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Edge extraction using a time-varying vortex beam in incoherent digital holography Yijie Pan,1,2,* Wei Jia,3 Junjie Yu,3 Kelly Dobson,2 Changhe Zhou,3 Yongtian Wang,1 and Ting-Chung Poon2,3 1

2

Beijing Engineering Research Center for Mixed Reality and Advanced Display, School of Optoelectronics, Beijing Institute of Technology, Beijing 100081, China

Bradley Department of Electrical and Computer Engineering, Virginia Polytechnic Institute and State University (Virginia Tech), Blacksburg, Virginia 24061, USA 3

Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China *Corresponding author: [email protected] Received April 28, 2014; revised June 5, 2014; accepted June 16, 2014; posted June 17, 2014 (Doc. ID 210978); published July 9, 2014

Edge extraction using a time-varying vortex beam (TV-VB) is demonstrated in optical scanning holography (OSH) operating in an incoherent mode. OSH is a two-pupil heterodyne scanning optical system. We propose that one of the pupil functions used is a delta function and the other pupil function is a spiral phase plate (SPP). The interference of these pupils creates a TV-VB to scan over an object to record the edge-only information of an object holographically. We also find that a reconstructed edge with better contrast is achieved by translating the SPP away from the pupil plane. Experimental results are compared with computer simulations and found to be in good agreement. © 2014 Optical Society of America OCIS codes: (100.2980) Image enhancement; (090.1995) Digital holography; (070.6110) Spatial filtering; (260.6042) Singular optics. http://dx.doi.org/10.1364/OL.39.004176

Optical edge contrast enhancement is a powerful image preprocessing tool in pattern recognition. The use of a spiral phase plate (SPP) to achieve a 2D radial Hilbert transformation has been studied extensively in coherent systems for isotropic and anisotropic edge detection [1–4] as well as some work in digital holography [5]. As for incoherent digital holography (IDH), there has been only a single attempt in edge extraction using the SPP [6]. There are only two existing techniques in IDH and the authors in Ref. [7] have employed the Fresnel incoherent correlation holography (FINCH) technique [7,8]. FINCH requires three phase-shifted records sequentially and is based on self-interference. Hence, bias buildup might become an issue if the object is of complicated features [9]. However, it can adapt easily to color recordings [10]. The other existing IDH technique is optical scanning holography (OSH) [11,12]. OSH is a two-pupil interaction optical system. It requires heterodyne scanning of the object and, therefore, there is no problem with the bias buildup as the holographic information is carried by the heterodyne frequency, which can be extracted using electronic filtering. In addition, the scanning method lends itself ideally to electronic multiplexing when multiple beams of different temporal frequencies are used for processing [13]. However, if video rate applications are needed, fast electronics and digital processing must be required. In this Letter, we demonstrate, for the first time to our knowledge, the use of OSH to record the edge-only information of an object holographically. A time-varying vortex beam (TV-VB) formed by making one of the pupil functions a delta function, and the other a SPP, is used to extract the edge-only information. The most interesting result is that low contrast resulting from a classical 2D radial Hilbert transformation can be strongly improved by translating the SPP away from the pupil plane for a distance. Figure 1 illustrates the OSH setup of the proposed method. The two beam splitters (BS1 and BS2 ) and the 0146-9592/14/144176-04$15.00/0

two mirrors (M1 and M2 ) form a Mach–Zehnder interferometer. In one arm of the interferometer, a SPP is placed on or at a distance away from the pupil plane specified by pupil function, p1 , which is on the front focal plane of lens L1 . In the other arm, a pinhole aperture is placed at p2 the pupil plane; that is the front focal plane of lens L2 . Two acousto-optic modulators, AOM1 and AOM2 , operating at frequencies Ω1 and Ω2 , are used to upshift the laser beam frequency, ω0 , to ω0  Ω1 and ω0  Ω2 , respectively, in the two arms of the interferometer. An object of complex amplitude, Γ0 x; y; z, is located at a distance z away from the back focal plane (BFP), which is the common back focal plane of lenses L1 and L2 , both

Fig. 1. OSH system setup. BS1 and BS2 , beam splitter; AOM1 and AOM2 , acousto-optic modulators; SPP, spiral phase plate; M1 and M2 , silver mirrors; M3 , x-y scanning mirror; PD1 and PD2 , photodectectors; ADC, analog-to-digital converter. © 2014 Optical Society of America

July 15, 2014 / Vol. 39, No. 14 / OPTICS LETTERS

with focal length f . Mirror M4 is used to direct the combined beams by beam splitter BS2 to photodetector PD2 to provide heterodyne frequency Ω1 − Ω2 as a reference signal to the lock-in amplifier. Also, the combination of the two beams is projected to an x-y scanning mirror, M3 , and used to scan over the object. Lens L3 collects all of the transmitted light (assuming a transmissive object for simplicity) onto photodectector PD1 to give a signal current, which is electronically processed by the lock-in amplifier to give the in-phase and quadrature signals, i.e., ic and is , respectively, as two outputs [11,12], ic and is , are the two currents carrying holographically processed information of the scanned object that are fed to an analog-to-digital converter (ADC) and then recorded by the PC. Once they are presented in the PC as 2D images after the 2D raster scan, they become the inphase and quadrature holograms, H I and H Q . From the two holograms, we can form a complex hologram in the PC, H C  H I  jH Q , that is free of the twin image upon holographic reconstruction [14]. If photodetector PD1 has a large active area, OSH is operating in the incoherent mode; i.e., the intensity of the object, jΓ0 x; y; zj2 , will be processed [15]. However, when photodetector PD1 is masked off with a pinhole and lens L3 is positioned as a Fourier transform lens with the object and photodetector PD1 being placed in its front and back focal planes, respectively, OSH is in the coherent mode [16]. In this Letter, OSH is operating in an incoherent mode as lens L3 collects all the transmitted energies onto the photodetector. According to [11], the complex hologram is given by 



z 2 HC  F k  k2y  FfjΓ0 x; y; zj g exp −j 2k0 x   f f ×p1 − kx ; − ky ; k0 k0 −1



2

(1)

when p2 is a delta function, where kx ; ky are spatial frequencies along the x and y directions, respectively. Now, when we take p1  1, the spectrum of the object is processed by a 2D complex chirp function. As a result, we have standard OSH; i.e., we simply have holographic recording of the object. Hence, we see that with p1 we can think of the spectrum of the object is first filtered by p1 and subsequently processed by the chirp function to generate holographic information of the processed spectrum. In the experiments that follow, we take p1 either a SPP or diffraction of the spiral phase to investigate preprocessing in the form of edge extraction. In passing, we mention such modifications of the pupils, commonly known as pupil engineering, have been investigated in the context of optical sectioning as well as enhanced depth resolution in OSH [17–19]. The experimental parameters used are as follows. We have used a He–Ne laser (15 mW, λ  632.8 nm, diameter of the Gaussian beam w  0.6 mm), two acousto-optic modulators (IntraAction AOM-40’s, with center frequencies 40.25 and 40 MHz, giving heterodyne frequency Ω1 − Ω2  25 kHz.), and two lenses of the same focal length (f  300 mm). Also, a 2 mm × 2 mm SPP whose topological charge equals one is used [20]. The object

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Fig. 2. (a) Object. The dotted line shows the 1D original signal used in the comparison of edge extraction performances in Fig. 6. (b) TV-VB (Media 1) used to scan the object to obtain the edge information of an object holographically.

is a 20 mm × 20 mm transparency located about z  868 mm from the BFP (see Fig. 1). We have performed two sets of experiments where we have taken d  0 mm, i.e., when the SPP is exactly at pupil plane p1 , and d  130 mm and is translated away from the focal plane (see Fig. 1). Figure 2(a) shows the transparency of a binary object, and Fig. 2(b) shows a snap shot of the scanning beam on the plane of the object, where d  130 mm. We shall call the scanning beam a TV-VB and it is shown in Media 1. The TV-VB is the result of the heterodyne interference between the plane wave (coming from the pinhole at pupil plane p2 ) and the beam diffracted from the BFP, where the field distribution on the BFP is given by the Fourier transform of p1 . The heterodyne frequency has been set to a few hertz in order to capture the motion of the vortex beam, as demonstrated in the media file. Figure 3 shows a simulation of the TV-VB, where we could observe detailed structure of the beam such as running fringes within the moving spiral (Media 2). Figure 4 shows the results for the case d  0 mm. Inphase and quadrature holograms are shown in Figs. 4(a) and 4(b), respectively. Figure 4(c) shows the reconstruction of complex hologram H C  H I  jH Q using Figs. 4(a) and 4(b), while Fig. 4(d) presents the simulation result. Figure 5 shows the results, correspondingly, as in Fig. 4, but for the case d  130 mm. In general, we observe that edge extraction has been achieved upon

Fig. 3. Snap shot of TV-VB (simulation). Fine fringes are seen within the spiral (Media 2).

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Fig. 4. Recording is performed when the spiral phase filter is on the pupil plane, i.e., d  0 mm. (a) In-phase hologram, H I , of Fig 2(a); (b) quadrature hologram, H Q , of Fig. 2(a); (c) reconstruction from the complex hologram H C  H I  jH Q ; and (d) numerical simulation result.

holographic reconstructions [see (c) and (d) of Figs. 4 and 5]. However, we note that for the case when the SPP is not exactly on the pupil plane, edge extraction with better contrast is achieved, as evident from Figs. 5(c) and 5(d). Figure 6 shows line traces across the reconstructed images of Figs. 4(d) and 5(d), as a comparison. The position of the line trace is shown in

Fig. 5. Recording is performed when the spiral phase filter is away from the pupil plane, i.e., d  130 mm. (a) In-phase hologram, H I , of Fig. 2(a); (b) quadrature holograms, H Q , of Fig. 2(a); (c) reconstruction from the complex hologram; and (d) numerical simulation result.

Fig. 6. Normalized intensities of the line trace across the reconstructed images in Figs. 4(d) and 5(d), where the red solid line represents the case when the SPP is not exactly on the pupil plane and the blue dashed line shows the result when the phase plate is on the pupil plane. The position of the line trace is shown in Fig. 2(a) with the dotted line over the original image.

Fig. 2(a) with the dotted line over the original image. The improvement over edge extraction in the case when the SPP is not on the pupil plane might be explained as follows. Let us first discuss edge extraction in a coherent 4-f system with a SPP as a filter on the pupil plane for spatial filtering [1]. A 1D signal along a radial direction is processed according to the 1D Hilbert transformation, i.e., the point spread function (PSF) takes on the form of 1∕πx, assuming the signal is along the x axis [21]. This PSF is an odd function with a long tail and could affect a nearby edge in the signal. Hence, not all extracted edges are of equal height and equal slope on two sides of the edge. These are evident from Fig. 6 (for example, see the extracted edge at pixel index 208 and the edge at pixel index 56, where the slope on the left edge is different from that on the right edge [see the dotted line]). In addition, extracted edges seem to ride on some dc bias, thereby further decreasing the contrast on the extracted edge. The condition of letting a SPP as a pupil function in the Fourier plane in a coherent system is equivalent to our case when the SPP is exactly on the pupil plane p1 , i.e., when d  0 in Fig. 1. When d ≠ 0, the pupil plane will have the distribution of a diffracted field from the incident of a Gaussian beam on the SPP. This field distribution is rather complicated and has been previously calculated [22]. One particularly interesting fact to point out is that this field distribution as a filtering function has ρ × Gaussian characteristics, where ρ is a spatial frequency variable corresponding to spatial variable r  x2  y2 1∕2 and such characteristics have been previously proposed to apodize the SPP to dampen down the sidelobes of the resulting 2D PSF [3]. Indeed, there has been a plethora of publications in recent years dealing with optical vortices with sidelobe suppression [23,24] in coherent optical systems. So far, our experimental and simulation results seem to suggest this explanation regarding sidelobe suppression, but for incoherent optical systems (see the solid line in Fig. 5). We

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Yijie Pan would like to thank the China Scholarship Council for financial support (No. 201206030006) on his visit to Virginia Tech.

Fig. 7. (a) Photograph of the Einstein transparency; (b) reconstruction from standard OSH; (c) reconstruction from TV-VB scanned gray scale image when d  130 mm, illustrating edge extraction.

plan to further investigate this in the future as the diffracted field from the incident of a Gaussian beam on a SPP is complex in general, and is more complicated as compared to a real characteristic of ρ × Gaussian used in [3] to suppress the sidelobes. As a final demonstration of edge extraction capabilities, Fig. 7 shows optical results of gray scale edge extraction. A transparency of a half-tone gray scale image of Einstein is shown in Fig. 7(a). A portion [red square in Fig. 7(a)] of the standard OSH result and the edge extracted image for d  130 mm are shown in Figs. 7(b) and 7(c), respectively. In conclusion, an edge extraction method using a TV-VB in incoherent digital holography is proposed and demonstrated. One pupil function of the OSH is a SPP or the diffracted distribution of a SPP. Both configurations achieve edge extraction, but the latter one obtains better contrast. This research is partially sponsored by the National Natural Science Foundation of China (No. 61235002).

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Edge extraction using a time-varying vortex beam in incoherent digital holography.

Edge extraction using a time-varying vortex beam (TV-VB) is demonstrated in optical scanning holography (OSH) operating in an incoherent mode. OSH is ...
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