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In recent years, there has been a significant transformation in the field of incoherent imaging with new possibilities of compressing three-dimensional (3D) information into a two-dimensional intensity distribution without two-beam interference ( TBI). Most incoherent 3D imagers without TBI are based on scattering by a random phase mask exhibit- ing sharp autocorrelation and low cross-correlation along the depth axis. Consequently, during reconstruction, high lateral and axial resolutions are obtained. Scattering based-Imaging requires a wasteful photon budget and is there- fore precluded in many power-sensitive applications. This study develops a proof-of-concept 3D incoherent imag- ing method using a rotating point spread function termed 3D Incoherent Imaging with Spiral Beams (3DI SB). The rotation speed of the point spread function (PSF) with displacement and the orbital angular momentum has been theoretically analyzed. The imaging characteristics of 3DI SB were compared with a direct imaging system using a dif- fractive lens, and the proposed system exhibited a higher focal depth than the direct imaging system. Different com- putational reconstruction methods such as the Lucy–Richardson algorithm (LRA), non-linear reconstruction (NLR), and the Lucy–Richardson–Rosen algorithm (LRRA) were compared. While LRRA performed better than both LRA and NLR for an ideal case, NLR performed better than both under real experimental conditions. Both single plane imaging, as well as synthetic 3D imaging, were demonstrated. We believe that the proposed approach might cause a paradigm shift in the current state-of-the-art incoherent imaging, fluorescence microscopy, and astronomical imaging. Keywords: Orbital angular momentum, Incoherent holography, Diffractive optics, Imaging, Microscopy Introduction and conoscopic holography [4] were some of the previ- Incoherent holography technologies have an interest- ously widely used architectures. Even though incoherent ing history [1]. In the beginning, incoherent hologra- holography is capable of broad applicability, the above phy systems had quite complicated architectures due bulky, complicated configurations hindered realizing its to spatial and temporal incoherence. Common path full potential. With the development of active devices, interferometers [2], rotational shear interferometers [3], the implementation of incoherent holography became relatively simpler. One well-known compact incoherent holography method was the Fresnel incoherent correla- *Correspondence: physics.vijay@gmail.com tion holography (FINCH) which supported two optical Vijayakumar Anand and Svetlana Khonina have contributed equally to channels in a single physical space and independently this work. modulated one of the channels allowing easy application Optical Sciences Center and ARC Training Centre in Surface Engineering for Advanced Materials (SEAM), School of Science, Computing to fluorescence microscopy [5]. Later, FINCH was found and Engineering Technologies, Swinburne University of Technology, to have a capability to achieve super-resolution by break- Hawthorn, Melbourne, VIC 3122, Australia ing the Lagrange invariant condition, which motivated Full list of author information is available at the end of the article © The Author(s) 2022. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. Anand et al. Nanoscale Research Letters (2022) 17:37 Page 2 of 13 further research and development in the area of inco- However, the method could not be applied directly for herent holography [6]. Today FINCH has evolved into multiplane imaging without multiplexing masks for the an advanced technique investigated by many prominent required planes. researchers across the globe. [7] In one of the recent attempts of implementing CAI While FINCH followed a natural path of evolution, methods in the infrared beamline of the Australian retaining its roots from classical holography, another Synchrotron, a better understanding of imaging con- branch of today’s incoherent holography called coded cepts was obtained [22]. The requirement of a high pho - aperture imaging (CAI) has been parallelly developed ton budget and the beam characteristics of Cassegrain based on computational optical methods. The CAI meth - objective lenses (COL) resulted in a new imaging path. ods have been indirect imaging approaches developed to The light diffracted from COL had four sharp inten - perform imaging with X-rays and Gamma rays, where sity peaks over a large depth, and so the autocorrelation manufacturing lenses for these wavelengths is challenging function was sharp along the depth axis while the cross- [8, 9]. In this direction, uniformly redundant array (URA) correlation was lower as the spacing between the spots [10], modified URA [11], spectral imaging methods [12] increased with distance. This is exactly the condition were developed consisting of special patterns for the needed for performing indirect 3D imaging. This obser - aperture masks. While the above are useful techniques, vation resulted in coining a new optical field condition a pure phase aperture has many interesting features, such for 3D imaging with CAI—sharp autocorrelation and low as the average speckle size equal to the diffraction-lim - cross-correlation along depth (SALCAD) field. There - ited spot size [13, 14]. By recording the speckle intensity fore, the scattered light is random SALCAD field, while patterns for an object and a point object followed by a the case with COL is a deterministic SALCAD field. For cross-correlation between the two intensity distributions, imaging 3D information using the SALCAD fields gener - the image of the object can be reconstructed closer to the ated by COL, a special reconstruction algorithm called resolution limit of conventional imaging. Lucy–Richardson–Rosen algorithm (LRRA) [23] was But unlike FINCH, CAI methods were initially devel- developed by combining the non-linear reconstruction oped to perform 2D imaging. Another main difference method (NLR) developed recently [24] and the well- between FINCH and CAI was that FINCH is intense known Lucy–Richardson algorithm (LRA) [25, 26]. on optical experiments and less intense in computa- The above study opened the possibility of implement - tional, but CAI was the opposite. In 2016, a holography ing deterministic SALCAD fields with energy concen - method called coded aperture correlation holography trated in a small area similar to conventional imaging but (COACH) was developed, which blurred the above dis- suitable for 3D imaging. In this direction, beams carrying tinct differences between FINCH and CAI [15]. COACH orbital angular momentum (OAM) exhibit interesting was intense in both optics as well as computational parts characteristics, including SALCAD property. An inter- which connected FINCH and CAI. COACH was able to esting family of beams is the OAM beams with rotat- encode not only 3D spatial but also spectral informa- ing intensity distributions along the depth axis [27–29]. tion effectively [16]. As the 3D imaging capabilities were There have been some reports on implementing rotating found to be unaffected even without two-beam interfer - point spread functions to improve the localization and ence conditions, COACH dropped the two-beam inter- sensing along the depth axis [30–34]. Most of the studies ference requirement and evolved into Interferenceless [30–32], employed a double helix pattern to improve 3D COACH (I-COACH) [17]. Later, many such techniques localization. In conventional fluorescence imaging, when based on I-COACH were reported, such as diffuserCam the fluorescent particles lie beyond the depth of field of [18] and scatter plate microscope [19] with different the imaging system, they appear blurred with a larger size computational reconstruction mechanisms. resulting in loss of resolution. By converting every point Even though the scattering mask-based CAI methods image into a double helix pattern, the depth of field is can encode spatial and spectral information in a single increased, and the location of the particle can be identi- shot, the photon budget requirement is enormous com- fied by the rotation angle of the two spots. This approach pared to conventional lens-based imagers [20]. This prob - improved the 3D localization of fluorescent particles. In lem prevents the implementation of such CAI methods to 2017, 3D imaging was demonstrated with a double helix power-sensitive applications such as astronomical imag- pattern using the widely used deconvolution algorithms, ing and fluorescence microscopy. A modified approach namely Weiner filter and the Lucy–Richardson algo - was developed to overcome this problem; Instead of rithm [33]. But the method was demonstrated only for scattering light uniformly, the light was collected and a short axial distance. In this study, we have investigated focused into a random array of points in the sensor plane the characteristics of 3D Incoherent Imaging using Spiral [21]. In this way, the efficiency of imaging was improved. Beam (3DI SB) and have demonstrated 3D imaging. Two A nand et al. Nanoscale Research Letters (2022) 17:37 Page 3 of 13 g(r , ϕ) = L(r)S(r , ϕ), (1) where L(r) = exp(−iπ r /(f )) is the DL in the MDOE and S(r , ϕ) is the spiral element in the following form: S(r, ϕ) = R (r) exp (i(2n − 1)ϕ), n (2) n=1 1, r < r ≤ r , n−1 n where R (r) = r = 2nf , n n 0, else. (3) where r is the radial coordinate and ϕ is the azimuthal coordinate, f is the focal length of the DL given as 1/f = 1/z + 1/z assuming paraxial approximation s h [35], λ is the wavelength and n is the order of circular zones, z and z are the object distance and image dis- Fig. 1 Optical configuration of 3DI SB and generation of phase s h distribution of the MDOE from a lens and spiral element tances, respectively. For a point object with unit ampli- tude located along the optical axis at z = z + Δ from the MDOE, where Δ is the defocus distance in the object plane, the complex amplitude after the MDOE using recently developed reconstruction algorithms, namely Kirchhoff-Fresnel integral is given as NLR [24] and LRRA [23] have been applied to expand 2π R the limits of 3D imaging [25, 26]. The manuscript con i πr E(ρ, θ, z) =− g(r, ϕ) exp i sists of five sections. In the second section, the design of z z the phase mask, theoretical analysis of beam propagation 0 0 to understand the rate of rotation, and OAM estimation iπ 2 2 exp ρ + r − 2ρr cos(ϕ − θ) rdrdϕ, were described. In the third section, simulation results of imaging are presented. The experimental results are pre (4) sented in the fourth section. The conclusion and future where R is the radius of the beam at the MDOE plane. perspectives of this research work are discussed in the Considering the focusing part of the MDOE (1), we can final section. rewrite expression (4) as follows: 2π R 2 2 Methods i iπρ iπr 1 1 2 E(ρ, θ, z) =− exp exp − The optical configuration of 3DI SB is shown in Fig. 1. z z z z h h s 0 0 Basically, this configuration includes the observed object, i2π a multifunctional diffractive optical element (MDOE) S(r, ϕ) exp − ρr cos(ϕ − θ) rdrdϕ. and the image sensor. This section consists of three sec (5) tions: design and analysis of MDOE, estimation of OAM, Obviously, for z = z we get a focused field, which is the and imaging methodology. Fourier transform of the spiral element S(r , ϕ) , and at z ≠ z we get a defocused field. Design and Analysis of MDOE When we substitute (2)–(3) into (5) we obtain: The MDOE that can generate a rotating intensity distri - bution has been adapted from Ref. [31]. This study uses 2π r 2 2 i iπρ iπr 1 1 spatially incoherent and temporally coherent illumina E(ρ, θ, z) =− exp × exp − z z z z h h s tion, so the design and calculation have been carried n=1 0 n−1 out for a single wavelength. The MDOE is formed by the i2π exp (i(2n − 1)ϕ) exp − ρr cos(ϕ − θ) rdrdϕ. modulo-2π phase addition of the phase distributions of a diffractive lens (DL) and a spiral element. The topological (6) charge distribution of the spiral element follows an arith For each ring of the spiral element S(r , ϕ) , the integration metic progression dependent upon the zone number of by d ϕ can be performed analytically: annular zones with equal areas starting from the center. Let us consider the MDOE in the following form: Anand et al. Nanoscale Research Letters (2022) 17:37 Page 4 of 13 Note that displacement Δ can be positive or negative, i2π iπρ 2n−1 E(ρ, θ, z) =− exp (−i) exp (i(2n − 1)θ) which means rotation in different directions. Moreover, z z h h n=1 n−1 for large values Δ, the rotation speed will differ depend - iπr 1 1 2π ing on the sign due to the nonlinearity of the expression exp − J ρr rdr. 2n−1 z z z s h (12). At small displacements, the rotation speed will be (7) linear: If the rings are sufficiently narrow, then instead of (7) we can approximately write: 2 2 2π iπρ π iπr (z − z) 2π (8) E(ρ, θ, z) ≈ exp r exp i(2n − 1) θ − exp J ρr . n 2n−1 n z z 2 zz z h h s h n=1 Let us rewrite expression (8) in a form convenient for dθ π f ≈ . analysis: (13) d� 2z 2π E(ρ, θ, z) ≈ A C (θ, z)r J ρr , n n 2n−1 n (9) h Calculation of the Orbital Angular Momentum (OAM) n=1 of the Field Generated by the Spiral Element 2π iπρ where A = exp , The normalized OAM of an arbitrary scalar field is z z h h iπr (z −z) π s n defined as follows [36]: C (θ, z) = exp i(2n − 1) θ − exp , 2 zz ∞ 2π z − z =− . Assuming a special 2f condition, z = z , � � s � � s h J ∂E(r, ϕ) z z f − z =− − . Expression (9) is a superposition of j = = Im E (r, ϕ) · rdrdϕ 2 z W ∂ϕ Bessel beams. The behavior of such beams during propa - 0 0 gation was studied in various works [27, 29, 32, 36], −1 ∞ 2π � � where it was shown that the rotation of the intensity of � � E (r, ϕ) · E(r, ϕ) rdrdϕ . the beam (9) is determined by the interference terms and the phase difference: 0 0 (14) 2 2 π� r − r n n 2 1 Since the OAM is an invariant characteristic of the field cos 2(n − n ) θ − − . 2 1 (10) 2 2zz s and is conserved during propagation and focusing, the quantity (15) can be calculated in any plane, including the Substituting the expressions for the radii of the rings initial plane (at z = 0): r = 2nf , instead of (10) we obtain: n R 2π � � � � ∂g(r, ϕ) π π f �(n − n ) 2 1 j = Im g (r, ϕ) · rdrdϕ cos 2(n − n ) θ − − . 2 1 (11) ∂ϕ 2 (z + �)z s s 0 0 (15) −1 R 2π � � Then the rotation speed of the beam intensity depending � � � � on the distance shift from object Δ will be (omitting the g(r, ϕ) rdrdϕ . smaller term): 0 0 dθ π f Since MDOE (1) is a purely phase element, the integral ≈ . (12) d� 2(z + �)z in the denominator of expression (15) corresponds to the s s energy of a bounded plane illuminating beam ε = πR . Considering that the focusing part of MDOE (1) does not A nand et al. Nanoscale Research Letters (2022) 17:37 Page 5 of 13 change the value of OAM, the integral in the numerator complicated 2D object consisting of multiple points, the of expression (15) can be calculated as follows: description of imaging systems with coherent and inco- R 2π r 2π � � � � � � ∂g(r, ϕ) ∂S(r, ϕ) ∗ ∗ ε = g (r, ϕ) · rdrdϕ = S (r, ϕ) · dϕ rdr ∂ϕ ∂ϕ n=1 0 0 n−1 0 (16) N N � � � � 2 2 = i2π 2n − 1 rdr = iπ 2n − 1 r − r . ( ) ( ) n n−1 n=1 n=1 n−1 Considering equality r = R n/N [27], we obtain the herent sources are completely different except for some following expression: specially imposed conditions of limited field of view and sparse distribution of points [37]. The current imaging configuration converts an object point into a rotating j = (2n − 1) = N . z (17) N intensity distribution with two main intensity peaks that n=1 rotate around the optical axis as a double helix. When u Th s, the normalized OAM of the beams generated by the object point’s location changes laterally, the intensity the N-zone spiral element (2)–(3) is equal to the number distribution shifts, and when the location changes axially, of zones N. then the intensity distribution rotates either clockwise or anticlockwise depending upon the direction of the axial Incoherent Imaging shift. The point spread function of the incoherent imag - In the previous sections, we described the design of the ing system is given as I = |E(ρ, θ, z)| . In this study, PSF DOE, theoretical analysis of the special beam, and esti- we are interested in creating an incoherent imaging sys- mation of OAM. The above studies were carried out for tem that is linear in intensity, and so for a 2D object O, a single point. This analysis for a single point is true for the intensity distribution is given as I = I ⊗ O , where O PSF both coherent as well as incoherent imaging systems. ‘⊗’ is the 2D convolutional operator. The image recon - However, when moving on from a single point to a struction is I = I ∗ I , where ‘*’ is the 2D correlation R O PSF Fig. 2 Simulated intensity distributions generated by the MDOE for z = a 15 cm, b 20 cm, c 25 cm, d 30 cm, e 35 cm, f 40 cm and g 45 cm. Simulated intensity distributions generated by the DL for z = h 15 cm, i 20 cm, j 25 cm, k 30 cm, l 35 cm, m 40 cm and n 45 cm. The white broken line is attached to (a–g) to indicate the rotation angle of the spiral beam. The cross-section of the autocorrelation of the intensity distributions (a–n) are compared in (o–u), where blue is for DL and red is for MDOE Anand et al. Nanoscale Research Letters (2022) 17:37 Page 6 of 13 facilitating indirect imaging over a long distance without needing a scattering mask. Simulation Results The simulation studies were carried out for the follow - ing specifications of the imaging system. Number of pix - els = 500 × 500, λ = 600 nm, mesh grid pixel size = 10 μm, z = 30 cm, f = 15 cm, z = 30 cm and D = 5 mm. The s h intensity distribution for z = 15 cm to 45 cm in steps of 5 cm for the current system and a reference system con- sisting of only a DL without a spiral element are shown in Fig. 3 Plot of normalized cross-correlation C (x = 0, y = 0,z) (red) and Fig. 2a–n, respectively. As seen from Fig. 2a–g, the rota- axial intensity of DL along the defocus distance tion appears to be faster when the defocus distance in the object plane Δ is smaller, and with a larger increase, the rotation slows down. Secondly, the rotation is faster when operator. Rewriting the above equation of I by substi- the point object is closer to the MDOE. The autocor - tuting the composition of I , gives; I = O ⊗ I ∗ I , R PSF PSF relation function is calculated for the case with MDOE where I ∗ I is the autocorrelation function that PSF PSF and DL, and plotted for the different object locations reconstructs the image function. The width of the auto - for MDOE (red) and diffractive lens (blue) in Fig. 2o–u, correlation function is twice that of the diffraction-lim - respectively. It is seen that except for the imaging condi- ited spot size, which is 1.22λz /D in the object plane, tion i.e., z = 30 cm, for all other cases, the autocorrelation assuming a regular correlation, also called matched fil - function obtained for the MDOE is sharper with lesser ter, is selected [38]. Alternative a correlation method background noise than the case with a DL. with phase-only filter [38] has a relatively sharper auto - The axial behavior is studied next. In this study, correlation function. Recently, the NLR method sharp- the simulated intensity distribution at z = 30 cm is ened the autocorrelation function with a width equal to cross-correlated with matched filter (along the trans - the diffraction-limited spot size [24]. A spherical lens verse coordinates) with other values from z = 15 cm to can create a sharp PSF at the sensor if the imaging con- 45 cm, i.e., C = I (z = 30 cm) ∗ I (z) . The plot of a PSF PSF dition 1 f = 1 z + 1 z is satisfied, where f, z , and z 1 2 1 2 C (x = 0, y = 0) for the MDOE and the intensity varia- are the focal length, point-lens distance, and lens-sensor tion for the DL are shown in Fig. 3. The FWHM of the distance, respectively. However, when the imaging condi- above profile is approximately the axial resolution of the tion is violated, the PSF sharpness decreases. In the pro- system ~ λ/NA , where NA is the numerical aperture. posed 3DI SB, the rotating PSF has sharp intensity peaks It is seen that the focal depth of the MDOE is longer over a relatively long distance. While the method does than that of the DL. In other words, the axial resolution not directly create a well-defined image on the sensor, of imaging using a DL is expected to be better than the the autocorrelation function is expected to be sharper, Fig. 4 Schematic of the LRRA. When α = β = 1, LRRA becomes LRA A nand et al. Nanoscale Research Letters (2022) 17:37 Page 7 of 13 Fig. 5 Simulated intensity distributions for the test object for MDOE and DL and reconstruction results from NLR, LRA and LRRA Anand et al. Nanoscale Research Letters (2022) 17:37 Page 8 of 13 Fig. 7 Reconstruction results using LRRA and NLR during a shift error of 0, 10, 20, 50 and 100 pixels 3DI SB method. The simulation study reveals two main The NLR is given as α β characteristics of imaging: lateral resolution and axial −1 ˜ ˜ ˜ ˜ I = F I exp i arg I I exp −i arg I R PSF PSF O O resolution. The lateral resolution given by the width of 2 where α and β are tuned between − 1 and 1, to obtain the the autocorrelation function is better for 3DI SB method minimum entropy given as than imaging using a DL except for the case when the S(p, q) = − φ(m, n) log [φ(m, n)]. imaging condition is satisfied. The axial resolution of 2 φ(m,n) = |C(m,n)|/ |C(m,n)| , where (m,n) are M N the 3DI SB method is lower than that of imaging using a the indexes of the correlation matrix, and C(m,n) is the DL. The videos of the variation in the intensity distribu - cross-correlation distribution. The LRRA is shown in tion for the DL and MDOE are given in Additional file 1: Fig. 4. The LRRA has three parameters namely α, β and Video S1 and Additional file 2: Videos S2, respectively. number of iterations. When α = 1, β = 1, the algorithm is The videos of the variation of the intensity distribution LRA. of the cross-correlation matrix C (x,y,z) for the MDOE The intensity distributions for the test object were when the distance was varied from z = 15 cm to 45 cm simulated for both DL and MDOE for distances from are given in Additional file 3: Videos S3. z = 15 cm to 45 cm, as shown in Fig. 5. The reconstruc - A test object with the words “Structured light” was tion results using NLR reconstruction (α = − 0.3, β = 0.6), selected for the simulation studies. In imaging with a LRA (iterations = 50) and LRRA (α = 0, β = 0.5 to 0.7, single DL, it is expected that the image resembles the object at the image plane and is blurred anywhere else. In the case of MDOE, the image of the object will prob- ably appear distorted for all distances. It is necessary to reconstruct the image by processing it with the PSF as described in “Incoherent Imaging” section. The pro - cessing can be achieved either using cross-correlation between two intensity matrices or by iteratively estimat- ing the maximum likelihood solution as in LRA [25, 26]. In some of the studies involving scattering and with Cas- segrain objective lenses, the NLR was found to exhibit a better quality of image reconstruction [22]. In a recent study [23], the LRRA was applied to distorted images recorded with a Cassegrain objective lens and found to perform better than both LRA and NLR. All three recon- Fig. 6 MSE map for the different cases of Fig. 5 with respect to the struction methods mentioned above have not yet been reference image of the test object obtained by direct imaging using DL applied to an exotic intensity distribution such as the rotating PSF. A nand et al. Nanoscale Research Letters (2022) 17:37 Page 9 of 13 Fig. 8 Schematic of the experimental setup. SLM—Spatial light modulator; f—focal length and D—diameter of the lens the test object using a DL as a reference. The MSE dis - iterations = 6) are shown in Fig. 5. In all these reconstruc- tribution for the different cases is shown in Fig. 6. An tion methods, each transverse plane is reconstructed important observation of LRRA was that it is highly with its own unique PSF. The optimal reconstruction (α sensitive to the relative locations of the I and I . The and β) for NLR was determined when the entropy of the PSF O reconstruction methods LRRA and NLR are compared reconstructed image was minimal. The optimal iteration next when there is a relative horizontal shift error of 5, 10, number for LRA and iteration number, α and β values of 20, and 50 pixels between I and I . The reconstruction LRRA were obtained when the difference between the PSF O results of NLR and LRRA are shown in Fig. 7. It is seen reconstructed image and test object was minimal. The that when there is a shift error, the performance of LRRA number of parameters to be optimized is one (number is significantly affected, while in the case of NLR, there is of iterations), two (α and β), and three (all the above) for only a shift in the location of the reconstructed informa- LRA, NLR, and LRRA, respectively. However, once the tion. The previous success of LRRA [23] on the data from values are optimized, for one case, the values remained the Australian synchrotron can be attributed to the lower almost the same when the experimental conditions were number of pixels in the image sensor (64 × 64 pixels), in not changed. It must be noted that LRRA is obtained by which the chances of error are lower. However, in most applying NLR to LRA, so LRRA does not require any imaging and holography experiments, the scientific cam - special workflow but just an additional parameter to era consists of at least 1 Megapixels. Consequently, LRRA optimize. The library of PSFs is computed by displaying may not be suitable for such cases without additional sta- a point object in the system input. Knowing its number bilization approaches to improve the resilience of LRRA in the library list for the PSF yielding the best in-focus in the presence of errors. image enables one to locate the observed object along the z-axis. Experiments It is seen that NLR always performs better than LRA The schematic of the experimental setup is shown in while LRRA performs better than both. One disadvan- Fig. 8. The same optical setup was used for both 2D tage of LRRA is that it involves a certain number of itera- and 3D experiments. An incoherent source (Thorlabs tions even though significantly lower than LRA, which LED625L, 12 mW, λ = 625 nm, ∆λ = 15 nm) was used can prevent the application of LRRA to real-time imag- for illuminating the object, elements 4, 5, and 6 (both ing as NLR. Another observation is that the performance digits and gratings) of group 5 of negative USAF tar- of LRA for MDOE is better than DL. The mean squared get were chosen as test object for the experiments. error (MSE) was calculated using the direct imaging of The SLM (Holoeye PLUTO, 1920 × 1080 pixels, 8 µm Anand et al. Nanoscale Research Letters (2022) 17:37 Page 10 of 13 Fig. 9 Experimentally recorded I , I , and the reconstruction results using NLR with (α = 0, β = 0.4) and a median filter for the MDOE and reference PSF O images obtained for DL. The scale bar is 0.5 mm pixel pitch, phase-only modulation) was used to modu- For 3D experiments, the fundamental properties of late the light beam by displaying the vortex phase mask incoherent imaging—the linearity and space invariance shown in Fig. 1, along with the lens function having a in intensity were exploited. It is necessary to understand focal length of 14 cm. The distance between the SLM what a 3D object is. When the object points exist along and the digital camera (Retiga R6-DCC3260M, pixel size the z direction, it is a 3D object. Some 3D objects have 4.54 μm × 4.54 μm) was 14 cm. A polarizer was used to points continuously along z while others have points dis- allow light only along the active axis of the SLM. A pin- tributed in only a few planes. In our study, we used the hole with a size of 15 µm was used to record the I s. 2D data recorded at different axial locations to test the PSF The location of the pinhole was shifted by Δ = 0, 1.4, 2.8, 3D imaging capabilities of the method. The images of and 4.4 cm and the PSF library was recorded. The USAF the intensity distributions from four separated planes at object was then mounted at exactly the same locations Δ = 0, 1.4, 2.8, and 4.4 cm are shown in the upper line of as the pinhole and the object intensity distributions were Fig. 10. The reconstructed images using the I recorded PSF recorded. The images of the recorded I and I (the at all four planes are shown in lines 2–5 of Fig. 10. It is PSF O system response to an object in the input) for Δ = 0, 1.4, seen that only the object information at the planes of the 2.8, and 4.4 cm and the reconstruction results using NLR I was in focus, while the information from other planes PSF (α = 0, β = 0.4) followed by the application of a median was blurred. The experiment described in Fig. 10 is iden- filter for MDOE are shown in Fig. 9. The reference images tical to recording the total intensity distribution simulta- recorded for DL Δ = 0, 1.4, 2.8, and 4.4 cm are also shown neously from multiple planes of an object. In the current in Fig. 9. As seen, the images obtained using MDOE are experiment, the out-of-focus and the in-focus intensities sharper in comparison to the ones obtained using DL. are computed in the computer instead of appearing on the sensor. A nand et al. Nanoscale Research Letters (2022) 17:37 Page 11 of 13 Fig. 10 Output images of the system in response to the resolution chart in the input, and reconstruction results using I recorded at the PSF respective planes of the objects found to be sharper than that of the case with DL. But Discussion and Conclusion upon applying NLR and LRRA, the correlation function In this study, a new incoherent imaging system, 3DI SB, became shaper for both cases. However, we expect that is investigated. The concept has been proposed as a pos - in practice, with various kinds of noise, the performance sible replacement for the existing scattering-based 3D of 3DI SB with any reconstruction method will be better imaging systems [17]. The 3DI SB approach has been because the intensity distribution is more concentrated studied only for a beam consisting of two intensity peaks on a small area along with its propagation than the case that rotate about the optical axis. Apparently, the two of the DL. Further studies are needed in both NLR and peaks exist over a long focal depth enabling a higher LRRA to improve the reconstruction results. We believe signal-to-noise ratio than scattering-based imagers and that our study will open a pathway for implementing dif- also a DL. The lateral resolution given by the autocorre - ferent incoherent exotic beams and structured light for lation function was found to be sharper for 3DI SB than multidimensional and multispectral imaging technolo- the case with a DL. The focal depth of the 3DI SB was gies [20]. found to be slightly higher than that of the imager using a diffractive lens. Three different reconstruction methods, namely LRA, NLR, and LRRA, were compared. LRRA Abbreviations and NLR were found to perform better than LRA. For 3DI SB: Three-dimensional incoherent imaging using spiral beams; CAI: Coded aperture imaging; COACH: Coded aperture correlation holography; DL: Diffrac- an ideal case, LRRA performed better than NLR, while tive lens; FINCH: Fresnel incoherent correlation holography; I-COACH: Interfer- for a practical case, NLR performed better than LRRA. enceless coded aperture correlation holography; LED: Light emitting diode; The results shown in Fig. 5 show an interesting result. LRA: Lucy–Richardson algorithm; LRRA : Lucy–Richardson–Rosen algorithm; MDOE: Multifunctional diffractive optical element; NLR: Non-linear reconstruc- When the autocorrelation function of 3DI SB and regular tion; OAM: Orbital angular momentum; PSF: Point spread function; SALCAD: imaging system with DL were compared, the former was Anand et al. 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Nanoscale Research Letters – Springer Journals
Published: Mar 24, 2022
Keywords: Orbital angular momentum; Incoherent holography; Diffractive optics; Imaging; Microscopy
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