Freeform Lens Research Articles

Differentiable design of freeform diffractive optical elements for beam shaping by representing phase distribution using multi-level B-splines

The generation of a specific laser beam profile on the work surface is key to various laser beam shaping tasks, relying heavily on diffractive optical elements (DOEs). Most beam-shaping DOEs are designed using iterative Fourier transform algorithms (IFTAs), which generally have slow convergence and prone to stagnate at local minima. Moreover, the microreliefs generated by IFTAs tend to be irregular, complicating manufacturing and causing uncontrolled scattering of light. We propose a differentiable DOE design method that applies a phase-smoothness constraint using multi-level B-splines. A multi-scale gradient-descent optimization strategy, naturally linked with the multi-level B-splines, is employed to robustly determine the optimized phase distribution that is fully continuous. This, in turn, can lead to more regular DOE microreliefs, which can simplify the fabrication process and be less sensitive to changes in wavelength and working distance. Furthermore, our method can also design a fully continuous freeform lens, distinguished from most freeform lens design approaches by its foundation in physical optics rather than geometrical optics. Simulation and experimental results of several design tasks demonstrate the effectiveness of the proposed method.

Research on the design of progressive addition multifocal defocused freeform lenses

Research on the design of progressive addition multifocal defocused freeform lenses

Terahertz quasi-optics imaging systems with easy zoom based on beam-shaping devices and a freeform lens.

Terahertz (THz) waves show outstanding application value in the nondestructive detection of hidden targets that are impenetrable to visible light. However, the uncertain location of hidden targets puts a higher demand on the zoom function of the THz quasi-optical systems, especially when the object is immovable. This paper proposes a continuous zoom system working in the THz band consisting of a negative axicon disk, a segmented axicon, and a freeform lens. The negative axicon disk and segmented axicon shape THz waves into annular beams with adjustable diameters, which are respectively focused at different positions by the zone-designed freeform lens. Both simulation and experimental results indicate that the zoom range of the system can reach 82 mm while maintaining an imaging resolution of 6 mm. Notably, continuous zooming is achieved by merely rotating the negative axicon disk, instead of the lens moving. Moreover, all devices in the system can be fabricated by 3D printing or machining. This approach offers the advantages of simple adjustment and low cost, providing, to our knowledge, a novel perspective for the design and application of THz quasi-optical imaging systems.

Portable Digital Lensless Holographic Microscope for fieldwork applications using a conventional smartphone sensor and a customized freeform lens

Portable Digital Lensless Holographic Microscope for fieldwork applications using a conventional smartphone sensor and a customized freeform lens

Miniature wide angle optical system with freeform lens

Miniature wide angle optical system with freeform lens

Design of wavefront detection system for eyeglasses based on transmissive phase measuring deflectometry using gamma error correction algorithm

With the increasingly widespread application of freeform lenses, the detection of lens dioptric power presents a challenge. To measure the surface dioptric power of freeform lenses, a wavefront detection system for eyeglass lenses based on transmissive phase measuring deflectometry has been developed. The system consists only of a monitor, an industrial camera, and the lens under test. In response to the phase errors introduced by nonlinear devices, a Frequency Domain Gamma Error Correction Algorithm based on Linear Regression (FDLR) is proposed to optimize the system detection process and enhance detection accuracy. In the experiments, the proposed system was used to measure the wavefront and surface dioptric power of both spherical lens and progressive addition lens. The results were compared with those obtained from a focimeter to verify the accuracy and reliability of the system. Furthermore, a comparison of the detection results for progressive addition lens before and after phase error correction was conducted to demonstrate the necessity and effectiveness of the proposed phase error correction algorithm.

Precision glass molding technology for XY polynomial freeform optical elements with simulations and experiments

Precision glass molding (PGM) technology, as an effective method for mass-producing glass lenses, is relatively mature in the molding process of aspheric lenses, but the glass molding technology for freeform optical elements is still in its infancy. For freeform optical elements, processing by conventional ultra-precision methods requires multiple processes and the resulting costs are high, while processing by PGM is efficient and inexpensive. Therefore, this paper investigates the molding technology of freeform lenses, the pre-compensation model of the freeform mold core is established, and predicts the residual stresses of freeform lenses after molding by the finite element method. Three different process parameters, molding temperature, molding rate and molding force, are verified. Experimental and simulation results show that the trends of residual stresses for the three process parameters are consistent. The optimal process parameters of the molding process are determined, under which the PV value of the molding lens is around 1.5µm. The experimental results show that the PV value of the molded lens is reduced to less than 1µm after using the pre-compensated mold core, which proves the validity of the pre-compensated model.

Differentiable design of a double-freeform lens with multi-level radial basis functions for extended source irradiance tailoring

Freeform optics are key for generating prescribed illumination patterns from given sources, which are crucial for solid-state lighting and machine vision illumination. There is an increasing demand for compact freeform optics, which presents a substantial challenge for current design methods since the source dimensions must be considered. Most current extended-source design methods, although requiring profound knowledge of optics and mathematics, focus on the modest goal of obtaining uniform irradiance distributions. We address a more challenging design problem of generating an irradiance distribution of arbitrary shape through a double-freeform lens that can fully encompass the extended source. We propose a differentiable design method whose uniqueness lies in the representation of the double-freeform surfaces using multi-level spherical radial basis functions, which has a natural link to a multi-scale optimization technique. In addition, we employ a sequential unconstrained minimization technology complemented with Lagrange multipliers that add key feasibility constraints on lens shape and size. The proposed method is flexible, general, and efficient in designing highly compact freeform lenses for generating both simple and complex irradiance distributions, as demonstrated through the design examples. This could enable a universal solution to the extended-source design problem.

Optical design of low-location illumination based on asymmetric double freeform surfaces.

The development of optical optics for low-location road lighting is a challenging problem in providing high luminance and uniformity of illumination and meeting many other specific requirements. This study proposes an optical design method of low-location illumination based on an asymmetric double freeform surface lens. The ray emitted from the light source is refracted and reflected through the different surface types to the corresponding area of the receiving surface. In the design example, the road has dual-side mounted luminaires and a width of 6m, and a height of 0.8m. Simulation results indicate that, compared with conventional high-pole streetlights, the luminance uniformity had increased from 0.60 to 0.66, the illuminance uniformity had improved from 0.75 to 0.86, and the glare had been reduced.

Optimization method for freeform progressive addition lenses using the coincident degree of weight distributions for power deviation and astigmatism

This study proposed an optimization method for freeform progressive addition lenses (PALs) based on the coincident degree of weight distributions (WD) for power deviation and astigmatism. Compared with the existing methods which were limited in optimizing WDs for power deviation and astigmatism, our proposed approach offers a more refined optimization. In the design phase, the power deviation and astigmatism of these lenses were evaluated using the existing surface shape. Compared with the prescriptions of the patients, the coincident degrees between the obtained distributions and prescriptions of the PAL power and astigmatism were calculated in the multi-view axis condition. Normalization processing of coincident degrees was performed, yielding the corresponding threshold value of WDs and optimizing the allocated coincident degrees. Based on a minimization error function model, two PALs were designed, simulated, machined, and evaluated using a commercial software. The optimized method reduced peripheral astigmatism and improved the optical properties of PALs. The proposed approach optimizes the freeform PALs and enhances their design optimization in optometry.

A varifocal augmented reality head‐up display using Alvarez freeform lenses

AbstractA crucial requirement of augmented reality head‐up displays (AR‐HUDs) is continuously adjustable virtual image distance (VID), which allows adaptation to various depths in road environments and thereby avoids visual fatigue. However, usual varifocal components for near‐eye displays are unavailable because AR‐HUDs require the varifocal component's aperture to be larger than 10 cm. This study considers the Alvarez lenses, which change the optical power by in‐plane sliding two freeform lenses. Under the paraxial assumption, classic Alvarez lenses can create a quadratic wavefront profile, but the large aperture and extensive diopter variation range required by AR‐HUDs lead to significant aberrations. Thus, the classic paraxial Alvarez lens design is extended by co‐optimizing Alvarez lenses with high‐order surface profiles and a primary freeform mirror. Therefore, a novel varifocal AR‐HUD containing Alvarez lenses with apertures larger than 15 cm is proposed. The AR‐HUD generates a varifocal plane whose VID can be continuously adjusted between 2.5 and 7.5 m, and another focal plane with a fixed VID at 7.5 m. In addition, merely one display panel is used for compactness. Finally, an AR‐HUD prototype with a reduced volume of 9.8 L was built. The expected varifocal performance and qualified imaging quality were experimentally verified through the field of view, VID, and image sharpness.

Freeform Liquid–Crystal Polarization Imaging Optics: A Practical Approach from Design to Fabrication

AbstractFreeform liquid–crystal polarization imaging optics (LPIOs) have vast potential for application in next‐generation interactive displays like augmented reality and virtual reality. However, tailoring the phase profile of a freeform LPIO for flexible aberration correction and diffraction efficiency control is still a big challenge due to the difficulties arising from the non‐zero étendue of an imaging system and the finite bandwidth of LPIOs. Here, a practical approach to creating freeform LPIOs from design to fabrication is proposed. A freeform polarization exposure is developed to produce a flexible, precise, and ultra‐high‐resolution photoalignment pattern for a freeform LPIO. A cooperative optimization is performed between the freeform LPIO and the freeform polarization exposure system to find the optimal phase of the freeform LPIO and the optimal surface profiles of the freeform lenses employed in the exposure system to produce freeform wavefronts. The optimal phase of the freeform LPIO is then achieved by superposition interference of the engineered freeform wavefronts via freeform polarization exposure. The effectiveness of the approach and the advantages of freeform LPIOs is experimentally demonstrated. The proposed approach will significantly promote the widespread applications of freeform LPIOs in the field of optics and photonics.

Fabrication of the freeform Fresnel lens by swinging-rotating diamond ruling

Freeform optics are widely applied across various industries, encompassing both imaging and non-imagining optical systems. To compact the freeform surface, the freeform Fresnel lens has been devised by optical designers. However, conventional slow slide servo techniques cannot effectively fabricate the intricate lens, due to its asymmetry and discontinuity. In this paper, a novel swinging-rotating diamond ruling method is proposed to fabricate a biconical freeform Fresnel lens, with the synchronized motions of X-, Z-, C- and A-axes. The morphed Archimedean driving spiral based on contour lines is constructed to avoid the tool-microstructure interference. The tool path generation algorithm is proposed to generate smooth and accurate tool path, encompassing an adaptive projection principle, a transition tool path interpolation method and a postprocessing method. Finally, the feasibility and superiority of the proposed method is validated through two case studies. It is demonstrated that the precise and efficient fabrication of the freeform Fresnel lens can be realized by swinging-rotating diamond ruling method.

Free-form lens design of LED extended light source based on improved weighted superposition optimization

Free-form surface has a wide range of applications in imaging and non-imaging optics, which can simplify the optical components, support structures required by the system and make the overall optical space more compact, but because of the large freedom of design, the design difficulty is greatly increased, how to meet the design requirements while improving efficiency, is the focus and difficulty of current research. Therefore, this paper proposed an improved optimization method for designing of free-form lens for light-emitting diode (LED) light distribution. Based on the lens matching curve obtained by weighted superposition, feedback optimization method was used to further improve the optimization possibility of lens. The weight factors of each surface are optimized by particle swarm optimization algorithm to minimize the evaluation function based on illuminance uniformity of the target surface. The illuminance uniformity on the target plane is optimized from 72.8% to 93.2% start form initial structure. The results show that the proposed algorithm has the advantages of high optimization efficiency and good optimization effect.

Optimizing freeform lenses for extended sources with algorithmic differentiable ray tracing and truncated hierarchical B-splines.

We propose a method for optimizing the geometry of a freeform lens to redirect the light emitted from an extended source into a desired irradiance distribution. We utilize a gradient-based optimization approach with MITSUBA 3, an algorithmic differentiable non-sequential ray tracer that allows us to obtain the gradients of the freeform surface parameters with respect to the produced irradiance distribution. To prevent the optimizer from getting trapped in local minima, we gradually increase the number of degrees of freedom of the surface by using Truncated Hierarchical B-splines (THB-splines) during optimization. The refinement locations are determined by analyzing the gradients of the surface vertices. We first design a freeform using a collimated beam (zero-etendue source) for a complex target distribution to demonstrate the method's effectiveness. Then, we demonstrate the ability of this approach to create a freeform that can project the light of an extended Lambertian source into a prescribed target distribution.

Design and analysis of a freeform surface lens coupler applied in hollow‐core photonic crystal fiber resonant optic gyros

AbstractIn this study, a novel freeform surface lens coupler is designed to realize the coupling functions in the hollow‐core photonic crystal fiber (HCPCF) resonator optical gyroscope. One surface of the lens is an ellipsoid. Coupling of the two HCPCF loop's ports can be directly accomplished via reflection on the ellipsoid surface. The other surface of the lens is the Biconic Zernike surface. Transmitting through the above two surfaces, the coupling between the intracavity and extracavity is completed. In this paper, simulations were performed based on two types of HCPCFs, a hollow‐core photonic bandgap fiber of HC19‐1550 and a state‐of‐the‐art nested antiresonant nodeless fiber. The volume of the lens coupler is calculated to be approximately 4 × 2 × 4 mm3. The simulation results show that the intracavity coupling loss can be as low as 0.011 dB, and the coupling loss between the intracavity and extracavity can be as low as 0.4 dB. Compared with the traditional fusion scheme between hollow‐core fiber and polarization‐maintaining fiber, the freeform surface lens coupler in this paper can achieve lower coupling loss and higher resonance fineness. Compared with other spatial coupling schemes, it has lower coupling loss and fewer coupling components.

Parameter-free Hubble constant from the quadruply lensed quasar SDSS J1004+4112

We present a free-form lens model for the multiply lensed quasar in the galaxy cluster SDSS J1004+4112. Our lens model draws minimal assumptions on the distribution of mass in the lens plane. We have paid particular attention to the model uncertainties on the predicted time delay originating from the particular configuration of model variables. Taking into account this uncertainty, we obtained a value of the Hubble constant of H0 = 74−13+9 km s−1 Mpc−1, which is consistent with recent independent estimates. The predicted time delay between the central image E and image C (the first to arrive) is ΔTE−C = 3200 ± 200 days for the estimated Hubble constant. Future measurements of ΔTE−C will allow for a tighter constraint to be imposed on H0 in this cluster-QSO system.

Process and Outcomes of Fitting Corneoscleral Profilometry-Driven Scleral Lenses for Patients With Ocular Surface Disease.

To assess the feasibility of obtaining cornea scleral profile (CSP) measurements using Scheimpflug imaging and report on the fitting process of free-form custom scleral lenses (SLs) for patients with ocular surface disease (OSD). This prospective study of patients fit with free-form SLs collected data on the following: demographics, indications for wear, corneal and scleral tomography, scan acquisition process, and SL fitting process. Cornea scleral profile scans were acquired on 15 eyes of nine patients. Mean scan time for right eyes was 10.7, and 9.7 min for left eyes. A mean of 2.9 follow-up visits were required to complete SL fitting, with a mean of 2.1 lenses ordered. One eye did not tolerate lens wear, and one eye could not be fit using the CSP scan because of insufficient data. The initial lens ordered was dispensed at the first follow-up visit for seven of the remaining 13 eyes, all of which were ultimately fit successfully in free-form lenses. In this study of profilometry-guided SL fitting for eyes with OSD and low magnitude corneal astigmatism, the number of lenses and follow-up visits required were similar to outcomes of previous studies that described the diagnostic approach to SL fitting. In addition, imaging technology does not negate the need for skilled clinical observation while fitting SLs.

Collimated flat-top beam shaper metasurface doublet based on the complex-amplitude constraint Gerchberg–Saxton algorithm

Abstract Collimated flat-top beam shapers primarily consisting of freeform lenses have a wide range of applications and pose challenges in terms of processing and integration when the diameter is less than millimeters. Metasurfaces represent a promising solution to planarize optics, can mimic any surface curvature without additional fabrication difficulty, and are suitable for flat-top optics. The conventional metasurface design approach relies on imparting the required phase using meta-atoms and encounters challenges in amplitude modulation due to near-field coupling and varying transmittances among meta-atoms with different phases, making the design of flat-top beam shapers difficult. In this paper, we propose a complex-amplitude constraint Gerchberg–Saxton algorithm for designing a collimated flat-top beam shaper metasurface doublet, which avoids the influence of strong near-field coupling on the amplitude distribution and simultaneously considers the amplitude-phase characteristics of the meta-atoms. A collimated flat-top beam with exceptional homogeneity U p of approximately 0.01, a wavefront error less than 0.1λ, and a transmittance higher than 86 % is experimentally obtained, comparable to commercial products based on freeform lenses. A collimated flat-top beam shaper metasurface doublet for generating flat-top beam is introduced, promoting efficient integration with laser systems.

Design of Freeform Surface Low-Distortion Automotive Lens Based on Point-by-Point Construction

Design of Freeform Surface Low-Distortion Automotive Lens Based on Point-by-Point Construction