Abstract

We analyze the potential efficiency of laser beam projection onto a remote object in atmosphere with incoherent and coherent phase-locked conformal-beam director systems composed of an adaptive array of fiber collimators. Adaptive optics compensation of turbulence-induced phase aberrations in these systems is performed at each fiber collimator. Our analysis is based on a derived expression for the atmospheric-averaged value of the mean square residual phase error as well as direct numerical simulations. Operation of both conformal-beam projection systems is compared for various adaptive system configurations characterized by the number of fiber collimators, the adaptive compensation resolution, and atmospheric turbulence conditions.

© 2008 Optical Society of America

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  1. M. A. Vorontsov and S. L. Lachinova, “Laser beam projection with adaptive array of fiber collimators. II. Analysis of atmospheric compensation efficiency,” J. Opt. Soc. Am. A 25, 1949-1959 (2008).
    [CrossRef]
  2. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, 1971).
  3. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).
  4. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).
  5. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372-1379 (1966).
    [CrossRef]
  6. F.Roddier, ed., Adaptive Optics in Astronomy (Cambridge U. Press, 1999).
    [CrossRef]
  7. R. K. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic, 1998).
  8. J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, 1998).
  9. M. A. Vorontsov and V. I. Shmalgauzen, The Principles of Adaptive Optics (Nauka, 1985).
  10. R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature 353, 144-146 (1991).
    [CrossRef]
  11. E. Kibblewhite, “Laser beacons for astronomy,” in Laser Guide Star Adaptive Optics, R.Q.Fugate, ed. (Philips Laboratory, Kirtland Air Force Base, 1992), pp. 24-36.
  12. N.Ageorges and C.Dainty, eds., Laser Guide Star Adaptive Optics for Astronomy (Kluwer Academic, 2000).
  13. T. R. O'Meara, “The multidither principle in adaptive optics,” J. Opt. Soc. Am. 67, 306-315 (1977).
    [CrossRef]
  14. M. A. Vorontsov, G. W. Carhart, and J. C. Ricklin, “Adaptive phase-distortion correction based on parallel gradient-descent optimization,” Opt. Lett. 22, 907-909 (1997).
    [CrossRef] [PubMed]
  15. M. A. Vorontsov, “Decoupled stochastic parallel gradient descent optimization for adaptive optics: Integrated approach for wave-front sensor information fusion,” J. Opt. Soc. Am. A 19, 356-368 (2002).
    [CrossRef]
  16. In this case the phase locking is similar to the piston-type aberration compensation in conventional adaptive optical systems with a segmented wavefront corrector. The difference is that in fiber-based conformal systems the phase shifts {vj(t)} can be introduced using fast (GHz rate) fiber-based phase shifters as shown in Fig. .
  17. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961)
  18. A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluids for very large Reynolds numbers,” Dokl. Akad. Nauk SSSR 30, 301-305 (1941) A. N. Kolmogorov,(in Russian) [English translation in Proc. R. Soc. London, Ser. A 434, 9-13 (1991)].
    [CrossRef]
  19. M. C. Roggemann and B. M. Welsh, Imaging through Turbulence (CRC Press, 1996)
  20. J. A. Fleck, J. R. Morris, and M. D. Feit, “Time dependent propagation of high energy laser beam through the atmosphere,” Appl. Phys. Lett. 11, 329-335 (1977).
  21. S. M. Flatte, G. Y. Wang, and J. Martin, “Irradiance variance of optical waves through atmospheric turbulence by numerical simulation and comparison with experiment,” J. Opt. Soc. Am. A 10, 2363-2370 (1993).
    [CrossRef]
  22. S. L. Lachinova and M. A. Vorontsov, “Performance analysis of an adaptive phase-locked tiled fiber array in atmospheric turbulence conditions,” Proc. SPIE 5895, 58950O (2005).
    [CrossRef]
  23. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207-211 (1976).
    [CrossRef]
  24. In the considered pupil-plane phase screen approximation, the target-plane field complex amplitude distribution A(r,z=L) represents the Fresnel diffraction integral, which is equivalent to the Fourier transform of the complex function A(r,z=0) multiplied by a quadratic phase exponential term exp(−ikr2/2L).
  25. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  26. Typical examples of pupil-plane phase screens and the corresponding target-plane intensity distributions can be found in .
  27. D. L. Fried and J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31, 2865-2882 (1992).
    [CrossRef] [PubMed]
  28. D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15, 2759-2768 (1998).
    [CrossRef]
  29. V. Aksenov, V. Banakh, and O. Tikhomirova, “Potential and vortex features of optical speckle fields and visualization of wave-front singularities,” Appl. Opt. 37, 4536-4540 (1998).
    [CrossRef]
  30. M. Minden, “Coherent coupling of a fiber amplifier array,” in Thirteenth Annual Solid State and Diode Laser Technology Review, SSDLTR 2000 Tech. Digest (Air Force Research Laboratory, 2000).
  31. J. Anderegg, S. J. Brosnan, M. E. Weber, H. Komine, and M. G. Wickham, “8-W coherently phased 4-element fiber array,” Proc. SPIE 4974, 1-6 (2003).
    [CrossRef]
  32. H. Bruesselbach, S. Wang, M. Minden, D. C. Jones, and M. Mangir, “Power-scalable phase-compensating fiber-array transceiver for laser communications through the atmosphere,” J. Opt. Soc. Am. B 22, 347-353 (2005).
    [CrossRef]
  33. L. Liu and M. A. Vorontsov, “Phase-locking of tiled fiber array using SPGD feedback controller,” Proc. SPIE 5895, 58950P (2005).
    [CrossRef]
  34. P. Sprangle, J. Penano, and A. Ting, “Incoherent combining of high-power fibers lasers for long-range directed energy applications. Interim Rept. May-Jun 2006,” NRL/MR/6790-06-8963 (Naval Research Laboratory, 2006).
  35. J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
    [CrossRef]
  36. J. E. Kansky, C. X. Yu, D. V. Murphy, S. E. J. Shaw, R. C. Lawrence, and C. Higgs, “Beam control of a 2D polarization maintaining fiber optic phased array with high-fiber count,” Proc. SPIE 6306, 63060G (2006).
    [CrossRef]
  37. T. M. Shay, “Theory of electronically phased coherent beam combination without a reference beam,” Opt. Express 14, 12188-12195 (2006).
    [CrossRef] [PubMed]
  38. T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, D. J. Nelson, and C. A. Lu, “Narrow linewidth coherent beam combining of optical fiber amplifier arrays,” Proc. SPIE 6451, 64511N (2007).
    [CrossRef]
  39. L. Liu, D. Loizos, M. A. Vorontsov, P. Sotiriadis, and G. Cauwenberghs, “Coherent combining of multiple beams with multi-dithering technique: 100KHz closed-loop compensation demonstration,” Proc. SPIE 6708, 67080D (2007).
    [CrossRef]
  40. L. Liu, M. A. Vorontsov, E. Polnau, T. Weyrauch, and L. A. Beresnev, “Adaptive phase-locked fiber array with wavefront phase tip-tilt compensation using piezoelectric fiber positioners,” Proc. SPIE 6708, 67080K (2007).
    [CrossRef]

2008

2007

T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, D. J. Nelson, and C. A. Lu, “Narrow linewidth coherent beam combining of optical fiber amplifier arrays,” Proc. SPIE 6451, 64511N (2007).
[CrossRef]

L. Liu, D. Loizos, M. A. Vorontsov, P. Sotiriadis, and G. Cauwenberghs, “Coherent combining of multiple beams with multi-dithering technique: 100KHz closed-loop compensation demonstration,” Proc. SPIE 6708, 67080D (2007).
[CrossRef]

L. Liu, M. A. Vorontsov, E. Polnau, T. Weyrauch, and L. A. Beresnev, “Adaptive phase-locked fiber array with wavefront phase tip-tilt compensation using piezoelectric fiber positioners,” Proc. SPIE 6708, 67080K (2007).
[CrossRef]

2006

P. Sprangle, J. Penano, and A. Ting, “Incoherent combining of high-power fibers lasers for long-range directed energy applications. Interim Rept. May-Jun 2006,” NRL/MR/6790-06-8963 (Naval Research Laboratory, 2006).

J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
[CrossRef]

J. E. Kansky, C. X. Yu, D. V. Murphy, S. E. J. Shaw, R. C. Lawrence, and C. Higgs, “Beam control of a 2D polarization maintaining fiber optic phased array with high-fiber count,” Proc. SPIE 6306, 63060G (2006).
[CrossRef]

T. M. Shay, “Theory of electronically phased coherent beam combination without a reference beam,” Opt. Express 14, 12188-12195 (2006).
[CrossRef] [PubMed]

2005

H. Bruesselbach, S. Wang, M. Minden, D. C. Jones, and M. Mangir, “Power-scalable phase-compensating fiber-array transceiver for laser communications through the atmosphere,” J. Opt. Soc. Am. B 22, 347-353 (2005).
[CrossRef]

L. Liu and M. A. Vorontsov, “Phase-locking of tiled fiber array using SPGD feedback controller,” Proc. SPIE 5895, 58950P (2005).
[CrossRef]

S. L. Lachinova and M. A. Vorontsov, “Performance analysis of an adaptive phase-locked tiled fiber array in atmospheric turbulence conditions,” Proc. SPIE 5895, 58950O (2005).
[CrossRef]

2003

J. Anderegg, S. J. Brosnan, M. E. Weber, H. Komine, and M. G. Wickham, “8-W coherently phased 4-element fiber array,” Proc. SPIE 4974, 1-6 (2003).
[CrossRef]

2002

2000

N.Ageorges and C.Dainty, eds., Laser Guide Star Adaptive Optics for Astronomy (Kluwer Academic, 2000).

M. Minden, “Coherent coupling of a fiber amplifier array,” in Thirteenth Annual Solid State and Diode Laser Technology Review, SSDLTR 2000 Tech. Digest (Air Force Research Laboratory, 2000).

1999

F.Roddier, ed., Adaptive Optics in Astronomy (Cambridge U. Press, 1999).
[CrossRef]

1998

R. K. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic, 1998).

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

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).

D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15, 2759-2768 (1998).
[CrossRef]

V. Aksenov, V. Banakh, and O. Tikhomirova, “Potential and vortex features of optical speckle fields and visualization of wave-front singularities,” Appl. Opt. 37, 4536-4540 (1998).
[CrossRef]

1997

1996

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

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

1993

1992

D. L. Fried and J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31, 2865-2882 (1992).
[CrossRef] [PubMed]

E. Kibblewhite, “Laser beacons for astronomy,” in Laser Guide Star Adaptive Optics, R.Q.Fugate, ed. (Philips Laboratory, Kirtland Air Force Base, 1992), pp. 24-36.

1991

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature 353, 144-146 (1991).
[CrossRef]

1985

M. A. Vorontsov and V. I. Shmalgauzen, The Principles of Adaptive Optics (Nauka, 1985).

1978

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

1977

T. R. O'Meara, “The multidither principle in adaptive optics,” J. Opt. Soc. Am. 67, 306-315 (1977).
[CrossRef]

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time dependent propagation of high energy laser beam through the atmosphere,” Appl. Phys. Lett. 11, 329-335 (1977).

1976

1971

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, 1971).

1966

1961

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

1941

A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluids for very large Reynolds numbers,” Dokl. Akad. Nauk SSSR 30, 301-305 (1941) A. N. Kolmogorov,(in Russian) [English translation in Proc. R. Soc. London, Ser. A 434, 9-13 (1991)].
[CrossRef]

Aksenov, V.

Ameer, G. A.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature 353, 144-146 (1991).
[CrossRef]

Anderegg, J.

J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
[CrossRef]

J. Anderegg, S. J. Brosnan, M. E. Weber, H. Komine, and M. G. Wickham, “8-W coherently phased 4-element fiber array,” Proc. SPIE 4974, 1-6 (2003).
[CrossRef]

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).

Baker, J. T.

T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, D. J. Nelson, and C. A. Lu, “Narrow linewidth coherent beam combining of optical fiber amplifier arrays,” Proc. SPIE 6451, 64511N (2007).
[CrossRef]

Banakh, V.

Benham, V.

T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, D. J. Nelson, and C. A. Lu, “Narrow linewidth coherent beam combining of optical fiber amplifier arrays,” Proc. SPIE 6451, 64511N (2007).
[CrossRef]

Beresnev, L. A.

L. Liu, M. A. Vorontsov, E. Polnau, T. Weyrauch, and L. A. Beresnev, “Adaptive phase-locked fiber array with wavefront phase tip-tilt compensation using piezoelectric fiber positioners,” Proc. SPIE 6708, 67080K (2007).
[CrossRef]

Boeke, B. R.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature 353, 144-146 (1991).
[CrossRef]

Brosnan, S.

J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
[CrossRef]

Brosnan, S. J.

J. Anderegg, S. J. Brosnan, M. E. Weber, H. Komine, and M. G. Wickham, “8-W coherently phased 4-element fiber array,” Proc. SPIE 4974, 1-6 (2003).
[CrossRef]

Browne, S. L.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature 353, 144-146 (1991).
[CrossRef]

Bruesselbach, H.

Carhart, G. W.

Cauwenberghs, G.

L. Liu, D. Loizos, M. A. Vorontsov, P. Sotiriadis, and G. Cauwenberghs, “Coherent combining of multiple beams with multi-dithering technique: 100KHz closed-loop compensation demonstration,” Proc. SPIE 6708, 67080D (2007).
[CrossRef]

Cheung, E.

J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
[CrossRef]

Epp, P.

J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
[CrossRef]

Feit, M. D.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time dependent propagation of high energy laser beam through the atmosphere,” Appl. Phys. Lett. 11, 329-335 (1977).

Flatte, S. M.

Fleck, J. A.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time dependent propagation of high energy laser beam through the atmosphere,” Appl. Phys. Lett. 11, 329-335 (1977).

Fried, D. L.

Fugate, R. Q.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature 353, 144-146 (1991).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Hammons, D.

J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
[CrossRef]

Hardy, J. W.

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

Higgs, C.

J. E. Kansky, C. X. Yu, D. V. Murphy, S. E. J. Shaw, R. C. Lawrence, and C. Higgs, “Beam control of a 2D polarization maintaining fiber optic phased array with high-fiber count,” Proc. SPIE 6306, 63060G (2006).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

Jones, D. C.

Kansky, J. E.

J. E. Kansky, C. X. Yu, D. V. Murphy, S. E. J. Shaw, R. C. Lawrence, and C. Higgs, “Beam control of a 2D polarization maintaining fiber optic phased array with high-fiber count,” Proc. SPIE 6306, 63060G (2006).
[CrossRef]

Kibblewhite, E.

E. Kibblewhite, “Laser beacons for astronomy,” in Laser Guide Star Adaptive Optics, R.Q.Fugate, ed. (Philips Laboratory, Kirtland Air Force Base, 1992), pp. 24-36.

Kolmogorov, A. N.

A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluids for very large Reynolds numbers,” Dokl. Akad. Nauk SSSR 30, 301-305 (1941) A. N. Kolmogorov,(in Russian) [English translation in Proc. R. Soc. London, Ser. A 434, 9-13 (1991)].
[CrossRef]

Komine, H.

J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
[CrossRef]

J. Anderegg, S. J. Brosnan, M. E. Weber, H. Komine, and M. G. Wickham, “8-W coherently phased 4-element fiber array,” Proc. SPIE 4974, 1-6 (2003).
[CrossRef]

Lachinova, S. L.

M. A. Vorontsov and S. L. Lachinova, “Laser beam projection with adaptive array of fiber collimators. II. Analysis of atmospheric compensation efficiency,” J. Opt. Soc. Am. A 25, 1949-1959 (2008).
[CrossRef]

S. L. Lachinova and M. A. Vorontsov, “Performance analysis of an adaptive phase-locked tiled fiber array in atmospheric turbulence conditions,” Proc. SPIE 5895, 58950O (2005).
[CrossRef]

Lawrence, R. C.

J. E. Kansky, C. X. Yu, D. V. Murphy, S. E. J. Shaw, R. C. Lawrence, and C. Higgs, “Beam control of a 2D polarization maintaining fiber optic phased array with high-fiber count,” Proc. SPIE 6306, 63060G (2006).
[CrossRef]

Liu, L.

L. Liu, D. Loizos, M. A. Vorontsov, P. Sotiriadis, and G. Cauwenberghs, “Coherent combining of multiple beams with multi-dithering technique: 100KHz closed-loop compensation demonstration,” Proc. SPIE 6708, 67080D (2007).
[CrossRef]

L. Liu, M. A. Vorontsov, E. Polnau, T. Weyrauch, and L. A. Beresnev, “Adaptive phase-locked fiber array with wavefront phase tip-tilt compensation using piezoelectric fiber positioners,” Proc. SPIE 6708, 67080K (2007).
[CrossRef]

L. Liu and M. A. Vorontsov, “Phase-locking of tiled fiber array using SPGD feedback controller,” Proc. SPIE 5895, 58950P (2005).
[CrossRef]

Loizos, D.

L. Liu, D. Loizos, M. A. Vorontsov, P. Sotiriadis, and G. Cauwenberghs, “Coherent combining of multiple beams with multi-dithering technique: 100KHz closed-loop compensation demonstration,” Proc. SPIE 6708, 67080D (2007).
[CrossRef]

Lu, C. A.

T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, D. J. Nelson, and C. A. Lu, “Narrow linewidth coherent beam combining of optical fiber amplifier arrays,” Proc. SPIE 6451, 64511N (2007).
[CrossRef]

Mangir, M.

Martin, J.

Minden, M.

H. Bruesselbach, S. Wang, M. Minden, D. C. Jones, and M. Mangir, “Power-scalable phase-compensating fiber-array transceiver for laser communications through the atmosphere,” J. Opt. Soc. Am. B 22, 347-353 (2005).
[CrossRef]

M. Minden, “Coherent coupling of a fiber amplifier array,” in Thirteenth Annual Solid State and Diode Laser Technology Review, SSDLTR 2000 Tech. Digest (Air Force Research Laboratory, 2000).

Morris, J. R.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time dependent propagation of high energy laser beam through the atmosphere,” Appl. Phys. Lett. 11, 329-335 (1977).

Murphy, D. V.

J. E. Kansky, C. X. Yu, D. V. Murphy, S. E. J. Shaw, R. C. Lawrence, and C. Higgs, “Beam control of a 2D polarization maintaining fiber optic phased array with high-fiber count,” Proc. SPIE 6306, 63060G (2006).
[CrossRef]

Nelson, D. J.

T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, D. J. Nelson, and C. A. Lu, “Narrow linewidth coherent beam combining of optical fiber amplifier arrays,” Proc. SPIE 6451, 64511N (2007).
[CrossRef]

Noll, R. J.

O'Meara, T. R.

Penano, J.

P. Sprangle, J. Penano, and A. Ting, “Incoherent combining of high-power fibers lasers for long-range directed energy applications. Interim Rept. May-Jun 2006,” NRL/MR/6790-06-8963 (Naval Research Laboratory, 2006).

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).

Pilkington, D.

T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, D. J. Nelson, and C. A. Lu, “Narrow linewidth coherent beam combining of optical fiber amplifier arrays,” Proc. SPIE 6451, 64511N (2007).
[CrossRef]

Polnau, E.

L. Liu, M. A. Vorontsov, E. Polnau, T. Weyrauch, and L. A. Beresnev, “Adaptive phase-locked fiber array with wavefront phase tip-tilt compensation using piezoelectric fiber positioners,” Proc. SPIE 6708, 67080K (2007).
[CrossRef]

Ricklin, J. C.

Roberts, P. H.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature 353, 144-146 (1991).
[CrossRef]

Roggemann, M. C.

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

Ruane, R. E.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature 353, 144-146 (1991).
[CrossRef]

Sanchez, A. D.

T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, D. J. Nelson, and C. A. Lu, “Narrow linewidth coherent beam combining of optical fiber amplifier arrays,” Proc. SPIE 6451, 64511N (2007).
[CrossRef]

Shaw, S. E. J.

J. E. Kansky, C. X. Yu, D. V. Murphy, S. E. J. Shaw, R. C. Lawrence, and C. Higgs, “Beam control of a 2D polarization maintaining fiber optic phased array with high-fiber count,” Proc. SPIE 6306, 63060G (2006).
[CrossRef]

Shay, T. M.

T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, D. J. Nelson, and C. A. Lu, “Narrow linewidth coherent beam combining of optical fiber amplifier arrays,” Proc. SPIE 6451, 64511N (2007).
[CrossRef]

T. M. Shay, “Theory of electronically phased coherent beam combination without a reference beam,” Opt. Express 14, 12188-12195 (2006).
[CrossRef] [PubMed]

Shmalgauzen, V. I.

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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In the considered pupil-plane phase screen approximation, the target-plane field complex amplitude distribution A(r,z=L) represents the Fresnel diffraction integral, which is equivalent to the Fourier transform of the complex function A(r,z=0) multiplied by a quadratic phase exponential term exp(−ikr2/2L).

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In this case the phase locking is similar to the piston-type aberration compensation in conventional adaptive optical systems with a segmented wavefront corrector. The difference is that in fiber-based conformal systems the phase shifts {vj(t)} can be introduced using fast (GHz rate) fiber-based phase shifters as shown in Fig. .

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

P. Sprangle, J. Penano, and A. Ting, “Incoherent combining of high-power fibers lasers for long-range directed energy applications. Interim Rept. May-Jun 2006,” NRL/MR/6790-06-8963 (Naval Research Laboratory, 2006).

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

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Figures (10)

Fig. 1
Fig. 1

Schematics of the conformal beam director based on an adaptive tiled fiber array: (a) Coherent and (b) incoherent conformal systems. The turbulence is represented by a pupil-plane phase-distorting layer.

Fig. 2
Fig. 2

Residual phase error ε C = [ ε C 2 ( N a o ) ] 1 2 for a phase-locked conformal coherent beam director with ideal compensation of N a o first Zernike aberrations as a function of D r 0 for a different number of subapertures N s u b : (a) phase locking only ( N a o = 0 ) , (b) phase locking and tip/tilt compensation ( N a o = 2 ) , and (c) Zernike aberration compensation in a conformal system with N s u b = 7 subapertures. Dashed curves in (a) and (b) correspond to the beam director with a monolithic aperture of diameter D ( N s u b = 1 ) .

Fig. 3
Fig. 3

Factor of the mean square residual phase error reduction α N s u b with increase in the number of phase-locked subapertures N s u b in the conformal system.

Fig. 4
Fig. 4

Atmospheric-averaged Strehl metric S t versus the ratio D r 0 for coherent phase-locked (a) and incoherent (b) conformal beam directors with a different number of subapertures N s u b and fixed total power. For both cases the conformal aperture fill factor f C = 1 , subaperture fill factor f s u b = 0.89 , and L = 0.1 L d i f , where L d i f = k D 2 8 is the diffraction length.

Fig. 5
Fig. 5

Atmospheric-averaged power-in-the-bucket metrics versus D r 0 ratio for coherent phase-locked J P I B c o h [(a) and (c)] and incoherent J P I B i n c o h [(b) and (d)] conformal beam directors with a different number of subapertures N s u b . Metrics are normalized by P 0 for the system configuration with P 0 = const. in (a) and (b) and by p 0 for the system with a fixed power per subaperture in (c) and (d). The bucket radius b T = 0.5 b T 0 , where b T 0 is the diffraction-limited beam radius at the target plane. The conformal aperture parameters are the same as in Fig. 4.

Fig. 6
Fig. 6

Gain factor G P L for the power-in-the-bucket metric J P I B ( b T = 0.5 b T 0 ) achieved by the use of coherent versus incoherent conformal systems with a different number of subapertures N s u b . The gain is shown as the function of the ratio D r 0 . The conformal aperture parameters are the same as in Fig. 4.

Fig. 7
Fig. 7

Gain factors G t i l t & P L c o h [(a)] and G t i l t i n c o h [(b)] for the power-in-the-bucket metrics achieved when tip/tilt wavefront phase aberrations are removed at each subaperture in a coherent phase-locked [(a)] and an incoherent [(b)] conformal system composed of N s u b subapertures. The gain is shown as the function of the ratio D r 0 . The conformal aperture parameters are the same as in Fig. 4.

Fig. 8
Fig. 8

Atmospheric-averaged power-in-the-bucket metrics versus ratio the D r 0 for coherent phase-locked J P I B c o h [(a) and (c)] and incoherent J P I B i n c o h [(b) and (d)] conformal beam directors with (solid curves) and without (dashed curves) tip/tilt phase aberration compensation at each of the N s u b system subapertures. Metrics are normalized by P 0 for the system configuration with P 0 = const. in (a) and (b) and by p 0 for the system with a fixed power per subaperture in (c) and (d). The conformal aperture parameters are the same as in Fig. 4.

Fig. 9
Fig. 9

Gain factors G A O c o h [(a)] and G A O i n c o h [(b)] (solid curves) for the power-in-the-bucket metrics achieved when tip/tilt and defocus wavefront phase aberrations are removed at each subaperture in a coherent phase-locked [(a)] and an incoherent [(b)] conformal systems composed of N s u b subapertures. The conformal aperture parameters are the same as in Fig. 4.

Fig. 10
Fig. 10

Efficiency comparison of coherent phase-locked (light-gray bars) and incoherent (dark-gray bars) conformal beam directors using atmospheric-averaged power-in-the-bucket metrics normalized by the total transmitted power P 0 for (a) N s u b = 7 , (b) N s u b = 19 , and (c) N s u b = 37 subapertures. The conformal aperture parameters are the same as in Fig. 4. In the bar diagrams (a)–(c), each set of bars corresponds to different D r 0 ratios (from left to right): D r 0 = 5 , 10, 15, and 20. Numbers at the tops of the bars denote fiber-collimator-array configurations with corresponding numbers of control channels at each subaperture N a o . For coherent systems, N a o = 0 , 2, and 3 indicate phase locking only (piston-type aberration compensation), phase locking and tip/tilt compensation, and phase locking plus tip/tilt plus defocus compensation, respectively. For incoherent systems, N a o = 0 , 2, and 3 designate no AO compensation, tip/tilt compensation only, and tip/tilt plus defocus aberration compensation, respectively.

Tables (2)

Tables Icon

Table 1 Zernike Aberration Coefficients α N a o a

Tables Icon

Table 2 Coefficient G M C [Eq. (24)] for Conformal Adaptive Systems with Different Numbers of Phase-Locked Subapertures N s u b and Control Channels for Zernike Aberration Compensation at Each Subaperture N a o

Equations (25)

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A j o u t ( ρ j , t ) = M ( ρ j ) A 0 ( ρ j ) exp [ i ϕ j o u t ( ρ j , t ) ] ,
ϕ j o u t ( ρ j , t ) = u j F ( ρ j ) + u j ( ρ j , t ) + Δ j o u t ( t ) + v j ( t ) ,
A o u t ( r , t ) = j = 1 N s u b A j o u t ( ρ j , t ) = j = 1 N s u b M ( ρ j ) A 0 ( ρ j ) exp { i [ u j t i l t ( ρ j ) + u j p + u j ( ρ j , t ) + v j ( t ) ] } ,
A j o u t ( ρ j , t ) = M ( ρ j ) A 0 ( ρ j ) exp { i [ u j t i l t ( ρ j ) + u j ( ρ j , t ) + Δ j o u t ( t ) ] } .
2 i k A ( r , z , t ) z = 2 A ( r , z , t ) + 2 k 2 n 1 ( r , z , t ) A ( r , z , t ) ,
2 i k A j ( r , z , t ) z = 2 A j ( r , z , t ) + 2 k 2 n 1 ( r , z , t ) A j ( r , z , t ) .
D Φ ( r ) = [ Φ 0 ( r ) Φ 0 ( r ) ] 2 = 6.88 ( r r 0 ) 5 3 ,
Φ 0 ( r ) = j = 1 N s u b Φ j ( ρ j ) = j = 1 N s u b [ Φ ̃ j ( ρ j ) + M ( ρ j ) Δ j a t ] ,
A ( r , z = 0 ) = j = 1 N s u b M ( ρ j ) A 0 ( ρ j ) exp { i [ u j t i l t ( ρ j ) + u j p + δ j ( ρ j ) + δ ¯ j ] } ,
A j ( r , z = 0 ) = M ( ρ j ) A 0 ( ρ j ) exp { i [ u j t i l t ( ρ j ) + δ j ( ρ j ) + Δ j o u t + Δ j a t ] } .
u ( r ) = j = 1 N s u b u j ( ρ j ) = j = 1 N s u b l = 1 N a o a j , l S l ( ρ j ) ,
Φ 0 ( r ) = j = 1 N s u b Φ j ( ρ j ) = j = 1 N s u b [ l = 1 b j , l S l ( ρ j ) + M ( ρ j ) Δ j a t ] ,
ε C 2 = 1 S C j = 1 N s u b Ω j [ δ j ( ρ j ) + M ( ρ j ) δ ¯ j ] 2 d 2 r ,
ε C 2 = 1 S C j = 1 N s u b Ω j δ j 2 ( ρ j ) d 2 r + 1 N s u b j = 1 N s u b δ ¯ j 2 .
ε C 2 = ε A O 2 + ε P L 2 = 1 N s u b j = 1 N s u b ε j 2 + ε P L 2 .
ε j 2 = 1 s s u b Ω j δ j 2 ( ρ j ) d 2 r = 1 s s u b Ω j [ u j ( ρ j ) + Φ ̃ j ( ρ j ) ] 2 d 2 r .
ε C 2 = ε a o 2 + ε P L 2 .
a j = b j , v j = Δ j a t .
ε a o 2 = l = N a o + 1 b l 2 ,
ε a o 2 ε a o 2 ( N a o ) = α N a o ( d r 0 ) 5 3 ,
ε C 2 ( N a o ) = α N a o ( d r 0 ) 5 3 + ε P L 2 .
ε M 2 ( N ) = α N ( D r 0 ) 5 3 .
ε C 2 ( N a o , N s u b ) = α N a o α N s u b ( D r 0 ) 5 3 ,
G M C = ε M 2 ( N ) ε C 2 ( N a o ) = α N α N a o α N s u b .
S t exp ( ε C 2 ) .

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