Focal Length Research Articles

Thermal lens focal length statistics for end-pumped solid-state lasers excited with different beam shapes

Thermal lens focal length statistics for end-pumped solid-state lasers excited with different beam shapes

Chromatic confocal displacement measurement using a double-sided phase diffractive lens

A chromatic confocal displacement measurement (CCDM) system using a double-sided phase diffractive lens (DPDL) is demonstrated. The DPDL is designed to produce a strong dispersion, minimize the optical structure and reduce the difficulty of fabrication and assembly, in addition to reducing the weight and size of the probe of the CCDM. A CCDM experimental system was set up based on the DPDL, and the results of the measurement experiments show that the axial measurement range of the CCDM system reaches 9.463 mm in the working wavelength range (500-700 nm), the relative error is less than 0.022% and the axial resolution near the designed focal length is better than 0.5 µm.

Exact Equations for the Back and Effective Focal Lengths of a Plano-Concave Thick Lens

In this work, we present the exact equations for the back and effective focal lengths of a plano-concave thick lens. These equations are derived through a detailed ray-tracing approach, calculating the meridional ray’s path parallel to the optical axis while taking into account lens thickness, refractive index, and surface curvature. Our analysis bridges the gap between paraxial approximations and exact ray behavior, highlighting how traditional models (e.g., thin-lens equations) can be extended to thick lenses. Additionally, we review recent advancements in negative refraction and surface phase manipulation techniques to demonstrate how our equations compare to modern numerical methods and metacoating-enhanced designs. This work provides a comprehensive and exact framework for understanding the optical properties of plano-concave thick lenses, offering new insights into their application in precision optics.

Four-level diffractive photon sieves by deep-UV femtosecond laser ablation

A growing demand for complex light manipulation and miniaturization of optics necessitates advanced optical elements, operating on light diffraction phenomena, capable not only of reshaping the intensity distribution but also integrating many optical functions in a compact, durable device. The prevailing fabrication methods for these elements often involve multi-step lithographic processes. In contrast, direct laser ablation offers a single-step, cost-effective, and maskless alternative. However, using solid-state laser systems’ fundamental wavelength (in the IR range) for ablation lacks the precise depth control required for multi-level diffractive optical element fabrication. In this paper, we present the first experimental proof that femtosecond direct laser ablation in the UV spectral range is a reliable method for fabricating diffractive optical elements. We demonstrate the high-quality production of compact photon sieve focusing elements with the shortest focal length ever reported at 9 mm. Furthermore, we report an efficiency of 3.3%, which, to our knowledge, is the highest for elements with such a small focal length and not far from the theoretical efficiency limit of 4.46% (considering perfect cylindrical ablated pits). Moreover, our fabricated elements focus light to a 2.3% smaller focal spot if compared to the refractive lens with the same parameters. This fabrication method shows great promise for advanced applications that require precise depth control in wide band gap materials, such as the fused quartz used in this study.

Super-oscillation sub-diffraction focusing with emulated atmospheric turbulence

The super-oscillation (SO) phenomena successfully applied to super-resolution optical telescopes have yet to study the existence of atmospheric turbulence (AT). In this paper, we first experimentally investigate the sub-diffraction focusing of the SO light field in the atmospheric-like turbulence. The 137-element adaptive optics (AO) system is utilized to correct the dynamic wavefront aberration produced by the AT and obtain the AO closed-loop RMS of the residual wavefront error ∼λ/15. For a proofed telescope (clear aperture 12 mm and focal length 1000 mm) @λ=632.8 nm, the FWHM of the experimental SO spot under the modest AT (wavefront error RMS ∼0.57λ) is about 0.79 times of the Airy spot compared with the non-AT result of 0.76 times, while the strong AT (wavefront error RMS ∼1.35λ) erases the sub-diffraction focusing effect of the SO field and the average side-lobe intensity increases significantly. This study is promising for super-resolving telescope, synthetic aperture for visible imaging, etc.

Cinematic Gaussians: Real‐Time HDR Radiance Fields with Depth of Field

AbstractRadiance field methods represent the state of the art in reconstructing complex scenes from multi‐view photos. However, these reconstructions often suffer from one or both of the following limitations: First, they typically represent scenes in low dynamic range (LDR), which restricts their use to evenly lit environments and hinders immersive viewing experiences. Secondly, their reliance on a pinhole camera model, assuming all scene elements are in focus in the input images, presents practical challenges and complicates refocusing during novel‐view synthesis. Addressing these limitations, we present a lightweight method based on 3D Gaussian Splatting that utilizes multi‐view LDR images of a scene with varying exposure times, apertures, and focus distances as input to reconstruct a high‐dynamic‐range (HDR) radiance field. By incorporating analytical convolutions of Gaussians based on a thin‐lens camera model as well as a tonemapping module, our reconstructions enable the rendering of HDR content with flexible refocusing capabilities. We demonstrate that our combined treatment of HDR and depth of field facilitates real‐time cinematic rendering, outperforming the state of the art.

Relationship Between Aberration Coefficients of an Optical Device and Its Focusing Property

The best focus point of a focused Gaussian beam subject to a phase aberration is generally shifted with respect to the focal plane of the focusing lens. This focus shift is attributed to a lensing effect that belongs to the phase aberration, which mean focal length can be determined from the aberration coefficients determined in the framework of a Zernike polynomial decomposition. In this paper, we have checked the validity of this procedure, already available in literature, applied to three aberration types: a pure primary spherical aberration, the Kerr effect induced by a Gaussian beam, and an axicon illuminated by a Gaussian beam. Note that usually, the mean focal length of an aberrated lens is based on the relation between the effective radius of curvature of the wavefront before and after the lens. However, in this paper, the focal length associated with the phase aberration under study is defined from the point of the best focus, where the diffracted intensity on the axis is the maximum.

A High-Quality and Convenient Camera Calibration Method Using a Single Image

Existing camera calibration methods using a single image have exhibited some limitations. These limitations include relying on large datasets, using inconveniently prepared calibration objects instead of commonly used planar patterns such as checkerboards, and requiring further improvement in accuracy. To address these issues, a high-quality and convenient camera calibration method is proposed, which only requires a single image of the commonly used planar checkerboard pattern. In the proposed method, a nonlinear objective function is derived by leveraging the linear distribution characteristics exhibited among corners. An algorithm based on enumeration theory is designed to minimize this function. It calibrates the first two radial distortion coefficients and principal points. The focal length and extrinsic parameters are linearly calibrated from the constraints provided by the linear projection model and the unit orthogonality of the rotation matrix. Additionally, a guideline is explored through theoretical analysis and numerical simulation to ensure calibration quality. The quality of the proposed method is evaluated by both simulated and real experiments, demonstrating its comparability with the well-known multi-image-based method and its superiority over advanced single-image-based methods.

Propagation characteristics of a circular Airyprime Gaussian beam in a gradient refractive index medium

In this paper, the focusing characteristics of a circular Airyprime Gaussian beam (CAPGB) propagating in a gradient refractive index (GRIN) medium is studied for the first time, to the best of our knowledge, and some interesting features are observed. We find that the CAPGB exhibits periodic focus–defocus behavior and completes a period propagation process with two focal points within a half variation period L/2 of the GRIN medium. Meanwhile, the CAPGB has singularity at the positions of z=(2j+1)L/4 on the optical axis. The focal lengths of bifocal points, the distance between two focal points, the focal intensity, and the focusing ability can be manipulated by beam parameters and the GRIN factor. It is noteworthy that the number (one or two) of focal points in one focusing period, and the focusing period or frequency of the CAPGB in the GRIN medium could be controlled by the beam distribution factor and GRIN factor, respectively. Moreover, the focusing ability of the CAPGB is much higher than that of a circular Airy Gaussian beam in the GRIN medium.

Optimal Design of Small-Aperture Optical Terminals for Free-Space Links

We present the generalised design of low-complexity, small-aperture optical terminals intended for kilometre-scale, terrestrial, free-space laser links between fixed and dynamic targets. The design features single-mode fibre coupling of the free-space beam, assisted by a fast-steering, tip/tilt mirror that enables first-order turbulence suppression and fine target tracking. The total power throughput over the free-space link and the scintillation index in fibre are optimised. The optimal tip/tilt correction bandwidth and range, aperture size, and focal length for a given link are derived using analytical atmospheric turbulence modelling and numerical simulations.

Liquid crystalline geometrical phase Alvarez-Lohmann lenses

Thin-film LC based geo-phase optics have previously been demonstrated to drastically reduce size, weight, and power requirements for large-aperture optical systems while providing non-mechanical functionality for discrete values – for example, digital switching of the beam steering angle. In this work, we present a series of geo-phase Alvarez-Lohmann lens systems capable of analog tuning of focal length through lateral translation or azimuthal rotation. For circularly polarized inputs, diffraction efficiencies greater than 90% were observed over the visible spectrum with some wavelengths exceeding 99% for lateral embodiments. Both cylindrical and spherical Alvarez-Lohman systems were fabricated with focal lengths varying from infinity to 110 mm with increasing lateral displacement. These LC-based geo-phase optical systems could enable a new generation of low-cost, high-performance, and ruggedized dynamic optical components.

Desensitization design of freeform off-axis TMA optical systems with configuration selection

The off-axis three-mirror anastigmatic (TMA) optical system has the advantages of eliminating all primary aberrations, having no aperture obscuration, no chromatic aberration, and good environmental adaptability, making it essential in high-performance optical instruments. Freeform surfaces, with their rich degrees of freedom (DOFs) in mathematical representation, positively affect the image quality and sensitivity of optical systems. When applied correctly to off-axis TMA optical systems, they can achieve fast F-numbers, large fields of view (FOVs), and long focal lengths. There are dozens of configurations for off-axis TMA optical systems, and freeform surfaces can be combined in numerous ways within these systems. This paper first analyzes the sensitivity of off-axis TMA optical systems with different configurations, selects the configuration with the lowest sensitivity, and combines it with a low sensitivity freeform surface combination to achieve high-performance freeform off-axis TMA optical systems with low sensitivity.

System of N Thin Coaxial Lenses

In geometrical optics, in a system of two thin coaxial lenses, there are several standard formulas, including “𝟏𝑭=𝟏𝒇𝟏+𝟏𝒇𝟐−𝒅𝒇𝟏𝒇𝟐”. The purpose of this paper is to generalize these formulas to the case of a system of an arbitrary number of thin lenses. In particular, this paper proves that the focal length Fn of a system of n thin coaxial lenses is given by 𝟏𝑭𝒏=Σ{(−𝟏)𝒎Π[(Σ𝟏𝒇𝒓𝒔𝒂𝒔𝒓𝒔=𝒂𝒔−𝟏+𝟏 𝟎=𝒂𝟎<𝒂𝟏<⋯<𝑎𝒎<𝒂𝒎+𝟏=𝒏;𝒅𝒏=𝟏 )𝒅𝒂𝒔]𝒎+𝟏𝒔=𝟏}𝒏−𝟏𝒎=𝟎, where, fr is the focal length of the rth lens, and dr is the distance between the rth lens and (r+1)th lens. For a fixed value of m, all combinations of values of the a’s (satisfying the condition “0 = a0 < a1 < … < am < am+1 = n”) are taken in the inner sum.

Orthogonal dispersion modeling of double-stage VIPA spectrometer in SBS system

Brillouin microscopy enables the assessment of the mechanical properties of biological tissues by mapping the Brillouin shift in three-dimensional (3D), all-optical, label-free, non-contact, and subcellular resolution. The virtually imaged phased array (VIPA) etalon is widely utilized for measuring Brillouin spectra owing to its superior light throughput, large angular dispersion, and rapid signal acquisition capabilities. The VIPA-based spectrometer plays a significant role in Brillouin microscopy, but it is highly sensitive to factors such as the tilt angle, beam radius, lens focal length, and so on. Here, we propose an orthogonal dispersion model based on paraxial wave theory. This model is employed to explore the output intensity distribution and dispersion characteristics of a double-stage VIPA spectrometer. To validate the proposed model, we develop a stimulated Brillouin scattering (SBS) system by leveraging the optimal spectrometer parameters to measure the Brillouin frequency shift in pure water. We demonstrate the capabilities of this model in mitigating spectral dispersion and enhancing spectral measurement accuracy by optimizing the spectrometer's system parameters. This work is pivotal for the future deployment of the double-stage VIPA spectrometer in SBS-based microscopy applications.

Low-cost, high-volume, manufacturable 0.88 NA multi-wavelength diffractive lens array for optical document security

We design, manufacture, and characterize a high-numerical-aperture (NA=0.88, f/0.27), multi-wavelength (480 nm, 550 nm, and 650 nm) multilevel diffractive microlens array (MLA). This MLA achieves multi-wavelength focusing with a depth of focus (DoF) twice the diffraction-limited value. Each microlens in the array is closely packed with a diameter of 70 µm and a focal length of 19 µm in air. The MLA is patterned on one surface of a polymer film via UV casting, positioning the focal plane at the distal end of the polymer film. Each microlens focuses light at three design wavelengths into a focal spot with an estimated FWHM of ∼310nm. By placing this MLA directly on a standard high-resolution banknote print (minimum feature width of 10–15 µm), we demonstrate color-integral imaging for anti-counterfeiting. In contrast, refractive MLAs cannot achieve high-NA, multi-wavelength focusing or extended DoF. The extended DoF of our MLA ensures reliable performance despite variations in the polymer film’s thickness. Our MLA, produced via UV casting, enables extremely low-cost, high-volume production, making it ideal for flat optics in banknotes and document security.

Experimental and numerical study of dissimilar fiber laser welding of martensitic AISI 1060 carbon steel with different configuration with austenitic 304 and ferritic 420 stainless steel

Welding with fiber laser for two distinct stainless steel types was performed to join with the martensitic carbon steel (AISI 1060) in order to assess the effect of laser welding operating parameters for two dissimilar materials on the weld characterization thorough experimental design method and numerical investigation. A full central composite design (CCD) matrix in accordance with the response surface methodology (RSM) was developed to represent the responses variations by equations including quadratic and nonlinear interaction terms. Different laser welding parameters were taken into account, such as laser power, linear speed of welding, focal distance of the beam, and beam deviation. The responses considered were the temperature field next to the melt pool border line, the maximum penetration depth of the fusion zone and the ultimate tensile strength of the dissimilar weld joint. Additionally, the variation of the microstructure fusion zone and adjacent areas and failure mechanism of the weld joint were evaluated. The experimental results indicate that the temperature field measured at the vicinity of the melt pool fusion line with both 304 and 420 stainless steel which primarily influenced by the incident power of the laser and linear velocity of beam, respectively. Additionally, the numerical simulation results are in good agreement with temperature experimental results at vicinity of fusion line where the temperature measured, experimentally. Apart from this, at the center of the fusion zone, the temperature clearly predicts via numerical simulation results which is not possible to assess experimentally. A clear comparison between the temperature distribution with different joint configuration illustrates that distinct relation between the temperature variation rate and different thermal conductivity coefficient of AISI 1060 with different AISI 304 and 420 joint configuration resulted different temperature distribution. The weld joint microstructure for 420 stainless steel–AISI 1060 steel joint at fusion boundary zone consists of columnar dendrites and skeletal columnar ferrite. At the center of fusion zone, a cellular-type structure is seen. For 304 stainless steel joint, the fusion zone has composed of a remarkable part of skeletal delta-ferrite and dendritic microstructure at austenite matrix microstructure. The fracture section of 304 steel has a ductile fracture and the depth of fracture dimples and cavities toward 304 stainless steel are higher than AISI 1060 steel.

Development of laser felling modes of gas-thermal coating

The use of copper and its alloys to create parts for metallurgical equipment is associated with an increase in abrasive wear and high-temperature corrosion. In this regard, there is a need to apply a protective coating. In particular, to prevent wear and premature chipping of the metal of copper tuyeres, the surface is hardened with a coating of zirconium dioxide stabilized with yttria oxide by thermal spraying in an air atmosphere. Due to the difference in the coefficient of thermal expansion of copper (at T = 300 K: 16.7 µm/m оС and at T = 750 K: 19.7 µm/m оС) and its low resistance to gas corrosion, the application of zirconium oxide (produced by a preapplied intermediate layer that plays a role in matching the coefficient of thermal expansion (CTE) between the copper base and the ceramic coating. In addition, the intermediate layer protects copper from gas corrosion. In this case, The use of copper and its alloys to create parts for metallurgical equipment is associated with an increase in abrasive wear and high-temperature corrosion. In this regard, there is a need to apply a protective coating. In particular, to prevent wear and premature chipping of the metal of copper tuyeres, the surface is hardened with a coating of zirconium dioxide stabilized with yttria oxide by thermal spraying in an air atmosphere. Due to the difference in the coefficient of thermal expansion of copper (at T = 300 K: 16.7 pm/m °G and at T = 750 K: 19.7 pm/m оG) and its low resistance to gas corrosion, the application of zirconium oxide (produced by a pre-applied intermediate layer that plays a role in matching the coefficient of thermal expansion (CTE) between the copper base and the ceramic coating. In addition, the intermediate layer protects copper from gas corrosion. In this case, nickel-based alloys were used as intermediate layers. The use of nickel as the basis of intermediate layers is due to the fact that copper and nickel form a continuous series of solid solutions, such as cupronickel or monel metal-like structures. This, in turn, assumes a smooth transition of thermophysical properties from copper to nickel alloy. To ensure increased adhesion of the transition layer to copper by increasing the area of mutual contact between copper and the sublayer (dagger penetration) and significantly increasing the homogeneity of the material of the intermediate layer made of a nickel alloy, laser melting of the intermediate sublayer (Ni–B–Si system) was used on a laser complex based on laser LS-5 with a power of 5 kW with a KUKA KR-60HA robot in an argon atmosphere. To test the modes, experiments were carried out on copper samples of a flat shape and a body of rotation. The optimal parameters for the process of melting flat samples were: processing speed 33 mm/s, power from 400 to 3900 W, focal length from 200 to 230 mm, pitch between tracks: 0.25, 0.5 and 1 mm. The optimal parameters for the process of melting rotating samples were: laser radiation power 400–450 W, processing step 0.125; 0.5, focal length from 200 to 210 mm.

Polarization-multiplexed zoom Moiré metalens for edge-enhanced imaging

Optical image processing with high operational efficiency has been applied as a pre-processing imaging system for image recognition. Edge-enhanced imaging as a high-efficiency optical image processing method is of great significance for feature extraction and target recognition. However, the edge-enhanced imaging system based on the 4F system and the spatial filter transforms mainly work under coherent light illumination conditions, without continuously zooming to track the spatial position of the target. Here, we demonstrate a polarization-multiplexed zoom Moiré metalens for edge-enhanced imaging under incoherent light illumination. Metalens is designed to generate polarization-dependent optical transfer functions that produce edge-enhanced images with a resolution of 1.2 µm by digital subtraction. Furthermore, continuous zoom at the range of 1-2× is realized by constructing a Moiré metalens composed of cascaded metasurfaces. The cascaded metasurfaces consist of two center-aligned dielectric metasurfaces, each with a Moiré phase sensitive to the rotation angle. By rotating the metasurface, the phase profile of the cascaded metasurfaces changes, and the effect of continuous zoom is realized. The focal length can be actively changed from 38 µm to 77 µm with the focusing efficiency of 50.3%. This metalens can be applied to machine vision, microscopic imaging, and promotes the development of multi-functional integrated optical systems.

Intraocular lens simulator using computational holographic display for cataract patients.

To develop and validate a holography based vision simulator for the demonstration of expected postoperative vision corresponding to monofocal and multifocal intraocular lenses (IOL) to cataract patients before surgery. An artificial eye model is used to measure the optical performance of different IOL types. The resultant aberrations and degradations are then modeled using phase holograms and shown to subjects on a holographic display. We measure the contrast and resolution loss, halos around the light sources, and point spread function (PSF) corresponding to three different IOLs. We tested the holography based vision simulator on 13 healthy subjects and 6 cataract patients. Monofocal, bifocal, and trifocal IOLs exhibited a contrast decrease of 5%, 42%, and 45% and a resolution limit of 4.49, 4.00, and 4.00 lp/mm (using 0.05 MTF criteria), respectively. Monofocal IOLs have the best resolution and contrast at the optimal focus distance, and multifocal lenses offer extended depth-of-field but exhibit prominent halos and reduced contrast/resolution. We confirmed that the visual functions of IOLs could be successfully modeled using phase holograms and simulated using a holographic display without using a physical IOL. Patients can experience the effects of different IOL options prior to surgery, which helps with IOL selection, expectation management, and patient satisfaction.

High beam quality and high peak power 1.6 µm deuterium Raman laser.

In this work, a 1.67 µm laser was generated in the pressurized deuterium pumped by a pulsed 1064 nm laser via the stimulated vibration + rotation Raman scattering (SV-RRS). By the optimization of the deuterium pressure, focal length, and polarization of the pump laser, a 66.2 mJ Raman laser was achieved with the configuration of 1.7 MPa deuterium pressure, 2245 mm focal length, and 220 mJ circularly polarized pump laser. Correspondingly, the photon conversion efficiency was 47.8%, the energy conversion efficiency was 30.5%, the pulse width was 5.4 ns, and the maximum peak power was 12.3 MW. In addition, SV-RRS was found to improve beam quality significantly: from Mx 2 = 2.67 and My 2 = 3.07 for the pump laser to Mx 2 = 1.42 and My 2 = 1.53 for the 1.67 µm Raman laser.