Abstract

We present experimental observations of chaotic phase synchronization of a sinusoidally modulated frequency-doubled Nd:YAG laser. Periodic and chaotic phase synchronization is detected using recurrence analysis and pseudo ensemble averaging applied to the observed intensity fluctuations and the modulated pump current.

© 2009 Optical Society of America

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  1. M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, Phys. Rev. Lett. 76, 1804 (1996).
    [CrossRef] [PubMed]
  2. U. Parlitz, L. Junge, W. Lauterborn, and L. Kocarev, Phys. Rev. E 54, 2115 (1996).
    [CrossRef]
  3. A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: a Universal Concept in Nonlinear Sciences (Cambridge U. Press, 2001).
    [CrossRef]
  4. C. M. Ticos, E. Rosa, W. B. Pardo, J. A. Walkenstein, and M. Monti, Phys. Rev. Lett. 85, 2929 (2000).
    [CrossRef] [PubMed]
  5. D. Maza, A. Vallone, H. Mancini, and S. Bocaletti, Phys. Rev. Lett. 85, 5567 (2000).
    [CrossRef]
  6. D. J. DeShazer, R. Breban, E. Ott, and R. Roy, Phys. Rev. Lett. 87, 044101 (2001).
    [CrossRef] [PubMed]
  7. E. Allaria, F. T. Arecchi, A. Di Garbo, and R. Meucci, Phys. Rev. Lett. 86, 791 (2001).
    [CrossRef] [PubMed]
  8. T. Baer, J. Opt. Soc. Am. B 3, 1175 (1986).
    [CrossRef]
  9. W. P. Risk, T. R. Gosnell, and A. V. Nurmikko, Compact Blue-Green Lasers (Cambridge U. Press, 2003).
    [CrossRef]
  10. C. Liu, R. Roy, H. D. I. Abarbanel, Z. Gills, and K. Nunes, Phys. Rev. E 55, 6483 (1997).
    [CrossRef]
  11. G. E. James, E. M. Harrell II, and R. Roy, Phys. Rev. A 41, 2778 (1990).
    [CrossRef] [PubMed]
  12. C. Bracikowski and R. Roy, Chaos 1, 49 (1991).
    [CrossRef] [PubMed]
  13. K. Wiesenfeld, C. Bracikowski, G. James, and R. Roy, Phys. Rev. Lett. 65, 1749 (1990).
    [CrossRef] [PubMed]
  14. K. Pyragas, F. Lange, T. Letz, J. Parisi, and A. Kittel, Phys. Rev. E 63, 016204 (2000).
    [CrossRef]
  15. A. Ahlborn and U. Parlitz, Phys. Rev. Lett. 93, 264101 (2004).
    [CrossRef]
  16. A. Ahlborn and U. Parlitz, Phys. Rev. Lett. 96, 034102 (2006).
    [CrossRef] [PubMed]
  17. A. Ahlborn and U. Parlitz, Opt. Lett. 31, 465 (2006).
    [CrossRef] [PubMed]
  18. M. G. Rosenblum, A. S. Pikovsky, J. Kurths, G. V. Osipov, I. Z. Kiss, and J. L. Hudson, Phys. Rev. Lett. 89, 264102 (2002).
    [CrossRef] [PubMed]
  19. M. C. Romano, M. Thiel, J. Kurths, I. Z. Kiss, and J. L. Hudson, Europhys. Lett. 71, 466 (2005).
    [CrossRef]
  20. A. S. Pikovsky, M. G. Rosenblum, G. V. Osipov, and J. Kurths, Physica D 104, 219 (1997).
    [CrossRef]

2006 (2)

A. Ahlborn and U. Parlitz, Phys. Rev. Lett. 96, 034102 (2006).
[CrossRef] [PubMed]

A. Ahlborn and U. Parlitz, Opt. Lett. 31, 465 (2006).
[CrossRef] [PubMed]

2005 (1)

M. C. Romano, M. Thiel, J. Kurths, I. Z. Kiss, and J. L. Hudson, Europhys. Lett. 71, 466 (2005).
[CrossRef]

2004 (1)

A. Ahlborn and U. Parlitz, Phys. Rev. Lett. 93, 264101 (2004).
[CrossRef]

2003 (1)

W. P. Risk, T. R. Gosnell, and A. V. Nurmikko, Compact Blue-Green Lasers (Cambridge U. Press, 2003).
[CrossRef]

2002 (1)

M. G. Rosenblum, A. S. Pikovsky, J. Kurths, G. V. Osipov, I. Z. Kiss, and J. L. Hudson, Phys. Rev. Lett. 89, 264102 (2002).
[CrossRef] [PubMed]

2001 (3)

D. J. DeShazer, R. Breban, E. Ott, and R. Roy, Phys. Rev. Lett. 87, 044101 (2001).
[CrossRef] [PubMed]

E. Allaria, F. T. Arecchi, A. Di Garbo, and R. Meucci, Phys. Rev. Lett. 86, 791 (2001).
[CrossRef] [PubMed]

A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: a Universal Concept in Nonlinear Sciences (Cambridge U. Press, 2001).
[CrossRef]

2000 (3)

C. M. Ticos, E. Rosa, W. B. Pardo, J. A. Walkenstein, and M. Monti, Phys. Rev. Lett. 85, 2929 (2000).
[CrossRef] [PubMed]

D. Maza, A. Vallone, H. Mancini, and S. Bocaletti, Phys. Rev. Lett. 85, 5567 (2000).
[CrossRef]

K. Pyragas, F. Lange, T. Letz, J. Parisi, and A. Kittel, Phys. Rev. E 63, 016204 (2000).
[CrossRef]

1997 (2)

A. S. Pikovsky, M. G. Rosenblum, G. V. Osipov, and J. Kurths, Physica D 104, 219 (1997).
[CrossRef]

C. Liu, R. Roy, H. D. I. Abarbanel, Z. Gills, and K. Nunes, Phys. Rev. E 55, 6483 (1997).
[CrossRef]

1996 (2)

M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, Phys. Rev. Lett. 76, 1804 (1996).
[CrossRef] [PubMed]

U. Parlitz, L. Junge, W. Lauterborn, and L. Kocarev, Phys. Rev. E 54, 2115 (1996).
[CrossRef]

1991 (1)

C. Bracikowski and R. Roy, Chaos 1, 49 (1991).
[CrossRef] [PubMed]

1990 (2)

K. Wiesenfeld, C. Bracikowski, G. James, and R. Roy, Phys. Rev. Lett. 65, 1749 (1990).
[CrossRef] [PubMed]

G. E. James, E. M. Harrell II, and R. Roy, Phys. Rev. A 41, 2778 (1990).
[CrossRef] [PubMed]

1986 (1)

Abarbanel, H. D. I.

C. Liu, R. Roy, H. D. I. Abarbanel, Z. Gills, and K. Nunes, Phys. Rev. E 55, 6483 (1997).
[CrossRef]

Ahlborn, A.

A. Ahlborn and U. Parlitz, Opt. Lett. 31, 465 (2006).
[CrossRef] [PubMed]

A. Ahlborn and U. Parlitz, Phys. Rev. Lett. 96, 034102 (2006).
[CrossRef] [PubMed]

A. Ahlborn and U. Parlitz, Phys. Rev. Lett. 93, 264101 (2004).
[CrossRef]

Allaria, E.

E. Allaria, F. T. Arecchi, A. Di Garbo, and R. Meucci, Phys. Rev. Lett. 86, 791 (2001).
[CrossRef] [PubMed]

Arecchi, F. T.

E. Allaria, F. T. Arecchi, A. Di Garbo, and R. Meucci, Phys. Rev. Lett. 86, 791 (2001).
[CrossRef] [PubMed]

Baer, T.

Bocaletti, S.

D. Maza, A. Vallone, H. Mancini, and S. Bocaletti, Phys. Rev. Lett. 85, 5567 (2000).
[CrossRef]

Bracikowski, C.

C. Bracikowski and R. Roy, Chaos 1, 49 (1991).
[CrossRef] [PubMed]

K. Wiesenfeld, C. Bracikowski, G. James, and R. Roy, Phys. Rev. Lett. 65, 1749 (1990).
[CrossRef] [PubMed]

Breban, R.

D. J. DeShazer, R. Breban, E. Ott, and R. Roy, Phys. Rev. Lett. 87, 044101 (2001).
[CrossRef] [PubMed]

DeShazer, D. J.

D. J. DeShazer, R. Breban, E. Ott, and R. Roy, Phys. Rev. Lett. 87, 044101 (2001).
[CrossRef] [PubMed]

Di Garbo, A.

E. Allaria, F. T. Arecchi, A. Di Garbo, and R. Meucci, Phys. Rev. Lett. 86, 791 (2001).
[CrossRef] [PubMed]

Gills, Z.

C. Liu, R. Roy, H. D. I. Abarbanel, Z. Gills, and K. Nunes, Phys. Rev. E 55, 6483 (1997).
[CrossRef]

Gosnell, T. R.

W. P. Risk, T. R. Gosnell, and A. V. Nurmikko, Compact Blue-Green Lasers (Cambridge U. Press, 2003).
[CrossRef]

Harrell, E. M.

G. E. James, E. M. Harrell II, and R. Roy, Phys. Rev. A 41, 2778 (1990).
[CrossRef] [PubMed]

Hudson, J. L.

M. C. Romano, M. Thiel, J. Kurths, I. Z. Kiss, and J. L. Hudson, Europhys. Lett. 71, 466 (2005).
[CrossRef]

M. G. Rosenblum, A. S. Pikovsky, J. Kurths, G. V. Osipov, I. Z. Kiss, and J. L. Hudson, Phys. Rev. Lett. 89, 264102 (2002).
[CrossRef] [PubMed]

James, G.

K. Wiesenfeld, C. Bracikowski, G. James, and R. Roy, Phys. Rev. Lett. 65, 1749 (1990).
[CrossRef] [PubMed]

James, G. E.

G. E. James, E. M. Harrell II, and R. Roy, Phys. Rev. A 41, 2778 (1990).
[CrossRef] [PubMed]

Junge, L.

U. Parlitz, L. Junge, W. Lauterborn, and L. Kocarev, Phys. Rev. E 54, 2115 (1996).
[CrossRef]

Kiss, I. Z.

M. C. Romano, M. Thiel, J. Kurths, I. Z. Kiss, and J. L. Hudson, Europhys. Lett. 71, 466 (2005).
[CrossRef]

M. G. Rosenblum, A. S. Pikovsky, J. Kurths, G. V. Osipov, I. Z. Kiss, and J. L. Hudson, Phys. Rev. Lett. 89, 264102 (2002).
[CrossRef] [PubMed]

Kittel, A.

K. Pyragas, F. Lange, T. Letz, J. Parisi, and A. Kittel, Phys. Rev. E 63, 016204 (2000).
[CrossRef]

Kocarev, L.

U. Parlitz, L. Junge, W. Lauterborn, and L. Kocarev, Phys. Rev. E 54, 2115 (1996).
[CrossRef]

Kurths, J.

M. C. Romano, M. Thiel, J. Kurths, I. Z. Kiss, and J. L. Hudson, Europhys. Lett. 71, 466 (2005).
[CrossRef]

M. G. Rosenblum, A. S. Pikovsky, J. Kurths, G. V. Osipov, I. Z. Kiss, and J. L. Hudson, Phys. Rev. Lett. 89, 264102 (2002).
[CrossRef] [PubMed]

A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: a Universal Concept in Nonlinear Sciences (Cambridge U. Press, 2001).
[CrossRef]

A. S. Pikovsky, M. G. Rosenblum, G. V. Osipov, and J. Kurths, Physica D 104, 219 (1997).
[CrossRef]

M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, Phys. Rev. Lett. 76, 1804 (1996).
[CrossRef] [PubMed]

Lange, F.

K. Pyragas, F. Lange, T. Letz, J. Parisi, and A. Kittel, Phys. Rev. E 63, 016204 (2000).
[CrossRef]

Lauterborn, W.

U. Parlitz, L. Junge, W. Lauterborn, and L. Kocarev, Phys. Rev. E 54, 2115 (1996).
[CrossRef]

Letz, T.

K. Pyragas, F. Lange, T. Letz, J. Parisi, and A. Kittel, Phys. Rev. E 63, 016204 (2000).
[CrossRef]

Liu, C.

C. Liu, R. Roy, H. D. I. Abarbanel, Z. Gills, and K. Nunes, Phys. Rev. E 55, 6483 (1997).
[CrossRef]

Mancini, H.

D. Maza, A. Vallone, H. Mancini, and S. Bocaletti, Phys. Rev. Lett. 85, 5567 (2000).
[CrossRef]

Maza, D.

D. Maza, A. Vallone, H. Mancini, and S. Bocaletti, Phys. Rev. Lett. 85, 5567 (2000).
[CrossRef]

Meucci, R.

E. Allaria, F. T. Arecchi, A. Di Garbo, and R. Meucci, Phys. Rev. Lett. 86, 791 (2001).
[CrossRef] [PubMed]

Monti, M.

C. M. Ticos, E. Rosa, W. B. Pardo, J. A. Walkenstein, and M. Monti, Phys. Rev. Lett. 85, 2929 (2000).
[CrossRef] [PubMed]

Nunes, K.

C. Liu, R. Roy, H. D. I. Abarbanel, Z. Gills, and K. Nunes, Phys. Rev. E 55, 6483 (1997).
[CrossRef]

Nurmikko, A. V.

W. P. Risk, T. R. Gosnell, and A. V. Nurmikko, Compact Blue-Green Lasers (Cambridge U. Press, 2003).
[CrossRef]

Osipov, G. V.

M. G. Rosenblum, A. S. Pikovsky, J. Kurths, G. V. Osipov, I. Z. Kiss, and J. L. Hudson, Phys. Rev. Lett. 89, 264102 (2002).
[CrossRef] [PubMed]

A. S. Pikovsky, M. G. Rosenblum, G. V. Osipov, and J. Kurths, Physica D 104, 219 (1997).
[CrossRef]

Ott, E.

D. J. DeShazer, R. Breban, E. Ott, and R. Roy, Phys. Rev. Lett. 87, 044101 (2001).
[CrossRef] [PubMed]

Pardo, W. B.

C. M. Ticos, E. Rosa, W. B. Pardo, J. A. Walkenstein, and M. Monti, Phys. Rev. Lett. 85, 2929 (2000).
[CrossRef] [PubMed]

Parisi, J.

K. Pyragas, F. Lange, T. Letz, J. Parisi, and A. Kittel, Phys. Rev. E 63, 016204 (2000).
[CrossRef]

Parlitz, U.

A. Ahlborn and U. Parlitz, Phys. Rev. Lett. 96, 034102 (2006).
[CrossRef] [PubMed]

A. Ahlborn and U. Parlitz, Opt. Lett. 31, 465 (2006).
[CrossRef] [PubMed]

A. Ahlborn and U. Parlitz, Phys. Rev. Lett. 93, 264101 (2004).
[CrossRef]

U. Parlitz, L. Junge, W. Lauterborn, and L. Kocarev, Phys. Rev. E 54, 2115 (1996).
[CrossRef]

Pikovsky, A.

A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: a Universal Concept in Nonlinear Sciences (Cambridge U. Press, 2001).
[CrossRef]

Pikovsky, A. S.

M. G. Rosenblum, A. S. Pikovsky, J. Kurths, G. V. Osipov, I. Z. Kiss, and J. L. Hudson, Phys. Rev. Lett. 89, 264102 (2002).
[CrossRef] [PubMed]

A. S. Pikovsky, M. G. Rosenblum, G. V. Osipov, and J. Kurths, Physica D 104, 219 (1997).
[CrossRef]

M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, Phys. Rev. Lett. 76, 1804 (1996).
[CrossRef] [PubMed]

Pyragas, K.

K. Pyragas, F. Lange, T. Letz, J. Parisi, and A. Kittel, Phys. Rev. E 63, 016204 (2000).
[CrossRef]

Risk, W. P.

W. P. Risk, T. R. Gosnell, and A. V. Nurmikko, Compact Blue-Green Lasers (Cambridge U. Press, 2003).
[CrossRef]

Romano, M. C.

M. C. Romano, M. Thiel, J. Kurths, I. Z. Kiss, and J. L. Hudson, Europhys. Lett. 71, 466 (2005).
[CrossRef]

Rosa, E.

C. M. Ticos, E. Rosa, W. B. Pardo, J. A. Walkenstein, and M. Monti, Phys. Rev. Lett. 85, 2929 (2000).
[CrossRef] [PubMed]

Rosenblum, M.

A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: a Universal Concept in Nonlinear Sciences (Cambridge U. Press, 2001).
[CrossRef]

Rosenblum, M. G.

M. G. Rosenblum, A. S. Pikovsky, J. Kurths, G. V. Osipov, I. Z. Kiss, and J. L. Hudson, Phys. Rev. Lett. 89, 264102 (2002).
[CrossRef] [PubMed]

A. S. Pikovsky, M. G. Rosenblum, G. V. Osipov, and J. Kurths, Physica D 104, 219 (1997).
[CrossRef]

M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, Phys. Rev. Lett. 76, 1804 (1996).
[CrossRef] [PubMed]

Roy, R.

D. J. DeShazer, R. Breban, E. Ott, and R. Roy, Phys. Rev. Lett. 87, 044101 (2001).
[CrossRef] [PubMed]

C. Liu, R. Roy, H. D. I. Abarbanel, Z. Gills, and K. Nunes, Phys. Rev. E 55, 6483 (1997).
[CrossRef]

C. Bracikowski and R. Roy, Chaos 1, 49 (1991).
[CrossRef] [PubMed]

G. E. James, E. M. Harrell II, and R. Roy, Phys. Rev. A 41, 2778 (1990).
[CrossRef] [PubMed]

K. Wiesenfeld, C. Bracikowski, G. James, and R. Roy, Phys. Rev. Lett. 65, 1749 (1990).
[CrossRef] [PubMed]

Thiel, M.

M. C. Romano, M. Thiel, J. Kurths, I. Z. Kiss, and J. L. Hudson, Europhys. Lett. 71, 466 (2005).
[CrossRef]

Ticos, C. M.

C. M. Ticos, E. Rosa, W. B. Pardo, J. A. Walkenstein, and M. Monti, Phys. Rev. Lett. 85, 2929 (2000).
[CrossRef] [PubMed]

Vallone, A.

D. Maza, A. Vallone, H. Mancini, and S. Bocaletti, Phys. Rev. Lett. 85, 5567 (2000).
[CrossRef]

Walkenstein, J. A.

C. M. Ticos, E. Rosa, W. B. Pardo, J. A. Walkenstein, and M. Monti, Phys. Rev. Lett. 85, 2929 (2000).
[CrossRef] [PubMed]

Wiesenfeld, K.

K. Wiesenfeld, C. Bracikowski, G. James, and R. Roy, Phys. Rev. Lett. 65, 1749 (1990).
[CrossRef] [PubMed]

Chaos (1)

C. Bracikowski and R. Roy, Chaos 1, 49 (1991).
[CrossRef] [PubMed]

Europhys. Lett. (1)

M. C. Romano, M. Thiel, J. Kurths, I. Z. Kiss, and J. L. Hudson, Europhys. Lett. 71, 466 (2005).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Lett. (1)

Phys. Rev. A (1)

G. E. James, E. M. Harrell II, and R. Roy, Phys. Rev. A 41, 2778 (1990).
[CrossRef] [PubMed]

Phys. Rev. E (3)

K. Pyragas, F. Lange, T. Letz, J. Parisi, and A. Kittel, Phys. Rev. E 63, 016204 (2000).
[CrossRef]

U. Parlitz, L. Junge, W. Lauterborn, and L. Kocarev, Phys. Rev. E 54, 2115 (1996).
[CrossRef]

C. Liu, R. Roy, H. D. I. Abarbanel, Z. Gills, and K. Nunes, Phys. Rev. E 55, 6483 (1997).
[CrossRef]

Phys. Rev. Lett. (9)

M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, Phys. Rev. Lett. 76, 1804 (1996).
[CrossRef] [PubMed]

C. M. Ticos, E. Rosa, W. B. Pardo, J. A. Walkenstein, and M. Monti, Phys. Rev. Lett. 85, 2929 (2000).
[CrossRef] [PubMed]

D. Maza, A. Vallone, H. Mancini, and S. Bocaletti, Phys. Rev. Lett. 85, 5567 (2000).
[CrossRef]

D. J. DeShazer, R. Breban, E. Ott, and R. Roy, Phys. Rev. Lett. 87, 044101 (2001).
[CrossRef] [PubMed]

E. Allaria, F. T. Arecchi, A. Di Garbo, and R. Meucci, Phys. Rev. Lett. 86, 791 (2001).
[CrossRef] [PubMed]

A. Ahlborn and U. Parlitz, Phys. Rev. Lett. 93, 264101 (2004).
[CrossRef]

A. Ahlborn and U. Parlitz, Phys. Rev. Lett. 96, 034102 (2006).
[CrossRef] [PubMed]

M. G. Rosenblum, A. S. Pikovsky, J. Kurths, G. V. Osipov, I. Z. Kiss, and J. L. Hudson, Phys. Rev. Lett. 89, 264102 (2002).
[CrossRef] [PubMed]

K. Wiesenfeld, C. Bracikowski, G. James, and R. Roy, Phys. Rev. Lett. 65, 1749 (1990).
[CrossRef] [PubMed]

Physica D (1)

A. S. Pikovsky, M. G. Rosenblum, G. V. Osipov, and J. Kurths, Physica D 104, 219 (1997).
[CrossRef]

Other (2)

W. P. Risk, T. R. Gosnell, and A. V. Nurmikko, Compact Blue-Green Lasers (Cambridge U. Press, 2003).
[CrossRef]

A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: a Universal Concept in Nonlinear Sciences (Cambridge U. Press, 2001).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Experimental setup. The laser’s pump current is modulated via a bias-T by a sinusoidal signal of adjustable frequency f and amplitude A. To analyze the laser’s response either the non-frequency-doubled infrared intensities ( 1064 nm ) or the frequency-doubled green intensity ( 532 nm ) are detected by photodiodes.

Fig. 2
Fig. 2

Time series showing an example of chaotic phase synchronization (top, laser intensity; bottom, pump current modulation).

Fig. 3
Fig. 3

(a) Recurrence P ( τ ) versus τ for the generator (thin line) and the laser intensity (thick line) for f = 1.1 MHz and a = 5.9 Vpp . Coincidence of τ values where maxima occur indicates phase synchronization. (b) Recurrence P ( τ ) versus τ for f = 1.26 MHz and a = 2 Vpp . Maxima of the thick curve (laser intensity) and the thin curve (periodic modulation) occur at different recurrence times τ. (c) Two-dimensional reconstruction of the intensity signal in terms of real and imaginary part of the analytic signal s. Light lines represent the full attractor, while dots constitute a Poincaré cross section obtained by stroboscopically sampling the intensity with the driving generator as trigger. The data are the same as in (a), and the fact that the stroboscopically sampled intensities cover some part of the full attractor only indicates phase synchronization. (d) Reconstructed attractor and Poincaré cross section for data used in (b). In contrast to (c) the sampled intensities (dots) cover the whole attractor, i.e., there exists no fixed phase relation.

Fig. 4
Fig. 4

Synchronization diagrams of the Nd:YAG laser whose pump current was modulated with frequency f and amplitude a. Synchronization occurs within the bright Arnold tongue around f 1 MHz and at some parameter combinations with periodic response (bright vertical stripes at about f 0.5 MHz and f 1 MHz ). (a) Cross-correlation coefficient Eq. (2) of the recurrence analysis. (b) Normalized pseudo ensemble average.

Equations (2)

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P ( τ ) = 1 N τ i = 1 N τ Θ ( ϵ x j x j + τ )
CRP = P ¯ L ( τ ) P ¯ G ( τ ) σ L σ G

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