1. Introduction

Gradient-index (GRIN) components are a class of micro-optical elements, which optical functionality is based on the variation of refractive index. The most commonly used are planar-surface optical elements where refractive index is changing in the direction perpendicular to the optical axis. In particular planar surface rod-type GRIN lenses with parabolic refractive index profile are an attractive approach for compact optical systems, as they can be easily integrated with other micro-optical components [1]. They have been applied in optical interconnects and fiber coupling systems [2], optical computing [3], and various endoscopic devices [4–6].

There are several proposed methods of GRIN microlens fabrication, and the most commonly used are ion-exchange, chemical vapor deposition (CVD) or neutron irradiation [7–9]. However, these techniques suffer from several drawbacks such as the inaccuracy of the refractive index distribution [10–11] and limited gradient of refractive index achievable, typically of the order of $\varDelta$n ∼0.1 per 250 µm [11]. Till now, the highest contrast obtained for the GRIN element developed using chemical vapor deposition was reported for the case of 1D multilayer, where refractive index gradient of $\varDelta$n = 0.25 per 6.5µm was obtained [12]. Additionally, these fabrication methods impose cylindrical symmetry of the manufactured optical components.

In 2009, we introduced the concept of nanostructured gradient-index (nGRIN) elements [13], and further developed it over the subsequent years [14–16]. Nanostructurization makes possible the creation of extremely large refractive index gradients, on the order of $\varDelta$n = 0.2 per 5µm in 2D structures and allows the development of any arbitrary refractive index profiles [14], such as optical vortices, axicons, or elliptical lenses. nGRIN approach provides fabrication of planar-surface gradient microlenses with diameters in the range of tens of micrometers, i.e., in the area dominated by refractive polymer microlenses with curved surfaces [17]. Moreover, nGRIN lenses can be nearly achromatic, when proper types of glasses are used [18].

The operation principle of nanostructured elements can be described by the effective medium theory (EMT) [5]. Let us assume that a nanostructured optical element is constructed of discrete sub-wavelength sized individual glass rods made of two, or more, different glasses. The rods are positioned in the structure according to the predetermined pattern. Nanostructured medium transmits light similarly to continuous refractive index material provided the diameter d of the individual rods is sufficiently small, i.e., d ≤ λ⁄(2π), where λ is the wavelength of the propagating light [13]. To parametrize structures, we introduced the nanostructured GRIN lens parameter n_G = λ/d. The nGRIN lenses, which are designed according to the EMT have n_G ≥ 2π, for a given operating wavelength. In this case their optical performance is similar to classical continuous-profile GRIN elements [17,18]. On the contrary, the optical parameters of lenses which have n_G values lower than 2π start to deteriorate with the growing size of the rods, since diffractive effects start to play a dominant role. Our initial tests show that this change is linear with the increase of the individual rod diameter. The limit of the lens efficiency can be derived from work of Mait at al. [19]. They stated that for the features size of the lens equal to 0.7λ the diffractive efficiency of the lens is equal to 78%. As diffractive effects are detrimental to the performance of the nGRIN lens, one can assume this as the upper limit of the rod diameter for the lens. Therefore, to satisfy the assumptions of the effective medium theory and to ensure good optical performance, the fabricated lenses are generally small (diameter < 30 µm), which hinders their practical use in industrial applications. To fully meet users expectations the nGRIN lenses diameters should be increased to the level of 50–100 µm.

The nanostructured GRIN elements are fabricated using a modified stack-and-draw technique, which is a similar method to the one commonly used for photonic crystal fiber fabrication. The process of the design of a nanostructured optical element begins with the determination of the desired refractive index distribution in the lens. This appointed profile will be a target function for the effective refractive index distribution of the glass nanostructure. Macroscopic glass rods made of two or more types of glass are manually assembled in the designed pattern, forming a so-called preform. The preform is then drawn using a fiber-drawing tower, and the pattern is reduced to its final size. In previous studies, we successfully used this technology to fabricate nGRIN lenses of 22.8 µm diameter and up to 115 rods at the diagonal (the total width of the entire optical element was 125 µm) [20]. According to the EMT these lenses are dedicated to work only at infrared wavelengths. Further increase of lens diameter requires a larger number of individual rods and simultaneously fulfill EMT boundary condition for a given wavelength (n_G ≥ 2π), [21]. An increase of number of individual rods over 12 thousands items increase dramatically a complexity of technological process due to limits of manual assembly of lens preforms and uniformity of heat distribution in lens preforms during drawing process.

As an alternative, we proposed a multiple step-index lens, consisting of seven concentric rings with subsequently increasing refractive index values [16]. Each of the rings was composed of a different type of so-called metarods, hexagonal structures arranged from 50 × 50 rods of 20 nm diameter, where the individual rods were fabricated from either NC21 or F2 glasses. By changing the amount of each type rods in the metarod, we were able to linearly vary the effective refractive index of the metarods between 1.5212 and 1.6068 and thus the effective refractive index of the corresponding ring. The fabricated nanostructured lens had a diameter of 100 µm in total and a length of 140 µm. For 633 nm wavelength, the lens working distance (wd) was equal to 40 µm, and the size of the focal spot was about 0.9 µm and 1.2 µm along the X and Y axes respectively. Although described approach allowed fabrication of nGRIN lens with large diameters, the fabrication process was complex and time-consuming. Instead of one stacking and one drawing process, 8 cycles were needed. Additionally, during the final stacking of the metarods, the possibility of introducing misplaced elements into the structure is increased as there are 7 different types of metarods that have to form the correct pattern. The fabricated nGRIN microlenses were suited for beam focusing only, they cannot be used for imaging applications.

There is also one more aspect related to fabrication of nanostructured optical components using stack and draw method. This method is based on assembly of a complex structure with rods and capillaries made of different glasses (a preform) and further drawing it at high temperature to integrate and scale down the assembled structure [14]. As a result, a structured, integrated fiber is obtained. Integration of individual rods is accompanied by ion diffusion processes between rods during preform drawing at high temperatures. The range of diffusion is in the order of single micrometres and strongly depends on temperature and heating time. Usually it is negligible since stack and draw technique is commonly used for development of photonic crystal fibres and imaging bundles. However, if we consider fiber structures with submicron-level feature size the scale of elements starts to comparable with the diffusion range. When diffusion between two nanorods of different glasses occurs, their chemical composition is changed in proximity of their border. If rods diameters are similar to the diffusion range, it mimics several rods with smaller diameters. Therefore, each individual rod can be treated as multiple rods with various refractive indexes. This way we can reduce the number of real rods required to assembly nanostructured lens, because light can ‘see’ every rod as multiple rods. This way EMT satisfied although the diameter of real rods is larger than imposed by EMT.

In this paper, we propose the development of large diameter nanostructured GRIN microlenses with a modified method where the temperature-controlled diffusion is applied for drawing preforms of nanostructured microlenses. The presence of high temperature during the drawing process stimulate the diffusion between different glass components. This phenomenon usually has a negligible influence on the performance of optical components manufactured from one type of homogenous glass or when the sizes of individual elements made of various glasses are much larger than any expected diffusion depth. As we have reported earlier for the case of nanostructured fibers, the process of diffusion might have a dramatic impact on, e.g., dispersion [22].

In this paper we show that limits in performance of nanostructured GRIN lenses related to the EMT can be bypassed when during drawing process a diffusion range is similar to the diameter of the individual rods in the nGRIN lens. In this case the ion distribution within the structure is locally averaged, and the refractive index profile resembles the one present in continuous gradient index lenses. This further results in enhancement of nGRIN lens optical performance. The proposed method allows to develop nanostructured GRIN microlenses with smaller amount of rods. This reduce a complexity of the fabrication, allow to develop larger microlenses over 100 µm of diameter and reduce short wavelength limits of performance related to EMT boundary condition. As a proof of concept of this technology, we fabricated GRIN lens with diameter 115 µm, that can be used for optical imaging and beam focusing.

2. Design and development of the nanostructured GRIN lens

The nanostructure of the proposed GRIN lens has been designed in such a way that the effective refractive index values change parabolically between the maximum located on the optical axis to the minimum on the edge of the aperture. The shape of the refractive index change can be described by the gradient constant g,

The internal structure of nGRIN microlens is determined by the distribution of low and high index nanorods inside the structure. The designing process begins with the random allocation of rods and the EMT serves to calculate local refractive index values. Then the algorithm of simulated annealing (SA) is used to rearrange the placement of the individual rods in the structure via the minimization of the cost function. The cost function is defined as a difference between effective refractive index distribution as showed in Figs. 1(a) and 1(b) for a given arrangement of the individual rods in the structure and the target refractive index distribution of the lens with continuous refractive index change (see Fig. 1(c)). In simulations and fabrication process we assume the hexagonal lattice of nanorods distribution. Such lattice ensures that circular rods remain in the same positions with respect to others during preform assembly and further fiber drawing process [7]. The designed structure of nGRIN microlens composed of NC42 and NC34 glass nanorods is shown in Fig. 1(d). The structure of the lens consists of 115 rods on the diagonal.

 

Fig. 1. a) Effective refractive index distribution in the nanostructured GRIN lens and b) the corresponding profile, c) refractive index distribution of the ideal, continuous GRIN lens, d) schematic of rods arrangement in nGRIN lens. Color bar denotes refractive index difference in the GRIN lens with respect to the background glass NC34

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The preform is assembled according to the design and drawn at a fiber drawing tower. We verified the quality of the fabricated structure using Scanning Electron Microscopy (SEM). The image presented in Fig. 2 confirms the good agreement between designed and fabricated nanostructured patterns.

 

Fig. 2. SEM image of the fabricated nGRIN lens, which design was based on a schematic from Fig. 1(d). The lens has 32 µm diameter on the diagonal. Dark and bright areas correspond to nanorods made of low and high index glasses with different oxide composition in the structure.

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For the structure fabrication, we used a pair of developed in house multicomponent glasses (thermal and optical glass parameters are described in Table 1). The refractive index difference is obtained by the differences in chemical composition of the glasses. Ideally, to process both glasses simultaneously, their thermal expansion coefficients and melting temperatures should be the same. Differences in these values can lead to distortions in the final structure, especially when the fibre is drawn at high process temperatures. Normally we assume that glasses are well-matched if the parameters for both glasses are within 5% range from each other. This means that this fabrication method can be used for any glass pair that have refractive index difference and similar expansion coefficients and melting temperatures and is not limited to glasses with similar chemical compositions and small difference in refractive indices.

Table 1. Thermal and optical properties of the glasses used in the lens fabrication

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3. Influence of diffusion on nGRIN lens performance

In the structure where no diffusion process occurs, the change in refractive index values at the interface between two different rods is a step function, without any transition area (see Fig. 3(a)). However, in the presence of diffusion, the migration of the elements causes the smearing of the refractive index profile, and the increase of effective rods diameter as shown in Fig. 3(b).

 

Fig. 3. The schematics of the influence of the diffusion process on the elemental composition and refractive index profile for the case of two closely located rods.

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We were unable to find publications concerning investigations of diffusion in multicomponent glasses, similar to ours. The process of diffusion has been studied both for temperatures below [24–27] and above glass transition temperatures [28], but only in less extensive glass systems such as e.g. SiO₂-Na₂O or SiO₂-B₂O₃. In this case of single doped silica, diffusion is simply related to the glass viscosity [28]. However, in multicomponent NC34/NC42 glasses composed of 8 different oxides (Table 2), the rate and range of diffusion of particular elements are different and might be interrelated [29]. In particular, the speed of diffusion depends on the concentration of individual elements, temperature, and weight of the molecules. Moreover, Fick's second law, which predicts how diffusion causes the concentration of ions in the glass to change with time, in its classical form is insufficient, and a more generalized matrix notation has been used to describe diffusion in dynamic situation near glass transition temperatures [30–32]. For the NC34/NC42 glass pair, the highest difference in concentration occurs for Ba and Pb, and diffusion of those two oxides should most strongly contribute to the refractive index change. At the same time the lightest oxides, for example, Li and Na are diffusing at a faster rate than the heavier ones like Pb and Ba, but their total influence on the refractive index change is expected to be smaller.

Table 2. Chemical composition of the glasses used in lens fabrication

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Among the fabrication process parameters, drawing temperature is the easiest to modify. In order to obtain distortion-free structure, this temperature has to be within the range of specific temperatures, namely hemisphere formation temperature and spreading temperature, of all involved glasses. Only when this condition is fulfilled glasses can be drawn together. In the case of considered NC34/NC42 glass pair, this temperature overlap window can be as high as 70°C (Table 1).

The easiest way to estimate the diffusion range is based on the analysis of SEM images. We compared the results for nGRIN lenses with diameters up to 32 µm, fabricated at drawing temperatures starting from 750°C, up to 830°C (Fig. 4). At the 750°C the diffusion range equals 81 ± 8 nm, then it increases to 200 ± 20 nm for 810°C and reaches 260 ± 26 nm for 830°C. Diffusion range is calculated based on high resolution SEM images. As diffused are we assume areas where a change of grey level in individual nanorods are observed (outer circles in Fig. 4(a)), while non-diffused area has uniform value of grey level (central circles in Fig. 4(a)). Clearly, with the increase of the process temperature, the diffused areas between individual glasses are expanding. As we show in Fig. 4(d) the relation between the effective diameter of the rods and the temperature is almost linear. The drawing temperature, however, cannot be increased much further. For the temperatures above 830°C, the tension of the fiber in the drawing tower was not sufficient to maintain circular cross-section of the drawn fiber. Therefore, we deemed 830°C as viable drawing temperature, with the longest possible diffusion range.

 

Fig. 4. SEM image of the diffused structure fabricated at various temperatures of the drawing process, (a) at 750°C, (b) at 790°C, (c) at 830°C. Inner circles denote non-diffused areas of high refractive index glass NC42. The outer circles denote area of diffusion between high index and low index component glasses (NC42 and NC34, respectively) in the lens structure. (d) Diffusion range as a function of temperature.

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Based on this information, we fabricated the second series of GRIN lenses drawn at the temperature of 830°C, with diameters equal 20 µm, 40 µm, and 115 µm. Using an EDS module installed in the SEM microscopy, we measured the chemical composition of the sample lenses. The EDS relies on the generation of X-rays with the spectrum unique to the given material composition. Since the interaction volume is larger than for classical SEM imaging, the resolution of the method is also smaller, and for similar nGRIN structures, it was estimated to be around 0.2 µm [24]. Light elements such as Li and Na are difficult to measure as valence electrons are involved in characteristic x-ray generation. Therefore, we were not able to confirm in EDS any change in the concentration of these elements. Nevertheless, the strongest contribution to the refractive index change is expected to come from heavy oxides, i.e. Pb and Ba.

EDS analysis reveals a change in Pb and Ba concentration along with the cross-section of the lens. For the lenses with the diameter of 20 µm and 40 µm (Figs. 5(a) and 5(b), respectively) the obtained concentration profiles are gradual and not binary as one would expect from a material consisting of two distinct glasses. This observation supports our claim that at 830°C diffusion range for glasses used is of the order of 1 um, which means it is greater than the diameter of individual rods used to compose the lens. Moreover, it also suggests that we have fabricated a new class of material – a compound glass with real refractive index change that is resembling an effective (not actual) profile of the binary stack of individual glass rods as shown in Fig. 1(b).

In the case of 115 µm diameter lens the measured concentration profiles are less smooth (Fig. 5(c)). However, even for this structure, we can notice that despite local concentration fluctuations, an averaged profile mirrors the designed effective refractive index distribution. With that in mind, we have used the lens with 115 µm diameter for further tests.

 

Fig. 5. EDS image of the concentration of the Pb, K and Ba oxides in the measured sample. (a) 20 µm lens, (b) 40 µm lens, 115 µm lens.

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4. Verification of the nGRIN lens performance

Next, we numerically verified the optical performance of the diffused nGRIN lens. Using the full-vector beam propagation method (BPM), we modeled light propagation through the 200 µm thick and 115 µm wide structure, with parabolic profile. In the simulations, the lens was positioned in free space and illuminated by the planar wave of λ = 658 nm wavelength. Three different situations were examined i) when there is no diffusion between glasses (Fig. 6(a)), ii) when the diffusion occurs (Figs. 6(b)) and 6(c)) when the ideal GRIN lens is used (Fig. 6(c)). Diffusion in the lens was assumed to have a simple Gaussian profile. This was necessary since using previous measurements (SEM and EDS) it is impossible to determine diffusion range separately for each of the oxides in the glass.

 

Fig. 6. nGRIN lens structures used in the BPM simulations. (a) – lens without diffusion, (b) – lens with diffusion pattern similar to the EDS images, (c) – ideal GRIN lens. Top row shows refractive index distributions, while the bottom row shows cross-sections along diagonal of the hexagon, drawn with the same resolution as the EDS image – 0.2 µm.

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According to BPM simulations the nGRIN lens without any diffusion had working distance of wd₁ = 1086 µm with 1/e² beam diameter at focal plane d₁ = 12.16 µm; diffusive nGRIN lens has focal plane at a working distance wd₂ = 1020 µm with 1/e² beam diameter at focal plane d₂ = 11.15 µm, while the lens with ideal refractive index profile has wd₃ = 1000 µm with 1/e² beam diameter at focal plane d₃ = 10.72 µm (Fig. 7). Intensity of focal spots for nGRIN microlenses are lower than in case of the ideal one. This is a result of larger diameter of focal spot but also Rayleigh scattering, which occurs in a medium with feature size smaller than the wavelength. Scattering losses for nGRIN microlens with diffusion are smaller since boundaries between nanorods are not sharp anymore. In general, we can predict that efficiency of nGRIN microlens will be lower than ideal ones.

 

Fig. 7. BPM modelling of light propagation through 200 µm long and 115 µm wide lenses, corresponding to structures presented in Fig. 6 respectively. In the left column we show intensity distributions along the propagation axis in free space after end facet of the nGRIN lens. In the right column there are intensity cross-sections at the focal plane.

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Afterward, we experimentally tested the focusing capabilities of 200 um long and 115 µm wide diffused nGRIN lens drawn at the temperature of 830°C. Schematic of the imaging setup is shown in Fig. 8. During the examination the lens was clamped to the high precision translation stage (Thorlabs Nanomax) and positioned in front of 40× microscope objective, used to collimate an output beam from diode laser (Thorlabs L658P040, 658 nm). The beam formed by the nGRIN lens was focused by the a second 40× microscope objective onto the CCD camera sensor, which had a resolution of 320 × 240 pixels and pixel size ∼1 µm². The distance between the CCD camera and the microscope objective was constant during the measurement. A series of 41 images were taken at different distances from the nGRIN lens facet, with the position changing with a step of 50 µm and a with the accuracy of 3 µm. The spatial resolution of each transversal image was 2.26 pixels/µm, which was determined by imaging the microscope calibration target with the same system at the same magnification. Later, the collected images were combined to give the longitudinal profile of the beam propagation behind end facet of the nGRIN lens (Fig. 9).

 

Fig. 8. Schematic of the setup used for characterization of nGRIN lens focusing properties.

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Fig. 9. The cross-sections of laser beams measured along the propagation axis for the 658 nm wavelength, behind the nGRIN microlens.

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The focal spot is observed at the distance of 1050 µm from the end facet of the nGRIN lens.

It is important to note that nGRIN microlens works properly for a short-wavelengths, as considered 658 nm illumination. For that wavelength the 1/e² beam diameter in focal spot equals 14.11 µm, and the working distance of the lens is 1050 µm. The Airy disk for diffraction limited lens characterized with the same parameters equals 7.75 µm. The comparison between the obtained experimental results and theoretical calculations performed for the same lens parameters is shown in Table 3.

Table 3. Comparison between simulation and experimental results for the tested nGRIN lens.

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Difference between measured and simulated working distance values is a result of errors related to determination of position of nGRIN surface during beam profile measurements (Fig. 9). An additional source of error is related to the limited accuracy of measurement of refractive indices of component glasses NC34 and NC42 (Table 1), further used in numerical modeling. The λ = 658 nm is much shorter wavelength than previously predicted by λ⁄(2π) EMT boundary condition for lens with 115 µm diameter. Furthermore, the diffusion for each individual nanorod in the simulated structure was approximated using a Gaussian profile, as it is impossible to accurately predict overall diffusion profile of an individual nanorod fabricated form multicomponent glass. Because of those difference both simulated and fabricated lenses have slightly different gradient constant g, resulting in different lens profiles. For measured lens the n_G = 0.2π at the λ = 658 nm, well below previously stated n_G ≥ 2π for the lens to be working properly, which confirms our claim of the fabrication of the new optical compound glass. However, it is important to note that performance of developed nGRIN microlenses is not diffraction limited since experimentally we obtain a focal spot 1.8 times larger than in case of ideal thin lens.

Finally, we have also verified imaging properties of the fabricated microlens. For imaging tests, we have used a nGRIN microlens with the same transversal geometrical parameters (diameter d = 115 µm, refractive index difference $\varDelta$n = 0.0072) and length equal to 240 µm that corresponds to 0.064 fraction of the lens pitch (Fig. 10). As a test pattern we used a ruler bar pattern with 10 µm scale, where each printed bar had the thickness of 3.2 µm (Fig. 11(a)). Both the nGRIN lens and test pattern was mounted on a high precision translation stage, with nGRIN lens positioned over the test pattern. The image of the test pattern formed by the nGRIN microlens was further magnified by a 100× microscope objective to record on the CCD camera sensor, which had a resolution of 320 × 240 pixels and pixel size ∼1 µm². A thermal white light source was used to illuminate the test pattern.

 

Fig. 10. Schematic of the setup used to verify the imaging properties of nGRIN microlenses.

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Fig. 11. Verification of imaging properties of nGRIN microlenses: (a) test pattern, (b) image of the test pattern obtained with nGRIN microlens with magnification 1.3.

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Using this experimental setup, we have obtained a clear, sharp images of test pattern with a magnification from 0.9 to 1.3 as shown in Fig. 11(b). The obtained images do not show geometric distortions and cover the area of the entire lens. A field of view for tested nGRIN lens depends on the magnification and is limited from 6670 µm² to 12800 µm² for magnification 1.3 and 0.9, respectively This is determined by a small diameter and numerical aperture of the microlens and relatively small distance between the test pattern and the input faced of the microlens.

5. Conclusions

We present a modified method for fabrication of large diameter nanostructured GRIN microlenses. Our previous approaches met either practical limits in size of nGRIN microlenses [18] or their quality of performance [15]. Here, we combine a nanostructurization with enhanced diffusion. This allowed for the increase of size of individual nanorods used in microlens structure beyond limits given by the effective medium theory. Increased temperature of drawing stimulated diffusion process between individual rods. As final we obtain locally smooth changes in refractive index and total refractive index distribution close to designed parabolic profile.

We show that we can control the diffusion process by selecting a proper temperature and remaining drawing parameters. Diffusion range in fabricated microlenses can be larger than individual rod diameter in the structure, which equals to 1 µm.

As a proof-of-concept, we have fabricated a nanostructured GRIN microlens with parabolic refractive index profile. For development of nanostructure we used two types of silicate glass rods with refractive index difference of $\varDelta$n = 0.0072 for 658 nm incident wavelength. The fabricated lens has diameter d = 115 µm and length l = 200 µm, which corresponds to 0.053 lens pitch. In this case a dimeter of individual nanorods is 1 µm. It is much larger than a diameter nanorod d = λ⁄(2π) = 0.1 µm allowed according to the boundary condition of the effective medium theory.

The test lens has good energy transfer and focusing properties. It has working distance of 1.05 mm, which corresponds to the focal length of 1.11 mm. The focal plane focused beam has a diameter d = 14.11 µm. Imaging tests were performed on lens with the same transversal geometrical parameters and length equal to 0.064 lens pitch. We have obtained a good quality imaging with magnification up to 1.3 and field of view of 6670 µm². The images are clear and do not show geometric distortions.

An enhancement our originally proposed fabrication process of nanostructured GRIN microlenses with controlled thermal diffusion allows to develop microlenses with a similar workload, larger diameters over 100 µm and good beam transmission and imaging properties.