Optical second harmonic generation in Yttrium Aluminum Borate single crystals (theoretical simulation and experiment)

1754-0429-1-8 1754-0429 Research article Optical second harmonic generation in Yttrium Aluminum Borate single crystals (theoretical simulation and experiment) Reshak H Ali maalidph@yahoo.co.uk Auluck S sauluck@gmail.com Majchrowski A i.kityk@ajd.czest.pl Kityk IV i.kityk@ajd.czest.pl

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 of Applied Physics, Military University of Technology, Kaliskiego 2, 00-908 Warsaw, Poland

Institue of Physics, J. Dlugosz University Czestochowa, Al. Armii Krajowej 13/15, Czestochowa, Poland, and Department of Chemistry, Silesian University of Technology, ul. Marcina Strzody 9, PL-44100 Gliwice, Poland

PMC Physics B 1754-0429 2008 1 1 8 http://www.physmathcentral.com/1754-0429/1/8 10.1186/1754-0429-1-8 0805.1671v1 25 10 2007 17 3 2008 17 3 2008 2008 Reshak et al. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Experimental measurements of the second order susceptibilities for the second harmonic generation are reported for YAl₃(BO₃)₄(YAB) single crystals for the two principal tensor components xyz and yyy. First principle's calculation of the linear and nonlinear optical susceptibilities for Yttrium Aluminum Borate YAl₃(BO₃)₄(YAB) crystal have been carried out 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 optical susceptibilities. The observed dependences of the second order susceptibilities for the static frequency limit and for the frequency may be a consequence of different contribution of electron-phonon interactions. The imaginary parts of the second order SHG susceptibility χ123(2)(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeE8aJnaaDaaaleaacqaIXaqmcqaIYaGmcqaIZaWmaeaacqGGOaakcqaIYaGmcqGGPaqkaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@35B5@, χ112(2)(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeE8aJnaaDaaaleaacqaIXaqmcqaIXaqmcqaIYaGmaeaacqGGOaakcqaIYaGmcqGGPaqkaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@35B1@, χ222(2)(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeE8aJnaaDaaaleaacqaIYaGmcqaIYaGmcqaIYaGmaeaacqGGOaakcqaIYaGmcqGGPaqkaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@35B5@, and χ213(2)(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeE8aJnaaDaaaleaacqaIYaGmcqaIXaqmcqaIZaWmaeaacqGGOaakcqaIYaGmcqGGPaqkaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@35B5@ are evaluated. We find that the 2ω inter-band and intra-band contributions to the real and imaginary parts of χijk(2)(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqedmvETj2BSbqee0evGueE0jxyaibaieYdOi=BI8qipeYdI8qiW7rqqrFfpeea0xe9LqFf0xc9q8qqaqFn0dXdHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeE8aJnaaDaaaleaacaWGPbGaamOAaiaadUgaaeaacaGGOaGaaGOmaiaacMcaaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@3752@ show opposite signs. The calculated second order susceptibilities are in reasonable good agreement with the experimental measurements.

PACS Codes: 71.15. Mb; 71.15.-m

I. Introduction

Yttrium Aluminium Borate YAl₃(BO₃)₄(YAB) belongs to a family of double borates which crystallize in the trigonal structure of the mineral huntite CaMg₃(CO₃)₄and belong to the space group R32 1. The general formula of these compounds is RX₃(BO₃)₄, where R = Y³⁺, Gd³⁺or any other lanthanide, and X = Al³⁺, Sc³⁺, Ga³⁺, Cr³⁺, Fe³⁺2. YAB is a non centro-symmetric crystal and 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 Y³⁺ions with other lanthanide ions, namely Nd³⁺, Yb³⁺, Dy³⁺and Er³⁺4 it is a promising material for laser applications. The nonlinear optical properties of this material in connection with lasing properties led to the construction of numerous systems generating red, green and blue light by self-frequency doubling effect 5. YAB can be obtained as nano crystallite powders by simple technological approaches 67. They also possess relatively large two-photon absorption 6, which makes them promising third order optical materials. At the same time they are good matrices for different rare earth ions 891011. The existing data is restrained by consideration of the local crystalline fields and the influence of the rare earth ions 12131415. Intrinsic defects also may play an important role in determining the optical susceptibilities 1617.

We feel that a reliable band structure will be of immense help in understanding the linear and nonlinear optical properties and show directions for technologists to obtain crystalline materials with desired optical properties. A reliable band structure calculation can help in determining the role of inter-band dipole matrix elements on the optical properties. It can give information on the dispersion of the bands in k-space and origin of the bands which are directly connected with the optical hyperpolarizabilities and susceptibilities 18. In the present work we report first principle'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 of the most accurate methods 2021 for the computation of the electronic structure of solids within density functional theory (DFT) 22. One specific feature of the borate crystals is the coexistence of the strong covalent and ionic chemical bonds, which provide relatively flat k-dispersion of the bands 23. Moreover there exists substantial anisotropy of the chemical bonds which substantially restrains the application of the pseudopotential method, even norm-conserving one 24. The aim of this paper is to understand the origin of birefringence and the high χ²(ω), using first principle's calculations.

In the Section 2 we present the computational and experimental details. Section 3 gives the results of the calculations and the measurements. The observed discrepancies are discussed following the band energy approach.

II. Computational and experimental details

YAB has the trigonal structure (Fig. 1) of the mineral huntite 1. YAB lattice provides suitable sites 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 and decomposes into YBO₃and AlBO₃. Therefore, it cannot be crystallized from stoichiometric melts. The application of high temperature solution growth (HTSG) method allows lowering the temperature of YAB crystallization below the temperature of the peritectic transformation.

Figure 1

(a). Yttrium Aluminium Borate YAl₃(BO₃)₄(YAB) crystal structure

(a). Yttrium Aluminium Borate YAl₃(BO₃)₄(YAB) crystal structure (b) Primitive unit cell, (c). Brillouin zone.

Self-consistent calculations of the electronic structure and optical properties based on the scalar relativistic full-potential linearized augmented plane wave method were carried out using the WIEN2K package 19. This is a very accurate and efficient scheme to solve the Kohn-Sham equation of density functional theory (DFT) 22. This is an implementation of the DFT with different possible approximations for the exchange-correlation (XC) potential. The XC is treated within the local density approximation (LDA) 25 and scalar relativistic equations are used to obtain self-consistency. The Kohn-Sham equations are solved using a basis of linear APW's. In the interstitial region the potential and the charge density are represented by Fourier series. In order to achieve energy convergence, the wave functions in the interstitial region were expanded in plane waves with a cut-off K_max= 9/R_MT, where R_MTdenotes the smallest atomic sphere radius and K_maxgives the magnitude of the largest K vector in the plane wave expansion. The R_MTare taken to be 2.14, 1.81, 1.28, and 1.28 atomic units (a.u.) for Y, Al, B and O respectively. The valence wave functions inside the spheres are expanded up to l_max= 10 while the charge density was Fourier expanded up to G_max= 14.

Self-consistency is obtained using 200 k⇀ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdUgaRzaaoaaaaa@2C6B@ points in the irreducible Brillouin zone (IBZ). We calculated the frequency dependent linear optical properties using 500 k⇀ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdUgaRzaaoaaaaa@2C6B@ points and nonlinear optical properties using 1400 k⇀ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdUgaRzaaoaaaaa@2C6B@ points in the IBZ. The self-consistent calculations are assumed to be converged when the total energy of the system is stable within 10^-5Ry.

The second order optical susceptibilities were measured by standard method (Fig. 2) using 10 ns Nd-YAG laser (Carat, Lviv, Ukraine, 2005) at the fundamental wavelength 1064 nm, with pulse repetition 7 Hz. The Glan polarizers were used for definition of the input and output directions to measure the different tensor components of the second order optical susceptibilities. The green interferometric filter was used to cut the output doubled frequency signal at 533 nm with respect to the fundamental ones. Detection was performed by fast response photodiodes connected with the GHz oscilloscope (NewPort). The crystals were cut in directions which allowed to do the measurements for two principal tensor components χ123(2)(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeE8aJnaaDaaaleaacqaIXaqmcqaIYaGmcqaIZaWmaeaacqGGOaakcqaIYaGmcqGGPaqkaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@35B5@ and χ222(2)(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeE8aJnaaDaaaleaacqaIYaGmcqaIYaGmcqaIYaGmaeaacqGGOaakcqaIYaGmcqGGPaqkaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@35B5@. The set-up allows us to achieve a precision of 0.08 pm/V for the second order susceptibility.

Figure 2

Second harmonic generation experimental setup (SHG): BS₁, BS₂-beam splitters, Ph₁, Ph₂-photodiodes, λ/2-half wave plate, P-Glan polarizer, A-Glan analyser, L-lens, RS-rotation stage, F-filter/s, PMT-photomultiplier tube

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 have trigonal symmetry, so we need to calculate two dielectric tensor components, corresponding to electric field E→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbdweafzaalaaaaa@2C1C@ perpendicular and parallel to the optical c-axis, to completely characterize the linear optical properties. These are ε2xx(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabew7aLnaaDaaaleaacqaIYaGmaeaacqWG4baEcqWG4baEaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@340F@ and ε2zz(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabew7aLnaaDaaaleaacqaIYaGmaeaacqWG6bGEcqWG6bGEaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@3417@, the imaginary parts of the frequency dependent dielectric function. We have performed calculations of the frequency-dependent dielectric function using the expressions 3132

ε 2 z z ( ω ) = 12 m ω 2 ∫ B Z ∑ | P n n ′ Z ( k ) | 2 d S k ∇ ω n n ′ ( k ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@5C41@

ε 2 x x ( ω ) = 6 m ω 2 ∫ B Z ∑ [ | P n n ′ X ( k ) | 2 + | P n n ′ Y ( k ) | 2 ] d S k ∇ ω n n ′ ( k ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@6A98@

The above expressions are written in atomic units with e²= 1/m = 2 and ħ = 1. where ω is the photon energy and PnnX(k) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabbcfaqnaaDaaaleaacqWGUbGBcqWGUbGBaeaacqWGybawaaGcdaqadaqaaiabdUgaRbGaayjkaiaawMcaaaaa@3340@ is the x component of the dipolar matrix elements between initial |nk⟩ and final |n'k⟩ states with their eigen-values E_n(k) and E_n'(k), respectively. ω_nn'(k) is the band energy difference ω_nn'(k) = E_n(k) - E_n'(k) and S_kis a constant energy surface S_{k =}{k; ω_nn'(k) = ω}.

Figure 3 shows the calculated imaginary part of the anisotropic frequency dependent dielectric function ε2xx(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabew7aLnaaDaaaleaacqaIYaGmaeaacqWG4baEcqWG4baEaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@340F@ and ε2zz(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabew7aLnaaDaaaleaacqaIYaGmaeaacqWG6bGEcqWG6bGEaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@3417@. Broadening is taken to be 0.04 eV. All the optical properties are scissors corrected 26 using a scissors correction of 0.6 eV. This value is the difference between the calculated (5.1 eV) and measured energy gap (5.7 eV). The calculated energy gap is smaller than the experimental gap as expected from an LDA calculation 27. It is well known that LDA calculations underestimate the energy gaps. A very simple way to overcome this drawback is to use the scissor correction, which merely makes the calculated energy gap equal to the experimental gap.

Figure 3

Calculated ε2⊥(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabew7aLnaaDaaaleaacqaIYaGmaeaacqGHLkIxaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@32CE@ (dark curve) and ε2II(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabew7aLnaaDaaaleaacqaIYaGmaeaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaeHbbjxAHXgaiqaacaWFjbGaa8xsaaaakmaabmaabaGaeqyYdChacaGLOaGaayzkaaaaaa@3C64@ (light curve)

From Figure 3, it can be seen that the optical absorption edges for ε2xx(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabew7aLnaaDaaaleaacqaIYaGmaeaacqWG4baEcqWG4baEaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@340F@ and ε2zz(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabew7aLnaaDaaaleaacqaIYaGmaeaacqWG6bGEcqWG6bGEaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@3417@ are located at 5.7 eV. Thereafter a small hump arises at around 6.0 eV. Looking at these spectra we note that ε2xx(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabew7aLnaaDaaaleaacqaIYaGmaeaacqWG4baEcqWG4baEaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@340F@ and ε2zz(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabew7aLnaaDaaaleaacqaIYaGmaeaacqWG6bGEcqWG6bGEaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@3417@ increases to reach the highest magnitude at around 7.5 eV for ε2xx(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabew7aLnaaDaaaleaacqaIYaGmaeaacqWG4baEcqWG4baEaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@340F@, and around 8.5 eV for ε2zz(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabew7aLnaaDaaaleaacqaIYaGmaeaacqWG6bGEcqWG6bGEaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@3417@. It is known that peaks in the optical response are determined by the electric-dipole transitions between the valence and conduction bands. These peaks can be identified from the band structure. The calculated band structure along certain symmetry directions is given in Figure 4. In order to identify these peaks we need to look at the optical transition dipole matrix elements. We mark the transitions, giving the major structure in ε2xx(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabew7aLnaaDaaaleaacqaIYaGmaeaacqWG4baEcqWG4baEaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@340F@ and ε2zz(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabew7aLnaaDaaaleaacqaIYaGmaeaacqWG6bGEcqWG6bGEaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@3417@ in the band structure diagram. These transitions are labeled according to the spectral peak positions in Figure 3. For simplicity we have labeled the transitions in Figure 4, as A, B, and C. The transitions (A) are responsible for the structures of ε2xx(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabew7aLnaaDaaaleaacqaIYaGmaeaacqWG4baEcqWG4baEaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@340F@ and ε2zz(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabew7aLnaaDaaaleaacqaIYaGmaeaacqWG6bGEcqWG6bGEaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@3417@ in the energy range 0.0–5.0 eV, the transitions (B) 5.0–10.0 eV, and the transitions (C) 10.0–14.0 eV.

Figure 4

The optical transitions shown on the band structure of YAB

The optical transitions shown on the band structure of YAB.

From the imaginary part of the dielectric function ε2xx(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabew7aLnaaDaaaleaacqaIYaGmaeaacqWG4baEcqWG4baEaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@340F@ and ε2zz(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabew7aLnaaDaaaleaacqaIYaGmaeaacqWG6bGEcqWG6bGEaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@3417@ the real part ε1xx(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabew7aLnaaDaaaleaacqaIXaqmaeaacqWG4baEcqWG4baEaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@340D@ and ε1zz(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabew7aLnaaDaaaleaacqaIXaqmaeaacqWG6bGEcqWG6bGEaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@3415@ is calculated by using of Kramers-Kronig relations 28. The results of our calculated ε1xx(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabew7aLnaaDaaaleaacqaIXaqmaeaacqWG4baEcqWG4baEaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@340D@ and ε1zz(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabew7aLnaaDaaaleaacqaIXaqmaeaacqWG6bGEcqWG6bGEaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@3415@ are shown in Figure 5. The calculated ε1xx(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabew7aLnaaDaaaleaacqaIXaqmaeaacqWG4baEcqWG4baEaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@340D@ is about 2.4 and ε1zz(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabew7aLnaaDaaaleaacqaIXaqmaeaacqWG6bGEcqWG6bGEaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@3415@ is about 2.5.

Figure 5

Calculated ε1⊥(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabew7aLnaaDaaaleaacqaIXaqmaeaacqGHLkIxaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@32CC@ (dark curve) and ε1II(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabew7aLnaaDaaaleaacqaIXaqmaeaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaeHbbjxAHXgaiqaacaWFjbGaa8xsaaaakmaabmaabaGaeqyYdChacaGLOaGaayzkaaaaaa@3C62@ (light curve)

These crystals show considerable anisotropy in the linear optical susceptibilities which favors large SHG susceptibilities. The birefringence is also important in fulfilling phase-matching conditions. The birefringence can be calculated from the linear response functions from which the anisotropy of the index of refraction is obtained. One can determine the value of the extraordinary and ordinary refraction indices. The birefringence is a difference between the extraordinary and ordinary refraction indices, Δn = n_e- n₀, where n_eis the index of refraction for an electric field oriented along the c-axis and n₀is the index of refraction for an electric field perpendicular to the c-axis. Figure 6, shows the birefringence Δn(ω) for this single crystal. The birefringence is important 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 optic parametric oscillators or frequency doublers and triplers. We note that the spectral feature of Δn(ω) shows strong oscillations around zero in the energy range up to 12.5 eV. Thereafter it drops to zero. We find that the calculated birefringence at zero energy is 0.025 in excellent agreement with our own measurement of 0.02. It is known that for the borates, the contribution of the electron-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 anisotropy for other borates 2324 one can conclude that the anisotropy caused by the chemical bonds is smaller than in the other borates 24.

Figure 6

Calculated Δn(ω)

Calculated Δn(ω).

B. Second order susceptibilities

The expressions of the complex second-order nonlinear optical susceptibility tensor χijk(2)(−2ω;ω;ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeE8aJnaaDaaaleaacqWGPbqAcqWGQbGAcqWGRbWAaeaacqGGOaakcqaIYaGmcqGGPaqkaaGcdaqadaqaaiabgkHiTiabikdaYiabeM8a3jabcUda7iabeM8a3jabcUda7iabeM8a3bGaayjkaiaawMcaaaaa@3E6B@ has been presented in previous works 3334. From the expressions we can obtain the three major contributions: the interband transitions χint⁡erijk(−2ω;ω,ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeE8aJnaaDaaaleaacyGGPbqAcqGGUbGBcqGG0baDcqWGLbqzcqWGYbGCaeaacqWGPbqAcqWGQbGAcqWGRbWAaaGcdaqadaqaaiabgkHiTiabikdaYiabeM8a3jabcUda7iabeM8a3jabcYcaSiabeM8a3bGaayjkaiaawMcaaaaa@4299@, the intraband transitions χint⁡raijk(−2ω;ω,ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeE8aJnaaDaaaleaacyGGPbqAcqGGUbGBcqGG0baDcqWGYbGCcqWGHbqyaeaacqWGPbqAcqWGQbGAcqWGRbWAaaGcdaqadaqaaiabgkHiTiabikdaYiabeM8a3jabcUda7iabeM8a3jabcYcaSiabeM8a3bGaayjkaiaawMcaaaaa@4291@ and the modulation of interband terms by intraband terms χmod⁡ijk(−2ω;ω,ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeE8aJnaaDaaaleaacyGGTbqBcqGGVbWBcqGGKbazaeaacqWGPbqAcqWGQbGAcqWGRbWAaaGcdaqadaqaaiabgkHiTiabikdaYiabeM8a3jabcUda7iabeM8a3jabcYcaSiabeM8a3bGaayjkaiaawMcaaaaa@3FC3@. These are

χ int ⁡ e r i j k ( − 2 ω ; ω , ω ) = e 3 ℏ 2 ∑ n m l ∫ d k → 4 π 3 r → n m i { r → m l j r → l n k } ( ω l n − ω m l ) { 2 f n m ( ω m n − 2 ω ) + f m l ( ω m l − ω ) + f l n ( ω l n − ω ) } MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@A699@

χ int ⁡ r a i j k ( − 2 ω ; ω , ω ) = e 3 ℏ 2 ∫ d k → 4 π 3 [ ∑ n m l ω n m r → n m i { r → m l j r → l n k } { f n l ω l n 2 ( ω l n − ω ) − f l m ω m l 2 ( ω m l − ω ) } − 8 i ∑ n m f n m r → n m i { Δ m n j r → n m k } ω m n 2 ( ω m n − 2 ω ) + 2 ∑ n m l f n m r → n m i { r → m l j r → l n k } ( ω m l − ω l n ) ω m n 2 ( ω m n − 2 ω ) ] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@0885@

χ mod ⁡ i j k ( − 2 ω ; ω , ω ) = e 3 2 ℏ 2 ∫ d k → 4 π 3 [ ∑ n m l f n m ω m n 2 ( ω m n − ω ) { ω n l r → l m i { r → m n j r → n l k } − ω l m r → n l i { r → l m j r → m n k } } − i ∑ n m f n m r → n m i { r → m n j Δ m n k } ω m n 2 ( ω m n − ω ) ] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@CDD1@

where 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 one valence band (v) and two conduction bands (c and c'), and the second transition vv'c, involving two valence bands (v and v') and one conduction band (c). The symbols are defined as Δnmi(k→)=ϑnni(k→)−ϑmmi(k→) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabfs5aenaaDaaaleaacqWGUbGBcqWGTbqBaeaacqWGPbqAaaGcdaqadaqaaiqbdUgaRzaalaaacaGLOaGaayzkaaGaeyypa0Jaeqy0dO0aa0baaSqaaiabd6gaUjabd6gaUbqaaiabdMgaPbaakmaabmaabaGafm4AaSMbaSaaaiaawIcacaGLPaaacqGHsislcqaHrpGsdaqhaaWcbaGaemyBa0MaemyBa0gabaGaemyAaKgaaOWaaeWaaeaacuWGRbWAgaWcaaGaayjkaiaawMcaaaaa@479C@ with ϑ→nmi MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiqbeg9akzaalaWaa0baaSqaaiabd6gaUjabd2gaTbqaaiabdMgaPbaaaaa@3101@ being the i component of the electron velocity given as ϑnmi(k→)=iωnm(k→)rnmi(k→) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeg9aknaaDaaaleaacqWGUbGBcqWGTbqBaeaacqWGPbqAaaGcdaqadaqaaiqbdUgaRzaalaaacaGLOaGaayzkaaGaeyypa0JaemyAaKMaeqyYdC3aaSbaaSqaaiabd6gaUjabd2gaTbqabaGcdaqadaqaaiqbdUgaRzaalaaacaGLOaGaayzkaaGaemOCai3aa0baaSqaaiabd6gaUjabd2gaTbqaaiabdMgaPbaakmaabmaabaGafm4AaSMbaSaaaiaawIcacaGLPaaaaaa@46DA@ and {rnmi(k→)rmlj(k→)}=12(rnmi(k→)rmlj(k→)+rnmj(k→)rmli(k→)) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=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@67A3@. The position matrix elements between states n and m, rnmi(k→) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdkhaYnaaDaaaleaacqWGUbGBcqWGTbqBaeaacqWGPbqAaaGcdaqadaqaaiqbdUgaRzaalaaacaGLOaGaayzkaaaaaa@33B8@, are calculated from the momentum matrix element Pnmi MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdcfaqnaaDaaaleaacqWGUbGBcqWGTbqBaeaacqWGPbqAaaaaaa@3070@ using the relation 35: rnmi(k→)=Pnmi(k→)imωnm(k→) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabdkhaYnaaDaaaleaacqWGUbGBcqWGTbqBaeaacqWGPbqAaaGcdaqadaqaaiqbdUgaRzaalaaacaGLOaGaayzkaaGaeyypa0tcfa4aaSaaaeaacqWGqbaudaqhaaqaaiabd6gaUjabd2gaTbqaaiabdMgaPbaadaqadaqaaiqbdUgaRzaalaaacaGLOaGaayzkaaaabaGaemyAaKMaemyBa0MaeqyYdC3aaSbaaeaacqWGUbGBcqWGTbqBaeqaamaabmaabaGafm4AaSMbaSaaaiaawIcacaGLPaaaaaaaaa@4832@, with the energy difference between the states n and m given by ħ ω_nm= ħ(ω_n- ω_m). f_nm= f_n- f_mis the difference of the Fermi distribution functions. i, j and k correspond to cartesian indices.

It has been demonstrated by Aspnes 36 that only one virtual-electron transitions (transitions between one valence band state and two conduction band states) give a significant contribution to the second-order tensor. Hence we ignore the virtual-hole contribution (transitions between two valence band states and one conduction band state) because it was found to be negative and more than an order of magnitude smaller than the virtual-electron contribution for these compounds. For simplicity we denote χijk(2)(−2ω;ω;ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeE8aJnaaDaaaleaacqWGPbqAcqWGQbGAcqWGRbWAaeaacqGGOaakcqaIYaGmcqGGPaqkaaGcdaqadaqaaiabgkHiTiabikdaYiabeM8a3jabcUda7iabeM8a3jabcUda7iabeM8a3bGaayjkaiaawMcaaaaa@3E6B@ by χijk(2)(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqedmvETj2BSbqee0evGueE0jxyaibaieYdOi=BI8qipeYdI8qiW7rqqrFfpeea0xe9LqFf0xc9q8qqaqFn0dXdHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeE8aJnaaDaaaleaacaWGPbGaamOAaiaadUgaaeaacaGGOaGaaGOmaiaacMcaaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@3752@.

We have measured the second order susceptibilities of YAB single crystal using Nd-YAG laser at the fundamental wavelength 1064 nm. Since the investigated crystals belong to the point group 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. These are χ123(2)(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeE8aJnaaDaaaleaacqaIXaqmcqaIYaGmcqaIZaWmaeaacqGGOaakcqaIYaGmcqGGPaqkaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@35B5@, χ112(2)(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeE8aJnaaDaaaleaacqaIXaqmcqaIXaqmcqaIYaGmaeaacqGGOaakcqaIYaGmcqGGPaqkaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@35B1@, χ222(2)(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeE8aJnaaDaaaleaacqaIYaGmcqaIYaGmcqaIYaGmaeaacqGGOaakcqaIYaGmcqGGPaqkaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@35B5@, χ213(2)(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeE8aJnaaDaaaleaacqaIYaGmcqaIXaqmcqaIZaWmaeaacqGGOaakcqaIYaGmcqGGPaqkaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@35B5@ and χ312(2)(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeE8aJnaaDaaaleaacqaIZaWmcqaIXaqmcqaIYaGmaeaacqGGOaakcqaIYaGmcqGGPaqkaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@35B5@. Here χijk(2)(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqedmvETj2BSbqee0evGueE0jxyaibaieYdOi=BI8qipeYdI8qiW7rqqrFfpeea0xe9LqFf0xc9q8qqaqFn0dXdHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeE8aJnaaDaaaleaacaWGPbGaamOAaiaadUgaaeaacaGGOaGaaGOmaiaacMcaaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@3752@ is the complex second-order nonlinear optical susceptibility tensor χijk(2)(−2ω;ω;ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeE8aJnaaDaaaleaacqWGPbqAcqWGQbGAcqWGRbWAaeaacqGGOaakcqaIYaGmcqGGPaqkaaGcdaqadaqaaiabgkHiTiabikdaYiabeM8a3jabcUda7iabeM8a3jabcUda7iabeM8a3bGaayjkaiaawMcaaaaa@3E6B@. The subscripts i, j, and k are Cartesian indices.

The calculated imaginary part of the second order SHG susceptibilities χ123(2)(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeE8aJnaaDaaaleaacqaIXaqmcqaIYaGmcqaIZaWmaeaacqGGOaakcqaIYaGmcqGGPaqkaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@35B5@, χ112(2)(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeE8aJnaaDaaaleaacqaIXaqmcqaIXaqmcqaIYaGmaeaacqGGOaakcqaIYaGmcqGGPaqkaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@35B1@, χ222(2)(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeE8aJnaaDaaaleaacqaIYaGmcqaIYaGmcqaIYaGmaeaacqGGOaakcqaIYaGmcqGGPaqkaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@35B5@, and χ213(2)(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeE8aJnaaDaaaleaacqaIYaGmcqaIXaqmcqaIZaWmaeaacqGGOaakcqaIYaGmcqGGPaqkaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@35B5@ are shown in Figures 7 and 8. We do not show the component χ312(2)(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeE8aJnaaDaaaleaacqaIZaWmcqaIXaqmcqaIYaGmaeaacqGGOaakcqaIYaGmcqGGPaqkaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@35B5@ because it is very small. Our calculation and measurement show that χ222(2)(ω) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeE8aJnaaDaaaleaacqaIYaGmcqaIYaGmcqaIYaGmaeaacqGGOaakcqaIYaGmcqGGPaqkaaGcdaqadaqaaiabeM8a3bGaayjkaiaawMcaaaaa@35B5@ is the dominant component which shows the largest total Re χijk(2)(0) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeE8aJnaaDaaaleaacqWGPbqAcqWGQbGAcqWGRbWAaeaacqGGOaakcqaIYaGmcqGGPaqkaaGcdaqadaqaaiabicdaWaGaayjkaiaawMcaaaaa@3617@ value compared to the other components (Table 1). A definite enhancement in the anisotropy on going from linear optical properties to the nonlinear optical properties is evident (Figures 7 and 8). It is well known that nonlinear optical susceptibilities are more sensitive to small changes in the band structure than the linear optical ones. Hence any anisotropy in the linear optical properties is enhanced more significantly in the nonlinear spectra.

Table 1

Calculated total, intra-band and inter-band contributions of Re χijk(2)(0) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeE8aJnaaDaaaleaacqWGPbqAcqWGQbGAcqWGRbWAaeaacqGGOaakcqaIYaGmcqGGPaqkaaGcdaqadaqaaiabicdaWaGaayjkaiaawMcaaaaa@3617@ in units of 10^-7esu, along with the measured χijk(2)(0) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaciaacaqabeaabeqacmaaaOqaaiabeE8aJnaaDaaaleaacqWGPbqAcqWGQbGAcqWGRbWAaeaacqGGOaakcqaIYaGmcqGGPaqkaaGcdaqadaqaaiabicdaWaGaayjkaiaawMcaaaaa@3617@ 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

0.008

-0.04

0.04

-0.0085

Total Re χ^ijk(0) pm/V

-0.5