Stokes Vector Formalism Based Second Harmonic ...

Stokes Vector Formalism Based Second Harmonic Generation Microscopy Jianjun Qiu*, Nirmal Mazumder, Han-Ruei Tsai, Chih-Wei Hu, and Fu-Jen Kao* Institute of Biophotonics, National Yang Ming University, Taipei, 11221, Taiwan Email: [email protected]; [email protected] ABSTRACT In this study, we have developed a four-channel Stokes vector formalism based second harmonic generation (SHG) microscopy to map and analyze SHG signal. A four-channel Stokesmeter setup is calibrated and integrated into a laser scanning microscope to measure and characterize the SH’s corresponding Stokes parameters. We are demonstrating the use of SH and its Stokes parameters to visualize the birefringence and crystalline orientation of KDP and collagen. We believe the developed method can reveal unprecedented information for biomedical and biomaterial studies. Key words: Stokes vector, polarization, second harmonic generation, KDP, collagen

1.

1. INTRODUCTION

Stokes vector was formulated by Sir George Gabriel Stokes to account for the polarization state of electromagnetic radiation1. When compared with Jones’ vector formalism, which is only applicable for fully polarized light with complex parameters in terms of electric field, Stokes vector consists of four intensities parameters, and is applicable to all polarization states of the electromagnetic wave2. It is defined as:

 S 0   I 0  I 90  S   I  I   90  S   1   0  S 2   I  4  I  4       S 3   I RCP  I LCP  ,

(1)

where S0 is the total intensity, S1 is the intensity difference between the linear polarized states at 0° and 90°, S2 is the intensity difference between the linear polarized states at 45° and -45°, and S3 is the intensity difference between the right and left circular polarized states, respectively. In biomedical studies, the Stokes vector formalism has been frequently used in describing the polarization state of the transmitted or scattering light from the specimen. It is often combined with the Mueller matrix formalism to measure the polarization state changes of scattered or transmitted light 3-12. Previously, the polarization state of light was measured in a one-channel configuration, which would take at least 4 measurements to obtain the four Stokes parameters. A few groups have reported simultaneous measurements of the four Stokes parameters using four-channel configurations, significantly reducing the acquisition time 13-17. Additionally, simultaneous measurement of the four Stokes parameters maintains the concurrency and the high sensitivity of the setup, which is critical for biomedical diagnosis. However, most applications of the Stokes vector-Muller matrix formalism are carried out in linear optical processes. In the case of nonlinear optics, the Muller matrix becomes highly complex. It is no longer a 4 by 4 matrix. Taking a second order nonlinear process as an example, the input Stokes vectors of polarization are 9 by 1 and the corresponding Mueller matrix would be 4 by 918. The following analysis and interpretation would be complicated and non-intuitive. As an alternative, we are focusing on the polarization state of the emitted light, to simplify the physical interpretation. Second harmonic generation (SHG) is a second order nonlinear optical process, in which two photons interacting with a nonlinear material are combined to form a new photon with half the wavelength of the initial photons 19. SH can be generated only from non-central-symmetrical structures, such as potassium dihydrogen phosphate (KDP) and collagen. KDP is a well

Single Molecule Spectroscopy and Superresolution Imaging V, edited by Jörg Enderlein, Zygmunt Karol Gryczynski, Rainer Erdmann, Felix Koberling, Ingo Gregor, Proc. of SPIE Vol. 8228, 82280C · © 2012 SPIE · CCC code: 1605-7422/12/$18 · doi: 10.1117/12.907199 Proc. of SPIE Vol. 8228 82280C-1 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 04/17/2013 Terms of Use: http://spiedl.org/terms

known material for its non-linear optical properties and is often used in optical modulators and SHG. Collagen is a chiral material with helix structure and presents widely in human body. There are many reports on the applications of SHG microscopy in characterizing the structural properties of different types of collagen 20-25. Many previous works are carried out by rotating a half-wave plate before the sample and detect the corresponding SH intensity changes. One major disadvantage of such measurement is that it is difficult to provide the information on the full polarization state of the SH signal at each pixel of the image. In this work, we are addressing this issue with the technique of a four-channel Stokes vector based polarization microscopy, using KDP micro-crystals and collagen as the demo specimens.

2.

2. METHODS AND MATERIALS

Figure 1 shows the schematic of the setup. In order to simultaneously retrieve the polarization states of the SH signals in a two-dimensional area, a four-channel configuration was integrated into a typical laser scanning microscope. A mode-locked Ti-sapphire laser (Mira Optima 900-F, Coherent) was used as the pulsed light source. The central wavelength of the laser was 800 nm and had a full-width-at-half-maximum (FWHM) spectral width of 10 nm, which gave a transform limited pulse width of ~120 fs. The pulse repetition rate is 76 MHz. The incident laser beam was modulated by a combination of waveplates (including a polarizer, a half-wave plate and/or a quarter-wave plate) in order to generate selected input polarization states required. The modulated laser beam was then passed through the scanning unit (FV300, Olympus) and the microscope (IX81, Olympus), and was focused by the objective lens (20, NA = 0.7, Olympus) on the sample to generate SH with a wavelength of 400 nm. In this study, to reduce depolarization effect caused by the objective lens, objective lens with not very high NA is selected26, 27. The scanned area in this work was approximately 100 m  100 m. The emitted light was then collimated by a condenser lens with NA of 0.5 and was filtered by an infrared-cut and UV-pass filter to eliminate the incident laser and the background light. Four-channel configuration

UVF iRcut

\,Irror Start igiial

Stop signal

Fig 1: Schematic of the four-channel Stokes vector polarimetry setup. The four-channel detection module is integrated into a typical laser scanning microscope. TCSPC: time-correlated single photon counting. BS: beam splitter. TDA: trigger diode assembly. HWP: half-wave plate. QWP: quarter-wave plate. IR cut: infrared cut. UVF: ultraviolet filter. FR: Fresnel rhomb. WP: Wollaston prism. PMT: photomultiplier tube.

In the detection part, the four-channel setup divided the SH signal into two parts with a beam splitter. The first part, which transmitted the beam splitter, passed through a Fresnel rhomb perpendicularly and was further divided into two beams of mutually orthogonal polarization by a Wallaston prism. The second part, which was reflected off the beams splitter, was also further divided into two parts of mutually orthogonal polarization by another Wallaston prism. In this four-channel configuration, the optical axes of the two Wallaston prisms were both set to 45 degrees relative to the principle optical axis. Since the Fresnel rhomb produces 90 degree retardation, the first part of the four-channel acts as a circular polarization

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analyzer. Conversely, the second part of the four-channel acts as a linear polarization analyzer. The output light beams from the four-channel configuration were coupled into four photon counting PMTs through four optical fibers with the same core diameters (1.5 mm). The signals collected by the PMTs were routed by a 4-channel time-correlated single photon counting (TCSPC) module (PicoHarp 300, PicoQuant). The TCSPC module recorded the position information of each emitted photon and thus two-dimensional intensity images could be reconstructed through the accumulating photons for each pixel 28. The relation between the detected intensities and the polarization state of the SH signal can be formulated by

I  A44  S out

(2)

In Eq. (2), I is the 4 by 1 intensity vector detected by TCSPC, A 4x4 is a 4 by 4 instrument matrix of the four-channel configuration, and Sout is the 4 by 1 Stokes vector of the SH. A4x4 was formulated by the eigenvalue calibration method (ECM), which was commonly used in measuring the polarization state changes of transmitted or scattering light 29. The system error or deviation of the four-channel configuration (A4x4 ) is characterized by the condition number C, which is defined as30

C  A44   A44 

1

(3)

In order to maintain high measurement accuracy, the condition number should be as low as possible. Since I is the detected intensity vector, and A4x4 is a constant matrix for a fixed configuration, Sout can thus be calculated as

S out   A44   I 1

(4)

The degree of polarization (DOP), degree of linear polarization (DOLP), degree of circular polarization (DOCP) and the anisotropy ratio of the SH light at each pixel of the scanning area were defined by following equations 31, 32:

S 0  S1  S 2  S 3 2

DOP 

2

DOCP  r

2

S0 S 0  S1  S 2 2

DOLP 

2

2

2

S0

(5)

S3 S0

I par  I perp I par  2 I perp

The expression of r in terms of Stokes parameters varies for different excitation polarization states. For linear horizontal excitation, r can be written as:

r

S1 3S 0  S1


(6)

We have developed a series of MATLAB programs to reconstruct the two-dimensional intensity images as well as the corresponding Stokes vector images and the polarization parameter images.

3.

3. RESULTS AND DISCUSSIONS

To demonstrate the performance of the four-channel configuration in measuring polarization states, we used air as the standard sample to measure the optical signals with known polarization states, and compared the reconstructed Stokes vector images with the expectations. A series of input polarization states were generated through rotating the half-wave plate, including linear horizontal polarization (LHP), linear vertical polarization (LVP), linear 45 degree polarization (L+45P) and right circular polarization (RCP). The corresponding Stokes vectors for these polarization states are [1, 1, 0, 0]’, [1, -1, 0, 0]’, [1, 0, 1, 0]’, [1, 0, 0, 1]’, respectively. Figure 2 shows the reconstructed Stokes vector images corresponding to various polarization states, which agree well with the expectations. The average value for each of the Stokes vector image was calculated and the relative error between the average and expectation value ranges from 1-9%. Such relative error was probably resulted from several factors, including manual operation error, imperfect coupling of the beams into fibers, imperfect properties of optical components. As shown, the edges in some of these images display certain non-uniformity, which is due to the imperfect coupling of the output beams into the fibers.

Fig. 2 Reconstructed Stokes vector images that correspond to various input polarization states.

The computed instrument matrix at 400 nm is:

A44

292.88  209.49  216.62  0.75  213.26  60.71 166.99 19.41  ,  267.48 84.15  78.93  209.49   72.30 32.95 247.98   250.91

(7)

which has a condition number of 2.89. Using the 4-channel setup, we performed SHG measurements using KDP powder and collagen as specimens. The wavelength of the laser was set to 800 nm. We first compared the SHG from KDP powder as a function of the incident laser powder and incident polarization angle. Since KDP produces strong SH signal, the excitation


powers of several milliwatts were used. As shown, the detected SH intensity displays power independent, and varies as a function of the input polarization angle. 1.2

1.0

>. 0.8 C

0

.8 0.6

=

i 0.4 0.2

0.0 0

50

100

150

200

250

300

350

Incident polarization angle (Degree)

Fig. 3 Normalized SH intensity as a function of incident polarization angle.

Fig. 4(a) shows the Stokes vector images reconstructed from the SHG coming out of the KDP powder. The incident laser was focused on the KDP micro-crystals with LHP incidence. In order to show the intensity distribution, the first image of the Stokes vector images displays as S0 image rather than a unit matrix. As shown in the S1/S0 and the S3/S0 images, the two blocks of KDP micro-crystals show significant difference of polarization states or orientation. For the smaller KDP microcrystal in the upper right hand corner, there are higher components of LHP and RCP. In contrast, there are higher components of LVP and LCP for the KDP micro-crystal in the lower left hand corner. The corresponding polarization parameter images are shown in Fig. 4(b). As shown in the DOP image, the degree of polarization is close to 1, indicating that most of the excited SH is polarized. The DOLP and the DOCP images demonstrate that there exist both linear and circular polarization states in the SHG from the KDP micro-crystals. The circular polarization is generated from the birefringence of the KDP. The anisotropy image, similar to the S1/S0 image, shows the orientation differences of the KDP micro-crystals. SI'S"

I)OP

11011'

s3's"

DOCP

AnisoIn,ps

I

+

4. (b)

(a)

Fig. 4 (a) Stokes vector images and (b) polarization parameter images of SHG from KDP micro-crystals.

As a chiral material, birefringence of collagen has also been observed from its corresponding SH Stokes vector and polarization parameter images, as shown in Fig. 5. To protect the collagen sample from being damaged by high excitation


power, the average power under the objective is controlled to around 20 mW. Similar to the results shown in Fig. 4, the polarization states of the SHG from collagen vary in different positions. SI'S"

Do,.

111)1 I'

rr OO(I

17

1.7

trill.

1t5

-I

(b)

(a)

Fig. 5 (a) Stokes vector images and (b) polarization parameter images from SH of collagen.

4.

4. CONCLUSION

In conclusion, we have developed a Stokes vector based module for nonlinear optical microscopy. The four-channel polarimetry configuration can perform simultaneous measurement of the four Stokes vector parameters of the second harmonic. We have demonstrated the use of SH polarization to visualize the birefringence and crystal orientation of KDP and collagen. The setup and the reconstruction method presented in this work could also be used in other nonlinear applications, especially the fluorescence microscopy to retrieve the spatiotemporal distribution of polarization states.

5.

ACKNOWLEDGEMENT

We appreciate greatly the kind help on implementing Stokes vector based methods from Professor Peter Török, Dr. Matthew Foreman and Dr. Carlos Romero from Imperial College in London, UK. The insightful discussion with Profs. Nees is crucial in interpreting the results. The authors would also like to thank the National Science Council, Taiwan (NSC99-2627-M-010002-, NSC98-2627-M-010-006-, NSC97-2112-M-010-002-MY3, and NSC98-2112-M-010-001-MY3), as well as the Ministry of Education, Taiwan under the “Aim for Top University” project for the generous support of the reported work.

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