Abstract

Theoretical analysis for the far-field diffraction of the coherent laser array (CLA) is presented. Based on the theoretical model of the propagation of CLA, the conditions and restrictions for coherent beam combination, two-dimensional steering, and dark-hollow beam generation are theoretically described in detail. With properly organized phase distributions and tilt control, the peak location of the far-field pattern of the CLA could shift two-dimensionally in a large scale of steering angle. With additional amplitude modulation, the far-field pattern could have special shapes. The simulated results agree with the theoretical analysis. It is a feasible way to realize all these applications by a CLA with well-arranged phase distributions and/or additional amplitude modulation.

© 2012 Optical Society of America

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References

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2010

Y. Zheng, X. Wang, F. Shen, and X. Li, “Generation of dark hollow beam via coherent combination based on adaptive optics,” Opt. Express 18, 26946–26958 (2010).
[CrossRef]

P. Zhou, X. Wang, Y. Ma, H. Ma, X. Xu, and Z. Liu, “Generation of a hollow beam by active phasing of a laser array using a stochastic parallel gradient descent algorithm,” J. Opt. 12, 015401 (2010).
[CrossRef]

J. Li, H. Zhang, and B. Lü, “Composite coherence vortices in a radial beam array propagating through atmospheric turbulence along a slant path,” J. Opt. 12, 065401 (2010).
[CrossRef]

2009

2008

2007

2006

2005

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, 6102U (2005).

F. Xiao, W. Hu, and A. Xu, “Optical phased-array beam steering controlled by wavelength,” Appl. Opt. 44, 5429–5433 (2005).
[CrossRef]

2004

2003

1996

1994

1993

1992

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, 6102U (2005).

Bergander, A.

Bowman, R.

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, 6102U (2005).

Browaeys, A.

C. Tuchendler, A. M. Lance, A. Browaeys, Y. R. P. Sortais, and P. Grangier, “Energy distribution and cooling of a single atom in an optical tweezer,” Phys. Rev. A 78, 033425 (2008).
[CrossRef]

Bu, J.

Burge, R. E.

Cai, Y.

Carberry, D.

Chai, L.

M. Hu, C. Wang, Y. Song, Y. Li, and L. Chai, “A hollow beam from a holey fiber,” Opt. Express 14, 4218–4224 (2006).

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, 6102U (2005).

Cheung, E. C.

Cordingley, J.

Dickey, F. M.

Epp, P.

Gahagan, K. T.

Gao, B. Z.

Ge, D.

Gibson, G.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2004).

Goodno, G. D.

Grangier, P.

C. Tuchendler, A. M. Lance, A. Browaeys, Y. R. P. Sortais, and P. Grangier, “Energy distribution and cooling of a single atom in an optical tweezer,” Phys. Rev. A 78, 033425 (2008).
[CrossRef]

Haliyo, S.

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, 6102U (2005).

Heckenberg, N. R.

Howland, D.

Hu, M.

M. Hu, C. Wang, Y. Song, Y. Li, and L. Chai, “A hollow beam from a holey fiber,” Opt. Express 14, 4218–4224 (2006).

Hu, W.

Injeyan, H.

Komine, H.

Kotlyar, V. V.

Kovalev, A. A.

Kudielka, K. H.

Lance, A. M.

C. Tuchendler, A. M. Lance, A. Browaeys, Y. R. P. Sortais, and P. Grangier, “Energy distribution and cooling of a single atom in an optical tweezer,” Phys. Rev. A 78, 033425 (2008).
[CrossRef]

Leeb, Walter R.

Li, J.

J. Li, H. Zhang, and B. Lü, “Composite coherence vortices in a radial beam array propagating through atmospheric turbulence along a slant path,” J. Opt. 12, 065401 (2010).
[CrossRef]

Li, X.

Li, Y.

M. Hu, C. Wang, Y. Song, Y. Li, and L. Chai, “A hollow beam from a holey fiber,” Opt. Express 14, 4218–4224 (2006).

Lin, Q.

Liu, Z.

P. Zhou, X. Wang, Y. Ma, H. Ma, X. Xu, and Z. Liu, “Generation of a hollow beam by active phasing of a laser array using a stochastic parallel gradient descent algorithm,” J. Opt. 12, 015401 (2010).
[CrossRef]

Long, W.

Lu, X.

Lü, B.

J. Li, H. Zhang, and B. Lü, “Composite coherence vortices in a radial beam array propagating through atmospheric turbulence along a slant path,” J. Opt. 12, 065401 (2010).
[CrossRef]

Ma, H.

P. Zhou, X. Wang, Y. Ma, H. Ma, X. Xu, and Z. Liu, “Generation of a hollow beam by active phasing of a laser array using a stochastic parallel gradient descent algorithm,” J. Opt. 12, 015401 (2010).
[CrossRef]

Ma, Y.

P. Zhou, X. Wang, Y. Ma, H. Ma, X. Xu, and Z. Liu, “Generation of a hollow beam by active phasing of a laser array using a stochastic parallel gradient descent algorithm,” J. Opt. 12, 015401 (2010).
[CrossRef]

McClellan, M.

McDuff, R.

McNaught, S. J.

Moh, K. J.

Neubert, W. M.

Pacoret, C.

Padgett, M.

Redmond, S.

Régnier, S.

Romero, L. A.

Scholtz, A. L.

Senthilkumaran, P.

Serati, S.

S. Serati and J. Stockley, “Phased array of phased arrays for free space optical communications,” in Proceedings of IEEE Conference on Aerospace (IEEE, 2003), pp. 1085–1093.

Shen, F.

Simpson, R.

Smith, C. P.

Sollee, J.

Song, Y.

M. Hu, C. Wang, Y. Song, Y. Li, and L. Chai, “A hollow beam from a holey fiber,” Opt. Express 14, 4218–4224 (2006).

Sortais, Y. R. P.

C. Tuchendler, A. M. Lance, A. Browaeys, Y. R. P. Sortais, and P. Grangier, “Energy distribution and cooling of a single atom in an optical tweezer,” Phys. Rev. A 78, 033425 (2008).
[CrossRef]

Stockley, J.

S. Serati and J. Stockley, “Phased array of phased arrays for free space optical communications,” in Proceedings of IEEE Conference on Aerospace (IEEE, 2003), pp. 1085–1093.

Svelto, O.

O. Svelto, Principles of Lasers, 4th ed. (Plenum, 1998).

Swartzlander, G. A.

Tuchendler, C.

C. Tuchendler, A. M. Lance, A. Browaeys, Y. R. P. Sortais, and P. Grangier, “Energy distribution and cooling of a single atom in an optical tweezer,” Phys. Rev. A 78, 033425 (2008).
[CrossRef]

Wang, C.

M. Hu, C. Wang, Y. Song, Y. Li, and L. Chai, “A hollow beam from a holey fiber,” Opt. Express 14, 4218–4224 (2006).

Wang, F.

Wang, L.

L. Wang, L. Wang, and S. Zhu, “Formation of optical vortices using coherent laser beam arrays,” Opt. Commun. 282, 1088–1094 (2009).
[CrossRef]

L. Wang, L. Wang, and S. Zhu, “Formation of optical vortices using coherent laser beam arrays,” Opt. Commun. 282, 1088–1094 (2009).
[CrossRef]

Wang, X.

P. Zhou, X. Wang, Y. Ma, H. Ma, X. Xu, and Z. Liu, “Generation of a hollow beam by active phasing of a laser array using a stochastic parallel gradient descent algorithm,” J. Opt. 12, 015401 (2010).
[CrossRef]

Y. Zheng, X. Wang, F. Shen, and X. Li, “Generation of dark hollow beam via coherent combination based on adaptive optics,” Opt. Express 18, 26946–26958 (2010).
[CrossRef]

Weber, M.

Weiss, S. B.

White, A. G.

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, 6102U (2005).

Xiao, F.

Xu, A.

Xu, X.

P. Zhou, X. Wang, Y. Ma, H. Ma, X. Xu, and Z. Liu, “Generation of a hollow beam by active phasing of a laser array using a stochastic parallel gradient descent algorithm,” J. Opt. 12, 015401 (2010).
[CrossRef]

Yuan, X.

Zhang, H.

J. Li, H. Zhang, and B. Lü, “Composite coherence vortices in a radial beam array propagating through atmospheric turbulence along a slant path,” J. Opt. 12, 065401 (2010).
[CrossRef]

Zheng, Y.

Zhou, P.

P. Zhou, X. Wang, Y. Ma, H. Ma, X. Xu, and Z. Liu, “Generation of a hollow beam by active phasing of a laser array using a stochastic parallel gradient descent algorithm,” J. Opt. 12, 015401 (2010).
[CrossRef]

Zhu, S.

L. Wang, L. Wang, and S. Zhu, “Formation of optical vortices using coherent laser beam arrays,” Opt. Commun. 282, 1088–1094 (2009).
[CrossRef]

Appl. Opt.

J. Opt.

P. Zhou, X. Wang, Y. Ma, H. Ma, X. Xu, and Z. Liu, “Generation of a hollow beam by active phasing of a laser array using a stochastic parallel gradient descent algorithm,” J. Opt. 12, 015401 (2010).
[CrossRef]

J. Li, H. Zhang, and B. Lü, “Composite coherence vortices in a radial beam array propagating through atmospheric turbulence along a slant path,” J. Opt. 12, 065401 (2010).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

L. Wang, L. Wang, and S. Zhu, “Formation of optical vortices using coherent laser beam arrays,” Opt. Commun. 282, 1088–1094 (2009).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

C. Tuchendler, A. M. Lance, A. Browaeys, Y. R. P. Sortais, and P. Grangier, “Energy distribution and cooling of a single atom in an optical tweezer,” Phys. Rev. A 78, 033425 (2008).
[CrossRef]

Proc. SPIE

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, 6102U (2005).

Other

S. Serati and J. Stockley, “Phased array of phased arrays for free space optical communications,” in Proceedings of IEEE Conference on Aerospace (IEEE, 2003), pp. 1085–1093.

O. Svelto, Principles of Lasers, 4th ed. (Plenum, 1998).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2004).

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Figures (12)

Fig. 1.
Fig. 1.

Arrangement of the coherent laser array with order m=2.

Fig. 2.
Fig. 2.

Distribution of the space modulation factor SF(x,y).

Fig. 3.
Fig. 3.

Normalized intensity distributions of (a) A(x,y), the diffraction pattern of a circle aperture with a radius w and (b) the far-field interference pattern of the CLA in Fig. 1 with N=7. The white lines present the one-dimensional intensity distributions at x=0 and y=0.

Fig. 4.
Fig. 4.

Peak locations of SF(xx0,yy0) marked with “+”.

Fig. 5.
Fig. 5.

(a) Efficient steering area (shadowed) and the peak locations of SF(x,y) marked with “+”; (b) Scope of efficient steering angle (shadowed) of the CLA in Fig. 1 with N=7.

Fig. 6.
Fig. 6.

One case of the far-field main lobe shifted to the upper left corner of the ESA, with α=β=0. (a) Initial phase distribution of the CLA and (b) the corresponding space modulator F(x,y). (c) Normalized far-field intensity distribution I(x,y) with the main lobe marked with a square, the pattern is got by the product of the space modulator in Fig. 6(b) and the aperture factor A(x,y) in Fig. 3(a). The crossed dashes are the referenced centric axes. The parameters are λ=632.8nm, f=120mm, d=16mm, w=7mm.

Fig. 7.
Fig. 7.

Long exposure of the far-field pattern when the main lobe shifts to the six corners of the ESA in Fig. 5(a). The crossed dashes are the referenced centric axes.

Fig. 8.
Fig. 8.

Efficient steering area (shadowed) of the CLA in Fig. 1 with N=7 when the CLA with tilts of α=pλ/d, β=qλ/3d, where, p=1, q=3.

Fig. 9.
Fig. 9.

Another case of the far-field main lobe shifted to the upper left corner of the ESA when the CLA has tilts of α=λ/d, β=3λ/3d. (a) Initial phase distribution of the CLA with 2π ignored. (b) The corresponding space factor F(x,y). (c) Normalized aperture factor A(x+αf,y+βf). (d) Normalized far-field intensity distribution I(x,y), which is the product of (b) and (c). Its main lobe is marked with a square. The crossed dashes are the referenced centric axes. The other calculation parameters are the same as in Fig. 6.

Fig. 10.
Fig. 10.

Long exposure of the pattern in far field when the main lobe shifts to the six corners of the ESA of the case α=λ/d, β=3λ/3d. The crossed dashes are the referenced centric axes.

Fig. 11.
Fig. 11.

Curves of (a) t1 and (b) t2 vary with the location y. The calculation parameters are φ=π, f=120mm, d=16mm, w=7mm, λ=632.8nm.

Fig. 12.
Fig. 12.

Calculated far-field intensity distributions and profiles of the DHBs formed in different values of t=0.300 (Curve 1), 0.218 (Curve 2), 0.168 (Curve 3), 0.135 (Curve 4), and 0.100 (Curve 5). The other calculation parameters are the same as in Fig. 11.

Equations (46)

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U(x,y,z)=iλBexp{ikz}sUs(u,v,0)·exp{ik2B[A(u2+v2)+D(x2+y2)2(ux+vy)]}dudv,
U(x,y,f)=iλfexp{ikf}exp{ik2f(x2+y2)}sUs(u,v,0)·exp{ikf(ux+vy)}dudv.
U(u,v,0)={Us(u,v,0),(u,v)s0,otherwise,
U(x,y,f)=iλfexp{ikf}exp{ik2f(x2+y2)}×U(u,v,0)exp{ikf(ux+vy)}dudv=iλfexp{ikf}exp{ik2f(x2+y2)}I{U(u,v,0)},
I{U(u,v,0)}=U(u,v,0)exp{ikf(ux+vy)}dudv=U(u,v,0)exp{i2π(ufX+vfY)}dudv|fX=x/λf,fY=y/λf.
Uj(u,v,0)=Mjexp{ik[αj(uaj)+βj(vbj)]+iϕj}g(uaj,vbj)=Mjexp{iϕj}t(uaj,vbj;αj,βj)g(uaj,vbj),
t(u,v;αj,βj)=exp{ik(αju+βjv)},
Uj(x,y,f)=iλfexp{ikf}exp{ik2f(x2+y2)}I{Uj(u,v,0)}=iMjλfexp{ikf+iϕj}exp{ik2f(x2+y2)}I{t(uaj,vbj;αj,βj)g(uaj,vbj)}.
I{t(uaj,vbj;αj,βj)g(uaj,vbj)}=g(uaj,vbj)exp{ik[αj(uaj)+βj(vbj)]}exp{i2πλf(ux+vy)}dudv=exp{ikf(ajx+bjy)}g(uaj,vbj)×exp{ikf[(x+αjf)(uaj)+(y+βjf)(vbj)]}d(uaj)d(vbj)=exp{ikf(ajx+bjy)}G(x+αjf,y+βjf)=exp{ikf(ajx+bjy)}·[G(x,y)*δ(x+αjf,y+βjf)],
G(x,y)=I{g(u,v)}.
U(x,y)=j=1NUj(x,y)=iλfexp{ikf}exp{ik2f(x2+y2)}×j=1NMjexp{iϕjikf(ajx+bjy)}[G(x,y)*δ(x+αjf,y+βjf)].
I(x,y)=U(x,y)U*(x,y).
{a1=b1=0,j=1,l=1,2,,m,n=j23l(l1),3l(l1)+2j3l(l+1)+1,aj=(2ln)d/2,bj=3nd/2,0n<l,aj=(3l2n)d/2,bj=3ld/2,ln<2l,aj=(ln)d/2,bj=3(3ln)d/2,2ln<3l,aj=(n5l)d/2,bj=3(n3l)d/2,3ln<4l,aj=(2n9l)d/2,bj=3ld/2,4ln<5l,aj=(n4l)d/2,bj=3(n6l)d/2,5ln<6l.
g(u,v)={1,u2+v2<w20,otherwise.
G(x,y)=I{g(u,v)}={πw2,x2+y2=0wλfx2+y2J1(2πwx2+y2λf),otherwise,
αj=α,βj=β,
U(x,y)=iλfexp{ikf}exp{ik2f(x2+y2)}G(x+αf,y+βf)j=1NMjexp{iϕjikf(ajx+bjy)},
I(x,y)=N2A(x+αf,y+βf)F(x,y),
A(x,y)=(λf)2|G(x,y)|2,F(x,y)=1N2j=1Np=1NMpMjexp{i(ϕjϕp)ikf[(ajap)x+(bjbp)y]}.
F(x,y)=SF(x,y)*δ(xx0,yy0)=SF(xx0,yy0),
SF(x,y)=N2j=1Np=1NMjMpexp{ikf[x(apaj)+y(bpbj)]}.
SF(x,y)=149{7+2j=17p>j7cos{kf[x(apaj)+y(bpbj)]}}=149[7+2(cos2kdxf+4cos3kdx2fcos3kdy2f+4coskdxf+2coskdxfcos3kdyf+8coskdx2fcos3kdy2f+2cos3kdyf)]=149[16cx2(cx2+cy2)+2(4cx21)(4cxcy1)1]=TF(cx,cy),
cos4θ=8cos4θ8cos2θ+1,cos3θ=4cos3θ3cosθ,cos2θ=2cos2θ1.
I(x,y)=N2A(x+αf,y+βf)SF(xx0,yy0).
I(0,0)=N2A(αf,βf)SF(x0,y0).
α=β=0,x0=y0=0.
I(x,y)=N2A(x,y)SF(x,y).
I(x,y)=N2A(x,y)SF(xx0,yy0).
TF(cx,cy)149{16cx2(cx2+cy2)+2(4cx21)[2(cx2+cy2)1]1}1.
cx=cy,cx2=cy2=1.
{xm=mλfd,m=0,±1,±2,,yn=nλf3d,n=0,±1,±2,,(1)(m+n)=1.
x02+y02(x0+δxj)2+(y0+δyj)2,j=1,2,,6,δxj=2λf3dcos(π3(j12)),δyj=2λf3dsin(π3(j12)).
α=pλd,β=qλ3d,
(x0xp)2+(y0yq)2(x0+δxjxp)2+(y0+δyjyq)2,j=1,2,,6,δxj=2λf3dcos(π3(j12)),δyj=2λf3dsin(π3(j12)).
M1=1,α=β=0,Mj=t>0,ϕj=ϕ1+φ;2j7,π<φπ,
F(x,y)=149[1+6t2+4t·g1(ξ,η)+4t2·g2(ξ,η)],
g1(ξ,η)=(cosξ+2cosξ2cos3η2)cosφ,g2(ξ,η)=4cosξ2cosξcos3η2+cos(3η)cos(2ξ)+cosξ+cosξcos(3η).
I(x,y)=A(x,y)[1+6t2+4t·g1(ξ,η)+4t2·g2(ξ,η)].
δI(x,y)=I(x,y)I(0,0)0,x2+y2<r2.
π2w4λ2f2[p2(η)t2+p1(η,φ)t+p0(η)]0,|η|<1.22πd/(d+w),
p2(η)=1π2w4|G(0,fηkd)|2(16cos23η2+16cos3η2+2)34,p1(η,φ)=1π2w4|G(0,fηkd)|2(8cos3η2+4)cosφ12cosφ,p0(η)=1π2w4|G(0,fηkd)|21,
G(0,fηkd)={πw2,η=02πwd|η|J1(w|η|d),otherwise.
t1,2=p1(η,φ)±p12(η,φ)4p2(η)p0(η)2p2(η).
p1(η,φ)/p2(η)<0,p12(η,φ)4p2(η)p0(η)>0.
I(0,0)=π2w4λ2f2(34t2+12tcosφ+1).
t3,4=12cosφ±(12cosφ)213668.

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