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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 1 — Jan. 14, 2013
  • pp: 238–244
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Design and demonstration of temporal cloaks with and without the time gap

Kedi Wu and Guo Ping Wang  »View Author Affiliations


Optics Express, Vol. 21, Issue 1, pp. 238-244 (2013)
http://dx.doi.org/10.1364/OE.21.000238


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Abstract

We introduce a Fourier analysis method to design temporal cloaks for hiding events in time domain. The cloaks are constructed with two linear time-invariant filters with different transfer functions, which can create a temporal gap and then closed it orderly, making any events occurring during the gap not detectable outside. We further reveal that even a no-gap temporal cloak can also hide events. All the analytical results are verified by fast Fourier transformation simulations.

© 2013 OSA

1. Introduction

Since the pioneering work on spatial invisibility cloaks for rendering any object in a specific region of space invisible [1

1. U. Leonhardt, “Optical Conformal Mapping,” Science 312(5781), 1777–1780 (2006). [CrossRef] [PubMed]

,2

2. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]

], rapid developments in artificial composite materials has allowed different kinds of spatial invisibility cloaks to be designed theoretically and realized experimentally [3

3. H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010). [CrossRef] [PubMed]

8

8. B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic Invisibility Cloak for Visible Light,” Phys. Rev. Lett. 106(3), 033901 (2011). [CrossRef] [PubMed]

]. On the other hand, another kind of distant cloaking devices based on complementary media has been proposed and numerically demonstrated to not only hide distant objects [9

9. Y. Lai, H. Y. Chen, Z.-Q. Zhang, and C. T. Chan, “Complementary Media Invisibility Cloak that Cloaks Objects at a Distance Outside the Cloaking Shell,” Phys. Rev. Lett. 102(9), 093901 (2009). [CrossRef] [PubMed]

] but also create illusions [10

10. Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z.-Q. Zhang, and C. T. Chan, “Illusion Optics: The Optical Transformation of an Object into Another Object,” Phys. Rev. Lett. 102(25), 253902 (2009). [CrossRef] [PubMed]

]. Recently, the concept of cloaking has even been theoretically and experimentally extended to cloaking in time, using a device that hides events occurring during a specific time window [11

11. M. W. McCall, A. Favaro, P. Kinsler, and A. Boardman, “A spacetime cloak, or a history editors,” J. Opt. 13(2), 024003 (2011). [CrossRef]

,12

12. M. Fridman, A. Farsi, Y. Okawachi, and A. L. Gaeta, “Demonstration of temporal cloaking,” Nature 481(7379), 62–65 (2012). [CrossRef] [PubMed]

].

2. Theoretical analysis

For an event (wave packet), it can be described as complex amplitude in time domain and frequency spectrum in frequency domain, respectively [18

18. B. H. Kolner, “Space-Time Duality and the Theory of Temporal Imaging,” IEEE J. Quantum Electron. 30(8), 1951–1963 (1994). [CrossRef]

,19

19. A. W. Lohmann and D. Mendlovic, “Temporal filtering with time lenses,” Appl. Opt. 31(29), 6212–6219 (1992). [CrossRef] [PubMed]

]
u(t)=12π+U(μ)exp(i2πμt)dμ=F1[U(μ)]
(1)
and
U(μ)=+u(t)exp(i2πμt)dt=F[u(t)]
(2)
respectively, where t is the time, μ=(ωω0)/2π is the normalized frequency with ω being the angular frequency and ω0 the center frequency, i is the imaginary unit, F and F1 represent the Fourier transform operator and inverse Fourier transform operator, respectively.

It is known from phase-shift theorem that a phase shift in frequency domain will lead to delay of an event in time domain [20

20. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (McGraw-Hill, 2005).

,21

21. R. Bracewell, The Fourier Transform and Its Applications, (McGraw Hill, 2000).

]. For example, to give an incident event u(t)a time delay Δt around the center time t0, one can add a phase shift exp(i2πμΔt) to its frequency spectrum U(μ). Therefore, to create a temporal gap, we can firstly synthesize a temporal filter with transfer function,
H1(μ)={exp(i2πμΔt),t>t0exp(+i2πμΔt),t<t0
(3)
in frequency domain. Then the transmission frequency spectrum of the incident event U(μ) after the filter becomes to
U'(μ)=U(μ)H1(μ)={U(μ)exp(i2πμΔt),t>t0U(μ)exp(+i2πμΔt),t<t0
(4)
Taking inverse Fourier transform on U'(μ) and employing shift theorem, we get the complex amplitude of transmission event in time domain as
u'(t)={u(tΔt),t>t0+Δt0,t0Δt<t<t0+Δtu(t+Δt),t<t0Δt
(5)
We see from Eq. (5) that u'(t) is not continuous now, and a temporal gap with width 2Δt is opened around the center time t0in time domain.

To close the opening time gap, we can synthesize another filter with transfer functionH2(μ)=H1(μ), where denotes the conjugation operator. In this case, the transfer function of the second filter can be written as
H2(μ)={exp(+i2πμΔt),t>t0exp(i2πμΔt),t<t0
(6)
Behind the second filter, transmission frequency spectrum of the incident event U(μ) becomes to
Uout(μ)=U'(μ)H2(μ)=U(μ)H1(μ)H2(μ)=CU(μ)
(7)
where C is a constant. Taking inverse Fourier transform on Uout(μ), we can get the output signal of event in time domain
uout(t)=F1[Uout(μ)]=Cu(t)
(8)
which is the same as that of the incident event u(t).

Comparing H2(μ) and H1(μ), we see that H2(μ)plays the role of closing the temporal gap opened by H1(μ), making the output signal continuous again and looked as nothing occurs in time domain. Therefore, when an interferential event ui(t) occurs during the temporal gapt0Δt<t<t0+Δt, from Eq. (5) we see that ui(t)u'(t)=0, meaning that ui(t) shows no effect on the output signal after the time window. Thus, the output signal uout(t) is still observed as u(t). However, when the interferential event ui(t) occurs outside the temporal gap, the output signal becomes into u'(t)ui(t)0[see Eq. (5)], indicating that interferential ui(t) is detectable.

The above temporal filters can be implemented by using a classical pulse shaping configuration consisted of a pair of gratings, lenses, and masks [20

20. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (McGraw-Hill, 2005).

,22

22. A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71(5), 1929–1960 (2000). [CrossRef]

], which was employed in a recent temporal cloak set-up [12

12. M. Fridman, A. Farsi, Y. Okawachi, and A. L. Gaeta, “Demonstration of temporal cloaking,” Nature 481(7379), 62–65 (2012). [CrossRef] [PubMed]

] as shown in Fig. 1
Fig. 1 Scheme of temporal filters consisted of a pair of gratings, lenses and mirrors.
. The role of the masks played is to modify the phase of frequency spectrum of the incident event through precise control over their thickness d and/or refractive index n. From Eq. (3) we see that when optical phase change is ±2πμΔt, then n and d should satisfy 2Δ(nd)=cΔt, where c is the speed of light in vacuum. By modulating n and/or dof the masks, we can open a time window by the first grating, lens, and mask system and then close the time window by the second grating, lens, and mask system orderly. For instance, choosing vacuum (n=1.0, constant) as the medium of the masks but changing the thickness of the first mask Δd=±3mm while the second mask Δd=3mm, we can create and then close a 20pstemporal gap in time domain in sequence. The thickness change of the masks can be realized by moving the two mirrors attached on the surfaces of the masks [12

12. M. Fridman, A. Farsi, Y. Okawachi, and A. L. Gaeta, “Demonstration of temporal cloaking,” Nature 481(7379), 62–65 (2012). [CrossRef] [PubMed]

]: One mirror reflects the frequency spectrum before t0 and the other reflects the frequency spectrum after t0. In this configuration, the incident event occurs before the first filter and the interferential event occurs between these two filters separately. After the incident event passing through the first filter, it interacts with interferential event. Then the observer can detect output signal behind the second filter. So incident event occurs earlier than interferential event and output signal is detected after interaction of two events.

3. Numerical simulations

To verify the above analytical results, we employ fast Fourier transform method [21

21. R. Bracewell, The Fourier Transform and Its Applications, (McGraw Hill, 2000).

,23

23. J. W. Cooley and J. W. Tukey, “An algorithm for the machine computation of the complex Fourier series,” Math. Comput. 19(90), 297–301 (1965). [CrossRef]

] to simulate an event occurs in the above filtering system. Since temporal filters act on a finite time slot, incident event is chosen occurring when filters are working. This avoids that staring or stopping filter will cause time gap not open. The incident event is set to be a Gauss beam u(t)=1.0sin(μ0t)exp[(t/40)2] with μ0=0.5THz[Fig. 2(a)
Fig. 2 Time-dependent complex amplitude distributions of (a) an incident event, (b) an interferential event occurs during the temporal gap, and the output signal as the cloak is (c) off and (d) on. Time-dependent complex amplitude distributions of (e) an interferential event occurs during the edge of temporal gap and (f) corresponding output signal.
]. The interferential event, read as ui(t)=1.0+20exp{[(t500)/10]2}[Fig. 2(b)], occurs behind the first temporal filter during the time interval 200ps (from400psto 600ps), where the two events are mixed together. Behind the second filter, we get the total output signal.

Figure 2(c) shows the output signal as the cloak is off [setting H1(μ)=H2(μ)=1, meaning that two filters are removed and no time window opens]. Obviously, the incident event [Fig. 2(a)] is disturbed by the interferential event. However, when two filters are employed (the time gap is orderly opened and closed), a completely different effect happened. The output signal [Fig. 2(d)] is exactly the same as that of the incident event and no any information of the interferential event involved.

To know the effect of an interferential event occurring at the edge of temporal gap but instead of during the gap, we simulate the output signal [Fig. 2(f)] as the interferential event is assumed to be ui'(t)=1.0+20exp{[(t400)/10]2}[see Fig. 2(e)], which indicates that the interferential event occurs during the time interval of from300ps to 500psbut instead of from 400psto 600ps. Comparing Fig. 2(f) and Fig. 2(a), we see that the output signal is also disturbed by the interferential event. Consequently, we conclude that only those interferential events occurred during the time gap can be canceled completely by such temporal filtering system.

Similar to distant spatial cloaks [9

9. Y. Lai, H. Y. Chen, Z.-Q. Zhang, and C. T. Chan, “Complementary Media Invisibility Cloak that Cloaks Objects at a Distance Outside the Cloaking Shell,” Phys. Rev. Lett. 102(9), 093901 (2009). [CrossRef] [PubMed]

,10

10. Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z.-Q. Zhang, and C. T. Chan, “Illusion Optics: The Optical Transformation of an Object into Another Object,” Phys. Rev. Lett. 102(25), 253902 (2009). [CrossRef] [PubMed]

], which can cloak objects at a distance outside the cloaks and be regarded as a conjugated spatial filter [14

14. K. Wu and G. P. Wang, “General insight into the complementary medium-based camouflage devices from Fourier optics,” Opt. Lett. 35(13), 2242–2244 (2010). [CrossRef] [PubMed]

17

17. Q. Cheng, K. Wu, and G. P. Wang, “All dielectric macroscopic cloaks for hiding objects and creating illusions at visible frequencies,” Opt. Express 19(23), 23240–23248 (2011). [CrossRef] [PubMed]

], we can also construct a temporal cloak without time window to conceal events. For an interferential eventUi(μ)in frequency domain, if there is a temporal filter with transfer function Hc(μ)=Ui(μ), then behind the filter, the eventUi(μ) will completely be compensated for and not be detectable. Consequently, only incident event U(μ) is measurable. In addition, to conceal an interferential event A(μ)Ui(μ) with amplitude changeA(μ), we can employ a temporal filter with transfer function Hc(μ)=Ui(μ)/A(μ) to compensate both phase and amplitude simultaneously.

Such conjugated temporal filter with transfer functionHc(μ) can also be constructed with a grating-lens-mask system, where the mask can theoretically be used to tune the transmission coefficient of the interferential event so as to synthesize such Hc(μ) [20

20. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (McGraw-Hill, 2005).

]. Here we introduce another technique to get a distant temporal cloak. For an interferential event occurs in a positive dispersive medium [right, Fig. 3(a)
Fig. 3 Scheme of the dispersion curves of media (a) for creating interferential events (left) and constructing conjugated temporal cloaks (right). Time-dependent complex amplitude distributions of (b) an interferential event, (c) mixed output signal of the incident event and interferential event, (d) conjugated temporal filter, and (e) output signal after filter is implemented for compensating for the interferential event, respectively.
], we can design a cloak constructed with a negative dispersive medium [left, Fig. 3(a)] to compensate for the effect of the interferential event. In this case, an interferential event occurs in a positive dispersive medium can be written as Ui(μ)=exp(14π2iμ2d2ndμ2z) with d2ndμ2>0 in frequency domain, while the corresponding conjugated temporal cloak should have a transfer function Hc(μ)=exp(14π2iμ2d2ndμ2z), which can be realized by using a negative dispersive medium withd2ndμ2<0.

In the simulation, supposing that during an incident event as that of Fig. 2(a) occurs, another interferential event with complex amplitude in frequency domain Ui(μ)=exp(1.0iμ2/3600) also occurs at t=456ps [For comparison, we show in Fig. 3 (b) the amplitude of the interferential event in time domain]. Figure 3(c) presents the simulated output signal. We can see that the output signal distinguishes from the incident event [Fig. 2(a)], meaning that the incident event is obviously disturbed by interferential event. However, as a temporal filter with transfer function Hc(μ)=exp(1.0iμ2/3600) [Fig. 3(d) shows its complex amplitude in time domain] at t=544ps is used to compensate for the interferential event, we see that the output signal [Fig. 3(e)] is exactly recovered into the form of the incident event [Fig. 2(a)], indicating that interferential event is concealed completely as nothing occurs.

4. Conclusions and discussions

To conclude, we have introduced a Fourier analysis method for constituting temporal cloaks to hide events in time domain. The cloaks can either be constructed by synthesizing two linear time-invariant filters with different transfer functions to open and close a temporal gap orderly, so that any events occurring during the gap are not detectable outside, or be designed based upon temporal compensation to make event outside the cloaks not detectable. All the analytical results are verified by numerical simulations. Our results may provide another way of constructing time cloaks for interesting applications in such fields as temporal imaging and sensing etc.

Acknowledgments

This work was supported by 973 Program (2011CB933600) and NSFC (Grants 60925020 and 11274247). K. D. W. is also supported by the academic award for excellent Ph.D. Candidates funded by Ministry of Education of China (Grant 5052011202009).

References and links

1.

U. Leonhardt, “Optical Conformal Mapping,” Science 312(5781), 1777–1780 (2006). [CrossRef] [PubMed]

2.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]

3.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010). [CrossRef] [PubMed]

4.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science 314(5801), 977–980 (2006). [CrossRef] [PubMed]

5.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009). [CrossRef] [PubMed]

6.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3(8), 461–463 (2009). [CrossRef]

7.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328(5976), 337–339 (2010). [CrossRef] [PubMed]

8.

B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic Invisibility Cloak for Visible Light,” Phys. Rev. Lett. 106(3), 033901 (2011). [CrossRef] [PubMed]

9.

Y. Lai, H. Y. Chen, Z.-Q. Zhang, and C. T. Chan, “Complementary Media Invisibility Cloak that Cloaks Objects at a Distance Outside the Cloaking Shell,” Phys. Rev. Lett. 102(9), 093901 (2009). [CrossRef] [PubMed]

10.

Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z.-Q. Zhang, and C. T. Chan, “Illusion Optics: The Optical Transformation of an Object into Another Object,” Phys. Rev. Lett. 102(25), 253902 (2009). [CrossRef] [PubMed]

11.

M. W. McCall, A. Favaro, P. Kinsler, and A. Boardman, “A spacetime cloak, or a history editors,” J. Opt. 13(2), 024003 (2011). [CrossRef]

12.

M. Fridman, A. Farsi, Y. Okawachi, and A. L. Gaeta, “Demonstration of temporal cloaking,” Nature 481(7379), 62–65 (2012). [CrossRef] [PubMed]

13.

A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. 43(4), 773–793 (1996). [CrossRef]

14.

K. Wu and G. P. Wang, “General insight into the complementary medium-based camouflage devices from Fourier optics,” Opt. Lett. 35(13), 2242–2244 (2010). [CrossRef] [PubMed]

15.

K. Wu, Q. Cheng, and G. P. Wang, “Fourier optics theory for invisibility cloaks,” J. Opt. Soc. Am. B 28(6), 1467–1474 (2011). [CrossRef]

16.

K. Wu and G. P. Wang, “Hiding objects and creating illusions above a carpet filter using a Fourier optics approach,” Opt. Express 18(19), 19894–19901 (2010). [CrossRef] [PubMed]

17.

Q. Cheng, K. Wu, and G. P. Wang, “All dielectric macroscopic cloaks for hiding objects and creating illusions at visible frequencies,” Opt. Express 19(23), 23240–23248 (2011). [CrossRef] [PubMed]

18.

B. H. Kolner, “Space-Time Duality and the Theory of Temporal Imaging,” IEEE J. Quantum Electron. 30(8), 1951–1963 (1994). [CrossRef]

19.

A. W. Lohmann and D. Mendlovic, “Temporal filtering with time lenses,” Appl. Opt. 31(29), 6212–6219 (1992). [CrossRef] [PubMed]

20.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (McGraw-Hill, 2005).

21.

R. Bracewell, The Fourier Transform and Its Applications, (McGraw Hill, 2000).

22.

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71(5), 1929–1960 (2000). [CrossRef]

23.

J. W. Cooley and J. W. Tukey, “An algorithm for the machine computation of the complex Fourier series,” Math. Comput. 19(90), 297–301 (1965). [CrossRef]

OCIS Codes
(070.0070) Fourier optics and signal processing : Fourier optics and signal processing
(230.3205) Optical devices : Invisibility cloaks

ToC Category:
Fourier Optics and Signal Processing

History
Original Manuscript: September 17, 2012
Revised Manuscript: December 2, 2012
Manuscript Accepted: December 12, 2012
Published: January 3, 2013

Citation
Kedi Wu and Guo Ping Wang, "Design and demonstration of temporal cloaks with and without the time gap," Opt. Express 21, 238-244 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-1-238


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References

  1. U. Leonhardt, “Optical Conformal Mapping,” Science312(5781), 1777–1780 (2006). [CrossRef] [PubMed]
  2. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science312(5781), 1780–1782 (2006). [CrossRef] [PubMed]
  3. H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater.9(5), 387–396 (2010). [CrossRef] [PubMed]
  4. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science314(5801), 977–980 (2006). [CrossRef] [PubMed]
  5. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater.8(7), 568–571 (2009). [CrossRef] [PubMed]
  6. L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics3(8), 461–463 (2009). [CrossRef]
  7. T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science328(5976), 337–339 (2010). [CrossRef] [PubMed]
  8. B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic Invisibility Cloak for Visible Light,” Phys. Rev. Lett.106(3), 033901 (2011). [CrossRef] [PubMed]
  9. Y. Lai, H. Y. Chen, Z.-Q. Zhang, and C. T. Chan, “Complementary Media Invisibility Cloak that Cloaks Objects at a Distance Outside the Cloaking Shell,” Phys. Rev. Lett.102(9), 093901 (2009). [CrossRef] [PubMed]
  10. Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z.-Q. Zhang, and C. T. Chan, “Illusion Optics: The Optical Transformation of an Object into Another Object,” Phys. Rev. Lett.102(25), 253902 (2009). [CrossRef] [PubMed]
  11. M. W. McCall, A. Favaro, P. Kinsler, and A. Boardman, “A spacetime cloak, or a history editors,” J. Opt.13(2), 024003 (2011). [CrossRef]
  12. M. Fridman, A. Farsi, Y. Okawachi, and A. L. Gaeta, “Demonstration of temporal cloaking,” Nature481(7379), 62–65 (2012). [CrossRef] [PubMed]
  13. A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt.43(4), 773–793 (1996). [CrossRef]
  14. K. Wu and G. P. Wang, “General insight into the complementary medium-based camouflage devices from Fourier optics,” Opt. Lett.35(13), 2242–2244 (2010). [CrossRef] [PubMed]
  15. K. Wu, Q. Cheng, and G. P. Wang, “Fourier optics theory for invisibility cloaks,” J. Opt. Soc. Am. B28(6), 1467–1474 (2011). [CrossRef]
  16. K. Wu and G. P. Wang, “Hiding objects and creating illusions above a carpet filter using a Fourier optics approach,” Opt. Express18(19), 19894–19901 (2010). [CrossRef] [PubMed]
  17. Q. Cheng, K. Wu, and G. P. Wang, “All dielectric macroscopic cloaks for hiding objects and creating illusions at visible frequencies,” Opt. Express19(23), 23240–23248 (2011). [CrossRef] [PubMed]
  18. B. H. Kolner, “Space-Time Duality and the Theory of Temporal Imaging,” IEEE J. Quantum Electron.30(8), 1951–1963 (1994). [CrossRef]
  19. A. W. Lohmann and D. Mendlovic, “Temporal filtering with time lenses,” Appl. Opt.31(29), 6212–6219 (1992). [CrossRef] [PubMed]
  20. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (McGraw-Hill, 2005).
  21. R. Bracewell, The Fourier Transform and Its Applications, (McGraw Hill, 2000).
  22. A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum.71(5), 1929–1960 (2000). [CrossRef]
  23. J. W. Cooley and J. W. Tukey, “An algorithm for the machine computation of the complex Fourier series,” Math. Comput.19(90), 297–301 (1965). [CrossRef]

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