Article

Oriental magic mirrors and the Laplacian image

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Abstract

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