【正文】 5 9 4 6 1 2 ( 1 )1 .5 9 4 6 5 7 6 1 .0 7p A? ? ?????????? (13) 計(jì)算結(jié)果和測(cè)量值 177。注意沒(méi)有必要知道 p的值。 . 例 2 計(jì)算從 0p? 到 01p?? 的距離 D，測(cè)量?jī)蓚€(gè)未知順序的相鄰邊緣 m和 m1的位置 Dm?和 1Dm?? 。我們知道，兩者之間的云母介質(zhì)基板具有折射率是 ，我們對(duì) Dm? 和 1Dm?? 進(jìn)行同步測(cè)量，得到 A? ? 和 1 ? ? ? （看圖 1，在這個(gè)例子中 0p? ， 01p?? ， Dm? ， 1Dm??的相對(duì)位置）。很明顯， m? p。因?yàn)?1363 ? 1310，所以我們的猜想是錯(cuò)誤的。 8 197。 是通過(guò) 0p? ， 01p?? 和 1Dm?? 帶入奇方程（ 1a）得到，我們得到 0 0 0 02 1 1 1( , , ) ( , , )DDm D D m D pDD? ? ? ? ? ?? ? ? ??。因此，他可以用于計(jì)算邊緣數(shù)來(lái)決定邊緣的（ mp）值。我們應(yīng)該看到，摻入了相變以后得到了精確的解，我們有充分的理由相信在計(jì)算過(guò)程中可以使用基質(zhì)擴(kuò)散折射率色散而忽略相變。由于接觸的波長(zhǎng)測(cè)量位置和系統(tǒng)的改變，上面的討論沒(méi)有完全忽視相位的變化，而是使用 D=0時(shí)的一個(gè)固定值來(lái)表述（這是貝利和他的同事得到的 8）。 for all separations, we show that in practice, for large separations, it is very hard to get to this Accuracy. In a SFA (surface force apparatus) [1] experiment, there is usually a three or fivelayer interferometer [2]. For simplicity we assume here a threelayer interferometer, although the analysis applies to any number of the transparent substrate layers (typically mica sheets) are brought into contact, the positions 0p? of the pthorder fringes is usually measured in the visible range of wavelengths. After the substrate surfaces are separated by a distance D, a threelayer interferometer is formed. The fringes shift to longer wavelengths and their new positions are given (see Ref. [3]) by2242 s i n( )2t a n( )4( 1 ) ( 1 ) c os ( )DPme d DP DPYDY??? ???? ???? ?? ???， , which, assuming no dispersion (we discuss this point later in the text), reduces to the more useful form 000102200112 si n( )12tan( )1( 1 ) c os( ) ( 1 )1pDPppm e dDP pDPppD????????? ??? ? ?????????? ? ??， where Y is the optical thickness of each substrate (we discuss later the difference between the optical and physical thicknesses) and .μ = μ /μ med, where μ (μγ or μβ ) andμ med are, respectively, the refractive indices of the mica and the intervening medium at . In Eq. (1a), + and ? refer to p odd and p even order fringes, respectively. Other ways of calculating thicknesses are available (see, for example, Ref. [4])。 is the refractive index of the mica at DP? (as noted in Eq. (1)). Substituting Eqs. (4) and (8) into Eq. (1), we obtainthe dispersive version of (1a) as ? ? 10000 1 2 1 202020t a n( ) 2 si n( ( 1 ) ( 1 ) ) 1 c os( ( 1 ) ( 1 ) ) ( 1 )p p p pppmed D D DP P Pp p pD ? ? ? ????? ? ? ? ? ?? ? ?? ? ? ? ???????? ? ? ? ? ? ? ? ????? （ 9） Equation (9) is the analogue of Eq. (1a) with a correction for the dispersion in the substrate refractive index, and as in Eq. (1a), + and ? refer to p odd and p even order fringes, respectively. Finally, we may write the analogue to Eq. (9) for any order of fringe, ? ? 10000 1 2 1 2112t a n( ) 2 si n( ( 1 ) ( 1 ) ) 1 c os( ( 1 ) ( 1 ) ) ( 1 )m m m mm e d m mD D Dm m mm m m mD ? ? ? ?? ? ? ?? ? ? ? ?? ? ?? ? ? ? ?????????? ? ? ? ? ? ? ? ????? （ 10） and then use Eq. (5) to replace the unknowns 0m? and 01m?? with the measured and 01p?? (and the calculated 02P?? in the case where the refractive index of the medium is not known). In certain situations, the integer number (m ? p) may not 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。, indeed, within the experimental error. Actually, one could use the measured value of 02P?? to determine one of the constants in Eq. (12) or, if we also measured 01p?? and 03p?? ,to get both constants. Note that it is not necessary to know the value of p, nor does it matter if p is odd or even—the equations are identical. Only for the separation measurements of the threelayer interferometer is it important to know whether p is odd or even (it happened to be odd for this speci?c example).We also note that this method can be used to calculate the contact wavelength of any fringe order, not just 02P?? . . Example 2 Calculating distances D from 0p? and 01p?? , and the measured positions Dm? and 1Dm?? of any two adjacent fringes of unknown order m and m? 1. A measured contact position of a fringe of unknown order p is at 0 ? ? and that of order p ? 1 is at 00 1 A? ? ? The substrate is brownish mica, whose refractive index is given by Eq. (12). The surfaces are well separated and we want to calculate the distance D between the two surfaces. We know that the medium between the two mica substrates has a refractive index of [14], and we perform simultaneous measurements of 0p? , 01p?? , Dm? , 1Dm?? in this example). For a quick estimate of D, one can use Eq. (6) to calculate the contact positions of various fringes p + 1,p + 2,... (see Example 1) and Eq. (1a) to calculate the distances of our threelayer interferometer until p + (p ? m) is found. A more accurate approach is to use Eqs. (5) and (6) for the calculation of the positions p + 1,p + 2,..., as described in Example 1, and Eq. (9) for the di