Biomedical Engineering Reference
In-Depth Information
1
Theoretical Framework
for SHG Microscopy:
A Matrix-Based Approach
1.1 Physical Overview ................................................................................4
Linear Polarizability at the Molecular Level • Nonlinear
Polarizability at the Molecular Level
1.2 Mathematical Formalism ....................................................................5
The Molecular Tensor in the Local Frame • Experimental
Observables and the Jones Tensor • Case 1: Azimuthal Rotation
Only • Case 2: Combined Azimuthal and Polar Rotation
1.3 Special Considerations....................................................................... 10
Corrections for the Local Dielectric Medium • Incorporating
the Influence of Focusing within the Paraxial
Approximation • Influence of Tight Focusing on the Local
Polarization: Beyond the Paraxial Approximation
1.4 Extension beyond the Thin Sample Limit....................................... 16
Phase Considerations from Dispersion • The Guoy Phase Shift
1.5 Summary..............................................................................................19
Acknowledgments ..........................................................................................19
References........................................................................................................19
Christopher M.
Dettmar
Purdue University
Garth J. Simpson
Purdue University
3The growing availability of turnkey ultrafast laser sources has fueled a revolution in optical micros-
copy, with second-harmonic generation (SHG) imaging emerging as a powerful probe for local order.
The coherent nature of SHG results in high specificity for certain classes of ordered structures, often
producing virtually no detectable background from disordered media. Consequently, image contrast
is unique from that achievable using conventional optical methods. SHG has been used, and promises
to provide continued insight, in the study of a wide variety of ordered systems, such as cells, organelles,
tissues, complex surfaces, and other chiral materials as described in more detail in Parts II and III of
this topic.
The key objective of this chapter is to provide an initial foundation for quantitatively connecting SHG
at the molecular level to the measured polarization-dependent image contrast in microscopy measure-
ments. A general matrix-based mathematical approach is presented that removes some of the conceptual
complexities of more conventional tensor-based methods. This approach has the additional advantage
of providing simple extendibility for incorporating increasing levels of complexity. Furthermore, it is
relatively simple to numerically invert to allow for either prediction or analysis. Several key limiting
cases commonly arising in SHG microscopy measurements are considered within this framework to
provide a reference point for more elaborate analyses.
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