1. Introduction

Conventional refractive lenses have a fairly large thickness, which limits the possibilities of their use in compact optical systems. This limitation can be overcome by using diffractive lenses (DLs) having the microrelief thickness comparable with the wavelength of the incident radiation. However, DLs are usually designed to operate at a single wavelength λ₀ and therefore have chromatic aberrations, which are significantly larger than those of conventional lenses. Indeed, the chromatic aberration (a change in the focus position caused by a change in wavelength) of a refractive lens is determined only by the material dispersion. In contrast, in the case of a DL, the focus position change is primarily due to a change in the angles of diffraction on different microrelief zones. These angles are proportional to the wavelength of the incident light, and their change results in a much stronger shift of the focus at varying wavelengths than the material dispersion.

Note that diffractive and refractive lenses have opposite chromatism. Therefore, combinations of refractive and diffractive elements are often used for the compensation of chromatic aberrations [1–3]. For reducing chromatic aberrations, the so-called harmonic diffraction lenses (HDLs) have also been proposed [4–9]. Compared with “ordinary” DLs, HDLs have a higher diffractive microrelief. The maximum height of a DL microrelief equals λ₀ /(n – 1), where n is the refractive index of the lens material, whereas the relief height of an HDL is M times greater. HDLs can be considered as an intermediate case between diffractive lenses (M=1) and thin refractive lenses, which can be obtained from HDLs at $M \gg 1$. HDLs have smaller chromatic aberrations and enable focusing several different wavelengths at the same focus using different diffraction orders. At the same time, the operating wavelengths of an HDL must satisfy a certain analytical expression, which depends on M and λ₀ [4–9]. Even at few-percent deviations of the incident wavelengths from the operating wavelengths satisfying this expression, a strong decrease in the focal peak intensity occurs. This means that the operating wavelengths of an HDL are related to each other and cannot be chosen arbitrarily.

In the past few years, several numerical methods have been proposed for the design of DLs operating at several prescribed wavelengths [10–19]. In order to distinguish such lenses from “ordinary” DLs operating at a single wavelength, we will refer to DLs operating at several wavelengths as spectral diffractive lenses (SDLs). The interest in this topic is mainly due to the prospects of using SDLs in a wide class of applications including the development of compact imaging systems for mobile devices and unmanned aerial vehicles [19].

The present authors believe that the most significant results on the design of SDLs were obtained in recent works [10–17]. In these works, the calculation of the SDL microrelief is carried out on the basis of the scalar diffraction theory using the direct binary search approach. This method takes into account the technological limitations (discretization and quantization of the diffractive relief) and enables obtaining a microrelief with a prescribed aspect ratio. It is important to note that the errors caused by the use of the scalar diffraction theory for the SDL design turn out to be small in most cases, even at high numerical apertures (at NA ∼ 0.8–0.9). Moreover, the results presented in the recent work [16] show that SDLs properly designed using the scalar diffraction theory exhibit better performance than metalenses designed using rigorous electromagnetic theory. It was demonstrated in [16] that SDLs are more efficient both for low and high numerical apertures. It is also important to note that the diffractive microrelief of the SDLs in [10–17] has a much simpler structure (from the fabrication point of view) than the metalenses [16], which usually consist of arrays of nanoresonators with varying parameters. Obviously, metalenses have a smaller thickness as compared with SDLs. The thickness of the nanoresonators forming a metalens can be by an order of magnitude smaller than the thickness of the diffractive relief of an SDL. However, a smaller thickness of the metalenses is not always an important advantage. This is due to the fact that both the thickness of an SDL and the thickness of a metalens usually turn out to be much smaller than the thickness of the substrate, on which the diffractive structures are fabricated and which is necessary for using them in a realistic optical setup.

The binary search method used for calculating the SDLs in Refs. [10–17] is described in detail in Refs. [11,14]. Despite the successful application of this method for designing various SDLs [10–17], it has several disadvantages. In particular, the method uses an objective function with adjustable parameters, the proper choice of which is a separate non-trivial problem. In addition, the convergence of the method substantially depends on the chosen initial approximation. Here, we propose a significantly simpler method for designing SDLs, which is non-iterative and also allows one to take into account the technological limitations associated with the fabrication of the diffractive microrelief. As examples, we design SDLs operating at three, five, and seven different wavelengths. The simulation results of the designed SDLs, as well as comparison with the known SDLs designed using the binary search approach, confirm high efficiency of the proposed method. For the experimental verification of the method, we fabricate an SDL operating at five wavelengths using direct laser writing in photoresist. The experimental results confirm high performance of the proposed method in practical problems of the SDL design.

2. Diffractive and harmonic lenses

Phase diffractive lenses (DLs) are usually designed for a single fixed wavelength. The microrelief of a DL with the focal length f designed for the wavelength λ₀ has the form

Expanding the complex transmission function of Eq. (2) in a Fourier series with respect to the variable $\xi = {\bmod _{2\pi }}[{{\varphi_\textrm{L}}(\rho )} ]\in [{0,2\pi } )$, one can easily obtain that at λ ≠ λ₀, the DL generates a set of diffraction orders corresponding to spherical beams with the foci [20,21]

Thus, it follows from Eq. (5) that when the wavelength deviates from the design value by Δλ, the DL focus shifts by ${\delta _f} = f - {f_1}(\lambda )= {{\Delta \lambda } \mathord{\left/ {\vphantom {{\Delta \lambda } {({{\lambda_0} + \Delta \lambda } )}}} \right.} {({{\lambda_0} + \Delta \lambda } )}} \cdot f$. According to Eq. (4), the fraction of the energy of the incident beam directed to this focus is given by

The chromatic aberration (focus shift caused by a change in wavelength) is the main disadvantage of the diffractive lens of Eq. (1). The chromatic aberration can be suppressed by increasing the height of the diffractive microrelief [4–8]. By replacing the function mod_2π[φ_L (ρ)] in Eq. (1) by the function mod_2πM [φ_L (ρ)], where M>1, M ∈ ℕ, one can obtain a diffractive lens with the focal distance f operating at the wavelength λ₀, which has the maximum relief height M times greater than that of the conventional DL. Such lenses are referred to as harmonic diffractive lenses (HDLs) [4]. The HDL microrelief is defined by

An HDL can be considered as an intermediate case between a conventional diffractive lens (M=1) and a thin refractive lens, which can be obtained from Eq. (6) at M = ⌊R²/(2λ₀ f)⌋, where ⌊·⌋ denotes the integer part of a number.

Let us consider the HDL performance at different wavelengths. The HDL complex transmission function at a wavelength λ ≠ λ₀ has the form

Despite the fact that an HDL allows one to direct an infinite number of wavelengths to the same focus, only several wavelengths close to the prescribed wavelength λ₀ (i.e. the wavelengths defined by Eq. (10) at m close to M) are of practical interest. A significant drawback of the HDL is that the position of the focus cannot be preserved at the wavelengths, which do not satisfy Eq. (10). Theoretically, by increasing the M value, one can approximate an arbitrary set of wavelengths in the vicinity of the wavelength λ₀ by the set of wavelengths λ_HDL,m defined by Eq. (10). However, the requirements for compactness and feasibility of diffractive lenses limit the possible M values. In particular, in the case when the diffractive microrelief is optically patterned in a resist layer, it is desirable to have the relief height not exceeding 5–6 µm. For example, at λ₀=600 nm and n=1.5 the M value must not exceed 5. For an HDL with M=5, the wavelengths λ_HDL,M+1=500 nm and λ_HDL,M−1=750 nm, which are closest to λ₀=λ_HDL,M, differ from λ₀ already by 100 and 150 nm, respectively. This example shows that in practical cases, the possibilities to approximate an arbitrary set of design wavelengths by the set of Eq. (10) are quite limited.

3. Design method

As it was noted above, the main disadvantage of the HDL is that it cannot be designed for an arbitrary set of wavelengths, which do not satisfy Eq. (10). In addition, the analysis of the HDL performance presented in the previous section neglects the dispersion of the lens material. If this dispersion is taken into account, the “energies” of the diffraction orders I_HDL,m(λ) in Eq. (9) are no longer unity at the wavelengths λ_HDL,m of Eq. (10) [8].

In this section, we consider a numerical method for the design of spectral diffractive lenses (SDLs), which preserve the position of the focus at several given wavelengths λ_k, k = 1, …, K. The proposed method is based on the minimization of an objective function describing the difference between the complex transmission functions of the designed SDL at the operating wavelengths and the complex transmission functions of lenses with the focal distance f designed for each of these wavelengths.

In order to explain the used objective function, let us first assume that the design wavelengths λ_k, k = 1, …, K satisfy Eq. (10) at a certain M, and the dispersion of the lens material can be neglected. In this case, an HDL defined by Eq. (6) will provide a solution to the problem of interest. According to Eq. (11), the following condition holds for an HDL:

Let us now proceed to the general case, when the wavelengths λ_k, k= 1, …, K do not satisfy Eq. (10). Assume that the radial profile of the sought-for SDL consists of N rectangular ridges with the same width Δ and different heights h_j, j= 1, …, N. Let us denote by

In the design of an SDL, it is necessary to take into account the technological limitations on the maximum microrelief height h_max and the number of the relief height levels Q ∈ ℕ. Let us assume that the relief heights h_j can take only the following Q values:

In this case, the h_j values minimizing the function (15) can be found by brute-force search:

Thus, the calculation of the SDL is carried out using Eq. (16). The present authors believe that this approach is significantly simpler than the algorithms based on direct binary search proposed in [10–14]. Let us reiterate that if the design wavelengths satisfy Eq. (10), our SDL design method (with the proper choice of h_max) leads to a harmonic diffractive lens. In this sense, the proposed approach generalizes HDLs to the case of arbitrary operating wavelengths and also allows one to take into account the material dispersion and technological restrictions (maximum relief height, radial discretization step and number of quantization levels). Owing to this, in certain cases, one can expect that Eqs. (8) and (9) describing the HDL focal distances and the energy distribution between the foci as functions of wavelengths would qualitatively describe the “spectral” behavior of an SDL.

Note that in Eqs. (15) and (16), complex transmission functions of paraxial lenses $L({\rho ;{\lambda_k}} )$ were used as “target” transmission functions. However, this is not a limitation of the method. If one needs to compensate for nonparaxial effects (spherical aberration), complex transmission functions of nonparaxial lenses should be used in Eq. (16):

4. Design examples: comparison with the binary search method

4.1 Spectral diffractive lens operating at three wavelengths

In order to assess the efficiency of the proposed method, we designed a cylindrical SDL operating at the wavelengths λ₁=460 nm, λ₂=540 nm, and λ₃=620 nm. The lens with the following parameters was designed: focal distance f=25 mm, numerical aperture NA=0.166, maximum microrelief height h_max=3 µm, radial discretization step Δ=3 µm, number of relief quantization levels Q=128. At NA=0.166, SDL aperture size equals $2R = 2f{\mathop{\rm tg}\nolimits} ({\arcsin NA} )= 8.42\textrm{ mm}$. The given parameters were chosen for comparison with Ref. [10], where an SDL with the same parameters was designed using the direct binary search approach. Let us note that in Ref. [10], SC1827 photoresist was used for the fabrication of the lens. In the present work, positive photoresist FP-3535 with a slightly lower refractive index is used as the lens material both for numerical simulations and for the fabrication of an SDL discussed in Section 5. In the calculations, the refractive index of the resist was defined by the Cauchy’s model $n\left( \lambda \right) = A + {B \mathord{\left/ {\vphantom {B {{\lambda ^2} + {C \mathord{\left/ {\vphantom {C {{\lambda ^4}}}} \right.} {{\lambda ^4}}}}}} \right.} {{\lambda ^2} + {C \mathord{\left/ {\vphantom {C {{\lambda ^4}}}} \right.} {{\lambda ^4}}}}}$ with the parameters $A = 1.631$, $B = 0.01267\textrm{ }\mu {\textrm{m}^2}$, and $C = 0.00118\textrm{ }\mu {\textrm{m}^4}$ fitted from ellipsometric measurements.

Despite the fact that the design method discussed above was formulated for a radial SDL, Eq. (16) can be applied for calculating a cylindrical SDL without any modifications. Such an SDL can be considered as a cross-section of the radial SDL along its diameter. Due to the symmetry of the problem, the microrelief height function h(ρ) is even, and therefore it is sufficient to calculate it at ρ∈[0, R). The SDL relief calculated using Eq. (16) at the parameters presented above is shown in Fig. 1. Similarly to Ref. [10], the intensity distributions generated by the designed SDL at the wavelengths λ_k, k=1, 2, 3 were calculated within the framework of the scalar diffraction theory using the “cylindrical” Fresnel–Kirchhoff integral:

 

Fig. 1. Microrelief profile of an SDL operating at three wavelengths λ₁=460 nm, λ₂=540 nm, and λ₃= 620 nm.

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Figure 2 shows the calculated distributions ${I_{\textrm{cyl,norm}}}({r;{\lambda_k}} )= {{{I_{\textrm{cyl}}}({r;{\lambda_k}} )} \mathord{\left/ {\vphantom {{{I_{\textrm{cyl}}}({r;{\lambda_k}} )} {{I_{\textrm{id,cyl}}}({0;{\lambda_k}} )}}} \right.} {{I_{\textrm{id,cyl}}}({0;{\lambda_k}} )}}$, which are normalized by the “ideal” focal intensities ${I_{\textrm{id,cyl}}}({0;{\lambda_k}} )= {{4{R^2}} \mathord{\left/ {\vphantom {{4{R^2}} {({{\lambda_k}f} )}}} \right.} {({{\lambda_k}f} )}}$. Due to symmetry, the distributions are shown only for non-negative $r \in [{0,2{\delta_{\textrm{cyl},k}}} )$, where ${\delta _{\textrm{cyl},k}} = {{{\lambda _k}f} \mathord{\left/ {\vphantom {{{\lambda_k}f} {({2R} )}}} \right.} {({2R} )}}$ are the sizes (half-widths) of the diffraction spots of a cylindrical lens. For the sake of comparison, Fig. 2 also shows the “ideal” intensity distributions (18). For ease of comparison, these distributions are shown with scaling coefficients ${{{I_{\textrm{cyl}}}\,({0;{\lambda_k}} )} \mathord{\left/ {\vphantom {{{I_{\textrm{cyl}}}\,({0;{\lambda_k}} )} {I_{\textrm{id,}\,\textrm{cyl}}^2({0;{\lambda_k}} )}}} \right.} {I_{\textrm{id,}\,\textrm{cyl}}^2({0;{\lambda_k}} )}}$ ensuring the same values of the normalized distributions of Eqs. (17) and (18) at the focus (at r=0). It is evident from Fig. 2 that the intensity distributions generated by the designed SDL are in a very good agreement with the target distributions defined by Eq. (18) (slight deviations can be seen only in the vicinity of secondary maxima).

 

Fig. 2. Normalized intensity distributions generated by the cylindrical SDL of Fig. 1 at three design wavelengths. Dashed lines show normalized “ideal” distributions generated by diffractive lenses designed separately for each of the wavelengths.

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Similarly to Ref. [10], let us define the SDL efficiency at wavelength λ_k as the fraction of energy of the incident beam, which is focused in the central diffraction lobe:

It is instructive to study the SDL efficiency as a function of wavelength and focal distance. At a certain wavelength λ and a focal distance f + Δf, the efficiency is defined similarly to Eq. (19) as the fraction of energy directed to the central diffraction lobe with the half-width ${\delta _{\textrm{cyl}}} = {{\lambda ({f + \Delta f} )} \mathord{\left/ {\vphantom {{\lambda ({f + \Delta f} )} {({2R} )}}} \right.} {({2R} )}}$. The calculated efficiency is shown in Fig. 3, which demonstrates three main maxima corresponding to three design wavelengths. Similarly to HDLs [4], a detuning from the design wavelengths results in a shift of the focus accompanied by a decrease in efficiency. It is also evident that the focal positions near each of the design wavelengths λ_k are well described by the expressions ${f_k}(\lambda )= f + \Delta f = {{{\lambda _k}f} \mathord{\left/ {\vphantom {{{\lambda_k}f} \lambda }} \right.} \lambda },\textrm{ }k = 1,2,3$ (dotted red curves in Fig. 3) completely similar to Eq. (8). Therefore, in the considered case, the “spectral” behavior of the designed SDL resembles the spectral behavior of an HDL.

 

Fig. 3. Efficiency of the SDL of Fig. 1 vs. wavelength and offset from the focus. Vertical dashed lines show the three design wavelength λ₁=460 nm, λ₂=540 nm, and λ₃= 620 nm. Dotted red curves show the estimates of the focal shifts Δf = f (λ_k – λ)/λ, k=1,2,3.

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Let us now estimate how the SDL efficiencies E_cyl,k depend on the maximum microrelief height h_max. As examples giving an idea of how an increase in h_max affects the SDL performance, we consider two values h_max=6 µm and h_max=12 µm corresponding to twofold and fourfold increase in the maximum relief height, respectively. For the designed SDLs, the average efficiencies amount to $\bar{E} = {1 \mathord{\left/ {\vphantom {1 3}} \right.} 3} \cdot \sum\nolimits_{k = 1}^3 {{E_{\textrm{cyl},k}}} = 0.76$ at h_max=6 µm and to $\bar{E} = 0.815$ at h_max=12 µm. At both h_max values, the efficiencies E_cyl,k at the three design wavelengths are approximately constant and differ by no more than 0.02 from the corresponding average values.

It is also of interest to study the performance of a radial SDL with the microrelief profile shown in Fig. 1. In the radial case, the generated intensity distributions calculated in the Fresnel–Kirchhoff approximation have the form [18]

Figure 4 shows the calculated radial distributions ${I_{\textrm{rad,norm}}}({r;{\lambda_k}} )= {{{I_{\textrm{rad}}}({r;{\lambda_k}} )} \mathord{\left/ {\vphantom {{{I_{\textrm{rad}}}({r;{\lambda_k}} )} {{I_{\textrm{id,rad}}}({0;{\lambda_k}} )}}} \right.} {{I_{\textrm{id,rad}}}({0;{\lambda_k}} )}}$, which are normalized by “ideal” focal intensities ${I_{\textrm{id,rad}}}({0;{\lambda_k}} )= {[{{{\pi R} \mathord{\left/ {\vphantom {{\pi R} {({{\lambda_k}f} )}}} \right.} {({{\lambda_k}f} )}}} ]^2}$, and the corresponding “ideal” distributions of Eq. (21) normalized similarly to the cylindrical case. As follows from Fig. 4, in this example, the generated distributions are visually indistinguishable from the “ideal” ones.

 

Fig. 4. Normalized intensity distributions generated by the radial SDL of Fig. 1 at three design wavelengths. Dashed lines show normalized “ideal” distributions generated by diffractive lenses designed separately for each of the wavelengths.

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Let us define the efficiency of the radial SDL at wavelength λ_k as

4.2 Spectral diffractive lens operating at seven wavelengths

As the next example, we designed a radial SDL for focusing the radiation with the following seven wavelengths: ${\lambda _k} = [{450 + 50(k - 1)} ]\textrm{ nm},\textrm{ }k = 1,\ldots ,7$. The calculation was carried out for the following parameters: f=1 mm, numerical aperture NA=0.18, maximum microrelief height h_max=2.6 µm, radial discretization step Δ=1 µm, number of relief quantization levels Q=128. At NA=0.18, SDL aperture size equals $2R = 2f{\mathop{\rm tg}\nolimits} ({\arcsin NA} )= 0.366\textrm{ mm}$. The given parameter values were chosen for comparison with Refs. [12,13], where a radial SDL having the same parameters was designed using the direct binary search approach. As in the previous example, FP-3535 photoresist was used as the SDL material.

The SDL relief calculated using Eq. (16) is shown in Fig. 5. The upper panels of Fig. 6 show the calculated intensity distributions ${I_{\textrm{rad,norm}}}({r;{\lambda_k}} ),\,\,r \in [{0,\,\,3{\delta_{\textrm{rad},k}}} )$ generated by the SDL at the design wavelengths. As before, these distributions are normalized by “ideal” focal intensities ${I_{\textrm{id,rad}}}({0;{\lambda_k}} )$. For the sake of comparison, “ideal” distributions of Eq. (21) normalized in the same way as in the previous example are also shown. For ease of comparison with the results of Refs. [12,13], lower panels of Fig. 6 show two-dimensional intensity distributions generated by the designed SDL. Let us note that in contrast to the upper panels, the 2D distributions presented in the lower panels of Fig. 6 are shown in the same scale for all the considered design wavelengths (in a 30×30 µm² square). From the upper panels of Fig. 6, it is evident that inside the diffraction spot [at $r \le {\delta _{\textrm{rad},k}} = 1.22{{{\lambda _k}f} \mathord{\left/ {\vphantom {{{\lambda_k}f} {({2R} )}}} \right.} {({2R} )}}$], the calculated distributions [Eq. (20)] are in good agreement with the “ideal” distributions [Eq. (21)]. Outside the diffraction spot, secondary maxima are present in the generated distributions, which are most noticeable at the wavelengths λ₆=700 nm and λ₇=750 nm. Despite this, the quality of the obtained distributions, in the opinion of the present authors, is significantly better than for a similar SDL designed in Refs. [12,13] (compare the lower panels of Fig. 6 and Fig. S1(d) in Ref. [13]). In particular, for the SDL presented in Refs. [12,13] and designed using the direct binary search approach, the resulting distributions at wavelengths λ₄=600 nm, λ₅=650 nm, and λ₇=750 nm are ring-shaped. The radii of the rings are noticeably larger than the radii of the diffraction spots δ_rad,k, and the maximum intensity at these rings exceeds the intensity values at the “geometrical” focus (at r=0). In contrast to the SDL presented in Refs. [12,13], the intensity maxima of the distributions generated by the SDL of Fig. 5 are located at r=0 for all the design wavelengths. The efficiency values E_rad,k [see Eq. (22)] for the SDL of Fig. 5 at the design wavelengths are approximately the same, and the average efficiency amounts to $\bar{E} = {1 \mathord{\left/ {\vphantom {1 7}} \right.} 7} \cdot \sum\nolimits_{k = 1}^7 {{E_{\textrm{rad},k}}} = 0.253$. Let us note that for the SDL designed using the direct binary search approach and presented in Refs. [12,13], significantly higher efficiency values are given. The average theoretical efficiency for this lens equals 0.472. However, it is not entirely clear from Refs. [12,13], how the diffraction spot size for this SDL was defined (especially in the case of ring-shaped intensity distributions), and, consequently, over what area the intensity distributions were integrated when calculating the efficiency values. In particular, in experimental studies presented in Refs. [12,13], the efficiency was defined in a form similar to Eq. (22), but the integration of the intensity distributions was performed over a circle with the radius δ_exp=9 µm. Such a radius significantly exceeds the diffraction spot size δ_rad,k, which even for the longest wavelength λ₇=750 nm is only δ_rad,7=2.5 µm. If we calculate the efficiencies of the designed SDL presented in Fig. 5 in a similar way and replace the upper integration limit in Eq. (23) by δ_exp=9 µm, the average efficiency of the SDL increases to 0.447. This value is close to the theoretical value of 0.472 presented in Refs. [12,13].

 

Fig. 5. Microrelief profile of an SDL operating at seven wavelengths λ_k=[450 + 50(k – 1)] nm, k=1,…,7.

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Fig. 6. Normalized intensity distributions generated by the radial SDL operating at seven design wavelengths. The upper panels show one-dimensional distributions (radial cross-sections) at r ∈ [0, 3δ_rad,k), the lower panels show two-dimensional distributions in a square with the size 30×30 µm². Dashed lines in the upper panels show normalized “ideal” distributions generated by diffractive lenses designed separately for each of the wavelengths.

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4.3 Small-aperture spectral diffractive lens

In Ref. [12], the binary search method was also applied to the design of an SDL with numerical aperture NA=0.05 and focal distance f=1 mm. At these parameters, the SDL aperture size equals 2R=0.1 mm. It is worth noting that in the case of the maximum relief height of 2.4 µm adopted in Ref. [12], the calculation of an SDL using Eq. (16) gives the relief that almost coincides with the smooth relief of a conventional thin refractive lens:

5. Experimental results

5.1 Design and fabrication of a spectral diffractive lens operating at five wavelengths

For the experimental investigation, we designed an SDL operating at five wavelengths: λ₁=450 nm, λ₂=500 nm, λ₃=540 nm, λ₄=580 nm, and λ₅=620 nm. An SDL with the following parameters was calculated: f=100 mm, aperture radius R=4 mm, maximum microrelief height h_max=6 µm, radial discretization step Δ =1 µm, number of relief quantization levels Q=256. These parameters were chosen according to the capabilities of the available technological equipment used for the SDL fabrication. As the lens material, FP-3535 photoresist was used. SDL relief calculated using Eq. (16) is shown in Fig. 7(a) with a solid line.

 

Fig. 7. (a) Calculated (solid line) and measured (dashed line) profiles of an SDL operating at five wavelengths. The SDL was fabricated using the direct laser writing technique. (b) Photograph of the fabricated SDL and (c) a fragment of the SDL microrelief measured using a Zygo New View 7300 white light interferometer.

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Figure 8(a) shows the calculated intensity distributions ${I_{\textrm{rad,norm}}}({r;{\lambda_k}} ),\,\,r \in [{0,\,\,3{\delta_{\textrm{rad},k}}} )$ generated at the design wavelengths. For the sake of comparison, “ideal” distributions described by Eq. (21) are also shown. All the presented distributions are normalized in the same way as described above in Section 4. It is evident from Fig. 8(a) that inside the diffraction spot [at $r \le {\delta _{\textrm{rad},k}} = 1.22{{{\lambda _k}f} \mathord{\left/ {\vphantom {{{\lambda_k}f} {({2R} )}}} \right.} {({2R} )}}$], the calculated distributions (20) are in good agreement with the “target” distributions (21). Outside the diffraction spot, small differences are observed between the calculated and ideal distributions. In particular, near the second theoretical maxima [near $r = 1.635{{{\lambda _k}f} \mathord{\left/ {\vphantom {{{\lambda_k}f} {({2R} )}}} \right.} {({2R} )}}$], slight differences are noticeable at λ₂=500 nm and λ₄=580 nm. For ease of comparison with the experimental results presented below, Figs. 8(b) and 8(c) also show two-dimensional intensity distributions generated by the lens. The 2D distributions are normalized by maximum intensity values and are shown in a 137×137 µm² square. Circles in Figs. 8(b) and 8(c) show the “theoretical” diffraction spots with the radii ${\delta _{\textrm{rad},k}}$. For visual clarity, in Fig. 8(c) the circles are shown at the normalized intensity level 0.6. From Figs. 8(b) and 8(c), it is clearly evident that at the design wavelengths, the SDL generates high-quality foci and the diameters of the focal spots are diffraction-limited. The theoretical efficiencies E_rad,k calculated using Eq. (22) are approximately constant at the design wavelengths. The average efficiency amounts to $\bar{E} = {1 \mathord{\left/ {\vphantom {1 5}} \right.} 5} \cdot \sum\nolimits_{k = 1}^5 {{E_{\textrm{rad},k}}} = 0.396$.

 

Fig. 8. Calculated intensity distributions generated by the designed SDL operating at five designed wavelengths [the SDL profile is shown in Fig. 7(a)]. (a) Radial cross-sections of the normalized intensity distributions at r ∈ [0, 3δ_rad,k). (b), (c) Two-dimensional normalized intensity distributions shown in a 137×137 µm² square region. The circles with the radii ${\delta _{\textrm{rad},k}}$ show the theoretical diffraction spot sizes.

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The designed SDL was fabricated by direct laser writing using the circular laser writing system CLWS-2014 [22]. The diffractive relief was patterned into a 6 µm layer of positive photoresist FP-3535, which was spin-coated on a quartz substrate. The photograph of the fabricated SDL and an interferometrically measured fragment of its relief are shown in Figs. 7(b) and 7(c), respectively. The profile of the diffractive microrelief of the fabricated SDL measured using a Tencor profilometer is shown with a dashed line in Fig. 7(a). From Fig. 7(a), it is evident that the profile of the fabricated SDL is in reasonable agreement with the required profile. The standard deviation σ between the calculated and the fabricated profiles amounts to 10.1%. The authors believe that the errors in the fabricated profile can be reduced by further refinement of the parameters of the laser writing process. However, as it is demonstrated by the experimental results presented below, even at σ = 10.1% the fabricated SDL works quite well.

5.2 Experimental investigation of the point spread functions

In order to measure the point spread functions of the fabricated SDL (i.e. the intensity distributions in the focal plane at the design wavelengths), the optical setup shown in Fig. 9 was implemented. This setup operates in the following way. A tunable laser 1 generates a beam with the required wavelength, which is focused by a lens 2 on a pinhole 3. Then, a lens 4 generates a collimated beam, which impinges on the investigated SDL 5. By adjusting the relative positions of lenses 2 and 4, a collimated beam with divergence angle less than 0.0001° was obtained using this setup. Micro-objective 6 constructs an enlarged image of the intensity distribution generated in the focal plane of the SDL on the matrix of a camera 7. By choosing the relative position of the SDL 5, micro-objective 6, and camera 7, it became possible to measure the generated intensity distributions with a sampling step of 0.57 µm (with the pixel size of the matrix being 3.2×3.2 µm²). Since the radii of the diffraction spots ${\delta _{\textrm{rad},k}}$ at the design wavelengths vary from 6.9 µm at λ₁=450 nm to 9.5 µm at λ₅=620 nm, such a sampling step is sufficient for adequate imaging of the generated intensity distributions.

 

Fig. 9. Optical setup implemented for measuring the point spread functions of the fabricated SDL: 1 – tunable laser NT-242, 2 – 20× microobjective, 3 – 10 µm pinhole, 4 – collimating lens, 5 – SDL, 6 – 6× micro-objective, 7 – Genie Nano M1240 Mono camera.

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The measured intensity distributions generated by the fabricated SDL are shown in Fig. 10. For ease of comparison with the calculated distributions shown in Figs. 8(b) and 8(c), the measured distributions are shown in the same square region and are also normalized by the respective maximum values. Similarly to Fig. 8, the circles in Fig. 10 show the “theoretical” diffraction spots with radii ${\delta _{\textrm{rad},k}} = 1.22{{{\lambda _k}f} \mathord{\left/ {\vphantom {{{\lambda_k}f} {({2R} )}}} \right.} {({2R} )}},\textrm{ }k = 1,\ldots ,5$. Figure 10 demonstrates that the fabricated SDL generates pronounced foci at the design wavelengths. The peak radii are in good agreement with the theoretical diffraction spot sizes ${\delta _{\textrm{rad},k}},\textrm{ }k = 1,\ldots ,5$. At the same time, from the comparison of Figs. 8 and 10, it can be seen that the measured focal peaks of the fabricated SDL are wider, and noticeable secondary maxima are present around the central peaks. These discrepancies are most likely caused by the errors in the fabrication of the SDL (see Fig. 7).

 

Fig. 10. Measured normalized intensity distributions generated by the fabricated SDL at the design wavelengths. The distributions are shown in a square with the size 137×137 µm², the circles show the theoretical sizes of the diffraction spots with the radii ${\delta _{\textrm{rad},k}}$.

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From the measured distributions, estimates of the efficiencies ${E_{\exp ,k}},\,\,k = 1,\ldots ,5$ of the fabricated SDL were obtained. As in the case of the theoretical efficiencies (22), the experimental efficiency ${E_{\exp ,k}}$ of the fabricated SDL at the wavelength ${\lambda _k}$ was determined as the ratio of the energy focused in the diffraction spot with the radius ${\delta _{\textrm{rad},k}}$ [shown with white circles in Fig. 10(а)] to the total incident energy. The obtained efficiency values amount to ${E_{\exp ,1}} = 0.246$ (at λ₁=450 nm), ${E_{\exp ,2}} = 0.087$ (at λ₂=500 nm), ${E_{\exp ,3}} = 0.093$ (at λ₃=540 nm), ${E_{\exp ,4}} = 0.161$ (at λ₄=580 nm), and ${E_{\exp ,5}} = 0.233$ (at λ₅=620 nm). These values are less than the theoretical estimates E_rad,k, which, as indicated in Subsection 5.1, are close to 0.4. We believe that this decrease in efficiency is mainly due to errors in the fabrication of the SDL profile. In particular, the results of numerical simulation of an SDL with the measured profile (not presented here) qualitatively confirm the measurement results. We consider the presented experimental results as proof-of-concept results confirming the manufacturability of the SDLs designed using the proposed approach. The refinement of the fabrication process in order to achieve higher efficiencies will be the subject of further research.

6. Conclusion

We proposed a method for designing spectral diffractive lenses (SDLs) having a fixed focus position at several different wavelengths. The method is based on minimizing an objective function describing the deviation of the complex transmission functions of the SDL from the complex transmission functions of diffractive lenses designed separately for each of the operating wavelengths. In the presented method, the SDL calculation is carried out without iterative minimization of the objective function. Therefore, this method is simpler than the direct binary search methods, which are widely used for the design of SDLs [10–17]. Using the proposed method, we designed SDLs operating at three, five and seven different wavelengths. The results of the numerical simulations of the calculated SDLs confirmed the formation of fixed-position focal spots at the design wavelengths. We compared the performance of the designed SDLs with the SDLs presented in Refs. [10–13] and designed using the direct binary search approach. The comparison results demonstrated that in the considered cases, the proposed design method is more efficient and provides better SDL performance.

For experimental verification of the proposed design method, an SDL operating at five wavelengths was fabricated and experimentally investigated. The diffractive relief of the lens was fabricated by direct laser writing in photoresist. The standard deviation between the designed and manufactured relief profiles amounted to 10.1%. The results of experimental studies demonstrated that despite this fabrication error, the SDL generates pronounced foci at all the design wavelengths. Moreover, the sizes of the formed focal peaks are close to the theoretical sizes of diffraction-limited spots.

The obtained results may find applications in the design and fabrication of novel flat diffractive lenses with reduced chromatic effects. Despite the fact that flat lenses are somewhat inferior to conventional refractive lenses in terms of efficiency, their use in real-world applications becomes possible owing to the rapidly developing computational imaging approach [23,24]. In this approach, the aberrations associated with the utilized flat optics are compensated by computational image post-processing (reconstruction).