Full-Dimensional Surface Characterization Based on Polarized Coherent Coded Aperture Correlation Holography

The coded aperture correlation holography (COACH) is a powerful three-dimensional imaging technique. However, the conventional COACH method can only restore the three-dimensional images of the samples under test, but cannot specify the physical characteristics. We propose a full-dimensional measurement method based on the polarized coded aperture correlation holography. Specifically, the dynamic phase and geometric phase distributions can be separated from two holograms associated with the horizontal and vertical polarization components, then the Stokes parameters, surface topography and reflectivity can be calculated simultaneously. This method achieves the same spatial resolution for the two terms, which greatly improves the reliability and practicability with respect to conventional measurement methods.

high integration and superior performance. At present, they are widely employed as meta-surfaces, sensors, MEMS devices etc in the fields of IC industry, aerospace and defense. In general, the non-destructive and accurate measurement of the surface topographies and physical properties is an urgent demand of promoting micro/nano-fabrication technologies and guaranteeing the functionalities of high-performance devices [1], [2]. Traditional probe-scanning instruments such as scanning tunneling microscopes and atomic force microscopes [3], and optical instruments such as confocal microscopes [4] and coherence-scanning interferometers [5] are gradually becoming inadequate for practical applications. The former would scratch the surfaces, and its measurement efficiency is very low. While the latter is limited in the measurement range and universality, and the accuracies in the lateral and vertical directions do not match each other. In addition, the beam reflected from the sample under test is usually disturbed and distorted, yielding measurement results of poor quality. However, digital holography [6], [7], [8], [9] can address the distortions of the object wave-fronts and it has the characteristics of noncontact, high-resolution and highly sensitive. Currently, digital holography has been applied to metrology [10], inspection [11], biomedical imaging [12], to name just a few application areas.
In digital holography field, an efficient, rapid, and wellestablished method is coded aperture correlation holography (COACH) [13]. In a COACH system, the emitted light from an object splits into two beams. One is modulated by chaotic coded phase mask (CPM), and the other propagates without modulation. The two mutually coherent beams interfere on the camera plane, where the recorded hologram is obtained as the accumulated interference patterns contributed from the entire object. Another hologram of a point object is recorded with the same CPM as before. The second hologram is used as the point spread function (PSF) in the digital reconstruction stage. Accordingly, the image is digitally reconstructed by correlating the object hologram with the point spread function. Originally, COACH was invented as an incoherent, self-interference digital holography technology and tremendously simplified optical holographic configurations to in-line and single-channel setups. It has the same lateral and axial resolutions as a regular imaging system, in addition to the capability of imaging 3D scenes. This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ Recently, to improve the capability to measure the topographies and achieve quantitative phase non-destructive imaging, Hai et al. modified the COACH system into a multimodal device that can operate under coherent as well as incoherent illuminations [14]. Consequently, the coherent holograms are empowered with an additional quantitative phase measurement capability. In this case, COACH is integrated with a standard Mach-Zehnder interferometer to record the phase distribution of the observed objects [15], [16], [17]. Specifically, it records off-axis hologram and removes twin image and the bias term in the spatial frequency spectrum. However, it should be noted that the off-axis optical configuration will result in low-resolution shadowing effect, which leads to a risk of losing some critical features or small features. Therefore, it is not suited to the measurement of micro/nano-structures.
The aforementioned methods only measure the surface topographies without considering the physical characteristics of the specimens. To fully specify the opto-electronics performances of micro/nano-devices, it is necessary to completely modulate and analyze the reflected/transmitted beam, including its amplitude, phase, and polarization state [18]. Usually, the polarization state is identified according to the intensity variations in different polarization directions via rotating polarizing elements, then the Stokes vector and Jones vector can be calculated [19], [20]. Considering that in actual applications, real-time measurement cannot be achieved because the polarizing elements need to be rotated multiple times. Azzam et al realized simultaneous measurement of all the four Stokes parameters using a photopolarimeter [21], [22]. However, these systems are complicated and require precise image matching, which inevitably introduce more systematic and numerical errors. Then, Coppola et al realized polarization imaging with polarized digital holography [23], [24]. It achieves high-speed polarization modulation by utilizing an optical rotatory device. However, the polarization states and surface topographies of the samples were measured separately, implying that the spatial resolutions of the two are not consistent, which affects the reliability of the characterization results.
In this present study, we aim to achieves simultaneous measurement for the physical properties and surface topographies of the micro/nano-structure. Therefore, a full-dimensional measurement method based on the polarized coded aperture correlation holography is proposed. The rest of the paper is organized as follows. The methodology of the proposed method is introduced in Section II. The effectiveness is experimentally verified in Section III, and finally, the paper is summarized in Section IV.

A. Coherently Illuminated COACH
In the COACH imaging, a hologram H OBJ associated with an object O(r) is recorded by a digital camera where * denotes convolution,r = (x, y) is the abscissa coordinate, A(r) is the complex amplitude satisfying the relation is the PSF of the optical system and |t(r)| 2 is recorded as the point spread hologram H PSF . Then, the object can be reconstructed by the cross-correlation between H OBJ and H PSF as follows, where ⊗ denotes 2D the cross-correlation. Contrarily, in a coherent imaging system, the linear convolution relationship holds true for the complex amplitude distribution, rather than the intensity distribution described in (1). Following [15], the COACH technology can be extended to a coherent imaging system by utilizing coherent light for illumination. Consequently, (1) can be rewritten as, where δ(·) denotes the Dirac function, k is the index of the dots and a k is a complex valued constant. k a k δ(r −r k ) and k |a k | 2 δ(r −r k ) are the expressions of PSF and H PSF in a coherent imaging system, respectively. To satisfy (3), a pseudo-randomly CPM generated by the Gerchberg-Saxton (GS) algorithm is used to modulate the optical field. There exists a Fourier relation between the CPM and the camera plane, and the image turns out to be multiple replications of the object. The number and distribution of the dots of the CPM directly determine those of the image replications. The distance between any two adjacent replications should be sufficiently large, so that no overlapping occurs between different image replications.
To specify the phase distribution of the sample under test, a standard Mach-Zehnder interferometer is integrated with the coded aperture imaging system, as illustrated in Fig. 1. While, phase shifting is demanded to separate the unwanted zero-order and twin image components. If the phase difference between the object wave and the reference wave is denoted as ϕ OR = ϕ O − ϕ R the resulting interferogram is, where A R is the complex amplitude of the reference beam, φ denotes the phase shift, namely 0, π/2, π and 3π/2 in practice.
After the unwanted zero-order and bias terms are eliminated by the four-step phase shifting, the image replications of the input object wave-front are retained. This term can be crosscorrelated with the corresponding PSF to reconstruct the object image according of (2) as follows, where (·) stands for the 2D Fourier transform. Then we have where h and ϕ denote the amplitude and phase of the Fourier transform of the PSF. The last approximation of (5) is valid under the assumption that the function −1 {h 2 } is a sharply peaked function, which is true for randomly distributed dots [17].

B. Characterization of Polarization States Based on Geometrical Phases
The geometric phase was first revealed by Pancharatnam [25]. Generally, the geometric phase can be used to indicate two characteristics of a beam, one is the change of the polarization state, and the other is the change of the beam propagation mode structure. We focus on the first one in this paper. To describe the polarization state of an optical field, Poincare proposed the Poincare sphere presentation method [26] based on the Stokes parameters, which can visually represent any polarization state. Fig. 3 presents a spherical coordinate system, where the horizontal and vertical linear polarization states (H, V) are separately designated as two poles. The azimuthal and polar angles are presented as 2ψ and 2χ, respectively. Since the Poincare sphere can conveniently describe the geometric phase caused by the change of the polarization state of the optical field. Therefore, we can represent the polarization information by the Poincare sphere and then establish a quantitative relationship between the geometric phase and the polarization state variations.
Assuming that the polarization state of the incident light is not identical with that of the output light passing through the optical system in Fig. 1. When two beams interfere with each other, in addition to the dynamic phase introduced by the optical path difference, there exists a geometric phase as well [27]. In this case, the conventional digital holography cannot distinguish them, and two holograms associated with different polarization directions are required. Fig. 2 shows the system configuration, which consists of a modified Mach-Zehnder interferometer with two reference beams. Specifically, a laser beam is expanded and collimated by a beam expander BE. After that, the collimated light is polarized by a polarizer P to an orientation of 45°. BS 1 splits the beam into a reference and an object arm. In the object arm, the sample under test is imaged to infinity by a microscopic objective and relayed toward the SLM by an optical relay consisting of L 1 and L 2 . The object wave-front is modulated by the CPM displayed on the SLM, and its polarization state needs to be specified. The reference beam is split into a horizontally polarized beam and a vertically polarized beam by a polarizing beam-splitter PBS 1 . Then the two beams are recombined by PBS 2 . Finally, the image sensor records the horizontal hologram and vertical hologram, respectively. Adding different phase shifts φ to the CPM, the two polarization components of the complex amplitude of the object, namely E H and E V , are obtained using the four-step phase shifting method.
According to Pancharatnam's theory [25], the two orthogonal components of the complex amplitude of the object can be expressed as, where OH denotes the central angle associated with the arc OH. Then the polar angle can be evaluated as Additionally, the dynamic phase and geometric phase can be written as where ϕ H and ϕ V denote the phase differences between the object beam and reference beam associated with the horizontal and vertical directions, respectively. ϕ Hd and ϕ Hg stand for the dynamic and geometric phases of the horizontal component, and ϕ V d and ϕ V g are those of the vertical component. The geometric phase of a polarization state H is half of the area of the geodesic triangle ROH, i.e., ϕ Hg = Ω/2. Similarly, we have ϕ V g = −2ψ + Ω/2. Thus, the azimuthal angle can be calculated by Given the two groups of complex amplitude distributions, the phases ϕ H and ϕ V can be obtained and then the former bracket on the right side of (10) can be easily calculated. While the value in the latter bracket is associated with the dynamic differences of the two groups of interference patterns. It is fixed for a particular optical system. Since the two reference waves are adjusted to have the same intensity, the azimuthal and polar angles can be estimated by the following relationships, Thus, the area Ω can be evaluated as a function of the angles 2ψ and 2χ, and the phases ϕ Hd and ϕ V d are calculated subsequently. As depicted in Fig. 3, the normalized Stokes parameters can be calculated by the azimuthal and polar angles. Therefore, the polarization states of the object beam can be described by the Stokes vector as follows, ⎧ ⎨ ⎩ S 1 = sin 2χ S 2 = cos 2χ cos 2ϕ where S 1 is the difference between the horizontal and vertical polarization components, S 2 is the difference between the polarization components in the direction of 45°and 135°, and S 3 is the difference between the right-handed and left-handed circular polarization components. Henceforth the Mueller matrix can be calculated from the Stokes parameters of the incident and reflected light, enabling the measurement of physical characteristics such as the dielectric constant and refractive index [28].

III. EXPERIMENTAL RESULTS AND DISCUSSIONS
To quantitatively verify the proposed method, a measurement system is established, as presented in Fig. 4. A He-Ne laser with a maximum output power of 41 mW and λ = 632.8 nm is adopted. The microscopic objective is OLYMPUS MPLFLN 10X with a numerical aperture of 0.25. The used camera is Manta G-419 with a resolution of 2048 × 2048 pixels, and the SLM is HOLOEYE PLUTO with 1920 × 1080 pixels. In addition, a lens L 1 with a focal length of 30 mm, and two lenses L 2 and L 3 with a focal length of 125 mm are applied, all of which have an aperture size of 25.4 mm. Several CPMs yielding different numbers of dots are tested to record H PSF , with a dot diameter of 30 μm.

A. Determine the Optimal Number of Dots
The optimal number of dots is determined first. We simply block the reference arm of the optical system and place a USAF 1951 resolution chart at the object plane. The light modulated by different CPMs is used to generate H PSF composed of different numbers of dots. According to the actual system configuration, we choose H PSF containing 8 dots, 15 dots, and 20 dots, respectively. Then, a part of the resolution chart is utilized as a target object. Fig. 5(a)-(c) illustrates the recorded holograms, in which case a special H PSF is designed so that the resulting H OBJ contains multiple duplicated images of the target object, indicated by the red solid-line circles. It can be found that the more dots contained, the lower the intensity is allocated to each dot, resulting in a darker hologram. In addition, the yellow circle indicates the signal from the unmodulated part of the CPM.  Finally, the image of the object is numerically reconstructed by cross-correlation between the object hologram H OBJ with H PSF . Fig. 5(d)-(f) presents their corresponding reconstruction images.
To conduct a reliable comparison, we evaluate the quality of each image in terms of Signal to Noise Ratio (SNR) and Structure Similarity Index Measure (SSIM), as presented in Table I. It can be found that the number of duplicated images on the camera plane has significant implications on the reconstructed image quality. Too many replications of the target object reduce the SNR level in the reconstructed images. At the same time, an H PSF consisting of fewer dots increases the out-of-focus image intensity and blurs the reconstructed image. The target object is better reconstructed when H PSF contains 15 dots. Fig. 6(a) and (b) show the original target object and the reconstructed image in the optimal condition, respectively. It can be seen from the partial views that the intensity of the target object can be reconstructed using the proposed method. Additionally, since the  autocorrelation of the point spread function is not sharp enough, the inherent background noise remains in the reconstruction results.

B. Measurement of Surface Topography
The experimental setup in Fig. 2 is slightly modified to demonstrate the performances of the proposed method in quantitative phase imaging. To this end, we place a reflective MEMS sample at the front focal plane of the microscopic objective. The dimension of the reflective object is different with that of the previously tested resolution chart. Therefore, an iris diaphragm is used to adjust the illuminated area of the reflective object to the same size as in the previous experiment. The surface topography of the MEMS sample is measured. Fig. 7(a) depicts the recorded hologram using the proposed method, which adopts a H PSF containing 15 randomly distributed dots. The magnified detail indicated with the red frame is presented at the right corner, where interference fringes can be observed. To restore the phase map of the object, a cross-correlation is calculated for the object wave-front with a corresponding H PSF . Fig. 7(b) depicts the reconstructed image, and Fig. 7(c) describes the reconstructed topography using the proposed method. Additionally, the measured result using a confocal microscope (CM) LEXT OSL 4000 is also provided for comparison. The measurement areas of the two methods are adjusted to be the same, and the comparison of the cross-sections are shown in Fig, 7(d). The reconstructed surface topography of the proposed method shows higher uniformity in comparison to the CM. The average step height measured by CM is about 257.3 nm and the step width are about 21.06 nm. While the step height and width of the proposed method are 272 nm and 23.03 nm. The two cross-sections are approximately coincident, which proves the reliability of the proposed method.

C. Specification of Stokes Parameters
Finally, the vertical and horizontal holograms of the sample are recorded, respectively. The geometric phase distribution of the sample can be easily obtained based on the dynamic phase distribution of the sample, as shown in Fig. 7. Therefore, the polarization state of the MEMS sample can be characterized. Fig. 8(a)-(c) shows the calculation results of the Stokes parameters. It is found that the Stokes parameters are different in the feature region and in the background region, which indicates that the physical properties of different areas are different. The polarization state of the incident light has been specified, and the corresponding normalized Stokes parameters are calculated. Therefore, the Jones matrix can be calculated from the Stokes parameters for the incident and reflected beams. According to the method proposed in [28], the average refractive indexes of the feature region and the background region are worked out as 1.681 and 1.513, respectively. However, there is a large amount of noise in both the measured surface topography and the Stokes parameters. This is mainly due to two reasons. First, the crosscorrelation algorithm is adopted for reconstruction, which requires the autocorrelation of H PSF to be small enough. However, this condition cannot be sufficiently satisfied. Second, a coherent laser is adopted for illumination, which introduces speckle noise. However, from the perspective of feature regions on the sample, the material characteristics of the object are revealed. It can identify the changes in material composition/features caused by doping, etching, annealing, phase transformation, and other processes and phenomena in micro/nano-fabrication. Moreover, combined with geometric topographies, it can more comprehensively characterize the materials.

IV. SUMMARY
In the proposed system, the polarized COACH is proposed for the full-dimensional surface characterization of micro/nanostructures. This method records the complex amplitudes of two polarization components to separate the geometric phase and dynamic phase distributions. An in-line configuration is adopted to obtain a higher spatial bandwidth. A random and sparse PSF is designed to improve the SNR of the recorded hologram. Additionally, the Stokes parameters of the sample can be calculated by establishing the quantitative relationship between the geometric phase and the polarization states. Therefore, the proposed method can measure the three-dimensional topography, reflectivity, and Stokes parameters at the same time, and with the same spatial resolution. Due to the high coherence of the laser beam and the limitations of the correlation reconstruction algorithm, the obtained Stokes parameters contain noise. Improving the stability and reliability of this method and comprehensive characterization of the optoelectronics functionalities of micro/nano structured devices will be investigated in the future.