Reflective dual field-of-view optical system based on the Alvarez principle.

A novel, to the best of our knowledge, dual-state reflective optical relay system based on the Alvarez system is proposed, which can be used for remote sensing applications. By keeping the image and pupil positions constant, it can be combined with a telescope to achieve two different magnifications. As a compact structure with only two moving parts, freeform optical mirrors and a nearly diffraction limited performance for the infrared wavelength 8 µm make it an attractive subsystem for space applications. Different design tradeoffs and the preferred layout properties are discussed in detail.

Extended aberration analysis in symmetry-free optical systems - part I: method of calculation.

The imaging performance evaluation of symmetry-free optical systems is of great importance during the correction process of optical design. However, due to the complexity and limitations of the available tools, the higher-order aberrations in the system cannot be well analyzed and are hard to control. In this paper, the theoretical background and the mathematical approach of a quantitative analysis method for surface-decomposed transverse aberration in the symmetry-free systems are introduced. With the mixed ray-tracing calculation in both real and paraxial cases, the implementations of full-order intrinsic/induced aberration, as well as the surface-additive Zernike coefficient fitting method are demonstrated. The applications of this method help assess the correction performance considering the relatively critical surfaces in an arbitrary off-axis system. The reliability and the accuracy of the method will be evaluated in part II with a test system. And as an illustration of the practical usage of the method for optical design, the corresponding applications on a group of lithographic systems will also be demonstrated.

Wigner function-based modeling and propagation of partially coherent light in optical systems with scattering surfaces.

Light scattering from residual manufacturing errors of optical surfaces has a large impact on the image quality of optical systems. Classical ray-based methods to simulate surface scattering in optical systems depend on statistical models of surface errors and neglect the wave properties of light, which prohibit the integration of statistical surface error models with beam propagation methods. Additionally, the impact of multiple scattering from different frequency components of surface errors cannot be easily modelled by existing methods. Here we analyze the impact of different frequency components of surface errors induced by diamond-turned surface grinding on image quality, and we propose a Wigner function-based approach in which light is modelled as partially coherent. In this unified model, by selecting the proper definition of light coherence, we can combine the statistical and deterministic models of surface errors, enabling efficient, simultaneous simulation of multiple scattering from high- and mid-spatial frequency (HSF and MSF, respectively) surface errors, as well as the interference and edge diffraction of light.

Improved correction by freeform surfaces in prism spectrometer concepts.

The correction of spatial resolution and distortion in imaging spectrometer systems is of great importance due to their significant impact on efficiency and quality. In this study, we analyze the corrective power of freeforms added at different positions in various spectrometer systems for high-performance requirements. The results show that the combination of a freeform prism and a second freeform close to the image has the best correction of distortion while preserving spot size.

Generalized chromatic aberrations in non-rotationally symmetric optical systems-Part I: mathematical approach.

In non-rotationally symmetric optical systems, chromatic aberrations must be defined in a generalized way with reference to the optical axis ray and optimal tilted parabasal image plane of the central wavelength. In this paper, the definition of generalized chromatic aberrations is clarified. The mathematical calculation in optical systems with arbitrary symmetry and surface shape is realized by ray- and wavefront-based methods, respectively, that are originally identical. The additivity of wavefront after each surface ensures the surface decomposition of chromatic aberrations. In the end, the influence of pupil aberration, which is of a higher order, is discussed. The consistency of our methods with Seidel aberrations in the lowest aberration order of the rotationally symmetric system and the application of our methods in diverse non-rotationally symmetric refractive systems will be addressed in detail in Part II.

Generalized chromatic aberrations in non-rotationally symmetric optical systems-Part II: applications.

In this paper, as a supplement to Part I, three examples will be provided to verify our methods. In the first example of a double Gauss system, the chromatic aberrations calculated by our methods and those given by the Seidel aberrations are compared. Then we will present two examples analyzing the generalized chromatic aberrations in refractive system with freeform surfaces by our methods. The stereomicroscope is utilized to demonstrate the feasibility and consistency of both ray- and wavefront-based methods. The anamorphic imaging objective lens proves that it is necessary to consider the pupil aberration when it cannot be neglected. In both examples, the surface decomposed a full-field display of the chromatic aberrations shows us that it is a convenient and powerful tool to analyze the generalized chromatic aberrations in nonrotationally symmetric refractive optical systems.

Flexible, fast, and benchmarked vectorial model for focused laser beams.

In-bulk processing of materials by laser radiation has largely evolved over the last decades and still opens up new scientific and industrial potentials. The development of any in-bulk processing application relies on the knowledge of laser propagation and especially the volumetric field distribution near the focus. Many commercial programs can simulate this, but, to adapt them, or to develop new methods, one usually must create a specific software. Besides, most of the time people also need to measure the actual field distribution near the focus to evaluate their assumptions in the simulation. To easily get access to this knowledge, we present our high-precision field distribution measuring method and release our in-house software InFocus [https://github.com/QF06/InFocus], under the Creative Commons 4.0 license. Our measurements provide 300 nm longitudinal resolution and diffraction limited lateral resolution. The in-house software allows fast vectorial analysis of the focused volumetric field distribution in bulk. Simulations of the linear propagation of light under different conditions (focusing optics, wavelength, spatial shape, and propagation medium) are in excellent agreement with propagation imaging experiments. The aberrations provoked by the refractive index mismatch as well as those induced by the focusing optics are both taken into account. The results indicate that our proposed model is suitable for the precise evaluation of energy deposition.

Efficient simulation of surface scattering in symmetry-free optical systems.

Simulation of scattering from optical surfaces is usually based on Monte Carlo methods in which the bidirectional scattering distribution function (BSDF) of the optical surfaces are sampled randomly by many rays, resulting in long calculation times. In order to accelerate the simulation, a quasi-analytical phase space model is proposed. In this model, few rays are traced from the object and image space to the target surface to determine the illumination and acceptance areas in phase space, where these areas can be conveniently coupled simultaneously in the spatial and angular domain. Since no random sampling is involved in the phase space model, no statistical noise perturbs the result and the surface scattering simulation can be greatly accelerated. Additionally, due to the use of real raytracing, the phase space model removes the limitation of paraxial approximation, which usually limits the accuracy of deterministic stray light analysis models. Meanwhile, by the discretization of the optical surfaces into subareas, this new approach is able to model freeform surfaces with arbitrary geometries, and space-variant BRDFs can be applied for different subareas of the optical surface.

Correction of 2D-telecentric scan systems with freeform surfaces.

For scanning systems the resolution, distortion as well as the telecentricity are important performance criteria. For two-dimensional scanning systems, scan mirrors deflecting in only one transverse direction are not allowing for telecentricity in x and y simultaneously in case of an axisymmetric system. It is possible to achieve two-dimensional telecentricity by splitting the pupils in x- and y-direction and shifting the principal planes in one dimension by changing the focal power using an anamorphic setup. However, for higher specifications concerning a large aperture and wide scanning angle, using cylindrical lenses are not enough to achieve a good system quality. It has been proved in many researches that freeform surfaces are effective to improve the resolution of systems without rotational symmetry. In this work, a systematic case study is presented to investigate the potential of freeform surfaces to improve the resolution, telecentricity, and distortion simultaneously. It is shown as a result that freeform surfaces offer large correction ability in all the three aspects concerning high specifications of 2D-telecentric anamorphic scan systems. This contribution provides the insight into the application of freeform surfaces in non-rotationally symmetric optical systems with refractive components.

Beam characterization by phase retrieval solving the transport-of-intensity-equation.

A new method for the characterization of coherent laser beams is proposed. It is based on the non-iterative solution of the transport-of-intensity-equation. The phase to recover is decomposed into paraxial properties of laser beams and a set of lateral shifted radial basis functions, which allows for the derivation of a direct solution of the phase by a least-squares fit without the need of an initial guess. The method is tested with synthetic data to deduce an accuracy metric. Additionally, two real laser beams are characterized. Including the real light source in terms of the reconstructed field allows for a more holistic simulation of optical systems.

Cascaded diffraction in optical systems. Part I: simulation model.

In every real optical system, the light beam or ray bundle is laterally limited by diaphragms or lens boundaries. Therefore, from a rigorous point of view, a diffraction calculation of the point spread function by assuming a truncation at the exit pupil is a simplification. If several boundaries at different $z$z positions inside a system truncate the rays, cascaded diffraction occurs and the field in the exit pupil is modified in amplitude and phase by the edge diffraction effects at all relevant boundaries. A simple model for a fast estimation of these effects and a rule of thumb for real optical systems are worked out in this work.

Cascaded diffraction in optical systems. Part II: example calculations.

Real imaging optical systems are in any case finite in diameter and therefore limit the ray cones coming from the various object points. The truncation of the ray cones generates edge diffraction effects, which have an impact on the point spread function and therefore on the resolution of the system. Traditionally real calculations of the diffraction effects are mostly simplified by assuming a truncation in the exit pupil only. In this hybrid approach, the light propagation therefore can be calculated by raytrace first, and the more cumbersome diffraction calculated is only performed between exit pupil and image plane. To come to a better quantitative understanding of the validity of this approximation, a more refined model is developed in an adjacent publication. Some example calculations for real setups and an overview about the order of magnitude of the cascaded diffraction effects are worked out in this contribution.

Application of Gaussian pulsed beam decomposition in modeling optical systems with diffraction grating.

A diffraction grating is one of the most commonly used components in ultrafast optical systems such as pulse stretchers and compressors. Hence, modeling the temporal dispersion and spatiotemporal distortions associated with the angular dispersion of a diffraction grating is very crucial for wave optical modeling of such systems. In this paper, the Gaussian pulsed beam decomposition (GPBD) method is extended to handle the propagation of ultrashort pulses, with arbitrary spatial and spectral profiles, through complex ultrashort pulse shaping systems containing diffraction gratings. Although the diffraction efficiencies are not rigorously computed, the GPBD method enables modeling of the large angular dispersion of idealized diffraction gratings without running into an impractically large number of spectral samples as in the case of Fourier-transform-based methods. The application of the extended method is demonstrated by performing the wave optical propagation of an ultrashort pulse through a single diffraction grating and then through a Treacy compressor system. By combining the Treacy compressor with the Martinez grating pair stretcher with internal lenses, the pulse shaping through a complete chirped pulse amplification (CPA) setup is modeled. Finally, the effects of using real dispersive lenses in the Martinez stretcher on the output pulse of the CPA setup are presented. For analysis of the output pulses, methods of computing the spatiotemporal and spatio-spectral amplitudes of the output pulse from the phase correct superposition of individual Gaussian pulsed beams are presented.

Gaussian pulsed beam decomposition for propagation of ultrashort pulses through optical systems.

Many applications of ultrashort laser pulses require manipulation and control of the pulse parameters by propagating them through different optical components before the target. This requires methods of simulating the pulse propagation taking into account all effects of dispersion, diffraction, and system aberrations. In this paper, we propose a method of propagating ultrashort pulses through a real optical system by using the Gaussian pulsed beam decomposition. An input pulse with arbitrary spatial and temporal (spectral) profiles is decomposed into a set of elementary Gaussian pulsed beams in the spatiospectral domain. The final scalar electric field of the ultrashort pulse after propagation is then obtained by performing the phase correct superposition of the electric fields all-Gaussian pulsed beams, which are propagated independently through the optical system. We demonstrate the application of the method by propagating an ultrashort pulse through a focusing aspherical lens with large chromatic aberration and a Bessel-X pulse generating axicon lens.

Spatially truncated Gaussian pulsed beam and its application in modeling diffraction of ultrashort pulses from hard apertures.

A new kind of pulsed beam, which we call a spatially truncated Gaussian pulsed beam, is defined to represent a Gaussian pulsed beam that is diffracted from a semi-infinite hard aperture. The analytical equations for the propagation of the spatially truncated Gaussian pulsed beam through a nonrotationally symmetric paraxial system with second-order dispersion is derived starting from the generalized spatiotemporal Huygens integral. The spatially truncated Gaussian pulsed beam is then combined with the conventional Gaussian pulsed beam decomposition method to enable the modeling of diffraction of a general ultrashort pulse from an arbitrarily shaped hard aperture. The accuracy of the analytical propagation equation derived for the propagation of the truncated Gaussian pulsed beam is evaluated by a numerical comparison with diffraction results obtained using the conventional pulse propagation method based on the Fourier transform algorithm. The application of the modified Gaussian pulsed beam decomposition method is demonstrated by propagating an ultrashort pulse after a circular aperture through a dispersive medium and a focusing aspherical lens with large chromatic aberration.

Propagation of truncated Gaussian beams and their application in modeling sharp-edge diffraction.

The two new kinds of truncated Gaussian beams, known as the half and quarter Gaussian beams, are defined as the product of the fundamental Gaussian beam with the Heaviside unit step function. Using the generalized Collins integral, the exact analytical propagation formulas are derived for the truncated Gaussian beams through paraxial optical systems. Combined with the Gaussian beam decomposition method, the truncated Gaussian beams are used to represent the sharp edges of a field after a hard aperture. The modified Gaussian beam decomposition method presented in this work enables the calculation of the diffraction of a given field from a hard aperture with an arbitrary shape. This solves one of the limitations of the conventional Gaussian beam decomposition method, which is its inability to accurately model the diffraction of fields with sharp edges, especially in the near field. The validity and accuracy of the proposed method are demonstrated using a few exemplary diffraction calculations.

Fast computation and characterization of perturbed Bessel-Gauss beams.

In this paper, an extended definition of the Strehl ratio is introduced for Bessel-Gauss (BGB) beams, which, in contrast to classical beams, are not defined at a single point but along the optical axis. For that purpose, the Fresnel diffraction integral is solved analytically to calculate the on-axis field of a BGB that is perturbed by spherical aberration, astigmatism, and coma. Certain approximations are introduced, discussed, and tested against a rigorous propagation method. The derived expression is applied to tolerancing of an example system.

Compact freeform illumination system design for pattern generation with extended light sources.

One of the major problems in freeform illumination design in a geometrical optics approximation is picture generation with extended light sources. In contrast to the freeform design with zero-étendue sources, the extension of the light source leads to the typical blurring effect and a contrast reduction of the required irradiance. This effect can be minimized by increasing the distance between the freeform surface and the light source, which according to étendue conservation, results in an impractically large projection optic. To tackle this problem, we propose a design concept consisting of the combination of a pattern-generating double freeform surface for collimated beam shaping, which is calculated for a zero-étendue light source, and an imaging projection system with a telecentric object space. The design concept works independently of the shape of the emission area of the light source and does not require a representation of the extended light source by several individual wavefronts. By interpreting the pattern blurring effect as a composition of a shift contribution and a distortion contribution, we show that both can be minimized simultaneously by an appropriate placement of the object plane of the imaging optics and by making the distance between both freeform surfaces as small as possible. This allows the calculation of compact, energy-efficient freeform illumination systems for picture generation with real extended light sources. We demonstrate the significant blurring reduction by designing a simple illumination system consisting of a collimation optic, a (zero-étendue) double freeform lens for collimated beam shaping, and a projection lens for the generation of the target distribution "Elaine" with an extended Lambertian emitter of 3 mm×3 mm extension and ±42 deg maximum opening angle. For a working distance to the projection system of 500 mm and a target area of 300 mm×300 mm, a relative blurring extension of 2% is estimated, compared to 23% for a single freeform projector with the same energy throughput and similar lateral extension. The influence of the doublefreeform thickness on the blurring reduction is demonstrated, and a summary of the design procedure for the developed design concept is given.

Ant colony optimization in lens design.

As the most widely used optimization algorithm in optical design, the damped least square (DLS) method is advantageous for its fast convergence and deterministic optimization path. However, results are strongly dependent on the initial system and problematic when it is stuck in a local minimum in the searching space. To overcome these disadvantages, a biology intelligence-based algorithm, ant colony optimization (ACO) is implemented and tested for optical design tasks. Additionally, the ACO method has a better performance considering glass optimization. Two ACO versions, which are proved applicable, are introduced and their feasibility is assessed based on some case studies.

General analysis and optimization strategy to suppress autofluorescence in microscope lenses.

With the development of fluorescence microcopy, the autofluorescence effect of optical glass in microscope lenses has become an important source of stray light which decreases the contrast of the image. However, the autofluorescence effect of various types of microscope lenses has not been thoroughly discussed and the dominant factors of this effect are not clear. Currently, the most commonly used analysis method of the autofluorescence effect is based on the volume scattering model and Monte Carlo raytracing, which is extremely time-consuming due to the large number of rays that need to be traced. In our previous work, we have presented an efficient phase-space-based simulation method, which significantly accelerates the simulation of the autofluorescence effect [Appl. Opt.58, 3589 (2019)APOPAI0003-693510.1364/AO.58.003589]. Here we apply this new method on different types of microscope lenses and perform a systematic analysis of the autofluorescence effect based on the simulation results. In order to obtain an overview of the autofluorescence effect of different types of microscope lenses, more than 100 microscope lenses with different numerical aperture (NA), magnification, working distance, and immersion medium are selected and the corresponding autofluorescence contribution from each element in the lenses is calculated. Following a systematic analysis of the simulation result, we find that the autofluorescence effect of a lens is dependent on the etendue and complexity of the system, while the most critical elements are usually found in the front and rear groups. After the origins of the autofluorescence contribution from these critical lens groups have been found, possibilities to reduce the autofluorescence intensity have been investigated. Finally, effective methods to reduce the autofluorescence effect are presented and compared. The major contribution of this work is a detailed analysis of the impact of different factors on the autofluorescence effect, based on which a guideline is provided for the optical designers as well as the users to design or to select an appropriate microscope lens with low autofluorescence intensity.