Abstract

Numerical studies of second-harmonic generation of beams with aberrations based on the solution of three-dimensional propagation equations including diffraction and walk-off effects are reported. Because of aberrations of the fundamental wave, the refractive index of an extraordinary wave, the phase-mismatch angle, etc. will vary across the aperture, and results of these studies show that these variations can lead to changes in conversion efficiency, transverse intensity distribution, and phase distortion of the second-harmonic beam. The second-moment radius and the divergent angle of a second-harmonic beam are presented. It has been found that second-harmonic beams are less divergent than fundamental waves, but the M2 factor can be either worse or better. The calculated results of different orders of Zernike aberrations are given.

© 2002 Optical Society of America

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References

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  1. P. Pliszka and P. P. Banerjee, “Nonlinear transverse effects in second-harmonic generation,” J. Opt. Soc. Am. B 10, 1810–1819 (1993).
    [CrossRef]
  2. P. Pliszka and P. P. Banerjee, “Self-phase modulation in quadratically nonlinear media,” J. Mod. Opt. 40, 1909–1916 (1993).
    [CrossRef]
  3. D. Eimerl, J. M. Auerbach, and P. W. Milonni, “Paraxial wave theory of second and third harmonic generation in uniaxial crystals. I. Narrowband pump fields,” J. Mod. Opt. 42, 1037–1067 (1995).
    [CrossRef]
  4. A. V. Smith and M. S. Browers, “Phase distortion in sum- and difference-frequency mixing in crystals,” J. Opt. Soc. Am. B 12, 49–57 (1995).
    [CrossRef]
  5. G. I. Stegeman, M. Sheik-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]
  6. P. W. Milonni, J. M. Auerbach, and D. Eimerl, “Frequency conversion modeling with spatially and temporally varying beams,” in Solid State Lasers for Application to Inertial Confinement Fusion (ICF), W. F. Krupke, ed., Proc. SPIE 2633, 230–241 (1997).
    [CrossRef]
  7. J. M. Auerbach, D. Eimerl, D. Milam, and P. W. Milonni, “Perturbation theory for electric-field amplitude and phase ripple transfer in frequency doubling and tripling,” Appl. Opt. 36, 606–612 (1997).
    [CrossRef] [PubMed]
  8. H. Jing, Z. Y. Dong, and J. W. Han, “Second-harmonic generation of phase aberrated laser beams by type I phase matching in uniaxial crystals,” Acta Opt. Sin. (to be published).
  9. H. Jing, Z. Y. Dong, and J. W. Han, “Second harmonic conversion-efficiency of aberrated laser beam,” Chin. J. Lasers (to be published).
  10. D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–379 (1966).
    [CrossRef]

1997 (2)

P. W. Milonni, J. M. Auerbach, and D. Eimerl, “Frequency conversion modeling with spatially and temporally varying beams,” in Solid State Lasers for Application to Inertial Confinement Fusion (ICF), W. F. Krupke, ed., Proc. SPIE 2633, 230–241 (1997).
[CrossRef]

J. M. Auerbach, D. Eimerl, D. Milam, and P. W. Milonni, “Perturbation theory for electric-field amplitude and phase ripple transfer in frequency doubling and tripling,” Appl. Opt. 36, 606–612 (1997).
[CrossRef] [PubMed]

1995 (2)

D. Eimerl, J. M. Auerbach, and P. W. Milonni, “Paraxial wave theory of second and third harmonic generation in uniaxial crystals. I. Narrowband pump fields,” J. Mod. Opt. 42, 1037–1067 (1995).
[CrossRef]

A. V. Smith and M. S. Browers, “Phase distortion in sum- and difference-frequency mixing in crystals,” J. Opt. Soc. Am. B 12, 49–57 (1995).
[CrossRef]

1993 (3)

1966 (1)

D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–379 (1966).
[CrossRef]

Ashkin, A.

D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–379 (1966).
[CrossRef]

Assanto, G.

Auerbach, J. M.

J. M. Auerbach, D. Eimerl, D. Milam, and P. W. Milonni, “Perturbation theory for electric-field amplitude and phase ripple transfer in frequency doubling and tripling,” Appl. Opt. 36, 606–612 (1997).
[CrossRef] [PubMed]

P. W. Milonni, J. M. Auerbach, and D. Eimerl, “Frequency conversion modeling with spatially and temporally varying beams,” in Solid State Lasers for Application to Inertial Confinement Fusion (ICF), W. F. Krupke, ed., Proc. SPIE 2633, 230–241 (1997).
[CrossRef]

D. Eimerl, J. M. Auerbach, and P. W. Milonni, “Paraxial wave theory of second and third harmonic generation in uniaxial crystals. I. Narrowband pump fields,” J. Mod. Opt. 42, 1037–1067 (1995).
[CrossRef]

Banerjee, P. P.

P. Pliszka and P. P. Banerjee, “Nonlinear transverse effects in second-harmonic generation,” J. Opt. Soc. Am. B 10, 1810–1819 (1993).
[CrossRef]

P. Pliszka and P. P. Banerjee, “Self-phase modulation in quadratically nonlinear media,” J. Mod. Opt. 40, 1909–1916 (1993).
[CrossRef]

Boyd, G. D.

D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–379 (1966).
[CrossRef]

Browers, M. S.

Eimerl, D.

J. M. Auerbach, D. Eimerl, D. Milam, and P. W. Milonni, “Perturbation theory for electric-field amplitude and phase ripple transfer in frequency doubling and tripling,” Appl. Opt. 36, 606–612 (1997).
[CrossRef] [PubMed]

P. W. Milonni, J. M. Auerbach, and D. Eimerl, “Frequency conversion modeling with spatially and temporally varying beams,” in Solid State Lasers for Application to Inertial Confinement Fusion (ICF), W. F. Krupke, ed., Proc. SPIE 2633, 230–241 (1997).
[CrossRef]

D. Eimerl, J. M. Auerbach, and P. W. Milonni, “Paraxial wave theory of second and third harmonic generation in uniaxial crystals. I. Narrowband pump fields,” J. Mod. Opt. 42, 1037–1067 (1995).
[CrossRef]

Kleinman, D. A.

D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–379 (1966).
[CrossRef]

Milam, D.

Milonni, P. W.

J. M. Auerbach, D. Eimerl, D. Milam, and P. W. Milonni, “Perturbation theory for electric-field amplitude and phase ripple transfer in frequency doubling and tripling,” Appl. Opt. 36, 606–612 (1997).
[CrossRef] [PubMed]

P. W. Milonni, J. M. Auerbach, and D. Eimerl, “Frequency conversion modeling with spatially and temporally varying beams,” in Solid State Lasers for Application to Inertial Confinement Fusion (ICF), W. F. Krupke, ed., Proc. SPIE 2633, 230–241 (1997).
[CrossRef]

D. Eimerl, J. M. Auerbach, and P. W. Milonni, “Paraxial wave theory of second and third harmonic generation in uniaxial crystals. I. Narrowband pump fields,” J. Mod. Opt. 42, 1037–1067 (1995).
[CrossRef]

Pliszka, P.

P. Pliszka and P. P. Banerjee, “Self-phase modulation in quadratically nonlinear media,” J. Mod. Opt. 40, 1909–1916 (1993).
[CrossRef]

P. Pliszka and P. P. Banerjee, “Nonlinear transverse effects in second-harmonic generation,” J. Opt. Soc. Am. B 10, 1810–1819 (1993).
[CrossRef]

Sheik-Bahae, M.

Smith, A. V.

Stegeman, G. I.

Van Stryland, E.

Appl. Opt. (1)

J. Mod. Opt. (2)

P. Pliszka and P. P. Banerjee, “Self-phase modulation in quadratically nonlinear media,” J. Mod. Opt. 40, 1909–1916 (1993).
[CrossRef]

D. Eimerl, J. M. Auerbach, and P. W. Milonni, “Paraxial wave theory of second and third harmonic generation in uniaxial crystals. I. Narrowband pump fields,” J. Mod. Opt. 42, 1037–1067 (1995).
[CrossRef]

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

Opt. Lett. (1)

Phys. Rev. (1)

D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–379 (1966).
[CrossRef]

Proc. SPIE (1)

P. W. Milonni, J. M. Auerbach, and D. Eimerl, “Frequency conversion modeling with spatially and temporally varying beams,” in Solid State Lasers for Application to Inertial Confinement Fusion (ICF), W. F. Krupke, ed., Proc. SPIE 2633, 230–241 (1997).
[CrossRef]

Other (2)

H. Jing, Z. Y. Dong, and J. W. Han, “Second-harmonic generation of phase aberrated laser beams by type I phase matching in uniaxial crystals,” Acta Opt. Sin. (to be published).

H. Jing, Z. Y. Dong, and J. W. Han, “Second harmonic conversion-efficiency of aberrated laser beam,” Chin. J. Lasers (to be published).

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Figures (17)

Fig. 1
Fig. 1

Principal dielectric axes XYZ and propagation system axes xyz for a uniaxial crystal (z, direction of propagation; Z, optic axis).

Fig. 2
Fig. 2

Transverse intensity distributions of the SHs of focused Gaussian beams: (a)–(d) From the experiments of Kleinman et al.10 (a), (c) Shown enlarged in (b). (d) Different focus conditions. (e)–(h) Corresponding simulated pictures.

Fig. 3
Fig. 3

Conversion efficiency for (a) Δk0=0, (b) Δk0=0.46 mm-1. Numbers on the curves are the Zernike orders of aberrations of the fundamental wave.

Fig. 4
Fig. 4

Intensity distribution and phase distortion of the SH (Δk0=0). (a), (b) Intensity distribution and phase, respectively, of the Gaussian fundamental wave without aberration. (c), (d) Super-Gaussian fundamental wave with no aberration.

Fig. 5
Fig. 5

Phase distortions of the SH for the Gaussian fundamental wave with aberrations (Δk0=0). n is the Zernike order of aberration of the fundamental wave. (a), (b), (c) Aberrations of fundamental wave when n=6, 7, 8, respectively. (d), (e), (f) Phase distortions of the SH corresponding to (a), (b), and (c), respectively. (g), (h), (i) Differences ΔΦ=Φfudam-0.5ΦSHG corresponding to (a), (b), and (c), respectively.

Fig. 6
Fig. 6

Intensity distribution and phase distortion of a SH with an apparent walk-off effect but without aberration (Δk0=0).

Fig. 7
Fig. 7

Phase distortions of the SH with aberrations and an apparent walk-off effect: (a) n=6, (b) n=7, (c) n=8. n is the Zernike order of aberration of the fundamental wave.

Fig. 8
Fig. 8

Divergent angles of the pump wave and the SH (Δk0=0) with aberrations (a) n=4, (b) n=5, (c) n=6, (d) n=7, (e) n=8, (f) n=9. n is the Zernike order of aberration of the fundamental wave.

Fig. 9
Fig. 9

Intensity distributions of the SH when the aberrations of the fundamental wave are all of the 8th Zernike aberration (Δk0=0). PV values of the aberrations at the fundamental wavelength are (a) 1, (b) 3, and (c) 5.

Fig. 10
Fig. 10

Second-moment radii of the SH (Δk0=0) in the x and y directions: (a) σx, (b) σy, respectively. The numbers on the curves are the Zernike orders of aberration of the fundamental wave.

Fig. 11
Fig. 11

Intensity distribution and phase distortion of the SH with phase mismatching (Δk0=0.46 mm-1) but without aberration.

Fig. 12
Fig. 12

Intensity distributions of the SH when the aberrations of the fundamental wave are all the 8th Zernike aberrations and Δk0=0.46 mm-1. PV values of the aberrations at the fundamental wavelength are (a) 0.2, (b) 0.6, (c) 1, (d) 1.4, and (f) 5.

Fig. 13
Fig. 13

(a), (b), (c) Differences ΔΦ=Φfudam-0.5ΦSHG that correspond to those of Figs. 12(a), 12(b), and 12(c), respectively. Intensity-dependent phase distortion can be clearly seen.

Fig. 14
Fig. 14

Divergent angles of the pump wave and the SH (Δk0=0.46 mm-1) with aberrations (a) n=4, (b) n=5, (c) n=6, (d) n=7, (e) n=8, (f) n=9. n is the Zernike order of aberration of the fundamental wave.

Fig. 15
Fig. 15

Second-moment radius of the SH (Δk0=0.46 mm-1) in the x and y directions: (a) σx, (b) σy, respectively. The numbers on the curves are the Zernike orders of aberration of the fundamental wave.

Fig. 16
Fig. 16

M2 factors of the pump wave and the SH (Δk0=0) with aberrations (a) n=4, (b) n=5, (c) n=6, (d) n=7, (e) n=8, (f) n=9. n is the Zernike order of aberration of the fundamental wave.

Fig. 17
Fig. 17

M2 factors of the pump wave and the SH (Δk0=0.46 mm-1) with aberrations (a) n=4, (b) n=5, (c) n=6, (d) n=7, (e) n=8, (f) n=9. n is the Zernike order of aberration of the fundamental wave.

Equations (29)

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E(r, t)=1/2 j=13 Aj(r)exp(-iwt)+c.c.
AX(x, y, z)=EX(x, y, z)exp(ino,wwz/c),
AY(x, y, z)=FY(x, y, z)exp(ino,wwz/c),
AX(x, y, z)=HX(x, y, z)exp[ine,w(θ)wz/c],
AY(x, y, z)=HY(x, y, z)exp[ine,w(θ)wz/c],
AZ(x, y, z)=HZ(x, y, z)exp[ine,w(θ)wz/c].
2x2+2y2F+2ino,w wc Fz=-w2c2χ¯F*H exp(iΔkz),
2Hx2+[1-β2w(θ)sin2 θ] 2Hy2+2ine,2w(θ) 2wc
×[1-β2w(θ)cos2 θ]1/2Hz+ρ2w(θ) Hy
=-(2w)22c2χ¯F2 exp(-iΔkz),
1ne,w2(θ)=cos2 θno,w2+sin2 θne,w2,
β2w(θ)=1-ne,2w2(θ)no,2w2,
Δk=2wc[ne,2w(θ)-no,w],
ρ2w(θ)=-β2w(θ)sin 2θ/21-β2w(θ)cos2 θ,
χ¯=-χ sin θ sin 2ϕ.
Z4=3[2(x2+y2)-1](defocus),
Z5=26xy(third-orderastigmatism),
Z6=6(x2-y2)(third-orderastigmatism),
Z7=22y[3(x2+y2)-2](coma),
Z8=22x[3(x2+y2)-2](coma),
Z9=22x(x2-3y2).
θx2=4λ2 -+-+(Sx-Sx¯)2I(Sx, Sy, z)dSxdSy-+-+ I(Sx, Sy, z)dSxdSy,
θy2=4λ2 -+-+(Sy-Sy¯)2I(Sx, Sy, z)dSxdSy-+-+ I(Sx, Sy, z)dSxdSy,
Sx¯=-+-+ SxI(Sx, Sy, z)dSxdSy-+-+ I(Sx, Sy, z)dSxdSy,
Sy¯=-+-+ SyI(Sx, Sy, z)dSxdSy-+-+ I(Sx, Sy, z)dSxdSy,
σx2=4 -+-+(x-x¯)2I(x, y, z)dxdyy-+-+ I(x, y, z)dxdy,
σy2=4 -+-+(y-y¯)2I(x, y, z)dxdyy-+-+ I(x, y, z)dxdy,
x¯=-+-+ xI(x, y, z)dxdyy-+-+ I(x, y, z)dxdy,
y¯=-+-+ yI(x, y, z)dxdyy-+-+ I(x, y, z)dxdy.

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