Yao-Xiong Huang: of of
Yao-Xiong Huang: of of
Yao-Xiong Huang: of of
J. Appl. Phys. 76 (5), 1 September 1994 0021-8979/94/76(5)/2575/7/$6.00 B 1994 American Institute of Physics 2575
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z
t reflecti wave
20
FIG. 2. (a) Angle of reflection 6,, and [b) angle of transmission 0, vs angle
of incidence oi for various values of /I and N,=O.22, N,=O.Si,
k,.(x+v,t)+k,,(y+v,t)+k&,t)-qt N,=O.837, and n=2. The angles on dashed lines are those of 180”-6’,.
(1)
= k&r+ v,t) + k,,.(y + v,t) + k,,(v,t) - o,t
= k,,(x + v,t) + kyt(y + v,t> + k,t(v,t) - qf, (1-n’P2)(k3))2+2 : j3(n2- l)(k.N)
where the subscripts i, r, and t denote the components for
incident, reflected, and transmitted waves, respectively. As +(l-/32j(kxN)2+~ (P2--n2)=0, (4)
Eq. (1) holds for all x, y, and t, it yields
where ~=(E/E~)~‘~,. and N=v/v is the unit vector in the
krr=kxf=kxi, (24 direction of v. Similarly, by combining Eqs. (2~) and (4), we
find that
i2b)
kzt=kdP,U+p, cos Wdl(l-@j,
i2’j Ot="i(l+P~ COSei-p&)/(1-&), (5b)
For our case, kxi=ko sin $i, k,i=O, kzi= -ko cos pi; here where
ko=Wi/C, and c=(E~,LQ) -1’2. Combining Eq. (2cj with the
dispersion relation for the reflected wave, s={C1-p,2)[‘Y2(n2-1)(1-Px sin Bi+,B, COS 8ij2
- sin2 ei] + ( 1 + /3, cos 8i)2}1’2,
and
we immediately obtain k,, and or as follows:
y= l/( 1- /!?2)?
k,,=ko(2P,+~s t’$+/?z cos f+)/(l-/3:), (34 At present, we have given all the components of wave
four vectors for both the reflected and transmitted waves.
WrzC0i(l+2P, c0S Bi+@)/(l-&j, i3b) From these results, we can conclude that, no matter how the
values of v and its three components v, , vY , and v, vary, the
where p=vIc, and Pa=vnlc (a=~, y, z). propagation vectors of the incident, reflected, and transmitted
The dispersion relation for the transmitted wave in the waves always lie on the same plane-the plane of incidence.
moving medium can be deduced following the same ways as In addition, both o, and k,, depend on & and ei. So, there
those described in Refs. 20 and 21, is Doppler shift in frequency for the reflected wave, but it
2576 J. Appl. Phys., Vol. 76, No. 5, 1 September 1994 Yao-Xiong Huang
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90 -0
0.2
0.4
70 0.6
0.8
0;
60
30
10
0.6 0
w 0.8
0.6
30 40 0.4
0: -0.2 c i 0.2
20 -0.4
30 $4
-0.6 -0.2
10 -0.8 -0.4
-0.6
0
10 -0.8
10 30 80 70 90
FIG. 3. (a) (9, and (b) .9, vs Bi for various values of p and e;
N,=N,,=N,=il.577, n=2. The angles on dashed lines are those of
lsoO-Or.
FIG. 4. (a) 0, and (b) 0, vs ei for various values of p and N,=O.837,
N,=OSl, N,=O.22, and n=2. The angles on dashed lines are those of
180”-e,.
occurs only in the case of u,# 0, and the angle of reflection
8, [=tan-‘(&.,/k,,)] is not equal to tli except for the case of
u,=O. Its dependence on Bi for various values of p in sev-
eral sets of N, , N,, , and N, is shown in Figs. 2-4, as the sin e,=sin t9,{sin2 &+[&(& c0S @i+l)’
solution of the modified law of reflection. It is noted that -2&(p, COS Bi+1)q+q2]l(1-p~)}-1’2
when p, is positive, 13~is smaller than 0i and drops to zero as
u, approaches the speed of light; but, for a negative p,, 0, is and
greater than Bi and becomes 90” at an incident angle Bi= 0,.
k,=ko{sii2 hJi+[&& cos Bi+ 1)’
According to Eq. (3a), 8,,=cos-‘[ - 2flJ( 1 + &)]. Beyond
this angle, k,, changes its sign, and the angle of reflection -2&(& COS 8i+l)q+q2]/(1-P~)2}1’2.
becomes greater than 90”. So readers should be aware that
the angles on the dashed lines in Figs. 2-4 are not e,‘s, but III. REFLECTION AND TRANSMISSION
those of 180”- 0,) instead. BiC is smaller as the magnitude of COEFFICIENTS (FOR TE WAVE)
]p,l increases and it exists only in the case of u,#O.
According to Eq. (5b), there also exists Doppler shift for Suppose the incident wave in the observer’s frame K
the transmitted wave in the case-of u,#O. However, different takes the form
from that in the reflected wave, the frequency shift in trans-
mitted wave depends not only on u, , but also on u, , uY , and
n. The angle of transmission &=tan-*](k,,lk,,)]. From Eqs. exp i(k,ix+kziz-cj>it), (84
(2a) and (5a), we can see that it is a function of Bi, n, u, , uY ,
and u,. The behavior of f3, vs 8i for various values of uX,
v,, , and uL is also shown in Figs. 2-4.
The relation between et and Bi now follows Snell’s law Hi=[z) =( 111 @JLxp i(k,ix+k,iz-tiit),
with modifications. According to the result of Eq. (5a), the
modified Snell’s law can be written as (W
where E. and Ho are constant amplitudes. By using the Lor-
k. sin &=k, sin 8,, 63 entz transformations for field vectors,“” one can find the
where real part of the incident wave in frame K’,
J. Appl. Phys., Vol. 76, No. 5, 1 September 1994 Yao-Xiong Huang 2577
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Eli = [ yp,, sin Si + ( 1 - y)N,N,]Ea , H:i=[ y sin Oi- YP,
Eb,=[y+y(& cos Oi-fl, sin t&)+(1-y)Ni]Eo, (9) - (1- y)N,(N, cos O,+N, sin Oi)]Ho.
n2Az(G3-G1)E:i+n2~2(G3-G1)E~i+[A2(n2A1-A3)+M2(it’M1-M3)]E~i
E:,= , (114
A2(A3-n2A22)+M2(M3--n2M2)+n2G2(G3-G2)
n2(G3-G1)E;i+(n2M1-M3)E;i+(n2M2-M3)E;,
El,= 2 W)
n2(Gz-‘%>
n2(G3-G1)E;,+(n2A1-A,)E;,+(n2A,-A,)E;,
E;,= 9 (llc)
n2U%-G3>
E;=C2(E;,+E;,),
E;,=E;,+E;,, w
E;,=E;,+E;,,
(134
H, _(G~-n2G~)H~i+(n2~,--M~)H~~+(n’M~-M~)H~,
2 Wb)
Y’ n2G2- G3
H, =(G,-n’G,)H:i+(n2Al-A3)H:i+(n*A2-Af)Hr,
XT 3 , (13c)
n-G2- G3
H;,=H;+H;,,
Gs=(y-l)N,(N, sin B,+N$)+P- y&W,
where
where L=k,,/k,, P=k,,lk,, U=wJoi, and W=w,/wi.
Ar=(y-l)N,(N, sin Si-Nz cos Bi)+sin Oi-ypX, Then, the real parts of the reflected and transmitted
waves in the observer’s frame K are obtained by applying the
inverse transformation to Eqs. (ll)-(14). They are written in
AZ= ( y- l)N,(N, sin Oi+N&) + sin ei- yfl,U, (15) the form as follows:
As=(y-l)N,(N,
Iwr=(y-l)N,,(N,
sin ciri+N,P)+sin
Bi)--y&,
(f$)=Y( ,,
2)+tl-dal($)-
M2=(y-
M3=(y-l)N,,(N,
l)N,(N, S~II &+N&)-
sin ~i+NJ’)-yP,W,
yfl,U, (16)
-YPO(
i ” yb,)
(;!), (18)
2578 J. Appl. Phys., Vol. 76, No. 5, 1 September 1994 ‘Yao-Xiong Huang
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_,/.-.-' -. -._- t-r-._.
-1.0 -0.11 -6.6 -0.4 -0.9. 0 0.2 0.4 0.6 0.6 1.0 -1.0 4.8 -0.6 -0.4 -0.1 0.2 0.4 0.6 0.6 1.0
B
FIG. 5. The reflection and transmission coefficients vs p when Bi= 0, n = 2, FIG. 6. The reflection and transmission coefficients vs ,B when 4=60”,
and (a) N,=O.22, N,=OSl, and N,=O.837; Ib) N,=N,=N,=O.577; n=2, and (a) N,=O.22, N,=O.Sl, N,=0.837; (b) N,=N,=N, =0.577;
(c) N,==O.837, N,=O:Sl, N,=O.22. (c) N,=0.837,Ny=0.51, Nz=0.22.
(2)=Y(
$I)
+(l-$a2(
sj (5;)
= @+;jW;j@ z (
-ypo[ Jy u’ :xj (zj, (19)
+?%
iyi’txj
( (Zij, (21)
where
al=N,E:,+NyE6,+N,E:,,
($j=($j+bh(;j a2=N,E:t+N,,Elt+N~~:I,
b*=N,H:,+N,H~,+NZH:,,
bz=NxH~,+N,,H;,+NzH;,.
+yEo(
-zy
1’f+j(Zj¶ (20)
From the results of Eqs. (18), (19), (20), and (21), with
those of Eqs. (9)~(17), we can see that in any case u,fO,
both the reflected and transmitted waves have components of
J. Appl. Phys., Vol. 76, No. 5, 1 September 1994 Yao-Xiong Huang 2579
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FIG, 7. The reflection and transmission coefficients vs the angle of inci- FIG. 8. The reflection and transmission coefficients vs the angle of inci-
dence 8, with /3=0.2, n=2, and N,=N,=N,=0.577. dence Bi With p=-0.2, n=2, and N,=Ny=Nz=0.577.
2580 J. Appl. Phys., Vol. 76, No. 5, 1 September 1994 Yao-Xiong Huang
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for the combination of all the cases. In other words, our, 4C. Yeh, J. Appl. Phys. 36, 3513 (1965).
expressions for wave four vectors, the modified law of re- ‘T. K. Shiozawa, K. Hazawa, and N. Kumagai, J. Appl. Phys. 38, 4459
(1967).
flection and Snell’s law, the reflected and transmitted wave “V. P. Pyati, J. Appl. Phys. 38, 652 (1967).
field vectors, and the reflection and transmission coefficients, 7P. Daly and H. Gruenberg, J. Appl. Phys. 38, 4486’11967).
can cover all the cases in which the medium is moving uni- sC. Yeh, J. Appl. Phys. 38, 5194 (1967).
formly with an arbitrary velocity; thus, it can be taken as the 9T. Shiozawa and N. Kumagai, Proc. IEEE 55, 1243 (1967).
“T. Shiozawa and K. Hazama, Radio Sci. 3, 569 (1968).
general solution for the problem. “J. M. Saca, J. Mod. Opt. 36, 1367 (1989).~
“D. Censor, Radio Sci. 4, 1079 (1969).
ACKNOWLEDGMENTS “M. Ohkubo, Electronic Commun. Jpn. (English trans.) 52-B, 125 (1969).
14K. Tanaka and K. Hazawa, Radio Sci. 7, 973 (1972).
This work was supported in part by the Chinese National “R. S. Mueller, Radio Sci. 22, 461 (1987).
Science foundation and an award for Chinese Outstanding 16C. Yeh, J. Appl. Phys. 37, 3079 (1966).
Young Professors from Chinese National Education Commit- t7Y. H. Pao and K. Hutter, Proc. IEEE 63, 1011 (1975).
“A good review on these aspects is given in J. Van Bladel, Relativity and
tee. The author would like to thank Q. C. Ho for some dis-
Engineering (Springer, Berlin, 1984), Chap. 5, and references therein.
cussions on the subject, and F. H. Huang for the drawing. I9 J. A. Kong, Theory of EZectromagnetic Waves (Wiley, New York, 1975).
‘OH C. Chen, Theory of Electromagnetic Wuves (McGraw-Hill, New York,
r W. Pauli, TIzeory of Relativity [Pergamon, New York, 1958). 1983).
aA. Sommerfeld, Opfik, 2nd ed. (Akademische, Leipzig, 1959). *r K. S. Kunz, J. Appl. Phys. 51, 873 (198Oj.
3C. T. Tai, URSI Spring Meeting, Washington, DC, 1965. “C. Meller, Theory of Relativity (Oxford University Press, L&don, 1959).
J. Appl. Phys., Vol. 76, No. 5, 1 September 1994 Yao-Xiong Huang 2581
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