Abstract

The developed iteration algorithm for simulation of lasers with an open resonator was employed in the study of transverse mode formation. The simulations of an axially symmetrical 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 model also includes an active medium homogeneous along the resonator axis that is represented by the formula for saturating gain. The calculations were performed for two types of lasers: with on-axis and off-axis gain maximum. In the first type of laser one can obtain either a principal mode or “multimode” generation. The latter means quasi-stationary generation with regular or chaotic oscillations. In the second type of laser high order single-mode generation is possible. Experimental results obtained on a fast axial flow 4 kW CO2 laser are also presented. They are in good agreement with the calculations.

© 2012 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2010 (1)

2008 (1)

2007 (1)

D. Toebaert, “An integrated approach to laser machine tool fabrication,” Laser User 49, 30–31 (2007).

2001 (1)

1992 (1)

1989 (2)

N. Takahashi, E. Tsuchida, and H. Sato, “Spatial variation of gain and saturation in a fast axial flow CO2 laser amplifier,” Appl. Opt. 28, 3725–3736 (1989).
[CrossRef]

E. Tsuchida and H. Sato, “Dependence of spatial gain distribution on gas-flow velocity and discharge current in a FAF CO2 laser amplifier,” Jpn. J. Appl. Phys. 28, 396–405(1989).
[CrossRef]

1970 (1)

1965 (1)

W. W. Rigrod, “Saturation effects in high-gain laser,” J. Appl. Phys. 36, 2487–2490 (1965).
[CrossRef]

1924 (1)

W. Schottky and J. Issendoff, “Über die Whärmewirkung kathodischer Gehäuseströme in Quecksilberentladungen,” Z. Phys. A Hadrons Nuclei 26, 85–94 (1924).

Anan’ev, Y. A.

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

Capjack, C. E.

Degnan, J. J.

Endo, Masamori

Grishaev, R. V.

Haus, H. A.

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

Hodgson, N.

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

Issendoff, J.

W. Schottky and J. Issendoff, “Über die Whärmewirkung kathodischer Gehäuseströme in Quecksilberentladungen,” Z. Phys. A Hadrons Nuclei 26, 85–94 (1924).

Katranji, Eugeny G.

Khilo, Anatol N.

Nikumb, S. K.

Niziev, V. G.

Raizer, Y. P.

Y. P. Raizer, Gas Discharge Physics (Springer Verlag, 1997).
[CrossRef]

Ramsay, I. A.

Reshef, H.

Rigrod, W. W.

W. W. Rigrod, “Saturation effects in high-gain laser,” J. Appl. Phys. 36, 2487–2490 (1965).
[CrossRef]

Ryzhevich, Anatol A.

Sato, H.

E. Tsuchida and H. Sato, “Dependence of spatial gain distribution on gas-flow velocity and discharge current in a FAF CO2 laser amplifier,” Jpn. J. Appl. Phys. 28, 396–405(1989).
[CrossRef]

N. Takahashi, E. Tsuchida, and H. Sato, “Spatial variation of gain and saturation in a fast axial flow CO2 laser amplifier,” Appl. Opt. 28, 3725–3736 (1989).
[CrossRef]

Schottky, W.

W. Schottky and J. Issendoff, “Über die Whärmewirkung kathodischer Gehäuseströme in Quecksilberentladungen,” Z. Phys. A Hadrons Nuclei 26, 85–94 (1924).

Seguin, H. J. J.

Siegman, A. E.

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

Takahashi, N.

Toebaert, D.

D. Toebaert, “An integrated approach to laser machine tool fabrication,” Laser User 49, 30–31 (2007).

Tsuchida, E.

N. Takahashi, E. Tsuchida, and H. Sato, “Spatial variation of gain and saturation in a fast axial flow CO2 laser amplifier,” Appl. Opt. 28, 3725–3736 (1989).
[CrossRef]

E. Tsuchida and H. Sato, “Dependence of spatial gain distribution on gas-flow velocity and discharge current in a FAF CO2 laser amplifier,” Jpn. J. Appl. Phys. 28, 396–405(1989).
[CrossRef]

Weber, H.

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

Yelden, E. F.

Appl. Opt. (4)

J. Appl. Phys. (1)

W. W. Rigrod, “Saturation effects in high-gain laser,” J. Appl. Phys. 36, 2487–2490 (1965).
[CrossRef]

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

Jpn. J. Appl. Phys. (1)

E. Tsuchida and H. Sato, “Dependence of spatial gain distribution on gas-flow velocity and discharge current in a FAF CO2 laser amplifier,” Jpn. J. Appl. Phys. 28, 396–405(1989).
[CrossRef]

Laser User (1)

D. Toebaert, “An integrated approach to laser machine tool fabrication,” Laser User 49, 30–31 (2007).

Opt. Lett. (1)

Z. Phys. A Hadrons Nuclei (1)

W. Schottky and J. Issendoff, “Über die Whärmewirkung kathodischer Gehäuseströme in Quecksilberentladungen,” Z. Phys. A Hadrons Nuclei 26, 85–94 (1924).

Other (5)

A. E. Siegman, Lasers (University Science Books, 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, 1992).

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

Y. P. Raizer, Gas Discharge Physics (Springer Verlag, 1997).
[CrossRef]

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Figures (10)

Fig. 1.
Fig. 1.

Evolution of the field amplitude at the mirror center in the laser for different mirror radii. Parameters: L·k=2,543,000, R1=R2=2L. SSG is presented by the formula α0·L=5.4·J0(2.405rrm).

Fig. 2.
Fig. 2.

Evolution of the field amplitude at the mirror center in the laser at “multimode” quasi-stationary generation. Parameters: rmk=6650, L·k=2,543,000. SSG is described by the following formula: α0·L=5.4·J0(2.405rrm). The upper picture (a) is for L/R1=L/R2=0.5, three-pass paraxial resonance conditions. In the center of (a) an enlarged section of field oscillations is shown. The lower figure (b) is for L/R1=L/R2=0.45. The conditions are out of paraxial resonance.

Fig. 3.
Fig. 3.

Radial distribution of SSG (left) and saturation irradiance (right). Curves 1, 2, 3, 4 correspond to α0L=2.88, 3.08, 3.38, 3.7 at the axis, respectively.

Fig. 4.
Fig. 4.

Evolution of the field amplitude at the mirror center in a symmetric resonator at low SSG. Parameters: rmk=10,000, L=254rm, R1=R2=2L. The curves are for different values of SSG: (a) α0L=2.88 (below the threshold of generation); (b) and (c) α0L=2.94 and 3.00, correspondingly (above the threshold of generation).

Fig. 5.
Fig. 5.

Evolution of the field amplitude at the mirror center in a symmetric resonator at SSG α0L(r/rm=0)=3.08. All other parameters are the same as in Fig. 4.

Fig. 6.
Fig. 6.

Evolution of the field amplitude at the mirror center in a symmetric resonator at SSG α0L(r/rm=0)=3.38 for both the curves. (a) All other parameters are the same as in Fig. 5. They correspond to the conditions of paraxial resonance. (b) Parameters of calculations: rmk=11,860, L=214rm, R1=R2=4.66L are out of the conditions of paraxial resonance.

Fig. 7.
Fig. 7.

Dependence of output power on reflectivity of the couple mirror (Rigrod’s formula [14]) as a result of numerical experiment. Parameters: rmk=10,000, L=254rm, R1=R2=2L. SSG α0L(r/rm=0)=3.08. Radial dependence of gain is presented in Fig. 3, curve 2.

Fig. 8.
Fig. 8.

Doughnut mode (a) and TEM10 mode (b) obtained experimentally. The small circle is located at the radius of minimum intensity of laser beam between peak and ring. The red color corresponds to maximum radiation intensity. The big circle shows the aperture size.

Fig. 9.
Fig. 9.

Radial dependences of the established field amplitude (up) and phase on the mirror (down). Parameters: rmk=6517, L=390rm, R1=R2=4.66·L. SSG α0L(r/rm=0)=3.4. Radial dependences of gain and saturation irradiance are presented in Fig. 3. The arrow points to the location of the phase drop.

Fig. 10.
Fig. 10.

Radial dependences of the established field amplitude (up) and phase on the mirror (down). Parameters: rmk=7500, L=339rm, R1=R2=4.66·L. SSG α0L(r/rm=0)=3.7. The radial dependences of gain and saturation irradiance are presented in Fig. 3. The arrows point to the location of the phase drop.

Equations (6)

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DEL(L,θ,r0)2πr0ikeikLLeikr022LJ0(kr0θ),
E(L,θ)0rmE0(r0)DEL(L,θ,r0)exp(ikr02R)dr0=2πeikLL0rmr0E0(r0)exp[ik(r022Lr02R)]J0(kr0θ)dr0.
α(r)=α0(r)1+I(r)/Isat(r),
E2n(L,θ)=2πeikLL0rmr0exp(ikr02L(1LR1))·exp[α0(r)·L1+[E1n1(r0)/Esat(r)]2]×(E1n1(r0)+Esn(r0))·km·J0(kr0θ)·dr0.
E1n1E2n[E1n1,R1,km]E1n[E2n,R2]E1n.
g1·g2=1+cosθ2;θ=2πKN;0KN/2,

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