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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 17 — Aug. 16, 2010
  • pp: 17756–17763
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Optical periodic code matching by single-shot broadband frequency-domain cross-correlation

Lev Chuntonov and Zohar Amitay  »View Author Affiliations


Optics Express, Vol. 18, Issue 17, pp. 17756-17763 (2010)
http://dx.doi.org/10.1364/OE.18.017756


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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.

© 2010 OSA

Code Division Multiple Access (CDMA) schemes are extensively employed in modern communication technologies in the rf-regime including, for example, the mobile 3G wide-band and Global Positioning System standards [1

1. L. Harte, R. Levine, and R. Kikta, 3G Wireless Demystified, (McGraw-Hill, New York, 2002).

]. It was recognized several decades ago that the developments of the communication technology in the rf are also applicable for optical communication [2

2. Optical Code Division Multiple Access, Fundamentals and Applications P. R. Prucnal ed. (CRC, Taylor & Francis, Boca Raton, 2006).

5

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

]. Among the advantages of the optical implementation of CDMA (OCDMA) are simplified decentralized networking, high information security, and high spectral efficiency. When OCDMA with shaped femtosecond pulses is considered, the codes are spread over the broad bandwidth of the pulses, where each code is converted into the optical characteristics of the different spectral components using pulse shaping techniques [6

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

]. The latter are experimentally implemented most powerfully in the frequency domain, with the resulting temporal waveforms of the shaped pulses also uniquely representing the imprinted codes. This highly efficient spread spectrum multiplexing strategy is potentially capable of supporting communication networks with a large number of users by employing code families that include a large number of codes with low correlation properties between the different codes [7

7. V. S. Pless, W. C. Huffman eds. Handbook of Coding Theory (Elsevier Science B. V., Amsterdam, 1998).

] such that their mutual interference is minimized and thus different users can be discriminated [8

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

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]

].

Specifically, there is a strong motivation to implement OCDMA communication using the state-of-the-art nested 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]

14

14. T. Helleseth, and P. V. Kumar, “Sequences with low correlation,” in Handbook of Coding Theory, V. S. Pless, W. C. Huffman eds. (Elsevier Science B. V., Amsterdam, 1998).

]. The numbers of codes in the 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

1. L. Harte, R. Levine, and R. Kikta, 3G Wireless Demystified, (McGraw-Hill, New York, 2002).

].

Optical measurements of correlation between codes have previously been demonstrated successfully using techniques that employ second harmonic generation (SHG) in periodically poled lithium niobate (PPLN) waveguides [3

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

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

,8

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

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]

]. However, these SHG-based techniques allow to measure the value of the correlation function only at the single point of zero shift (delay) between the compared codes. Such single-point measurement does not allow to fully benefit from the special properties of the 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.

Our new optical code matching technique is based on a single-shot measurement of the values of the correlation function between two codes at all possible shifts (delays) between them. The two codes are encoded into the spectral phases of two broadband phase-shaped femtosecond pulses. As illustrated in Fig. 1(a)
Fig. 1 The experimental setup and four-wave mixing excitation scheme.
, the two pulses are mixed in a nonlinear medium (such as fused silica glass) together with an additional (amplitude-shaped) narrowband pulse in the experimental BOXCAR configuration [15

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

]. The spectrum of the resulting background-free FWM 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 interfering excitation pathways that are photo-induced by the different pulses and lead to the generation of the FWM signal are schematically illustrated in Fig. 1(b). For identical pulses they interfere constructively, while for non-identical pulses they interfere destructively. Hence, high contrast is achieved between the cross-correlation function of non-identical codes and the auto-correlation function of identical codes, which allows to determine with a very high reliability whether the two encoded sequences are identical or not. If they are identical, the periodicity is then obtained. Our present experiments are conducted using pulses of 800-nm originating from a Ti:Sapphire femtosecond laser system. However, there is no limitation to implement the technique presented here at the standard communication wavelength of 1550 nm.

As mentioned above, we consider the families of periodic cyclically distinct pseudo-random sequences sk(t) over the alphabet of integers of size p, 0sk(t) p-1, with arithmetic modulo p, where p≥2. The cross-correlation function between two sequences is defined as
c(k,r,τ)=t=0L1ωpsk(t+τ)sr(t),
(1)
where k and r are the indexes of the sequences in the set of size M, 0τ L-1, L is the period of the sequence, and ωp=ei2π/p is the p-th root of unity. The merit factor commonly used to quantify the performance of a given code family is the maximal value Cmaxof the cross-correlation function over all the possible delays τ, i.e., Cmax = max{c(k,r,τ)} excluding the cases where simultaneously k = r and τ = 0. The values at these latter cases correspond to the values of the auto-correlation functions at zero delay, for whichCmax=L for any code in the family. We thus demand CmaxL in order to ensure the non-intercepted communication without interference between different users of the network. The nested chain low-correlation code families S(0) ⊆S(1) ⊆S(2) ⊆... obey the desired requirements [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]

14

14. T. Helleseth, and P. V. Kumar, “Sequences with low correlation,” in Handbook of Coding Theory, V. S. Pless, W. C. Huffman eds. (Elsevier Science B. V., Amsterdam, 1998).

]. For a given code period L, with a given alphabet size p, the code family of higher order includes more code sequences and the corresponding value of Cmax increases. However, as the code period L increases, the number of available codes increases, while the value of Cmax decreases. The Cmax values of the S-families have a well defined bounds and distribution, and thus can be properly tailored to the implementation needs by the choice of L and the order of the family.

As is common in OCDMA with broadband femtosecond laser pulses, the imprinting of the codes into the pulses is experimentally implemented by femtosecond pulse shaping techniques. We follow the design and method of Weiner and Heritage [6

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

]. The corresponding optical setup incorporates pixilated liquid crystal spatial light modulator (SLM) for controlling the electric field E(ω)=|E(ω)|eiΦ(ω) of the different spectral components of the laser pulse, with |E(ω)| and Φ(ω) being, respectively, the spectral amplitude and phase at frequency 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 Δω. The similarity between the representation of integer code sequences sk(τ) in the complex plane and the phase factor of the electric field is used to convert the codes into the relative phases of the broadband pulse spectral components: Φ(ω)=(2π/p)sk(ω). Once the encoding process is completed at the transmitter site, measurement of the cross-correlation function c(k,r,τ) is the important step to be performed at the receiver or router site.

The cross-correlation function c(k,r,τ) between the two codes sk and sr, which are converted into the phases Φk(ω) and Φr(ω) of the pulses k and r, corresponds to the cross-correlation function ξ(k,r,Δ) between the electric fields Ek(ω) and Er(ω):
ξ(k,r,Δ)=Ek(ω)Er(ω+Δ)dω=|Ek(ω)|·|Er(ω+Δ)|ei[Φk(ω)Φr(ω+Δ)]dω.
Neglecting the spectral envelope profiles|Ek,r(ω)|, ξ(k,r,Δ) is formally equivalent to c(k,r,τ) (see Eq. (1). They differ only by the fact that Δ is a continuous variable representing the detuning between spectral components of Ek(ω)andEr(ω+Δ), while τ is a discrete integer variable representing the relative delay (shift) between the two codes.

Physically, ξ(k,r,Δ) interferes (via Raman-type transitions) all the possible two-photon pathways of one absorbed photon and one emitted photon that their frequencies differ by Δ. Suppose that the continuous axis of frequency ω is divided into small bins of size Δω and each bin is associated with specific value of t (the index of the code component; see above), such that the phase Φ(ω)of the electric field E(ω) is constant across each bin. For simplicity, we analyze first the case where the spectral envelope changes extremely slow and can be approximated by a constant: |Ek,r(ω)|1. Similar toc(k,r,τ), ξ(k,r,Δ) obtains its maximal values when Δ=nΔωL if 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 ΔnΔωL or k≠r the interference between the two-photon pathways is destructive and ξ(k,r,Δ)ξ(k,k,nΔωL). Overall, the profile of ξ(k,r,Δ) has peaks corresponding to the cases of the constructive interference that appear with the spacing associated with the period of ΔωL. If the interference is destructive, for the case of flat spectral profiles ofEk,r(ω), the bounds on the value of ξ(k,r,Δ) are obtained within the framework of the Number Theory [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]

14

14. T. Helleseth, and P. V. Kumar, “Sequences with low correlation,” in Handbook of Coding Theory, V. S. Pless, W. C. Huffman eds. (Elsevier Science B. V., Amsterdam, 1998).

]. For the proper choice of the code family, this destructive interference is kept sufficiently low and enables clear discrimination between the matching (k = r) and non-matching (k≠r) codes.

Experimentally, in order to measure the entire correlation function in a single-shot measurement one needs to employ four-wave mixing process utilizing a medium of third-order susceptibilityχ(3). The corresponding interaction involves two absorbed photons and one emitted photon that originate from the laser pulses as well as a fourth emitted photon that is generated by the non-linear polarization of the medium. The latter is the one being detected as the experimental signal:
Esig(ω)Epump(ω)dωEk(ω)Er(ω+ωω)dω.
In order to avoid the broadening of the interference-pattern of interest between Ek(ω) and Er(ω) due to the presence of an additional broadbandEpump(ω), the latter should be of a narrow bandwidth. Let us assume that
Epumpnarrow(ω)=Epump(ω)δ(ω0ω)dω.
Then, we obtain
Esig(ω0+Δ)Epump(ω0)Ek(ω)Er(ω+Δ)dω=Epump(ω0)|Ek(ω)||Er(ω+Δ)|ei[Φk(ω)Φr(ω+Δ)]dω.
(2)
Therefore, overall, we implement the measurement of the correlation function between codes using a frequency-domain measurement of the spectrumIsig|Esig(ω)|2. In the case Epump is of a finite bandwidth, the measured Isig(ω) corresponds to the cross-correlation function ξ(k,r,Δ) convoluted with the spectral profile ofEpump.

Our experimental setup utilizes 800-nm femtosecond laser pulses of 65-fs pulse duration (spectral FWHM of 15 nm) split into the three beams schematically shown in Fig. 1. Two beams pass through a 4f phase-shaping optical set-up incorporating 640-pixel liquid-crystal SLM that its optical design allows to independently assign different code to each beam. The experimental shaping resolutions corresponding to the two beams are, respectively, Δω = 0.110 and 0.136 nm per pixel. The close matching between these two resolutions is of high importance for the successful code recognition. Thus, in our case, the corresponding pixel binning is of, respectively, 5 and 4 pixels per bin. The third beam is amplitude-shaped using similar 4f shaping set-up to yield narrow spectrum with 0.9-nm FWHM. The resulting three beams, carrying 5-μJ pulses, are focused at zero delay using BOXCAR geometrical configuration into a piece of fused silica having 1-mm thickness. The spatially-isolated output FWM signal is collected and its spectrum is measured using a CCD camera coupled to a spectrometer (0.1-nm resolution). The intensity of our focused pulses is about 5 × 108 W/cm2 and the single-shot FWM output signal is of a total flux of about 107 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 108-W/cm2 magnitude and thus the only difference from the present situation will essentially be the reduction of the signal output flux to about 104 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

16. R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses, (Kluwer Academic Publishers, Norwell, 2000).

]. Thus, standard low-energy Ti:Sapphire femtosecond oscillators are also suitable sources for implementing the new technique presented here.

In the present work we use the quaternary (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

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]

14

14. T. Helleseth, and P. V. Kumar, “Sequences with low correlation,” in Handbook of Coding Theory, V. S. Pless, W. C. Huffman eds. (Elsevier Science B. V., Amsterdam, 1998).

]. The chosen period of the codes is L = 2l−1 = 7, where l = 3 is the order of the generating polynomial. The size of the family S(0) is M = L + 2 = 9 with |Cmax| = L+1+13.8. The size of the family S(1) is M = (L + 2)(L + 1) = 72, including also the nested S(0) family. Formally, the corresponding value of |Cmax| for this case is |Cmax| = 2L+1+16.66; however, for the specific case of l = 3, the occurrence of this value is zero and thus the effective value is |Cmax| = 5 [13

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]

].

Figure 2
Fig. 2 Typical experimental spectrum of Isig(ω) for the cases of identical [k = r, auto-correlation, black line] and different codes [k≠r. cross-correlation, red line]. The distance between the peaks for the case of k = r corresponds to the periodΔL.
presents typical measured spectrum Isig(ω) for cases of 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 byΔL=nΔωL, where n is an integer determined by the period of the code. This is in high contrast to the Isig(ω)spectrum obtained for the cases of destructive interference among the pathways associated with the cross-correlation cases of different codes, i.e. 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
Fig. 3 Histograms of the measured code period ΔL obtained for 432 different codes belonging to the family S(1) for six different values of the programmed code period (corresponding to n = 3-8; see text). The distributions reflect the signal-to-noise experimental ratio: (a) 3.3±0.8, (b) 4.0±0.13, (c) 5.0±0.12, (d) 6.0±0.08, (e) 7.0±0.1, (f) 7.9±0.13. Overall, the correct recognition of the code period is for the 98% of the sequences.
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(ΔL/ΔωL). The results demonstrate successful recognition of the period for 98% of the encoded codes. As seen from Fig. 3, the 2% of false recognition correspond mostly to the cases of the shortest and longest periods. For the cases of short period, the limiting factor is the spectral width of the narrowband pump pulse that, as described above, leads to the smearing of the spectral features of the FWM spectrum. For the cases of long period, the total spectral width of pulses Ek,r(ω) becomes important: if the pulse spectrum is not broad enough, the number of constructively-interfering pathways is not large enough and the contrast between the cases of k = r and k≠r is not sufficient for the proper recognition.

The codes with different periods are easily discriminated based on the measurement of Isig(ω) due to the lack of constructive interferences among the pathways leading to the emitted signal. For the experimental discrimination between different codes with the same period it is enough to measure the signal at the frequencies of the auto-correlation peaks, i.e., measuring Isig(ω0+mΔL) with m = 0, 1, 2 etc. The corresponding results for the code families S(0) and S(1) are presented, respectively, in Figs. 4(a)
Fig. 4 Two-dimensional color maps of the normalized correlation measurements between different code pairs k (x-axis) and r (y-axis). The diagonal corresponds to the auto-correlation cases. Each line is normalized by the value of its auto-correlation case. (a) Results for the family S(0), l = 3. (b) Results for the family S(1), l = 3. See the text for details.
and 4(b) as two-dimensional color maps. The color represents the normalized value of the summation mIsig(ω0+mΔL) for different code pairs 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 |Ek,r(ω)| is almost flat and does not vary significantly across the spectrum, these auto-correlation values on the diagonal are all of the same value. This can be deduced directly from Eq. (2). Since the pulses used in the present work are of a Gaussian spectrum and thus deviate from the ideal case, the measured auto-correlation values on the diagonal vary from one code to the other. Hence, the maps of Fig. 4 present the measured values after the normalization of each line in the map by the value measured for its corresponding auto-correlation cell on the diagonal (thus all the presented values on the diagonal are one). This normalization procedure is equivalent to performing a preceding calibration measurement of the different actual auto-correlation values. Thus, it does not reduce from the generality of our results. We have also verified that this normalization does not reduce from the generality of our results and their analysis by conducting numerical calculations that simulate both our experiment and the ideal-case experiment of flat pulse spectrum. It is also important to mention that it is experimentally feasible to achieve conditions that are very close to those of the ideal case described above by utilizing amplitude shaping simultaneously with the phase shaping to yield flat spectral envelope of the pulses.

So, we have analyzed the cross-correlation values for the cases of 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 |Cmax|/L obtained in these measurements is 0.62±0.1. In the corresponding ideal case the value is |Cmax|/L = 0.54, as evaluated from the Number Theory for the suitable choice of code parameters (l = 3, p = 4).

For the S(1)'s codes, we obtain a correct identification of 99.4% of the codes with a signal threshold of 95% of the auto-correlation signal [see Fig. 4(b)]. Also here, this is in agreement with the results of our numerical simulations. As this threshold is rather high, it is important to note that our simulations show that the use of 30-fs femtosecond pulses having 33-nm spectral width allows to lower the corresponding threshold to a value of 85%. In the ideal case, the corresponding threshold can be further reduced to a value of 75%. Regarding the |Cmax|/L value, the present corresponding experimental mean value is 0.88±0.16, while the ideal theoretical value is 0.71. Last, it worth also mentioning that for pulses with spectral envelope profile that does not vary significantly over a bandwidth of 50 nm, our simulations show that one expects threshold values as low as 20%.

In conclusion, we have introduced and experimentally demonstrated a new technique for all-optical code matching based on a single-shot frequency-domain cross-correlation measurement. In the present work we have only demonstrated proof-of-the-concept experiments using codes that belong to the nested S-families of low orders, relatively small alphabet, and having short period. Considering the high reliability of the 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

This research was supported by The James Franck Program in Laser Matter Interaction and by The P. and E. Nathan Research Fund.

References and Links

1.

L. Harte, R. Levine, and R. Kikta, 3G Wireless Demystified, (McGraw-Hill, New York, 2002).

2.

Optical Code Division Multiple Access, Fundamentals and Applications P. R. Prucnal ed. (CRC, Taylor & Francis, Boca Raton, 2006).

3.

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

4.

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]

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]

7.

V. S. Pless, W. C. Huffman eds. Handbook of Coding Theory (Elsevier Science B. V., Amsterdam, 1998).

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]

9.

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]

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. 29(16), 1864–1866 (2004). [CrossRef] [PubMed]

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]

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]

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]

14.

T. Helleseth, and P. V. Kumar, “Sequences with low correlation,” in Handbook of Coding Theory, V. S. Pless, W. C. Huffman eds. (Elsevier Science B. V., Amsterdam, 1998).

15.

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

16.

R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses, (Kluwer Academic Publishers, Norwell, 2000).

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

  1. L. Harte, R. Levine, and R. Kikta, 3G Wireless Demystified, (McGraw-Hill, New York, 2002).
  2. Optical Code Division Multiple Access, Fundamentals and Applications P. R. Prucnal ed. (CRC, Taylor & Francis, Boca Raton, 2006).
  3. A. M. Weiner, Z. Jiang, and D. E. Leaird, “Spectrally phase-coded O-CDMA,” J. Opt. Netw. 6(6), 728–755 (2007). [CrossRef]
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