Resolution enhancement in nonlinear interferenceless COACH with point response of subdiffraction limit patterns

Mani Ratnam Rai; A. Vijayakumar; Yusuke Ogura; Joseph Rosen

doi:10.1364/OE.27.000391

1. Introduction

The search for imaging technologies with an improved lateral resolution is ongoing for about two centuries [1–3]. The main obstacle towards reaching higher imaging resolution in the optical regime is the diffraction limit [4]. Higher resolving power can be achieved by use of electromagnetic sources with smaller wavelengths, resulting in an improved resolution limit. This research direction of resolution improvement has inspired the developments of microscopes in the spectral regime of ultraviolet [5] and x-ray [6]. However, such sources are limited to certain applications. Techniques, such as structured illumination, suffer from other limitations, as several images of the same object illuminated by several different gratings must be captured to improve the resolution of a single image [7]. Notable fluorescence imaging techniques such as stochastic optical reconstruction microscopy (STORM) [8] and stimulated emission depletion microscopy (STED) [9], are limited for imaging fluorescent specimens in a specific time-scheme. Other computational techniques such as Fourier ptychography [10], pixel super-resolution [11], synthetic aperture [12], and compressive imaging [13] make use of aperture engineering and digital post-processing. However, the computational load, complexity and low time resolution limit the use of these techniques to static scenes.

Incoherent digital holography techniques such as Fresnel incoherent correlation holography (FINCH) [14–16], Fourier incoherent single channel holography (FISCH) [17] and self-interference digital holography (SIDH) [18], have better imaging resolution than direct imaging with the same numerical aperture. FINCH has also been used as a platform for applying other super-resolution techniques such as synthetic aperture system [19], structured illumination microscopy [20] and imaging with an input diffuser [21]. However, the need for two-beam interference with a perfect overlap between them decreases the robustness of FINCH. In 2016, an incoherent digital holography technique termed coded aperture correlation holography (COACH) was developed to overcome the axial resolution limit of FINCH [22]. COACH and FINCH belong to the same category of self-interference holography systems, but with the dissimilarity of having a different optical aperture. In FINCH, the object wave is modulated by a quadratic phase mask, while in COACH the same wave is modulated by a pseudorandom coded phase mask (CPM). The pseudorandom CPM improves the axial resolution of COACH compared to FINCH, while exhibiting a lower lateral resolution than FINCH. The optical characteristics of COACH was found to be similar to that of lens-based direct imaging, but with the inherent capability of 3D imaging typical to general digital holography.

Unlike FINCH, in COACH the 3D location of the object is encoded not only in the phase of the object wave modulated by the CPM but also in its intensity distribution. This property has enabled to record holograms without two-beam interference, a fundamental requirement for recording phase distributions. Since two-beam interference is no longer necessary to record holograms of 3D objects, a simpler version of COACH called interferenceless COACH (I-COACH) could be applied [23]. I-COACH exhibits the same resolution characteristics as that of COACH and of lens-based direct imaging. Moreover, I-COACH requires at least two camera shots to reconstruct an image of the object with an acceptable signal-to-noise ratio (SNR). I-COACH with an adaptive non-linear reconstruction technique has been developed recently to reconstruct objects with a single camera shot, without compromising the SNR [24,25].

In general, the image in I-COACH is reconstructed by a cross-correlation between two response patterns, the response to the object and the response to a single point. Therefore, the smallest image point that can be resolved has the size of the correlation path, which is equal to the smallest feature size in each of the correlated patterns. The smallest feature size on the recorded pattern is inversely proportional to the diameter of the system aperture. Hence, in ordinary circumstances, the resolution limits of I-COACH and direct imaging system are the same and are inversely proportional to the diameter of the system aperture. On the other hand, if the smallest feature size of the response patterns can be reduced without increasing the aperture size and the NA, one can improve the resolution of I-COACH in comparison to a direct imaging system with the same NA. Recently, a modified Gerchberg-Saxton algorithm (GSA) [26] has been developed to design diffractive fan-out elements for the generation of subdiffraction spots in a single as well as multiple planes [27,28]. In the present study, we propose to use this technique of creating subdiffraction spots by a single CPM. The goal of this procedure is to obtain point response pattern on the camera in a shape of a bunch of spots, each of which is smaller than a regular diffraction-limited spot. In other words, we propose to engineer the CPMs of I-COACH for generating subdiffraction spots with smaller feature size than can be achieved by a regular CPM. The expected benefit of such technique is some reduction in the correlation path and thus improvement of the lateral resolution beyond the diffraction limit, without increasing the system aperture size and its NA. Hence, we term the new technique Resolution Enhanced COACH (RE-COACH). The technique has many advantages compared to other super resolution techniques. It helps to retain the interferenceless and motionless characteristics of I-COACH with a configuration as simple as that of regular lens-based imaging. Moreover, the technique does not demand any time-consuming multiple intensity recordings and complex computational procedures. Note that the techniques of I-COACH [23,24] and of generating subdiffraction spots [27,28] are connected by the length of the correlation path. Explicitly, the modified method of creating subdiffraction spots originally suggested in [27,28] reduces the characteristic correlation path of the intensity responses in I-COACH [23,24]. Hence, the shorter correlation paths are translated to a resolution improvement in comparison to direct and to traditional I-COACH imaging.

2. Methodology

The optical configuration of RE-COACH is identical to that of I-COACH shown in Fig. 1, and the two systems differ only by the CPMs displayed on the spatial light modulator (SLM). The principle of I-COACH is described as follows. Light from an incoherent source critically illuminates an object using a refractive lens L₁. The light diffracted from the object is collimated by a refractive lens L₂, and the collimated light is incident on the SLM. The CPM synthesized using the modified GSA is displayed on the SLM, and the light modulated by the CPM is recorded by an image sensor. The I-COACH system is calibrated using a point object, which is axially translated along the Z-axis and the corresponding point spread hologram (PSH) library is recorded [23]. Following the calibration phase, an object is placed inside the defined 3D space and an object intensity pattern is recorded. The 3D object image is reconstructed by non-linear cross-correlation of the object intensity pattern with the PSH library [24].

Fig. 1 Optical configuration of I-COACH. L₁, L₂ – Refractive lenses: P- Polarizer; SLM - Spatial light modulator; DL – Diffractive lens; CPM - Coded phase mask.

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2.1 Synthesis of coded phase mask

The RE-COACH system contains a CPM that generates subdiffraction-limit spots at randomly designated positions. The CPMs are synthesized using our previously developed algorithm, which is based on the GSA with our own constraints [27]. An important operation of our constraints is to control the phases of the spots. When the phase of a spot differs by π from the phase of its adjacent spots, the spot size can be reduced by destructive interference between the spots.

To achieve notable resolution enhancement, some subdiffraction-limit spots should be arranged with a low density. Under such condition, most spots are isolated from others, and the spot size tends to be large because destructive interference rarely occurs. To overcome this problem, sets of 3 × 3 spots are arranged randomly over the Fourier plane of the CPM. With this strategy, the central spot within 3 × 3 spots is surrounded by up to 4 spots with an opposite phase on its left, right, top, and bottom, and thus the central spot becomes substantially narrower.

In the design, the total number of pixels is 8192 × 8192. For the pixel size on the CPM plane of 8.0 μm, the pixel size on the spot-generation plane is calculated to be 3.9 μm, assuming the focal length of the Fourier lens is 40 cm and the wavelength is 635 nm. The shape of a CPM is a circle with a radius of 512 pixels. The pixels out of the CPM contain zero values. The spot-generation plane is divided into the spot-area and the surrounding area, which is prepared to increase the flexibility of the optical-pattern generation within the spot-area [27]. The spot-area, which is placed at the center of the spot-generation plane, is further divided into 512 × 512 cells each of which consists of 6 × 6 pixels. Accordingly, the size of the spot-area is 3072 × 3072 pixels. The cell size of 23.3 μm, is equivalent to ⁓ 73% of the full width at half maximum, 31.9 μm, of the diffraction limit spot, namely an Airy disc generated using the experimental system with a clear disk-shape aperture with a radius of 512 pixels. Only the area of the central 256 × 256 cells is used to generate spots for avoiding influence of the noisy optical pattern at the surrounding area outside the central area of 3072 × 3072 pixels. In addition, spots can be arranged at every two cells; thus, the number of positions where spots can be placed is 128 × 128.

We designed several CPMs by changing the spot density to investigate the performance and property of RE-COACH. Here the spot density is defined as the number of arranged spots over the number of potential spot-positions (128 × 128). For example, when 164 sets of 3 × 3 spots are arranged, the spot density is approximately 164 × (3 × 3)/(128 × 128) = 0.01 × (3 × 3) or 0.09. The CPMs were designed as a phase-only and Fourier-transform type masks. In the experiments, the phase distribution of a diffractive Fourier lens with a focal length of z_h is added into the phase of the designed CPM.

2.2 Theoretical analysis

The theoretical analysis is carried out for a point object and extended by a superposition for a multipoint object. A point object located at $({\bar{r}}_{s}, - z_{s}) = (x_{s}, y_{s}, - z_{s})$ emits light with an amplitude $\sqrt{I_{s}}$ . The complex amplitude just before lens L₂ is given as $\sqrt{I_{s}} C_{0} Q (1 / z_{s}) L ({\bar{r}}_{s} / z_{s})$ , where Q and L represent the quadratic and linear phase functions, given by $Q (a) = \exp [i π a λ^{- 1} (x^{2} + y^{2})]$ and $L (\bar{s} / z) = \exp [i 2 π {(λ z)}^{- 1} (s_{x} x + s_{y} y)],$ respectively and C₀ is a complex constant. The complex amplitude just after lens L₂ with a focal length of f₂, is given by $\sqrt{I_{s}} C_{0} Q (1 / z_{s}) L ({\bar{r}}_{s} / z_{s}) Q (- 1 / f_{2})$ . Assuming that the distance d between the lens L₂ and the SLM is negligibly small, the complex amplitude after the SLM can be expressed as $\sqrt{I_{s}} C_{0} Q (1 / z_{s} - 1 / f_{2}) L ({\bar{r}}_{s} / z_{s}) \exp [i Φ (\bar{r})] Q (- 1 / z_{h}),$ where $Φ (\bar{r})$ is the CPM synthesized using the modified GSA and z_h is the focal length of the diffractive lens attached to the CPM. The CPM is multiplied by the diffractive lens Q(−1/z_h) to match accurately the GSA condition of Fourier relation between the SLM and the sensor planes. Therefore, the intensity pattern on the image sensor located at a distance of z_h from the SLM is given by,

I_{P S H} ({\bar{r}}_{0}; {\bar{r}}_{s}, z_{s}) = | \sqrt{I_{s}} C_{0} L (\frac{{\bar{r}}_{s}}{z_{s}}) \exp [i Φ (\bar{r})] Q (\frac{1}{z_{s}} - \frac{1}{f_{2}} - \frac{1}{z_{h}}) {* Q (\frac{1}{z_{h}}) |}^{2},

where the symbol '*' represents a two-dimensional convolution and

{\bar{r}}_{0} = (u, v)

is the transverse location vector on the plane of the image sensor. Equation (1) can be reduced into a Fourier transform as the following [4],

\begin{array}{l} I_{P S H} ({\bar{r}}_{0}; {\bar{r}}_{s}, z_{s}) = {| ν [\frac{1}{λ z_{F}}] ℑ {\sqrt{I_{s}} C_{0} L (\frac{{\bar{r}}_{s}}{z_{s}}) Q (- ζ) \exp [i Φ (\bar{r})]} |}^{2} \\ = I_{P S H} ({\bar{r}}_{0} - \frac{z_{h}}{z_{s}} {\bar{r}}_{s}; 0, z_{s}), \begin{matrix}  \end{matrix} \begin{matrix}  \end{matrix} \begin{matrix}  \end{matrix} \end{array}

where

ζ = \frac{(z_{s} - f_{2}) (z_{s} f_{2} + z_{s} z_{h} - z_{h} f_{2})}{{(z_{s} f_{2})}^{2}}, z_{F} = \frac{z_{s} f_{2} z_{h}}{z_{s} f_{2} + z_{s} z_{h} - z_{h} f_{2}}

ℑ

is the operator of 2D Fourier transform and ν is the scaling operator defined by the relation ν[α]f(x) = f(αx). The intensity on the sensor plane is a shifted version (by

{\bar{r}}_{s}

z_h/z_s) of the intensity response for a point object located on the optical axis

({\bar{r}}_{s} = 0)

. Equation (3) indicates that for z_s = f₂ the value of ζ is zero and thus

Q (- ζ) = 1

in Eq. (2). In other words, the PSH for a point located at (0,0,-z_s = -f₂) is a scaled Fourier transform of the CPM. Therefore, the CPM is synthesized by the GSA to yield the subdiffraction spots only for points located at the input plane z_s = f₂. The CPM cannot guarantee a response of subdiffraction spots for points outside this plane of z_s = f₂. Consequently, one can conclude that the expected gain of resolution is guaranteed only for 2D objects displayed on the front focal plane of lens L₂. Other lateral planes of a 3D object can still be imaged since according to Eq. (2) the shift-invariance property is maintained regardless of the values of ζ and z_s, but the gain of resolution is no longer guaranteed.

A 2D object illuminated by a spatially incoherent light and located at the same distance z_s from the lens L₂ can be considered as a collection of N uncorrelated object points given as,

o ({\bar{r}}_{s}) = \sum_{j}^{N} a_{j} δ (\bar{r} - {\bar{r}}_{s, j}) .

Since the system is linear, the intensity distribution for the 2D object on the sensor plane is a sum of all the shifted point responses, given by,

I_{O B J} ({\bar{r}}_{0}; z_{s}) = \sum_{j} a_{j} I_{P S H} ({\bar{r}}_{0} - \frac{z_{h}}{z_{s}} {\bar{r}}_{s, j}; 0, z_{s})

In the first report of I-COACH [23], three intensity recordings with different pseudorandom phase masks for the point object, as well as for the object, were captured in order to remove the bias and the background noise during reconstruction. Later a phase-only filtering technique was employed to reduce the background noise further [29]. In the subsequent studies [30–32], the optical configuration of I-COACH was improved, and the number of intensity recordings was decreased to two. As mentioned in the introduction, recently, a non-linear reconstruction (NLR) technique was developed, which reduced the number of intensity recordings to one [24]. In the NLR technique, the magnitudes, $| {\hat{I}}_{O B J} |$ and $| {\hat{I}}_{P S H} |$ , are raised to the power of o and r, respectively, where ${\hat{I}}_{O B J}$ and ${\hat{I}}_{P S H}$ are the Fourier transforms of I_OBJ and I_PSH given in Eqs. (2) and (5), respectively. In the NLR optimization procedure, only the spectral magnitudes of the object, and of the reconstructing function, are raised to the power of o and r, while the phase information remains intact. For an object of a point located at $({\bar{r}}_{s}, z_{s})$ , the Fourier transform of the cross-correlation between the object and the PSH-based reconstructing function is,

\begin{array}{l} \hat{C} = ℑ {{I^{'}}_{O B J}^{} \otimes {I^{'}}_{P S H}^{}} = {\hat{I}}^{'}_{O B J}^{} \cdot {\hat{I}}^{'}_{P S H}^{*} \\ = {| {\hat{I}}_{P S H}^{} |}^{o} \exp [i (φ_{P S H} + 2 π z_{h} {\bar{r}}_{s} \cdot \bar{ν} / z_{s})] {| {\hat{I}}_{P S H}^{} |}^{r} \exp [- i φ_{P S H}] \\ = {| {\hat{I}}_{P S H}^{} |}^{o} \exp [i 2 π z_{h} {\bar{r}}_{s} \cdot \bar{ν} / z_{s}] {| {\hat{I}}_{P S H}^{} |}^{r}, \begin{matrix}  \end{matrix} \begin{matrix}  \end{matrix} \begin{matrix}  \end{matrix} \end{array}

where

{\hat{I}}^{'}_{O B J}^{}

and

{\hat{I}}^{'}_{P S H}^{}

are the Fourier transforms of

{I^{'}}_{O B J}^{}

and

{I^{'}}_{P S H}^{}

, respectively.

{I^{'}}_{O B J}^{}

and

{I^{'}}_{P S H}^{}

are the patterns

I_{O B J}^{}

and

I_{P S H}^{}

, respectively, after the nonlinear operations described above. Since the object is a point, its ideal reconstruction is a delta function. Hence theoretically, the condition o + r = 0 yields uniform

\hat{C}

and consequently a reconstructed image of an ideal point. However, the noise in any practical system produces other optimal values for o and r, which usually do not satisfy the above-mentioned condition. Note that the procedure expressed in Eq. (6) is valid for a point object and because of the non-linearity of the power raising, one cannot show the validity of this procedure for general multipoint object. Nevertheless, many experiments in various conditions indicate that practically the NLR method works quite well for an arbitrary multipoint object.

In order to find the optimal values of o and r, we change them, in some predefined step, in the range $(- 1 \leq o \leq 1, - 1 \leq r \leq 1)$ . The entropy is calculated for each value of o and r and the optimal reconstruction in the sense of minimum entropy value is extracted. Entropy is a blind figure-of-merit and therefore it is not necessary to a priory recognize the object before its image is reconstructed. The entropy corresponding to the normalized intensity distribution function $ϕ (m, n)$ of the output reconstructed matrix R(m,n) is given as,

S (o, r) = - \sum_{M} \sum_{N} ϕ (m, n) \log [ϕ (m, n)],

where m,n are the pixel coordinates of the image and

ϕ (m, n)

is given by,

ϕ (m, n) = \frac{{| R (m, n) |}^{}}{{\sum_{M} \sum_{N} | R (m, n) |}^{}} .

The NLR reconstruction procedure is adaptive to the various experimental conditions and thus it finds optimal values of o and r for each experiment.

3. Experiments and results

Different experiments described in the next subsections were carried out to evaluate the capabilities of the method of resolution enhancement in I-COACH.

3.1 2D Super-resolution imaging

The experimental verification of the proposed RE-COACH was carried out using a digital holography setup shown in Fig. 2. The experimental setup consists of two illumination channels with identical LEDs emitting light at a wavelength of 635 nm (Thorlabs LED635L, 170 mW, λ = 635 nm, Δλ = 15 nm). Two identical lenses L_1A and L_1B were used to critically illuminate the objects. In channel 1, a pinhole with a size of 10 μm was mounted for capturing the PSH. In channel 2, Groups 5, 6 and 7 of United States Air Force (USAF) resolution target was illuminated using the second LED. The object and the pinhole were positioned at the front focal plane of lens L₂ at a distance of 17.5 cm from L₂. The distance between the lens L_{2} and the phase-only reflective SLM (Holoeye PLUTO, 1920 × 1080 pixels, 8 μm pixel pitch, phase-only modulation) was 20.5 cm. The distance between the lens L₂ and the beam splitter BS2 was 15 cm. The distance between the SLM and the image sensor (Thorlabs 8051-M-USB, 3296 × 2472 pixels, 5.5 μm pixel pitch, monochrome) was z_h = 40 cm. A bandpass filter with (λ_c = 632.8 nm and Δλ = 5 nm) was mounted between the SLM and the image sensor.

Fig. 2 Experimental setup. BS1 and BS2 – Beam splitters; SLM – Spatial light modulator; USAF – United States Air Force; L_1A, L_1B and L₂ – Refractive lenses; LED1 and LED2 – Identical light emitting diodes; CPM – Coded phase mask; QPM – Quadratic phase mask; BPF – Band pass filter (λ_c = 632.8 nm and Δλ = 5 nm); P- Polarizer; ⦿- Polarization orientation perpendicular to the plane of the page.

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The phase pattern displayed on the SLM is obtained by modulo-2π phase addition of the CPM with a quadratic phase mask (QPM) of 40 cm focal length. Each CPM is synthesized by the modified GSA and the QPM is used to satisfy the Fourier relation of the GSA between the SLM and the sensor plane. The PSHs were recorded using CPMs with different spot densities. The images of the CPM, PSH and object holograms for spot densities 0.009 and 0.09 are shown in Fig. 3. Image reconstructions for different spot densities using NLR are shown in Fig. 4. In order to determine the optimal spot density in the sense of maximum resolution, we consider several measures. The plots of averaged line profile are obtained by averaging one horizontal and one vertical gratings of the same USAF object. The dip percentage of minimum from maximum and visibility are calculated and shown in Fig. 5, for the element 2 of group 6 (indicated on Fig. 4 by red frames) with a grating line spacing of 6.96 μm. From Fig. 5, it is seen that the CPM synthesized with a spot density = 0.036 has the maximum lateral resolution. A comparison of the resolution limits between three methods of imaging is demonstrated in Fig. 6. The resolution limit for RE-COACH, regular I-COACH and direct imaging are 6.96 μm (Group 6 element 2), 7.82 μm (Group 6 element 1) and 8.77 μm (Group 5 element 6), respectively as shown in red boxes of Fig. 6. RE-COACH has shown an improvement in lateral resolution by a factor of 1.26 against direct imaging and 1.12 against the regular I-COACH respectively.

Fig. 3 Images of the CPM, PSH and object Hologram for spot densities of 0.009 and 0.09.

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Fig. 4 Reconstruction results of USAF Target (Group 6 element 2) for spot densities 0.009, 0.018, 0.036, 0.054, 0.072 and 0.09. Minimum resolvable feature is shown in red box.

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Fig. 5 Line Profile, Visibility Plot and dip of minima percentage from maxima for different spot densities.

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Fig. 6 Object reconstruction results for RE-COACH, regular I-COACH and direct imaging. The minimum resolvable feature is shown in the red box.

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3.2 3D imaging

For the experiment of 3D imaging, a pinhole with a diameter of 10 μm was used. The PSHs for the optimal spot density of 0.036 at two different axial locations of the pinhole, separated by a distance of 4 mm, were recorded. Two objects: object ‘4’ (Group 4 & Plane A) and ‘1’ (Group 3 element 1 & Plane B) of United States Air Force (USAF) were mounted in the channels 1 and 2 respectively at the same axial locations of the pinhole where the PSHs were recorded. The 3D image was reconstructed by a cross-correlation of the same object hologram with the PSHs of the two planes A and B using the NLR technique. Figure 7 shows the reconstruction results for different axial planes. When plane A is reconstructed, the object ‘4’ is in focus and the other object ‘1’ is defocused and vice versa in the other case of plane B. Hence, by this experiment, it is concluded that the 3D imaging capability of regular I-COACH is retained in the RE-COACH technique.

Fig. 7 Reconstruction results of (a) plane A, (b) plane B separated by a distance of 4 mm from a single camera shot of RE-COACH. (c) Direct imaging when plane A was in focus.

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In an additional experiment, the axial response of RE-COACH was measured and compared against the response of direct imaging. In this experiment, the PSH was recorded for a pinhole located at z = 0, and the location of the pinhole was varied between z = −2 cm to z = 2 cm in steps of 0.5 cm. The recorded PSHs I_PSH(z) were cross-correlated with I_PSH(z = 0) using the non-linear filter (r = 0.3, o = 0.6) and the correlation results are plotted against the axial location of the pinhole as shown in Fig. 8. The graph is compared with that of direct imaging using the same point object and z distances. The axial width was determined by calculating the FWHM of the normalized response plots as shown in Fig. 8. The axial width of RE-COACH was found to be 1.653 mm whereas the direct imaging has an axial resolution of 2.12 mm, which indicate that the axial response of RE-COACH is narrower by a factor of 1.28 than the direct imaging.

Fig. 8 Plots of the axial response curves for RE-COACH (blue) and direct imaging (red).

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3.3 Imaging of greyscale objects

From application point of view, the objects used in real-world imaging are not always binary. In this section, we have imaged greyscale objects using the RE-COACH technique. To display a greyscale object, we used an amplitude SLM (Holoeye LC2012, 1024 × 768, 36 μm pixel pitch) with greyscale values from 200 to 256 in step of 8. The amplitude SLM was mounted in one of the channels of the experimental setup and the hologram of spot density of 0.036 was recorded. A reconstructed image and direct imaging results of the object are shown in Fig. 9. As seen in Fig. 9, RE-COACH can reconstruct the objects of all grey scale values that can be registered in the direct imaging.

Fig. 9 Reconstruction results of the RE-COACH method and direct imaging for greyscale objects.

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4. Summary and conclusions

A modified Gerchberg–Saxton algorithm which generates subdiffraction spot arrays have been implemented in the RE-COACH system for resolution enhancement. In this system, subdiffraction spots were randomly distributed over the sensor area, where each array contains 3 × 3 subdiffraction spots. CPMs with different spot densities from 0.009 to 0.09 were synthesized and the corresponding object and point-spread holograms were recorded and then reconstructed using the NLR technique. To determine the spot density having best reconstruction result, visibility plot, Rayleigh dip and line profile for the minimum resolvable object were plotted and the reconstruction result for the spot density of 0.036 has demonstrated the highest resolution.

The RE-COACH method was compared to the regular I-COACH and direct imaging and the minimum feature resolved were 6.96 μm, 7.82 μm and 8.77 μm, respectively. These results show that the RE-COACH method has a resolution enhancement of about 1.26 over direct imaging and 1.12 over the conventional I-COACH. In further experimentation, 3D imaging property and reconstruction results for greyscale objects were tested. Hence, it can be concluded that RE-COACH method has all intrinsic properties of regular I-COACH, but with the additional advantage of imaging with improved resolution.

As a final remark, let us briefly regard to the well-known incoherent digital holography technique, called FINCH mentioned in the introduction with its special resolution properties [14–16]. FINCH has a superior lateral resolution of about 1.5 times higher than an equivalent direct imaging system [16]. However, it has been shown that FINCH has lower axial resolution and therefore it is difficult to use FINCH for imaging thick objects. COACH and I-COACH were developed to improve the lower axial resolution problem of FINCH at the expense of some loss of the higher lateral resolution. With this advancement of the RE-COACH technique, improved lateral resolution can be achieved together with other advantages; The 3D scene is captured by a single camera shot, and there is no need to any interferometric setup. The axial resolution of RE-COACH, as reflected from the axial response of Fig. 8, is at least as good as the direct imaging. Hence, in comparison to FINCH and other self-interference digital holography methods, RE-COACH is more robust, with a higher power efficiency, better time resolution and relaxed temporal coherence requirements. With all these benefits, we believe that RE-COACH technique can be used for 3D biomedical imaging, fluorescence microscopy and 3D imaging through scattering media.

Funding

Israel Science Foundation (ISF) (Grant No. 1669/16) and the Israel Ministry of Science and Technology (MOST).

Acknowledgment

Part of this study was done during a Research Stay of JR at the Alfried Krupp Wissenschaftskolleg Greifswald.

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Resolution enhancement in nonlinear interferenceless COACH with point response of subdiffraction limit patterns

Abstract

1. Introduction

2. Methodology

2.1 Synthesis of coded phase mask

2.2 Theoretical analysis

3. Experiments and results

3.1 2D Super-resolution imaging

3.2 3D imaging

3.3 Imaging of greyscale objects

4. Summary and conclusions

Funding

Acknowledgment

References

Cited By

Figures (9)

Equations (8)

Optics Express