Abstract

We describe compression and expansion of the time–bandwidth product of signals and present tools to design optical data compression and expansion systems that solve bottlenecks in the real-time capture and generation of wideband data. Applications of this analog photonic transformation include more efficient ways to sample, digitize, and store optical data. Time–bandwidth engineering is enabled by the recently introduced Stretched Modulation (SM) Distribution function, a mathematical tool that describes the bandwidth and temporal duration of signals after arbitrary phase and amplitude transformations. We demonstrate design of time–bandwidth engineering systems in both near-field and far-field regimes that employ engineered group delay (GD), and we derive closed-form mathematical equations governing the operation of such systems. These equations identify an important criterion for the maximum curvature of warped GD that must be met to achieve time–bandwidth compression. We also show application of the SM Distribution to benchmark different GD profiles and to the analysis of tolerance to system nonidealities, such as GD ripples.

© 2014 Optical Society of America

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  14. F. Qian, Q. Song, E. Tien, S. K. Kalyoncu, and O. Boyraz, “Real-time optical imaging and tracking of micron-sized particles,” Opt. Commun. 282, 4672–4675 (2009).
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2014

B. Jalali and M. H. Asghari, “Anamorphic stretch transform; putting the squeeze on big data,” Opt. Photon. News 25(2), 24–31 (2014).
[CrossRef]

M. H. Asghari and B. Jalali, “Discrete anamorphic transform for image compression,” IEEE Signal Process. Lett. 21, 829–833 (2014).
[CrossRef]

M. H. Asghari and B. Jalali, “Experimental demonstration of optical real-time data compression,” Appl. Phys. Lett. 104, 111101 (2014).
[CrossRef]

2013

2012

B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation,” Sci. Rep. 2, 882 (2012).
[CrossRef]

2011

2010

M. H. Asghari, Y. Park, and J. Azaña, “Complex-field measurement of ultrafast dynamic optical waveforms based on real-time spectral interferometry,” Opt. Express 18, 16526–16538 (2010).
[CrossRef]

B. Jalali, D. R. Solli, K. Goda, K. Tsia, and C. Ropers, “Real-time measurements, rare events, and photon economics,” Eur. Phys. J. Spec. Top. 185, 145–157 (2010).
[CrossRef]

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790–795 (2010).
[CrossRef]

2009

N. Akhmediev, A. Ankiewicz, and M. Taki, “Waves that appear from nowhere and disappear without a trace,” Phys. Lett. A 373, 675–678 (2009).
[CrossRef]

K. Goda, K. K. Tsia, and B. Jalali, “Serial time-encoded amplified imaging for real-time observation of fast dynamic phenomena,” Nature 458, 1145–1149 (2009).
[CrossRef]

F. Qian, Q. Song, E. Tien, S. K. Kalyoncu, and O. Boyraz, “Real-time optical imaging and tracking of micron-sized particles,” Opt. Commun. 282, 4672–4675 (2009).
[CrossRef]

Y. Okawachi, R. Salem, M. A. Foster, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “High-resolution spectroscopy using a frequency magnifier,” Opt. Express 17, 5691–5697 (2009).
[CrossRef]

2008

J. Azana, Y. Park, T.-J. Ahn, and F. Li, “Simple and highly sensitive optical pulse characterization method based on electro-optic spectral signal differentiation,” Opt. Lett. 33, 437–439 (2008).
[CrossRef]

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, 81–84 (2008).
[CrossRef]

2007

2003

J. D. McKinney, D. S. Seo, and A. M. Weiner, “Photonically assisted generation of continuous arbitrary millemetre electromagnetic waveforms,” Electron. Lett. 39, 309–311 (2003).
[CrossRef]

Y. Han and B. Jalali, “Photonic time-stretched analog-to-digital converter: Fundamental concepts and practical considerations,” J. Lightwave Technol. 21, 3085–3103 (2003).
[CrossRef]

1999

P. V. Kelkar, F. Coppinger, A. S. Bhushan, and B. Jalali, “Time-domain optical sensing,” Electron. Lett. 35, 1661–1662 (1999).
[CrossRef]

1977

J. B. Allen, “Short time spectral analysis, synthesis, and modification by discrete Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. 25, 235–238 (1977).
[CrossRef]

1932

E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Ahn, T.-J.

Akhmediev, N.

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790–795 (2010).
[CrossRef]

N. Akhmediev, A. Ankiewicz, and M. Taki, “Waves that appear from nowhere and disappear without a trace,” Phys. Lett. A 373, 675–678 (2009).
[CrossRef]

Allen, J. B.

J. B. Allen, “Short time spectral analysis, synthesis, and modification by discrete Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. 25, 235–238 (1977).
[CrossRef]

Andersen, J. K.

R. M. Fortenberry, W. V. Sorin, H. Lin, S. A. Newton, J. K. Andersen, and M. N. Islam, “Low-power ultrashort optical pulse characterization using linear dispersion,” in Conference on Optical Fiber Communication (1997), pp. 290–291.

Ankiewicz, A.

N. Akhmediev, A. Ankiewicz, and M. Taki, “Waves that appear from nowhere and disappear without a trace,” Phys. Lett. A 373, 675–678 (2009).
[CrossRef]

Asghari, M. H.

B. Jalali and M. H. Asghari, “Anamorphic stretch transform; putting the squeeze on big data,” Opt. Photon. News 25(2), 24–31 (2014).
[CrossRef]

M. H. Asghari and B. Jalali, “Experimental demonstration of optical real-time data compression,” Appl. Phys. Lett. 104, 111101 (2014).
[CrossRef]

M. H. Asghari and B. Jalali, “Discrete anamorphic transform for image compression,” IEEE Signal Process. Lett. 21, 829–833 (2014).
[CrossRef]

M. H. Asghari and B. Jalali, “Anamorphic transformation and its application to time-bandwidth compression,” Appl. Opt. 52, 6735–6743 (2013).
[CrossRef]

M. H. Asghari, Y. Park, and J. Azaña, “Complex-field measurement of ultrafast dynamic optical waveforms based on real-time spectral interferometry,” Opt. Express 18, 16526–16538 (2010).
[CrossRef]

M. H. Asghari and B. Jalali, “Anamorphic transformation and its application to time-bandwidth compression,” arXiv:1307.0137v4 (2013).

M. H. Asghari and B. Jalali, “Warped time lens in temporal imaging for optical real-time data compression,” Chin. Sci. Bull., doi:10.1007/s11434-014-0352-0 (to be published).
[CrossRef]

M. H. Asghari and B. Jalali, “Anamorphic temporal imaging using a warped time lens,” presented at the Conference on Lasers and Electro-Optics, San Jose, Calif., 8–14 June2014.

Azana, J.

Azaña, J.

Bhushan, A. S.

P. V. Kelkar, F. Coppinger, A. S. Bhushan, and B. Jalali, “Time-domain optical sensing,” Electron. Lett. 35, 1661–1662 (1999).
[CrossRef]

Boyraz, O.

F. Qian, Q. Song, E. Tien, S. K. Kalyoncu, and O. Boyraz, “Real-time optical imaging and tracking of micron-sized particles,” Opt. Commun. 282, 4672–4675 (2009).
[CrossRef]

Broaddus, D. H.

D. H. Broaddus, M. A. Foster, O. Kuzucu, K. W. Koch, and A. L. Gaeta, “Ultrafast, single-shot phase and amplitude measurement via a temporal imaging approach,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2010), paper CMK6.

Coppinger, F.

P. V. Kelkar, F. Coppinger, A. S. Bhushan, and B. Jalali, “Time-domain optical sensing,” Electron. Lett. 35, 1661–1662 (1999).
[CrossRef]

Dias, F.

B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation,” Sci. Rep. 2, 882 (2012).
[CrossRef]

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790–795 (2010).
[CrossRef]

Dudley, J. M.

B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation,” Sci. Rep. 2, 882 (2012).
[CrossRef]

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790–795 (2010).
[CrossRef]

Fatome, J.

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790–795 (2010).
[CrossRef]

Finot, C.

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790–795 (2010).
[CrossRef]

Fortenberry, R. M.

R. M. Fortenberry, W. V. Sorin, H. Lin, S. A. Newton, J. K. Andersen, and M. N. Islam, “Low-power ultrashort optical pulse characterization using linear dispersion,” in Conference on Optical Fiber Communication (1997), pp. 290–291.

Foster, M. A.

Y. Okawachi, R. Salem, M. A. Foster, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “High-resolution spectroscopy using a frequency magnifier,” Opt. Express 17, 5691–5697 (2009).
[CrossRef]

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, 81–84 (2008).
[CrossRef]

D. H. Broaddus, M. A. Foster, O. Kuzucu, K. W. Koch, and A. L. Gaeta, “Ultrafast, single-shot phase and amplitude measurement via a temporal imaging approach,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2010), paper CMK6.

Gaeta, A. L.

Y. Okawachi, R. Salem, M. A. Foster, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “High-resolution spectroscopy using a frequency magnifier,” Opt. Express 17, 5691–5697 (2009).
[CrossRef]

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, 81–84 (2008).
[CrossRef]

D. H. Broaddus, M. A. Foster, O. Kuzucu, K. W. Koch, and A. L. Gaeta, “Ultrafast, single-shot phase and amplitude measurement via a temporal imaging approach,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2010), paper CMK6.

Galt, S.

Genty, G.

B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation,” Sci. Rep. 2, 882 (2012).
[CrossRef]

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790–795 (2010).
[CrossRef]

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, 81–84 (2008).
[CrossRef]

Goda, K.

B. Jalali, D. R. Solli, K. Goda, K. Tsia, and C. Ropers, “Real-time measurements, rare events, and photon economics,” Eur. Phys. J. Spec. Top. 185, 145–157 (2010).
[CrossRef]

K. Goda, K. K. Tsia, and B. Jalali, “Serial time-encoded amplified imaging for real-time observation of fast dynamic phenomena,” Nature 458, 1145–1149 (2009).
[CrossRef]

Han, Y.

Islam, M. N.

R. M. Fortenberry, W. V. Sorin, H. Lin, S. A. Newton, J. K. Andersen, and M. N. Islam, “Low-power ultrashort optical pulse characterization using linear dispersion,” in Conference on Optical Fiber Communication (1997), pp. 290–291.

Jalali, B.

M. H. Asghari and B. Jalali, “Discrete anamorphic transform for image compression,” IEEE Signal Process. Lett. 21, 829–833 (2014).
[CrossRef]

M. H. Asghari and B. Jalali, “Experimental demonstration of optical real-time data compression,” Appl. Phys. Lett. 104, 111101 (2014).
[CrossRef]

B. Jalali and M. H. Asghari, “Anamorphic stretch transform; putting the squeeze on big data,” Opt. Photon. News 25(2), 24–31 (2014).
[CrossRef]

M. H. Asghari and B. Jalali, “Anamorphic transformation and its application to time-bandwidth compression,” Appl. Opt. 52, 6735–6743 (2013).
[CrossRef]

B. Jalali, D. R. Solli, K. Goda, K. Tsia, and C. Ropers, “Real-time measurements, rare events, and photon economics,” Eur. Phys. J. Spec. Top. 185, 145–157 (2010).
[CrossRef]

K. Goda, K. K. Tsia, and B. Jalali, “Serial time-encoded amplified imaging for real-time observation of fast dynamic phenomena,” Nature 458, 1145–1149 (2009).
[CrossRef]

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[CrossRef]

Y. Han and B. Jalali, “Photonic time-stretched analog-to-digital converter: Fundamental concepts and practical considerations,” J. Lightwave Technol. 21, 3085–3103 (2003).
[CrossRef]

P. V. Kelkar, F. Coppinger, A. S. Bhushan, and B. Jalali, “Time-domain optical sensing,” Electron. Lett. 35, 1661–1662 (1999).
[CrossRef]

B. Jalali, P. Kelkar, and V. Saxena, “Photonic arbitrary waveform generator,” in Proceedings of 14th IEEE Annual Meeting (2001), pp. 253–254.

M. H. Asghari and B. Jalali, “Anamorphic transformation and its application to time-bandwidth compression,” arXiv:1307.0137v4 (2013).

M. H. Asghari and B. Jalali, “Anamorphic temporal imaging using a warped time lens,” presented at the Conference on Lasers and Electro-Optics, San Jose, Calif., 8–14 June2014.

M. H. Asghari and B. Jalali, “Warped time lens in temporal imaging for optical real-time data compression,” Chin. Sci. Bull., doi:10.1007/s11434-014-0352-0 (to be published).
[CrossRef]

Kalyoncu, S. K.

F. Qian, Q. Song, E. Tien, S. K. Kalyoncu, and O. Boyraz, “Real-time optical imaging and tracking of micron-sized particles,” Opt. Commun. 282, 4672–4675 (2009).
[CrossRef]

Kelkar, P.

B. Jalali, P. Kelkar, and V. Saxena, “Photonic arbitrary waveform generator,” in Proceedings of 14th IEEE Annual Meeting (2001), pp. 253–254.

Kelkar, P. V.

P. V. Kelkar, F. Coppinger, A. S. Bhushan, and B. Jalali, “Time-domain optical sensing,” Electron. Lett. 35, 1661–1662 (1999).
[CrossRef]

Kibler, B.

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790–795 (2010).
[CrossRef]

Koch, K. W.

D. H. Broaddus, M. A. Foster, O. Kuzucu, K. W. Koch, and A. L. Gaeta, “Ultrafast, single-shot phase and amplitude measurement via a temporal imaging approach,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2010), paper CMK6.

Koonath, P.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[CrossRef]

Kudlinski, A.

B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation,” Sci. Rep. 2, 882 (2012).
[CrossRef]

Kuzucu, O.

D. H. Broaddus, M. A. Foster, O. Kuzucu, K. W. Koch, and A. L. Gaeta, “Ultrafast, single-shot phase and amplitude measurement via a temporal imaging approach,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2010), paper CMK6.

Lacourt, P. A.

B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation,” Sci. Rep. 2, 882 (2012).
[CrossRef]

Larger, L.

B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation,” Sci. Rep. 2, 882 (2012).
[CrossRef]

Li, F.

Lin, H.

R. M. Fortenberry, W. V. Sorin, H. Lin, S. A. Newton, J. K. Andersen, and M. N. Islam, “Low-power ultrashort optical pulse characterization using linear dispersion,” in Conference on Optical Fiber Communication (1997), pp. 290–291.

Lipson, M.

Y. Okawachi, R. Salem, M. A. Foster, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “High-resolution spectroscopy using a frequency magnifier,” Opt. Express 17, 5691–5697 (2009).
[CrossRef]

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, 81–84 (2008).
[CrossRef]

McKinney, J. D.

J. D. McKinney, D. S. Seo, and A. M. Weiner, “Photonically assisted generation of continuous arbitrary millemetre electromagnetic waveforms,” Electron. Lett. 39, 309–311 (2003).
[CrossRef]

Merolla, J. M.

B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation,” Sci. Rep. 2, 882 (2012).
[CrossRef]

Millot, G.

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790–795 (2010).
[CrossRef]

Mussot, A.

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Y. Okawachi, R. Salem, M. A. Foster, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “High-resolution spectroscopy using a frequency magnifier,” Opt. Express 17, 5691–5697 (2009).
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B. Jalali, P. Kelkar, and V. Saxena, “Photonic arbitrary waveform generator,” in Proceedings of 14th IEEE Annual Meeting (2001), pp. 253–254.

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B. Jalali, D. R. Solli, K. Goda, K. Tsia, and C. Ropers, “Real-time measurements, rare events, and photon economics,” Eur. Phys. J. Spec. Top. 185, 145–157 (2010).
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B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation,” Sci. Rep. 2, 882 (2012).
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B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation,” Sci. Rep. 2, 882 (2012).
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B. Jalali, D. R. Solli, K. Goda, K. Tsia, and C. Ropers, “Real-time measurements, rare events, and photon economics,” Eur. Phys. J. Spec. Top. 185, 145–157 (2010).
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B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation,” Sci. Rep. 2, 882 (2012).
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[CrossRef]

Eur. Phys. J. Spec. Top.

B. Jalali, D. R. Solli, K. Goda, K. Tsia, and C. Ropers, “Real-time measurements, rare events, and photon economics,” Eur. Phys. J. Spec. Top. 185, 145–157 (2010).
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M. H. Asghari and B. Jalali, “Discrete anamorphic transform for image compression,” IEEE Signal Process. Lett. 21, 829–833 (2014).
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D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
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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, 81–84 (2008).
[CrossRef]

K. Goda, K. K. Tsia, and B. Jalali, “Serial time-encoded amplified imaging for real-time observation of fast dynamic phenomena,” Nature 458, 1145–1149 (2009).
[CrossRef]

Opt. Commun.

F. Qian, Q. Song, E. Tien, S. K. Kalyoncu, and O. Boyraz, “Real-time optical imaging and tracking of micron-sized particles,” Opt. Commun. 282, 4672–4675 (2009).
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B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation,” Sci. Rep. 2, 882 (2012).
[CrossRef]

Other

R. M. Fortenberry, W. V. Sorin, H. Lin, S. A. Newton, J. K. Andersen, and M. N. Islam, “Low-power ultrashort optical pulse characterization using linear dispersion,” in Conference on Optical Fiber Communication (1997), pp. 290–291.

B. Jalali, P. Kelkar, and V. Saxena, “Photonic arbitrary waveform generator,” in Proceedings of 14th IEEE Annual Meeting (2001), pp. 253–254.

M. H. Asghari and B. Jalali, “Anamorphic transformation and its application to time-bandwidth compression,” arXiv:1307.0137v4 (2013).

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D. H. Broaddus, M. A. Foster, O. Kuzucu, K. W. Koch, and A. L. Gaeta, “Ultrafast, single-shot phase and amplitude measurement via a temporal imaging approach,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2010), paper CMK6.

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Figures (13)

Fig. 1.
Fig. 1.

System block diagrams for two applications of TBE of optical signals. (a) Real-time measurement. In such a system, optical signal bandwidth is compressed to match the backend digitizer speed, and, at the same time, the volume of data is reduced. After capturing and postprocessing, the input signal is digitally reconstructed by backpropagation. (b) Wideband waveform generation. In such a system, the transformation increases the TBP of a synthesized optical waveform.

Fig. 2.
Fig. 2.

S M Distribution is a mathematical tool to design and benchmark optical TBE systems. At time = 0 (horizontal axis), the magnitude of the S M function represents the intensity modulation bandwidth, and its half-extent along the time axis is the record length. The top and bottom plots are qualitative and show the magnitude of S M Distribution of input and output signals in a system with a sublinear GD profile. The S M Distribution shows how the TBP of the signal intensity can be engineered.

Fig. 3.
Fig. 3.

Three different qualitative GD profiles that can be used to engineer the TBP of optical signals. S M Distribution plots corresponding to these profiles are shown in Fig. 5.

Fig. 4.
Fig. 4.

Arbitrary input signal in (a) time and (b) spectral domains used in this paper.

Fig. 5.
Fig. 5.

S M Distribution is used to design TBE systems with engineered TBP. The magnitude of S M is shown here. At Time = 0 (horizontal axis), it represents the modulation bandwidth. The half-extent along the vertical direction is the record length. Here we have compared the Distribution for three cases: time–bandwidth (a) conserved, (b) compressed, and (c) expanded. In each case, we have shown operation in the near field and the far field. The input signal is shown in Fig. 4. The qualitative GD profile corresponding to each case is shown in Fig. 3.

Fig. 6.
Fig. 6.

The magnitude of the S M Distribution describes the output signal duration and modulation bandwidth after passing through a TBE system.

Fig. 7.
Fig. 7.

(a) Three different GD profiles to engineer the TBP of optical signals. A = 2.81 ns , B = 0.22 ns , and k is the dispersion strength variable. (b) Simulated output TBP as a function of dispersion strength, k , compared to the calculated TBP using Eq. (11). Examples of S M Distributions in three regions for the case of linear GD are shown in Fig. 8.

Fig. 8.
Fig. 8.

Examples of S M Distribution in three regions shown in Fig. 7(b) for a kernel with linear GD.

Fig. 9.
Fig. 9.

(a) System block diagram for a simple experimentally demonstrated analog optical TBE (compression and expansion) system [22]. In this implementation, the desired phase kernel profile for TBE is obtained using a CFBG. The transformed signal has both phase and amplitude requiring complex field recovery before reconstruction in the digital domain. (b) Example of designed grating period profile to realize time–bandwidth compression using a sub-linear GD profile. (c) Experimentally measured GD profile of the fabricated chirp grating compared to the target sub-linear GD profile using the grating design in (b). This system is able to operate analog optical real-time data compression.

Fig. 10.
Fig. 10.

Four different GD profiles used to study the effect of GD ripples on the performance of TBE systems. We consider operation in (a) and (b) the far field as well as (c) and (d) the near field. In each regime we consider two cases: GD ripples of less than ± 4 % and ± 8 % of the maximum GD. S M Distribution plots corresponding to these profiles are shown in Figs. 11 and 12.

Fig. 11.
Fig. 11.

S M Distribution can be used to analyze the effect of nonidealities on the performance of TBE systems. Here we have compared S M plots for two systems operating in the far field with (a) 4% and (b) 8% GD ripples. The GD profiles are shown in Figs. 10(a) and 10(b). The large ripple in the right side of the GD profiles in Figs. 10(a) and 10(b) results in the new track seen in their S M plots, indicated with triangles. The small ripple in the left side of GD profiles in Figs. 10(a) and 10(b) results in the new track seen in their S M plots, indicated with squares. This figure shows that the output modulation bandwidth in the far-field regime is determined mainly by the GD ripples at lower frequencies.

Fig. 12.
Fig. 12.

Comparison of S M Distribution plots for two systems operating in the near field with (a) 4% and (b) 8% GD ripples. The GD profiles are shown in Figs. 10(c) and 10(d). This figure shows that the output modulation bandwidth in the far-field regime is determined mainly by the GD ripples at higher frequencies.

Fig. 13.
Fig. 13.

(a) Block diagram of an optical anamorphic temporal imaging system for TBE using a warped optical time lens [27,28]. In this technique the signal is mixed with a local oscillator with nonlinear instantaneous frequency. (b) Qualitative S M Distribution plots before (top) and after (bottom) anamorphic temporal imaging showing how to engineer the TBP using anamorphic temporal imaging. Using this method, spectral resolution is increased without proportional increase in the bandwidth, i.e., time–bandwidth compression. An anamorphic temporal imaging system can be also used to increase the TBP.

Equations (12)

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S M ( ω , t ) = E ˜ i ( ω 1 ) E ˜ i * ( ω 1 + ω ) K ˜ ( ω 1 ) K ˜ * ( ω 1 + ω ) e j · ω 1 · t d ω 1 ,
S M ( ω , 0 ) = F T { | E o ( t ) | 2 } ,
K ˜ ( ω ) = e j · β ( ω ) .
| S M , out ( ω , t ) | = | E ˜ i ( ω 1 ω 2 ) E ˜ i * ( ω 1 + ω 2 ) · e j ( β ( ω 1 ω 2 ) β ( ω 1 + ω 2 ) ) e j · ω 1 · t d ω 1 | .
β ( ω 1 ω / 2 ) β ( ω 1 + ω / 2 ) = 2 · ω 1 · β ( ω / 2 ) ω 1 3 3 · β ( ω / 2 ) + = n = 0 2 · ω 1 2 n + 1 ( 2 n + 1 ) ! 2 n τ ( ω 1 ) ω 1 2 n | ω 1 = ω / 2 .
β ( ω 1 ω / 2 ) β ( ω 1 + ω / 2 ) 2 · ω 1 · τ ( ω / 2 ) .
| S M , out ( ω , t ) | = | S M , in ( ω , t + 2 · τ ( ω / 2 ) ) | ,
T out = T in + 2 τ ( Δ ω / 2 ) .
T in + 2 τ ( Δ ω m / 4 ) = 0 Δ ω m = 4 · τ 1 ( T in / 2 ) .
Δ ω m = min { 4 · τ 1 ( T in 2 ) , 2 · Δ ω } .
TBP out = ( T in + 2 · τ ( Δ ω / 2 ) ) · min { 4 · τ 1 ( T in 2 ) , 2 · Δ ω } .
Linear G D : τ ( ω ) = k · A · B · ω , Super-linear G D : τ ( ω ) = k · A · tan ( B / 2 · ω ) , Sub-linear G D : τ ( ω ) = k · A · tan 1 ( B · ω ) ,

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