Optical second harmonic generation in Yttrium Aluminum Borate single crystals (theoretical simulation and experiment)
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Optical second harmonic generation in Yttrium Aluminum Borate single crystals (theoretical simulation and experiment)
Ali H Reshak* , S Auluck , A Majchrowski and IV Kityk Address: Institute of Physical Biology-South Bohemia University, Institute of System Biology and Ecology-Academy of Sciences –Nove Hrady 37333, Czech Republic, Physics Department, Indian Institute of Technology, Kanpur (UP) 247667, India, Institute ofApplied Physics, Military University of Technology, Kaliskiego 2, 00-908 Warsaw, Poland and Institue of Physics, J. DlugoszUniversity Czestochowa, Al. Armii Krajowej 13/15, Czestochowa, Poland, and Department of Chemistry, Silesian University ofTechnology, ul. Marcina Strzody 9, PL-44100 Gliwice, PolandEmail: Ali H Reshak* - [email protected]; S Auluck - [email protected]; A Majchrowski - [email protected]; IV Kityk - [email protected]* Corresponding author
Abstract
Experimental measurements of the second order susceptibilities for the second harmonicgeneration are reported for YAl (BO ) (YAB) single crystals for the two principal tensorcomponents xyz and yyy. First principle's calculation of the linear and nonlinear opticalsusceptibilities for Yttrium Aluminum Borate YAl (BO ) (YAB) crystal have been carriedout within a framework of the full-potential linear augmented plane wave (FP-LAPW)method. Our calculations show a large anisotropy of the linear and nonlinear opticalsusceptibilities. The observed dependences of the second order susceptibilities for thestatic frequency limit and for the frequency may be a consequence of different contributionof electron-phonon interactions. The imaginary parts of the second order SHGsusceptibility , , , and are evaluated. We findthat the 2 ω inter-band and intra-band contributions to the real and imaginary parts of show opposite signs. The calculated second order susceptibilities are inreasonable good agreement with the experimental measurements. PACS Codes:
I. Introduction
Yttrium Aluminium Borate YAl (BO ) (YAB) belongs to a family of double borates which crys-tallize in the trigonal structure of the mineral huntite CaMg (CO ) and belong to the space Published: 17 March 2008
PMC Physics B χ ω ( ) χ ω ( ) χ ω ( ) χ ω ( ) χ ω ijk ( )2 ( ) MC Physics B (page number not for citation purposes) group R32 [1]. The general formula of these compounds is RX (BO ) , where R = Y , Gd or anyother lanthanide, and X = Al , Sc , Ga , Cr , Fe [2]. YAB is a non centro-symmetric crystaland as early as in 1974 it was reported to be a very effective second-harmonic generating material[3]. Furthermore, owing to its good chemical stability and possibility of substituting Y ions withother lanthanide ions, namely Nd , Yb , Dy and Er [4] it is a promising material for laserapplications. The nonlinear optical properties of this material in connection with lasing proper-ties led to the construction of numerous systems generating red, green and blue light by self-fre-quency doubling effect [5]. YAB can be obtained as nano crystallite powders by simpletechnological approaches [6,7]. They also possess relatively large two-photon absorption [6],which makes them promising third order optical materials. At the same time they are good matri-ces for different rare earth ions [8-11]. The existing data is restrained by consideration of the localcrystalline fields and the influence of the rare earth ions [12-15]. Intrinsic defects also may playan important role in determining the optical susceptibilities [16,17].We feel that a reliable band structure will be of immense help in understanding the linear andnonlinear optical properties and show directions for technologists to obtain crystalline materialswith desired optical properties. A reliable band structure calculation can help in determining therole of inter-band dipole matrix elements on the optical properties. It can give information onthe dispersion of the bands in k -space and origin of the bands which are directly connected withthe optical hyperpolarizabilities and susceptibilities [18]. In the present work we report first prin-ciple's calculation of the linear and nonlinear optical susceptibilities for YAB using the state-of-the-art full potential linear augmented plane wave method [19] which has proven to be one ofthe most accurate methods [20,21] for the computation of the electronic structure of solidswithin density functional theory (DFT) [22]. One specific feature of the borate crystals is the coex-istence of the strong covalent and ionic chemical bonds, which provide relatively flat k -disper-sion of the bands [23]. Moreover there exists substantial anisotropy of the chemical bonds whichsubstantially restrains the application of the pseudopotential method, even norm-conservingone [24]. The aim of this paper is to understand the origin of birefringence and the high χ ( ω ),using first principle's calculations.In the Section 2 we present the computational and experimental details. Section 3 gives theresults of the calculations and the measurements. The observed discrepancies are discussed fol-lowing the band energy approach. II. Computational and experimental details
YAB has the trigonal structure (Fig. 1) of the mineral huntite [1]. YAB lattice provides suitablesites for rare-earth or transition metal ions doping [8]. The lattice parameters of YAB crystal are a= b = 9.295 Å, c = 7.243 Å, α = γ = 90° and β = 120°. YAB melts incongruently at 1280°C anddecomposes into YBO and AlBO . Therefore, it cannot be crystallized from stoichiometric melts. MC Physics B (page number not for citation purposes) The application of high temperature solution growth (HTSG) method allows lowering the tem-perature of YAB crystallization below the temperature of the peritectic transformation.Self-consistent calculations of the electronic structure and optical properties based on the sca-lar relativistic full-potential linearized augmented plane wave method were carried out using theWIEN2K package [19]. This is a very accurate and efficient scheme to solve the Kohn-Sham equa-tion of density functional theory (DFT) [22]. This is an implementation of the DFT with differentpossible approximations for the exchange-correlation (XC) potential. The XC is treated within (a). Yttrium Aluminium Borate YAl (BO ) (YAB) crystal structure Figure 1 (a). Yttrium Aluminium Borate YAl (BO ) (YAB) crystal structure (b) Primitive unit cell, (c). Brillouin zone. MC Physics B (page number not for citation purposes) the local density approximation (LDA) [25] and scalar relativistic equations are used to obtainself-consistency. The Kohn-Sham equations are solved using a basis of linear APW's. In the inter-stitial region the potential and the charge density are represented by Fourier series. In order toachieve energy convergence, the wave functions in the interstitial region were expanded in planewaves with a cut-off K max = 9/ R MT , where R MT denotes the smallest atomic sphere radius and K max gives the magnitude of the largest K vector in the plane wave expansion. The R MT are taken to be2.14, 1.81, 1.28, and 1.28 atomic units (a.u.) for Y, Al, B and O respectively. The valence wavefunctions inside the spheres are expanded up to l max = 10 while the charge density was Fourierexpanded up to G max = 14.Self-consistency is obtained using 200 points in the irreducible Brillouin zone (IBZ). Wecalculated the frequency dependent linear optical properties using 500 points and nonlinearoptical properties using 1400 points in the IBZ. The self-consistent calculations are assumedto be converged when the total energy of the system is stable within 10 -5 Ry.The second order optical susceptibilities were measured by standard method (Fig. 2) using 10ns Nd-YAG laser (Carat, Lviv, Ukraine, 2005) at the fundamental wavelength 1064 nm, withpulse repetition 7 Hz. The Glan polarizers were used for definition of the input and output direc-tions to measure the different tensor components of the second order optical susceptibilities. Thegreen interferometric filter was used to cut the output doubled frequency signal at 533 nm withrespect to the fundamental ones. Detection was performed by fast response photodiodes con-nected with the GHz oscilloscope (NewPort). The crystals were cut in directions which allowedto do the measurements for two principal tensor components and . The set-up allows us to achieve a precision of 0.08 pm/V for the second order susceptibility.
III. Results and Discussion
A. First order optical susceptibilities and birefringence
We first consider the linear optical properties of the YAB crystal. The investigated crystals havetrigonal symmetry, so we need to calculate two dielectric tensor components, corresponding toelectric field perpendicular and parallel to the optical c-axis, to completely characterize the lin-ear optical properties. These are and , the imaginary parts of the frequencydependent dielectric function. We have performed calculations of the frequency-dependent die-lectric function using the expressions [31,32] (cid:75) k (cid:75) k (cid:75) k χ ω ( ) χ ω ( ) (cid:71) E ε ω xx ( ) ε ω zz ( ) MC Physics B (page number not for citation purposes) The above expressions are written in atomic units with e = 1/m = 2 and ħ = 1. where ω is thephoton energy and is the x component of the dipolar matrix elements between initial| nk (cid:2) and final | n ' k (cid:2) states with their eigen-values E n ( k ) and E n ' ( k ), respectively. ω nn ' ( k ) is the bandenergy difference ω nn ' ( k ) = E n ( k ) - E n ' ( k ) and S k is a constant energy surface S k = { k ; ω nn ' ( k ) = ω }.Figure 3 shows the calculated imaginary part of the anisotropic frequency dependent dielectricfunction and . Broadening is taken to be 0.04 eV. All the optical properties arescissors corrected [26] using a scissors correction of 0.6 eV. This value is the difference betweenthe calculated (5.1 eV) and measured energy gap (5.7 eV). The calculated energy gap is smallerthan the experimental gap as expected from an LDA calculation [27]. It is well known that LDAcalculations underestimate the energy gaps. A very simple way to overcome this drawback is touse the scissor correction, which merely makes the calculated energy gap equal to the experimen-tal gap.From Figure 3, it can be seen that the optical absorption edges for and arelocated at 5.7 eV. Thereafter a small hump arises at around 6.0 eV. Looking at these spectra we ε ω ω ω
122 2 zz ( ) = ′ ( ) ∇ ′ ( ) ∑∫ m PnnZ k dSknn k BZ (1) ε ω ω ω xx BZ m PnnX k PnnY k dSknn k ( ) = ′ ( ) + ′ ( ) ⎡⎣⎢⎢ ⎤⎦⎥⎥∇ ′ ( ) ∑∫ (2) P nnX k ( ) ε ω xx ( ) ε ω zz ( ) ε ω xx ( ) ε ω zz ( ) Second harmonic generation experimental setup (SHG): BS , BS -beam splitters, Ph , Ph -photodiodes, λ /2-half wave plate, P-Glan polarizer, A-Glan analyser, L-lens, RS-rotation stage, F-filter/s, PMT-photomultiplier tube Figure 2
Second harmonic generation experimental setup (SHG): BS , BS -beam splitters, Ph , Ph -photodiodes, λ /2-half wave plate, P-Glan polarizer, A-Glan analyser, L-lens, RS-rotation stage, F-filter/s, PMT-photomultiplier tube. MC Physics B (page number not for citation purposes) note that and increases to reach the highest magnitude at around 7.5 eV for, and around 8.5 eV for . It is known that peaks in the optical response are deter-mined by the electric-dipole transitions between the valence and conduction bands. These peakscan be identified from the band structure. The calculated band structure along certain symmetrydirections is given in Figure 4. In order to identify these peaks we need to look at the optical tran-sition dipole matrix elements. We mark the transitions, giving the major structure in and in the band structure diagram. These transitions are labeled according to the spectralpeak positions in Figure 3. For simplicity we have labeled the transitions in Figure 4, as A, B, andC. The transitions (A) are responsible for the structures of and in the energyrange 0.0–5.0 eV, the transitions (B) 5.0–10.0 eV, and the transitions (C) 10.0–14.0 eV.From the imaginary part of the dielectric function and the real part and is calculated by using of Kramers-Kronig relations [28]. The results of our calculated and are shown in Figure 5. The calculated is about 2.4 and isabout 2.5. ε ω xx ( ) ε ω zz ( ) ε ω xx ( ) ε ω zz ( ) ε ω xx ( ) ε ω zz ( ) ε ω xx ( ) ε ω zz ( ) ε ω xx ( ) ε ω zz ( ) ε ω xx ( ) ε ω zz ( ) ε ω xx ( ) ε ω zz ( ) ε ω xx ( ) ε ω zz ( ) Calculated (dark curve) and (light curve)
Figure 3
Calculated (dark curve) and (light curve). ε ω ⊥ ( ) ε ω II ( ) MC Physics B (page number not for citation purposes) The optical transitions shown on the band structure of YAB
Figure 4
The optical transitions shown on the band structure of YAB.Calculated (dark curve) and (light curve)
Figure 5
Calculated (dark curve) and (light curve). ε ω ⊥ ( ) ε ω II ( ) MC Physics B (page number not for citation purposes) These crystals show considerable anisotropy in the linear optical susceptibilities which favorslarge SHG susceptibilities. The birefringence is also important in fulfilling phase-matching con-ditions. The birefringence can be calculated from the linear response functions from which theanisotropy of the index of refraction is obtained. One can determine the value of the extraordi-nary and ordinary refraction indices. The birefringence is a difference between the extraordinaryand ordinary refraction indices, Δ n = n e - n , where n e is the index of refraction for an electric fieldoriented along the c -axis and n is the index of refraction for an electric field perpendicular to the c -axis. Figure 6, shows the birefringence Δ n ( ω ) for this single crystal. The birefringence is impor-tant only in the non-absorbing region, which is below the energy gap. In the absorption region,the absorption will make it difficult for these compounds to be used as nonlinear crystals in opticparametric oscillators or frequency doublers and triplers. We note that the spectral feature of Δ n ( ω ) shows strong oscillations around zero in the energy range up to 12.5 eV. Thereafter it dropsto zero. We find that the calculated birefringence at zero energy is 0.025 in excellent agreementwith our own measurement of 0.02. It is known that for the borates, the contribution of the elec-tron-phonon interaction to the dielectric dispersion may be neglected for the SHG effects [29]contrary to the linear electro-optics Pockels effect. Comparing these dependences with the ani-sotropy for other borates [23,24] one can conclude that the anisotropy caused by the chemicalbonds is smaller than in the other borates [24]. Calculated Δ n ( ω ) Figure 6
Calculated Δ n ( ω ). MC Physics B (page number not for citation purposes) B. Second order susceptibilities
The expressions of the complex second-order nonlinear optical susceptibility tensor has been presented in previous works [33,34]. From the expressions we canobtain the three major contributions: the interband transitions , the intrabandtransitions and the modulation of interband terms by intraband terms. These arewhere n ≠ m ≠ l . Here n denotes the valence states, m the conduction states and l denotes all states( l ≠ m , n ). There are two kinds of transitions which take place one of them vcc' , involving onevalence band ( v ) and two conduction bands ( c and c' ), and the second transition vv'c , involvingtwo valence bands ( v and v' ) and one conduction band ( c ). The symbols are defined as with being the i component of the electron velocity given as and . The χ ω ω ω ijk ( ) ; ; − ( ) χ ω ω ω int ; , erijk − ( ) χ ω ω ω int ; , raijk − ( ) χ ω ω ω mod ; , ijk − ( ) χ ω ω ω π ω ω int ; , erijk e dk rnmi rmlj rlnkln ml fn − ( ) = { } − ( ) (cid:61) (cid:71) (cid:71) (cid:71) (cid:71) mmmn fmlml flnln nml ω ω ω ω ω ω− ( ) + − ( ) + − ( ) ⎧⎨⎪⎩⎪ ⎫⎬⎪⎭⎪ ∫∑ (3) χ ω ω ω π ω ω ω int ; , raijk nm nmi mlj lnk e dk r r r fnlln l − ( ) = { } (cid:61) (cid:71) (cid:71) (cid:71) (cid:71) nn flmml mli fnmrnmi mnj r nml − ( ) − − ( ) ⎧⎨⎪⎩⎪ ⎫⎬⎪⎭⎪⎡⎣⎢⎢⎢− ∑∫ ω ω ω ω (cid:71) (cid:71) Δ nnmkmn mn fnmrnmi rmlj rlnk ml lnmn mn { } − ( ) + { } − ( ) −ω ω ω ω ωω ω (cid:71) (cid:71) (cid:71) ωω ( ) ⎤⎦⎥⎥⎥⎥ ∑∑ nmlnm (4) χ ω ω ω π ω ω ω ω mod ; , ijk nl lmi mnj n e dk fnmmn mn r r r − ( ) = − ( ) (cid:61) (cid:71) (cid:71) (cid:71) (cid:71) llk lm nli lmj mnknml r r ri fnmrnmi rmnj mnk { } − { } { } ⎡⎣⎢⎢− { } ∑∫ ω (cid:71) (cid:71) (cid:71)(cid:71) (cid:71) Δωω ω ω mn mn nm − ( ) ⎤⎦⎥⎥⎥⎥ ∑ (5) Δ nmi nni mmi k k k (cid:71) (cid:71) (cid:71) ( ) = ( ) − ( ) ϑ ϑ (cid:71) ϑ nmi ϑ ω nmi nm nmi k i k r k (cid:71) (cid:71) (cid:71) ( ) = ( ) ( ) r k r k r k r k r k r k nmi mlj nmi mlj nmj mli (cid:71) (cid:71) (cid:71) (cid:71) (cid:71) (cid:71) ( ) ( ) { } = ( ) ( ) + ( ) ( ) ( ) MC Physics B (page number not for citation purposes) position matrix elements between states n and m , , are calculated from the momentummatrix element using the relation [35]: , with the energy differencebetween the states n and m given by ħ ω nm = ħ ( ω n - ω m ). f nm = f n - f m is the difference of the Fermidistribution functions. i , j and k correspond to cartesian indices.It has been demonstrated by Aspnes [36] that only one virtual-electron transitions (transitionsbetween one valence band state and two conduction band states) give a significant contributionto the second-order tensor. Hence we ignore the virtual-hole contribution (transitions betweentwo valence band states and one conduction band state) because it was found to be negative andmore than an order of magnitude smaller than the virtual-electron contribution for these com-pounds. For simplicity we denote by .We have measured the second order susceptibilities of YAB single crystal using Nd-YAG laserat the fundamental wavelength 1064 nm. Since the investigated crystals belong to the pointgroup R32 there are only five independent components of the SHG tensor, namely, the 123, 112,222, 213 and 312 components (1, 2, and 3 refer to the x, y and z axes, respectively) [30]. Theseare , , , and . Here is the complex sec-ond-order nonlinear optical susceptibility tensor . The subscripts i, j , and k areCartesian indices.The calculated imaginary part of the second order SHG susceptibilities , ,, and are shown in Figures 7 and 8. We do not show the component because it is very small. Our calculation and measurement show that is thedominant component which shows the largest total Re value compared to the othercomponents (Table 1). A definite enhancement in the anisotropy on going from linear opticalproperties to the nonlinear optical properties is evident (Figures 7 and 8). It is well known thatnonlinear optical susceptibilities are more sensitive to small changes in the band structure thanthe linear optical ones. Hence any anisotropy in the linear optical properties is enhanced moresignificantly in the nonlinear spectra. r k nmi (cid:71) ( ) P nmi r k nmi Pnmi kim nm k (cid:71) (cid:71) (cid:71) ( ) = ( )( ) ω χ ω ω ω ijk ( ) ; ; − ( ) χ ω ijk ( )2 ( ) χ ω ( ) χ ω ( ) χ ω ( ) χ ω ( ) χ ω ( ) χ ω ijk ( )2 ( ) χ ω ω ω ijk ( ) ; ; − ( ) χ ω ( ) χ ω ( ) χ ω ( ) χ ω ( ) χ ω ( ) χ ω ( ) χ ijk ( )2 ( ) MC Physics B (page number not for citation purposes) Calculated Im (dark curve) and Im (light curve), all Im χ (2) ( ω ) are multiplied by 10 -7 , and in esu units Figure 7
Calculated Im (dark curve) and Im (light curve), all Im χ (2) ( ω ) are multiplied by 10 -7 , and in esu units. χ ω ( ) χ ω ( ) Calculated Im (dark curve) and Im (light curve), all Im χ (2) ( ω ) are multiplied by 10 -7 , and in esu units Figure 8
Calculated Im (dark curve) and Im (light curve), all Im χ (2) ( ω ) are multiplied by 10 -7 , and in esu units. χ ω ( ) χ ω ( ) MC Physics B (page number not for citation purposes) In Figure 9, we show the 2 ω inter-band and intra-band contributions to Im . Wenote the opposite signs of the two contributions throughout the frequency range. We have calcu-lated the total complex susceptibility for ,... and . The real part of the domi-nant component is shown in Figure 10. The zero-frequency limit of all components is listed inTable 1.From above we can see the total second order susceptibility determining SHG is zero belowhalf the band gap. The 2 ω terms start contributing at energies ~1/2 E g and the ω terms for energyvalues above E g . In the low energy regime ( ≤ χ ω ( ) χ ω ( ) χ ω ( ) Calculated Im along with the intra (2 ω ) and inter (2 ω )-band contributions Figure 9
Calculated Im along with the intra (2 ω ) and inter (2 ω )-band contributions. All Im χ (2) ( ω ) are multiplied by 10 -7 , and in esu units. χ ω ( ) Table 1: Calculated total, intra-band and inter-band contributions of Re in units of 10 -7 esu, along with the measured in units of pm/V. Component 123 112 222 213 Re χ ijk (0) total -0.0011 -0.009 0.01 0.0002 Re χ ijk (0)int er -0.007 0.035 -0.038 0.006 Re χ ijk (0)int ra Total Re χ ijk (0) pm/V -0.5 -0.7 0.8 0.5 Experimental SHG χ ijk (0) (pm/V) 0.12 1.03 χχ ijk ( )2 (( )) χχ ijk ( )2 (( )) MC Physics B (page number not for citation purposes) ω contributions. Beyond 5.7 eV (values of the fundamental energy gaps) the major contributioncomes from the ω term.One could expect that the structures in Im could be understood from the features of ε ( ω ) . Unlike the linear optical spectra, the features in the SHG susceptibility are very difficult toidentify from the band structure because of the presence of 2 ω and ω terms. But we use of thelinear optical spectra to identify the different resonance leading to various features in the SHGspectra. The first spectral band in Im between 0.0–5.0 eV is mainly originated from 2 ω resonance and arises from the first structure in ε ( ω ). The second band between 5.0–7.0 eV isassociated with interference between the ω resonance and 2 ω resonance and is associated withhigh structure in ε ( ω ). The last structure from 7.0–8.0 eV is mainly due to ω resonance and isassociated with the tail in ε ( ω ).From an experimental viewpoint, one of the quantities of interest is the magnitude of SHG(proportional to the second order susceptibility). We present the absolute values of, and in Figure 11. The first peak for these compo-nents are located at 2 ω = 5.31 and 5.11 eV with the peak values of (0.052 and 0.081) × 10 -7 esu, χ ω ijk ( )2 ( ) χ ω ( ) χ ω χ ω ( ) = ( ) χ ω χ ω ( ) = ( ) Calculated Re along with the intra (2 ω ) and inter (2 ω )-band contributions Figure 10
Calculated Re along with the intra (2 ω ) and inter (2 ω )-band contributions. All Im χ (2) ( ω ) are multiplied by 10 -7 , and in esu units. χ ω ( ) MC Physics B (page number not for citation purposes) respectively. To evaluate the performed calculations we have done the measurements of the abso-lute value of and for the YAB single crystals for the Nd-YAG laser wavelength1064 nm and we have revealed the corresponding values equal to about (0.042 and 0.061)×10 -7 esu, respectively confirming sufficiently good agreement. The calculated second order suscepti-bilities show substantially good agreement with the measured one. IV. Conclusion
We have performed experimental measurements of the second order susceptibilities for the sec-ond harmonic generation for the YAl (BO ) (YAB) single crystals for the two principal tensorcomponents xyz and yyy. We have reported a first principle's calculation of the linear and non-linear optical susceptibilities using the FP-LAPW method within a framework of DFT. Our calcu-lations show that YAB possesses a direct energy band gap of about 5.1 eV located at Γ point ofthe Brillouin zone. This is smaller than the experimental value of 5.7 eV. The calculated imagi-nary and real parts of the second order SHG susceptibility and were found to be in reasonable agreement with the measurements. Wenote that any anisotropy in the linear optical susceptibilities will significantly enhance the non-linear optical susceptibilities. Our calculations show that the 2 ω inter-band and intra-band con- χ ω ( ) χ ω ( ) χ ω χ ω ( ) = ( ) χ ω χ ω ( ) = ( ) Calculated absolute value of (dark curve) and (light curve)
Figure 11
Calculated absolute value of (dark curve) and (light curve). All absolute values of χ (2) ( ω ) are mul-tiplied by 10 -7 , and in esu units. χ ω ( ) χ ω ( ) MC Physics B (page number not for citation purposes) tributions to the real and imaginary parts of show opposite signs. This fact may be usedin future for molecular engineering of the crystals in the desirable directions. Acknowledgements
The authors would like to thank the Institute of Physical Biology and Institute of System Biology and Ecology-Computer Center for providing the computational facilities. This work was supported from the institutionalresearch concept of the Institute of Physical Biology, UFB (No. MSM6007665808), and the Institute of SystemBiology and Ecology, ASCR (No. AVOZ60870520).
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