Abstract

We provide an approximate but simple analytical solution to the radial distribution of deposited energy in a diode-array-pumped laser rod, subject to some assumptions that are naturally fulfilled for most applications of practical interest. The solution is useful to survey quickly irradiance distributions for a wide variety of pumping geometries and to find the radially most uniform energy deposition. We find that the radial deposition profile, as well as the pump light absorption efficiency, is largely controlled by just two dimensionless parameters: the number of absorption depths and the ratio of the width of the unabsorbed pump beam at the rod center divided by the rod radius. A side-by-side comparison with a numerical model is given. Results describing the best achievable trade-off between absorption efficiency and pumping uniformity are presented in the form of a recipe that can be followed without studying our research in detail. Finally, the model equations are applied to a practical side-pumped geometry.

© 1996 Optical Society of America

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References

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  1. W. Koechner, Solid State Laser Engineering, 3rd ed. (Springer-Verlag, New York, 1992), particularly Chap. 3.6.5, “Laser diode pumped systems,” and references therein.
  2. N. MacKinnon, B. D. Sinclair, W. Sibbett, S. N. Jenny, I. T. Jenks, “Ultracompact, laser diode array pumped, Nd:YVO4/KDP frequency doubled composite-material micro-chip laser,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 156.
  3. B. J. Comaskey, G. F. Albrecht, R. J. Beach, S. P. Velsko, S. B. Sutton, S. C. Mitchell, C. S. Petty, K. S. Jancaitis, W. J. Bennett, B. L. Freitas, R. W. Solarz, “A one kilowatt average power diode pumped Nd:YAG folded zig-zag slab laser,” in Diode Pumping of Average-Power Solid State Laser, G. F. Albrecht, R. J. Beach, S. P. Velsko, eds., Proc. SPIE1865, 9–11 (1993).
  4. J. J. Kasinski, W. Hughes, D. DiBiase, P. Bournes, 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]
  5. F. Salin, J. Squire, “Gain guiding in solid state laser,” Opt. Lett. 17, 1352–1354 (1992).
    [CrossRef] [PubMed]
  6. G. Cerullo, S. DeSilversti, V. Magni, “High efficiency, 40 W CW Nd:YLF laser with large TEM00 mode,” Opt. Commun. 93, 77–81 (1992).
    [CrossRef]
  7. D. DeSilvestri, P. Laporta, V. Magni, O. Svelto, “Solid state unstable resonators with tapered reflectivity mirror: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
    [CrossRef]
  8. B. Comaskey, G. F. Albrecht, S. P. Velsko, B. D. Moran, “24 watts average power at 537 mm from an externally frequency doubled Q-switched diode pumped Nd:YOS laser oscillator,” Appl. Opt. 33, 6377–6382 (1994).
    [CrossRef] [PubMed]

1994 (1)

1992 (3)

F. Salin, J. Squire, “Gain guiding in solid state laser,” Opt. Lett. 17, 1352–1354 (1992).
[CrossRef] [PubMed]

J. J. Kasinski, W. Hughes, D. DiBiase, P. Bournes, 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]

G. Cerullo, S. DeSilversti, V. Magni, “High efficiency, 40 W CW Nd:YLF laser with large TEM00 mode,” Opt. Commun. 93, 77–81 (1992).
[CrossRef]

1988 (1)

D. DeSilvestri, P. Laporta, V. Magni, O. Svelto, “Solid state unstable resonators with tapered reflectivity mirror: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

Albrecht, G. F.

B. Comaskey, G. F. Albrecht, S. P. Velsko, B. D. Moran, “24 watts average power at 537 mm from an externally frequency doubled Q-switched diode pumped Nd:YOS laser oscillator,” Appl. Opt. 33, 6377–6382 (1994).
[CrossRef] [PubMed]

B. J. Comaskey, G. F. Albrecht, R. J. Beach, S. P. Velsko, S. B. Sutton, S. C. Mitchell, C. S. Petty, K. S. Jancaitis, W. J. Bennett, B. L. Freitas, R. W. Solarz, “A one kilowatt average power diode pumped Nd:YAG folded zig-zag slab laser,” in Diode Pumping of Average-Power Solid State Laser, G. F. Albrecht, R. J. Beach, S. P. Velsko, eds., Proc. SPIE1865, 9–11 (1993).

Beach, R. J.

B. J. Comaskey, G. F. Albrecht, R. J. Beach, S. P. Velsko, S. B. Sutton, S. C. Mitchell, C. S. Petty, K. S. Jancaitis, W. J. Bennett, B. L. Freitas, R. W. Solarz, “A one kilowatt average power diode pumped Nd:YAG folded zig-zag slab laser,” in Diode Pumping of Average-Power Solid State Laser, G. F. Albrecht, R. J. Beach, S. P. Velsko, eds., Proc. SPIE1865, 9–11 (1993).

Bennett, W. J.

B. J. Comaskey, G. F. Albrecht, R. J. Beach, S. P. Velsko, S. B. Sutton, S. C. Mitchell, C. S. Petty, K. S. Jancaitis, W. J. Bennett, B. L. Freitas, R. W. Solarz, “A one kilowatt average power diode pumped Nd:YAG folded zig-zag slab laser,” in Diode Pumping of Average-Power Solid State Laser, G. F. Albrecht, R. J. Beach, S. P. Velsko, eds., Proc. SPIE1865, 9–11 (1993).

Bournes, P.

J. J. Kasinski, W. Hughes, D. DiBiase, P. Bournes, 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]

Burnham, R.

J. J. Kasinski, W. Hughes, D. DiBiase, P. Bournes, 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]

Cerullo, G.

G. Cerullo, S. DeSilversti, V. Magni, “High efficiency, 40 W CW Nd:YLF laser with large TEM00 mode,” Opt. Commun. 93, 77–81 (1992).
[CrossRef]

Comaskey, B.

Comaskey, B. J.

B. J. Comaskey, G. F. Albrecht, R. J. Beach, S. P. Velsko, S. B. Sutton, S. C. Mitchell, C. S. Petty, K. S. Jancaitis, W. J. Bennett, B. L. Freitas, R. W. Solarz, “A one kilowatt average power diode pumped Nd:YAG folded zig-zag slab laser,” in Diode Pumping of Average-Power Solid State Laser, G. F. Albrecht, R. J. Beach, S. P. Velsko, eds., Proc. SPIE1865, 9–11 (1993).

DeSilversti, S.

G. Cerullo, S. DeSilversti, V. Magni, “High efficiency, 40 W CW Nd:YLF laser with large TEM00 mode,” Opt. Commun. 93, 77–81 (1992).
[CrossRef]

DeSilvestri, D.

D. DeSilvestri, P. Laporta, V. Magni, O. Svelto, “Solid state unstable resonators with tapered reflectivity mirror: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

DiBiase, D.

J. J. Kasinski, W. Hughes, D. DiBiase, P. Bournes, 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]

Freitas, B. L.

B. J. Comaskey, G. F. Albrecht, R. J. Beach, S. P. Velsko, S. B. Sutton, S. C. Mitchell, C. S. Petty, K. S. Jancaitis, W. J. Bennett, B. L. Freitas, R. W. Solarz, “A one kilowatt average power diode pumped Nd:YAG folded zig-zag slab laser,” in Diode Pumping of Average-Power Solid State Laser, G. F. Albrecht, R. J. Beach, S. P. Velsko, eds., Proc. SPIE1865, 9–11 (1993).

Hughes, W.

J. J. Kasinski, W. Hughes, D. DiBiase, P. Bournes, 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]

Jancaitis, K. S.

B. J. Comaskey, G. F. Albrecht, R. J. Beach, S. P. Velsko, S. B. Sutton, S. C. Mitchell, C. S. Petty, K. S. Jancaitis, W. J. Bennett, B. L. Freitas, R. W. Solarz, “A one kilowatt average power diode pumped Nd:YAG folded zig-zag slab laser,” in Diode Pumping of Average-Power Solid State Laser, G. F. Albrecht, R. J. Beach, S. P. Velsko, eds., Proc. SPIE1865, 9–11 (1993).

Jenks, I. T.

N. MacKinnon, B. D. Sinclair, W. Sibbett, S. N. Jenny, I. T. Jenks, “Ultracompact, laser diode array pumped, Nd:YVO4/KDP frequency doubled composite-material micro-chip laser,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 156.

Jenny, S. N.

N. MacKinnon, B. D. Sinclair, W. Sibbett, S. N. Jenny, I. T. Jenks, “Ultracompact, laser diode array pumped, Nd:YVO4/KDP frequency doubled composite-material micro-chip laser,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 156.

Kasinski, J. J.

J. J. Kasinski, W. Hughes, D. DiBiase, P. Bournes, 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]

Koechner, W.

W. Koechner, Solid State Laser Engineering, 3rd ed. (Springer-Verlag, New York, 1992), particularly Chap. 3.6.5, “Laser diode pumped systems,” and references therein.

Laporta, P.

D. DeSilvestri, P. Laporta, V. Magni, O. Svelto, “Solid state unstable resonators with tapered reflectivity mirror: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

MacKinnon, N.

N. MacKinnon, B. D. Sinclair, W. Sibbett, S. N. Jenny, I. T. Jenks, “Ultracompact, laser diode array pumped, Nd:YVO4/KDP frequency doubled composite-material micro-chip laser,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 156.

Magni, V.

G. Cerullo, S. DeSilversti, V. Magni, “High efficiency, 40 W CW Nd:YLF laser with large TEM00 mode,” Opt. Commun. 93, 77–81 (1992).
[CrossRef]

D. DeSilvestri, P. Laporta, V. Magni, O. Svelto, “Solid state unstable resonators with tapered reflectivity mirror: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

Mitchell, S. C.

B. J. Comaskey, G. F. Albrecht, R. J. Beach, S. P. Velsko, S. B. Sutton, S. C. Mitchell, C. S. Petty, K. S. Jancaitis, W. J. Bennett, B. L. Freitas, R. W. Solarz, “A one kilowatt average power diode pumped Nd:YAG folded zig-zag slab laser,” in Diode Pumping of Average-Power Solid State Laser, G. F. Albrecht, R. J. Beach, S. P. Velsko, eds., Proc. SPIE1865, 9–11 (1993).

Moran, B. D.

Petty, C. S.

B. J. Comaskey, G. F. Albrecht, R. J. Beach, S. P. Velsko, S. B. Sutton, S. C. Mitchell, C. S. Petty, K. S. Jancaitis, W. J. Bennett, B. L. Freitas, R. W. Solarz, “A one kilowatt average power diode pumped Nd:YAG folded zig-zag slab laser,” in Diode Pumping of Average-Power Solid State Laser, G. F. Albrecht, R. J. Beach, S. P. Velsko, eds., Proc. SPIE1865, 9–11 (1993).

Salin, F.

Sibbett, W.

N. MacKinnon, B. D. Sinclair, W. Sibbett, S. N. Jenny, I. T. Jenks, “Ultracompact, laser diode array pumped, Nd:YVO4/KDP frequency doubled composite-material micro-chip laser,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 156.

Sinclair, B. D.

N. MacKinnon, B. D. Sinclair, W. Sibbett, S. N. Jenny, I. T. Jenks, “Ultracompact, laser diode array pumped, Nd:YVO4/KDP frequency doubled composite-material micro-chip laser,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 156.

Solarz, R. W.

B. J. Comaskey, G. F. Albrecht, R. J. Beach, S. P. Velsko, S. B. Sutton, S. C. Mitchell, C. S. Petty, K. S. Jancaitis, W. J. Bennett, B. L. Freitas, R. W. Solarz, “A one kilowatt average power diode pumped Nd:YAG folded zig-zag slab laser,” in Diode Pumping of Average-Power Solid State Laser, G. F. Albrecht, R. J. Beach, S. P. Velsko, eds., Proc. SPIE1865, 9–11 (1993).

Squire, J.

Sutton, S. B.

B. J. Comaskey, G. F. Albrecht, R. J. Beach, S. P. Velsko, S. B. Sutton, S. C. Mitchell, C. S. Petty, K. S. Jancaitis, W. J. Bennett, B. L. Freitas, R. W. Solarz, “A one kilowatt average power diode pumped Nd:YAG folded zig-zag slab laser,” in Diode Pumping of Average-Power Solid State Laser, G. F. Albrecht, R. J. Beach, S. P. Velsko, eds., Proc. SPIE1865, 9–11 (1993).

Svelto, O.

D. DeSilvestri, P. Laporta, V. Magni, O. Svelto, “Solid state unstable resonators with tapered reflectivity mirror: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

Velsko, S. P.

B. Comaskey, G. F. Albrecht, S. P. Velsko, B. D. Moran, “24 watts average power at 537 mm from an externally frequency doubled Q-switched diode pumped Nd:YOS laser oscillator,” Appl. Opt. 33, 6377–6382 (1994).
[CrossRef] [PubMed]

B. J. Comaskey, G. F. Albrecht, R. J. Beach, S. P. Velsko, S. B. Sutton, S. C. Mitchell, C. S. Petty, K. S. Jancaitis, W. J. Bennett, B. L. Freitas, R. W. Solarz, “A one kilowatt average power diode pumped Nd:YAG folded zig-zag slab laser,” in Diode Pumping of Average-Power Solid State Laser, G. F. Albrecht, R. J. Beach, S. P. Velsko, eds., Proc. SPIE1865, 9–11 (1993).

Appl. Opt. (1)

IEEE J. Quantum Electron. (2)

D. DeSilvestri, P. Laporta, V. Magni, O. Svelto, “Solid state unstable resonators with tapered reflectivity mirror: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

J. J. Kasinski, W. Hughes, D. DiBiase, P. Bournes, 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]

Opt. Commun. (1)

G. Cerullo, S. DeSilversti, V. Magni, “High efficiency, 40 W CW Nd:YLF laser with large TEM00 mode,” Opt. Commun. 93, 77–81 (1992).
[CrossRef]

Opt. Lett. (1)

Other (3)

W. Koechner, Solid State Laser Engineering, 3rd ed. (Springer-Verlag, New York, 1992), particularly Chap. 3.6.5, “Laser diode pumped systems,” and references therein.

N. MacKinnon, B. D. Sinclair, W. Sibbett, S. N. Jenny, I. T. Jenks, “Ultracompact, laser diode array pumped, Nd:YVO4/KDP frequency doubled composite-material micro-chip laser,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 156.

B. J. Comaskey, G. F. Albrecht, R. J. Beach, S. P. Velsko, S. B. Sutton, S. C. Mitchell, C. S. Petty, K. S. Jancaitis, W. J. Bennett, B. L. Freitas, R. W. Solarz, “A one kilowatt average power diode pumped Nd:YAG folded zig-zag slab laser,” in Diode Pumping of Average-Power Solid State Laser, G. F. Albrecht, R. J. Beach, S. P. Velsko, eds., Proc. SPIE1865, 9–11 (1993).

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Figures (14)

Fig. 1
Fig. 1

Source and rod geometry and nomenclature used in the development of the inner solution.

Fig. 2
Fig. 2

Depiction of the four possible effective source configurations, including the validity range of the ratio of source radius and rod radius (RR).

Fig. 3
Fig. 3

Characteristic ray trace for several stripes in the ideal radial diode.

Fig. 4
Fig. 4

Comparison of the analytical models with the numerical model for Rα = 1.5 and for three θe spread angles of (a) 7°, (b) 14°, (c) 28°.

Fig. 5
Fig. 5

Comparison of the analytical models with the numerical model for Rα = 3 and for three θe spread angles of (a) 7°, (b) 14°, (c) 28°.

Fig. 6
Fig. 6

Comparison of numerical solutions for Rα = 3, θe = 14 degrees, and for converging and diverging sources. The solid curve represents the diverging source (RR = 1.5) and the circles represent the converging source (RR = −1.5).

Fig. 7
Fig. 7

Variation of the uniformity parameter σ with the absorption length (Rα) and the product RRθe. The locus of minimum values of the uniformity parameter, as a function of RRθe, is given by the filled circles.

Fig. 8
Fig. 8

Effect of source spread angle (φ) and absorption length (Rα) on the absorption efficiency as a function of RR. Curves for spread angles (φ) of 20° and 40° are given for absorption lengths (Rα) of 0.5, 1, and 2.

Fig. 9
Fig. 9

For uniformity optimized absorption length, the variation of the absorption efficiency (ɛ) with RR for source spread angles of 5°, 10°, 15°, 20°, 25°, 35°, and 45°.

Fig. 10
Fig. 10

For uniformity optimized absorption length, the variation of the absorption efficiency (ɛ) with source spread angle (φ) for RR values of source spread angles of 1°, 1.5°, 2°, 3°, 4°, 5°, 10°, and 20°.

Fig. 11
Fig. 11

Typical lens-based geometry used to achieve uniform illumination and the proper energy spread at the rod center.

Fig. 12
Fig. 12

Isolated picture of a single diode panel including a geometric trace of rays showing the spread at the rod center: (a) the entire geometry, (b) the rod isolated.

Fig. 13
Fig. 13

Comparison of the numerical solution with the outer-region solution and several spread angle selections for the inner-region solution.

Fig. 14
Fig. 14

Normalized energy-density contours of (a) four sources in the Fig. 7 configuration, (b) an axially multiplexed arrangement that creates eight-fold symmetry.

Equations (32)

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I ( X , Y ) = 2 π I 0 w exp [ - 2 ( X ) 2 w 2 ] ,
w 2 = w 0 2 [ 1 + ( λ Y π w 0 2 ) 2 ] .
I ( X , Y ) = 2 π I 0 w 0 exp [ - 2 π 2 w 0 2 X 2 π 2 w 0 4 + λ 2 ( Y - R S ) 2 - α D ] [ π 2 w 0 4 + λ 2 ( Y - R S ) 2 ] 1 / 2 .
I ( x ) = 2 / π I 0 w 0 exp [ - 2 R 00 2 x 2 cos 2 ( ϕ ) 1 + θ e 2 ( x R 00 sin ( ϕ ) - R s 0 ) 2 / 2 - α D ] [ 1 + 1 2 θ e 2 ( x R 00 sin ( ϕ ) - R s 0 ) 2 ] 1 / 2 ,
D = R 0 [ 1 - x sin ( ϕ ) R R R R + x 2 cos 2 ( ϕ ) ( 1 R R - 1 2 ) ] .
I r = 4 I w π R 00 exp ( - R α ) ( R R 2 R 00 2 θ e 2 + 2 ) 1 / 2 ( C x 2 + 1 ) ,
C = 1 4 R R ( R R 2 R 00 2 θ e 2 + 2 ) 2 ( R R { R R 4 R 00 4 R α θ e 4 ( R α + 1 ) + 2 R R 2 R 00 2 θ e 2 [ R 00 2 ( θ e 2 - 4 ) + 2 R α ( R α + 1 ) ] - 2 [ R 00 2 ( θ e 2 + 8 ) - 2 R α ( R α + 1 ) ] } - 4 R α ( R R 2 R 00 2 θ e 2 + 2 ) s i g n ( R R ) ) .
I a = I G ( 2 π R 00 ) exp ( - R α ) ( 1 + 1 2 R s 0 2 θ e 2 ) 1 / 2 ,
I = 4 I w π exp ( - R α ) [ ( R R θ e ) 2 + 2 R 00 2 ] 1 / 2 × { x 2 [ R α ( R R θ e ) 2 ( R α + 1 ) + 2 ( θ e 2 - 4 ) 4 ( R R θ e ) 2 + 1 R 00 2 ( 8 - 5 θ e 2 2 ( R R θ e ) 4 - R α ( R R θ e ) 2 R R ) ] + 1 } .
I r = 4 π I w exp ( - R α ) R R θ e × [ x 2 R α ( R R θ e ) 2 ( R α + 1 ) + 2 ( θ e 2 - 4 ) 4 ( R R θ e ) 2 + 1 ] .
I r R α = 0 = 4 π I w R R θ e [ x 2 ( θ e 2 - 4 ) 2 R R 2 θ e 2 + 1 ] .
0 2 π sin ( n ϕ - ϕ r ) sin 2 ( ϕ ) d ϕ ,
0 2 π sin ( n ϕ - ϕ r ) sin m ( ϕ ) d ϕ
I = I w { ( 1 / x ) exp [ - R α ( 1 - x ) ] + ( 1 / x ) × exp [ - R α ( 1 + x - 2 r c ) ] } .
I R = I w { ( 1 / x ) exp [ - R α ( 1 - x ) ] + ( 1 / x ) × exp [ - R α ( 1 + x ) ] } .
I p = I w [ 1 + exp ( - 2 R α ) ]
I E = I w x [ 6 π exp ( - R α ) × ( { x 2 [ R R 2 R α ( R α + 1 ) + 2 ] + 4 R R 2 } 3 / 2 18 R R 2 ) ] .
x t = 1 6 R R θ e .
x t F = 0.41 R R θ e + 0.045 ( R R θ e ) 2 + 0.05 R α R R θ e .
σ = 0 1 ( I - I ¯ ) 2 d x I 0 2 = 0 x t ( I inner - I ¯ ) 2 d x + x t 1 ( I outer - I ¯ ) 2 d x I 0 2 ,
I ¯ = 0 1 I d x .
D φ = 2 ( R 0 2 - R s 2 sin 2 φ ) 1 / 2 .
D avg = 2 R 0 φ 0 0 φ 0 ( 1 - R R 2 sin 2 φ ) 1 / 2 d φ .
φ 0 = sin - 1 ( 1 / R R ) ,
ɛ = 1 - exp ( - α D a v g ) ,
ɛ = sin - 1 ( 1 / R R ) φ 0 { 1 - exp [ - 2 R α sin - 1 ( 1 / R R ) × 0 sin - 1 ( 1 / R R ) ( 1 - R R 2 sin 2 φ ) 1 / 2 d φ ] } ,
ɛ = 1 - exp ( - R α { sin - 1 ( R R φ 0 ) R R φ 0 + [ 1 - ( R R φ 0 ) 2 ] 1 / 2 } ) .
ɛ = R R sin - 1 ( 1 / R R ) R R φ 0 { 1 - exp [ - π R α 2 R R sin - 1 ( 1 / R R ) ] } .
ɛ mvc = 1 - exp { - sin - 1 ( R R φ 0 ) / ( R R φ 0 ) + [ 1 - ( R R φ 0 ) 2 ] 1 / 2 0.537 + 0.083 ( R R φ 0 ) 5 / 3 - 0.401 exp ( - R R φ 0 ) } ,
I ^ ( x ) = 1 2 π 0 2 π I ( x , θ ) d θ .
D = [ ( x c - x L ) 2 + ( y c - y L ) 2 ] 1 / 2 .
D = R o R R x 2 cos 2 ϕ + R R 2 { - 2 R R [ x 2 cos 2 ϕ ( x sin ϕ - R R ) + R R 2 x sin ϕ ] [ x 2 cos 2 ϕ ( 1 - R R 2 ) + R R ] 1 / 2 + [ x 2 sin 2 ϕ - 2 R R x sin ϕ ] x 4 cos 4 ϕ + x 2 cos 2 ϕ R R 2 [ x 2 ( cos 2 ϕ + 2 sin 2 ϕ ) + 1 - 2 R R x sin ϕ ] + R R 4 [ x 2 sin 2 ϕ - x 2 cos 2 ϕ + 1 ] } 1 / 2 .

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