Abstract

We demonstrate that it is possible to perform a fractional Fourier transform of an incident pulse with a continuously variable degree of fractionality, using a dispersive and nonlinear Kerr medium. This medium acts as a linear waveguide for the optical pulse if its intensity is small and if the refractive index is nonlinearly time-dependently modified by a simultaneously launched bright soliton. For optical pulses the dispersive and nonlinear medium is perfectly equivalent to a graded-index refractive medium for optical beams.

© 1999 Optical Society of America

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References

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  1. D. Dragoman, K.-H. Brenner, M. Dragoman, J. Bähr, and U. Krackhardt, Opt. Lett. 23, 1499 (1998).
    [CrossRef]
  2. A. W. Lohmann and B. H. Soffer, J. Opt. Soc. Am. A 11, 1798 (1994).
    [CrossRef]
  3. S. Abe and J. T. Sheridan, Opt. Commun. 137, 214 (1997).
    [CrossRef]
  4. B. H. Kolner, IEEE J. Quantum Electron. 30, 1951 (1994).
    [CrossRef]
  5. D. Dragoman and M. Dragoman, Appl. Opt. 35, 7025 (1996).
    [CrossRef] [PubMed]
  6. G. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1989).
  7. V. Namias, J. Inst. Math. Its Appl. 25, 241 (1980).
    [CrossRef]
  8. D. J. Kane and R. Trebino, Opt. Lett. 18, 823 (1993).
    [CrossRef] [PubMed]
  9. D. J. Kane and R. Trebino, IEEE J. Quantum Electron. 29, 571 (1993).
    [CrossRef]
  10. A. Kwok, L. Jusinski, M. A. Krumbügel, J. N. Sweetser, D. N. Fittinghoff, and R. Trebino, IEEE J. Sel. Top. Quantum Electron. 4, 271 (1998).
    [CrossRef]
  11. M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wong, Opt. Lett. 23, 2041 (1993).
    [CrossRef]

1998

D. Dragoman, K.-H. Brenner, M. Dragoman, J. Bähr, and U. Krackhardt, Opt. Lett. 23, 1499 (1998).
[CrossRef]

A. Kwok, L. Jusinski, M. A. Krumbügel, J. N. Sweetser, D. N. Fittinghoff, and R. Trebino, IEEE J. Sel. Top. Quantum Electron. 4, 271 (1998).
[CrossRef]

1997

S. Abe and J. T. Sheridan, Opt. Commun. 137, 214 (1997).
[CrossRef]

1996

1994

B. H. Kolner, IEEE J. Quantum Electron. 30, 1951 (1994).
[CrossRef]

A. W. Lohmann and B. H. Soffer, J. Opt. Soc. Am. A 11, 1798 (1994).
[CrossRef]

1993

D. J. Kane and R. Trebino, Opt. Lett. 18, 823 (1993).
[CrossRef] [PubMed]

D. J. Kane and R. Trebino, IEEE J. Quantum Electron. 29, 571 (1993).
[CrossRef]

M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wong, Opt. Lett. 23, 2041 (1993).
[CrossRef]

1980

V. Namias, J. Inst. Math. Its Appl. 25, 241 (1980).
[CrossRef]

Abe, S.

S. Abe and J. T. Sheridan, Opt. Commun. 137, 214 (1997).
[CrossRef]

Agrawal, G.

G. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1989).

Bähr, J.

Beck, M.

M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wong, Opt. Lett. 23, 2041 (1993).
[CrossRef]

Brenner, K.-H.

Dragoman, D.

Dragoman, M.

Fittinghoff, D. N.

A. Kwok, L. Jusinski, M. A. Krumbügel, J. N. Sweetser, D. N. Fittinghoff, and R. Trebino, IEEE J. Sel. Top. Quantum Electron. 4, 271 (1998).
[CrossRef]

Jusinski, L.

A. Kwok, L. Jusinski, M. A. Krumbügel, J. N. Sweetser, D. N. Fittinghoff, and R. Trebino, IEEE J. Sel. Top. Quantum Electron. 4, 271 (1998).
[CrossRef]

Kane, D. J.

D. J. Kane and R. Trebino, IEEE J. Quantum Electron. 29, 571 (1993).
[CrossRef]

D. J. Kane and R. Trebino, Opt. Lett. 18, 823 (1993).
[CrossRef] [PubMed]

Kolner, B. H.

B. H. Kolner, IEEE J. Quantum Electron. 30, 1951 (1994).
[CrossRef]

Krackhardt, U.

Krumbügel, M. A.

A. Kwok, L. Jusinski, M. A. Krumbügel, J. N. Sweetser, D. N. Fittinghoff, and R. Trebino, IEEE J. Sel. Top. Quantum Electron. 4, 271 (1998).
[CrossRef]

Kwok, A.

A. Kwok, L. Jusinski, M. A. Krumbügel, J. N. Sweetser, D. N. Fittinghoff, and R. Trebino, IEEE J. Sel. Top. Quantum Electron. 4, 271 (1998).
[CrossRef]

Lohmann, A. W.

Namias, V.

V. Namias, J. Inst. Math. Its Appl. 25, 241 (1980).
[CrossRef]

Raymer, M. G.

M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wong, Opt. Lett. 23, 2041 (1993).
[CrossRef]

Sheridan, J. T.

S. Abe and J. T. Sheridan, Opt. Commun. 137, 214 (1997).
[CrossRef]

Soffer, B. H.

Sweetser, J. N.

A. Kwok, L. Jusinski, M. A. Krumbügel, J. N. Sweetser, D. N. Fittinghoff, and R. Trebino, IEEE J. Sel. Top. Quantum Electron. 4, 271 (1998).
[CrossRef]

Trebino, R.

A. Kwok, L. Jusinski, M. A. Krumbügel, J. N. Sweetser, D. N. Fittinghoff, and R. Trebino, IEEE J. Sel. Top. Quantum Electron. 4, 271 (1998).
[CrossRef]

D. J. Kane and R. Trebino, IEEE J. Quantum Electron. 29, 571 (1993).
[CrossRef]

D. J. Kane and R. Trebino, Opt. Lett. 18, 823 (1993).
[CrossRef] [PubMed]

Walmsley, I. A.

M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wong, Opt. Lett. 23, 2041 (1993).
[CrossRef]

Wong, V.

M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wong, Opt. Lett. 23, 2041 (1993).
[CrossRef]

Appl. Opt.

IEEE J. Quantum Electron.

B. H. Kolner, IEEE J. Quantum Electron. 30, 1951 (1994).
[CrossRef]

D. J. Kane and R. Trebino, IEEE J. Quantum Electron. 29, 571 (1993).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

A. Kwok, L. Jusinski, M. A. Krumbügel, J. N. Sweetser, D. N. Fittinghoff, and R. Trebino, IEEE J. Sel. Top. Quantum Electron. 4, 271 (1998).
[CrossRef]

J. Inst. Math. Its Appl.

V. Namias, J. Inst. Math. Its Appl. 25, 241 (1980).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

S. Abe and J. T. Sheridan, Opt. Commun. 137, 214 (1997).
[CrossRef]

Opt. Lett.

Other

G. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1989).

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Figures (2)

Fig. 1
Fig. 1

Schematic diagram of an H-rod microlens and ray propagation in its meridional and sagittal planes, between the input and the output planes.

Fig. 2
Fig. 2

Schematic diagram of a dispersive and nonlinear medium that is used to perform a FRFT with a continuously variable degree of fractionality for a pulse that is exiting the medium through the oblique plane PP.

Equations (6)

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

d2dx2-β2-k02n2xφx=d2dx2-β2-k02n021-A2x2φx=0,
iz-β222t2+k0Δntut, z=0,
st=2β/γ1/2sech2β/β21/2t.
d2ψdt2-2β2β-k0Δntψ=d2ψdt2-2β2β-2β sech22β/β21/2tψ=0.
φoz,x¯i=φxiexpik02n0AcosAzsinAzxi-x¯i2dxi=1KF2Az/πx¯iφxiexpik0n0Ax¯icosAz-1sinAz,
Fαuf=Kfxi×expik0n0Au2+xi2cosαπ22 sinαπ2-uxisinαπ2dxi.

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