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We introduce a kind of radially polarized partially coherent (RPPC) beam with a prescribed non-uniform correlation function, called a radially polarized cosine non-uniformly correlated (RPCNUC) beam. Based on the extended Huygens–Fresnel principle, we study the propagation properties in free space and in a turbulent atmosphere. Unlike RPPC beams with uniform coherence, RPCNUC beams possess the invariance of dark hollow cores and radial polarization, and exhibit self-focusing properties. In a turbulent atmosphere, the intensity distribution demonstrates self-healing properties over a certain propagation distance. We also investigate how to adjust the beam parameters to reduce the turbulence-induced degradation in detail.

Lithography-free layered dielectric media plays an important role in photonic platforms and opens up new possibilities in the design of devices with large-scale compatibilities. In this work, we propose a hybrid structure consisting of phononic thin layer and Weyl semimetallic photonic crystals. Manipulation of electromagnetic waves and the interactions between different optical modes are presented, which is highly demanded for both fundamental research and practical applications. The weak and strong coupling effects result in hybrid polariton modes, of which the dispersion can be quantitatively described by a coupled harmonic oscillator model. In the Kretschmann coupling proposal, when the incident angle is smaller than the critical angle (θ c = 25∘) of total internal reflection, a weak coupling is generated. However, when the incident angle is larger than θ c , anticrossing resluts from strong coupling can be observed. We believe that the use of easily manufactured WSM elements and the tunable hybridization of multiple optical modes will enable the manipulation of light-matter interactions with more flexibility in the mid-infrared range, and can significantly improve the functional properties of various devices.

We introduce a partially coherent beam, called a multi-hyperbolic sine-correlated (MHSC) beam, by employing a multi-hyperbolic sine function to modulate the spectral degree of coherence. Based on the extended Huygens–Fresnel principle and second-order moments of the Wigner distribution function, we derive the analytical expressions for the spectral intensity, the root-mean-square (rms) angular width and the M 2 factor in turbulent atmosphere. Numerical results show that the intensity profile, which keeps the dark-hollow invariant in free space, will be gradually destroyed by the turbulence along the propagation distance. We believe that the MHSC beams have significant advantage over the hyperbolic sine-correlated beams for reducing the turbulence-induced degradation, especially for the MHSC beams with a higher beam order N . The effects of the beams parameters and the turbulent atmosphere on the beam quality are analyzed in detail.

With the development of modern precision optical fabrication and measurement technologies, optical freeform surfaces have been widely employed in different applications, especially in off-axis refl...

We introduce a class of planar random source generating ring-shaped array profiles in the far field. The model for spatial correlation function of this new source is composed by the Fourier transform of the Laguerre-Gaussian array function with circular symmetry. The analytical expressions for the spectral density (average intensity) in free space and atmospheric turbulence are derived. The effect of the source parameters on the propagation properties are investigated in detail. Numerical results show that the source can produce a ring-shaped optical array profile consisting of dark-hollow-shaped or Gaussian-like-shaped beamlets at some distance, and then keep shape-invariant for long distances in free space. The number of rings, the position of beamlets, and other features can be conveniently controlled by changing the initial source parameters. In addition, increasing the relative separation distance of each beamlet or decreasing the radius of beamlets can reduce the turbulence-induced degradation.

By utilizing the extended Huygens-Fresnel principle, we derive the analytical formulas for the cross-spectral density matrix elements of a radially polarized multi-Gaussian Schell-model (RPMGSM) beam propagating in oceanic turbulence. Effects of beam parameters and oceanic turbulence parameters on the propagation properties of RPMGSM beams are investigated in detail by numerical simulation. Our results show that the RPMGSM beam with larger beam order M has an advantage over the radially polarized Gaussian Schell-model beam for reducing turbulence-induced degradation. Compared with temperature-induced turbulence fluctuation, the salinity-induced turbulence fluctuation makes a greater contribution to the influence on propagation properties of RPMGSM beams.

Head-mounted display (HMD) system is a research hotspot in virtual reality, which has continuous and rapid development in training, medical treatment, aerospace, and entertainment. For the more comfortable immersive experience, the requirements of compact structure, large exit pupil and low F number should be met simultaneously in HMD system. With the use of x-y polynomial surface, an off-axis head-mounted display system with two freeform reflectors is achieved. The HMD system demonstrates a 10 mm exit pupil, a 23°×16° field of view, an f/# number of 3.0 and an eye relief larger than 15 mm. Based on the tolerance analysis of surface figure errors and position location, we present the on-machine measurement and compensation for enhancing machining accuracy, and the frame and mirror integration for high assembling accuracy. Finally, the HMD system was fabricated and the imaging quality was evaluated. In comparison with the results, the effectiveness of the on-machine measurement and compensation are validated, as well as the reliability of tolerance analysis and mechanical structure.

In this paper, a direct design method for an off-axis two-mirror telecentric scanning system with a linear field of view (FOV) is proposed. A single freeform mirror structure is firstly considered, in which the aberration free geometry of the off-axis parabolic (OAP) surface is leveraged to provide the focusing function and build surface contour of the sub-region on the mirror for each FOV. Multiple OAP surfaces for construction of the freeform mirror are located at an OAP base to satisfy the telecentric condition. The imaging distortion of this single freeform mirror structure is analyzed and found unavoidable due to the unsymmetrical geometry of the OAP base. A freeform reflective corrector is supplemented, and it is constructed from multiple plane surfaces located at a curved base to fulfill the f-theta scanning geometry. Thus, a two-mirror structure composed of one freeform primary mirror and one freeform reflective corrector is established. Each plane-OAP surfaces pair corresponds to a specific FOV. These multiple OAP surfaces and multiple plane surfaces are then expanded and mixed respectively, to construct the freeform primary mirror and freeform reflective corrector. An f-theta two-mirror freeform scanning system with ± 10.4° linear FOV is designed using the proposed construction method. The design result is diffraction-limited, and a scanning error less than 5 μm and telecentricity angle less than 0.2° are achieved.

A method for measuring the radius of curvature with a pinhole point diffraction interferometer (PDI) is proposed. Using the wavefront difference method and the Gaussian imaging equation, the longitudinal displacement of the converging rays passing through a standard plane-parallel-plate sample in PDI interference cavity is determined. Based on this longitudinal displacement, a precise formula for radius of curvature calculation is deduced. An experimental system for radius of curvature measurements is set up to verify the principle. With this testing system, the radius of curvature of spherical mirrors and the surface figure can be measured in a higher precision simultaneously. Some sources of uncertainty in measurement are discussed based on detailed error analysis. The experimental results indicate that the measurement accuracy Δ R / R 0 is in the order of 10−4.

We present a compact dual-view endoscope objective lens with a field of view (FOV) of ± 80°and F/# of 3.4. The endoscope consists of two optical configurations for increasing FOV within the volume constraint. The front view configuration is a fisheye lens with a FOV of ± 55°, and the side view configuration is a panoramic annular lens that covers the remaining FOV. The two configurations are combined by a hybrid lens that consists of center refractive portion and side catadioptric portion. Both the front and rear surfaces of the hybrid lens are aspherized with the use of annularly stitched Q-type aspheres. Thus, a compact endoscope is successfully implemented with fewer lenses, with a total length of 11.5 mm and a maximum diameter of 5.5 mm. The modulation transfer function at 167 lp/mm is above 0.4 over the entire FOV. The relative illumination is more than 0.65 and the optical distortion is within 10%. Moreover, the near telecentric condition is fulfilled and supports constant magnification focusing.

The classic Czerny-Turner spectrometer consists of a plane grating and two spherical mirrors. The optical path geometry adopted for incident and grating dispersed light is off-axis reflection, so the spherical collimating and focusing mirrors introduce coma and astigmatism. The conventional configuration is asymmetrical for coma automatic compensation, but suffers from astigmatism. We substitute the off-axis parabolic (OAP) surfaces for spherical surfaces of the collimating mirror and each sub-region of the focusing mirror, to achieve an aberration free configuration. The multiple OAP surfaces are then expanded and mixed, to construct a freeform surface integrating the collimating and focusing mirrors into a single element. Results show that a 0.1 nm spectral resolution is achieved over a bandwidth of 400 nm centered at 800 nm, in the designed spectrometer comprised of a plane grating and one freeform mirror. The construction method is advantageous to integrated optic design, and the resulting freeform mirror spectrometer is compact, and simplifies manufacture and alignment.

The precision single point diamond turning technique has been a promising technology for generating small and medium-sized freeform optical elements with high surface quality. In this paper, we present an extremely off-axis freeform optical system with a large 10.0 mm pupil diameter and a low 3.0 F-number over a wide 28° field of view. It is composed of two freeform aluminum mirrors, which are fabricated efficiently by the single point diamond turning machine. The manufacturing strategy and parameters are estimated rationally and comprehensively, based on the freeform surface characters. The freeform aluminum mirror surface can reach submicron surface accuracy and achieve nanometer surface roughness. The final assembled prototype of the off-axis two-mirror freeform display optical system has the advantages of compactness, a broad spectrum, and good display imaging performance.

A white-light microscopy interferometer was developed for measurement of the 3D profile and roughness.10X, 20X and 50X Mirau interference microscope objectives with the numerical aperture of 0.3, 0.4 and 0.55 were designed, manufactured and then provided as the accessories. Thickness deviation between beam splitter plate and reference mirror plate as well as the numerical aperture will both affect the contrast of interference fringe, according to optical modeling and image evaluation. The former would generate dispersion and then decrease the fringe contrast, while the latter would not produce dispersion separately but impact the amount of dispersion when thickness deviation exists, and their influence on fringe contrast was based on the expression of white-light interference intensity. Simulations for interference fringes from Mirau interference microscope objectives with different NA and thickness deviation were implemented, demonstrated that the fringe contrast will be falling with NA and thickness deviation increasing. A standard step with the nominal step value of 110 nm was used to calibrate the white-light microscopy interferometer, showing that less than1nm deviation can be reached.

We address here the problem of the number of freeform surfaces and the degrees-of-freedom they provide in optics design. We analyze the degrees-of-freedom lost when prescribing a mapping between object-and-image or source-and-receiver.

Modern advanced manufacturing and testing technologies allow the application of freeform optical elements. Compared with traditional spherical surfaces, an optical freeform surface has more degrees of freedom in optical design and provides substantially improved imaging performance. In freeform optics, the representation technique of a freeform surface has been a fundamental and key research topic in recent years. Moreover, it has a close relationship with other aspects of the design, manufacturing, testing, and application of optical freeform surfaces. Improvements in freeform surface representation techniques will make a significant contribution to the further development of freeform optics. We present a detailed review of the different types of optical freeform surface representation techniques and their applications and discuss their properties and differences. Additionally, we analyze the future trends of optical freeform surface representation techniques.

A combination method using numerical orthogonal polynomials and overlapping averaging is presented for freeform surface estimation. The whole effective aperture of freeform surface is decomposed into multiple overlapped subapertures. The corresponding local surface over each subaperture is reconstructed by numerical orthogonal polynomials suited for general shaped aperture. Then, the whole freeform surface is obtained by overlapping averaging approach from multiple local surfaces. The performance of the presented combination method is discussed and demonstrated by examples and further verified by an experiment. The results show that the combination method could reach nanometer accuracy. Meanwhile, the local deformations of freeform surface can be characterized finely.

Four orthogonal polynomials for reconstructing a wavefront over a square aperture based on the modal method are currently available, namely, the 2D Chebyshev polynomials, 2D Legendre polynomials, Zernike square polynomials and Numerical polynomials. They are all orthogonal over the full unit square domain. 2D Chebyshev polynomials are defined by the product of Chebyshev polynomials in x and y variables, as are 2D Legendre polynomials. Zernike square polynomials are derived by the Gram-Schmidt orthogonalization process, where the integration region across the full unit square is circumscribed outside the unit circle. Numerical polynomials are obtained by numerical calculation. The presented study is to compare these four orthogonal polynomials by theoretical analysis and numerical experiments from the aspects of reconstruction accuracy, remaining errors, and robustness. Results show that the Numerical orthogonal polynomial is superior to the other three polynomials because of its high accuracy and robustness even in the case of a wavefront with incomplete data.

Freeform complex surfaces with steep variations in local areas cannot be fitted by analytical functions with high accuracy in the whole aperture in optics. A framework with a combination of Zernike polynomials as base function with radial basis function method is presented to improve reconstruction accuracy for freeform complex surfaces with circular aperture. The algorithm of the framework is analyzed, and the performance of the method is investigated and verified by numerical experiments. The proposed method is accurate and highly adaptable, which could be applied in modern optical manufacturing and measurement.

Wavefront aberration function is modified and expressed by combination of Bi-Zernike polynomials in rotationally symmetric optical systems. The field and pupil parameters of wavefront aberration function are separated and described respectively by orthonormal Zernike polynomials.

Wavefront estimation from the slope-based sensing metrologies zis important in modern optical testing. A numerical orthogonal transformation method is proposed for deriving the numerical orthogonal gradient polynomials as numerical orthogonal basis functions for directly fitting the measured slope data and then converting to the wavefront in a straightforward way in the modal approach. The presented method can be employed in the wavefront estimation from its slopes over the general shaped aperture. Moreover, the numerical orthogonal transformation method could be applied to the wavefront estimation from its slope measurements over the dynamic varying aperture. The performance of the numerical orthogonal transformation method is discussed, demonstrated and verified by the examples. They indicate that the presented method is valid, accurate and easily implemented for wavefront estimation from its slopes.

In practical optical measurements, the wavefront data are recorded by pixelated imaging sensors. The closed-form analytical base polynomial will lose its orthogonality in the discrete wavefront database. For a wavefront with an irregularly shaped aperture, the corresponding analytical base polynomials are laboriously derived. The use of numerical orthogonal polynomials for reconstructing a wavefront with a general shaped aperture over the discrete data points is presented. Numerical polynomials are orthogonal over the discrete data points regardless of the boundary shape of the aperture. The performance of numerical orthogonal polynomials is confirmed by theoretical analysis and experiments. The results demonstrate the adaptability, validity, and accuracy of numerical orthogonal polynomials for estimating the wavefront over a general shaped aperture from regular boundary to an irregular boundary.

We present a refractive index measurement method by measuring the longitudinal displacement of the converging rays after passing through a plane-parallel-plate sample. Using the wavefront difference method, the error source from the divergence angle of the beams is eliminated, and an analytical formula for refractive index calculation is derived. With a Fizeau interferometer, an experimental system for plane-parallel-plate sample testing is proposed to verify the principle. Experimental results indicate that the order of its accuracy is 10−4.