Com m e nt o n the "Corrected F res n e l Coefficie nts for Lossy Mate ria l s" Abstract
It is shown that the "corrected" Fresnel reflection and transmission coefficients derived recently by Cann i n g [1] using a complex transm ission wave vector approach and i nvolvi ng a real true angle of refraction are identical to the trad itional coefficients based on a com plex angle of refractio n . Keywords: E lectromagnetic p ropagation ; electromag netic p ropagation i n absorbing med i a ; d ielectric losses; electromagn etic reflectio n ; electromagnetic refractio n ;
A,17 e iPi or , E-i - y-"'o
1. I ntrod uction
C
(1)
onsider a uniform plane wave in medium 1 impinging obliquely on medium 2, characterized by finite conduc tivity and occupying the region z � O. Medium 1 is described by a wavenumber A. angular frequency,
(i),
= �sl PI ' (i)
expressed in terms of the
the real permittivity,
sl '
and the real
permeability, PI . Medium 2 is a lossy dielectric with complex wavenumber given by
fJ2
where
c
fields in medium 1 and the transmitted fields in medium 2 are given as follows:
= �s2P2 (i)
(2)
is the speed of light in vacuum,
P2 r
Ht =
is the relative
permeability of medium 2 (assumed to be real), and
sZr
suppressing an exp ( -i{i)t ) time dependence. The reflected
sir
-
and
are respectively the real and imaginary parts of the com
plex relative permittivity of the second medium.
-.
(3)
- VxEt· 1
-
l{i)P2
Here,
2. Trad iti o n a l Approach 2 . 1 TE Polarization It is assumed that the incident electric-plane field in medium 1 is linearly polarized in the positive y direction, and propagates along the incident wave vector
Pi = A. ( sin Bix + cos BiZ) , where Bi
is the angle of incidence
and the caret sign denotes a unit vector. The incident electric field and its corresponding magnetic field are thus given as
and
are respectively the reflection and transmission wave vectors, involving the real angle of reflection, Br , and the complex angle of refraction, Bt;
and
TTE
IEEE Antennas and Propagation Magazine, Vol. 53 , No. 4, August 20 1 1
r TE
is the Fresnel reflection coefficient,
is the Fresnel transmission coefficient.
161
Following the standard approach [2] of equating the tan gential components of the total electric and magnetic fields at the interface z 0 results in the equality of the angle of reflec tion and the angle of incidence, the "complex" Snell 's law,
(11)
=
(4)
2 . 3 True Ang le of Refraction
and the Fresnel reflection and transmission coefficients
From the complex Snell 's law given in Equation (4), it follows that
(5)
(6)
( 1 2) ==
2.2 TM Polarizati on
where p and q are real quantities, and q
It is assumed that the incident magnetic field in medium 1 is linearly polarized in the positive y direction, and propagates along the incident wave vector
pj
=
p + iq , > O. Consequently,
the exponential term associated with the refracted fields in either polarization can be rewritten as
Pt ( sin OJx + cos OJ z) . The
incident magnetic field and its corresponding electric field are thus given as follows :
H-i -- yAHoe (pj." ,
_
-e
(7)
( 1 3)
-qz e j(xPI sin Bj+pz ) .
The surfaces of constant amplitude are defined as
The reflected fields in medium 1 and the transmitted fields in medium 2 are expressed as
=
refraction is then defined as follows : ,/, 'I'
(8)
qz
constant; thus, the wave decays along the positive z direction. On the other hand, the surfaces of constant phase are given by the planes xPt sin OJ + pz = constant. The true (real) angle of
I
-tan _
(PI ) sin OJ
p
.
( 1 4)
Using the notation introduced in Equation ( 1 3), the Fresnel coefficients for both polarizations are rewritten as follows: (9)
where case.
A'
r TE
Pt, Or ' and 0t are defined as in the TE-polarization
Equating the tangential components of the total electric and magnetic fields at the interface z 0 yields the equality of the angle of reflection and the angle of incidence, Snell's law (cf. Equation (4)), and the Fresnel reflection and transmission coefficients : =
( 1 0)
1 62
TTE
=
=
r TM
J.i2 Pt cos OJ - J.il (p + iq ) J.i2 Pt cos 0i + J.il (p + iq ) '
( 1 5)
,
( 1 6)
J.i2Pt cos OJ + J.il ( P+iq ) Pt cos OJ - � ( p+iq )
=
....!:....
&2
Pt cos OJ + � ( P + iq )
----
---
&2
c-:: cosOi
" &IJ.i1
.
OJ&IJ.i2 -2- - ( p + zq ) P2
( 1 7)
IEEE Antennas and Propagation Magazine, Vol. 53 , No. 4, August 20 1 1
(2 1 )
( 1 8)
Equating the tangential components o f the total electric and magnetic fields at the boundary z yields the "corrected" Fresnel reflection and transmission coefficients [ 1 ] :
=0
The second form in Equations ( 1 7) and ( 1 8) will prove useful in the following section. 3. Can n i n g 's Approach The traditional approach summarized in Section 2 suffers from the appearance of the complex angle of refraction, the
(}t,
physical meaning of which can be understood only by means of the indirect derivation of the "true" angle of refraction. Various authors (cf., e.g., [ 1 , 3]) have tried to circumvent the aforementioned deficiency by means of ab initio introduction of a real true angle of refraction. The incident and reflected fields for both polarizations are defined as in the traditional case. The main difference is in the refracting fields. For both polarizations, the transmitted wave vector is defined as
pt(2) =kl +ik2 ' where both vectors kl,2
(24)
are real. The expo
nential term associated with the refracted fields of either polarization is then written as ( 1 9) Equating the tangential components of the total electric and magnetic fields at the interface z again results in the equal ity of the angle of reflection and the angle of incidence. However, Snell's law requires special attention. Specifically, one has
(25)
=0
0
(2 )
4. Com parison of the Trad iti o n a l Approach with Can n i n g 's Approach
at
zo:;: = z =O. r
-
(
However,
) - zx ( z x r )
== z' o r z' -
,
'
-
= zx ( z x r ) - ,
'
Within the framework of the traditional approach, the refracted field and the reflection and transmission Fresnel coefficients for both the TE and TM polarizations depend on the computations of the real quantities p and q using Equation ( 1 2), viz.,
-
zo:;: =O. It follows from Equation (20 ) that (pjxz-Ptxz)=O, or [pjxz-(kl+ik2)xzJ=0 , from which one obtains fiJ. sin (}j = kl sin () + ik2 sin If/; kl,2 = Ikl,21. For k2 =F- 0, the angle If/ must equal zero. Thus, k2 =k2z. In addition, Snell's law assumes the form PI sin (}j =kl sin ()
�pi - fiJ.2 sin 2 (}j =� J.i280 (82r +i8Zr ) - fiJ.2 sin 2 (}j , oi
at the interface
involving the real true angle of refraction, (). This is made clearer from the refraction exponential form
(26)
=p + iq ,
together with the true angle of refraction, r/J, defined in Equa tion ( 1 3). On the other hand, in Canning's approach, the refracted field and the reflection and transmission Fresnel coefficients for both the TE and TM polarizations depend on the computations of the real quantities and the real angle of
IEEE Antennas and Propagation Magazine, Vol. 53 , No. 4, August 20 1 1
k1,2
1 63
refraction. () . These computations are effected by means of the dispersion relationship
JJ?). JJ?) pi =
and Snell's law. More
explicitly, from the three coupled equations
determined by the traditional approach and Canning's approach are identical. Of course, it is known that the real refraction vector,
(27a)
(27b) (27c) A comparison of the refraction exponential terms given in Equations ( 1 3) and (2 1 ) indicates that q k2 and p kl cos () . =
=
Introducing the latter into Equation (26) and squaring both sides, one obtains
coincides with the direction of the time-averaged transmitted Poynting vector in the case of the TE polarization, but not for the TM polarization [4] . As a consequence, the notion of a "true" angle of refraction should only be associated with the real refraction vector
kl
(on the plane of incidence), which is
always perpendicular to the surfaces of equal phase.
6 . Referen ces
(28)
1 . F. X. Canning, "Corrected Fresnel Coefficients for Lossy Materials," IEEE International Symposium on Antennas and Propagation, Spokane, WA, 20 1 1 , pp. 2 1 33-2 1 36.
or
2. J. A.
Stratton, Electromagnetic Theory, New York, McGraw Hill, 1 94 1 .
(29)
which is precisely the dispersion relation. Thus, both the tra ditional approach and Canning's approach are self-consistent using the identifications q k2 ' P kl cos () , and () ¢J =
=
=
•
5. Concl u d i n g Remarks
Based on the analysis in this note, the ''true'' angles of refraction, the reflection and transmission Fresnel coefficients, and, as result, all the fields for both TE and TM polarizations
1 64
3 . A . R . Melnyk and M . J. Harrison, "Theory of Optical Exci tation of Plasmons in Metals," Phys. Rev. B, 2, 1 970, pp. 835850. 4 . R. De Roo and Chen-To Tai, "Plane Wave Reflection and Refraction Involving a Finite Conducting Medium," IEEE Antennas and Propagation Magazine, 45, 5, October 2003 , pp. 54-6 1 .
Ioannis M . Besieris Bradley Department of Electrical and Computer Engineering Virginia Polytechnic Institute and State University Blacksburg VA 24060, USA E-mail:
[email protected]
IEEE Antennas and Propagation Magazine, Vol. 53, No. 4, August 20 1 1
