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A method of easy fabrication of 2D light-induced nonlinear photonic lattices in self-defocusing LiNbO3:Fe crystal
Available online at www.sciencedirect.com
Optical Materials 30 (2007) 527–531 www.elsevier.com/locate/optmat
A method of easy fabrication of 2D light-induced nonlinear photonic lattices in self-defocusing LiNbO3:Fe crystal Nan Zhu, Zhaohong Liu, Ru Guo *, Si-Min Liu Photonics Center, College of Physics Science, Nankai University, Tianjin 300071, PR China Received 6 July 2006; received in revised form 7 November 2006; accepted 18 December 2006 Available online 27 February 2007
Abstract Using a Fourier transform method we fabricate experimentally the square and hexagonal two dimensional light-induce nonlinear photonic lattices of over 20 · 20 segments with different period in self-defocusing LiNbO3:Fe crystal. The experimental setup of this method is very simple, the photonic lattices are quite regular and robust, and the period of the lattice can be dominated easily. 2007 Elsevier B.V. All rights reserved. PACS: 42.30.Kq; 42.65.Wi; 42.70.Nq; 42.82.Et Keywords: Nonlinear photonic lattice; Photorefractive LiNbO3:Fe crystal; Fourier transform lens
1. Introduction Closely spaced nonlinear period optical systems, such as photonic lattices and waveguide arrays, have attracted substantial research interest in the past several years owing to their strong link with the emerging science and technology of nonlinear photonic crystals and their novel possibilities to control light propagation, steering and trapping [1,2]. In such array structures, the collective behavior of wave propagation exhibits intriguing phenomena that also occur in other discrete systems, such as biological molecules [3], solid-state systems [4] and Bose–Einstein condensates [5]. Apart from their value for fundamental research, nonlinear waveguide arrays are of interest because of their potential applications for signal processing and information technology [6,7]. Recently, several methods for fabricating photonic lattices by optical irradiation in various materials such as UV epoxy [8], glass [9], and photorefractive materials [10–15] have been proposed. The light-induced pho-
*
Corresponding author. Tel.: +86 02223498011; fax: +86 02223506238. E-mail address: [email protected] (R. Guo).
0925-3467/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optmat.2006.12.010
tonic lattices based on photorefractive effect can be created at millwatt-optical-power levels. A LiNbO3:Fe crystal is a self-defocusing photorefractive material where the light induces a negative refractive index change with the order of 104 [10,11]. This index change is sufficient to create the photonic lattices which can be kept for a few months in the dark room. In 1996, Matoba et al. [12] proposed a theoretical model to fabricate the 2D waveguide arrays. Lights emitted from four coherent point sources pass through a Fourier transform lens and form the four coherent plane waves. They interfere in the photorefractive crystal and optically induce the 2D waveguide arrays. However, it is difficult to create four intensive coherent point sources, so their experiments were still carried out using the Mach-Zehnder configuration. Motivated by this idea, we proposed an experimental method of easy fabrication of 2D photonic lattices. The key of this method is using four little holes instead of four point sources and a Fourier lens with long focus. We would like to call it a Fourier-transform method. In this letter, we report on experimental fabrication of the 2D light-induced nonlinear photonic lattices in the self-defocusing LiNbO3:Fe crystal using this method. The square
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N. Zhu et al. / Optical Materials 30 (2007) 527–531
and hexagonal photonic lattices of over 20 · 20 segments with different periods are fabricated. Compared with the previous fabrication method (the interference of coherent beams [13] and amplitude modulation of a partially spatially coherent beam [14,15]), our experimental setup is simple, and the period of the photonic lattices can be dominated easily. The photonic lattice is quite regular and robust because they are written based on interference array beams through the self-defocusing LiNbO3:Fe crystal with the photovoltaic saturable nonlinearity.
"
! ! #2 pffiffiffi pffiffiffi p 2ax p 2ay Iðx; yÞ ¼ 4 cos pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ cos pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k f 2 þ 2a2 k f 2 þ 2a2 ð1Þ where a is the distance between the two little holes, k is the wavelength and f is the focal length of the Fourier-transform lens L2. Obviously, the period of the photonic lattice can be given by j¼k
2. Experimental details The experimental setup to fabricate the 2D light-induced nonlinear photonic lattice is shown in Fig. 1a. A laser beam derived from a CW frequency-doubled Nd:YAG laser at k = 532 nm initially passed through a spatial filter (SF) and was expanded and perfectly collimated by lens L1 (f1 = 135 mm). Then the broad beam with a plane wave front illuminates onto an amplitude mask with four little holes, which is placed in the front focus plane (x 0 –y 0 )of Fourier lens L2 (f1 = 650 mm). The diameter d and spacing a of the four little holes are 0.7 mm and 8 mm, respectively, their central positions are (x 0 ± a,y 0 ) and (x 0 ,y 0 ± a) on the amplitude mask, as shown in Fig. 1b. Diffracting beams emitted from four little holes are transformed into four coherent plane waves by the Fourier lens L2. The intensity distribution of the interference pattern in the x–y plane behind the lens L2 can be described as
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi f 2 þ 2a2 = 2a
ð2Þ
These interference array beams are uniform along the direction of the optical axis within the interference region of the four plane waves because the four little holes are coplanar and the layout is symmetric. These regular array beams with the intensity about 4.2 MW/cm2 illuminate at the input face of the crystal and propagate through it, then optically induce nonlinear index change in the self-defocusing LiNbO3:Fe crystal. The LiNbO3:Fe crystal (doped with 0.02 wt%, the absorption coefficient a = 4.58/cm) has dimensions of 5 mm · 5 mm · 10 mm, and array beams always propagate along the 10-mm side. To exploit the dominant electro-optic component r33 of our crystal, the laser beam is linearly polarized to the crystal ^c axis. We let the crystal’s optical c-axis parallel to the x-direction. And in order to trail off the effect of the nonlinear anisotropy of the crystal, the principal axis (x00 and y00 ) of the 2D square lattice orient in the 45 directions relative to the x(//c) and y axes, as shown in Fig. 1b. The spatial inten-
Fig. 1. (a) Experimental setup for fabrication of a photonic lattice: SF, spatial filter; BS, beam slitter; LN, LiNbO3:Fe crystal. (b) Schematic diagram of an optical system of Fourier transform.
N. Zhu et al. / Optical Materials 30 (2007) 527–531
sity profile at either the front or the back face of the crystal is imaged onto a CCD camera by moving lens L3 (f3 = 50 mm), respectively. The He–Ne laser is used to probe the refractive index change of the light-induced nonlinear photonic lattices. First, we create a stable and regular square photonic lattice using the Fourier transform method. At the output face of the crystal, the transverse pattern of the beam consists of about 20 · 20 intensity pixels with 25 lm separation. The diameter of the intensity pixels is 12 lm. Because of magnification in imaging to the CCD camera, only part of the array (7 · 7) can be recorded. Fig. 2a depicts the pattern of the output beam which is just as the input patterns, when nonlinearity was not present at time t = 0. This means the interference array beam experience robust linear propagation inside the whole crystal. After 5 min illumination time, the nonlinearity of the crystal is enough to form light-induced photonic lattice, and a satisfactory image at the output face of the crystal was observed (Fig. 2b). Fig. 2c shows output photograph of the probe beam, which is an expanded and collimated He–Ne laser beam. The photorefractive photovoltaic LiNbO3:Fe crystal has self-defocusing saturable nonlinearity [10,11], when LiNbO3:Fe crystal is irradiated by writing light, the direction of the light-induced space charge field in the bright region of periodic structures is parallel to self-polarization direction of the crystal, so a refractiveindex change through the Pockels effect is less than zero
529
(Dn < 0), while the direction of the field in the dark region of the periodic structure is inverse. Thus, the waveguides in the photonic lattices are formed in the dark region as the refractive index change Dn > 0. Once the nonlinear waveguide arrays are formed, it is possible to use a probe beam to test their guiding properties. The probe light in the waveguides forms the bright regions that correspond to dark regions of writing light intensity, while the probe dark regions correspond to bright regions of writing light intensity. We use a sign ‘+’ to represent the same position in Fig. 2a–c for confirming the intensity inversion. A He–Ne laser (k = 632.8 nm) as probe light is focused onto the front face of the crystal. Fig. 3a and b show the intensity profiles of a focused beam at the input face of the crystal and its liner diffraction at the output face of the crystal without the photonic lattice, respectively. After the photonic lattice is created in the crystal, most of (about 60%) the energy of the probe beam is guided by a waveguide when the test beam impinges on the center of a waveguide in the photonic lattice, and the other part of energy is coupled from the center towards the diagonal directions of the array (Fig. 3c). This discrete diffraction results from that the adjacent waveguides in lattice are not fully isolated from one another, but are connected through a grid of narrow equipotential ‘backbones’ [13] as shown in Fig. 2d, which can allow power to leak slowly along this grid. It is worth pointing out that the waveguides in the photonic lattice, which is created from a Nd:YAG laser (k = 532 nm), can
Fig. 2. 2 D nonlinear photonic lattices. Shown are intensity patterns of (a) output diffraction-free beam at t = 0 without nonlinearity; (b) with nonlinearity; (c) guidance of a broad beam into all waveguide channels; (d) three-dimensional intensity plot of (c).
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N. Zhu et al. / Optical Materials 30 (2007) 527–531
Fig. 3. The intensity patterns of a focused probe beam into 2D nonlinear photonic lattices: (a) input profile; (b) diffraction profile output without array of 2D waveguides; (c) output profile when a focused probe beam impinges on the center of a waveguide in the array. Top: 2D transverse patterns; Bottom: corresponding 3D transverse patterns.
be used to guide an intensive coherent laser at longer wavelengths without being damaged, and this photonic lattice can even be fixed in the crystal permanently. According to the Eq. (2), the period of the photonic lattices can be easily controlled by changing the distance of the two nearest little holes on the mask. Fig. 4 shows two photonic lattices with periods of 30 lm (top) and 20 lm (bottom). The patterns of photonic lattice of the output
beam are shown at the left column and the photographs probed by the He–Ne laser are shown at the right column. As can be seen from Fig. 4b and d, the broad probe beam gets split into several channels fitting into the light-induced waveguides and propagates exactly along the light-induced waveguides. Thus, the photonic lattices with different periods are easily created in LiNbO3:Fe crystal by present method.
Fig. 4. The square photonic lattices with periods of 30 lm (top) and 20 lm (bottom). The left column is the output profile of the writing beam and right column is the output profile of the probe beam using the He–Ne laser.
Fig. 5. The hexagonal photonic lattices with periods of 25 lm (top) and 20 lm (bottom). The left column is the output profile of the writing beam and the right column is the output profile of the probe beam using the He– Ne laser.
N. Zhu et al. / Optical Materials 30 (2007) 527–531
Next, if we set four coherent point sources on the mask to distribute as the rhombus, hexagonal photonic lattice can also be fabricated. Typical results are shown in Fig. 5. Fig. 5 shows two photonic lattices with periods of 25 lm (top) and 20 lm (bottom). The patterns of the output beam are shown at the left column and the photograph probed by the He–Ne laser shown at the right column. As can be seen from the right column of Fig. 5, the lightinduced photonic lattices are elliptical which is due to the separation between two nearest little holes on the mask along the x 0 axis being different from that along the y 0 axis, so the waveguide formed in the dark regions become elliptical type. However, we consider that this property described above can not effect the applications of these hexagonal photonic lattices. 3. Conclusion In this letter, we use the Fourier transform method to fabricate variety of photonic lattices in self-defocusing LiNbO3:Fe crystal. The advantage of this method is that the experimental setup is very simple; the photonic lattices can be easily dominated and are quite regular and robust. Apart from its potential applications in signal processing and information technology, these light-induced nonlinear photonic lattices might offer an easy method to study the discrete solitons and bandgap solitons in nonlinear photonic lattices.
531
Acknowledgement This research was supported by the National Natured Science Foundation of China (Grants: 60278006, 60378013 and 10474047). References [1] D.N. Christodoulides, F. Lederer, Y. Silberg, Nature 424 (2003) 807. [2] S. Mingaleev, Y. Kivshar, Opt. Photonics News 13 (2002) 48. [3] A.S. Davydov, Biology and Quantum Mechanics, Pergamon, Oxford, 1982. [4] A.J. Sievers, S. Takeno, Phys. Rev. Lett. 61 (1988) 970. [5] A. Trombettoni, A. Smerzi, Phys. Rev. Lett. 86 (2001) 2353. [6] C. Bosshard, P.V. Mamyshev, G.I. Stegeman, Opt. Lett. 19 (1994) 90. [7] W. Krolikowski, Y.S. Kivshar, J. Opt. Soc. Am. B 13 (1996) 876. [8] S.J. Frisken, Opt. Lett. 18 (1993) 1035. [9] K.M. Davis, K. Miura, N. Sugimoto, K. Hirao, Opt. Lett. 21 (1996) 1729. [10] F. Chen, M. Stepic, C.E. Ruter, D. Runde, D. Kip, V. Shandarov, O. Manela, M. Segev, Opt. Exp. 13 (2005) 4314. [11] Tao Song, Simin Liu, Ru Guo, Zhaohong Liu, Nan Zhu, Yuanmei Gao, Opt. Exp. 14 (5) (2006) 1924. [12] O. Matoba, K. Itoh, Y. Ichioka, Opt. Lett. 21 (1996) 122. [13] J.W. Fleischer, M. Segev, N.K. Efremidis, D.N. Christodoulides, Nature 422 (2003) 147. [14] H. Martin, E.D. Eugenieva, Z. Chen, D.N. Christodoulides, Phys. Rev. Lett. 92 (2004) 123902. [15] Z. Chen, H. Martin, E.D. Eugenieva, J. Xu, A. Bezryadina, Phys. Rev. Lett. 92 (2004) 143902.
Optical Materials 30 (2007) 527–531 www.elsevier.com/locate/optmat
A method of easy fabrication of 2D light-induced nonlinear photonic lattices in self-defocusing LiNbO3:Fe crystal Nan Zhu, Zhaohong Liu, Ru Guo *, Si-Min Liu Photonics Center, College of Physics Science, Nankai University, Tianjin 300071, PR China Received 6 July 2006; received in revised form 7 November 2006; accepted 18 December 2006 Available online 27 February 2007
Abstract Using a Fourier transform method we fabricate experimentally the square and hexagonal two dimensional light-induce nonlinear photonic lattices of over 20 · 20 segments with different period in self-defocusing LiNbO3:Fe crystal. The experimental setup of this method is very simple, the photonic lattices are quite regular and robust, and the period of the lattice can be dominated easily. 2007 Elsevier B.V. All rights reserved. PACS: 42.30.Kq; 42.65.Wi; 42.70.Nq; 42.82.Et Keywords: Nonlinear photonic lattice; Photorefractive LiNbO3:Fe crystal; Fourier transform lens
1. Introduction Closely spaced nonlinear period optical systems, such as photonic lattices and waveguide arrays, have attracted substantial research interest in the past several years owing to their strong link with the emerging science and technology of nonlinear photonic crystals and their novel possibilities to control light propagation, steering and trapping [1,2]. In such array structures, the collective behavior of wave propagation exhibits intriguing phenomena that also occur in other discrete systems, such as biological molecules [3], solid-state systems [4] and Bose–Einstein condensates [5]. Apart from their value for fundamental research, nonlinear waveguide arrays are of interest because of their potential applications for signal processing and information technology [6,7]. Recently, several methods for fabricating photonic lattices by optical irradiation in various materials such as UV epoxy [8], glass [9], and photorefractive materials [10–15] have been proposed. The light-induced pho-
*
Corresponding author. Tel.: +86 02223498011; fax: +86 02223506238. E-mail address: [email protected] (R. Guo).
0925-3467/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optmat.2006.12.010
tonic lattices based on photorefractive effect can be created at millwatt-optical-power levels. A LiNbO3:Fe crystal is a self-defocusing photorefractive material where the light induces a negative refractive index change with the order of 104 [10,11]. This index change is sufficient to create the photonic lattices which can be kept for a few months in the dark room. In 1996, Matoba et al. [12] proposed a theoretical model to fabricate the 2D waveguide arrays. Lights emitted from four coherent point sources pass through a Fourier transform lens and form the four coherent plane waves. They interfere in the photorefractive crystal and optically induce the 2D waveguide arrays. However, it is difficult to create four intensive coherent point sources, so their experiments were still carried out using the Mach-Zehnder configuration. Motivated by this idea, we proposed an experimental method of easy fabrication of 2D photonic lattices. The key of this method is using four little holes instead of four point sources and a Fourier lens with long focus. We would like to call it a Fourier-transform method. In this letter, we report on experimental fabrication of the 2D light-induced nonlinear photonic lattices in the self-defocusing LiNbO3:Fe crystal using this method. The square
528
N. Zhu et al. / Optical Materials 30 (2007) 527–531
and hexagonal photonic lattices of over 20 · 20 segments with different periods are fabricated. Compared with the previous fabrication method (the interference of coherent beams [13] and amplitude modulation of a partially spatially coherent beam [14,15]), our experimental setup is simple, and the period of the photonic lattices can be dominated easily. The photonic lattice is quite regular and robust because they are written based on interference array beams through the self-defocusing LiNbO3:Fe crystal with the photovoltaic saturable nonlinearity.
"
! ! #2 pffiffiffi pffiffiffi p 2ax p 2ay Iðx; yÞ ¼ 4 cos pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ cos pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k f 2 þ 2a2 k f 2 þ 2a2 ð1Þ where a is the distance between the two little holes, k is the wavelength and f is the focal length of the Fourier-transform lens L2. Obviously, the period of the photonic lattice can be given by j¼k
2. Experimental details The experimental setup to fabricate the 2D light-induced nonlinear photonic lattice is shown in Fig. 1a. A laser beam derived from a CW frequency-doubled Nd:YAG laser at k = 532 nm initially passed through a spatial filter (SF) and was expanded and perfectly collimated by lens L1 (f1 = 135 mm). Then the broad beam with a plane wave front illuminates onto an amplitude mask with four little holes, which is placed in the front focus plane (x 0 –y 0 )of Fourier lens L2 (f1 = 650 mm). The diameter d and spacing a of the four little holes are 0.7 mm and 8 mm, respectively, their central positions are (x 0 ± a,y 0 ) and (x 0 ,y 0 ± a) on the amplitude mask, as shown in Fig. 1b. Diffracting beams emitted from four little holes are transformed into four coherent plane waves by the Fourier lens L2. The intensity distribution of the interference pattern in the x–y plane behind the lens L2 can be described as
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi f 2 þ 2a2 = 2a
ð2Þ
These interference array beams are uniform along the direction of the optical axis within the interference region of the four plane waves because the four little holes are coplanar and the layout is symmetric. These regular array beams with the intensity about 4.2 MW/cm2 illuminate at the input face of the crystal and propagate through it, then optically induce nonlinear index change in the self-defocusing LiNbO3:Fe crystal. The LiNbO3:Fe crystal (doped with 0.02 wt%, the absorption coefficient a = 4.58/cm) has dimensions of 5 mm · 5 mm · 10 mm, and array beams always propagate along the 10-mm side. To exploit the dominant electro-optic component r33 of our crystal, the laser beam is linearly polarized to the crystal ^c axis. We let the crystal’s optical c-axis parallel to the x-direction. And in order to trail off the effect of the nonlinear anisotropy of the crystal, the principal axis (x00 and y00 ) of the 2D square lattice orient in the 45 directions relative to the x(//c) and y axes, as shown in Fig. 1b. The spatial inten-
Fig. 1. (a) Experimental setup for fabrication of a photonic lattice: SF, spatial filter; BS, beam slitter; LN, LiNbO3:Fe crystal. (b) Schematic diagram of an optical system of Fourier transform.
N. Zhu et al. / Optical Materials 30 (2007) 527–531
sity profile at either the front or the back face of the crystal is imaged onto a CCD camera by moving lens L3 (f3 = 50 mm), respectively. The He–Ne laser is used to probe the refractive index change of the light-induced nonlinear photonic lattices. First, we create a stable and regular square photonic lattice using the Fourier transform method. At the output face of the crystal, the transverse pattern of the beam consists of about 20 · 20 intensity pixels with 25 lm separation. The diameter of the intensity pixels is 12 lm. Because of magnification in imaging to the CCD camera, only part of the array (7 · 7) can be recorded. Fig. 2a depicts the pattern of the output beam which is just as the input patterns, when nonlinearity was not present at time t = 0. This means the interference array beam experience robust linear propagation inside the whole crystal. After 5 min illumination time, the nonlinearity of the crystal is enough to form light-induced photonic lattice, and a satisfactory image at the output face of the crystal was observed (Fig. 2b). Fig. 2c shows output photograph of the probe beam, which is an expanded and collimated He–Ne laser beam. The photorefractive photovoltaic LiNbO3:Fe crystal has self-defocusing saturable nonlinearity [10,11], when LiNbO3:Fe crystal is irradiated by writing light, the direction of the light-induced space charge field in the bright region of periodic structures is parallel to self-polarization direction of the crystal, so a refractiveindex change through the Pockels effect is less than zero
529
(Dn < 0), while the direction of the field in the dark region of the periodic structure is inverse. Thus, the waveguides in the photonic lattices are formed in the dark region as the refractive index change Dn > 0. Once the nonlinear waveguide arrays are formed, it is possible to use a probe beam to test their guiding properties. The probe light in the waveguides forms the bright regions that correspond to dark regions of writing light intensity, while the probe dark regions correspond to bright regions of writing light intensity. We use a sign ‘+’ to represent the same position in Fig. 2a–c for confirming the intensity inversion. A He–Ne laser (k = 632.8 nm) as probe light is focused onto the front face of the crystal. Fig. 3a and b show the intensity profiles of a focused beam at the input face of the crystal and its liner diffraction at the output face of the crystal without the photonic lattice, respectively. After the photonic lattice is created in the crystal, most of (about 60%) the energy of the probe beam is guided by a waveguide when the test beam impinges on the center of a waveguide in the photonic lattice, and the other part of energy is coupled from the center towards the diagonal directions of the array (Fig. 3c). This discrete diffraction results from that the adjacent waveguides in lattice are not fully isolated from one another, but are connected through a grid of narrow equipotential ‘backbones’ [13] as shown in Fig. 2d, which can allow power to leak slowly along this grid. It is worth pointing out that the waveguides in the photonic lattice, which is created from a Nd:YAG laser (k = 532 nm), can
Fig. 2. 2 D nonlinear photonic lattices. Shown are intensity patterns of (a) output diffraction-free beam at t = 0 without nonlinearity; (b) with nonlinearity; (c) guidance of a broad beam into all waveguide channels; (d) three-dimensional intensity plot of (c).
530
N. Zhu et al. / Optical Materials 30 (2007) 527–531
Fig. 3. The intensity patterns of a focused probe beam into 2D nonlinear photonic lattices: (a) input profile; (b) diffraction profile output without array of 2D waveguides; (c) output profile when a focused probe beam impinges on the center of a waveguide in the array. Top: 2D transverse patterns; Bottom: corresponding 3D transverse patterns.
be used to guide an intensive coherent laser at longer wavelengths without being damaged, and this photonic lattice can even be fixed in the crystal permanently. According to the Eq. (2), the period of the photonic lattices can be easily controlled by changing the distance of the two nearest little holes on the mask. Fig. 4 shows two photonic lattices with periods of 30 lm (top) and 20 lm (bottom). The patterns of photonic lattice of the output
beam are shown at the left column and the photographs probed by the He–Ne laser are shown at the right column. As can be seen from Fig. 4b and d, the broad probe beam gets split into several channels fitting into the light-induced waveguides and propagates exactly along the light-induced waveguides. Thus, the photonic lattices with different periods are easily created in LiNbO3:Fe crystal by present method.
Fig. 4. The square photonic lattices with periods of 30 lm (top) and 20 lm (bottom). The left column is the output profile of the writing beam and right column is the output profile of the probe beam using the He–Ne laser.
Fig. 5. The hexagonal photonic lattices with periods of 25 lm (top) and 20 lm (bottom). The left column is the output profile of the writing beam and the right column is the output profile of the probe beam using the He– Ne laser.
N. Zhu et al. / Optical Materials 30 (2007) 527–531
Next, if we set four coherent point sources on the mask to distribute as the rhombus, hexagonal photonic lattice can also be fabricated. Typical results are shown in Fig. 5. Fig. 5 shows two photonic lattices with periods of 25 lm (top) and 20 lm (bottom). The patterns of the output beam are shown at the left column and the photograph probed by the He–Ne laser shown at the right column. As can be seen from the right column of Fig. 5, the lightinduced photonic lattices are elliptical which is due to the separation between two nearest little holes on the mask along the x 0 axis being different from that along the y 0 axis, so the waveguide formed in the dark regions become elliptical type. However, we consider that this property described above can not effect the applications of these hexagonal photonic lattices. 3. Conclusion In this letter, we use the Fourier transform method to fabricate variety of photonic lattices in self-defocusing LiNbO3:Fe crystal. The advantage of this method is that the experimental setup is very simple; the photonic lattices can be easily dominated and are quite regular and robust. Apart from its potential applications in signal processing and information technology, these light-induced nonlinear photonic lattices might offer an easy method to study the discrete solitons and bandgap solitons in nonlinear photonic lattices.
531
Acknowledgement This research was supported by the National Natured Science Foundation of China (Grants: 60278006, 60378013 and 10474047). References [1] D.N. Christodoulides, F. Lederer, Y. Silberg, Nature 424 (2003) 807. [2] S. Mingaleev, Y. Kivshar, Opt. Photonics News 13 (2002) 48. [3] A.S. Davydov, Biology and Quantum Mechanics, Pergamon, Oxford, 1982. [4] A.J. Sievers, S. Takeno, Phys. Rev. Lett. 61 (1988) 970. [5] A. Trombettoni, A. Smerzi, Phys. Rev. Lett. 86 (2001) 2353. [6] C. Bosshard, P.V. Mamyshev, G.I. Stegeman, Opt. Lett. 19 (1994) 90. [7] W. Krolikowski, Y.S. Kivshar, J. Opt. Soc. Am. B 13 (1996) 876. [8] S.J. Frisken, Opt. Lett. 18 (1993) 1035. [9] K.M. Davis, K. Miura, N. Sugimoto, K. Hirao, Opt. Lett. 21 (1996) 1729. [10] F. Chen, M. Stepic, C.E. Ruter, D. Runde, D. Kip, V. Shandarov, O. Manela, M. Segev, Opt. Exp. 13 (2005) 4314. [11] Tao Song, Simin Liu, Ru Guo, Zhaohong Liu, Nan Zhu, Yuanmei Gao, Opt. Exp. 14 (5) (2006) 1924. [12] O. Matoba, K. Itoh, Y. Ichioka, Opt. Lett. 21 (1996) 122. [13] J.W. Fleischer, M. Segev, N.K. Efremidis, D.N. Christodoulides, Nature 422 (2003) 147. [14] H. Martin, E.D. Eugenieva, Z. Chen, D.N. Christodoulides, Phys. Rev. Lett. 92 (2004) 123902. [15] Z. Chen, H. Martin, E.D. Eugenieva, J. Xu, A. Bezryadina, Phys. Rev. Lett. 92 (2004) 143902.