Abstract

We demonstrate high-power Cr2+:ZnSe master oscillator power amplifier (MOPA) pure continuous wave (CW) laser systems with output power of 14 W and amplifier gain greater than 2X. In addition, we develop a theoretical model for this type of amplification and show single-knob tunability at high powers over 400 nm.

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References

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  1. L. D. DeLoach, R. H. Page, G. D. Wilke, S. A. Payne, and W. F. Krupke, “Transition metal-doped zinc chalcogenides: spectroscopy and laser demonstration of a new class of gain media,” IEEE J. Quantum Electron. 32(6), 885–895 (1996).
    [CrossRef]
  2. R. H. Page, K. I. Schaffers, L. D. DeLoach, G. D. Wilke, F. D. Patel, J. B. Tassano, S. A. Payne, W. F. Krupke, K. T. Chen, and A. Burger, “Cr2+-doped zinc chalcogenides as efficient, widely tunable mid-infrared lasers,” IEEE J. Quantum Electron. 33(4), 609–619 (1997).
    [CrossRef]
  3. I. T. Sorokina, E. Sorokin, S. Mirov, V. Fedorov, V. Badikov, V. Panyutin, and K. I. Schaffers, “Broadly tunable compact continuous-wave Cr2+:ZnS laser,” Opt. Lett. 27(12), 1040–1042 (2002).
    [CrossRef]
  4. U. Hömmerich, X. Wu, V. R. Davis, S. B. Trivedi, K. Grasza, R. J. Chen, and S. Kutcher, “Demonstration of room-temperature laser action at 2.5 mum from Cr2+:Cd0.85Mn0.15Te,” Opt. Lett. 22(15), 1180–1182 (1997).
    [CrossRef] [PubMed]
  5. J. McKay, K. L. Schepler, and G. C. Catella, “Efficient grating-tuned mid-infrared Cr2+:CdSe laser,” Opt. Lett. 24(22), 1575–1577 (1999).
    [CrossRef]
  6. U. Demirbas and A. Sennaroglu, “Intracavity-pumped Cr2+:ZnSe laser with ultrabroad tuning range between 1880 and 3100 nm,” Opt. Lett. 31(15), 2293–2295 (2006).
    [CrossRef] [PubMed]
  7. I. T. Sorokina, “Cr2+-doped II-VI materials for lasers and nonlinear optics,” Opt. Mater. 26(4), 395–412 (2004).
    [CrossRef]
  8. I. S. Moskalev, V. V. Fedorov, S. B. Mirov, P. A. Berry, and K. L. Schepler, “12-Watt CW Polycrystalline Cr2+:ZnSe Laser Pumped by Tm-fiber Laser,” in Advanced Solid State Photonics(Optical Society of America, Denver, CO, 2008), p. WB30.
  9. T. J. Carrig, G. J. Wagner, W. J. Alford, and A. Zakel, “Chromium-doped chalcogenide lasers,” in Solid State Lasers and Amplifiers, A. Sennaroglu, J. G. Fujimoto, and C. R. Pollock, eds. (SPIE, Bellingham, WA, 2004), pp. 74–82.
  10. G. J. Wagner, B. G. Tiemann, W. J. Alford, and T. J. Carrig, “Single-Frequency Cr:ZnSe Laser,” in Advanced Solid-State Photonics(Optical Society of America, 2004), p. WB12.
  11. E. Sorokin, I. T. Sorokina, M. S. Mirov, V. V. Fedorov, I. S. Moskalev, and S. B. Mirov, “Ultrabroad Continuous-Wave Tuning of Ceramic Cr:ZnSe and Cr:ZnS Lasers,” in Advanced Solid-State Photonics(Optical Society of America, 2010), p. AMC2.
  12. I. S. Moskalev, V. V. Fedorov, and S. B. Mirov, “10-watt, pure continuous-wave, polycrystalline Cr2+:ZnS laser,” Opt. Express 17(4), 2048–2056 (2009).
    [CrossRef] [PubMed]
  13. R. J. Harris, G. T. Johnston, G. A. Kepple, P. C. Krok, and H. Mukai, “Infrared thermooptic coefficient measurement of polycrystalline ZnSe, ZnS, CdTe, CaF2, and BaF2, single crystal KCI, and TI-20 glass,” Appl. Opt. 16(2), 436–438 (1977).
    [CrossRef] [PubMed]
  14. D. M. Simanovskii, H. A. Schwettman, H. Lee, and A. J. Welch, “Midinfrared Optical Breakdown in Transparent Dielectrics,” Phys. Rev. Lett. 91(10), 107601 (2003).
    [CrossRef] [PubMed]
  15. K. L. Schepler, R. D. Peterson, P. A. Berry, and J. B. McKay, “Thermal Effects in Cr2+:ZnSe Thin Disk Lasers,” IEEE J. Sel. Top. Quantum Electron. 11(3), 713–720 (2005).
    [CrossRef]
  16. P. A. Berry, and K. L. Schepler, “Cr2+:ZnSe master oscillator / power amplifier for improved power scaling,” in Solid State Lasers XIX: Technology and Devices(SPIE, San Francisco, California, USA, 2010), pp. 75781L–75711.
  17. A. Sennaroglu, U. Demirbas, A. Kurt, and M. Somer, “Concentration dependence of fluorescence and lasing efficiency in Cr2+:ZnSe lasers,” Opt. Mater. 29(6), 703–708 (2007).
    [CrossRef]
  18. H. Kogelnik, E. Ippen, A. Dienes, and C. Shank, “Astigmatically compensated cavities for CW dye lasers,” IEEE J. Quantum Electron. 8(3), 373–379 (1972).
    [CrossRef]

2009

2007

A. Sennaroglu, U. Demirbas, A. Kurt, and M. Somer, “Concentration dependence of fluorescence and lasing efficiency in Cr2+:ZnSe lasers,” Opt. Mater. 29(6), 703–708 (2007).
[CrossRef]

2006

2005

K. L. Schepler, R. D. Peterson, P. A. Berry, and J. B. McKay, “Thermal Effects in Cr2+:ZnSe Thin Disk Lasers,” IEEE J. Sel. Top. Quantum Electron. 11(3), 713–720 (2005).
[CrossRef]

2004

I. T. Sorokina, “Cr2+-doped II-VI materials for lasers and nonlinear optics,” Opt. Mater. 26(4), 395–412 (2004).
[CrossRef]

2003

D. M. Simanovskii, H. A. Schwettman, H. Lee, and A. J. Welch, “Midinfrared Optical Breakdown in Transparent Dielectrics,” Phys. Rev. Lett. 91(10), 107601 (2003).
[CrossRef] [PubMed]

2002

1999

1997

U. Hömmerich, X. Wu, V. R. Davis, S. B. Trivedi, K. Grasza, R. J. Chen, and S. Kutcher, “Demonstration of room-temperature laser action at 2.5 mum from Cr2+:Cd0.85Mn0.15Te,” Opt. Lett. 22(15), 1180–1182 (1997).
[CrossRef] [PubMed]

R. H. Page, K. I. Schaffers, L. D. DeLoach, G. D. Wilke, F. D. Patel, J. B. Tassano, S. A. Payne, W. F. Krupke, K. T. Chen, and A. Burger, “Cr2+-doped zinc chalcogenides as efficient, widely tunable mid-infrared lasers,” IEEE J. Quantum Electron. 33(4), 609–619 (1997).
[CrossRef]

1996

L. D. DeLoach, R. H. Page, G. D. Wilke, S. A. Payne, and W. F. Krupke, “Transition metal-doped zinc chalcogenides: spectroscopy and laser demonstration of a new class of gain media,” IEEE J. Quantum Electron. 32(6), 885–895 (1996).
[CrossRef]

1977

1972

H. Kogelnik, E. Ippen, A. Dienes, and C. Shank, “Astigmatically compensated cavities for CW dye lasers,” IEEE J. Quantum Electron. 8(3), 373–379 (1972).
[CrossRef]

Badikov, V.

Berry, P. A.

K. L. Schepler, R. D. Peterson, P. A. Berry, and J. B. McKay, “Thermal Effects in Cr2+:ZnSe Thin Disk Lasers,” IEEE J. Sel. Top. Quantum Electron. 11(3), 713–720 (2005).
[CrossRef]

Burger, A.

R. H. Page, K. I. Schaffers, L. D. DeLoach, G. D. Wilke, F. D. Patel, J. B. Tassano, S. A. Payne, W. F. Krupke, K. T. Chen, and A. Burger, “Cr2+-doped zinc chalcogenides as efficient, widely tunable mid-infrared lasers,” IEEE J. Quantum Electron. 33(4), 609–619 (1997).
[CrossRef]

Catella, G. C.

Chen, K. T.

R. H. Page, K. I. Schaffers, L. D. DeLoach, G. D. Wilke, F. D. Patel, J. B. Tassano, S. A. Payne, W. F. Krupke, K. T. Chen, and A. Burger, “Cr2+-doped zinc chalcogenides as efficient, widely tunable mid-infrared lasers,” IEEE J. Quantum Electron. 33(4), 609–619 (1997).
[CrossRef]

Chen, R. J.

Davis, V. R.

DeLoach, L. D.

R. H. Page, K. I. Schaffers, L. D. DeLoach, G. D. Wilke, F. D. Patel, J. B. Tassano, S. A. Payne, W. F. Krupke, K. T. Chen, and A. Burger, “Cr2+-doped zinc chalcogenides as efficient, widely tunable mid-infrared lasers,” IEEE J. Quantum Electron. 33(4), 609–619 (1997).
[CrossRef]

L. D. DeLoach, R. H. Page, G. D. Wilke, S. A. Payne, and W. F. Krupke, “Transition metal-doped zinc chalcogenides: spectroscopy and laser demonstration of a new class of gain media,” IEEE J. Quantum Electron. 32(6), 885–895 (1996).
[CrossRef]

Demirbas, U.

A. Sennaroglu, U. Demirbas, A. Kurt, and M. Somer, “Concentration dependence of fluorescence and lasing efficiency in Cr2+:ZnSe lasers,” Opt. Mater. 29(6), 703–708 (2007).
[CrossRef]

U. Demirbas and A. Sennaroglu, “Intracavity-pumped Cr2+:ZnSe laser with ultrabroad tuning range between 1880 and 3100 nm,” Opt. Lett. 31(15), 2293–2295 (2006).
[CrossRef] [PubMed]

Dienes, A.

H. Kogelnik, E. Ippen, A. Dienes, and C. Shank, “Astigmatically compensated cavities for CW dye lasers,” IEEE J. Quantum Electron. 8(3), 373–379 (1972).
[CrossRef]

Fedorov, V.

Fedorov, V. V.

Grasza, K.

Harris, R. J.

Hömmerich, U.

Ippen, E.

H. Kogelnik, E. Ippen, A. Dienes, and C. Shank, “Astigmatically compensated cavities for CW dye lasers,” IEEE J. Quantum Electron. 8(3), 373–379 (1972).
[CrossRef]

Johnston, G. T.

Kepple, G. A.

Kogelnik, H.

H. Kogelnik, E. Ippen, A. Dienes, and C. Shank, “Astigmatically compensated cavities for CW dye lasers,” IEEE J. Quantum Electron. 8(3), 373–379 (1972).
[CrossRef]

Krok, P. C.

Krupke, W. F.

R. H. Page, K. I. Schaffers, L. D. DeLoach, G. D. Wilke, F. D. Patel, J. B. Tassano, S. A. Payne, W. F. Krupke, K. T. Chen, and A. Burger, “Cr2+-doped zinc chalcogenides as efficient, widely tunable mid-infrared lasers,” IEEE J. Quantum Electron. 33(4), 609–619 (1997).
[CrossRef]

L. D. DeLoach, R. H. Page, G. D. Wilke, S. A. Payne, and W. F. Krupke, “Transition metal-doped zinc chalcogenides: spectroscopy and laser demonstration of a new class of gain media,” IEEE J. Quantum Electron. 32(6), 885–895 (1996).
[CrossRef]

Kurt, A.

A. Sennaroglu, U. Demirbas, A. Kurt, and M. Somer, “Concentration dependence of fluorescence and lasing efficiency in Cr2+:ZnSe lasers,” Opt. Mater. 29(6), 703–708 (2007).
[CrossRef]

Kutcher, S.

Lee, H.

D. M. Simanovskii, H. A. Schwettman, H. Lee, and A. J. Welch, “Midinfrared Optical Breakdown in Transparent Dielectrics,” Phys. Rev. Lett. 91(10), 107601 (2003).
[CrossRef] [PubMed]

McKay, J.

McKay, J. B.

K. L. Schepler, R. D. Peterson, P. A. Berry, and J. B. McKay, “Thermal Effects in Cr2+:ZnSe Thin Disk Lasers,” IEEE J. Sel. Top. Quantum Electron. 11(3), 713–720 (2005).
[CrossRef]

Mirov, S.

Mirov, S. B.

Moskalev, I. S.

Mukai, H.

Page, R. H.

R. H. Page, K. I. Schaffers, L. D. DeLoach, G. D. Wilke, F. D. Patel, J. B. Tassano, S. A. Payne, W. F. Krupke, K. T. Chen, and A. Burger, “Cr2+-doped zinc chalcogenides as efficient, widely tunable mid-infrared lasers,” IEEE J. Quantum Electron. 33(4), 609–619 (1997).
[CrossRef]

L. D. DeLoach, R. H. Page, G. D. Wilke, S. A. Payne, and W. F. Krupke, “Transition metal-doped zinc chalcogenides: spectroscopy and laser demonstration of a new class of gain media,” IEEE J. Quantum Electron. 32(6), 885–895 (1996).
[CrossRef]

Panyutin, V.

Patel, F. D.

R. H. Page, K. I. Schaffers, L. D. DeLoach, G. D. Wilke, F. D. Patel, J. B. Tassano, S. A. Payne, W. F. Krupke, K. T. Chen, and A. Burger, “Cr2+-doped zinc chalcogenides as efficient, widely tunable mid-infrared lasers,” IEEE J. Quantum Electron. 33(4), 609–619 (1997).
[CrossRef]

Payne, S. A.

R. H. Page, K. I. Schaffers, L. D. DeLoach, G. D. Wilke, F. D. Patel, J. B. Tassano, S. A. Payne, W. F. Krupke, K. T. Chen, and A. Burger, “Cr2+-doped zinc chalcogenides as efficient, widely tunable mid-infrared lasers,” IEEE J. Quantum Electron. 33(4), 609–619 (1997).
[CrossRef]

L. D. DeLoach, R. H. Page, G. D. Wilke, S. A. Payne, and W. F. Krupke, “Transition metal-doped zinc chalcogenides: spectroscopy and laser demonstration of a new class of gain media,” IEEE J. Quantum Electron. 32(6), 885–895 (1996).
[CrossRef]

Peterson, R. D.

K. L. Schepler, R. D. Peterson, P. A. Berry, and J. B. McKay, “Thermal Effects in Cr2+:ZnSe Thin Disk Lasers,” IEEE J. Sel. Top. Quantum Electron. 11(3), 713–720 (2005).
[CrossRef]

Schaffers, K. I.

I. T. Sorokina, E. Sorokin, S. Mirov, V. Fedorov, V. Badikov, V. Panyutin, and K. I. Schaffers, “Broadly tunable compact continuous-wave Cr2+:ZnS laser,” Opt. Lett. 27(12), 1040–1042 (2002).
[CrossRef]

R. H. Page, K. I. Schaffers, L. D. DeLoach, G. D. Wilke, F. D. Patel, J. B. Tassano, S. A. Payne, W. F. Krupke, K. T. Chen, and A. Burger, “Cr2+-doped zinc chalcogenides as efficient, widely tunable mid-infrared lasers,” IEEE J. Quantum Electron. 33(4), 609–619 (1997).
[CrossRef]

Schepler, K. L.

K. L. Schepler, R. D. Peterson, P. A. Berry, and J. B. McKay, “Thermal Effects in Cr2+:ZnSe Thin Disk Lasers,” IEEE J. Sel. Top. Quantum Electron. 11(3), 713–720 (2005).
[CrossRef]

J. McKay, K. L. Schepler, and G. C. Catella, “Efficient grating-tuned mid-infrared Cr2+:CdSe laser,” Opt. Lett. 24(22), 1575–1577 (1999).
[CrossRef]

Schwettman, H. A.

D. M. Simanovskii, H. A. Schwettman, H. Lee, and A. J. Welch, “Midinfrared Optical Breakdown in Transparent Dielectrics,” Phys. Rev. Lett. 91(10), 107601 (2003).
[CrossRef] [PubMed]

Sennaroglu, A.

A. Sennaroglu, U. Demirbas, A. Kurt, and M. Somer, “Concentration dependence of fluorescence and lasing efficiency in Cr2+:ZnSe lasers,” Opt. Mater. 29(6), 703–708 (2007).
[CrossRef]

U. Demirbas and A. Sennaroglu, “Intracavity-pumped Cr2+:ZnSe laser with ultrabroad tuning range between 1880 and 3100 nm,” Opt. Lett. 31(15), 2293–2295 (2006).
[CrossRef] [PubMed]

Shank, C.

H. Kogelnik, E. Ippen, A. Dienes, and C. Shank, “Astigmatically compensated cavities for CW dye lasers,” IEEE J. Quantum Electron. 8(3), 373–379 (1972).
[CrossRef]

Simanovskii, D. M.

D. M. Simanovskii, H. A. Schwettman, H. Lee, and A. J. Welch, “Midinfrared Optical Breakdown in Transparent Dielectrics,” Phys. Rev. Lett. 91(10), 107601 (2003).
[CrossRef] [PubMed]

Somer, M.

A. Sennaroglu, U. Demirbas, A. Kurt, and M. Somer, “Concentration dependence of fluorescence and lasing efficiency in Cr2+:ZnSe lasers,” Opt. Mater. 29(6), 703–708 (2007).
[CrossRef]

Sorokin, E.

Sorokina, I. T.

Tassano, J. B.

R. H. Page, K. I. Schaffers, L. D. DeLoach, G. D. Wilke, F. D. Patel, J. B. Tassano, S. A. Payne, W. F. Krupke, K. T. Chen, and A. Burger, “Cr2+-doped zinc chalcogenides as efficient, widely tunable mid-infrared lasers,” IEEE J. Quantum Electron. 33(4), 609–619 (1997).
[CrossRef]

Trivedi, S. B.

Welch, A. J.

D. M. Simanovskii, H. A. Schwettman, H. Lee, and A. J. Welch, “Midinfrared Optical Breakdown in Transparent Dielectrics,” Phys. Rev. Lett. 91(10), 107601 (2003).
[CrossRef] [PubMed]

Wilke, G. D.

R. H. Page, K. I. Schaffers, L. D. DeLoach, G. D. Wilke, F. D. Patel, J. B. Tassano, S. A. Payne, W. F. Krupke, K. T. Chen, and A. Burger, “Cr2+-doped zinc chalcogenides as efficient, widely tunable mid-infrared lasers,” IEEE J. Quantum Electron. 33(4), 609–619 (1997).
[CrossRef]

L. D. DeLoach, R. H. Page, G. D. Wilke, S. A. Payne, and W. F. Krupke, “Transition metal-doped zinc chalcogenides: spectroscopy and laser demonstration of a new class of gain media,” IEEE J. Quantum Electron. 32(6), 885–895 (1996).
[CrossRef]

Wu, X.

Appl. Opt.

IEEE J. Quantum Electron.

H. Kogelnik, E. Ippen, A. Dienes, and C. Shank, “Astigmatically compensated cavities for CW dye lasers,” IEEE J. Quantum Electron. 8(3), 373–379 (1972).
[CrossRef]

L. D. DeLoach, R. H. Page, G. D. Wilke, S. A. Payne, and W. F. Krupke, “Transition metal-doped zinc chalcogenides: spectroscopy and laser demonstration of a new class of gain media,” IEEE J. Quantum Electron. 32(6), 885–895 (1996).
[CrossRef]

R. H. Page, K. I. Schaffers, L. D. DeLoach, G. D. Wilke, F. D. Patel, J. B. Tassano, S. A. Payne, W. F. Krupke, K. T. Chen, and A. Burger, “Cr2+-doped zinc chalcogenides as efficient, widely tunable mid-infrared lasers,” IEEE J. Quantum Electron. 33(4), 609–619 (1997).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

K. L. Schepler, R. D. Peterson, P. A. Berry, and J. B. McKay, “Thermal Effects in Cr2+:ZnSe Thin Disk Lasers,” IEEE J. Sel. Top. Quantum Electron. 11(3), 713–720 (2005).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Mater.

I. T. Sorokina, “Cr2+-doped II-VI materials for lasers and nonlinear optics,” Opt. Mater. 26(4), 395–412 (2004).
[CrossRef]

A. Sennaroglu, U. Demirbas, A. Kurt, and M. Somer, “Concentration dependence of fluorescence and lasing efficiency in Cr2+:ZnSe lasers,” Opt. Mater. 29(6), 703–708 (2007).
[CrossRef]

Phys. Rev. Lett.

D. M. Simanovskii, H. A. Schwettman, H. Lee, and A. J. Welch, “Midinfrared Optical Breakdown in Transparent Dielectrics,” Phys. Rev. Lett. 91(10), 107601 (2003).
[CrossRef] [PubMed]

Other

P. A. Berry, and K. L. Schepler, “Cr2+:ZnSe master oscillator / power amplifier for improved power scaling,” in Solid State Lasers XIX: Technology and Devices(SPIE, San Francisco, California, USA, 2010), pp. 75781L–75711.

I. S. Moskalev, V. V. Fedorov, S. B. Mirov, P. A. Berry, and K. L. Schepler, “12-Watt CW Polycrystalline Cr2+:ZnSe Laser Pumped by Tm-fiber Laser,” in Advanced Solid State Photonics(Optical Society of America, Denver, CO, 2008), p. WB30.

T. J. Carrig, G. J. Wagner, W. J. Alford, and A. Zakel, “Chromium-doped chalcogenide lasers,” in Solid State Lasers and Amplifiers, A. Sennaroglu, J. G. Fujimoto, and C. R. Pollock, eds. (SPIE, Bellingham, WA, 2004), pp. 74–82.

G. J. Wagner, B. G. Tiemann, W. J. Alford, and T. J. Carrig, “Single-Frequency Cr:ZnSe Laser,” in Advanced Solid-State Photonics(Optical Society of America, 2004), p. WB12.

E. Sorokin, I. T. Sorokina, M. S. Mirov, V. V. Fedorov, I. S. Moskalev, and S. B. Mirov, “Ultrabroad Continuous-Wave Tuning of Ceramic Cr:ZnSe and Cr:ZnS Lasers,” in Advanced Solid-State Photonics(Optical Society of America, 2010), p. AMC2.

Supplementary Material (1)

» Media 1: AVI (1732 KB)     

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Figures (7)

Fig. 1
Fig. 1

Example model predictions showing (left) input signal growth and pump absorption (with signal present and without), over the crystal length for 5 W input signal and (top) 5 W pump and (bottom) 25 W pump, all at 80 µm spot size. Also shown (right) are amplification comparisons for 5 and 10 W input signal powers with varying spot size as a function of pump power.

Fig. 2
Fig. 2

Astigmatically-compensated Z-cavity design which can use either a single collimated-leg design (dashed) or a double-collimated leg design. OC = output coupler, LP = laser pump lens, FE = functional element.

Fig. 3
Fig. 3

A) Normal-incidence L-cavity scheme which can use either an HR curved mirror (M1) or a combination of planar HR (M2) and positive lens (L1). B) Power amplifier design where LMO and LPA are chosen for optimal overlap of pump and signal beams.

Fig. 4
Fig. 4

Power scaling ability of MOPA configurations. Experimental data (discrete points) is compared with model predictions for La - and Z-cavities, both with 80 µm and 100 μm spot sizes, respectively.

Fig. 5
Fig. 5

Output spectrum of free-running Lb -cavity as a function of output power showing a shift to higher wavelengths at increased power.

Fig. 6
Fig. 6

Tunable output power of Z-cavity over 400 nm which was limited by mirror reflectivities and grating efficiency.

Fig. 7
Fig. 7

Single-frame excerpt (left) from video showing near-field beam profile changes of Lb -cavity as a function of decreasing pump power. (Media 1) starts with maximum pump power and ends at zero pump power. Transitions between different modes cause instability in power output (right).

Equations (11)

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

n * ( t , z , r ) t = I p ( t , z , r ) h ν p ( σ p a n g ( t , z , r ) σ p e n * ( t , z , r ) ) I s ( t , z , r ) σ s e h ν s n * ( t , z , r ) n * ( t , z , r ) τ
I p ( z , r ) z = I p ( z , r ) [ σ p a ( n 0 n * ( z , r ) ) + σ p e n * ( z , r ) γ p ]
I s ( z , r ) z = I s ( z , r ) ( σ s e n * ( z , r ) γ s )
n * ( z , r ) = I p ( z , r ) h ν p σ p a n 0 I p ( z , r ) h ν p ( σ p a + σ p e ) + I s ( z , r ) h ν s σ s e + 1 τ
lim I p ( n * ) = σ p a σ p a + σ p e n 0
J p ( z , r ) = I p ( z , r ) h ν p ( σ p a + σ p e ) τ = I p ( z , r ) I p , s a t
J s ( z , r ) = I s ( z , r ) h ν s σ s e τ = I s ( z , r ) I s , s a t
n * ( z ) = n 0 σ p a ( σ p a + σ p e ) J p ( z ) J p ( z ) + J s ( z ) + 1
d J p ( z ) d z = J p ( z ) [ σ p a n 0 + ( σ p a + σ p e ) n * ( z ) ]
d J s ( z ) d z = J s ( z ) σ s e n * ( z )
I ( r , δ , N ) = exp ( 2 r 2 N δ 2 N )

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