## Abstract

We present a new method for recording off-axis digital Fourier holograms of three-dimensional objects under spatially incoherent illumination. The method is implemented by modifying the optical configuration of triangular interferometer. The recording properties and 3D reconstruction ability of the proposed method are investigated theoretically and experimentally. Multicolor holographic recording and reconstruction of spatially incoherent illuminated object are achieved by using the proposed off-axis Fourier triangular interferometer and monochromatic digital camera. Only three holograms are sufficient to rebuild a color image without zero-order and twin image disturbing effect. Combining with some image fusion skills during reconstruction, the reconstructed color images with satisfied quality are demonstrated.

© 2014 Optical Society of America

## 1. Introduction

Incoherent holography has been considered a valuable technique for 3D imaging since the possibility of making holograms of objects illuminated with incoherent light was proposed by Mertz and Young [1]. Based on the wave nature of light, the incoherent holography is usually explained as the superposition of Fresnel zone patterns. Each Fresnel zone pattern is formed by the interference of twin beams originating from the same point on an incoherent object [2]. The theory and practice of incoherent holography have been noteworthily developed in recent years [3–12]. Several schemes have been presented for incoherent holographic recording such as triangular holography [4], conoscopic holography [5], Fresnel incoherent correlation holography (FINCH) [6, 7] and so on [8–11]. As it is known, three dimensional color imaging has great importance in digital holography and color digital holography have been paid more attention [12–14]. However, the triangular color holography presented in this paper should be differently categorized because incoherent holographic recording set-up and spatially incoherent light source are employed.

The combination of incoherent holography and color holography is an attractive research focus as it expanding the modality of both the techniques. Fluorescence color holography has been realized by FINCH which recording the phase-shifting holograms for each fluorescent color and further combining them in computer to produce a 3D color image [7]. Full color holographic recording of object under natural light illumination is implemented by an optical system based on self-interference with different curvature [8]. The recording speed of the systems is also limited as a large number of exposures are required for the color imaging. The temporal resolution is decreased by the use of phase-shifting technology in the above two incoherent color holographic imaging techniques. Triangular Holography (TH) is another typical method to achieve holographic recording of spatially incoherent objects [4]. Some valuable research works have been presented in recent years [15–17]. The resolution, suppression of twin image and phase error analysis has already been paid attention. However any other performances of triangular holography have not been exploited yet, especially as a potential technique of 3D color imaging.

In this paper, the triangular optical configuration are modified by titling the mirror and introducing another Fourier transform lens, which results in recording off-axis Fourier digital holograms. Under single wavelength illumination, 3D properties of the incoherent object can be retrieved from one single-shot hologram using the proposed method. Furthermore, we present color holographic imaging of 3D incoherent objects using the novel off-axis Fourier triangular holography. Three holograms are recorded respectively when the object is illuminated by tricolor (red, green and blue) light. The three quasi-monochromatic holograms are resized and reconstructed, and three quasi-monochromatic reconstructed images with identical size are acquired, and the color image of the original object is acquired by synthesizing the three reconstructed images. We coin our method Incoherent Fourier Triangular Color digital Holography (IFTCH). Some image fusion skill during reconstruction are combined with IFTCH, the reconstructed color images with satisfied quality are demonstrated. Fourier digital holograms have some inherent advantages such as relatively easy and fast reconstruction process and making the most efficient use of the space-bandwidth product of the hologram [2]. Furthermore, in Fourier digital holography the transverse position of the reconstructed images can be accurately assured and making it easier to synchronize the resized reconstructed images in color holography [14]. Therefore, we believe that our method holds some unique advantages for color holographic imaging of 3D incoherent objects with its capability of recording off-axis Fourier color holograms under incoherent illumination and retrieving 3D information.

## 2. Principle and methodology

In the IFTCH system, spatially incoherent narrowband light sources with low temporal coherence are employed to illuminate the object. The simplified schematic of the IFTCH system is shown in Fig. 1, where Fig. 1(a) is the basic optical configuration and Fig. 1(b) is the adaptation of IFTCH for detail analysis. Light scattered from the object is collimated by the lens L0, continues to propagate through a triangular interferometer by clockwise and counterclockwise passage respectively, which is composed of polarizing beam splitter PBS, lens L1 and L2. The light travels along clockwise and counter-clockwise are shown as blue solid and red dashed line, respectively. By introducing the polarizer P1 and PBS, the intensity ratio between the light travels through the two passages can be adjusted. The two beams are combined by the PBS again and interfering with each other after passing through the polarizer P2. The interference pattern is imaged onto the CCD plane by using a positive lens L3.

To briefly, an off-axis point source is considered as the object as shown in Fig. 1(b), which has an amplitude of *A*_{s} and is positioned at (*x _{s}*,

*y*

_{s},

*z*

_{s}), a distance

*z*

_{s}to the lens L0. The light field over the L0 plane has the form of

*Q*(

*b*) = exp[

*jπbλ*

^{−1}(

*x*

^{2}+

*y*

^{2})],

*L$(\overrightarrow{b})$*= exp[

*j2πλ*

^{−1}(

*b*+

_{x}x*b*)] and ${\overrightarrow{r}}_{s}$ = (

_{y}y*x*,

_{s}*y*).

_{s}*c*(${\overrightarrow{r}}_{s}$,

*z*) =

_{s}*z*

_{s}^{-}^{1}exp(

*j2πλ*

^{−1}

*z*)exp[

_{s}*jπλ*

^{−1}

*z*

_{s}^{−1}(

*x*

_{s}^{2}+

*y*

_{s}^{2})] is a complex constant dependent on the longitudinal and lateral position of the point object,

*λ*is the central wavelength of the recording light. After passing through the lens L0 and a free space propagation of distance

*f*

_{0}, the field on the P1 plane has the form of

*c*(${\overrightarrow{r}}_{s}$,

^{′}*z*) is a complex constant and

_{s}*f*=

_{s}*z*

_{s}f_{0}/(

*f*

_{0}-

*z*). The light is divided into two parts by PBS, where this two parts of light passing through a 4

_{s}*f*system along opposite directions. The light field on P1 plane is imaged onto P2 plane, with magnification of

*α*= -

*f*

_{1}/

*f*

_{2}by clockwise light passage (solid blue line in Fig. 1) and magnification of 1/

*α*= -

*f*

_{2}/

*f*

_{1}by counterclockwise light passage (dashed red line in Fig. 1) through the 4

*f*system, respectively. On P2 plane, we have two radially sheared copies of the field

*u*, which can be described as

_{i}*u*and

_{c}*u*

_{cc}*L*(${\overrightarrow{r}}_{c}$, ${\overrightarrow{r}}_{s}$) and

*L*(${\overrightarrow{r}}_{cc}$, ${\overrightarrow{r}}_{s}$) are the linear phase functions that are introduced by the tilt of M1,

*c*

^{′′}(${\overrightarrow{r}}_{s}$,

*z*) and

_{s}*c*

^{′′′}(${\overrightarrow{r}}_{s}$,

*z*) are complex constants. The interference pattern of the two fields on P2 plane is imaged onto CCD with unit magnification using L3, and the intensity on CCD plane is given by

_{s}*L*($\overrightarrow{r}$,${\overrightarrow{r}}_{s}$) is the linear phase function that introduced by the tilted mirror M1,

_{c}*c*

_{1}and

*c*

_{2}are constants;

*c*.

*c*. is the complex conjugate of the left term inside the brace. For a complex object composed of many spatially incoherent points, the recorded hologram is a incoherent summation over all object points’ contribution. Under the quasi-monochromatic approximation of the light source, the recorded hologram is

If the object point is located at the front focal plane of lens L0 (*z _{s}* =

*f*

_{0}), then Eq. (5) is reduced to

According to Eq. (7), the linear phase function *L*(${\overrightarrow{r}}_{s}$) is just depends on ${\overrightarrow{r}}_{s}$ and its amplitude is proportion to the intensity *A _{s}*

^{2}of the point source. That is, this special relationship between the object and the recorded hologram exactly defines a 2D Fourier transform (FT) of the points, thus Fourier holograms are recorded. The points located at the front focal plane of the L0 can be reconstructed by calculating the 2D inverse FT of the hologram. Furthermore, the reconstructed image can be spatially separated from its twin image and the zero order in the frequency spectrum domain owe to the additional linear phase function

*L*($\overrightarrow{r}$,${\overrightarrow{r}}_{s}$). The points located at other planes (

_{c}*z*≠

_{s}*f*

_{0}) can be reconstructed by calculating the 2D inverse FT and an additional Fresnel propagation. This procedure can be expressed as

^{−1}denotes the 2D inverse FT.

*β*is a scaling operator, and if a Fourier transforming lens of focal length

*f*is used to optically reconstruct the hologram,

_{r}*β*= 1/

*λ*, where the

_{r}f_{r}*λ*is the central wavelength of the reconstruction light. The reconstruction distance

_{r}*z*can be calculated as

_{r}It can be deduced from Eq. (5) that, the reconstructed image and its twin image are in focus respectively when the sign of Eq. (9) is chosen as negative and positive.

As it is known that, color holography can be achieved by synthesizing three holograms of the same object captured using tricolor (red, green and blue) light source. In order to obtain an accurately fused result, the reconstructed images of the tricolor holograms should be identical in size. As the magnification of the system is related to the wavelength and the recording distance, to ensure the reconstructed images have the same size. If the holograms are assumed to be squared and composed of *N* × *N* pixels, the following equation should be satisfied

*N*

_{1}

^{2},

*N*

_{2}

^{2}and

*N*

_{3}

^{2}are the pixel number of the three quasi-monochromatic holograms after being padded with zeros.

*λ*

_{1},

*λ*

_{2}and

*λ*

_{3}are the recording wavelengths;

*D*

_{1},

*D*

_{2}and

*D*

_{3}are the corresponding recording distances. Furthermore, the quality of the color synthesized reconstructed image is affected by the spatially mismatch of the three quasi-monochromatic images. This distortion is inherently caused by the zero padding procedure which stretching the three holograms in the frequency domain, and the chromatic aberrations induced by the optical elements. To ensure good quality of the reconstructed color images during synthesis, the green and blue images are shifted to achieve a better overlap with the red image. And the quantitative values of shift are evaluated by Correlation Coefficients (CC) method, where the CC of two images

*I*

_{1}and

*I*

_{2}is defined as

*r*[18]

*Iv*

_{1}and

*Iv*

_{2}is the vectorized form of

*I*

_{1}and

*I*

_{2}respectively;

*n*is the length of the vector

*Iv*

_{1}and

*Iv*

_{2};

*σ*(

*Iv*) and

*E*(

*Iv*) are the standard deviation and average value operators, respectively.

To demonstrate the 3D reconstruction ability and the validity for color holographic imaging of the IFTCH, the experiments on recording off-axis Fourier holograms of 3D incoherent objects under single-wavelength illumination and three different color illuminating light, are implemented respectively.

## 3. Experiments

The IFTCH system is implemented using an optical setup shown in Fig. 2. A Newport arc lamp (Oriel Research Arc Lamp Source 66477, 150 W) is used as the light source and a Thorlabs DCU224M CCD (1280 × 1024 pixels, 4.65μm pixel pitch, and monochrome) is used to record the holograms. Other parameters in the system are *f*_{0} = 150mm, *f*_{1} = 150mm, *f*_{2} = 175mm, *f*_{3} = 100mm. The light source is filtered through an interference filter (Newport 10BPF10-532, 532nm central wavelength, 10nm full width at half maximum bandwidth). And illuminates two dices (four dots plane of the front dice and three dots plane of the behind dice are face to the CCD) being placed at *z _{s}* =

*f*

_{0}= 150mm. The P1 and P2 are inserted by 45° respect to the optical axis.

Single-shot hologram is captured and digitally reconstructed based on Eq. (8). The reconstructed image can be spatially separated from the zero order and the twin image. Thus the speed of the reconstruction process of IFTCH is very fast by simply calculating the 2D fast FT of the holograms and then extracting the + 1 order frequency spectrum component with an appropriate filter. Part of the recorded hologram and its reconstructed image are shown in the Fig. 3(a) and Fig. 3(b), respectively. Specklelike noise in the Fig. 3(b) is mainly caused by the snap-shot noise of CCD. In order to obtain high quality reconstructed images, multiple holograms are recorded sequentially and the reconstructed images are superposed. The improved reconstructed image by superposing 10 reconstructed images is shown in Fig. 3(c). Comparing with Fig. 3(b), the signal to noise ratio (SNR) is nearly doubled. The dependence of the SNR on the number of the images being superposed is shown in Fig. 3(d).

To demonstrate the IFTCH capability of maintaining 3D information, hologram is recorded with a die positioned at *z _{s}* = 50mm. The hologram is shown in Fig. 4(a), and Fig. 4(b) is the 2D FT of the recorded hologram. The real image ( + 1 order) and its twin image (−1 order) are both obviously out of focus in Fig. 4(b) due to the existence distance between the object plane and the front focal plane of lens L0. The reconstructed image at the best plane of focus of real image is shown in Fig. 4(c), after being propagated an additional distance of

*z*digitally, where

_{r}*z*is calculated according to Eq. (9). The twin image is in focus as shown in Fig. 4(d) if the additional propagation distance is -

_{r}*z*.

_{r}As it is mentioned in the introduction, the transverse position of the reconstructed images from Fourier digital hologram can be accurately assured and making it easier to synchronize the resized reconstructed images in color holography. Additionally, the proposed IFTCH present added advantages for the application of color holography owe to the capability of recording incoherent off-axis Fourier holograms. Fourier holograms are recorded and reconstructed with three different color illuminating light, respectively. The color image of the original object is acquired by resizing and synthesizing the quasi-monochromatic reconstructed images. In the experiments, the recorded hologram is transformed to a square matrix (1024 × 1024) to avoid the vision aberration, and the pixel numbers of the three quasi-monochromatic holograms are calculated according to Eq. (10). As the lenses in the optical setup are wavelength sensitive, the recording distances (distance from CCD to L3) for different color illuminations are changed during the recording procedure to ensure the accurate Fourier transform relationship between the object and the holograms. In the configuration of Fig. 2, the red channel is recorded at the distance of *D*_{1} = 66mm with central wavelength *λ*_{1} = 650nm and Full Width at Half Maximum (FWHM) bandwidth of 25nm, the green channel is recorded at the distance of *D*_{2} = 68mm with wavelength *λ*_{2} = 533nm and FWHM bandwidth of 10nm, the blue channel is recorded at the distance of *D*_{3} = 70mm with wavelength *λ*_{3} = 465nm and FWHM bandwidth of 30nm. After padded with zeros, the size of the hologram is 1227 × 1227 pixels for red channel, 1140 × 1140 pixels for green channel and 1024 × 1024 pixels for blue channel.

To achieve a better overlap of the three quasi-monochromatic images, the Correlation Coefficient (CC) r being defined as Eq. (11) is optimized by shifting the images. The CC between the Green and Red images is maximized by shifting the Green image along the horizontal direction, as shown in Fig. 5(a). Optimized vertical shift value is evaluated and shown in Fig. 5(b) using the optimized horizontal shift value. The same procedure is implemented between the Blue and Red images as shown in Fig. 5(c) and Fig. 5(d). The optimized Green-Red and Blue-Red shift values are used to achieve a better overlapping for synthesizing the color image. The reconstructed images at each wavelength and the fused color image are shown in Fig. 6.

## 4. Conclusion and discussion

We have proposed and demonstrated a method of realizing 3D color by incoherent off-axis Fourier triangular digital holography, which is coined as IFTCH. The theoretical analysis and the experimental results show the 3D reconstruction abilities and advantages for color imaging of the method. Color holographic imaging of the spatially incoherent object was achieved by IFTCH. As the quality of the reconstructed images is restricted by the shot noise of the digital camera, ten holograms is capture and the corresponding reconstructed images are averaged. Actually, this kind of noise can be suppressed effectively by adopting a cooled camera.

The reconstruction procedure was investigated in detail. Quasi-monochromatic reconstructed images with same size were acquired, as holograms for each color channel were padded with zeros according to the corresponding illumination wavelengths and recording distances. To ensure good quality of the reconstructed color images and achieving a better overlap during synthesis, the spatial positions of the quasi-monochromatic images were shifted. The quantitative shift values were evaluated by maximizing the CC value. The proposed methods shows it great potential in some practical applications such as three-dimensional spectral imaging, color holographic imaging for dynamic samples and the three-dimensional fluorescence imaging.

## Acknowledgment

This work was partially supported by National Science Foundation of China under Grant No. 61107002 & No. 61077004, the Scientific Research Project from Beijing Municipal Education Committee under Grant No.km201410005032, Beijing Municipal Party Committee Organization Department Q0006111201401 and the Basic Research Foundation of Beijing University of Technology under Grant No. X4006111201301.

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