31:Tbc0b,viewgeo = svg` `32:Tc67e,transfer = svg ` `33:T8bdb,realimg = svg ``34:[] 2f:{"title":"Benton Math","cells":[{"id":0,"value":"md`# Benton Math`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2,"value":"md `This notebook is an interactive synthesis of Stephen Benton's model for calculating how to make a rainbow hologram, hence the name *Benton Math*. [H1](#glossary) and [H2](#glossary) cameras are created to record a monochromatic hologram, reconstructed by white light. The typical scenario for using Benton math, is to define the colour of your hologram (for instance, yellow ), knowing that you are using a particular colour laser (such as, red ). The steps below are carried out to convert the red colour of the laser into a yellow hologram that the viewer sees.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":690,"value":"md`Two major calculations used to perform this conversion, are known as the [Grating Equation](#glossary) and the [Lens Law](#gloaary). Before doing these calculations, it is essential to paint a picture of what is known as the *reconstruction geometry* which defines how the viewer \"sees\" a hologram, and how white light reconstructs the image.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":667,"value":"html `

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":9,"value":"viewg = md`## Reconstruction Geometry`","pinned":false,"hidden":false,"mode":"ojs"},{"id":913,"value":"md`Viewing the hologram`","pinned":false,"hidden":false,"mode":"ojs"},{"id":60,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":343,"value":"DOM.download(serialize(viewgeo), null, \"Download as SVG\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":93,"value":"html `

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":43,"value":"grating = md `## Grating Equation`","pinned":false,"hidden":false,"mode":"ojs"},{"id":69,"value":"md `Used to calculate the reference angle for the final hologram ([H2](#glossary)) given the viewing parameters.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":11,"value":"tex.block `\\sin \\space Reference \\space \\theta = \\sin \\space Desired \\space Reconstruction \\space Angle \\space \\theta \\bigl( \\frac{Laser \\space \\lambda}{Desired \\space \\lambda} \\bigr)`","pinned":false,"hidden":false,"mode":"ojs"},{"id":71,"value":"md `Sin of reference is the sine of the reference angle for [H2](#glossary). Sin of desired reconstruction is the desired reconstruction angle of [H2](#glossary). Laser lambda is the wavelength of the laser used in making H2. Desired lambda is the desired colour seen at the normal of [H2](#glossary).`","pinned":false,"hidden":false,"mode":"ojs"},{"id":41,"value":"html `

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":39,"value":"llaw = md `## Lens Law`","pinned":false,"hidden":false,"mode":"ojs"},{"id":5,"value":"tex.block `\\frac{1}{F} = \\frac{1}{Rr} + \\frac{1}{Rc}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":25,"value":"md`F is the focal length. Rr (ray real) is the viewing distance. Rc (ray conjugate) is the illumination distance.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":91,"value":"html `

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":95,"value":"html `

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":101,"value":"tgeo = md`## Transfer Geometry`","pinned":false,"hidden":false,"mode":"ojs"},{"id":916,"value":"md`H1 to H2 transfer`","pinned":false,"hidden":false,"mode":"ojs"},{"id":104,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":333,"value":"DOM.download(serialize(transfer), null, \"Download as SVG\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":686,"value":"html`

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":114,"value":"step1 = md `__Step 1:__ Calculating the focal length for the final hologram (H2), using our model: `","pinned":false,"hidden":false,"mode":"ojs"},{"id":450,"value":"viewof Rc = {\n const div = html`Illumination Distance: cm
`;\n const text = div.querySelector(\"[type=text]\");\n div.value = text.value = 150;\n text.oninput = () => div.value = text.value;\n return div;\n}\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":458,"value":"viewof Rr = {\n const div = html`Viewing Distance: cm
`;\n const text = div.querySelector(\"[type=text]\");\n div.value = text.value = 75;\n text.oninput = () => div.value = text.value;\n return div;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":275,"value":"tex.block `\\frac{1}{F} = \\frac{1}{${Rr}} + \\frac{1}{${Rc}}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":602,"value":"F = 1/(1/Rc + 1/Rr)","pinned":false,"hidden":false,"mode":"ojs"},{"id":277,"value":"md`__F_yellow__ = ${1/(1/Rc + 1/Rr)} cm`","pinned":false,"hidden":false,"mode":"ojs"},{"id":684,"value":"html`

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":280,"value":"step2 = md `__Step 2:__ Determine the correct focal length for red, (a HeNe laser will be used to transfer H2):`","pinned":false,"hidden":false,"mode":"ojs"},{"id":463,"value":"viewof L1 = {\n const div = html`Desired Colour: nm (Lambda 1)
`;\n const text = div.querySelector(\"[type=number]\");\n div.value = text.value = 575;\n text.oninput = () => div.value = text.value;\n return div;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":748,"value":"html` `","pinned":false,"hidden":false,"mode":"ojs"},{"id":468,"value":"viewof L2 = {\n const div = html`Laser Wavelength: nm (Lambda 2)
`;\n const text = div.querySelector(\"[type=number]\");\n div.value = text.value = 632.8;\n text.oninput = () => div.value = text.value;\n return div;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":750,"value":"html` `","pinned":false,"hidden":false,"mode":"ojs"},{"id":282,"value":"tex.block`\\frac{1}{F_{red}} = \\bigl(\\frac{\\lambda2}{\\lambda1}\\bigr)\\bigl(\\frac{1}{F_{yellow}}\\bigr) \\space | \\space \\frac{1}{F_{red}} = \\bigl(\\frac{${L2}}{${L1}}\\bigr)\\bigl(\\frac{1}{${F}}\\bigr)`","pinned":false,"hidden":false,"mode":"ojs"},{"id":580,"value":"F2 = 1/((L2/L1) * (1/F))","pinned":false,"hidden":false,"mode":"ojs"},{"id":311,"value":"md`__F_red__ = ${F2.toFixed(1)} cm`","pinned":false,"hidden":false,"mode":"ojs"},{"id":618,"value":"html`

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":316,"value":"step3 = md`__Step 3:__ Determine the correct H1-H2 distance, given the red focal length. Assume a collimated reconstruction beam with a cylindrical lens at a distance (in cm) from H1.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":626,"value":"viewof Rc1 = {\n const div = html`Cylindrical Lens to H1 distance: cm`;\n const text = div.querySelector(\"[type=text]\");\n div.value = text.value = 300;\n text.oninput = () => div.value = text.value;\n return div;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":319,"value":"md `Isolate the unknown, ${tex `\\frac{1}{Rv}`}, by adding ${tex `\\frac{1}{Rc}`} to each side of the equation.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":323,"value":"tex.block `\\frac{1}{F_{red}} = \\frac{1}{Rv} - \\frac{1}{Rc}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":326,"value":"tex.block `\\frac{1}{Rv} = \\frac{1}{Rc} + \\frac{1}{F_{red}}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":328,"value":"md `Where an Rc is the distance from the cylindrical lens. Rv is the real image projection, and therefore the image plane.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":609,"value":"html`

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":611,"value":"tex.block `\\frac{1}{${F2.toFixed(1)}} = \\frac{1}{Rv} - \\frac{1}{${Rc1}}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":613,"value":"tex.block `\\frac{1}{Rv} = \\frac{1}{${Rc1}} + \\frac{1}{${F2.toFixed(1)}}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":648,"value":"Rv = 1/(1/Rc1 + 1/F2.toFixed(1))","pinned":false,"hidden":false,"mode":"ojs"},{"id":615,"value":"html`

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":363,"value":"step4 = md`__Step 4:__ Calculate H1 to object distance given a reference beam that is diverging from a distance (in cm) from H1.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":367,"value":"tex.block `\\frac{1}{F} = \\frac{1}{Rv} + \\frac{1}{Rc}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":370,"value":"md`where:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":373,"value":"md`Rv is the H1-H2 distance. Rc is the spatial filter distance. F is the H1 - object distance.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":634,"value":"html`

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":675,"value":"mgeo = md`## Master Geometry`","pinned":false,"hidden":false,"mode":"ojs"},{"id":920,"value":"md`Making of H1`","pinned":false,"hidden":false,"mode":"ojs"},{"id":354,"value":"$33","pinned":false,"hidden":false,"mode":"ojs"},{"id":361,"value":"DOM.download(serialize(realimg), null, \"Download as SVG\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":105,"value":"html`

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":640,"value":"viewof Rc2 = {\n const div = html`Spatial filter distance: cm`;\n const text = div.querySelector(\"[type=text]\");\n div.value = text.value = 300;\n text.oninput = () => div.value = text.value;\n return div;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":643,"value":"viewof F3 = {\n const div = html`The distance from the subject to H1: cm`;\n const text = div.querySelector(\"[type=text]\");\n div.value = text.value = 35;\n text.oninput = () => div.value = text.value;\n return div;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":636,"value":"tex.block `\\frac{1}{${F3}} = \\frac{1}{Rv} + \\frac{1}{${Rc2}}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":654,"value":"Rv2 = 1/(1/F3 - 1/Rc2)","pinned":false,"hidden":false,"mode":"ojs"},{"id":637,"value":"html`

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":376,"value":"result = md`## Processed Result`","pinned":false,"hidden":false,"mode":"ojs"},{"id":378,"value":"md`Once processed, H2's emulsion will shrink altering the reconstruction angle. Tip H2 away from the reference beam by 7% of the reference angle.` ","pinned":false,"hidden":false,"mode":"ojs"},{"id":381,"value":"tex.block`.07 x 51 \\space degrees = approx. \\space 3.5 \\space degrees`","pinned":false,"hidden":false,"mode":"ojs"},{"id":386,"value":"md`The scene in H2 will be magnified as follows:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":388,"value":"tex.block`\\frac{H2-H1 \\space distance}{H1-Object \\space distance} = percentage \\space magnification`","pinned":false,"hidden":false,"mode":"ojs"},{"id":392,"value":"tex.block`\\frac{39.6}{35} = 13`","pinned":false,"hidden":false,"mode":"ojs"},{"id":364,"value":"html`

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":198,"value":"md`## References`","pinned":false,"hidden":false,"mode":"ojs"},{"id":202,"value":"md `\nPepper, A. (1989). Holographic Space: a generalised graphic definition. Leonardo (1989), 295-298. [Link](https://muse.jhu.edu/article/609435/pdf) \\n\nPoissant, L. & Nelson, L. (2001). New Media Dictionary: Part IV: Holography. Leonardo 34(4), 370-380. [Link](https://muse.jhu.edu/article/19728)\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":349,"value":"html `

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":51,"value":"glossary = md `## Glossary`","pinned":false,"hidden":false,"mode":"ojs"},{"id":217,"value":"md`_As defined in the New Media Dictionary (2001)_`","pinned":false,"hidden":false,"mode":"ojs"},{"id":295,"value":"md`H1—The symbol used to designate a single-step hologram used as a holographic master. Denisyuk’s reflection holograms and shadowgrams are of the H1 type. Most H1 transmission holograms could only be viewed with a laser until Stephen Benton invented the rainbow hologram (1969), which uses an H1 hologram in which the vertical parallax has been blocked to record a second hologram that, in turn, can be viewed with white light, while maintaining considerable brightness and significant depth of field.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":297,"value":"md`H2—The symbol used to designate a copy hologram. The technique used for copy transmission holography (rainbow holograms—also called “Benton holography” after its inventor), revolutionized the artistic potential of holography in 1969. With this technique, holography was freed from total dependence on the laser as an essential component of the viewing process; artists could conduct aesthetic research working with the prismatic colors of the light spectrum.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":215,"value":"md`ORTHOSCOPIC IMAGE—An image that has the same appearance of depth and viewing angle as the original 3D image. An orthoscopic image may be real or virtual.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":213,"value":"md`PSEUDOSCOPIC IMAGE—An image in which the scene depth (or viewing angle) is inverted, i.e. switched from front to back and from left to right. Looking at a pseudoscopic image can be compared to looking at the mold of a sculpted object (negative image). In holography, such an image is produced by “folding” the hologram on itself, on its vertical axis. A pseudoscopic image may be real or virtual. What was real becomes virtual, and vice versa. When a viewer moves sideways, the foreground elements of the image move in relation to the background, in the direction opposite to that in which the viewer is moving. Pseudoscopic images move away from the viewer. Both transmission and reflection holograms use pseudoscopic images.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":211,"value":"md`REAL IMAGE—The immaterial image that appears in front of the holographic film or plate. Some large holograms (e.g. rainbow holograms) project real images 2 or 3 m in front of the film. A real image may be orthoscopic or pseudoscopic. Shadowgrams produced using a single step transmission technique (H1) and visible with white light always produce a pseudoscopic real image.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":219,"value":"md`VIRTUAL IMAGE—An image produced on the back plane\nof a mirror or hologram recording medium (e.g. on the side opposite the light source). In the case of a hologram, there is no left-right inversion of the subject image. A virtual image is usually orthoscopic, but may also be pseudoscopic.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":404,"value":"html`

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":406,"value":"serialize = {\n const xmlns = \"http://www.w3.org/2000/xmlns/\";\n const xlinkns = \"http://www.w3.org/1999/xlink\";\n const svgns = \"http://www.w3.org/2000/svg\";\n return function serialize(svg) {\n svg = svg.cloneNode(true);\n const fragment = window.location.href + \"#\";\n const walker = document.createTreeWalker(svg, NodeFilter.SHOW_ELEMENT, null, false);\n while (walker.nextNode()) {\n for (const attr of walker.currentNode.attributes) {\n if (attr.value.includes(fragment)) {\n attr.value = attr.value.replace(fragment, \"#\");\n }\n }\n }\n svg.setAttributeNS(xmlns, \"xmlns\", svgns);\n svg.setAttributeNS(xmlns, \"xmlns:xlink\", xlinkns);\n const serializer = new window.XMLSerializer;\n const string = serializer.serializeToString(svg);\n return new Blob([string], {type: \"image/svg+xml\"});\n };\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":739,"value":"import {x, color, wavelengthToColor} from \"@magfoto/wavelengths-and-spectral-colours\"","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W34","stdlib":"1","titling":"auto"}