Abstract

Sylvester’s theorem is often applied to problems involving light propagation through periodic optical systems represented by unimodular 2 × 2 transfer matrices. We extend this theorem to apply to broader classes of optics-related matrices. These matrices may be 2 × 2 or take on an important augmented 3 × 3 form. The results, which are summarized in tabular form, are useful for the analysis and the synthesis of a variety of optical systems, such as those that contain periodic distributed-feedback lasers, lossy birefringent filters, periodic pulse compressors, and misaligned lenses and mirrors. The results are also applicable to other types of system such as periodic electric circuits with intracavity independent sources, high-energy particle accelerators, and periodic computer graphics manipulations that may include object translation. As an example, we use the 3 × 3 form of Sylvester’s theorem to examine Gaussian beam propagation in a misaligned resonator.

© 1995 Optical Society of America

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References

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  1. A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, New York, 1975).
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    [CrossRef]
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  7. L. W. Casperson, P. M. Scheinert, “Multipass resonators for annular gain lasers,” Opt. Quantum. Electron. 13, 193–199 (1981).
    [CrossRef]
  8. L. W. Casperson, “Beam propagation in periodic quadratic-index waveguides,” Appl. Opt. 24, 4395–4403 (1985).
    [CrossRef] [PubMed]
  9. D. A. Edwards, H. J. Syphers, An Introduction to the Physics of High Energy Accelerators (Wiley, New York, 1993), pp. 60–68.
  10. P. J. Bryant, K. Johnson, The Principles of Circular Accelerators and Storage Rings (Cambridge U. Press, Cambridge, 1993), p. 37.
  11. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).
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  15. J. Hong, W. Huang, T. Makino, “On the transfer matrix method for distributed-feedback waveguide devices,” J. Lightwave Technol. 10, 1860–1868 (1992).
    [CrossRef]
  16. J. Capmany, M. A. Muriel, “A new transfer matrix formalism for the analysis of fiber ring resonators: compound coupled structures for FDMA demultiplexing,” J. Lightwave Technol. 8, 1904–1919 (1990).
    [CrossRef]
  17. O. E. Martinez, “Matrix formalism for pulse compressors,” IEEE J. Quantum Electron. 24, 2530–2536 (1988).
    [CrossRef]
  18. O. E. Martinez, “Matrix formalism for dispersive laser cavities,” IEEE J. Quantum Electron. 25, 296–300 (1989).
    [CrossRef]
  19. D. A. Edwards, H. J. Syphers, An Introduction to the Physics of High Energy Accelerators (Wiley, New York, 1993), pp. 88–94.
  20. R. A. Plastock, G. Kalley, Theory and Problems of Computer Graphics, Schaum’s Outline Series (McGraw-Hill, New York, 1986), pp. 82–87.
  21. J. J. Sylvester, “Sur les puissances et les racines de substitutions lineaires,” C. R. Acad. Sci. XCIV, 55–59 (1882); also published in The Collected Mathematical Papers of James Joseph Sylvester (Cambridge U. Press, Cambridge, 1912), Vol. 4, pp. 562–564.
  22. P. I. Richards, Manual of Mathematical Physics (Pergamon, New York, 1959), pp. 311–312.
  23. R. A. Horn, C. R. Johnson, Topics in Matrix Analysis (Cambridge U. Press, Cambridge, 1991), p. 401.
  24. S. Barnett, Matrices, Methods and Applications (Clarendon, Oxford, 1990), p. 234.
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  26. Ref. 14, p. 67.
  27. A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989), p. 124.
  28. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984), pp. 138–139.
  29. P. Yeh, A. Yariv, C.-S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 67, 423–438 (1977).
    [CrossRef]
  30. H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
  31. H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
    [CrossRef]
  32. P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988), pp. 476–477.
  33. R. A. Fraser, W. J. Duncan, A. R. Collar, Elementary Matrices and Some Applications to Dynamics and Differential Equations (Cambridge U. Press, Cambridge, 1963), pp. 83–87.
  34. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964), pp. 771–802; see also M. R. Spiegel, Mathematical Handbook of Formulas and Tables, Schaum’s Outline Series (McGraw-Hill, New York, 1968), pp. 157–159.
  35. L. W. Casperson, “Gaussian light beams in inhomogeneous media,” Appl. Opt. 12, 2434–2441 (1973).
    [CrossRef] [PubMed]
  36. J. T. Verdeyen, Laser Electronics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1989), pp. 38–48.

1994

1992

J. Hong, W. Huang, T. Makino, “On the transfer matrix method for distributed-feedback waveguide devices,” J. Lightwave Technol. 10, 1860–1868 (1992).
[CrossRef]

1990

J. Capmany, M. A. Muriel, “A new transfer matrix formalism for the analysis of fiber ring resonators: compound coupled structures for FDMA demultiplexing,” J. Lightwave Technol. 8, 1904–1919 (1990).
[CrossRef]

1989

O. E. Martinez, “Matrix formalism for dispersive laser cavities,” IEEE J. Quantum Electron. 25, 296–300 (1989).
[CrossRef]

1988

O. E. Martinez, “Matrix formalism for pulse compressors,” IEEE J. Quantum Electron. 24, 2530–2536 (1988).
[CrossRef]

1985

1982

1981

L. W. Casperson, P. M. Scheinert, “Multipass resonators for annular gain lasers,” Opt. Quantum. Electron. 13, 193–199 (1981).
[CrossRef]

1977

1975

1973

1969

1966

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

1965

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).

1961

J. R. Pierce, “Modes in sequences of lenses,” Proc. Natl. Acad. Sci. USA 47, 1803–1813 (1961).
[CrossRef]

1942

1941

1882

J. J. Sylvester, “Sur les puissances et les racines de substitutions lineaires,” C. R. Acad. Sci. XCIV, 55–59 (1882); also published in The Collected Mathematical Papers of James Joseph Sylvester (Cambridge U. Press, Cambridge, 1912), Vol. 4, pp. 562–564.

1858

A. Cayley, “A memoir on the theory of matrices,” Philos. Trans. R. Soc. London CXLVIII, 17–27 (1858); also published in The Collected Mathematical Papers of Arthur Cayley (Cambridge U. Press, Cambridge, 1889), Vol. 2, pp. 475–496.

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964), pp. 771–802; see also M. R. Spiegel, Mathematical Handbook of Formulas and Tables, Schaum’s Outline Series (McGraw-Hill, New York, 1968), pp. 157–159.

Arnaud, J. A.

Barnett, S.

S. Barnett, Matrices, Methods and Applications (Clarendon, Oxford, 1990), p. 234.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), pp. 51–70.

Bryant, P. J.

P. J. Bryant, K. Johnson, The Principles of Circular Accelerators and Storage Rings (Cambridge U. Press, Cambridge, 1993), p. 37.

Burch, J. M.

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, New York, 1975).

Capmany, J.

J. Capmany, M. A. Muriel, “A new transfer matrix formalism for the analysis of fiber ring resonators: compound coupled structures for FDMA demultiplexing,” J. Lightwave Technol. 8, 1904–1919 (1990).
[CrossRef]

Casperson, L. W.

Cayley, A.

A. Cayley, “A memoir on the theory of matrices,” Philos. Trans. R. Soc. London CXLVIII, 17–27 (1858); also published in The Collected Mathematical Papers of Arthur Cayley (Cambridge U. Press, Cambridge, 1889), Vol. 2, pp. 475–496.

Collar, A. R.

R. A. Fraser, W. J. Duncan, A. R. Collar, Elementary Matrices and Some Applications to Dynamics and Differential Equations (Cambridge U. Press, Cambridge, 1963), pp. 83–87.

Duncan, W. J.

R. A. Fraser, W. J. Duncan, A. R. Collar, Elementary Matrices and Some Applications to Dynamics and Differential Equations (Cambridge U. Press, Cambridge, 1963), pp. 83–87.

Eberly, J. H.

P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988), pp. 476–477.

Edwards, D. A.

D. A. Edwards, H. J. Syphers, An Introduction to the Physics of High Energy Accelerators (Wiley, New York, 1993), pp. 88–94.

D. A. Edwards, H. J. Syphers, An Introduction to the Physics of High Energy Accelerators (Wiley, New York, 1993), pp. 60–68.

Fraser, R. A.

R. A. Fraser, W. J. Duncan, A. R. Collar, Elementary Matrices and Some Applications to Dynamics and Differential Equations (Cambridge U. Press, Cambridge, 1963), pp. 83–87.

Gerrard, A.

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, New York, 1975).

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984), pp. 138–139.

Hong, C.-S.

Hong, J.

J. Hong, W. Huang, T. Makino, “On the transfer matrix method for distributed-feedback waveguide devices,” J. Lightwave Technol. 10, 1860–1868 (1992).
[CrossRef]

Horn, R. A.

R. A. Horn, C. R. Johnson, Topics in Matrix Analysis (Cambridge U. Press, Cambridge, 1991), p. 401.

Huang, W.

J. Hong, W. Huang, T. Makino, “On the transfer matrix method for distributed-feedback waveguide devices,” J. Lightwave Technol. 10, 1860–1868 (1992).
[CrossRef]

Johnson, C. R.

R. A. Horn, C. R. Johnson, Topics in Matrix Analysis (Cambridge U. Press, Cambridge, 1991), p. 401.

Johnson, K.

P. J. Bryant, K. Johnson, The Principles of Circular Accelerators and Storage Rings (Cambridge U. Press, Cambridge, 1993), p. 37.

Jones, R. C.

Kalley, G.

R. A. Plastock, G. Kalley, Theory and Problems of Computer Graphics, Schaum’s Outline Series (McGraw-Hill, New York, 1986), pp. 82–87.

Kogelnik, H.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).

Li, T.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Lunnam, S. D.

Makino, T.

J. Hong, W. Huang, T. Makino, “On the transfer matrix method for distributed-feedback waveguide devices,” J. Lightwave Technol. 10, 1860–1868 (1992).
[CrossRef]

Martinez, O. E.

O. E. Martinez, “Matrix formalism for dispersive laser cavities,” IEEE J. Quantum Electron. 25, 296–300 (1989).
[CrossRef]

O. E. Martinez, “Matrix formalism for pulse compressors,” IEEE J. Quantum Electron. 24, 2530–2536 (1988).
[CrossRef]

Milonni, P. W.

P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988), pp. 476–477.

Muriel, M. A.

J. Capmany, M. A. Muriel, “A new transfer matrix formalism for the analysis of fiber ring resonators: compound coupled structures for FDMA demultiplexing,” J. Lightwave Technol. 8, 1904–1919 (1990).
[CrossRef]

Pierce, J. R.

J. R. Pierce, “Modes in sequences of lenses,” Proc. Natl. Acad. Sci. USA 47, 1803–1813 (1961).
[CrossRef]

Plastock, R. A.

R. A. Plastock, G. Kalley, Theory and Problems of Computer Graphics, Schaum’s Outline Series (McGraw-Hill, New York, 1986), pp. 82–87.

Richards, P. I.

P. I. Richards, Manual of Mathematical Physics (Pergamon, New York, 1959), pp. 311–312.

Scheinert, P. M.

L. W. Casperson, P. M. Scheinert, “Multipass resonators for annular gain lasers,” Opt. Quantum. Electron. 13, 193–199 (1981).
[CrossRef]

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964), pp. 771–802; see also M. R. Spiegel, Mathematical Handbook of Formulas and Tables, Schaum’s Outline Series (McGraw-Hill, New York, 1968), pp. 157–159.

Sylvester, J. J.

J. J. Sylvester, “Sur les puissances et les racines de substitutions lineaires,” C. R. Acad. Sci. XCIV, 55–59 (1882); also published in The Collected Mathematical Papers of James Joseph Sylvester (Cambridge U. Press, Cambridge, 1912), Vol. 4, pp. 562–564.

Syphers, H. J.

D. A. Edwards, H. J. Syphers, An Introduction to the Physics of High Energy Accelerators (Wiley, New York, 1993), pp. 60–68.

D. A. Edwards, H. J. Syphers, An Introduction to the Physics of High Energy Accelerators (Wiley, New York, 1993), pp. 88–94.

Tovar, A. A.

Verdeyen, J. T.

J. T. Verdeyen, Laser Electronics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1989), pp. 38–48.

White, J. U.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), pp. 51–70.

Yariv, A.

P. Yeh, A. Yariv, C.-S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 67, 423–438 (1977).
[CrossRef]

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989), p. 124.

Yeh, P.

Appl. Opt.

Bell Syst. Tech. J.

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).

C. R. Acad. Sci.

J. J. Sylvester, “Sur les puissances et les racines de substitutions lineaires,” C. R. Acad. Sci. XCIV, 55–59 (1882); also published in The Collected Mathematical Papers of James Joseph Sylvester (Cambridge U. Press, Cambridge, 1912), Vol. 4, pp. 562–564.

IEEE J. Quantum Electron.

O. E. Martinez, “Matrix formalism for pulse compressors,” IEEE J. Quantum Electron. 24, 2530–2536 (1988).
[CrossRef]

O. E. Martinez, “Matrix formalism for dispersive laser cavities,” IEEE J. Quantum Electron. 25, 296–300 (1989).
[CrossRef]

J. Lightwave Technol.

J. Hong, W. Huang, T. Makino, “On the transfer matrix method for distributed-feedback waveguide devices,” J. Lightwave Technol. 10, 1860–1868 (1992).
[CrossRef]

J. Capmany, M. A. Muriel, “A new transfer matrix formalism for the analysis of fiber ring resonators: compound coupled structures for FDMA demultiplexing,” J. Lightwave Technol. 8, 1904–1919 (1990).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Quantum. Electron.

L. W. Casperson, P. M. Scheinert, “Multipass resonators for annular gain lasers,” Opt. Quantum. Electron. 13, 193–199 (1981).
[CrossRef]

Philos. Trans. R. Soc. London

A. Cayley, “A memoir on the theory of matrices,” Philos. Trans. R. Soc. London CXLVIII, 17–27 (1858); also published in The Collected Mathematical Papers of Arthur Cayley (Cambridge U. Press, Cambridge, 1889), Vol. 2, pp. 475–496.

Proc. IEEE

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Proc. Natl. Acad. Sci. USA

J. R. Pierce, “Modes in sequences of lenses,” Proc. Natl. Acad. Sci. USA 47, 1803–1813 (1961).
[CrossRef]

Other

D. A. Edwards, H. J. Syphers, An Introduction to the Physics of High Energy Accelerators (Wiley, New York, 1993), pp. 60–68.

P. J. Bryant, K. Johnson, The Principles of Circular Accelerators and Storage Rings (Cambridge U. Press, Cambridge, 1993), p. 37.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Ref. 14, p. 67.

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989), p. 124.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984), pp. 138–139.

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, New York, 1975).

D. A. Edwards, H. J. Syphers, An Introduction to the Physics of High Energy Accelerators (Wiley, New York, 1993), pp. 88–94.

R. A. Plastock, G. Kalley, Theory and Problems of Computer Graphics, Schaum’s Outline Series (McGraw-Hill, New York, 1986), pp. 82–87.

P. I. Richards, Manual of Mathematical Physics (Pergamon, New York, 1959), pp. 311–312.

R. A. Horn, C. R. Johnson, Topics in Matrix Analysis (Cambridge U. Press, Cambridge, 1991), p. 401.

S. Barnett, Matrices, Methods and Applications (Clarendon, Oxford, 1990), p. 234.

P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988), pp. 476–477.

R. A. Fraser, W. J. Duncan, A. R. Collar, Elementary Matrices and Some Applications to Dynamics and Differential Equations (Cambridge U. Press, Cambridge, 1963), pp. 83–87.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964), pp. 771–802; see also M. R. Spiegel, Mathematical Handbook of Formulas and Tables, Schaum’s Outline Series (McGraw-Hill, New York, 1968), pp. 157–159.

J. T. Verdeyen, Laser Electronics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1989), pp. 38–48.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), pp. 51–70.

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Tables (2)

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Table 1 Generalized 2 × 2 Sylvester Theorems

Tables Icon

Table 2 Generalized 3 × 3 Sylvester Theorems

Equations (133)

Equations on this page are rendered with MathJax. Learn more.

p ( λ ) T - λ I = 0.
T s = j = 1 n ( λ j s i j n λ i I - T λ i - λ j ) .
[ A B C D ] s = j = 1 2 ( λ j s i j 2 1 λ i - λ j [ λ i - A - B - C - λ i - D ] ) ,
= λ 1 s λ 2 - λ 1 [ λ 2 - A - B - C λ 2 - D ] + λ 2 s λ 1 - λ 2 [ λ 1 - A - B - C λ 1 - D ] ,
= [ A λ 2 s - λ 1 s λ 2 - λ 1 + λ 1 λ 2 λ 1 s - 1 - λ 2 s - 1 λ 2 - λ 1 B λ 2 s - λ 1 s λ 2 - λ 1 C λ 2 s - λ 1 s λ 2 - λ 1 D λ 2 s - λ 1 s λ 2 - λ 1 + λ 1 λ 2 λ 1 s - 1 - λ 2 s - 1 λ 2 - λ 1 ] .
p ( λ ) = ( A - λ ) ( D - λ ) - B C = 0.
λ 2 - ( A + D ) λ + ( A D - B C ) = 0.
cos θ A + D 2
λ 1 , 2 = exp ( ± i θ ) ,
[ A B C D ] s = 1 sin θ × [ A sin ( s θ ) - sin [ ( s - 1 ) θ ] B sin ( s θ ) C sin ( s θ ) D sin ( s θ ) - sin [ ( s - 1 ) θ ] ] ,
[ A B C D ] s = [ A U s - 1 ( x ) - U s - 2 ( x ) B U s - 1 ( x ) C U s - 1 ( x ) D U s - 1 ( x ) - U s - 2 ( x ) ] ,
x ½ ( A + D ) ,
U s ( x ) = sin [ ( s + 1 ) cos - 1 ( x ) ] ( 1 - x 2 ) 1 / 2 .
U 0 ( x ) = 1 , U 1 ( x ) = 2 x , U 2 ( x ) = 4 x 2 - 1 , U 3 ( x ) = 8 x 3 - 4 x , U 4 ( x ) = 16 x 4 - 12 x 2 + 1 , U 5 ( x ) = 32 x 5 - 32 x 3 + 6 x , U 6 ( x ) = 64 x 6 - 80 x 4 + 24 x 2 - 1.
U n + 1 - 2 x U n + U n - 1 = 0.
θ = - i ln { A + D 2 + [ ( A + D 2 ) 2 - 1 ] 1 / 2 } .
( a + i b ) 1 / 2 = [ ( a 2 + b 2 ) 1 / 2 + a 2 ] 1 / 2 + i ( sgn b ) [ ( a 2 + b 2 ) 1 / 2 - a 2 ] 1 / 2 ,
ln ( a + i b ) = ln [ ( a 2 + b 2 ) 1 / 2 ] + i [ tan - 1 ( b / a ) ] .
sin ( a + i b ) = sin a cosh b + i cos a sinh b .
[ A B C D ] s = 1 sinh ϕ [ A sinh ( s ϕ ) - sinh [ ( s - 1 ) ϕ ] B sinh ( s ϕ ) C sinh ( s ϕ ) D sinh ( s ϕ ) - sinh [ ( s - 1 ) ϕ ] ] ,
cosh ϕ = A + D 2 .
[ cos θ χ sin θ - χ - 1 sin θ cos θ ] s = [ cos ( s θ ) χ sin ( s θ ) - χ - 1 sin ( s θ ) cos ( s θ ) ] .
[ 1 χ 1 0 1 ] s = [ 1 s χ 1 0 1 ] ,
[ 1 0 χ 2 1 ] s = [ 1 0 s χ 2 1 ] .
[ χ 1 0 0 χ 2 ] s = [ χ 1 s 0 0 χ 2 s ] .
[ 0 χ 1 χ 2 0 ] 2 s = ( χ 1 χ 2 ) s [ 1 0 0 1 ] ,
[ 0 χ 1 χ 2 0 ] 2 s + 1 = ( χ 1 χ 2 ) s [ 0 χ 1 χ 2 0 ] ,
[ 1 0 0 1 ] = [ [ 1 0 - 3 / l 1 ] [ 1 l 0 1 ] ] 3 = [ ± [ 1 0 0 1 ] ] 2 = [ ± [ 1 0 0 - 1 ] ] 2 = [ 0 γ γ - 1 0 ] 2 = - [ 0 γ - γ - 1 0 ] 2 ,
T s = [ 1 0 0 1 ] ,
A sin ( s θ ) - sin [ ( s - 1 ) θ ] sin θ = 1 ,
C sin ( s θ ) sin θ = 0 ,
B sin ( s θ ) sin θ = 0 ,
D sin ( s θ ) - sin [ ( s - 1 ) θ ] sin θ = 1 ,
sin ( s θ ) = 0 ,
sin θ 0.
cos ( s θ ) = 1.
s θ = 2 k π ,
0 k s / 2.
cos ( s θ ) = ( A D - B C ) - s / 2 .
cos θ = ½ ( A + D ) ( A D - B C ) - 1 / 2
I = [ cos ϕ γ sin ϕ γ - 1 sin ϕ - cos ϕ ] .
[ A B C D ] s = τ s / 2 [ A τ - 1 / 2 B τ - 1 / 2 C τ - 1 / 2 D τ - 1 / 2 ] s .
[ A B C D ] s = τ ( s - 1 ) / 2 sin θ [ A sin ( s θ ) = - τ sin [ ( s - 1 ) θ ] B sin ( s θ ) C sin ( s θ ) D sin ( s θ ) - τ sin [ ( s - 1 ) θ ] ] .
cos θ ½ ( A + D ) τ - 1 / 2 .
[ A B C D ] s = ( A + D ) s - 1 [ A B C D ] .
[ A B C D ] 1 / 4 = 1 1 / 4 δ ( 2 τ 1 / 4 + δ ) 1 / 2 × [ A + τ + τ 1 / 4 δ B C D + τ + τ 1 / 4 δ ] ,
δ A + D + 2 τ .
[ X s Y s 1 ] = [ A B E C D F 0 0 1 ] s [ X 0 Y 0 1 ] [ A s B s E s C s D s F s 0 0 1 ] [ X 0 Y 0 1 ] .
A s E + B s F + E s = A E s + B F s + E ,
C s E + D s F + F s = C E s + D F s + F .
[ A B E C D F 0 0 1 ] s = 1 sin θ [ A sin ( s θ ) - sin [ ( s - 1 ) θ ] B sin ( s θ ) E s C sin ( s θ ) D sin ( s θ ) - sin [ ( s - 1 ) θ ] F s 0 0 sin θ ] ,
E s ( ( A - 1 ) sin ( s θ ) + ( D - 1 ) { sin [ ( s - 1 ) θ ] + sin θ } 2 ( cos θ - 1 ) ) E + { sin ( s θ ) - sin [ ( s - 1 ) θ ] - sin θ 2 ( cos θ - 1 ) } B F ,
F s { sin ( s θ ) - sin [ ( s - 1 ) θ ] - sin θ 2 ( cos θ - 1 ) } C E + ( ( D - 1 ) sin ( s θ ) + ( A - 1 ) { sin [ ( s - 1 ) θ ] + sin θ } 2 ( cos θ - 1 ) ) F .
[ A B E C D F 0 0 1 ] s = τ ( s - 1 ) / 2 sin θ [ A sin ( s θ ) - τ sin [ ( s - 1 ) θ ] B sin ( s θ ) E s C sin ( s θ ) D sin ( s θ ) - τ sin [ ( s - 1 ) θ ] F s 0 0 sin θ ] .
cos θ ½ ( A + D ) τ - 1 / 2 .
E s ( ( A τ - 1 / 2 - 1 ) sin ( s θ ) + ( D τ - 1 / 2 - 1 ) { sin [ ( s - 1 ) θ ] + sin θ } 2 ( cos θ - 1 ) ) E + { sin ( s θ ) - sin [ ( s - 1 ) θ ] - sin θ 2 τ ( cos θ - 1 ) } B F ,
F s { sin ( s θ ) - sin [ ( s - 1 ) θ ] - sin θ 2 τ ( cos θ - 1 ) } C E + ( ( D τ - 1 / 2 - 1 ) sin ( s θ ) + ( A τ - 1 / 2 - 1 ) { sin [ ( s - 1 ) θ ] + sin θ } 2 ( cos θ - 1 ) ) F .
[ A B F C D F 0 0 1 ] s = ( A + D ) s - 1 [ A B E s C D F s 0 0 ( A + D ) 1 - s ] ,
E s 1 ( A + D ) s - 1 { E + [ ( A + D ) s - 1 - 1 A + D - 1 ] ( A E + B F ) } ,
F s 1 ( A + D ) s - 1 { F + [ ( A + D ) s - 1 - 1 A + D - 1 ] ( C E + D F ) } .
[ cos θ χ sin θ E - χ - 1 sin θ cos θ F 0 0 1 ] s = [ cos ( s θ ) χ sin ( s θ ) E s - χ - 1 sin ( s θ ) cos ( s θ ) F s 0 0 1 ] ,
E s 1 + cos θ 2 [ sin ( s θ ) sin θ + 1 - cos ( s θ ) 1 + cos θ ] E + χ sin θ 2 [ 1 - cos ( s θ ) 1 - cos θ - sin ( s θ ) sin θ ] F ,
F s 1 + cos θ 2 [ sin ( s θ ) sin θ + 1 - cos ( s θ ) 1 + cos θ ] F - χ - 1 sin θ 2 [ 1 - cos ( s θ ) 1 - cos θ - sin ( s θ ) sin θ ] E .
[ 1 B E 0 1 F 0 0 1 ] s = [ 1 s B s E + s ( s - 1 ) B F / 2 0 1 s F 0 0 1 ] ,
[ 1 0 E C 1 F 0 0 1 ] s = [ 1 0 s E s C 1 s F + s ( s - 1 ) C E / 2 0 0 1 ] .
T = [ 1 d 0 0 1 0 0 0 1 ] [ 1 0 0 - 2 / R 1 0 0 0 1 ] [ 1 d 0 0 1 0 0 0 1 ] × [ 1 0 0 0 1 tan 2 ϕ 0 0 1 ]
= [ 1 - 2 d / R 2 d ( 1 - d / R ) 2 d ( 1 - d / R ) tan 2 ϕ - 2 / R 1 - 2 d / R ( 1 - 2 d / R ) tan 2 ϕ 0 0 1 ] ,
r s = 1 + cos θ 2 [ sin ( s θ ) sin θ + 1 - cos ( s θ ) 1 + cos θ ] E + sin θ 2 [ 1 - cos ( s θ ) 1 - cos θ - sin ( s θ ) sin θ ] χ F ,
B F = A E ,
2 r s r 1 = sin ( s θ ) sin θ + 1 - cos ( s θ ) 1 - cos θ ,
r s = r av + r max sin ( s θ + α ) ,
T = M - 1 Λ M .
T s = M - 1 Λ s M
Λ s = [ λ 1 0 0 λ 2 ] s = [ λ 1 s 0 0 λ 2 s ] .
[ M 1 M 2 M 3 M 4 ] [ A B C D ] = [ λ 1 0 0 λ 2 ] [ M 1 M 2 M 3 M 4 ] .
[ A - λ 1 C B D - λ 1 ] [ M 1 M 2 ] = 0 ,
[ A - λ 2 C B D - λ 2 ] [ M 3 M 4 ] = 0 .
λ 1 , 2 2 - ( A + D ) λ 1 , 2 + 1 = 0 ,
λ 1 , 2 = exp ( ± i θ ) ,
cos θ A + D 2
M 2 = ( λ 1 - A ) M 1 C ,
M 4 = ( λ 2 - A ) M 3 C .
M = [ 1 ( λ 1 - A ) / C 1 ( λ 2 - A ) / C ] ,
M - 1 = C λ 2 - λ 1 [ ( λ 2 - A ) / C - ( λ 1 - A ) / C - 1 1 ] .
T s = C λ 2 - λ 1 [ ( λ 2 - A ) / C - ( λ 1 - A ) / C - 1 1 ] [ λ 1 s 0 0 λ 2 s ] [ 1 ( λ 1 - A ) / C 1 ( λ 2 - A ) / C ]
= 1 λ 2 - λ 1 [ λ 1 s ( λ 2 - A ) - λ 2 s ( λ 1 - A ) ( λ 1 - A ) ( λ 2 - A ) ( λ 2 s - λ 1 s ) / C C ( λ 2 s - λ 1 s ) λ 2 s ( λ 2 - A ) - λ 1 s ( λ 1 - A ) ] .
λ 1 + λ 2 = A + D ,
λ 2 s - λ 1 s λ 2 - λ 1 = sin ( s θ ) sin θ .
[ A B C D ] s = 1 sin θ × [ A sin ( s θ ) - sin [ ( s - 1 ) θ ] B sin ( s θ ) C sin ( s θ ) D sin ( s θ ) - sin [ ( s - 1 ) θ ] ] ,
A D - B C = 1 ,
[ A B C D ] s = 1 sin θ × [ A sin ( s θ ) - sin [ ( s - 1 ) θ ] B sin ( s θ ) C sin ( s θ ) D sin ( s θ ) - sin [ ( s - 1 ) θ ] ] ,
cos θ A + D 2 .
[ A B C D ] s + 1 = [ A B C D ] [ A B C D ] s
= 1 sin θ [ A B C D ] [ A sin ( s θ ) - sin [ ( s - 1 ) θ ] B sin ( s θ ) C sin ( s θ ) D sin ( s θ ) - sin [ ( s - 1 ) θ ] ] .
= 1 sin θ [ ( A 2 + B C ) sin ( s θ ) - A sin [ ( s - 1 ) θ ] B ( A + D ) sin ( s θ ) - B sin [ ( s - 1 ) θ ] C ( A + D ) sin ( s θ ) - C sin [ ( s - 1 ) θ ] ( B C + D 2 ) sin ( s θ ) - D sin [ ( s - 1 ) θ ] ] .
A s + 1 = ( A 2 + B C ) sin ( s θ ) - A sin [ ( s - 1 ) θ ]
= A ( A + D ) sin ( s θ ) - A [ sin ( s θ ) cos θ - cos ( s θ ) sin θ ] - sin ( s θ )
= A [ sin ( s θ ) cos θ + cos ( s θ ) sin θ ] - sin ( s θ )
= A sin [ ( s + 1 ) θ ] - sin ( s θ ) ,
B s + 1 = B ( A + D ) sin ( s θ ) - B sin [ ( s - 1 ) θ ]
= 2 B sin ( s θ ) cos θ - B [ sin ( s θ ) cos θ - cos ( s θ ) sin θ ]
= B [ sin ( s θ ) cos θ + cos ( s θ ) sin θ ]
= B sin [ ( s + 1 ) θ ] ,
[ A B C D ] s + 1 = 1 sin θ × [ A sin [ ( s + 1 ) θ ] - sin ( s θ ) B sin [ ( s + 1 ) θ ] C sin [ ( s + 1 ) θ ] D sin [ ( s + 1 ) θ ] - sin ( s θ ) ] .
[ A B C D ] s [ A s B s C s D s ] = 1 sin θ × [ A sin ( s θ ) - sin [ ( s - 1 ) θ ] B sin ( s θ ) C sin ( s θ ) D sin ( s θ ) - sin [ ( s - 1 ) θ ] ] ,
cos θ A + D 2 .
( sin θ ) [ A s B s C s D s ] = sin ( s θ ) [ A B C D ] - sin [ ( s - 1 ) θ ] [ 1 0 0 1 ] .
[ A B C D ] = 1 sin ( s θ ) × [ A s sin θ + sin [ ( s - 1 ) θ ] B s sin θ C s sin θ D s sin θ + sin [ ( s - 1 ) θ ] ] .
A s + D s 2 = [ ( A + D ) / 2 ] sin ( s θ ) - sin [ ( s - 1 ) θ ] sin θ
= ( cos θ ) sin ( s θ ) - sin [ ( s - 1 ) θ ] sin θ
= cos ( s θ ) ,
ϕ = s θ
[ A B C D ] = [ A s B s C s D s ] 1 / s = 1 sin ϕ [ A s sin ( ϕ / s ) - sin [ ( 1 / s - 1 ) ϕ ] B s sin ( ϕ / s ) C s sin ( ϕ / s ) D s sin ( ϕ / s ) - sin [ ( 1 / s - 1 ) ϕ ] ] ,
cos ϕ A s + D s 2 .
( T n ) m = T n m ,
( T n ) 1 / s = T n / s .
T n = [ A n B n C n D n ] = 1 sin θ × [ A sin ( n θ ) - sin [ ( n - 1 ) θ ] B sin ( n θ ) C sin ( n θ ) D sin ( n θ ) - sin [ ( n - 1 ) θ ] ] ,
cos θ A + D 2 .
( T n ) m = [ A n m B n m C n m D n m ] = 1 sin ϕ [ A n sin ( m ϕ ) - sin [ ( m - 1 ) ϕ ] B n sin ( m ϕ ) C n sin ( m ϕ ) D n sin ( m ϕ ) - sin [ ( m - 1 ) ϕ ] ] ,
cos ϕ A n + D n 2 .
cos ϕ = [ ( A + D ) / 2 ] sin ( n θ ) - sin [ ( n - 1 ) θ ] sin θ
= ( cos θ ) sin ( n θ ) - sin [ ( n - 1 ) θ ] sin θ
= cos ( n θ ) .
ϕ = n θ .
A n m = A n sin ( m ϕ ) - sin [ ( m - 1 ) ϕ ] sin ϕ
= { A sin ( n θ ) - sin [ ( n - 1 ) θ ] sin θ } sin ( n m θ ) - sin [ ( m - 1 ) n θ ] sin ( n θ )
= A sin ( n m θ ) sin θ - sin [ ( n m - 1 ) θ ] sin θ - sin [ ( n - 1 ) θ ] sin ( n m θ ) + ( sin θ ) sin [ ( m - 1 ) n θ ] - sin ( n θ ) sin [ ( n m - 1 ) θ ] sin ( n θ ) sin θ
= A sin ( n m θ ) sin θ - sin [ ( n m - 1 ) θ ] sin θ ,
B n m = B n sin ( m ϕ ) sin ϕ
= B sin ( n θ ) sin θ sin ( n m θ ) sin ( n θ )
= B sin ( n m θ ) sin θ .
( T n ) m = 1 sin θ [ A sin ( n m θ ) - sin [ ( n m - 1 ) θ ] B sin ( n m θ ) C sin ( n m θ ) D sin ( n m θ ) - sin [ ( n m - 1 ) θ ] ] ,
cos θ A + D 2 .

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