Issue 
J. Eur. Opt. SocietyRapid Publ.
Volume 20, Number 2, 2024
EOSAM 2023



Article Number  32  
Number of page(s)  11  
DOI  https://doi.org/10.1051/jeos/2024026  
Published online  20 August 2024 
Review Article
Analysis of the recording of Fibonacci lenses on photopolymers with 3D diffusion model
^{1}
Instituto Universitario de Física Aplicada a las Ciencias y las Tecnologías, Universidad de Alicante, Apartado 99, Alicante E03080, Spain
^{2}
Departamento de Física, Ingeniería de Sistemas y Teoría de la Señal, Universidad de Alicante, Apartado 99, Alicante E03080, Spain
^{*} Corresponding author: juanc.bravo@ua.es
Received:
13
November
2023
Accepted:
13
May
2024
In this work a threedimensional diffusion model is used to model photopolymers as a recording media. This model allows us to predict the properties of the Diffractive Optical Elements (DOEs) once we recorded into the photopolymer. This model had never been tested with more complex elements, such as multifocal diffractive lenses, as presented in the following in this work. In addition, the model includes; the estimation of the refractive index modulation, the lowpass filtering effect due to the experimental optical setup, and the evolution of the transverse intensity distribution. In this way, the selection of the appropriate material characteristics depending on the intended DOE application is made possible. Specifically, an acrylamidebased PVA/AA photopolymer is simulated using the proposed model. Moreover, coverplating and index matching systems are considered together to avoid the effects of thickness variation. Furthermore, in order to compare their properties using the proposed model, we focus on Fibonacci lenses (FL), a type of bifocal lenses. This allows us to evaluate the dependence of the focii intensity on the polymerisation rate, the diffusivity parameter, lowpass filtering effect and the use of the index matching system for these lenses. This enables us to know the recording parameters in order to produce this type of multifocal diffractive lenses with higher quality and precision.
Key words: Diffractive optical elements / Photopolymers / Diffractive lenses / Diffusion model / Spatial light modulator / 4F system / PVA/AA / Intraocular lenses / Ophthalmology application / Lowpass filtering
© The Author(s), published by EDP Sciences, 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Diffractive Optical Elements (DOEs) [1] have a wide range of practical applications in fields such as diffractive microoptics [2], medical laser treatments [3], and solar energy concentrators [4], among others. Due to their unique properties and versatility, DOEs are commonly employed to manipulate light and obtain desired light patterns.
Photopolymeric materials offer an ideal platform for recording DOEs due to to their favorable phase modulation characteristics, as we have recently successfully modeled [5, 6]. This model reproduce the 3D phase image formation in the photopolymer [7, 8], considering factors like nonlocal polymerisation, depthdependent light attenuation, diffusion process and variations in polymerisation rates, as addressed in [9]. Furthermore, it includes lowpass filtering to account for the recording optical configuration. The model presented also describes photopolymerisation in three dimensions, showing a more complex and realistic diffusion process, as it is shown in [10]. It also introduce coverplating together with index matching techniques to avoid surface variation effects and to simulate the experimental results. The present work is divided in two main parts: experimental and numerical results. The numerical aspect can be further subdivided into three distinct stages. Firstly, we simulate the intensity pattern of a complex DOE, specifically focusing on Fibonacci Lenses (FL) [11–14]. This type of lens exhibit bifocal behaviour and present the capacity to achieve two distinct and reduced focal lengths by either increasing the lens order or decreasing the radius [15]. Secondly, we calculate the refractive index modulation resulting from the polymerisation process using our numerical diffusion model, thereby obtaining monomer and polymer concentration profiles over time. Finally, we use Fresnel propagation to illustrate the effects on the evolution of axial transverse intensity distribution generated by a zone plate composed of Fibonacci lens. Following the acquisition of numerical results, we establish an experimental setup for the production and subsequent recording of polyvinyl alcoholacrylamide (PVA/AA) based photopolymer samples. This photopolymer recently was demonstrated its capability to be used as DOE recording media and present better behavior at low spatial frequency than other photopolymers like HPDLC or NPC [16, 17]. Firstly blazed gratings were recorded, then some DOEs such as axicon, fork and blaze gratings, were recorded and fabricated [9]. The next step is to record and optimizing more complex DOEs. To achieve this, we use a Liquid Crystal on Silicon (LCoS) Spatial Light Modulator (SLM) [18, 19] to obtain the desired amplitude recording intensity of these Fibonacci Lenses and a 4F system to record the DOE, which will enable us to mass fabrication. Notably, our setup allows realtime adjustments of the dimensions and magnification of the final recorded lens through the SLM and optical 4F system. This capability facilitates realtime monitoring during the recording process, underscoring the predictive capabilities of our diffusion model in relation to the experimental results.
Once these stages are shown, we present numerical results relating to various parameters. These results include insights into the nonlinear response of the polymerisation rate concerning the recording intensity, the influence of different diffusivity values in PVA/AA solution samples, and the utilization of coverplating techniques. Consequently, this work allow us to explore the influence of the material properties in the recording of Fibonacci lenses. These lenses represent the fusion between the mathematical Fibonacci sequence and the practical functionality of bifocal diffractive lenses, thus forming a novel intersection within the field of optics. This unique combination not only redefines the visual experience by optimizing near and far vision correction but also opens doors to innovative applications in fields such as ophthalmology, including intraocular lenses.
2 Theory
2.1 Theoretical Diffusion Model
In general, DOEs formation in photopolymers depends on several factors; the monomer and polymer concentration, M and P respectively, the polymerisation rate, , the molecules diffusion inside the recording media, D, due to Fick’s Law and the creation of holes during the polymerisation process H. The holes are the primary origin of the thickness variation. We assumed that the fast swelling of the illuminated areas (initially there is rapid shrinkage in the illuminated areas), is due to the mass transport through the surface, clearly faster than the diffusion in the bulk of the material, and the surface tension forces [20]. Therefore, the equations that govern this model are:(1) (2) (3)
In this work, both the monomer and holes diffusivity in the polymerisation process are consider as constants, , respectively. On the other hand, represents the holes rate generation, which is assumed proportional to , the polymerisation rate, which also depends on the reaction rate and the recording intensity. This dependence is given by:(4)
, where is the recording intensity, is the rate constant, γ is the relationship between intensity and polymerisation rate, is the intensity depth attenuation coefficient due to light absorption and is the attenuation of the polymerisation due to the Trommsdorff’s effect [10]. To solve these differential equations we use FiniteDifferences Method (FDM). Thus, in our formulation these equations can be written as:(5) (6) (7)
where the superscript l represent the temporal step, while the spatial steps in coordinate are denoted by the subscripts , respectively. In the other hand, due to the use of a numerical method, the CourantFriedrichsLewy condition or stability criterion must be satisfied:(8)
To obtain the refractive index modulation during the recording process, it is necessary to calculate monomer and polymer concentrations during the exposure time. Therefore, average refractive index can be measured using LorentzLorenz equation as follows:(9)
In this context, the subscripts m, p, h and b denote the refractive indices of the monomer, polymer, holes and binder, respectively. On the other hand, the refractive index of interest is denoted as n, representing the average refractive index of the photopolymer. This average refractive index depends on the concentrations of monomer and polymer, as well as the initial monomer concentration, denoted as . Hence, n is influenced by both spatial and temporal variations in these concentrations.
On the other hand, local thickness variation in x and y are determined by the volume fraction of the holes:(10)
where d is the thickness of the sample layer, is the intrinsic thickness (that can change due the monomer diffusion), and is the shrinkage due to the holes. In order to avoid this thickness variation due to the holes creation and the relief effects we used a non soluble paraffin liquid with refractive index of 1.4679 that fits perfectly the PVA/AA sample. So the phase change depends on:(11)
and the holes refractive index, , is 1 when indexmatching system is not used, and 1.4679 when we use the paraffin, which is liquid at room temperature.
2.2 Fibonacci Lenses
The Fibonacci sequence is a recursive set of numbers that obeys:(12)
where φ is the golden ratio. In [11] is presented a binary generating function, for a πphase of the FL as follows:(13)
,where and is the normalized radial coordinate, with a the radius of the lens and , plus this interval is segmented into subintervals of length . , denotes the floor function of x, which yields the largest integer less than or equal to x as its output. In Figure 1, the generation of a binary Fibonacci function is presented. This function obeys the recurring set . Then, a binary Fibonacci phase function is constructed. Finally, the radial profile is generated by mapping the axial reduced coordinate with the lens radius, a. To achieve this, we first establish two seed or starter sets, and . Subsequently, following the recurrence law, set is constructed. The construction proceeds to the desired order, as depicted in Figure 1, which is illustrated up to the 6th order. Once the final set, , is constructed, we normalize its length to unity and establish the correspondence between the radial normalized coordinate ζ and the lens radius a.
Fig. 1 Generation of 6order Fibonacci Lens. In the left side, set construction is presented, and in the right side, axial mapping function is shown. 
3 Experimental Setup
3.1 Photopolymeric solution
The photopolymeric media is a PolyvinilalcoholAcrylamide (PVA/AA) solution based on water. PVA is used as support polymer or binder, triethanolamine (TEA) as electron donor, acrylamide (AA) as monomer, Yellow Eosine (YE) as a dye and N,N’ methylbisacrylamide (BMA) as a crosslinker. In Figure 2, a photochemistry scheme of the photopolymerization reaction is shown to facilitate comprehension of the role played by each compound.
Fig. 2 Schematic of photochemistry of photopolymerisation reaction. (a) The dye absorbs a photon from the incident light with wavelength . (b) Through a redox reaction the dye transfers its energy to the initiator, thereby activating it. (c) The photopolymerisation is initiated when the activated radical generator binds to a monomer. The generation of radicals is propagated along the chain to form a polymer chain [21]. (d) At the photopolymerisation reaction forms a dense mesh in the illuminated areas causing shrinkage. A monomer gradient forms. (e) After a certain exposure time, , diffusion of monomers cause the illuminated areas to swell, as it is presented in [22]. An index matching liquid evens out the surface relief. 
On the other hand, in Table 1, the different compounds and their proportions are presented.
Formula used for elaborating the photopolymeric solution to recording DOEs.
This photopolymeric solution is sensitive to green light ( nm) and transparent for red light nm) due to the addition of Yellowish Eosine (YE). Subsequently, the next step involves depositing the PVA/AA solution onto glass substrates of , typically employing a volume of 1500 to achieve a thickness ranging from 100 to 130 . The drying process typically requires an operational window with a temperature of 24°C and a humidity of 60%. It is necessary to ensure that this deposition is conducted on a flat levelled surface to minimize thickness variations across the sample. Furthermore, it is essential to allow water vapor dissipation from the sample, avoiding complete enclosure during the drying process. Regarding the polymerisation reaction, there are two distinct mechanisms to induce phase modulation within the photopolymer. These mechanisms involve either alterations in surface height (thickness) or modulation of refractive indices. To minimize thickness variation an “index matching system” is implemented. A brief scheme of this system is presented in Figure 3. This system entails the utilization of a liquid paraffin with a refractive index of , closely matching that of the compounds present in the PVA/AA photopolymer solution. Additionally, a coverplate is employed to physically restricting the surface. This system affords us the capability to maintain a consistent photopolymer thickness, and preserve the recorded DOE, while ensuring that phase modulation is contingent solely upon refractive index modulation.
Fig. 3 Schematic of indexmatching system. (a) after 24 hours of drying, paraffin and coverplate glass are put over the PVA/AA sample. (b) The final sample is ready to be placed into the experimental setup presented in Figure 4. (c) Fibonacci lens about 2000 of radius is recorded into the sample. 
3.2 Setup used for recording FL
Figure 4 represents the scheme of the setup used, which includes a spatial light modulator (SLM) based on liquid crystal on silicon (LCoS) to generate FL patterns working in amplitude mode. In this setup, we can distinguish two distinct arms. The recording arm is formed by the green beam provided by a 533 laser. This recording light is directed towards the SLM based on LCoS which has a resolution of and a 8 pixel pitch. Two Linear Polarizers (LP) are used in and to linearly polarize the green beam since the SLM operates in amplitudeonly mode. After modulating the recording beam, a 4F system is used to reproduce the image in the PVA/AA sample. This enable us to adjust the final magnification of the recorded lenses. This takes importance in the diffraction efficiency, since the resolution of the white and black rings are crucial for the correct formation of the multifocal phase lens. In our work, we employed a 2:3 image scale by using lenses L4 and L3. Thus, reducing the size of the image enables the recording of higher spatial frequencies. Therefore, lenses with shorter focal lengths can be recorded compared to previous works [8]. The other arm utilizes a laser with a wavelength of 632 for the readout process. Thanks to a red filter (RF), only the light from the reading can reaches the CCD camera positioned on an axial scanning platform, which is used to locate the two focal points of a FL. The use of a CCD camera enables us to acquire detailed information about the diffraction pattern at each step along the axial platform with high resolution. On the other hand, the use of diaphragm D3 enables us to filter high frequencies and speckles. The final resolution of the lens depends on the diameter of the diaphragm, the alignment of the recording and readout beams, and the lens aperture. Regarding the diaphragm size, diameters smaller than 0.6 adversely affect the final diffraction pattern recorded by the lens.
Fig. 4 Setup used to record FL and to analyze in real time the formation and the different focal points. Where D is a diaphragm, L denotes a lens, SF is a spatial filter, BS is a beam splitter, M is a mirror, LP is a linear polarizer, and RF is a red filter. 
4 Results and discussion
We choose Fibonacci lenses for their complexity and practical applications to test the model’s accuracy. We simulated the recording of a FL profile after 150 of exposure time and a material with a thickness of . As it is shown in [11], FL produce two focal points. To validate this, an analysis will be conducted on the variation of axial intensity distribution, considering the influence of the factor γ from equation (4).
4.1 Comparison between experimental and numerical results
Once the FL intensity pattern, shown in Figure 1, is projected onto the photopolymeric material, the diffusion process begins. This leads to a modulation of the refractive index during the exposure time in the sample section (Figure 5a). The use of a 4F system for forming images of these lenses in the material plane implies a nonideal formation. The resolution of the optical system is affected by the use of a diaphragm between the lenses, as well as the size and numerical aperture of the lenses themselves. These optical components act as a lowpass filter, eliminating the higher spatial frequencies. This has a direct result on the modulation of the refractive index of the lens, as shown in Figure 5b). Therefore, the refractive index modulation profile is smoothed in the areas where more gradient exists.
Fig. 5 In (a), the refractive index modulation due to the diffusion process during the recording of the FL phase into the photopolymer is represented. Therefore, in (b), a refractive index modulation slice is represented for the ideal lens and lowpass filtering. Red and blue curves represent the ideal refractive index modulation and lowpass effect, respectively. 
Once the refractive index modulation, , has been obtained, a plane wave is propagated through the material to simulate its reconstruction. The axial irradiance distribution produced by an FL and its associated Fresnel Zone Plate (FZP) can be calculated with the FresnelKirchhoff integral:(14)
Where is the axial reduced coordinate, nm, is equal to the phase change in equation (11), z the axial distance from the material and ζ is the input plane spatial coordinate. Figure 6 presents a heat map representing the intensity distribution for a transverse cross section of the area close to the Fibonacci lens. The relationship between the two focal points is given by . Therefore, foci approaches the axial positions cm and cm. Moreover, the farther focal point is broader than the close one, as predicted in [11]. In addition to this, the broadening relation shows that the far focii lobe, in , is times broader than the second lobe in .
Fig. 6 Evolution of the axial and transverse intensity distribution for a Fibonacci Lens with a radius of mm and the result obtained using the proposed diffusion model and the experimental results. Heat map plot corresponds to the intensity distribution of an ideal FL. 
This result is also obtained using the diffusion model, and the main difference between the curves lies in the decrease of the intensity of the shortest focal point . This decrease is due to the lowpass filtering of the 4F configuration. The higher spatial frequencies disappear and affect the intensity and energy of the lowest focal point, . Furthermore, it is important to note that the proposed diffusion model aims to accurately simulate the real photopolymerization process, thus revealing the fundamental properties of a Fibonacci lens.
The experimental data, presented in Figure 6, were obtained using a 12.3megapixel color camera with a CMOS sensor. We were able to observe the formation of the focal points in realtime while recording the lens. At each time step, corresponding to each spatial step of the linear stage on which the camera is positioned, we computed the average intensity of diffracted light at the center of the lens. However, experimentally the intensity maximums are not as sharp as the numerical results due to the small relative transverse width of the foci. This transverse width is around 10 microns in diameter, see heat map in Figure 6, which affects the average intensity calculation. Nevertheless, both focal points can be located. The blue line, labeled as Diffusion Model, in Figure 6 is the result obtained in equation (14) after numerically determining the average refractive index. The red dotted line, labeled as Ideal, corresponds to the diffraction pattern with FresnelKirchhoff integral but using the Fibonacci phase function (13) instead of the phase change due to refractive index modulation from equation (11).
We observed that both experimental focal points fall within the range predicted by the diffusion model and the ideal result. Furthermore, the ratio of the width of the longer focal point, , and the shorter focal point, , corresponds to the relationship .
4.2 Influence of the parameter γLg
The parameter γ represents the nonlinearity of the polymerization rate with the incident light into the material [23]. This factor relates diffusion with photopolymerization process, showing the saturation possibility of the phase modulation rate. Figure 7 shows the numerical influence of nonlinearity on diffraction efficiency at each focal point for various exposure times. In Figure 8, we present the numerical results for each γ value. The focal points are clearly visible, showing for , the diffraction efficiency is higher than that for a linear relationship, such as . This result demonstrates that the photopolymer phase modulation depends on the exposure, which, in turn, relies on irradiance and time. The intensity of the green laser beam (532 nm) used for the recording process was 1.5 , with a diffusivity value of .
Fig. 7 Diffraction efficiency at each focal point for different values of exposure times and γ. Results obtained by the numerical diffusion model including lowpass filtering. 
Fig. 8 Numerical simulation of the evolution of the axial and transverse numerical intensity distribution after 60 seconds of recording produced by a FL for different values of γ obtained by the diffusion model and the ideal result (represented in red dotted line). The heatmap above corresponds to the simulation for for an ideal lens. 
4.3 Influence of the diffusivity parameter D_{m}Lg
The diffusivity value of the PVA/AA photopolymer depends on several factors, including the proportions of the compounds, environmental temperature, and humidity, as studied in [24]. The mean diffusivity values fall within the range of . Therefore, we can calculate how the diffraction efficiency of FL is affected by changes in diffusivity using the proposed model. In Figure 9, the simulated normalized intensity in each focal point is represented for different exposure times and for different values of mean photopolymer diffusivity. As it is shown, for higher values of the mean diffusivity parameter, the numerical results reach higher diffraction efficiency values for lower exposure times. Despite this, for lower values such as 0.02 , there are less perturbations due to there are less phase profile changes than for higher values of . This distribution is due to the liquid behavior of the photopolymer [25]. When the recording media is more liquid (higher mean diffusivity), the photopolymerization process takes less time to achieve the estimated diffraction efficiency. Hence, for lower values of mean diffusivity, the axial irradiance distribution decreases in general terms because the diffusion process has not yet completed. So, these numerical simulations allow us to fit the exposure time of recording this lenses for achieving higher diffraction efficiency. Showing that only 60 seconds of recording for an irradiance of is enough to obtain higher diffraction efficiencies for higher diffusivities; around 80% for the large focal lenght, , (9(a)). Approximately 60% diffraction efficiency is achieved for the shorter, , as depicted in Figure 9(b). It is important to mention the impact of lowpass filtering, which directly influences the diffraction efficiency of the focal points. For the shorter focal point, , the diffraction efficiency is lower than the longer focal point regardless of the diffusivity. This is due to the elimination of the higher spatial frequencies with lowpass filtering. On the other hand, extending lens recording for a prolonged period of time, the diffraction efficiency reaches its maximum level for lower values of mean diffusivity, which may be slightly higher than for higher mean diffusivities. Thus, if the mean diffusivity is small and the exposure time is long enough, the lens image can be formed more accurately and reaching higher diffraction efficiencies while avoiding overmodulation.
Fig. 9 Numerical simulation of the intensity distribution at each focal point produced by a FL for different exposure times and for different values of with lowpass filtering. The intensity pattern of the FL used was . 
4.4 Influence of the coverplate
The application of coverplating and index matching techniques enables precise control over the thickness of photopolymeric material. In Figure 10, we present the numerical normalized intensity of each focal point, for (a) and for (b) as a function of exposure time, comparing results with and without the implementation of the coverplating technique. The numerical data shows that, for the same exposure time, higher diffraction efficiency can be achieved when using lower values of the mean diffusivity parameter, This is particularly significant when using the coverplate. Consequently, these techniques take into account that, when the photopolymer’s diffusivity is low, the effects of thickness variation and refractive index modulation are negligible. Furthermore, Figure 10 underscores the significance of exposure time in the case where coverplating is not employed. In general terms, the use of the indexmatching technique allows us to achieve higher diffraction efficiencies with shorter exposure times in every focii. The energy distribution behavior across the exposure time is similar in both cases; however, the main difference lies in the higher diffraction efficiency achieved with the cover plating application, along with a higher mean diffusivity parameter. This behaviour can be understood through the physics of the indexmatching and coverplating system. Avoiding thickness variation permits only modulation in the refractive index of the photopolymer when the incident light is recorded. Nevertheless, indexmatching reduces the measured monomer diffusion by more than a factor of 10 [20]. This result is coherent when considering the rapid rates of surface recovery and monomer diffusion through the surface.
Fig. 10 Evolution of the diffracted intensity at each focal point for a FL with ). Each curve corresponds to a different value of mean diffusivity, , with lowpass filtering, and with or without coverplating. The label “no CP” in the green and blue dotted curves indicate the cases where coverplating technique was not used. 
5 Conclusions
Fibonacci lenses have been recorded in PVA/AA based photopolymers. Upon obtaining theoretical results using a 3D diffusion model with lowpass filtering of the amplitude image of these lenses, a value for γ can be proposed. This value is established by relating the recording intensity to the polymerisation rate. Accordingly, lower values of γ yield higher diffraction efficiencies, which explains the correlation between exposure and phase modulation. On the other hand, it is crucial to emphasize the significance of employing index matching and coverplating techniques. Particularly, for lower mean diffusivity values, these techniques carry great importance to achieve higher diffraction efficiency within reduced exposure times. Nevertheless, when considering longer exposure times in the absence of coverplating, variations in thickness become more significant. Furthermore, the experimental setup and indexmatching system proposed enable us to efficiently massproduce and preserve these types of lenses. Consequently, the diffusion model has the capability to fit parameters such as exposure time, diffusivity, and recording light intensity, thus yielding results similar to experimental findings. Another conclusion drawn from this work is the predictive capability of the model when studying the diffraction efficiency of the focal points. Incorporating the lowpass filtering intrinsic to the experimental optical system into the model allows us to obtain significantly more realistic results. The elimination of higher spatial frequencies has a direct effect on the energy distribution of the diffracted light. Thus, distributing more energy to the longest focal point than to the shortest one, in line with what was obtained experimentally for a Fibonacci lens.
To sump up, the experimental results obtained clearly show the bifocal behaviour of these lenses in spite of the narrow diffracted light spot. It should be noted that the 3D model does not simulate perfect illumination. This is due to the consideration of a lowpass filtering in the amplitude image to replicate the effect of the experimental setup. Therefore, resulting in the observed intensity distributions of the two focal points in both experimental and simulated data.
This study represents an ongoing experimental investigation with a significant potential. Bifocal diffractive lenses have a broad spectrum of applications in various fields, such as ophthalmology and Xray microscopy. For example, in the field of ophthalmology, Fibonacci lenses can be used as intraocular progressive lenses to effectively manage presbyopia in patients.
Funding
Funded by the “Generalitat Valenciana” (Spain) (IDIFEDER/2021/014, cofunded by EU through FEDER Programme; PROMETEO/2021/006 and INVEST/2022/419 financed by Next Generation EU), “Ministerio de Ciencia e Innovación” (Spain) (PID2021123124OBI00 and PID2019106601RBI00).
Conflicts of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Data availability statement
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Author contribution statement
Following the taxonomy of CRediT, the contributions of each author are as follows:
J. C. Bravo contributed to the conceptualization, formal analysis, investigation, methodology, resources, software development, data curation, visualization, and writing of the original draft as well as the review and editing of the manuscript.
J. J. SirventVerdú participated in the conceptualization, formal analysis, investigation, and visualization.
J. C. GarcíaVázquez was responsible for data curation and visualization.
A. PérezBernabeu also handled data curation and visualization.
J. ColominaMartínez contributed to the conceptualization, formal analysis, software development, and visualization.
R. Fernández was involved in project administration, conceptualization, investigation, resources, supervision, validation, and visualization.
A. Márquez focused on funding acquisition, investigation, resources, supervision, validation, and visualization.
S. Gallego was responsible for funding acquisition, project administration, conceptualization, investigation, methodology, resources, supervision, validation, and visualization.
References
 Kazanskiy N. (2018) Opt. Tec. Tel. 10774, 206. [Google Scholar]
 Rossi M., Kunz R., Herzig H.P. (1995) Appl. Opt. 34, 5996. [NASA ADS] [CrossRef] [Google Scholar]
 Kim H., Hwang J.K., Jung M., Choi J., Kang H.W. (2020) Biomed. Opt. Exp. 11, 7286. [CrossRef] [Google Scholar]
 Huang Q., Wang J., Quan B., Zhang Q., Zhang D., Li D., Meng Q., Pan L., Wang Y., Yang G. (2013) Appl. Opt. 52, 2312. [NASA ADS] [CrossRef] [Google Scholar]
 Lawrence J., O’Neill F. (2001) J. Sheridan. Optik. 112, 449. [CrossRef] [Google Scholar]
 Fernández R., Gallego S., Márquez A., NavarroFuster V., Beléndez A. (2016) Materials. 9, 195. [Google Scholar]
 Bowley C., Crawford G. (2000) Appl. Phys. Lett. 76, 2235. [NASA ADS] [CrossRef] [Google Scholar]
 R. Fernández, S. Gallego Rico, A. Márquez, J. Francés, C. Neipp, D. Puerto, E.M. Calzado, I. Pascual Villalobos, A. Beléndez. Proc. SPIE. 11367: 113671E (49) (2020). [Google Scholar]
 Fernández R., Gallego S., Márquez A., Neipp C., Calzado E., Francés J., MoralesVidal M., Beléndez A. (2019) Polymers. 11, 1920. [CrossRef] [Google Scholar]
 Kelly J.V., O’Neill F.T., Sheridan J.T., Neipp C., Gallego S., Ortuño M. (2005) J. Opt. Soc. Am. B 22, 407. [NASA ADS] [CrossRef] [Google Scholar]
 Monsoriu J.A., Calatayud A., Remon L., Furlan W.D., Saavedra G., Andrés P. (2013) IEEE Photonics J. 5, 3400106. [CrossRef] [Google Scholar]
 Ferrando V., Calatayud A., Andrés P., Torroba R., Furlan W.D., Monsoriu J.A. (2014) IEEE Photonics J. 6, 1. [CrossRef] [Google Scholar]
 Calatayud A., Ferrando V., Remón L., Furlan W.D., J.A. (2013) Monsoriu, Opt. Exp. 21, 10234. [CrossRef] [Google Scholar]
 Cheng S., Liu M., Xia T., Tao S. (2018) Laser Phys. 28. [Google Scholar]
 Ke J., Zhang J. (2016) Opt. Commun. 368, 34. [CrossRef] [Google Scholar]
 Fernández R., Gallego S., Márquez A., Francés J., Martínez F., Pascual I., Beléndez A. (2018) Opt. Mater. 76, 295. [CrossRef] [Google Scholar]
 Infusino M., De Luca A., Barna V., Caputo R., Umeton C. (2012) Opt. Exp. 20, 23138. [CrossRef] [Google Scholar]
 G. Lazarev, P.J. Chen, J. Strauss, N. Fontaine, A. Forbes, Opt. Exp. (2019). [Google Scholar]
 Yzuel M.J., Campos J., Márquez A., Escalera J.C., Davis J.A., Iemmi C., Ledesma S. (2000) Appl. Opt. 39, 6034. [NASA ADS] [CrossRef] [Google Scholar]
 Gallego S., Fernández R., Márquez A., Ortuño M., Neipp C., Gleeson M.R., Sheridan J.T., Beléndez A. (2015) Opt. Lett. 40, 3221. [NASA ADS] [CrossRef] [Google Scholar]
 Weiser M., Bruder F., Fäcke T., Hönel D., Jurbergs D., Rölle T. (2010) Macromol. Symposia 296, 133. [CrossRef] [Google Scholar]
 Wu D., Huang Y., Zhang Q., Wang P., Pei Y., Zhao Z., Fang D. (2022) Journal of the Mechanics and Phys, Solids 162. [Google Scholar]
 Gallego S., Márquez A., Guardiola F.J., Riquelme M., Fernández R., Pascual I., Beléndez A. (2013) Opt. Exp. 21, 10995. [NASA ADS] [CrossRef] [Google Scholar]
 Gallego S., Márquez A., Ortuño M., Francés J., Marini S., Beléndez A., Pascual I. (2011) Opt. Exp. 19, 10896. [NASA ADS] [CrossRef] [Google Scholar]
 Close C.E., Gleeson M., Mooney D.A., Sheridan J.T. (2011) JOSA B. 28, 842. [NASA ADS] [CrossRef] [Google Scholar]
All Tables
All Figures
Fig. 1 Generation of 6order Fibonacci Lens. In the left side, set construction is presented, and in the right side, axial mapping function is shown. 

In the text 
Fig. 2 Schematic of photochemistry of photopolymerisation reaction. (a) The dye absorbs a photon from the incident light with wavelength . (b) Through a redox reaction the dye transfers its energy to the initiator, thereby activating it. (c) The photopolymerisation is initiated when the activated radical generator binds to a monomer. The generation of radicals is propagated along the chain to form a polymer chain [21]. (d) At the photopolymerisation reaction forms a dense mesh in the illuminated areas causing shrinkage. A monomer gradient forms. (e) After a certain exposure time, , diffusion of monomers cause the illuminated areas to swell, as it is presented in [22]. An index matching liquid evens out the surface relief. 

In the text 
Fig. 3 Schematic of indexmatching system. (a) after 24 hours of drying, paraffin and coverplate glass are put over the PVA/AA sample. (b) The final sample is ready to be placed into the experimental setup presented in Figure 4. (c) Fibonacci lens about 2000 of radius is recorded into the sample. 

In the text 
Fig. 4 Setup used to record FL and to analyze in real time the formation and the different focal points. Where D is a diaphragm, L denotes a lens, SF is a spatial filter, BS is a beam splitter, M is a mirror, LP is a linear polarizer, and RF is a red filter. 

In the text 
Fig. 5 In (a), the refractive index modulation due to the diffusion process during the recording of the FL phase into the photopolymer is represented. Therefore, in (b), a refractive index modulation slice is represented for the ideal lens and lowpass filtering. Red and blue curves represent the ideal refractive index modulation and lowpass effect, respectively. 

In the text 
Fig. 6 Evolution of the axial and transverse intensity distribution for a Fibonacci Lens with a radius of mm and the result obtained using the proposed diffusion model and the experimental results. Heat map plot corresponds to the intensity distribution of an ideal FL. 

In the text 
Fig. 7 Diffraction efficiency at each focal point for different values of exposure times and γ. Results obtained by the numerical diffusion model including lowpass filtering. 

In the text 
Fig. 8 Numerical simulation of the evolution of the axial and transverse numerical intensity distribution after 60 seconds of recording produced by a FL for different values of γ obtained by the diffusion model and the ideal result (represented in red dotted line). The heatmap above corresponds to the simulation for for an ideal lens. 

In the text 
Fig. 9 Numerical simulation of the intensity distribution at each focal point produced by a FL for different exposure times and for different values of with lowpass filtering. The intensity pattern of the FL used was . 

In the text 
Fig. 10 Evolution of the diffracted intensity at each focal point for a FL with ). Each curve corresponds to a different value of mean diffusivity, , with lowpass filtering, and with or without coverplating. The label “no CP” in the green and blue dotted curves indicate the cases where coverplating technique was not used. 

In the text 
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