{"id":1238,"date":"2017-08-03T11:18:24","date_gmt":"2017-08-03T02:18:24","guid":{"rendered":"http:\/\/atelier.bonryu.com\/?page_id=1238"},"modified":"2017-10-29T21:06:13","modified_gmt":"2017-10-29T12:06:13","slug":"rem4-optimumdia","status":"publish","type":"page","link":"https:\/\/atelier.bonryu.com\/en\/welcome\/lensless\/phphoto-l\/rem4-optimumdia\/","title":{"rendered":"Appendix_4: Optimum Diameter of Pinhole"},"content":{"rendered":"<p style=\"text-align: right;\"><span style=\"color: #000000;\">[mathjax]<br \/><ul class=\"bogo-language-switcher list-view\"><li class=\"en-US en current first\"><span class=\"bogoflags bogoflags-us\"><\/span> <span class=\"bogo-language-name\"><a rel=\"alternate\" hreflang=\"en-US\" href=\"https:\/\/atelier.bonryu.com\/en\/wp-json\/wp\/v2\/pages\/1238\/\" title=\"English\" class=\"current\" aria-current=\"page\">English<\/a><\/span><\/li>\n<li class=\"ja last\"><span class=\"bogoflags bogoflags-jp\"><\/span> <span class=\"bogo-language-name\"><a rel=\"alternate\" hreflang=\"ja\" href=\"https:\/\/atelier.bonryu.com\/wp-json\/wp\/v2\/pages\/1238\/\" title=\"Japanese\">\u65e5\u672c\u8a9e<\/a><\/span><\/li>\n<\/ul><\/span><\/p>\n<h3 class=\"paragraph_style_5\"><span style=\"color: #339966;\"><strong><span class=\"style_1\">Huygens\u2019 Principle<\/span><\/strong><\/span><\/h3>\n<p class=\"paragraph_style_5\">In order to understand why there is the\u00a0<strong>optimum value for a pinhole diameter<\/strong>, we consider the mechanism of the pinhole phenomenon by using a simple model.\u00a0 We analyze the problem on the basis of the famous <strong>Huygens\u2019 principle <\/strong>(1690, Holland, Christiaan Huygens, 1629 -1695).\u00a0 It should be noted that, as by the <strong>original Huygens\u2019 principle<\/strong> the diffraction phenomenon of a light wave cannot be explained properly, the <strong>principle of Huygens and Fresnel<\/strong> is used for the analysis <strong>as the &#8220;Huygens&#8217; Principle&#8221;<\/strong>, where the effect of the interference is taken into account appropriately.<\/p>\n<p class=\"paragraph_style_5\">According to the Huygens\u2019 principle, secondary spherical waves are emitted from points on the surface wavefront of an incident light wave and the envelop of the secondary spherical waves form a new surface wavefront (figure below).\u00a0 Therefore, a light wave with a flat surface (a plane wave) remains a plane wave without change in every point endlessly.\u00a0<\/p>\n<p><strong><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-732 size-full\" src=\"http:\/\/atelier.bonryu.com\/wordpress\/wp-content\/uploads\/2017\/06\/PH_E01_PlaneWs.jpg\" alt=\"\" width=\"173\" height=\"187\" \/><\/strong><\/p>\n<p class=\"paragraph_style_6\"><strong>Huygens\u2019 principle<br \/><\/strong><em>A new surface wavefront is formed by an envelope (a red line) of secondary\u00a0spherical\u00a0waves each of which is emitted from a point on a previous surface wavefront as a source.<\/em><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>According to the Huygens\u2019 principle, secondary spherical waves are emitted from points on a surface wavefront of an incident light wave and the envelop of the secondary waves form a new surface wavefront (above figure).\u00a0 Therefore, a light wave with a flat surface (a plane wave) remains a plane wave without change in every point.\u00a0<\/p>\n<p>However, when the plane wave encounters a wall with a very small pinhole, a new secondary spherical wave is emitted from the pinhole as a point source (figure below).\u00a0 The new wavefront is that of the secondary wave itself because the pinhole is very small and the source is considered mathematically as a \u00a0point without an area.\u00a0 By applying the Huygens\u2019 principle to this spherical wave, it is understandable that the light wave after passing through the pinhole remains still a spherical wave in every point.\u00a0 In this case the \u201dpinhole phenomenon\u201c cannot be observed because the light wave does not go straight to the traveling direction of the incident plane wave.<\/p>\n<p>&nbsp;<\/p>\n<p class=\"paragraph_style_8\"><strong><a href=\"http:\/\/atelier.bonryu.com\/welcome\/lensless\/phphoto-l\/rem4-optimumdia\/sphericalwave\/\"><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-1542 size-full\" src=\"http:\/\/atelier.bonryu.com\/wordpress\/wp-content\/uploads\/2017\/08\/SphericalWave.jpg\" alt=\"\" width=\"394\" height=\"284\" srcset=\"https:\/\/atelier.bonryu.com\/wordpress\/wp-content\/uploads\/2017\/08\/SphericalWave.jpg 394w, https:\/\/atelier.bonryu.com\/wordpress\/wp-content\/uploads\/2017\/08\/SphericalWave-300x216.jpg 300w\" sizes=\"auto, (max-width: 394px) 100vw, 394px\" \/><\/a>Small pinhole case<\/strong><br \/><em>When a diameter of a pinhole is extremely small a light of a plane wave passing through the pinhole becomes a spherical wave.<\/em><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p class=\"paragraph_style_8\"><strong><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-1543 size-full\" src=\"http:\/\/atelier.bonryu.com\/wordpress\/wp-content\/uploads\/2017\/08\/PlaneWave.jpg\" alt=\"\" width=\"357\" height=\"289\" srcset=\"https:\/\/atelier.bonryu.com\/wordpress\/wp-content\/uploads\/2017\/08\/PlaneWave.jpg 357w, https:\/\/atelier.bonryu.com\/wordpress\/wp-content\/uploads\/2017\/08\/PlaneWave-300x243.jpg 300w\" sizes=\"auto, (max-width: 357px) 100vw, 357px\" \/>Large pinhole case<\/strong><br \/><em>When a diameter of a pinhole is very large a light of a plane wave passing through the pinhole remains a plane wave.<\/em><\/p>\n<p><strong>\u00a0<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<h3><span style=\"color: #339966;\">Smallness of a Pinhole<\/span><\/h3>\n<p>Then, how small is a diameter of an\u00a0<strong>extremely small pinhole<\/strong>? We consider the length on the basis of the wave length of the visible light wave, i.e., \\(400 &#8211; 700 nm\\) (\\(1 nm = 1\/1000000 mm\\)). \u00a0This gives a criterion whether the light wave after passing through a pinhole becomes a spherical wave or a plane wave. \u00a0However, the distance between a pinhole and an imaging screen is also important when we consider if the image on the screen will blur after passing through the pinhole due to the diffraction phenomenon. \u00a0When we use a pinhole camera with this distance of \\(50 mm\\) the effect of the diffraction phenomenon will be observed for a pinhole with the diameter of \\(0.1 mm\\) (\\(=100,000 nm\\)). \u00a0This length is 200 times as large as wavelength of visible light. \u00a0It should be remarked that even a pinhole with such a<strong> &#8220;large diameter&#8221;<\/strong> is an <strong>extremely small pinhole<\/strong>!! \u00a0If a diameter of a pinhole is very large in comparison with the wavelength of light, a plane wave of light entering the pinhole proceeds rectilinearly as a plane wave after passing through the pinhole. \u00a0You may be disinclined against the fact that a light ray with a diameter of, for example, \\(0.3 mm\\) is a plane wave, but this length is about 1000 times as large as wavelength of the visible light and the light ray is surely a plane wave.<\/p>\n<h3><span style=\"color: #339966;\"><strong><span class=\"style_1\">Behavior of a Diffracted Light Wave<\/span><\/strong><\/span><\/h3>\n<p>It may be instructive to display the dependence of the behavior of the light beam passing through a pinhole on its size. \u00a0We consider the behavior of a light wave with a wavelength \\(\\lambda\\) of \\(550 nm\\)\u00a0emitted from a point source at infinity. \u00a0For simplicity we assume that the light wave passes through a slit instead of a pinhole and calculate the wave propagation on the basis of the Fresnel diffraction.\u00a0 Contour plots of intensity distribution (white: high intensity, black: low intensity) of the light for cases of different slit widths, \\(d=0.05, 0.1, 0.15\\), and \\(0.3 mm\\) are shown in the following figure. \u00a0Pink bands denote paths of the light wave when the diffraction is not taken into account. \u00a0Though both the units of the vertical and the horizontal axes are millimeters in this figure, it should be noted that the horizontal axis is extremely expanded.\u00a0 Because the effect of the diffraction is more noticeable in the distance from the pinhole plate and in the case of small pinhole, an image projected on a distant screen becomes more sharper with increasing width of the slit.<\/p>\n<p><span class=\"style_7\">\u3000<\/span> <span class=\"style_7\"><strong><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-741 size-full\" src=\"http:\/\/atelier.bonryu.com\/wordpress\/wp-content\/uploads\/2017\/06\/ph_proc_a.jpg\" alt=\"\" width=\"529\" height=\"237\" srcset=\"https:\/\/atelier.bonryu.com\/wordpress\/wp-content\/uploads\/2017\/06\/ph_proc_a.jpg 529w, https:\/\/atelier.bonryu.com\/wordpress\/wp-content\/uploads\/2017\/06\/ph_proc_a-300x134.jpg 300w\" sizes=\"auto, (max-width: 529px) 100vw, 529px\" \/><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-742 size-full\" src=\"http:\/\/atelier.bonryu.com\/wordpress\/wp-content\/uploads\/2017\/06\/ph_proc_b.jpg\" alt=\"\" width=\"529\" height=\"276\" srcset=\"https:\/\/atelier.bonryu.com\/wordpress\/wp-content\/uploads\/2017\/06\/ph_proc_b.jpg 529w, https:\/\/atelier.bonryu.com\/wordpress\/wp-content\/uploads\/2017\/06\/ph_proc_b-300x157.jpg 300w\" sizes=\"auto, (max-width: 529px) 100vw, 529px\" \/><\/strong><\/span> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p class=\"paragraph_style_6\"><strong>Dependence of diffracted light intensity on pinhole size (slit width)<\/strong>\u00a0<br \/><em>In this figure a slit is located on the horizontal axis and the incident light comes from beneath.\u00a0 The pink band denotes the light beam without the diffraction phenomenon.\u00a0 Sub-figures (a), (b), (c), and (d) show the cases with the slit width \\(d=0.05, 0.1, 0.15\\), and \\(0.3 mm\\), respectively.<\/em><\/p>\n<p>From the above figure it is not difficult to understand that the diffracted light diverges rapidly in the case of a narrow slit as \\(d=0.05 mm\\) but by using a wider slit as \\(d=0.15 mm\\) the width of the light beam is confined within the width of the slit at the distance of \\(100 mm\\) from the pinhole plane.\u00a0 We made calculation of the distribution of the light intensity at the focal plane for the case of the slit width of \\(d=0.15mm\\) (figure below).\u00a0 It is also clearly seen that the minimum beam width is attained for \\(d=0.15 mm\\).<\/p>\n<p>&nbsp;<\/p>\n<div id=\"id21\" class=\"style_SkipStroke_3 shape-with-text\">\n<div class=\"text-content graphic_textbox_layout_style_default_External_290_72\">\n<div class=\"graphic_textbox_layout_style_default\">\n<p class=\"paragraph_style_7\"><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-743 size-full\" src=\"http:\/\/atelier.bonryu.com\/wordpress\/wp-content\/uploads\/2017\/06\/PH_E06_Slit100.jpg\" alt=\"\" width=\"288\" height=\"177\" \/><strong><span class=\"style_3\">Distribution of the light intensity on the imaging plane (f=100 mm)<\/span> <span class=\"style_4\">for different slit widths, \\(d=0.05, 0.1, 0.15\\), and \\(0.3 mm\\).<\/span><\/strong><\/p>\n<p class=\"paragraph_style_7\">\u00a0<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div id=\"id22\" class=\"style_SkipStroke_3 shape-with-text\">\n<div class=\"text-content graphic_textbox_layout_style_default_External_632_119\">\n<div class=\"graphic_textbox_layout_style_default\">\n<h3 class=\"paragraph paragraph_style_11\"><span class=\"style_5\" style=\"color: #339966;\">Optimum diameter of a pinhole<\/span><\/h3>\n<div class=\"paragraph paragraph_style_11\">A candidate of the optimum pinhole diameter is the <strong>diameter of the central circle of a zone plate<\/strong>, $$d=2a=2\\sqrt{\\lambda f}$$. \u00a0This formula is obtained by assuming that a light passing through the edge of the pinhole is negated by interference with the light from the center of the pinhole. \u00a0But this assumption does not seem to have a substantial reason, and it is not easy to justify this equation as a formula to determine the optimum pinhole diameter.\u00a0 Next, we consider the diameter on the basis of the Fraunhofer diffraction formula.\u00a0 Exactly speaking, it is questionable whether the equation of the Fraunhofer diffraction can be applied or not to this problem.\u00a0 However, the equation is not so bad anyway and may give a good approximate result in some measure.\u00a0 On the basis of the Fraunhofer diffraction one can derive an equation describing the intensity distribution of the diffracted light on the focal plane.\u00a0 The result shows that in the central circle the light is most intense and concentric circler zones surround the central circle, where the light intensity is very weak.\u00a0 The radius of the central circle is derived as $$r \\cong 3.832 \\frac{\\lambda f} {2 \\pi a}$$. \u00a0For the \u00a0optimization of pinhole diameter we impose a condition that the radius of the central circle of the image is equal to the radius of the pinhole (\\(r=a\\)).\u00a0 Then the pinhole radius is given as follows. \u00a0$$a \\cong \\sqrt{0.6098 \\lambda f} \\cong 0.78 \\sqrt{\\lambda f}, \u00a0 d=2a\\cong 1.56 \\sqrt{\\lambda f}$$<\/div>\n<p class=\"paragraph_style_11\">This equation is the same one as that in the main text.\u00a0 There are more other formulas with different coefficient. \u00a0But differences are not important. \u00a0It is important that the diameter of the pinhole is proportional to \\(\\sqrt{\\lambda}$$\/. As there are a certain finite extent of the range of wavelength of the visible light the optimum value of diameter of a pinhole becomes different for different choice of wavelength. \u00a0But such a difference is not an obstacle to take a photograph.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div>\u00a0<\/div>\n<div>\n<p class=\"paragraph_style_6\">\u00a0<\/p>\n<p class=\"paragraph_style_6\"><strong><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-1541 size-full\" src=\"http:\/\/atelier.bonryu.com\/wordpress\/wp-content\/uploads\/2017\/08\/LightIntensDist.jpg\" alt=\"\" width=\"478\" height=\"455\" srcset=\"https:\/\/atelier.bonryu.com\/wordpress\/wp-content\/uploads\/2017\/08\/LightIntensDist.jpg 478w, https:\/\/atelier.bonryu.com\/wordpress\/wp-content\/uploads\/2017\/08\/LightIntensDist-300x286.jpg 300w\" sizes=\"auto, (max-width: 478px) 100vw, 478px\" \/>Determination of the optimum pinhole diameter<br \/><\/strong><em>Determine the pinhole diameter from the diameter of the central circle of the distribution of the light diffracted by a circular pinhole.<\/em><\/p>\n<p class=\"paragraph_style_6\"><strong>\u00a0<\/strong><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>[mathjax] Huygens\u2019 Principle In order to understand why there is the\u00a0optimum value for a pinhole diameter, we consider the mechanism of the pinhole phenomenon by using a simple model.\u00a0 We analyze the problem on the basis of the famous Huygens\u2019 principle (1690, Holland, Christiaan Huygens, 1629 -1695).\u00a0 It should be noted that, as by the &hellip; <a href=\"https:\/\/atelier.bonryu.com\/en\/welcome\/lensless\/phphoto-l\/rem4-optimumdia\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Appendix_4: Optimum Diameter of Pinhole<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":892,"menu_order":139,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_locale":"en_US","_original_post":"671","footnotes":""},"class_list":["post-1238","page","type-page","status-publish","hentry","en-US"],"_links":{"self":[{"href":"https:\/\/atelier.bonryu.com\/wp-json\/wp\/v2\/pages\/1238","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/atelier.bonryu.com\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/atelier.bonryu.com\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/atelier.bonryu.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/atelier.bonryu.com\/wp-json\/wp\/v2\/comments?post=1238"}],"version-history":[{"count":10,"href":"https:\/\/atelier.bonryu.com\/wp-json\/wp\/v2\/pages\/1238\/revisions"}],"predecessor-version":[{"id":1613,"href":"https:\/\/atelier.bonryu.com\/wp-json\/wp\/v2\/pages\/1238\/revisions\/1613"}],"up":[{"embeddable":true,"href":"https:\/\/atelier.bonryu.com\/wp-json\/wp\/v2\/pages\/892"}],"wp:attachment":[{"href":"https:\/\/atelier.bonryu.com\/wp-json\/wp\/v2\/media?parent=1238"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}