- Email: [email protected]
Multiplane imaging and depth-of-focus extending in digital holography by a single-shot digital hologram
Optics Communications 286 (2013) 117–122
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Multiplane imaging and depth-of-focus extending in digital holography by a single-shot digital hologram Weiqing Pan Department of Physics, Zhejiang Science and Technology University, Hangzhou, Zhejiang 310023, China
a r t i c l e i n f o
abstract
Article history: Received 7 March 2012 Received in revised form 19 August 2012 Accepted 4 September 2012 Available online 18 September 2012
Limited depth of field is the main drawback of conventional microscopy that prevents observation of thick semi-transparent objects with all their features in focus, only a portion of the imaged volume along the optical axis is in good focus at once. The paper presents a novel reconstruction algorithm to image multiple planes at different depths simultaneously and realize extended focused imaging. A shift parameter that accounts for the coordinate system’s transverse displacement of the image plane at different depths is introduced in the diffraction integral kernel. Combination of the diffraction integral kernel with different shift values and reconstruction depths yields multiplane imaging resolution in a single reconstruction. Moreover an extended depth-of-focus method is also presented through modifying the proposed multiplane imaging algorithm. Description of the method and experimental results are reported. & 2012 Elsevier B.V. All rights reserved.
Keywords: Digital holography Multiplane imaging Depth-of-focus extending
1. Introduction Holography is an established technique for recording and reconstructing real-world three-dimensional (3D) objects. Digital holography (DH) is a way to replace the hologram plate by a CCD array and to substitute the optical reconstruction process by a numerical calculation in the computer [1]. Digital holography enables the advantage of focus adjustment numerically after the measurement and avoiding the mechanical focus tracking during the measurement [2]. On the other hand, the depth-of-focus of the digital holography is limited. The depth-of-focus range for a reconstruction using the Fresnel approximation is of the order of a millimeter. For objects having 3D extension, scenes containing multiple objects or containing multiple object features located at different depths, only some portions of the object can be in good focus in each of those planes. In addition, with an increase of lateral resolution using an objective lens in digital holographic microscopy, due to their high numerical aperture, the depth-of-focus is greatly squeezed [3]. Different strategies have been found to solve the problem of the limited depth-of-focus [4–6]. For example, in the classical optical microscopy the extended depth-of-focus imaging can be obtained by merging several images of the same object in which each of them are partially focalized [7]. Another method is the use of special cubic phase plates to extend the depth-of-focus [8]. However the above techniques have generally not considered the case of DH. In fact, a hologram contains the necessary information to extract the image planes in the volume by changing the
E-mail address: [email protected] 0030-4018/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2012.09.013
reconstruction distance. Using the unique numerical focusing capability of DH, many algorithms have been given to manage with much flexibility the depth-of-focus for microscopic objects as MEMS and biological samples [9–12]. In many fields of science such as imaging particle fields or medical imaging, it is desirable to simultaneously image multiplane at different depths [13,14]. A first way makes use of lens array in which each lens can image a single plane in focus. However this method has many limitations [15]. Afterward Blanchard and Greenaway proposed a smart approach in which a diffraction grating has been adopted in the optical setup to split the propagating optical field in three diffraction orders and the wavefield resulting from the three diffraction orders can form three images at different planes. A furthermore investigation has been published in which a quadratic deformed grating has been used for multiple imaging and three dimensional particle tracking in biological microscopy with nanometer resolution along the optical axis [16]. Recently, Melania et al. proposed that the synthetic diffraction grating can be included into the numerical reconstruction of DH to image three planes at different depths simultaneously. Moreover they demonstrated that the adoption of a deformed diffraction grating can be exploited in multi-wavelength DH [17].
2. The algorithm of multiplane imaging In digital holography, the reconstruction algorithm is usually composed of two steps: complex amplitude retrieval and its backpropagation to the object plane. For numerical backpropagation, the Fresnel algorithm and the angular spectrum (AS) algorithm are
118
W. Pan / Optics Communications 286 (2013) 117–122
Multiple image planes Hologram Object Hm,n ( fx,fy)
Reconstruction distance zmn Z1,1
Z2,1
Z1,2
Z2,2
M
N
∑ ∑ Hm,n ( fx, fy) m=1 n=1
Fig. 1. Conceptual flow chart describing how the multiplane imaging is obtained by a digital holography approach.
widely used. The Fresnel approximation is limited to small numerical aperture (NA), resulting in a limited spatial resolution. In principle, an exact scalar propagation of light between two planes can be calculated by the AS algorithm. In this paper the multiplane imaging and depth-of-focus extending in digital holography will be realized by modification of the AS algorithm. Fig. 1 illustrates a scheme for simultaneously imaging multiple objects z-planes onto a single image plane. In the object volume, there are multiple objects or multiple object features located at different depths. Using the interference between an object wave and a reference wave, the information of the 3D object are recorded to be the digital hologram by a CCD camera. Then by modifying the reconstruction algorithm Hm,n(fx, fy), for the fixed reconstruction distance zm,n, the digital hologram can be reconstructed on the image plane at the desired transverse spatial shift (xm, yn) from the image center. For each reconstruction distance zm,n, one single image plane of the volume is reconstructed. Due to the limited depth-of-focus, only some section of the object volume will be in focus. Of course it is possible to obtain the entire volume by reconstructing a number of image planes in the volume of interest along the z-axis, and with the desired longitudinal resolution Dz. In this way a stack of images at different depths and different transverse spatial shifts can be easily obtained. Then by overlapping the stack of images, multiple object planes at different depths can be simultaneously imaged onto a single image plane. Of course, by properly modifying the transfer function of diffraction propagation Hm,n(fx, fy) with the suitable input parameters of (xm, yn, zm,n), the multiplane imaging process, this can be realized by one reconstruction process. In the following, we will derive the special transfer function for simultaneous imaging of different focal planes. According to the angular spectrum theorem, the Rayleigh– Sommerfeld diffraction integral is written in the spatial frequency domain as Uðf x ,f y ; zÞ ¼ Uðf x ,f y ; 0ÞHðf x ,f y ; zÞ
l
exp ½j2pðf x xm þ f y yn Þ
ð3Þ
here xm and yn are the spatial shifts for the reconstruction image at distance zm,n, the subscript m¼1,2,3,y,M and n¼1,2,3,y,N means the blocks’ serial numbers for column and row. Letting the size of reconstruction field of view is Lx Ly and the size of blocks is Dx¼Lx/M, Dy¼ Ly/N, then xm and yn can be expressed as ( xm ¼ Dx m M 2þ 1 ð4Þ 1 yn ¼ Dy n N þ 2 Corresponding to the above spatial shift (xm, yn), the reconstruction distance zm,n from the recording plane is given by zm,n ¼ ½ðn1ÞM þ m
MN Dz þz0 2
ð5Þ
where z0 is the basic reconstruction distance for coarse locating the object position, Dz is the distance of multiple image planes along z-axis which can be easily changed and used to finely focus the object features. From Eq. (5) we can see that the recording distance zm,n is an arithmetic serial and the common difference is Dz. By substituting Eqs. (4) and (5) into Eq. (3) and neglecting the constant-phase element exp ððj2p=lÞzm,n Þ we can get the modified transfer function as 2
2
Hm,n ðf x ,f y Þ ¼ Cðf x ,f y Þexp fjpm½lDzðf x þ f y Þ þ2f x Dxg
ð1Þ
where fx and fy are the spatial frequencies; U(fx, fy;0) is the Fourier transform of the diffraction object field which has been derived from the digital hologram; U(fx, fy;z) is the Fourier transform of the reconstruction image field at distance z from the hologram; and H(fx, fy;z) is the transfer function for the diffraction propagation and is explicitly expressed as h i 2p 2 2 Hðf x ,f y ; zÞ ¼ exp j z exp jplzðf x þ f y Þ ð2Þ
l
By inputting the reconstruction distance z, multiple images at different depth-of-focus can be reconstructed on different image planes. To separate the reconstruction images on the image plane, we divide the reconstruction field of view into M N blocks (see Fig. 1), and then the reconstructed object fields at different depths will be imaged on the different blocks. Accordingly the transfer function of Eq. (2) must be modified with a linear phase factor as h i 2p 2 2 Hm,n ðf x ,f y Þ ¼ exp j zm,n exp jplzm,n ðf x þ f y Þ
2
2
exp fjpn½lM Dzðf x þ f y Þ þ 2f y Dyg
ð6Þ
where Mð2 þNÞDz 2 2 ðf x þ f y Þg 2z0 exp fjp½ðM þ 1ÞDxf x þ ðN þ 1ÞDyf y g
Cðf x ,f y Þ ¼ exp fjplz0 ½1
ð7Þ
By using Eq. (6) a serial of image planes with transverse separations can be reconstructed at different depths. To simultaneously image the multiple images on a single image plane, we combine the serial of
W. Pan / Optics Communications 286 (2013) 117–122
reconstruction image planes, i.e. serial of transfer functions Eq. (6) as M X N X
HMN ðf x ,f y Þ ¼
Hm,n ðf x ,f y Þ
m¼1n¼1
¼ Cðf x ,f y Þ
M X
N X
resulting image with a greater depth-of-focus than any of the individually reconstructed images. Therefore the corresponding modified transfer function is derived as HMN ðf x ,f y Þ ¼
2
2
8 < exp fjplz0 ðf 2x þ f 2y Þg½exp ðjMNQ =2Þexp ðjMNQ =2Þ=½exp ðjQ Þ1,
exp fjpm½lDzðf x þ f y Þ þ2f x Dxg
m¼1
119
2
: MNexp fjplz0 ðf 2x þ f 2y Þg,
2
exp fjpn½lM Dzðf x þf y Þ þ 2f y Dyg
ð8Þ
ð12Þ
n¼1
Then by using the summation formula of geometric progression, the above summations can be simplified as the following: HM ðf x ,f y Þ ¼
M X
2
2
exp fjpm½lDzðf x þ f y Þ þ 2f x Dxg
m¼1
( exp ðjQÞ½1exp ðjMQ Þ ¼
1exp ðjQ Þ
,
Q a0
M,Q ¼ 0
where
2
2
Q ðf x ,f y Þ ¼ p½lDzðf x þ f y Þ þ 2f x Dx
ð9Þ HN ðf x ,f y Þ ¼
N X
2
2
exp fjpn½lM Dzðf x þf y Þ þ 2f y Dyg
n¼1
¼
8 ^ < exp ðjQ^ Þ½1exp ðjNQÞ , 1exp ðjQ^ Þ
Q^ a0
where
2 2 Q^ ðf x ,f y Þ ¼ p½lMDzðf x þ f y Þ þ 2f y Dy
: N, Q^ ¼ 0
ð10Þ Therefore combining Eqs. (8)–(10) the modified transfer function HMN(fx, fy) used for simultaneous imaging of different focal planes can be written with a simplified formula as HMN ðf x ,f y Þ ¼ Cðf x ,f y ÞHM ðf x ,f y ÞHN ðf x ,f y Þ
ð11Þ
3. Reconstruction formula for depth-of-focus extension According to the above proposed method, if setting Dx¼0, Dy¼0 and using the modified transfer function as Eq. (11), the multiple complex images simultaneously reconstructed at different focus distances will be coherently superposed to give a
Q a0
Q ¼0
lDzðf 2x þ f 2y Þ.
Comparing Eq. (12) with Eq. (2), a where Q ðf x ,f y Þ ¼ p phase function with parameters MN and Dz is introduced by the proposed method to modify the Rayleigh–Sommerfeld diffraction transfer function. By selecting the appropriate parameters MN and Dz, the focusing characteristic of the reconstruction process will be improved. For evaluating it, the normalized intensity distribution near the focus point z0 ¼ 0 is proposed. Fig. 2(a) and (b) separately shows the axial intensity and the transverse intensity on focal plane variation with Dz¼0, 0.05, 0.1, 0.5 where MN¼6. From this we can see that when Dz is greater than 0.05 the intensity distributions on axis and on focal plane are like Sinc functions. It means the reconstruction function will appear as multi-focus phenomena. However when Dz¼0, i.e. Q¼0 it degrades to the intensity distribution of the normal Rayleigh– Sommerfeld reconstruction function. Therefore to realize the depth-of-focus extension reconstruction we must insure selecting a relative small Dz to avoid appearing multi-focus phenomena. Fig. 2(c) and (d) presents the axial and transverse intensity variations with MN¼ 30, 50 and 70 where Dz¼0.05. From Fig. 2(c) we can see that with the proposed reconstruction function the axial intensity distributions are more plane than with the normal Rayleigh–Sommerfeld reconstruction function i.e. Dz¼0. This means that the proposed method can extend the depth-of-focus of reconstructed image. Moreover with the parameter of MN increasing the depth-of-focus will be enlarged as shown in Fig. 2(c). However from Fig. 2(d) we can see that the spread width of the transverse intensity on focal plane is also enlarged with the parameter of MN. This illuminates that the transverse resolution of the proposed method will be decreased with the depth-of-focus extension.
Fig. 2. The variation of normalized intensity distribution near the focal point with parameters Dz and MN: (a) the axial intensity and (b) the transverse intensity on focal plane variation with Dz where MN¼ 6, (c) the axial intensity and (d) the transverse intensity on focal plane variation with MN where Dz ¼ 0.05.
120
W. Pan / Optics Communications 286 (2013) 117–122
4. Experiments To demonstrate our technique, we performed different experiments. In the first case, four different fibers of cotton are positioned at different distances from the CCD array respectively. The CCD used has 1392 1040 pixels (pixel size of 9.3 mm 9.3 mm), the working wavelength is l ¼0.6328 mm. A digital hologram in lens-less configuration is recorded. To simultaneously obtain the four fibers at different depths in-focus on a single image, we performed the numerical reconstruction of the corresponding hologram used by Eq. (11) with the parameters setting as z0 ¼ 67 mm, M¼2, N¼2, Dz¼2.5 mm. The amplitude of the obtained reconstruction is shown in Fig. 3. We can see that the four fibers of cotton are simultaneously in focus imaged on four different blocks of the reconstruction field of view. Each fiber is in focus at distances of 69.5 mm, 67 mm, 64.5 mm and 62 mm. And the in focus fiber has been denoted by a magnifier icon. For the same hologram the reconstruction results obtained with the depth-of-focus extension algorithm of Eq. (12) are shown in Fig. 4. In this extended focused reconstruction z0 ¼ 67 mm, Dz¼ 0.05 mm and MN is varying between 50, 100, 200. It is clear that in the normal Rayleigh–Sommerfeld reconstruction image (Fig. 4(a)) only the straight bias fiber is in focus while the other three fibers are blurred and out of focus. Since a coherent light is used in the digital holography, the out of focus areas at the sharp edges of fibers show highly visible diffraction fringes. However in the extended focused results as Fig. 4(b–d), more fibers are focused and the diffraction fringes at the edges of fibers are disappearing with MN increment and the corresponding depth-of-focus enlargement. Finally when
Z1,1=-69.5mm
Z2,1=-67mm
Z1,2=-64.5mm
Z1,2=-62mm
Fig. 3. The amplitude reconstruction of the four fibers of cotton, each fiber is in focus at different distances and they have been denoted by a magnifier icon.
MN¼200 and the corresponding depth-of-focus is about 10 mm (200 0.05) in Fig. 4(d), the four fibers are almost simultaneously focused and the diffraction pattern has been completely eliminated. Fig. 5 shows the comparison among the normalized amplitude profiles measured along the red lines defined in Fig. 4(a–d). It is clear that the three pulses are the fiber positions and the fringes around the pulses are aroused by out of focus imaging. With the reconstruction parameter MN increment and depth-of-focus extension the image contrast ratios are improving. The diffraction fringes around the three fibers are also suppressed and the sunken top of out focus fiber pulses are reducing. Therefore by this technique we can obtain an extended focused image with all portions of a thick object in one view. As a further experiment we applied the method to holograms of a biological sample. The specimen is formed by two microorganism slices that are transverse slices of male ascaris lumbricoide and paramecium at different depths. In this case, a 10 microscopic objective is inserted in the interferometer used in microscope configuration. Fig. 6 shows the 2 2 multiplane amplitude and phase reconstruction at a distance z0 ¼ 30 mm and with the focus plane interval Dz ¼10 mm. The reconstruction image at the top left corner of the field view corresponds to a distance z1,1 ¼ 40 mm at which the paramecium cell indicated by the white arrow is in good focus, while the one at the bottom right corner corresponds to a depth z2,2 ¼ 10 mm where the male ascaris lumbricoides transverse slice also denoted by white arrows are well visible. The other two reconstruction images respectively at distance z2,1 ¼ 30 mm and z1,2 ¼ 20 mm are the Fresnel diffraction patterns of the two microorganism specimens. For the transparent cells containing features of varying refractive indices, the phase reconstruction as shown in Fig. 6(b) can be used, in which the focused transparent cells are denoted by white
Fig. 5. The reconstruction result without focus extension (a) and the focus extension results (b), (c) and (d) with Dz¼ 0.05 mm and different MN numbers.
3
1 2
Fig. 4. The reconstruction result without focus extension (a) and the extended focused results (b), (c) and (d) with Dz ¼0.05 mm and different MN numbers. (a) Dz¼ 0, MN¼ 1, (b) Dz ¼0.05, MN ¼50, (c) Dz ¼ 0.05, MN¼100 and (d) Dz ¼0.05, MN ¼200. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
W. Pan / Optics Communications 286 (2013) 117–122
Z1,1=-40mm
Z2,1=-30mm
Z1,1=-40mm
Z1,2=-20mm
Z2,2=-10mm
Z1,2=-20mm
121
Z2,1=-30mm
Z2,2=-10mm
Fig. 6. The 2 2 multiplane reconstruction at a distance z0 ¼ 30 mm and with the focus plane interval Dz ¼10 mm: (a) the amplitude reconstruction and (b) the phase reconstruction image at which the elements indicated by the white arrows are in focus.
Z1,1=-40mm
Z2,1=-35mm
Z3 1=-30mm
Z1,1=-40mm
Z2,1=-35mm
Z3,1=-30mm
Z1,2=-25mm
Z2,2=-20mm
Z3 2=-15mm
Z1,2=-25mm
Z2,2=-20mm
Z3,2=-15mm
Z1,3=-10m
Z2,3=-5mm
Z3,3=0mm
Z1,3=-10m
Z2,3=-5mm
Z3,3=0mm
Phase
Amplitude
Fig. 7. The 3 3 multiplane reconstruction at a distance z0 ¼ 20 mm and with the focus plane interval Dz¼ 5 mm: (a) the amplitude reconstruction and (b) the phase reconstruction image.
Fig. 8. The reconstruction result of microorganism specimen without focus extension (a) and the extended focused results (b), (c) and (d) with Dz¼ 0.05 mm and different MN numbers. (a) Dz¼ 0, MN¼ 1, (b) Dz ¼0.05, MN¼200, (c) Dz ¼0.05, MN¼ 400 and (d) Dz ¼0.05, MN¼ 800.
arrows. Therefore, by this technique, we were able to image in a single reconstruction multiple different object planes allowing us to see cells lying on different planes simultaneously. Obviously it is possible to simultaneously reconstruct more multiplane images at different depths by changing the focus interval Dz and the object plane number M N. In Fig. 7 we demonstrate that up to nine images can be simultaneously obtained by setting the parameters as z0 ¼20 mm, Dz ¼5 mm and M N ¼3 3. Since the multiplane imaging transfer function Eq. (11) is an analytical formula and no loop computation is needed, the computation speed is not affected by the increment object plane number M N. Therefore in the above case of
M N ¼3 3, the computational time is one-ninth in respect to the case of separated reconstructions, thus increasing the computational efficiency. The results of extended focused amplitude and phase image reconstruction with Eq. (12) are given in Fig. 8(b–d). One can see that the diffraction fringes and noise have been efficiently removed from the background with the parameter MN increment compared to the partially focused image (a). By the comparison of some elements denoted with white arrows in the reconstruction images, one can see that the elements’ size is decreasing and the edge is sharping with MN increment. This means the focused depth is extended and the elements are in focus since the
122
W. Pan / Optics Communications 286 (2013) 117–122
elements’ width has the smallest size when they are in their plane of focus. Comparing the paramecium cell highlighted by white ellipse in Fig. 8(a) and (d), it is clear that the Rayleigh– Sommerfeld reconstruction (Fig. 8(a)) blurred the cell due to the limited focus depth of reconstruction and the details of the inner part of the cell is invisible. However in the reconstruction with the proposed extended focused algorithm, i.e. Fig. 8(d) the paramecium cell is focused and the inner cell core illustrated with black arrow is clearly visible.
5. Conclusion In conclusion, the paper has proposed a novel reconstruction algorithm to image multiple planes at different depths simultaneously and realize extended focused imaging. That allows to observe simultaneously and in a single field of view multiple planes in good focus by a single reconstruction step. It is possible to have flexible numerical focus at any plane in the imaged volume and it is important to outline that computation is less heavy in this case. In addition the imaged planes can be changed easily by changing the parameters setting in the numerical reconstruction. Moreover, a method based on the multiplane imaging algorithm to extend the depth-of-focus using a single-shot digital hologram without any mechanical scanning or specially designed optical components is also proposed. The results show that the reconstruction’s depth-of-focus will be enlarged with the parameter of MN increment. However the transverse resolution of the proposed method will be decreased with the depth-of-focus extension.
Acknowledgment This work was supported by the National Natural Science Foundation of China (No. 51005212), the Science and Technology Planning Project of Zhejiang Province, China (2010C31095) and Beforehand Research Project of ZUST (F703108A01). References ¨ [1] U. Schnars, W. Juptner, Applied Optics 33 (1994) 179. [2] P. Ferraro, G. Coppola, S. De Nicola, A. Finizio, G. Pierattini, Optics Letters 28 (2003) 1257. [3] Conor P. McElhinney, Bryan M. Hennelly, Thomas J. Naughton, Applied Optics 47 (2008) D71. [4] H. Zhao, Q. Li, H. Feng, Optics Letters 33 (11) (2008) 1171. [5] S. Bagheri, B. Javidi, Optics Letters 33 (7) (2008) 757. [6] R.J. Pieper, A. Korpel, Applied Optics 22 (10) (1983) 1449. [7] G. Mikula, Z. Jaroszewicz, A. Kolodziejczyk, K. Petelczyc, M. Sypek, Optics Express 15 (15) (2007) 9184. [8] E.R. Dowski Jr., W.T. Cathey, Applied Optics 34 (11) (1995) 1859. [9] T. Ikeda, G. Popescu, R.R. Dasari, M.S. Feld, Optics Letters 30 (2005) 1165. [10] S.J. Jeong, C.K. Hong, Applied Optics 47 (16) (2008) 3064. [11] P. Ferraro, S. Grilli, D. Alfieri, S. De Nicola, A. Finizio, G. Pierattini, B. Javidi, G. Coppola, V. Striano, Optics Express 13 (18) (2005) 6738. [12] C.P. McElhinney, B.M. Hennelly, T.J. Naughton, Applied Optics 47 (19) (2008) D71. [13] W. Xu, M.H. Jericho, H.J. Kreuzer, I.A. Meinertzhagen, Optics Letters 28 (3) (2003) 164. [14] L. Holtzer, T. Meckel, T. Schmidt, Applied Physics Letters 90 (5) (2007) 053902. [15] P.J. Verveer, J. Swoger, F. Pampaloni, K. Greger, M. Marcello, E.H.K. Stelzer, Nature Methods 4 (4) (2007) 311. [16] Paul A. Dalgarno, Heather I.C. Dalgarno, Aure´lie Putoud, Robert Lambert, Lynn Paterson, David C. Logan, David P. Towers, Richard J. Warburton, Alan H. Greenaway, Optics Express 18 (2010) 877. [17] Melania Paturzo, A. Finizio, Pietro Ferraro, Journal of Display Technology 7 (1) (2011) 24.
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Multiplane imaging and depth-of-focus extending in digital holography by a single-shot digital hologram Weiqing Pan Department of Physics, Zhejiang Science and Technology University, Hangzhou, Zhejiang 310023, China
a r t i c l e i n f o
abstract
Article history: Received 7 March 2012 Received in revised form 19 August 2012 Accepted 4 September 2012 Available online 18 September 2012
Limited depth of field is the main drawback of conventional microscopy that prevents observation of thick semi-transparent objects with all their features in focus, only a portion of the imaged volume along the optical axis is in good focus at once. The paper presents a novel reconstruction algorithm to image multiple planes at different depths simultaneously and realize extended focused imaging. A shift parameter that accounts for the coordinate system’s transverse displacement of the image plane at different depths is introduced in the diffraction integral kernel. Combination of the diffraction integral kernel with different shift values and reconstruction depths yields multiplane imaging resolution in a single reconstruction. Moreover an extended depth-of-focus method is also presented through modifying the proposed multiplane imaging algorithm. Description of the method and experimental results are reported. & 2012 Elsevier B.V. All rights reserved.
Keywords: Digital holography Multiplane imaging Depth-of-focus extending
1. Introduction Holography is an established technique for recording and reconstructing real-world three-dimensional (3D) objects. Digital holography (DH) is a way to replace the hologram plate by a CCD array and to substitute the optical reconstruction process by a numerical calculation in the computer [1]. Digital holography enables the advantage of focus adjustment numerically after the measurement and avoiding the mechanical focus tracking during the measurement [2]. On the other hand, the depth-of-focus of the digital holography is limited. The depth-of-focus range for a reconstruction using the Fresnel approximation is of the order of a millimeter. For objects having 3D extension, scenes containing multiple objects or containing multiple object features located at different depths, only some portions of the object can be in good focus in each of those planes. In addition, with an increase of lateral resolution using an objective lens in digital holographic microscopy, due to their high numerical aperture, the depth-of-focus is greatly squeezed [3]. Different strategies have been found to solve the problem of the limited depth-of-focus [4–6]. For example, in the classical optical microscopy the extended depth-of-focus imaging can be obtained by merging several images of the same object in which each of them are partially focalized [7]. Another method is the use of special cubic phase plates to extend the depth-of-focus [8]. However the above techniques have generally not considered the case of DH. In fact, a hologram contains the necessary information to extract the image planes in the volume by changing the
E-mail address: [email protected] 0030-4018/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2012.09.013
reconstruction distance. Using the unique numerical focusing capability of DH, many algorithms have been given to manage with much flexibility the depth-of-focus for microscopic objects as MEMS and biological samples [9–12]. In many fields of science such as imaging particle fields or medical imaging, it is desirable to simultaneously image multiplane at different depths [13,14]. A first way makes use of lens array in which each lens can image a single plane in focus. However this method has many limitations [15]. Afterward Blanchard and Greenaway proposed a smart approach in which a diffraction grating has been adopted in the optical setup to split the propagating optical field in three diffraction orders and the wavefield resulting from the three diffraction orders can form three images at different planes. A furthermore investigation has been published in which a quadratic deformed grating has been used for multiple imaging and three dimensional particle tracking in biological microscopy with nanometer resolution along the optical axis [16]. Recently, Melania et al. proposed that the synthetic diffraction grating can be included into the numerical reconstruction of DH to image three planes at different depths simultaneously. Moreover they demonstrated that the adoption of a deformed diffraction grating can be exploited in multi-wavelength DH [17].
2. The algorithm of multiplane imaging In digital holography, the reconstruction algorithm is usually composed of two steps: complex amplitude retrieval and its backpropagation to the object plane. For numerical backpropagation, the Fresnel algorithm and the angular spectrum (AS) algorithm are
118
W. Pan / Optics Communications 286 (2013) 117–122
Multiple image planes Hologram Object Hm,n ( fx,fy)
Reconstruction distance zmn Z1,1
Z2,1
Z1,2
Z2,2
M
N
∑ ∑ Hm,n ( fx, fy) m=1 n=1
Fig. 1. Conceptual flow chart describing how the multiplane imaging is obtained by a digital holography approach.
widely used. The Fresnel approximation is limited to small numerical aperture (NA), resulting in a limited spatial resolution. In principle, an exact scalar propagation of light between two planes can be calculated by the AS algorithm. In this paper the multiplane imaging and depth-of-focus extending in digital holography will be realized by modification of the AS algorithm. Fig. 1 illustrates a scheme for simultaneously imaging multiple objects z-planes onto a single image plane. In the object volume, there are multiple objects or multiple object features located at different depths. Using the interference between an object wave and a reference wave, the information of the 3D object are recorded to be the digital hologram by a CCD camera. Then by modifying the reconstruction algorithm Hm,n(fx, fy), for the fixed reconstruction distance zm,n, the digital hologram can be reconstructed on the image plane at the desired transverse spatial shift (xm, yn) from the image center. For each reconstruction distance zm,n, one single image plane of the volume is reconstructed. Due to the limited depth-of-focus, only some section of the object volume will be in focus. Of course it is possible to obtain the entire volume by reconstructing a number of image planes in the volume of interest along the z-axis, and with the desired longitudinal resolution Dz. In this way a stack of images at different depths and different transverse spatial shifts can be easily obtained. Then by overlapping the stack of images, multiple object planes at different depths can be simultaneously imaged onto a single image plane. Of course, by properly modifying the transfer function of diffraction propagation Hm,n(fx, fy) with the suitable input parameters of (xm, yn, zm,n), the multiplane imaging process, this can be realized by one reconstruction process. In the following, we will derive the special transfer function for simultaneous imaging of different focal planes. According to the angular spectrum theorem, the Rayleigh– Sommerfeld diffraction integral is written in the spatial frequency domain as Uðf x ,f y ; zÞ ¼ Uðf x ,f y ; 0ÞHðf x ,f y ; zÞ
l
exp ½j2pðf x xm þ f y yn Þ
ð3Þ
here xm and yn are the spatial shifts for the reconstruction image at distance zm,n, the subscript m¼1,2,3,y,M and n¼1,2,3,y,N means the blocks’ serial numbers for column and row. Letting the size of reconstruction field of view is Lx Ly and the size of blocks is Dx¼Lx/M, Dy¼ Ly/N, then xm and yn can be expressed as ( xm ¼ Dx m M 2þ 1 ð4Þ 1 yn ¼ Dy n N þ 2 Corresponding to the above spatial shift (xm, yn), the reconstruction distance zm,n from the recording plane is given by zm,n ¼ ½ðn1ÞM þ m
MN Dz þz0 2
ð5Þ
where z0 is the basic reconstruction distance for coarse locating the object position, Dz is the distance of multiple image planes along z-axis which can be easily changed and used to finely focus the object features. From Eq. (5) we can see that the recording distance zm,n is an arithmetic serial and the common difference is Dz. By substituting Eqs. (4) and (5) into Eq. (3) and neglecting the constant-phase element exp ððj2p=lÞzm,n Þ we can get the modified transfer function as 2
2
Hm,n ðf x ,f y Þ ¼ Cðf x ,f y Þexp fjpm½lDzðf x þ f y Þ þ2f x Dxg
ð1Þ
where fx and fy are the spatial frequencies; U(fx, fy;0) is the Fourier transform of the diffraction object field which has been derived from the digital hologram; U(fx, fy;z) is the Fourier transform of the reconstruction image field at distance z from the hologram; and H(fx, fy;z) is the transfer function for the diffraction propagation and is explicitly expressed as h i 2p 2 2 Hðf x ,f y ; zÞ ¼ exp j z exp jplzðf x þ f y Þ ð2Þ
l
By inputting the reconstruction distance z, multiple images at different depth-of-focus can be reconstructed on different image planes. To separate the reconstruction images on the image plane, we divide the reconstruction field of view into M N blocks (see Fig. 1), and then the reconstructed object fields at different depths will be imaged on the different blocks. Accordingly the transfer function of Eq. (2) must be modified with a linear phase factor as h i 2p 2 2 Hm,n ðf x ,f y Þ ¼ exp j zm,n exp jplzm,n ðf x þ f y Þ
2
2
exp fjpn½lM Dzðf x þ f y Þ þ 2f y Dyg
ð6Þ
where Mð2 þNÞDz 2 2 ðf x þ f y Þg 2z0 exp fjp½ðM þ 1ÞDxf x þ ðN þ 1ÞDyf y g
Cðf x ,f y Þ ¼ exp fjplz0 ½1
ð7Þ
By using Eq. (6) a serial of image planes with transverse separations can be reconstructed at different depths. To simultaneously image the multiple images on a single image plane, we combine the serial of
W. Pan / Optics Communications 286 (2013) 117–122
reconstruction image planes, i.e. serial of transfer functions Eq. (6) as M X N X
HMN ðf x ,f y Þ ¼
Hm,n ðf x ,f y Þ
m¼1n¼1
¼ Cðf x ,f y Þ
M X
N X
resulting image with a greater depth-of-focus than any of the individually reconstructed images. Therefore the corresponding modified transfer function is derived as HMN ðf x ,f y Þ ¼
2
2
8 < exp fjplz0 ðf 2x þ f 2y Þg½exp ðjMNQ =2Þexp ðjMNQ =2Þ=½exp ðjQ Þ1,
exp fjpm½lDzðf x þ f y Þ þ2f x Dxg
m¼1
119
2
: MNexp fjplz0 ðf 2x þ f 2y Þg,
2
exp fjpn½lM Dzðf x þf y Þ þ 2f y Dyg
ð8Þ
ð12Þ
n¼1
Then by using the summation formula of geometric progression, the above summations can be simplified as the following: HM ðf x ,f y Þ ¼
M X
2
2
exp fjpm½lDzðf x þ f y Þ þ 2f x Dxg
m¼1
( exp ðjQÞ½1exp ðjMQ Þ ¼
1exp ðjQ Þ
,
Q a0
M,Q ¼ 0
where
2
2
Q ðf x ,f y Þ ¼ p½lDzðf x þ f y Þ þ 2f x Dx
ð9Þ HN ðf x ,f y Þ ¼
N X
2
2
exp fjpn½lM Dzðf x þf y Þ þ 2f y Dyg
n¼1
¼
8 ^ < exp ðjQ^ Þ½1exp ðjNQÞ , 1exp ðjQ^ Þ
Q^ a0
where
2 2 Q^ ðf x ,f y Þ ¼ p½lMDzðf x þ f y Þ þ 2f y Dy
: N, Q^ ¼ 0
ð10Þ Therefore combining Eqs. (8)–(10) the modified transfer function HMN(fx, fy) used for simultaneous imaging of different focal planes can be written with a simplified formula as HMN ðf x ,f y Þ ¼ Cðf x ,f y ÞHM ðf x ,f y ÞHN ðf x ,f y Þ
ð11Þ
3. Reconstruction formula for depth-of-focus extension According to the above proposed method, if setting Dx¼0, Dy¼0 and using the modified transfer function as Eq. (11), the multiple complex images simultaneously reconstructed at different focus distances will be coherently superposed to give a
Q a0
Q ¼0
lDzðf 2x þ f 2y Þ.
Comparing Eq. (12) with Eq. (2), a where Q ðf x ,f y Þ ¼ p phase function with parameters MN and Dz is introduced by the proposed method to modify the Rayleigh–Sommerfeld diffraction transfer function. By selecting the appropriate parameters MN and Dz, the focusing characteristic of the reconstruction process will be improved. For evaluating it, the normalized intensity distribution near the focus point z0 ¼ 0 is proposed. Fig. 2(a) and (b) separately shows the axial intensity and the transverse intensity on focal plane variation with Dz¼0, 0.05, 0.1, 0.5 where MN¼6. From this we can see that when Dz is greater than 0.05 the intensity distributions on axis and on focal plane are like Sinc functions. It means the reconstruction function will appear as multi-focus phenomena. However when Dz¼0, i.e. Q¼0 it degrades to the intensity distribution of the normal Rayleigh– Sommerfeld reconstruction function. Therefore to realize the depth-of-focus extension reconstruction we must insure selecting a relative small Dz to avoid appearing multi-focus phenomena. Fig. 2(c) and (d) presents the axial and transverse intensity variations with MN¼ 30, 50 and 70 where Dz¼0.05. From Fig. 2(c) we can see that with the proposed reconstruction function the axial intensity distributions are more plane than with the normal Rayleigh–Sommerfeld reconstruction function i.e. Dz¼0. This means that the proposed method can extend the depth-of-focus of reconstructed image. Moreover with the parameter of MN increasing the depth-of-focus will be enlarged as shown in Fig. 2(c). However from Fig. 2(d) we can see that the spread width of the transverse intensity on focal plane is also enlarged with the parameter of MN. This illuminates that the transverse resolution of the proposed method will be decreased with the depth-of-focus extension.
Fig. 2. The variation of normalized intensity distribution near the focal point with parameters Dz and MN: (a) the axial intensity and (b) the transverse intensity on focal plane variation with Dz where MN¼ 6, (c) the axial intensity and (d) the transverse intensity on focal plane variation with MN where Dz ¼ 0.05.
120
W. Pan / Optics Communications 286 (2013) 117–122
4. Experiments To demonstrate our technique, we performed different experiments. In the first case, four different fibers of cotton are positioned at different distances from the CCD array respectively. The CCD used has 1392 1040 pixels (pixel size of 9.3 mm 9.3 mm), the working wavelength is l ¼0.6328 mm. A digital hologram in lens-less configuration is recorded. To simultaneously obtain the four fibers at different depths in-focus on a single image, we performed the numerical reconstruction of the corresponding hologram used by Eq. (11) with the parameters setting as z0 ¼ 67 mm, M¼2, N¼2, Dz¼2.5 mm. The amplitude of the obtained reconstruction is shown in Fig. 3. We can see that the four fibers of cotton are simultaneously in focus imaged on four different blocks of the reconstruction field of view. Each fiber is in focus at distances of 69.5 mm, 67 mm, 64.5 mm and 62 mm. And the in focus fiber has been denoted by a magnifier icon. For the same hologram the reconstruction results obtained with the depth-of-focus extension algorithm of Eq. (12) are shown in Fig. 4. In this extended focused reconstruction z0 ¼ 67 mm, Dz¼ 0.05 mm and MN is varying between 50, 100, 200. It is clear that in the normal Rayleigh–Sommerfeld reconstruction image (Fig. 4(a)) only the straight bias fiber is in focus while the other three fibers are blurred and out of focus. Since a coherent light is used in the digital holography, the out of focus areas at the sharp edges of fibers show highly visible diffraction fringes. However in the extended focused results as Fig. 4(b–d), more fibers are focused and the diffraction fringes at the edges of fibers are disappearing with MN increment and the corresponding depth-of-focus enlargement. Finally when
Z1,1=-69.5mm
Z2,1=-67mm
Z1,2=-64.5mm
Z1,2=-62mm
Fig. 3. The amplitude reconstruction of the four fibers of cotton, each fiber is in focus at different distances and they have been denoted by a magnifier icon.
MN¼200 and the corresponding depth-of-focus is about 10 mm (200 0.05) in Fig. 4(d), the four fibers are almost simultaneously focused and the diffraction pattern has been completely eliminated. Fig. 5 shows the comparison among the normalized amplitude profiles measured along the red lines defined in Fig. 4(a–d). It is clear that the three pulses are the fiber positions and the fringes around the pulses are aroused by out of focus imaging. With the reconstruction parameter MN increment and depth-of-focus extension the image contrast ratios are improving. The diffraction fringes around the three fibers are also suppressed and the sunken top of out focus fiber pulses are reducing. Therefore by this technique we can obtain an extended focused image with all portions of a thick object in one view. As a further experiment we applied the method to holograms of a biological sample. The specimen is formed by two microorganism slices that are transverse slices of male ascaris lumbricoide and paramecium at different depths. In this case, a 10 microscopic objective is inserted in the interferometer used in microscope configuration. Fig. 6 shows the 2 2 multiplane amplitude and phase reconstruction at a distance z0 ¼ 30 mm and with the focus plane interval Dz ¼10 mm. The reconstruction image at the top left corner of the field view corresponds to a distance z1,1 ¼ 40 mm at which the paramecium cell indicated by the white arrow is in good focus, while the one at the bottom right corner corresponds to a depth z2,2 ¼ 10 mm where the male ascaris lumbricoides transverse slice also denoted by white arrows are well visible. The other two reconstruction images respectively at distance z2,1 ¼ 30 mm and z1,2 ¼ 20 mm are the Fresnel diffraction patterns of the two microorganism specimens. For the transparent cells containing features of varying refractive indices, the phase reconstruction as shown in Fig. 6(b) can be used, in which the focused transparent cells are denoted by white
Fig. 5. The reconstruction result without focus extension (a) and the focus extension results (b), (c) and (d) with Dz¼ 0.05 mm and different MN numbers.
3
1 2
Fig. 4. The reconstruction result without focus extension (a) and the extended focused results (b), (c) and (d) with Dz ¼0.05 mm and different MN numbers. (a) Dz¼ 0, MN¼ 1, (b) Dz ¼0.05, MN ¼50, (c) Dz ¼ 0.05, MN¼100 and (d) Dz ¼0.05, MN ¼200. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
W. Pan / Optics Communications 286 (2013) 117–122
Z1,1=-40mm
Z2,1=-30mm
Z1,1=-40mm
Z1,2=-20mm
Z2,2=-10mm
Z1,2=-20mm
121
Z2,1=-30mm
Z2,2=-10mm
Fig. 6. The 2 2 multiplane reconstruction at a distance z0 ¼ 30 mm and with the focus plane interval Dz ¼10 mm: (a) the amplitude reconstruction and (b) the phase reconstruction image at which the elements indicated by the white arrows are in focus.
Z1,1=-40mm
Z2,1=-35mm
Z3 1=-30mm
Z1,1=-40mm
Z2,1=-35mm
Z3,1=-30mm
Z1,2=-25mm
Z2,2=-20mm
Z3 2=-15mm
Z1,2=-25mm
Z2,2=-20mm
Z3,2=-15mm
Z1,3=-10m
Z2,3=-5mm
Z3,3=0mm
Z1,3=-10m
Z2,3=-5mm
Z3,3=0mm
Phase
Amplitude
Fig. 7. The 3 3 multiplane reconstruction at a distance z0 ¼ 20 mm and with the focus plane interval Dz¼ 5 mm: (a) the amplitude reconstruction and (b) the phase reconstruction image.
Fig. 8. The reconstruction result of microorganism specimen without focus extension (a) and the extended focused results (b), (c) and (d) with Dz¼ 0.05 mm and different MN numbers. (a) Dz¼ 0, MN¼ 1, (b) Dz ¼0.05, MN¼200, (c) Dz ¼0.05, MN¼ 400 and (d) Dz ¼0.05, MN¼ 800.
arrows. Therefore, by this technique, we were able to image in a single reconstruction multiple different object planes allowing us to see cells lying on different planes simultaneously. Obviously it is possible to simultaneously reconstruct more multiplane images at different depths by changing the focus interval Dz and the object plane number M N. In Fig. 7 we demonstrate that up to nine images can be simultaneously obtained by setting the parameters as z0 ¼20 mm, Dz ¼5 mm and M N ¼3 3. Since the multiplane imaging transfer function Eq. (11) is an analytical formula and no loop computation is needed, the computation speed is not affected by the increment object plane number M N. Therefore in the above case of
M N ¼3 3, the computational time is one-ninth in respect to the case of separated reconstructions, thus increasing the computational efficiency. The results of extended focused amplitude and phase image reconstruction with Eq. (12) are given in Fig. 8(b–d). One can see that the diffraction fringes and noise have been efficiently removed from the background with the parameter MN increment compared to the partially focused image (a). By the comparison of some elements denoted with white arrows in the reconstruction images, one can see that the elements’ size is decreasing and the edge is sharping with MN increment. This means the focused depth is extended and the elements are in focus since the
122
W. Pan / Optics Communications 286 (2013) 117–122
elements’ width has the smallest size when they are in their plane of focus. Comparing the paramecium cell highlighted by white ellipse in Fig. 8(a) and (d), it is clear that the Rayleigh– Sommerfeld reconstruction (Fig. 8(a)) blurred the cell due to the limited focus depth of reconstruction and the details of the inner part of the cell is invisible. However in the reconstruction with the proposed extended focused algorithm, i.e. Fig. 8(d) the paramecium cell is focused and the inner cell core illustrated with black arrow is clearly visible.
5. Conclusion In conclusion, the paper has proposed a novel reconstruction algorithm to image multiple planes at different depths simultaneously and realize extended focused imaging. That allows to observe simultaneously and in a single field of view multiple planes in good focus by a single reconstruction step. It is possible to have flexible numerical focus at any plane in the imaged volume and it is important to outline that computation is less heavy in this case. In addition the imaged planes can be changed easily by changing the parameters setting in the numerical reconstruction. Moreover, a method based on the multiplane imaging algorithm to extend the depth-of-focus using a single-shot digital hologram without any mechanical scanning or specially designed optical components is also proposed. The results show that the reconstruction’s depth-of-focus will be enlarged with the parameter of MN increment. However the transverse resolution of the proposed method will be decreased with the depth-of-focus extension.
Acknowledgment This work was supported by the National Natural Science Foundation of China (No. 51005212), the Science and Technology Planning Project of Zhejiang Province, China (2010C31095) and Beforehand Research Project of ZUST (F703108A01). References ¨ [1] U. Schnars, W. Juptner, Applied Optics 33 (1994) 179. [2] P. Ferraro, G. Coppola, S. De Nicola, A. Finizio, G. Pierattini, Optics Letters 28 (2003) 1257. [3] Conor P. McElhinney, Bryan M. Hennelly, Thomas J. Naughton, Applied Optics 47 (2008) D71. [4] H. Zhao, Q. Li, H. Feng, Optics Letters 33 (11) (2008) 1171. [5] S. Bagheri, B. Javidi, Optics Letters 33 (7) (2008) 757. [6] R.J. Pieper, A. Korpel, Applied Optics 22 (10) (1983) 1449. [7] G. Mikula, Z. Jaroszewicz, A. Kolodziejczyk, K. Petelczyc, M. Sypek, Optics Express 15 (15) (2007) 9184. [8] E.R. Dowski Jr., W.T. Cathey, Applied Optics 34 (11) (1995) 1859. [9] T. Ikeda, G. Popescu, R.R. Dasari, M.S. Feld, Optics Letters 30 (2005) 1165. [10] S.J. Jeong, C.K. Hong, Applied Optics 47 (16) (2008) 3064. [11] P. Ferraro, S. Grilli, D. Alfieri, S. De Nicola, A. Finizio, G. Pierattini, B. Javidi, G. Coppola, V. Striano, Optics Express 13 (18) (2005) 6738. [12] C.P. McElhinney, B.M. Hennelly, T.J. Naughton, Applied Optics 47 (19) (2008) D71. [13] W. Xu, M.H. Jericho, H.J. Kreuzer, I.A. Meinertzhagen, Optics Letters 28 (3) (2003) 164. [14] L. Holtzer, T. Meckel, T. Schmidt, Applied Physics Letters 90 (5) (2007) 053902. [15] P.J. Verveer, J. Swoger, F. Pampaloni, K. Greger, M. Marcello, E.H.K. Stelzer, Nature Methods 4 (4) (2007) 311. [16] Paul A. Dalgarno, Heather I.C. Dalgarno, Aure´lie Putoud, Robert Lambert, Lynn Paterson, David C. Logan, David P. Towers, Richard J. Warburton, Alan H. Greenaway, Optics Express 18 (2010) 877. [17] Melania Paturzo, A. Finizio, Pietro Ferraro, Journal of Display Technology 7 (1) (2011) 24.