The behavior of holographic filters and gratings is described by coupled wavetheory.35

Chapter 6, Problem 7-24

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The behavior of holographic filters and gratings is described by coupled wavetheory.35 The Bragg wavelength Ab for a holographic optical clement is given byAb = 2ndcos Iiwhere n is the refractive index of the grating material; d is the grating period, orspacing; and Ii is the angle of incidence of the beam of radiation.(a) Create a spreadsheet to calculate the Bragg wavelength for a holographicgrating with a grating spacing of 17.1 flm and a refractive index of 1.53 atangles of incidence from 0 to 90.(b) At what angle is the Bragg wavelength 462 nm?(c) The Bragg wavelength is sometimes called the "playback wavelength." Why?(d) What is the historical significance of the term "Bragg wavelength?"(e) A tunable volume holographic filter has been developed for communicationsapplications.'6 Thc filter is said to be tunable over the wavelength range 2ndto 2d~. If the filter has a Bragg wavelength of 1550 nm and a gratingspacing of 535 nm, calculate the angular tuning range of the filter.(f) Find the Bragg wavelength for a grating with a spacing of 211.5 nm, assumingthat the grating is made of the same material. Find the wavelength range overwhich this grating may be tuned.(g) Discuss potential spectroscopic applications of tunable holographic filters.(h) When a volume grating is created in a holographic film, the refractive indexof the film material is varied by an amount t1n, which is referred to as therefractive-index modulation. In the ideal case, this quantity is given by:in = ~2twhere Ais the wavelength of the laser forming the interference pattern, and tis the thickness of the film. Calculate the refractive-index modulation in a film38 flm thick with a laser wavelength of 633 nm.(i) For real gratings, the refractive-index modulation isAsin-IV7)~:in = -----7Ttwhere D, is the grating efficiency. Derive an expression for the ideal gratingefficiency, and determine its value.(j) Calculate the efficiency for a grating 7.5 flm thick if its refractive-index modulationis 0.030 at a wavelength of 633 nm.(k) Holographic films for fabricating filters and gratings are available commercially,but they can be made in the laboratory using common chemicals. Use asearch engine to find a recipe for dichromated gelatin for holographic films.Describe the fabrication of the tilms, the chemistry of the process, and how thefilms are recorded.