【正文】 be available or measurable. Since we have two different equations (10) for odd and even fringes, we can use the measured Dm? and 1Dm?? and simply guess a value for (p ? m). Then, using Eq. (5) with 0p? and 01p?? we calculate the contact positions which correspond to fringes m, m ? 1, and m ? 2. Apparently, only if the guess is correct will one obtain the same separation D using both forms of Eq.(10) for even and for odd fringes. In practice, however,this method (which anyway works only for a threelayer interferometer) can be done only for very small (p ? m) values, while for large (p ?m) other factors such as the dispersion of the medium 181。ngstromerrors in the wavelength measurement can produce a micrometerscale error in the separation. This paper describes an easy and accurate way of calculating the separation from any order of fringes at any surface separation. From Eq. (1a) it is clear that if the refractive indices of the (mica) substrate and the medium are known then the separation between the surfaces D can be obtained from 0p? ， DP? and 01p?? . In the case where the medium refractive index μ is not known [10,11], there is a need to solve two equationswith two unknowns, D and μmed. The two equations are Eq. (1a) for an odd fringe and Eq. (1a) for an even fringe. In order to solve these simultaneous equations, we need to know the location of two adjacent fringes, DP? and 1DP?? at finite D and the ―contact‖ positions of three adjacent fringes, 0p? , 01p?? , and 02P?? , at D = 0. The limitation of the above ―conventional‖method is that DP? , 1Dp?? , 0p? , 01p?? , and 02P?? need to be in the visible for them to be measured. This limits the range of measurable separations D obtained using this approach to D 1 μm for typical substrate thicknesses (thicker substrates would increase the range but decrease the accuracy of the measurements). In this paper we describe a procedure that allows accurate and unambiguous measurements of distance and refractive index to be made using the contact position of only two adjacent fringes, 0p? and 01p?? , and—more importantly—using two adjacent fringes of any order, Dm? and 1Dm?? , and not necessarily DP? and 1Dp?? . 1. General procedure for measuring D andμ med of films of arbitrary thickness The ―contact‖ positions, 0p? correspond to a singlelayer interferometer. A onelayer interferometer made of a substrate with a refractive indexμ and physical thickness Y, obeys the relation 2μ Y (2)= pλ , p = 1, 2, 3, . . .,∞, whereλ is the wavelength that corresponds to constructive interference and p is a natural number (the fringe order). In practice there is never a onelayer interferometer, since the reflecting layer results in a phase change of the light at the substrate–reflector interface (say a mica–silver interface). Such a phase change for the mica–silver interface reflection has been studied [12], and it has been shown that it can be viewed as an apparent small change in the optical thickness of the substrate to Y instead of Y [13]: 2μ Y = pλ , p = 1, 2, 3, . . . ,∞. (2a) It is intuitive that the solution to the proposed problem may be provided using this equation. Specifically, using Eq. (2a) to substitute the thickness of the layer Y for the contact position of the fringe in Eq. (1a), we should be able to relate one contact fringe position to any other contact fringe position. Hence, the problem bees one of how to express a single layer interferometer in terms of a relation between different fringe orders (rather than the presentation of Eq. (2a)). Using Eq. (2a) and assuming constant Y [13], we write for three arbitrary fringes of order p, p ?1, and m 02 ppYp??? ， （ 3a） 0112 ( 1)ppYp?????? (3b) 02 mmYm??? (3c) where we writeμ p to emphasize the dispersive nature of the substrate, ., that each refractive indexμ p corresponds to a wavelength 0p? . Equations (3a) and (3b) have just two unknowns, Y and p, and solving for them we get 010 11 1 ppppp???????? （ 4） From Eqs. (3a) and (3c) and using Eq. (4) we obtain 001011 ( )( 1 )pmm ppppppm???????????? ? ? （ 5） Equation (5) relates the position of the mth fringe to that of the pth fringe as a function of the number of fringes (p?m) between them, as well as five other parameters that need to be known, includingμ m. Note that (p ? m) is an integer, which may be positive, negative, or zero. Now, we may use any fringe in order to use Eq. (1a) for the calculation of separations and refractive indices: we simply need to count the number of fringes from m to p, put this number as (p ? m) into Eq. (5), and calculate a new set of contact positions 0m? , 01m?? , and 02m?? . As we show later in the worked Example 1, the value ofμ m can be obtained by substitution of an approximated λ m given by Eq. (6) into Eq. (12), which we introduce below. Experimentally, if a measurement is done at a large separation on a fringe Dm? whose contact position 0m? is not visible, one way of finding (p ? m) (after pleting a measurement using the mth fringe) is to bring the surfaces to substrate–substrate contact while counting the passing fringes until one finds the pfringe. This number is the integer (p ? m). Equation (5) requires knowledge ofμ m. To obtain this, note that Eq. (5) has a very weak dependence on the refractive index of the substrate. Neglecting the dispersion, for example, results in 0000 111 ( ) (