Page 264 - Modern Optical Engineering The Design of Optical Systems

244 Chapter Eleven

Since most calculations are carried out at normal incidence ( j 0)
and for nonabsorbing materials (K j 0), one may ordinarily use u j
n j N j .
The subscript notation is j 0 for the substrate, j 1 for the first
film, j 2 for the second, etc., j p 1 for the last film and j p for
the final medium, which is usually air. For each film g j , the effective
optical thickness, in radians, is computed from

2 n jt j cos j
g (11.11)
j

where is the wavelength of light for which the calculation is made.
Starting with E 1 E 0 1.0 and H 1 u 0 E 0 u 0 , the following

equations are applied iteratively at each surface, with the subscript
j advancing from j 1 to j p 1.
iH j
E E cos g sin g (11.12)
j 1 j j j
u
j
H iu E sin g H cos g (11.13)
j 1 j j j j j
where i 1 and the other terms have been defined above. Readers
familiar with matrix notation may prefer to manipulate the equivalent
matrix form
cos g j u j sin g j (11.14)
i
E j 1
E j

H 1
j iu sin g j cos g j H j
j
When Eqs. 11.12 and 11.13 (or 11.14) have been applied to the entire
stack, we have the values of E p and H p , which will generally be complex
numbers of the form z x iy. These are substituted into
1 H p
E E x iy (11.15)
p p 2 2
2 u
p
1 H p
E E x iy (11.16)
p p 1 1
2 u p
and the reflectance of the thin-film system is found from

E p 2
R (11.17)

E
p