Xolography for linear volumetric 3D printing.

The range of applications for additive manufacturing is expanding quickly, including mass production of athletic footwear parts1, dental ceramics2 and aerospace components3 as well as fabrication of microfluidics4, medical devices5, and artificial organs6. The light-induced additive manufacturing techniques7 used are particularly successful owing to their high spatial and temporal control, but such techniques still share the common motifs of pointwise or layered generation, as do stereolithography8, laser powder bed fusion9, and continuous liquid interface production10 and its successors11,12. Volumetric 3D printing13-20 is the next step onward from sequential additive manufacturing methods. Here we introduce xolography, a dual colour technique using photoswitchable photoinitiators to induce local polymerization inside a confined monomer volume upon linear excitation by intersecting light beams of different wavelengths. We demonstrate this concept with a volumetric printer designed to generate three-dimensional objects with complex structural features as well as mechanical and optical functions. Compared to state-of-the-art volumetric printing methods, our technique has a resolution about ten times higher than computed axial lithography without feedback optimization, and a volume generation rate four to five orders of magnitude higher than two-photon photopolymerization. We expect this technology to transform rapid volumetric production for objects at the nanoscopic to macroscopic length scales.

Measuring refractive index of glass by using speckle.

A method to measure the refractive index of an optically flat, regularly shaped slab of glass using speckle correlation-based techniques is reported. The intensity of the diffraction field of the diffuser is captured by a CCD both with and without the glass present. As the position of the peak correlation coefficient is quantitatively related to the change in optical path length arising due to the presence of the glass, the refractive index of the glass can be evaluated by cross-correlating the two captured images. The theoretical correlation function that describes the effects of such an optical path length change is discussed, and the resulting speckle decorrelation function derived. Two glass samples are measured to demonstrate the accuracy and robustness of the proposed technique.

A Review of Hologram Storage and Self-Written Waveguides Formation in Photopolymer Media.

Photopolymer materials have received a great deal of attention because they are inexpensive, self-processing materials that are extremely versatile, offering many advantages over more traditional materials. To achieve their full potential, there is significant value in understanding the photophysical and photochemical processes taking place within such materials. This paper includes a brief review of recent attempts to more fully understand what is needed to optimize the performance of photopolymer materials for Holographic Data Storage (HDS) and Self-Written Waveguides (SWWs) applications. Specifically, we aim to discuss the evolution of our understanding of what takes place inside these materials and what happens during photopolymerization process, with the objective of further improving the performance of such materials. Starting with a review of the photosensitizer absorptivity, a dye model combining the associated electromagnetics and photochemical kinetics is presented. Thereafter, the optimization of photopolymer materials for HDS and SWWs applications is reviewed. It is clear that many promising materials are being developed for the next generation optical applications media.

Fresnel transform as a projection onto a Nijboer-Zernike basis set.

The Fresnel transform is widely used in optics to calculate the free-space propagation of paraxial fields. Generally, there is no analytical solution for the Fresnel transform; therefore, the numerical methods are used often. In this Letter, we propose a new semi-analytical method to calculate the Fresnel transform, which is based on an extended Nijboer-Zernike theory. We calculate two examples to investigate how the sampling rate and maximal number of Zernike polynomials affect the accuracy of our results, and then use this method to calculate the reconstruction of two different kinds of holograms. At the end, we discuss the advantages and disadvantages of our method.

Numerical solution of nonparaxial scalar diffraction integrals for focused fields.

In this paper, we present sampling conditions for fast-Fourier-transform-based field propagations. The input field and the propagation kernel are analyzed in a combined manner to derive sampling criteria that guarantee accurate calculation results in the output plane. These sampling criteria are also applicable to the propagation of general fields. For focal field calculations, geometrical optics is used to obtain a priori knowledge about the input and output fields. This a priori knowledge is used to determine an optimum balance between computational load and calculation accuracy. In a numerical example, correct results are obtained even though both the input field and the propagation kernel are sampled below the Nyquist rate. Finally, we show how chirp z-transform-based zoom-algorithms may be analyzed using the same techniques.

Fast nonparaxial scalar focal field calculations.

An efficient algorithm for calculating nonparaxial scalar field distributions in the focal region of a lens is discussed. The algorithm is based on fast Fourier transform implementations of the first Rayleigh-Sommerfeld diffraction integral and assumes that the input field at the pupil plane has a larger extent than the field in the focal region. A sampling grid is defined over a finite region in the output plane and referred to as a tile. The input field is divided into multiple separate spatial regions of the size of the output tile. Finally, the input tiles are added coherently to form a summed tile, which is propagated to the output plane. Since only a single tile is propagated, there are significant reductions of computational load and memory requirements. This method is combined either with a subpixel sampling technique or with a chirp z-transform to realize smaller sampling intervals in the output plane than in the input plane. For a given example the resulting methods enable a speedup of approximately 800× in comparison to the normal angular spectrum method, while the memory requirements are reduced by more than 99%.

Numerical calculation of the Fresnel transform.

In this paper, we address the problem of calculating Fresnel diffraction integrals using a finite number of uniformly spaced samples. General and simple sampling rules of thumb are derived that allow the user to calculate the distribution for any propagation distance. It is shown how these rules can be extended to fast-Fourier-transform-based algorithms to increase calculation efficiency. A comparison with other theoretical approaches is made.

Paraxial light distribution in the focal region of a lens: a comparison of several analytical solutions and a numerical result.

The distribution of the complex field in the focal region of a lens is a classical optical diffraction problem. Today, it remains of significant theoretical importance for understanding the properties of imaging systems. In the paraxial regime, it is possible to find analytical solutions in the neighborhood of the focus, when a plane wave is incident on a focusing lens whose finite extent is limited by a circular aperture. For example, in Born and Wolf's treatment of this problem, two different, but mathematically equivalent analytical solutions, are presented that describe the 3D field distribution using infinite sums of [Formula: see text] and [Formula: see text] type Lommel functions. An alternative solution expresses the distribution in terms of Zernike polynomials, and was presented by Nijboer in 1947. More recently, Cao derived an alternative analytical solution by expanding the Fresnel kernel using a Taylor series expansion. In practical calculations, however, only a finite number of terms from these infinite series expansions is actually used to calculate the distribution in the focal region. In this manuscript, we compare and contrast each of these different solutions to a numerically calculated result, paying particular attention to how quickly each solution converges for a range of different spatial locations behind the focusing lens. We also examine the time taken to calculate each of the analytical solutions. The numerical solution is calculated in a polar coordinate system and is semi-analytic. The integration over the angle is solved analytically, while the radial coordinate is sampled with a sampling interval of [Formula: see text] and then numerically integrated. This produces an infinite set of replicas in the diffraction plane, that are located in circular rings centered at the optical axis and each with radii given by [Formula: see text], where [Formula: see text] is the replica order. These circular replicas are shown to be fundamentally different from the replicas that arise in a Cartesian coordinate system.

K speckle: space-time correlation function of doubly scattered light in an imaging system.

The scattering of coherent monochromatic light at an optically rough surface, such as a diffuser, produces a speckle field, which is usually described by reference to its statistical properties. For example, the real and imaginary parts of a fully developed speckle field can be modeled as a random circular Gaussian process. When such a speckle field is used to illuminate a second diffuser, the statistics of the resulting doubly scattered field are in general no longer Gaussian, but rather follow a K distribution. In this paper we determine the space-time correlation function of such a doubly scattered speckle field that has been imaged by a single lens system. A space-time correlation function is derived that contains four separate terms; similar to the Gaussian case it contains an average DC term and a fluctuating AC term. However, in addition there are two terms that are related to contributions from each of the diffusers independently. We examine how our space-time correlation function varies as the diffusers are rotated at different speeds and as the point spread function of the imaging system is changed. A series of numerical simulations are used to confirm different aspects of the theoretical analysis. We then finish with a discussion of our results and some potential applications, including controlling spatial coherence and speckle reduction.

Filtering role of the sensor pixel in Fourier and Fresnel digital holography.

Digital holography is a modern imaging technique whereby a propagated object wave interferes with a known (spherical or plane) reference wave at a plane where a digital sensor is situated. The resulting intensity distribution is recorded by a CCD or CMOS sensor array to produce a digital hologram. This digital hologram can be processed in several ways to isolate the real image term. Using a propagation algorithm, the object wave can be numerically reconstructed from this real image term. Several factors limit the performance of such imaging systems, such as the finite extent of the sensor array and the finite size of the equally spaced sensor pixels, which average the light intensity incident upon them. Theoretical results indicate that in a Fresnel-based system the role of these finite-size pixels is to attenuate higher spatial frequencies by convolving the reconstructed signal with a rectangular function of equal size to the light-sensitive area of the pixel. However, when a spherical reference wave is used, as is the case with "lensless" Fourier-based systems, spatial frequencies will not be attenuated; rather it is the complex amplitude of the reconstructed signal that will be attenuated. In this manuscript we explore this question in more detail, providing new theoretical and experimental results. By assuming a fully developed speckle field for the object wave, we examine the first-order statistical distributions for the integrated intensity of the object wave, and the interference term, using numerical simulations. We show that the statistical distribution of the interference term can be changed, by varying the sphericity of the reference wave. Experimental results are provided where we compare the performance of a Fresnel and Fourier holographic system as a function of pixel size.

Speckle suppression by doubly scattering systems.

Speckle suppression in a two-diffuser system is examined. An analytical expression for the speckle space-time correlation function is derived, so that the speckle suppression mechanism can be investigated statistically. The grain size of the speckle field illuminating the second diffuser has a major impact on the speckle contrast after temporal averaging. It is shown that, when both the diffusers are rotating, the one with the lower rotating speed determines the period of the speckle correlation function. The coherent length of the averaged speckle intensity is shown to equal the mean speckle size of the individual speckle pattern before averaging. Numerical and experimental results are presented to verify our analysis in the context of speckle reduction.

Two-step phase-shift interferometry with known but arbitrary reference waves: a graphical interpretation.

There are many applications in biology and metrology where it is important to be able to measure both the amplitude and phase of an optical wave field. There are several different techniques for making this type of measurement, including digital holography and phase retrieval methods. In this paper we propose an analytical generalization of this two-step phase-shifting algorithm. We investigate how to reconstruct the object signal if both reference waves are different in phase and amplitude. The resulting equations produce two different solutions and hence an ambiguity remains as to the correct solution. Because of the complexity of the generalized analytical expressions we propose a graphical-vectorial method for solution of this ambiguity problem. Combining our graphical method with a constraint on the amplitude of the object field we can unambiguously determine the correct result. The results of the simulation are presented and discussed.

Speckle orientation in paraxial optical systems.

The statistical properties of speckles in paraxial optical systems depend on the system parameters. In particular, the speckle orientation and the lateral dependence (x and y) of the longitudinal speckle size can vary significantly. For example, the off-axis longitudinal correlation length remains equal to the on-axis size for speckles in a Fourier transform system, while it decreases dramatically as the observation position moves off axis in a Fresnel system. In this paper, we review the speckle correlation function in general linear canonical transform (LCT) systems, clearly demonstrating that speckle properties can be controlled by introducing different optical components, i.e., lenses and sections of free space. Using a series of numerical simulations, we examine how the correlation function changes for some typical LCT systems. The integrating effect of the camera pixel and the impact this has on the measured first- and second-order statistics of the speckle intensities is also examined theoretically. A series of experimental results are then presented to confirm several of these predictions. First, the effect the pixel size has on the measured first-order speckle statistics is demonstrated, and second, the orientation of speckles in a Fourier transform system is measured, showing that the speckles lie parallel to the optical axis.

Twin removal in digital holography using diffuse illumination.

A method to numerically remove the twin image for inline digital holography, using multiple digital holograms, is discussed. Each individual hologram is recorded by using a statistically independent speckle field to illuminate the object. If the holograms are recorded in this manner and then numerically reconstructed, the twin image appears as a different speckle pattern in each of the reconstructions. By performing speckle-reduction techniques the presence of the twin image can be greatly reduced. A theoretical model is developed, and experimental results are presented that validate this approach. We show experimentally that the dc object intensity term can also be removed by using this technique.

Three-dimensional speckle size in generalized optical systems with limiting apertures.

Correlation properties of speckle fields at the output of quadratic phase systems with hard square and circular apertures are examined. Using the linear canonical transform and ABCD ray matrix techniques to describe these general optical systems, we first derive analytical formulas for determining axial and lateral speckle sizes. Then using a numerical technique, we extend the analysis so that the correlation properties of nonaxial speckles can also be considered. Using some simple optical systems as examples, we demonstrate how this approach may be conveniently applied. The results of this analysis apply broadly both to the design of metrology systems and to speckle control schemes.

Numerical sampling rules for paraxial regime pulse diffraction calculations.

Sampling rules for numerically calculating ultrashort pulse fields are discussed. Such pulses are not monochromatic but rather have a finite spectral distribution about some central (temporal) frequency. Accordingly, the diffraction pattern for many spectral components must be considered. From a numerical implementation viewpoint, one may ask how many of these spectral components are needed to accurately calculate the pulse field. Using an analytical expression for the Fresnel diffraction from a 1-D slit, we examine this question by varying the number of contributing spectral components. We show how undersampling the spectral profile produces erroneous numerical artifacts (aliasing) in the spatial-temporal domain. A guideline, based on graphical considerations, is proposed that determines appropriate sampling conditions. We show that there is a relationship between this sampling rule and a diffraction wave that emerges from the aperture edge; comparisons are drawn with boundary diffraction waves. Numerical results for 2-D square and circular apertures are presented and discussed, and a potentially time-saving calculation technique that relates pulse distributions in different z planes is described.

Controlling speckle using lenses and free space.

The correlation properties of speckle fields are studied for general paraxial systems. The previous studies on lateral and longitudinal speckle size for the case of free-space propagation (Fresnel transform) are generalized to the case of the linear canonical transform. These results have implications for the control of speckle size, through appropriate design of optical systems, with particular relevance for speckle interferometry.

Fundamental diffraction limitations in a paraxial 4-f imaging system with coherent and incoherent illumination.

In the usual model of an imaging system, only the effects of the aperture stop are considered in determining diffraction-limited system performance. In fact, diffraction at other stops--those associated with different lens elements, for example--can also affect system performance and cause the imaging to be space variant, even in the absence of vignetting in the conventional ray optics sense. For the 4-f imaging system investigated in this paper, the severity of the space variance depends on the relative sizes of the two lens stops and the aperture stops. If the diameters of the lenses are equal, the aperture of the first lens has a greater effect on system performance than does that of the second.

Generalized Yamaguchi correlation factor for coherent quadratic phase speckle metrology systems with an aperture.

In speckle-based metrology systems, a finite range of possible motion or deformation can be measured. When coherent imaging systems with a single limiting aperture are used in speckle metrology, the observed decorrelation effects that ultimately define this range are described by the well-known Yamaguchi correlation factor. We extend this result to all coherent quadratic phase paraxial optical systems with a single aperture and provide experimental results to support our theoretical conclusions.

Paraxial speckle-based metrology systems with an aperture.

Digital speckle photography can be used in the analysis of surface motion in combination with an optical linear canonical transform (LCT). Previously [D. P. Kelly et al. Appl. Opt.44, 2720 (2005)] it has been shown that optical fractional Fourier transforms (OFRTs) can be used to vary the range and sensitivity of speckle-based metrology systems, allowing the measurement of both the magnitude and direction of tilting (rotation) and translation motion simultaneously, provided that the motion is captured in two separate OFRT domains. This requires two bulk optical systems. We extend the OFRT analysis to more general LCT systems with a single limiting aperture. The effect of a limiting aperture in LCT systems is examined in more detail by deriving a generalized Yamaguchi correlation factor. We demonstrate the benefits of using an LCT approach to metrology design. Using this technique, we show that by varying the curvature of the illuminating field, we can effectively change the output domain. From a practical perspective this means that estimation of the motion of a target can be achieved by using one bulk optical system and different illuminating conditions. Experimental results are provided to support our theoretical analysis.