Abstract

To estimate the probability distributions of power fades, we consider two basic types of disturbance in electromagnetic wave propagation through atmospheric turbulence: wave-front intensity fluctuations and wave-front distortion. We assess the reduction in the cumulative probability of losses caused by these two effects through spatial diversity by using a multiaperture receiver configuration. Degradations in receiver performance are determined with fractal techniques used to simulate the turbulence-induced wave-front phase distortion, and a log normal model is assumed for the collected power fluctuations.

© 1997 Optical Society of America

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References

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  1. R. E. Hufnagel, “Propagation through atmospheric turbulence,” in Infrared Handbook (Office of Naval Research, Department of the Navy, Arlington, Va., 1978).
  2. D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am. 55, 1427–1435 (1965).
    [CrossRef]
  3. S. Rosenberg, M. C. Teich, “Photocounting array receivers for optical communication through the lognormal atmospheric channel. 2: Optimum and suboptimum receiver performance for binary signaling,” Appl. Opt. 12, 2625–2635 (1973).
    [CrossRef] [PubMed]
  4. J. H. Churnside, M. C. McIntyre, “Averaged threshold receiver for direct detection of optical communications through the lognormal atmospheric channel,” Appl. Opt. 16, 2669–2676 (1977).
    [CrossRef] [PubMed]
  5. A. Belmonte, A. Comerón, J. Bará, J. A. Rubio, “Averaging of collected-power fluctuations by a multiaperture receiver system,” Opt. Eng. 35, 2775–2778 (1996).
    [CrossRef]
  6. A Belmonte, A Comerón, J. Bará, J. A. Rubio, E. Fernández, P. Menéndez–Valdés, “The impact of the point-spread function on the performance of a multiple-aperture optical ground station,” in Atmospheric Propagation and Remote Sensing IV, J. C. Dainty, ed. Proc. SPIE2471, 324–334 (1995).
  7. D. L. Fried, “Aperture averaging of scintillation,” J. Opt. Soc. Am. 57, 169–175 (1967).
    [CrossRef]
  8. V. I. Tatarski, Wave Propagation in a Turbulent Medium (Dover, New York, 1967).
  9. J. C. Dainty, B. M. Levine, B. K. Brames, K. A. O’Donnell, “Measurements of the wavelength dependence and other properties of stellar scintillation at Mauna Kea, Hawaii,” Appl. Opt. 21, 127–145 (1978).
  10. S. H. Reiger, “Starlight scintillations and atmospheric turbulence,” Astron. J. 68, 395–406 (1963).
    [CrossRef]
  11. E. Jakeman, G. Parry, E. R. Pike, P. N. Pusey, “The twinkling of stars,” Contemp. Phys. 19, 127–145 (1978).
    [CrossRef]
  12. A Belmonte, “Effects of the atmospheric turbulence on the propagation of electromagnetic waves,” (in Spanish), Ph.D. dissertation (Polytechnic University of Catalonia, Barcelona, Spain, 1995).
  13. H. T. Yura, W. G. McKinley, “Optical scintillation statistics for IR ground-to-space laser communication systems,” Appl. Opt. 22, 3353–3358 (1983).
    [CrossRef] [PubMed]
  14. B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” in Image Processing (Pacific Grove), J. C. Urbach, ed., Proc. SPIE74, 225–233 (1976).
    [CrossRef]
  15. N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
    [CrossRef]
  16. C. Schwartz, G. Baum, E. N. Ribak, “Turbulent-degraded wave fronts as fractal surfaces,” J. Opt. Soc. Am. A 11, 444–451 (1994).
    [CrossRef]
  17. R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
    [CrossRef]
  18. R. J. Noll, “Zernike polynomials and atmosphere turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  19. H. O. Peitgen, D. Saupe, eds., The Science of Fractal Images (Springer–Verlag, New York, 1988).
  20. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

1996 (1)

A. Belmonte, A. Comerón, J. Bará, J. A. Rubio, “Averaging of collected-power fluctuations by a multiaperture receiver system,” Opt. Eng. 35, 2775–2778 (1996).
[CrossRef]

1994 (1)

1992 (1)

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

1990 (1)

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

1983 (1)

1978 (2)

1977 (1)

1976 (1)

1973 (1)

1967 (1)

1965 (1)

1963 (1)

S. H. Reiger, “Starlight scintillations and atmospheric turbulence,” Astron. J. 68, 395–406 (1963).
[CrossRef]

Bará, J.

A. Belmonte, A. Comerón, J. Bará, J. A. Rubio, “Averaging of collected-power fluctuations by a multiaperture receiver system,” Opt. Eng. 35, 2775–2778 (1996).
[CrossRef]

A Belmonte, A Comerón, J. Bará, J. A. Rubio, E. Fernández, P. Menéndez–Valdés, “The impact of the point-spread function on the performance of a multiple-aperture optical ground station,” in Atmospheric Propagation and Remote Sensing IV, J. C. Dainty, ed. Proc. SPIE2471, 324–334 (1995).

Baum, G.

Belmonte, A

A Belmonte, A Comerón, J. Bará, J. A. Rubio, E. Fernández, P. Menéndez–Valdés, “The impact of the point-spread function on the performance of a multiple-aperture optical ground station,” in Atmospheric Propagation and Remote Sensing IV, J. C. Dainty, ed. Proc. SPIE2471, 324–334 (1995).

A Belmonte, “Effects of the atmospheric turbulence on the propagation of electromagnetic waves,” (in Spanish), Ph.D. dissertation (Polytechnic University of Catalonia, Barcelona, Spain, 1995).

Belmonte, A.

A. Belmonte, A. Comerón, J. Bará, J. A. Rubio, “Averaging of collected-power fluctuations by a multiaperture receiver system,” Opt. Eng. 35, 2775–2778 (1996).
[CrossRef]

Brames, B. K.

Churnside, J. H.

Comerón, A

A Belmonte, A Comerón, J. Bará, J. A. Rubio, E. Fernández, P. Menéndez–Valdés, “The impact of the point-spread function on the performance of a multiple-aperture optical ground station,” in Atmospheric Propagation and Remote Sensing IV, J. C. Dainty, ed. Proc. SPIE2471, 324–334 (1995).

Comerón, A.

A. Belmonte, A. Comerón, J. Bará, J. A. Rubio, “Averaging of collected-power fluctuations by a multiaperture receiver system,” Opt. Eng. 35, 2775–2778 (1996).
[CrossRef]

Dainty, J. C.

Fernández, E.

A Belmonte, A Comerón, J. Bará, J. A. Rubio, E. Fernández, P. Menéndez–Valdés, “The impact of the point-spread function on the performance of a multiple-aperture optical ground station,” in Atmospheric Propagation and Remote Sensing IV, J. C. Dainty, ed. Proc. SPIE2471, 324–334 (1995).

Fried, D. L.

Glindemann, A.

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Goodman, J. W.

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

Hufnagel, R. E.

R. E. Hufnagel, “Propagation through atmospheric turbulence,” in Infrared Handbook (Office of Naval Research, Department of the Navy, Arlington, Va., 1978).

Jakeman, E.

E. Jakeman, G. Parry, E. R. Pike, P. N. Pusey, “The twinkling of stars,” Contemp. Phys. 19, 127–145 (1978).
[CrossRef]

Lane, R. G.

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Levine, B. M.

McGlamery, B. L.

B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” in Image Processing (Pacific Grove), J. C. Urbach, ed., Proc. SPIE74, 225–233 (1976).
[CrossRef]

McIntyre, M. C.

McKinley, W. G.

Menéndez–Valdés, P.

A Belmonte, A Comerón, J. Bará, J. A. Rubio, E. Fernández, P. Menéndez–Valdés, “The impact of the point-spread function on the performance of a multiple-aperture optical ground station,” in Atmospheric Propagation and Remote Sensing IV, J. C. Dainty, ed. Proc. SPIE2471, 324–334 (1995).

Noll, R. J.

O’Donnell, K. A.

Parry, G.

E. Jakeman, G. Parry, E. R. Pike, P. N. Pusey, “The twinkling of stars,” Contemp. Phys. 19, 127–145 (1978).
[CrossRef]

Pike, E. R.

E. Jakeman, G. Parry, E. R. Pike, P. N. Pusey, “The twinkling of stars,” Contemp. Phys. 19, 127–145 (1978).
[CrossRef]

Pusey, P. N.

E. Jakeman, G. Parry, E. R. Pike, P. N. Pusey, “The twinkling of stars,” Contemp. Phys. 19, 127–145 (1978).
[CrossRef]

Reiger, S. H.

S. H. Reiger, “Starlight scintillations and atmospheric turbulence,” Astron. J. 68, 395–406 (1963).
[CrossRef]

Ribak, E. N.

Roddier, N.

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

Rosenberg, S.

Rubio, J. A.

A. Belmonte, A. Comerón, J. Bará, J. A. Rubio, “Averaging of collected-power fluctuations by a multiaperture receiver system,” Opt. Eng. 35, 2775–2778 (1996).
[CrossRef]

A Belmonte, A Comerón, J. Bará, J. A. Rubio, E. Fernández, P. Menéndez–Valdés, “The impact of the point-spread function on the performance of a multiple-aperture optical ground station,” in Atmospheric Propagation and Remote Sensing IV, J. C. Dainty, ed. Proc. SPIE2471, 324–334 (1995).

Schwartz, C.

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (Dover, New York, 1967).

Teich, M. C.

Yura, H. T.

Appl. Opt. (4)

Astron. J. (1)

S. H. Reiger, “Starlight scintillations and atmospheric turbulence,” Astron. J. 68, 395–406 (1963).
[CrossRef]

Contemp. Phys. (1)

E. Jakeman, G. Parry, E. R. Pike, P. N. Pusey, “The twinkling of stars,” Contemp. Phys. 19, 127–145 (1978).
[CrossRef]

J. Opt. Soc. Am. (3)

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

Opt. Eng. (2)

A. Belmonte, A. Comerón, J. Bará, J. A. Rubio, “Averaging of collected-power fluctuations by a multiaperture receiver system,” Opt. Eng. 35, 2775–2778 (1996).
[CrossRef]

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

Waves Random Media (1)

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Other (7)

H. O. Peitgen, D. Saupe, eds., The Science of Fractal Images (Springer–Verlag, New York, 1988).

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

A Belmonte, “Effects of the atmospheric turbulence on the propagation of electromagnetic waves,” (in Spanish), Ph.D. dissertation (Polytechnic University of Catalonia, Barcelona, Spain, 1995).

B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” in Image Processing (Pacific Grove), J. C. Urbach, ed., Proc. SPIE74, 225–233 (1976).
[CrossRef]

A Belmonte, A Comerón, J. Bará, J. A. Rubio, E. Fernández, P. Menéndez–Valdés, “The impact of the point-spread function on the performance of a multiple-aperture optical ground station,” in Atmospheric Propagation and Remote Sensing IV, J. C. Dainty, ed. Proc. SPIE2471, 324–334 (1995).

V. I. Tatarski, Wave Propagation in a Turbulent Medium (Dover, New York, 1967).

R. E. Hufnagel, “Propagation through atmospheric turbulence,” in Infrared Handbook (Office of Naval Research, Department of the Navy, Arlington, Va., 1978).

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Figures (8)

Fig. 1
Fig. 1

Geometry of circular apertures with a center-to-center separation |rirj| and an individual aperture diameter D.

Fig. 2
Fig. 2

Intensity normalized covariance function versus normalized spatial separation for spherical wave-propagation geometry. The corresponding collected power covariance is shown for two aperture diameters D. λ is the wavelength and L is the propagation path length, yielding a λ L intensity correlation length when, as in this case, the covariance function is computed from Rytov’s approximation.

Fig. 3
Fig. 3

Probability that the power-fluctuation-induced loss level on the abscissa is exceeded as a function of the number of subapertures arranged in a row. The atmospheric path is characterized by the intensity variance of σI2 = 0.1 and the normalized intensity covariance function shown in Fig. 2. D = λ L / 2 and l = λ L, where D is the individual aperture diameter and l is the center-to-center distance between contiguous subapertures.

Fig. 4
Fig. 4

Probability that a given power-fluctuation-induced loss level on the abscissa is exceeded for (a) a two-aperture array and (b) a square array of 2 × 2 subapertures. The same turbulence assumptions as in Fig. 3 are made for the atmospheric path, D = λ L / 2. Calculations are shown as a function of the distance l between the subapertures.

Fig. 5
Fig. 5

Example of a fractal wave front generated by an improved version of the midpoint displacement algorithm. Given the simulated instantaneous wave front, we can compute the instantaneous point-spread function using a simple Fourier transform.

Fig. 6
Fig. 6

Probability that the loss on the abscissa is exceeded owing to point-spread function distortion for several numbers of subapertures. Turbulence conditions yielding r0/D = 0.1 and individual apertures characterized by a 10−4-rad field of view and a diameter of D = λ L / 2, with a center-to-center distance between them of l = λ L, are assumed.

Fig. 7
Fig. 7

Probability that a given spot-distortion-induced loss level on the abscissa is exceeded for (a) a two-aperture array and (b) a 2 × 2 square array of apertures. The same assumptions as in Fig. 6 are made for the atmospheric turbulence and the individual apertures. Calculations are shown for several distances l between the apertures.

Fig. 8
Fig. 8

Probability that the loss on the abscissa is exceeded owing to the combined effect of scintillation and focal-spot distortion: (a) a two-subaperture array with the center-to-center separation of l = λ L and (b) a 2 × 2 square array of subapertures lying on the vertices of a square of l = λ L each side. In both cases individual subapertures characterized by a 10−4-rad field of view, a diameter of D = λ L / 2, and turbulence conditions yielding r0/D = 0.1, an intensity variance of σI2 = 0.1, and the normalized intensity covariance function shown in Fig. 2 are assumed. The results are compared with the probability of total losses for a single subaperture with an area equivalent to one, four, or eight times the total array area.

Equations (17)

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P = i = 1 N S i W ( ρ i - r i ) I ( ρ i ) d ρ i
σ P 2 = i = 1 N j = 1 N C P ( ρ i - r j ) .
C P ( ρ = S K W ( ρ ) C I ( ρ + ρ ) d ρ .
σ P 2 = C P ( 0 ) = S K W ( ρ ) C I ( ρ ) d ρ ,
χ P = ln ( P / P 0 ) ,
p χ P ( χ P ) = 1 ( 2 π σ χ P 2 ) 1 / 2 exp [ - ( χ P - χ P 0 ) 2 2 σ χ P ] ,
p P ( P ) = 1 ( 2 π σ χ P ) 1 / 2 1 2 P exp { - 1 2 σ χ P [ 1 2 ln ( P P 0 ) + σ χ P 2 ] } ,
σ P 2 = P 0 2 [ exp ( σ χ P 2 ) - 1 ] .
L P = - 10 log ( P / P 0 ) .
D S ( ρ ) = 6.88 ( ρ / r 0 ) 5 / 3 ,
D S ( ρ ) = 2 0 Φ S ( K ) [ 1 - exp ( j K · ρ ) ] d K ,
Φ S ( K ) = 0.023 r 0 - 5 / 3 K - 11 / 3 .
D B ( ρ ) ρ 2 H
Φ B ( K ) K - ( 2 H + E ) ,
F = E - H + 1 ,
I S ( x , y ) = 1 / ( f λ ) 2 F x / f λ , y f λ { W ( u , v ) exp [ j S ( u , v ) ] } 2 ,
L S = - 10 log [ 1 - x 2 + y 2 R D 2 I S ( x , y ) d x d y ] .

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