Abstract

We derive general boundary conditions for optical rectification of Gaussian beams and discuss the effect on cascaded optical rectification and the linear electro-optic effect as well as terahertz generation. These conditions are applied to degenerate four-wave mixing in a novel geometry in which we experimentally demonstrate, for the first time to our knowledge, that the combination of optical rectification of a single pump beam and the linear electro-optic effect produces a refractive-index grating in the transparent spectral region of the inorganic crystal KNbO3. The intensity of a probe beam diffracted by this grating was determined and corresponds to the theoretically expected result obtained from the known material parameters.

© 2001 Optical Society of America

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  1. M. Bass, P. A. Franken, J. F. Ward, and G. Weinreich, “Optical rectification,” Phys. Rev. Lett. 9, 446–448 (1962).
    [CrossRef]
  2. B. N. Morozov, “Optical rectification in ammonium dihydrogen phosphate and ammonium oxalate,” Opt. Spectrosc. 27, 164–166 (1969).
  3. P. N. Zanadvorov, V. M. Moldavskaya, and V. M. Zhukov, “Optical rectification and piezoelectric effect in a lithium niobate crystal,” Sov. Phys. Solid State 11, 1156–1157 (1969).
  4. M. Zgonik and P. Günter, “Cascaded nonlinearities in optical four-wave mixing,” J. Opt. Soc. Am. B 13, 570–576 (1996).
    [CrossRef]
  5. I. Biaggio, “Nonlocal contributions to degenerate four-wave mixing in noncentrosymmetric materials,” Phys. Rev. Lett. 82, 193–196 (1999).
    [CrossRef]
  6. Ch. Bosshard, I. Biaggio, St. Fischer, S. Follonier, and P. Günter, “Cascaded contributions to degenerate four-wave mixing in an acentric organic crystal,” Opt. Lett. 24, 196–198 (1999).
    [CrossRef]
  7. Ch. Bosshard, R. Spreiter, M. Zgonik, and P. Günter, “Kerr nonlinearity via cascaded optical rectification and the linear electro-optic effect,” Phys. Rev. Lett. 74, 2816–2819 (1995).
    [CrossRef] [PubMed]
  8. G. V. Krivoshchekov and V. I. Stroganov, “Effect of double refraction in crystals on the optical rectification effect,” Opt. Spectrosc. 28, 653–654 (1970).
  9. C. Flytzanis and N. Bloembergen, “Infrared dispersion of third-order susceptibilities in dielectrics: retardation effects,” Prog. Quantum Electron. 4, 271–300 (1977).
    [CrossRef]
  10. M. C. Nuss and J. Orenstein, “Terahertz time-domain spectroscopy,” in Millimeter and Submillimeter Wave Spectroscopy of Solids, G. Grüner, eds., Vol. 74 of Topics in Applied Physics (Springer Verlag, Berlin, 1998), pp. 7–50.
    [CrossRef]
  11. R. W. Hellwarth, “Generation of time-reversed wave fronts by nonlinear refraction,” J. Opt. Soc. Am. 67, 1–3 (1977).
    [CrossRef]
  12. F. P. Strohkendl, L. R. Dalton, R. W. Hellwarth, H. W. Sarkas, and Z. H. Kafafi, “Phase-mismatched degenerate four-wave mixing: complex third-order susceptibility tensor elements of C60 at 768 nm,” J. Opt. Soc. Am. B 14, 92–98 (1997).
    [CrossRef]
  13. M. Zgonik, R. Schlesser, I. Biaggio, E. Voit, J. Tscherry, and P. Günter, “Materials constant of KNbO3 relevant for electro- and acousto-optics,” J. Appl. Phys. 74, 1287–1297 (1993).
    [CrossRef]

1999 (2)

1997 (1)

1996 (1)

1995 (1)

Ch. Bosshard, R. Spreiter, M. Zgonik, and P. Günter, “Kerr nonlinearity via cascaded optical rectification and the linear electro-optic effect,” Phys. Rev. Lett. 74, 2816–2819 (1995).
[CrossRef] [PubMed]

1993 (1)

M. Zgonik, R. Schlesser, I. Biaggio, E. Voit, J. Tscherry, and P. Günter, “Materials constant of KNbO3 relevant for electro- and acousto-optics,” J. Appl. Phys. 74, 1287–1297 (1993).
[CrossRef]

1977 (2)

C. Flytzanis and N. Bloembergen, “Infrared dispersion of third-order susceptibilities in dielectrics: retardation effects,” Prog. Quantum Electron. 4, 271–300 (1977).
[CrossRef]

R. W. Hellwarth, “Generation of time-reversed wave fronts by nonlinear refraction,” J. Opt. Soc. Am. 67, 1–3 (1977).
[CrossRef]

1970 (1)

G. V. Krivoshchekov and V. I. Stroganov, “Effect of double refraction in crystals on the optical rectification effect,” Opt. Spectrosc. 28, 653–654 (1970).

1969 (2)

B. N. Morozov, “Optical rectification in ammonium dihydrogen phosphate and ammonium oxalate,” Opt. Spectrosc. 27, 164–166 (1969).

P. N. Zanadvorov, V. M. Moldavskaya, and V. M. Zhukov, “Optical rectification and piezoelectric effect in a lithium niobate crystal,” Sov. Phys. Solid State 11, 1156–1157 (1969).

1962 (1)

M. Bass, P. A. Franken, J. F. Ward, and G. Weinreich, “Optical rectification,” Phys. Rev. Lett. 9, 446–448 (1962).
[CrossRef]

Bass, M.

M. Bass, P. A. Franken, J. F. Ward, and G. Weinreich, “Optical rectification,” Phys. Rev. Lett. 9, 446–448 (1962).
[CrossRef]

Biaggio, I.

I. Biaggio, “Nonlocal contributions to degenerate four-wave mixing in noncentrosymmetric materials,” Phys. Rev. Lett. 82, 193–196 (1999).
[CrossRef]

Ch. Bosshard, I. Biaggio, St. Fischer, S. Follonier, and P. Günter, “Cascaded contributions to degenerate four-wave mixing in an acentric organic crystal,” Opt. Lett. 24, 196–198 (1999).
[CrossRef]

M. Zgonik, R. Schlesser, I. Biaggio, E. Voit, J. Tscherry, and P. Günter, “Materials constant of KNbO3 relevant for electro- and acousto-optics,” J. Appl. Phys. 74, 1287–1297 (1993).
[CrossRef]

Bloembergen, N.

C. Flytzanis and N. Bloembergen, “Infrared dispersion of third-order susceptibilities in dielectrics: retardation effects,” Prog. Quantum Electron. 4, 271–300 (1977).
[CrossRef]

Bosshard, Ch.

Ch. Bosshard, I. Biaggio, St. Fischer, S. Follonier, and P. Günter, “Cascaded contributions to degenerate four-wave mixing in an acentric organic crystal,” Opt. Lett. 24, 196–198 (1999).
[CrossRef]

Ch. Bosshard, R. Spreiter, M. Zgonik, and P. Günter, “Kerr nonlinearity via cascaded optical rectification and the linear electro-optic effect,” Phys. Rev. Lett. 74, 2816–2819 (1995).
[CrossRef] [PubMed]

Dalton, L. R.

Fischer, St.

Flytzanis, C.

C. Flytzanis and N. Bloembergen, “Infrared dispersion of third-order susceptibilities in dielectrics: retardation effects,” Prog. Quantum Electron. 4, 271–300 (1977).
[CrossRef]

Follonier, S.

Franken, P. A.

M. Bass, P. A. Franken, J. F. Ward, and G. Weinreich, “Optical rectification,” Phys. Rev. Lett. 9, 446–448 (1962).
[CrossRef]

Günter, P.

Ch. Bosshard, I. Biaggio, St. Fischer, S. Follonier, and P. Günter, “Cascaded contributions to degenerate four-wave mixing in an acentric organic crystal,” Opt. Lett. 24, 196–198 (1999).
[CrossRef]

M. Zgonik and P. Günter, “Cascaded nonlinearities in optical four-wave mixing,” J. Opt. Soc. Am. B 13, 570–576 (1996).
[CrossRef]

Ch. Bosshard, R. Spreiter, M. Zgonik, and P. Günter, “Kerr nonlinearity via cascaded optical rectification and the linear electro-optic effect,” Phys. Rev. Lett. 74, 2816–2819 (1995).
[CrossRef] [PubMed]

M. Zgonik, R. Schlesser, I. Biaggio, E. Voit, J. Tscherry, and P. Günter, “Materials constant of KNbO3 relevant for electro- and acousto-optics,” J. Appl. Phys. 74, 1287–1297 (1993).
[CrossRef]

Hellwarth, R. W.

Kafafi, Z. H.

Krivoshchekov, G. V.

G. V. Krivoshchekov and V. I. Stroganov, “Effect of double refraction in crystals on the optical rectification effect,” Opt. Spectrosc. 28, 653–654 (1970).

Moldavskaya, V. M.

P. N. Zanadvorov, V. M. Moldavskaya, and V. M. Zhukov, “Optical rectification and piezoelectric effect in a lithium niobate crystal,” Sov. Phys. Solid State 11, 1156–1157 (1969).

Morozov, B. N.

B. N. Morozov, “Optical rectification in ammonium dihydrogen phosphate and ammonium oxalate,” Opt. Spectrosc. 27, 164–166 (1969).

Sarkas, H. W.

Schlesser, R.

M. Zgonik, R. Schlesser, I. Biaggio, E. Voit, J. Tscherry, and P. Günter, “Materials constant of KNbO3 relevant for electro- and acousto-optics,” J. Appl. Phys. 74, 1287–1297 (1993).
[CrossRef]

Spreiter, R.

Ch. Bosshard, R. Spreiter, M. Zgonik, and P. Günter, “Kerr nonlinearity via cascaded optical rectification and the linear electro-optic effect,” Phys. Rev. Lett. 74, 2816–2819 (1995).
[CrossRef] [PubMed]

Stroganov, V. I.

G. V. Krivoshchekov and V. I. Stroganov, “Effect of double refraction in crystals on the optical rectification effect,” Opt. Spectrosc. 28, 653–654 (1970).

Strohkendl, F. P.

Tscherry, J.

M. Zgonik, R. Schlesser, I. Biaggio, E. Voit, J. Tscherry, and P. Günter, “Materials constant of KNbO3 relevant for electro- and acousto-optics,” J. Appl. Phys. 74, 1287–1297 (1993).
[CrossRef]

Voit, E.

M. Zgonik, R. Schlesser, I. Biaggio, E. Voit, J. Tscherry, and P. Günter, “Materials constant of KNbO3 relevant for electro- and acousto-optics,” J. Appl. Phys. 74, 1287–1297 (1993).
[CrossRef]

Ward, J. F.

M. Bass, P. A. Franken, J. F. Ward, and G. Weinreich, “Optical rectification,” Phys. Rev. Lett. 9, 446–448 (1962).
[CrossRef]

Weinreich, G.

M. Bass, P. A. Franken, J. F. Ward, and G. Weinreich, “Optical rectification,” Phys. Rev. Lett. 9, 446–448 (1962).
[CrossRef]

Zanadvorov, P. N.

P. N. Zanadvorov, V. M. Moldavskaya, and V. M. Zhukov, “Optical rectification and piezoelectric effect in a lithium niobate crystal,” Sov. Phys. Solid State 11, 1156–1157 (1969).

Zgonik, M.

M. Zgonik and P. Günter, “Cascaded nonlinearities in optical four-wave mixing,” J. Opt. Soc. Am. B 13, 570–576 (1996).
[CrossRef]

Ch. Bosshard, R. Spreiter, M. Zgonik, and P. Günter, “Kerr nonlinearity via cascaded optical rectification and the linear electro-optic effect,” Phys. Rev. Lett. 74, 2816–2819 (1995).
[CrossRef] [PubMed]

M. Zgonik, R. Schlesser, I. Biaggio, E. Voit, J. Tscherry, and P. Günter, “Materials constant of KNbO3 relevant for electro- and acousto-optics,” J. Appl. Phys. 74, 1287–1297 (1993).
[CrossRef]

Zhukov, V. M.

P. N. Zanadvorov, V. M. Moldavskaya, and V. M. Zhukov, “Optical rectification and piezoelectric effect in a lithium niobate crystal,” Sov. Phys. Solid State 11, 1156–1157 (1969).

J. Appl. Phys. (1)

M. Zgonik, R. Schlesser, I. Biaggio, E. Voit, J. Tscherry, and P. Günter, “Materials constant of KNbO3 relevant for electro- and acousto-optics,” J. Appl. Phys. 74, 1287–1297 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (2)

Opt. Lett. (1)

Opt. Spectrosc. (2)

G. V. Krivoshchekov and V. I. Stroganov, “Effect of double refraction in crystals on the optical rectification effect,” Opt. Spectrosc. 28, 653–654 (1970).

B. N. Morozov, “Optical rectification in ammonium dihydrogen phosphate and ammonium oxalate,” Opt. Spectrosc. 27, 164–166 (1969).

Phys. Rev. Lett. (3)

I. Biaggio, “Nonlocal contributions to degenerate four-wave mixing in noncentrosymmetric materials,” Phys. Rev. Lett. 82, 193–196 (1999).
[CrossRef]

Ch. Bosshard, R. Spreiter, M. Zgonik, and P. Günter, “Kerr nonlinearity via cascaded optical rectification and the linear electro-optic effect,” Phys. Rev. Lett. 74, 2816–2819 (1995).
[CrossRef] [PubMed]

M. Bass, P. A. Franken, J. F. Ward, and G. Weinreich, “Optical rectification,” Phys. Rev. Lett. 9, 446–448 (1962).
[CrossRef]

Prog. Quantum Electron. (1)

C. Flytzanis and N. Bloembergen, “Infrared dispersion of third-order susceptibilities in dielectrics: retardation effects,” Prog. Quantum Electron. 4, 271–300 (1977).
[CrossRef]

Sov. Phys. Solid State (1)

P. N. Zanadvorov, V. M. Moldavskaya, and V. M. Zhukov, “Optical rectification and piezoelectric effect in a lithium niobate crystal,” Sov. Phys. Solid State 11, 1156–1157 (1969).

Other (1)

M. C. Nuss and J. Orenstein, “Terahertz time-domain spectroscopy,” in Millimeter and Submillimeter Wave Spectroscopy of Solids, G. Grüner, eds., Vol. 74 of Topics in Applied Physics (Springer Verlag, Berlin, 1998), pp. 7–50.
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Resulting electro-static potential (gray scale) and electric field (arrows) of optical rectification of a Gaussian beam that is polarized along the x axis. The circle indicates the beam diameter 2w0.

Fig. 2
Fig. 2

Depolarization factor as a function of the dielectric constant and for the geometric factors K=1 (one-dimensional beam) and K=0.49 (Gaussian beam).

Fig. 3
Fig. 3

(a) Classical geometry of degenerate four-wave mixing for the measurement of the third-order susceptibility χ2323(3)(-ω,-ω, ω, ω). (b) The corresponding phase-matching (PM) condition (Δk=k4+k3-k2-k1=0) for the various third-order nonlinear optical susceptibilities in a birefringent material illustrated with normal surfaces (left) and wave-vector summation (center).

Fig. 4
Fig. 4

(a) Novel geometry for degenerate four-wave mixing. The pump beam produces a polarization grating by optical rectification, which is probed by a second beam incident under the Bragg angle of the grating. (b) The corresponding phase-matching condition (Δk=k4+k3-k2-k1=0) for this experiment illustrated with normal surfaces (left) and wave-vector summation (center).

Fig. 5
Fig. 5

Measurement of the diffracted intensity as a function of the probe-beam delay. The solid curve shows the theoretical behavior assuming an instantaneous response.

Equations (34)

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Pi(t)=Pi0+ε0[χij(1)Ej(t)+χijk(2)Ej(t)Ek(t)+χijkl(3)Ej(t)Ek(t)El(t)+].
E(t)=(1/2)[Eω exp(-iωt)+c.c.].
P(t)=12 k[Pωk exp(-iωkt)+c.c.].
Pi(0)=ε0(1/2)χijk(2)(0, ω,-ω)EjωEkω*
0=divD=div(ε0E(0)+Ptot(0)).
Ptot,i(0)=Pi(0)+ε0χij(1)Ej(0),
0=div(ε0ε_E(0)+P(0)),
Δ1n2ij=rijkEk+RijklEkEl+,
Δ1n2ij=rijkPk(0)ε0(εkk+2)+Ek(0).
P1(0)=(1/2)ε0χ111(0, ω,-ω)|E1(x, y)|2,
rijk(ω)=-2ni2nj2χijk(-ω, ω, 0).
E1(x, y)=E1 exp-x2+y22w0,
P(0)(x, y, z)
=exP(0) exp-x2+y2w02(-L/2zL/2),0(|z|>L/2)
div E(0)=-1ε0ε div P(0).
E(0)=-grad U.
ΔU=1ε0 div P(0)ε=1ε0 2xP(0)εw02exp-x2+y2w0=1ε0ρq.
U=-14πε0  ρq(r)|r-r|d3r,
ρq=2xP(0)εw02exp-x2+y2w0.
U(x, y)=12πε0 --ρq(x, y)×asinhL2(x-x2)+(y-y2)dxdy.
E(0)(x, y)=-12πε0 --(x-x)(y-y) Lx exp(x2+y)w022w02[(x-x)2+(y-y)2]3/21+L24[(x-x)2+(y-y)2] dxdy.
Ex(0)=-Px(0)εε0K,K=0.490,Lw01,
Ptot(0)=ε0χE(0)+P(0)=P(0)(1-K)+Kε=P(0)×Rdepol,
Pi(ω, k4)=ε0(3/2)χijkl(3)(-ω,-ω, ω, ω,-k4,-k3, k2, k1)·Ej*(-ω,-k3)Ek(ω, k2)El(ω, k1),
P2OR(x)=ε0χ232(2)|E3E2|×cos2πλ(n2-n3)x+Δϕ0.
χ2323,eff(3)=χ2323,dir(3)+χ2323,casc(3)=χ2323,dir(3)+23 (χ232(2))2ε2+2×Rdepol,
PiOR(r)=(1/2)ε0χijk(2)(0,-ω, ω)Ej*(-ω,-k2)Ek(ω, k3),
=(1/2)ε0χijk(2)(0,-ω, ω)|Ej(-ω)||Ek(ω)|×cos[(k2-k3)r+Δϕ0],
χijkl,eff(3)=χijkl,dir(3)+23 χikm(2)(-ω, ω, 0)χmjl(2)(-0,-ω, ω)εmm+2×Cm×Rdepo,
Cm=1transversecase-2/εmmlongitudinalcase,
I4(i)=256ω2εo2c4 32χijkl,eff(3)2L2exp-α1(i) L2exp-α2(j) L2exp-α3(k) L2exp-α4(l) L2(n1(i)+1)2(n2(j)+1)2(n3(k)+1)2(n4(l)+1)22I1(j)I2(k)I3(l),
I4=256ω2εo2c4 32χ2323,eff(3)2Leff2×1(n1+1)2(n2+1)2(n3+1)2(n4+1)2I1I2I3,
I4=64ω2εo2c4 32χ2323,eff(3)2×Leff2 1(n45°+1)4(n2+1)2(n3+1)2I1Ip2,
η(τ)=E4(τ)E1=AE1=AE1 I4(t, τ)dt=256ω2εo2c4(n45°+1)4(n2+1)2(n3+1)23π3wo4τp2×Leff232χ2323,eff(3)2Ep2 exp-2τ23τp2.

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