  Using Density Functional Theory (DFT) method we compute linear optical
absorption spectra and nonlinear optical susceptibilities of hydrogen
passivated armchair and zigzag Phosphorous Nanoribbons (aPNR and zPNR) as well
as \alpha-phase phosphorous monolayer. We observe that: (a) Crystallographic
direction has a strong effect on the band edge absorption which causes optical
anisotropy as well as a red shift of absorption spectra by increasing the
nanoribbon width. (b) The absorption values are in the order of $10^{5}
cm^{-1}$ which are similar to the experimentally measured values. (c) There is
two orders of magnitude enhancement of the 2nd order nonlinear optical
susceptibility, $\chi^{(2)}$, in nanoribbons which emanates from breaking the
centro-symmetric structure of a monolayer phosphorene by hydrogen surface
terminations. (d) Chief among our results is that the 3rd order susceptibility,
$\chi^{(3)}$, for phosphorene monolayer and nanoribbons are about $~10^{-13}$
esu ($~10^{-21} \frac{m^{2}}{V^{2}}$) which are in close agreement with
experimentally reported values as well as a recently calculated value based on
semi-analytic method. This strongly supports reliability of our method in
calculating nonlinear optical susceptibilities of phosphorene and in general
other nanostructures. Enhanced 2nd order optical nonlinearity in phosphorene
promises better second harmonic and frequency difference (THz) generation for
photonics applications.
