Well unless they have come up with some way new since I looked into it...
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They take a long sillica tube heated to 1500 C and pass an oxygen metal halide gas mixture through it.
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A chemical reaction occurs in the gas and glass "soot" is formed on the inside of the tube.
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The tube is rotated while the heater is moved to and fro along the tube and the soot forms a thin layer of silica glass. (The rotation and heater movement ensure that the layer is of a consistant thickness.)
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The first layer is the cladding and then, by changing the constituents of the incomming gas to alter the refractive index of the next layer, the core is placed. (graded index fiber is produced by carefully, and continuously, changing the constiuents of the gas.)
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The temperature is then increased to about 1800 C and the tube is collapsed to form a solid rod called a preform.
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The preform is heated to about 1200 C and pulled into a fine strand. (Laser guages are used to monitor the thickness of the fiber.)