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Optical Coatings  241

        Using Eqs. 11.7 and 11.8 for r 1 and r 2 , Eq. 11.3 can be solved for the
        thickness which yields a minimum reflectance. As the preceding
        discussion would lead one to expect, this occurs when the optical thick-
        ness of the film is one-quarter wavelength, that is,

                                     n t                            (11.9)
                                      1 1  4
        At normal incidence the reflectivity of a quarter-wave film is thus equal to

                                                2                 (11.10a)
                                            2
                                      n   n )
                                       2
                                   (n 0
                                            1
                                            2
                                   (n n   n )
                                     0
                                            1
                                       2
        and the film index which will produce a zero reflectance is
                                           0
                                   n     n n                      (11.10b)
                                    1        2
        Thus, to produce a coating which will completely eliminate reflections at
        an air-glass surface, a quarter-wave coating of a material whose index is
        the square root of the index of the glass is required. Magnesium fluoride
        (MgF 2 ) with an index of 1.38 is used for this purpose; its ability to form
        a hard durable film which will withstand weathering and frequent clean-
        ing is the prime reason for its use, despite the fact that its index is higher
        than the optimum value for almost all optical glasses. Equation 11.10b
        indicates that the magnesium fluoride, with its index of 1.38, would be
        an ideal low-reflection coating material for a substrate with an index of
            2
        1.38   1.904. Thus it is a much more efficient low-reflection coating for
        high-index glass than for ordinary glass of a lower index. The measured
        white light reflection of a low-reflection coating on various index materi-
        als is shown in Fig. 11.3.












                                         Figure 11.3 The measured reflec-
                                         tion of white light from an
                                         uncoated surface and from a sur-
                                         face coated with a quarter-wave
                                         MgF 2 low-reflection coating, as a
                                         function of the index of the base
                                         material.
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