Abstract

New techniques to produce spatiotemporal phase modulation without the use of electro-optic devices are proposed and discussed. By using the nonlinear second-order effect in crystal it is possible to change the amplitude modulation of a pump wave into the phase of a signal wave. To that end, we propose the use of a well-known cascading configuration for which the phase mismatch is high. Analytical results of spatial and temporal incoherent phase modulation are developed with the correlation function formalisms. Furthermore, highly accurate expansions of signal phase and intensity are derived. The effects of group-velocity difference, group-velocity dispersion, and diffraction on the change of amplitude into phase modulation are studied. Finally, an experimental demonstration of a KDP crystal with a sinusoidal pump modulation that creates sinusoidal phase modulation is proposed.

© 2000 Optical Society of America

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References

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  1. M. Andre, “Megajoule solid state laser for ICF applications,” in Technical Committee Meeting on Drivers and Ignition Facilities for Inertial Fusion, J. Coutant, ed., Proceedings of the International Atomic Energy Agency (Commissariat à l’Ènergie Atomique/Direction des Applications Militaires, Limeil-Valenton, France, 1995), pp. 77–78.
  2. J. E. Rothenberg, “Comparison of beam-smoothing methods for direct-drive inertial confinement fusion,” J. Opt. Soc. Am. B 14, 1664–1671 (1997).
    [CrossRef]
  3. R. H. Lehmberg and S. P. Obenschain, “Use of induced spatial incoherency for uniform illumination,” Opt. Commun. 46, 27–31 (1983).
    [CrossRef]
  4. J. Garnier, L. Videau, C. Gouedard, and A. Migus, “Statistical analysis for beam smoothing and some applications,” J. Opt. Soc. Am. A 14, 1928–1937 (1997).
    [CrossRef]
  5. D. Veron, G. Thiell, and C. Gouedard, “Optical smoothing of the high power laser Phebus Nd-glass laser using the multimode optical fiber technique,” Opt. Commun. 97, 259–271 (1993).
    [CrossRef]
  6. J. E. Rothenberg, “Two-dimensional smoothing by spectral dispersion for direct-drive inertial confinement fusion,” in Solid State Lasers for Application to Inertial Confinement Fusion (ICF), W. F. Krupke, ed., Proc. SPIE 2633, 634–644 (1995).
    [CrossRef]
  7. J. Garnier, J. P. Fouque, L. Videau, C. Gouédard, and A. Migus, “Amplification of broadband incoherent light in homogeneously broadened media in the presence of Kerr nonlinearity,” J. Opt. Soc. Am. B 14, 2563–2569 (1997).
    [CrossRef]
  8. G. Stegeman, M. Shiek-Bahae, E. Van Stryland, and G. Assanto, “Large nonlinear phase shifts in second-order nonlinear-optical processes,” Opt. Lett. 18, 13–15 (1993).
    [CrossRef] [PubMed]
  9. D. C. Hutchings, J. S. Aitchison, and C. N. Ironside, “All-optical switching based on nondegenerate phase shifts from a cascaded second-order nonlinearity,” Opt. Lett. 18, 793–795 (1993).
    [CrossRef] [PubMed]
  10. I. Buchvarov, S. Saltiel, Ch. Iglev, and K. Koynov, “Intensity dependent change of the polarization state as a result of non-linear phase shift in type II frequency doubling crystals,” Opt. Commun. 141, 173–179 (1997).
    [CrossRef]
  11. R. DeSalvo, D. J. Hagan, G. Stegeman, and E. W. Van Stryland, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett. 18, 574–576 (1993).
  12. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  13. D. Middleton, Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).
  14. J. T. Manassah, “Self-phase modulation of incoherent light revisited,” Opt. Lett. 16, 1439–1441 (1991).
    [CrossRef]
  15. J. W. Goodman, “Statistical properties of speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed., Vol. 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1984), pp. 9–75.
  16. A. Yariv, Quantum Electronics (Wiley, New York, 1988).
  17. J. T. Manassah, “Phase modulation in second-order nonlinear-optical processes,” Phys. Rev. A 42, 4085–4101 (1990).
    [CrossRef]
  18. A. C. L. Boscheron, “Etude de nouvelles configurations de conversion de frequence pour l’optimisation des lasers de haute puissance,” Ph.D. dissertation (University of Paris XI, Orsay, France, 1996).
  19. A. Kobyakov and F. Lederer, “Cascading of quadratic nonlinearities: an analytical study,” Phys. Rev. A 54, 3455–3471 (1996).
    [CrossRef] [PubMed]
  20. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).
  21. P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists (Springer-Verlag, Berlin, 1954).
  22. A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures (North-Holland, Amsterdam, 1978).

1997 (4)

1996 (1)

A. Kobyakov and F. Lederer, “Cascading of quadratic nonlinearities: an analytical study,” Phys. Rev. A 54, 3455–3471 (1996).
[CrossRef] [PubMed]

1995 (1)

J. E. Rothenberg, “Two-dimensional smoothing by spectral dispersion for direct-drive inertial confinement fusion,” in Solid State Lasers for Application to Inertial Confinement Fusion (ICF), W. F. Krupke, ed., Proc. SPIE 2633, 634–644 (1995).
[CrossRef]

1993 (4)

1991 (1)

1990 (1)

J. T. Manassah, “Phase modulation in second-order nonlinear-optical processes,” Phys. Rev. A 42, 4085–4101 (1990).
[CrossRef]

1983 (1)

R. H. Lehmberg and S. P. Obenschain, “Use of induced spatial incoherency for uniform illumination,” Opt. Commun. 46, 27–31 (1983).
[CrossRef]

Aitchison, J. S.

Assanto, G.

Buchvarov, I.

I. Buchvarov, S. Saltiel, Ch. Iglev, and K. Koynov, “Intensity dependent change of the polarization state as a result of non-linear phase shift in type II frequency doubling crystals,” Opt. Commun. 141, 173–179 (1997).
[CrossRef]

DeSalvo, R.

Fouque, J. P.

Garnier, J.

Gouedard, C.

J. Garnier, L. Videau, C. Gouedard, and A. Migus, “Statistical analysis for beam smoothing and some applications,” J. Opt. Soc. Am. A 14, 1928–1937 (1997).
[CrossRef]

D. Veron, G. Thiell, and C. Gouedard, “Optical smoothing of the high power laser Phebus Nd-glass laser using the multimode optical fiber technique,” Opt. Commun. 97, 259–271 (1993).
[CrossRef]

Gouédard, C.

Hagan, D. J.

Hutchings, D. C.

Iglev, Ch.

I. Buchvarov, S. Saltiel, Ch. Iglev, and K. Koynov, “Intensity dependent change of the polarization state as a result of non-linear phase shift in type II frequency doubling crystals,” Opt. Commun. 141, 173–179 (1997).
[CrossRef]

Ironside, C. N.

Kobyakov, A.

A. Kobyakov and F. Lederer, “Cascading of quadratic nonlinearities: an analytical study,” Phys. Rev. A 54, 3455–3471 (1996).
[CrossRef] [PubMed]

Koynov, K.

I. Buchvarov, S. Saltiel, Ch. Iglev, and K. Koynov, “Intensity dependent change of the polarization state as a result of non-linear phase shift in type II frequency doubling crystals,” Opt. Commun. 141, 173–179 (1997).
[CrossRef]

Lederer, F.

A. Kobyakov and F. Lederer, “Cascading of quadratic nonlinearities: an analytical study,” Phys. Rev. A 54, 3455–3471 (1996).
[CrossRef] [PubMed]

Lehmberg, R. H.

R. H. Lehmberg and S. P. Obenschain, “Use of induced spatial incoherency for uniform illumination,” Opt. Commun. 46, 27–31 (1983).
[CrossRef]

Manassah, J. T.

J. T. Manassah, “Self-phase modulation of incoherent light revisited,” Opt. Lett. 16, 1439–1441 (1991).
[CrossRef]

J. T. Manassah, “Phase modulation in second-order nonlinear-optical processes,” Phys. Rev. A 42, 4085–4101 (1990).
[CrossRef]

Migus, A.

Obenschain, S. P.

R. H. Lehmberg and S. P. Obenschain, “Use of induced spatial incoherency for uniform illumination,” Opt. Commun. 46, 27–31 (1983).
[CrossRef]

Rothenberg, J. E.

J. E. Rothenberg, “Comparison of beam-smoothing methods for direct-drive inertial confinement fusion,” J. Opt. Soc. Am. B 14, 1664–1671 (1997).
[CrossRef]

J. E. Rothenberg, “Two-dimensional smoothing by spectral dispersion for direct-drive inertial confinement fusion,” in Solid State Lasers for Application to Inertial Confinement Fusion (ICF), W. F. Krupke, ed., Proc. SPIE 2633, 634–644 (1995).
[CrossRef]

Saltiel, S.

I. Buchvarov, S. Saltiel, Ch. Iglev, and K. Koynov, “Intensity dependent change of the polarization state as a result of non-linear phase shift in type II frequency doubling crystals,” Opt. Commun. 141, 173–179 (1997).
[CrossRef]

Shiek-Bahae, M.

Stegeman, G.

Thiell, G.

D. Veron, G. Thiell, and C. Gouedard, “Optical smoothing of the high power laser Phebus Nd-glass laser using the multimode optical fiber technique,” Opt. Commun. 97, 259–271 (1993).
[CrossRef]

Van Stryland, E.

Van Stryland, E. W.

Veron, D.

D. Veron, G. Thiell, and C. Gouedard, “Optical smoothing of the high power laser Phebus Nd-glass laser using the multimode optical fiber technique,” Opt. Commun. 97, 259–271 (1993).
[CrossRef]

Videau, L.

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

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

Opt. Commun. (3)

I. Buchvarov, S. Saltiel, Ch. Iglev, and K. Koynov, “Intensity dependent change of the polarization state as a result of non-linear phase shift in type II frequency doubling crystals,” Opt. Commun. 141, 173–179 (1997).
[CrossRef]

R. H. Lehmberg and S. P. Obenschain, “Use of induced spatial incoherency for uniform illumination,” Opt. Commun. 46, 27–31 (1983).
[CrossRef]

D. Veron, G. Thiell, and C. Gouedard, “Optical smoothing of the high power laser Phebus Nd-glass laser using the multimode optical fiber technique,” Opt. Commun. 97, 259–271 (1993).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. A (2)

J. T. Manassah, “Phase modulation in second-order nonlinear-optical processes,” Phys. Rev. A 42, 4085–4101 (1990).
[CrossRef]

A. Kobyakov and F. Lederer, “Cascading of quadratic nonlinearities: an analytical study,” Phys. Rev. A 54, 3455–3471 (1996).
[CrossRef] [PubMed]

Proc. SPIE (1)

J. E. Rothenberg, “Two-dimensional smoothing by spectral dispersion for direct-drive inertial confinement fusion,” in Solid State Lasers for Application to Inertial Confinement Fusion (ICF), W. F. Krupke, ed., Proc. SPIE 2633, 634–644 (1995).
[CrossRef]

Other (9)

J. W. Goodman, “Statistical properties of speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed., Vol. 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1984), pp. 9–75.

A. Yariv, Quantum Electronics (Wiley, New York, 1988).

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists (Springer-Verlag, Berlin, 1954).

A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures (North-Holland, Amsterdam, 1978).

A. C. L. Boscheron, “Etude de nouvelles configurations de conversion de frequence pour l’optimisation des lasers de haute puissance,” Ph.D. dissertation (University of Paris XI, Orsay, France, 1996).

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

D. Middleton, Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).

M. Andre, “Megajoule solid state laser for ICF applications,” in Technical Committee Meeting on Drivers and Ignition Facilities for Inertial Fusion, J. Coutant, ed., Proceedings of the International Atomic Energy Agency (Commissariat à l’Ènergie Atomique/Direction des Applications Militaires, Limeil-Valenton, France, 1995), pp. 77–78.

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Figures (6)

Fig. 1
Fig. 1

Schematic of the setup for the conversion of amplitude modulation into phase modulation.

Fig. 2
Fig. 2

(a) Correlation function and (b) spectral intensity for the signal and the pump waves. Solid curves, pump functions; dashed curves, signal wave for B=1; dotted curves, signal wave for B=2. The pump correlation function is a Gaussian function with a coherence time tc of 1 ps. We removed the Dirac function from the spectra.

Fig. 3
Fig. 3

(a) Nonlinear phase and (b) intensity depletion as functions of the ratio z/L in a type I configuration. Solid curves, numerical results; dotted curves, obtained with a first-order expansion; dashed curves, obtained with a second-order expansion. L=1 cm, I=25 GW/cm2, Pc=1 GW, Δk=30 cm-1.

Fig. 4
Fig. 4

(a) Nonlinear phase and (b) intensity depletion of the signal wave as functions of the pump intensity in a type II configuration. Solid curves, numerical results; dotted curves, obtained with a first-order expansion; dashed curves, obtained with a second-order expansion. L=1 cm, Is0=0.1 GW/cm2, Pc=1 GW, Δk=40 cm-1.

Fig. 5
Fig. 5

(a) Correlation function and (b) spectral intensity for the signal wave in the same configuration as Fig. 2. Solid curves, without the group-velocity difference effect; dotted curves, B2Δsv2z2/6tc2=0.2.

Fig. 6
Fig. 6

(a) Nonlinear phase and (b) normalized signal intensity as a function of time (in picoseconds). The pump intensity is sinusoidal, with vm=10 GHz and Ip=2.5 GW/cm2. The crystal is assumed to be a 2-cm KDP crystal in a type II configuration. L=2 cm, Pc=1 GW, Δk=19.3 cm-1, σs=-0.03ps2/m.

Equations (78)

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ϕNL(x, y, t)=-2Ip(x, y, t)LΔkPc,
Pc=0cnsnpnhλsλp8π2deff2,
I˜p(ν)=-+ fp(t)exp(-2iπνt)dt.
fp(t)=1IpEp(0)Ep*(t)=exp(-t2/2tc2),
I˜p(ν)=I˜p exp(-ν2/2νc2),
fs(t)=11+B(L)2[1-fp2(t)],
B(L)=2IpLΔkPc.
I˜s(ν)=-+fs(t)exp(-2iπνt)dt=αδ(ν)+(1-α)Isc˜(ν),
Isc˜(ν)=-+ fp2(t)1+B(L)2[1-fp2(t)]exp(-2iπνt)dt.
Ej=12n(ωj)c01/2Ej expi(kjz-ωjt)+c.c.
Esz=iωsωp1PcEp*Eh exp(iΔkz),
Epz=iωpωs1PcEs*Eh exp(iΔkz),
Ehz=i ωhωpωs1PcEpEs exp(-iΔkz),
Is,p(z)=Is0,p0-8Is0Ip0PcΔk2sin2[ϕ(z)],
ϕs,p(z)=-2Ip0,s0PcΔk2{Δkz-sin[2ϕ(z)]},
Ii(z)=Ii-4Ii2PcΔk2sin2[ϕ(z)],
ϕi(z)=-IiPcΔk2{Δkz-sin[2ϕ(z)]}.
Esz=-Δsv Est+iωsωp1PcEp*Eh exp(iΔkz),
Epz=iωpωs1PcEs*Eh exp(iΔkz),
Ehz=-Δhv Eht+i ωhωpωs1PcEpEs exp(-iΔkz),
Es(z, t)z=-Δsv Es(z, t)t-i 2Ip(z, t)ΔkPcEs(z, t),
ϕ(z, t)=-2ΔkPc0z Ip(t-Δsvz)dz.
fs(t)=fs0(t)1+Δsv2z26tc2B(z)2fs02[g1(t/tc)+B(z)2g2(t/tc)]+OΔsvztc3,
f˜p(u)=fp(utc),
g1(u)=-f˜p(0)+f˜p2(u)+f˜pf˜p(u),
g2(u)=f˜p(0)[f˜p2(u)-1]+f˜p2(u)+f˜pf˜p(u)+3f˜p2f˜p2(u)-f˜p3f˜p(u).
I˜s(ν)=-+ fs(t)exp(-2iπνt)dt=αδ(ν)+(1-α)Isc˜(ν),
α=11+B(L)21-Δsν2L26tc2B(L)21+B(L)2f˜p(0).
Esz=-iσs 2Est2+iωsωp1PcEp*Eh exp(iΔkz),
Epz=-iσp 2Ept2+iωpωs1PcEs*Eh exp(iΔkz),
Ehz=-iσh 2Eht2+i ωhωpωs1PcEpEs exp(-iΔkz),
Es(z, t)z=-iσs 2Es(z, t)t2-i 2Ip(z, t)ΔkPcEs(z, t),
fs(t)=fs0(t)1-2σsL3tc2B(L)3f˜p(0)[1-f˜p(t/tc)2]fs0(t)+OσsLtc22.
α=11+B(L)21-2σsL3tc2B(L)31+B(L)2f˜p(0).
Cs2(L)=|Es|4-|Es|22|Es|22.
Cs(L)=B(L) |σs|Ltc2[3/2f˜p(0)2+2f˜(4)(0)]1/2+OσsLtc22.
fp(x)=1IpEp(0,0)Ep*(x, y)=exp[-(x2+y2)/2rc2],
Esz=-Δsx Esx+iωsωp1PcEp*Eh exp(iΔkz),
Epz=iωpωs-1PcEs*Eh exp(iΔkz),
Ehz=-Δhx Ehx+i ωhωpωs1PcEpEs exp(-iΔkz),
Esz=-iσsdif2Esx2+2Esy2+iωsωp1PcEp*Eh exp(iΔkz),
Epz=-iσpdif2Epx2+2Epy2+iωpωs1PcEs*Eh exp(iΔkz),
Ehz=-iσhdif2Ehx2+2Ehy2+i ωhωpωs1PcEpEs exp(-iΔkz),
Ep(t)=2Ip cos(2πνmt),
Es(t)=exp i[B0(L)cos(4πνmt)].
B(L)=B0(L)sinc(2πνmΔsvL),
Cs(L)=4π2B(L)|σs|Lνm2+O[(4π2σsLνm2)2].
Is,p(z)=Ns0,p0-Nb sn2NcItPc1/2m ωs,pωhIt,
Ih(z)=Nb sn2zNcItPc1/2z, mIt,
ϕs,p=ϕs0,p0+12Δkz-PcItNc1/2 Δk2×NbNs0,p0, amzNcItPc1/2m, m,
ϕh=ϕs0+ϕp0+π2Δkz,
N2-Ns0+Np0+14PcItΔk2N+Ns0Np0=0,
sn-1(x, m)=0x dt[(1-t2)(1-mt2)]1/2,
am(u, m)=arcsin[sn(u, m)].
(n, ϕ, m)=0ϕ dθ(1-n sin2 θ)[(1-m sin2 θ)]1/2.
Nc=γ4+2-4βγ1-8γ-64βγ3(β+4)+512βγ4(3β+4),
Nb=4βγ1-8γ+64βγ3(β+4)-512βγ4(3β+4),
m=16βγ21-16γ+512βγ4(6+β),
ϕ(z)=zNcItPc1/2=Δkz21+4γ-8γ2(1+β)+32γ3(1+3β)-160γ4(1+6β+β2).
sn(u, m)=sin(u)-¼m[u-½ sin(2u)]×cos(u)+O(m2),
(n, ϕ, m)=l=0+j=0lnl(-1)jt2l(ϕ)-1/2jmnj,
Is,p(z)=Is0,p0-8Is0Ip0Itγ-1(1-8γ-1)×sin2Δkz2[1+4γ-1-8γ-2(1+β)]+O(γ-3),
Ih(z)=16Is0Ip0Itγ-1(1-8γ-1)×sin2Δkz2[1+4γ-1-8γ-2(1+β)]+O(γ-3),
ϕs,p(z)=Δkz2[hs,p(1)γ-1+hs,p(2)γ-2+hs,p(3)γ-3+hs,p(4)γ-4]+O(γ-5),
hs,p(1)=-4Np0,s0 t2(ϕ)ϕ,
hs,p(2)=-8(β-4Np0,s0) t2(ϕ)ϕ+2Np0,s02 t4(ϕ)ϕ,
hs,p(3)=-322[Np0,s0(4+β)-2β] t2(ϕ)ϕ+Np0,s0(β-8Np0,s0) t4(ϕ)ϕ+2Np0,s03 t6(ϕ)ϕ.
X=[Re(E˜s), Im(E˜s), Re(E˜p), Im(E˜p),Re(E˜h), Im(E˜h)],
F(X, ζ)={ωs/ωp[-X3X5 sin(ζ)-X4X6 sin(ζ)-X3X6 cos(ζ)+X4X5 cos(ζ)]ωs/ωp[X3X5 cos(ζ)+X4X6 cos(ζ)-X3X6 sin(ζ)+X4X5 sin(ζ)]ωp/ωs[-X1X5 sin(ζ)-X2X6 sin(ζ)-X1X6 cos(ζ)+X2X5 cos(ζ)]ωp/ωs[X1X5 cos(ζ)+X2X6 cos(ζ)-X1X6 sin(ζ)+X2X5 sin(ζ)]ωh/ωpωs[X1X3 sin(ζ)-X2X4 sin(ζ)-X1X4 cos(ζ)-X2X3 cos(ζ)]ωh/ωpωs[X1X3 cos(ζ)-X2X4 cos(ζ)+X1X4 sin(ζ)+X2X3 sin(ζ)]}.
Xδz=1δFXδ, zδ2+GXδt, 2Xδt2
Xz=b(X)+GXt, 2Xt2
bj(x) -i=1d 1Z00Z0 dζ 0ζ dηFi(x, ζ) Fjxi(x, η).
X¯δ(z, t)=Xδ(z, t)+δf1Xδ(z, t), zδ2+δ2f2Xδ(z, t), zδ2,
f1(x, ζ) -0ζ F(x, η)dη,
f2(x, ζ) -0ζ[F(x, η)]f1(x, η)dη+ζb(x).
X¯δ(z, t)z=1δFXδ, zδ2+1δf1ζXδ, zδ2+FXδ, zδ2xf1Xδ, zδ2+f2ζXδ, zδ2+G(Xtδ, Xttδ)+O(δ),
Xδ(z, t)-X0(t)
=0z b[Xδ(z, t)]+G[Xtδ(z, t), Xttδ(z, t)]dz+O(δ).

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