Abstract

The evolution of the optical power associated with the Cartesian components of a paraxial beam propagating along the optical axis in a uniaxial crystal is investigated. The energy exchange is found to undergo a saturation that is due to both diffraction and coupling between the x- and y-field components; for linearly polarized circularly symmetric input beams, the asymptotic power exchange always amounts to a quarter of the total power. The general results are applied to the case of astigmatic Gaussian beams, which admits of a fully analytical description. The case of finite length crystals is also considered.

© 2002 Optical Society of America

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References

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  1. H. C. Chen, Theory of Electromagnetic Waves (McGraw-Hill, New York, 1983).
  2. R. M. Herrero, J. M. Movilla, P. M. Mejias, “Beam propagation through uniaxial anisotropic media: global changes in the spatial profile,” J. Opt. Soc. Am. A 18, 2009–2014 (2001).
    [CrossRef]
  3. J. A. Fleck, M. D. Feit, “Beam propagation in uniaxial anisotropic media,” J. Opt. Soc. Am. 73, 920–926 (1983).
    [CrossRef]
  4. A. Ciattoni, B. Crosignani, P. Di Porto, “Paraxial vector theory of propagation in uniaxially anisotropic media,” J. Opt. Soc. Am. A 18, 1656–1661 (2001).
    [CrossRef]
  5. A. Ciattoni, G. Cincotti, C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A 19, 792–796 (2002).
    [CrossRef]
  6. U. Hempelmann, H. Herrmann, G. Mrozynski, V. Reimann, W. Sohler, “Integrated optical proton exchanged TM-pass polarizers in LiNbO3: modeling and experimental performance,” J. Lightwave Technol. 13, 1750–1759 (1995).
    [CrossRef]
  7. R. Wolfe, R. A. Lieberman, V. J. Fratello, R. E. Scotti, N. Kopylov, “Etch-tuned ridged waveguide magneto-optic isolator,” Appl. Phys. Lett. 56, 426–428 (1990).
    [CrossRef]
  8. O. Zhuromoskyy, M. Lohmeyer, N. Bahlmann, H. Dotsch, P. Hertel, A. F. Popkov, “Analysis of polarization independent Mach-Zehnder-type integrated optical isolator,” J. Lightwave Technol. 17, 1200–1205 (1999).
    [CrossRef]
  9. Y. Suzuki, H. Iwamura, T. Miyazawa, O. Mikami, “A novel waveguide polarization mode splitter using refractive index changes induced by superlattice disordering,” IEEE J. Quantum Electron. 30, 1794–1799 (1994).
    [CrossRef]
  10. G. Cincotti, A. Ciattoni, C. Palma, “Hermite-Gauss beams in uniaxially anisotropic crystals,” IEEE J. Quantum Electron. 12, 1517–1524 (2001).
    [CrossRef]

2002

2001

1999

1995

U. Hempelmann, H. Herrmann, G. Mrozynski, V. Reimann, W. Sohler, “Integrated optical proton exchanged TM-pass polarizers in LiNbO3: modeling and experimental performance,” J. Lightwave Technol. 13, 1750–1759 (1995).
[CrossRef]

1994

Y. Suzuki, H. Iwamura, T. Miyazawa, O. Mikami, “A novel waveguide polarization mode splitter using refractive index changes induced by superlattice disordering,” IEEE J. Quantum Electron. 30, 1794–1799 (1994).
[CrossRef]

1990

R. Wolfe, R. A. Lieberman, V. J. Fratello, R. E. Scotti, N. Kopylov, “Etch-tuned ridged waveguide magneto-optic isolator,” Appl. Phys. Lett. 56, 426–428 (1990).
[CrossRef]

1983

Bahlmann, N.

Chen, H. C.

H. C. Chen, Theory of Electromagnetic Waves (McGraw-Hill, New York, 1983).

Ciattoni, A.

Cincotti, G.

A. Ciattoni, G. Cincotti, C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A 19, 792–796 (2002).
[CrossRef]

G. Cincotti, A. Ciattoni, C. Palma, “Hermite-Gauss beams in uniaxially anisotropic crystals,” IEEE J. Quantum Electron. 12, 1517–1524 (2001).
[CrossRef]

Crosignani, B.

Di Porto, P.

Dotsch, H.

Feit, M. D.

Fleck, J. A.

Fratello, V. J.

R. Wolfe, R. A. Lieberman, V. J. Fratello, R. E. Scotti, N. Kopylov, “Etch-tuned ridged waveguide magneto-optic isolator,” Appl. Phys. Lett. 56, 426–428 (1990).
[CrossRef]

Hempelmann, U.

U. Hempelmann, H. Herrmann, G. Mrozynski, V. Reimann, W. Sohler, “Integrated optical proton exchanged TM-pass polarizers in LiNbO3: modeling and experimental performance,” J. Lightwave Technol. 13, 1750–1759 (1995).
[CrossRef]

Herrero, R. M.

Herrmann, H.

U. Hempelmann, H. Herrmann, G. Mrozynski, V. Reimann, W. Sohler, “Integrated optical proton exchanged TM-pass polarizers in LiNbO3: modeling and experimental performance,” J. Lightwave Technol. 13, 1750–1759 (1995).
[CrossRef]

Hertel, P.

Iwamura, H.

Y. Suzuki, H. Iwamura, T. Miyazawa, O. Mikami, “A novel waveguide polarization mode splitter using refractive index changes induced by superlattice disordering,” IEEE J. Quantum Electron. 30, 1794–1799 (1994).
[CrossRef]

Kopylov, N.

R. Wolfe, R. A. Lieberman, V. J. Fratello, R. E. Scotti, N. Kopylov, “Etch-tuned ridged waveguide magneto-optic isolator,” Appl. Phys. Lett. 56, 426–428 (1990).
[CrossRef]

Lieberman, R. A.

R. Wolfe, R. A. Lieberman, V. J. Fratello, R. E. Scotti, N. Kopylov, “Etch-tuned ridged waveguide magneto-optic isolator,” Appl. Phys. Lett. 56, 426–428 (1990).
[CrossRef]

Lohmeyer, M.

Mejias, P. M.

Mikami, O.

Y. Suzuki, H. Iwamura, T. Miyazawa, O. Mikami, “A novel waveguide polarization mode splitter using refractive index changes induced by superlattice disordering,” IEEE J. Quantum Electron. 30, 1794–1799 (1994).
[CrossRef]

Miyazawa, T.

Y. Suzuki, H. Iwamura, T. Miyazawa, O. Mikami, “A novel waveguide polarization mode splitter using refractive index changes induced by superlattice disordering,” IEEE J. Quantum Electron. 30, 1794–1799 (1994).
[CrossRef]

Movilla, J. M.

Mrozynski, G.

U. Hempelmann, H. Herrmann, G. Mrozynski, V. Reimann, W. Sohler, “Integrated optical proton exchanged TM-pass polarizers in LiNbO3: modeling and experimental performance,” J. Lightwave Technol. 13, 1750–1759 (1995).
[CrossRef]

Palma, C.

A. Ciattoni, G. Cincotti, C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A 19, 792–796 (2002).
[CrossRef]

G. Cincotti, A. Ciattoni, C. Palma, “Hermite-Gauss beams in uniaxially anisotropic crystals,” IEEE J. Quantum Electron. 12, 1517–1524 (2001).
[CrossRef]

Popkov, A. F.

Reimann, V.

U. Hempelmann, H. Herrmann, G. Mrozynski, V. Reimann, W. Sohler, “Integrated optical proton exchanged TM-pass polarizers in LiNbO3: modeling and experimental performance,” J. Lightwave Technol. 13, 1750–1759 (1995).
[CrossRef]

Scotti, R. E.

R. Wolfe, R. A. Lieberman, V. J. Fratello, R. E. Scotti, N. Kopylov, “Etch-tuned ridged waveguide magneto-optic isolator,” Appl. Phys. Lett. 56, 426–428 (1990).
[CrossRef]

Sohler, W.

U. Hempelmann, H. Herrmann, G. Mrozynski, V. Reimann, W. Sohler, “Integrated optical proton exchanged TM-pass polarizers in LiNbO3: modeling and experimental performance,” J. Lightwave Technol. 13, 1750–1759 (1995).
[CrossRef]

Suzuki, Y.

Y. Suzuki, H. Iwamura, T. Miyazawa, O. Mikami, “A novel waveguide polarization mode splitter using refractive index changes induced by superlattice disordering,” IEEE J. Quantum Electron. 30, 1794–1799 (1994).
[CrossRef]

Wolfe, R.

R. Wolfe, R. A. Lieberman, V. J. Fratello, R. E. Scotti, N. Kopylov, “Etch-tuned ridged waveguide magneto-optic isolator,” Appl. Phys. Lett. 56, 426–428 (1990).
[CrossRef]

Zhuromoskyy, O.

Appl. Phys. Lett.

R. Wolfe, R. A. Lieberman, V. J. Fratello, R. E. Scotti, N. Kopylov, “Etch-tuned ridged waveguide magneto-optic isolator,” Appl. Phys. Lett. 56, 426–428 (1990).
[CrossRef]

IEEE J. Quantum Electron.

Y. Suzuki, H. Iwamura, T. Miyazawa, O. Mikami, “A novel waveguide polarization mode splitter using refractive index changes induced by superlattice disordering,” IEEE J. Quantum Electron. 30, 1794–1799 (1994).
[CrossRef]

G. Cincotti, A. Ciattoni, C. Palma, “Hermite-Gauss beams in uniaxially anisotropic crystals,” IEEE J. Quantum Electron. 12, 1517–1524 (2001).
[CrossRef]

J. Lightwave Technol.

U. Hempelmann, H. Herrmann, G. Mrozynski, V. Reimann, W. Sohler, “Integrated optical proton exchanged TM-pass polarizers in LiNbO3: modeling and experimental performance,” J. Lightwave Technol. 13, 1750–1759 (1995).
[CrossRef]

O. Zhuromoskyy, M. Lohmeyer, N. Bahlmann, H. Dotsch, P. Hertel, A. F. Popkov, “Analysis of polarization independent Mach-Zehnder-type integrated optical isolator,” J. Lightwave Technol. 17, 1200–1205 (1999).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Other

H. C. Chen, Theory of Electromagnetic Waves (McGraw-Hill, New York, 1983).

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Figures (2)

Fig. 1
Fig. 1

Plot of ΔW/Wx(0) as a function of the dimensionless parameter ζ=|Δ|/k0no(sx2+sy2)z for χ=-1, γ=0.1, 0.2, 0.35, 0.5, 0.7, and 1 (γ=sy/sx). The power exchange is a monotonic increasing function of z and reaches a finite value for z+ (saturation).

Fig. 2
Fig. 2

Plot of Q0/Wx(0) as a function of γ=sy/sx. The fundamental circularly symmetric Gaussian beam (for which γ=1) maximizes the power exchange.

Equations (63)

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E(r, z)=exp(ik0noz)[Ao(r, z)+Ae(r, z)],
Ao(r, z)=d2k×expik·r-iz2k0nok2PˆoA˜(k),
Ae(r, z)=d2k×expik·r-inoz2k0ne2k2PˆeA˜(k),
Pˆo=1k2 ky2-kxky-kxkykx2,Pˆe=1k2 kx2kxkykxkyky2,
Aˆ(k)=1(2π)2 d2r exp(-ik·r)E(r, 0).
Wj(z)d2r|Ej(r, z)|2=d2rE(r, z)sˆjE(r, z),
sˆx=1000,sˆy=0001,
Wj(z)=(2π)2d2kE˜(k, z)sˆjEˆ(k, z),
E˜(k, z)=1(2π)2 d2r exp(-ik·r)E(r, z)=exp(ik0noz)exp-iz2k0nok2Pˆo+exp-inoz2k0ne2k2PˆeA˜(k).
Wj(z)=(2π)2d2kA˜(k)(PˆosˆjPˆo+PˆesˆjPˆe)A˜(k)+(2π)2d2kA˜(k)exp-izΔ2k0nok2PˆosˆjPˆe+expizΔ2k0nok2PˆesˆjPˆoAˆ(k),
Δ=no2-ne2ne2.
Pˆosˆj=sˆjPˆo+σj kxkyk2 01-10,
Pˆesˆj=sˆjPˆe-σj kxkyk2 01-10,
Wx(z)=Wx(0)-[Q0-Q(z)]W˙x(0)-ΔW(z),
Wy(z)=Wy(0)+[Q0-Q(z)]Wy(0)+ΔW(z),
Q0=(2π)2d2k kxky(k2)2 A˜(k)×2kxkyky2-kx2ky2-kx2-2kxkyA˜(k),
Q(z)=(2π)2d2k kxky(k2)2 coszΔ2k0nok2×A˜(k)2kxkyky2-kx2ky2-kx2-2kxkyA˜(k)+(2π)2d2k kxkyk2 sinzΔ2k0nok2×A˜(k)0-ii0A˜(k),
ΔW(z)=Q0-Q(z).
Wx(z)+Wy(z)=Wx(0)+Wy(0),
limz+ ΔW(z)=Q0,
Wx()=Wx(0)-Q0,
Wy()=Wy(0)+Q0,
dWxdz=-dWydz=-Δk0no×Im d2r Ex(r, z)x Ey*(r, z)y,
Wj()=(2π)2d2kA˜(k)[PˆosˆjPˆo+PˆesˆjPˆe]A˜(k),
Wx(z)=Wx(0)-ΔW(z),
Wy(z)=ΔW(z),
Q0=(2π)2d2k2kxkyk22|A˜x(k)|2,
Q(z)=(2π)2d2k2kxkyk22×coszΔ2k0nok2|A˜x(k)|2,
ΔW(z)=Q0-Q(z).
d2kkx2-ky2k22|A˜x(k)|2>0,
Q0<12Wx(0),
|Q(z)|(2π)2d2k2kxkyk22|Aˆx(k)|2=Q0.
Q0=14Wx(0).
E(r, 0)=E0 exp-x22σx2-y22σy2eˆx,
Wx(0)=d2r|Ex(r, 0)|2=πE02σxσy.
A˜(k)=E0 σxσy2π exp-12(σx2kx2+σy2ky2)eˆx,
Q(z)=Wx(0) 2σxσyπ Re d2kkxkyk22×exp-(σx2kx2+σy2ky2)+i zΔ2k0nok2.
Q(z)=Wx(0) Reσxσyσx2+σy2-i Δ2k0noz 1F×1+(1-F+χ1-F)22(1-F)+χ(2-F)1-F,
F=σx2-σy2σx2+σy2+i Δ2k0noz2.
Q0=Q(0)=Wx(0) sxsyσx2+σy2
Q(z)=Wx(0)4 11+Δ2k0noσ2z2,
ΔW(z)w1z+w2z2,
w1=(2π)2 Δ2k0no d2kkxkyA˜(k)0i-i0A˜(k),
w2=(2π)2Δ2k0no2d2kkxkyA˜(k)×2kxkyky2-kx2ky2-kx2-2kxkyA˜(k).
zz0=4πno|Δ| sλs,
w1=Δ2k0no d2rE(r, 0)0-ii0xy2E(r, 0),
w2=Δ2k0no2d2rE(r, 0)×2xyy2-x2y2-x2-2xyxy2E(r, 0),
dWjdz=(2π)2Δ2ik0no d2kA˜(k)k2exp-izΔ2k0nok2PˆosˆjPˆe-expizΔ2k0nok2PˆesˆjPˆoA˜(k).
T(z)=-Δk0no Im d2r Ex(r, z)x Ey*(r, z)y,
T(z)=(2π)2Δ2ik0no d2kkxkyE˜(k, z)01-10E˜(k, z).
Pˆo01-10Pˆe=k2kxkyPˆosˆxPˆe,
Pˆe01-10Pˆo=-k2kxkyPˆesˆxPˆo,
Pˆo01-10Pˆo=Pˆe01-10Pˆe=0,
J(β)=d2kkxkyk22 exp-(sx2kx2+sy2ky2)+i sx2β2k2,
J(β)=0dkk02πdθ cos2 θ sin2 θ exp-k2sx2 cos2 θ+sy2 sin2 θ-i sx2β2.
J(β)=02πdθ cos2 θ sin2 θsx2 cos2 θ+sy2 sin2 θ-i sx2β2.
J(β)=i4sx2 Γdt 1t3×(t4-1)2(1-γ2)t4+2(1+γ2-iβ)t2+(1-γ2),
t1,2=±-(1+γ2-iβ)+(1+γ2-iβ)2-(1-γ2)21-γ21/2,
t3,4=±-(1+γ2-iβ)-(1+γ2-iβ)2-(1-γ2)21-γ21/2
J(β)=2πi[Res(t=0)+Res(t=t1)+Res(t=t2)],Δ<0,
J(βk)=2πi[Res(t=0)+Res(t=t3)+Res(t=t4)],Δ>0.
Res(t=0)=2(1+γ2-iθ)(1-γ2)2,
Res(t=t1,2,3,4)=14t1,2,3,44 (t1,2,3,44-1)2(1-γ2)t1,2,3,42+(1+γ2-iθ).

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