Dual-plane in-line digital holography based on liquid crystal on silicon spatial light modulator Spozmai Panezai,1 Dayong Wang,1,* Jie Zhao,2 Yunxin Wang,1 and Lu Rong1 1
Institute of Information Photonics Technology and College of Applied Sciences, Beijing University of Technology, Beijing 100124, China 2
The Pilot College, Beijing University of Technology, Beijing 101101, China *Corresponding author: [email protected] Received 16 May 2014; revised 30 June 2014; accepted 2 July 2014; posted 3 July 2014 (Doc. ID 212120); published 1 August 2014
A dual-plane in-line digital holographic method is proposed with a liquid crystal on silicon (LCOS) spatial light modulator (SLM) for recording holograms at two slightly displaced planes. The computer-generated chirp-like complex reflectance is displayed on the LCOS SLM to adapt the object beam at two planes for recording two holograms processed to eliminate the DC term and twin image accurately; no mechanical components or manual operation during data acquisition is required. The proposed approach improves the speed, accuracy, and stability of the experiment. Computer simulation and experiments for both amplitude and phase objects are carried out to validate the proposed method. © 2014 Optical Society of America OCIS codes: (090.1995) Digital holography; (130.4110) Modulators; (100.3010) Image reconstruction techniques. http://dx.doi.org/10.1364/AO.53.00G105
1. Introduction
Digital holography (DH) is one of the most important and basic techniques for recording and reconstructing the complete complex field of an object wavefront. A hologram is an interference pattern of a beam diffracted from the object and an undiffracted reference beam. There are two ways to insert the reference beam: off-axis [1,2] and on-axis configuration [3]. The reconstruction of the hologram results in three diffraction components: the DC term, object field, and its conjugate, which are overlapped in the case of the on-axis configuration and are spatially separated in the off-axis one. In off-axis configuration, the three diffraction components are separated due to a reference beam that propagates at a small angle with respect to the object beam. The required object’s wavefront is isolated by performing a spatial1559-128X/14/27G105-06$15.00/0 © 2014 Optical Society of America
frequency filtering. As a single hologram is sufficient for numerical reconstruction of the object information, this technique is suitable for the acquisition of dynamic processes, such as live cells in growth media. However, due to the angle between object and reference beams, the available space–bandwidth product (SBP) of the detector cannot be used efficiently. Therefore, the higher spatial-frequency components of the object information might be lost, which results in lower spatial resolution [4]. However, on-axis DH has a larger field of view and SBP than off-axis DH. To solve the twin-image problem of on-axis DH, various efforts have been made such as using phase-shifting, in which [5] three or four holograms of the same object are recorded by introducing stepwise phase retardations in the reference beam. This process enables the removal of the DC and the twin image by digitally processing the phase-shifted holograms. However, a relatively long time is required to apply the temporal phase shifts and acquire the holograms sequentially, which limits 20 September 2014 / Vol. 53, No. 27 / APPLIED OPTICS
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its applications. In addition, it has been shown that calibration errors in phase-shifting and system fluctuations between the frames degrade the quality of the reconstructed image [6–8]. Phase-retrieval algorithms have also been used for digital reconstruction of in-line holograms, but it is limited by the size of the object [9]. Dual-plane in-line DH is also suggested when two holograms are recorded at two different planes separated by a small distance perpendicular to the propagation direction and later reconstructed using a method of Fourier-domain processing [10–12]. Although this method involves simple computation and requires two holograms, it still needs a precise moving driver for recording holograms at two planes because if perfect alignment in the DH system is not achieved, a defocused twin image would superpose on the reconstructed image. Choosing the distance between two planes is also important but is limited due to the dynamic range of the translation stage [13]. In order to decrease the acquisition time in the dual-plane method, two charge-coupled devices (CCDs) are used for recording holograms, but it increases the pre-experimental efforts in the sense of proper alignment of CCDs at two different planes because even a small alignment error of one pixel will lead to a rough result [14]. In this paper, we proposed a dual-plane in-line digital holographic method based on a computergenerated chirp-like complex reflectance written on a LCOS spatial light modulator (SLM) in which the radius of the complex function is changed to reflect the object beam at two planes separated by a very small distance. The advantage of this kind of computer-generated dual-plane holographic technique is that the hologram recordings can be controlled digitally so that no precise mechanical moving element or pre-experimental alignment is required. Additionally, it is worth noting that the holograms’ acquisition and reconstruction process is very fast and stable, and the choice of distance between them is not limited. The article is arranged as follows: in Section 2, the basic principle is described. In Section 3, the computer simulations are carried out to verify the proposed method, whereas experimental results are presented in Section 4, and finally the conclusions are drawn in Section 5. 2. Principle
The schematic of the basic principle is shown in Fig. 1.
The principle of the proposed technique is based on the LCOS SLM, which is used to display two complex patterns, one after the other, as shown in Fig. 1, and are called complex pattern 1 and complex pattern 2. The laser beam diffracted from the object is made incident on the LCOS SLM, where complex pattern 1 is displayed. After reflection, the object beam seems to be coming from plane 1 rather than the original object position for the recording of the first hologram. The reference beam is a plane wave, which is not shown in Fig. 1 for simplicity. In order to record the second hologram at distance dz from the first one, complex pattern 2 is displayed on the LCOS. Now the object beam, after reflection from LCOS, seems to be coming from plane 2 at a distance dz from plane 1. Both the holograms are recorded for the object beam being reflected from LCOS at a fixed CCD position. Consider the object wave after free-space propagation over a distance z is made incident on LCOS and can be written in the form of a Fresnel propagation integral [15] as follows:
ZZ jk 2 xi y2i O0 x0 ; y0 2z ∞ jk 2 2 x y0 × exp 2z 0 jk × exp − xi x0 yi y0 dx0 dy0 ; z
Oi xi ; yi exp
where x0 ; y0 is the object plane and xi ; yi is the LCOS plane. Meeser et al. [16] used a reflectance function written on an SLM in order to adopt the reference beam according to the varying position of the object in off-axis digital lensless Fourier holography. In this paper, the same complex reflectance function is adopted for recording holograms at two separate planes located at distances z and z dz from the object plane, given as follows: Ri xi ; yi exp
−jkdzi x2i y2i jkdzi . 2zz dzi
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(2)
The modified object beam after reflection from LCOS is O0i xi ; yi Oi xi ; yi · Rxi ; yi ;
Fig. 1. Basic scheme for the explanation of the dual-plane in-line DH based on the LCOS SLM.
(1)
(3)
which appears to travel distance z dz instead of distance z. For both holograms, the object beam is reflected off the LCOS in order to suppress the noise during recording (dzi dz1 , dz2 ), and the distance between them is 0.05 mm. The modified object beam is propagated toward CCD by using the Rayleigh–Sommerfeld diffraction formula [15] with parameter distance z given as follows:
QO0i xi ;yi ;z −
1 2π
ZZ
O0i xi ;yi
∂ expjkri dxi dyi ∂z ri
O1 xi ;yi IFFT
Oi X;Y;z dzi Oi xi ;yi ⊗ hxi ;yi ;X;Y;z dzi ; (4) where Q is an operator corresponding to the Rayleigh–Sommerfeld transform; hxi ; yi ; X; Y; z dzi is the propagation operator over distance z dzi , i 1, 2, ⊗; and X; Y represents the convolution function and CCD plane. At the CCD plane, the object beam interferes with the reference beam (plane wave) of the unit amplitude. The two holograms are recorded by a fixed CCD without changing its position as given below: I 1 X; Y; z dz1 jO1 X; Y; z dz1 1j2 ; 2
I 2 X; Y; z dz2 jO2 X; Y; z dz2 1j :
(5)
(6)
The method of subtraction of average intensity of the recorded hologram, is used to suppress the zeroorder diffraction image, while keeping the object and the reference beam amplitudes comparable. After modification, Eqs. (5) and (6) took the following form: I 1 X;Y;z dz1 ≈ O1 X;Y; z dz1 O1 X; Y;z dz1 ; (7)
I 2 X; Y;z dz2 ≈ O2 X;Y;z dz2 O2 X;Y; z dz2 : (8) For the suppression of twin images, hologram 2 as described in Eq. (8) is propagated through a distance dz (distance between the two holograms) by using Rayleigh–Sommerfeld propagation operator Q. Considering the assumption as elaborated in [13], hologram 2 is subtracted from Eq. (7):
FFTΔIX;Y ; Hf X ;f Y ;zdz1 1−Hf X ;f Y ;2dz (11)
where FFT and IFFT are the Fourier transform and inverse Fourier transform; and Hf X ; f Y ; z dz1 and Hf X ; f Y ; 2dz are the transfer functions that are calculated numerically by using real parameters of recorded holograms, given as Hf X ;f Y ;zdz1 exp−jkzdz1 1−λ2 f 2X −λ2 f 2Y 1∕2 ; (12)
Hf X ; f Y ; 2dz exp−2jkdz1 − λ2 f 2X − λ2 f 2Y 1∕2 ; (13) where k 2π∕λ is the wavenumber and λ is the wavelength of light; and f X and f Y are the spatial frequency components along the x and y directions, respectively. The object information is then reconstructed by taking the inverse Fresnel transform of O1 xi ; yi over distance z. The validity of the method is checked by computer simulation and experimental implication. 3. Simulation
To check the validity of the proposed method, a simulation was carried out with a resolution test target being used as an object. It is an 8-bit, 256 pixel × 256 pixel image located in the middle of a 512 pixel × 512 pixel background (the value of the background pixels is 0), as shown in Fig. 2(a). The distance from the CCD to the nearest image plane is z 120 mm. We assumed the working wavelength is a green laser light of 532 nm, and pixel pitch of the CCD is
ΔIX; Y O1 X;Y;z dz1 − QfO2 X;Y; z dz2 ;dzg O1 xi ;yi ⊗ hxi ; yi ;X;Y; z dz1 − O1 xi ;yi ⊗ hxi ; yi ;X;Y; z dz1 ⊗ hxi ; yi ;X;Y; 2dz: (9) By taking the Fourier transform of the above equation, we get the following equation: FFTΔIX; Y O1 f xi ; f yi Hf X ; f Y ; z dz1 × f1 − Hf X ; f Y ; 2dzg
(10)
Finally, the object beam is calculated numerically using
Fig. 2. Simulation results for the USAF resolution chart, (a) object, (b) hologram at distance of 120.1 mm, (c) hologram at distance of 120.15 mm, and (d) reconstructed result. 20 September 2014 / Vol. 53, No. 27 / APPLIED OPTICS
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4.65 μm × 4.65 μm according to the real parameters. The object beam is propagated at a distance of 120 mm by a Fresnel diffraction formula toward a LCOS SLM. On LCOS, the complex reflectance pattern as given in Eq. (2) is displayed and consists of 512 pixels × 512 pixels with the pixel pitch of 8 μm × 8 μm. Two patterns, one with dz1 0.1 mm and the other with dz2 0.15 mm, are displayed on LCOS, as shown in Fig. 1 as complex pattern 1 and complex pattern 2. The object beam is multiplied with these patterns and is propagated toward CCD by using the Rayleigh–Sommerfeld diffraction formula. At the CCD plane, the reference beam of unit amplitude interferes with the object beam, and the two holograms, recorded for pattern dz1 0.1 mm and for pattern dz2 0.15 mm, are shown in Figs. 2(a) and 2(b). It means that the two holograms are recorded as being separated by a small distance of 0.05 mm by fixed CCD position, and later both are processed using a dual-plane algorithm. The reconstructed result as shown in Fig. 2(d) is very clear and without error, which confirms the experimental applicability of the proposed method. 4. Experimental Results
For the experimental demonstration of the proposed method, the Mach–Zehnder interferometer, as shown in Fig. 3, is used. Linearly polarized light from Cobolt Samba 532 nm laser is spatially filtered and collimated by a beam expender (BE) and is passed through half-wave plate (HWP1) in order to get the same intensity in both arms of the interferometer. Thereafter, the beam is divided by a polarizing beam splitter (PBS) into two beams; the one transmitted through HWP2 and sample is called the object beam, while the other beam serves as the reference beam. The sample is an amplitude object, an element of Groups 0 and 1 of a U.S. Air Force resolution chart. The diffracted light from the object is made incident on LCOS by a BS. After reflection from LCOS, the light is interfered with by the plane wave using the same BS. The parallel-aligned nematic liquidcrystal SLM (LCOS, HED6010XXX Holoeye) is used in the experiment with a resolution of 1920 pixels × 1080 pixels with a pixel pitch of 8 μm and fill factor of 87%.
Fig. 3. Setup of the dual-plane in-line DH recording system. HWP1 and HWP2, half-wave plates; BS, beam splitter; PBS, polarizing beam splitter; and M, mirror. G108
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The two holograms are recorded at distances of z dz1 118.1 mm and z dz2 118.15 mm from the image plane by a CCD with square pixels of size 4.65 μm, as shown in Figs. 4(a) and 4(b). The reconstructed image is shown in Fig. 4(c), which depicts the clearly resolved element 2 of Group 1. The corresponding width of the line is 223 μm, and the normalized intensity distribution of the pixels along the red line in Fig. 4(c) is shown in Fig. 4(d). It is important to note that the applicability of the reconstruction algorithm assumes that the reference beam is a known constant distribution. This requires that the reference wavefront, at planes 1 and 2, should be a plane wave and can be obtained by careful spatial filtering and collimation. If dz is equal to an integer multiple of λ, the phase at these two planes would theoretically have the same distribution. However, this requirement is difficult to fulfill for practical reasons [13], but in the case of LCOS SLM-based dual-plane in-line DH, it can be achieved easily by digital means. Consequently, the two-plane reference wavefronts will typically have a constant phase difference. The experimental results for a complex object that is a laser-etched grating of 140 μm period and 21 μm groove depth are depicted in Fig. 5. Figure 5(a) shows the wrapped reconstructed result using the proposed method, whereas Fig. 5(b) shows the image recorded by the “WYKONT110 optical-profiling system”. Figure 5(c) shows the 3D view of a portion of Fig. 5(a) in the yellow dashed box, and Fig. 5(d) shows a comparison of the profile curves obtained from the two methods: dual-plane in-line DH and laser optical-profiling system. The experimentally measured period of grating using the proposed method is 139.5 μm, which is very close to the 140 μm measured by the laser optical-profiling
Fig. 4. Experimental results of an amplitude object (USAF resolution chart): (a) hologram at plane z dz1 , (b) hologram at plane z dz2 , (c) reconstructed amplitude image, and (d) normalized pixel intensity values in arbitrary units along the direction of the line shown in (c) respectively.
5. Conclusion
In this paper, a dual-plane in-line digital holographic method is introduced based on a LCOS SLM for recording holograms at two slightly displaced planes. The suggested method is more effective to eliminate the DC term and twin image in the reconstructed in-line digital hologram. The practical advantage of LCOS-based dual-plane in-line DH is that it eliminates the need to accurately change the position of the object relative to the camera or to change the camera’s position; this advantage removes mechanical alignment issues such as misalignments due to nonorthogonalities and tilts in experimental setup that can arise from the use of the two-plane method. The reported approach also has the capability of enhancing the hologram acquisition rate because it is controlled digitally. Two holograms are recorded after reflection from the LCOS with the same function displayed on it, but only by creating a small distance between them so it reduces the noise during the recording process due to the LCOS function. We believe that the dual-plane in-line digitalholographic method based on a LCOS SLM will become useful in real-time analysis. This work is financially supported by the National Natural Science Foundation of China (Nos. 61307010 and 61205010), Beijing Municipal Natural Science Foundation (No. 1122004), China Postdoctoral Research Foundation (No. 2013M540828), Beijing Postdoctoral Research Foundation (No. 2013ZZ-17), Research Fund for the Doctoral Program of Higher Education of China (No. 20121103120003), and Beijing University Fund for Scientific Research of Doctor (No. X0006111201102). References
Fig. 5. Experimental results of a complex object (grating): (a) reconstructed phase image of grating by dual-plane in-line DH, (b) phase image by laser-profiler system, (c) 3D view of portion of (a) in the yellow dashed box, and (d) comparison of the curves obtained from the two methods.
system. The measured height profile is 10 μm, whereas by the laser optical-profiling system, it is 20 μm as clearly evident from the comparison curves shown in Fig. 5(d). The roughness in the curve of the optical-profiling system is attributed to the steepness of the grating because the reflected light could not properly reach the detector, whereas the dualplane in-line holography did not suffer. Although the curve obtained by dual-plane in-line DH is very smooth in comparison to that of the optical-profiling system, it does not get the real value of the height profile because the grating’s height profile exceeds the illumination wavelength used for recording the digital hologram and can be solved by using dual wavelengths [17].
1. E. N. Leith and J. Upatnieks, “Wavefront reconstruction with continuous-tone objects,” J. Opt. Soc. Am. 53, 1377–1381 (1963). 2. E. N. Leith and J. Upatnieks, “Wavefront reconstruction with diffuse illumination and three dimensional objects,” J. Opt. Soc. Am. 54, 1295–1301 (1964). 3. D. Gabor and W. Goss, “Interference microscope with total wavefront reconstruction,” J. Opt. Soc. Am. 56, 849–856 (1966). 4. L. Xu, X. Peng, Z. Guo, J. Miao, and A. Asundi, “Imaging analysis of digital holography,” Opt. Express 13, 2444–2452 (2005). 5. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22, 1268–1270 (1997). 6. C. P. Brophy, “Effect of intensity error correlation on the computed phase of the phase-shifting interferometry,” J. Opt. Soc. Am. A 7, 537–541 (1990). 7. J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wavefront measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983). 8. C. S. Gao, L. Zhang, H. T. Wang, J. Liao, and Y. Y. Zhu, “Phaseshifting error and its elimination in phase-shifting digital holography,” Opt. Lett. 27, 1687–1689 (2002). 9. L. Rong, F. Pan, W. Xiao, Y. Li, and F. Wang, “Twin image elimination from two in-line holograms via phase retrieval,” Chin. Opt. Lett. 10, 0609021 (2012). 10. Y. Zhang and X. Zhang, “Reconstruction of a complex object from two in-line holograms,” Opt. Express 11, 572–578 (2003). 20 September 2014 / Vol. 53, No. 27 / APPLIED OPTICS
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11. Y. Zhang, G. Pedrini, W. Osten, and H. J. Tiziani, “Reconstruction of in-line digital holograms from two intensity measurements,” Opt. Lett. 29, 1787–1789 (2004). 12. B. Das and C. Yelleswarapu, “Dual plane in-line digital holographic microscopy,” Opt. Lett. 35, 3426–3428 (2010). 13. G. Situ, J. P. Ryle, U. Gopinathan, and J. T. Sheridan, “Generalized in-line digital holographic technique based on intensity measurements at two different planes,” Appl. Opt. 47, 711–717 (2008). 14. H. C. Lee, S. H. Kim, and D.-S. Kim, “Two step on-axis digital holography using dual-channel Mach–Zehnder
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interferometer and matched filter algorithm,” J. Opt. Soc. Korea 14, 363–367 (2010). 15. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2004). 16. T. Meeser, C. Falldorf, C. V. Kopylow, and R. B. Bergmann, “Reference wave adaptation in digital lensless Fourier holography by means of a spatial light modulator,” Proc. SPIE 8082, 808206 (2011). 17. J. Gass, A. Dakoff, and M. K. Kim, “Phase imaging without 2π ambiguity by multi-wavelength digital holography,” Opt. Express 28, 1141–1143 (2003).
Institute of Information Photonics Technology and College of Applied Sciences, Beijing University of Technology, Beijing 100124, China 2
The Pilot College, Beijing University of Technology, Beijing 101101, China *Corresponding author: [email protected] Received 16 May 2014; revised 30 June 2014; accepted 2 July 2014; posted 3 July 2014 (Doc. ID 212120); published 1 August 2014
A dual-plane in-line digital holographic method is proposed with a liquid crystal on silicon (LCOS) spatial light modulator (SLM) for recording holograms at two slightly displaced planes. The computer-generated chirp-like complex reflectance is displayed on the LCOS SLM to adapt the object beam at two planes for recording two holograms processed to eliminate the DC term and twin image accurately; no mechanical components or manual operation during data acquisition is required. The proposed approach improves the speed, accuracy, and stability of the experiment. Computer simulation and experiments for both amplitude and phase objects are carried out to validate the proposed method. © 2014 Optical Society of America OCIS codes: (090.1995) Digital holography; (130.4110) Modulators; (100.3010) Image reconstruction techniques. http://dx.doi.org/10.1364/AO.53.00G105
1. Introduction
Digital holography (DH) is one of the most important and basic techniques for recording and reconstructing the complete complex field of an object wavefront. A hologram is an interference pattern of a beam diffracted from the object and an undiffracted reference beam. There are two ways to insert the reference beam: off-axis [1,2] and on-axis configuration [3]. The reconstruction of the hologram results in three diffraction components: the DC term, object field, and its conjugate, which are overlapped in the case of the on-axis configuration and are spatially separated in the off-axis one. In off-axis configuration, the three diffraction components are separated due to a reference beam that propagates at a small angle with respect to the object beam. The required object’s wavefront is isolated by performing a spatial1559-128X/14/27G105-06$15.00/0 © 2014 Optical Society of America
frequency filtering. As a single hologram is sufficient for numerical reconstruction of the object information, this technique is suitable for the acquisition of dynamic processes, such as live cells in growth media. However, due to the angle between object and reference beams, the available space–bandwidth product (SBP) of the detector cannot be used efficiently. Therefore, the higher spatial-frequency components of the object information might be lost, which results in lower spatial resolution [4]. However, on-axis DH has a larger field of view and SBP than off-axis DH. To solve the twin-image problem of on-axis DH, various efforts have been made such as using phase-shifting, in which [5] three or four holograms of the same object are recorded by introducing stepwise phase retardations in the reference beam. This process enables the removal of the DC and the twin image by digitally processing the phase-shifted holograms. However, a relatively long time is required to apply the temporal phase shifts and acquire the holograms sequentially, which limits 20 September 2014 / Vol. 53, No. 27 / APPLIED OPTICS
G105
its applications. In addition, it has been shown that calibration errors in phase-shifting and system fluctuations between the frames degrade the quality of the reconstructed image [6–8]. Phase-retrieval algorithms have also been used for digital reconstruction of in-line holograms, but it is limited by the size of the object [9]. Dual-plane in-line DH is also suggested when two holograms are recorded at two different planes separated by a small distance perpendicular to the propagation direction and later reconstructed using a method of Fourier-domain processing [10–12]. Although this method involves simple computation and requires two holograms, it still needs a precise moving driver for recording holograms at two planes because if perfect alignment in the DH system is not achieved, a defocused twin image would superpose on the reconstructed image. Choosing the distance between two planes is also important but is limited due to the dynamic range of the translation stage [13]. In order to decrease the acquisition time in the dual-plane method, two charge-coupled devices (CCDs) are used for recording holograms, but it increases the pre-experimental efforts in the sense of proper alignment of CCDs at two different planes because even a small alignment error of one pixel will lead to a rough result [14]. In this paper, we proposed a dual-plane in-line digital holographic method based on a computergenerated chirp-like complex reflectance written on a LCOS spatial light modulator (SLM) in which the radius of the complex function is changed to reflect the object beam at two planes separated by a very small distance. The advantage of this kind of computer-generated dual-plane holographic technique is that the hologram recordings can be controlled digitally so that no precise mechanical moving element or pre-experimental alignment is required. Additionally, it is worth noting that the holograms’ acquisition and reconstruction process is very fast and stable, and the choice of distance between them is not limited. The article is arranged as follows: in Section 2, the basic principle is described. In Section 3, the computer simulations are carried out to verify the proposed method, whereas experimental results are presented in Section 4, and finally the conclusions are drawn in Section 5. 2. Principle
The schematic of the basic principle is shown in Fig. 1.
The principle of the proposed technique is based on the LCOS SLM, which is used to display two complex patterns, one after the other, as shown in Fig. 1, and are called complex pattern 1 and complex pattern 2. The laser beam diffracted from the object is made incident on the LCOS SLM, where complex pattern 1 is displayed. After reflection, the object beam seems to be coming from plane 1 rather than the original object position for the recording of the first hologram. The reference beam is a plane wave, which is not shown in Fig. 1 for simplicity. In order to record the second hologram at distance dz from the first one, complex pattern 2 is displayed on the LCOS. Now the object beam, after reflection from LCOS, seems to be coming from plane 2 at a distance dz from plane 1. Both the holograms are recorded for the object beam being reflected from LCOS at a fixed CCD position. Consider the object wave after free-space propagation over a distance z is made incident on LCOS and can be written in the form of a Fresnel propagation integral [15] as follows:
ZZ jk 2 xi y2i O0 x0 ; y0 2z ∞ jk 2 2 x y0 × exp 2z 0 jk × exp − xi x0 yi y0 dx0 dy0 ; z
Oi xi ; yi exp
where x0 ; y0 is the object plane and xi ; yi is the LCOS plane. Meeser et al. [16] used a reflectance function written on an SLM in order to adopt the reference beam according to the varying position of the object in off-axis digital lensless Fourier holography. In this paper, the same complex reflectance function is adopted for recording holograms at two separate planes located at distances z and z dz from the object plane, given as follows: Ri xi ; yi exp
−jkdzi x2i y2i jkdzi . 2zz dzi
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(2)
The modified object beam after reflection from LCOS is O0i xi ; yi Oi xi ; yi · Rxi ; yi ;
Fig. 1. Basic scheme for the explanation of the dual-plane in-line DH based on the LCOS SLM.
(1)
(3)
which appears to travel distance z dz instead of distance z. For both holograms, the object beam is reflected off the LCOS in order to suppress the noise during recording (dzi dz1 , dz2 ), and the distance between them is 0.05 mm. The modified object beam is propagated toward CCD by using the Rayleigh–Sommerfeld diffraction formula [15] with parameter distance z given as follows:
QO0i xi ;yi ;z −
1 2π
ZZ
O0i xi ;yi
∂ expjkri dxi dyi ∂z ri
O1 xi ;yi IFFT
Oi X;Y;z dzi Oi xi ;yi ⊗ hxi ;yi ;X;Y;z dzi ; (4) where Q is an operator corresponding to the Rayleigh–Sommerfeld transform; hxi ; yi ; X; Y; z dzi is the propagation operator over distance z dzi , i 1, 2, ⊗; and X; Y represents the convolution function and CCD plane. At the CCD plane, the object beam interferes with the reference beam (plane wave) of the unit amplitude. The two holograms are recorded by a fixed CCD without changing its position as given below: I 1 X; Y; z dz1 jO1 X; Y; z dz1 1j2 ; 2
I 2 X; Y; z dz2 jO2 X; Y; z dz2 1j :
(5)
(6)
The method of subtraction of average intensity of the recorded hologram, is used to suppress the zeroorder diffraction image, while keeping the object and the reference beam amplitudes comparable. After modification, Eqs. (5) and (6) took the following form: I 1 X;Y;z dz1 ≈ O1 X;Y; z dz1 O1 X; Y;z dz1 ; (7)
I 2 X; Y;z dz2 ≈ O2 X;Y;z dz2 O2 X;Y; z dz2 : (8) For the suppression of twin images, hologram 2 as described in Eq. (8) is propagated through a distance dz (distance between the two holograms) by using Rayleigh–Sommerfeld propagation operator Q. Considering the assumption as elaborated in [13], hologram 2 is subtracted from Eq. (7):
FFTΔIX;Y ; Hf X ;f Y ;zdz1 1−Hf X ;f Y ;2dz (11)
where FFT and IFFT are the Fourier transform and inverse Fourier transform; and Hf X ; f Y ; z dz1 and Hf X ; f Y ; 2dz are the transfer functions that are calculated numerically by using real parameters of recorded holograms, given as Hf X ;f Y ;zdz1 exp−jkzdz1 1−λ2 f 2X −λ2 f 2Y 1∕2 ; (12)
Hf X ; f Y ; 2dz exp−2jkdz1 − λ2 f 2X − λ2 f 2Y 1∕2 ; (13) where k 2π∕λ is the wavenumber and λ is the wavelength of light; and f X and f Y are the spatial frequency components along the x and y directions, respectively. The object information is then reconstructed by taking the inverse Fresnel transform of O1 xi ; yi over distance z. The validity of the method is checked by computer simulation and experimental implication. 3. Simulation
To check the validity of the proposed method, a simulation was carried out with a resolution test target being used as an object. It is an 8-bit, 256 pixel × 256 pixel image located in the middle of a 512 pixel × 512 pixel background (the value of the background pixels is 0), as shown in Fig. 2(a). The distance from the CCD to the nearest image plane is z 120 mm. We assumed the working wavelength is a green laser light of 532 nm, and pixel pitch of the CCD is
ΔIX; Y O1 X;Y;z dz1 − QfO2 X;Y; z dz2 ;dzg O1 xi ;yi ⊗ hxi ; yi ;X;Y; z dz1 − O1 xi ;yi ⊗ hxi ; yi ;X;Y; z dz1 ⊗ hxi ; yi ;X;Y; 2dz: (9) By taking the Fourier transform of the above equation, we get the following equation: FFTΔIX; Y O1 f xi ; f yi Hf X ; f Y ; z dz1 × f1 − Hf X ; f Y ; 2dzg
(10)
Finally, the object beam is calculated numerically using
Fig. 2. Simulation results for the USAF resolution chart, (a) object, (b) hologram at distance of 120.1 mm, (c) hologram at distance of 120.15 mm, and (d) reconstructed result. 20 September 2014 / Vol. 53, No. 27 / APPLIED OPTICS
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4.65 μm × 4.65 μm according to the real parameters. The object beam is propagated at a distance of 120 mm by a Fresnel diffraction formula toward a LCOS SLM. On LCOS, the complex reflectance pattern as given in Eq. (2) is displayed and consists of 512 pixels × 512 pixels with the pixel pitch of 8 μm × 8 μm. Two patterns, one with dz1 0.1 mm and the other with dz2 0.15 mm, are displayed on LCOS, as shown in Fig. 1 as complex pattern 1 and complex pattern 2. The object beam is multiplied with these patterns and is propagated toward CCD by using the Rayleigh–Sommerfeld diffraction formula. At the CCD plane, the reference beam of unit amplitude interferes with the object beam, and the two holograms, recorded for pattern dz1 0.1 mm and for pattern dz2 0.15 mm, are shown in Figs. 2(a) and 2(b). It means that the two holograms are recorded as being separated by a small distance of 0.05 mm by fixed CCD position, and later both are processed using a dual-plane algorithm. The reconstructed result as shown in Fig. 2(d) is very clear and without error, which confirms the experimental applicability of the proposed method. 4. Experimental Results
For the experimental demonstration of the proposed method, the Mach–Zehnder interferometer, as shown in Fig. 3, is used. Linearly polarized light from Cobolt Samba 532 nm laser is spatially filtered and collimated by a beam expender (BE) and is passed through half-wave plate (HWP1) in order to get the same intensity in both arms of the interferometer. Thereafter, the beam is divided by a polarizing beam splitter (PBS) into two beams; the one transmitted through HWP2 and sample is called the object beam, while the other beam serves as the reference beam. The sample is an amplitude object, an element of Groups 0 and 1 of a U.S. Air Force resolution chart. The diffracted light from the object is made incident on LCOS by a BS. After reflection from LCOS, the light is interfered with by the plane wave using the same BS. The parallel-aligned nematic liquidcrystal SLM (LCOS, HED6010XXX Holoeye) is used in the experiment with a resolution of 1920 pixels × 1080 pixels with a pixel pitch of 8 μm and fill factor of 87%.
Fig. 3. Setup of the dual-plane in-line DH recording system. HWP1 and HWP2, half-wave plates; BS, beam splitter; PBS, polarizing beam splitter; and M, mirror. G108
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The two holograms are recorded at distances of z dz1 118.1 mm and z dz2 118.15 mm from the image plane by a CCD with square pixels of size 4.65 μm, as shown in Figs. 4(a) and 4(b). The reconstructed image is shown in Fig. 4(c), which depicts the clearly resolved element 2 of Group 1. The corresponding width of the line is 223 μm, and the normalized intensity distribution of the pixels along the red line in Fig. 4(c) is shown in Fig. 4(d). It is important to note that the applicability of the reconstruction algorithm assumes that the reference beam is a known constant distribution. This requires that the reference wavefront, at planes 1 and 2, should be a plane wave and can be obtained by careful spatial filtering and collimation. If dz is equal to an integer multiple of λ, the phase at these two planes would theoretically have the same distribution. However, this requirement is difficult to fulfill for practical reasons [13], but in the case of LCOS SLM-based dual-plane in-line DH, it can be achieved easily by digital means. Consequently, the two-plane reference wavefronts will typically have a constant phase difference. The experimental results for a complex object that is a laser-etched grating of 140 μm period and 21 μm groove depth are depicted in Fig. 5. Figure 5(a) shows the wrapped reconstructed result using the proposed method, whereas Fig. 5(b) shows the image recorded by the “WYKONT110 optical-profiling system”. Figure 5(c) shows the 3D view of a portion of Fig. 5(a) in the yellow dashed box, and Fig. 5(d) shows a comparison of the profile curves obtained from the two methods: dual-plane in-line DH and laser optical-profiling system. The experimentally measured period of grating using the proposed method is 139.5 μm, which is very close to the 140 μm measured by the laser optical-profiling
Fig. 4. Experimental results of an amplitude object (USAF resolution chart): (a) hologram at plane z dz1 , (b) hologram at plane z dz2 , (c) reconstructed amplitude image, and (d) normalized pixel intensity values in arbitrary units along the direction of the line shown in (c) respectively.
5. Conclusion
In this paper, a dual-plane in-line digital holographic method is introduced based on a LCOS SLM for recording holograms at two slightly displaced planes. The suggested method is more effective to eliminate the DC term and twin image in the reconstructed in-line digital hologram. The practical advantage of LCOS-based dual-plane in-line DH is that it eliminates the need to accurately change the position of the object relative to the camera or to change the camera’s position; this advantage removes mechanical alignment issues such as misalignments due to nonorthogonalities and tilts in experimental setup that can arise from the use of the two-plane method. The reported approach also has the capability of enhancing the hologram acquisition rate because it is controlled digitally. Two holograms are recorded after reflection from the LCOS with the same function displayed on it, but only by creating a small distance between them so it reduces the noise during the recording process due to the LCOS function. We believe that the dual-plane in-line digitalholographic method based on a LCOS SLM will become useful in real-time analysis. This work is financially supported by the National Natural Science Foundation of China (Nos. 61307010 and 61205010), Beijing Municipal Natural Science Foundation (No. 1122004), China Postdoctoral Research Foundation (No. 2013M540828), Beijing Postdoctoral Research Foundation (No. 2013ZZ-17), Research Fund for the Doctoral Program of Higher Education of China (No. 20121103120003), and Beijing University Fund for Scientific Research of Doctor (No. X0006111201102). References
Fig. 5. Experimental results of a complex object (grating): (a) reconstructed phase image of grating by dual-plane in-line DH, (b) phase image by laser-profiler system, (c) 3D view of portion of (a) in the yellow dashed box, and (d) comparison of the curves obtained from the two methods.
system. The measured height profile is 10 μm, whereas by the laser optical-profiling system, it is 20 μm as clearly evident from the comparison curves shown in Fig. 5(d). The roughness in the curve of the optical-profiling system is attributed to the steepness of the grating because the reflected light could not properly reach the detector, whereas the dualplane in-line holography did not suffer. Although the curve obtained by dual-plane in-line DH is very smooth in comparison to that of the optical-profiling system, it does not get the real value of the height profile because the grating’s height profile exceeds the illumination wavelength used for recording the digital hologram and can be solved by using dual wavelengths [17].
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