The ideas of incoherent holography were conceived after the invention of coherent-light holography and their concepts seems indirectly related to it. In this work, we adopt an approach based on statistical optics to describe the process of recording of an incoherent-object hologram as a complex spatial coherence function. A Sagnac radial shearing interferometer is used for the correlation of optical fields and a Pockels cell is used to phase shift the interfering fields with the objective to quantify and to retrieve the spatial coherence function.
©2013 Optical Society of America
Since the invention of on-axis holography by Gabor , various attempts have been made to record a hologram of an incoherently illuminated or self-luminous object. The pioneering ideas from Mertz and Young  and later from Lohmann  were followed by a variety of experimental geometries that enables one to record the hologram of an incoherent object [4–12]. Generally, in incoherent holography, a copy of such an object is made with the help of an appropriate interferometer. The light emitted by each point on the object could interfere only with its counterpart in its copy. Depending on whether the observation plane lies in the Fresnel or Fourier domain, the field from each point of an incoherently illuminated object upon interfering with its counterpart from the copy, produces a distinct Fresnel zone plate and a sinusoidal fringe pattern respectively. The incoherent addition of such interference patterns from various points on the object generates the Fresnel or Fourier transform of the object which is termed as hologram of the object. Unlike in the case of coherent-light holography, due to the incoherent addition of the fringe patterns caused by all object points, an increasing dc bias is generated in the recorded hologram as the number of points on the object increases. This required recording devices with high dynamic range . The further development of incoherent holography was dedicated to remove the dc bias and conjugate image problem using phase shifting techniques [13–15]. With the availability of high resolution pixelated spatial light modulators (SLM), interferometry and phase shifting are implemented using dynamic diffractive optical elements displayed on them to record object information in Fresnel and Fourier regimes [16,17]. The above mentioned ideas of incoherent holography were conceived after the invention of coherent-light holography and their concepts seems indirectly related to it. Here we adopt an approach based on statistical optics for describing the recording of incoherent-object hologram as a complex spatial coherence function. Whereas the interference patterns resulting from all object points are incoherently added, the complex spatial coherence functions represented by those interference patterns are coherently superposed.
We first clarify how our approach is related to some of the existing techniques of holography. Long before the invention of holography, the Michelson stellar interferometer was proposed and implemented for the numerical reconstruction of a self-luminous object from its spatial coherence function detected with an interferometer . When applied to astronomical objects, it reconstructs an intensity distribution from the recorded coherence function. As alternative to incoherent holography, techniques such as gamma-holography , visible cone-beam tomography  and coherence holography [21–23] based on the principle of the Michelson stellar interferometer were also proposed for recording the 3-D information of incoherent objects [24,25]. In gamma-holography, exclusive use is made of spatially incoherent light for both recording and reconstruction of a hologram using a wavefront folding / rotational shear interferometer (RSI). The gamma-hologram is a real valued function encoded with spatial coherence function at far-field of the incoherently illuminated object. Later by recording a set of phase shifted interferograms for such geometry, the complex valued spatial coherence function itself is measured and used for reconstructing the object in visible cone-beam tomography. In coherence holography, even though the hologram is recorded using coherent-light, it is reconstructed with spatially incoherent light. In this case, the reconstructed object is also a complex valued function as it is represented by the spatial coherence function. In any case, these techniques emphasize that whenever the use of incoherent light is made for recording or reconstructing a hologram, the information recorded or reconstructed corresponds to the spatial coherence function of the field rather than the field itself.
In this work, we make use of the robust scheme provided by coherence holography for recording the incoherent-object hologram as spatial coherence function even with temporally broad band light using the Sagnac radial shearing interferometer . A Pockels cell is used for phase shifting the interfering fields with the objective to quantify and retrieve the spatial coherence function.
The principle behind the proposed technique is based on the van Cittert-Zernike theorem [26,27]. According to that theorem, the spatial coherence function measured at far-field from an incoherent source is proportional to the Fourier transform of the source’s planar intensity distribution. The theorem was later generalized and it was shown that it is well applicable to an extended source having an arbitrary surface . For holography and 3-dimensional imaging, we are specifically interested in a source distributed itself in 3-D space. Attempts have been made to extend the theorem for 3-D incoherent source . We will show that we do not require any complicated mathematical assumptions different to the basic concept of the van Cittert-Zernike theorem.
Consider an incoherently illuminated or self-luminous object emitting light with spectral scattering densityas shown in Fig. 1 . Let us assume that the section of the object lies on the x-y plane located at distance z = z’. is the spectral component of the optical field corresponding to wavelength emitted from each object point in this plane with as its instantaneous phase. The Fourier transform of this field performed with a lens L having a focal length f and being placed at z = f can be written asEq. (2), we can writeEq. (3), we have used the fact that the object is spatially incoherent for which we can assume a 2-D delta function for the object points on the z’ plane such thatFig. 1. However, the point sampling by sequential scanning is impractical for fast and mechanics-free measurement. To simultaneously obtain a 2-D correlation map with correlation lengths covering the required range of and, we propose the use of Sagnac radial shearing interferometer with a telescopic lens system with magnification [23,28]. With and due to the radial shear and, we measure as . The 3-D intensity distribution of the object can be retrieved in a way similar to back propagating a complex field using diffraction formula, given by:
3. Experiment and results
As shown in Fig. 2 (a) , the field distribution at the back focal plane of L1 with a focal length of 50mm is directed into a properly designed common-path Sagnac radial shearing interferometer comprising of a polarizing beam splitter (PBS) and mirrors M1, M2 and M3. The PBS at the input of the interferometer splits the incoming beam of light into two counter propagating beams with orthogonal states of polarization. The telescopic system with magnification = f3/f2 = 1.067, formed by lenses L2 (focal length f2 = 150mm) and L3 (focal length f3 = 160mm), introduced inside the interferometer gives a radial shear between the counter propagating beams as they travel through interferometer. At the output of the interferometer, where the focal planes of L2 and L3 meet, we have two radially-sheared copies of the field that are imaged onto an 8-bit CCD (Hitachi KP-2FA) with unit magnification using lens L4 with a focal length of 60mm. A polarizer P placed at the input of the interferometer makes the light polarized and also helps to balance the intensity of the radially sheared beams. Using an analyzer A with its axis kept at 45° to the orientation of the polarization of the two beams, interference between the two orthogonally polarized beams was achieved. Its fringe contrast and phase represent.
Due to the common path nature of the Sagnac interferometer, we achieve the interference of the optical field even with low temporal coherence light without using any interference filter. We use the electro-optic effect of Pockels cell (PC)  for phase-shifting orthogonally polarized radially sheared beams using a longitudinal KD*P crystal (LEYSOP model EM 510). We operate the Pockels cell in linear range and record 5 phase-shifted interferograms with equal phase steps and measure the fringe contrast and fringe phase using 5-step phase shifting algorithm .
Four different objects as shown in the top left corner in Fig. 2 (a) are chosen for presenting our results. Object 1 representing a self-luminous object consists of two light emitting diodes (LEDs), Luxeon Star LXHL-MMIC and LXHL-MMID having spectral width at half maximum of about 35nm at wavelength = 530nm kept at different depths in the front focal region of lens L1. Object 2 and object 3 representing a positive and negative transparent objects respectively consists of corresponding transparencies of numeral 3 having a size of about 2mm are placed at z = 1mm. They are illuminated by the LED, LXHL-MMID from behind. Object 4 representing an incoherently illuminated object is a metal spring illuminated with the LED from its right side. Figures 2(b)-2(e) show one of the interferograms recorded with objects 1, 2, 3 and 4 respectively. Figures 2(f)-2(i) show the corresponding fringe contrast and Figs. 2(j)-2(m) corresponding fringe phase jointly representing the complex spatial coherence function at the back focal plane of lens L1. The photographs of the 4 objects with different sections of the object in focus are shown in Figs. 3(a) -3(g) for comparison with the results presented in Fig. 4 .
Figures 4(a)-4(d) show single-frame excerpts from videos (Media 1, Media 2, Media 3, and Media 4) and describe a combined image of amplitude and phase of the reconstructed objects given by representing objects 1-4. The media files show the amplitude and phase of the object in x-y plane as we vary z from -5mm to + 5mm during the reconstruction process.
We presented the recording of an incoherent-object hologram as a complex spatial coherence function using a Sagnac radial-shearing interferometer, a Pockels cell and an 8-bit camera. 3-D object reconstruction can be achieved even in outdoor environment due to the inherent stability provided by the common path interferometer. Due to the implementation of phase shift using a Pockels cell, the system is mechanics free and has a potential for automated fast measurement applicable to dynamic situations. By changing the shearing parameter, we can tailor the range of measured coherence function depending on the object under test. This could enable one to use it as coherence zooming microscope.
D N Naik gratefully acknowledges financial support of the Alexander von Humboldt foundation.
References and links
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