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Accepted: Jun. 20, 2019

Posted: Sep. 6, 2019

Published Online: Sep. 6, 2019

The Author Email: Xiaona Yan (xnyan@staff.shu.edu.cn)

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Xiaoyan Wang, Xiaona Yan, Kailong Jin, Ye Dai, Zuanming Jin, Xihua Yang, Guohong Ma. Generation of femtosecond dual pulses by a transverse standing wave in a volume holographic grating[J]. Chinese Optics Letters, 2019, 17(11): 113201

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 $n=n0+Δncos(Kx),$ (1)

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 $er0(t)=exp(−jω0t−t2T2),$ (2)

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 $Er0(ω)=12π∫−∞∞er0(t)exp(jωt)dt=T2πexp[−T2(ω−ω0)24].$ (3)

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 $E(ω,z,x)=Er′(ω,z,x)+Es′(ω,z,x)=Er(ω,z)exp(−jρ·r)+Es(ω,z)exp(−jσ·r),$ (4)

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 $dEr(ω,z)dz=−jκEs(ω,z)cosθ,dEs(ω,z)dz+jδEs(ω,z)cosθ=−jκEr(ω,z)cosθ,$ (5)

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 $Er(ω,z)=r1exp(jγ1z)+r2exp(jγ2z),Es(ω,z)=s1exp(jγ1z)+s2exp(jγ2z),$ (6)

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 $γ1,2=−δ2cosθs±12(δcosθs)2+4κ2cosθscosθ,$ (7)

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 $r1=1−r2=γ2γ2−γ1Er0(ω),s1=−s2=κ(γ2−γ1)cosθEr0(ω),$ (8)

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 $Er′(ω,z,x)=[r1exp(−jq0z(1)z)+r2exp(−jq0z(2)z)]⁢exp(−jq0xx)=Er1+Er2,Es′(ω,z,x)=[s1exp(−jq0z(1)z)+s2exp(−jq0z(2)z)]⁢exp[−j(q0x−K)x]=Es1+Es2,$ (9)

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 $Ii(t,z,x)=|12π∫−∞+∞Ei′(ω,z,x)exp(−jωt)dω|2, i=r,s,$ (10)

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 $Ers1(ω,z,x)=Er1+Es1={r1exp(−jq0xx)+s1exp[−j(q0x−K)x]}exp(−jq0z(1)z),Ers2(ω,z,x)=Er2+Es2={r2exp(−jq0xx)+s2exp[−j(q0x−K)x]}exp(−jq0z(2)z).$ (11)

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 $I(t,z,x)=|12π∫−∞∞[Ers1(ω,z,x)+Ers2(ω,z,x)]exp(−jωt)dω|2.$ (12)

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 $Ers1∝−j2sin(12Kx)exp(−jq0z(1)z),Ers2∝2cos(12Kx)exp(−jq0z(2)z).$ (13)

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 $Irs1∝1−cos(Kx),Irs2∝1+cos(Kx).$ (14)

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 $vgz(1,2)=∂ω∂q0z(1,2)|ω=ω0=2c·cosθ2n(cosθ)2∓Δn,$ (15)

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 $Δt=dvgz(2)−dvgz(1)=d·Δnccosθ.$ (16)