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OPTICS LETTERS / Vol. 40, No. 10 / May 15, 2015
Coherence experiments in single-pixel digital holography Jung-Ping Liu,1,* Chia-Hao Guo,1 Wei-Jen Hsiao,1 Ting-Chung Poon,1,2 and Peter Tsang3 1 2 3
Department of Photonics, Feng Chia University, 100 Wenhwa Rd., Seatwen, Taichung 40724, Taiwan
Bradley Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, Virginia 24061, USA
Department of Electronic Engineering, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong *Corresponding author:
[email protected] Received February 16, 2015; revised April 11, 2015; accepted April 20, 2015; posted April 23, 2015 (Doc. ID 234767); published May 13, 2015
In optical scanning holography (OSH), the coherence properties of the acquired holograms depend on the singlepixel size, i.e., the active area of the photodetector. For the first time, to the best of our knowledge, we have demonstrated coherent, partial coherent, and incoherent three-dimensional (3D) imaging by experiment in such a singlepixel digital holographic recording system. We have found, for the incoherent mode of OSH, in which the detector of the largest active area is applied, the 3D location of a diffusely reflecting object can be successfully retrieved without speckle noise. For the partial coherent mode employing a smaller pixel size of the detector, significant speckles and randomly distributed bright spots appear among the reconstructed images. For the coherent mode of OSH when the size of the pixel is vanishingly small, the bright spots disappear. However, the speckle remains and the signalto-noise ratio is low. © 2015 Optical Society of America OCIS codes: (090.1995) Digital holography; (110.6880) Three-dimensional image acquisition. http://dx.doi.org/10.1364/OL.40.002366
Optical scanning holography (OSH) is a single-pixel digital holographic recording technique for threedimensional (3D) imaging [1–3]. In OSH, a heterodynestructured beam, which is composed of two coherent beams with a temporal frequency difference, is applied to raster scan a 3D object. The light scattered from the object is detected by a single-pixel photodetector, and the electric signal from the detector is demodulated by a lock-in amplifier, eventually generating a complex hologram in a computer. One of OSH’s important properties is that the digital holographic system can be operated in the coherent mode or in the incoherent mode, depending on the pixel size of the detector [4–6]. When the pixel size of the detector is infinitesimal, the OSH system is operated in the coherent mode because the complex amplitude of the 3D object is coded in the hologram. The object phase can be quantitatively measured from the reconstructed image [7,8]. OSH can be also operated in the incoherent mode by using a detector with infinite size of the pixel. In this mode, it is the intensity distribution of the 3D object to be coded in the hologram. Although the phase of the object is lost, the speckle is efficiently suppressed in this operation mode and the signal-to-noise ratio (SNR) is improved [7,9]. In the practical implementation of OSH, one usually applies a large pixel detector for recording an incoherent hologram and a pinhole against the detector for recording a coherent hologram. The size of the pixel of the photodetector will indeed affect the coherence property of the acquired hologram according to the OSH theory [3–6]. In this Letter, we report for the first time, to the best of our knowledge, on a series of experiments to demonstrate the effect of the pixel size of the photodetector on the acquired holograms. We will also address some important issues in OSH from the experimental point of view. The experimental setup is illustrated in Fig. 1. First, a linearly polarized He–Ne laser beam (λ 0.6328 μm) passes through a sawtooth-wave-driven electro-optic modulator (EOM) to generate two light modes (s-wave and p-wave) with a frequency difference νs . The two 0146-9592/15/102366-04$15.00/0
modes of light are separated by a polarizing beam splitter (PBS1). One of the beams is collimated, and the other is first collimated and focused to the front focal plane of lens L2. The two beams are recombined as a heterodyne beam by a beam splitter (BS). One of the heterodyne beams is directly measured by photodetector PD2 as a reference input of the lock-in amplifier. The other heterodyne beam is projected to the object target by L2. A polarizing beam splitter PBS2 and a quarter-wave plate (QWP) are inserted between L2 and the object. By this way, the p-wave is converted to a circularly polarized beam on the object plane. Similarly, the retroreflected object light is converted to an s-wave, which can be reflected by
Fig. 1. Schematic of the experimental setup. HWP, half-wave plate; EOM, electro-optic modulator; PBS, polarizing beam splitter; M, mirror; BE, beam splitter; L, lens; BS, beam splitter; PD, photodetector; QWP, quarter-wave plate. © 2015 Optical Society of America
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PBS2. The reflected light is converged by lens L3 (with focal length f 3 10 cm), and detected by photodetector PD1 at the back focal plane of L3. A variable iris or a pinhole is positioned against PD1 to effectuate various sizes of the pixel of the detector. The object to be scanned is a toy car (size 2.5 cm × 1.5 cm × 1.2 cm), which is a real 3D diffusely reflecting object. The toy car is mounted on a two-axis motorized stage, and it is moved relative to the heterodyne beam. The signals from PD1 and PD2 are sent to the lock-in amplifier to remove the carrier at frequency νs , and to demodulate the phase and amplitude of the signal. After scanning, all the acquired data are arranged as a final complex hologram in the PC. In our experimental condition, the heterodyne beam consists of a plane wave and a spherical wave of different temporal frequencies, and the complex hologram acquired in the coherent mode is expressed as [2,10] Z H c x; y
he x; y; z⊙Rx; y; zdz;
(1)
where ⊙ denotes the correlation calculation, Rx; y; z stands for the complex amplitude reflectance of a planar object located at a distance z from the focal plane of lens L2, and j −jπ 2 2 exp x y : he x; y; z λz λz
(2)
In the incoherent mode, the complex hologram is expressed as Z H ic x; y
he x; y; z⊙jRx; y; zj2 dz:
(3)
Therefore, it is the intensity response of the object to be recorded in the hologram. Note that we have modeled a 3D object as a collection of planar objects along the depth z of the object. Hence, the integration over z in Eq. (3) represents the holographic recording of a 3D object. To begin with, we have used a photodiode detector (Thorlabs: FDS100) with an active area or the pixel size of 3.6 mm × 3.6 mm as PD1. The numerical aperture of the spherical scanning beam is about 0.014, and the separation between adjacent sampling points (i.e., pixel pitch of the hologram) is Δ 22 μm. The hologram size is 1028 × 700 pixels Lx × Ly 22.6 mm × 15.4 mm. Figure 2(a) shows the amplitude and Fig. 2(b) shows
Fig. 2. (a) Amplitude and (b) phase modulus of the acquired hologram in the incoherent mode.
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the phase of the acquired hologram. Since there is no speckle among the hologram, the acquired hologram is referred to as the incoherent hologram henceforth. It is noted that the phase [Fig. 2(b)] is flat among the object region where the surface is relatively flat, which is not the case in conventional digital holography. This phenomenon is explained as follows. In the incoherent mode of OSH [Eq. (3)], each object point is encoded in the hologram with a response function (RF), he x; y; z. As shown in the left side of Fig. 3, for a shorter object distance z, the fringe frequency of its corresponding RF is higher, and vice versa. Note that the center phase of the RF is independent of the object distance z because the phase of the object has been removed. Thus, as all response functions of the object points are accumulated to a final hologram, the phase variation at the center of the hologram is small, as shown in the right side of Fig. 3. This is also the reason the phase map in Fig. 2(b) is relatively flat. To verify that the 3D information has been recorded, the incoherent hologram is reconstructed by propagation [10,11], E r x; y; zr hx; y; zr ⊗ Hx; y;
(4)
where hx; y; zr he x; y; zr exp−j2πzr ∕λ is the freespace impulse response, ⊗ denotes the convolution calculation, and E r x; y; zr is the diffracted optical field at zr from the hologram. Hx; y is either the coherent hologram or the incoherent hologram given by Eqs. (1) or (3), respectively. For incoherent imaging, and according to Eqs. (3) and (4), the reconstructed field at the object plane is proportional to the object’s intensity response, i.e., E r ∝ jRx; y; zr j2 . As a result, the amplitude of reconstructed field jE r j is always shown as the reconstructed image henceforth unless indicated otherwise. Figure 4(a) shows the reconstructed image (zr 63 mm). The enlarged portion of the reconstructed images at the front side (zr 53 mm) and the back side (zr 73 mm) are shown in Figs. 4(b)–4(e), which provides an apparent difference between the focus view and the defocus view, which gives the evidence that the 3D information of a diffusely reflecting object can be successfully recorded by the incoherent mode of OSH. We have conducted another series of experiments, in which a circular iris or pinhole is positioned against PD1 to simulate a detector with variable pixel size.
Fig. 3. Relationship between object distance z and the response function (RF).
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Fig. 4. (a) Reconstructed full-size image of the incoherent hologram at zr 63 mm. (b), (c) Selected portions of the reconstructed image at zr 53 mm. (d), (e) Selected portions of the reconstructed image at zr 73 mm. The selected portions are marked by two squares in (a). (See Media 1.)
The diameters of the iris (or pinhole) are 3 mm, 1.5 mm, 150 μm, 50 μm, 15 μm, and 5 μm. When the pinhole diameter was smaller than 150 μm, the signal was too weak to detect. In such a case we replaced the photodiode detector by a photomultiplier tube (PMT, Thorlabs: PMM02). The acquired holograms are shown in Fig. 5. As the detector size is being reduced, the phase variance at the center of the hologram becomes larger. Figure 6 shows the reconstructed images at zr 63 mm of these holograms. When the pixel size was reduced to 3 mm and then to 1.5 mm diameter, the speckle noise becomes more and more prominent [see Figs. 6(a) and 6(b)]. As the pixel size was further reduced, the reconstructed images contain not only speckles but also show randomly distributed bright spots [Figs. 6(c)–6(e)]. These bright spots alter the features of the image and thus result in serious error on the reconstructed images. This phenomenon could be due to the filtering effect of the size of the pixel [12], which is verified by experiment for the first time, to the best of our knowledge, and reported in this Letter. In the general theory, the hologram acquired
Fig. 5. Amplitude (left column) and phase modulus (right column) of the holograms with pixel diameters of (a) 3 mm, (b) 1.5 mm, (c) 150 μm, (d) 50 μm, (e) 15 μm, and (f) 5 μm.
by OSH with a mask mx; y against the pixel detector can be expressed as [12] Z H OSH x; y he x; y; z⊙Rx; y; zRF x; y; zdz; (5) where f2 −f 3 kx −f 3 ky : ; RF x; y; z F −1 F fR x; y; zg × 32 m k0 k0 k0 (6) Here, RF represents the action of filtering of the complex conjugated spectrum due to mask mx; y. Accordingly,
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Fig. 6. 63 mm (b) 1.5 and (f)
Reconstructed images [(a)–(e): jEr j; (f): jE r j2 ] at z of holograms with pixel diameters of (a) 3 mm, mm, (c) 150 μm (see Media 2), (d) 50 μm, (e) 15 μm, 5 μm (see Media 3).
we can explain Eq. (5) as the object information Rx; y; z is being apodized by RF . Hence, the hologram is usually being filtered when a finite pixel size is present unless two extreme cases occur, namely, the coherent mode of OSH [Eq. (1)] and the incoherent mode of OSH [Eq. (3)]. The critical pixel diameter for achieving incoherent mode of OSH is λ0 f 3 ∕Δ 2.9 mm [12]. Our experimental results agree with the theoretical prediction. Finally, when the diameter of the pixel size was reduced to the smallest (5 μm), the signal in this condition is very weak and the SNR is low. Nevertheless, the toy car can be still reconstructed, as shown in Fig. 6(f). Now the hologram is recorded in the coherent mode because the 5 μm pixel size is smaller than the critical diameter 2λ0 f 3 ∕Lx 5.6 μm [12]. According to Eqs. (1) and (4), the reconstructed field at the object plane is proportional to the object’s complex amplitude, i.e., Er ∝ Rx; y; zr . This is also the reason that the reconstructed image shown in Fig. 6(f) is the square of the absolute value of the reconstructed field, jE r j2 . The bright spot disappears but the speckles exist as in standard coherent imaging. In conclusion, in this Letter we have acquired digital holograms by scanning of a real 3D diffusely reflecting object. We have, for the first time, to the best of our knowledge, demonstrated that a 3D diffusely reflecting object can be successfully recorded and reconstructed by the incoherent mode of OSH, illustrating 3D reconstruction. We have also demonstrated that the
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reduced pixel size of the digital holographic system will seriously degrade the quality of the reconstructed image. As a result, the pixel size of the detector must be carefully determined prior to recording for the type of application intended. Finally, we have, also for the first time, to the best of our knowledge, acquired a coherent hologram of a 3D diffusely reflecting object. There are several shortcomings on the coherent mode OSH. First, the SNR and the resolution of the coherent-mode hologram are not as good as that of the incoherent-mode hologram. In addition, the axial location of the single-pixel detector must be precisely located at the back focal plane of lens L3. Otherwise, the pixel will allow not only the light from the same direction but also from the other directions. As a result, a low-quality hologram like Fig. 5(e) will be acquired. On the other hand, the incoherent mode of OSH has been proved to be able to record the 3D information of the object, and its optical alignment is very easy. Thus, the incoherent mode of OSH should be applied to record a diffusely reflecting object for 3D imaging applications. However, it should be indicated that the incoherent mode of OSH cannot be directly used for holographic 3D display. When the incoherent hologram [Eq. (3)] is directly used to address a complexmodulation spatial light modulator, the viewer will see an image with intensity distribution proportional to jE r j2 ∝ jRx; y; zj4 , not the real intensity distribution jRx; y; zj2 . A contrast-enhanced image, i.e., the intensity distribution is raised to the power of 2, will be reconstructed. This aspect is worthy of further investigation. This work is sponsored by MOST of Taiwan under contract number 103-2221-E-035-037-MY3. The authors thank Mr. Sheng-Yen Wang in preparing some of the experiments. References 1. T.-C. Poon, M. H. Wu, K. Shinoda, and Y. Suzuki, Proc. IEEE 84, 753 (1996). 2. T.-C. Poon, Optical Scanning Holography with MATLAB (Springer, 2007). 3. T.-C. Poon, J. Opt. Soc. Korea 13, 406 (2009). 4. T.-C. Poon and G. Indebetouw, Appl. Opt. 42, 1485 (2003). 5. G. Indebetouw, P. Klysubun, T. Kim, and T.-C. Poon, J. Opt. Soc. Am. A 17, 380 (2000). 6. J. Swoger, M. Martinez-Corral, J. Huisken, and E. H. K. Stelzer, J. Opt. Soc. Am. A 19, 1910 (2002). 7. G. Indebetouw and W. Zhong, J. Opt. Soc. Am. A 23, 1699 (2006). 8. G. Indebetouw, Y. Tada, and J. Leacock, Biomed. Eng. Online 5, 63 (2006). 9. Y. S. Kim, T. Kim, S. S. Woo, H. Kang, T.-C. Poon, and C. Zhou, Opt. Express 21, 8183 (2013). 10. T.-C. Poon and J.-P. Liu, Introduction to Modern Digital Holography with MATLAB (Cambridge University, 2014). 11. J.-P. Liu, J. Opt. Soc. Am. A 29, 1956 (2012). 12. J.-P. Liu, Appl. Opt. 54, A59 (2015).
