Abstract

Lens ducts are simple optical devices that have found application in the coupling of pump radiation from extended two-dimensional semiconductor laser diode arrays into solid-state laser gain media. The operation of these devices relies on the combined effects of lensing at their curved input surface and channeling by total internal reflection off their canted planar sides, to contain and couple semiconductor diode laser light efficiently to the input face of a solid-state laser crystal or glass. The lens duct provides a robust method for amplifying the irradiance of laser diode array pump sources and has made possible a scalable diode end-pumping architecture that offers the opportunity to expand significantly the number of ions and transitions that can be practically engaged in diode-pumped solid-state laser systems. An analytic model that describes the transfer efficiency of lens ducts and aids in the optimization of their design is presented.

© 1996 Optical Society of America

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  1. G. L. Harnagel, M. Vakili, K. R. Anderson, D. P. Worland, J. G. Endriz, D. R. Scifres, “High-duty cycle, high-power two-dimensional laser diode arrays,” Electron. Lett. 29, 1008–1010 (1993).
    [CrossRef]
  2. R. Beach, W. Benett, B. Freitas, D. Mundinger, B. Comaskey, R. Solarz, M. A. Emanuel, “Modular microchannel cooled heatsinks for high average power laser diode arrays,” IEEE J. Quantum Electron. 28, 966–976 (1992).
    [CrossRef]
  3. B. Zhou, T. J. Kane, G. J. Dixon, B. L. Byer, “Efficient, frequency-stable laser-diode-pumped Nd:YAG laser,” Opt. Lett. 10, 62–64 (1985).
    [CrossRef] [PubMed]
  4. T. Y. Fan, R. L. Byer, “Diode laser-pumped solid state lasers,” IEEE J. Quantum Electron. 24, 895–912 (1988).
    [CrossRef]
  5. W. F. Krupke, L. L. Chase, “Ground-state depleted solid state lasers: principles, characteristics, and scaling,” Opt. Quantum Electron. 22, S1–S22 (1989).
    [CrossRef]
  6. T. Y. Fan, A. Sanchez, W. E. Defoe, “Scalable, end-pumped, diode-laser-pumped laser,” Opt. Lett. 14, 1057–1059 (1989).
    [CrossRef] [PubMed]
  7. T. S. Rose, J. S. Swenson, R. A. Fields, “High-efficiency longitudinal diode bar pumping of solid-state lasers,” in Advanced Solid-State Lasers, L. L. Chase, A. A. Pinto, eds., Vol. 13 of OSA Proceedings (Optical Society of America, Washington, D.C., 1992), pp. 186–189.
  8. S. C. Tidwell, J. F. Seamans, M. S. Bowers, A. K. Cousins, “Scaling CW diode-end-pumped Nd:YAG lasers to high average powers,” IEEE J. Quantum Electron. 28, 997–1009 (1992).
    [CrossRef]
  9. L. E. Holder, C. K. Kennedy, L. Long, G. Dube, “One joule per Q-switched pulse diode-pumped laser,” IEEE J. Quantum Electron. 28, 986–991 (1992).
    [CrossRef]
  10. B. Comaskey, G. F. Albrecht, S. Velsko, R. Beach, S. Sutton, B. Moran, C. Petty, “24-W average power in the green 100% solid-state laser system,” in Conference on Lasers and Electro-Optics, Vol. 11 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 248–250.
  11. P. Lacovara, P. Gleckman, R. Holman, R. Winston, “Nonimaging concentrators for diode-pumped slab lasers,” in Nonimaging Optics: Maximum Efficiency Light Transfer, R. L. Holman, R. Winston, eds., Proc. SPIE 1528, 135–141 (1991).
  12. R. Beach, J. Davin, S. Mitchell, W. Benett, B. Freitas, R. W. Solarz, P. Avizonis, “Passively Q-switched transverse-diode-pumped Nd3+:YLF laser oscillator,” Opt. Lett. 17, 124–126 (1992).
    [CrossRef] [PubMed]
  13. B. J. Comaskey, R. Beach, G. Albrecht, W. J. Benett, B. L. Freitas, C. Petty, D. VanLue, D. Mundinger, R. W. Solarz, “High power diode pumped slab laser,” IEEE J. Quantum Electron. 28, 992–996 (1992).
    [CrossRef]
  14. J. J. Kasinski, W. Hughes, D. DiBiase, P. Bourne, R. Burnham, “One joule output from a diode-array-pumped Nd:YAG laser with side-pumped rod geometry,” IEEE J. Quantum Electron. 28, 977–985 (1992).
    [CrossRef]
  15. R. Beach, P. Reichert, W. Benett, B. Freitas, S. Mitchell, S. Velsko, J. Davin, R. Solarz, “Scalable diode-end-pumping technology applied to a 100-mJ Q-switched Nd3+:YLF laser oscillator,” Opt. Lett. 18, 1326–1328 (1993).
    [CrossRef] [PubMed]
  16. M. D. Perry, S. A. Payne, T. Ditmire, R. Beach, G. J. Quarles, W. Ignatuk, R. Olsen, J. Weston, “Better materials trigger, Cr:LiSAF laser development,” Laser Focus World, September1993, pp. 82–92.
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  18. R. J. Beach, M. A. Emanuel, B. L. Freitas, J. A. Skidmore, N. W. Carlson, W. J. Benett, R. W. Solarz, “Applications of microlens-conditioned laser diode arrays,” in Micro-Optics/Micromechanics and Laser Scanning and Shaping, M. E. Motamedi, L. Beiser, eds., Proc. SPIE 2383, 283–297 (1995).
  19. R. J. Beach, W. J. Benett, “Lensing duct,” U.S. patent5,307,430 (26April1994).
  20. R. J. Beach, M. A. Emanuel, W. J. Benett, B. L. Freitas, D. Ciarlo, N. W. Carlson, S. B. Sutton, J. A. Skidmore, R. W. Solarz, “Improved performance of high average power semiconductor arrays for applications in diode pumped solid state lasers,” in Laser Diode Technology and Applications VI, P. C. Chen, L. A. Johnson, H. Temkin, eds., Proc. SPIE 2148, 13–29 (1994).
  21. G. Feugnet, C. Bussac, C. Larat, M. Schwarz, J. P. Pochelle, “High efficiency TEM00 laser longitudinally pumped by a high power array,” in Advanced Solid-State Lasers, Vol. 24 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1995) pp. 82–84.
  22. W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, New York, 1989), App. A.
  23. G. P. Agrawal, N. K. Dutta, Long-Wavelength Semiconductor Lasers (Von Nostrand Reinhold, New York, 1986), p. 60.
  24. S. A. Payne, L. K. Smith, R. J. Beach, B. C. Chai, J. T. Tassano, L. D. DeLoach, W. L. Kway, R. W. Solarz, W. F. Krupke, “Properties of Cr:LiSrAlF6 crystals for laser operation,” Appl. Opt. 33, 5526–5536 (1994).
    [CrossRef] [PubMed]

1994 (1)

1993 (3)

R. Beach, P. Reichert, W. Benett, B. Freitas, S. Mitchell, S. Velsko, J. Davin, R. Solarz, “Scalable diode-end-pumping technology applied to a 100-mJ Q-switched Nd3+:YLF laser oscillator,” Opt. Lett. 18, 1326–1328 (1993).
[CrossRef] [PubMed]

M. D. Perry, S. A. Payne, T. Ditmire, R. Beach, G. J. Quarles, W. Ignatuk, R. Olsen, J. Weston, “Better materials trigger, Cr:LiSAF laser development,” Laser Focus World, September1993, pp. 82–92.

G. L. Harnagel, M. Vakili, K. R. Anderson, D. P. Worland, J. G. Endriz, D. R. Scifres, “High-duty cycle, high-power two-dimensional laser diode arrays,” Electron. Lett. 29, 1008–1010 (1993).
[CrossRef]

1992 (6)

R. Beach, W. Benett, B. Freitas, D. Mundinger, B. Comaskey, R. Solarz, M. A. Emanuel, “Modular microchannel cooled heatsinks for high average power laser diode arrays,” IEEE J. Quantum Electron. 28, 966–976 (1992).
[CrossRef]

S. C. Tidwell, J. F. Seamans, M. S. Bowers, A. K. Cousins, “Scaling CW diode-end-pumped Nd:YAG lasers to high average powers,” IEEE J. Quantum Electron. 28, 997–1009 (1992).
[CrossRef]

L. E. Holder, C. K. Kennedy, L. Long, G. Dube, “One joule per Q-switched pulse diode-pumped laser,” IEEE J. Quantum Electron. 28, 986–991 (1992).
[CrossRef]

R. Beach, J. Davin, S. Mitchell, W. Benett, B. Freitas, R. W. Solarz, P. Avizonis, “Passively Q-switched transverse-diode-pumped Nd3+:YLF laser oscillator,” Opt. Lett. 17, 124–126 (1992).
[CrossRef] [PubMed]

B. J. Comaskey, R. Beach, G. Albrecht, W. J. Benett, B. L. Freitas, C. Petty, D. VanLue, D. Mundinger, R. W. Solarz, “High power diode pumped slab laser,” IEEE J. Quantum Electron. 28, 992–996 (1992).
[CrossRef]

J. J. Kasinski, W. Hughes, D. DiBiase, P. Bourne, R. Burnham, “One joule output from a diode-array-pumped Nd:YAG laser with side-pumped rod geometry,” IEEE J. Quantum Electron. 28, 977–985 (1992).
[CrossRef]

1991 (1)

1989 (2)

W. F. Krupke, L. L. Chase, “Ground-state depleted solid state lasers: principles, characteristics, and scaling,” Opt. Quantum Electron. 22, S1–S22 (1989).
[CrossRef]

T. Y. Fan, A. Sanchez, W. E. Defoe, “Scalable, end-pumped, diode-laser-pumped laser,” Opt. Lett. 14, 1057–1059 (1989).
[CrossRef] [PubMed]

1988 (1)

T. Y. Fan, R. L. Byer, “Diode laser-pumped solid state lasers,” IEEE J. Quantum Electron. 24, 895–912 (1988).
[CrossRef]

1985 (1)

Agrawal, G. P.

G. P. Agrawal, N. K. Dutta, Long-Wavelength Semiconductor Lasers (Von Nostrand Reinhold, New York, 1986), p. 60.

Albrecht, G.

B. J. Comaskey, R. Beach, G. Albrecht, W. J. Benett, B. L. Freitas, C. Petty, D. VanLue, D. Mundinger, R. W. Solarz, “High power diode pumped slab laser,” IEEE J. Quantum Electron. 28, 992–996 (1992).
[CrossRef]

Albrecht, G. F.

B. Comaskey, G. F. Albrecht, S. Velsko, R. Beach, S. Sutton, B. Moran, C. Petty, “24-W average power in the green 100% solid-state laser system,” in Conference on Lasers and Electro-Optics, Vol. 11 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 248–250.

Anderson, K. R.

G. L. Harnagel, M. Vakili, K. R. Anderson, D. P. Worland, J. G. Endriz, D. R. Scifres, “High-duty cycle, high-power two-dimensional laser diode arrays,” Electron. Lett. 29, 1008–1010 (1993).
[CrossRef]

Avizonis, P.

Baer, T. M.

Beach, R.

R. Beach, P. Reichert, W. Benett, B. Freitas, S. Mitchell, S. Velsko, J. Davin, R. Solarz, “Scalable diode-end-pumping technology applied to a 100-mJ Q-switched Nd3+:YLF laser oscillator,” Opt. Lett. 18, 1326–1328 (1993).
[CrossRef] [PubMed]

M. D. Perry, S. A. Payne, T. Ditmire, R. Beach, G. J. Quarles, W. Ignatuk, R. Olsen, J. Weston, “Better materials trigger, Cr:LiSAF laser development,” Laser Focus World, September1993, pp. 82–92.

R. Beach, J. Davin, S. Mitchell, W. Benett, B. Freitas, R. W. Solarz, P. Avizonis, “Passively Q-switched transverse-diode-pumped Nd3+:YLF laser oscillator,” Opt. Lett. 17, 124–126 (1992).
[CrossRef] [PubMed]

B. J. Comaskey, R. Beach, G. Albrecht, W. J. Benett, B. L. Freitas, C. Petty, D. VanLue, D. Mundinger, R. W. Solarz, “High power diode pumped slab laser,” IEEE J. Quantum Electron. 28, 992–996 (1992).
[CrossRef]

R. Beach, W. Benett, B. Freitas, D. Mundinger, B. Comaskey, R. Solarz, M. A. Emanuel, “Modular microchannel cooled heatsinks for high average power laser diode arrays,” IEEE J. Quantum Electron. 28, 966–976 (1992).
[CrossRef]

B. Comaskey, G. F. Albrecht, S. Velsko, R. Beach, S. Sutton, B. Moran, C. Petty, “24-W average power in the green 100% solid-state laser system,” in Conference on Lasers and Electro-Optics, Vol. 11 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 248–250.

Beach, R. J.

S. A. Payne, L. K. Smith, R. J. Beach, B. C. Chai, J. T. Tassano, L. D. DeLoach, W. L. Kway, R. W. Solarz, W. F. Krupke, “Properties of Cr:LiSrAlF6 crystals for laser operation,” Appl. Opt. 33, 5526–5536 (1994).
[CrossRef] [PubMed]

R. J. Beach, M. A. Emanuel, B. L. Freitas, J. A. Skidmore, N. W. Carlson, W. J. Benett, R. W. Solarz, “Applications of microlens-conditioned laser diode arrays,” in Micro-Optics/Micromechanics and Laser Scanning and Shaping, M. E. Motamedi, L. Beiser, eds., Proc. SPIE 2383, 283–297 (1995).

R. J. Beach, W. J. Benett, “Lensing duct,” U.S. patent5,307,430 (26April1994).

R. J. Beach, M. A. Emanuel, W. J. Benett, B. L. Freitas, D. Ciarlo, N. W. Carlson, S. B. Sutton, J. A. Skidmore, R. W. Solarz, “Improved performance of high average power semiconductor arrays for applications in diode pumped solid state lasers,” in Laser Diode Technology and Applications VI, P. C. Chen, L. A. Johnson, H. Temkin, eds., Proc. SPIE 2148, 13–29 (1994).

Benett, W.

Benett, W. J.

B. J. Comaskey, R. Beach, G. Albrecht, W. J. Benett, B. L. Freitas, C. Petty, D. VanLue, D. Mundinger, R. W. Solarz, “High power diode pumped slab laser,” IEEE J. Quantum Electron. 28, 992–996 (1992).
[CrossRef]

R. J. Beach, W. J. Benett, “Lensing duct,” U.S. patent5,307,430 (26April1994).

R. J. Beach, M. A. Emanuel, B. L. Freitas, J. A. Skidmore, N. W. Carlson, W. J. Benett, R. W. Solarz, “Applications of microlens-conditioned laser diode arrays,” in Micro-Optics/Micromechanics and Laser Scanning and Shaping, M. E. Motamedi, L. Beiser, eds., Proc. SPIE 2383, 283–297 (1995).

R. J. Beach, M. A. Emanuel, W. J. Benett, B. L. Freitas, D. Ciarlo, N. W. Carlson, S. B. Sutton, J. A. Skidmore, R. W. Solarz, “Improved performance of high average power semiconductor arrays for applications in diode pumped solid state lasers,” in Laser Diode Technology and Applications VI, P. C. Chen, L. A. Johnson, H. Temkin, eds., Proc. SPIE 2148, 13–29 (1994).

Bourne, P.

J. J. Kasinski, W. Hughes, D. DiBiase, P. Bourne, R. Burnham, “One joule output from a diode-array-pumped Nd:YAG laser with side-pumped rod geometry,” IEEE J. Quantum Electron. 28, 977–985 (1992).
[CrossRef]

Bowers, M. S.

S. C. Tidwell, J. F. Seamans, M. S. Bowers, A. K. Cousins, “Scaling CW diode-end-pumped Nd:YAG lasers to high average powers,” IEEE J. Quantum Electron. 28, 997–1009 (1992).
[CrossRef]

Burnham, R.

J. J. Kasinski, W. Hughes, D. DiBiase, P. Bourne, R. Burnham, “One joule output from a diode-array-pumped Nd:YAG laser with side-pumped rod geometry,” IEEE J. Quantum Electron. 28, 977–985 (1992).
[CrossRef]

Bussac, C.

G. Feugnet, C. Bussac, C. Larat, M. Schwarz, J. P. Pochelle, “High efficiency TEM00 laser longitudinally pumped by a high power array,” in Advanced Solid-State Lasers, Vol. 24 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1995) pp. 82–84.

Byer, B. L.

Byer, R. L.

T. Y. Fan, R. L. Byer, “Diode laser-pumped solid state lasers,” IEEE J. Quantum Electron. 24, 895–912 (1988).
[CrossRef]

Carlson, N. W.

R. J. Beach, M. A. Emanuel, W. J. Benett, B. L. Freitas, D. Ciarlo, N. W. Carlson, S. B. Sutton, J. A. Skidmore, R. W. Solarz, “Improved performance of high average power semiconductor arrays for applications in diode pumped solid state lasers,” in Laser Diode Technology and Applications VI, P. C. Chen, L. A. Johnson, H. Temkin, eds., Proc. SPIE 2148, 13–29 (1994).

R. J. Beach, M. A. Emanuel, B. L. Freitas, J. A. Skidmore, N. W. Carlson, W. J. Benett, R. W. Solarz, “Applications of microlens-conditioned laser diode arrays,” in Micro-Optics/Micromechanics and Laser Scanning and Shaping, M. E. Motamedi, L. Beiser, eds., Proc. SPIE 2383, 283–297 (1995).

Chai, B. C.

Chase, L. L.

W. F. Krupke, L. L. Chase, “Ground-state depleted solid state lasers: principles, characteristics, and scaling,” Opt. Quantum Electron. 22, S1–S22 (1989).
[CrossRef]

Ciarlo, D.

R. J. Beach, M. A. Emanuel, W. J. Benett, B. L. Freitas, D. Ciarlo, N. W. Carlson, S. B. Sutton, J. A. Skidmore, R. W. Solarz, “Improved performance of high average power semiconductor arrays for applications in diode pumped solid state lasers,” in Laser Diode Technology and Applications VI, P. C. Chen, L. A. Johnson, H. Temkin, eds., Proc. SPIE 2148, 13–29 (1994).

Comaskey, B.

R. Beach, W. Benett, B. Freitas, D. Mundinger, B. Comaskey, R. Solarz, M. A. Emanuel, “Modular microchannel cooled heatsinks for high average power laser diode arrays,” IEEE J. Quantum Electron. 28, 966–976 (1992).
[CrossRef]

B. Comaskey, G. F. Albrecht, S. Velsko, R. Beach, S. Sutton, B. Moran, C. Petty, “24-W average power in the green 100% solid-state laser system,” in Conference on Lasers and Electro-Optics, Vol. 11 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 248–250.

Comaskey, B. J.

B. J. Comaskey, R. Beach, G. Albrecht, W. J. Benett, B. L. Freitas, C. Petty, D. VanLue, D. Mundinger, R. W. Solarz, “High power diode pumped slab laser,” IEEE J. Quantum Electron. 28, 992–996 (1992).
[CrossRef]

Cousins, A. K.

S. C. Tidwell, J. F. Seamans, M. S. Bowers, A. K. Cousins, “Scaling CW diode-end-pumped Nd:YAG lasers to high average powers,” IEEE J. Quantum Electron. 28, 997–1009 (1992).
[CrossRef]

Davin, J.

Defoe, W. E.

DeLoach, L. D.

DiBiase, D.

J. J. Kasinski, W. Hughes, D. DiBiase, P. Bourne, R. Burnham, “One joule output from a diode-array-pumped Nd:YAG laser with side-pumped rod geometry,” IEEE J. Quantum Electron. 28, 977–985 (1992).
[CrossRef]

Ditmire, T.

M. D. Perry, S. A. Payne, T. Ditmire, R. Beach, G. J. Quarles, W. Ignatuk, R. Olsen, J. Weston, “Better materials trigger, Cr:LiSAF laser development,” Laser Focus World, September1993, pp. 82–92.

Dixon, G. J.

Dube, G.

L. E. Holder, C. K. Kennedy, L. Long, G. Dube, “One joule per Q-switched pulse diode-pumped laser,” IEEE J. Quantum Electron. 28, 986–991 (1992).
[CrossRef]

Dutta, N. K.

G. P. Agrawal, N. K. Dutta, Long-Wavelength Semiconductor Lasers (Von Nostrand Reinhold, New York, 1986), p. 60.

Emanuel, M. A.

R. Beach, W. Benett, B. Freitas, D. Mundinger, B. Comaskey, R. Solarz, M. A. Emanuel, “Modular microchannel cooled heatsinks for high average power laser diode arrays,” IEEE J. Quantum Electron. 28, 966–976 (1992).
[CrossRef]

R. J. Beach, M. A. Emanuel, B. L. Freitas, J. A. Skidmore, N. W. Carlson, W. J. Benett, R. W. Solarz, “Applications of microlens-conditioned laser diode arrays,” in Micro-Optics/Micromechanics and Laser Scanning and Shaping, M. E. Motamedi, L. Beiser, eds., Proc. SPIE 2383, 283–297 (1995).

R. J. Beach, M. A. Emanuel, W. J. Benett, B. L. Freitas, D. Ciarlo, N. W. Carlson, S. B. Sutton, J. A. Skidmore, R. W. Solarz, “Improved performance of high average power semiconductor arrays for applications in diode pumped solid state lasers,” in Laser Diode Technology and Applications VI, P. C. Chen, L. A. Johnson, H. Temkin, eds., Proc. SPIE 2148, 13–29 (1994).

Endriz, J. G.

G. L. Harnagel, M. Vakili, K. R. Anderson, D. P. Worland, J. G. Endriz, D. R. Scifres, “High-duty cycle, high-power two-dimensional laser diode arrays,” Electron. Lett. 29, 1008–1010 (1993).
[CrossRef]

Fan, T. Y.

T. Y. Fan, A. Sanchez, W. E. Defoe, “Scalable, end-pumped, diode-laser-pumped laser,” Opt. Lett. 14, 1057–1059 (1989).
[CrossRef] [PubMed]

T. Y. Fan, R. L. Byer, “Diode laser-pumped solid state lasers,” IEEE J. Quantum Electron. 24, 895–912 (1988).
[CrossRef]

Feugnet, G.

G. Feugnet, C. Bussac, C. Larat, M. Schwarz, J. P. Pochelle, “High efficiency TEM00 laser longitudinally pumped by a high power array,” in Advanced Solid-State Lasers, Vol. 24 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1995) pp. 82–84.

Fields, R. A.

T. S. Rose, J. S. Swenson, R. A. Fields, “High-efficiency longitudinal diode bar pumping of solid-state lasers,” in Advanced Solid-State Lasers, L. L. Chase, A. A. Pinto, eds., Vol. 13 of OSA Proceedings (Optical Society of America, Washington, D.C., 1992), pp. 186–189.

Freitas, B.

Freitas, B. L.

B. J. Comaskey, R. Beach, G. Albrecht, W. J. Benett, B. L. Freitas, C. Petty, D. VanLue, D. Mundinger, R. W. Solarz, “High power diode pumped slab laser,” IEEE J. Quantum Electron. 28, 992–996 (1992).
[CrossRef]

R. J. Beach, M. A. Emanuel, W. J. Benett, B. L. Freitas, D. Ciarlo, N. W. Carlson, S. B. Sutton, J. A. Skidmore, R. W. Solarz, “Improved performance of high average power semiconductor arrays for applications in diode pumped solid state lasers,” in Laser Diode Technology and Applications VI, P. C. Chen, L. A. Johnson, H. Temkin, eds., Proc. SPIE 2148, 13–29 (1994).

R. J. Beach, M. A. Emanuel, B. L. Freitas, J. A. Skidmore, N. W. Carlson, W. J. Benett, R. W. Solarz, “Applications of microlens-conditioned laser diode arrays,” in Micro-Optics/Micromechanics and Laser Scanning and Shaping, M. E. Motamedi, L. Beiser, eds., Proc. SPIE 2383, 283–297 (1995).

Gleckman, P.

P. Lacovara, P. Gleckman, R. Holman, R. Winston, “Nonimaging concentrators for diode-pumped slab lasers,” in Nonimaging Optics: Maximum Efficiency Light Transfer, R. L. Holman, R. Winston, eds., Proc. SPIE 1528, 135–141 (1991).

Harnagel, G. L.

G. L. Harnagel, M. Vakili, K. R. Anderson, D. P. Worland, J. G. Endriz, D. R. Scifres, “High-duty cycle, high-power two-dimensional laser diode arrays,” Electron. Lett. 29, 1008–1010 (1993).
[CrossRef]

Holder, L. E.

L. E. Holder, C. K. Kennedy, L. Long, G. Dube, “One joule per Q-switched pulse diode-pumped laser,” IEEE J. Quantum Electron. 28, 986–991 (1992).
[CrossRef]

Holman, R.

P. Lacovara, P. Gleckman, R. Holman, R. Winston, “Nonimaging concentrators for diode-pumped slab lasers,” in Nonimaging Optics: Maximum Efficiency Light Transfer, R. L. Holman, R. Winston, eds., Proc. SPIE 1528, 135–141 (1991).

Hughes, W.

J. J. Kasinski, W. Hughes, D. DiBiase, P. Bourne, R. Burnham, “One joule output from a diode-array-pumped Nd:YAG laser with side-pumped rod geometry,” IEEE J. Quantum Electron. 28, 977–985 (1992).
[CrossRef]

Ignatuk, W.

M. D. Perry, S. A. Payne, T. Ditmire, R. Beach, G. J. Quarles, W. Ignatuk, R. Olsen, J. Weston, “Better materials trigger, Cr:LiSAF laser development,” Laser Focus World, September1993, pp. 82–92.

Kane, T. J.

Kasinski, J. J.

J. J. Kasinski, W. Hughes, D. DiBiase, P. Bourne, R. Burnham, “One joule output from a diode-array-pumped Nd:YAG laser with side-pumped rod geometry,” IEEE J. Quantum Electron. 28, 977–985 (1992).
[CrossRef]

Kennedy, C. K.

L. E. Holder, C. K. Kennedy, L. Long, G. Dube, “One joule per Q-switched pulse diode-pumped laser,” IEEE J. Quantum Electron. 28, 986–991 (1992).
[CrossRef]

Krupke, W. F.

Kway, W. L.

Lacovara, P.

P. Lacovara, P. Gleckman, R. Holman, R. Winston, “Nonimaging concentrators for diode-pumped slab lasers,” in Nonimaging Optics: Maximum Efficiency Light Transfer, R. L. Holman, R. Winston, eds., Proc. SPIE 1528, 135–141 (1991).

Larat, C.

G. Feugnet, C. Bussac, C. Larat, M. Schwarz, J. P. Pochelle, “High efficiency TEM00 laser longitudinally pumped by a high power array,” in Advanced Solid-State Lasers, Vol. 24 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1995) pp. 82–84.

Long, L.

L. E. Holder, C. K. Kennedy, L. Long, G. Dube, “One joule per Q-switched pulse diode-pumped laser,” IEEE J. Quantum Electron. 28, 986–991 (1992).
[CrossRef]

Mitchell, S.

Moran, B.

B. Comaskey, G. F. Albrecht, S. Velsko, R. Beach, S. Sutton, B. Moran, C. Petty, “24-W average power in the green 100% solid-state laser system,” in Conference on Lasers and Electro-Optics, Vol. 11 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 248–250.

Mundinger, D.

B. J. Comaskey, R. Beach, G. Albrecht, W. J. Benett, B. L. Freitas, C. Petty, D. VanLue, D. Mundinger, R. W. Solarz, “High power diode pumped slab laser,” IEEE J. Quantum Electron. 28, 992–996 (1992).
[CrossRef]

R. Beach, W. Benett, B. Freitas, D. Mundinger, B. Comaskey, R. Solarz, M. A. Emanuel, “Modular microchannel cooled heatsinks for high average power laser diode arrays,” IEEE J. Quantum Electron. 28, 966–976 (1992).
[CrossRef]

Olsen, R.

M. D. Perry, S. A. Payne, T. Ditmire, R. Beach, G. J. Quarles, W. Ignatuk, R. Olsen, J. Weston, “Better materials trigger, Cr:LiSAF laser development,” Laser Focus World, September1993, pp. 82–92.

Payne, S. A.

S. A. Payne, L. K. Smith, R. J. Beach, B. C. Chai, J. T. Tassano, L. D. DeLoach, W. L. Kway, R. W. Solarz, W. F. Krupke, “Properties of Cr:LiSrAlF6 crystals for laser operation,” Appl. Opt. 33, 5526–5536 (1994).
[CrossRef] [PubMed]

M. D. Perry, S. A. Payne, T. Ditmire, R. Beach, G. J. Quarles, W. Ignatuk, R. Olsen, J. Weston, “Better materials trigger, Cr:LiSAF laser development,” Laser Focus World, September1993, pp. 82–92.

Perry, M. D.

M. D. Perry, S. A. Payne, T. Ditmire, R. Beach, G. J. Quarles, W. Ignatuk, R. Olsen, J. Weston, “Better materials trigger, Cr:LiSAF laser development,” Laser Focus World, September1993, pp. 82–92.

Petty, C.

B. J. Comaskey, R. Beach, G. Albrecht, W. J. Benett, B. L. Freitas, C. Petty, D. VanLue, D. Mundinger, R. W. Solarz, “High power diode pumped slab laser,” IEEE J. Quantum Electron. 28, 992–996 (1992).
[CrossRef]

B. Comaskey, G. F. Albrecht, S. Velsko, R. Beach, S. Sutton, B. Moran, C. Petty, “24-W average power in the green 100% solid-state laser system,” in Conference on Lasers and Electro-Optics, Vol. 11 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 248–250.

Pochelle, J. P.

G. Feugnet, C. Bussac, C. Larat, M. Schwarz, J. P. Pochelle, “High efficiency TEM00 laser longitudinally pumped by a high power array,” in Advanced Solid-State Lasers, Vol. 24 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1995) pp. 82–84.

Quarles, G. J.

M. D. Perry, S. A. Payne, T. Ditmire, R. Beach, G. J. Quarles, W. Ignatuk, R. Olsen, J. Weston, “Better materials trigger, Cr:LiSAF laser development,” Laser Focus World, September1993, pp. 82–92.

Reichert, P.

Rose, T. S.

T. S. Rose, J. S. Swenson, R. A. Fields, “High-efficiency longitudinal diode bar pumping of solid-state lasers,” in Advanced Solid-State Lasers, L. L. Chase, A. A. Pinto, eds., Vol. 13 of OSA Proceedings (Optical Society of America, Washington, D.C., 1992), pp. 186–189.

Sanchez, A.

Schwarz, M.

G. Feugnet, C. Bussac, C. Larat, M. Schwarz, J. P. Pochelle, “High efficiency TEM00 laser longitudinally pumped by a high power array,” in Advanced Solid-State Lasers, Vol. 24 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1995) pp. 82–84.

Scifres, D. R.

G. L. Harnagel, M. Vakili, K. R. Anderson, D. P. Worland, J. G. Endriz, D. R. Scifres, “High-duty cycle, high-power two-dimensional laser diode arrays,” Electron. Lett. 29, 1008–1010 (1993).
[CrossRef]

Seamans, J. F.

S. C. Tidwell, J. F. Seamans, M. S. Bowers, A. K. Cousins, “Scaling CW diode-end-pumped Nd:YAG lasers to high average powers,” IEEE J. Quantum Electron. 28, 997–1009 (1992).
[CrossRef]

Skidmore, J. A.

R. J. Beach, M. A. Emanuel, W. J. Benett, B. L. Freitas, D. Ciarlo, N. W. Carlson, S. B. Sutton, J. A. Skidmore, R. W. Solarz, “Improved performance of high average power semiconductor arrays for applications in diode pumped solid state lasers,” in Laser Diode Technology and Applications VI, P. C. Chen, L. A. Johnson, H. Temkin, eds., Proc. SPIE 2148, 13–29 (1994).

R. J. Beach, M. A. Emanuel, B. L. Freitas, J. A. Skidmore, N. W. Carlson, W. J. Benett, R. W. Solarz, “Applications of microlens-conditioned laser diode arrays,” in Micro-Optics/Micromechanics and Laser Scanning and Shaping, M. E. Motamedi, L. Beiser, eds., Proc. SPIE 2383, 283–297 (1995).

Smith, L. K.

Snyder, J. J.

Solarz, R.

R. Beach, P. Reichert, W. Benett, B. Freitas, S. Mitchell, S. Velsko, J. Davin, R. Solarz, “Scalable diode-end-pumping technology applied to a 100-mJ Q-switched Nd3+:YLF laser oscillator,” Opt. Lett. 18, 1326–1328 (1993).
[CrossRef] [PubMed]

R. Beach, W. Benett, B. Freitas, D. Mundinger, B. Comaskey, R. Solarz, M. A. Emanuel, “Modular microchannel cooled heatsinks for high average power laser diode arrays,” IEEE J. Quantum Electron. 28, 966–976 (1992).
[CrossRef]

Solarz, R. W.

S. A. Payne, L. K. Smith, R. J. Beach, B. C. Chai, J. T. Tassano, L. D. DeLoach, W. L. Kway, R. W. Solarz, W. F. Krupke, “Properties of Cr:LiSrAlF6 crystals for laser operation,” Appl. Opt. 33, 5526–5536 (1994).
[CrossRef] [PubMed]

R. Beach, J. Davin, S. Mitchell, W. Benett, B. Freitas, R. W. Solarz, P. Avizonis, “Passively Q-switched transverse-diode-pumped Nd3+:YLF laser oscillator,” Opt. Lett. 17, 124–126 (1992).
[CrossRef] [PubMed]

B. J. Comaskey, R. Beach, G. Albrecht, W. J. Benett, B. L. Freitas, C. Petty, D. VanLue, D. Mundinger, R. W. Solarz, “High power diode pumped slab laser,” IEEE J. Quantum Electron. 28, 992–996 (1992).
[CrossRef]

R. J. Beach, M. A. Emanuel, W. J. Benett, B. L. Freitas, D. Ciarlo, N. W. Carlson, S. B. Sutton, J. A. Skidmore, R. W. Solarz, “Improved performance of high average power semiconductor arrays for applications in diode pumped solid state lasers,” in Laser Diode Technology and Applications VI, P. C. Chen, L. A. Johnson, H. Temkin, eds., Proc. SPIE 2148, 13–29 (1994).

R. J. Beach, M. A. Emanuel, B. L. Freitas, J. A. Skidmore, N. W. Carlson, W. J. Benett, R. W. Solarz, “Applications of microlens-conditioned laser diode arrays,” in Micro-Optics/Micromechanics and Laser Scanning and Shaping, M. E. Motamedi, L. Beiser, eds., Proc. SPIE 2383, 283–297 (1995).

Sutton, S.

B. Comaskey, G. F. Albrecht, S. Velsko, R. Beach, S. Sutton, B. Moran, C. Petty, “24-W average power in the green 100% solid-state laser system,” in Conference on Lasers and Electro-Optics, Vol. 11 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 248–250.

Sutton, S. B.

R. J. Beach, M. A. Emanuel, W. J. Benett, B. L. Freitas, D. Ciarlo, N. W. Carlson, S. B. Sutton, J. A. Skidmore, R. W. Solarz, “Improved performance of high average power semiconductor arrays for applications in diode pumped solid state lasers,” in Laser Diode Technology and Applications VI, P. C. Chen, L. A. Johnson, H. Temkin, eds., Proc. SPIE 2148, 13–29 (1994).

Swenson, J. S.

T. S. Rose, J. S. Swenson, R. A. Fields, “High-efficiency longitudinal diode bar pumping of solid-state lasers,” in Advanced Solid-State Lasers, L. L. Chase, A. A. Pinto, eds., Vol. 13 of OSA Proceedings (Optical Society of America, Washington, D.C., 1992), pp. 186–189.

Tassano, J. T.

Tidwell, S. C.

S. C. Tidwell, J. F. Seamans, M. S. Bowers, A. K. Cousins, “Scaling CW diode-end-pumped Nd:YAG lasers to high average powers,” IEEE J. Quantum Electron. 28, 997–1009 (1992).
[CrossRef]

Vakili, M.

G. L. Harnagel, M. Vakili, K. R. Anderson, D. P. Worland, J. G. Endriz, D. R. Scifres, “High-duty cycle, high-power two-dimensional laser diode arrays,” Electron. Lett. 29, 1008–1010 (1993).
[CrossRef]

VanLue, D.

B. J. Comaskey, R. Beach, G. Albrecht, W. J. Benett, B. L. Freitas, C. Petty, D. VanLue, D. Mundinger, R. W. Solarz, “High power diode pumped slab laser,” IEEE J. Quantum Electron. 28, 992–996 (1992).
[CrossRef]

Velsko, S.

R. Beach, P. Reichert, W. Benett, B. Freitas, S. Mitchell, S. Velsko, J. Davin, R. Solarz, “Scalable diode-end-pumping technology applied to a 100-mJ Q-switched Nd3+:YLF laser oscillator,” Opt. Lett. 18, 1326–1328 (1993).
[CrossRef] [PubMed]

B. Comaskey, G. F. Albrecht, S. Velsko, R. Beach, S. Sutton, B. Moran, C. Petty, “24-W average power in the green 100% solid-state laser system,” in Conference on Lasers and Electro-Optics, Vol. 11 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 248–250.

Welford, W. T.

W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, New York, 1989), App. A.

Weston, J.

M. D. Perry, S. A. Payne, T. Ditmire, R. Beach, G. J. Quarles, W. Ignatuk, R. Olsen, J. Weston, “Better materials trigger, Cr:LiSAF laser development,” Laser Focus World, September1993, pp. 82–92.

Winston, R.

W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, New York, 1989), App. A.

P. Lacovara, P. Gleckman, R. Holman, R. Winston, “Nonimaging concentrators for diode-pumped slab lasers,” in Nonimaging Optics: Maximum Efficiency Light Transfer, R. L. Holman, R. Winston, eds., Proc. SPIE 1528, 135–141 (1991).

Worland, D. P.

G. L. Harnagel, M. Vakili, K. R. Anderson, D. P. Worland, J. G. Endriz, D. R. Scifres, “High-duty cycle, high-power two-dimensional laser diode arrays,” Electron. Lett. 29, 1008–1010 (1993).
[CrossRef]

Zhou, B.

Appl. Opt. (2)

Electron. Lett. (1)

G. L. Harnagel, M. Vakili, K. R. Anderson, D. P. Worland, J. G. Endriz, D. R. Scifres, “High-duty cycle, high-power two-dimensional laser diode arrays,” Electron. Lett. 29, 1008–1010 (1993).
[CrossRef]

IEEE J. Quantum Electron. (6)

R. Beach, W. Benett, B. Freitas, D. Mundinger, B. Comaskey, R. Solarz, M. A. Emanuel, “Modular microchannel cooled heatsinks for high average power laser diode arrays,” IEEE J. Quantum Electron. 28, 966–976 (1992).
[CrossRef]

T. Y. Fan, R. L. Byer, “Diode laser-pumped solid state lasers,” IEEE J. Quantum Electron. 24, 895–912 (1988).
[CrossRef]

S. C. Tidwell, J. F. Seamans, M. S. Bowers, A. K. Cousins, “Scaling CW diode-end-pumped Nd:YAG lasers to high average powers,” IEEE J. Quantum Electron. 28, 997–1009 (1992).
[CrossRef]

L. E. Holder, C. K. Kennedy, L. Long, G. Dube, “One joule per Q-switched pulse diode-pumped laser,” IEEE J. Quantum Electron. 28, 986–991 (1992).
[CrossRef]

B. J. Comaskey, R. Beach, G. Albrecht, W. J. Benett, B. L. Freitas, C. Petty, D. VanLue, D. Mundinger, R. W. Solarz, “High power diode pumped slab laser,” IEEE J. Quantum Electron. 28, 992–996 (1992).
[CrossRef]

J. J. Kasinski, W. Hughes, D. DiBiase, P. Bourne, R. Burnham, “One joule output from a diode-array-pumped Nd:YAG laser with side-pumped rod geometry,” IEEE J. Quantum Electron. 28, 977–985 (1992).
[CrossRef]

Laser Focus World (1)

M. D. Perry, S. A. Payne, T. Ditmire, R. Beach, G. J. Quarles, W. Ignatuk, R. Olsen, J. Weston, “Better materials trigger, Cr:LiSAF laser development,” Laser Focus World, September1993, pp. 82–92.

Opt. Lett. (4)

Opt. Quantum Electron. (1)

W. F. Krupke, L. L. Chase, “Ground-state depleted solid state lasers: principles, characteristics, and scaling,” Opt. Quantum Electron. 22, S1–S22 (1989).
[CrossRef]

Other (9)

B. Comaskey, G. F. Albrecht, S. Velsko, R. Beach, S. Sutton, B. Moran, C. Petty, “24-W average power in the green 100% solid-state laser system,” in Conference on Lasers and Electro-Optics, Vol. 11 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 248–250.

P. Lacovara, P. Gleckman, R. Holman, R. Winston, “Nonimaging concentrators for diode-pumped slab lasers,” in Nonimaging Optics: Maximum Efficiency Light Transfer, R. L. Holman, R. Winston, eds., Proc. SPIE 1528, 135–141 (1991).

T. S. Rose, J. S. Swenson, R. A. Fields, “High-efficiency longitudinal diode bar pumping of solid-state lasers,” in Advanced Solid-State Lasers, L. L. Chase, A. A. Pinto, eds., Vol. 13 of OSA Proceedings (Optical Society of America, Washington, D.C., 1992), pp. 186–189.

R. J. Beach, M. A. Emanuel, B. L. Freitas, J. A. Skidmore, N. W. Carlson, W. J. Benett, R. W. Solarz, “Applications of microlens-conditioned laser diode arrays,” in Micro-Optics/Micromechanics and Laser Scanning and Shaping, M. E. Motamedi, L. Beiser, eds., Proc. SPIE 2383, 283–297 (1995).

R. J. Beach, W. J. Benett, “Lensing duct,” U.S. patent5,307,430 (26April1994).

R. J. Beach, M. A. Emanuel, W. J. Benett, B. L. Freitas, D. Ciarlo, N. W. Carlson, S. B. Sutton, J. A. Skidmore, R. W. Solarz, “Improved performance of high average power semiconductor arrays for applications in diode pumped solid state lasers,” in Laser Diode Technology and Applications VI, P. C. Chen, L. A. Johnson, H. Temkin, eds., Proc. SPIE 2148, 13–29 (1994).

G. Feugnet, C. Bussac, C. Larat, M. Schwarz, J. P. Pochelle, “High efficiency TEM00 laser longitudinally pumped by a high power array,” in Advanced Solid-State Lasers, Vol. 24 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1995) pp. 82–84.

W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, New York, 1989), App. A.

G. P. Agrawal, N. K. Dutta, Long-Wavelength Semiconductor Lasers (Von Nostrand Reinhold, New York, 1986), p. 60.

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Figures (9)

Fig. 1
Fig. 1

Schematic of a lens duct delivering the radiation from an extended microlens conditioned two-dimensional laser diode array into a laser rod. There is a small air gap between the output end of the lens duct and the laser rod input face to accommodate a dichroic coating on the laser rod, which transmits the pump radiation but reflects the laser radiation. The inset shows the output end of the lens duct, which is square in cross section and sized so as to just inscribe the circular end of the laser rod.

Fig. 2
Fig. 2

Schematic diagram of the placement of a cylindrical microlens on an LLNL-developed silicon microchannel-cooled laser diode array module.

Fig. 3
Fig. 3

Illustration of the focusing of a normally incident plane wave at a distance of rn/(n − 1) into a material having an input radius of curvature r and an index of refraction n.

Fig. 4
Fig. 4

Two-dimensional view of a lens duct, illustrating the method-of-images construction used in calculating the transfer efficiency of radiation from a diode array into a laser rod. Shown is a bundle of rays that bounces by total internal reflection off one of the lens duct’s canted planar sides, as well as the related direct bundle of rays generated by the use of the method of images.

Fig. 5
Fig. 5

Three-dimensional lens duct and its method-of-images construction, which in this case leads to a torus surface.

Fig. 6
Fig. 6

Plot of transfer efficiency as a function of lens duct length, l d , from a two-dimensional diode array into a 1/4-in. (0.635-cm) YAG rod as shown in Fig. 1. The values of the various diode array, lens duct, and laser rod parameters used in the calculation and as called out in Fig. 1 are h a = 12 cm, w a = 1.5 cm, θ s = 160 mRad, θ f = 20 mRad, r d = 12 cm, h d = 12.2 cm, w d = 1.7 cm, s d = 0.44 cm. The refractive index of the lens duct is assumed to be 1.5, the refractive index of the rod is 1.82, and all surfaces are assumed to be uncoated.

Fig. 7
Fig. 7

For the uncoated lens duct considered in Fig. 6 at its optimized length, l d , of 33 cm, this is a plot of transfer efficiency into an uncoated YAG rod as a function of the length of the side of the square output end of the duct, s d , for three different values of the fast axis conditioned divergence.

Fig. 8
Fig. 8

Same data plotted in Fig. 7, but in a different format. Here the transfer efficiency through the uncoated lens duct and into the uncoated YAG rod is plotted against irradiance magnification for the three values of θ f considered.

Fig. 9
Fig. 9

Lens delivery system constructed from f/2 cylindrical lenses that performs the same function as the lens duct described in the text.

Equations (37)

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l = r n n - 1 .
l d = ( 0.92 ) r d n d n d - 1 ,
D ( z , θ z ) = { 1 h a π θ 0 exp [ - ( θ z θ 0 ) 2 ] for z h a 2 0 otherwise .
d T = { [ 1 - ( n d - 1 n d + 1 ) 2 ] D [ z d , n d θ z + ( n d - 1 ) z d r d ] [ d s · ( - e r ) n d ( y c - y d , z c - z d ) ] T duct air rod if d s · ( - e r ) > 0 0 otherwise ,
θ z = sin - 1 [ z c - z d ( y c - y d , z c - z d ) ]
e r = ( y c - y d , z c - z d ) ( y c - y d , z c - z d )
[ z d θ inside duct ] = [ 1 0 - n d - 1 n d 1 r d 1 n d ] ( z d θ outside duct ) .
T duct air = 1 - [ sin ( θ d - θ a ) sin ( θ d + θ a ) ] 2 ,
T duct air rod = ( 1 - [ sin ( θ d - θ a ) sin ( θ d + θ a ) ] 2 ) ( 1 - [ sin ( θ a - θ r ) sin ( θ a + θ r ) ] 2 ) ,
η = h A 2 < z D < h A 2 d z D semicircle d T .
x t = ( r o + r i cos θ ) cos ϕ , y t = ( r o + r i cos θ ) sin ϕ , z t = r i sin θ .
- π 2 θ π 2 - π ϕ 0.
r i = s d l d h d - s d ,
r o = s d I d w d - s d - r i .
d Σ = | i j k x t ϕ y t ϕ z t ϕ x t θ y t θ z t θ | d θ d ϕ ,
D ( x , z , θ x , θ z ) = 1 w a h a π θ s θ f exp { - [ ( θ x θ s ) 2 + ( θ z θ f ) 2 ] } .
d T = { [ 1 - ( n d - 1 n d + 1 ) 2 ] D [ x d , z d , n d θ x + ( n d - 1 ) x d r d , n d θ z + ( n d - 1 ) z d r d ) [ d Σ · ( - e r ) n d 2 ( x c - x d , y c - y d , z c - z d ) 2 ] T duct air rod if d Σ · ( - e r ) > 0 0 otherwise ,
θ x = sin - 1 [ x t - x d ( x t - x d , y t - y d , z t - z d ) ]
θ z = sin - 1 [ z t - z d ( x t - x d , y t - y d , z t - z d ) ]
e r = ( x t - x d , y t - y d , z t - z d ) ( x t - x d , y t - y d , z t - z d )
e t = d Σ d Σ ,
θ d = cos - 1 ( - e r · e t ) .
θ a = sin - 1 [ n d sin ( θ d ) ] ,
θ r = sin - 1 [ sin ( θ a ) n r ] ,
e E = z × e r z × e r .
e s - pol = e t × e r e t × e r ,
e p - pol = e r × e s - pol ,
T duct air rod = ( e E · e s - pol ) 2 { 1 - [ sin ( θ a - θ d ) sin ( θ a + θ d ) ] 2 } × { 1 - [ sin ( θ r - θ a ) sin ( θ r + θ a ) ] 2 } + ( e E · e p - pol ) 2 { 1 - [ tan ( θ a - θ d ) tan ( θ a + θ d ) ] 2 } × { 1 - [ tan ( θ r - θ a ) tan ( θ r + θ a ) ] 2 } .
η net = - w d 2 x d w d 2 - h d 2 z d h d / 2 d x d d z d torus surface : - π 2 θ π 2 - π ϕ 0 d T .
r d = h a .
l d = ( 0.92 ) r d n d n d - 1 = ( 0.92 ) 12 cm 1.5 1.5 - 1 = 33.1 cm .
η net = η AR coated system η Fresnel .
M I = A diode aperture A lens duct output end η net ,
θ x x = F s θ x , θ z z = F f θ z .
D focal plane ( x , z ) = - w d 2 x d w d 2 - h d 2 z d h d 2 d x d d z d diode emission angles d θ x d θ z × D ( x d , z d , θ x , θ z ) δ ( x - F s θ x ) δ ( z - F f θ z ) ,
D focal plane ( x , z ) = 1 π F s θ s F f θ f exp { - [ ( x F s θ s ) 2 + ( z F f θ f ) 2 ] } .
η lens system = z = - r rod r rod d z x = - ( r rod 2 - z 2 ) 1 / 2 ( r rod 2 - z 2 ) 1 / 2 d x D focal plane ( x , z ) .

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