Abstract

The first-order Rytov approximation predicts that the probability-density function for intensity fluctuations in a random medium should be a log normal distribution. Here the corrections to this prediction that arise from the second term in the Rytov series are explored. The primary effect is to skew the distribution so as to favor values that are less than the average. This effect is controlled by the Rytov variance β02 alone, and the predicted distribution contains no adjustable parameters. The theoretical result is compared with numerical simulations for weak scattering of plane and spherical waves. The agreement is quite good unless the intensity fluctuations are very large or very small relative to the mean irradiance. In those ranges, the predictions require additional terms in the Rytov expansion.

© 2001 Optical Society of America

Full Article  |  PDF Article

Errata

Albert D. Wheelon, "Skewed distribution of irradiance predicted by the second-order Rytov approximation: errata," J. Opt. Soc. Am. A 19, 414-414 (2002)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-19-2-414

References

  • View by:
  • |
  • |
  • |

  1. S. M. Rytov, “Diffraction of light by ultrasonic waves,” Izv. Akad. Nauk SSSR Ser. Fiz. 2, 223–259 (1937).
  2. A. M. Obukhov, “The influence of weak atmospheric inhomogeneities upon the propagation of sound and light,” Izv. Akad. Nauk. SSSR Ser. Geogr. Geofiz. 2, 155–165 (1953); English translation by W. C. Hoffman, published as (U.S. Air Force Project RAND, Santa Monica, Calif., July28, 1955).
  3. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Office, Springfield, Va.1971), p. 292.
  4. V. I. Tatarskii, “Second approximation in the problem of wave propagation in media with random irregularities,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 5, 490–507 (1962); [Sov. Radiophys. 5, 164–202 (1962)].
  5. H. T. Yura, “Optical propagation through a turbulent medium,” J. Opt. Soc. Am. A 59, 111–112 (1969).
    [CrossRef]
  6. R. A. Schmeltzer, “Means, variances and covariances for laser propagation through a random medium,” Q. Appl. Math. 24, 339–354 (1967).
  7. Ref. 3, pp. 253–258.
  8. H. T. Yura, C. C. Sung, S. F. Clifford, R. J. Hill, “Second order Rytov approximation,” J. Opt. Soc. Am. 73, 500–502 (1983).
    [CrossRef]
  9. A. D. Wheelon, Electromagnetic Scintillation, Vol. One: Geometrical Optics (Cambridge U. Press, Cambridge, UK, 2001), Problem 3, Chap. 2.
  10. A. N. Malakhov, Cumulant Analysis of Random Non-Gaussian Processes and Their Transformations (in Russian, Soviet Radio, Moscow, 1978). Unfortunately, there appears to be no English translation of this valuable reference.
  11. R. J. Hill, G. R. Ochs, “Inner-scale dependence of scintillation variances measured in weak scintillation,” J. Opt. Soc. Am. A 9, 1406–1411 (1992).
    [CrossRef]
  12. J. M. Martin, S. M. Flatté, “Intensity images andstatistics from numerical simulation of wave propagation in 3-D random media,” Appl. Opt. 27, 2111–2126 (1988).
    [CrossRef] [PubMed]
  13. S. M. Flatté, C. Bracher, G. Y. Wang, “Probability-density functions of irradiance for waves in atmospheric turbulence calculated by numerical simulation,” J. Opt. Soc. Am. A 11, 2080–2092 (1994).
    [CrossRef]
  14. R. J. Hill, R. G. Frehlich, “Probability distribution of irradiance for the onset of strong scintillation,” J. Opt. Soc. Am. A 14, 1530–1540 (1997).
    [CrossRef]
  15. R. J. Hill, R. G. Frehlich, W. D. Otto, “The probability distribution of irradiance scintillation,” (National Technical Information Service, Springfield, Va.1997).
  16. R. G. Frehlich (Cooperative Institute for Research in Environmental Sciences, University of Colorado at Boulder, Campus Box 216, Boulder, Colorado 80309-1149, rgf@cires.colorado.edu) (personal communication, November26, 1997).
  17. R. M. Manning, “Beam wave propagation within the second Rytov perturbation approximation,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 39, 423–436 (1996).
  18. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, Cambridge, UK, 1952).

1997 (1)

1996 (1)

R. M. Manning, “Beam wave propagation within the second Rytov perturbation approximation,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 39, 423–436 (1996).

1994 (1)

1992 (1)

1988 (1)

1983 (1)

1969 (1)

H. T. Yura, “Optical propagation through a turbulent medium,” J. Opt. Soc. Am. A 59, 111–112 (1969).
[CrossRef]

1967 (1)

R. A. Schmeltzer, “Means, variances and covariances for laser propagation through a random medium,” Q. Appl. Math. 24, 339–354 (1967).

1962 (1)

V. I. Tatarskii, “Second approximation in the problem of wave propagation in media with random irregularities,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 5, 490–507 (1962); [Sov. Radiophys. 5, 164–202 (1962)].

1953 (1)

A. M. Obukhov, “The influence of weak atmospheric inhomogeneities upon the propagation of sound and light,” Izv. Akad. Nauk. SSSR Ser. Geogr. Geofiz. 2, 155–165 (1953); English translation by W. C. Hoffman, published as (U.S. Air Force Project RAND, Santa Monica, Calif., July28, 1955).

1937 (1)

S. M. Rytov, “Diffraction of light by ultrasonic waves,” Izv. Akad. Nauk SSSR Ser. Fiz. 2, 223–259 (1937).

Bracher, C.

Clifford, S. F.

Flatté, S. M.

Frehlich, R. G.

R. J. Hill, R. G. Frehlich, “Probability distribution of irradiance for the onset of strong scintillation,” J. Opt. Soc. Am. A 14, 1530–1540 (1997).
[CrossRef]

R. J. Hill, R. G. Frehlich, W. D. Otto, “The probability distribution of irradiance scintillation,” (National Technical Information Service, Springfield, Va.1997).

R. G. Frehlich (Cooperative Institute for Research in Environmental Sciences, University of Colorado at Boulder, Campus Box 216, Boulder, Colorado 80309-1149, rgf@cires.colorado.edu) (personal communication, November26, 1997).

Hill, R. J.

Malakhov, A. N.

A. N. Malakhov, Cumulant Analysis of Random Non-Gaussian Processes and Their Transformations (in Russian, Soviet Radio, Moscow, 1978). Unfortunately, there appears to be no English translation of this valuable reference.

Manning, R. M.

R. M. Manning, “Beam wave propagation within the second Rytov perturbation approximation,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 39, 423–436 (1996).

Martin, J. M.

Obukhov, A. M.

A. M. Obukhov, “The influence of weak atmospheric inhomogeneities upon the propagation of sound and light,” Izv. Akad. Nauk. SSSR Ser. Geogr. Geofiz. 2, 155–165 (1953); English translation by W. C. Hoffman, published as (U.S. Air Force Project RAND, Santa Monica, Calif., July28, 1955).

Ochs, G. R.

Otto, W. D.

R. J. Hill, R. G. Frehlich, W. D. Otto, “The probability distribution of irradiance scintillation,” (National Technical Information Service, Springfield, Va.1997).

Rytov, S. M.

S. M. Rytov, “Diffraction of light by ultrasonic waves,” Izv. Akad. Nauk SSSR Ser. Fiz. 2, 223–259 (1937).

Schmeltzer, R. A.

R. A. Schmeltzer, “Means, variances and covariances for laser propagation through a random medium,” Q. Appl. Math. 24, 339–354 (1967).

Sung, C. C.

Tatarskii, V. I.

V. I. Tatarskii, “Second approximation in the problem of wave propagation in media with random irregularities,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 5, 490–507 (1962); [Sov. Radiophys. 5, 164–202 (1962)].

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Office, Springfield, Va.1971), p. 292.

Wang, G. Y.

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, Cambridge, UK, 1952).

Wheelon, A. D.

A. D. Wheelon, Electromagnetic Scintillation, Vol. One: Geometrical Optics (Cambridge U. Press, Cambridge, UK, 2001), Problem 3, Chap. 2.

Yura, H. T.

H. T. Yura, C. C. Sung, S. F. Clifford, R. J. Hill, “Second order Rytov approximation,” J. Opt. Soc. Am. 73, 500–502 (1983).
[CrossRef]

H. T. Yura, “Optical propagation through a turbulent medium,” J. Opt. Soc. Am. A 59, 111–112 (1969).
[CrossRef]

Appl. Opt. (1)

Izv. Akad. Nauk SSSR Ser. Fiz. (1)

S. M. Rytov, “Diffraction of light by ultrasonic waves,” Izv. Akad. Nauk SSSR Ser. Fiz. 2, 223–259 (1937).

Izv. Akad. Nauk. SSSR Ser. Geogr. Geofiz. (1)

A. M. Obukhov, “The influence of weak atmospheric inhomogeneities upon the propagation of sound and light,” Izv. Akad. Nauk. SSSR Ser. Geogr. Geofiz. 2, 155–165 (1953); English translation by W. C. Hoffman, published as (U.S. Air Force Project RAND, Santa Monica, Calif., July28, 1955).

Izv. Vyssh. Uchebn. Zaved. Radiofiz. (2)

V. I. Tatarskii, “Second approximation in the problem of wave propagation in media with random irregularities,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 5, 490–507 (1962); [Sov. Radiophys. 5, 164–202 (1962)].

R. M. Manning, “Beam wave propagation within the second Rytov perturbation approximation,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 39, 423–436 (1996).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Q. Appl. Math. (1)

R. A. Schmeltzer, “Means, variances and covariances for laser propagation through a random medium,” Q. Appl. Math. 24, 339–354 (1967).

Other (7)

Ref. 3, pp. 253–258.

A. D. Wheelon, Electromagnetic Scintillation, Vol. One: Geometrical Optics (Cambridge U. Press, Cambridge, UK, 2001), Problem 3, Chap. 2.

A. N. Malakhov, Cumulant Analysis of Random Non-Gaussian Processes and Their Transformations (in Russian, Soviet Radio, Moscow, 1978). Unfortunately, there appears to be no English translation of this valuable reference.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Office, Springfield, Va.1971), p. 292.

R. J. Hill, R. G. Frehlich, W. D. Otto, “The probability distribution of irradiance scintillation,” (National Technical Information Service, Springfield, Va.1997).

R. G. Frehlich (Cooperative Institute for Research in Environmental Sciences, University of Colorado at Boulder, Campus Box 216, Boulder, Colorado 80309-1149, rgf@cires.colorado.edu) (personal communication, November26, 1997).

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, Cambridge, UK, 1952).

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

PDF of irradiance predicted by the second Rytov approximation for three negative values of the skewness parameter. The PDF is scaled with the Rytov deviation β0, as indicated by Eq. (37). The independent variable plotted along the horizontal axis is identified in Eq. (40). The log normal distribution is plotted for reference and corresponds to γ=0.

Fig. 2
Fig. 2

Comparison of the predicted distribution of irradiance with the numerical simulation of a spherical wave for β02=0.06. A log normal distribution is shown for reference and corresponds to γ=0.

Fig. 3
Fig. 3

Comparison of the predicted distribution of irradiance with the numerical simulation of a plane wave for β02=0.20. A log normal distribution is shown for reference and corresponds to γ=0.

Fig. 4
Fig. 4

Geometry for analyzing the double forward scattering of a plane wave in a random medium.

Equations (110)

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

2E+k2[1+Δ(r, t)]E=source,
E(R)=E0(R)exp[Ψ(R)].
2E0+k2E0=source.
E0(R)=exp[ψ0(R)].
2Ψ+2ψ0·Ψ+(Ψ)2+k2Δ(r, t)=0.
Ψ(R)=ψ1(R)+ψ2(R)+.
2ψ1+2ψ0·ψ1+k2Δ(r, t)=0,2ψ2+2ψ0·ψ2+(ψ1)2=0.
2Q+2ψ0·Q=f(R)
Q(R)= d3rG(R, r)f(r)exp[ψ0(r)-ψ0(R)],
G(R, r)=14π|R-r| exp(i|R-r|).
Q(R)= d3rG(R, r)f(r) E0(r)E0(R).
ψ1(R)=-k2  d3rG(R, r)Δ(r) E0(r)E0(R).
E1(R)=E0(R)exp-k2  d3rG(R, r)Δ(r) E0(r)E0(R).
ψ1(R)=a(R)+ib(R).
ψ2(R)=- d3rG(R, r) E0(r)E0(R)[ψ1(r)]2.
E2(R)=E0(R)exp- d3rG(R, r) E0(r)E0(R)×[k2Δ(r)+(ψ1)2].
ψ2=ϕ-12ψ12.
2ϕ+2ψ0·ϕ=-k2Δψ1.
ϕ(R)=-k2  d3rG(R, r) E0(r)E0(R)Δ(r)ψ1(r).
ϕ(R)=k4  d3r1G(R, r1)Δ(r1)× d3r2G(r1, r2)Δ(r2) E0(r2)E0(R).
ϕ(R)=c(R)+id(R).
E2(R)=E0(R)exp[a+c-12a2+12b2+i(b+d-ab)].
I=I0 exp(2a+2c-a2+b2).
H=log(I/I0)=2a+2c-a2+b2.
P(H)=12π - dq exp(-iqH)exp(iqH).
exp(iqH)=exp(iqK1-12q2K2-i6q3K3).
K1=H,
K2=H2-H2,
K3=H3-3HH2+2H3.
c=-12(a2+b2),
a2=χ2=τ2,b2=ϕ2=σ2, ab=χϕ=ρστ.
K1=2a+2c-a2+b2=-2a2=-2τ2.
K2=(2a+2c-a2+b2)2-2a+2c-a2+b22=4τ2+O(τ4).
K3=12(2a2c+a2b2-a4+2τ4).
Planewaves:K3=-11.142τpl4,Sphericalwaves:K3=-5.6854τsph4.
Planewaves:τpl2=0.307Cn2R11/6k7/6,Sphericalwaves:τsph2=0.124Cn2R11/6k7/6.
P(H)=12π - dq exp-iq(H-K1)-12q2K2-i6q3K3.
u=H-K1K2.
x=qK2
P(u, γ)=1πK2 0 dx cos(xu+γu3)exp(-12x2).
γ=K36(K2)3/2
P(u, 0)=1πK2 0dx cos(xu)exp(-12x2)=1πK2 exp(-12u2),
β02=(δI/I0)2=4χ2=4ρ2 sothatβ0=2τ.
u=1β0 logII0-12β02.
log[β0P(u, γ)]=log1π 0 dx cos(xu+γu3)×exp(-12x2).
Planewaves:γ=-0.232τpl =-0.116β0(pl),Sphericalwaves:γ=-0.118τsph=-0.059β0(sph).
Sphericalwave:γ=-0.059×0.245=-0.01456.
Planewave:γ=-0.116×0.4472=-0.0519.
- d3rG(R, r)E0(r)[ψ1(r)]2=-12E0(R)ψ12(R)+k2  d3rG(R, r)E0(r)Δ(r)ψ1(r).
 d3r(R2+k2)G(R, r){k2E0(r)Δ(r)ψ1(r)+[ψ1(r)2]}=12(R2+k2)[E0(R)ψ12(R)].
2(AB)=A2B+2A·B+B2A
k2E0Δψ1+(ψ1)2=12[ψ12(2+k2)E0+4ψ1E0·ψ1+2(ψ12ψ1+(ψ1)2)]
E0ψ1[2ψ1+2 ln E0·ψ1-k2Δ]=0.
I=I0exp[iq(2a+2c-a2+b2)]|q=-i.
I=I0 exp(iqK1-12q2K2)|q=-i,
=I0 exp(K1+12K2).
K1=2a+2c-a2+b2,
=2c-a2+b2,
K2=(2a+2c-a2+b2)2-2a+2c-a2+b22,
=4a2+O(τ4).
I=I0 exp(2c+a2+b2),
c=-12[a2+b2].
K3=12[2a2c+a2b2-a4+2τ4].
a4=3τ4,a2b2=σ2τ2(1+2ρ2),
K3=24[a2c+σ2τ2(1+2ρ2)-τ4].
a(R)=-k2  d3rA(R, r)Δ(r),
A(R, r)=RG(R, r) E0(r)E0(R),
c(R)=k4  d3r  d3rC(R, r,r)Δ(r)Δ(r),
C(R, r, r)=RG(R,r)G(r, r) E0(r)E0(R).
a2c=k8  d3r1  d3r2C(R, r1, r2)× d3r3A(R, r3) d3r4A(R, r4)×Δ(r1)Δ(r2)Δ(r3)Δ(r4).
Δ1Δ2Δ3Δ4=Δ1Δ2Δ3Δ4+Δ1Δ3Δ2Δ4+Δ1Δ4Δ2Δ3.
ca2=-12(τ2+σ2)τ2.
K3=24[a2c+(ρστ)2-τ4],
a2c=2k8  d3r1  d3r2  d3r3  d3r4C(R, r1, r2)×A(R, r3)A(R, r4)[Δ1Δ3Δ2Δ4+Δ1Δ4Δ2Δ3].
ΔiΔj= d3κΦ(k)exp[iκ·(ri-rj)].
a2c=2k8  d3κΦ(κ) d3ηΦ(η)×D(-κ)D(-η)M(κ, η),
D(κ)= d3rA(R, r)exp(iκ·r),
M(κ, η)= d3r1  d3r2C(R, r1, r2)×exp[i(κ·r1+η·r2)].
E0(R)=E0 exp(ikz).
G(r2, r1)=14π|z2-z1| expikz2-z1-r12+r22-2r1r2 cos(ϕ1-ϕ2)2(z2-z1).
A(R, r)=14π(R-z) coskr22(R-z)
D(κ)=12k 0R dz sinkr2(R-z)2kexp(iκzz).
C(R, r1, r2)=expikr222(R-z2)+r12+r22-2r1r2 cos(ϕ1-ϕ2)2(z2-z1)+c.c.32π2(R-z1)(z2-z1),
κ=ixκr cos ω+iyκr sin ω+izκz,
η=ixηr cos ν+iyηr sin ν+izηz.
 d3κΦ(κ) d3ηΦ(η) - dκz0 dκrκr02π dωΦ(κr2+κz2)×- dηz 0 dηrηr 02π dνΦ(ηr2+ηz2).
I=02π dω 02π dνM(κ, η)=0R dz2 0 dr2r2 02π dϕ2 0z2 dz1 0 dr1r1×02π dϕ1 02π dω 02π dν exp i[κzz2+κrr2 cos(ϕ2-ω)]exp i[ηzz1+ηrr1 cos(ϕ1-ν)]×expikr222(R-z2)+r12+r22-2r1r2 cos(ϕ1-ϕ2)2(z2-z1)+c.c.32π2(R-z1)(z2-z1).
I=116 0R dz2 0z2 dz1 exp i[(κzz2+ηzz1)](R-z1)(z2-z1)×0 dr2r2 0 dr1r1×J0(κrr2)J0(ηrr1)02π dϕ2 02π dϕ1×expikr222(R-z2)+r12+r22-2r1r2 cos(ϕ1-ϕ2)2(z2-z1)+c.c..
I=4π2 0R dz2 0z2 dz1 exp[i(κzz2+ηzz1)](R-z1)(z2-z1)×0 dr2r2 0 dr1r1J0kr1r2z2-z1×J0(κrr2)J0(ηrr1)expikr222(R-z2)+r12+r222(z2-z1)+c.c..
0dxxJ0(bx)J0(cx)exp(-a2x2)=12a2I0bc2a2exp-b2+c24a2.
I=-π2k2 0R dz2 0z2 dz1 exp i[κzz2+ηzz1]×J0krηr(R-z2)k ×cosκr2(R-z2)+ηr2(R-z1)2k.
a2c=-π22k4 0R dz2 0z2 dz1 0R dz3 0R dz4×- dκz0 dκrκrΦ(κr2+κz2)×- dηz0 dηrηrΦ(ηr2+ηz2)×sinκr2(R-z3)2ksinηr2(R-z4)2k×cosκr2(R-z2)+ηr2(R-z1)2k×J0krηr(R-z2)k ×exp{i[κz(z3-z1)+ηz(z2-z4)]}.
I=0R dz2 0z2 dz1 0R dz3 0R dz4×exp{i[κz(z3-z1)+ηz(z2-z4)]}=4π2δ(κz)δ(ηz)0R dz2 0z2 dz1, z3=z1,z4=z2.
x=R-z2,y=R-z1,
a2c=-2π4k4 0R dx 0x dy 0 dκκΦ(κ)×0 dηηΦ(η)J0kηxk×sinκ2y2ksinη2x2k×cosκ2y+η2x2k.
(ρστ)2-τ4=2π4k4 0R dx 0x dy 0 dκκΦ(κ)×0 dηηΦ(η)×sinκ2y2ksinη2x2kcosκ2y+η2x2k.
K3=48π4k4R2 01 dw 0w dv 0 dκκΦ(κ)×0 dηηΦ(η)sinκ2Rw2k×sinη2Rv2kcosκ2Rw+η2Rv2k×1-J0R2κηvk.
Φ(κ)=0.132Cn2κ11/3 F(kl0),0<jκ<,
s2=Rκ2wk,t2=Rη2vk,
K3=48(0.132π2Cn2k7/6R11/6)2Npl,
Npl=01dww5/6 0w dvv5/6 0 dss8/3 0 dtt8/3 sins22sint22×coss2+t22[1-J0(stv/w)].
Npl=1211 0 dss8/3 0s dtt8/3 sins22sint22×coss2+t2201 dpp8/3[1-J0(pst)].
Npl=-0.0129.
0.132π2Cn2k7/6R11/6=4.242τpl2,
K3=-11.142τpl4.
a2c=-2(0.132π2Cn2k7/6R11/6)2Nsph,
Nsph=01 dw 0w dv 0 dss8/3 0 dtt8/3×sins2w(1-w)2ksint2v(1-v)2k×J0stv(1-v)k×coss2w(1-w)+t2v(1-v)2k.
Nsph=-0.01076.
0.132π2Cn2kγ7/6R11/6=10.4913τsph2.
Spherical:K3=-5.6854τsph4,

Metrics