Egor'evsk College of Civilian Aviation Technology, 140303 Egor'evsk, Moscow Region, Russia. Submitted November 11, 1996; resubmitted October 28, 1997.
TECHNICAL PHYSICS
VOLUME 44, NUMBER 5
MAY 1999
Propagation of a monochromatic electromagnetic plane wave in a medium with nonsimple motion V. O. Gladyshev Egor’evsk College of Civilian Aviation Technology, 140303 Egor’evsk, Moscow Region, Russia
~Submitted November 11, 1996; resubmitted October 28, 1997! Zh. Tekh. Fiz. 69, 97–100 ~May 1999!
An exact analytical solution is obtained for the trajectory of the wave vector of a monochromatic electromagnetic plane wave in a medium with nonsimple motion. It is shown that the spatial dragging of the electromagnetic wave by the moving medium can be described correctly in the general case only if relativistic terms of order b 2 are taken into account. © 1999 American Institute of Physics. @S1063-7842~99!01705-5#
INTRODUCTION
FORMULATION OF THE PROBLEM
The solution of the dispersion relation for the propagation of a monochromatic electromagnetic plane wave in a moving medium reduces to finding the wave vector k2 of the electromagnetic wave in the medium. For propagation of electromagnetic radiation in a medium with nonsimple motion it is necessary to obtain a solution for each local region along the entire propagation trajectory of the wave, since the wave vector at each point of the trajectory depends on the velocity vector u2 of the medium.1 This dependence is a consequence of the fact that the Fizeau effect is a particular case of the spatial dragging of the light by a moving medium2 and can be investigated experimentally. For example, in experiments on laser ranging of a space vehicle ~SV! it was found that the Fizeau effect has an appreciable influence on the direction of the laser beam passing through a moving quartz reflector.3,4 Since the effect should appear in interferometric measurements at comparatively low velocities of the propagation medium of the electromagnetic wave, it should affect the results of a wide class of experiments. The investigation of this phenomenon is also of interest from a different standpoint, since it can be viewed as a precise test of the electrodynamics of moving media in which the interaction of an electromagnetic wave with the moving matter can be analyzed on an atomic scale on the basis of macroscopic observations. For this reason there naturally arises the problem of finding analytical equations describing the trajectory of the wave vector in a medium with nonsimple ~nonrectilinear! motion. We note that in the literature the description of the effect of the motion of the medium on the propagation of electromagnetic radiation is ordinarily limited to approximate calculations. This is admissible in calculations of the Fizeau effect, which can be described quite accurately taking into account only the first-order relativistic terms. However, this simplification is unacceptable for subtle phenomenon such as the curvature of the trajectory of an electromagnetic wave, whose magnitude is determined by second-order relativistic terms.
Let us consider an inertial coordinate system in which a medium with permittivity « 1 and magnetic permeability m 1 is at rest in the half space Z,0 and a medium with « 2 and m 2 , which are measured in its own coordinate system, moves with arbitrary velocity u2 in the half space Z.0. A tangential discontinuity of the velocity exists at the interface between the media. We assume that the velocity field is constant along the Y axis. Neglecting the dispersion of the moving medium and taking « 1 m 1 51 the wave vector k2 in the second medium can be expressed as5
1063-7842/99/44(5)/4/$15.00
k 2x 5
v0 sin q 0 , c
k 2z 5
1 v0 $ 2 k 2 b 2z g 22 ~ 12 b 2x sin q 0 ! 2 c 12 k 2 b 2z g 22
~1a!
6 ~ cos2 q 0 ~ 12 k 22 b 2z g 22 ! 1 k 2 g 22 ~ 12 b 2x sinq 0 ! 2 ! 1/2 % , ~1b! 2 2 where g 22 2 512( b 2z 1 b 2x ), k 2 5« 2 m 2 21, b 2x 5u 2x /c, b 2z 5u 2z /c, u2 is the velocity vector of the second medium, q 0 is the angle of incidence of the electromagnetic wave on the interface of the two media, v 0 is the angular frequency of the electromagnetic wave, c is the speed of light in vacuum, and the sign is chosen so that the wave propagates away from the interface between the two media. The form of the expression for k 2z imposes a restriction on the function b 2 (x,z) for which analytical solutions of the equation of the trajectory of the wave vector in the medium exist. Using the function ~1b! in searching for a solution of the form z5 f (x) leads to an implicit integral equation x (x,z) z5 * 0max f(x,z)dx, which does not have an analytical solution in the general case.6 However, there is an analytically solvable case where the spatial character of the dragging of the light is most naturally manifested. Let the velocity u 2 be a function of the coordinates x and z of the form
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© 1999 American Institute of Physics
Tech. Phys. 44 (5), May 1999
b 22 5
v2 c2
~ R 0 2z ! 2 1
v2 c2
V. O. Gladyshev
~2!
x 2,
which corresponds to rotation with angular velocity v relative to the center (0,R 0 ). This function determines the parameters u 2x 5 v (R 0 2z) and u 2z 5 v x as functions of the independent coordinates. The function ~2! with both components requires the use of numerical methods, which was done in Ref. 2, since analytical methods lead to computational errors, due to the dropping of terms in the series expansion, and so on, that are larger than the effect under investigation. On the other hand the tangential component of the velocity of the medium influences the spatial character of the dragging of the light ~the curvature of the trajectory of the wave vector!. For this reason this is the most interesting case for studying the spatial dragging of light by a moving medium. Let only a tangential component b 2x be present in the moving medium and b 2z 50. This corresponds to a shear flow with velocity varying linearly with distance from the boundary. Then the angle of refraction q 2 can be written as tan2 q 2 ~ z ! 5 5
S D k 2x k 2z
g 2 ~ 12 b 2x ~ z !! 1 ~ n 22 21 !~ 12 ab x ~ z !! 2
, ~3!
where a 5sinq0 , g 5cosq0 , n 2 5 A« 2 m 2 , and b x (z) 5u 2x (z)/c. The wave vector k 0 52 p /l 0 of the incident electromagnetic wave satisfies the relation k 0 @1/R 0 , which makes it possible to use the solution of the wave equation for an electromagnetic plane wave undergoing a tangential velocity discontinuity at the interface between the media4 for each local region of the medium. We shall be interested in the equation describing the trajectory of k2 , i.e., the analytical dependence x5 f (z). Evidently, thedifference relation Dx i 5tan q 2 ~ z i ! Dz i , where Dx i 5x i 2x i21 , Dz i 5z i 2z i21 , , and x i 5 ( ij51 Dx j , holds for each local region of the medium. Then a relation between the instantaneous coordinates can be obtained by summing and switching to an integral in the limit Dz i →0
E
z
0
tan q 2 ~ z ! dz.
We shall seek the solution of Eq. ~4! in a general form. For this, we substitute expression ~3! and switch to the new variable b x . After transformations we obtain x5 t
E
b2~ b
b1
x 21 ! d b x
AG 4 ~ b x !
~5!
,
where
t5
c v
a
A12n 22 a 2
,
G 4 ~ b x ! 5 ~ a2 b x !~ b2 b x !~ b x 2c !~ b x 2d ! ,
b x1,25
a ~ 12n 22 ! 6 g 2 n 2 12n 22 a 2
a5 b x1 ,
b52c51,
, d5 b x2 .
J S5
E
dbx ~ b x 21 ! S AG 4 ~ b x !
.
~6!
Then the coordinate x can be expressed as b
x5 t ~ J 22 22J 21 ! u b 2 . 1
~7!
In order to reduce J 22 to tabulated integrals, its order must be increased. Let us expand G 4 ( b x ) in powers of ( b x 21) G 4 ~ b x ! 5b 0 ~ b x 21 ! 4 1b 1 ~ b x 21 ! 3 1b 2 ~ b x 21 ! 2 1b 3 ~ b x 21 ! 1b 4 , where
z i 5 ( ij51 Dz j
x5
ANALYTICAL SOLUTION OF THE EQUATION FOR THE TRAJECTORY OF THE WAVE VECTOR OF AN ELECTROMAGNETIC WAVE
The expression contains the root of a quartic polynomial and it can be shown that Eq. ~5! can be represented as a composition of elliptic integrals. The integration limits are determined from the expression for b 2x (z) for the initial and final coordinates z 1 and z 2 of the trajectory of the wave vector. We introduce the notation
2
a 2 ~ 12 b 2x ~ z !!
567
~4!
A characteristic feature of this expression is that the limits can be set arbitrarily, but we do not have precise information about the point where the trajectory intersects, for example, a prescribed cylindrical surface of radius R 0 . Therefore, generally speaking, the upper limit of the integral is variable. We also note that the expression for the angle of refraction is exact and contains quadratic terms. This is of fundamental significance for studying the spatial dragging of light by a moving medium.
b 0 51,
b 1 542 ~ b x1 1 b x2 ! ,
b 2 551 b x b x2 23 ~ b x1 1 b x2 ! , b 3 52 ~ 11 b x1 b x2 2 ~ b x1 1 b x2 !! , and b 4 50. Integrating the first derivative of the product
AG 4 ~ b x !~ b x 21 ! 2S , we obtain a recurrence relation that makes it possible to lower the order of the elliptic integral b 0 ~ 22S ! J S23 1 1
b1 ~ 322S ! J S22 1b 2 ~ 12S ! J S21 2
b3 ~ 122S ! J S 2b 4 SJ S11 5 AG 4 ~ b x !~ b x 21 ! 2S , 2
S51,2,3, . . . .
~8!
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Tech. Phys. 44 (5), May 1999
V. O. Gladyshev
Using Eq. ~8! with S51 and b 4 50 we obtain an expression for J 22
S
D
1 2 AG 4 ~ b x ! 2b 1 J 21 1b 3 J 1 . J 22 5 2b 0 b x 21
~9!
Substituting expression ~9! into Eq. ~7!, the expression for x becomes x5
S
t 2 AG 4 ~ b x ! 1b 3 J 1 1 ~ 4b 0 2b 1 ! J 21 2b 0 b x 21
DU
b2
.
~10!
b1
We note that a.b> b x .c.d, and we introduce the notations I 15
E
b xd b x
bx
c
AG ~ b x ! 4
I 25
,
E
dbx
bx
c
AG ~ b x ! 4
b
I 3 52J 1 u c x .
,
Then Eq. ~10! can be expressed in terms of tabulated integrals
S
t 2 AG 4 ~ b x ! 2b 3 I 3 1 ~ 4b 0 2b 1 !~ I 1 2I 2 ! x5 2b 0 b x 21
DU
b2
I 2 52gF ~ w ,k ! , I 3 52gq @~ c2d ! P ~ w ,n 2 ,k ! 1 ~ 12c ! F ~ w ,k !# , 1
Aa2c) ~ b2d !
,
q5
1 ~ 12c !~ 12d !
,
~11!
where F( w ,k) and P( w ,n i ,k) are normal elliptic integrals of the first and third kinds, to which correspond the characteristics n 1 and n 2 , the amplitude w , and the modulus k given by n 15
b2c , b2d
w 5sin
21
n 25
~ b2c !~ 12d ! , ~ b2d !~ 12c !
A
~ b2d !~ b x 2c ! , ~ b2c !~ b x 2d !
k5
A
x5 t c 1 P ~ w ,n 2 ,k ! 2c 2 P ~ w ,n 1 ,k ! 2c 3 F ~ w ,k ! 2
AG 4 ~ b x ! 12 b x
DU
,
where 1 c 1 5 ~ 11 b x2 !~ 12 b x1 ! , t
D w 5 A12k 2 sin2 w ,
k5sin a .
The use of this expression decreases the interpolation error because the tables of F( w ,k) and E( w ,k) is more accurate. ESTIMATION OF THE EFFECT OF TERMS OF ORDER b 2
Here it is appropriate to raise the question of whether or not terms of order b 2 are needed to describe the threedimensional Fizeau effect. Indeed, the electrodynamics formulas for the longitudinal Fizeau effect are linear in the velocity of the medium and terms of order b 2 are negligible, though they are present in the exact solution of the dispersion relation. However, for the three-dimensional Fizeau effect, i.e., when a transverse component of dragging of the wave appears, it may be necessary to take account of the terms with b 2 . To justify this assertion we shall obtain an expression for the path length of the light beam in a medium with a chosen law of motion neglecting b 2 . In the geometricoptics approximation the path length of the light beam can be written as S5
EA 0
11tan2 q 2 ~ z ! dz,
~13!
where tan2 q 2 >
a2 g 2 1 ~ n 22 21 !~ 122 ab x !
~3 !
8
.
Changing to the new variable and performing the transformation, we obtain
b2 b1
P ~ w ,n 2 51, k ! 5F ~ w ,k ! 2sec2 a ~ E ~ w ,k ! 2tan w D w ! ,
z
~ b2c !~ a2d ! . ~ a2c !~ b2d !
Finally, substituting the coefficients a, b, c, and d we obtain
S
Comparing the computational results obtained with Eq. ~12!, using tables of elliptic integrals,7 with the results of direct numerical calculations using Eq. ~4! shows that the accuracy of the analytical calculations depends on the resolution of the tables of elliptic integrals and that interpolation must be used. Nonetheless, the expression obtained is exact, and it is desirable to construct more accurate tables of elliptic integrals in the parameter range corresponding to the experimental data. It is also evident that n 2 51 for any d. In this case P( w ,n 2 ,k) can be expressed in terms of elliptic integrals of the first and second kinds:
, b1
I 1 52g @~ c2d ! P ~ w ,n 1 ,k ! 1dF ~ w ,k !# ,
g5
t5 A~ 11 b x1 !~ 12 b x2 ! .
~12!
S52
c v
EA b2
b1
a2 b x dbx , b2 b x
~14!
where a5
n 22 2 a ~ n 22 21 !
,
b5
n 22 2 a 2 2 a ~ n 22 21 !
.
1 c 2 5 ~ 11 b x2 !~ b x2 1 b x1 ! , t
Performing the integral in Eq. ~14! gives
1 c 3 5 @ 2 ~ 12 b x1 ! 1 ~ 12 b x2 !~ b x2 1 b x1 !# , t
S5
S U U
11 c c ~ a2b ! 1 ln 1 2 v 2 21 c 21
DU
2
, 1
~15!
Tech. Phys. 44 (5), May 1999
V. O. Gladyshev
where
c5
A
a2 b x , b2 b x
c 1,25
A
a2 b x1,2 . b2 b x1,2
For v 50 the length of the trajectory up to intersection with the straight line z5R 0 can be written as S 0 5R 0 / cosq2 , where q 2 can be found from Snell’s law. Then the difference of the path length S, taking account of only terms of order b , and the path length S 0 without rotation will approximately characterize the magnitude of the Fizeau effect in a moving medium. Numerical calculations show that the curvature of the propagation trajectory of an electromagnetic wave in a medium with rotation6 is of the same order of smallness as the error in the calculations performed using Eq. ~15!. This confirms that b 2 must be taken into account in order to describe correctly the spatial dragging of electromagnetic radiation by a moving medium. CONCLUSIONS
The Fizeau effect is usually characterized by the magnitude of the drift of the phase velocity of the superposition field of the excitation and secondary electromagnetic waves in a moving medium. It is also convenient to use as the characteristic of the longitudinal dragging of the electromagnetic wave the phase difference between the waves which have passed through the moving medium in opposite directions. An additional effect appears in the case of three-
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dimensional dragging of an electromagnetic wave — the deflection of the trajectory of the wave vector of the superposition wave in the medium. The analytical solutions of the equation for the trajectory of the wave vector in a moving medium can be used to describe this phenomenon. In closing, we note that the solution of Eq. ~4! can be represented as a composition of elliptic integrals not only for the chosen law of motion of the medium. This opens up the possibility of using analytical methods to describe the trajectory of a wave vector in media with more complicated laws of motion. This work was performed as part of the Scientific and Technical Program between Institutions of Higher Education ‘‘Fundamental Research at Technical Institutions of Higher Education in Russia.’’ 1
S. Solimeno, B. Coroziniani, and P. DiPorto, Guiding, Diffraction, and Confinement of Optical Radiation @Academic Press, New York, 1986; Mir, Moscow, 1989, 664 pp.#. 2 V. O. Gladyshev, JETP Lett. 58, 569 ~1993!. 3 V. P. Vasil’ev, V. A. Grishmanovski, L. F. Pliev, and T. P. Startsev, JETP Lett. 55, 316 ~1992!. 4 V. P. Vasil’ev, L. I. Gusev, J. J. Dengan, and V. D. Shargorodski, Radiotekh., No. 4, 80 ~1966!. 5 B. M. Bolotovski and S. N. Stolyarov, Usp. Fiz. Nauk 159, 155 ~1989! @Sov. Phys. Usp. 32, 813 ~1989!#. 6 V. O. Gladyshev, Pis’ma Zh. Tekh. Fiz. 19~19!, 23 ~1993! @Tech. Phys. Lett. 19~10!, 611 ~1993!#. 7 M. Abramowitz and I. Stegun ~Eds.!, Handbook of Mathematical Functions @Dover, New York, 1965; Nauka, Moscow, 1979, 832 pp.#. Translated by M. E. Alferieff
