• High Power Laser Science and Engineering
  • Vol. 12, Issue 6, 06000e85 (2024)
Efim Khazanov*
Author Affiliations
  • Gaponov-Grekhov Institute of Applied Physics of the Russian Academy of Sciences, Niznij Novgorod, Russia
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    DOI: 10.1017/hpl.2024.58 Cite this Article Set citation alerts
    Efim Khazanov, "Dependence of the focal intensity of a femtosecond laser pulse on the non-flatness of compressor diffraction gratings," High Power Laser Sci. Eng. 12, 06000e85 (2024) Copy Citation Text show less

    Abstract

    An analytical expression for the focal intensity of a laser pulse was obtained for an asymmetric out-of-plane compressor with gratings of arbitrary surface shape. The focal intensity is most strongly affected by the linear angular chirp caused by the spatial shift of different frequencies on the second and third gratings. The chirp can be eliminated by simply rotating the fourth grating by an optimal angle, which significantly reduces the requirements for the grating quality. It is shown that the decrease in the focal intensity depends on the product of the grating surface root mean square and pulse spectrum bandwidth. With low-quality gratings, spectrum narrowing would not reduce focal intensity; contrariwise, it may even slightly increase it.
    E0(ω,\boldsymbolr)=eiφin(ω)+iφD(ω)eiHyeiφ0(ω,\boldsymbolr)|E0(ω,\boldsymbolr)|, ((1))

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    H=ωcsinγ1. ((2))

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    Ereflected(ω,\boldsymbolr)=eiφn(ω,\boldsymbolr)Eincident(ω,\boldsymbolr). ((3))

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    φn(ω,\boldsymbolr)=ωcdn(ω,\boldsymbolr), ((4))

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    dn(ω,\boldsymbolr)=cosγ1,2(cosθ1,2(ω)+cosα1,2)hn(xcosα1,2;ycosγ1,2), ((5))

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    sinθ1,2(ω)=2πcωcosγ1,2N1,2+sinα1,2. ((6))

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    Ej(t,\boldsymbolr)=Ej(ω,\boldsymbolr)eiωtdω. ((7))

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    Ej(ω,\boldsymbolκ)=Ej(ω,\boldsymbolr)ei\boldsymbolκ\boldsymbolrd\boldsymbolr. ((8))

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    Ψp1,2(ω,kx,ky)=L1,2kzx(cosθ~1,2+cos(α1,2±atankxkz)), ((9))

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    sinθ~1,2(ω,kx,ky)=2πkzxN1,2+sin(α1,2±atankxkz). ((10))

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    Ψp1,2(ω,\boldsymbolκ)=Ψ1,2(ω)+\boldsymbolR1,2(ω)\boldsymbolκ, ((11))

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    Ψ1,2(ω)=Ψp1,2(ω,\boldsymbolκ=\boldsymbolκ1,2)\boldsymbolR1,2(ω)\boldsymbolκ1,2, ((12))

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    \boldsymbolR1,2(ω)Ψp1,2(ω,\boldsymbolκ=\boldsymbolκ1,2)\boldsymbolκ=±L(sin(θ1,2(ω)+α1,2)cosθ1,2(ω)tanγ1,21+cos(θ1,2(ω)+α1,2)cosθ1,2(ω)). ((13))

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    E1(ω,\boldsymbolr)=eiφ1(ω,\boldsymbolr)E0(ω,\boldsymbolr), ((14))

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    E2(ω,\boldsymbolκ)=eiΨp1(ω,\boldsymbolκ)eiφ1(ω,\boldsymbolr)E0(ω,\boldsymbolr)ei\boldsymbolκ\boldsymbolrd\boldsymbolr. ((15))

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    E3(ω,\boldsymbolr)=eiφ23(ω,\boldsymbolr\boldsymbolR1(ω0))E2(ω,\boldsymbolr), ((16))

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    E3(ω,\boldsymbolκ)=eiΨ1(ω)ei\boldsymbolκ\boldsymbolR1(ω)d\boldsymbolreiφ23(ω,\boldsymbolr+\boldsymbolρ1(ω))ei\boldsymbolκ\boldsymbolreiφ1(ω,\boldsymbolr)E0(ω,\boldsymbolr), ((17))

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    \boldsymbolρ1,2(ω)=\boldsymbolR1,2(ω)\boldsymbolR1,2(ω0) ((18))

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    E4(ω,\boldsymbolκ)=eiΨp2(ω,\boldsymbolκ)E3(ω,\boldsymbolκ). ((19))

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    E4(ω,\boldsymbolr)=eiΨ2(ω)eiΨ1(ω)eiφ23(ω,\boldsymbolr\boldsymbolR12(ω)+\boldsymbolρ1(ω))eiφ1(ω,\boldsymbolr\boldsymbolR12(ω))×E0(ω,\boldsymbolr\boldsymbolR12(ω)), ((20))

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    \boldsymbolR12(ω)=\boldsymbolR1(ω)+\boldsymbolR2(ω). ((21))

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    E5(ω,\boldsymbolr)=E4(ω,\boldsymbolr)eiφ4(\boldsymbolr\boldsymbolR12(ω0))ei\boldsymbolε(ω)(\boldsymbolr\boldsymbolR12(ω0)), ((22))

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    εx=ωccosγ2cosα2(cosα2+cosθ2(ω))δx,εy=ωc(cosθ2(ω)+cosα2)δy. ((23))

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    E6(ω,\boldsymbolr)=E5(ω,\boldsymbolr)eiφam(ω,\boldsymbolr\boldsymbolR12(ω0)). ((24))

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    E6(ω,\boldsymbolκ)=eiΨ12(ω)eiφin(ω)+iφD(ω)ei\boldsymbolκ\boldsymbolR12(ω)×d\boldsymbolrei\boldsymbolκ\boldsymbolrei\boldsymbolε(ω)\boldsymbolreiHyeiϕ(ω,\boldsymbolr)E0(ω,\boldsymbolr), ((25))

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    Ψ12(ω)=Ψ1(ω)+Ψ2(ω)+\boldsymbolε(ω)(\boldsymbolR12(ω)\boldsymbolR12(ω0)), ((26))

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    ϕ(ω,\boldsymbolr)=φ0(ω,\boldsymbolr)+φ1(ω,\boldsymbolr)+φ23(ω,\boldsymbolr+\boldsymbolρ1(ω))+φ4(ω,\boldsymbolr+\boldsymbolρ1(ω)+\boldsymbolρ2(ω))+φam(ω,\boldsymbolr+\boldsymbolρ1(ω)+\boldsymbolρ2(ω)). ((27))

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    E6(ω,\boldsymbolκ)=eiΨ12(ω)eiφin(ω)+iφD(ω)eiΨaber(ω)ei\boldsymbolκ\boldsymbolR12(ω)×d\boldsymbolrei\boldsymbolκ\boldsymbolrei\boldsymbolε(ω)\boldsymbolreiHyeiϕ23(ω,\boldsymbolr)+iϕall(ω,\boldsymbolr)E0(ω,\boldsymbolr), ((28))

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    ϕ23(ω,\boldsymbolr)=ωcj=23(dj(ω0,\boldsymbolr+\boldsymbolρ1(ω))dj(ω0,\boldsymbolr+\boldsymbolρ1(ω)+\boldsymbolρ2(ω))dj(ω0,\boldsymbolρ1(ω))+dj(ω0,\boldsymbolρ1(ω)+\boldsymbolρ2(ω))), ((29))

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    ϕall(ω,\boldsymbolr)=ωcj=14(dj(ω,\boldsymbolr)dj(ω0,\boldsymbolr)), ((30))

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    Ψaber(ω)=ωcj=23(dj(ω0,\boldsymbolρ1(ω))dj(ω0,\boldsymbolρ1(ω)+\boldsymbolρ2(ω))). ((31))

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    2ham(\boldsymbolr)=j=14dj(ω0,\boldsymbolr)cωφ0(ω,\boldsymbolr), ((32))

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    Ef(ω)=E6(ω,κx=εx(ω0),κy=εy(ω0)H). ((33))

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    Ef(Ω)=d\boldsymbolreiμ(Ω)\boldsymbolreiϕ23(Ω,\boldsymbolr)+iϕall(Ω,\boldsymbolr)E0(Ω,\boldsymbolr), ((34))

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    μ(Ω)=\boldsymbolε(Ω)\boldsymbolε(Ω0)=Ωω0\boldsymbolAμ+Cμ(Ωω0)2\boldsymbolAμ+O((Ωω0)3), ((35))

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    \boldsymbolAμ=k0w1+cos(α2+β2)cosα2cosβ2(δxcosγ2δycosα2),Cμ=(sinα2sinβ2)22cos2β2(1+cos(α2+β2)), ((36))

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    φD(Ω)=φin(Ω)Ψ1(Ω)Ψ2(Ω)+HR12y(Ω)Ψaber(Ω)+μ(Ω)(\boldsymbolρ1(Ω)+\boldsymbolρ2(Ω)). ((37))

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    E0(Ω,x,y)∣=∣E00eξ2μex2νey2ν, ((38))

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    Ef(T)=∣E00Δωw2dξeiξTeξ2μ×d\boldsymboluex2νey2νei(μ(ξ)\boldsymbolu+ϕ23(ξ,\boldsymbolu)+ϕall(ξ,\boldsymbolu)). ((39))

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    St=|Ef(Tmax)|2|Ef(T=0;μ=ϕ23=ϕall=0)|2, ((40))

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    hn(\boldsymbolr)=cω0(Zn0z20(\boldsymbolr)+Zn2z22(\boldsymbolr)+Zn2z22(\boldsymbolr)). ((41))

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    ϕ23(ξ,\boldsymbolu)=(ξΔL2w\boldsymbolq+(ξΔ)2L2w\boldsymbolq~)\boldsymbolu+O(L2wΔ3), ((42))

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    ϕall(ω,x,y)=ξΔ(Px2+Qy2+Sxy)+O(Δ2), ((43))

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    \boldsymbolq=(\boldsymbolZ^2+\boldsymbolZ^3)\boldsymbolA,\boldsymbolq~=(\boldsymbolZ^2+\boldsymbolZ^3)(\boldsymbolA\boldsymbolB), ((44))

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    \boldsymbolA=(cosα2cos3β2(sinα2sinβ2)tanγ2(sinα2sinβ2)2cos3β2),\boldsymbolB=(cosα2cos3β2(sinα2sinβ2)(132sinβ2sinα2sinβ2cos2β2)tanγ2(sinα2sinβ2)2cos3β2(1+cosβ2+3(sinα2sinβ2)2cos2β2)), ((45))

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    \boldsymbolZ^n=cosγ1,2(cosβ1,2+cosα1,2)×(2(Zn023+Zn26)cosα1,26Zn2cosα1,26Zn2cosγ1,22(Zn023Zn26)cosγ1,2), ((46))

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    P=n=14pn,Q=n=14qn,S=n=14sn, ((47))

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    (pnqnsn)=cosγ1,2tanβ1,2(sinα1,2sinβ1,2)6×(1cos2α1,2(Zn02+Zn2)1cos2γ1,2(Zn02Zn2)1cosα1,2cosγ1,2Zn2). ((48))

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    Ef(T)=∣E00Δωw2dξeiξTeξ2μ×d\boldsymboluex2νey2νcos(ξΔL2w\boldsymbolqu), ((49))

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    St=1Γ(32ν)Γ(12ν)Γ(32μ)Γ(12μ)Δ2(L2w)2\boldsymbolq\boldsymbol2, ((50))

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    \boldsymbolAμ=L2w\boldsymbolq. ((51))

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    (δxδy)=1k0wL2wcosα2cosβ21+cos(α2+β2)(qxcosγ2qycosα2). ((52))

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    Ef(T)=|E00|Δωw2dξeξ2μd\boldsymboluex2νey2ν×cos(ξΔ(Px2+Qy2+Sxy)+(ξΔ)2L2w\boldsymbolsuξT), ((53))

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    Stcomp=1G23Gall, ((54))

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    G23=Δ4(L2w)2Γ(52μ)Γ(32ν)Γ(12μ)Γ(12ν)\boldsymbols2, ((55))

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    Gall=Δ2Γ(32μ)Γ2(12μ)Γ2(12ν)×{(P2+Q2)(Γ(12μ)Γ(52ν)Γ(12ν)Γ(32μ)Γ2(32ν))+S2Γ(12μ)Γ2(32ν)}. ((56))

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    σ2=14w2wwdxww(14n=14hn(\boldsymbolr))2dy. ((57))

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    Efim Khazanov, "Dependence of the focal intensity of a femtosecond laser pulse on the non-flatness of compressor diffraction gratings," High Power Laser Sci. Eng. 12, 06000e85 (2024)
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