## Abstract

The possibility and the efficiency of using a single diffractive lens to achromatize and apochromatize micro-objectives with plastic lenses are shown. In addition, recommendations are given on assembling the starting configurations of the objectives and calculating the design parameters required for subsequent optimization. It is also shown that achievable optical performance of achromatic and apochromatic micro-objectives with plastic lenses satisfy the qualifying standards for cell-phone objectives and closed-circuit television (CCTV) cameras.

© 2010 Optical Society of America

## 1. Introduction

As is well known, the first-order chromatism correction of refractive optical systems within the limited choice of optical materials can be achieved by means of diffractive lenses (DL) (see [1, 2, 3, 4, 5], for example). One of the possible ways to achromatize or apochromatize an optical system with the chromatism above the required level is shown in [6]. It implies the usage of a diffractive–refractive correction unit, consisting of a DL and one or two refractive lenses (RLs). The correction unit can be assembled using both additional lenses and the optical system’s own RLs, located near the aperture stop.

In the present paper, recommendations are given for rational assembling the micro-objective configuration that allows achieving achromatization or even apochromatization by means of merely a single additional DL. These recommendations also open the way for further optimization because they give starting design values. Both the recommendations and the calculations are given for photo- and cell-phone micro-objectives and closed-circuit television (CCTV) systems of security cameras working in day/night vision; that is, in the spectral interval that includes visible and near-infrared ranges from ${\lambda}_{\mathrm{min}}=0.4\text{}\mathrm{\mu m}$ to ${\lambda}_{\mathrm{max}}=0.9\text{}\mathrm{\mu m}$. Because of the low-cost requirements and high optical performance of the given micro-objective class, the usage of optical plastic materials seems advisable to produce their elements. Modern shaping methods based on precision punching allow replicating plastic lenses with aspheric surfaces and, as required, applying a diffractive microrelief [7, 8].

## 2. Achromat Configuration and Optimization Results

Based on the analysis of commercial micro-objectives of the class in question offered by different manufacturers [9], their basic optical performance can be taken as default values as follows: the back focal length ${f}^{\prime}\approx 3.7\text{}\mathrm{mm}$, relative aperture $1:2.4$, and field angle in object space $2\omega =60\xb0$. This optical performance can be reached if the triplet was used as the starting configuration of the achromat. Such a triplet can consist of lenses made from the same crownlike plastic and having aspheric surfaces.

In the visible range, i.e., between the blue $F-$ and red $C-$ spectral lines of hydrogen (${\lambda}_{\mathrm{min}}={\lambda}_{\mathrm{F}}=0.48613\text{}\mathrm{\mu m}$ and ${\lambda}_{\mathrm{max}}={\lambda}_{\mathrm{C}}=0.65626\text{}\mathrm{\mu m}$), it is possible to be limited to achromatization that provides the equality of system image distances at the edges of the selected spectral range (${s}_{{\lambda}_{\mathrm{min}}}^{\prime}={s}_{{\lambda}_{\mathrm{max}}}^{\prime}$), due to the small width of a secondary spectrum. In a wider spectral range that includes the visible and near-infrared ranges, the full exclusion of a primary chromatism influence to an objective’s resolution is only possible via apochromatization. It should provide equality of system image distances on three wavelengths ${\lambda}_{\mathrm{min}}<\overline{\lambda}<{\lambda}_{\mathrm{max}}$, that is, ${s}_{{\lambda}_{\mathrm{min}}}^{\prime}={s}_{\overline{\lambda}}^{\prime}={s}_{{\lambda}_{\mathrm{max}}}^{\prime}$.

The refractive indices and the Abbe numbers, as well as all other optical parameters in this paper, are presented for the yellow d-line of helium (${\lambda}_{\mathrm{d}}=0.58756\text{}\mathrm{\mu m}$) that, due to rigid requirements for the camera’s resolution in day vision, was used for apochromatization alongside ${\lambda}_{\mathrm{min}}=0.4\text{}\mathrm{\mu m}$ and ${\lambda}_{\mathrm{max}}=0.9\text{}\mathrm{\mu m}$); that is, $\overline{\lambda}={\lambda}_{\mathrm{d}}$. Polymethyl methacrylate (PMMA), i.e., acrylic (${n}_{\mathrm{d}}=1.491756$, ${\nu}_{\mathrm{d}}=57.4408$) and polycarbonate (PC) (${n}_{\mathrm{d}}=1.585470$; ${\nu}_{\mathrm{d}}=29.9092$) were chosen as a crownlike optical material and a flintlike material, respectively.

To achromatize the triplet with all RLs made of PMMA, it proved quite sufficient to include a single DL into the configuration [10]. Figure 1 and Table 1 show the lens drawing and optimized lens listing for the four-lens achromat, respectively. In Table 1 and subsequent tables, ${\alpha}_{i}$, *m*, and ${A}_{j}$ are parameters of aspheric surfaces and the DL’s circular micro structure applied to the back side of the aspheric surface of the first RL. The above parameters appear in equations that are used to describe optical ele ments, in particular, in the optical design program ZEMAX [11].

The sag or *z* coordinate of the even aspheric surface is given by the following expression:

*c*is the curvature (the reciprocal of the radius),

*ρ*is the radial coordinate, and ${\alpha}_{i}(i=2,3,\dots )$ are polynomial coefficients.

The equation of the DL structure spatial frequency (its units are cycles/millimeter) takes the form

Thus, DL adds phase to the ray according to the following polynomial expansion: where*ρ*is the radial distance from the optical axis and

*m*is the diffraction order.

The DL with the structure described by Eqs. (2, 3) has optical power determined by ${A}_{1}$ and *m*:

The analysis of aberration curves of a micro- objective with the design values given in Table 1 has shown it possible to combine achromatization with the significant reduction of the secondary spectrum by restricting variation of the back focal length in the visible spectral range (${\lambda}_{\mathrm{min}}=0.48613\text{}\mathrm{\mu m}$, ${\lambda}_{\mathrm{max}}=0.65626\text{}\mathrm{\mu m}$) by a magnitude of $\mathrm{\Delta}{s}_{\mathrm{F}}^{\prime}\le 7.5\text{}\mathrm{\mu m}$.

It was also found possible to significantly lower monochromatic aberrations and the spherochro matism. As a result, the polychromatic resolution of this objective at the relative aperture $1:2.4$ is $93\text{cycles}/\mathrm{mm}$ across the field angle $2\omega =60\xb0$ with contrast $K=0.5$, and, with $K=0.76$, it never goes below $50\text{cycles}/\mathrm{mm}$. The distortion within the entire field of vision does not exceed $\pm 0.7\%$. However, this objective has a notable drawback. It is the quite wide range of field angles in the image space that leads to a heterogeneous lighting of the multielement photodetector surface. Indeed, incidence angle of the main rays onto peripheral elements of the photodetector is ${{\omega}^{\prime}}_{\mathrm{max}}=1{.1\omega}_{\mathrm{max}}$, that is, $33\xb0$. The above disadvantage can mostly be avoided by including an additional RL as a terminal lens. Figure 2 and Table 2 show the lens drawing and optimized lens listing for the five-lens achromat, respectively.

Ray aberration, wavefront spherical aberration, and polychromatic modulation transfer function (MTF) curves for this micro-objective are shown in Figs. 3, 4, 5, 6. The MTF has been calculated in a paraxial image plane. As follows from these figures, the resulting plane-achromat at the relative aperture $1:2.4$ provides the resolution of $125\text{cycles}/\mathrm{mm}$ across the field angle $2\omega \le 60\xb0$ at a contrast no lower than 0.5, and $100\text{cycles}/\mathrm{mm}$ at a contrast of 0.59. The range of changes of the back focal length in the visible spectral range does not exceed $6.61\text{}\mathrm{\mu m}$, and the distortion is lower than $\pm 1\%$. Incidence angle of the main rays onto peripheral elements of the photodetector is ${\omega}_{\mathrm{max}}^{\prime}=0.66{\omega}_{\mathrm{max}}$, that is, $19.8\xb0$.

## 3. Design and Apochromat Optimization Results

The studies have shown that apochromatization is achievable with an additional DL included in the optical scheme of a micro-objective; however, the scheme should satisfy several conditions. First, along with positive lenses made of PMMA, it should include at least one negative lens made of PC that would be the place for a DL. Second, prior to the inclusion of a DL, the starting configuration should provide a way to limit the focal length drop within the selected spectral range to the value equal to the distance along the optical axis between the first zeros of intensity in the three-dimensional light distribution near focus [12]:

*f*-number. Also, the absolute value of the radius of the Petzval surface, on which an apochromatic optical system could form a stigmatic image [12], should satisfy the following condition (in case of $2\omega =60\xb0$):

To these requirements it is necessary to add the ones of restriction of the coma and astigmatism, which should be carried out at the surface curvatures of the lenses acceptable from the point of view of ray heights. All the above-mentioned requirements can already be fulfilled by a simple triplet. In particular, this is conditioned by the possibility of using aspheric refractive surfaces, and also by the fact that spherical aberration correction is not required for the starting configuration because a DL will be used in the design. However, the residual spherochromatism of the achromat based on a triplet notably limits the resolution of a micro-objective due to the extended spectral range and necessitates using a scheme with four RLs, each one with two aspheric surfaces. The starting design values of the scheme (surface radii, lens thicknesses, and air gaps, and also parameters for even aspheric surfaces) can be found by using any known method of dimension or aberration calculation). However, a pseudo-ray method has been used by the authors that implies a ray-tracing approximation with different infinitesimal orders [13].

After placing a DL on the front surface of the second RL and further optimization, the final configuration and design values of the five-lens apochromatic micro-objective have been obtained (see Fig. 7).

The main advantages of a five-lens apochromat are a low level of monochromatic aberrations, a strict apochromatization with a small tertiary spectrum, a virtually zero distortion, and a wide spectral range polychromatic resolution completely comparable with a four-lens achromat resolution. Unfortunately, the incidence angle of the main rays onto peripheral elements of the photodetector is unacceptable (${{\omega}^{\prime}}_{\mathrm{max}}=1.{13\omega}_{\mathrm{max}}\approx 34\xb0$).

By replacing three RLs of a micro-objective with two-lens cemented elements, it was possible to satisfy the above-mentioned requirements and, simultaneously, to limit the ratio of field angles in the image space and object space over 2.8 times (i.e., reduce it to ${{\omega}^{\prime}}_{\mathrm{max}}=0.{4\omega}_{\mathrm{max}}$) and thereby approximate the ray path to telecentric parameters ${\omega}_{\mathrm{max}}^{\prime}\approx 12\xb0$. At the same time, the DL was moved onto the back refracting surface of the first cemented element. The optimization of this eight-lens configuration with only four aspheric surfaces has led to a plane- apo-chromat with the focal length ${f}^{\prime}=3.7\text{}\mathrm{mm}$ that provides a resolution of $100\text{cycles}/\mathrm{mm}$ at the relative aperture $1:2.4$ and a contrast of no less than 0.5 within the field angle range $2\omega \le 60\xb0$. The range of changes of the back focal length does not exceed $6.9\text{}\mathrm{\mu m}$ in the range from ${\lambda}_{\mathrm{min}}=0.4\text{}\mathrm{\mu m}$ to ${\lambda}_{\mathrm{max}}=0.9\text{}\mathrm{\mu m}$, the lateral color is comparable with the Airy disk, and the distortion is less than $\pm 1\%$. Figure 8 and Table 3 show the lens drawing and the optimized lens listing for the eight-lens plane-apochromat, respectively. Ray aberrations, wavefront spherical aberration, and MTF curves calculated in the paraxial image plane are shown in Figs. 9, 10, 11, 12. These curves demonstrate the capabilities of the eight-lens plane-apochromat.

Notice that the DL structures of all micro-objectives presented in this paper have a low-frequency microstructure, which allows their manufacturing with two-layer relief-phase structures. For this purpose, the same optical plastics, i.e., PMMA and PC, can be used. Use of PMMA and PC warrants nearly 100% of the DL’s diffraction efficiency within the entire spectral range [14].

## 4. Conclusion

To summarize the results of the present paper, the following conclusions can be made.

- An achromatization in the visible range with the low level of a secondary spectrum is possible via the inclusion of a single DL in a micro-objective having all its RLs made of the same crownlike plastic.
- The inclusion of a single DL in a micro-objective having its RLs made of two types of crownlike or flintlike plastic allows an apochromatization in a wide range of wavelengths, including the visual and the near-infrared ranges.
- Recommendations are given by the authors that allow assembling the starting configurations of objectives and calculation of the design values required for further optimization.
- The attainable optical performance of achromatic and apochromatic micro-objectives with plastic lenses satisfy the qualifying standards for cell-phone objectives and CCTV cameras.

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