• Advanced Photonics Nexus
  • Vol. 2, Issue 2, 024001 (2023)
Xin Wang1、2、3、†, Zilong Zhang1、2、3、*, Xing Fu4、5, Adnan Khan6, Suyi Zhao1、2、3, Yuan Gao1、2、3, Yuchen Jie1、2、3, Wei He1、2、3, Xiaotian Li1、2、3, Qiang Liu4、5、*, and Changming Zhao1、2、3
Author Affiliations
  • 1Beijing Institute of Technology, School of Optics and Photonics, Beijing, China
  • 2Ministry of Education, Key Laboratory of Photoelectronic Imaging Technology and System, Beijing, China
  • 3Ministry of Industry and Information Technology, Key Laboratory of Photonics Information Technology, Beijing, China
  • 4Tsinghua University, Ministry of Education, Key Laboratory of Photonic Control Technology, Beijing, China
  • 5Tsinghua University, State Key Laboratory of Precision Measurement Technology and Instruments, Department of Precision Instrument, Beijing, China
  • 6Nankai University, School of Physics, Key Laboratory of Weak Light Nonlinear Photonics, Tianjin, China
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    DOI: 10.1117/1.APN.2.2.024001 Cite this Article Set citation alerts
    Xin Wang, Zilong Zhang, Xing Fu, Adnan Khan, Suyi Zhao, Yuan Gao, Yuchen Jie, Wei He, Xiaotian Li, Qiang Liu, Changming Zhao. Evolution on spatial patterns of structured laser beams: from spontaneous organization to multiple transformations[J]. Advanced Photonics Nexus, 2023, 2(2): 024001 Copy Citation Text show less

    Abstract

    Spatial patterns are a significant characteristic of lasers. The knowledge of spatial patterns of structured laser beams is rapidly expanding, along with the progress of studies on laser physics and technology. Particularly in the last decades, owing to the in-depth attention on structured light with multiple degrees of freedom, the research on spatial and spatiotemporal structures of laser beams has been promptly developed. Such beams have hatched various breakthroughs in many fields, including imaging, microscopy, metrology, communication, optical trapping, and quantum information processing. Here, we would like to provide an overview of the extensive research on several areas relevant to spatial patterns of structured laser beams, from spontaneous organization to multiple transformations. These include the early theory of beam pattern formation based on the Maxwell–Bloch equations, the recent eigenmodes superposition theory based on the time-averaged Helmholtz equations, the beam patterns extension of ultrafast lasers to the spatiotemporal beam structures, and the structural transformations in the nonlinear frequency conversion process of structured beams.
    {Et=(iωc+κ)E+κP+idκ2EPt=γP+γEDDt=γ[(DD0)+12(E*P+P*E)],

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    Eτ=(D01)Ei(βd2)Eg(βd2)2EE|E|2,

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    {Eτ=(D1)Ei(βd2)Eg(βd2)2EDτ=γ[(DD0)+|E|2D].

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    2E+k2E=0,

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    k=0m+n(2i)kPk(nk,mk)(0)Hn+mk(x)Hk(y)=2m+n×{(1)mm!(x+iy)nmLmnm(x2+y2)for  nm(1)nn!(xiy)mnLmmn(x2+y2)for  m>n,

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    Pk(nk,mk)(0)=(1)k2kk!dkdtk[(1t)n(1+t)m]|t=0,

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    LGp,±l(x,y,z)=K=0m+n(±i)Kb(n,m,K)·HGm+nK,K(x,y,z),

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    b(n.m,K)=[(NK)!K!2Nn!m!]1/21K!dKdtK[(1t)n(1+t)m]|t=0,

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    HLGn,m(r,z|α)=12N1n!m!exp(π|r|2w)HLn,m(rπw|α)×exp[ikz+ikr22Ri(m+n+1)Ψ(z)],

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    |ψn,m.lN,P,Q=12N/2K=0NeiKϕ(NK)12|ψn+QK,m,lPK,

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    HIGp,m±=IGp,me(ξ,η,ε)±iIGp,mo(ξ,η,ε),

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    SHENn,m(x,y,z|β,γ)=K=0NeiβKb(n,m,K),{(i)KIGN,NKe(x,y,z|ε=2/tan2γ,for  (1)K=1(i)KIGN,NK+1o(x,y,z|ε=2/tan2γ,for  (1)K1.

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    Etot=m,nam,nXGm,n(·)*exp[iϕm,n+ikz+ikx2+y2R(z)iqψ(z)],

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    Etot(x,y,z)=exp[iω¯cz+iω¯cx2+y2R(z)¯iq¯ψ(z)]·m,nam,nXGm,n(·)*exp(iϕm,n),

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    {ω¯=ω0+m,nam,nΔωnm,nam,nΔωn=ωnω0=c(kx2+ky2)2kz.

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    C^=F^(x,y)F^(ω)SA^(x,y,t)P^(x,y,t),

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    Ei+1(x,y,t)=C^Ei(x,y,t),

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    G(ρ,ϕ)=FT{gR(r)eilθ}=2π(i)leilθHl{gR(r)},

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    E(ρ,ϕ)=G(ρ,ϕ)exp(ikzziωt).

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    Ez=iω2ε0cnPNLeiΔkz,

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    {P(2)(ω1)=2ε0χ(2)(ω1;ω3,ω2)·E3E2*P(2)(ω2)=2ε0χ(2)(ω2;ω3,ω1)·E3E1*P(2)(ω3)=2ε0χ(2)(ω3;ω1,ω2)·E1E2,

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    {dE1(z)dz=iω1cn1χ(2)(ω1;ω3,ω2)·E3(z)E2*(z)eiΔkzdE2(z)dz=iω2cn2χ(2)(ω2;ω3,ω1)·E3(z)E1*(z)eiΔkzdE3(z)dz=iω3cn3χ(2)(ω3;ω1,ω2)·E1(z)E2(z)eiΔkz.

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    ESFG{(LG0l+eiθLG0l)*LG0lLG02l+eiθLGl0(LG0l1+eiθLG0l2)*LG0l2LG0l1+l2+eiθLGl20(LG0l+eiθ1LG0l)*(LG0l+eiθ2LG0l)LG02l+ei(θ1+θ2)LG02l+(eiθ1+eiθ2)LGl*0,

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    ESHG(r,φ)[XGm1,n1(r,φ)+eiϕXGm2,n2(r,φ)]×[XGm1,n1(r,φ)+eiϕXGm2,n2(r,φ)]=XGm1,n12(r,φ)+2eiϕXGm1,n1(r,φ)XGm2,n2(r,φ)+ei2ϕXGm2,n22(r,φ).

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    Xin Wang, Zilong Zhang, Xing Fu, Adnan Khan, Suyi Zhao, Yuan Gao, Yuchen Jie, Wei He, Xiaotian Li, Qiang Liu, Changming Zhao. Evolution on spatial patterns of structured laser beams: from spontaneous organization to multiple transformations[J]. Advanced Photonics Nexus, 2023, 2(2): 024001
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