Abstract

By numerically solving the nonlinear field equations, we simulate second-harmonic generation by laser pulses within a nonlinear medium without making the usual slowly-varying-amplitude approximation, an approximation which may fail when laser pulses of moderate intensity or ultrashort duration are used to drive a nonlinear process. Under these conditions we show that a backward-traveling, second-harmonic wave is created, and that the magnitude of this wave is indicative of the breakdown of the slowly-varying-amplitude approximation. Conditions necessary for experimental detection of this wave are discussed.

© 2003 Optical Society of America

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References

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  1. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
    [CrossRef]
  2. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).
  3. R. W. Boyd, Nonlinear Optics (Academic, Boston, Mass., 1992).
  4. S. E. Harris, “Proposed backward wave oscillation in the infrared,” Appl. Phys. Lett. 9, 114–116 (1966).
    [CrossRef]
  5. T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
    [CrossRef]
  6. L. W. Casperson, “Field-equation approximations and amplification in high gain lasers: numeric results,” Phys. Rev. A 44, 3291–3304 (1991).
    [CrossRef] [PubMed]
  7. L. W. Casperson, “Field-equation approximations and amplification in high gain lasers: analytic results,” Phys. Rev. A 44, 3305–3316 (1991).
    [CrossRef] [PubMed]
  8. L. W. Casperson, “Field-equation approximation and the dynamics of high-gain lasers,” Phys. Rev. A 43, 5057–5067 (1991).
    [CrossRef] [PubMed]
  9. S. Hughes, “Breakdown of the area theorem: carrier-wave rabi flopping of femtosecond optical pulses,” Phys. Rev. Lett. 81, 3363–3366 (1998).
    [CrossRef]
  10. F. Bloch and A. J. Siegert, “Magnetic resonance for nonrotating fields,” Phys. Rev. 57, 522–527 (1940).
    [CrossRef]
  11. For a discussion of the rotating-wave approximation in the context of optical phenomena see L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (New York, Dover, 1987), Chap. 2.
  12. Y. J. Ding, J. U. Kang, and J. B. Khurgin, “Theory of backward second-harmonic and third-harmonic generation using laser pulses in quasi-phase-matched second-order nonlinear medium,” IEEE J. Quantum Electron. 34, 966–974 (1998).
    [CrossRef]
  13. X. Gu, R. Y. Korotkov, Y. J. Ding, J. U. Kang, and J. B. Khurgin, “Backward second-harmonic generation in periodically poled lithium niobate,” J. Opt. Soc. Am. B 15, 1561–1566 (1998).
    [CrossRef]
  14. X. H. Gu, M. Makarov, Y. J. Ding, J. B. Khurgin, and W. P. Risk, “Backward second-harmonic and third-harmonic generation in a periodically poled potassium titanyl phosphate waveguide,” Opt. Lett. 24, 127–129 (1999).
    [CrossRef]

1999 (1)

1998 (3)

Y. J. Ding, J. U. Kang, and J. B. Khurgin, “Theory of backward second-harmonic and third-harmonic generation using laser pulses in quasi-phase-matched second-order nonlinear medium,” IEEE J. Quantum Electron. 34, 966–974 (1998).
[CrossRef]

X. Gu, R. Y. Korotkov, Y. J. Ding, J. U. Kang, and J. B. Khurgin, “Backward second-harmonic generation in periodically poled lithium niobate,” J. Opt. Soc. Am. B 15, 1561–1566 (1998).
[CrossRef]

S. Hughes, “Breakdown of the area theorem: carrier-wave rabi flopping of femtosecond optical pulses,” Phys. Rev. Lett. 81, 3363–3366 (1998).
[CrossRef]

1997 (1)

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[CrossRef]

1991 (3)

L. W. Casperson, “Field-equation approximations and amplification in high gain lasers: numeric results,” Phys. Rev. A 44, 3291–3304 (1991).
[CrossRef] [PubMed]

L. W. Casperson, “Field-equation approximations and amplification in high gain lasers: analytic results,” Phys. Rev. A 44, 3305–3316 (1991).
[CrossRef] [PubMed]

L. W. Casperson, “Field-equation approximation and the dynamics of high-gain lasers,” Phys. Rev. A 43, 5057–5067 (1991).
[CrossRef] [PubMed]

1966 (1)

S. E. Harris, “Proposed backward wave oscillation in the infrared,” Appl. Phys. Lett. 9, 114–116 (1966).
[CrossRef]

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

1940 (1)

F. Bloch and A. J. Siegert, “Magnetic resonance for nonrotating fields,” Phys. Rev. 57, 522–527 (1940).
[CrossRef]

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Bloch, F.

F. Bloch and A. J. Siegert, “Magnetic resonance for nonrotating fields,” Phys. Rev. 57, 522–527 (1940).
[CrossRef]

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Brabec, T.

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[CrossRef]

Casperson, L. W.

L. W. Casperson, “Field-equation approximations and amplification in high gain lasers: numeric results,” Phys. Rev. A 44, 3291–3304 (1991).
[CrossRef] [PubMed]

L. W. Casperson, “Field-equation approximation and the dynamics of high-gain lasers,” Phys. Rev. A 43, 5057–5067 (1991).
[CrossRef] [PubMed]

L. W. Casperson, “Field-equation approximations and amplification in high gain lasers: analytic results,” Phys. Rev. A 44, 3305–3316 (1991).
[CrossRef] [PubMed]

Ding, Y. J.

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Gu, X.

Gu, X. H.

Harris, S. E.

S. E. Harris, “Proposed backward wave oscillation in the infrared,” Appl. Phys. Lett. 9, 114–116 (1966).
[CrossRef]

Hughes, S.

S. Hughes, “Breakdown of the area theorem: carrier-wave rabi flopping of femtosecond optical pulses,” Phys. Rev. Lett. 81, 3363–3366 (1998).
[CrossRef]

Kang, J. U.

X. Gu, R. Y. Korotkov, Y. J. Ding, J. U. Kang, and J. B. Khurgin, “Backward second-harmonic generation in periodically poled lithium niobate,” J. Opt. Soc. Am. B 15, 1561–1566 (1998).
[CrossRef]

Y. J. Ding, J. U. Kang, and J. B. Khurgin, “Theory of backward second-harmonic and third-harmonic generation using laser pulses in quasi-phase-matched second-order nonlinear medium,” IEEE J. Quantum Electron. 34, 966–974 (1998).
[CrossRef]

Khurgin, J. B.

Korotkov, R. Y.

Krausz, F.

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[CrossRef]

Makarov, M.

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Risk, W. P.

Siegert, A. J.

F. Bloch and A. J. Siegert, “Magnetic resonance for nonrotating fields,” Phys. Rev. 57, 522–527 (1940).
[CrossRef]

Appl. Phys. Lett. (1)

S. E. Harris, “Proposed backward wave oscillation in the infrared,” Appl. Phys. Lett. 9, 114–116 (1966).
[CrossRef]

IEEE J. Quantum Electron. (1)

Y. J. Ding, J. U. Kang, and J. B. Khurgin, “Theory of backward second-harmonic and third-harmonic generation using laser pulses in quasi-phase-matched second-order nonlinear medium,” IEEE J. Quantum Electron. 34, 966–974 (1998).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Lett. (1)

Phys. Rev. (2)

F. Bloch and A. J. Siegert, “Magnetic resonance for nonrotating fields,” Phys. Rev. 57, 522–527 (1940).
[CrossRef]

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Phys. Rev. A (3)

L. W. Casperson, “Field-equation approximations and amplification in high gain lasers: numeric results,” Phys. Rev. A 44, 3291–3304 (1991).
[CrossRef] [PubMed]

L. W. Casperson, “Field-equation approximations and amplification in high gain lasers: analytic results,” Phys. Rev. A 44, 3305–3316 (1991).
[CrossRef] [PubMed]

L. W. Casperson, “Field-equation approximation and the dynamics of high-gain lasers,” Phys. Rev. A 43, 5057–5067 (1991).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

S. Hughes, “Breakdown of the area theorem: carrier-wave rabi flopping of femtosecond optical pulses,” Phys. Rev. Lett. 81, 3363–3366 (1998).
[CrossRef]

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[CrossRef]

Other (3)

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

R. W. Boyd, Nonlinear Optics (Academic, Boston, Mass., 1992).

For a discussion of the rotating-wave approximation in the context of optical phenomena see L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (New York, Dover, 1987), Chap. 2.

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Figures (3)

Fig. 1
Fig. 1

Box represents the nonlinear medium and the fundamental wave is incident from the left. As discussed in the text, the eight boundary conditions applied for the numeric analysis are (1) I1-(L)=0, (2) I2-(L)=0, (3) [I2+(L)]2=1-[I1+(L)]2, (4) I2+(0)=0, (5) ϕ1(L)=0, (6) ϕ2(L)=0, (7) θ=2ϕ1-ϕ2=?, and (8) I1+(L)=? A question mark denotes a boundary condition that is unknown and must be found self-consistently.

Fig. 2
Fig. 2

(a) Forward- and (b) backward-traveling waves plotted as a function of ξ for a1=0.01. The growth of a backward-traveling, second-harmonic wave is evident in (b) as is the existence of a backward-traveling wave at the fundamental frequency. If the nonlinear medium is taken to be lithium niobate (deff5×10-8 esu) then for typical laser parameters (ω1=2.36×1015 Hz, I=1012W/cm2) the unitless length ξ=3 corresponds to a crystal length of 18 μm. Note that if the the SVAA approximation is invoked, no backward-traveling waves exist.

Fig. 3
Fig. 3

Backward SHG efficiency versus parameter a1. At pump-laser intensities above 1010W/cm2 the backward second harmonic can be detected. The calculation is truncated for laser intensities greater than 1013W/cm2 since damage of the nonlinear medium would be expected for higher intensities. The intensity axis is calculated assuming lithium niobate as the nonlinear medium.

Equations (44)

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××E˜+n2c22E˜t2=-4πc22P˜NLt2.
E˜1(z,t)=E1(z)exp(-iω1t)+c.c.,
E1(z)=A1(z)exp(ik1z).
P1(z)=4deffE1*(z)E2(z),
P2(z)=2deffE12(z).
d2A1dz2+2ik1dA1dz=-16deffπω12c2 A1*A2exp(-iΔkz),
d2A2dz2+2ik2dA2dz=-8deffπω22c2 A12exp(iΔkz),
l=18πω1deffn12n2c32πI1/2,
-a1u1+a1u1(ϕ1)2+u1ϕ1
=u1u2cos(Δsξ+2ϕ1-ϕ2),
2a1u1ϕ1+a1u1ϕ1+u1
=u1u2sin(Δsξ+2ϕ1-ϕ2),
-a2u2+a2u2(ϕ2)2+u2ϕ2
=u12cos(Δsξ+2ϕ1-ϕ2),
2a2u2ϕ2+a2u2ϕ2+u2
=-u12sin(Δsξ+2ϕ1-ϕ2),
E˜(z, t)z=-1cB˜(z, t)t,
E˜(z, t)z=iωμ1c H˜(z, t),
H˜i(z, t)=1μ11/2Hi(z)exp(-iωit)=1μ11/2Mi(z)exp(ikiz)exp(-iωit).
2aA(ξ)+iA(ξ)=iM(ξ).
2aAr(ξ)+2aiAi(ξ)+iAr(ξ)-Ai(ξ)
=iMr(ξ)-Mi(ξ).
Mr(ξ)=2aAi(ξ)+Ar(ξ),
Mi(ξ)=-2aAr(ξ)+Ai(ξ),
E˜(z,t)=E+(z)exp(ikz-iωt)+E-(z)exp(-ikz-iωt)=[E+(z)+E-(z)exp(-2ikz)]exp(ikz-iωt).
H˜(z, t)=1μ11/2[H+(z)+H-(z)exp(-2ikz)]×exp(ikz-iωt).
A(z)=E+(z)+E-(z)exp(-2ikz),
M(z)=H+(z)+H-(z)exp(-2ikz).
E+(z)=H+(z),
E-(z)=-H-(z),
A+(z)=M+(z),
A-(z)=-M-(z).
A(z)=A+(z)+A-(z)exp(-2ikz),
M(z)=A+(z)-A-(z)exp(-2ikz).
A+(z)=12 [A(z)+M(z)],
A-(z)=12 [A(z)-M(z)]exp(2ikz),
M+(z)=12 [M(z)+A(z)],
M-(z)=12 [M(z)-A(z)]exp(2ikz).
I±=c4π (Ar±Mr±+Ai±Mi±).
I+(ξ)=c4π14 [Ar(ξ)+Mr(ξ)]2+14 [Ai(ξ)+Mi(ξ)]2,
I-(ξ)=c4π14 [Ar(ξ)-Mr(ξ)]2+14 [Ai(ξ)-Mi(ξ)]2.
I+(ξ)=c4π {[Ar(ξ)+aAi(ξ)]2+[Ai(ξ)-aAr(ξ)]2},
I-(ξ)=c4π {[aAi(ξ)]2+[aAr(ξ)]2}.
I=n14n2c32π3deff2 a12,

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