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Abstract
We introduce a novel approach in optical engineering by combining Dammann gratings with binary Fresnel zone plates to create a unique hybrid optical element with enhanced energetic efficiency of its focal spots. Traditionally, binary Fresnel zone plates focus light at multiple points with varying intensities, while Dammann gratings are renowned for their efficient and uniform light splitting capabilities. Our innovation lies in merging these two elements and generating a binary circular Dammann (varying along the radial direction) Fresnel zone plate that concentrates most of the incident light into a small and desired number of focused points with equal intensities, rather than distributing light’s energy non-equally across multiple points. This novel design significantly enhances the efficiency and precision of light manipulation. It opens new possibilities in applications requiring high-intensity focal points, such as in advanced medical imaging and in accurate scientific measurements. By redefining the conventional roles of these optical elements, our research contributes an advancement to the field, paving the way for innovative solutions in various optical applications.
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1. Introduction
The concept of binary Fresnel zone plates (FZPs) represents a significant advancement in the field of optics, offering a unique approach to light manipulation and focusing. These devices are grounded in the principles of diffraction and interference, fundamental phenomena in wave physics [1]. Fresnel's initial work into the study of wave optics led to the development of the Fresnel equations, which describe the behavior of light as it traverses different media. This foundation paved the way for the conceptualization of the Fresnel zone plate, which exploits the principles of constructive and destructive interference to focus light [1]. Unlike traditional lenses that rely on refraction, FZPs use a series of concentric rings, typically alternating between opaque and transparent zones, to diffract light. These rings correspond to the Fresnel zones, each of which causes light to converge or diverge, thereby focusing or defocusing the wavefront [2].
In practical applications, FZPs have been employed in a wide array of fields ranging from microscopy to astronomical telescopes. Their ability to focus light without the chromatic aberrations caused by material dispersion typical of refractive lenses as well as their feasibility for implementing binary and flat optical structures, makes them particularly useful in ophthalmic as well as in imaging applications. Lensless imaging with FZPs [3], currently represents a hot topic of research.
One significant application of binary FZPs is in X-ray microscopy. Due to the difficulty in fabricating lenses that can refract X-rays effectively, FZPs offer an alternative method to focus X-rays for high-resolution imaging [4,5]. This application is particularly relevant in medical imaging and materials science, where the high penetrating power of X-rays is harnessed for detailed internal imaging [6,7]. With respect to X Ray imaging, it is also noteworthy to highlight that FZPs have been successfully employed to implement achromatic focusing [8]. In astronomy, FZPs have been explored as means to focus and image distant astronomical objects. Their lightweight and potentially large aperture size compared to traditional glass lenses make them attractive for space telescopes [9,10]. The ability of FZPs to focus light across a broad spectrum also allows for multi-wavelength observations, crucial in astrophysical studies [11]. Another flourishing area of application for FZPs is in the field of photonic integrated circuits (PICs). As the demand for faster and more efficient optical communication systems grows, FZPs provide a means to manipulate light at a microscale, crucial for the development of compact and high-performance PICs [12–14]. Recent advancements in nanofabrication techniques have further expanded the potential of FZPs. By fabricating zone plates with nanoscale precision, researchers have achieved unprecedented focusing capabilities, opening new avenues in nanoscopy and nanolithography [15]. Recent work on refractive index manipulation with nanoscale xy precision by tip-enhanced femtosecond pulsed laser irradiation may further expand this field of research [16].
Despite their advantages, the fabrication and utilization of FZPs also present a series of challenges. One significant issue is the efficiency of these plates. Because of their diffractive nature, a significant portion of the incident light is lost or scattered, leading to lower overall efficiency compared to refractive lenses [17]. Researchers are actively exploring ways to enhance the efficiency of FZPs, for example, by employing phase-correcting techniques or optimizing the geometry of the zones [18,19,20]. This is especially relevant in the case of binary FZPs where the incident energy is divided between a large number of focal planes, many of which have no specific role for the application at hand.
The Dammann grating, a binary grating optimized for splitting an incident beam into multiple output beams, has been extensively studied and utilized in various applications due to its ability to generate equal-intensity, equal-distanced multiple beams [21,22]. This feature makes it an attractive candidate for integration with FZPs, which can focus light but struggle with issues of efficiency. Traditional FZPs, while effective in generating several axial focal points, often suffer from unequal intensities among these focal points. The introduction of Dammann gratings into the circular phase plate, forming the Dammann zone plate (DZP), addresses this issue by producing a series of axial focal points with equal intensity [23]. This equalization of intensity is crucial for applications where uniformity in the focal plane is paramount. The DZP, by virtue of its design, ensures that the light intensity is consistent across all focal points, thus enhancing their efficiency and utility in applications such as three-dimensional imaging and optical communications [24]. Moreover, the DZP can generate energy distribution within a smaller number of focal points as might be desired in some applications, especially those associated with medicine. Given the above, the combination of Dammann gratings with Fresnel zone plates presents a unique opportunity to enhance the capabilities and efficiency of a wide palette of optical systems.
The versatility of Dammann gratings, when combined with Fresnel zone plates, broadens the spectrum of potential applications of the two technologies. From splitting signals in fiber optics to improving femtosecond laser pulse characterization, the application range of DZPs extends to various fields in photonics and beyond. This versatility is crucial for driving innovation in areas where traditional optical elements fall short.
The innovation presented in our paper lies in the unique combination of Dammann gratings with Fresnel zone plates, not merely to create a series of equally intense focal points, but rather to concentrate most of the incident light into a single, highly defined focal point. This approach marks a significant departure from the traditional applications of Dammann gratings and Fresnel zone plates, where the primary objective has often been to distribute light uniformly across multiple (transversal in regular cases and axial in our case) points. Our methodology leverages the inherent strengths of Dammann gratings in light distribution and Fresnel zone plates in focusing capabilities, but with a refined focus on maximizing the intensity at a singular point. This targeted concentration of light is achieved by manipulating the patterns of the Dammann grating in conjunction with the focusing geometry of the Fresnel zone plate, resulting in a hybrid optical element that directs a predominant portion of light to a specific focal point. This novel design presents an important advancement in optical engineering, offering enhanced capabilities for applications that demand high-intensity focal points, such as in laser machining, medical imaging, and precise scientific measurements. By redefining the conventional roles of Dammann gratings and Fresnel zone plates, our paper introduces an innovative paradigm in the manipulation and controlling of light in various optical technologies.
2. Fundamental principles
Designing a new circular Dammann Fresnel zone plate requires combining two key technologies: Dammann gratings and Fresnel zone plates and using a different approach of focusing most of the incoming light into a single point instead of spreading it evenly across many points. The goal of this design is to understand and effectively merge the unique properties of both components to create a device that can precisely focus light.
Dammann gratings are specialized optical tools designed for evenly splitting light into multiple beams. These gratings work on the far-field diffraction principle, meaning they manipulate light at a distance, and are used in technologies like optical fiber communications, laser pulse shaping, and creating three-dimensional light arrays. Essentially, these gratings are phase-only modulators optimized for equal-intensity spot generation at different diffraction orders.
We consider a Dammann grating with the phase value of 0 or π and a period of T. The coordinates of the phase jump points of the 1D grating in each period are al, bl (l = 1, 2, … L), where L is the number of phase jump points in the positive region of x0 (the structure is symmetric around x0 = 0). The corresponding transmittance function is written as [25].
Fresnel zone plates are specialized lenses designed to focus light. They consist of many closely spaced rings, called zones, and are effective in concentrating light into specific points along their main axis. While these lenses can create multiple focal points, the light’s intensity often varies between these points, especially at higher orders.
Among different roles, the Dammann grating can also serve as a binary amplitude grating, exhibiting a similar structure as in other use-cases, but with transmission values of 0 and 1.
If we consider a small zone plate compared to the focal length, the central radius of every ring can be approximated as
where $n$ is the number of the ring having radius of $r_n$ and $f$ is the focal length of the zone plate. $\lambda $ is the optical wavelength.Following this, a binary amplitude grating can be regarded as a square wave profile with a period of $2\lambda f$ when expressed as a function of ${r^2}$, where r is the radius. Considering the transmission cross section of an amplitude binary Fresnel zone plate as a function of its radius square, t(${r^2}$) as showed in Fig. 1:
Fig. 1. Transmission cross section of an amplitude binary Fresnel zone plate as a function of its radius square.
Thus, due to the periodicity of t(r2) it can be expressed using Fourier series (t(r2) is a continuous and a periodic function and thus it could be developed to Fourier series):
If a single period of $t({{r^2}} )$ is rectangular, as depicted in Fig. 1, then An can be expressed as :
3. Design integration
The circular Dammann Fresnel zone plate (CDFZP) integrates the principles of both Dammann gratings and Fresnel zone plates and aims to concentrate the majority of incident light into a single, highly defined focal point. The challenge lies in aligning the uniform light distribution capabilities of the Dammann gratings with the focusing geometry of the Fresnel zone plate. Theoretically, this involves manipulating the Dammann grating patterns to work cohesively with the concentric ring structure of the Fresnel zone plates. This integration aims to manipulate the Dammann grating patterns with the Fresnel zones to ensure that light is efficiently focused. This is done by designing the transition points of the Dammann grating so that the light from the multiple diffraction orders of the Dammann grating is constructively interfered at the focal point of the Fresnel zone plate. This involves adjusting the grating period and the design of the Dammann grating to ensure that the majority of the light beams converge at the Fresnel zone plate’s focal point.
The first step is to optimize the design for the wavelength of interest, as the focal properties are wavelength dependent. For our experiments we chose λ = 532 nm, a wavelength of interest in diverse applications relevant for FZPs, e.g. [26,27]. Next, we chose the focal length of our CDFZP to be f = 3 cm. Taking these into account we determine that the period of the square wave profile when expressed as a function of ${r^2}$ is 0.03192 mm2.
So, assuming we know the transition points for the Dammann grating we can use the following equation to determine the transition points for the CDFZP.
In the next section we discuss how to obtain the transition points for the Dammann grating.
4. Focusing and energetic efficiency
The proposed hybrid design maximizes the light intensity at a single focal point through the precise optimization of the CDFZP’s transition points and the FZP’s focusing geometry.
The design must ensure that most of the incident light passes through the CDFZP and is directed and concentrated at the focal point. In order to achieve this, we introduce two key parameters ${\eta _1}$ and ${\eta _2}$, which are central to the design and functioning of this element. The parameter ${\eta _1}$ represents the energy of the first diffraction order relative to the other diffraction orders. This ratio is crucial as it determines the strength of focus at a single point. In the context of our hybrid grating, maximizing this ratio assures us that a significant portion of the incident light is concentrated at the desired focal point, enhancing the focusing contrast. We define ${\eta _1}$ as:
The ${\eta _2}$ parameter measures the energetic efficiency of the hybrid grating and is indicative of how effectively the hybrid grating utilizes the incident light to achieve the desired focusing effect at single focus (at focal length of f, which is the energy of the first Fourier coefficient A1). ${\eta _2}$ can be expressed as follows:
By combining these two parameters in a product we can account for both the (i) focusing effect and (ii) the energetic efficiency of the hybrid design in the parameter $\eta = {\eta _1}{\eta _2}$.
In the context of optimizing the imaging properties of the hybrid design, a parameter sweep of the transition points, designated as x1, x2 and x3, was conducted (we assumed only 3 transitions points per period of the Dammann). These parameters were constrained withing the range 0 to 1, adhering to the inequality x1 < x2 < x3. This constraint ensures a methodical exploration of the parameter space, facilitating a thorough investigation of the hybrid grating’s behavior under varied values of the transition points.
For each unique triplet of transition point values, the corresponding Fourier coefficients were computed. Following this, $\eta $ was calculated for each set of parameters.
The optimization process was directed towards identifying the set of parameters that yield the maximal value of $\eta $, which represent the most optimal configuration for the hybrid grating. The process included search over all possible options within given range of values imposed by the physical constraints of the problem. This optimal set was further refined by the condition that the difference between any two consecutive parameters must exceed 0.05. This constraint was imposed to avoid overpassing the maximum resolution during the fabrication process which included printing the mask on a slide with a high-resolution printer.
5. Results
In Fig. 2, we present the characteristics and specifications of the CDFZP with the highest ability to focus 532 nm light at a focal length of 3 cm.
Fig. 2. Schematic representation of the computed CDFZP with superior focusing ability. The diameter of the CDFZP is 1.6 mm.
The transition points for the CDFZP play a crucial role in its focusing efficiency. For the CDFZP in Fig. 2 the transition points are ${x_1} = \; 0.05263$, ${x_2} = \; 0.10526$, ${x_3} = \; 0.52631$.
The Fourier coefficients of the CDFZP are presented in Table 1.
Table 1. Normalized Fourier coefficients for the CDFZPs with the highest score
These coefficients are normalized relative to the A0 coefficient (no focusing ability), facilitating an easier and more effective comparison of their relative magnitude. The efficiency of the CDFZP is quantified by the parameter $\eta $, calculated to be 37.28. The worst $\eta $ value that was obtained during the optimization process was 0.00403.
Note that the values for $\eta $ are not the energetic efficiency in percentages but rather the product between the expressions of Eqs. (7) and (8).
Obviously the higher the value of $\eta $ is the better it is (more energy in the first diffraction order and in the transmission of the grating in general). The optimization process showed an improvement of about 4 orders of magnitude between the worst and the best values. The plot of a single period of the optimized grating can be seen in Fig. 3.
Fig. 3. Single period of the optimized CDFZP grating plotting it transmission versus r2 (r is the radial coordinate).
6. Simulation
Next, we employed thorough simulations with the diffractsim Python library [28] to emulate the propagation of 532 nm light through CDFZPs with the characteristics discussed in the previous section. This computational model was essential to evaluate and validate the optical behavior of the CDFZPs. The results, depicted in Fig. 4, illustrate the focusing properties of the CDFZPs, highlighting their performance.
Fig. 4. The focusing properties through the CDFZP are showcased by the optical rays’ paths (a) and the intensity (b) of the outbound 532 nm light. The simulated lens has a focal length of 3 cm and a diameter of 1.6 mm.
Figure 4(a) presents the electric field distribution as the plane wave emerges from the CDFZP. The grating effectively modifies the phase of the incoming light, steering it towards a single main focal point, instead of multiple focal points like in the case of a regular FZP. Figure 4(b) complements this by displaying the intensity of the electric field, highlighting the focal point. Notably, the simulations confirm the design’s intentions: most of the incoming light is concentrated in a single point (namely A1, i.e. the focal length of f) demonstrating a clear difference from the multifocal patterns typically associated with a conventional Fresnel zone plate. The single-point focus emphasizes the precision with which the CDFZP can direct light, thereby confirming its enhanced capabilities for imaging, optical fabrication, and others.
Moreover, the focal length, as determined from the peak intensity on the intensity profile, is measured as 3 cm. This is in accordance with the specified design parameters for the CDFZP at the given wavelength of 532 nm. Overall, the performed simulations indicate that the CDFZP with a diameter of 1.6 mm demonstrates a significant advantage in terms of focusing efficiency and accuracy, compared to other evaluated versions. These simulations substantiate the important potential held by the proposed CDFZP in applications where precise light manipulation is crucial, such as in optical imaging systems and photonics devices.
7. Imaging properties
Next, we performed further simulations with the diffractsim [28] Python library to evaluate the imaging properties of the proposed CDFZP in comparison to a traditional lens. The main aim was to analyze the CDFZP’s ability to manipulate light in an energy-efficiency manner, as well as to assess its potential to enhance the imaging quality in terms of resolution and contrast.
Test images, as shown in Fig. 5, were simulated for two basic optical imaging systems. One is equipped with the CDFZP as its imaging lens and the other one with a traditional refractive lens. By employing as ground-truth a reference image, we evaluated the imaging quality obtained by both optical imaging systems.
Fig. 5. Imaging capabilities of the CDFZP compared to a traditional lens. (a) reference image, (b) imaging through a traditional lens, and (c) imaging through a CDFZP.
Figure 5(b) illustrates the imaging result through a conventional lens. As expected, the conventional lens displays a predictable diffraction pattern with standard resolution and contrast profile.
In contrast, Fig. 5(c) presents the image captured through the CDFZP. The CDFZP’s unique structure results in a more uniform intensity distribution across the image but lower contrast. The presence of the 0th diffraction order (A0 Fourier coefficient, which represents no focusing), inherent to the CDFZP design, contributes to a baseline intensity that reduces the overall contrast in the image. Note that interference fringes are clearly visible in Fig. 5(c) because we have considered a single wavelength for the spectral line and because the optical element had a width of 50µm. This was not the case for the experimental situation.
Comparing the two results, one can observe that the resolution of the image obtained via the CDFZP is similar to the conventional lens, as in both cases the highest spatial frequency (the smallest discernible spatial feature) are identical. Importantly, the simulations corroborated the energy efficiency of the CDFZP, intrinsic to their design. By directing more photons into the 1st diffraction order (A1 Fourier coefficient), the CDFZP maximizes the energy utilized for imaging, making it a potentially valuable component in systems where energy efficiency is paramount.
The through focus modulation transfer function (MTF) is a critical metric for assessing the optical performance of imaging systems [29]. It provides a quantitative measure of the system’s ability to reproduce contrast from the object to the image as a function of spatial frequency. In our study, we computed the through focus MTF curves for two distinct optical components: a conventional lens and the CDFZP (Fig. 6).
Fig. 6. Through focus MTS chart for the conventional lens (blue) and CDFZP (orange). Since the conventional lens is mono-focal, its through focus MTF is zero everywhere except in the proximity of z = 3 cm (the focal length of the lens).
8. Experimental results
In this section we further investigate the focusing capabilities of the proposed CDFZP, by employing an experimental optical setup by which a collimated 532 nm laser beam was directed through the designed CDFZP. The setup itself included only a single imaging element as described in the numerical simulation, presented in section 7. The element was printed on a slide and had 20 periods. The imaging element was either CDFZP or a traditional refractive imaging lens. The beam profile was reconstructed based on a series of images captured at 1 mm intervals along the z-axis, providing insight into the propagation characteristics of the laser beam through air.
Figure 7 illustrates the beam propagation profile along the z-axis, offering a clear view of the beam’s trajectory in free space.
Figure 8 presents a comparative analysis of the laser beam profile with and without the CDFZP in place. The image on the left shows the unfocused beam’s broad and diffused profile. In stark contrast, the image on the right, acquired with the CDFZP in place, reveals a focused beam, demonstrating the CDFZP’s effectiveness in beam shaping and focusing.
Note that in the defocused beam of the 'without mask' configuration, Fig. 8 left, interference fringes are generated inside the spot beam due to the glass plate being positioned on top of the detection array (in the camera) and since the illumination beam was a coherent laser beam.
Figure 9 provides a detailed examination of the beam’s focal spot, indicated by a red dashed line. The lower portion of the figure displays the intensity profile with the red line denoting the full width at half maximum (FWHM) of the spot. The FWHM values of 18.43 µm and 19.15 µm along the y and x axes, respectively, quantify the focusing precision of the CDFZP. Those numbers are very close to the theoretical resolution limit of λF# where λ is the optical wavelength (532 nm) and F# is the F-number of the imaging element (that in our case was about 19 since the illumination’s spot diameter was about 1.6 mm and the focal length is 3 cm). These experiments confirm the computational predictions for the CDFZP’s ability to concentrate a laser beam to a focal spot.
Fig. 9. The laser beam illuminated the CDFZP and then propagated along the z-axis in free space. The beam's focal spot is indicated by the red dashed line appearing in the upper row with y-z and x-z cross section of the diffracted intensity. The cross section marked by the red dashed line in the upper row is presented in the lower row. The red line marking the width of the point spread function on the profile image represents the FWHM of the spot, and its corresponding value is provided nearby.
9. Discussions and conclusions
In this work, we introduced the circular Dammann Fresnel zone plate (CDFZP), a novel optical element that marks an advancement in optical engineering. This innovation stems from the unique integration of Dammann gratings with Fresnel zone plates. Unlike conventional approaches that distribute light non-uniformly across multiple points, the CDFZP channels concentrate most of the incident light into a, tunable, small number of focusing points, that are intensely equalized. This design significantly enhances the efficiency and precision of light manipulation, rendering the proposed optical element as a highly valuable tool in fields where precise light control is paramount.
We employed rigorous simulations and experimental work, that collectively thoroughly demonstrate the CDFZP’s ability to focus light with high intensity and precision. The empirical evidence gathered from the experimental work fully aligns to the simulation results, and highlights the potential of CDFZP’s to become a potential transformative tool in optical engineering.
Looking forward, there are several avenues for future research that will be worth pursuing. Optimization of the CDFZP’s design parameters for various wavelengths and applications can further enhance its versatility and efficiency. Furthermore, investigating the potential of the CDFZP in other fields, such as astronomical telescopes or nanoscopy, could lead to new discoveries and innovations. Additionally, combining the CDFZP with emerging technologies in nanofabrication and photonics might pave the way for development of even more sophisticated optical systems.
A practical example of an important application in which the proposed design could be very useful relates to ophthalmic usage. In Ref. [30] a novel technique involves ablation of the cornea epithelium for removing the need in glasses. Only a binary amplitude ablation is possible in that application, and thus the concept presented in this paper is very suitable for that application since energetic efficiency and amount of energy diverted towards A1 should be as high as possible.
To this end, the use of artificial intelligence (AI) to forecast CDFZP configurations that are optimal for specific applications, could significantly aid the penetration of this novel technology in real-life applications. For instance, a specific cost function related to certain features in the image quality could be defined dependent on the specific application and the AI algorithm could optimize the zone plate structure to minimize the defined cost function. In addition, in some applications where binary non real structure could be feasible for realization or in applications where the structure could even be non-binary, the AI algorithms could take the new constrains over the structure into account when performing the optimization process.
Acknowledgments
SGS acknowledges the support of the UEFISCDI RO-NO-2019-0601 MEDYCONAI Grant. GAS acknowledges the support of the Ministry for Research, Innovation and Digitization CNCS-UEFISCDI, project number PCE-119, PNIII-P4-PCE-2021-0444. ZZ acknowledges the support of the Ministry of Innovation Science and Technology for grant # 3-18137.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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