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Investigation of the systematic axial measurement error caused by the space variance effect in digital holography
Optics and Lasers in Engineering 112 (2019) 16–25
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Investigation of the systematic axial measurement error caused by the space variance effect in digital holography Yan Hao a, Chiyue Liu a, Jun long a, Ping Cai a,∗, Qian Kemao b, Anand Asundi c a
Department of Instrument Science and Engineering, School of EIEE, Shanghai Jiao Tong University, Shanghai 200240, China School of Computer Science and Engineering, Nanyang Technological University, Singapore 639798, Singapore c School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798, Singapore b
a b s t r a c t Digital holography (DH) is one of the most promising quantitative phase measurement techniques and has been successfully used in 3D imaging and measurement. One of its attractive advantages is its excellent theoretical axial measurement accuracy of better than 1 nanometer. However, in practice, the axial accuracy has been quoted to be in the range of tens nanometers limited by the axial errors existing in DH system. In order to improve the axial measurement accuracy to approach the theoretical value, it is necessary to identify error sources and then reduce the errors according to their properties. In this paper, the space-variance effect of digital holography system is investigated and demonstrated to be an important systematic axial measurement error (SAME) source, especially for features with high frequency. The properties of the space-variant SAME are investigated through simulations and experiments. The object position, object height, object frequency content and object-CCD distance are found to be related to the space-variant SAME. Careful and appropriate placement of the object according to its features is thus necessary to reduce such SAME in a DH system. Based on the investigation, the guideline to appropriately position an object according to its properties is provided in this work.
1. Introduction Digital holography (DH) [1–3] provides an easy way to obtain quantitative phase distribution containing depth information of an object and allows 3D imaging of the object. DH has thus been widely applied in microscopic 3D measurement and imaging such as 3D cell imaging [4], 3D micro-structure measurement [5], 3D micro-particles tracking [6], etc. The theoretical systematic axial measurement error (SAME) of DH is smaller than 1 nanometers, where the quantization effect of CCD pixel is considered to be the only error source [4]. However, in practice, the SAME is dramatically increased to tens of nanometers [5–9], implying the existence of other axial error sources. The phase error caused by lens aberrations has been studied [10–12], which does not happen in lensless DH. In [13], The CCD aperture size has been shown to be an important contributor to the SAME, but the error it caused is smaller than that from practical experiments [13], hinting that there exist other unrevealed error sources. The space-variance effect (SVE) has been found to affect the lateral resolution [14,15], but its influence on the SAME remains unknown and thus examined in this paper. Lensless DH system is considered so that the phase aberrations of microscope objectives and other lenses are all excluded. Through simulation and experiments, the SAME caused by the SVE will be confirmed, from which, how to position the object being measured is discussed. The rest of the paper is organized as follows. In Section 2, the principle of DH and the SVE are briefly introduced. In Section 3, the impact of ∗
the SVE on the SAME is examined for both point and step objects through simulation, and the results are analyzed. In Section 4, the experiments are performed to validate our analysis. In Section 5, the conclusions of this work are drawn. 2. Digital holography and its space-variance effect In this section, the principle of off-axis DH and its SVE are briefly introduced. 2.1. Principle of DH As shown in Fig. 1, off-axis DH includes two processes: digital recording and numerical reconstruction. In the digital recording process, a light wave illuminates the object. The object wave O is reflected by an opaque object or transmits through a transparent object, and carries the object information. The reference wave R illuminates the CCD with an angle of 𝜃. The two waves interfere to generate the hologram 𝐼𝐻 (x, y) at the CCD plane with the following resulting intensity: 𝐼𝐻 (x, y) = |𝑂(𝑥, 𝑦) + 𝑅(𝑥, 𝑦)|2 = |𝑂(𝑥, 𝑦)|2 + |𝑅(𝑥, 𝑦)|2 + 𝑅(𝑥, 𝑦)𝑂∗ (𝑥, 𝑦) + 𝑅∗ (𝑥, 𝑦)𝑂(𝑥, 𝑦)
(1)
where ∗ denotes the complex conjugate. After the sampling and digitalization by the CCD camera, a digital hologram is formed. In the numerical reconstruction process, the third or the fourth terms of Eq. (1) can be filtered out numerically due to their spatial
Corresponding author. E-mail address: [email protected] (P. Cai).
https://doi.org/10.1016/j.optlaseng.2018.06.009 Received 1 February 2018; Received in revised form 18 May 2018; Accepted 12 June 2018 0143-8166/© 2018 Published by Elsevier Ltd.
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Fig. 1. An illustration of an off-axis DH system. 𝑗𝜋 2 at the CCD plane can be written as exp[ 𝜆𝑑 (𝑥 + 𝑦2 )], which will be limited by the finite CCD sensing area 𝐿𝑥 × 𝐿𝑦 , resulting in the following truncated wavefront, ] [ ( ) ( ) ) 𝑗𝜋 ( 2 𝑦 𝑥 𝑊𝐴 (𝑥, 𝑦) = exp rect . (6) 𝑥 + 𝑦2 × rect 𝜆𝑑 𝐿𝑥 𝐿𝑦
Similarly, the wavefront from point B located at (𝑥, 𝑦) = (𝑥0 , 𝑦0 ) can be written as } { [ ( ) ( ) )2 ( )2 ] 𝑗𝜋 ( 𝑥 𝑥 𝑥 − 𝑥0 + 𝑦 − 𝑦0 𝑊𝐵 (𝑥, 𝑦) = exp × rect rect 𝜆𝑑 𝐿𝑥 𝐿𝑦 ( ) ≠ 𝑊𝐴 𝑥 − 𝑥0, 𝑦 − 𝑦0 , (7) which shows that the recording process is space-variant. The physical process of light diffraction follows that the light containing frequency 𝑓 of an object deflects from its incident direction with an angle α with a relation of 𝑓 = 𝑠𝑖𝑛𝛼 . For the point A located at (𝑥, 𝑦) = 𝜆 (0, 0), the diffracted lights which could be collected by CCD are in the an-
Fig. 2. An illustration of the space-variance effect of DH. The propagation and recording of the wavefronts from two points A and B.
−𝐿 ∕2−𝑥
𝐿
𝑥 direction while the bandwidths of point B is (− 2𝜆𝑑 −
𝐿𝑦
𝐼𝑚[𝛾(𝑥, 𝑦)] . 𝑅𝑒[𝛾(𝑥, 𝑦)]
𝐿 ∕2−𝑦
𝐿𝑦
𝑦0 , 𝜆𝑑 2𝜆𝑑
𝑦0 ) 𝜆𝑑
𝑥0 𝐿𝑥 , 𝜆𝑑 2𝜆𝑑
−
𝑥0 ) 𝜆𝑑
in 𝑥
direction and (− 2𝜆𝑑 − − in 𝑦 direction respectively. It is noticed that both bandwidths are finite but they are not the same. The finite bandwidths indicate that the CCD truncation only retains partial information of an object point. The information loss contributes to the SAME, which has been examined in [13]. The different bandwidths indicate that different points suffer different information loss and thus the ultimate SAME is also space-variant. It is thus necessary and significant to examine the severity of the non-uniformity of SAME. If this error variation is small, then the object can be freely positioned within the object plane of the DH system. Otherwise, the object positioning has to carefully follow a guideline to avoid the occurrence of severe space-variant SAME.
where 𝜆 is the laser wavelength; 𝑑 is the reconstruction distance;𝑓𝑥 and 𝑓𝑦 are the coordinates in the spectrum domain. The intensity and phase can be calculated as
𝜑(𝑥, 𝑦) = 𝑎𝑟𝑐𝑡𝑎𝑛
−𝐿 ∕2−𝑦
𝐿 ∕2−𝑥
𝑡𝑎𝑛𝛼 ∈ [ 𝑥 𝑑 0 , 𝑥 𝑑 0 ] in 𝑥 direction and 𝑡𝑎𝑛𝛽 ∈ [ 𝑦 𝑑 0 , 𝑦 𝑑 0 ] in 𝑦 direction respectively. In practice, the distance 𝑑 is usually quite large than the CCD widths 𝐿𝑥 and 𝐿𝑦 such that we have the approximation of 𝑠𝑖𝑛𝛼 ≅ 𝑡𝑎𝑛𝛼 and 𝑠𝑖𝑛𝛽 ≅ 𝑡𝑎𝑛𝛽. Hence the bandwidths of point 𝐿 𝐿 𝐿𝑥 𝐿𝑥 A recorded by CCD are (− 2𝜆𝑑 , 2𝜆𝑑 ) in 𝑥 direction and (− 2𝜆𝑑𝑦 , 2𝜆𝑑𝑦 ) in 𝑦
where and −1 denotes Fourier and inverse Fourier transforms respectively; 𝐺(𝑓𝑥 , 𝑓𝑦 ) is the transfer function of an optical system expressed as [ ] √ ( ) ( )2 ( )2 𝑑 𝐺 𝑓𝑥 , 𝑓𝑦 = exp 𝑗2𝜋 1 − 𝜆𝑓𝑥 − 𝜆𝑓𝑦 (3) 𝜆
𝐼 (𝑥, 𝑦) = |𝛾(𝑥, 𝑦)|2 ,
𝐿 ∕2 𝐿 ∕2
𝐿 ∕2 𝐿 ∕2
gle range of 𝑡𝑎𝑛𝛼 ∈ [− 𝑥𝑑 , 𝑥𝑑 ] in 𝑥 direction and 𝑡𝑎𝑛𝛽 ∈ [− 𝑦𝑑 , 𝑦𝑑 ] in 𝑦 direction respectively. However, for the point B located at (𝑥, 𝑦) = (𝑥0 , 𝑦0 ), the diffracted lights collected by CCD are in the angle range of
carrier introduced by the off-axis geometry. We take the fourth term 𝑅∗ (𝑥, 𝑦)𝑂(𝑥, 𝑦) for example. A quantitative object wavefield at the object plane can be computed from the fourth term of Eq. (1) by multiplying the reference wave 𝑅(𝑥, 𝑦) followed by Fresnel diffraction integration as [16,17]: { { } ( )} 𝛾(𝑥, 𝑦) = −1 𝑅(𝑥, 𝑦)[𝑅∗ (𝑥, 𝑦)𝑂(𝑥, 𝑦)] × 𝐺 𝑓𝑥 , 𝑓𝑦 (2)
(4)
(5)
From phase 𝜑(𝑥, 𝑦), the desired object height in the axial direction can be obtained. 2.2. Space-variance effect of digital holography
3. Simulation investigation of the space-variance effect on system axial measurement error
The SVE exists in any practical systems and DH is not an exception. Two point sources A and B at different locations in Fig. 2 are used for explanation. Point A is located at (𝑥, 𝑦) = (0, 0). Its propagated wavefront
In this section, the impact of the SVE on the SAME is investigated by simulation. Simulation is adopted because the phase retrieval involving recording and reconstruction is a non-linear process and is difficult for 17
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Fig. 3. The PSF images at different spatial positions. (a) the 3D height profile of the PSF at the center of object plane; (b) the section across the center of the PSF in (a) along x axis; (c) the 3D height profile of the PSF at −1.4 mm; (d) the section across the center of the PSF in of (c) along x axis; (e) the variations of axial error of PSF with respect to the positions and height.
theoretical analysis. Nevertheless, the simulation results will be compared and supported by real experiments. Two types of objects, point sources and step objects, are investigated. The point spread function (PSF) is a good indication of the performance of DH system. Therefore, the impact of the SVE on SAME of the PSF is firstly investigated. The ideal PSF of a DH system is 𝛿(𝑥, 𝑦), but in a practical system it has a space-variant spread, due to the bandwidthlimit and space-variance discussed in Section 2.2. Consequently, a spatially varying SAME is resulted. Thus the spatial variation of the PSF should be examined. Given a point object 𝛿(𝑥 − 𝑥0 , 𝑦 − 𝑦0 ) and the reconstructed object point𝛾(𝑥, 𝑦), it is readily to find that the PSF can be determined as ( ) PSF(𝑥0 ,𝑦0 ) (𝑥, 𝑦) = 𝛾 𝑥 + 𝑥0 , 𝑦 + 𝑦0 .
We consider a point object with a height of 100 nm with the following simulation parameters: the CCD size is 𝐿𝑥 = 3.72mm, 𝐿𝑦 = 4.76 mm; the wavelength 𝜆 is 633 nm; the reconstruction distance 𝑑 is 123 mm; the pixel size is T = 4.65 μm; an off-axis lensless geometry is adopted where the carrier frequency is a = 53.76 × 103 Hz determined by the angle 𝜃. We first put the point object at (𝑥0 , 𝑦0 ) = (0, 0) mm, i.e., the height of such point object is 𝑧(𝑥, 𝑦) = 100 × 𝛿(𝑥, 𝑦) nm. The reconstructed point object, PSF(0,0) (𝑥, 𝑦), is shown in Fig. 3(a) with a central section plotted at Fig. 3(b). This PSF is like a spread delta function mainly due to the limited CCD size. Next, we put the point object at (𝑥0 , 𝑦0 ) = (−1.4, 0)mm, i.e., the height of point object is 𝑧(𝑥, 𝑦) = 100 × 𝛿(𝑥 + 1.4, 𝑦) nm, and reconstruct the object again. The reconstructed function is then shifted by 1.4 mm horizontally according to Eq. (8) and obtain PSF(−1.4,0) (𝑥, 𝑦),
(8) 18
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Fig. 4. An illustration of the space-variant SAME of a step object. (a) An example of the true image of a step object of 100 nm height; (b) the reconstructed image of the focused step with 𝑥0 = 0 mm; (c) the SAME with 𝑥0 = 0 mm; (d) the reconstructed image of the focused step with 𝑥0 = −1.3 mm; (e) the SAME with 𝑥0 = −1.3 mm.
as shown in Fig. 3(c) and (d). As expected, first, PSF(−1.4,0) (𝑥, 𝑦) ≠ PSF(0,0) (𝑥, 𝑦), verifying the SVE of the DH system; second, PSF(−1.4,0) (𝑥, 𝑦) is more distorted from a delta function and is asymmetrical, showing lower imaging quality. In fact, for (𝑥0 , 𝑦0 ) = (0, 0), the reconstructed object height error is 16.6 nm, but for (𝑥0 , 𝑦0 ) = (−1.4, 0), the error increases to 42.1 nm, which is too large in a high precision measurement of an object of 100 nm height. For the point sources with different heights, the axial errors are calculated as compared in Fig. 3(e). It is seen that larger height has larger axial errors and the non-uniformity of the space-
variant axial error is more severe. Therefore, the space-variant SAME exists and the non-uniformity of the space-variant SAME of PSF is severe and cannot be ignored. The object positioning has to be carefully considered in order to avoid the occurrence of such severe SAME. In practice, most of the objects are not isolated points but of finite size. Among objects of finite size, step objects are investigated in this work because they are often seen in real imaging, contain rich frequency information, and usually have poor imaging quality. We define a step
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Fig. 5. The quantitative evaluations of the impact of step heights on the space-variant SAME. (a) the PAE and (b) the MAE of step region with different heights H ranging from −600 nm to 600 nm and at five different locations 𝑥0 = 1.2, 0.6, 0, − 0.6, − 1.2 mm; (c) the PAE and (d) the MAE of the step region; (e) the PAE and (f) the MAE of the flat region at different locations 𝑥0 ranging from −1.4 mm to 1.4 mm and of different heights H = 40, 100, 160, 220 and 280 nm.
object of height H as below: [ ] { 𝐻 , (𝑥, 𝑦) ∈ −2.38, 𝑥0 × [−2.38, 2.38] 𝑧(𝑥, 𝑦) = , 0, otherwise
±2.38mm, while the last one at 𝑥 = 𝑥0 is variable when 𝑥0 changes. The SVE on this varying step is concerned and investigated. The object with 𝑥0 = 0 and 𝐻 = 100 nm is shown in Fig. 4(a) as an example, and its reconstructed height images is shown in Fig. 4(b). For clarity, the reconstruction error is also shown in Fig. 4(c). The concerned edge at 𝑥0 = 0 mm has an axial error of about 30.47 nm. We then re-
(9)
where the unit of x and y is mm and that for z is nm. This object contains four step edges, among which, three are fixed at 𝑥 = −2.38mm and 𝑦 = 20
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Fig. 6. The quantitative evaluations of the impact of CCD size on the space-variant SAME of step objects. (a) the PAE; (b) the MAE at the step region and (c) the PAE; (d) the MAE at the flat region at different locations 𝑥0 ranging from −1.4 mm to 1.4 mm and of different CCD sizes Lx = 3.72, 5.12, 6.51, 7.91 and 9.30 mm.
locate this edge at 𝑥0 = − 1.3 mm, with the reconstructed height image and the reconstruction error shown in Fig. 4(d) and (e), respectively. This time, the axial error of the concerned edge decreases to 11.49 nm. The non-uniformity of SAME reaching 18.98 nm is large in the measurement of a jump height of 100 nm, which demonstrates the SVE as an important SAME contributor. For the flat region of the step excluding the jump site where only low frequency information is involved, the maximum error of Fig. 4(c) and (e) are 1.70 nm and 5.36 nm, respectively. This is because the flat regions have very low frequency to be adequately captured by CCD with little information loss. To obtain a more systematic view of SVE, a series of detailed simulation are carried out. The SAME is quantitatively examined using the peak axial error (PAE) and the mean axial error (MAE), defined as the peak error and mean absolute error of a region, respectively. Two regions are considered: one is the step region including 50 pixels to the right and 50 pixels to the left of the step, and the other is a flat region excluding 50 pixels near the step and object boarders. First, the impact of step height on the space-variant SAME is investigated. (i) For five typical edge locations 𝑥0 , the step height H changes in a range from −600 nm to 600 nm. Both PAE and MAE of the step region are shown in Fig. 5(a) and (b), respectively. Periodicity is observed with a period of λ∕2 . This is reasonable because, when the height changes λ∕2, the optical path changes λ in our simulation, then the optical wavefront remains the same, and consequently the phase error is the same. This is a good property so that one does not need to concern the height amount of a specimen. Among all height values, the height values gen-
erating the optical path lengths of integral multiple of wavelength 𝑁λ have the smallest axial error. (ii) For five representative height values, the edge location changes in a range from −1.4 mm to 1.4 mm with an increment of 4.65 μm, along x-axis in object plane. The axial error results are shown in Fig. 5(c) and (d), where the SVE is clearly observed. Take the plot of height 160 nm in Fig. 5(c) as an example, the PAE value changes from 5 nm to about 70 nm at different locations which reveals the serious impact of SVE. Furthermore, it is shown in Fig. 5(c) and (d) that steps of different heights have quite different space-variant axial error performances. Hence, the SAME is not only space-variant but also height-variant. These phenomena indicate that SVE is an important SAME source and careful and appropriate positioning of the object in experiments is necessary. Generally speaking, the overall error is smallest at 𝑥0 = 0. (iii) As observed from Fig. 5(e) and (f), the PAE and MAE of the flat region are also space-variant. However the amplitude of the axial error is much smaller and the non-uniformity is less obvious than those of the jump region which indicates the jump site containing high frequency information contents has larger SAME than the flat region containing lower frequency information contents. The reason is that, when two objects containing different frequency contents pass the same bandwidth-limit DH filter, the loss of the object containing low frequency contents is less than the loss of the object with high frequency contents. More information loss results in larger space-variant SAME. Thus, the space-variant SAME is related to object frequency components and the features with high frequencies are more vulnerable to the space-variant SAME. 21
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Fig. 7. The quantitative evaluations of the impact of the distance on the space-variant SAME. (a) the PAE; (b) the MAE at the step region and (c) the PAE; (d) the MAE at the flat region of different positions with 𝑥0 ranging from −1.4 mm to 1.4 mm and with different object distances d = 30, 60, 120, 240, 480 mm.
Second, the impact of CCD size on the space-variant SAME is investigated. For five representative CCD sizes Lx, the edge location 𝑥0 changes in a range from −1.4 mm to 1.4 mm with an increment of 4.65 μm, along x-axis in object plane. The axial error results are shown in Fig. 6. The PAE and MAE of the step region are shown in Fig. 6(a) and (b), respectively. The PAE and MAE of the flat region are shown in Fig. 6(c) and (d), respectively. Increasing the size of CCD reduces the space-variant SAME. This is because increasing the CCD size increases the bandwidth of the DH filter and thus alleviates the information loss and the resulting space-variant SAME duo to filtering. As observed from Fig. 6(c) and (d), the PAE and MAE of the flat region is much smaller than those of the jump region in Fig. 6(a) and (b), which verifies that the space-variant SAME is related to object frequency components and the features with high frequencies are more vulnerable to the space-variant SAME. Third, the impact of the distance between object and CCD on the space-variant SAME is examined. For five representative distances d, the edge location 𝑥0 changes in a range from −1.4 mm to 1.4 mm with an increment of 4.65 μm, along x-axis in object plane. The PAE and MAE of the step region are shown in Fig. 7(a) and (b), respectively, while those of the flat region are shown in Fig. 7(c) and (d), respectively. As increasing the distance between object and CCD is equivalent to decreasing the CCD size, and thus the observed phenomenon in Fig. 7 is similar to that in Fig. 6. Reducing the distance between object and CCD, if possible in practice, is encouraged. According to this simulation investigation, the non-uniformity of the space-variant SAME is seriously affected by object position, object sur-
face profile property, CCD size, and object-CCD distance. Accordingly, the following are suggestions for DH practice: (i) Generally speaking, the object should be positioned on the optical axis perpendicular to and passing through the CCD center, as the objects spectrum is best conserved in this location; (ii) If the object has both flats and jumps (or both continuous and discontinuous parts), the jumps or discontinuous parts should be positioned on the optical axis as they are more vulnerable to spectrum information loss; (iii) If possible, a larger CCD is prefer, although it will increase the cost of the system; (iv) If possible, the object-CCD distance should be made small to increase the equivalent CCD size. 4. Experiment investigation of the space-variant system axial measurement error In this section, the space-variant SMAE is investigated by experiments. The first specimen is a US air force (USAF) target which is a standard resolution chart as shown in Fig. 8(a). The heights of the coated bars are all 100 nm, making the USAF target a good and readily available sample to study the axial measurement. In this experiment, Group 2 Element 3 (G2E3) is used for detailed examination. In our experiment, 56 holograms with the G2E3’s location shifting from −1.3 mm to 1.3 mm with an interval of 0.047 mm are recorded with a lensless DH system [18]. The height images of G2E3 at dif22
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Fig. 8. The quantitative evaluation of space-variant SAME for the USAF target in experiment. (a) the image of the specimen: USAF target; holograms with different locations 𝑥0 of (b) −0.77 mm; (c) 0 mm; (d) 0.77 mm; the reconstructed height images and profiles (insets) of G2E3 with object location 𝑥0 of (e) −0.77 mm; (f) 0 mm; (g) 0.77 mm; (h) the PAE; (i) the MAE at the step region and (j) the PAE; (k) the MAE at the flat region of different values of 𝑥0 ranging from −1.3 mm to 1.3 mm.
ferent locations are reconstructed. Three of the 56 holograms at locations −0.77 mm, 0, 0.77 mm are shown in Fig. 8(b), (c) and (d), respectively. Their corresponding reconstructed height images are presented in Fig. 8(e), (f) and (g), respectively. The corresponding height profiles of G2E3 are drawn in the insets. The height profiles show that the reconstructions and the corresponding SAMEs are space-variant. Such SVE of the SAME is obvious especially at the jump site of the step where high frequencies present. The non-uniformity of SAME among Fig. 8(e), (f) and (g) reaches about 20 nm which is large in the measurement of a step height of 100 nm and demonstrates the SVE as an important SAME contributor in experiment.
To quantitatively evaluate the SAME, the PAE and MAE of the right step region and a flat region at the different locations are calculated and shown in Fig. 8(h), (i), (j) and (k). It is seen that, as the step shifts from −1.3 mm to 1.3 mm, the PAE changes with ringing in a range from 10 nm to 70 nm at different locations which quantitatively verifies the existence of severe space-variant SAME. The non-uniformity of PAE reaching 60 nm is large in the measurement of a step of 100 nm height. This indicates that SVE is an important SAME source and appropriate positioning of the object is necessary. It is observed the PAE and MAE of the flat region in Fig. 8(j) and (k) are also space-variant. However, their amplitudes are much smaller and 23
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Fig. 9. The quantitative evaluation of space variant SAME for a step object on a MEMS target in experiment. (a) The hologram of the MEMS specimen; (b) the reconstructed intensity image of (a); (c) the reconstructed phase image of (a); (d) the PAE; (e) the MAE at the step region and (f) the PAE; (g) the MAE at the flat region of different locations 𝑥0 ranging from-0.8 mm to 0.8 mm. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
the non-uniformity is less obvious than those of the jump region which demonstrate the jump site containing high frequency information contents has larger SAME than the flat region containing lower frequency information contents. These results agree with the simulation investigation in last section that the space-variant SAME is related to object frequency components and the features with high frequencies are more vulnerable to the space-variant SAME. We further investigate another MEMS specimen which is also a step object. The hologram and the reconstructed intensity image and phase image with the step located at the center are shown in Fig. 9(a), (b) and (c). In this experiment, a region of interest (ROI) on the object, highlighted by the blue square in Fig. 9(c), is further examined. In the experiment, 112 holograms with the ROI’s location shifting from −1.3 mm to 1.3 mm with an interval of 0.023 mm are recorded
with the same lensless DH system [18]. The PAE and MAE of the step region and a flat region in the ROI at the different locations are calculated and shown in Fig. 9(d), (e), (f) and (g). It is seen that, as the step shifts from −1.3 mm to 1.3 mm, the PAE changes with ringing in a range from 60 nm to 140 nm at different locations which quantitatively verifies the existence of severe space-variant SAME. The non-uniformity of PAE reaching 80 nm is large which proves SVE is an important SAME source. The above two experiments verify the existence and the severe nonuniformity of the space-variant SAME which indicates that the appropriate and careful positioning of the object according to its frequency contents is quite necessary in order to suppress such space variant SAME.
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5. Conclusions
[5] Marquet P, Rappaz B, Magistretti PJ, Cuche E, Emery Y, Colomb T, et al. Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy. Opt Lett 2005;30(March):468–70. [6] Colomb T, Cuche E, Charriere F, Kuhn J, Aspert N, Montfort F, et al. Automatic procedure for aberration compensation in digital holographic microscopy and applications to specimen shape compensation. Appl Opt 2006;45(February):851–63. [7] Rappaz B, Marquet P, Cuche E, Emery Y, Depeursinge C, Magistretti PJ. Measurement of the integral refractive index and dynamic cell morphometry of living cells with digital holographic microscopy. Opt Express 2005;13(November):9361–73. [8] Charriere F, Kuhn J, Colomb T, Montfort F, Cuche E, Emery Y, et al. Characterization of microlenses by digital holographic microscopy. Appl Opt 2006;45(, February):829–35. [9] Mann CJ, Yu LF, Lo CM, Kim MK. High-resolution quantitative phase-contrast microscopy by digital holography. Opt Express 2005;13(October):8693–8. [10] Zhang YZ, Wang DY, Wang YX, Tao SQ. Automatic compensation of total phase aberrations in digital holographic biological imaging. Chin Phys Lett 2011;28(November). [11] Colomb T, Kuhn JK, Charriere F, Depeursinge C, Marquet P, Aspert N. Total aberrations compensation in digital holographic microscopy with a reference conjugated hologram. Opt Express 2006;14(May):4300–6. [12] Cui HK, Wang DY, Wang YX, Zhao J, Zhang YZ. Phase aberration compensation by spectrum centering in digital holographic microscopy. Opt Commun 2011;284(August):4152–5. [13] Hao Y, Asundi A. Impact of charge-coupled device size on axial measurement error in digital holographic system. Opt Lett 2013;38(April):1194–6. [14] Hao Y, Asundi A. resolution analysis of a digital holography system. Appl Opt 2011;50(January):11 10. [15] Kelly DP, Hennelly BM, Pandey N, Naughton TJ, Rhodes WT. Resolution limits in practical digital holographic systems. Opt Eng 2009;48(, September). [16] Colomb T, Montfort F, Kuhn J, Aspert N, Cuche E, Marian A, et al. Numerical parametric lens for shifting, magnification, and complete aberration compensation in digital holographic microscopy. J Opt Soc Am A Opt Image Sci Vis 2006;23(December):3177–90. [17] Hao Y, Asundi A. Comparison of digital holographic microscope and confocal microscope methods for characterization of micro-optical diffractive components. Ninth international symposium on laser metrology, pts 1 and 2, 7155; 2008. [18] Hao Y, Asundi A. Studies on aperture synthesis in digital Fresnel holography. Opt Lasers Eng 2012;50(April):556–62.
In this work, the impact of space-variance effect on the systematic axial measurement error (phase error) in DH is investigated by both simulation and experiment. The results show that the space-variance effect has an important impact on the systematic axial measurement error (SAME). Furthermore, it is found that the features of higher frequency information components are more vulnerable to such spacevariant SAME than those of lower frequency information. The influences of object height, the size of CCD and CCD-object distance on such spacevariant SAME are all investigated. According to our results, careful and appropriate positioning of the object is necessary to suppress such spacevariant SAME. The guideline to appropriately position an object according to its properties is also provided in this work. Acknowledgment This work was supported by the National key R&D Program of China (2016YFF0200700), the National Natural Science Foundation, China (61405111,61502295), and Shanghai Engineering Research Center for Intelligent diagnosis and treatment instrument(15DZ2252000). References [1] Schnars U, Juptner W. Direct recording of holograms by a CCD target and numerical reconstruction. Appl Opt 1994;33(January):179–81. [2] Schnars U, Juptner WPO. Digital recording and numerical reconstruction of holograms. Meas Sci Technol 2002;13(September):R85–R101. [3] Cuche E, Bevilacqua F, Depeursinge C. Digital holography for quantitative phase– contrast imaging. Opt Lett 1999;24(March):291–3. [4] Pandey N, Hennelly B. Quantization noise and its reduction in lensless Fourier digital holography. Appl Opt 2011;50(, March):B58–70.
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Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Investigation of the systematic axial measurement error caused by the space variance effect in digital holography Yan Hao a, Chiyue Liu a, Jun long a, Ping Cai a,∗, Qian Kemao b, Anand Asundi c a
Department of Instrument Science and Engineering, School of EIEE, Shanghai Jiao Tong University, Shanghai 200240, China School of Computer Science and Engineering, Nanyang Technological University, Singapore 639798, Singapore c School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798, Singapore b
a b s t r a c t Digital holography (DH) is one of the most promising quantitative phase measurement techniques and has been successfully used in 3D imaging and measurement. One of its attractive advantages is its excellent theoretical axial measurement accuracy of better than 1 nanometer. However, in practice, the axial accuracy has been quoted to be in the range of tens nanometers limited by the axial errors existing in DH system. In order to improve the axial measurement accuracy to approach the theoretical value, it is necessary to identify error sources and then reduce the errors according to their properties. In this paper, the space-variance effect of digital holography system is investigated and demonstrated to be an important systematic axial measurement error (SAME) source, especially for features with high frequency. The properties of the space-variant SAME are investigated through simulations and experiments. The object position, object height, object frequency content and object-CCD distance are found to be related to the space-variant SAME. Careful and appropriate placement of the object according to its features is thus necessary to reduce such SAME in a DH system. Based on the investigation, the guideline to appropriately position an object according to its properties is provided in this work.
1. Introduction Digital holography (DH) [1–3] provides an easy way to obtain quantitative phase distribution containing depth information of an object and allows 3D imaging of the object. DH has thus been widely applied in microscopic 3D measurement and imaging such as 3D cell imaging [4], 3D micro-structure measurement [5], 3D micro-particles tracking [6], etc. The theoretical systematic axial measurement error (SAME) of DH is smaller than 1 nanometers, where the quantization effect of CCD pixel is considered to be the only error source [4]. However, in practice, the SAME is dramatically increased to tens of nanometers [5–9], implying the existence of other axial error sources. The phase error caused by lens aberrations has been studied [10–12], which does not happen in lensless DH. In [13], The CCD aperture size has been shown to be an important contributor to the SAME, but the error it caused is smaller than that from practical experiments [13], hinting that there exist other unrevealed error sources. The space-variance effect (SVE) has been found to affect the lateral resolution [14,15], but its influence on the SAME remains unknown and thus examined in this paper. Lensless DH system is considered so that the phase aberrations of microscope objectives and other lenses are all excluded. Through simulation and experiments, the SAME caused by the SVE will be confirmed, from which, how to position the object being measured is discussed. The rest of the paper is organized as follows. In Section 2, the principle of DH and the SVE are briefly introduced. In Section 3, the impact of ∗
the SVE on the SAME is examined for both point and step objects through simulation, and the results are analyzed. In Section 4, the experiments are performed to validate our analysis. In Section 5, the conclusions of this work are drawn. 2. Digital holography and its space-variance effect In this section, the principle of off-axis DH and its SVE are briefly introduced. 2.1. Principle of DH As shown in Fig. 1, off-axis DH includes two processes: digital recording and numerical reconstruction. In the digital recording process, a light wave illuminates the object. The object wave O is reflected by an opaque object or transmits through a transparent object, and carries the object information. The reference wave R illuminates the CCD with an angle of 𝜃. The two waves interfere to generate the hologram 𝐼𝐻 (x, y) at the CCD plane with the following resulting intensity: 𝐼𝐻 (x, y) = |𝑂(𝑥, 𝑦) + 𝑅(𝑥, 𝑦)|2 = |𝑂(𝑥, 𝑦)|2 + |𝑅(𝑥, 𝑦)|2 + 𝑅(𝑥, 𝑦)𝑂∗ (𝑥, 𝑦) + 𝑅∗ (𝑥, 𝑦)𝑂(𝑥, 𝑦)
(1)
where ∗ denotes the complex conjugate. After the sampling and digitalization by the CCD camera, a digital hologram is formed. In the numerical reconstruction process, the third or the fourth terms of Eq. (1) can be filtered out numerically due to their spatial
Corresponding author. E-mail address: [email protected] (P. Cai).
https://doi.org/10.1016/j.optlaseng.2018.06.009 Received 1 February 2018; Received in revised form 18 May 2018; Accepted 12 June 2018 0143-8166/© 2018 Published by Elsevier Ltd.
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Fig. 1. An illustration of an off-axis DH system. 𝑗𝜋 2 at the CCD plane can be written as exp[ 𝜆𝑑 (𝑥 + 𝑦2 )], which will be limited by the finite CCD sensing area 𝐿𝑥 × 𝐿𝑦 , resulting in the following truncated wavefront, ] [ ( ) ( ) ) 𝑗𝜋 ( 2 𝑦 𝑥 𝑊𝐴 (𝑥, 𝑦) = exp rect . (6) 𝑥 + 𝑦2 × rect 𝜆𝑑 𝐿𝑥 𝐿𝑦
Similarly, the wavefront from point B located at (𝑥, 𝑦) = (𝑥0 , 𝑦0 ) can be written as } { [ ( ) ( ) )2 ( )2 ] 𝑗𝜋 ( 𝑥 𝑥 𝑥 − 𝑥0 + 𝑦 − 𝑦0 𝑊𝐵 (𝑥, 𝑦) = exp × rect rect 𝜆𝑑 𝐿𝑥 𝐿𝑦 ( ) ≠ 𝑊𝐴 𝑥 − 𝑥0, 𝑦 − 𝑦0 , (7) which shows that the recording process is space-variant. The physical process of light diffraction follows that the light containing frequency 𝑓 of an object deflects from its incident direction with an angle α with a relation of 𝑓 = 𝑠𝑖𝑛𝛼 . For the point A located at (𝑥, 𝑦) = 𝜆 (0, 0), the diffracted lights which could be collected by CCD are in the an-
Fig. 2. An illustration of the space-variance effect of DH. The propagation and recording of the wavefronts from two points A and B.
−𝐿 ∕2−𝑥
𝐿
𝑥 direction while the bandwidths of point B is (− 2𝜆𝑑 −
𝐿𝑦
𝐼𝑚[𝛾(𝑥, 𝑦)] . 𝑅𝑒[𝛾(𝑥, 𝑦)]
𝐿 ∕2−𝑦
𝐿𝑦
𝑦0 , 𝜆𝑑 2𝜆𝑑
𝑦0 ) 𝜆𝑑
𝑥0 𝐿𝑥 , 𝜆𝑑 2𝜆𝑑
−
𝑥0 ) 𝜆𝑑
in 𝑥
direction and (− 2𝜆𝑑 − − in 𝑦 direction respectively. It is noticed that both bandwidths are finite but they are not the same. The finite bandwidths indicate that the CCD truncation only retains partial information of an object point. The information loss contributes to the SAME, which has been examined in [13]. The different bandwidths indicate that different points suffer different information loss and thus the ultimate SAME is also space-variant. It is thus necessary and significant to examine the severity of the non-uniformity of SAME. If this error variation is small, then the object can be freely positioned within the object plane of the DH system. Otherwise, the object positioning has to carefully follow a guideline to avoid the occurrence of severe space-variant SAME.
where 𝜆 is the laser wavelength; 𝑑 is the reconstruction distance;𝑓𝑥 and 𝑓𝑦 are the coordinates in the spectrum domain. The intensity and phase can be calculated as
𝜑(𝑥, 𝑦) = 𝑎𝑟𝑐𝑡𝑎𝑛
−𝐿 ∕2−𝑦
𝐿 ∕2−𝑥
𝑡𝑎𝑛𝛼 ∈ [ 𝑥 𝑑 0 , 𝑥 𝑑 0 ] in 𝑥 direction and 𝑡𝑎𝑛𝛽 ∈ [ 𝑦 𝑑 0 , 𝑦 𝑑 0 ] in 𝑦 direction respectively. In practice, the distance 𝑑 is usually quite large than the CCD widths 𝐿𝑥 and 𝐿𝑦 such that we have the approximation of 𝑠𝑖𝑛𝛼 ≅ 𝑡𝑎𝑛𝛼 and 𝑠𝑖𝑛𝛽 ≅ 𝑡𝑎𝑛𝛽. Hence the bandwidths of point 𝐿 𝐿 𝐿𝑥 𝐿𝑥 A recorded by CCD are (− 2𝜆𝑑 , 2𝜆𝑑 ) in 𝑥 direction and (− 2𝜆𝑑𝑦 , 2𝜆𝑑𝑦 ) in 𝑦
where and −1 denotes Fourier and inverse Fourier transforms respectively; 𝐺(𝑓𝑥 , 𝑓𝑦 ) is the transfer function of an optical system expressed as [ ] √ ( ) ( )2 ( )2 𝑑 𝐺 𝑓𝑥 , 𝑓𝑦 = exp 𝑗2𝜋 1 − 𝜆𝑓𝑥 − 𝜆𝑓𝑦 (3) 𝜆
𝐼 (𝑥, 𝑦) = |𝛾(𝑥, 𝑦)|2 ,
𝐿 ∕2 𝐿 ∕2
𝐿 ∕2 𝐿 ∕2
gle range of 𝑡𝑎𝑛𝛼 ∈ [− 𝑥𝑑 , 𝑥𝑑 ] in 𝑥 direction and 𝑡𝑎𝑛𝛽 ∈ [− 𝑦𝑑 , 𝑦𝑑 ] in 𝑦 direction respectively. However, for the point B located at (𝑥, 𝑦) = (𝑥0 , 𝑦0 ), the diffracted lights collected by CCD are in the angle range of
carrier introduced by the off-axis geometry. We take the fourth term 𝑅∗ (𝑥, 𝑦)𝑂(𝑥, 𝑦) for example. A quantitative object wavefield at the object plane can be computed from the fourth term of Eq. (1) by multiplying the reference wave 𝑅(𝑥, 𝑦) followed by Fresnel diffraction integration as [16,17]: { { } ( )} 𝛾(𝑥, 𝑦) = −1 𝑅(𝑥, 𝑦)[𝑅∗ (𝑥, 𝑦)𝑂(𝑥, 𝑦)] × 𝐺 𝑓𝑥 , 𝑓𝑦 (2)
(4)
(5)
From phase 𝜑(𝑥, 𝑦), the desired object height in the axial direction can be obtained. 2.2. Space-variance effect of digital holography
3. Simulation investigation of the space-variance effect on system axial measurement error
The SVE exists in any practical systems and DH is not an exception. Two point sources A and B at different locations in Fig. 2 are used for explanation. Point A is located at (𝑥, 𝑦) = (0, 0). Its propagated wavefront
In this section, the impact of the SVE on the SAME is investigated by simulation. Simulation is adopted because the phase retrieval involving recording and reconstruction is a non-linear process and is difficult for 17
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Fig. 3. The PSF images at different spatial positions. (a) the 3D height profile of the PSF at the center of object plane; (b) the section across the center of the PSF in (a) along x axis; (c) the 3D height profile of the PSF at −1.4 mm; (d) the section across the center of the PSF in of (c) along x axis; (e) the variations of axial error of PSF with respect to the positions and height.
theoretical analysis. Nevertheless, the simulation results will be compared and supported by real experiments. Two types of objects, point sources and step objects, are investigated. The point spread function (PSF) is a good indication of the performance of DH system. Therefore, the impact of the SVE on SAME of the PSF is firstly investigated. The ideal PSF of a DH system is 𝛿(𝑥, 𝑦), but in a practical system it has a space-variant spread, due to the bandwidthlimit and space-variance discussed in Section 2.2. Consequently, a spatially varying SAME is resulted. Thus the spatial variation of the PSF should be examined. Given a point object 𝛿(𝑥 − 𝑥0 , 𝑦 − 𝑦0 ) and the reconstructed object point𝛾(𝑥, 𝑦), it is readily to find that the PSF can be determined as ( ) PSF(𝑥0 ,𝑦0 ) (𝑥, 𝑦) = 𝛾 𝑥 + 𝑥0 , 𝑦 + 𝑦0 .
We consider a point object with a height of 100 nm with the following simulation parameters: the CCD size is 𝐿𝑥 = 3.72mm, 𝐿𝑦 = 4.76 mm; the wavelength 𝜆 is 633 nm; the reconstruction distance 𝑑 is 123 mm; the pixel size is T = 4.65 μm; an off-axis lensless geometry is adopted where the carrier frequency is a = 53.76 × 103 Hz determined by the angle 𝜃. We first put the point object at (𝑥0 , 𝑦0 ) = (0, 0) mm, i.e., the height of such point object is 𝑧(𝑥, 𝑦) = 100 × 𝛿(𝑥, 𝑦) nm. The reconstructed point object, PSF(0,0) (𝑥, 𝑦), is shown in Fig. 3(a) with a central section plotted at Fig. 3(b). This PSF is like a spread delta function mainly due to the limited CCD size. Next, we put the point object at (𝑥0 , 𝑦0 ) = (−1.4, 0)mm, i.e., the height of point object is 𝑧(𝑥, 𝑦) = 100 × 𝛿(𝑥 + 1.4, 𝑦) nm, and reconstruct the object again. The reconstructed function is then shifted by 1.4 mm horizontally according to Eq. (8) and obtain PSF(−1.4,0) (𝑥, 𝑦),
(8) 18
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Fig. 4. An illustration of the space-variant SAME of a step object. (a) An example of the true image of a step object of 100 nm height; (b) the reconstructed image of the focused step with 𝑥0 = 0 mm; (c) the SAME with 𝑥0 = 0 mm; (d) the reconstructed image of the focused step with 𝑥0 = −1.3 mm; (e) the SAME with 𝑥0 = −1.3 mm.
as shown in Fig. 3(c) and (d). As expected, first, PSF(−1.4,0) (𝑥, 𝑦) ≠ PSF(0,0) (𝑥, 𝑦), verifying the SVE of the DH system; second, PSF(−1.4,0) (𝑥, 𝑦) is more distorted from a delta function and is asymmetrical, showing lower imaging quality. In fact, for (𝑥0 , 𝑦0 ) = (0, 0), the reconstructed object height error is 16.6 nm, but for (𝑥0 , 𝑦0 ) = (−1.4, 0), the error increases to 42.1 nm, which is too large in a high precision measurement of an object of 100 nm height. For the point sources with different heights, the axial errors are calculated as compared in Fig. 3(e). It is seen that larger height has larger axial errors and the non-uniformity of the space-
variant axial error is more severe. Therefore, the space-variant SAME exists and the non-uniformity of the space-variant SAME of PSF is severe and cannot be ignored. The object positioning has to be carefully considered in order to avoid the occurrence of such severe SAME. In practice, most of the objects are not isolated points but of finite size. Among objects of finite size, step objects are investigated in this work because they are often seen in real imaging, contain rich frequency information, and usually have poor imaging quality. We define a step
19
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Fig. 5. The quantitative evaluations of the impact of step heights on the space-variant SAME. (a) the PAE and (b) the MAE of step region with different heights H ranging from −600 nm to 600 nm and at five different locations 𝑥0 = 1.2, 0.6, 0, − 0.6, − 1.2 mm; (c) the PAE and (d) the MAE of the step region; (e) the PAE and (f) the MAE of the flat region at different locations 𝑥0 ranging from −1.4 mm to 1.4 mm and of different heights H = 40, 100, 160, 220 and 280 nm.
object of height H as below: [ ] { 𝐻 , (𝑥, 𝑦) ∈ −2.38, 𝑥0 × [−2.38, 2.38] 𝑧(𝑥, 𝑦) = , 0, otherwise
±2.38mm, while the last one at 𝑥 = 𝑥0 is variable when 𝑥0 changes. The SVE on this varying step is concerned and investigated. The object with 𝑥0 = 0 and 𝐻 = 100 nm is shown in Fig. 4(a) as an example, and its reconstructed height images is shown in Fig. 4(b). For clarity, the reconstruction error is also shown in Fig. 4(c). The concerned edge at 𝑥0 = 0 mm has an axial error of about 30.47 nm. We then re-
(9)
where the unit of x and y is mm and that for z is nm. This object contains four step edges, among which, three are fixed at 𝑥 = −2.38mm and 𝑦 = 20
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Fig. 6. The quantitative evaluations of the impact of CCD size on the space-variant SAME of step objects. (a) the PAE; (b) the MAE at the step region and (c) the PAE; (d) the MAE at the flat region at different locations 𝑥0 ranging from −1.4 mm to 1.4 mm and of different CCD sizes Lx = 3.72, 5.12, 6.51, 7.91 and 9.30 mm.
locate this edge at 𝑥0 = − 1.3 mm, with the reconstructed height image and the reconstruction error shown in Fig. 4(d) and (e), respectively. This time, the axial error of the concerned edge decreases to 11.49 nm. The non-uniformity of SAME reaching 18.98 nm is large in the measurement of a jump height of 100 nm, which demonstrates the SVE as an important SAME contributor. For the flat region of the step excluding the jump site where only low frequency information is involved, the maximum error of Fig. 4(c) and (e) are 1.70 nm and 5.36 nm, respectively. This is because the flat regions have very low frequency to be adequately captured by CCD with little information loss. To obtain a more systematic view of SVE, a series of detailed simulation are carried out. The SAME is quantitatively examined using the peak axial error (PAE) and the mean axial error (MAE), defined as the peak error and mean absolute error of a region, respectively. Two regions are considered: one is the step region including 50 pixels to the right and 50 pixels to the left of the step, and the other is a flat region excluding 50 pixels near the step and object boarders. First, the impact of step height on the space-variant SAME is investigated. (i) For five typical edge locations 𝑥0 , the step height H changes in a range from −600 nm to 600 nm. Both PAE and MAE of the step region are shown in Fig. 5(a) and (b), respectively. Periodicity is observed with a period of λ∕2 . This is reasonable because, when the height changes λ∕2, the optical path changes λ in our simulation, then the optical wavefront remains the same, and consequently the phase error is the same. This is a good property so that one does not need to concern the height amount of a specimen. Among all height values, the height values gen-
erating the optical path lengths of integral multiple of wavelength 𝑁λ have the smallest axial error. (ii) For five representative height values, the edge location changes in a range from −1.4 mm to 1.4 mm with an increment of 4.65 μm, along x-axis in object plane. The axial error results are shown in Fig. 5(c) and (d), where the SVE is clearly observed. Take the plot of height 160 nm in Fig. 5(c) as an example, the PAE value changes from 5 nm to about 70 nm at different locations which reveals the serious impact of SVE. Furthermore, it is shown in Fig. 5(c) and (d) that steps of different heights have quite different space-variant axial error performances. Hence, the SAME is not only space-variant but also height-variant. These phenomena indicate that SVE is an important SAME source and careful and appropriate positioning of the object in experiments is necessary. Generally speaking, the overall error is smallest at 𝑥0 = 0. (iii) As observed from Fig. 5(e) and (f), the PAE and MAE of the flat region are also space-variant. However the amplitude of the axial error is much smaller and the non-uniformity is less obvious than those of the jump region which indicates the jump site containing high frequency information contents has larger SAME than the flat region containing lower frequency information contents. The reason is that, when two objects containing different frequency contents pass the same bandwidth-limit DH filter, the loss of the object containing low frequency contents is less than the loss of the object with high frequency contents. More information loss results in larger space-variant SAME. Thus, the space-variant SAME is related to object frequency components and the features with high frequencies are more vulnerable to the space-variant SAME. 21
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Fig. 7. The quantitative evaluations of the impact of the distance on the space-variant SAME. (a) the PAE; (b) the MAE at the step region and (c) the PAE; (d) the MAE at the flat region of different positions with 𝑥0 ranging from −1.4 mm to 1.4 mm and with different object distances d = 30, 60, 120, 240, 480 mm.
Second, the impact of CCD size on the space-variant SAME is investigated. For five representative CCD sizes Lx, the edge location 𝑥0 changes in a range from −1.4 mm to 1.4 mm with an increment of 4.65 μm, along x-axis in object plane. The axial error results are shown in Fig. 6. The PAE and MAE of the step region are shown in Fig. 6(a) and (b), respectively. The PAE and MAE of the flat region are shown in Fig. 6(c) and (d), respectively. Increasing the size of CCD reduces the space-variant SAME. This is because increasing the CCD size increases the bandwidth of the DH filter and thus alleviates the information loss and the resulting space-variant SAME duo to filtering. As observed from Fig. 6(c) and (d), the PAE and MAE of the flat region is much smaller than those of the jump region in Fig. 6(a) and (b), which verifies that the space-variant SAME is related to object frequency components and the features with high frequencies are more vulnerable to the space-variant SAME. Third, the impact of the distance between object and CCD on the space-variant SAME is examined. For five representative distances d, the edge location 𝑥0 changes in a range from −1.4 mm to 1.4 mm with an increment of 4.65 μm, along x-axis in object plane. The PAE and MAE of the step region are shown in Fig. 7(a) and (b), respectively, while those of the flat region are shown in Fig. 7(c) and (d), respectively. As increasing the distance between object and CCD is equivalent to decreasing the CCD size, and thus the observed phenomenon in Fig. 7 is similar to that in Fig. 6. Reducing the distance between object and CCD, if possible in practice, is encouraged. According to this simulation investigation, the non-uniformity of the space-variant SAME is seriously affected by object position, object sur-
face profile property, CCD size, and object-CCD distance. Accordingly, the following are suggestions for DH practice: (i) Generally speaking, the object should be positioned on the optical axis perpendicular to and passing through the CCD center, as the objects spectrum is best conserved in this location; (ii) If the object has both flats and jumps (or both continuous and discontinuous parts), the jumps or discontinuous parts should be positioned on the optical axis as they are more vulnerable to spectrum information loss; (iii) If possible, a larger CCD is prefer, although it will increase the cost of the system; (iv) If possible, the object-CCD distance should be made small to increase the equivalent CCD size. 4. Experiment investigation of the space-variant system axial measurement error In this section, the space-variant SMAE is investigated by experiments. The first specimen is a US air force (USAF) target which is a standard resolution chart as shown in Fig. 8(a). The heights of the coated bars are all 100 nm, making the USAF target a good and readily available sample to study the axial measurement. In this experiment, Group 2 Element 3 (G2E3) is used for detailed examination. In our experiment, 56 holograms with the G2E3’s location shifting from −1.3 mm to 1.3 mm with an interval of 0.047 mm are recorded with a lensless DH system [18]. The height images of G2E3 at dif22
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Fig. 8. The quantitative evaluation of space-variant SAME for the USAF target in experiment. (a) the image of the specimen: USAF target; holograms with different locations 𝑥0 of (b) −0.77 mm; (c) 0 mm; (d) 0.77 mm; the reconstructed height images and profiles (insets) of G2E3 with object location 𝑥0 of (e) −0.77 mm; (f) 0 mm; (g) 0.77 mm; (h) the PAE; (i) the MAE at the step region and (j) the PAE; (k) the MAE at the flat region of different values of 𝑥0 ranging from −1.3 mm to 1.3 mm.
ferent locations are reconstructed. Three of the 56 holograms at locations −0.77 mm, 0, 0.77 mm are shown in Fig. 8(b), (c) and (d), respectively. Their corresponding reconstructed height images are presented in Fig. 8(e), (f) and (g), respectively. The corresponding height profiles of G2E3 are drawn in the insets. The height profiles show that the reconstructions and the corresponding SAMEs are space-variant. Such SVE of the SAME is obvious especially at the jump site of the step where high frequencies present. The non-uniformity of SAME among Fig. 8(e), (f) and (g) reaches about 20 nm which is large in the measurement of a step height of 100 nm and demonstrates the SVE as an important SAME contributor in experiment.
To quantitatively evaluate the SAME, the PAE and MAE of the right step region and a flat region at the different locations are calculated and shown in Fig. 8(h), (i), (j) and (k). It is seen that, as the step shifts from −1.3 mm to 1.3 mm, the PAE changes with ringing in a range from 10 nm to 70 nm at different locations which quantitatively verifies the existence of severe space-variant SAME. The non-uniformity of PAE reaching 60 nm is large in the measurement of a step of 100 nm height. This indicates that SVE is an important SAME source and appropriate positioning of the object is necessary. It is observed the PAE and MAE of the flat region in Fig. 8(j) and (k) are also space-variant. However, their amplitudes are much smaller and 23
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Fig. 9. The quantitative evaluation of space variant SAME for a step object on a MEMS target in experiment. (a) The hologram of the MEMS specimen; (b) the reconstructed intensity image of (a); (c) the reconstructed phase image of (a); (d) the PAE; (e) the MAE at the step region and (f) the PAE; (g) the MAE at the flat region of different locations 𝑥0 ranging from-0.8 mm to 0.8 mm. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
the non-uniformity is less obvious than those of the jump region which demonstrate the jump site containing high frequency information contents has larger SAME than the flat region containing lower frequency information contents. These results agree with the simulation investigation in last section that the space-variant SAME is related to object frequency components and the features with high frequencies are more vulnerable to the space-variant SAME. We further investigate another MEMS specimen which is also a step object. The hologram and the reconstructed intensity image and phase image with the step located at the center are shown in Fig. 9(a), (b) and (c). In this experiment, a region of interest (ROI) on the object, highlighted by the blue square in Fig. 9(c), is further examined. In the experiment, 112 holograms with the ROI’s location shifting from −1.3 mm to 1.3 mm with an interval of 0.023 mm are recorded
with the same lensless DH system [18]. The PAE and MAE of the step region and a flat region in the ROI at the different locations are calculated and shown in Fig. 9(d), (e), (f) and (g). It is seen that, as the step shifts from −1.3 mm to 1.3 mm, the PAE changes with ringing in a range from 60 nm to 140 nm at different locations which quantitatively verifies the existence of severe space-variant SAME. The non-uniformity of PAE reaching 80 nm is large which proves SVE is an important SAME source. The above two experiments verify the existence and the severe nonuniformity of the space-variant SAME which indicates that the appropriate and careful positioning of the object according to its frequency contents is quite necessary in order to suppress such space variant SAME.
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5. Conclusions
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In this work, the impact of space-variance effect on the systematic axial measurement error (phase error) in DH is investigated by both simulation and experiment. The results show that the space-variance effect has an important impact on the systematic axial measurement error (SAME). Furthermore, it is found that the features of higher frequency information components are more vulnerable to such spacevariant SAME than those of lower frequency information. The influences of object height, the size of CCD and CCD-object distance on such spacevariant SAME are all investigated. According to our results, careful and appropriate positioning of the object is necessary to suppress such spacevariant SAME. The guideline to appropriately position an object according to its properties is also provided in this work. Acknowledgment This work was supported by the National key R&D Program of China (2016YFF0200700), the National Natural Science Foundation, China (61405111,61502295), and Shanghai Engineering Research Center for Intelligent diagnosis and treatment instrument(15DZ2252000). References [1] Schnars U, Juptner W. Direct recording of holograms by a CCD target and numerical reconstruction. Appl Opt 1994;33(January):179–81. [2] Schnars U, Juptner WPO. Digital recording and numerical reconstruction of holograms. Meas Sci Technol 2002;13(September):R85–R101. [3] Cuche E, Bevilacqua F, Depeursinge C. Digital holography for quantitative phase– contrast imaging. Opt Lett 1999;24(March):291–3. [4] Pandey N, Hennelly B. Quantization noise and its reduction in lensless Fourier digital holography. Appl Opt 2011;50(, March):B58–70.
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