Abstract

An approach for scaling end-pumped rod lasers to high output powers by employing a crystal with a continuously varying, nearly hyperbolic dopant concentration profile, resulting in a homogenization of the longitudinal temperature distribution, is presented. The crystal is characterized by determining its dopant concentration profile with an absorption measurement. The fluorescence upon spectrally narrow excitation is recorded, indicating the role of quenching of the upper laser level at high dopant concentration. The on-axis temperature distribution is calculated by employing a Fourier-Bessel approach for solving the stationary heat conduction equation, taking the temperature dependence of the heat conductivity and the dependence of the heat fraction on the dopant concentration into account. Experimentally, a maximum output power of 187W at an optical-to-optical efficiency of 53 % has been demonstrated.

© 2008 Optical Society of America

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References

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  • Publication
  1. S. C. Tidwell, J. F. Seamans, M. S. Bowers, and A. K. Cousins, “Scaling CW Diode-End-Pumped Nd:YAG Lasers to High Average Powers,” IEEE J. Quantum Electron. 28, 997–1009 (1992).
    [Crossref]
  2. R. Wilhelm, M. Frede, and D. Kracht, “Power Scaling of End-Pumped Solid-State Rod Lasers by Longitudinal Dopant Concentration Gradients,” IEEE J. Quantum Electron. 44, 232–244 (2008).
    [Crossref]
  3. M. Frede, R. Wilhelm, M. Brendel, C. Fallnich, F. Seifert, B. Willke, and K. Danzmann, “High power fundamental mode Nd:YAG laser with efficient birefringence compensation,” Opt. Express 12, 3581–3589 (2004), http://www.opticsexpress.org/abstract.cfm?URI=oe-12-15-3581.
    [Crossref] [PubMed]
  4. D. Kracht, R. Wilhelm, M. Frede, K. Duprç, and L. Ackermann, “407 W End-Pumped Multi-Segmented Nd:YAG Laser,” Opt. Express 13, 10140–10144 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-25-10140.
    [Crossref] [PubMed]
  5. W. Koechner, Solid-State Laser Engineering (Springer, New York, 1996).
  6. S. Guy, C. L. Bonner, D. P. Shepherd, D. C. Hanna, A. C. Tropper, and B. Ferrand, “High-Inversion Densities in Nd:YAG: Upconversion and Bleaching,” IEEE J. Quantum Electron. 34, 900–909 (1998).
    [Crossref]
  7. R. C. Powell, Physics of Solid-State Laser Materials (Springer, New York, 1998).
    [Crossref]
  8. A. A. Kaminskij, Laser Crystals: Their Physics and Properties (Springer, Berlin, 1981).
  9. D. C. Brown, “Heat, Fluorescence, and Stimulated-Emission Power Densities and Fractions in Nd:YAG,” IEEE J. Quantum Electron. 34, 560–572 (1998).
    [Crossref]
  10. D. T. C. Hurle, Handbook of Crystal Growth, Vol. 2A (North-Holland, Amsterdam, 1994).
  11. K. Becker, “Einkristallzüchtung” in Ullmans Enzylopädie der technischen Chemie (Verlag Chemie, Weinheim, 1978).
  12. G. Bitz, Investigation of Correlations between the Optical Properties and the Laser Specific Parameters of Laser-Active Solid-State Materials (PhD thesis, Universität Kaiserslautern, 2001).
    [PubMed]
  13. V. Lupei, A. Lupei, S. Georgescu, and C. Ionescu, “Energy Transfer Between Nd3+ Ions in YAG,” Opt. Commun. 60, 59–63 (1986).
    [Crossref]
  14. T. Y. Fan, “Heat Generation in Nd:YAG and Yb:YAG,” IEEE J. Quantum Electron.  29, 1457–1459 (1993).
    [Crossref]
  15. R. Wilhelm, M. Frede, D. Freiburg, D. Kracht, and C. Fallnich, “Thermal Design of Segmented Rod Laser Crystals,” in Advanced Solid-State Photonics 2005 Technical Digest on CD-ROM (The Optical Society of America, Washington, DC, 2005), paper MB46.
  16. C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hügel, “A 1-kW CW Thin Disc Laser,” IEEE J. Sel. Top. Quantum Electron.  6, 650–657 (2000).
    [Crossref]
  17. Y. Sato, J. Akiyama, and T. Taira, “Novel Model on Thermal Conductivity in Laser Media: Dependence on Rare-Earth Concentration,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies (The Optical Society of America, Washington, DC, 2008), paper CtuQ7.

2008 (1)

R. Wilhelm, M. Frede, and D. Kracht, “Power Scaling of End-Pumped Solid-State Rod Lasers by Longitudinal Dopant Concentration Gradients,” IEEE J. Quantum Electron. 44, 232–244 (2008).
[Crossref]

2005 (1)

2004 (1)

2000 (1)

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hügel, “A 1-kW CW Thin Disc Laser,” IEEE J. Sel. Top. Quantum Electron.  6, 650–657 (2000).
[Crossref]

1998 (2)

S. Guy, C. L. Bonner, D. P. Shepherd, D. C. Hanna, A. C. Tropper, and B. Ferrand, “High-Inversion Densities in Nd:YAG: Upconversion and Bleaching,” IEEE J. Quantum Electron. 34, 900–909 (1998).
[Crossref]

D. C. Brown, “Heat, Fluorescence, and Stimulated-Emission Power Densities and Fractions in Nd:YAG,” IEEE J. Quantum Electron. 34, 560–572 (1998).
[Crossref]

1993 (1)

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

1992 (1)

S. C. Tidwell, J. F. Seamans, M. S. Bowers, and A. K. Cousins, “Scaling CW Diode-End-Pumped Nd:YAG Lasers to High Average Powers,” IEEE J. Quantum Electron. 28, 997–1009 (1992).
[Crossref]

1986 (1)

V. Lupei, A. Lupei, S. Georgescu, and C. Ionescu, “Energy Transfer Between Nd3+ Ions in YAG,” Opt. Commun. 60, 59–63 (1986).
[Crossref]

Ackermann, L.

Akiyama, J.

Y. Sato, J. Akiyama, and T. Taira, “Novel Model on Thermal Conductivity in Laser Media: Dependence on Rare-Earth Concentration,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies (The Optical Society of America, Washington, DC, 2008), paper CtuQ7.

Becker, K.

K. Becker, “Einkristallzüchtung” in Ullmans Enzylopädie der technischen Chemie (Verlag Chemie, Weinheim, 1978).

Bitz, G.

G. Bitz, Investigation of Correlations between the Optical Properties and the Laser Specific Parameters of Laser-Active Solid-State Materials (PhD thesis, Universität Kaiserslautern, 2001).
[PubMed]

Bonner, C. L.

S. Guy, C. L. Bonner, D. P. Shepherd, D. C. Hanna, A. C. Tropper, and B. Ferrand, “High-Inversion Densities in Nd:YAG: Upconversion and Bleaching,” IEEE J. Quantum Electron. 34, 900–909 (1998).
[Crossref]

Bowers, M. S.

S. C. Tidwell, J. F. Seamans, M. S. Bowers, and A. K. Cousins, “Scaling CW Diode-End-Pumped Nd:YAG Lasers to High Average Powers,” IEEE J. Quantum Electron. 28, 997–1009 (1992).
[Crossref]

Brendel, M.

Brown, D. C.

D. C. Brown, “Heat, Fluorescence, and Stimulated-Emission Power Densities and Fractions in Nd:YAG,” IEEE J. Quantum Electron. 34, 560–572 (1998).
[Crossref]

Contag, K.

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hügel, “A 1-kW CW Thin Disc Laser,” IEEE J. Sel. Top. Quantum Electron.  6, 650–657 (2000).
[Crossref]

Cousins, A. K.

S. C. Tidwell, J. F. Seamans, M. S. Bowers, and A. K. Cousins, “Scaling CW Diode-End-Pumped Nd:YAG Lasers to High Average Powers,” IEEE J. Quantum Electron. 28, 997–1009 (1992).
[Crossref]

Danzmann, K.

Duprç, K.

Fallnich, C.

M. Frede, R. Wilhelm, M. Brendel, C. Fallnich, F. Seifert, B. Willke, and K. Danzmann, “High power fundamental mode Nd:YAG laser with efficient birefringence compensation,” Opt. Express 12, 3581–3589 (2004), http://www.opticsexpress.org/abstract.cfm?URI=oe-12-15-3581.
[Crossref] [PubMed]

R. Wilhelm, M. Frede, D. Freiburg, D. Kracht, and C. Fallnich, “Thermal Design of Segmented Rod Laser Crystals,” in Advanced Solid-State Photonics 2005 Technical Digest on CD-ROM (The Optical Society of America, Washington, DC, 2005), paper MB46.

Fan, T. Y.

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

Ferrand, B.

S. Guy, C. L. Bonner, D. P. Shepherd, D. C. Hanna, A. C. Tropper, and B. Ferrand, “High-Inversion Densities in Nd:YAG: Upconversion and Bleaching,” IEEE J. Quantum Electron. 34, 900–909 (1998).
[Crossref]

Frede, M.

R. Wilhelm, M. Frede, and D. Kracht, “Power Scaling of End-Pumped Solid-State Rod Lasers by Longitudinal Dopant Concentration Gradients,” IEEE J. Quantum Electron. 44, 232–244 (2008).
[Crossref]

D. Kracht, R. Wilhelm, M. Frede, K. Duprç, and L. Ackermann, “407 W End-Pumped Multi-Segmented Nd:YAG Laser,” Opt. Express 13, 10140–10144 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-25-10140.
[Crossref] [PubMed]

M. Frede, R. Wilhelm, M. Brendel, C. Fallnich, F. Seifert, B. Willke, and K. Danzmann, “High power fundamental mode Nd:YAG laser with efficient birefringence compensation,” Opt. Express 12, 3581–3589 (2004), http://www.opticsexpress.org/abstract.cfm?URI=oe-12-15-3581.
[Crossref] [PubMed]

R. Wilhelm, M. Frede, D. Freiburg, D. Kracht, and C. Fallnich, “Thermal Design of Segmented Rod Laser Crystals,” in Advanced Solid-State Photonics 2005 Technical Digest on CD-ROM (The Optical Society of America, Washington, DC, 2005), paper MB46.

Freiburg, D.

R. Wilhelm, M. Frede, D. Freiburg, D. Kracht, and C. Fallnich, “Thermal Design of Segmented Rod Laser Crystals,” in Advanced Solid-State Photonics 2005 Technical Digest on CD-ROM (The Optical Society of America, Washington, DC, 2005), paper MB46.

Georgescu, S.

V. Lupei, A. Lupei, S. Georgescu, and C. Ionescu, “Energy Transfer Between Nd3+ Ions in YAG,” Opt. Commun. 60, 59–63 (1986).
[Crossref]

Giesen, A.

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hügel, “A 1-kW CW Thin Disc Laser,” IEEE J. Sel. Top. Quantum Electron.  6, 650–657 (2000).
[Crossref]

Guy, S.

S. Guy, C. L. Bonner, D. P. Shepherd, D. C. Hanna, A. C. Tropper, and B. Ferrand, “High-Inversion Densities in Nd:YAG: Upconversion and Bleaching,” IEEE J. Quantum Electron. 34, 900–909 (1998).
[Crossref]

Hanna, D. C.

S. Guy, C. L. Bonner, D. P. Shepherd, D. C. Hanna, A. C. Tropper, and B. Ferrand, “High-Inversion Densities in Nd:YAG: Upconversion and Bleaching,” IEEE J. Quantum Electron. 34, 900–909 (1998).
[Crossref]

Hügel, H.

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hügel, “A 1-kW CW Thin Disc Laser,” IEEE J. Sel. Top. Quantum Electron.  6, 650–657 (2000).
[Crossref]

Hurle, D. T. C.

D. T. C. Hurle, Handbook of Crystal Growth, Vol. 2A (North-Holland, Amsterdam, 1994).

Ionescu, C.

V. Lupei, A. Lupei, S. Georgescu, and C. Ionescu, “Energy Transfer Between Nd3+ Ions in YAG,” Opt. Commun. 60, 59–63 (1986).
[Crossref]

Kaminskij, A. A.

A. A. Kaminskij, Laser Crystals: Their Physics and Properties (Springer, Berlin, 1981).

Koechner, W.

W. Koechner, Solid-State Laser Engineering (Springer, New York, 1996).

Kracht, D.

R. Wilhelm, M. Frede, and D. Kracht, “Power Scaling of End-Pumped Solid-State Rod Lasers by Longitudinal Dopant Concentration Gradients,” IEEE J. Quantum Electron. 44, 232–244 (2008).
[Crossref]

D. Kracht, R. Wilhelm, M. Frede, K. Duprç, and L. Ackermann, “407 W End-Pumped Multi-Segmented Nd:YAG Laser,” Opt. Express 13, 10140–10144 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-25-10140.
[Crossref] [PubMed]

R. Wilhelm, M. Frede, D. Freiburg, D. Kracht, and C. Fallnich, “Thermal Design of Segmented Rod Laser Crystals,” in Advanced Solid-State Photonics 2005 Technical Digest on CD-ROM (The Optical Society of America, Washington, DC, 2005), paper MB46.

Larionov, M.

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hügel, “A 1-kW CW Thin Disc Laser,” IEEE J. Sel. Top. Quantum Electron.  6, 650–657 (2000).
[Crossref]

Lupei, A.

V. Lupei, A. Lupei, S. Georgescu, and C. Ionescu, “Energy Transfer Between Nd3+ Ions in YAG,” Opt. Commun. 60, 59–63 (1986).
[Crossref]

Lupei, V.

V. Lupei, A. Lupei, S. Georgescu, and C. Ionescu, “Energy Transfer Between Nd3+ Ions in YAG,” Opt. Commun. 60, 59–63 (1986).
[Crossref]

Powell, R. C.

R. C. Powell, Physics of Solid-State Laser Materials (Springer, New York, 1998).
[Crossref]

Sato, Y.

Y. Sato, J. Akiyama, and T. Taira, “Novel Model on Thermal Conductivity in Laser Media: Dependence on Rare-Earth Concentration,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies (The Optical Society of America, Washington, DC, 2008), paper CtuQ7.

Seamans, J. F.

S. C. Tidwell, J. F. Seamans, M. S. Bowers, and A. K. Cousins, “Scaling CW Diode-End-Pumped Nd:YAG Lasers to High Average Powers,” IEEE J. Quantum Electron. 28, 997–1009 (1992).
[Crossref]

Seifert, F.

Shepherd, D. P.

S. Guy, C. L. Bonner, D. P. Shepherd, D. C. Hanna, A. C. Tropper, and B. Ferrand, “High-Inversion Densities in Nd:YAG: Upconversion and Bleaching,” IEEE J. Quantum Electron. 34, 900–909 (1998).
[Crossref]

Stewen, C.

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hügel, “A 1-kW CW Thin Disc Laser,” IEEE J. Sel. Top. Quantum Electron.  6, 650–657 (2000).
[Crossref]

Taira, T.

Y. Sato, J. Akiyama, and T. Taira, “Novel Model on Thermal Conductivity in Laser Media: Dependence on Rare-Earth Concentration,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies (The Optical Society of America, Washington, DC, 2008), paper CtuQ7.

Tidwell, S. C.

S. C. Tidwell, J. F. Seamans, M. S. Bowers, and A. K. Cousins, “Scaling CW Diode-End-Pumped Nd:YAG Lasers to High Average Powers,” IEEE J. Quantum Electron. 28, 997–1009 (1992).
[Crossref]

Tropper, A. C.

S. Guy, C. L. Bonner, D. P. Shepherd, D. C. Hanna, A. C. Tropper, and B. Ferrand, “High-Inversion Densities in Nd:YAG: Upconversion and Bleaching,” IEEE J. Quantum Electron. 34, 900–909 (1998).
[Crossref]

Wilhelm, R.

R. Wilhelm, M. Frede, and D. Kracht, “Power Scaling of End-Pumped Solid-State Rod Lasers by Longitudinal Dopant Concentration Gradients,” IEEE J. Quantum Electron. 44, 232–244 (2008).
[Crossref]

D. Kracht, R. Wilhelm, M. Frede, K. Duprç, and L. Ackermann, “407 W End-Pumped Multi-Segmented Nd:YAG Laser,” Opt. Express 13, 10140–10144 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-25-10140.
[Crossref] [PubMed]

M. Frede, R. Wilhelm, M. Brendel, C. Fallnich, F. Seifert, B. Willke, and K. Danzmann, “High power fundamental mode Nd:YAG laser with efficient birefringence compensation,” Opt. Express 12, 3581–3589 (2004), http://www.opticsexpress.org/abstract.cfm?URI=oe-12-15-3581.
[Crossref] [PubMed]

R. Wilhelm, M. Frede, D. Freiburg, D. Kracht, and C. Fallnich, “Thermal Design of Segmented Rod Laser Crystals,” in Advanced Solid-State Photonics 2005 Technical Digest on CD-ROM (The Optical Society of America, Washington, DC, 2005), paper MB46.

Willke, B.

IEEE J. Quantum Electron (1)

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

IEEE J. Quantum Electron. (4)

S. Guy, C. L. Bonner, D. P. Shepherd, D. C. Hanna, A. C. Tropper, and B. Ferrand, “High-Inversion Densities in Nd:YAG: Upconversion and Bleaching,” IEEE J. Quantum Electron. 34, 900–909 (1998).
[Crossref]

S. C. Tidwell, J. F. Seamans, M. S. Bowers, and A. K. Cousins, “Scaling CW Diode-End-Pumped Nd:YAG Lasers to High Average Powers,” IEEE J. Quantum Electron. 28, 997–1009 (1992).
[Crossref]

R. Wilhelm, M. Frede, and D. Kracht, “Power Scaling of End-Pumped Solid-State Rod Lasers by Longitudinal Dopant Concentration Gradients,” IEEE J. Quantum Electron. 44, 232–244 (2008).
[Crossref]

D. C. Brown, “Heat, Fluorescence, and Stimulated-Emission Power Densities and Fractions in Nd:YAG,” IEEE J. Quantum Electron. 34, 560–572 (1998).
[Crossref]

IEEE J. Sel. Top. Quantum Electron (1)

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hügel, “A 1-kW CW Thin Disc Laser,” IEEE J. Sel. Top. Quantum Electron.  6, 650–657 (2000).
[Crossref]

Opt. Commun. (1)

V. Lupei, A. Lupei, S. Georgescu, and C. Ionescu, “Energy Transfer Between Nd3+ Ions in YAG,” Opt. Commun. 60, 59–63 (1986).
[Crossref]

Opt. Express (2)

Other (8)

W. Koechner, Solid-State Laser Engineering (Springer, New York, 1996).

D. T. C. Hurle, Handbook of Crystal Growth, Vol. 2A (North-Holland, Amsterdam, 1994).

K. Becker, “Einkristallzüchtung” in Ullmans Enzylopädie der technischen Chemie (Verlag Chemie, Weinheim, 1978).

G. Bitz, Investigation of Correlations between the Optical Properties and the Laser Specific Parameters of Laser-Active Solid-State Materials (PhD thesis, Universität Kaiserslautern, 2001).
[PubMed]

Y. Sato, J. Akiyama, and T. Taira, “Novel Model on Thermal Conductivity in Laser Media: Dependence on Rare-Earth Concentration,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies (The Optical Society of America, Washington, DC, 2008), paper CtuQ7.

R. C. Powell, Physics of Solid-State Laser Materials (Springer, New York, 1998).
[Crossref]

A. A. Kaminskij, Laser Crystals: Their Physics and Properties (Springer, Berlin, 1981).

R. Wilhelm, M. Frede, D. Freiburg, D. Kracht, and C. Fallnich, “Thermal Design of Segmented Rod Laser Crystals,” in Advanced Solid-State Photonics 2005 Technical Digest on CD-ROM (The Optical Society of America, Washington, DC, 2005), paper MB46.

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Figures (6)

Fig. 1.
Fig. 1.

Determination of the dopant concentration profile of the crystal: (a) experimental scheme of the absorption measurement; (b) geometry for determining the average optical path length.

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Fig. 2.
Fig. 2.

(a) Longitudinal dopant concentration profile from absorption measurements and hyperbolic fit. The optimum profiles resulting from solving (10) for different pump wavelengths and absorption efficiencies are shown for comaprison (b) Optimum starting and final dopant concentration values against pump wavelength for 2.5 nm spectral width (FWHM). The measured dopant concentration profile is close to the optimum distribution for a pump wavelength of 802 nm. The absorption efficiency is limited to 80%.

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Fig. 3.
Fig. 3.

(a) Experimental for determining the longitudinal fluorescence distribution. (b) Longitudinal fluorescence distribution upon spectrally narrow excitation at 803 nm.

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Fig. 4.
Fig. 4.

(a) On-axis temperature distribution for 440 W of pump power with laser action. Dashed lines with unity pump-metastable quantum efficiency, solid line with dopant concentration dependent pump-metastable quantum efficiency. (b) Maximum applicable pump power before reaching fracture limit of 130 MPa. Solid lines: rod with hyperbolic dopant concentration profile, dashed lines: uniformly doped rod with same absorption efficiency at 803 nm.

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Fig. 5.
Fig. 5.

Schematic setup of the laser system.

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Fig. 6.
Fig. 6.

Multimode laser output power against absorbed pump power.

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Tables (1)

Tables Icon

Table 1. Coefficients c 1 to c 4 used in (11) for describing the optimum dopant concentration profile. Bold lines are shown in figure 2 a. The fitting parameters used in the temperature calculation to described experimentally obtained data are given in the last line for comparison.

View Table

Equations (15)

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

σ = 2 α E η h 8 π K ( 1 − ν ) dP dz
q ( r , z ) = P pump η h ( z ) 2 π ∫ 0 R dr r f ( r ) α eff ( z ) exp ( − ∫ 0 z dz ′ α eff ( z ′ ) ) f ( r ) ,
α eff ( z ) = ∫ 0 ∞ d λ P λ ( λ , z ) α ( λ , z ) ∫ 0 ∞ d λ P λ ( λ , z ) ,
d P ( z ) dz = − α eff ( z ) · P ( z ) = const . = − P ( 0 ) ξ ⇒ P ( z ) = P ( 0 ) ( 1 − ξ · z ) .
α eff ( z ) = ξ 1 − ξ z ,
ξ = dP dz ∣ max P pump .
L p = 2 · R 2 − r 2 n 2 .
L ¯ p = 4 π w 2 ( 1 − exp ( − 2 R 2 ⁄ w 2 ) ) ∫ 0 R ∫ 0 2 π d ϕ d ρ ρ exp ( − 2 ρ 2 w 2 ) R 2 − ρ 2 sin 2 ϕ n 2 ,
T = exp ( − ∫ 0 L dz α ( z ) ) .
d P ( z ) dz = C ( z ) C 0 ∫ 0 ∞ d λ α 0 ( λ ) P λ ( λ , z ) ∫ 0 ∞ d λ P λ ( λ , z ) = 1 − T L
C opt ( z ) = c 1 1 − c 2 z + c 3 z 1 − c 4 z .
τ ( ρ ) = τ rad exp ( − b ρ c )
I fluor ∝ ρ ( z ) I pump ( z ) τ ρ τ rad
( ∂ 2 ∂ r 2 + 1 r ∂ ∂ r + ∂ 2 ∂ z 2 ) T ( r , z ) = − 1 K ( T ) [ q ( r , z ) + ∂ K ∂ T ( ( ∂ T ∂ r ) 2 + ( ∂ T ∂ z ) 2 ) ] .
C ( z ) = c 1 1 − c 2 z ,

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