Abstract

By use of matrix-based techniques it is shown how the space–bandwidth product (SBP) of a signal, as indicated by the location of the signal energy in the Wigner distribution function, can be tracked through any quadratic-phase optical system whose operation is described by the linear canonical transform. Then, applying the regular uniform sampling criteria imposed by the SBP and linking the criteria explicitly to a decomposition of the optical matrix of the system, it is shown how numerical algorithms (employing interpolation and decimation), which exhibit both invertibility and additivity, can be implemented. Algorithms appearing in the literature for a variety of transforms (Fresnel, fractional Fourier) are shown to be special cases of our general approach. The method is shown to allow the existing algorithms to be optimized and is also shown to permit the invention of many new algorithms.

© 2005 Optical Society of America

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References

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  1. T. Erseghe, P. Kraniauskas, G. Cariolaro, “Unified fractional Fourier transform and sampling theorem,” IEEE Trans. Signal Process. 47, 3419–3423 (1999).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  4. F. J. Marinho, L. M. Bernardo, “Numerical calculation of fractional Fourier transforms with a single fast Fourier transform algorithm,” J. Opt. Soc. Am. A 15, 2111–2116 (1998).
    [CrossRef]
  5. D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
    [CrossRef]
  6. M. Sypek, “Light propagation in the Fresnel region. New numerical approach,” Opt. Commun. 116, 43–48 (1995).
    [CrossRef]
  7. D. Mas, J. Perez, C. Hernandez, C. Vazquez, J. J. Miret, C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2005 (1)

2003 (1)

D. Mas, J. Perez, C. Hernandez, C. Vazquez, J. J. Miret, C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[CrossRef]

2000 (2)

1999 (2)

T. Erseghe, P. Kraniauskas, G. Cariolaro, “Unified fractional Fourier transform and sampling theorem,” IEEE Trans. Signal Process. 47, 3419–3423 (1999).
[CrossRef]

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

1998 (1)

1997 (3)

1996 (3)

1995 (1)

M. Sypek, “Light propagation in the Fresnel region. New numerical approach,” Opt. Commun. 116, 43–48 (1995).
[CrossRef]

1994 (2)

S. Abe, J. T. Sheridan and , “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach,” J. Phys. A 27, 4179–4187 (1994);Corrigenda, 7937–7938 (1994).
[CrossRef]

S. Abe, J. T. Sheridan, “Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation,” Opt. Lett. 19, 1801–1803 (1994).
[CrossRef] [PubMed]

1993 (1)

1982 (1)

1981 (1)

R. E. Crochiere, L. R. Rabiner, “Interpolation and decimation of digital signals—A tutorial review,” Proc. IEEE 69, 300–331 (1981).
[CrossRef]

1979 (1)

1974 (1)

1965 (1)

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

1932 (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Abe, S.

S. Abe, J. T. Sheridan and , “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach,” J. Phys. A 27, 4179–4187 (1994);Corrigenda, 7937–7938 (1994).
[CrossRef]

S. Abe, J. T. Sheridan, “Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation,” Opt. Lett. 19, 1801–1803 (1994).
[CrossRef] [PubMed]

Arikan, O.

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
[CrossRef]

M. J. Bastiaans, “Application of the Wigner distribution function in optics,” in The Wigner Distribution—Theory and Applications in Signal Processing, W. Mecklenbrauker and F. Hlawatsch, eds. (Elsevier Science, Amsterdam, 1997).

Bernardo, L. M.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

F. J. Marinho, L. M. Bernardo, “Numerical calculation of fractional Fourier transforms with a single fast Fourier transform algorithm,” J. Opt. Soc. Am. A 15, 2111–2116 (1998).
[CrossRef]

Bihari, B.

Bozdagi, G.

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

Cariolaro, G.

T. Erseghe, P. Kraniauskas, G. Cariolaro, “Unified fractional Fourier transform and sampling theorem,” IEEE Trans. Signal Process. 47, 3419–3423 (1999).
[CrossRef]

Chen, R. T.

Cooley, J. W.

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Crochiere, R. E.

R. E. Crochiere, L. R. Rabiner, “Interpolation and decimation of digital signals—A tutorial review,” Proc. IEEE 69, 300–331 (1981).
[CrossRef]

Deng, X.

Dorsch, R. G.

Erseghe, T.

T. Erseghe, P. Kraniauskas, G. Cariolaro, “Unified fractional Fourier transform and sampling theorem,” IEEE Trans. Signal Process. 47, 3419–3423 (1999).
[CrossRef]

Ferreira, C.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevsky, C. Ferreira, “Space–bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13, 470–473 (1996).
[CrossRef]

Gang, J.

Garcia, J.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

J. Garcia, D. Mas, R. G. Dorsch, “Fractional Fourier transform calculation through the fast Fourier transform algorithm,” Appl. Opt. 35, 7013–7018 (1996).
[CrossRef]

Goodman, J.

J. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Hennelly, B. M.

Hernandez, C.

D. Mas, J. Perez, C. Hernandez, C. Vazquez, J. J. Miret, C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[CrossRef]

Illueca, C.

D. Mas, J. Perez, C. Hernandez, C. Vazquez, J. J. Miret, C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[CrossRef]

Konforti, N.

D. Mendlovic, Z. Zalevsky, N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

Kraniauskas, P.

T. Erseghe, P. Kraniauskas, G. Cariolaro, “Unified fractional Fourier transform and sampling theorem,” IEEE Trans. Signal Process. 47, 3419–3423 (1999).
[CrossRef]

Kutay, M. A.

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, Hoboken, N.J., 2001).

Lohmann, A. W.

Marinho, F.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Marinho, F. J.

Mas, D.

D. Mas, J. Perez, C. Hernandez, C. Vazquez, J. J. Miret, C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[CrossRef]

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

J. Garcia, D. Mas, R. G. Dorsch, “Fractional Fourier transform calculation through the fast Fourier transform algorithm,” Appl. Opt. 35, 7013–7018 (1996).
[CrossRef]

Mendlovic, D.

Miret, J. J.

D. Mas, J. Perez, C. Hernandez, C. Vazquez, J. J. Miret, C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[CrossRef]

Nazarathy, M.

Ozaktas, H. M.

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, Hoboken, N.J., 2001).

Papoulis, A.

Perez, J.

D. Mas, J. Perez, C. Hernandez, C. Vazquez, J. J. Miret, C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[CrossRef]

Rabiner, L. R.

R. E. Crochiere, L. R. Rabiner, “Interpolation and decimation of digital signals—A tutorial review,” Proc. IEEE 69, 300–331 (1981).
[CrossRef]

Rhodes, W. T.

W. T. Rhodes, “Light tubes, Wigner diagrams and optical signal propagation simulation,” in Optical Information Processing: A Tribute to Adolf Lohmann, H. J. Caulfield, ed. (SPIE Press, Bellingham, Wash., 2002), pp. 343–356.

W. T. Rhodes, “Numerical simulation of Fresnel-regime wave propagation: the light tube model,” in Wave-Optical Systems Engineering, F. Wyrowski, ed., Proc. SPIE4436, 21–26 (2001).
[CrossRef]

Shamir, J.

Sheridan, J. T.

Sypek, M.

M. Sypek, “Light propagation in the Fresnel region. New numerical approach,” Opt. Commun. 116, 43–48 (1995).
[CrossRef]

Tukey, J. W.

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Vazquez, C.

D. Mas, J. Perez, C. Hernandez, C. Vazquez, J. J. Miret, C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[CrossRef]

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Zalevsky, Z.

Zhao, F.

Appl. Opt. (1)

IEEE Trans. Signal Process. (2)

T. Erseghe, P. Kraniauskas, G. Cariolaro, “Unified fractional Fourier transform and sampling theorem,” IEEE Trans. Signal Process. 47, 3419–3423 (1999).
[CrossRef]

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

J. Mod. Opt. (1)

D. Mendlovic, Z. Zalevsky, N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (8)

J. Phys. A (1)

S. Abe, J. T. Sheridan and , “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach,” J. Phys. A 27, 4179–4187 (1994);Corrigenda, 7937–7938 (1994).
[CrossRef]

Math. Comput. (1)

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Opt. Commun. (3)

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

M. Sypek, “Light propagation in the Fresnel region. New numerical approach,” Opt. Commun. 116, 43–48 (1995).
[CrossRef]

D. Mas, J. Perez, C. Hernandez, C. Vazquez, J. J. Miret, C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Proc. IEEE (1)

R. E. Crochiere, L. R. Rabiner, “Interpolation and decimation of digital signals—A tutorial review,” Proc. IEEE 69, 300–331 (1981).
[CrossRef]

Other (6)

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977).

M. J. Bastiaans, “Application of the Wigner distribution function in optics,” in The Wigner Distribution—Theory and Applications in Signal Processing, W. Mecklenbrauker and F. Hlawatsch, eds. (Elsevier Science, Amsterdam, 1997).

J. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, Hoboken, N.J., 2001).

W. T. Rhodes, “Light tubes, Wigner diagrams and optical signal propagation simulation,” in Optical Information Processing: A Tribute to Adolf Lohmann, H. J. Caulfield, ed. (SPIE Press, Bellingham, Wash., 2002), pp. 343–356.

W. T. Rhodes, “Numerical simulation of Fresnel-regime wave propagation: the light tube model,” in Wave-Optical Systems Engineering, F. Wyrowski, ed., Proc. SPIE4436, 21–26 (2001).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

WDF of a signal before and after different types of LCT are applied to the signal: (a) WDF of the original signal, (b) WDF of the signal after it has been Fourier transformed, (c) WDF of the signal after it has been fractional Fourier transformed, (d) WDF of the Fresnel transformed signal, (e) WDF of the signal after it has been chirp multiplied, (f) WDF of the magnified signal.

Fig. 2
Fig. 2

WDF of a signal with bounded signal energy (a) before and (b) after application of an arbitrary LCT.

Equations (62)

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

W u ( x , k ) = ψ { u ( x ) } ( x , k ) = u ( x ξ 2 ) u * ( x ξ 2 ) exp ( j 2 π k ξ ) d ξ ,
u ( x ) u * ( 0 ) = W u ( x 2 , k ) exp ( j 2 π k x ) d k ,
u ( 0 ) 2 = W u ( 0 , k ) d k .
W u ( x , k ) = U ( k ξ 2 ) U * ( k ξ 2 ) exp ( j 2 π x ξ ) d ξ .
u ( x ) 0 , x > W 0 2 ,
U ( k ) = u ( x ) exp ( j 2 π k x ) d x 0 , k > B 0 2 ,
W 0 2 W 0 2 u ( x ) 2 d x = η E ,
B 0 2 B 0 2 U ( k ) 2 d k = η E ,
E = u ( x ) 2 d x = U ( k ) 2 d k .
δ x 1 B 0 .
N = W 0 δ x W 0 B 0 .
u α , β , γ ( x ) = L α , β , γ { u ( x ) } ( x ) = exp ( j π 4 ) β u ( x ) × exp [ j π ( α x 2 2 β x x + γ x 2 ) ] d x ,
W ( x , k ) W ( a x + b k , c x + d k ) ,
[ x k ] = L [ x k ] = [ a b c d ] [ x k ] = [ γ β 1 β β + α γ β α β ] [ x k ] .
S = [ x 1 x 2 x 3 x 4 k 1 k 2 k 3 k 4 ] ,
S = L S = [ x 1 x 2 x 3 x 4 k 1 k 2 k 3 k 4 ] = [ a x 1 + b k 1 a x 2 + b k 2 a x 3 + b k 3 a x 4 + b k 4 c x 1 + d k 1 c x 2 + d k 2 c x 3 + d k 3 c x 4 + d k 4 ] ,
D = [ 1 1 1 0 0 0 1 0 0 1 1 0 0 1 0 1 0 1 0 0 1 0 1 1 ] .
E = [ W 0 B 0 ] = Max S D = Max [ x 1 x 2 x 1 x 3 x 1 x 4 x 2 x 3 x 2 x 4 x 3 x 4 k 1 k 2 k 1 k 3 k 1 k 4 k 2 k 3 k 2 k 4 k 3 k 4 ] ,
N 0 = 1 2 E T R E = 1 2 [ W 0 B 0 ] [ 0 1 1 0 ] [ W 0 B 0 ] = W 0 B 0 .
E = [ W n B n ] = Max S D = Max [ a ( x 1 x 2 ) + b ( k 1 k 2 ) a ( x 1 x 3 ) + b ( k 1 k 3 ) a ( x 1 x 4 ) + b ( k 1 k 4 ) a ( x 2 x 3 ) + b ( k 2 k 3 ) c ( x 1 x 2 ) + d ( k 1 k 2 ) c ( x 1 x 3 ) + d ( k 1 k 3 ) c ( x 1 x 4 ) + d ( k 1 k 4 ) c ( x 2 x 3 ) + d ( k 2 k 3 ) ]
[ a ( x 2 x 4 ) + b ( k 2 k 4 ) a ( x 3 x 4 ) + b ( k 3 k 4 ) c ( x 2 x 4 ) + d ( k 2 k 4 ) c ( x 3 x 4 ) + d ( k 3 k 4 ) ] .
[ x 1 x 2 x 3 x 4 k 1 k 2 k 3 k 4 ] = [ W 0 2 W 0 2 W 0 2 W 0 2 B 0 2 B 0 2 B 0 2 B 0 2 ] .
E = [ W n B n ] = Max [ a W 0 b B 0 a W 0 b B 0 b B 0 + a W 0 c W 0 d B 0 c W 0 d B 0 d B 0 + c W 0 ] .
F { u ( x ) } ( k ) = u ( x ) exp ( j 2 π λ f x k ) d x ,
[ a b c d ] = [ 0 λ f 1 λ f 0 ] .
E = [ W n B n ] = [ λ f B 0 W 0 λ f ] ,
N n = W n B n = λ f B 0 W 0 λ f = W 0 B 0 = N 0 .
F p { u ( x ) } ( x p ) = u ( x ) exp { j π λ q [ x 2 tan ( p π 2 ) 2 x x p sin ( p π 2 ) + x p 2 tan ( p π 2 ) ] } d x ,
[ a b c d ] = [ cos ( p π 2 ) λ q sin ( p π 2 ) sin ( p π 2 ) λ q cos ( p π 2 ) ] .
E = [ W n B n ] = [ B 0 λ q sin ( p π 2 ) + W 0 cos ( p π 2 ) B 0 cos ( p π 2 ) + W 0 sin ( p π 2 ) λ q ] ,
N n = W n B n = N 0 + ( 1 / 2 ) sin ( p π ) ( λ q B 0 2 + W 0 2 λ q ) .
FST z { u ( x ) } ( x z ) = u ( x ) exp [ j π λ z ( x 2 2 x x z + x z 2 ) ] d x ,
[ a b c d ] = [ 1 λ z 0 1 ] .
E = [ W n B n ] = [ λ z B 0 + W 0 B 0 ] .
N n = N 0 + λ z B 0 2 .
C M { u ( x ) } ( x ) = u ( x ) exp ( j π λ f x 2 ) δ ( x x ) d x = u ( x ) exp ( j π λ f x 2 ) ,
ψ { u ( x ) g ( x ) } ( x , k ) = W u ( x , k k ) W g ( x , k ) d k .
[ a b c d ] = [ 1 0 1 λ f 1 ] .
E = Max [ ( x 1 x 2 ) ( x 1 x 3 ) ( x 1 x 4 ) ( x 2 x 3 ) ( x 2 x 1 ) λ f + ( k 1 k 2 ) ( x 3 x 1 ) λ f + ( k 1 k 3 ) ( x 4 x 1 ) λ f + ( k 1 k 4 ) ( x 3 x 2 ) λ f + ( k 2 k 3 ) ]
[ ( x 2 x 4 ) ( x 3 x 4 ) ( x 4 x 2 ) λ f + ( k 2 k 4 ) ( x 4 x 3 ) λ f + ( k 3 k 4 ) ] ,
E = [ W n B n ] = [ W 0 B 0 + W 0 λ f ] .
N n = N 0 + W 0 2 λ f
M M { u ( x ) } ( x ) = 1 M u ( x ) δ ( x x M ) d x = 1 M u ( x M ) ,
δ ( x ) = lim y { y 1 2 exp [ j π ( x 2 y 1 4 ) ] } ,
[ a b c d ] = [ M 0 0 1 M ] .
E = [ W n B n ] = [ M W 0 B 0 M ] .
N n = ( M W 0 ) ( B 0 M ) = N 0 .
δ x M δ x ,
δ k δ k M .
f ( n δ x ) f ( n δ x ) exp [ j 2 π λ f ( n δ x ) 2 ] ,
[ 0 λ f 1 λ f 0 ] = [ λ f 0 0 1 λ f ] [ 0 1 1 0 ] .
[ 1 λ z 0 1 ] = [ 1 0 1 λ z 1 ] [ λ z 0 0 1 λ z ] [ 0 1 1 0 ] [ 1 0 1 λ z 1 ] = M 4 M 3 M 2 M 1 .
[ 1 λ z 0 1 ] = [ 0 1 1 0 ] [ 1 0 λ z 1 ] [ 0 1 1 0 ] = M 3 M 2 M 1 .
[ 1 λ z 0 1 ] = [ M 0 0 1 M ] [ 1 0 1 λ f 2 1 ] [ 1 λ z T 0 1 ] [ 1 0 1 λ f 1 1 ] = M 4 M 3 M 2 M 1 ,
[ cos ϕ λ q sin ϕ sin ϕ λ q cos ϕ ] = [ 1 0 1 λ q tan ϕ 1 ] [ λ q sin ϕ 0 0 1 λ q sin ϕ ] × [ 0 1 1 0 ] [ 1 0 1 λ q tan ϕ 1 ] = M 4 M 3 M 2 M 1 ,
[ cos ϕ sin ϕ sin ϕ cos ϕ ] = [ 1 0 T S 1 ] [ 0 1 1 0 ] [ 1 0 1 S 1 ] [ 0 1 1 0 ] [ 1 0 T S 1 ] = M 5 M 4 M 3 M 2 M 1 ,
[ cos ϕ λ q sin ϕ sin ϕ λ q cos ϕ ] = [ 1 0 1 λ f 1 ] [ 0 1 1 0 ] [ 1 0 λ z 1 ] × [ 0 1 1 0 ] [ 1 0 1 λ f 1 ] = M 5 M 4 M 3 M 2 M 1
[ cos ( p π 2 ) λ q sin ( p π 2 ) sin ( p π 2 ) λ q cos ( p π 2 ) ] = [ 0 λ q 1 λ q 0 ] × [ cos [ ( p + 1 ) π 2 ] λ q sin [ ( p + 1 ) π 2 ] sin [ ( p + 1 ) π 2 ] λ q cos [ ( p + 1 ) π 2 ] ] .
[ 1 λ z 0 1 ] = [ 1 cos ϕ 0 0 cos ϕ ] [ 1 0 tan ϕ λ q 1 ] × [ cos ϕ λ q sin ϕ sin ϕ λ q cos ϕ ] = M 3 M 2 M 1 ,
[ 1 λ z 0 1 ] [ 1 0 1 λ z c 1 ] = M 2 M 1 .
[ a b c d ] = [ 1 0 c a 1 ] [ a 0 0 1 a ] [ 1 b a 0 1 ] = M 3 M 2 M 1 .
[ a b c d ] = [ 1 0 1 λ f 1 ] [ m 0 0 1 m ] [ cos ϕ λ q sin ϕ sin ϕ λ q cos ϕ ] = M 3 M 2 M 1 ,

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