## 1.

## Introduction

Optical imaging techniques such as confocal laser scanning and multiphoton imaging have revolutionized biological optical microscopy.^{1, 2} In specific applications, polarization microscopy can provide optical structure information that is absent from intensity imaging alone.^{3, 4, 5} Recently, second-harmonic generation (SHG) microscopy has become an important imaging modality in biological sciences.^{6, 7} Biological structures such as muscles, tendons, and corneas are strong generators of second-harmonic signals.^{8, 9} With a pulsed femtosecond laser as the excitation source, SHG microscopy has been demonstrated to be an effective, minimally invasive imaging tool for biomedical studies in a wide array of areas including the interaction of cell with extracellular matrix, cancer growth, keratoconus, and skin thermal damage.
^{10, 11, 12, 13} In addition to morphological information, the polarization dependence of the SHG signal can provide structural information below the resolution of optical microscopy.^{8} Although SHG produces strong forward scattering signals, *in vivo* studies with optical microscopes usually require epi-illuminated detection. In this configuration, a dichroic mirror is required to separate the excitation source and the emission signal.^{2} Since the reflective properties of dichroic mirrors can alter the polarization ellipticity of the incident excitation light, polarization studies in an SHG laser scanning microscope are often performed by fixing the excitation polarization while rotating the samples.^{8, 14} However, the sample rotation approach requires the precise alignment of the observation area to the center of the sample rotation stage, and this causes increased difficulties for *in vivo* microscopic observation, which often requires specialized animal-mounting devices.^{15, 16} In addition, subsequent analysis of polarization-resolved images is complicated by processing specimen images at different angular orientations. Therefore, we propose a generalized method using wave plates to compensate for the ellipticity altering effects of optical components such as that caused by a dichroic mirror. Mathematical analysis was performed to demonstrate the application of this approach in delivering linearly polarized excitation light of the desired direction onto the sample. We also obtained the polarization dependence of the SHG signal of a rat tail tendon and a mouse leg muscle. The second-order susceptibility tensor element ratios derived from our data are compared with previously obtained results. It was shown that nonlinear susceptibility can be used to infer molecular structures of biological samples.^{14, 17, 18} Our work has potential application in monitoring structural changes of biological samples and may help diagnose diseases associated with second-harmonic generating tissues such as collagens, muscles, and lipids. Our methodology is especially useful in microendoscopy applications where polarization-controlling optical components can not be easily positioned between the dichroic mirror and the focusing objective.

## 2.

## Mathematical Analysis

The design of our ellipticity compensation scheme is shown in Fig. 1 . Consider an optical component with a phase retardation $\delta $ for light polarized along the $x$ axis with respect to light polarized along the $y$ axis. Moreover, the electric field transmission (or reflection) coefficients along the $x$ and $y$ axes are ${\eta}_{x}$ and ${\eta}_{y}$ , respectively (the transmission or reflection ratio parameter $\gamma $ is defined to be $\gamma ={\eta}_{x}\u2215{\eta}_{y}$ ). The angles of the optical axes of all optical components are measured relative to the $x$ axis. As a result, the transmission or reflection of linearly polarized light by such an optical component would result in elliptically polarized light. To obtain linearly polarized light with the desired output polarization angle $\theta $ on reflecting from or transmitting through such an optical component, one can deliver onto the optical component a corresponding elliptically polarized light with particular ellipticity and polarization direction $\beta $ to offset the polarization ellipticity altering effect of the optical component.

As shown in Fig. 1, ellipticity compensation can be achieved by the use of a half-wave plate (HWP) and a quarter-wave plate (QWP) at the respective angles of
$\alpha \u22152$
and
$\beta $
(Senarmont compensator).^{19} Although the ellipticity altering effect of an optical component can be corrected for using a linear polarizer and a QWP, the rotation of a linear polarizer can result in excitation beam translation and contribute to variations in scan field nonuniformity. Therefore, we used a HWP and linearly polarized light as input instead of the combination of a linear polarizer and circularly polarized light to achieve ellipticity compensation. Mathematically, our approach involves the derivation of a relationship between the angles
$\alpha $
,
$\beta $
, and
$\theta $
and the physical parameters
$\delta $
and
$\gamma $
. We show that for a given set of
$\delta $
and
$\gamma $
, unique angular combinations of
$\alpha $
and
$\beta $
can be used to obtain the desired output polarization angle
$\theta $
.

When a linearly polarized light along the $x$ axis with angular frequency $\omega $ passes through a HWP oriented at an angle $\alpha \u22152$ relative to the $x$ axis, the electric field can be written as

## Eq. 1

$$\mathbf{E}={E}_{0}\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\alpha \phantom{\rule{0.2em}{0ex}}\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\omega t\widehat{\mathbf{x}}+{E}_{0}\phantom{\rule{0.2em}{0ex}}\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\alpha \phantom{\rule{0.2em}{0ex}}\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\omega t\widehat{\mathbf{y}},$$## Eq. 2

$$\mathbf{E}={E}_{0}(\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\beta \phantom{\rule{0.2em}{0ex}}\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\phi \phantom{\rule{0.2em}{0ex}}\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\omega t-\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\beta \phantom{\rule{0.2em}{0ex}}\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\phi \phantom{\rule{0.2em}{0ex}}\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\omega t)\widehat{\mathbf{x}}+{E}_{0}(\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\beta \phantom{\rule{0.2em}{0ex}}\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\phi \phantom{\rule{0.2em}{0ex}}\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\omega t+\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\beta \phantom{\rule{0.2em}{0ex}}\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\phi \phantom{\rule{0.2em}{0ex}}\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\omega t)\widehat{\mathbf{y}},$$## Eq. 3

$$\mathbf{E}={E}_{0}(\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\beta \phantom{\rule{0.2em}{0ex}}\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\phi \phantom{\rule{0.2em}{0ex}}\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\omega t-\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\beta \phantom{\rule{0.2em}{0ex}}\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\phi \phantom{\rule{0.2em}{0ex}}\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\omega t)\widehat{\mathbf{x}}+\gamma {E}_{0}[\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\beta \phantom{\rule{0.2em}{0ex}}\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\phi \phantom{\rule{0.2em}{0ex}}\mathrm{sin}(\omega t+\delta )+\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\beta \phantom{\rule{0.2em}{0ex}}\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\phi \phantom{\rule{0.2em}{0ex}}\mathrm{cos}(\omega t+\delta )]\widehat{\mathbf{y}}.$$## Eq. 4

$$\mathbf{E}={E}_{0}({d}_{1}\phantom{\rule{0.2em}{0ex}}\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\omega t+{d}_{2}\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\omega t)\widehat{\mathbf{x}}+{E}_{0}({d}_{3}\phantom{\rule{0.2em}{0ex}}\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\omega t+{d}_{4}\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\omega t)\widehat{\mathbf{y}},$$## Eq. 5

$$\mathrm{tan}\phantom{\rule{0.2em}{0ex}}{\phi}_{\pm}=\frac{1\pm {(1+{\mathrm{sin}}^{2}\phantom{\rule{0.2em}{0ex}}2\beta \phantom{\rule{0.2em}{0ex}}{\mathrm{tan}}^{2}\phantom{\rule{0.2em}{0ex}}\delta )}^{1\u22152}}{\mathrm{sin}\phantom{\rule{0.2em}{0ex}}2\beta \phantom{\rule{0.2em}{0ex}}\mathrm{tan}\phantom{\rule{0.2em}{0ex}}\delta}.$$The absence of $\gamma $ in Eq. 5 shows that the ratio of the transmission or reflection ratio parameter has no effect on the linearity of incident polarization. Only the phase retardation $\delta $ can influence the ellipticity of the output polarization. Since the product of the two solutions $\left(\mathrm{tan}\phantom{\rule{0.2em}{0ex}}{\phi}_{+}\right)\left(\mathrm{tan}\phantom{\rule{0.2em}{0ex}}{\phi}_{-}\right)=-1$ , the two unique angles ${\phi}_{+}$ and ${\phi}_{-}$ differ by $90\phantom{\rule{0.3em}{0ex}}\mathrm{deg}$ . For an optical component with a phase retardation $\delta $ , for every QWP orientation, there are two corresponding $\alpha $ angles, within $180\phantom{\rule{0.3em}{0ex}}\mathrm{deg}$ , that will result in a linearly polarized output. Figure 2a shows the generalized solution of relative polarization angle $\phi $ for various combinations of the QWP angle $\beta $ and the phase retardation angle $\delta $ .

To determine the QWP angle $\beta $ , we substitute the solution for $\phi $ into the relation $\mathrm{tan}\phantom{\rule{0.2em}{0ex}}\theta ={E}_{y}\u2215{E}_{x}$ , and obtain the output polarization equation as a function of $\gamma $ , $\beta $ , and $\delta $ .

## Eq. 6

$$\mathrm{tan}\phantom{\rule{0.2em}{0ex}}\theta =\gamma \phantom{\rule{0.2em}{0ex}}\mathrm{tan}\phantom{\rule{0.2em}{0ex}}\beta \phantom{\rule{0.2em}{0ex}}\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\delta -\gamma \phantom{\rule{0.2em}{0ex}}\mathrm{tan}\phantom{\rule{0.2em}{0ex}}\phi \phantom{\rule{0.2em}{0ex}}\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\delta .$$## Eq. 7

$$\mathrm{tan}\phantom{\rule{0.2em}{0ex}}{\beta}_{\pm}=\frac{{\gamma}^{2}-{\mathrm{tan}}^{2}\phantom{\rule{0.2em}{0ex}}\theta \pm {({\gamma}^{4}+{\mathrm{tan}}^{4}\phantom{\rule{0.2em}{0ex}}\theta +2{\gamma}^{2}\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\phantom{\rule{0.2em}{0ex}}2\delta \phantom{\rule{0.2em}{0ex}}{\mathrm{tan}}^{2}\phantom{\rule{0.2em}{0ex}}\theta )}^{1\u22152}}{-2\gamma \phantom{\rule{0.2em}{0ex}}\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\delta \phantom{\rule{0.2em}{0ex}}\mathrm{tan}\phantom{\rule{0.2em}{0ex}}\theta}.$$In practice, with known values of the two parameters $\delta $ and $\gamma $ , we can use Eq. 7 to obtain the needed QWP angle for the desired output polarization angle. For example, to obtain an output polarization angle of $60\phantom{\rule{0.3em}{0ex}}\mathrm{deg}$ , the QWP needs to be $45\phantom{\rule{0.3em}{0ex}}\mathrm{deg}$ . From there, the QWP angle can be used in Eq. 5 to obtain the corresponding relative polarization angle, which is $158\phantom{\rule{0.3em}{0ex}}\mathrm{deg}$ in this case.

Since $\gamma $ is variable, the relative output power will be different for different output polarization orientations. Note that the input power of linearly polarized light, ${P}_{i}$ (proportional to ${E}^{2}$ ) remains unchanged as light passed through HWP and QWP. To obtain a relation between the output polarization orientation and power, we must know the actual transmission or reflection efficiencies rather than their ratio. After transmitting through or reflecting from the optical component, the electric field obeys the relation ${\mathrm{tan}}^{2}\theta ={E}_{y}^{2}\u2215{E}_{x}^{2}$ . As a result, the output power can be expressed as

## Eq. 8

$${P}_{0}=\frac{{E}_{x}^{2}+{E}_{y}^{2}}{[{({E}_{x}\u2215{\eta}_{x})}^{2}+{({E}_{y}\u2215{\eta}_{y})}^{2}]}{P}_{i}.$$## Eq. 9

$${P}_{0}={\eta}_{x}^{2}{\gamma}^{2}\left(\frac{1+{\mathrm{tan}}^{2}\phantom{\rule{0.2em}{0ex}}\theta}{{\gamma}^{2}+{\mathrm{tan}}^{2}\phantom{\rule{0.2em}{0ex}}\theta}\right){P}_{i}.$$## 3.

## Equipment, Material, and Method

The homebuilt epi-illuminated laser scanning SHG microscope used in this study is similar to one that has been previously described.^{20} The excitation source is a pulsed ti:sapphire laser (Tsunami, Spectral Physics, Mountain View, California) tuned to
$780\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
. The main dichroic filter used in the system is a short-pass dichroic mirror (700dcspxruv-2p, Chroma Technology, Rockingham, Vermont), and a narrow-band pass filter (HQ390-22m-2p, Chroma Technology) is used to select the SHG signal
$(390\pm 10\phantom{\rule{0.3em}{0ex}}\mathrm{nm})$
. We used a Fluor,
$40\times \u2215\mathrm{NA}\phantom{\rule{0.3em}{0ex}}0.8$
water immersion objective (Nikon, Japan). The use of an NA 0.8 objective is motivated by the fact that a previous study showed that objectives with higher NA values than 0.8 can alter the polarization of the excitation light.^{21} The lateral resolution of our SHG system is estimated to be^{22}
$0.42\phantom{\rule{0.3em}{0ex}}\mathrm{\mu}\mathrm{m}$
.

Experimentally, a linear polarizer and a power meter were used to verify the output polarization state. In our approach, for every 10-deg change in the QWP angle, we rotated the HWP and the linear polarizer in front of the power meter to determine the angular combination that results in linearly polarized output. The angular range of the QWP is 0 to $180\phantom{\rule{0.3em}{0ex}}\mathrm{deg}$ and that of the HWP is 0 to $90\phantom{\rule{0.3em}{0ex}}\mathrm{deg}$ . Since the HWP requires only $90\phantom{\rule{0.3em}{0ex}}\mathrm{deg}$ to rotate the polarization of linearly polarized light within the $180\phantom{\rule{0.3em}{0ex}}\mathrm{deg}$ range, the equation of HWP angle $\alpha \u22152$ is $\alpha \u22152=(\phi +\beta )\u22152$ . The angular positions of the QWP, the HWP, and the linear polarizer were recorded after satisfying the condition of at least an 100:1 rejection of the cross-polarization intensities as the linear polarizer was rotated.

The rat tail tendon sample used in our experiment came from an 8-week-old rat. After the rat was sacrificed, its tail was cut and stored at $-80\xb0\mathrm{C}$ until right before experimentation. As in previous studies, we found that the process of freezing and thawing do not affect SHG imaging (our unpublished data and Ref. 23). At room temperature, a small piece of rat tail tendon around $2\phantom{\rule{0.3em}{0ex}}\mathrm{cm}$ in length was cut and immersed in phosphate-buffered saline solution. The wet tendon was then placed on a glass slide and sealed with a cover glass. The mouse leg muscle specimen came from a 6-week-old mouse after sacrifice and preparation was similar as for rat tail tendon.

We used a theoretical model with variable parameters to investigate the excitation polarization SHG properties of fibril of cylindrical symmetry.^{23} In analyzing the polarization resolved SHG images, we adopted the following second order polarization model

## Eq. 10

$$\mathbf{P}=a\mathbf{s}{(\mathbf{s}\cdot \mathbf{E})}^{2}+b\mathbf{s}(\mathbf{E}\cdot \mathbf{E})+c\mathbf{E}(\mathbf{s}\cdot \mathbf{E}).$$^{14}The dependence of SHG intensity on the incident polarization can be described as

## Eq. 11

$$I\propto {E}_{0}^{4}[{({d}_{31}\phantom{\rule{0.2em}{0ex}}{\mathrm{sin}}^{2}\phantom{\rule{0.2em}{0ex}}\varphi +{d}_{33}\phantom{\rule{0.2em}{0ex}}{\mathrm{cos}}^{2}\phantom{\rule{0.2em}{0ex}}\varphi )}^{2}+{\left({d}_{15}\phantom{\rule{0.2em}{0ex}}\mathrm{sin}\phantom{\rule{0.2em}{0ex}}2\varphi \right)}^{2}],$$^{14, 24}To verify the validity of our approach, the ellipticity compensation method was applied to see if the sample-rotating data fits our results well. Least-squares fitting was performed with the IDL program (ITT Visual Information Solutions). We selected 21 evenly spaced linear polarization output angles and used them to obtain excitation polarization-resolved SHG images of rat tail tendon and mouse leg muscle. The $55\times 55\phantom{\rule{0.3em}{0ex}}\mathrm{\mu}\mathrm{m}$ image area was scanned at $256\times 256$ pixels resolution. To minimize sample birefringence effects, the SHG images were acquired at approximately $5\phantom{\rule{0.3em}{0ex}}\mathrm{\mu}\mathrm{m}$ below the specimen surface. For each angle, we scanned the specimens three times and averaged the images in forming a final set of images for second-order susceptibility analysis. Graphically, the SHG intensity variation within the analyzed regions is indicated by error bars.

## 4.

## Results and Discussion

One task we performed was to verify the relationship between the QWP angles, HWP angles, output polarization angles, and relative output power. Figures 3a, 3b, 3c show the experimental results (dots) and the theoretical prediction (curved lines) of our results. The curves were obtained from the mathematical calculation using the independently obtained phase retardation angle and the $x$ and $y$ reflection coefficients of the main dichroic mirror. For the dichroic mirror used in our study, the phase retardation was determined to be $-52\phantom{\rule{0.3em}{0ex}}\mathrm{deg}$ and the ratio of reflection coefficients ( $x$ to $y$ axes) of the electric field was determined to be 0.97. These values were obtained using different angles of linearly polarized light incident onto the dichroic mirror and measuring the ellipticity of reflected excitation source. Figure 3a shows the combinations of relative HWP and QWP angles that achieve linearly polarized output, and the result shown in Fig. 3b is the linear polarization angles with the associated QWP angles.

Furthermore, we compared the angular dependence of fluorescence intensity with the theoretical prediction, by using an aqueous fluorescein sample. Due to the two-photon fluorescence excitation process, the fluorescence intensity $F$ , which depends quadratically on the excitation intensity, is given by

## Eq. 12

$$F\propto {\gamma}^{4}{\left(\frac{1+{\mathrm{tan}}^{2}\phantom{\rule{0.2em}{0ex}}\theta}{{\gamma}^{2}+{\mathrm{tan}}^{2}\phantom{\rule{0.2em}{0ex}}\theta}\right)}^{2}.$$To validate our approach, we scanned the rat tail tendon and the mouse leg muscle, which is mainly composed of type I collagen fibers and myofibrils. Figure 4a
presents the SHG images of tendon fibrils scanned using the excitation of four different linear polarization angles, as indicated by the arrow directions. From the figure, it is clear that the SHG signal varies significantly with the different excitation polarization angles. The SHG intensity tends to be higher when the direction of the linear polarization is paralleled to the fiber and decreases as the polarization direction becomes perpendicular to the fibers. In using Eq. 11 to analyze the excitation polarization-resolved SHG images, we obtained results similar to the graph shown in Fig. 5a
. The data in Fig. 5 correspond to the average intensity of a
$5\times 5\text{-}\mathrm{pixel}$
square shown by the small black box A in the center of Fig. 4b, and error bars are the standard deviations of intensity within 25 pixels in area. Previous result assumed that
$c=2b$
in Eq. 11 and
$r=b\u2215a$
, which is the only variable of normalized SHG intensity.^{23} Our model fitting leads to the susceptibility tensor element ratio of
$r=-0.70\pm 0.05$
, consistent with
$r=-0.71\pm 0.05$
obtained by sample rotation (our unpublished data) and –0.73 obtained by Stoller
^{23} Note in Fig. 5 that the angles shown on the
$x$
axis are measured counterclockwise with respect to the orientation of the fiber, which at our selected square is
$13\phantom{\rule{0.3em}{0ex}}\mathrm{deg}$
counterclockwise relative to the vertical direction of the image. Figure 5b is the fitting of the SHG signal in the same area without wave plates compensation, where the polarization is assumed to be unaffected by the dichroic mirror and is rotated by HWP. In our experiment, the intensity variation is asymmetric and model fitting gives
$r=-0.52\pm 0.02$
, which deviates significantly from previous results. To further test our polarization ellipticity compensation method, we also performed analysis at both large area (
$30\times 30$
pixels) and four corners, whose positions are shown by black boxes in Fig. 4b, and the results are shown by Table 1
.

## Table 1

The γ fitting results at different areas within the SHG image.

A (small area) | B | C | D | E | A (large area) | |
---|---|---|---|---|---|---|

With compensation | $-0.70\pm 0.05$ | $-0.73\pm 0.03$ | $-0.69\pm 0.03$ | $-0.72\pm 0.04$ | $-0.74\pm 0.05$ | $-0.72\pm 0.04$ |

Without compensation | $-0.52\pm 0.02$ | $-0.52\pm 0.01$ | $-0.53\pm 0.02$ | $-0.53\pm 0.02$ | $-0.51\pm 0.02$ | $-0.53\pm 0.02$ |

It is clear that, with our polarization ellipticity compensation scheme, the fitting result of $\gamma $ agrees well at different locations within the SHG image. One advantage of our compensation method is that we can analyze a relatively small area when rotating the polarization of light at the scale of a single fibril.

In addition, we scanned and tested our methodology in the SHG imaging of the mouse leg, which is another fibrous tissue capable of producing intense SHG signal. Figure 6a
shows the SHG intensity change at different excitation polarization angles. As performed with the rat tail tendon sample, we performed the parameter fitting in Eq. 11 on the muscle sample, and the result is shown in Fig. 7
. The muscle SHG comes from myofibril, and it was shown^{8} that the fitting parameters can be used to derive the pitch angle
$\theta $
of myosin rod by the relation
${\mathrm{tan}}^{2}\theta =2{d}_{31}\u2215{d}_{33}$
. In this manner, the myosin rod pitch angle was determined to be
$61.2\phantom{\rule{0.3em}{0ex}}\mathrm{deg}$
, as measured by Plotnikov
^{8}

Our analysis result is shown in Table 2 . In all regions analyzed, the derived values of the pitch angle are closer to previous published results when the ellipticity scheme was applied.

## Table 2

Fitted pitch angles at different areas within the SHG image.

A (small area) | B | C | D | E | A (large area) | |
---|---|---|---|---|---|---|

With compensation (deg) | $61.7\pm 1.3$ | $61.4\pm 1.3$ | $61.5\pm 1.5$ | $61.0\pm 1.3$ | $60.9\pm 1.7$ | $60.6\pm 1.4$ |

Without compensation (deg) | $55.4\pm 2.1$ | $55.3\pm 1.9$ | $55.6\pm 1.7$ | $56.1\pm 2.0$ | $55.4\pm 2.1$ | $56.6\pm 1.6$ |

Based on these results, we conclude that our approach can be used to achieve polarization-resolved SHG imaging without specimen rotation and that the polarization ellipticity altering properties of optical components such as the dichroics can be corrected for. This work shows that our method of using polarization rotation of excitation light enables us to more easily analyze data and with more precision.

## 5.

## Conclusion

We performed the necessary mathematical analysis and devised an experimental approach for compensating the ellipticity altering effect of an optical component using a QWP and a HWP. We demonstrated that linearly polarized output with the controlled polarization angle perpendicular to the direction of propagation can be achieved for excitation-polarization-resolved SHG microscopy. For an optical component with known phase retardation $\delta $ and ratio of $s$ - and $p$ - waves transmission or reflection coefficient $\gamma $ , we can determine the angular positions of the added wave plates to achieve output with arbitrary linear polarization angles. The rotation of wave plates can be automated using motorized rotary stages controlled by computer. One can envision the combination of our methodology and a fast-scanning, video-rate SHG microscope that can greatly reduce the data acquisition in achieving specimen-fixed, excitation-polarization-resolved SHG microscopy. Experimentally, our approach was used to achieve excitation-polarization-resolved SHG imaging of type I collagen fibers in a rat tail tendon and myofibrils in a mouse leg muscle, and the results demonstrate that polarized SHG microscopy can be achieved without specimen rotation. Our method is applicable for delivering the desired polarization of excitation light in cases where there are optical components that contribute to ellipticity-altering effects by properly controlling the relative angular orientations between a HWP and a QWP. Our methodology is invaluable for excitation-polarization-resolved SHG experiments in a number of ways. First, image processing and the potential artifacts associated with the sample rotation approach are eliminated. Furthermore, our approach can be conveniently applied to microendoscopy applications where polarization-compensating components can not be easily positioned between the dichroic mirror and the focusing objective. Finally, in principle, our methodology is generally applicable to correcting the ellipticity-altering artifacts in deep-tissue, excitation-polarization-resolved SHG microscopy provided that the specimen birefringence properties are known. This technique has potential biological and medical applications, such as monitoring molecule changes in tissues and organs with a strong SHG signal and fluorescence polarization measurement in providing orientation information of fluorescent molecules in systems such as cell membranes.

## Acknowledgments

We acknowledge the support of the National Research Program for Genomic Medicine (NRPGM) and the National Science Council of Taiwan (NSC-95-3112-B-002-018). This work was completed using the Optical Molecular Imaging Microscopy Core Facility (A5) of NRPGM.