Abstract

Achromatic phase matching (APM), a favorable method in increasing phase matching bandwidth, is explored for second harmonic generation of ultrashort pulses based on a typical grating telescope system. A set of coupled equations incorporating angular dispersion is constructed in the space–time domain. An analytic solution with a pump of tilting pulsed Gaussian beam to the equations is given under the undepleted pump approximation. With the aid of matrix formalism, some properties of the conversion are demonstrated. Though a maximal phase matching bandwidth is obtained, angular dispersion makes the conversion active only around the geometrical focus. Theory shows that APM does not require an overall pulse-front matching in the conversion process.

© 2013 Optical Society of America

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  1. G. Szabo and Z. Bor, “Broadband frequency doubler for femtosecond pulses,” Appl. Phys. B 50, 51–54 (1990).
    [CrossRef]
  2. O. E. Martinez, “Achromatic phase matching for second harmonic generation of femtosecond pulses,” IEEE J. Quantum Electron. 25, 2464–2468 (1989).
    [CrossRef]
  3. T. R. Zhang, H. R. Choo, and M. C. Downer, “Phase and group velocity matching for second harmonic generation for femtosecond pulses,” Appl. Opt. 29, 3927–3933 (1990).
    [CrossRef]
  4. P. Baum, S. Lochbrunner, and E. Riedle, “Tunable sub-10-fs ultraviolet pulses generated by achromatic frequency doubling,” Opt. Lett. 29, 1686–1688 (2004).
    [CrossRef]
  5. O. E. Martinez, “Pulse distortions in tilted pulse schemes for ultrashort pulses,” Opt. Commun. 59, 229–232 (1986).
    [CrossRef]
  6. Z. Bor, B. Racz, G. Szabo, M. Hilbert, and H. A. Hazim, “Femtosecond pulse front tilt caused by angular-dispersion,” Opt. Eng. 32, 2501–2504 (1993).
    [CrossRef]
  7. J. Hebling, “Derivation of the pulse front tilt caused by angular dispersion,” Opt. Quantum Electron. 28, 1759–1763 (1996).
    [CrossRef]
  8. C. Dorrer, E. M. Kosik, and I. A. Walmsley, “Spatio-temporal characterization of ultrashort optical pulses using two-dimensional shearing interferometry,” Appl. Phys. B 74, s209–s217 (2002).
    [CrossRef]
  9. P. D. Trapani, A. Andreoni, C. Solcia, P. Foggi, R. Danielius, A. Dubietis, and A. Piskarskas, “Matching of group velocities in three-wave parametric interaction with femtosecond pulses and application to traveling-wave generators,” J. Opt. Soc. Am. B 12, 2237–2244 (1995).
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  12. H. Liu, W. Zhao, Y. Yang, H. Wang, Y. Wang, and G. Chen, “Matching of both group-velocity and pulse-front for ultrabroadband three-wave-mixing with noncollinear angularly dispersed geometry,” Appl. Phys. B 82, 585–594 (2006).
    [CrossRef]
  13. A. M. Schober, M. Charbonneau-Lefort, and M. M. Fejer, “Broadband quasi-phase-matched second-harmonic generation of ultrashort optical pulses with spectral angular dispersion,” J. Opt. Soc. Am. B 22, 1699–1713 (2005).
    [CrossRef]
  14. O. E. Martinez, “Grating and prism compressors in the case of finite beam size,” J. Opt. Soc. Am. B 3, 929–934 (1986).
    [CrossRef]
  15. O. E. Martinez, “Matrix formalism for pulse compressors,” IEEE J. Quantum Electron. 24, 2530–2536 (1988).
    [CrossRef]
  16. A. G. Kostenbauder, “Ray-pulse matrices: a rational treatment for dispersive optical Systems,” IEEE J. Quantum Electron. 26, 1148–1157 (1990).
    [CrossRef]
  17. S. Akturk, X. Gu, E. Zeek, and R. Trebino, “Pulse-front tilt caused by spatial and temporal chirp,” Opt. Express 12, 4399–4410 (2004).
    [CrossRef]
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    [CrossRef]
  19. J. J. Huang, X. Y. Hu, D. M. Ren, and Y. C. Qu, “Simulation of three-wave interactions for focused beams with an assistant curvilinear coordinate system,” J. Opt. Soc. Am. B 23, 1312–1322 (2006).
    [CrossRef]
  20. R. W. Boyd, Nonlinear Optics (Elsevier, 2008).
  21. J. J. Huang and L. Y. Zhang, “Transformation of few-cycle ultrashort pulsed Gaussian beams by an angular disperser,” J. Phys. B 43, 175601 (2010).
    [CrossRef]

2010 (1)

J. J. Huang and L. Y. Zhang, “Transformation of few-cycle ultrashort pulsed Gaussian beams by an angular disperser,” J. Phys. B 43, 175601 (2010).
[CrossRef]

2006 (2)

H. Liu, W. Zhao, Y. Yang, H. Wang, Y. Wang, and G. Chen, “Matching of both group-velocity and pulse-front for ultrabroadband three-wave-mixing with noncollinear angularly dispersed geometry,” Appl. Phys. B 82, 585–594 (2006).
[CrossRef]

J. J. Huang, X. Y. Hu, D. M. Ren, and Y. C. Qu, “Simulation of three-wave interactions for focused beams with an assistant curvilinear coordinate system,” J. Opt. Soc. Am. B 23, 1312–1322 (2006).
[CrossRef]

2005 (2)

2004 (2)

2002 (1)

C. Dorrer, E. M. Kosik, and I. A. Walmsley, “Spatio-temporal characterization of ultrashort optical pulses using two-dimensional shearing interferometry,” Appl. Phys. B 74, s209–s217 (2002).
[CrossRef]

2001 (1)

1996 (1)

J. Hebling, “Derivation of the pulse front tilt caused by angular dispersion,” Opt. Quantum Electron. 28, 1759–1763 (1996).
[CrossRef]

1995 (1)

1993 (1)

Z. Bor, B. Racz, G. Szabo, M. Hilbert, and H. A. Hazim, “Femtosecond pulse front tilt caused by angular-dispersion,” Opt. Eng. 32, 2501–2504 (1993).
[CrossRef]

1990 (3)

G. Szabo and Z. Bor, “Broadband frequency doubler for femtosecond pulses,” Appl. Phys. B 50, 51–54 (1990).
[CrossRef]

A. G. Kostenbauder, “Ray-pulse matrices: a rational treatment for dispersive optical Systems,” IEEE J. Quantum Electron. 26, 1148–1157 (1990).
[CrossRef]

T. R. Zhang, H. R. Choo, and M. C. Downer, “Phase and group velocity matching for second harmonic generation for femtosecond pulses,” Appl. Opt. 29, 3927–3933 (1990).
[CrossRef]

1989 (1)

O. E. Martinez, “Achromatic phase matching for second harmonic generation of femtosecond pulses,” IEEE J. Quantum Electron. 25, 2464–2468 (1989).
[CrossRef]

1988 (1)

O. E. Martinez, “Matrix formalism for pulse compressors,” IEEE J. Quantum Electron. 24, 2530–2536 (1988).
[CrossRef]

1986 (2)

O. E. Martinez, “Pulse distortions in tilted pulse schemes for ultrashort pulses,” Opt. Commun. 59, 229–232 (1986).
[CrossRef]

O. E. Martinez, “Grating and prism compressors in the case of finite beam size,” J. Opt. Soc. Am. B 3, 929–934 (1986).
[CrossRef]

Akturk, S.

Andreoni, A.

Baum, P.

Bor, Z.

Z. Bor, B. Racz, G. Szabo, M. Hilbert, and H. A. Hazim, “Femtosecond pulse front tilt caused by angular-dispersion,” Opt. Eng. 32, 2501–2504 (1993).
[CrossRef]

G. Szabo and Z. Bor, “Broadband frequency doubler for femtosecond pulses,” Appl. Phys. B 50, 51–54 (1990).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Elsevier, 2008).

Charbonneau-Lefort, M.

Chen, G.

H. Liu, W. Zhao, Y. Yang, H. Wang, Y. Wang, and G. Chen, “Matching of both group-velocity and pulse-front for ultrabroadband three-wave-mixing with noncollinear angularly dispersed geometry,” Appl. Phys. B 82, 585–594 (2006).
[CrossRef]

Choo, H. R.

Danielius, R.

Dorrer, C.

C. Dorrer, E. M. Kosik, and I. A. Walmsley, “Spatio-temporal characterization of ultrashort optical pulses using two-dimensional shearing interferometry,” Appl. Phys. B 74, s209–s217 (2002).
[CrossRef]

Downer, M. C.

Dubietis, A.

Fejer, M. M.

Foggi, P.

Gabolde, P.

Gu, X.

Hazim, H. A.

Z. Bor, B. Racz, G. Szabo, M. Hilbert, and H. A. Hazim, “Femtosecond pulse front tilt caused by angular-dispersion,” Opt. Eng. 32, 2501–2504 (1993).
[CrossRef]

Hebling, J.

J. Hebling, “Derivation of the pulse front tilt caused by angular dispersion,” Opt. Quantum Electron. 28, 1759–1763 (1996).
[CrossRef]

Hilbert, M.

Z. Bor, B. Racz, G. Szabo, M. Hilbert, and H. A. Hazim, “Femtosecond pulse front tilt caused by angular-dispersion,” Opt. Eng. 32, 2501–2504 (1993).
[CrossRef]

Hu, X. Y.

Huang, J. J.

J. J. Huang and L. Y. Zhang, “Transformation of few-cycle ultrashort pulsed Gaussian beams by an angular disperser,” J. Phys. B 43, 175601 (2010).
[CrossRef]

J. J. Huang, X. Y. Hu, D. M. Ren, and Y. C. Qu, “Simulation of three-wave interactions for focused beams with an assistant curvilinear coordinate system,” J. Opt. Soc. Am. B 23, 1312–1322 (2006).
[CrossRef]

Kobayashi, T.

T. Kobayashi, Femtosecond optical frequency comb: Principle, Operation and Applications (Springer, 2005).

Kosik, E. M.

C. Dorrer, E. M. Kosik, and I. A. Walmsley, “Spatio-temporal characterization of ultrashort optical pulses using two-dimensional shearing interferometry,” Appl. Phys. B 74, s209–s217 (2002).
[CrossRef]

Kostenbauder, A. G.

A. G. Kostenbauder, “Ray-pulse matrices: a rational treatment for dispersive optical Systems,” IEEE J. Quantum Electron. 26, 1148–1157 (1990).
[CrossRef]

Liu, H.

H. Liu, W. Zhao, Y. Yang, H. Wang, Y. Wang, and G. Chen, “Matching of both group-velocity and pulse-front for ultrabroadband three-wave-mixing with noncollinear angularly dispersed geometry,” Appl. Phys. B 82, 585–594 (2006).
[CrossRef]

Lochbrunner, S.

Martinez, O. E.

O. E. Martinez, “Achromatic phase matching for second harmonic generation of femtosecond pulses,” IEEE J. Quantum Electron. 25, 2464–2468 (1989).
[CrossRef]

O. E. Martinez, “Matrix formalism for pulse compressors,” IEEE J. Quantum Electron. 24, 2530–2536 (1988).
[CrossRef]

O. E. Martinez, “Grating and prism compressors in the case of finite beam size,” J. Opt. Soc. Am. B 3, 929–934 (1986).
[CrossRef]

O. E. Martinez, “Pulse distortions in tilted pulse schemes for ultrashort pulses,” Opt. Commun. 59, 229–232 (1986).
[CrossRef]

Piskarskas, A.

Qu, Y. C.

Racz, B.

Z. Bor, B. Racz, G. Szabo, M. Hilbert, and H. A. Hazim, “Femtosecond pulse front tilt caused by angular-dispersion,” Opt. Eng. 32, 2501–2504 (1993).
[CrossRef]

Ren, D. M.

Riedle, E.

Schober, A. M.

Smith, A. V.

Solcia, C.

Szabo, G.

Z. Bor, B. Racz, G. Szabo, M. Hilbert, and H. A. Hazim, “Femtosecond pulse front tilt caused by angular-dispersion,” Opt. Eng. 32, 2501–2504 (1993).
[CrossRef]

G. Szabo and Z. Bor, “Broadband frequency doubler for femtosecond pulses,” Appl. Phys. B 50, 51–54 (1990).
[CrossRef]

Trapani, P. D.

Trebino, R.

Walmsley, I. A.

C. Dorrer, E. M. Kosik, and I. A. Walmsley, “Spatio-temporal characterization of ultrashort optical pulses using two-dimensional shearing interferometry,” Appl. Phys. B 74, s209–s217 (2002).
[CrossRef]

Wang, H.

H. Liu, W. Zhao, Y. Yang, H. Wang, Y. Wang, and G. Chen, “Matching of both group-velocity and pulse-front for ultrabroadband three-wave-mixing with noncollinear angularly dispersed geometry,” Appl. Phys. B 82, 585–594 (2006).
[CrossRef]

Wang, Y.

H. Liu, W. Zhao, Y. Yang, H. Wang, Y. Wang, and G. Chen, “Matching of both group-velocity and pulse-front for ultrabroadband three-wave-mixing with noncollinear angularly dispersed geometry,” Appl. Phys. B 82, 585–594 (2006).
[CrossRef]

Yang, Y.

H. Liu, W. Zhao, Y. Yang, H. Wang, Y. Wang, and G. Chen, “Matching of both group-velocity and pulse-front for ultrabroadband three-wave-mixing with noncollinear angularly dispersed geometry,” Appl. Phys. B 82, 585–594 (2006).
[CrossRef]

Zeek, E.

Zhang, L. Y.

J. J. Huang and L. Y. Zhang, “Transformation of few-cycle ultrashort pulsed Gaussian beams by an angular disperser,” J. Phys. B 43, 175601 (2010).
[CrossRef]

Zhang, T. R.

Zhao, W.

H. Liu, W. Zhao, Y. Yang, H. Wang, Y. Wang, and G. Chen, “Matching of both group-velocity and pulse-front for ultrabroadband three-wave-mixing with noncollinear angularly dispersed geometry,” Appl. Phys. B 82, 585–594 (2006).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (3)

H. Liu, W. Zhao, Y. Yang, H. Wang, Y. Wang, and G. Chen, “Matching of both group-velocity and pulse-front for ultrabroadband three-wave-mixing with noncollinear angularly dispersed geometry,” Appl. Phys. B 82, 585–594 (2006).
[CrossRef]

G. Szabo and Z. Bor, “Broadband frequency doubler for femtosecond pulses,” Appl. Phys. B 50, 51–54 (1990).
[CrossRef]

C. Dorrer, E. M. Kosik, and I. A. Walmsley, “Spatio-temporal characterization of ultrashort optical pulses using two-dimensional shearing interferometry,” Appl. Phys. B 74, s209–s217 (2002).
[CrossRef]

IEEE J. Quantum Electron. (3)

O. E. Martinez, “Matrix formalism for pulse compressors,” IEEE J. Quantum Electron. 24, 2530–2536 (1988).
[CrossRef]

A. G. Kostenbauder, “Ray-pulse matrices: a rational treatment for dispersive optical Systems,” IEEE J. Quantum Electron. 26, 1148–1157 (1990).
[CrossRef]

O. E. Martinez, “Achromatic phase matching for second harmonic generation of femtosecond pulses,” IEEE J. Quantum Electron. 25, 2464–2468 (1989).
[CrossRef]

J. Opt. Soc. Am. B (4)

J. Phys. B (1)

J. J. Huang and L. Y. Zhang, “Transformation of few-cycle ultrashort pulsed Gaussian beams by an angular disperser,” J. Phys. B 43, 175601 (2010).
[CrossRef]

Opt. Commun. (1)

O. E. Martinez, “Pulse distortions in tilted pulse schemes for ultrashort pulses,” Opt. Commun. 59, 229–232 (1986).
[CrossRef]

Opt. Eng. (1)

Z. Bor, B. Racz, G. Szabo, M. Hilbert, and H. A. Hazim, “Femtosecond pulse front tilt caused by angular-dispersion,” Opt. Eng. 32, 2501–2504 (1993).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Opt. Quantum Electron. (1)

J. Hebling, “Derivation of the pulse front tilt caused by angular dispersion,” Opt. Quantum Electron. 28, 1759–1763 (1996).
[CrossRef]

Other (2)

T. Kobayashi, Femtosecond optical frequency comb: Principle, Operation and Applications (Springer, 2005).

R. W. Boyd, Nonlinear Optics (Elsevier, 2008).

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Figures (6)

Fig. 1.
Fig. 1.

Demonstration of PFM-SHG for ultrashort pulses (shown as narrow envelopes) with a tilting angle ψ in a FH pulse and a lateral walk-off angle ρ in a generated SH pulse, where the PFM denotes a match of their group velocities in the propagation direction upon neglecting diffraction and dispersion.

Fig. 2.
Fig. 2.

Schematic diagram of a setup of APM-SHG of ultrashort pulses, where G1, G2 are gratings and L1, L2 are thin lenses together with a nonlinear crystal (NC). The beam waist radii and q parameters are labeled in three zones, which will be used in the following paragraph.

Fig. 3.
Fig. 3.

Sampled rays with different angular frequencies associated with two coordinate systems, where the grating direction denotes the orientation of optical axis.

Fig. 4.
Fig. 4.

Evaluation of the efficiency factor D(s,d) upon parameter s for several values of d.

Fig. 5.
Fig. 5.

Evaluation of the efficiency factor D(s,d) upon parameter d, where three sampled points are shown as A for 100 fs, B for 50 fs, and C for 30 fs of pulse width.

Fig. 6.
Fig. 6.

Illustration of SHG of ultrashort pulses with varying PFT due to diffraction. Two SH pulse envelopes generated at A and B, respectively, are drawn with a schematic compensating system which separates FH and SH pulses, but generally is unable to rectify both of the two pulses at the same time.

Equations (57)

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Δk2k1k2=0,
Δk(Ω,θ)=2k1(Ω)k2(Ω,θ)=0.
βdθdω=Δkω(Δkθ)1.
Δkω=2δu,δuu21u11,
Δkθ=k2θ=2Ωcne1neneθ=k2ρ,
β(0)=2δuk2ρ|ω=0=δuk1ρ|ω=0.
(xθtΩ)out=(AB0ECD0FGH1I0001)(xθtΩ)in,
FθoutΩin=cΩ0GA=cΩ0toutxout,
F=θoutΩin=nedθdω|ω=0.
tanψlimΔt0ΔzΔx=zx.
zx=zttx=u1tx.
tanψ=u1δuρ.
u1cosψ=u2cos(ψρ),
tanψ=k1u1β(0).
(Δ+2ρj2xz)E˜j+kj2E˜j=Ωj2ε0c2P˜jNL,
E˜j(r,z,ωj)=E0A˜j(r,z,ωj)exp(iΩjtikj0z)
Aj(r,z,t)=A˜j(r,z,ωj)exp[iωjtikj0(zz)]dωj,
PjNL(r,z,t)=P˜jNL(r,z,ωj)exp[iωjtikj0(zz)]dωj,
Ej(r,z,t)=Ej0Aj(r,z,t)exp(iΩjtikj0z)
2kj0i[z+(ρj+tanΔθ)x]A˜j+(kj2kj02+Δ)A˜j=Ωj2ε0Ej0c2P˜jNL,
[z+(ρj+βjωj)x+iΔ2kj0]A˜j+i(ωjvj+αjωj22)A˜j=iΩj2P˜jNL2ε0kj0Ej0c2.
1/vjdkj/dωj,αjd2kj/dωj2+1/(uj2kj0).
(z+β1ω1x+i2k102x2)A˜1+i(ω1v1+α1ω122)A˜1=iσA1*A2˜exp(iΔkz),
[z+(ρ+β2ω2)x+i2k202x2]A˜2+i(ω2v2+α2ω222)A˜2=iσA˜12exp(iΔkz),
zz=z(cosΔθ1)xsinΔθ12(βjωj)2z+βjωjx
B˜j(x,z,ωj)A˜j(x,z,ωj)exp(12ikj0βj2ωj2z),
(f*g)(x)f(y)g(xy)dy,
(z+β1ω1x+i2k102x2)B˜1+i(ω1v1+η1ω122)B˜1=iσ[B˜2(ω)exp(iΔkβ2xω)]*B˜1*exp[iΔkz],
[z+(ρ+β2ω2)x+i2k202x2]B˜2+i(ω2v2+η2ω222)B˜2=iσB˜1*B˜1exp(iΔkβ2xω2)
(ziβ12τx+i2k102x2)B1iη122τ2B1=iσB1*B2(,τ+Δkβ2x)exp(iΔkz),
[z+(ρiβ2τ)x+i2k202x2]B2+δvB2τiη222B2τ2=iσB12(,τΔkβ2x)exp(iΔkz),
Bj(x,z,τ)=B˜j(x,z,ωj)exp(iωjτ)dω.
A1z+1u1A1t+i2k102A1x2iα122A1t2=iσA1*A2exp(iΔkz),
A2z+1u2A2t+i2k202A2x2+ρA2xiα222A2t2=iσA12exp(iΔkz),
A2z+i2k202A2xw2iα222A2τ2=iσA12exp(iΔkz).
A^2(kx,z,ω)=12πA2(xw,z,τ)exp(ikxxwiωτ)dxwdτ,
A2(xw,z,τ)=A^2(kx,z,ω)exp(iωτikxxw)dkxdω.
A2(x,L,τ)=iσQ1P0q1(L)0Lexp[ik0xw2/q1(L)]q1(z)p1(z)[p1(L)+2iΔα(Lz)]×exp(2{τ2(z)xwk0β0[1LzCq1(L)]}2q1(L)[p1(L)+2iΔα(Lz)])dz,
MII=(β02βG2b0β02b1βG2zeff002b1βG2β0f22b0b1βG2β002πβG22β0πλ02b0β0πb1λ0zeff100001),
G2=(m200001/m202πβG24πm2βG2/λ00100001).
A2(x,0,t)=0L(Q1Q2q1(z)p1(z)p2(z))1/2exp{2[tΔt(z)]2p2(z)}×iσ0P0q21/2exp{ik0[x+ξ1(z)]2+ξ2(z)q2}dz,
σ0=σ(Toutw1n1/w2)1/2,ξ0=1/[Cq1(L)+D],
ξ1(z)=ρ(Lz)ξ0,ξ2(z)=ρ2(Lz)2(Aξ0ξ02),
p2(z)=p1(zC)+2i[α1(LzC)+Δα(Lz)]
A2(x,0,t)=iσ0P0(Q1Q2q2p2)1/2exp(2t2p2)0L[p1(z)]1/2×[q1(z)]1/2exp{ik0[x+ξ1(z)]2+ξ2(z)q2}dz.
A2(x,0,t)=iσ0L(Q1P0q2p2)1/2exp(2t2p2ik0x2q2)×2{1+2iP0[η1(LzC)+γC]}1/2+{12iP0(η1zCγC)}1/2.
D(s,d)|2[1+id(1s)]1/2+[1isd]1/2|2
G1=(m100001/m102πβG12πm1βG1/λ00100001),
MI=(m1a1a0zeff/m102πzeffβ0m1/(n1f1)a0/(n1m1)02πβ12πm1βG1/λ0012πα1z0001),
Ein=E0(Q0P0q0p0)1/2exp(ik0x22q0τ2p0),
E1=E0(im1Q0P02πλ0a0zeffzα1q0p0)1/2exp(ik0xin22q0τin2p0)×exp[iπλ0(xinxτinτ)Σ(xinxτinτ)]dxindτin,
Σ=(m12(2πβG12zλ0α1a1zeffa0)m1a0zeffm1βG1zα1m1a0zeff1n1zeff0m1βG1zα10λ02πα1z),
E1=E10(Q1P0q1p1)1/2exp[ik0x22q1(τεx)2p1],
εk0β0(1zeff/q1),γCγ0+α1zC.
p1(z)P0+2i(γC+n1η1zeffk0β02zeff2/q1).
tanψ=k0β0u1{1+zeff[Im(p1)ZR1Re(p1)1R1]},
1z1+1z=1f1.

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