Abstract

The cavity axes change when the output mirror is misaligned because these cavities possess a common output mirror. We use 4×4 augmented matrices to establish new cavity axes, which result in different line deviations for different cavities. These deviations increase the diffraction losses and decrease the output power. The coordinate transforms are used to solve the output light fields because the output beams do not have a common propagation direction. We show that the shapes of the combined laser beams change when the output mirror is misaligned. These studies provide references for the design and application of CO2 lasers.

© 2009 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  8. Y. Xu, Y. Li, T. Feng, and Y. Qiu, “Influence of misalignment of the output-mirror of axisymmetric-structural CO2 laser on self-phase-locking,” Opt. Laser Technol. 41, 897-902(2009).
    [CrossRef]
  9. Y. Qiu, Y. Li, Y. Xu, F. Fu, and W. Guo, “The influence of misaligned holophote of axisymmetrical-fold combination CO2 laser on phase-locking,” Opt. Laser Technol. 41, 70-80 (2009).
    [CrossRef]
  10. B. Lv, Laser Optics (Sichuan U. Press, 1992), in Chinese.
  11. Y. Xu, Y. Li, T. Feng, Y.I Qiu, F. Fu, and W. Guo, “Phase-locking principle of axisymmetric structural CO2 laser and theoretical study of the influences of parameters-changes on phase-locking,” J. Opt. Soc. Am. B 25, 1303-1311 (2008).
    [CrossRef]
  12. Y. Xu, Y. Li, Y. Qiu, T. Feng, F. Fu, and W. Guo, “Influences of misalignment of control-mirror of axisymmetric-structural CO2 laser on phase-locking,” Appl. Opt. 47, 6291-6299 (2008).
    [CrossRef] [PubMed]
  13. Y. Xu, Y. Li, B. Zhang, T. Feng, and Y. Qiu, “Theoretical study of misalignment analysis of the holophote and self-phase-locking of axisymmetrical-structural CO2 laser,” Opt. Lasers Eng. 47, 782-792 (2009).
    [CrossRef]

2009 (3)

Y. Xu, Y. Li, T. Feng, and Y. Qiu, “Influence of misalignment of the output-mirror of axisymmetric-structural CO2 laser on self-phase-locking,” Opt. Laser Technol. 41, 897-902(2009).
[CrossRef]

Y. Qiu, Y. Li, Y. Xu, F. Fu, and W. Guo, “The influence of misaligned holophote of axisymmetrical-fold combination CO2 laser on phase-locking,” Opt. Laser Technol. 41, 70-80 (2009).
[CrossRef]

Y. Xu, Y. Li, B. Zhang, T. Feng, and Y. Qiu, “Theoretical study of misalignment analysis of the holophote and self-phase-locking of axisymmetrical-structural CO2 laser,” Opt. Lasers Eng. 47, 782-792 (2009).
[CrossRef]

2008 (3)

2007 (1)

Y. Cai and X. Lv, “Propagation of Bessel and Bessel-Gaussian beams through an unapertured or apertured misaligned paraxial optical systems,” Opt. Commun. 274, 1-7 (2007).
[CrossRef]

2006 (1)

J. Liu, Y. Li, J. Guo, M. Chen, Y. Yang, D. Xu, Z. Liu, and L. Zhang, “Research on near field distributions of axisymmetric folded-combined CO2 laser,” Opt. Eng. 45, 074201(2006).
[CrossRef]

2005 (1)

Y. Li, J. Liu, M. Chen, and J. Guo, “Axisymmetric-fold combination laser resonator,” Opt. Eng. 44, 064204 (2005).
[CrossRef]

2001 (1)

1988 (1)

1986 (1)

Banyasz, I.

Cai, Y.

Y. Cai and X. Lv, “Propagation of Bessel and Bessel-Gaussian beams through an unapertured or apertured misaligned paraxial optical systems,” Opt. Commun. 274, 1-7 (2007).
[CrossRef]

Chen, M.

J. Liu, Y. Li, J. Guo, M. Chen, Y. Yang, D. Xu, Z. Liu, and L. Zhang, “Research on near field distributions of axisymmetric folded-combined CO2 laser,” Opt. Eng. 45, 074201(2006).
[CrossRef]

Y. Li, J. Liu, M. Chen, and J. Guo, “Axisymmetric-fold combination laser resonator,” Opt. Eng. 44, 064204 (2005).
[CrossRef]

Feng, T.

Y. Xu, Y. Li, B. Zhang, T. Feng, and Y. Qiu, “Theoretical study of misalignment analysis of the holophote and self-phase-locking of axisymmetrical-structural CO2 laser,” Opt. Lasers Eng. 47, 782-792 (2009).
[CrossRef]

Y. Xu, Y. Li, T. Feng, and Y. Qiu, “Influence of misalignment of the output-mirror of axisymmetric-structural CO2 laser on self-phase-locking,” Opt. Laser Technol. 41, 897-902(2009).
[CrossRef]

Y. Xu, Y. Li, Y. Qiu, T. Feng, F. Fu, and W. Guo, “Influences of misalignment of control-mirror of axisymmetric-structural CO2 laser on phase-locking,” Appl. Opt. 47, 6291-6299 (2008).
[CrossRef] [PubMed]

X. Liu, Y. Li, T. Yan, T. Feng, and S. Gao, “Experimental research on axisymmetrical-fold combination CO2 laser,” Opt. Commun. 281, 3553-3556 (2008).
[CrossRef]

Y. Xu, Y. Li, T. Feng, Y.I Qiu, F. Fu, and W. Guo, “Phase-locking principle of axisymmetric structural CO2 laser and theoretical study of the influences of parameters-changes on phase-locking,” J. Opt. Soc. Am. B 25, 1303-1311 (2008).
[CrossRef]

Fu, F.

Gao, S.

X. Liu, Y. Li, T. Yan, T. Feng, and S. Gao, “Experimental research on axisymmetrical-fold combination CO2 laser,” Opt. Commun. 281, 3553-3556 (2008).
[CrossRef]

Guo, J.

J. Liu, Y. Li, J. Guo, M. Chen, Y. Yang, D. Xu, Z. Liu, and L. Zhang, “Research on near field distributions of axisymmetric folded-combined CO2 laser,” Opt. Eng. 45, 074201(2006).
[CrossRef]

Y. Li, J. Liu, M. Chen, and J. Guo, “Axisymmetric-fold combination laser resonator,” Opt. Eng. 44, 064204 (2005).
[CrossRef]

Guo, W.

Kiss, G.

Li, Y.

Y. Qiu, Y. Li, Y. Xu, F. Fu, and W. Guo, “The influence of misaligned holophote of axisymmetrical-fold combination CO2 laser on phase-locking,” Opt. Laser Technol. 41, 70-80 (2009).
[CrossRef]

Y. Xu, Y. Li, B. Zhang, T. Feng, and Y. Qiu, “Theoretical study of misalignment analysis of the holophote and self-phase-locking of axisymmetrical-structural CO2 laser,” Opt. Lasers Eng. 47, 782-792 (2009).
[CrossRef]

Y. Xu, Y. Li, T. Feng, and Y. Qiu, “Influence of misalignment of the output-mirror of axisymmetric-structural CO2 laser on self-phase-locking,” Opt. Laser Technol. 41, 897-902(2009).
[CrossRef]

Y. Xu, Y. Li, Y. Qiu, T. Feng, F. Fu, and W. Guo, “Influences of misalignment of control-mirror of axisymmetric-structural CO2 laser on phase-locking,” Appl. Opt. 47, 6291-6299 (2008).
[CrossRef] [PubMed]

X. Liu, Y. Li, T. Yan, T. Feng, and S. Gao, “Experimental research on axisymmetrical-fold combination CO2 laser,” Opt. Commun. 281, 3553-3556 (2008).
[CrossRef]

Y. Xu, Y. Li, T. Feng, Y.I Qiu, F. Fu, and W. Guo, “Phase-locking principle of axisymmetric structural CO2 laser and theoretical study of the influences of parameters-changes on phase-locking,” J. Opt. Soc. Am. B 25, 1303-1311 (2008).
[CrossRef]

J. Liu, Y. Li, J. Guo, M. Chen, Y. Yang, D. Xu, Z. Liu, and L. Zhang, “Research on near field distributions of axisymmetric folded-combined CO2 laser,” Opt. Eng. 45, 074201(2006).
[CrossRef]

Y. Li, J. Liu, M. Chen, and J. Guo, “Axisymmetric-fold combination laser resonator,” Opt. Eng. 44, 064204 (2005).
[CrossRef]

Liu, J.

J. Liu, Y. Li, J. Guo, M. Chen, Y. Yang, D. Xu, Z. Liu, and L. Zhang, “Research on near field distributions of axisymmetric folded-combined CO2 laser,” Opt. Eng. 45, 074201(2006).
[CrossRef]

Y. Li, J. Liu, M. Chen, and J. Guo, “Axisymmetric-fold combination laser resonator,” Opt. Eng. 44, 064204 (2005).
[CrossRef]

Liu, Q.

Liu, X.

X. Liu, Y. Li, T. Yan, T. Feng, and S. Gao, “Experimental research on axisymmetrical-fold combination CO2 laser,” Opt. Commun. 281, 3553-3556 (2008).
[CrossRef]

Liu, Z.

J. Liu, Y. Li, J. Guo, M. Chen, Y. Yang, D. Xu, Z. Liu, and L. Zhang, “Research on near field distributions of axisymmetric folded-combined CO2 laser,” Opt. Eng. 45, 074201(2006).
[CrossRef]

Lv, B.

B. Lv, Laser Optics (Sichuan U. Press, 1992), in Chinese.

Lv, X.

Y. Cai and X. Lv, “Propagation of Bessel and Bessel-Gaussian beams through an unapertured or apertured misaligned paraxial optical systems,” Opt. Commun. 274, 1-7 (2007).
[CrossRef]

McMillan, C. F.

Ohba, R.

Qiu, Y.

Y. Xu, Y. Li, T. Feng, and Y. Qiu, “Influence of misalignment of the output-mirror of axisymmetric-structural CO2 laser on self-phase-locking,” Opt. Laser Technol. 41, 897-902(2009).
[CrossRef]

Y. Qiu, Y. Li, Y. Xu, F. Fu, and W. Guo, “The influence of misaligned holophote of axisymmetrical-fold combination CO2 laser on phase-locking,” Opt. Laser Technol. 41, 70-80 (2009).
[CrossRef]

Y. Xu, Y. Li, B. Zhang, T. Feng, and Y. Qiu, “Theoretical study of misalignment analysis of the holophote and self-phase-locking of axisymmetrical-structural CO2 laser,” Opt. Lasers Eng. 47, 782-792 (2009).
[CrossRef]

Y. Xu, Y. Li, Y. Qiu, T. Feng, F. Fu, and W. Guo, “Influences of misalignment of control-mirror of axisymmetric-structural CO2 laser on phase-locking,” Appl. Opt. 47, 6291-6299 (2008).
[CrossRef] [PubMed]

Qiu, Y.I

Varga, P.

Xu, D.

J. Liu, Y. Li, J. Guo, M. Chen, Y. Yang, D. Xu, Z. Liu, and L. Zhang, “Research on near field distributions of axisymmetric folded-combined CO2 laser,” Opt. Eng. 45, 074201(2006).
[CrossRef]

Xu, Y.

Y. Xu, Y. Li, T. Feng, and Y. Qiu, “Influence of misalignment of the output-mirror of axisymmetric-structural CO2 laser on self-phase-locking,” Opt. Laser Technol. 41, 897-902(2009).
[CrossRef]

Y. Qiu, Y. Li, Y. Xu, F. Fu, and W. Guo, “The influence of misaligned holophote of axisymmetrical-fold combination CO2 laser on phase-locking,” Opt. Laser Technol. 41, 70-80 (2009).
[CrossRef]

Y. Xu, Y. Li, B. Zhang, T. Feng, and Y. Qiu, “Theoretical study of misalignment analysis of the holophote and self-phase-locking of axisymmetrical-structural CO2 laser,” Opt. Lasers Eng. 47, 782-792 (2009).
[CrossRef]

Y. Xu, Y. Li, T. Feng, Y.I Qiu, F. Fu, and W. Guo, “Phase-locking principle of axisymmetric structural CO2 laser and theoretical study of the influences of parameters-changes on phase-locking,” J. Opt. Soc. Am. B 25, 1303-1311 (2008).
[CrossRef]

Y. Xu, Y. Li, Y. Qiu, T. Feng, F. Fu, and W. Guo, “Influences of misalignment of control-mirror of axisymmetric-structural CO2 laser on phase-locking,” Appl. Opt. 47, 6291-6299 (2008).
[CrossRef] [PubMed]

Yan, T.

X. Liu, Y. Li, T. Yan, T. Feng, and S. Gao, “Experimental research on axisymmetrical-fold combination CO2 laser,” Opt. Commun. 281, 3553-3556 (2008).
[CrossRef]

Yang, Y.

J. Liu, Y. Li, J. Guo, M. Chen, Y. Yang, D. Xu, Z. Liu, and L. Zhang, “Research on near field distributions of axisymmetric folded-combined CO2 laser,” Opt. Eng. 45, 074201(2006).
[CrossRef]

Zhang, B.

Y. Xu, Y. Li, B. Zhang, T. Feng, and Y. Qiu, “Theoretical study of misalignment analysis of the holophote and self-phase-locking of axisymmetrical-structural CO2 laser,” Opt. Lasers Eng. 47, 782-792 (2009).
[CrossRef]

Zhang, L.

J. Liu, Y. Li, J. Guo, M. Chen, Y. Yang, D. Xu, Z. Liu, and L. Zhang, “Research on near field distributions of axisymmetric folded-combined CO2 laser,” Opt. Eng. 45, 074201(2006).
[CrossRef]

Appl. Opt. (4)

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

Opt. Commun. (2)

X. Liu, Y. Li, T. Yan, T. Feng, and S. Gao, “Experimental research on axisymmetrical-fold combination CO2 laser,” Opt. Commun. 281, 3553-3556 (2008).
[CrossRef]

Y. Cai and X. Lv, “Propagation of Bessel and Bessel-Gaussian beams through an unapertured or apertured misaligned paraxial optical systems,” Opt. Commun. 274, 1-7 (2007).
[CrossRef]

Opt. Eng. (2)

Y. Li, J. Liu, M. Chen, and J. Guo, “Axisymmetric-fold combination laser resonator,” Opt. Eng. 44, 064204 (2005).
[CrossRef]

J. Liu, Y. Li, J. Guo, M. Chen, Y. Yang, D. Xu, Z. Liu, and L. Zhang, “Research on near field distributions of axisymmetric folded-combined CO2 laser,” Opt. Eng. 45, 074201(2006).
[CrossRef]

Opt. Laser Technol. (2)

Y. Xu, Y. Li, T. Feng, and Y. Qiu, “Influence of misalignment of the output-mirror of axisymmetric-structural CO2 laser on self-phase-locking,” Opt. Laser Technol. 41, 897-902(2009).
[CrossRef]

Y. Qiu, Y. Li, Y. Xu, F. Fu, and W. Guo, “The influence of misaligned holophote of axisymmetrical-fold combination CO2 laser on phase-locking,” Opt. Laser Technol. 41, 70-80 (2009).
[CrossRef]

Opt. Lasers Eng. (1)

Y. Xu, Y. Li, B. Zhang, T. Feng, and Y. Qiu, “Theoretical study of misalignment analysis of the holophote and self-phase-locking of axisymmetrical-structural CO2 laser,” Opt. Lasers Eng. 47, 782-792 (2009).
[CrossRef]

Other (1)

B. Lv, Laser Optics (Sichuan U. Press, 1992), in Chinese.

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Figures (2)

Fig. 1
Fig. 1

Structure of the ASFC CO 2 laser with a misaligned output mirror.

Fig. 2
Fig. 2

Light intensities and light spot profiles with the dif ferent conditions: (a)  Z = 0.03 m , α = 1 s, L i = 1.6 m , ρ i = 2.2 m , β = arcsin ( 3 / 150 ) ; (b)  Z = 0.03 m , α = 10 s, L i = 1.5 m , ρ i = 2 m , β = arcsin ( 3 / 150 ) ; (c)  Z = 0.03 m , α = 20 s, L i = 1.5 m , ρ i = 2 m , β = arcsin ( 3 / 150 ) ; (d)  Z = 0.05 m , α = 20 s, L i = 1.5 m , ρ i = 2 m , β = arcsin ( 4 / 150 ) ; (e)  Z = 0.07 m , α = 30 s, L i = 1.5 m , ρ i = 2 m , β = arcsin ( 4 / 150 ) .

Tables (3)

Tables Icon

Table 1 Values of Line Deviations X x i , Y y i , and r i

Tables Icon

Table 2 Line Deviations r 5 , 6 for Different Cavity Lengths and Mirror Curvatures when α = 20 s

Tables Icon

Table 3 Diffraction Loss and Output Power

Equations (16)

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( P p i θ p i 1 1 ) = ( 1 0 0 0 0 1 0 2 ε p i ± 1 0 0 1 0 0 0 0 1 ) ( 1 L i 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) ( 1 0 0 0 2 ρ i 1 0 0 0 0 1 0 0 0 0 1 ) ( 1 L i 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) × ( 1 0 0 0 0 1 0 2 ε p i 0 0 1 0 0 0 0 1 ) ( 1 L i 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) ( 1 0 0 0 2 ρ i 1 0 0 0 0 1 0 0 0 0 1 ) ( 1 L i 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) ( P p i θ p i 1 1 ) ,
( P p 0 θ p 0 1 1 ) = ( 1 0 0 0 0 1 0 2 ε p 0 0 0 1 0 0 0 0 1 ) ( 1 L 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) ( 1 0 0 0 2 ρ 0 1 0 0 0 0 1 0 0 0 0 1 ) ( 1 L 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) ( P p 0 θ p 0 1 1 ) ,
θ p 0 = ε p 0 , P p 0 = ( L 0 ρ 0 ) ε p 0 ,
θ p i = ε p i ± 1 , P p i = 2 L i ρ i 2 ε p i ± 1 ρ i 2 ε p i ,
r i = X x i 2 + Y y i 2 .
L 00 ( r i ) = exp [ 2 ( R r i ) 2 ω 2 ] ,
Δ L 00 = L 00 ( 0 ) L 00 ( r i ) .
P o = 14.4 π ( 1 α T ) 1 / 2 T { g 0 L 7 , 8 + ln [ ( 1 α T ) ( 1 α T T ) ] 1 / 2 } [ ( 1 α T ) 1 / 2 + ( 1 α T T ) 1 / 2 ] [ 1 ( 1 α T ) ( 1 α T T ) 1 / 2 ] ,
ε n ( x n , y n , z n ) = A 0 ω 0 ω n ( z n ) exp ( x n 2 + y n 2 ω n 2 ( z n ) ) exp { i [ k z n + k ( x n 2 + y n 2 ) 2 R n ( z n ) φ n ( z n ) ] } ,
R n ( z n ) = z n [ 1 + ( π ω 0 2 λ z n ) 2 ] , ω n 2 ( z n ) = ω 0 2 [ 1 + ( λ z n π ω 0 2 ) 2 ] , φ n ( z n ) = arctan ( λ z n π ω 0 2 ) .
x 0 = X X x 0 , y 0 = Y Y y 0 , z 0 = Z ,
x 1 , 2 = X X x 1 , 2 , y 1 , 2 = ( Y Y y 1 , 2 ) cos β ± Z sin β , z 1 , 2 = Z cos β ( Y Y y 1 , 2 ) sin β ,
y 3 , 4 = Y Y y 3 , 4 , x 3 , 4 = ( X X x 3 , 4 ) cos β ± Z sin β , z 3 , 4 = Z cos β ( X X x 3 , 4 ) sin β ,
x 5 , 6 = ( X X x 5 , 6 ) sin π 4 ( Y Y y 5 , 6 ) cos π 4 , y 5 , 6 = [ ( X X x 5 , 6 ) sin π 4 + ( Y Y y 5 , 6 ) cos π 4 ] cos β ± Z sin β , z 5 , 6 = Z cos β [ ( X X x 5 , 6 ) sin π 4 + ( Y Y y 5 , 6 ) cos π 4 ] sin β ,
y 7 , 8 = ( X X x 7 , 8 ) sin π 4 + ( Y Y y 7 , 8 ) cos π 4 , x 7 , 8 = [ ( X X x 7 , 8 ) sin π 4 ( Y Y y 7 , 8 ) cos π 4 ] cos β ± Z sin β , z 7 , 8 = Z cos β [ ( X X x 7 , 8 ) sin π 4 ( Y Y y 7 , 8 ) cos π 4 ] sin β ,
I n ( X , Y ) = ε 0 ε 0 * + ( ε 1 + ε 2 ) ( ε 1 + ε 2 ) * + ( ε 3 + ε 4 ) ( ε 3 + ε 4 ) * + ( ε 5 + ε 6 ) ( ε 5 + ε 6 ) * + ( ε 7 + ε 8 ) ( ε 7 + ε 8 ) * .

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