1. Introduction

Diffractive optical elements (DOEs) produce the ideal diffraction efficiency distribution by generating phase change [1]. Whereas refractive optical elements are too bulky to be used in lightweight and compact systems, DOEs—with their arbitrary phase distribution [2], thin element property [3], and special dispersion ability [4,5]—are applied in infrared systems [6], microscopes systems [7], and vision systems [8–10].

The phase distribution of DOEs causes light beams to split into multiple diffraction orders. Based on this performance, Multifocal diffractive optical elements (MDOEs) can produce different focal lengths, thus achieving multifocal imaging. MDOEs are commonly utilized in the design of ophthalmic lenses, especially intraocular lenses (IOLs) [11–14]. Conventional MDOE consist of superimposed diffractive structures with different period radii from the inside to the outside of the lens, which yield a varying diopter and thus produce a multifocal image [15–17].

However, the structural design of MDOEs is responsible for imaging defects because each diffraction ring can only take part of the imaging effect when an MDOE with a combination of different diffractive structures is applied. The transition region leads to stray light and to interference between adjacent focal points, causing image quality to deteriorate. The imaging in this type of design is limited by the size of the aperture: when the aperture of the system is reduced, this type of design will not make all the diffractive regions effective [17]. For example, in diffractive IOL designs, when the pupil contracts in a bright environment, the edge diffractive structures do not participate in imaging, which may cause incomplete realization of the design function [18]. When the pupil is dilated under low-light conditions, more diffractive regions with different designs direct light to the retina simultaneously; therefore, diffractive multifocal lenses have the potential to produce halos and glare [19]. They also lead to a decrease in contrast sensitivity [20]. Overall, a method to design high-diffraction-efficiency MDOEs without destroying the diffraction periodicity must be discovered.

Furthermore, owing to their superior performance and thin element property, MDOEs can be designed with other functions, such as broadband. Typically, the DOE method increases the design freedom required to realize broadband by designing multilayer diffractive optical elements. We previously studied the imaging characteristics of broadband multilayer diffractive optics elements and realized high-diffraction-efficiency imaging with broadband [21]. However, for applications such as ophthalmology systems, multilayer MDOEs cannot be used because of the limited design space. Hence, it is necessary to create a model to design single-layer DOEs with a high degree of freedom to achieve functions such as broadband imaging. Peng et al. [22] used the modified particle swarm optimization algorithm to identify the diffraction phase structure of the max point spread function (PSF) and achieve high-quality broadband imaging. Kim et al. [23] proposed to divide the diffraction plane into M × N small regions. This approach provided high-quality broadband imaging based on the direct binary search (DBS) method using the Fresnel diffraction principle. Doskolovich et al. [24] proposed a diffraction design that generates multiple fixed focal points at different wavelengths, by dividing the diffraction plane into N small regions. Each region used an algorithm to obtain the optimal solution, thus, the design model of broadband imaging was obtained. Overall, most studies achieved excellent broadband imaging on single-layer diffractive optical elements through direct laser writing technique either with complex step structures changing in one period or with dividing small regions discarded periodicity. However, for typical applications of single-layer MDOEs such as ophthalmic lenses, the most common manufacturing method is still single point diamond machining. Unfortunately, as we illustrated in another study, step DOE processing with single-point diamond has high requirements in terms of tool type and radius; further limitations arise from the additional shadow effects caused by the diffractive step structure [25]. Therefore, processing of the complex diffractive step structures with single point diamond is extremely difficult. A smooth continuous diffraction structure can be the key to improve this situation. Moreover, a smooth curved design is also more advantageous for lenses implanted in the human eye such as IOL. Owing to there are no visible protrusions which may cause postoperative adverse effects, the design is more compatible with the human eyes [26]. In addition, the accumulation of removed lens epithelial cells (LECs) and inflammatory cells is the main cause of recurrence after IOL surgery [27]. A smoother curved phase profile is less likely to accumulate impurities in the grooves.

Considering the above views, a facile method of MDOE design with high freedom and a smooth profile while maintaining the periodicity would be desirable. Accordingly, we proposed the continuous variable curve with multi-subperiods (CVCMS) method. The detailed design process of broadband MDOE systems using CVCMS method is proposed and discussed through model and parametric analyses. The proposed method offers significantly better freedom of design, high diffraction efficiency, excellent broadband performance, and smooth diffraction profile. The method can also be used for realizing other functions that are difficult to achieve with single-layer MDOEs. The remainder of the paper is structured as follows. Section 2 analyzes and discusses the scalar diffraction theory (SDT) to establish the basis of the study. Section 3 details the complete design process of the CVCMS method based on the SDT and analyzes the method. Section 4 derives the evaluation function of the diffraction efficiency for broadband MDOEs. Subsequently, a multifocal intraocular lens with broadband capability is designed, and the MDOE produced using the CVCMS method is compared with conventional multifocal designs. Section 5 summarizes the study.

2. Principle of diffractive optical elements (DOEs)

In this section, we analyzed the SDT as the basis for the rest of the study from the perspective of wavefront phase transformation. Figure 1(a) illustrates the wavefront transformation of DOE. The wavefront determines the focal length, which is expressed as:

 

Fig. 1. Schematic of diffractive optical elements (DOEs): (a) Wavefront of DOEs; (b) actual profile of DOEs; (c) optical path diagram; (d) diffraction efficiency of monofocal DOEs with different harmonic numbers at different wavelengths; (e) diffraction efficiencies of bifocal DOEs with different phase delays at different wavelengths; (f) diffraction efficiency of an example multifocal DOE at different wavelengths.

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The diffractive optical element achieves an arbitrary light distribution through a period of phase change. The phase profile is expressed as:

The diffraction efficiency can be derived from the Fourier expansion of the lens transmittance function as follows:

Then, the diffraction efficiency is expressed as:

The variation of the refractive index of most diffractive optical element materials with the wavelength is slow, i.e., $\frac{{({{{n^{\prime}}_\lambda } - {n_\lambda }} )}}{{({{{n^{\prime}}_{{\lambda_0}}} - {n_{{\lambda_0}}}} )}} \to 1$. Thus, the effect of the material can be neglected. Dispersion constant can be simplified as $\mu = \frac{{{\lambda _0}}}{\lambda }$. Then, Eq. (7) is rewritten as:

As depicted in Fig. 1(c), different orders of light with different diffraction efficiencies are directed to their respective focal points.

For the conventional DOE design, the phase distribution is determined by the phase delay β. Different phase delays correspond to different light distribution (Figs. 1(d)–(f)). The diffraction efficiency at different orders changes with β. In Fig. 1(d), when β is 1, the diffraction efficiency at the design wavelength reaches 1. When the phase delay is set to β+M, at this time the phase $\varphi = 2\pi (\beta + M)\gamma $, where M is called the harmonic number. Accordingly, such DOEs are known as harmonic diffractive optical elements (HDOEs), which have a smaller dispersion range and are typically used in optical systems to correct chromatic aberration. Several studies have used HDOEs to design diffractive lenses that performed better than their conventional counterparts. However, as Fig. 1(d) shows, the use of HDOEs also poses problems for imaging over a wide range of wavelength. When the value β varies between 0 and 1, the distribution of light changes between two diffraction orders. When β is 0.5, the distribution between diffraction orders 0 and 1 is 1:1, which is the common bifocal DOE design with equal diffraction efficiency. As indicated in Fig. 1(e), the broadband nature of the bifocal design is degraded compared to the monofocal design, leading to poor imaging over the broadband spectrum. Figure 1(f) illustrates a trifocal design with design ideas from Ref. [28]. However, as Fig. 1(f) shown, this design is inadequate for imaging over the broadband, and will cause degradation of imaging quality when the wavelength change. Considering Fig. 1(d)-(f), the broadband performance of DOEs declines as more focal points are applied. Thus, we have described and analyzed monofocal, bifocal and multifocal diffractive optical elements under SDT. Based on these principles, we designed the CVCMS method in Section 3 and a specific system using the CVCMS method in Section 4.

3. Construction and analysis of CVCMS method

In this section, we detail the design method, CVCMS, for adding degrees of freedom and smoothening the phase profile of single-layer DOEs. As depicted in Fig. 2, the construction of the model is divided into three steps. First, the diffraction phase-profile model of a single-layer DOE must be improved. An efficient initial structural model is essential for the subsequent optimization of the structure. Therefore, in Step 1, we transformed the diffraction phase profile to smoothen it and make the profile parameters more favorable for subsequent derivations.

 

Fig. 2. Continuous variable curve with multi-subperiods (CVCMS) design process.

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Second, we performed a regional division within the same period. As described in Section 1, multifocal design based on the superposition of different DOE structures causes degradation of the image quality. Therefore, in Step 2, we establish a subperiods optimization method for the structure proposed in Step 1 to improve the freedom of design while maintaining periodicity.

Finally, for a diffractive structure, as noted in Section 1, a continuous smooth phase profile in each period is beneficial both in manufacturing and applications. Hence, in Step 3, we propose a continuous curve design method to smoothen the curve obtained in Step 2 and make the connections of each subperiod more continuous.

In summary, in Section 3.1, we establish a method to change the diffraction phase profile. In Section 3.2, we provide a method to create multiple subperiods within the same period. In Section 3.3, we provide a method to make phase curve continuous and smooth, and demonstrate the CVCMS method. The entire design model is constructed and analyzed in three steps.

3.1 Step 1-Change of diffraction phase profile

From Section 2, the Fourier series of transmittance function of DOEs can be expressed as:

The choice of the initial structure is crucial. At this point, as observed in Eq. (9), the diffraction efficiency is determined by the maximum phase delay and phase profile. The phase profile of a conventional diffractive structure is a fixed step curve, we transformed the profile curve into a complex function using more operands. The first step is to transform the conventional step structure curve into a Sin/Cos-type function, which has a period of 2π and a more curved phase profile. It is consistent with the diffraction periodicity and easily accepts more degrees of freedom. In summary, the Fourier series for the change curve is expressed as:

However, for this model, the problems stemming from the limitation of the structural degrees of freedom must be solved. the equal energy distribution in the –1 and +1 orders by Eq. (10) leads to a restricted design; meanwhile, when we attempted to increase the diffraction efficiency at ±1 orders, the diffraction efficiencies at ±1 orders could only be increased to approximately 0.338, and lead to a significant decrease of diffraction efficiency at 0 order. For example, if β = 1.7, the diffraction efficiency of the ±1 order reaches the almost maximum designable value. However, the diffraction efficiency at the 0th order is only 0.15 and the broadband performance is poor. Hence, it is also inconvenient to design a DOE with a high diffraction efficiency at an arbitrary order.

3.2 Step 2-Method of multi-subperiods

As explained in Section 1, the diffraction efficiency of traditional multifocal DOEs is limited by the size of the applied diffraction area. Therefore, only structural changes within a period can achieve consistent imaging effects from inside to outside. Accordingly, we improved the form of the phase expression within the period mentioned in Section 3.1 to maintain periodicity. We divided the whole period into k weighted regions. w represents the design proportion of each subperiod. The selection of w depends on the design and solution requirements. c_m can be expressed as:

Because the diffraction efficiency is expressed as ${|{{c_m}} |^2}$, in Eq. (11), the subperiods are not independent and all affect each other. Moreover, no regions can be arbitrarily adjusted, which means that the sequential order of adjustment has an impact on the diffraction efficiency.

3.3 Step 3-Method of continuous curve

Application of the method derived in Section 3.2 would inevitably cause discontinuities between subperiods. Combining the analysis of the above two steps, we proposed a continuous curve approach. Under Cos transformation, we defined t_k as the continuous factor and added the corresponding t_k to each subperiod in Eq. (11), obtaining Eq. (12). After change, the maximum and minimum values of Cos remained ±1, showing that t_k has no impact on the maximum phase delay of each subperiod in Eq. (11). Therefore, the continuous-curve design can be achieved while still supplementing more operands. c_m that with continuous factor t_k can be expressed as:

The first period should preferably be set in the center, so that the initial point is well determined and processed, i.e., t₁ = 0. We make the diffraction phase profile have equal values at each subperiod edge. Hence,

Then, t₂, t₃, …t_kcan be expressed as:

To ensure the validity of the equation, we need to ensure that

Values outside this region are discarded. Substituting Eq. (14) into Eq. (12) yields

Thus, we completed the construction of the CVCMS method. To demonstrate the degrees of freedom of the model, we illustrate an example of two subperiods in Fig. 3. As indicated in Eq. (1), different orders correspond to different distances. We utilized this approach to design a multifocus IOL, which propose a distribution of the diffraction efficiencies at the far, intermediate, and near focal points with the orders of –1, 0, and 1 (The specific system design is explained in Section 4.2). We derived the relationship between the phase delays of two subperiods and the diffraction efficiency at subperiod division proportion w₁of 0.3, 0.5, or 0.7. The three various w₁ in Fig. 3 are analyzed to illustrate the diverse diffraction efficiency distributions at different subperiod division proportions. The design results, compared to the problems listed with Eq. (10), break the diffraction efficiency value of ±1 orders that must be equal, and lead to the arbitrary diffraction efficiency distribution at different orders. Unlike structures designed with conventional design, the structure designed with CVCMS has continuity, i.e., the curve remains continuous over a period (examples are available in Figs. 2 and 7). The brown region in Fig. 3 represents the area where Eq. (15) is not satisfied, i.e., the region where a continuous curve could not be designed. The diffraction efficiency distribution varies with the size of the diffraction subregion and phase delay distribution, which corroborates the assertion in Section 3.2 that the subperiods affect one another and that the sequence of each subperiod cannot be manipulated. By accepting different input values, the design model can achieve an adjustable phase distribution with the continuous curve. A further increase in the number of subperiods helps increase the degrees of freedom for further design. This will be discussed in the subsequent section.

 

Fig. 3. Results of CVCMS-designed MDOE between phase delays of two subperiods and diffraction efficiency. (a) Subperiod division proportion w₁ = 0.3; (b) subperiod division proportion w₁ = 0.5; (c) subperiod division proportion w₁ = 0.7.

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4. Design example and discussion

In this section, we demonstrated a specific application of the proposed method. We first propose the evaluation method for broadband MDOEs. Next, a broadband multifocal IOL is designed based on the CVCMS method, and CVCMS method is discussed with other design methods.

4.1 Evaluation of diffraction efficiency with broadband multifocal design

Evaluation of broadband MDOEs differs from that of conventional single-focus diffractive optical elements in that excessive diffraction efficiency at a single order is considered detrimental because it affects the magnitude of the efficiency at other orders. MDOEs need to achieve a balanced high diffraction efficiency for broadband imaging. With the proposed design model, an arbitrary diffraction-efficiency distribution can be achieved by applying a simple loop solution. Nevertheless, the design results for the DOE should be evaluated. Owing to the large variation in diffraction efficiency, conventional evaluation methods like the polychromatic integral diffraction efficiency (PIDE) are not applicable to broadband MDOEs. Therefore, we propose using two equations for evaluation instead. First, the degree of separation is expressed as:

4.2 Evaluation of broadband diffractive optical system design

We designed a multifocal intraocular lens system based on the structure of the human eye using the Liou–Brennan model [29]. The Liou–Brennan model consists of a cornea, pupil, crystalline lens, and retina. In this model, the crystalline lens uses the gradient refractive index to better simulate the refractive index of the natural crystalline lens in the real human eye. The specific parameters related to this model are listed in Table 1. The human eye focuses on the far field, while focusing on the other distances depend on the modulation of the crystalline lens.

Table 1. Liou–Brennan model of the human eye^a

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When the natural crystalline lens aged, it was removed and placed into the designed diffractive multifocal intraocular lenses. The initial design parameters of the IOL are listed in Table 2. After adding the trifocal MDOE using ±1 and 0 orders, we set the additional diffractive diopters to –1.66D, 0D, 1.66D at the far, intermediate, and near points. Then, the diffractive IOL has powers of 18.7D at the far point (similar to the crystalline lens in the Liou–Brennan model), 20.36D at the intermediate point, and 22.02D at the near point. Different design bands will cause different optimization effects. In this design, we analyzed the wavelength that the human eye normally accepts (500–600 nm). The design process is illustrated in Fig. 4.

 

Fig. 4. Intraocular lens design (a) Liou–Brennan model of the eye (adapted from Ref. [29], Fig. 4). (b) Designed optical path diagram (red: Far point, green: Intermediate point, blue: Near point). (c) MTF results at design wavelength.

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Table 2. Intraocular lens (IOL) design parameters

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The final optimization result is summarized in Table 3. The modulation transfer function (MTF) at 80lp/mm of this model reached 0.3295, 0.3705, and 0.3945 at the design wavelength and achieved broadband MTF at the intermediate point as well as the near point; the broadband performance was relatively poor at the far point but still within the acceptable range.

Table 3. Modulation transfer function (MTF) design

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4.3 Results and discussion

During the design of multifocal lenses, the allocation of multifocal points depends on the requirement of the application. For the design in this section, we aimed at optimizing the far point to demonstrate the arbitrary manipulability of the proposed method, which helps people whose work requires far-point vision. The adjustable diffraction efficiency at specific orders is useful for different design objectives. However, arbitrary phase distribution also poses certain design challenges.

In addition, for a diffractive optical system, the PSF is determined by the diffraction efficiency size and MTF size. For MDOEs, the PSF is expressed as:

As indicated in Eqs. (22) and (23), the final PSF is related to the diffraction efficiency. Optical software such as Code V and Zemax OpticsStudio normally set the diffraction efficiency to 1, because diffraction efficiencies with different orders are not considered in conventional monofocal lens designs. However, for MDOEs, as indicated in Eq. (22), a reasonable allocation of multifocus diffraction efficiency can help improve the overall imaging quality. In the system design proposed in Section 4.2, the MTF of the far point decreased more than that of the other two focal points when the system deviated from the design wavelength. Therefore, for this system, the diffraction efficiency at the far point should be increased.

To this end, we describe the design of a three-subperiods MDOE in this section. Different waveband requirements lead to different solution results. In this design, we solved for the best broadband design over the 500-600 nm waveband. The broadband evaluation function Eqs. (18), (19), (20) and (21) are given in Section 4.1 as the solution conditions. The parameters were solved by a simple loop solution. We first determined three equal subperiods in the split ratios of 0.33, 0.66, and 1, respectively. We then set the diffraction efficiencies at the intermediate point (0 order) and near point (1 order) to be equal as possible while maintaining the maximum at the optimized far point (–1 order), which is impossible to design through Eq. (10), and maintaining the broadband performance at all three focal points. The result parameters are shown in Table 4. The diffraction efficiency distribution is presented in Fig. 5 and Table 6.

 

Fig. 5. Diffraction efficiency of MDOE design using the CVCMS method.

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Table 4. MDOE parameter results with CVCMS design

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From Fig. 5, we achieved the case of almost equal diffraction efficiency at the intermediate and near focal points, with a focus on the optimization of far focal point. Then, to demonstrate the superiority of the model, we compared the proposed design with other multifocal design methods in terms of the diffraction efficiency, broadband performance evaluation, and actual profile in the following analysis.

To validate the comparison, we attempted to design similar diffraction efficiencies at the far point using different design methods. The comparison group design A was based on the superposition of different diffraction structures designed from inside to outside, which are common [14,16,17]. We simulated the ratio of incident light energy on the central and edge areas, which was 1:1. The region for far- and near-point imaging at the center radius spanned 0–2.121 mm. The region for the far- and intermediate-point imaging at the edge radius spanned 2.121–3 mm. The specific parameters are listed in Table 5. The comparison group design B was solved through a step structure through Eq. (24) using the design method of Ref. [28]. The specific parameters related to this are also listed in Table 5.

First, we compared design A with the proposed design. We programmed a distribution with a similar diffraction efficiency as the proposed design at the far point. We observed that, at the other two focal points, the diffraction efficiency decreased significantly with the proposed design, and the overall diffraction efficiency utilization rate also differed notably. For multifocal diffraction structures, on the one hand, the overall utilization rate of diffraction efficiency determines the light energy distribution at the utilized focal points. On the other hand, because the unutilized light leads to phenomena such as glare, which also lead to the degradation of imaging quality, the overall diffraction efficiency utilization rate must be improved. For broadband performance, according to Table 6, the ${\bar{p}_m}$ values at the near and far points of design A were significantly lower than those with the proposed design but at the intermediate points were marginally higher. Notably, as the diffraction efficiency of design A was lower than that in the proposed design at design wavelength, the actual effect of intermediate points was smaller. In summary, the CVCMS design and design A exhibited similar broadband performances at the intermediate point. At the other two points, the proposed design was significantly better than design A in terms of both diffraction efficiency and broadband performance. Furthermore, the size of the utilized diffraction area significantly affected the performance of design A: when the radius was smaller than a certain threshold (< 2.121 mm), the diffraction structure at the edge responsible for the intermediate focal point could not be utilized, i.e., the intermediate diffraction efficiency was 0. Moreover, the overall performance resembled that of a bifocal lens. Combined with the above, our design’s performance is superior to that of design A.

Table 5. Parameters of comparative designs A and B

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Second, design B was compared with the proposed design. A diffraction efficiency similar to that achieved by the proposed method was also designed at the far point. As indicated in Table 6, the near- and intermediate-point diffraction efficiencies of design B were smaller than ours, and the overall diffraction efficiency was lower. Moreover, the diffraction efficiency of design B decreased rapidly in the waveband. Especially, at the intermediate and near points, design B produced weak imaging results in the broadband range. The evaluation function (Eq. (19)) reveals that its far, intermediate, and near ${\bar{p}_m}$ values were small compared to those of the proposed design. Design B exemplifies a general problem with this class of design with weak broadband imaging, which is also illustrated in Fig. 1(f). This phenomenon is visually represented in Fig. 6 through an evaluation of the degree of separation. As illustrated in Fig. 6, the separation of the two designs at the far point focus was extremely similar. This was probably because we made design B to closely emulate our design on the far point at the beginning of the design. At the other two distances, the degree of separation of our design was much less than that of design B. Combined with the analysis of ${\bar{p}_m}$, the results demonstrate simultaneous stable broadband imaging at multifocal points achieved by the proposed design compared to design B.

Table 6. Results with different design approaches

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Fig. 6. Degree of separation comparison between design B and CVCMS method.

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We evaluated the profile optimization between design B and the proposed design. The phase profile using the proposed CVCMS model at design wavelength can be expressed as

 

Fig. 7. DOE phase profile comparison between design B and CVCMS method. (a) Result by design B. (b) Result by CVCMS method.

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Finally, we elaborate on the adjustable diffraction efficiency improvement on both the PSF at any given order and the systems imaging quality. For the same DOE system, first, the design of three uniformly irradiated focal points was selected using Eq. (10) with β = 1.4376 as the control group. We compared it with the proposed CVCMS design in term of the PSF at far points. As illustrated in Fig. 8, the PSF of the proposed design at each wavelength is improved than that of control group. The growth rates of the entire pixel area with the PSF value at the far focal points were 1.1940, 1.1887, 1.1854, and 1.1777 at 500, 530, 550, and 600 nm, respectively. Figure 8 yields PSF is enhanced by the diffraction efficiency. Combined with Eqs. (22) and (23), the enhanced PSF, i.e., the enhanced MTF alleviate the relatively poor imaging performance at far point which shows in Section 4.2, as well as the lack of consideration of diffraction efficiencies at different orders by optical software. Meanwhile, we reiterate that our model can achieve a high diffraction efficiency at a far point without affecting the performance of the other two focal points (as a counterexample, the poorer broadband performance at the other two focal points of design B causes a reduction in the broadband PSF at the other two focal points). As the proposed design has a relatively high diffraction efficiency at the other two focal points and the IOL system itself has a high MTF at them (Table 3), through design, the quality of imaging at three points is high enough to satisfy the design requirements. In other words, the imaging quality of system is enhanced by the modulation of the diffraction efficiency. This also demonstrates that designing MDOEs that can achieve an arbitrary diffraction efficiency distribution can help systems achieve higher imaging quality.

 

Fig. 8. Schematic of the PSF at far focal point of an actual diffraction system (normalized by the maximum value), PSF values from left to right are compared at 500, 530, 550, and 600 nm. The PSF is calculated using a Fourier transform with a trifocal uniform design (a) or the proposed CVCMS design (b). The optical system uses the design in Section 4.2.

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In summary, based on the multifocal diffraction efficiency distribution required by the actual application and the evaluation of imaging quality of a diffractive optical system, we determined the diffraction efficiency distribution needed to design MDOEs. Next, according to the CVCMS method, we built the specific model design of MDOE. Finally, the design results were evaluated on five criteria: diffraction efficiency value of each wavelength, degree of separation, broadband stability, overall diffraction efficiency value, and optimal profile. After the three steps, we established the overall design process of MDOEs, which exhibited better performance than the conventional ones, and obtained a multifocal IOL design with high broadband diffraction efficiency and optimized profile. The method has excellent generality and plays an exemplary role in the design prospects of MDOEs.

5. Conclusion

In this study, we proposed the CVCMS method for designing MDOEs with arbitrary diffraction efficiency distribution and a smooth phase profile in each diffractive ring while retaining the periodicity. The method was implemented on a broadband multifocal intraocular lens design for demonstration and analyzed. The study was conducted as follows: