Abstract

We develop a wave-coupling theory of the linear electro-optic effect, which includes linear absorption of the electro-optic medium. A general solution of the output complex amplitudes of two independent polarized components of a light wave is obtained for a light wave propagating along any direction with an external dc electric field along an arbitrary direction in any linear absorbent electro-optic medium. We use a LiNbO3 crystal as an example for studying the influences of linear absorption on the phases and amplitudes of the two wave components as well as the half-wave voltage and the extinction ratio of electro-optic modulation. © 2005 Optical Society of America.

© 2005 Optical Society of America

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References

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  1. N. C. J. van der Valk, T. Wenckebach, and P. C. M. Planken, “Full mathematical description of electro-optic detection in optically isotropic crystals,” J. Opt. Soc. Am. B  21, 622–631 (2004).
    [CrossRef]
  2. A. Yacoubian and P. K. Das, “Digital-to-analog conversion using electrooptic modulators,” IEEE Photonics Technol. Lett.  15, 117–119 (2003).
    [CrossRef]
  3. B.-E. Benkelfat, E.-H. Horache, Q. Zou, and B. Vinouze, “An electro-optic modulation technique for direct and accurate measurement of birefringence,” Opt. Commun.  221, 271–278 (2003).
    [CrossRef]
  4. J. A. Ferrari, E. Garbusi, and E. M. Frins, “Modified Michelson interferometer with electrooptic phase control,” Opt. Commun.  209, 245–253 (2002).
    [CrossRef]
  5. V. Bernard, C. Daussy, G. Nogues, L. Constantin, P. E. Durand, A. Amy-Klein, A. Van Lerberghe, and C. Chardonnet, “CO2 laser stabilization to 0.1-Hz level using external electrooptic modulation,” IEEE J. Quantum Electron.  33, 1282–1287 (1997).
    [CrossRef]
  6. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light wares in a nonlineardielectric,” Phys. Rev.  127, 1918–1939 (1962).
    [CrossRef]
  7. W. L. She and W. K. Lee, “Wave coupling theory of linear electrooptic effect,” Opt. Commun.  195, 303–311 (2001).
    [CrossRef]
  8. A. Yariv, Quantum Electronics (Wiley, 1988), pp. 298–338.
  9. A. M. Glass, D. von der Linde, and T. J. Negran, “High-voltage bulk photovoltaic effect and the photorefrac-tive process in LiNbO3,” Appl. Phys. Lett.  25, 233–235 (1974).
    [CrossRef]

2004 (1)

2003 (2)

A. Yacoubian and P. K. Das, “Digital-to-analog conversion using electrooptic modulators,” IEEE Photonics Technol. Lett.  15, 117–119 (2003).
[CrossRef]

B.-E. Benkelfat, E.-H. Horache, Q. Zou, and B. Vinouze, “An electro-optic modulation technique for direct and accurate measurement of birefringence,” Opt. Commun.  221, 271–278 (2003).
[CrossRef]

2002 (1)

J. A. Ferrari, E. Garbusi, and E. M. Frins, “Modified Michelson interferometer with electrooptic phase control,” Opt. Commun.  209, 245–253 (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]

1997 (1)

V. Bernard, C. Daussy, G. Nogues, L. Constantin, P. E. Durand, A. Amy-Klein, A. Van Lerberghe, and C. Chardonnet, “CO2 laser stabilization to 0.1-Hz level using external electrooptic modulation,” IEEE J. Quantum Electron.  33, 1282–1287 (1997).
[CrossRef]

1974 (1)

A. M. Glass, D. von der Linde, and T. J. Negran, “High-voltage bulk photovoltaic effect and the photorefrac-tive process in LiNbO3,” Appl. Phys. Lett.  25, 233–235 (1974).
[CrossRef]

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light wares in a nonlineardielectric,” Phys. Rev.  127, 1918–1939 (1962).
[CrossRef]

Amy-Klein, A.

V. Bernard, C. Daussy, G. Nogues, L. Constantin, P. E. Durand, A. Amy-Klein, A. Van Lerberghe, and C. Chardonnet, “CO2 laser stabilization to 0.1-Hz level using external electrooptic modulation,” IEEE J. Quantum Electron.  33, 1282–1287 (1997).
[CrossRef]

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light wares in a nonlineardielectric,” Phys. Rev.  127, 1918–1939 (1962).
[CrossRef]

Benkelfat, B.-E.

B.-E. Benkelfat, E.-H. Horache, Q. Zou, and B. Vinouze, “An electro-optic modulation technique for direct and accurate measurement of birefringence,” Opt. Commun.  221, 271–278 (2003).
[CrossRef]

Bernard, V.

V. Bernard, C. Daussy, G. Nogues, L. Constantin, P. E. Durand, A. Amy-Klein, A. Van Lerberghe, and C. Chardonnet, “CO2 laser stabilization to 0.1-Hz level using external electrooptic modulation,” IEEE J. Quantum Electron.  33, 1282–1287 (1997).
[CrossRef]

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light wares in a nonlineardielectric,” Phys. Rev.  127, 1918–1939 (1962).
[CrossRef]

Chardonnet, C.

V. Bernard, C. Daussy, G. Nogues, L. Constantin, P. E. Durand, A. Amy-Klein, A. Van Lerberghe, and C. Chardonnet, “CO2 laser stabilization to 0.1-Hz level using external electrooptic modulation,” IEEE J. Quantum Electron.  33, 1282–1287 (1997).
[CrossRef]

Constantin, L.

V. Bernard, C. Daussy, G. Nogues, L. Constantin, P. E. Durand, A. Amy-Klein, A. Van Lerberghe, and C. Chardonnet, “CO2 laser stabilization to 0.1-Hz level using external electrooptic modulation,” IEEE J. Quantum Electron.  33, 1282–1287 (1997).
[CrossRef]

Das, P. K.

A. Yacoubian and P. K. Das, “Digital-to-analog conversion using electrooptic modulators,” IEEE Photonics Technol. Lett.  15, 117–119 (2003).
[CrossRef]

Daussy, C.

V. Bernard, C. Daussy, G. Nogues, L. Constantin, P. E. Durand, A. Amy-Klein, A. Van Lerberghe, and C. Chardonnet, “CO2 laser stabilization to 0.1-Hz level using external electrooptic modulation,” IEEE J. Quantum Electron.  33, 1282–1287 (1997).
[CrossRef]

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light wares in a nonlineardielectric,” Phys. Rev.  127, 1918–1939 (1962).
[CrossRef]

Durand, P. E.

V. Bernard, C. Daussy, G. Nogues, L. Constantin, P. E. Durand, A. Amy-Klein, A. Van Lerberghe, and C. Chardonnet, “CO2 laser stabilization to 0.1-Hz level using external electrooptic modulation,” IEEE J. Quantum Electron.  33, 1282–1287 (1997).
[CrossRef]

Ferrari, J. A.

J. A. Ferrari, E. Garbusi, and E. M. Frins, “Modified Michelson interferometer with electrooptic phase control,” Opt. Commun.  209, 245–253 (2002).
[CrossRef]

Frins, E. M.

J. A. Ferrari, E. Garbusi, and E. M. Frins, “Modified Michelson interferometer with electrooptic phase control,” Opt. Commun.  209, 245–253 (2002).
[CrossRef]

Garbusi, E.

J. A. Ferrari, E. Garbusi, and E. M. Frins, “Modified Michelson interferometer with electrooptic phase control,” Opt. Commun.  209, 245–253 (2002).
[CrossRef]

Glass, A. M.

A. M. Glass, D. von der Linde, and T. J. Negran, “High-voltage bulk photovoltaic effect and the photorefrac-tive process in LiNbO3,” Appl. Phys. Lett.  25, 233–235 (1974).
[CrossRef]

Horache, E.-H.

B.-E. Benkelfat, E.-H. Horache, Q. Zou, and B. Vinouze, “An electro-optic modulation technique for direct and accurate measurement of birefringence,” Opt. Commun.  221, 271–278 (2003).
[CrossRef]

Lee, W. K.

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

Negran, T. J.

A. M. Glass, D. von der Linde, and T. J. Negran, “High-voltage bulk photovoltaic effect and the photorefrac-tive process in LiNbO3,” Appl. Phys. Lett.  25, 233–235 (1974).
[CrossRef]

Nogues, G.

V. Bernard, C. Daussy, G. Nogues, L. Constantin, P. E. Durand, A. Amy-Klein, A. Van Lerberghe, and C. Chardonnet, “CO2 laser stabilization to 0.1-Hz level using external electrooptic modulation,” IEEE J. Quantum Electron.  33, 1282–1287 (1997).
[CrossRef]

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light wares in a nonlineardielectric,” Phys. Rev.  127, 1918–1939 (1962).
[CrossRef]

Planken, P. C.

She, W. L.

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

van der Valk, N. C.

Van Lerberghe, A.

V. Bernard, C. Daussy, G. Nogues, L. Constantin, P. E. Durand, A. Amy-Klein, A. Van Lerberghe, and C. Chardonnet, “CO2 laser stabilization to 0.1-Hz level using external electrooptic modulation,” IEEE J. Quantum Electron.  33, 1282–1287 (1997).
[CrossRef]

Vinouze, B.

B.-E. Benkelfat, E.-H. Horache, Q. Zou, and B. Vinouze, “An electro-optic modulation technique for direct and accurate measurement of birefringence,” Opt. Commun.  221, 271–278 (2003).
[CrossRef]

von der Linde, D.

A. M. Glass, D. von der Linde, and T. J. Negran, “High-voltage bulk photovoltaic effect and the photorefrac-tive process in LiNbO3,” Appl. Phys. Lett.  25, 233–235 (1974).
[CrossRef]

Wenckebach, T.

Yacoubian, A.

A. Yacoubian and P. K. Das, “Digital-to-analog conversion using electrooptic modulators,” IEEE Photonics Technol. Lett.  15, 117–119 (2003).
[CrossRef]

Yariv, A.

A. Yariv, Quantum Electronics (Wiley, 1988), pp. 298–338.

Zou, Q.

B.-E. Benkelfat, E.-H. Horache, Q. Zou, and B. Vinouze, “An electro-optic modulation technique for direct and accurate measurement of birefringence,” Opt. Commun.  221, 271–278 (2003).
[CrossRef]

Appl. Phys. Lett. (1)

A. M. Glass, D. von der Linde, and T. J. Negran, “High-voltage bulk photovoltaic effect and the photorefrac-tive process in LiNbO3,” Appl. Phys. Lett.  25, 233–235 (1974).
[CrossRef]

IEEE J. Quantum Electron. (1)

V. Bernard, C. Daussy, G. Nogues, L. Constantin, P. E. Durand, A. Amy-Klein, A. Van Lerberghe, and C. Chardonnet, “CO2 laser stabilization to 0.1-Hz level using external electrooptic modulation,” IEEE J. Quantum Electron.  33, 1282–1287 (1997).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

A. Yacoubian and P. K. Das, “Digital-to-analog conversion using electrooptic modulators,” IEEE Photonics Technol. Lett.  15, 117–119 (2003).
[CrossRef]

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

Opt. Commun. (3)

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

B.-E. Benkelfat, E.-H. Horache, Q. Zou, and B. Vinouze, “An electro-optic modulation technique for direct and accurate measurement of birefringence,” Opt. Commun.  221, 271–278 (2003).
[CrossRef]

J. A. Ferrari, E. Garbusi, and E. M. Frins, “Modified Michelson interferometer with electrooptic phase control,” Opt. Commun.  209, 245–253 (2002).
[CrossRef]

Phys. Rev. (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light wares in a nonlineardielectric,” Phys. Rev.  127, 1918–1939 (1962).
[CrossRef]

Other (1)

A. Yariv, Quantum Electronics (Wiley, 1988), pp. 298–338.

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Figures (11)

Fig. 1
Fig. 1

ϕ 1 as a function of α 1 and α 2 .

Fig. 2
Fig. 2

ϕ 2 as a function of α 1 and α 2 .

Fig. 3
Fig. 3

ϕ 2 ϕ 1 as a function of α 1 and α 2 .

Fig. 4
Fig. 4

Electric field (corresponding to the half-wave voltage) of electro-optic modulation as a function of α 1 and α 2 .

Fig. 5
Fig. 5

Functional dependence of Δ ϕ 1 on E 0 .

Fig. 6
Fig. 6

Functional dependence of Δ ϕ 2 on E 0 .

Fig. 7
Fig. 7

Amplitude of E 1 L as a function of α 1 and α 2 .

Fig. 8
Fig. 8

Amplitude of E 2 L as a functionof α 1 and α 2 .

Fig. 9
Fig. 9

Output intensity versus the absorption coefficients α 1 and α 2 with E 0 = 2000 V cm .

Fig. 10
Fig. 10

Output intensity versus the absorption coefficients α 1 and α 2 with E 0 = 0 V cm .

Fig. 11
Fig. 11

Extinction ratio (ER) corresponding to both E p and E 0 = 0 versus the absorption coefficients α 1 and α 2 .

Equations (37)

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[ E ( t ) ] 2 E ( t ) + μ 0 σ E ( t ) t + 1 c 2 2 [ ϵ E ( t ) ] t 2 = μ 0 2 P NLS ( t ) t 2 ,
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 1 ( r ) = E 1 ( r ) a , E 2 ( r ) = E 2 ( r ) b , E ( 0 ) = E 0 c ,
d E 1 ( r ) dr = i d 1 E 2 ( r ) exp ( i Δ k r ) i d 2 E 1 ( r ) α 1 2 E 1 ( r ) ,
d E 2 ( r ) d r = i d 3 E 1 ( r ) exp ( i Δ k r ) i d 4 E 2 ( r ) α 2 2 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 eff 3 E 0 ,
r eff 1 = j , k , l ( ϵ j j ϵ k k ) ( a j r j k l b k c l ) ,
r eff 2 = j , k , l ( ϵ j j ϵ k k ) ( a j r j k l a k c l ) ,
r eff 3 = j , k , l ( ϵ j j ϵ k k ) ( b j r j k l b k c 1 ) ,
Δ k = k 2 k 1 ,
E 1 ( ω ) = E 1 ( r ) exp ( i k 1 r ) = ρ 1 ( r ) exp [ i ( k 1 + β ) r + i ϕ 1 ( r ) α 1 + α 2 4 r ] ,
E 2 ( ω ) = E 2 ( r ) exp ( i k 2 r ) = ρ 2 ( r ) exp [ i ( k 1 + β ) r + i ϕ 2 ( r ) α 1 + α 2 4 r ] ,
ρ 1 = { E 1 2 ( 0 ) cos 2 ( μ r ) + [ γ E 1 ( 0 ) d 1 E 2 ( 0 ) μ ] 2 sin 2 ( μ r ) } 1 2 ,
ϕ 1 ( r ) = arg [ E 1 ( 0 ) cos ( μ r ) + i γ E 1 ( 0 ) d 1 E 2 ( 0 ) μ sin ( μ r ) ] ,
ρ 2 ( r ) = { E 2 2 ( 0 ) cos 2 ( μ r ) + [ γ E 2 ( 0 ) + d 3 E 1 ( 0 ) μ ] 2 sin 2 ( μ r ) } 1 2 ,
ϕ 2 ( r ) = arg [ E 2 ( 0 ) sin ( μ r ) + i γ E 2 ( 0 ) d 3 E 1 ( 0 ) 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 .
I out = ρ 1 2 ( L ) + ρ 2 2 ( L ) 2 ρ 1 ( L ) ρ 2 ( L ) cos [ ϕ 1 ( L ) ϕ 2 ( L ) ] 2 exp ( α 1 + α 2 2 L ) ,
E 1 ( ω ) = ρ 1 ( r ) exp [ i ( k 1 + β ) r + i ϕ 1 ( r ) α 1 + α 2 4 r ] ,
E 2 ( ω ) = ρ 2 ( r ) exp [ i ( k 1 + β ) r + i ϕ 2 ( r ) α 1 + α 2 4 r ] ,
ρ 1 ( r ) = ( 1 2 + m 1 2 ) 1 2 ,
ϕ 1 ( r ) = arg ( 1 + i m 1 ) ,
ρ 2 ( r ) = ( 2 2 + m 2 2 ) 1 2 ,
ϕ 2 ( r ) = arg ( 2 + i m 2 ) ,
1 = t E 1 ( 0 ) cos ( p r ) 2 + s [ u E 1 ( 0 ) d 1 p E 2 ( 0 ) ] cos ( p r ) t [ ν E 1 ( 0 ) + d 1 q E 2 ( 0 ) ] sin ( p r ) 2 ( p 2 + q 2 ) ,
m 1 = s E 1 ( 0 ) sin ( p r ) 2 + t [ u E 1 ( 0 ) d 1 p E 2 ( 0 ) ] sin ( p r ) + s [ ν E 1 ( 0 ) + d 1 q E 2 ( 0 ) ] cos ( p r ) 2 ( p 2 + q 2 ) ,
2 = t E 2 ( 0 ) cos ( p r ) 2 + s [ u E 2 ( 0 ) d 3 p E 1 ( 0 ) ] cos ( p r ) + t [ ν E 2 ( 0 ) d 3 q E 1 ( 0 ) ] sin ( p r ) 2 ( p 2 + q 2 ) ,
m 2 = s E 2 ( 0 ) sin ( p r ) 2 + t [ u E 2 ( 0 ) d 3 p E 1 ( 0 ) ] sin ( p r ) + s [ ν E 2 ( 0 ) + d 3 q E 1 ( 0 ) ] cos ( p r ) 2 ( p 2 + q 2 ) ,
u = ( p γ + q α ) , ν = ( p α q γ ) ,
s = exp ( q r ) exp ( q r ) , t = exp ( q r ) + exp ( q r ) ,
p = 2 γ α { 2 ( γ 2 α 2 + d 1 d 3 ) + 2 [ ( γ 2 α 2 + d 1 d 3 ) 2 + 4 γ 2 α 2 ] 1 2 } 1 2 ,
q = 1 2 { 2 ( γ 2 α 2 + d 1 d 3 ) + 2 [ ( γ 2 α 2 + d 1 d 3 ) 2 + 4 γ 2 α 2 ] 1 2 } 1 2 ,
α = α 1 α 2 4 .
I out = ρ 1 2 ( L ) + ρ 2 2 ( L ) 2 ρ 1 ( L ) ρ 2 ( L ) cos [ ϕ 1 ( L ) ϕ 2 ( L ) ] 2 exp ( α 1 + α 2 2 L ) .

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