Abstract

In order to conveniently and precisely estimate the spot size of laser modes in continuously single-end-pumped solid-state lasers, a ray matrix for the thermal-induced lenslike medium with exponential heat distribution along the axial direction is, for the first time to our best knowledge, derived under assumptions of only radial heat flow and uniform pump spot size. An equivalent optical model is also developed and especially modified for a lenslike gain medium in a continuously single-end-pumped solid-state laser. The result is verified by discussing how it fits the practical cases much better than the previous simplified model and by numerical simulation in a practice problem. This work is especially important in cases with a relatively long gain medium, which has been used more and more frequently in recent years.

© 2011 Optical Society of America

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References

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  1. V. Magni, “Resonators for solid-state lasers with large-volume fundamental mode and high alignment stability,” Appl. Opt. 25, 107–117 (1986).
    [CrossRef] [PubMed]
  2. H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
  3. Y. E. Young, S. D. Setzler, T. M. Pollak, and E. P. Chicklis, “Optical parametric oscillator pumped at 1645 nm by a 9 W, fiber-laser-pumped, Q-switched Er:YAG laser,” in Advanced Solid-State Photonics, OSA Technical Digest (Optical Society of America, 2004), paper TuC2.
  4. H. Y. Zhu, G. Zhang, Y. M. Duan, C. H. Huang, and Y. Wei, “Compact continuous-wave Nd:YVO4 laser with self-Raman conversion and sum frequency generation,” Chin. Phys. Lett. 28, 054202 (2011).
    [CrossRef]
  5. C. Du, S. Ruan, H. Zhang, Y. Yu, F. Zeng, J. Wang, and M. Jiang, “A 13.3 W laser-diode-array end-pumped Nd:GdYVO4 continuous-wave laser at 1.34 μm,” Appl. Phys. B 80, 45–48(2005).
    [CrossRef]
  6. S. Y. Zhang, X. J. Cheng, L. Xu, and J. Q. Xu, “Power scaling of continuous-wave diode-end pump Tm:LiLuF4 slab laser,” Laser Phys. Lett. 6, 856–859 (2009).
    [CrossRef]
  7. Y. F. Chen, T. M. Huang, C. C. Liao, Y. P. Lan, and S. C. Wang, “Efficient high-power diode-end-pumped TEM00Nd:YVO4 laser,” IEEE Photon. Technol. Lett. 11, 1241–1243 (1999).
    [CrossRef]
  8. L. Huang, M. Gong, Q. Liu, P. Yan, and H. Zhang, “A novel prism beam-shaping laser diode bar end-pumped TEM00 mode Nd:YVO4 laser,” Laser Phys. 20, 1949–1953 (2010).
    [CrossRef]
  9. J. Yu, H. X. Meng, and T. F. Jin, “Modeling of thermal lensing in cw end-pumped solid-state lasers,” Proc. SPIE 2889, 171–177 (1996).
    [CrossRef]
  10. W. Koechner, Solid-State Laser Engineering, 2nd ed.(Academic, 1988).
  11. M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56, 1831–1833 (1990).
    [CrossRef]
  12. 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]
  13. S. B. Sutton and G. F. Albrecht, “Optical distortion in end-pumped solid-state rod lasers,” Appl. Opt. 32, 5256–5269(1993).
    [CrossRef] [PubMed]
  14. Z. Xiong, Z. G. Li, W. L. Huang, and G. C. Lim, “Detailed investigation of thermal effects in longitudinally diode-pumped Nd:YVO4 lasers,” IEEE J. Quantum Electron. 39, 979–986(2003).
    [CrossRef]

2011

H. Y. Zhu, G. Zhang, Y. M. Duan, C. H. Huang, and Y. Wei, “Compact continuous-wave Nd:YVO4 laser with self-Raman conversion and sum frequency generation,” Chin. Phys. Lett. 28, 054202 (2011).
[CrossRef]

2010

L. Huang, M. Gong, Q. Liu, P. Yan, and H. Zhang, “A novel prism beam-shaping laser diode bar end-pumped TEM00 mode Nd:YVO4 laser,” Laser Phys. 20, 1949–1953 (2010).
[CrossRef]

2009

S. Y. Zhang, X. J. Cheng, L. Xu, and J. Q. Xu, “Power scaling of continuous-wave diode-end pump Tm:LiLuF4 slab laser,” Laser Phys. Lett. 6, 856–859 (2009).
[CrossRef]

2005

C. Du, S. Ruan, H. Zhang, Y. Yu, F. Zeng, J. Wang, and M. Jiang, “A 13.3 W laser-diode-array end-pumped Nd:GdYVO4 continuous-wave laser at 1.34 μm,” Appl. Phys. B 80, 45–48(2005).
[CrossRef]

2003

Z. Xiong, Z. G. Li, W. L. Huang, and G. C. Lim, “Detailed investigation of thermal effects in longitudinally diode-pumped Nd:YVO4 lasers,” IEEE J. Quantum Electron. 39, 979–986(2003).
[CrossRef]

1999

Y. F. Chen, T. M. Huang, C. C. Liao, Y. P. Lan, and S. C. Wang, “Efficient high-power diode-end-pumped TEM00Nd:YVO4 laser,” IEEE Photon. Technol. Lett. 11, 1241–1243 (1999).
[CrossRef]

1996

J. Yu, H. X. Meng, and T. F. Jin, “Modeling of thermal lensing in cw end-pumped solid-state lasers,” Proc. SPIE 2889, 171–177 (1996).
[CrossRef]

1993

1992

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]

1990

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56, 1831–1833 (1990).
[CrossRef]

1986

1965

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).

Albrecht, G. F.

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]

Chen, Y. F.

Y. F. Chen, T. M. Huang, C. C. Liao, Y. P. Lan, and S. C. Wang, “Efficient high-power diode-end-pumped TEM00Nd:YVO4 laser,” IEEE Photon. Technol. Lett. 11, 1241–1243 (1999).
[CrossRef]

Cheng, X. J.

S. Y. Zhang, X. J. Cheng, L. Xu, and J. Q. Xu, “Power scaling of continuous-wave diode-end pump Tm:LiLuF4 slab laser,” Laser Phys. Lett. 6, 856–859 (2009).
[CrossRef]

Chicklis, E. P.

Y. E. Young, S. D. Setzler, T. M. Pollak, and E. P. Chicklis, “Optical parametric oscillator pumped at 1645 nm by a 9 W, fiber-laser-pumped, Q-switched Er:YAG laser,” in Advanced Solid-State Photonics, OSA Technical Digest (Optical Society of America, 2004), paper TuC2.

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]

Du, C.

C. Du, S. Ruan, H. Zhang, Y. Yu, F. Zeng, J. Wang, and M. Jiang, “A 13.3 W laser-diode-array end-pumped Nd:GdYVO4 continuous-wave laser at 1.34 μm,” Appl. Phys. B 80, 45–48(2005).
[CrossRef]

Duan, Y. M.

H. Y. Zhu, G. Zhang, Y. M. Duan, C. H. Huang, and Y. Wei, “Compact continuous-wave Nd:YVO4 laser with self-Raman conversion and sum frequency generation,” Chin. Phys. Lett. 28, 054202 (2011).
[CrossRef]

Fields, R. A.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56, 1831–1833 (1990).
[CrossRef]

Fincher, C. L.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56, 1831–1833 (1990).
[CrossRef]

Gong, M.

L. Huang, M. Gong, Q. Liu, P. Yan, and H. Zhang, “A novel prism beam-shaping laser diode bar end-pumped TEM00 mode Nd:YVO4 laser,” Laser Phys. 20, 1949–1953 (2010).
[CrossRef]

Huang, C. H.

H. Y. Zhu, G. Zhang, Y. M. Duan, C. H. Huang, and Y. Wei, “Compact continuous-wave Nd:YVO4 laser with self-Raman conversion and sum frequency generation,” Chin. Phys. Lett. 28, 054202 (2011).
[CrossRef]

Huang, L.

L. Huang, M. Gong, Q. Liu, P. Yan, and H. Zhang, “A novel prism beam-shaping laser diode bar end-pumped TEM00 mode Nd:YVO4 laser,” Laser Phys. 20, 1949–1953 (2010).
[CrossRef]

Huang, T. M.

Y. F. Chen, T. M. Huang, C. C. Liao, Y. P. Lan, and S. C. Wang, “Efficient high-power diode-end-pumped TEM00Nd:YVO4 laser,” IEEE Photon. Technol. Lett. 11, 1241–1243 (1999).
[CrossRef]

Huang, W. L.

Z. Xiong, Z. G. Li, W. L. Huang, and G. C. Lim, “Detailed investigation of thermal effects in longitudinally diode-pumped Nd:YVO4 lasers,” IEEE J. Quantum Electron. 39, 979–986(2003).
[CrossRef]

Innocenzi, M. E.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56, 1831–1833 (1990).
[CrossRef]

Jiang, M.

C. Du, S. Ruan, H. Zhang, Y. Yu, F. Zeng, J. Wang, and M. Jiang, “A 13.3 W laser-diode-array end-pumped Nd:GdYVO4 continuous-wave laser at 1.34 μm,” Appl. Phys. B 80, 45–48(2005).
[CrossRef]

Jin, T. F.

J. Yu, H. X. Meng, and T. F. Jin, “Modeling of thermal lensing in cw end-pumped solid-state lasers,” Proc. SPIE 2889, 171–177 (1996).
[CrossRef]

Koechner, W.

W. Koechner, Solid-State Laser Engineering, 2nd ed.(Academic, 1988).

Kogelnik, H.

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).

Lan, Y. P.

Y. F. Chen, T. M. Huang, C. C. Liao, Y. P. Lan, and S. C. Wang, “Efficient high-power diode-end-pumped TEM00Nd:YVO4 laser,” IEEE Photon. Technol. Lett. 11, 1241–1243 (1999).
[CrossRef]

Li, Z. G.

Z. Xiong, Z. G. Li, W. L. Huang, and G. C. Lim, “Detailed investigation of thermal effects in longitudinally diode-pumped Nd:YVO4 lasers,” IEEE J. Quantum Electron. 39, 979–986(2003).
[CrossRef]

Liao, C. C.

Y. F. Chen, T. M. Huang, C. C. Liao, Y. P. Lan, and S. C. Wang, “Efficient high-power diode-end-pumped TEM00Nd:YVO4 laser,” IEEE Photon. Technol. Lett. 11, 1241–1243 (1999).
[CrossRef]

Lim, G. C.

Z. Xiong, Z. G. Li, W. L. Huang, and G. C. Lim, “Detailed investigation of thermal effects in longitudinally diode-pumped Nd:YVO4 lasers,” IEEE J. Quantum Electron. 39, 979–986(2003).
[CrossRef]

Liu, Q.

L. Huang, M. Gong, Q. Liu, P. Yan, and H. Zhang, “A novel prism beam-shaping laser diode bar end-pumped TEM00 mode Nd:YVO4 laser,” Laser Phys. 20, 1949–1953 (2010).
[CrossRef]

Magni, V.

Meng, H. X.

J. Yu, H. X. Meng, and T. F. Jin, “Modeling of thermal lensing in cw end-pumped solid-state lasers,” Proc. SPIE 2889, 171–177 (1996).
[CrossRef]

Pollak, T. M.

Y. E. Young, S. D. Setzler, T. M. Pollak, and E. P. Chicklis, “Optical parametric oscillator pumped at 1645 nm by a 9 W, fiber-laser-pumped, Q-switched Er:YAG laser,” in Advanced Solid-State Photonics, OSA Technical Digest (Optical Society of America, 2004), paper TuC2.

Ruan, S.

C. Du, S. Ruan, H. Zhang, Y. Yu, F. Zeng, J. Wang, and M. Jiang, “A 13.3 W laser-diode-array end-pumped Nd:GdYVO4 continuous-wave laser at 1.34 μm,” Appl. Phys. B 80, 45–48(2005).
[CrossRef]

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]

Setzler, S. D.

Y. E. Young, S. D. Setzler, T. M. Pollak, and E. P. Chicklis, “Optical parametric oscillator pumped at 1645 nm by a 9 W, fiber-laser-pumped, Q-switched Er:YAG laser,” in Advanced Solid-State Photonics, OSA Technical Digest (Optical Society of America, 2004), paper TuC2.

Sutton, S. B.

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]

Wang, J.

C. Du, S. Ruan, H. Zhang, Y. Yu, F. Zeng, J. Wang, and M. Jiang, “A 13.3 W laser-diode-array end-pumped Nd:GdYVO4 continuous-wave laser at 1.34 μm,” Appl. Phys. B 80, 45–48(2005).
[CrossRef]

Wang, S. C.

Y. F. Chen, T. M. Huang, C. C. Liao, Y. P. Lan, and S. C. Wang, “Efficient high-power diode-end-pumped TEM00Nd:YVO4 laser,” IEEE Photon. Technol. Lett. 11, 1241–1243 (1999).
[CrossRef]

Wei, Y.

H. Y. Zhu, G. Zhang, Y. M. Duan, C. H. Huang, and Y. Wei, “Compact continuous-wave Nd:YVO4 laser with self-Raman conversion and sum frequency generation,” Chin. Phys. Lett. 28, 054202 (2011).
[CrossRef]

Xiong, Z.

Z. Xiong, Z. G. Li, W. L. Huang, and G. C. Lim, “Detailed investigation of thermal effects in longitudinally diode-pumped Nd:YVO4 lasers,” IEEE J. Quantum Electron. 39, 979–986(2003).
[CrossRef]

Xu, J. Q.

S. Y. Zhang, X. J. Cheng, L. Xu, and J. Q. Xu, “Power scaling of continuous-wave diode-end pump Tm:LiLuF4 slab laser,” Laser Phys. Lett. 6, 856–859 (2009).
[CrossRef]

Xu, L.

S. Y. Zhang, X. J. Cheng, L. Xu, and J. Q. Xu, “Power scaling of continuous-wave diode-end pump Tm:LiLuF4 slab laser,” Laser Phys. Lett. 6, 856–859 (2009).
[CrossRef]

Yan, P.

L. Huang, M. Gong, Q. Liu, P. Yan, and H. Zhang, “A novel prism beam-shaping laser diode bar end-pumped TEM00 mode Nd:YVO4 laser,” Laser Phys. 20, 1949–1953 (2010).
[CrossRef]

Young, Y. E.

Y. E. Young, S. D. Setzler, T. M. Pollak, and E. P. Chicklis, “Optical parametric oscillator pumped at 1645 nm by a 9 W, fiber-laser-pumped, Q-switched Er:YAG laser,” in Advanced Solid-State Photonics, OSA Technical Digest (Optical Society of America, 2004), paper TuC2.

Yu, J.

J. Yu, H. X. Meng, and T. F. Jin, “Modeling of thermal lensing in cw end-pumped solid-state lasers,” Proc. SPIE 2889, 171–177 (1996).
[CrossRef]

Yu, Y.

C. Du, S. Ruan, H. Zhang, Y. Yu, F. Zeng, J. Wang, and M. Jiang, “A 13.3 W laser-diode-array end-pumped Nd:GdYVO4 continuous-wave laser at 1.34 μm,” Appl. Phys. B 80, 45–48(2005).
[CrossRef]

Yura, H. T.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56, 1831–1833 (1990).
[CrossRef]

Zeng, F.

C. Du, S. Ruan, H. Zhang, Y. Yu, F. Zeng, J. Wang, and M. Jiang, “A 13.3 W laser-diode-array end-pumped Nd:GdYVO4 continuous-wave laser at 1.34 μm,” Appl. Phys. B 80, 45–48(2005).
[CrossRef]

Zhang, G.

H. Y. Zhu, G. Zhang, Y. M. Duan, C. H. Huang, and Y. Wei, “Compact continuous-wave Nd:YVO4 laser with self-Raman conversion and sum frequency generation,” Chin. Phys. Lett. 28, 054202 (2011).
[CrossRef]

Zhang, H.

L. Huang, M. Gong, Q. Liu, P. Yan, and H. Zhang, “A novel prism beam-shaping laser diode bar end-pumped TEM00 mode Nd:YVO4 laser,” Laser Phys. 20, 1949–1953 (2010).
[CrossRef]

C. Du, S. Ruan, H. Zhang, Y. Yu, F. Zeng, J. Wang, and M. Jiang, “A 13.3 W laser-diode-array end-pumped Nd:GdYVO4 continuous-wave laser at 1.34 μm,” Appl. Phys. B 80, 45–48(2005).
[CrossRef]

Zhang, S. Y.

S. Y. Zhang, X. J. Cheng, L. Xu, and J. Q. Xu, “Power scaling of continuous-wave diode-end pump Tm:LiLuF4 slab laser,” Laser Phys. Lett. 6, 856–859 (2009).
[CrossRef]

Zhu, H. Y.

H. Y. Zhu, G. Zhang, Y. M. Duan, C. H. Huang, and Y. Wei, “Compact continuous-wave Nd:YVO4 laser with self-Raman conversion and sum frequency generation,” Chin. Phys. Lett. 28, 054202 (2011).
[CrossRef]

Appl. Opt.

Appl. Phys. B

C. Du, S. Ruan, H. Zhang, Y. Yu, F. Zeng, J. Wang, and M. Jiang, “A 13.3 W laser-diode-array end-pumped Nd:GdYVO4 continuous-wave laser at 1.34 μm,” Appl. Phys. B 80, 45–48(2005).
[CrossRef]

Appl. Phys. Lett.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56, 1831–1833 (1990).
[CrossRef]

Bell Syst. Tech. J.

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).

Chin. Phys. Lett.

H. Y. Zhu, G. Zhang, Y. M. Duan, C. H. Huang, and Y. Wei, “Compact continuous-wave Nd:YVO4 laser with self-Raman conversion and sum frequency generation,” Chin. Phys. Lett. 28, 054202 (2011).
[CrossRef]

IEEE J. Quantum Electron.

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]

Z. Xiong, Z. G. Li, W. L. Huang, and G. C. Lim, “Detailed investigation of thermal effects in longitudinally diode-pumped Nd:YVO4 lasers,” IEEE J. Quantum Electron. 39, 979–986(2003).
[CrossRef]

IEEE Photon. Technol. Lett.

Y. F. Chen, T. M. Huang, C. C. Liao, Y. P. Lan, and S. C. Wang, “Efficient high-power diode-end-pumped TEM00Nd:YVO4 laser,” IEEE Photon. Technol. Lett. 11, 1241–1243 (1999).
[CrossRef]

Laser Phys.

L. Huang, M. Gong, Q. Liu, P. Yan, and H. Zhang, “A novel prism beam-shaping laser diode bar end-pumped TEM00 mode Nd:YVO4 laser,” Laser Phys. 20, 1949–1953 (2010).
[CrossRef]

Laser Phys. Lett.

S. Y. Zhang, X. J. Cheng, L. Xu, and J. Q. Xu, “Power scaling of continuous-wave diode-end pump Tm:LiLuF4 slab laser,” Laser Phys. Lett. 6, 856–859 (2009).
[CrossRef]

Proc. SPIE

J. Yu, H. X. Meng, and T. F. Jin, “Modeling of thermal lensing in cw end-pumped solid-state lasers,” Proc. SPIE 2889, 171–177 (1996).
[CrossRef]

Other

W. Koechner, Solid-State Laser Engineering, 2nd ed.(Academic, 1988).

Y. E. Young, S. D. Setzler, T. M. Pollak, and E. P. Chicklis, “Optical parametric oscillator pumped at 1645 nm by a 9 W, fiber-laser-pumped, Q-switched Er:YAG laser,” in Advanced Solid-State Photonics, OSA Technical Digest (Optical Society of America, 2004), paper TuC2.

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Figures (8)

Fig. 1
Fig. 1

Kogelnik’s equivalent optical model.

Fig. 2
Fig. 2

Model for lenslike gain medium with uneven heat generation along the axial direction.

Fig. 3
Fig. 3

Range of a , b, c , and d .

Fig. 4
Fig. 4

Equivalent optical model for ray matrix S.

Fig. 5
Fig. 5

Modified equivalent optical Model I.

Fig. 6
Fig. 6

Modified equivalent optical Model II.

Fig. 7
Fig. 7

Spot sizes of the fundamental laser mode on the incident face by the first kind of numerical simulation and comparison with those by Kogelnik’s model and the modified Model II: circles, Kogelnik’s model; pentagrams, modified Model II; regular triangles, simulation with F : f de = 1 ; squares, simulation with F : f de = 2 .

Fig. 8
Fig. 8

Spot sizes of the fundamental laser mode on the incident face by the second kind of numerical simulation and comparison with those by Kogelnik’s model and the modified Model II: circles, Kogelnik’s model; pentagrams, the modified Model II; squares, simulation with distance of L / 8 between the pump waist and incident face; regular triangles, simulation with the distance of L / 4 ; inverted triangles, simulation with distance of 3 L / 8 .

Equations (73)

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

f = b 2 n 0 sin 2 L b ,
h = b 2 n 0 tan L b ,
Δ L = 2 h = b n 0 tan L n b .
f m = e α m L n f 0 ,
f m = e 0 m f 0 .
S 0 = ( 1 0 1 f 0 1 ) .
S 1 = ( 1 Δ L 0 1 ) ( 1 0 e 0 f 0 1 ) = ( 1 e 0 Δ L f 0 Δ L e 0 1 f 0 1 ) .
S m = ( 1 e 0 m Δ L f 0 Δ L e 0 m 1 f 0 1 ) .
S = i = 0 n 1 S i .
S = ( A B C D ) = ( 1 a n 1 Δ L f 0 + o ( Δ L f 0 ) b n 1 , 1 Δ L b n 1 , 2 Δ L 2 f 0 + o ( Δ L f 0 ) Δ L c n 1 , 1 1 f 0 + c n 1 , 2 Δ L f 0 2 + o ( Δ L f 0 ) 1 f 0 1 d n 1 Δ L f 0 + o ( Δ L f 0 ) ) ,
a n 1 = e 0 e 0 n ( 1 e 0 ) 2 ( n 1 ) e 0 n 1 e 0 ,
b n 1 , 1 = n 1 ,
b n 1 , 2 = ( n 2 ) ( e 0 + e 0 n ) ( 1 e 0 ) 2 2 e 0 2 2 e 0 n ( 1 e 0 ) 3 ,
c n 1 , 1 = 1 e 0 n 1 e 0 ,
c n 1 , 2 = e 0 n e 0 n ( 1 e 0 ) 2 + e 0 3 e 0 2 n + 1 ( 1 e 0 ) 3 ( 1 + e 0 ) ,
d n 1 = n 1 1 e 0 e 0 e 0 n ( 1 e 0 ) 2 .
1 f total = 1 f di + 1 f bi + 1 f de ,
1 f lm = 1 f di + 1 f bi .
( 1 0 1 f lm 1 ) = i = 0 n 1 ( 1 0 e 0 i f 0 1 ) .
f 0 = 1 e 0 n 1 e 0 f lm .
S = ( 1 ( 1 ln T T 1 T ) L n 0 f lm L n 0 + [ 1 + T ( 1 T ) ln T + 2 ( ln T ) 2 ] L 2 n 0 2 f lm 1 f lm [ T ( 1 T ) 2 + 1 + T 2 ( 1 T ) ln T ] L n 0 f lm 2 1 ( 1 ln T + 1 1 T ) L n 0 f lm ) ,
S = ( 1 a L n 0 f lm L n 0 ( 1 + b L n 0 f lm ) 1 f lm ( 1 + c L n 0 f lm ) 1 d L n 0 f lm ) ,
S ( 1 ( 1 ln T T 1 T ) L n 0 f lm L n 0 [ 1 ( 1 ln T T 1 T ) ( 1 ln T + 1 1 T ) L n 0 f lm ] 1 f lm 1 ( 1 ln T + 1 1 T ) L n 0 f lm ) .
S ( 1 D 1 0 1 ) ( 1 0 1 F 1 ) ( 1 D 2 0 1 ) ,
D 1 = ( 1 ln T T 1 T ) L n 0 ,
F = f lm ,
D 2 = ( 1 ln T + 1 1 T ) L n 0 .
h = L 2 n 0 .
S M I ( 1 0 1 f de 1 ) ( 1 D 1 0 1 ) ( 1 0 1 F 1 ) ( 1 D 2 0 1 ) .
S Tlens = ( 1 0 1 f de 1 ) ( 1 D 1 0 1 ) ( 1 0 1 F 1 ) .
S Tlens = ( 1 D 1 0 1 ) ( 1 0 1 F 1 ) ( 1 D 2 0 1 ) ,
1 F = 1 F + 1 f de D 1 F f de ,
D 1 = D 1 F F ,
D 2 = D 1 F f de ,
S M I I ( 1 D 1 0 1 ) ( 1 0 1 F 1 ) ( 1 D 2 + D 2 0 1 ) .
D 1 D 2 D 1 2 D 1 F .
D 1 D 2 D 1 2 .
D 1 = ( 1 ln T T 1 T ) L 2 n 0 ,
D 2 + D 2 = ( 1 ln T + 2 T 1 T ) L 2 n 0 .
F = f me ,
S = ( 1 1 2 L n 0 f lm L n 0 1 6 L 2 n 0 2 f lm 1 f lm + 1 6 L n 0 f lm 2 1 1 2 L n 0 f lm ) ,
S = ( 1 L n 0 1 f lm 1 L n 0 f lm ) .
S = ( 1 0 1 f lm 1 ) ( 1 L n 0 0 1 ) .
S 0 S 1 = ( 1 e 0 Δ L f 0 Δ L ( 1 + e 0 ) 1 f 0 + e 0 Δ L f 0 2 1 Δ L f 0 ) ,
S 0 S 1 S 2 = ( 1 ( e 0 + 2 e 0 2 ) Δ L f 0 + o ( Δ L f 0 ) 2 Δ L e 0 Δ L 2 f 0 + o ( Δ L f 0 ) Δ L ( 1 + e 0 + e 0 2 ) 1 f 0 + ( e 0 + 2 e 0 2 + e 0 3 ) Δ L f 0 2 + o ( Δ L f 0 ) 1 f 0 1 ( 2 + e 0 ) Δ L f 0 + o ( Δ L f 0 ) ) ,
S 0 S 1 S 2 S 3 = ( 1 a 3 Δ L f 0 + o ( Δ L f 0 ) b 3 , 1 Δ L b 3 , 2 Δ L 2 f 0 + o ( Δ L f 0 ) Δ L c 3 , 1 1 f 0 + c 3 , 2 Δ L f 0 2 + o ( Δ L f 0 ) 1 f 0 1 d 3 Δ L f 0 + o ( Δ L f 0 ) ) ,
a 3 = e 0 + 2 e 0 2 + 3 e 0 3 ,
b 3 , 1 = 3 ,
b 3 , 2 = 2 e 0 + 2 e 0 2 ,
c 3 , 1 = 1 + e 0 + e 0 2 + e 0 3 ,
c 3 , 2 = e 0 + 2 e 0 2 + 4 e 0 3 + 2 e 0 4 + e 0 5 ,
d 3 = 3 + 2 e 0 + e 0 2 .
S m = ( 1 a m Δ L f 0 + o ( Δ L f 0 ) b m , 1 Δ L b m , 2 Δ L 2 f 0 + o ( Δ L f 0 ) Δ L c m , 1 1 f 0 + c m , 2 Δ L f 0 2 + o ( Δ L f 0 ) 1 f 0 1 d m Δ L f 0 + o ( Δ L f 0 ) ) ,
a m = e 0 + 2 e 0 2 + 3 e 0 3 + + m e 0 m ,
b m , 1 = m ,
c m , 1 = 1 + e 0 + e 0 2 + e 0 3 + + e 0 m .
S m + 1 = S m S m + 1 = ( 1 a m Δ L f 0 + o ( Δ L f 0 ) b m , 1 Δ L b m , 2 Δ L 2 f 0 + o ( Δ L f 0 ) Δ L c m , 1 1 f 0 + c m , 2 Δ L f 0 2 + o ( Δ L f 0 ) 1 f 0 1 d m Δ L f 0 + o ( Δ L f 0 ) ) ( 1 e 0 m + 1 Δ L f 0 Δ L e 0 m + 1 1 f 0 1 ) = ( 1 [ a m + ( b m , 1 + 1 ) e 0 m + 1 ] Δ L f 0 + o ( Δ L f 0 ) ( b m , 1 + 1 ) Δ L ( a m + b m , 2 ) Δ L 2 f 0 + o ( Δ L f 0 ) Δ L ( c m , 1 + e 0 m + 1 ) 1 f 0 + [ c m , 2 + ( c m , 1 + d m ) e 0 m + 1 ] Δ L f 0 2 + o ( Δ L f 0 ) 1 f 0 1 ( d m + c m , 1 ) Δ L f 0 + o ( Δ L f 0 ) ) .
a m + 1 = a m + ( b m , 1 + 1 ) e 0 m + 1 = a m + ( m + 1 ) e 0 m + 1 = a m + 1 ,
b m + 1 , 1 = b m , 1 + 1 = b m + 1 , 1 ,
c m + 1 , 1 = c m , 1 + e 0 m + 1 = c m + 1 , 1 .
b m + 1 , 2 = a m + b m , 2 ,
c m + 1 , 2 = c m , 2 + ( c m , 1 + d m ) e 0 m + 1 ,
d m + 1 = d m + c m , 1 .
b m + 1 , 2 b m , 2 = a m ,
c m + 1 , 2 c m , 2 = ( c m , 1 + d m ) e 0 m + 1 ,
d m + 1 d m = c m , 1 .
b 0 , 2 = c 0 , 2 = d 0 = 0.
b m , 2 = ( m 1 ) ( e 0 + e 0 m + 1 ) ( 1 e 0 ) 2 2 e 0 2 2 e 0 m + 1 ( 1 e 0 ) 3 ,
c m , 2 = e 0 ( m + 1 ) e 0 m + 1 ( 1 e 0 ) 2 + e 0 3 e 0 2 m + 3 ( 1 e 0 ) 3 ( 1 + e 0 ) ,
d m = m 1 e 0 e 0 e 0 m + 1 ( 1 e 0 ) 2 .
A = 1 [ e 0 e 0 n ( 1 e 0 ) 2 ( n 1 ) e 0 n 1 e 0 ] Δ L f 0 + o ( Δ L f 0 ) .
A = 1 b ( 1 T ) n 0 f lm [ ( e α L n T ) tan L n b 1 e α L n ( n 1 ) T tan L n b ] + o [ ( 1 e α L n ) tan L n b ] .
lim n A = 1 ( 1 ln T T 1 T ) L n 0 f lm .

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