Abstract

Any general first-order optical system can be represented using ${\textbf{S}} \in Sp(4,\mathbb{R})$, where $Sp(4,\mathbb{R})$ is the symplectic group with real entries in four dimensions. We prove that any ${\textbf{S}} \in Sp(4,\mathbb{R})$ can be realized using not more than 18 thin lenses of arbitrary focal length and seven unit distance. New identities that realize ${\textbf{S}} = {S_1} \oplus {S_2}$, where ${S_1}, {S_2} \in Sp(2,\mathbb{R})$, are obtained. Also, it is proved any ${\textbf{S}}$ of the form ${S_1} \oplus {S_2}$ can be realized using a maximum of eight thin lenses of arbitrary focal length and three unit distance. Moreover, decompositions for examples such as differential magnifier, partial Fourier transform, and inverse partial Fourier transform are also provided. A “gadget” is proposed that can realize any ${\textbf{S}} \in Sp(4,\mathbb{R})$ using thin lens transformations—which can be realized through the use of eight spatial light modulators (SLMs) and seven unit distance. Experimental limitation imposed by SLMs while realizing thin lens transformations is also outlined. The justification for the choice of unit distance according to the availability of thin lenses in a lab is given too.

© 2020 Optical Society of America

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References

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2020 (1)

2018 (3)

T. Malhotra, R. Gutiérrez-Cuevas, J. Hassett, M. R. Dennis, A. N. Vamivakas, and M. A. Alonso, “Measuring geometric phase without interferometry,” Phys. Rev. Lett. 120, 233602 (2018).
[Crossref]

P. A. A. Yasir and J. S. Ivan, “Estimation of phases with dislocations in paraxial wave fields from intensity measurements,” Phys. Rev. A 97, 023817 (2018).
[Crossref]

X. Gu, M. Krenn, M. Erhard, and A. Zeilinger, “Gouy phase radial mode sorter for light: concepts and experiments,” Phys. Rev. Lett. 120, 103601 (2018).
[Crossref]

2017 (2)

Y. Zhou, M. Mirhosseini, D. Fu, J. Zhao, S. M. H. Rafsanjani, A. E. Willner, and R. W. Boyd, “Sorting photons by radial quantum number,” Phys. Rev. Lett. 119, 263602 (2017).
[Crossref]

P. A. A. Yasir and J. S. Ivan, “Realization of first-order optical systems using thin lenses of positive focal length,” J. Opt. Soc. Am. A 34, 2007–2012 (2017).
[Crossref]

2016 (1)

2014 (1)

2010 (1)

2009 (1)

O. Matoba, T. Nomura, E. Perez-Cabre, M. S. Millan, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97, 1128–1148 (2009).
[Crossref]

2003 (2)

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[Crossref]

H. Xu, “An SVD-like matrix decomposition and its applications,” Linear Algebra Appl. 368, 1–24 (2003).
[Crossref]

2000 (1)

1999 (1)

1995 (1)

Arvind, B. Dutta, N. Mukunda, and R. Simon, “The real symplectic groups in quantum mechanics and optics,” Pramana 45, 471–497 (1995).
[Crossref]

1994 (1)

R. Simon, N. Mukunda, and B. Dutta, “Quantum-noise matrix for multimode systems: U(n) invariance, squeezing, and normal forms,” Phys. Rev. A 49, 1567–1583 (1994).
[Crossref]

1993 (1)

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

1990 (1)

R. Simon and N. Mukunda, “Minimal three component SU(2) gadget for polarization optics,” Phys. Lett. A 143, 165–169 (1990).
[Crossref]

1988 (1)

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 3028–3038 (1988).
[Crossref]

1985 (1)

E. C. G. Sudarshan, N. Mukunda, and R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[Crossref]

1983 (1)

1982 (1)

1981 (3)

1980 (3)

M. Nazarathy and J. Shamir, “Fourier optics described by operator algebra,” J. Opt. Soc. Am. 70, 150–159 (1980).
[Crossref]

H. H. Arsenault, “A matrix representation for non-symmetrical optical systems,” J. Opt. 11, 87–91 (1980).
[Crossref]

H. H. Arsenault, “Generalization of the principal plane concept in matrix optics,” Am. J. Phys. 48, 397–399 (1980).
[Crossref]

1971 (1)

M. Moshinsky and C. Quesne, “Linear canonical transformations and their unitary representations,” J. Math. Phys. 12, 1772–1780 (1971).
[Crossref]

1970 (1)

1966 (1)

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

Alonso, M. A.

T. Malhotra, R. Gutiérrez-Cuevas, J. Hassett, M. R. Dennis, A. N. Vamivakas, and M. A. Alonso, “Measuring geometric phase without interferometry,” Phys. Rev. Lett. 120, 233602 (2018).
[Crossref]

Arsenault, H. H.

H. H. Arsenault and B. Macukow, “Factorization of the transfer matrix for symmetrical optical systems,” J. Opt. Soc. Am. 73, 1350–1359 (1983).
[Crossref]

H. H. Arsenault, “A matrix representation for non-symmetrical optical systems,” J. Opt. 11, 87–91 (1980).
[Crossref]

H. H. Arsenault, “Generalization of the principal plane concept in matrix optics,” Am. J. Phys. 48, 397–399 (1980).
[Crossref]

Arvind,

Arvind, B. Dutta, N. Mukunda, and R. Simon, “The real symplectic groups in quantum mechanics and optics,” Pramana 45, 471–497 (1995).
[Crossref]

Asokan, S.

Bacry, H.

H. Bacry and M. Cadilhac, “Metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
[Crossref]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

Biedenharn, L. C.

L. C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Physics: Theory and Application (Addison-Wesley, 1981).

Boyd, R. W.

Y. Zhou, M. Mirhosseini, D. Fu, J. Zhao, S. M. H. Rafsanjani, A. E. Willner, and R. W. Boyd, “Sorting photons by radial quantum number,” Phys. Rev. Lett. 119, 263602 (2017).
[Crossref]

Cadilhac, M.

H. Bacry and M. Cadilhac, “Metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
[Crossref]

Casperson, L. W.

Collins, S. A.

Courtial, J.

Crawford, P. R.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[Crossref]

Dennis, M. R.

T. Malhotra, R. Gutiérrez-Cuevas, J. Hassett, M. R. Dennis, A. N. Vamivakas, and M. A. Alonso, “Measuring geometric phase without interferometry,” Phys. Rev. Lett. 120, 233602 (2018).
[Crossref]

Dutta, B.

Arvind, B. Dutta, N. Mukunda, and R. Simon, “The real symplectic groups in quantum mechanics and optics,” Pramana 45, 471–497 (1995).
[Crossref]

R. Simon, N. Mukunda, and B. Dutta, “Quantum-noise matrix for multimode systems: U(n) invariance, squeezing, and normal forms,” Phys. Rev. A 49, 1567–1583 (1994).
[Crossref]

Erhard, M.

X. Gu, M. Krenn, M. Erhard, and A. Zeilinger, “Gouy phase radial mode sorter for light: concepts and experiments,” Phys. Rev. Lett. 120, 103601 (2018).
[Crossref]

Forbes, A.

A. Forbes, Laser Beam Propagation (CRC Press, 2014).

Fu, D.

Y. Zhou, M. Mirhosseini, D. Fu, J. Zhao, S. M. H. Rafsanjani, A. E. Willner, and R. W. Boyd, “Sorting photons by radial quantum number,” Phys. Rev. Lett. 119, 263602 (2017).
[Crossref]

Galvez, E. J.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[Crossref]

Goldstein, H.

H. Goldstein, Classical Mechanics (Addison-Wesley, 1980).

Gu, X.

X. Gu, M. Krenn, M. Erhard, and A. Zeilinger, “Gouy phase radial mode sorter for light: concepts and experiments,” Phys. Rev. Lett. 120, 103601 (2018).
[Crossref]

Gutiérrez-Cuevas, R.

T. Malhotra, R. Gutiérrez-Cuevas, J. Hassett, M. R. Dennis, A. N. Vamivakas, and M. A. Alonso, “Measuring geometric phase without interferometry,” Phys. Rev. Lett. 120, 233602 (2018).
[Crossref]

Habraken, S. J. M.

Haglin, P. J.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[Crossref]

Hamermesh, M.

M. Hamermesh, Group Theory and Its Application to Physical Problems (Courier Corporation, 2012).

Hassett, J.

T. Malhotra, R. Gutiérrez-Cuevas, J. Hassett, M. R. Dennis, A. N. Vamivakas, and M. A. Alonso, “Measuring geometric phase without interferometry,” Phys. Rev. Lett. 120, 233602 (2018).
[Crossref]

Hecht, E.

E. Hecht, Optics (Pearson, 2017).

Horn, R. A.

R. A. Horn and C. R. Johnson, “Norms for vectors and matrices,” in Matrix Analysis (Cambridge University, 2013), pp. 313–386.

Ivan, J. S.

Javidi, B.

O. Matoba, T. Nomura, E. Perez-Cabre, M. S. Millan, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97, 1128–1148 (2009).
[Crossref]

Johnson, C. R.

R. A. Horn and C. R. Johnson, “Norms for vectors and matrices,” in Matrix Analysis (Cambridge University, 2013), pp. 313–386.

Kogelnik, H.

Krenn, M.

X. Gu, M. Krenn, M. Erhard, and A. Zeilinger, “Gouy phase radial mode sorter for light: concepts and experiments,” Phys. Rev. Lett. 120, 103601 (2018).
[Crossref]

Kutay, M. A.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform: With Applications in Optics and Signal Processing (Wiley, 2001).

H. M. Ozaktas, M. A. Kutay, and D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” in Advances in Imaging and Electron Physics (1999), Vol. 106, pp. 239–291.

Li, T.

Li, Y.-M.

Louck, J. D.

L. C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Physics: Theory and Application (Addison-Wesley, 1981).

Macukow, B.

Malhotra, T.

T. Malhotra, R. Gutiérrez-Cuevas, J. Hassett, M. R. Dennis, A. N. Vamivakas, and M. A. Alonso, “Measuring geometric phase without interferometry,” Phys. Rev. Lett. 120, 233602 (2018).
[Crossref]

Matoba, O.

O. Matoba, T. Nomura, E. Perez-Cabre, M. S. Millan, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97, 1128–1148 (2009).
[Crossref]

Mendlovic, D.

H. M. Ozaktas and D. Mendlovic, “Fractional Fourier transforms and their optical implementation. II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
[Crossref]

H. M. Ozaktas, M. A. Kutay, and D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” in Advances in Imaging and Electron Physics (1999), Vol. 106, pp. 239–291.

Millan, M. S.

O. Matoba, T. Nomura, E. Perez-Cabre, M. S. Millan, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97, 1128–1148 (2009).
[Crossref]

Mirhosseini, M.

Y. Zhou, M. Mirhosseini, D. Fu, J. Zhao, S. M. H. Rafsanjani, A. E. Willner, and R. W. Boyd, “Sorting photons by radial quantum number,” Phys. Rev. Lett. 119, 263602 (2017).
[Crossref]

Moshinsky, M.

M. Moshinsky and C. Quesne, “Linear canonical transformations and their unitary representations,” J. Math. Phys. 12, 1772–1780 (1971).
[Crossref]

Mukunda, N.

Arvind, B. Dutta, N. Mukunda, and R. Simon, “The real symplectic groups in quantum mechanics and optics,” Pramana 45, 471–497 (1995).
[Crossref]

R. Simon, N. Mukunda, and B. Dutta, “Quantum-noise matrix for multimode systems: U(n) invariance, squeezing, and normal forms,” Phys. Rev. A 49, 1567–1583 (1994).
[Crossref]

R. Simon and N. Mukunda, “Minimal three component SU(2) gadget for polarization optics,” Phys. Lett. A 143, 165–169 (1990).
[Crossref]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 3028–3038 (1988).
[Crossref]

E. C. G. Sudarshan, N. Mukunda, and R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[Crossref]

Nazarathy, M.

Nienhuis, G.

Nomura, T.

O. Matoba, T. Nomura, E. Perez-Cabre, M. S. Millan, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97, 1128–1148 (2009).
[Crossref]

Ozaktas, H. M.

H. M. Ozaktas and D. Mendlovic, “Fractional Fourier transforms and their optical implementation. II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
[Crossref]

H. M. Ozaktas, M. A. Kutay, and D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” in Advances in Imaging and Electron Physics (1999), Vol. 106, pp. 239–291.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform: With Applications in Optics and Signal Processing (Wiley, 2001).

Padgett, M. J.

Perez-Cabre, E.

O. Matoba, T. Nomura, E. Perez-Cabre, M. S. Millan, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97, 1128–1148 (2009).
[Crossref]

Pysher, M. J.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[Crossref]

Quesne, C.

M. Moshinsky and C. Quesne, “Linear canonical transformations and their unitary representations,” J. Math. Phys. 12, 1772–1780 (1971).
[Crossref]

Rafsanjani, S. M. H.

Y. Zhou, M. Mirhosseini, D. Fu, J. Zhao, S. M. H. Rafsanjani, A. E. Willner, and R. W. Boyd, “Sorting photons by radial quantum number,” Phys. Rev. Lett. 119, 263602 (2017).
[Crossref]

Sakurai, J. J.

J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, 1994).

Shamir, J.

Simon, R.

R. Simon and K. B. Wolf, “Structure of the set of paraxial optical systems,” J. Opt. Soc. Am. A 17, 342–355 (2000).
[Crossref]

Arvind, B. Dutta, N. Mukunda, and R. Simon, “The real symplectic groups in quantum mechanics and optics,” Pramana 45, 471–497 (1995).
[Crossref]

R. Simon, N. Mukunda, and B. Dutta, “Quantum-noise matrix for multimode systems: U(n) invariance, squeezing, and normal forms,” Phys. Rev. A 49, 1567–1583 (1994).
[Crossref]

R. Simon and N. Mukunda, “Minimal three component SU(2) gadget for polarization optics,” Phys. Lett. A 143, 165–169 (1990).
[Crossref]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 3028–3038 (1988).
[Crossref]

E. C. G. Sudarshan, N. Mukunda, and R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[Crossref]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

Stoler, D.

Sudarshan, E. C. G.

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 3028–3038 (1988).
[Crossref]

E. C. G. Sudarshan, N. Mukunda, and R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[Crossref]

Sztul, H. I.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[Crossref]

Vamivakas, A. N.

T. Malhotra, R. Gutiérrez-Cuevas, J. Hassett, M. R. Dennis, A. N. Vamivakas, and M. A. Alonso, “Measuring geometric phase without interferometry,” Phys. Rev. Lett. 120, 233602 (2018).
[Crossref]

Wang, R.

Wei, D.

Williams, R. E.

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[Crossref]

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Figures (1)

Fig. 1.
Fig. 1. Planes ${P_1}$, ${P_2}$, $\ldots$, ${P_8}$ denote eight transverse planes that are one unit distance apart. While ${P_1}$ is the input plane, ${P_8}$ is the output plane. Thin lenses, corresponding to the realization of ${{\textbf{J}}_{1a}}(\pi + 2{\beta _2})$ [see Eqs. (43) and (44)], are placed in planes ${P_1}$, ${P_2}$, and ${P_3}$. Thin lenses realizing ${\textbf{E}}^\prime $ [see Eqs. (44) and (46)] are available in ${P_3}$, ${P_4}$, ${P_5}$, and ${P_6}$. The red lines at ${P_3}$ and ${P_6}$ denote that ${\textbf{E}}^\prime $ was obtained by rotating the optical setup realizing ${\textbf{E}}$ [see Eq. (46)] through an angle ${\gamma _2}$ clockwise. Thin lenses corresponding to the realization ${\textbf{J}}{^\prime _{1a}}(\pi - 2{\beta _1})$ [see Eqs. (44) and (45)] are placed in ${P_6}$, ${P_7}$, and ${P_8}$. The blue lines at ${P_6}$ and ${P_8}$ denote that ${\textbf{J}}{^\prime _{1a}}(\pi - 2{\beta _1})$ was obtained by rotating the optical setup realizing ${{\textbf{J}}_{1a}}(\pi - 2{\beta _1})$ [see Eq. (45)] through an angle ${\gamma _1} - {\gamma _2}$ anticlockwise. Finally, since the remaining matrix ${{\textbf{J}}_3}({\gamma _1} - {\gamma _2})$ [see Eqs. (24) and (44)] rotates the phase space column vector ${[x,y]^T}$ through an angle ${\gamma _1} - {\gamma _2}$ anticlockwise about the $z$ axis, we need to orient the output plane of the device measuring the light field intensity through the same amount clockwise (see the green line in ${P_8}$). Here the angles are not to scale.

Tables (1)

Tables Icon

Table 1. List of All Possible Symplectic Matrices of the Form S 1 S 2 [See Eq. (28)] That Can Be Realized Using Any One of I 1 , I 2 , I 3 , and I 4 a

Equations (66)

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ψ ( x ; z ) U ψ ( x ; z ) ,
U ( S ) ξ ^ U ( S ) = S 1 ξ ^ ,
S Σ S T = Σ ,
Σ = [ 0 1 1 0 ] [ 0 1 1 0 ] ,
V = [ 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 ] .
U ( S ~ ) χ ^ U ( S ~ ) = S ~ 1 χ ^ ,
S ~ Ω S ~ T = Ω ,
Ω = [ 0 𝟙 2 𝟙 2 0 ] .
S = V S ~ V T .
exp ( i π / 2 ) λ | det s ~ 2 | exp [ i κ ( 1 2 x 1 T s ~ 2 1 s ~ 4 x 1 x 1 T s ~ 2 1 x + 1 2 x T s ~ 1 s ~ 2 1 x ) ] ,
S ~ = [ 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 ] [ s ~ 1 s ~ 2 s ~ 3 s ~ 4 ] ,
ψ 2 ( x 1 ) = 1 det s ~ 1 exp ( i κ 2 x 1 T s ~ 3 s ~ 1 1 x 1 ) ψ 1 ( s ~ 1 1 x 1 ) .
S = [ s ~ 11 s ~ 13 s ~ 12 s ~ 14 s ~ 31 s ~ 33 s ~ 32 s ~ 34 s ~ 21 s ~ 23 s ~ 22 s ~ 24 s ~ 41 s ~ 43 s ~ 42 s ~ 44 ] .
F d = exp [ i d κ 2 ( p ^ x 2 + p ^ y 2 ) ] ,
F ( d ) = F ( d ) F ( d ) , where F ( d ) = [ 1 d 0 1 ] .
L f = exp [ i κ 2 f ( x ^ 2 + y ^ 2 ) ] L f x L f y ,
L ( f ) = L ( f ) L ( f ) , where L ( f ) = [ 1 0 1 / f 1 ] .
L x ( f ) = L ( f ) 𝟙 2 and L y ( f ) = 𝟙 2 L ( f ) .
S = R 1 MR 2 T ,
R = J 0 ( α ) J 1 ( β ) J 3 ( γ ) J 1 ( δ ) ,
J 0 ( α ) = R α R α ,
J 1 ( α ) = R α R α ,
J 2 ( α ) = [ ( cos α ) 𝟙 2 ( sin α ) σ 1 ( sin α ) σ 1 ( cos α ) 𝟙 2 ] ,
J 3 ( α ) = [ ( cos α ) 𝟙 2 ( sin α ) 𝟙 2 ( sin α ) 𝟙 2 ( cos α ) 𝟙 2 ] ,
R α = [ cos α sin α sin α cos α ] ,
σ d = [ 0 d 1 / d 0 ] .
M = diag { a , 1 / a , b , 1 / b } M a M b ,
S = S 1 S 2 [ a 1 b 1 c 1 d 1 ] [ a 2 b 2 c 2 d 2 ]
S 1 S 2 = L x ( b 1 1 + d 1 b 1 ) L y ( b 2 1 + d 2 b 2 ) F ( 1 ) × L x ( 1 2 b 1 ) L y ( 1 2 b 2 ) F ( 1 ) L x ( b 1 1 + a 1 b 1 ) × L y ( b 2 1 + a 2 b 2 ) .
J 1 ( π / 2 ) = L x ( 1 2 ) F ( 1 ) L x ( 1 3 ) L y ( 1 ) F ( 1 ) L x ( 1 2 ) .
S 1 S 2 = L x ( a 1 1 c 1 + a 1 ) L y ( a 2 1 c 2 + a 2 ) F ( 1 ) × L x ( 1 2 + a 1 ) L y ( 1 2 + a 2 ) F ( 1 ) L x ( a 1 1 + b 1 + 2 a 1 ) × L y ( a 2 1 + b 2 + 2 a 2 ) F ( 1 ) L ( 1 ) .
M = L x ( a 1 + a ) L y ( b 1 + b ) F ( 1 ) L x ( 1 2 + a ) L y ( 1 2 + b ) × F ( 1 ) L x ( a 1 + 2 a ) L y ( b 1 + 2 b ) F ( 1 ) L ( 1 ) .
S 1 S 2 = L x ( a 1 1 c 1 + a 1 ) L y ( b 2 1 d 2 + 2 b 2 ) F ( 1 ) × L x ( 1 2 + a 1 ) L y ( 1 3 ) F ( 1 ) L x ( a 1 1 + b 1 + 2 a 1 ) × L y ( 1 b 2 + 3 ) F ( 1 ) L x ( 1 ) L y ( b 2 1 a 2 + b 2 ) .
P y ( d ) = L x ( 1 2 ) L y ( d 1 + 2 d ) F ( 1 ) L ( 1 3 ) F ( 1 ) L x ( 1 3 ) × L y ( 1 d + 3 ) F ( 1 ) L x ( 1 ) L y ( d 1 + d ) ,
P y 1 ( d ) = L x ( 1 2 ) L y ( d 2 d 1 ) F ( 1 ) L ( 1 3 ) F ( 1 ) L x ( 1 3 ) × L y ( 1 3 d ) F ( 1 ) L x ( 1 ) L y ( d d 1 ) .
S 1 S 2 = L x ( b 1 1 d 1 + 2 b 1 ) L y ( a 2 1 c 2 + a 2 ) F ( 1 ) × L x ( 1 3 ) L y ( 1 2 + a 2 ) F ( 1 ) L x ( 1 b 1 + 3 ) × L y ( a 2 1 + b 2 + 2 a 2 ) F ( 1 ) L x ( b 1 1 a 1 + b 1 ) L y ( 1 ) .
S = [ J 1 ( β 1 ) J 3 ( γ 1 ) J 1 ( δ 1 ) J 0 ( α 1 ) ] M [ J 0 ( α 2 ) J 1 ( δ 2 ) × J 3 ( γ 2 ) J 1 ( β 2 ) ] = J 3 ( γ 1 γ 2 ) J 1 ( β 1 ) D J 1 ( β 2 ) ,
J 1 ( β 1 ) = J 3 ( γ 2 γ 1 ) J 1 ( β 1 ) J 3 ( γ 1 γ 2 ) ,
D = J 3 ( γ 2 ) D J 3 ( γ 2 ) ,
D = J 1 ( δ 1 ) J 0 ( α 1 ) M J 0 ( α 2 ) J 1 ( δ 2 ) .
J 1 ( β ) = L x ( sin β 1 + cos β + sin β ) L y ( sin β 1 + cos β sin β ) F ( 1 ) × L x ( 1 2 + sin β ) L y ( 1 2 sin β ) F ( 1 ) × L x ( sin β 1 + cos β + sin β ) L y ( sin β 1 + cos β sin β ) .
J 1 ( β ) = J 0 ( π + β ) [ 𝟙 2 0 0 R π 2 β ] J 0 ( π + β ) J 1 a ( π 2 β ) ,
J 1 a ( π 2 β ) = L x ( 1 2 ) L y ( sin 2 β 1 cos 2 β + sin 2 β ) F ( 1 ) × L x ( 1 2 ) L y ( 1 2 + sin 2 β ) F ( 1 ) × L y ( sin 2 β 1 cos 2 β + sin 2 β ) .
S = J 3 ( γ 1 γ 2 ) J 1 a ( π 2 β 1 ) E J 1 a ( π + 2 β 2 ) ,
J 1 a ( π 2 β 1 ) = J 3 ( γ 2 γ 1 ) J 1 a ( π 2 β 1 ) J 3 ( γ 1 γ 2 ) ,
E = J 3 ( γ 2 ) E J 3 ( γ 2 ) ,
E = J 0 ( π + β 1 ) D J 0 ( π β 2 ) .
R = J 0 ( α ) J 3 ( β ) J 1 ( γ ) J 3 ( δ ) = J 3 ( β + δ ) × J 3 ( δ ) J 0 ( α ) J 1 ( γ ) J 3 ( δ ) .
J 2 ( α ) = J 3 ( π 4 ) J 1 ( α ) J 3 ( π 4 ) .
a 1 + a f 2 .
MS M 1 = [ M a s 1 M a 1 M a s 2 M a 1 M a s 3 M a 1 M a s 4 M a 1 ] .
M a s 1 M a 1 = [ s 11 a 2 s 12 s 21 / a 2 s 22 ] ,
[ α β γ δ ] = X F ( 1 + δ β + 1 ) L ( 1 ) F ( 2 + β ) L ( 1 ) × F ( 1 + α β + 1 ) Y ,
X = F ( 1 ) L ( 1 ) F ( 1 ) ,
Y = F ( 2 ) L ( 1 ) F ( 3 ) L ( 1 ) F ( 2 ) ,
X F ( d ) Y = L ( 1 / d ) , and X L ( d ) Y = F ( 1 / d )
[ α β γ δ ] = L ( β 1 + δ + β ) F ( 1 ) L ( 1 2 + β ) F ( 1 ) × L ( β 1 + α + β ) .
[ a b c d ] = L ( b 1 + d b ) × F ( 1 ) L ( 1 2 b ) F ( 1 ) L ( b 1 + a b ) .
X [ α β γ δ ] Y = X [ F ( 1 + β δ + 1 ) L ( 1 ) F ( 2 + δ ) L ( 1 ) × F ( 2 + 1 + γ δ ) L ( 1 ) F ( 1 ) ] Y .
[ δ γ β α ] = L ( δ 1 + β + δ ) F ( 1 ) L ( 1 2 + δ ) F ( 1 ) × L ( δ 1 + γ + 2 δ ) F ( 1 ) L ( 1 ) .
[ a b c d ] = L ( a 1 c + a ) F ( 1 ) L ( 1 2 + a ) F ( 1 ) × L ( a 1 + b + 2 a ) F ( 1 ) L ( 1 ) .
[ a b c d ] = L ( b 1 d + 2 b ) F ( 1 ) L ( 1 3 ) F ( 1 ) L ( 1 b + 3 ) × F ( 1 ) L ( b 1 a + b ) .
[ Tr ( S T S ) ] 2 Tr ( Q 1 T Q 1 ) 2 Tr ( Q 2 T Q 2 ) 2 ,
R α = L ( 1 1 + cot α 2 ) F ( 1 ) L ( 1 2 + sin α ) F ( 1 ) × L ( 1 1 + cot α 2 ) .
R α = L ( 1 1 + cot ( π 4 + α 2 ) ) F ( 1 ) L ( 1 2 + cos α ) F ( 1 ) × L ( 1 2 + cot ( π 4 + α 2 ) ) F ( 1 ) L ( 1 ) .
L ( f ) = L ( f 1 + 2 f ) F ( 1 ) L ( 1 3 ) F ( 1 ) L ( 1 3 ) F ( 1 ) L ( 1 ) .

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