Chapter 5 . Polarization

5 .8 PHYSICAL OPTICS
they are representative of intenal reflections in materials of refractive index n 0 3 . 33 and
1 . 25 , respectively , when the other medium is air ( n 1 1) .
The intensity transmission coef ficients T s and T p are obtained from the Poynting vector
and for nonabsorbing materials are
T s 1 R s n 1 cos θ 1
t 2 s 4 n 0 n 1 cos θ 0 cos θ 1
n 0 cos θ 0 ( n 0 cos θ 0 n 1 cos θ 1 ) sin 2 θ 0 sin 2 θ 1
2 sin 2 ( θ 0 θ 1 )
(23)
T p 1 R p n 1 cos θ 1
t 2 p 4 n 0 n 1 cos θ 0 cos θ 1
n 0 cos θ 0 ( n 1 cos θ 0 n 0 cos θ 1 ) 2
sin 2 θ 0 sin 2 θ 1
sin 2 ( θ 0 θ 1 ) cos 2 ( θ 0 θ 1 )
(24)
These coef ficients are for light passing through a single boundary and hence are of limited
usefulness . In actual cases , the light is transmitted through a slab of material where there
are two boundaries , generally multiple reflections within the material , and sometimes
interference ef fects when the boundaries are smooth and plane-parallel .
The intensity transmission coef ficient T sample for a slab of transparent material in air is
given by the well-known Airy equation 5 when the sample has smooth , plane-parallel sides
and coherent multiple reflections occur within it :
where
1
T sample
1 [4 R s ,p/ (1 R s ,p) 2 ] sin 2
2 π n 1 d cos θ 1
(25)
(26)
The values of R s and R p can be determined from Eqs . (20) to (22) ; d is the sample
thickness , the wavelength , n 1 the refractive index of the material , and θ 1 the angle of
refraction . Equation (25) holds for all angles of incidence including the Brewster angle ,
where R p 0 [see Eq . (48)] . The Airy equation predicts that at a given angle of incidence
the transmission of the sample will vary from a maximum value of 1 to a minimum value of
(1 R s ,p ) 2 / (1 R s ,p ) 2 as the wavelength or the thickness is changed . If the sample is very
thick , the oscillations in the transmittance will occur at wavelengths very close together
and hence will be unresolved . A complete oscillation occurs every time changes by π , so
that the wavelength interval between oscillations is
2
2 n 1 d cos θ 1
(27)
An an example , a sample 1 mm thick with an index of 1 . 5 at 5000 Å will have transmission
maxima separated by 0 . 83 Å when measured at normal incidence (cos θ 1 1) . These
maxima would not be resolved by most commercial spectrophotometers . In such a case ,
one would be measuring the average transmission T sample , av :
T sample , av 1 R s ,p
1 R s ,p
(28)
For nonabsorbing materials , this is the same value as that which would be obtained if the
multiply reflected beams did not coherently interfere within the sample . If the sample is
wedge-shaped , so that no multiply reflected beams contribute to the measured transmittance
, T sample is simply T 2 s or T 2 p and can be calculated from Eq . (23) or (24) .
When the material is absorbing , i . e ., has a complex refractive index , it is not so easy to