Abstract

We generalize the traditional concept of temporal optical interferometry to any degree of freedom of a coherent optical field. By identifying the structure of a unitary optical transformation that we designate the generalized phase operator, we enable optical interferometry to be carried out in any modal basis describing a degree of freedom. The structure of the generalized phase operator is that of a fractional optical transform, thus establishing the connection between fractional transforms, optical interferometry, and modal analysis.

© 2012 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
  2. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001).
  3. B. J. Smith, B. Killett, M. G. Raymer, K. Banaszek, and I. A. Walmsley, Opt. Lett. 30, 3365 (2005).
    [CrossRef]
  4. L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003).
  5. R. Zambrini and S. M. Barnett, Phys. Rev. Lett. 96, 113901 (2006).
    [CrossRef]
  6. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
    [CrossRef]
  7. H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, E. Yaoc, and J. Courtial, Opt. Commun. 223, 117 (2003).
    [CrossRef]
  8. V. Namias, IMA J. Appl. Math. 26, 187 (1980).
    [CrossRef]
  9. T. Alieva and M. J. Bastiaans, Opt. Lett. 24, 1206 (1999).
    [CrossRef]
  10. A. F. Abouraddy, T. M. Yarnall, and B. E. A. Saleh, Opt. Lett. 36, 4683 (2011).
    [CrossRef]
  11. X. Xue, H. Wei, and A. G. Kirk, Opt. Lett. 26, 1746 (2001).
    [CrossRef]
  12. H. Sasada and M. Okamoto, Phys. Rev. A 68, 012323 (2003).
    [CrossRef]
  13. A. F. Abouraddy, T. Yarnall, B. E. A. Saleh, and M. C. Teich, Phys. Rev. A 75, 052114 (2007).
    [CrossRef]
  14. T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, Phys. Rev. Lett. 99, 170408 (2007).
    [CrossRef]
  15. T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, Phys. Rev. Lett. 99, 250502 (2007).
    [CrossRef]

2011 (1)

2007 (3)

A. F. Abouraddy, T. Yarnall, B. E. A. Saleh, and M. C. Teich, Phys. Rev. A 75, 052114 (2007).
[CrossRef]

T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, Phys. Rev. Lett. 99, 170408 (2007).
[CrossRef]

T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, Phys. Rev. Lett. 99, 250502 (2007).
[CrossRef]

2006 (1)

R. Zambrini and S. M. Barnett, Phys. Rev. Lett. 96, 113901 (2006).
[CrossRef]

2005 (1)

2003 (2)

H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, E. Yaoc, and J. Courtial, Opt. Commun. 223, 117 (2003).
[CrossRef]

H. Sasada and M. Okamoto, Phys. Rev. A 68, 012323 (2003).
[CrossRef]

2002 (1)

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef]

2001 (1)

1999 (1)

1980 (1)

V. Namias, IMA J. Appl. Math. 26, 187 (1980).
[CrossRef]

Abouraddy, A. F.

A. F. Abouraddy, T. M. Yarnall, and B. E. A. Saleh, Opt. Lett. 36, 4683 (2011).
[CrossRef]

A. F. Abouraddy, T. Yarnall, B. E. A. Saleh, and M. C. Teich, Phys. Rev. A 75, 052114 (2007).
[CrossRef]

T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, Phys. Rev. Lett. 99, 170408 (2007).
[CrossRef]

T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, Phys. Rev. Lett. 99, 250502 (2007).
[CrossRef]

Alieva, T.

Allen, L.

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003).

Banaszek, K.

Barnett, S. M.

R. Zambrini and S. M. Barnett, Phys. Rev. Lett. 96, 113901 (2006).
[CrossRef]

H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, E. Yaoc, and J. Courtial, Opt. Commun. 223, 117 (2003).
[CrossRef]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef]

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003).

Bastiaans, M. J.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Courtial, J.

H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, E. Yaoc, and J. Courtial, Opt. Commun. 223, 117 (2003).
[CrossRef]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef]

Franke-Arnold, S.

H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, E. Yaoc, and J. Courtial, Opt. Commun. 223, 117 (2003).
[CrossRef]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef]

Killett, B.

Kirk, A. G.

Kutay, M. A.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001).

Leach, J.

H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, E. Yaoc, and J. Courtial, Opt. Commun. 223, 117 (2003).
[CrossRef]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef]

Namias, V.

V. Namias, IMA J. Appl. Math. 26, 187 (1980).
[CrossRef]

Okamoto, M.

H. Sasada and M. Okamoto, Phys. Rev. A 68, 012323 (2003).
[CrossRef]

Ozaktas, H. M.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001).

Padgett, M. J.

H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, E. Yaoc, and J. Courtial, Opt. Commun. 223, 117 (2003).
[CrossRef]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef]

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003).

Raymer, M. G.

Saleh, B. E. A.

A. F. Abouraddy, T. M. Yarnall, and B. E. A. Saleh, Opt. Lett. 36, 4683 (2011).
[CrossRef]

A. F. Abouraddy, T. Yarnall, B. E. A. Saleh, and M. C. Teich, Phys. Rev. A 75, 052114 (2007).
[CrossRef]

T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, Phys. Rev. Lett. 99, 250502 (2007).
[CrossRef]

T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, Phys. Rev. Lett. 99, 170408 (2007).
[CrossRef]

Sasada, H.

H. Sasada and M. Okamoto, Phys. Rev. A 68, 012323 (2003).
[CrossRef]

Smith, B. J.

Teich, M. C.

A. F. Abouraddy, T. Yarnall, B. E. A. Saleh, and M. C. Teich, Phys. Rev. A 75, 052114 (2007).
[CrossRef]

T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, Phys. Rev. Lett. 99, 170408 (2007).
[CrossRef]

T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, Phys. Rev. Lett. 99, 250502 (2007).
[CrossRef]

Walmsley, I. A.

Wei, H.

H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, E. Yaoc, and J. Courtial, Opt. Commun. 223, 117 (2003).
[CrossRef]

X. Xue, H. Wei, and A. G. Kirk, Opt. Lett. 26, 1746 (2001).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Xue, X.

H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, E. Yaoc, and J. Courtial, Opt. Commun. 223, 117 (2003).
[CrossRef]

X. Xue, H. Wei, and A. G. Kirk, Opt. Lett. 26, 1746 (2001).
[CrossRef]

Yaoc, E.

H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, E. Yaoc, and J. Courtial, Opt. Commun. 223, 117 (2003).
[CrossRef]

Yarnall, T.

T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, Phys. Rev. Lett. 99, 170408 (2007).
[CrossRef]

A. F. Abouraddy, T. Yarnall, B. E. A. Saleh, and M. C. Teich, Phys. Rev. A 75, 052114 (2007).
[CrossRef]

T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, Phys. Rev. Lett. 99, 250502 (2007).
[CrossRef]

Yarnall, T. M.

Zalevsky, Z.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001).

Zambrini, R.

R. Zambrini and S. M. Barnett, Phys. Rev. Lett. 96, 113901 (2006).
[CrossRef]

IMA J. Appl. Math. (1)

V. Namias, IMA J. Appl. Math. 26, 187 (1980).
[CrossRef]

Opt. Commun. (1)

H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, E. Yaoc, and J. Courtial, Opt. Commun. 223, 117 (2003).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. A (2)

H. Sasada and M. Okamoto, Phys. Rev. A 68, 012323 (2003).
[CrossRef]

A. F. Abouraddy, T. Yarnall, B. E. A. Saleh, and M. C. Teich, Phys. Rev. A 75, 052114 (2007).
[CrossRef]

Phys. Rev. Lett. (4)

T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, Phys. Rev. Lett. 99, 170408 (2007).
[CrossRef]

T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, Phys. Rev. Lett. 99, 250502 (2007).
[CrossRef]

R. Zambrini and S. M. Barnett, Phys. Rev. Lett. 96, 113901 (2006).
[CrossRef]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef]

Other (3)

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003).

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform (Wiley, 2001).

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Figures (3)

Fig. 1.
Fig. 1.

Generalized interferometry using a balanced MZI with (a) a transformation Λ in one arm for analyzing one DoF, or (b)  Λ 1 and Λ 2 for analyzing two DoFs.

Fig. 2.
Fig. 2.

(a) Intensity I ( x , y ) = | E ( x , y ) | 2 ( 25.6 w o × 25.6 w o ; w o is the Gaussian beam width parameter). Insets show the two superposed modes. (b)–(c) 2D interferogram and modal analysis in the OAM-LG basis and (d)–(e) in the HG basis. The ranges of α and β are [ 0 , 2 π ] and [ 0 , 1 ] in (b) and [ 0 , 4 ] and [ 0 , 4 ] in (d), respectively.

Fig. 3.
Fig. 3.

(a) Intensity I ( x , t ) = | E ( x , t ) | 2 of three superposed pulsed modes with a 1 = a 2 = a 3 , t 1 τ o = 3 , t 2 τ o = 0 , t 3 τ o = 3 , T 1 τ o = 1 , T 2 τ o = 1.2 , T 3 τ o = 1.5 (see text); x o is the Gaussian beam width parameter. The insets are the superposed pulsed modes; the white dotted lines are the pulse centers; (b) spatially integrated temporal autocorrelation of each pulse separately and the superposed pulse; (c) 2D interferogram P ( α , τ ) ; (d) hybrid discrete-continuous modal analysis in the HG basis and time.

Tables (1)

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Table 1. Realizations of GPOs for Generalized Interferometry

Equations (3)

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Λ ( x , x ; α ) = n e i n α ψ n ( x ) ψ n * ( x ) .
P ( α ) = 1 + d x E ( x ) E o * ( x ; α ) = 1 + n | c n | 2 cos n α .
P ( α , β ) = 1 + d x d y E 1 ( x , y ; α ) E 2 * ( x , y ; β ) = 1 + n m | c n m | 2 cos ( n α m β ) ;

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