## Abstract

We present a spatially incoherent dual path Fourier holographic system. Conceptually it is similar to Fourier incoherent single channel holography (FISCH). Although our incoherent off-axis Fourier holographic (IOFH) system does not have the robustness of a single channel system, it has three advantages over FISCH, with two being quite obvious from setup. First, no SLM is required, thus making the system simple and cost-effective. Second, it is capable of high light throughput because in FISCH, the use of SLM reduces light intensity in half by splitting one beam into two; furthermore, an analyzer is required to create interference which also reduces light intensity. The third advantage, which makes this IOFH system applicable even for on-axis samples (as opposed to samples in a half plane as is necessary for FISCH), is achieved by tilting one mirror. Here we demonstrate our system with a sample in half plane as in FISCH for different axial positions, and then by placing the object on an optical axis and tilting one mirror. The reconstructed images demonstrate holographic capabilities of our IOFH system for both on-axis and half plane sample locations.

© 2016 Optical Society of America

## 1. Introduction

Holography, first proposed by Gabor [1], allows us to capture the three dimensional information of an object. Initially, holography was mainly applied using coherent light sources, but lately incoherent holography is also being used.

Among the many incoherent holography techniques, one relatively slow method captures images from different directions to compute a hologram [2–4]. The requirement for capturing many images makes its use limited to non-moving systems. One type of incoherent Fresnel holography, known as scanning holography, generates a hologram by scanning an object with the Fresnel Zone Plate (FZP) pattern and then integrates the fluorescence produced for each scanning position by using a point detector, thus giving a correlation between FZP and the object [5,6]. The scanning process also makes this method slow, precluding its use for moving objects.

Two other methods for generating holograms are Fresnel Incoherent Correlation Holography (FINCH) and Fourier Incoherent Single Channel Holography (FISCH) [7–10]. Although FINCH does not require scanning, as is necessary for optical scanning holography, this type of imaging requires the capture of at least three holograms with different phase factors to produce one complex hologram free from zero-order and twin image disturbances using the phase shift method [11]. Fourier holography [12] has the advantage of having an increased space-bandwidth product [13]. Fourier incoherent single channel holography (FISCH) [14] was recently proposed and an enhanced version was described by reducing the optical path difference by utilizing two SLMs [15]. For both FINCH and FISCH the use of SLM to split a beam of light originating from one point into two beams with different curvatures gives them single channel robustness. Single channel systems are easy to align and have the advantage of robust system performance under various environmental changes (for example, a vibration-isolating environment is not necessary for creating a hologram, as is the case with dual-path systems). Two other methods for Fourier incoherent holography are based on radial and rotational shearing interferometry [16–18]. FISCH has combined the advantages of both radial shear interferometry (while retaining three-dimensional information) and rotational shear interferometry (both interfering beams have the same magnification) [15].

FINCH has already been implemented using a dual path setup without using SLM, at the cost of single channel robustness [19]. That dual path system also requires three holograms with different phases which are captured by a moving plane mirror. An improved version of this technique overcomes the limitation of acquiring three or more holograms to compute a complex hologram, and is known as single-shot self-interference incoherent digital holography (SS-SIDH) [20]. It requires only one hologram, and spatial filtering [21] in the Fourier domain is used to eliminate twin image and zero-order biased term contributions.

Here we describe a dual path incoherent off-axis Fourier holographic (IOFH) system. Conceptually it is similar to FISCH, although the use of dual paths makes it less robust than a single channel system. Still there are several advantages because 1) SLM is unnecessary, making the setup simple and cost-effective; 2) it is capable of high light throughput because in FISCH the use of SLM to split a single beam into two reduces light by fifty percent; also, an analyzer is required to create interference between orthogonal beams, causing further light reduction; 3) by tilting one mirror, this system becomes useful for on-axis samples, which was not possible with FISCH because of restrictions for placing objects in a half plane. As compared to SS-SIDH, which is equivalent to FINCH used to capture Fresnel holograms, our IOFH system is equivalent to FISCH that captures Fourier holograms with the advantage of increased space bandwidth product [13]. We believe that the potential of our IOFH system for capturing on-axis Fourier incoherent hologram will make it useful for Fourier incoherent holographic microscopy. The proposed system is shown in Fig. 1.

## 2. System design and methodology

An incoherent light from LED was used to illuminate the negative resolution target as shown in Fig. 1. Light scattered from the target, after passing through a band pass filter, was modified by lens *L _{o}*. The temporally filtered but spatially incoherent light, was split by a beam splitter. One light beam, after passing through lens

*L*

_{1}, is reflected back by mirror

*M*

_{2}, which is present at a focal distance from

*L*

_{1}. After passing through the lens, the reflected light finally reaches the camera. The other half beam, after being reflected from mirror

*M*

_{1}, interferes with the first beam at the camera.

To demonstrate this system, we considered a diverging beam coming from an off-axis point
$p\left(\overrightarrow{{r}_{s}},{z}_{s}\right)$ where
$\overrightarrow{{r}_{s}}$ is the radial position of point and *z _{s}* is the distance of point from lens

*L*. The light field just after the lens

_{o}*L*is given by

_{o}*Q*(

*t*) =

*exp*[

*i*2

*πtλ*

^{−1}(

*x*

^{2}+

*y*

^{2})] and

*L*(

*t*) =

*exp*[

*i*2

*πtλ*

^{−1}(

*t*

_{x}*x*+

*t*)] are the quadratic and linear phase function,

_{y}y*A*is the amplitude and $c\left(\overrightarrow{{r}_{s}},{z}_{s}\right)$ is a complex constant.

_{s}*L*

_{1}and is reflected by

*M*

_{1}on the surface of camera is given by:

*d*=

*d*

_{1}+ 2

*d*

_{3}+

*d*

_{4}and $*Q\left(\frac{1}{d}\right)$ represents a two dimensional convolution that propagates the field by distance

*d*in free space. Now by applying the methods described in [22], the above Eq. (3) can be simplified as follows:

The other beam reflected from the beam splitter, after passing through the lens *L*_{1}, is reflected back by mirror *M*_{2} placed at its focus. This reflected beam at the surface of the camera is given by:

*d*

_{12}=

*d*

_{1}+

*d*

_{2}and

*d*

_{24}=

*d*

_{2}+

*d*

_{4}. In this equation

*B*is the field just after the lens

*L*. So the first convolution $*Q\left(\frac{1}{{d}_{12}}\right)$ gives the field just before lens

_{o}*L*

_{1}, then product $\cdot Q\left(\frac{-1}{f}\right)$ propagates field through lens

*L*

_{1}of focal length

*f*. The second convolution $*Q\left(\frac{1}{2f}\right)$ describes light field propagation up to mirror

*M*

_{2}then back to the same plane, where product $\cdot Q\left(\frac{-1}{f}\right)$ shift field through lens

*L*as shown in Fig. 1, and further convolution $*Q\left(\frac{1}{{d}_{24}}\right)$ results in a field on the surface of camera at distance

*d*

_{24}from lens. Here we describe the step by step simplification of Eq. (5).

The intensity captured by the camera resulting from interference of two beams is given by

This equation also includes the contributions from twin image and zero-order terms, but because of the off-axis position of the object they appear in the reconstruction plane at different locations. Later, we will also include the effects of tilting one mirror which results in separation of these images even for on-axis objects.

Here
${B}_{1}^{*}$ and
${B}_{2}^{*}$ are complex conjugates of *B*_{1} and *B*_{2}, respectively. Next, to show that the hologram is captured in the Fourier plane we analyzed the third term of Eq. (15) at *z _{s}* =

*f*. At this location

_{o}*b*= −

*f*, and both ${f}_{s}=\frac{{f}_{o}{z}_{s}}{{f}_{o}-{z}_{s}}$ and $F=\frac{-f(b+2f)}{(b+f)}$ approach infinity. Therefore, two quadratic terms in

*B*

_{1}and ${B}_{2}^{*}$ do not contribute and only one linear term, which is a function of the object position and its intensity is proportional to

*A*

_{s}^{2}. These contributions all together define the Fourier transform hologram. The total hologram is an incoherent summation of contribution by all points and is given by

One advantage of the Fourier transform hologram is that it requires a one-step reconstruction process when the object is at focus; otherwise, it requires further prorogation after inverse Fourier transform when the object is not at focus, as explained in FISCH [14]. Next, in order to incorporate the tilt effect of mirror *M*_{1} only one linear term is added to Eq. (4) making the system equivalent to off-axis. This additional linear phase term, according to the Fourier shift theorem, results in separation of all three terms upon reconstruction even when the object is on-axis.

## 3. Results and discussion

The implementation of our system is shown in Fig. 1. A white mounted LED (MWWHL3, Thor-labs, NJ, USA) was used to illuminate the negative resolution chart (RC). The other parameters of the systems are *f _{o}* = 200

*mm*,

*d*

_{1}= 23

*mm*,

*d*

_{2}= 23

*mm*,

*d*

_{3}= 175

*mm*,

*d*

_{4}= 30

*mm*and

*f*= 150

*mm*. The light, after passing through the resolution chart, was filtered using a band pass filter (

*λ*= 600

*nm*, Δ

*λ*= 10

*nm*).

First, we placed the resolution chart in the half plane and recorded two holograms one at focal length *z _{s}* =

*f*= 200

_{o}*mm*and the other at

*z*= 175

_{s}*mm*. As explained in FISCH and above, when the object is at focal length, each point of the object is mapped in a hologram plane with linear phase $L\left(\frac{2\overrightarrow{{r}_{s}}}{{f}_{o}}\right)$ dependence which, upon inverse Fourier transform, results in shift of the reconstructed image and twin image at $\left(\frac{\pm 2{r}_{s}{f}_{r}}{{f}_{o}}\right)$ where

*f*is focal length of the reconstruction lens. The reconstruction of object placed at focus requires only inverse Fourier transform, whereas for out of focus position it needs further field propagation in free space after performing inverse Fourier transform. This further propagation through the distance ${z}_{r}=\left(\frac{F+{d}_{24}-{f}_{s}-d}{(F+{d}_{24})({f}_{s}+d)}\right){f}_{r}^{2}$ is required because of quadratic term contribution in Eq. (16) for off focus position more reconstruction process details are mentioned in [14].

_{r}The first hologram captured with RC in the half plane at *z _{s}* =

*f*= 200

_{o}*mm*and its reconstructed image is shown in Fig. 2, where both image and twin image are in focus.

To demonstrate the three-dimensional imaging capabilities of our IOFH system the second hologram of the object in the half plane at *z _{s}* = 175

*mm*and its reconstruction are shown in Fig. 3. The captured hologram is shown partially in Fig. 3(a). After reconstruction and depending on the propagation direction, only the image or twin image is in focus, as shown in Figs. 3(b) and 3(c).

In order to elaborate the potential of the IOFH system to capture holograms of on-axis samples, we first placed the object on-axis at *z _{s}* =

*f*= 200

_{o}*mm*, with the resulting hologram and its reconstruction shown in Fig. 4. It is clear from the blurriness of Fig. 4(b) that all the three terms are at the same location. Next, we introduced a tilt in one beam by tilting mirror

*M*

_{1}(1°) and this contributed an additional linear phase which, upon reconstruction, shifts the three terms at different locations. A portion of the captured hologram, after tilting the mirror, is shown in Fig. 4(c). It is clear from the image reconstruction in Fig. 4(d) that both the image and its twin image are in focus at different locations in the same plane.

To demonstrate the on-axis three-dimensional holographic capability of our system we have captured the hologram partially shown in Fig. 5(a). This image was captured with RC on-axis at *z _{s}* = 175

*mm*, and it is clear from the image reconstruction in Figs. 5(b) and 5(c) that both the image and twin image are in focus in different planes.

## 4. Conclusions

A method to capture Fourier incoherent holograms, based on the principal of FISCH but without using SLM, was demonstrated. Despite lacking single channel robustness, our IOFH system is simple, cost-effective, and has the additional benefit of having high light throughput. The extra capability of our IOFH system for capturing on-axis holograms was also demonstrated. We believe that our simpler IOFH system has great potential for some practical applications such as incoherent holographic microscopy, adaptive optics for compensating aberrations in optical path, imaging for dynamic samples due to its high temporal resolution, and three dimensional fluorescence imaging.

## Funding

New Growth Engine Industry Project of the Ministry of Knowledge and Economy (No.10047579); National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2015R1A2A2A03005382); the GIST Research Institute (GRI) in 2016.

## Acknowledgments

The authors thank Prof. Michael Ye at the Gwangju Institute of Science and Technology (GIST) for carefully proofreading the manuscript.

## References and links

**1. **D. Gabor, “A new microscopic principle,” Nature **161**(4098), 777–778 (1948). [CrossRef] [PubMed]

**2. **Y. Li, D. Abookasis, and J. Rosen, “Computer-generated holograms of three-dimensional realistic objects recorded without wave interference,” Appl. Opt. **40**(7), 2864–2870 (2001). [CrossRef]

**3. **Y. Sando, M. Itoh, and T. Yatagai, “Holographic three-dimensional display synthesized from three-dimensional Fourier spectra of real existing objects,” Opt. Lett. **28**(24), 2518–2520 (2003). [CrossRef] [PubMed]

**4. **N. T. Shaked and J. Rosen, “Multiple-viewpoint projection holograms synthesized by spatially incoherent correlation with broadband functions,” J. Opt. Soc. Am. A **25**(8), 2129–2138 (2008). [CrossRef]

**5. **T. C. Poon and A. Korpel, “Optical transfer function of an acousto-optic heterodyning image processor,” Opt. Lett. **4**(10), 317–319 (1979). [CrossRef] [PubMed]

**6. **J. Rosen, G. Indebetouw, and G. Brooker, “Homodyne scanning holography,” Opt. Express **14**(10), 4280–4285 (2006). [CrossRef]

**7. **J. Rosen and G. Brooker, “Digital spatially incoherent Fresnel holography,” Opt. Lett. **32**(8), 912–914 (2007). [CrossRef] [PubMed]

**8. **G. Brooker, N. Siegel, V. Wang, and J. Rosen, “Optimal resolution in Fresnel incoherent correlation holographic fluorescence microscopy,” Opt. Express **19**(6), 5047–5062 (2011). [CrossRef] [PubMed]

**9. **J. Rosen, N. Siegel, and G. Brooker, “Theoretical and experimental demonstration of resolution beyond the Rayleigh limit by FINCH fluorescence microscopic imaging,” Opt. Express **19**(27), 26249–26268 (2011). [CrossRef]

**10. **B. Katz, J. Rosen, R. Kelner, and G. Brooker, “Enhanced resolution and throughput of Fresnel incoherent correlation holography (FINCH) using dual diffractive lenses on a spatial light modulator (SLM),” Opt. Express **20**(8), 9109–9121 (2012). [CrossRef] [PubMed]

**11. **I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Express **22**(16), 1268–1270 (1997).

**12. **G. W. Stroke and R. C. Restrick III, “Holography with spatially noncoherent light,” Appl. Phys. Lett. **7**(9), 229–231 (1965). [CrossRef]

**13. **J. W. Goodman, *Introduction to Fourier Optics* (Roberts and Company Publishers, 2005).

**14. **R. Kelner and J. Rosen, “Spatially incoherent single channel digital Fourier holography,” Opt. Lett. **37**(13), 3723–3725 (2012). [CrossRef] [PubMed]

**15. **R. Kelner, J. Rosen, and G. Brooker, “Enhanced resolution in Fourier incoherent single channel holography (FISCH) with reduced optical path difference,” Opt. Express **21**(17), 20131–20144 (2013). [CrossRef] [PubMed]

**16. **D. N. Naik, G. Pedrini, and W. Osten, “Recording of incoherent-object hologram as complex spatial coherence function using Sagnac radial shearing interferometer and a Pockels cell,” Opt. Express **21**(4), 3990–3995 (2013). [CrossRef] [PubMed]

**17. **O. Bryngdahl and A. Lohmann, “Variable magnification in incoherent holography,” Appl. Opt. **9**(1), 231–232 (1970). [CrossRef] [PubMed]

**18. **Y. Wan, T. Man, and D. Wang, “Incoherent off-axis Fourier triangular color holograph,” Opt. Express **22**(7), 8565–8573 (2014). [CrossRef] [PubMed]

**19. **M. K. Kim, “Adaptive optics by incoherent digital holography,” Opt. Lett. **37**(13), 2694–2696 (2012). [CrossRef] [PubMed]

**20. **J. Hong and M. K. Kim, “Single-shot self-interference incoherent digital holography using off-axis configuration,” Opt. Lett. **38**(23), 5196–5199 (2013). [CrossRef] [PubMed]

**21. **E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. **39**(23), 4070–4075 (2000). [CrossRef]

**22. **B. Katz and J. Rosen, “Super-resolution in incoherent optical imaging using synthetic aperture with Fresnel elements,” Opt. Express **18**(2), 962–972 (2010). [CrossRef] [PubMed]