Abstract

The irradiance of a partially coherent light propagated under the influence of multiple random effects is shown to be the convolution of the irradiance propagated in a vacuum with the system’s point spread function representing the random effects. This is true regardless of whether the propagation is far-field or not. We also show that the far-field irradiance of any laser system, regardless of complexity, can be expressed in terms of three basic parameters; laser power, field area, and a pupil factor. A general analytical formula for the far-field irradiance distribution for partially coherent laser sources of any complexity is derived. The formula includes multiple random effects including strong turbulence, random beam jitter, partial coherence, in addition to laser system pupil effects. An efficient matrix based numerical solution is also developed to verify the accuracy of the formula. Applications to the propagation of clipped Gaussian or flat-top beams with an obscuration, both as a single beam or an array of beams, are shown to give accurate results over the whole range of weak to strong turbulence as compared to numerical modeling.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article

Corrections

Sami A. Shakir, Timothy T. Clark, Daniel S. Cargill, and Richard Carreras, "Far-field propagation of partially coherent laser light in random mediums: erratum," Opt. Express 26, 21019-21019 (2018)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-26-16-21019

OSA Recommended Articles
General wave optics propagation scaling law

Sami A. Shakir, Thomas M. Dolash, Mark Spencer, Richard Berdine, Daniel S. Cargill, and Richard Carreras
J. Opt. Soc. Am. A 33(12) 2477-2484 (2016)

Propagation characteristics of a partially coherent self-shifting beam in random media

Yuyan Wang, Zhangrong Mei, Ming Zhang, and Yonghua Mao
Appl. Opt. 59(7) 1834-1840 (2020)

Analyzing the propagation behavior of scintillation index and bit error rate of a partially coherent flat-topped laser beam in oceanic turbulence

Masoud Yousefi, Shole Golmohammady, Ahmad Mashal, and Fatemeh Dabbagh Kashani
J. Opt. Soc. Am. A 32(11) 1982-1992 (2015)

References

  • View by:
  • |
  • |
  • |

  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  2. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley & Sons, New York, 1978).
  3. J. W. Goodman, Statistical Optics (Wiley & Sons, New York, 1985).
  4. B. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, New York, 1991).
  5. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd. ed. (SPIE Optical Engineering, Bellingham, Wash., 2005).
  6. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, Piscataway, NJ, 1997); [previously published as Vols I & II by Academic, New York (1978)].
  7. F. Wang and O. Korotkova, “Convolution approach for beam propagation in random media,” Opt. Lett. 41(7), 1546–1549 (2016).
    [Crossref] [PubMed]
  8. R. S. Lawrence and J. W. Strohbehn, “A survey of clear air propagation effects relevant to optical communications,” Proc. IEEE 58(10), 1523–1545 (1970).
    [Crossref]
  9. J. W. Strohbehn, ed., Laser Beam Propagation in the Atmosphere (Springer, New York, 1978).
  10. H. Weichel, Laser Beam Propagation in the Atmosphere (SPIE Optical Engineering, Bellingham, WA, 1990).
  11. W. P. Brown, “Second moment of a wave propagating in a random medium,” J. Opt. Soc. Am. 61(8), 1051–1059 (1971).
    [Crossref]
  12. R. Lutomirski and H. Yura, “Wave structure function and mutual coherence function of an optical wave in a turbulent atmosphere,” J. Opt. Soc. Am. 61(4), 482–487 (1971).
    [Crossref]
  13. S. M. Wandzura, “Meaning of quadratic structure functions,” J. Opt. Soc. Am. 70(6), 745–747 (1980).
    [Crossref]
  14. G. Gbur, “Partially coherent beam propagation in atmospheric turbulence [invited],” J. Opt. Soc. Am. A 31(9), 2038–2045 (2014).
    [Crossref] [PubMed]
  15. J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19(9), 1794–1802 (2002).
    [Crossref] [PubMed]
  16. A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982).
    [Crossref]
  17. Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review [invited],” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014).
    [Crossref] [PubMed]
  18. F. Wang, X. Liu, and Y. Cai, “Propagation of partially coherent beam in turbulent atmosphere: a review (Invited Review),” Prog. Electromagnetics Res. 150, 123–143 (2015).
    [Crossref]
  19. T. L. Ho and M. J. Beran, “Propagation of the Fourth-Order coherence function in a random medium,” J. Opt. Soc. Am. 58(10), 1335–1341 (1968).
    [Crossref]
  20. M. Charnotskii, “Common omissions and misconceptions of wave propagation in turbulence: discussion,” J. Opt. Soc. Am. A 29(5), 711–721 (2012).
    [Crossref] [PubMed]
  21. S. A. Shakir, T. M. Dolash, M. Spencer, R. Berdine, D. S. Cargill, and R. Carreras, “General wave optics propagation scaling law,” J. Opt. Soc. Am. A 33(12), 2477–2484 (2016).
    [Crossref] [PubMed]
  22. S. A. Shakir, D. L. Fried, E. A. Pease, T. J. Brennan, and T. M. Dolash, “Efficient matrix approach to optical wave propagation and Linear Canonical Transforms,” Opt. Express 23(20), 26853–26862 (2015).
    [Crossref] [PubMed]

2016 (2)

2015 (2)

S. A. Shakir, D. L. Fried, E. A. Pease, T. J. Brennan, and T. M. Dolash, “Efficient matrix approach to optical wave propagation and Linear Canonical Transforms,” Opt. Express 23(20), 26853–26862 (2015).
[Crossref] [PubMed]

F. Wang, X. Liu, and Y. Cai, “Propagation of partially coherent beam in turbulent atmosphere: a review (Invited Review),” Prog. Electromagnetics Res. 150, 123–143 (2015).
[Crossref]

2014 (2)

2012 (1)

2002 (1)

1982 (1)

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982).
[Crossref]

1980 (1)

1971 (2)

1970 (1)

R. S. Lawrence and J. W. Strohbehn, “A survey of clear air propagation effects relevant to optical communications,” Proc. IEEE 58(10), 1523–1545 (1970).
[Crossref]

1968 (1)

Beran, M. J.

Berdine, R.

Brennan, T. J.

Brown, W. P.

Cai, Y.

F. Wang, X. Liu, and Y. Cai, “Propagation of partially coherent beam in turbulent atmosphere: a review (Invited Review),” Prog. Electromagnetics Res. 150, 123–143 (2015).
[Crossref]

Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review [invited],” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014).
[Crossref] [PubMed]

Cargill, D. S.

Carreras, R.

Charnotskii, M.

Chen, Y.

Davidson, F. M.

Dolash, T. M.

Friberg, A. T.

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982).
[Crossref]

Fried, D. L.

Gbur, G.

Ho, T. L.

Korotkova, O.

Lawrence, R. S.

R. S. Lawrence and J. W. Strohbehn, “A survey of clear air propagation effects relevant to optical communications,” Proc. IEEE 58(10), 1523–1545 (1970).
[Crossref]

Liu, X.

F. Wang, X. Liu, and Y. Cai, “Propagation of partially coherent beam in turbulent atmosphere: a review (Invited Review),” Prog. Electromagnetics Res. 150, 123–143 (2015).
[Crossref]

Lutomirski, R.

Pease, E. A.

Ricklin, J. C.

Shakir, S. A.

Spencer, M.

Strohbehn, J. W.

R. S. Lawrence and J. W. Strohbehn, “A survey of clear air propagation effects relevant to optical communications,” Proc. IEEE 58(10), 1523–1545 (1970).
[Crossref]

Sudol, R. J.

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982).
[Crossref]

Wandzura, S. M.

Wang, F.

Yura, H.

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (5)

Opt. Commun. (1)

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Proc. IEEE (1)

R. S. Lawrence and J. W. Strohbehn, “A survey of clear air propagation effects relevant to optical communications,” Proc. IEEE 58(10), 1523–1545 (1970).
[Crossref]

Prog. Electromagnetics Res. (1)

F. Wang, X. Liu, and Y. Cai, “Propagation of partially coherent beam in turbulent atmosphere: a review (Invited Review),” Prog. Electromagnetics Res. 150, 123–143 (2015).
[Crossref]

Other (8)

J. W. Strohbehn, ed., Laser Beam Propagation in the Atmosphere (Springer, New York, 1978).

H. Weichel, Laser Beam Propagation in the Atmosphere (SPIE Optical Engineering, Bellingham, WA, 1990).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley & Sons, New York, 1978).

J. W. Goodman, Statistical Optics (Wiley & Sons, New York, 1985).

B. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, New York, 1991).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd. ed. (SPIE Optical Engineering, Bellingham, Wash., 2005).

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, Piscataway, NJ, 1997); [previously published as Vols I & II by Academic, New York (1978)].

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 (4)

Fig. 1
Fig. 1 Comparison of the far-field irradiance distribution for a uniform beam (flat-top) with an obscuration to a Gaussian beam (dashed blue line). Both have the same wavelength, field area Ao and laser power PL, hence the same peak-far-field irradiance. (see text for details).
Fig. 2
Fig. 2 Peak mean far-field irradiance as a function of propagation range for three different source cases: a flat-top beam, a Gaussian beam, and a flat-top with obscuration. Parameters { P L , F o , A o } are chosen to have the same product.
Fig. 3
Fig. 3 (a) Peak far-field irradiance vs. propagation range for three turbulence strengths. (b) Irradiance profile vs. x-axis at a range of 10 km. In both (a) an (b) solid curves represent numerical simulation using XEMA code and crosses are predictions of Eq. (30).
Fig. 4
Fig. 4 The peak far-field irradiance vs. propagation range for an array of four flat-top beams with an individual beam radius of 5cm and a wavelength of 1.0μm. The three turbulence scenarios correspond to a refractive index structure constant values C n 2 =0 , 10 15 m 2/3 and 5x 10 15 m 2/3 . Solid curves represent numerical simulation using XEMA and crosses are predictions of Eq. (30)

Tables (1)

Tables Icon

Table 1 Field effective area and pupil factor for various laser types and geometries

Equations (40)

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

U(r,z)= U o ( ρ 1 ,0)G( ρ 1 ,r,z) exp( ψ( ρ 1 ,r,z) ) d 2 ρ
G( ρ 1 ,r,z)= i λz exp( i2πz/λ )exp[ i π λz | ρ 1 r | 2 ]
I(r,z) = U(r,z) U * (r,z) = U ( ρ 1 ) o U o * ( ρ 2 )G(r, ρ 1 ,z) G * (r, ρ 2 ,z) M 2 [ | ρ 2 ρ 1 |,z ] d 2 ρ 1 d 2 ρ 2
M 2 (| ρ 2 ρ 1 |,z)= exp[ ψ(r, ρ 1 ,z)+ ψ (r, ρ 2 ,z) ]
M 2 (| ρ 2 ρ 1 |)=exp[ 1 2 D sph (| ρ 2 ρ 1 |) ]
ψ( ρ 1 ,r,z)= m=1 M ψ m ( ρ 1 ,r,z)
M 2 (ρ)=exp[ 1 2 m=1 M D m (ρ/ ρ m ) ]= m=1 M MT F m (ρ/ ρ m )
MT F jitter (ρ)=exp[ ( ρ/ ρ jitter ) 2 ]
MT F turb (ρ)=exp[ ( ρ/ ρ sph ) 5/3 ]
ρ sph ( z )= [ 1.46 k 2 0 z C n 2 (z') ( z'/z ) 5/3 dz' ] 3/5 , l o ρ sph L o .
U ' o ( ρ 1 )= U o ( ρ 1 )exp[ iϕ( ρ 1 ) ]
MT F PC (ρ)=exp[ ( ρ/ ρ PC ) 2 ]
I(r,z) = I o (r,z) P M (r,z)
P M (r,z)=F.T. { m=1 M MT F m (ρ/ ρ m ) } r/λz
I(r,z) = 1 λ 2 z 2 U ˜ ( ρ 1 ) o U ˜ o * ( ρ 2 ) exp[ 1 2 m=1 M D m ( | ρ 2 ρ 1 | ) ] exp[ i2πr( ρ 1 ρ 2 ) ] d 2 ρ 1 d 2 ρ 2
U ˜ o (ρ)= U o (ρ)exp[ iπ ρ 2 / λ 2 z 2 ]
I(r,z) = 1 λ 2 z 2 F( ρ ) M 2 ( ρ )exp[ i2πρr/λz ] d 2 ρ
F( ρ )= U ˜ ( ρ 1 ) o U ˜ o * ( ρ 1 ρ) d 2 ρ 1
I o (r,z)= 1 λ 2 z 2 F( ρ ) exp[ i2πρr/λz ] d 2 ρ.
I pk (z)= P L F o 2 A o λ 2 z 2
I Pk (z)= 1 λ 2 z 2 | U O ( x 1 , y 1 )p( x 1 , y 1 )d x 1 d y 1 | 2
A o = 1 P L [ | U O ( x 1 , y 1 ) |d x 1 d y 1 ] 2
F o =| U O ( x 1 , y 1 )p( x 1 , y 1 )d x 1 d y 1 |/ | U O ( x 1 , y 1 ) |d x 1 d y 1
I pk_FF (z)= P L A o F o 2 S/ λ 2 z 2
I(0,z) = 1 λ 2 z 2 F( ρ ) exp[ 1 2 m=1 M D m ( ρ ) ] d 2 ρ
I(0,z) = 1 λ 2 z 2 exp[ 1 2 m=1 M D m ( ρ c ) ] F( ρ ) d 2 ρ
I(r,z)= 2 P L π W eff (z) 2 Sexp[ 2 r 2 / W eff (z) 2 ]
W eff (z)= w o { ( 1z/ f L ) 2 + ( λz/π w o 2 ) 2 } 1/2 S 1/2
S=1+ A o / A GS + A o / A jit +0.9 ( A o / A turb ) 5/6 +0.175 ( A o / A turb ) 5/4
I(r,z)=[ P L F o 2 A o S/ λ 2 z 2 ]exp[ 2 r 2 / W eff (z) 2 ]
W eff (z)=λz/ 0.5π A o S
U= H T U o H
I = J T ( Q.D )J
J(x, ρ x )= Δ ρ x λz sinc( xΔ ρ x λz )exp[ i 2π λz ρ x x ]
I(r,z) = ( 1/λz ) 2 d 2 ρ d 2 R exp[ 2 R 2 w o 2 ρ 2 w o 2 ] exp[ m=1 M ( ρ/ ρ m ) α m ] x exp[ i2π λz (1z/ f L )R·ρ i2π λz r·ρ ]
I( r,z ) = 2π P L λ 2 z 2 0 ρ J o ( 2πrρ/λz )exp[ π 2 w z 2 2 λ 2 z 2 ρ 2 ] exp[ m=1 M ( ρ/ ρ m ) α m ]
I( r,z ) = 4 P L π w z 2 0 t J o ( 2 2 rt/ w z )exp[ t 2 ] exp[ m=1 M ( ta/ ρ m ) α m ]
0 t J o ( bt )exp[ c 2 t 2 ] dt= 1 2 c 2 exp[ b 2 /4 c 2 ]
I( r,z ) = 2 P L π W eff 2 exp[ 2 r 2 / W eff 2 ]
W eff 2 = w o 2 [ ( 1z/ f L ) 2 + ( λz/π w o 2 ) 2 ]/S

Metrics