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Elimination of zero-order diffraction in digital off-axis holography
Optics Communications 240 (2004) 261–267 www.elsevier.com/locate/optcom
Elimination of zero-order diffraction in digital off-axis holography Yimo Zhang, Qieni Lu¨ *, Baozhen Ge Key Laboratory of Opto-electronics Information and Technical Science, College of Opto-electronics and Precision Instrument Engineering, Ministry of Education, Tianjin University, Tianjin 300072, PR China Received 21 February 2004; received in revised form 17 May 2004; accepted 18 June 2004
Abstract A simple experimental method of eliminating zero-order diffraction in the reconstructed image of off-axis digital holography is presented. Holographic diffraction grating acting as a beam splitter, an off-axis holography system is formed. The holograms of object with different recording parameters are obtained by adjusting the reflecting mirror in the recording optical system to vary the incidence orientation of the object beam in CCD to introduce a phase shift. The zero-order image can be eliminated by numerically processing the holograms of object with different recording parameters. The theoretical analyses have been done in detail and the experimental results are also given. The zero-order image eliminated, the area of reconstructed image increases remarkably, the image quality can then be significantly improved and the better resolution obtained. The experimental results show that the method presented in this paper is feasible, simple in optical system and easy in operation and data processing. 2004 Published by Elsevier B.V. PACS: 42.79.Yd Keywords: Digital off-axis holography; Zero-order diffraction; Holographic grating
1. Introduction Digital holography, digital recording of hologram and numerical reconstruction of the wave fields, offers an alternative recording and recon-
*
Corresponding author. Tel.: +86-22-274-02594; fax: +8622-274-04547. E-mail address: [email protected] (Q. Lu¨). 0030-4018/$ - see front matter 2004 Published by Elsevier B.V. doi:10.1016/j.optcom.2004.06.040
struction in holography. In comparison with classical holography, the main advantage of digital holography is that intensity as well as phase information of a holographically stored wavefront can be achieved directly in the numerical reconstruction process, and that chemical processing is no necessary, thus increasing the flexibility and speed of the experimental process. With the development of high-resolution CCD and fast computer, it has become possible, and been widely applied in
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Y. Zhang et al. / Optics Communications 240 (2004) 261–267
various fields of science and engineering, for example, deformation measurement [1–3], holographic microscopy [4,5], etc. As in classical holography, off-axis geometries is generally used in digital holography, and its reconstructed image contains a zero-order image, a virtual image and a real image. Even if these three terms appear at different locations in the reconstructed images, the virtual image and the zero-order image need to be eliminated. Because the total number of sampling points of the reconstructed image is equal to the total number of pixels of the image sensor, the virtual image and the zeroorder image limit the size of the real image area. Eliminating the two undesirable images results in an enhancement of image quality. Two kinds of method have been proposed for this purpose. One kind of method is based on digital processing method. For example, Cuche et al. [6] proposed that zero-order image and virtual image can be digitally eliminated by means of filtering their associated spatial frequencies in the computed Fourier transform of the hologram. In [7] space-domain modulation of detected hologram was used to suppress zero-order and/or virtual image. The other kind of method based on experimental method was presented by Takaki et al. [8], which eliminates zero-order and virtual image by two shutters and one phase-modulator. In this paper, we propose the experimental method of eliminating the zero-order image and describe it in detail. The holographic grating is used to configure an off-axis holographic recording system. The two holograms recorded with different recording parameters are obtained and numerically subtracted to eliminate the zero-order image. The recording parameter with the different orientation of the object beam is gained by adjusting the reflecting mirror. Some experimental results are also given.
2. Hologram digital recording and numerical reconstruction The diagram of the hologram recording optical system we design is shown in Fig. 1. The holographic diffraction grating of low frequency is used,
plane-wave
object
(x,y)
M
+1 order
O
θg R
zero-order grating
CCD
Fig. 1. Schematic diagram of the recording system of off-axis hologram by CCD. O: object wave; R: reference wave; M: reflecting mirror.
functioning as beam splitter. The monochrome plane parallel light wave illuminates holographic grating normally; through grating diffraction, the three different plane parallel light waves, namely, the zero-order and the ±1 order diffraction waves, are generated. The +1 order diffraction waves illuminate the object and then are reflected onto the hologram plane by reflecting mirror M, acting as the object wave. The zero-order diffraction waves propagate perpendicular to the hologram plane oxy, acting as the reference wave. Assume that the complex-amplitude distribution of the object beam on the hologram plane is denoted by O(x,y) = jO(x,y)jexp jh (x,y), and the complexamplitude distribution of the plane reference beam on the hologram plane is expressed as r(x,y) = R. k is the wavelength of the incident light. The intensity distribution of the hologram is given by 2
Iðx; yÞ ¼ jOðx; yÞj þ R2 þ RjOðx; yÞj exp jhðx; yÞ þ RjOðx; yÞj exp½jhðx; yÞ;
ð1Þ
where superscript stands for the complex conjugate, jO(x,y)j2 is the intensity of the object wave and R2 is the intensity of the reference wave. The third and the fourth terms on the right-hand side of Eq. (1) are the interference terms. A digital hologram is recorded by a CCD sensor and transmitted to a computer by frame grabber card, and then, numerically reconstructed in computer. The digital hologram I(k,l) can be expressed as x y Iðk; lÞ ¼ Iðx; yÞrect ; MDx N Dy
M=2 X
N =2 X
k¼M=2 l¼N =2
dðx kDx; y lDyÞ:
ð2Þ
Y. Zhang et al. / Optics Communications 240 (2004) 261–267
The holographic reconstruction, in physical terms, is to reconstruct the wavefront at the object, i.e., to obtain the three-dimensional structure of the object from the two-dimensional hologram. The reconstructed wavefront u(m,n) can be obtained in the observation plane o 0 x 0 y 0 by computing the discrete Fresnel integral of the digital hologram intensity I(k,l),
exp j 2p d jp 2 02 2 02 k ðm Dx þ n Dy Þ uðm; nÞ ¼ exp kd jkd p FFTfIðk; lÞ exp j ðk 2 Dx2 þ l2 Dy 2 Þg; kd ð3Þ where k,l,m,n are integers, m = 0,1,. . .,M1; n = 0,1,. . .,N1, M · N is the number of light-sensitive pixels on CCD, Dx and Dy are the distances between neighboring pixels on a CCD in the horizontal and vertical directions, Dx 0 and Dy 0 are the pixel size in the reconstructed image, Dx 0 = kd/MDx, Dy 0 = kd/NDy, FFT is the fast-Fouriertransform operator, d is the distance between the hologram and the observation. In Eq. (3), the reconstruction wavefront u(m,n) consists of three terms: the zero-order image, the real image and the conjugate image. The undiffracted reconstruction wave forms a zero-order image which is located in the center and is the big and bright spot, and corresponds to the first two terms on the right-hand side of Eq. (1). The conjugate image and the real image are located at the two sides of the reconstructed image, respectively, and correspond to the interference terms on the right-hand side of Eq. (1).
3. The method to eliminate zero-order diffraction To eliminate the zero-order image, we need to eliminate the first two terms on the right-hand side of Eq. (1). In our system, the zero-order diffraction on the right-hand side of Eq. (1) can be eliminated by phase-shifting method of phase-modulated object beams. Adjust the reflecting mirror M, and change the incidence orientation of the object beam, then, the complex-amplitude distribution of the object beam
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on the hologram plane is denoted by O 0 (x,y) = jO 0 (x,y)j · exp jh 0 (x,y) without losing generality. The intensity distribution of the hologram is described as 2
I 0 ðx; yÞ ¼ jO0 ðx; yÞj þ R2 þ RjO0 ðx; yÞj exp jh0 ðx; yÞ þ RjO0 ðx; yÞj exp½jh0 ðx; yÞ:
ð4Þ
Subtracting Eq. (4) from Eq. (1), we obtain 2
2
DIðx; yÞ ¼ fjOðx; yÞj jO0 ðx; yÞj g þ RfjOðx; yÞj exp jhðx; yÞ jO0 ðx; yÞj exp jh0 ðx; yÞg þ RfjOðx; yÞj exp½jhðx; yÞ jO0 ðx; yÞj exp½jh0 ðx; yÞg:
ð5Þ
In Eq. (5), jO(x,y)j2jO 0 (x,y)j2 denotes the residual intensity distribution of the object wave. By defining jO0 ðx; yÞj=jOðx; yÞj ¼ V ; Dhðx; yÞ ¼ h0 ðx; yÞ hðx; yÞ:
ð6Þ
Eq. (5) can be rewritten as DIðx; yÞ ¼ jOðx; yÞj2 ð1 V 2 Þ þ ROðx; yÞ ½1 V exp jDhðx; yÞ þ RO ðx; yÞ ½1 V expfjDhðx; yÞg:
ð7Þ
If h 0 (x,y) „ h(x,y), namely, Dh(x,y) „ 0, then, DI(x,y) „ 0. Replace I(k,l) with DI(x,y) in Eq. (3), when jO(x,y)jjO 0 (x,y)j „ 0, the reconstruction wavefront u(m,n) contains five terms: the zero-order diffraction which corresponds to the first term on the right-hand side of Eq. (5), the two real image terms and the two conjugate image terms which correspond to the second terms and the third terms on the right-hand side of Eq. (5). When jO(x,y)jjO 0 (x,y)j = 0, the zero-order image has been eliminated if h 0 (x,y) „ h(x,y). Eq. (7) can be rewritten as DIðx; yÞ ¼ ROðx; yÞ½1 exp jDhðx; yÞ þ RO ðx; yÞ½1 expfjDhðx; yÞg:
ð8Þ
Therefore, the reconstruction wavefront u(m,n) contains four terms: the two real image terms and the two conjugate image terms which correspond to the second terms and the third terms
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on the right-hand side of Eq. (5). If Dh(x,y) = constant, [1exp jDh(x,y)] and [1exp {jDh(x,y)}] is also constant, then the reconstruction wavefront u(m,n) contains only two terms: the real image term and the conjugate image term which correspond to the third and the fourth terms on the right-hand side of Eq. (1). The zero-order image has been eliminated. Owing to the phase difference of the two digital holograms, the interference between the two real image terms of two digital holograms occur, and so does the two conjugate image terms at the same time. The fringe spacing depends on the phase difference Dh(x,y). The bigger Dh(x,y), the smaller fringe spacing.
4. Experiment results The experimental system shown in Fig. 1 is built. The He–Ne laser with the power of 10 mW, k = 632.8 nm, is used as the light source. A plane parallel light wave is produced after collimating lens. To obtain high diffraction efficiency and small diffraction angle, a sinusoidal phase holographic grating of low frequency made in our laboratory is used in the experiment and the diffraction angle of this grating is hg = 1.45. A transparent letter ‘‘T’’ with the size of 1 mm · 2 mm written in a dark photographic slide is used as the object in the experiments. The hologram is recorded with a CCD sensor (DALSA 1M30 camera, 1024 · 1024 pixels; pixel size 12 lm · 12 lm). The distance between the CCD and the object is 685 mm.
Fig. 2(a) shows a hologram recorded with the experimental set-up shown in Fig. 1. Fig. 2(b) shows the reconstructed image obtained by numerical reconstruction of hologram shown in Fig. 2(a). The reconstruction distance d = 685 mm. The bright square in the center of Fig. 2(b) is the undiffracted reconstruction wave (zero-order) and contains 348 · 348 pixels. And the twin images are mostly superposed on the bright zero-order diffraction and cannot be almost resolved. So it is impossible to obtain useful information about the object if the zero-order diffraction is not eliminated. After varying the recording parameter by adjusting reflecting mirror M and changing the incidence angle of object beam, the digital recording hologram and the reconstructed image are shown in Fig. 3(a) and (b), respectively. Fig. 4(a) shows the computed two-dimensional Fourier spectrum of the subtracted hologram obtained by subtracting Fig. 3(a) from Fig. 2(a). The spatial frequencies located in the center of Fig. 4(a) is the residual intensity distribution of the object wave, which is FFTjO(x,y)j2 jO 0 (x,y)j2 = d(fx,fy); the spatial frequencies of the interference terms: two real image terms (RO and RO 0 ) and two conjugate image term (RO and RO 0 ) are located symmetrically with respect to the center of Fig. 4(a). It can be seen from Fig. 4(a) that the different terms of interference pattern produces well-separated contribution in Fig. 2(b) and Fig. 3(b). In Fig. 2(b) and Fig. 3(b), spatial frequencies corresponding to the zero-order of diffraction are located in the center of the Fourier plane, and spatial frequencies of the interference terms RO and RO are located
Fig. 2. (a) Digital recording off-axis hologram. (b) The amplitude-contrast image obtained by numerical reconstruction of (a).
Y. Zhang et al. / Optics Communications 240 (2004) 261–267
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Fig. 3. (a) Digital recording off-axis hologram after adjusting reflecting mirror. (b) The amplitude-contrast image obtained by numerical reconstruction of (a).
significantly improved and the real image is visible. The reconstructed image located in the center of Fig. 4(b) is the image of residual intensity distribution of the object wave. Due to the phase difference of two subtracted digital holograms, the real image and the virtual image are interference fringe pattern and fringe spacing depends on Dh(x,y). Fig. 5(a) and (b) shows the computed two-dimensional Fourier spectrum and the numerically reconstructed image of the subtracted hologram, respectively. It can be seen from Fig. 5(a) that the phase difference of the two subtracted hologram is bigger than that of Fig. 4(a). So the smaller fringe spacing is in Fig. 5(b). Fig. 4. (a) The two-dimensional Fourier spectrum of the hologram obtained by subtracting Fig. 3(a) from Fig. 2(a). (b) The amplitude-contrast image obtained by numerical reconstructed image of hologram obtained by subtracting Fig. 3(a) from Fig. 2(a).
symmetrically with respect to the center of the Fourier plane and their distances to the center depend on the incidence angle h(x,y) of the object beam. As the zero-order diffraction on the righthand side of Eq. (1) has been mostly subtracted in Fig. 4(a), the spatial frequency of the zero-order diffraction, which is located in the center, has almost disappeared in Fig. 4(a). Using the Eq. (3), the amplitude-contrast image of subtracted hologram is obtained as shown in Fig. 4(b). The bright square in the center has been mostly removed; as a consequence, the resolution of the real image is
Fig. 5. (a) The two-dimensional Fourier spectrum of hologram obtained by the subtracted hologram. (b) The amplitudecontrast image obtained by numerical reconstructed image of the subtracted hologram.
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Fig. 6(a) shows the computed two-dimensional Fourier spectrum of the contrast hologram obtained by subtracting alternative two digital holograms which are recorded with the experimental set-up shown in Fig. 1. The spatial frequency of the zero-order diffraction, which is located in the center, has disappeared in Fig. 6(a). This means jO(x,y)j = jO 0 (x,y)j, and the zero-order diffraction on the right-hand side of Eq. (1) has been subtracted in Fig. 6(a). The reconstructed image obtained by numerical reconstruction of the subtracted hologram is shown in Fig. 6(b). The bright square in the center of reconstructed image has been removed and the real image and the virtual image shown in Fig. 6(b) are very clear and the interference fringe pattern can not be seen. At the same time, the frequency shifting in Fig. 6(a) is zero. This means the frequency terms in Fig. 6(a) have only two terms (RO, RO), and the reconstructed image has the real image and the virtual image, which has been validated in Fig. 6(b). This shows that the phase difference between two digital holograms to be subtracted is constant, namely, Dh(x,y) is constant. After two digital holograms with a constant phase difference subtract, the reconstructed image has only the real
Fig. 7. The numerical reconstructed image of the subtracted hologram of the test target.
image and the virtual image. The experimental results correspond with theoretical analysis of Eq. (6). It is proved that the zero-order diffraction of the reconstructed image can be eliminated when two digital holograms with a constant phase difference subtract, this result of which is just of interest and has practical value. The numerical reconstructed image of the subtracted hologram of the test target with a resolution of 100 lm and a size of 1 · 1 mm2 is shown in Fig. 7 in which the zero-order has been removed and the real image and the virtual image can be observed.
5. Conclusion
Fig. 6. Elimination of the zero-order diffraction in digital offaxis holography: (a) the two-dimensional Fourier spectrum of the subtracted hologram, (b) the amplitude-contrast image obtained by numerical reconstruction of the subtracted hologram.
A simple experimental method has been presented for eliminating the zero-order diffraction in the off-axis holography. This method, substantially, uses phase-shifting technique. It is easy to obtain the phase shifting to the object beam through adjusting reflecting mirror. The advantage of this method is that the optical setup and the experimental procedure are simple and the result is satisfactory. As the zero-order image is filtered, the contrast of the reconstructed image is enhanced and the image quality is improved. Holographic diffraction grating acts as the beam splitter. If the object is illuminated by the
Y. Zhang et al. / Optics Communications 240 (2004) 261–267
zero-order diffraction waves, the +1 order diffraction waves act as a plane reference wave. The zero-order diffraction on the right-hand side of Eq. (1) can also be eliminated by the phase-shifting method of phase-modulated reference beams. Acknowledgement The paper is supported by National Nature Science Foundation of China (No. 60077028). The Nature Science Fund of Tianjin Opto-electronic Sci. and Tec. Center and The Lab Opto-electronic Information Sci. and Tec. Ministry of Education, P.R.China.
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Elimination of zero-order diffraction in digital off-axis holography Yimo Zhang, Qieni Lu¨ *, Baozhen Ge Key Laboratory of Opto-electronics Information and Technical Science, College of Opto-electronics and Precision Instrument Engineering, Ministry of Education, Tianjin University, Tianjin 300072, PR China Received 21 February 2004; received in revised form 17 May 2004; accepted 18 June 2004
Abstract A simple experimental method of eliminating zero-order diffraction in the reconstructed image of off-axis digital holography is presented. Holographic diffraction grating acting as a beam splitter, an off-axis holography system is formed. The holograms of object with different recording parameters are obtained by adjusting the reflecting mirror in the recording optical system to vary the incidence orientation of the object beam in CCD to introduce a phase shift. The zero-order image can be eliminated by numerically processing the holograms of object with different recording parameters. The theoretical analyses have been done in detail and the experimental results are also given. The zero-order image eliminated, the area of reconstructed image increases remarkably, the image quality can then be significantly improved and the better resolution obtained. The experimental results show that the method presented in this paper is feasible, simple in optical system and easy in operation and data processing. 2004 Published by Elsevier B.V. PACS: 42.79.Yd Keywords: Digital off-axis holography; Zero-order diffraction; Holographic grating
1. Introduction Digital holography, digital recording of hologram and numerical reconstruction of the wave fields, offers an alternative recording and recon-
*
Corresponding author. Tel.: +86-22-274-02594; fax: +8622-274-04547. E-mail address: [email protected] (Q. Lu¨). 0030-4018/$ - see front matter 2004 Published by Elsevier B.V. doi:10.1016/j.optcom.2004.06.040
struction in holography. In comparison with classical holography, the main advantage of digital holography is that intensity as well as phase information of a holographically stored wavefront can be achieved directly in the numerical reconstruction process, and that chemical processing is no necessary, thus increasing the flexibility and speed of the experimental process. With the development of high-resolution CCD and fast computer, it has become possible, and been widely applied in
262
Y. Zhang et al. / Optics Communications 240 (2004) 261–267
various fields of science and engineering, for example, deformation measurement [1–3], holographic microscopy [4,5], etc. As in classical holography, off-axis geometries is generally used in digital holography, and its reconstructed image contains a zero-order image, a virtual image and a real image. Even if these three terms appear at different locations in the reconstructed images, the virtual image and the zero-order image need to be eliminated. Because the total number of sampling points of the reconstructed image is equal to the total number of pixels of the image sensor, the virtual image and the zeroorder image limit the size of the real image area. Eliminating the two undesirable images results in an enhancement of image quality. Two kinds of method have been proposed for this purpose. One kind of method is based on digital processing method. For example, Cuche et al. [6] proposed that zero-order image and virtual image can be digitally eliminated by means of filtering their associated spatial frequencies in the computed Fourier transform of the hologram. In [7] space-domain modulation of detected hologram was used to suppress zero-order and/or virtual image. The other kind of method based on experimental method was presented by Takaki et al. [8], which eliminates zero-order and virtual image by two shutters and one phase-modulator. In this paper, we propose the experimental method of eliminating the zero-order image and describe it in detail. The holographic grating is used to configure an off-axis holographic recording system. The two holograms recorded with different recording parameters are obtained and numerically subtracted to eliminate the zero-order image. The recording parameter with the different orientation of the object beam is gained by adjusting the reflecting mirror. Some experimental results are also given.
2. Hologram digital recording and numerical reconstruction The diagram of the hologram recording optical system we design is shown in Fig. 1. The holographic diffraction grating of low frequency is used,
plane-wave
object
(x,y)
M
+1 order
O
θg R
zero-order grating
CCD
Fig. 1. Schematic diagram of the recording system of off-axis hologram by CCD. O: object wave; R: reference wave; M: reflecting mirror.
functioning as beam splitter. The monochrome plane parallel light wave illuminates holographic grating normally; through grating diffraction, the three different plane parallel light waves, namely, the zero-order and the ±1 order diffraction waves, are generated. The +1 order diffraction waves illuminate the object and then are reflected onto the hologram plane by reflecting mirror M, acting as the object wave. The zero-order diffraction waves propagate perpendicular to the hologram plane oxy, acting as the reference wave. Assume that the complex-amplitude distribution of the object beam on the hologram plane is denoted by O(x,y) = jO(x,y)jexp jh (x,y), and the complexamplitude distribution of the plane reference beam on the hologram plane is expressed as r(x,y) = R. k is the wavelength of the incident light. The intensity distribution of the hologram is given by 2
Iðx; yÞ ¼ jOðx; yÞj þ R2 þ RjOðx; yÞj exp jhðx; yÞ þ RjOðx; yÞj exp½jhðx; yÞ;
ð1Þ
where superscript stands for the complex conjugate, jO(x,y)j2 is the intensity of the object wave and R2 is the intensity of the reference wave. The third and the fourth terms on the right-hand side of Eq. (1) are the interference terms. A digital hologram is recorded by a CCD sensor and transmitted to a computer by frame grabber card, and then, numerically reconstructed in computer. The digital hologram I(k,l) can be expressed as x y Iðk; lÞ ¼ Iðx; yÞrect ; MDx N Dy
M=2 X
N =2 X
k¼M=2 l¼N =2
dðx kDx; y lDyÞ:
ð2Þ
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The holographic reconstruction, in physical terms, is to reconstruct the wavefront at the object, i.e., to obtain the three-dimensional structure of the object from the two-dimensional hologram. The reconstructed wavefront u(m,n) can be obtained in the observation plane o 0 x 0 y 0 by computing the discrete Fresnel integral of the digital hologram intensity I(k,l),
exp j 2p d jp 2 02 2 02 k ðm Dx þ n Dy Þ uðm; nÞ ¼ exp kd jkd p FFTfIðk; lÞ exp j ðk 2 Dx2 þ l2 Dy 2 Þg; kd ð3Þ where k,l,m,n are integers, m = 0,1,. . .,M1; n = 0,1,. . .,N1, M · N is the number of light-sensitive pixels on CCD, Dx and Dy are the distances between neighboring pixels on a CCD in the horizontal and vertical directions, Dx 0 and Dy 0 are the pixel size in the reconstructed image, Dx 0 = kd/MDx, Dy 0 = kd/NDy, FFT is the fast-Fouriertransform operator, d is the distance between the hologram and the observation. In Eq. (3), the reconstruction wavefront u(m,n) consists of three terms: the zero-order image, the real image and the conjugate image. The undiffracted reconstruction wave forms a zero-order image which is located in the center and is the big and bright spot, and corresponds to the first two terms on the right-hand side of Eq. (1). The conjugate image and the real image are located at the two sides of the reconstructed image, respectively, and correspond to the interference terms on the right-hand side of Eq. (1).
3. The method to eliminate zero-order diffraction To eliminate the zero-order image, we need to eliminate the first two terms on the right-hand side of Eq. (1). In our system, the zero-order diffraction on the right-hand side of Eq. (1) can be eliminated by phase-shifting method of phase-modulated object beams. Adjust the reflecting mirror M, and change the incidence orientation of the object beam, then, the complex-amplitude distribution of the object beam
263
on the hologram plane is denoted by O 0 (x,y) = jO 0 (x,y)j · exp jh 0 (x,y) without losing generality. The intensity distribution of the hologram is described as 2
I 0 ðx; yÞ ¼ jO0 ðx; yÞj þ R2 þ RjO0 ðx; yÞj exp jh0 ðx; yÞ þ RjO0 ðx; yÞj exp½jh0 ðx; yÞ:
ð4Þ
Subtracting Eq. (4) from Eq. (1), we obtain 2
2
DIðx; yÞ ¼ fjOðx; yÞj jO0 ðx; yÞj g þ RfjOðx; yÞj exp jhðx; yÞ jO0 ðx; yÞj exp jh0 ðx; yÞg þ RfjOðx; yÞj exp½jhðx; yÞ jO0 ðx; yÞj exp½jh0 ðx; yÞg:
ð5Þ
In Eq. (5), jO(x,y)j2jO 0 (x,y)j2 denotes the residual intensity distribution of the object wave. By defining jO0 ðx; yÞj=jOðx; yÞj ¼ V ; Dhðx; yÞ ¼ h0 ðx; yÞ hðx; yÞ:
ð6Þ
Eq. (5) can be rewritten as DIðx; yÞ ¼ jOðx; yÞj2 ð1 V 2 Þ þ ROðx; yÞ ½1 V exp jDhðx; yÞ þ RO ðx; yÞ ½1 V expfjDhðx; yÞg:
ð7Þ
If h 0 (x,y) „ h(x,y), namely, Dh(x,y) „ 0, then, DI(x,y) „ 0. Replace I(k,l) with DI(x,y) in Eq. (3), when jO(x,y)jjO 0 (x,y)j „ 0, the reconstruction wavefront u(m,n) contains five terms: the zero-order diffraction which corresponds to the first term on the right-hand side of Eq. (5), the two real image terms and the two conjugate image terms which correspond to the second terms and the third terms on the right-hand side of Eq. (5). When jO(x,y)jjO 0 (x,y)j = 0, the zero-order image has been eliminated if h 0 (x,y) „ h(x,y). Eq. (7) can be rewritten as DIðx; yÞ ¼ ROðx; yÞ½1 exp jDhðx; yÞ þ RO ðx; yÞ½1 expfjDhðx; yÞg:
ð8Þ
Therefore, the reconstruction wavefront u(m,n) contains four terms: the two real image terms and the two conjugate image terms which correspond to the second terms and the third terms
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on the right-hand side of Eq. (5). If Dh(x,y) = constant, [1exp jDh(x,y)] and [1exp {jDh(x,y)}] is also constant, then the reconstruction wavefront u(m,n) contains only two terms: the real image term and the conjugate image term which correspond to the third and the fourth terms on the right-hand side of Eq. (1). The zero-order image has been eliminated. Owing to the phase difference of the two digital holograms, the interference between the two real image terms of two digital holograms occur, and so does the two conjugate image terms at the same time. The fringe spacing depends on the phase difference Dh(x,y). The bigger Dh(x,y), the smaller fringe spacing.
4. Experiment results The experimental system shown in Fig. 1 is built. The He–Ne laser with the power of 10 mW, k = 632.8 nm, is used as the light source. A plane parallel light wave is produced after collimating lens. To obtain high diffraction efficiency and small diffraction angle, a sinusoidal phase holographic grating of low frequency made in our laboratory is used in the experiment and the diffraction angle of this grating is hg = 1.45. A transparent letter ‘‘T’’ with the size of 1 mm · 2 mm written in a dark photographic slide is used as the object in the experiments. The hologram is recorded with a CCD sensor (DALSA 1M30 camera, 1024 · 1024 pixels; pixel size 12 lm · 12 lm). The distance between the CCD and the object is 685 mm.
Fig. 2(a) shows a hologram recorded with the experimental set-up shown in Fig. 1. Fig. 2(b) shows the reconstructed image obtained by numerical reconstruction of hologram shown in Fig. 2(a). The reconstruction distance d = 685 mm. The bright square in the center of Fig. 2(b) is the undiffracted reconstruction wave (zero-order) and contains 348 · 348 pixels. And the twin images are mostly superposed on the bright zero-order diffraction and cannot be almost resolved. So it is impossible to obtain useful information about the object if the zero-order diffraction is not eliminated. After varying the recording parameter by adjusting reflecting mirror M and changing the incidence angle of object beam, the digital recording hologram and the reconstructed image are shown in Fig. 3(a) and (b), respectively. Fig. 4(a) shows the computed two-dimensional Fourier spectrum of the subtracted hologram obtained by subtracting Fig. 3(a) from Fig. 2(a). The spatial frequencies located in the center of Fig. 4(a) is the residual intensity distribution of the object wave, which is FFTjO(x,y)j2 jO 0 (x,y)j2 = d(fx,fy); the spatial frequencies of the interference terms: two real image terms (RO and RO 0 ) and two conjugate image term (RO and RO 0 ) are located symmetrically with respect to the center of Fig. 4(a). It can be seen from Fig. 4(a) that the different terms of interference pattern produces well-separated contribution in Fig. 2(b) and Fig. 3(b). In Fig. 2(b) and Fig. 3(b), spatial frequencies corresponding to the zero-order of diffraction are located in the center of the Fourier plane, and spatial frequencies of the interference terms RO and RO are located
Fig. 2. (a) Digital recording off-axis hologram. (b) The amplitude-contrast image obtained by numerical reconstruction of (a).
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Fig. 3. (a) Digital recording off-axis hologram after adjusting reflecting mirror. (b) The amplitude-contrast image obtained by numerical reconstruction of (a).
significantly improved and the real image is visible. The reconstructed image located in the center of Fig. 4(b) is the image of residual intensity distribution of the object wave. Due to the phase difference of two subtracted digital holograms, the real image and the virtual image are interference fringe pattern and fringe spacing depends on Dh(x,y). Fig. 5(a) and (b) shows the computed two-dimensional Fourier spectrum and the numerically reconstructed image of the subtracted hologram, respectively. It can be seen from Fig. 5(a) that the phase difference of the two subtracted hologram is bigger than that of Fig. 4(a). So the smaller fringe spacing is in Fig. 5(b). Fig. 4. (a) The two-dimensional Fourier spectrum of the hologram obtained by subtracting Fig. 3(a) from Fig. 2(a). (b) The amplitude-contrast image obtained by numerical reconstructed image of hologram obtained by subtracting Fig. 3(a) from Fig. 2(a).
symmetrically with respect to the center of the Fourier plane and their distances to the center depend on the incidence angle h(x,y) of the object beam. As the zero-order diffraction on the righthand side of Eq. (1) has been mostly subtracted in Fig. 4(a), the spatial frequency of the zero-order diffraction, which is located in the center, has almost disappeared in Fig. 4(a). Using the Eq. (3), the amplitude-contrast image of subtracted hologram is obtained as shown in Fig. 4(b). The bright square in the center has been mostly removed; as a consequence, the resolution of the real image is
Fig. 5. (a) The two-dimensional Fourier spectrum of hologram obtained by the subtracted hologram. (b) The amplitudecontrast image obtained by numerical reconstructed image of the subtracted hologram.
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Fig. 6(a) shows the computed two-dimensional Fourier spectrum of the contrast hologram obtained by subtracting alternative two digital holograms which are recorded with the experimental set-up shown in Fig. 1. The spatial frequency of the zero-order diffraction, which is located in the center, has disappeared in Fig. 6(a). This means jO(x,y)j = jO 0 (x,y)j, and the zero-order diffraction on the right-hand side of Eq. (1) has been subtracted in Fig. 6(a). The reconstructed image obtained by numerical reconstruction of the subtracted hologram is shown in Fig. 6(b). The bright square in the center of reconstructed image has been removed and the real image and the virtual image shown in Fig. 6(b) are very clear and the interference fringe pattern can not be seen. At the same time, the frequency shifting in Fig. 6(a) is zero. This means the frequency terms in Fig. 6(a) have only two terms (RO, RO), and the reconstructed image has the real image and the virtual image, which has been validated in Fig. 6(b). This shows that the phase difference between two digital holograms to be subtracted is constant, namely, Dh(x,y) is constant. After two digital holograms with a constant phase difference subtract, the reconstructed image has only the real
Fig. 7. The numerical reconstructed image of the subtracted hologram of the test target.
image and the virtual image. The experimental results correspond with theoretical analysis of Eq. (6). It is proved that the zero-order diffraction of the reconstructed image can be eliminated when two digital holograms with a constant phase difference subtract, this result of which is just of interest and has practical value. The numerical reconstructed image of the subtracted hologram of the test target with a resolution of 100 lm and a size of 1 · 1 mm2 is shown in Fig. 7 in which the zero-order has been removed and the real image and the virtual image can be observed.
5. Conclusion
Fig. 6. Elimination of the zero-order diffraction in digital offaxis holography: (a) the two-dimensional Fourier spectrum of the subtracted hologram, (b) the amplitude-contrast image obtained by numerical reconstruction of the subtracted hologram.
A simple experimental method has been presented for eliminating the zero-order diffraction in the off-axis holography. This method, substantially, uses phase-shifting technique. It is easy to obtain the phase shifting to the object beam through adjusting reflecting mirror. The advantage of this method is that the optical setup and the experimental procedure are simple and the result is satisfactory. As the zero-order image is filtered, the contrast of the reconstructed image is enhanced and the image quality is improved. Holographic diffraction grating acts as the beam splitter. If the object is illuminated by the
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zero-order diffraction waves, the +1 order diffraction waves act as a plane reference wave. The zero-order diffraction on the right-hand side of Eq. (1) can also be eliminated by the phase-shifting method of phase-modulated reference beams. Acknowledgement The paper is supported by National Nature Science Foundation of China (No. 60077028). The Nature Science Fund of Tianjin Opto-electronic Sci. and Tec. Center and The Lab Opto-electronic Information Sci. and Tec. Ministry of Education, P.R.China.
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