## Symmetric and antisymmetric solitons in finite lattices |

Optics Express, Vol. 19, Issue 18, pp. 17179-17188 (2011)

http://dx.doi.org/10.1364/OE.19.017179

Acrobat PDF (2011 KB)

### Abstract

We propose a simple model for the realization of symmetrically and antisymmetrically shape-preserving nonlinear waves with nonvanishing intensities at infinity. A finite lattice embedded into a defocusing saturable medium can support various families of novel solitons, including out-of-phase and in-phase solitons with symmetric and antisymmetric profiles. Although the lattice is finite, the existence and stability of solitons depend strongly on the band-gap structure of the corresponding infinite lattice. Saturable nonlinearity enhances the pedestal height and renormalized energy flow of solitons evidently. In particular, increasing the lattice site number or saturation degree of nonlinearity can considerably suppresses the instability of solitons. In addition, we find two branches of in-phase solitons in finite lattices and one branch of them can be dynamically stable. Our findings may provide a helpful hint for linking the solitons supported by infinite and finite lattices.

© 2011 OSA

3. L. Stenflo, “Theory of nonlinear plasma surface waves,” Phys. Scr. **T63**, 59–62 (1996). [CrossRef]

4. I. L. Garanovich, A. A. Sukhorukov, and Y. S. Kivshar, “Surface multi-gap vector solitons,” Opt. Express **14**, 4780–4785 (2006). [CrossRef] [PubMed]

6. S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, “Observation of discrete surface solitons,” Phys. Rev. Lett. **96**, 063901 (2006). [CrossRef] [PubMed]

7. M. I. Molina, I. L. Garanovich, A. A. Sukhorukov, and Y. S. Kivshar, “Discrete surface solitons in semi-infinite binary waveguide arrays,” Opt. Lett. **31**, 2332–2334 (2006). [CrossRef] [PubMed]

11. Y. He, D. Mihalache, and B. Hu, “Soliton drift, rebound, penetration, and trapping at the interface between media with uniform and spatially modulated nonlinearities,” Opt. Lett. **35**, 1716–1718 (2010). [CrossRef] [PubMed]

12. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface gap solitons,” Phys. Rev. Lett. **96**, 073901 (2006). [CrossRef] [PubMed]

13. C. R. Rosberg, D. N. Neshev, W. Krolikowski, A. Mitchell, R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Observation of surface gap solitons in semi-infinite waveguide arrays,” Phys. Rev. Lett. **97**, 083901 (2006). [CrossRef] [PubMed]

14. Y. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. **298**, 81–197 (1998). [CrossRef]

18. J. Belmonte-Beitia and J. Cuevas, “Existence of dark solitons in a class of stationary nonlinear Schrödinger equations with periodically modulated nonlinearity and periodic asymptotics,” J. Math. Phys. **52**, 032702 (2011). [CrossRef]

19. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. **463**, 1–126 (2008). [CrossRef]

20. Y. S. Kivshar and B. A. Malomed, “Raman-induced optical shocks in nonlinear fibers,” Opt. Lett. **18**, 485–487 (1993). [CrossRef] [PubMed]

21. S. Wabnitz, “Chiral polarization solitons in elliptically birefringent spun optical fibers,” Opt. Lett. **34**, 908–910 (2009). [CrossRef] [PubMed]

20. Y. S. Kivshar and B. A. Malomed, “Raman-induced optical shocks in nonlinear fibers,” Opt. Lett. **18**, 485–487 (1993). [CrossRef] [PubMed]

21. S. Wabnitz, “Chiral polarization solitons in elliptically birefringent spun optical fibers,” Opt. Lett. **34**, 908–910 (2009). [CrossRef] [PubMed]

24. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface lattice kink solitons,” Opt. Express **14**, 12365–12372 (2006). [CrossRef] [PubMed]

25. C. Huang, J. Zheng, S. Zhong, and L. Dong, “Interface kink solitons in defocusing saturable nonlinear media,” Opt. Commun. **284**, 4225–4228 (2011). [CrossRef]

24. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface lattice kink solitons,” Opt. Express **14**, 12365–12372 (2006). [CrossRef] [PubMed]

26. F. Ye, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Bragg guiding of domainlike nonlinear modes and kink arrays in lower-index core structures,” Opt. Lett. **33**, 1288–1290 (2008). [CrossRef] [PubMed]

25. C. Huang, J. Zheng, S. Zhong, and L. Dong, “Interface kink solitons in defocusing saturable nonlinear media,” Opt. Commun. **284**, 4225–4228 (2011). [CrossRef]

*z*axis in a finite optical lattice embedded into a defocusing saturable medium. The evolution of laser beam is governed by the nonlinear Schrödinger equation for the normalized field amplitude

*q*: where the transverse

*x*and longitudinal

*z*coordinates are normalized to the width and diffraction length of beam, respectively;

*s*stands for the saturation parameter;

*p*characterizes the lattice depth and the function

*R*(

*x*) describes the profile of linear refractive index modulation.

*q*(

*x,z*) =

*w*(

*x*)exp(

*ibz*), where

*w*(

*x*) is a real function depicting the soliton profiles and

*b*is a nonlinear propagation constant. Substituting the expression into Eq. (1), one obtains an ordinary differential equation. Note that the Neumann boundary conditions are applied for numerically solving the soliton profiles. Solitons are characterized by the propagation constant

*b*, saturation parameter

*s*, lattice distribution

*R*, lattice depth

*p*, lattice frequency Ω and lattice site number

*n*. Without loss of generality, we vary

*b,s,n, p*and fix lattice frequency Ω in numerical analysis.

*q*(

*x,z*) = [

*w*(

*x*) +

*u*(

*x*) +

*iv*(

*x*)]exp(

*ibz*), here

*u,v*≪

*w*stand for the real and imaginary parts of perturbation, which may grow with a complex index

*δ*during propagation. Substituting the perturbed solution into Eq. (1) and linearizing it around the stationary solution

*w*yield a system of coupled equations for perturbations

*u*and

*v*: which can be solved numerically. Solitons are stable only when all real parts of

*δ*equal zero.

*x*→ ±∞ and decaying oscillatory tails in lattice modulated region. According to Eq. (1), one can deduce that symmetric solitons drop off from two pedestals whose height is determined by

*w*(

*x*→ ±∞) = [−

*b*/(1+

*sb*)]

^{1/2}, which also gives out the existence condition of solitons, i. e., solitons can exist only when the relations

*b*≤ 0 and 1 +

*sb*> 0 (or

*b*> −1/

*s*) are satisfied simultaneously. To quantitatively depict the energy flow of the wave fronts and oscillatory tails trapped in lattices, we define the concept of “renormalized energy flow” based on the profiles of solitons [24

24. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface lattice kink solitons,” Opt. Express **14**, 12365–12372 (2006). [CrossRef] [PubMed]

*H*(

*x*) = 0 in lattice modulated region and 1 elsewhere.

*R*(

*x*) = cos

^{2}(Ω

*x*) for |

*x*| =

*nπ*/2Ω and

*R*(

*x*)

*=*0 otherwise as a linear refractive index modulation. Here, Ω is the modulation frequency;

*n*is an odd positive integer and denotes the lattice site number.

*b*inside the existence region

*b*≤

_{low}*b*≤

*b*and weakens with the decrease of

_{upp}*b*. Soliton will degenerate into chirped truncated Bloch wave [27

27. J. Wang, J. Yang, T. J. Alexander, and Y. S. Kivshar, “Truncated-Bloch-wave solitons in optical lattices,” Phys. Rev. A **79**, 043610 (2009). [CrossRef]

*s*[Fig. 1(c)]. Yet, the saturation parameter does not affect the localization of solitons in lattices evidently. One can also infer from the relation

*w*(

*x*→ ±∞) = [−

*b*/(1 +

*sb*)]

^{1/2}that the pedestal height of solitons approaches to infinity when propagation constant goes to its lower cutoff. For example,

*w*(

*x*→ ±∞) = [−

*b*/(1 +

*sb*)]

^{1/2}→ ∞ when

*b*→ −2 for

*s*= 0.5.

25. C. Huang, J. Zheng, S. Zhong, and L. Dong, “Interface kink solitons in defocusing saturable nonlinear media,” Opt. Commun. **284**, 4225–4228 (2011). [CrossRef]

*R*(

*x*) = cos

^{2}(Ω

*x*) in Fig. 2(a). The finite gaps expand with the growth of lattice depth. As is well known, localized solitons in infinite lattices always reside in the gaps while Bloch modes exist in the bands. Solitons can exist in both semi-infinite gap and finite gaps in periodically modulated focusing media. For defocusing media with imprinted optical lattices, solitons can only be found in finite gaps [19

19. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. **463**, 1–126 (2008). [CrossRef]

*b*=

_{upp}*b*+ 1/

_{low}*s*when the lattice depth exceeds a critical value [Fig. 2(b)]. Deviation of this relation at small

*p*is due to the fact that the upper cutoff of propagation constant is restricted by the upper edge of the first gap of lattice spectrum. Out-of-phase solitons stop to exist when

*b*approaches to the upper edge of the second band. Nonlinear waves will degenerate into linear ones when

_{upp}*b*→

*b*. Namely, out-of-phase solitons are nonlinear modes bifurcating from the linear modes. Unlike out-of-phase solitons, the existence domain of in-phase solitons expands with the lattice depth firstly and shrinks later when the lattice depth exceeds a critical value [Fig. 2(c)]. Note that the hoofed existence domain does not satisfy the relation

_{upp}*b*=

_{upp}*b*+ 1/

_{low}*s*.

*U*of out-of-phase solitons is a monotonically decreasing function of the propagation constant

_{r}*b*. It vanishes at the upper cutoffs and approaches to infinity at the lower cutoffs [Fig. 2(d)]. Meanwhile,

*U*increases with the growth of the lattice site number

_{r}*n*. The inset plot illustrates the dependence of the existence domain of out-of-phase solitons on the saturation parameter. Solitons with higher renormalized energy flow can be found in the vicinity of

*b*in the high-saturation regime. Comparing with the kink solitons in defocusing cubic media with imprinted semi-infinite lattices [24

_{upp}**14**, 12365–12372 (2006). [CrossRef] [PubMed]

*b*–

*s*plane. Out-of-phase solitons are almost completely stable in the first gap. There exists a narrow instability area near the lower edge of the first gap for small saturation parameters. Solitons shaped as chirped truncated Bloch waves in the second band are completely unstable. On the other hand, solitons are completely stable when the saturation parameter exceeds a critical value. This property illustrates that the saturation of the nonlinear media can be utilized to suppress the instability of solitons. To understand the influence of the lattice site number on the existence and stability of the symmetric waves, we exhaustively solve the stationary solutions of solitons and comprehensively conduct linear stability analysis on them. The results are summarized in Fig. 3(b). The instability domain of out-of-phase solitons in the first gap shrinks rapidly with the growth the lattice sites and vanishes when the lattice site number exceeds a critical value (

*n*= 25) which constitutes one of our central results. That is to say, one can improve the stability of symmetric out-of-phase solitons by increasing the lattice site number. Yet, the growth of lattice site number cannot change the instability of solitons residing in the second band.

_{cr}*b*→

*b*. These inflexions indicate that in-phase solitons have threshold energy flow. In other words, such solitons are purely nonlinear modes and cannot be bifurcated from linear modes, which differs from out-of-phase solitons [Fig. 2(d)].

_{upp}**14**, 12365–12372 (2006). [CrossRef] [PubMed]

*b*= −1,

*n*= 13,

*p*= 4 is plotted in Fig. 4(c). To verify the linear stability analysis results, we extensively perform propagation simulations of solitons by the split-step Fourier method. Typical stable and unstable propagation examples of out-of-phase and in-phase solitons are presented in Fig. 4(d)-4(f). Note the soliton shown in Fig. 4(f) suffers weak oscillatory instability since the eigen-spectrum is complex and has small real parts [Re(

*δ*)

*= 0.052].*

_{max}14. Y. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. **298**, 81–197 (1998). [CrossRef]

*R*(

*x*) = 1 – cos(Ω

*x*) for |

*x*| ≤

*nπ*/Ω and

*R*(

*x*) = 0 otherwise. In the following, discussions, we set Ω = 4,

*p*= 2 in numerical analysis. Even

*n*is needed for dark-like antisymmetric solitons.

*P*= 11.26) [inset in Fig. 5(b)]. The main features, such as the location of oscillatory peaks and localization of oscillatory tails, are similar to those of symmetric solitons. The amplitudes at the symmetric center of out-of-phase and in-phase dark-like solitons are zero which is different from those of symmetric waves [Fig. 5(c) and 5(d)].

_{th}*n*< 5) do not exhibit periodicity. The corresponding band-gap structure of infinite lattice exerts no influence on the solitons. With the increase of lattice sites, periodicity becomes stronger and nonlinear modes are partly affected by the band-gap of infinite lattices. It is impossible to realize symmetric and antisymmetric solitons with nonvanishing amplitudes in infinite lattices. However, our results show that solitons strongly “feel” the restriction of infinite lattices when the lattice site number is larger (e.g.

*n*= 12). Thus, we deem that finite lattices should support diverse types of

*localized*solitons, such as fundamental, dipole, triple solitons in 1D nonlinear systems and multipole, vortex, vector solitons in 2D nonlinear systems.

## Acknowledgments

## References and links

1. | R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, |

2. | G. Maugin, |

3. | L. Stenflo, “Theory of nonlinear plasma surface waves,” Phys. Scr. |

4. | I. L. Garanovich, A. A. Sukhorukov, and Y. S. Kivshar, “Surface multi-gap vector solitons,” Opt. Express |

5. | K. G. Makris, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Hache, “Discrete surface solitons,” Opt. Lett. |

6. | S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, “Observation of discrete surface solitons,” Phys. Rev. Lett. |

7. | M. I. Molina, I. L. Garanovich, A. A. Sukhorukov, and Y. S. Kivshar, “Discrete surface solitons in semi-infinite binary waveguide arrays,” Opt. Lett. |

8. | E. Smirnov, M. Stepić, C. E. Rüter, D. Kip, and V. Shandarov, “Observation of staggered surface solitary waves in one-dimensional waveguide arrays,” Opt. Lett. |

9. | Y. Kominis, A. Papadopoulos, and K. Hizanidis, “Surface solitons in waveguide arrays: analytical solutions,” Opt. Express |

10. | Y. Kominis and K. Hizanidis, “Power-dependent reflection, transmission, and trapping dynamics of lattice solitons at interfaces,” Phys. Rev. Lett. |

11. | Y. He, D. Mihalache, and B. Hu, “Soliton drift, rebound, penetration, and trapping at the interface between media with uniform and spatially modulated nonlinearities,” Opt. Lett. |

12. | Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface gap solitons,” Phys. Rev. Lett. |

13. | C. R. Rosberg, D. N. Neshev, W. Krolikowski, A. Mitchell, R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Observation of surface gap solitons in semi-infinite waveguide arrays,” Phys. Rev. Lett. |

14. | Y. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. |

15. | V. V. Konotop and S. Takeno, “Stationary dark localized modes: discrete nonlinear Schrödinger equations,” Phys. Rev. E |

16. | P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, “Matter-wave dark solitons in optical lattices,” J. Opt. B |

17. | Y. Kominis and K. Hizanidis, “Lattice solitons in self-defocusing optical media: analytical solutions of the nonlinear Kronig-Penney model,” Opt. Lett. |

18. | J. Belmonte-Beitia and J. Cuevas, “Existence of dark solitons in a class of stationary nonlinear Schrödinger equations with periodically modulated nonlinearity and periodic asymptotics,” J. Math. Phys. |

19. | F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. |

20. | Y. S. Kivshar and B. A. Malomed, “Raman-induced optical shocks in nonlinear fibers,” Opt. Lett. |

21. | S. Wabnitz, “Chiral polarization solitons in elliptically birefringent spun optical fibers,” Opt. Lett. |

22. | G. Agrawal, |

23. | S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, |

24. | Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface lattice kink solitons,” Opt. Express |

25. | C. Huang, J. Zheng, S. Zhong, and L. Dong, “Interface kink solitons in defocusing saturable nonlinear media,” Opt. Commun. |

26. | F. Ye, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Bragg guiding of domainlike nonlinear modes and kink arrays in lower-index core structures,” Opt. Lett. |

27. | J. Wang, J. Yang, T. J. Alexander, and Y. S. Kivshar, “Truncated-Bloch-wave solitons in optical lattices,” Phys. Rev. A |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(190.6135) Nonlinear optics : Spatial solitons

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: July 6, 2011

Manuscript Accepted: August 5, 2011

Published: August 17, 2011

**Citation**

Shunsheng Zhong, Changming Huang, Chunyan Li, and Liangwei Dong, "Symmetric and antisymmetric solitons in finite lattices," Opt. Express **19**, 17179-17188 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-18-17179

Sort: Year | Journal | Reset

### References

- R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations (Academic Press, 1982).
- G. Maugin, Nonlinear Waves in Elastic Crystals (Oxford University Press, 2000).
- L. Stenflo, “Theory of nonlinear plasma surface waves,” Phys. Scr. T63, 59–62 (1996). [CrossRef]
- I. L. Garanovich, A. A. Sukhorukov, and Y. S. Kivshar, “Surface multi-gap vector solitons,” Opt. Express 14, 4780–4785 (2006). [CrossRef] [PubMed]
- K. G. Makris, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Hache, “Discrete surface solitons,” Opt. Lett. 30, 2466–2468 (2005). [CrossRef] [PubMed]
- S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, “Observation of discrete surface solitons,” Phys. Rev. Lett. 96, 063901 (2006). [CrossRef] [PubMed]
- M. I. Molina, I. L. Garanovich, A. A. Sukhorukov, and Y. S. Kivshar, “Discrete surface solitons in semi-infinite binary waveguide arrays,” Opt. Lett. 31, 2332–2334 (2006). [CrossRef] [PubMed]
- E. Smirnov, M. Stepić, C. E. Rüter, D. Kip, and V. Shandarov, “Observation of staggered surface solitary waves in one-dimensional waveguide arrays,” Opt. Lett. 31, 2338–2340 (2006). [CrossRef] [PubMed]
- Y. Kominis, A. Papadopoulos, and K. Hizanidis, “Surface solitons in waveguide arrays: analytical solutions,” Opt. Express 15, 10041–10051 (2007). [CrossRef] [PubMed]
- Y. Kominis and K. Hizanidis, “Power-dependent reflection, transmission, and trapping dynamics of lattice solitons at interfaces,” Phys. Rev. Lett. 102, 133903 (2009). [CrossRef] [PubMed]
- Y. He, D. Mihalache, and B. Hu, “Soliton drift, rebound, penetration, and trapping at the interface between media with uniform and spatially modulated nonlinearities,” Opt. Lett. 35, 1716–1718 (2010). [CrossRef] [PubMed]
- Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface gap solitons,” Phys. Rev. Lett. 96, 073901 (2006). [CrossRef] [PubMed]
- C. R. Rosberg, D. N. Neshev, W. Krolikowski, A. Mitchell, R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Observation of surface gap solitons in semi-infinite waveguide arrays,” Phys. Rev. Lett. 97, 083901 (2006). [CrossRef] [PubMed]
- Y. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 81–197 (1998). [CrossRef]
- V. V. Konotop and S. Takeno, “Stationary dark localized modes: discrete nonlinear Schrödinger equations,” Phys. Rev. E 60, 1001–1008 (1999). [CrossRef]
- P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, “Matter-wave dark solitons in optical lattices,” J. Opt. B 6, S309–S317 (2004). [CrossRef]
- Y. Kominis and K. Hizanidis, “Lattice solitons in self-defocusing optical media: analytical solutions of the nonlinear Kronig-Penney model,” Opt. Lett. 31, 2888–2890 (2006). [CrossRef] [PubMed]
- J. Belmonte-Beitia and J. Cuevas, “Existence of dark solitons in a class of stationary nonlinear Schrödinger equations with periodically modulated nonlinearity and periodic asymptotics,” J. Math. Phys. 52, 032702 (2011). [CrossRef]
- F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008). [CrossRef]
- Y. S. Kivshar and B. A. Malomed, “Raman-induced optical shocks in nonlinear fibers,” Opt. Lett. 18, 485–487 (1993). [CrossRef] [PubMed]
- S. Wabnitz, “Chiral polarization solitons in elliptically birefringent spun optical fibers,” Opt. Lett. 34, 908–910 (2009). [CrossRef] [PubMed]
- G. Agrawal, Nonlinear Fiber Optics (Academic Press, 1995).
- S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (Izdatel Nauka, 1988).
- Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface lattice kink solitons,” Opt. Express 14, 12365–12372 (2006). [CrossRef] [PubMed]
- C. Huang, J. Zheng, S. Zhong, and L. Dong, “Interface kink solitons in defocusing saturable nonlinear media,” Opt. Commun. 284, 4225–4228 (2011). [CrossRef]
- F. Ye, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Bragg guiding of domainlike nonlinear modes and kink arrays in lower-index core structures,” Opt. Lett. 33, 1288–1290 (2008). [CrossRef] [PubMed]
- J. Wang, J. Yang, T. J. Alexander, and Y. S. Kivshar, “Truncated-Bloch-wave solitons in optical lattices,” Phys. Rev. A 79, 043610 (2009). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.