## Abstract

We present a novel fabrication technology for nano-structured graded index micro-optical components, based on the stack-and-draw method used for photonic crystal fibres. These discrete structures can be described with an effective refractive index distribution. Furthermore we present spherical nano-structured microlenses with a flat facet fabricated with this method and designed using an algorithm based on the Maxwell-Garnett mixing formula. Finally we show theoretical verification by using FDTD simulations for a nano-structured lens as well as experimental data obtained in the microwave regime.

©2009 Optical Society of America

## 1. Introduction

Gradient index components are a class of micro optical elements which obtain their optical functionality by changes of the refractive index without changing the surface profile. They are most commonly used as gradient index (GRIN) lenses [1]. The typical approach to fabricate radial gradient index materials is based on ion exchange [2], or by forming the gradient by leaching or by stuffing in a sol-gel formed glass [3]. These technologies allow only monotonic refractive index distributions with only very small contrasts.

Here we present a high volume fabrication technology for nano-structured dielectric elements which can be considered as gradient index materials with arbitrary index distributions within the limits of the refractive index constants of about Δ*n* ≈ 0.3 and a maximum gradient of about Δ*n* ≈ 0.3 per μm . They can be designed not only as microlenses but also as general arbitrary diffractive and refractive elements.

The technology introduced in this paper is based on the fabrication method for photonic crystal fibres (PCF), in particular all solid PCFs [4]. The optical functionality is achieved by assembling two materials in a binary subwavelength pattern which shows the behavior of a continuous gradient index material. The obtained micro-optical components have a flat facet and a profile of the refractive index dependent on the material distribution. Possible applications include microlenses and microlens arrays as well as general phase correctors. The feature sizes of the elements can range down to 100nm and less. Unlike conventional diffusion based methods for fabricating gradient index materials our approach allows us to obtain arbitrary nonmonotonic refractive index profiles and it is also possible to create microlens arrays with 100% fill factor in one fabrication procedure.

The focusing effect of gradient index photonic crystals has recently been reported for 2D photonic crystals [5]. In this approach the focusing is related to the resonance between adjacent holes with varying distance between the holes. The approach taken in this paper differs from this description: here we treat the pattern of a binary structure with very small features compared to the wavelength as an effective medium. The focusing effect can then be explained by an effective gradient index profile.

Dielectric structures with an unclustered distribution of the refractive index and feature sizes small compared to the wavelength of the incident light can be treated as a continuous gradient index material by applying the Maxwell-Garnett mixing formula.

## 2. Design of nano-structured gradient index micro-optical components

In this section we introduce a new description of arbitrary two dimensional dielectric structures with out of plane illumination. We are considering non absorbing discrete dielectric structures assembled from materials with different refractive indices and feature sizes that are small compared to the wavelength. Locally, on the wavelength scale, the materials are statistically distributed which results in an effective refractive index that can be calculated by averaging the refractive index over a certain neighbourhood. On the whole structure the fill factor of each material has a defined distribution to achieve the desired index profile. The analysis of these structures is based on the Maxwell Garnett mixing formula [6]. The optical properties of material compounds with small feature sizes compared to the wavelength can be treated as an effective yield of the compound’s properties such as the fill factor of each material and the shape of the elements. In the case of a circular or hexagonal pixel shape the random medium model can be applied to calculate the effective permittivity:

where *ε*
_{1} and *ε*
_{2} are the permittivities of the two glasses, < *ε* >= *ε*
_{2} + *f*(*ε*
_{1} - *ε*
_{2}) and f is the fill factor of *ε*
_{1}. Due to the very similar permittivities of the glasses used in the presented fabrication technology the difference between *ε _{eff}* and <

*ε*> is in the order of 10

^{-4}

*ε*and therefore we can assume

_{eff}*ε*= <

_{eff}*ε*> for our calculations. To obtain the effective permittivity distribution from the pixelated binary pattern, the effective permittivity for each pixel is calculated by averaging over a neighbourhood of about 1.5

*λ*or less which results in a continuous index distribution that can be treated as a gradient index material.

A strict upper limit for the accuracy of this approximation cannot be made as it depends on several factors such as the pattern and the index contrast of the materials. We could however show this approximation to be accurate in most cases for pixel sizes of about *λ*/10 and smaller. The experimental results described in section 6 show focusing for an element with feature sizes of *λ*/5, however the focus is not diffraction limited. In this case a rigorous treatment such as the Fourier modal method [7, 8] or the FDTD method [9] has to be applied to calculate the field propagation accurately.

A good approach for the search of the required material distribution is simulated annealing [10], a Monte Carlo method similar to the direct binary search, with the advantage that local minima of the cost function are more likely to be avoided. The cost function gives an indication of the quality of the pattern by calculating how well the current index distribution matches the target. We consider the standard deviation of the averaged index distribution to the desired index pattern as cost function:

where *n _{eff}* (

*p*) =

*ε*

^{2}

_{eff}is the effective index at point p, N(p) is the desired index value respectively and P is the total number of rods of which the structure is assembled.

The index profile of a radial GRIN lens is typically parabolic [11]:

where *n _{max}* is the maximum refractive at the center of the lens,

*r*is the diameter and a is a constant. The pitch length of the GRIN lens is then given as

*p*= 2

*πa*

^{-0.5}.

The design process starts with a random material distribution for the pixels. In each iteration step the material of one random pixel is changed and the new cost function *c _{NEW}* is calculated. The change is accepted if

*c*<

_{NEW}*c*and with a probability of

_{OLD}*p*=

*k*· exp{ - (

*c*-

_{NEW}*c*)/

_{OLD}*T*(

*I*)} if

*c*≥

_{NEW}*c*where k is a constant and T(I) is a parameter that is decreased exponentially with the number of iteration steps.

_{OLD}Figure 1(a) shows the result for a design of a spherical microlens. The effective index distribution calculated with equation 1 is very close to the desired parabolic index profile.

## 3. Verification of the Maxwell-Garnett approach using FDTD simulations

To verify our concept of developing gradient index elements by mixing materials we performed a sequence of 3D FDTD simulations comparing the light propagation in an ideal gradient index lens and in the nano-structured lens shown in Fig. 1(a). As refractive indices we used data from thermally matched glasses that are used for the fabrication of such structures. The two glasses are NC21A, a borosilicate glass developed in house, and F2, a lead-silicate glass supplied by SCHOTT AG. For the exact optical properties refer to the paper by Lorenc et al [12]. The red areas in Fig. 1 represent F2 glass (*n _{D}* = 1.60), the blue areas represent NC21A glass (

*n*= 1.51) respectively. The diameter of the lens is 10 μm with a feature size of 200 nm. The simulations were performed with a plane wave illumination with

_{D}*λ*= 1 μm , a spatial step size of 40nm, and a temporal step size of 6.67·10

^{-17}s. The computational space is terminated by the perfectly matched layer boundary condition [13]. Figure 2(a) shows the light propagation inside the nano structured lens (Fig. 1(a)) for the first 50 μm . The simulated results show very good agreement with the simulation of an ideal GRIN lens (Fig. 2(b)). The quarter pitch is 24.96 μm for the ideal lens and 24.24 μm for the nano-structured lens. Taking the value a=0.005 for the ideal grin lens the expected quarter pitch

*p*

_{1/4}can be calculated with

*p*

_{1/4}= 0.5

*πa*

^{-0.5}= 22.28 μm .

Figures 3(a) and 3(b) show the intensity distribution of the focal plane for the structured lens and the ideal GRIN lens respectively. The logarithmic scale reveals small differences in the first sidelobe, the central spot however is nearly identical. Figure 3(c) shows a cross section of the field intensity through the focal plane on a linear scale. The non-spherical assembly of the lens introduces a small asymmetry between the x- and the y-axis and a slightly lower peak intensity than an ideal GRIN lens with a smooth parabolic index distribution.

These simulation results suggest that the effective medium formula that has been used to design this lens gives an adequate description for the nano-structured diffractive elements introduced in this paper. Possible sources of errors are imperfections in the convergence of the design algorithm, the size of the features as well as numerical errors of the FDTD simulations.

## 4. Fabrication of nano-structures

The fabrication of the nano structured elements is based on the stack-and-draw technique used for photonic crystal fibres [14, 15, 16]. For the elements considered in this paper, rods of two different types of glass with matched thermal and mechanical properties are used. This method has been used successfully for the fabrication of all-solid photonic bandgap fibres [14]. Rather than drawing photonic crystal fibres with feature size typically in the order of a few μm we fabricate non-guiding structures with a much larger diameter and feature sizes of 200nm and less. These structures are cut to discs of a few hundred μm thickness for use as optical GRIN elements like lenses and beam correctors.

The fabrication of a nano structured optical element starts from the assembly of a macroscopic glass preform with the same structure as it is required in the final optical element although much larger in scale. These patterns can consist of a few hundreds up to a few thousand glass rods with final diameter of *λ*/10 or smaller for designs in the effective medium domain (Fig. 4(a)). The stacking of the first preform can be done either by hand or, for larger structures, a robot can be used to perform this standard pick-and-place routine [17].

At present only two different glasses are included in the structures although the use of further materials is possible. The glass rods, which are of similar diameter, can vary in refractive index but must be thermally matched, i.e. the coefficient of thermal expansion, glass transition temperature and softening point should be close. When these conditions are fulfilled the nano structured element has minimal internal tension and similar mechanical properties to a monolithic slab of glass.

In the next step, the preform is processed to generate an intermediate preform of about 2 mm diameter by using the fibre drawing tower (Fig. 4(b)). The set of intermediate performs are stacked together to form the final pattern, e.g. a single lens or micro-lens array (Fig. 4(c)). In this step it is possible to stack identical preforms to create an array of the structure or combine different intermediate preforms to one larger structure. If necessary the stack and draw steps can be repeated.

In the last step the final structure with nanometer feature size is generated by using the fibre drawing tower (Fig. 4(d)).

Finally, the nano structured rod is cut into discs and with a thickness dependent on the design functionality (Fig. 4(e)). The possible feature sizes of elements fabricated with this method range from an upper end of a few micrometers down to nanometers, although for features sizes below 10nm we would expect diffusion effects to soften the transitions between areas with different refractive indices. This effect does not pose a problem for our purpose since we assume an effective refractive index distribution by averaging the refractive index over the neighbourhood of each point. The final product is then cut to disks and polished from the top and bottom side which results in a single layer with an arbitrary pattern of the two materials.

## 5. Verification of technology for the fabrication of nano-structured micro-optical components

To verify the stack-and-draw technology we created the nano-structured gradient index lens that is shown in Fig. 1(a). The structure is assembled from F2 and NC21A. The rods are arranged in a hexagonal lattice with 50 rods in diameter. The final array is a hexagonal pattern of micro-lenses with 6 μm diameter each. The pictures show a small section of the microlens array which contains hundreds of identical lenses

Figure 5(a) shows an image of the first preform after drawing to a diameter of 1.9mm. Figure 5(b) shows the second intermediate preform which was obtained by stacking several rods from the first preform into a matrix. The pictures in Fig. 5 were obtained with an optical microscope. Figure 6 shows images of the final spherical micro-lens obtained by a phase contrast microscope. The images were taken with a microscope objective with NA=0.9. Pictures 6(a), 6(b) and 6(c) show the lens array at its final stage with a diameter of 6 μm. To enhance the visibility of the contrast one lens is displayed as a false color plot (Fig. 6(b)) and a cross-section through the center (Fig. 6(c)). The larger lens still shows the structure from the assembled pattern whereas the measurement of the smaller lens shows the expected parabolic phase distribution. With an optical microscope with NA=0.9 it is impossible to resolve the features with a size of 120nm of the smaller lens. Taking only the limited resolution of about 500 nm into accoubt can not explain the parabolic phase profile. Thus the measured phase distribution is a combination of the effective index and the limited resolution of the microscope objective. The lens was not tested for optical performance since its diameter is too small with respect to the wavelength to perform qantitive measurements.

The GRIN lens shown in Fig. 6 has a refractive index contrast of 0.09 between the two glasses. Taking a combination of another pair of thermally matched glasses, a lantanium based soft glass (n=1.825) and a silicate based glass (n=1.525), increases the possible index contrast to Δn=0.3.

## 6. Experimental verification in the microwave range

In this section we present experimental data for a scaled up lens, designed for the microwave range (*λ*=3 cm), to provide evidence for our theoretical results. Preliminary tests on the optical lens confirm the theoretical analysis. A detailed optical analysis of the nano-structured microlens will be published in a separate paper.

Figure 7(a) shows the design obtained with the algorithm of section 2. For experimental investigation we used rods with a diameter of 6mm and tubes with an outer diameter of 6mm and an inner diameter of 5mm. The experiments were performed at *λ* = 30*mm* giving a feature size of *λ*/5. In a hexagonal pattern the solid circular rods give a fill factor of f=0.907, the tubes give f=0.277 respectively. Taking the refractive index n=1.4 of PTFE at microwave frequencies [18] and the material fill factor of the rods and tubes we can calculate the two effective indices n _{rod} = 1.376 and n_{tube} = 1.094. The ratio of wavelength to feature size is very low so we would not expect a perfect GRIN behaviour. However this is the best ratio that is practically accomplishable. As Fig. 8 shows, the microwave lens displayed in Fig. 7(c) shows relatively good GRIN characteristics even for feature sizes of *λ*/5.

For the experiments a network analyser was used which measures the amplitude as well as the phase of the field. The green line in Fig. 8(a) shows a cross section of the measured phase immediately after the lens and a parabolic fit that would be expected for a perfect lens. The difference to the parabolic phase distribution results in a larger spot size in the focus but we expect it to improve with smaller feature sizes. Figures 8(b) and 8(c) show the measured intensity in the focal plane. The focusing effect of the field is clearly visible although the spot size is about twice the diffraction limit. Reasons for this are the feature size which is over the generally assumed limit of the mixing approximation and the amplitude profile of the microwave source which varied across the element.

## 7. Conclusion

We introduced a concept for the design and analysis of two dimensional nano-structured dielectric components by applying a formula based on the Maxwell-Garnett mixing formula. We showed evidence that these structures behave as gradient index materials by comparing FDTD simulations of such a nano-structured lens with an ideal gradient index lens. We introduced a fabrication technology to create these structures with a feature size of 200nm and below. Microscopic pictures of several stages of the fabrication are shown. Finally we gave experimental evidence by applying the method to a microwave lens. The investigated component showed the behaviour of a spherical lens.

## Acknowledgments

We gratefully acknowledge the UKEPSRC (Basic Technology grant No GR/S85764) for supporting this research. We thank Dariusz Pysz and Ryszard Stepien from the Institute of Electronic Materials Technology, Warsaw for fabricating the nano-structured lens and Dr. J. S. Hong for kindly offering his lab for microwave experiments. We also thank Dr. W. Saj for support with the FDTD simulations.

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