## Abstract

We present a rigorous electromagnetic design and analysis of two-dimensional diffractive lenses (DLs) with different axial resolution and high lateral resolution. Without paraxial approximation, focusing characteristics of two kinds of DL, one with a long focal depth and a high lateral resolution, the other with high axial resolution and high lateral resolution, for *f*-numbers of 0.6, 1.0, 1.5, and 2.0 have been determined including the actual focal depth, the ratio between the focal depth of the designed DL and the focal depth of the conventional quadratic lens, and the spot size of the central lobe at the actual focal plane. Numerical and graphic results show that the designed DLs indeed have a long focal depth and a high lateral resolution, or high axial resolution and high lateral resolution by use of different preset focal depths.

© 2003 Optical Society of America

## 1. Introduction

In recent years, various diffractive lenses (DLs) with low axial resolution (i.e., a long focal depth) and high lateral resolution have been extensively studied because of their wide variety of uses such as in high-precision optical alignment systems, profile measurement systems, and optical disk readout systems [1,2]. Meanwhile, with the development of confocal scanning microscopy systems, diffractive focal elements with high axial resolution (i.e., a short focal depth) as well as high lateral resolution are needed when one deals with imaging systems and serves to prompt increased development in multidimensional imaging [3,4]. Focal elements that can decrease the laser writing spot size and increase the read optics resolution are another important design for optical data storage setups. Refinements in micro- photolithography, laser-beam writing, inductively coupled plasma writing, reactive ion etching, and electron-beam writing have made possible the production of diffractive optical elements (DOEs) with smaller features. Obtaining accurate focusing performances of DOEs depends on a fully rigorous solution of the electromagnetic equation. However, most designs and analyses for determining the axial and lateral resolution of DLs have been in paraxial approximation and based on scalar diffractive analysis methods. Torok and Wilson[5] have described a rigorous theory for axial resolution in confocal microscopes and have given a proper, rigorous, and theoretical description of the axial response. More recently, Ye[6] and Dong and Liu[7] designed and analyzed diffractive axilenses with a long focal depth and high lateral resolution based on the rigorous electromagnetic method, known as the boundary-element method, and obtained good results. However, these researchers did not study how to construct phase distribution for lenses with high axial resolution and high lateral resolution based on rigorous diffractive theory. To overcome these limitations and motivated by the wide applications of DLs with different axial resolution and high lateral resolution, we present, what is to the best of our knowledge, the first special phase functions in nonparaxial approximation for two kinds of DL, one with low axial resolution (a long focal depth) and high lateral resolution, and the other with high axial resolution (a short focal depth) and high lateral resolution on the basis of a rigorous electromagnetic computational model. Compared with other computational electromagnetic methods, such as the finite-element method [8], the boundary-element method [9], and the hybrid finite element-boundary element method [10], the finite-difference time-domain (FDTD) [11–13] method does not need to solve a large system of equations and can have a linear computational dependence on the number of solution points in the computational region. For these reasons, it is computationally efficient and well suited for the calculation of DLs both in the time domain and in the spectral domain. We numerically analyze and design these two kinds of DL by using the FDTD method and our work is focused on two-dimensional, continuous profile DLs. In comparison with conventional lenses, the results of rigorous numerical simulations have shown that these two kinds of DL can have the required focusing characteristics. In our study we used the focal depth over which the intensity is greater than 80% of maximum intensity along the right axis (i.e., the y axis) to describe the axial resolution, and we used the diffraction-limited spot size to describe the lateral resolution.

In Section 2 we discuss the basic formulas used in our study. In Section 3 we provide the results of nonparaxial, rigorous designs of two kinds of DL with different axial resolution and high lateral resolution. Finally, a brief conclusion and our contribution are given in Section 4.

## 2. Theoretical formulas

Interaction between the incident field and the DL is analyzed rigorously by use of the FDTD method that can be described as follows. In our study, Maxwell's equations can be written as

$$\frac{\partial {H}_{x}}{\partial t}=-\frac{1}{\mu}\left(\frac{\partial {E}_{z}}{\partial y}+{\sigma}^{*}{H}_{x}\right)$$

$$\frac{\partial {H}_{y}}{\partial t}=\frac{1}{\mu}\left(\frac{\partial {E}_{z}}{\partial x}-{\sigma}^{*}{H}_{y}\right)$$

where *E _{z}*,

*H*,

_{x}*H*are the electric field and magnetic field components, respectively, which are the functions of

_{y}*x, y*(space components), and

*t*(time component),

*σ*and

*σ** are the medium’s electric conductivity and magnetic conductivity, respectively. In general, the medium of DLs is not magnetic, so the relationship is satisfied as follows:

*µ*

_{0}=

*µ*,

*σ**=0. By use of Yee’s grids and applying the central different expressions, Eq. (1) can be calculated by a digital computer. Because the extent of the FDTD region (a computational region in the FDTD method) is limited and to be able to simulate unbounded free space, absorbing boundary conditions must be along the FDTD region in which the DL is totally embedded. The perfectly matched layer absorbing boundary conditions proposed by Berenger [14] were used for our study. The normally incident TE polarization plane wave that is supposedly incident from medium to air is introduced along the connecting boundary by use of a totally scattered field algorithm [15], and the technique of a semi-infinite FDTD analysis was adopted for our study [16]. To reduce the computational time, the height of the FDTD region should be limited to nearly the same height as the DL profiles. In our study, the field distribution just above the DLs calculated by the FDTD method could be propagated from the near field (the output plane) to the far field (the observation plane) by use of the angular spectrum (AS) propagation method [12,13].

In free space, each AS component propagates to the observation plane (*y*=*y*
_{0}) making a phase delay caused by the transform function as in Eq. (2):

where *f _{x}* and

*cos*(

*α*) are the AS spatial frequency and the direction cosine in the

*x*direction, respectively. By use of powerful tools such as the FDTD method and the AS propagation method, we can design and analyze two kinds of DL as described in the following section.

## 3. Numerical experiments

#### 3.1 Profile distribution of diffractive lenses

In a nonparaxial approximation, the profile function of DLs can be written as

where *n*
_{1} and *n*
_{2} are the refractive indices of the medium and the air of lenses, respectively; *n*
_{1}>*n*
_{2}; *λ*=*λ*
_{0}/*n*
_{2}; *λ*
_{0} is the incident wavelength in vacuum; *m* is the number of zones; *D* and *f* are the diameter and the focal length of DLs, respectively. The focal length is usually a constant for conventional DLs, but, to design DLs with different axial resolutions, we set the focal length as a continuous function that was suggested by Davidson *et al.* [2]:

where *f*
_{0} and *d _{f}* are the beginning focal length and the preset focal depth. So substituting Eq. (4) into Eq. (3), we obtained the profile distribution of DLs in the nonparaxial approximation as

As can be seen, Eq. (5) reduces to a conventional DL profile distribution when the preset focal depth *df* equals zero. By using different preset focal depths, we obtained a DL with a different actual focal depth, which can be verified as described below.

#### 3.2 Numerical simulations

We define
$\overline{df}$
and
$\overline{d{f}_{0}}$
as the actual focal depth of the designed DL and that of a conventional lens, respectively. Meanwhile, to evaluate the axial resolution and the lateral resolution, we calculated the actual depth
$\overline{df}$
, the ratio
$Ra=\frac{\overline{df}}{\overline{d{f}_{0}}}$
(i.e., the relative focal depth [7]), and the spot size of the focused beam in the actual focal plane. The lens was assumed to be fabricated in glass (refractive index *n*
_{1}=1.5) and the outside medium is air (refractive index *n*
_{2}=1.0). The FDTD method requires sampling the calculated region at λ/20 intervals, and wavelength λ in air is 1 *µm*. The *f*-number of a DL is defined as the ratio of its beginning focal length *f*
_{0} to its diameter *D* (*f*/#=*f*
_{0}/*D*). We designed four families of DLs with *f*-numbers of 0.6, 1.0, 1.5, and 2.0, all having a 12-*µm* diameter, resulting in DL focal lengths of 7.2, 12, 18, and 24 *µm*. To obtain lenses with a different axial resolution, we set the preset focal length of *df* as ±0.3*f*
_{0} and ±0.6*f*
_{0}. With a positive value we obtained a DL with low axial resolution and with a negative value we obtained a DL with high axial resolution. So a higher axial resolution means a smaller (<1) *Ra* value, and a lower axial resolution means a larger (>1) *Ra* value.

We calculated the axial direction intensity distribution of a DL for *f*/1.0 by using the FDTD method and the AS method as shown in Fig. 1 for different values of preset focal depths *df*. For comparison, the axial intensity distribution of a conventional DL (*df*=0 *µm*) was also calculated and is represented by curve a in Fig.1. Curve b corresponds to *df*=-3.6 *µm* and curve c corresponds to *df*=+3.6 *µm* (0.3*f*
_{0}). It is obvious that the shape of curve b is narrower than that of curve a, and the shape of curve c is wider than that of curve a. This means that, when setting a positive *df*, we can design a DL with a longer focal depth, i.e., a lower axial resolution, and this conclusion is in good agreement with that in Ref. 7. In contrast, when the *df* is negative, the designed DL will have a shorter focal depth (i.e., a higher axial resolution). The focal depths of curves a, b, and c are 4.2, 3.4, and 6.8 *µm*, respectively. Values of *Ra* for two lenses are 0.80 and 1.62. The actual focal planes, meaning the position that corresponds to the maximal intensity peak for curves a, b, and c in Fig. 1, are located at *y*=10.2, 7.9, and 11.3 *µm*, respectively. The lateral intensity distributions at the focal planes for these three DLs are shown in Fig. 2. The sizes of the central lobes are 1.14, 0.97, and 1.24 *µm*, which are almost the same as the diffraction-limited spot size *λf*
_{0}/*D*=1 *µm*. The lateral intensity profiles of the designed DLs at different observation planes within the focal depth region are plotted in Fig. 3: (a)
*df*=-3.6 *µm* and (b)
*df*=+3.6 *µm* (dashed curves correspond to *y*=6.6 and 8.6 *µm*, solid curves represent the actual focal plane, and the dotted curves correspond to *y*=9.8 and 15.0 *µm*). The plots clearly demonstrate that the designed DLs represent good focusing characteristics, indicating that the total intensity profiles of the electric fields are always focused on the central lobe, which illustrates that the designed DLs have high lateral resolutions. To obtain a regional view of both the axial resolution and the lateral resolution, we display the propagation plots of electric field intensity over a region surrounded by focal points in Fig. 4: a, *df*=-3.6 *µm*; b, *df*=0 *µm* (a conventional lens); c, *df*=+3.6 *µm*. The bright regions correspond to high field values, and the dark regions correspond to low values. As meant by figures, we believe that our designed DLs have special functions of different axial resolutions and high lateral resolutions.

We set the *f*-number at *f*/2.0 for this design, and the important features of the DL remained unchanged. Figure 5 shows the axial direction intensity of the DL for *f*/2.0 with a different value for *df*: curve a, a conventional DL (*df*=0 *µm*); curve b, *df*=-14.4 *µm;* curve c, +14.4 *µm* (0.6*f*
_{0}). The actual focal depths for three DLs are 11.1, 4.1, and 17.2 µm, and the *Ra* values for curves b and c are 0.37 and 1.55, respectively.

Compared with the focusing characteristic of a conventional DL, it is evident that the designed DLs can be obtained with different focal depths according to the value of the preset focal depth *df*. The actual focal planes for curves a, b, and c appear at *y*=17.3, 10.0, and 22.5 *µm*, respectively. The lateral intensity distributions at the actual focal planes of three DLs are shown in Fig. 6, and the spot sizes of the central lobes are 1.00, 1.59, and 2.10 *µm*, which are similar to a diffraction-limited spot size of 2 *µm*. To observe the lateral intensity distribution on the plane for different distances from the DL surface, we calculated the lateral intensity of DLs for different depths *df* at three different distances from the DL, respectively. The results are shown in Fig. 7: (a)
*df*=-14.4 *µm* (dashed curve, *y*=9.3 *µm;* solid curves, the actual focal plane; dotted curve, *y*=12.4 *µm*); (b)
*df*=+14.4 *µm* (dashed curve, *y*=17.9 *µm;* solid curve, the actual focal plane; dotted curve, *y*=32.7 *µm*). As an initial example, the plots evidently illustrate that the designed DLs can achieve long focal depths with high lateral resolution if we set a positive preset focal depth, as well as high axial resolution with high lateral resolution if we set a negative preset focal depth. Also, to present a global view of the DL focusing performance, propagation plots of the electric-field intensity are displayed in Fig. 8.

Finally, to compare the focusing characteristics of different designs of DLs with different *f*-numbers (*f*/# of 0.6, 1.0,1.5, and 2) and with different preset focal depths (**±**0.3 *f*
_{0} and **±**0.6 *f*
_{0}), we summarized the numerical results in Table 1, including the actual focal depth, the value of relative focal depth *Ra*, the position of the actual focal plane, and the spot size of the central lobe at the actual focal plane. It is obvious that the actual focal depth of the DL increases when the preset focal depth increases, and the axial resolution of the DL increases when the preset focal depth is reduced for a larger *f*-number (i.e., *f*/#>1). A smaller preset focal depth cannot always produce a DL with a higher axial resolution for a smaller *f*-number (i.e., *f*/#<=1), and we are currently investigating this. We believe that this inexplicable difference is not caused by computational precision, because we set the space interval at λ/40, recalculated the DL for a small *f*-number, and obtained almost the same results: *f*/#=0.6 and
$\overline{df}$
of 2.1, 1.4, and 2.4 *µm* for the space interval of λ/20;
$\overline{df}$
of 2.2, 1.4, and 2.4 *µm* for λ/40. For *f*/#=1.0, the
$\overline{df}$
calculated with the space interval of λ/20 is the same as that calculated with the space interval of λ/40. In addition, the focusing performance of a small *f*-number DL will deteriorate because of some factors such as the diffractive structure shadow and the large obliquity factor for off-axis angles [17]. However, as has been shown, with a negative preset focal depth, we can indeed produce a DL with high axial resolution and high lateral resolution simultaneously.

## 4. Conclusion

By using the rigorous FDTD electromagnetic computational model, we have presented the nonparaxial design and analyses of two kinds of DL: one has a long focal depth with high lateral resolution, the other has high axial resolution and high lateral resolution. We designed diffractive lenses with *f*-numbers of 0.6, 1.0, 1.5, and 2.0 by using different preset focal depths. In comparison with a conventional lens, the results of rigorous numerical simulations illustrate that these two kinds of DL can achieve the required focusing characteristics. The combination of numerical and graphic results have shown that the designed DLs have long focal depths if we use a positive preset focal depth and have high axial resolution if we use a negative preset focal depth. In general, the focal depth of DLs of all four *f*-numbers increases when the preset focal depth increases, and the focal depth of DLs whose *f*-numbers are larger than 1 decreases when the preset focal depth decreases. However, a smaller preset focal depth cannot always produce the DL with a shorter focal depth when the *f*-number of the DL is less than 1. On the basis of the design and analysis results, our method could prove to be a useful technique for the rigorous design of DLs with different axial resolution and high lateral resolution. These DLs might also be useful for practical applications such as in optical disk readout systems and in confocal scanning microscopy systems.

## Acknowledgments

This research is supported by National Research Fund for Fundamental Key Projects No. 973 (G19990330).

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