Consequences of brain refractive index mismatch for two-photon microscopy 

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OCT signal strength is sensitive to refractive-index-induced defocus

For the measurement of the refractive index of the sample, the sensitivity of high-NA OCT to defocus aberration was exploited (Figure 4-1). When imaging with a correctly aligned system at the surface of a biological sample, the focal plane of the microscope objective coincides with the center of the coherence volume defining the sectioning volume of the OCT system, leading to good imaging quality, limited only by diffraction and speckle (Figure 4-1a). The relative arm length is defined to be zero in this case. When imaging deeper into a sample with a phase refractive index n’ larger than that of the immersion medium n, refraction at the surface causes the actual focus zA of the objective to be enlarged and shifted deeper into the sample with respect to the nominal focus zN (Figure 4-1b). At the same time, the coherence volume penetrates the tissue slower than the nominal focus due to the increased group refractive index n’g (defined as n’g = c/v’g with c being the vacuum speed of light and v’g the group velocity in the sample) (Figure 4-1b). This discrepancy between the plane imaged with OCT, defined by the position of the coherence volume (CV), and the actual focus results in a reduction of interference contrast (i.e. in a loss in OCT signal). This signal loss is therefore related to the optical properties of the sample and for a proper analysis of this effect, we have to take n’ and its dispersion into account and differentiate between n’ and n’g of the sample.

Dispersion and high NA complicate the refractive index measurement

From the nominal focus zN and the optimal reference arm length , the refractive index n’ of the sample can, in principle, be inferred. Previous calculations of the relationship between n’ and  at a given zN have focused on paraxial or marginal ray analysis and were only used in conjunction with low NA (i.e. below 0.2). Even though some papers presented formulae differentiating between n’ and n’g, calculations were always performed assuming zero dispersion in all media involved (i.e. n = ng, n’ = n’g). (Tearney, et al., 1995; Knuettel & Boehlau-Godau, 2000; Alexandrov, et al., 2003; Zvyagin, et al., 2003).
While these approximations greatly simplify the calculations, they are certainly inadequate for the NA 0.8 objectives used in this work. The use of high-NA objectives increases the defocus aberration and, therefore, the sensitivity to the refractive index, at the expense of a more complicated calculation. Even monochromatically, the point of best focus in presence of a refractive index mismatch is no longer described correctly by the paraxial theory in the case of high NA objectives. Since the immersion water, which has a significant level of dispersion in the near infrared wavelength region, served as a reference medium, we felt it necessary to revisit the issue of the influence of dispersion on the refractive index determination. The experimental precision of our data goes beyond the 1-2% limit presented in previous work (Tearney, et al., 1995; Knuettel & Boehlau-Godau, 2000; Alexandrov, et al., 2003; Zvyagin, et al., 2003; Labiau, et al., 2009), which means dispersion can no longer be neglected.
In principle, the sample should be described as a volume with a certain mean refractive index n’, a given amount of dispersion n'() and a random homogeneous distribution of scatterers. The light of all different wavelengths and incident angles is focused by the microscope objective into this volume, where each scatterer produces secondary spherical waves whose amplitudes and phases depend on the amplitude and phase of the incoming waves at the position of the scatterer. These secondary waves will re-enter the microscope objective and ultimately be transmitted to the detector, where each wave interferes with the corresponding wave from the reference arm. The intensity measured by the detector is the absolute square of the sum of all of these waves. Without  refractive index mismatch, a scatterer at the geometric focus will produce secondary waves which are perfectly in phase with the waves in the reference arm, independent of angle and wavelength, maximizing constructive interference. Any effect which causes a dephasing for some of the waves will decrease interference and, thereby, OCT signal.

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Modeling the image formation process in high-NA OCT

When both NA and dispersion become important, the concept of group velocity is not helpful anymore since wave components with different wavelengths will not only propagate with different speeds, but will also be refracted by different angles. The only rigorous solution is to examine directly the total intensity I on a given pixel of the detector after integrating the amplitude of the electromagnetic field E over all possible wavelengths and incident and reflection angles and summing over both interferometer arms:

Table of contents :

1 Introduction
1.1 We want to image the brain
1.2 Optical aberrations cause imperfect imaging
1.3 Aberrations can be thought of as a phase term on the wavefront
1.4 Many imaging systems can be limited by aberrations
1.4.1 Wide-field microscopy
1.4.2 Confocal microscopy
1.4.3 Two-photon microscopy
1.4.4 Structured illumination microscopy
1.4.5 PALM/FPALM/STORM
1.4.6 Stimulated emission depletion
1.4.7 Optical coherence tomography
1.5 Astronomy uses direct wavefront measurement
1.5.1 Why direct wavefront sensing is hard in microscopy
1.6 The sample refractive index sets the scale for aberrations
1.7 Our refractive index measurements led us to develop deep-OCM
1.8 Deep rat brain imaging is limited by aberrations
1.9 Image analysis allows indirect wavefront measurement
1.9.1 Metric-based, imaging-model-agnostic methods
1.9.2 Metric-based modal wavefront sensing
1.9.3 Pupil segmentation
1.9.4 Phase diversity
2 Deep-OCM
2.1 Details of the setup
2.2 Animal preparation and treatment
2.3 High-speed in vivo rat brain imaging shows blood flow
2.4 Importance of defocus correction for high-NA OCT and OCM
3 Myelin imaging with deep-OCM
3.1 Deep-OCM shows myelin
3.1.1 Sensitivity to fiber orientation
3.2 Imaging myelin fibers in vivo in cortex
3.3 Myelin imaging in the peripheral nervous system
3.4 Discussion
4 Rat brain refractive index
4.1 Introduction
4.2 Measuring refractive index using high-NA OCT
4.2.1 OCT signal strength is sensitive to refractive-index-induced defocus .
4.2.2 Dispersion and high NA complicate the refractive index measurement .
4.2.3 Modeling the image formation process in high-NA OCT
4.2.4 Assuming a dispersion function allows calculation of the refractive index
4.2.5 Choosing a suitable metric increases penetration depth
4.3 Results
4.3.1 Rat brain refractive index as a function of rat age
4.3.2 Importance of dispersion and high NA
4.4 Discussion
4.4.1 A model of defocus in OCT taking high NA and dispersion into account
4.4.2 Value and (non-)dependence of brain refractive index
4.4.3 Limits to the measuring precision
4.4.4 Potential systematic errors
4.4.5 Comparison with recent measurements
5 Consequences of brain refractive index mismatch for two-photon microscopy 
5.1 Discussion
6 Maximum-A-Posteriori Focus and Stigmation (MAPFoSt)
6.1 Introduction
6.2 Materials and methods
6.2.1 The MAPFoSt algorithm
6.2.2 Data analysis
6.2.3 Experiments
6.2.4 Simulating image pairs
6.3 Results
6.3.1 Simulations show bias-free aberration estimation
6.3.2 SEM imaging experiments
6.4 Discussion
7 General Discussion
7.1 Correcting aberrations adds complexity
7.2 Defocus correction in high-NA OCM is worth it
7.3 Two-photon rat-brain imaging suffers from spherical aberration
7.4 The race is still on
8 Acknowledgements
9 Appendices
9.1 Deep-OCM motor placement
9.2 Derivations for MAPFoSt
9.2.1 Calculating the MTF and its derivatives
9.2.2 Calculating the MAPFoSt posterior and profile posterior
9.3 The heuristic SEM autofocus and auto-stigmation algorithm
9.4 Modal wavefront sensing for SEM
10 Literature
11 List of Acronyms
12 French summary / Résumé substantiel de cette thèse

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