Abstract

We developed an iteration algorithm for open resonator simulation and employed it in studying the dynamics of mode formation. Simulations of an axially symmetrical empty resonator rely on an analytical description of radiation diffraction from a narrow ring. Reflection of an incident wave with a specified amplitude-phase distribution from the mirror is calculated by the Green function method. The process of mode formation is characterized by relaxation oscillations of various frequencies depending on the resonator parameters. The evolution of the relaxation oscillation amplitude can be aperiodic in nature, or it can occur as beats of a different frequency. It has been shown that there is a consistency between the known conditions of paraxial resonance obtained in the approximation of geometric optics and the aperiodic processes of evolution of relaxation oscillation amplitude in mode forming. An investigation has been performed on the factors affecting the time of mode formation. The possibility has been shown for multipass mode suppression and TEM10 mode generation by the use of an absorber mask on the resonator mirror.

© 2010 Optical Society of America

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    [CrossRef] [PubMed]
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2010 (1)

C. H. Chen, P. Y. Huang, and C. W. Kuo, “Geometric modes outside the multi-bouncing fundamental Gaussian beam model,” J. Opt. 12, 015708 (2010).
[CrossRef]

2008 (1)

2007 (1)

2005 (2)

2003 (1)

2001 (3)

H. H. Wu and W. F. Hsieh, “Observations of multipass transverse modes in an axially pumped solid-state laser with different fractionally degenerate resonator configurations,” J.  Opt. Soc. Am. B 18, 7–12 (2001).
[CrossRef]

J. Dingjan, M. P. Exter, and J. P. Woerdman, “Geometric modes in a single-frequency Nd:YVO4 laser,” Opt. Commun. 188, 345–351 (2001).
[CrossRef]

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

1996 (1)

X. Li, Y. Chen, T. Li, Y. He, Y. Gao, H. Cao, andS. Deng, “Development of red internal mirror He–Ne lasers with near-critical concave–convex stable resonator,” Proc. SPIE 2889, 358–366 (1996).
[CrossRef]

1992 (1)

1970 (1)

1966 (1)

1961 (1)

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

Anan’ev, Y. A.

Y. A. Anan’ev, Laser Resonators and the Beam Divergence Problem (Institute of Physics Publishing, 1992).

Anatol, A. R.

Anatol, N. K.

Cao, H.

X. Li, Y. Chen, T. Li, Y. He, Y. Gao, H. Cao, andS. Deng, “Development of red internal mirror He–Ne lasers with near-critical concave–convex stable resonator,” Proc. SPIE 2889, 358–366 (1996).
[CrossRef]

Capjack, C. E.

Chen, C. H.

Chen, Y.

X. Li, Y. Chen, T. Li, Y. He, Y. Gao, H. Cao, andS. Deng, “Development of red internal mirror He–Ne lasers with near-critical concave–convex stable resonator,” Proc. SPIE 2889, 358–366 (1996).
[CrossRef]

Chiu, C. F.

Crosignani, B.

S. Solimeno, B. Crosignani, and P. Guiding DiPorto, Diffraction and Confinement of Optical Radiation (Academic, 1986).

Davidson, N.

Degnan, J. J.

Deng, S.

X. Li, Y. Chen, T. Li, Y. He, Y. Gao, H. Cao, andS. Deng, “Development of red internal mirror He–Ne lasers with near-critical concave–convex stable resonator,” Proc. SPIE 2889, 358–366 (1996).
[CrossRef]

Dingjan, J.

J. Dingjan, M. P. Exter, and J. P. Woerdman, “Geometric modes in a single-frequency Nd:YVO4 laser,” Opt. Commun. 188, 345–351 (2001).
[CrossRef]

DiPorto, P. Guiding

S. Solimeno, B. Crosignani, and P. Guiding DiPorto, Diffraction and Confinement of Optical Radiation (Academic, 1986).

Endo, M.

Exter, M. P.

J. Dingjan, M. P. Exter, and J. P. Woerdman, “Geometric modes in a single-frequency Nd:YVO4 laser,” Opt. Commun. 188, 345–351 (2001).
[CrossRef]

Fox, A. G.

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

Friesem, A. A.

Gao, Y.

X. Li, Y. Chen, T. Li, Y. He, Y. Gao, H. Cao, andS. Deng, “Development of red internal mirror He–Ne lasers with near-critical concave–convex stable resonator,” Proc. SPIE 2889, 358–366 (1996).
[CrossRef]

Haus, H. A.

H. A. HausWaves and Fields in Optoelectronics (Prentice-Hall, 1984).

He, Y.

X. Li, Y. Chen, T. Li, Y. He, Y. Gao, H. Cao, andS. Deng, “Development of red internal mirror He–Ne lasers with near-critical concave–convex stable resonator,” Proc. SPIE 2889, 358–366 (1996).
[CrossRef]

Hodgson, N.

N. Hodgson and H. Weber, Optical Resonators: Fundamentals, Advanced Concepts and Applications (Springer Verlag, 1997).

Hsieh, W. F.

C. H. Chen, P. T. Tai, M. D. Wei, and W. F. Hsieh, “Multibeam-waist modes in an end-pumped Nd-YVO4 laser,” J. Opt. Soc. Am. B 20, 1220–1226 (2003).
[CrossRef]

H. H. Wu and W. F. Hsieh, “Observations of multipass transverse modes in an axially pumped solid-state laser with different fractionally degenerate resonator configurations,” J.  Opt. Soc. Am. B 18, 7–12 (2001).
[CrossRef]

Huang, P. Y.

C. H. Chen, P. Y. Huang, and C. W. Kuo, “Geometric modes outside the multi-bouncing fundamental Gaussian beam model,” J. Opt. 12, 015708 (2010).
[CrossRef]

Ishaaya, A. A.

Katranji, G. E.

Kogelnik, H.

Kuo, C. W.

C. H. Chen, P. Y. Huang, and C. W. Kuo, “Geometric modes outside the multi-bouncing fundamental Gaussian beam model,” J. Opt. 12, 015708 (2010).
[CrossRef]

Li, T.

X. Li, Y. Chen, T. Li, Y. He, Y. Gao, H. Cao, andS. Deng, “Development of red internal mirror He–Ne lasers with near-critical concave–convex stable resonator,” Proc. SPIE 2889, 358–366 (1996).
[CrossRef]

H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
[CrossRef] [PubMed]

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

Li, X.

X. Li, Y. Chen, T. Li, Y. He, Y. Gao, H. Cao, andS. Deng, “Development of red internal mirror He–Ne lasers with near-critical concave–convex stable resonator,” Proc. SPIE 2889, 358–366 (1996).
[CrossRef]

Nesterov, A. V.

A. V. Nesterov and V. G. Niziev, “Vector solution of the diffraction task using the Hertz vector,” Phys. Rev. E 71, 046608(2005).
[CrossRef]

Nikumb, S. K.

Niziev, V. G.

A. V. Nesterov and V. G. Niziev, “Vector solution of the diffraction task using the Hertz vector,” Phys. Rev. E 71, 046608(2005).
[CrossRef]

Ramsay, I. A.

Reshef, H.

Seguin, H. J. J.

Siegman, A. E.

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

Solimeno, S.

S. Solimeno, B. Crosignani, and P. Guiding DiPorto, Diffraction and Confinement of Optical Radiation (Academic, 1986).

Tai, P. T.

Weber, H.

N. Hodgson and H. Weber, Optical Resonators: Fundamentals, Advanced Concepts and Applications (Springer Verlag, 1997).

Wei, M. D.

Woerdman, J. P.

J. Dingjan, M. P. Exter, and J. P. Woerdman, “Geometric modes in a single-frequency Nd:YVO4 laser,” Opt. Commun. 188, 345–351 (2001).
[CrossRef]

Wu, H. H.

H. H. Wu and W. F. Hsieh, “Observations of multipass transverse modes in an axially pumped solid-state laser with different fractionally degenerate resonator configurations,” J.  Opt. Soc. Am. B 18, 7–12 (2001).
[CrossRef]

Yelden, E. F.

Appl. Opt. (3)

Bell Syst. Tech. J. (1)

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

J. Opt. (1)

C. H. Chen, P. Y. Huang, and C. W. Kuo, “Geometric modes outside the multi-bouncing fundamental Gaussian beam model,” J. Opt. 12, 015708 (2010).
[CrossRef]

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

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

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

H. H. Wu and W. F. Hsieh, “Observations of multipass transverse modes in an axially pumped solid-state laser with different fractionally degenerate resonator configurations,” J.  Opt. Soc. Am. B 18, 7–12 (2001).
[CrossRef]

Opt. Commun. (1)

J. Dingjan, M. P. Exter, and J. P. Woerdman, “Geometric modes in a single-frequency Nd:YVO4 laser,” Opt. Commun. 188, 345–351 (2001).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. E (1)

A. V. Nesterov and V. G. Niziev, “Vector solution of the diffraction task using the Hertz vector,” Phys. Rev. E 71, 046608(2005).
[CrossRef]

Proc. SPIE (1)

X. Li, Y. Chen, T. Li, Y. He, Y. Gao, H. Cao, andS. Deng, “Development of red internal mirror He–Ne lasers with near-critical concave–convex stable resonator,” Proc. SPIE 2889, 358–366 (1996).
[CrossRef]

Other (5)

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

N. Hodgson and H. Weber, Optical Resonators: Fundamentals, Advanced Concepts and Applications (Springer Verlag, 1997).

Y. A. Anan’ev, Laser Resonators and the Beam Divergence Problem (Institute of Physics Publishing, 1992).

H. A. HausWaves and Fields in Optoelectronics (Prentice-Hall, 1984).

S. Solimeno, B. Crosignani, and P. Guiding DiPorto, Diffraction and Confinement of Optical Radiation (Academic, 1986).

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Figures (10)

Fig. 1
Fig. 1

Explanation of the calculation procedure. The field can be specified (or calculated) on one of the three surfaces shown.

Fig. 2
Fig. 2

Amplitude of the field reflected from the mirror at its axis as a function of the distance from the mirror. Dimensionless parameters: r m · k = 2.2 × 10 3 and R · k = 3.52 × 10 5 (k is the wave vector).

Fig. 3
Fig. 3

Amplitude of the field reflected from the mirror on the sphere of the L radius passing through the focus of the mirror as a function of the radius. Calculated parameters: r m · k = 2 × 10 3 , L · k = 2 × 10 5 , and R = 2 · L .

Fig. 4
Fig. 4

Radius dependence of the field amplitude at the mirror. The column of figures to the right shows the number of bounces. Dimensionless parameters: r m · k = 3 × 10 3 and R 1 = R 2 = 2 · L . Two groups of curves correspond to different resonator lengths: a, L = 200 · r m and b, L = 500 · r m . The initial field amplitude is uniform; the front is flat.

Fig. 5
Fig. 5

Field amplitude on the second mirror at r = 0.14 · r m . Parameters: r m · k = 2900 and L = 195 · r m . The initial field amplitude is uniform; the front is flat. a, R 1 = R 2 = 2.5 · L , b, R 1 = 2.5 · L , R 2 = , and c R 1 = 2.5 · L , R 2 = 2.5 · L .

Fig. 6
Fig. 6

Evolution of the field amplitude on the second mirror at r = 0.14 · r m . Parameters: r m · k = 2900 , L = 2.12 · r m , and R 1 = R 2 = 2.6 · L . The curves are for different initial field distributions. For a–c, the initial wavefront is flat. a, The field amplitude is uniform. b, The distribution is a Bessel function J 0 ( 2.405 · r / r m ) . c, The distribution is a Gaussian function exp ( r 2 / w 0 2 ) , w 0 / r m = 0.413 . d, The field distribution is the same as in the previous case. The wavefront coincides with the mirror surface.

Fig. 7
Fig. 7

Evolution of the field amplitude on the second mirror at r = 0.14 · r m . Parameters: r m · k = 2900 and L = 195 · r m . The second mirror is flat. The initial field amplitude is uniform; the front is flat. a, R 1 = 2.0 · L , b, R 1 = 2.1 · L , and c, R 1 = 2.3 · L .

Fig. 8
Fig. 8

Examples of paraxial resonances. The field amplitude at the mirror center as a function of the number of bounces. The points on the curve correspond to consequent reflections from the mirror. a, Seven-pass resonance at g 1 = g 2 = 0.222 . b, Five-pass resonance at g 1 = g 2 = 0.309 . c, Five-pass resonance at g 1 = g 2 = 0.809 . d, Three-pass resonance at g 1 = g 2 = 0.5 .

Fig. 9
Fig. 9

Effect of the kind of the initial field distribution upon the mode formation. Temporal evolution of the field at the center of the mirror a, b, radial distribution of the field in the c, d, steady state mode. Parameters: r m · k = 2450 , L = 231 · r m , and g 1 = g 2 = 0.309 . a, The initial wave has a wavefront matching the mirror surface and the field distribution in the form of the TEM 10 mode at w 0 = 0.445 · r m . b, The same parameters as in a, except for the wavefront; here it is flat.

Fig. 10
Fig. 10

Effect of the absorbing mask at the mirror upon mode formation. a, Temporal evolution of the field at the center of the mirror and b, radial distribution of the field in the steady state mode. Parameters: r m · k = 2450 , L = 231 · r m , and g 1 = g 2 = 0.309 . The initial field amplitude is uniform; the front is flat. The radius of the absorbing ring is w 1 / 2 = 0.31 · r m ; it coincides with the zero field location for the TEM 10 mode. The area of the absorbing ring comprises 1% of that of the mirror.

Equations (13)

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DEL ( L , θ , r 0 ) 2 π r 0 i k e i k L L e i k r 0 2 2 L J 0 ( k r 0 θ ) .
E ( L , θ ) 0 r m E 0 ( r 0 ) DEL ( L , θ , r 0 ) exp ( i k r 0 2 R ) d r 0 = 2 π e i k L L 0 r m r 0 E 0 ( r 0 ) exp [ i k 2 ( r 0 2 L 2 r 0 2 R ) ] J 0 ( k r 0 θ ) d r 0 .
exp ( i k r 0 2 L ( 1 L R i ) ) ,
| E | F · sin y y ; y = π 2 F · ( 1 2 L / R ) ; F = r m 2 λ L .
E 2 F J 1 ( x ) x ; x = 2 π F r r m ; F = r m 2 λ L .
g 1 · g 2 = 1 + cos θ 2 ; θ = 2 π K N ; 0 K N / 2 ,
E ( r ) = 0 2 π 0 r ( G ( | r r | ) z E ( ρ , φ , z ) E ( ρ , φ , z ) z G ( | r r | ) ) ρ · d φ · d r ,
E = i k 0 2 π 0 r m GE 0 ( ρ ) · ( 1 z | r r | ) ρ · d φ · d ρ .
| r r | r ( 1 x · x + y · y r 2 + x 2 + y 2 2 · r 2 ) = r ( 1 ρ r sin θ cos ( φ φ ) + ρ 2 2 · r 2 ) .
E ( r ) = 2 i k 0 r m 0 2 π GE 0 ( ρ ) · ρ · d φ · d ρ = 2 i k 0 r m E 0 ( ρ ) 0 2 π e i k | r r | | r r | ρ · d φ · d ρ 2 i k e i k r r 0 r m E 0 ( ρ ) 0 2 π e i k ρ sin θ cos ( φ φ ) e i k ρ 2 2 r ρ · d φ · d ρ = 2 i k e i k r r 0 r m E 0 ( ρ ) e i k ρ 2 2 r J 0 ( k ρ sin θ ) ρ d ρ .
DEL ( L , θ , r 0 ) 2 π r 0 i k e i k L L e i k r 0 2 2 L J 0 ( k r 0 θ ) .
E ( L , θ ) 0 r m r 0 E 0 ( r 0 ) DEL ( L , θ , r 0 ) exp ( i k r 0 2 R ) d r 0 .
DEA ( L , θ , r 0 ) 2 π r 0 i k e i k L L e i k r 0 2 2 L J 1 ( k r 0 θ ) .

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