Abstract

An iterative threshold model for a pulsed singly resonant Gaussian-reflectivity-mirror (GRM) confocal unstable optical parametric oscillator (OPO) has been proposed. It is found that OPO threshold is determined by important parameters such as GRM central reflectance, Gaussian reflectivity profile, cavity magnification factor, cavity physical length, crystal length, pump pulsewidth. It is demonstrated that this model can be extended to plane-parallel resonator or uniform-reflectivity-mirror (URM) unstable resonator when some specific values are taken. Experimental results show excellent agreement with values calculated from theoretical model. Both theoretical calculations and experimental data illustrate that GRM is a useful solution to reduce threshold of unstable OPO.

©2005 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Threshold studies of pulsed confocal unstable optical parametric oscillators

Mali Gong, Shanshan Zou, Gang Chen, Ping Yan, Qiang Liu, and Lei Huang
Opt. Express 12(13) 2932-2944 (2004)

Unstable resonator optical parametric oscillator based on quasi-phase-matched RbTiOAsO4

Göran Hansson, Håkan Karlsson, and Fredrik Laurell
Appl. Opt. 40(30) 5446-5451 (2001)

References

  • View by:
  • |
  • |
  • |

  1. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), pp. 858–913.
  2. W. A. Neuman and S. P. Velsko, “Effect of cavity design on optical parametric oscillator performance,” in Advanced Solid-State Lasers, A. Payne and C. R. Pollock, eds., Vol.1 of OSA Trends in Optics and Photonics Series (Optical. Society of. America, Washington, D. C., 1996), pp.179–181.
  3. M. K. Brown and M. S. Bowers, “High energy, near diffraction limited output from optical parametric oscillators using unstable resonators,” in Solid State Lasers VI, R. Scheps, ed., Proc. SPIE 2986, 113–122 (1997).
  4. S. Pearl, Y. Ehrlich, S. Fastig, and S. Rosenwaks,” Nearly diffraction-limited signal generated by a lower beam-quality pump in an optical parametric oscillator,” App. Opt. 42, 1048–1051 (2003).
    [Crossref]
  5. G. Hansson, H. Karlsson, and F. Laurell, “Unstable resonator optical parametric oscillator based on quasi-phase-matched RbTiOAsO4,” App. Opt. 40, 5446–5451 (2001).
    [Crossref]
  6. B. C. Johnson, V. J. Newell, J. B. Clark, and E. S. McPhee, “Narrow-bandwidth low-divergence optical parametric oscillator for nonlinear frequency-conversion applications,” J. Opt. Soc. Am. B 12, 2122–2127 (1995).
    [Crossref]
  7. J. N. Farmer, M. S. Bowers, and W. S. Schaprf, “High brightness eyesafe optical parametric oscillator using confocal unstable resonators,” in Advanced Solid-State Lasers, M. M. Fejer, H. Injeyan, and U. Keller, eds., Vol. 26 of OSA Trends in Optics and Photonics Series (Optical. Society of. America, Washington, D. C., 1999), pp.567–571.
  8. M. Morin, “Graded reflectivity mirror unstable laser resonators,” Opt. Quantum Electron. 29, 819–66 (1997).
    [Crossref]
  9. Kevin J. Snell, Nathalie McCarthy, Miche Piche, and Pierre Lavign, “Single transverse mode oscillation from an unstable resonator Nd:YAG laser using a variable reflectivity mirror,” Opt. Commun. 65, 377–382 (1988).
    [Crossref]
  10. Nathalie McCarthy and Pierre Lavigne, “Large-size Gaussian mode in unstable resonators using Gaussian mirrors,” Opt. Lett. 10, 553–555 (1985).
    [Crossref] [PubMed]
  11. Suresh Chandra, Toomas H. Allik, and J. Andrew Hutchinson, “Nonconfocal unstable resonator for solid-state dye lasers based on a gradient-reflectivity mirror,” Opt. Lett. 20, 2387–2389 (1995).
    [Crossref] [PubMed]
  12. S. Chandr, T. H. Allik, J. A. Hutchinson, and M. S. Bowers, “Improved OPO brightness with a GRM nonconfocal unstable resonator,” in Advanced Solid-State Lasers, S. A. Payne and C. R. Pollock, eds., Vol.1 of OSA Trends in Optics and Photonics Series (Optical. Society of. America, Washington, D. C., 1996), pp.177–178.
  13. S. J. Brosnan and R. L. Byer, “Optical parametric oscillator threshold and linewidth studies,” IEEE J. of Quantum Electron. QE-15, 415–431 (1979).
    [Crossref]
  14. M. Gong, S. Zou, G. Chen, P. Yan, Q. Liu, and L. Huang, “Threshold studies of pulsed confocal unstable optical parametric oscillators,” Opt. Express 12, 2932–2944 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-13-2932.
    [Crossref] [PubMed]
  15. W. J. Alford and A. V. Smith, “Wavelength variation of the second-order nonlinear coefficients of KNbO3, KTiOPO4, KTiOAsO4, LiNbO3, LiIO3, β -BaB2O4, KH2PO4, and LiB3O5 crystals: a test of Miller wavelength scaling,” J. Opt. Soc. Am. B 18, 524–533 (2001).
    [Crossref]

2004 (1)

2003 (1)

S. Pearl, Y. Ehrlich, S. Fastig, and S. Rosenwaks,” Nearly diffraction-limited signal generated by a lower beam-quality pump in an optical parametric oscillator,” App. Opt. 42, 1048–1051 (2003).
[Crossref]

2001 (2)

1997 (2)

M. Morin, “Graded reflectivity mirror unstable laser resonators,” Opt. Quantum Electron. 29, 819–66 (1997).
[Crossref]

M. K. Brown and M. S. Bowers, “High energy, near diffraction limited output from optical parametric oscillators using unstable resonators,” in Solid State Lasers VI, R. Scheps, ed., Proc. SPIE 2986, 113–122 (1997).

1995 (2)

1988 (1)

Kevin J. Snell, Nathalie McCarthy, Miche Piche, and Pierre Lavign, “Single transverse mode oscillation from an unstable resonator Nd:YAG laser using a variable reflectivity mirror,” Opt. Commun. 65, 377–382 (1988).
[Crossref]

1985 (1)

1979 (1)

S. J. Brosnan and R. L. Byer, “Optical parametric oscillator threshold and linewidth studies,” IEEE J. of Quantum Electron. QE-15, 415–431 (1979).
[Crossref]

Alford, W. J.

Allik, T. H.

S. Chandr, T. H. Allik, J. A. Hutchinson, and M. S. Bowers, “Improved OPO brightness with a GRM nonconfocal unstable resonator,” in Advanced Solid-State Lasers, S. A. Payne and C. R. Pollock, eds., Vol.1 of OSA Trends in Optics and Photonics Series (Optical. Society of. America, Washington, D. C., 1996), pp.177–178.

Allik, Toomas H.

Bowers, M. S.

M. K. Brown and M. S. Bowers, “High energy, near diffraction limited output from optical parametric oscillators using unstable resonators,” in Solid State Lasers VI, R. Scheps, ed., Proc. SPIE 2986, 113–122 (1997).

J. N. Farmer, M. S. Bowers, and W. S. Schaprf, “High brightness eyesafe optical parametric oscillator using confocal unstable resonators,” in Advanced Solid-State Lasers, M. M. Fejer, H. Injeyan, and U. Keller, eds., Vol. 26 of OSA Trends in Optics and Photonics Series (Optical. Society of. America, Washington, D. C., 1999), pp.567–571.

S. Chandr, T. H. Allik, J. A. Hutchinson, and M. S. Bowers, “Improved OPO brightness with a GRM nonconfocal unstable resonator,” in Advanced Solid-State Lasers, S. A. Payne and C. R. Pollock, eds., Vol.1 of OSA Trends in Optics and Photonics Series (Optical. Society of. America, Washington, D. C., 1996), pp.177–178.

Brosnan, S. J.

S. J. Brosnan and R. L. Byer, “Optical parametric oscillator threshold and linewidth studies,” IEEE J. of Quantum Electron. QE-15, 415–431 (1979).
[Crossref]

Brown, M. K.

M. K. Brown and M. S. Bowers, “High energy, near diffraction limited output from optical parametric oscillators using unstable resonators,” in Solid State Lasers VI, R. Scheps, ed., Proc. SPIE 2986, 113–122 (1997).

Byer, R. L.

S. J. Brosnan and R. L. Byer, “Optical parametric oscillator threshold and linewidth studies,” IEEE J. of Quantum Electron. QE-15, 415–431 (1979).
[Crossref]

Chandr, S.

S. Chandr, T. H. Allik, J. A. Hutchinson, and M. S. Bowers, “Improved OPO brightness with a GRM nonconfocal unstable resonator,” in Advanced Solid-State Lasers, S. A. Payne and C. R. Pollock, eds., Vol.1 of OSA Trends in Optics and Photonics Series (Optical. Society of. America, Washington, D. C., 1996), pp.177–178.

Chandra, Suresh

Chen, G.

Clark, J. B.

Ehrlich, Y.

S. Pearl, Y. Ehrlich, S. Fastig, and S. Rosenwaks,” Nearly diffraction-limited signal generated by a lower beam-quality pump in an optical parametric oscillator,” App. Opt. 42, 1048–1051 (2003).
[Crossref]

Farmer, J. N.

J. N. Farmer, M. S. Bowers, and W. S. Schaprf, “High brightness eyesafe optical parametric oscillator using confocal unstable resonators,” in Advanced Solid-State Lasers, M. M. Fejer, H. Injeyan, and U. Keller, eds., Vol. 26 of OSA Trends in Optics and Photonics Series (Optical. Society of. America, Washington, D. C., 1999), pp.567–571.

Fastig, S.

S. Pearl, Y. Ehrlich, S. Fastig, and S. Rosenwaks,” Nearly diffraction-limited signal generated by a lower beam-quality pump in an optical parametric oscillator,” App. Opt. 42, 1048–1051 (2003).
[Crossref]

Gong, M.

Hansson, G.

G. Hansson, H. Karlsson, and F. Laurell, “Unstable resonator optical parametric oscillator based on quasi-phase-matched RbTiOAsO4,” App. Opt. 40, 5446–5451 (2001).
[Crossref]

Huang, L.

Hutchinson, J. A.

S. Chandr, T. H. Allik, J. A. Hutchinson, and M. S. Bowers, “Improved OPO brightness with a GRM nonconfocal unstable resonator,” in Advanced Solid-State Lasers, S. A. Payne and C. R. Pollock, eds., Vol.1 of OSA Trends in Optics and Photonics Series (Optical. Society of. America, Washington, D. C., 1996), pp.177–178.

Hutchinson, J. Andrew

Johnson, B. C.

Karlsson, H.

G. Hansson, H. Karlsson, and F. Laurell, “Unstable resonator optical parametric oscillator based on quasi-phase-matched RbTiOAsO4,” App. Opt. 40, 5446–5451 (2001).
[Crossref]

Laurell, F.

G. Hansson, H. Karlsson, and F. Laurell, “Unstable resonator optical parametric oscillator based on quasi-phase-matched RbTiOAsO4,” App. Opt. 40, 5446–5451 (2001).
[Crossref]

Lavign, Pierre

Kevin J. Snell, Nathalie McCarthy, Miche Piche, and Pierre Lavign, “Single transverse mode oscillation from an unstable resonator Nd:YAG laser using a variable reflectivity mirror,” Opt. Commun. 65, 377–382 (1988).
[Crossref]

Lavigne, Pierre

Liu, Q.

McCarthy, Nathalie

Kevin J. Snell, Nathalie McCarthy, Miche Piche, and Pierre Lavign, “Single transverse mode oscillation from an unstable resonator Nd:YAG laser using a variable reflectivity mirror,” Opt. Commun. 65, 377–382 (1988).
[Crossref]

Nathalie McCarthy and Pierre Lavigne, “Large-size Gaussian mode in unstable resonators using Gaussian mirrors,” Opt. Lett. 10, 553–555 (1985).
[Crossref] [PubMed]

McPhee, E. S.

Morin, M.

M. Morin, “Graded reflectivity mirror unstable laser resonators,” Opt. Quantum Electron. 29, 819–66 (1997).
[Crossref]

Neuman, W. A.

W. A. Neuman and S. P. Velsko, “Effect of cavity design on optical parametric oscillator performance,” in Advanced Solid-State Lasers, A. Payne and C. R. Pollock, eds., Vol.1 of OSA Trends in Optics and Photonics Series (Optical. Society of. America, Washington, D. C., 1996), pp.179–181.

Newell, V. J.

Pearl, S.

S. Pearl, Y. Ehrlich, S. Fastig, and S. Rosenwaks,” Nearly diffraction-limited signal generated by a lower beam-quality pump in an optical parametric oscillator,” App. Opt. 42, 1048–1051 (2003).
[Crossref]

Piche, Miche

Kevin J. Snell, Nathalie McCarthy, Miche Piche, and Pierre Lavign, “Single transverse mode oscillation from an unstable resonator Nd:YAG laser using a variable reflectivity mirror,” Opt. Commun. 65, 377–382 (1988).
[Crossref]

Rosenwaks, S.

S. Pearl, Y. Ehrlich, S. Fastig, and S. Rosenwaks,” Nearly diffraction-limited signal generated by a lower beam-quality pump in an optical parametric oscillator,” App. Opt. 42, 1048–1051 (2003).
[Crossref]

Schaprf, W. S.

J. N. Farmer, M. S. Bowers, and W. S. Schaprf, “High brightness eyesafe optical parametric oscillator using confocal unstable resonators,” in Advanced Solid-State Lasers, M. M. Fejer, H. Injeyan, and U. Keller, eds., Vol. 26 of OSA Trends in Optics and Photonics Series (Optical. Society of. America, Washington, D. C., 1999), pp.567–571.

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), pp. 858–913.

Smith, A. V.

Snell, Kevin J.

Kevin J. Snell, Nathalie McCarthy, Miche Piche, and Pierre Lavign, “Single transverse mode oscillation from an unstable resonator Nd:YAG laser using a variable reflectivity mirror,” Opt. Commun. 65, 377–382 (1988).
[Crossref]

Velsko, S. P.

W. A. Neuman and S. P. Velsko, “Effect of cavity design on optical parametric oscillator performance,” in Advanced Solid-State Lasers, A. Payne and C. R. Pollock, eds., Vol.1 of OSA Trends in Optics and Photonics Series (Optical. Society of. America, Washington, D. C., 1996), pp.179–181.

Yan, P.

Zou, S.

App. Opt. (2)

S. Pearl, Y. Ehrlich, S. Fastig, and S. Rosenwaks,” Nearly diffraction-limited signal generated by a lower beam-quality pump in an optical parametric oscillator,” App. Opt. 42, 1048–1051 (2003).
[Crossref]

G. Hansson, H. Karlsson, and F. Laurell, “Unstable resonator optical parametric oscillator based on quasi-phase-matched RbTiOAsO4,” App. Opt. 40, 5446–5451 (2001).
[Crossref]

IEEE J. of Quantum Electron. (1)

S. J. Brosnan and R. L. Byer, “Optical parametric oscillator threshold and linewidth studies,” IEEE J. of Quantum Electron. QE-15, 415–431 (1979).
[Crossref]

in Solid State Lasers VI (1)

M. K. Brown and M. S. Bowers, “High energy, near diffraction limited output from optical parametric oscillators using unstable resonators,” in Solid State Lasers VI, R. Scheps, ed., Proc. SPIE 2986, 113–122 (1997).

J. Opt. Soc. Am. B (2)

Opt. Commun. (1)

Kevin J. Snell, Nathalie McCarthy, Miche Piche, and Pierre Lavign, “Single transverse mode oscillation from an unstable resonator Nd:YAG laser using a variable reflectivity mirror,” Opt. Commun. 65, 377–382 (1988).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Opt. Quantum Electron. (1)

M. Morin, “Graded reflectivity mirror unstable laser resonators,” Opt. Quantum Electron. 29, 819–66 (1997).
[Crossref]

Other (4)

S. Chandr, T. H. Allik, J. A. Hutchinson, and M. S. Bowers, “Improved OPO brightness with a GRM nonconfocal unstable resonator,” in Advanced Solid-State Lasers, S. A. Payne and C. R. Pollock, eds., Vol.1 of OSA Trends in Optics and Photonics Series (Optical. Society of. America, Washington, D. C., 1996), pp.177–178.

J. N. Farmer, M. S. Bowers, and W. S. Schaprf, “High brightness eyesafe optical parametric oscillator using confocal unstable resonators,” in Advanced Solid-State Lasers, M. M. Fejer, H. Injeyan, and U. Keller, eds., Vol. 26 of OSA Trends in Optics and Photonics Series (Optical. Society of. America, Washington, D. C., 1999), pp.567–571.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), pp. 858–913.

W. A. Neuman and S. P. Velsko, “Effect of cavity design on optical parametric oscillator performance,” in Advanced Solid-State Lasers, A. Payne and C. R. Pollock, eds., Vol.1 of OSA Trends in Optics and Photonics Series (Optical. Society of. America, Washington, D. C., 1996), pp.179–181.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1.
Fig. 1. Confocal unstable singly resonant OPO. The input mirror M1 is a concave mirror, which is highly reflecting at the signal wavelength and highly transmitting at the pump and idler wavelengths. The output coupler M2 is a convex mirror, which is highly reflective or transmitting at the pump wavelength, highly transmitting at the idler wavelength, and has a Gauss-profile signal reflectance.
Fig. 2.
Fig. 2. Threshold energy versus central signal reflectance of output coupler for a double-pass pumped OPO. Solid lines show results of our theoretical model for various cavity magnification factors. The GRM plane-parallel cavity result is plotted at M=1.00. Dashed line shows the result of corresponding URM unstable OPO with equal effective output coupling for the M=1.20 case. L=54 mm, lc =20 mm, 2rp =2.8 mm, w=1.45 mm, T=13.5 ns.
Fig. 3.
Fig. 3. Threshold energy versus the 1/e radius of the Gaussian signal reflectivity profile of output coupler for a double-pass pumped OPO. Solid lines show results of our theoretical model for various cavity magnification factors. L=54 mm, lc =20 mm, 2rp =2.8 mm, Rmax =0.85, T=13.5 ns.
Fig. 4.
Fig. 4. Threshold energy versus cavity physical length for a double-pass pumped OPO. Solid lines show results of our theoretical model for various cavity magnification factors. Dashed line shows the result of corresponding URM unstable OPO with equal effective output coupling for the M=1.20 case. lc =20 mm, 2rp =2.8 mm, Rmax =0.85, w=1.45 mm, T=13.5 ns.
Fig. 5.
Fig. 5. Threshold energy versus pump pulsewidth for a double-pass pumped OPO. Solid lines show results of our theoretical model for various cavity magnification factors. L=54 mm, lc =20 mm, 2rp =2.8 mm, Rmax =0.85, w=1.45 mm.
Fig. 6.
Fig. 6. Threshold energy versus crystal length for a double-pass pumped OPO. Solid lines show results of our theoretical model for various cavity magnification factors. L=60 mm, 2rp =2.8 mm, Rmax =0.85, w=1.45 mm, T=13.5 ns.

Tables (1)

Tables Icon

Table 1. Experimental threshold energy values of three cavities compared with results from the theoretical model

Equations (32)

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

R ( r ) = R max e ( r w ) 2
E s ( r , z ) z + α s E s ( r , z ) = i N s E p ( r , z ) E i * ( r , z ) e i Δ k r r e i Δ k z z
E i ( r , z ) z + α i E i ( r , z ) = i N i E p ( r , z ) E s * ( r , z ) e i Δ k r r e i Δ k z z
N j = ω j d eff n j c 1 + θ j 2 , j = s , i
E s ( M s ( l c ) r , l c ) = e α l c E s ( r , 0 ) M s ( l c ) cosh ( β 0 l c e ( r 2 r p ) 2 )
β 0 = 2 N s N i n p c ε 0 I p 0 e ( t τ p ) 2
P m 0 = 0 ( 1 2 n s c ε 0 ) . E start ( r ) 2 · 2 π r dr = ( 1 2 n s c ε 0 ) · E start 0 2 · π r s 2
E s f ( r , l c ) = E start ( r ) · e α l c cosh ( β f l c e ( r 2 r p ) 2 ) , r : 0 r max
β f = 2 N s f N i f n p c ε 0 I p 0 e ( t τ p ) 2
N j f = ω j d eff c n j , j = s , i
E sr b ( r ) = R max · e ( r 2 w ) 2 E s f ( r , l c ) , r : 0 r max
E sround ( r ) = 1 M e α l c E sr b ( r M )
= E start 0 R max e 2 α l c M cosh ( β f l c e ( r 2 Mr p ) 2 ) · e ( r 2 Mr s ) 2 · e ( r 2 Mw ) 2 , r : 0 Mr max
P m = 0 Mr max ( 1 2 n s c ε 0 ) · E sround ( r ) 2 2 π r dr
= R max M 2 e 4 α l c ( 1 2 n s c ε 0 ) · 2 π · E start 0 2 · 0 Mr max e ( r Mr s ) 2 e ( r Mw ) 2 cosh 2 ( β f l c e ( r 2 Mr p ) 2 ) r dr
P m = P m 1 { R max e 4 α l c [ 1 4 ( r 1 r s ) 2 e 2 β f l c ( 1 e ( r max r 1 ) 2 ) + 1 2 ( r 2 r s ) 2 ( 1 e ( r max r 2 ) 2 ) ] }
{ 1 r 1 2 = 1 w 2 + 1 r s 2 + β f l c r p 2 1 r 2 2 = 1 w 2 + 1 r s 2
E s b ( r , 0 ) = 1 M 1 E sr b ( r M 1 ) , r : 0 M 1 r max
β b = 2 N s b N i b n p c ε 0 · γ 0 2 I p 0 M 1 2 e ( t τ p ) 2
N j b = ω j d eff n j c 1 + θ j 2 , j = s , i
E s b ( r , l c ) = { E s b ( r M 2 , 0 ) M 2 e α l c cosh ( β b e ( r 2 M 2 M 1 r p ) 2 ) r : 0 r max E s b ( r M 2 , 0 ) M 2 e α l c r : r max M 1 M 2 r max
E sround ( r ) = 1 M 3 E s b ( r M 3 , l c )
= { E start 0 R max e 2 α l c M e ( r 2 M ) 2 ( 1 w 2 + 1 r s 2 ) cosh ( β f l c e ( r 2 Mr p ) 2 ) cosh ( β b l c e ( r 2 Mr p ) 2 ) r : 0 M 3 r max E start 0 R max e 2 α l c M e ( r 2 M ) 2 ( 1 w 2 + 1 r s 2 ) cosh ( β f l c e ( r 2 Mr p ) 2 ) r : M 3 r max Mr max
P m = 0 Mr max ( 1 2 n s c ε 0 ) E sround ( r ) 2 2 π r dr
= [ 0 M 3 r max e r 2 M 2 ( 1 w 2 + 1 r s 2 ) cosh 2 ( β f l c e ( r 2 Mr p ) 2 ) cosh 2 ( β b l c e ( r 2 Mr p ) 2 ) r dr
+ M 3 r max Mr max e r 2 M 2 ( 1 w 2 + 1 r s 2 ) cosh 2 ( β f l c e ( r 2 Mr p ) 2 ) r dr ] · ( 1 2 n s c ε 0 ) · 2 π E start 0 2 R max e 4 α l c M 2
β b = γ β f · [ 1 2 ( MR 2 n s ) 2 r 2 ]
P m = P m 1 { R max e 4 α l c [ e 2 β f l c ( 1 + γ ) 16 r 1 2 r s 2 ( 1 e ( M 3 r max Mr 1 ) 2 ) + e 2 β f l c γ 8 r 2 2 r s 2 ( 1 e ( M 3 r max Mr 2 ) 2 ) +
e 2 β f l c 8 r 3 2 r s 2 ( 1 + e ( M 3 r max Mr 3 ) 2 2 e ( r max r 3 ) 2 ) + 1 4 r 4 2 r s 2 ( 1 + e ( M 3 r max Mr 4 ) 2 2 e ( r max r 4 ) 2 ) ] }
{ 1 r 1 2 = 1 w 2 + 1 r s 2 + β f l c ( 1 + γ ) r p 2 + 4 β f l c γ ( R 2 n s ) 2 1 r 2 2 = 1 w 2 + 1 r s 2 + β f l c γ r p 2 + 4 β f l c γ ( R 2 n 2 ) 2 1 r 3 2 = 1 w 2 + 1 r s 2 + β f l c r p 2 1 r 4 2 = 1 w 2 + 1 r s 2
Q = 0 0 I p ( r , t ) dt 2 π r dr = I p 0 π 1.5 r p 2 τ p
R eff = R max M 2

Metrics