Abstract

In a strongly turbulent medium, the scintillation index of flat-topped Gaussian beams is derived and evaluated. In the formulation, unified solution of Rytov method is utilized. Our results correctly reduce to the existing strong turbulence scintillation index of the Gaussian beam, and naturally to spherical and plane wave scintillations. Another checkpoint of our result is the scintillation index of flat-topped Gaussian beams in weak turbulence. Regardless of the order of flatness, scintillations of flat-topped Gaussian beams in strong turbulence are found to be determined mainly by the small-scale effects. For large-sized beams in moderate and strongly turbulent medium, flatter beams exhibit smaller scintillations.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).
  2. V. A. Banakh, G. M. Krekov, V. L. Mironov, S. S. Khmelevtsov, and R. S. Tsvik, “Focused-laser-beam scintillations in the turbulent atmosphere,” J. Opt. Soc. Am. 64, 516–518 (1974).
    [CrossRef]
  3. R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
    [CrossRef]
  4. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vol.  2.
  5. G. Y. Patrushev, “Fluctuations of the field of a wave beam on reflection in a turbulent atmosphere,” Sov. J. Quantum Electron. 8, 1315–1318 (1978).
    [CrossRef]
  6. R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” Proc. IEEE 68, 1424–1443 (1980).
    [CrossRef]
  7. J. C. Leader, “Beam-intensity fluctuations in atmospheric turbulence,” J. Opt. Soc. Am. 71, 542–558 (1981).
    [CrossRef]
  8. W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A 11, 2719–2726 (1994).
    [CrossRef]
  9. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001).
    [CrossRef]
  10. L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave,” Waves Random Media 11, 271–291 (2001).
    [CrossRef]
  11. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).
    [CrossRef]
  12. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
    [CrossRef]
  13. Y. Li, “Light beams with flat-topped profiles,” Opt. Lett. 27, 1007–1009 (2002).
    [CrossRef]
  14. Y. Li, “New expressions for flat-topped light beams,” Opt. Commun. 206, 225–234 (2002).
    [CrossRef]
  15. Y. Cai and Q. Lin, “Light beams with elliptical flat-topped profiles,” J. Opt. A: Pure Appl. Opt. 6, 390–395 (2004).
    [CrossRef]
  16. Y. Baykal and H. T. Eyyuboğlu, “Scintillation index of flat-topped-Gaussian beams,” Appl. Opt. 45, 3793–3797 (2006).
    [CrossRef] [PubMed]
  17. D. C. Cowan, J. Recolons, L. C. Andrews, and C. Y. Young, “Propagation of flattened Gaussian beams in the atmosphere: a comparison of theory with a computer simulation model,” Proc. SPIE 6215, 62150B (2006).
    [CrossRef]
  18. Y. Baykal and H. T. Eyyuboğlu, “Scintillations of incoherent flat-topped Gaussian source field in turbulence,” Appl. Opt. 46, 5044–5050 (2007).
    [CrossRef] [PubMed]
  19. R. L. Fante, “Some physical insights into beam propagation in strong turbulence,” Radio Sci. 15, 757–762 (1980).
    [CrossRef]
  20. R. J. Hill, “Theory of saturation of optical scintillation by strong turbulence: plane-wave variance and covariance and spherical-wave covariance,” J. Opt. Soc. Am. 72, 212–222 (1982).
    [CrossRef]
  21. L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16, 1417–1429 (1999).
    [CrossRef]
  22. I. B. Djordjevic, G. T. Djordjevic, “On the communication over strong atmospheric turbulence channels by adaptive modulation and coding,” Opt. Express 17, 18250–18262 (2009).
    [CrossRef] [PubMed]
  23. Y. Ren, A. Dang, B. Luo, and H. Guo, “Capacities for long-distance free-space optical links under beam wander effects,” IEEE Photon. Technol. Lett. 22, 1069–1071 (2010).
    [CrossRef]
  24. H. Gerçekcioğlu, Y. Baykal, and C. Nakiboğlu, “Annular beam scintillations in strong turbulence,” J. Opt. Soc. Am. A 27, 1834–1839 (2010).
    [CrossRef]
  25. Y. Baykal, “Correlation and structure functions of Hermite-sinusoidal-Gaussian laser beams in a turbulent atmosphere,” J. Opt. Soc. Am. A 21, 1290–1299 (2004).
    [CrossRef]
  26. M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillations in a turbulent medium,” Waves Random Media 4, 243–273(1994).
    [CrossRef]
  27. Y. Baykal, H. T. Eyyuboğlu, and Y. Cai, “Scintillations of partially coherent multiple Gaussian beams in turbulence,” Appl. Opt. 48, 1943–1954 (2009).
    [CrossRef] [PubMed]

2010

Y. Ren, A. Dang, B. Luo, and H. Guo, “Capacities for long-distance free-space optical links under beam wander effects,” IEEE Photon. Technol. Lett. 22, 1069–1071 (2010).
[CrossRef]

H. Gerçekcioğlu, Y. Baykal, and C. Nakiboğlu, “Annular beam scintillations in strong turbulence,” J. Opt. Soc. Am. A 27, 1834–1839 (2010).
[CrossRef]

2009

2007

2006

Y. Baykal and H. T. Eyyuboğlu, “Scintillation index of flat-topped-Gaussian beams,” Appl. Opt. 45, 3793–3797 (2006).
[CrossRef] [PubMed]

D. C. Cowan, J. Recolons, L. C. Andrews, and C. Y. Young, “Propagation of flattened Gaussian beams in the atmosphere: a comparison of theory with a computer simulation model,” Proc. SPIE 6215, 62150B (2006).
[CrossRef]

2004

Y. Cai and Q. Lin, “Light beams with elliptical flat-topped profiles,” J. Opt. A: Pure Appl. Opt. 6, 390–395 (2004).
[CrossRef]

Y. Baykal, “Correlation and structure functions of Hermite-sinusoidal-Gaussian laser beams in a turbulent atmosphere,” J. Opt. Soc. Am. A 21, 1290–1299 (2004).
[CrossRef]

2002

Y. Li, “Light beams with flat-topped profiles,” Opt. Lett. 27, 1007–1009 (2002).
[CrossRef]

Y. Li, “New expressions for flat-topped light beams,” Opt. Commun. 206, 225–234 (2002).
[CrossRef]

2001

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave,” Waves Random Media 11, 271–291 (2001).
[CrossRef]

1999

1994

M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillations in a turbulent medium,” Waves Random Media 4, 243–273(1994).
[CrossRef]

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A 11, 2719–2726 (1994).
[CrossRef]

1982

1981

1980

R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” Proc. IEEE 68, 1424–1443 (1980).
[CrossRef]

R. L. Fante, “Some physical insights into beam propagation in strong turbulence,” Radio Sci. 15, 757–762 (1980).
[CrossRef]

1978

G. Y. Patrushev, “Fluctuations of the field of a wave beam on reflection in a turbulent atmosphere,” Sov. J. Quantum Electron. 8, 1315–1318 (1978).
[CrossRef]

1975

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
[CrossRef]

1974

Al-Habash, M. A.

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave,” Waves Random Media 11, 271–291 (2001).
[CrossRef]

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16, 1417–1429 (1999).
[CrossRef]

Andrews, L. C.

D. C. Cowan, J. Recolons, L. C. Andrews, and C. Y. Young, “Propagation of flattened Gaussian beams in the atmosphere: a comparison of theory with a computer simulation model,” Proc. SPIE 6215, 62150B (2006).
[CrossRef]

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave,” Waves Random Media 11, 271–291 (2001).
[CrossRef]

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16, 1417–1429 (1999).
[CrossRef]

W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A 11, 2719–2726 (1994).
[CrossRef]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).
[CrossRef]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001).
[CrossRef]

Banakh, V. A.

Baykal, Y.

Cai, Y.

Charnotskii, M. I.

M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillations in a turbulent medium,” Waves Random Media 4, 243–273(1994).
[CrossRef]

Cowan, D. C.

D. C. Cowan, J. Recolons, L. C. Andrews, and C. Y. Young, “Propagation of flattened Gaussian beams in the atmosphere: a comparison of theory with a computer simulation model,” Proc. SPIE 6215, 62150B (2006).
[CrossRef]

Dang, A.

Y. Ren, A. Dang, B. Luo, and H. Guo, “Capacities for long-distance free-space optical links under beam wander effects,” IEEE Photon. Technol. Lett. 22, 1069–1071 (2010).
[CrossRef]

Djordjevic, G. T.

Djordjevic, I. B.

Eyyuboglu, H. T.

Fante, R. L.

R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” Proc. IEEE 68, 1424–1443 (1980).
[CrossRef]

R. L. Fante, “Some physical insights into beam propagation in strong turbulence,” Radio Sci. 15, 757–762 (1980).
[CrossRef]

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
[CrossRef]

Gerçekcioglu, H.

Gori, F.

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

Guo, H.

Y. Ren, A. Dang, B. Luo, and H. Guo, “Capacities for long-distance free-space optical links under beam wander effects,” IEEE Photon. Technol. Lett. 22, 1069–1071 (2010).
[CrossRef]

Hill, R. J.

Hopen, C. Y.

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave,” Waves Random Media 11, 271–291 (2001).
[CrossRef]

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16, 1417–1429 (1999).
[CrossRef]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vol.  2.

Khmelevtsov, S. S.

Krekov, G. M.

Leader, J. C.

Li, Y.

Y. Li, “New expressions for flat-topped light beams,” Opt. Commun. 206, 225–234 (2002).
[CrossRef]

Y. Li, “Light beams with flat-topped profiles,” Opt. Lett. 27, 1007–1009 (2002).
[CrossRef]

Lin, Q.

Y. Cai and Q. Lin, “Light beams with elliptical flat-topped profiles,” J. Opt. A: Pure Appl. Opt. 6, 390–395 (2004).
[CrossRef]

Luo, B.

Y. Ren, A. Dang, B. Luo, and H. Guo, “Capacities for long-distance free-space optical links under beam wander effects,” IEEE Photon. Technol. Lett. 22, 1069–1071 (2010).
[CrossRef]

Miller, W. B.

Mironov, V. L.

Nakiboglu, C.

Patrushev, G. Y.

G. Y. Patrushev, “Fluctuations of the field of a wave beam on reflection in a turbulent atmosphere,” Sov. J. Quantum Electron. 8, 1315–1318 (1978).
[CrossRef]

Phillips, R. L.

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave,” Waves Random Media 11, 271–291 (2001).
[CrossRef]

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16, 1417–1429 (1999).
[CrossRef]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).
[CrossRef]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001).
[CrossRef]

Recolons, J.

D. C. Cowan, J. Recolons, L. C. Andrews, and C. Y. Young, “Propagation of flattened Gaussian beams in the atmosphere: a comparison of theory with a computer simulation model,” Proc. SPIE 6215, 62150B (2006).
[CrossRef]

Ren, Y.

Y. Ren, A. Dang, B. Luo, and H. Guo, “Capacities for long-distance free-space optical links under beam wander effects,” IEEE Photon. Technol. Lett. 22, 1069–1071 (2010).
[CrossRef]

Ricklin, J. C.

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

Tsvik, R. S.

Young, C. Y.

D. C. Cowan, J. Recolons, L. C. Andrews, and C. Y. Young, “Propagation of flattened Gaussian beams in the atmosphere: a comparison of theory with a computer simulation model,” Proc. SPIE 6215, 62150B (2006).
[CrossRef]

Appl. Opt.

IEEE Photon. Technol. Lett.

Y. Ren, A. Dang, B. Luo, and H. Guo, “Capacities for long-distance free-space optical links under beam wander effects,” IEEE Photon. Technol. Lett. 22, 1069–1071 (2010).
[CrossRef]

J. Opt. A: Pure Appl. Opt.

Y. Cai and Q. Lin, “Light beams with elliptical flat-topped profiles,” J. Opt. A: Pure Appl. Opt. 6, 390–395 (2004).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

Y. Li, “New expressions for flat-topped light beams,” Opt. Commun. 206, 225–234 (2002).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. IEEE

R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” Proc. IEEE 68, 1424–1443 (1980).
[CrossRef]

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
[CrossRef]

Proc. SPIE

D. C. Cowan, J. Recolons, L. C. Andrews, and C. Y. Young, “Propagation of flattened Gaussian beams in the atmosphere: a comparison of theory with a computer simulation model,” Proc. SPIE 6215, 62150B (2006).
[CrossRef]

Radio Sci.

R. L. Fante, “Some physical insights into beam propagation in strong turbulence,” Radio Sci. 15, 757–762 (1980).
[CrossRef]

Sov. J. Quantum Electron.

G. Y. Patrushev, “Fluctuations of the field of a wave beam on reflection in a turbulent atmosphere,” Sov. J. Quantum Electron. 8, 1315–1318 (1978).
[CrossRef]

Waves Random Media

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave,” Waves Random Media 11, 271–291 (2001).
[CrossRef]

M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillations in a turbulent medium,” Waves Random Media 4, 243–273(1994).
[CrossRef]

Other

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).
[CrossRef]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001).
[CrossRef]

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vol.  2.

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 (7)

Fig. 1
Fig. 1

Scintillation index in strong turbulence versus the square root of the Rytov plane wave scintillation index for plane, spherical, and flat-topped Gaussian beams ( N = 1 ) with α s = 1 cm and α s = 2 cm .

Fig. 2
Fig. 2

Scintillation index in strong turbulence of flat-topped Gaussian beams at N = 1 , 5 , 15 and α s = 1 cm with the large- and small-scale scintillation indices versus (a)  α R , (b)  N Fn 1 .

Fig. 3
Fig. 3

Scintillation index in strong turbulence of flat-topped Gaussian beams at N = 1 , 5 , 10 , 15 and α s = 2 cm with the large- and small-scale scintillation indices versus (a)  α R , (b)  N Fn 1 .

Fig. 4
Fig. 4

Scintillation index in strong turbulence of flat-topped beams at N = 5 at selected values of source size.

Fig. 5
Fig. 5

Scintillation index in strong turbulence of flat-topped beams with N = 10 at selected values of source size.

Fig. 6
Fig. 6

Scintillation index in strong turbulence versus the source size of flat-topped Gaussian beams with N = 1 , 5 , 10 , 15 at L = 1 km .

Fig. 7
Fig. 7

Scintillation index in strong turbulence versus the source size of flat-topped Gaussian beams with N = 1 , 5 , 10 , 15 at L = 2 km .

Equations (12)

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

m wt 2 = 4 π Re { 0 L d η 0 κ d κ 0 2 π d θ [ G 1 ( η , κ , θ ) + G 2 ( η , κ , θ ) ] Φ n ( κ ) } .
G 1 ( L , η , κ ) = k 2 D 2 ( L ) n 1 = 1 N n 2 = 1 N ( 1 ) n 1 + n 2 ( 1 + i α n 1 L ) ( 1 + i α n 2 L ) ( N n 1 ) ( N n 2 ) exp [ i ( L η ) ( 1 + i α n 1 η ) κ 2 2 k ( 1 + i α n 1 L ) ] × exp [ i ( L η ) ( 1 + i α n 2 η ) κ 2 2 k ( 1 + i α n 2 L ) ]
G 2 ( L , η , κ ) = k 2 | D ( L ) | 2 n 1 = 1 N n 2 = 1 N ( 1 ) n 1 + n 2 ( N n 1 ) ( N n 2 ) 1 ( 1 + i α n 1 L ) ( 1 i α n 2 L ) exp [ i ( L η ) ( 1 + i α n 1 η ) κ 2 2 k ( 1 + i α n 1 L ) ] × exp [ i ( L η ) ( 1 i α n 2 η ) κ 2 2 k ( 1 i α n 2 L ) ] ,
D ( L ) = n = 1 N ( 1 ) n 1 ( N n ) 1 ( 1 + i α n L ) ,
u s ( s x , s y ) = n = 1 N ( 1 ) n 1 N ( N n ) exp { [ n ( s x 2 + s y 2 ) 2 α s 2 ] } ,
Φ n , e ( κ ) = 0.033 C n 2 κ 11 / 3 [ exp ( κ 2 κ x 2 ) + κ 11 / 3 ( κ 2 + κ y 2 ) 11 / 6 ] ,
m LS 2 = 8.7 k 2 C n 2 Re { D 2 ( L ) n 1 = 1 N n 2 = 1 N ( 1 ) n 1 + n 2 ( N n 1 ) ( N n 2 ) 1 ( 1 + i α n 1 L ) ( 1 + i α n 2 L ) × 0 L d η [ i ( L η ) 2 k ( 1 + i α n 1 η 1 + i α n 1 L + 1 + i α n 2 η 1 + i α n 2 L ) + 0.5 ( 1 κ x n 1 2 + 1 κ x n 2 2 ) ] 5 / 6 | D ( L ) | 2 n 1 = 1 N n 2 = 1 N ( 1 ) n 1 + n 2 ( N n 1 ) ( N n 2 ) 1 ( 1 + i α n 1 L ) ( 1 i α n 2 L ) × 0 L d η [ i ( L η ) 2 k ( 1 + i α n 1 η 1 + i α n 1 L - 1 i α n 2 η 1 i α n 2 L ) + 0.5 ( 1 κ x n 1 2 + 1 κ x n 2 2 ) ] 5 / 6 } ,
m SS 2 = 1.39 k 2 C n 2 Re { | D ( L ) | 2 n 1 = 1 N n 2 = 1 N ( 1 ) n 1 + n 2 ( 1 + i α n 1 L ) ( 1 i α n 2 L ) ( N n 1 ) ( N n 2 ) n = 0 Γ ( n + 5 / 6 ) ( κ y n 1 κ y n 2 ) n 5 / 6 × 0 L d η ( i ( L η ) ( 1 + i α n 1 η ) 2 k ( 1 + i α n 1 L ) + i ( L η ) ( 1 i α n 2 η ) 2 k ( 1 i α n 2 L ) ) n D 2 ( L ) n 1 = 1 N n 2 = 1 N ( 1 ) n 1 + n 2 ( 1 + i α n 1 L ) ( 1 + i α n 2 L ) ( N n 1 ) × ( N n 2 ) n = 0 Γ ( n + 5 / 6 ) ( κ y n 1 κ y n 2 ) n 5 / 6 0 L d η ( i ( L η ) ( 1 + i α n 1 η ) 2 k ( 1 + i α n 1 L ) i ( L η ) ( 1 + i α n 2 η ) 2 k ( 1 + i α n 2 L ) ) n } .
κ x = B 0.5 { β ( σ R m ) 12 / 7 + 1.12 β σ R 12 / 5 [ 1 + 2.17 ( B 2 α s 4 + B 2 ) ] 6 / 7 } 1 / 2 ,
κ y = B 0.5 [ 3 ( σ R m ) 12 / 5 + 2.07 σ R 12 / 5 ] 0 . 5 ,
β = [ 1 3 0.5 B 2 α s 4 + B 2 + 0.2 ( B 2 α s 4 + B 2 ) 2 ] 6 / 7 , and B = L / k .
m 2 = exp ( m LS 2 + m SS 2 ) 1.

Metrics