Theoretical Framework for SHG Microscopy: A Matrix-Based Approach - Second Harmonic Generation Imaging - page 7

In Equation 1.3, the six unique elements of the Jones tensor have been written as a vector, with the
Jones vector of the doubled light given by the product of this vector with a matrix derived from the ele-
ments of the incident Jones vector for e L ω . This form is certainly not the only way to write this relationship,
but the use of the vectorized form of the tensor has distinct practical advantages in subsequent steps.
Experimentally, image contrast in SHG microscopy depends on detection of either the intensities of
the different polarization components or the fields themselves as functions of the incident polarization
state(s). Precise knowledge of the Jones tensor allows prediction of the experimental observables within
the image. Under appropriate conditions, the problem can also be inverted to recover the Jones tensor
elements from polarization-dependent SHG measurements. 6b,7,8 The scope of the current work is focused
primarily on providing a general framework for relating any experimental observables back to the Jones
tensor and ultimately the local tensor introduced in the preceding section.

1.2.3 case 1: Azimuthal Rotation only
The strategy taken here to connect the local and laboratory responses will be to start with the simplest
cases initially neglecting some of the more subtle interactions, then to systematically introduce increas-
ingly general frameworks. Consistent with this approach, we will consider first the simplest case of
plane-wave excitation of a point source located in vacuum.
Even within this simple context, the local frame is generally not perfectly oriented with the laboratory
frame in SHG microscopy measurements. In most cases, the local uniaxial axis is tilted significantly
from the horizontal and vertical image axes, with that orientation changing as a function of position
within the field of view. Two obvious strategies emerge for interpreting polarization-dependent mea-
surements in such instances: (1) mathematically rotate the molecular tensor to the laboratory frame to
interpret the laboratory-frame measurements, or (2) rotate the laboratory frame optical fields to the local
frame, then rotate the generated local polarization back to the laboratory frame. The two approaches are
mathematically equivalent, but the first is arguably easier to implement with respect to book-keeping
and will be the only one presented herein.
Considering a system with local uniaxial symmetry, only the two rotation angles ϕ and θ are required
for projecting the local tensor onto the laboratory frame (Figure 1.3). In the case of fibers, the uniaxial
axis runs parallel to the fiber axis. For surfaces and membranes, the uniaxial axis is normal to the surface.
Often, the uniaxial axis lies close to the X − Y plane of the image, such that the rotation can be reason-
ably approximated as depending only on ϕ (i.e., assuming θ = π/2). In this limiting case, the coordinate

Z

z ′

Y

ψ

X

θ

y ′

φ

x ′

FIgurE 1.3 Depiction of the Euler angles involved in rotating a point from the XYZ coordinate system to the
x ′ y ′ z ′ coordinate system or vice versa . ϕ is the azimuthal angle, θ is the polar angle, and ψ is the twist angle.