We represent a field (either electric or magnetic) of the incident, reflected, and
transmitted (or refracted) waves by F i , F r , and F t , respectively. We call a dielectric
of the incidence side (and, hence, reflection side) D1 and another dielectric of the
transmission side D2. The fields are described by
F i = F i ε i e
i k i ÁxÀωt
ð
Þ ,
ð8:6Þ
F r = F r ε r e
i k r ÁxÀωt
ð
Þ ,
ð8:7Þ
F t = F t ε t e
i k t ÁxÀωt
ð
Þ ,
ð8:8Þ
where F i , F r , and F t denote an amplitude of the field; ε i , ε r , and ε t represent a unit
vector of polarization direction, i.e., the direction along which the field oscillates; k i ,
k r , and k t are wavenumber vectors such that k i ⊥ ε i , k r ⊥ ε r , and k t ⊥ ε t . These
wavenumber vectors represent the propagation directions of individual waves. In
(8.6) to (8.8), indices of i, r, and t stand for incidence, reflection, and transmission,
respectively.
Let x s be an arbitrary position vector at the interface between the dielectrics. Also,
let t be a unit vector paralleling the interface. Thus, tangential components of the
field are described as
F i t = F i t Á ε i
ð
Þe
i k i Áx s Àωt
ð
Þ ,
ð8:9Þ
F r t = F r t Á ε r
ð
Þe
i k r Áx s Àωt
ð
Þ ,
ð8:10Þ
F t t = F t t Á ε t
ð
Þe
i k t Áx s Àωt
ð
Þ
:
ð8:11Þ
Note that F i t and F r t represent the field in D1 just close to the interface and that F t t
denotes the field in D2 just close to the interface. Thus, in light of (8.4) and (8.5), we
have
F i t þ F r t ¼ F t t :
ð8:12Þ
Notice that (8.12) holds with any position x s and any time t.
Let us think of elementary calculation of exponential functions or exponential
polynomials and the relationship between individual coefficients and exponents.
With respect to two functions e
ikx and e
ik
0 x , we have two alternatives according to a
value Wronskian takes. Here, Wronskian W is expressed as
W ¼
e
ikx
e
ik
0 x
e
ikx
À Á 0
e
ik
0 x
À Á 0
¼ Ài k À k
0
ð
Þe
i kþk
0
ð
Þx
:
ð8:13Þ
(i) W 6 ¼ 0 if and only if k 6 ¼ k
0 . In this case, e
ikx and e
ik
0 x are said to be linearly
independent. That is, on condition of k 6 ¼ k
0
, for any x we have
298
8 Reflection and Transmission of Electromagnetic Waves in Dielectric Media
transmitted (or refracted) waves by F i , F r , and F t , respectively. We call a dielectric
of the incidence side (and, hence, reflection side) D1 and another dielectric of the
transmission side D2. The fields are described by
F i = F i ε i e
i k i ÁxÀωt
ð
Þ ,
ð8:6Þ
F r = F r ε r e
i k r ÁxÀωt
ð
Þ ,
ð8:7Þ
F t = F t ε t e
i k t ÁxÀωt
ð
Þ ,
ð8:8Þ
where F i , F r , and F t denote an amplitude of the field; ε i , ε r , and ε t represent a unit
vector of polarization direction, i.e., the direction along which the field oscillates; k i ,
k r , and k t are wavenumber vectors such that k i ⊥ ε i , k r ⊥ ε r , and k t ⊥ ε t . These
wavenumber vectors represent the propagation directions of individual waves. In
(8.6) to (8.8), indices of i, r, and t stand for incidence, reflection, and transmission,
respectively.
Let x s be an arbitrary position vector at the interface between the dielectrics. Also,
let t be a unit vector paralleling the interface. Thus, tangential components of the
field are described as
F i t = F i t Á ε i
ð
Þe
i k i Áx s Àωt
ð
Þ ,
ð8:9Þ
F r t = F r t Á ε r
ð
Þe
i k r Áx s Àωt
ð
Þ ,
ð8:10Þ
F t t = F t t Á ε t
ð
Þe
i k t Áx s Àωt
ð
Þ
:
ð8:11Þ
Note that F i t and F r t represent the field in D1 just close to the interface and that F t t
denotes the field in D2 just close to the interface. Thus, in light of (8.4) and (8.5), we
have
F i t þ F r t ¼ F t t :
ð8:12Þ
Notice that (8.12) holds with any position x s and any time t.
Let us think of elementary calculation of exponential functions or exponential
polynomials and the relationship between individual coefficients and exponents.
With respect to two functions e
ikx and e
ik
0 x , we have two alternatives according to a
value Wronskian takes. Here, Wronskian W is expressed as
W ¼
e
ikx
e
ik
0 x
e
ikx
À Á 0
e
ik
0 x
À Á 0
¼ Ài k À k
0
ð
Þe
i kþk
0
ð
Þx
:
ð8:13Þ
(i) W 6 ¼ 0 if and only if k 6 ¼ k
0 . In this case, e
ikx and e
ik
0 x are said to be linearly
independent. That is, on condition of k 6 ¼ k
0
, for any x we have
298
8 Reflection and Transmission of Electromagnetic Waves in Dielectric Media