## Optical periodic code matching by single-shot broadband frequency-domain cross-correlation |

Optics Express, Vol. 18, Issue 17, pp. 17756-17763 (2010)

http://dx.doi.org/10.1364/OE.18.017756

Acrobat PDF (943 KB)

### Abstract

We introduce and experimentally demonstrate a simple and reliable optical technique for matching between two periodic numerical sequences based on optical single-shot measurement of their broadband cross-correlation function in the frequency domain. Each sequence is optically encoded into the shape of the different broadband femtosecond pulse using pulse-shaping techniques. The two corresponding shaped pulses are mixed in a nonlinear medium together with an additional (amplitude-shaped) narrowband pulse. The spectrum of the resulting four-wave mixing signal is measured to provide the cross-correlation function of the two encoded sequences. For identical sequences it is the auto-correlation function that is being measured, allowing also the identification of the sequence period. The high contrast achieved here between cross-correlation and auto-correlation functions allows to determine with a very high reliability whether the two encoded sequences are identical or not. The demonstrated technique might be employed in an optical implementation of CDMA communication protocol.

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5. A. M. Weiner, “Fourier information optics for the ultrafast time domain,” Appl. Opt. **47**(4), A88–A96 (2008). [CrossRef] [PubMed]

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

8. Z. Zheng and A. M. Weiner, “Spectral phase correlation of coded femtosecond pulses by second-harmonic generation in thick nonlinear crystals,” Opt. Lett. **25**(13), 984–986 (2000). [CrossRef]

11. Z. Jiang, D. S. Seo, S.-D. Yang, D. E. Leaird, R. V. Roussev, C. Langrock, M. M. Fejer, and A. M. Weiner, “Low-power high-contrast coded waveform discrimination at 10 GHz via nonlinear processing,” IEEE Photon. Technol. Lett. **16**(7), 1778–1780 (2004). [CrossRef]

*S*code families, which incorporate periodic cyclically-distinct pseudo-random sequences over arbitrary extended alphabets and have very important advantages over other code families [12

12. S. Boztas, A. R. Hammons, and P. V. Kumar, “4-phase sequences with near-optimum correlation properties,” IEEE Trans. Inf. Theory **38**(3), 1101–1113 (1992). [CrossRef]

*S*-families as well as the properties of the correlation functions between codes are determined by the length of the code and the size of the alphabet being used. They are analytically derived by the Number Theory that puts well-defined bounds on the values of the auto- and cross-correlation functions at different shifts (delays) between the compared codes. The use of these

*S*-families, as compared to other families that are of binary codes, has already lead to significant improvements in the capacity and reliability of CDMA networks in the rf regime [1].

3. A. M. Weiner, Z. Jiang, and D. E. Leaird, “Spectrally phase-coded O-CDMA,” J. Opt. Netw. **6**(6), 728–755 (2007). [CrossRef]

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

8. Z. Zheng and A. M. Weiner, “Spectral phase correlation of coded femtosecond pulses by second-harmonic generation in thick nonlinear crystals,” Opt. Lett. **25**(13), 984–986 (2000). [CrossRef]

11. Z. Jiang, D. S. Seo, S.-D. Yang, D. E. Leaird, R. V. Roussev, C. Langrock, M. M. Fejer, and A. M. Weiner, “Low-power high-contrast coded waveform discrimination at 10 GHz via nonlinear processing,” IEEE Photon. Technol. Lett. **16**(7), 1778–1780 (2004). [CrossRef]

*S*-families. For example, only multiple-point measurement of the full auto-correlation function allows the identification of the periodicity of the code. Such periodicity measurement introduces an additional characteristic that can be used, for instance, for routing incoming information to different parts of the network (even without the identification of the code itself, just by splitting the incoming code signal and measuring its auto-correlation function). Also, generally, the signal-to-noise ratio is reduced and thus the reliability is improved when the code comparison is based on multiple measured values rather than on a single measured value. In the present work we introduce and experimentally demonstrate a simple, efficient, and reliable optical technique for measuring the full (multiple-point) cross-correlation function between pairs of codes belonging to these

*S*families of interest. The new technique is based on the third-order nonlinear process of four-wave mixing (FWM) in simple media such as fused silica glass. We believe that this new technique will significantly contribute to further improvement in the performance (capacity and reliability) of OCDMA networks.

15. A. C. Eckbreth, “BOXCARS: Crossed-beam phase-matched CARS generation in gases,” Appl. Phys. Lett. **32**(7), 421–423 (1978). [CrossRef]

*p*, 0

*p*-1, with arithmetic modulo

*p*, where

*p*≥2. The cross-correlation function between two sequences is defined aswhere

*k*and

*r*are the indexes of the sequences in the set of size

*M*, 0

*L*-1,

*L*is the period of the sequence, and

*p*-th root of unity. The merit factor commonly used to quantify the performance of a given code family is the maximal value

*τ*, i.e.,

*k = r*and

*τ*= 0. The values at these latter cases correspond to the values of the auto-correlation functions at zero delay, for which

12. S. Boztas, A. R. Hammons, and P. V. Kumar, “4-phase sequences with near-optimum correlation properties,” IEEE Trans. Inf. Theory **38**(3), 1101–1113 (1992). [CrossRef]

*L*, with a given alphabet size

*p*, the code family of higher order includes more code sequences and the corresponding value of

*L*increases, the number of available codes increases, while the value of

*L*and the order of the family.

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

*ω*. The phase-shaping is implemented via control over the voltage applied at different pixels of liquid crystal, which induces different phase retardation of the transmitted light. The spectral width of each pixel is

*k*and

*r*, corresponds to the cross-correlation function

*Δ*is a continuous variable representing the detuning between spectral components of

*τ*is a discrete integer variable representing the relative delay (shift) between the two codes.

*Δ*. Suppose that the continuous axis of frequency

*ω*is divided into small bins of size

*t*(the index of the code component; see above), such that the phase

*k*=

*r*, as the interference between all the corresponding two-photon pathways is fully constructive. Here

*n*= 0,1,2… and

*L*is the period of the code. For

*k*≠r the interference between the two-photon pathways is destructive and

12. S. Boztas, A. R. Hammons, and P. V. Kumar, “4-phase sequences with near-optimum correlation properties,” IEEE Trans. Inf. Theory **38**(3), 1101–1113 (1992). [CrossRef]

*k*=

*r*) and non-matching (

*k*≠r) codes.

^{8}W/cm

^{2}and the single-shot FWM output signal is of a total flux of about 10

^{7}photons, which is by far enough for measuring the corresponding spectrum. It is important to mention that the above implementation can also be extended to pulses with energies down-to several nano-joules (nJ). This can be achieved by proper optimization of the optical setup, such that the intensities of the focused beams will stay of the 10

^{8}-W/cm

^{2}magnitude and thus the only difference from the present situation will essentially be the reduction of the signal output flux to about 10

^{4}photons. This latter photon flux is experimentally appropriate for spectral measurement, as is demonstrated, for example, in frequency-resolved optical gating (FROG) measurements of nJ pulses [16]. Thus, standard low-energy Ti:Sapphire femtosecond oscillators are also suitable sources for implementing the new technique presented here.

*p*= 4) codes that belong to the nested families S(0) and S(1) and are generated using corresponding linear shift registers constructed upon the primitive polynomials [12

**38**(3), 1101–1113 (1992). [CrossRef]

*L*= 2

*−1 = 7, where*

^{l}*l*= 3 is the order of the generating polynomial. The size of the family S(0) is

*M*=

*L +*2 = 9 with

*M =*(

*L*+ 2)(

*L*+ 1) = 72, including also the nested S(0) family. Formally, the corresponding value of

*l*= 3, the occurrence of this value is zero and thus the effective value is

13. P. V. Kumar, T. Helleseth, A. R. Calderbank, and A. R. Hammons, “Large families of quaternary sequences with low correlation,” IEEE Trans. Inf. Theory **42**(2), 579–592 (1996). [CrossRef]

*k = r*(black curve) and

*k*≠r (red curve). The highly pronounced peaks observed in the spectrum correspond to the constructive interference between the photo-induced pathways associated with the auto-correlation case of identical codes, i.e.,

*k = r*. The auto-correlation peaks are separated by

*n*is an integer determined by the period of the code. This is in high contrast to the

*k*≠r. The measured distance between the auto-correlation peaks is used for determining the code period. The corresponding results for the codes belonging to the family S(1) are shown in Fig. 3 for several cases of the programmed code period. The results are presented as histograms of the retrieved period given as the retrieved value of

*n*obtained via the relation

*n*= round

*k = r*and

*k*≠r is not sufficient for the proper recognition.

*m*= 0, 1, 2 etc. The corresponding results for the code families S(0) and S(1) are presented, respectively, in Figs. 4(a) and 4(b) as two-dimensional color maps. The color represents the normalized value of the summation

*k*(

*x*-axis) and

*r*(

*y*-axis), where the values on the diagonal correspond to the auto-correlation cases of

*r = k*. In the ideal case, where the spectrum of the probe pulse is a delta-function and the spectral envelope

*r*≠k as compared to the corresponding case of

*r = k*. As the signal corresponding to

*r = k*is sufficiently higher than for

*r*≠k, we obtain a correct recognition of 100% of the S(0)'s codes with a signal threshold of 80% of the auto-correlation signal [see Fig. 4(a)]. This value also agrees very well with the value obtained from our numerical simulations of the present experiment. It worth also mentioning that the experimental mean value of

*l*= 3,

*p*= 4).

*S*-families codes, the experimental extension to the cases of

*S*-families of higher orders and over larger alphabets is rather straightforward. Experimentally, longer codes imply pulses of wider bandwidth and/or pulse-shaping setup of higher resolution. The corresponding femtosecond technology is widely available. As can be concluded from the results presented above, proper optimization of the experimental conditions and parameters will also allow to achieve low threshold levels as required for applicative use. We believe that this new technique will significantly contribute to the improvement in the performance of OCDMA networks.

## Acknowledgements

## References and Links

1. | L. Harte, R. Levine, and R. Kikta, |

2. | Optical Code Division Multiple Access, |

3. | A. M. Weiner, Z. Jiang, and D. E. Leaird, “Spectrally phase-coded O-CDMA,” J. Opt. Netw. |

4. | J. P. Heritage and A. M. Weiner, “Advances in spectral optical code-division multiple-access communications,” IEEE J. Sel. Top. Quantum Electron. |

5. | A. M. Weiner, “Fourier information optics for the ultrafast time domain,” Appl. Opt. |

6. | A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. |

7. | V. S. Pless, W. C. Huffman eds. |

8. | Z. Zheng and A. M. Weiner, “Spectral phase correlation of coded femtosecond pulses by second-harmonic generation in thick nonlinear crystals,” Opt. Lett. |

9. | Z. Zheng, A. M. Weiner, K. R. Parameswaran, M. H. Chou, and M. M. Fejer, “Low-power spectral phase correlator using periodically poled LiNbO |

10. | D. S. Seo, Z. Jiang, D. E. Leaird, and A. M. Weiner, “Pulse shaper in a loop: demonstration of cascadable ultrafast all-optical code translation,” Opt. Lett. |

11. | Z. Jiang, D. S. Seo, S.-D. Yang, D. E. Leaird, R. V. Roussev, C. Langrock, M. M. Fejer, and A. M. Weiner, “Low-power high-contrast coded waveform discrimination at 10 GHz via nonlinear processing,” IEEE Photon. Technol. Lett. |

12. | S. Boztas, A. R. Hammons, and P. V. Kumar, “4-phase sequences with near-optimum correlation properties,” IEEE Trans. Inf. Theory |

13. | P. V. Kumar, T. Helleseth, A. R. Calderbank, and A. R. Hammons, “Large families of quaternary sequences with low correlation,” IEEE Trans. Inf. Theory |

14. | T. Helleseth, and P. V. Kumar, “Sequences with low correlation,” in |

15. | A. C. Eckbreth, “BOXCARS: Crossed-beam phase-matched CARS generation in gases,” Appl. Phys. Lett. |

16. | R. Trebino, |

**OCIS Codes**

(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing

(200.3050) Optics in computing : Information processing

(060.4251) Fiber optics and optical communications : Networks, assignment and routing algorithms

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: March 24, 2010

Revised Manuscript: July 9, 2010

Manuscript Accepted: July 12, 2010

Published: August 3, 2010

**Citation**

Lev Chuntonov and Zohar Amitay, "Optical periodic code matching by single-shot broadband frequency-domain cross-correlation," Opt. Express **18**, 17756-17763 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-17-17756

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### References

- L. Harte, R. Levine, and R. Kikta, 3G Wireless Demystified, (McGraw-Hill, New York, 2002).
- Optical Code Division Multiple Access, Fundamentals and Applications P. R. Prucnal ed. (CRC, Taylor & Francis, Boca Raton, 2006).
- A. M. Weiner, Z. Jiang, and D. E. Leaird, “Spectrally phase-coded O-CDMA,” J. Opt. Netw. 6(6), 728–755 (2007). [CrossRef]
- J. P. Heritage and A. M. Weiner, “Advances in spectral optical code-division multiple-access communications,” IEEE J. Sel. Top. Quantum Electron. 13(5), 1351–1369 (2007). [CrossRef]
- A. M. Weiner, “Fourier information optics for the ultrafast time domain,” Appl. Opt. 47(4), A88–A96 (2008). [CrossRef] [PubMed]
- A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71(5), 1929–1960 (2000). [CrossRef]
- V. S. Pless and W. C. Huffman eds., Handbook of Coding Theory (Elsevier Science B. V., Amsterdam, 1998).
- Z. Zheng and A. M. Weiner, “Spectral phase correlation of coded femtosecond pulses by second-harmonic generation in thick nonlinear crystals,” Opt. Lett. 25(13), 984–986 (2000). [CrossRef]
- Z. Zheng, A. M. Weiner, K. R. Parameswaran, M. H. Chou, and M. M. Fejer, “Low-power spectral phase correlator using periodically poled LiNbO3 waveguides,” IEEE Photon. Technol. Lett. 13(4), 376–378 (2001). [CrossRef]
- D. S. Seo, Z. Jiang, D. E. Leaird, and A. M. Weiner, “Pulse shaper in a loop: demonstration of cascadable ultrafast all-optical code translation,” Opt. Lett. 29(16), 1864–1866 (2004). [CrossRef] [PubMed]
- Z. Jiang, D. S. Seo, S.-D. Yang, D. E. Leaird, R. V. Roussev, C. Langrock, M. M. Fejer, and A. M. Weiner, “Low-power high-contrast coded waveform discrimination at 10 GHz via nonlinear processing,” IEEE Photon. Technol. Lett. 16(7), 1778–1780 (2004). [CrossRef]
- S. Boztas, A. R. Hammons, and P. V. Kumar, “4-phase sequences with near-optimum correlation properties,” IEEE Trans. Inf. Theory 38(3), 1101–1113 (1992). [CrossRef]
- P. V. Kumar, T. Helleseth, A. R. Calderbank, and A. R. Hammons, “Large families of quaternary sequences with low correlation,” IEEE Trans. Inf. Theory 42(2), 579–592 (1996). [CrossRef]
- T. Helleseth and P. V. Kumar, “Sequences with low correlation,” in Handbook of Coding Theory, V. S. Pless and W. C. Huffman eds., (Elsevier Science B. V., Amsterdam, 1998).
- A. C. Eckbreth, “BOXCARS: Crossed-beam phase-matched CARS generation in gases,” Appl. Phys. Lett. 32(7), 421–423 (1978). [CrossRef]
- R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses, (Kluwer Academic Publishers, Norwell, 2000).

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