## Abstract

Second harmonic generation microscopy(SHGM) has become widely used to image biological samples. Due to the complexity of biological samples, more and more effort has been put on polarization imaging in SHGM technology to uncover their structures. In this work, we put forward a novel stitching method based on careful mathematical calculation, and accomplish it by rotating laser polarization. We first show its validity in imaging a perfectly synthesized bio-origin polymer poly (3-hyroxybutyrate-co-3-hydroxyhexanoate) (PHBHHx). Then, we test its power by getting a true image of fibrillar collagen structure of rat-tail tendon.

© 2006 Optical Society of America

## 1. Introduction

Second harmonic generation microscopy(SHGM) is an emerging microscopic technique for a wide range of material, biological, and medical investigations. Advances in the developments of SHGM have provided researchers with novel means by which non-invasive visualization of non-biological and biological specimens can be achieved with high penetration and high spatial resolution. Second Harmonic Generation (SHG) is a second-order nonlinear optical process and thus requires an environment without a center of symmetry to produce signals, and is known to leave no energy deposition to the interacted matters due to their virtual energy conservation characteristic, that is, the emitted SHG photon energy is the same as the total absorbed excitation photon energy. This provides the optical non-invasive nature desirable for microscopy applications. In history, SHG was first demonstrated by Franken in crystalline quartz in 1961 [1]. In 1974, Hellwarth first integrated SHG into an optical microscope to visualize the microscopic crystal structure in polycrystalline ZnSe [2]. In 1979, SHG began to be used as a optical probe to characterize biological specimen collagen by Freund [3]. And in 2003, perfect 3-D SHG images of collagen were obtained by Cox and his coworkers [4].

Due to its coherent nature during the generation processes, SHG is sensitive to the local biological structures and is particular useful for structural studies. A review of recent and not so recent work reveals that angular dependence of SHG in collagen has been well appreciated, characterized and exploited for structural information and tissue organization [5, 6, 7, 8, 9]. In this work, we present a new SHG technique for studying collagen: imaging collagen fibers by reconstructing three complementary images which are generated at different laser polarization angles. Our technique allows us to image collagen without the interference of laser polarization on microscopic scales in the final reconstructed picture. To test our theory, we experimentally apply it to Poly (3-hydroxybutyrate-co-3-hydroxyhexanoate) (PHBHHx), a kind of biological origin material which shares the same symmetry as collagen in microscopic structure but with much more macroscopic uniformity. Then, this new imaging method is implemented on fibrillar collagen from rat-tail tendon to obtain a real image of collagen.

## 2. Theory

#### 2.1. SHG in collagen

Due to the complexity of biological tissue, calculating SHG intensity in biological physics is much more difficult than in crystal or surface physics. But under long-wavelength approximation, which means that optical SHG averages molecular properties over dimensions of the order of light wavelength, so that all local asymmetries disappear, collagen can be reduced to a relative simple mathematical model, which can be exactly calculated using classic electromagnetic theory. Based on this approximation, extensive discussions on an analytical model for SHG in collagen have been well developed previously [10, 11].

For collagen with cylindrical symmetry, the intensity of second harmonic light is given by [10]

Where *I _{SHG}* is the second harmonic intensity,

**P**is the second-order nonlinear polarization,

**s**is the unit vector along the symmetry axis,

**k**is the laser propagation direction. Let us assume the fiber is located in the x-y plane; and is oriented along the x axis. The laser beam propagates along the z axis and polarized at an angle

*α*with x axis. This geometry is shown in Fig. 1. And from it, we can obtain an expression of the second harmonic intensity from Eq. (1) as a function of α.

Here *γ=b/a; γ* is a fundamental parameter of the second-order nonlinear susceptibility tensor. It is the ratio of the tensor’s two independent elements.

#### 2.2. The principles of reconstruction of complementary images

From Eq. (2), we can see that the intensity of SHG depends not only on the intrinsic parameter *γ*, but also on angle *α*, which relates to researcher’s choice for laser polarization. Previous works have used this angular dependence to reveal the structure and orientation of collagen fibers. But no one has pointed out how to exhibit the complete information of collagen fibers in only one SHG image. One may think of using circularly polarized laser as an alternative light source to wipe off the angle *α* influence intuitively. But virtually it is not feasible.(Note: Since the output of most femtosecond laser for SHGMnowadays is linearly polarized light, which can be easily adapted to be circularly polarized by simply inserting a λ/4 wave plate into the light pass, but is difficult to become randomly polarized light by such straightforward procedures, based on the considerations of practicability and simplification of the experimental apparatus, we here only focus on circular polarized light case, and not take randomly polarized light case into account.)

In confocal system, let the pixel dwell time per scan equal to Δ*t*, the initial angle between the electric vector and the axis of collagen equal to *α*
_{0}. During each pixel dwell time, the angle Δ*φ* that the electric vector has rotated can be derived:

f is the frequency of the incident light. From Eq. (2) and Eq. (3), we can get the total SHG intensity that sample has emitted

In the equation, B is a constant. After integration,

$$+\frac{1}{4}\left(1+6\gamma +8{\gamma}^{2}\right){\mathrm{sin}2\alpha \mid}_{{\alpha}_{0}}^{{\alpha}_{0}+\Delta \varphi}+\frac{1}{32}\left(1+4\gamma \right){\mathrm{sin}4\alpha \mid}_{{\alpha}_{0}}^{{\alpha}_{0}+\Delta \varphi}]$$

Eq. (5) tells us that only if Δ*φ*=*nπ*, n is an integer, can the second and third items on the right of equation vanish, which is hard to control during scanning because of the fluctuation of laser frequency.

In this work, based on polarization imaging of linearly polarized light, we put forward a new imaging method: stitching complementary images together, to wipe off the influence of laser polarization and get a real image of collagen fibers. We rotate the polarization of laser *π*/3 a time, and perform three independent measurements to get three complementary images. Then we stitch these obtained images together in the manner of intensity summation. Here, we will prove that these three images are complementary, and after combining them together, we can eliminate the influence of *α* dependence.

Let the angle between laser polarization and the axis of fibrillar collagen in the first measurement equal to *α*; we rotate the half wave plate to make this angle become *α+π*/3, *α*+2*π*/3 in the second and third measurement respectively. Second harmonic intensity in these three images are:

$${I}_{2}=A\left[\frac{1}{8}\left(1+20\gamma +40{\gamma}^{2}\right)+\frac{1}{2}\left(1+6\gamma +8{\gamma}^{2}\right)\mathrm{cos}2\left(\alpha +\frac{\pi}{3}\right)+\frac{1}{8}\left(1+4\gamma \right)\mathrm{cos}4\left(\alpha +\frac{\pi}{3}\right)\right]$$

$${I}_{3}=A\left[\frac{1}{8}\left(1+20\gamma +40{\gamma}^{2}\right)+\frac{1}{2}\left(1+6\gamma +8{\gamma}^{2}\right)\mathrm{cos}2\left(\alpha +\frac{2\pi}{3}\right)+\frac{1}{8}\left(1+4\gamma \right)\mathrm{cos}4\left(\alpha +\frac{2\pi}{3}\right)\right]$$

When they are combined in a manner of intensity addition, the total second harmonic intensity in the stitched image is:

$$=A\{\frac{3}{8}\left(1+20\gamma +40{\gamma}^{2}\right)+\frac{1}{2}\left(1+6\gamma +8{\gamma}^{2}\right)\left[\mathrm{cos}2\alpha +\mathrm{cos}\left(2\alpha +\frac{2\pi}{3}\right)+\mathrm{cos}\left(2\alpha +\frac{4\pi}{3}\right)\right]$$

$$+\frac{1}{8}\left(1+4\gamma \right)\left[\mathrm{cos}4\alpha +\mathrm{cos}\left(4\alpha +\frac{4\pi}{3}\right)+\mathrm{cos}\left(4\alpha +\frac{8\pi}{3}\right)\right]\}$$

$$=\frac{3A}{8}\left(1+20\gamma +40{\gamma}^{2}\right)$$

For collagen whose *γ* typical value is -0.7, we describe the dependence of three successive images and combined image on angle *α* in Fig. 2. From Eq. (4) and Fig. 2 we can see that in the final stitched image, second harmonic intensity does not depend on angle *α* any more, and only relates to second order nonlinear polarization susceptibility *γ* of collagen. In a word, these three images are complementary and through this mathematical “trick”, we successfully eliminate the influence of polarization of laser and get a true image what researchers want.

## 3. Materials and methods

#### 3.1. Scanning microscope and laser

The SHG imaging experiments were performed on modified Bio-Rad MRC1024 multiphoton laser scanning microscope (Bio-Rad MRC 1024MP). The whole setup is illustrated in Fig. 3.

The laser system is an argon ion pumped femtosecond titanium sapphire oscillator(Tsunami, Spectra Physics, USA), characterized by a pulse width of 80 fs at a repetition rate of 75 MHz at 810 nm. Average powers at the sample were about 50mW. Because SHG is a coherent process, the signal wave copropagates with the laser and is collected in the forward direction. The objective (Nikon 20×) has working distance f=11mm, numerical aperture N.A.=0.4. We used an objective with a relatively low numerical aperture to ensure that the polarization state of the laser beam was not substantially altered in the focus. The transmitted second harmonic signal was firstly collected by a condenser, then passed through a band-pass filter(mid-wavelength: 405.16nm, FWHM: 8nm), and was finally transferred to electric signals by a photomultiplier. In optical path, we use a Glan Polarizer lens to guarantee that output laser is purely linear polarized, and we use half wave plate to rotate the polarization of laser to the angles we need.

#### 3.2. PHBHHx preparations

Poly(R-3-hydroxybutyrate-co-R-3-hydroxyhexanoate) random copolymers with 3- hydroxyhexanoate content of 5 mol% (PHBHHx) was synthesized by Aeromonas hydrophila 4AK4 and its mutant strains [12]. White powder was obtained after purification, which was carried out via precipitation from chloroform solution by adding heptane. The powder sandwiched between two glass slides were melted on a hot stage at 210°C. Then it was transferred to another stage preset at 90°C for isothermal crystallization. After overnight crystallization, the sample was slowly cooled down to room temperature and used for further SHG imaging.

#### 3.3. Fibrillar collagen preparations

We got fibrillar collagen from rat-tail tendon. A rat-tail tendon, dissected from the tails of ICR(Internal Control Region) rats, was immediately used after being taken out. Individual tendon fascicles were removed from the tendon bundles by a dissecting microscope. Typical fascicles were several centimeters long, but had a diameter of only a few tenths of a millimeter. Fascicles were gently compressed between two glass slides in order to provide a more uniform surface normal to the laser beam. The fibers were not stretched, allowing the naturally crimped structure to remain.

## 4. Results and discussion

To test our reconstruction method, we firstly apply it to PHBHHx, a kind of biological origin material which is a strong second harmonic generator. The wavelength of incident laser in our experiment is 810nm, and due to the permutation of molecules in PHBHHx along its growing direction, PHBHHx can also be considered as with cylindrical symmetry along its growing axis under long-wavelength approximation just like collagen. But unlike the complexity of biological structures, we can precisely process PHBHHX to the shape we want. In our experiment, all PHBHHx fibers are arranged in radius direction (growing direction) to form a radiation-like plate, which has perfect center symmetry. The PHBHHx plate generates SHG signal periodically along radius direction, and thus form dark and bright circular structure. Figure 4 shows what we obtain by our stitching method. Figures 4(a), 4(b), 4(c) are obtained under laser polarization of *α, α*+*π*/3, *α*+2*π*/3 with respect to the radius of plate, and 4(d) is a stitched image by summing over former separate ones. From an arbitrary separate image, We can see that SHG generating part is only about one third of whole plate. And after stitching them together, the complete circular structure of PHBHHx is perfectly presented. (Note: Birefringence which exists because of the thickness of our plate affects the brightness of certain regions, but it is not a decisive factor in our experiment because it is relative small compared to main illuminating parts.)

Next, we imply our reconstruction method on collagen form rat-tail tendon. Collagen is the most important building block in the entire animal world, is known to consist of three parallel, interwined, polar helices. It is diverse in its structural and functional properties. Because collagen molecules are organized naturally into structures on the scale of the wavelength of light and lack a center of inversion symmetry, they are able to generate second harmonic light. In our experiment, fibrillar collagen parallel to each other, and are screwed along axis direction. Figure 5 shows our experimental results. Just as showed with PHBHHx, stitched image Fig. 5(d) contains the complete structure information of entire fibrillar collagen, in which intensity of SHG generating parts does not relate to laser polarization any more, only reflecting the intrinsic characters of the sample. In the final stitched image, we can also see that there are repetitive bands of overlap among all three polarization rotations. This is because the laser scanning mirrors can introduce ellipticity downstream of the polarizer and half wave plate, due to the non 45 degree incidence angles inherent in such a scanning system, which is especially true at large fields of view, as used here. But the effect of such an elliptical component on the quality of final stitched image is small, because the measurement of the polarization of incident light at the sample shows that they can still be treated as linearly polarized light at good approximation (*P*≈0.9).

In a word, the stitched image overcomes the *α* dependence puzzle when one wants to wipe off the influence of laser polarization in only one SHG image. Thus, it provides another useful method in biological and medical imaging research.

## 5. Conclusions

SHG imaging method has been widely used in biological and medical research as an effective probe to explore the internal structure of biological materials. Our work put forward a new imaging method-reconstruction of complementary images. By stitching complementary images, we have eliminated the angle dependence of images when using linear polarized laser, and showed true SHG images of collagen. This will provide biologists and medical researchers another useful visualization tool to explore the nature of living samples. Recent work shows that cell spindles can also be seen by SHGM [13, 14]. As spindles are composed of microtubule, which also can be considered as having cylindrical symmetry along their axes. We believe better results can be got if combined with our work. In addition, our work here mainly focuses on the collagen with cylindrical symmetry; for biological samples with other symmetry, we believe similar ways also exist in eliminating the influence of laser polarization in one image.

## Acknowledgments

This work is supported by National 973 Project (No. 001CB510307) of China and Natural Science Foundation of China (No. 10574081), and we would like to thank Prof. Minyan Yao for useful discussions.

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