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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 11 — May. 28, 2007
  • pp: 6894–6899
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Two-dimensional high-precision fiber waveguide arrays for coherent light propagation

U. Röpke, H. Bartelt, S. Unger, K. Schuster, and J. Kobelke  »View Author Affiliations


Optics Express, Vol. 15, Issue 11, pp. 6894-6899 (2007)
http://dx.doi.org/10.1364/OE.15.006894


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Abstract

Fiber waveguide arrays can be applied as a very useful tool for the investigation of effects in discrete optics. The observation of coherent propagation in such discrete waveguide arrays requires, however, high structural precision and great material homogeneity. The fabrication of such a fiber array with close tolerances compared to conventional fiber technology is discussed. Linear propagation effects are modeled for an ideal fiber waveguide array and are compared with experimental results. The good agreement of these results with each other indicates the applicability of such fiber waveguide arrays in studying linear and non-linear properties in discrete optics.

© 2007 Optical Society of America

1. Introduction

In recent years there has been an increasing interest in the fabrication and investigation of artificial optical media. Materials composed of weakly coupled waveguides, which are arranged in a periodic structure (waveguide arrays), exhibit unique propagation properties which differ strongly from light field evolution in isotropic media. They constitute a useful model system for studying properties of linear and nonlinear discrete systems, e.g. specific field structures, coupling effects, localization, spatial solitons, and the interplay of propagation, dissipation (gain) and discreteness [1–3

1. J. W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. K. Efremidis, “Spatial photonics in nonlinear waveguide arrays,” Opt. Express 13, 1780–1796 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-6-1780. [CrossRef] [PubMed]

]. Moreover, interest in such materials has been stimulated by their application potential, e. g. for the switching or routing of signals [4

4. D. N. Christodoulides and E. D. Eugenieva, “Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays,” Phys. Rev. Lett. 87, 2339011-4 (2001). [CrossRef]

, 5

5. J. Proctor and J. N. Kutz, “Nonlinear mode-coupling for passive mode-locking: application of waveguide arrays, dual-core fibers, and/or fiber arrays,” Opt. Express 13, 8933–8950 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-22-8933. [CrossRef] [PubMed]

] and the generation or amplification of light [6–8

6. Y. Huo and P. K. Cheo, “Analysis of transverse mode competition and selection in multicore fiber lasers,” J. Opt. Soc. Am. B 22, 2345–2349 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=josab-22-11-2345. [CrossRef]

].

One-dimensional waveguide arrays have been fabricated by planar technologies as integrated optical elements in AlGaAs and polymers. They form the basis of extensive studies in linear and nonlinear effects in discrete optics of one-dimensional arrays (see e. g. [2

2. T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett. 88, 0939011-4 (2002). [CrossRef]

]).

Uniform two-dimensional waveguide arrays have already been successfully prepared in bulk materials using photorefractive effects [9

9. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003). [CrossRef] [PubMed]

], or femtosecond laser pulse inscription [10–12

10. S. Nolte, M. Will, J. Burghoff, and A. Tuennermann, “Femtosecond waveguide writing: a new avenue to threedimensional integrated optics,” Appl. Phys. A. 77, 109–111 (2003). [CrossRef]

], but the array lengths are restricted to less than 10 cm. New fiber technologies developed for microstructured fibers and photonic crystal fibers offer a good basis for the preparation of large two-dimensional waveguide arrays with long interaction lengths in comparison to waveguide arrays in bulk materials. However, the realized fiber arrays behaved optically more like a random system, even for rather regular waveguide lattice structures [13

13. T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tuennermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett. 93, 053901 (2004). [CrossRef] [PubMed]

, 14

14. T. Pertsch, A. Chipouline, S. Nolte, F. Lederer, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, U. Peschel, and A. Tünnermann, “From diffusive to coherent light propagation in disordered nonlinear fiber arrays,” in Photonic Metamaterials: From Random to Periodic, Technical Digest (CD) (Optical Society of America, 2006), paper WA7, http://www.opticsinfobase.org/abstract.cfm?URI=META-2006-WA7

]. As a consequence, high specific requirements concerning material homogeneity and structural dimensions have to be fulfilled to achieve conditions suitable for the observation of non-stochastic effects of discrete optics. Fiber waveguide arrays have also been fabricated as dissipative arrays consisting of rare earth doped cores. So far only small arrays of 6 to 19 cores have been studied with the focus directed to the in-phase supermode selection by phase locking at the output facet of the fiber and not to the specific properties of discrete optics [7

7. L. Michaille, C. R. Bennett, D. M. Taylor, T. J. Shepherd, J. Broeng, H. R. Simonsen, and A. Petersson, “Phase locking and supermode selection in multicore photonic crystal fiber lasers with a large doped area,” Opt. Lett. 30, 1668–1670 (2005) http://www.opticsinfobase.org/abstract.cfm?URI=ol-30-13-1668. [CrossRef] [PubMed]

, 8

8. L. Li, A. Schülzgen, S. Chen, V. L. Temyanko, J. V. Moloney, and N. Peyghambarian, “Phase locking and in-phase supermode selection in monolithic multicore fiber lasers,” Opt. Lett. 31, 2577–2579 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=ol-31-17-2577. [CrossRef] [PubMed]

].

In the following sections, we will discuss the requirements for optical fiber waveguide arrays. We will then describe the fabrication process of high-precision two-dimensional passive waveguide arrays with parameters well beyond conventional fiber technology. The linear propagation properties within such a fiber waveguide array will then be analyzed and compared with experimental results.

2. Fiber-optic arrays: theoretical background and requirements

The requirements for a fiber waveguide array with coherently coupled waveguides can be derived from model calculations. We describe the light propagation in weakly guiding and weakly coupled waveguides by a perturbation theory [15

15. A. W. Snyder and J. D. Love, “Optical waveguide theory”, (Chapman & Hall, 1983).

]. The undisturbed system consists of a hexagonal array of N uncoupled identical circular step-index waveguides with cladding index n 0, index step relative to the core index Δn 0, core radius R and core distances Λ. The coupling of light energy between two neighboring cores can then be considered as a first-order perturbation effect. The propagation length for a total energy exchange is denoted as coupling length Lc (cf. [15

15. A. W. Snyder and J. D. Love, “Optical waveguide theory”, (Chapman & Hall, 1983).

]),

LC=λ04Δn0AE02dAARE0(x+Λ,y)E0(x,y)dA.
(1)

Here, λ 0 is the vacuum wavelength and E 0(x,y) the circular symmetric field distribution of the undisturbed fundamental mode in a waveguide cross-section with coordinates x and y, normalized to a power of 1W. The integrals in the numerator and denominator are taken over the total area A∞ and the core area AR, respectively. Because of the weak coupling, a variation of Λ as well as the coupling between cores with distances greater than Λ can be neglected as perturbations of higher order.

The complex amplitudes of the unperturbed mode fields of the waveguides are combined in an N-dimensional vector function u(z) of the propagation length z, with the squared modulus |ui|2 as the power guided by the waveguide i. In first perturbation order it holds

iλ02πdudz+(MD+λ04LCMC)u=n21WAeffdiag(ui2)u+,
(2)

with

MD=Δn0diag(δniΔn0AE02dA+2πRδRiE0(R)2+ΓiARE0(x+Λ,y)2dA)AE02dA.
(3)

Aeff is the effective core area and n 2 is the nonlinear coefficient (2.5∙10-20 m2/W in silica). With diag(Xi) a diagonal matrix is denoted with elements δij Xi, where i, j are waveguide index numbers and δij is the Kronecker delta. On the right hand side of Eq. (2), many different physical effects may be included that influence light propagation (nonlinearity, dispersion, loss, gain, …). Here, only the nonlinear Kerr effect is given to assess the order of magnitude of interesting effects [16

16. G. P. Agrawal, “Nonlinear fiber optics”, 2nd ed., (Academic Press, 1995)

]. The second term on the left hand side of Eq. (2) describes the first-order influence of the real array structure on light propagation. The elements of the coupling matrix Mc are equal to 1 if their indices refer to adjacent waveguides; otherwise they are zero. This matrix describes the array topology including the boundary. All diagonal elements vanish. In contrast to this matrix, the deviation matrix MD, Eq. (3), contains only diagonal elements, which represent first-order corrections of the individual effective mode indices of the waveguides due to deviations of the index step δni, to deviations of the core radius δRi, and to the number of neighboring cores Γi. While the last-named effect is negligibly small, the former two are critical for the coherent coupling of light within the array. Introduced by variations within the technology processes of making waveguide arrays, they cause phase mismatches between coupling waveguides and impair their energy exchange. Thus, even arrays with a seemingly excellent hexagonal structure may optically behave as a random system rather [14

14. T. Pertsch, A. Chipouline, S. Nolte, F. Lederer, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, U. Peschel, and A. Tünnermann, “From diffusive to coherent light propagation in disordered nonlinear fiber arrays,” in Photonic Metamaterials: From Random to Periodic, Technical Digest (CD) (Optical Society of America, 2006), paper WA7, http://www.opticsinfobase.org/abstract.cfm?URI=META-2006-WA7

].

It is a challenging task to fabricate fiber-optic waveguide arrays that implement such coherent coupling and nonlinear effects at the same scale of propagation length without being compromised by fabrication statistics. In order to prevent material damage, we specify a maximum peak pulse intensity of 1015 W/m2. Then the Kerr effect corresponds to an index increase by 2∙10-5. The index splitting of array supermodes by the coupling can be estimated as 3λ0/4LC , which is comparable to the Kerr-effect if Lc ≈ 58 mm at λ0 = 155 μm. This means that the components of MD (representing the influence of variations of refractive index and core radius) should be kept well below 10-5 in order to maintain coherent coupling. The fabrication tolerance of present-day fiber technology based on telecommunication monomode preforms exceeds this value by a factor of about 10 [13

13. T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tuennermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett. 93, 053901 (2004). [CrossRef] [PubMed]

, 17

17. K.P. Hansen and J. Broeng, “High -Power Photonic Crystal Fiber Lasers”, Photonics Spectra, 82–89 (2006)

]. A practically achievable index variation δn in high-purity silica is about 5∙10-6. Even with such a high-grade material, a core diameter tolerance of 0.1% is needed to meet the requirements of coherent light propagation with coupling lengths of several cm.

3. Fabrication and characterization of high-precision fiber-optic waveguide arrays

Fiber-optic waveguide arrays with high structural precision were fabricated using the stack-and-draw technique. At first a precision circular step index fiber rod was prepared (rod-in-tube technique, precision grinding and drawing) and cut into elements of approx. 30 cm length. In a second step, these rod elements were assembled to obtain a packed preform. This stack of rods was enclosed by buffer elements made from the cladding material. The buffer area was designed to adapt the hexagonal rod pattern of the array to a circular outside jacketing. Furthermore, it optically isolates the waveguide array from the surrounding area. After a consolidation process (collapse of cavities), the packed preform is drawn out to form the fiber waveguide array in the final step.

The preform was made using high-grade silica products from Heraeus-Tenevo (core material: F300 high-purity silica rods; cladding material: F320 fluorine-doped silica tubes, jacketing material: silica tube F300). A small index step Δn 0 ≈ 1.2 ∙10-3 between core and cladding of the single waveguides results in a relatively low numerical aperture and, thus, in a large monomode core area. A core radius of R = 9.5 μm was chosen to guarantee stable single-mode operation of the uncoupled waveguides at a wavelength of λ0 = 1.55 μm, and a core distance of Λ = 34 μm was chosen to obtain a coupling length of Lc ≈ 57 mm.

Fig. 1. Cross-sections of a high-precision waveguide array: white light transmission microscopy (left), reflected-light/differential interference contrast microscopy image of an etched cleaved facet (right).

Fig. 1 (left) shows the cross-section of a fabricated array with 61 waveguide elements. The outer fiber diameter is 585 μm (R = 9.49 μm, Λ = 33.9 μm). The excellent structural quality of the waveguide array, including boundary, can be observed in the micrograph of an etched end facet of the waveguide array, Fig. 1 (right). The variance of core distances is below 1%, and the differences in the shape and size of cores are beyond the resolution limit of our structural analysis.

4. Model calculations and experimental investigations

The arrays are characterized using cw-light in the linear optical regime. In this case the right hand side of (2) vanishes. For the analysis of light propagation in experimentally implemented arrays with MD ≠ 0, the supermodes and the coupling patterns of ideal arrays with MD = 0 are helpful tools, and Eq. (2) can then be used in a simplified form:

iλ02πdudz+(λ04LCMC)u=0.
(4)

The light distribution vector u after a propagation length z is obtained by scalar multiplication of the input vector u(0) by a matrix exponential function of z,

u(z)=exp(iπz2LCMC)u(0).
(5)

For the 61-core array shown in Fig. 1 the eigenvalues of the 61 supermodes and the light distributions of the 9 supermodes with highest symmetry (60°) are presented in Fig. 2. The supermodes with the lowest symmetry (180°) occur in pairs (degenerated eigenvalues). The supermodes account for the array boundary. Thus the light energy is confined to the array region.

Fig. 2. Eigenvalues of the supermodes of a 61-core array (left), and field distribution of the supermodes with the highest symmetry (right). The colors (yellow and blue) indicate the sign of the real-valued amplitudes ui.

A useful method to experimentally analyze weakly coupled waveguide arrays is the measurement of the output intensity pattern for various excitation conditions and propagation lengths. In Fig. 3 a/c, experimental results are shown in pseudocolor intensity representation, which were obtained from the sample of Fig. 1 with λ0 = 1.53 μm and two array lengths, zA = 26.5 mm and 55 mm. For excitation, light was launched in specific single cores at the input facet (central excitation and excitation at boundary positions).

The experimental results are compared with distributions computed for an ideal array by means of Eq. (5) as shown in Fig. 3 b/d, where ζ= zA/Lc is used as a model parameter. In the case of central excitation, all supermodes shown in Fig. 2 are excited, and the change in the propagating light distribution can be interpreted as their beating patterns. For shorter propagation lengths, Fig. 3 a/b, the spreading of light demonstrates the light diffraction in regular discrete optical materials. At larger propagation lengths, Fig. 3 c/d, the boundary effects change this diffraction process to confine the light energy within the array.

For a quantitative comparison with the model, the proportion of the individual waveguides on the total power was determined from the measured output intensity patterns. The values of ζ were estimated from an analysis with the method of least squares as ζ = 0.42 (zA = 26.5 mm) and ζ = 0.85 (zA = 55 mm) with an error of ±5%. From this results a coupling length of the array of Lc = zA/ζ = 65 mm ±3 mm and an index splitting of the supermodes of 3λ0/4Lc = 1.8∙10-5 was obtained. The difference relative to the designed coupling length 57 mm can be explained by the measuring wavelength used and a 3% higher index step Δn 0 compared to the value assumed for the model calculation.

Fig. 3. Output distribution of light launched at specific cores: array center (upper row), boundary corner (middle row) and middle of a boundary line (lower row); columns (a) and (c) show experimental results for sample lengths zA = 26.5 mm and 55 mm, respectively; columns (b) and (d) show computer simulations of an ideal array with ζ = 0.42 and 0.85, respectively.

The measured intensity patterns, especially for central excitation, show small disturbances due to waveguide deviations (MD ≠ 0). However, the light propagation is dominated by coherent coupling at least for lengths of 60 mm. Such fiber-optic waveguide arrays would therefore be suitable as model systems for studying the nonlinear dynamics of short pulses in discrete optic systems.

5. Conclusions

It has been shown that it is possible, with an optimized fiber technology, to fabricate silica-based fiber waveguide arrays with a high structural precision and material homogeneity in order to allow observation of three fundamental mechanisms on the same scale of propagation length: the coupling of waveguides, the nonlinear Kerr effect and the spatial coherence of light propagation. The investigated array is composed of 61 waveguides with a coupling length of 65 mm at a wavelength of 1.53 μm. Model calculations and experimental results are well in agreement concerning waveguide diffraction and coherent coupling. With the achieved precision of the fiber waveguide array, influences of nonlinear effects on propagation will become accessible to quantitative investigation. Further understanding of propagation in such discrete optical systems could open manifold application opportunities, e.g., for beam and pulse shaping and for switching of light within such waveguides using cubic nonlinearities.

Acknowledgments

This work was funded by the DFG within the research unit “Nonlinear Dynamics”. Fruitful discussions with the partners in the research unit are gratefully acknowledged.

References and links

1.

J. W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. K. Efremidis, “Spatial photonics in nonlinear waveguide arrays,” Opt. Express 13, 1780–1796 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-6-1780. [CrossRef] [PubMed]

2.

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett. 88, 0939011-4 (2002). [CrossRef]

3.

A. Szameit, D. Blömer, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, “Discrete Nonlinear Localization in Femtosecond Laser Written Waveguides in Fused Silica,” Opt. Express 13, 10552–10557 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-26-10552. [CrossRef] [PubMed]

4.

D. N. Christodoulides and E. D. Eugenieva, “Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays,” Phys. Rev. Lett. 87, 2339011-4 (2001). [CrossRef]

5.

J. Proctor and J. N. Kutz, “Nonlinear mode-coupling for passive mode-locking: application of waveguide arrays, dual-core fibers, and/or fiber arrays,” Opt. Express 13, 8933–8950 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-22-8933. [CrossRef] [PubMed]

6.

Y. Huo and P. K. Cheo, “Analysis of transverse mode competition and selection in multicore fiber lasers,” J. Opt. Soc. Am. B 22, 2345–2349 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=josab-22-11-2345. [CrossRef]

7.

L. Michaille, C. R. Bennett, D. M. Taylor, T. J. Shepherd, J. Broeng, H. R. Simonsen, and A. Petersson, “Phase locking and supermode selection in multicore photonic crystal fiber lasers with a large doped area,” Opt. Lett. 30, 1668–1670 (2005) http://www.opticsinfobase.org/abstract.cfm?URI=ol-30-13-1668. [CrossRef] [PubMed]

8.

L. Li, A. Schülzgen, S. Chen, V. L. Temyanko, J. V. Moloney, and N. Peyghambarian, “Phase locking and in-phase supermode selection in monolithic multicore fiber lasers,” Opt. Lett. 31, 2577–2579 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=ol-31-17-2577. [CrossRef] [PubMed]

9.

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003). [CrossRef] [PubMed]

10.

S. Nolte, M. Will, J. Burghoff, and A. Tuennermann, “Femtosecond waveguide writing: a new avenue to threedimensional integrated optics,” Appl. Phys. A. 77, 109–111 (2003). [CrossRef]

11.

A. Szameit, D. Bloemer, J. Burghoff, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, “Hexagonal waveguide arrays written with fs-laser pulses,” Appl. Phys. B. 82, 507–512 (2006). [CrossRef]

12.

A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, “Two-dimensional soliton in cubic fs laser written waveguide arrays in fused silica,” Opt. Express 14, 6055–6062 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-13-6055. [CrossRef] [PubMed]

13.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tuennermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett. 93, 053901 (2004). [CrossRef] [PubMed]

14.

T. Pertsch, A. Chipouline, S. Nolte, F. Lederer, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, U. Peschel, and A. Tünnermann, “From diffusive to coherent light propagation in disordered nonlinear fiber arrays,” in Photonic Metamaterials: From Random to Periodic, Technical Digest (CD) (Optical Society of America, 2006), paper WA7, http://www.opticsinfobase.org/abstract.cfm?URI=META-2006-WA7

15.

A. W. Snyder and J. D. Love, “Optical waveguide theory”, (Chapman & Hall, 1983).

16.

G. P. Agrawal, “Nonlinear fiber optics”, 2nd ed., (Academic Press, 1995)

17.

K.P. Hansen and J. Broeng, “High -Power Photonic Crystal Fiber Lasers”, Photonics Spectra, 82–89 (2006)

OCIS Codes
(060.2270) Fiber optics and optical communications : Fiber characterization
(060.2280) Fiber optics and optical communications : Fiber design and fabrication
(060.2290) Fiber optics and optical communications : Fiber materials
(060.2310) Fiber optics and optical communications : Fiber optics
(230.3990) Optical devices : Micro-optical devices
(230.4000) Optical devices : Microstructure fabrication

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: March 5, 2007
Revised Manuscript: May 15, 2007
Manuscript Accepted: May 15, 2007
Published: May 21, 2007

Citation
U. Röpke, H. Bartelt, S. Unger, K. Schuster, and J. Kobelke, "Two-dimensional high-precision fiber waveguide arrays for coherent light propagation," Opt. Express 15, 6894-6899 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-11-6894


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References

  1. J. W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. K. Efremidis, "Spatial photonics in nonlinear waveguide arrays," Opt. Express 13, 1780-1796 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-6-1780. [CrossRef] [PubMed]
  2. T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, "Anomalous refraction and diffraction in discrete optical systems," Phys. Rev. Lett. 88, 0939011-4 (2002). [CrossRef]
  3. A. Szameit, D. Blömer, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, "Discrete Nonlinear Localization in Femtosecond Laser Written Waveguides in Fused Silica," Opt. Express 13, 10552-10557 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-26-10552. [CrossRef] [PubMed]
  4. D. N. Christodoulides and E. D. Eugenieva, "Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays," Phys. Rev. Lett. 87, 2339011-4 (2001). [CrossRef]
  5. J. Proctor and J. N. Kutz, "Nonlinear mode-coupling for passive mode-locking: application of waveguide arrays, dual-core fibers, and/or fiber arrays," Opt. Express 13, 8933-8950 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-22-8933. [CrossRef] [PubMed]
  6. Y. Huo and P. K. Cheo, "Analysis of transverse mode competition and selection in multicore fiber lasers," J. Opt. Soc. Am. B 22, 2345-2349 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=josab-22-11-2345. [CrossRef]
  7. L. Michaille, C. R. Bennett, D. M. Taylor, T. J. Shepherd, J. Broeng, H. R. Simonsen, and A. Petersson, "Phase locking and supermode selection in multicore photonic crystal fiber lasers with a large doped area," Opt. Lett. 30, 1668-1670 (2005) http://www.opticsinfobase.org/abstract.cfm?URI=ol-30-13-1668. [CrossRef] [PubMed]
  8. L. Li, A. Schülzgen, S. Chen, V. L. Temyanko, J. V. Moloney, and N. Peyghambarian, "Phase locking and in-phase supermode selection in monolithic multicore fiber lasers," Opt. Lett. 31, 2577-2579 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=ol-31-17-2577. [CrossRef] [PubMed]
  9. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices," Nature 422, 147-150 (2003). [CrossRef] [PubMed]
  10. S. Nolte, M. Will, J. Burghoff, and A. Tuennermann, "Femtosecond waveguide writing: a new avenue to threedimensional integrated optics," Appl. Phys. A. 77, 109-111 (2003). [CrossRef]
  11. A. Szameit, D. Bloemer, J. Burghoff, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, "Hexagonal waveguide arrays written with fs-laser pulses," Appl. Phys. B. 82, 507-512 (2006). [CrossRef]
  12. A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, "Two-dimensional soliton in cubic fs laser written waveguide arrays in fused silica," Opt. Express 14, 6055-6062 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-13-6055. [CrossRef] [PubMed]
  13. T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tuennermann, and F. Lederer, "Nonlinearity and disorder in fiber arrays," Phys. Rev. Lett. 93, 053901 (2004). [CrossRef] [PubMed]
  14. T. Pertsch, A. Chipouline, S. Nolte, F. Lederer, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, U. Peschel, and A. Tünnermann, "From diffusive to coherent light propagation in disordered nonlinear fiber arrays," in Photonic Metamaterials: From Random to Periodic, Technical Digest (CD) (Optical Society of America, 2006), paper WA7, http://www.opticsinfobase.org/abstract.cfm?URI=META-2006-WA7
  15. A. W. Snyder, J. D. Love, "Optical waveguide theory", (Chapman & Hall, 1983).
  16. G. P. Agrawal, "Nonlinear fiber optics", 2nd ed., (Academic Press, 1995)
  17. K.P. Hansen, J. Broeng, " High -Power Photonic Crystal Fiber Lasers", Photonics Spectra, 82-89 (2006)

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