Mapping polymer birefringence in three-dimensions using a polarizing microscope with oblique illumination. Michael Shribak and Rudolf Oldenbourg.
Copyright 2004 Society of Photo-Optical Instrumentation Engineers. This paper was published in Proceedings of SPIE, vol. 5462, p. 57-67, (2004) and is made available as an electronic reprint (preprint) with permission of SPIE. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper are prohibited.
Mapping polymer birefringence in three-dimensions using a polarizing microscope with oblique illumination Michael Shribak and Rudolf Oldenbourg Marine Biological Laboratory, Woods Hole, MA 02543, USA ABSTRACT We have been developing a polarized light microscope with liquid crystal universal compensator and circular polarizer (the LC-PolScope) for recording images, which are independent of the orientation of birefringent objects. Separate images show the retardance and the slow axis azimuth distributions of the in-plane birefringence of the focused region in the specimen. However the measured (apparent) retardance still depends on the angle between the crystal optic axis and the axis of the illuminating beam of light. If the illuminating beam is close to parallel to the optic axis the measured retardance value decreases dramatically and becomes zero when the two axes are parallel. The description of birefringent objects oriented in 3-dimensional space requires the introduction of two additional parameters: the principal retardance and the inclination angle. Together with the azimuth angle they completely characterize the birefringence properties of a specimen, assuming the specimen has a uniaxial optical indicatrix. We devised a new technique for measuring the three birefringence parameters without moving the specimen. For exploring the out-of-plane birefringence the new instrument which is based on the LC-PolScope technique contains an additional spatial light modulator, implemented here as a liquid crystal mask. The mask is located in the aperture plane of the condenser lens. Partial occlusion of the condenser aperture changes the direction of the central ray of the cone of light converging on the specimen. So we can obtain the retardance and azimuth images using different sets of illumination rays. For experimental verification we used a biological object called an aster. An aster consists of nearly parallel arrays of microtubules, a stiff biopolymer, radiating from a common organizing center called a centrosome. The object is spherically symmetric, and its 3 dimensional distribution of birefringence orientation can be predicted. Experimental results have shown the developed polarizing microscope can successfully be used for imaging and measuring three-dimensional orientation of birefringent objects.
INTRODUCTION A standard polarizing microscope with crossed linear polarizers shows a picture in which the image contrast of a birefringent object depends on both, the amount of relative retardation induced by the object and the orientation of its principal axis. For example, if the orientation of the principal axis is parallel to one of the polarizers, then the birefringent object becomes invisible. For best visibility, the principal axis must be oriented at 45° to the polarizers. In addition, the image intensity of the object is proportional to the square of the sine of the relative phase shift between the ordinary and the extraordinary ray passing through the object. The addition of a waveplate as compensator to the optical train provides the opportunity to improve the contrast and measure the relative retardation, but the principal restriction of orientation dependent and non-linear image contrast remains. The advent of new electro-optical devices, electronic imaging and digital image processing have allowed to create techniques that independently measure the relative retardation, also called retardance, and the orientation, also called azimuth, of birefringent objects. Measurement results are simultaneously obtained for all birefringent objects in the field of view and are displayed as separate images, one is called retardance image, the other azimuth image. These new types of polarizing microscopes typically incorporate electronically adjustable polarizers and waveplates and a video camera with digital image processing system and custom designed software. For example, optical systems have been reported that use either fixed linear polarizers and variable liquid crystal waveplates [1], or a rotating linear polarizer and fixed circular polarizer [2]. In either case, a combination of adjustable polarizers and waveplates is used to create a sequence of polarization states for illuminating the specimen and for analyzing the light that is transmitted or reflected by the specimen. In quick succession, several raw images of the specimen are recorded at predetermined settings of the polarizers and waveplates and the images are processed in order to calculate the retardance and azimuth pictures of the
Biophotonics Micro- and Nano-Imaging, edited by Dario Anselmetti, Proceedings of SPIE Vol. 5462 (SPIE, Bellingham, WA, 2004) 1605-7422/04/$15 · doi: 10.1117/12.549050
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specimen [3]. Instead of generating the raw images sequentially, it is also possible to generate them in parallel by using a non-polarizing beamsplitter with a set of polarization analyzers in the imaging path [4]. The advanced techniques are successfully used in industrial, research and clinical applications, including studies of the cell cytoskeleton [5] and for in vitro fertilization procedures [6]. However, the retardance measured with these techniques still depends on the angle between the crystal optic axis of the birefringent object and the axis of the beam illuminating the object. If these two axes are nearly parallel to each other the measured retardance value decreases dramatically and becomes zero for parallel axes. In principle, for changing this angle it is possible to use a universal stage, originally invented by E.S. Fedorov more than 100 years ago [7]. With this apparatus the specimen, supported between two glass hemispheres, may be manually rotated through measured angles about vertical and horizontal axes. Hence, the specimen can be rotated to a position in which the optic axis of the birefringent object under observation is perpendicular to the illumination beam and the maximum retardance is measured. The universal stage, however, is relatively bulky and can accommodate only long working distance, low resolution optics. While it allows to measure precise angles and the maximum retardance of an object, it can do so only for relatively large objects, one object at a time. Therefore, the universal stage has found use mainly in geological applications for the analysis of thin rock sections containing birefringent crystals that are at least a few microns in diameter. We are now introducing a new technique that rotates the illumination beam instead of the specimen, combining in one instrument high numerical aperture optics, an aperture scanning device, and a universal compensator for measuring specimen birefringence with high sensitivity and high resolution. The so-called Scanned Aperture Pol-Scope is able to measure the maximum retardance of a uniaxial birefringent object, regardless of how it is oriented, and it can determine the two angles that characterize the orientation of the object in 3-dimensional space. In addition, the new technique measures all these parameters for every resolved specimen point simultaneously, requiring only a few seconds for measuring up to a million specimen points.
EXPERIMENTAL SETUP A schematic of the Scanned Aperture Pol-Scope is shown in Fig. 1. The light is produced by the high-pressure mercury arc lamp and filtered by a green narrowband filter (546 nm, 12-nm FWHM, Omega, Brattleboro, Vermont). We used also an Ellis fiber optic light scrambler (Technical Video, Port Townsend, WA), which isn’t shown in the picture. The optical set-up includes a custom made aperture scanning device built with liquid crystal devices. The scanner implements two independent functions: an amplitude mask and a polarization state generator each of which is computer controlled. The amplitude mask with 8 pie-shaped sectors is used to partially block the aperture thereby steering the cone of light illuminating the specimen. The mask is created by a pair of parallel linear polarizers, P1 and P2, with the segmented liquid crystal cell LCM placed between. We use the azimuth of the first polarizer as a reference orientation. The azimuth of the cell is 45°. It introduces full or half-wave retardation in order to transmit or block light passing through a sector. The set of linear polarizer P2 and pair of variable retardation plates LCA and LCB with azimuths 45° and 0° allows to generate a polarized beam with any ellipticity and azimuth of the major axis. Such an arrangement can be called a universal compensator or complete polarization state generator [1, 8]. A similar arrangement was proposed by Yamaguchi and Hasunuma in 1967 using a circular polarizer and KDP Z-cut crystals instead of liquid crystal devices [9, 10]. The sectored universal compensator is used as both a polarization rectifier [11] and a birefringence measurement device [3], which works in pair with the left circular analyzer. All components of the scanning device are bonded together and form a 7 mm thick optical flat with an approximate diameter of 25 mm. The liquid crystal based scanning device was designed in collaboration with Cambridge Research and Instrumentation, Woburn, MA, which manufactured the device and its electronic control circuitry. The computer software for hardware control and image analysis was developed inhouse.
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Fig. 1. Schematic of optical (left) and electronic components of the Scanned Aperture Pol-Scope. The optical set-up is based on a traditional light microscope enhanced by an aperture scanning device and an analyzer for circularly polarized light. The scanning device includes the liquid crystal devices LCA, LCB, and LCM. All liquid crystal devices are variable linear retarder plates with 8 or 9 individually controlled sectors. LCM is sandwiched between two linear polarizers and is used to block or transmit light passing through 8 pie-shaped sectors. A pair of sectors in LCA and LCB arranged in series function as a universal compensator for controlling the polarization state of the passing light. The computer controlled scanning device is placed in the front aperture plane of the condenser lens.
The Scanned Aperture Pol-Scope in our laboratory is implemented on an inverted microscope Zeiss Axiovert 200M. The condenser mount was custom modified to add the scanning device in the aperture plane of the 1.4 NA oil immersion condenser lens. For recording the aster images shown next we used a Plan-NEOUFLUAR 100x/1.30 oil immersion objective.
MICROTUBULE BIREFRINGENCE For a description of the optical polarization properties of microtubules we use the form birefringence model, which was developed by Wiener in 1912 [12]. A similar approach was taken by Oldenbourg et al. [13] for estimating the birefringence of single and bundled microtubules measured with a high NA imaging system. Here we present a different derivation of the theoretical result. Wiener considered dielectric rods embedded in a media with different refractive index. He has shown that the rod assembly behaves as a positive uniaxial crystal, with its optics axis parallel to the axes of the rods ( n e > n o ). Because the difference between the extraordinary and ordinary refractive indices is positive, the slow axis corresponds to the vibration direction with extraordinary refractive index, and the fast axis corresponds to vibration direction with ordinary refractive index. Wiener’s equations can be written as:
no = n2
2
2
2
2
( f (n
2
2
2
2
2
)f,
n1 + n 2 + f n1 n 2 n1 + n 2
ne = n2
1
(n 1+
1
n2
n2
2
2
n2
), ) (1)
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where n1 and n 2 are refractive indices of the rods and media, respectively, and f is the fraction of the total volume, that the rods occupy. Using the above formulas we obtain the following simple formula describing the birefringence n e n o for the case f
