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Accuracy concerns in digital speckle photography combined with Fresnel digital holographic interferometry
Optics and Lasers in Engineering 104 (2018) 84–89
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Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Accuracy concerns in digital speckle photography combined with Fresnel digital holographic interferometry Yuchen Zhao a, Redouane Zemmamouche b,c, Jean-François Vandenrijt a, Marc P. Georges a,∗ a
Centre Spatial de Liège-Université de Liège, Liege Science Park, B-4031, Angleur, Belgium Université Ferhat Abbas, Sétif1, Institut d’Optique et Mécanique de Précision, Laboratoire d’Optique Appliquée, Avenue Said Boukhrissa, Sétif 19000, Algeria c University Med Boudiaf, Faculty of Technology, Mechanical Department, Msila 28000, Algeria b
a r t i c l e
i n f o
Keywords: Digital holography Digital speckle photography Nondestructive testing Metrology
a b s t r a c t A combination of digital holographic interferometry (DHI) and digital speckle photography (DSP) allows in-plane and out-of-plane displacement measurement between two states of an object. The former can be determined by correlating the two speckle patterns whereas the latter is given by the phase difference obtained from DHI. We show that the amplitude of numerically reconstructed object wavefront obtained from Fresnel in-line digital holography (DH), in combination with phase shifting techniques, can be used as speckle patterns in DSP. The accuracy of in-plane measurement is improved after correcting the phase errors induced by reference wave during reconstruction process. Furthermore, unlike conventional imaging system, Fresnel DH offers the possibility to resize the pixel size of speckle patterns situated on the reconstruction plane under the same optical configuration simply by zero-padding the hologram. The flexibility of speckle size adjustment in Fresnel DH ensures the accuracy of estimation result using DSP. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Digital speckle photography (DSP) is a simple and robust method for tracking in-plane deformation. The displacement is determined by numerically cross-correlating the speckle patterns generated on the surface of a scattering object under coherent illumination at the initial and deformed states [1]. A displacement over several tens of speckle diameter can be obtained with subpixel accuracy if crucial parameters such as speckle size and finite window size are properly defined when using DSP [2]. DSP can be combined with interferometry techniques based on speckle effect. Interferometry techniques including electronic speckle pattern interferometry (ESPI) (also named digital speckle pattern interferometry and TV–holography) and speckle shearing interferometry (also known as shearography) allow a measurement of out-of-plane displacement fields in their usual geometric configuration where the object illumination is in the same direction as the field of view of the observation system [3]. Implementing DSP on these interferometry techniques is useful for obtaining simultaneously in-plane and out-of-plane displacements, which have always been of great interest for industrial non-contact metrology applications [4–8]. One necessary condition for obtaining accurate results from DSP is that the speckle size should be larger than two pixels to meet sampling requirement [2]. In all these
∗
Corresponding author. E-mail address: [email protected] (M.P. Georges).
http://dx.doi.org/10.1016/j.optlaseng.2017.09.011 Received 30 April 2017; Received in revised form 31 July 2017; Accepted 8 September 2017 Available online 14 September 2017 0143-8166/© 2017 Elsevier Ltd. All rights reserved.
cases, speckle patterns are imaged onto a CCD camera, which can be sufficiently resolved by correctly setting the imaging aperture. Digital holographic interferometry (DHI) is another technique which allows measuring out-of-plane movements based on digital holography (DH). Differing from other interferometry techniques, DH records the holograms and reconstructs numerically the object wavefield based on diffraction theory [9,10]. The phase difference of object at two instants relates to the out-of-plane displacements located on each point of the object. On the other hand, the reconstructed amplitude of a diffused object also contains speckles and thus can be used for in-plane displacement measurement. A combination of DHI and DSP provides access to threedimensional displacement. However, compared with the DSP based on imaged speckle patterns, large bias and standard deviation errors are reported in earlier works when applying DSP on DH reconstructed images [11,12]. In this paper, we present a combination of DSP and phase-stepping Fresnel digital holography, making it possible to measure in-plane and out-of-plane displacement of a large object at long working distance. Our objective is to investigate the causes of low accuracy in in-plane displacement measurement reported in [11,12]. We demonstrate that phase errors caused by slightly divergent reference beam contribute to a complementary speckle displacement. Moreover, the speckle features in the reconstructed plane are not resolved enough to match the speckle
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Optics and Lasers in Engineering 104 (2018) 84–89
size requirement for DSP, which leads to systematic errors in in-plane displacement. We show experimentally that the in-plane displacement measurement can be significantly improved after these two sources of error are corrected. In Section 2 we recall the theoretical basis of DHI and DSP, based on which sources of error existing in in-plane displacement measurement are derived analytically. In Section 3 we describe the setup. In Section 4, we present different experimental results. Each source of error is examined and corrected. Finally some conclusions and discussion are presented in the last section.
2. Theoretical analysis 2.1. Digital holography interferometry combined with digital speckle photography
Fig. 1. Schematic of the experimental setup.
DH consists of recording the holograms with an array sensor and numerically reconstructing the original wavefield at given distances based on the recorded holograms. During the recording process, reference wave UR (𝜉, 𝜂) interferes with object wavefield UO (𝜉, 𝜂) in the plane of CCD detector (𝜉, 𝜂), forming interference patterns, i.e., holograms IH (𝜉, 𝜂) given by: 𝐼𝐻 =|𝑈𝑅 |2 + |𝑈𝑂 |2 + 𝑈 ∗ 𝑅 𝑈𝑂 + 𝑈𝑅 𝑈 ∗ 𝑂 ,
𝑈𝑂 (𝑚, 𝑛, 𝑧) =
×
𝑛𝜋 2 )| , 2
(1)
𝐴𝑂 (𝑚, 𝑛) =|𝑈𝑂 (𝑚, 𝑛)|,
(6)
and
[ ] 𝜑𝑂 (𝑚, 𝑛) = arg 𝑈𝑂 (𝑚, 𝑛) .
(7) ( ) For two states of the object where a displacement 𝑑⃗ 𝑑𝑥 , 𝑑𝑦 , 𝑑𝑧 has taken place, after reconstruction using Eqs. (5)–(7), the difference between the phase at initial state 𝜑O, ini and the phase at displaced state 𝜑O, disp is related to optical path difference caused by applied displacement given by Δ𝜑𝑂 =𝜑𝑂, 𝑑𝑖𝑠𝑝 − 𝜑𝑂, 𝑖𝑛𝑖 = 𝑑⃗ ⋅ 𝑆⃗ ,
(2)
(8)
where 𝑆⃗ stands for the sensitivity vector which is determined by the directions of illumination and observation [3]. In accordance with the geometrical setup depicted in Fig. 1, Eq. (8) can be written with the illuminating angle 𝜃(m, n) on each point of object:
(3)
2𝜋 (9) [sin 𝜃(𝑚, 𝑛)𝑑𝑥 + (1 + cos 𝜃(𝑚, 𝑛))𝑑𝑧 ]. 𝜆 When applying DHI, the amplitude images AO, disp (m, n) and AO, ini (m, n) which carry the speckle patterns of two states are also obtained during reconstruction. The amplitude images can be processed by DSP for inplane displacement measurement. DSP technique consists of computing the speckle displacement between two digitized speckle patterns. More precisely, in the algorithm proposed by Shjödahl [1], DSP computes the cross-correlation of subimages (typically 32 × 32 pixels) taken at the same location in two speckle patterns. Locating the correlation peak gives the displacement measurement with subpixel accuracy. Extensive research has been carried out on the accuracy concerns of DSP such as the influence of finite window size, speckle size and measurement range based on imaging systems [1,2,15], which should be re-adapted into DH configurations. Δ𝜑𝑂 (𝑚, 𝑛) =
If the complex amplitude distribution of the reference wavefront UR (𝜉, 𝜂) is known, the object wavefield located in the object plane (x, y) at distance z from the hologram plane, UO (x, y, z), can be retrieved by back propagating the processed hologram HPS (𝜉, 𝜂) according to scalar diffraction theory. Under the paraxial approximation, the reconstructed wavefield at object plane UO (x, y, z) can be calculated by the Fresnel transform [3,9,10]: ( ) [ )] i 2𝜋 𝜋 ( 2 exp −i 𝑧 exp −i 𝑥 + 𝑦2 𝜆𝑧 𝜆 𝜆𝑧 [ )] 𝜋 ( 2 𝜉 + 𝜂2 × 𝑈𝑅 (𝜉, 𝜂)𝐻𝑃 𝑆 (𝜉, 𝜂) exp −i ∬ 𝜆𝑧 [ ] 2𝜋 × exp i (𝑥𝜉 + 𝑦𝜂) 𝑑 𝜉𝑑 𝜂. 𝜆𝑧
(5)
Comparing with the definition of Inverse Discrete Fourier Transformation (IDFT), the sum part in Eq. (5) can be rewritten as IDFT of the [ ( )] product 𝑈𝑅 𝐻𝑃 𝑆 exp −i𝜋∕𝜆𝑧 𝑘2 Δ𝜉 2 + 𝑙2 Δ𝜂 2 , which can be calculated efficiently with fast Fourier transform (FFT) algorithm. The reconstruction of HPS with Eq. (5) gives the both the amplitude AO and the phase map 𝜑O of the observed object at distance z, where
𝐻𝑃 𝑆 (𝜉, 𝜂) =
𝑈𝑂 (𝑥, 𝑦, 𝑧) =
𝑈𝑅 (𝑘, 𝑙)𝐻𝑃 𝑆 (𝑘, 𝑙)
[ [ ( )] )] 𝜋 ( 2 2 𝑘𝑚 𝑙𝑛 × exp −i + . 𝑘 Δ𝜉 + 𝑙2 Δ𝜂 2 exp i2𝜋 𝜆𝑧 𝑀 𝑁
and ) ( )] 1 [( 𝐼 (𝜉, 𝜂) − 𝐼𝐻,2 (𝜉, 𝜂) − i 𝐼𝐻,1 (𝜉, 𝜂) − 𝐼𝐻,3 (𝜉, 𝜂) 4 𝐻,0 = 𝑈𝑅∗ (𝜉, 𝜂)𝑈𝑂 (𝜉, 𝜂).
𝑀−1 ∑ 𝑁−1 ∑ 𝑘=0 𝑙=0
where the asterisk denotes complex conjugates and the dependency to the coordinates (𝜉, 𝜂) is omitted for simplicity. Eq. (1) contains four terms corresponding to three diffraction orders: the 0 order is composed of terms |𝑈𝑅 |2 + |𝑈𝑂 |2 , the +1 order U∗ R UO which contains initial information of object wavefield, and –1 order UR U∗ O which is also known as twin image. The +1 order is of interest for reconstructing the object wavefield. Under in-line configuration, the unwanted diffraction orders superimposed on the +1 order can be eliminated through applying phase-shifting (PS) techniques [13]. This technique requires several captures of holograms whereby corresponding phase shifts are introduced between object and reference arms. For instance, a four-step PS algorithm proposed in [14] aimed at forming a new compound hologram HPS (𝜉, 𝜂) from 4 phase-shifted holograms IH, n (𝜉, 𝜂) with n = 0, 1, 2, 3: 𝐼𝐻,𝑛 (𝜉, 𝜂) =|𝑈𝑅 (𝜉, 𝜂) + 𝑈𝑂 (𝜉, 𝜂) exp(−i
[ ( )] ( ) i 2𝜋 𝜋 𝑚2 𝑛2 exp −i 𝑧 exp −i + 𝜆𝑧 𝜆 𝜆𝑧 𝑀 2 Δ𝜉 2 𝑁 2 Δ𝜂 2
(4)
Considering that the hologram is sampled by the array sensor on M × N pixels with dimensions Δ𝜉 × Δ𝜂, it is therefore necessary to digitize Eq. (4), yielding the computation of object wavefield on M × N discrete points, i.e.,
2.2. Influence of phase errors of DH on DSP measurement The reconstruction of object wavefield can be fully achieved during DH on the premise of reference wave parameters in the detector plane 85
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during the recording process. Ideally in Fresnel in-line DH, the propagated object wave interferes with a plane reference wave: ∗ 𝐻𝑃 𝑆, 𝑖𝑑𝑒𝑎𝑙 (𝜉, 𝜂) =𝑈𝑅, 𝑖𝑑𝑒𝑎𝑙 (𝜉, 𝜂)𝑈𝑂 (𝜉, 𝜂),
adjustment under a fixed optical configuration is more flexible thanks to zero-padding in numerical reconstruction, which we discuss hereafter. In Fresnel DH, the pixel resolution in the reconstruction plane Δx and Δy is given by
(10)
with 𝑈𝑅, 𝑖𝑑𝑒𝑎𝑙 (𝜉, 𝜂) = 𝑅0 . In actual experimental condition, in spite of collimation system adopted to the reference beam, small mismatches still remain. In addition, there exist other wavefront aberrations which are recorded on the hologram. The actual reference beam interfering with object wave contains a phase variation factor exp [i𝜑𝜀 (𝜉, 𝜂)] over the detector plane [16,17]: 𝐻𝑃 𝑆, 𝑎𝑐𝑡𝑢𝑎𝑙 (𝜉, 𝜂) = =
Δ𝑥 =
𝜎𝑥 =
𝜆𝑧 . 𝑁Δ𝜂
(14)
𝜆𝑧 ; 𝐷𝑥
𝜎𝑦 =
𝜆𝑧 . 𝐷𝑦
(15)
In the case of DH the rays come directly on the CCD sensor with dimensions 𝐷𝑥 × 𝐷𝑦 = 𝑀 Δ𝜉 × 𝑁 Δ𝜂. Combining Eqs. (14) and (15), we find that
(11)
𝜎𝑥 = Δ𝑥;
where the model introduced in [16] is employed hereby to describe the phase variation. The quadratic terms a𝜉 2 and b𝜂 2 in Eq. (11) result from the slight divergence of reference beam; c𝜉 and d𝜂 correspond to an arbitrary tilt of reference beam. Higher order of phase errors are omitted here for simplicity. The existence of uncorrected phase errors affects not only the quality of reconstructed image, but also the speckle displacement measured in the reconstruction plane. When a displacement of dx along x direction is applied on the object, the distribution of diffracted object wave in the hologram plane UO (𝜉, 𝜂) is shifted d𝜉 along 𝜉 direction. In the ideal case, the holo) ( gram 𝐻𝑃 𝑆, 𝑖𝑑𝑒𝑎𝑙 𝜉 − 𝑑𝜉 , 𝜂 records the interference between UO (𝜉, 𝜂) and ( ( ) ) 𝑈𝑅, 𝑖𝑑𝑒𝑎𝑙 𝜉 − 𝑑𝜉 , 𝜂 . The reconstruction of 𝐻𝑃 𝑆, 𝑖𝑑𝑒𝑎𝑙 𝜉 − 𝑑𝜉 , 𝜂 gives the information of initial object.
𝜎𝑦 = Δ𝑦.
(16)
This signifies that the speckle grain on the object occupies one pixel in the reconstruction plane which cannot meet the sampling criterion. To circumvent this, we can pad the raw hologram to 2M × 2N or even larger size with zeros. Although zero-padding does neither add any information into hologram nor change the intrinsic resolution, it helps decrease the pixel size in reconstruction plane [10]. One speckle grain can thus be represented by more pixels. The effect of zero-padding in the application of DSP will be shown later in this paper. 3. Experimental setup The optical setup used here is a classical in-line Fresnel holographic configuration, depicted in Fig. 1. The laser is a diode pumped solid state Nd:YAG laser with a wavelength of 532 nm and power of 400 mW from Coherent, Inc. The CCD camera is a Micam VHR 1000 with a pixel size of 8.6 × 8.3 μm2 . 480 × 480 pixels window is used for hologram recording. A beamsplitter (BS) divides the laser beam into a reference and an object beam. The reference beam is reflected by a mirror placed on a piezo-mount (M-PZT) for introducing the phase-shifting during recording. It is expanded by a beam expander which is constituted by an objective lens and spatial filter (SF) and followed by a collimation lens L1. The reference beam reaches the CCD detector through the beam combiner (BC). The object beam is expanded by lens L2 to illuminate the object with illuminating angle 𝜃 = 4.3◦ . The object is an aluminum plate covered with scattering white powder, mounted on a controllable translation and rotation stages at a distance z of 1400 mm from the camera sensor. The region of interest (ROI) on the illuminated object during measurement is 5.5 × 5.5 cm2 . The reference beam interferes with the wave reflected by the object on the surface of CCD detector.
[ ( ( )]} ) ) 𝜋 ( 2 𝜉 + 𝜂2 𝑈𝑂 (𝑥, 𝑦, 𝑧) = ℱ 𝑈𝑅, 𝑖𝑑𝑒𝑎𝑙 𝜉 − 𝑑𝜉 , 𝜂 𝐻𝑃 𝑆, 𝑖𝑑𝑒𝑎𝑙 𝜉 − 𝑑𝜉 , 𝜂 exp −i 𝜆𝑧 { [ )]} 𝜋 ( 2 = ℱ −1 𝑅20 𝑈𝑂 (𝜉, 𝜂) exp −i (12) 𝜉 + 𝜂2 𝜆𝑧 −1
Δ𝑦 =
The speckle size 𝜎 x and 𝜎 y observed on the object through an aperture of dimension Dx × Dy at a distance z from the latter is given by
∗ 𝑈𝑅, 𝑎𝑐𝑡𝑢𝑎𝑙 (𝜉, 𝜂)𝑈𝑂 (𝜉, 𝜂) ∗ 𝑈𝑅, 𝑖𝑑𝑒𝑎𝑙 (𝜉, 𝜂) exp[−i𝜑𝜀 (𝜉, 𝜂)]𝑈𝑂 (𝜉, 𝜂)
)] [ ( = 𝐻𝑃 𝑆, 𝑖𝑑𝑒𝑎𝑙 (𝜉, 𝜂) exp −i 𝑎𝜉 2 + 𝑏𝜂 2 + 𝑐𝜉 + 𝑑𝜂 ,
𝜆𝑧 ; 𝑀Δ𝜉
{
) ( If the hologram 𝐻𝑃 𝑆, 𝑎𝑐𝑡𝑢𝑎𝑙 𝜉 − 𝑑𝜉 , 𝜂 which contains uncorrected phase error is reconstructed in the same manner as in Eq. (12): { [ ( ( )]} ) ) 𝜋 ( 2 𝜉 + 𝜂2 𝑈𝑂′ (𝑥, 𝑦, 𝑧) = ℱ −1 𝑈𝑅, 𝑖𝑑𝑒𝑎𝑙 𝜉 − 𝑑𝜉 , 𝜂 𝐻𝑃 𝑆, 𝑎𝑐𝑡𝑢𝑎𝑙 𝜉 − 𝑑𝜉 , 𝜂 exp −i 𝜆𝑧 { [ ( )]} ) 𝜋 ( 2 𝜉 + 𝜂2 = ℱ −1 𝑅20 𝑈𝑂 (𝜉, 𝜂) exp[−𝑖𝜑𝜀 𝜉 − 𝑑𝜉 , 𝜂 ] exp −i 𝜆𝑧 ( ) ( ) 𝑎𝑑𝜉 𝑐 𝑑 = 𝑈𝑂 (𝑥, 𝑦, 𝑧) ⊗ 𝛿 𝑥 − ⊗𝛿 𝑦− + 2𝜋 𝜋 2𝜋 { [ ]} (13) ⊗ℱ −1 exp −i(𝑎𝜉 2 + 𝑎𝑑𝜉2 + 𝑏𝜂 2 − 𝑐𝑑𝜉 )
The inverse Fourier transform of phase error factor contains a Dirac delta term 𝛿(𝑥 − 2𝑐𝜋 + 𝑎𝑑𝜉 ∕𝜋) which contributes to a complementary displacement on the reconstructed images 𝑈𝑂′ (𝑥, 𝑦, 𝑧) of two relatively displaced states (d𝜉 ≠ 0). The speckle displacement 𝑑𝑥′ found in reconstruction plane does not correspond to original displacement of object dx . The same situation occurs if displacement took place along y direction. Thus a correction of reference wavefront by retrieving the quadratic and higher orders of 𝜑𝜀 (𝜉, 𝜂) is necessary prior to taking reconstructed amplitude map for in-plane displacement measurement. For quadratic terms, parameters a and b can be estimated by optimizing the sharpness of reconstructed image [13,17]. For more precise correction, the higher order aberrations also need to be retrieved by applying aberration correction methods which can be found in [17].
4. Results 4.1. Effect of phase error correction on in-plane displacement measurement We have first studied how phase errors existing in DH reconstruction affect the DSP measurement result. To avoid the errors induced by undersampling which will be further discussed in the next subsection, the holograms of 480 × 480 pixels hereby have been properly padded to 1152 × 1152 pixels. The size of subimage is 32 × 32 pixels. The result is shown in Fig. 2. The object is first replaced by a grid target for observing the quality of reconstructed image. Reconstructing the recorded hologram with Eq. (10), the reconstructed image is degraded on the horizontal direction, shown in Fig. 2(a). Fig. 2(b) shows the DSP in-plane displacement measurement results under the uncorrected reconstruction. The measurement error is plotted in Fig. 2(c). All the 13 displacements are severely underestimated. Then we compensated the quadratic phase errors by optimizing the sharpness of reconstructed images. The focal plane detection criterion demonstrated by Dubois et al. [19] is applied here to quantify the image sharpness. For amplitude-contrast object, the integral of amplitude
2.3. Speckle size adjustment in DH The speckle size has a decisive effect in DSP accuracy. First of all, the range of measurable displacement in DSP is limited by the decorrelation of speckle patterns to several tens of pixel size [18]. Besides, the speckle size should be greater than 2 pixels to satisfy the Nyquist sampling criterion. The consequence of undersampling is introducing a systematic error that pushes the estimate towards the closest integral pixel value [2]. In the cases where speckle patterns are imaged, the speckle should be resized by adjusting the F-number of imaging system. In DH, speckle size 86
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Fig. 2. (a) Zoomed-in reconstructed grid target. (b) DSP measurement and (c) measurement error under reconstruction without quadratic phase error correction. (d) Zoomed-in reconstructed grid target. (e) DSP measurement and (f) measurement error under reconstruction where quadratic phase error correction is applied. (g) Ratio of measured slope over actual slope (blue) and the value of critical function Md (red) versus the value of parameter a. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Table 1 Comparison of DSP measurement results under various zero-padding levels. Zero-padding N
480 480 480 480 960 960 960 960 1152 1152 1152 1152 1440 1440 1440 1440
Speckle size (px) 𝜎 x
Applied Translation (μm) dx
Applied Translation (px) kx
Average result
1 1 1 1 2 2 2 2 2.4 2.4 2.4 2.4 3 3 3 3
50 100 150 200 50 100 150 200 50 100 150 200 50 100 150 200
0.321 0.642 0.963 1.283 0.642 1.283 1.925 2.567 0.770 1.540 2.310 3.080 0.963 1.925 2.889 3.850
0.047 0.951 0.999 1.043 0.688 1.200 1.906 2.619 0.787 1.520 2.296 3.067 0.911 1.889 2.888 3.841
modulus (Md) is considered as a critical function which reaches minimum when optimum sharpness is achieved. Fig. 2(d) shows the image reconstructed by the corrected reference wavefront with parameter 𝑎 = 1.098 × 10−4 of Eq. (11). Compared to Fig. 2(a), the sharpness of reconstructed image is significantly improved. Fig. 2(e) shows the DSP results under corrected reconstruction with the same set of displacements. The measured displacements match the actual ones. Measurement error plotted in Fig. 2(f) confirms that the bias error is eliminated. In Fig. 2(g) we plotted the ratio of measured slope over actual slope as well as the value of critical function Md versus the value of parameter a. The ratio reaches 1 when the minimum of Md is found at 𝑎 = 1.098 × 10−4 .
std (px) sk
(px) 𝒌𝒙
Average result
std (μm) sd
(μm) 𝒅𝒎 0.041 0.045 0.029 0.052 0.041 0.036 0.037 0.073 0.042 0.074 0.069 0.059 0.067 0.071 0.092 0.102
7.026 148.1 155.6 162.5 53.60 93.53 152.8 204.1 51.10 98.70 149.1 199.2 47.30 98.13 149.9 199.5
6.394 3.554 4.512 8.125 3.377 1.403 2.903 5.715 2.708 2.369 4.622 3.785 3.500 1.864 1.499 5.367
The results obtained from correlating 81 pairs of subimages give the average result (𝑘𝑥 ) and the standard deviation (std) (sk ). The results under different zero-padding conditions N are summarized in Table. 1. When images are directly reconstructed by the recorded holograms without zero-padding, one speckle possesses one pixel as discussed in previous section. In this case, the systematic errors caused by undersampling described in [2] are clearly observed. Compared with the actual displacement kx , the measured result 𝑘𝑥 contains a drift toward the nearest integral pixel value. Up to 0.5 pixel of error can be introduced if DSP is combined with DH without zero-padding. Converting back to the real world coordinate, the error can be disastrous when the displacement is small (7.026 ± 6.394 μm for a translation of 50 μm). Then the holograms are padded with zeros to meet the Nyquist criterion. Fig. 3(a) shows a zoom on a given portion of the reconstructed image corresponding to a zone of approximately 7 × 7 mm2 on the object with different zero-padding applied. Insets show the subimages (32 × 32 pixels) of each speckle pattern where each single speckle can be identified. From the speckle patterns, we observe that the speckle is better resolved after a larger zero-padding. The absolute values of measured residuals relative to theoretical values of applied translation are plotted in Fig. 3(b). We confirm that when the number of pixels per speckle exceeds 2, this sys-
4.2. Effect of zero-padding on in-plane displacement measurement We have studied the effect of zero-padding on the ability of DH to generate speckle pattern images that are usable in DSP. The phase errors during reconstruction have been eliminated. Four different pure in-plane translations dx are applied on the object. They are converted to pixels kx with corresponding speckle size 𝜎 x on the reconstruction plane and measured with DSP algorithm. When the size of hologram is 480 × 480 pixels, measured area contains 9 × 9 subimages of 32 × 32 pixels. 87
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Fig. 3. Zero-padding effects. (a) Zoom of reconstructed Speckle pattern under different zero-padding levels. Insets on the top-left corner: subimages (32 × 32 pixels) under each case. (b) The absolute value of measured residuals relative to theoretical values of applied translation under different zero-padding cases.
Fig. 4. Simultaneous measurement by DHI and DSP. (a) Compound hologram after zero-padding. (b) Amplitude after reconstruction from (a). (c) In-plane displacement obtained by DSP on the basis of central part of (b). (d) Phase difference of two states when mixed movement is applied. (e) Computational result of phase difference caused by bulk in-plane displacement achieved from (c). (f) Restoration of phase difference caused by out-of-plane displacement. (g) Phase difference when only translation is applied (pure in-plane displacement). (h) Phase difference when only rotation is applied (pure out-of-plane displacement).
tematic error is successfully circumvented. In classical DSP, it is also proved that the random error, i.e., the std of measurement increases with the speckle size [2]. It is the same case here, when increasing the zero-padding value, the std sk increases. The zero-padding case 1152 × 1152 gives a satisfactory result when the subimage is 32 × 32 pixels.
recorded prior to the object movement and the second ones after. Afterwards zero-padding is applied with the suitable level, here with a final pixel number of 1152 × 1152 (Fig. 4(a)). The zero-padded compound holograms are injected in the Fresnel calculation Eq. (5) with corrected reference wavefront. The amplitude and phase of the reconstructed object wave are retrieved. Fig. 4(b) shows the ROI of reconstructed amplitude. Fig. 4(c) shows the in-plane displacement obtained by applying DSP on the central part of (Fig. 4(b)) of two states. A 0.4 pixel shift of speckle patterns is obtained which corresponds to 31.5 μm. This is measurable thanks to the subpixel accuracies of cross-correlation algorithm. The DHI result is shown in Fig. 4(d–f): from both phases computed through the Fresnel integral, before and after movement, their difference gives a phase map modulo 2𝜋, shown in Fig. 4(d). The in-plane displacement is far beyond the out-of-plane displacement which induces unwanted fringes when slight variations of sensitivity vector occur. The
4.3. Application in three-dimensional displacement We considered to validate the effectiveness and accuracy of the proposed method for mixed three-dimensional displacement in this section. The example is shown in Fig. 4. The movement of object combines a rotation of 6 arcseconds around an axis parallel to y-axis given by the encoders of rotation stage and confirmed by a separate theodolite and a translation of 30 μm along x-axis given by the encoders of translation stage with accuracy of 1 μm. First series of phase-shifted holograms are 88
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phase difference induced by in-plane displacement should be estimated in order to extract the out-of-plane displacement. With knowledge of inplane displacement obtained from DSP (31.5 μm) and sensitivity vector on each point of ROI on object calculated from geometrical arrangement of holographic setup, we calculated phase difference caused by pure inplane displacement (Fig. 4(e)). After subtracting Fig. 4(e) from Fig. 4(d), the phase difference caused by out-of-plane is shown in Fig. 4(f), corresponding to 1.5 μm of out-of-plane displacement between two borders of illuminated area, which is equivalent to a rotation angle of 5.5 arcseconds. To confirm the separation of phase difference, we applied the above-mentioned in-plane translation and out-of-plane rotation separately in two individual measurements. The phase differences caused by each motion are demonstrated in Fig. 4(g) and Fig. 4(h) respectively. Out-of-plane displacement obtained from Fig. 4(h) is 1.6 μm, corresponding to 6.0 arcseconds of rotation. Comparing Figs. 4(e) and (g), we conclude that the estimation of in-plane displacement and sensitivity vector contains bias error which further contributes to error in out-ofplane displacement measurement result.
view which will eventually cause unwanted fringes. Although the outof-plane displacement can be extracted after knowing the in-plane displacement, the accuracy of out-of-plane displacement decreases when the phase difference contributed by in-plane displacement increases. When the in-plane displacement exceeds the speckle size, it is still measurable by DSP. Meanwhile, for DHI out-of-plane measurement, the appearance of large in-plane displacement causes the decorrelation and loss of fringes. Fringe retrieval techniques proposed by compensating the speckle shift using measured in-plane displacement [5,7] are not applicable here since the recovered fringes are dominantly caused by large in-plane displacements (roughly 1000 times larger than measurable out-of-plane displacements). Therefore out-of-plane displacement measured by DHI is no longer exploitable under such circumstances. Further consideration is needed for this application. Acknowledgments The authors would like to acknowledge Professor Mikael Sjödahl and Professor Pascal Picart for fruitful discussions. Redouane Zemmamouche gratefully acknowledges the Ministry of Higher Education and Scientific Research of Algeria, the University of M’SILA, and the Institut National d’Optique et Mécanique de Precision of Sétif University for providing the grant for his internship at Centre Spatial de Liège (Université de Liège).
5. Discussion and conclusions A combination of DSP with DHI has been investigated for the application of in-plane and out-of-plane displacement measurement. The speckle patterns required for applying DSP were obtained from the amplitude images reconstructed by DH principle under Fresnel in-line DH configuration. The phase-shifting was requested for extracting the object image without being disturbed by other diffraction orders. We showed analytically and experimentally that the intrinsic properties of DH reconstruction contribute to inaccurate in-plane displacement measurement results. A complementary speckle movement emerged when phase errors were introduced into reconstruction process. The phase errors originated from slightly uncollimated reference wave or higher order aberrations. Consequently, the speckle displacement calculated from DSP can no longer represent the in-plane displacement of object. Phase errors should be corrected prior to applying DSP on reconstructed images. Another concern is the speckle size. We have shown that the speckle on the reconstructed plane occupied only one pixel, which could not satisfy the Nyquist criterion. As a result, the DSP measurement contained systematic error which is a drift towards the closest integral pixel value. The speckle size can be redefined by zero-padding the recorded hologram under the same optical configuration. We found that exploitable DSP result is obtained after the sampling condition is satisfied. This method provided reliable in-plane and out-of-plane displacement measurement after these two sources of error are corrected. A mixed displacement measurement was conducted. The out-of-plane displacement can be extracted based on the knowledge of in-plane displacement given by DSP measurement. We discussed the limitation of simultaneous 3D displacement based on the proposed method. First of all, the range of measurable displacements is different between the out-of-plane ones measured by DHI and the in-plane ones measured by DSP. The situation is totally similar to the case of ESPI combined with DSP previously reported in [1,4,5,8]. For the out-of-plane displacement measured by DHI, we have the typical measurement range of interferometry which is related to the laser wavelength and the number of fringes that can be resolved [3,9,10]. The zero-padding applied during reconstruction does not add any intrinsic information, hence does not have impact on the out-of-plane measurement range. In the case of DSP, the in-plane measurement range is several tens of speckle diameter with subpixel accuracies as reported for conventional DSP [1,2,4]. The intrinsic accuracy also depends on the magnification of reconstructed plane. Furthermore, extra attention should be paid on the accuracy of out-of-plane displacement measurement when measurable mixed displacement occurs. In Fresnel DH setup, slight variations of sensitivity vector are inevitable due to large field-of-
References [1] Sjödahl M, Benckert LR. Electronic speckle photography: analysis of an algorithm giving the displacement with subpixel accuracy. Appl Opt 1993;32(13):2278–84. [2] Sjödahl M, Benckert LR. Systematic and random errors in electronic speckle photography. Appl Opt 1994;33(31):7461–71. [3] Kreis T. Handbook of holographic interferometry: optical and digital methods. John Wiley & Sons; 2006. [4] Sjödahl M, Saldner HO. Three-dimensional deformation field measurements with simultaneous tvholography and electronic speckle photography. Appl Opt 1997;36(16):3645–8. [5] Andersson A, Runnemalm A, Sjödahl M. Digital speckle-pattern interferometry: fringe retrieval for large in-plane deformations with digital speckle photography. Appl Opt 1999;38(25):5408–12. [6] Groves RM, Fu S, James SW, Tatam RP. Single-axis combined shearography and digital speckle photography instrument for full surface strain characterization. Opt Eng 2005;44(2). 025602–025602 [7] Gren P. Pulsed tv holography combined with digital speckle photography restores lost interference phase. Appl Opt 2001;40(14):2304–9. [8] Zemmamouche R, Vandenrijt J-F, Medjahed A, Oliveira Id, Georges MP. Use of specklegrams background terms for speckle photography combined with phase-shifting electronic speckle pattern interferometry. Opt Eng 2015;54(8). 084110–084110. [9] Schnars U, Falldorf C, Watson J, Jüptner W. Digital holography and wavefront sensing: principles, techniques and applications. 2nd. Springer Publishing Company, Incorporated; 2014. [10] Picart P, Gross M, Marquet P. Basic fundamentals of digital holography. New Techniques Digital Holography 2015:1–66. [11] Yan H, Pan B. Three-dimensional displacement measurement based on the combination of digital holography and digital image correlation. Opt Lett 2014;39(17):5166–9. [12] Singh AK, Pedrini G, Peng X, Osten W. Nanoscale measurement of in-plane and out-of-plane displacements of microscopic object by sensor fusion. Opt Eng 2016;55(12). 121722–121722. [13] Yamaguchi I, ichi Kato J, Ohta S, Mizuno J. Image formation in phase-shifting digital holography and applications to microscopy. Appl Opt 2001;40(34):6177–86. [14] De Nicola S, Ferraro P, Finizio A, Pierattini G. Wave front reconstruction of fresnel off-axis holograms with compensation of aberrations by means of phase-shifting digital holography. Opt Lasers Eng 2002;37(4):331–40. [15] Sjödahl M. Accuracy in electronic speckle photography. Appl Opt 1997;36(13):2875–85. [16] Khodadad D, Bergström P, Hällstig E, Sjödahl M. Fast and robust automatic calibration for single-shot dual-wavelength digital holography based on speckle displacements. Appl Opt 2015;54(16):5003–10. [17] Tippie AE. Aberration correction in digital holography. University of Rochester; 2012. Ph.D. thesis. [18] Chen D, Chiang F. Optimal sampling and range of measurement in displacement-only laser-speckle correlation. Exp Mech 1992;32(2):145–53. [19] Dubois F, Schockaert C, Callens N, Yourassowsky C. Focus plane detection criteria in digital holography microscopy by amplitude analysis. Opt Express 2006;14(13):5895–908.
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Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Accuracy concerns in digital speckle photography combined with Fresnel digital holographic interferometry Yuchen Zhao a, Redouane Zemmamouche b,c, Jean-François Vandenrijt a, Marc P. Georges a,∗ a
Centre Spatial de Liège-Université de Liège, Liege Science Park, B-4031, Angleur, Belgium Université Ferhat Abbas, Sétif1, Institut d’Optique et Mécanique de Précision, Laboratoire d’Optique Appliquée, Avenue Said Boukhrissa, Sétif 19000, Algeria c University Med Boudiaf, Faculty of Technology, Mechanical Department, Msila 28000, Algeria b
a r t i c l e
i n f o
Keywords: Digital holography Digital speckle photography Nondestructive testing Metrology
a b s t r a c t A combination of digital holographic interferometry (DHI) and digital speckle photography (DSP) allows in-plane and out-of-plane displacement measurement between two states of an object. The former can be determined by correlating the two speckle patterns whereas the latter is given by the phase difference obtained from DHI. We show that the amplitude of numerically reconstructed object wavefront obtained from Fresnel in-line digital holography (DH), in combination with phase shifting techniques, can be used as speckle patterns in DSP. The accuracy of in-plane measurement is improved after correcting the phase errors induced by reference wave during reconstruction process. Furthermore, unlike conventional imaging system, Fresnel DH offers the possibility to resize the pixel size of speckle patterns situated on the reconstruction plane under the same optical configuration simply by zero-padding the hologram. The flexibility of speckle size adjustment in Fresnel DH ensures the accuracy of estimation result using DSP. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Digital speckle photography (DSP) is a simple and robust method for tracking in-plane deformation. The displacement is determined by numerically cross-correlating the speckle patterns generated on the surface of a scattering object under coherent illumination at the initial and deformed states [1]. A displacement over several tens of speckle diameter can be obtained with subpixel accuracy if crucial parameters such as speckle size and finite window size are properly defined when using DSP [2]. DSP can be combined with interferometry techniques based on speckle effect. Interferometry techniques including electronic speckle pattern interferometry (ESPI) (also named digital speckle pattern interferometry and TV–holography) and speckle shearing interferometry (also known as shearography) allow a measurement of out-of-plane displacement fields in their usual geometric configuration where the object illumination is in the same direction as the field of view of the observation system [3]. Implementing DSP on these interferometry techniques is useful for obtaining simultaneously in-plane and out-of-plane displacements, which have always been of great interest for industrial non-contact metrology applications [4–8]. One necessary condition for obtaining accurate results from DSP is that the speckle size should be larger than two pixels to meet sampling requirement [2]. In all these
∗
Corresponding author. E-mail address: [email protected] (M.P. Georges).
http://dx.doi.org/10.1016/j.optlaseng.2017.09.011 Received 30 April 2017; Received in revised form 31 July 2017; Accepted 8 September 2017 Available online 14 September 2017 0143-8166/© 2017 Elsevier Ltd. All rights reserved.
cases, speckle patterns are imaged onto a CCD camera, which can be sufficiently resolved by correctly setting the imaging aperture. Digital holographic interferometry (DHI) is another technique which allows measuring out-of-plane movements based on digital holography (DH). Differing from other interferometry techniques, DH records the holograms and reconstructs numerically the object wavefield based on diffraction theory [9,10]. The phase difference of object at two instants relates to the out-of-plane displacements located on each point of the object. On the other hand, the reconstructed amplitude of a diffused object also contains speckles and thus can be used for in-plane displacement measurement. A combination of DHI and DSP provides access to threedimensional displacement. However, compared with the DSP based on imaged speckle patterns, large bias and standard deviation errors are reported in earlier works when applying DSP on DH reconstructed images [11,12]. In this paper, we present a combination of DSP and phase-stepping Fresnel digital holography, making it possible to measure in-plane and out-of-plane displacement of a large object at long working distance. Our objective is to investigate the causes of low accuracy in in-plane displacement measurement reported in [11,12]. We demonstrate that phase errors caused by slightly divergent reference beam contribute to a complementary speckle displacement. Moreover, the speckle features in the reconstructed plane are not resolved enough to match the speckle
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size requirement for DSP, which leads to systematic errors in in-plane displacement. We show experimentally that the in-plane displacement measurement can be significantly improved after these two sources of error are corrected. In Section 2 we recall the theoretical basis of DHI and DSP, based on which sources of error existing in in-plane displacement measurement are derived analytically. In Section 3 we describe the setup. In Section 4, we present different experimental results. Each source of error is examined and corrected. Finally some conclusions and discussion are presented in the last section.
2. Theoretical analysis 2.1. Digital holography interferometry combined with digital speckle photography
Fig. 1. Schematic of the experimental setup.
DH consists of recording the holograms with an array sensor and numerically reconstructing the original wavefield at given distances based on the recorded holograms. During the recording process, reference wave UR (𝜉, 𝜂) interferes with object wavefield UO (𝜉, 𝜂) in the plane of CCD detector (𝜉, 𝜂), forming interference patterns, i.e., holograms IH (𝜉, 𝜂) given by: 𝐼𝐻 =|𝑈𝑅 |2 + |𝑈𝑂 |2 + 𝑈 ∗ 𝑅 𝑈𝑂 + 𝑈𝑅 𝑈 ∗ 𝑂 ,
𝑈𝑂 (𝑚, 𝑛, 𝑧) =
×
𝑛𝜋 2 )| , 2
(1)
𝐴𝑂 (𝑚, 𝑛) =|𝑈𝑂 (𝑚, 𝑛)|,
(6)
and
[ ] 𝜑𝑂 (𝑚, 𝑛) = arg 𝑈𝑂 (𝑚, 𝑛) .
(7) ( ) For two states of the object where a displacement 𝑑⃗ 𝑑𝑥 , 𝑑𝑦 , 𝑑𝑧 has taken place, after reconstruction using Eqs. (5)–(7), the difference between the phase at initial state 𝜑O, ini and the phase at displaced state 𝜑O, disp is related to optical path difference caused by applied displacement given by Δ𝜑𝑂 =𝜑𝑂, 𝑑𝑖𝑠𝑝 − 𝜑𝑂, 𝑖𝑛𝑖 = 𝑑⃗ ⋅ 𝑆⃗ ,
(2)
(8)
where 𝑆⃗ stands for the sensitivity vector which is determined by the directions of illumination and observation [3]. In accordance with the geometrical setup depicted in Fig. 1, Eq. (8) can be written with the illuminating angle 𝜃(m, n) on each point of object:
(3)
2𝜋 (9) [sin 𝜃(𝑚, 𝑛)𝑑𝑥 + (1 + cos 𝜃(𝑚, 𝑛))𝑑𝑧 ]. 𝜆 When applying DHI, the amplitude images AO, disp (m, n) and AO, ini (m, n) which carry the speckle patterns of two states are also obtained during reconstruction. The amplitude images can be processed by DSP for inplane displacement measurement. DSP technique consists of computing the speckle displacement between two digitized speckle patterns. More precisely, in the algorithm proposed by Shjödahl [1], DSP computes the cross-correlation of subimages (typically 32 × 32 pixels) taken at the same location in two speckle patterns. Locating the correlation peak gives the displacement measurement with subpixel accuracy. Extensive research has been carried out on the accuracy concerns of DSP such as the influence of finite window size, speckle size and measurement range based on imaging systems [1,2,15], which should be re-adapted into DH configurations. Δ𝜑𝑂 (𝑚, 𝑛) =
If the complex amplitude distribution of the reference wavefront UR (𝜉, 𝜂) is known, the object wavefield located in the object plane (x, y) at distance z from the hologram plane, UO (x, y, z), can be retrieved by back propagating the processed hologram HPS (𝜉, 𝜂) according to scalar diffraction theory. Under the paraxial approximation, the reconstructed wavefield at object plane UO (x, y, z) can be calculated by the Fresnel transform [3,9,10]: ( ) [ )] i 2𝜋 𝜋 ( 2 exp −i 𝑧 exp −i 𝑥 + 𝑦2 𝜆𝑧 𝜆 𝜆𝑧 [ )] 𝜋 ( 2 𝜉 + 𝜂2 × 𝑈𝑅 (𝜉, 𝜂)𝐻𝑃 𝑆 (𝜉, 𝜂) exp −i ∬ 𝜆𝑧 [ ] 2𝜋 × exp i (𝑥𝜉 + 𝑦𝜂) 𝑑 𝜉𝑑 𝜂. 𝜆𝑧
(5)
Comparing with the definition of Inverse Discrete Fourier Transformation (IDFT), the sum part in Eq. (5) can be rewritten as IDFT of the [ ( )] product 𝑈𝑅 𝐻𝑃 𝑆 exp −i𝜋∕𝜆𝑧 𝑘2 Δ𝜉 2 + 𝑙2 Δ𝜂 2 , which can be calculated efficiently with fast Fourier transform (FFT) algorithm. The reconstruction of HPS with Eq. (5) gives the both the amplitude AO and the phase map 𝜑O of the observed object at distance z, where
𝐻𝑃 𝑆 (𝜉, 𝜂) =
𝑈𝑂 (𝑥, 𝑦, 𝑧) =
𝑈𝑅 (𝑘, 𝑙)𝐻𝑃 𝑆 (𝑘, 𝑙)
[ [ ( )] )] 𝜋 ( 2 2 𝑘𝑚 𝑙𝑛 × exp −i + . 𝑘 Δ𝜉 + 𝑙2 Δ𝜂 2 exp i2𝜋 𝜆𝑧 𝑀 𝑁
and ) ( )] 1 [( 𝐼 (𝜉, 𝜂) − 𝐼𝐻,2 (𝜉, 𝜂) − i 𝐼𝐻,1 (𝜉, 𝜂) − 𝐼𝐻,3 (𝜉, 𝜂) 4 𝐻,0 = 𝑈𝑅∗ (𝜉, 𝜂)𝑈𝑂 (𝜉, 𝜂).
𝑀−1 ∑ 𝑁−1 ∑ 𝑘=0 𝑙=0
where the asterisk denotes complex conjugates and the dependency to the coordinates (𝜉, 𝜂) is omitted for simplicity. Eq. (1) contains four terms corresponding to three diffraction orders: the 0 order is composed of terms |𝑈𝑅 |2 + |𝑈𝑂 |2 , the +1 order U∗ R UO which contains initial information of object wavefield, and –1 order UR U∗ O which is also known as twin image. The +1 order is of interest for reconstructing the object wavefield. Under in-line configuration, the unwanted diffraction orders superimposed on the +1 order can be eliminated through applying phase-shifting (PS) techniques [13]. This technique requires several captures of holograms whereby corresponding phase shifts are introduced between object and reference arms. For instance, a four-step PS algorithm proposed in [14] aimed at forming a new compound hologram HPS (𝜉, 𝜂) from 4 phase-shifted holograms IH, n (𝜉, 𝜂) with n = 0, 1, 2, 3: 𝐼𝐻,𝑛 (𝜉, 𝜂) =|𝑈𝑅 (𝜉, 𝜂) + 𝑈𝑂 (𝜉, 𝜂) exp(−i
[ ( )] ( ) i 2𝜋 𝜋 𝑚2 𝑛2 exp −i 𝑧 exp −i + 𝜆𝑧 𝜆 𝜆𝑧 𝑀 2 Δ𝜉 2 𝑁 2 Δ𝜂 2
(4)
Considering that the hologram is sampled by the array sensor on M × N pixels with dimensions Δ𝜉 × Δ𝜂, it is therefore necessary to digitize Eq. (4), yielding the computation of object wavefield on M × N discrete points, i.e.,
2.2. Influence of phase errors of DH on DSP measurement The reconstruction of object wavefield can be fully achieved during DH on the premise of reference wave parameters in the detector plane 85
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during the recording process. Ideally in Fresnel in-line DH, the propagated object wave interferes with a plane reference wave: ∗ 𝐻𝑃 𝑆, 𝑖𝑑𝑒𝑎𝑙 (𝜉, 𝜂) =𝑈𝑅, 𝑖𝑑𝑒𝑎𝑙 (𝜉, 𝜂)𝑈𝑂 (𝜉, 𝜂),
adjustment under a fixed optical configuration is more flexible thanks to zero-padding in numerical reconstruction, which we discuss hereafter. In Fresnel DH, the pixel resolution in the reconstruction plane Δx and Δy is given by
(10)
with 𝑈𝑅, 𝑖𝑑𝑒𝑎𝑙 (𝜉, 𝜂) = 𝑅0 . In actual experimental condition, in spite of collimation system adopted to the reference beam, small mismatches still remain. In addition, there exist other wavefront aberrations which are recorded on the hologram. The actual reference beam interfering with object wave contains a phase variation factor exp [i𝜑𝜀 (𝜉, 𝜂)] over the detector plane [16,17]: 𝐻𝑃 𝑆, 𝑎𝑐𝑡𝑢𝑎𝑙 (𝜉, 𝜂) = =
Δ𝑥 =
𝜎𝑥 =
𝜆𝑧 . 𝑁Δ𝜂
(14)
𝜆𝑧 ; 𝐷𝑥
𝜎𝑦 =
𝜆𝑧 . 𝐷𝑦
(15)
In the case of DH the rays come directly on the CCD sensor with dimensions 𝐷𝑥 × 𝐷𝑦 = 𝑀 Δ𝜉 × 𝑁 Δ𝜂. Combining Eqs. (14) and (15), we find that
(11)
𝜎𝑥 = Δ𝑥;
where the model introduced in [16] is employed hereby to describe the phase variation. The quadratic terms a𝜉 2 and b𝜂 2 in Eq. (11) result from the slight divergence of reference beam; c𝜉 and d𝜂 correspond to an arbitrary tilt of reference beam. Higher order of phase errors are omitted here for simplicity. The existence of uncorrected phase errors affects not only the quality of reconstructed image, but also the speckle displacement measured in the reconstruction plane. When a displacement of dx along x direction is applied on the object, the distribution of diffracted object wave in the hologram plane UO (𝜉, 𝜂) is shifted d𝜉 along 𝜉 direction. In the ideal case, the holo) ( gram 𝐻𝑃 𝑆, 𝑖𝑑𝑒𝑎𝑙 𝜉 − 𝑑𝜉 , 𝜂 records the interference between UO (𝜉, 𝜂) and ( ( ) ) 𝑈𝑅, 𝑖𝑑𝑒𝑎𝑙 𝜉 − 𝑑𝜉 , 𝜂 . The reconstruction of 𝐻𝑃 𝑆, 𝑖𝑑𝑒𝑎𝑙 𝜉 − 𝑑𝜉 , 𝜂 gives the information of initial object.
𝜎𝑦 = Δ𝑦.
(16)
This signifies that the speckle grain on the object occupies one pixel in the reconstruction plane which cannot meet the sampling criterion. To circumvent this, we can pad the raw hologram to 2M × 2N or even larger size with zeros. Although zero-padding does neither add any information into hologram nor change the intrinsic resolution, it helps decrease the pixel size in reconstruction plane [10]. One speckle grain can thus be represented by more pixels. The effect of zero-padding in the application of DSP will be shown later in this paper. 3. Experimental setup The optical setup used here is a classical in-line Fresnel holographic configuration, depicted in Fig. 1. The laser is a diode pumped solid state Nd:YAG laser with a wavelength of 532 nm and power of 400 mW from Coherent, Inc. The CCD camera is a Micam VHR 1000 with a pixel size of 8.6 × 8.3 μm2 . 480 × 480 pixels window is used for hologram recording. A beamsplitter (BS) divides the laser beam into a reference and an object beam. The reference beam is reflected by a mirror placed on a piezo-mount (M-PZT) for introducing the phase-shifting during recording. It is expanded by a beam expander which is constituted by an objective lens and spatial filter (SF) and followed by a collimation lens L1. The reference beam reaches the CCD detector through the beam combiner (BC). The object beam is expanded by lens L2 to illuminate the object with illuminating angle 𝜃 = 4.3◦ . The object is an aluminum plate covered with scattering white powder, mounted on a controllable translation and rotation stages at a distance z of 1400 mm from the camera sensor. The region of interest (ROI) on the illuminated object during measurement is 5.5 × 5.5 cm2 . The reference beam interferes with the wave reflected by the object on the surface of CCD detector.
[ ( ( )]} ) ) 𝜋 ( 2 𝜉 + 𝜂2 𝑈𝑂 (𝑥, 𝑦, 𝑧) = ℱ 𝑈𝑅, 𝑖𝑑𝑒𝑎𝑙 𝜉 − 𝑑𝜉 , 𝜂 𝐻𝑃 𝑆, 𝑖𝑑𝑒𝑎𝑙 𝜉 − 𝑑𝜉 , 𝜂 exp −i 𝜆𝑧 { [ )]} 𝜋 ( 2 = ℱ −1 𝑅20 𝑈𝑂 (𝜉, 𝜂) exp −i (12) 𝜉 + 𝜂2 𝜆𝑧 −1
Δ𝑦 =
The speckle size 𝜎 x and 𝜎 y observed on the object through an aperture of dimension Dx × Dy at a distance z from the latter is given by
∗ 𝑈𝑅, 𝑎𝑐𝑡𝑢𝑎𝑙 (𝜉, 𝜂)𝑈𝑂 (𝜉, 𝜂) ∗ 𝑈𝑅, 𝑖𝑑𝑒𝑎𝑙 (𝜉, 𝜂) exp[−i𝜑𝜀 (𝜉, 𝜂)]𝑈𝑂 (𝜉, 𝜂)
)] [ ( = 𝐻𝑃 𝑆, 𝑖𝑑𝑒𝑎𝑙 (𝜉, 𝜂) exp −i 𝑎𝜉 2 + 𝑏𝜂 2 + 𝑐𝜉 + 𝑑𝜂 ,
𝜆𝑧 ; 𝑀Δ𝜉
{
) ( If the hologram 𝐻𝑃 𝑆, 𝑎𝑐𝑡𝑢𝑎𝑙 𝜉 − 𝑑𝜉 , 𝜂 which contains uncorrected phase error is reconstructed in the same manner as in Eq. (12): { [ ( ( )]} ) ) 𝜋 ( 2 𝜉 + 𝜂2 𝑈𝑂′ (𝑥, 𝑦, 𝑧) = ℱ −1 𝑈𝑅, 𝑖𝑑𝑒𝑎𝑙 𝜉 − 𝑑𝜉 , 𝜂 𝐻𝑃 𝑆, 𝑎𝑐𝑡𝑢𝑎𝑙 𝜉 − 𝑑𝜉 , 𝜂 exp −i 𝜆𝑧 { [ ( )]} ) 𝜋 ( 2 𝜉 + 𝜂2 = ℱ −1 𝑅20 𝑈𝑂 (𝜉, 𝜂) exp[−𝑖𝜑𝜀 𝜉 − 𝑑𝜉 , 𝜂 ] exp −i 𝜆𝑧 ( ) ( ) 𝑎𝑑𝜉 𝑐 𝑑 = 𝑈𝑂 (𝑥, 𝑦, 𝑧) ⊗ 𝛿 𝑥 − ⊗𝛿 𝑦− + 2𝜋 𝜋 2𝜋 { [ ]} (13) ⊗ℱ −1 exp −i(𝑎𝜉 2 + 𝑎𝑑𝜉2 + 𝑏𝜂 2 − 𝑐𝑑𝜉 )
The inverse Fourier transform of phase error factor contains a Dirac delta term 𝛿(𝑥 − 2𝑐𝜋 + 𝑎𝑑𝜉 ∕𝜋) which contributes to a complementary displacement on the reconstructed images 𝑈𝑂′ (𝑥, 𝑦, 𝑧) of two relatively displaced states (d𝜉 ≠ 0). The speckle displacement 𝑑𝑥′ found in reconstruction plane does not correspond to original displacement of object dx . The same situation occurs if displacement took place along y direction. Thus a correction of reference wavefront by retrieving the quadratic and higher orders of 𝜑𝜀 (𝜉, 𝜂) is necessary prior to taking reconstructed amplitude map for in-plane displacement measurement. For quadratic terms, parameters a and b can be estimated by optimizing the sharpness of reconstructed image [13,17]. For more precise correction, the higher order aberrations also need to be retrieved by applying aberration correction methods which can be found in [17].
4. Results 4.1. Effect of phase error correction on in-plane displacement measurement We have first studied how phase errors existing in DH reconstruction affect the DSP measurement result. To avoid the errors induced by undersampling which will be further discussed in the next subsection, the holograms of 480 × 480 pixels hereby have been properly padded to 1152 × 1152 pixels. The size of subimage is 32 × 32 pixels. The result is shown in Fig. 2. The object is first replaced by a grid target for observing the quality of reconstructed image. Reconstructing the recorded hologram with Eq. (10), the reconstructed image is degraded on the horizontal direction, shown in Fig. 2(a). Fig. 2(b) shows the DSP in-plane displacement measurement results under the uncorrected reconstruction. The measurement error is plotted in Fig. 2(c). All the 13 displacements are severely underestimated. Then we compensated the quadratic phase errors by optimizing the sharpness of reconstructed images. The focal plane detection criterion demonstrated by Dubois et al. [19] is applied here to quantify the image sharpness. For amplitude-contrast object, the integral of amplitude
2.3. Speckle size adjustment in DH The speckle size has a decisive effect in DSP accuracy. First of all, the range of measurable displacement in DSP is limited by the decorrelation of speckle patterns to several tens of pixel size [18]. Besides, the speckle size should be greater than 2 pixels to satisfy the Nyquist sampling criterion. The consequence of undersampling is introducing a systematic error that pushes the estimate towards the closest integral pixel value [2]. In the cases where speckle patterns are imaged, the speckle should be resized by adjusting the F-number of imaging system. In DH, speckle size 86
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Fig. 2. (a) Zoomed-in reconstructed grid target. (b) DSP measurement and (c) measurement error under reconstruction without quadratic phase error correction. (d) Zoomed-in reconstructed grid target. (e) DSP measurement and (f) measurement error under reconstruction where quadratic phase error correction is applied. (g) Ratio of measured slope over actual slope (blue) and the value of critical function Md (red) versus the value of parameter a. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Table 1 Comparison of DSP measurement results under various zero-padding levels. Zero-padding N
480 480 480 480 960 960 960 960 1152 1152 1152 1152 1440 1440 1440 1440
Speckle size (px) 𝜎 x
Applied Translation (μm) dx
Applied Translation (px) kx
Average result
1 1 1 1 2 2 2 2 2.4 2.4 2.4 2.4 3 3 3 3
50 100 150 200 50 100 150 200 50 100 150 200 50 100 150 200
0.321 0.642 0.963 1.283 0.642 1.283 1.925 2.567 0.770 1.540 2.310 3.080 0.963 1.925 2.889 3.850
0.047 0.951 0.999 1.043 0.688 1.200 1.906 2.619 0.787 1.520 2.296 3.067 0.911 1.889 2.888 3.841
modulus (Md) is considered as a critical function which reaches minimum when optimum sharpness is achieved. Fig. 2(d) shows the image reconstructed by the corrected reference wavefront with parameter 𝑎 = 1.098 × 10−4 of Eq. (11). Compared to Fig. 2(a), the sharpness of reconstructed image is significantly improved. Fig. 2(e) shows the DSP results under corrected reconstruction with the same set of displacements. The measured displacements match the actual ones. Measurement error plotted in Fig. 2(f) confirms that the bias error is eliminated. In Fig. 2(g) we plotted the ratio of measured slope over actual slope as well as the value of critical function Md versus the value of parameter a. The ratio reaches 1 when the minimum of Md is found at 𝑎 = 1.098 × 10−4 .
std (px) sk
(px) 𝒌𝒙
Average result
std (μm) sd
(μm) 𝒅𝒎 0.041 0.045 0.029 0.052 0.041 0.036 0.037 0.073 0.042 0.074 0.069 0.059 0.067 0.071 0.092 0.102
7.026 148.1 155.6 162.5 53.60 93.53 152.8 204.1 51.10 98.70 149.1 199.2 47.30 98.13 149.9 199.5
6.394 3.554 4.512 8.125 3.377 1.403 2.903 5.715 2.708 2.369 4.622 3.785 3.500 1.864 1.499 5.367
The results obtained from correlating 81 pairs of subimages give the average result (𝑘𝑥 ) and the standard deviation (std) (sk ). The results under different zero-padding conditions N are summarized in Table. 1. When images are directly reconstructed by the recorded holograms without zero-padding, one speckle possesses one pixel as discussed in previous section. In this case, the systematic errors caused by undersampling described in [2] are clearly observed. Compared with the actual displacement kx , the measured result 𝑘𝑥 contains a drift toward the nearest integral pixel value. Up to 0.5 pixel of error can be introduced if DSP is combined with DH without zero-padding. Converting back to the real world coordinate, the error can be disastrous when the displacement is small (7.026 ± 6.394 μm for a translation of 50 μm). Then the holograms are padded with zeros to meet the Nyquist criterion. Fig. 3(a) shows a zoom on a given portion of the reconstructed image corresponding to a zone of approximately 7 × 7 mm2 on the object with different zero-padding applied. Insets show the subimages (32 × 32 pixels) of each speckle pattern where each single speckle can be identified. From the speckle patterns, we observe that the speckle is better resolved after a larger zero-padding. The absolute values of measured residuals relative to theoretical values of applied translation are plotted in Fig. 3(b). We confirm that when the number of pixels per speckle exceeds 2, this sys-
4.2. Effect of zero-padding on in-plane displacement measurement We have studied the effect of zero-padding on the ability of DH to generate speckle pattern images that are usable in DSP. The phase errors during reconstruction have been eliminated. Four different pure in-plane translations dx are applied on the object. They are converted to pixels kx with corresponding speckle size 𝜎 x on the reconstruction plane and measured with DSP algorithm. When the size of hologram is 480 × 480 pixels, measured area contains 9 × 9 subimages of 32 × 32 pixels. 87
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Fig. 3. Zero-padding effects. (a) Zoom of reconstructed Speckle pattern under different zero-padding levels. Insets on the top-left corner: subimages (32 × 32 pixels) under each case. (b) The absolute value of measured residuals relative to theoretical values of applied translation under different zero-padding cases.
Fig. 4. Simultaneous measurement by DHI and DSP. (a) Compound hologram after zero-padding. (b) Amplitude after reconstruction from (a). (c) In-plane displacement obtained by DSP on the basis of central part of (b). (d) Phase difference of two states when mixed movement is applied. (e) Computational result of phase difference caused by bulk in-plane displacement achieved from (c). (f) Restoration of phase difference caused by out-of-plane displacement. (g) Phase difference when only translation is applied (pure in-plane displacement). (h) Phase difference when only rotation is applied (pure out-of-plane displacement).
tematic error is successfully circumvented. In classical DSP, it is also proved that the random error, i.e., the std of measurement increases with the speckle size [2]. It is the same case here, when increasing the zero-padding value, the std sk increases. The zero-padding case 1152 × 1152 gives a satisfactory result when the subimage is 32 × 32 pixels.
recorded prior to the object movement and the second ones after. Afterwards zero-padding is applied with the suitable level, here with a final pixel number of 1152 × 1152 (Fig. 4(a)). The zero-padded compound holograms are injected in the Fresnel calculation Eq. (5) with corrected reference wavefront. The amplitude and phase of the reconstructed object wave are retrieved. Fig. 4(b) shows the ROI of reconstructed amplitude. Fig. 4(c) shows the in-plane displacement obtained by applying DSP on the central part of (Fig. 4(b)) of two states. A 0.4 pixel shift of speckle patterns is obtained which corresponds to 31.5 μm. This is measurable thanks to the subpixel accuracies of cross-correlation algorithm. The DHI result is shown in Fig. 4(d–f): from both phases computed through the Fresnel integral, before and after movement, their difference gives a phase map modulo 2𝜋, shown in Fig. 4(d). The in-plane displacement is far beyond the out-of-plane displacement which induces unwanted fringes when slight variations of sensitivity vector occur. The
4.3. Application in three-dimensional displacement We considered to validate the effectiveness and accuracy of the proposed method for mixed three-dimensional displacement in this section. The example is shown in Fig. 4. The movement of object combines a rotation of 6 arcseconds around an axis parallel to y-axis given by the encoders of rotation stage and confirmed by a separate theodolite and a translation of 30 μm along x-axis given by the encoders of translation stage with accuracy of 1 μm. First series of phase-shifted holograms are 88
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phase difference induced by in-plane displacement should be estimated in order to extract the out-of-plane displacement. With knowledge of inplane displacement obtained from DSP (31.5 μm) and sensitivity vector on each point of ROI on object calculated from geometrical arrangement of holographic setup, we calculated phase difference caused by pure inplane displacement (Fig. 4(e)). After subtracting Fig. 4(e) from Fig. 4(d), the phase difference caused by out-of-plane is shown in Fig. 4(f), corresponding to 1.5 μm of out-of-plane displacement between two borders of illuminated area, which is equivalent to a rotation angle of 5.5 arcseconds. To confirm the separation of phase difference, we applied the above-mentioned in-plane translation and out-of-plane rotation separately in two individual measurements. The phase differences caused by each motion are demonstrated in Fig. 4(g) and Fig. 4(h) respectively. Out-of-plane displacement obtained from Fig. 4(h) is 1.6 μm, corresponding to 6.0 arcseconds of rotation. Comparing Figs. 4(e) and (g), we conclude that the estimation of in-plane displacement and sensitivity vector contains bias error which further contributes to error in out-ofplane displacement measurement result.
view which will eventually cause unwanted fringes. Although the outof-plane displacement can be extracted after knowing the in-plane displacement, the accuracy of out-of-plane displacement decreases when the phase difference contributed by in-plane displacement increases. When the in-plane displacement exceeds the speckle size, it is still measurable by DSP. Meanwhile, for DHI out-of-plane measurement, the appearance of large in-plane displacement causes the decorrelation and loss of fringes. Fringe retrieval techniques proposed by compensating the speckle shift using measured in-plane displacement [5,7] are not applicable here since the recovered fringes are dominantly caused by large in-plane displacements (roughly 1000 times larger than measurable out-of-plane displacements). Therefore out-of-plane displacement measured by DHI is no longer exploitable under such circumstances. Further consideration is needed for this application. Acknowledgments The authors would like to acknowledge Professor Mikael Sjödahl and Professor Pascal Picart for fruitful discussions. Redouane Zemmamouche gratefully acknowledges the Ministry of Higher Education and Scientific Research of Algeria, the University of M’SILA, and the Institut National d’Optique et Mécanique de Precision of Sétif University for providing the grant for his internship at Centre Spatial de Liège (Université de Liège).
5. Discussion and conclusions A combination of DSP with DHI has been investigated for the application of in-plane and out-of-plane displacement measurement. The speckle patterns required for applying DSP were obtained from the amplitude images reconstructed by DH principle under Fresnel in-line DH configuration. The phase-shifting was requested for extracting the object image without being disturbed by other diffraction orders. We showed analytically and experimentally that the intrinsic properties of DH reconstruction contribute to inaccurate in-plane displacement measurement results. A complementary speckle movement emerged when phase errors were introduced into reconstruction process. The phase errors originated from slightly uncollimated reference wave or higher order aberrations. Consequently, the speckle displacement calculated from DSP can no longer represent the in-plane displacement of object. Phase errors should be corrected prior to applying DSP on reconstructed images. Another concern is the speckle size. We have shown that the speckle on the reconstructed plane occupied only one pixel, which could not satisfy the Nyquist criterion. As a result, the DSP measurement contained systematic error which is a drift towards the closest integral pixel value. The speckle size can be redefined by zero-padding the recorded hologram under the same optical configuration. We found that exploitable DSP result is obtained after the sampling condition is satisfied. This method provided reliable in-plane and out-of-plane displacement measurement after these two sources of error are corrected. A mixed displacement measurement was conducted. The out-of-plane displacement can be extracted based on the knowledge of in-plane displacement given by DSP measurement. We discussed the limitation of simultaneous 3D displacement based on the proposed method. First of all, the range of measurable displacements is different between the out-of-plane ones measured by DHI and the in-plane ones measured by DSP. The situation is totally similar to the case of ESPI combined with DSP previously reported in [1,4,5,8]. For the out-of-plane displacement measured by DHI, we have the typical measurement range of interferometry which is related to the laser wavelength and the number of fringes that can be resolved [3,9,10]. The zero-padding applied during reconstruction does not add any intrinsic information, hence does not have impact on the out-of-plane measurement range. In the case of DSP, the in-plane measurement range is several tens of speckle diameter with subpixel accuracies as reported for conventional DSP [1,2,4]. The intrinsic accuracy also depends on the magnification of reconstructed plane. Furthermore, extra attention should be paid on the accuracy of out-of-plane displacement measurement when measurable mixed displacement occurs. In Fresnel DH setup, slight variations of sensitivity vector are inevitable due to large field-of-
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