Multilayer interference filters having various transmission characteristics are described and compared. The emphasis is on filters with narrow stop bands such as might be used to eliminate the hazard from a laser beam. Four types of filters are considered: (1) quarter-wave stacks of two dielectric materials having matching layers one-eighth wavelength thick; (2) quarter-wave stacks of two dielectric materials having all layers of the same optical thickness (including the end layers); (3) quarter-wave stacks wherein all layers are of the same optical thickness, but the refractive indices of the layers may all be different to achieve equal reflection ripples in the passband; and (4) multilayer stacks of two dielectric materials wherein each layer may be of a different optical thickness to achieve nearly equal reflection ripples in the passband. The new formulas presented give the bandwidths between nulls of all the various filters as well as the bandwidths between equal-ripple points of the equal-ripple filters. Explicit formulas are stated for the ripple envelopes of filter types (1) and (2), and for the ripple heights of equal-ripple filters of types (3) and (4). A first-order design procedure based on the theory of linear arrays is given and evaluated by working numerical examples; general design criteria are presented to establish the validity of the first-order theory.
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