- Email: [email protected]
Phase-shifting in-line digital holography on a digital micro-mirror device
ARTICLE IN PRESS Optics and Lasers in Engineering 47 (2009) 896–901
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Phase-shifting in-line digital holography on a digital micro-mirror device Wenjing Zhou a,, Qiangsheng Xu a, Yingjie Yu a, Anand Asundi b a b
Department of Precision Mechanical Engineering, Shanghai University, No.149, Yanchang Road, Shanghai 200072, PR China School of Mechanical & Aerospace Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singapore
a r t i c l e in f o
a b s t r a c t
Article history: Received 26 November 2008 Received in revised form 18 February 2009 Accepted 20 February 2009 Available online 21 May 2009
A digital phase-shifting in-line holographic system based on the single coherence beam is developed. A series of phase-shifting fringes are generated by computer and outputted by a digital micro-mirror device (DMD). These fringes modulate the coherence beam because of the intensity modulation ability of DMD. In this work, the reconstructed conjugate image would not appear because of applying the phase-shifting algorithm. And calibration for the value of the optical phase shift is not necessary. An experiment based on a lens-less digital in-line micro-holographic setup with a phase grating specimen is conducted to demonstrate the validity of the present method. Crown Copyright & 2009 Published by Elsevier Ltd. All rights reserved.
Keywords: Digital in-line holography Phase-shifting fringe Digital micro-mirror device Phase grating
1. Introduction Though digital in-line holography is fazed by the reconstructed twin images, people try to apply it to many different areas [1–5]. Compared with off-axis holography, traditional in-line holography proposed by Gabor [6] shows two remarkable traits: highreconstruction resolution because of utilizing adequately the recording array area of CCD, and simple experimental setup with single illuminating beam. But its fatal disadvantages is also well known—three reconstruction images (two conjugate image and zero order one) always overlap each other, a focused real image always accompanies with a defocused virtual image and vice versa. This existed twin image debases excessive quality of the reconstructed images of in-line hologram which limits its effective application. In early years, the phase-shifting algorithm was firstly introduced to calculate iteratively the object wave on the hologram plane. These works were developed well by Yamaguchi [7,8]. Later, the method based on the point diffraction proposed by Kreuzer et al. [2] was validated effectively by application in biological specimen. Its difference with Gabor in-line holographic experimental setup [6] is that the point light source (spherical beam) is applied to illuminate the transparent sample, the interference between the directly transmission wave from the specimen (as reference wave) and the scattered wave from the specimen (object wave) forms the in-line hologram. At the same time, other in-line hologram without the specimen needs to be recorded as the reference hologram. This approach follows
Corresponding author. Tel./fax: +86 21 56334108.
E-mail address: [email protected] (W. Zhou).
along the marked advantages of in-line holography as simplify and high resolution. But its reconstruction algorithm is complex. The specimen must be high transparent and smaller than the size of sensitive area of CCD. In this letter, we introduced a digital micro-mirror device (DMD) into a digital phase-shifting in-line holographic interferential system. A DMD is applied to modulate the spherical coherence light source, then, a group of phase-shifting fringe singles is obtained. The modulated light source illuminates the tested specimen to form a group of phase-shifting in-line holograms. Here the general phase-shift algorithm can be used to obtain the object wave information of hologram on hologram plane. At last, the original object information on object plane can be reconstructed numerically by using the diffraction theory. The proposed approach keeps similar advantages of the digital in-line holography, such as simplify and high resolution, and avoids the phase-shifting aberration because of introducing phase-shifting fringe by a DMD. In addition, it has no special limitation for the tested specimen. The phase-shifting interferometry (PSI) is a well established and widely used technique. Yamaguchi [7] presented the in-line phase-shifting digital holography. It can finish deformation measurement very well because of the phase-shifting algorithm. In presented in-line holographic experimental setup, a reference wave was introduced. Its angle with the object wave is zero. A piezoelectric transducer (PZT) is widely employed as a phase shifter to change the phase of the reference wave, but accurate calibration of the voltages to the PZT is necessary. Ishii et al. [9,10] presented a PST with a laser diode (LD) source. The phase shift was performed by changing the injection current of LD in an unbalanced interferometer. The value of the phase shift given by
0143-8166/$ - see front matter Crown Copyright & 2009 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2009.02.008
ARTICLE IN PRESS W. Zhou et al. / Optics and Lasers in Engineering 47 (2009) 896–901
the LD in each interferometer must still be calibrated, because it depends on the optical path difference of the unbalanced interferometer. Youichi Bitou [11] developed a digital phaseshifting polarization interferometer that uses an electrically addressed spatial light modulator (EA-SLM). By use of the polarization property of a LCD-coupled parallel-aligned nematic liquid-crystal spatial light modulator (PAL-SLM), two orthogonally polarized beams, of which one is diffracted and the other is nondiffracted (only reflected) from the phase grating displayed on the LCD-coupled PAL-SLM, are introduced into a Michelson-type polarization interferometer. The optical phase of the diffraction
897
beam is digitally shifted by a phase shift of the binary phase grating, having no mechanical actuator such as a PZT. And calibration for the value of the optical phase shift is not necessary. Although the presented method in this paper used fringe projecting technique [12] as reference, they only have the same way to record information of object wave, it is clear they applied different demodulation approach to reconstruct the recorded object. In this paper, the principle of the proposed method is described. The phase-shifting in-line transmission holographic system is designed and built. At last, the experiment results are presented with a phase grating as a specimen. The validity is testified.
2. Proposed phase-shifting in-line digital hologram processing technique Phase-shifting fringes generated on a DMD illuminate the object to form the in-line hologram. Firstly, the object wave on the hologram plane (at the position of CCD) can be calculated by Fig. 1. Schematic of phase-shifting in-line holography on a DMD.
Fig. 2. Sketch of structure of a micro-mirror device.
Fig. 3. Two setups to test the amplitude modulation capability of DMD. (a) Projecting the laser source onto surface of DMD with 661. (b) Projecting the laser source onto surface of DMD with 901.
Fig. 4. Characteristic curve for the amplitude modulation capability of DMD with setup shown in Fig. 3a. (a) Characteristic curve of ‘‘on’’ diffracted beam. (b) Characteristic curve of ‘‘off’’ diffracted beam.
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W. Zhou et al. / Optics and Lasers in Engineering 47 (2009) 896–901
phase-shifting algorithm. Furthermore, the original object wave, or the object wave at other plane, can be reconstructed according to the diffraction theory. Fig. 1 shows a typical schematic of the setup.
In order to improve the precision, dual-holograms processing technique is adopted [13]. Four phase-shifting in-line holograms without an object are recorded, and their reconstructed information on the hologram plane can be expressed as Rðx; ZÞ ¼ C R ½I1 I3 þ iðI2 I4 Þ
2.1. Calculating the object wave on the hologram plane with phaseshifting algorithm
Thus, the noise can be easily eliminated by doing as follows: bðx; ZÞ ¼
One four-step phase-shifting algorithm is needed in this experimental work. Four phase-shifted fringe patterns are generated in the computer and exported on the DMD. The CCD records four modulated in-line holograms, which can be expressed as h pi m ¼ 0; 1; 2; 3 (1) Im ðx; ZÞ ¼ A2O þ A2r þ 2AO Ar cos jo ðx; ZÞ m 2 According to the four-step phase-shifting algorithm, the object wave on the hologram plane can be calculated as Oðx; ZÞ ¼ C O ½I1 I3 þ iðI2 I4 Þ
(2)
where CO is constant.
(3)
jOðx; ZÞj expfi½a tanðOðx; ZÞÞ a tanðRðx; ZÞÞg jRðx; ZÞj
(4)
2.2. Reconstructing the original object wave based on diffraction theory Based on diffraction theory, the original object wave c(x,y) is given as ZZ 1 expfikrg cðx; yÞ ¼ dx dZ (5) bðx; ZÞ r il qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 where r ¼ d þ ðx xÞ2 þ ðy ZÞ2 , d is the distance between CCD and the object. An amplitude-contrast image |c(x,y)|, and a phase-contrast image the atan[c(x,y)] can be calculated using either the Fresnel approximation or convolution algorithms [14]. In this paper the latter is applied.
3. Design of phase-shifting in-line digital holographic system 3.1. Describing structure and characteristic of DMD A DMD is composed of a series reflective micro-mirror unit. Each micro-mirror is riveted at two shores shown in Fig. 2. When DMD works, each reflective micro-mirror will sway with three angles: +yr, yr and 01. Micro-mirror is set as ‘‘on’’ state for +yr ¼ 121 or ‘‘off’’ state for yr ¼ 121. These sways action combined with the binary pulse width modulation (PWM) technique can accurately control the gray level of light. Thus a DMD is illuminated by a coherent light beam, one direct-reflected beam from the micro-surface of DMD, and two diffraction beams
Fig. 5. Characteristic curve for the amplitude modulation capability of DMD with setup shown in Fig. 3b. (a) Characteristic curve of ‘‘on’’ diffracted beam. (b) Characteristic curve of ‘‘off’’ diffracted beam.
Fig. 6. Phase-shifting in-line holographic system for transparent object. 1. Semiconductor, 2. Diaphragm, 3. DMD, 4. Phase-shifting fringes, 5. Object, 6. Reflected beam, 7. ‘‘On’’ diffracted beam, 8. ‘‘Off’’ diffracted beam, 9. CCD.
ARTICLE IN PRESS W. Zhou et al. / Optics and Lasers in Engineering 47 (2009) 896–901
(‘‘on’’ and ‘‘off’’ diffraction arrays) from micro-mirrors will be outputted. Now DMD has been used widely in digital light processing (DLP), high distinct TV (HDTV), micro-display and digital RAM, because of its high resolution, high aperture ratio, good gray level, good contrast, and fast response speed [15]. To make the best use of the amplitude modulation capability of DMD, the incidence direction and angle need satisfy some criterion conditions: i.e. the incident direction should be perpendicular to the diagonal of the micro-mirror. The incidence angle should be 661 to the surface of DMD. Fig. 3a and b shows two setups to test the amplitude modulation capability of DMD with 661 and 901 between the projecting light and the reflect plane of DMD, respectively. Twenty-five frames digital grid single (0–255) are inputted continuously into DMD and the grid difference between two adjacent frames is 10. Then the corresponding outputted grid maps are collected and their mean grid values are
899
calculated. Figs. 4 and 5 are the final tested results which show the characteristic curve for the amplitude modulation capability of DMD, in which the dot curve represents the original data and the line represents the fitted one. Figs. 4a and 5a show the characteristic curve of ‘‘on’’ diffracted beam of DMD, Figs. 4b and 5b show the one of ‘‘off’’ diffracted beam of DMD. Compared Fig. 4 with Fig. 5, it is clear that the line modulation capability of DMD for amplitude is better when projecting the laser source onto DMD with 661, while the line modulation capability of ‘‘on’’ diffracted beam of DMD is better than one of ‘‘off’’ diffracted beam of DMD. Thus, in this paper, the setup shown in Fig. 3a is selected to build the designed phase-shifting in-line transmission holographic system, and the zero order of ‘‘on’’ diffracted beam array of DMD is used as the illuminating light source. Actually, if setting properly the grid range modulated by DMD, the two setups can be applied with similarly effect.
Fig. 7. Phase-shifting in-line holographic system for reflective object. 1. Semiconductor laser, 2. Diaphragm, 3. Polarizer, 4. DMD, 5. Phase-shifting fringes, 6. PBS, 7. 1/4 wave plate, 8. Object. 9. CCD.
Fig. 8. Photo of the experimental setup. 1. Semiconductor laser, 2. Adjustable diaphragm, 3. DMD, 4. Sample, 5. CCD.
Fig. 9. Four-step phase-shifting in-line digital holograms. (a) Holograms with object. (b) Holograms without object.
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3.2. Designed phase-shifting in-line transmission holographic system Fig. 6 presents the designed phase-shifting in-line transmission, based on a lens-less magnification principle [13], holographic system. A semiconductor laser beam passes through an adjustable diaphragm to illuminate the DMD at an incidence angle of 661. The light wave is modulated by the phase-shifting fringes generated on the DMD. The zero-order light of ‘‘on’’ diffracted beam array of DMD directly illuminates the tested specimen and CCD records the transmission in-line hologram. It is necessary to adjust the distances between semiconductor, DMD and CCD to ensure no superposition of diffraction order and that no other order impinges on the CCD. Fig. 7 is the design for reflective object.
4. Results and discussion Fig. 8 presents the experimental system. It is comprised of a semiconductor laser (l ¼ 660 nm, adjustable power: 0–35 mW), an adjustable diaphragm (diameter from 0 to 20 mm), a DMD (1024*768 pixels with a pixel size of 13.68 mm and an overturn angle: 121), a tested specimen of sinusoidal phase grating
(T ¼ 100, 0.33 mm depth), a high resolution CCD (1392*1040 pixels with a pixel size 4.65*4.65 mm). Fig. 9a presents the four phase-shifting holograms of the phase grating, while Fig. 9b presents the holograms without the object. The scale of each hologram is 1392*1024 pixels. The reconstruction distance is 496.6 mm, the amplification of optical system is about 2.2 [13], and the results processed by the proposed method in this paper can be seen in Fig. 10. The lateral resolution of the reconstructed image along y-axis is 4.65 mm because of a convolution algorithms being applied [14]. The period of the reconstructed phase grate can also be calculated 4:65 1040 99:92 mm 22 2:2 where 4.65, 1040 are the lateral resolution and the pixel number of the reconstructed phase grate along y-axis, respectively. 22, 2.2 are the period number of the reconstructed phase grate and the amplification of optical system, respectively. Then 20 line traces along y-axis, one of them is shown in Fig. 10b, are equidistantly taken from the 3D-distribution of profile of the reconstructed phase grating shown in Fig. 10a. The average depth of the phase grating by using 20 line traces is about 0.3 mm.
5. Conclusions This paper proposes a novel phase-shifting in-line holographic system based on the single laser beam. A DMD is used to generate the phase-shifted fringes. This work applies a lens-less digital phase-shifting in-line micro-holographic setup. Two groups of in-line holograms, one with a phase grating specimen, and one with nothing object, are digitally recorded. These holograms are numerically reconstructed by using the four-step phase-shifting algorithm and the diffraction theory. The effectiveness of this presented approach has been demonstrated by the experiments results. The method in this paper can be applied to the measurement of both transparent and reflective objects by suitable arrangement of the optical setup. Specially, if both ‘‘on’’ diffracted beam and ‘‘off’’ diffracted beam of DMD can be utilized at the same time, the other novel phase-shift in-line holographic system based on two beams can be built, which named the orthogonal phase-shifting in-line holographic recording system in our work. It can record the holograms of object from two directions which are vertical to each other, and then try to obtain the more or all field reconstruction information of object, and also try to improve the lateral resolution of the numerical reconstruction of hologram. These works will be presented in other paper.
Acknowledgments This was supported by the Science and Technology Commission of the Shanghai Municipality (No. 075115001), National Natural Science Foundation (No. 60772124), Shanghai Leading Academic Discipline, PR China.
Appendix A. Supplementary Materials Fig. 10. Reconstructed results. (a) Reconstructed 3D image from the holograms. (b) A line trace of 3D image.
The online version of this article contains additional supplementary data. Please visit doi:10.1016/j.optlaseng.2009.02.008
ARTICLE IN PRESS W. Zhou et al. / Optics and Lasers in Engineering 47 (2009) 896–901
References [1] Murata S, Yasuda N. Potential of digital holography in particle measurement. Opt Laser Technol 2000;32:567–74. [2] Kreuzer HJ, Jericho MJ, Meinertzhagen IA, Xu WB. Digital in-line holography with photons and electrons. J Phys: Condens Matter 2001;13: 10,729–40. [3] Pan G, Meng H. Digital holography of particle fields: reconstruction by use of complex amplitude. Appl Opt 2003;42:827–33. [4] Jericho SK, Garcia-Sucerquia J, Xuv W, Jericho MH, Kreuzer HJ. Submersible digital in-line holographic microscope. Opt Lett 2006;31:1211–3. [5] Nomura T, Javidi B, Murata S. Polarization imaging of a 3D object by use of onaxis phase-shifting digital holography. Opt Lett 2007;32:481–3. [6] Gabor D. A new microscopic principle. Nature 1948;161:777–8. [7] Yamaguchi I, Zhang T. Phase-shifting digital holography. Opt Lett 1997; 22:1268–70.
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[8] Yamaguchi I, Kato J, Ohta S, Mizuno J. Image formation in phase-shifting digital holography and applications to microscopy. Appl Opt 2001;40: 6177–86. [9] Ishii Y, Chen J, Murata K. Digital phase-measuring interferometry with a tunable laser. Opt Lett 1987;12:233–5. [10] Ishii Y. Wavelength-tunable laser-diode interferometer. Opt Rev 1999;6:273. [11] Bitou Y. Digital phase-shifting interferometer with an electrically addressed liquid-crystal spatial light modulator. Opt Lett 2003;28:1576–8. [12] Srinivasan V, Liu HC, Halioua M. Automated phase-measuring profilometry: a phase mapping approach. Appl Opt 1985;24:185–8. [13] Zhou WJ, Yu YJ, Asundi A. Study on aberration suppressing methods in digital micro-holography. Opt Laser Eng 2009;47:264–70. [14] Schnars U, Juptner WPO. Digital recording and numerical reconstruction of holograms. Meas Sci Technol 2002;13:85–101. [15] DMD0.7 XGA 12* DDR DMD discorveryTM. Product preview data sheet [Z], TI DN 2503686, 2005.
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Phase-shifting in-line digital holography on a digital micro-mirror device Wenjing Zhou a,, Qiangsheng Xu a, Yingjie Yu a, Anand Asundi b a b
Department of Precision Mechanical Engineering, Shanghai University, No.149, Yanchang Road, Shanghai 200072, PR China School of Mechanical & Aerospace Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singapore
a r t i c l e in f o
a b s t r a c t
Article history: Received 26 November 2008 Received in revised form 18 February 2009 Accepted 20 February 2009 Available online 21 May 2009
A digital phase-shifting in-line holographic system based on the single coherence beam is developed. A series of phase-shifting fringes are generated by computer and outputted by a digital micro-mirror device (DMD). These fringes modulate the coherence beam because of the intensity modulation ability of DMD. In this work, the reconstructed conjugate image would not appear because of applying the phase-shifting algorithm. And calibration for the value of the optical phase shift is not necessary. An experiment based on a lens-less digital in-line micro-holographic setup with a phase grating specimen is conducted to demonstrate the validity of the present method. Crown Copyright & 2009 Published by Elsevier Ltd. All rights reserved.
Keywords: Digital in-line holography Phase-shifting fringe Digital micro-mirror device Phase grating
1. Introduction Though digital in-line holography is fazed by the reconstructed twin images, people try to apply it to many different areas [1–5]. Compared with off-axis holography, traditional in-line holography proposed by Gabor [6] shows two remarkable traits: highreconstruction resolution because of utilizing adequately the recording array area of CCD, and simple experimental setup with single illuminating beam. But its fatal disadvantages is also well known—three reconstruction images (two conjugate image and zero order one) always overlap each other, a focused real image always accompanies with a defocused virtual image and vice versa. This existed twin image debases excessive quality of the reconstructed images of in-line hologram which limits its effective application. In early years, the phase-shifting algorithm was firstly introduced to calculate iteratively the object wave on the hologram plane. These works were developed well by Yamaguchi [7,8]. Later, the method based on the point diffraction proposed by Kreuzer et al. [2] was validated effectively by application in biological specimen. Its difference with Gabor in-line holographic experimental setup [6] is that the point light source (spherical beam) is applied to illuminate the transparent sample, the interference between the directly transmission wave from the specimen (as reference wave) and the scattered wave from the specimen (object wave) forms the in-line hologram. At the same time, other in-line hologram without the specimen needs to be recorded as the reference hologram. This approach follows
Corresponding author. Tel./fax: +86 21 56334108.
E-mail address: [email protected] (W. Zhou).
along the marked advantages of in-line holography as simplify and high resolution. But its reconstruction algorithm is complex. The specimen must be high transparent and smaller than the size of sensitive area of CCD. In this letter, we introduced a digital micro-mirror device (DMD) into a digital phase-shifting in-line holographic interferential system. A DMD is applied to modulate the spherical coherence light source, then, a group of phase-shifting fringe singles is obtained. The modulated light source illuminates the tested specimen to form a group of phase-shifting in-line holograms. Here the general phase-shift algorithm can be used to obtain the object wave information of hologram on hologram plane. At last, the original object information on object plane can be reconstructed numerically by using the diffraction theory. The proposed approach keeps similar advantages of the digital in-line holography, such as simplify and high resolution, and avoids the phase-shifting aberration because of introducing phase-shifting fringe by a DMD. In addition, it has no special limitation for the tested specimen. The phase-shifting interferometry (PSI) is a well established and widely used technique. Yamaguchi [7] presented the in-line phase-shifting digital holography. It can finish deformation measurement very well because of the phase-shifting algorithm. In presented in-line holographic experimental setup, a reference wave was introduced. Its angle with the object wave is zero. A piezoelectric transducer (PZT) is widely employed as a phase shifter to change the phase of the reference wave, but accurate calibration of the voltages to the PZT is necessary. Ishii et al. [9,10] presented a PST with a laser diode (LD) source. The phase shift was performed by changing the injection current of LD in an unbalanced interferometer. The value of the phase shift given by
0143-8166/$ - see front matter Crown Copyright & 2009 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2009.02.008
ARTICLE IN PRESS W. Zhou et al. / Optics and Lasers in Engineering 47 (2009) 896–901
the LD in each interferometer must still be calibrated, because it depends on the optical path difference of the unbalanced interferometer. Youichi Bitou [11] developed a digital phaseshifting polarization interferometer that uses an electrically addressed spatial light modulator (EA-SLM). By use of the polarization property of a LCD-coupled parallel-aligned nematic liquid-crystal spatial light modulator (PAL-SLM), two orthogonally polarized beams, of which one is diffracted and the other is nondiffracted (only reflected) from the phase grating displayed on the LCD-coupled PAL-SLM, are introduced into a Michelson-type polarization interferometer. The optical phase of the diffraction
897
beam is digitally shifted by a phase shift of the binary phase grating, having no mechanical actuator such as a PZT. And calibration for the value of the optical phase shift is not necessary. Although the presented method in this paper used fringe projecting technique [12] as reference, they only have the same way to record information of object wave, it is clear they applied different demodulation approach to reconstruct the recorded object. In this paper, the principle of the proposed method is described. The phase-shifting in-line transmission holographic system is designed and built. At last, the experiment results are presented with a phase grating as a specimen. The validity is testified.
2. Proposed phase-shifting in-line digital hologram processing technique Phase-shifting fringes generated on a DMD illuminate the object to form the in-line hologram. Firstly, the object wave on the hologram plane (at the position of CCD) can be calculated by Fig. 1. Schematic of phase-shifting in-line holography on a DMD.
Fig. 2. Sketch of structure of a micro-mirror device.
Fig. 3. Two setups to test the amplitude modulation capability of DMD. (a) Projecting the laser source onto surface of DMD with 661. (b) Projecting the laser source onto surface of DMD with 901.
Fig. 4. Characteristic curve for the amplitude modulation capability of DMD with setup shown in Fig. 3a. (a) Characteristic curve of ‘‘on’’ diffracted beam. (b) Characteristic curve of ‘‘off’’ diffracted beam.
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phase-shifting algorithm. Furthermore, the original object wave, or the object wave at other plane, can be reconstructed according to the diffraction theory. Fig. 1 shows a typical schematic of the setup.
In order to improve the precision, dual-holograms processing technique is adopted [13]. Four phase-shifting in-line holograms without an object are recorded, and their reconstructed information on the hologram plane can be expressed as Rðx; ZÞ ¼ C R ½I1 I3 þ iðI2 I4 Þ
2.1. Calculating the object wave on the hologram plane with phaseshifting algorithm
Thus, the noise can be easily eliminated by doing as follows: bðx; ZÞ ¼
One four-step phase-shifting algorithm is needed in this experimental work. Four phase-shifted fringe patterns are generated in the computer and exported on the DMD. The CCD records four modulated in-line holograms, which can be expressed as h pi m ¼ 0; 1; 2; 3 (1) Im ðx; ZÞ ¼ A2O þ A2r þ 2AO Ar cos jo ðx; ZÞ m 2 According to the four-step phase-shifting algorithm, the object wave on the hologram plane can be calculated as Oðx; ZÞ ¼ C O ½I1 I3 þ iðI2 I4 Þ
(2)
where CO is constant.
(3)
jOðx; ZÞj expfi½a tanðOðx; ZÞÞ a tanðRðx; ZÞÞg jRðx; ZÞj
(4)
2.2. Reconstructing the original object wave based on diffraction theory Based on diffraction theory, the original object wave c(x,y) is given as ZZ 1 expfikrg cðx; yÞ ¼ dx dZ (5) bðx; ZÞ r il qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 where r ¼ d þ ðx xÞ2 þ ðy ZÞ2 , d is the distance between CCD and the object. An amplitude-contrast image |c(x,y)|, and a phase-contrast image the atan[c(x,y)] can be calculated using either the Fresnel approximation or convolution algorithms [14]. In this paper the latter is applied.
3. Design of phase-shifting in-line digital holographic system 3.1. Describing structure and characteristic of DMD A DMD is composed of a series reflective micro-mirror unit. Each micro-mirror is riveted at two shores shown in Fig. 2. When DMD works, each reflective micro-mirror will sway with three angles: +yr, yr and 01. Micro-mirror is set as ‘‘on’’ state for +yr ¼ 121 or ‘‘off’’ state for yr ¼ 121. These sways action combined with the binary pulse width modulation (PWM) technique can accurately control the gray level of light. Thus a DMD is illuminated by a coherent light beam, one direct-reflected beam from the micro-surface of DMD, and two diffraction beams
Fig. 5. Characteristic curve for the amplitude modulation capability of DMD with setup shown in Fig. 3b. (a) Characteristic curve of ‘‘on’’ diffracted beam. (b) Characteristic curve of ‘‘off’’ diffracted beam.
Fig. 6. Phase-shifting in-line holographic system for transparent object. 1. Semiconductor, 2. Diaphragm, 3. DMD, 4. Phase-shifting fringes, 5. Object, 6. Reflected beam, 7. ‘‘On’’ diffracted beam, 8. ‘‘Off’’ diffracted beam, 9. CCD.
ARTICLE IN PRESS W. Zhou et al. / Optics and Lasers in Engineering 47 (2009) 896–901
(‘‘on’’ and ‘‘off’’ diffraction arrays) from micro-mirrors will be outputted. Now DMD has been used widely in digital light processing (DLP), high distinct TV (HDTV), micro-display and digital RAM, because of its high resolution, high aperture ratio, good gray level, good contrast, and fast response speed [15]. To make the best use of the amplitude modulation capability of DMD, the incidence direction and angle need satisfy some criterion conditions: i.e. the incident direction should be perpendicular to the diagonal of the micro-mirror. The incidence angle should be 661 to the surface of DMD. Fig. 3a and b shows two setups to test the amplitude modulation capability of DMD with 661 and 901 between the projecting light and the reflect plane of DMD, respectively. Twenty-five frames digital grid single (0–255) are inputted continuously into DMD and the grid difference between two adjacent frames is 10. Then the corresponding outputted grid maps are collected and their mean grid values are
899
calculated. Figs. 4 and 5 are the final tested results which show the characteristic curve for the amplitude modulation capability of DMD, in which the dot curve represents the original data and the line represents the fitted one. Figs. 4a and 5a show the characteristic curve of ‘‘on’’ diffracted beam of DMD, Figs. 4b and 5b show the one of ‘‘off’’ diffracted beam of DMD. Compared Fig. 4 with Fig. 5, it is clear that the line modulation capability of DMD for amplitude is better when projecting the laser source onto DMD with 661, while the line modulation capability of ‘‘on’’ diffracted beam of DMD is better than one of ‘‘off’’ diffracted beam of DMD. Thus, in this paper, the setup shown in Fig. 3a is selected to build the designed phase-shifting in-line transmission holographic system, and the zero order of ‘‘on’’ diffracted beam array of DMD is used as the illuminating light source. Actually, if setting properly the grid range modulated by DMD, the two setups can be applied with similarly effect.
Fig. 7. Phase-shifting in-line holographic system for reflective object. 1. Semiconductor laser, 2. Diaphragm, 3. Polarizer, 4. DMD, 5. Phase-shifting fringes, 6. PBS, 7. 1/4 wave plate, 8. Object. 9. CCD.
Fig. 8. Photo of the experimental setup. 1. Semiconductor laser, 2. Adjustable diaphragm, 3. DMD, 4. Sample, 5. CCD.
Fig. 9. Four-step phase-shifting in-line digital holograms. (a) Holograms with object. (b) Holograms without object.
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3.2. Designed phase-shifting in-line transmission holographic system Fig. 6 presents the designed phase-shifting in-line transmission, based on a lens-less magnification principle [13], holographic system. A semiconductor laser beam passes through an adjustable diaphragm to illuminate the DMD at an incidence angle of 661. The light wave is modulated by the phase-shifting fringes generated on the DMD. The zero-order light of ‘‘on’’ diffracted beam array of DMD directly illuminates the tested specimen and CCD records the transmission in-line hologram. It is necessary to adjust the distances between semiconductor, DMD and CCD to ensure no superposition of diffraction order and that no other order impinges on the CCD. Fig. 7 is the design for reflective object.
4. Results and discussion Fig. 8 presents the experimental system. It is comprised of a semiconductor laser (l ¼ 660 nm, adjustable power: 0–35 mW), an adjustable diaphragm (diameter from 0 to 20 mm), a DMD (1024*768 pixels with a pixel size of 13.68 mm and an overturn angle: 121), a tested specimen of sinusoidal phase grating
(T ¼ 100, 0.33 mm depth), a high resolution CCD (1392*1040 pixels with a pixel size 4.65*4.65 mm). Fig. 9a presents the four phase-shifting holograms of the phase grating, while Fig. 9b presents the holograms without the object. The scale of each hologram is 1392*1024 pixels. The reconstruction distance is 496.6 mm, the amplification of optical system is about 2.2 [13], and the results processed by the proposed method in this paper can be seen in Fig. 10. The lateral resolution of the reconstructed image along y-axis is 4.65 mm because of a convolution algorithms being applied [14]. The period of the reconstructed phase grate can also be calculated 4:65 1040 99:92 mm 22 2:2 where 4.65, 1040 are the lateral resolution and the pixel number of the reconstructed phase grate along y-axis, respectively. 22, 2.2 are the period number of the reconstructed phase grate and the amplification of optical system, respectively. Then 20 line traces along y-axis, one of them is shown in Fig. 10b, are equidistantly taken from the 3D-distribution of profile of the reconstructed phase grating shown in Fig. 10a. The average depth of the phase grating by using 20 line traces is about 0.3 mm.
5. Conclusions This paper proposes a novel phase-shifting in-line holographic system based on the single laser beam. A DMD is used to generate the phase-shifted fringes. This work applies a lens-less digital phase-shifting in-line micro-holographic setup. Two groups of in-line holograms, one with a phase grating specimen, and one with nothing object, are digitally recorded. These holograms are numerically reconstructed by using the four-step phase-shifting algorithm and the diffraction theory. The effectiveness of this presented approach has been demonstrated by the experiments results. The method in this paper can be applied to the measurement of both transparent and reflective objects by suitable arrangement of the optical setup. Specially, if both ‘‘on’’ diffracted beam and ‘‘off’’ diffracted beam of DMD can be utilized at the same time, the other novel phase-shift in-line holographic system based on two beams can be built, which named the orthogonal phase-shifting in-line holographic recording system in our work. It can record the holograms of object from two directions which are vertical to each other, and then try to obtain the more or all field reconstruction information of object, and also try to improve the lateral resolution of the numerical reconstruction of hologram. These works will be presented in other paper.
Acknowledgments This was supported by the Science and Technology Commission of the Shanghai Municipality (No. 075115001), National Natural Science Foundation (No. 60772124), Shanghai Leading Academic Discipline, PR China.
Appendix A. Supplementary Materials Fig. 10. Reconstructed results. (a) Reconstructed 3D image from the holograms. (b) A line trace of 3D image.
The online version of this article contains additional supplementary data. Please visit doi:10.1016/j.optlaseng.2009.02.008
ARTICLE IN PRESS W. Zhou et al. / Optics and Lasers in Engineering 47 (2009) 896–901
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