
Second Harmonic Generation Imaging8
transformation to express a local frame vector, v, or point in the laboratory frame, v′, is performed using
a simple rotation matrix.
′
′
=
−
⋅
v
v
v
v
x
y
x
y
cos sin
sin cos
φ φ
φ φ
(1.4)
is expression can be rewritten more concisely.
(1.5)
Transformation of a tensor is performed using the same basic approach, but with considerably more
elements that must be rotated. If we consider rst the most general set of eight nonzero possible elements
within both the laboratory tensor (six of which are unique in SHG) and the molecular tensor when the
z-axis lies within the X−Y plane, the rotation mat ...