Abstract

Coherent beam combination (CBC) of fiber lasers based on self-imaging properties of a strongly confined tapered waveguide (SCTW) is studied systematically. Analytical formulas are derived for the positions, amplitudes, and phases of the N self-images at the output of a SCTW, which are important for quantitative analysis of waveguide-based CBC. The formulas are verified with numerical examples by a finite difference beam propagation method (FDBPM) and the errors of the analytical expressions are studied. This study shows that the analytical formulas agree well with the FDBPM simulation results when the taper angle is less than 1.4° and the phase distortion is less than λ/10. The relative errors increase as the taper angle increases. Based on the theoretical model and the FDBPM, we simulated the CBC of fiber laser array and compared the CBC based on the tapered waveguide with that based on the nontapered one. The effects of input beam number, aperture fill factor, and taper angle on the combination performance have been studied. The study reveals that a beam which has near-diffraction limited beam quality (M21.41) and a single beam without side lobe in the far field can be achieved with tapered-waveguide-based CBC. It is shown that beam quality depends on input beam number, aperture fill factor, and taper angle. There exists a best fill factor which will increase as input beam number increases. The tolerance of the system on the fill factor and the taper angle is studied, which is 0.45<t<0.67 and θ<0.8°, respectively. The results may be useful for practical, high-power fiber laser systems.

© 2013 Chinese Laser Press

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber laser: current status and future perspectives,” J. Opt. Soc. Am. B 27, B63–B92 (2010).
    [CrossRef]
  2. J. R. Leger, J. Nilsson, J. P. Huignard, A. P. Napartovich, T. M. Shay, and A. Shirakawa, “Laser beam combining and fiber laser systems,” IEEE J. Sel. Top. Quantum Electron. 15, 237–239 (2009).
    [CrossRef]
  3. T. Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11, 567–577 (2005).
    [CrossRef]
  4. T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, “High-power phase locking of a fiber amplifier array,” Proc. SPIE 7195, 71951M (2009).
    [CrossRef]
  5. B. He, Q. H. Lou, J. Zhou, J. Dong, Y. Wei, D. Xue, Y. Qi, Z. Su, L. Li, and F. Zhang, “High power coherent beam combination from two fiber lasers,” Opt. Express 14, 2721–2726 (2006).
    [CrossRef]
  6. J. Wang, K. Duan, and Y. Wang, “Experimental study of coherent beam combining of two fiber lasers,” Acta Phys. Sin. 57, 5627–5631 (2008).
  7. P. Zhou, Z. J. Liu, X. L. Wang, Y. X. Ma, H. T. Ma, X. J. Xu, and S. F. Guo, “Coherent beam combining of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15, 248–256 (2009).
    [CrossRef]
  8. 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, 6708K (2007).
  9. 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]
  10. P. Zhou, Y. Ma, X. Wang, H. Ma, J. Wang, X. Xu, and Z. Liu, “Coherent beam combination of a hexagonal distributed high power fiber amplifier array,” Appl. Opt. 48, 6537–6540 (2009).
    [CrossRef]
  11. E. C. Cheung, J. G. Ho, G. D. Goodno, R. R. Rice, J. Rothenberg, P. Thielen, M. Weber, and M. Wickham, “Diffractive-optics-based beam combination of a phase-locked fiber laser array,” Opt. Lett. 33, 354–356 (2008).
    [CrossRef]
  12. R. Uberna, A. Bratcher, and B. G. Tiemann, “Coherent polarization beam combination,” IEEE J. Quantum Electron. 46, 1191–1196 (2010).
    [CrossRef]
  13. S. E. Christensen and O. Koski, “2-Dimensional waveguide coherent beam combiner,” in Advanced Solid-State Photonics, OSA Technical Digest (CD) (Optical Society of America, 2007), paper WC1.
  14. R. Uberna, A. Bratcher, T. G. Alley, A. D. Sanchez, A. S. Flores, and B. Pulford, “Coherent combination of high power fiber amplifiers in a two-dimensional re-imaging waveguide,” Opt. Express 18, 13547–13553 (2010).
    [CrossRef]
  15. W. S. C. Chang, Fundamentals of Guided-Wave Optoelectronic Devices (Cambridge University, 2010).
  16. R. Ulrich and T. Kamiya, “Resolution of self-images in planar optical waveguides,” J. Opt. Soc. Am. 68, 583–592 (1978).
    [CrossRef]
  17. M. Bachmann, P. A. Besse, and H. Melchior, “General self-imaging properties in N×N multimode interference couplers including phase relations,” Appl. Opt. 33, 3905–3911 (1994).
    [CrossRef]
  18. S. He, X. Ao, and V. Romanov, “General properties of N×M self-images in a strongly confined rectangular waveguide,” Appl. Opt. 42, 4855–4859 (2003).
    [CrossRef]
  19. H. J. Baker, J. R. Lee, and D. R. Hall, “Self-imaging and high-beam-quality operation in multi-mode planar waveguide optical amplifiers,” Opt. Express 10, 297–302 (2002).
    [CrossRef]
  20. R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical technique for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
    [CrossRef]
  21. R. Scarmozzino and R. M. Osgood, “Comparison of finite-difference and Fourier-transform solutions of the parabolic wave equation with emphasis on integrated-optics applications,” J. Opt. Soc. Am. A 8, 724–731 (1991).
    [CrossRef]
  22. Y. Chung and N. Dagli, “An assessment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335–1339 (1990).
    [CrossRef]
  23. Y. Chung and N. Dagli, “Modeling of guided-wave optical components with efficient finite-difference beam propagation methods,” in Antennas and Propagation Society International Symposium (IEEE, 1992), pp. 248–251.
  24. C. Vassallo, Optical Waveguide Concepts (Elsevier, 1991).
  25. C. Vassallo, “Interest of improved three-point formulas for finite-difference modeling of optical devices,” J. Opt. Soc. Am. A 14, 3273–3284 (1997).
    [CrossRef]
  26. K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis (Wiley, 2001).
  27. C. Chen, Foundations for Guided-Wave Optics (Wiley, 2007).
  28. G. R. Hadley, “Transparent boundary condition for the beam propagation method,” Opt. Lett. 16, 624–626 (1991).
    [CrossRef]
  29. G. R. Hadley, “Transparent boundary condition for the beam propagation method,” IEEE J. Quantum Electron. 28, 363–370 (1992).
    [CrossRef]
  30. A. E. Siegman, “How to (maybe) measure laser beam quality,” in Diode Pumped Solid State Lasers: Applications and Issues, M. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics (Optical Society of America, 1998), paper MQ1.
  31. T. Poon and T. Kim, Engineering Optics with MATLAB (World Scientific, 2006).
  32. J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010).

2010 (3)

2009 (4)

J. R. Leger, J. Nilsson, J. P. Huignard, A. P. Napartovich, T. M. Shay, and A. Shirakawa, “Laser beam combining and fiber laser systems,” IEEE J. Sel. Top. Quantum Electron. 15, 237–239 (2009).
[CrossRef]

T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, “High-power phase locking of a fiber amplifier array,” Proc. SPIE 7195, 71951M (2009).
[CrossRef]

P. Zhou, Z. J. Liu, X. L. Wang, Y. X. Ma, H. T. Ma, X. J. Xu, and S. F. Guo, “Coherent beam combining of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15, 248–256 (2009).
[CrossRef]

P. Zhou, Y. Ma, X. Wang, H. Ma, J. Wang, X. Xu, and Z. Liu, “Coherent beam combination of a hexagonal distributed high power fiber amplifier array,” Appl. Opt. 48, 6537–6540 (2009).
[CrossRef]

2008 (2)

2007 (1)

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, 6708K (2007).

2006 (2)

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]

B. He, Q. H. Lou, J. Zhou, J. Dong, Y. Wei, D. Xue, Y. Qi, Z. Su, L. Li, and F. Zhang, “High power coherent beam combination from two fiber lasers,” Opt. Express 14, 2721–2726 (2006).
[CrossRef]

2005 (1)

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

2003 (1)

2002 (1)

2000 (1)

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical technique for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[CrossRef]

1997 (1)

1994 (1)

1992 (1)

G. R. Hadley, “Transparent boundary condition for the beam propagation method,” IEEE J. Quantum Electron. 28, 363–370 (1992).
[CrossRef]

1991 (2)

1990 (1)

Y. Chung and N. Dagli, “An assessment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335–1339 (1990).
[CrossRef]

1978 (1)

Alley, T. G.

Ao, X.

Bachmann, M.

Baker, H. J.

Baker, J. T.

T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, “High-power phase locking of a fiber amplifier array,” Proc. SPIE 7195, 71951M (2009).
[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, 6708K (2007).

Besse, P. A.

Bratcher, A.

Bronder, T. J.

T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, “High-power phase locking of a fiber amplifier array,” Proc. SPIE 7195, 71951M (2009).
[CrossRef]

Chang, W. S. C.

W. S. C. Chang, Fundamentals of Guided-Wave Optoelectronic Devices (Cambridge University, 2010).

Chen, C.

C. Chen, Foundations for Guided-Wave Optics (Wiley, 2007).

Cheung, E. C.

Christensen, S. E.

S. E. Christensen and O. Koski, “2-Dimensional waveguide coherent beam combiner,” in Advanced Solid-State Photonics, OSA Technical Digest (CD) (Optical Society of America, 2007), paper WC1.

Chung, Y.

Y. Chung and N. Dagli, “An assessment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335–1339 (1990).
[CrossRef]

Y. Chung and N. Dagli, “Modeling of guided-wave optical components with efficient finite-difference beam propagation methods,” in Antennas and Propagation Society International Symposium (IEEE, 1992), pp. 248–251.

Clarkson, W. A.

Dagli, N.

Y. Chung and N. Dagli, “An assessment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335–1339 (1990).
[CrossRef]

Y. Chung and N. Dagli, “Modeling of guided-wave optical components with efficient finite-difference beam propagation methods,” in Antennas and Propagation Society International Symposium (IEEE, 1992), pp. 248–251.

Dong, J.

Duan, K.

J. Wang, K. Duan, and Y. Wang, “Experimental study of coherent beam combining of two fiber lasers,” Acta Phys. Sin. 57, 5627–5631 (2008).

Fan, T. Y.

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

Flores, A. S.

Gallant, D.

T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, “High-power phase locking of a fiber amplifier array,” Proc. SPIE 7195, 71951M (2009).
[CrossRef]

Goodno, G. D.

Gopinath, A.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical technique for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[CrossRef]

Guo, S. F.

P. Zhou, Z. J. Liu, X. L. Wang, Y. X. Ma, H. T. Ma, X. J. Xu, and S. F. Guo, “Coherent beam combining of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15, 248–256 (2009).
[CrossRef]

Hadley, G. R.

G. R. Hadley, “Transparent boundary condition for the beam propagation method,” IEEE J. Quantum Electron. 28, 363–370 (1992).
[CrossRef]

G. R. Hadley, “Transparent boundary condition for the beam propagation method,” Opt. Lett. 16, 624–626 (1991).
[CrossRef]

Hall, D. R.

He, B.

He, S.

Helfert, S.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical technique for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[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]

Ho, J. G.

Huignard, J. P.

J. R. Leger, J. Nilsson, J. P. Huignard, A. P. Napartovich, T. M. Shay, and A. Shirakawa, “Laser beam combining and fiber laser systems,” IEEE J. Sel. Top. Quantum Electron. 15, 237–239 (2009).
[CrossRef]

Kamiya, T.

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]

Kawano, K.

K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis (Wiley, 2001).

Kim, T.

T. Poon and T. Kim, Engineering Optics with MATLAB (World Scientific, 2006).

Kitoh, T.

K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis (Wiley, 2001).

Koski, O.

S. E. Christensen and O. Koski, “2-Dimensional waveguide coherent beam combiner,” in Advanced Solid-State Photonics, OSA Technical Digest (CD) (Optical Society of America, 2007), paper WC1.

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]

Lee, J. R.

Leger, J. R.

J. R. Leger, J. Nilsson, J. P. Huignard, A. P. Napartovich, T. M. Shay, and A. Shirakawa, “Laser beam combining and fiber laser systems,” IEEE J. Sel. Top. Quantum Electron. 15, 237–239 (2009).
[CrossRef]

Li, L.

Liu, L.

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, 6708K (2007).

Liu, Z.

Liu, Z. J.

P. Zhou, Z. J. Liu, X. L. Wang, Y. X. Ma, H. T. Ma, X. J. Xu, and S. F. Guo, “Coherent beam combining of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15, 248–256 (2009).
[CrossRef]

Lou, Q. H.

Lu, C. A.

T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, “High-power phase locking of a fiber amplifier array,” Proc. SPIE 7195, 71951M (2009).
[CrossRef]

Lucero, A.

T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, “High-power phase locking of a fiber amplifier array,” Proc. SPIE 7195, 71951M (2009).
[CrossRef]

Ma, H.

Ma, H. T.

P. Zhou, Z. J. Liu, X. L. Wang, Y. X. Ma, H. T. Ma, X. J. Xu, and S. F. Guo, “Coherent beam combining of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15, 248–256 (2009).
[CrossRef]

Ma, Y.

Ma, Y. X.

P. Zhou, Z. J. Liu, X. L. Wang, Y. X. Ma, H. T. Ma, X. J. Xu, and S. F. Guo, “Coherent beam combining of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15, 248–256 (2009).
[CrossRef]

Melchior, H.

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]

Napartovich, A. P.

J. R. Leger, J. Nilsson, J. P. Huignard, A. P. Napartovich, T. M. Shay, and A. Shirakawa, “Laser beam combining and fiber laser systems,” IEEE J. Sel. Top. Quantum Electron. 15, 237–239 (2009).
[CrossRef]

Nilsson, J.

D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber laser: current status and future perspectives,” J. Opt. Soc. Am. B 27, B63–B92 (2010).
[CrossRef]

J. R. Leger, J. Nilsson, J. P. Huignard, A. P. Napartovich, T. M. Shay, and A. Shirakawa, “Laser beam combining and fiber laser systems,” IEEE J. Sel. Top. Quantum Electron. 15, 237–239 (2009).
[CrossRef]

Osgood, R. M.

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, 6708K (2007).

Poon, T.

T. Poon and T. Kim, Engineering Optics with MATLAB (World Scientific, 2006).

Pregla, R.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical technique for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[CrossRef]

Pulford, B.

R. Uberna, A. Bratcher, T. G. Alley, A. D. Sanchez, A. S. Flores, and B. Pulford, “Coherent combination of high power fiber amplifiers in a two-dimensional re-imaging waveguide,” Opt. Express 18, 13547–13553 (2010).
[CrossRef]

T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, “High-power phase locking of a fiber amplifier array,” Proc. SPIE 7195, 71951M (2009).
[CrossRef]

Qi, Y.

Rice, R. R.

Richardson, D. J.

Robin, C. A.

T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, “High-power phase locking of a fiber amplifier array,” Proc. SPIE 7195, 71951M (2009).
[CrossRef]

Romanov, V.

Rothenberg, J.

Sanchez, A. D.

R. Uberna, A. Bratcher, T. G. Alley, A. D. Sanchez, A. S. Flores, and B. Pulford, “Coherent combination of high power fiber amplifiers in a two-dimensional re-imaging waveguide,” Opt. Express 18, 13547–13553 (2010).
[CrossRef]

T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, “High-power phase locking of a fiber amplifier array,” Proc. SPIE 7195, 71951M (2009).
[CrossRef]

Scarmozzino, R.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical technique for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[CrossRef]

R. Scarmozzino and R. M. Osgood, “Comparison of finite-difference and Fourier-transform solutions of the parabolic wave equation with emphasis on integrated-optics applications,” J. Opt. Soc. Am. A 8, 724–731 (1991).
[CrossRef]

Schmidt, J. D.

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

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, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, “High-power phase locking of a fiber amplifier array,” Proc. SPIE 7195, 71951M (2009).
[CrossRef]

J. R. Leger, J. Nilsson, J. P. Huignard, A. P. Napartovich, T. M. Shay, and A. Shirakawa, “Laser beam combining and fiber laser systems,” IEEE J. Sel. Top. Quantum Electron. 15, 237–239 (2009).
[CrossRef]

Shirakawa, A.

J. R. Leger, J. Nilsson, J. P. Huignard, A. P. Napartovich, T. M. Shay, and A. Shirakawa, “Laser beam combining and fiber laser systems,” IEEE J. Sel. Top. Quantum Electron. 15, 237–239 (2009).
[CrossRef]

Siegman, A. E.

A. E. Siegman, “How to (maybe) measure laser beam quality,” in Diode Pumped Solid State Lasers: Applications and Issues, M. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics (Optical Society of America, 1998), paper MQ1.

Su, Z.

Thielen, P.

Tiemann, B. G.

R. Uberna, A. Bratcher, and B. G. Tiemann, “Coherent polarization beam combination,” IEEE J. Quantum Electron. 46, 1191–1196 (2010).
[CrossRef]

Uberna, R.

Ulrich, R.

Vassallo, C.

Vergien, C. L.

T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, “High-power phase locking of a fiber amplifier array,” Proc. SPIE 7195, 71951M (2009).
[CrossRef]

Vorontsov, M. 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, 6708K (2007).

Wang, J.

P. Zhou, Y. Ma, X. Wang, H. Ma, J. Wang, X. Xu, and Z. Liu, “Coherent beam combination of a hexagonal distributed high power fiber amplifier array,” Appl. Opt. 48, 6537–6540 (2009).
[CrossRef]

J. Wang, K. Duan, and Y. Wang, “Experimental study of coherent beam combining of two fiber lasers,” Acta Phys. Sin. 57, 5627–5631 (2008).

Wang, X.

Wang, X. L.

P. Zhou, Z. J. Liu, X. L. Wang, Y. X. Ma, H. T. Ma, X. J. Xu, and S. F. Guo, “Coherent beam combining of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15, 248–256 (2009).
[CrossRef]

Wang, Y.

J. Wang, K. Duan, and Y. Wang, “Experimental study of coherent beam combining of two fiber lasers,” Acta Phys. Sin. 57, 5627–5631 (2008).

Weber, M.

Wei, Y.

Weyrauch, T.

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, 6708K (2007).

Wickham, M.

Xu, X.

Xu, X. J.

P. Zhou, Z. J. Liu, X. L. Wang, Y. X. Ma, H. T. Ma, X. J. Xu, and S. F. Guo, “Coherent beam combining of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15, 248–256 (2009).
[CrossRef]

Xue, D.

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]

Zerinque, C.

T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, “High-power phase locking of a fiber amplifier array,” Proc. SPIE 7195, 71951M (2009).
[CrossRef]

Zhang, F.

Zhou, J.

Zhou, P.

P. Zhou, Z. J. Liu, X. L. Wang, Y. X. Ma, H. T. Ma, X. J. Xu, and S. F. Guo, “Coherent beam combining of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15, 248–256 (2009).
[CrossRef]

P. Zhou, Y. Ma, X. Wang, H. Ma, J. Wang, X. Xu, and Z. Liu, “Coherent beam combination of a hexagonal distributed high power fiber amplifier array,” Appl. Opt. 48, 6537–6540 (2009).
[CrossRef]

Acta Phys. Sin. (1)

J. Wang, K. Duan, and Y. Wang, “Experimental study of coherent beam combining of two fiber lasers,” Acta Phys. Sin. 57, 5627–5631 (2008).

Appl. Opt. (3)

IEEE J. Quantum Electron. (3)

R. Uberna, A. Bratcher, and B. G. Tiemann, “Coherent polarization beam combination,” IEEE J. Quantum Electron. 46, 1191–1196 (2010).
[CrossRef]

Y. Chung and N. Dagli, “An assessment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335–1339 (1990).
[CrossRef]

G. R. Hadley, “Transparent boundary condition for the beam propagation method,” IEEE J. Quantum Electron. 28, 363–370 (1992).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (4)

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical technique for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[CrossRef]

P. Zhou, Z. J. Liu, X. L. Wang, Y. X. Ma, H. T. Ma, X. J. Xu, and S. F. Guo, “Coherent beam combining of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15, 248–256 (2009).
[CrossRef]

J. R. Leger, J. Nilsson, J. P. Huignard, A. P. Napartovich, T. M. Shay, and A. Shirakawa, “Laser beam combining and fiber laser systems,” IEEE J. Sel. Top. Quantum Electron. 15, 237–239 (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]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

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

Opt. Express (3)

Opt. Lett. (2)

Proc. SPIE (3)

T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, “High-power phase locking of a fiber amplifier array,” Proc. SPIE 7195, 71951M (2009).
[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, 6708K (2007).

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]

Other (9)

S. E. Christensen and O. Koski, “2-Dimensional waveguide coherent beam combiner,” in Advanced Solid-State Photonics, OSA Technical Digest (CD) (Optical Society of America, 2007), paper WC1.

W. S. C. Chang, Fundamentals of Guided-Wave Optoelectronic Devices (Cambridge University, 2010).

K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis (Wiley, 2001).

C. Chen, Foundations for Guided-Wave Optics (Wiley, 2007).

Y. Chung and N. Dagli, “Modeling of guided-wave optical components with efficient finite-difference beam propagation methods,” in Antennas and Propagation Society International Symposium (IEEE, 1992), pp. 248–251.

C. Vassallo, Optical Waveguide Concepts (Elsevier, 1991).

A. E. Siegman, “How to (maybe) measure laser beam quality,” in Diode Pumped Solid State Lasers: Applications and Issues, M. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics (Optical Society of America, 1998), paper MQ1.

T. Poon and T. Kim, Engineering Optics with MATLAB (World Scientific, 2006).

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

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 (12)

Fig. 1.
Fig. 1.

Schematic diagram of the waveguide-based CBC system.

Fig. 2.
Fig. 2.

Schematic diagram of a tapered waveguide.

Fig. 3.
Fig. 3.

Amplitude distribution in the SCTW for Gaussian beam propagation. (a) Offset input Gaussian beam and (b) axially aligned input Gaussian beam.

Fig. 4.
Fig. 4.

Field distribution at the output of the SCTW. (a), (c), (e), and (g) are for the self-imaging of the laser beam; (b), (d), (f), and (h) are for the combination application.

Fig. 5.
Fig. 5.

Field distributions at the output of the SCTW for different taper angles. (a), (c) Amplitude distribution and (b), (d) Phase distribution.

Fig. 6.
Fig. 6.

3×3 fiber laser array.

Fig. 7.
Fig. 7.

Transverse intensity distribution of 3×3 fiber lasers.

Fig. 8.
Fig. 8.

Transverse intensity distribution of 3×3 fiber lasers.

Fig. 9.
Fig. 9.

Optimal designation of the system and the results. (a) M2 as a function of t, (b) near-field distribution for the optimum t, and (c) far-field intensity distribution for different t.

Fig. 10.
Fig. 10.

M2 as a function of t for different beam array.

Fig. 11.
Fig. 11.

Dependence on taper angle. N=2, W1=50μm.

Fig. 12.
Fig. 12.

Combining based on 2D SCWT. (a) Schematic diagram of a 2D tapered waveguide, (b) intensity distribution of input laser beams, and (c) intensity distribution of output laser beams.

Tables (4)

Tables Icon

Table 1. Parameters of the Tapered Waveguide

Tables Icon

Table 2. Parameters for Setting of 3×3 Fiber Laser Array

Tables Icon

Table 3. Parameters of the Square-Cross Tapered Waveguide

Tables Icon

Table 4. Results of Two Kinds of CBC Systems

Equations (37)

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

f(x,0)=i=0a2i(0)E2i(x,0),ai(0)=2W10W1f(x)Ei*(x,0)dx.
Ei(x,z)=sin[π(i+1)xW(z)],i=0,1,2,,
W(z)=(W12ztanθ)+2(λ2π)(ncn)2σ(n2nc2)1/2,
kxi(z)=(i+1)π/W(z),
βi2(z)=n2k02kxi2(z).
βi(z)nk0kxi2(z)/2nk0,
β0(z)nk0Δβ02(z)/8,Δβ02(z)=β0(z)β2(z)8π22nk0W2(z),
β2i(z)β0(z)i(i+1)Δβ02(z)2,
8π22nk00L0dzW2(z)=2π.
L0=W0λnW0+2tanθ,
fout(x,LN)=i=0a2i(LN)E2i(x,LN)exp(j0LNβ2i(z)dz).
fout(x,LN)=i=0a2i(LN)A2iE2i(x,LN),
A2i=exp[j[nk0π22nk0W0(W02tanθLN)]LN+jπLN(nW0λ)(W02tanθLN)i(i+1)].
A0=exp[j[nk0π22nk0W0(W02tanθLN)]LN],
A2i=A2(i1)exp{jπLN(nW0λ)(W02tanθLN)2i}.
B2i=1Cq=0N1exp(jΦ2i,q),
Φ2i,q=π(2i+1)xqW(LN)φq,
xq=(2q+1N)1NW(LN)2,
φq=q(Nq1)1Nπ,
C=exp{j[nk0π22nk0W0(W02tanθLN)]LN}q=0N1exp[jπxqW(LN)jφq].
Φ2i,q1=Φ2(i1),q+π1N(2i).
B2i=B2(i1)exp(jπ1N2i).
LN(nW0λ)(W02tanθLN)=1N,
LN=nW02Nλ+2ntanθW0,W(LN)=W02ntanθW02Nλ+2ntanθW0,
A2i=B2i=1Cq=0N1exp(jπ(2i+1)xqW(LN)jφq).
A2i=B2i=1Cq=0N1exp(jπ(2i+1)xqW(LN)jφq).
A2iE2i(x,LN)=12j{A2iexp[jπ(2i+1)xW(LN)]A2iexp[jπ(2i+1)xW(LN)]}.
A2iE2i(x,LN)=1Cq=0N112j{exp[jπ(2i+1)xxqW(LN)]exp[jπ(2i+1)xxqW(LN)]}exp(jφq).=1Cq=0N1E2i(xxq,LN)exp(jφq).
fout(x,LN)=1Ci=0a2i(LN)q=0N1E2i(xxq,LN)exp(jφq),=1Cq=0N1exp(jφq)i=0a2i(LN)E2i(xxq,LN).
fout(x,LN)=1Cq=0N1f(xxq,LN)exp(jφq).
ζ=|LN_simulationLN_analytical|LN_simulation×100%,
PIB=hhhhI(r⃗,z)dxdyI(r⃗,z)dxdy,
M2=w2(z)w02(λ/πw0)2(zz0)
w(z)=2σ(z),
w0=2σ(0),
σ2(z)=(xx0)2I(x,z)dxI(x,z)dx,
x0=xI(x,z)dxI(x,z)dx.

Metrics