Abstract

A procedure for the synthesis of birefringent networks having arbitrarily prescribed transfer functions is presented. The basic network configuration consists of n identical cascaded birefringent crystals between an input and an output polarizer. The crystals are cut with their optic axes perpendicular to their length. The variables determined by the synthesis procedure are the angles of the optic axes of the crystals and the angle of the output polarizer. Any transfer function which is periodic with frequency and whose corresponding impulse response is real and causal can, in theory, be realized. A network of n crystals allows the approximation of a desired function by (n+1) terms of a Fourier exponential series. Bandwidths of less than 1 Å appear possible.

© 1964 Optical Society of America

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References

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  1. H. Pohlack, Jenaer Jahrbuch,  1962, p. 181 (in German).
  2. L. Young, J. Opt. Soc. Am. 51, 967 (1961).
    [Crossref]
  3. J. S. Seeley, Proc. Phys. Soc. (London) 78, 998 (1961).
    [Crossref]
  4. R. J. Pegis, J. Opt. Soc. Am. 51, 1255 (1961).
    [Crossref]
  5. If biaxial crystals are used, crystals in the monoclinic and triclinic systems will probably not be satisfactory since the directions of their principal axes are dependent upon temperature and wavelength.
  6. H. G. Jerrard, J. Opt. Soc. Am. 38, 35 (1948).
    [Crossref]
  7. L. Mertz, J. Opt. Soc. Am. 50 (June1960) (advertisement facing p. xii).
  8. S. E. Harris and E. O. Ammann, Proc. IEEE 52, 411 (1964).
    [Crossref]
  9. B. Lyot, Compt. Rend. 197, 1593 (1933).
  10. I. Solc, Czech. J. Phys. 3, 366 (1953);Czech. J. Phys. 4, 607, 669 (1954);Czech. J. Phys. 5, 114 (1955).
  11. J. W. Evans, J. Opt. Soc. Am. 39, 229 (1949).
    [Crossref]
  12. J. W. Evans, J. Opt. Soc. Am. 48, 142 (1958).
    [Crossref]
  13. Y. Öhman, Nature141, 157 (1938);Nature 141, 291 (1938);Pop. Astron. Tidskrift, No. 1–2,  11, 27 (1938).
    [Crossref]
  14. J. W. Evans, Publ. Astron. Soc. Pacific 52, 305 (1940).
    [Crossref]
  15. J. W. Evans, Ciencia Invest. (Buenos Aires) 3, 365 (1947).
  16. B. H. Billings, J. Opt. Soc. Am. 37, 738 (1947).
    [Crossref] [PubMed]
  17. W. H. Steel, R. N. Smartt, and R. G. Giovanelli, Australian J. Phys. 14, 201 (1961).
    [Crossref]
  18. J. W. Evans, Appl. Opt. 2, 193 (1963).
    [Crossref]
  19. J. A. Aseltine, Transform Method in Linear System Analysis (McGraw-Hill Book Company, Inc., New York, 1958).
  20. E. A. Guillemin, Theory of Linear Physical Systems (John Wiley & Sons, Inc., New York, 1963), p. 430.
  21. Ref. 20, p. 408.
  22. D. C. Murdoch, Linear Algebra for Undergraduates (John and Wiley & Sons, Inc., New York, 1947), p. 50–51.
  23. S. E. Harris, Appl. Phys. Letters 2, 47 (1963).
    [Crossref]
  24. R. C. Jones, J. Opt. Soc. Am. 31, 488 (1941).
    [Crossref]

1964 (1)

S. E. Harris and E. O. Ammann, Proc. IEEE 52, 411 (1964).
[Crossref]

1963 (2)

J. W. Evans, Appl. Opt. 2, 193 (1963).
[Crossref]

S. E. Harris, Appl. Phys. Letters 2, 47 (1963).
[Crossref]

1961 (4)

W. H. Steel, R. N. Smartt, and R. G. Giovanelli, Australian J. Phys. 14, 201 (1961).
[Crossref]

L. Young, J. Opt. Soc. Am. 51, 967 (1961).
[Crossref]

J. S. Seeley, Proc. Phys. Soc. (London) 78, 998 (1961).
[Crossref]

R. J. Pegis, J. Opt. Soc. Am. 51, 1255 (1961).
[Crossref]

1960 (1)

L. Mertz, J. Opt. Soc. Am. 50 (June1960) (advertisement facing p. xii).

1958 (1)

1953 (1)

I. Solc, Czech. J. Phys. 3, 366 (1953);Czech. J. Phys. 4, 607, 669 (1954);Czech. J. Phys. 5, 114 (1955).

1949 (1)

1948 (1)

1947 (2)

J. W. Evans, Ciencia Invest. (Buenos Aires) 3, 365 (1947).

B. H. Billings, J. Opt. Soc. Am. 37, 738 (1947).
[Crossref] [PubMed]

1941 (1)

1940 (1)

J. W. Evans, Publ. Astron. Soc. Pacific 52, 305 (1940).
[Crossref]

1933 (1)

B. Lyot, Compt. Rend. 197, 1593 (1933).

Ammann, E. O.

S. E. Harris and E. O. Ammann, Proc. IEEE 52, 411 (1964).
[Crossref]

Aseltine, J. A.

J. A. Aseltine, Transform Method in Linear System Analysis (McGraw-Hill Book Company, Inc., New York, 1958).

Billings, B. H.

Evans, J. W.

J. W. Evans, Appl. Opt. 2, 193 (1963).
[Crossref]

J. W. Evans, J. Opt. Soc. Am. 48, 142 (1958).
[Crossref]

J. W. Evans, J. Opt. Soc. Am. 39, 229 (1949).
[Crossref]

J. W. Evans, Ciencia Invest. (Buenos Aires) 3, 365 (1947).

J. W. Evans, Publ. Astron. Soc. Pacific 52, 305 (1940).
[Crossref]

Giovanelli, R. G.

W. H. Steel, R. N. Smartt, and R. G. Giovanelli, Australian J. Phys. 14, 201 (1961).
[Crossref]

Guillemin, E. A.

E. A. Guillemin, Theory of Linear Physical Systems (John Wiley & Sons, Inc., New York, 1963), p. 430.

Harris, S. E.

S. E. Harris and E. O. Ammann, Proc. IEEE 52, 411 (1964).
[Crossref]

S. E. Harris, Appl. Phys. Letters 2, 47 (1963).
[Crossref]

Jerrard, H. G.

Jones, R. C.

Lyot, B.

B. Lyot, Compt. Rend. 197, 1593 (1933).

Mertz, L.

L. Mertz, J. Opt. Soc. Am. 50 (June1960) (advertisement facing p. xii).

Murdoch, D. C.

D. C. Murdoch, Linear Algebra for Undergraduates (John and Wiley & Sons, Inc., New York, 1947), p. 50–51.

Öhman, Y.

Y. Öhman, Nature141, 157 (1938);Nature 141, 291 (1938);Pop. Astron. Tidskrift, No. 1–2,  11, 27 (1938).
[Crossref]

Pegis, R. J.

Pohlack, H.

H. Pohlack, Jenaer Jahrbuch,  1962, p. 181 (in German).

Seeley, J. S.

J. S. Seeley, Proc. Phys. Soc. (London) 78, 998 (1961).
[Crossref]

Smartt, R. N.

W. H. Steel, R. N. Smartt, and R. G. Giovanelli, Australian J. Phys. 14, 201 (1961).
[Crossref]

Solc, I.

I. Solc, Czech. J. Phys. 3, 366 (1953);Czech. J. Phys. 4, 607, 669 (1954);Czech. J. Phys. 5, 114 (1955).

Steel, W. H.

W. H. Steel, R. N. Smartt, and R. G. Giovanelli, Australian J. Phys. 14, 201 (1961).
[Crossref]

Young, L.

Appl. Opt. (1)

Appl. Phys. Letters (1)

S. E. Harris, Appl. Phys. Letters 2, 47 (1963).
[Crossref]

Australian J. Phys. (1)

W. H. Steel, R. N. Smartt, and R. G. Giovanelli, Australian J. Phys. 14, 201 (1961).
[Crossref]

Ciencia Invest. (Buenos Aires) (1)

J. W. Evans, Ciencia Invest. (Buenos Aires) 3, 365 (1947).

Compt. Rend. (1)

B. Lyot, Compt. Rend. 197, 1593 (1933).

Czech. J. Phys. (1)

I. Solc, Czech. J. Phys. 3, 366 (1953);Czech. J. Phys. 4, 607, 669 (1954);Czech. J. Phys. 5, 114 (1955).

J. Opt. Soc. Am. (8)

Jenaer Jahrbuch (1)

H. Pohlack, Jenaer Jahrbuch,  1962, p. 181 (in German).

Proc. IEEE (1)

S. E. Harris and E. O. Ammann, Proc. IEEE 52, 411 (1964).
[Crossref]

Proc. Phys. Soc. (London) (1)

J. S. Seeley, Proc. Phys. Soc. (London) 78, 998 (1961).
[Crossref]

Publ. Astron. Soc. Pacific (1)

J. W. Evans, Publ. Astron. Soc. Pacific 52, 305 (1940).
[Crossref]

Other (6)

J. A. Aseltine, Transform Method in Linear System Analysis (McGraw-Hill Book Company, Inc., New York, 1958).

E. A. Guillemin, Theory of Linear Physical Systems (John Wiley & Sons, Inc., New York, 1963), p. 430.

Ref. 20, p. 408.

D. C. Murdoch, Linear Algebra for Undergraduates (John and Wiley & Sons, Inc., New York, 1947), p. 50–51.

If biaxial crystals are used, crystals in the monoclinic and triclinic systems will probably not be satisfactory since the directions of their principal axes are dependent upon temperature and wavelength.

Y. Öhman, Nature141, 157 (1938);Nature 141, 291 (1938);Pop. Astron. Tidskrift, No. 1–2,  11, 27 (1938).
[Crossref]

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Figures (11)

F. 1
F. 1

Basic configuration of optical network (four stages). Polarizers are shown shaded.

F. 2
F. 2

Four-stage Lyot filter. Polarizers are shown shaded.

F. 3
F. 3

Four-stage Solc fan filter.

F. 4
F. 4

Impulse response of a single bire-fringent crystal.

F. 5
F. 5

Impulse response of several birefringent crystals.

F. 6
F. 6

Impulse responses and corresponding transfer functions for a network whose impulse response is (a) g(t), and (b) g(t) sampled.

F. 7
F. 7

Periodicity of network response for several types of birefringent crystals. Q: quartz, Δn = 0.009; M: mica, Δn = 0.04; C: calcite, Δn = 0.17; S: sodium nitrate, Δn = 0.24.

F. 8
F. 8

Summary of impulse notation: Impulse pyramid for a two-stage network. Top: input; next to top: output from first crystal; next to bottom: output from second crystal; bottom: output from polarizer. Solid strokes: polarized along fast axis of crystal. Broken strokes: polarized along slow axis of crystal.

F. 9
F. 9

n-stage network. Compare with two-stage network in Fig. 8.

Fig. 10
Fig. 10

Angle conventions used in the synthesis procedure: (a) output polarizer; (b) relative crystal angles; (c) input polarizer.

F. 11
F. 11

Ideal and approximating transfer functions of example. Ideal transfer function is shown by dotted line and approximating transfer function by solid line.

Tables (3)

Tables Icon

Table I Related sets of Di and their corresponding θi.

Tables Icon

Table II The 16 real sets of Di.

Tables Icon

Table III Summary of results of example.

Equations (77)

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t S t F = L Δ η / c ,
C ( ω ) = C 0 + C 1 e i a ω + C 2 e i 2 a ω + + C n e ina ω = k = 0 n C k e ika ω .
C ( t ) = C 0 δ ( t ) + C 1 δ ( t a ) + C 2 δ ( t 2 a ) + + C n δ ( t n a ) = k = 0 n C k δ ( t k a ) .
Number of crystals necessary q periodicity bandwidth ,
θ 1 = ϕ 1 , θ 2 = ϕ 2 ϕ 1 , θ n = ϕ n ϕ n 1 , θ p = ϕ p ϕ n .
F 2 ( t ) = F 0 2 δ ( t ) + F 1 2 δ ( t a ) ,
S 2 ( t ) = S 1 2 δ ( t a ) + S 2 2 δ ( t 2 a ) .
D ( ω ) D * ( ω ) = ( I 0 0 ) 2 C ( ω ) C * ( ω ) .
D ( ω ) = D 0 + D 1 e i a ω + D 2 e i 2 a ω + + D n e ina ω .
D ( t ) = D 0 + D 1 δ ( t a ) + D 2 δ ( t 2 a ) + + D n δ ( t n a ) .
[ F i n S i n ] = [ sin θ p cos θ p cos θ p sin θ p ] [ C i D i ] ,
F n n = S 0 n = 0 .
tan θ p = D n / C n
tan θ p = C 0 / D 0 .
[ F 0 1 S 1 1 ] = [ sin θ 1 cos θ 1 ] [ I 0 0 ] ,
( F 0 2 F 1 2 S 1 2 S 2 2 ) = ( cos θ 2 0 0 sin θ 2 sin θ 2 0 0 cos θ 2 ) [ F 0 1 S 1 1 ] ,
( F 0 3 F 1 3 F 2 3 S 1 3 S 2 3 S 3 3 ) = ( cos θ 3 0 0 0 0 cos θ 3 sin θ 3 0 0 0 0 sin θ 3 sin θ 3 0 0 0 0 sin θ 3 cos θ 3 0 0 0 0 cos θ 3 ) ( F 0 2 F 1 2 S 1 2 S 2 2 ) .
( F 0 i F 1 i F 2 i F i 3 i F i 2 i F i 1 i S 1 i S 2 i S 3 i S i 2 i S i 1 i S i i ) ( cos θ i 0 0 0 0 0 0 cos θ i 0 0 0 0 0 0 cos θ i 0 0 0 0 0 0 sin θ i 0 0 0 0 0 0 sin θ i 0 0 0 0 0 0 sin θ i sin θ i 0 0 0 0 0 0 sin θ i 0 0 0 0 0 0 sin θ i 0 0 0 0 0 0 cos θ i 0 0 0 0 0 0 cos θ i 0 0 0 0 0 0 cos θ i ) ( F 0 i 1 F 1 i 1 F 2 i 1 F i 3 i 1 F i 2 i 1 S 1 i 1 S 2 i 1 S 3 i 1 S 4 i 1 S i 2 i 1 S i 1 i 1 )
( cos θ 3 0 0 0 0 cos θ 3 sin θ 3 0 0 0 0 sin θ 3 sin θ 3 0 0 0 0 sin θ 3 cos θ 3 0 0 0 0 cos θ 3 )
( cos θ 3 0 0 0 F 0 3 0 cos θ 3 sin θ 3 0 F 1 3 0 0 0 sin θ 3 F 2 3 sin θ 3 0 0 0 S 1 3 0 sin θ 3 cos θ 3 0 S 2 3 0 0 0 cos θ 3 S 3 3 ) .
tan θ 3 = ( F 2 3 / S 3 3 )
F 0 3 F 2 3 + S 1 3 S 3 3 = 0 .
tan θ 1 = ( F 0 1 / S 1 1 ) ,
( F 0 1 ) 2 + ( S 1 1 ) 2 = ( I 0 0 ) 2 ,
tan θ 2 = ( F 1 2 / S 2 2 ) ,
F 0 2 F 1 2 + S 1 2 S 2 2 = 0 ,
tan θ i = ( F i 1 i / S i i ) ,
F 0 i F i 1 i + S 1 i S i i = 0 .
K ( ω ) = 4 / π 2 [ ( 1 / 25 ) e + i 5 a ω + ( 1 / 9 ) e + i 3 a ω + e i a ω + e i a ω + ( 1 / 9 ) e i 3 a ω + ( 1 / 25 ) e i 5 a ω ] ,
C ( ω ) = e i 5 a ω K ( ω ) = 4 / π 2 [ 1 / 25 + ( 1 / 9 ) e i 2 a ω + e i 4 a ω + e i 6 a ω + ( 1 / 9 ) e i 8 a ω + ( 1 / 25 ) e i 10 a ω ] .
C ( ω ) = 0.01621 + 0.04503 e i b ω + 0.40528 e i 2 b ω + 0.40528 e i 3 b ω + 0.0450 e i 4 b ω + 0.01621 e i 5 b ω .
| D ( ω ) | 2 = D ( ω ) D * ( ω ) = ( I 0 0 ) 2 C ( ω ) C * ( ω ) , = ( I 0 0 ) 2 0.33309 0.40443 cos b ω 0.09928 cos 2 b ω 0.03034 cos 3 b ω 0.00292 cos 4 b ω 0.000526 cos 5 b ω .
| D ( ω ) | 2 = 0.66691 0.40443 cos b ω 0.09928 cos 2 b ω 0.03034 cos 3 b ω 0.00292 cos 4 b ω 0.000526 cos 5 b ω .
0.00263 x 5 0.00146 x 4 0.01517 x 3 0.04964 x 2 0.20222 x + 0.66691 0.20222 x 1 0.04964 x 2 0.01517 x 3 0.00146 x 4 0.000263 x 5 = 0 .
B 5 = A 5 = 0.00263 , B 4 = A 4 = 0.00146 , B 3 = A 3 5 A 5 = 0.01385 , B 2 = A 2 4 A 4 = 0.04380 , B 1 = A 1 + 5 A 5 3 A 3 = 0.15803 , B 0 = A 0 + 2 A 4 2 A 2 = + 0.76327 .
0.000263 y 5 0.00146 y 4 0.01385 y 3 0.04380 y 2 0.15803 y + 0.76327 = 0 .
y 1 = 4.07379 + i 3.93269 , y 2 = 4.07379 i 3.93269 , y 3 = 0.18957 + i 6.39532 , y 4 = 0.18957 i 6.39532 , y 5 = 2.21289 .
x 1 = 3.95066 + i 4.05920 , x 2 = 3.95066 i 4.05920 , x 3 = 0.18525 + i 6.54791 , x 4 = 0.18525 i 6.54791 , x 5 = 1.57997 ; x 1 1 = 0.12313 i 0.12652 , x 2 1 = 0.12313 i 0.12652 , x 3 1 = 0.00432 i 0.15260 , x 4 1 = 0.00432 + i 0.15260 , x 5 1 = 0.63293 .
( x x 1 ) ( x x 2 ) ( x x 3 ) ( x x 4 ) ( x x 5 ) .
x 5 + 5.95085 x 4 + 60.16845 x 3 + 213.29090 x 2 + 859.85121 x 2175.20862 .
D 0 = 2175.20862 q , D 1 = 859.85121 q , D 2 = 213.29090 q , D 3 = 60.16845 q , D 4 = 5.95085 q , D 5 = q .
q = ± 3.47586 × 10 4 .
D 0 = 0.75607 , D 1 = 0.29887 , D 2 = 0.07414 , D 3 = 0.02091 , D 4 = 0.002068 , D 5 = 0.000348 .
tan θ p = D 5 / C 5 = 0.02144 ,
θ p = 1 ° 1 4 .
( F 0 5 F 1 5 F 2 5 F 3 5 F 4 5 ) = ( 0.75625 0.29784 0.06543 0.01222 0.00110 ) · ( S 1 5 S 2 5 S 3 5 S 4 5 S 5 5 ) = ( 0.05143 0.40678 0.40564 0.04507 0.01621 ) .
tan θ 5 = ( F 4 5 / S 5 5 ) = 0.06799 ,
θ 5 = 3 ° 5 3 .
( F 0 4 F 1 4 F 2 4 F 3 4 F 4 4 ) = 1 { ( F 4 5 ) 2 + ( S 5 5 ) 2 } 1 2 ( F 0 5 S 1 5 F 1 5 S 2 5 F 2 5 S 3 5 F 3 5 S 4 5 F 4 5 S 5 5 ) [ S 5 5 F 4 5 ] = ( 0.75799 0.26955 0.03777 0.00913 0 ) ,
( S 0 4 S 1 4 S 2 4 S 3 4 S 4 4 ) = 1 { ( F 4 5 ) 2 + ( S 5 5 ) 2 } 1 2 ( F 0 5 S 1 5 F 1 5 S 2 5 F 2 5 S 3 5 F 3 5 S 4 5 F 4 5 S 5 5 ) [ F 4 5 S 5 5 ] = ( 0 0.42605 0.40914 0.04579 0.01625 ) .
( F 0 3 F 1 3 F 2 3 F 3 3 ) = 1 { ( F 3 4 ) 2 + ( S 4 4 ) 2 } 1 2 ( F 0 4 S 1 4 F 1 4 S 2 4 F 2 4 S 3 4 F 3 4 S 4 4 ) [ S 4 4 F 3 4 ] ( 0.86952 0.03454 0.01049 0 ) , ( S 0 3 S 1 3 S 2 3 S 3 3 ) = 1 { ( F 3 4 ) 2 + ( S 4 4 ) 2 } 1 2 ( F 0 4 S 1 4 F 1 4 S 2 4 F 2 4 S 3 4 F 3 4 S 4 4 ) [ F 3 4 S 4 4 ] = ( 0 0.48873 0.05842 0.01864 ) . tan θ 3 = ( F 2 3 / S 3 3 ) = 0.56268 , θ 3 = 29 ° 2 0 . ( F 0 2 F 1 2 F 2 2 ) = 1 { ( F 2 3 ) 2 + ( S 3 3 ) 2 } 1 2 ( F 0 3 S 1 3 F 1 3 S 2 3 F 2 3 S 3 3 ) [ S 3 3 F 2 3 ] = ( 0.99746 0.00145 0 ) , ( S 0 2 S 1 2 S 2 2 ) = 1 { ( F 2 3 ) 2 + ( S 3 3 ) 2 } 1 2 ( F 0 3 S 1 3 F 1 3 S 2 3 F 2 3 S 3 3 ) [ F 2 3 S 3 3 ] = ( 0 0.06785 0.02139 ) . tan θ 2 = ( F 1 3 / S 2 2 ) = 0.06802 , θ 2 = 3 ° 5 3 . [ F 0 1 F 1 1 ] = 1 { ( F 1 2 ) 2 + ( S 2 2 ) 2 } 1 2 [ F 0 2 S 1 2 F 1 2 S 2 2 ] [ S 2 2 F 1 2 ] = [ 0.99977 0 ] , [ S 0 1 S 1 1 ] = 1 { ( F 1 2 ) 2 + ( S 2 2 ) 2 } 1 2 [ F 0 2 S 1 2 F 1 2 S 2 2 ] [ F 1 2 S 2 2 ] = [ 0 0.02144 ] . tan θ 1 = ( F 0 1 / S 1 1 ) = 46.62959 , θ 1 = 88 ° 4 6 ,
C tuned ( ω ) = k = 0 n C k e i k ( a ω θ ) = C untuned ( ω θ / a ) .
| D ( ω ) | 2 = A 0 + 2 A 1 cos a ω + + 2 A n cos n a ω .
| D ( ω ) | 2 = A n e ina ω + A n 1 e i ( n 1 ) a ω + + A 1 e i a ω + A 0 + A 1 e i a ω + + A n 1 e i ( n 1 ) a ω + A n e ina ω .
| D ( ω ) | 2 = ( D 0 + D 1 e i a ω + D 2 e i 2 a ω + + D n e ina ω ) × ( D 0 + D 1 e i a ω + + D n e ina ω ) .
| D ( ω ) | 2 = D ( ω ) D * ( ω ) .
A n x n + A n 1 x n 1 + + A 1 x + A 0 + A 1 x 1 + + A n 1 x ( n 1 ) + A n x n = 0 .
B n ( x + x 1 ) n + B n 1 ( x + x 1 ) n 1 + + B 0 = 0 ,
x + x 1 = y
B n y n + B n 1 y n 1 + + B 0 = 0 .
x 1 , 1 / x 1 , x 2 , 1 / x 2 , x n . 1 / x n .
( x x 1 ) ( x 1 / x 2 ) ( x x 3 ) ( x x n ) = x n + d n 1 x n 1 + + d 2 x 2 + d 1 x + d 0 .
D 0 = q d 0 , D 1 = q d 1 , D n = d q n .
q 2 ( d 0 2 + d 1 2 + + d n 2 ) = A 0 ,
F 4 ( ω ) F 4 * ( ω ) + S 4 ( ω ) S 4 * ( ω ) = ( I 0 0 ) 2 .
( F 0 4 ) 2 + ( F 1 4 ) 2 + ( F 2 4 ) 2 + ( F 3 4 ) 2 + ( S 1 4 ) 2 + ( S 2 4 ) 2 + ( S 3 4 ) 2 + ( S 4 4 ) 2 = ( I 0 0 ) 2 ,
F 0 4 F 1 4 + F 1 4 F 2 4 + F 2 4 F 3 4 + S 1 4 S 2 4 + S 2 4 S 3 4 + S 3 4 S 4 4 = 0 ,
F 0 4 F 2 4 + F 1 4 F 3 4 + S 1 4 S 3 4 + S 2 4 S 4 4 = 0 ,
F 0 4 F 3 4 + S 1 4 S 4 4 = 0 .
( F 0 i ) 2 + ( F 1 i ) 2 + + ( F i 1 i ) 2 + ( S 1 i ) 2 + ( S 2 i ) 2 + + ( S i i ) 2 = ( I 0 0 ) 2 ,
F 0 i F 1 i + F 1 i F 2 i + + F i 2 i F i 1 i + S 1 i S 2 i + S 2 i S 3 i + + S i 1 i S i i = 0 ,
F 0 i F 2 i + + F i 3 i F i 1 i + S 1 i S 3 i + + S i 2 i S i i = 0 , F 0 i F i 1 i + S 1 i S i i = 0 .
( C 0 ) 2 + ( C 1 ) 2 + + ( C n ) 2 + ( D 0 ) 2 + ( D 1 ) 2 + + ( D n ) 2 = ( I 0 0 ) 2 ,
C 0 C 1 + C 1 C 2 + + C n 1 C n + D 0 D 1 + D 1 D 2 + + D n 1 D n = 0 ,
C 0 C 2 + C 1 C 3 + + C n 2 C n + D 0 D 2 + D 1 D 3 + + D n 2 D n = 0 , C 0 C n + D 0 D n = 0 .
( F 0 i 1 F 1 i 1 F 2 i 1 F i 1 i 1 ) = 1 { ( F i 1 i ) 2 + ( S i i ) 2 } 1 2 ( F 0 i S 1 i F 1 i S 2 i F i 1 i S i i ) [ S i i F i 1 i ] .
( S 0 i 1 S 1 i 1 S i 1 i 1 ) = 1 { ( F i 1 i ) 2 + ( S i i ) 2 } 1 2 ( F 0 i S 1 i F 1 i S 2 i F i 1 i S i i ) [ F i 1 i S i i ] .