Investigation of symmetry of photorefractive effect in LiNbO<sub>3</sub>

doi:10.1364/OE.15.010782

« Show journal navigation

Investigation of symmetry of photorefractive effect in LiNbO₃

Ivan Turek and Norbert Tarjányi »View Author Affiliations

Optics Express, Vol. 15, Issue 17, pp. 10782-10788 (2007)
http://dx.doi.org/10.1364/OE.15.010782

View Full Text Article

Acrobat PDF (591 KB)

Journal Search
Article Lookup

Article Tools

Citations

Alert me when this article is cited
Export Citation/Save

Article Navigation

▸ Article Outline
– Abstract
1. Introduction
2. Basic relations
3. Arrangement of the experiment
4. Comparison of the experimental results with standardly used model
4. Conclusion
– References and links

Article Tools
Full Text
Article Info
References
Cited By
Figures
Metrics
Download PDF

Viewing Equations in the
HTML Article

Optimal Viewing Recommendations

Full Text
Article Info
References (16)
Cited By
Figures (5)
Metrics

Abstract

The investigation of the light-induced changes of refractive index in a LiNbO3: Fe sample in case of strip-like illumination is presented in the contribution. The changes of the refractive index are visualized by interferograms of the sample obtained for various orientation of the illuminated strip and various polarization of the light used during interferogram creation. The investigation shows that character of the dependences of the refractive index on the coordinate perpendicular to the illuminated strip is different for different strip orientation. It indicates the possibility that for different orientation of gradient of the illumination the different mechanisms are responsible for changes of the refractive index.

1. Introduction

The photorefractive effect, i.e. the effect of change of the refractive index due to inhomogeneous illumination is of interest in numerous applications. There are papers devoted to utilization of the effect for optical data storage [1

Q. Gao and R. Kostuk, “Cross - talk noise and storage capacity of holographic memories with a LiNbO₃ crystal in the open - circuit condition,” Appl. Opt. 37, 929–936 (1998). [CrossRef]

, 2

J. Ashley, et al., “Holographic data storage,” IBM J. Res. Develop. 44, 341–368 (2000). [CrossRef]

], fabrication of devices for signal processing [3

A. K. Zajtsev, S. H. Lin, and K. Y. Hsu, “Sidelobe suppression of spectral response in holographic optical filter,” Opt. Commun. 190, 103–108 (2001). [CrossRef]

, 4

S.-F. Chen, C. S. Wu, and C.-C. Sun, “Design for a high dense wavelength division multiplexer based on volume holographic gratings,” Opt. Eng. 43, 2028–2033 (2004). [CrossRef]

] or optical wave guiding including solitons [5

S. Mailis, C. Riziotis, I. T. Wellington, P. G. R. Smith, C. B. E. Gawith, and R. W. Eason, “Direct ultraviolet writing of channel waveguides in congruent lithium niobate single crystals,” Opt. Lett. 28, 1433–1435 (2003). [CrossRef] [PubMed]

–7

M. Paturzo, L. Miccio, S. De Nicola, P. De Natale, and P. Ferraro, “Amplitude and phase reconstruction of photorefractive spatial bright-soliton in LiNbO₃ during its dynamic formation by digital holography,” Opt. Express 15, 8243–8251 (2007). [CrossRef] [PubMed]

]. The effect was explained as a result of photovoltaic effect and electrooptic effect soon after its first observation [8

A. Ashkin, G. D. Boyd, J. M. Dziedzic, R. G. Smith, A. A. Ballman, J. J. Levinstein, and K. Nassau, “Optically - induced refractive index inhomogeneities in LiNbO₃ and LiTaO₃ ,” Appl. Phys. Lett. 9, 72–74 (1966). [CrossRef]

–10

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

]. Continuing investigation during next decades provided the more detailed insight into the field of the photorefractivity [e.g. 11

B. I. Sturman, F. Agulló-López, M. Carrascosa, and L. Solymar, “On microscopic description of photorefractive phenomena,” Appl. Phys. B 68, 1013–1020 (1999). [CrossRef]

]. The photovoltaic effect and electrooptic effect are the basis of photorefractivity models used in literature till now. Our previous investigation of the effect in LiNbO₃ [12

I. Turek and N. Tarjányi, “Interference imaging of photorefractive record in thin sample of LiNbO₃ crystal,” Proc. SPIE 5945, 59450J-1 (2005).

, 13

I. Turek and N. Tarjányi, “The photorefractive effect in LiNbO₃ crystals,” 15th Czech-Polish-Slovak Conference on Wave and Quantum Aspects of Contemporary Optics, Proc. SPIE 6609, 660906 (2007).

] indicates that the symmetry of the spatial (coordinate) dependence of the refractive index modulation (of the photorefractive “record”) differs from the symmetry expected when the electrooptic effect is the mechanism responsible for the refractive index modulation. However, that investigation was performed only for illumination with gradient parallel with optical axis of the crystal, i.e. with z-direction. In order to obtain more complete information about the photorefractive effect we performed the “records” of strip-like illumination with its gradient oriented also in other directions in the crystal.

2. Basic relations

In general [e.g. 11

B. I. Sturman, F. Agulló-López, M. Carrascosa, and L. Solymar, “On microscopic description of photorefractive phenomena,” Appl. Phys. B 68, 1013–1020 (1999). [CrossRef]

, 14

P. Yeh, Introduction to photorefractive nonlinear optics , (John Wiley & Sons, Inc. New York, 1993) p.27.

], it is accepted that the internal electric current caused by a non-homogeneous illumination of the sample of a photorefractive material and the space charge field created by the current are essential for the existence of the photorefractive effect. The total current consists of the diffusion, photogalvanic (often named bulk photovoltaic) and drift currents and is given by equation

\vec{j} = e \vec{\vec{D}} grad (n_{c}) + \vec{β} I (\vec{r}) - e n_{c} \vec{\vec{μ}} \vec{E},

(1)

where e is the elementary charge, n_c is the density of electrons in the conduction band, β⃗ is the photogalvanic constant, I(r⃗) is the spatially dependent sample illumination, E⃗ is the internal electric field and µ⃗ the electron mobility, respectively. The internal electric field is produced by the space charge ρ, the density of which is given by equation

\frac{d ρ}{d t} = - div (\vec{j}),

(2)

and E⃗ is determined by ρ through the Maxwell equation

div (E_{i}) = η_{i, j} ρ,

(3)

where η_i,j is impermitivity tensor and i=1,2,3.

It follows from the equations above that the directional symmetry (the anisotropy) of the space charge field creation is given by symmetry of η_i,j, β_i and the illumination I(r_i) (because diffusion constants (and the carrier’s mobility) are usually assumed to be equal for each direction, i.e. D_ij=Dδ_ij).

In the accepted model of photorefractivity it is supposed that the refractive index is modulated by the electric field trough the linear electrooptic effect. It means that the refractive index can be expressed through impermitivity tensor

η_{ij} = η_{0 ij} + r_{i jk} E_{k},

(4)

where η_ij and η_0ij are elements of impermitivity tensor in presence and without electric field, r_ijk are elements of electrooptic tensor and E_k are components of the electric field influencing the refractive index. It means that the anisotropy of the effect depends also on symmetry of electrooptical tensor r_ijk .

The refractive index of waves propagating in various directions and possessing different polarizations can be expressed through the equation for refractive index ellipsoid [14

P. Yeh, Introduction to photorefractive nonlinear optics , (John Wiley & Sons, Inc. New York, 1993) p.27.

, 15

M. Born and E. Wolf, Principles of Optics , (Cambridge University Press, United Kingdom, 2002), pp.799–808.

]

\sum_{i, j} η_{i j} x_{i} x_{j} = 1 .

(5)

Since r_{i j k} and η_{i j} are symmetric with respect to i and j, i.e. r_{i j k} =r_{j i k} and η_{i j} =η_{j i} , the shortened (“matrix”) notation can be used and Eq. (4) can be rewritten as

η_{α} = η_{0 α} + r_{α k} E_{k},

(6)

where α=1 for i,j=1,1 ; α=2 for i,j=2,2 ; α=3 for i,j=3,3 ; α=4 for i,j=2,3; α=5 for i,j=2,3 and ; α=6 for i,j=1,2.

Using this expression the equation for index ellipsoid (5) can be rewritten [14

P. Yeh, Introduction to photorefractive nonlinear optics , (John Wiley & Sons, Inc. New York, 1993) p.27.

] into

(\frac{1}{n_{x}^{2}} + r_{1 k} E_{k}) x^{2} + (\frac{1}{n_{y}^{2}} + r_{2 k} E_{k}) y^{2} + (\frac{1}{n_{z}^{2}} + r_{3 k} E_{k}) z^{2} +

+ 2 y z r_{4 k} E_{k} + 2 x z r_{5 k} E_{k} + 2 x y r_{6 k} E_{k} = 1,

(7)

where n_x, n_y and n_z are the principal axes of refractive index ellipsoid in case when no electric field is applied. The intersection of the index ellipsoid with plane perpendicular to wave vector of propagating beam (“reading” beam used for visualization of the refractive index distribution) gives (in general) the ellipse which semi-axes are coincident with refractive indices of the ordinary and extraordinary polarizations of the reading beam. When the internal field E_k and values of r_αk are known, according to Eq. (4) the Eq. (7) describes the anisotropy of the refractive index.

Character of the current j determines not only anisotropy of the photorefractive effect, but also the symmetry of the spatial dependence of the refractive index. It follows from the known rate equations [14

P. Yeh, Introduction to photorefractive nonlinear optics , (John Wiley & Sons, Inc. New York, 1993) p.27.

]

\frac{\partial}{\partial t} (n_{c} + n_{d}) = \frac{1}{e} div (\vec{j}),

(8)

where n_d is the density of the electrons on donor centers (or traps).

For example: when the current results from the carrier diffusion only (dominantly) and the illumination is a symmetric function of only one coordinate (z in the following) then the increasing of the total charge density is described by a symmetric function, because ∂j/∂z is proportional to ∂²nc/∂z² and symmetry of n_c correlates (when the linear approximation is valid) with symmetry of I(z). On the other hand, when j is given by external electric field or photogalvanic effect symmetry, the total charge density distribution should be antisymmetric because ∂j/∂z is proportional to the first derivative of n_c as the derivation changes the even function into odd function and vice versa. So the symmetry of the internal electric field (and also changes of refractive index due to electrooptic effect) should differ from symmetry of charge distribution, because it is an integral of charge density (Eq.(3)). These examples illustrate that the knowledge of symmetry of refractive index modulation can give some information about the mechanism of its creation.

3. Arrangement of the experiment

For the experimental investigation of the symmetry of photorefractive effect we used an aperiodic illumination because the symmetry of the mechanism manifests itself more expressive in this case [12

I. Turek and N. Tarjányi, “Interference imaging of photorefractive record in thin sample of LiNbO₃ crystal,” Proc. SPIE 5945, 59450J-1 (2005).

, 13

]. In order to obtain results which are easily to interpret, we used (approximately) one-dimensional strip-like illumination. Such illumination was applied on sample of LiNbO₃: Fe (1cm×1cm×1cm) with polished two pairs of surfaces (oriented in z and y crystallographic directions). The sample was illuminated by blue beam of Ar laser (λ=488 nm) with Gaussian diameter of about 3 mm. The beam was restricted by a thin slit (with width of about 0.5 mm) put just in front of the sample surface (Fig. 1.). The used intensity of the writing beam was usually from 5 to 20 mW/mm² and exposure time usually from 100 to 900 seconds.

Fig. 1. Realization of the strip-like illumination.

In general, there are several techniques of imaging but taking into account the character of the created photorefractive record a technique employing the Mach-Zehnder interferometer was used [i.e. 16

M. de Angelis, S. De Nicola, A. Finizio, G. Pierattini, P. Ferraro, S. Pelli, G. Righini, and S. Sebastiani, “Digital-holography refractive-index-profile measurement of phase gratings,” Appl. Phys. Lett. 88, 111114–111116 (2006). [CrossRef]

] (Fig. 2.).

Fig. 2. Visualization of the refractive index changes. M - mirror, SM - semitransparent mirror, S - sample.

The mirrors of the interferometer were adjustable in order to produce (with homogeneous samples) an interference field with interference fringes parallel to the gradient of the refractive index modulation. An inhomogeneity of the refractive index in the sample causes the shift of the interference fringes in direction perpendicular to the fringes. So (when the inhomogeneity is one-dimensional), the shape of the fringes images the shape of spatial dependence of the refractive index.

The area with decreased or increased refractive index (depending on the mirror position) can cause the same deformation of the interference fringes. So the sign of the refractive index changes was checked by putting a plate with a very thin “drop” of a gelatin as an equivalent of area with increased refractive index (increasing of the optical path).

The interference field was projected onto a screen by an objective with focal length of 50 mm which provided the magnification sufficient for observation and taking a photograph. The objective was adjusted to the proper position in order to project the optical field just behind the rear surface of the sample and the field of the reference beam in the equivalent distance.

To minimize the influence of the illumination on the refractive index during reading the refractive index inhomogeneity (during reading the record) the He-Ne laser (λ=633 nm) with intensity smaller than 1 mW/mm² was used. All experiments (concerning writing and reading) were performed at room temperature.

4. Comparison of the experimental results with standardly used model

The electrooptic tensor of LiNbO₃ crystals (in matrix notation) is [14

P. Yeh, Introduction to photorefractive nonlinear optics , (John Wiley & Sons, Inc. New York, 1993) p.27.

, 15

M. Born and E. Wolf, Principles of Optics , (Cambridge University Press, United Kingdom, 2002), pp.799–808.

]

r_{α k} = (\begin{matrix} 0 & - r_{22} & r_{13} \\ 0 & r_{22} & r_{13} \\ 0 & 0 & r_{33} \\ 0 & r_{51} & 0 \\ r_{51} & 0 & 0 \\ {- r}_{22} & 0 & 0 \end{matrix})

(9)

where r₂₂=6.8 10^-12 m/V, r₅₁=32.6 10^-12 m/V, r₁₃=9.6 10^-12 m/V and r₃₃=30.9 10^-12 m/V. Supposing that the electric field is parallel to gradient of illumination intensity and refractive index modulation is due to electrooptic effect Eq. (7) and matrix of electrooptic coefficient according to Eq. (9) allows to express the refractive indices for orientation of the electric field in the direction of crystallographic axes.

When E is parallel with z-axis, the index ellipsoid in LiNbO₃ according Eqs. (7) and (9) has the form:

(\frac{1}{n_{o}^{2}} + r_{13} E_{3}) x^{2} + (\frac{1}{n_{o}^{2}} + r_{13} E_{3}) y^{2} (\frac{1}{n_{e}^{2}} + r_{33} E_{3}) z^{2} - 1 = 0 .

(10)

The intersection of the (x z) plane with the index ellipsoid determined by Eq. (10) for field E large enough in order the refractive index changes to be seen in the graph is drawn in Fig. 3. For comparison also the intersection when no electric field is applied is drawn in this figure. The interferograms visualizing the changes of refractive index resulting from the strip-like illumination perpendicular to z-axis (with gradient of the intensity (mainly) in z-direction) are also presented in the figure. The presented interferograms were obtained with He-Ne laser beam polarized in z- and x-directions.

Fig. 3. (a). The intersection of the index ellipsoid with xz plane when electric field is parallel with z-axis (E=10⁷ V/cm; red curve) and without the field (blue curve). Interferograms showing the changes of refractive index when reading beam is polarized in z-direction (b). and x-direction (c). The blue dashed curves mark the illumination region. Intensity of the writing beam (in the middle of the illuminated part) was 5 mWmm^-2 and sample was exposed to light for 100 seconds.

The interferograms presented in the Fig. 3. show that the illumination with gradient in z-direction really causes changes of the refractive index for both polarizations of the beam propagating in y-direction as expected from the considered model based on electrooptic effect. Number of interference fringes between the fringes of the same interference order (in Fig. 3(b), 3(c) the number of the fringes between the dashed yellow lines) shows the change of the phase of the optical wave (Δϕ_e and Δϕ_o) caused by the illumination. The relationship between change of phase of the optical wave and change of the optical path allows to determine the change of the refractive index. The changes of the refractive index, according to obtained interferograms, are about 2.5·10^-4 and 1·10^-4 for extraordinary and ordinary beams, respectively. The same changes have been observed for both polarizations of the writing laser beam. The ratio of amplitudes of the refractive index modulation determined in this way corresponds (approximately) to the ratio of the relevant elements of the electrooptic tensor. However, as it can be seen from this figure the function which expresses the spatial dependence of the refractive index changes n(z) is symmetric with respect to plane of symmetry of the illumination distribution, what is in disagreement with the symmetry of the function following from the model mentioned in the paragraph.

When the electric field is parallel with y-axis, then Eq. (7) has the form:

(\frac{1}{n_{o}^{2}} - r_{22} E_{2}) x^{2} + (\frac{1}{n_{o}^{2}} + r_{22} E_{2}) y^{2} + \frac{1}{n_{o}^{2}} z^{2} + 2 y z r_{51} E_{2} - 1 = 0

(11)

The intersections of (x y) plane with index ellipsoid in this case and the corresponding interferograms are presented in Fig. 4.

Fig. 4. The intersection of the index ellipsoid with xy plane when electric field is parallel with y-axis (E=10⁷ V/cm; red curve) and when E is zero (blue curve). (b) and (c) interferograms showing the changes of refractive index trough the change of the fringe’s shape and the change of intensity. Reading beam is polarized in x-direction. The exposure time was 900 seconds and intensity of writing beam was 5 mWmm^-2.

When the electric field is parallel with x-axis, then Eq. (7) has the form:

\frac{1}{n_{o}^{2}} x^{2} + \frac{1}{n_{o}^{2}} y^{2} + \frac{1}{n_{e}^{2}} z^{2} + 2 x z r_{51} E_{1} - 2 x y r_{22} E_{1} - 1 = 0

(12)

The intersections of (x z) plane with the index ellipsoid for the case of electric field being in x-direction and the relevant interferograms are drawn in Fig. 5.

Fig. 5. The intersection of the index ellipsoid with xz plane when electric field is parallel with x-axis (E=10⁷V/cm; red curve) and when it is zero (blue curve). Interferograms showing the changes of phase (refractive index) when reading beam is polarized in z-direction (b) and x-direction (c). The exposure time was 600 seconds and intensity of writing beam was 5 mWmm-²

The interferograms presented in Fig. 5. show that the shape of the refractive index modulation differs from the shape when the illumination gradient was oriented in z-direction (Fig. 3). The main difference is that the changes of refractive index (Fig. 5) has the same sign (negative) in all places along normal to the illuminated strip (along x-axis), while at the strip perpendicular to z-axis (Fig. 3) the refractive index change is positive outside the illuminated strip and negative inside it. The difference of the function expressing the dependence of the refractive index modulation indicates that other mechanisms take place when the illumination gradient is in x-direction or in z-direction. Moreover, the semi-axes of the index ellipsoid following from the model based on electrooptic effect for electric field parallel with y-axis are not in direction of x- and z-axes. So the interference fringes when the reading beam is polarized in that direction should be split into fringes with polarization oriented in direction of the ellipsoid what has not been observed.

4. Conclusion

The performed investigation of the symmetry and anisotropy of the refractive index inhomogeneities shows that the electrooptic effect itself can not describe all features of the photorefractive effect and other mechanisms influencing the permitivity of the sample should be taken into account (for example the dependence of refractive index on the donor occupancy or changes of concentration of any compensating ions due to their mobility and internal electric field). Maybe also the quadratic electrooptic effect may take place in process of the refractive index change. However, finding the real reason of the all observed features needs additional investigation.

Acknowledgment

This work was supported by Slovak Science and Technology Assistance Agency under contract No. APVT-20-013504.

References and Links

1.	Q. Gao and R. Kostuk, “Cross - talk noise and storage capacity of holographic memories with a LiNbO₃ crystal in the open - circuit condition,” Appl. Opt. 37, 929–936 (1998). [CrossRef]
2.	J. Ashley, et al., “Holographic data storage,” IBM J. Res. Develop. 44, 341–368 (2000). [CrossRef]
3.	A. K. Zajtsev, S. H. Lin, and K. Y. Hsu, “Sidelobe suppression of spectral response in holographic optical filter,” Opt. Commun. 190, 103–108 (2001). [CrossRef]
4.	S.-F. Chen, C. S. Wu, and C.-C. Sun, “Design for a high dense wavelength division multiplexer based on volume holographic gratings,” Opt. Eng. 43, 2028–2033 (2004). [CrossRef]
5.	S. Mailis, C. Riziotis, I. T. Wellington, P. G. R. Smith, C. B. E. Gawith, and R. W. Eason, “Direct ultraviolet writing of channel waveguides in congruent lithium niobate single crystals,” Opt. Lett. 28, 1433–1435 (2003). [CrossRef] [PubMed]
6.	G. Couton, H. Maillotte, R. Giust, and M. Chauvet, “Formation of reconfigurable singlemode channel waveguides in LiNbO₃ using spatial solitons,” Electron. Lett. 39, 286–287 (2003). [CrossRef]
7.	M. Paturzo, L. Miccio, S. De Nicola, P. De Natale, and P. Ferraro, “Amplitude and phase reconstruction of photorefractive spatial bright-soliton in LiNbO₃ during its dynamic formation by digital holography,” Opt. Express 15, 8243–8251 (2007). [CrossRef] [PubMed]
8.	A. Ashkin, G. D. Boyd, J. M. Dziedzic, R. G. Smith, A. A. Ballman, J. J. Levinstein, and K. Nassau, “Optically - induced refractive index inhomogeneities in LiNbO₃ and LiTaO₃ ,” Appl. Phys. Lett. 9, 72–74 (1966). [CrossRef]
9.	F. S. Chen, “Optically induced change of refractive indices in LiNbO₃ and LiTaO₃ ,” J. Appl. Phys. 40, 3389–3396 (1969). [CrossRef]
10.	A. M. Glass, D. von der Linde, and T. J. Negran, “High-voltage bulk photovoltaic effect and the photorefractive process in LiNbO₃ ,” Appl. Phys. Lett. 25, 233–235 (1974). [CrossRef]
11.	B. I. Sturman, F. Agulló-López, M. Carrascosa, and L. Solymar, “On microscopic description of photorefractive phenomena,” Appl. Phys. B 68, 1013–1020 (1999). [CrossRef]
12.	I. Turek and N. Tarjányi, “Interference imaging of photorefractive record in thin sample of LiNbO₃ crystal,” Proc. SPIE 5945, 59450J-1 (2005).
13.	I. Turek and N. Tarjányi, “The photorefractive effect in LiNbO₃ crystals,” 15th Czech-Polish-Slovak Conference on Wave and Quantum Aspects of Contemporary Optics, Proc. SPIE 6609, 660906 (2007).
14.	P. Yeh, Introduction to photorefractive nonlinear optics , (John Wiley & Sons, Inc. New York, 1993) p.27.
15.	M. Born and E. Wolf, Principles of Optics , (Cambridge University Press, United Kingdom, 2002), pp.799–808.
16.	M. de Angelis, S. De Nicola, A. Finizio, G. Pierattini, P. Ferraro, S. Pelli, G. Righini, and S. Sebastiani, “Digital-holography refractive-index-profile measurement of phase gratings,” Appl. Phys. Lett. 88, 111114–111116 (2006). [CrossRef]

OCIS Codes
(160.3730) Materials : Lithium niobate
(160.5320) Materials : Photorefractive materials
(260.3160) Physical optics : Interference

ToC Category:
Materials

History
Original Manuscript: May 24, 2007
Revised Manuscript: July 20, 2007
Manuscript Accepted: August 7, 2007
Published: August 10, 2007

Citation
Ivan Turek and Norbert Tarjányi, "Investigation of symmetry of photorefractive effect in LiNbO₃," Opt. Express 15, 10782-10788 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-17-10782

Sort: Author | Year | Journal | Reset

References

Q. Gao and R. Kostuk, "Cross - talk noise and storage capacity of holographic memories with a LiNbO₃ crystal in the open - circuit condition," Appl. Opt. 37, 929-936 (1998). [CrossRef]
J. Ashley, et al., "Holographic data storage," IBM J. Res. Develop. 44, 341-368 (2000). [CrossRef]
A. K. Zajtsev, S. H. Lin, and K. Y. Hsu, "Sidelobe suppression of spectral response in holographic optical filter," Opt. Commun. 190, 103-108 (2001). [CrossRef]
S.-F. Chen, C. S. Wu, and C.-C. Sun, "Design for a high dense wavelength division multiplexer based on volume holographic gratings," Opt. Eng. 43, 2028 - 2033 (2004). [CrossRef]
S. Mailis, C. Riziotis, I. T. Wellington, P. G. R. Smith, C. B. E. Gawith, and R. W. Eason, "Direct ultraviolet writing of channel waveguides in congruent lithium niobate single crystals," Opt. Lett. 28, 1433-1435 (2003). [CrossRef] [PubMed]
G. Couton, H. Maillotte, R. Giust, and M. Chauvet, "Formation of reconfigurable singlemode channel waveguides in LiNbO₃ using spatial solitons," Electron. Lett. 39, 286-287 (2003). [CrossRef]
M. Paturzo, L. Miccio, S. De Nicola, P. De Natale, and P. Ferraro, "Amplitude and phase reconstruction of photorefractive spatial bright-soliton in LiNbO3 during its dynamic formation by digital holography," Opt. Express 15, 8243-8251 (2007). [CrossRef] [PubMed]
A. Ashkin, G. D. Boyd, J. M. Dziedzic, R. G. Smith, A. A. Ballman, J. J. Levinstein, and K. Nassau, "Optically - induced refractive index inhomogeneities in LiNbO₃ and LiTaO₃," Appl. Phys. Lett. 9, 72 -74 (1966). [CrossRef]
F. S. Chen, "Optically induced change of refractive indices in LiNbO₃ and LiTaO₃," J. Appl. Phys. 40, 3389-3396 (1969). [CrossRef]
A. M. Glass, D. von der Linde, and T. J. Negran, "High-voltage bulk photovoltaic effect and the photorefractive process in LiNbO₃," Appl. Phys. Lett. 25, 233-235 (1974). [CrossRef]
B. I. Sturman, F. Agulló-López, M. Carrascosa, and L. Solymar, "On microscopic description of photorefractive phenomena," Appl. Phys. B 68, 1013-1020 (1999). [CrossRef]
I. Turek and N. Tarjányi, "Interference imaging of photorefractive record in thin sample of LiNbO₃ crystal," Proc. SPIE 5945, 59450J-1 (2005).
I. Turek and N. Tarjányi, "The photorefractive effect in LiNbO₃ crystals," 15th Czech-Polish-Slovak Conference on Wave and Quantum Aspects of Contemporary Optics, Proc. SPIE 6609, 660906 (2007).
P. Yeh, Introduction to photorefractive nonlinear optics, (John Wiley & Sons, Inc. New York, 1993) p. 27.
M. Born and E. Wolf, Principles of Optics, (Cambridge University Press, United Kingdom, 2002), pp. 799-808.
M. de Angelis, S. De Nicola, A. Finizio, G. Pierattini, P. Ferraro, S. Pelli, G. Righini, and S. Sebastiani, "Digital-holography refractive-index-profile measurement of phase gratings," Appl. Phys. Lett. 88, 111114-111116 (2006). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Click here to see a list of articles that cite this paper

Figures


Fig. 1.	Fig. 2.	Fig. 3.


Fig. 4.	Fig. 5.

« Previous Article | Next Article »

OSA is a member of CrossRef.

OpticsInfoBase

Optics Express

Investigation of symmetry of photorefractive effect in LiNbO₃

Article Tools