A theory based on the two-band tight-binding approximation for $\ensuremath{\pi}$ electrons is developed to describe the second-order nonlinear optical (NLO) properties of arrays of uniformly sized and well-aligned boron-nitride single-walled nanotubes (BN-SWNTs) with a zigzag achiral structure. It is assumed that the coherent light beam at frequency $\ensuremath{\omega}$, incident upon the nanotube sample, is linearly polarized along the symmetry axis of the nanotubes. The long-axis NLO susceptibility ${\ensuremath{\chi}}^{(2)}(\ensuremath{\omega})$ of those nanotubes is calculated within the independent nanotube approximation and in neglecting local-field effects. Using the perturbation-theory formalism in the crystal-momentum representation, we derive an explicit analytic expression for the ${\ensuremath{\chi}}^{(2)}(\ensuremath{\omega})$ and apply it to study three distinct second-order NLO effects possible in the BN-SWNTs due to their noncentrosymmetric structure---namely, second-harmonic generation (SHG), linear electro-optical (LEO) effect, and nonlinear optical rectification (NOR). The theory is illustrated by numerical model calculations of the SHG, LEO, and NOR susceptibility spectra for several representative BN-SWNT ensembles consisting of large-diameter nanotubes. The calculated SHG spectra are found to be dominated by the highly peaked $2\ensuremath{\omega}$ resonance at half the band-gap energy of the BN-SWNTs, where the absorption of light is negligible. Distinct features are also found in the LEO and NOR susceptibility spectra, e.g., a sudden switching of the susceptibility from a positive peak value to a negative peak one in the near vicinity of the fundamental absorption edge. A fairly large magnitude of those susceptibilities, reaching the order of ${10}^{\ensuremath{-}7}\text{ }\text{esu}$ under off-resonant conditions and up to ${10}^{\ensuremath{-}6}\text{ }\text{esu}$ in the resonant case, suggests that BN-SWNTs are a promising material for various electro-optical device applications.
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