Abstract

The diffraction of plane electromagnetic waves by a slit in an infinitely thin, perfectly conducting screen between two media with different dielectric constants is studied using rigorous diffraction theory. Both the case where the electric vector is parallel to slit and the case where the magnetic vector is parallel to slit are examined. The problem is formulated in elliptic cylinder coordinates and solved in terms of Mathieu functions. The explicit determination of the diffracted wave-expansion coefficients leads to solving infinite systems of complex linear equations. The long-wave (Rayleigh scattering) region is studied in detail. The scattered intensity at infinity, transmission coefficient, and backscatter coefficient are evaluated. Finally, numerical results are presented for some special cases.

© 1963 Optical Society of America

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References

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  1. M. Fizeau, Ann. Chim. 63, 385 (1861).
  2. Rayleigh, Phil. Mag. 43, 259 (1897).
  3. Rayleigh, Phil. Mag. 14, 350 (1907).
  4. Rayleigh, Proc. Roy. Soc. (London) A89194 (1913).
  5. P. M. Morse and P. J. Rubenstein, Phys. Rev. 54, 895 (1938).
    [Crossref]
  6. H. Hönl, A. W. Maue, and K. Westpfahl, “Theorie der Beugung,” in Encyclopedia of Physics, edited by S. Flügge (Springer-Verlag, Berlin, 1961), Vol. XXV/1, pp. 418–453.
  7. J. Meixner, New York University Institute of Mathematical Sciences, Division of Electromagnetic Research, Research Report EM-68, 1954.
  8. J. Meixner and F. W. Schäfke, Mathieusche Funktionen und Sphäroidfunktionen (Springer-Verlag, Berlin, 1954).
    [Crossref]
  9. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., New York, 1953), Part II.
  10. L. D. Landau and E. M. Liftshitz, Electrodynamics of Continuous Media (Pergamon Press, Ltd., London, 1961).
  11. C. J. Bouwkamp, Physica 12, 467 (1946).
    [Crossref]
  12. J. Meixner, Ann. Physik 6, 2 (1949).
  13. Methods similar to this have been employed by the author in various water wave problems. See, for example, R. Barakat, J. Fluid. Mech. 13, 540 (1962).
    [Crossref]
  14. Reference 9, p 1430.
  15. R. Barakat, A. Houston, and E. Levin, J. Math. and Phys. (to he published).

1962 (1)

Methods similar to this have been employed by the author in various water wave problems. See, for example, R. Barakat, J. Fluid. Mech. 13, 540 (1962).
[Crossref]

1949 (1)

J. Meixner, Ann. Physik 6, 2 (1949).

1946 (1)

C. J. Bouwkamp, Physica 12, 467 (1946).
[Crossref]

1938 (1)

P. M. Morse and P. J. Rubenstein, Phys. Rev. 54, 895 (1938).
[Crossref]

1913 (1)

Rayleigh, Proc. Roy. Soc. (London) A89194 (1913).

1907 (1)

Rayleigh, Phil. Mag. 14, 350 (1907).

1897 (1)

Rayleigh, Phil. Mag. 43, 259 (1897).

1861 (1)

M. Fizeau, Ann. Chim. 63, 385 (1861).

Barakat, R.

Methods similar to this have been employed by the author in various water wave problems. See, for example, R. Barakat, J. Fluid. Mech. 13, 540 (1962).
[Crossref]

R. Barakat, A. Houston, and E. Levin, J. Math. and Phys. (to he published).

Bouwkamp, C. J.

C. J. Bouwkamp, Physica 12, 467 (1946).
[Crossref]

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., New York, 1953), Part II.

Fizeau, M.

M. Fizeau, Ann. Chim. 63, 385 (1861).

Hönl, H.

H. Hönl, A. W. Maue, and K. Westpfahl, “Theorie der Beugung,” in Encyclopedia of Physics, edited by S. Flügge (Springer-Verlag, Berlin, 1961), Vol. XXV/1, pp. 418–453.

Houston, A.

R. Barakat, A. Houston, and E. Levin, J. Math. and Phys. (to he published).

Landau, L. D.

L. D. Landau and E. M. Liftshitz, Electrodynamics of Continuous Media (Pergamon Press, Ltd., London, 1961).

Levin, E.

R. Barakat, A. Houston, and E. Levin, J. Math. and Phys. (to he published).

Liftshitz, E. M.

L. D. Landau and E. M. Liftshitz, Electrodynamics of Continuous Media (Pergamon Press, Ltd., London, 1961).

Maue, A. W.

H. Hönl, A. W. Maue, and K. Westpfahl, “Theorie der Beugung,” in Encyclopedia of Physics, edited by S. Flügge (Springer-Verlag, Berlin, 1961), Vol. XXV/1, pp. 418–453.

Meixner, J.

J. Meixner, Ann. Physik 6, 2 (1949).

J. Meixner, New York University Institute of Mathematical Sciences, Division of Electromagnetic Research, Research Report EM-68, 1954.

J. Meixner and F. W. Schäfke, Mathieusche Funktionen und Sphäroidfunktionen (Springer-Verlag, Berlin, 1954).
[Crossref]

Morse, P. M.

P. M. Morse and P. J. Rubenstein, Phys. Rev. 54, 895 (1938).
[Crossref]

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., New York, 1953), Part II.

Rayleigh,

Rayleigh, Proc. Roy. Soc. (London) A89194 (1913).

Rayleigh, Phil. Mag. 14, 350 (1907).

Rayleigh, Phil. Mag. 43, 259 (1897).

Rubenstein, P. J.

P. M. Morse and P. J. Rubenstein, Phys. Rev. 54, 895 (1938).
[Crossref]

Schäfke, F. W.

J. Meixner and F. W. Schäfke, Mathieusche Funktionen und Sphäroidfunktionen (Springer-Verlag, Berlin, 1954).
[Crossref]

Westpfahl, K.

H. Hönl, A. W. Maue, and K. Westpfahl, “Theorie der Beugung,” in Encyclopedia of Physics, edited by S. Flügge (Springer-Verlag, Berlin, 1961), Vol. XXV/1, pp. 418–453.

Ann. Chim. (1)

M. Fizeau, Ann. Chim. 63, 385 (1861).

Ann. Physik (1)

J. Meixner, Ann. Physik 6, 2 (1949).

J. Fluid. Mech. (1)

Methods similar to this have been employed by the author in various water wave problems. See, for example, R. Barakat, J. Fluid. Mech. 13, 540 (1962).
[Crossref]

Phil. Mag. (2)

Rayleigh, Phil. Mag. 43, 259 (1897).

Rayleigh, Phil. Mag. 14, 350 (1907).

Phys. Rev. (1)

P. M. Morse and P. J. Rubenstein, Phys. Rev. 54, 895 (1938).
[Crossref]

Physica (1)

C. J. Bouwkamp, Physica 12, 467 (1946).
[Crossref]

Proc. Roy. Soc. (London) (1)

Rayleigh, Proc. Roy. Soc. (London) A89194 (1913).

Other (7)

H. Hönl, A. W. Maue, and K. Westpfahl, “Theorie der Beugung,” in Encyclopedia of Physics, edited by S. Flügge (Springer-Verlag, Berlin, 1961), Vol. XXV/1, pp. 418–453.

J. Meixner, New York University Institute of Mathematical Sciences, Division of Electromagnetic Research, Research Report EM-68, 1954.

J. Meixner and F. W. Schäfke, Mathieusche Funktionen und Sphäroidfunktionen (Springer-Verlag, Berlin, 1954).
[Crossref]

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., New York, 1953), Part II.

L. D. Landau and E. M. Liftshitz, Electrodynamics of Continuous Media (Pergamon Press, Ltd., London, 1961).

Reference 9, p 1430.

R. Barakat, A. Houston, and E. Levin, J. Math. and Phys. (to he published).

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Figures (13)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

Elliptic cylinder coordinate system employed in problem.

Fig. 3
Fig. 3

Scattered intensity at infinity (electric vector case) for various values of h2 at h1=1.0. The direction of wave propagation is normal to the slit.

Fig. 4
Fig. 4

Scattered intensity at infinity (electric vector case) for various values of h2 at h1=1.0. The direction of wave propagation is normal to the slit.

Fig. 5
Fig. 5

Scattered intensity at infinity (electric vector case) for various values of h2 at h1=1.0. The direction of wave propagation is normal to the slit.

Fig. 6
Fig. 6

Electric vector transmission coefficient Te′ as a function of the angle of incidence and h2 for h1=1.0.

Fig. 7
Fig. 7

Electric vector backscatter coefficient Re′ as a function of the angle of incidence and h2 for h1=1.0.

Fig. 8
Fig. 8

Scattered intensity at infinity for normal incidence (magnetic vector case) in Region 1 for values of h2 less than h1=1.0.

Fig. 9
Fig. 9

Scattered intensity at infinity for normal incidence (magnetic vector case) in Region 1 for values of h2 greater than h1=1.0.

Fig. 10
Fig. 10

Scattered intensity at infinity for normal incidence (magnetic vector case) in Region 2 for values of h2 less than h1=1.0.

Fig. 11
Fig. 11

Scattered intensity at infinity for normal incidence (magnetic vector case) in Region 2 for values of h2 greater than h1=1.0.

Fig. 12
Fig. 12

Magnetic vector transmission coefficient Tm′ as a function of the angle of incidence and h2 for h1=1.0.

Fig. 13
Fig. 13

Magnetic vector backscatter coefficient Rm′ as a function of the angle of incidence and h2 for h1=1.0.

Tables (11)

Tables Icon

Table I Electric vector expansion coefficients B1, B3 for h1=1 and α=180° (normal incidence).

Tables Icon

Table II Square of absolute value of electric vector expansion coefficients A1, B1, as a function of the angle of incidence α for h1=1.

Tables Icon

Table III Square of absolute value of electric vector expansion coefficients A2, B2 as a function of the angle of incidence α for h1=1.

Tables Icon

Table IV Scattered intensity at infinity Ω1e(h1,α) for normal incidence (α=−90°) in Region 1.

Tables Icon

Table V Scattered intensity at infinity Ω2e(h2,α) for normal incidence (α=−90°) in Region 2.

Tables Icon

Table VI Electric vector transmission coefficient Te′ as a function of h2 and α h1=1.0.

Tables Icon

Table VII Electric vector backscatter coefficient Re′ as a function of h2 and g for h1=1.0.

Tables Icon

Table VIII Scattered intensity at infinity Ω1m(h1,α) for normal incidence (α=−90°) in Region 1.

Tables Icon

Table IX Scattered intensity at infinity Ω2m(h2,α) for normal incidence (α=−90°) in Region 2.

Tables Icon

Table X Magnetic vector transmission coefficient Tm′ as a function of h2 and α for h1=1.0.

Tables Icon

Table XI Magnetic vector backscatter coefficient Rm′ as a function of h2 and α for h1=1.0.

Equations (104)

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x = a cosh u cos v ( u 0 ) , y = a sinh u sin v ( 0 v 2 π ) , z = z .
i ω μ H ˜ = curl E ˜ ; div E ˜ = 0 , - i ω E ˜ = curl H ˜ ; div H ˜ = 0 ,
( 2 F / u 2 ) + ( 2 F / v 2 ) + h 2 ( cosh 2 u - cos 2 v ) F = 0 ,
( d 2 F 1 / d u 2 ) + ( h 2 cosh 2 u - b ) F 1 = 0 , ( d 2 F 2 / d v 2 ) - ( h 2 cos 2 v - b ) F 2 = 0 ,
E z / y = i ω H x ;             E z / x = - i ω H y .
( 2 x 2 + 2 y 2 ) E z = i ω ( H x y - H y x ) = - ω 2 E z .
E z i + E z r + E z d = - E z d ,
E z d y 0 = - E z d y 0             ( x > a , y = 0 ) .
μ 2 ( / y ) [ E z i + E z r + E z d ] = μ 1 ( / y ) E z d ,
E z i = ( 8 π ) 1 2 n = 0 i n N n J e n ( h 1 , u ) S e n ( h 1 , v ) S e n ( h 1 , α ) + ( 8 π ) 1 2 n = 1 i n N n J o n ( h 1 , u ) S o n ( h 1 , v ) S o n ( h 1 , α ) ,
E z i = e i k 1 ( x cos α + y sin α ) ,
k 1 = 2 π / λ 1 = ω ( 1 μ 1 ) 1 2 ,
E z r = e i k 1 ( x cos α - y sin α ) ,
E z r = ( 8 π ) 1 2 n = 0 i n N n J e n ( h 1 , u ) S e n ( h 1 , v ) S e n ( h 1 , α ) - ( 8 π ) 1 2 n = 1 i n N n J o n ( h 1 , u ) S o n ( h 1 , v ) S o n ( h 1 , α ) .
H o n ( 1 ) ( h , u ) H o n ( h , u ) = J o n ( h , u ) + i N o n ( h , u ) ,
lim ( h cosh u ) H o n ( 1 ) ( h , u ) ~ ( h cosh u ) - 1 2 exp { i [ h cosh u - 1 4 ( 2 n + 1 ) ] π } .
E d = n = 1 A n H o n ( h 1 , u ) S o n ( h 1 , v )             ( y 0 ) = n = 1 B n H o n ( h 2 , u ) S o n ( h 2 , v )             ( y 0 ) ,
S o n ( h , 0 ) = S o n ( h , π ) = 0.
n = 1 A n H o n ( h 1 , 0 ) S o n ( h 1 , v ) = - n = 1 B n H o n ( h 2 , 0 ) S o n ( h 2 , v ) .
y = 1 a sin v u | w = 0 ,
2 ( 8 π ) 1 2 n = 1 i n N n J o n ( h 1 , 0 ) S o n ( h 1 , v ) S o n ( h 1 , α ) + n = 1 A n H o n ( h 1 , 0 ) S o n ( h 1 , v ) = μ 1 μ 2 n = 1 B n H o n ( h 2 , 0 ) S o n ( h 2 , v ) .
0 2 π S o n ( h 1 , v ) S o m ( h 1 , v ) d v = δ m n N n ( h 1 ) ,
R m n ( h 1 , h 2 ) = 0 2 π S o m ( h 1 , v ) S o n ( h 2 , v ) d v ,
A m N m ( h 1 ) H o m ( h 1 , 0 ) = - n = 1 B n R m n ( h 1 , h 2 ) H o n ( h 2 , 0 ) .
2 ( 8 π ) 1 2 i m J o m ( h 1 , 0 ) S o m ( h 1 , α ) + A m N m ( h 1 ) H o m ( h 1 , 0 )             ( m = 1 , 2 , ) = μ 1 μ 2 n = 1 B n R m n ( h 1 , h 2 ) H o n ( h 2 , 0 ) .
2 ( 8 π ) 1 2 i m J o m ( h 1 , 0 ) S o m ( h 1 , α ) = n = 1 B n [ R m n ( h 1 , h 2 ) μ 1 μ 2 H o n ( h 2 , 0 ) + H o m ( h 1 , 0 ) H o m ( h 1 , 0 ) H o n ( h 2 , 0 ) ] .
R m n ( h 1 , h 2 ) = 0             { m = even n = odd             { m = odd m = even ;
S o m ( h , 3 π / 2 ) = 0             ( m = even ) ,
H z / y = - i ω E x ;             H z / x = i ω E y ,
[ ( 2 / x 2 ) + ( 2 / y 2 ) ] H z + k 2 H z = 0 ,
H z i + H z r + H z d = H z d ,
2 ( / y ) [ H z i + H z r + H z d ] = 1 ( / y ) H z d .
( / y ) H z = 0.
H d = n = 0 A n H e n ( h 1 , u ) S e n ( h 1 , v )             ( y 0 ) = n = 0 B n H e n ( h 2 , u ) S e n ( h 2 , v )             ( y 0 ) .
H e n ( 1 ) ( h , u ) H e n ( h , u ) = J e n ( h , u ) + i N e n ( h , u ) ,
H i = H r = ( 8 π ) 1 2 n = 0 i n N n J e n ( h 1 , 0 ) S e n ( h 1 , v ) S e ( h 1 , α ) .
2 ( 8 π ) 1 2 n = 0 i n N n J e n ( h 1 , 0 ) S e n ( h 1 , v ) S e n ( h 1 , α ) + n = 0 A n H e n ( h 1 , 0 ) S e n ( h 1 , v ) = n = 0 B n H e n ( h 2 , 0 ) S e n ( h 2 , v ) .
y = 1 a sin v u | u = 0 ;
n = 0 A n H e n ( h 1 , 0 ) S e n ( h 1 , v ) = - 1 2 n = 0 B n H e n ( h 2 , 0 ) S e n ( h 2 , v ) ,
0 2 π S e m ( h , v ) S e n ( h , v ) d v = N n ( h ) δ m n ,
2 ( 8 π ) 1 2 i m J e m ( h 1 , 0 ) S e m ( h 1 , α ) + A m H e m ( h 1 , 0 ) N m = n = 0 B n H e n ( h 2 , 0 ) Q m n ( h 1 , h 2 ) ,
A m H e m ( h 1 , 0 ) N m = - 1 2 n = 0 B n H e n ( h 2 , 0 ) Q m n ( h 1 , h 2 ) ,
Q m n ( h 1 , h 2 ) = 0 2 π S e m ( h 1 , v ) S e n ( h 2 , v ) d v .
2 ( 8 π ) 1 2 i m J e m ( h 1 , 0 ) S e m ( h 1 , α ) = n = 0 B n Q m n ( h 1 , h 2 ) × [ H e n ( h 2 , 0 ) + ( 1 2 ) H e n ( h 2 , 0 ) H e m ( h 1 , 0 ) H e m ( h 1 , 0 ) ] .
2 ( 8 π ) 1 2 i J o 1 ( h 1 , 0 ) S o 1 ( h 1 , α ) = B 1 R 11 { H o 1 ( h 2 , 0 ) + H o 1 ( h 1 , 0 ) H o 1 ( h 1 , 0 ) H o 1 ( h 2 , 0 ) } .
H o 1 ( h , 0 ) = J o 1 ( h , 0 ) + i N o 1 ( h , 0 ) ~ - i 2 h ( 2 π ) 1 2             ( h 1 ) ,
H o 1 ( h , 0 ) ~ h 2 ( π 2 ) 1 2 + i 2 h ( 2 π ) 1 2             ( h 1 ) ,
S o 1 ( h , α ) ~ sin α             ( h 1 ) ,
R 11 ( h 1 , h 2 ) ~ π             ( h 1 , h 2 1 ) .
4 q h 1 2 sin α = B 1 [ 8 ( 2 / π ) 1 2 - i q 2 h 1 2 ( 1 + q 2 ) ( π / 2 ) 1 2 ] ,
- q 2 h 1 4 sin 2 α = B 2 [ π h 1 4 ( 1 + q 4 ) + 8 i ] .
A 1 N 1 ( h 1 ) H o 1 ( h 1 , 0 ) = - B 1 R 11 H o 1 ( h 2 , 0 ) .
A 1 = - ( 1 / q ) B 1 .
2 ( 8 π ) 1 2 J e 0 ( h 1 , 0 ) S e 0 ( h 1 , α ) = B 0 Q 00 [ H e 0 ( h 2 , 0 ) + q 2 H e 0 ( h 1 , 0 ) H e 0 ( h 2 , 0 ) H e 0 ( h 1 , 0 ) ] .
H e 0 ( h , 0 ) = J e 0 ( h , 0 ) + i N e 0 ( h , 0 ) ~ ( π 2 ) 1 2 + i ( 2 π ) 1 2 log ( γ h 4 )             ( γ = 1.781 ) ,
H e 0 ( h , 0 ) ~ i ( 2 / π ) 1 2 ,
S e 0 ( h , α ) ~ 1 ,             Q 00 ( h 1 , h 2 ) ~ 2 π .
4 π = 2 π B 0 [ ( π 2 ) 1 2 ( 1 + q 2 ) + i ( 2 π ) 1 2 × { log γ q h 1 4 + q 2 log γ h 1 4 } ] ,
( 2 π ) 1 2 = B 0 [ ( π / 2 ) ( 1 + q 2 ) + i { ( 1 + q 2 ) log ( γ q h 1 / 4 ) - q 2 log q } ] .
A 0 H e 0 ( h 1 , 0 ) N 0 ( h 1 ) = - q 2 B 0 H e 0 ( h 2 , 0 ) Q 00 .
N 0 ( h 1 ) ~ 2 π ,             Q 00 ( h 1 , h 2 ) ~ 2 π ;
A 0 = - q 2 B 0 .
lim u cosh u = r / a ,
ϒ 1 e = E d 1 2 = a h 1 r n = 1 m = 1 A n A m * S o n ( h 1 , v ) S o m ( h 1 , v ) ,
ϒ 2 e = E d 2 2 = a h 2 r n = 1 m = 1 B n B m * S o n ( h 2 , v ) S o m ( h 2 , v ) ,
ϒ 1 e ~ ( a / h 1 r ) A 1 2 sin 2 v ,
ϒ 2 e ~ ( a / q h 1 r ) B 1 2 sin 2 v .
ϒ 2 e ~ [ a sin 2 v 8 r ] 16 q 2 h 1 4 sin 2 α ( 128 / π ) + q 4 h 1 4 ( π / 2 ) ( 1 + q 2 ) 2 ~ ( a π q h 1 3 / 8 r ) sin 2 v sin 2 α .
ϒ 1 e ~ ( a π h 1 3 / 8 r q 2 ) sin 2 v sin 2 α .
ϒ 2 e ϒ 1 e = ( h 2 h 1 ) 3 = ( n 1 n 2 ) 3 = ( 2 1 ) 3 2 .
ϒ 1 m = H d 1 2 = a h 1 r n = 0 m = 0 A n A m * S e n ( h 1 , v ) S e m ( h 1 , v ) ,
ϒ 2 m = H d 2 2 = a h 2 r n = 0 m = 0 B n B m * S e n ( h 2 , v ) S e m ( h 2 , v ) .
ϒ 1 m ~ 2 ( a / r ) ( π / q h 1 ) q 5 [ ( π / 2 ) ( 1 + q 2 ) ] 2 + [ ( 1 + q 2 ) log ( γ q h 1 / 4 ) - q 2 log q ] 2 ,
ϒ 2 m ~ 2 ( a / r ) ( π / q h 1 ) [ ( π / 2 ) ( 1 + q 2 ) ] 2 + [ ( 1 + q 2 ) log ( γ q h 1 / 4 ) - q 2 log q ] 2 ,
ϒ 2 m ϒ 1 m = ( h 1 h 2 ) 5 = ( n 2 n 1 ) 5 = ( 1 2 ) 5 2 .
ϒ m ~ ( a / r ) ( 2 π / h ) π 2 + 4 log 2 ( γ h / 4 ) .
T = 1 ( 1 / r ) π 2 π ϒ 2 d v .
π 2 π S e n ( h , v ) S e m ( h , v ) d v = 1 2 N n ( h ) δ m n ,
π 2 π S o n ( h , v ) S o m ( h , v ) d v = 1 2 N n ( h ) δ m n ,
T e = a h 2 n = 1 1 2 N n ( h 2 ) B n 2 ,
T m = a h 2 n = 0 1 2 N n ( h 2 ) B n 2 .
T e ~ ( π 2 a q h 1 3 / 16 ) sin 2 α ,
T m ~ ( π 2 a / q h 1 ) [ ( π / 2 ) ( 1 + q 2 ) ] 2 + [ ( 1 + q 2 ) log ( γ q h 1 / 4 ) - q 2 log q ] 2 .
R = 1 ( 1 / r ) 0 π ϒ 1 d v .
R e = a h 1 n = 1 1 2 N n ( h 1 ) A n 2 ,
R m = a h 1 n = 0 1 2 N n ( h 1 ) A n 2 ,
R e ~ ( π 2 a h 1 3 / 16 q 2 ) sin 2 α ,
R m ~ ( a q 5 π 2 / q h 1 ) [ ( π / 2 ) ( 1 + q 2 ) ] 2 + [ ( 1 + q 2 ) log ( γ q h 1 / 4 ) - q 2 log q ] 2 .
T e R e = ( h 2 h 1 ) 3 = ( n 1 n 2 ) 3 = ( 2 1 ) 3 2 ,
T m R m = ( h 1 h 2 ) 5 = ( n 2 n 1 ) 5 = ( 1 2 ) 5 2 .
T e R m = ( n 1 / n 2 ) 8 T m R e ,
T e = T e / a ;             R e = R e / a .
T m = T m / a ;             R m = R m / a .
E < ;             H ~ 0 ( D - 1 2 ) .
H < ;             E ~ 0 ( D - 1 2 ) .
R m n ( h 1 , h 2 ) = 0 2 π S o m ( h 1 , v ) S o n ( h 2 , v ) d v .
R m n ( h 1 , h 2 ) = k = 1 l = 1 D o 2 k m ( h 1 ) D o 2 l n ( h 2 ) × 0 2 π sin 2 k v sin 2 v d v             ( m , n = even ) = π k = 1 D o 2 k m ( h 1 ) D o 2 k n ( h 2 ) .
R m n ( h 1 , h 2 ) = k = 0 l = 0 D o 2 k + 1 m ( h 1 ) D o 2 l + 1 n ( h 2 ) × 0 2 π sin ( 2 k + 1 ) v sin ( 2 l + 1 ) v d v             ( m , n = odd ) = π k = 0 D o 2 k + 1 m ( h 1 ) D o 2 k + 1 n ( h 2 ) .
R m n ( h 1 , h 2 ) = 0             { m = even n = odd             { m = odd n = even .
Q m n ( h 1 , h 2 ) = 2 π D e 0 m ( h 1 ) D e 0 n ( h 2 ) + π k = 1 D e 2 k m ( h 1 ) D e 2 k n ( h 2 ) ,
Q m n ( h 1 , h 2 ) = π k = 0 D e 2 k + 1 m ( h 1 ) D e 2 k + 1 n ( h 2 ) ,
Q m n ( h 1 , h 2 ) = 0
D o 1 1 ( h ) ~ 1 + ( 3 h 2 / 32 ) ;             D e 0 0 ( h ) ~ 1 + 1 8 h 2 ,
R 11 ( h 1 , h 2 ) ~ π ;             Q 00 ( h 1 , h 2 ) ~ π .