## Abstract

We investigate a two-dimensional low F-number dual micro-axilens array with binary structures based on a rigorous electromagnetic theory. The focal characteristics of a binary dual micro-axilens array (BDMA), including axial performances (focal depth and focal shift) and transverse performances (focal spot size and diffraction efficiency), have been analyzed in detail for different F-numbers, different incident polarization (TE and TM) waves, and different distances between micro-axilens. Numerical results reveal that the interference effect of a BDMA is not very evident, which is useful for building a BDMA with a high fill factor, and the focal characteristics of a BDMA are sensitive to the polarization of an incident wave. The comparative results have also shown that the diffraction efficiency of a BDMA will increase and the focal spot size of a BDMS will decrease when the F-number increases, for both TE polarization and TM polarization, respectively. It is expected that this investigation will provide useful insight into the design of micro-optical elements with high integration.

© 2011 OSA

## 1. Introduction

With developments in micro-fabrication technology––such as laser beam writing, electron beam writing, and reactive ion etching––micro optical elements with small characteristic sizes and high refinement are available [1,2]. Recently, various micro-axilenses and micro-axicons with continuous profiles have been studied in order to modulate the incident light wave to produce a constant intensity distribution in the focal region [3–8]. These elements have wide applications, such as high-precision alignment, nano-particle guiding, and scanning. Micro-axilenses that have characteristic scales comparable to or smaller than the incident wavelength are referred to as subwavelength micro-axilenses; binary subwavelength micro-axilenses whose profiles consist of binary features that are less than a wavelength have many merits, like high fill factor, easy design, easy integration, and easy fabrication using modern micro-photolithography and fabrication techniques [9–11]. Due to micro-axilenses’ binary subwavelength feature sizes, the electromagnetic scattering effect is more important than the light diffraction, so a rigorous electromagnetic theory should be used. Two-dimensional binary subwavelength diffractive lenses for implementing long focal depth and high transverse resolution based on a rigorous electromagnetic theory have been explored [10]. Otherwise, in our previous study we analyzed the beam-focusing characteristics (focal shift, focal depth, focal spot size, and diffraction efficiency) of micro diffractive lenses (micro-axilens) with binary subwavelength structures based on electromagnetic theory in detail [12]. Little work, however, has been done on analyzing the focusing characteristics of focused micro-axilens arrays with binary profiles––for instance, a binary dual micro-axilens array (BDMA). At the same time, with the growing demands of industrial applications including high-speed parallel optical switching networks, optical displays, and infrared photo-detection, there are more requirements for micro-axilens arrays with high resolution and a high signal-to-noise-ratio. So, a micro-axilens array with both a low F-number (the ratio of lens focal length to its diameter) and a high fill factor are continuously pursued [13,14]. Motivated by practical requirements and the advantages of a BDMA, in this paper we analyze the focal characteristics of a BDMA, including axial performances (focal depth and focal shift) and transverse performances (focal spot size and diffraction efficiency) in detail for different F-numbers, different incident polarization (TE and TM) waves, and different distances between binary micro-axilens by using a two-dimensional finite-difference time-domain (FDTD) method. Numerical results reveal that the interference effect of a BDMA is not very evident, which is useful for building a BDMA with high fill factors, and the focal characteristics of a BDMA are sensitive to the polarization of incident wave, which can be used for polarization sensors.

This paper is organized as follows. In Section 2 we give the fundamental formula used in our study. In Section 3 we analyze the dual-array system with different F-numbers, different incident polarization waves, and different distances between binary micro-axilens based on rigorous electromagnetic theory. Finally, brief conclusions and our contribution are included in Section 4.

## 2. Theoretical formulae and analysis model

We focus on an analysis of a two-dimensional (2-D) BDMA with the normally incident illumination, so in a Cartesian coordinate system for TE polarization, Maxwell’s equation can be written as Eq. (1a):

*x, y*(space components), and

*t*(time component); and $\sigma \text{\hspace{0.17em} and \hspace{0.17em}}{\sigma}^{*}\text{\hspace{0.05em} \hspace{0.05em}}$are the medium’s electric conductivity and magnetic conductivity, respectively. In general, the medium of a BDMA is not magnetic, so the relationship is satisfied as follows:${\mu}_{0}=\mu ,\text{\hspace{0.17em} \hspace{0.17em}}{\sigma}^{*}=0\text{\hspace{0.05em} \hspace{0.05em}}$. Similarly, for TM polarization the equation can be written as Eq. (1b):

By using general Yee’s grids [15], Eq. (1) can be solved by the FDTD method [16].

The 2-D electromagnetic scattering problem in the system of a binary dual micro-axilens array (BDMA) is schematically shown in Fig. 1
, where *l* is the distance between the two binary micro-axilenses and D is the diameter of the individual lens. We assume that a uniform plane wave is incident from the lower surface of the BDMA in the uniform dielectric medium, which fills the lower half space as the substrate; and, as the incident wave suffers from wavefront modulation at the surface of the BDMA with the same material as the substrate, it results in the focusing of the incident light in the free space. To determine the light field distribution, we apply the FDTD method to calculate the whole field transportation process.

In the non-paraxial approximation and for normal incidence, the profile function of a micro-axilens array can be written as follows [6,10]:

*m*is the number of zones; and D and

*f*are the diameter and the geometrical focal length of a lens, respectively. “±” represents the two micro-axilens placed symmetrically at the two sides of the

*y*axis.

In order to procure a BDMA binary structure, we approximate the continuous profile given by Eq. (2) to a piecewise-linear profile and then encode the individual linear segments as binary structures [17], as shown in Fig. 1.

In the next section, we will analyze the focusing performances of a BDMA, including focal depth, focal shift, focal spot size, diffraction efficiency for different F-numbers, different incident polarization waves, and different distances between binary micro-axilens, respectively.

## 3. Analysis and results

In our analysis, the refractive indices *n*
_{1} and *n*
_{2} are taken as 1.5 for glass material and 1.0 for free space (air), respectively. The material of lenses and substrates is the same (glass), and the thickness of substrates is assumed as infinite. The size of the aperture of the micro-axilens D is equal to 10 μm. A plane wave with wavelength *λ* = 1 μm in free space is normally incident on the BDMA along the *y* axis with TE or TM polarization, as shown in Fig. 1.

Meanwhile, we define focal depth as a region over which the intensity is greater than 80% of maximum intensity along the optical axis (*x* axis), and we define focal shift as a value that denotes the difference between the position of the maximum irradiance *y*
_{max} along the optical axis and the geometrical focal length *f* as defined in Eq. (2). It can be written as

Focal spot size is defined as the minimum-to-minimum width of the main lobe of the intensity profile on the real focal plane (*y*
_{max}, not the geometry focal plane, due to focal shift). The diffraction efficiency *η*, which expresses the focusing ability of a BDMA, is defined by the ratio of intensity in the main lobes on the real focal plane to the incident intensity.

Now we survey the interference effect in a binary dual micro-axilens array system. Each micro-axilens has an F-number 0.3 and a diameter of 10 μm, and so the geometrical focal length (multiplied F-number by diameter) is 3 μm. It is assumed that the two axilenses are placed abutting each other with zero distance between them (i.e., *l* = 0 μm) and they have the same parameters and are symmetrically placed on the two sides of the *y* axis. The total field distribution is calculated by the FDTD method as shown in Fig. 2
, which can give a global review of focusing characteristics for the BDMA with different incidence polarization waves (here (a) for TE polarization and (b) for TM polarization, respectively). It is clearly seen that two focused spots appear in the focal region, and the interference effect is not obvious between each binary micro-axilens, even for zero space between them, but the field distribution is different with TE polarization and TM polarization.

In order to reveal the focusing performance of the BDMA for TE and TM polarization, we give the optical field distribution of a normalized intensity distribution for TE and TM waves on both the axial and lateral directions, respectively, as shown in Fig. 3
––(a) axial intensity distribution along *y* axis; (b) lateral intensity distribution at the real focal plane (*y*
_{max} = 2.49 μm for TE polarization and *y*
_{max} = 2.40 μm for TM polarization, respectively). It is clear that there are certain distinctions for different polarization waves. To denote these distinctions more clearly, analysis results are listed as follows: the focal depth of the BDMA is 1.06 μm for TE polarization and 1.27 μm for TM polarization, respectively. The geometrical focal length is 3 μm, but the real focal length is shorter than the geometrical one, and there is focal shift, obviously. The focal shift Δf is 0.51 μm for the TE wave and 0.60 μm for the TM wave, respectively. The focal spot size and diffraction efficiency are 1.17 μm and 54.2% for the TE wave and 1.38 μm and 46.1% for the TM wave, respectively. From these results, we can basically see that the BDMA can get two good focused regions clearly, and meanwhile the focusing characteristics of the BDMA are sensitive to the polarization of incidence waves.

This can be explained. Basically, the binary sub-wavelength profiles of a BDMA form many micro-waveguide structures, which cannot be treated as phase-only modulations due to their subwavelength scales, and micro-waveguides modulate incident electromagnetic waves to produce a local resonance effect, which has been analyzed for binary blazed gratings [18–20]. The BDMA has aperiodic waveguides to produce a non-uniform resonance domain, and then it modulates a different incident polarization wave with different resonance effects, so the focusing characteristics of the BDMA are different with different polarization forms.

In order to compare the focusing characteristics of a lens with continuous profiles for TE and TM polarization, we give intensity distributions on both axial and lateral directions with the same F-number and diameter, as shown in Figs. 3(c) and 3(d). As we can see, two spots have been focused clearly in the focal region, but the interference effect is not obvious between each axilens with continuous profile. The focusing characteristics of the BDMA are more sensitive to the polarization of incidence waves than those of continuous ones due to the vector diffraction nature of binary structures like the micro-waveguiding effect, the shading effect, multiple scattering, and so on. Besides, binary structures have many advantages, such as high fill factor, easy design for beam control, and easy fabrication.

A BDMA with a low F-number and high fill factor is very useful for integration systems, so next we will change the distance between dual binary micro-axilenses to analyze their focusing characteristics. In this study, the spacing is varied for *l* = 0.0 to 10 μm for different polarization waves, and all other parameters are the same as in the previous case except for the spacing. The numerical results are summarized in Table 1
, including real focal length, focal shift Δf, focal spot size, and diffraction efficiency, respectively.

Meanwhile, all these analysis parameters have been plotted as a function of spacing *l*, as shown in Figs. 4(a)–(d)
, which are for focal shift, focal depth, focal spot size, and diffraction efficiency, respectively. It is clear that the focusing characteristics of the BDMA are obviously sensitive to the polarization of incidence waves. However, the interference effect is not very obvious between two focused beams, and the diffraction efficiency of the BDMA is 54.2% and 55.6% for distance *l* = 0 μm and *l* = 10 μm for TE polarization waves, respectively. In Figs. 4(a)–(d), we can also see that the parameters are slightly oscillating when the distance is increased. Compared with the diameter of a lens, the maximum distance is not so long and is only equal to the diameter; the interaction of two lenses clearly exists due to the vector diffraction nature of binary subwavelength structures, such as multiple electromagnetic scattering and micro-waveguides. For a global view, the total field distribution of a BDMA with space between = 10 μm is shown in Fig. 5
––(a) for TE polarization and (b) for TM polarization, respectively.

In order to analyze focal performances of the BDMA further, and considering that changes in F-numbers without changing micro-axilens diameters will affect performances, BDMAs with different F-numbers (0.4, 0.6, and 0.8) have been investigated with diameter 10 μm and different spaces between *l* (0, 2, and 4 μm). The analysis results, such as focal depth, focal shift, focal spot size, and diffraction efficiency, have been summarized in Table 2
for different incidence polarization waves. As shown in Table 2, differences of BDMA focusing characteristics between different incidence polarizations are very clear. Besides, it is evident that focal depth, focal spot size, and diffraction efficiency increase as F-numbers increase, and the focal shift represents some oscillation. Compared with the former case (F-number = 0.3), we can see diffraction efficiency is low for a BDMA with a lower F-number due to BDMA special profile structures. With the same diameter of a BDMA, when the F-number is lower, the profile structures of the BDMA lens are more complex, and there are more micro-waveguides with different feature sizes on the profile and more electromagnetic shadowing effects [21]. So, the relationship between local resonance effects, multiple electromagnetic scattering, and evanescent waves will make for lower diffraction efficiency. However, binary structures of a BDMA, like aspect-ratios and profile heights, can be designed carefully to get higher diffraction efficiency with a lower F-number.

## 4. Conclusion

We present the rigorous electromagnetic analysis of a BDMA by using a two-dimensional finite-difference time-domain method. The focal characteristics of a BDMA, including focal depth, focal shift, focal spot size, and diffraction efficiency have been analyzed in detail for different F-numbers (0.3, 0.4, 0.6, and 0.8), different incident polarization (TE and TM) waves, and different distances between micro-axilens based on the rigorous electromagnetic theory. The lateral and axial intensity distribution of a BDMA for TE polarization and TM polarization are calculated and compared for different spaces between dual binary micro-axilenses. The comparative results have shown that the interference effect of a BDMA is not very evident, which is useful for building a BDMA with a high fill factor, and the focal characteristics of a BDMA are sensitive to the polarization of incident waves, which can be used for polarization sensors. The numerical results have also shown that the diffraction efficiency of a BDMA will increase and the focal spot size of BDMS will decrease when the F-number increases, for both TE polarization and TM polarization, respectively. We believe that our study results will be useful in fields of application, analysis, and design of micro-axilens arrays with both low F-numbers and high fill factors and will provide useful insight into the design of micro-optical elements with high integration.

## Acknowledgments

This research is supported by the Specialized Research Fund for the Doctoral Program of Higher Education (20091102120020) and the NSFC (No. 50875015).

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