Abstract

Temporal holograms can be realized by temporal amplitude-only modulation devices and used for generation and processing of complex (amplitude and phase) time-domain signals. Based on the temporal hologram concept, we numerically and experimentally demonstrate a novel design for linear optical pulse compression using temporal modulation of continuous-wave light combined with dispersion. The newly introduced scheme overcomes the undesired background problem that is intrinsic to designs based on temporal zone plates, while also offering an energy efficiency of ~25%. This pulse compression scheme can ideally provide an arbitrarily high time-bandwidth product using a low peak-power modulation driving signal, though in practice it is limited by the achievable modulation bandwidth and dispersion amount.

© 2015 Optical Society of America

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References

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  1. B. H. Kolner, “Active pulse compression using an integrated electro-optic phase modulator,” Appl. Phys. Lett. 52(14), 1122–1124 (1988).
    [Crossref]
  2. J. van Howe, J. Hansryd, and C. Xu, “Multiwavelength pulse generator using time-lens compression,” Opt. Lett. 29(13), 1470–1472 (2004).
    [Crossref] [PubMed]
  3. K. Wang and C. Xu, “Wavelength-tunable high-energy soliton pulse generation from a large-mode-area fiber pumped by a time-lens source,” Opt. Lett. 36(6), 942–944 (2011).
    [Crossref] [PubMed]
  4. T. Jannson and J. Jannson, “Temporal self-imaging effect in single-mode fibers,” J. Opt. Soc. Am. 71, 1373–1376 (1981).
  5. Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53(11), 1057–1060 (1984).
    [Crossref]
  6. B. H. Kolner, “Generalization of the concepts of focal length and f-number to space and time,” J. Opt. Soc. Am. A 11(12), 3229–3234 (1994).
    [Crossref]
  7. C. V. Bennett, R. P. Scott, and B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up-conversion time microscope,” Appl. Phys. Lett. 65(20), 2513–2515 (1994).
    [Crossref]
  8. B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30(8), 1951–1963 (1994).
    [Crossref]
  9. J. Wigmore and D. Grischkowsky, “Temporal compression of light,” IEEE J. Quantum Electron. 14(4), 310–315 (1978).
    [Crossref]
  10. J. Bjorkholm, E. Turner, and D. Pearson, “Conversion of cw light into a train of subnanosecond pulses using frequency modulation and the dispersion of a near‐resonant atomic vapor,” Appl. Phys. Lett. 26(10), 564–566 (1975).
    [Crossref]
  11. M. Kauffman, W. Banyai, A. Godil, and D. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64(3), 270–272 (1994).
    [Crossref]
  12. B. H. Kolner, “Electro-optic time lenses for shaping and imaging optical waveforms,” in Broadband Optical Modulators, Antao Chen and E. J. Murphy, eds. (CRC Press, 2011), pp. 427–454.
  13. M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature 456(7218), 81–84 (2008).
    [Crossref] [PubMed]
  14. L. E. Munioz-Camuniez, V. Torres-Company, J. Lancis, J. Ojeda-Castañeda, and P. Andrés, “Electro-optic time lens with an extended time aperture,” J. Opt. Soc. Am. B 27(10), 2110–2115 (2010).
    [Crossref]
  15. M. T. Flores-Arias, L. Chantada, C. Bao, M. V. Pérez, and C. Gómez-Reino, “Temporal zone plate,” J. Opt. Soc. Am. A 25(12), 3077–3082 (2008).
    [Crossref] [PubMed]
  16. B. Li, M. Li, S. Lou, and J. Azaña, “Linear optical pulse compression based on temporal zone plates,” Opt. Express 21(14), 16814–16830 (2013).
    [Crossref] [PubMed]
  17. B. Li, S. Lou, and J. Azaña, “Novel temporal zone plate designs with improved energy efficiency and noise performance,” J. Lightwave Technol. 32, 4201–4207 (2014).
  18. J. F. Heanue, M. C. Bashaw, and L. Hesselink, “Volume holographic storage and retrieval of digital data,” Science 265(5173), 749–752 (1994).
    [Crossref] [PubMed]
  19. T. Zhang and I. Yamaguchi, “Three-dimensional microscopy with phase-shifting digital holography,” Opt. Lett. 23(15), 1221–1223 (1998).
    [Crossref] [PubMed]
  20. G. Pedrini and H. J. Tiziani, “Short-coherence digital microscopy by use of a lensless holographic imaging system,” Appl. Opt. 41(22), 4489–4496 (2002).
    [Crossref] [PubMed]
  21. M. R. Fernández-Ruiz, M. Li, and J. Azaña, “Time-domain holograms for generation and processing of temporal complex information by intensity-only modulation processes,” Opt. Express 21(8), 10314–10323 (2013).
    [Crossref] [PubMed]
  22. C. E. Shannon, “Communication In The Presence Of Noise,” Proc. IEEE 86(2), 447–457 (1998).
    [Crossref]

2014 (1)

2013 (2)

2011 (1)

2010 (1)

2008 (2)

M. T. Flores-Arias, L. Chantada, C. Bao, M. V. Pérez, and C. Gómez-Reino, “Temporal zone plate,” J. Opt. Soc. Am. A 25(12), 3077–3082 (2008).
[Crossref] [PubMed]

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature 456(7218), 81–84 (2008).
[Crossref] [PubMed]

2004 (1)

2002 (1)

1998 (2)

1994 (5)

B. H. Kolner, “Generalization of the concepts of focal length and f-number to space and time,” J. Opt. Soc. Am. A 11(12), 3229–3234 (1994).
[Crossref]

C. V. Bennett, R. P. Scott, and B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up-conversion time microscope,” Appl. Phys. Lett. 65(20), 2513–2515 (1994).
[Crossref]

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30(8), 1951–1963 (1994).
[Crossref]

J. F. Heanue, M. C. Bashaw, and L. Hesselink, “Volume holographic storage and retrieval of digital data,” Science 265(5173), 749–752 (1994).
[Crossref] [PubMed]

M. Kauffman, W. Banyai, A. Godil, and D. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64(3), 270–272 (1994).
[Crossref]

1988 (1)

B. H. Kolner, “Active pulse compression using an integrated electro-optic phase modulator,” Appl. Phys. Lett. 52(14), 1122–1124 (1988).
[Crossref]

1984 (1)

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53(11), 1057–1060 (1984).
[Crossref]

1981 (1)

1978 (1)

J. Wigmore and D. Grischkowsky, “Temporal compression of light,” IEEE J. Quantum Electron. 14(4), 310–315 (1978).
[Crossref]

1975 (1)

J. Bjorkholm, E. Turner, and D. Pearson, “Conversion of cw light into a train of subnanosecond pulses using frequency modulation and the dispersion of a near‐resonant atomic vapor,” Appl. Phys. Lett. 26(10), 564–566 (1975).
[Crossref]

Andrés, P.

Arinaga, S.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53(11), 1057–1060 (1984).
[Crossref]

Azaña, J.

Banyai, W.

M. Kauffman, W. Banyai, A. Godil, and D. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64(3), 270–272 (1994).
[Crossref]

Bao, C.

Bashaw, M. C.

J. F. Heanue, M. C. Bashaw, and L. Hesselink, “Volume holographic storage and retrieval of digital data,” Science 265(5173), 749–752 (1994).
[Crossref] [PubMed]

Bennett, C. V.

C. V. Bennett, R. P. Scott, and B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up-conversion time microscope,” Appl. Phys. Lett. 65(20), 2513–2515 (1994).
[Crossref]

Bjorkholm, J.

J. Bjorkholm, E. Turner, and D. Pearson, “Conversion of cw light into a train of subnanosecond pulses using frequency modulation and the dispersion of a near‐resonant atomic vapor,” Appl. Phys. Lett. 26(10), 564–566 (1975).
[Crossref]

Bloom, D.

M. Kauffman, W. Banyai, A. Godil, and D. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64(3), 270–272 (1994).
[Crossref]

Chantada, L.

Fernández-Ruiz, M. R.

Flores-Arias, M. T.

Foster, M. A.

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature 456(7218), 81–84 (2008).
[Crossref] [PubMed]

Gaeta, A. L.

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature 456(7218), 81–84 (2008).
[Crossref] [PubMed]

Geraghty, D. F.

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature 456(7218), 81–84 (2008).
[Crossref] [PubMed]

Godil, A.

M. Kauffman, W. Banyai, A. Godil, and D. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64(3), 270–272 (1994).
[Crossref]

Gómez-Reino, C.

Grischkowsky, D.

J. Wigmore and D. Grischkowsky, “Temporal compression of light,” IEEE J. Quantum Electron. 14(4), 310–315 (1978).
[Crossref]

Hansryd, J.

Heanue, J. F.

J. F. Heanue, M. C. Bashaw, and L. Hesselink, “Volume holographic storage and retrieval of digital data,” Science 265(5173), 749–752 (1994).
[Crossref] [PubMed]

Hesselink, L.

J. F. Heanue, M. C. Bashaw, and L. Hesselink, “Volume holographic storage and retrieval of digital data,” Science 265(5173), 749–752 (1994).
[Crossref] [PubMed]

Jannson, J.

Jannson, T.

Kato, Y.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53(11), 1057–1060 (1984).
[Crossref]

Kauffman, M.

M. Kauffman, W. Banyai, A. Godil, and D. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64(3), 270–272 (1994).
[Crossref]

Kitagawa, Y.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53(11), 1057–1060 (1984).
[Crossref]

Kolner, B. H.

B. H. Kolner, “Generalization of the concepts of focal length and f-number to space and time,” J. Opt. Soc. Am. A 11(12), 3229–3234 (1994).
[Crossref]

C. V. Bennett, R. P. Scott, and B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up-conversion time microscope,” Appl. Phys. Lett. 65(20), 2513–2515 (1994).
[Crossref]

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30(8), 1951–1963 (1994).
[Crossref]

B. H. Kolner, “Active pulse compression using an integrated electro-optic phase modulator,” Appl. Phys. Lett. 52(14), 1122–1124 (1988).
[Crossref]

Lancis, J.

Li, B.

Li, M.

Lipson, M.

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature 456(7218), 81–84 (2008).
[Crossref] [PubMed]

Lou, S.

Mima, K.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53(11), 1057–1060 (1984).
[Crossref]

Miyanaga, N.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53(11), 1057–1060 (1984).
[Crossref]

Munioz-Camuniez, L. E.

Nakatsuka, M.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53(11), 1057–1060 (1984).
[Crossref]

Ojeda-Castañeda, J.

Pearson, D.

J. Bjorkholm, E. Turner, and D. Pearson, “Conversion of cw light into a train of subnanosecond pulses using frequency modulation and the dispersion of a near‐resonant atomic vapor,” Appl. Phys. Lett. 26(10), 564–566 (1975).
[Crossref]

Pedrini, G.

Pérez, M. V.

Salem, R.

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature 456(7218), 81–84 (2008).
[Crossref] [PubMed]

Scott, R. P.

C. V. Bennett, R. P. Scott, and B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up-conversion time microscope,” Appl. Phys. Lett. 65(20), 2513–2515 (1994).
[Crossref]

Shannon, C. E.

C. E. Shannon, “Communication In The Presence Of Noise,” Proc. IEEE 86(2), 447–457 (1998).
[Crossref]

Tiziani, H. J.

Torres-Company, V.

Turner, E.

J. Bjorkholm, E. Turner, and D. Pearson, “Conversion of cw light into a train of subnanosecond pulses using frequency modulation and the dispersion of a near‐resonant atomic vapor,” Appl. Phys. Lett. 26(10), 564–566 (1975).
[Crossref]

Turner-Foster, A. C.

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature 456(7218), 81–84 (2008).
[Crossref] [PubMed]

van Howe, J.

Wang, K.

Wigmore, J.

J. Wigmore and D. Grischkowsky, “Temporal compression of light,” IEEE J. Quantum Electron. 14(4), 310–315 (1978).
[Crossref]

Xu, C.

Yamaguchi, I.

Yamanaka, C.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53(11), 1057–1060 (1984).
[Crossref]

Zhang, T.

Appl. Opt. (1)

Appl. Phys. Lett. (4)

B. H. Kolner, “Active pulse compression using an integrated electro-optic phase modulator,” Appl. Phys. Lett. 52(14), 1122–1124 (1988).
[Crossref]

C. V. Bennett, R. P. Scott, and B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up-conversion time microscope,” Appl. Phys. Lett. 65(20), 2513–2515 (1994).
[Crossref]

J. Bjorkholm, E. Turner, and D. Pearson, “Conversion of cw light into a train of subnanosecond pulses using frequency modulation and the dispersion of a near‐resonant atomic vapor,” Appl. Phys. Lett. 26(10), 564–566 (1975).
[Crossref]

M. Kauffman, W. Banyai, A. Godil, and D. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64(3), 270–272 (1994).
[Crossref]

IEEE J. Quantum Electron. (2)

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30(8), 1951–1963 (1994).
[Crossref]

J. Wigmore and D. Grischkowsky, “Temporal compression of light,” IEEE J. Quantum Electron. 14(4), 310–315 (1978).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

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

Nature (1)

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature 456(7218), 81–84 (2008).
[Crossref] [PubMed]

Opt. Express (2)

Opt. Lett. (3)

Phys. Rev. Lett. (1)

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53(11), 1057–1060 (1984).
[Crossref]

Proc. IEEE (1)

C. E. Shannon, “Communication In The Presence Of Noise,” Proc. IEEE 86(2), 447–457 (1998).
[Crossref]

Science (1)

J. F. Heanue, M. C. Bashaw, and L. Hesselink, “Volume holographic storage and retrieval of digital data,” Science 265(5173), 749–752 (1994).
[Crossref] [PubMed]

Other (1)

B. H. Kolner, “Electro-optic time lenses for shaping and imaging optical waveforms,” in Broadband Optical Modulators, Antao Chen and E. J. Murphy, eds. (CRC Press, 2011), pp. 427–454.

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Figures (15)

Fig. 1
Fig. 1 Scheme for linear optical pulse compression using the temporal hologram concept, with the terminology used in the text.
Fig. 2
Fig. 2 Numerically calculated (thick orange lines) and measured (thin black lines) electronic waveforms for temporal amplitude modulation in the implemented temporal holograms. (a)-(d) correspond to four temporal holograms with different (increasing) temporal apertures.
Fig. 3
Fig. 3 Numerically calculated (thick orange lines) and measured (thin black lines) optical intensity waveforms after temporal amplitude modulation. (a)-(d) correspond to the four temporal holograms with different temporal apertures shown in Figs. 2(a)-2(d).
Fig. 4
Fig. 4 Numerically calculated (thick orange lines) and measured (thin black lines) spectra of the optical waveforms after temporal amplitude modulation. (a)-(d) correspond to the four temporal holograms with different temporal apertures shown in Figs. 2(a)-2(d).
Fig. 5
Fig. 5 Numerically calculated (thick orange lines) and measured (thin black lines) optical spectra after edge-pass filtering. (a)-(d) correspond to the four temporal holograms with different temporal apertures shown in Figs. 2(a)-2(d).
Fig. 6
Fig. 6 Numerically calculated (thick orange lines) and measured (thin black lines) output optical waveforms after LCFBG. (a)-(d) correspond to the four temporal holograms with different temporal apertures shown in Figs. 2(a)-2(d).
Fig. 7
Fig. 7 Numerically calculated (thick orange lines) and measured (thin black lines) electronic waveforms for temporal amplitude modulation in the implemented temporal holograms. (a)-(c) correspond to three temporal holograms with different (increasing) frequency shifts.
Fig. 8
Fig. 8 Numerically calculated (thick orange lines) and measured (thin black lines) optical intensity waveforms after temporal amplitude modulation. (a)-(c) correspond to the three temporal holograms with different frequency shifts shown in Figs. 7(a)-7(c).
Fig. 9
Fig. 9 Numerically calculated (thick orange lines) and measured (thin black lines) spectra of the optical modulated waveforms. (a)-(c) correspond to the three temporal holograms with different frequency shifts shown in Figs. 7(a)-7(c).
Fig. 10
Fig. 10 Numerically calculated (thick orange lines) and measured (thin black lines) spectra of the optical spectra after edge-pass filtering. (a)-(c) correspond to the three temporal holograms with different frequency shifts shown in Figs. 7(a)-7(c).
Fig. 11
Fig. 11 Numerically calculated (thick orange lines) and measured (thin black lines) output optical waveforms after LCFBG. (a)-(c) correspond to the three temporal holograms with different frequency shifts shown in Figs. 7(a)-7(c).
Fig. 12
Fig. 12 Numerically calculated electronic waveforms for temporal amplitude modulation (a-c) and for temporal phase modulation (d). (a)-(d) correspond to the implemented TIZP (a), TAZP (b), temporal hologram (c) and time lens (d), respectively.
Fig. 13
Fig. 13 Numerically calculated optical intensity waveforms after temporal amplitude modulation (a-c) and temporal phase modulation (d). (a)-(d) correspond to the implemented TIZP (a), TAZP (b), temporal hologram (c) and time lens (d), respectively.
Fig. 14
Fig. 14 Numerically calculated output optical waveforms after LCFBG. (a)-(d) correspond to the implemented TIZP (a), TAZP (b), temporal hologram (c) and time lens (d), respectively, as shown in Fig. 12. To help visualizing the output signal background, (e)-(g), respectively corresponding to (a)-(d), show the intensity profiles in log scale.
Fig. 15
Fig. 15 Numerically calculated output optical spectra. (a)-(d) correspond to the implemented TIZP (a), TAZP (b), temporal hologram (c) and time lens (d), respectively, as shown in Fig. 12. The inset in (a) shows a zoom of the bottom of the compressed pulse spectrum.

Tables (1)

Tables Icon

Table 1 The performance of various linear pulse compression schemes.

Equations (14)

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A( t )= FT -1 [ A ˜ ( ω ) ]= 1 2π A ˜ ( ω )exp( jωt )dω ,
A ˜ ( ω )=FT[ A( t ) ]= A( t )exp( jωt )dt ,
A Mod ( t )= A s0 ( t )cos[ ω i t+ ϕ 0 ( t ) ] = 1 2 A s0 ( t )exp[ j ϕ 0 ( t ) ]exp( j ω i t )+ 1 2 A s0 ( t )exp[ j ϕ 0 ( t ) ]exp( j ω i t ) = 1 2 E s ( t )exp( j ω i t )+ 1 2 E s ( t )exp( j ω i t ),
A ˜ Mod ( ω )=FT[ A Mod ( t ) ] = 1 4π FT[ E s ( t ) ]δ( ω ω i )+ 1 4π FT[ E s ( t ) ]δ( ω+ ω i ).
E Fil ( t )= FT -1 [ E ˜ Fil ( ω ) ] = FT -1 [ 1 4π FT[ E s ( t ) ]δ( ω ω i ) ] = 1 2 E s ( t )exp( j ω i t ).
E Ch ( t )= A s0 ( t )exp( j t 2 2 Φ ¨ 0 ).
A Mod ( t )= A s0 ( t )cos( ω i t t 2 2 Φ ¨ 0 ).
A ˜ Mod ( ω )=FT[ A Mod ( t ) ] = 1 4π FT[ E Ch ( t ) ]δ( ω ω i )+ 1 4π FT[ E Ch ( t ) ]δ( ω+ ω i ).
ΔT2| Φ ¨ 0 ω i |.
f s 1 π ( | ω i |+ ΔT 2| Φ ¨ 0 | ) ΔT π| Φ ¨ 0 | .
δt= 1 Δf = 2π| Φ ¨ 0 | ΔT 2 f s .
TBP= ΔT δt = Δ T 2 2π| Φ ¨ 0 | π| Φ ¨ 0 | f s 2 2 .
E Fil ( t )= FT 1 { 1 4π FT[ E Ch ( t ) ]δ( ω ω i ) } = 1 2 E Ch ( t )exp( j ω i t ).
A Out ( τ )= E Fil ( t )[ h time exp( j t 2 2 Φ ¨ 0 ) ] = 1 2 h time exp[ j ( τ+ Φ ¨ 0 ω i ) 2 2 Φ ¨ 0 ] A ˜ s0 ( τ Φ ¨ 0 ),

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