Abstract

This paper investigates atmospheric array tilt and its effect on target-in-the-loop optical phased array (OPA) performance. Assuming a direct-solve, piston-only-phase-compensation OPA, two expressions for the atmospheric array tilt variance are derived using Mellin transform techniques. The first—the “full” array tilt variance—is germane when the OPA is sensitive to atmospheric tilt and is shown to significantly impact OPA target-plane intensity. The second—the Zernike-tilt-removed array tilt variance—is relevant when a separate system compensates for atmospheric tilt (the more likely scenario) and is shown to negligibly affect OPA performance. To show how atmospheric array tilt errors affect target-plane intensity, moments of the far-zone (or focused) array intensity, as functions of the array tilt variance, are derived and discussed. Lastly, Monte Carlo simulation results are presented to validate the theoretical array tilt variance expressions.

Full Article  |  PDF Article
OSA Recommended Articles
Rigorous investigation of the array-tilt aberration for hexagonal, optical phased arrays

Milo W. Hyde, Jason E. Wyman, and Glenn A. Tyler
Appl. Opt. 53(11) 2416-2424 (2014)

Comparative efficiency analysis of fiber-array and conventional beam director systems in volume turbulence

Mikhail Vorontsov, Grigory Filimonov, Vladimir Ovchinnikov, Ernst Polnau, Svetlana Lachinova, Thomas Weyrauch, and Joseph Mangano
Appl. Opt. 55(15) 4170-4185 (2016)

Coherent optical array receivers for the mitigation of atmospheric turbulence and speckle effects

Philip Gatt, Thomas P. Costello, Dean A. Heimmermann, Diana C. Castellanos, Arthur R. Weeks, and C. Martin Stickley
Appl. Opt. 35(30) 5999-6009 (1996)

References

  • View by:
  • |
  • |
  • |

  1. A. Brignon, ed., Coherent Laser Beam Combining (Wiley-VCH, 2013).
  2. B. N. Pulford, “LOCSET phase locking: operation, diagnostics, and applications,” Ph.D. thesis (The University of New Mexico, 2011).
  3. T. M. Shay, “Theory of electronically phased coherent beam combination without a reference beam,” Opt. Express 14, 12188–12195 (2006).
    [Crossref]
  4. M. A. Vorontsov, T. Weyrauch, L. A. Beresnev, G. W. Carhart, L. Liu, and K. Aschenbach, “Adaptive array of phase-locked fiber collimators: analysis and experimental demonstration,” IEEE J. Sel. Top. Quantum Electron. 15, 269–280 (2009).
    [Crossref]
  5. M. Vorontsov, G. Filimonov, V. Ovchinnikov, E. Polnau, S. Lachinova, T. Weyrauch, and J. Mangano, “Comparative efficiency analysis of fiber-array and conventional beam director systems in volume turbulence,” Appl. Opt. 55, 4170–4185 (2016).
    [Crossref]
  6. G. D. Goodno and J. E. Rothenberg, “Atmospheric propagation and combining of high power lasers: comment,” Appl. Opt. 55, 8335–8337 (2016).
    [Crossref]
  7. M. A. Vorontsov and T. Weyrauch, “High-power lasers for directed-energy applications: comment,” Appl. Opt. 55, 9950–9953 (2016).
    [Crossref]
  8. S. J. McNaught, P. A. Thielen, L. N. Adams, J. G. Ho, A. M. Johnson, J. P. Machan, J. E. Rothenberg, C. C. Shih, D. M. Shimabukuro, M. P. Wacks, M. E. Weber, and G. D. Goodno, “Scalable coherent combining of kilowatt fiber amplifiers into a 2.4-kW beam,” IEEE J. Sel. Top. Quantum Electron. 20, 174–181 (2014).
    [Crossref]
  9. A. Flores, T. Ehrehreich, R. Holten, B. Anderson, and I. Dajani, “Multi-kW coherent combining of fiber lasers seeded with pseudo random phase modulated light,” Proc. SPIE 9728, 97281Y (2016).
    [Crossref]
  10. P. Sprangle, B. Hafizi, A. Ting, and R. Fischer, “High-power lasers for directed-energy applications,” Appl. Opt. 54, F201–F209 (2015).
    [Crossref]
  11. W. Nelson, P. Sprangle, and C. C. Davis, “Atmospheric propagation and combining of high-power lasers,” Appl. Opt. 55, 1757–1764 (2016).
    [Crossref]
  12. M. A. Vorontsov and S. L. Lachinova, “Laser beam projection with adaptive array of fiber collimators. I. Basic considerations for analysis,” J. Opt. Soc. Am. A 25, 1949–1959 (2008).
    [Crossref]
  13. S. L. Lachinova and M. A. Vorontsov, “Laser beam projection with adaptive array of fiber collimators. II. Analysis of atmospheric compensation efficiency,” J. Opt. Soc. Am. A 25, 1960–1973 (2008).
    [Crossref]
  14. M. A. Vorontsov, “Speckle effects in target-in-the-loop laser beam projection systems,” Adv. Opt. Technol. 2, 369–395 (2013).
    [Crossref]
  15. T. Weyrauch, M. Vorontsov, J. Mangano, V. Ovchinnikov, D. Bricker, E. Polnau, and A. Rostov, “Deep turbulence effects mitigation with coherent combining of 21 laser beams over 7  km,” Opt. Lett. 41, 840–843 (2016).
    [Crossref]
  16. V. Jolivet, P. Bourdon, B. Bennal, L. Lombard, D. Goular, E. Pourtal, G. Canat, Y. Jaouen, B. Moreau, and O. Vasseur, “Beam shaping of single-mode and multimode fiber amplifier arrays for propagation through atmospheric turbulence,” IEEE J. Sel. Top. Quantum Electron. 15, 257–268 (2009).
    [Crossref]
  17. Y. Ma, P. Zhou, R. Tao, L. Si, and Z. Liu, “Target-in-the-loop coherent beam combination of 100  W level fiber laser array based on an extended target with a scattering surface,” Opt. Lett. 38, 1019–1021 (2013).
    [Crossref]
  18. R. Tao, Y. Ma, L. Si, X. Dong, P. Zhou, and Z. Liu, “Target-in-the-loop high-power adaptive phase-locked fiber laser array using single-frequency dithering technique,” Appl. Phys. B 105, 285–291 (2011).
    [Crossref]
  19. G. A. Tyler, “Accommodation of speckle in object-based phasing,” J. Opt. Soc. Am. A 29, 722–733 (2012).
    [Crossref]
  20. In the case of SPGD, the metric that is maximized or minimized to yield phase estimates is typically based on the statistics of the received speckle pattern [1, 14]. The target-induced or speckle phases are not directly estimated.
  21. J. F. Riker, G. A. Tyler, and J. L. Vaughn, “Speckle imaging from an array,” Proc. SPIE 9982, 99820J (2016).
    [Crossref]
  22. M. W. Hyde, J. E. Wyman, and G. A. Tyler, “Rigorous investigation of the array-tilt aberration for hexagonal, optical phased arrays,” Appl. Opt. 53, 2416–2424 (2014).
    [Crossref]
  23. V. N. Mahajan, Optical Imaging and Aberrations, Part III: Wavefront Analysis (SPIE, 2013).
  24. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [Crossref]
  25. M. F. Spencer and M. W. Hyde, “An investigation of stair mode in optical phased arrays using tiled apertures,” Proc. SPIE 8520, 852006 (2012).
    [Crossref]
  26. R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms, 2nd ed. (SPIE, 2007).
  27. M. C. Roggemann and B. M. Welsh, Imaging Through Turbulence (CRC Press, 1996).
  28. L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE, 2005).
  29. J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University, 1998).
  30. R. K. Tyson, Principles of Adaptive Optics, 4th ed. (CRC Press, 2015).
  31. Under normal operating conditions (i.e., d<r0), an initially uncorrected array will produce a focused spot on the target with a diameter proportional to λz/d, where z is the distance to the target. The angle this spot subtends when viewed from the array is λ/d. Since the array uses the scattered return from this spot to phase on the target, θ0 must be greater than λ/d, otherwise array performance is limited by θ0 and not diffraction.
  32. R. A. Motes, S. A. Shakir, and R. W. Berdine, Introduction to High-Power Fiber Lasers, 2nd ed. (Directed Energy Professional Society, 2013).
  33. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).
  34. J. W. Goodman, Statistical Optics, 2nd ed. (Wiley, 2015).
  35. D. M. Winker, “Effect of a finite outer scale on the Zernike decomposition of atmospheric optical turbulence,” J. Opt. Soc. Am. A 8, 1568–1573 (1991).
    [Crossref]
  36. J. P. Bos, V. S. Rao Gudimetla, and J. D. Schmidt, “Differential piston phase variance in non-Kolmogorov atmospheres,” J. Opt. Soc. Am. A 34, 1433–1440 (2017).
    [Crossref]
  37. J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010).
  38. J. P. Bos, M. C. Roggemann, and V. S. Rao Gudimetla, “Anisotropic non-Kolmogorov turbulence phase screens with variable orientation,” Appl. Opt. 54, 2039–2045 (2015).
    [Crossref]
  39. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).
  40. M. W. Hyde, J. E. McCrae, and G. A. Tyler, “Target-based coherent beam combining of an optical phased array fed by a broadband laser source,” J. Mod. Opt. 64, 2149–2156 (2017).
    [Crossref]
  41. M. W. Hyde, G. A. Tyler, and C. R. Garcia, “Target-in-the-loop phasing of a fiber laser array fed by a linewidth-broadened master oscillator,” Proc. SPIE 10192, 101920K (2017).
    [Crossref]
  42. The small difference between Zernike tilt and gradient tilt—measured by a tracking system employing a centroid sensor—is ignored [26].
  43. J. Wyman and M. W. Hyde, “Detection and correction of stair mode across an optical phased array,” in IEEE Aerospace Conference (2014), pp. 1–10.
  44. J. E. McCrae and S. T. Fiorino, “Simulation of array tilt effects in laser phased arrays,” in IEEE Aerospace Conference (2016), pp. 1–7.

2017 (3)

J. P. Bos, V. S. Rao Gudimetla, and J. D. Schmidt, “Differential piston phase variance in non-Kolmogorov atmospheres,” J. Opt. Soc. Am. A 34, 1433–1440 (2017).
[Crossref]

M. W. Hyde, J. E. McCrae, and G. A. Tyler, “Target-based coherent beam combining of an optical phased array fed by a broadband laser source,” J. Mod. Opt. 64, 2149–2156 (2017).
[Crossref]

M. W. Hyde, G. A. Tyler, and C. R. Garcia, “Target-in-the-loop phasing of a fiber laser array fed by a linewidth-broadened master oscillator,” Proc. SPIE 10192, 101920K (2017).
[Crossref]

2016 (7)

2015 (2)

2014 (2)

M. W. Hyde, J. E. Wyman, and G. A. Tyler, “Rigorous investigation of the array-tilt aberration for hexagonal, optical phased arrays,” Appl. Opt. 53, 2416–2424 (2014).
[Crossref]

S. J. McNaught, P. A. Thielen, L. N. Adams, J. G. Ho, A. M. Johnson, J. P. Machan, J. E. Rothenberg, C. C. Shih, D. M. Shimabukuro, M. P. Wacks, M. E. Weber, and G. D. Goodno, “Scalable coherent combining of kilowatt fiber amplifiers into a 2.4-kW beam,” IEEE J. Sel. Top. Quantum Electron. 20, 174–181 (2014).
[Crossref]

2013 (2)

2012 (2)

G. A. Tyler, “Accommodation of speckle in object-based phasing,” J. Opt. Soc. Am. A 29, 722–733 (2012).
[Crossref]

M. F. Spencer and M. W. Hyde, “An investigation of stair mode in optical phased arrays using tiled apertures,” Proc. SPIE 8520, 852006 (2012).
[Crossref]

2011 (1)

R. Tao, Y. Ma, L. Si, X. Dong, P. Zhou, and Z. Liu, “Target-in-the-loop high-power adaptive phase-locked fiber laser array using single-frequency dithering technique,” Appl. Phys. B 105, 285–291 (2011).
[Crossref]

2009 (2)

V. Jolivet, P. Bourdon, B. Bennal, L. Lombard, D. Goular, E. Pourtal, G. Canat, Y. Jaouen, B. Moreau, and O. Vasseur, “Beam shaping of single-mode and multimode fiber amplifier arrays for propagation through atmospheric turbulence,” IEEE J. Sel. Top. Quantum Electron. 15, 257–268 (2009).
[Crossref]

M. A. Vorontsov, T. Weyrauch, L. A. Beresnev, G. W. Carhart, L. Liu, and K. Aschenbach, “Adaptive array of phase-locked fiber collimators: analysis and experimental demonstration,” IEEE J. Sel. Top. Quantum Electron. 15, 269–280 (2009).
[Crossref]

2008 (2)

2006 (1)

1991 (1)

1976 (1)

Adams, L. N.

S. J. McNaught, P. A. Thielen, L. N. Adams, J. G. Ho, A. M. Johnson, J. P. Machan, J. E. Rothenberg, C. C. Shih, D. M. Shimabukuro, M. P. Wacks, M. E. Weber, and G. D. Goodno, “Scalable coherent combining of kilowatt fiber amplifiers into a 2.4-kW beam,” IEEE J. Sel. Top. Quantum Electron. 20, 174–181 (2014).
[Crossref]

Anderson, B.

A. Flores, T. Ehrehreich, R. Holten, B. Anderson, and I. Dajani, “Multi-kW coherent combining of fiber lasers seeded with pseudo random phase modulated light,” Proc. SPIE 9728, 97281Y (2016).
[Crossref]

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE, 2005).

Aschenbach, K.

M. A. Vorontsov, T. Weyrauch, L. A. Beresnev, G. W. Carhart, L. Liu, and K. Aschenbach, “Adaptive array of phase-locked fiber collimators: analysis and experimental demonstration,” IEEE J. Sel. Top. Quantum Electron. 15, 269–280 (2009).
[Crossref]

Bennal, B.

V. Jolivet, P. Bourdon, B. Bennal, L. Lombard, D. Goular, E. Pourtal, G. Canat, Y. Jaouen, B. Moreau, and O. Vasseur, “Beam shaping of single-mode and multimode fiber amplifier arrays for propagation through atmospheric turbulence,” IEEE J. Sel. Top. Quantum Electron. 15, 257–268 (2009).
[Crossref]

Berdine, R. W.

R. A. Motes, S. A. Shakir, and R. W. Berdine, Introduction to High-Power Fiber Lasers, 2nd ed. (Directed Energy Professional Society, 2013).

Beresnev, L. A.

M. A. Vorontsov, T. Weyrauch, L. A. Beresnev, G. W. Carhart, L. Liu, and K. Aschenbach, “Adaptive array of phase-locked fiber collimators: analysis and experimental demonstration,” IEEE J. Sel. Top. Quantum Electron. 15, 269–280 (2009).
[Crossref]

Bos, J. P.

Bourdon, P.

V. Jolivet, P. Bourdon, B. Bennal, L. Lombard, D. Goular, E. Pourtal, G. Canat, Y. Jaouen, B. Moreau, and O. Vasseur, “Beam shaping of single-mode and multimode fiber amplifier arrays for propagation through atmospheric turbulence,” IEEE J. Sel. Top. Quantum Electron. 15, 257–268 (2009).
[Crossref]

Bricker, D.

Canat, G.

V. Jolivet, P. Bourdon, B. Bennal, L. Lombard, D. Goular, E. Pourtal, G. Canat, Y. Jaouen, B. Moreau, and O. Vasseur, “Beam shaping of single-mode and multimode fiber amplifier arrays for propagation through atmospheric turbulence,” IEEE J. Sel. Top. Quantum Electron. 15, 257–268 (2009).
[Crossref]

Carhart, G. W.

M. A. Vorontsov, T. Weyrauch, L. A. Beresnev, G. W. Carhart, L. Liu, and K. Aschenbach, “Adaptive array of phase-locked fiber collimators: analysis and experimental demonstration,” IEEE J. Sel. Top. Quantum Electron. 15, 269–280 (2009).
[Crossref]

Dajani, I.

A. Flores, T. Ehrehreich, R. Holten, B. Anderson, and I. Dajani, “Multi-kW coherent combining of fiber lasers seeded with pseudo random phase modulated light,” Proc. SPIE 9728, 97281Y (2016).
[Crossref]

Davis, C. C.

Dong, X.

R. Tao, Y. Ma, L. Si, X. Dong, P. Zhou, and Z. Liu, “Target-in-the-loop high-power adaptive phase-locked fiber laser array using single-frequency dithering technique,” Appl. Phys. B 105, 285–291 (2011).
[Crossref]

Ehrehreich, T.

A. Flores, T. Ehrehreich, R. Holten, B. Anderson, and I. Dajani, “Multi-kW coherent combining of fiber lasers seeded with pseudo random phase modulated light,” Proc. SPIE 9728, 97281Y (2016).
[Crossref]

Filimonov, G.

Fiorino, S. T.

J. E. McCrae and S. T. Fiorino, “Simulation of array tilt effects in laser phased arrays,” in IEEE Aerospace Conference (2016), pp. 1–7.

Fischer, R.

Flores, A.

A. Flores, T. Ehrehreich, R. Holten, B. Anderson, and I. Dajani, “Multi-kW coherent combining of fiber lasers seeded with pseudo random phase modulated light,” Proc. SPIE 9728, 97281Y (2016).
[Crossref]

Garcia, C. R.

M. W. Hyde, G. A. Tyler, and C. R. Garcia, “Target-in-the-loop phasing of a fiber laser array fed by a linewidth-broadened master oscillator,” Proc. SPIE 10192, 101920K (2017).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

J. W. Goodman, Statistical Optics, 2nd ed. (Wiley, 2015).

Goodno, G. D.

G. D. Goodno and J. E. Rothenberg, “Atmospheric propagation and combining of high power lasers: comment,” Appl. Opt. 55, 8335–8337 (2016).
[Crossref]

S. J. McNaught, P. A. Thielen, L. N. Adams, J. G. Ho, A. M. Johnson, J. P. Machan, J. E. Rothenberg, C. C. Shih, D. M. Shimabukuro, M. P. Wacks, M. E. Weber, and G. D. Goodno, “Scalable coherent combining of kilowatt fiber amplifiers into a 2.4-kW beam,” IEEE J. Sel. Top. Quantum Electron. 20, 174–181 (2014).
[Crossref]

Goular, D.

V. Jolivet, P. Bourdon, B. Bennal, L. Lombard, D. Goular, E. Pourtal, G. Canat, Y. Jaouen, B. Moreau, and O. Vasseur, “Beam shaping of single-mode and multimode fiber amplifier arrays for propagation through atmospheric turbulence,” IEEE J. Sel. Top. Quantum Electron. 15, 257–268 (2009).
[Crossref]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).

Hafizi, B.

Hardy, J. W.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University, 1998).

Ho, J. G.

S. J. McNaught, P. A. Thielen, L. N. Adams, J. G. Ho, A. M. Johnson, J. P. Machan, J. E. Rothenberg, C. C. Shih, D. M. Shimabukuro, M. P. Wacks, M. E. Weber, and G. D. Goodno, “Scalable coherent combining of kilowatt fiber amplifiers into a 2.4-kW beam,” IEEE J. Sel. Top. Quantum Electron. 20, 174–181 (2014).
[Crossref]

Holten, R.

A. Flores, T. Ehrehreich, R. Holten, B. Anderson, and I. Dajani, “Multi-kW coherent combining of fiber lasers seeded with pseudo random phase modulated light,” Proc. SPIE 9728, 97281Y (2016).
[Crossref]

Hyde, M. W.

M. W. Hyde, J. E. McCrae, and G. A. Tyler, “Target-based coherent beam combining of an optical phased array fed by a broadband laser source,” J. Mod. Opt. 64, 2149–2156 (2017).
[Crossref]

M. W. Hyde, G. A. Tyler, and C. R. Garcia, “Target-in-the-loop phasing of a fiber laser array fed by a linewidth-broadened master oscillator,” Proc. SPIE 10192, 101920K (2017).
[Crossref]

M. W. Hyde, J. E. Wyman, and G. A. Tyler, “Rigorous investigation of the array-tilt aberration for hexagonal, optical phased arrays,” Appl. Opt. 53, 2416–2424 (2014).
[Crossref]

M. F. Spencer and M. W. Hyde, “An investigation of stair mode in optical phased arrays using tiled apertures,” Proc. SPIE 8520, 852006 (2012).
[Crossref]

J. Wyman and M. W. Hyde, “Detection and correction of stair mode across an optical phased array,” in IEEE Aerospace Conference (2014), pp. 1–10.

Jaouen, Y.

V. Jolivet, P. Bourdon, B. Bennal, L. Lombard, D. Goular, E. Pourtal, G. Canat, Y. Jaouen, B. Moreau, and O. Vasseur, “Beam shaping of single-mode and multimode fiber amplifier arrays for propagation through atmospheric turbulence,” IEEE J. Sel. Top. Quantum Electron. 15, 257–268 (2009).
[Crossref]

Johnson, A. M.

S. J. McNaught, P. A. Thielen, L. N. Adams, J. G. Ho, A. M. Johnson, J. P. Machan, J. E. Rothenberg, C. C. Shih, D. M. Shimabukuro, M. P. Wacks, M. E. Weber, and G. D. Goodno, “Scalable coherent combining of kilowatt fiber amplifiers into a 2.4-kW beam,” IEEE J. Sel. Top. Quantum Electron. 20, 174–181 (2014).
[Crossref]

Jolivet, V.

V. Jolivet, P. Bourdon, B. Bennal, L. Lombard, D. Goular, E. Pourtal, G. Canat, Y. Jaouen, B. Moreau, and O. Vasseur, “Beam shaping of single-mode and multimode fiber amplifier arrays for propagation through atmospheric turbulence,” IEEE J. Sel. Top. Quantum Electron. 15, 257–268 (2009).
[Crossref]

Lachinova, S.

Lachinova, S. L.

Liu, L.

M. A. Vorontsov, T. Weyrauch, L. A. Beresnev, G. W. Carhart, L. Liu, and K. Aschenbach, “Adaptive array of phase-locked fiber collimators: analysis and experimental demonstration,” IEEE J. Sel. Top. Quantum Electron. 15, 269–280 (2009).
[Crossref]

Liu, Z.

Y. Ma, P. Zhou, R. Tao, L. Si, and Z. Liu, “Target-in-the-loop coherent beam combination of 100  W level fiber laser array based on an extended target with a scattering surface,” Opt. Lett. 38, 1019–1021 (2013).
[Crossref]

R. Tao, Y. Ma, L. Si, X. Dong, P. Zhou, and Z. Liu, “Target-in-the-loop high-power adaptive phase-locked fiber laser array using single-frequency dithering technique,” Appl. Phys. B 105, 285–291 (2011).
[Crossref]

Lombard, L.

V. Jolivet, P. Bourdon, B. Bennal, L. Lombard, D. Goular, E. Pourtal, G. Canat, Y. Jaouen, B. Moreau, and O. Vasseur, “Beam shaping of single-mode and multimode fiber amplifier arrays for propagation through atmospheric turbulence,” IEEE J. Sel. Top. Quantum Electron. 15, 257–268 (2009).
[Crossref]

Ma, Y.

Y. Ma, P. Zhou, R. Tao, L. Si, and Z. Liu, “Target-in-the-loop coherent beam combination of 100  W level fiber laser array based on an extended target with a scattering surface,” Opt. Lett. 38, 1019–1021 (2013).
[Crossref]

R. Tao, Y. Ma, L. Si, X. Dong, P. Zhou, and Z. Liu, “Target-in-the-loop high-power adaptive phase-locked fiber laser array using single-frequency dithering technique,” Appl. Phys. B 105, 285–291 (2011).
[Crossref]

Machan, J. P.

S. J. McNaught, P. A. Thielen, L. N. Adams, J. G. Ho, A. M. Johnson, J. P. Machan, J. E. Rothenberg, C. C. Shih, D. M. Shimabukuro, M. P. Wacks, M. E. Weber, and G. D. Goodno, “Scalable coherent combining of kilowatt fiber amplifiers into a 2.4-kW beam,” IEEE J. Sel. Top. Quantum Electron. 20, 174–181 (2014).
[Crossref]

Mahajan, V. N.

V. N. Mahajan, Optical Imaging and Aberrations, Part III: Wavefront Analysis (SPIE, 2013).

Mangano, J.

McCrae, J. E.

M. W. Hyde, J. E. McCrae, and G. A. Tyler, “Target-based coherent beam combining of an optical phased array fed by a broadband laser source,” J. Mod. Opt. 64, 2149–2156 (2017).
[Crossref]

J. E. McCrae and S. T. Fiorino, “Simulation of array tilt effects in laser phased arrays,” in IEEE Aerospace Conference (2016), pp. 1–7.

McNaught, S. J.

S. J. McNaught, P. A. Thielen, L. N. Adams, J. G. Ho, A. M. Johnson, J. P. Machan, J. E. Rothenberg, C. C. Shih, D. M. Shimabukuro, M. P. Wacks, M. E. Weber, and G. D. Goodno, “Scalable coherent combining of kilowatt fiber amplifiers into a 2.4-kW beam,” IEEE J. Sel. Top. Quantum Electron. 20, 174–181 (2014).
[Crossref]

Moreau, B.

V. Jolivet, P. Bourdon, B. Bennal, L. Lombard, D. Goular, E. Pourtal, G. Canat, Y. Jaouen, B. Moreau, and O. Vasseur, “Beam shaping of single-mode and multimode fiber amplifier arrays for propagation through atmospheric turbulence,” IEEE J. Sel. Top. Quantum Electron. 15, 257–268 (2009).
[Crossref]

Motes, R. A.

R. A. Motes, S. A. Shakir, and R. W. Berdine, Introduction to High-Power Fiber Lasers, 2nd ed. (Directed Energy Professional Society, 2013).

Nelson, W.

Noll, R. J.

Ovchinnikov, V.

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE, 2005).

Polnau, E.

Pourtal, E.

V. Jolivet, P. Bourdon, B. Bennal, L. Lombard, D. Goular, E. Pourtal, G. Canat, Y. Jaouen, B. Moreau, and O. Vasseur, “Beam shaping of single-mode and multimode fiber amplifier arrays for propagation through atmospheric turbulence,” IEEE J. Sel. Top. Quantum Electron. 15, 257–268 (2009).
[Crossref]

Pulford, B. N.

B. N. Pulford, “LOCSET phase locking: operation, diagnostics, and applications,” Ph.D. thesis (The University of New Mexico, 2011).

Rao Gudimetla, V. S.

Riker, J. F.

J. F. Riker, G. A. Tyler, and J. L. Vaughn, “Speckle imaging from an array,” Proc. SPIE 9982, 99820J (2016).
[Crossref]

Roggemann, M. C.

Rostov, A.

Rothenberg, J. E.

G. D. Goodno and J. E. Rothenberg, “Atmospheric propagation and combining of high power lasers: comment,” Appl. Opt. 55, 8335–8337 (2016).
[Crossref]

S. J. McNaught, P. A. Thielen, L. N. Adams, J. G. Ho, A. M. Johnson, J. P. Machan, J. E. Rothenberg, C. C. Shih, D. M. Shimabukuro, M. P. Wacks, M. E. Weber, and G. D. Goodno, “Scalable coherent combining of kilowatt fiber amplifiers into a 2.4-kW beam,” IEEE J. Sel. Top. Quantum Electron. 20, 174–181 (2014).
[Crossref]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).

Sasiela, R. J.

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms, 2nd ed. (SPIE, 2007).

Schmidt, J. D.

Shakir, S. A.

R. A. Motes, S. A. Shakir, and R. W. Berdine, Introduction to High-Power Fiber Lasers, 2nd ed. (Directed Energy Professional Society, 2013).

Shay, T. M.

Shih, C. C.

S. J. McNaught, P. A. Thielen, L. N. Adams, J. G. Ho, A. M. Johnson, J. P. Machan, J. E. Rothenberg, C. C. Shih, D. M. Shimabukuro, M. P. Wacks, M. E. Weber, and G. D. Goodno, “Scalable coherent combining of kilowatt fiber amplifiers into a 2.4-kW beam,” IEEE J. Sel. Top. Quantum Electron. 20, 174–181 (2014).
[Crossref]

Shimabukuro, D. M.

S. J. McNaught, P. A. Thielen, L. N. Adams, J. G. Ho, A. M. Johnson, J. P. Machan, J. E. Rothenberg, C. C. Shih, D. M. Shimabukuro, M. P. Wacks, M. E. Weber, and G. D. Goodno, “Scalable coherent combining of kilowatt fiber amplifiers into a 2.4-kW beam,” IEEE J. Sel. Top. Quantum Electron. 20, 174–181 (2014).
[Crossref]

Si, L.

Y. Ma, P. Zhou, R. Tao, L. Si, and Z. Liu, “Target-in-the-loop coherent beam combination of 100  W level fiber laser array based on an extended target with a scattering surface,” Opt. Lett. 38, 1019–1021 (2013).
[Crossref]

R. Tao, Y. Ma, L. Si, X. Dong, P. Zhou, and Z. Liu, “Target-in-the-loop high-power adaptive phase-locked fiber laser array using single-frequency dithering technique,” Appl. Phys. B 105, 285–291 (2011).
[Crossref]

Spencer, M. F.

M. F. Spencer and M. W. Hyde, “An investigation of stair mode in optical phased arrays using tiled apertures,” Proc. SPIE 8520, 852006 (2012).
[Crossref]

Sprangle, P.

Tao, R.

Y. Ma, P. Zhou, R. Tao, L. Si, and Z. Liu, “Target-in-the-loop coherent beam combination of 100  W level fiber laser array based on an extended target with a scattering surface,” Opt. Lett. 38, 1019–1021 (2013).
[Crossref]

R. Tao, Y. Ma, L. Si, X. Dong, P. Zhou, and Z. Liu, “Target-in-the-loop high-power adaptive phase-locked fiber laser array using single-frequency dithering technique,” Appl. Phys. B 105, 285–291 (2011).
[Crossref]

Thielen, P. A.

S. J. McNaught, P. A. Thielen, L. N. Adams, J. G. Ho, A. M. Johnson, J. P. Machan, J. E. Rothenberg, C. C. Shih, D. M. Shimabukuro, M. P. Wacks, M. E. Weber, and G. D. Goodno, “Scalable coherent combining of kilowatt fiber amplifiers into a 2.4-kW beam,” IEEE J. Sel. Top. Quantum Electron. 20, 174–181 (2014).
[Crossref]

Ting, A.

Tyler, G. A.

M. W. Hyde, J. E. McCrae, and G. A. Tyler, “Target-based coherent beam combining of an optical phased array fed by a broadband laser source,” J. Mod. Opt. 64, 2149–2156 (2017).
[Crossref]

M. W. Hyde, G. A. Tyler, and C. R. Garcia, “Target-in-the-loop phasing of a fiber laser array fed by a linewidth-broadened master oscillator,” Proc. SPIE 10192, 101920K (2017).
[Crossref]

J. F. Riker, G. A. Tyler, and J. L. Vaughn, “Speckle imaging from an array,” Proc. SPIE 9982, 99820J (2016).
[Crossref]

M. W. Hyde, J. E. Wyman, and G. A. Tyler, “Rigorous investigation of the array-tilt aberration for hexagonal, optical phased arrays,” Appl. Opt. 53, 2416–2424 (2014).
[Crossref]

G. A. Tyler, “Accommodation of speckle in object-based phasing,” J. Opt. Soc. Am. A 29, 722–733 (2012).
[Crossref]

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics, 4th ed. (CRC Press, 2015).

Vasseur, O.

V. Jolivet, P. Bourdon, B. Bennal, L. Lombard, D. Goular, E. Pourtal, G. Canat, Y. Jaouen, B. Moreau, and O. Vasseur, “Beam shaping of single-mode and multimode fiber amplifier arrays for propagation through atmospheric turbulence,” IEEE J. Sel. Top. Quantum Electron. 15, 257–268 (2009).
[Crossref]

Vaughn, J. L.

J. F. Riker, G. A. Tyler, and J. L. Vaughn, “Speckle imaging from an array,” Proc. SPIE 9982, 99820J (2016).
[Crossref]

Vorontsov, M.

Vorontsov, M. A.

Wacks, M. P.

S. J. McNaught, P. A. Thielen, L. N. Adams, J. G. Ho, A. M. Johnson, J. P. Machan, J. E. Rothenberg, C. C. Shih, D. M. Shimabukuro, M. P. Wacks, M. E. Weber, and G. D. Goodno, “Scalable coherent combining of kilowatt fiber amplifiers into a 2.4-kW beam,” IEEE J. Sel. Top. Quantum Electron. 20, 174–181 (2014).
[Crossref]

Weber, M. E.

S. J. McNaught, P. A. Thielen, L. N. Adams, J. G. Ho, A. M. Johnson, J. P. Machan, J. E. Rothenberg, C. C. Shih, D. M. Shimabukuro, M. P. Wacks, M. E. Weber, and G. D. Goodno, “Scalable coherent combining of kilowatt fiber amplifiers into a 2.4-kW beam,” IEEE J. Sel. Top. Quantum Electron. 20, 174–181 (2014).
[Crossref]

Welsh, B. M.

M. C. Roggemann and B. M. Welsh, Imaging Through Turbulence (CRC Press, 1996).

Weyrauch, T.

Winker, D. M.

Wyman, J.

J. Wyman and M. W. Hyde, “Detection and correction of stair mode across an optical phased array,” in IEEE Aerospace Conference (2014), pp. 1–10.

Wyman, J. E.

Zhou, P.

Y. Ma, P. Zhou, R. Tao, L. Si, and Z. Liu, “Target-in-the-loop coherent beam combination of 100  W level fiber laser array based on an extended target with a scattering surface,” Opt. Lett. 38, 1019–1021 (2013).
[Crossref]

R. Tao, Y. Ma, L. Si, X. Dong, P. Zhou, and Z. Liu, “Target-in-the-loop high-power adaptive phase-locked fiber laser array using single-frequency dithering technique,” Appl. Phys. B 105, 285–291 (2011).
[Crossref]

Adv. Opt. Technol. (1)

M. A. Vorontsov, “Speckle effects in target-in-the-loop laser beam projection systems,” Adv. Opt. Technol. 2, 369–395 (2013).
[Crossref]

Appl. Opt. (7)

Appl. Phys. B (1)

R. Tao, Y. Ma, L. Si, X. Dong, P. Zhou, and Z. Liu, “Target-in-the-loop high-power adaptive phase-locked fiber laser array using single-frequency dithering technique,” Appl. Phys. B 105, 285–291 (2011).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (3)

S. J. McNaught, P. A. Thielen, L. N. Adams, J. G. Ho, A. M. Johnson, J. P. Machan, J. E. Rothenberg, C. C. Shih, D. M. Shimabukuro, M. P. Wacks, M. E. Weber, and G. D. Goodno, “Scalable coherent combining of kilowatt fiber amplifiers into a 2.4-kW beam,” IEEE J. Sel. Top. Quantum Electron. 20, 174–181 (2014).
[Crossref]

M. A. Vorontsov, T. Weyrauch, L. A. Beresnev, G. W. Carhart, L. Liu, and K. Aschenbach, “Adaptive array of phase-locked fiber collimators: analysis and experimental demonstration,” IEEE J. Sel. Top. Quantum Electron. 15, 269–280 (2009).
[Crossref]

V. Jolivet, P. Bourdon, B. Bennal, L. Lombard, D. Goular, E. Pourtal, G. Canat, Y. Jaouen, B. Moreau, and O. Vasseur, “Beam shaping of single-mode and multimode fiber amplifier arrays for propagation through atmospheric turbulence,” IEEE J. Sel. Top. Quantum Electron. 15, 257–268 (2009).
[Crossref]

J. Mod. Opt. (1)

M. W. Hyde, J. E. McCrae, and G. A. Tyler, “Target-based coherent beam combining of an optical phased array fed by a broadband laser source,” J. Mod. Opt. 64, 2149–2156 (2017).
[Crossref]

J. Opt. Soc. Am. (1)

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

Opt. Express (1)

Opt. Lett. (2)

Proc. SPIE (4)

A. Flores, T. Ehrehreich, R. Holten, B. Anderson, and I. Dajani, “Multi-kW coherent combining of fiber lasers seeded with pseudo random phase modulated light,” Proc. SPIE 9728, 97281Y (2016).
[Crossref]

M. F. Spencer and M. W. Hyde, “An investigation of stair mode in optical phased arrays using tiled apertures,” Proc. SPIE 8520, 852006 (2012).
[Crossref]

M. W. Hyde, G. A. Tyler, and C. R. Garcia, “Target-in-the-loop phasing of a fiber laser array fed by a linewidth-broadened master oscillator,” Proc. SPIE 10192, 101920K (2017).
[Crossref]

J. F. Riker, G. A. Tyler, and J. L. Vaughn, “Speckle imaging from an array,” Proc. SPIE 9982, 99820J (2016).
[Crossref]

Other (18)

The small difference between Zernike tilt and gradient tilt—measured by a tracking system employing a centroid sensor—is ignored [26].

J. Wyman and M. W. Hyde, “Detection and correction of stair mode across an optical phased array,” in IEEE Aerospace Conference (2014), pp. 1–10.

J. E. McCrae and S. T. Fiorino, “Simulation of array tilt effects in laser phased arrays,” in IEEE Aerospace Conference (2016), pp. 1–7.

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms, 2nd ed. (SPIE, 2007).

M. C. Roggemann and B. M. Welsh, Imaging Through Turbulence (CRC Press, 1996).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE, 2005).

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University, 1998).

R. K. Tyson, Principles of Adaptive Optics, 4th ed. (CRC Press, 2015).

Under normal operating conditions (i.e., d<r0), an initially uncorrected array will produce a focused spot on the target with a diameter proportional to λz/d, where z is the distance to the target. The angle this spot subtends when viewed from the array is λ/d. Since the array uses the scattered return from this spot to phase on the target, θ0 must be greater than λ/d, otherwise array performance is limited by θ0 and not diffraction.

R. A. Motes, S. A. Shakir, and R. W. Berdine, Introduction to High-Power Fiber Lasers, 2nd ed. (Directed Energy Professional Society, 2013).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

J. W. Goodman, Statistical Optics, 2nd ed. (Wiley, 2015).

J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).

In the case of SPGD, the metric that is maximized or minimized to yield phase estimates is typically based on the statistics of the received speckle pattern [1, 14]. The target-induced or speckle phases are not directly estimated.

V. N. Mahajan, Optical Imaging and Aberrations, Part III: Wavefront Analysis (SPIE, 2013).

A. Brignon, ed., Coherent Laser Beam Combining (Wiley-VCH, 2013).

B. N. Pulford, “LOCSET phase locking: operation, diagnostics, and applications,” Ph.D. thesis (The University of New Mexico, 2011).

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 (5)

Fig. 1.
Fig. 1. Seven-element hexagonal array composed of identical circular elements of diameter d and center-to-center spacing D . R 7 is the radius of the circle that circumscribes the array.
Fig. 2.
Fig. 2. Mean target-plane intensities using (a)  | W y | 2 = 0.0284    waves 2 and (b)  | W y , tr | 2 = 9.9985 × 10 5    waves 2 , respectively. These array tilt variances were computed assuming that d = r 0 = 0.1    m , L 0 = 100    m , d / D = 0.95 , and N = 7 . The white circle in both images marks the edge of the Airy disk for a single array element with λ = 1    μm .
Fig. 3.
Fig. 3. Array tilt variances for an N = 7 element array plotted versus fill factor d / D —(a) “full” array tilt variance | W y | 2 and (b) Zernike-tilt-removed array tilt variance | W y , tr | 2 . The solid traces are the theoretical variances; the symbols are the variances obtained from the simulation. d / r 0 values are differentiated by the color of the trace or symbol.
Fig. 4.
Fig. 4. Array tilt variances for an N = 19 element array plotted versus fill factor d / D —(a) “full” array tilt variance | W y | 2 and (b) Zernike-tilt-removed array tilt variance | W y , tr | 2 . The solid traces are the theoretical variances; the symbols are the variances obtained from the simulation. d / r 0 values are differentiated by the color of the trace or symbol.
Fig. 5.
Fig. 5. Array tilt variances for an N = 37 element array plotted versus fill factor d / D —(a) “full” array tilt variance | W y | 2 and (b) Zernike-tilt-removed array tilt variance | W y , tr | 2 . The solid traces are the theoretical variances; the symbols are the variances obtained from the simulation. d / r 0 values are differentiated by the color of the trace or symbol.

Equations (40)

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

R N = L D + d 2 ,
N = 1 + 6 l = 1 L l .
ϕ = 2 π R W , [ ϕ 1 ϕ N ] = 2 π [ 2 D 3 x 1 2 D y 1 2 D 3 x N 2 D y N ] [ W x W y ] ,
ϕ i = 1 π ( d / 2 ) 2 circ ( | ρ ρ i | d / 2 ) ϕ ( ρ ) d 2 ρ ,
ϕ ϕ = ( 2 π ) 2 R W W R ,
W W = 1 ( 2 π ) 2 ( R R ) 1 R ϕ ϕ R ( R R ) 1 .
ϕ i ϕ j * = 1 A 2 P i ( ρ 1 ) P j * ( ρ 2 ) ϕ ( ρ 1 ) ϕ * ( ρ 2 ) d 2 ρ 1 d 2 ρ 2 ,
ϕ i ϕ j * = 1 A 2 Γ ϕ ( t ) P i ( s ) P j * ( s t ) d 2 s d 2 t .
P ˜ i ( f ) = P i ( s ) exp ( j 2 π f · s ) d 2 s , P i ( s ) = P ˜ i ( f ) exp ( j 2 π f · s ) d 2 f ,
ϕ i ϕ j * = 1 A 2 P ˜ i ( f ) ϕ ϕ ( f ) P ˜ j * ( f ) d 2 f ,
ϕ ϕ ( f ) = 5 π 8 / 3 11 [ 24 5 Γ ( 6 5 ) ] 5 / 6 Γ ( 17 / 6 ) Γ ( 1 / 6 ) r 0 5 / 3 ( f 2 + f 0 2 ) 11 / 6 0.023 r 0 5 / 3 ( f 2 + f 0 2 ) 11 / 6 ,
ϕ i ϕ j * = 0.046 A f 0 11 / 3 r 0 5 / 3 0 d f f J 1 2 ( f ) J 0 ( ρ i j d / 2 f ) × [ 1 + ( 1 π d f 0 ) 2 f 2 ] 11 / 6 ,
| ϕ i | 2 = 0.0863 ( L 0 r 0 ) 5 / 3 F 3 2 ( 3 2 , 1 ; 1 6 , 3 , 2 ; π 2 d 2 L 0 2 ) 1.0324 ( d r 0 ) 5 / 3 F 2 1 ( 7 3 ; 17 6 , 23 6 ; π 2 d 2 L 0 2 ) ,
ϕ i ϕ j * = 0.0863 ( L 0 r 0 ) 5 / 3 n = 0 ( π ρ i j / L 0 ) 2 n ( 1 ) n ( 1 / 6 ) n × F 3 2 ( 3 2 , n + 1 ; n + 1 6 , 3 , 2 ; π 2 d 2 L 0 2 ) 3.4419 ( ρ i j r 0 ) 5 / 3 n = 0 ( π ρ i j / L 0 ) 2 n ( 1 ) n ( 11 / 6 ) n × F 2 3 ( 3 2 , n 5 6 , n 5 6 ; 3 , 2 ; d 2 ρ i j 2 ) ,
W W [ 0.0852 0 0 0.0284 ] .
D i j = | ϕ i ϕ j | 2 = 2 ( ϕ i 2 ϕ i ϕ j ) ,
ϕ i , tr = ϕ i a 2 Z 2 , i a 3 Z 3 , i .
a k = W ( ρ ) Z k ( ρ ) ϕ ( ρ ) d 2 ρ , W ( ρ ) = { 1 π R N 2 ρ < R N 0 ρ > R N ,
ϕ i , tr ϕ j , tr * = ϕ i ϕ j * + 4 R N 2 ( x i x j | a 2 | 2 + y i y j | a 3 | 2 + x i y j a 2 a 3 * + x j y i a 2 * a 3 ) 2 R N ( x i a 2 ϕ j * + y i a 3 ϕ j * + x j a 2 * ϕ i + y j a 3 * ϕ i ) ,
a k a l * = W ( ρ 1 ) Z k ( ρ 1 ) W * ( ρ 2 ) Z l * ( ρ 2 ) × Γ ϕ ( ρ 1 ρ 2 ) d 2 ρ 1 d 2 ρ 2 , a k ϕ i * = 1 A W ( ρ 1 ) Z k ( ρ 1 ) P i * ( ρ 2 ) × Γ ϕ ( ρ 1 ρ 2 ) d 2 ρ 1 d 2 ρ 2 ,
| a k | 2 = 0.092 π R N 2 r 0 5 / 3 f 0 11 / 3 0 d f f J 2 2 ( f ) × [ 1 + ( 1 2 π f 0 R N ) 2 f 2 ] 11 / 6 , [ a 2 ϕ i * a 3 ϕ i * ] = [ cos θ i sin θ i ] 0.092 ( d / 2 ) π R N r 0 5 / 3 f 0 11 / 3 0 d f f × J 1 ( d / 2 R N f ) J 1 ( ρ i R N f ) J 2 ( f ) [ 1 + ( 1 2 π f 0 R N ) 2 f 2 ] 11 / 6 ,
| a 2 | 2 = | a 3 | 2 = | a T | 2 = 1.4251 ( R N r 0 ) 5 / 3 F 3 2 ( 7 3 , 11 6 ; 5 6 , 17 6 , 29 6 ; 4 π 2 R N 2 L 0 2 ) 2.5557 ( R N L 0 ) 1 / 3 ( R N r 0 ) 5 / 3 F 3 2 ( 5 2 , 2 ; 5 , 3 , 7 6 ; 4 π 2 R N 2 L 0 2 ) .
[ a 2 ϕ i * a 3 ϕ i * ] = 2 R N [ x i y i ] I i I i = 1.5645 ( R N r 0 ) 5 / 3 × m = 0 n = 0 ( d 2 R N ) 2 m ( ρ i R N ) 2 n ( 1 6 ) m + n ( 11 6 ) m + n ( 2 ) m ( 1 ) m ( 2 ) n ( 1 ) n × F 2 1 ( 11 6 ; m n + 5 6 , m n + 17 6 ; π 2 R N 2 L 0 2 ) 2.5557 ( R N L 0 ) 1 / 3 ( R N r 0 ) 5 / 3 × m = 0 n = 0 ( π d 2 L 0 ) 2 m ( π ρ i L 0 ) 2 n ( 2 ) m + n ( 2 ) m ( 1 ) m ( 2 ) n ( 1 ) n ( 7 6 ) m + n × F 2 1 ( m + n + 2 ; m + n + 7 6 , 3 ; π 2 R N 2 L 0 2 ) .
ϕ i , tr ϕ j , tr * = ϕ i ϕ j * + 4 R N 2 ( x i x j + y i y j ) ( | a T | 2 I i I j ) .
U ( ρ ) = i = 1 N P i ( ρ ) exp ( j ϕ i ) , ϕ i = 2 π 2 D 3 x i W x + 2 π 2 D y i W y .
I ( ρ , z ) = [ A λ z ] 2 jinc 2 ( k z d 2 ρ ) × i = 1 N j = 1 N exp [ j ( ϕ i ϕ j ) ] exp [ j k z ( ρ i ρ j ) · ρ ] ,
I ( ρ , z ) = [ A λ z ] 2 jinc 2 ( k z d 2 ρ ) × i = 1 N j = 1 N exp ( 8 π 2 D 2 ρ i j 2 | W y | 2 ) exp ( j k z ρ i j · ρ ) ,
I 2 ( ρ , z ) = [ A λ z ] 4 jinc 4 ( k z d 2 ρ ) × i = 1 N j = 1 N m = 1 N n = 1 N exp [ j ( ϕ i ϕ j + ϕ m ϕ n ) ] × exp ( j k z ρ i j · ρ ) exp ( j k z ρ m n · ρ ) .
I 2 ( ρ , z ) = [ A λ z ] 4 jinc 4 ( k z d 2 ρ ) × i = 1 N j = 1 N m = 1 N n = 1 N exp ( 8 π 2 D 2 ρ i j m n 2 | W y | 2 ) × exp ( j k z ρ i j m n · ρ ) ,
σ I 2 ( ρ , z ) = I 2 ( ρ , z ) [ I ( ρ , z ) ] 2 .
ϕ TIL = ϕ tele + ϕ atm + ϕ tar , at , ϕ loc = ϕ tele ,
ϕ tele = ϕ tele , ho + ϕ tele , at , ϕ atm = ϕ atm , ho + ϕ atm , at ,
ϕ loc ϕ TIL = ϕ atm , ho ϕ atm , at ϕ tar , at .
W at ( ϕ loc ϕ TIL ) = ϕ atm , at ϕ tar , at .
ϕ cmd = ϕ TIL + W at ( ϕ loc ϕ TIL ) = ϕ tele + ϕ atm , ho .
ϕ res = ϕ true ϕ cmd = ϕ tele + ϕ atm ϕ tele ϕ atm , ho = ϕ atm , at .
Z k , i = 1 A P i ( ρ ) Z k ( ρ ) d 2 ρ = π R N 2 A P i ( ρ ) W ( ρ ) Z k ( ρ ) d 2 ρ .
[ Z 2 , i Z 3 , i ] = 4 R N d / 2 [ cos θ i sin θ i ] 0 d f f J 1 ( 2 π f d 2 ) J 1 ( 2 π f ρ i ) × J 2 ( 2 π f R N ) .
[ Z 2 , i Z 3 , i ] = 2 ρ i R N [ cos θ i sin θ i ] m = 0 n = 0 ( 1 ) m ( 1 ) n m ! n ! ( d 2 R N ) 2 m × ( ρ i R N ) 2 n Γ ( m + n + 2 ) Γ ( m n + 1 ) Γ ( m + 2 ) Γ ( n + 2 ) .
[ Z 2 , i Z 3 , i ] = 2 ρ i R N [ cos θ i sin θ i ] = 2 R N [ x i y i ] .

Metrics