Abstract

The non-Hermitian operators of the ideal nonorthogonal multilayer optical polarizers are spectrally analyzed in the framework of skew-angular biorthonormal vector bases. It is shown that these polarizers correspond to skew projectors and their operators are generated by skew projectors, exactly as the canonical ideal polarizers correspond to Hermitian projectors. Thus the common feature of all the polarizers (Hermitian and non-Hermitian) is that their “nuclei” are (orthogonal or skew) projectors—the generating projectors. It is shown that if these nonorthogonal polarizers are looked upon as variable devices, two kinds of degeneracy may occur for suitable values of the inner parameter of the device: The corresponding operators may become normal (more precisely, Hermitian) or, on the contrary, very pathological—defective and singular. In the first case their eigenvectors and biorthogonal conjugate eigenvectors collapse into a unique pair of eigenvectors; in the second case their eigenvectors (as well as their biorthogonal conjugates) collapse into a single vector.

© 2006 Optical Society of America

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  1. M. V. Berry and S. Klein, "Geometric phases from stacks of crystal plates," J. Mod. Opt. 43, 165-180 (1996).
    [CrossRef]
  2. W. A. Shurcliff, Polarized Light (Harvard U. Press, 1962).
  3. B. Higman, Applied Group-Theoretic and Matrix Methods (Clarendon, 1955).
  4. C. Lanczos, Applied Analysis (Pitman, 1957).
  5. P. Lancaster and M. Tismenetsky, The Theory of Matrices (Academic, 1985).
  6. M. C. Pease, Methods of Matrix Algebra (Academic, 1965).
  7. K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, "Atom waves in crystals of light," Phys. Rev. Lett. 77, 4980-4983 (1996).
    [CrossRef] [PubMed]
  8. M. V. Berry and M. R. Dennis, "The optical singularities of birefringent dichroic chiral crystal," Proc. R. Soc. London, Ser. A 459, 1261-1292 (2003).
    [CrossRef]
  9. M. V. Berry and D. H. J. O'Dell, "Diffraction by volume gratings with imaginary potentials," J. Phys. A 31, 2093-2101 (1998).
    [CrossRef]
  10. W. D. Heiss, M. Müller, and I. Rotter, "Collectivity, phase transitions, and exceptional points in open quantum systems," Phys. Rev. E 58, 2894-2901 (1998).
    [CrossRef]
  11. T. Stehmann, W. D. Heiss, and F. G. Scholtz, "Observation of exceptional points in electronic circuits," J. Phys. A 37, 7813-7819 (2004).
    [CrossRef]
  12. M. Philipp, P. von Brentano, G. Pascovici, and A. Richter, "Frequency and width crossing of two interacting resonances in a microwave cavity," Phys. Rev. E 62, 1922-1926 (2000).
    [CrossRef]
  13. W. D. Heiss, "Repulsion of resonant states and exceptional points," Phys. Rev. E 61, 929-932 (2000).
    [CrossRef]
  14. F. Keck, H. J. Korsch, and S. Mossmann, "Unfolding a diabolic point: a generalized crossing scenario," J. Phys. A 36, 2125-2137 (2003).
    [CrossRef]
  15. M. V. Berry, "Physics of nonhermitian degeneracies," Czech. J. Phys. 55, 1039-1046 (2004).
  16. W. D. Heiss, "Exceptional points—their universal occurrence and their physical significance," Czech. J. Phys. 55, 1091-1099 (2004).
  17. A. P. Seyranian, O. N. Kirillov, and A. A. Mailybaev, "Coupling of eigenvalues of complex matrices at diabolic and exceptional points," J. Phys. A 38, 1723-1740 (2005).
    [CrossRef]
  18. T. Kato, Perturbation Theory for Linear Operators (Springer, 1966).
  19. A. Messiah, Mécanique Quantique (Dunod, 1964).
  20. T. Tudor, "Operational form of the theory of polarization optical devices: I. Spectral theory of the basic devices," Optik (Stuttgart) 114, 539-547 (2003).
  21. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1996).
  22. T. Tudor, "Generalized observables in polarization optics," J. Phys. A 36, 9567-9590 (2003).
    [CrossRef]
  23. M. V. Berry, "Mode degeneracies and the Petermann excess-noise factor for unstable lasers," J. Mod. Opt. 50, 63-81 (2003).
  24. E. B. Davies and J. T. Lewis, "An operatorial approach to quantum probability," Commun. Math. Phys. 17, 239-260 (1970).
    [CrossRef]
  25. M. V. Berry and M. R. Dennis, "Black polarization sandwiches as square roots of zero," J. Opt. A, Pure Appl. Opt. 6, S24-S25 (2004).
    [CrossRef]
  26. C. Whitney, "Pauli-algebraic operators in polarization optics," J. Opt. Soc. Am. 61, 1207-1213 (1971).
    [CrossRef]
  27. R. Bhandari, "Halfwave retarder for all polarization states," Appl. Opt. 36, 2799-2801 (1997).
    [CrossRef] [PubMed]

2005 (1)

A. P. Seyranian, O. N. Kirillov, and A. A. Mailybaev, "Coupling of eigenvalues of complex matrices at diabolic and exceptional points," J. Phys. A 38, 1723-1740 (2005).
[CrossRef]

2004 (4)

M. V. Berry and M. R. Dennis, "Black polarization sandwiches as square roots of zero," J. Opt. A, Pure Appl. Opt. 6, S24-S25 (2004).
[CrossRef]

T. Stehmann, W. D. Heiss, and F. G. Scholtz, "Observation of exceptional points in electronic circuits," J. Phys. A 37, 7813-7819 (2004).
[CrossRef]

M. V. Berry, "Physics of nonhermitian degeneracies," Czech. J. Phys. 55, 1039-1046 (2004).

W. D. Heiss, "Exceptional points—their universal occurrence and their physical significance," Czech. J. Phys. 55, 1091-1099 (2004).

2003 (5)

M. V. Berry and M. R. Dennis, "The optical singularities of birefringent dichroic chiral crystal," Proc. R. Soc. London, Ser. A 459, 1261-1292 (2003).
[CrossRef]

F. Keck, H. J. Korsch, and S. Mossmann, "Unfolding a diabolic point: a generalized crossing scenario," J. Phys. A 36, 2125-2137 (2003).
[CrossRef]

T. Tudor, "Operational form of the theory of polarization optical devices: I. Spectral theory of the basic devices," Optik (Stuttgart) 114, 539-547 (2003).

T. Tudor, "Generalized observables in polarization optics," J. Phys. A 36, 9567-9590 (2003).
[CrossRef]

M. V. Berry, "Mode degeneracies and the Petermann excess-noise factor for unstable lasers," J. Mod. Opt. 50, 63-81 (2003).

2000 (2)

M. Philipp, P. von Brentano, G. Pascovici, and A. Richter, "Frequency and width crossing of two interacting resonances in a microwave cavity," Phys. Rev. E 62, 1922-1926 (2000).
[CrossRef]

W. D. Heiss, "Repulsion of resonant states and exceptional points," Phys. Rev. E 61, 929-932 (2000).
[CrossRef]

1998 (2)

M. V. Berry and D. H. J. O'Dell, "Diffraction by volume gratings with imaginary potentials," J. Phys. A 31, 2093-2101 (1998).
[CrossRef]

W. D. Heiss, M. Müller, and I. Rotter, "Collectivity, phase transitions, and exceptional points in open quantum systems," Phys. Rev. E 58, 2894-2901 (1998).
[CrossRef]

1997 (1)

1996 (2)

M. V. Berry and S. Klein, "Geometric phases from stacks of crystal plates," J. Mod. Opt. 43, 165-180 (1996).
[CrossRef]

K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, "Atom waves in crystals of light," Phys. Rev. Lett. 77, 4980-4983 (1996).
[CrossRef] [PubMed]

1971 (1)

1970 (1)

E. B. Davies and J. T. Lewis, "An operatorial approach to quantum probability," Commun. Math. Phys. 17, 239-260 (1970).
[CrossRef]

Abfalterer, R.

K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, "Atom waves in crystals of light," Phys. Rev. Lett. 77, 4980-4983 (1996).
[CrossRef] [PubMed]

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1996).

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1996).

Bernet, S.

K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, "Atom waves in crystals of light," Phys. Rev. Lett. 77, 4980-4983 (1996).
[CrossRef] [PubMed]

Berry, M. V.

M. V. Berry, "Physics of nonhermitian degeneracies," Czech. J. Phys. 55, 1039-1046 (2004).

M. V. Berry and M. R. Dennis, "Black polarization sandwiches as square roots of zero," J. Opt. A, Pure Appl. Opt. 6, S24-S25 (2004).
[CrossRef]

M. V. Berry, "Mode degeneracies and the Petermann excess-noise factor for unstable lasers," J. Mod. Opt. 50, 63-81 (2003).

M. V. Berry and M. R. Dennis, "The optical singularities of birefringent dichroic chiral crystal," Proc. R. Soc. London, Ser. A 459, 1261-1292 (2003).
[CrossRef]

M. V. Berry and D. H. J. O'Dell, "Diffraction by volume gratings with imaginary potentials," J. Phys. A 31, 2093-2101 (1998).
[CrossRef]

M. V. Berry and S. Klein, "Geometric phases from stacks of crystal plates," J. Mod. Opt. 43, 165-180 (1996).
[CrossRef]

Bhandari, R.

Davies, E. B.

E. B. Davies and J. T. Lewis, "An operatorial approach to quantum probability," Commun. Math. Phys. 17, 239-260 (1970).
[CrossRef]

Dennis, M. R.

M. V. Berry and M. R. Dennis, "Black polarization sandwiches as square roots of zero," J. Opt. A, Pure Appl. Opt. 6, S24-S25 (2004).
[CrossRef]

M. V. Berry and M. R. Dennis, "The optical singularities of birefringent dichroic chiral crystal," Proc. R. Soc. London, Ser. A 459, 1261-1292 (2003).
[CrossRef]

Heiss, W. D.

W. D. Heiss, "Exceptional points—their universal occurrence and their physical significance," Czech. J. Phys. 55, 1091-1099 (2004).

T. Stehmann, W. D. Heiss, and F. G. Scholtz, "Observation of exceptional points in electronic circuits," J. Phys. A 37, 7813-7819 (2004).
[CrossRef]

W. D. Heiss, "Repulsion of resonant states and exceptional points," Phys. Rev. E 61, 929-932 (2000).
[CrossRef]

W. D. Heiss, M. Müller, and I. Rotter, "Collectivity, phase transitions, and exceptional points in open quantum systems," Phys. Rev. E 58, 2894-2901 (1998).
[CrossRef]

Higman, B.

B. Higman, Applied Group-Theoretic and Matrix Methods (Clarendon, 1955).

Kato, T.

T. Kato, Perturbation Theory for Linear Operators (Springer, 1966).

Keck, F.

F. Keck, H. J. Korsch, and S. Mossmann, "Unfolding a diabolic point: a generalized crossing scenario," J. Phys. A 36, 2125-2137 (2003).
[CrossRef]

Kirillov, O. N.

A. P. Seyranian, O. N. Kirillov, and A. A. Mailybaev, "Coupling of eigenvalues of complex matrices at diabolic and exceptional points," J. Phys. A 38, 1723-1740 (2005).
[CrossRef]

Klein, S.

M. V. Berry and S. Klein, "Geometric phases from stacks of crystal plates," J. Mod. Opt. 43, 165-180 (1996).
[CrossRef]

Korsch, H. J.

F. Keck, H. J. Korsch, and S. Mossmann, "Unfolding a diabolic point: a generalized crossing scenario," J. Phys. A 36, 2125-2137 (2003).
[CrossRef]

Lancaster, P.

P. Lancaster and M. Tismenetsky, The Theory of Matrices (Academic, 1985).

Lanczos, C.

C. Lanczos, Applied Analysis (Pitman, 1957).

Lewis, J. T.

E. B. Davies and J. T. Lewis, "An operatorial approach to quantum probability," Commun. Math. Phys. 17, 239-260 (1970).
[CrossRef]

Mailybaev, A. A.

A. P. Seyranian, O. N. Kirillov, and A. A. Mailybaev, "Coupling of eigenvalues of complex matrices at diabolic and exceptional points," J. Phys. A 38, 1723-1740 (2005).
[CrossRef]

Messiah, A.

A. Messiah, Mécanique Quantique (Dunod, 1964).

Mossmann, S.

F. Keck, H. J. Korsch, and S. Mossmann, "Unfolding a diabolic point: a generalized crossing scenario," J. Phys. A 36, 2125-2137 (2003).
[CrossRef]

Müller, M.

W. D. Heiss, M. Müller, and I. Rotter, "Collectivity, phase transitions, and exceptional points in open quantum systems," Phys. Rev. E 58, 2894-2901 (1998).
[CrossRef]

Oberthaler, K.

K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, "Atom waves in crystals of light," Phys. Rev. Lett. 77, 4980-4983 (1996).
[CrossRef] [PubMed]

O'Dell, D. H. J.

M. V. Berry and D. H. J. O'Dell, "Diffraction by volume gratings with imaginary potentials," J. Phys. A 31, 2093-2101 (1998).
[CrossRef]

Pascovici, G.

M. Philipp, P. von Brentano, G. Pascovici, and A. Richter, "Frequency and width crossing of two interacting resonances in a microwave cavity," Phys. Rev. E 62, 1922-1926 (2000).
[CrossRef]

Pease, M. C.

M. C. Pease, Methods of Matrix Algebra (Academic, 1965).

Philipp, M.

M. Philipp, P. von Brentano, G. Pascovici, and A. Richter, "Frequency and width crossing of two interacting resonances in a microwave cavity," Phys. Rev. E 62, 1922-1926 (2000).
[CrossRef]

Richter, A.

M. Philipp, P. von Brentano, G. Pascovici, and A. Richter, "Frequency and width crossing of two interacting resonances in a microwave cavity," Phys. Rev. E 62, 1922-1926 (2000).
[CrossRef]

Rotter, I.

W. D. Heiss, M. Müller, and I. Rotter, "Collectivity, phase transitions, and exceptional points in open quantum systems," Phys. Rev. E 58, 2894-2901 (1998).
[CrossRef]

Schmiedmayer, J.

K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, "Atom waves in crystals of light," Phys. Rev. Lett. 77, 4980-4983 (1996).
[CrossRef] [PubMed]

Scholtz, F. G.

T. Stehmann, W. D. Heiss, and F. G. Scholtz, "Observation of exceptional points in electronic circuits," J. Phys. A 37, 7813-7819 (2004).
[CrossRef]

Seyranian, A. P.

A. P. Seyranian, O. N. Kirillov, and A. A. Mailybaev, "Coupling of eigenvalues of complex matrices at diabolic and exceptional points," J. Phys. A 38, 1723-1740 (2005).
[CrossRef]

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light (Harvard U. Press, 1962).

Stehmann, T.

T. Stehmann, W. D. Heiss, and F. G. Scholtz, "Observation of exceptional points in electronic circuits," J. Phys. A 37, 7813-7819 (2004).
[CrossRef]

Tismenetsky, M.

P. Lancaster and M. Tismenetsky, The Theory of Matrices (Academic, 1985).

Tudor, T.

T. Tudor, "Operational form of the theory of polarization optical devices: I. Spectral theory of the basic devices," Optik (Stuttgart) 114, 539-547 (2003).

T. Tudor, "Generalized observables in polarization optics," J. Phys. A 36, 9567-9590 (2003).
[CrossRef]

von Brentano, P.

M. Philipp, P. von Brentano, G. Pascovici, and A. Richter, "Frequency and width crossing of two interacting resonances in a microwave cavity," Phys. Rev. E 62, 1922-1926 (2000).
[CrossRef]

Whitney, C.

Zeilinger, A.

K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, "Atom waves in crystals of light," Phys. Rev. Lett. 77, 4980-4983 (1996).
[CrossRef] [PubMed]

Appl. Opt. (1)

Commun. Math. Phys. (1)

E. B. Davies and J. T. Lewis, "An operatorial approach to quantum probability," Commun. Math. Phys. 17, 239-260 (1970).
[CrossRef]

Czech. J. Phys. (2)

M. V. Berry, "Physics of nonhermitian degeneracies," Czech. J. Phys. 55, 1039-1046 (2004).

W. D. Heiss, "Exceptional points—their universal occurrence and their physical significance," Czech. J. Phys. 55, 1091-1099 (2004).

J. Mod. Opt. (2)

M. V. Berry and S. Klein, "Geometric phases from stacks of crystal plates," J. Mod. Opt. 43, 165-180 (1996).
[CrossRef]

M. V. Berry, "Mode degeneracies and the Petermann excess-noise factor for unstable lasers," J. Mod. Opt. 50, 63-81 (2003).

J. Opt. A, Pure Appl. Opt. (1)

M. V. Berry and M. R. Dennis, "Black polarization sandwiches as square roots of zero," J. Opt. A, Pure Appl. Opt. 6, S24-S25 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Phys. A (5)

F. Keck, H. J. Korsch, and S. Mossmann, "Unfolding a diabolic point: a generalized crossing scenario," J. Phys. A 36, 2125-2137 (2003).
[CrossRef]

T. Tudor, "Generalized observables in polarization optics," J. Phys. A 36, 9567-9590 (2003).
[CrossRef]

M. V. Berry and D. H. J. O'Dell, "Diffraction by volume gratings with imaginary potentials," J. Phys. A 31, 2093-2101 (1998).
[CrossRef]

A. P. Seyranian, O. N. Kirillov, and A. A. Mailybaev, "Coupling of eigenvalues of complex matrices at diabolic and exceptional points," J. Phys. A 38, 1723-1740 (2005).
[CrossRef]

T. Stehmann, W. D. Heiss, and F. G. Scholtz, "Observation of exceptional points in electronic circuits," J. Phys. A 37, 7813-7819 (2004).
[CrossRef]

Optik (Stuttgart) (1)

T. Tudor, "Operational form of the theory of polarization optical devices: I. Spectral theory of the basic devices," Optik (Stuttgart) 114, 539-547 (2003).

Phys. Rev. E (3)

M. Philipp, P. von Brentano, G. Pascovici, and A. Richter, "Frequency and width crossing of two interacting resonances in a microwave cavity," Phys. Rev. E 62, 1922-1926 (2000).
[CrossRef]

W. D. Heiss, "Repulsion of resonant states and exceptional points," Phys. Rev. E 61, 929-932 (2000).
[CrossRef]

W. D. Heiss, M. Müller, and I. Rotter, "Collectivity, phase transitions, and exceptional points in open quantum systems," Phys. Rev. E 58, 2894-2901 (1998).
[CrossRef]

Phys. Rev. Lett. (1)

K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, "Atom waves in crystals of light," Phys. Rev. Lett. 77, 4980-4983 (1996).
[CrossRef] [PubMed]

Proc. R. Soc. London, Ser. A (1)

M. V. Berry and M. R. Dennis, "The optical singularities of birefringent dichroic chiral crystal," Proc. R. Soc. London, Ser. A 459, 1261-1292 (2003).
[CrossRef]

Other (8)

W. A. Shurcliff, Polarized Light (Harvard U. Press, 1962).

B. Higman, Applied Group-Theoretic and Matrix Methods (Clarendon, 1955).

C. Lanczos, Applied Analysis (Pitman, 1957).

P. Lancaster and M. Tismenetsky, The Theory of Matrices (Academic, 1985).

M. C. Pease, Methods of Matrix Algebra (Academic, 1965).

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1996).

T. Kato, Perturbation Theory for Linear Operators (Springer, 1966).

A. Messiah, Mécanique Quantique (Dunod, 1964).

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Figures (4)

Fig. 1
Fig. 1

Biorthonormal eigensystem corresponding to the operator P 1 .

Fig. 2
Fig. 2

Skew projector T 1 gives the projection of the state vector S on E 1 = P θ along E 2 = P y .

Fig. 3
Fig. 3

Poincaré axes of the non-Hermitian elliptical polarizer (41).

Fig. 4
Fig. 4

Poincaré axes of the non-Hermitian linear polarizer (12).

Equations (79)

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

D = i , j = 1 2 l i j S i S j = i , j = 1 2 S i D S j S i S j ,
D = i = 1 2 λ i S i S i ,
D = λ M E M E M + λ m E m E m ,
P P θ = P θ P θ ,
P R = R R ,
R P θ ( δ ) = e i δ 2 P θ P θ + e i δ 2 P θ + 90 P θ + 90 .
P P θ = cos 2 θ P x P x + sin θ cos θ [ P x P y + P y P x ] + sin 2 θ P y P y ,
R P θ ( δ ) = ( cos δ 2 + i cos 2 θ sin δ 2 ) P x P x + i sin 2 θ sin δ 2 [ P x P y + P y P x ] + ( cos δ 2 i cos 2 θ sin δ 2 ) P y P y .
F j E i = δ i j .
E i F i
D = λ 1 E 1 F 1 + λ 2 E 2 F 2 .
P = P P θ P P x = P θ P θ P x P x = cos θ P θ P x .
E 1 = P θ , with λ 1 = cos 2 θ ,
E 2 = P y , with λ 2 = 0 .
F 1 E 2 = F 1 P y = 0 F 1 = a P x ,
F 1 E 1 = F 1 P θ = 1 a * P x P θ = a * cos θ = 1 , a = 1 cos θ ,
F 1 = 1 cos θ P x .
F 2 = 1 cos θ P θ + 90 .
T 1 = E 1 F 1 = 1 cos θ P θ P x ,
T 2 = E 2 F 2 = 1 cos θ P y P θ + 90 .
T 1 T 2 = 0 ,
T 1 + T 2 = 1 cos θ { P θ P x + P y P θ + 90 } = 1 cos θ { [ cos θ P x + sin θ P y ] P x + P y [ sin θ < P x + cos θ < P y ] } = P x P x + P y P y = P 1 + P 2 = 1 ,
T 1 S = 1 cos θ P θ P x S = P x S cos θ P θ .
P = λ 1 T 1 ,
P = R P θ 2 ( π ) P P x = [ i cos θ P x P x + i sin θ P x P y + i sin θ P y P x i cos θ P y P y ] P x P x
= i cos θ P x P x + i sin θ P y P x .
cos θ P x + sin θ P y = P θ ,
P = i P θ P x .
E 1 = P θ , with λ 1 = i P x P θ = e i π 2 cos θ ,
E 2 = P y , with λ 2 = 0 .
F 1 E 2 = F 1 P y = 0 , F 1 E 1 = F 1 P θ = 1 ,
F 1 = 1 cos θ P x ,
F 2 E 1 = F 2 P θ = 0 , F 2 E 2 = F 2 P y = 1 ,
F 2 = 1 cos θ P θ + 90 .
T 1 = E 1 F 1 = 1 cos θ P θ P x ,
T 2 = E 2 F 2 = 1 cos θ P y P θ + 90 .
P = λ 1 T 1 .
C = R P x ( π 2 ) P P 45 .
C = ( e i π 4 P x P x + e i π 4 P y P y ) P 45 P 45 = e i π 4 ( P x P 45 P x + e i π 2 P y P 45 P y ) P 45 = e i π 4 1 2 ( P x i P y ) P 45 = e i π 4 L P 45 .
E 1 = L , with λ 1 = e i π 4 P 45 L = 1 2 ,
E 2 = P 45 , with λ 2 = 0 .
F 1 E 1 = a * P 45 L = a * e i π 4 2 = 1 a = 2 e i π 4 ,
F 1 = 2 e i π 4 P 45 ,
F 2 E 2 = b * R P 45 = b * e i π 4 2 = 1 b = 2 e i π 4 ,
F 2 = 2 e i π 4 R .
T 1 = E 1 F 1 = 2 e i π 4 L P 45 ,
T 2 = E 2 F 2 = 2 e i π 4 P 45 R .
E = R P x ( π 2 ) P P θ .
E = ( e i π 4 P x P x + e i π 4 P y P y ) P θ P θ = e i π 4 ( cos θ P x i sin θ P y ) P θ = e i π 4 E P θ ,
E = cos θ P x i sin θ P y .
E 1 = E ,
λ 1 = e i π 4 P θ E = e i π 4 ( cos 2 θ i sin 2 θ ) ,
E 2 = P θ 90 , with λ 2 = 0 .
F 1 E 2 = 0 F 1 = a P θ ,
F 1 E 1 = 1 a * P θ ( cos θ P x i sin θ P y ) = a * ( cos 2 θ i sin 2 θ ) = 1 ,
F 1 = 1 cos 2 θ + i sin 2 θ P θ .
F 2 E 1 = 0 F 2 = b E = b ( sin θ P x + i cos θ P y )
F 2 E 2 = 1 b * ( sin 2 + i cos 2 θ ) = 1 ,
F 2 = 1 sin 2 θ i cos 2 θ ( sin θ P x + i cos θ P y ) .
T 1 = E 1 F 1 = 1 cos 2 θ i sin 2 θ E P θ ,
T 2 = E 2 F 2 = 1 sin 2 θ + i cos 2 θ P 90 θ E .
λ 1 T 1 = e i π 4 ( cos 2 θ i sin 2 θ ) 1 cos 2 θ i sin 2 θ E P θ = E .
E 1 F 1 , E 2 F 2 , E 1 E 2 .
ρ = E 1 E 2 2 .
E 1 = E 2 , F 1 F 2 .
E 1 = P x , E 2 = e i π P y ,
F 1 = P x , F 2 = e i π P y .
E 1 = L , E 2 = P 45 ,
F 1 = 2 e i π 4 P 45 , F 2 = 2 e i π 4 R .
E 1 = e i π 2 P y , E 2 = P x ,
F 1 = e i π 2 P y , F 2 = P x ,
E 1 = P x , E 2 = P y ,
F 1 = P x , F 2 = P y ;
E 1 = P θ , E 2 = P y ,
F 1 = 1 cos θ P x , F 2 = 1 cos θ P θ + 90 ,
E 1 = P y , E 2 = P y ,
F 1 P x , F 2 P x ,
E 1 = P x , E 2 = P y ,
F 1 P x , F 2 P y ,

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