Abstract

A new kind of optical self-defocusing is described that in steady state is independent of optical beam power and is strongly asymmetric. The physical mechanism responsible is the photorefractive effect. We present a theory that explains the observed dependence of this self-defocusing on polarization, angle of incidence, beam size, and crystal orientation. Experimental results, using a single-domain crystal of BaTiO3, are presented that show excellent quantitative agreement with the theory. Possible device applications are discussed, including an optical diode and a low-power bistable device with permanent memory.

© 1982 Optical Society of America

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References

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  1. J. Feinberg and R. W. Hellwarth, “Phase-conjugating mirror with continuous-wave gain,” Opt. Lett. 5, 519–521 (1980); Opt. Lett. 6, 257 (1981).
    [CrossRef] [PubMed]
  2. R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
    [CrossRef]
  3. G. A. Askaryan, “Effects of the gradient of a strong electromagnetic beam on electrons and atoms,” Sov. Phys. JETP 15, 1088–1090 (1962).
  4. A. Ashkin, G. D. Boyd, J. M. Dziedzig, R. G. Smith, A. A. Ballman, J. J. Levinstein, and K. Nassau, “Optically induced refractive index inhomogeneities in LiNbO3and LiTaO3,” Appl. Phys. Lett. 9, 72–74 (1966).
    [CrossRef]
  5. F. S. Chen, “A laser-induced inhomogeneity of refractive indices in KTN,” J. Appl. Phys. 38, 3418–3420 (1967).
    [CrossRef]
  6. M. G. Moharam and L. Young, “Hologram writing by the photorefractive effect and Gaussian beams at constant applied voltage,” J. Appl. Phys. 47, 4048–4051 (1976).
    [CrossRef]
  7. J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980); J. Appl. Phys. 52, 537 (1981).
    [CrossRef]
  8. A. R. Johnston and J. M. Weingart, “Determination of the low-frequency linear electro-optic effect in tetragonal BaTiO3,” J. Opt. Soc. Am. 55, 828–834 (1965); A. R. Johnston, “The strain-free electrooptic effect in single crystal barium titanate,” Appl. Phys. Lett 7, 195–198 (1965); Appl. Phys. Lett 8, 54 (1966); I. P. Kaminow, “Barium titanate light phase modulator,” Appl. Phys. Lett. 7, 123–125 (1965); “Barium titanate modulator II,” Appl. Phys. Lett. 8, 305–307 (1966).
    [CrossRef]
  9. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [CrossRef]

1980 (2)

J. Feinberg and R. W. Hellwarth, “Phase-conjugating mirror with continuous-wave gain,” Opt. Lett. 5, 519–521 (1980); Opt. Lett. 6, 257 (1981).
[CrossRef] [PubMed]

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980); J. Appl. Phys. 52, 537 (1981).
[CrossRef]

1976 (1)

M. G. Moharam and L. Young, “Hologram writing by the photorefractive effect and Gaussian beams at constant applied voltage,” J. Appl. Phys. 47, 4048–4051 (1976).
[CrossRef]

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

1967 (1)

F. S. Chen, “A laser-induced inhomogeneity of refractive indices in KTN,” J. Appl. Phys. 38, 3418–3420 (1967).
[CrossRef]

1966 (1)

A. Ashkin, G. D. Boyd, J. M. Dziedzig, R. G. Smith, A. A. Ballman, J. J. Levinstein, and K. Nassau, “Optically induced refractive index inhomogeneities in LiNbO3and LiTaO3,” Appl. Phys. Lett. 9, 72–74 (1966).
[CrossRef]

1965 (1)

1964 (1)

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[CrossRef]

1962 (1)

G. A. Askaryan, “Effects of the gradient of a strong electromagnetic beam on electrons and atoms,” Sov. Phys. JETP 15, 1088–1090 (1962).

Ashkin, A.

A. Ashkin, G. D. Boyd, J. M. Dziedzig, R. G. Smith, A. A. Ballman, J. J. Levinstein, and K. Nassau, “Optically induced refractive index inhomogeneities in LiNbO3and LiTaO3,” Appl. Phys. Lett. 9, 72–74 (1966).
[CrossRef]

Askaryan, G. A.

G. A. Askaryan, “Effects of the gradient of a strong electromagnetic beam on electrons and atoms,” Sov. Phys. JETP 15, 1088–1090 (1962).

Ballman, A. A.

A. Ashkin, G. D. Boyd, J. M. Dziedzig, R. G. Smith, A. A. Ballman, J. J. Levinstein, and K. Nassau, “Optically induced refractive index inhomogeneities in LiNbO3and LiTaO3,” Appl. Phys. Lett. 9, 72–74 (1966).
[CrossRef]

Boyd, G. D.

A. Ashkin, G. D. Boyd, J. M. Dziedzig, R. G. Smith, A. A. Ballman, J. J. Levinstein, and K. Nassau, “Optically induced refractive index inhomogeneities in LiNbO3and LiTaO3,” Appl. Phys. Lett. 9, 72–74 (1966).
[CrossRef]

Chen, F. S.

F. S. Chen, “A laser-induced inhomogeneity of refractive indices in KTN,” J. Appl. Phys. 38, 3418–3420 (1967).
[CrossRef]

Chiao, R. Y.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[CrossRef]

Dziedzig, J. M.

A. Ashkin, G. D. Boyd, J. M. Dziedzig, R. G. Smith, A. A. Ballman, J. J. Levinstein, and K. Nassau, “Optically induced refractive index inhomogeneities in LiNbO3and LiTaO3,” Appl. Phys. Lett. 9, 72–74 (1966).
[CrossRef]

Feinberg, J.

J. Feinberg and R. W. Hellwarth, “Phase-conjugating mirror with continuous-wave gain,” Opt. Lett. 5, 519–521 (1980); Opt. Lett. 6, 257 (1981).
[CrossRef] [PubMed]

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980); J. Appl. Phys. 52, 537 (1981).
[CrossRef]

Garmire, E.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[CrossRef]

Heiman, D.

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980); J. Appl. Phys. 52, 537 (1981).
[CrossRef]

Hellwarth, R. W.

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980); J. Appl. Phys. 52, 537 (1981).
[CrossRef]

J. Feinberg and R. W. Hellwarth, “Phase-conjugating mirror with continuous-wave gain,” Opt. Lett. 5, 519–521 (1980); Opt. Lett. 6, 257 (1981).
[CrossRef] [PubMed]

Johnston, A. R.

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

Levinstein, J. J.

A. Ashkin, G. D. Boyd, J. M. Dziedzig, R. G. Smith, A. A. Ballman, J. J. Levinstein, and K. Nassau, “Optically induced refractive index inhomogeneities in LiNbO3and LiTaO3,” Appl. Phys. Lett. 9, 72–74 (1966).
[CrossRef]

Moharam, M. G.

M. G. Moharam and L. Young, “Hologram writing by the photorefractive effect and Gaussian beams at constant applied voltage,” J. Appl. Phys. 47, 4048–4051 (1976).
[CrossRef]

Nassau, K.

A. Ashkin, G. D. Boyd, J. M. Dziedzig, R. G. Smith, A. A. Ballman, J. J. Levinstein, and K. Nassau, “Optically induced refractive index inhomogeneities in LiNbO3and LiTaO3,” Appl. Phys. Lett. 9, 72–74 (1966).
[CrossRef]

Smith, R. G.

A. Ashkin, G. D. Boyd, J. M. Dziedzig, R. G. Smith, A. A. Ballman, J. J. Levinstein, and K. Nassau, “Optically induced refractive index inhomogeneities in LiNbO3and LiTaO3,” Appl. Phys. Lett. 9, 72–74 (1966).
[CrossRef]

Tanguay, A. R.

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980); J. Appl. Phys. 52, 537 (1981).
[CrossRef]

Townes, C. H.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[CrossRef]

Weingart, J. M.

Young, L.

M. G. Moharam and L. Young, “Hologram writing by the photorefractive effect and Gaussian beams at constant applied voltage,” J. Appl. Phys. 47, 4048–4051 (1976).
[CrossRef]

Appl. Phys. Lett. (1)

A. Ashkin, G. D. Boyd, J. M. Dziedzig, R. G. Smith, A. A. Ballman, J. J. Levinstein, and K. Nassau, “Optically induced refractive index inhomogeneities in LiNbO3and LiTaO3,” Appl. Phys. Lett. 9, 72–74 (1966).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

J. Appl. Phys. (3)

F. S. Chen, “A laser-induced inhomogeneity of refractive indices in KTN,” J. Appl. Phys. 38, 3418–3420 (1967).
[CrossRef]

M. G. Moharam and L. Young, “Hologram writing by the photorefractive effect and Gaussian beams at constant applied voltage,” J. Appl. Phys. 47, 4048–4051 (1976).
[CrossRef]

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980); J. Appl. Phys. 52, 537 (1981).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Lett. (1)

Phys. Rev. Lett. (1)

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[CrossRef]

Sov. Phys. JETP (1)

G. A. Askaryan, “Effects of the gradient of a strong electromagnetic beam on electrons and atoms,” Sov. Phys. JETP 15, 1088–1090 (1962).

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Figures (5)

Fig. 1
Fig. 1

Geometry for optical beam fanning. The z axis is taken to be the direction of propagation of the incident beam in the crystal. The y axis is chosen to lie in the plane formed by the crystal’s optic axis (c axis) and the z axis. The direction of beam fanning is determined by the direction of the c axis.

Fig. 2
Fig. 2

Plots of the intensity of the incident Gaussian beam with 1/e beam diameter 2w0, the steady-state charge distribution in the sample arising from charge migration, the electrostatic field that is due to these charges, and the refractive-index change caused by this electric field. Note that the migrating charges, which have positive charge in BaTiO3, accumulate in the less intense wings of the intensity curve. The refractive-index change deflects the incident beam and causes it to fan out. The index change (bottom plot) is linear with distance in the central region and can be thought of as a prism that deflects the incident beam to one side (here, to the right).

Fig. 3
Fig. 3

Optical beam fanning in a single-domain crystal of BaTiO3. The crystal is the small rectangle inside the larger square, which is an oil-filled cuvette. The beam enters the crystal from the left and emerges into a glass cell filled with I2 vapor on the right. Fluorescence from the I2 makes the exit beam visible. The c axis of the crystal lies in the plane of the page and is pointed in the 10 o’clock direction. In both photographs the incident beam is polarized in the plane of the page, making it an extraordinary ray in the crystal. This produces the strong beam fanning visible in the top photograph. In the bottom photograph the crystal was vibrated slightly during the exposure, thereby washing out the index pattern and eliminating all beam fanning. Beam fanning also disappears if the polarization of the incident beam is rotated by 90° to make it an ordinary ray in the crystal. The width of the transmitted beam appears broadened by radiative trapping in the I2 cell. The stripes seen in the fan are probably due to scratches on the exit face of the crystal.

Fig. 4
Fig. 4

Experimental and theoretical plots of the far-field beam-fanning intensities in the forward direction for various angles θ between the incident beam and the c axis of the crystal. Note that the intensity peak shifts to the right as θ is increased and that an interference dip appears on the opposite side. The intensity fan visible in Fig. 3 is far to the right here and off the scale of these plots. The fanning-strength parameter S is computed for each θ from Eq. (17) and substituted into a diffraction integral to produce the theoretical curves.

Fig. 5
Fig. 5

A proposed optical bistable device with erasable memory. In state (a) strong beam fanning prevents any of the incident beam from hitting the mirror. In state (b) the beam returning from the mirror inhibits beam fanning. If the path between the crystal and the mirror is interrupted, the device will be set in state (a) and will remain there if the interruption is removed. If the crystal is temporarily flooded with a uniform beam (not shown), it will be reset to state (b). If the incident beam is turned off, the crystal will remember which state it is in and will resume it when the incident beam is turned on again.

Equations (19)

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X ( x ) = - 0 ω · R · E ( x ) · ω ,
E ( x ) = Re ( i m q k B T e k 0 q ^ α + i f · q ^ 1 + i α · f + α 2 e i q · x ) .
k 0 = [ N e 2 / ( 0 k B T ) ] 1 / 2 .
m q = 2 I q d 2 ( q ) / - + I q d 2 ( q ) ,
I q = 1 2 π + + e - i q · x I ( x ) d 2 ( x )
I ( x , y , z = 0 ) = G ( x , y , z = 0 ) = I exp [ - ( x 2 + y 2 ) / w 0 2 ] ,
E ( x ) = 1 2 π - + Re { 2 i k B T π q 0 2 e q ^ [ 1 + q 2 / k 0 2 ] × exp ( - q 2 q 0 2 + i q · x ) } ) d 2 ( q ) ,
E ( x ) = k B T e w 0 2 T G ( x ) / G 0 = - 2 k B T ( x x ^ + y y ^ ) e π w 0 2 exp - [ ( x 2 + y 2 ) / w 0 2 ] ,
G 0 - + G ( x , y , z = 0 ) d 2 ( x ) .
E ( x ) ( 1 + f 2 ) E ( x ) E 0 = 0 + E E 0 ( x ) ,
E E 0 ( x ) = - 1 2 π 0 2 π d β ( E 0 · q ^ ) q d d S [ s Φ ( 1 , 3 2 , - s 2 w 0 2 ) ] ,
Φ ( a , b , t ) = 1 + a b t + a b ( a + 1 ) ( b + 1 ) t 2 2 ! + a ( a + 1 ) ( a + 2 ) b ( b + 1 ) ( b + 2 ) t 3 3 ! + .
n ( x ) = ( 2 n 0 ) - 1 e * · X · ê ,
n ORD ( y ) = - [ E ( x ) · y ^ ] cos θ n o 3 2 r 13 ,
n EXT ( y ) = - [ E ( x ) · y ^ ] cos θ 2 n ( θ ) ( n o 4 r 13 sin 2 θ ) + 2 n o 2 n e 2 r 42 sin 2 θ + n e 4 r 33 cos 2 θ ) .
I ( y ) FAR - FIELD | - + U ( y ) T ( y ) exp ( - i y y k / f ) d y | 2 ,
T ( y ) = exp [ i k l n ( y ) ]
k l n ( y ) = S y w 0 exp ( - y 2 / w 0 2 ) ,
S k B T k L π w 0 e × ( n 0 4 r 13 sin 2 θ + 2 n 0 2 n e 2 r 42 sin 2 θ + n e 4 r 33 cos 2 θ ) ,