linear effect, a polynomial approximation can be made to fit an
experimentally measured Ta-E curve.
A polynomial representation provides a more accurate approximation
over a wider range of exposure than possible with a linear
representation. Such representations have been studied with
polynomials of degree three being the most common choice.64-66 Using
a third order polynomial, the transmittance can be expressed as
Ta = Co + C1 ET + C2 ET2 + C3 ET3 (6.9)
where ET = Edc + Eac. Since Edc is slowly varying, equation 6.9 can
be written
Ta = Ao + A1 Eac + A2 Eac2 + A3 Eac3 (6.10)
where the Ai depend on the average local exposure Edc. Only the ac
portion of the exposing light causes fringes which will diffract light
in the resultant hologram.
Each power of Eac will diffract light to a unique angle or
diffractive order. The terms in equation 6.10 can be written
Ta = TO + TI + T2 + T3 (6.11)
where TO is a real quantity and T1 contains all those terms which
contribute to the first diffraction order. T2 and T3 contain only
terms which appear in the second and third diffracted orders. Thus TI
can be separated spatially from all other contributions and contains
the filter term which is desired. This term when expanded can be
written
Ta = [(CI + 2C2Er + 3C3Er2) + (2C2Es + 9C3ErEs + 3c3Es2)]Eac
(6.12)