Abstract

A recently published sparse spectrum (SS) model of the phase front perturbations by atmospheric turbulence [J. Opt. Soc. Am. A 30, 479 (2013)] is based on the trigonometric series with discrete random support. The SS model enables fewer computational efforts, while preserving the wide range of scales typically associated with turbulence perturbations. We present an improved version of the SS model that accurately reproduces the power-law spectral density of the phase fluctuations in the arbitrary wide spectral band. We examine the higher-order statistics of the SS phase samples for four versions of the SS model. We also present the calculations of the long-exposure Strehl numbers and scintillation index for the different versions of the SS model. A nonoverlapping SS model with a log-uniform partition emerges as the most appropriate for the atmospheric turbulence representation.

© 2013 Optical Society of America

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References

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  1. M. I. Charnotskii, “Sparse spectrum model for a turbulent phase,” J. Opt. Soc. Am. A 30, 479–488 (2013).
    [CrossRef]
  2. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).
  3. M. Charnotskii, “Sparse spectrum model for the turbulent phase simulations,” Proc. SPIE 8732, 873208 (2013).
    [CrossRef]
  4. R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88, 541–662 (1978).
    [CrossRef]
  5. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).
  6. M. Charnotskii, “Sparse spectrum model of the sea surface,” Proceedings of the 30th International Conference on Ocean, Offshore and Arctic Engineering, Rotterdam, The Netherlands, 2011, paper 49958.
  7. M. Charnotskii, “Sparse spectrum model of the sea surface elevations,” Proceedings of the 22nd International Offshore and Polar Engineering Conference, Rhodes, Greece, 2012, pp. 655–659.
  8. M. I. Charnotskii, “Statistical modeling of the point spread function for imaging through turbulence,” Opt. Eng. 51, 101706 (2012).
    [CrossRef]
  9. K. S. Gochelashvili and V. I. Shishov, “Focused irradiance beyond a layer of turbulent atmosphere,” Opt. Acta 19, 327–332 (1972).
    [CrossRef]

2013 (2)

M. I. Charnotskii, “Sparse spectrum model for a turbulent phase,” J. Opt. Soc. Am. A 30, 479–488 (2013).
[CrossRef]

M. Charnotskii, “Sparse spectrum model for the turbulent phase simulations,” Proc. SPIE 8732, 873208 (2013).
[CrossRef]

2012 (1)

M. I. Charnotskii, “Statistical modeling of the point spread function for imaging through turbulence,” Opt. Eng. 51, 101706 (2012).
[CrossRef]

1978 (1)

R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88, 541–662 (1978).
[CrossRef]

1972 (1)

K. S. Gochelashvili and V. I. Shishov, “Focused irradiance beyond a layer of turbulent atmosphere,” Opt. Acta 19, 327–332 (1972).
[CrossRef]

Andrews, L. C.

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

Charnotskii, M.

M. Charnotskii, “Sparse spectrum model for the turbulent phase simulations,” Proc. SPIE 8732, 873208 (2013).
[CrossRef]

M. Charnotskii, “Sparse spectrum model of the sea surface,” Proceedings of the 30th International Conference on Ocean, Offshore and Arctic Engineering, Rotterdam, The Netherlands, 2011, paper 49958.

M. Charnotskii, “Sparse spectrum model of the sea surface elevations,” Proceedings of the 22nd International Offshore and Polar Engineering Conference, Rhodes, Greece, 2012, pp. 655–659.

Charnotskii, M. I.

M. I. Charnotskii, “Sparse spectrum model for a turbulent phase,” J. Opt. Soc. Am. A 30, 479–488 (2013).
[CrossRef]

M. I. Charnotskii, “Statistical modeling of the point spread function for imaging through turbulence,” Opt. Eng. 51, 101706 (2012).
[CrossRef]

Gochelashvili, K. S.

K. S. Gochelashvili and V. I. Shishov, “Focused irradiance beyond a layer of turbulent atmosphere,” Opt. Acta 19, 327–332 (1972).
[CrossRef]

Hill, R. J.

R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88, 541–662 (1978).
[CrossRef]

Phillips, R. L.

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

Shishov, V. I.

K. S. Gochelashvili and V. I. Shishov, “Focused irradiance beyond a layer of turbulent atmosphere,” Opt. Acta 19, 327–332 (1972).
[CrossRef]

Tatarskii, V. I.

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

J. Fluid Mech. (1)

R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88, 541–662 (1978).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

K. S. Gochelashvili and V. I. Shishov, “Focused irradiance beyond a layer of turbulent atmosphere,” Opt. Acta 19, 327–332 (1972).
[CrossRef]

Opt. Eng. (1)

M. I. Charnotskii, “Statistical modeling of the point spread function for imaging through turbulence,” Opt. Eng. 51, 101706 (2012).
[CrossRef]

Proc. SPIE (1)

M. Charnotskii, “Sparse spectrum model for the turbulent phase simulations,” Proc. SPIE 8732, 873208 (2013).
[CrossRef]

Other (4)

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

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

M. Charnotskii, “Sparse spectrum model of the sea surface,” Proceedings of the 30th International Conference on Ocean, Offshore and Arctic Engineering, Rotterdam, The Netherlands, 2011, paper 49958.

M. Charnotskii, “Sparse spectrum model of the sea surface elevations,” Proceedings of the 22nd International Offshore and Polar Engineering Conference, Rhodes, Greece, 2012, pp. 655–659.

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Figures (26)

Fig. 1.
Fig. 1.

Normalized structure function for α=5/3, calculated by the integration of Eq. (7), with inner–outer scale ratio ε=0.01 (heavy solid curve) quadratic asymptote, Eq. (8) (long-dashed curve), saturation asymptote, Eq. (9) (short-dashed curve); 5/3 power structure function, Eq. (10) (thin solid curve).

Fig. 2.
Fig. 2.

Same as Fig. 1, but for inner–outer scale ratio ε=0.001.

Fig. 3.
Fig. 3.

Same as Fig. 1, but for inner–outer scale ratio ε=0.0001.

Fig. 4.
Fig. 4.

Normalized phase structure function for ε=104 and different values of the spectral exponent α, calculated by numerical integration of Eq. (7).

Fig. 5.
Fig. 5.

Normalized phase structure function derived from 250 phase samples generated by the uniform PM. N=500.

Fig. 6.
Fig. 6.

Normalized phase structure function derived from 250 phase samples generated by the log-uniform PM. N=500.

Fig. 7.
Fig. 7.

Normalized phase structure function derived from 250 phase samples generated by the equal energy PM. N=500.

Fig. 8.
Fig. 8.

Normalized phase structure function derived from 250 phase samples generated by the OM. N=500.

Fig. 9.
Fig. 9.

Normalized phase structure functions averaged over a large number of phase samples generated by the PM and OM with 10 components.

Fig. 10.
Fig. 10.

Normalized phase structure functions averaged over a large number of phase samples generated by the PM and OM with 100 components.

Fig. 11.
Fig. 11.

Normalized phase structure function averaged over a large number of phase samples generated by the PM and OM with 5000 components.

Fig. 12.
Fig. 12.

Normalized fourth moment of phase difference [Eq. (28)] for the uniform PM and different numbers of spectral components.

Fig. 13.
Fig. 13.

Normalized fourth moment of phase difference [Eq. (28)] for the log-uniform PM and different numbers of spectral components.

Fig. 14.
Fig. 14.

Normalized fourth moment of phase difference [Eq. (28)] for the equal energy PM and different numbers of spectral components.

Fig. 15.
Fig. 15.

Normalized fourth moment of phase difference [Eq. (28)], for the OM and different numbers of spectral components.

Fig. 16.
Fig. 16.

Average Strehl number calculated according to Eq. (31) for the inner-to-outer scale ratio ε=105. Heavy solid curve, BLPL spectrum [Eq. (5)]; diamond-marked curve, power-law structure function [Eq. (1)] for α=5/3; dot-dashed line, asymptote [Eq. (32)]; short-dashed line, asymptote [Eq. (33)].

Fig. 17.
Fig. 17.

Same as Fig. 16, but for ε=104.

Fig. 18.
Fig. 18.

Same as Fig. 16, but for ε=103.

Fig. 19.
Fig. 19.

Average Strehl number for the uniform PM. The parameter is the number of spectral components for the phase. Thin dot-dashed and short-dashed curves are, respectively, the average Strehl numbers calculated by Eq. (31) for the BLPL spectrum [Eq. (5)] and power-law structure function [Eq. (1)] with α=5/3.

Fig. 20.
Fig. 20.

Same as Fig. 19, but for the log-uniform PM.

Fig. 21.
Fig. 21.

Same as Fig. 19, but for the equal energy PM.

Fig. 22.
Fig. 22.

Same as Fig. 19, but for the OM.

Fig. 23.
Fig. 23.

SI in the image of the point source calculated from the samples based on the uniform PM.

Fig. 24.
Fig. 24.

SI in the image of the point source calculated from the samples based on the log-uniform PM.

Fig. 25.
Fig. 25.

SI in the image of the point source calculated from the samples based on the equal energy PM.

Fig. 26.
Fig. 26.

SI in the image of the point source calculated from the samples based on the OM.

Tables (1)

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Table 1. Spectral Partitions Used in Calculations

Equations (34)

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D(r⃗)=[S(R⃗+r⃗)S(R⃗)]2=(rrC)α,1<α<2,
D(r⃗)=2d2KΦ(K⃗)[1cos(K⃗·r⃗)],
Φ(K⃗)=C(α)rCαK2α,
C(α)=α2α2Γ(1+α/2)πΓ(1α/2).
Φ(K⃗)={C(α)rCαK2α,K0<K<k00,0<K<K0k0<k.
S2=d2KΦ(K⃗)=2πC(α)α[(K0rC)α(k0rC)α].
d(y)=D(r=k01y)D()=α(εα1)ε1dtt1α[1J0(ty)],εK0k01.
d(y<1)α(1ε2α)4(2α)(εα1)y2.
d(yε1)1+2αε32y322π(1εα)sin(εyπ4).
d(1<y<ε1)α4πC(εα1)yα.
S(r⃗)=Ren=1Nanexp(iK⃗n·r⃗),
an=0,anam=0,anam*=snδmn.
P{K⃗nK⃗+dK⃗}=pn(K⃗)dK⃗.
D(r⃗)=[S(R⃗+r⃗)S(R⃗)]2{an,K⃗n}=n=1Nsn[1cos(K⃗n·r⃗)]{K⃗n}=d2Kn=1Nsnpn(K⃗)[1cos(K⃗·r⃗)].
n=1Nsnpn(K⃗)=2Φ(K⃗).
pn(K⃗)d2K=pn(K,φ)kdkdφ=pn(K)dkdφ2π.
n=1Nsnpn(K)=4πC(α)rCαK1α,K0Kk0.
K0<K1<K2<<KN1<KN=k0,
snpn(K)=4πC(α)rCαK1α,Kn1KKn,1nN.
sn=4πC(α)αrCα(Kn1αKnα),1nN,
pn(K)=αKn1αKnαK1α,Kn1KKn,1nN.
K=[Knα+(Kn1αKnα)ξ]1α,
sn=s=4πC(α)NαrCα(K0αk0α),1nN.
pn(K)=p(K)=αK0αk0αK1α,K0Kk0,1nN,
K=[k0α+(K0αk0α)ξ]1α,
B={K⃗n·r⃗m},
S(r⃗m)=Re[A·exp(iB)].
σD2(r⃗)=[S(r⃗0)S(r⃗0+r⃗)]4[S(r⃗0)S(r⃗0+r⃗)]22[S(r⃗0)S(r⃗0+r⃗)]22.
u(rIM)=Cd2rA(r)exp[ikr·rIM+iS(r)],
St#=1Σ2d2Rd2ρA(R⃗+12ρ⃗)A(R⃗12ρ⃗)×exp[iS(R⃗+12ρ⃗)iS(R⃗12ρ⃗)],
St#=1Σ2d2ρCA(ρ⃗)exp[12D(ρ⃗)],CA(ρ⃗)=d2RA(R⃗+12ρ⃗)A(R⃗12ρ⃗).
St#Γ(1+2α)22+2α[D(2a)]2α=Γ(1+2α)22αrC2a2.
St#2(2α)πC(α)rCαk0α2a2.
σI2=St#2St#2St#2.

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