Abstract

We derive a new model and simulation technique called “Dynamic Multimode Analysis (DMA)” to simulate the 3-dimensional dynamic behavior of a laser. A Gaussian mode analysis is used to obtain resonator eigenmodes taking into account thermal aberrations. These modes are coupled by a set of rate equations to describe the dynamic behavior of the individual modes for cw and Q-switched lasers. Our approach analyzes mode competition and provides a detailed description of the laser beam in terms of output power, beam quality factor M2, and pulse shape. Comparison of experimental data with our simulation results provides new insight into the effect of mode competition and the operation of Q-switched lasers.

© 2009 OSA

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References

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  1. P. A. Bélanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. 16(4), 196–198 (1991).
    [CrossRef] [PubMed]
  2. A. E. Siegman, Lasers, (University Science Books, Mill Valley, 1986).
  3. O. Svelto, Principles of lasers (Springer, New York, 1998).
  4. A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–458 (1961).
  5. X. Zhang, S. Zhao, Q. Wang, B. Ozygus, and H. Weber, “Modeling of diode-pumped actively Q-switched lasers,” IEEE J. Quantum Electron. 35(12), 1912–1918 (1999).
    [CrossRef]
  6. O. A. Louchev, Y. Urata, and S. Wada, “Numerical simulation and optimization of Q-switched 2 mum Tm,Ho:YLF laser,” Opt. Express 15(7), 3940–3947 (2007).
    [CrossRef] [PubMed]
  7. G. Li, S. Zhao, H. Zhao, K. Yang, and S. Ding, “Rate equations and solutions of a laser-diode end-pumped passively Q-switched intracavity doubling laser by taking into account intracavity laser spatial distribution,” Opt. Commun. 234(1-6), 321–328 (2004).
    [CrossRef]
  8. K. Altmann, C. Pflaum, and D. Seider, “Three-dimensional finite element computation of laser cavity eigenmodes [corrected],” Appl. Opt. 43(9), 1892–1901 (2004).
    [CrossRef] [PubMed]
  9. B. Heubeck, C. Pflaum, and G. Steinle, “New finite elements for large-scale simulation of optical Waves,” SIAM J. Sci. Comput. 31(2), 1063–1081 (2009).
    [CrossRef]
  10. B. Heubeck and C. Pflaum, “Numerical simulation of multiple modes in solid state lasers,” Proc. SPIE 6190, 61900X (2006).
    [CrossRef]
  11. LAS-CAD GmbH Munich, Germany, http://www.las-cad.com/ .
  12. R. Paschotta, “Beam quality deterioration of lasers caused by intracavity beam distortions,” Opt. Express 14(13), 6069–6074 (2006).
    [CrossRef] [PubMed]
  13. W. Koechner, Solid-state laser engineering, (Springer, Berlin, 2006).
  14. T. Y. Fan, “Heat generation in Nd:YAG and Yb:YAG,” IEEE J. Quantum Electron. 29(6), 1457–1459 (1993).
    [CrossRef]
  15. Q. Zhang, B. Ozygus, and H. Weber, “Degeneration effects in laser cavities,” Eur. Phys. J. Appl. Phys. 6(3), 293–298 (1999).
    [CrossRef]
  16. S. Wang, X. Wang, T. Riesbeck, and H. J. Eichler, “Thermal lensing effects in pulsed end pumped Nd lasers at 940 nm,” Proc. SPIE 7194, 71940J (2009).
    [CrossRef]
  17. R. J. LeVeque, Finite volume methods for hyperbolic problems (Cambridge Texts in Applied Mathematics, 2007).
  18. E. Hairer, and G. Wanner, Solving ordinary differential Eqs. (2): Stiff and differential-algebraic problems, (Springer, Berlin, 1996).
  19. A. E. Siegman and S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29(4), 1212–1217 (1993).
    [CrossRef]
  20. International Organization for Standardization EN ISO 11146:2000 and EN ISO 11146:1999,” Laser and laser-related equipment. Test methods for laser beam parameters. Beam width, divergence angle and beam propagation factor,” (ISO, Geneva, 2000).
  21. F. Träger, Springer Handbook of Lasers and Optics (Springer, Berlin, 2007), Chap. 11.2.
  22. J. J. Degnan, “Theory of the Optimally Coupled Q-Switched Laser,” IEEE J. Quantum Electron. 25(2), 214–220 (1989).
    [CrossRef]
  23. R. B. Kay and D. Poulios, “Q-Switched Rate Equations for Diode Side-Pumped Slab and Zigzag Slab Lasers Including Gaussian Beam Shapes,” IEEE J. Quantum Electron. 41(10), 1278–1284 (2005).
    [CrossRef]

2009 (2)

B. Heubeck, C. Pflaum, and G. Steinle, “New finite elements for large-scale simulation of optical Waves,” SIAM J. Sci. Comput. 31(2), 1063–1081 (2009).
[CrossRef]

S. Wang, X. Wang, T. Riesbeck, and H. J. Eichler, “Thermal lensing effects in pulsed end pumped Nd lasers at 940 nm,” Proc. SPIE 7194, 71940J (2009).
[CrossRef]

2007 (1)

2006 (2)

B. Heubeck and C. Pflaum, “Numerical simulation of multiple modes in solid state lasers,” Proc. SPIE 6190, 61900X (2006).
[CrossRef]

R. Paschotta, “Beam quality deterioration of lasers caused by intracavity beam distortions,” Opt. Express 14(13), 6069–6074 (2006).
[CrossRef] [PubMed]

2005 (1)

R. B. Kay and D. Poulios, “Q-Switched Rate Equations for Diode Side-Pumped Slab and Zigzag Slab Lasers Including Gaussian Beam Shapes,” IEEE J. Quantum Electron. 41(10), 1278–1284 (2005).
[CrossRef]

2004 (2)

G. Li, S. Zhao, H. Zhao, K. Yang, and S. Ding, “Rate equations and solutions of a laser-diode end-pumped passively Q-switched intracavity doubling laser by taking into account intracavity laser spatial distribution,” Opt. Commun. 234(1-6), 321–328 (2004).
[CrossRef]

K. Altmann, C. Pflaum, and D. Seider, “Three-dimensional finite element computation of laser cavity eigenmodes [corrected],” Appl. Opt. 43(9), 1892–1901 (2004).
[CrossRef] [PubMed]

1999 (2)

X. Zhang, S. Zhao, Q. Wang, B. Ozygus, and H. Weber, “Modeling of diode-pumped actively Q-switched lasers,” IEEE J. Quantum Electron. 35(12), 1912–1918 (1999).
[CrossRef]

Q. Zhang, B. Ozygus, and H. Weber, “Degeneration effects in laser cavities,” Eur. Phys. J. Appl. Phys. 6(3), 293–298 (1999).
[CrossRef]

1993 (2)

A. E. Siegman and S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29(4), 1212–1217 (1993).
[CrossRef]

T. Y. Fan, “Heat generation in Nd:YAG and Yb:YAG,” IEEE J. Quantum Electron. 29(6), 1457–1459 (1993).
[CrossRef]

1991 (1)

1989 (1)

J. J. Degnan, “Theory of the Optimally Coupled Q-Switched Laser,” IEEE J. Quantum Electron. 25(2), 214–220 (1989).
[CrossRef]

1961 (1)

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

Altmann, K.

Bélanger, P. A.

Degnan, J. J.

J. J. Degnan, “Theory of the Optimally Coupled Q-Switched Laser,” IEEE J. Quantum Electron. 25(2), 214–220 (1989).
[CrossRef]

Ding, S.

G. Li, S. Zhao, H. Zhao, K. Yang, and S. Ding, “Rate equations and solutions of a laser-diode end-pumped passively Q-switched intracavity doubling laser by taking into account intracavity laser spatial distribution,” Opt. Commun. 234(1-6), 321–328 (2004).
[CrossRef]

Eichler, H. J.

S. Wang, X. Wang, T. Riesbeck, and H. J. Eichler, “Thermal lensing effects in pulsed end pumped Nd lasers at 940 nm,” Proc. SPIE 7194, 71940J (2009).
[CrossRef]

Fan, T. Y.

T. Y. Fan, “Heat generation in Nd:YAG and Yb:YAG,” IEEE J. Quantum Electron. 29(6), 1457–1459 (1993).
[CrossRef]

Fox, A. G.

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

Heubeck, B.

B. Heubeck, C. Pflaum, and G. Steinle, “New finite elements for large-scale simulation of optical Waves,” SIAM J. Sci. Comput. 31(2), 1063–1081 (2009).
[CrossRef]

B. Heubeck and C. Pflaum, “Numerical simulation of multiple modes in solid state lasers,” Proc. SPIE 6190, 61900X (2006).
[CrossRef]

Kay, R. B.

R. B. Kay and D. Poulios, “Q-Switched Rate Equations for Diode Side-Pumped Slab and Zigzag Slab Lasers Including Gaussian Beam Shapes,” IEEE J. Quantum Electron. 41(10), 1278–1284 (2005).
[CrossRef]

Li, G.

G. Li, S. Zhao, H. Zhao, K. Yang, and S. Ding, “Rate equations and solutions of a laser-diode end-pumped passively Q-switched intracavity doubling laser by taking into account intracavity laser spatial distribution,” Opt. Commun. 234(1-6), 321–328 (2004).
[CrossRef]

Li, T.

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

Louchev, O. A.

Ozygus, B.

X. Zhang, S. Zhao, Q. Wang, B. Ozygus, and H. Weber, “Modeling of diode-pumped actively Q-switched lasers,” IEEE J. Quantum Electron. 35(12), 1912–1918 (1999).
[CrossRef]

Q. Zhang, B. Ozygus, and H. Weber, “Degeneration effects in laser cavities,” Eur. Phys. J. Appl. Phys. 6(3), 293–298 (1999).
[CrossRef]

Paschotta, R.

Pflaum, C.

B. Heubeck, C. Pflaum, and G. Steinle, “New finite elements for large-scale simulation of optical Waves,” SIAM J. Sci. Comput. 31(2), 1063–1081 (2009).
[CrossRef]

B. Heubeck and C. Pflaum, “Numerical simulation of multiple modes in solid state lasers,” Proc. SPIE 6190, 61900X (2006).
[CrossRef]

K. Altmann, C. Pflaum, and D. Seider, “Three-dimensional finite element computation of laser cavity eigenmodes [corrected],” Appl. Opt. 43(9), 1892–1901 (2004).
[CrossRef] [PubMed]

Poulios, D.

R. B. Kay and D. Poulios, “Q-Switched Rate Equations for Diode Side-Pumped Slab and Zigzag Slab Lasers Including Gaussian Beam Shapes,” IEEE J. Quantum Electron. 41(10), 1278–1284 (2005).
[CrossRef]

Riesbeck, T.

S. Wang, X. Wang, T. Riesbeck, and H. J. Eichler, “Thermal lensing effects in pulsed end pumped Nd lasers at 940 nm,” Proc. SPIE 7194, 71940J (2009).
[CrossRef]

Seider, D.

Siegman, A. E.

A. E. Siegman and S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29(4), 1212–1217 (1993).
[CrossRef]

Steinle, G.

B. Heubeck, C. Pflaum, and G. Steinle, “New finite elements for large-scale simulation of optical Waves,” SIAM J. Sci. Comput. 31(2), 1063–1081 (2009).
[CrossRef]

Townsend, S. W.

A. E. Siegman and S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29(4), 1212–1217 (1993).
[CrossRef]

Urata, Y.

Wada, S.

Wang, Q.

X. Zhang, S. Zhao, Q. Wang, B. Ozygus, and H. Weber, “Modeling of diode-pumped actively Q-switched lasers,” IEEE J. Quantum Electron. 35(12), 1912–1918 (1999).
[CrossRef]

Wang, S.

S. Wang, X. Wang, T. Riesbeck, and H. J. Eichler, “Thermal lensing effects in pulsed end pumped Nd lasers at 940 nm,” Proc. SPIE 7194, 71940J (2009).
[CrossRef]

Wang, X.

S. Wang, X. Wang, T. Riesbeck, and H. J. Eichler, “Thermal lensing effects in pulsed end pumped Nd lasers at 940 nm,” Proc. SPIE 7194, 71940J (2009).
[CrossRef]

Weber, H.

Q. Zhang, B. Ozygus, and H. Weber, “Degeneration effects in laser cavities,” Eur. Phys. J. Appl. Phys. 6(3), 293–298 (1999).
[CrossRef]

X. Zhang, S. Zhao, Q. Wang, B. Ozygus, and H. Weber, “Modeling of diode-pumped actively Q-switched lasers,” IEEE J. Quantum Electron. 35(12), 1912–1918 (1999).
[CrossRef]

Yang, K.

G. Li, S. Zhao, H. Zhao, K. Yang, and S. Ding, “Rate equations and solutions of a laser-diode end-pumped passively Q-switched intracavity doubling laser by taking into account intracavity laser spatial distribution,” Opt. Commun. 234(1-6), 321–328 (2004).
[CrossRef]

Zhang, Q.

Q. Zhang, B. Ozygus, and H. Weber, “Degeneration effects in laser cavities,” Eur. Phys. J. Appl. Phys. 6(3), 293–298 (1999).
[CrossRef]

Zhang, X.

X. Zhang, S. Zhao, Q. Wang, B. Ozygus, and H. Weber, “Modeling of diode-pumped actively Q-switched lasers,” IEEE J. Quantum Electron. 35(12), 1912–1918 (1999).
[CrossRef]

Zhao, H.

G. Li, S. Zhao, H. Zhao, K. Yang, and S. Ding, “Rate equations and solutions of a laser-diode end-pumped passively Q-switched intracavity doubling laser by taking into account intracavity laser spatial distribution,” Opt. Commun. 234(1-6), 321–328 (2004).
[CrossRef]

Zhao, S.

G. Li, S. Zhao, H. Zhao, K. Yang, and S. Ding, “Rate equations and solutions of a laser-diode end-pumped passively Q-switched intracavity doubling laser by taking into account intracavity laser spatial distribution,” Opt. Commun. 234(1-6), 321–328 (2004).
[CrossRef]

X. Zhang, S. Zhao, Q. Wang, B. Ozygus, and H. Weber, “Modeling of diode-pumped actively Q-switched lasers,” IEEE J. Quantum Electron. 35(12), 1912–1918 (1999).
[CrossRef]

Appl. Opt. (1)

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).

Eur. Phys. J. Appl. Phys. (1)

Q. Zhang, B. Ozygus, and H. Weber, “Degeneration effects in laser cavities,” Eur. Phys. J. Appl. Phys. 6(3), 293–298 (1999).
[CrossRef]

IEEE J. Quantum Electron. (5)

T. Y. Fan, “Heat generation in Nd:YAG and Yb:YAG,” IEEE J. Quantum Electron. 29(6), 1457–1459 (1993).
[CrossRef]

X. Zhang, S. Zhao, Q. Wang, B. Ozygus, and H. Weber, “Modeling of diode-pumped actively Q-switched lasers,” IEEE J. Quantum Electron. 35(12), 1912–1918 (1999).
[CrossRef]

A. E. Siegman and S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29(4), 1212–1217 (1993).
[CrossRef]

J. J. Degnan, “Theory of the Optimally Coupled Q-Switched Laser,” IEEE J. Quantum Electron. 25(2), 214–220 (1989).
[CrossRef]

R. B. Kay and D. Poulios, “Q-Switched Rate Equations for Diode Side-Pumped Slab and Zigzag Slab Lasers Including Gaussian Beam Shapes,” IEEE J. Quantum Electron. 41(10), 1278–1284 (2005).
[CrossRef]

Opt. Commun. (1)

G. Li, S. Zhao, H. Zhao, K. Yang, and S. Ding, “Rate equations and solutions of a laser-diode end-pumped passively Q-switched intracavity doubling laser by taking into account intracavity laser spatial distribution,” Opt. Commun. 234(1-6), 321–328 (2004).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Proc. SPIE (2)

B. Heubeck and C. Pflaum, “Numerical simulation of multiple modes in solid state lasers,” Proc. SPIE 6190, 61900X (2006).
[CrossRef]

S. Wang, X. Wang, T. Riesbeck, and H. J. Eichler, “Thermal lensing effects in pulsed end pumped Nd lasers at 940 nm,” Proc. SPIE 7194, 71940J (2009).
[CrossRef]

SIAM J. Sci. Comput. (1)

B. Heubeck, C. Pflaum, and G. Steinle, “New finite elements for large-scale simulation of optical Waves,” SIAM J. Sci. Comput. 31(2), 1063–1081 (2009).
[CrossRef]

Other (8)

LAS-CAD GmbH Munich, Germany, http://www.las-cad.com/ .

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

O. Svelto, Principles of lasers (Springer, New York, 1998).

R. J. LeVeque, Finite volume methods for hyperbolic problems (Cambridge Texts in Applied Mathematics, 2007).

E. Hairer, and G. Wanner, Solving ordinary differential Eqs. (2): Stiff and differential-algebraic problems, (Springer, Berlin, 1996).

W. Koechner, Solid-state laser engineering, (Springer, Berlin, 2006).

International Organization for Standardization EN ISO 11146:2000 and EN ISO 11146:1999,” Laser and laser-related equipment. Test methods for laser beam parameters. Beam width, divergence angle and beam propagation factor,” (ISO, Geneva, 2000).

F. Träger, Springer Handbook of Lasers and Optics (Springer, Berlin, 2007), Chap. 11.2.

Supplementary Material (4)

» Media 1: MOV (1420 KB)     
» Media 2: MOV (915 KB)     
» Media 3: MOV (733 KB)     
» Media 4: MOV (576 KB)     

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Figures (12)

Fig. 1
Fig. 1

Resonator sketch.

Fig. 2
Fig. 2

Crystal decomposition in several Gaussian ducts.

Fig. 3
Fig. 3

Time scheme for active Q-switching.

Fig. 4
Fig. 4

3-dimensional view of the absorbed pump power density in the crystal. Absorption mainly occurs in the red areas close to the crystal end faces.

Fig. 5
Fig. 5

Simulated pulse shape in Q-switched mode.

Fig. 6
Fig. 6

Simulated and measured values of average output power in Q-switched mode.

Fig. 7
Fig. 7

Comparison of simulated and measured values of pulse duration (FWHM).

Fig. 8
Fig. 8

Dynamic behavior of output power and beam quality in cw-operation.

Fig. 9
Fig. 9

Dynamic behavior of output power and beam quality for Q-switched operation.

Fig. 10
Fig. 10

(a) Transverse beam profile in cw operation (Media 1). (b) 3-dimensional view of the cw distribution of the population inversion density (Media 2).

Fig. 11
Fig. 11

Dynamic behavior of the Q-switched laser during the numerical pulse period which is denoted as period II in Fig. 3. (a) Transverse beam profile at maximum output power during Q-switched operation (Media 3). (b) 3-dimensional view of the population inversion density at the end of a Q-switch period (Media 4).

Fig. 12
Fig. 12

Dynamic behavior of population inversion.

Tables (1)

Tables Icon

Table 1 To compare the laser beam in cw and Q-switched operation, average values for the output power of the resonator eigenmodes, the maximum and averaged overall output power, and the beam quality are given.

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

N(t,x)t=W(t,x)N(t,x)N(t,x)τf+Rp(t,x)(NtotN)Ntot,
Φ(t)t =ΩaW(t,x)N(t,x)dVΦ(t)τc.
W(t,x)=σcncrystωρ(t,x),
Φ(t)=1ωΩρ(t,x)dx
ρcryst(t,x,y,zI)=ncrystρout(t,x,y,zI),
ρ(t,x)=Φ(t)n(z)Veffω|u(t,x)|2
Veff=Ωn(z)|u(t,x)|2 dx
Φ(t)t=σcVeffΦ(t)ΩaN(t,x)|u(t,x)|2d(x)Φ(t)τc.
ρ(t,x)=i=1Mρi(t,x).
Φ(t)=i=1MΦi(t).
ρi(t,x)=Φi(t)n(z)Veffω|ui(x)|2
W(t,x)=σcVeffi=1M(Φi|ui|2)
Φit=σcVeffΦiΩaN|ui|2d(x)Φiτc,
Nt=σcVeffNi=1M(Φi|ui|2)Nτf+Rp(NtotN)Ntot
I=(l,j,k);l,j=1,.,mx,y;k=1,.,mz.
Φit=σcVeffΦiJNJcJ|ui|2d(x)Φiτc,
NIt=σcVeffNIi=1M(ΦicI|ui|2d(x))NIτf+RpcINtotd(x)NINtot,
Ltot(t)=LReslog(1LQS(t)),
τc(t) =τrLtot(t).
Pout,i(t)=log(rr)c2lopt(ω)Φi(t),
Paverage,i=1T0TPout,i(t)dt,
Pout(t)=iPout,i(t),
Paverage=iPaverage,i=0TPout(t)dt.
Mx/y2(t)=i(2px/y,i+1)Pout,i(t)Pout(t).
Mx/y2 =0TMx/y2(t)·Pout(t)dt0TPout(t)dt

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