Abstract

Self-imaging properties of fiber lasers in a strongly confined waveguide (SCW) and their application in coherent beam combination (CBC) are studied theoretically. Analytical formulas are derived for the positions, amplitudes, and phases of the N images at the end of an SCW, which is important for quantitative analysis of waveguide CBC. The formulas are verified with experimental results and numerical simulation of a finite difference beam propagation method (BPM). The error of our analytical formulas is less than 6%, which can be reduced to less than 1.5% with Goos–Hahnchen penetration depth considered. Based on the theoretical model and BPM, we studied the combination of two laser beams based on an SCW. The effects of the waveguide refractive index and Gaussian beam waist are studied. We also simulated the CBC of nine and 16 fiber lasers, and a single beam without side lobes was achieved.

© 2012 Optical Society of America

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References

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2010

2009

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]

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. 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]

2008

2007

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

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

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

2003

2002

2000

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

1994

1992

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

1991

1990

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

1978

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, 67080K (2007).
[CrossRef]

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 Series (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 Proceedings of IEEE Antennas and Propagation Society International Symposium, 1992, Vol. 1 (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 Proceedings of IEEE Antennas and Propagation Society International Symposium, 1992, Vol. 1 (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).

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 Series (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, 67080K (2007).
[CrossRef]

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]

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguide(Academic, 1974).

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, 67080K (2007).
[CrossRef]

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, 13547–13533 (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, 13547–13533 (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]

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, Vol. 17 of Trends in Optics and Photonics Series (OSA, 1998), pp. 184–199.

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, 67080K (2007).
[CrossRef]

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, 67080K (2007).
[CrossRef]

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.

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Figures (9)

Fig. 1.
Fig. 1.

Schematic diagram of the waveguide-based CBC system.

Fig. 2.
Fig. 2.

Schematic diagram of a planar waveguide.

Fig. 3.
Fig. 3.

Amplitude distribution in the SCW for propagation through an imaging length.

Fig. 4.
Fig. 4.

Amplitude distribution in the SCW.

Fig. 5.
Fig. 5.

M2 versus waveguide length.

Fig. 6.
Fig. 6.

Array of 3×3 fiber lasers.

Fig. 7.
Fig. 7.

Amplitude distribution of 3×3 fiber lasers.

Fig. 8.
Fig. 8.

Amplitude distribution of 4×4 fiber lasers.

Fig. 9.
Fig. 9.

M2 versus waveguide length.

Tables (3)

Tables Icon

Table 1. Relative Phases Predicted by Our Analytical Formulas and the Corresponding Values Calculated by the BPM

Tables Icon

Table 2. Transmission Power for Different Types of Waveguides

Tables Icon

Table 3. Parameter Settings of the 3×3 Fiber Laser Array

Equations (26)

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

L=4nW2λ.
W=W0+2(λ2π)(ncn)2σ(n2nc2)1/2,
Ei(x)=sin[π(i+1)xW],i=0,1,2,,
kxi=(i+1)π/W.
βi2=n2k02kxi2.
βink0kxi2/2nk0
β0nk0Δβ02/8,Δβ02=β0β28π22nk0W2,
β2iβ0i(i+1)Δβ022,
f(x)=i=0a2iE2i(x),ai=2W0Wf(x)Ei*(x)dx.
LNM=MN2Lc=MN2πΔβ02,
fout(x)=i=0a2iE2i(x)exp(jβ2iLNM).
fout(x)=i=0a2iA2iE2i(x),
A2i=exp[jβ0LNM+jπMNi(i+1)].
A0=exp(jβ0LNM),A2i=A2(i1)exp(jπMN2i).
B2i=1Cq=0N1exp(jΦ2i,q),
Φ2i,q=π(2i+1)xqWφq,
xq=(2q+1N)MNW2,
φq=q(Nq1)MNπ,
C=exp(jβ0LNM)q=0N1exp[jπxqWjφq].
Φ2i,q1=Φ2(i1),q+πMN(2i).
B2i=B2(i1)exp[jπMN2i].
A2i=B2i=1Cq=0N1exp(jπ(2i+1)xqWjφq).
A2i=B2i=1Cq=0N1exp(jπ(2i+1)xqWjϕq).
A2iE2i(x)=12j{A2iexp[jπ(2i+1)xW]exp[jπ(2i+1)xW]}.
A2iE2i(x)=1Cq=0N112j{exp[jπ(2i+1)xxqW]exp[jπ(2i+1)xxqW]}exp(jφq)=1Cq=0N1E2i(xxq)exp(jφq).
fout(x)=1Cq=0N1f(xxq)exp(jφq).

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