FM4F.4.pdf
FiO/LS Technical Digest © OSA 2012
Digital Spatially Incoherent Fourier Holography 1,2
Roy Kelner1 and Joseph Rosen2 Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel 1
[email protected],
[email protected]
Abstract: We present a new method for recording digital Fourier holograms under incoherent illumination. A single exposure captured by a digital camera is sufficient to record a real-valued hologram which encodes complete 3D objects. OCIS codes: (090.0090) Holography; (090.1995) Digital holography; (110.6880) Three-dimensional image acquisition; (100.3010) Image reconstruction techniques; (070.6120) Spatial light modulators.
1. Introduction In recent years we have witnessed noteworthy achievements of incoherent holography techniques such as optical scanning holography (OSH) [1] and Fresnel incoherent correlation holography (FINCH) [2]. The latter has fundamental system robustness since it is based on a single channel incoherent interferometer and because it does not require any scanning nor mechanical movement. However, though a single FINCH image contains the complete 3D information of an object, at least three images are required to solve the twin image problem [2]. We present a new method for recording digital Fourier holograms under incoherent illumination. By recording a Fourier hologram [3] of a half plane (or space) the twin image problem is avoided and the object can be reconstructed from a single exposure. We coin our method Fourier incoherent single channel holography (FISCH). 2. System description A schematic view of a FISCH recorder is shown in Fig. 1. A white-light source illuminates a 3D object. Then, light scattered from the object propagates through a single channel (or arm) incoherent interferometer. Eventually, interference pattern are captured by the CCD sensor.
RC
L0 f0
SLM 2
SLM 1
d0
f1
2 f1
f1
d1
P2 CCD
zs
P1
Fig. 1. Schematic of the FISCH recorder: SLM, spatial light modulator; CCD, charge-coupled device; P1 and P2, polarizers; RC, resolution chart.
Consider a point source object positioned at the point ( xs , ys , zs ), a distance z s from the lens L0 . A tilted diverging spherical wave of the form T ( x, y ) c(rs , zs ) L(rs / z s )Q(1 / zs ) is induced over that lens, where rs ( xs , ys ) , c ( rs , z s ) is a complex valued constant dependent on the position and intensity of the point source, and L(s ) exp[i2 1 (sx x sy y)] and Q( s) exp[i s 1 ( x 2 y 2 )] are the linear and the quadratic phase functions, respectively, in which is defined as the central wavelength. A diffractive optical element of the form Q(1/ f1 ) is displayed on the spatial light modulators (SLMs). The SLMs are polarization sensitive, so by introducing the linear polarizer P1 the single channel optical apparatus is effectively split into two sub-channels (see [2] for a detailed explanation). In one sub-channel the SLMs function as converging diffractive lenses with a focal length f1 , while in the other they do not alter the wave. The second polarizer P 2 rejoins the two sub-channels. For simplicity, we continue the analysis for the special case of zs f 0 . That is, the point source object is positioned at the front focal plane of the lens L0 . This lens transforms the spherical wave T ( x, y ) into a tilted plane wave of the simpler form T ( x, y ) c(rs , f 0 ) L(rs / f 0 ). The wave at the unmodulated interferometer sub-channel simply propagates a distance of d 0 2 f1 d1 before arriving to the CCD plane where it is then equal to: r D1 ( x, y; rs , f 0 ) c1 (rs , f 0 ) L s . (1) f0 The tilted plane wave at the modulated interferometer sub-channel undergoes a 180 inversion by the two converging diffractive lens of focal length f1 displayed on the SLMs and separated by a distance of 2 f1 so that upon arrival to the CCD plane it is equal to: r 1 1 rs 1 1 1 D2 ( x, y; rs , f 0 ) c2 (rs , f 0 ) L s Q ·Q Q ·Q Q c2 (rs , f 0 ) L , (2) 2 f1 f1 d1 f0 d 0 f1 f0
FM4F.4.pdf
FiO/LS Technical Digest © OSA 2012
where Q(1 / zd ) denotes a Fresnel propagation of a wave to a distance of zd (mathematically, denotes a 2D convolution) and ·Q ( 1 / f l ) denotes the influence of a converging lens of focal length f l upon a wave propagating through it. Eq. (2) has been simplified based on a mathematical justification presented in [4]. Finally, the recorded intensity over the CCD plane is equal to: 2rs 2 2 2 I ( x, y; rs , f 0 ) D1 ( x, y; rs , f 0 ) D2 ( x, y; rs , f 0 ) c '1 (rs , f 0 ) c '2 (rs , f 0 ) [c '1 (rs , f 0 )c '2 (rs , f 0 )·L c.c.], (3) f0 where c.c. is the complex conjugate of the left term inside the square brackets. For a point of arbitrary z s the recorded intensity is I ( x, y; rs , zs ) . We refrain from giving here the detailed expression for the sake of brevity. Since each point source is only spatially coherent to itself, the recorded hologram due to many point sources is simply a summation of all point source contributions. That is, the recorded hologram is H ( x, y ) I ( x, y; rs , z s )dxs dys dzs . Note that according to Eq. (3) all source points located at the front focal plane of lens L0 can be reconstructed by a simple calculation of the inverse Fourier transform (FT) of H ( x, y ). Hence, we have a digital Fourier hologram. Points located on other planes may be out of focus, and can be reconstructed by applying additional Fresnel propagation of the form 1{H ( x, y )} Q(1/ zr ) , where 1 is the inverse FT and z r is the reconstruction distance. By recording a Fourier hologram the twin image problem is avoided if the object is properly positioned inside a half plane (e.g., ys 0 ). 3. Experiments and results
A FISCH system based on Fig. 1 was implemented using a Holoeye PLUTO SLM ( 1920 x 1080 pixels, 8 m pixel pitch, phase only modulation) and a PixelFly CCD ( 1280 x 1024 pixels, 6.7 m pixel pitch, monochrome). Other parameters of the system were: f 0 30cm, d 0 13cm and f1 17.5cm. Since the SLM is of a reflective type a small and negligible angle was introduced between it and the optical axis. A mirror was introduced at a distance of f1 after SLM 1 , so that it also functioned as SLM 2 . Finally, a beam splitter was introduced before SLM 1 to allow us to record information at the designated CCD position (Fig. 1) using a simple two-lens imaging system. Note that the above modification does not alter the working concept of the proposed system. The resolution test chart (Edmund Optics Negative NBS 1963A) was set at two different z s positions and was illuminated using a 150W Halogen lamp light filtered through a 650 20nm band pass filter. The hologram reconstructions are presented in Fig. 2. The first hologram was recorded with the resolution test chart positioned at zs f 0 30cm. Its reconstruction (Fig. 2 (a)) is simply the inverse discrete FT of the hologram. Notice that both the image and its twin are in sharp focus. The second hologram was recorded with the resolution test chart positioned at z s f 0 5cm 25cm. Three reconstruction planes are shown in Fig. 2 (b), (c) and (d) with a positive, zero and negative z r values, respectively, and clearly demonstrate the refocusing capability of the system. Note that the zero order term was not filtered in any manner. Yet, the images gray levels were linearly determined with accordance to the resolution chart image.
Fig. 2. Hologram reconstruction: (a) zs 30cm; (b)-(d) zs 25cm.
4. Summary
A new method for recording spatially incoherent digital Fourier hologram, FISCH, has been presented. It is based on a single channel incoherent interferometer and does not require any scanning nor mechanical movement. The results demonstrate the viability of FISCH and its ability to maintain the complete 3D information of an object. 5. References [1] T.-C. Poon, "Optical scanning holography - a review of recent progress," J. Opt. Soc. Korea 13, 406-415 (2009). [2] G. Brooker, N. Siegel, V. Wang, and J. Rosen, "Optimal resolution in Fresnel incoherent correlation holographic fluorescence microscopy," Opt. Express 19, 5047-5062 (2011). [3] G. W. Stroke and R. C. Restrick, "Holography with spatially noncoherent light," Appl. Phys. Lett. 7, 229 (1965). [4] B. Katz and J. Rosen, "Super-Resolution in Incoherent Optical Imaging using Synthetic Aperture with Fresnel Elements," Opt. Express 18, 962–972 (2010).
