Abstract

Using the space–time duality theory, we introduce and analyze theoretically the temporal equivalent of the spatial self-imaging phenomena under point-source illumination (spherical wave-front illumination). This temporal effect can be produced by passing a given periodic optical pulse sequence through a time lens followed by a dispersive medium. The time lens (quadratic phase modulator) implements the time-domain equivalent of spherical wave-front illumination. Based on this temporal effect, we demonstrate that a system composed of a time lens followed by a dispersive medium can be configured to operate over a periodic optical pulse sequence of finite duration (i) as a conventional temporal imaging system, providing a distortionless temporal compression or expansion of the original pulse sequence, or (ii) as an advanced temporal imaging system, combining the capabilities of a conventional imaging system with those of the fractional temporal self-imaging effect, i.e., multiplication of the original pulse-repetition rate.

© 2003 Optical Society of America

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  1. E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5, 454–458 (1969).
    [CrossRef]
  2. T. Jannson and J. Jannson, “Temporal self-imaging effect in single-mode fibers,” J. Opt. Soc. Am. 71, 1373–1376 (1981).
  3. T. Jannson, “Real-time Fourier transformation in dispersive optical fibers,” Opt. Lett. 8, 232–234 (1983).
    [CrossRef] [PubMed]
  4. B. H. Kolner and M. Nazarathy, “Temporal imaging with a time lens,” Opt. Lett. 14, 630–632 (1989).
    [CrossRef] [PubMed]
  5. B. H. Kolner and M. Nazarathy, “Temporal imaging with a time lens: erratum,” Opt. Lett. 15, 655 (1990).
    [CrossRef]
  6. A. W. Lohman and D. Mendlovic, “Temporal filtering with time lenses,” Appl. Opt. 31, 6212–6219 (1992).
    [CrossRef]
  7. A. A. Godil, B. A. Auld, and D. M. Bloom, “Time lens producing 1.9 ps optical pulses,” Appl. Phys. Lett. 62, 1047–1049 (1993).
    [CrossRef]
  8. A. Papoulis, “Pulse compression, fiber communications, and diffraction: a unified approach,” J. Opt. Soc. Am. A 11, 3–13 (1994).
    [CrossRef]
  9. B. H. Kolner, “Space–time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1963 (1994).
    [CrossRef]
  10. P. Naulleau and E. Leith, “Stretch, time lenses and incoherent time imaging,” Appl. Opt. 34, 4119–4128 (1995).
    [CrossRef] [PubMed]
  11. F. Mitschke and U. Morgner, “The temporal Talbot effect,” Opt. Photon. News 9(6), 45–47 (1998).
    [CrossRef]
  12. C. V. Bennett and B. H. Kolner, “Upconversion time microscope demonstrating 103× magnification of femtosecond waveforms,” Opt. Lett. 24, 783–785 (1999).
    [CrossRef]
  13. C. V. Bennett and B. H. Kolner, “Principles of parametric temporal imaging. I. System configurations,” IEEE J. Quantum Electron. 36, 430–437 (2000).
    [CrossRef]
  14. C. V. Bennett and B. H. Kolner, “Principles of parametric temporal imaging. II. System performance,” IEEE J. Quantum Electron. 36, 649–655 (2000).
    [CrossRef]
  15. M. A. Muriel, J. Azaña, and A. Carballar, “Real-time Fourier transformer based on fiber gratings,” Opt. Lett. 24, 1–3 (1999).
    [CrossRef]
  16. J. Azaña, L. R. Chen, M. A. Muriel, and P. W. E. Smith, “Experimental demonstration of real-time Fourier transforma-tion using linearly chirped fibre Bragg gratings,” Electron. Lett. 35, 2223–2224 (1999).
    [CrossRef]
  17. J. Azaña and M. A. Muriel, “Technique for multiplying the repetition rates of periodic trains of pulses by means of a temporal self-imaging effect in chirped fiber gratings,” Opt. Lett. 24, 1672–1674 (1999).
    [CrossRef]
  18. J. Azaña and M. A. Muriel, “Temporal Talbot effect in fiber gratings and its applications,” Appl. Opt. 38, 6700–6704 (1999).
    [CrossRef]
  19. J. Azaña and M. A. Muriel, “Real-time optical spectrum analysis based on the time-space duality in chirped fiber gratings,” IEEE J. Quantum Electron. 36, 517–526 (2000).
    [CrossRef]
  20. N. K. Berger, B. Levit, S. Atkins, and B. Fischer, “Time-lens-based spectral analysis of optical pulses by electrooptic phase modulation,” Electron. Lett. 36, 1644–1646 (2000).
    [CrossRef]
  21. J. Azaña and M. A. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. 7, 728–744 (2001).
    [CrossRef]
  22. K. Patorski, “The self-imaging phenomenon and its applications,” Progress in Optics XXVII, E. Wolf, ed. (Elsevier, Amsterdam, The Netherlands, ), 19891–108.
  23. M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
    [CrossRef]
  24. S. Longhi, M. Marano, P. Laporta, O. Svelto, M. Belmonte, B. Agogliati, L. Arcangeli, V. Pruneri, M. N. Zervas, and M. Ibsen, “40-GHz pulse-train generation at 1.5 μm with a chirped fiber grating as a frequency multiplier,” Opt. Lett. 25, 1481–1483 (2000).
    [CrossRef]
  25. J. M. Cowley and A. F. Moodie, “Fourier images. I. The point source,” Proc. Phys. Soc. London 70, 486–496 (1957).
    [CrossRef]
  26. J. M. Cowley and A. F. Moodie, “Fourier images. II. The out-of-focus patterns,” Proc. Phys. Soc. London 70, 497–504 (1957).
    [CrossRef]
  27. A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill, New York, 1987).
  28. J. Azaña, “Temporal self-imaging effects of periodic optical pulse sequences of finite duration,” J. Opt. Soc. Am. B 20, 83–90 (2003).
    [CrossRef]

2003 (1)

2001 (1)

J. Azaña and M. A. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. 7, 728–744 (2001).
[CrossRef]

2000 (5)

J. Azaña and M. A. Muriel, “Real-time optical spectrum analysis based on the time-space duality in chirped fiber gratings,” IEEE J. Quantum Electron. 36, 517–526 (2000).
[CrossRef]

N. K. Berger, B. Levit, S. Atkins, and B. Fischer, “Time-lens-based spectral analysis of optical pulses by electrooptic phase modulation,” Electron. Lett. 36, 1644–1646 (2000).
[CrossRef]

S. Longhi, M. Marano, P. Laporta, O. Svelto, M. Belmonte, B. Agogliati, L. Arcangeli, V. Pruneri, M. N. Zervas, and M. Ibsen, “40-GHz pulse-train generation at 1.5 μm with a chirped fiber grating as a frequency multiplier,” Opt. Lett. 25, 1481–1483 (2000).
[CrossRef]

C. V. Bennett and B. H. Kolner, “Principles of parametric temporal imaging. I. System configurations,” IEEE J. Quantum Electron. 36, 430–437 (2000).
[CrossRef]

C. V. Bennett and B. H. Kolner, “Principles of parametric temporal imaging. II. System performance,” IEEE J. Quantum Electron. 36, 649–655 (2000).
[CrossRef]

1999 (5)

1998 (1)

F. Mitschke and U. Morgner, “The temporal Talbot effect,” Opt. Photon. News 9(6), 45–47 (1998).
[CrossRef]

1996 (1)

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[CrossRef]

1995 (1)

1994 (2)

A. Papoulis, “Pulse compression, fiber communications, and diffraction: a unified approach,” J. Opt. Soc. Am. A 11, 3–13 (1994).
[CrossRef]

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

1993 (1)

A. A. Godil, B. A. Auld, and D. M. Bloom, “Time lens producing 1.9 ps optical pulses,” Appl. Phys. Lett. 62, 1047–1049 (1993).
[CrossRef]

1992 (1)

1990 (1)

1989 (1)

1983 (1)

1981 (1)

1969 (1)

E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5, 454–458 (1969).
[CrossRef]

1957 (2)

J. M. Cowley and A. F. Moodie, “Fourier images. I. The point source,” Proc. Phys. Soc. London 70, 486–496 (1957).
[CrossRef]

J. M. Cowley and A. F. Moodie, “Fourier images. II. The out-of-focus patterns,” Proc. Phys. Soc. London 70, 497–504 (1957).
[CrossRef]

Agogliati, B.

Arcangeli, L.

Atkins, S.

N. K. Berger, B. Levit, S. Atkins, and B. Fischer, “Time-lens-based spectral analysis of optical pulses by electrooptic phase modulation,” Electron. Lett. 36, 1644–1646 (2000).
[CrossRef]

Auld, B. A.

A. A. Godil, B. A. Auld, and D. M. Bloom, “Time lens producing 1.9 ps optical pulses,” Appl. Phys. Lett. 62, 1047–1049 (1993).
[CrossRef]

Azaña, J.

J. Azaña, “Temporal self-imaging effects of periodic optical pulse sequences of finite duration,” J. Opt. Soc. Am. B 20, 83–90 (2003).
[CrossRef]

J. Azaña and M. A. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. 7, 728–744 (2001).
[CrossRef]

J. Azaña and M. A. Muriel, “Real-time optical spectrum analysis based on the time-space duality in chirped fiber gratings,” IEEE J. Quantum Electron. 36, 517–526 (2000).
[CrossRef]

M. A. Muriel, J. Azaña, and A. Carballar, “Real-time Fourier transformer based on fiber gratings,” Opt. Lett. 24, 1–3 (1999).
[CrossRef]

J. Azaña, L. R. Chen, M. A. Muriel, and P. W. E. Smith, “Experimental demonstration of real-time Fourier transforma-tion using linearly chirped fibre Bragg gratings,” Electron. Lett. 35, 2223–2224 (1999).
[CrossRef]

J. Azaña and M. A. Muriel, “Technique for multiplying the repetition rates of periodic trains of pulses by means of a temporal self-imaging effect in chirped fiber gratings,” Opt. Lett. 24, 1672–1674 (1999).
[CrossRef]

J. Azaña and M. A. Muriel, “Temporal Talbot effect in fiber gratings and its applications,” Appl. Opt. 38, 6700–6704 (1999).
[CrossRef]

Belmonte, M.

Bennett, C. V.

C. V. Bennett and B. H. Kolner, “Principles of parametric temporal imaging. II. System performance,” IEEE J. Quantum Electron. 36, 649–655 (2000).
[CrossRef]

C. V. Bennett and B. H. Kolner, “Principles of parametric temporal imaging. I. System configurations,” IEEE J. Quantum Electron. 36, 430–437 (2000).
[CrossRef]

C. V. Bennett and B. H. Kolner, “Upconversion time microscope demonstrating 103× magnification of femtosecond waveforms,” Opt. Lett. 24, 783–785 (1999).
[CrossRef]

Berger, N. K.

N. K. Berger, B. Levit, S. Atkins, and B. Fischer, “Time-lens-based spectral analysis of optical pulses by electrooptic phase modulation,” Electron. Lett. 36, 1644–1646 (2000).
[CrossRef]

Berry, M. V.

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[CrossRef]

Bloom, D. M.

A. A. Godil, B. A. Auld, and D. M. Bloom, “Time lens producing 1.9 ps optical pulses,” Appl. Phys. Lett. 62, 1047–1049 (1993).
[CrossRef]

Carballar, A.

Chen, L. R.

J. Azaña, L. R. Chen, M. A. Muriel, and P. W. E. Smith, “Experimental demonstration of real-time Fourier transforma-tion using linearly chirped fibre Bragg gratings,” Electron. Lett. 35, 2223–2224 (1999).
[CrossRef]

Cowley, J. M.

J. M. Cowley and A. F. Moodie, “Fourier images. I. The point source,” Proc. Phys. Soc. London 70, 486–496 (1957).
[CrossRef]

J. M. Cowley and A. F. Moodie, “Fourier images. II. The out-of-focus patterns,” Proc. Phys. Soc. London 70, 497–504 (1957).
[CrossRef]

Fischer, B.

N. K. Berger, B. Levit, S. Atkins, and B. Fischer, “Time-lens-based spectral analysis of optical pulses by electrooptic phase modulation,” Electron. Lett. 36, 1644–1646 (2000).
[CrossRef]

Godil, A. A.

A. A. Godil, B. A. Auld, and D. M. Bloom, “Time lens producing 1.9 ps optical pulses,” Appl. Phys. Lett. 62, 1047–1049 (1993).
[CrossRef]

Ibsen, M.

Jannson, J.

Jannson, T.

Klein, S.

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[CrossRef]

Kolner, B. H.

C. V. Bennett and B. H. Kolner, “Principles of parametric temporal imaging. II. System performance,” IEEE J. Quantum Electron. 36, 649–655 (2000).
[CrossRef]

C. V. Bennett and B. H. Kolner, “Principles of parametric temporal imaging. I. System configurations,” IEEE J. Quantum Electron. 36, 430–437 (2000).
[CrossRef]

C. V. Bennett and B. H. Kolner, “Upconversion time microscope demonstrating 103× magnification of femtosecond waveforms,” Opt. Lett. 24, 783–785 (1999).
[CrossRef]

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

B. H. Kolner and M. Nazarathy, “Temporal imaging with a time lens: erratum,” Opt. Lett. 15, 655 (1990).
[CrossRef]

B. H. Kolner and M. Nazarathy, “Temporal imaging with a time lens,” Opt. Lett. 14, 630–632 (1989).
[CrossRef] [PubMed]

Laporta, P.

Leith, E.

Levit, B.

N. K. Berger, B. Levit, S. Atkins, and B. Fischer, “Time-lens-based spectral analysis of optical pulses by electrooptic phase modulation,” Electron. Lett. 36, 1644–1646 (2000).
[CrossRef]

Lohman, A. W.

Longhi, S.

Marano, M.

Mendlovic, D.

Mitschke, F.

F. Mitschke and U. Morgner, “The temporal Talbot effect,” Opt. Photon. News 9(6), 45–47 (1998).
[CrossRef]

Moodie, A. F.

J. M. Cowley and A. F. Moodie, “Fourier images. II. The out-of-focus patterns,” Proc. Phys. Soc. London 70, 497–504 (1957).
[CrossRef]

J. M. Cowley and A. F. Moodie, “Fourier images. I. The point source,” Proc. Phys. Soc. London 70, 486–496 (1957).
[CrossRef]

Morgner, U.

F. Mitschke and U. Morgner, “The temporal Talbot effect,” Opt. Photon. News 9(6), 45–47 (1998).
[CrossRef]

Muriel, M. A.

J. Azaña and M. A. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. 7, 728–744 (2001).
[CrossRef]

J. Azaña and M. A. Muriel, “Real-time optical spectrum analysis based on the time-space duality in chirped fiber gratings,” IEEE J. Quantum Electron. 36, 517–526 (2000).
[CrossRef]

M. A. Muriel, J. Azaña, and A. Carballar, “Real-time Fourier transformer based on fiber gratings,” Opt. Lett. 24, 1–3 (1999).
[CrossRef]

J. Azaña, L. R. Chen, M. A. Muriel, and P. W. E. Smith, “Experimental demonstration of real-time Fourier transforma-tion using linearly chirped fibre Bragg gratings,” Electron. Lett. 35, 2223–2224 (1999).
[CrossRef]

J. Azaña and M. A. Muriel, “Temporal Talbot effect in fiber gratings and its applications,” Appl. Opt. 38, 6700–6704 (1999).
[CrossRef]

J. Azaña and M. A. Muriel, “Technique for multiplying the repetition rates of periodic trains of pulses by means of a temporal self-imaging effect in chirped fiber gratings,” Opt. Lett. 24, 1672–1674 (1999).
[CrossRef]

Naulleau, P.

Nazarathy, M.

Papoulis, A.

Pruneri, V.

Smith, P. W. E.

J. Azaña, L. R. Chen, M. A. Muriel, and P. W. E. Smith, “Experimental demonstration of real-time Fourier transforma-tion using linearly chirped fibre Bragg gratings,” Electron. Lett. 35, 2223–2224 (1999).
[CrossRef]

Svelto, O.

Treacy, E. B.

E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5, 454–458 (1969).
[CrossRef]

Zervas, M. N.

Appl. Opt. (3)

Appl. Phys. Lett. (1)

A. A. Godil, B. A. Auld, and D. M. Bloom, “Time lens producing 1.9 ps optical pulses,” Appl. Phys. Lett. 62, 1047–1049 (1993).
[CrossRef]

Electron. Lett. (2)

J. Azaña, L. R. Chen, M. A. Muriel, and P. W. E. Smith, “Experimental demonstration of real-time Fourier transforma-tion using linearly chirped fibre Bragg gratings,” Electron. Lett. 35, 2223–2224 (1999).
[CrossRef]

N. K. Berger, B. Levit, S. Atkins, and B. Fischer, “Time-lens-based spectral analysis of optical pulses by electrooptic phase modulation,” Electron. Lett. 36, 1644–1646 (2000).
[CrossRef]

IEEE J. Quantum Electron. (5)

J. Azaña and M. A. Muriel, “Real-time optical spectrum analysis based on the time-space duality in chirped fiber gratings,” IEEE J. Quantum Electron. 36, 517–526 (2000).
[CrossRef]

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

C. V. Bennett and B. H. Kolner, “Principles of parametric temporal imaging. I. System configurations,” IEEE J. Quantum Electron. 36, 430–437 (2000).
[CrossRef]

C. V. Bennett and B. H. Kolner, “Principles of parametric temporal imaging. II. System performance,” IEEE J. Quantum Electron. 36, 649–655 (2000).
[CrossRef]

E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5, 454–458 (1969).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

J. Azaña and M. A. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. 7, 728–744 (2001).
[CrossRef]

J. Mod. Opt. (1)

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Opt. Lett. (7)

Opt. Photon. News (1)

F. Mitschke and U. Morgner, “The temporal Talbot effect,” Opt. Photon. News 9(6), 45–47 (1998).
[CrossRef]

Proc. Phys. Soc. London (2)

J. M. Cowley and A. F. Moodie, “Fourier images. I. The point source,” Proc. Phys. Soc. London 70, 486–496 (1957).
[CrossRef]

J. M. Cowley and A. F. Moodie, “Fourier images. II. The out-of-focus patterns,” Proc. Phys. Soc. London 70, 497–504 (1957).
[CrossRef]

Other (2)

A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill, New York, 1987).

K. Patorski, “The self-imaging phenomenon and its applications,” Progress in Optics XXVII, E. Wolf, ed. (Elsevier, Amsterdam, The Netherlands, ), 19891–108.

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Figures (3)

Fig. 1
Fig. 1

(a) General self-imaging problem (point-source illumination). (b) Spatial equivalent of the general self-imaging problem. (c) Temporal equivalent of the general self-imaging problem. The representations show the case of the integer self-imaging effect.

Fig. 2
Fig. 2

(a) General temporal self-imaging problem. (b) Equivalent conventional temporal self-imaging problem. The representations show the case of the integer self-imaging effect.

Fig. 3
Fig. 3

Numerical example of a temporal self-imaging system (Mt=-1/5, m=2). Upper plot: input pulse sequence. The dashed curve represents the temporal envelope of the sequence. Bottom plot: output pulse sequence. The dashed curve represents the scaled (compressed and reversed) version of the input temporal envelope. The inset shows an individual pulse of the input sequence (dashed curve, time scale at the top) and an individual pulse of the output sequence (solid curve, time scale at the bottom).

Tables (3)

Tables Icon

Table 1 Mathematical Equivalencies in Space–Time Duality Theory

Tables Icon

Table 2 Solutions of the General Temporal Self-Imaging Problem

Tables Icon

Table 3 Possible Temporal Imaging Configurations of the Time-Lens/Dispersion System

Equations (53)

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

z=N X2Mxλ,
Mx=(R+z)R
z=NmX2Mxλ,
l(t)exp[jϕ(t)]expj ϕ¨t2 t2,
H(ω)exp[jΦ(ω)]expj Φ¨ω2 ω2,
h(t)exp-j 12Φ¨ω t2.
a1(t)=q=-+a0(t-qT1)=p=-+apexpjp 2πT1 t,
ap=T1dta0(t)exp-jp 2πT1 t.
a2(t)=[a1(t)l(t)] * h(t),
a2(t)p=-p=+apexpjp 2πT1 t×expj ϕ¨t2 t2 * exp-j t22Φ¨ω.
a2(t)p=-+ap-+dτ expjp 2πT1 τ×expj ϕ¨t2 τ2exp-j (t-τ)22Φ¨ω=exp-j t22Φ¨ωp=-+apOp(t),
Op(t)=-+dτ expjp 2πT1 τ×expjϕ¨tΦ¨ω-12Φ¨ωτ2expj tτΦ¨ω.
exp(-ατ2)Iπαexp-ω24α,
expjϕ¨tΦ¨ω-12Φ¨ωτ2
Ij 2πΦ¨ωϕ¨tΦ¨ω-1exp-j Φ¨ωω22(ϕ¨tΦ¨ω-1)ω=-t/Φ¨ω.
δt=2πpΦ¨ω/T1.
Op(t)exp-j (t+δt)22Φ¨ω(ϕ¨tΦ¨ω-1)=expj 12Φ¨ω(1-ϕ¨tΦ¨ω) t2×expj Φ¨ω2(1-ϕ¨tΦ¨ω)p 2πT12×expjp 2πT1t(1-ϕ¨tΦ¨ω).
a2(t)expj ϕ¨t2Mt t2p=-+apexpj Φ¨ω2Mtp 2πT12×expjp 2πT1tMt,
Mt=1-ϕ¨tΦ¨ω.
Φ¨ω=Φ¨ωMt,
t=tMt.
a2(t)expj Mtϕ¨t2 t2a20(t),
a20(t)=p=-+apexpj Φ¨ω2p 2πT12expjp 2πT1 t.
exp[j(Φ¨ω/2)ω2]|ω=p(2π/T1)=exp{j(Φ¨ω/2)[p(2π/T1)]2}.
|Φ¨ω|=NmT122π,
a20(t)p=-+exp(jφp)a0[t-p(T1/m)],
|Φ¨ω|=N T122π |Mt|,
I2(t)I1(t/Mt)=I0(t/Mt) * q=-+δ(t-qMtT1).
|Φ¨ω|=NmT122π |Mt|,
I2(t)I0(t/Mt) * q=-+δ[t-(Mt/m)T1].
a1(t)=e(t)a1(t)=e(t)q=-+a0(t-qT1)=e(t)p=-+apexpjp 2πT1 t.
a2(t)=[a1(t)l(t)] * h(t)=[e(t)a1(t)l(t)] * h(t)=p=-+ape(t)l(t)expjp 2πT1 t * h(t),
a2(t)exp-j t22Φ¨ωp=-+apOp(t),
Op(t)=-+dτe(τ)expjp 2πT1 τ×expjϕ¨tΦ¨ω-12Φ¨ωτ2expj tτΦ¨ω.
Op(t)=Eω=-tΦ¨ω * -+dτ expjp 2πT1 τ×expjϕ¨tΦ¨ω-12Φ¨ωτ2expj tτΦ¨ω=E(ω=-t/Φ¨ω) * Op(t),
a2(t)exp-j t22Φ¨ω E(ω=-t/Φ¨ω) * p=-+apOp(t)=exp-j t22Φ¨ω E(ω=-t/Φ¨ω) * expj t22Φ¨ω ×exp-j t22Φ¨ωp=-+apOp(t)exp-j t22Φ¨ω ×E(ω=-t/Φ¨ω) * expj t22Φ¨ωa2(t),
a2(t)exp-j Mtt22Φ¨ω×E(ω=-t/Φ¨ω) * expj t22Φ¨ωa20(t).
a2(t)E(ω=-t/Φ¨ω) * expj t22Φ¨ω×p=-+exp(jφp)a0[t-p(T1/m)],
a2(t)E(ω=-t/Φ¨ω) * p=-+exp(jωipt)×exp(jφp)a0[t-p(T1/m)],
ωip=ωi(t=pT1/m)=pT1mΦ¨ω.
A2(ω)e(t=Φ¨ωω)×p=-+exp(jφp)exp[j(pT1/m)(ω-ωip)]×A0(ω-ωip),
e(Φ¨ωω)e(Φ¨ωωip)et=p T1m
A2(ω)p=-+exp(jφp)e[t=p(T1/m)]×exp[j(pT1/m)(ω-ωip)]A0(ω-ωip),
a2(t)p=-+exp(jφp)e[t=p(T1/m)]×exp(jωipt)a0[t-p(T1/m)].
I2(t)=|a2(t)|2p=-+|e[t=p(T1/m)]|2I0
×[t-p(T1/m)]
=|e(t)|2p=-+I0[t-p(T1/m)],
I2(t)|e(t/Mt)|2
×I0(t/Mt) * p=-+δ[t-(Mt/m)T1]
=|e(t/Mt)|2I2(t),
Φ¨ωΔωenvΔt0.
ΔtenvKenv2πNmT12Δt0,
NpulsesKenv2πNmT1Δt0.

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