Phase is a fundamental resource for optical imaging but cannot be directly observed with intensity measurements. The existing methods to quantify a phase distribution rely on complex devices and structures and lead to difficulties of optical alignment and adjustment. We experimentally demonstrate a phase mining method based on the so-called adjustable spatial differentiation, by analyzing the polarization of light reflection from a single planar dielectric interface. Introducing an adjustable bias, we create a virtual light source to render the measured images with a shadow-cast effect. From the virtual shadowed images, we can further recover the phase distribution of a transparent object with the accuracy of 0.05λ RMS. Without any dependence on wavelength or material dispersion, this method directly stems from the intrinsic properties of light and can be generally extended to a broad frequency range. |
1.IntroductionPhase distribution plays the key role in biological and x-ray imaging, where the information on objects or structures is mainly stored in phase rather than intensity variations. Phase distribution cannot be directly retrieved from conversional intensity measurements. Zernike invented the phase contrast microscopy1 to render the transparent objects but without the ability to quantify the phase distribution. Later, in order to enhance the image contrast, Nomarski prisms were created for differential interference contrast (DIC) imaging,2 which offers the phase gradient information. Based on Fourier optics, a differentiation filter3–5 or spiral phase filter6–9 in 4f system can visualize the phase of optical wavefront. Further, various quantitative phase measurement technologies10 based on the beam interference11–16 and propagation17–20 were developed for mapping the optical thickness of specimens. However, these methods usually rely on complex modulation devices in the spatial or spatial frequency domains, resulting in the difficulties of optical alignment and adjustment. In recent years, optical analog computing of spatial differentiation has attracted great attention, which enables an entire image process on a single shot.21–31 Various optical analog computing devices were designed for edge detection.32–44 In particular, Silva et al. theoretically proposed realization of optical mathematical operations with metamaterials, as well as the Green’s function slabs.21 Later, a subwavelength-scale plasmonic differentiator of a 50-nm-thick silver layer was experimentally demonstrated and applied for real-time edge detection.32 Most recently, spatial differentiation methods were developed based on the geometric phases, including the Rytov–Vladimirskii–Berry phase40 and the Pancharatnam–Berry phase.41,45 So far, the current application of optical spatial differentiation is only limited to edge detection. As the fundamental difficulty to determine the phase of electric fields by measuring their intensities, one cannot determine the sign of the differentiation signal and hence recover the phase distributions. In this article, we propose an adjustable spatial differentiation to characterize and quantitatively recover the phase distribution, just by simply analyzing the polarization of light reflection from a single planar dielectric interface. By investigating the field transformation between two linear cross polarizations, we experimentally demonstrate that the light reflection enables an optical analog computing of spatial differentiation with adjustable direction. We show that this effect is related to the angular Goos–Hänchen (GH) shift46–48 and the Imbert–Fedorov (IF) shift.49,50 Furthermore, by tuning a uniform constant background as the bias, we create a virtual light source to render the measured images with a shadow-cast effect and further quantify the phase distribution of a coherent field within accuracy. Without complex modulation devices,3–9,11–14 our method offers great simplicity and flexibility, which can circumvent the fabrication of complex structures and also the difficulties of optical alignment and adjustment. Importantly, since the proposed method is irrelevant to resonance or material dispersion, it works at general wavelengths with large temporal bandwidths, which is very suitable for high-throughput real-time image processing. 2.Adjustable Spatial DifferentiationSchematically shown as Fig. 1, the proposed phase mining scheme only includes a dielectric interface as a planar reflector and two polarizers P1 and P2, where the incident light is assuming a phase distribution . Next we show that such a light reflection process can transform the original uniform intensity image with invisible phase structure into a structured contrast image, by optical computing of spatial differentiation to the incident electric field. Moreover, the direction of the spatial differentiation can be further adjusted with different combinations of the polarizer orientation angles. To specifically depict the transformation, we decompose the incident (output) beam into a series of plane waves as according to spatial Fourier transform. Then, the spatial transformation between the measured field and the original one is determined by a spatial spectral transfer function . We denote the orientation angles of the polarizers P1 and P2 as and , respectively. The spatial spectral transfer function can be written as (see Sec. 1 in the Supplemental Material) where and are the Fresnel reflection coefficients of the - and -polarized plane waves, respectively. In Eq. (1), the third term is induced by the opposite -directional displacements for left- and right-handed circularly polarized beams, which is known as the IF shift.40,51 By considering the GH effect, we expand the () around the incident angle as , where is the Fresnel reflection coefficient for the central plane wave, and is the wavevector number in vacuum. We note that here such a spatial dispersion during partial reflection on the dielectric interface purely leads to an angular GH shift.46–48For a partial reflection process on a dielectric interface, we can control the orientation angle of the polarizers P1 and P2 to satisfy the cross-polarization condition In such a condition, the P2 polarizer is oriented orthogonally to the polarization of the reflected central plane wave, which is first polarized by P1 and then reflected by the dielectric interface with an incident angle . Under the cross-polarization condition, the spatial spectral transfer function becomes where and are two coefficients as and . Equation (3) shows that the spatial spectral transfer function is linearly dependent on the spatial frequencies and , which corresponds to the computation of a directional differentiation in the spatial domain: . We note that the directional spatial differentiation occurs in every oblique partial reflection case on a dielectric interface, but hardly in total internal reflection or metallic reflection cases where the complex and prevent the satisfaction of Eq. (2). Besides, since and in Eq. (3) are also complex, the directional differentiation no longer exists unless is purely real.Considering the original field , the output field is which offers a differential contrast image of the phase object along direction defined as . Besides, the coefficient in Eq. (4) varies with different directions (see Sec. 3 in the Supplemental Material), which is a quantitatively important coefficient when utilizing spatial differentiation along different directions in the meantime.We note that the direction , described by an angle as shown in Fig. 1, is continuously adjustable with different values of and . Under a certain incident angle, the direction angle varies with different pairs of and that satisfy the cross-polarization condition. We note that only when the incident angle is smaller than Brewster angle, can cover a complete range from 0 deg to 180 deg. However, when the incident angle is larger than Brewster angle, both and become negative, and hence the coefficients and are always negative and in a limited range so that the corresponding values of are also limited. 3.Bias Introduction and Phase MiningWe note that even though the edges of phase objects can be detected through the directional differentiation along direction , the sign of the differentiation cannot be distinguished since the measured intensity is proportional to . It means the edges cannot be determined as the ridges or the troughs in the phase structure. In order to determine the sign of the differentiation, we add a uniform constant background as a bias into the proposed spatial differentiation and generate contrast images with shadow-cast effect. We show that the bias can be easily introduced and adjusted in the proposed phase quantifying scheme, in comparison to the other methods, such as traditional DIC microscopy2,52 or spiral phase contrast microscopy. The uniform constant background is introduced by breaking the spatial differentiation requirement of Eq. (2), that is, by rotating the polarizers for a small angle deviating from the cross-polarization condition. In this case, Eq. (2) is changed to where is a constant and continuously adjustable with and .Following the same way as Eqs. (3) and (4), after introducing the uniform constant background, the measured reflected image is changed to (see the details in Sec. 4 of the Supplemental Material) and becomes shadow-cast. The sign of can be determined from the change of intensity. For example, with a positive bias , the intensity appears brighter (darker) in the areas with positive (negative) directional derivative . The biased image exhibits a shadow-cast effect, which appears as illuminated by a virtual light source, as schematically shown in Fig. 4(c). Finally, we can recover the phase distribution with both the determined sign and the absolute value of . We implement the retrieval of phase distribution through a two-dimensional (2-D) Fourier method.53 It first requires two orthogonal directional derivatives, for example, the partial derivatives and . Then, we combine them as a complex distribution . In spatial frequency domain, its 2-D Fourier transform is . Therefore, the phase distribution can be recovered as 4.Experimental DemonstrationTo demonstrate the proposed phase quantifying, we use a green laser source with wavelength and a BK7 glass slab with refractive index 1.5195 for reflection. According to Eqs. (1)–(3), we first simulate the adjustable range of under every different incident angle and the chosen orientation angle of P1. As shown in Fig. 2(a), the direction angle can be fully adjustable from 0 deg to 180 deg only when the incident angle is smaller than the Brewster angle (black dashed line). In order to show a complete adjustable range from 0 deg to 180 deg, we select an incident angle [the white dashed line in Fig. 2(a)]. We next experimentally demonstrate spatial differentiation for the direction angles , 90 deg, 135 deg, and 180 deg, by choosing the appropriate orientation angles and (see specific values in Sec. 5 of the Supplemental Material). By the requirement of Eq. (2), Fig. 2(b) gives the specific values of and . The points c to f in Figs. 2(a) and 2(b) correspond to the cases with these four direction angles , respectively. As shown in Figs. 3(a)–3(d), the experiment demonstrates the edge-enhanced differential contrast imaging for a disc phase distribution [shown as the inset in Fig. 3(a)], with and for the differentiation direction , 90 deg, 135 deg, and 180 deg. Figures 3(a)–3(d) clearly exhibit the edges of the disc pattern as a circle, except the parts that are parallel to the differentiation direction. These results indeed show the adjustability of the direction of spatial differentiation. For an accurate evaluation of the spatial differentiation direction, we experimentally measure the spatial spectral transfer functions as shown in Figs. 3(e)–3(h) (see the detailed procedures in Sec. 2 of the Supplemental Material). As Eq. (3) expects, Figs. 3(e)–3(h) indeed exhibit linear dependences on both and . For each case, the measured direction angles are 45.11 deg, 90.01 deg, 134.91 deg, and 180.00 deg, respectively, which are determined by fitting the gradient directions of the experimentally measured spatial spectral transfer functions (see Sec. 3 of the Supplemental Material). The measured results are shown as the arrows in Figs. 3(e)–3(h), respectively. For comparison, the theoretical spatial spectral transfer functions are calculated from Eq. (3) and shown in Figs. 3(i)–3(l), respectively; these functions agree well with the experimental ones. With the figure of merit (FOM) defined in Ref. 54, even though the bandwidth and gain of spatial differentiation vary with the differentiation direction, the FOM of the vertical one is 0.679. We note that specifically when or 90 deg, the spatial differentiation is only from the IF shift and exhibits vertically as or 180 deg. It becomes totally horizontal for , where the spatial differentiation is purely induced by the angular GH shift since the coefficient of the vertical spatial differentiation is . Next, we experimentally demonstrate the biased imaging for a transparent phase object, by introducing a uniform constant background as the bias. According to Eq. (5), we first simulate the bias value as shown in Fig. 4(a), where the black dashed line corresponds to , i.e., the cross-polarization condition is satisfied. In order to demonstrate the biased effect, we generate the phase distribution of the incident light with a reflective phase-only spatial light modulator (SLM). An epithelial cell’s image55 in Fig. 4(b) is loaded into SLM, with a prior quantitative phase distribution. We introduce the bias by only controlling the value of deviating from the black dashed line to red triangle points in Fig. 4(a). The specific deviation angles of are 9 deg, 5 deg, 2 deg, and 1 deg, corresponding to bias values , 0.0151, 0.0100, and 0.0054, respectively (see Sec. 5 of the Supplemental Material). As shown in Figs. 4(d)–4(g), the results show the biased contrast images, which seem to result from a virtual light source obliquely illuminating the object along different directions [schematically shown in Fig. 4(c)]. These shadow-cast images result from the positive biases that render the ridges and the troughs in phase distribution as the bright edges and the shadowed ones, respectively. With a positive and very small bias as a perturbation, we can determine the sign of the spatial differentiation . The absolute value of can be obtained from the nonbias case, and therefore we experimentally acquire the first-order directional derivatives of (see the details in Sec. 6 in the Supplemental Material). Figures 5(a) and 5(b) show two directional derivatives of the incident phase distribution shown as Fig. 4(b) along - and -directions, respectively. For comparison, the ideal spatial differentiations along - and -directions are calculated as Figs. 5(c) and 5(d), respectively. These experimental results clearly coincide well with the calculated ones, indicating great accuracy of the performance. Finally, we recover the phase from the obtained directional derivatives through the 2-D Fourier algorithm. Figure 5(e) shows the result of the recovered phase distribution, which coincides with the original incident one [Fig. 5(f)]. The root mean square (RMS) between the recovered result and the original distribution is calculated as , which can be further reduced with an optimized image system. In this way, we not only enhance the contrast to make transparent objects visible but also successfully recover its original phase distribution. 5.ConclusionWe experimentally demonstrated a phase mining method only by analyzing the polarization of light reflection from a single planar dielectric interface. The direction of the spatial differentiation is continuously adjustable assuring differential contrast enhancement along different directions. Importantly, the method easily introduces a bias and generates shadow-cast differential contrast images. Based on this bias scheme, we determine the sign of the spatial differentiation of the phase distribution and finally recover the original phase information for a uniform-intensity image. We note that the present method is also suitable for birefringent specimens with advantages over conventional DIC, because the illumination happens before polarization analysis in light reflection on a dielectric interface. This planar-interface scheme is much simpler and almost costless in comparison to the methods with complex Nomarski prisms or SLMs. Without the requirement of rotating the objects, the adjustability of differentiation direction works in situ and hence circumvents the difficulties and errors by using image registration. This method also works under a partially coherent illumination. For example, here a partially coherent beam is used for illumination of the phase object, in order to eliminate speckles and enhance the image quality. Our experimental results confirm that the proposed method is independent of material and wavelength, and therefore, by x-ray or electron polarization analysis,56,57 it opens a new avenue to quantify the phase in x-ray or electron microscopy imaging. AcknowledgmentsThe authors acknowledge funding through the National Natural Science Foundation of China (NSFC Grants Nos. 91850108 and 61675179), the National Key Research and Development Program of China (Grant No. 2017YFA0205700), the Open Foundation of the State Key Laboratory of Modern Optical Instrumentation, and the Open Research Program of Key Laboratory of 3D Micro/Nano Fabrication and Characterization of Zhejiang Province. Z.R., T.Z., and J.H. are named inventors on a number of related patent applications related to this work. ReferencesF. Zernike,
“How I discovered phase contrast,”
Science, 121
(3141), 345
–349
(1955). https://doi.org/10.1126/science.121.3141.345 SCIEAS 0036-8075 Google Scholar
R. Allen, G. David and G. Nomarski,
“The Zeiss-Nomarski differential interference equipment for transmitted-light microscopy,”
Z. Wiss. Mikrosk. Mikrosk. Tech., 69
(4), 193
–221
(1969). Google Scholar
H. Furuhashi, K. Matsuda and C. P. Grover,
“Visualization of phase objects by use of a differentiation filter,”
Appl. Opt., 42
(2), 218
–226
(2003). https://doi.org/10.1364/AO.42.000218 APOPAI 0003-6935 Google Scholar
D. Schmidt et al.,
“Optical wavefront differentiation: wavefront sensing for solar adaptive optics based on a LCD,”
Proc. SPIE, 6584 658408
(2007). https://doi.org/10.1117/12.722640 PSISDG 0277-786X Google Scholar
H. Furuhashi et al.,
“Phase measurement of optical wavefront by an SLM differentiation filter,”
(2009). Google Scholar
J. A. Davis et al.,
“Image processing with the radial Hilbert transform: theory and experiments,”
Opt. Lett., 25
(2), 99
–101
(2000). https://doi.org/10.1364/OL.25.000099 OPLEDP 0146-9592 Google Scholar
S. Fürhapter et al.,
“Spiral phase contrast imaging in microscopy,”
Opt. Express, 13
(3), 689
–694
(2005). https://doi.org/10.1364/OPEX.13.000689 OPEXFF 1094-4087 Google Scholar
A. Jesacher et al.,
“Shadow effects in spiral phase contrast microscopy,”
Phys. Rev. Lett., 94
(23), 233902
(2005). https://doi.org/10.1103/PhysRevLett.94.233902 PRLTAO 0031-9007 Google Scholar
X. Qiu et al.,
“Spiral phase contrast imaging in nonlinear optics: seeing phase objects using invisible illumination,”
Optica, 5
(2), 208
–212
(2018). https://doi.org/10.1364/OPTICA.5.000208 Google Scholar
Y. Park, C. Depeursinge and G. Popescu,
“Quantitative phase imaging in biomedicine,”
Nat. Photonics, 12
(10), 578
–589
(2018). https://doi.org/10.1038/s41566-018-0253-x NPAHBY 1749-4885 Google Scholar
T. Ikeda et al.,
“Hilbert phase microscopy for investigating fast dynamics in transparent systems,”
Opt. Lett., 30
(10), 1165
–1167
(2005). https://doi.org/10.1364/OL.30.001165 OPLEDP 0146-9592 Google Scholar
G. Popescu et al.,
“Diffraction phase microscopy for quantifying cell structure and dynamics,”
Opt. Lett., 31
(6), 775
–777
(2006). https://doi.org/10.1364/OL.31.000775 OPLEDP 0146-9592 Google Scholar
T. J. McIntyre et al.,
“Differential interference contrast imaging using a spatial light modulator,”
Opt. Lett., 34
(19), 2988
–2990
(2009). https://doi.org/10.1364/OL.34.002988 OPLEDP 0146-9592 Google Scholar
C. Zheng et al.,
“Digital micromirror device-based common-path quantitative phase imaging,”
Opt. Lett., 42
(7), 1448
–1451
(2017). https://doi.org/10.1364/OL.42.001448 OPLEDP 0146-9592 Google Scholar
P. Ferraro et al.,
“Quantitative phase-contrast microscopy by a lateral shear approach to digital holographic image reconstruction,”
Opt. Lett., 31
(10), 1405
–1407
(2006). https://doi.org/10.1364/OL.31.001405 OPLEDP 0146-9592 Google Scholar
C. J. Mann et al.,
“High-resolution quantitative phase-contrast microscopy by digital holography,”
Opt. Express, 13
(22), 8693
–8698
(2005). https://doi.org/10.1364/OPEX.13.008693 OPEXFF 1094-4087 Google Scholar
A. Barty et al.,
“Quantitative optical phase microscopy,”
Opt. Lett., 23
(11), 817
–819
(1998). https://doi.org/10.1364/OL.23.000817 OPLEDP 0146-9592 Google Scholar
H. M. L. Faulkner and J. Rodenburg,
“Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,”
Phys. Rev. Lett., 93
(2), 023903
(2004). https://doi.org/10.1103/PhysRevLett.93.023903 PRLTAO 0031-9007 Google Scholar
S. S. Kou et al.,
“Transport-of-intensity approach to differential interference contrast (TI-DIC) microscopy for quantitative phase imaging,”
Opt. Lett., 35
(3), 447
–449
(2010). https://doi.org/10.1364/OL.35.000447 OPLEDP 0146-9592 Google Scholar
C. Zuo et al.,
“Noninterferometric single-shot quantitative phase microscopy,”
Opt. Lett., 38
(18), 3538
–3541
(2013). https://doi.org/10.1364/OL.38.003538 OPLEDP 0146-9592 Google Scholar
A. Silva et al.,
“Performing mathematical operations with metamaterials,”
Science, 343
(6167), 160
–163
(2014). https://doi.org/10.1126/science.1242818 SCIEAS 0036-8075 Google Scholar
A. Pors, M. G. Nielsen and S. I. Bozhevolnyi,
“Analog computing using reflective plasmonic metasurfaces,”
Nano Lett., 15
(1), 791
–797
(2014). https://doi.org/10.1021/nl5047297 NALEFD 1530-6984 Google Scholar
L. L. Doskolovich et al.,
“Spatial differentiation of optical beams using phase-shifted Bragg grating,”
Opt. Lett., 39
(5), 1278
–1281
(2014). https://doi.org/10.1364/OL.39.001278 OPLEDP 0146-9592 Google Scholar
Z. Ruan,
“Spatial mode control of surface plasmon polariton excitation with gain medium: from spatial differentiator to integrator,”
Opt. Lett., 40
(4), 601
–604
(2015). https://doi.org/10.1364/OL.40.000601 OPLEDP 0146-9592 Google Scholar
S. Abdollahramezani et al.,
“Analog computing using graphene-based metalines,”
Opt. Lett., 40
(22), 5239
–5242
(2015). https://doi.org/10.1364/OL.40.005239 OPLEDP 0146-9592 Google Scholar
A. Chizari et al.,
“Analog optical computing based on a dielectric meta-reflect array,”
Opt. Lett., 41
(15), 3451
–3454
(2016). https://doi.org/10.1364/OL.41.003451 OPLEDP 0146-9592 Google Scholar
A. Youssefi et al.,
“Analog computing by Brewster effect,”
Opt. Lett., 41
(15), 3467
–3470
(2016). https://doi.org/10.1364/OL.41.003467 OPLEDP 0146-9592 Google Scholar
Y. Hwang and T. J. Davis,
“Optical metasurfaces for subwavelength difference operations,”
Appl. Phys. Lett., 109
(18), 181101
(2016). https://doi.org/10.1063/1.4966666 APPLAB 0003-6951 Google Scholar
Y. Fang, Y. Lou and Z. Ruan,
“On-grating graphene surface plasmons enabling spatial differentiation in the terahertz region,”
Opt. Lett., 42
(19), 3840
–3843
(2017). https://doi.org/10.1364/OL.42.003840 OPLEDP 0146-9592 Google Scholar
W. Wu et al.,
“Multilayered analog optical differentiating device: performance analysis on structural parameters,”
Opt. Lett., 42
(24), 5270
–5273
(2017). https://doi.org/10.1364/OL.42.005270 OPLEDP 0146-9592 Google Scholar
Y. Hwang et al.,
“Plasmonic circuit for second-order spatial differentiation at the subwavelength scale,”
Opt. Express, 26
(6), 7368
–7375
(2018). https://doi.org/10.1364/OE.26.007368 OPEXFF 1094-4087 Google Scholar
T. Zhu et al.,
“Plasmonic computing of spatial differentiation,”
Nat. Commun., 8 15391
(2017). https://doi.org/10.1038/ncomms15391 NCAOBW 2041-1723 Google Scholar
J. Zhang, Q. Ying and Z. Ruan,
“Time response of plasmonic spatial differentiators,”
Opt. Lett., 44
(18), 4511
–4514
(2018). https://doi.org/10.1364/OL.44.004511 Google Scholar
Y. Fang and Z. Ruan,
“Optical spatial differentiator for a synthetic three-dimensional optical field,”
Opt. Lett., 43
(23), 5893
–5896
(2018). https://doi.org/10.1364/OL.43.005893 OPLEDP 0146-9592 Google Scholar
A. Saba et al.,
“Two dimensional edge detection by guided mode resonant metasurface,”
IEEE Photonics Technol. Lett., 30 853
–856
(2018). https://doi.org/10.1109/LPT.2018.2820045 IPTLEL 1041-1135 Google Scholar
Z. Dong et al.,
“Optical spatial differentiator based on subwavelength high-contrast gratings,”
Appl. Phys. Lett., 112
(18), 181102
(2018). https://doi.org/10.1063/1.5026309 APPLAB 0003-6951 Google Scholar
A. Roberts, D. E. Gómez and T. J. Davis,
“Optical image processing with metasurface dark modes,”
J. Opt. Soc. Am. A, 35 1575
–1584
(2018). https://doi.org/10.1364/JOSAA.35.001575 JOAOD6 0740-3232 Google Scholar
C. Guo et al.,
“Photonic crystal slab Laplace operator for image differentiation,”
Optica, 5
(3), 251
–256
(2018). https://doi.org/10.1364/OPTICA.5.000251 Google Scholar
H. Kwon et al.,
“Nonlocal metasurfaces for optical signal processing,”
Phys. Rev. Lett., 121 173004
(2018). https://doi.org/10.1103/PhysRevLett.121.173004 PRLTAO 0031-9007 Google Scholar
T. Zhu et al.,
“Generalized spatial differentiation from the spin hall effect of light and its application in image processing of edge detection,”
Phys. Rev. Appl., 11 034043
(2019). https://doi.org/10.1103/PhysRevApplied.11.034043 PRAHB2 2331-7019 Google Scholar
J. Zhou et al.,
“Optical edge detection based on high-efficiency dielectric metasurface,”
Proc. Natl. Acad. Sci. U. S. A., 116
(23), 11137
–11140
(2019). https://doi.org/10.1073/pnas.1820636116 PNASA6 0027-8424 Google Scholar
C. Guo et al.,
“Isotropic wavevector domain image filters by a photonic crystal slab device,”
J. Opt. Soc. Am. A, 35 1685
–1691
(2018). https://doi.org/10.1364/JOSAA.35.001685 JOAOD6 0740-3232 Google Scholar
A. Momeni et al.,
“Generalized optical signal processing based on multioperator metasurfaces synthesized by susceptibility tensors,”
Phys. Rev. Appl., 11 064042
(2019). https://doi.org/10.1103/PhysRevApplied.11.064042 PRAHB2 2331-7019 Google Scholar
T. J. Davis et al.,
“Metasurfaces with asymmetric optical transfer functions for optical signal processing,”
Phys. Rev. Lett., 123 013901
(2019). https://doi.org/10.1103/PhysRevLett.123.013901 PRLTAO 0031-9007 Google Scholar
L. A. Alemán-Castaneda et al.,
“Shearing interferometry via geometric phase,”
Optica, 6 396
–399
(2019). https://doi.org/10.1364/OPTICA.6.000396 Google Scholar
J. W. Ra, H. Bertoni and L. Felsen,
“Reflection and transmission of beams at a dielectric interface,”
SIAM J. Appl. Math., 24
(3), 396
–413
(1973). https://doi.org/10.1137/0124041 SMJMAP 0036-1399 Google Scholar
C. C. Chan and T. Tamir,
“Angular shift of a Gaussian beam reflected near the Brewster angle,”
Opt. Lett., 10
(8), 378
–380
(1985). https://doi.org/10.1364/OL.10.000378 OPLEDP 0146-9592 Google Scholar
M. Merano et al.,
“Observing angular deviations in the specular reflection of a light beam,”
Nat. Photonics, 3
(6), 337
–340
(2009). https://doi.org/10.1038/nphoton.2009.75 NPAHBY 1749-4885 Google Scholar
F. I. Fedorov,
“K teorii polnogo otrazheniya,”
Dokl. Akad. Nauk SSSR, 105
(3), 465
–468
(1955). Google Scholar
C. Imbert,
“Calculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized light beam,”
Phys. Rev. D, 5 787
–796
(1972). https://doi.org/10.1103/PhysRevD.5.787 Google Scholar
O. Hosten and P. Kwiat,
“Observation of the spin Hall effect of light via weak measurements,”
Science, 319
(5864), 787
–790
(2008). https://doi.org/10.1126/science.1152697 SCIEAS 0036-8075 Google Scholar
N. T. Shaked, Z. Zalevsky and L. L. Satterwhite, Biomedical Optical Phase Microscopy and Nanoscopy, Academic Press, Elsevier
(2012). Google Scholar
M. R. Arnison et al.,
“Linear phase imaging using differential interference contrast microscopy,”
J. Microsc., 214
(1), 7
–12
(2004). https://doi.org/10.1111/j.0022-2720.2004.01293.x JMICAR 0022-2720 Google Scholar
P. Karimi, A. Khavasi and S. S. M. Khaleghi,
“Fundamental limit for gain and resolution in analog optical edge detection,”
Opt. Express, 28
(2), 898
–911
(2020). https://doi.org/10.1364/OE.379492 OPEXFF 1094-4087 Google Scholar
M. W. Davidson,
“mCerulean fused to the tyrosine kinase C-Src,”
http://zeiss-campus.magnet.fsu.edu/galleries/static/fpfusions/csrc.html Google Scholar
V. A. Belyakov and V. E. Dmitrienko,
“Polarization phenomena in x-ray optics,”
Sov. Phys. Usp., 32
(8), 697
–719
(1989). https://doi.org/10.1070/PU1989v032n08ABEH002748 SOPUAP 0038-5670 Google Scholar
M. Scheinfein et al.,
“Scanning electron microscopy with polarization analysis (SEMPA),”
Rev. Sci. Instrum., 61
(10), 2501
–2527
(1990). https://doi.org/10.1063/1.1141908 RSINAK 0034-6748 Google Scholar
BiographyZhichao Ruan is a professor at the Department of Physics and professor by courtesy at the College of Optical Science and Engineering, Zhejiang University. He received his PhD in applied physics from the Royal Institute of Technology (KTH), Sweden, in 2007. He has been a research associate at Ginzton Laboratory, Stanford University, and a senior engineer at KLA-Tencor Corp. His research interests include surface plasmon, metamaterial, and nanostructure applications in optical analogy computing, solar cell, and transparent electrode. |