Abstract

Simulation and experimental improvement of a pulsed Cr,Tm,Ho:YAG (CTH:YAG) laser is presented. In order to simulate the CTH-Laser a generalized version of the Dynamic Mode Analysis (gDMA) is introduced, which includes an abstract formalism to describe arbitrary rate equations. This novel version of DMA enables the coupling between individual modes of the resonator and the complex excitation dynamics of the CTH state system. With the proposed method gDMA a full 3D simulation was conducted and the beam quality of the generated pulses could be calculated for various crystal diameters. Based upon the simulation results the crystal diameter was decreased in experiment. This reduction led to an improvement of M2 from 36 to 27, which is in good agreement with the experimental results. Additionally, the pulse energy depending on the pump power exhibits a close agreement with the experimental measurements. Moreover, the strength of each interionic mechanism in Cr,Tm,Ho:YAG is analyzed and the back transfer from Holmium to Thulium is identified to be the most dominant loss source for stimulated emission at 2090 nm. All in all, the presented extension of DMA represents an accurate and efficient method to simulate the amplification of higher order modes in gain media with strong interionic mechanisms.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. N. P. Barnes, B. M. Walsh, and E. D. Filer, “Ho:Ho upconversion: applications to Ho lasers,” J. Opt. Soc. Am. B 20, 1212–1219 (2003).
    [Crossref]
  2. B. M. Walsh, G. W. Grew, and N. P. Barnes, “Energy levels and intensity parameters of Ho3+ ions in Y3Al5O12 and Lu3Al5O12,” J. Phys. Chem. Solids 67, 1567–1582 (2006).
    [Crossref]
  3. R. Springer, I. Alexeev, J. Heberle, and C. Pflaum, “Numerical simulation of short laser pulse amplification,” J. Opt. Soc. Am. B 36, 717–727 (2019).
    [Crossref]
  4. H. Huang, J. Huang, Y. Ge, H. Zheng, W. Weng, H. Wu, and W. Lin, “2.1 μm composite Tm/Ho:YAG laser,” Opt. Lett. 43, 1271–1274 (2018).
    [Crossref] [PubMed]
  5. T. Y. Fan, G. Huber, R. L. Byer, and P. Mitzscherlich, “Spectroscopy and diode laser-pumped operation of Tm, Ho:YAG,” IEEE J. Quantum Electron. 24, 924–933 (1988).
    [Crossref]
  6. M. G. Jani, N. P. Barnes, and K. E. Murray, “Flash-lamp-pumped Ho:Tm:Cr:YAG and Ho:Tm:Er:YLF lasers: experimental results of a single, long pulse length comparison,” Appl. Opt. 36, 3357–3362 (1997).
    [Crossref] [PubMed]
  7. B. Fei, W. Guo, J. Huang, Q. Huang, J. Chen, J. Li, W. Chen, G. Zhang, and Y. Cao, “Spectroscopic properties and energy transfers in Cr, Tm, Ho triple-doped Y3Al5O12 transparent ceramics,” Opt. Mater. Express 3, 2037–2044 (2013).
    [Crossref]
  8. Y. Kalisky, J. Kagan, D. Sagie, A. Brenier, C. Pedrini, and G. Boulon, “Spectroscopic properties, energy transfer, and laser operation of pulsed holmium lasers,” J. Appl. Phys. 70, 4095–4100 (1991).
    [Crossref]
  9. M. Wohlmuth, C. Pflaum, K. Altmann, M. Paster, and C. Hahn, “Dynamic multimode analysis of q-switched solid state laser cavities,” Opt. Express 17, 17303–17316 (2009).
    [Crossref] [PubMed]
  10. K. H. Kim, Y. S. Choi, N. P. Barnes, R. V. Hess, C. H. Bair, and P. Brockman, “Investigation of 2.1-μm lasing properties of Ho:Tm:Cr:YAG crystals under flash-lamp pumping at various operating conditions,” Appl. Opt. 32, 2066–2074 (1993).
    [Crossref] [PubMed]
  11. D. Bruneau, S. Delmonte, and J. Pelon, “Modeling of Tm, Ho:YAG and Tm, Ho:YLF 2-μm lasers and calculation of extractable energies,” Appl. Opt. 37, 8406–8419 (1998).
    [Crossref]
  12. G. Armagan, B. D. Bartolo, and A. Buonchristiani, “Spectroscopic investigation of Cr to Tm energy transfer in yttrium aluminum garnet crystals,” J. Lumin. 44, 129–139 (1989).
    [Crossref]
  13. M. Falconieri, A. Lanzi, G. Salvetti, and A. Toncelli, “Fluorescence dynamics in an optically-excited Tm, Ho:YAG crystal,” Opt. Mater. 7, 135–143 (1997).
    [Crossref]
  14. G. Rustad and K. Stenersen, “Modeling of laser-pumped Tm and Ho lasers accounting for upconversion and ground-state depletion,” IEEE J. Quantum Electron. 32, 1645–1656 (1996).
    [Crossref]
  15. “Advanced Software for Laser Design,” http://www.asldweb.com/ .
  16. O. Svelto, Principles of Lasers (Springer, 2009).
  17. R. Springer, T. Hannan, and C. Pflaum, “Influence of interionic energy transfer mechanisms in Tm, Ho:YAG on the maximum extractable energy in regenerative amplifiers,” (2018).
  18. R. R. Petrin, M. G. Jani, R. C. Powell, and M. Kokta, “Spectral dynamics of laser-pumped Y3Al5O12:Tm, Ho lasers,” Opt. Mater. 1, 111–124 (1992).
    [Crossref]
  19. S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. F. Krupke, “Infrared cross-section measurements for crystals doped with Er3+, Tm3+, and Ho3+,” IEEE J. Quantum Electron. 28, 2619–2630 (1992).
    [Crossref]
  20. NASA, “Database Lasers,” http://web.archive.org/web/20050404033633/http://aesd.larc.nasa.gov:80/GL/laser/spectra/spectra.htm . (Accessed: December 28, 2018).
  21. M. Wohlmuth, “Dynamic multimode analysis of solid-state lasers,” Dissertation, Friedrich-Alexander Universität Erlangen-Nürnberg (2011).

2019 (1)

2018 (1)

2013 (1)

2009 (1)

2006 (1)

B. M. Walsh, G. W. Grew, and N. P. Barnes, “Energy levels and intensity parameters of Ho3+ ions in Y3Al5O12 and Lu3Al5O12,” J. Phys. Chem. Solids 67, 1567–1582 (2006).
[Crossref]

2003 (1)

1998 (1)

1997 (2)

M. G. Jani, N. P. Barnes, and K. E. Murray, “Flash-lamp-pumped Ho:Tm:Cr:YAG and Ho:Tm:Er:YLF lasers: experimental results of a single, long pulse length comparison,” Appl. Opt. 36, 3357–3362 (1997).
[Crossref] [PubMed]

M. Falconieri, A. Lanzi, G. Salvetti, and A. Toncelli, “Fluorescence dynamics in an optically-excited Tm, Ho:YAG crystal,” Opt. Mater. 7, 135–143 (1997).
[Crossref]

1996 (1)

G. Rustad and K. Stenersen, “Modeling of laser-pumped Tm and Ho lasers accounting for upconversion and ground-state depletion,” IEEE J. Quantum Electron. 32, 1645–1656 (1996).
[Crossref]

1993 (1)

1992 (2)

R. R. Petrin, M. G. Jani, R. C. Powell, and M. Kokta, “Spectral dynamics of laser-pumped Y3Al5O12:Tm, Ho lasers,” Opt. Mater. 1, 111–124 (1992).
[Crossref]

S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. F. Krupke, “Infrared cross-section measurements for crystals doped with Er3+, Tm3+, and Ho3+,” IEEE J. Quantum Electron. 28, 2619–2630 (1992).
[Crossref]

1991 (1)

Y. Kalisky, J. Kagan, D. Sagie, A. Brenier, C. Pedrini, and G. Boulon, “Spectroscopic properties, energy transfer, and laser operation of pulsed holmium lasers,” J. Appl. Phys. 70, 4095–4100 (1991).
[Crossref]

1989 (1)

G. Armagan, B. D. Bartolo, and A. Buonchristiani, “Spectroscopic investigation of Cr to Tm energy transfer in yttrium aluminum garnet crystals,” J. Lumin. 44, 129–139 (1989).
[Crossref]

1988 (1)

T. Y. Fan, G. Huber, R. L. Byer, and P. Mitzscherlich, “Spectroscopy and diode laser-pumped operation of Tm, Ho:YAG,” IEEE J. Quantum Electron. 24, 924–933 (1988).
[Crossref]

Alexeev, I.

Altmann, K.

Armagan, G.

G. Armagan, B. D. Bartolo, and A. Buonchristiani, “Spectroscopic investigation of Cr to Tm energy transfer in yttrium aluminum garnet crystals,” J. Lumin. 44, 129–139 (1989).
[Crossref]

Bair, C. H.

Barnes, N. P.

Bartolo, B. D.

G. Armagan, B. D. Bartolo, and A. Buonchristiani, “Spectroscopic investigation of Cr to Tm energy transfer in yttrium aluminum garnet crystals,” J. Lumin. 44, 129–139 (1989).
[Crossref]

Boulon, G.

Y. Kalisky, J. Kagan, D. Sagie, A. Brenier, C. Pedrini, and G. Boulon, “Spectroscopic properties, energy transfer, and laser operation of pulsed holmium lasers,” J. Appl. Phys. 70, 4095–4100 (1991).
[Crossref]

Brenier, A.

Y. Kalisky, J. Kagan, D. Sagie, A. Brenier, C. Pedrini, and G. Boulon, “Spectroscopic properties, energy transfer, and laser operation of pulsed holmium lasers,” J. Appl. Phys. 70, 4095–4100 (1991).
[Crossref]

Brockman, P.

Bruneau, D.

Buonchristiani, A.

G. Armagan, B. D. Bartolo, and A. Buonchristiani, “Spectroscopic investigation of Cr to Tm energy transfer in yttrium aluminum garnet crystals,” J. Lumin. 44, 129–139 (1989).
[Crossref]

Byer, R. L.

T. Y. Fan, G. Huber, R. L. Byer, and P. Mitzscherlich, “Spectroscopy and diode laser-pumped operation of Tm, Ho:YAG,” IEEE J. Quantum Electron. 24, 924–933 (1988).
[Crossref]

Cao, Y.

Chase, L. L.

S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. F. Krupke, “Infrared cross-section measurements for crystals doped with Er3+, Tm3+, and Ho3+,” IEEE J. Quantum Electron. 28, 2619–2630 (1992).
[Crossref]

Chen, J.

Chen, W.

Choi, Y. S.

Delmonte, S.

Falconieri, M.

M. Falconieri, A. Lanzi, G. Salvetti, and A. Toncelli, “Fluorescence dynamics in an optically-excited Tm, Ho:YAG crystal,” Opt. Mater. 7, 135–143 (1997).
[Crossref]

Fan, T. Y.

T. Y. Fan, G. Huber, R. L. Byer, and P. Mitzscherlich, “Spectroscopy and diode laser-pumped operation of Tm, Ho:YAG,” IEEE J. Quantum Electron. 24, 924–933 (1988).
[Crossref]

Fei, B.

Filer, E. D.

Ge, Y.

Grew, G. W.

B. M. Walsh, G. W. Grew, and N. P. Barnes, “Energy levels and intensity parameters of Ho3+ ions in Y3Al5O12 and Lu3Al5O12,” J. Phys. Chem. Solids 67, 1567–1582 (2006).
[Crossref]

Guo, W.

Hahn, C.

Hannan, T.

R. Springer, T. Hannan, and C. Pflaum, “Influence of interionic energy transfer mechanisms in Tm, Ho:YAG on the maximum extractable energy in regenerative amplifiers,” (2018).

Heberle, J.

Hess, R. V.

Huang, H.

Huang, J.

Huang, Q.

Huber, G.

T. Y. Fan, G. Huber, R. L. Byer, and P. Mitzscherlich, “Spectroscopy and diode laser-pumped operation of Tm, Ho:YAG,” IEEE J. Quantum Electron. 24, 924–933 (1988).
[Crossref]

Jani, M. G.

Kagan, J.

Y. Kalisky, J. Kagan, D. Sagie, A. Brenier, C. Pedrini, and G. Boulon, “Spectroscopic properties, energy transfer, and laser operation of pulsed holmium lasers,” J. Appl. Phys. 70, 4095–4100 (1991).
[Crossref]

Kalisky, Y.

Y. Kalisky, J. Kagan, D. Sagie, A. Brenier, C. Pedrini, and G. Boulon, “Spectroscopic properties, energy transfer, and laser operation of pulsed holmium lasers,” J. Appl. Phys. 70, 4095–4100 (1991).
[Crossref]

Kim, K. H.

Kokta, M.

R. R. Petrin, M. G. Jani, R. C. Powell, and M. Kokta, “Spectral dynamics of laser-pumped Y3Al5O12:Tm, Ho lasers,” Opt. Mater. 1, 111–124 (1992).
[Crossref]

Krupke, W. F.

S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. F. Krupke, “Infrared cross-section measurements for crystals doped with Er3+, Tm3+, and Ho3+,” IEEE J. Quantum Electron. 28, 2619–2630 (1992).
[Crossref]

Kway, W. L.

S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. F. Krupke, “Infrared cross-section measurements for crystals doped with Er3+, Tm3+, and Ho3+,” IEEE J. Quantum Electron. 28, 2619–2630 (1992).
[Crossref]

Lanzi, A.

M. Falconieri, A. Lanzi, G. Salvetti, and A. Toncelli, “Fluorescence dynamics in an optically-excited Tm, Ho:YAG crystal,” Opt. Mater. 7, 135–143 (1997).
[Crossref]

Li, J.

Lin, W.

Mitzscherlich, P.

T. Y. Fan, G. Huber, R. L. Byer, and P. Mitzscherlich, “Spectroscopy and diode laser-pumped operation of Tm, Ho:YAG,” IEEE J. Quantum Electron. 24, 924–933 (1988).
[Crossref]

Murray, K. E.

Paster, M.

Payne, S. A.

S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. F. Krupke, “Infrared cross-section measurements for crystals doped with Er3+, Tm3+, and Ho3+,” IEEE J. Quantum Electron. 28, 2619–2630 (1992).
[Crossref]

Pedrini, C.

Y. Kalisky, J. Kagan, D. Sagie, A. Brenier, C. Pedrini, and G. Boulon, “Spectroscopic properties, energy transfer, and laser operation of pulsed holmium lasers,” J. Appl. Phys. 70, 4095–4100 (1991).
[Crossref]

Pelon, J.

Petrin, R. R.

R. R. Petrin, M. G. Jani, R. C. Powell, and M. Kokta, “Spectral dynamics of laser-pumped Y3Al5O12:Tm, Ho lasers,” Opt. Mater. 1, 111–124 (1992).
[Crossref]

Pflaum, C.

Powell, R. C.

R. R. Petrin, M. G. Jani, R. C. Powell, and M. Kokta, “Spectral dynamics of laser-pumped Y3Al5O12:Tm, Ho lasers,” Opt. Mater. 1, 111–124 (1992).
[Crossref]

Rustad, G.

G. Rustad and K. Stenersen, “Modeling of laser-pumped Tm and Ho lasers accounting for upconversion and ground-state depletion,” IEEE J. Quantum Electron. 32, 1645–1656 (1996).
[Crossref]

Sagie, D.

Y. Kalisky, J. Kagan, D. Sagie, A. Brenier, C. Pedrini, and G. Boulon, “Spectroscopic properties, energy transfer, and laser operation of pulsed holmium lasers,” J. Appl. Phys. 70, 4095–4100 (1991).
[Crossref]

Salvetti, G.

M. Falconieri, A. Lanzi, G. Salvetti, and A. Toncelli, “Fluorescence dynamics in an optically-excited Tm, Ho:YAG crystal,” Opt. Mater. 7, 135–143 (1997).
[Crossref]

Smith, L. K.

S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. F. Krupke, “Infrared cross-section measurements for crystals doped with Er3+, Tm3+, and Ho3+,” IEEE J. Quantum Electron. 28, 2619–2630 (1992).
[Crossref]

Springer, R.

R. Springer, I. Alexeev, J. Heberle, and C. Pflaum, “Numerical simulation of short laser pulse amplification,” J. Opt. Soc. Am. B 36, 717–727 (2019).
[Crossref]

R. Springer, T. Hannan, and C. Pflaum, “Influence of interionic energy transfer mechanisms in Tm, Ho:YAG on the maximum extractable energy in regenerative amplifiers,” (2018).

Stenersen, K.

G. Rustad and K. Stenersen, “Modeling of laser-pumped Tm and Ho lasers accounting for upconversion and ground-state depletion,” IEEE J. Quantum Electron. 32, 1645–1656 (1996).
[Crossref]

Svelto, O.

O. Svelto, Principles of Lasers (Springer, 2009).

Toncelli, A.

M. Falconieri, A. Lanzi, G. Salvetti, and A. Toncelli, “Fluorescence dynamics in an optically-excited Tm, Ho:YAG crystal,” Opt. Mater. 7, 135–143 (1997).
[Crossref]

Walsh, B. M.

B. M. Walsh, G. W. Grew, and N. P. Barnes, “Energy levels and intensity parameters of Ho3+ ions in Y3Al5O12 and Lu3Al5O12,” J. Phys. Chem. Solids 67, 1567–1582 (2006).
[Crossref]

N. P. Barnes, B. M. Walsh, and E. D. Filer, “Ho:Ho upconversion: applications to Ho lasers,” J. Opt. Soc. Am. B 20, 1212–1219 (2003).
[Crossref]

Weng, W.

Wohlmuth, M.

M. Wohlmuth, C. Pflaum, K. Altmann, M. Paster, and C. Hahn, “Dynamic multimode analysis of q-switched solid state laser cavities,” Opt. Express 17, 17303–17316 (2009).
[Crossref] [PubMed]

M. Wohlmuth, “Dynamic multimode analysis of solid-state lasers,” Dissertation, Friedrich-Alexander Universität Erlangen-Nürnberg (2011).

Wu, H.

Zhang, G.

Zheng, H.

Appl. Opt. (3)

IEEE J. Quantum Electron. (3)

G. Rustad and K. Stenersen, “Modeling of laser-pumped Tm and Ho lasers accounting for upconversion and ground-state depletion,” IEEE J. Quantum Electron. 32, 1645–1656 (1996).
[Crossref]

T. Y. Fan, G. Huber, R. L. Byer, and P. Mitzscherlich, “Spectroscopy and diode laser-pumped operation of Tm, Ho:YAG,” IEEE J. Quantum Electron. 24, 924–933 (1988).
[Crossref]

S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. F. Krupke, “Infrared cross-section measurements for crystals doped with Er3+, Tm3+, and Ho3+,” IEEE J. Quantum Electron. 28, 2619–2630 (1992).
[Crossref]

J. Appl. Phys. (1)

Y. Kalisky, J. Kagan, D. Sagie, A. Brenier, C. Pedrini, and G. Boulon, “Spectroscopic properties, energy transfer, and laser operation of pulsed holmium lasers,” J. Appl. Phys. 70, 4095–4100 (1991).
[Crossref]

J. Lumin. (1)

G. Armagan, B. D. Bartolo, and A. Buonchristiani, “Spectroscopic investigation of Cr to Tm energy transfer in yttrium aluminum garnet crystals,” J. Lumin. 44, 129–139 (1989).
[Crossref]

J. Opt. Soc. Am. B (2)

J. Phys. Chem. Solids (1)

B. M. Walsh, G. W. Grew, and N. P. Barnes, “Energy levels and intensity parameters of Ho3+ ions in Y3Al5O12 and Lu3Al5O12,” J. Phys. Chem. Solids 67, 1567–1582 (2006).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Opt. Mater. (2)

M. Falconieri, A. Lanzi, G. Salvetti, and A. Toncelli, “Fluorescence dynamics in an optically-excited Tm, Ho:YAG crystal,” Opt. Mater. 7, 135–143 (1997).
[Crossref]

R. R. Petrin, M. G. Jani, R. C. Powell, and M. Kokta, “Spectral dynamics of laser-pumped Y3Al5O12:Tm, Ho lasers,” Opt. Mater. 1, 111–124 (1992).
[Crossref]

Opt. Mater. Express (1)

Other (5)

“Advanced Software for Laser Design,” http://www.asldweb.com/ .

O. Svelto, Principles of Lasers (Springer, 2009).

R. Springer, T. Hannan, and C. Pflaum, “Influence of interionic energy transfer mechanisms in Tm, Ho:YAG on the maximum extractable energy in regenerative amplifiers,” (2018).

NASA, “Database Lasers,” http://web.archive.org/web/20050404033633/http://aesd.larc.nasa.gov:80/GL/laser/spectra/spectra.htm . (Accessed: December 28, 2018).

M. Wohlmuth, “Dynamic multimode analysis of solid-state lasers,” Dissertation, Friedrich-Alexander Universität Erlangen-Nürnberg (2011).

Supplementary Material (1)

NameDescription
» Visualization 1       Visualization 1 shows the mode competition of super imposed Hermite-Gaussian modes in a flashlamp pumped Cr,Tm,Ho:YAG resonator. Here, the highly dynamic amplifcation process of more than 250 modes can be seen, which starts from 90 microseconds. Afte

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Figures (10)

Fig. 1
Fig. 1 Reconstructed pump cavity in simulation. The Flashlamp is located at the left hand side and is represented with the yellow cylinder. The Cr,Tm,Ho:YAG rod is placed parallel to the flashlamp axis.
Fig. 2
Fig. 2 The absorption spectrum of Cr3+ ions [7] matches the spectral range of to the measured flashlamp emission spectrum.
Fig. 3
Fig. 3 Excitation and relaxation mechanisms in Cr,Tm,Ho:YAG: a) Pump absorption Pabs b) Net (effective) Energy Transfer E13,12,1,11 c) Upconversion U2124 d) Upconversion U2123 e) Cross Relaxation C4212 f)Energy Transfer E4,1,6,10 g) Energy Transfer E2168 h) Energy Transfer E2156 i) Energy Transfer E6512 j) Upconversion U6568 k) Stimulated Emission at 2090 nm. The blue color highlights interionic mechanism that involve ions of the same species. Spontaneous emission is included in all calculations but not displayed here.
Fig. 4
Fig. 4 The proposed gDMA method is obtained by coupling the former DMA with the suggested abstract formalism for arbitrary rate equations. The gDMA is required to compute mode structure, mode competition and beam quality of the CTH-Laser.
Fig. 5
Fig. 5 The simulation of the CTH-Laser with a 4 mm diameter rod exhibits a close match to the experimental measurements. However, the exact cross sections for stimulated emission and absorption are not known and reported differently in literature [7,20].
Fig. 6
Fig. 6 Simulated Data: An improved beam quality is obtained by decreasing the diameter of the Cr,Tm,Ho:YAG crystal from 4 mm (blue line) to 3.5 mm (orange line). At the same time, the pulse power remains constant.
Fig. 7
Fig. 7 Figures 7(a) to 7(f) display simulated siTEM modes in the Cr,Tm,Ho:YAG crystal. Each siTEM mode represents a superposition of 100 Hermite-Gauss modes. Modes of higher order exhibit strong intensity regions close to the jacket surface of the crystal.
Fig. 8
Fig. 8 Results from the time-dependent simulation of the crystal’s focal length for 182 J flashlamp pulse energy.
Fig. 9
Fig. 9 The absolute temperature gradient increases with decreasing crystal diameter, which leads to stronger thermal lensing. Reducing the crystal diameter improves the beam quality but thermal lensing gets more dominant. This leads to an unstable resonator below a diameter of 3 mm
Fig. 10
Fig. 10 a) State excitation is shifted from Cr3+ via Tm3+ to Ho3+.The population on each state has been normalized to the doping density of the respective ion species. b) The transfer from Tm3+ to Ho3+ is the dominant mechanism for the excitation of the upper lasing state in Ho3+.

Tables (5)

Tables Icon

Table 1 Coefficients of Interionic Mechanisms in Cr,Tm,Ho:YAG

Tables Icon

Table 2 Fluorescent (Fl.) lifetimes that are not listed have values in the μs range.

Tables Icon

Table 3 Calculation Module and chosen Simulation Parameters

Tables Icon

Table 4 The simulated focal length from thermal lensing is in good agreement with the experimental measurement. All focal lengths are given in mm with respect to x- and y-direction (x,y).

Tables Icon

Table 5 A crystal diameter less than 3 mm results in an instable resonator. In experiment a M x , y 2 value of (39.6/31.6) and (26.7/27.4) was measured for 4 mm and 3.5 mm crystal diameter, respectively.

Equations (33)

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

N i t = F i s { 1 , , S } i s = i σ i s , j s s N i + s { 1 , , S } j s = i σ i s , j s s N i for i = 0 , , L 1 .
V eff = Ω resonator n ( z ) | u m | 2 d V = 0 L n ( z ) d z ,
Φ m t = c V eff Φ m Ω crystal ( σ e N i σ a N j ) | u m | 2 d V Φ m τ c ,
σ ˜ e : = σ e c V eff , σ ˜ a : = σ a c V eff .
N i t = + m | u m | 2 Φ m ( σ ˜ e N i + σ ˜ a N j ) = σ i , j e N i + σ i , j a N j
N j t = + m | u m | 2 Φ m ( σ ˜ e N i σ ˜ a N j ) = + σ i , j e N i σ i , j a N j .
N k t = + W E N j N i = + σ i , k N j where σ i , k = W E N i
N i t = W E N j N i = σ i , k N j where σ i , k = W E N i
N j t = W E N i N j = σ j , l N i where σ j , l = W E N j
N l t = + W E N i N j = + σ j , l N i where σ j , l = W E N j .
2 E 1 t : = N 13 t = R pump k 13 , 12 , 1 , 11 N 13 N 1 N 13 τ 13
4 A 2 t : = N 12 t = R pump + k 13 , 12 , 1 , 11 N 13 N 1 + N 13 τ 13
3 F 2 , 3 t : = N 11 t = k 13 , 12 , 1 , 11 N 13 N 1 N 11 τ 11
5 F 2 t : = N 10 t = k 4 , 1 , 6 , 10 N 4 N 6 + N 11 τ 11 N 10 τ 10
5 F 5 t : = N 9 t = N 10 τ 10 N 9 τ 9
5 I 5 t : = N 8 t = k 2168 N 2 N 6 + k 6568 N 6 2 + N 9 τ 9 N 8 τ 8
5 I 6 t : = N 7 t = N 8 τ 8 N 7 τ 7
5 I 7 t : = N 6 t = σ e c n N 6 + σ a c n N 5 k 4 , 1 , 6 , 10 N 4 N 6 + k 2156 N 2 N 5 k 6512 N 6 N 1 k 2168 N 2 N 6 2 k 6568 N 6 2 + N 7 τ 7 N 6 τ 6
5 I 8 t : = N 5 t = σ e c n N 6 σ a c n N 5 k 2156 N 2 N 5 + k 6512 N 6 N 1 k 6568 N 6 2 + N 6 τ 6
3 H 4 t : = N 4 t = k 4212 N 4 N 1 k 4 , 1 , 6 , 10 N 4 N 10 + k 2124 N 2 2 N 4 τ 4
3 H 5 t : = N 3 t = k 2123 N 2 2 + N 4 τ 4 N 3 τ 3
3 F 4 t : = N 2 t = 2 k 2124 N 2 2 2 k 2123 N 2 2 + 2 k 4212 N 4 N 1 k 2156 N 2 N 5 + k 6512 N 1 N 6 k 2168 N 6 N 2 + N 3 τ 3 N 2 τ 2
3 H 6 t : = N 1 t = k 2124 N 2 2 + k 2123 N 2 2 k 4212 N 4 N 1 + k 2156 N 2 N 5 k 6512 N 1 N 6 + k 4 , 1 , 6 , 10 N 4 N 6 + k 2168 N 2 N 6 k 13 , 12 , 1 , 11 N 13 N 1 + N 2 τ 2 .
P heat = η heat , eff λ pump λ laser P abs ( λ pump ) ( 1 λ pump λ laser ) + λ pump > λ laser P abs ( λ pump ) .
R pump = λ λ laser P abs ( λ ) / E photon ( λ )
M x 2 ( t ) = i M x , i 2 P out , i ( t ) P out ( t ) ,
M y 2 ( t ) = i M y , i 2 P out , i ( t ) P out ( t ) .
M x , p 2 = 2 p + 1 ,
M y , q 2 = 2 q + 1
| u i g i , l i ( x , z ) | 2 = 1 g i k = l i l i + g i 1 | TEM k ( x , z ) | 2
| siTEM i j ( x ) | 2 = | u i g i , l i ( x , z ) | | u j g j , l j ( y , z ) |
| u 0 10 , 0 ( x , z ) | 2 = 1 10 k = 0 0 + 10 1 | TEM k ( x , z ) | 2 ,
| u 4 10 , 36 ( y , z ) | 2 = 1 10 k = 36 36 + 10 1 | TEM k ( y , z ) | 2 ,