OPSqLattice[x_, y_, XsiGL_]  :=  
      Sum[ 1 * Exp[I*n*2*Pi*y]*
          Exp[-(x - n)^2/(2  XsiGL^2)],  {n, -10^2, 
          10^2}]

OPSqLattice[0,  .23, 1] // N

Plot3D[ 
      Abs[OPSqLattice[x, y,  .4]],  {x, -2, 2},  {y, -2, 2},  
      PlotPoints -> 50]

ContourPlot[ 
      Abs[OPSqLattice[x, y,  .4]],  {x, -2, 2},  {y, -2, 2},  
      PlotPoints -> 50]

Plot[ Abs[OPSqLattice[x, 0,  .4]],  {x, -2, 2}]

Plot[ Abs[OPSqLattice[0, y,  .4]],  {y, -2, 2}]

OPTriLattice[x_, y_, XsiGL_]  :=  
      Sum[ If[EvenQ[n], 1, I] * Exp[I*n*2*Pi*y]*
          Exp[-(x - n)^2/(2  XsiGL^2)],  {n, -10^2, 
          10^2}]

OPTriLattice[0,  .23, 1] // N

Plot3D[ 
      Abs[OPTriLattice[x, y,  .4]],  {x, -2, 2},  {y, -2, 2},  
      PlotPoints -> 50]

ContourPlot[ 
      Abs[OPTriLattice[x, y,  .4]],  {x, -2, 2},  {y, -2, 2},  
      PlotPoints -> 50]