Abstract

The effect of the polarization state on electro-optic coupling is studied by using the wave coupling theory of the linear electro-optic effect. The numerical results show that the polarization state obviously influences the electro-optic coupling. The conditions for realizing perfect coupling are emphasized. As an application of perfect coupling, a novel polarization rotator, which can rotate the polarization of a light beam with an arbitrary angle but keep the output intensity unchanged, is presented.

© 2006 Optical Society of America

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References

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  1. R. L. Sutherland, "Electro-Optic Effects," in Handbook of Nonlinear Optics (Marcel Dekker, 2003), pp. 843-946.
  2. F. S. Chen, "Modulators for optical communications," Proc. IEEE. 58, 1440-1457 (1970).
    [CrossRef]
  3. B. Ferguson and X. C. Zhang, "Materials for terahertz science and technology," Nature Mater. 1, 26-33 (2002).
    [CrossRef]
  4. Q. Wu and X. C. Zhang, "Free-space electro-optic sampling of terahertz beams," Appl. Phys. Lett. 67, 3523-3525 (1995).
    [CrossRef]
  5. M. J. Gunning and R. E. Raab, "Algebraic determination of the principal refractive indices and axes in the electro-optic effect," Appl. Opt. 37, 8438-8447 (1998).
    [CrossRef]
  6. D. F. Nelson, "General solution for the electro-optic effect," J. Opt. Soc. Am. 65, 1144-1151 (1975).
    [CrossRef]
  7. A. Yariv, "Coupled-mode theory for guided-wave optics," IEEE J. Quantum. Electron. QE-9, 919-933 (1973).
    [CrossRef]
  8. W. L. She and W. K. Lee, "Wave coupling theory of linear electrooptic effect," Opt. Commun. 195, 303-311 (2001).
    [CrossRef]
  9. D. D. Wu, H. B. Chen, W. L. She, and W. K. Lee, "Wave coupling theory of linear electrooptic effect in a linear absorbent medium," J. Opt. Soc. Am. B 22, 2366-2371 (2005).
    [CrossRef]
  10. A. Yariv, Quantum Electronics (Wiley, 1988).

2005 (1)

2002 (1)

B. Ferguson and X. C. Zhang, "Materials for terahertz science and technology," Nature Mater. 1, 26-33 (2002).
[CrossRef]

2001 (1)

W. L. She and W. K. Lee, "Wave coupling theory of linear electrooptic effect," Opt. Commun. 195, 303-311 (2001).
[CrossRef]

1998 (1)

1995 (1)

Q. Wu and X. C. Zhang, "Free-space electro-optic sampling of terahertz beams," Appl. Phys. Lett. 67, 3523-3525 (1995).
[CrossRef]

1975 (1)

1973 (1)

A. Yariv, "Coupled-mode theory for guided-wave optics," IEEE J. Quantum. Electron. QE-9, 919-933 (1973).
[CrossRef]

1970 (1)

F. S. Chen, "Modulators for optical communications," Proc. IEEE. 58, 1440-1457 (1970).
[CrossRef]

Chen, F. S.

F. S. Chen, "Modulators for optical communications," Proc. IEEE. 58, 1440-1457 (1970).
[CrossRef]

Chen, H. B.

Ferguson, B.

B. Ferguson and X. C. Zhang, "Materials for terahertz science and technology," Nature Mater. 1, 26-33 (2002).
[CrossRef]

Gunning, M. J.

Lee, W. K.

Nelson, D. F.

Raab, R. E.

She, W. L.

Sutherland, R. L.

R. L. Sutherland, "Electro-Optic Effects," in Handbook of Nonlinear Optics (Marcel Dekker, 2003), pp. 843-946.

Wu, D. D.

Wu, Q.

Q. Wu and X. C. Zhang, "Free-space electro-optic sampling of terahertz beams," Appl. Phys. Lett. 67, 3523-3525 (1995).
[CrossRef]

Yariv, A.

A. Yariv, "Coupled-mode theory for guided-wave optics," IEEE J. Quantum. Electron. QE-9, 919-933 (1973).
[CrossRef]

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

Zhang, X. C.

B. Ferguson and X. C. Zhang, "Materials for terahertz science and technology," Nature Mater. 1, 26-33 (2002).
[CrossRef]

Q. Wu and X. C. Zhang, "Free-space electro-optic sampling of terahertz beams," Appl. Phys. Lett. 67, 3523-3525 (1995).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

Q. Wu and X. C. Zhang, "Free-space electro-optic sampling of terahertz beams," Appl. Phys. Lett. 67, 3523-3525 (1995).
[CrossRef]

IEEE J. Quantum. Electron. (1)

A. Yariv, "Coupled-mode theory for guided-wave optics," IEEE J. Quantum. Electron. QE-9, 919-933 (1973).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Nature Mater. (1)

B. Ferguson and X. C. Zhang, "Materials for terahertz science and technology," Nature Mater. 1, 26-33 (2002).
[CrossRef]

Opt. Commun. (1)

W. L. She and W. K. Lee, "Wave coupling theory of linear electrooptic effect," Opt. Commun. 195, 303-311 (2001).
[CrossRef]

Proc. IEEE. (1)

F. S. Chen, "Modulators for optical communications," Proc. IEEE. 58, 1440-1457 (1970).
[CrossRef]

Other (2)

R. L. Sutherland, "Electro-Optic Effects," in Handbook of Nonlinear Optics (Marcel Dekker, 2003), pp. 843-946.

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

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Figures (6)

Fig. 1
Fig. 1

Effect of external field E 0 on the output powers of E 1 (solid curve) and E 2 (dashed curve) when (a) δ = 0 , (b) δ = 0.3 π , (c) δ = 0.5 π , and (d) δ = 0.7 π .

Fig. 2
Fig. 2

Effect of external field E 0 on ϕ 1 ( L ) (solid curve) and ϕ 2 ( L ) (dashed curve) when (a) δ = 0 , (b) δ = 0.3 π , (c) δ = 0.5 π , and (d) δ = 0.7 π .

Fig. 3
Fig. 3

Configuration of the polarization rotator.

Fig. 4
Fig. 4

Relations of the various unit vectors concerned: r ^ l 1 and r ^ l 2 are the slow and the fast axes of the first λ / 4 plate, x and y are the directions of the x axis and the y axis of the crystal, E ^ 1 and E ^ 2 are the directions of the two perpendicular components of light in the crystal, E ^ in is the polarization direction of the incident light, and c is the unit direction of the external field.

Fig. 5
Fig. 5

Effect of external field E 0 on the output powers of E 1 (solid curve) and E 2 (dashed curve).

Fig. 6
Fig. 6

Effect of external field E 0 on the polarization angle θ.

Equations (28)

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E ( t ) = E ( 0 ) + [ 1 2 E ( ω ) exp ( i ω t ) + c.c . ] ,
E ( ω ) = E 1 ( ω ) + E 2 ( ω ) = E 1 ( r ) exp ( i k 1 r ) + E 2 ( r ) exp ( i k 2 r ) ,
E ( ω ) = E 1 ( ω ) + E 2 ( ω ) = E 1 ( r ) exp ( i k 1 r ) + E 2 ( r ) exp ( i k 2 r ) .
E 1 ( r ) = E 1 ( r ) a ,    E 2 ( r ) = E 2 ( r ) b ,    E ( 0 ) = E 0 c ,
r eff 1 = j , k , l ( ε j j ε k k ) ( a j r j k l b k c l ) ,
r eff2 = j , k , l ( ε j j ε k k ) ( a j r j k l a k c l ) ,
r eff3 = j , k , l ( ε j j ε k k ) ( b j r j k l b k c l ) ,
d E 1 ( r ) d r = i d 1 E 2 ( r ) e i Δ k r i d 2 E 1 ( r ) ,
d E 2 ( r ) d r = i d 3 E 1 ( r ) e i Δ k r i d 4 E 2 ( r ) ,
d 1 = k 0 2 n 1 r eff 1 E 0 ,    d 2 = k 0 2 n 1 r eff 2 E 0 ,
d 3 = k 0 2 n 2 r eff 1 E 0 ,    d 4 = k 0 2 n 2 r eff3 E 0 ,
E 1 ( ω ) = E 1 ( r ) exp ( i k 1 r ) = ρ 1 ( r ) exp [ i ( k 1 + β ) r + i ϕ 1 ( r ) ] ,
E 2 ( ω ) = E 2 ( r ) exp ( i k 2 r ) = ρ 2 ( r ) exp ( i δ ) exp [ i ( k 1 + β ) r + i ϕ 2 ( r ) ] ,
ρ 1 ( r ) = { [ E 10 cos ( μ r ) + d 1 μ sin δ E 20 sin ( μ r ) ] 2 + [ γ E 10 d 1 cos δ E 20 μ ] 2 sin 2 ( μ r ) } 1 / 2 ,
ϕ 1 ( r ) = arg [ E 10 cos ( μ r ) + d 1 μ sin δ E 20 sin ( μ r ) + i γ E 10 d 1 cos δ E 20 μ sin ( μ r ) ] ,
ρ 2 ( r ) = { [ E 20 cos ( μ r ) d 3 μ sin δ E 10 sin ( μ r ) ] 2 + [ γ E 20 + d 3 cos δ E 10 μ ] 2 sin 2 ( μ r ) } 1 / 2 ,
ϕ 2 ( r ) = arg [ E 20 cos ( μ r ) d 3 μ sin δ E 10 sin ( μ r ) + i γ E 20 d 3 cos δ E 10 μ sin ( μ r ) ] ,
μ = 1 2 [ ( Δ k + d 2 d 4 ) 2 + 4 d 1 d 3 ] 1 / 2 ,
γ = d 4 d 2 Δ k 2 ,    β = Δ k d 2 d 4 2 .
E 1 ( ω ) E 10 exp [ i ( k 1 d 2 ) r ] ,
E 2 ( ω ) E 20 exp ( i δ ) exp [ i ( k 2 d 4 ) r ] .
E 1 ( r ) = E 10 cos ( d 1 r ) + E 20 sin ( d 1 r ) ,
E 2 ( r ) = i [ E 20 cos ( d 1 r ) E 10 sin ( d 1 r ) ] .
E in = cos ( φ ) i ^ + sin ( φ ) j ^ ,
E p 1 = sin ( φ + π 8 ) r ^ l 1 + i cos ( φ + π 8 ) r ^ l 2 .
E c = sin ( φ + π 8 + d 1 L ) r ^ l 1 + i cos ( φ + π 8 + d 1 L ) r ^ l 2 ,
E out = sin ( φ + π 8 + d 1 L ) r ^ l 1 + cos ( φ + π 8 + d 1 L ) r ^ l 2 .
E out = cos ( φ + d 1 L ) i ^ + sin ( φ + d 1 L ) j ^ .

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