Spatially incoherent off-axis Fourier holography without using spatial light modulator (SLM)

Dilband Muhammad; Cuong M. Nguyen; Jihoon Lee; Hyuk-Sang Kwon

doi:10.1364/OE.24.022097

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.

Fig. 1 Schematic of the dual path incoherent off-axis Fourier holographic (IOFH) system, RC resolution chart, BPF band pass filter, L_o, L₁ lenses with focal length f_o, f, BS beam spliter, M₁, M₂ mirrors, CCD charge-coupled device.

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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 L₁, is reflected back by mirror M₂, which is present at a focal distance from L₁. After passing through the lens, the reflected light finally reaches the camera. The other half beam, after being reflected from mirror M₁, interferes with the first beam at the camera.

To demonstrate this system, we considered a diverging beam coming from an off-axis point $p (\vec{r_{s}}, z_{s})$ where $\vec{r_{s}}$ is the radial position of point and z_s is the distance of point from lens L_o. The light field just after the lens L_o is given by

B (x, y; \vec{r_{s}}, z_{s}) = A_{s} c (\vec{r_{s}}, z_{s}) Q (\frac{1}{z_{s}}) L (\frac{- \vec{r_{s}}}{z_{s}}) \cdot Q (\frac{- 1}{f_{o}})

where Q(t) = exp[i2πtλ⁻¹(x² + y²)] and L(t) = exp[i2πtλ⁻¹(t_x x + t_yy)] are the quadratic and linear phase function, A_s is the amplitude and

c (\vec{r_{s}}, z_{s})

is a complex constant.

B (x, y; \vec{r_{s}} z_{s}) = A_{s} c (\vec{r_{s}}, z_{s}) L (\frac{- \vec{r_{s}}}{z_{s}}) Q (\frac{f_{o} - z_{s}}{f_{o} z_{s}}) = A_{s} c (\vec{r_{s}}, z_{s}) L (\frac{- \vec{r_{s}}}{z_{s}}) Q (\frac{1}{f_{s}})

where

f_{s} = \frac{f_{o} z_{s}}{f_{o} - z_{s}}

. The one beam which does not pass through L₁ and is reflected by M₁ on the surface of camera is given by:

B_{1} (x, y; \vec{r_{s}}, z_{s}) = A_{s} c (\vec{r_{s}}, z_{s}) L (\frac{- \vec{r_{s}}}{z_{s}}) Q (\frac{1}{f_{s}}) * Q (\frac{1}{d})

where d = d₁ + 2d₃ + d₄ and

* Q (\frac{1}{d})

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:

B_{1} (x, y; \vec{r_{s}}, z_{s}) = A_{s} c^{'} (\vec{r_{s}}, z_{s}) Q (\frac{1}{f_{s} + d}) L (\frac{- \vec{r_{s}} f_{s}}{z_{s} (f_{s} + d)})

The other beam reflected from the beam splitter, after passing through the lens L₁, is reflected back by mirror M₂ placed at its focus. This reflected beam at the surface of the camera is given by:

B_{2} (x, y; \vec{r_{s}}, z_{s}) = B * Q (\frac{1}{d_{12}}) \cdot Q (\frac{- 1}{2 f}) * Q (\frac{1}{f}) * Q (\frac{1}{d_{24}})

where d₁₂ = d₁ + d₂ and d₂₄ = d₂ + d₄. In this equation B is the field just after the lens L_o. So the first convolution

* Q (\frac{1}{d_{12}})

gives the field just before lens L₁, then product

\cdot Q (\frac{- 1}{f})

propagates field through lens L₁ of focal length f. The second convolution

* Q (\frac{1}{2 f})

describes light field propagation up to mirror M₂ then back to the same plane, where product

\cdot Q (\frac{- 1}{f})

shift field through lens L as shown in Fig. 1, and further convolution

* Q (\frac{1}{d_{24}})

results in a field on the surface of camera at distance d₂₄ from lens. Here we describe the step by step simplification of Eq. (5).

B_{2} (x, y; \vec{r_{s}}, z_{s}) = [A_{s} c^{'} (\vec{r_{s}}, z_{s}) L (\frac{- \vec{r_{s}}}{z_{s}}) Q (\frac{1}{f_{s}}) * Q (\frac{1}{d_{12}}) \cdot Q (\frac{- 1}{2 f}) * Q (\frac{1}{2 f}) \cdot Q (\frac{- 1}{f}) * Q (\frac{1}{d_{24}})]

B_{2} (x, y; \vec{r_{s}}, z_{s}) = [A_{s} c^{'} (\vec{r_{s}}, z_{s}) L (\frac{- \vec{r_{s}} f_{s}}{z_{s} (f_{s} + d_{12})}) Q (\frac{1}{f_{s} + d_{12}})] \cdot Q (\frac{- 1}{f}) * Q (\frac{1}{2 f}) \cdot Q (\frac{- 1}{f}) * Q (\frac{1}{d_{24}})

B_{2} (x, y; \vec{r_{s}}, z_{s}) = [A_{s} c^{'} (\vec{r_{s}}, z_{s}) L (\frac{- \vec{r_{s}} f_{s}}{z_{s} (f_{s} + d_{12})}) Q (\frac{f - f_{s} - d_{12}}{f (f_{s} + d_{12})})] * (Q (\frac{1}{2 f}) \cdot Q (\frac{- 1}{f}) * Q (\frac{1}{d_{24}})

B_{2} (x, y; \vec{r_{s}}, z_{s}) = [A_{s} c^{'} (\vec{r_{s}}, z_{s}) L (a) Q (\frac{1}{2 f})] * Q (\frac{1}{2 f}) \cdot Q (\frac{- 1}{f}) * Q (\frac{1}{d_{24}})

where

a = \frac{- \vec{r_{s}} f_{s}}{z_{s} (f_{s} + d_{12})}

and

\frac{1}{b} = \frac{f - f_{s} - d_{12}}{f (f_{s} + d_{12})}

.

B_{2} (x, y; \vec{r_{s}}, z_{s}) = [A_{s} c^{'} (\vec{r_{s}}, z_{s}) L (\frac{a b}{a + 2 f}) Q (\frac{1}{b + 2 f})] \cdot Q (\frac{- 1}{f}) * Q (\frac{1}{d_{24}})

B_{2} (x, y; \vec{r_{s}}, z_{s}) = [A_{s} c^{'} (\vec{r_{s}}, z_{s}) L (\frac{a b}{a + 2 f}) Q (\frac{- (b + f)}{f (b + 2 f)})] * Q (\frac{1}{d_{24}})

B_{2} (x, y; \vec{r_{s}}, z_{s}) = [A_{s} c^{'} (\vec{r_{s}}, z_{s}) L (E) Q (\frac{1}{F})] * Q (\frac{1}{d_{24}})

where

E = \frac{a b}{a + 2 f}

and

\frac{1}{F} = \frac{- (b + f)}{f (b + 2 f)}

.

B_{2} (x, y; \vec{r_{s}}, z_{s}) = A_{s} c^{'} (\vec{r_{s}}, z_{s}) L (\frac{E F}{F + d_{24}}) Q (\frac{1}{F + d_{24}})

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

I (x, y; \vec{r_{s}}, z_{s}) = {| B_{1} + B_{2} |}^{2}

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.

I (x, y; \vec{r_{s}}, z_{s}) = {A_{s}}^{2} ({c_{1}}^{2} + {c_{2}}^{2}) + B_{1} {B_{2}}^{*} + B_{2} {B_{1}}^{*}

Here $B_{1}^{*}$ and $B_{2}^{*}$ are complex conjugates of B₁ and B₂, 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_o. At this location b = − f, and both $f_{s} = \frac{f_{o} z_{s}}{f_{o} - z_{s}}$ and $F = \frac{- f (b + 2 f)}{(b + f)}$ approach infinity. Therefore, two quadratic terms in B₁ 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_s². 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

H (x, y) = ∭ I (x, y; \vec{r_{s}}, Z_{s}) d x_{s} d y_{s} d z_{s}

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 M₁ 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 = 200mm, d₁ = 23mm, d₂ = 23mm, d₃ = 175mm, d₄ = 30mm and f = 150mm. The light, after passing through the resolution chart, was filtered using a band pass filter (λ = 600nm, Δλ = 10nm).

First, we placed the resolution chart in the half plane and recorded two holograms one at focal length z_s = f_o = 200mm and the other at z_s = 175mm. 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 $L (\frac{2 \vec{r_{s}}}{f_{o}})$ dependence which, upon inverse Fourier transform, results in shift of the reconstructed image and twin image at $(\frac{\pm 2 r_{s} f_{r}}{f_{o}})$ where f_r 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 $z_{r} = (\frac{F + d_{24} - f_{s} - d}{(F + d_{24}) (f_{s} + d)}) f_{r}^{2}$ is required because of quadratic term contribution in Eq. (16) for off focus position more reconstruction process details are mentioned in [14].

The first hologram captured with RC in the half plane at z_s = f_o = 200mm and its reconstructed image is shown in Fig. 2, where both image and twin image are in focus.

Fig. 2 (a) A portion of the hologram captured with resolution chart (RC) at z_s = 200mm in half plane, (b) and its inverse Fourier transform, where both image and twin image are in focus.

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To demonstrate the three-dimensional imaging capabilities of our IOFH system the second hologram of the object in the half plane at z_s = 175mm 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).

Fig. 3 (a) A portion of the hologram with RC at z_s = 175mm (off focus) in half plane, in (b) and (c) its reconstruction depending on propagation direction if either image or twin image is in focus.

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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_o = 200mm, 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 M₁ (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.

Fig. 4 (a) A partial hologram with RC on-axis at z_s = 200mm (at focus) and (b) its inverse Fourier Transform, image and twin image coincide with zero order term. (c) A portion of the hologram at the same location as in (a) but with one tilted beam. (d) Inverse Fourier transform of (c), both image and twin image are in focus in the same plane at different locations depending on tilt.

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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 = 175mm, 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.

Fig. 5 (a) A portion of the hologram with on-axis RC at z_s = 175mm (off focus), in (b) and (c) its reconstruction depending on propagation direction either image or twin image is in focus

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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.

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Spatially incoherent off-axis Fourier holography without using spatial light modulator (SLM)

Abstract

1. Introduction

2. System design and methodology

3. Results and discussion

4. Conclusions

Funding

Acknowledgments

References and links

Cited By

Figures (5)

Equations (16)

Optics Express