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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 14 — Jul. 9, 2007
  • pp: 8812–8817
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Spectral Talbot effect in sampled fiber Bragg gratings with super-periodic structures

Xi-Hua Zou, Wei Pan, Bin Luo, Meng-Yao Wang, and Wei-Li Zhang  »View Author Affiliations


Optics Express, Vol. 15, Issue 14, pp. 8812-8817 (2007)
http://dx.doi.org/10.1364/OE.15.008812


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Abstract

Due to the equivalence between an integral multiple of 2π and zero in the phase space, a general configuration of sampled fiber Bragg gratings (SFBGs) with super-periodic structures has been introduced and investigated. These super-periodic structures can be used to implement spectral Talbot effect with a large degree of freedom, as long as any one of the three parameters involved in quadratic phase profile is an even number. Then the phase profiles of such SFBGs are analyzed in detail. Although their phase increments are constant or non-constant periodic functions in different cases, theoretical analysis and simulations show that the obtained filtering characteristics are the same. In contrast to uniform SFBGs with identical sampling period, multiplied filtering channels and similar group-delay characteristic are achieved for these SFBGs with super-periodic structures.

© 2007 Optical Society of America

1. Introduction

Sampled fiber Bragg gratings (SFBGs) have attracted considerable attention due to their versatile applications in multi-channel filters [1–10

1. M. Ibsen, M. K. Durkin, M. J. Cole, and R. I. Laming, “Sinc-sampled fiber Bragg gratings for identical multiple wavelength operation,” IEEE Photon. Technol. Lett. 10, 842–844 (1998). [CrossRef]

], dispersion compensators [10–12

10. H. Li, Y. Sheng, Y. Li, and J. E. Rothenberg, “Phased-only sampled fiber Bragg gratings for high-channel-count chromatic dispersion compensation,” J. Lightwave Technol. 21, 2074–2083 (2003). [CrossRef]

], and optical code division multiple access (OCDMA) technique. As for dense multi-channel filters, the so-called spectral Talbot effect (i.e., spectral self-imaging effect) is a promising choice. The spectral Talbot effect, a member of Talbot effect family, is able to create exact images of the original object in specific spectral (or wavelength) domain. Via “strongly chirped SFBGs” [2

2. X. F. Chen, C. C. Fan, Y. Luo, S. Z. Xie, and S. Hu, “Novel flat multichannel filter based on strongly chirped sampled fiber Bragg grating,” IEEE Photon. Technol. Lett. 12, 1501–1503 (2000). [CrossRef]

], this effect was first used for channel multiplication. Then the concept “spectral Talbot effect” for SFBGs was definitely proposed in Ref [4

4. C. Wang, J. Azaña, and L. R. Chen, “Spectral Talbot-like phenomena in one-dimensional photonic bandgap structures,” Opt. Lett. 29, 1590–1592 (2004). [CrossRef] [PubMed]

]. Based on their work concerning temporal Talbot effect [13

13. J. Azaña and M. A. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. 7, 728–744 (2001). [CrossRef]

], Azaña, Chen, and Wang have conducted systematized and comprehensive investigations about the spectral Talbot effect [4–6

4. C. Wang, J. Azaña, and L. R. Chen, “Spectral Talbot-like phenomena in one-dimensional photonic bandgap structures,” Opt. Lett. 29, 1590–1592 (2004). [CrossRef] [PubMed]

]. Further, this effect can be obtained by the combination of chirp effect and phase shifts as well [7

7. Y. T. Dai, X. F. Chen, X. Xu, C. Fan, and S. Z. Xie, “High channel-count comb filter based on chirped sampled fiber Bragg grating and phase shift,” IEEE Photon. Technol. Lett. 17, 1040–1042 (2005). [CrossRef]

, 12

12. Y. T. Dai, X. F. Chen, J. Sun, and S. Z. Xie, “Wideband multichannel dispersion compensation based on a strongly chirped sampled Bragg grating and phase shifts”, Opt. Lett. 31, 311–313 (2006). [CrossRef] [PubMed]

].

Usually, the phase profiles of aforementioned approaches vary continuously along the whole grating; i.e., the profiles are non-periodic. Recently, we have proposed a periodically chirped SFBG consisting of many identical, repeating super-periodic structures for implementing spectral Talbot effect [8

8. X. H. Zou, W. Pan, B. Luo, Z. M. Qin, M. Y. Wang, and W. L. Zhang, “Periodically chirped sampled fiber Bragg gratings for multichannel comb filter,” IEEE Photon. Technol. Lett. 18, 1371–1373 (2006). [CrossRef]

]. Also, the multiple-phase-shift (MPS) technique [9

9. Y. Nasu and S. Yamashita, “Densification of sampled fiber Bragg gratings using multiple phase shift (MPS) technique,” J. Lightwave Technol. 23, 1808–1817 (2005). [CrossRef]

] can be regarded as one kind of super-periodic structures.

After the introduction of “super-periodic structure,” it is necessary to further carry out a general and systematized investigation on SFBGs with such structures, which is similar to the research on periodic Talbot filters for pulse multiplication [14

14. J. Azaña and S. Gupta, “Complete family of periodic Talbot filter for pulse repetition rate multiplication,” Opt. Express 14, 4270–4279 (2006). [CrossRef] [PubMed]

]. For this purpose, we propose a general configuration for SFBGs with super-periodic structures, and this general configuration allows a large degree of freedom to obtain spectral Talbot effect. The rest of this paper is organized as follows. In section 2, a general phase condition for spectral Talbot effect is derived and analyzed, which provides theoretical origin for periodic phase profile. Then, in Section 3, we are devoted to derive equivalent phase transition points for super-periodic structures. In order to get transition points, the parity of parameters involved in phase profile should be specifically designed. Then detailed analysis and discussion on phase profile, phase increment (phase condition) and spectral Talbot phenomena are given out. Finally, we draw a compact conclusion in section 4.

2. General phase condition for spectral Talbot effect

For an arbitrary FBG, we can express its refractive-index modulation profile Δn(z) by

Δn(z)=Δnk=+u(zkP)Re{1+expj[β0z+θ(z)]},
(1)

where Δn is the “dc” index modulation (the fringe visibility is specified as 1), u(z) and θ(z) represent the sampling function and the phase profile, β 0=2π0 is the central spatial frequency, and Λ0 is the central grating period. As for Talbot effect, the phase profile θ(z) is always the decisive factor. Aided by Taylor expansion formula, θ(z) can be written as

θ(z)=θ2z22+θ3z36+θ4z424+=r=2+θrzr(r!),
(2)

where θr is the r-th derivative of θ(z) at z = 0 . Meanwhile, we can get a general phase condition for spectral Talbot effect. In Ref. [13

13. J. Azaña and M. A. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. 7, 728–744 (2001). [CrossRef]

], authors have already analyzed phase conditions for integer and fractional Talbot effect, respectively. According to those phase conditions, we introduce a phase increment “θ(k) to express a general phase condition for both integer and fractional effect:

Δθ(k)=θ[z=(k+m)P]θ(z=kP)=Nkπ,
(3)

where Nk is an integer determined by k, m (a positive integer) is the multiplying factor of Talbot effect, and P is the sampling period. If m = 1, Eq. (3) corresponds to integer Talbot effect; if m = 2, 3, 4, ⋯, it stands for fractional effect. Then we return to the phase profile, Eq. (2). There are so many terms on its right side that it is hard to derive each θr if we treat all terms as an entity. For simplicity, by means of treating each term as an independent part to satisfy Eq. (3) respectively, θr can be easily obtained as:

Nk,rπ=θrr!×mi=0r1ki(k+m)(r1i),θr=π(r!)Prm×Nk,ri=0r1ki(k+m)(r1i)=π(r!)Prm×Nk,r,
(4)

where Nk,r is an integer determined by k and r As a coefficient of Taylor series, each θr should be a constant along z-axis. Namely θr is required to be independent of the variation of k. To eliminate the influence of k, we replace Nk,r with a fixed number Nk,r ́. Consequently, θr can be expressed by Eq. (4).

3. Super-periodic structures for implementing spectral Talbot effect

In Section 2, we have derived a general phase condition and corresponding Taylor coefficients for spectral Talbot effect. They enable us to design conventional chirped-SFBGs conveniently. However, when it comes to the SFBGs with super-periodic structures, a series of extra equivalent transition points are needed to achieve spectral Talbot effect.

3.1 Phase transition points for super-periodic structures

First of all, we have to state an equivalent transition transform in the phase space. That is, an integral multiple of 2π (2) is equivalent to zero, where N is an integer. With this equivalence, the value of Eq. (1) keeps unchanged when we replace θ(z) = 2 with θ(z) = 0 . Assuming that these transition points are located at sampling points

θ(z=tP)=2,
(5)

we can view these points as starting points of new super-periodic structures, where t is an integer. Such new super-periodic structures are reproductions of the original structure around z = 0 . In this way, such a SFBG comprises identical, repeating super-periodic structures, the so-called SFBG with super-periodic structures.

Here, we set the period of super-periodic structure as (s×m)P, where s is a positive integer. When only quadratic phase profile is considerer, Eq. (5) can be rewritten as

θ[z=(s×m)P]=(s2mNk,2)π=2.
(6)

A simple analysis of Eq. (6) reveals that (s 2 mN k,2 ́) must be an even number to satisfy Eq. (6). As long as any parameter in the set (s, m, N k,2 ́) is an even number, we can find suitable transition points. Note that Eq. (6) provides a general configuration for super-periodic structures. As reported in Ref [8

8. X. H. Zou, W. Pan, B. Luo, Z. M. Qin, M. Y. Wang, and W. L. Zhang, “Periodically chirped sampled fiber Bragg gratings for multichannel comb filter,” IEEE Photon. Technol. Lett. 18, 1371–1373 (2006). [CrossRef]

], only the parity of N k,2 ́ has been taken into consideration. Here, besides N k,2 ́, m can be adjusted properly to obtain transition points. What is more, a new parameter s is introduced to get transition points for the first time, which allows a larger degree of freedom than before to obtain Talbot effect. When s > 1, the corresponding phase increment is greatly different from those investigated previously [6–9

6. J. Azaña, C. Wang, and L. R. Chen, “Spectral self-imaging phenomena in sampled Bragg gratings,” J. Opt. Soc. Am. B 22, 1829–1841 (2005). [CrossRef]

, 14

14. J. Azaña and S. Gupta, “Complete family of periodic Talbot filter for pulse repetition rate multiplication,” Opt. Express 14, 4270–4279 (2006). [CrossRef] [PubMed]

] and we will give a detailed description next. In brief, the general configuration here accounts for a full-scale expansion of super-periodic structures.

Once super-periodic structures are adopted, “θ(k) is not a linear function any more, but a periodic function. The simplest example is that “θ(k) is equal to a constant, and this constant may be zero or mN k,2 ́ π according to Eq. (7) derived through the combination of Eqs. (2), (3) and (4).

Δθ(k)=Nk,2(2k+m)π.
(7)

On the other hand, “θ(k) can be a non-constant periodic function. There are probably a variety of such periodic functions and we will show a schematic diagram about such functions of certain case later in this work.

3.2 Concrete super-periodic structures and simulations

On the basis of the derivations and analysis above, we classify the general configuration of super-periodic structures into two cases.

Fig. 1. Phase profiles of (a) conventional chirped-SFBG, (b) proposed SFBG with and “θ(k) = 0, and (c) proposed SFBG with “θ(k) = mN k,2 ́ π; (d) chirp effect corresponding to phase profile (a).
Fig. 2. Filtering characteristics of SFBGs with super-periodic structures (s =1, N k,2 ́ =1, m = 2)

First of all, when s = 1 is offered, either m or N k,2 ́ is required to be an even number, so that spectral Talbot effect can be realized. In this case, “θ(k) is zero or mN k,2 ́ π, and detailed phase profiles are presented in Fig. 1. Fig. 1(a) illustrates the quadratic phase profile of conventional chirped-SFBG. For SFBGs with super-periodic structures, the phase profile for “θ(k) = 0 is shown in Fig. 1(b), while the phase profile for “θ(k) = mN k,2 ́ π is illustrated in Fig. 1(c). We can see that each period of phase profiles originates from the section around z = 0 . It is convenient to derive a corresponding chirp effect Λ(z) for Fig. 1(b), which is indicated in Fig. 1(d). However, it is not easy to get a resulting chirp effect for Fig. 1(c), and now we just introduce such a phase profile in theory.

Secondly, there exists a more complicated case when s > 1. If s = 3, 5, 7, 9⋯, one between m and N k,2 ́ is required to be an even number, which is identical to the case s = 1. More importantly, if s = 2, 4, 6, 8⋯, the spectral Talbot phenomenon can always be realized, regardless of the parity of m or N k,2 ́.

As predicted previously, the phase increment here is a non-constant periodic function. In this case, the period of phase profile [Fig. 3(a)] is (s×m)P, not (mP). According to Fig. 3(a), we are capable of deriving a periodic function “θ(k) shown in Fig. 3(b), which might be the most remarkable difference between the two cases (s = 1 and s > 1).

Fig. 3. Schematic diagram of (a) phase profile and (b) phase condition (phase increment) for SFBGs with super-periodic structures when s > 1.

Similarly, spectral Talbot phenomena are obtainable through numerical simulation. In Fig. 4, the channel spacing has reduced by m times and the group-delay characteristic is nearly identical with that of uniform SFBG. In addition, such derived parameter sets can be utilized to design temporal, periodic Talbot filters [14

14. J. Azaña and S. Gupta, “Complete family of periodic Talbot filter for pulse repetition rate multiplication,” Opt. Express 14, 4270–4279 (2006). [CrossRef] [PubMed]

].

3.3 Discussions

By combining the result of Ref. [14

14. J. Azaña and S. Gupta, “Complete family of periodic Talbot filter for pulse repetition rate multiplication,” Opt. Express 14, 4270–4279 (2006). [CrossRef] [PubMed]

] and ours, both temporal and spectral Talbot effect can be realized by super-periodic structures. The phase profile required by equivalent transition transform is the same and the fabrication process may be a bit complex. On the other hand, such SFBGs with super-periodic structures possess several outstanding features. Like conventional chirped-SFBGs, these SFBGs are with high energetic efficiency and tunable control of channel spacing (i.e., spacing is not being fixed solely by sampling period) [2

2. X. F. Chen, C. C. Fan, Y. Luo, S. Z. Xie, and S. Hu, “Novel flat multichannel filter based on strongly chirped sampled fiber Bragg grating,” IEEE Photon. Technol. Lett. 12, 1501–1503 (2000). [CrossRef]

, 4–7

4. C. Wang, J. Azaña, and L. R. Chen, “Spectral Talbot-like phenomena in one-dimensional photonic bandgap structures,” Opt. Lett. 29, 1590–1592 (2004). [CrossRef] [PubMed]

]. More importantly, compared with conventional chirped-SFBGs, these proposed SFBGs can provide a larger degree of freedom to design multi-channel filters. Three parameters in the set (s, m, Nk,2 ́), rather than only m, can be adjusted properly. As far as we know, this obtained degree of freedom is the largest one to implement spectra Talbot effect. Furthermore, proposed SFBGs allow a larger tolerance on chirp coefficient because super-periodic structures can bring with a relatively lower deviation on phase profile [8

8. X. H. Zou, W. Pan, B. Luo, Z. M. Qin, M. Y. Wang, and W. L. Zhang, “Periodically chirped sampled fiber Bragg gratings for multichannel comb filter,” IEEE Photon. Technol. Lett. 18, 1371–1373 (2006). [CrossRef]

].

Fig. 4. Filtering characteristics of proposed SFBGs with different parameters: (a) s=2, N k,2 ́ =1, m =3, (b) s = 3, N k,2 ́ =1, m = 2.

4. Conclusion

We have introduced a general configuration of SFBGs with super-periodic structures for achieving spectral Talbot effect. As long as any parameter of the set (s, m, Nk,2 ́) is an even number, a series of equivalent transition points can be determined, which corresponds to new super-periodic structures. On the basis of this principle, we succeed in achieving the spectral Talbot phenomena with a large degree of freedom. In the meantime, the phase profile and phase condition (phase increment) with different parameters are analyzed. The phase increment is zero or non-zero constant if s = 1, while the increment is a non-constant periodic function if s > 1.

Acknowledgments

This work is supported by the NNSFC by -90201011, Project of EMC -105148, and Fund of BOTCN - KF2006. The authors would like to thank Prof. J. Azaña and Prof. X. F. Chen for their help.

References and links

1.

M. Ibsen, M. K. Durkin, M. J. Cole, and R. I. Laming, “Sinc-sampled fiber Bragg gratings for identical multiple wavelength operation,” IEEE Photon. Technol. Lett. 10, 842–844 (1998). [CrossRef]

2.

X. F. Chen, C. C. Fan, Y. Luo, S. Z. Xie, and S. Hu, “Novel flat multichannel filter based on strongly chirped sampled fiber Bragg grating,” IEEE Photon. Technol. Lett. 12, 1501–1503 (2000). [CrossRef]

3.

A. V. Buryak, K. Y. Kolossovski, and D. Y. Stepanov, “Optimization of refractive index sampling for multichannel fiber Bragg gratings,” IEEE J. Quantum Electron. 39, 91–98 (2003). [CrossRef]

4.

C. Wang, J. Azaña, and L. R. Chen, “Spectral Talbot-like phenomena in one-dimensional photonic bandgap structures,” Opt. Lett. 29, 1590–1592 (2004). [CrossRef] [PubMed]

5.

L. R. Chen and J. Azaña, “Spectral Talbot phenomena in sampled arbitrarily chirped Bragg gratings,” Opt. Commun. 250, 302–308 (2005). [CrossRef]

6.

J. Azaña, C. Wang, and L. R. Chen, “Spectral self-imaging phenomena in sampled Bragg gratings,” J. Opt. Soc. Am. B 22, 1829–1841 (2005). [CrossRef]

7.

Y. T. Dai, X. F. Chen, X. Xu, C. Fan, and S. Z. Xie, “High channel-count comb filter based on chirped sampled fiber Bragg grating and phase shift,” IEEE Photon. Technol. Lett. 17, 1040–1042 (2005). [CrossRef]

8.

X. H. Zou, W. Pan, B. Luo, Z. M. Qin, M. Y. Wang, and W. L. Zhang, “Periodically chirped sampled fiber Bragg gratings for multichannel comb filter,” IEEE Photon. Technol. Lett. 18, 1371–1373 (2006). [CrossRef]

9.

Y. Nasu and S. Yamashita, “Densification of sampled fiber Bragg gratings using multiple phase shift (MPS) technique,” J. Lightwave Technol. 23, 1808–1817 (2005). [CrossRef]

10.

H. Li, Y. Sheng, Y. Li, and J. E. Rothenberg, “Phased-only sampled fiber Bragg gratings for high-channel-count chromatic dispersion compensation,” J. Lightwave Technol. 21, 2074–2083 (2003). [CrossRef]

11.

F. Ouellette, P. A. Krug, T. Stephens, G. Dhosi, and B. Eggleton, “Broadband and WDM dispersion compensation using chirped sampled fiber Bragg gratings,” Electron. Lett. 31, 899–901 (1995). [CrossRef]

12.

Y. T. Dai, X. F. Chen, J. Sun, and S. Z. Xie, “Wideband multichannel dispersion compensation based on a strongly chirped sampled Bragg grating and phase shifts”, Opt. Lett. 31, 311–313 (2006). [CrossRef] [PubMed]

13.

J. Azaña and M. A. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. 7, 728–744 (2001). [CrossRef]

14.

J. Azaña and S. Gupta, “Complete family of periodic Talbot filter for pulse repetition rate multiplication,” Opt. Express 14, 4270–4279 (2006). [CrossRef] [PubMed]

15.

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997). [CrossRef]

OCIS Codes
(050.2770) Diffraction and gratings : Gratings
(060.2340) Fiber optics and optical communications : Fiber optics components
(070.6760) Fourier optics and signal processing : Talbot and self-imaging effects

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: April 30, 2007
Revised Manuscript: June 8, 2007
Manuscript Accepted: June 8, 2007
Published: June 28, 2007

Citation
Xi-Hua Zou, Wei Pan, Bin Luo, Meng-Yao Wang, and Wei-Li Zhang, "Spectral Talbot effect in sampled fiber Bragg gratings with super-periodic structures," Opt. Express 15, 8812-8817 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-14-8812


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References

  1. M. Ibsen, M. K. Durkin, M. J. Cole, and R. I. Laming, "Sinc-sampled fiber Bragg gratings for identical multiple wavelength operation," IEEE Photon. Technol. Lett. 10, 842-844 (1998). [CrossRef]
  2. X. F. Chen, C. C. Fan, Y. Luo, S. Z. Xie, and S. Hu, "Novel flat multichannel filter based on strongly chirped sampled fiber Bragg grating," IEEE Photon. Technol. Lett. 12, 1501-1503 (2000). [CrossRef]
  3. A. V. Buryak, K. Y. Kolossovski, and D. Y. Stepanov, "Optimization of refractive index sampling for multichannel fiber Bragg gratings," IEEE J. Quantum Electron. 39, 91-98 (2003). [CrossRef]
  4. C. Wang, J. Azaña, and L. R. Chen, "Spectral Talbot-like phenomena in one-dimensional photonic bandgap structures," Opt. Lett. 29, 1590-1592 (2004). [CrossRef] [PubMed]
  5. L. R. Chen and J. Azaña, "Spectral Talbot phenomena in sampled arbitrarily chirped Bragg gratings," Opt. Commun. 250, 302-308 (2005). [CrossRef]
  6. J. Azaña, C. Wang, and L. R. Chen, "Spectral self-imaging phenomena in sampled Bragg gratings," J. Opt. Soc. Am. B 22, 1829-1841 (2005). [CrossRef]
  7. Y. T. Dai, X. F. Chen, X. Xu, C. Fan, and S. Z. Xie, "High channel-count comb filter based on chirped sampled fiber Bragg grating and phase shift," IEEE Photon. Technol. Lett. 17, 1040-1042 (2005). [CrossRef]
  8. X. H. Zou, W. Pan, B. Luo, Z. M. Qin, M. Y. Wang, and W. L. Zhang, "Periodically chirped sampled fiber Bragg gratings for multichannel comb filter," IEEE Photon. Technol. Lett. 18, 1371-1373 (2006). [CrossRef]
  9. Y. Nasu and S. Yamashita, "Densification of sampled fiber Bragg gratings using multiple phase shift (MPS) technique," J. Lightwave Technol. 23, 1808-1817 (2005). [CrossRef]
  10. H. Li, Y. Sheng, Y. Li, and J. E. Rothenberg, "Phased-only sampled fiber Bragg gratings for high-channel-count chromatic dispersion compensation," J. Lightwave Technol. 21, 2074-2083 (2003). [CrossRef]
  11. F. Ouellette, P. A. Krug, T. Stephens, G. Dhosi, and B. Eggleton, "Broadband and WDM dispersion compensation using chirped sampled fiber Bragg gratings," Electron. Lett. 31, 899-901 (1995). [CrossRef]
  12. Y. T. Dai, X. F. Chen, J. Sun, and S. Z. Xie, "Wideband multichannel dispersion compensation based on a strongly chirped sampled Bragg grating and phase shifts," Opt. Lett. 31, 311-313 (2006). [CrossRef] [PubMed]
  13. J. Azaña and M. A. Muriel, "Temporal self-imaging effects: theory and application for multiplying pulse repetition rates," IEEE J. Sel. Top. Quantum Electron. 7, 728-744 (2001). [CrossRef]
  14. J. Azaña and S. Gupta, "Complete family of periodic Talbot filter for pulse repetition rate multiplication," Opt. Express 14, 4270-4279 (2006). [CrossRef] [PubMed]
  15. T. Erdogan, "Fiber grating spectra," J. Lightwave Technol. 15, 1277-1294 (1997). [CrossRef]

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