Abstract

On the basis of space–time duality, we propose experimental setups to implement the cross-ambiguity function optically in space and time in one and two dimensions. In space the cross-ambiguity is shown to be related to the coupling efficiency between butt-joined optical waveguides. In time it is related to the spectrogram or the frequency-resolved optical gating techniques for the characterization of optical pulses.

© 1999 Optical Society of America

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References

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  1. P. M. Woodward, Probability and Information Theory with Application to Radar (Pergamon, London, 1953).
  2. J. Tu, S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
    [CrossRef]
  3. H. H. Szu, J. A. Blodgett, “Wigner distribution and ambiguity function,” in Optics in Four Dimensions—1980 (American Institute of Physics, New York, 1981), pp. 355–381.
  4. L. Cohen, “Time–frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).
    [CrossRef]
  5. B. H. Kolner, “Space–time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1963 (1994).
    [CrossRef]
  6. D. Dragoman, M. Dragoman, “Wigner transform implementation in the time–frequency domain,” Appl. Opt. 35, 7025–7030 (1996).
    [CrossRef] [PubMed]
  7. K. W. DeLong, D. N. Fittinghoff, R. Trebino, “Practical issues in ultrashort-pulse measurements using frequency-resolved optical gating,” IEEE J. Quantum Electron. 32, 1253–1264 (1996).
    [CrossRef]
  8. S. I. Hosain, J.-P. Meunier, Z. H. Wang, “Coupling efficiency of butt-joined planar waveguides with simultaneous tilt and transverse offset,” J. Lightwave Technol. 14, 901–906 (1996).
    [CrossRef]
  9. R. K. Luneburg, Mathematical Theory of Optics (University of California at Berkeley, Berkeley Calif., 1966).
  10. D. Onciul, “Efficiency of light launching into waveguides: a phase space approach,” Optik (Stuttgart) 96, 20–24 (1994).
  11. D. Dragoman, “Wigner distribution function representation of the coupling coefficient,” Appl. Opt. 34, 6758–6763 (1995).
    [CrossRef] [PubMed]
  12. D. Dragoman, M. Dragoman, “Integrated-optic devices characterization with the Wigner transform,” IEEE J. Select. Topics Quantum Electron. 2, 181–186 (1996).
    [CrossRef]
  13. K. H. Brenner, K. Wodkiewicz, “The time-dependent physical spectrum of light and the Wigner distribution function,” Opt. Commun. 43, 103–106 (1982).
    [CrossRef]
  14. H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
    [CrossRef]
  15. M. J. Bastiaans, “Wigner distribution function display: a supplement to ambiguity function display using a single 1-D input,” Appl. Opt. 19, 192–195 (1980).
    [CrossRef] [PubMed]
  16. R. Bamler, H. Glünder, “The Wigner distribution function of two-dimensional signals coherent-optical generation and display,” Opt. Acta 30, 1789–1803 (1983).
    [CrossRef]
  17. E. B. Treacy, “Measurement and interpretation of dynamic spectrogram of picosecond light pulses,” J. Appl. Phys. 42, 3848–3858 (1971).
    [CrossRef]
  18. A. A. Godil, B. A. Auld, D. M. Bloom, “Picosecond time-lenses,” IEEE J. Quantum Electron. 30, 827–837 (1994).
    [CrossRef]
  19. N. Agrawal, M. Wegener, “Ultrafast graded-gap electron transfer optical modulator structure,” Appl. Phys. Lett. 65, 685–687 (1994).
    [CrossRef]

1997 (1)

J. Tu, S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
[CrossRef]

1996 (4)

K. W. DeLong, D. N. Fittinghoff, R. Trebino, “Practical issues in ultrashort-pulse measurements using frequency-resolved optical gating,” IEEE J. Quantum Electron. 32, 1253–1264 (1996).
[CrossRef]

S. I. Hosain, J.-P. Meunier, Z. H. Wang, “Coupling efficiency of butt-joined planar waveguides with simultaneous tilt and transverse offset,” J. Lightwave Technol. 14, 901–906 (1996).
[CrossRef]

D. Dragoman, M. Dragoman, “Integrated-optic devices characterization with the Wigner transform,” IEEE J. Select. Topics Quantum Electron. 2, 181–186 (1996).
[CrossRef]

D. Dragoman, M. Dragoman, “Wigner transform implementation in the time–frequency domain,” Appl. Opt. 35, 7025–7030 (1996).
[CrossRef] [PubMed]

1995 (1)

1994 (4)

B. H. Kolner, “Space–time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1963 (1994).
[CrossRef]

A. A. Godil, B. A. Auld, D. M. Bloom, “Picosecond time-lenses,” IEEE J. Quantum Electron. 30, 827–837 (1994).
[CrossRef]

N. Agrawal, M. Wegener, “Ultrafast graded-gap electron transfer optical modulator structure,” Appl. Phys. Lett. 65, 685–687 (1994).
[CrossRef]

D. Onciul, “Efficiency of light launching into waveguides: a phase space approach,” Optik (Stuttgart) 96, 20–24 (1994).

1989 (1)

L. Cohen, “Time–frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).
[CrossRef]

1983 (1)

R. Bamler, H. Glünder, “The Wigner distribution function of two-dimensional signals coherent-optical generation and display,” Opt. Acta 30, 1789–1803 (1983).
[CrossRef]

1982 (1)

K. H. Brenner, K. Wodkiewicz, “The time-dependent physical spectrum of light and the Wigner distribution function,” Opt. Commun. 43, 103–106 (1982).
[CrossRef]

1980 (2)

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

M. J. Bastiaans, “Wigner distribution function display: a supplement to ambiguity function display using a single 1-D input,” Appl. Opt. 19, 192–195 (1980).
[CrossRef] [PubMed]

1971 (1)

E. B. Treacy, “Measurement and interpretation of dynamic spectrogram of picosecond light pulses,” J. Appl. Phys. 42, 3848–3858 (1971).
[CrossRef]

Agrawal, N.

N. Agrawal, M. Wegener, “Ultrafast graded-gap electron transfer optical modulator structure,” Appl. Phys. Lett. 65, 685–687 (1994).
[CrossRef]

Auld, B. A.

A. A. Godil, B. A. Auld, D. M. Bloom, “Picosecond time-lenses,” IEEE J. Quantum Electron. 30, 827–837 (1994).
[CrossRef]

Bamler, R.

R. Bamler, H. Glünder, “The Wigner distribution function of two-dimensional signals coherent-optical generation and display,” Opt. Acta 30, 1789–1803 (1983).
[CrossRef]

Bartelt, H. O.

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Bastiaans, M. J.

Blodgett, J. A.

H. H. Szu, J. A. Blodgett, “Wigner distribution and ambiguity function,” in Optics in Four Dimensions—1980 (American Institute of Physics, New York, 1981), pp. 355–381.

Bloom, D. M.

A. A. Godil, B. A. Auld, D. M. Bloom, “Picosecond time-lenses,” IEEE J. Quantum Electron. 30, 827–837 (1994).
[CrossRef]

Brenner, K. H.

K. H. Brenner, K. Wodkiewicz, “The time-dependent physical spectrum of light and the Wigner distribution function,” Opt. Commun. 43, 103–106 (1982).
[CrossRef]

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Cohen, L.

L. Cohen, “Time–frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).
[CrossRef]

DeLong, K. W.

K. W. DeLong, D. N. Fittinghoff, R. Trebino, “Practical issues in ultrashort-pulse measurements using frequency-resolved optical gating,” IEEE J. Quantum Electron. 32, 1253–1264 (1996).
[CrossRef]

Dragoman, D.

Dragoman, M.

D. Dragoman, M. Dragoman, “Wigner transform implementation in the time–frequency domain,” Appl. Opt. 35, 7025–7030 (1996).
[CrossRef] [PubMed]

D. Dragoman, M. Dragoman, “Integrated-optic devices characterization with the Wigner transform,” IEEE J. Select. Topics Quantum Electron. 2, 181–186 (1996).
[CrossRef]

Fittinghoff, D. N.

K. W. DeLong, D. N. Fittinghoff, R. Trebino, “Practical issues in ultrashort-pulse measurements using frequency-resolved optical gating,” IEEE J. Quantum Electron. 32, 1253–1264 (1996).
[CrossRef]

Glünder, H.

R. Bamler, H. Glünder, “The Wigner distribution function of two-dimensional signals coherent-optical generation and display,” Opt. Acta 30, 1789–1803 (1983).
[CrossRef]

Godil, A. A.

A. A. Godil, B. A. Auld, D. M. Bloom, “Picosecond time-lenses,” IEEE J. Quantum Electron. 30, 827–837 (1994).
[CrossRef]

Hosain, S. I.

S. I. Hosain, J.-P. Meunier, Z. H. Wang, “Coupling efficiency of butt-joined planar waveguides with simultaneous tilt and transverse offset,” J. Lightwave Technol. 14, 901–906 (1996).
[CrossRef]

Kolner, B. H.

B. H. Kolner, “Space–time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1963 (1994).
[CrossRef]

Lohmann, A. W.

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California at Berkeley, Berkeley Calif., 1966).

Meunier, J.-P.

S. I. Hosain, J.-P. Meunier, Z. H. Wang, “Coupling efficiency of butt-joined planar waveguides with simultaneous tilt and transverse offset,” J. Lightwave Technol. 14, 901–906 (1996).
[CrossRef]

Onciul, D.

D. Onciul, “Efficiency of light launching into waveguides: a phase space approach,” Optik (Stuttgart) 96, 20–24 (1994).

Szu, H. H.

H. H. Szu, J. A. Blodgett, “Wigner distribution and ambiguity function,” in Optics in Four Dimensions—1980 (American Institute of Physics, New York, 1981), pp. 355–381.

Tamura, S.

J. Tu, S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
[CrossRef]

Treacy, E. B.

E. B. Treacy, “Measurement and interpretation of dynamic spectrogram of picosecond light pulses,” J. Appl. Phys. 42, 3848–3858 (1971).
[CrossRef]

Trebino, R.

K. W. DeLong, D. N. Fittinghoff, R. Trebino, “Practical issues in ultrashort-pulse measurements using frequency-resolved optical gating,” IEEE J. Quantum Electron. 32, 1253–1264 (1996).
[CrossRef]

Tu, J.

J. Tu, S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
[CrossRef]

Wang, Z. H.

S. I. Hosain, J.-P. Meunier, Z. H. Wang, “Coupling efficiency of butt-joined planar waveguides with simultaneous tilt and transverse offset,” J. Lightwave Technol. 14, 901–906 (1996).
[CrossRef]

Wegener, M.

N. Agrawal, M. Wegener, “Ultrafast graded-gap electron transfer optical modulator structure,” Appl. Phys. Lett. 65, 685–687 (1994).
[CrossRef]

Wodkiewicz, K.

K. H. Brenner, K. Wodkiewicz, “The time-dependent physical spectrum of light and the Wigner distribution function,” Opt. Commun. 43, 103–106 (1982).
[CrossRef]

Woodward, P. M.

P. M. Woodward, Probability and Information Theory with Application to Radar (Pergamon, London, 1953).

Appl. Opt. (3)

Appl. Phys. Lett. (1)

N. Agrawal, M. Wegener, “Ultrafast graded-gap electron transfer optical modulator structure,” Appl. Phys. Lett. 65, 685–687 (1994).
[CrossRef]

IEEE J. Quantum Electron. (3)

A. A. Godil, B. A. Auld, D. M. Bloom, “Picosecond time-lenses,” IEEE J. Quantum Electron. 30, 827–837 (1994).
[CrossRef]

B. H. Kolner, “Space–time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1963 (1994).
[CrossRef]

K. W. DeLong, D. N. Fittinghoff, R. Trebino, “Practical issues in ultrashort-pulse measurements using frequency-resolved optical gating,” IEEE J. Quantum Electron. 32, 1253–1264 (1996).
[CrossRef]

IEEE J. Select. Topics Quantum Electron. (1)

D. Dragoman, M. Dragoman, “Integrated-optic devices characterization with the Wigner transform,” IEEE J. Select. Topics Quantum Electron. 2, 181–186 (1996).
[CrossRef]

J. Appl. Phys. (1)

E. B. Treacy, “Measurement and interpretation of dynamic spectrogram of picosecond light pulses,” J. Appl. Phys. 42, 3848–3858 (1971).
[CrossRef]

J. Lightwave Technol. (1)

S. I. Hosain, J.-P. Meunier, Z. H. Wang, “Coupling efficiency of butt-joined planar waveguides with simultaneous tilt and transverse offset,” J. Lightwave Technol. 14, 901–906 (1996).
[CrossRef]

Opt. Acta (1)

R. Bamler, H. Glünder, “The Wigner distribution function of two-dimensional signals coherent-optical generation and display,” Opt. Acta 30, 1789–1803 (1983).
[CrossRef]

Opt. Commun. (2)

K. H. Brenner, K. Wodkiewicz, “The time-dependent physical spectrum of light and the Wigner distribution function,” Opt. Commun. 43, 103–106 (1982).
[CrossRef]

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Optik (Stuttgart) (1)

D. Onciul, “Efficiency of light launching into waveguides: a phase space approach,” Optik (Stuttgart) 96, 20–24 (1994).

Phys. Rev. E (1)

J. Tu, S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
[CrossRef]

Proc. IEEE (1)

L. Cohen, “Time–frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).
[CrossRef]

Other (3)

P. M. Woodward, Probability and Information Theory with Application to Radar (Pergamon, London, 1953).

H. H. Szu, J. A. Blodgett, “Wigner distribution and ambiguity function,” in Optics in Four Dimensions—1980 (American Institute of Physics, New York, 1981), pp. 355–381.

R. K. Luneburg, Mathematical Theory of Optics (University of California at Berkeley, Berkeley Calif., 1966).

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Figures (4)

Fig. 1
Fig. 1

Setup for the implementation of the 1-D spatial cross-AF.

Fig. 2
Fig. 2

Setup for the implementation of the 2-D spatial cross-AF.

Fig. 3
Fig. 3

Setup for the implementation of the 1-D temporal cross-AF.

Fig. 4
Fig. 4

Setup for the implementation of the 2-D temporal cross-AF.

Tables (1)

Tables Icon

Table 1 Space–Time Analogy for 1-D Fields

Equations (13)

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ηD,p= φ1x+D2φ2*x-D2expik0pxdx2,
ηDx,Dy,px,py= φ1x+Dx2, y+Dy2φ2*x-Dx2, y-Dy2×expik0pxx+pyydxdy2,
Aφx, y, qx, qy= φx+x2, y+y2φ*x-x2,y-y2expik0qxx+qyydxdy,
Aφ1φ2x, y, qx, qy= φ1x+x2, y+y2φ2*x-x2,y-y2expik0qxx+qyydxdy,
ηDx,Dy,px,py=|Aφ1φ2Dx, Dy, px, py|2,
ηD,p=|Aφ1φ2D, p|2.
η=η0,0,0,0=k024π2  Wφ1x, y, qx, qyWφ2x, y, qx, qydxdydqxdqy,
Wφx, y, qx, qy= φx+x2, y+y2φ*x-x2,y-y2expik0qxx+qyydxdy.
Wφ1φ2x, y, qx, qy= φ1x+x2, y+y2φ2*x-x2,y-y2expik0qxx+qyydxdy,
ηDx,Dy,px,py=k024π2  Wφ1x, y, qx, qyWφ2x-Dx,y-Dy, qx-px, qy-pydxdydqxdqy.
 Wφ1x, y, qx, qyWφ2x-x0, y-y0, qx-qx0,qy-qy0dxdydqxdqy=4π2k02 φ1x, yφ2x-x0, y-y0×expik0qx0x+qy0ydxdy2.
ηT,Ω= φ1t+T2φ2*t-T2expiΩtdt2=|Aφ1φ2T, Ω|2.
ηTx,Ty,Ωx,Ωy= φ1tx+Tx2, ty+Ty2φ2*tx-Tx2,ty-Ty2expiΩxtx+Ωytydtxdty2=|Aφ1φ2Tx, Ty, Ωx, Ωy|2.

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