Abstract

We present a new model for cw, Yb-doped, bulk crystal lasers emitting on a three-level laser transition near 980 nm. This model takes into account the saturation of the absorption and the spatial evolution of the pump and laser beams inside the crystal, which are two important points in three-level lasers. We have validated our numerical model by building an efficient cw Yb:S-FAP (ytterbium:strontium-fluoroapatite) laser emitting at 985 nm and pumped at 899 nm by a Ti:sapphire laser. Our model can be used to define the parameters that optimize the laser output power at 985 nm, such as the relative sizes of the pump and laser beams, the optimum output coupler, or characteristics of the crystal (length, Yb concentration). By adapting this model to diode pumping, we have established the conditions to obtain cw, efficient laser operation at 985 nm of an Yb:S-FAP crystal pumped by a standard 1×100-μm2 laser diode.

© 2005 Optical Society of America

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    [CrossRef]

2002

C. Lim and Y. Izawa, "Modeling of end-pumped, cw, quasi-three-level lasers," IEEE J. Quantum Electron. 38, 306-311 (2002).
[CrossRef]

2001

K. Shaffers, J. Tassano, P. Waide, S. Payne, and J. Morris, "Progress in the growth of Yb:S-FAP laser crystals," J. Cryst. Growth 225, 449-453 (2001).
[CrossRef]

2000

F. Augé, F. Druon, F. Balembois, P. Georges, A. Brun, F. Mougel, G. Aka, and D. Vivien, "Theoretical and experimental investigations of a diode-pumped, quasi-three-level laser: the Yb3+-doped Ca4(GdO(BO3)3 (Yb:GdCOB) laser," IEEE J. Quantum Electron. 36, 598-606 (2000).
[CrossRef]

A. J. Bayramian, C. Bibeau, R. J. Beach, C. D. Marshall, S. A. Payne, and W. F. Krupke, "Three-level Q-switched laser operation of ytterbium-doped Sr5(PO4)3F at 985 nm," Opt. Lett. 25, 622-624 (2000).
[CrossRef]

G. L. Bourdet, "Theoretical investigation of quasi-three-level, longitudinally pumped, continuous-wave lasers," Appl. Opt. 39, 966-971 (2000).
[CrossRef]

1998

1997

1994

L. D. DeLoach, S. A. Payne, L. K. Smith, W. L. Kway, and W. F. Krupke, "Laser and spectroscopic properties of Sr5(PO4)3F:Yb," J. Opt. Soc. Am. B 11, 269-275 (1994).
[CrossRef]

D. S. Sumida and T. Y. Fan, "Effect of radiation trapping on fluorescence lifetime and emission cross section measurements in solid-state laser media," Opt. Lett. 19, 1343-1345 (1994).
[CrossRef] [PubMed]

S. A. Payne, L. D. DeLoach, L. K. Smith, W. L. Kway, J. B. Tassano, W. F. Krupke, B. H. T. Chai, and G. Loutts, "Ytterbium-doped apatite-structure crystals: a new class of laser materials," J. Appl. Phys. 76, 497-503 (1994).
[CrossRef]

C. Barnard, P. Mylinski, J. Chrostowski, and M. Kavehrad, "Analytical model for rare-earth-doped fiber amplifiers and lasers," IEEE J. Quantum Electron. 30, 1817-1830 (1994).
[CrossRef]

1993

L. D. DeLoach, S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. F. Krupke, "Evaluation of absorption andemission properties of Yb3+-doped crystals for laser applications," IEEE J. Quantum Electron. 29, 1179-1190 (1993).
[CrossRef]

1992

L. D. Merkle, A. Pinto, H. R. Verdun, and B. McIntosh, "Laser action from Mn5+ in Ba3(VO4)2," Appl. Phys. Lett. 61, 2386-2388 (1992).
[CrossRef]

1991

C. Randy Giles and E. Desurvire, "Modeling erbium-doped fiber amplifiers," J. Lightwave Technol. 9, 271-283 (1991).
[CrossRef]

1990

T. Y. Fan and A. Sanchez, "Pump source requirements for end-pumped lasers," IEEE J. Quantum Electron. 26, 311-316 (1990).
[CrossRef]

1988

1987

T. Y. Fan and R. L. Byer, "Modeling and cw operation of a quasi-three-level 946 nm Nd:YAG laser," IEEE J. Quantum Electron. 23, 605-611 (1987).
[CrossRef]

1963

W. W. Rigrod, "Gain saturation and output power of optical masers," J. Appl. Phys. 34, 2602-2609 (1963).
[CrossRef]

1960

T. Maiman, "Stimulated optical radiation in ruby," Nature 4736, 493-494 (1960).
[CrossRef]

Aka, G.

F. Augé, F. Druon, F. Balembois, P. Georges, A. Brun, F. Mougel, G. Aka, and D. Vivien, "Theoretical and experimental investigations of a diode-pumped, quasi-three-level laser: the Yb3+-doped Ca4(GdO(BO3)3 (Yb:GdCOB) laser," IEEE J. Quantum Electron. 36, 598-606 (2000).
[CrossRef]

Armitage, J.

Augé, F.

F. Augé, F. Druon, F. Balembois, P. Georges, A. Brun, F. Mougel, G. Aka, and D. Vivien, "Theoretical and experimental investigations of a diode-pumped, quasi-three-level laser: the Yb3+-doped Ca4(GdO(BO3)3 (Yb:GdCOB) laser," IEEE J. Quantum Electron. 36, 598-606 (2000).
[CrossRef]

Balembois, F.

F. Augé, F. Druon, F. Balembois, P. Georges, A. Brun, F. Mougel, G. Aka, and D. Vivien, "Theoretical and experimental investigations of a diode-pumped, quasi-three-level laser: the Yb3+-doped Ca4(GdO(BO3)3 (Yb:GdCOB) laser," IEEE J. Quantum Electron. 36, 598-606 (2000).
[CrossRef]

Barnard, C.

C. Barnard, P. Mylinski, J. Chrostowski, and M. Kavehrad, "Analytical model for rare-earth-doped fiber amplifiers and lasers," IEEE J. Quantum Electron. 30, 1817-1830 (1994).
[CrossRef]

Bayramian, A. J.

Beach, R. J.

Bibeau, C.

Bourdet, G. L.

Brun, A.

F. Augé, F. Druon, F. Balembois, P. Georges, A. Brun, F. Mougel, G. Aka, and D. Vivien, "Theoretical and experimental investigations of a diode-pumped, quasi-three-level laser: the Yb3+-doped Ca4(GdO(BO3)3 (Yb:GdCOB) laser," IEEE J. Quantum Electron. 36, 598-606 (2000).
[CrossRef]

Byer, R. L.

T. Y. Fan and R. L. Byer, "Modeling and cw operation of a quasi-three-level 946 nm Nd:YAG laser," IEEE J. Quantum Electron. 23, 605-611 (1987).
[CrossRef]

Chai, B. H. T.

S. A. Payne, L. D. DeLoach, L. K. Smith, W. L. Kway, J. B. Tassano, W. F. Krupke, B. H. T. Chai, and G. Loutts, "Ytterbium-doped apatite-structure crystals: a new class of laser materials," J. Appl. Phys. 76, 497-503 (1994).
[CrossRef]

Chase, L. L.

L. D. DeLoach, S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. F. Krupke, "Evaluation of absorption andemission properties of Yb3+-doped crystals for laser applications," IEEE J. Quantum Electron. 29, 1179-1190 (1993).
[CrossRef]

Chrostowski, J.

C. Barnard, P. Mylinski, J. Chrostowski, and M. Kavehrad, "Analytical model for rare-earth-doped fiber amplifiers and lasers," IEEE J. Quantum Electron. 30, 1817-1830 (1994).
[CrossRef]

DeLoach, L. D.

S. A. Payne, L. D. DeLoach, L. K. Smith, W. L. Kway, J. B. Tassano, W. F. Krupke, B. H. T. Chai, and G. Loutts, "Ytterbium-doped apatite-structure crystals: a new class of laser materials," J. Appl. Phys. 76, 497-503 (1994).
[CrossRef]

L. D. DeLoach, S. A. Payne, L. K. Smith, W. L. Kway, and W. F. Krupke, "Laser and spectroscopic properties of Sr5(PO4)3F:Yb," J. Opt. Soc. Am. B 11, 269-275 (1994).
[CrossRef]

L. D. DeLoach, S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. F. Krupke, "Evaluation of absorption andemission properties of Yb3+-doped crystals for laser applications," IEEE J. Quantum Electron. 29, 1179-1190 (1993).
[CrossRef]

Desurvire, E.

C. Randy Giles and E. Desurvire, "Modeling erbium-doped fiber amplifiers," J. Lightwave Technol. 9, 271-283 (1991).
[CrossRef]

Druon, F.

F. Augé, F. Druon, F. Balembois, P. Georges, A. Brun, F. Mougel, G. Aka, and D. Vivien, "Theoretical and experimental investigations of a diode-pumped, quasi-three-level laser: the Yb3+-doped Ca4(GdO(BO3)3 (Yb:GdCOB) laser," IEEE J. Quantum Electron. 36, 598-606 (2000).
[CrossRef]

Fan, T. Y.

Fan , T. Y.

T. Y. Fan and A. Sanchez, "Pump source requirements for end-pumped lasers," IEEE J. Quantum Electron. 26, 311-316 (1990).
[CrossRef]

T. Y. Fan and R. L. Byer, "Modeling and cw operation of a quasi-three-level 946 nm Nd:YAG laser," IEEE J. Quantum Electron. 23, 605-611 (1987).
[CrossRef]

Georges, P.

F. Augé, F. Druon, F. Balembois, P. Georges, A. Brun, F. Mougel, G. Aka, and D. Vivien, "Theoretical and experimental investigations of a diode-pumped, quasi-three-level laser: the Yb3+-doped Ca4(GdO(BO3)3 (Yb:GdCOB) laser," IEEE J. Quantum Electron. 36, 598-606 (2000).
[CrossRef]

Giles , C. Randy

C. Randy Giles and E. Desurvire, "Modeling erbium-doped fiber amplifiers," J. Lightwave Technol. 9, 271-283 (1991).
[CrossRef]

Hanna, D. C.

Izawa, Y.

C. Lim and Y. Izawa, "Modeling of end-pumped, cw, quasi-three-level lasers," IEEE J. Quantum Electron. 38, 306-311 (2002).
[CrossRef]

Kavehrad, M.

C. Barnard, P. Mylinski, J. Chrostowski, and M. Kavehrad, "Analytical model for rare-earth-doped fiber amplifiers and lasers," IEEE J. Quantum Electron. 30, 1817-1830 (1994).
[CrossRef]

Krupke, W. F.

A. J. Bayramian, C. Bibeau, R. J. Beach, C. D. Marshall, S. A. Payne, and W. F. Krupke, "Three-level Q-switched laser operation of ytterbium-doped Sr5(PO4)3F at 985 nm," Opt. Lett. 25, 622-624 (2000).
[CrossRef]

L. D. DeLoach, S. A. Payne, L. K. Smith, W. L. Kway, and W. F. Krupke, "Laser and spectroscopic properties of Sr5(PO4)3F:Yb," J. Opt. Soc. Am. B 11, 269-275 (1994).
[CrossRef]

S. A. Payne, L. D. DeLoach, L. K. Smith, W. L. Kway, J. B. Tassano, W. F. Krupke, B. H. T. Chai, and G. Loutts, "Ytterbium-doped apatite-structure crystals: a new class of laser materials," J. Appl. Phys. 76, 497-503 (1994).
[CrossRef]

L. D. DeLoach, S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. F. Krupke, "Evaluation of absorption andemission properties of Yb3+-doped crystals for laser applications," IEEE J. Quantum Electron. 29, 1179-1190 (1993).
[CrossRef]

Kuleshov, N. V.

Kway, W. L.

L. D. DeLoach, S. A. Payne, L. K. Smith, W. L. Kway, and W. F. Krupke, "Laser and spectroscopic properties of Sr5(PO4)3F:Yb," J. Opt. Soc. Am. B 11, 269-275 (1994).
[CrossRef]

S. A. Payne, L. D. DeLoach, L. K. Smith, W. L. Kway, J. B. Tassano, W. F. Krupke, B. H. T. Chai, and G. Loutts, "Ytterbium-doped apatite-structure crystals: a new class of laser materials," J. Appl. Phys. 76, 497-503 (1994).
[CrossRef]

L. D. DeLoach, S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. F. Krupke, "Evaluation of absorption andemission properties of Yb3+-doped crystals for laser applications," IEEE J. Quantum Electron. 29, 1179-1190 (1993).
[CrossRef]

Lim , C.

C. Lim and Y. Izawa, "Modeling of end-pumped, cw, quasi-three-level lasers," IEEE J. Quantum Electron. 38, 306-311 (2002).
[CrossRef]

Loutts, G.

S. A. Payne, L. D. DeLoach, L. K. Smith, W. L. Kway, J. B. Tassano, W. F. Krupke, B. H. T. Chai, and G. Loutts, "Ytterbium-doped apatite-structure crystals: a new class of laser materials," J. Appl. Phys. 76, 497-503 (1994).
[CrossRef]

Maiman, T.

T. Maiman, "Stimulated optical radiation in ruby," Nature 4736, 493-494 (1960).
[CrossRef]

Marshall, C. D.

McIntosh, B.

L. D. Merkle, A. Pinto, H. R. Verdun, and B. McIntosh, "Laser action from Mn5+ in Ba3(VO4)2," Appl. Phys. Lett. 61, 2386-2388 (1992).
[CrossRef]

Merkle, L. D.

L. D. Merkle, A. Pinto, H. R. Verdun, and B. McIntosh, "Laser action from Mn5+ in Ba3(VO4)2," Appl. Phys. Lett. 61, 2386-2388 (1992).
[CrossRef]

Mikhailov, V. P.

Minelly, J. D.

Morris, J.

K. Shaffers, J. Tassano, P. Waide, S. Payne, and J. Morris, "Progress in the growth of Yb:S-FAP laser crystals," J. Cryst. Growth 225, 449-453 (2001).
[CrossRef]

Mougel, F.

F. Augé, F. Druon, F. Balembois, P. Georges, A. Brun, F. Mougel, G. Aka, and D. Vivien, "Theoretical and experimental investigations of a diode-pumped, quasi-three-level laser: the Yb3+-doped Ca4(GdO(BO3)3 (Yb:GdCOB) laser," IEEE J. Quantum Electron. 36, 598-606 (2000).
[CrossRef]

Mylinski, P.

C. Barnard, P. Mylinski, J. Chrostowski, and M. Kavehrad, "Analytical model for rare-earth-doped fiber amplifiers and lasers," IEEE J. Quantum Electron. 30, 1817-1830 (1994).
[CrossRef]

Nilsson, J.

Paschotta, R.

Payne, S.

K. Shaffers, J. Tassano, P. Waide, S. Payne, and J. Morris, "Progress in the growth of Yb:S-FAP laser crystals," J. Cryst. Growth 225, 449-453 (2001).
[CrossRef]

Payne, S. A.

A. J. Bayramian, C. Bibeau, R. J. Beach, C. D. Marshall, S. A. Payne, and W. F. Krupke, "Three-level Q-switched laser operation of ytterbium-doped Sr5(PO4)3F at 985 nm," Opt. Lett. 25, 622-624 (2000).
[CrossRef]

L. D. DeLoach, S. A. Payne, L. K. Smith, W. L. Kway, and W. F. Krupke, "Laser and spectroscopic properties of Sr5(PO4)3F:Yb," J. Opt. Soc. Am. B 11, 269-275 (1994).
[CrossRef]

S. A. Payne, L. D. DeLoach, L. K. Smith, W. L. Kway, J. B. Tassano, W. F. Krupke, B. H. T. Chai, and G. Loutts, "Ytterbium-doped apatite-structure crystals: a new class of laser materials," J. Appl. Phys. 76, 497-503 (1994).
[CrossRef]

L. D. DeLoach, S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. F. Krupke, "Evaluation of absorption andemission properties of Yb3+-doped crystals for laser applications," IEEE J. Quantum Electron. 29, 1179-1190 (1993).
[CrossRef]

Pinto, A.

L. D. Merkle, A. Pinto, H. R. Verdun, and B. McIntosh, "Laser action from Mn5+ in Ba3(VO4)2," Appl. Phys. Lett. 61, 2386-2388 (1992).
[CrossRef]

Podlipensky, A. A.

Rigrod, W. W.

W. W. Rigrod, "Gain saturation and output power of optical masers," J. Appl. Phys. 34, 2602-2609 (1963).
[CrossRef]

Risk, W. P.

Sanchez, A.

T. Y. Fan and A. Sanchez, "Pump source requirements for end-pumped lasers," IEEE J. Quantum Electron. 26, 311-316 (1990).
[CrossRef]

Shaffers, K.

K. Shaffers, J. Tassano, P. Waide, S. Payne, and J. Morris, "Progress in the growth of Yb:S-FAP laser crystals," J. Cryst. Growth 225, 449-453 (2001).
[CrossRef]

Smith, L. K.

S. A. Payne, L. D. DeLoach, L. K. Smith, W. L. Kway, J. B. Tassano, W. F. Krupke, B. H. T. Chai, and G. Loutts, "Ytterbium-doped apatite-structure crystals: a new class of laser materials," J. Appl. Phys. 76, 497-503 (1994).
[CrossRef]

L. D. DeLoach, S. A. Payne, L. K. Smith, W. L. Kway, and W. F. Krupke, "Laser and spectroscopic properties of Sr5(PO4)3F:Yb," J. Opt. Soc. Am. B 11, 269-275 (1994).
[CrossRef]

L. D. DeLoach, S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. F. Krupke, "Evaluation of absorption andemission properties of Yb3+-doped crystals for laser applications," IEEE J. Quantum Electron. 29, 1179-1190 (1993).
[CrossRef]

Sumida , D. S.

Tassano, J.

K. Shaffers, J. Tassano, P. Waide, S. Payne, and J. Morris, "Progress in the growth of Yb:S-FAP laser crystals," J. Cryst. Growth 225, 449-453 (2001).
[CrossRef]

Tassano, J. B.

S. A. Payne, L. D. DeLoach, L. K. Smith, W. L. Kway, J. B. Tassano, W. F. Krupke, B. H. T. Chai, and G. Loutts, "Ytterbium-doped apatite-structure crystals: a new class of laser materials," J. Appl. Phys. 76, 497-503 (1994).
[CrossRef]

Tropper, A. C.

Verdun, H. R.

L. D. Merkle, A. Pinto, H. R. Verdun, and B. McIntosh, "Laser action from Mn5+ in Ba3(VO4)2," Appl. Phys. Lett. 61, 2386-2388 (1992).
[CrossRef]

Vivien, D.

F. Augé, F. Druon, F. Balembois, P. Georges, A. Brun, F. Mougel, G. Aka, and D. Vivien, "Theoretical and experimental investigations of a diode-pumped, quasi-three-level laser: the Yb3+-doped Ca4(GdO(BO3)3 (Yb:GdCOB) laser," IEEE J. Quantum Electron. 36, 598-606 (2000).
[CrossRef]

Waide, P.

K. Shaffers, J. Tassano, P. Waide, S. Payne, and J. Morris, "Progress in the growth of Yb:S-FAP laser crystals," J. Cryst. Growth 225, 449-453 (2001).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

L. D. Merkle, A. Pinto, H. R. Verdun, and B. McIntosh, "Laser action from Mn5+ in Ba3(VO4)2," Appl. Phys. Lett. 61, 2386-2388 (1992).
[CrossRef]

IEEE J. Quantum Electron.

C. Barnard, P. Mylinski, J. Chrostowski, and M. Kavehrad, "Analytical model for rare-earth-doped fiber amplifiers and lasers," IEEE J. Quantum Electron. 30, 1817-1830 (1994).
[CrossRef]

T. Y. Fan and R. L. Byer, "Modeling and cw operation of a quasi-three-level 946 nm Nd:YAG laser," IEEE J. Quantum Electron. 23, 605-611 (1987).
[CrossRef]

L. D. DeLoach, S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. F. Krupke, "Evaluation of absorption andemission properties of Yb3+-doped crystals for laser applications," IEEE J. Quantum Electron. 29, 1179-1190 (1993).
[CrossRef]

C. Lim and Y. Izawa, "Modeling of end-pumped, cw, quasi-three-level lasers," IEEE J. Quantum Electron. 38, 306-311 (2002).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Energy diagram of Yb in S-FAP host showing the pump and three-level laser transitions and the notations used in the model. The quasi-three-level transition that is usually studied is in dashed line.

Fig. 2
Fig. 2

Laser cavity and notations.

Fig. 3
Fig. 3

Theoretical absorption of the pump in the crystal (described in Table 1) versus incident pump power in the nonsaturated regime, in the saturated regime without laser effect, and in the saturated regime with laser effect. Comparison is shown with the experimental data obtained in the nonsaturated regime and in the saturated regime without laser effect.

Fig. 4
Fig. 4

Gain G versus incident pump power for a single pass and a double pass of the pump in the crystal (described in Table 1).

Fig. 5
Fig. 5

Example of theoretical output power at 985 nm versus incident pump power for a single pass and a double pass of the pump in the crystal.

Fig. 6
Fig. 6

Experimental laser output power at 985 nm versus incident pump power for two different output couplers T=7% and T=22%; comparison is shown with the model. In both cases the passive losses L are adjusted at 2.5%.

Fig. 7
Fig. 7

Experimental gain at laser threshold versus incident pump power (triangles) and comparison with the model (solid curve).

Fig. 8
Fig. 8

Search for the optimum size of the laser beam inside the crystal with a double pass of the pump for different crystal length l. Pump beam waist radius is 80 µm, doping level is 1.9×1019 ions/cm3, incident pump power is 1.5 W, transmission of the output coupler is 5%, and passive losses are 1%.

Fig. 9
Fig. 9

Search for the optimum output coupler for different values of incident pump power Pp. The crystal used for the simulation is described in Table 1. The passive losses L are 1%.

Fig. 10
Fig. 10

Output power at 985 nm versus crystal length with a double pass of the pump for different pump powers Pp (left axis). Optimum laser beam waist ωopt was determined by Fig. 8 at the maximum available pump power of 1.5 W versus crystal length (right axis). For each crystal length the output power is calculated with ωopt. Yb concentration N is 1.9×1019 ions/cm3, pump beam waist radius is 80 µm, transmission of the output coupler is 5%, and passive losses are 1%.

Fig. 11
Fig. 11

Influence of the Yb concentration on the output power with a double pass of the pump for different crystal length l. Incident pump power is 1.5 W, laser beam waist radius is 50 µm, pump beam waist radius is 80 µm, passive losses are 1%, and transmission of the output coupler is 5%.

Fig. 12
Fig. 12

Comparison of the optimized theoretical performance at 985 nm obtained with a Ti:sapphire pumping and a diode pumping in the double-pass configuration. The optimized Yb:S-FAP crystals are respectively described in Table 3 and Table 4. The diode used for the numerical simulations is a laser diode with a 1×100-μm2 emitting area delivering a maximum output power of 2 W at 899 nm. The FWHM of the spectrum is 2 nm. In the direction parallel to the diode junction the beam quality factor M2 is 15.

Fig. 13
Fig. 13

Solid curve represents the output power at 985 nm versus wavelength of the Ti:sapphire pump for an incident pump power of 1.45 W. Circles, experimental data; triangles, theoretical data. Comparison is shown with the absorption spectrum (dashed curve).

Tables (4)

Tables Icon

Table 1 Parameters of the Yb:S-FAP Crystal Used in the Experiment and in Sections 2 and 3 and Subsections 4.A and 4.B

Tables Icon

Table 2 Estimation of the Transparency Pump Intensity Itr for Different Yb-Doped Materials

Tables Icon

Table 3 Optimized Parameters of the Yb:S-FAP Crystal for Emission at 985 nm with Double-Pass Ti:sapphire Pumpinga

Tables Icon

Table 4 Optimized Parameters of the Yb:S-FAP Crystal for Laser Emission at 985 nm with Double-Pass Laser Diode Pumping

Equations (25)

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N=N1+N2.
I(r, z)=(2-T)Ic(r, z),
Ic(r, z)=2PoutTπωc2(z) exp-2r2ωc2(z).
ωc(z)=ωc01+λπnωc02(z-zc0)21/2.
g=σel N2-σal N1.
g=NσelσapλphcIp-σalτσapλphcIp+(σel+σal)λhcI+1τ.
Itr=hcλpσalσelσap1τ.
G=1+0ldz0rc4g(r, z)ωc2(z) exp-2r2ωc2(z)rdr2.
G=1(1-T)(1-L),
dIpdz=-αpIp,
αp=σapN1-σepN2.
dIpdz=-NσapσelI+σap/τσapIp+(σel+σal)I+1/τIp.
Ip(r, z)=Ip0(z)exp-2r2ωp2(z).
ωp(z)=ωp01+Mp2λπnωp02(z-zp0)21/2.
dIp0dz=-αp01+(Ip0/Ipsat)Ip0.
Ip0(0)=hcλp2Pinπωp2(0).
Ip(r, 0)=Ip0(0)exp[-2r2/ωp2(0)].
Ip0(z)=Ip0+(z)+Ip0-(z).
Ip0+Ip0-=Kp.
Ip0=Ip0++KpIp0+.
ωpx(z)=ωp0x1+M2λpπnωp0x2(z-zp0x)21/2.
ωpy(z)=ωp0y1+λpπnωp0y2(z-zp0y)21/2.
Ip(x, y, z)=Ip0(z)exp-2x2ωpx2(z)exp-2y2ωpy2(z).
G=1+0ldz0xc0yc2g(x, y, z)πωc2(z)×exp-2(x2+y2)ωc2(z)dxdy2,
σap,m=iλp,iσap,iiλp,i.

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