Abstract

We propose and demonstrate time-domain equivalents of spatial zone plates, namely temporal zone plates, as alternatives to conventional time lenses. Both temporal intensity zone plates, based on intensity-only temporal modulation, and temporal phase zone plates, based on phase-only temporal modulation, are introduced and studied. Temporal zone plates do not exhibit the limiting tradeoff between temporal aperture and frequency bandwidth (temporal resolution) of conventional linear time lenses. As a result, these zone plates can be ideally designed to offer a time-bandwidth product (TBP) as large as desired, practically limited by the achievable temporal modulation bandwidth (limiting the temporal resolution) and the amount of dispersion needed in the target processing systems (limiting the temporal aperture). We numerically and experimentally demonstrate linear optical pulse compression by using temporal zone plates based on linear electro-optic temporal modulation followed by fiber-optics dispersion. In the pulse-compression experiment based on temporal phase zone plates, we achieve a resolution of ~25.5 ps over a temporal aperture of ~5.77 ns, representing an experimental TBP larger than 226 using a phase-modulation amplitude of only ~0.8π rad. We also numerically study the potential of these devices to achieve temporal imaging of optical waveforms and present a comparative analysis on the performance of different temporal intensity and phase zone plates.

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    [CrossRef]

2012

2010

2008

M. T. Flores-Arias, L. Chantada, C. Bao, M. V. Pérez, and C. Gómez-Reino, “Temporal zone plate,” J. Opt. Soc. Am. A25(12), 3077–3082 (2008).
[CrossRef] [PubMed]

R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “Optical time lens based on four-wave mixing on a silicon chip,” Opt. Lett.33(10), 1047–1049 (2008), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-33-10-1047 .
[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,” Nature456(7218), 81–84 (2008).
[CrossRef] [PubMed]

T. T. Ng, F. Parmigiani, M. Ibsen, Z. Zhang, P. Petropoulos, and D. J. Richardson, “Compensation of linear distortions by using XPM with parabolic pulses as a time lens,” IEEE Photon. Technol. Lett.20(13), 1097–1099 (2008).
[CrossRef]

2006

R. Llorente, R. Clavero, and J. Marti, “Performance analysis of polarimetric PMD monitoring by real-time optical Fourier transformers,” IEEE Photon. Technol. Lett.18(12), 1383–1385 (2006).
[CrossRef]

2004

2001

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(4), 728–744 (2001).
[CrossRef]

1999

1998

C. E. Shannon, “Communication In The Presence Of Noise,” Proc. IEEE86(2), 447–457 (1998).
[CrossRef]

1997

1994

B. H. Kolner, “Generalization of the concepts of focal length and f-number to space and time,” J. Opt. Soc. Am. A11(12), 3229–3234 (1994).
[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]

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]

1984

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

1978

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

1975

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]

1974

1968

1967

1966

1898

R. W. Wood, “LIII. Phase-reversal zone-plates, and diffraction-telescopes,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 45, 511-522 (1898).

Andres, 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.

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(4), 728–744 (2001).
[CrossRef]

M. A. Muriel, J. Azaña, and A. Carballar, “Real-time Fourier transformer based on fiber gratings,” Opt. Lett.24(1), 1–3 (1999), http://ol.osa.org/abstract.cfm?URI=ol-24-1-1 .
[CrossRef] [PubMed]

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.

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]

Burdet, N.

Carballar, A.

Chantada, L.

Chu, Y. S.

Clavero, R.

R. Llorente, R. Clavero, and J. Marti, “Performance analysis of polarimetric PMD monitoring by real-time optical Fourier transformers,” IEEE Photon. Technol. Lett.18(12), 1383–1385 (2006).
[CrossRef]

Collins, L. F.

Flores-Arias, M. T.

Fontaine, N. K.

R. P. Scott, N. K. Fontaine, D. J. Geisler, and S. Yoo, “Frequency-to-time-assisted interferometry for full-field optical waveform measurements with picosecond resolution and microsecond record lengths,” IEEE Photon. J.4(3), 748–758 (2012).
[CrossRef]

Foster, M. A.

Gaeta, A. L.

Geisler, D. J.

R. P. Scott, N. K. Fontaine, D. J. Geisler, and S. Yoo, “Frequency-to-time-assisted interferometry for full-field optical waveform measurements with picosecond resolution and microsecond record lengths,” IEEE Photon. J.4(3), 748–758 (2012).
[CrossRef]

Geraghty, D. F.

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.

Harder, R.

Horman, M. H.

Huang, X.

Ibsen, M.

T. T. Ng, F. Parmigiani, M. Ibsen, Z. Zhang, P. Petropoulos, and D. J. Richardson, “Compensation of linear distortions by using XPM with parabolic pulses as a time lens,” IEEE Photon. Technol. Lett.20(13), 1097–1099 (2008).
[CrossRef]

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]

Kirz, J.

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, “The pinhole time camera,” J. Opt. Soc. Am. A14(12), 3349–3357 (1997).
[CrossRef]

B. H. Kolner, “Generalization of the concepts of focal length and f-number to space and time,” J. Opt. Soc. Am. A11(12), 3229–3234 (1994).
[CrossRef]

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

Kumar, K.

Lancis, J.

Legnini, D.

Lipson, M.

Llorente, R.

R. Llorente, R. Clavero, and J. Marti, “Performance analysis of polarimetric PMD monitoring by real-time optical Fourier transformers,” IEEE Photon. Technol. Lett.18(12), 1383–1385 (2006).
[CrossRef]

Marti, J.

R. Llorente, R. Clavero, and J. Marti, “Performance analysis of polarimetric PMD monitoring by real-time optical Fourier transformers,” IEEE Photon. Technol. Lett.18(12), 1383–1385 (2006).
[CrossRef]

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]

Morrison, G. R.

Munioz-Camuniez, L. E.

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(4), 728–744 (2001).
[CrossRef]

M. A. Muriel, J. Azaña, and A. Carballar, “Real-time Fourier transformer based on fiber gratings,” Opt. Lett.24(1), 1–3 (1999), http://ol.osa.org/abstract.cfm?URI=ol-24-1-1 .
[CrossRef] [PubMed]

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]

Ng, T. T.

T. T. Ng, F. Parmigiani, M. Ibsen, Z. Zhang, P. Petropoulos, and D. J. Richardson, “Compensation of linear distortions by using XPM with parabolic pulses as a time lens,” IEEE Photon. Technol. Lett.20(13), 1097–1099 (2008).
[CrossRef]

Ojeda-Castaneda, J.

Parmigiani, F.

T. T. Ng, F. Parmigiani, M. Ibsen, Z. Zhang, P. Petropoulos, and D. J. Richardson, “Compensation of linear distortions by using XPM with parabolic pulses as a time lens,” IEEE Photon. Technol. Lett.20(13), 1097–1099 (2008).
[CrossRef]

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]

Pérez, M. V.

Peterson, I.

Petropoulos, P.

T. T. Ng, F. Parmigiani, M. Ibsen, Z. Zhang, P. Petropoulos, and D. J. Richardson, “Compensation of linear distortions by using XPM with parabolic pulses as a time lens,” IEEE Photon. Technol. Lett.20(13), 1097–1099 (2008).
[CrossRef]

Richardson, D. J.

T. T. Ng, F. Parmigiani, M. Ibsen, Z. Zhang, P. Petropoulos, and D. J. Richardson, “Compensation of linear distortions by using XPM with parabolic pulses as a time lens,” IEEE Photon. Technol. Lett.20(13), 1097–1099 (2008).
[CrossRef]

Robinson, I. K.

Salem, R.

Scott, R. P.

R. P. Scott, N. K. Fontaine, D. J. Geisler, and S. Yoo, “Frequency-to-time-assisted interferometry for full-field optical waveform measurements with picosecond resolution and microsecond record lengths,” IEEE Photon. J.4(3), 748–758 (2012).
[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]

Shannon, C. E.

C. E. Shannon, “Communication In The Presence Of Noise,” Proc. IEEE86(2), 447–457 (1998).
[CrossRef]

Torres-Company, V.

Turner, A. C.

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,” Nature456(7218), 81–84 (2008).
[CrossRef] [PubMed]

van Howe, J.

Vine, D. J.

Waldman, G. S.

Wang, L.

Wang, Y.

Wigmore, J.

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

Wojcik, M.

Wood, R. W.

R. W. Wood, “LIII. Phase-reversal zone-plates, and diffraction-telescopes,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 45, 511-522 (1898).

Xu, C.

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]

Yoo, S.

R. P. Scott, N. K. Fontaine, D. J. Geisler, and S. Yoo, “Frequency-to-time-assisted interferometry for full-field optical waveform measurements with picosecond resolution and microsecond record lengths,” IEEE Photon. J.4(3), 748–758 (2012).
[CrossRef]

Zhang, X.

Zhang, Z.

T. T. Ng, F. Parmigiani, M. Ibsen, Z. Zhang, P. Petropoulos, and D. J. Richardson, “Compensation of linear distortions by using XPM with parabolic pulses as a time lens,” IEEE Photon. Technol. Lett.20(13), 1097–1099 (2008).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

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.

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]

IEEE J. Sel. Top. Quantum Electron.

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(4), 728–744 (2001).
[CrossRef]

IEEE Photon. J.

R. P. Scott, N. K. Fontaine, D. J. Geisler, and S. Yoo, “Frequency-to-time-assisted interferometry for full-field optical waveform measurements with picosecond resolution and microsecond record lengths,” IEEE Photon. J.4(3), 748–758 (2012).
[CrossRef]

IEEE Photon. Technol. Lett.

T. T. Ng, F. Parmigiani, M. Ibsen, Z. Zhang, P. Petropoulos, and D. J. Richardson, “Compensation of linear distortions by using XPM with parabolic pulses as a time lens,” IEEE Photon. Technol. Lett.20(13), 1097–1099 (2008).
[CrossRef]

R. Llorente, R. Clavero, and J. Marti, “Performance analysis of polarimetric PMD monitoring by real-time optical Fourier transformers,” IEEE Photon. Technol. Lett.18(12), 1383–1385 (2006).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

LIII. Phase-reversal zone-plates, and diffraction-telescopes

R. W. Wood, “LIII. Phase-reversal zone-plates, and diffraction-telescopes,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 45, 511-522 (1898).

Nature

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature456(7218), 81–84 (2008).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

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

C. E. Shannon, “Communication In The Presence Of Noise,” Proc. IEEE86(2), 447–457 (1998).
[CrossRef]

Other

R. W. Wood, “Zone-plate,” in Physical optics, (The Macmillan Company, New York, 1934).

L. Rayleigh, “Wave Theory of Light,” in Encyclopedia Britannica, 9th ed., 24, 429 (1888).

B. H. Kolner, Broadband Optical Modulators (CRC Press, 2011), Chap. 19.

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Figures (16)

Fig. 1
Fig. 1

Space-time duality. (a) Light focusing by a spatial (intensity) zone plate. (b) Pulse compression by a temporal intensity zone plate.

Fig. 2
Fig. 2

Experimental scheme for linear optical pulse compression using the temporal GZP concept, with the terminology used in the text.

Fig. 3
Fig. 3

Ideal optical waveforms (left vertical axis), measured electronic waveforms (right vertical axis), and measured optical waveforms (left vertical axis) for temporal intensity modulation in the implemented temporal GZPs when the used dispersion values are (a) 10000 ps/nm and (b) 6667 ps/nm, respectively. All optical waveforms are represented in normalized units.

Fig. 4
Fig. 4

Spectra of the CW light and the modulated light for two regimes in which dispersion values of (a) 10000 ps/nm and (b) 6667 ps/nm are used, respectively. The spectra are measured by an optical spectrum analyzer, which has a resolution of 0.01 nm.

Fig. 5
Fig. 5

Temporally compressed intensity pulse waveforms in the ideal case and experiment using temporal GZPs when the used dispersion values are (a) 10000 ps/nm and (b) 6667 ps/nm, respectively. All waveforms are represented in normalized units.

Fig. 6
Fig. 6

The intensity-modulation profiles of temporal FZPs and temporal GZPs when the used dispersion values are (a) 10000 ps/nm and (b) 6667 ps/nm, respectively. The output compressed pulses in (c), (e) and (d), (f) correspond to (a) and (b), respectively.

Fig. 7
Fig. 7

The spectra of the modulating signals shown in Fig. 6. Figure 7(a) and 7(b) correspond to Fig. 6(a) and 6(b), respectively. Figure 7(c) and 7(d) show a zoom around the bottom of Fig. 7(a) and 7(b), respectively.

Fig. 8
Fig. 8

Space-time duality. (a) Light focusing by a spatial phase zone plate. (b) Pulse compression by a temporal phase zone plate.

Fig. 9
Fig. 9

(a) The light-collecting efficiency of temporal GPZP. The optimum phase-modulation amplitude for each order n is indicated by arrows. (b) The light-collecting efficiency of temporal GPZP and temporal RWZP, respectively.

Fig. 10
Fig. 10

Experimental scheme for linear optical pulse compression using the GPZP concept.

Fig. 11
Fig. 11

Ideal and experimentally measured electronic waveforms for temporal phase modulation in the implemented temporal GPZPs for orders (a) n = 1, (b) n = 2, and (c) n = 3.

Fig. 12
Fig. 12

Spectra of the CW light and the light modulated by temporal GPZPs for orders (a) n = 1, (b) n = 2, and (c) n = 3. The spectra are measured by an optical spectrum analyzer, which has a resolution of 0.01 nm.

Fig. 13
Fig. 13

Temporally compressed intensity waveforms in the ideal case, simulation, and experiment using temporal GPZPs of orders (a) n = 1, (b) n = 2, and (c) n = 3. (d)-(f) show a closer view of the compressed optical pulses in (a)-(c). All waveforms are represented in normalized units.

Fig. 14
Fig. 14

Phase-modulation profiles for temporal RWZPs and temporal GPZPs of orders (a) n = 1, (b) n = 2, and (c) n = 3. The outputs in (d)-(f) correspond to (a)-(c), respectively. The left insets in (d)-(f) show a closer view of the output compressed optical pulses. The right insets in (d)-(f) are normalized spectra of the output compressed optical pulses.

Fig. 15
Fig. 15

Temporal imaging system, where the optical carrier frequency of input optical signal is 1550 nm, dispersion ϕ1″ and ϕ2″ are 10000 ps/nm and 2000 ps/nm, respectively.

Fig. 16
Fig. 16

The temporal intensity profiles (a)-(d) and spectra (e)-(h) of the output optical signals when the time lens in Fig. 15 is replaced with different zone plates.

Equations (20)

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A F ( t )=1/2 +( 1/2 )sgn[ cos( a t 2 ) ],
A G ( t )=1/2 +( 1/2 )cos( a t 2 ),
A F ( t )= n= [ sin( nπ /2 ) / nπ ]exp( jna t 2 ) ,
A G ( t )= n=1 1 ( 1/2 n 2 /4 )exp( jna t 2 ) ,
η F = [ sin( nπ /2 ) / nπ ] 2 ,
η G = ( 1/2 n 2 /4 ) 2 .
δ τ F 3.5π / | n | f s ,
δ τ G 1/ | n | f s .
TB P F = Δ t F / δ τ F | n | f s 2 / 12.6| a |π ,
TB P G = Δ t G / δ τ G | n |π f s 2 / | a | ,
ϕ RW ( t )=π/2 +( π/2 )sgn[ cos( a t 2 ) ],
ϕ GP ( t )= Γ 0 cos( a t 2 )
H RW ( t )=exp[ j ϕ RW ( t ) ]=2 n=,n0 [ sin( nπ /2 ) / nπ ] exp( jna t 2 ),
H GP ( t )=exp[ j ϕ GP ( t ) ]= n= j n J n ( Γ 0 )exp( jna t 2 ) ,
η RW =4 [ sin( nπ /2 ) / nπ ] 2 ,
η G = [ J n ( Γ 0 ) ] 2 .
δ τ RW 3.5π / | n | f s ,
δ τ GP 1/ | n | f s .
TB P RW = Δ t RW / δ τ RW | n | f s 2 / 12.6| a |π ,
TB P GP = Δ t GP / δ τ GP π| n | f s 2 / | a |

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