Abstract

Optical sum-frequency mixing offers an efficient way to generate coherent light at wavelengths that are not directly accessible by lasing transitions. The efficiency of these mixing processes can be enhanced by using elliptical rather than spherical Gaussian beams. In this paper, an expression for the nonlinear conversion efficiency in these processes is derived. Numerical results are presented that allow maximizing the obtainable output power by an optimization of the input beam parameters.

© 2012 Optical Society of America

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References

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  1. P. A. Franken, A. E. Hill, C. W. Peters, and G. W. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
    [CrossRef]
  2. D. A. Kleinman, “Theory of second harmonic generation of light,” Phys. Rev. 128, 1761–1775 (1962).
    [CrossRef]
  3. G. D. Boyd, A. Ashkin, J. M. Dziedzic, and D. A. Kleinman, “Second-harmonic generation of light with double refraction,” Phys. Rev. 137, A1305–A1320 (1965).
    [CrossRef]
  4. A. Ashkin, G. D. Boyd, and J. M. Dziedzic, “Resonant optical second harmonic generation and mixing,” IEEE J. Quantum Electron. 2, 109–124 (1966).
    [CrossRef]
  5. G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597 (1968).
    [CrossRef]
  6. T. Freegarde, J. Coutts, J. Walz, D. Leibfried, and T. W. Hänsch, “General analysis of Type I second-harmonic generation with elliptical Gaussian beams,” J. Opt. Soc. Am. B 14, 2010–2016 (1997).
    [CrossRef]
  7. 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]
  8. F. Bréhat and B. Wyncke, “Calculation of double-refraction walk-off angle along the phase-matching directions in non-linear biaxial crystals,” J. Phys. B At. Mol. Opt. Phys. 22, 1891–1898 (1989).
    [CrossRef]
  9. W. P. Risk, T. R. Gosnell, and A. V. Nurmikko, Compact Blue-Green Lasers (Cambridge University, 2003).

1997

1989

F. Bréhat and B. Wyncke, “Calculation of double-refraction walk-off angle along the phase-matching directions in non-linear biaxial crystals,” J. Phys. B At. Mol. Opt. Phys. 22, 1891–1898 (1989).
[CrossRef]

1968

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

1966

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]

A. Ashkin, G. D. Boyd, and J. M. Dziedzic, “Resonant optical second harmonic generation and mixing,” IEEE J. Quantum Electron. 2, 109–124 (1966).
[CrossRef]

1965

G. D. Boyd, A. Ashkin, J. M. Dziedzic, and D. A. Kleinman, “Second-harmonic generation of light with double refraction,” Phys. Rev. 137, A1305–A1320 (1965).
[CrossRef]

1962

D. A. Kleinman, “Theory of second harmonic generation of light,” Phys. Rev. 128, 1761–1775 (1962).
[CrossRef]

1961

P. A. Franken, A. E. Hill, C. W. Peters, and G. W. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Ashkin, A.

A. Ashkin, G. D. Boyd, and J. M. Dziedzic, “Resonant optical second harmonic generation and mixing,” IEEE J. Quantum Electron. 2, 109–124 (1966).
[CrossRef]

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]

G. D. Boyd, A. Ashkin, J. M. Dziedzic, and D. A. Kleinman, “Second-harmonic generation of light with double refraction,” Phys. Rev. 137, A1305–A1320 (1965).
[CrossRef]

Boyd, G. D.

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

A. Ashkin, G. D. Boyd, and J. M. Dziedzic, “Resonant optical second harmonic generation and mixing,” IEEE J. Quantum Electron. 2, 109–124 (1966).
[CrossRef]

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]

G. D. Boyd, A. Ashkin, J. M. Dziedzic, and D. A. Kleinman, “Second-harmonic generation of light with double refraction,” Phys. Rev. 137, A1305–A1320 (1965).
[CrossRef]

Bréhat, F.

F. Bréhat and B. Wyncke, “Calculation of double-refraction walk-off angle along the phase-matching directions in non-linear biaxial crystals,” J. Phys. B At. Mol. Opt. Phys. 22, 1891–1898 (1989).
[CrossRef]

Coutts, J.

Dziedzic, J. M.

A. Ashkin, G. D. Boyd, and J. M. Dziedzic, “Resonant optical second harmonic generation and mixing,” IEEE J. Quantum Electron. 2, 109–124 (1966).
[CrossRef]

G. D. Boyd, A. Ashkin, J. M. Dziedzic, and D. A. Kleinman, “Second-harmonic generation of light with double refraction,” Phys. Rev. 137, A1305–A1320 (1965).
[CrossRef]

Franken, P. A.

P. A. Franken, A. E. Hill, C. W. Peters, and G. W. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Freegarde, T.

Gosnell, T. R.

W. P. Risk, T. R. Gosnell, and A. V. Nurmikko, Compact Blue-Green Lasers (Cambridge University, 2003).

Hänsch, T. W.

Hill, A. E.

P. A. Franken, A. E. Hill, C. W. Peters, and G. W. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Kleinman, D. A.

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

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]

G. D. Boyd, A. Ashkin, J. M. Dziedzic, and D. A. Kleinman, “Second-harmonic generation of light with double refraction,” Phys. Rev. 137, A1305–A1320 (1965).
[CrossRef]

D. A. Kleinman, “Theory of second harmonic generation of light,” Phys. Rev. 128, 1761–1775 (1962).
[CrossRef]

Leibfried, D.

Nurmikko, A. V.

W. P. Risk, T. R. Gosnell, and A. V. Nurmikko, Compact Blue-Green Lasers (Cambridge University, 2003).

Peters, C. W.

P. A. Franken, A. E. Hill, C. W. Peters, and G. W. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Risk, W. P.

W. P. Risk, T. R. Gosnell, and A. V. Nurmikko, Compact Blue-Green Lasers (Cambridge University, 2003).

Walz, J.

Weinreich, G. W.

P. A. Franken, A. E. Hill, C. W. Peters, and G. W. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Wyncke, B.

F. Bréhat and B. Wyncke, “Calculation of double-refraction walk-off angle along the phase-matching directions in non-linear biaxial crystals,” J. Phys. B At. Mol. Opt. Phys. 22, 1891–1898 (1989).
[CrossRef]

IEEE J. Quantum Electron.

A. Ashkin, G. D. Boyd, and J. M. Dziedzic, “Resonant optical second harmonic generation and mixing,” IEEE J. Quantum Electron. 2, 109–124 (1966).
[CrossRef]

J. Appl. Phys.

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. B At. Mol. Opt. Phys.

F. Bréhat and B. Wyncke, “Calculation of double-refraction walk-off angle along the phase-matching directions in non-linear biaxial crystals,” J. Phys. B At. Mol. Opt. Phys. 22, 1891–1898 (1989).
[CrossRef]

Phys. Rev.

D. A. Kleinman, “Theory of second harmonic generation of light,” Phys. Rev. 128, 1761–1775 (1962).
[CrossRef]

G. D. Boyd, A. Ashkin, J. M. Dziedzic, and D. A. Kleinman, “Second-harmonic generation of light with double refraction,” Phys. Rev. 137, A1305–A1320 (1965).
[CrossRef]

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]

Phys. Rev. Lett.

P. A. Franken, A. E. Hill, C. W. Peters, and G. W. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Other

W. P. Risk, T. R. Gosnell, and A. V. Nurmikko, Compact Blue-Green Lasers (Cambridge University, 2003).

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Figures (3)

Fig. 1.
Fig. 1.

Dependence of the Boyd–Kleinman factor hopt on the focusing parameters ξ1 and ξ2 for different wave vector ratios. Top left, k1=k2; top right, 2k1=k2; bottom left, 3k1=k2; bottom right, 4k1=k2.

Fig. 2.
Fig. 2.

Dependence of the Boyd–Kleinman factor hopt on the focus position parameters μ1 and μ2 for different wave vector ratios. Top left, k1=k2; top right, 2k1=k2; bottom left, 3k1=k2; bottom right, 4k1=k2.

Fig. 3.
Fig. 3.

Optimized focusing parameters ξjmopt(j{1,2},m{x,y}) of the two input beams dependent on the double-refraction parameter B for different wave vector ratios. Top left, k1=k2; top right, 2k1=k2; bottom left, 3k1=k2; bottom right, 4k1=k2. On the right axes, the conversion enhancement over conventional spherical input beams is displayed.

Equations (19)

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Ej(x,y,z)=Ej0exp[ikjzαj2z]exp[(x2/wjx21+iτjx+y2/wjy21+iτjy)]((1+iτjx)(1+iτjy))1/2,j{1,2}withτjx=2bjx(zfjx),andτjy=2bjy(zfjy).
P3(x,y,z)=2ϵ0deffE1(x,y,z)E2(x,y,z)L(z){L(z)=10zlL(z)=0otherwise.
P˜3(kx,ky,z)=exp[α32(lz)](2π)2dxdyP3(x,y,z)ei(kxx+kyy)ei(kzγ3kx)z.
kz=k32(kx2+ky2)k3(1kx2+ky22k32),
P˜3=ϵ0deffE10E20exp[α32l]exp[(α+iΔk)z]exp[i(kx2+ky22k3+γ3kx)z]π((1+iτ1x)(1+iτ2x)(1+iτ1y)(1+iτ2y))1/2T×dxdyexp((x2((w1x2(1+iτ1x))1+(w2x2(1+iτ2x))1)1/2Ax+y2((w1y2(1+iτ1y))1+(w2y2(1+iτ2y))1)1/2Ay)ei(kxx+kyy),
E˜3(kx,ky,l)=iω32ϵ0n3ccos2γ30ldzP˜3(kx,ky,z),
P3=ϵ0n3c2dxdy|E3(x,y,l)|2=ϵ0n3c2dkxdky|E˜3(kx,ky,l)|2.
τm()=2(z()fm)/bm,m{x,y},
bm=((b1m+b2m)2+4(f1mf2m)2)k1k2+2b1mb2m(k12+k22)2(k1+k2)(b1mk1+b2mk2),
fm=b1mf2mk1+b2mf1mk2b1mk1+b2mk2,
P3=2(ω1+ω2)2deff2P1P2ϵ0c3n1n2n3πcos4γ3k1k2lh,
h=e^Rexp[αl]exp[μαl]×dτxdτxexp[κx(τx+τx)+iσx(τxτx)]exp[γ32bx28(τxτx)2Ax+Ax*izzk1+k2](1+iτx)(1iτx)+Qx(1+i(e^τx+Δ))(1i(e^τx+Δ))+Qy.
R=12lk1b1xb1yk2b2xb2y(b1xk1+b2xk2)(b1yk1+b2yk2),
Qm=((b1mb2m)2+4(f1mf2m)2)2k12k22(((b1m+b2m)2+4(f1mf2m)2)k1k2+2b1mb2m(k12+k22))2.
B=γ32lk1+k22
P3=16ω32deff2P1P2eα3lϵ0c3n1n2n3πcos4γ3wjmdzdzeα(z+z)+iΔk(zz)×A(τ⃗)A*(τ⃗)B(τ⃗)B*(τ⃗)T(τ⃗)T*(τ⃗)A(τ⃗)+A*(τ⃗)izzk1+k2Cx(τ⃗)+Cx*(τ⃗)B(τ⃗)+B*(τ⃗)izzk1+k2Cy(τ⃗)+Cy*(τ⃗)×exp[γ32(zz)2A(τ⃗)+A*(τ⃗)izzk1+k2].
τm()=2bm(z()fm),
AA*BB*TT*=(wjm)2(Cx+Cx*)(Cy+Cy*)Px[(1+iτx)(1iτx)+Qx]Py[(1+iτy)(1iτy)+Qy]
Pm=(b1m2k1k2+(b2m2+4(f1mf2m)2)k1k2+2b1mb2m(k12+k1k2+k22))24k12k22(k1+k2)2(b1mk1+b2mk2),

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