Theory of Image Formation in Non-linear Optical Microscopy
Creators: name: Van Der Kolk, Jarno Nicolaas
Publisher: name: Université d'Ottawa / University of Ottawa
Year: 2017
Subjects: Nonlinear Optics
Microscopy
Second Harmonic Generation
Coherent Anti-Stokes Raman Scattering
Stimulated Raman Scattering
Index of Refraction
Simulation
Collagen fibrils
Computational
Phase

description: Nonlinear optical microscopy is a collection of very powerful imaging techniques. Linear optical microscopes probe the refractive index and absorption, which both stem from the first-order linear electric susceptibility. Especially in biological tissue, the variation in the refractive index is often small and the tissue is, in many cases, transparant. Nonlinear optical microscopes on the other hand probe the nonlinear higher-order susceptibilities, which can be chemically sensitive, leading to the capability to achieve label-free imaging. Nonlinear optical microscopes have been in development for more than thirty years and they are based on numerous nonlinear optical processes. The ones I will concentrate on in this thesis are second harmonic generation (SHG), coherent anti-Stokes Raman scattering (CARS), and stimulated Raman Scattering (SRS). The first technique is commonly used to image collagen as those molecules have a particularly large second-order nonlinear susceptibility due to their chiral structure. CARS and SRS on the other hand are often used because they resonantly target vibrational resonances in molecules, giving rise to the aforementioned label-free imaging. Deep understanding of the nonlinear imaging process is crucial to the interpretation of the images these techniques produce. Computational tools are exceptionally suited for this task as they allow studying the electromagnetic field anywhere in the sample as well as the far-field, and one can change any of the material properties to study their effect. One such tool is finite-difference time-domain (FDTD) that our group developed for nonlinear optical microscopy simulations. It is a direct discretization of Maxwell's equation. While computationally costly, it does allow any arbitrary shaped sample to be simulated. The sample can have frequency dependent refractive indexes, and also nonlinear media with third-order nonlinearities such as Kerr media and Raman-active media, but also second-order nonlinearities for SHG. The code is designed in such a way that it can run on thousands of CPUs on a wide variety of compute cluster which allows our group to obtain nanoscale resolution. Another computational tool I use is the free-space Green's function solution to the Helmholtz equation, which can be used to calculate the Hertz vector in the frequency domain, both in the near- and far-field, based on the induced nonlinear polarization. The electric field is then calculated from this Hertz vector. This technique is much faster then FDTD and also allows for arbitrary shapes of the nonlinear electric susceptibility in the sample. However, it assumes a homogeneous refractive index throughout the entire spatial domain and requires complete knowledge of the input beam or beams that induce the nonlinear polarization. In this thesis, I use these tools to study the image formation process of various nonlinear optical processes mentioned earlier. For example, I study the effect of an inhomogeneous refractive index on the images produced by these microscopes. In literature the index of refraction is almost always assumed to be homogeneous, because, as mentioned before, the inhomogeneity of the refractive index is often small. However, I show that these small differences in the index of refraction can have a significant effect on the measured far-field intensity signal. For example, in SRS and CARS images, the measured signal can increase by an order of magnitude depending on the index mismatch and structure of the sample. Additionally, significant shifts in perceived position occur. Even nonresonant nonlinear signals can be evoked purely through a mismatch in linear refractive index. Computational modelling can also help reveal additional detail. As SHG is a coherent process, subwavelength information can be inferred through the phase information. Our experimental collaborators built an interferometric SHG (I-SHG) microscope for exactly that purpose. We used this to image collagen fibrils, which are all aligned in a parallel fashion. However, because collagen fibrils have a chiral molecular structure, they can point either ``up'' or ``down''. Using my Green's function simulations of the SHG imaging process of collagen fibrils, I was able to predict the standard deviation in the measured phase and link it to the orientation of collagen fibrils in the focal spot of the probing laser beam, even though the diameters are far below the minimum resolvable capabilities of the microscope. We found that the ``upwards'' fibrils make up 46--53% of the sample. Even with a normal SHG microscope that does not measures phase, additional subresolution information is obtainable. With our collaborators we measured the ratio of the forward SHG intensity signal to that in the backward direction and with my simulations, we are able to link this to the fibril diameters in collagen tissue. Thus we inferred that the fibril diameter increases as a function of tissue depth. Furthermore, a computational technique called ptychography is able to retrieve phase information without an interferometric reference beam. Additionally, it increases resolution to the theoretical limit, independent of the laser focal spot size, and corrects for distortions in the input beam as well. I have developed this technique for use with nonlinear optical microscopy and was able to show it is a viable alternative to I-SHG by imaging simulated rat tail tendon at the diffraction limit while retrieving the orientation of the fibrils through the phase of the SHG signal. I also implemented the algorithm for CARS, where the phase information can be used to greatly increase the signal-to-noise ratio by reducing the nonresonant background radiation that results from competing nonlinear optical processes. I showed an example of this by imaging a simulated fibroblast cell where the CARS process was tuned to the lipid droplets inside of the cell. I am currently in talk with experimentalists to apply this theoretical technique to experiments as that would further demonstrate the impact of my work. Finally, keeping in theme with the collagen fibrils, I show that the ratio of the forward SHG signal to the backward signal, the F/B ratio, is affected by a mismatch in the refractive index for fibrils larger than 100nm. This measure is an indicator of fibril diameter and thus important for making qualitative predictions. Single fibrils are generally too small to be significantly affected by near-field effects, but the bigger fibrils can be. Fibrils in rat tail tendon have a distribution of fibrils diameters and the large fibrils occur infrequent. However, I found that the large fibrils are largely responsible for the forward as well as backward signal, thus refractive index mismatches still affect the F/B ratio significantly despite their infrequency. The F/B ratio for a collection of fibrils placed in a n=1.47 medium was found to be 31.8±0.7% higher than for those in a n=1.33 medium. Our experimental colleagues have done preliminary measurements on mouse tail tendon where they found an increase of 40±20%, in line with the value of 28.1±0.6% that I found for simulations with mouse tail tendon. In conclusion, the theoretical tools I have used in my thesis have provided me with the ability to study nonlinear optical image formation processes with a level of detail that would be near-impossible to do experimentally. I have used this ability to show how refractive index mismatches, such as those found in biological tissue, can significantly distort the far-field intensity signals. I have shown this for SRS and CARS where the far-field intensity signal appeared an order-of-magnitude larger compared to the same sample without a refractive index mismatch with the background medium. Additionally, shifts in the perceived position of the object under investigation were observed and I showed the presence of a nonresonant background signal in AM-SRS. Likewise I showed that in the SHG imaging of collagen fibrils significant changes in the F/B ratio can occur. All of these effects have important implications as these types of images as biomedical researches rely on the correct interpretation of nonlinear optical microscopy images for both research and diagnostics. Apart from showing the effect of a refractive index mismatch, I have also shown that computation modelling can be used to infer subwavelength features in SHG imaging experiments of collagen fibril such as fibril orientation and fibril diameter. These methods have the potential to aid medical researchers as changes in the structure of collagen are often an early indicator of diseases such as osteoarthritis. Finally, I showed that the ptychography algorithm I developed for nonlinear optical microscopy is able to retrieve phase information of the nonlinear electric susceptibility in SHG and CARS imaging while also enhancing the resolution and correcting for distortions in the input beams. I can also use much larger laser spot sizes than in conventional experiments without compromising the obtained resolution, thus fewer measurements are required. The technique is not limited to SHG and CARS either; it will work for other nonlinear optical processes as well. Experimental verification of nonlinear ptychography will be done soon. This technique has to potential to significantly improve current imaging techniques since access to the phase information allows one to observe additional information about the sample as we showed with the I-SHG microscope.