Abstract

The symmetry theorems on the complete forward and backward scattering Mueller matrices for light scattering from a single dielectric scatterer (as opposed to an ensemble of scatterers) are systematically and thoroughly analyzed. Symmetry operations considered include discrete rotations about the incident direction and mirror planes not coinciding with the scattering plane. For forward scattering we find sixteen different symmetry shapes (not including the totally asymmetric one), which may be classified into five symmetry classes, with identical reductions in the forward scattering matrices for all symmetry shapes that fall into the same symmetry class. For backward scattering we find only four different symmetry shapes, which may be classified into only two symmetry classes. The forward scattering symmetry theorems also lead to a symmetry theorem on the total extinction cross section. Based on the conclusions of this work it should be possible to design quick and nondestructive methods for the identification of certain small objects, when suitable partial information about the objects to be identified is already available. A promising practical example is given.

© 1987 Optical Society of America

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References

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  1. C.-R. Hu, G. W. Kattawar, M. E. Parkin, “Complete Mueller Matrix Calculations for Light Scattering from Dielectric Cubes of Dimensions of the Order of a Wavelength,” in Proceedings, CRDC 1984 Scientific Conference on Obscuration and Aerosol Research, R. H. Kohl, D. Stroud, Eds. (R. H. Kohl & Associates, Tullahoma, TN 37388, 1985), p. 307;see also G. W. Kattawar, C.-R. Hu, M. E. Parkin, “Mueller Matrix Calculations for Dielectric Cubes: Comparison with Experiments,” in Proceedings, 1985 Scientific Conference on Obscuration and Aerosal Research, R. H. Kohl, Ed. (R. H. Kohl & Associates, Tullahoma, TN 37388, 1986), p. 537; and to be published.
  2. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957), Chap. 5.
  3. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  4. These nine relations appear to have been derived explicitly for the first time by K. D. Abhyankar, A. L. Fymat, “Relations Between the Elements of the Phase Matrix for Scattering,” [J. Math. Phys. 10, 1935 (1969)];see also E. S. Fry, G. W. Kattawar, “Relationships Between Elements of the Stokes Matrix,” Appl. Opt. 20, 2811 (1981);J. W. Hovenier, H. C. van de Hulst, C. V. M. van der Mee, “Conditions for the Elements of the Scattering Matrix,” Astron. Astrophys. 157, 301 (1986). The relations derived in these references are for arbitrary scattering angle ϕ, whereas the relations presented here are specialized to ϕ = 0 [and π in Eqs. (33a)–(33l)] and, therefore, can exhibit the explicit ϕ dependence.
    [CrossRef] [PubMed]
  5. M. Hamermesh, Group Theory (Addison-Wesley, Reading, MA, 1962), Chap. 2;and V. Heine, Group Theory in Quantum Mechanics (Pergamon, Oxford, 1960), Chap. 3, Sec. 16 and Appendix J.
  6. The symmetry shape (8b) with m = 3 is easily emulated with a cubic scatterer at corner-on incidence. With some further deletions of subcubes starting with this configuration one can also emulate the symmetry shape (12b) with m = 3. We have numerical results for the former case but not the latter.
  7. W. T. Keeton, Biological Sciences (Norton, New York, 1980).

1969 (1)

These nine relations appear to have been derived explicitly for the first time by K. D. Abhyankar, A. L. Fymat, “Relations Between the Elements of the Phase Matrix for Scattering,” [J. Math. Phys. 10, 1935 (1969)];see also E. S. Fry, G. W. Kattawar, “Relationships Between Elements of the Stokes Matrix,” Appl. Opt. 20, 2811 (1981);J. W. Hovenier, H. C. van de Hulst, C. V. M. van der Mee, “Conditions for the Elements of the Scattering Matrix,” Astron. Astrophys. 157, 301 (1986). The relations derived in these references are for arbitrary scattering angle ϕ, whereas the relations presented here are specialized to ϕ = 0 [and π in Eqs. (33a)–(33l)] and, therefore, can exhibit the explicit ϕ dependence.
[CrossRef] [PubMed]

Abhyankar, K. D.

These nine relations appear to have been derived explicitly for the first time by K. D. Abhyankar, A. L. Fymat, “Relations Between the Elements of the Phase Matrix for Scattering,” [J. Math. Phys. 10, 1935 (1969)];see also E. S. Fry, G. W. Kattawar, “Relationships Between Elements of the Stokes Matrix,” Appl. Opt. 20, 2811 (1981);J. W. Hovenier, H. C. van de Hulst, C. V. M. van der Mee, “Conditions for the Elements of the Scattering Matrix,” Astron. Astrophys. 157, 301 (1986). The relations derived in these references are for arbitrary scattering angle ϕ, whereas the relations presented here are specialized to ϕ = 0 [and π in Eqs. (33a)–(33l)] and, therefore, can exhibit the explicit ϕ dependence.
[CrossRef] [PubMed]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Fymat, A. L.

These nine relations appear to have been derived explicitly for the first time by K. D. Abhyankar, A. L. Fymat, “Relations Between the Elements of the Phase Matrix for Scattering,” [J. Math. Phys. 10, 1935 (1969)];see also E. S. Fry, G. W. Kattawar, “Relationships Between Elements of the Stokes Matrix,” Appl. Opt. 20, 2811 (1981);J. W. Hovenier, H. C. van de Hulst, C. V. M. van der Mee, “Conditions for the Elements of the Scattering Matrix,” Astron. Astrophys. 157, 301 (1986). The relations derived in these references are for arbitrary scattering angle ϕ, whereas the relations presented here are specialized to ϕ = 0 [and π in Eqs. (33a)–(33l)] and, therefore, can exhibit the explicit ϕ dependence.
[CrossRef] [PubMed]

Hamermesh, M.

M. Hamermesh, Group Theory (Addison-Wesley, Reading, MA, 1962), Chap. 2;and V. Heine, Group Theory in Quantum Mechanics (Pergamon, Oxford, 1960), Chap. 3, Sec. 16 and Appendix J.

Hu, C.-R.

C.-R. Hu, G. W. Kattawar, M. E. Parkin, “Complete Mueller Matrix Calculations for Light Scattering from Dielectric Cubes of Dimensions of the Order of a Wavelength,” in Proceedings, CRDC 1984 Scientific Conference on Obscuration and Aerosol Research, R. H. Kohl, D. Stroud, Eds. (R. H. Kohl & Associates, Tullahoma, TN 37388, 1985), p. 307;see also G. W. Kattawar, C.-R. Hu, M. E. Parkin, “Mueller Matrix Calculations for Dielectric Cubes: Comparison with Experiments,” in Proceedings, 1985 Scientific Conference on Obscuration and Aerosal Research, R. H. Kohl, Ed. (R. H. Kohl & Associates, Tullahoma, TN 37388, 1986), p. 537; and to be published.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Kattawar, G. W.

C.-R. Hu, G. W. Kattawar, M. E. Parkin, “Complete Mueller Matrix Calculations for Light Scattering from Dielectric Cubes of Dimensions of the Order of a Wavelength,” in Proceedings, CRDC 1984 Scientific Conference on Obscuration and Aerosol Research, R. H. Kohl, D. Stroud, Eds. (R. H. Kohl & Associates, Tullahoma, TN 37388, 1985), p. 307;see also G. W. Kattawar, C.-R. Hu, M. E. Parkin, “Mueller Matrix Calculations for Dielectric Cubes: Comparison with Experiments,” in Proceedings, 1985 Scientific Conference on Obscuration and Aerosal Research, R. H. Kohl, Ed. (R. H. Kohl & Associates, Tullahoma, TN 37388, 1986), p. 537; and to be published.

Keeton, W. T.

W. T. Keeton, Biological Sciences (Norton, New York, 1980).

Parkin, M. E.

C.-R. Hu, G. W. Kattawar, M. E. Parkin, “Complete Mueller Matrix Calculations for Light Scattering from Dielectric Cubes of Dimensions of the Order of a Wavelength,” in Proceedings, CRDC 1984 Scientific Conference on Obscuration and Aerosol Research, R. H. Kohl, D. Stroud, Eds. (R. H. Kohl & Associates, Tullahoma, TN 37388, 1985), p. 307;see also G. W. Kattawar, C.-R. Hu, M. E. Parkin, “Mueller Matrix Calculations for Dielectric Cubes: Comparison with Experiments,” in Proceedings, 1985 Scientific Conference on Obscuration and Aerosal Research, R. H. Kohl, Ed. (R. H. Kohl & Associates, Tullahoma, TN 37388, 1986), p. 537; and to be published.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957), Chap. 5.

J. Math. Phys. (1)

These nine relations appear to have been derived explicitly for the first time by K. D. Abhyankar, A. L. Fymat, “Relations Between the Elements of the Phase Matrix for Scattering,” [J. Math. Phys. 10, 1935 (1969)];see also E. S. Fry, G. W. Kattawar, “Relationships Between Elements of the Stokes Matrix,” Appl. Opt. 20, 2811 (1981);J. W. Hovenier, H. C. van de Hulst, C. V. M. van der Mee, “Conditions for the Elements of the Scattering Matrix,” Astron. Astrophys. 157, 301 (1986). The relations derived in these references are for arbitrary scattering angle ϕ, whereas the relations presented here are specialized to ϕ = 0 [and π in Eqs. (33a)–(33l)] and, therefore, can exhibit the explicit ϕ dependence.
[CrossRef] [PubMed]

Other (6)

M. Hamermesh, Group Theory (Addison-Wesley, Reading, MA, 1962), Chap. 2;and V. Heine, Group Theory in Quantum Mechanics (Pergamon, Oxford, 1960), Chap. 3, Sec. 16 and Appendix J.

The symmetry shape (8b) with m = 3 is easily emulated with a cubic scatterer at corner-on incidence. With some further deletions of subcubes starting with this configuration one can also emulate the symmetry shape (12b) with m = 3. We have numerical results for the former case but not the latter.

W. T. Keeton, Biological Sciences (Norton, New York, 1980).

C.-R. Hu, G. W. Kattawar, M. E. Parkin, “Complete Mueller Matrix Calculations for Light Scattering from Dielectric Cubes of Dimensions of the Order of a Wavelength,” in Proceedings, CRDC 1984 Scientific Conference on Obscuration and Aerosol Research, R. H. Kohl, D. Stroud, Eds. (R. H. Kohl & Associates, Tullahoma, TN 37388, 1985), p. 307;see also G. W. Kattawar, C.-R. Hu, M. E. Parkin, “Mueller Matrix Calculations for Dielectric Cubes: Comparison with Experiments,” in Proceedings, 1985 Scientific Conference on Obscuration and Aerosal Research, R. H. Kohl, Ed. (R. H. Kohl & Associates, Tullahoma, TN 37388, 1986), p. 537; and to be published.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957), Chap. 5.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

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Figures (4)

Fig. 1
Fig. 1

Choice of coordinates and conventions is the same as that of Ref. 3. In particular, the incident light beam is along the z axis; the scattering plane makes an angle ϕ with the x-z plane; the scattered light beam lies in this plane and makes an angle θ with the z axis. The directions of the unit vectors êi and êi as shown allow the incident electric field to be resolved into the parallel (E‖i) and the perpendicular (Ei) components; and the directions of the unit vectors e‖s and es as shown allow the scattered electric field to be similarly resolved into the components E‖s and Es.

Fig. 2
Fig. 2

One representative object is drawn for each of the sixteen forward scattering symmetry shapes. The symmetry operations have been indicated on each figure with self-explanatory symbols.

Fig.3
Fig.3

Full sixteen elements of the normalized forward (a) and backward (b) scattering Mueller matrices plotted as functions of ϕ from 0 to 180° are given for the case where the deletions of subcubes from a face-on cubic scatterer are such that the remaining object has the forward scattering symmetry shape (8b) for m = 2. The subcubes deleted are 1, 2, 3, 4, 13, 14, 15, 16 in the first layer with similar deletions in the second layer and 49, 52, 53, 56, 57, 60, 61, 64 in the fourth layer with similar deletions in the third layer. The edge length of the starting cube in units of the wavelength has been taken to be 0.8, and the complex index of refraction m of the cube has been taken to be 1.6 + 0.005i.

Fig. 4
Fig. 4

Similar plots as Fig. 3, except that the deletions of the subcubes are such that the remaining object has the symmetry shape (6). The subcubes deleted are 1, 2, 4, 8, 9, 13, 15, 16 with similar deletions in the second layer and 49, 51, 52, 53, 60, 61, 62, 64 with similar deletions in the third layer. The edge length of the starting cube and the complex index of reflection are taken to be the same as in Fig. 3.

Tables (1)

Tables Icon

Table I Classification of Symmetry Shapes for Forward and Backward Scattering

Equations (210)

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[ E s ( θ , ϕ ) E s ( θ , ϕ ) ] = exp [ ik ( r z ) ] ikr [ S 2 ( θ , ϕ ) S 3 ( θ , ϕ ) S 4 ( θ , ϕ ) S 1 ( θ , ϕ ) ] ( E i E i ) ,
( I s Q s U s V s ) = 1 k 2 r 2 ( S 11 S 12 S 13 S 14 S 21 S 22 S 23 S 24 S 31 S 32 S 33 S 34 S 41 S 42 S 43 S 44 ) ( I i Q i U i V i )
S i 0 ( ϕ ) S i ( θ = 0 , ϕ ) , S ij 0 ( ϕ ) S i j ( θ = 0 , ϕ ) ,
S i π ( ϕ ) S i ( θ = π , ϕ ) , S ij π ( ϕ ) S i j ( θ = π , ϕ ) .
( E xs E ys ) θ = 0 = 1 ikr ( C 1 + A C 2 + B C 2 + B C 1 A ) ( E xi E yi ) ,
( E s E s ) θ = 0 = ( cos ϕ sin ϕ sin ϕ cos ϕ ) ( E xs E ys ) θ = 0 ,
( E xi E yi ) = ( cos ϕ sin ϕ sin ϕ cos ϕ ) ( E i E i ) ,
( S 20 S 30 S 40 S 10 ) = ( C 1 + A cos 2 ϕ + B sin 2 ϕ A sin 2 ϕ B cos 2 ϕ C 2 A sin 2 ϕ B cos 2 ϕ + C 2 C 1 A cos 2 ϕ B sin 2 ϕ )
S 110 = ( | C 1 | 2 + | C 2 | 2 + | A | 2 + | B | 2 ) ,
S 120 = 2 Re [ C 1 * ( A cos 2 ϕ + B sin 2 ϕ ) + C 2 * ( A sin 2 ϕ B cos 2 ϕ ) ] ,
S 210 = 2 Re [ C 1 * ( A cos 2 ϕ + B sin 2 ϕ ) C 2 * ( A sin 2 ϕ B cos 2 ϕ ) ] ,
S 130 = 2 Re [ C 1 * ( A sin 2 ϕ B cos 2 ϕ ) C 2 * ( A cos 2 ϕ + B sin 2 ϕ ) ] ,
S 310 = 2 Re [ C 1 * ( A sin 2 ϕ B cos 2 ϕ ) + C 2 * ( A cos 2 ϕ + B sin 2 ϕ ) ] ,
S 140 = 2 Im ( C 1 * C 2 + A * B ) ,
S 410 = 2 Im ( C 1 * C 2 A * B ) ,
S 220 = [ | C 1 | 2 | C 2 | 2 ( | A | 2 | B | 2 ) × cos 4 ϕ 2 Re ( A * B ) sin 4 ϕ ] ,
S 230 = Re [ 2 C 1 * C 2 + ( | A | 2 | B | 2 ) sin 4 ϕ 2 A * B cos 4 ϕ ] ,
S 320 = Re [ 2 C 1 * C 2 + ( | A | 2 | B | 2 ) sin 4 ϕ 2 A * B cos 4 ϕ ] ,
S 240 = 2 Im [ C 1 * ( A sin 2 ϕ B cos 2 ϕ ) + C 2 * ( A cos 2 ϕ + B sin 2 ϕ ) ] ,
S 420 = 2 Im [ C 1 * ( A sin 2 ϕ B cos 2 ϕ ) C 2 * ( A cos 2 ϕ + B sin 2 ϕ ) ] ,
S 330 = [ | C 1 | 2 | C 2 | 2 ( | A | 2 | B | 2 ) × cos 4 ϕ 2 Re ( A * B ) sin 4 ϕ ] ,
S 340 = 2 Im [ C 1 * ( A cos 2 ϕ + B sin 2 ϕ ) C 2 * ( A sin 2 ϕ B cos 2 ϕ ) ] ,
S 430 = 2 Im [ C 1 * ( A cos 2 ϕ + B sin 2 ϕ ) + C 2 * ( A sin 2 ϕ B cos 2 ϕ ) ] ,
S 440 = ( | C 1 | 2 + | C 2 | 2 | A | 2 | B | 2 ) ,
| C 1 * A | 2 = | C 1 | 2 | A | 2 , | C 1 * B | 2 = | C 1 | 2 | B | 2 ,
| C 2 * A | 2 = | C 2 | 2 | A | 2 , | C 2 * B | 2 = | C 2 | 2 | B | 2 ,
| C 1 * C 2 | 2 = | C 1 | 2 | C 2 | 2 , | A * B | 2 = | A | 2 | B | 2 ,
Arg C 1 * A C 1 * B = Arg A B * = Arg C 2 * A C 2 * B ,
Arg C 1 * A C 2 * B = Arg C 1 * C 2 ( = Arg C 1 * B C 2 * B ) ,
| C 1 | 2 ( S 110 + S 220 + S 330 + S 440 ) / 4 ,
| C 2 | 2 ( S 110 S 220 S 330 + S 440 ) / 4 ,
C 1 * C 2 ( S 230 + S 320 + i S 140 + i S 410 ) / 4 ,
| A | 2 [ 2 ( S 110 S 440 ) + ( S 220 S 330 ) × cos 4 ϕ + ( S 230 + S 320 ) sin 4 ϕ ] / 8 ,
| B | 2 [ 2 ( S 110 S 440 ) ( S 220 S 330 ) × cos 4 ϕ ( S 230 + S 320 ) sin 4 ϕ ] / 8 ,
A * B [ ( S 220 S 330 ) sin 4 ϕ ( S 230 + S 320 ) × cos 4 ϕ + 2 i ( S 140 S 410 ) ] / 8 ,
C 1 * A { [ ( S 120 + S 210 ) + i ( S 340 S 430 ) ] cos 2 ϕ + [ ( S 130 + S 310 ) + i ( S 240 + S 420 ) ] sin 2 ϕ } / 8 ,
C 1 * B { [ ( S 120 + S 210 ) + i ( S 340 S 430 ) ] sin 2 ϕ [ ( S 130 + S 310 ) + i ( S 240 + S 420 ) ] cos 2 ϕ } / 8 ,
C 2 * A { [ ( S 130 + S 310 ) + i ( S 240 S 420 ) ] cos 2 ϕ + [ ( S 120 S 210 ) i ( S 340 + S 430 ) ] sin 2 ϕ } / 8 ,
C 2 * B { [ ( S 130 + S 310 ) i ( S 240 + S 420 ) ] sin 2 ϕ [ ( S 120 S 210 ) i ( S 340 + S 430 ) ] cos 2 ϕ } / 8 .
( E x s E y s ) θ = 0 = 1 ikr ( C 1 + A cos 2 ϕ 0 + B sin 2 ϕ 0 C 2 A sin 2 ϕ 0 + B cos 2 ϕ 0 C 2 A cos 2 ϕ 0 + B cos 2 ϕ 0 C 1 A cos 2 ϕ 0 B sin 2 ϕ 0 ) ( E x i E y i ) .
( S 20 S 30 S 40 S 10 ) = [ C 1 + D cos 2 ( ϕ ϕ 0 ) D sin 2 ( ϕ ϕ 0 ) D sin 2 ( ϕ ϕ 0 ) C 1 D cos 2 ( ϕ ϕ 0 ) ] .
( C 1 + A cos 2 γ + B sin 2 γ C 2 A sin 2 γ + B cos 2 γ C 2 A sin 2 γ + B cos 2 γ C 1 A cos 2 γ B sin 2 γ ) .
A ( 1 cos 2 γ ) B sin 2 γ = 0 , A sin 2 γ + B ( 1 cos 2 γ ) = 0 .
( S 20 S 30 S 40 S 10 ) = ( C 1 C 2 C 2 C 1 ) ,
( S 20 S 30 S 40 S 10 ) = [ C 1 + E cos 2 ( ϕ ϕ 1 ) E sin 2 ( ϕ ϕ 1 ) C 2 E sin 2 ( ϕ ϕ 1 ) + C 2 C 1 E cos 2 ( ϕ ϕ 1 ) ] .
( S 20 S 40 S 30 S 10 ) .
( S 20 S 30 S 40 S 10 ) = ( C 1 + A cos 2 ϕ + B sin 2 ϕ A sin 2 ϕ B cos 2 ϕ A sin 2 ϕ B cos 2 ϕ C 1 A cos 2 ϕ B sin 2 ϕ ) .
( σ ϕ 0 C 2 , ϕ 1 ) 2 = [ C ( 2 ϕ 0 2 ϕ 1 ) , z σ z ] 2 = C ( 4 ϕ 0 4 ϕ 1 ) , z ,
[ σ z , σ ϕ 0 ] = [ σ z , C n , z ] = 0 ,
σ ϕ 0 C n , z = C n , z 1 σ ϕ 0 ,
σ ϕ 0 σ ϕ 0 = C ( 2 ϕ 0 2 ϕ 0 ) , z ,
C γ , z C γ , z = C ( γ + γ ) , z ,
σ z σ z = 1 ,
( a ) σ ϕ 0 C n , z , ( b ) σ ϕ 0 σ z , ( c ) C n , z σ z , ( d ) σ ϕ 0 C n , z σ z .
S 110 = | C 1 | 2 + | D | 2 ,
S 120 = S 210 = ( C 1 * D + C 1 D * ) cos 2 ( ϕ ϕ 0 ) ,
S 130 = S 310 = ( C 1 * D + C 1 D * ) sin 2 ( ϕ ϕ 0 ) ,
S 140 = S 410 = 0 ,
S 220 = | C 1 | 2 + | D | 2 cos 4 ( ϕ ϕ 0 ) ,
S 230 = S 320 = | D | 2 sin 4 ( ϕ ϕ 0 ) ,
S 240 = S 420 = i ( C 1 * D C 1 D * ) sin 2 ( ϕ ϕ 0 ) ,
S 330 = | C 1 | 2 | D | 2 cos 4 ( ϕ ϕ 0 ) ,
S 340 = S 430 = i ( C 1 * D C 1 D * ) cos 2 ( ϕ ϕ 0 ) ,
S 440 = | C 1 | 2 | D | 2 ,
S 120 = S 210 , S 130 = S 310 , S 140 = S 410 = 0 ,
S 230 = S 320 , S 240 = S 420 , S 340 = S 430 ,
S 110 + S 440 = S 220 + S 330 ,
( S 110 S 440 ) 2 = ( S 220 S 330 ) 2 + 4 S 230 2 ,
tan 4 ( ϕ ϕ 0 ) = 2 S 230 / ( S 220 S 330 ) ,
tan 2 ( ϕ ϕ 0 ) = S 130 / S 120 = S 240 / S 340 ,
| C 1 * D | 2 = | C 1 | 2 | D | 2 ,
( C 1 * D ) 2 [ ( S 120 + i S 340 ) 2 + ( S 130 i S 240 ) 2 ] / 4 ,
| C 1 | 2 ( S 110 + S 440 ) / 2 ,
| D | 2 ( S 110 S 440 ) / 2 .
( | C 1 | 2 + | D | 2 ( C 1 * D + C 1 D * ) 0 0 ( C 1 * D + C 1 D * ) | C 1 | 2 + | D | 2 0 0 0 0 | C 1 | 2 | D | 2 i ( C 1 * D C 1 D * ) 0 0 i ( C 1 * D C 1 D * ) | C 1 | 2 | D | 2 ) .
( | C 1 | 2 + | D | 2 0 ( C 1 * D + C 1 D * ) 0 0 | C 1 | 2 | D | 2 0 ( C 1 * D C 1 D * ) ( C 1 * D + C 1 D * ) 0 | C 1 | 2 + | D | 2 0 0 i ( C 1 * D C 1 D * ) 0 | C 1 | 2 | D | 2 ) .
( | C 1 | 2 + | C 2 | 2 0 0 i ( C 1 * C 2 C 1 C 2 * ) 0 | C 1 | 2 | C 2 | 2 ( C 1 * C 2 + C 1 C 2 * ) 0 0 ( C 1 * C 2 + C 1 C 2 * ) | C 1 | 2 | C 2 | 2 0 i ( C 1 * C 2 C 1 C 2 * ) 0 0 | C 1 | 2 + | C 2 | 2 ) ,
S 110 = S 440 , S 220 = S 330 ,
S 140 = S 410 , S 230 = S 320 ,
S 320 2 + S 140 2 = S 110 2 S 220 2 .
| C 1 | 2 = ( S 110 + S 220 ) / 2 ,
| C 2 | 2 = ( S 110 S 220 ) / 2 ,
C 1 * C 2 = ( S 320 + i S 140 ) / 2 ,
S 110 = | C 1 | 2 + | C 2 | 2 + | D | 2 ,
S 120 = ( C 1 * D + C 1 D * ) cos 2 ( ϕ ϕ 1 ) + ( C 2 * D + C 2 D * ) sin 2 ( ϕ ϕ 1 ) ,
S 210 = ( C 1 * D + C 1 D * ) cos 2 ( ϕ ϕ 1 ) ( C 2 * D + C 2 D * ) sin 2 ( ϕ ϕ 1 ) ,
S 130 = ( C 1 * D + C 1 D * ) sin 2 ( ϕ ϕ 1 ) ( C 2 * D + C 2 D * ) cos 2 ( ϕ ϕ 1 ) ,
S 310 = ( C 1 * D + C 1 D * ) sin 2 ( ϕ ϕ 1 ) + ( C 2 * D + C 2 D * ) cos 2 ( ϕ ϕ 1 ) ,
S 140 = S 410 = i ( C 1 * C 2 C 1 C 2 * ) ,
S 220 = | C 1 | 2 | C 2 | 2 + | D | 2 cos 4 ( ϕ ϕ 1 ) ,
S 230 = ( C 1 * C 2 + C 1 C 2 * ) + | D | 2 sin 4 ( ϕ ϕ 1 ) ,
S 320 = ( C 1 * C 2 + C 1 C 2 * ) + | D | 2 sin 4 ( ϕ ϕ 1 ) ,
S 240 = i ( C 1 * D C 1 D * ) sin 2 ( ϕ ϕ 1 ) + i ( C 2 * D C 2 D * ) cos 2 ( ϕ ϕ 1 ) ,
S 420 = i ( C 1 * D C 1 D * ) sin 2 ( ϕ ϕ 1 ) + i ( C 2 * D C 2 D * ) cos 2 ( ϕ ϕ 1 ) ,
S 330 = | C 1 | 2 | C 2 | 2 | D | 2 cos 4 ( ϕ ϕ 1 ) ,
S 340 = i ( C 1 * D C 1 D * ) cos 2 ( ϕ ϕ 1 ) + i ( C 2 * D C 2 D * ) sin 2 ( ϕ ϕ 1 ) ,
S 430 = i ( C 1 * D C 1 D * ) cos 2 ( ϕ ϕ 1 ) + i ( C 2 * D C 2 D * ) sin 2 ( ϕ ϕ 1 ) ,
S 440 = | C 1 | 2 + | C 2 | 2 | D | 2 ,
S 140 = S 410 ,
( S 110 S 440 ) 2 = ( S 220 S 330 ) 2 + ( S 320 S 230 ) 2 ,
tan 4 ( ϕ ϕ 1 ) = ( S 320 + S 230 ) / ( S 220 S 330 ) ,
tan 2 ( ϕ ϕ 1 ) = ( S 310 + S 130 ) / ( S 120 S 210 )
= ( S 420 + S 240 ) / ( S 340 S 430 )
= ( S 120 S 210 ) / ( S 310 S 130 )
= ( S 340 + S 430 ) / ( S 420 + S 240 ) ,
| C 1 * C 2 | 2 = | C 1 | 2 | C 2 | 2 ,
| C 1 * D | 2 = | C 1 | 2 | D | 2 , | C 2 * D | 2 = | C 2 | 2 | D | 2 ,
Arg [ ( C 1 * D ) 2 / ( C 2 * D ) 2 ] = Arg ( C 1 * C 2 ) 2 ,
| C 1 | 2 ( S 110 + S 440 + S 220 + S 330 ) / 4 ,
| C 2 | 2 ( S 110 + S 440 S 220 S 330 ) / 4 ,
C 1 * C 2 [ ( S 320 S 230 ) + 2 i S 140 ] / 4 ,
( C 1 * D ) 2 { [ ( S 120 + S 210 ) + i ( S 340 S 430 ) ] 2 + [ ( S 310 + S 130 ) + i ( S 420 S 240 ) ] 2 } / 16 ,
( C 2 * D ) 2 { [ ( S 120 S 210 ) i ( S 340 + S 430 ) ] 2 + [ ( S 310 S 130 ) i ( S 420 + S 240 ) ] 2 } / 16 .
S 110 = | C 1 | 2 + | A | 2 + | B | 2 ,
S 120 = S 210 = 2 Re [ C 1 * ( A cos 2 ϕ + B sin 2 ϕ ) ] ,
S 130 = S 310 = 2 Re [ C 1 * ( A sin 2 ϕ B cos 2 ϕ ) ] ,
S 140 = S 410 = i ( AB * BA * ) ,
S 220 = | C 1 | 2 + ( | A | 2 | B | 2 ) cos 4 ϕ + ( AB * + BA * ) sin 4 ϕ ,
S 230 = S 320 = ( | A | 2 | B | 2 ) sin 4 ϕ ( AB * + BA * ) cos 4 ϕ ,
S 240 = S 420 = 2 Im [ C 1 * ( A sin 2 ϕ B cos 2 ϕ ) ] ,
S 330 = | C 1 | 2 ( | A 2 | B | 2 ) cos 4 ϕ ( AB * + BA * ) sin 4 ϕ ,
S 340 = S 430 = 2 Im [ C 1 * ( A cos 2 ϕ + B sin 2 ϕ ) ] ,
S 440 = | C 1 | 2 | A | 2 | B | 2 ,
S 120 = S 210 , S 130 = S 310 , S 140 = S 410 ,
S 230 = S 320 , S 240 = S 420 , S 340 = S 430 ,
S 220 + S 330 = S 110 + S 440 ,
| C 1 * A | 2 = | C 1 | 2 | A | 2 , | C 1 * B | 2 = | C 1 | 2 | B | 2 ,
| A * B | 2 = | A | 2 | B | 2 ,
Arg [ ( C 1 * B ) 2 / ( C 1 * A ) 2 ] = Arg ( A * B ) 2 ,
| C 1 | 2 ( S 110 + S 440 ) / 2 ,
| A | 2 [ ( S 110 S 440 ) + ( S 220 S 330 ) × cos 4 ϕ + 2 S 230 sin 4 ϕ ] / 4 ,
| B | 2 [ ( S 110 S 440 ) ( S 220 S 330 ) × cos 4 ϕ 2 S 230 sin 4 ϕ ] / 4 ,
A * B [ ( S 220 S 330 ) sin 4 ϕ 2 S 230 cos 4 ϕ + 2 i S 140 ] / 4 ,
C 1 * A [ ( S 120 + i S 340 ) cos 2 ϕ + ( S 130 + i S 420 ) sin 2 ϕ ] / 2 ,
C 1 * B [ ( S 120 + i S 340 ) sin 2 ϕ ( S 130 + i S 420 ) cos 2 ϕ ] / 2 .
( S 20 S 30 S 40 S 10 ) = C 1 ( 1 0 0 1 ) ,
| C 1 | 2 ( 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) .
( S 2 π S 3 π S 4 π S 1 π ) = ( C 1 A cos 2 ϕ B sin 2 ϕ A sin 2 ϕ + B cos 2 ϕ A sin 2 ϕ B cos 2 ϕ C 1 A cos 2 ϕ B sin 2 ϕ ) ,
( E xs E ys ) ϕ = π = exp ( 2 ikr ) ikr ( C 1 + A B B C 1 A ) ( E xi C yi ) ,
S 11 π = | C 1 | 2 + | A | 2 + | B | 2 ,
S 12 π = S 21 π = 2 Re [ C 1 * ( A cos 2 ϕ + B sin 2 ϕ ) ] ,
S 13 π = S 31 π = 2 Re [ C 1 * ( A sin 2 ϕ B cos 2 ϕ ) ] ,
S 14 π = S 41 π = 2 Im ( A * B ) ,
S 22 π = | C 1 | 2 + ( | A | 2 | B | 2 ) cos 4 ϕ ( A * B + A B * ) sin 4 ϕ ,
S 23 π = S 32 π = ( | A | 2 | B | 2 ) sin 4 ϕ ( A * B + A B * ) cos 4 ϕ ,
S 24 π = S 42 π = 2 Im [ C 1 * ( A sin 2 ϕ B cos 2 ϕ ) ] ,
S 33 π = | C 1 | 2 + ( | A | 2 | B | 2 ) cos 4 ϕ + ( A * B + A B * ) sin 4 ϕ ,
S 34 π = S 43 π = 2 Im [ C 1 * ( A cos 2 ϕ + B sin 2 ϕ ) ] ,
S 44 π = | C 1 | 2 + | A | 2 + | B | 2
S 12 π = S 21 π , S 13 π = S 31 π , S 14 π = S 41 π ,
S 23 π = S 32 π , S 24 π = S 42 π , S 34 π = S 43 π ,
S 22 π S 33 π = S 11 π S 44 π ,
| C 1 * A | 2 = | C 1 | 2 | A | 2 , | C 1 * B | 2 = | C 1 | 2 | B | 2 ,
| A * B | 2 = | A | 2 | B | 2 ,
Arg [ ( C 1 * B ) 2 / ( C 1 * A ) 2 ] = Arg ( A * B ) 2 ,
| C 1 | 2 ( S 11 π S 44 π ) / 2 ,
| A | 2 [ ( S 11 π + S 44 π ) + ( S 22 π + S 33 π ) cos 4 ϕ + 2 S 23 π sin 4 ϕ ] / 4 ,
| B | 2 [ ( S 11 π + S 44 π ) ( S 22 π + S 33 π ) cos 4 ϕ 2 S 23 π sin 4 ϕ ] / 4 ,
A * B [ ( S 22 π + S 33 π ) sin 4 ϕ 2 S 23 π cos 4 ϕ + 2 i S 14 π ] / 4 ,
C 1 * A [ ( S 12 π i S 34 π ) cos 2 ϕ + ( S 13 π i S 42 π ) sin 2 ϕ ] / 4 ,
C 1 * B [ ( S 12 π i S 34 π ) sin 2 ϕ ( S 13 π i S 42 π ) cos 2 ϕ ] / 4 .
( S 2 π S 3 π S 4 π S 1 π ) = [ C 1 D cos 2 ( ϕ ϕ 0 ) D sin 2 ( ϕ ϕ 0 ) D sin 2 ( ϕ ϕ 0 ) C 1 D cos 2 ( ϕ ϕ 0 ) ] .
( S 2 π S 3 π S 4 π S 1 π ) = C 1 ( 1 0 0 1 ) .
S 11 π = | C 1 | 2 + | D | 2 ,
S 12 π = S 21 π = ( C 1 * D + C 1 D * ) cos 2 ( ϕ ϕ 0 ) ,
S 13 π = S 31 π = ( C 1 * D + C 1 D * ) sin 2 ( ϕ ϕ 0 ) ,
S 14 π = S 41 π = 0 ,
S 22 π = | C 1 | 2 + | D | 2 cos 4 ( ϕ ϕ 0 ) ,
S 23 π = S 32 π = | D | 2 cos 4 ( ϕ ϕ 0 ) ,
S 24 π = S 42 π = i ( C 1 * D C 1 D * ) sin 2 ( ϕ ϕ 0 ) ,
S 33 π = | C 1 | 2 + | D | 2 cos 4 ( ϕ ϕ 0 ) ,
S 34 π = S 43 π = i ( C 1 * D C 1 D * ) cos 2 ( ϕ ϕ 0 ) ,
S 44 π = | C 1 | 2 + | D | 2 ,
S 12 π = S 21 π , S 13 π = S 31 π , S 14 π = S 41 π = 0 ,
S 23 π = S 32 π , S 24 π = S 42 π , S 34 π = S 43 π ,
S 11 π S 44 π = S 22 π S 33 π ,
( S 11 π + S 44 π ) 2 = ( S 22 π + S 33 π ) 2 + 4 S 23 π 2 ,
tan 4 ( ϕ ϕ 0 ) = 2 S 23 π / ( S 22 π + S 33 π ) ,
tan 2 ( ϕ ϕ 0 ) = S 13 π / S 12 π = S 24 π / S 34 π ,
| C 1 * D | 2 = | C 1 | 2 | D | 2 ,
| C 1 | 2 ( S 11 π S 44 π ) / 2 ,
| D | 2 ( S 11 π + S 44 π ) / 2 ,
( C 1 * D ) 2 [ ( S 12 π i S 24 π ) 2 + ( S 13 π i S 24 π ) 2 ] / 4 .
| C 1 | 2 ( 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) .
e ̂ α = cos α e ̂ x + sin α e ̂ y = cos ( ϕ α ) e ̂ i + sin ( ϕ α ) e ̂ i
( E i E i ) = E 0 [ cos ( ϕ α ) sin ( ϕ α ) ] .
X α = [ S 2 cos ( ϕ α ) + S 3 sin ( ϕ α ) ] e ̂ s + [ S 4 cos ( ϕ α ) + S 1 sin ( ϕ α ) ] e ̂ s .
C ext ( α ) = ( 4 π / k 2 ) Re { ( X α e ̂ α ) θ = 0 ϕ } = ( 4 π / k 2 ) Re { C 1 + A cos 2 α + B sin 2 α } ,
C ext ( α ) = ( 4 π / k 2 ) Re { C 1 + D cos 2 ( α ϕ 0 ) } .
C ext ( α ) = ( 4 π / k 2 ) Re C 1 ,
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