is γk
⃗
i
c-v
≤ω'≤γk
⃗
i
c+v
, or γ
1-β
ω≤ω'≤γ
1+β
ω.
Suppose there exist two suitable media 1 and 2, and
their refractive indexes satisfy the following
conditions: the ω' is smaller but greater than γ(1−β)ω,
and the refractive indexes of the two media meet
n
f1
(ω')<n
f2
(ω'). In other word, the relative refractive
index n
f21
>1. With the increase of ω', when ω' is
greater but less than γ(1+β)ω, the refractive indexes
of the two media can meet n
f1
(ω')>n
f2
(ω'), that is, the
relative refractive index n
f21
<1. There is a special case
of n
f21
=1 between the two cases. When the media are
determined, both n
f1
(ω'), n
f2
(ω') and n
f21
are uniquely
determined by ω', while ω' is determined by ω and v⃗
according to Eq.(1). Hence v⃗ is bound to affect the
laws of half-wave loss, critical angle of total
reflection, and Brewster’s angle. However, the
relations of refractive indexes and wave frequencies
are so complex in fact, and it is difficult to have a
unified theoretical formula to describe for different
media. This is another interesting subject valuable to
investigate deeply in the next.
6 CONCLUSION
In this paper, the problem of electromagnetic waves
transmission at high-speed interface moving in any
direction has been studied extensively. Firstly, the
case of incidence from vacuum to medium is
discussed, and the reflection and refraction laws are
obtained by Lorentz transformation, Snell’s law and
Fresnel’s law, which are different vastly from the
laws in static system. Besides the half-wave loss, total
reflection and Brewster’s law are analyzed.
Furthermore, the case of incidence from one medium
to another is also discussed. In the specific
application, according to the method and theoretical
formula in this paper, the final results needed can be
obtained by numerical calculation by substituting
concrete conditions like ω, n
f
and v⃗.
ACKNOWLEDGMENTS
This work is supported by Research Project on
Undergraduate Teaching Reform of Higher
Education of Shandong Province (Z2021036), Group
Theory Curriculum Ideological and Political
Education Demonstration Project of Shandong
Province, Research Project on Teaching Reform of
Postgraduate Education of Shandong Province
(SDYJG19038), Quality Course Construction of
Colleges and Universities in Shandong Province
(JPKC201311), and Shandong Students’ innovation
and entrepreneurship training program 2023.
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