# Randomly Multiplexed Diffractive Lens and Axicon for Spatial and Spectral Imaging

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^{2}

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## Abstract

**:**

## 1. Introduction

_{O}= O$\otimes $I

_{PSF}, where “$\otimes $” is a 2D convolutional operator. The image of the object O’ is reconstructed using a cross-correlation between I

_{PSF}and I

_{O}, O’ = I

_{O}*I

_{PSF}, where “$*$” is the 2D correlation operator. In the earlier studies [1,2,3,4,5,6,7], the image reconstruction in FINCH was achieved using Fresnel backpropagation and requires at least three camera recordings to synthesize a complex hologram for which the reconstructing quadratic phase function (I

_{PSF}) was optimal. However, the quadratic phase function is not an optimal function for reconstructing object information from an amplitude-only single camera recording. In the proposed method, a single camera shot becomes sufficient by adapting the non-linear filter approach of I-COACH. In summary, the similarity in the above three methods of COACH, I-COACH, and FINCH is that they are all linear systems regarding intensity such that the same reconstruction mechanism via cross-correlation can be applied to all of them. The differences in the above methods exist in the beam properties, which create the self-interference distribution and the resulting different imaging characteristics [17].

## 2. Methodology

_{2}from the HDOE is given as a convolution of the complex amplitude after HDOE with $Q\left(1/{z}_{2}\right)=\mathrm{exp}\left(j\pi {r}^{2}/\lambda {z}_{2}\right)$. The intensity at the image sensor is given as:

_{2}, this component will generate a random diffraction pattern characteristic of the random multiplexer. The second component consists of the axicon phase and the quadratic phase with the random multiplexing function $\left[1-f\left(x,y\right)\right]$, which will result in a Bessel-like function with random multiplexing noise when propagated to z

_{2}. The intensity distribution of a complicated object can be given as I

_{O}= O$\otimes $I

_{PSF}. The image of the object O’ is reconstructed using a cross-correlation between I

_{PSF}and I

_{O}, O’ = O$\otimes $I

_{PSF}*I

_{PSF}. Alternatively, the image can be considered as the object sampled by the autocorrelation function I

_{PSF}*I

_{PSF}. The regular cross-correlation I

_{O}*I

_{PSF}is also called the matched filter [16,18]. The other versions of the matched filter, including the Weiner filter and the phase-only filter, have better performances and resolutions. A recent study concluded that the non-linear filter has the best performance in terms of the signal-to-noise ratio and resolution [28]. The reconstruction using a non-linear filter can be expressed as:

## 3. Experiments and Results

_{1}= 10 cm. The angle of the axicon was approximately α = 0.6° with a period of Λ ≈ 60 μm. The random phase function f(x,y) was synthesized using the Gerchberg–Saxton algorithm with β = 0.1 radians [8,20]. The HDOE was designed as a binary element with only two-phase levels. The HDOE was fabricated using electron beam lithography (EBL; Raith 150

^{2}, RAITH, Dortmund, Germany) in a PMMA 950K (A7) resist (Microchem, Round Rock, TX, USA) on indium tin oxide (ITO)-coated glass substrates with a thickness of 1.1 mm and developed using methyl isobutyl ketone (MIBK) and isopropyl alcohol (IPA) solution (Microchem, Round Rock, TX, USA). The electron beam dose was 150 μC/cm

^{2}. A write field of 100 μm was used with stitching to fabricate a 5 mm size element. The optical microscope images of the HDOE are shown in Figure 2a,b. The results indicate no stitching errors. The thickness of the PMMA resist was found to be around 700 nm, which was close to the expected value of λ/2(n

_{r}− 1) = 617 nm (n

_{r}is the refractive index of the resist ≈ 1.5), and achieved the maximum efficiency of 40.5% for a binary element. The outermost zone width was approximately 12 μm. However, the random multiplexing generated features smaller than 12 μm at various locations.

_{c}= 617 nm, full width at half maximum (FWHM) = 18 nm; M530L3, λ

_{c}= 530 nm, FWHM = 33 nm), which were spatially incoherent and had low temporal coherence owing to the larger values of the FWHMs. It must be noted that the previous studies [1,2,3,4,5,6] were carried out with temporally coherent sources with a FWHM of 1 nm. A pinhole with a diameter of 20 µm was used to simulate the point object. Two standard objects, namely the United States Air Force (USAF) resolution target (group 2, element 6, 7.13 lp/mm) and the National Bureau of Standards (NBS) resolution target (7.1 lp/mm), were used to generate two plane objects with different thicknesses and wavelengths. The schematic of the two-channel experimental setup is shown in Figure 3. The light from the two LEDs were collected using two identical lenses L with a focal length of 10 cm that critically illuminated the two objects. The light that formed the two channels were combined using a beam splitter (BS). The light that diffracted from objects was modulated using the HDOE and the self-interference intensity distribution was recorded using an image sensor (Thorlabs DCU223M (Thorlabs, Newton, NJ, USA), 1024 × 768 pixels, pixel size = 4.65 μm).

_{1}= 10 cm and z

_{2}= 10 cm, as shown in Figure 4c,d, respectively. In the next experiment, the two objects (USAF and NBS objects) were mounted at the same distance z

_{1}= 10 cm and critically illuminated using red and green LEDs, respectively. The image of the hologram is shown in Figure 4e. The reconstruction results (p = 0, q = 0.63) of the hologram using the red and green point spread functions are shown in Figure 4f,g, respectively. It is seen that the point spread holograms only successfully reconstructed the objects illuminated by the same wavelength, while the object illuminated by a different wavelength was not reconstructed.

_{1}= 6 cm to 16 cm in steps of 2 cm. From the reconstruction results shown in Figure 5, it was observed that the system could distinctly resolve the axial and spectral information.

## 4. Discussion

## 5. Summary, Conclusions, and Outlook

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Optical configuration of Fresnel incoherent correlation holography (FINCH) with a randomly multiplexed axicon and a Fresnel zone lens. A conical wave (green) and a plane wave (blue) is generated for every object point. The two beams have the same diameter. The creation of the hybrid diffractive optical element (HDOE) from a binary axicon, a binary Fresnel zone lens, binary random matrix, and an inverse binary random matrix is shown.

**Figure 2.**Optical microscope images of the (

**a**) central and (

**b**) outermost part of the HDOE fabricated using electron beam lithography. The dark blue color indicates the resist remained and the light blue color indicates the resist was removed.

**Figure 3.**Schematic of the two-channel experimental setup. NBS: National Bureau of Standards, USAF: United States Air Force.

**Figure 4.**Direct imaging of the (

**a**) USAF and (

**b**) NBS objects. Point spread holograms recorded for (

**c**) λ

_{c}= 617 nm and (

**d**) λ

_{c}= 530 nm. (

**e**) Object hologram recorded when the USAF and NBS objects were at the same distance from the HDOE. (

**f**,

**g**) The reconstruction results using (

**c**) and (

**d**), respectively. The scale corresponds to the pixel intensity recorded by the image sensor.

**Figure 5.**Four-dimensional reconstruction results demonstrated using holograms of two plane objects of different thicknesses (d = 0, 5, and 10 cm) and two different wavelengths. The point spread functions reconstructed the depth-specific and wavelength-specific information with the maximum focus while the information from other planes and wavelengths were blurred with decreased intensities. The blue border lines indicate the reconstruction results. The green border lines and red border lines indicate the point spread function of the green and red wavelengths and the yellow square boxes indicate the cases with the best focus.

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**MDPI and ACS Style**

Anand, V.; Katkus, T.; Juodkazis, S.
Randomly Multiplexed Diffractive Lens and Axicon for Spatial and Spectral Imaging. *Micromachines* **2020**, *11*, 437.
https://doi.org/10.3390/mi11040437

**AMA Style**

Anand V, Katkus T, Juodkazis S.
Randomly Multiplexed Diffractive Lens and Axicon for Spatial and Spectral Imaging. *Micromachines*. 2020; 11(4):437.
https://doi.org/10.3390/mi11040437

**Chicago/Turabian Style**

Anand, Vijayakumar, Tomas Katkus, and Saulius Juodkazis.
2020. "Randomly Multiplexed Diffractive Lens and Axicon for Spatial and Spectral Imaging" *Micromachines* 11, no. 4: 437.
https://doi.org/10.3390/mi11040437