Abstract

Current Strehl ratio models for actively compensated free-space optical communications terminals do not accurately predict system performance under strong turbulence conditions as they are based on weak turbulence theory. For evaluation of compensated systems, we present an approach for simulating the Strehl ratio with both low-order (tip/tilt) and higher-order (adaptive optics) correction. Our simulation results are then compared to the published models and their range of turbulence validity is assessed. Finally, we propose a new Strehl ratio model and antenna gain equation that are valid for general turbulence conditions independent of the degree of compensation.

© 2014 Optical Society of America

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References

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2012 (1)

J. C. Juarez, D. W. Young, R. A. Venkat, D. M. Brown, A. M. Brown, R. L. Oberc, J. E. Sluz, H. A. Pike, L. B. Stotts, “Analysis of link performance for the FOENEX laser communications system,” Proc. SPIE 8380, 838007 (2012).
[Crossref]

2011 (2)

H. R. Burris, M. S. Ferraro, W. Freeman, P. G. Goetz, R. Mahon, C. I. Moore, J. L. Murphy, J. Overfield, W. S. Rabinovich, W. R. Smith, M. R. Suite, L. M. Thomas, B. B. Xu, “Tactical network demonstration with free space lasercomm,” Proc. SPIE 7923, 792305 (2011).
[Crossref]

J. C. Juarez, D. W. Young, J. E. Sluz, L. B. Stotts, “High-sensitivity DPSK receiver for high-bandwidth free-space optical communication links,” Opt. Express 19(11), 10789–10796 (2011).
[Crossref] [PubMed]

2009 (1)

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, D. W. Young, “Hybrid Optical RF Airborne Communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[Crossref]

2006 (2)

D. S. Grinch, J. A. Cunningham, D. S. Fisher, “Laser system for cooperative pointing and tracking of moving terminals over long distance,” Proc. SPIE 6238, 623803 (2006).
[Crossref]

L. C. Andrews, R. L. Phillips, R. J. Sasiela, R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45(7), 076001 (2006).
[Crossref]

2002 (1)

T. Weyrauch, M. A. Vorontsov, J. Gowens, T. G. Bifano, “Fiber coupling with adaptive optics for free-space optical communication,” Proc. SPIE 4489, 177–184 (2002).
[Crossref]

1998 (2)

F. Roddier, “Maximum gain and efficiency of adaptive optics systems,” Publ. Astron. Soc. Pac. 110(749), 837–840 (1998).
[Crossref]

A. J. Lambert, D. Fraser, “Linear systems approach to simulation of optical diffraction,” Appl. Opt. 37(34), 7933–7939 (1998).
[Crossref] [PubMed]

1997 (1)

B. M. Welsh, “A Fourier series based atmospheric phase screen generator for simulating anisoplanatic geometries and temporal evolution,” Proc. SPIE 3125, 327–338 (1997).
[Crossref]

1995 (1)

1992 (1)

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2(3), 209–224 (1992).
[Crossref]

1976 (1)

1974 (1)

1972 (1)

1967 (1)

Andrews, L. C.

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, D. W. Young, “Hybrid Optical RF Airborne Communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[Crossref]

L. C. Andrews, R. L. Phillips, R. J. Sasiela, R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45(7), 076001 (2006).
[Crossref]

Bifano, T. G.

T. Weyrauch, M. A. Vorontsov, J. Gowens, T. G. Bifano, “Fiber coupling with adaptive optics for free-space optical communication,” Proc. SPIE 4489, 177–184 (2002).
[Crossref]

Brown, A. M.

J. C. Juarez, D. W. Young, R. A. Venkat, D. M. Brown, A. M. Brown, R. L. Oberc, J. E. Sluz, H. A. Pike, L. B. Stotts, “Analysis of link performance for the FOENEX laser communications system,” Proc. SPIE 8380, 838007 (2012).
[Crossref]

Brown, D. M.

J. C. Juarez, D. W. Young, R. A. Venkat, D. M. Brown, A. M. Brown, R. L. Oberc, J. E. Sluz, H. A. Pike, L. B. Stotts, “Analysis of link performance for the FOENEX laser communications system,” Proc. SPIE 8380, 838007 (2012).
[Crossref]

Burris, H. R.

H. R. Burris, M. S. Ferraro, W. Freeman, P. G. Goetz, R. Mahon, C. I. Moore, J. L. Murphy, J. Overfield, W. S. Rabinovich, W. R. Smith, M. R. Suite, L. M. Thomas, B. B. Xu, “Tactical network demonstration with free space lasercomm,” Proc. SPIE 7923, 792305 (2011).
[Crossref]

Cherry, P. C.

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, D. W. Young, “Hybrid Optical RF Airborne Communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[Crossref]

Coles, W. A.

Cunningham, J. A.

D. S. Grinch, J. A. Cunningham, D. S. Fisher, “Laser system for cooperative pointing and tracking of moving terminals over long distance,” Proc. SPIE 6238, 623803 (2006).
[Crossref]

Dainty, J. C.

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2(3), 209–224 (1992).
[Crossref]

Degnan, J. J.

Ferraro, M. S.

H. R. Burris, M. S. Ferraro, W. Freeman, P. G. Goetz, R. Mahon, C. I. Moore, J. L. Murphy, J. Overfield, W. S. Rabinovich, W. R. Smith, M. R. Suite, L. M. Thomas, B. B. Xu, “Tactical network demonstration with free space lasercomm,” Proc. SPIE 7923, 792305 (2011).
[Crossref]

Filice, J. P.

Fisher, D. S.

D. S. Grinch, J. A. Cunningham, D. S. Fisher, “Laser system for cooperative pointing and tracking of moving terminals over long distance,” Proc. SPIE 6238, 623803 (2006).
[Crossref]

Foshee, J. J.

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, D. W. Young, “Hybrid Optical RF Airborne Communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[Crossref]

Fraser, D.

Freeman, W.

H. R. Burris, M. S. Ferraro, W. Freeman, P. G. Goetz, R. Mahon, C. I. Moore, J. L. Murphy, J. Overfield, W. S. Rabinovich, W. R. Smith, M. R. Suite, L. M. Thomas, B. B. Xu, “Tactical network demonstration with free space lasercomm,” Proc. SPIE 7923, 792305 (2011).
[Crossref]

Frehlich, R. G.

Fried, D. L.

Glindemann, A.

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2(3), 209–224 (1992).
[Crossref]

Goetz, P. G.

H. R. Burris, M. S. Ferraro, W. Freeman, P. G. Goetz, R. Mahon, C. I. Moore, J. L. Murphy, J. Overfield, W. S. Rabinovich, W. R. Smith, M. R. Suite, L. M. Thomas, B. B. Xu, “Tactical network demonstration with free space lasercomm,” Proc. SPIE 7923, 792305 (2011).
[Crossref]

Gowens, J.

T. Weyrauch, M. A. Vorontsov, J. Gowens, T. G. Bifano, “Fiber coupling with adaptive optics for free-space optical communication,” Proc. SPIE 4489, 177–184 (2002).
[Crossref]

Grinch, D. S.

D. S. Grinch, J. A. Cunningham, D. S. Fisher, “Laser system for cooperative pointing and tracking of moving terminals over long distance,” Proc. SPIE 6238, 623803 (2006).
[Crossref]

Juarez, J. C.

J. C. Juarez, D. W. Young, R. A. Venkat, D. M. Brown, A. M. Brown, R. L. Oberc, J. E. Sluz, H. A. Pike, L. B. Stotts, “Analysis of link performance for the FOENEX laser communications system,” Proc. SPIE 8380, 838007 (2012).
[Crossref]

J. C. Juarez, D. W. Young, J. E. Sluz, L. B. Stotts, “High-sensitivity DPSK receiver for high-bandwidth free-space optical communication links,” Opt. Express 19(11), 10789–10796 (2011).
[Crossref] [PubMed]

Klein, B. J.

Kolodzy, P. J.

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, D. W. Young, “Hybrid Optical RF Airborne Communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[Crossref]

Lambert, A. J.

Lane, R. G.

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2(3), 209–224 (1992).
[Crossref]

Mahon, R.

H. R. Burris, M. S. Ferraro, W. Freeman, P. G. Goetz, R. Mahon, C. I. Moore, J. L. Murphy, J. Overfield, W. S. Rabinovich, W. R. Smith, M. R. Suite, L. M. Thomas, B. B. Xu, “Tactical network demonstration with free space lasercomm,” Proc. SPIE 7923, 792305 (2011).
[Crossref]

McGlamery, B. L.

McIntire, W. K.

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, D. W. Young, “Hybrid Optical RF Airborne Communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[Crossref]

Moore, C. I.

H. R. Burris, M. S. Ferraro, W. Freeman, P. G. Goetz, R. Mahon, C. I. Moore, J. L. Murphy, J. Overfield, W. S. Rabinovich, W. R. Smith, M. R. Suite, L. M. Thomas, B. B. Xu, “Tactical network demonstration with free space lasercomm,” Proc. SPIE 7923, 792305 (2011).
[Crossref]

Murphy, J. L.

H. R. Burris, M. S. Ferraro, W. Freeman, P. G. Goetz, R. Mahon, C. I. Moore, J. L. Murphy, J. Overfield, W. S. Rabinovich, W. R. Smith, M. R. Suite, L. M. Thomas, B. B. Xu, “Tactical network demonstration with free space lasercomm,” Proc. SPIE 7923, 792305 (2011).
[Crossref]

Noll, R. J.

Northcott, M.

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, D. W. Young, “Hybrid Optical RF Airborne Communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[Crossref]

Oberc, R. L.

J. C. Juarez, D. W. Young, R. A. Venkat, D. M. Brown, A. M. Brown, R. L. Oberc, J. E. Sluz, H. A. Pike, L. B. Stotts, “Analysis of link performance for the FOENEX laser communications system,” Proc. SPIE 8380, 838007 (2012).
[Crossref]

Overfield, J.

H. R. Burris, M. S. Ferraro, W. Freeman, P. G. Goetz, R. Mahon, C. I. Moore, J. L. Murphy, J. Overfield, W. S. Rabinovich, W. R. Smith, M. R. Suite, L. M. Thomas, B. B. Xu, “Tactical network demonstration with free space lasercomm,” Proc. SPIE 7923, 792305 (2011).
[Crossref]

Parenti, R. R.

L. C. Andrews, R. L. Phillips, R. J. Sasiela, R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45(7), 076001 (2006).
[Crossref]

Phillips, R. L.

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, D. W. Young, “Hybrid Optical RF Airborne Communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[Crossref]

L. C. Andrews, R. L. Phillips, R. J. Sasiela, R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45(7), 076001 (2006).
[Crossref]

Pike, H. A.

J. C. Juarez, D. W. Young, R. A. Venkat, D. M. Brown, A. M. Brown, R. L. Oberc, J. E. Sluz, H. A. Pike, L. B. Stotts, “Analysis of link performance for the FOENEX laser communications system,” Proc. SPIE 8380, 838007 (2012).
[Crossref]

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, D. W. Young, “Hybrid Optical RF Airborne Communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[Crossref]

Rabinovich, W. S.

H. R. Burris, M. S. Ferraro, W. Freeman, P. G. Goetz, R. Mahon, C. I. Moore, J. L. Murphy, J. Overfield, W. S. Rabinovich, W. R. Smith, M. R. Suite, L. M. Thomas, B. B. Xu, “Tactical network demonstration with free space lasercomm,” Proc. SPIE 7923, 792305 (2011).
[Crossref]

Roddier, F.

F. Roddier, “Maximum gain and efficiency of adaptive optics systems,” Publ. Astron. Soc. Pac. 110(749), 837–840 (1998).
[Crossref]

Sasiela, R. J.

L. C. Andrews, R. L. Phillips, R. J. Sasiela, R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45(7), 076001 (2006).
[Crossref]

Sluz, J. E.

J. C. Juarez, D. W. Young, R. A. Venkat, D. M. Brown, A. M. Brown, R. L. Oberc, J. E. Sluz, H. A. Pike, L. B. Stotts, “Analysis of link performance for the FOENEX laser communications system,” Proc. SPIE 8380, 838007 (2012).
[Crossref]

J. C. Juarez, D. W. Young, J. E. Sluz, L. B. Stotts, “High-sensitivity DPSK receiver for high-bandwidth free-space optical communication links,” Opt. Express 19(11), 10789–10796 (2011).
[Crossref] [PubMed]

Smith, W. R.

H. R. Burris, M. S. Ferraro, W. Freeman, P. G. Goetz, R. Mahon, C. I. Moore, J. L. Murphy, J. Overfield, W. S. Rabinovich, W. R. Smith, M. R. Suite, L. M. Thomas, B. B. Xu, “Tactical network demonstration with free space lasercomm,” Proc. SPIE 7923, 792305 (2011).
[Crossref]

Stadler, B.

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, D. W. Young, “Hybrid Optical RF Airborne Communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[Crossref]

Stotts, L. B.

J. C. Juarez, D. W. Young, R. A. Venkat, D. M. Brown, A. M. Brown, R. L. Oberc, J. E. Sluz, H. A. Pike, L. B. Stotts, “Analysis of link performance for the FOENEX laser communications system,” Proc. SPIE 8380, 838007 (2012).
[Crossref]

J. C. Juarez, D. W. Young, J. E. Sluz, L. B. Stotts, “High-sensitivity DPSK receiver for high-bandwidth free-space optical communication links,” Opt. Express 19(11), 10789–10796 (2011).
[Crossref] [PubMed]

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, D. W. Young, “Hybrid Optical RF Airborne Communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[Crossref]

Suite, M. R.

H. R. Burris, M. S. Ferraro, W. Freeman, P. G. Goetz, R. Mahon, C. I. Moore, J. L. Murphy, J. Overfield, W. S. Rabinovich, W. R. Smith, M. R. Suite, L. M. Thomas, B. B. Xu, “Tactical network demonstration with free space lasercomm,” Proc. SPIE 7923, 792305 (2011).
[Crossref]

Thomas, L. M.

H. R. Burris, M. S. Ferraro, W. Freeman, P. G. Goetz, R. Mahon, C. I. Moore, J. L. Murphy, J. Overfield, W. S. Rabinovich, W. R. Smith, M. R. Suite, L. M. Thomas, B. B. Xu, “Tactical network demonstration with free space lasercomm,” Proc. SPIE 7923, 792305 (2011).
[Crossref]

Venkat, R. A.

J. C. Juarez, D. W. Young, R. A. Venkat, D. M. Brown, A. M. Brown, R. L. Oberc, J. E. Sluz, H. A. Pike, L. B. Stotts, “Analysis of link performance for the FOENEX laser communications system,” Proc. SPIE 8380, 838007 (2012).
[Crossref]

Vorontsov, M. A.

T. Weyrauch, M. A. Vorontsov, J. Gowens, T. G. Bifano, “Fiber coupling with adaptive optics for free-space optical communication,” Proc. SPIE 4489, 177–184 (2002).
[Crossref]

Welsh, B. M.

B. M. Welsh, “A Fourier series based atmospheric phase screen generator for simulating anisoplanatic geometries and temporal evolution,” Proc. SPIE 3125, 327–338 (1997).
[Crossref]

Weyrauch, T.

T. Weyrauch, M. A. Vorontsov, J. Gowens, T. G. Bifano, “Fiber coupling with adaptive optics for free-space optical communication,” Proc. SPIE 4489, 177–184 (2002).
[Crossref]

Xu, B. B.

H. R. Burris, M. S. Ferraro, W. Freeman, P. G. Goetz, R. Mahon, C. I. Moore, J. L. Murphy, J. Overfield, W. S. Rabinovich, W. R. Smith, M. R. Suite, L. M. Thomas, B. B. Xu, “Tactical network demonstration with free space lasercomm,” Proc. SPIE 7923, 792305 (2011).
[Crossref]

Yadlowsky, M.

Young, D. W.

J. C. Juarez, D. W. Young, R. A. Venkat, D. M. Brown, A. M. Brown, R. L. Oberc, J. E. Sluz, H. A. Pike, L. B. Stotts, “Analysis of link performance for the FOENEX laser communications system,” Proc. SPIE 8380, 838007 (2012).
[Crossref]

J. C. Juarez, D. W. Young, J. E. Sluz, L. B. Stotts, “High-sensitivity DPSK receiver for high-bandwidth free-space optical communication links,” Opt. Express 19(11), 10789–10796 (2011).
[Crossref] [PubMed]

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, D. W. Young, “Hybrid Optical RF Airborne Communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[Crossref]

Yura, H. T.

Appl. Opt. (3)

J. Opt. Soc. Am. (3)

Opt. Eng. (1)

L. C. Andrews, R. L. Phillips, R. J. Sasiela, R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45(7), 076001 (2006).
[Crossref]

Opt. Express (1)

Proc. IEEE (1)

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, D. W. Young, “Hybrid Optical RF Airborne Communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[Crossref]

Proc. SPIE (5)

T. Weyrauch, M. A. Vorontsov, J. Gowens, T. G. Bifano, “Fiber coupling with adaptive optics for free-space optical communication,” Proc. SPIE 4489, 177–184 (2002).
[Crossref]

J. C. Juarez, D. W. Young, R. A. Venkat, D. M. Brown, A. M. Brown, R. L. Oberc, J. E. Sluz, H. A. Pike, L. B. Stotts, “Analysis of link performance for the FOENEX laser communications system,” Proc. SPIE 8380, 838007 (2012).
[Crossref]

D. S. Grinch, J. A. Cunningham, D. S. Fisher, “Laser system for cooperative pointing and tracking of moving terminals over long distance,” Proc. SPIE 6238, 623803 (2006).
[Crossref]

H. R. Burris, M. S. Ferraro, W. Freeman, P. G. Goetz, R. Mahon, C. I. Moore, J. L. Murphy, J. Overfield, W. S. Rabinovich, W. R. Smith, M. R. Suite, L. M. Thomas, B. B. Xu, “Tactical network demonstration with free space lasercomm,” Proc. SPIE 7923, 792305 (2011).
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[Crossref]

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[Crossref]

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[Crossref]

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Figures (6)

Fig. 1
Fig. 1 Pupil plane simulations of (a) random aberrated wavefront with r0 = 2.5 cm; (b) Zernike decomposition of first 3 modes of the aberrated wavefront; (c) Zernike decomposition of first 18 modes of the aberrated wavefront.
Fig. 2
Fig. 2 Focal plane intensities for (a) an uncompensated wavefront with r0 = 2.5 cm; (b) wavefront with tip/tilt compensation, and; (c) a wavefront with the first 18 Zernike modes of compensation.
Fig. 3
Fig. 3 Comparison of Strehl ratio theory and simulation results for uncorrected and tip/tilt corrected wavefronts. The solid lines represent theoretical models while solid markers represent simulation results. Maréchal – Maréchal approximation; Uncom. A&P Weak – Uncompensated weak turbulence (Andrews & Phillips), Uncom A&P Gen. – Uncompensated general turbulence (Andrews & Phillips); TT Roddier – Tip/tilt compensation (Roddier); TT A&P Sim – Tip/tilt compensation (Andrews & Phillips Simulation); TT A&P – Tip/tilt compensation (Andrews & Phillips general turbulence); SR Uncom – Simulation Strehl ratio, uncompensated; SR TT – Simulation Strehl ratio, tip/tilt compensated
Fig. 4
Fig. 4 Comparison of Strehl ratio theory and simulation results for 6 and 21 Zernike modes or actuators channels of correction. The solid lines represent theoretical models while the dotted lines represent simulation results.
Fig. 5
Fig. 5 Comparison of Strehl ratio simulation results and proposed model for compensation with 3, 10, 21, and 36 Zernike modes or actuators channels. The solid lines represent theoretical models while the dotted lines represent simulation results.
Fig. 6
Fig. 6 Total antenna gain vs. aperture diameter at λ = 1.55 μm for different compensation levels with turbulence of r0 = 1 cm.

Equations (29)

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r o = [ 0.423 k 2 0 L C n 2 (z) ( z/L ) 5/3 dz ] 3/5
I o =P π D 2 4 λ 2 f 2 ,
σ 2 = 1 π 0 2π 0 1 [ W(r,θ) W ¯ ] 2 rdrdθ,
Sexp( σ 2 ),
σ 2 =1.03(D/ r o ) 5/3 .
S= [1+ σ 2 ] 1 =[1+(D/ r o ) 5/3 ] 1 , D / r o <1 .
S[1+(D/ r o ) 5/3 ] 6/5 , 0D / r o <.
P(x,y)= P A (x,y) e i2πW(x,y) ,
Z (r,θ) i ={ 2(n+1) R n m (r) G m (θ),m0 R n m (r),m=0 ,
R n m (r)= s=0 (nm) /2 (1) s (ns)! s!( n+m 2 s )!( nm 2 s )! r n2s ,
G m (θ)={ sin(mθ),iodd cos(mθ),ieven .
W = j=1 a j Z j (r,θ) ,
a i = 0 2π 0 1 W(r,θ) Z i (r,θ)rdrdθ 0 2π 0 1 Z i 2 (r,θ)rdrdθ .
W C = j=1 J a j Z j .
σ J 2 = 1 π 0 2π 0 1 [ W(r,θ) W C ] 2 rdrdθ.
σ J 2 = W (r,θ) 2 j=2 J | a j | 2 ,
σ J 2 0.2944 J 3 /2 (D/ r o ) 5/3 .
σ J 2 k(D/ r o ) 5/3 N 5/6 ,
S com exp( k (D/ r o ) 5/3 N 5/6 ),
S com [1+k (D/ r o ) 5/3 N 5/6 ] 6/5 , 0D / r o <.
S tt { 1+[ 0.983 0.856 1+0.0071 (D/ r o ) 5/3 ]× (D/ r o ) 5/3 } 6/5 , 0D / r o <.
( y 2 n 2 y 2 )=( A B C D )( y 1 n 1 y 1 ),
U(A r 2 )= 1 iλB e iπβ r 2 2 [U( r 1 ) e iπα r 1 2 ],
ϕ(x,y)= Ψ( f x , f y ) e i2π( f x x+ f y y) d f x df , y
ϕ(x,y)= n= m= c n,m exp[i2π( f x n x+ f y m y)],
| c n,m | 2 = Φ ϕ ( f x n , f y m )Δ f x n Δ f y m .
S com [1+ σ J 2 +0.5 σ J 4 +0.167 σ J 6 ] 6/5 + [ k 2 1 +( k 2 +0.35) (D/ r o ) 5/3 ] 1.3 , 0D / r o <.
G a =10 log 10 ( πD λ ) 2 (dBi).
G Tot =10 log 10 [ ( πD λ ) 2 × S com ](dBi).

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