Abstract

Control methods and system architectures that can be used for locking in phase of multiple laser beams that are generated at the transmitter aperture plane of a coherent fiber-collimator array system (pupil-plane phase locking) are considered. In the proposed and analyzed phase-locking techniques, sensing of the piston phase differences is performed using interference of periphery (tail) sections of the laser beams prior to their clipping by the fiber-collimator transmitter apertures. This obscuration-free sensing technique eliminates the need for a beam splitter being directly located inside the optical train of the transmitted beams—one of the major drawbacks of large-aperture and/or high-power fiber-array systems. Numerical simulation results demonstrate efficiency of the proposed phase-locking methods.

© 2010 Optical Society of America

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2010 (2)

A. Kobyakov, M. Sauer, and D. Chowdhury, “Stimulated Brillouin scattering in optical fibers,” Adv. Opt. Photon. 2, 1–59 (2010).
[CrossRef]

V. I. Kovalev, Department of Physics, Heriot-Watt University, Edinburgh EH14 4AS, UK (personal communication, 2010).

2009 (7)

P. F. McManamon, P. J. Bos, M. J. Escuti, J. Heikenfeld, S. Serati, H. Xie, and E. A. Watson, “A review of phased array steering for narrow-band electrooptical systems,” Proc. IEEE 97, 1078–1096 (2009).
[CrossRef]

J.R.Leger, J.Nilsson, J.P.Huignard, A.P.Napartovich, T.M.Shay, and A.Shirakawa, eds., Special issue, “Laser beam combining and fiber laser systems,” IEEE J. Sel. Top. Quantum Electron.15, 237–470 (2009).

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]

P. Sprangle, A. Ting, J. Penano, R. Fischer, and B. Hafizi, “Incoherent combining and atmospheric propagation of high-power fiber lasers for directed-energy applications,” IEEE J. Quantum Electron. 45, 138–148 (2009).
[CrossRef]

R. A. Motes and R. W. Berdine, Introduction to High Power Fiber Lasers (Directed Energy Professional Society, 2009).

Z. Chen, J. Hou, P. Zhou, X. Wang, X. Xu, Z. Jiang, and Z. Liu, “Mutual injection locking and coherent combining of three individual fiber lasers,” Opt. Commun. 282, 60–63 (2009).
[CrossRef]

J. Ichikawa, “LiNbO3 based optical devices for fiber communication,” J. China Univ. Posts Telecommun. 16, 12–15 (2009).
[CrossRef]

2008 (3)

2007 (4)

S. J. Augst, J. K. Ranka, T. Y. Fan, and A. Sanchez, “Beam combining of ytterbium fiber amplifiers (Invited),” J. Opt. Soc. Am. B 24, 1707–1715 (2007).
[CrossRef]

T. H. Loftus, A. M. Thomas, P. R. Hoffman, M. Norsen, R. Royse, A. Liu, and E. C. Honea, “Spectrally beam-combined fiber lasers for high-average-power applications,” IEEE J. Sel. Top. Quantum Electron. 13, 487–497 (2007).
[CrossRef]

B. Lei and Y. Feng, “Phase locking of an array of three fiber lasers by an all-fiber coupling loop,” Opt. Express 15, 17114–17119 (2007).
[CrossRef] [PubMed]

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University Press, 2007).

2006 (3)

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 (2)

H. Bruesselbach, D. C. Jones, M. S. Mangir, M. Minden, and J. L. Rogers, “Self-organized coherence in fiber laser arrays,” Opt. Lett. 30, 1339–1341 (2005).
[CrossRef] [PubMed]

T. Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11, 567–577 (2005).
[CrossRef]

2003 (2)

A. F. Glova, “Phase locking of optically coupled lasers,” Quantum Electron. 33, 283–306 (2003).
[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]

2002 (1)

2001 (1)

2000 (3)

M. A. Vorontsov, G. W. Carhart, M. Cohen, and G. Cauwenberghs, “Adaptive optics based on analog parallel stochastic optimization: analysis and experimental demonstration,” J. Opt. Soc. Am. A 17, 1440–1453 (2000).
[CrossRef]

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A review of lithium niobate modulators for fiber-optic communications systems,” IEEE J. Sel. Top. Quantum Electron. 6, 69–82 (2000).
[CrossRef]

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

1997 (1)

1996 (2)

W. S. Levine, ed., The Control Handbook (CRC Press, 1996).

K. H. Kudielka, A. Kalmar, and W. R. Leeb, “Design and breadboarding of a phased telescope array for free-space laser communications,” in Proceedings of IEEE International Symposium on Phased Array Systems and Technology (IEEE, 1996), pp. 419–424.
[CrossRef]

1994 (2)

C. D. Nabors, “Effects of phase errors on coherent emitter arrays,” Appl. Opt. 33, 2284–2289 (1994).
[CrossRef] [PubMed]

B. M. ter Haar Romey, ed., Geometry-Driven Diffusion in Computer Vision (Kluwer Academic, 1994).

1993 (1)

J. R. Leger, “External methods of phase locking and coherent beam addition of diode lasers,” in Surface Emitting Semiconductor Lasers and Arrays, G.A.Evans and J.M.Hammer, eds. (Academic, 1993), pp. 379–433.

1985 (1)

M. A. Vorontsov and V. I. Shmal’hauzen, Principles of Adaptive Optics (Nauka, 1985).

1980 (1)

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980).

1977 (1)

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]

Andrusyak, O.

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]

Attanasio, D. V.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A review of lithium niobate modulators for fiber-optic communications systems,” IEEE J. Sel. Top. Quantum Electron. 6, 69–82 (2000).
[CrossRef]

Augst, S. J.

Berdine, R. W.

R. A. Motes and R. W. Berdine, Introduction to High Power Fiber Lasers (Directed Energy Professional Society, 2009).

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]

E. W. Justh, M. A. Vorontsov, G. W. Carhart, L. A. Beresnev, and P. S. Krishnaprasad, “Adaptive optics with advanced phase-contrast techniques. II. High-resolution wave-front control,” J. Opt. Soc. Am. A 18, 1300–1311 (2001).
[CrossRef]

M. A. Vorontsov, G. W. Carhart, L. A. Beresnev, T. Weyrauch, and S. L. Lachinova, “Phase-locking system for fiber collimator array,” Invention Disclosure, U.S. Army Research Laboratory, University of Maryland (June 2009).

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980).

Bos, P. J.

P. F. McManamon, P. J. Bos, M. J. Escuti, J. Heikenfeld, S. Serati, H. Xie, and E. A. Watson, “A review of phased array steering for narrow-band electrooptical systems,” Proc. IEEE 97, 1078–1096 (2009).
[CrossRef]

Bossi, D. E.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A review of lithium niobate modulators for fiber-optic communications systems,” IEEE J. Sel. Top. Quantum Electron. 6, 69–82 (2000).
[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]

Bruesselbach, H.

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]

E. W. Justh, M. A. Vorontsov, G. W. Carhart, L. A. Beresnev, and P. S. Krishnaprasad, “Adaptive optics with advanced phase-contrast techniques. II. High-resolution wave-front control,” J. Opt. Soc. Am. A 18, 1300–1311 (2001).
[CrossRef]

M. A. Vorontsov, G. W. Carhart, M. Cohen, and G. Cauwenberghs, “Adaptive optics based on analog parallel stochastic optimization: analysis and experimental demonstration,” J. Opt. Soc. Am. A 17, 1440–1453 (2000).
[CrossRef]

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]

M. A. Vorontsov, G. W. Carhart, L. A. Beresnev, T. Weyrauch, and S. L. Lachinova, “Phase-locking system for fiber collimator array,” Invention Disclosure, U.S. Army Research Laboratory, University of Maryland (June 2009).

Cauwenberghs, G.

Chen, Z.

Z. Chen, J. Hou, P. Zhou, X. Wang, X. Xu, Z. Jiang, and Z. Liu, “Mutual injection locking and coherent combining of three individual fiber lasers,” Opt. Commun. 282, 60–63 (2009).
[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]

Chowdhury, D.

Ciapurin, I.

Cohen, M.

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]

Escuti, M. J.

P. F. McManamon, P. J. Bos, M. J. Escuti, J. Heikenfeld, S. Serati, H. Xie, and E. A. Watson, “A review of phased array steering for narrow-band electrooptical systems,” Proc. IEEE 97, 1078–1096 (2009).
[CrossRef]

Fan, T. Y.

S. J. Augst, J. K. Ranka, T. Y. Fan, and A. Sanchez, “Beam combining of ytterbium fiber amplifiers (Invited),” J. Opt. Soc. Am. B 24, 1707–1715 (2007).
[CrossRef]

T. Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11, 567–577 (2005).
[CrossRef]

Feng, Y.

Fischer, R.

P. Sprangle, A. Ting, J. Penano, R. Fischer, and B. Hafizi, “Incoherent combining and atmospheric propagation of high-power fiber lasers for directed-energy applications,” IEEE J. Quantum Electron. 45, 138–148 (2009).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University Press, 2007).

Fritz, D. J.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A review of lithium niobate modulators for fiber-optic communications systems,” IEEE J. Sel. Top. Quantum Electron. 6, 69–82 (2000).
[CrossRef]

Glebov, L.

Glova, A. F.

A. F. Glova, “Phase locking of optically coupled lasers,” Quantum Electron. 33, 283–306 (2003).
[CrossRef]

Hafizi, B.

P. Sprangle, A. Ting, J. Penano, R. Fischer, and B. Hafizi, “Incoherent combining and atmospheric propagation of high-power fiber lasers for directed-energy applications,” IEEE J. Quantum Electron. 45, 138–148 (2009).
[CrossRef]

Hallemeier, P. F.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A review of lithium niobate modulators for fiber-optic communications systems,” IEEE J. Sel. Top. Quantum Electron. 6, 69–82 (2000).
[CrossRef]

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]

Heikenfeld, J.

P. F. McManamon, P. J. Bos, M. J. Escuti, J. Heikenfeld, S. Serati, H. Xie, and E. A. Watson, “A review of phased array steering for narrow-band electrooptical systems,” Proc. IEEE 97, 1078–1096 (2009).
[CrossRef]

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]

Hoffman, P. R.

T. H. Loftus, A. M. Thomas, P. R. Hoffman, M. Norsen, R. Royse, A. Liu, and E. C. Honea, “Spectrally beam-combined fiber lasers for high-average-power applications,” IEEE J. Sel. Top. Quantum Electron. 13, 487–497 (2007).
[CrossRef]

Honea, E. C.

T. H. Loftus, A. M. Thomas, P. R. Hoffman, M. Norsen, R. Royse, A. Liu, and E. C. Honea, “Spectrally beam-combined fiber lasers for high-average-power applications,” IEEE J. Sel. Top. Quantum Electron. 13, 487–497 (2007).
[CrossRef]

Hou, J.

Z. Chen, J. Hou, P. Zhou, X. Wang, X. Xu, Z. Jiang, and Z. Liu, “Mutual injection locking and coherent combining of three individual fiber lasers,” Opt. Commun. 282, 60–63 (2009).
[CrossRef]

Ichikawa, J.

J. Ichikawa, “LiNbO3 based optical devices for fiber communication,” J. China Univ. Posts Telecommun. 16, 12–15 (2009).
[CrossRef]

Jiang, Z.

Z. Chen, J. Hou, P. Zhou, X. Wang, X. Xu, Z. Jiang, and Z. Liu, “Mutual injection locking and coherent combining of three individual fiber lasers,” Opt. Commun. 282, 60–63 (2009).
[CrossRef]

Jones, D. C.

Justh, E. W.

Kalmar, A.

K. H. Kudielka, A. Kalmar, and W. R. Leeb, “Design and breadboarding of a phased telescope array for free-space laser communications,” in Proceedings of IEEE International Symposium on Phased Array Systems and Technology (IEEE, 1996), pp. 419–424.
[CrossRef]

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]

Kissa, K. M.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A review of lithium niobate modulators for fiber-optic communications systems,” IEEE J. Sel. Top. Quantum Electron. 6, 69–82 (2000).
[CrossRef]

Kobyakov, A.

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]

Kovalev, V. I.

V. I. Kovalev, Department of Physics, Heriot-Watt University, Edinburgh EH14 4AS, UK (personal communication, 2010).

Krishnaprasad, P. S.

Kudielka, K. H.

K. H. Kudielka, A. Kalmar, and W. R. Leeb, “Design and breadboarding of a phased telescope array for free-space laser communications,” in Proceedings of IEEE International Symposium on Phased Array Systems and Technology (IEEE, 1996), pp. 419–424.
[CrossRef]

Lachinova, S. L.

Lafaw, D. A.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A review of lithium niobate modulators for fiber-optic communications systems,” IEEE J. Sel. Top. Quantum Electron. 6, 69–82 (2000).
[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]

Leeb, W. R.

K. H. Kudielka, A. Kalmar, and W. R. Leeb, “Design and breadboarding of a phased telescope array for free-space laser communications,” in Proceedings of IEEE International Symposium on Phased Array Systems and Technology (IEEE, 1996), pp. 419–424.
[CrossRef]

Leger, J. R.

J. R. Leger, “External methods of phase locking and coherent beam addition of diode lasers,” in Surface Emitting Semiconductor Lasers and Arrays, G.A.Evans and J.M.Hammer, eds. (Academic, 1993), pp. 379–433.

Lei, B.

Levine, W. S.

W. S. Levine, ed., The Control Handbook (CRC Press, 1996).

Liu, A.

T. H. Loftus, A. M. Thomas, P. R. Hoffman, M. Norsen, R. Royse, A. Liu, and E. C. Honea, “Spectrally beam-combined fiber lasers for high-average-power applications,” IEEE J. Sel. Top. Quantum Electron. 13, 487–497 (2007).
[CrossRef]

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.

Z. Chen, J. Hou, P. Zhou, X. Wang, X. Xu, Z. Jiang, and Z. Liu, “Mutual injection locking and coherent combining of three individual fiber lasers,” Opt. Commun. 282, 60–63 (2009).
[CrossRef]

Loftus, T. H.

T. H. Loftus, A. M. Thomas, P. R. Hoffman, M. Norsen, R. Royse, A. Liu, and E. C. Honea, “Spectrally beam-combined fiber lasers for high-average-power applications,” IEEE J. Sel. Top. Quantum Electron. 13, 487–497 (2007).
[CrossRef]

Maack, D.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A review of lithium niobate modulators for fiber-optic communications systems,” IEEE J. Sel. Top. Quantum Electron. 6, 69–82 (2000).
[CrossRef]

Mangir, M. S.

McBrien, G. J.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A review of lithium niobate modulators for fiber-optic communications systems,” IEEE J. Sel. Top. Quantum Electron. 6, 69–82 (2000).
[CrossRef]

McManamon, P. F.

P. F. McManamon, P. J. Bos, M. J. Escuti, J. Heikenfeld, S. Serati, H. Xie, and E. A. Watson, “A review of phased array steering for narrow-band electrooptical systems,” Proc. IEEE 97, 1078–1096 (2009).
[CrossRef]

Minden, M.

H. Bruesselbach, D. C. Jones, M. S. Mangir, M. Minden, and J. L. Rogers, “Self-organized coherence in fiber laser arrays,” Opt. Lett. 30, 1339–1341 (2005).
[CrossRef] [PubMed]

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

Motes, R. A.

R. A. Motes and R. W. Berdine, Introduction to High Power Fiber Lasers (Directed Energy Professional Society, 2009).

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]

Murphy, E. J.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A review of lithium niobate modulators for fiber-optic communications systems,” IEEE J. Sel. Top. Quantum Electron. 6, 69–82 (2000).
[CrossRef]

Nabors, C. D.

Norsen, M.

T. H. Loftus, A. M. Thomas, P. R. Hoffman, M. Norsen, R. Royse, A. Liu, and E. C. Honea, “Spectrally beam-combined fiber lasers for high-average-power applications,” IEEE J. Sel. Top. Quantum Electron. 13, 487–497 (2007).
[CrossRef]

O’Meara, T. R.

Penano, J.

P. Sprangle, A. Ting, J. Penano, R. Fischer, and B. Hafizi, “Incoherent combining and atmospheric propagation of high-power fiber lasers for directed-energy applications,” IEEE J. Quantum Electron. 45, 138–148 (2009).
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Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University Press, 2007).

Ranka, J. K.

Ricklin, J. C.

Rogers, J. L.

Royse, R.

T. H. Loftus, A. M. Thomas, P. R. Hoffman, M. Norsen, R. Royse, A. Liu, and E. C. Honea, “Spectrally beam-combined fiber lasers for high-average-power applications,” IEEE J. Sel. Top. Quantum Electron. 13, 487–497 (2007).
[CrossRef]

Sanchez, A.

Sauer, M.

Serati, S.

P. F. McManamon, P. J. Bos, M. J. Escuti, J. Heikenfeld, S. Serati, H. Xie, and E. A. Watson, “A review of phased array steering for narrow-band electrooptical systems,” Proc. IEEE 97, 1078–1096 (2009).
[CrossRef]

Sevian, A.

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.

Shmal’hauzen, V. I.

M. A. Vorontsov and V. I. Shmal’hauzen, Principles of Adaptive Optics (Nauka, 1985).

Smirnov, V.

Sprangle, P.

P. Sprangle, A. Ting, J. Penano, R. Fischer, and B. Hafizi, “Incoherent combining and atmospheric propagation of high-power fiber lasers for directed-energy applications,” IEEE J. Quantum Electron. 45, 138–148 (2009).
[CrossRef]

ter Haar Romey, B. M.

B. M. ter Haar Romey, ed., Geometry-Driven Diffusion in Computer Vision (Kluwer Academic, 1994).

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University Press, 2007).

Thomas, A. M.

T. H. Loftus, A. M. Thomas, P. R. Hoffman, M. Norsen, R. Royse, A. Liu, and E. C. Honea, “Spectrally beam-combined fiber lasers for high-average-power applications,” IEEE J. Sel. Top. Quantum Electron. 13, 487–497 (2007).
[CrossRef]

Ting, A.

P. Sprangle, A. Ting, J. Penano, R. Fischer, and B. Hafizi, “Incoherent combining and atmospheric propagation of high-power fiber lasers for directed-energy applications,” IEEE J. Quantum Electron. 45, 138–148 (2009).
[CrossRef]

Venus, G.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University Press, 2007).

Vorontsov, M. 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]

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]

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]

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]

E. W. Justh, M. A. Vorontsov, G. W. Carhart, L. A. Beresnev, and P. S. Krishnaprasad, “Adaptive optics with advanced phase-contrast techniques. II. High-resolution wave-front control,” J. Opt. Soc. Am. A 18, 1300–1311 (2001).
[CrossRef]

M. A. Vorontsov, G. W. Carhart, M. Cohen, and G. Cauwenberghs, “Adaptive optics based on analog parallel stochastic optimization: analysis and experimental demonstration,” J. Opt. Soc. Am. A 17, 1440–1453 (2000).
[CrossRef]

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]

M. A. Vorontsov and V. I. Shmal’hauzen, Principles of Adaptive Optics (Nauka, 1985).

M. A. Vorontsov, G. W. Carhart, L. A. Beresnev, T. Weyrauch, and S. L. Lachinova, “Phase-locking system for fiber collimator array,” Invention Disclosure, U.S. Army Research Laboratory, University of Maryland (June 2009).

Wang, X.

Z. Chen, J. Hou, P. Zhou, X. Wang, X. Xu, Z. Jiang, and Z. Liu, “Mutual injection locking and coherent combining of three individual fiber lasers,” Opt. Commun. 282, 60–63 (2009).
[CrossRef]

Watson, E. A.

P. F. McManamon, P. J. Bos, M. J. Escuti, J. Heikenfeld, S. Serati, H. Xie, and E. A. Watson, “A review of phased array steering for narrow-band electrooptical systems,” Proc. IEEE 97, 1078–1096 (2009).
[CrossRef]

Weber, M.

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]

Weber, M. E.

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]

Weyrauch, T.

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]

M. A. Vorontsov, G. W. Carhart, L. A. Beresnev, T. Weyrauch, and S. L. Lachinova, “Phase-locking system for fiber collimator array,” Invention Disclosure, U.S. Army Research Laboratory, University of Maryland (June 2009).

Wickham, M.

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]

Wickham, M. G.

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]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980).

Wooten, E. L.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A review of lithium niobate modulators for fiber-optic communications systems,” IEEE J. Sel. Top. Quantum Electron. 6, 69–82 (2000).
[CrossRef]

Xie, H.

P. F. McManamon, P. J. Bos, M. J. Escuti, J. Heikenfeld, S. Serati, H. Xie, and E. A. Watson, “A review of phased array steering for narrow-band electrooptical systems,” Proc. IEEE 97, 1078–1096 (2009).
[CrossRef]

Xu, X.

Z. Chen, J. Hou, P. Zhou, X. Wang, X. Xu, Z. Jiang, and Z. Liu, “Mutual injection locking and coherent combining of three individual fiber lasers,” Opt. Commun. 282, 60–63 (2009).
[CrossRef]

Yi-Yan, A.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A review of lithium niobate modulators for fiber-optic communications systems,” IEEE J. Sel. Top. Quantum Electron. 6, 69–82 (2000).
[CrossRef]

Yu, C. X.

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]

Zhou, P.

Z. Chen, J. Hou, P. Zhou, X. Wang, X. Xu, Z. Jiang, and Z. Liu, “Mutual injection locking and coherent combining of three individual fiber lasers,” Opt. Commun. 282, 60–63 (2009).
[CrossRef]

Adv. Opt. Photon. (1)

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

P. Sprangle, A. Ting, J. Penano, R. Fischer, and B. Hafizi, “Incoherent combining and atmospheric propagation of high-power fiber lasers for directed-energy applications,” IEEE J. Quantum Electron. 45, 138–148 (2009).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (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]

T. Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11, 567–577 (2005).
[CrossRef]

T. H. Loftus, A. M. Thomas, P. R. Hoffman, M. Norsen, R. Royse, A. Liu, and E. C. Honea, “Spectrally beam-combined fiber lasers for high-average-power applications,” IEEE J. Sel. Top. Quantum Electron. 13, 487–497 (2007).
[CrossRef]

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A review of lithium niobate modulators for fiber-optic communications systems,” IEEE J. Sel. Top. Quantum Electron. 6, 69–82 (2000).
[CrossRef]

J. China Univ. Posts Telecommun. (1)

J. Ichikawa, “LiNbO3 based optical devices for fiber communication,” J. China Univ. Posts Telecommun. 16, 12–15 (2009).
[CrossRef]

J. Opt. Soc. Am. (1)

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

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

Opt. Commun. (1)

Z. Chen, J. Hou, P. Zhou, X. Wang, X. Xu, Z. Jiang, and Z. Liu, “Mutual injection locking and coherent combining of three individual fiber lasers,” Opt. Commun. 282, 60–63 (2009).
[CrossRef]

Opt. Express (2)

Opt. Lett. (3)

Proc. IEEE (1)

P. F. McManamon, P. J. Bos, M. J. Escuti, J. Heikenfeld, S. Serati, H. Xie, and E. A. Watson, “A review of phased array steering for narrow-band electrooptical systems,” Proc. IEEE 97, 1078–1096 (2009).
[CrossRef]

Proc. SPIE (3)

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]

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]

Quantum Electron. (1)

A. F. Glova, “Phase locking of optically coupled lasers,” Quantum Electron. 33, 283–306 (2003).
[CrossRef]

Other (13)

J. R. Leger, “External methods of phase locking and coherent beam addition of diode lasers,” in Surface Emitting Semiconductor Lasers and Arrays, G.A.Evans and J.M.Hammer, eds. (Academic, 1993), pp. 379–433.

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

J.R.Leger, J.Nilsson, J.P.Huignard, A.P.Napartovich, T.M.Shay, and A.Shirakawa, eds., Special issue, “Laser beam combining and fiber laser systems,” IEEE J. Sel. Top. Quantum Electron.15, 237–470 (2009).

K. H. Kudielka, A. Kalmar, and W. R. Leeb, “Design and breadboarding of a phased telescope array for free-space laser communications,” in Proceedings of IEEE International Symposium on Phased Array Systems and Technology (IEEE, 1996), pp. 419–424.
[CrossRef]

V. I. Kovalev, Department of Physics, Heriot-Watt University, Edinburgh EH14 4AS, UK (personal communication, 2010).

EOSPACE, http://www.eospace.com/.

R. A. Motes and R. W. Berdine, Introduction to High Power Fiber Lasers (Directed Energy Professional Society, 2009).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University Press, 2007).

W. S. Levine, ed., The Control Handbook (CRC Press, 1996).

M. A. Vorontsov and V. I. Shmal’hauzen, Principles of Adaptive Optics (Nauka, 1985).

B. M. ter Haar Romey, ed., Geometry-Driven Diffusion in Computer Vision (Kluwer Academic, 1994).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980).

M. A. Vorontsov, G. W. Carhart, L. A. Beresnev, T. Weyrauch, and S. L. Lachinova, “Phase-locking system for fiber collimator array,” Invention Disclosure, U.S. Army Research Laboratory, University of Maryland (June 2009).

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Figures (9)

Fig. 1
Fig. 1

Notional schematics of laser transmitter (beam director) based on (a) sparse (conformal) array of fiber collimators with a single laser source and (b) pupil-plane incoherent combining of laser beams originating from multiple laser sources.

Fig. 2
Fig. 2

Pupil-plane phase-locking control system basic architectures: (a) notional schematic, (b) phase-locking receiver system with a coherent reference wave, and (c) phase-locking receiver with focal-plane beam combining. The systems are based on either electronic or optical feedback loop.

Fig. 3
Fig. 3

Principal schematics of pupil-plane phase-locking optical systems based on optical path difference stabilization with (a) heterodyne signal detection and (b) multidithering control techniques. Here ω, , LPF, and PID denote, respectively, dither signal generation, signal summation, low-pass electrical filtering, and proportional-integral-derivative control.

Fig. 4
Fig. 4

Sensing of piston phases using interference of tail sections of two beamlets in fiber-collimator array systems: (a) schematic illustration of two-tail interference, (b) fiber-array system composed of three clusters (A, B, and C) coupled by two-tail sensors, (c) two-tail sensing of piston phases in a cluster A composed of seven fiber collimators with hexagonal arrangement, and (d) coupling of three hexagonal clusters using two-tail interference sensors. Black squares in (b)–(d) denote point-size photodetectors.

Fig. 5
Fig. 5

Phase locking based on the gradient-flow optimization control in a fiber-array system with two-tail sensors for (a), (b) a single cluster and (c), (d) two coupled clusters shown in the top-right inserts in (b) and (d). Temporal dynamics of phase errors { δ j ( t ) } from random initial conditions { δ j ( 0 ) } in (a) and (c) and the ensemble-averaged phase-locking error standard deviations σ δ ( t ) and σ δ A + B ( t ) in (b) and (d) are obtained by numerical integration of Eqs. (5) with γ j = 1 , j = 2 , , 7 , and Eqs. (7, 8) with γ j A = γ j B = 1 , j = 1 , , 7 , and γ A , B = γ B , A = 0.5 , respectively. The reference phase δ 1 is shown by the horizontal line in (a). The insert in (c) illustrates an initial stage of the phase-locking process.

Fig. 6
Fig. 6

Pupil-plane phase locking of the fiber-array cluster system as in Fig. 4c with two-tail sensors using optimization of (a), (c) the combined and (b), (d) the local metrics. Dynamics of the residual piston phase errors { δ j ( t ) } , j = 1 , , 7 , from an identical set of random initial conditions in (a) and (b) and the ensemble-averaged phase-locking performance metrics J ̂ ( t ) and σ δ ( t ) in (c) and (d) are obtained using the control algorithms (13, 14), respectively. Reference phase δ 1 is shown by the horizontal lines in (a), (b).

Fig. 7
Fig. 7

Phase locking of a fiber array [shown in the insert in (a)] using local metrics obtained with three-tail sensors. Time dependences of the ensemble-averaged phase-locking performance measures J ̂ ( t ) and σ δ ( t ) using (a) optimization of the combined metric with SPGD (dashed lines) and local metrics with D-SPGD (solid lines) control algorithms and (b) the gradient-flow optimization algorithm (19).

Fig. 8
Fig. 8

Phase locking based on focal-plane beam-tail interference sensors: (a) notional schematic of the focal-plane beam-tail sensor and (b) geometries of off-axis focusing mirrors (M) corresponding to two- and three-tail focal-plane sensors. Grayscale images are examples of focal-plane intensity distributions obtained using the parameters of the experimental fiber array reported in [3]. Small circles at their centers show photodetector apertures.

Fig. 9
Fig. 9

Phase locking with focal-plane beam-tail sensors: (a) schematic of the combined focusing element for three-tail focal-plane sensing used in numerical simulations (left), phase pattern of the complex transfer function argument in Eq. (21) corresponding to three off-axis parabolic mirrors with a common focal spot (top-right), intensity distribution of the beam tails at the CFE plane (middle-right), and intensity distribution at the focal plane (bottom-right); (b) metrics J ̂ and σ δ versus the iteration number n for three different photodetector aperture radii: b PD = 0 (solid lines), b PD = b Airy 2 (dashed lines), and b PD = b Airy (dot-dashed lines). The dashed circle in (a) indicates the Airy disk of diameter 2 b Airy .

Equations (37)

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τ d u j ( t ) d t = γ j J j [ δ 1 ( t ) , , δ j ( t ) , , δ N sub ( t ) ]
( j = 1 , , N sub ) .
u j ( n + 1 ) = u j ( n ) + γ ( n ) Δ J ( n ) Δ u j ( n ) ( j = 1 , , N sub ) ,
J 1 , 2 ( t ) = J 1 , 2 [ δ 1 ( t ) , δ 2 ( t ) ] = I 1 + I 2 + 2 I 1 I 2 cos [ δ 2 ( t ) δ 1 ( t ) + ζ 1 , 2 ] ,
J 1 , j ( t ) = J 1 , j [ δ 1 ( t ) , δ j ( t ) ] = I 1 + I j + 2 I 1 I j cos [ δ j ( t ) δ 1 ( t ) ]
( j = 2 , , N 0 ) ,
τ d δ j ( t ) d t = γ j J 1 , j [ δ 1 ( t ) , δ j ( t ) ] ( j = 2 , , N 0 ) ,
σ δ 2 ( t ) = 1 N 0 1 j = 2 N 0 = 7 [ δ j ( t ) δ 1 ] 2 .
τ d δ j A ( t ) d t = γ j A sin [ δ 1 A ( t ) δ j A ( t ) ] + κ j i γ A , B sin [ δ k B ( t ) δ j A ( t ) ]
( j = 2 , , N 0 A ) ,
τ d δ 1 A ( t ) d t = 1 N 0 A 1 j = 2 N 0 A γ j A sin [ δ 1 A ( t ) δ j A ( t ) ] ,
τ d δ j B ( t ) d t = γ j B sin [ δ 1 B ( t ) δ j B ( t ) ] + κ j k γ B , A sin [ δ i A ( t ) δ j B ( t ) ]
( j = 2 , , N 0 B ) ,
τ d δ 1 B ( t ) d t = 1 N 0 B 1 j = 2 N 0 B κ j B sin [ δ 1 B ( t ) δ j B ( t ) ] .
δ j A ( t ) = mod 2 π [ δ 1 A ( t ) ] ( j = 2 , , N 0 A ) ,
δ j B ( t ) = mod 2 π [ δ 1 B ( t ) ] ( j = 2 , , N 0 B ) ,
δ 1 A ( t ) = mod 2 π [ δ 1 B ( t ) ] .
τ d δ i A ( t ) d t = γ A , B sin [ δ k B ( t ) δ i A ( t ) ] ,
τ d δ k B ( t ) d t = γ A , B sin [ δ i A ( t ) δ k B ( t ) ] .
J l , k ( t ) = J l , k [ δ l ( t ) , δ k ( t ) ] = I l + I k + 2 I l I k cos [ δ k ( t ) δ l ( t ) ]
( k = 1 , , N 0 ; k l ) .
J Σ ( t ) = k > l M J l , k [ δ l ( t ) , δ k ( t ) ] .
δ j ( n + 1 ) = δ j ( n ) + γ Σ ( n ) Δ J Σ ( n ) Δ δ j ( n ) ( j = 2 , , N 0 ; n = 1 , ) ,
δ j ( n + 1 ) = δ j ( n ) + γ j ( n ) Δ J 1 , j ( n ) Δ δ j ( n ) ( j = 2 , , N 0 ; n = 1 , ) ,
J l , k , m ( t ) = J l , k , m [ δ l ( t ) , δ k ( t ) , δ m ( t ) ] = | I l 1 2 exp [ i δ l ( t ) ] + I k 1 2 exp [ i δ k ( t ) ] + I m 1 2 exp [ i δ m ( t ) ] | 2 ,
J l , k , m [ δ l ( t ) , δ k ( t ) , δ m ( t ) ] = I l | 1 + μ k exp [ i δ k ( t ) i δ l ( t ) ] + μ m exp [ i δ m ( t ) i δ l ( t ) ] | 2 = 2 I l { μ 0 2 + μ k cos [ δ k ( t ) δ l ( t ) ] + μ m cos [ δ m ( t ) δ l ( t ) ] + μ k μ m cos [ δ k ( t ) δ m ( t ) ] } ,
J Σ ( t ) = m > k > l K J l , k , m [ δ l ( t ) , δ k ( t ) , δ m ( t ) ] + p > q M J q , p [ δ q ( t ) , δ p ( t ) ] .
J Σ = J 1 , 2 , 3 ( δ 1 , δ 2 , δ 3 ) + J 1 , 4 , 5 ( δ 1 , δ 4 , δ 5 ) + J 1 , 6 , 7 ( δ 1 , δ 6 , δ 7 ) .
τ d δ j ( t ) d t = γ j J 1 , j , j + 1 δ j = γ j { sin [ δ 1 ( t ) δ j ( t ) ] + sin [ δ j + 1 ( t ) δ j ( t ) ] } ,
τ d δ j + 1 ( t ) d t = γ j + 1 J 1 , j , j + 1 δ j + 1 = γ j + 1 { sin [ δ 1 ( t ) δ j + 1 ( t ) ] + sin [ δ j ( t ) δ j + 1 ( t ) ] } ,
A j ( r r j , t ) = A 0 ( r r j ) exp [ i k | r r j | 2 2 f + i δ j ( t ) ]
( j = 1 , , N sub ) ,
T l CFE ( r , r j , r k , t ) = V l ( r r k ) exp [ i φ ( r r j , f ) + i φ ( r r k , f CFE ) ] ,
A l CFE ( r , r j , r k , t ) = A 0 ( r r j ) V l ( r r k ) × exp [ i φ ( r r k , f CFE ) + i δ j ( t ) ] A ̃ l CFE ( r , r j , r k ) exp [ i δ j ( t ) ] ,
A l PD ( r , r j , r k , t ) = i k 2 π f CFE exp ( i k f CFE ) A l CFE ( r , r j , r k , t ) exp [ i φ ( r r , f CFE ) ] d r A ̃ l PD ( r , r j , r k ) exp [ i δ j ( t ) ] ,
I k PD ( r , t ) = | l k = 1 N n g A ̃ l k PD ( r , r k ) exp [ i δ l k ( t ) ] | 2 ,
J k ( t ) = S PD I k PD ( r , t ) d 2 ( r r k ) ,

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