } × exp [ j 2 L m π M λ 1 ( λ f X ) 2 ( λ f Y ) 2 ] } ,
T 1 ( x 1 , y 1 ) = exp ( j k θ 0 x 1     2 + y 1     2 ) .
E out ( x , y ) = F 1 { F { E 2 ( x 2 , y 2 ) T 2 ( x 2 , y 2 ) } × exp [ j 2 L m π M λ 1 ( λ f X ) 2 ( λ f Y ) 2 ] } ,
T 2 ( x 2 , y 2 ) = exp [ j k ( x 2     2 + y 2     2 ) / R ] ,

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