Objective Traditional diffraction lenses can manipulate wavefronts by adjusting the refractive index of materials and surface profiles of devices, which are widely applied in optical imaging. However, such imaging capabilities are usually constrained by limitations of low focusing efficiency, a single operating wavelength, and chromatic aberration. To suppress chromatic aberration and improve image quality, it is commonly used by combining positive and negative refractive index lenses. Nevertheless, this scheme hinders the miniaturization and integration of optical systems. Although monolithic refractive-diffractive hybrid lenses provide an alternative method to enhance system integration for achromatic performance, complex structural design restricts their further applications. Recently, metasurfaces offer another methodology for realization of broadband achromatic, but the high processing precision in micro-nanofabrication is always required. Consequently, there is still a great challenge for designing diffractive lenses with simple structure, low processing difficulty, and achromatism.

Methods For conventional direct binary search (DBS) algorithm to optimize diffraction lenses, the random initial structures are commonly used, whereas it commonly leads to slow convergence, sensitivity to the starting state. Here, a modified DBS algorithm is proposed for designing three-wavelength multi-order achromatic diffraction microlenses by directly tuning the discrete surface-relief heights. Utilizing the radially symmetric initial height distribution, such approach can reduce the randomness and improve computational stability and corresponding main procedures are as follows: First, the ideal continuous phase distribution of achromatic focusing lens with a parabolic phase profile can be generated; Second, such continuous phase is quantized into discrete phase corresponding to different step-heights for optimizing multilevel diffractive lens by iterative perturbations of local phase; Third, a radially symmetric initial structure determined by modulo function is introduced to replace the conventional random initialization, and then the step heights corresponding to different wavelengths are alternately distributed at distinct positions. Finally, a figure-of-merit (FOM) is adopted to evaluate the performance of the designed lens and guides the iterative updates of the step-height distributions. As a proof of concept, the 8-step and 16-step diffraction lenses (with a diameter, focal length, and numerical aperture of 560 μm, 10 mm, and 0.028, respectively) are designed and analyzed both theoretically and through simulation.

Results and Discussions For pre-optimization, the average focusing efficiencies of the diffractive lenses with 8 phase levels (L=8) and 16 phase levels (L=16) are 77.04% and 78.82% at the wavelengths of 1.064 μm, 1.300 μm, and 1.550 μm, respectively. Therefore, the average focusing efficiencies of both lenses are less than 79% affected by chromatic aberration in theory. To improve the focusing efficiency and suppress chromatic aberration, a modified DBS algorithm is employed to optimize the step-height distribution. For post-optimization, the focusing efficiencies of diffraction lens (L=8) can reach 87.92%, 94.32%, and 90.73% (the efficiency is 90.99%) at 1.064 μm, 1.30 μm, and 1.55 μm after 4000 iterations, respectively, which is improved 13.95% compared to pre-optimization in theory. When L is changed to 16, the efficiencies exceed 90.01%, 96.63%, and 90.73% (at 1.064 μm, 1.300 μm, and 1.550 μm) respectively, and corresponding average value surpasses 92.46% (the improvement is 13.64% compared to pre-optimization) under the same conditions. These results indicate that the proposed achromatic diffraction lens maintains higher performance with the proposed modified DBS algorithm in theory. To further validate the achromatic performance, numerical simulations are conducted using CST microwave studio. For the diffraction lens with L=8 and 16, the focusing efficiencies at 1.064 μm are 87.17% and 89.53% in simulation, respectively. Moreover, it can be increased to 89.10% and 96.24% at 1.30 μm, while the efficiencies are 89.10% and 89.08% at 1.55 μm, respectively. Therefore, corresponding average efficiencies can reach 90.09% (L=8) and 91.62% (L=16), respectively. Note that the slight difference between theoretical prediction and numerical simulation may be attributed to the ideal assumptions (neglected the interactions between structures) in the theory. Furthermore, the point spread functions (PSFs) at the focal plane for different wavelengths are employed to characterize the achromatic performance of the designed diffraction lens. The simulated full width at half maximums (FWHM) are 25.8 μm, 26.9 μm, and 33.4 μm (L=8) and 26.9 μm, 26.9 μm, and 33.4 μm (L=16) at 1.064 μm, 1.300 μm, and 1.550 μm, respectively, which are agree well with the theoretical results of 21.8 μm, 26.6 μm, and 32.2 μm and close to their diffraction limits of 19.00 μm, 23.22 μm, and 27.69 μm at different wavelengths. Totally, both the 8-step and 16-step diffraction lenses possess an average focusing efficiency of ~79% (at 1.064 μm, 1.30 μm, and 1.55 μm) before optimization. However, the average efficiencies can exceed 90.99% and 92.46%, (improvements of 13.95% and 13.64%), respectively, based on the improved DBS algorithm (after 4000 iterations) in theory. Moreover, it has also demonstrated that the FWHM of the 8-step and 16-step diffraction lenses can reach the diffraction limit at three wavelengths, exhibiting high imaging performance.

Conclusions Here, a new methodology is proposed for achieving three-wavelength achromatism, and may offer advantages in reduction of device complexity, requirements of machining accuracy, and the manufacturing costs. These results may be further extended to diverse applications, such as three-wavelength imaging systems, optical computing, lightweight collimators, optical communications, biomedical imaging, and laser processing.