The Effect of a Parabolic Apodizer on Improving the Imaging of Optical Systems with Coma and Astigmatism Aberrations

Abstract

We present the results of improving resolution in the imaging of two closely spaced point sources with an optical system under the influence of apodization and different types of aberrations. In particular, we consider the effect of coma and astigmatism, which are well-known aberrations that can deteriorate the resolution of an optical imaging system. Furthermore, a parabolic apodizer was included in an optical system to improve its imaging capabilities. We found that the two-point imaging performance of an optical system with a parabolic apodizer strongly depends on the coherence conditions of incident light. Furthermore, to analyze the efficiency of the parabolic apodizer, we compared the results of two-point imaging obtained with apodized and unapodized optical systems for distances between the two-point sources, less than or equal to the diffraction limit of an optical system. Moreover, the results of imaging the USAF chart with a parabolic apodizer are presented to show the apodizer’s efficacy in single-object imaging. Our results can be applied to the imaging of closely moving structures in microscopy, resolving dense spectral lines in spectroscopy experiments, and developing systems useful for resolving the images of closely associated far-distance objects in astronomical observations.

1. Introduction

High-resolution optical imaging in the presence of various aberrations is significant in modern imaging technologies. Developing a high-performance optical imaging system that can provide expected output images in the presence of different optical aberrations with different techniques and methods is important and technically feasible. Generally, engineering the point spread function (PSF) of optical tools is a technique employed in various potential applications [1,2,3,4,5,6]. A simple way to engineer the PSF of optical tools is achieved with pupil filters [7,8]. Apodization is one of the important approaches or methods presented to improve the performance of optical imaging systems. In this context, selecting an amplitude or phase apodizer with a suitable optical transmission can significantly improve the resolution and contrast that are highly desirable in imaging applications. The resolution of an optical focusing system is restricted due to adverse effects introduced by various optical aberrations and diffraction from the apertures. These effects alter the resultant intensity distribution in an image plane and impose a fundamental limit on the minimum resolvable distance between the two points (objects). In such cases, the Rayleigh resolution criterion has been applied to estimate the resolving power of optical imaging systems under light illumination with varying wave coherency [9,10,11]. In other words, if the diffraction patterns of two-point sources do not coincide and are easily distinguishable, they are said to be well-resolved. If they are closer in distance, their diffraction patterns overlap strongly and are said to be unresolved. According to the modified Rayleigh criterion [9], the two monochromatic indistinguishable point sources can be resolved when the intensity at the midpoint is 25–30% less than the maximum intensity obtained in the resultant composite intensity distribution produced by the circular aperture in an image plane. Furthermore, this concept has also been extended to an optical system with a Gaussian aperture illuminated by a partially coherent light [12]. Generally, aberrations induce phase errors in the wavefront when the light field passes through various components and causes the redistribution of the light field in the focal region of an optical system [13,14,15,16]. A simple and easy way to deal with an optical system with aberrations is to alter the aperture function in the pupil plane that modifies the light transmittance, which modulates the point spread function (PSF) into desired shapes [13,14,15,16,17]. Our previous works demonstrated an aperture engineering technique presented as amplitude apodization applied across the pupil plane that improved the two-point resolution of optical imaging systems under different degrees of coherence light illumination [18,19]. All of these investigations proved that apodization is an efficient approach for tailoring the light intensity distribution in the focal plane of an optical system. Several research works have established the significance of various apodization filters and analytical tools in enhancing or altering the resolution of optical imaging systems [20,21,22,23,24,25]. However, the field is still growing to find more efficient, simple, and cost-effective methods to improve the resolution of an optical imaging system. Recently, phase-only pupil filters characterized as apodization filters combined with discrete adaptive optics have been used as a simple approach for simultaneously compensating aberrations and reshaping the PSF [26].

In the present work, our investigations mainly focused on improving the resolution in the imaging of two equal intensity point sources formed by an optical imaging system that includes a parabolic apodizer, which works effectively under the influence of coma and astigmatism aberrations and varying coherence of point sources. Imaging two closely associated point sources under the influence of aberrations is a challenging objective for optical instruments. Finding suitable apodization for resolving two overlapping point sources under the influence of radially asymmetric aberrations is complex work, and no studies have been reported in this direction. A study reported that spherical aberration has shown a significant effect on the imaging performance or two-point resolution of optical imaging systems [19]. Please note that the amplitude response of the optical system with coma and astigmatism is asymmetric, whereas the optical system with spherical aberration yields a symmetric amplitude response. Therefore, the complexity involved in separating two-point sources in the image plane is greater in the presence of coma and astigmatism relative to spherical aberration. However, the resolving power of the optical imaging system under coma and astigmatism will be enhanced through the use of suitable apodization filters, which is a significant advancement that we achieved in this work. Generally, coma is an off-axis aberration that creates comet-like intensity distribution translated away from the optical axis. The lens pupil with a coma aberration forms light field distributions with a sharp center, and the edges of the light field happen to be blurred. A lens pupil with astigmatism can focus a light field at different planes located off the optical axis. The imaging of two overlapping point sources under the influence of such aberrations is subject to undesired effects in the image plane. A suitable apodization filter that was selected for dealing with this hard problem is a parabolic apodizer that aims to improve the imaging performance of optical systems by tailoring the resultant intensity distributions of two overlapping point sources. Therefore, we explored the efficacy of the parabolic apodizer in enhancing the two-point resolution or resolving power of the optical imaging system under the strong effect of these asymmetric aberrations. Furthermore, the role of varying coherence conditions has also been analyzed. To understand the effects in the image plane, we calculated the composite intensity distributions of two overlapping point sources formed by an optical imaging system under various situations and coherence conditions. Moreover, we showed the efficiency of the parabolic apodizer in single-object imaging by numerically calculating the images of the USAF chart under coma and astigmatism. However, this is not a primary objective for the present study to achieve. For better USAF imaging, apodization filters whose optical transmission should be opposite to the parabolic apodizer are highly desirable in investigations.

2. Theoretical Descriptions

The optical field in the focal plane or observation plane of a two-dimensional optical system can be given as:

$$G(\rho ,\theta )\hspace{0.17em}=-\frac{ik}{2\pi f}{\displaystyle \underset{0}{\overset{R}{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}A(r)\hspace{0.17em}g(r,\phi )\exp \left[-i\frac{k}{f}r\rho \cos (\theta -\phi )\right]r\mathrm{d}r\mathrm{d}\phi }},$$

(1)

A(r) is the input field distribution of a plane wave, f is the focal length of a lens pupil, $k=2\pi /\lambda$ is the wavenumber, ‘λ’ is the wavelength of an incident light field, and g(r,φ) is the pupil function of an optical system. ‘r’ is the distance of the point on the exit pupil system from the axis and ‘R’ is the radius of the focusing system in the optical scheme. Therefore, the amplitude transmittance of the proposed amplitude filter at the exit pupil varies with the radial coordinate ‘r’. Coma and astigmatism are aberrations investigated for which the wavefront in the exit pupil alters with radial distance (r) and polar angle (φ). Therefore, these aberrations yield an asymmetrical amplitude response, as shown in the resultant optical images (Section 3.2). Here, the light fields of two-point sources are separated by a distance equal to or less than the diffraction limit of the optical system. For simplicity, we have considered equal intensity point sources. When the light field distributions of two-point sources either coherently or incoherently interfere, evaluating the Rayleigh criterion is not an accurate assessment for two-point resolution, as it will not reveal essential characteristics and features of the diffracted light field. Therefore, it is necessary to compute and evaluate the diffracted light field distribution to assess two-point imaging under the illumination of coherent and incoherent lights.

To include the effect of parabolic apodization on the imaging of an optical system, we have considered a two-dimensional radially symmetric amplitude filter, whose functional form can be written as:

$$P(r)={\left(r/R\right)}^2$$

(2)

To better understand the functionality of the parabolic apodizer, the amplitude transmittance as a function of the radial coordinate (r) is shown in Section 3.2. We introduced optical aberrations in the pupil plane of an apodized optical system to study the effect of aberrations, such as astigmatism and coma. The apodizer is assumed to coincide with the lens at the exit pupil of the optical imaging system, elected and oriented to provide the effect of aberrations and apodization together. Therefore, all effects of the incident wave coherency, aberration, and apodization are evaluated in the pupil function term.

For astigmatism, we introduced the following functional form:

$$g_a(r,\phi )=\exp \left[iw2\pi {\left(r/R\right)}^2{\sin }^2(\phi )\right].$$

(3)

For coma, we introduced the following functional form:

$$g_c(r,\phi )=\exp \left[iw2\pi {\left(r/R\right)}^3{\sin }^3(\phi )\right].$$

(4)

We performed the study for w = 0.3. Here, ‘w’ is the constant that controls the wave aberration. It is clear that for higher values of ‘w’, the aberration effect increases, resulting in the resolving power of the optical system degrading more, and the apodization effect becomes insignificant. However, for USAF calculations, we investigated the parabolic apodizer efficiency for different ‘w’ values representing apodizer performance underneath different amounts of aberrations.

However, if the lens is illuminated by a Gaussian wave, with a beam waist of ${\sigma }_0$ and field distribution of the form, $\exp \left(-\frac{x^2}{2{\sigma }_0^2}\right)$ the radius of the focal spot would be ${\sigma }_{0f}$ [10,11] evaluated with the following expression:

$$\sqrt{2\pi {\sigma }_0^2}\exp \left(-\frac{u^2}{2{\sigma }_{0f}^2}\right), {\sigma }_{0f}=\frac{\lambda f}{2\pi {\sigma }_0}.$$

(5)

3. Numerical Modeling

For the two-point resolution study, we used the following parameters of an optical system for the numerical calculations: λ = 532 nm, R = 1 mm, f = 300 mm. Following the standard diffraction theory [10,11], we evaluate the diffraction-limited spot size radius ${\rho }_0\approx 0.08 \mathrm{mm}$ for illumination with a plane wave. The resultant intensity distribution in the focal plane can be utilized in calculating the diffraction limit of an optical imaging system in the absence of apodization and aberrations.

3.1. Modulation Transfer Function (MTF)

The modulation transfer function (MTF) is an important parameter commonly used to compare the performance of optical systems. It explains the ability to transfer the contrast from the object to the image at a specific resolution. Therefore, to understand the performance of an optical system under the effect of aberrations and the selected apodization technique, we calculated the MTFs, as shown in Figure 1. Generally, the modulation transfer function is calculated from the point spread function (PSF) of an optical imaging system under different optical situations. Calculating the MTF helps to understand the relevance of the proposed apodization that can be applied to separate the diffraction patterns of two overlapping point sources. As illustrated in Figure 1, the value of the MTF for an apodized optical system is higher at a higher spatial frequency. Thus, the amplitude filter (apodizer) provides a higher spatial frequency under the diffraction limit subject to altering the imaging of the optical system, particularly while dealing with the two-point resolution problem [19]. In the presence of astigmatism and coma aberrations, the parabolic amplitude filter also provides a similar effect (we investigated the direction of φ = π/2). Note that these plots are obtained by considering an incoherent optical system. However, this apodization mechanism is not a good choice for the classical imaging system when we use images. However, this approach is realized to perform better than other apodization mechanisms investigated for separating overlapping point sources [18,19]; the investigation details are reported in Section 3.2.

3.2. Results and Discussion

To understand the performance of a parabolic apodizer in single-object imaging, we performed numerical calculations to generate USAF images formed by the optical system in the presence of a parabolic apodizer and optical aberrations. For imaging the object plane (USAF chart), we considered the following apodized optical system underneath the different optical situations addressed in this section.

The wavelength is set as λ = 532 nm to illuminate the USAF chart inserted in the object plane. The schematic of the optical system is shown in Figure 2. The lens L₁ is located at a distance equal to its focal length (f₁ = 300 mm) and collimates the diverging wavefront from the object plane. The lens L₂ is located at the f₁ + f₂ distance from the lens L₁ and at the f₂ distance from the image plane, which is the focal length of the lens L₂. The apodizer is assumed to coincide with the lens L₂, which is the exit pupil of the apodized optical system. The light transmittance through the exit pupil of an optical imaging system is determined by the amplitude transmittance of the parabolic apodizer, the lens L₂ focal length (f₂ = f₁), and the illuminating wavelength λ.

Next, the optical scheme shown in Figure 2 can obtain the composite intensity distributions of two closely associated point sources with parabolic apodization, where object 2 represents the input fields and image 2 represents the output field, which is an important aspect of the present study. The lens L₁ collimates the light field from two-point sources located at the object plane (the input light field is created with diffraction patterns of two-point sources that are overlapped in close vicinity) and directs the resultant light field towards the exit pupil of the imaging system. The light field distribution across the exit pupil can be modified by the parabolic apodizer, and the resultant light field is focused by the lens (L₂) into the image plane. The apodized light field distribution focused on the Camera sensor can be qualitatively analyzed. Therefore, the resultant light intensity distributions are detected in the form of two-point images (or diffraction patterns) that are said to be well-resolved or separated from each other.

Generally, employing the apodizer suppresses optical sidelobes or engineering the PSF of the resultant diffracted field at the exit pupil of the optical system. This effect suppresses the portion of the light field or modifies the spatial intensity distribution of the light field diffused in the imaging plane. For example, a well-known Gaussian apodizer is characterized for far-field and PSF engineering [12,27], unlike a parabolic apodizer [18]. With a parabolic apodizer, the light transmittance increases radially at the exit pupil of an optical system. This apodizer has been proven effective in separating the images of two closely overlapped point sources because the apodizer can reduce the focal spot in all directions regardless of high-intensity optical sidelobes in an image plane. This apodizer shows some effectiveness in single-object imaging in the presence of aberrations, and it is trivial. However, to quantify these minor improvements in the imaging of a standard USAF object, we calculated the diffraction efficiency in different situations, as demonstrated in Figure 3. The diffraction efficiency (η) is the ratio of the transmitted light intensity through the apodizer into the image plane to the light intensity incident on the apodizer from the object plane. The calculated results are evident in understanding the overall relevance of the parabolic apodizer in single-object imaging. We performed the study for different values of w, and this is the parameter that controls the amount or magnitude of coma and astigmatism aberration. This study shows that the amount of light flux diffused into the image plane decreases with parabolic apodization. Therefore, the diffraction efficiency was evaluated with small values in all of these observations. The top row of Figure 3 represents output images of the USAF object formed by the apodized optical system under the influence of coma aberration.

The parabolic apodizer attenuates the low-frequency components of the light field that pass through its center and blocks a significant portion of the USAF image with coma aberration, as seen in Figure 3. Note that the output images included in the top row confirm that the diffraction efficiency (η) considerably improved by coma relatively with the output images with astigmatism depicted in the bottom row of Figure 3. In the presence of coma aberration, apodized output images were found with higher diffraction efficiency (η), and a similar trend was noticed for different w values. However, an apodized output image formed without aberrations has a low diffraction efficiency (η) equal to 0.86%, as shown in Figure 2. Specifically, in the case of astigmatism aberration, even though there is a high loss in intensity and diffraction efficiency for output images formed by the apodized optical system, the resultant output images preserve most features of an original USAF chart or the object, as illustrated in Figure 3. Therefore, the parabolic apodizer was proven to be effective to a certain extent in the imaging of an object underneath astigmatism aberration. Note that apodizers useful in PSF engineering can modify the output of an optical system by suppressing sidelobes or sharpening the central peak or zero-order maximum and are recommended for achieving high-resolution images of an object such as ‘USAF’.

Next, we investigated how well images of two overlapping points are resolved or separated, and this approach is described as two-point resolution. According to the Rayleigh resolution criterion [9], two-point sources are said to be just resolved when the central peak of one diffraction pattern coincides with the first minimum of the other diffraction pattern. However, for best characterizing two-point resolution under the illumination of coherent and incoherent light, the Rayleigh resolution criterion is insufficient to disclose essential characteristics of the resulting light field distribution in the focal region of an optical system under the influence of aberrations and wave coherency conditions. This crucial problem in physical optics is solved by computing composite image intensity distributions of two overlapping point sources that are mutually incoherent or coherent with each other, as achieved in the present study. Computing these composite intensity distributions under different situations is an important aspect of the present study. Following Equation (1), amplitude apodization was applied to assess the composite intensity distributions of two-point sources formed by an optical system in the presence of coma and astigmatism aberrations. The resultant composite intensity distributions are attained and are displayed in Figure 4, Figure 5 and Figure 6. The standard approach has been followed to calculate composite intensity distributions for coherent and incoherent cases [18,19]. Based on this approach, we controlled the degree of coherence for two-point sources radiating with equal intensities [18,19].

Figure 4 illustrates the composite intensity distributions of two mutually incoherent point sources formed by an optical system, where two points are separated by a distance equal to the diffraction limit (ρ₀~0.08 mm) [28]. In the presence of aberrations, the two-point sources are well-resolved or separated by the apodized optical system relative to an unapodized optical system, as revealed in Figure 4. The results shown in the first column of Figure 4 represent composite intensity distributions formed by the aberration-free optical system. In this case, the performance of the apodized optical system in separating images of two overlapping point sources (Figure 4g) remains identical to the ability of an unapodized optical system, as shown in Figure 4a. Moreover, the effect of amplitude apodization in resolving the images of two-point sources formed by an optical system suffering from aberrations is presented in the second and third columns of Figure 4. For unapodized pupils, the two-point sources are regarded as just resolved under astigmatism aberration, as evidenced by a shallow dip in the resultant intensity distribution illustrated in Figure 4b,e. A parabolic apodization across the pupil plane redistributes the resultant intensity distribution in an image plane, producing a clear separation evidenced by a shallow dip located midway between two zero-order maxima, thus gaining a lower-intensity value than that of the dip detected in the unapodized case, and the resultant apodized composite intensity distribution is displayed in Figure 4h,k. In this case, two-point sources are regarded as well-resolved. However, the resultant composite intensity distribution appeared with non-zero first minima, and the enhanced sidelobe region is an apparent effect triggered by parabolic apodization. A similar trend was noticed for the two-point sources formed by an apodized optical system with coma aberration. With a parabolic apodizer, the composite intensity distribution of two-point sources formed in an image plane is substantially redistributed, and the same is shown in Figure 4i,l. The presence of a shallow dip with the lowest intensity value in the composite intensity distribution manifests that the two-point sources are well-resolved relative to the unapodized composite intensity distribution of two-point sources as illustrated in Figure 4c,f. Specifically, under coma aberration, two-point sources formed with a parabolic apodizer are considered super-resolved or extremely separated regardless of the enhanced sidelobe region (first-order diffraction maximum with increased intensity) detected in the resultant composite intensity distribution. These observations show that the proposed amplitude apodization technique has proven effective in separating images of two-point sources under various optical situations.

Figure 5 shows composite intensity distributions of two mutually incoherent point sources, separated by the distance (0.8ρ₀~0.065 mm), which is less than the diffraction limit (ρ₀~0.08 mm). The resultant composite intensity distributions illustrated in Figure 5 reveal that the apodized optical system separated images of two-point sources under astigmatism and coma aberrations, even though the two-point sources are closely spaced, i.e., strongly overlapped diffraction patterns. In Figure 5, the first column represents composite intensity distributions of two-point sources formed by the aberration-free optical system. It can be observed that the separation of two-point images formed by the apodized optical system shown in Figure 5g remains analogous to the separation of two-point images found in an image plane of an unapodized optical system as shown in Figure 5a. A dip in resultant composite intensity distributions with the same lower intensity is shown in Figure 5d,j, which establishes similar performance in both cases, and two-point sources are regarded as just resolved. However, the performance of a parabolic apodization in resolving the two-point sources separated by a distance less than the diffraction limit (ρ₀) proved to be effective with aberrations, and corresponding results are shown in the second and third columns of Figure 5.

In the presence of astigmatism, the peak area of the composite intensity distribution is relatively flat, representing no indication of separated images, as illustrated in Figure 5b,e. Therefore, in this case, two-point sources are regarded as unresolved. However, with parabolic apodization, the performance of an optical imaging system was improved in forming the spatially redistributed intensity distribution consisting of two zero-order maxima with non-zero first minima. In this case, a clear dip is noticed across the peak area of the resultant intensity distributions, and the dip observed is steeper than that of the unapodized case, as depicted in Figure 5h,k. In this case, two-point sources are said to be resolved regardless of the enhanced sidelobe region. A similar trend was noticed in the composite intensity distributions formed by the apodized optical system with coma aberration. Figure 5c shows the 2D image, and Figure 5f displays a curve plot representing unapodized composite intensity distributions for easy comparison with the apodized case. With the parabolic apodization, intensity distributions corresponding to the zero-order maximum are substantially redistributed at the cost of enhanced sidelobe regions in the image plane, and the same is shown in Figure 5i,l. However, in this case, the shallow dip formed in the resulting composite distribution establishes the separation of two-point sources in the image plane and the shallow dip located midway between two zero-order maxima of the apodized composite intensity distribution. As a result, two-point sources are said to be well-resolved or highly separated regardless of high sidelobe levels. Finally, it is essential to understand that the parabolic apodization technique is potentially important to provide beneficial imaging results under such optical situations.

Figure 6 illustrates composite intensity distributions of two-point sources that are mutually coherent, separated by the distance equal to the diffraction limit (ρ₀). In Figure 6, the results displayed in the first column represent composite intensity distributions formed by an aberration-free optical system. In particular, for an unapodized optical system without aberrations, images of two mutually coherent point sources varying with equal intensities show a tendency to separate or resolve. This effect is established with a clear presence of a dip located midway between composite intensity distributions, as shown in Figure 6a,d. In contrast, the composite intensity distribution with parabolic apodization is shown in Figure 6g,j. In this case, a shallow dip was observed midway between the resultant composite intensity distribution, indicating a separation of two-point sources in the resultant intensity distribution. These results established that the two coherent point sources are well-separated or resolved regardless of increasing sidelobe intensities, which is the latter property. Note that the intensity levels of sidelobes are relatively high under the influence of coma compared to the unapodized optical system with astigmatism. However, these optical sidelobe intensities are increased using parabolic apodization.

For an unapodized optical system with astigmatism, two-point sources showed a tendency to resolve or separate, as illustrated in Figure 6b,e. However, this situation corresponds to separation, or the presence of a dip with a lower intensity value was improved for the unapodized optical system with coma aberration, and the same is demonstrated in the resultant composite intensity distribution shown in Figure 6c,f. By applying parabolic apodization, intensity profiles corresponding to two zero-order maxima in the composite intensity distribution are redistributed, as evidenced by a shallow dip in the resultant intensity distribution detected in an image plane, and this effect is illustrated in Figure 6k,l. Hence, the diffracted patterns of two overlapping point sources are regarded as well-resolved or separated, and corresponding 2D images are shown in Figure 6h,i. Moreover, with apodization and coma aberration, the light flux enclosed in two zero-order maxima was significantly reduced while separating the mutually coherent diffraction patterns of two-point sources. Finally, it was concluded that applying a parabolic apodizer proved to be effective in separating the individual diffracted patterns of two-point sources formed by an incoherent optical system relative to a coherent optical system. However, the improvements that we achieved in the case of a coherent optical system with parabolic apodization and aberrations are desirable in imaging applications to obtain beneficial results, and they are also important in developing coherent imaging tools.

The variation in dip intensities as a function of distance separation between two coherent or incoherent point sources under the influence of aberrations and apodization has been quantitatively investigated. The corresponding results are illustrated in Figure 7. The composite image intensity distribution of two-point sources formed by the optical imaging system has the lowest intensity dip, proving that two-point sources are well separated in the image plane. The dip intensity values are subject to change linearly with the distance separations w.r.to the diffraction limit in both incoherent and coherent cases under different optical situations. The variation in the dip with lower intensity values across the resultant intensity distributions of two-point sources under incoherent illumination (Figure 7a) is significant relative to the variation of dip intensities across intensity distributions of two-point sources under coherent illumination (Figure 7b). This justifies that parabolic apodization is more effective in bringing significant changes in the performance of incoherent optical imaging systems than that of coherent optical imaging systems while resolving two closely associated point sources. Specifically, Figure 7a illustrates the dip intensity values in the resultant composite intensity distributions formed by an incoherent optical system in the presence of coma and astigmatism aberrations. With the apodization and coma (red-dashed line), the dip intensity value changes from ~0.35 to ~0.085 when two-point sources are separated by a distance equal to ~0.08 mm or 80 µm (diffraction limit). Due to the appearance of a shallow dip, the separation between two-point sources remains excellent, i.e., well-resolved two-point sources are formed in the image plane. When the distance between two-point sources approaches ~0.065 mm or 65 µm (less than the diffraction limit), the dip lower intensity changes from ~0.595 to ~0.21. These observations conclude that a dip with lower intensities in the resultant image intensity distribution establishes diffracted patterns of two-point sources separated in an image plane. Therefore, two mutual point sources formed by an incoherent optical system are regarded as well-resolved. It is clear that with parabolic apodization, the imaging performance of an optical system under the effect of coma and astigmatism aberrations significantly improved.

Furthermore, with astigmatism and apodization (green-dashed line), the dip with the lower intensity value changes from ~0.675 to ~0.492 for a distance separation equal to ~0.08 mm or 80 µm. The dip with the lower intensity value changes from ~0.972 to ~0.794 as the distance separation between overlapping two-point sources is set as ~0.065 mm or 65 µm. A similar trend is noticed for the coherent optical system, and the variation in dip intensities for different distance separations is shown in Figure 7b. For example, with the coma and parabolic apodization (red-dashed line), when a distance separation equals ~0.08 mm or 80 µm, a dip is found in the resultant composite intensity distribution, and its lower intensity approaches 0.080. However, in the presence of astigmatism, the dip with the lower intensity obtains the value of ~0.928, which was altered to ~0.707 when employing parabolic apodization. From these observations, it was realized that the impact of parabolic apodization on improving the imaging performance of the aberrated incoherent optical system is superior to that of a coherent optical system with aberrations. However, improvements in the performance of a coherent optical imaging system achieved with parabolic apodization and aberrations are potentially important to developing apodized optical imaging systems that can be applied in coherent imaging experiments.

4. Conclusions

Parabolic apodization and aberrations are highly effective in improving the imaging of two closely associated point objects, i.e., whose individual diffraction patterns closely overlapped. In the presence of coma and astigmatism aberrations, parabolic apodization attained significant developments across the resultant image intensity distributions of two-point sources detected in the image plane. Apodized optical systems were applied for the direct imaging of two overlapping point sources under coherent and incoherent light illumination. It was found that the imaging performance of an aberrated optical system strongly depends on parabolic apodization and the degree of spatial coherence of the incident light. The figure results and curve plots presented in this work have proven that parabolic apodization is more effective in improving the two-point resolution of incoherent optical systems than that of coherent optical systems. However, the improvements obtained in the coherent case are potentially important to developing more promising apodization systems in coherent optics. For USAF imaging, in the presence of astigmatism, even though there is a high loss in intensity and diffraction efficiency for output images formed by the apodized optical system, the resultant output images preserve most features of an original USAF chart or the object. Therefore, the parabolic apodizer was proven to be effective to a certain extent in imaging USAF in the presence of astigmatism aberration. However, these USAF results were trivial in understanding the efficacy of the parabolic apodizer. Mainly, parabolic apodization worked effectively in improving the two-point resolution in both coherent and incoherent cases, so it is anticipated that this apodizer will improve the two-point resolution of aberrated optical imaging systems under partially coherent illumination. The numerically simulated results presented in this study pave new ways for the design of pupil functions with suitable optical transmission functions that can be utilized to meet specific imaging applications and imaging conditions. Parabolic apodization across the objective lens of an optical microscope may improve the imaging of closely associated or moving structures in biological specimens. The parabolic apodizer can be useful in resolving closely moving or located bright objects in Astronomical observations. Specifically, the optical transmission profile of a parabolic apodizer is simple and easy to develop with well-established fabrication technologies. Furthermore, we plan to fabricate the parabolic apodizer filters to investigate their efficacy in real-time coherent and incoherent imaging experiments.