Abstract

We show that for nonnormal singular operators corresponding to the nonorthogonal polarizers, the unitary polar component is constituted by two partial isometries, one of them “active” and the other “hidden” or “mute.” For each such operator there exists a complementary one, corresponding also to a nonorthogonal polarizer, which has the same unitary polar component and whose partial isometries reverse their roles.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. J. Gil, Eur. Phys. J. Appl. Phys. 40, 1 (2007).
    [CrossRef]
  2. R. C. Jones, J. Opt. Soc. Am. 32, 486 (1942).
    [CrossRef]
  3. S. Pancharatnam, Proc. Indian Acad. Sci. A42, 86 (1955).
  4. S. N. Savenkov, O. I. Sydoruk, and R. S. Muttiah, Appl. Opt. 46, 6700 (2007).
    [CrossRef]
  5. S.-Y. Lu and R. A. Chipman, J. Opt. Soc. Am. A 11, 766 (1994).
    [CrossRef]
  6. M. V. Berry and M. R. Dennis, J. Opt. A 6, S24 (2004).
    [CrossRef]
  7. O. Sydoruk and S. N. Savenkov, J. Opt. 12, 035702 (2010).
    [CrossRef]
  8. T. Tudor, J. Opt. Soc. Am. A 23, 1513 (2006).
    [CrossRef]
  9. C. Whitney, J. Opt. Soc. Am. 61, 1207 (1971).
    [CrossRef]
  10. J. J. Gil and E. Bernabeu, Optik 76, 67 (1987).
  11. S.-Y. Lu and R. A. Chipman, J. Opt. Soc. Am. A 13, 1106 (1996).
    [CrossRef]
  12. S. N. Savenkov and O. I. Sydoruk, J. Opt. Soc. Am. A 22, 1447 (2005).
    [CrossRef]
  13. J. J. Gil, J. Opt. Soc. Am. A 30, 701 (2013).
    [CrossRef]
  14. E. B. Davies and J. T. Lewis, Commun. Math. Phys. 17, 239 (1970).
    [CrossRef]
  15. R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University, 1985).
  16. T. Kato, Perturbation Theory for Linear Operators (Springer, 1966).
  17. T. Tudor, J. Phys. A 36, 9577 (2003).
    [CrossRef]
  18. O. V. Angelsky, S. G. Hanson, C. Yu. Zenkova, M. P. Gorsky, and N. V. Gorodyns’ka, Opt. Express 17, 15623 (2009).
    [CrossRef]
  19. W. M. de Muynck, J. Phys. A 31, 431 (1998).
    [CrossRef]
  20. P. Busch, P. J. Lahti, and P. Mittelstaedt, The Quantum Theory of Measurement (Springer, 1996).

2013

2010

O. Sydoruk and S. N. Savenkov, J. Opt. 12, 035702 (2010).
[CrossRef]

2009

2007

2006

2005

2004

M. V. Berry and M. R. Dennis, J. Opt. A 6, S24 (2004).
[CrossRef]

2003

T. Tudor, J. Phys. A 36, 9577 (2003).
[CrossRef]

1998

W. M. de Muynck, J. Phys. A 31, 431 (1998).
[CrossRef]

1996

1994

1987

J. J. Gil and E. Bernabeu, Optik 76, 67 (1987).

1971

1970

E. B. Davies and J. T. Lewis, Commun. Math. Phys. 17, 239 (1970).
[CrossRef]

1955

S. Pancharatnam, Proc. Indian Acad. Sci. A42, 86 (1955).

1942

Angelsky, O. V.

Bernabeu, E.

J. J. Gil and E. Bernabeu, Optik 76, 67 (1987).

Berry, M. V.

M. V. Berry and M. R. Dennis, J. Opt. A 6, S24 (2004).
[CrossRef]

Busch, P.

P. Busch, P. J. Lahti, and P. Mittelstaedt, The Quantum Theory of Measurement (Springer, 1996).

Chipman, R. A.

Davies, E. B.

E. B. Davies and J. T. Lewis, Commun. Math. Phys. 17, 239 (1970).
[CrossRef]

de Muynck, W. M.

W. M. de Muynck, J. Phys. A 31, 431 (1998).
[CrossRef]

Dennis, M. R.

M. V. Berry and M. R. Dennis, J. Opt. A 6, S24 (2004).
[CrossRef]

Gil, J. J.

J. J. Gil, J. Opt. Soc. Am. A 30, 701 (2013).
[CrossRef]

J. J. Gil, Eur. Phys. J. Appl. Phys. 40, 1 (2007).
[CrossRef]

J. J. Gil and E. Bernabeu, Optik 76, 67 (1987).

Gorodyns’ka, N. V.

Gorsky, M. P.

Hanson, S. G.

Horn, R. A.

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University, 1985).

Johnson, C. R.

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University, 1985).

Jones, R. C.

Kato, T.

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

Lahti, P. J.

P. Busch, P. J. Lahti, and P. Mittelstaedt, The Quantum Theory of Measurement (Springer, 1996).

Lewis, J. T.

E. B. Davies and J. T. Lewis, Commun. Math. Phys. 17, 239 (1970).
[CrossRef]

Lu, S.-Y.

Mittelstaedt, P.

P. Busch, P. J. Lahti, and P. Mittelstaedt, The Quantum Theory of Measurement (Springer, 1996).

Muttiah, R. S.

Pancharatnam, S.

S. Pancharatnam, Proc. Indian Acad. Sci. A42, 86 (1955).

Savenkov, S. N.

Sydoruk, O.

O. Sydoruk and S. N. Savenkov, J. Opt. 12, 035702 (2010).
[CrossRef]

Sydoruk, O. I.

Tudor, T.

Whitney, C.

Zenkova, C. Yu.

Appl. Opt.

Commun. Math. Phys.

E. B. Davies and J. T. Lewis, Commun. Math. Phys. 17, 239 (1970).
[CrossRef]

Eur. Phys. J. Appl. Phys.

J. J. Gil, Eur. Phys. J. Appl. Phys. 40, 1 (2007).
[CrossRef]

J. Opt.

O. Sydoruk and S. N. Savenkov, J. Opt. 12, 035702 (2010).
[CrossRef]

J. Opt. A

M. V. Berry and M. R. Dennis, J. Opt. A 6, S24 (2004).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. A

T. Tudor, J. Phys. A 36, 9577 (2003).
[CrossRef]

W. M. de Muynck, J. Phys. A 31, 431 (1998).
[CrossRef]

Opt. Express

Optik

J. J. Gil and E. Bernabeu, Optik 76, 67 (1987).

Proc. Indian Acad. Sci.

S. Pancharatnam, Proc. Indian Acad. Sci. A42, 86 (1955).

Other

P. Busch, P. J. Lahti, and P. Mittelstaedt, The Quantum Theory of Measurement (Springer, 1996).

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University, 1985).

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (1)

Fig. 1.
Fig. 1.

Two pairs of singular unit vectors of the operator P.

Equations (55)

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

P=P|PθP|Px=|PθPθ|PxPx|=cosθ|PθPx|,
|Pθ,withλ1=cosθPx|Pθ=cos2θ,
|Py,withλ2=0.
P|S=cosθ|PθPx|S=cosθPx|S|Pθ,
P=UHr=HlU=UPP=PPU,
P=cosθ|PxPθ|.
Hr=PP=cosθ|PxPx|,
Hl=PP=cosθ|PθPθ|.
cosθ|PθPx|=Ucosθ|PxPx|,U=|PθPx|,
UU=|PθPx|PxPθ|=|PθPθ|=P|PθI.
U1=|PθPx|,
U1|aPx=a|PθU1|aPx=a2=a|Px.
U2=|Pθ+90°Py|.
U=U1+U2=|PθPx|+|Pθ+90°Py|
UU=[|PθPx|+|Pθ+90°Py|][PxPθ|+|PyPθ+90°|]=|PθPθ|+|Pθ+90°Pθ+90°|=I.
P=cosθ|PθPx|+0|Pθ+90°Py|.
P=UHr=[|PθPx|+|Pθ+90°Py|]cosθ|PxPx|=cosθ|PθPx|.
Q=cosθ|Pθ+90°Py|.
Q=cosθ|Pθ+90°Py|=|Pθ+90°Pθ+90°|PyPy|=P|Pθ+90°P|Py.
|Pθ+90°,withλ1=cosθPy|Pθ+90°=cos2θ,
|Px,withλ2=0.
Q|S=cosθPy|S|Pθ+90°.
Q=UHr=HlU=UQQ=QQU,
Q=cosθ|PyPθ+90°|.
Hr=QQ=cosθ|PyPy|,
Hl=QQ=cosθ|Pθ+90°Pθ+90°|.
HrHr=0,HlHl=0.
cosθ|Pθ+90°Py|=Ucosθ|PyPy|,U=|Pθ+90°Py|,
U=U1+U2=U1+U2=|PθPx|+|Pθ+90°Py|,
U2=U1=|PθPx|
UHr=[|Pθ+90°Py|+|PθPx|]cosθ|PyPy|=cosθ|Pθ+90°Py|=Q.
Q=cosθ|Pθ+90°Py|+0|PθPx|.
P|Px+P|Py=|PxPx|+|PyPy|=I.
C=R|Px(π/2)P|P45°.
C=|LP45°|,
|L,withλ1=P45°|L=1/2,
|P45°,withλ2=0.
C=UHr=HlU
Hr=CC=|P45°P45°|,
Hl=CC=|LL|,
U=|LP45°|+|RP45°|.
(|LP45°|+|RP45°|)|P45°P45°|=|LL|(|LP45°|+|RP45°|)=|LP45°|.
U1=|LP45°|,
U2=|RP45°|,
U1a|P45°=a2=a|P45°.
K=|RP45°|=R|Px(π/2)P|P45°.
|R,withλ1=1/2,
|P45°,withλ2=0.
Hr=KK=|P45°P45°|,
Hl=KK=|RR|
U1=U2=|RP45°|,
U2=U1=|LP45°|.
E=R|Px(π/2)P|Pθ.
F=R|Px(π/2)P|Pθ+90°.
P|Px=|PxPx|,P|Py=|PyPy|.

Metrics