# thin film interference equation

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Each half wavelength has been numbered, so we can keep track of it. In butterflies, the thin-film optics are visible when the wing itself is not covered by pigmented wing scales, which is the case in the blue wing spots of the Aglais io butterfly. Light that is transmitted reaches the bottom surface and may once again be transmitted or reflected. The name is a bit misleading, because the structure in the pattern observed is dominated by interference effects. i [5], The first production of thin-film coatings occurred quite by accident. 0 He noticed that pressing the barbule would shift the color toward the blue, while swelling it with a chemical would shift it toward the red. Figure A shows a wave incident on a thin film. The oil may have an index of refraction near 1.5 and the water has an index of 1.33. l 71 0 obj <>/Filter/FlateDecode/ID[<41878BF1257D17E0BA2AE71A92914EAE><8014C2F7F8E0D7478B003AB9DDC20DE4>]/Index[58 24]/Info 57 0 R/Length 75/Prev 77163/Root 59 0 R/Size 82/Type/XRef/W[1 2 1]>>stream ( This phase shift occurs if the refractive index of the medium the light is travelling through is less than the refractive index of the material it is striking. Because a half wavelength fits in the film, the peaks of one reflected wave line up precisely with the troughs of the other (and vice versa), so the waves cancel. g . ⁡ Part of the wave is reflected off the top surface of the film; note that this reflected wave is flipped by 180°, so peaks are now troughs and troughs are now peaks. θ m . The thickness of the film is very small at the point of contact and gradually increases from the centre outwards. Thus, the film shown in these figures reflects more strongly at the wavelength of the light beam in the first figure, and less strongly at that of the beam in the second figure. A wave is reflected in phase when it tries to enter a medium with a faster speed of light (when passing from a medium with a higher index of refraction to a medium with a lower index of refraction). %%EOF ��Q���{��i�]7t�mG��8ɳ7\��97�����U��j��(;pS��x�����ZlHU��î�v� ��Ôv�-�K_�v�H����' ( is the film thickness, Thin Film Interference When light hits a surface, it can be absorbed, transmitted or reflected. !���*HN=o�ܣv��!=�l��c���xW�}��~S�,���v�W��б�֢��6�Jq[�Um���M%a�0��U9�K���U��ܕ��CRo�����FPM��=�JV;�� ȏ�\$lX��}b�?����*eʿ��ᮍ�j� �ըJ#�|\�iA'Km�Ϳ�O�/4�j� > Young's contribution went largely unnoticed until the work of Augustin Fresnel, who helped to establish the wave theory of light in 1816. = n w If, on the other hand, the reflected beams have opposite phase, the resulting beam is attenuated (as in the second figure). For waves incident normal to the film, the path length difference is twice the film thickness (, For waves incident at an angle to the normal, the path length difference is longer by a factor of, A path length difference equal to a whole number of wavelengths (, A path length difference equal to an odd number of half wavelengths (. i If the distance traveled by beam A is an odd integer multiple of the half wavelength of light in the film, the beams destructively interfere (as in the second figure). Additionally, a phase shift of 180° or Various possible film configurations and the related equations are explained in more detail in the examples below. Interference of transmitted light is completely constructive for these films. t This means none of the wave energy is reflected back into the first medium. {\displaystyle n_{\rm {air}}1} for quality notes on quantum mechanics and laser refer my site www.vtuphysics.blogspot.com. shift. i The equations for interference will be the same. Generally, in dealing with thin-film interference the key wavelength is the wavelength in the film itself. In optics, a thin film is a layer of material with thickness in the sub-nanometer to micron range. Draw DM normal to BP and CN normal to BD. %PDF-1.5 %���� The large number of interfering rays produces an interferometer with extremely high resolution, somewhat like the multiple slits of a diffraction grating increase its resolution. l < θ n endstream endobj 59 0 obj <> endobj 60 0 obj <> endobj 61 0 obj <>stream In the discussion below it is assumed that the incident and reflected rays are perpendicular to the interfaces. [1] The glossy appearance of buttercup flowers is also due to a thin film[2][3] as well as the shiny breast feathers of the bird of paradise. ) A film thickness of 1/4 the wavelength in the film results in destructive interference (this is derived below). w r i (Thin film interference does not violate the law of conservation of energy. m Given, µ = 4/3, λ1 = 6.1 x 10-7 m λ2 = 6.0 x 10-7m solution: Sin i = 4/5 µ = Sin i / Sin r Sin r = sin i / µ = ⅘ / 4/3 = sin r = ⅗ = 0.6 cos r = √1 – sin2r = 0.8 Given two consecutive fringes are overlapping, nλ1 = (n + 1)… If thin film interference is involved, take note of the path length difference between the two rays that interfere. I have been working on creating a Blender simulation for thin-film interference using the complex versions of the Fresnel and interference equations. As light strikes the surface of a film it is either transmitted or reflected at the upper surface. Therefore, the colors observed are rarely those of the rainbow, but browns, golds, turquoises, teals, bright blues, purples, and magentas. [5] After the invention of the Fabry–Perot interferometer, in 1899, the mechanisms of thin-film interference could be demonstrated on a larger scale. The other contribution to this shift can be either 0 or , depending on what happens when it reflects (this reflection occurs at point b on the diagram). [6], "Functional optics of glossy buttercup flowers", "Iridescent flowers? t In the diagram above, let's say that the light leaving the edge of the slit (ray 1) arrives at the screen half a wavelength out of phase with the light leaving the middle of the slit (ray 5). The other colors do not reflect as intensely as red light, so the film looks red. A good method for analyzing a thin-film problem involves these steps: Step 1. Calculate the thickness of the film. Referring to the ray diagram above, the OPD between the two waves is the following: Using Snell's law, , the reflected waves will be completely out of phase and will destructively interfere. Animation of interference of waves coming from two point sources. The diagram looks a little complicated at first glance, but it really is straightforward once you understand what it shows. In 1919, Lord Rayleigh proposed that the bright, changing colors were not caused by dyes or pigments, but by microscopic structures, which he termed "structural colors. The shape of the diffraction pattern is determined by the width (W) of the slits, while the shape of the interference pattern is determined by d, the distance between the slits. n When the thickness of the film is an odd multiple of one quarter-wavelength of the light on it, the reflected waves from both surfaces interfere to cancel each other.