Abstract

A new theory of Wood’s anomalies is presented which is based on a guided wave approach rather than the customary multiple scattering procedure. This approach provides both new insight and a method of calculation. It is shown that two distinct types of anomalies may exist: a Rayleigh wavelength type due to the emergence of a new spectal order at grazing angle, and a resonance type which is related to the guided complex waves supportable by the grating. A general theoretical treatment is presented which makes use of a surface reactance to take into account the standing waves in the grating grooves, and which derives the locations and detailed shapes of the anomalies. Rigorous results are obtained for a specific example; the amplitudes of all of the spectral orders are determined explicitly, and the Wood’s anomaly effects are demonstrated clearly in graphical form for a variety of cases.

© 1965 Optical Society of America

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References

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  1. R. W. Wood, Phil. Mag. 4, 396 (1902).
  2. C. H. Palmer, J. Opt. Soc. Am. 42, 269 (1952).
    [CrossRef]
  3. J. E. Stewart, W. S. Gallaway, Appl. Opt. 4, 421 (1962).
    [CrossRef]
  4. V. Twersky, Inst. Radio Engrs. Trans. AP-4, 330 (1956).
  5. R. F. Millar, Can. J. Phys. 39, 81 (1961).
    [CrossRef]
  6. V. Twersky, J. Res. Natl. Bur. Std. 64D, 715 (1960).
  7. L. R. Ingersoll, Astrophys. J. 51, 129 (1920).
    [CrossRef]
  8. J. Strong, Phys. Rev. 49, 291 (1936).
    [CrossRef]
  9. R. W. Wood, Phil. Mag. 23, 310 (1912).
  10. Lord Rayleigh, Proc Roy. Soc. (London) A79, 399 (1907).
  11. U. Fano, Ann. Phys. 32, 393 (1938).
    [CrossRef]
  12. K. Artmann, Z. Phys. 119, 529 (1942).
    [CrossRef]
  13. R. W. Wood, Phys. Rev. 48, 928 (1935).
    [CrossRef]
  14. H. C. Palmer, J. Opt. Soc. Am. 46, 50 (1956).
    [CrossRef]
  15. H. C. Palmer, J. Opt. Soc. Am. 51, 1438 (1961).
    [CrossRef]
  16. B. A. Lippmann, J. Opt. Soc. Am. 43, 408 (1953).
    [CrossRef]
  17. B. A. Lippmann, A. Oppenheim, Tech. Res. Group N.Y. (1954).
  18. S. N. Karp, J. Radlow, Inst. Radio Engrs. Trans. AP-4, 654 (1956).
  19. R. F. Millar, Can. J. Phys. 39, 104 (1961).
    [CrossRef]
  20. V. Twersky, J. Appl. Phys. 23, 1099 (1952).
    [CrossRef]
  21. V. Twersky, Rept. EDL-M105 Sylvania (1957).
  22. J. E. Burke, V. Twersky, Rept. EDL-E44 Sylvania (1960).
  23. V. Twersky, J. Opt. Soc. Am. 52, 145 (1962).
    [CrossRef]
  24. V. Twersky, Inst. Radio Engrs. Trans. AP-10, 737 (1962).
  25. A. A. Oliner, A. Hessel, Inst. Radio Engrs. Trans. AP-7, Spec. Suppl. S201 (1959).
  26. A. Hessel, Rept. PIBMRI 825-60, Polytechnic Institute of Brooklyn (1960).
  27. T. Tamir, A. A. Oliner, Proc. Inst. Elec. Engrs. (London) 110, 310, 325 (1963).
    [CrossRef]
  28. J. Meixner, F. W. Schaefke, Mathieusche Funktionen und Sphaeroidfunktionen (Springer-Verlag, Berlin1954), pp. 89–93.

1963 (1)

T. Tamir, A. A. Oliner, Proc. Inst. Elec. Engrs. (London) 110, 310, 325 (1963).
[CrossRef]

1962 (3)

V. Twersky, Inst. Radio Engrs. Trans. AP-10, 737 (1962).

J. E. Stewart, W. S. Gallaway, Appl. Opt. 4, 421 (1962).
[CrossRef]

V. Twersky, J. Opt. Soc. Am. 52, 145 (1962).
[CrossRef]

1961 (3)

H. C. Palmer, J. Opt. Soc. Am. 51, 1438 (1961).
[CrossRef]

R. F. Millar, Can. J. Phys. 39, 81 (1961).
[CrossRef]

R. F. Millar, Can. J. Phys. 39, 104 (1961).
[CrossRef]

1960 (1)

V. Twersky, J. Res. Natl. Bur. Std. 64D, 715 (1960).

1959 (1)

A. A. Oliner, A. Hessel, Inst. Radio Engrs. Trans. AP-7, Spec. Suppl. S201 (1959).

1956 (3)

V. Twersky, Inst. Radio Engrs. Trans. AP-4, 330 (1956).

S. N. Karp, J. Radlow, Inst. Radio Engrs. Trans. AP-4, 654 (1956).

H. C. Palmer, J. Opt. Soc. Am. 46, 50 (1956).
[CrossRef]

1954 (1)

B. A. Lippmann, A. Oppenheim, Tech. Res. Group N.Y. (1954).

1953 (1)

1952 (2)

V. Twersky, J. Appl. Phys. 23, 1099 (1952).
[CrossRef]

C. H. Palmer, J. Opt. Soc. Am. 42, 269 (1952).
[CrossRef]

1942 (1)

K. Artmann, Z. Phys. 119, 529 (1942).
[CrossRef]

1938 (1)

U. Fano, Ann. Phys. 32, 393 (1938).
[CrossRef]

1936 (1)

J. Strong, Phys. Rev. 49, 291 (1936).
[CrossRef]

1935 (1)

R. W. Wood, Phys. Rev. 48, 928 (1935).
[CrossRef]

1920 (1)

L. R. Ingersoll, Astrophys. J. 51, 129 (1920).
[CrossRef]

1912 (1)

R. W. Wood, Phil. Mag. 23, 310 (1912).

1907 (1)

Lord Rayleigh, Proc Roy. Soc. (London) A79, 399 (1907).

1902 (1)

R. W. Wood, Phil. Mag. 4, 396 (1902).

Artmann, K.

K. Artmann, Z. Phys. 119, 529 (1942).
[CrossRef]

Burke, J. E.

J. E. Burke, V. Twersky, Rept. EDL-E44 Sylvania (1960).

Fano, U.

U. Fano, Ann. Phys. 32, 393 (1938).
[CrossRef]

Gallaway, W. S.

J. E. Stewart, W. S. Gallaway, Appl. Opt. 4, 421 (1962).
[CrossRef]

Hessel, A.

A. A. Oliner, A. Hessel, Inst. Radio Engrs. Trans. AP-7, Spec. Suppl. S201 (1959).

A. Hessel, Rept. PIBMRI 825-60, Polytechnic Institute of Brooklyn (1960).

Ingersoll, L. R.

L. R. Ingersoll, Astrophys. J. 51, 129 (1920).
[CrossRef]

Karp, S. N.

S. N. Karp, J. Radlow, Inst. Radio Engrs. Trans. AP-4, 654 (1956).

Lippmann, B. A.

B. A. Lippmann, A. Oppenheim, Tech. Res. Group N.Y. (1954).

B. A. Lippmann, J. Opt. Soc. Am. 43, 408 (1953).
[CrossRef]

Meixner, J.

J. Meixner, F. W. Schaefke, Mathieusche Funktionen und Sphaeroidfunktionen (Springer-Verlag, Berlin1954), pp. 89–93.

Millar, R. F.

R. F. Millar, Can. J. Phys. 39, 104 (1961).
[CrossRef]

R. F. Millar, Can. J. Phys. 39, 81 (1961).
[CrossRef]

Oliner, A. A.

T. Tamir, A. A. Oliner, Proc. Inst. Elec. Engrs. (London) 110, 310, 325 (1963).
[CrossRef]

A. A. Oliner, A. Hessel, Inst. Radio Engrs. Trans. AP-7, Spec. Suppl. S201 (1959).

Oppenheim, A.

B. A. Lippmann, A. Oppenheim, Tech. Res. Group N.Y. (1954).

Palmer, C. H.

Palmer, H. C.

Radlow, J.

S. N. Karp, J. Radlow, Inst. Radio Engrs. Trans. AP-4, 654 (1956).

Rayleigh, Lord

Lord Rayleigh, Proc Roy. Soc. (London) A79, 399 (1907).

Schaefke, F. W.

J. Meixner, F. W. Schaefke, Mathieusche Funktionen und Sphaeroidfunktionen (Springer-Verlag, Berlin1954), pp. 89–93.

Stewart, J. E.

J. E. Stewart, W. S. Gallaway, Appl. Opt. 4, 421 (1962).
[CrossRef]

Strong, J.

J. Strong, Phys. Rev. 49, 291 (1936).
[CrossRef]

Tamir, T.

T. Tamir, A. A. Oliner, Proc. Inst. Elec. Engrs. (London) 110, 310, 325 (1963).
[CrossRef]

Twersky, V.

V. Twersky, Inst. Radio Engrs. Trans. AP-10, 737 (1962).

V. Twersky, J. Opt. Soc. Am. 52, 145 (1962).
[CrossRef]

V. Twersky, J. Res. Natl. Bur. Std. 64D, 715 (1960).

V. Twersky, Inst. Radio Engrs. Trans. AP-4, 330 (1956).

V. Twersky, J. Appl. Phys. 23, 1099 (1952).
[CrossRef]

J. E. Burke, V. Twersky, Rept. EDL-E44 Sylvania (1960).

V. Twersky, Rept. EDL-M105 Sylvania (1957).

Wood, R. W.

R. W. Wood, Phys. Rev. 48, 928 (1935).
[CrossRef]

R. W. Wood, Phil. Mag. 23, 310 (1912).

R. W. Wood, Phil. Mag. 4, 396 (1902).

Ann. Phys. (1)

U. Fano, Ann. Phys. 32, 393 (1938).
[CrossRef]

Appl. Opt. (1)

J. E. Stewart, W. S. Gallaway, Appl. Opt. 4, 421 (1962).
[CrossRef]

Astrophys. J. (1)

L. R. Ingersoll, Astrophys. J. 51, 129 (1920).
[CrossRef]

Can. J. Phys. (2)

R. F. Millar, Can. J. Phys. 39, 81 (1961).
[CrossRef]

R. F. Millar, Can. J. Phys. 39, 104 (1961).
[CrossRef]

Inst. Radio Engrs. Trans. (4)

V. Twersky, Inst. Radio Engrs. Trans. AP-4, 330 (1956).

V. Twersky, Inst. Radio Engrs. Trans. AP-10, 737 (1962).

A. A. Oliner, A. Hessel, Inst. Radio Engrs. Trans. AP-7, Spec. Suppl. S201 (1959).

S. N. Karp, J. Radlow, Inst. Radio Engrs. Trans. AP-4, 654 (1956).

J. Appl. Phys. (1)

V. Twersky, J. Appl. Phys. 23, 1099 (1952).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Res. Natl. Bur. Std. (1)

V. Twersky, J. Res. Natl. Bur. Std. 64D, 715 (1960).

Phil. Mag. (2)

R. W. Wood, Phil. Mag. 23, 310 (1912).

R. W. Wood, Phil. Mag. 4, 396 (1902).

Phys. Rev. (2)

J. Strong, Phys. Rev. 49, 291 (1936).
[CrossRef]

R. W. Wood, Phys. Rev. 48, 928 (1935).
[CrossRef]

Proc Roy. Soc. (London) (1)

Lord Rayleigh, Proc Roy. Soc. (London) A79, 399 (1907).

Proc. Inst. Elec. Engrs. (London) (1)

T. Tamir, A. A. Oliner, Proc. Inst. Elec. Engrs. (London) 110, 310, 325 (1963).
[CrossRef]

Tech. Res. Group N.Y. (1)

B. A. Lippmann, A. Oppenheim, Tech. Res. Group N.Y. (1954).

Z. Phys. (1)

K. Artmann, Z. Phys. 119, 529 (1942).
[CrossRef]

Other (4)

V. Twersky, Rept. EDL-M105 Sylvania (1957).

J. E. Burke, V. Twersky, Rept. EDL-E44 Sylvania (1960).

J. Meixner, F. W. Schaefke, Mathieusche Funktionen und Sphaeroidfunktionen (Springer-Verlag, Berlin1954), pp. 89–93.

A. Hessel, Rept. PIBMRI 825-60, Polytechnic Institute of Brooklyn (1960).

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Figures (25)

Fig. 1
Fig. 1

Reflected and diffracted spectral orders produced by plane-wave incidence on a reflection grating.

Fig. 2
Fig. 2

A spectrogram exhibiting Wood’s anomalies (reproduced from Wood13).

Fig. 3
Fig. 3

Classes of reflection gratings which exhibit Wood’s anomalies.

Fig. 4
Fig. 4

A multimode waveguide terminated by a resonant cavity—an analogy to a diffraction grating.

Fig. 5
Fig. 5

Example of correspondence between the fields of a diffracted plane wave and a leaky wave: (a) plane-wave case; (b) leaky-wave case.

Fig. 6
Fig. 6

An S-polarized plane wave incident on a plane with a periodically varying surface reactance.

Fig. 7
Fig. 7

Equivalent surface reactance vs angle of incidence for a single propagating spectral order (reflected wave only). (The parameter is λ/d.)

Fig. 8
Fig. 8

Phase of I0 vs angle of incidence for a single propagating spectral order. (The parameter is λ/d.)

Fig. 9
Fig. 9

Equivalent surface reactance vs angle of incidence for a single propagating spectral order. (The parameter is M.)

Fig. 10
Fig. 10

Phase of I0 vs angle of incidence for a single propagating spectral order. (The parameter is M.)

Fig. 11
Fig. 11

Relative magnitude of the n = 0 spectral order vs angle of incidence for two propagating orders.

Fig. 12
Fig. 12

Relative magnitude of the n = − 1 spectral order vs angle of incidence for two propagating orders.

Fig. 13
Fig. 13

Phase of the n = − 1 spectral order vs angle of incidence for two propagating orders.

Fig. 14
Fig. 14

Relative powers in the n = −1 and n = 0 propagating spectral orders vs sinθ (θ; = angle of incidence).

Fig. 15
Fig. 15

Relative amplitude of the n = 1 resonant sp ectral order vs sinθ (θ = angle of incidence).

Fig. 16
Fig. 16

Comparison of calculations for the magnitude of the n = − 1 spectral order for a resonance anomaly: solid curve computed from exact solution, crosses obtained from Eq. (32) using only the knowledge of the positions of the null and the complex pole.

Fig. 17
Fig. 17

Relative amplitude of the n = 1 resonant spectral order vs sinθ for a larger value of M.

Fig. 18
Fig. 18

Relative powers in the n = 0 and n = −1 propagating spectral orders vs sinθ for a larger value of M.

Fig. 19
Fig. 19

Relative power in the n = − 1 spectral order vs λ/d, showing a double anomaly and a pronounced resonance near a Rayleigh wavelength.

Fig. 20
Fig. 20

Relative power in the n = 0 spectral order vs λ/d for the case of Fig. 19.

Fig. 21
Fig. 21

Phase of the n = − 1 spectral order vs λ/d for the case of Fig. 19.

Fig. 22
Fig. 22

Relative intensity of the n = 0 spectral order vs λ/d for the case of two propagating spectral orders: accuracy considerations.

Fig. 23
Fig. 23

Relative amplitude of the n = −3 resonant spectral order vs λ/d for the case of four propagating spectral orders.

Fig. 24
Fig. 24

Relative power in the n = − 2 propagating spectral order vs λ/d for the case of Fig. 23.

Fig. 25
Fig. 25

Relative power in the n = − 1 propagating spectral order vs λ/d for the case of Fig. 23.

Equations (127)

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n λ g 2 > l > ( 2 n 1 ) λ g 4 , n = 1 , 2 , . . . .
H i ( x , z ) = H e i k s x e i κ 0 z ,
κ 0 = ( k 2 k s 2 ) 1 / 2 ,
k s = k sin θ .
Z s ( x ) = E x ( x , 0 ) H y ( x , 0 ) .
Z s ( x ) = ν = Z ν s e i ( 2 π / d ) ν x ,
Z s ν = Z s * ν .
Z s 00 + κ ω = 0 ,
κ = ( k 2 k s 2 ) 1 / 2 = i ( k s 2 k 2 ) 1 / 2 = i α , α > 0 .
H s w = H 0 e i k s x e α z , z 0 ,
( 2 + k 2 ) H y = 0 , z > 0 ,
H s ( x , z ) = n = I n ( k s ) e i [ k s + ( 2 π n / d ) ] x e i κ n z , for z 0 , n = 0 , ± 1 , ± 2 , . . . ,
κ n = [ k 2 ( k s + 2 π n d ) 2 ] 1 / 2 .
κ n = i [ ( k s + 2 π n d ) 2 k 2 ] 1 / 2 , for k 2 < ( k s + 2 π n d ) 2 .
[ Z s ( x ) H y ( x , z ) + 1 i ω H y ( x , z ) z ] z = 0 = 0 .
n = κ n ω I ¯ ¯ n e i ( 2 π / d ) n x 2 H κ 0 ω + m = ν = Z ν s Î m e i ( 2 π / d ) ( ν + m ) x = 0 ,
Î 0 = I 0 + H , Î m = I m , m 0 .
n = m = ( Z n m s + κ n ω δ n m ) Î m e i ( 2 π / d ) n x = 2 κ 0 ω H .
m = ( Z n m s + κ n ω δ n m ) Î m = 2 κ 0 ω H δ n 0 ,
δ n m = 0 , m n , 1 , m = n , n = 0 , ± 1 , ± 2 , . . . ,
[ Z 0 s + κ 2 ω Z 1 s Z 2 s Z 3 s Z 4 s Z 1 s Z 0 s + κ 1 ω Z 1 s Z 2 s Z 3 s . . . Z 2 s Z 1 s Z 0 s + κ 0 ω Z 1 s Z 2 s Z 3 s Z 2 s Z 1 s Z 0 s + κ 1 ω Z 1 s Z 4 s Z 3 s Z 2 s Z 1 s Z 0 s + κ 2 ω ] [ Î 2 Î 1 Î 0 Î 1 Î 2 ] = [ 0 0 2 κ 0 ω H 0 0 ] .
( Z ) Î = V .
Î n = Δ n Δ ,
( a ) D n = Z 0 s + κ n ω = 0 ,
( b ) κ n = [ k 2 ( k s + 2 π n d ) 2 ] 1 / 2 = 0 .
Z s 0 + κ n ω ,
Z s 00 + κ n ω ,
H s w ( x , z ) = H 0 e i k s x e i κ n z = H 0 e i k s x exp ( i ω Z s 00 z ) ,
( Z ) = ( D + Z ) ,
( D ) i j = ( Z 0 s + κ i ω ) δ i j ,
( 1 + D 1 Z ) Î = ( D ) 1 V ,
Δ ( k , k s , 0 ) = ν = ( Z s 00 + κ ν ω ) ν = f ν .
Δ ( k , k s , 0 ) = 0
Δ ( k 0 , k s , l ) = 0
t = [ k 2 ( k s + 2 π n d ) 2 ] 1 / 2 .
Δ ( k , k s , l ) = Δ ̂ ( k , t , l )
Δ ̂ ( k 0 , t 0 , 0 ) = 0
Δ ̂ ( k 0 , t , l ) = 0
k s p = β + i α ,
k s + 2 π n d β .
Î n = I n = Δ n Δ = Δ n ( k s ) ( k s β i α ) g ( k s ) .
I n C n k s β i α
| I n | | C n | ( k s β ) 2 + α 2 .
Q = β 2 α .
Î n 1 = I n 1 = Δ n 1 Δ C n 1 ( k s β 1 i α 1 ) ( k s β i α )
| I n 1 | C n 1 [ ( k s β 1 ) 2 + α 1 2 ( k s β ) 2 + α 2 ] 1 / 2 .
β 1 k = sin θ min ,
| I n 1 | 2 k s = 0
( k s β 1 ) ( k s β ) ( β 1 β ) = ( k s β ) α 1 2 ( k s β 1 ) α 2 .
k s max k = sin θ max = β k + ( α k ) 2 β k β 1 k .
| I n | k s = k s | Δ n Δ | = | Δ Δ n k s Δ n Δ k s Δ 2 | .
D n Z s 0 + ω μ κ n .
Y s 0 + κ n ω μ ,
Z s ( x ) = Z s ( 1 + M cos 2 π x d ) ,
Z s 0 = Z s = Z s 00 Z s 1 = Z s 1 = Z s M 2 Z s ν = 0 , ν 0 , ± 1 .
[ Z s + κ 2 ω Z s M 2 0 0 0 Z s M 2 Z s + κ 1 ω Z s M 2 0 0 . . . 0 Z s M 2 Z s + κ 0 ω Z s M 2 0 0 0 Z s M 2 Z s + κ 1 ω Z s M 2 0 0 0 Z s M 2 Z s + κ 2 ω ] [ Î 2 Î 1 Î 0 Î 1 Î 2 ] = [ 0 0 2 κ 0 ω H 0 0 ]
Î n + 1 + D ¯ n Î n + Î n 1 = 0 , n = 1 , 2 , . . . and n = 1 , 2 , . . .
Î 1 + D ¯ 0 Î 0 + Î 1 = V 0 ,
D ¯ n = 2 M { 1 + 1 Z s [ 1 ( k s k + 2 π n k d ) 2 ] 1 / 2 } = 2 M { 1 + 1 Z s [ 1 ( sin θ + n λ d ) 2 ] 1 / 2 }
V 0 = 4 M 1 Z s [ 1 ( k s k ) 2 ] 1 / 2 = 4 M cos θ Z s ,
Z s = Z s μ / .
Î n Î n + 1 = 1 D ¯ n + Î n + 1 Î n , n = 1 , 2 , . . . ,
Î n Î n 1 = 1 | D ¯ n 1 | | D ¯ n + 1 1 | | D ¯ n + 2 . . . = A n , n = 1 , 2 , . . . ,
Î 1 Î 0 = 1 | D ¯ 1 1 | | D ¯ 2 1 | | D ¯ 3 . . . = A 1 .
Î n Î n + 1 = 1 | D ¯ n 1 | | D ¯ n 1 1 | | D ¯ n 2 . . . = B n , n = 1 , 2 , . . . ,
Î 1 Î 0 = 1 | D ¯ 1 1 | | D ¯ 2 . . . = B 1 .
Î 0 = V 0 D ¯ 0 + A 1 + B 1 ,
Î 1 = Î 0 A 1
Î n = Î 0 ν = 1 n A ν , n = 1 , 2 , . . .
Î 1 = Î 0 B 1
Î n = Î 0 ν = 1 n B ν , n = 1 , 2 , . . .
Î 0 H = ( 2 / Z s ) cos θ 1 + 1 Z s cos θ M 2 4 ( 1 | D 1 M 2 / 4 | | D 2 . . . + 1 | D 1 M 2 / 4 | | D 2 . . . )
I 0 H = cos θ Z s 1 + M 2 4 [ 1 | D 1 M 2 / 4 | | D 2 . . . + 1 | D 1 M 2 / 4 | | D 2 . . . ] cos θ Z s + 1 M 2 4 [ 1 | D 1 M 2 / 4 | | D 2 . . . + 1 | D 1 M 2 / 4 | | D 2 . . . ]
Î 1 = I 1 = [ M / 2 | D 1 M 2 / 4 | | D 2 . . . ] Î 0
Î 1 = I 1 = ( M / 2 | D 1 M 2 / 4 | | D 2 . . . ) Î 0 ,
I n I n 1 = M / 2 | D n M 2 / 4 | | D n + 1 . . . , n = 1 , 2 , . . .
I n I n + 1 = M / 2 | D n M 2 / 4 | | D n 1 . . . , n = 1 , 2 , . . . ,
D n = 1 + 1 Z s [ 1 ( sin θ + n λ d ) 2 ] 1 / 2 = M 2 D ¯ n .
Δ = | D 2 M 2 0 0 0 M 2 D 1 M 2 0 0 . . . 0 M 2 D 0 M 2 0 . . . 0 0 M 2 D 1 M 2 0 0 0 M 2 D 2 | ,
I 1 = | D 2 M 2 0 0 0 0 M 2 D 1 M 2 0 0 0 0 M 2 D 0 V ¯ 0 0 0 0 M 2 0 M 2 0 0 0 0 0 D 2 M 2 0 0 0 0 M 2 D 3 | , Δ
V ¯ = 2 κ 0 ω H Z s .
I 1 V ¯ = M 2 | D 2 D 3 . . . | | D 1 D 2 . . . | Δ ,
| D n M 2 . . . M 2 D n + 1 | and | D n M 2 . . . M 2 D n 1 | .
I 1 V ¯ = M 2 | D 2 D 3 . . . | | D 1 D 2 . . . | | D 2 D 3 . . . | | D 1 D 0 . . . | M 2 4 | D 3 D 4 . . . | | D 0 D 1 . . . | = M 2 | D 2 D 3 . . . | | D 3 D 4 . . . | | D 1 D 2 . . . | | D 0 D 1 . . . | | D 2 D 3 . . . | | D 3 D 4 . . . | | D 1 D 0 . . . | | D 0 D 1 . . . | M 2 4 .
| D 2 D 3 . . . | | D 3 D 4 . . . | = D 2 | D 3 D 4 . . . | M 2 4 | D 4 D 5 . . . | | D 3 D 4 . . . | = D 2 M 2 4 | D 4 D 5 . . . | | D 3 D 4 . . . | = D 2 M 2 4 | D 4 D 5 . . . | D 3 | D 4 D 5 . . . | M 2 4 | D 5 D 6 . . . | = D 2 M 2 / 4 | D 3 M 2 / 4 | | D 4 . . .
| D 1 D 2 . . . | | D 0 D 1 . . . | = 1 | D 0 M 2 / 4 | | D 1 . . .
| D 1 D 0 . . . | | D 0 D 1 . . . | = D 1 M 2 / 4 | D 0 M 2 / 4 | | D 1 . . .
I 1 V ¯ M 2 ( D 2 M 2 / 4 | D 3 M 2 / 4 | | D 4 . . . ) ( 1 | D 0 M 2 / 4 | | D 1 M 2 / 4 | | D 2 . . . ) ( D 2 M 2 / 4 | D 3 . . . ) ( D 1 M 2 / 4 | D 0 M 2 / 4 | | D 1 . . . ) M 2 4 .
I 1 H = M 2 ( 1 | D 1 M 2 / 4 | | D 2 . . . ) 2 κ 0 ω Z a D 0 M 2 / 4 | | D 1 M 2 / 4 | | D 2 . . . M 2 / 4 | D 1 M 2 / 4 | D 2 . . .
I 1 H = M 2 ( D 2 M 2 / 4 | | D 3 . . . ) ( 1 | D 0 M 2 / 4 | | D 1 . . . ) ( 1 | D 1 M 2 / 4 | | D 0 . . . ) 2 κ 0 ω Z s ( D 2 M 2 / 4 | | D 3 . . . M 2 / 4 | D 1 M 2 / 4 | | D 0 M 2 / 4 | | D 1 . . . )
D 2 M 2 / 4 D 3 = 0 .
D 2 M 2 / 4 D 3 M 2 / 4 D 1 = 0 .
D 2 = 1 1 X s [ ( k s k 4 π k d ) 2 1 ] 1 / 2 ,
k s 0 k = ( 1 + X s 2 ) 1 / 2 + 4 π k d ,
k s = k s 0 + δ ,
[ ( k s 0 k 4 π k d ) 2 + 2 δ k ( k s 0 k 4 π k d ) 1 ] 1 / 2 X s + δ k ( k s 0 k 4 π k d ) X s
D 2 ( k s 0 + δ ) δ k ( 1 + X s 2 ) 1 / 2 X s 2 .
δ m k = M 2 4 X s 2 ( 1 + X s 2 ) 1 / 2 1 1 X s [ ( 1 + X s 2 + λ d ) 2 1 ] 1 / 2
δ p k 1 + X s 2 X s 2 = M 2 4 { 1 1 1 X s [ ( 1 + X s 2 + λ d ) 2 1 ] 1 / 2 } + 1 1 i X s [ 1 ( 1 + X s 2 λ d ) 2 ] 1 / 2 } .
Re [ δ p k δ m k ] = ( M 2 / 4 ) X s 2 ( 1 + X s 2 ) 1 / 2 × 1 1 + 1 X s 2 { 1 [ ( 1 + X s 2 ) 1 / 2 λ d ] 2 } = X s 4 ( 1 + X s 2 ) 1 / 2 × M 2 / 4 ( λ d ) [ 2 ( 1 + X s 2 ) 1 / 2 λ d ] .
I 1 H = M 2 ( 1 | D 1 M 2 / 4 | | D 2 . . . ) ( D 1 M 2 / 4 | D 2 . . . ) ( 1 | D 0 M 2 / 4 | | D 1 . . . ) 2 κ 0 ω Z s ( D 1 M 2 / 4 | D 2 . . . M 2 / 4 | D 0 . . . ) .
D 1 M 2 / 4 D 2 = 0
D 1 M 2 / 4 D 2 M 2 / 4 D 0 = 0 .
D 1 = 1 1 X s [ ( k s k + 2 π 2 k d ) 1 ] 1 / 2 ,
k s 0 k = 1 + X s 2 2 π k d ,
D 1 ( k s 0 + δ ) = δ k 1 + X 2 X s 2 ,
Re [ δ p k δ m k ] = X 4 1 + X s 2 M 2 / 4 ( λ d ) ( 2 1 + X s 2 λ d ) ,
I 0 k s = Î 0 k s = Î 0 = F V 0 V 0 [ D ¯ 0 + A 1 + B 1 ] F 2
F = D ¯ 0 + A 1 + B 1 .
B 1 = 1 G 2 G = B 1 2 ( D ¯ 1 . . . ) .
D ¯ 1 = k s ( 2 M { 1 + i X s [ 1 ( k s k 2 π k d ) 2 ] } 1 / 2 ) = i X s 2 M k ( k s k 2 π k d ) 1 ( k s k 2 π k d ) 2 ,
B 1 = i X s 2 B 1 2 M k ( 2 π k d k s k ) 1 ( k s k 2 π k d ) 2 .
I 0 k s = Î 0 V 0 B 1 F 2 = Î 0 B 1 F ,
| I 0 | 2 k s = 2 Re [ I 0 * I 0 k s ] = 2 Re [ I 0 * Î 0 B 1 F * | F | 2 ]
| I 0 | 2 k s = 2 i B 1 | F | 2 I m [ F * ( | I 0 | 2 + H I 0 * ) ] .
I 1 k s = B 1 Î 0 + B 1 Î 0 .
I 1 k s = B 1 Î 0 F ( D ¯ 0 + A 1 ) .
| I 1 | 2 k s = 2 Re [ I 1 * I 1 k s ] = 2 | Î 0 | 2 | B 1 | 2 | F | 2 Re [ B 1 D ¯ 0 ] ,
D n = 0
D m = 0 , n m
Γ c = I 0 H = i cos θ X s [ 1 + M 2 ( A 1 + B 1 ) ] i cos θ X s + [ 1 + M 2 ( A 1 + B 1 ) ] .
Γ c = 1 + i X eq 1 i X eq .
X eq = X s cos θ [ 1 + M 2 ( A 1 + B 1 ) ] .
X eq = X s cos θ ,
sin θ = λ d ( 1 + X s 2 ) 1 / 2 ,
κ 0 k | I 0 | 2 + κ 1 k | I 1 | 2 = κ 0 k | H | 2 .
β k = 1.203 , α k = 0.00688 ,

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