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Theoretical background of polarization-resolved SHG microscopy
Introduction to multiphoton microscopy
Multiphoton microscopy (MPM), also known as nonlinear optical microscopy, is a class of optical imaging techniques which rely on various nonlinear processes occurring in nonlinear media. Since it was introduced in a milestone paper of Webb, Denk and Strickler [98], multiphoton microscopy gained more and more popularity in cellular and tissue biology because of its intrinsic 3D-capability, various available endogenous contrasts in tissues, and its small invasivity. In this section, some general features of multiphoton microscopy will be treated in section 2.1, followed by more detailed description of major contrasts used in MPM. Second harmonic generation (SHG) is a dominant topic of the present work, so it will be discussed in much more details and occupies the entire section 2.2. Tensorial response of collagen with respect to polarized excitation will be treated in the final section 2.3.
Principles and contrast mechanisms of multiphoton microscopy
By definition, multiphoton microscopy implements nonlinear optical interactions. This nonlinearity confers several advantages to multiphoton microscopy, such as optical sec-tioning, deep tissue penetration and small phototoxicity. However, these advantages have a price to pay, which is, notably, the need in extremely intensive electric field to be created in the sample.
The large variety of nonlinear optical interactions gives rise to a number of contrasts that can be used in nonlinear microscopy. In this section we will present two-photon excited fluorescence (2PEF), third harmonic generation (THG) and coherent anti-Stokes Raman scattering (CARS). Simplified Jablonski diagrams for these processes are shown in the Fig. 2.1. Second harmonic generation (Fig. 2.1 a) will be briefly introduced as well, but its detailed description is given in the next section.
Making intense fields
Classically, the appearance of nonlinear effects in optics signifies that electron response to sinusoidal excitation field becomes anharmonic. It happens when the perturbation field is large enough to be comparable with typical atomic electric field, which is Eat = e/a20 ∼ 1011 V/m [99]. A beam of a continuous laser light of 1 W, focused in a 0.5 μm × 0.5 μm spot will produce a field intensity of only about 107 V/m. Reaching higher intensities not only presents a technical challenge, but will also lead to energy flux and heat deposition much stronger than what biological tissues can tolerate. However, this problem is circumvented by using ultrashort pulse lasers. It allows one to increase peak intensities necessary for nonlinear processes to occur, while maintaining low average intensity. For example, a 100 mW 100 fs-pulsed laser with 80 MHz rate can provide peak intensity even larger than the atomic field, thus enabling all types of nonlinear interactions up to those leading to destruction of the tissue. Since the apparition of turn-key femtosecond lasers, such as a mode-locked Ti:Sa laser, nonlinear microscopy is gaining more and more popularity.
Optical sectioning
An outstanding and well-known property of multiphoton microscopy is its intrinsic optical sectioning. Sectioning means that the imaging system is able to resolve structures in the axial direction.
Lets consider a tightly focused beam in a medium with bulk fluo rescence and discuss the z-resolution for single-photon excitation and for two-photon excitation. Single-photon excited fluorescence (1PEF) scales linearly with the incident intensity, while 2PEF scales as the square of the incident intensity [99]. For a small volume of the excitation cone,
the number of photons emitted will be dN ω (z) ∝ E2 for 1PEF, and dN 2ω (z) ∝ E4 for dSdz dSdz
2PEF, where E is the excitation field. As the fluence is constant along the propagation direction, i.e. E2dS = C (we assume that there is no beam attenuation along the propagation), we can roughly take E(z) S(z) = C, E and S being the effective area of the beam section at z and the average field across this section. Thus, the number of photons emitted from a slab of thickness dz is dNω(z) ∝ dz E2dS = Cdz for 1PEF, and dN 2ω ˜ (z) ∝ dz/S(z) for 2PEF. For 1PEF, each slab of uniformly fluorescent sample will produce the same number of photons. The integral dNω(z)dz diverges, and one cannot say where a detected photon comes from.
Unlike conventional fluorescence microscopy, confocal microscopy can provide axial resolution. The excitation is still linear, but the detection path is modified by introducing a pinhole, which privileges the detection only of those photons that were emitted near the focal plane. The synergetic effect of tight focusing, which provides high lateral resolution, and the pinhole responsible for axial sectioning, ensures 3D-capability.
More detailed comparison of resolution and PSF between 1PEF and 2PEF can be found in [100].
Two-photon excited fluorescence (2PEF)
2PEF microscopy is far the most widespread nonlinear microscopy technique, and it was first made to be commercially available by Bio-Rad in 1996. Nowadays, a number of manu-facturers (Zeiss, Leica, Nikon, Olympus, …) produce commercial 2PEF microscopes. The process of simultaneous two-photon absorption was first described by Maria Goeppert-Mayer in her PhD thesis in 1931, and the first microscope was built in 1990 by Webb, Denk and Strickler in their lab at Cornell University [98].
Excitation by two photons with subsequent fluorescence (2PEF) is a nonlinear optical process, which is based on the probability of a fluorophore to be excited by simultaneous absorption of two photons in a single quantum event. Each photon carries about half of the energy necessary for the excitation of the fluorophore. The excitation is followed by the emission of a single fluorescence photon with energy lower than the excitation energy due to the Stokes shift, and typically higher than the energy of either photon used to excite the molecule. The diagram illustrating this process is shown in the Fig. 2.1 c. This process is dependent on the imaginary part of the third-order nonlinear susceptibility tensor, and the probability of 2PEF scales as the square of incident intensity [99].
Since 2PEF is a fluorescence imaging technique with 3D-resolution, exactly as the more conventional confocal microscopy, it is informative to compare their properties.
Typically, all the vast library of fluorophores used in confocal microscopy can be directly used for 2PEF. However, for many fluorophores their two-photon absorption spectra differ significantly from their single-photon counterpart due to different selection rules for single- and two-photon transitions. Moreover, two-photon excitation spectra are usually wider. On one hand, it can lead to the loss of selectivity if it is achieved by using different excitation wavelength to excite different fluorophores. On the other hand, while absorption spectra widen and overlap, the respective fluorescence emission spectra remain relatively well separated. It allows for simultaneous excitation of several fluorophores by a single wavelength and selectivity is achieved by choosing different spectral windows for fluorescence emission detection.
For a chosen wavelength and numerical aperture, the resolution of 2PEF is similar to that of confocal microscopy. While the confocal microscopy outperforms 2PEF by a factor of 2 at small depth in weakly-scattering tissue, in thick and scattering tissues 2PEF evens the score. First, the scattering in tissue is significantly higher for the single-photon excita-tion wavelength λ than for two-photon excitation wavelength 2λ (for the same transition energy of the molecule), which results in more deteriorated PSF for confocal imaging. Secondly, scattering leads to lower fluorescence signal, which requires to further open pinhole, thus accepting photons from a wider layer near the focus and deteriorating the resolution.
Due to low phototoxicity, which is restrained to the focal volume, two-photon mi-croscopy permits to further increase incident intensity and to efficiently excite endoge-nous fluorophores, which are generally much weaker than dedicated fluorescent dyes. The most common endogenous fluorophores in biological tissues are aromatic amino acids (tryptophan, tyrosine and phenylalanine), electron carriers (NADH, FAD) and structural proteins such as elastin and collagen. However, collagen fluorescence is weak compared to that of elastin, and it can be visualized much more efficiently by SHG microscopy.
An example of 2PEF image is displayed in Fig. 2.3. It shows a lung tissue affected by idiopathic lung fibrosis (ILF), which is a rare disease characterized by apparition of large collagen producing areas known as fibrotic foci. Two-photon excited fluorescence from endogenous fluorophores is shown in red, and green represents second harmonic generation from collagen fibers (see next section).
Second and third harmonic generation
Second harmonic generation was the first nonlinear optical process experimentally ob-served. It was demonstrated in 1961 by Franken et al. [101] soon after the invention of lasers. It has several applications across different areas of physics, and is often used in lasers to obtain new wavelengths. However, this manuscript focuses exclusively on SHG application as a contrast in a multiphoton microscopy that is called second harmonic mi-croscopy. SHG is specific to highly ordered structures such as those adopted by fibrillar collagen, and can be used in unstained biological tissues. SHG is the central topic of this manuscript, and its detailed description will be done in the next section.
Another promising contrast used in multiphoton microscopy is third harmonic genera-tion (THG). It is a coherent nonlinear process of simultaneous scattering of three photons to produce a single photon with the energy equal to the sum of the three incident photons.
The technique relies on pulsed light with wavelength from 0.9 to 1.5 μm, produced by infrared lasers or optical parametric oscillators (OPOs). Since it is a coherent process, its interpretation could be more difficult than that of incoherent 2PEF. The detected photons outcome depends on several factors, such as scatterers geometry, the form of radiation diagram and the scattering properties of the tissue. For example, there is no THG signal from a bulk sample with uniform nonzero third-order susceptibility χ(3), due to the Gouy phase anomaly and resulting coherent cancellation of THG signal from different parts of the focal volume [99, 102, 103]. On the other hand, a high contrast is observed on the interfaces of two media with different χ(3), such as water-oil interfaces [104, 105].
Since THG can visualize oil-water interfaces, it can reveal outer bilipid layer and nucleus surface, along with smaller organelles such as mitochondria [104]. In the context of this manuscript, it is noteworthy that THG was used to image interfaces between subsequent collagen layers in corneal stroma [106]. It was attributed to different collagen fibril orientation in each collagen slab, resulting in different third-order susceptibilities χ(3) on the either sides of the interface [106].
Coherent anti-Stokes Raman Scattering
Coherent anti-Stokes scattering (CARS) in a scanning microscope was first demonstrated in 1982 by Duncan et al. [107], but it received no development until the end of the century, when it started to gain popularity. CARS relies on four-wave mixing, as shown in the Fig. 2.1 d, which is a third-order nonlinear process. This technique is more difficult to implement, since it requires 2 different excitation beams to be spatially aligned and temporally synchronized with high precision. The first is the pump field ωp which is involved twice (two red upward arrows in the Fig. 2.1 d) and the second is Stokes field ωs (smaller red downward arrow) involved once in the scattering event. This method allows for probing vibrational levels with efficient enhancement by the resonance at ωp ωs. CARS microscopy also possesses an intrinsic 3D resolution, as its efficiency is proportional to Iω2p Iωs that is sufficient to ensure the optical sectioning (see subsection 2.1.2).
The advantage of CARS consists in chemical specificity attained with intrinsic axial resolution. Indeed, 3D-confinement of the excitation volume enables microscopy capa-bilities, while changing ωs provides spectral scanning of vibrational levels in the studied sample. However, in practice, obtaining pixel-wise spectra is time consuming, and often CARS is used with ωs set to a fixed vibrational frequency. For instance, it can be C-H bond stretching band at 2840 cm−1, which allows for imaging of C-H rich lipid bodies.
Second Harmonic Generation microscopy
In this section we will first describe general principles of second harmonic generation in a medium, paying particular attention to the role of coherence in SHG process, and to the notion of resolution in SHG. After that, we will discuss the capability of collagen and its assemblies to efficiently generate second harmonic radiation.
Mechanisms and principles
Physical origins
SHG is a nonlinear optical phenomenon where two photons at the same wavelength are scattered by a single molecule to produce a photon at half the wavelength. SHG is a coherent and instantaneous process (in contrast to 2PEF), which means that the phase of the generated wave is strictly related to that of the incident field. In a more general case, the induced dipole p in such a molecule can be written as p = [α]E + [β]EE + [γ]EEE + … (2.2)
Here, E is the incident field, [α] is the linear polarizability of the molecule, and [β] and [γ] are first and second hyperpolarizabilities. The first term corresponds to the linear scattering, the second is responsible for SHG, and the third term governs THG. In the most general case, the polarizability and first and second hyperpolarizabilities are tensors with two, three and four dimensions respectively.
Harmonics generation can be illustrated by Lorentz oscillator model, treating the interaction of an electromagnetic wave with a bound electron. According to this model, an electron driven by the excitation field starts to oscillate and to generate the secondary wave. For a harmonic potential, the oscillation is sinusoidal with the same frequency ω as the driving field. However, for large excitation fields, the real potential can differ significantly from harmonic one, which influences the oscillation signature. More precisely, such an oscillation contains contributions of harmonic frequencies, i.e. 2ω, 3ω,… which are radiated more or less efficiently (see Fig. 2.4.)
The role of coherence in SHG
The coherent nature is the most prominent feature of SHG, which determines the rest of its properties. On one hand, the coherent amplification of signal from ordered and well-aligned structures makes that they are the only ones to efficiently produce SHG. It accomplishes the most important role of SHG in tissue microscopy, which is the specific visualization of fibrillar collagens, tubulin microtubules, and sarcomeres in muscles with-out staining. On the other hand, this property significantly impedes both qualitative and quantitative interpretation of the SHG images, and even compromises the notion of resolution in SHG microscopy, as will be shown in the following item.
To illustrate the role of coherence, lets consider a simple example of the nonlinear scattering in two different media with the same number of scatterers within a small volume. The first volume contains harmonophores aligned along a specific direction, while in the second volume, the harmonophores are randomly oriented (see Fig. 2.5).
Table of contents :
1 Collagen
1.1 Structure and synthesis
1.1.1 Collagen types
1.1.2 Collagen type I structure and synthesis
1.1.3 Hierarchical organization of collagen assemblies
1.2 Tendon biomechanics
1.3 Collagen visualization techniques and challenges
1.3.1 Conventional visualization techniques
1.3.1.1 Electron microscopy
1.3.1.2 Atomic force microscopy
1.3.1.3 Polarized light microscopy
1.3.1.4 Optical coherence tomography
1.3.1.5 Histological and immunohistochemistry staining
1.3.2 Challenges in collagen visualization
1.4 Conclusion
2 Theoretical background of polarization-resolved SHG microscopy
2.1 Principles and contrast mechanisms of multiphoton microscopy
2.1.1 Making intense fields
2.1.2 Optical sectioning
2.1.3 Two-photon excited fluorescence (2PEF)
2.1.4 Second and third harmonic generation
2.1.5 Coherent anti-Stokes Raman Scattering
2.2 Second Harmonic Generation microscopy
2.2.1 Mechanisms and principles
2.2.1.1 Physical origins
2.2.1.2 The role of coherence in SHG
2.2.1.3 Resolution: comparison between SHG microscopy and other techniques
2.2.2 Origin of SHG in collagen
2.3 Collagen tensorial response
2.3.1 Tensorial formalism of medium polarization
2.3.2 Symmetries of collagen assemblies and nonlinear response tensors
2.3.3 Second harmonic response of collagen at molecular and fibrillar scale
2.3.4 P-SHG signal for C∞v symmetry
2.3.5 Tensorial response variation versus disorder in collagen fascicle
2.4 Conclusion
3 Linear optical effects in polarization-resolved SHG microscopy
3.1 Experimental setup for simultaneous SHG/2PEF imaging
3.2 Experimental artefacts in thick anisotropic tissue
3.2.1 Polarization-resolved Second Harmonic microscopy in anisotropic thick tissues
3.3 Numerical simulations
3.3.1 Overview of SHG simulation in tendon
3.3.2 Focal field calculation in birefringent media
3.3.2.1 Field propagation in a uniform medium. Angular spectrum representation
3.3.2.2 Field in a birefringent medium
3.3.2.3 Boundary conditions between isotropic and birefringent media
3.3.2.4 Results and discussion
3.3.3 SH radiation in tendon
3.3.3.1 Radiation of a punctual dipole in a birefringent medium
3.3.3.2 Radiation integral
3.3.3.3 Simplification of the calculation using relative order of magnitude and symmetry of SH radiation components
3.3.3.4 Results: angular radiation diagrams
3.3.3.5 Results: total SHG intensity polarization diagrams
3.3.3.6 Discussion of the simulated ρ and
3.3.4 SHG simulations in cornea
3.3.4.1 Introduction
3.3.4.2 In vivo structural imaging of the cornea by polarizationresolved second harmonic microscopy
3.4 Discussion
4 Tendon biomechanics
4.1 Experimental setup
4.2 Proof of concept of biomechanical assays coupled with SHG imaging
4.2.1 Introduction
4.2.2 Monitoring micrometer-scale collagen organization in rat-tail tendon upon mechanical strain using second harmonic microscopy
4.3 Varying fibril ordering in tendon
4.3.1 Introduction
4.3.2 Polarization-Resolved Second-Harmonic Generation in Tendon upon Mechanical Stretching
4.4 Discussion
