Abstract

Integrated-optical All-Pass Filters are of interest for their potential compactness and economy of production. For broadband applications, the number of APFs involved can be as large as 50. To find optima for all the large number of parameters involved, we need a fast and efficient algorithm based on recursive equations. APF design algorithms based on complex cepstrum are proposed in digital signal processing. In this paper, we enhance these algorithms to efficiently fit the differential phase profile required for in-line broadband Polarization Mode Dispersion and Polarization Dependent Loss compensation.

© 2004 Optical Society of America

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References

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  1. C. Madsen and J. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach. New York, Wiley (1999).
  2. C.K. Madsen, G. Lenz, A.J. Bruce, M.A. Capuzzo, L.T. Gomez, and R.E. Scotti, “Integrated All-Pass Filters for Tunable Dispersion and Dispersion Slope Compensation,” IEEE Photon. Technol. Lett. 11, 1623–1625 (1999).
    [Crossref]
  3. C.K. Madsen and P. Oswald, “Optical filter architecture for approximating any 2×2 unitary matrix,” Opt. Lett. 28, 534–536 (2003).
    [Crossref] [PubMed]
  4. P.B. Phua, H. A. Haus, and E. P. Ippen, “All-Frequency PMD Compensator In Feed-Forward Scheme,” J.of Lightwave Technol. 22, 1280–1289 (2004).
    [Crossref]
  5. P.B. Phua and E. P. Ippen, “A Deterministic Broadband Polarization-Dependent-Loss Compensator,” To appear in J. Lightwave Technol..
  6. G.R. Reddy and M.N.S. Swamy, “Digital All-Pass Filter Design Through Discrete Hilbert Transform,” Proc. IEEE Int. Conf. Acoustics, Speech Signal Processing, 646–649 (1990)
  7. K. Rajamani and Y.S. Lai, “A Novel Method for Designing Allpass Digital Filters,” IEEE Signal Processing Lett. 6, 207–209 (1999)
    [Crossref]
  8. G.J. Dolecek and J.D. Carmona, “Digital All-Pass filter design method based on Complex Cepstrums,” Electron. Lett. 39, 695–697 (2003).
    [Crossref]
  9. A.V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, Prentice Hall (1989)
  10. B.P. Bogert, M.J.R. Healy, and J.W. Tukey, “The Quefrency Analysis of Time Series for Echoes: Cepstrum, Pseudoautocovariance, Cross-Cepstrum, and Saphe Cracking,” Proc. Symposium Time Series Analysis,M. Rosenblatt, Ed., John Wiley and Sons, New York, 209–243 (1963).
  11. T. Barwicz, M.A. Popovic, P.T. Rakich, M.R. Watts, H.A. Haus, E.P. Ippen, and H.I. Smith, “Microring-resonator-based add-drop filters in SiN: fabrication and analysis,” Opt. Express 12, 1437–1442 (2004).
    [Crossref] [PubMed]

2004 (2)

2003 (2)

C.K. Madsen and P. Oswald, “Optical filter architecture for approximating any 2×2 unitary matrix,” Opt. Lett. 28, 534–536 (2003).
[Crossref] [PubMed]

G.J. Dolecek and J.D. Carmona, “Digital All-Pass filter design method based on Complex Cepstrums,” Electron. Lett. 39, 695–697 (2003).
[Crossref]

1999 (2)

K. Rajamani and Y.S. Lai, “A Novel Method for Designing Allpass Digital Filters,” IEEE Signal Processing Lett. 6, 207–209 (1999)
[Crossref]

C.K. Madsen, G. Lenz, A.J. Bruce, M.A. Capuzzo, L.T. Gomez, and R.E. Scotti, “Integrated All-Pass Filters for Tunable Dispersion and Dispersion Slope Compensation,” IEEE Photon. Technol. Lett. 11, 1623–1625 (1999).
[Crossref]

1990 (1)

G.R. Reddy and M.N.S. Swamy, “Digital All-Pass Filter Design Through Discrete Hilbert Transform,” Proc. IEEE Int. Conf. Acoustics, Speech Signal Processing, 646–649 (1990)

1963 (1)

B.P. Bogert, M.J.R. Healy, and J.W. Tukey, “The Quefrency Analysis of Time Series for Echoes: Cepstrum, Pseudoautocovariance, Cross-Cepstrum, and Saphe Cracking,” Proc. Symposium Time Series Analysis,M. Rosenblatt, Ed., John Wiley and Sons, New York, 209–243 (1963).

Barwicz, T.

Bogert, B.P.

B.P. Bogert, M.J.R. Healy, and J.W. Tukey, “The Quefrency Analysis of Time Series for Echoes: Cepstrum, Pseudoautocovariance, Cross-Cepstrum, and Saphe Cracking,” Proc. Symposium Time Series Analysis,M. Rosenblatt, Ed., John Wiley and Sons, New York, 209–243 (1963).

Bruce, A.J.

C.K. Madsen, G. Lenz, A.J. Bruce, M.A. Capuzzo, L.T. Gomez, and R.E. Scotti, “Integrated All-Pass Filters for Tunable Dispersion and Dispersion Slope Compensation,” IEEE Photon. Technol. Lett. 11, 1623–1625 (1999).
[Crossref]

Capuzzo, M.A.

C.K. Madsen, G. Lenz, A.J. Bruce, M.A. Capuzzo, L.T. Gomez, and R.E. Scotti, “Integrated All-Pass Filters for Tunable Dispersion and Dispersion Slope Compensation,” IEEE Photon. Technol. Lett. 11, 1623–1625 (1999).
[Crossref]

Carmona, J.D.

G.J. Dolecek and J.D. Carmona, “Digital All-Pass filter design method based on Complex Cepstrums,” Electron. Lett. 39, 695–697 (2003).
[Crossref]

Dolecek, G.J.

G.J. Dolecek and J.D. Carmona, “Digital All-Pass filter design method based on Complex Cepstrums,” Electron. Lett. 39, 695–697 (2003).
[Crossref]

Gomez, L.T.

C.K. Madsen, G. Lenz, A.J. Bruce, M.A. Capuzzo, L.T. Gomez, and R.E. Scotti, “Integrated All-Pass Filters for Tunable Dispersion and Dispersion Slope Compensation,” IEEE Photon. Technol. Lett. 11, 1623–1625 (1999).
[Crossref]

Haus, H. A.

P.B. Phua, H. A. Haus, and E. P. Ippen, “All-Frequency PMD Compensator In Feed-Forward Scheme,” J.of Lightwave Technol. 22, 1280–1289 (2004).
[Crossref]

Haus, H.A.

Healy, M.J.R.

B.P. Bogert, M.J.R. Healy, and J.W. Tukey, “The Quefrency Analysis of Time Series for Echoes: Cepstrum, Pseudoautocovariance, Cross-Cepstrum, and Saphe Cracking,” Proc. Symposium Time Series Analysis,M. Rosenblatt, Ed., John Wiley and Sons, New York, 209–243 (1963).

Ippen, E. P.

P.B. Phua, H. A. Haus, and E. P. Ippen, “All-Frequency PMD Compensator In Feed-Forward Scheme,” J.of Lightwave Technol. 22, 1280–1289 (2004).
[Crossref]

P.B. Phua and E. P. Ippen, “A Deterministic Broadband Polarization-Dependent-Loss Compensator,” To appear in J. Lightwave Technol..

Ippen, E.P.

Lai, Y.S.

K. Rajamani and Y.S. Lai, “A Novel Method for Designing Allpass Digital Filters,” IEEE Signal Processing Lett. 6, 207–209 (1999)
[Crossref]

Lenz, G.

C.K. Madsen, G. Lenz, A.J. Bruce, M.A. Capuzzo, L.T. Gomez, and R.E. Scotti, “Integrated All-Pass Filters for Tunable Dispersion and Dispersion Slope Compensation,” IEEE Photon. Technol. Lett. 11, 1623–1625 (1999).
[Crossref]

Madsen, C.

C. Madsen and J. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach. New York, Wiley (1999).

Madsen, C.K.

C.K. Madsen and P. Oswald, “Optical filter architecture for approximating any 2×2 unitary matrix,” Opt. Lett. 28, 534–536 (2003).
[Crossref] [PubMed]

C.K. Madsen, G. Lenz, A.J. Bruce, M.A. Capuzzo, L.T. Gomez, and R.E. Scotti, “Integrated All-Pass Filters for Tunable Dispersion and Dispersion Slope Compensation,” IEEE Photon. Technol. Lett. 11, 1623–1625 (1999).
[Crossref]

Oppenheim, A.V.

A.V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, Prentice Hall (1989)

Oswald, P.

Phua, P.B.

P.B. Phua, H. A. Haus, and E. P. Ippen, “All-Frequency PMD Compensator In Feed-Forward Scheme,” J.of Lightwave Technol. 22, 1280–1289 (2004).
[Crossref]

P.B. Phua and E. P. Ippen, “A Deterministic Broadband Polarization-Dependent-Loss Compensator,” To appear in J. Lightwave Technol..

Popovic, M.A.

Rajamani, K.

K. Rajamani and Y.S. Lai, “A Novel Method for Designing Allpass Digital Filters,” IEEE Signal Processing Lett. 6, 207–209 (1999)
[Crossref]

Rakich, P.T.

Reddy, G.R.

G.R. Reddy and M.N.S. Swamy, “Digital All-Pass Filter Design Through Discrete Hilbert Transform,” Proc. IEEE Int. Conf. Acoustics, Speech Signal Processing, 646–649 (1990)

Schafer, R. W.

A.V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, Prentice Hall (1989)

Scotti, R.E.

C.K. Madsen, G. Lenz, A.J. Bruce, M.A. Capuzzo, L.T. Gomez, and R.E. Scotti, “Integrated All-Pass Filters for Tunable Dispersion and Dispersion Slope Compensation,” IEEE Photon. Technol. Lett. 11, 1623–1625 (1999).
[Crossref]

Smith, H.I.

Swamy, M.N.S.

G.R. Reddy and M.N.S. Swamy, “Digital All-Pass Filter Design Through Discrete Hilbert Transform,” Proc. IEEE Int. Conf. Acoustics, Speech Signal Processing, 646–649 (1990)

Tukey, J.W.

B.P. Bogert, M.J.R. Healy, and J.W. Tukey, “The Quefrency Analysis of Time Series for Echoes: Cepstrum, Pseudoautocovariance, Cross-Cepstrum, and Saphe Cracking,” Proc. Symposium Time Series Analysis,M. Rosenblatt, Ed., John Wiley and Sons, New York, 209–243 (1963).

Watts, M.R.

Zhao, J.

C. Madsen and J. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach. New York, Wiley (1999).

Electron. Lett. (1)

G.J. Dolecek and J.D. Carmona, “Digital All-Pass filter design method based on Complex Cepstrums,” Electron. Lett. 39, 695–697 (2003).
[Crossref]

IEEE Photon. Technol. Lett. (1)

C.K. Madsen, G. Lenz, A.J. Bruce, M.A. Capuzzo, L.T. Gomez, and R.E. Scotti, “Integrated All-Pass Filters for Tunable Dispersion and Dispersion Slope Compensation,” IEEE Photon. Technol. Lett. 11, 1623–1625 (1999).
[Crossref]

IEEE Signal Processing Lett. (1)

K. Rajamani and Y.S. Lai, “A Novel Method for Designing Allpass Digital Filters,” IEEE Signal Processing Lett. 6, 207–209 (1999)
[Crossref]

J.of Lightwave Technol. (1)

P.B. Phua, H. A. Haus, and E. P. Ippen, “All-Frequency PMD Compensator In Feed-Forward Scheme,” J.of Lightwave Technol. 22, 1280–1289 (2004).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Proc. IEEE Int. Conf. Acoustics, Speech Signal Processing (1)

G.R. Reddy and M.N.S. Swamy, “Digital All-Pass Filter Design Through Discrete Hilbert Transform,” Proc. IEEE Int. Conf. Acoustics, Speech Signal Processing, 646–649 (1990)

Other (4)

P.B. Phua and E. P. Ippen, “A Deterministic Broadband Polarization-Dependent-Loss Compensator,” To appear in J. Lightwave Technol..

A.V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, Prentice Hall (1989)

B.P. Bogert, M.J.R. Healy, and J.W. Tukey, “The Quefrency Analysis of Time Series for Echoes: Cepstrum, Pseudoautocovariance, Cross-Cepstrum, and Saphe Cracking,” Proc. Symposium Time Series Analysis,M. Rosenblatt, Ed., John Wiley and Sons, New York, 209–243 (1963).

C. Madsen and J. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach. New York, Wiley (1999).

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Figures (3)

Fig. 1.
Fig. 1.

A common building block in the broadband PMD compensator

Fig. 2.
Fig. 2.

A random sample of the rotation angle profile θ(ω) required for one of the frequency-dependent polarization rotators used in the broadband PMD compensator. The desired rotation angle profile of θ(ω) is shown by the solid curve while the profile approximated by the APFs is shown by the dashed-curve. The number of APFs used for each polarization arms is 10 for (a), 20 for (b) and 30 for (c).

Fig. 3.
Fig. 3.

The cumulative probability distribution of the Bit Error Rate (BER) curve with and without PMD compensation.

Equations (23)

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H ( z ) = z N N ( z ) D ( z ) = z N n = 0 N a n * z n n = 0 N a n z n
H ( e j ω ) = e j ω N n = 0 N a n * e j ω n n = 0 N a n e j ω n
ϕ H ( ω ) = N ω + ϕ N ( ω ) ϕ D ( ω ) = N ω 2 ϕ D ( ω )
τ H ( ω ) = d ϕ H ( ω ) d ω = N + 2 ϕ D ( ω )
τ D ( ω ) = τ H ( ω ) 2 + N 2
Φ Vert ( ω ) Φ Hor ( ω ) = θ ( ω )
τ Vert ( ω ) τ Hor ( ω ) = τ DGD ( ω )
τ Vert ( ω ) + τ Hor ( ω ) = N Vert + N Hor
τ Vert ( ω ) = N Vert + N Hor 2 + τ DGD ( ω ) 2
τ Hor ( ω ) = N Vert + N Hor 2 τ DGD ( ω ) 2
τ DV ( ω ) = τ Vert ( ω ) 2 + N Vert 2
τ DV even ( ω ) = τ DV ( ω ) + τ DV ( ω ) 2
τ DV odd ( ω ) = τ DV ( ω ) τ DV ( ω ) 2
D ̂ ( ω ) = ln D ( ω ) = c ( 0 ) + k = 1 c ( k ) e j k ω
ϕ ( ω ) + 2 υ π = k = 1 Re [ c ( k ) ] sin k ω + k = 1 Im [ c ( k ) ] cos k ω
τ ( ω ) = k = 1 k Re [ c ( k ) ] cos k ω + k = 1 k Im [ c ( k ) ] sin k ω
ln D ( ω ) = c ( 0 ) + k = 1 Re [ c ( k ) ] cos k ω + k = 1 Im [ c ( k ) ] sin k ω
τ DV even ( ω ) = k = 1 k Re [ c ( k ) ] cos k ω
τ DV odd ( ω ) = k = 1 k Im [ c ( k ) ] sin k ω
a n = k = 0 n ( k n ) c ( k ) a n k n > 0
d D ̂ ( z ) d z = 1 D ( z ) d D ( z ) d z
n x [ n ] z Transform z d X ( z ) d z
x 1 [ n ] * x 2 [ n ] z Transform X 1 ( Z ) . X 2 ( Z )

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