{"id":1498,"date":"2017-10-27T16:32:16","date_gmt":"2017-10-27T16:32:16","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/thin-film-interference\/"},"modified":"2017-11-08T03:27:19","modified_gmt":"2017-11-08T03:27:19","slug":"thin-film-interference","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/thin-film-interference\/","title":{"raw":"Thin Film Interference","rendered":"Thin Film Interference"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Discuss the rainbow formation by thin films.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id1169736805140\">The bright colors seen in an oil slick floating on water or in a sunlit soap bubble are caused by interference. The brightest colors are those that interfere constructively. This interference is between light reflected from different surfaces of a thin film; thus, the effect is known as <span data-type=\"term\" id=\"import-auto-id1169738035684\">thin film interference<\/span>. As noticed before, interference effects are most prominent when light interacts with something having a size similar to its wavelength. A thin film is one having a thickness [latex]t[\/latex] smaller than a few times the wavelength of light, [latex]\\lambda [\/latex]. Since color is associated indirectly with [latex]\\lambda [\/latex] and since all interference depends in some way on the ratio of [latex]\\lambda [\/latex] to the size of the object involved, we should expect to see different colors for different thicknesses of a film, as in <a href=\"\/contents\/6cd5c4f3-f154-4617-912b-c12f47dd6429@5#fs-id2871611\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"fs-id2871611\">\n<div class=\"bc-figcaption figcaption\">These soap bubbles exhibit brilliant colors when exposed to sunlight. (credit: Scott Robinson, Flickr)<\/div>\n<p><span data-type=\"media\" id=\"fs-id1704308\" data-alt=\"Soap bubbles reflecting mostly purple and blue light with some regions of orange.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_28_00_02a.jpg\" data-media-type=\"image\/jpg\" alt=\"Soap bubbles reflecting mostly purple and blue light with some regions of orange.\" width=\"300\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1169737818762\">What causes thin film interference? <a href=\"#import-auto-id1169737818442\" class=\"autogenerated-content\">(Figure)<\/a> shows how light reflected from the top and bottom surfaces of a film can interfere. Incident light is only partially reflected from the top surface of the film (ray 1). The remainder enters the film and is itself partially reflected from the bottom surface. Part of the light reflected from the bottom surface can emerge from the top of the film (ray 2) and interfere with light reflected from the top (ray 1). Since the ray that enters the film travels a greater distance, it may be in or out of phase with the ray reflected from the top. However, consider for a moment, again, the bubbles in <a href=\"\/contents\/6cd5c4f3-f154-4617-912b-c12f47dd6429@5#fs-id2871611\" class=\"autogenerated-content\">(Figure)<\/a>. The bubbles are darkest where they are thinnest. Furthermore, if you observe a soap bubble carefully, you will note it gets dark at the point where it breaks. For very thin films, the difference in path lengths of ray 1 and ray 2 in <a href=\"#import-auto-id1169737818442\" class=\"autogenerated-content\">(Figure)<\/a> is negligible; so why should they interfere destructively and not constructively? The answer is that a phase change can occur upon reflection. The rule is as follows:<\/p>\n<p id=\"fs-id1169738238309\"><strong>When light reflects from a medium having an index of refraction greater than that of the medium in which it is traveling, a [latex]\\text{180\u00ba}[\/latex] phase change (or a [latex]\\lambda \/2[\/latex]<br>\n     shift) occurs.<\/strong><\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1169737818442\">\n<div class=\"bc-figcaption figcaption\">Light striking a thin film is partially reflected (ray 1) and partially refracted at the top surface. The refracted ray is partially reflected at the bottom surface and emerges as ray 2. These rays will interfere in a way that depends on the thickness of the film and the indices of refraction of the various media.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1169737918080\" data-alt=\"The figure shows three materials, or media, stacked one upon the other. The topmost medium is labeled n one, the next is labeled n two and its thickness is t, and the lowest is labeled n three. A light ray labeled incident light starts in the n one medium and propagates down and to the right to strike the n one n two interface. The ray gets partially reflected and partially refracted. The partially reflected ray is labeled ray one. The refracted ray continues downward in the n two medium and is reflected back up from the n two n three interface. This reflected ray, labeled ray two, refracts again upon passing up through the n two n one interface and continues upward parallel to ray one. Ray one and ray two then enter an observer\u2019s eye.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_28_07_01a.jpg\" data-media-type=\"image\/jpg\" alt=\"The figure shows three materials, or media, stacked one upon the other. The topmost medium is labeled n one, the next is labeled n two and its thickness is t, and the lowest is labeled n three. A light ray labeled incident light starts in the n one medium and propagates down and to the right to strike the n one n two interface. The ray gets partially reflected and partially refracted. The partially reflected ray is labeled ray one. The refracted ray continues downward in the n two medium and is reflected back up from the n two n three interface. This reflected ray, labeled ray two, refracts again upon passing up through the n two n one interface and continues upward parallel to ray one. Ray one and ray two then enter an observer\u2019s eye.\" width=\"216\"><\/span><\/p><\/div>\n<p>If the film in <a href=\"#import-auto-id1169737818442\" class=\"autogenerated-content\">(Figure)<\/a> is a soap bubble (essentially water with air on both sides), then there is a [latex]\\lambda \/2[\/latex] shift for ray 1 and none for ray 2. Thus, when the film is very thin, the path length difference between the two rays is negligible, they are exactly out of phase, and destructive interference will occur at all wavelengths and so the soap bubble will be dark here.<\/p>\n<p id=\"import-auto-id1169738085117\">The thickness of the film relative to the wavelength of light is the other crucial factor in thin film interference. Ray 2 in <a href=\"#import-auto-id1169737818442\" class=\"autogenerated-content\">(Figure)<\/a> travels a greater distance than ray 1. For light incident perpendicular to the surface, ray 2 travels a distance approximately [latex]2t[\/latex] farther than ray 1. When this distance is an integral or half-integral multiple of the wavelength in the medium ([latex]{\\lambda }_{n}=\\lambda \/n[\/latex], where [latex]\\lambda [\/latex] is the wavelength in vacuum and [latex]n[\/latex] is the index of refraction), constructive or destructive interference occurs, depending also on whether there is a phase change in either ray.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1169738090922\">\n<div data-type=\"title\" class=\"title\">Calculating Non-reflective Lens Coating Using Thin Film Interference<\/div>\n<p id=\"import-auto-id1169738079772\">Sophisticated cameras use a series of several lenses. Light can reflect from the surfaces of these various lenses and degrade image clarity. To limit these reflections, lenses are coated with a thin layer of magnesium fluoride that causes destructive thin film interference. What is the thinnest this film can be, if its index of refraction is 1.38 and it is designed to limit the reflection of 550-nm light, normally the most intense visible wavelength? The index of refraction of glass is 1.52.<\/p>\n<p id=\"import-auto-id1169738086089\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id1169738083234\">Refer to <a href=\"#import-auto-id1169737818442\" class=\"autogenerated-content\">(Figure)<\/a> and use [latex]{n}_{1}=\\text{100}[\/latex] for air, [latex]{n}_{2}=1\\text{.}\\text{38}[\/latex], and [latex]{n}_{3}=1\\text{.}\\text{52}[\/latex]. Both ray 1 and ray 2 will have a [latex]\\lambda \/2[\/latex] shift upon reflection. Thus, to obtain destructive interference, ray 2 will need to travel a half wavelength farther than ray 1. For rays incident perpendicularly, the path length difference is [latex]2t[\/latex].<\/p>\n<p id=\"import-auto-id1169738060616\"><strong>Solution<\/strong><\/p>\n<p>To obtain destructive interference here,<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]2t=\\frac{{\\lambda }_{{n}_{2}}}{2}\\text{,}[\/latex]<\/div>\n<p id=\"import-auto-id1169737804641\">where [latex]{\\lambda }_{{n}_{2}}[\/latex] is the wavelength in the film and is given by [latex]{\\lambda }_{{n}_{2}}=\\frac{\\lambda }{{n}_{2}}[\/latex].<\/p>\n<p id=\"import-auto-id1169738074136\">Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-416\">[latex]2t=\\frac{\\lambda \/{n}_{2}}{2}.[\/latex]<\/div>\n<p id=\"import-auto-id1169738052013\">Solving for [latex]t[\/latex] and entering known values yields<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\begin{array}{lll}t&amp; =&amp; \\frac{\\lambda \/{n}_{2}}{4}=\\frac{\\left(\\text{550 nm}\\right)\/\\text{1.38}}{4}\\\\ &amp; =&amp; \\text{99.6 nm.}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id1169737952148\"><strong>Discussion<\/strong><\/p>\n<p>Films such as the one in this example are most effective in producing destructive interference when the thinnest layer is used, since light over a broader range of incident angles will be reduced in intensity. These films are called non-reflective coatings; this is only an approximately correct description, though, since other wavelengths will only be partially cancelled. Non-reflective coatings are used in car windows and sunglasses.<\/p>\n<\/div>\n<p id=\"import-auto-id1169737966586\">Thin film interference is most constructive or most destructive when the path length difference for the two rays is an integral or half-integral wavelength, respectively. That is, for rays incident perpendicularly, [latex]2t={\\lambda }_{n},\\phantom{\\rule{0.25em}{0ex}}{2\\lambda }_{n},\\phantom{\\rule{0.25em}{0ex}}{3\\lambda }_{n},\\dots [\/latex] or [latex]2t={\\lambda }_{n}\/2,\\phantom{\\rule{0.25em}{0ex}}{3\\lambda }_{n}\/2,\\phantom{\\rule{0.25em}{0ex}}{5\\lambda }_{n}\/2,\\dots [\/latex]. To know whether interference is constructive or destructive, you must also determine if there is a phase change upon reflection. Thin film interference thus depends on film thickness, the wavelength of light, and the refractive indices. For white light incident on a film that varies in thickness, you will observe rainbow colors of constructive interference for various wavelengths as the thickness varies.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1169736763149\">\n<div data-type=\"title\" class=\"title\">Soap Bubbles: More Than One Thickness can be Constructive<\/div>\n<p id=\"import-auto-id1169737757640\">(a) What are the three smallest thicknesses of a soap bubble that produce constructive interference for red light with a wavelength of 650 nm? The index of refraction of soap is taken to be the same as that of water. (b) What three smallest thicknesses will give destructive interference?<\/p>\n<p id=\"import-auto-id1169737042137\"><strong>Strategy and Concept<\/strong><\/p>\n<p id=\"import-auto-id1169738211471\">Use <a href=\"#import-auto-id1169737818442\" class=\"autogenerated-content\">(Figure)<\/a> to visualize the bubble. Note that [latex]{n}_{1}={n}_{3}=1\\text{.}\\text{00}[\/latex] for air, and [latex]{n}_{2}=1\\text{.}\\text{333}[\/latex] for soap (equivalent to water). There is a [latex]\\lambda \/2[\/latex] shift for ray 1 reflected from the top surface of the bubble, and no shift for ray 2 reflected from the bottom surface. To get constructive interference, then, the path length difference ([latex]2t[\/latex]) must be a half-integral multiple of the wavelength\u2014the first three being [latex]{\\lambda }_{n}\/2,\\phantom{\\rule{0.25em}{0ex}}{3\\lambda }_{n}\/2[\/latex], and [latex]{5\\lambda }_{n}\/2[\/latex]. To get destructive interference, the path length difference must be an integral multiple of the wavelength\u2014the first three being [latex]0,\\phantom{\\rule{0.25em}{0ex}}{\\lambda }_{n}[\/latex], and [latex]{2\\lambda }_{n}[\/latex].<\/p>\n<p id=\"import-auto-id1169738005796\"><strong>Solution for (a)<\/strong><\/p>\n<p id=\"import-auto-id1169737788882\"><em data-effect=\"italics\">Constructive interference<\/em> occurs here when<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]2{t}_{\\text{c}}=\\frac{{\\lambda }_{n}}{2},\\phantom{\\rule{0.25em}{0ex}}\\frac{{3\\lambda }_{n}}{2},\\phantom{\\rule{0.25em}{0ex}}\\frac{{5\\lambda }_{n}}{2}\\text{,}\\dots .[\/latex]<\/div>\n<p id=\"import-auto-id1169736800518\">The smallest constructive thickness [latex]{t}_{\\text{c}}[\/latex] thus is<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-889\">[latex]\\begin{array}{lll}{t}_{\\text{c}}&amp; =&amp; \\frac{{\\lambda }_{n}}{4}=\\frac{\\lambda \/n}{4}=\\frac{\\left(\\text{650}\\phantom{\\rule{0.25em}{0ex}}\\text{nm}\\right)\/1\\text{.}\\text{333}}{4}\\\\ &amp; =&amp; \\text{122 nm.}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id1169738220151\">The next thickness that gives constructive interference is [latex]{t\\prime }_{\\text{c}}={3\\lambda }_{n}\/4[\/latex], so that<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{t\\prime }_{\\text{c}}=\\text{366 nm.}[\/latex]<\/div>\n<p id=\"import-auto-id1169738136931\">Finally, the third thickness producing constructive interference is [latex]{t\\prime \\prime }_{\\text{c}}\\le {5\\lambda }_{n}\/4[\/latex], so that<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{t\\prime \\prime }_{\\text{c}}=\\text{610}\\phantom{\\rule{0.25em}{0ex}}\\text{nm.}[\/latex]<\/div>\n<p id=\"import-auto-id1169738145668\"><strong>Solution for (b)<\/strong><\/p>\n<p id=\"import-auto-id1169737979425\">For <em data-effect=\"italics\">destructive interference<\/em>, the path length difference here is an integral multiple of the wavelength. The first occurs for zero thickness, since there is a phase change at the top surface. That is,<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{t}_{\\text{d}}=0.[\/latex]<\/div>\n<p id=\"import-auto-id1169738074282\">The first non-zero thickness producing destructive interference is<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]2{t\\prime }_{\\text{d}}={\\lambda }_{n}.[\/latex]<\/div>\n<p id=\"import-auto-id1169736588480\">Substituting known values gives<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\begin{array}{lll}{t\\prime }_{\\text{d}}&amp; =&amp; \\frac{\\lambda {}_{n}\\text{}}{2}=\\frac{\\lambda \/n}{2}=\\frac{\\left(\\text{650}\\phantom{\\rule{0.25em}{0ex}}\\text{nm}\\right)\/1\\text{.}\\text{333}}{2}\\\\ &amp; =&amp; \\text{244 nm.}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id1169738212215\">Finally, the third destructive thickness is [latex]2{t\\prime \\prime }_{\\text{d}}={2\\lambda }_{n}[\/latex], so that<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-670\">[latex]\\begin{array}{lll}{t\\prime \\prime }_{\\text{d}}&amp; =&amp; {\\lambda }_{n}=\\frac{\\lambda }{n}=\\frac{\\text{650}\\phantom{\\rule{0.25em}{0ex}}\\text{nm}}{1\\text{.}\\text{333}}\\\\ &amp; =&amp; \\text{488 nm.}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id1169737874215\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id1169737795299\">If the bubble was illuminated with pure red light, we would see bright and dark bands at very uniform increases in thickness. First would be a dark band at 0 thickness, then bright at 122 nm thickness, then dark at 244 nm, bright at 366 nm, dark at 488 nm, and bright at 610 nm. If the bubble varied smoothly in thickness, like a smooth wedge, then the bands would be evenly spaced.<\/p>\n<\/div>\n<p id=\"import-auto-id1169736614823\">Another example of thin film interference can be seen when microscope slides are separated (see <a href=\"#import-auto-id1169738244250\" class=\"autogenerated-content\">(Figure)<\/a>). The slides are very flat, so that the wedge of air between them increases in thickness very uniformly. A phase change occurs at the second surface but not the first, and so there is a dark band where the slides touch. The rainbow colors of constructive interference repeat, going from violet to red again and again as the distance between the slides increases. As the layer of air increases, the bands become more difficult to see, because slight changes in incident angle have greater effects on path length differences. If pure-wavelength light instead of white light is used, then bright and dark bands are obtained rather than repeating rainbow colors.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1169738244250\">\n<div class=\"bc-figcaption figcaption\">(a) The rainbow color bands are produced by thin film interference in the air between the two glass slides. (b) Schematic of the paths taken by rays in the wedge of air between the slides.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1169736629490\" data-alt=\"Figure A shows two microscope slides that have been pressed together. Multicolor swirling rainbow bands are visible coming from the slides. Figure B shows a cross section of two glass slides stacked one on top of the other. The lower slide is horizontal and the upper slide is tilted up at an angle that is larger than the actual angle between slides would be. Two rays come from above and impinge upon the slides. Their refraction and partial reflection is shown at each glass air interface.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_28_07_02a.jpg\" data-media-type=\"image\/jpg\" alt=\"Figure A shows two microscope slides that have been pressed together. Multicolor swirling rainbow bands are visible coming from the slides. Figure B shows a cross section of two glass slides stacked one on top of the other. The lower slide is horizontal and the upper slide is tilted up at an angle that is larger than the actual angle between slides would be. Two rays come from above and impinge upon the slides. Their refraction and partial reflection is shown at each glass air interface.\" width=\"400\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1169737927160\">An important application of thin film interference is found in the manufacturing of optical instruments. A lens or mirror can be compared with a master as it is being ground, allowing it to be shaped to an accuracy of less than a wavelength over its entire surface. <a href=\"#fs-id1169736610625\" class=\"autogenerated-content\">(Figure)<\/a> illustrates the phenomenon called Newton\u2019s rings, which occurs when the plane surfaces of two lenses are placed together. (The circular bands are called Newton\u2019s rings because Isaac Newton described them and their use in detail. Newton did not discover them; Robert Hooke did, and Newton did not believe they were due to the wave character of light.) Each successive ring of a given color indicates an increase of only one wavelength in the distance between the lens and the blank, so that great precision can be obtained. Once the lens is perfect, there will be no rings.<\/p>\n<div class=\"bc-figure figure\" id=\"fs-id1169736610625\">\n<div class=\"bc-figcaption figcaption\">\u201cNewton's rings\u201d interference fringes are produced when two plano-convex lenses are placed together with their plane surfaces in contact. The rings are created by interference between the light reflected off the two surfaces as a result of a slight gap between them, indicating that these surfaces are not precisely plane but are slightly convex. (credit: Ulf Seifert, Wikimedia Commons)<\/div>\n<p><span data-type=\"media\" id=\"fs-id1169738035379\" data-alt=\"This figure shows rainbow-colored concentric rings obtained when two plano-convex lenses are placed together with their flat surfaces in contact.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_28_07_03a.jpg\" data-media-type=\"image\/jpg\" alt=\"This figure shows rainbow-colored concentric rings obtained when two plano-convex lenses are placed together with their flat surfaces in contact.\" width=\"200\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1169737813232\">The wings of certain moths and butterflies have nearly iridescent colors due to thin film interference. In addition to pigmentation, the wing\u2019s color is affected greatly by constructive interference of certain wavelengths reflected from its film-coated surface. Car manufacturers are offering special paint jobs that use thin film interference to produce colors that change with angle. This expensive option is based on variation of thin film path length differences with angle. Security features on credit cards, banknotes, driving licenses and similar items prone to forgery use thin film interference, diffraction gratings, or holograms. Australia led the way with dollar bills printed on polymer with a diffraction grating security feature making the currency difficult to forge. Other countries such as New Zealand and Taiwan are using similar technologies, while the United States currency includes a thin film interference effect.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1169737904590\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Making Connections: Take-Home Experiment\u2014Thin Film Interference<\/div>\n<p id=\"import-auto-id1169737763576\">One feature of thin film interference and diffraction gratings is that the pattern shifts as you change the angle at which you look or move your head. Find examples of thin film interference and gratings around you. Explain how the patterns change for each specific example. Find examples where the thickness changes giving rise to changing colors. If you can find two microscope slides, then try observing the effect shown in <a href=\"#import-auto-id1169738244250\" class=\"autogenerated-content\">(Figure)<\/a>. Try separating one end of the two slides with a hair or maybe a thin piece of paper and observe the effect.<\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169737988256\">\n<h1 data-type=\"title\">Problem-Solving Strategies for Wave Optics<\/h1>\n<p id=\"import-auto-id1169738107852\"><strong>Step 1.<\/strong><em data-effect=\"italics\">Examine the situation to determine that interference is involved<\/em>. Identify whether slits or thin film interference are considered in the problem.<\/p>\n<p><strong>Step 2.<\/strong><em data-effect=\"italics\">If slits are involved<\/em>, note that diffraction gratings and double slits produce very similar interference patterns, but that gratings have narrower (sharper) maxima. Single slit patterns are characterized by a large central maximum and smaller maxima to the sides.<\/p>\n<p id=\"import-auto-id1169737761861\"><strong>Step 3.<\/strong><em data-effect=\"italics\">If thin film interference is involved, take note of the path length difference between the two rays that interfere<\/em>. Be certain to use the wavelength in the medium involved, since it differs from the wavelength in vacuum. Note also that there is an additional [latex]\\lambda \/2[\/latex] phase shift when light reflects from a medium with a greater index of refraction.<\/p>\n<p><strong>Step 4.<\/strong><em data-effect=\"italics\">Identify exactly what needs to be determined in the problem (identify the unknowns)<\/em>. A written list is useful. Draw a diagram of the situation. Labeling the diagram is useful.<\/p>\n<p id=\"import-auto-id1169736864048\"><strong>Step 5.<\/strong><em data-effect=\"italics\">Make a list of what is given or can be inferred from the problem as stated (identify the knowns)<\/em>.<\/p>\n<p id=\"import-auto-id1169738116480\"><strong>Step 6.<\/strong><em data-effect=\"italics\">Solve the appropriate equation for the quantity to be determined (the unknown), and enter the knowns<\/em>. Slits, gratings, and the Rayleigh limit involve equations.<\/p>\n<p id=\"import-auto-id1169737813576\"><strong>Step 7.<\/strong><em data-effect=\"italics\">For thin film interference, you will have constructive interference for a total shift that is an integral number of wavelengths. You will have destructive interference for a total shift of a half-integral number of wavelengths<\/em>. Always keep in mind that crest to crest is constructive whereas crest to trough is destructive.<\/p>\n<p id=\"import-auto-id1169737861300\"><strong>Step 8.<\/strong><em data-effect=\"italics\">Check to see if the answer is reasonable: Does it make sense?<\/em> Angles in interference patterns cannot be greater than [latex]\\text{90\u00ba}[\/latex], for example.<\/p>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul>\n<li id=\"import-auto-id1169737785898\">Thin film interference occurs between the light reflected from the top and bottom surfaces of a film. In addition to the path length difference, there can be a phase change.<\/li>\n<li id=\"import-auto-id1169736645684\">When light reflects from a medium having an index of refraction greater than that of the medium in which it is traveling, a [latex]\\text{180\u00ba}[\/latex] phase change (or a [latex]\\lambda \/2[\/latex] shift) occurs.<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id1169738034459\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169737725267\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169737994783\">\n<p id=\"import-auto-id1169737949657\">What effect does increasing the wedge angle have on the spacing of interference fringes? If the wedge angle is too large, fringes are not observed. Why?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169738110184\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169735706971\">\n<p id=\"import-auto-id1169737852676\">How is the difference in paths taken by two originally in-phase light waves related to whether they interfere constructively or destructively? How can this be affected by reflection? By refraction?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169737935684\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169738055870\">\n<p id=\"import-auto-id1169737850049\">Is there a phase change in the light reflected from either surface of a contact lens floating on a person\u2019s tear layer? The index of refraction of the lens is about 1.5, and its top surface is dry.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169738006717\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169737994677\">\n<p id=\"import-auto-id1169737778532\">In placing a sample on a microscope slide, a glass cover is placed over a water drop on the glass slide. Light incident from above can reflect from the top and bottom of the glass cover and from the glass slide below the water drop. At which surfaces will there be a phase change in the reflected light?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169736581954\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169737970134\">\n<p id=\"import-auto-id1169737812726\">Answer the above question if the fluid between the two pieces of crown glass is carbon disulfide.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169736606633\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169737701857\">\n<p id=\"import-auto-id1169738232888\">While contemplating the food value of a slice of ham, you notice a rainbow of color reflected from its moist surface. Explain its origin.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169736772198\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\">\n<p>An inventor notices that a soap bubble is dark at its thinnest and realizes that destructive interference is taking place for all wavelengths. How could she use this knowledge to make a non-reflective coating for lenses that is effective at all wavelengths? That is, what limits would there be on the index of refraction and thickness of the coating? How might this be impractical?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169738045669\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169737803309\">\n<p id=\"import-auto-id1169737999648\">A non-reflective coating like the one described in <a href=\"#fs-id1169738090922\" class=\"autogenerated-content\">(Figure)<\/a> works ideally for a single wavelength and for perpendicular incidence. What happens for other wavelengths and other incident directions? Be specific.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169737777453\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\">\n<p id=\"import-auto-id1169738162964\">Why is it much more difficult to see interference fringes for light reflected from a thick piece of glass than from a thin film? Would it be easier if monochromatic light were used?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id1169738134093\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169738246614\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169738244998\">\n<p>A soap bubble is 100 nm thick and illuminated by white light incident perpendicular to its surface. What wavelength and color of visible light is most constructively reflected, assuming the same index of refraction as water?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1169738135767\">\n<p id=\"import-auto-id1169737802646\">532 nm (green)<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169738028271\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169736713736\">\n<p id=\"import-auto-id1169737799271\">An oil slick on water is 120 nm thick and illuminated by white light incident perpendicular to its surface. What color does the oil appear (what is the most constructively reflected wavelength), given its index of refraction is 1.40?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169737897068\">\n<p id=\"import-auto-id1169738188366\">Calculate the minimum thickness of an oil slick on water that appears blue when illuminated by white light perpendicular to its surface. Take the blue wavelength to be 470 nm and the index of refraction of oil to be 1.40.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1169737799446\">\n<p id=\"import-auto-id1169736708130\">83.9 nm<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169736624201\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169736693588\">\n<p id=\"import-auto-id1169736581639\">Find the minimum thickness of a soap bubble that appears red when illuminated by white light perpendicular to its surface. Take the wavelength to be 680 nm, and assume the same index of refraction as water.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169736675909\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\">\n<p id=\"import-auto-id1169737787716\">A film of soapy water ([latex]n=1\\text{.}\\text{33}[\/latex]) on top of a plastic cutting board has a thickness of 233 nm. What color is most strongly reflected if it is illuminated perpendicular to its surface?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1169738208568\">\n<p id=\"import-auto-id1169736619360\">620 nm (orange)<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169737132388\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169738186795\">\n<p id=\"import-auto-id1169737788404\">What are the three smallest non-zero thicknesses of soapy water ([latex]n=1\\text{.}\\text{33}[\/latex]) on Plexiglas if it appears green (constructively reflecting 520-nm light) when illuminated perpendicularly by white light? Explicitly show how you follow the steps in <a href=\"#fs-id1169737988256\">Problem Solving Strategies for Wave Optics<\/a>.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169738247055\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169738249231\">\n<p id=\"import-auto-id1169736708026\">Suppose you have a lens system that is to be used primarily for 700-nm red light. What is the second thinnest coating of fluorite (magnesium fluoride) that would be non-reflective for this wavelength?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1169738054339\">\n<p>380 nm<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169735571594\">\n<p id=\"import-auto-id1169738076684\">(a) As a soap bubble thins it becomes dark, because the path length difference becomes small compared with the wavelength of light and there is a phase shift at the top surface. If it becomes dark when the path length difference is less than one-fourth the wavelength, what is the thickest the bubble can be and appear dark at all visible wavelengths? Assume the same index of refraction as water. (b) Discuss the fragility of the film considering the thickness found.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169737802684\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169737861298\">\n<p id=\"import-auto-id1169737861105\">A film of oil on water will appear dark when it is very thin, because the path length difference becomes small compared with the wavelength of light and there is a phase shift at the top surface. If it becomes dark when the path length difference is less than one-fourth the wavelength, what is the thickest the oil can be and appear dark at all visible wavelengths? Oil has an index of refraction of 1.40.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1169737965953\">\n<p>33.9 nm<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169738117735\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169737713189\">\n<p><a href=\"#import-auto-id1169738244250\" class=\"autogenerated-content\">(Figure)<\/a> shows two glass slides illuminated by pure-wavelength light incident perpendicularly. The top slide touches the bottom slide at one end and rests on a 0.100-mm-diameter hair at the other end, forming a wedge of air. (a) How far apart are the dark bands, if the slides are 7.50 cm long and 589-nm light is used? (b) Is there any difference if the slides are made from crown or flint glass? Explain.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169738246205\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169738092976\">\n<p id=\"import-auto-id1169737050952\"><a href=\"#import-auto-id1169738244250\" class=\"autogenerated-content\">(Figure)<\/a> shows two 7.50-cm-long glass slides illuminated by pure 589-nm wavelength light incident perpendicularly. The top slide touches the bottom slide at one end and rests on some debris at the other end, forming a wedge of air. How thick is the debris, if the dark bands are 1.00 mm apart?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1169737973966\">\n<p id=\"import-auto-id1169738116428\">[latex]4\\text{.}\\text{42}\u00d7{\\text{10}}^{-5}\\phantom{\\rule{0.25em}{0ex}}\\text{m}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169737761659\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169736739921\">\n<p id=\"import-auto-id1169737931422\">Repeat <a href=\"#fs-id1169738246614\" class=\"autogenerated-content\">(Figure)<\/a>, but take the light to be incident at a [latex]\\text{45\u00ba}[\/latex] angle.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169736660356\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169736669384\">\n<p id=\"import-auto-id1169737909085\">Repeat <a href=\"#fs-id1169738028271\" class=\"autogenerated-content\">(Figure)<\/a>, but take the light to be incident at a [latex]\\text{45\u00ba}[\/latex] angle.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1169737756588\">\n<p>The oil film will appear black, since the reflected light is not in the visible part of the spectrum.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169737700998\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169738132137\">\n<p id=\"import-auto-id1169738077411\"><strong>Unreasonable Results<\/strong><\/p>\n<p id=\"import-auto-id1169737796054\">To save money on making military aircraft invisible to radar, an inventor decides to coat them with a non-reflective material having an index of refraction of 1.20, which is between that of air and the surface of the plane. This, he reasons, should be much cheaper than designing Stealth bombers. (a) What thickness should the coating be to inhibit the reflection of 4.00-cm wavelength radar? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id1169738036749\">\n<dt>thin film interference<\/dt>\n<dd id=\"fs-id1169736739972\">interference between light reflected from different surfaces of a thin film<\/dd>\n<\/dl>\n<\/div>\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Discuss the rainbow formation by thin films.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id1169736805140\">The bright colors seen in an oil slick floating on water or in a sunlit soap bubble are caused by interference. The brightest colors are those that interfere constructively. This interference is between light reflected from different surfaces of a thin film; thus, the effect is known as <span data-type=\"term\" id=\"import-auto-id1169738035684\">thin film interference<\/span>. As noticed before, interference effects are most prominent when light interacts with something having a size similar to its wavelength. A thin film is one having a thickness <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/> smaller than a few times the wavelength of light, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-167ba1af36068a5016ffce6c6a2d3499_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: 0px;\" \/>. Since color is associated indirectly with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-167ba1af36068a5016ffce6c6a2d3499_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: 0px;\" \/> and since all interference depends in some way on the ratio of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-167ba1af36068a5016ffce6c6a2d3499_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: 0px;\" \/> to the size of the object involved, we should expect to see different colors for different thicknesses of a film, as in <a href=\"\/contents\/6cd5c4f3-f154-4617-912b-c12f47dd6429@5#fs-id2871611\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"fs-id2871611\">\n<div class=\"bc-figcaption figcaption\">These soap bubbles exhibit brilliant colors when exposed to sunlight. (credit: Scott Robinson, Flickr)<\/div>\n<p><span data-type=\"media\" id=\"fs-id1704308\" data-alt=\"Soap bubbles reflecting mostly purple and blue light with some regions of orange.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_28_00_02a.jpg\" data-media-type=\"image\/jpg\" alt=\"Soap bubbles reflecting mostly purple and blue light with some regions of orange.\" width=\"300\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1169737818762\">What causes thin film interference? <a href=\"#import-auto-id1169737818442\" class=\"autogenerated-content\">(Figure)<\/a> shows how light reflected from the top and bottom surfaces of a film can interfere. Incident light is only partially reflected from the top surface of the film (ray 1). The remainder enters the film and is itself partially reflected from the bottom surface. Part of the light reflected from the bottom surface can emerge from the top of the film (ray 2) and interfere with light reflected from the top (ray 1). Since the ray that enters the film travels a greater distance, it may be in or out of phase with the ray reflected from the top. However, consider for a moment, again, the bubbles in <a href=\"\/contents\/6cd5c4f3-f154-4617-912b-c12f47dd6429@5#fs-id2871611\" class=\"autogenerated-content\">(Figure)<\/a>. The bubbles are darkest where they are thinnest. Furthermore, if you observe a soap bubble carefully, you will note it gets dark at the point where it breaks. For very thin films, the difference in path lengths of ray 1 and ray 2 in <a href=\"#import-auto-id1169737818442\" class=\"autogenerated-content\">(Figure)<\/a> is negligible; so why should they interfere destructively and not constructively? The answer is that a phase change can occur upon reflection. The rule is as follows:<\/p>\n<p id=\"fs-id1169738238309\"><strong>When light reflects from a medium having an index of refraction greater than that of the medium in which it is traveling, a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d99e5df9b7f723e173ac5bd5364c01e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#56;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: -1px;\" \/> phase change (or a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-08246831fcb3aaf5611c696383443844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#47;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"27\" style=\"vertical-align: -5px;\" \/><br \/>\n     shift) occurs.<\/strong><\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1169737818442\">\n<div class=\"bc-figcaption figcaption\">Light striking a thin film is partially reflected (ray 1) and partially refracted at the top surface. The refracted ray is partially reflected at the bottom surface and emerges as ray 2. These rays will interfere in a way that depends on the thickness of the film and the indices of refraction of the various media.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1169737918080\" data-alt=\"The figure shows three materials, or media, stacked one upon the other. The topmost medium is labeled n one, the next is labeled n two and its thickness is t, and the lowest is labeled n three. A light ray labeled incident light starts in the n one medium and propagates down and to the right to strike the n one n two interface. The ray gets partially reflected and partially refracted. The partially reflected ray is labeled ray one. The refracted ray continues downward in the n two medium and is reflected back up from the n two n three interface. This reflected ray, labeled ray two, refracts again upon passing up through the n two n one interface and continues upward parallel to ray one. Ray one and ray two then enter an observer\u2019s eye.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_28_07_01a.jpg\" data-media-type=\"image\/jpg\" alt=\"The figure shows three materials, or media, stacked one upon the other. The topmost medium is labeled n one, the next is labeled n two and its thickness is t, and the lowest is labeled n three. A light ray labeled incident light starts in the n one medium and propagates down and to the right to strike the n one n two interface. The ray gets partially reflected and partially refracted. The partially reflected ray is labeled ray one. The refracted ray continues downward in the n two medium and is reflected back up from the n two n three interface. This reflected ray, labeled ray two, refracts again upon passing up through the n two n one interface and continues upward parallel to ray one. Ray one and ray two then enter an observer\u2019s eye.\" width=\"216\" \/><\/span><\/p>\n<\/div>\n<p>If the film in <a href=\"#import-auto-id1169737818442\" class=\"autogenerated-content\">(Figure)<\/a> is a soap bubble (essentially water with air on both sides), then there is a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-08246831fcb3aaf5611c696383443844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#47;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"27\" style=\"vertical-align: -5px;\" \/> shift for ray 1 and none for ray 2. Thus, when the film is very thin, the path length difference between the two rays is negligible, they are exactly out of phase, and destructive interference will occur at all wavelengths and so the soap bubble will be dark here.<\/p>\n<p id=\"import-auto-id1169738085117\">The thickness of the film relative to the wavelength of light is the other crucial factor in thin film interference. Ray 2 in <a href=\"#import-auto-id1169737818442\" class=\"autogenerated-content\">(Figure)<\/a> travels a greater distance than ray 1. For light incident perpendicular to the surface, ray 2 travels a distance approximately <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-388d9ae1fdf29c1b93b9dbc474e03c08_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: 0px;\" \/> farther than ray 1. When this distance is an integral or half-integral multiple of the wavelength in the medium (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-cd17186e3159ac48c1ad65244260683c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#47;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -5px;\" \/>, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-167ba1af36068a5016ffce6c6a2d3499_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: 0px;\" \/> is the wavelength in vacuum and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> is the index of refraction), constructive or destructive interference occurs, depending also on whether there is a phase change in either ray.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1169738090922\">\n<div data-type=\"title\" class=\"title\">Calculating Non-reflective Lens Coating Using Thin Film Interference<\/div>\n<p id=\"import-auto-id1169738079772\">Sophisticated cameras use a series of several lenses. Light can reflect from the surfaces of these various lenses and degrade image clarity. To limit these reflections, lenses are coated with a thin layer of magnesium fluoride that causes destructive thin film interference. What is the thinnest this film can be, if its index of refraction is 1.38 and it is designed to limit the reflection of 550-nm light, normally the most intense visible wavelength? The index of refraction of glass is 1.52.<\/p>\n<p id=\"import-auto-id1169738086089\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id1169738083234\">Refer to <a href=\"#import-auto-id1169737818442\" class=\"autogenerated-content\">(Figure)<\/a> and use <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c25b3e8a7bbd6cdf8b588d91ee80070d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#95;&#123;&#49;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"69\" style=\"vertical-align: -4px;\" \/> for air, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-83077c4d97708bd9fbb6894a4305b99b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#95;&#123;&#50;&#125;&#61;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"74\" style=\"vertical-align: -3px;\" \/>, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7b1cc233a58874bc7011dc194d94d3f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#95;&#123;&#51;&#125;&#61;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"73\" style=\"vertical-align: -3px;\" \/>. Both ray 1 and ray 2 will have a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-08246831fcb3aaf5611c696383443844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#47;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"27\" style=\"vertical-align: -5px;\" \/> shift upon reflection. Thus, to obtain destructive interference, ray 2 will need to travel a half wavelength farther than ray 1. For rays incident perpendicularly, the path length difference is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-388d9ae1fdf29c1b93b9dbc474e03c08_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<p id=\"import-auto-id1169738060616\"><strong>Solution<\/strong><\/p>\n<p>To obtain destructive interference here,<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5f02c91eebe72b701510726a0fdb60d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#116;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#125;&#95;&#123;&#123;&#110;&#125;&#95;&#123;&#50;&#125;&#125;&#125;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"70\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1169737804641\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5bee3e53ef51386b9179ec0362fbd7a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#125;&#95;&#123;&#123;&#110;&#125;&#95;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"24\" style=\"vertical-align: -4px;\" \/> is the wavelength in the film and is given by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7703faf0207ae6f8f6b6d628615bbc17_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#125;&#95;&#123;&#123;&#110;&#125;&#95;&#123;&#50;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#125;&#123;&#123;&#110;&#125;&#95;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"68\" style=\"vertical-align: -8px;\" \/>.<\/p>\n<p id=\"import-auto-id1169738074136\">Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-416\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d59894df517f589f61e34f9c87c688e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#116;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#47;&#123;&#110;&#125;&#95;&#123;&#50;&#125;&#125;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"77\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1169738052013\">Solving for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/> and entering known values yields<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-60cf39a5f1cd2dff8241f3518a3a5adb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#116;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#47;&#123;&#110;&#125;&#95;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#53;&#48;&#32;&#110;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#47;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#46;&#51;&#56;&#125;&#125;&#123;&#52;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#57;&#46;&#54;&#32;&#110;&#109;&#46;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"199\" style=\"vertical-align: -12px;\" \/><\/div>\n<p id=\"import-auto-id1169737952148\"><strong>Discussion<\/strong><\/p>\n<p>Films such as the one in this example are most effective in producing destructive interference when the thinnest layer is used, since light over a broader range of incident angles will be reduced in intensity. These films are called non-reflective coatings; this is only an approximately correct description, though, since other wavelengths will only be partially cancelled. Non-reflective coatings are used in car windows and sunglasses.<\/p>\n<\/div>\n<p id=\"import-auto-id1169737966586\">Thin film interference is most constructive or most destructive when the path length difference for the two rays is an integral or half-integral wavelength, respectively. That is, for rays incident perpendicularly, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-63a21a86625977c39d75ed0aa652f8ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#116;&#61;&#123;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#125;&#95;&#123;&#110;&#125;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#50;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#125;&#95;&#123;&#110;&#125;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#51;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#125;&#95;&#123;&#110;&#125;&#44;&#92;&#100;&#111;&#116;&#115;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"167\" style=\"vertical-align: -4px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-410b2dc5317862621f57b184cb1a9fc8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#116;&#61;&#123;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#125;&#95;&#123;&#110;&#125;&#47;&#50;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#51;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#125;&#95;&#123;&#110;&#125;&#47;&#50;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#53;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#125;&#95;&#123;&#110;&#125;&#47;&#50;&#44;&#92;&#100;&#111;&#116;&#115;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"221\" style=\"vertical-align: -5px;\" \/>. To know whether interference is constructive or destructive, you must also determine if there is a phase change upon reflection. Thin film interference thus depends on film thickness, the wavelength of light, and the refractive indices. For white light incident on a film that varies in thickness, you will observe rainbow colors of constructive interference for various wavelengths as the thickness varies.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1169736763149\">\n<div data-type=\"title\" class=\"title\">Soap Bubbles: More Than One Thickness can be Constructive<\/div>\n<p id=\"import-auto-id1169737757640\">(a) What are the three smallest thicknesses of a soap bubble that produce constructive interference for red light with a wavelength of 650 nm? The index of refraction of soap is taken to be the same as that of water. (b) What three smallest thicknesses will give destructive interference?<\/p>\n<p id=\"import-auto-id1169737042137\"><strong>Strategy and Concept<\/strong><\/p>\n<p id=\"import-auto-id1169738211471\">Use <a href=\"#import-auto-id1169737818442\" class=\"autogenerated-content\">(Figure)<\/a> to visualize the bubble. Note that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4a29d40e45fa6b59c5ddd5408e74af4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#95;&#123;&#49;&#125;&#61;&#123;&#110;&#125;&#95;&#123;&#51;&#125;&#61;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"116\" style=\"vertical-align: -4px;\" \/> for air, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dee0e87a69e745ae619f954f6beddf08_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#95;&#123;&#50;&#125;&#61;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#51;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"83\" style=\"vertical-align: -3px;\" \/> for soap (equivalent to water). There is a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-08246831fcb3aaf5611c696383443844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#47;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"27\" style=\"vertical-align: -5px;\" \/> shift for ray 1 reflected from the top surface of the bubble, and no shift for ray 2 reflected from the bottom surface. To get constructive interference, then, the path length difference (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-388d9ae1fdf29c1b93b9dbc474e03c08_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: 0px;\" \/>) must be a half-integral multiple of the wavelength\u2014the first three being <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8aecd654b236b6c06f400a12bcea4b0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#125;&#95;&#123;&#110;&#125;&#47;&#50;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#51;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#125;&#95;&#123;&#110;&#125;&#47;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"95\" style=\"vertical-align: -5px;\" \/>, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7acf25e4913a1ca8e1fe5e96ef106de1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#53;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#125;&#95;&#123;&#110;&#125;&#47;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -5px;\" \/>. To get destructive interference, the path length difference must be an integral multiple of the wavelength\u2014the first three being <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d6d1861d264eaba6bdf4eacba330a144_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#125;&#95;&#123;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"39\" style=\"vertical-align: -4px;\" \/>, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ecea0c7b060606fe9b85064e088f64db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#50;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#125;&#95;&#123;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"27\" style=\"vertical-align: -3px;\" \/>.<\/p>\n<p id=\"import-auto-id1169738005796\"><strong>Solution for (a)<\/strong><\/p>\n<p id=\"import-auto-id1169737788882\"><em data-effect=\"italics\">Constructive interference<\/em> occurs here when<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c3f08394e1088b882a2d1ae3841c4ba7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#116;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#125;&#95;&#123;&#110;&#125;&#125;&#123;&#50;&#125;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#51;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#125;&#95;&#123;&#110;&#125;&#125;&#123;&#50;&#125;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#53;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#125;&#95;&#123;&#110;&#125;&#125;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;&#92;&#100;&#111;&#116;&#115;&#32;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"180\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1169736800518\">The smallest constructive thickness <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f8ce61b1df1dd1ab684136b27323c2f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#116;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"12\" style=\"vertical-align: -3px;\" \/> thus is<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-889\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-57b81833660d6e75ed4c4d0c23ed45cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#123;&#116;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#125;&#125;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#125;&#95;&#123;&#110;&#125;&#125;&#123;&#52;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#47;&#110;&#125;&#123;&#52;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#53;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#47;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#51;&#51;&#125;&#125;&#123;&#52;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#50;&#50;&#32;&#110;&#109;&#46;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"250\" style=\"vertical-align: -13px;\" \/><\/div>\n<p id=\"import-auto-id1169738220151\">The next thickness that gives constructive interference is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-28b9692e0cf8efba4f71af5977fc50a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#116;&#92;&#112;&#114;&#105;&#109;&#101;&#32;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#125;&#125;&#61;&#123;&#51;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#125;&#95;&#123;&#110;&#125;&#47;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"88\" style=\"vertical-align: -5px;\" \/>, so that<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-124fb6af6eaa56fed1c899bd928bc39e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#116;&#92;&#112;&#114;&#105;&#109;&#101;&#32;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#125;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#54;&#54;&#32;&#110;&#109;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"103\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"import-auto-id1169738136931\">Finally, the third thickness producing constructive interference is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-119b6d5176e7a210815643975cfae289_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#116;&#92;&#112;&#114;&#105;&#109;&#101;&#32;&#92;&#112;&#114;&#105;&#109;&#101;&#32;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#125;&#125;&#92;&#108;&#101;&#32;&#123;&#53;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#125;&#95;&#123;&#110;&#125;&#47;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"93\" style=\"vertical-align: -5px;\" \/>, so that<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-aea2242dfe43fd81cc2418ac17155bf0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#116;&#92;&#112;&#114;&#105;&#109;&#101;&#32;&#92;&#112;&#114;&#105;&#109;&#101;&#32;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#125;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#49;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#109;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"107\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"import-auto-id1169738145668\"><strong>Solution for (b)<\/strong><\/p>\n<p id=\"import-auto-id1169737979425\">For <em data-effect=\"italics\">destructive interference<\/em>, the path length difference here is an integral multiple of the wavelength. The first occurs for zero thickness, since there is a phase change at the top surface. That is,<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8d774d49423f1cc86ab44dafa6b67483_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#116;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#100;&#125;&#125;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"51\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"import-auto-id1169738074282\">The first non-zero thickness producing destructive interference is<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8d8bdf3a44082c1b708af408dd75380d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#116;&#92;&#112;&#114;&#105;&#109;&#101;&#32;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#100;&#125;&#125;&#61;&#123;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#125;&#95;&#123;&#110;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"76\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"import-auto-id1169736588480\">Substituting known values gives<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5043575da79f49ef7114dbecde4ea3ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#123;&#116;&#92;&#112;&#114;&#105;&#109;&#101;&#32;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#100;&#125;&#125;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#123;&#125;&#95;&#123;&#110;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#125;&#123;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#47;&#110;&#125;&#123;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#53;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#47;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#51;&#51;&#125;&#125;&#123;&#50;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#52;&#52;&#32;&#110;&#109;&#46;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"257\" style=\"vertical-align: -13px;\" \/><\/div>\n<p id=\"import-auto-id1169738212215\">Finally, the third destructive thickness is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e93b3da718a6ad5131582a2e75d54d8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#116;&#92;&#112;&#114;&#105;&#109;&#101;&#32;&#92;&#112;&#114;&#105;&#109;&#101;&#32;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#100;&#125;&#125;&#61;&#123;&#50;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#125;&#95;&#123;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"84\" style=\"vertical-align: -3px;\" \/>, so that<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-670\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7e5942f019a9269c0f6f5cf72d3b31bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#123;&#116;&#92;&#112;&#114;&#105;&#109;&#101;&#32;&#92;&#112;&#114;&#105;&#109;&#101;&#32;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#100;&#125;&#125;&#38;&#32;&#61;&#38;&#32;&#123;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#125;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#53;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#109;&#125;&#125;&#123;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#51;&#51;&#125;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#56;&#56;&#32;&#110;&#109;&#46;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"197\" style=\"vertical-align: -12px;\" \/><\/div>\n<p id=\"import-auto-id1169737874215\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id1169737795299\">If the bubble was illuminated with pure red light, we would see bright and dark bands at very uniform increases in thickness. First would be a dark band at 0 thickness, then bright at 122 nm thickness, then dark at 244 nm, bright at 366 nm, dark at 488 nm, and bright at 610 nm. If the bubble varied smoothly in thickness, like a smooth wedge, then the bands would be evenly spaced.<\/p>\n<\/div>\n<p id=\"import-auto-id1169736614823\">Another example of thin film interference can be seen when microscope slides are separated (see <a href=\"#import-auto-id1169738244250\" class=\"autogenerated-content\">(Figure)<\/a>). The slides are very flat, so that the wedge of air between them increases in thickness very uniformly. A phase change occurs at the second surface but not the first, and so there is a dark band where the slides touch. The rainbow colors of constructive interference repeat, going from violet to red again and again as the distance between the slides increases. As the layer of air increases, the bands become more difficult to see, because slight changes in incident angle have greater effects on path length differences. If pure-wavelength light instead of white light is used, then bright and dark bands are obtained rather than repeating rainbow colors.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1169738244250\">\n<div class=\"bc-figcaption figcaption\">(a) The rainbow color bands are produced by thin film interference in the air between the two glass slides. (b) Schematic of the paths taken by rays in the wedge of air between the slides.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1169736629490\" data-alt=\"Figure A shows two microscope slides that have been pressed together. Multicolor swirling rainbow bands are visible coming from the slides. Figure B shows a cross section of two glass slides stacked one on top of the other. The lower slide is horizontal and the upper slide is tilted up at an angle that is larger than the actual angle between slides would be. Two rays come from above and impinge upon the slides. Their refraction and partial reflection is shown at each glass air interface.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_28_07_02a.jpg\" data-media-type=\"image\/jpg\" alt=\"Figure A shows two microscope slides that have been pressed together. Multicolor swirling rainbow bands are visible coming from the slides. Figure B shows a cross section of two glass slides stacked one on top of the other. The lower slide is horizontal and the upper slide is tilted up at an angle that is larger than the actual angle between slides would be. Two rays come from above and impinge upon the slides. Their refraction and partial reflection is shown at each glass air interface.\" width=\"400\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1169737927160\">An important application of thin film interference is found in the manufacturing of optical instruments. A lens or mirror can be compared with a master as it is being ground, allowing it to be shaped to an accuracy of less than a wavelength over its entire surface. <a href=\"#fs-id1169736610625\" class=\"autogenerated-content\">(Figure)<\/a> illustrates the phenomenon called Newton\u2019s rings, which occurs when the plane surfaces of two lenses are placed together. (The circular bands are called Newton\u2019s rings because Isaac Newton described them and their use in detail. Newton did not discover them; Robert Hooke did, and Newton did not believe they were due to the wave character of light.) Each successive ring of a given color indicates an increase of only one wavelength in the distance between the lens and the blank, so that great precision can be obtained. Once the lens is perfect, there will be no rings.<\/p>\n<div class=\"bc-figure figure\" id=\"fs-id1169736610625\">\n<div class=\"bc-figcaption figcaption\">\u201cNewton&#8217;s rings\u201d interference fringes are produced when two plano-convex lenses are placed together with their plane surfaces in contact. The rings are created by interference between the light reflected off the two surfaces as a result of a slight gap between them, indicating that these surfaces are not precisely plane but are slightly convex. (credit: Ulf Seifert, Wikimedia Commons)<\/div>\n<p><span data-type=\"media\" id=\"fs-id1169738035379\" data-alt=\"This figure shows rainbow-colored concentric rings obtained when two plano-convex lenses are placed together with their flat surfaces in contact.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_28_07_03a.jpg\" data-media-type=\"image\/jpg\" alt=\"This figure shows rainbow-colored concentric rings obtained when two plano-convex lenses are placed together with their flat surfaces in contact.\" width=\"200\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1169737813232\">The wings of certain moths and butterflies have nearly iridescent colors due to thin film interference. In addition to pigmentation, the wing\u2019s color is affected greatly by constructive interference of certain wavelengths reflected from its film-coated surface. Car manufacturers are offering special paint jobs that use thin film interference to produce colors that change with angle. This expensive option is based on variation of thin film path length differences with angle. Security features on credit cards, banknotes, driving licenses and similar items prone to forgery use thin film interference, diffraction gratings, or holograms. Australia led the way with dollar bills printed on polymer with a diffraction grating security feature making the currency difficult to forge. Other countries such as New Zealand and Taiwan are using similar technologies, while the United States currency includes a thin film interference effect.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1169737904590\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Making Connections: Take-Home Experiment\u2014Thin Film Interference<\/div>\n<p id=\"import-auto-id1169737763576\">One feature of thin film interference and diffraction gratings is that the pattern shifts as you change the angle at which you look or move your head. Find examples of thin film interference and gratings around you. Explain how the patterns change for each specific example. Find examples where the thickness changes giving rise to changing colors. If you can find two microscope slides, then try observing the effect shown in <a href=\"#import-auto-id1169738244250\" class=\"autogenerated-content\">(Figure)<\/a>. Try separating one end of the two slides with a hair or maybe a thin piece of paper and observe the effect.<\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169737988256\">\n<h1 data-type=\"title\">Problem-Solving Strategies for Wave Optics<\/h1>\n<p id=\"import-auto-id1169738107852\"><strong>Step 1.<\/strong><em data-effect=\"italics\">Examine the situation to determine that interference is involved<\/em>. Identify whether slits or thin film interference are considered in the problem.<\/p>\n<p><strong>Step 2.<\/strong><em data-effect=\"italics\">If slits are involved<\/em>, note that diffraction gratings and double slits produce very similar interference patterns, but that gratings have narrower (sharper) maxima. Single slit patterns are characterized by a large central maximum and smaller maxima to the sides.<\/p>\n<p id=\"import-auto-id1169737761861\"><strong>Step 3.<\/strong><em data-effect=\"italics\">If thin film interference is involved, take note of the path length difference between the two rays that interfere<\/em>. Be certain to use the wavelength in the medium involved, since it differs from the wavelength in vacuum. Note also that there is an additional <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-08246831fcb3aaf5611c696383443844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#47;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"27\" style=\"vertical-align: -5px;\" \/> phase shift when light reflects from a medium with a greater index of refraction.<\/p>\n<p><strong>Step 4.<\/strong><em data-effect=\"italics\">Identify exactly what needs to be determined in the problem (identify the unknowns)<\/em>. A written list is useful. Draw a diagram of the situation. Labeling the diagram is useful.<\/p>\n<p id=\"import-auto-id1169736864048\"><strong>Step 5.<\/strong><em data-effect=\"italics\">Make a list of what is given or can be inferred from the problem as stated (identify the knowns)<\/em>.<\/p>\n<p id=\"import-auto-id1169738116480\"><strong>Step 6.<\/strong><em data-effect=\"italics\">Solve the appropriate equation for the quantity to be determined (the unknown), and enter the knowns<\/em>. Slits, gratings, and the Rayleigh limit involve equations.<\/p>\n<p id=\"import-auto-id1169737813576\"><strong>Step 7.<\/strong><em data-effect=\"italics\">For thin film interference, you will have constructive interference for a total shift that is an integral number of wavelengths. You will have destructive interference for a total shift of a half-integral number of wavelengths<\/em>. Always keep in mind that crest to crest is constructive whereas crest to trough is destructive.<\/p>\n<p id=\"import-auto-id1169737861300\"><strong>Step 8.<\/strong><em data-effect=\"italics\">Check to see if the answer is reasonable: Does it make sense?<\/em> Angles in interference patterns cannot be greater than <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0ef094c705f55f76b4993ff72af9e73f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>, for example.<\/p>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul>\n<li id=\"import-auto-id1169737785898\">Thin film interference occurs between the light reflected from the top and bottom surfaces of a film. In addition to the path length difference, there can be a phase change.<\/li>\n<li id=\"import-auto-id1169736645684\">When light reflects from a medium having an index of refraction greater than that of the medium in which it is traveling, a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d99e5df9b7f723e173ac5bd5364c01e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#56;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: -1px;\" \/> phase change (or a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-08246831fcb3aaf5611c696383443844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#47;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"27\" style=\"vertical-align: -5px;\" \/> shift) occurs.<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id1169738034459\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169737725267\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169737994783\">\n<p id=\"import-auto-id1169737949657\">What effect does increasing the wedge angle have on the spacing of interference fringes? If the wedge angle is too large, fringes are not observed. Why?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169738110184\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169735706971\">\n<p id=\"import-auto-id1169737852676\">How is the difference in paths taken by two originally in-phase light waves related to whether they interfere constructively or destructively? How can this be affected by reflection? By refraction?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169737935684\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169738055870\">\n<p id=\"import-auto-id1169737850049\">Is there a phase change in the light reflected from either surface of a contact lens floating on a person\u2019s tear layer? The index of refraction of the lens is about 1.5, and its top surface is dry.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169738006717\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169737994677\">\n<p id=\"import-auto-id1169737778532\">In placing a sample on a microscope slide, a glass cover is placed over a water drop on the glass slide. Light incident from above can reflect from the top and bottom of the glass cover and from the glass slide below the water drop. At which surfaces will there be a phase change in the reflected light?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169736581954\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169737970134\">\n<p id=\"import-auto-id1169737812726\">Answer the above question if the fluid between the two pieces of crown glass is carbon disulfide.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169736606633\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169737701857\">\n<p id=\"import-auto-id1169738232888\">While contemplating the food value of a slice of ham, you notice a rainbow of color reflected from its moist surface. Explain its origin.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169736772198\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\">\n<p>An inventor notices that a soap bubble is dark at its thinnest and realizes that destructive interference is taking place for all wavelengths. How could she use this knowledge to make a non-reflective coating for lenses that is effective at all wavelengths? That is, what limits would there be on the index of refraction and thickness of the coating? How might this be impractical?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169738045669\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169737803309\">\n<p id=\"import-auto-id1169737999648\">A non-reflective coating like the one described in <a href=\"#fs-id1169738090922\" class=\"autogenerated-content\">(Figure)<\/a> works ideally for a single wavelength and for perpendicular incidence. What happens for other wavelengths and other incident directions? Be specific.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169737777453\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\">\n<p id=\"import-auto-id1169738162964\">Why is it much more difficult to see interference fringes for light reflected from a thick piece of glass than from a thin film? Would it be easier if monochromatic light were used?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id1169738134093\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169738246614\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169738244998\">\n<p>A soap bubble is 100 nm thick and illuminated by white light incident perpendicular to its surface. What wavelength and color of visible light is most constructively reflected, assuming the same index of refraction as water?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1169738135767\">\n<p id=\"import-auto-id1169737802646\">532 nm (green)<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169738028271\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169736713736\">\n<p id=\"import-auto-id1169737799271\">An oil slick on water is 120 nm thick and illuminated by white light incident perpendicular to its surface. What color does the oil appear (what is the most constructively reflected wavelength), given its index of refraction is 1.40?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169737897068\">\n<p id=\"import-auto-id1169738188366\">Calculate the minimum thickness of an oil slick on water that appears blue when illuminated by white light perpendicular to its surface. Take the blue wavelength to be 470 nm and the index of refraction of oil to be 1.40.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1169737799446\">\n<p id=\"import-auto-id1169736708130\">83.9 nm<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169736624201\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169736693588\">\n<p id=\"import-auto-id1169736581639\">Find the minimum thickness of a soap bubble that appears red when illuminated by white light perpendicular to its surface. Take the wavelength to be 680 nm, and assume the same index of refraction as water.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169736675909\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\">\n<p id=\"import-auto-id1169737787716\">A film of soapy water (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c1356c6cfc7c04c7ac28bc446f8032c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"66\" style=\"vertical-align: -1px;\" \/>) on top of a plastic cutting board has a thickness of 233 nm. What color is most strongly reflected if it is illuminated perpendicular to its surface?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1169738208568\">\n<p id=\"import-auto-id1169736619360\">620 nm (orange)<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169737132388\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169738186795\">\n<p id=\"import-auto-id1169737788404\">What are the three smallest non-zero thicknesses of soapy water (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c1356c6cfc7c04c7ac28bc446f8032c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"66\" style=\"vertical-align: -1px;\" \/>) on Plexiglas if it appears green (constructively reflecting 520-nm light) when illuminated perpendicularly by white light? Explicitly show how you follow the steps in <a href=\"#fs-id1169737988256\">Problem Solving Strategies for Wave Optics<\/a>.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169738247055\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169738249231\">\n<p id=\"import-auto-id1169736708026\">Suppose you have a lens system that is to be used primarily for 700-nm red light. What is the second thinnest coating of fluorite (magnesium fluoride) that would be non-reflective for this wavelength?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1169738054339\">\n<p>380 nm<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169735571594\">\n<p id=\"import-auto-id1169738076684\">(a) As a soap bubble thins it becomes dark, because the path length difference becomes small compared with the wavelength of light and there is a phase shift at the top surface. If it becomes dark when the path length difference is less than one-fourth the wavelength, what is the thickest the bubble can be and appear dark at all visible wavelengths? Assume the same index of refraction as water. (b) Discuss the fragility of the film considering the thickness found.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169737802684\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169737861298\">\n<p id=\"import-auto-id1169737861105\">A film of oil on water will appear dark when it is very thin, because the path length difference becomes small compared with the wavelength of light and there is a phase shift at the top surface. If it becomes dark when the path length difference is less than one-fourth the wavelength, what is the thickest the oil can be and appear dark at all visible wavelengths? Oil has an index of refraction of 1.40.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1169737965953\">\n<p>33.9 nm<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169738117735\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169737713189\">\n<p><a href=\"#import-auto-id1169738244250\" class=\"autogenerated-content\">(Figure)<\/a> shows two glass slides illuminated by pure-wavelength light incident perpendicularly. The top slide touches the bottom slide at one end and rests on a 0.100-mm-diameter hair at the other end, forming a wedge of air. (a) How far apart are the dark bands, if the slides are 7.50 cm long and 589-nm light is used? (b) Is there any difference if the slides are made from crown or flint glass? Explain.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169738246205\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169738092976\">\n<p id=\"import-auto-id1169737050952\"><a href=\"#import-auto-id1169738244250\" class=\"autogenerated-content\">(Figure)<\/a> shows two 7.50-cm-long glass slides illuminated by pure 589-nm wavelength light incident perpendicularly. The top slide touches the bottom slide at one end and rests on some debris at the other end, forming a wedge of air. How thick is the debris, if the dark bands are 1.00 mm apart?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1169737973966\">\n<p id=\"import-auto-id1169738116428\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4511204e03ebf47f46284a52f6709d93_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#50;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#53;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"87\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169737761659\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169736739921\">\n<p id=\"import-auto-id1169737931422\">Repeat <a href=\"#fs-id1169738246614\" class=\"autogenerated-content\">(Figure)<\/a>, but take the light to be incident at a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1142d1c44cfaf3459c45a3d6cc399899_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#53;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"17\" style=\"vertical-align: -1px;\" \/> angle.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169736660356\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169736669384\">\n<p id=\"import-auto-id1169737909085\">Repeat <a href=\"#fs-id1169738028271\" class=\"autogenerated-content\">(Figure)<\/a>, but take the light to be incident at a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1142d1c44cfaf3459c45a3d6cc399899_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#53;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"17\" style=\"vertical-align: -1px;\" \/> angle.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1169737756588\">\n<p>The oil film will appear black, since the reflected light is not in the visible part of the spectrum.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1169737700998\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1169738132137\">\n<p id=\"import-auto-id1169738077411\"><strong>Unreasonable Results<\/strong><\/p>\n<p id=\"import-auto-id1169737796054\">To save money on making military aircraft invisible to radar, an inventor decides to coat them with a non-reflective material having an index of refraction of 1.20, which is between that of air and the surface of the plane. This, he reasons, should be much cheaper than designing Stealth bombers. (a) What thickness should the coating be to inhibit the reflection of 4.00-cm wavelength radar? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id1169738036749\">\n<dt>thin film interference<\/dt>\n<dd id=\"fs-id1169736739972\">interference between light reflected from different surfaces of a thin film<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":211,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"all-rights-reserved"},"chapter-type":[],"contributor":[],"license":[56],"class_list":["post-1498","chapter","type-chapter","status-publish","hentry","license-all-rights-reserved"],"part":1446,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1498","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/users\/211"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1498\/revisions"}],"predecessor-version":[{"id":1499,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1498\/revisions\/1499"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/parts\/1446"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1498\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/media?parent=1498"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=1498"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/contributor?post=1498"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/license?post=1498"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}