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The pattern embossed on the back of an oriental magic mirror appears in the patch of light projected onto a screen from its apparently featureless reflecting surface. In reality, the embossed pattern is reproduced in low relief on the front, and analysis shows that the projected image results from pre-focal ray deviation. In this interesting regime of geometrical optics, the image intensity is given simply by the Laplacian of the height function of the relief. For patterns consisting of steps, this predicts a characteristic effect, confirmed by observation: the image of each step exhibits a bright line on the low side and a dark line on the high side. Laplacian-image analysis of a magic-mirror image indicates that steps on the reflecting surface are about 400 nm high and laterally smoothed by about 0.5 mm.

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... Previous approaches to modelling the image formation (Berry, 2006;Gitin, 2009) were aimed at finding correspondence between the back surface height map and the intensity map of the projected Makyoh image. These studies have pointed out the edge-detection property (Berry, 2006) or attempted to determine the point spread function linking the back pattern elevation and the Makyoh image intensity (Gitin, 2009); both works were based on the Laplacian approximation of imaging and a convolution-based backside pattern transfer model. ...

... Previous approaches to modelling the image formation (Berry, 2006;Gitin, 2009) were aimed at finding correspondence between the back surface height map and the intensity map of the projected Makyoh image. These studies have pointed out the edge-detection property (Berry, 2006) or attempted to determine the point spread function linking the back pattern elevation and the Makyoh image intensity (Gitin, 2009); both works were based on the Laplacian approximation of imaging and a convolution-based backside pattern transfer model. A somehow different approach (Gamo, 1984;Hibino et al., 1990) defines a mechano-optical modulation transfer function for periodic back relief on the analogy of the optical modulation transfer function. ...

... For modelling the image formation, we consider a globally flat mirror. The back-to-front pattern transfer can be described by the convolution (smoothing) with a Gaussian (Berry, 2006;Gitin, 2009): ...

The imaging of Japanese magic mirrors (Makyohs) is discussed by comparing the visual images of the backside relief pattern under different illumination conditions and the projected Makyoh image, rather than considering the back relief pattern as the input. It is also hypothesised that for a large number of magic mirrors, the Makyoh imaging is beyond the Laplacian regime under optimum viewing conditions, which accentuates the bright regions in the image; these correspond to reflecting highlights or step edges in the visual image. Published magic mirror images support this hypothesis. Simulations of the visual and Makyoh images are also presented.

... Oriental magic mirrors ('Makyoh', after their Japanese name) [1,2] have received much interest from the optics community since the 19 th century [3][4][5][6]. Such a mirror is an essentially flat or slightly convex mirror made of bronze with a backside relief pattern. ...

... Efforts to quantitatively model the image forming as a function of Page 1 of 7 AUTHOR SUBMITTED MANUSCRIPT -JOPT-109849. R1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 A c c e p t e d M a n u s c r i p t [3]. He treated the pattern transfer process as smoothing the abrupt step into the error function. ...

... The degree of the non-linearity can be expressed by the ratio of the Gaussian to the mean curvature related terms as -LK/H. If L is much smaller than the local surface curvature radii, the image intensity can be approximated by the Laplacian of the surface relief pattern [3,18] as ...

A model of magic-mirror (Makyoh) imaging with the mirror backside relief as the input is described. The mechanical pattern transfer is modelled by a convolution approach while the optical imaging using a nonlinear geometrical optical model, more general than previous approaches. Characteristic features of the imaging is analysed and the linear approximation is examined. Diffraction effects are also treated. Characteristic features for the ancient mirror and applications in semiconductor wafer inspection are discussed.

... The mechanism behind ancient magic mirrors from China and Japan was not understood until the 20th century, despite the earliest creations of these artistic pieces dating back to 2000 BC [1]. The cast bronze mirrors presented as normal mirrors while viewing one's reflection. ...

... The magic mirror effect can be quantitatively explained through standard diffraction theory [3]. However, the final understanding of how images are formed from the magic mirrors was derived in 2005 by Sir Michael Berry [1]. Here, it was shown that the intensity of the image is given to the first-order approximation by the Laplacian of the height of surface reliefs on the mirror. ...

... Specifically, the surface should be "smooth" enough, with gentle variations, such that caustics are not formed before the image appears. It is shown that the intensity of the Laplacian image is given in terms of the height of the surface relief, h, by I Laplacian Mirror (r, Z) 1 + Z∇ 2 h(r) [1]. Here, Z = 2D/M and r = R/M are the rescaled distance along the propagation direction and transverse position from the center of the mirror, respectively, normalized to the magnification of the convex mirror M. D and R are the distances from the mirror and transverse position of the image, respectively. ...

Magic windows (or mirrors) consist of optical devices with a surface deformation or thickness distribution devised in such a way to form a desired image. The associated image intensity distribution has been shown in previous works to be related to the Laplacian of the height of the surface relief. Exploiting the Laplacian theory to calculate the needed phase pattern, we experimentally realize such devices with flat optics employing optical polarization-wavefront coupling, which represent a new paradigm for light manipulation. The desired pattern and experimental specifications for designing the flat optics was implemented with a reconfigurable spatial light modulator, which acted as the magic mirror. The flat plate, an optical polarization-wavefront coupler, is then fabricated by spatially structuring nematic liquid crystals. The plate is used to demonstrate the concept of a polarization-switchable magic window, where, depending on the input circular polarization handedness, one can display either the desired image or the image resulting from the negative of the window’s phase.

... The mechanism behind ancient magic mirrors from China and Japan was not understood until the 20th century, despite the earliest creations of these artistic pieces dating back to 2000 BC [1]. The cast bronze mirrors presented as normal mirrors while viewing ones reflection. ...

... The mathematical description and final understanding of how the magic mirrors worked was derived in 2005 by Sir Michael Berry [1]. Here, it was shown that the intensity of the image is given to the first order approximation by the Laplacian of the height of surface reliefs on the mirror. ...

... Specifically, the surface should be "smooth" enough, with gentle variations, such that caustics are not formed before the image appears. It is shown that the intensity of the Laplacian image is given in terms of the height of the surface relief, h, by I Laplacian Mirror (r, Z) 1 + Z ∇ 2 h(r) [1]. Here, Z = 2D/M and r = R/M are the distance along the propagation direction and transverse position from the centre of the mirror respectively, normalized to the magnification of the convex mirror M. D and R are the distance from the mirror and transverse position of the image, respectively. ...

Magic windows (or mirrors) consist of optical devices with a surface deformation or thickness distribution devised in such a way to form a desired image. The associated image intensity distribution has been shown to be related to the Laplacian of the height of the surface relief. We experimentally realize such devices with flat optics employing optical spin-to-orbital angular momentum coupling, which represent a new paradigm for light manipulation. The desired pattern and experimental specifications for designing the flat optics was implemented with a re-configurable spatial light modulator which acted as the magic mirror. The flat plate, optical spin-to-orbital angular momentum coupler, is then fabricated by spatially structuring nematic liquid crystals. The plate is used to demonstrate the concept of a polarization-switchable magic window, where, depending on the input circular polarization handedness, one can display either the desired image or the image resulting from the negative of the window's phase.

... Such an optical element has a freeform surface, whose effect on the light passing thorough it is defined by the value of the Laplace operator of its sur face relief. In order to calculate the surface relief reproducing the desired intensity distribution one thus needs to solve the Poisson's equation [43][44][45]. ...

... The relatively simple intensity pattern of a ring was chosen for a test window. By solving Poisson's equation in the manner given in [43][44][45], one can calculate the corresponding surface relief ( figure 9(a)). It should be noted here that the absolute values of the surface depth do not play an important role; it is the aspect ratio of XY size to depth that defines the local radius of curvature and hence whether the Laplacian condi tion is fulfilled [43,44]. ...

... By solving Poisson's equation in the manner given in [43][44][45], one can calculate the corresponding surface relief ( figure 9(a)). It should be noted here that the absolute values of the surface depth do not play an important role; it is the aspect ratio of XY size to depth that defines the local radius of curvature and hence whether the Laplacian condi tion is fulfilled [43,44]. Here, if the etching depth that can be achieved by this method is of the order of several microme tres, the linear dimensions of the window will be of several millimetres. ...

In this work, it is shown that controllable increases in chemical reactivity of borosilicate glass can be induced through spatially resolved femtosecond laser irradiation at fluence values significantly lower than the damage threshold. The hydrofluoric acid etch rate has been found to be closely correlated to the reduction in optical transmission of the glass at 488 nm, which is, in turn, governed by the production of boron–oxygen hole centers. The combination of laser irradiation below the ablation threshold followed by chemical etching is shown to yield surfaces that have a roughness lower than those achieved by either laser or chemical etching alone. Application of this effect to the manufacture of freeform Laplacian optics is demonstrated.

... The basic principle of FOEs has been described in [5], when Berry examined the mode of operation of an ancient device called "magic mirror", a seemingly smooth slightly concave mirror, projecting sharp images at a distant plane upon illumination with sunlight. It turned out that the projected images correspond basically to the Laplace operation (i.e. the twodimensional curvature) of the surface profile [5,6], i.e. the surfaces are basically constructed as a solution of the inhomogeneous Poisson Equation using the desired image as the inhomogeneous part. ...

... The basic principle of FOEs has been described in [5], when Berry examined the mode of operation of an ancient device called "magic mirror", a seemingly smooth slightly concave mirror, projecting sharp images at a distant plane upon illumination with sunlight. It turned out that the projected images correspond basically to the Laplace operation (i.e. the twodimensional curvature) of the surface profile [5,6], i.e. the surfaces are basically constructed as a solution of the inhomogeneous Poisson Equation using the desired image as the inhomogeneous part. These "Laplacian images" develop at larger reconstruction distances into caustic structures (i.e. ...

... We basically follow the method described in [7], but use a different numerical implementation. Basically, FOE calculation is based on the fact that the reconstructed image (in the ray optical regime, i.e. close enough to the FOE) is mainly the Laplacian (i.e. the two-dimensional curvature) of the phase profile of the FOE [5,13]. This is due to the fact that the local curvature of the FOE surface produces a corresponding positive or negative lensing effect, which results in amplification or attenuation of the image intensity in the respective areas, forming the desired image in the pre-caustic region. ...

Modern liquid crystal spatial light modulators (SLMs) are capable of shifting the optical path length by some microns, which corresponds to phase shifts of several multiples of 2π. We use this capability to display freeform optical elements (FOEs) on a SLM, as largely smooth phase variations with only a small number of wrapping lines. These FOEs can be programmed to generate so-called caustic intensity distributions, which may be real images reconstructed at a selected position in front of the SLM surface. In contrast to standard diffractive structures, reconstruction of the freeform images is non-dispersive (i.e. white light images can be programmed), free of speckle, and its efficiency does not depend on the wavelength. These features promise novel applications in image projection, and various application fields of SLMs in microscopy.

... The basic principle of FOEs has been described in [5], when Berry examined the mode of operation of an ancient device called "magic mirror", a seemingly smooth slightly concave mirror, projecting sharp images at a distant plane upon illumination with sunlight. It turned out that the projected images correspond basically to the Laplace operation (i.e. the twodimensional curvature) of the surface profile [5,6], i.e. the surfaces are basically constructed as a solution of the inhomogeneous Poisson Equation using the desired image as the inhomogeneous part. ...

... The basic principle of FOEs has been described in [5], when Berry examined the mode of operation of an ancient device called "magic mirror", a seemingly smooth slightly concave mirror, projecting sharp images at a distant plane upon illumination with sunlight. It turned out that the projected images correspond basically to the Laplace operation (i.e. the twodimensional curvature) of the surface profile [5,6], i.e. the surfaces are basically constructed as a solution of the inhomogeneous Poisson Equation using the desired image as the inhomogeneous part. These "Laplacian images" develop at larger reconstruction distances into caustic structures (i.e. ...

... We basically follow the method described in [7], but use a different numerical implementation. Basically, FOE calculation is based on the fact that the reconstructed image (in the ray optical regime, i.e. close enough to the FOE) is mainly the Laplacian (i.e. the two-dimensional curvature) of the phase profile of the FOE [5,13]. This is due to the fact that the local curvature of the FOE surface produces a corresponding positive or negative lensing effect, which results in amplification or attenuation of the image intensity in the respective areas, forming the desired image in the pre-caustic region. ...

Modern liquid crystal spatial light modulators (SLMs) are capable of shifting the optical path length by some microns, which corresponds to phase shifts of several multiples of 2π. We use this capability to display freeform optical elements (FOEs) on a SLM, as largely smooth phase variations with only a small number of wrapping lines. These FOEs can be programmed to generate so-called caustic intensity distributions, which may be real images reconstructed at a selected position in front of the SLM surface. In contrast to standard diffractive structures, reconstruction of the freeform images is non-dispersive (i.e. white light images can be programmed), free of speckle, and its efficiency does not depend on the wavelength. These features promise novel applications in image projection, and various application fields of SLMs in microscopy.

... Perhaps the earliest example of PCI being put into practice involves the art and demonstration of so-called 'magic mirrors' [5][6][7] that date back to at least the fifth century CE in China and apparently even earlier to the Han Dynasty (206 BCE-24 CE) [7,8], and, the performance of which remained steeped in mystery until the twentieth century. The basic feature of these magic mirrors lay in the fact that a cast or embossed design on the back surface of these bronze mirrors became visible when the highly polished and smooth convex front surface of the mirror was illuminated by direct sunlight and the reflected beam from the front surface of the mirror projected on to a wall. ...

... Although the properties of magic mirrors were known in the West from 1832, a proper explanation for their imaging properties baffled scientists for around 100 years until 1933 when W. H. Bragg [5,6,9] provided a satisfactory explanation, in general, physical terms for their properties. The underlying explanation was due to height distortions in the front surface of the mirror, believed caused by strains induced when polishing the front surface of the mirror as a result of the structure on the back surface. ...

... The underlying explanation was due to height distortions in the front surface of the mirror, believed caused by strains induced when polishing the front surface of the mirror as a result of the structure on the back surface. These minute height distortions led to visible light images of the back-surface structure being produced in the reflected beam via Fresnel diffraction in the near-field regime [6]. ...

This review provides a brief overview, albeit from a somewhat personal perspective, of the evolution and key features of various hard X-ray phase-contrast imaging (PCI) methods of current interest in connection with translation to a wide range of imaging applications. Although such methods have already found wide-ranging applications using synchrotron sources, application to dynamic studies in a laboratory/clinical context, for example for in vivo imaging, has been slow due to the current limitations in the brilliance of compact laboratory sources and the availability of suitable high-performance X-ray detectors. On the theoretical side, promising new PCI methods are evolving which can record both components of the phase gradient in a single exposure and which can accept a relatively large spectral bandpass. In order to help to identify the most promising paths forward, we make some suggestions as to how the various PCI methods might be compared for performance with a particular view to identifying those which are the most efficient, given the fact that source performance is currently a key limiting factor on the improved performance and applicability of PCI systems, especially in the context of dynamic sample studies. The rapid ongoing development of both suitable improved sources and detectors gives strong encouragement to the view that hard X-ray PCI methods are poised for improved performance and an even wider range of applications in the near future.

... Projecting a parallel beam onto the surface, a reflected pattern corresponding to the back pattern appears on a distant screen due to the focusing/defocusing action of the surface relief pattern as if the mirror was transparent. Berry has shown, based on a solid geometrical optical analysis [2], that the image of such a mirror is the Laplacian of the surface height profile, provided that the screen-to-mirror distance is much lower than the curvature radius of any of the surface features. ...

... (4) Our derivation is as follows: if L is small, we can approximate equation (1) as f(r) = r, and, consequently, use r instead of f on the left-hand side of equation (2). That is, the topological mapping due to surface slopes is neglected. ...

... (The absolute value sign has been dropped from the denominator since it is relevant only in the caustic regime.) We thus obtained the same formula as Berry did [2] (in fact, Berry used a 'reduced mirror-to-screen distance' D which incorporates a mean curvature of the whole mirror and point source illumination; for our case of plane mirror and parallel beam D equals 2L). ...

Berry has shown (2006 Eur. J. Phys. 27 109–18) that the image of an oriental magic mirror (an essentially flat mirror with small surface relief) is the Laplacian of the surface relief for low-curvature features. In this note, an alternative derivation is presented and the physical meaning of the used approximations is explained.

... We make some remarks on Berry's paper [Eur. Berry just presented a comprehensive explanation of the physical part for the optics of the magic mirror [1]. The story related to this strange metal mirror could be traced in [2][3][4]. ...

... apart and was absolutely certain that his explanation was correct [6]. Other attempts could be traced in [1][2]. ...

... In such an interesting regime of geometrical optics, the image intensity could be given simply by the Laplacian of the height function of the relief as Berry demonstrated in [1]. For instance, Berry used the error function (cf. ...

We make some remarks on Berry's paper [{\it Eur. J. Phys.} 27 (2006) 109-118].

... In addition to the mathematical community, the Monge-Ampére equation has also been broadly studied in many applied fields such as elasticity, geometric optics, and image processing. See [14] and [46]. Today such free-form optics are important in illumination applications. ...

... subject to K c = f (14) where B, G are from the boundary condition and H is from the smoothness condition. Note that we need to justify that the minimization has a solution. ...

We use trivariate spline functions for the numerical solution of the Dirichlet problem of the 3D elliptic Monge-Ampére equation. Mainly we use the spline collocation method introduced in [SIAM J. Numerical Analysis, 2405-2434,2022] to numerically solve iterative Poisson equations and use an averaged algorithm to ensure the convergence of the iterations. We shall also establish the rate of convergence under a sufficient condition and provide some numerical evidence to show the numerical rates. Then we present many computational results to demonstrate that this approach works very well. In particular, we tested many known convex solutions as well as nonconvex solutions over convex and nonconvex domains and compared them with several existing numerical methods to show the efficiency and effectiveness of our approach.

... While the front surface is being polished, uneven pressures from the back surface profiles transfer the back surface pattern onto the front surface in the form of gentle invisible surface relief. [22,23] Prior works [24][25][26][27][28][29] have shown to estimate the irradiance pattern of the oriental magic mirror (or window) by taking the Laplacian of the gentle surface relief. While studying surface perturbations, a similar phenomenon was observed. ...

... A surface perturbation within the tolerance range acts as the gentle surface relief of the freeform surface. Consequently, the Laplacian relationship found in the oriental magic mirrors [24][25][26][27][28][29] can also be expanded to surface perturbations and utilized to predict the corresponding target irradiance. ...

This paper presents a methodology for illumination optics tolerancing. Specifically, we investigate tolerancing deformation of a single freeform surface under a point source illumination. Through investigation, we recognized and report here three surface-deformation characteristics that build tolerancing intuitions. First, we show that positive and negative irradiance changes occur together as a consequence of flux conservation. Then, we demonstrate a linear relationship between the steepness of the surface perturbation and the magnitude of the irradiance change. Lastly, we show that the Laplacian magic mirror concept (M. V. Berry [Eur. J. Phys. 27, 109 (2006)10.1088/0143-0807/27/1/012]) can be expanded to surface deformation tolerancing. Utilizing these surface deformation characteristics, we propose a fast and predictable tolerancing method for a sequential illumination optic.

... Presently, however, there is no direct way of intuitively interpreting MEM contrast of a given general specimen. Here, we present a theory of Laplacian image contrast (see Berry 2006) in MEM that is an approximation of the geometrical theory, yet applicable to a wide range of practical imaging situations. The advantage of the theory is that the image contrast can be interpreted in terms of the Laplacian of an effective two-dimensional phase object that is directly related to the near-surface microfield. ...

... This is an important result for the intuitive interpretation of MEM contrast of surface topography. Laplacian imaging is widely encountered in many contexts ranging from X-ray imaging (Paganin 2006) to oriental magic mirrors (Berry 2006) and their modern equivalent in Makyoh topography (Riesz 2000). It is also known as out of focus contrast in TEM of thin specimens (Lynch et al. 1975;Cowley 1995;Spence 2003). ...

We discuss an intuitive approach to interpreting mirror electron microscopy (MEM) images, whereby image contrast is primarily caused by the Laplacian of small height or potential variations across a sample surface. This variation is blurred slightly to account for the interaction of the electrons with the electrical potential away from the surface. The method is derived from the established geometrical theory of MEM contrast, and whilst it loses quantitative accuracy outside its domain of validity, it retains a simplicity that enables rapid interpretation of MEM images. A strong parallel exists between this method and out of focus contrast in transmission electron microscopy (TEM), which allows a number of extensions to be made, such as including the effects of spherical and chromatic aberration.

... This means that the second derivative of the shape of the surface perturbation can be used to understand the irradiance distribution change. This Laplacian relationship has also been used to explain how light reflected from 'Oriental Magic Mirrors' provides an image of the surface relief on the backside of the mirror [7]. ...

... The theory behind the method is well studied (Saines and Tomilin, 1999;Berry, 2005;Riesz, 2011); its sensitivity is sufficient to, e.g., inspect semiconductor wafers for irregularities (Kugimiya, 1988;Blaustein and Hahn, 1989;Hahn et al., 1990;Szabó et al., 1995;von Finck et al., 2009;Tobisch et al., 2012;Hologenix, 2020). ...

Deflectometry as a technique to assess reflective surfaces has now existed for some 40 years. Its different aspects and variations have been studied in multiple theses and research articles; reviews are available for certain subtopics. Still a field of active development with many unsolved problems, deflectometry now encompasses a large variety of application domains, hardware setup types, and processing workflows for different purposes, and spans a range from qualitative defect inspection of large vehicles to precision measurements of microscopic optics. Over these years, many exciting developments have accumulated in the underlying theory, in the systems design, and in the implementation specifics. This diversity of topics is difficult to grasp for experts and non-experts alike and may present an obstacle to a wider acceptance of deflectometry as a useful tool for research and industrial applications. This paper presents an attempt to summarize the status of deflectometry and to map relations between its notable branches. Its aim is to provide a communication basis for experienced practitioners and also to offer a convenient entry point for those interested in learning about the method. The list of references introduces some prominent trends and established research groups in order to facilitate further self-directed exploration.

... What made these mirrors interesting was that when illuminated by the sun light, they reflected the pattern engraved on the back (Fig. 4.6,a). Despite the still existing controversy on how people managed to craft these artifacts, their reflective properties come from small bumps transferred from the back of the mirror to the polished front face [148,149]. When the light is reflected by these bumps it deviates in a way that perfectly reproduces the pattern engraved. ...

In this thesis I will explain the properties of intersubband polaritons by a theoretical and experimental point of view. After a brief introduction to explain some general aspects of the intersubband polariton generation I will describe an experimental campaign with the aim to verify the possibility to create a polariton laser. The emission of light from a
polaritonic state lacks a clear experimental verification, therefore this experiment may be the first observation of a polariton laser. The possibility to use graphene to design a better optical resonance will be largely discussed. In the end I will illustrate a concept to manipulate the light with 3D printed windows. The final objective being the design
of windows able to remove aberration from laser source (in particular TeraHertz sources).

... The theory behind the method is well studied [118][119][120]; its sensitivity is sufficient to e.g. inspect semiconductor wafers for irregularities [121][122][123][124][125][126][127]. ...

Deflectometry as a technical approach to assessing reflective surfaces has now existed for almost 40 years. Different aspects and variations of the method have been studied in multiple theses and research articles, and reviews are also becoming available for certain subtopics. Still a field of active development with many unsolved problems, deflectometry now encompasses a large variety of application domains, hardware setup types, and processing workflows designed for different purposes, and spans a range from qualitative defect inspection of large vehicles to precision measurements of microscopic optics. Over these years, many exciting developments have accumulated in the underlying theory, in the systems design, and in the implementation specifics. This diversity of topics is difficult to grasp for experts and non-experts alike and may present an obstacle to a wider acceptance of deflectometry as a useful tool in other research fields and in the industry. This paper presents an attempt to summarize the status of deflectometry, and to map relations between its notable "spin-off" branches. The intention of the paper is to provide a common communication basis for practitioners and at the same time to offer a convenient entry point for those interested in learning and using the method. The list of references is extensive but definitely not exhaustive, introducing some prominent trends and established research groups in order to facilitate further self-directed exploration by the reader.

... The design of freeform surfaces is categorized as an inverse problem and presents some computational challenges. Conventionally, a caustic-projection system has a smooth optical surface that refracts light to form an envelope of light rays, as shown in Fig. 1(a) [2,3]. As presented in Ref. [4], causticprojection holographic optical elements (HOEs) with a freeform optical function produce a caustic optical field within a thin and flat HOE film, where the HOE replicates the optical function of arbitrary caustic-projection freeform surfaces holographically, as shown in Fig. 1(b). ...

We propose a holographic printing technique for generating highly efficient large-deflection-angle freeform holographic optical elements (HOEs). For industrial device applications, the optical efficiency and deflection angle of HOEs are critical. To fabricate a high-frequency volume grating in a hogel, we design an optomechanical hogel recording system with a high angle deflection capability, which contrasts with the conventional printing scheme, the wavefront holographic printing technique featuring a paraxial deflection angle. With the proposed system, a large-deflection-angle HOE is experimentally demonstrated, and short-throw holographic caustic projection patterns are realized.

... work is more like one of the many Han dynasty innovations, the "magic mirror": see for example [50]. These mirrors concentrate or thin out the rays, but the rays do not meet or create singular points. ...

... Dans ce chapitre nous avons établi un résultat d'estimation en norme [14,16]. Aujourd'hui de telles surfaces de forme libre sont très importantes dans les applications d'éclairage. ...

Résumé en françaisCe travail porte sur une analyse d’erreur a priori pour une méthode d’éléments finis mixtes de l’équation elliptique de Monge-Ampère et uneanalyse d’erreur a posteriori pour une méthode d’éléments finis mixtes du problème couplé stationnaire Navier-Stokes/Darcy. Dans un premiertemps, nous avons considéré : Problème de Monge-Ampère. Pour ce problème nous avons suivi les étapes suivantes :1. Nous avons écrit une formulation faible mixte.2. Ce problème faible est approché par un problème discret dont lesespaces sont construits à l’aide d’éléments finis mixtes linéaire.L’existence de solution pour le problème discret s’obtient à l’aide du théorème du point fixe de Brouwer. La méthode d’éléments finisest celle du gradient reconstruit. Cette méthode est détaillée dans la thèse.3. Pour prouver la convergence de la méthode, nous avons établi unemajoration d’erreur a priori entre la solution du problème continu et celle du problème discret.4. Un résultat de régularité elliptique est établi qui a servi dans la majorationd’erreur a priori. Et dans un deuxième temps, nous avons considéré : Problème couplé stationnaire Navier-Stokes/DarcyPour ce problème nous avons adopté une démarche analogue à celle du problème précédent.1. Dans un premier temps, nous avons écrit une formulation faible mixtedu problème continu.2. Pour ce problème, nous avons proposé une discrétisation conforme àl’aide d’éléments finis mixtes : celui de Bernardi-Raugel et Raviart- Thomas pour la vitesse et les fonctions constantes par morceaux pourla pression.3. Pour l’analyse d’erreur a posteriori, nous avons construit de nouveauxindicateurs d’erreur. Nous avons prouvé que cette famille d’indicateursd’erreur est fiable et efficace.

... The experimental demonstration of this relationship supports the theoretical model published by Nicolas and Garca [19], which explains the intensity modulation in term of surface curvature error. Older models with visible light [20] and whose analytical solutions can be easily applied to the X-ray regime are also consistent with the present observations. ...

A parallel paper [Berujon, Cojocaru, Piault, Celestre, Roth, Barrett & Ziegler (2020), J. Synchrotron Rad.27, 284–292] reviewed theoretically some of the available processing schemes for X-ray wavefront sensing based on random modulation. Shown here are experimental applications of the technique for characterizing both refractive and reflective optical components. These fast and accurate X-ray at-wavelength metrology methods can assist the manufacture of X-ray optics that transport X-ray beams with a minimum amount of wavefront distortion. It is also recalled how such methods can facilitate online optimization of active optics.

... This relationship linking the intensity modulations to the mirror curvatures in the mm −1 spatial frequency scale with amplitude of the order of the wavelength actually supports the theoretical model published by Nicolas and Garca [50] and older models with visible light [51]. ...

X-ray near-field speckle-based phase-sensing approaches provide efficient means to characterise optical elements. Here, we present a theoretical review of several of these speckle methods in the frame of optical characterisation and provide a generalization of the concept. As we also demonstrate experimentally in another paper, the methods theoretically developed here can be applied with different beams and optics and within a variety of situations where at-wavelength metrology is desired. By understanding the differences between the various processing methods, it is possible to find and implement the best suited approach for each metrology scenario.

... Recently, there has been strong interest in freeform lens design, both for general lighting applications and also to generate images from caustics [4]. In the latter application, we can distinguish between discrete optimization methods that work on a pixelated version of the problem (e.g. ...

Phase-only light modulation shows great promise for many imaging applications, including future projection displays. While images can be formed efficiently by avoiding per-pixel attenuation of light most projection efforts utilizing phase-only modulators are based on holographic principles which rely on interference of coherent laser light and a Fourier lens. Limitations of this type of an approach include scaling to higher power as well as visible artifacts such as speckle and image noise.
We propose an alternative approach: operating the spatial phase modulator with broadband illumination by treating it as a programmable freeform lens. We describe a simple optimization approach for generating phase modulation patterns or freeform lenses that, when illuminated by a collimated, broadband light source, will project a pre-defined caustic image on a designated image plane. The optimization procedure is based on a simple geometric optics image formation model and can be implemented computationally efficient. We perform simulations and show early experimental results that suggest that the implementation on a phase-only modulator can create structured light fields suitable, for example, for efficient illumination of a spatial light modulator (SLM) within a traditional projector. In an alternative application, the algorithm provides a fast way to compute geometries for static, freeform lens manufacturing.

... The Chinese magic mirror is a circular, metallic, hand polished mirror having a relief pattern on its back side. The back side relief pattern is normally not visible in the front polished surface except when the front surface of the mirror reflects light on a wall/surface; the reflection shows the pattern on the back side of the mirror [9][10][11]. Several attempts have been made to replicate the magic mirror through modern machining and polishing techniques. ...

A nano-texturing method in single point diamond turning using backside patterned workpiece is presented. The back side of the workpiece is pre-machined to first create a pattern. The front side is then diamond turned on an ultra-precision lathe. After machining down to a certain thickness, periodic bumps and valleys that mirror the back side pattern start to appear on the front diamond machined surface. The periodic wavy/bumpy surfaces have nanometer depths, and possess mirror finish. The results suggest that this technique provides an alternative method to create optical features that are conventionally developed using tool-spindle synchronized cutting motions.

... Already 2000 years ago reflectors projecting images, called Chinese magic mirrors, have been hand-crafted of bronze in China and Japan, but the recipe has been lost and reconstructed several times over the ages; see [7] and [46]. Today such free-form optics are important in illumination applications. ...

The inverse reflector problem arises in geometrical nonimaging optics: Given
a light source and a target, the question is how to design a reflecting
free-form surface such that a desired light density distribution is generated
on the target, e.g., a projected image on a screen. This optical problem can
mathematically be understood as a problem of optimal transport and equivalently
be expressed by a secondary boundary value problem of the Monge-Amp\`ere
equation, which consists of a highly nonlinear partial differential equation of
second order and constraints. In our approach the Monge-Amp\`ere equation is
numerically solved using a collocation method based on tensor-product
B-splines, in which nested iteration techniques are applied to ensure the
convergence of the nonlinear solver and to speed up the calculation. In the
numerical method special care has to be taken for the constraint: It enters the
discrete problem formulation via a Picard-type iteration. Numerical results are
presented as well for benchmark problems for the standard Monge-Amp\`ere
equation as for the inverse reflector problem for various images. The designed
reflector surfaces are validated by a forward simulation using ray tracing.

... 4). Damit konnte die Hypothese von Berry verifiziert werden, dass die Intensität im Bild proportional zur Laplace-gefilterten Höhenkarte der Topographie ist [14]. Es gibt übrigens verschiedene Spiegeltypen: je nach Herstellungsverfahren bildet sich die Struktur der Vorderseite negativ oder positiv ab. ...

... The illuminance is infinite (caustic limit) if L equals either (2C min ) -1 or (2C max ) -1 ; that is, when the screen is in focus of a surface area element. It can be shown using a rigorous geometrical optical analysis [7] but it follows [8] also from Eqs. (1) and (2) that if the surface curvatures are negligible compared to L all over the surface, the image intensity can be approximated as ...

Imaging properties of the Makyoh imaging of periodic and quasi-periodic surfaces are analysed using a spectral approach based on a geometrical optical model. It is shown that in spite of the nonlinear nature of imaging, the Fourier representation can be used with some restrictions and limitations. Ray-tracing simulations are performed to illustrate the results.

... work is more like one of the many Han dynasty innovations, the "magic mirror": see for example [50]. These mirrors concentrate or thin out the rays, but the rays do not meet or create singular points. ...

Acoustic waves can force fluids and small objects along the direction of sound propagation (streaming). Potential applications include forcing the flow of fluid and small particles for drug delivery into the brain, or of underground fluids, such as oil, toward a borehole collection point. Well-focused beams are needed for amplitudes to be large enough to be effective. Time-reversal acoustics (TRA) permits focusing of sound in heterogeneous media, but requires a sensor near a point of desired focus. There is a need, addressed in this talk, to send signals off the grid of available sensors so that forces may be delivered in regions between receivers, and no receiver need be at a point where focus is desired. The Acoustic Shepherd introduces interpolation or data matching of TRA-derived Green functions for this purpose. One particular scheme of geometric-mean interpolation is analyzed, and the curve along which the focus moves from one receiver to another as the interpolating weights are varied is shown. General approximation methods from statistical estimation theory are also described. These methods are expected to work better in heterogeneous media (having unknown acoustic properties) than well-known beam forming methods designed for homogeneous media.

We use trivariate spline functions for the numerical solution of the Dirichlet problem of the 3D elliptic Monge-Amp\'ere equation. Mainly we use the spline collocation method introduced in \cite{LL21} to numerically solve iterative Poisson equations and use an averaged algorithm to ensure the convergence of the iterations. We shall also establish the rate of convergence under a sufficient condition and provide some numerical evidence to show the numerical rates. Then we present many computational results to demonstrate that this approach works very well. In particular, we tested many known convex solutions as well as nonconvex solutions over convex and nonconvex domains and compared them with several existing numerical methods to show the efficiency and effectiveness of our approach.

Projection was invented by nature – witness the eye with its lens that projects an image of the world on the retina. Projection in this book (usually) signifies an optical process in which a two-dimensional transparency is illuminated so that the light passing through it is focused by a refracting lens (dioptric lens, in the literature of the seventeenth century) to throw an image on a screen. Aristotle (384–322 BCE) noted that a beam of light, when it passed through a small opening no matter its shape, formed a circular image (Mannoni 2000, p. 4). This phenomenon is the basis for the camera obscura, which projects an image of the daylight world through a pinhole aperture into a dark room onto a white wall or screen, as described below and in the Chap. 7. In addition to a pinhole or a lens, projection with a mirror (catoptric projection) is possible and it may well be that projection technology was first attained in China more than 4600 years ago based on a truly uncommon phenomenon.

The effect of non-flatness of semiconductor wafers on characteristics of manufactured devices is shown through defocusing of an image of a topological layout of a structure being formed and through reduction of resolution at photolithographic processing. For quality control of non-flatness the Makyoh method is widely used. However, it does not allow obtaining quantitative characteristics of observed defects, which essentially restricts its application. The objective of this work has been developing of a calculation method for dimensions of topographic defects of wafers having semiconductor structures formed on them, which has allowed determining acceptability criteria for wafers, depending on defects dimensions and conducting their timely penalization.
A calculation method under development is based on deduction of relationships linking distortion of image elements to curvature of local sections of a semiconductor wafer that has formed structures. These structures have been considered to be image finite elements and within this range the curvature radius has been assumed to be constant. Sequential calculation of deviation of element ends from ideal plane based on determining their curvature radius has allowed obtaining geometry of a target surface in a set range of elements. Conditions of image formation and requirements to structures have been determined.
Analytical expressions relating a deviation value of elements of a light-to-dark image with surface geometry have been obtained. This allows conducting effective quantitative control of observed topographic defects both under production and research conditions. Examples of calculation of topographic defects of semiconductor silicon wafers have been provided. Comparison of the obtained results with the data obtained by conventional methods has shown their complete conformity.

Outgoing cylindrical waves scattered by a disk, or emerging from a source inside it, are represented by Hankel functions of order m. For large m, these waves decay rapidly outside the disk and resemble radially evanescent surface waves travelling around it. But they eventually leak weakly away, in a manner described accurately by the asymptotics of the Hankel function. The transition occurs at radial distance m (in wavelength units), which constitutes a circular caustic from which the radiation leaking out, described by the streamlines, appears to issue tangentially. In the evanescent region, the streamlines form spirals, whose windings get exponentially closer nearer the disk. These insights are intended to help graduate students demystify mathematics associated with scattering theory.

http://dx.doi.org/10.5007/2175-7941.2016v33n2p355
O artigo apresenta uma revisão bibliográfica acerca da reflexão luminosa. Foram revisados 152 artigos sobre o tema, publicados em revistas acadêmicas brasileiras e internacionais voltadas de forma específica para o ensino de física, entre 1995 e 2015. O diálogo entre os artigos foi estabelecido a partir de dez categorias de classificação dos textos, de acordo com o seu subtema específico: princípios da reflexão, refletância, velocidade da luz, reflexão difusa, imagens reais e virtuais, espelhos planos, espelhos curvos, associações de espelhos, polarização e espelhos relativísticos. A revisão evidenciou grande concentração de esforços de pesquisa em aplicações de espelhos curvos e planos. Por outro lado, temas como refletância, velocidade da luz, espelhos relativísticos e a relação entre os fenômenos da reflexão e da polarização mostraram carência de trabalhos. A revisão mostrou que esses quatro subtemas não são usuais nas publicações de autores brasileiros, demonstrando possibilidades de pesquisas futuras. O uso de materiais cotidianos em atividades experimentais também foi evidenciado, principalmente em artigos voltados para o tratamento da óptica no ensino médio, fato que contribui para a continuidade de pesquisas sobre o tema. Ressalta-se ainda que a celebração do Ano Internacional da Luz (2015) não parece ter alavancado de forma substancial as publicações sobre o tema.

Bridging the terahertz gap requires synergism between the microwave and photonic communities. Advances in each of these two communities are often complementary but sometimes overlooked. One example is the manipulation of waves, known as anomalous diffraction via the use of metamaterials. This is achieved by controlling the surface impedance of each pixel at the interface of two different materials or alternately the phase and magnitude of the wave diffracted off the pixel. In this paper, a review of developments in wave manipulation from microwave to optical frequencies is presented, together with our new results in the terahertz regime. Generation of phase curves for pixel design requires a priori information on material properties at terahertz frequencies. Fabrication of terahertz devices entails micromachining in the clean room while their experimental validation demands both amplitude and phase information. Through judicial selection of practices in microwave and photonic communities, we can further the exploration of wave phenomena at terahertz frequencies.

A transparent sheet, flat to unaided vision but with a gentle surface relief, can concentrate light onto a screen with intensity reproducing any desired image: the sheet is a 'magic window'. When the ray deflections are sufficiently small that there are no caustics between the window and the screen, the image intensity is the Laplacian function of the relief height function - a very simple approximation to general freeform optics. Therefore the desired relief is obtained by solving Poisson's equation. Numerical simulations indicate that the Laplacian image approximation will apply to realistic situations.

Cinema projectors need to compete with home theater displays in terms of image quality. High frame rate and spatial resolution as well as stereoscopic 3D are common features today, but even the most advanced cinema projectors lack in-scene contrast and, more important, high peak luminance, both of which are essential perceptual attributes of images appearing realistic. At the same time, HDR image statistics suggest that the average image intensity in a controlled ambient viewing environment such as the cinema can be as low as 1% for cinematic HDR content and not often higher than 18%, middle gray in photography. Traditional projection systems form images and colors by blocking the source light from a lamp, therefore attenuating between 99% and 82% of light, on average. This inefficient use of light poses significant challenges for achieving higher peak brightness levels.
In this work, we propose a new projector architecture built around commercially available components, in which light can be steered to form images. The gain in system efficiency significantly reduces the total cost of ownership of a projector (fewer components and lower operating cost), and at the same time increases peak luminance and improves black level beyond what is practically achievable with incumbent projector technologies. At the heart of this computational display technology is a new projector hardware design using phase modulation in combination with a new optimization algorithm that is capable of on-the-fly computation of freeform lens surfaces.

The Chinese magic mirror is an ancient convex bronze mirror, it reflects parallel light rays to form a unique image within
the reflected patch of light by altering the reflected ray paths. Using Phase Measuring Reflectometry (PMR), surface
irregularities of a micron range were found to be present on the mirror; these irregularities concentrate and disperse
reflected light rays, giving rise to brighter and darker patches on the reflected image, forming a contrast, allowing the
unique pattern to be observed. To ascertain location and nature of the surface defects that come in forms of indentations
and raised platforms, other measurement techniques were employed. Reverse engineering then facilitated the exploration
of reproduction of a very own original Chinese Magic Mirror with the use of optical principles behind the mirror.

In this study, we propose a new approach for displaying images and controlling light spaces using actively deformable mirrors. This approach enables programming the mirror-reflected light from sunlight or other parallel/point light sources to create arbitrary light spaces in various scenarios.

Non-linearity in Makyoh (magic-mirror) imaging is analysed using a
geometrical optical approach. The sources of non-linearity are
identified as (1) a topological mapping of the imaged surface due to
surface gradients, (2) the hyperbolic-like dependence of the image
intensity on the local curvatures, and (3) the quadratic dependence of
the intensity due to local Gaussian surface curvatures. Criteria for an
approximate linear imaging are given and the relevance to
Makyoh-topography image evaluation is discussed.

Zusammenfassung
In der Prozess- und Qualitätskontrolle spielen die Erkennung und Einstufung von Oberflächenstrukturen und Defekten eine immer wichtigere Rolle. Mit der Methode des Makyoh-Imaging (“Ma-Kyoh”, jap. für magischer Spiegel) steht ein leistungsstarkes, hochempfindliches Verfahren zur Verfügung, bei dem lokale konvexe und konkave Strukturen unter kollimierter Beleuchtung die Intensitätsverteilung im Reflex beeinflussen. Anhand einer am Fraunhofer IOF realisierten Messanordnung wird gezeigt, dass über den Reflex sowohl zuverlässig als auch schnell qualitative und quantitative Rückschlüsse auf Oberflächenstrukturen und Defekte möglich sind. Ergebnisse untersuchter Feinstrukturen auf Wafern und polierten Glasoberflächen werden vorgestellt.

The sensitivity of Makyoh (magic-mirror) imaging is analysed semi-quantitatively using geometrical optical approach. A simple three-scale surface model is introduced for the framework of the analysis. Vision physiological effects on the perceived sensitivity and pattern detectability are discussed. The sensitivity is given as a function of the surface feature's lateral size, height and the instrumental parameters for a given contrast discriminating level. The surface roughness is taken into account through speckle effects. The effect of the finite lateral size of the light source is analysed as well.

The interaction of waves with inhomogeneous media leads to the natural focusing of light, the channelling of waves into stable caustics. We have extended natural focusing to x rays, observing caustics in topographs of ferroelectric lithium niobate. Voltage across domains of reversed polarity induces perturbations to the local crystal planes, producing dramatic variations in the images. Ray tracing shows a "catastrophic" discontinuity, causing bright focal lines. Analysis reveals details of boundary strains and local ferroelectric properties. Controlled focusing could be extended to designed domain patterns to probe microstructural properties, and also to a type of voltage-controlled ferroelectric optics for x rays. (c) 2007 American Institute of Physics.

Image formation in a quasi-linear isoplanar system consisting of a plane-parallel layer of bronze (a "magic mirror") and a plane-parallel layer of free space (air) is described. The exhaustive characteristic of the quasi-linear isoplanar system is performed with a point spread function, where the role of an incoming signal from a point source is investigated with a local camber (or a hollow) on the back of the bronze mirror. Note that the point spread function of the image system should be as close as possible to a Dirac delta function. The quasi-linear isoplanar imaging magic-mirror-layer-of-space system should map a point source input signal (local camber on the back surface of a bronze mirror) to a point output signal (a light point on the screen). At a certain parity between the thickness of the layer of bronze and the thickness of the layer of free space, this linear isoplanar system forms the image with a very large depth of field.

A theory of edge detection is presented. The analysis proceeds in two parts. (1) Intensity changes, which occur in a natural image over a wide range of scales, are detected separately at different scales. An appropriate filter for this purpose at a given scale is found to be the second derivative of a Gaussian, and it is shown that, provided some simple conditions are satisfied, these primary filters need not be orientation-dependent. Thus, intensity changes at a given scale are best detected by finding the zero values of delta 2G(x,y)*I(x,y) for image I, where G(x,y) is a two-dimensional Gaussian distribution and delta 2 is the Laplacian. The intensity changes thus discovered in each of the channels are then represented by oriented primitives called zero-crossing segments, and evidence is given that this representation is complete. (2) Intensity changes in images arise from surface discontinuities or from reflectance or illumination boundaries, and these all have the property that they are spatially. Because of this, the zero-crossing segments from the different channels are not independent, and rules are deduced for combining them into a description of the image. This description is called the raw primal sketch. The theory explains several basic psychophysical findings, and the operation of forming oriented zero-crossing segments from the output of centre-surround delta 2G filters acting on the image forms the basis for a physiological model of simple cells (see Marr & Ullman 1979).

Publisher Summary This chapter discusses the morphologies of caustics and their diffraction patterns. In catastrophe optics, wave motion is viewed in terms of the contrast and interplay among the morphologies of three extreme regimes. Firstly, if the wavelength λ is small in comparison with scales of variation of diffracting objects or refracting media, the wavefield is dominated by the caustics and associated diffraction patterns. Secondly, when waves propagate in environments which can be modeled by a hierarchy of scales extending to the infinitely small, caustics cannot occur and the limit λ → 0 is not geometrical optics. And thirdly, when waves are explored on the scale of λ, the principal features are wavefronts, which are dominated by their singularities in the form of lines in space. The chapter also discusses the diffraction catastrophes that both clothe and underlie caustics. Each structurally stable caustic has its characteristic diffraction pattern, whose wave function has an integral representation in terms of the standard polynomial describing the corresponding catastrophe. The diffraction catastrophes constitute a new hierarchy of functions, different from the special functions of analysis. The newest application of catastrophe optics is to random short waves, whose statistical properties are determined by the random caustic structure.

Chinese "magic" mirrors are made from bronze with the front side a mirror and the reverse side a molded image. When light is reflected from the mirror,the image on the reverse side appears. Discusses reflections of conventional mirrors, possible explanations for the magic mirror phenomenon, and applications of the phenomenon to semiconductor wafers. (MDH)

The word, 'Ma-kyoh', literally means 'magic-mirror' or 'wonder-mirror'. It is just a simple flat mirror made mainly of bronze and completely featureless when being observed directly. However, a bright image of Buddha engraved on the back of the mirror appears on a dark temple wall when it reflects the light of the Sun or the Moon. This ancient holy mirror has recently been applied to the evaluation of Si wafers, and, nowadays, the application has spread over various fields of technology related to or eager for super-flat mirror surfaces, like hard discs, optical discs, glass substrates and the like. The 'Makyoh' technology is now almost completely re-defined and is being used in many fields in the modern high-tech world with good results.

The Japanese mirror must, from three points of view, attract the notice of foreigners sojourning in that country—its prominence in the temples, the important feature it forms in the limited furniture of a Japanese household, and the wonderful property (which has apparently

Scitation is the online home of leading journals and conference proceedings from AIP Publishing and AIP Member Societies

We examine the structure of five Chinese magic
mirror replicas using a special imaging technique developed by the
authors. All mirrors are found to have a two-layered structure. The
reflecting surface that gives rise to a projected magic pattern on the
screen is hidden under a polished half-reflecting top layer. An
alternative method of making the magic mirror using ancient
technology has been proposed. Finally, we suggest a simple method
of reconstructing a mirror replica in the laboratory.

The technique based on the concept of the ancient Chinese holy brass mirror, the magic mirror or 'Makyoh', is utilised to inspect and characterise the surfaces of polished GaAs wafers. This non-destructive method was first introduced to examine the polished surfaces of silicon wafers and epitaxial layers. Undulations and latent damage on the mirror-like polished surfaces are easily visualised over the entire substrate in real time. Thus study extends the magic mirror method to inspect III-V compound crystals, Some polished GaAs, GaP and InP wafers are examined and discussed. Saw marks, dimples, polish scratches, texture defects and other damage, e.g. twin lines, facet boundaries and striations, hidden in the polished surface are observed over the entire wafer. Comparison between etched wafers and unetched wafers observed by the magic mirror and revealing similar deformations is discussed.

Makyoh, the “Magic Mirror” is a very useful tool with which to visualize local irregularities of the surface of a mirror-like polished silicon wafer. Using the Makyoh method, bright and dark spots are visible in the image of wafers projected on an instrument screen. However, these spots have not yet been correlated to specific defects on a wafer surface. First, local convex-type defects on the mirror-like surface of silicon wafers are observed with Makyoh, a flatness tester and other micro surface measurement systems. Next, we report our attempts to create actrual convex and concave shapes on silicon wafer surfaces by intentionally varying polishing conditions, and which shapes are observed. We also discuss the relationship between silicon wafer flatness and the Makyoh magic mirror image.

A geometrical optical model for the image formation of Makyoh (or magic-mirror) topography is presented. General relations are given on the optical settings. The basic equations of the imaging are derived. Fundamental features of the imaging are pointed out, and optimum working conditions are established. Simulations of the image of an isolated defect (hillock or depression) and a periodic (sinusoidal) surface are presented for different optical settings. Supporting experimental images are presented.

An awe-inspiring mirror used for centuries in religions of the Far East has been polished and brushed up to become a powerful scientific tool. This “magic mirror”, very simple and non-destructive, transforms latent damage, scratches, waviness and other flaws on mirror-like surfaces into visual images using the “Makyoh” principle. The technique detects undulations of a few nm over a distance of about 0.5 mm. It has recently been used to characterize highly finished mirror-polished large diameter Si wafers for ULSI applications, replacing the laborious and unstable naked eye wafer inspection lines. Our studies have shown that the technique is also very useful for monitoring surface sensitive IC processes such as epitaxy, chemical vapor deposition, rapid thermal process, ion implantation, etc.

A method to detect, locate, and estimate edges in a
one-dimensional signal is presented. It is inherently more accurate than
all previous schemes as it explicitly models and corrects interaction
between nearby edges. The method is iterative with initial estimation of
edges provided by the zero crossings of the signal convolved with
Laplacian of Gaussian (LoG) filter. The necessary computations
necessitate knowledge of this convolved output only in a neighborhood
around each zero crossing and in most cases, could be performed locally
by independent parallel processors. Results on one-dimensional slices
extracted from real images, and on images which have been proposed
independently in the row and column directions are shown. An analysis of
the method is provided including issues of complexity and convergence,
and directions of future research are outlined

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