Non-invasive surface profile measurement using a unitary transformation subspace approach in digital holography

Jagadesh Ramaiah; Jagadesh Ramaiah; Ankur Vishnoi; Ankur Vishnoi; Rajshekhar Gannavarpu; Rajshekhar Gannavarpu

doi:10.1364/OPTCON.451357

1. Introduction

Measurement of nanoscale surface profile is an important problem in applications like material characterization [1], defect inspection [2], precision engineering [3], and semiconductor wafer monitoring [4]. For surface profile measurement, some of the prominent techniques include the probe based instruments such as atomic force microscope (AFM) [5–7], scanning electron microscope (SEM) [8–10] and stylus profilometer [11–13], and optical interferometric techniques such as scanning white light interferometer [14–16], phase-shifting interferometer [17,18], off-axis low coherence interferometers [19–21], Fizeau interferometer [22] and diffraction phase microscope [23–25]. Due to their invasive nature and low throughput ascribed to point or line scanning operations, the probe based techniques are difficult to apply for interrogating large surface areas and fast non-contact metrology. On the other hand optical interferometric techniques offer the flexibility of non-invasive and high throughput measurements. However, techniques such as scanning white light interferometer and phase-shifting interferometer require multiple scans and phase-shifted frames, and thus, are not practically suitable for dynamic measurements. Diffraction phase microscope uses a common-path interferometric setup for imaging, which imparts robustness against external disturbances; though the technique requires careful spatial filtering using optical elements such as pinhole or spatial light modulator [24]. Recently, digital holographic microscopy (DHM) [26,27] has emerged as a prominent technique in non-invasive surface metrology. Digital holography relies on the interference of a coherent reference wave and wave scattered from the test specimen to form a hologram. The hologram is then digitally processed using numerical reconstruction procedure to obtain the complex object wavefield [28]. The technique offers several merits such as full-field measurement, simple operation, feasibility of digital recording and processing of holograms and decent throughput. The main quantity of interest in digital holographic microscopy for profilometry applications is the phase of the complex object wavefield since it directly provides information about the surface profile of the test specimen. Hence, several methods have been proposed for phase retrieval [29] in digital holography such as phase-shifting [30–32], wavelet transform (WT) [33,34], windowed Fourier transform [35–37], state-space methods [38–40], polynomial modeling methods [41,42] and machine learning approaches [43–45]. For most of these methods, detrimental factors such as noise and non-uniform intensity fluctuations constitute major impediments for phase retrieval. These intensity fluctuations can be caused by factors such as irregular illumination, non-uniform surface reflection from the test sample and image abnormalities such as corrupted pixels. In this work, we aim to address these challenges by applying a unitary transformation based parameter estimation method for non-contact surface profile measurement. The unitary transformation approach relies on signal subspace processing [46] and has been utilized for diverse applications in wireless communication [47], deformation testing [48], radar range estimation [49] and adaptive signal processing [50]. For our study, we investigate the method in the context of digital holographic microscopy and demonstrate its practical applicability for reliable estimation of surface profiles of multiple micro-structures on a standard calibration target. To the best of our knowledge, the proposed unitary transformation approach capable of handling the challenges of noise and non-uniform amplitude fluctuations has not been hitherto applied for nanoscale surface profile measurement in digital holographic microscopy. The paper is organized as follows. The details about the digital holographic microscopy experimental setup for surface profile measurement and the associated phase estimation method are given in section 2. The simulation and experimental results are outlined in section 3, followed by discussions and conclusions.

2. Experimental setup and signal processing

The schematic diagram of the digital holographic microscopy setup in an epi-illumination configuration is shown in Fig. 1. A laser (Coherent Sapphire LP) is used as the light source (LS) in the experiment. The beam originating at LS is expanded using a beam expander (BE) and further split into two beams using a beam splitter (BS1), as shown in the figure. The beam reflected from the mirror (M1) is used to illuminate the test sample, whereas the beam reflected from the mirror (M2) is used as a reference beam. The beam scattered from the test sample is magnified using a microscope objective (MO) and subsequently interferes with the reference beam using a combining beam splitter (BS2). An off-axis configuration is realized by introducing a small tilt angle between the reference and the magnified object beam. The interference pattern formed by the two beams, or the hologram, is captured using a CMOS camera (Allied Vision: Mako U-503B). In addition, a reference hologram is also recorded for aberration correction [51] by illuminating only the plain or background surface of the test sample.

Fig. 1. Schematic depiction of the experiment setup.

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Each hologram is numerically reconstructed using the angular spectrum approach [52] in order to obtain a digital representation of the complex object wavefield. The parameters used for reconstruction include wavelength of 532 nm, reconstruction distance of 0.1 mm, image size of 2592 $\times$ 1944 pixels and the pixel size of 2.2 $\mu m$ $\times$ 2.2 $\mu m$. Aberration due to microscopic objective was removed using the reference conjugation approach [51] where we multiply the conjugate of the reconstructed object wavefield signal from reference hologram with the reconstructed object wavefield signal from hologram corresponding to the test sample with surface features. Other aberrations due to external disturbances and vibrations were removed using a background line-fitting method [53]. Next, our goal is to extract the phase information encoded in the reconstructed object wavefield signal and accordingly, we formulate the signal processing approach as outlined below.

Mathematically, the reconstructed complex object wavefield signal can be represented in the following form,

(1)$$\boldsymbol{\Gamma}_o(m,n) = A_o(m,n) e^{j\phi_o(m,n)} + \eta_o(m,n)$$

where $\phi _o$ is the phase, $A_o$ is the amplitude, $\eta _o$ is the additive white Gaussian noise (AWGN) and $(m,n)$ are indices along row and column representing the location of a pixel. In our approach, a small window is selected around each pixel such that the phase inside the window can be modeled as a linear function of the following form,

(2)$$\phi(p,q) = \alpha_0 + \alpha_pp + \alpha_qq$$

where $p,q \in [-L,\ldots,L-1,L]$. Using the above model, the signal data inside a window can be expressed using the following equation,

(3)$$\boldsymbol{\Gamma}_w(p,q) = A e^{j\phi(p,q)} + \eta(p,q).$$

Note that the amplitude term $A$ is assumed to behave as a constant within the given window. Also, the AWGN noise term in the window is represented as $\eta$ with variance denoted by $\sigma ^2$. It is evident that the phase measurement problem is now modeled as a parameter estimation problem where the linear coefficients are the desired parameters.

To elucidate the theory of the proposed method, we initially ignore the noise term in our analysis for the sake of mathematical simplicity. The proposed model is generalized later to include the effect of noise. Thus, the above equation can be written using vector notation of the following form,

(4)$$\boldsymbol{\Gamma'} = Ae^{j\alpha_0}\boldsymbol{\Gamma}$$

where

(5)$$\boldsymbol{\Gamma} = \textbf{a}_M(\alpha_q) \textbf{a}^T_M(\alpha_p)$$

and "$(\cdot )^T$" indicates the transpose operation, the parameter $M=2L+1$ denotes the vector size and

\begin{array}{r}\textbf{a}_M(x) = \left[ e^{{-}j \left(\frac{M-1}{2}\right) x} ,\ \cdots ,\ e^{{-}jx} ,\ 1 ,\ e^{jx} ,\right.\\ \left. \cdots ,\ e^{j \left(\frac{M-1}{2}\right) x} \right]^T \end{array}.

In the above expression, considering the center element of $\boldsymbol {\Gamma }$ as the reference, it is clear that the matrix is conjugate centrosymmetric (or centro-Hermitian), which is mathematically stated as $\boldsymbol {\Pi }_M\boldsymbol {\Gamma }\boldsymbol {\Pi }_M = \boldsymbol {\Gamma }^*$ with $(.)^*$ denoting the complex conjugate operation and $\boldsymbol {\Pi }_M$ representing an exchange matrix of size $M\times M$ having ones in anti-diagonal elements and zeros everywhere else. Based on this property, the complex matrix $\boldsymbol {\Gamma }$ can be transformed into a real matrix by multiplying it with another centro-Hermitian matrix, which is defined as follows [46]:

(6)$$\textbf{K}_{M} = \frac{1}{\sqrt{2}} \begin{bmatrix} \textbf{I}_{M/2} & j\textbf{I}_{M/2} \\ \boldsymbol{\Pi}_{M/2} & -j\boldsymbol{\Pi}_{M/2} \end{bmatrix}$$

if $M$ is even, and

(7)$$\textbf{K}_{M} = \frac{1}{\sqrt{2}} \begin{bmatrix} \textbf{I}_{(M-1)/2} & \textbf{0}_{(M-1)/2} & j\textbf{I}_{(M-1)/2} \\ \textbf{0}^T_{(M-1)/2} & \sqrt{2} & \textbf{0}^T_{(M-1)/2} \\ \boldsymbol{\Pi}_{(M-1)/2} & \textbf{0}_{(M-1)/2} & -j\boldsymbol{\Pi}_{(M-1)/2} \end{bmatrix}$$

if $M$ is odd. In the above equations, the notation $\textbf {0}_k$ represents a column zero vector of size $k$ and $\textbf {I}_k$ represents an identity matrix of size $k$. As the matrix $\textbf {K}_M$ is unitary, we apply unitary transformation of the signal $\boldsymbol {\Gamma }$ using the following operation,

(8)$$\begin{aligned} \textbf{R} &= \textbf{K}_M^H \boldsymbol{\Gamma} \textbf{K}_M^*\\ &= \textbf{d}_M(\alpha_q) \textbf{d}_M^T(\alpha_p) \end{aligned}$$

where "$(\cdot )^H$" indicates Hermitian transpose operation and,

(9)$$\begin{array}{r}\textbf{d}_M(x) = \sqrt{2} \left[cos\left(\frac{M-1}{2}x\right),\ \cdots,\ cos(x), \frac{1}{\sqrt{2}},\right.\\ \left. -sin\left(\frac{M-1}{2}x\right),\ \cdots,\ -sin(x) \right]^T \end{array}.$$

Further, the matrix $\boldsymbol {\Gamma }$ in Eq. (5) also satisfies invariance property which is stated as

(10)$$e^{j\alpha_q}\textbf{J}_1 \boldsymbol{\Gamma} = \textbf{J}_2 \boldsymbol{\Gamma}$$

where $\textbf {J}_1$ and $\textbf {J}_2$ are matrices of size $(M-1)\times M$, given as,

(11)$$\begin{aligned}\textbf{J}_1 = \left[\textbf{I}_{M-1} \quad \textbf{0}_{M-1} \right]\\ \textbf{J}_2 = \left[\textbf{0}_{M-1} \quad \textbf{I}_{M-1} \right] \end{aligned}.$$

Here, the matrices $\textbf {J}_1$ and $\textbf {J}_2$ are used to create sub-arrays by omitting the last and first rows of $\boldsymbol {\Gamma }$ respectively. Since $\textbf {K}_M$ is unitary, we obtain,

(12)$$\begin{aligned}e^{j\alpha_q}\textbf{J}_1 \textbf{K}_M\textbf{K}_M^H \boldsymbol{\Gamma}\textbf{K}_M^* &= \textbf{J}_2 \textbf{K}_M\textbf{K}_M^H \boldsymbol{\Gamma} \textbf{K}_M^*\\ e^{j\alpha_q}\textbf{J}_1 \textbf{K}_M \textbf{R} &= \textbf{J}_2 \textbf{K}_M \textbf{R} \end{aligned}.$$

Pre-multiplying $\textbf {K}_{M-1}^H$ on both sides will result in the invariance relationship as follows,

(13)$$e^{j\alpha_q} \textbf{K}_{M-1}^H \textbf{J}_1 \textbf{K}_M \textbf{R} = \textbf{K}_{M-1}^H \textbf{J}_2 \textbf{K}_M \textbf{R}.$$

Now, using the relations $\boldsymbol {\Pi }_{M-1}\textbf {J}_2\boldsymbol {\Pi }_M=\textbf {J}_1$, $\textbf {K}_{M-1}^H\boldsymbol {\Pi }_{M-1} = \textbf {K}_{M-1}^T$, $\boldsymbol {\Pi }_M\textbf {K}_M=\textbf {K}_M^*$ and $\boldsymbol {\Pi }_M\boldsymbol {\Pi }_M=\textbf {I}_M$, we obtain

(14)$$\begin{aligned}\textbf{K}_{M-1}^H \textbf{J}_2 \textbf{K}_M &= \textbf{K}_{M-1}^H \boldsymbol{\Pi}_{M-1}\boldsymbol{\Pi}_{M-1} \textbf{J}_2 \boldsymbol{\Pi}_M\boldsymbol{\Pi}_M \textbf{K}_M\\ &= \textbf{K}_{M-1}^T \textbf{J}_1 \textbf{K}_M^*\\ &= \left( \textbf{K}_{M-1}^H \textbf{J}_1 \textbf{K}_M \right)^* \end{aligned}.$$

Letting $\textbf {Z}_1$ and $\textbf {Z}_2$ to be the real and imaginary parts of $\textbf {K}_{M-1}^H \textbf {J}_2 \textbf {K}_M$, the Eq. (13) can be written as

(15)$$e^{j\frac{\alpha_q}{2}} (\textbf{Z}_1-j\textbf{Z}_2) \textbf{R} = e^{{-}j\frac{\alpha_q}{2}} (\textbf{Z}_1+j\textbf{Z}_2) \textbf{R}$$

which reduces to

(16)$$\tan\left(\frac{\alpha_q}{2}\right) \textbf{Z}_1 \textbf{R} = \textbf{Z}_2 \textbf{R}.$$

Following similar analysis using the invariance property relating two sub-arrays created by omitting last and first columns of $\boldsymbol {\Gamma }$, we obtain,

(17)$$\tan\left(\frac{\alpha_p}{2}\right) \textbf{R} \textbf{Z}_1^T = \textbf{R} \textbf{Z}_2 ^T.$$

In the presence of noise, the separation of signal and noise components can be ascertained by representing $\textbf {R}$ in vector form as given by

(18)$$\textbf{r} = \textit{vec}(\textbf{R})$$

where $\textit {vec}(.)$ denotes the operation of vertically stacking the columns of the given matrix one below other. Consequently, using the relation $\textit {vec}(\textbf {ABC}) = (\textbf {C}^T \otimes \textbf {A})\textit {vec}(\textbf {B})$ where "$\otimes$" indicates the Kronecker product, Eqs. (16) and (17) can be written as,

(19)$$\begin{aligned}tan\left(\frac{\alpha_q}{2}\right) (\textbf{I}_M\otimes\textbf{Z}_1)\textbf{r} &= (\textbf{I}_M\otimes\textbf{Z}_2) \textbf{r}\\ tan\left(\frac{\alpha_p}{2}\right) (\textbf{Z}_1\otimes\textbf{I}_M)\textbf{r} &= (\textbf{Z}_2\otimes\textbf{I}_M) \textbf{r} \end{aligned}.$$

It can be noted that in the presence of noise, the matrix $\textbf {R}$ in Eq. (8) may contain complex elements. Therefore, the vector $\textbf {r}$ representing the signal component in the measured data can be computed as the dominant (corresponding to largest singular value) left singular vector of the matrix

(20)$$Y = [\Re(\textbf{r})\ \Im(\textbf{r})]$$

where "$\Re (\cdot )$" and "$\Im (\cdot )$" denote real and imaginary parts of a complex vector. If we consider $\textbf {L}_s$ to be the dominant left singular vector, then values of $\alpha _p$ and $\alpha _q$ can be computed as

(21)$$\begin{aligned} \alpha_p &= 2tan^{{-}1}\left[(\textbf{Z}_{p1}\textbf{L}_s)^+\textbf{Z}_{p2}\textbf{L}_s\right]\\ \alpha_q &= 2tan^{{-}1}\left[(\textbf{Z}_{q1}\textbf{L}_s)^+\textbf{Z}_{q2}\textbf{L}_s\right] \end{aligned}$$

where $\textbf {Z}_{p1} = \textbf {Z}_1\otimes \textbf {I}_M$, $\textbf {Z}_{p2} = \textbf {Z}_2\otimes \textbf {I}_M$, $\textbf {Z}_{q1} = \textbf {I}_M\otimes \textbf {Z}_1$, $\textbf {Z}_{q2} = \textbf {I}_M\otimes \textbf {Z}_2$ and $(\cdot )^+$ denotes Moore–Penrose pseudo-inverse operation. Subsequently, the value of $\alpha _0$ can be computed using

(22)$$\alpha_0 = arg(\overline{\boldsymbol{\Gamma}_w e^{{-}j(\alpha_pp+\alpha_qq)}})$$

with $\overline {(.)}$ representing the mean operation. Using these coefficients, the phase at the pixel around which the window is centered can be computed using Eq. (2). The process is repeated for all pixels to obtain the overall phase map. In case of $2\pi$ phase jumps, we apply simple phase unwrapping [54] to obtain a smooth phase map. The main advantage of the unitary transformation obtained via the unitary matrix $\textbf {K}_{M}$ in our analysis is to exploit the centro-Hermitian nature of the complex wavefield signal within a given window. This leads to reduced computational complexity while estimating the phase coefficient parameters and better immunity against noise due to efficient separation of the signal and noise components [46].

To summarize, the phase estimation process using unitary transformation based proposed method is as follows:

1. Select a symmetric window $\boldsymbol {\Gamma }_w$ of size $M\times M$ around a given pixel.
2. Compute the dominant left singular vectors of the matrix $Y$ shown in Eq. (20).
3. Calculate the values of unknown coefficients $\alpha _0$, $\alpha _p$ and $\alpha _q$ using Eqs. (21) and (22).
4. Using the values of these coefficients, estimate the phase inside the window.
5. Repeat the above steps for all pixels to get the overall phase.

3. Results

3.1 Simulations

We simulated noisy complex wavefield signal of the form given in Eq. (1) with additive white Gaussian noise. The image size was 256 $\times$ 256 pixels and signal to noise ratio (SNR) was 5 dB for our simulations. We also simulated two different types of non-uniform amplitude fluctuations in the complex signal by separately assigning higher and lower amplitude values across different regions. Next, we applied the proposed method with parameter $M=11$ for phase retrieval from the simulated noisy signal. For comparison, we also estimated the phase map using state of the art wavelet transform (WT) phase method [34], windowed Fourier transform method [35] and state-space method based on Extended Kalman Filter (EKF) [39]. For the wavelet transform, we used the wavelet ridge approach with Mexican hat mother wavelet and zero rotation angle to extract the phase map. For the windowed Fourier transform method, we used the windowed Fourier ridge algorithm with the parameters $wxl=wyl=-0.5, wxh=wyh=0.5,wxi=wyi=0.01,\sigma _x=\sigma _y=4$ to retrieve the phase. All computations were performed on a workstation containing Intel Xeon processor (E5-1660) with clock frequency 3.20 Gigahertz and 160 Gigabyte memory.

In Fig. 2(a), we show the real part of simulated noisy complex signal exhibiting non-uniform amplitude variations with distinct brighter and darker pixels. The phase map (in radians) estimated using the proposed method is shown in Fig. 2(b) and the corresponding absolute estimation error is shown in Fig. 2(c). Comparatively, phase maps obtained using wavelet transform method, EKF method and windowed Fourier transform method are shown in parts (d,f,h) of Fig. 2. Their corresponding phase estimation errors are given in parts (e,g,i) of Fig. 2. In Fig. 3(a), we show the real part of another simulated noisy complex signal exhibiting non-uniform amplitude variations with different set of brighter and darker pixels. The phase maps estimated using the proposed, wavelet transform, EKF and windowed Fourier transform methods are shown in parts (b,d,f,h) of Fig. 3. The corresponding phase estimation errors are depicted in parts (c,e,g,i) of Fig. 3. From the estimation errors, we can infer that the proposed method offers better phase estimation accuracy than the standard methods.

Fig. 2. (a) Real part of the noisy simulated complex wavefield signal with non-uniform amplitude fluctuations. Estimated phase in radians using the (b) proposed, (d) wavelet transform, (f) EKF and (h) windowed Fourier transform methods. Corresponding phase estimation errors in (c), (e), (g) and (i).

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Fig. 3. (a) Real part of the another noisy simulated complex wavefield signal with non-uniform amplitude fluctuations. Estimated phase in radians using the (b) proposed, (d) wavelet transform, (f) EKF and (h) windowed Fourier transform methods. Corresponding phase estimation errors in (c), (e), (g) and (i).

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Additionally, we computed the root mean square errors (RMSE) for phase estimation using the different methods. The RMSE values at various noise levels are plotted in Fig. 4. In part (a) of Fig. 4, the RMSE versus SNR plot is shown for the case of non-uniform amplitude fluctuation depicted in Fig. 2(a). In part (b) of Fig. 4, the RMSE versus SNR plot is shown for the case of non-uniform amplitude fluctuation depicted in Fig. 3(a). All the RMSE values in these plots were computed by taking average over ten iterations to account for randomness. These results show that the proposed method offers better phase estimation accuracy for broad range of noise levels, and is also robust under severe noise or low SNR conditions. Regarding computational performance, the computational time required for phase extraction was about 25 seconds for the proposed method, 5 seconds for the wavelet transform method, 120 seconds for the EKF method, and 78 seconds for the windowed Fourier transform.

Fig. 4. Root mean square errors plotted against different SNR values for the two types of non-uniform amplitudes.

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3.2 Experiments

To illustrate the practical utility of the proposed method, we measured the profile of surface structures located on a standard calibration target (TGXYZ02, Mikromasch). The test target sample features silicon dioxide structures on a silicon substrate, and these microscopic structures resemble pillars and arrows with manufacturer specified surface height of 100 nm with a tolerance of 3 nm. For the micro-pillar structure, we used a microscope objective (Newport M-40X) with magnification value of 40 and numerical aperture (NA) value of 0.65 for imaging. For the arrow structure, we used a microscope objective (Newport M-20X) with magnification value of 20 and numerical aperture (NA) value of 0.40 for imaging. The phase corresponding to each recorded interferogram was estimated using the proposed method. Subsequently, the surface profile map of the object under test was computed using the following relation [55],

(23)$$S = \frac{\lambda}{4\pi (n-1)}\phi$$

where $S$ is the surface profile map, $\lambda$ is the wavelength, $n$ is the refractive index of the silicon dioxide structure and $\phi$ is the estimated phase map in radians. We computed the phase maps using the proposed method, wavelet transform method, EKF method and windowed Fourier transform method. For the proposed method, we used window size parameter $M=25$ in the experimental analysis. To remove the effect of any phase discontinuities, we used a simple phase unwrapping algorithm [54] for the experimental images.

The hologram corresponding to micro-pillar structures is shown in Fig. 5(a). The surface profile or height map estimated using the proposed method is depicted in Fig. 5(b). The colorbar indicates the surface profile variation in nanometers in the Fig. 5(b). For comparison, the surface profile maps using the wavelet transform method, EKF method and windowed Fourier transform method are shown in parts (c-e) of Fig. 5. Similarly, the hologram corresponding to arrow structure is shown in Fig. 6(a). The surface profile or height map estimated using the proposed method is depicted in Fig. 6(b). For comparison, the surface profile maps using the wavelet transform method, EKF method and windowed Fourier transform method are shown in parts (c-e) of Fig. 6. From these figures, we observe that the presence of non-uniform intensity variations coupled with noise can lead to significant deterioration of estimation accuracy for the standard methods. Regarding computational performance, the computational time required for phase extraction corresponding to micro-pillar structure hologram with size 965 (vertical) $\times$ 720 (horizontal) pixels was about 414 seconds for the proposed method, 93 seconds for the wavelet transform method, 1322 seconds for the EKF method, and 1224 seconds for the windowed Fourier transform method. Similarly, the computational time required for phase extraction corresponding to arrow structure hologram with size 610 (vertical) $\times$ 910 (horizontal) pixels was about 331 seconds for the proposed method, 65 seconds for the wavelet transform method, 1052 seconds for the EKF method, and 1260 seconds for the windowed Fourier transform method.

Fig. 5. (a) Experimentally recorded hologram with micro-pillar structures. Estimated surface profile map in nanometers using (b) proposed method, (c) wavelet transform method, (d) EKF method and (e) windowed Fourier transform method.

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Fig. 6. (a) Experimentally recorded hologram with arrow structure. Estimated surface profile map in nanometers using (b) proposed method, (c) wavelet transform method, (d) EKF method and (e) windowed Fourier transform method.

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The height of the micro-pillar structure is computed by plotting the normalized histogram of the estimated surface map as shown in Fig. 7(a). In the histogram plot, the horizontal axis represents the height values and the vertical axis denotes the normalized pixel count. Evidently, there are two distinct peaks that correspond to the micro-pillar structure and the planar background. The distance between the two peaks on the plot gives the quantitative estimate of the height of the structure. For ease of visualization, the origin of the plot on the horizontal axis is suitably shifted such that it coincides with the first peak, and the location of the second peak gives the height estimate. Similarly, the hologram corresponding to arrow structure is given in Fig. 7(b). It is evident that the height estimates obtained using the proposed method are close to the manufacturer specified value for both the micro-pillar and arrow structures. For further testing, we also imaged four micro-pillar structure regions in the test sample using our experimental setup. Next, we computed the height values from surface profile histogram plots using the proposed method corresponding to these regions, and calculated the mean height and standard deviation. Thus, the proposed method offered a measured height value of 95.75 nm (mean) $\pm$ 1.92 nm (standard deviation) against manufacturer specified height value of 100 nm with tolerance of 3 nm. From these results, we infer that the proposed method shows good ability for surface profile measurement under experimental conditions.

Fig. 7. Histograms of surface profile maps for (a) micro-pillar structure and (b) arrow structure.

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4. Discussion

The simulation and experimental results demonstrate the effectiveness of the proposed method for surface profile measurement in digital holographic microscopy. In particular, the proposed method offers good robustness against noise and non-uniform amplitude fluctuations in the hologram. From simulation results, we infer that the proposed method exhibits root mean square error values less than 1 radians for phase estimation for a broad range of noise levels. In addition, we observe that the experimental holograms are affected by non-uniform intensity variations, and the proposed method provides decent height map estimates in this case. The main bottleneck in the proposed method is the computation of singular vectors of each window matrix, which is a computationally intensive operation. The challenge of high computational cost could be addressed using the parallel computing features of graphics processing units which have emerged as rapid fringe processing tools [56–58] for precision optical metrology. In addition, we note that the choice of window size affects the accuracy and computational performance of the proposed method. Within a small window, the linear phase approximation used in the proposed method has high validity. Especially, in presence of rapid phase variations corresponding to sharp edges, the linear phase approximation is only true for a small window. However, this comes at the cost of high noise susceptibility caused by less number of signal data captured by the small window. Conversely, a large window ensures that the method is less sensitive to noise, which however leads to high computational burden. Hence, there is a general trade-off between noise robustness and computational performance, and an optimal window strategy would be explored for future work. By heuristic observation, we found that a window size in the range of approximately 1 to 10 percent of the image size gives decent results for the proposed method.

5. Conclusion

In this article, we proposed a robust method for nanoscale surface profile measurement in digital holographic microscopy. The method shows good robustness against noise and non-uniform amplitude fluctuations. We believe that the method has potential for non-contact optical metrology and non-destructive testing.

Funding

Department of Science and Technology, Ministry of Science and Technology, India (DST/NM/NT/2018/2).

Disclosures

The authors declare that there are no conflicts of interest related to this paper.

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Non-invasive surface profile measurement using a unitary transformation subspace approach in digital holography

Abstract

1. Introduction

2. Experimental setup and signal processing

3. Results

3.1 Simulations

3.2 Experiments

4. Discussion

5. Conclusion

Funding

Disclosures

Data Availability

References

Data Availability

Cited By

Figures (7)

Equations (24)

Optics Continuum

Jagadesh Ramaiah	https://orcid.org/0000-0002-2350-0265
Ankur Vishnoi	https://orcid.org/0000-0002-0136-3378
Rajshekhar Gannavarpu	https://orcid.org/0000-0002-4277-590X