Abstract

We demonstrate the feasibility of fully coherent reconstruction of the complex envelope of arbitrary optical fields while using an incoherent source as a local oscillator (LO). The reconstruction relies on a signal processing procedure that we describe, and the only requirement from the system is that the receiver’s electrical bandwidth and sampling rate are at least twice as high as the bandwidth of the received signal and of the LO. The proposed scheme is particularly attractive in spectral regions where no high-quality lasers are available.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. N. Tesla, Lecture Before the New York Academy of Sciences-April 6, 1897, Leland I. Anderson, ed., Twenty-First Century Books, pp. 73–74 (1994).
  2. The only relevant exception is that of the differential receiver, which we discuss later in the paper.
  3. M. Bertolotti, The history of laserInstitute of Physics Publishing, London (2005).
  4. K. Kikuchi, “Fundamentals of Coherent Optical Fiber Communications,” J. Lightwave Technol. 34, 157–179 (2016).
    [Crossref]
  5. H.-W. Hübers, S. G. Pavlov, A. D. Semenov, R. Köhler, L. Mahler, A. Tredicucci, H. E. Beere, D. A. Ritchie, and E. H. Linfield, “Terahertz quantum cascade laser as local oscillator in a heterodyne receiver,” Opt. Express 13, 5890–5896 (2005).
    [Crossref] [PubMed]
  6. H.-W. Hübers, “Terahertz Heterodyne Receivers,” IEEE Journal of Selected Topics in Quantum Electronics 14, 378–391 (2008).
    [Crossref]
  7. This implies that the lowest sampling rate of Bs = (Ba + Bb)/2 is achieved when Δν = (Ba − Bb)/2.
  8. Obviously, the roles of a(t) and b(t) are symmetric, meaning that the two waveforms can also be obtained from the zeros of Iab (t) and Ibb (t) = |b(t)|2.
  9. This procedure will rigorously reproduce the correct waveforms only as long as none of the zeros are common to both a(t) and b(t) (or to a(t) and b* (t)). Yet the probability of this happening with unrelated waveforms is too low to be of any practical relevance.
  10. The reason that it is the inverse and not the direct transform has to do with opposite sign conventions between our definition of the Fourier transform (adopted from the optics literature) and the definition used in digital signal processing applications.
  11. Notice that the inability to reconstruct the constant phase offset is not unique to the proposed reconstruction method. The constant phase is not recoverable also in standard homodyne and heterodyne receivers.
  12. A. Mecozzi and M. Shtaif, “Coherent detection with an incoherent local oscillator: supplementary material,” figshare (2018), http://dx.doi.org/10.6084/m9.figshare.7064774
  13. M. Petkovic, “Iterative Methods for Simultaneous Inclusion of Polynomial Zeros,” Lecture Notes in Mathematics Volume 1387, Springer-VerlagBerlin Heidelberg (1989).
    [Crossref]
  14. H. W. Kuhn, “The Hungarian Method for the assignment problem,” Naval Research Logistics Quarterly 2, 83–97 (1955).
    [Crossref]
  15. J. Munkres, “Algorithms for the Assignment and Transportation Problems,” Journal of the Society for Industrial and Applied Mathematics 5, 32–38 (1957).
    [Crossref]
  16. J. Bijsterbosch and A. Volgenan, “Solving the Rectangular assignment problem and applications,” Annals of Operations Research 181, 443–462 (2010).
    [Crossref]
  17. N. Kikuchi, K. Mandai, S. Sasaki, and K. Sekine, “Proposal and first experimental demonstration of digital incoherent optical field detector for chromatic dispersion compensation,” in Proceedings of European Conference on Optical Communications 2006, Post-deadline Paper Th4.4.4.
  18. M. Nazarathy, X. Liu, L. Christen, Y. Lize, and A. Wilner, “Self-coherent optical detection of multisymbol differential phase-shift-keyed transmission,” J. Lightwave Technol. 26, 1921–1934 (2008).
    [Crossref]
  19. Liu Xiang, S. Chandrasekhar, and A. Leven, “Digital self-coherent detection,” Opt. Express 16, 792–803 (2008)
    [Crossref]
  20. A. Magen and O. Amrani, “Approaching coherent performance in differential detection via diversity,” Opt. Express 23, 4529–4538 (2015).
    [Crossref] [PubMed]
  21. Some insight can be obtained by considering the example where a(t) is periodic with period T . Then the zeros of a(t − Td) are obtained by rotating the zeros of a(t) by an angle of 2πTd /T , and the pairs of zeros are perfectly distinguishable even when Td /T is a small fraction of unity (e.g. Td ~ 0.1T).
  22. S. Betti, G. De Marchis, and E. Iannone, “Polarization modulated direct detection optical transmission systems,” J. Lightwave Technol. 10, 1985–1997 (1992).
    [Crossref]
  23. Di Che, An Li Xi Chen, Hu Qian, and W. Shieh, “Rejuvenating direct modulation and direct detection for modern optical communications,” Opt. Commun. 409, 86–93 (2017).
    [Crossref]
  24. It should be stressed that this equivalence is only in terms of the measured quantities and not in terms of the processing that we apply to them.
  25. H. Khodakarami, Di Che, and W. Shieh, “Information Capacity of Polarization-Modulated and Directly Detected Optical Systems Dominated by Amplified Spontaneous Emission Noise,” J. Lightwave Technol. 35, 2797–2802 (2017).
    [Crossref]
  26. J. B. Hough, M. Krishnapur, Y. Peres, and B. Virag, Zeros of Gaussian analytic functions and determinantal point processes (University Lecture Series, 2009) vol. 51.
  27. A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing(Prentice Hall, 1999).
  28. This is because we define the Z-extension as in Eq. (3), in a way consistent with a definition of Z-transform as Σk ak zk . With the most conventional definition of Z-transform Σk ak z−k the zeros of the minimum phase waveform would be all inside the unit circle.
  29. When a zero of Iaa (t) falls exactly on the unit circle, it must be a double zero because it coincides with its inverse conjugate. One such zero can be included in Z_aa.

2017 (2)

Di Che, An Li Xi Chen, Hu Qian, and W. Shieh, “Rejuvenating direct modulation and direct detection for modern optical communications,” Opt. Commun. 409, 86–93 (2017).
[Crossref]

H. Khodakarami, Di Che, and W. Shieh, “Information Capacity of Polarization-Modulated and Directly Detected Optical Systems Dominated by Amplified Spontaneous Emission Noise,” J. Lightwave Technol. 35, 2797–2802 (2017).
[Crossref]

2016 (1)

2015 (1)

2010 (1)

J. Bijsterbosch and A. Volgenan, “Solving the Rectangular assignment problem and applications,” Annals of Operations Research 181, 443–462 (2010).
[Crossref]

2008 (3)

2005 (1)

1992 (1)

S. Betti, G. De Marchis, and E. Iannone, “Polarization modulated direct detection optical transmission systems,” J. Lightwave Technol. 10, 1985–1997 (1992).
[Crossref]

1957 (1)

J. Munkres, “Algorithms for the Assignment and Transportation Problems,” Journal of the Society for Industrial and Applied Mathematics 5, 32–38 (1957).
[Crossref]

1955 (1)

H. W. Kuhn, “The Hungarian Method for the assignment problem,” Naval Research Logistics Quarterly 2, 83–97 (1955).
[Crossref]

Amrani, O.

Beere, H. E.

Bertolotti, M.

M. Bertolotti, The history of laserInstitute of Physics Publishing, London (2005).

Betti, S.

S. Betti, G. De Marchis, and E. Iannone, “Polarization modulated direct detection optical transmission systems,” J. Lightwave Technol. 10, 1985–1997 (1992).
[Crossref]

Bijsterbosch, J.

J. Bijsterbosch and A. Volgenan, “Solving the Rectangular assignment problem and applications,” Annals of Operations Research 181, 443–462 (2010).
[Crossref]

Buck, J. R.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing(Prentice Hall, 1999).

Chandrasekhar, S.

Che, Di

H. Khodakarami, Di Che, and W. Shieh, “Information Capacity of Polarization-Modulated and Directly Detected Optical Systems Dominated by Amplified Spontaneous Emission Noise,” J. Lightwave Technol. 35, 2797–2802 (2017).
[Crossref]

Di Che, An Li Xi Chen, Hu Qian, and W. Shieh, “Rejuvenating direct modulation and direct detection for modern optical communications,” Opt. Commun. 409, 86–93 (2017).
[Crossref]

Christen, L.

Hough, J. B.

J. B. Hough, M. Krishnapur, Y. Peres, and B. Virag, Zeros of Gaussian analytic functions and determinantal point processes (University Lecture Series, 2009) vol. 51.

Hübers, H.-W.

Iannone, E.

S. Betti, G. De Marchis, and E. Iannone, “Polarization modulated direct detection optical transmission systems,” J. Lightwave Technol. 10, 1985–1997 (1992).
[Crossref]

Khodakarami, H.

Kikuchi, K.

Kikuchi, N.

N. Kikuchi, K. Mandai, S. Sasaki, and K. Sekine, “Proposal and first experimental demonstration of digital incoherent optical field detector for chromatic dispersion compensation,” in Proceedings of European Conference on Optical Communications 2006, Post-deadline Paper Th4.4.4.

Köhler, R.

Krishnapur, M.

J. B. Hough, M. Krishnapur, Y. Peres, and B. Virag, Zeros of Gaussian analytic functions and determinantal point processes (University Lecture Series, 2009) vol. 51.

Kuhn, H. W.

H. W. Kuhn, “The Hungarian Method for the assignment problem,” Naval Research Logistics Quarterly 2, 83–97 (1955).
[Crossref]

Leven, A.

Linfield, E. H.

Liu, X.

Lize, Y.

Magen, A.

Mahler, L.

Mandai, K.

N. Kikuchi, K. Mandai, S. Sasaki, and K. Sekine, “Proposal and first experimental demonstration of digital incoherent optical field detector for chromatic dispersion compensation,” in Proceedings of European Conference on Optical Communications 2006, Post-deadline Paper Th4.4.4.

Marchis, G. De

S. Betti, G. De Marchis, and E. Iannone, “Polarization modulated direct detection optical transmission systems,” J. Lightwave Technol. 10, 1985–1997 (1992).
[Crossref]

Munkres, J.

J. Munkres, “Algorithms for the Assignment and Transportation Problems,” Journal of the Society for Industrial and Applied Mathematics 5, 32–38 (1957).
[Crossref]

Nazarathy, M.

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing(Prentice Hall, 1999).

Pavlov, S. G.

Peres, Y.

J. B. Hough, M. Krishnapur, Y. Peres, and B. Virag, Zeros of Gaussian analytic functions and determinantal point processes (University Lecture Series, 2009) vol. 51.

Petkovic, M.

M. Petkovic, “Iterative Methods for Simultaneous Inclusion of Polynomial Zeros,” Lecture Notes in Mathematics Volume 1387, Springer-VerlagBerlin Heidelberg (1989).
[Crossref]

Qian, Hu

Di Che, An Li Xi Chen, Hu Qian, and W. Shieh, “Rejuvenating direct modulation and direct detection for modern optical communications,” Opt. Commun. 409, 86–93 (2017).
[Crossref]

Ritchie, D. A.

Sasaki, S.

N. Kikuchi, K. Mandai, S. Sasaki, and K. Sekine, “Proposal and first experimental demonstration of digital incoherent optical field detector for chromatic dispersion compensation,” in Proceedings of European Conference on Optical Communications 2006, Post-deadline Paper Th4.4.4.

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing(Prentice Hall, 1999).

Sekine, K.

N. Kikuchi, K. Mandai, S. Sasaki, and K. Sekine, “Proposal and first experimental demonstration of digital incoherent optical field detector for chromatic dispersion compensation,” in Proceedings of European Conference on Optical Communications 2006, Post-deadline Paper Th4.4.4.

Semenov, A. D.

Shieh, W.

Di Che, An Li Xi Chen, Hu Qian, and W. Shieh, “Rejuvenating direct modulation and direct detection for modern optical communications,” Opt. Commun. 409, 86–93 (2017).
[Crossref]

H. Khodakarami, Di Che, and W. Shieh, “Information Capacity of Polarization-Modulated and Directly Detected Optical Systems Dominated by Amplified Spontaneous Emission Noise,” J. Lightwave Technol. 35, 2797–2802 (2017).
[Crossref]

Tesla, N.

N. Tesla, Lecture Before the New York Academy of Sciences-April 6, 1897, Leland I. Anderson, ed., Twenty-First Century Books, pp. 73–74 (1994).

Tredicucci, A.

Virag, B.

J. B. Hough, M. Krishnapur, Y. Peres, and B. Virag, Zeros of Gaussian analytic functions and determinantal point processes (University Lecture Series, 2009) vol. 51.

Volgenan, A.

J. Bijsterbosch and A. Volgenan, “Solving the Rectangular assignment problem and applications,” Annals of Operations Research 181, 443–462 (2010).
[Crossref]

Wilner, A.

Xi Chen, An Li

Di Che, An Li Xi Chen, Hu Qian, and W. Shieh, “Rejuvenating direct modulation and direct detection for modern optical communications,” Opt. Commun. 409, 86–93 (2017).
[Crossref]

Xiang, Liu

Annals of Operations Research (1)

J. Bijsterbosch and A. Volgenan, “Solving the Rectangular assignment problem and applications,” Annals of Operations Research 181, 443–462 (2010).
[Crossref]

IEEE Journal of Selected Topics in Quantum Electronics (1)

H.-W. Hübers, “Terahertz Heterodyne Receivers,” IEEE Journal of Selected Topics in Quantum Electronics 14, 378–391 (2008).
[Crossref]

J. Lightwave Technol. (4)

Journal of the Society for Industrial and Applied Mathematics (1)

J. Munkres, “Algorithms for the Assignment and Transportation Problems,” Journal of the Society for Industrial and Applied Mathematics 5, 32–38 (1957).
[Crossref]

Naval Research Logistics Quarterly (1)

H. W. Kuhn, “The Hungarian Method for the assignment problem,” Naval Research Logistics Quarterly 2, 83–97 (1955).
[Crossref]

Opt. Commun. (1)

Di Che, An Li Xi Chen, Hu Qian, and W. Shieh, “Rejuvenating direct modulation and direct detection for modern optical communications,” Opt. Commun. 409, 86–93 (2017).
[Crossref]

Opt. Express (3)

Other (17)

N. Tesla, Lecture Before the New York Academy of Sciences-April 6, 1897, Leland I. Anderson, ed., Twenty-First Century Books, pp. 73–74 (1994).

The only relevant exception is that of the differential receiver, which we discuss later in the paper.

M. Bertolotti, The history of laserInstitute of Physics Publishing, London (2005).

N. Kikuchi, K. Mandai, S. Sasaki, and K. Sekine, “Proposal and first experimental demonstration of digital incoherent optical field detector for chromatic dispersion compensation,” in Proceedings of European Conference on Optical Communications 2006, Post-deadline Paper Th4.4.4.

This implies that the lowest sampling rate of Bs = (Ba + Bb)/2 is achieved when Δν = (Ba − Bb)/2.

Obviously, the roles of a(t) and b(t) are symmetric, meaning that the two waveforms can also be obtained from the zeros of Iab (t) and Ibb (t) = |b(t)|2.

This procedure will rigorously reproduce the correct waveforms only as long as none of the zeros are common to both a(t) and b(t) (or to a(t) and b* (t)). Yet the probability of this happening with unrelated waveforms is too low to be of any practical relevance.

The reason that it is the inverse and not the direct transform has to do with opposite sign conventions between our definition of the Fourier transform (adopted from the optics literature) and the definition used in digital signal processing applications.

Notice that the inability to reconstruct the constant phase offset is not unique to the proposed reconstruction method. The constant phase is not recoverable also in standard homodyne and heterodyne receivers.

A. Mecozzi and M. Shtaif, “Coherent detection with an incoherent local oscillator: supplementary material,” figshare (2018), http://dx.doi.org/10.6084/m9.figshare.7064774

M. Petkovic, “Iterative Methods for Simultaneous Inclusion of Polynomial Zeros,” Lecture Notes in Mathematics Volume 1387, Springer-VerlagBerlin Heidelberg (1989).
[Crossref]

Some insight can be obtained by considering the example where a(t) is periodic with period T . Then the zeros of a(t − Td) are obtained by rotating the zeros of a(t) by an angle of 2πTd /T , and the pairs of zeros are perfectly distinguishable even when Td /T is a small fraction of unity (e.g. Td ~ 0.1T).

J. B. Hough, M. Krishnapur, Y. Peres, and B. Virag, Zeros of Gaussian analytic functions and determinantal point processes (University Lecture Series, 2009) vol. 51.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing(Prentice Hall, 1999).

This is because we define the Z-extension as in Eq. (3), in a way consistent with a definition of Z-transform as Σk ak zk . With the most conventional definition of Z-transform Σk ak z−k the zeros of the minimum phase waveform would be all inside the unit circle.

When a zero of Iaa (t) falls exactly on the unit circle, it must be a double zero because it coincides with its inverse conjugate. One such zero can be included in Z_aa.

It should be stressed that this equivalence is only in terms of the measured quantities and not in terms of the processing that we apply to them.

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Figures (6)

Fig. 1
Fig. 1 Schematic of the proposed receiver. The received signal a(t) exp (−iνat) and the local oscillator b(t) exp (−iνbt) are mixed in a standard homodyne receiver whose electric outputs are the real and imaginary parts of the complex signal Iab(t) = a(t)b(t). This signal together with the intensity Iaa(t) = |a(t)|2, directly measured with a photo-diode (PD), are fed into a digital signal processing (DSP) unit whose outputs are the digitized versions of the complex envelopes a(t) and b (t) up to an arbitrary, time independent phase offset θ. Both a(t) and b(t) are arbitrary complex band-limited signals.
Fig. 2
Fig. 2 The real (a) and imaginary (b) parts of a(t) and b(t), the detected field shown with solid blue lines, the reconstructed fields with dashed red lines. (c) Shows the power spectra of the two waveforms and (d) shows the zeros of Iaa(t) (crosses and stars) and of Iab (t)(circles and squares). The blue stars and the red circles are the zeros of a(t) in Iaa(t)and Iab (t) respectively, the blue crosses are the inverse conjugate of the zeros of a(t) in Iaa(t) and the red squares are the inverse conjugates of the zeros of b(t) in Iab(t). The reconstruction of a(t) and b(t) on the basis of the their zeros as found in (d) is perfect, as long as the intensities are correctly normalized and the constant phase-offset is eliminated (as in traditional homodyne). The signals where generated with Ba = Bb = 16/T.
Fig. 3
Fig. 3 The phase of the original signal (solid blue lines), the reconstructed phase (dashed red lines), and the phase reconstruction error (blue "+" signs). The intensity waveform multiplied by the time windowing function w2(t) is shown for reference by the purple dashed curve. The left (a) and right (b) panels relate to the reconstruction of a(t) and b(t), respectively.
Fig. 4
Fig. 4 Time domain profiles of the real and imaginary parts of the signals a(t) and b(t) after removing the effect of the time-windowing and discarding the temporal tails outside the range t ∈[T/2 − T0, T/2 + T0]. The original and reconstructed waveforms are marked by the solid blue and dashed red curves, respectively, and they are almost indistinguishable in the figures (demonstrating the quality of the reconstruction).
Fig. 5
Fig. 5 The zeros of Iaa(t) and of Iab (t). Blue star: zeros of a(t) in Iaa(t); blue crosses: zeros of a* (t) in Iaa(t); red circles: zeros of a(t) in Iab(t); red squares: zeros of b* (t) in Iab(t). The figure in (b) is a zoomed in version of (a).
Fig. 6
Fig. 6 The probability that ρ< 1 − Δ as a function of Δ with for a(t) (left panel) and b(t)right panel. The curves represent different noise to signal ratios (i.e. the ratio between the variance of the imbalance noise na(t) and the average power of a(t)). Blue circles show Pn/Pa = 10−4, red squares show Pn/Pa = 10−6, and black stars show Pn/Pa = 0.

Equations (8)

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

a ( t ) = k = 0 M a 1 F a ( k ) e i k Ω t ,
F a ( k ) = 1 T 0 T a ( t ) e i k Ω t d t .
Z a ( Z ) = k = 0 M a 1 F a ( k ) Z k .
Z a ( Z ) = F a ( 0 ) k = 1 M a 1 ( Z a , k ) k = 1 M a 1 ( Z Z a , k ) .
I a x ( t ) = k = M x + 1 M a 1 F a x ( k ) e i k Ω t ,
Z I a x ( Z ) = 1 Z M x 1 k = 0 M a + M x 2 F a x ( k M x + 1 ) Z k ,
ρ = | a * ( t ) a ^ ( t ) d t | a ( t ) | 2 d t | a ^ ( t ) | 2 d t | ,
Z a * ( Z ) = [ k = 0 M a 1 F a ( k ) ( 1 Z * ) k ] * = [ Z a ( 1 / Z * ) ] * .

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