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Diffractive lens design for optimized focusing Xiaowen Wan, Bing Shen, and Rajesh Menon* Department of Electrical and Computer Engineering, University of Utah, Salt Lake City, Utah 84112, USA *Corresponding author: [email protected] Received June 20, 2014; revised October 9, 2014; accepted October 10, 2014; posted October 10, 2014 (Doc. ID 214377); published November 3, 2014 We utilized nonlinear optimization to design phase-only diffractive lenses that focus light to a spot, whose width is smaller than that dictated by the far-field diffraction limit. Although scalar-diffraction theory was utilized for the design, careful comparisons against rigorous finite-difference time-domain simulations confirm the superfocusing effect. We were able to design a lens with a focal spot size that is 25% smaller than that formed by a conventional lens of the same numerical aperture. An optimization strategy that allows one to design such lenses is clearly explained. Furthermore, we performed careful simulations to elucidate the effects of fabrication errors and defocus on the performance of such optimized lenses. Since these lenses are thin, binary, and planar, large uniform arrays could be readily fabricated enabling important applications in microscopy and lithography. © 2014 Optical Society of America OCIS codes: (050.1965) Diffractive lenses; (050.1970) Diffractive optics; (230.3990) Micro-optical devices; (050.1380) Binary optics. http://dx.doi.org/10.1364/JOSAA.31.000B27
1. INTRODUCTION Diffractive optics generate complex electromagnetic fields by manipulating the amplitude and phase of incident wavefronts [1]. The geometric configuration of the diffractive element can be specifically designed to achieve the desired intensity or phase patterns at the observation planes [2–4]. Conventional scalar-diffraction theory can be applied to predict the field distribution with good accuracy [1]. Compared to a diffractive element that imposes amplitude modulation, a phase-only diffractive optic has the advantage of low loss and, thereby, better photon throughput. This benefits applications that either require high power density, such as photolithography [5], or have a low photon budget, such as fluorescence microscopy [6]. Diffractive elements, which require only two levels of phase shifts (so-called binary diffractive optics), are significantly easier to fabricate. Although these optics suffer from inherent chromatic aberration [7], they are useful for singlewavelength applications. Usually, the phase shift is introduced by an optical path difference between alternate regions [2]. The zone plate is an exemplary device in the family of diffractive optical elements. Since its first demonstration by Soret in 1875 [8], it has been exploited for a variety of applications including multicolor confocal imaging [7], x-ray microscopy [9], superresolution data storage [10], parallel photolithography [11–13], conversion of the focus spot profile of laser beams [14], focusing neutrons [15], and millimeterwave antennas [16]. In traditional projection photolithography [17,18] and confocal microscopy [19], multiple complex lenses are used to tightly focus the light into a diffraction-limited focal spot or a diffraction-limited image. Replacing bulky complex refractive optics by thin diffractive lenses can enable compact, lightweight systems, while maintaining a high numerical aperture (NA). The zone plate is essentially composed of a number of concentric ring-shaped zones. If the zones are made of absorbing 1084-7529/14/120B27-07$15.00/0
material, one gets an amplitude zone plate. On the other hand, if the material is transparent, the optic is a phase zone plate. The conventional on-axis zone plate is circularly symmetric, and can be fully described by the radii and the transmission function of the individual zones. In the past, parametric search methods have been applied to modify the conventional zone plate to reduce the width of the focal spot [10,11]. Reductions of both the central spot size at focus and controlling of the side lobe intensity were achieved. Nevertheless, the results in Ref. [10] are limited by the available degrees of freedom and the small field of view. Besides, the multiphase elements in Refs. [10,11] are not as easily fabricated and replicated as their binary-phase counterparts. No analysis on chromatic aberration, defocus, and fabrication error tolerance was given in Refs. [10,11]. In this article, we extend this previous approach to lenses with many more binary zones, and utilize a new optimization strategy to achieve even smaller focal spots than before. Specifically, we present a binary-phase diffractive-lens (BPDL) design, which is capable of focusing light with minimized central spot size for moderate and high NAs. In the following sections, the model of the diffractive optic and a nonlinear optimization approach are explained in detail. The best design demonstrates a spot size defined as the full width at half-maximum (FWHM) of 196 nm (much smaller than the far-field diffraction limit of 244 nm) at λ 400 nm. We further compare our scalar-diffraction simulations to finite difference time-domain (FDTD) simulations. Finally, we perform a careful analysis of the impact of fabrication errors and defocus on the performance of these BPDLs.
2. DESIGN The basic schematic of the proposed BPDL design is illustrated by a cross-sectional view in Fig. 1. Note that our lenses are rotationally symmetric. An incident plane wave is used © 2014 Optical Society of America
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such that the amplitude is constant across the lens. Adjacent zones of concentric rings experience a relative phase shift, described by ψ 2πh∕λn − 1, where h is the step height between neighboring zones, λ is the wavelength of incident light, and n is the real part of the refractive index of the zone plate material at wavelength λ. In this case, both the phase shift ψ (or equivalently, the zone height, h) and radii of the zones r 1 ; r 2 ;…r M are treated as design variables to be optimized. Note that planar fabrication processes can enable large-array, highly uniform fabrication of such phase-only diffractive optics in dielectric materials [20]. Also, photocurable nanoimprint lithography can be used to achieve mass production of high-resolution BPDLs [21, 22]. In Fig. 1, ρ and ρ0 are the radial coordinates in the zone plate plane and the focal plane, respectively, and z represents the direction of light propagation. The BPDL can be described by a circular-symmetric transmission function [2]: 8 < eiψ Tρ 0 : 1
9 r 2m < ρ ≤ r 2m1 ; m ∈ f0; 1; 2; 3; …g = ; ρ > rM ; elsewhere
(1)
in which r m is the radius of the mth zone, and m is a set of positive integers bounded by M. M is the total number of zones. The well-known Fresnel–Kirchhoff formulation of the scalar-diffraction theory is employed to model the propagation of light from the phase zone plate to the observation plane [1]. This scalar model allows for fast computation with reasonable accuracy, which is later validated by rigorous FDTD simulations [23]. The typical time for optimizing the BPDL is about 3 h, which is much faster than that of full-wave electromagnetic simulation. The intensity distribution in the plane of observation at distance d along the optical axis is given by Z Z 2π 1 ∞ γρ0 ; z; λ; f ⃗r; hg ρdρ dϕTρ; f ⃗r; hg iλ 0 0 h pi exp i 2πλ ρ2 ρ02 − 2ρρ0 cosϕ z2 p × ρ2 ρ02 − 2ρρ0 cosϕ z2 2 z × 1 p : (2) 2 02 0 2 ρ ρ − 2ρρ cosϕ z In the past, a genetic algorithm has been successfully deployed for optimizing phase zone plates to generate optical nulls without phase singularities [2]. The aim of our optimization, as stated previously, is to achieve a smaller focal spot compared to a normal zone plate. The cost function used for optimization is defined as Z ρ0 1 Ef ⃗r; hg −w1 · γρ0 ; λ; f⃗r; hgρ0 dρ0 w2 0 Z ρ0 2 · γρ0 ; λ; f⃗r; hgρ0 dρ0 w3 · FWHM: ρ01
(3)
Equation (3) contains three terms, indicating contributions from the total energy in the central region of the focal spot integrated from 0 to ρ01 , the total energy in the peripheral region of the focal spot integrated from ρ01 to ρ02 , and the FWHM of the focal spot, respectively. Since we expect to concentrate light in the focal spot with narrower width, the sign of each
Fig. 1. Schematic of the BPDL and design methodology. The design variables are the radii of the zones, r 1 ; r 2 ;…; r M , and the height of the zones, h. Intensity distributions in the focal plane illustrate the design requirement for a BPDL with tight focusing at illumination wavelength, λ.
term is selected such that the cost function is minimized during optimization. And w1 , w2 , and w3 are positive weights for the three terms. Delicate balancing among these values is critical in achieving optimal results, which is discussed in Section 4. In addition, physical constraints Eqs. (4)–(6) have to be imposed to ensure that the zones are retained in the correct order and that the width of each zone is greater than Δ during optimization. Here, Δ is basically the minimum feature that can be patterned. We assume that electron-beam lithography (EBL) will be utilized to pattern nanometer-scale structures [20]: rp > rq
∀M ≥ p > q ≥ 1;
fp; qg ∈ I;
r p − r p−1 > Δ > 0 ∀M ≥ p > 1; r 1 > 2Δ:
p ∈ I;
(4)
(5)
(6)
Both the height, h, and radii, r 1 ; r 2 ; …r M , of zones are optimization variables. The genetic algorithm is an ideal choice for solving the complex problem of free-space propagation dictated by a highly nonlinear function Eq. (1) bounded by the cost function Eq. (3) and additional constraints Eqs. (4)–(6). It is a mathematical abstraction of natural selection [24, 25]. The algorithm begins with an initial population and continues in an iterative manner [2]. Perturbing a conventional binaryphase zone plate generates the initial population. The perturbations are chosen from a Gaussian distribution of zero mean and standard deviations 50λ and 15λ for the zone radii and the zone height, respectively. During each iteration, a new population is generated. Each individual within the population is evaluated in terms of the defined cost function, based on which they are ranked. The individuals in the next generation are selected based on the principle that the individuals with the lowest cost-function values in the current generation are retained and the others are derived from adding random perturbations of the variables to the retained individuals. Both the number of individuals in the population (either 25 or 50) and the number of iterations (either 25 or 50) dictate the performance of the eventually optimized BPDL design.
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This impact is described in Section 4. All perturbations are subject to the constraints expressed in Eq. (3).
3. RESULTS Figure 2 shows the results of the best BPDL design optimized for minimized FWHM of the focal spot and high central spot intensity. The design wavelength is λ 400 nm. The optimization strategy to reach this particular lens design is detailed in the following section, where optimization parameters and steps are discussed. The grayscale plot in Fig. 2(a) visualizes the piecewise transmission function of a lens after 50 iterations. A series of concentric rings (or zones) with alternating phase shifts are obvious. It consists of 150 zones with outer radius of 57.4 μm. The focal length of the designed lens is 40 μm, corresponding to NA 0.8207. Specific design parameters (radii and zone height) are shown in Table 1. The minimum zone width, Δ, is 200 nm, which can be readily fabricated by most EBL tools. Here, we assume that the diffractive lens is made of polymethylmethacrylate (PMMA), a widely used dielectric with excellent transparency and easy processes
Fig. 2. (a) Transmission function of the optimized BPDL. NA 0.8207, focal length 40 μm. The other optimization parameters are described in the text. Within the aperture of the BPDL, the gray rings are phase shifted with respect to the white ones; their relative height difference, h, is 384 nm. Outside the BPDL is opaque. (b) Radial intensity distributions of the optimized BPDL focal spot and the conventional zone plate focal spot (black solid line). Blue and red solid lines correspond to the simulation results calculated using the scalar method and the FDTD method, respectively. The inset shows a magnified view of the focal spot.
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Table 1. h r1 r2 r3 r4 r5 r6 r7 r8 r9 r 10 r 11 r 12 r 13 r 14 r 15 r 16 r 17 r 18 r 19 r 20 r 21 r 22 r 23 r 24 r 25 r 26 r 27 r 28 r 29 r 30 r 31 r 32 r 33 r 34 r 35 r 36 r 37 r 38 r 39 r 40 r 41 r 42 r 43 r 44 r 45 r 46 r 47 r 48 r 49 r 50
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Radii and Height of the Zones in Units of λ (λ 400 nm)
0.9597 10.0797 12.0695 18.4904 19.1648 24.3048 26.5862 28.6980 30.1720 31.3693 33.0078 33.6092 35.1020 35.6962 36.8804 38.6879 39.9747 43.1657 44.1931 45.6668 46.7944 48.0975 48.8126 49.3126 50.1845 50.7066 51.6614 52.6360 53.6640 56.0704 56.7136 57.5843 58.8412 59.4965 60.8368 61.5956 62.6772 63.5222 64.4333 65.6052 66.2145 66.7698 67.4215 67.9440 69.6539 70.7615 71.4830 72.2704 73.2885 74.2822 74.7822
r 51 r 52 r 53 r 54 r 55 r 56 r 57 r 58 r 59 r 60 r 61 r 62 r 63 r 64 r 65 r 66 r 67 r 68 r 69 r 70 r 71 r 72 r 73 r 74 r 75 r 76 r 77 r 78 r 79 r 80 r 81 r 82 r 83 r 84 r 85 r 86 r 87 r 88 r 89 r 90 r 91 r 92 r 93 r 94 r 95 r 96 r 97 r 98 r 99 r 100 r 101
75.7164 76.7098 77.6211 78.3223 79.1194 79.8964 80.6487 81.6154 82.1791 83.0206 83.9554 84.7735 85.3034 86.3196 87.0068 87.8666 88.3700 89.2380 89.8504 90.5420 91.3399 92.2853 92.9537 93.5712 94.1637 94.9364 95.7130 96.5719 97.1983 97.8045 98.3555 98.8555 99.9170 100.9016 101.4902 102.3614 102.9500 103.5945 104.2601 105.0311 105.7210 106.2853 106.8638 107.5675 108.5673 109.0918 109.7822 110.5180 111.2237 111.9300 112.4300
r 102 r 103 r 104 r 105 r 106 r 107 r 108 r 109 r 110 r 111 r 112 r 113 r 114 r 115 r 116 r 117 r 118 r 119 r 120 r 121 r 122 r 123 r 124 r 125 r 126 r 127 r 128 r 129 r 130 r 131 r 132 r 133 r 134 r 135 r 136 r 137 r 138 r 139 r 140 r 141 r 142 r 143 r 144 r 145 r 146 r 147 r 148 r 149 r 150
113.1907 113.7388 114.3833 115.0669 115.8429 116.6045 117.1607 117.6607 118.3847 118.8847 119.6855 120.4629 121.0467 121.7310 122.3396 122.9411 123.4411 124.3293 124.8784 125.4784 126.1632 126.6632 127.4819 128.1237 128.8671 129.4002 129.9002 130.6152 131.2793 131.8115 132.5869 133.0870 133.8729 134.3729 135.0090 135.5091 136.1413 136.9486 137.5175 138.0176 138.5785 139.2747 139.8130 140.6156 141.2537 142.0038 142.5038 143.0303 143.5803
for patterning and replication. The index of refraction of PMMA at λ 400 nm is measured to be 1.51 by spectroscopic ellipsometry. The weighting factors, w1 , w2 , and w3 are chosen to emphasize each term during optimization. These are discussed in the next section. The normalized radial intensity distribution in the focal plane, also referred to as the point-spread-function (PSF) of the diffractive lens, is calculated by both scalar-diffraction theory and the FDTD method. Here, circularly polarized light is utilized in the FDTD simulation because the BPDL is rotationally symmetric. They are plotted together for comparison
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in Fig. 2(b). The PSFs derived from Eq. (2) and the FDTD simulation match quite well. Although only scalar theory is utilized during optimization, the FDTD simulation confirms that it is a reasonable approximation. The relative error is around 7.7% based on the inset of Fig. 2(b). The FWHMs of the focal spots predicted by scalar theory and FDTD are 196 and 194 nm, respectively. This is smaller than the spot size of the diffraction-limited Airy disk given by λ∕2NA 244 nm [10]. The calculated normalized intensity distribution of the diffraction-limited focal spot (of a conventional phase zone plate of the same NA) is also plotted in Fig. 2(b). Although Ref. [10] states that it is difficult to attain both tight focus and reduced side lobes at the same time by uniform phase shift in zone plate design, here we demonstrate that it is feasible to achieve this by expanding the degrees of optimization freedom and by choosing the weighting factors judiciously. As can be seen in Fig. 2(b) (blue line), the intensity of the side lobes is less than 20% (10% calculated by FDTD) of that of the peak. This is comparable with results by the multiphase diffractive superresolution element shown in Ref. [10], although the FWHM shown here is much smaller. Also, the available field-of-view is significantly larger. As shown in Fig. 2(b), everywhere with ρ0 > 2λ is sufficiently suppressed. Moreover, our binary-phase device has much smaller dimension compared to the one optimized in Ref. [11], which has a diameter of about 5 mm. This compactness will enable applications in which an array of zone plates is utilized for large-area lithography [12].
4. OPTIMIZATION STRATEGY In this section, we describe how to select the optimization parameters in each step to ultimately reach the best lens design shown above. In order to understand this, we present a simpler BPDL design with just 40 zones. As previously mentioned, both the number of individuals in each generation (or population size) and the maximum number of iterations (or the number of generations) are changeable. Additionally, the integration ranges (ρ01 and ρ02 ) in the cost function [Eq. (3)] also play a critical role in the optimization process. To start with, we ignore the third term of FWHM in the cost function, while retaining the first and the second terms of power integration over focal and peripheral regions. In Table 2, we summarize the FWHMs of the calculated PSFs of different optimized BPDL designs by using various parameters. The ratios between the first and the second terms (w1 ∶w2 ) are 1∶100 and 1∶1000. We consider three different kinds of integration range ρ02 while ρ01 is fixed to be 0.1/NA: (1) ρ02 0.5λ∕NA, (2) ρ02 λ∕NA, and (3) ρ02 2λ∕NA. Several important conclusions can be extracted from Table 2. The larger the population size and the higher the number of generations, the smaller the FWHM. This is intuitive from the
natural evolution point of view, since having more perturbations offers more chances for improvements, and more generations tend to evolve to stronger and more advanced individuals. Comparing each column of Table 2, it can be concluded that the smallest integration range of ρ02 leads to the tightest focusing. Finally, a moderate weighting factor ratio (1∶100) may result in a better result than a more exaggerated ratio (1∶1000). These analyses provide confidence that we can achieve lens design with optimal performance by having a population of 50 individuals, 50 generations, ρ01 0.1λ∕NA, ρ02 0.5λ∕NA, and w1 ∶w2 1∶100. In the first step above, all the other variables are adjusted for the best optimization results, except that the number of zones is fixed to be 40. To further optimize the BPDL design, we increased the zone number to 80, 100, 120, and 150 with all the other variables above remaining at the best combination. That is, the population size, the generations, and the w1 ∶w2 ratio are chosen to be 50, 50, and 1∶100, respectively. Meanwhile, the integration range ρ01 and ρ02 is fixed to be 0.1λ∕NA and 0.5λ∕NA. More generations and larger population size could potentially lead to better results, but the improvement may not be significant. Figure 3 plots the FWHM as a function of NA for the optimized BPDLs and a diffraction-limited focal spot. When the zone number is 150, the FWHM of the calculated PSF is decreased to 0.49λ (196 nm), which indicates a very tight BPDL focus compared to a zone plate. More generations or larger population size could potentially lead to even better results. A 150-zone lens optimized with population size of 50 after 75 generations achieved a FWHM of 0.46λ (184 nm 75% of 244 nm). Note that the FWHM is not taken into account in the cost function yet. So now, as the third step, we show the results of two distinct scenarios: one is to further optimize the lens with the previously optimized design as the initial population, and the other is to optimize with the conventional zone plate as the initial population (the same as that in the two steps above). Here, only the third term (FWHM) in the cost function is utilized (w1 0, w2 0, w3 1). All the other parameters are obtained from the best option in Table 2. The FWHMs of the simulated PSFs are listed in Table 3. Compared with the best result in Table 2, employing FWHM as the cost function enables us to further improve the focusing performance of the lens (from 0.67λ to 0.62λ). On the contrary, we found that if only the FWHM is used in the cost function from the very beginning, it is challenging to reach this design. Finally, cost-function modification and initial population modification are applied to check whether any further improvement can be achieved based on the best optimization results in step two above (0.49λ). With optimized data as the initial population in the genetic-algorithm optimization, and with the cost function modified to be FWHM, the final optimized value of FWHM is 0.4850λ. Therefore, when the
Table 2. FWHMs of the Focal Spots for Different Optimized BPDL Designs with Varied Parameters ρ02 0.5∕NA w1 ∶w2 25 25 50 50
generations generations generations generations
25 50 25 50
population population population population
size size size size
ρ02 1∕NA
ρ02 2∕NA
1∶100
1∶1000
1∶100
1∶1000
1∶100
1∶1000
0.8150 0.7900 0.7850 0.6700
0.8200 0.7900 0.7700 0.7100
0.8200 0.8550 0.8150 0.8400
0.8700 0.8550 0.8305 0.7950
0.8950 0.9000 0.9000 0.9000
0.9000 0.8950 0.9000 0.8950
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Fig. 3. FWHMs of the diffraction-limited focal spot (green solid line) and the focal spot generated by the optimized BPDL (blue solid line) versus different numbers of zones, corresponding to different NAs.
Table 3. Effects of the Varied Cost Function and/or Initial Population on the FWHMs of the Focal Spot for Best-Optimized BPDL Design in the First Step Initial Population Calculated Based on Zone Plate Parameters
Previous Optimized Data as Initial Population
0.8400
0.6200
FWHM
zone number is increased to 150, cost-function modification and initial-population modification do not provide much improvement. This step provides reasonable improvement if the zone number (NA) is smaller. To summarize, we propose the following optimization strategy. First, execute optimization with the conventional zone plate as the basis for the initial population and with intensity integration terms in the cost function. Second, improve the best result from the first step by increasing the zone number of the BPDL with all the other variables remaining the same. Third, depending on the zone number of the BPDL, costfunction modification (taking the FWHM term only) and initial population modification may be used to further improve the focusing performance of the BPDL design. Combining both global and local search algorithms may provide even better solutions [11]. Multiphase elements may be considered as an alternative [10,11], but they require more complicated patterning.
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Fig. 4. FWHM of the PSF of the optimized BPDL at various illumination wavelengths, calculated by both scalar theory (blue solid line) and FDTD (MEEP) (red squares). The intensity patterns (by FDTD) at three wavelengths at the observation plane are shown as insets.
However, within this 12 nm bandwidth, the FWHM is always lower than the diffraction limit of 244 nm. Interestingly, between the wavelengths of 398 and 401 nm, the variation of the FWHM is below 1 nm. This demonstrates its relatively good tolerance to potential instability of the light source. In addition, the scalar plot matches the FDTD results well, further proving its appropriateness in predicting the PSFs. The light intensity distributions from FDTD simulations are shown as insets in Fig. 4. B. Defocus The impact of defocus can be characterized by sweeping the observation plane through the focus of the BPDL. The FWHM at various defocus distances is plotted in Fig. 5. Simulation results obtained by scalar diffraction and by FDTD are shown for comparison. Between −2.5λ and 1.5λ defocus, the FWHM is increased by only 8 nm, which is 4% of the FWHM at the focal plane. Specifically, the spot size change can be limited to 1 nm from −1λ to 0.25λ defocus. This is somewhat better
5. DISCUSSION A. Bandwidth Although the optimized BPDL is designed for singlewavelength operation, it is necessary to understand its spectral response, since, in practice, the light source may suffer from limited spectral coherence and peak wavelength instability. Figure 4 shows the FWHMs of the focal spots at different incident wavelengths from 394 to 406 nm, calculated by both scalar-diffraction theory (blue solid line) and the FDTD method (squares). The FWHM reaches its minimum at 400 nm and becomes worse at both shorter and longer wavelengths.
Fig. 5. FWHMs of the optimized BPDL at various observation planes near focus, calculated using scalar theory (blue solid line) and using FDTD (red squares). The intensity patterns (by FDTD) at three observation planes are shown as insets.
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its robustness in terms of chromatic aberrations, defocus, and fabrication errors. Such a superfocusing lens can be applied in optical lithography and microscopy.
ACKNOWLEDGMENTS Discussions with Hank Smith are gratefully acknowledged. The research was partially funded by AFOSR STTR award no. 12RSE299.
REFERENCES
Fig. 6. Percentage change in FWHM of the optimized BPDL under simulated fabrication errors. The errors are simulated as random perturbations from normal distributions with standard deviations of radii and height given by σ r and σ h , respectively.
than the depth of focus given by scalar-diffraction theory, λ∕NA2 ∼ 1.49λ. Note that the best results of the scalar and the FDTD simulations are located at different defocus. This can be due to the interpolation errors when determining the optimized FWHMs in both simulation methods and/or the uncertainty in determining the zero plane of the zone plate from which the focal length is calculated in the FDTD model. The insets in Fig. 5 show the focal spots at different defocus. C. Fabrication Error Tolerance The BPDL proposed here can be fabricated by traditional planar processes. Therefore, it becomes important to analyze the focusing performance of the lens under different fabrication errors. For simulation purposes, errors are introduced as perturbations to both zone radii r 1 ; r 2 ; …r M and height, h. The perturbations are randomly generated from a normal distribution with zero mean and standard deviations of σ r and σ h for radii and height, respectively. To quantify the effect, Fig. 6 shows the percentage change in FWHM calculated using scalar theory at different σ r and σ h values. As expected, greater standard deviations in radii and height lead to larger changes in FWHM. Nevertheless, even with σ r 0.35λ 140 nm and σ h 0.04λ 16 nm, it is possible to suppress the change in FWHM to less than 4% (or 0.04 nm × 196 nm 7.8 nm). It tends to be more robust than multiphase diffractive superresolution elements since only two phase states are incorporated. Moreover, fabrication error tolerance can be readily incorporated as part of the cost function when optimizing the diffractive lens design so that a more robust device may be obtained [2].
6. CONCLUSION In this paper, we described a nonlinear optimization technique for designing BPDLs using scalar-diffraction theory in conjunction with a genetic algorithm. With a judiciously defined cost function and optimization strategy, we obtained a lens that is capable of sub-diffraction-limited focusing (FWHM 196 nm for λ 400 nm and NA 0.8207). The result was validated by rigorous FDTD simulations. Further analyses show
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software package for electromagnetic simulation by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010). 24. M. Mitchell, An Introduction to Genetic Algorithms (Massachusetts Institute of Technology, 1996). 25. The Genetic Algorithm and Direct Search toolbox in MATLAB, http://www.mathworks.com/access/helpdesk/help/ toolbox/gads/.
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Diffractive lens design for optimized focusing Xiaowen Wan, Bing Shen, and Rajesh Menon* Department of Electrical and Computer Engineering, University of Utah, Salt Lake City, Utah 84112, USA *Corresponding author: [email protected] Received June 20, 2014; revised October 9, 2014; accepted October 10, 2014; posted October 10, 2014 (Doc. ID 214377); published November 3, 2014 We utilized nonlinear optimization to design phase-only diffractive lenses that focus light to a spot, whose width is smaller than that dictated by the far-field diffraction limit. Although scalar-diffraction theory was utilized for the design, careful comparisons against rigorous finite-difference time-domain simulations confirm the superfocusing effect. We were able to design a lens with a focal spot size that is 25% smaller than that formed by a conventional lens of the same numerical aperture. An optimization strategy that allows one to design such lenses is clearly explained. Furthermore, we performed careful simulations to elucidate the effects of fabrication errors and defocus on the performance of such optimized lenses. Since these lenses are thin, binary, and planar, large uniform arrays could be readily fabricated enabling important applications in microscopy and lithography. © 2014 Optical Society of America OCIS codes: (050.1965) Diffractive lenses; (050.1970) Diffractive optics; (230.3990) Micro-optical devices; (050.1380) Binary optics. http://dx.doi.org/10.1364/JOSAA.31.000B27
1. INTRODUCTION Diffractive optics generate complex electromagnetic fields by manipulating the amplitude and phase of incident wavefronts [1]. The geometric configuration of the diffractive element can be specifically designed to achieve the desired intensity or phase patterns at the observation planes [2–4]. Conventional scalar-diffraction theory can be applied to predict the field distribution with good accuracy [1]. Compared to a diffractive element that imposes amplitude modulation, a phase-only diffractive optic has the advantage of low loss and, thereby, better photon throughput. This benefits applications that either require high power density, such as photolithography [5], or have a low photon budget, such as fluorescence microscopy [6]. Diffractive elements, which require only two levels of phase shifts (so-called binary diffractive optics), are significantly easier to fabricate. Although these optics suffer from inherent chromatic aberration [7], they are useful for singlewavelength applications. Usually, the phase shift is introduced by an optical path difference between alternate regions [2]. The zone plate is an exemplary device in the family of diffractive optical elements. Since its first demonstration by Soret in 1875 [8], it has been exploited for a variety of applications including multicolor confocal imaging [7], x-ray microscopy [9], superresolution data storage [10], parallel photolithography [11–13], conversion of the focus spot profile of laser beams [14], focusing neutrons [15], and millimeterwave antennas [16]. In traditional projection photolithography [17,18] and confocal microscopy [19], multiple complex lenses are used to tightly focus the light into a diffraction-limited focal spot or a diffraction-limited image. Replacing bulky complex refractive optics by thin diffractive lenses can enable compact, lightweight systems, while maintaining a high numerical aperture (NA). The zone plate is essentially composed of a number of concentric ring-shaped zones. If the zones are made of absorbing 1084-7529/14/120B27-07$15.00/0
material, one gets an amplitude zone plate. On the other hand, if the material is transparent, the optic is a phase zone plate. The conventional on-axis zone plate is circularly symmetric, and can be fully described by the radii and the transmission function of the individual zones. In the past, parametric search methods have been applied to modify the conventional zone plate to reduce the width of the focal spot [10,11]. Reductions of both the central spot size at focus and controlling of the side lobe intensity were achieved. Nevertheless, the results in Ref. [10] are limited by the available degrees of freedom and the small field of view. Besides, the multiphase elements in Refs. [10,11] are not as easily fabricated and replicated as their binary-phase counterparts. No analysis on chromatic aberration, defocus, and fabrication error tolerance was given in Refs. [10,11]. In this article, we extend this previous approach to lenses with many more binary zones, and utilize a new optimization strategy to achieve even smaller focal spots than before. Specifically, we present a binary-phase diffractive-lens (BPDL) design, which is capable of focusing light with minimized central spot size for moderate and high NAs. In the following sections, the model of the diffractive optic and a nonlinear optimization approach are explained in detail. The best design demonstrates a spot size defined as the full width at half-maximum (FWHM) of 196 nm (much smaller than the far-field diffraction limit of 244 nm) at λ 400 nm. We further compare our scalar-diffraction simulations to finite difference time-domain (FDTD) simulations. Finally, we perform a careful analysis of the impact of fabrication errors and defocus on the performance of these BPDLs.
2. DESIGN The basic schematic of the proposed BPDL design is illustrated by a cross-sectional view in Fig. 1. Note that our lenses are rotationally symmetric. An incident plane wave is used © 2014 Optical Society of America
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such that the amplitude is constant across the lens. Adjacent zones of concentric rings experience a relative phase shift, described by ψ 2πh∕λn − 1, where h is the step height between neighboring zones, λ is the wavelength of incident light, and n is the real part of the refractive index of the zone plate material at wavelength λ. In this case, both the phase shift ψ (or equivalently, the zone height, h) and radii of the zones r 1 ; r 2 ;…r M are treated as design variables to be optimized. Note that planar fabrication processes can enable large-array, highly uniform fabrication of such phase-only diffractive optics in dielectric materials [20]. Also, photocurable nanoimprint lithography can be used to achieve mass production of high-resolution BPDLs [21, 22]. In Fig. 1, ρ and ρ0 are the radial coordinates in the zone plate plane and the focal plane, respectively, and z represents the direction of light propagation. The BPDL can be described by a circular-symmetric transmission function [2]: 8 < eiψ Tρ 0 : 1
9 r 2m < ρ ≤ r 2m1 ; m ∈ f0; 1; 2; 3; …g = ; ρ > rM ; elsewhere
(1)
in which r m is the radius of the mth zone, and m is a set of positive integers bounded by M. M is the total number of zones. The well-known Fresnel–Kirchhoff formulation of the scalar-diffraction theory is employed to model the propagation of light from the phase zone plate to the observation plane [1]. This scalar model allows for fast computation with reasonable accuracy, which is later validated by rigorous FDTD simulations [23]. The typical time for optimizing the BPDL is about 3 h, which is much faster than that of full-wave electromagnetic simulation. The intensity distribution in the plane of observation at distance d along the optical axis is given by Z Z 2π 1 ∞ γρ0 ; z; λ; f ⃗r; hg ρdρ dϕTρ; f ⃗r; hg iλ 0 0 h pi exp i 2πλ ρ2 ρ02 − 2ρρ0 cosϕ z2 p × ρ2 ρ02 − 2ρρ0 cosϕ z2 2 z × 1 p : (2) 2 02 0 2 ρ ρ − 2ρρ cosϕ z In the past, a genetic algorithm has been successfully deployed for optimizing phase zone plates to generate optical nulls without phase singularities [2]. The aim of our optimization, as stated previously, is to achieve a smaller focal spot compared to a normal zone plate. The cost function used for optimization is defined as Z ρ0 1 Ef ⃗r; hg −w1 · γρ0 ; λ; f⃗r; hgρ0 dρ0 w2 0 Z ρ0 2 · γρ0 ; λ; f⃗r; hgρ0 dρ0 w3 · FWHM: ρ01
(3)
Equation (3) contains three terms, indicating contributions from the total energy in the central region of the focal spot integrated from 0 to ρ01 , the total energy in the peripheral region of the focal spot integrated from ρ01 to ρ02 , and the FWHM of the focal spot, respectively. Since we expect to concentrate light in the focal spot with narrower width, the sign of each
Fig. 1. Schematic of the BPDL and design methodology. The design variables are the radii of the zones, r 1 ; r 2 ;…; r M , and the height of the zones, h. Intensity distributions in the focal plane illustrate the design requirement for a BPDL with tight focusing at illumination wavelength, λ.
term is selected such that the cost function is minimized during optimization. And w1 , w2 , and w3 are positive weights for the three terms. Delicate balancing among these values is critical in achieving optimal results, which is discussed in Section 4. In addition, physical constraints Eqs. (4)–(6) have to be imposed to ensure that the zones are retained in the correct order and that the width of each zone is greater than Δ during optimization. Here, Δ is basically the minimum feature that can be patterned. We assume that electron-beam lithography (EBL) will be utilized to pattern nanometer-scale structures [20]: rp > rq
∀M ≥ p > q ≥ 1;
fp; qg ∈ I;
r p − r p−1 > Δ > 0 ∀M ≥ p > 1; r 1 > 2Δ:
p ∈ I;
(4)
(5)
(6)
Both the height, h, and radii, r 1 ; r 2 ; …r M , of zones are optimization variables. The genetic algorithm is an ideal choice for solving the complex problem of free-space propagation dictated by a highly nonlinear function Eq. (1) bounded by the cost function Eq. (3) and additional constraints Eqs. (4)–(6). It is a mathematical abstraction of natural selection [24, 25]. The algorithm begins with an initial population and continues in an iterative manner [2]. Perturbing a conventional binaryphase zone plate generates the initial population. The perturbations are chosen from a Gaussian distribution of zero mean and standard deviations 50λ and 15λ for the zone radii and the zone height, respectively. During each iteration, a new population is generated. Each individual within the population is evaluated in terms of the defined cost function, based on which they are ranked. The individuals in the next generation are selected based on the principle that the individuals with the lowest cost-function values in the current generation are retained and the others are derived from adding random perturbations of the variables to the retained individuals. Both the number of individuals in the population (either 25 or 50) and the number of iterations (either 25 or 50) dictate the performance of the eventually optimized BPDL design.
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This impact is described in Section 4. All perturbations are subject to the constraints expressed in Eq. (3).
3. RESULTS Figure 2 shows the results of the best BPDL design optimized for minimized FWHM of the focal spot and high central spot intensity. The design wavelength is λ 400 nm. The optimization strategy to reach this particular lens design is detailed in the following section, where optimization parameters and steps are discussed. The grayscale plot in Fig. 2(a) visualizes the piecewise transmission function of a lens after 50 iterations. A series of concentric rings (or zones) with alternating phase shifts are obvious. It consists of 150 zones with outer radius of 57.4 μm. The focal length of the designed lens is 40 μm, corresponding to NA 0.8207. Specific design parameters (radii and zone height) are shown in Table 1. The minimum zone width, Δ, is 200 nm, which can be readily fabricated by most EBL tools. Here, we assume that the diffractive lens is made of polymethylmethacrylate (PMMA), a widely used dielectric with excellent transparency and easy processes
Fig. 2. (a) Transmission function of the optimized BPDL. NA 0.8207, focal length 40 μm. The other optimization parameters are described in the text. Within the aperture of the BPDL, the gray rings are phase shifted with respect to the white ones; their relative height difference, h, is 384 nm. Outside the BPDL is opaque. (b) Radial intensity distributions of the optimized BPDL focal spot and the conventional zone plate focal spot (black solid line). Blue and red solid lines correspond to the simulation results calculated using the scalar method and the FDTD method, respectively. The inset shows a magnified view of the focal spot.
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Table 1. h r1 r2 r3 r4 r5 r6 r7 r8 r9 r 10 r 11 r 12 r 13 r 14 r 15 r 16 r 17 r 18 r 19 r 20 r 21 r 22 r 23 r 24 r 25 r 26 r 27 r 28 r 29 r 30 r 31 r 32 r 33 r 34 r 35 r 36 r 37 r 38 r 39 r 40 r 41 r 42 r 43 r 44 r 45 r 46 r 47 r 48 r 49 r 50
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Radii and Height of the Zones in Units of λ (λ 400 nm)
0.9597 10.0797 12.0695 18.4904 19.1648 24.3048 26.5862 28.6980 30.1720 31.3693 33.0078 33.6092 35.1020 35.6962 36.8804 38.6879 39.9747 43.1657 44.1931 45.6668 46.7944 48.0975 48.8126 49.3126 50.1845 50.7066 51.6614 52.6360 53.6640 56.0704 56.7136 57.5843 58.8412 59.4965 60.8368 61.5956 62.6772 63.5222 64.4333 65.6052 66.2145 66.7698 67.4215 67.9440 69.6539 70.7615 71.4830 72.2704 73.2885 74.2822 74.7822
r 51 r 52 r 53 r 54 r 55 r 56 r 57 r 58 r 59 r 60 r 61 r 62 r 63 r 64 r 65 r 66 r 67 r 68 r 69 r 70 r 71 r 72 r 73 r 74 r 75 r 76 r 77 r 78 r 79 r 80 r 81 r 82 r 83 r 84 r 85 r 86 r 87 r 88 r 89 r 90 r 91 r 92 r 93 r 94 r 95 r 96 r 97 r 98 r 99 r 100 r 101
75.7164 76.7098 77.6211 78.3223 79.1194 79.8964 80.6487 81.6154 82.1791 83.0206 83.9554 84.7735 85.3034 86.3196 87.0068 87.8666 88.3700 89.2380 89.8504 90.5420 91.3399 92.2853 92.9537 93.5712 94.1637 94.9364 95.7130 96.5719 97.1983 97.8045 98.3555 98.8555 99.9170 100.9016 101.4902 102.3614 102.9500 103.5945 104.2601 105.0311 105.7210 106.2853 106.8638 107.5675 108.5673 109.0918 109.7822 110.5180 111.2237 111.9300 112.4300
r 102 r 103 r 104 r 105 r 106 r 107 r 108 r 109 r 110 r 111 r 112 r 113 r 114 r 115 r 116 r 117 r 118 r 119 r 120 r 121 r 122 r 123 r 124 r 125 r 126 r 127 r 128 r 129 r 130 r 131 r 132 r 133 r 134 r 135 r 136 r 137 r 138 r 139 r 140 r 141 r 142 r 143 r 144 r 145 r 146 r 147 r 148 r 149 r 150
113.1907 113.7388 114.3833 115.0669 115.8429 116.6045 117.1607 117.6607 118.3847 118.8847 119.6855 120.4629 121.0467 121.7310 122.3396 122.9411 123.4411 124.3293 124.8784 125.4784 126.1632 126.6632 127.4819 128.1237 128.8671 129.4002 129.9002 130.6152 131.2793 131.8115 132.5869 133.0870 133.8729 134.3729 135.0090 135.5091 136.1413 136.9486 137.5175 138.0176 138.5785 139.2747 139.8130 140.6156 141.2537 142.0038 142.5038 143.0303 143.5803
for patterning and replication. The index of refraction of PMMA at λ 400 nm is measured to be 1.51 by spectroscopic ellipsometry. The weighting factors, w1 , w2 , and w3 are chosen to emphasize each term during optimization. These are discussed in the next section. The normalized radial intensity distribution in the focal plane, also referred to as the point-spread-function (PSF) of the diffractive lens, is calculated by both scalar-diffraction theory and the FDTD method. Here, circularly polarized light is utilized in the FDTD simulation because the BPDL is rotationally symmetric. They are plotted together for comparison
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in Fig. 2(b). The PSFs derived from Eq. (2) and the FDTD simulation match quite well. Although only scalar theory is utilized during optimization, the FDTD simulation confirms that it is a reasonable approximation. The relative error is around 7.7% based on the inset of Fig. 2(b). The FWHMs of the focal spots predicted by scalar theory and FDTD are 196 and 194 nm, respectively. This is smaller than the spot size of the diffraction-limited Airy disk given by λ∕2NA 244 nm [10]. The calculated normalized intensity distribution of the diffraction-limited focal spot (of a conventional phase zone plate of the same NA) is also plotted in Fig. 2(b). Although Ref. [10] states that it is difficult to attain both tight focus and reduced side lobes at the same time by uniform phase shift in zone plate design, here we demonstrate that it is feasible to achieve this by expanding the degrees of optimization freedom and by choosing the weighting factors judiciously. As can be seen in Fig. 2(b) (blue line), the intensity of the side lobes is less than 20% (10% calculated by FDTD) of that of the peak. This is comparable with results by the multiphase diffractive superresolution element shown in Ref. [10], although the FWHM shown here is much smaller. Also, the available field-of-view is significantly larger. As shown in Fig. 2(b), everywhere with ρ0 > 2λ is sufficiently suppressed. Moreover, our binary-phase device has much smaller dimension compared to the one optimized in Ref. [11], which has a diameter of about 5 mm. This compactness will enable applications in which an array of zone plates is utilized for large-area lithography [12].
4. OPTIMIZATION STRATEGY In this section, we describe how to select the optimization parameters in each step to ultimately reach the best lens design shown above. In order to understand this, we present a simpler BPDL design with just 40 zones. As previously mentioned, both the number of individuals in each generation (or population size) and the maximum number of iterations (or the number of generations) are changeable. Additionally, the integration ranges (ρ01 and ρ02 ) in the cost function [Eq. (3)] also play a critical role in the optimization process. To start with, we ignore the third term of FWHM in the cost function, while retaining the first and the second terms of power integration over focal and peripheral regions. In Table 2, we summarize the FWHMs of the calculated PSFs of different optimized BPDL designs by using various parameters. The ratios between the first and the second terms (w1 ∶w2 ) are 1∶100 and 1∶1000. We consider three different kinds of integration range ρ02 while ρ01 is fixed to be 0.1/NA: (1) ρ02 0.5λ∕NA, (2) ρ02 λ∕NA, and (3) ρ02 2λ∕NA. Several important conclusions can be extracted from Table 2. The larger the population size and the higher the number of generations, the smaller the FWHM. This is intuitive from the
natural evolution point of view, since having more perturbations offers more chances for improvements, and more generations tend to evolve to stronger and more advanced individuals. Comparing each column of Table 2, it can be concluded that the smallest integration range of ρ02 leads to the tightest focusing. Finally, a moderate weighting factor ratio (1∶100) may result in a better result than a more exaggerated ratio (1∶1000). These analyses provide confidence that we can achieve lens design with optimal performance by having a population of 50 individuals, 50 generations, ρ01 0.1λ∕NA, ρ02 0.5λ∕NA, and w1 ∶w2 1∶100. In the first step above, all the other variables are adjusted for the best optimization results, except that the number of zones is fixed to be 40. To further optimize the BPDL design, we increased the zone number to 80, 100, 120, and 150 with all the other variables above remaining at the best combination. That is, the population size, the generations, and the w1 ∶w2 ratio are chosen to be 50, 50, and 1∶100, respectively. Meanwhile, the integration range ρ01 and ρ02 is fixed to be 0.1λ∕NA and 0.5λ∕NA. More generations and larger population size could potentially lead to better results, but the improvement may not be significant. Figure 3 plots the FWHM as a function of NA for the optimized BPDLs and a diffraction-limited focal spot. When the zone number is 150, the FWHM of the calculated PSF is decreased to 0.49λ (196 nm), which indicates a very tight BPDL focus compared to a zone plate. More generations or larger population size could potentially lead to even better results. A 150-zone lens optimized with population size of 50 after 75 generations achieved a FWHM of 0.46λ (184 nm 75% of 244 nm). Note that the FWHM is not taken into account in the cost function yet. So now, as the third step, we show the results of two distinct scenarios: one is to further optimize the lens with the previously optimized design as the initial population, and the other is to optimize with the conventional zone plate as the initial population (the same as that in the two steps above). Here, only the third term (FWHM) in the cost function is utilized (w1 0, w2 0, w3 1). All the other parameters are obtained from the best option in Table 2. The FWHMs of the simulated PSFs are listed in Table 3. Compared with the best result in Table 2, employing FWHM as the cost function enables us to further improve the focusing performance of the lens (from 0.67λ to 0.62λ). On the contrary, we found that if only the FWHM is used in the cost function from the very beginning, it is challenging to reach this design. Finally, cost-function modification and initial population modification are applied to check whether any further improvement can be achieved based on the best optimization results in step two above (0.49λ). With optimized data as the initial population in the genetic-algorithm optimization, and with the cost function modified to be FWHM, the final optimized value of FWHM is 0.4850λ. Therefore, when the
Table 2. FWHMs of the Focal Spots for Different Optimized BPDL Designs with Varied Parameters ρ02 0.5∕NA w1 ∶w2 25 25 50 50
generations generations generations generations
25 50 25 50
population population population population
size size size size
ρ02 1∕NA
ρ02 2∕NA
1∶100
1∶1000
1∶100
1∶1000
1∶100
1∶1000
0.8150 0.7900 0.7850 0.6700
0.8200 0.7900 0.7700 0.7100
0.8200 0.8550 0.8150 0.8400
0.8700 0.8550 0.8305 0.7950
0.8950 0.9000 0.9000 0.9000
0.9000 0.8950 0.9000 0.8950
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Fig. 3. FWHMs of the diffraction-limited focal spot (green solid line) and the focal spot generated by the optimized BPDL (blue solid line) versus different numbers of zones, corresponding to different NAs.
Table 3. Effects of the Varied Cost Function and/or Initial Population on the FWHMs of the Focal Spot for Best-Optimized BPDL Design in the First Step Initial Population Calculated Based on Zone Plate Parameters
Previous Optimized Data as Initial Population
0.8400
0.6200
FWHM
zone number is increased to 150, cost-function modification and initial-population modification do not provide much improvement. This step provides reasonable improvement if the zone number (NA) is smaller. To summarize, we propose the following optimization strategy. First, execute optimization with the conventional zone plate as the basis for the initial population and with intensity integration terms in the cost function. Second, improve the best result from the first step by increasing the zone number of the BPDL with all the other variables remaining the same. Third, depending on the zone number of the BPDL, costfunction modification (taking the FWHM term only) and initial population modification may be used to further improve the focusing performance of the BPDL design. Combining both global and local search algorithms may provide even better solutions [11]. Multiphase elements may be considered as an alternative [10,11], but they require more complicated patterning.
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Fig. 4. FWHM of the PSF of the optimized BPDL at various illumination wavelengths, calculated by both scalar theory (blue solid line) and FDTD (MEEP) (red squares). The intensity patterns (by FDTD) at three wavelengths at the observation plane are shown as insets.
However, within this 12 nm bandwidth, the FWHM is always lower than the diffraction limit of 244 nm. Interestingly, between the wavelengths of 398 and 401 nm, the variation of the FWHM is below 1 nm. This demonstrates its relatively good tolerance to potential instability of the light source. In addition, the scalar plot matches the FDTD results well, further proving its appropriateness in predicting the PSFs. The light intensity distributions from FDTD simulations are shown as insets in Fig. 4. B. Defocus The impact of defocus can be characterized by sweeping the observation plane through the focus of the BPDL. The FWHM at various defocus distances is plotted in Fig. 5. Simulation results obtained by scalar diffraction and by FDTD are shown for comparison. Between −2.5λ and 1.5λ defocus, the FWHM is increased by only 8 nm, which is 4% of the FWHM at the focal plane. Specifically, the spot size change can be limited to 1 nm from −1λ to 0.25λ defocus. This is somewhat better
5. DISCUSSION A. Bandwidth Although the optimized BPDL is designed for singlewavelength operation, it is necessary to understand its spectral response, since, in practice, the light source may suffer from limited spectral coherence and peak wavelength instability. Figure 4 shows the FWHMs of the focal spots at different incident wavelengths from 394 to 406 nm, calculated by both scalar-diffraction theory (blue solid line) and the FDTD method (squares). The FWHM reaches its minimum at 400 nm and becomes worse at both shorter and longer wavelengths.
Fig. 5. FWHMs of the optimized BPDL at various observation planes near focus, calculated using scalar theory (blue solid line) and using FDTD (red squares). The intensity patterns (by FDTD) at three observation planes are shown as insets.
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its robustness in terms of chromatic aberrations, defocus, and fabrication errors. Such a superfocusing lens can be applied in optical lithography and microscopy.
ACKNOWLEDGMENTS Discussions with Hank Smith are gratefully acknowledged. The research was partially funded by AFOSR STTR award no. 12RSE299.
REFERENCES
Fig. 6. Percentage change in FWHM of the optimized BPDL under simulated fabrication errors. The errors are simulated as random perturbations from normal distributions with standard deviations of radii and height given by σ r and σ h , respectively.
than the depth of focus given by scalar-diffraction theory, λ∕NA2 ∼ 1.49λ. Note that the best results of the scalar and the FDTD simulations are located at different defocus. This can be due to the interpolation errors when determining the optimized FWHMs in both simulation methods and/or the uncertainty in determining the zero plane of the zone plate from which the focal length is calculated in the FDTD model. The insets in Fig. 5 show the focal spots at different defocus. C. Fabrication Error Tolerance The BPDL proposed here can be fabricated by traditional planar processes. Therefore, it becomes important to analyze the focusing performance of the lens under different fabrication errors. For simulation purposes, errors are introduced as perturbations to both zone radii r 1 ; r 2 ; …r M and height, h. The perturbations are randomly generated from a normal distribution with zero mean and standard deviations of σ r and σ h for radii and height, respectively. To quantify the effect, Fig. 6 shows the percentage change in FWHM calculated using scalar theory at different σ r and σ h values. As expected, greater standard deviations in radii and height lead to larger changes in FWHM. Nevertheless, even with σ r 0.35λ 140 nm and σ h 0.04λ 16 nm, it is possible to suppress the change in FWHM to less than 4% (or 0.04 nm × 196 nm 7.8 nm). It tends to be more robust than multiphase diffractive superresolution elements since only two phase states are incorporated. Moreover, fabrication error tolerance can be readily incorporated as part of the cost function when optimizing the diffractive lens design so that a more robust device may be obtained [2].
6. CONCLUSION In this paper, we described a nonlinear optimization technique for designing BPDLs using scalar-diffraction theory in conjunction with a genetic algorithm. With a judiciously defined cost function and optimization strategy, we obtained a lens that is capable of sub-diffraction-limited focusing (FWHM 196 nm for λ 400 nm and NA 0.8207). The result was validated by rigorous FDTD simulations. Further analyses show
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