Abstract

An elliptic Gaussian optical beam (EGB) in a Kerr-law nonlinear graded-index rod lens is treated as two dependent optical beams. Two coupled differential equations of the dimensionless beam-width parameters of two beams in the rod lens are derived by a variational approach and then solved for what is to my knowledge the first time. Investigations of the propagation and the collapse of the EGB in the rod lens and the transformation of the EGB by the rod lens are presented. It is concluded that the properties of propagation, collapse, and transformation are largely determined by the power and initial ellipticity of the incident EGB. The field derived also applies to the EGB propagating in a nonlinear graded-index fiber.

© 2000 Optical Society of America

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References

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  1. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989).
  2. G. I. Stegeman and R. H. Stolen, “Waveguides and fibers for nonlinear optics,” J. Opt. Soc. Am. B 6, 652–662 (1989).
    [CrossRef]
  3. P. Tran, “All-optical switching with a nonlinear chiral photonic bandgap structure,” J. Opt. Soc. Am. B 16, 70–73 (1999).
    [CrossRef]
  4. E. Desurvire, Erbium-Doped Fiber Amplifiers: Principles and Applications (Wiley, New York, 1994).
  5. R. A. Betts, T. Tjugiarto, Y. L. Xue, and P. L. Chu, “Nonlinear refractive index in erbium doped optical fiber: theory and experiment,” IEEE J. Quantum Electron. 27, 908–913 (1991).
    [CrossRef]
  6. R. A. Sammut and C. Pask, “Gaussian and equivalent-step-index approximations for nonlinear waveguides,” J. Opt. Soc. Am. B 8, 395–402 (1991).
    [CrossRef]
  7. M. Karlsson and D. Anderson, “Super-Gaussian approximation of the fundamental radial mode in nonlinear parabolic-index optical fibers,” J. Opt. Soc. Am. B 9, 1558–1562 (1992).
    [CrossRef]
  8. Z. Chen and H. Lai, “Imaging properties of Gaussian beams with a nonlinear graded-index rod lens,” J. Opt. Soc. Am. B 9, 2248–2251 (1992).
    [CrossRef]
  9. Z. Chen, X. Chen, and H. Lai, “Effect of beam power on imaging characteristics of Gaussian beams with a defocusing GRIN rod lens,” IEE Proc. J. 139, 309–312 (1992).
  10. L. Gagnon and C. Paré, “Nonlinear radiation modes connected to parabolic graded-index profiles by the lens transformation,” J. Opt. Soc. Am. A 8, 601–607 (1991).
    [CrossRef]
  11. F. Cornolti, M. Lucchesi, and B. Zambon, “Elliptic Gaussian beam self-focusing in nonlinear media,” Opt. Commun. 75, 129–135 (1990).
    [CrossRef]
  12. C. R. Giuliano, J. H. Marburger, and A. Yariv, “Enhancement of self-focusing threshold in sapphire with elliptical beams,” Appl. Phys. Lett. 21, 58–60 (1972).
    [CrossRef]
  13. S. Konar and A. Sengupta, “Propagation of an elliptic Gaussian laser beam in a medium with saturable nonlinearity,” J. Opt. Soc. Am. B 11, 1644–1646 (1994).
    [CrossRef]
  14. S. M. Mian, B. Taheri, and J. P. Wicksted, “Effects of beam ellipticity on Z-scan measurement,” J. Opt. Soc. Am. B 13, 856–863 (1996).
    [CrossRef]
  15. A. Yariv, Introduction to Optical Electronics (Holt, Rinehart & Winston, New York, 1976), Chaps. 2 and 3.
  16. F. H. Berkshire and J. D. Gibbon, “Collapse in the n-dimensional nonlinear Schrödinger equation—a parallel with Sundman’s results in the N-body problem,” Stud. Appl. Math. 69, 229–262 (1983).
  17. J. R. Ray and J. L. Reid, “More exact invariants for the time-dependent harmonic oscillator,” Phys. Lett. 71A, 317–318 (1979).
    [CrossRef]

1999

1996

1994

1992

1991

1990

F. Cornolti, M. Lucchesi, and B. Zambon, “Elliptic Gaussian beam self-focusing in nonlinear media,” Opt. Commun. 75, 129–135 (1990).
[CrossRef]

1989

1983

F. H. Berkshire and J. D. Gibbon, “Collapse in the n-dimensional nonlinear Schrödinger equation—a parallel with Sundman’s results in the N-body problem,” Stud. Appl. Math. 69, 229–262 (1983).

1979

J. R. Ray and J. L. Reid, “More exact invariants for the time-dependent harmonic oscillator,” Phys. Lett. 71A, 317–318 (1979).
[CrossRef]

1972

C. R. Giuliano, J. H. Marburger, and A. Yariv, “Enhancement of self-focusing threshold in sapphire with elliptical beams,” Appl. Phys. Lett. 21, 58–60 (1972).
[CrossRef]

Anderson, D.

Berkshire, F. H.

F. H. Berkshire and J. D. Gibbon, “Collapse in the n-dimensional nonlinear Schrödinger equation—a parallel with Sundman’s results in the N-body problem,” Stud. Appl. Math. 69, 229–262 (1983).

Betts, R. A.

R. A. Betts, T. Tjugiarto, Y. L. Xue, and P. L. Chu, “Nonlinear refractive index in erbium doped optical fiber: theory and experiment,” IEEE J. Quantum Electron. 27, 908–913 (1991).
[CrossRef]

Chen, X.

Z. Chen, X. Chen, and H. Lai, “Effect of beam power on imaging characteristics of Gaussian beams with a defocusing GRIN rod lens,” IEE Proc. J. 139, 309–312 (1992).

Chen, Z.

Z. Chen and H. Lai, “Imaging properties of Gaussian beams with a nonlinear graded-index rod lens,” J. Opt. Soc. Am. B 9, 2248–2251 (1992).
[CrossRef]

Z. Chen, X. Chen, and H. Lai, “Effect of beam power on imaging characteristics of Gaussian beams with a defocusing GRIN rod lens,” IEE Proc. J. 139, 309–312 (1992).

Chu, P. L.

R. A. Betts, T. Tjugiarto, Y. L. Xue, and P. L. Chu, “Nonlinear refractive index in erbium doped optical fiber: theory and experiment,” IEEE J. Quantum Electron. 27, 908–913 (1991).
[CrossRef]

Cornolti, F.

F. Cornolti, M. Lucchesi, and B. Zambon, “Elliptic Gaussian beam self-focusing in nonlinear media,” Opt. Commun. 75, 129–135 (1990).
[CrossRef]

Gagnon, L.

Gibbon, J. D.

F. H. Berkshire and J. D. Gibbon, “Collapse in the n-dimensional nonlinear Schrödinger equation—a parallel with Sundman’s results in the N-body problem,” Stud. Appl. Math. 69, 229–262 (1983).

Giuliano, C. R.

C. R. Giuliano, J. H. Marburger, and A. Yariv, “Enhancement of self-focusing threshold in sapphire with elliptical beams,” Appl. Phys. Lett. 21, 58–60 (1972).
[CrossRef]

Karlsson, M.

Konar, S.

Lai, H.

Z. Chen and H. Lai, “Imaging properties of Gaussian beams with a nonlinear graded-index rod lens,” J. Opt. Soc. Am. B 9, 2248–2251 (1992).
[CrossRef]

Z. Chen, X. Chen, and H. Lai, “Effect of beam power on imaging characteristics of Gaussian beams with a defocusing GRIN rod lens,” IEE Proc. J. 139, 309–312 (1992).

Lucchesi, M.

F. Cornolti, M. Lucchesi, and B. Zambon, “Elliptic Gaussian beam self-focusing in nonlinear media,” Opt. Commun. 75, 129–135 (1990).
[CrossRef]

Marburger, J. H.

C. R. Giuliano, J. H. Marburger, and A. Yariv, “Enhancement of self-focusing threshold in sapphire with elliptical beams,” Appl. Phys. Lett. 21, 58–60 (1972).
[CrossRef]

Mian, S. M.

Paré, C.

Pask, C.

Ray, J. R.

J. R. Ray and J. L. Reid, “More exact invariants for the time-dependent harmonic oscillator,” Phys. Lett. 71A, 317–318 (1979).
[CrossRef]

Reid, J. L.

J. R. Ray and J. L. Reid, “More exact invariants for the time-dependent harmonic oscillator,” Phys. Lett. 71A, 317–318 (1979).
[CrossRef]

Sammut, R. A.

Sengupta, A.

Stegeman, G. I.

Stolen, R. H.

Taheri, B.

Tjugiarto, T.

R. A. Betts, T. Tjugiarto, Y. L. Xue, and P. L. Chu, “Nonlinear refractive index in erbium doped optical fiber: theory and experiment,” IEEE J. Quantum Electron. 27, 908–913 (1991).
[CrossRef]

Tran, P.

Wicksted, J. P.

Xue, Y. L.

R. A. Betts, T. Tjugiarto, Y. L. Xue, and P. L. Chu, “Nonlinear refractive index in erbium doped optical fiber: theory and experiment,” IEEE J. Quantum Electron. 27, 908–913 (1991).
[CrossRef]

Yariv, A.

C. R. Giuliano, J. H. Marburger, and A. Yariv, “Enhancement of self-focusing threshold in sapphire with elliptical beams,” Appl. Phys. Lett. 21, 58–60 (1972).
[CrossRef]

Zambon, B.

F. Cornolti, M. Lucchesi, and B. Zambon, “Elliptic Gaussian beam self-focusing in nonlinear media,” Opt. Commun. 75, 129–135 (1990).
[CrossRef]

Appl. Phys. Lett.

C. R. Giuliano, J. H. Marburger, and A. Yariv, “Enhancement of self-focusing threshold in sapphire with elliptical beams,” Appl. Phys. Lett. 21, 58–60 (1972).
[CrossRef]

IEE Proc. J.

Z. Chen, X. Chen, and H. Lai, “Effect of beam power on imaging characteristics of Gaussian beams with a defocusing GRIN rod lens,” IEE Proc. J. 139, 309–312 (1992).

IEEE J. Quantum Electron.

R. A. Betts, T. Tjugiarto, Y. L. Xue, and P. L. Chu, “Nonlinear refractive index in erbium doped optical fiber: theory and experiment,” IEEE J. Quantum Electron. 27, 908–913 (1991).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

F. Cornolti, M. Lucchesi, and B. Zambon, “Elliptic Gaussian beam self-focusing in nonlinear media,” Opt. Commun. 75, 129–135 (1990).
[CrossRef]

Phys. Lett.

J. R. Ray and J. L. Reid, “More exact invariants for the time-dependent harmonic oscillator,” Phys. Lett. 71A, 317–318 (1979).
[CrossRef]

Stud. Appl. Math.

F. H. Berkshire and J. D. Gibbon, “Collapse in the n-dimensional nonlinear Schrödinger equation—a parallel with Sundman’s results in the N-body problem,” Stud. Appl. Math. 69, 229–262 (1983).

Other

E. Desurvire, Erbium-Doped Fiber Amplifiers: Principles and Applications (Wiley, New York, 1994).

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart & Winston, New York, 1976), Chaps. 2 and 3.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989).

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Figures (8)

Fig. 1
Fig. 1

xz beam for an EGB passing through a Kerr-law nonlinear GRIN rod lens.

Fig. 2
Fig. 2

yz beam for an EGB passing through a Kerr-law nonlinear GRIN rod lens.

Fig. 3
Fig. 3

Normalized collapse power relative to normalized propagation distance AZ/π for L0x=L0y=0 and w0xw0y=900λ2. A, w0x=w0y=30λ; B, w0x=45λ, w0y=20λ; C, w0x=90λ, w0y=10λ.

Fig. 4
Fig. 4

G and θ in a rod lens relative to Az/π for Q=0.7, w0x=45λ, w0y=20λ, and L0x=L0y=0.

Fig. 5
Fig. 5

Beam-width radii of xz and yz beams in a rod lens relative to Az/π for Q=0.7, w0x=45λ, w0y=20λ, and L0x=L0y=0.

Fig. 6
Fig. 6

G and beam-width radii of xz and yz beams in a rod lens relative to Az/π for Q=0.5, w0x=45λ, w0y=20λ, and L0x=L0y=0.

Fig. 7
Fig. 7

Beam-width radii of xz and yz beams in a rod lens relative to Az/π for w0x=45λ, w0y=20λ, and L0x=L0y=0. A, Q=1.38; B, Q=1.5; C, Q=2.

Fig. 8
Fig. 8

Positions of new waists and linear magnifications of waists for xz and yz beams after transformation relative to L0 (L0=L0x=L0y) for d=3.5 mm, w0x=30λ, and w0y=40λ. A, Q=0; B, Q=0.7; C, Q=1.2.

Equations (48)

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w1x=w0x(1+L0x2/Z0x2)1/2,
w1y=w0y(1+L0y2/Z0y2)1/2,
ρ1x=L0x/[nco(L0x2+Z0x2)],
ρ1y=L0y/[nco(L0y2+Z0y2)]
n(x, y)=n0(x, y)+n2I,
n02(x, y)=nco2[1-A(x2+y2)],A>0
E=φ(x, y, z)exp(-ikz),k=2πnco/λ,
n2(x, y)=n02(x, y)+α|φ|2,α=nco2n2cε0,
2x2+2y2φ-2ik φz-Ak2(x2+y2)φ
+αk2nco2|φ|2φ=0.
J=i2φ φ*z-φ* φz-12kφxφ*x+φyφ*y-12Ak(x2+y2)φφ*+αk4nco2(φφ*)2.
Jφ*-xJφi*-yJφj*-zJφk*=0,
φ(x, y, z)=φ0[ωx(z)ωy(z)]1/2×exp-x2w1x2ωx2(z)-y2w1y2ωy2(z)-i k2ρx(z)x2-i k2ρy(z)y2+iϕ(z),
ωx(0)=ωy(0)=1,
ρx(0)=ρ1x,ρy(0)=ρ1y,
J=Jdxdy=-πφ02 w1x w1y16-8 dφ(z)dz+kw1x2ωx2(z)×dρx(z)dz+ρx2(z)+A+kw1y2ωy2(z)×dρy(z)dz+ρy2(z)+A+4k1w1x2ωx2(z)+1w1y2ωy2(z)-αkφ02nco2ωx(z)ωy(z).
Jξi-ddzJ(dξi/dz)=0
d2ωxdz2+Aωx=B11ωx3-w1xw1yQωyωx2,
d2ωydz2+Aωy=B21ωy3-w1yw1xQωxωy2,
ρx=1ωxdωxdz,ρy=1ωydωydz,
dφdz=1kw1x2ωx2+1kw1y2ωy2-3Qkw1xw1yωxωy,
P=12ncocε0φφ* dx dy,
Pc=λ2/(2πncon2),Q=P/Pc,
G=G0(x2+y2)|φ|2dx dy,
d2Gdz2=4H,
H=G021k2dφdx2+dφdy2-A(x2+y2)|φ|2-α2nco2|φ|4dx dy,
dHdz=-A dGdz,
G=w1x2ωx2+w1y2ωy2,
H=12w1x2dωxdz2-Aωx2+B1ωx2+w1y2dωydz2-Aωy2+B2ωy2-8Qk2w1xw1yωxωy.
H=-AG+D1,
D1=12w1x2(ρ1x2+A+B1)+w1y2(ρ1y2+A+B2)-8Qk2w1xw1y,
G=w1x2+w1y2-D1Acos(2Az)+w1x2ρ1x+w1y2ρ1yAsin(2Az)+D1A.
IV=ωx dωydz-ωy dωxdz2+B1 ωyωxωyωx-2Q w1xw1y+B2 ωxωyωxωy-2Q w1yw1x.
w1xωx=G cos θ,w1yωy=G sin θ
dθ[D2-T(θ)]1/2=D3dzG.
T(θ)=B1w1x4 (tan θ)(tan θ-2Q)+B2w1y4 (cot θ)(cot θ-2Q),
D2=w1x2w1y2(ρ1x-ρ1y)2+T(θ0),
tan θ0=w1y/w1x,
D3=+1,ρyρx-1,ρy<ρx.
wx(z)=w1xωx(z),wy(z)=w1yωy(z),
ρx=12GdGdz-dθdztan θ,
ρy=12GdGdz+dθdzcot θ.
Lx=-π2nco w1x4ωx4(d)ρx(d)/[π2nco2w1x4ωx4(d)ρx2(d)+λ2],
Ly=-π2nco w1y4ωy4(d)ρy(d)/[π2nco2w1y4ωy4(d)ρy2(d)+λ2],
w3x={λ2Lx2+π2w1x4ωx4[1+ncoρx(d)Lx]2}1/2/[πw1xωx(d)],
w3y={λ2Ly2+π2w1y4ωy4[1+ncoρy(d)Ly]2}1/2/[πw1yωy(d)],
Mx=w3x/w0x,My=w3y/w0y.
Q=q2B2×-A(q2+1)+q2ρ1x2+ρ1y2+B2(q2+1)q2+2A A(q2+1)+(q2ρ1x+ρ1y)sin(2AZ)1-cos(2AZ).

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