ABSTRACTT FFT FFT-' Proc. of SPIE Vol U-1

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1 Phase econstuction stategies in phase-locking system based on multi-apetue wave font sensos P.A. Semenov, S.D. Pol skikh Shvabe-Reseach, Moscow, Russian Fedeation; ABSTRACTT System of coheent beam combining with multi-apetue wave font senso based on the Gechbeg-Saxton algoithm is consideed. Diffeent system achitectues and eduction methods based on block algoithms ae poposed. The system is simulated numeically fo a diffeent numbe of phased channels. Keywods: multichannel lase systems, phase-locking, Gechbeg-Saxton algoithm, global optimization.. INTRODUCTION Nowadays phase synchonization methods of seveal lases ae widely applied in the systems with high aveage powe and naowness. Thee have been moe than 20 technical solutions fo ealizing coheent beam combining []. Among diffeent appoaches of coheent combining the most significant ones ae methods based on active contol of each lase in the system. The contol is usually conducted by distibuted adaptive optical systems (DAOS). In most pactical casess DAOS is an adaptive system of apetue sensing which is based on optimization (maximization) of image shapnesss function [2]. The altenative system is the one with efeence beam and wave font senso (WFS) based on hologaphic and heteodyne pinciples [3]. Howeve, such systems ae moe difficult to constuct. It is also impossible to use taditional adaptive optics wave font sensos in this multi-apetuee system, because its wok is based on the pinciple of building a continuous chat of phase abeations. This pape consides the system of coheent combining with active feedback of multi-apetue wave font sensoss (MWFS) without suppoting beam goup based on Gechbeg-Saxton algoithm [4]. This algoithm allows to econstuct the complex fields on lens apetue and on its focal plane by measuing its intensity distibutions. Stuctual scheme of the Gechbeg-Saxton algoithm iteation pocedue is shown on Fig.. ;) exp(ip FFT i exp(t<pé f ) exp(icp' <= FFT-' M() exp(i(pé Figue. Stuctual scheme of the Gechbeg-Saxton algoithm. 2st Intenational Symposium on Atmospheic and Ocean Optics: Atmospheic Physics, edited by G. G. Matvienko, O. A. Romanovskii, Poc. of SPIE Vol. 9680, 96800U 205 SPIE CCC code: X/5/$8 doi: 0.7/ Poc. of SPIE Vol U- Downloaded Fom: on /27/205 Tems of Use:

2 Fo the phase chosen as the initial appoximation, and the given measued field intensity distibution in the input plane A, the complex amplitude in the outpu plane B is calculated. Then, modulus of the field amplitude calculated in the output plane is eplaced by the measued intensity distibution in this plane. Then, the backwad beam popagation fom the output to the input plane is calculated and the amplitude in the input plane is eplaced by the measued intensity distibution in the input plane and the calculated phase is taken as the next appoximation. Mathematically, the Gechbeg Saxton algoithm is witten in the fom of the following iteation pocedue: 0) whee ϕ ( ( ξ) is the initial estimate of the phase at the apetue; P and P2 ae pojection opeatos of the functions E E 0 ( ξ ) = ( 0 M ( ξ ) exp[ iϕ ) (ξ )]; E ( ξ E k ) = P FT{ P FT [ E ( )]}, k=,,2, 2 Stuctue scheme of active phase-locking system based on the Gechbeg-Saxton algoithm is shown on Fig.2. E() and onto sets of functions with given modulus; FT and FT ae the diect and invese Fouie tansfoms, espectively. k ξ () _ L 4 I v I I- L2 -. i MWFS Figue 2. Block scheme of the phase-locking system ( maste oscillato; 2 phase modulatos; 3 fibe amplifies; 4 focal plane; 5,6 CCD-cameas; 7 computing device; L,L2 lenses). Optical signal fom maste oscillato splits to the phase modulatos and afte passing though fibe amplifies the main beam focusess on object. The second beam foms contol signals fo phase modulatos and divides into two sub beams fo intensity egistation in focal and apetue planes. Measued intensities ae sent to computing device whee iteated algoithm defines phase distibution. The contol system based on this distibution foms contol signals fo phase modulatos. Poc. of SPIE Vol U-2 Downloaded Fom: on /27/205 Tems of Use:

3 2. PHASE RECONSTRUCTION STRATEGY The typical peculiaity of using this algoithm in phase estoation is the pesence of stagnation states, o so-called extemums [5]. In these conditions, convegence of the algoithm can be guaanteed by using global optimization methods. The choice of the optimum method fo a specific poblem is affected by diffeent factos caused by the specificity of the poblem. Fo the Gechbeg Saxton algoithm, this specificity is the apidity and ease of local optimization when thee is no necessity fo bulky calculations of patial deivatives. Indeed, the simplicity of pefoming the pojection opeations P and P2 (the obtained modulus is eplaced by a known one) ensues a high speed of pefoming iteations almost egadless of the dimension of the poblem. In connection with this, when constucting the global optimization pocedue, it is suitable to focus on the appoach that is based on multiple finding local exteme values fom diffeent initial points positioned andomly on the entie optimization set and choosing the best one of them. The algoithm of this type is called the andom multistat. In addition to apidness of pefoming iteations, its doubtless advantage, is the possibility of the simultaneous finding local exteme values fom diffeent initial points, which makes it possible to implement it in paallel computes consisting of single type pocessos pefoming simila opeations, the main one of which is the fast Fouie tansfom. In this case, the time fo finding the global exteme value does not exceed the time fo pefoming an individual local optimization pocedue. As shown in [6], fo the poblem of phase synchonization in a multichannel lase system based on this algoithm, the numbe of paallel pocessos inceases linealy with the numbe of channels. Howeve with futhe inceasing of the numbe of channels, the paallel pocessos numbe does not affect on the esult. In this case it is necessay to use some dimension eduction methods, eithe by mathematical level, o in hadwae. 3. BLOCK PHASE SYNCHRONISATION SYSTEM One of the possible mechanisms fo the eduction can be based on the block pinciple. In this case, the common channel system is divided into blocks of seveal channels with paallel phase estoation in each block. Mathematical modeling shows that fo a hexagonal system constuction the base numbe of channels in the block should not exceed 7. One of the Gechbeg-Saxton algoithm peculiaities is that the phase infomation estoation in each channel occus accuate within constant phase shift in the block. The additional phase coss-linking between blocks is needed. In this case two vaiants ae possible: i. System division into blocks with common channels fo seveal blocks and coss-linking between common channels. ii. Division into blocks without common channels and conducting exta iteation pocedue of entie system. System using coss-linking seems to be moe attactive, because it does not equie additional iteation pocedue of the whole system. Howeve, this system is moe complicated, as the issue of optimal adiation sepaation in common channels is not esolved yet. We will conside the convegence of the Gechbeg-Saxton algoithm fo multichannel lase system with independent blocks. Conside the total numbe of sub-apetues in hexagonal systems [7], and ewite it in the following way: n n( n + ) N = + 6 l = + 3n ( n + ) = 7 + 6, (2) l= 2 whee, n is intege numbe. Accoding to (2), in the case of hexagonal system constuction, it is convenient to divide n(n + ) system on independent blocks with 6 channels and one block with 7 channels. 2 Fist, we will conside the 37 channel lase system. Figue 3 shows the division example in this case. Poc. of SPIE Vol U-3 Downloaded Fom: on /27/205 Tems of Use:

4 Figue 3. Block division example of 37 channel lase system. Phase etieval esults of 37 channel lase system ae shown on figue 4. The etieval quality was estimated by the standadized modulus eo δ = 2 /2 k { [ E k ( ξ ) M( ξ)] d ξ}, δ (3) whee δ is the modulus eo at the fist iteation. With this division, the time equied to synchonize a single block emains at the same level as fo the 7 channels (about 5 iteations). Futhe numbe of channels incease does not esult at synchonization time only the numbe of paallel pocessos inceases (-2 in each block). The total numbe of iteations foms with the maximum numbe of iteations fo 7 channel system and the n-channel system, whee n is the numbe of blocks. o 20 Figue 4. Gechbeg-Saxton algoithm convegence ( 2 stage block system; 2 system without blocks). \ Poc. of SPIE Vol U-4 Downloaded Fom: on /27/205 Tems of Use:

5 Accoding to this, figue 5 shows numbe of convegence iteations against the numbe of channels. Nit 200 z o L N Figue 5. Numbe of convegence iteations Nit against the numbe of channels N ( block system; 2 system without blocks). 4. CONCLUSIONS The pape descibes an appoach to phase synchonization in multichannel lase systems based on iteation image econstuctionn Gechbeg-Saxton algoithm. The pesence of so-called stagnationn states is a distinctive featue of this algoithm. Methods of global optimization allows to eliminate this estiction and make effective ecovey stategy, povides guaanteed convegence of the algoithm. And block-eduction pinciple makes possible to combine a lage numbe of lase channels. REFERENCES [] Liu, Z.J., Zhou, P., Xu, X.J., Wang, X.L., Ma, Y.X., Coheent beam combining of high powe fibe lases: Pogesss and pospect, Sci. China Tech. Sci. 56(7), (203). [2] Voontsov, M.A., Weyauch, T., Beesnev, L. A., Cahat, G.W., Liu, L., Aschenbach. K., Adaptive Aay of Phase-Locked Fibe Collimatos: Analysis and Expeimental Demonstatio n, IEEE 5(2), (2009). [3] Fan, T.Y., Lase beam combination fo high-powe, high-adiance souces, IEEE (3) ), (2005). [4] Gechbeg, R.W., Saxton, W.O., A pactical algoithm fo the detemination of phase fom image and diffaction plane pictues, Optik (Stuttgat) 35(2), (972). [5] Pol skikh, S.D., The phase poblem: Analysis of local extema and image econstuction algoithms, J. Comm. Tech. Electonics 53( (2), (2008). [6] Pol skikh, S.D., Semenov, P.A., Adaptive Optical System Based on the Gechbeg Saxton algoithm fo Phase Locking of Single Mode Lase Radiatos, Atmospheic and Oceanic Optics 27(4), (204). [7] Voontsov, M.A., Lachinova, S.L.., Lase beam pojection with adaptive aay of fibe collimatos,, J. Opt. Soc. Am. A. 25(8), (2008). Poc. of SPIE Vol U-5 Downloaded Fom: on /27/205 Tems of Use: