Abstract

Spectrally modulated Airy-based pulses peak amplitude modulation (PAM) in linear dispersive media is investigated, designed, and numerically simulated. As it is shown here, it is possible to design the spectral modulation of the initial Airy-based pulses to obtain a pre-defined PAM profile as the pulse propagates. Although optical pulses self-amplitude modulation is a well-known effect under non-linear propagation, the designed Airy-based pulses exhibit PAM under linear dispersive propagation. This extraordinary linear propagation property can be applied in many kinds of dispersive media, enabling its use in a broad range of experiments and applications.

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References

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  1. M. Berry and N. Balazs, “Nonspreading wave packets,” Am. J. Phys 47, 264–267 (1979).
    [Crossref]
  2. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy airy beams,” Opt. Lett. 32, 979–981 (2007).
    [Crossref] [PubMed]
  3. G. Siviloglou, J. Broky, A. Dogariu, and D. Christodoulides, “Observation of accelerating airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
    [Crossref]
  4. I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. 106, 213903 (2011).
    [Crossref] [PubMed]
  5. A. Lotti, D. Faccio, A. Couairon, D. Papazoglou, P. Panagiotopoulos, D. Abdollahpour, and S. Tzortzakis, “Stationary nonlinear airy beams,” Physical Review A 84, 021807 (2011).
    [Crossref]
  6. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–bessel wave packets as versatile linear light bullets,” Nat. Photonics 4, 103–106 (2010).
    [Crossref]
  7. D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105, 253901 (2010).
    [Crossref]
  8. Y. Fattal, A. Rudnick, and D. M. Marom, “Soliton shedding from airy pulses in kerr media,” Opt. Express 19, 17298–17307 (2011).
    [Crossref] [PubMed]
  9. A. Rudnick and D. M. Marom, “Airy-soliton interactions in kerr media,” Opt. Express 19, 25570–25582 (2011).
    [Crossref]
  10. C. Ament, P. Polynkin, and J. V. Moloney, “Supercontinuum generation with femtosecond self-healing airy pulses,” Phys. Rev. Lett. 107, 243901 (2011).
    [Crossref]
  11. Y. Hu, M. Li, D. Bongiovanni, M. Clerici, J. Yao, Z. Chen, J. Azaña, and R. Morandotti, “Spectrum to distance mapping via nonlinear airy pulses,” Opt. Lett. 38, 380–382 (2013).
    [Crossref] [PubMed]
  12. M. A. Preciado and M. A. Muriel, “Metodo y sistema para la transmision de pulsos opticos a traves de medios dispersivos,” Spain patent Es2364935(2010).
  13. M. A. Preciado and M. A. Muriel, “Band-limited airy pulses for invariant propagation in single mode fibers,” J. Lightwave Technol. 30, 3660–3666 (2012).
    [Crossref]
  14. M. A. Preciado and K. Sugden, “Proposal and design of airy-based rocket pulses for invariant propagation in lossy dispersive media,” Opt. Lett. 37, 4970–4972 (2012).
    [Crossref] [PubMed]
  15. Y. S. Kivshar and G. Agrawal, Optical Solitons: From Fibers to Photonic Crystals(Academic press, 2003).
  16. I. Kaminer, Y. Lumer, M. Segev, and D. N. Christodoulides, “Causality effects on accelerating light pulses,” Opt. Express 19, 23132–23139 (2011).
    [Crossref] [PubMed]
  17. M. Potasek and G. Agrawal, “Self-amplitude-modulation of optical pulses in nonlinear dispersive fibers,” Phys. Rev. A. 36, 3862 (1987).
    [Crossref] [PubMed]
  18. O. Vallee and M. Soares, Airy Functions and Applications to Physics(Imperial College, 2004).
  19. J. Azaña, “Time-frequency (wigner) analysis of linear and nonlinear pulse propagation in optical fibers,” EURASIP J. Appl. Sig. Processing 2005, 1554–1565 (2005).
    [Crossref]
  20. J. Azaña and M. A. Muriel, “Study of optical pulses-fiber gratings interaction by means of joint time-frequency signal representations,” J. Lightwave Technol. 21, 2931 (2003).
    [Crossref]
  21. ITU-T, Optical Fibres, Cables and Systems(ITU, 2009).
  22. A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications (The Oxford Series in Electrical and Computer Engineering)(Oxford University Press, Inc., 2006).
  23. M. Ibsen and R. Feced, “Fiber bragg gratings for pure dispersion-slope compensation,” Opt. Lett. 28, 980–982 (2003).
    [Crossref] [PubMed]
  24. M. A. Preciado, V. Garcia-Munoz, and M. A. Muriel, “Grating design of oppositely chirped fbgs for pulse shaping,” IEEE Photon. Technol. Lett. 19, 435–437 (2007).
    [Crossref]
  25. M. A. Preciado, X. Shu, and K. Sugden, “Proposal and design of phase-modulated fiber gratings in transmission for pulse shaping,” Opt. Lett. 38, 70–72 (2013).
    [Crossref] [PubMed]
  26. A. M. Weiner, S. Enguehard, and B. Hatfield, “Femtosecond optical pulse shaping and processing,” Prog. Quantum Electron. 19, 161–238 (1995).
    [Crossref]

2013 (2)

2012 (2)

2011 (6)

I. Kaminer, Y. Lumer, M. Segev, and D. N. Christodoulides, “Causality effects on accelerating light pulses,” Opt. Express 19, 23132–23139 (2011).
[Crossref] [PubMed]

I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. 106, 213903 (2011).
[Crossref] [PubMed]

A. Lotti, D. Faccio, A. Couairon, D. Papazoglou, P. Panagiotopoulos, D. Abdollahpour, and S. Tzortzakis, “Stationary nonlinear airy beams,” Physical Review A 84, 021807 (2011).
[Crossref]

Y. Fattal, A. Rudnick, and D. M. Marom, “Soliton shedding from airy pulses in kerr media,” Opt. Express 19, 17298–17307 (2011).
[Crossref] [PubMed]

A. Rudnick and D. M. Marom, “Airy-soliton interactions in kerr media,” Opt. Express 19, 25570–25582 (2011).
[Crossref]

C. Ament, P. Polynkin, and J. V. Moloney, “Supercontinuum generation with femtosecond self-healing airy pulses,” Phys. Rev. Lett. 107, 243901 (2011).
[Crossref]

2010 (3)

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–bessel wave packets as versatile linear light bullets,” Nat. Photonics 4, 103–106 (2010).
[Crossref]

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105, 253901 (2010).
[Crossref]

M. A. Preciado and M. A. Muriel, “Metodo y sistema para la transmision de pulsos opticos a traves de medios dispersivos,” Spain patent Es2364935(2010).

2007 (3)

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy airy beams,” Opt. Lett. 32, 979–981 (2007).
[Crossref] [PubMed]

G. Siviloglou, J. Broky, A. Dogariu, and D. Christodoulides, “Observation of accelerating airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

M. A. Preciado, V. Garcia-Munoz, and M. A. Muriel, “Grating design of oppositely chirped fbgs for pulse shaping,” IEEE Photon. Technol. Lett. 19, 435–437 (2007).
[Crossref]

2005 (1)

J. Azaña, “Time-frequency (wigner) analysis of linear and nonlinear pulse propagation in optical fibers,” EURASIP J. Appl. Sig. Processing 2005, 1554–1565 (2005).
[Crossref]

2003 (2)

1995 (1)

A. M. Weiner, S. Enguehard, and B. Hatfield, “Femtosecond optical pulse shaping and processing,” Prog. Quantum Electron. 19, 161–238 (1995).
[Crossref]

1987 (1)

M. Potasek and G. Agrawal, “Self-amplitude-modulation of optical pulses in nonlinear dispersive fibers,” Phys. Rev. A. 36, 3862 (1987).
[Crossref] [PubMed]

1979 (1)

M. Berry and N. Balazs, “Nonspreading wave packets,” Am. J. Phys 47, 264–267 (1979).
[Crossref]

Abdollahpour, D.

A. Lotti, D. Faccio, A. Couairon, D. Papazoglou, P. Panagiotopoulos, D. Abdollahpour, and S. Tzortzakis, “Stationary nonlinear airy beams,” Physical Review A 84, 021807 (2011).
[Crossref]

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105, 253901 (2010).
[Crossref]

Agrawal, G.

M. Potasek and G. Agrawal, “Self-amplitude-modulation of optical pulses in nonlinear dispersive fibers,” Phys. Rev. A. 36, 3862 (1987).
[Crossref] [PubMed]

Y. S. Kivshar and G. Agrawal, Optical Solitons: From Fibers to Photonic Crystals(Academic press, 2003).

Ament, C.

C. Ament, P. Polynkin, and J. V. Moloney, “Supercontinuum generation with femtosecond self-healing airy pulses,” Phys. Rev. Lett. 107, 243901 (2011).
[Crossref]

Azaña, J.

Balazs, N.

M. Berry and N. Balazs, “Nonspreading wave packets,” Am. J. Phys 47, 264–267 (1979).
[Crossref]

Berry, M.

M. Berry and N. Balazs, “Nonspreading wave packets,” Am. J. Phys 47, 264–267 (1979).
[Crossref]

Bongiovanni, D.

Broky, J.

G. Siviloglou, J. Broky, A. Dogariu, and D. Christodoulides, “Observation of accelerating airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

Chen, Z.

Chong, A.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–bessel wave packets as versatile linear light bullets,” Nat. Photonics 4, 103–106 (2010).
[Crossref]

Christodoulides, D.

G. Siviloglou, J. Broky, A. Dogariu, and D. Christodoulides, “Observation of accelerating airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

Christodoulides, D. N.

I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. 106, 213903 (2011).
[Crossref] [PubMed]

I. Kaminer, Y. Lumer, M. Segev, and D. N. Christodoulides, “Causality effects on accelerating light pulses,” Opt. Express 19, 23132–23139 (2011).
[Crossref] [PubMed]

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–bessel wave packets as versatile linear light bullets,” Nat. Photonics 4, 103–106 (2010).
[Crossref]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy airy beams,” Opt. Lett. 32, 979–981 (2007).
[Crossref] [PubMed]

Clerici, M.

Couairon, A.

A. Lotti, D. Faccio, A. Couairon, D. Papazoglou, P. Panagiotopoulos, D. Abdollahpour, and S. Tzortzakis, “Stationary nonlinear airy beams,” Physical Review A 84, 021807 (2011).
[Crossref]

Dogariu, A.

G. Siviloglou, J. Broky, A. Dogariu, and D. Christodoulides, “Observation of accelerating airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

Enguehard, S.

A. M. Weiner, S. Enguehard, and B. Hatfield, “Femtosecond optical pulse shaping and processing,” Prog. Quantum Electron. 19, 161–238 (1995).
[Crossref]

Faccio, D.

A. Lotti, D. Faccio, A. Couairon, D. Papazoglou, P. Panagiotopoulos, D. Abdollahpour, and S. Tzortzakis, “Stationary nonlinear airy beams,” Physical Review A 84, 021807 (2011).
[Crossref]

Fattal, Y.

Feced, R.

Garcia-Munoz, V.

M. A. Preciado, V. Garcia-Munoz, and M. A. Muriel, “Grating design of oppositely chirped fbgs for pulse shaping,” IEEE Photon. Technol. Lett. 19, 435–437 (2007).
[Crossref]

Hatfield, B.

A. M. Weiner, S. Enguehard, and B. Hatfield, “Femtosecond optical pulse shaping and processing,” Prog. Quantum Electron. 19, 161–238 (1995).
[Crossref]

Hu, Y.

Ibsen, M.

Kaminer, I.

I. Kaminer, Y. Lumer, M. Segev, and D. N. Christodoulides, “Causality effects on accelerating light pulses,” Opt. Express 19, 23132–23139 (2011).
[Crossref] [PubMed]

I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. 106, 213903 (2011).
[Crossref] [PubMed]

Kivshar, Y. S.

Y. S. Kivshar and G. Agrawal, Optical Solitons: From Fibers to Photonic Crystals(Academic press, 2003).

Li, M.

Lotti, A.

A. Lotti, D. Faccio, A. Couairon, D. Papazoglou, P. Panagiotopoulos, D. Abdollahpour, and S. Tzortzakis, “Stationary nonlinear airy beams,” Physical Review A 84, 021807 (2011).
[Crossref]

Lumer, Y.

Marom, D. M.

Moloney, J. V.

C. Ament, P. Polynkin, and J. V. Moloney, “Supercontinuum generation with femtosecond self-healing airy pulses,” Phys. Rev. Lett. 107, 243901 (2011).
[Crossref]

Morandotti, R.

Muriel, M. A.

M. A. Preciado and M. A. Muriel, “Band-limited airy pulses for invariant propagation in single mode fibers,” J. Lightwave Technol. 30, 3660–3666 (2012).
[Crossref]

M. A. Preciado and M. A. Muriel, “Metodo y sistema para la transmision de pulsos opticos a traves de medios dispersivos,” Spain patent Es2364935(2010).

M. A. Preciado, V. Garcia-Munoz, and M. A. Muriel, “Grating design of oppositely chirped fbgs for pulse shaping,” IEEE Photon. Technol. Lett. 19, 435–437 (2007).
[Crossref]

J. Azaña and M. A. Muriel, “Study of optical pulses-fiber gratings interaction by means of joint time-frequency signal representations,” J. Lightwave Technol. 21, 2931 (2003).
[Crossref]

Panagiotopoulos, P.

A. Lotti, D. Faccio, A. Couairon, D. Papazoglou, P. Panagiotopoulos, D. Abdollahpour, and S. Tzortzakis, “Stationary nonlinear airy beams,” Physical Review A 84, 021807 (2011).
[Crossref]

Papazoglou, D.

A. Lotti, D. Faccio, A. Couairon, D. Papazoglou, P. Panagiotopoulos, D. Abdollahpour, and S. Tzortzakis, “Stationary nonlinear airy beams,” Physical Review A 84, 021807 (2011).
[Crossref]

Papazoglou, D. G.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105, 253901 (2010).
[Crossref]

Polynkin, P.

C. Ament, P. Polynkin, and J. V. Moloney, “Supercontinuum generation with femtosecond self-healing airy pulses,” Phys. Rev. Lett. 107, 243901 (2011).
[Crossref]

Potasek, M.

M. Potasek and G. Agrawal, “Self-amplitude-modulation of optical pulses in nonlinear dispersive fibers,” Phys. Rev. A. 36, 3862 (1987).
[Crossref] [PubMed]

Preciado, M. A.

Renninger, W. H.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–bessel wave packets as versatile linear light bullets,” Nat. Photonics 4, 103–106 (2010).
[Crossref]

Rudnick, A.

Segev, M.

I. Kaminer, Y. Lumer, M. Segev, and D. N. Christodoulides, “Causality effects on accelerating light pulses,” Opt. Express 19, 23132–23139 (2011).
[Crossref] [PubMed]

I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. 106, 213903 (2011).
[Crossref] [PubMed]

Shu, X.

Siviloglou, G.

G. Siviloglou, J. Broky, A. Dogariu, and D. Christodoulides, “Observation of accelerating airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

Siviloglou, G. A.

Soares, M.

O. Vallee and M. Soares, Airy Functions and Applications to Physics(Imperial College, 2004).

Sugden, K.

Suntsov, S.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105, 253901 (2010).
[Crossref]

Tzortzakis, S.

A. Lotti, D. Faccio, A. Couairon, D. Papazoglou, P. Panagiotopoulos, D. Abdollahpour, and S. Tzortzakis, “Stationary nonlinear airy beams,” Physical Review A 84, 021807 (2011).
[Crossref]

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105, 253901 (2010).
[Crossref]

Vallee, O.

O. Vallee and M. Soares, Airy Functions and Applications to Physics(Imperial College, 2004).

Weiner, A. M.

A. M. Weiner, S. Enguehard, and B. Hatfield, “Femtosecond optical pulse shaping and processing,” Prog. Quantum Electron. 19, 161–238 (1995).
[Crossref]

Wise, F. W.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–bessel wave packets as versatile linear light bullets,” Nat. Photonics 4, 103–106 (2010).
[Crossref]

Yao, J.

Yariv, A.

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications (The Oxford Series in Electrical and Computer Engineering)(Oxford University Press, Inc., 2006).

Yeh, P.

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications (The Oxford Series in Electrical and Computer Engineering)(Oxford University Press, Inc., 2006).

Am. J. Phys (1)

M. Berry and N. Balazs, “Nonspreading wave packets,” Am. J. Phys 47, 264–267 (1979).
[Crossref]

EURASIP J. Appl. Sig. Processing (1)

J. Azaña, “Time-frequency (wigner) analysis of linear and nonlinear pulse propagation in optical fibers,” EURASIP J. Appl. Sig. Processing 2005, 1554–1565 (2005).
[Crossref]

IEEE Photon. Technol. Lett. (1)

M. A. Preciado, V. Garcia-Munoz, and M. A. Muriel, “Grating design of oppositely chirped fbgs for pulse shaping,” IEEE Photon. Technol. Lett. 19, 435–437 (2007).
[Crossref]

J. Lightwave Technol. (2)

Nat. Photonics (1)

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–bessel wave packets as versatile linear light bullets,” Nat. Photonics 4, 103–106 (2010).
[Crossref]

Opt. Express (3)

Opt. Lett. (5)

Phys. Rev. A. (1)

M. Potasek and G. Agrawal, “Self-amplitude-modulation of optical pulses in nonlinear dispersive fibers,” Phys. Rev. A. 36, 3862 (1987).
[Crossref] [PubMed]

Phys. Rev. Lett. (4)

G. Siviloglou, J. Broky, A. Dogariu, and D. Christodoulides, “Observation of accelerating airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. 106, 213903 (2011).
[Crossref] [PubMed]

C. Ament, P. Polynkin, and J. V. Moloney, “Supercontinuum generation with femtosecond self-healing airy pulses,” Phys. Rev. Lett. 107, 243901 (2011).
[Crossref]

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105, 253901 (2010).
[Crossref]

Physical Review A (1)

A. Lotti, D. Faccio, A. Couairon, D. Papazoglou, P. Panagiotopoulos, D. Abdollahpour, and S. Tzortzakis, “Stationary nonlinear airy beams,” Physical Review A 84, 021807 (2011).
[Crossref]

Prog. Quantum Electron. (1)

A. M. Weiner, S. Enguehard, and B. Hatfield, “Femtosecond optical pulse shaping and processing,” Prog. Quantum Electron. 19, 161–238 (1995).
[Crossref]

Spain patent (1)

M. A. Preciado and M. A. Muriel, “Metodo y sistema para la transmision de pulsos opticos a traves de medios dispersivos,” Spain patent Es2364935(2010).

Other (4)

Y. S. Kivshar and G. Agrawal, Optical Solitons: From Fibers to Photonic Crystals(Academic press, 2003).

ITU-T, Optical Fibres, Cables and Systems(ITU, 2009).

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications (The Oxford Series in Electrical and Computer Engineering)(Oxford University Press, Inc., 2006).

O. Vallee and M. Soares, Airy Functions and Applications to Physics(Imperial College, 2004).

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Figures (6)

Fig. 1
Fig. 1

Illustrative representation of the linear dispersive PAM of the Airy-based pulse (example of oscillatory pre-defined PAM). The propagation delay is not represented for a clear visualization of the PAM.

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Fig. 2
Fig. 2

Time-frequency range of the main lobe of the Airy-based pulse, temporally centred at t = Δt(z) within a temporal width FWHMt, and spectrally centred at ω = Δω(z) within a spectral width δω. This spectral range is the integration interval used to approximate the main lobe intensity peak by applying the Parseval theorem.

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Fig. 3
Fig. 3

Four pre-defined PAM(z) profiles (red-dotted), and the numerically obtained propagated pulse peak intensity, using r = 0.2 (blue-dashed), r = 0.1 (blue-solid), and r = 0.05 (blue-dash-dotted), for examples from (a) to (d), in a propagation path of L=10 km.

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Fig. 4
Fig. 4

Main lobe temporal width of the Airy-based pulse for examples from (a) to (d), using r = 0.2 (blue-dashed), r = 0.1 (blue-solid), and r = 0.05 (blue-dash-dotted), in a propagation path of L=10 km.

Download Full Size | PPT Slide | PDF

Fig. 5
Fig. 5

Energy spectral density |AM(z)|2 = |M(z)|2 of the Airy-based pulse for examples from (a) to (d), for r = 0.1.

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Fig. 6
Fig. 6

Color map representation of the evolution of the temporal intensity of the propagated Airy-based pulse for examples from (a) to (d), in a propagation path of L=10 km.The propagation delay is not represented for a clear visualization of the PAM.

Download Full Size | PPT Slide | PDF

Equations (13)

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

F prop ( ω , z ) = A ( ω ) H D ( ω , z ) = A ( ω − Δ ω ( z ) ) exp ( − j ( Δ t ( z ) ω + ϕ ( z ) ) )
Δ ω ( z ) = β 2 z 6 ξ
A M ( ω ) = M ( ω ) A ( ω ) = M ( ω ) exp ( j ξ ω 3 )
F M , prop ( ω , z ) = A M ( ω ) H D ( ω , z ) H A ( ω , z ) = = M ( ω ) A ( ω ) H D ( ω , z ) H A ( ω , z ) = = M ( ω ) H A ( ω , z ) A ( ω − Δ ω ( z ) ) e − j ( Δ t ( z ) ω + ϕ ( z ) )
δ ω ≅ FWHM ω = 4 log ( 2 ) / FWHM t = | 1.2 ξ − 1 / 3 |
E m l ( z ) ≈ ∫ Δ ω ( z ) − δ ω 2 Δ ω ( z ) + δ ω 2 | F M , prop ( ω , z ) | 2 d ω = = ∫ δ ω 2 δ ω 2 | F M , prop ( ω + Δ ω ( z ) , z ) | 2 d ω ≈ ≈ δ ω | F M , prop ( Δ ω ( z ) , z ) | 2 = δ ω SAM ( z )
PAM ( z ) = | F M , prop ( Δ ω ( z ) , z ) | 2 = | M ( Δ ω ( z ) ) H A ( Δ ω ( z ) , z ) | 2
I ml ( z ) ∝ E ml ( z ) ≈ δ ω PAM ( z )
M ( ω ) = PAM ( z = 6 ω ξ β 2 ) | H A ( ω , z = 6 ω ξ β 2 ) |
B = | β 2 L 6 ξ |
H A ( z , ω ) = 10 − α ( ω ) ( z + L / 2 ) 20
A M ( ω ) = PAM ( z = 6 ω ξ β 2 ) 10 α ( ω ) 20 ( 6 ω ξ β 2 + L 2 ) e j ξ ω 3
| ξ | = | β 2 L r 7.2 | 3 2

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