Abstract

We analyze and simulate the generation of mid-IR radiation from the propagation of bichromatic laser pulses in transparent, centrosymmetric dielectrics. The process relies on using the beatwave associated with the bichromatic pulse to seed four-wave parametric amplification in the mid-IR. We derive propagation equations describing the evolution of the pump waves and scattered waves including the effects of third-order nonlinearity, dispersion, and finite laser spot size. An expression for the growth rate of the scattered waves due to four-wave mixing is derived in the limit of negligible pump depletion and is characterized for various transparent dielectric materials. For fused silica, it is found that a bichromatic pump with wavelengths near 1 μm can generate forward-directed radiation near 3 μm. Fully explicit particle-in-cell modeling shows exponential growth and high conversion efficiency to mid-IR and visible radiation when the beat-wave-generated frequency comb is tuned to overlap the gain band of the four-wave amplification process.

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  19. D. Gordon, M. Helle, and J. R. Peñano, “Fully explicit nonlinear optics model in a particle-in-cell framework,” NRL Tech.l Rep. NRL/MR/6790-13-9459 (Naval Research Laboratory, 2013).

2011 (1)

2009 (1)

2007 (2)

2002 (1)

2001 (1)

C. Kapetanakos, B. Hafizi, P. Sprangle, R. Hubbard, and A. Ting, “Progress in the development of a high average power ultra-broadband infrared radiation source,” IEEE J. Quantum Electron. 37, 641–652 (2001).
[CrossRef]

2000 (1)

1999 (1)

C. Kapetanakos, B. Hafizi, H. Milehberg, P. Sprangle, R. Hubbard, and A. Ting, “Generation of high-average-power ultrabroad-band infrared pulses,” IEEE J. Quantum Electron. 35, 565–576 (1999).
[CrossRef]

1998 (1)

A. Brodeur and S. L. Chin, “Band-gap dependence of the ultrafast white-light continuum,” Phys. Rev. Lett. 80, 4406–4409 (1998).
[CrossRef]

1997 (1)

1994 (1)

1992 (1)

1984 (1)

1970 (1)

R. R. Alfano and S. L. Shapiro, “Emission in the region 4000 to 7000 Å via four-photon coupling in glass,” Phys. Rev. Lett. 24, 584–587 (1970).
[CrossRef]

Agrawal, G.

G. Agrawal, Nonlinear Fiber Optics (Academic, 1995).

Alfano, R. R.

R. R. Alfano and S. L. Shapiro, “Emission in the region 4000 to 7000 Å via four-photon coupling in glass,” Phys. Rev. Lett. 24, 584–587 (1970).
[CrossRef]

Alisauskas, S.

André, Y.-B.

Andriukaitis, G.

Balciunas, T.

Baltuska, A.

Boyd, R.

R. Boyd, Nonlinear Optics (Academic, 2003).

Brida, D.

Brodeur, A.

A. Brodeur and S. L. Chin, “Band-gap dependence of the ultrafast white-light continuum,” Phys. Rev. Lett. 80, 4406–4409 (1998).
[CrossRef]

Cerullo, G.

Chen, M.-C.

Chin, S. L.

A. Brodeur and S. L. Chin, “Band-gap dependence of the ultrafast white-light continuum,” Phys. Rev. Lett. 80, 4406–4409 (1998).
[CrossRef]

Cirmi, G.

Couairon, A.

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47–189 (2007).
[CrossRef]

Ebrahim-Zadeh, M.

M. Ebrahim-Zadeh and I. Sorokina, Mid-Infrared Coherent Sources and Applications (Springer, 2008).

Esarey, E.

P. Sprangle, A. Ting, E. Esarey, and R. Hubbard, “Active remote sensing using ultra broadband radiation,” Tech. Rep. NRL/MR/6790-97-7885 (Naval Research Laboratory, 1997).

Franco, M.

Franco, M. A.

Golovchenko, E. A.

Gordon, D.

D. Gordon, M. Helle, and J. R. Peñano, “Fully explicit nonlinear optics model in a particle-in-cell framework,” NRL Tech.l Rep. NRL/MR/6790-13-9459 (Naval Research Laboratory, 2013).

Grillon, G.

Hafizi, B.

C. Kapetanakos, B. Hafizi, P. Sprangle, R. Hubbard, and A. Ting, “Progress in the development of a high average power ultra-broadband infrared radiation source,” IEEE J. Quantum Electron. 37, 641–652 (2001).
[CrossRef]

C. Kapetanakos, B. Hafizi, H. Milehberg, P. Sprangle, R. Hubbard, and A. Ting, “Generation of high-average-power ultrabroad-band infrared pulses,” IEEE J. Quantum Electron. 35, 565–576 (1999).
[CrossRef]

Helle, M.

D. Gordon, M. Helle, and J. R. Peñano, “Fully explicit nonlinear optics model in a particle-in-cell framework,” NRL Tech.l Rep. NRL/MR/6790-13-9459 (Naval Research Laboratory, 2013).

Herrmann, J.

Hubbard, R.

C. Kapetanakos, B. Hafizi, P. Sprangle, R. Hubbard, and A. Ting, “Progress in the development of a high average power ultra-broadband infrared radiation source,” IEEE J. Quantum Electron. 37, 641–652 (2001).
[CrossRef]

C. Kapetanakos, B. Hafizi, H. Milehberg, P. Sprangle, R. Hubbard, and A. Ting, “Generation of high-average-power ultrabroad-band infrared pulses,” IEEE J. Quantum Electron. 35, 565–576 (1999).
[CrossRef]

P. Sprangle, A. Ting, E. Esarey, and R. Hubbard, “Active remote sensing using ultra broadband radiation,” Tech. Rep. NRL/MR/6790-97-7885 (Naval Research Laboratory, 1997).

Husakou, A.

Jauregui, C.

Kapetanakos, C.

C. Kapetanakos, B. Hafizi, P. Sprangle, R. Hubbard, and A. Ting, “Progress in the development of a high average power ultra-broadband infrared radiation source,” IEEE J. Quantum Electron. 37, 641–652 (2001).
[CrossRef]

C. Kapetanakos, B. Hafizi, H. Milehberg, P. Sprangle, R. Hubbard, and A. Ting, “Generation of high-average-power ultrabroad-band infrared pulses,” IEEE J. Quantum Electron. 35, 565–576 (1999).
[CrossRef]

Kapteyn, H.

Kasparian, J.

Limpert, J.

Manzoni, C.

Marangoni, M.

Milehberg, H.

C. Kapetanakos, B. Hafizi, H. Milehberg, P. Sprangle, R. Hubbard, and A. Ting, “Generation of high-average-power ultrabroad-band infrared pulses,” IEEE J. Quantum Electron. 35, 565–576 (1999).
[CrossRef]

Mondelain, D.

Murnane, M.

Mysyrowicz, A.

Nibbering, E. T. J.

Niedermeier, S.

Nodop, D.

Peñano, J. R.

D. Gordon, M. Helle, and J. R. Peñano, “Fully explicit nonlinear optics model in a particle-in-cell framework,” NRL Tech.l Rep. NRL/MR/6790-13-9459 (Naval Research Laboratory, 2013).

Pilipetskii, A. N.

Popmintchev, T.

Prade, B.

Prade, B. S.

Pugzlys, A.

Rodriguez, M.

Rothenberg, J. E.

Sauerbrey, R.

Schimpf, D.

Shapiro, S. L.

R. R. Alfano and S. L. Shapiro, “Emission in the region 4000 to 7000 Å via four-photon coupling in glass,” Phys. Rev. Lett. 24, 584–587 (1970).
[CrossRef]

Shen, Y. R.

Silvestri, S. D.

Sorokina, I.

M. Ebrahim-Zadeh and I. Sorokina, Mid-Infrared Coherent Sources and Applications (Springer, 2008).

Sprangle, P.

C. Kapetanakos, B. Hafizi, P. Sprangle, R. Hubbard, and A. Ting, “Progress in the development of a high average power ultra-broadband infrared radiation source,” IEEE J. Quantum Electron. 37, 641–652 (2001).
[CrossRef]

C. Kapetanakos, B. Hafizi, H. Milehberg, P. Sprangle, R. Hubbard, and A. Ting, “Generation of high-average-power ultrabroad-band infrared pulses,” IEEE J. Quantum Electron. 35, 565–576 (1999).
[CrossRef]

P. Sprangle, A. Ting, E. Esarey, and R. Hubbard, “Active remote sensing using ultra broadband radiation,” Tech. Rep. NRL/MR/6790-97-7885 (Naval Research Laboratory, 1997).

Ting, A.

C. Kapetanakos, B. Hafizi, P. Sprangle, R. Hubbard, and A. Ting, “Progress in the development of a high average power ultra-broadband infrared radiation source,” IEEE J. Quantum Electron. 37, 641–652 (2001).
[CrossRef]

C. Kapetanakos, B. Hafizi, H. Milehberg, P. Sprangle, R. Hubbard, and A. Ting, “Generation of high-average-power ultrabroad-band infrared pulses,” IEEE J. Quantum Electron. 35, 565–576 (1999).
[CrossRef]

P. Sprangle, A. Ting, E. Esarey, and R. Hubbard, “Active remote sensing using ultra broadband radiation,” Tech. Rep. NRL/MR/6790-97-7885 (Naval Research Laboratory, 1997).

Tünnermann, A.

Tzortzakis, S.

Wille, H.

Wolf, J.-P.

Wöste, L.

Yang, G.

Yu, J.

IEEE J. Quantum Electron. (2)

C. Kapetanakos, B. Hafizi, H. Milehberg, P. Sprangle, R. Hubbard, and A. Ting, “Generation of high-average-power ultrabroad-band infrared pulses,” IEEE J. Quantum Electron. 35, 565–576 (1999).
[CrossRef]

C. Kapetanakos, B. Hafizi, P. Sprangle, R. Hubbard, and A. Ting, “Progress in the development of a high average power ultra-broadband infrared radiation source,” IEEE J. Quantum Electron. 37, 641–652 (2001).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Express (1)

Opt. Lett. (5)

Phys. Rep. (1)

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47–189 (2007).
[CrossRef]

Phys. Rev. Lett. (2)

R. R. Alfano and S. L. Shapiro, “Emission in the region 4000 to 7000 Å via four-photon coupling in glass,” Phys. Rev. Lett. 24, 584–587 (1970).
[CrossRef]

A. Brodeur and S. L. Chin, “Band-gap dependence of the ultrafast white-light continuum,” Phys. Rev. Lett. 80, 4406–4409 (1998).
[CrossRef]

Other (5)

M. Ebrahim-Zadeh and I. Sorokina, Mid-Infrared Coherent Sources and Applications (Springer, 2008).

R. Boyd, Nonlinear Optics (Academic, 2003).

D. Gordon, M. Helle, and J. R. Peñano, “Fully explicit nonlinear optics model in a particle-in-cell framework,” NRL Tech.l Rep. NRL/MR/6790-13-9459 (Naval Research Laboratory, 2013).

P. Sprangle, A. Ting, E. Esarey, and R. Hubbard, “Active remote sensing using ultra broadband radiation,” Tech. Rep. NRL/MR/6790-97-7885 (Naval Research Laboratory, 1997).

G. Agrawal, Nonlinear Fiber Optics (Academic, 1995).

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Figures (11)

Fig. 1.
Fig. 1.

Schematic diagram of four-wave mixing in a centrosymmetric dielectric to produce visible and mid-IR radiation.

Fig. 2.
Fig. 2.

(a) Growth rate [Eq. (18)] and (b) total gain in 1 cm of propagation [Eq. (19)] versus wavelength and propagation angle for propagation in fused silica. The pump laser pulse has wavelengths of 950 and 1160 nm, spot size 0.1 mm, duration 5 psec, and a total energy of 0.6 mJ.

Fig. 3.
Fig. 3.

(a) Growth rate [Eq. (18)] and (b) total gain in 1 cm of propagation [Eq. (19)] versus wavelength and propagation angle for propagation in magnesium flouride. The pump laser pulse has wavelengths of 1064 and 1220 nm, spot size 0.1 mm, duration 5 psec, and a total energy of 0.6 mJ.

Fig. 4.
Fig. 4.

IR wavelength versus central pump wavelength for fused silica (dashed curve) and magnesium flouride (solid curve). The frequency separation of the pumps is held constant at δω/ω0=2.2%.

Fig. 5.
Fig. 5.

UBG spectrum [Eq. (22), solid curve, arbitrary units] at z=0.4cm for propagation in fused silica of a pump pulse with λ1=950nm and λ2=1155nm, I1=I2=0.3TW/cm2, and τ0=1psec. Dashed curve shows the corresponding growth rate as a function of photon energy for on-axis four-wave amplification.

Fig. 6.
Fig. 6.

Phase velocity versus vacuum wavelength for propagation in fused silica. The solid curve is obtained from the Sellmeier formula; the dots are obtained using our fully explicit nonlinear simulation code.

Fig. 7.
Fig. 7.

Laser fluence at z=12mm in the IR (solid curve) and the visible regime (dashed curve) in fused silica as a function of pump wavelength, λ2, for λ1=950nm. The intensity of each frequency component is 0.37TW/cm2.

Fig. 8.
Fig. 8.

Simulated spectral intensity after propagating 1.2 cm in fused silica. Pump wavelengths are 950 and 1160 nm. The intensity of each pump frequency component is 0.37TW/cm2. The pump pulse duration is 5.2 psec.

Fig. 9.
Fig. 9.

Laser fluence at z=12mm in the IR (solid curve) and the visible regime (dashed curve) in MgF2 as a function of pump wavelength, λ2, for λ1=1064nm. The intensity of each frequency component is 0.3TW/cm2.

Fig. 10.
Fig. 10.

Fluence in mid-IR band as a function of propagation distance z for SiO2 (solid curve) and MgF2 (dashed curve) corresponding to the resonant conditions of Figs. 7 and 9, respectively.

Fig. 11.
Fig. 11.

Simulation results showing temporal profiles of intensity for various frequency components of the laser pulse after propagating 1.2 cm in fused silica. (a) Pump pulses with the dashed and solid curves denoting λ1=950nm and λ2=1161nm, respectively. (b) Corresponding signals in the IR (solid curve) and visible regime (dashed curve), respectively.

Equations (32)

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E(x,z,t)=j=1412Aj(z,t)exp[iψj(x,z,t)]y^+c.c.,
E˜(kx,z,ω)12πdxdtexp[i(kxxωt)]E(x,z,t),
PNL(x,z,t)=j=1412PNL,j(z,t)exp[iψj(x,z,t)]y^+c.c.
[2z2kx2+β2(ω)]E˜=4πω2c2P˜NL,
12πdkxdωexp[i(kxxωt)]β2(ω)E˜(kx,z,ω)j14exp[iψj(x,z,t)]n=0Bn(ωj)2n!(i)nntnAj(z,t)y^,
[(z+ikzj)2kxj2+n=0Bn(ωj)n!(i)nntn]Aj=4πc2(tiωj)2PNL,j.
PNL,1=34χ3,1[(|A1|2+2k1|Ak|2)A1+2A2*A3A4exp(iΔψ)],
PNL,2=34χ3,2[(|A2|2+2k2|Ak|2)A2+2A1*A3A4exp(iΔψ)],
PNL,3=34χ3,3[(|A3|2+2k3|Ak|2)A3+2A1A2A4*exp(iΔψ)],
PNL,4=34χ3,4[(|A4|2+2k4|Ak|2)A4+2A1A2A3*exp(iΔψ)],
[(1+ikzjvoτ)2ikzjη+β02(ωj)(kzj2+kxj2)+F(ωj)]Aj=4πc2(τiωj)2PNL,j,
F(ωj)=2i[β0(ωj)β1(ωj)kzjvo]τ+[1vo2β12(ωj)β0(ωj)β2(ωj)]2τ2+n=3Bn(ωj)n!(i)nnτn
A1η=i3πω122kz1c2χ3,1(|A1|2+2|A2|2)A1,A2η=i3πω222kz2c2χ3,2(|A2|2+2|A1|2)A2.
A1(η)=a10exp(iκ1η),A2(η)=a20exp(iκ2η),
κ1=3πω122kz1c2χ3,1(|a10|2+2|a20|2),κ2=3πω222kz2c2χ3,2(|a20|2+2|a10|2).
ω1+ω2ω3ω4=0,
[η+α32+ikx322kz3+iKD,3]A3=i3πω32kz3c2χ3,3[(|a10|2+|a20|2)A3+a10a20A4*exp[i(κ1+κ2+ΔkL)η]],
[η+α42+ikx422kz4+iKD,4]A4=i3πω42kz4c2χ3,4[(|a10|2+|a20|2)A4+a10a20A3*exp[i(κ1+κ2+ΔkL)η]],
iKD,jF(ωj)2ikzj=[β1(ωj)1vo]τ+i2[β2(ωj)1kzj(1vo2β12(ωj))]2τ2i2kzjn=3Bn(ωj)n!(i)nnτn,
A^(Ω)=12πdτexp(iΩτ)A(τ),
κj=4πωj2kzjc2χ3,j(|a10|2+|a20|2)kx22kzjKD,j(ωj,iΩ)+iαj2
ηu^3=i3πω32kz3c2χ3,3a10a20u^4*exp[iΔkη],
ηu^4*=i3πω42kz4c2χ3,4a10a20u^3exp[iΔk*η],
Δk=κ1+κ2κ3κ4+ΔkL
2η2u^3iΔkηu^3γ02u^3=0,
γ02=12π2ω32ω42kz3kz4c4χ3,3χ3,4a102a202.
Γ=iΔk2±(γ02Δk24)1/2,
N(θ,ω,ω0)=Re[Γ]0Z0exp[2z2/Zslip2(θ,ω,ω0)]dz,
zA=iγ|A|2A,
A(z,τ)=A0cos(Δωτ)exp[ikNL(τ)z][θ(τ+τL2)θ(ττL2)],
kNL(τ)=γA022[1+cos(2Δωτ)][θ(τ+τL2)θ(ττL2)]2,
A˜(z,ω)=12πA0eiγA02/2n=(i)nJn(γA022z)τL2×{sinc[(ω+(2n+1)Δω)τL2]+sinc[(ω+(2n1)Δω)τL2]}.

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