- Email: [email protected]
Phase aberration compensation by spectrum centering in digital holographic microscopy
Optics Communications 284 (2011) 4152–4155
Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Phase aberration compensation by spectrum centering in digital holographic microscopy Huakun Cui a, b, Dayong Wang a, b,⁎, Yunxin Wang b, Jie Zhao b, Yizhuo Zhang b a b
Institute of Information Photonics Technology, Beijing University of Technology, Beijing 100124, China College of Applied Sciences, Beijing University of Technology, Beijing 100124, China
a r t i c l e
i n f o
Article history: Received 24 November 2010 Received in revised form 18 March 2011 Accepted 2 May 2011 Available online 17 May 2011 Keywords: Digital holographic microscopy Aberration compensation Spectrum centering
a b s t r a c t Firstly, an approach is proposed for eliminating the tilt phase aberration introduced by the tilt reference wave. It is based on the argument distribution of the hologram spectrum to locate exactly the position of the carrier frequency of the virtual image. This tilt aberration will be corrected by shifting the filtered hologram spectrum to the coordinate origin of the frequency domain, without knowledge of the focal length of the imaging lens or distances in the setup. Then the subsequent quadratic phase aberration compensation is performed only by adjusting a single parameter. The method is demonstrated by the phase contrast imaging of the cervical carcinoma cells. Crown Copyright © 2011 Published by Elsevier B.V. All rights reserved.
1. Introduction The digital holographic microscopy (DHM) is a powerful technique for the real-time quantitative amplitude and phase contrast imaging [1–10]. The phase contrast imaging is important for a wide range of applications. In particular, the off-axis DHM allows the separation of the different diffraction image from a single hologram. As a result of the off-axis geometry, the tilt phase distortion in the reconstructed plane is introduced. Several approaches have been proposed to remove the aberration. Cuche et al. [11] described a method using a single hologram, which involved the computation of a digital replica of the reference wave depending on two reconstruction parameters. Ferraro et al. [12] proposed a double-exposure technique to compensate completely the inherent wave front curvature in the quantitative phase contrast imaging, while it needed to record two holograms with and without the samples respectively. Then a subtraction procedure between the two holograms was necessary. Colomb et al. [13,14] compensated the tilt aberration by recording a hologram corresponding to a blank image to compute the first-order parameters directly from the hologram. However, it might be limited by selecting profiles or areas known to be flat in the hologram plane. Miccio et al. [15] performed a two-dimensional fitting with the Zernike polynomials of the reconstructed unwrapped phase, while the application of the method might be suitable only for the special case of thin objects. In this paper, we propose a new, to our knowledge, approach that uses a single hologram to totally eliminate the tilt phase aberration
without knowledge of any optical parameter. This aberration is compensated by shifting the filtered hologram spectrum to the center of the frequency domain, which is based on the argument distribution of the hologram spectrum. Cuche et al. [16] and Colomb et al. [14] studied the spectrum centering for the tilt phase aberration elimination, which is based on the modulus distribution of the hologram spectrum. However, the method has not been demonstrated in the experiment. As Colomb et al. explained, there existed two large difficulties to apply the method based on detecting the position of the modulus maximum: limitation by the pixel accuracy and the central frequency spreading. Thus it is impossible to center the frequency region of interest by the simple maximum-modulus detection. Here by analyzing the characters of the argument distribution in the hologram frequency domain, we find that there always exists a maximum among the argument distribution. Then it is possible to locate exactly the center of the target spectrum. As it will be shown, the position of the argument extremum corresponding to the carrier frequency is possible to be detected. The spectrum location can be used not only to compensate the tilt phase aberration, but also to make the subsequent quadratic aberration compensation simpler because only a single parameter needs to be adjusted. Furthermore, the spectrum location is very important for the image synthesis in the color digital holography and super-resolution digital holography. Finally, this proposed approach is demonstrated experimentally by the phase contrast imaging of the cervical carcinoma cells in digital holographic microscopy. 2. Principle
⁎ Corresponding author. Tel.: + 86 10 6739 1741. E-mail address: [email protected] (D. Wang).
The schematic diagram for recording the pre-magnification digital hologram in transmission is depicted in Fig. 1, where d1 and d2 are the
0030-4018/$ – see front matter. Crown Copyright © 2011 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.05.013
H. Cui et al. / Optics Communications 284 (2011) 4152–4155
Fig. 1. Schematic diagram for recording the pre-magnification digital hologram.
object distance and image distance, MO is the microscope objective, and the CCD is set before the magnified image of the object at a distance d. The coordinates (xo,yo), (x,y) and (xi,yi) describe the object plane, the hologram plane and the image plane, respectively. Suppose the complex amplitude distribution of the object is O0(x0,y0), and the object is illuminated by a plane wave and the transmitted light is collected by a MO that produces a wave front in the hologram plane called object wave O(x,y). The interference between the object wave O and the plane reference wave R produces the hologram intensity 2
2
H = jOj + jRj + OR + O R;
ð1Þ
where * denotes the complex conjugate. Without loss of generality, the virtual image term OR* is what we are interested in. The spectrum of the term can be calculated by
U + 1 fx ; fy
cosα cosβ ; fy + ; = FT OR = FT fOg⊗δ fx + λ λ
ð2Þ
where, FT{⋅} represents the Fourier transformation, ⊗ denotes the convolution, λ is the wavelength of the laser source, α and β are the incidence angles of the reference wave with respect to the propagation direction of the object wave, (fx, fy) denotes the coordinate in the spectrum domain. It is noted that the target spectrum is the convolution of the δ function and the spectrum of the object. The position of the δ function represents the carrier frequency of the term. Under the paraxial approximation, the spectrum of the object can be deduced by using the Fresnel diffraction integral as follows
jk 2 2 ⊗FT x +y FT fOðx; yÞg = FT exp − 8 2d <
jkA 2 2 FT O0 ðx0 ; y0 Þexp x0 + y0 : d x 2 f = 2 ;f
j
x
λdd1
y
9 = ; d y ; = 2
ð3Þ
λdd1
where, A defines the distance relationship A=
−d22 + dd1 + dd2 : dd21
4153
the complex function. The argument distribution may contain some important information and has not yet been developed. Usually, there is no analytic solution in Eq. (3). Here by analyzing the characters of the argument distribution in the hologram frequency domain based on the experimental data, it is found that there always exists a maximum among the argument distribution. Then it is possible to exactly fix the specific position of the argument extremum corresponding to the carrier frequency. After the spectrum centering, an inverse Fourier transform is operated on the resulting spectrum to produce the complex amplitude distribution H ' (x,y), which is the exact complex field of the object wave in the hologram plane. Then by propagating H ' (x,y) from the hologram plane to the image plane through the free space, the reconstructed diffraction field can be obtained as follows
jk 2 jk 2 2 2 x + yi FT H0 ðx; yÞexp x +y Oi ðxi ; yi Þ = exp xi y 2d i 2d ;f = i fx = λd y λd ð5Þ x y jk 1 2 2 x + yi ; = O0 − i ; − i exp 2 f ⋅M i M M
where, f is the focal length of the MO, and M = d2/d1 is the magnification of the optical system. It is noted that there is a limitation on the minimum value of the distance d, which depends on the wavelength and parameters of the CCD array [9]. Usually, f is known after the MO is decided. As it is shown, the tilt phase aberration has been compensated. There is only the quadric phase curvature introduced by the MO left, which can be compensated by multiplication of the image wave field with the digital phase mask as follows [11] jk 1 2 2 xi + yi : Φðxi ; yi Þ = exp − 2 f ⋅M
ð6Þ
Then only a single parameter M must be adjusted to compensate the phase curvature. 3. Experiment It is demonstrated below that the phase aberration can be compensated by spectrum centering based on the argument distribution of the hologram spectrum. Fig. 2 shows the optical configuration of a pre-magnification DHM. The light transmitted by the specimen is collected by a MO and interferes with the tilt plane reference wave on the CCD by means of a beam splitter (BS). The laser wavelength λ is 532 nm and the MO is a 25× objective with a focal length f = 10.13 mm and numerical aperture NA = 0.4. The CCD detector is made of 1392 × 1040 square pixels of 6.45 μm size. The specimen is the cervical carcinoma cells. In order to reduce the computation load, the hologram is cut to 801 × 801 pixels as shown in Fig. 3. Fig. 3(b), (c) and (d) is the
ð4Þ
Reference wave
As it is shown, in order to compensate the tilt phase aberration, the two parameters α and β are needed to be adjusted or measured in the experiment. As a result of the tilt reference wave, the carrier frequencies of the real or virtual images are not in the center of the spectrum but have a spatial separation. If the position of the carrier frequency is obtained, the spectrum of the virtual image term will be filtered and exactly shifted to the coordinate origin of the hologram frequency domain which can compensate the tilt phase aberration. Usually, the modulus image of the spectrum is used to locate the center position of the carrier frequency, but it is difficult to exactly center the frequency region of interest by the simple maximummodulus detection. As we know from Eqs. (2) and (3), the spectrum is
CCD
MO
Object Specimen
wave
BS
Fig. 2. Optical configuration of a pre-magnification DHM.
4154
H. Cui et al. / Optics Communications 284 (2011) 4152–4155
b
a
c
d
Fig. 3. (a) Digital hologram of the cervical carcinoma cells. (b) Modulus image, (c) argument image and (d) three-dimensional argument distribution of the hologram spectra.
modulus image, argument image and three-dimensional argument distribution of the hologram spectrum, respectively. We can see that it is hard to detect the position of the carrier spectra of the virtual image from the modulus image as shown in Fig. 3(b), because there is no obvious characteristic point. Fortunately, there always exists a maximum on the argument distribution of the hologram spectrum as shown in Fig. 3(c) and (d). In order to overcome the influence of the noise and assure the high accuracy of the position, the least squares surface fitting method is applied to fit the argument distribution of the interesting spectra region (shown in the dotted circle of Fig. 3(c)). Then the extremum point of the fitted surface can be found, whose coordinate here is (488,623). After the manual spatial filtering and being moved to the coordinate origin of the hologram frequency domain, the modulus image and argument image of the carrier spectra of the virtual image term are illustrated in Fig. 4(a) and (b). And Fig. 4(c) shows the modulus of the object wave in the hologram plane which is retrieved by the inverse Fourier transformation of the resulting spectra. The image diffraction field in the reconstructed plane at distance d = 690 mm from the hologram plane is calculated. And the quadratic phase aberration is compensated by multiplication of the reconstructed wave front with the digital phase mask shown in Eq. (6). The values of
a
b
the reconstructed distance d and the magnification of the imaging lens M are fixed for a given experimental configuration but must be modified exactly. Here the autofocus methods are used to adjust d and the M value is properly adjusted until the curved fringes disappear in the reconstructed image as shown in Fig. 5(a) and (b). Fig. 5 shows the phase contrast images in the reconstructed plane. From Fig. 5(a), we can see that there is only the quadric phase curvature left, and the tilt phase aberration has be totally eliminated. After compensating the quadric curvature introduced by the MO, the proper phase contrast images are obtained as shown in Fig. 5(b) and (c). 4. Conclusion In conclusion, a new method is proposed to locate the center of the target spectrum among the hologram spectrum, which is based on the argument distribution of the hologram spectrum. By the exact spectrum centering, the tilt phase aberration introduced by the tilt reference wave can be eliminated almost totally with a single hologram, without knowledge of the optical parameters in the experiment. Then the subsequent quadratic phase aberration compensation is performed only by adjusting a single parameter. It is demonstrated by the phase contrast imaging of the cervical carcinoma cells in pre-magnification DHM.
c
Fig. 4. (a) Modulus image and (b) argument image of the spectra after manual spatial filtering and shifting to the center. (c) Modulus of the object wave in the hologram plane after the inverse Fourier transform of the resulting spectra.
H. Cui et al. / Optics Communications 284 (2011) 4152–4155
a
b
4155
c
Fig. 5. Phase contrast image (mod 2π) with the size of 81 × 81 obtained (a) without digital phase mask, and (b) with proper digital phase mask (M = 74.5). (c) The three-dimensional phase distribution of the reconstructed phase.
Acknowledgment This work is financially supported by the National Natural Science Foundation of China under grant (no. 61077004), Science Foundation of Education Commission of Beijing under grant (no. KZ200910005001), and Innovative Talent and Team Development for Serving Beijing. References [1] U. Schnars, W. Jüptner, Appl. Opt. 33 (1994) 179. [2] I. Yamaguchi, T. Zhang, Opt. Lett. 22 (1997) 1268. [3] D. Mas, J. Garcia, C. Ferreira, L.M. Bernardo, F. Marinho, Opt. Commun. 164 (1999) 233. [4] I. Yamaguchi, T. Matsumura, J. Kato, Opt. Lett. 27 (2002) 1108. [5] J. Garcia-Sucerquia, W. Xu, S.K. Jericho, P. Klages, M.H. Jericho, H.J. Kreuzer, Appl. Opt. 45 (2006) 836.
[6] N. Lue, W. Choi, G. Popescu, T. Ikeda, R.R. Dasari, K. Badizadegan, M.S. Feld, Appl. Opt. 46 (2007) 1836. [7] B. Kemper, G. von Bally, Appl. Opt. 47 (2008) A52. [8] J. Li, P. Tankam, Z. Peng, P. Picart, Opt. Lett. 34 (2009) 572. [9] M.K. Kim, J. Opt. Soc. Korea 14 (2010) 77. [10] J. Jang, C.Y. Bae, J.K. Park, J.C. Ye, Opt. Lett. 35 (2010) 514. [11] E. Cuche, P. Marquet, C. Depeursinge, Appl. Opt. 38 (1999) 6994. [12] P. Ferraro, S.D. Nicola, A. Finizio, G. Coppola, S. Grilli, C. Magro, G. Pierattini, Appl. Opt. 42 (2003) 1938. [13] T. Colomb, J. Kühn, F. Charrière, C. Depeursinge, P. Marquet, N. Aspert, Opt. Express 14 (2006) 4300. [14] T. Colomb, F. Montfort, J. Kühn, N. Aspert, E. Cuche, A. Marian, F. Charrière, S. Bourquin, P. Marquet, C. Depeursinge, J. Opt. Soc. Am. A 23 (2006) 3177. [15] L. Miccio, D. Alfieri, S. Grilli, P. Ferraro, A. Finizio, L.D. Petrocellis, S.D. Nicola, Appl. Phys. Lett. 90 (2007) 041104. [16] E. Cuche, P. Marquet, C. Depeursinge, Appl. Opt. 39 (2000) 4070.
Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Phase aberration compensation by spectrum centering in digital holographic microscopy Huakun Cui a, b, Dayong Wang a, b,⁎, Yunxin Wang b, Jie Zhao b, Yizhuo Zhang b a b
Institute of Information Photonics Technology, Beijing University of Technology, Beijing 100124, China College of Applied Sciences, Beijing University of Technology, Beijing 100124, China
a r t i c l e
i n f o
Article history: Received 24 November 2010 Received in revised form 18 March 2011 Accepted 2 May 2011 Available online 17 May 2011 Keywords: Digital holographic microscopy Aberration compensation Spectrum centering
a b s t r a c t Firstly, an approach is proposed for eliminating the tilt phase aberration introduced by the tilt reference wave. It is based on the argument distribution of the hologram spectrum to locate exactly the position of the carrier frequency of the virtual image. This tilt aberration will be corrected by shifting the filtered hologram spectrum to the coordinate origin of the frequency domain, without knowledge of the focal length of the imaging lens or distances in the setup. Then the subsequent quadratic phase aberration compensation is performed only by adjusting a single parameter. The method is demonstrated by the phase contrast imaging of the cervical carcinoma cells. Crown Copyright © 2011 Published by Elsevier B.V. All rights reserved.
1. Introduction The digital holographic microscopy (DHM) is a powerful technique for the real-time quantitative amplitude and phase contrast imaging [1–10]. The phase contrast imaging is important for a wide range of applications. In particular, the off-axis DHM allows the separation of the different diffraction image from a single hologram. As a result of the off-axis geometry, the tilt phase distortion in the reconstructed plane is introduced. Several approaches have been proposed to remove the aberration. Cuche et al. [11] described a method using a single hologram, which involved the computation of a digital replica of the reference wave depending on two reconstruction parameters. Ferraro et al. [12] proposed a double-exposure technique to compensate completely the inherent wave front curvature in the quantitative phase contrast imaging, while it needed to record two holograms with and without the samples respectively. Then a subtraction procedure between the two holograms was necessary. Colomb et al. [13,14] compensated the tilt aberration by recording a hologram corresponding to a blank image to compute the first-order parameters directly from the hologram. However, it might be limited by selecting profiles or areas known to be flat in the hologram plane. Miccio et al. [15] performed a two-dimensional fitting with the Zernike polynomials of the reconstructed unwrapped phase, while the application of the method might be suitable only for the special case of thin objects. In this paper, we propose a new, to our knowledge, approach that uses a single hologram to totally eliminate the tilt phase aberration
without knowledge of any optical parameter. This aberration is compensated by shifting the filtered hologram spectrum to the center of the frequency domain, which is based on the argument distribution of the hologram spectrum. Cuche et al. [16] and Colomb et al. [14] studied the spectrum centering for the tilt phase aberration elimination, which is based on the modulus distribution of the hologram spectrum. However, the method has not been demonstrated in the experiment. As Colomb et al. explained, there existed two large difficulties to apply the method based on detecting the position of the modulus maximum: limitation by the pixel accuracy and the central frequency spreading. Thus it is impossible to center the frequency region of interest by the simple maximum-modulus detection. Here by analyzing the characters of the argument distribution in the hologram frequency domain, we find that there always exists a maximum among the argument distribution. Then it is possible to locate exactly the center of the target spectrum. As it will be shown, the position of the argument extremum corresponding to the carrier frequency is possible to be detected. The spectrum location can be used not only to compensate the tilt phase aberration, but also to make the subsequent quadratic aberration compensation simpler because only a single parameter needs to be adjusted. Furthermore, the spectrum location is very important for the image synthesis in the color digital holography and super-resolution digital holography. Finally, this proposed approach is demonstrated experimentally by the phase contrast imaging of the cervical carcinoma cells in digital holographic microscopy. 2. Principle
⁎ Corresponding author. Tel.: + 86 10 6739 1741. E-mail address: [email protected] (D. Wang).
The schematic diagram for recording the pre-magnification digital hologram in transmission is depicted in Fig. 1, where d1 and d2 are the
0030-4018/$ – see front matter. Crown Copyright © 2011 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.05.013
H. Cui et al. / Optics Communications 284 (2011) 4152–4155
Fig. 1. Schematic diagram for recording the pre-magnification digital hologram.
object distance and image distance, MO is the microscope objective, and the CCD is set before the magnified image of the object at a distance d. The coordinates (xo,yo), (x,y) and (xi,yi) describe the object plane, the hologram plane and the image plane, respectively. Suppose the complex amplitude distribution of the object is O0(x0,y0), and the object is illuminated by a plane wave and the transmitted light is collected by a MO that produces a wave front in the hologram plane called object wave O(x,y). The interference between the object wave O and the plane reference wave R produces the hologram intensity 2
2
H = jOj + jRj + OR + O R;
ð1Þ
where * denotes the complex conjugate. Without loss of generality, the virtual image term OR* is what we are interested in. The spectrum of the term can be calculated by
U + 1 fx ; fy
cosα cosβ ; fy + ; = FT OR = FT fOg⊗δ fx + λ λ
ð2Þ
where, FT{⋅} represents the Fourier transformation, ⊗ denotes the convolution, λ is the wavelength of the laser source, α and β are the incidence angles of the reference wave with respect to the propagation direction of the object wave, (fx, fy) denotes the coordinate in the spectrum domain. It is noted that the target spectrum is the convolution of the δ function and the spectrum of the object. The position of the δ function represents the carrier frequency of the term. Under the paraxial approximation, the spectrum of the object can be deduced by using the Fresnel diffraction integral as follows
jk 2 2 ⊗FT x +y FT fOðx; yÞg = FT exp − 8 2d <
jkA 2 2 FT O0 ðx0 ; y0 Þexp x0 + y0 : d x 2 f = 2 ;f
j
x
λdd1
y
9 = ; d y ; = 2
ð3Þ
λdd1
where, A defines the distance relationship A=
−d22 + dd1 + dd2 : dd21
4153
the complex function. The argument distribution may contain some important information and has not yet been developed. Usually, there is no analytic solution in Eq. (3). Here by analyzing the characters of the argument distribution in the hologram frequency domain based on the experimental data, it is found that there always exists a maximum among the argument distribution. Then it is possible to exactly fix the specific position of the argument extremum corresponding to the carrier frequency. After the spectrum centering, an inverse Fourier transform is operated on the resulting spectrum to produce the complex amplitude distribution H ' (x,y), which is the exact complex field of the object wave in the hologram plane. Then by propagating H ' (x,y) from the hologram plane to the image plane through the free space, the reconstructed diffraction field can be obtained as follows
jk 2 jk 2 2 2 x + yi FT H0 ðx; yÞexp x +y Oi ðxi ; yi Þ = exp xi y 2d i 2d ;f = i fx = λd y λd ð5Þ x y jk 1 2 2 x + yi ; = O0 − i ; − i exp 2 f ⋅M i M M
where, f is the focal length of the MO, and M = d2/d1 is the magnification of the optical system. It is noted that there is a limitation on the minimum value of the distance d, which depends on the wavelength and parameters of the CCD array [9]. Usually, f is known after the MO is decided. As it is shown, the tilt phase aberration has been compensated. There is only the quadric phase curvature introduced by the MO left, which can be compensated by multiplication of the image wave field with the digital phase mask as follows [11] jk 1 2 2 xi + yi : Φðxi ; yi Þ = exp − 2 f ⋅M
ð6Þ
Then only a single parameter M must be adjusted to compensate the phase curvature. 3. Experiment It is demonstrated below that the phase aberration can be compensated by spectrum centering based on the argument distribution of the hologram spectrum. Fig. 2 shows the optical configuration of a pre-magnification DHM. The light transmitted by the specimen is collected by a MO and interferes with the tilt plane reference wave on the CCD by means of a beam splitter (BS). The laser wavelength λ is 532 nm and the MO is a 25× objective with a focal length f = 10.13 mm and numerical aperture NA = 0.4. The CCD detector is made of 1392 × 1040 square pixels of 6.45 μm size. The specimen is the cervical carcinoma cells. In order to reduce the computation load, the hologram is cut to 801 × 801 pixels as shown in Fig. 3. Fig. 3(b), (c) and (d) is the
ð4Þ
Reference wave
As it is shown, in order to compensate the tilt phase aberration, the two parameters α and β are needed to be adjusted or measured in the experiment. As a result of the tilt reference wave, the carrier frequencies of the real or virtual images are not in the center of the spectrum but have a spatial separation. If the position of the carrier frequency is obtained, the spectrum of the virtual image term will be filtered and exactly shifted to the coordinate origin of the hologram frequency domain which can compensate the tilt phase aberration. Usually, the modulus image of the spectrum is used to locate the center position of the carrier frequency, but it is difficult to exactly center the frequency region of interest by the simple maximummodulus detection. As we know from Eqs. (2) and (3), the spectrum is
CCD
MO
Object Specimen
wave
BS
Fig. 2. Optical configuration of a pre-magnification DHM.
4154
H. Cui et al. / Optics Communications 284 (2011) 4152–4155
b
a
c
d
Fig. 3. (a) Digital hologram of the cervical carcinoma cells. (b) Modulus image, (c) argument image and (d) three-dimensional argument distribution of the hologram spectra.
modulus image, argument image and three-dimensional argument distribution of the hologram spectrum, respectively. We can see that it is hard to detect the position of the carrier spectra of the virtual image from the modulus image as shown in Fig. 3(b), because there is no obvious characteristic point. Fortunately, there always exists a maximum on the argument distribution of the hologram spectrum as shown in Fig. 3(c) and (d). In order to overcome the influence of the noise and assure the high accuracy of the position, the least squares surface fitting method is applied to fit the argument distribution of the interesting spectra region (shown in the dotted circle of Fig. 3(c)). Then the extremum point of the fitted surface can be found, whose coordinate here is (488,623). After the manual spatial filtering and being moved to the coordinate origin of the hologram frequency domain, the modulus image and argument image of the carrier spectra of the virtual image term are illustrated in Fig. 4(a) and (b). And Fig. 4(c) shows the modulus of the object wave in the hologram plane which is retrieved by the inverse Fourier transformation of the resulting spectra. The image diffraction field in the reconstructed plane at distance d = 690 mm from the hologram plane is calculated. And the quadratic phase aberration is compensated by multiplication of the reconstructed wave front with the digital phase mask shown in Eq. (6). The values of
a
b
the reconstructed distance d and the magnification of the imaging lens M are fixed for a given experimental configuration but must be modified exactly. Here the autofocus methods are used to adjust d and the M value is properly adjusted until the curved fringes disappear in the reconstructed image as shown in Fig. 5(a) and (b). Fig. 5 shows the phase contrast images in the reconstructed plane. From Fig. 5(a), we can see that there is only the quadric phase curvature left, and the tilt phase aberration has be totally eliminated. After compensating the quadric curvature introduced by the MO, the proper phase contrast images are obtained as shown in Fig. 5(b) and (c). 4. Conclusion In conclusion, a new method is proposed to locate the center of the target spectrum among the hologram spectrum, which is based on the argument distribution of the hologram spectrum. By the exact spectrum centering, the tilt phase aberration introduced by the tilt reference wave can be eliminated almost totally with a single hologram, without knowledge of the optical parameters in the experiment. Then the subsequent quadratic phase aberration compensation is performed only by adjusting a single parameter. It is demonstrated by the phase contrast imaging of the cervical carcinoma cells in pre-magnification DHM.
c
Fig. 4. (a) Modulus image and (b) argument image of the spectra after manual spatial filtering and shifting to the center. (c) Modulus of the object wave in the hologram plane after the inverse Fourier transform of the resulting spectra.
H. Cui et al. / Optics Communications 284 (2011) 4152–4155
a
b
4155
c
Fig. 5. Phase contrast image (mod 2π) with the size of 81 × 81 obtained (a) without digital phase mask, and (b) with proper digital phase mask (M = 74.5). (c) The three-dimensional phase distribution of the reconstructed phase.
Acknowledgment This work is financially supported by the National Natural Science Foundation of China under grant (no. 61077004), Science Foundation of Education Commission of Beijing under grant (no. KZ200910005001), and Innovative Talent and Team Development for Serving Beijing. References [1] U. Schnars, W. Jüptner, Appl. Opt. 33 (1994) 179. [2] I. Yamaguchi, T. Zhang, Opt. Lett. 22 (1997) 1268. [3] D. Mas, J. Garcia, C. Ferreira, L.M. Bernardo, F. Marinho, Opt. Commun. 164 (1999) 233. [4] I. Yamaguchi, T. Matsumura, J. Kato, Opt. Lett. 27 (2002) 1108. [5] J. Garcia-Sucerquia, W. Xu, S.K. Jericho, P. Klages, M.H. Jericho, H.J. Kreuzer, Appl. Opt. 45 (2006) 836.
[6] N. Lue, W. Choi, G. Popescu, T. Ikeda, R.R. Dasari, K. Badizadegan, M.S. Feld, Appl. Opt. 46 (2007) 1836. [7] B. Kemper, G. von Bally, Appl. Opt. 47 (2008) A52. [8] J. Li, P. Tankam, Z. Peng, P. Picart, Opt. Lett. 34 (2009) 572. [9] M.K. Kim, J. Opt. Soc. Korea 14 (2010) 77. [10] J. Jang, C.Y. Bae, J.K. Park, J.C. Ye, Opt. Lett. 35 (2010) 514. [11] E. Cuche, P. Marquet, C. Depeursinge, Appl. Opt. 38 (1999) 6994. [12] P. Ferraro, S.D. Nicola, A. Finizio, G. Coppola, S. Grilli, C. Magro, G. Pierattini, Appl. Opt. 42 (2003) 1938. [13] T. Colomb, J. Kühn, F. Charrière, C. Depeursinge, P. Marquet, N. Aspert, Opt. Express 14 (2006) 4300. [14] T. Colomb, F. Montfort, J. Kühn, N. Aspert, E. Cuche, A. Marian, F. Charrière, S. Bourquin, P. Marquet, C. Depeursinge, J. Opt. Soc. Am. A 23 (2006) 3177. [15] L. Miccio, D. Alfieri, S. Grilli, P. Ferraro, A. Finizio, L.D. Petrocellis, S.D. Nicola, Appl. Phys. Lett. 90 (2007) 041104. [16] E. Cuche, P. Marquet, C. Depeursinge, Appl. Opt. 39 (2000) 4070.