Coherent Gradient Sensing Microscopy (Micro-CGS): a microscale curvature detection technique

Transcription

1 Coherent Gradient Sensing Microscopy (Micro-CGS): a microscale curvature detection technique Maxim Budyansky, Christopher Madormo, Jamie L. Maciaszek, and George Lykotrafitis * Department of Mechanical Engineering, University of Connecticut, 191 Auditorium Road, Unit 3139, Storrs-Mansfield, CT, 0669, USA * Corresponding author: George Lykotrafitis. Department of Mechanical Engineering The University of Connecticut 191 Auditorium Road, Unit 3139 Storrs, CT , USA Tel. : ; fax.: address: gelyko@engr.uconn.edu Abstract: Micro-CGS, a full-field, real-time, micro-interferometric technique for quantitative imaging is introduced. Micro-CGS extends the capabilities of the large-scale shearing interferometric technique of Coherent Gradient Sensing (CGS) in the micro-scale. The experimental setup is detailed and resulting interferograms are shown. A digital image processing program is created to compute specimen curvature from recorded fringe patterns. The experimental method is validated by the successful measurement of the curvature of 30-micron glass microspheres. Keywords: Interferometric imaging; Laser optics; Microscopy; Diffraction gratings

2 1. Introduction A majority of the optical techniques currently used in microscopy, such as Differential Interference Contrast (DIC), Dark-Field (DF), and Phase Contrast (PC) microscopy, are qualitative in nature [1-3]. Recently, there has been an emergence of quantitative optical techniques that obtain spatial and temporal information in the micro-scale including Hilbert phase microscopy [4, 5], holography [6], wavefront sensing [7], and quantitative DIC [8]. In this framework, we developed Micro-CGS, a micro-interferometric technique that offers a noninvasive direct measurement of the gradient of specimen thickness. A digital image processing program is then used to determine the specimen curvature. Classical CGS is a well-established large-scale quantitative optical interferometric technique which has been successfully employed in the field of fracture mechanics as well as in the measurement of residual stresses in thin films [9-16]. CGS can be used in two operating modes, transmission and reflection. The working principle of CGS is shown in Fig. 1. A collimated laser beam with a planar wave front passes through a specimen. Due to the difference between the refractive index of the specimen and the refractive index of the environment, the initially planar wave front is deformed. This deformation is directly correlated to the thickness of the specimen, assuming a constant refractive index throughout the specimen. Two high-density diffraction gratings, G 1 and G, with lines directed along the x 1 -axis laterally shear the beam along x -axis (see Fig. 1). When the incident wave front S(x 1,x ), encounters the first diffraction grating, G 1, it is diffracted into several wave fronts E 0, E 1, E -1, etc. For aesthetic purposes only E -1, E 0, E 1 are shown. The resulting wave fronts are again diffracted at the second diffraction grating, G. This produces an additional set of wave fronts, E 1,1, E 1,0, E 1,-1, E 0,1, E 0,0, E 0,-1, E -1,1, E -1,0, E -1,-1, etc. for each incident wave front. A focusing lens is then used to combine parallel wave fronts at the filter plane. A diffraction pattern is produced consisting of several diffraction spots D 1, D 0, D -1, etc. The single diffraction spot D 1 or D -1, which contains information on the gradient of the optical thickness of the specimen [9, 10], is isolated and the generated fringe pattern is then imaged by a CCD camera.

3 . Experimental Setup Micro-CGS is based upon the introduction of a microscopic objective into the Classical CGS experimental setup. A schematic of the Micro-CGS setup can be seen in Fig. and a picture of the actual experimental optical setup is displayed in Fig. 3. A collimated 635-nm He-Ne laser is spatially filtered and passed through a specimen. A microscope objective is introduced into the optical path for magnification purposes. The objective used was an Olympus infinitely corrected 40x microscope objective lens. Upon exiting the microscope objective, the beam is re-collimated, and directed through the pair of Ronchi diffraction gratings of 40 lines/mm. In the experimental setup shown in Fig. 3, the beam is passed through a 10 expander before entering the gratings. The expander was implemented when imaging microspheres with diameters in the range of 10 to 50 μm. After the D +1 spot is filtered through an iris, a CCD camera, Casio EX-F1, is focused onto the image plane to capture the generated interferogram. The orientation of the diffraction gratings dictates the axis along which the setup will measure curvature. For the current setup the gratings are oriented along the x 1 direction and thus they provide surface gradient information along the x -axis. In order to acquire a complete description of the sample s curvature, a second optical path with the pair of gratings oriented perpendicular to the x 1 direction has to be implemented. 3. Digital Image Processing The digital image processing software utilized to derive the curvature of the specimen from the recorded interferogram was coded in MATLAB (The MathWorks, Natick, MA). To validate the methodology, a spherical wave front is created by passing a collimated laser beam through a plano-convex lens - in air - of known focal length f mm and a known radius of curvature f 1. The wave front then passes through the pair of gratings and an interferogram corresponding to the x axis surface gradients is generated. The algorithm begins by reading in the interferogram captured by the CCD camera, which creates a picture comprised of pixel by pixel intensity distributions (Fig. 4a). The intensity distribution of the CGS interferogram can be expressed in a complex form as

4 I( x, x ) a( x, x ) c( x, x ) c ( x, x ), (1) * i ( x, ) Where (, ) ( (, ) 1 x c x x b x x e ) / and 1 1 * * c ( x1, x ) is the complex conjugate of c ( x1, x). The program then converts the image to the two-dimensional Fourier domain by employing D Fast- Fourier transform. The resulting Fourier transform magnitude in the frequency domain can be seen in Fig. 4b. The intensity distribution in the Fourier domain is written as I(, ) A(, ) C(, ) C (, ), () * where 1 and are the spatial frequencies in the Fourier domain. A ( 1, ) contains the constant- and low-frequency information related to gradual changes of the intensity background * (Fig. 4c). C, ) is the complex conjugate of C 1, ) and essentially contains the same ( 1 ( information. High-frequency information is associated with abrupt changes of the intensity background as shown in Fig. 4d. Band pass filtering is used in order to eliminate A 1, ) and ( * either C 1, ) or C, ). The inverse Fourier transform of the remaining term is ( employed to derive ( 1 * c x x or c x, x 1, obtained by using the following expression 1. The full-field wrapped phase distribution is then 1 Im[ c( x1, x)] ( x1, x) tan (3) Re[ c( x, x )] 1 The wrapped phase is shown in Fig. 5a while a typical single column of the wrapped phase image is displayed in Fig. 5b. The values oscillate approximately between -π and π. After obtaining the wrapped phase, a phase unwrapping algorithm is implemented to generate the unwrapped phase, which is displayed in Fig. 6a accompanied by a single column representation shown in Fig. 6b. The unwrapped phase is then shifted so that the zero order fringe is centered about the wrapped phase axis. This particular case established linearity in fringe order for the

5 plano-convex lens which has constant curvature. The phase change x 1, x ) is related to the ( thickness h ( x 1, x) of the lens via the expression ( x1, x) ( n / ) h( x1, x), where is the wave length of the laser beam, and n 0.5 is the refractive index contrast between the planoconvex lens n 1.53 and the air n 1.0 to obtain the curvature of the specimen, the unwrapped phase is differentiated with respect to the x direction according to Eq. (4) for CGS in transmission [9, 10, 13]. h( x, x ) p N x d n D 1 where h x x is the thickness of the specimen, p is the grating pitch, N is the change in the 1, number of fringes, d is the distance between the diffraction gratings, and D is the distance between the fringes. From Eq. (4), the mean experimental curvature was found to be approximately mm -1. A plot of the ratio of the experimental vs. the known curvature 1 1 ( f1) mm is shown in Fig. 7. The calculated curvature is within 1% of the analytical value. We finally note that micro-cgs is not sensitive to rigid body motion because it directly measures the gradient of the specimen surface topography. (4) 4. Results A sample of 30.1 µm ±.1 µm diameter glass microspheres (Thermo-Scientific), and a refractive index of n 1. 51, in aqueous solution ( n 1. 33) was imaged utilizing the aforementioned experimental setup. An image taken of a group of beads can be seen in Fig. 8a. For validation purposes, a single bead was isolated and imaged. The corresponding interferogram captured in the CCD camera is shown in Fig. 8b. The increase in the number of fringes is due to the increased distance between the gratings. In this way a larger number of experimental points were used in the calculation of the curvature. We note that because the beads were magnified before the diffraction gratings, a magnification factor of 343 was implemented into Eq. (4). The magnification factor was obtained by measuring the diameter of the image of a bead and then comparing it with the value obtained via an optical microscope.

6 A rectangular cross section was taken from the center of a bead and analyzed by the digital image processing program, described in the previous section, to calculate the mean experimental curvature. We used only the central part of the image because Eq. (4) assumes that the surface can be approximated by a parabolic function which is true only at the central part of the bead. The wrapped phase of the rectangular cross section is shown in Fig. (9a). The developed phase unwrapping algorithm was employed to obtain the unwrapped phase, shown in Fig. (9b). The resulting curvature measurement is illustrated in Fig. 10. The mean curvature obtained from the image processing program was m 1 which corresponds to a 8.9 μm diameter. The result was independently verified by measuring the diameter of the bead on the recorded image. 5. Conclusions The full-field, real-time, optical microinterferometric technique of micro-cgs has been presented. The experimental method was validated by measuring the curvature of glass microspheres of 30.1 µm ±.1 µm diameters. The accuracy of the results signifies that micro- CGS can be used to successfully obtain curvature of static micro-specimens. Furthermore, the optical system produced repeatable fringe interferograms within the beads, signifying the successful curvature detection capability of micro-cgs. While the technique was applied in a static experiment, the only limitation on the response rate is set by the recording speed of the CCD camera meaning that micro-cgs can be employed in the detection of the changing of micro-curvature during dynamic phenomena. In particular, we anticipate applying it as a diagnostic tool to distinguish between normal and abnormal cells based on their different thermal vibration topography. For example, malaria infected RBCs also show a two-fold increase in their elasticity during the maturation of the parasite [17] and they portray a different thermal vibration footprint compared to normal RBCs [18]. Also, RBCs from patients with the sickle cell trait which show a two-fold increase in their Young s modulus while their morphology is similar to normal RBCs [19]. Thus, micro-cgs measurements can be used as novel biomarkers in addition to traditional methods such as cytomorphological and immunohistochemical diagnosis. Especially in cases of morphological overlap, the mechanical signature of cells can be the decisive factor in distinguishing between normal and abnormal cells.

7 Acknowledgements We would like to acknowledge Ares J. Rosakis of the California Institute of Technology for very helpful discussions on Coherent Gradient Sensing and providing us with the Ronchi gratings used in the experimental setup. This work was supported by UCRF Large Grant J980.

8 Figure captions Figure 1. (Color on the Web only) Working principle of CGS. Figure. (Color on the Web only) Micro-CGS. Figure 3. Micro-CGS experimental setup. Figure 4. (a) Cropped interferogram for 75mm diameter plano-convex lens. (b) Fourier Transform magnitude of the interferogram shown in Fig. 4a in the spatial frequency domain 1,. (c) Gradual changes of the intensity background in Fig. 4a related to constant- and low-frequency information. (d) Rapid changes of the intensity background in Fig. 4a related to high-frequency information. Figure 5. (Color on the Web only) (a) Full-field wrapped phase corresponding to Fig. 4a, (b) Wrapped phase in radians vs. distance along the x -axis shown in Fig. 5a. Figure 6. (Color on the Web only) (a) Full-field unwrapped phase. (b) Unwrapped fringe order vs. distance on the x -axis shown in Fig. 6a. Figure 7. (Color on the Web only) Experimental vs. analytical curvature ratio of a plano-convex lens. Figure 8. (a) Interferogram of a group of 30.1 µm ±.1 µm glass microspheres. (b) Interferogram of an isolated glass microsphere. Figure 9. (a) Full-field wrapped phase. (b) Full-field unwrapped phase. Figure 10. Curvature distribution corresponding to Fig. 9b.

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