Theoretical Framework for SHG Microscopy: A Matrix-Based Approach - Second Harmonic Generation Imaging - page 9

The expression in Equation 1.7 provides an initial handle to connect the local and laboratory frames
in SHG microscopy measurements, offering a route for interpreting local structure and orientation
information from polarization-dependent SHG measurements. Interestingly, all chiral-specific effects
linked to the β xyz and related tensor elements disappear in this orientation. Because the generated polar-
ization is orthogonal to the two driving fields for β xyz and its permutations, the only nonlinear polariza-
tion allowed is polarized along the direction of beam propagation (i.e., along z ).
Using this set of expressions, the far-field polarization state of the doubled light expressed as a Jones
vector can be related directly back to the azimuthal rotation angle of the sample within the field of view
ϕ , the four unique elements within the local uniaxial tensor β (2) , and the incident far-field fundamental
polarization state through Equations 1.3 and 1.7.
However, the genesis of the laboratory frame tensor describing the experimental observables in such a
relatively simple form required the use of several assumptions and approximations that may or may not
hold in many microscopy measurements, including: (i) the assumption of the local z -axis lying within
the image plane, (ii) the assumption of plane-wave excitation (as opposed to a tightly focused beam), (iii)
the neglect of local field corrections from dielectric screening, and (iv) the assumption of a point SHG
source, each of which is considered in subsequent sections.

1.2.4 case 2: combined Azimuthal and Polar Rotation
The situation becomes a bit more interesting when considering both polar (in and out of the image
plane) and azimuthal (rotation within the image plane) orientation of the local z- axis, depicted in Figure
1.3. However, this case also represents the more common situation in practice, in which the local axis
is not perfectly aligned within the image plane. Rotation of an object out-of-plane in a fixed laboratory
reference frame can be treated by including a tilt angle rotation, θ , with the total rotation matrix given
by the matrix product of the two individual operations.

−

cos φ sin φ 0

0

cos θ 0 sin θ cos cos θ φ − sin φ

φ

sin cos θ φ

R ( , ) θ φ = sin φ cos φ ⋅ 0 1 0 = cos θ sin φ cos sin sin θ φ (1.9)

0 0 1 − sin θ 0 cos θ − sin θ 0 cos θ

Working through the coordinate transformations using the equalities present for SHG of uniaxial
assemblies β yyz = β yzy = β xzx = β xxz , β zyy = β zxx , and β xyz = β xzy = − β yzx = − β yxz yields the following set of ori-
entational angles connecting the local frame to the macroscopic observables.

χ
χ
χ
χ
χ
χ

0 2 ( Hi Ag + ) Hi Ag + Fi

J HHH ,

Db Aa 2 Fh Fh Fh β

β yyz

− −

J HHV , xyz

,

2 Da − 2 Fg Hh Ai + Fg

J HVV ,

= ⋅

(1.10)

− 2 Db − 2 Fh Hh Af + F h β

β

J VHH , zyy

− Da Ab − 2 Fg − Fg Fg

J VVH , zzz

l eff

0 2( Hf + Ah Hf ) + Ah Ff

J VVV ,

L

The same shorthand notation for the orientational averages is used, with capital letters indicating the
trigonometric functions of the polar tilt angle, θ, and lowercase letters indicating the identical trigonomet-
ric functions of the azimuthal rotation angle, ϕ . Assuming a polar tilt angle of θ = π /2, consistent with the
local uniaxial z -axis oriented within the image plane, recovers the expression in Equation 1.7.
The expressions above provide a starting point for connecting local structure and orientation to the
experimental observables measured in the laboratory frame. If one knows the molecular tensor a priori ,