Abstract

We present a novel phase locking scheme for the coherent combination of beam arrays in the filled aperture configuration. Employing a phase dithering mechanism for the different beams similar to LOCSET, dithering frequencies for sequential combination steps are reused. By applying an additional phase alternating scheme, this allows for the use of standard synchronized multichannel lock-in electronics for phase locking a large number of channels even when the frequency bandwidth of the employed phase actuators is limited.

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References

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  1. T. Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11(3), 567–577 (2005).
    [Crossref]
  2. T. Hansch and B. Couillaud, “Laser frequency stabilization by polarization spectroscopy of a reflecting reference cavity,” Opt. Commun. 35(3), 441–444 (1980).
    [Crossref]
  3. A. Klenke, S. Breitkopf, M. Kienel, T. Gottschall, T. Eidam, S. Hädrich, J. Rothhardt, J. Limpert, and A. Tünnermann, “530 W, 1.3 mJ, four-channel coherently combined femtosecond fiber chirped-pulse amplification system,” Opt. Lett. 38(13), 2283–2285 (2013).
    [Crossref] [PubMed]
  4. T. M. Shay, “Theory of electronically phased coherent beam combination without a reference beam,” Opt. Express 14(25), 12188–12195 (2006).
    [Crossref] [PubMed]
  5. M. A. Vorontsov, G. W. Carhart, and J. C. Ricklin, “Adaptive phase-distortion correction based on parallel gradient-descent optimization,” Opt. Lett. 22(12), 907–909 (1997).
    [Crossref] [PubMed]
  6. A. Klenke, M. Wojdyr, M. Müller, M. Kienel, T. Eidam, H. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Large-pitch multicore fiber for coherent combination of ultrashort pulses,” in 2015 European Conference on Lasers and Electro-Optics - European Quantum Electronics Conference (Optical Society of America, 2015), paper CJ_1_2.
  7. A. Klenke, E. Seise, J. Limpert, and A. Tünnermann, “Basic considerations on coherent combining of ultrashort laser pulses,” Opt. Express 19(25), 25379–25387 (2011).
    [Crossref] [PubMed]
  8. Y. Ma, X. Wang, J. Leng, H. Xiao, X. Dong, J. Zhu, W. Du, P. Zhou, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination of 1.08 kW fiber amplifier array using single frequency dithering technique,” Opt. Lett. 36(6), 951–953 (2011).
    [Crossref] [PubMed]
  9. M. Jiang, R. Su, Z. Zhang, Y. Ma, X. Wang, and P. Zhou, “Coherent beam combining of fiber lasers using a CDMA-based single-frequency dithering technique,” Appl. Opt. 56(15), 4255–4260 (2017).
    [Crossref] [PubMed]
  10. H. K. Ahn and H. J. Kong, “Cascaded multi-dithering theory for coherent beam combining of multiplexed beam elements,” Opt. Express 23(9), 12407–12413 (2015).
    [Crossref] [PubMed]
  11. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1964).

2017 (1)

2015 (1)

2013 (1)

2011 (2)

2006 (1)

2005 (1)

T. Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11(3), 567–577 (2005).
[Crossref]

1997 (1)

1980 (1)

T. Hansch and B. Couillaud, “Laser frequency stabilization by polarization spectroscopy of a reflecting reference cavity,” Opt. Commun. 35(3), 441–444 (1980).
[Crossref]

Ahn, H. K.

Breitkopf, S.

Carhart, G. W.

Couillaud, B.

T. Hansch and B. Couillaud, “Laser frequency stabilization by polarization spectroscopy of a reflecting reference cavity,” Opt. Commun. 35(3), 441–444 (1980).
[Crossref]

Dong, X.

Du, W.

Eidam, T.

Fan, T. Y.

T. Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11(3), 567–577 (2005).
[Crossref]

Gottschall, T.

Hädrich, S.

Hansch, T.

T. Hansch and B. Couillaud, “Laser frequency stabilization by polarization spectroscopy of a reflecting reference cavity,” Opt. Commun. 35(3), 441–444 (1980).
[Crossref]

Jiang, M.

Kienel, M.

Klenke, A.

Kong, H. J.

Leng, J.

Limpert, J.

Liu, Z.

Ma, Y.

Ricklin, J. C.

Rothhardt, J.

Seise, E.

Shay, T. M.

Si, L.

Su, R.

Tünnermann, A.

Vorontsov, M. A.

Wang, X.

Xiao, H.

Xu, X.

Zhang, Z.

Zhao, Y.

Zhou, P.

Zhu, J.

Appl. Opt. (1)

IEEE J. Sel. Top. Quantum Electron. (1)

T. Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11(3), 567–577 (2005).
[Crossref]

Opt. Commun. (1)

T. Hansch and B. Couillaud, “Laser frequency stabilization by polarization spectroscopy of a reflecting reference cavity,” Opt. Commun. 35(3), 441–444 (1980).
[Crossref]

Opt. Express (3)

Opt. Lett. (3)

Other (2)

A. Klenke, M. Wojdyr, M. Müller, M. Kienel, T. Eidam, H. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Large-pitch multicore fiber for coherent combination of ultrashort pulses,” in 2015 European Conference on Lasers and Electro-Optics - European Quantum Electronics Conference (Optical Society of America, 2015), paper CJ_1_2.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1964).

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Figures (3)

Fig. 1
Fig. 1 a) Beam combination scheme for a one-dimensional beam array for four beams (E1-E4). The dashed lines are the non-combining parts of the beams that form the rejection port. Possible photodiode detector positions for the combined powers are in this rejection port (ED2-ED4) and behind the HR-mirror (ED2*-ED3*, ED4* could be retrieved by using a fraction of the combined beam), b) Combination scheme for a two-dimensional beam array
Fig. 2
Fig. 2 Example phase offsets (red) for a four channel system and the required movement of the phase actuators (blue) for sequential phase locking: (a) the fourth channel has a positive phase offset and the actuator is supposed to decrease it to optimize for best combination efficiency and (b) the third channel has a positive phase offset. The actuators are supposed to decrease the phase offset of the third channel, but increase it for the fourth channel to adapt to the combined beam of the first three channels.
Fig. 3
Fig. 3 a) Example configuration of applied phase dither for a two-dimensional 4x4 actuator array with two different frequencies f1 and f2 and a phase offset alternating scheme indicated by the darker and lighter colors. b) Time dependence of the normalized combined power for the run with the highest convergence time of 1000 numerical simulations starting with random initial phases for each beam). c) Simulation of the LOCSET scheme for the same beam array combination with different phase dither frequencies for each actuator.

Tables (1)

Tables Icon

Table 1 Sample channel configuration for the phase alternating scheme

Equations (7)

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E comb N = r=1 N 1 N E r , E 1 = E 0 , E r = E 0 exp(i( ϕ mod (t)+ ϕ r )) 1 2 E DN* = T HR E comb N = T HR ( 1 N E N + 1 N r=1 n E r ) I DN* | E DN* | 2 = T HR ( 1 N | E N | 2 1 N r=1,s=1 n ( E r * E s +cc) + 1 N r=1 n ( E N * E r +cc) ) = T HR ( const+ 1 N r=2 n 2 | E 0 | 2 cos( ϕ mod (t)+ ϕ r ) + 1 N 2 | E 0 | 2 cos( ϕ mod (t)+ ϕ N ) )
S N τ>> 1 ωmod 1 τ 0 τ I D sin( ω mod t)dt= 2 N | E 0 | 2 J 1 (β)( r=2 n sin( ϕ r ) +sin( ϕ N ) )
E comb N = r=1 N 1 N E r , E 1 = E 0 , E r = E 0 exp(i( ϕ mod (t)+ ϕ r )) E DN = N1 N E N 1 N r=1 n 1 N1 E r I DN | E DN | 2 =( N1 N | E N | 2 + 1 N(N1) r=1,s=1 n ( E r * E s +cc ) 1 N r=1 n ( E N * E r +cc ) ) =const+( 1 N(N1) r=2 n 2 | E N | 2 cos( ϕ mod (t)+ ϕ r ) 1 N 2 | E N | 2 cos( ϕ mod (t)+ ϕ N ) )
S N τ>> 1 ωmod 1 τ 0 τ I DN sin( ω mod t)dt= 2 N | E 0 | 2 J 1 (β)( 1 N1 r=2 n sin( ϕ r ) sin( ϕ N ) )
E 1 = E 0 , E +r = E 0 exp(i( ϕ mod (t)+ ϕ r )), E r = E 0 exp(i( ϕ mod (t)+ ϕ r )) E +N = E 0 exp(i( ϕ mod (t)+ ϕ +N )) E n = 1 n ( E 1 + r E +r + r E r ) | E n | 2 = 1 n ( | E 0 | 2 + | r E +r | 2 + | i E r | 2 +( ( r E +r * )( r E r ) ) +( E 0 * r E +r +cc )+( E 0 * r E r +cc ) ) =const+ 1 n ( ( E 0 * r E +r +cc )+( E 0 * r E r +cc ) +2 r s | E +r | | E s |cos( 2 ϕ mod ( t )+ ϕ +r ϕ s ) ) =const+ 1 n | E 0 | 2 ( 2( r cos( ϕ mod ( t )+ ϕ +r ) )+2( r cos( ϕ mod ( t )+ ϕ r ) ) +2 r s cos( 2 ϕ mod ( t )+ ϕ +r ϕ s ) ) I DN | E DN | 2 = | 1 N1 E +N 1 N E 2 | 2 =const+ 1 N | E n | 2 N1 N 1 n | E 0 | 2 ( 2cos( ϕ mod ( t )+ ϕ +N ) +2 r cos( 2 ϕ mod ( t )+ ϕ +N ϕ r ) )
I DN const+ 1 N 1 N1 | E 0 | 2 ( 2( r 2 ϕ mod ( t )+ ϕ +r )+2( r ϕ mod ( t )+ ϕ r ) 4 r s ( ϕ +r ϕ s ) ϕ mod ( t )+ ) 1 N | E 0 | 2 ( 2 ϕ mod ( t ) ϕ +N 4 r ( ϕ +N ϕ r ) ϕ mod ( t ) ) const 1 N 1 N1 | E 0 | 2 ( ( 2( N1 ) ) r ϕ mod ( t ) ϕ +r )( 2( N1 ) )( ( r ϕ mod ( t )+ ϕ r ) ) + 1 N | E 0 | 2 ( 2 ϕ mod ( t ) ϕ +N +4( N 2 1 ) ϕ +N ϕ mod ( t )4 r ϕ mod ( t ) ϕ r ) const 1 N | E 0 | 2 ( 2( r ϕ +r )+2( r ϕ r )2( N1 ) ϕ +N ) ϕ mod ( t ) S +N τ>> 1 ω mod 1 N | E 0 | 2 β 2 ( ( r ϕ +r )+( r ϕ r )( N1 ) ϕ +N )
| E D | 2 = | N1 N E N 1 N E 2 | 2 =const+ 1 N | E n | 2 N1 N 1 n | E 0 | 2 ( 2cos( ϕ mod ( t )+ ϕ +N ) +2( r cos( 2 ϕ mod ( t )+ ϕ N ϕ +r ) ) ) | E D | 2 const+ 1 N 1 N1 | E 0 | 2 ( 2( r ϕ mod ( t ) ϕ +r )+2( r ϕ mod ( t ) ϕ r ) 4 r s ( ϕ +r ϕ s ) ϕ mod ( t ) ) 1 N | E 0 | 2 ( 2 ϕ mod ( t ) ϕ N +4 r ( ϕ N ϕ +r ) ϕ mod ( t ) ) const 1 N | E 0 | 2 ( 2N N1 ( r ϕ +r ) 2N N1 ( r ϕ r )+2N ϕ N ) ϕ mod ( t ) S N τ>> 1 ω mod 1 N | E 0 | 2 β 2 ( 2N N1 ( r ϕ +r )+ N N1 ( r ϕ r )N ϕ N )

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