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Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 24 — Nov. 28, 2005
  • pp: 9721–9728
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Spatial modes in a PCF fiber generated continuum

Y. Vidne and M. Rosenbluh  »View Author Affiliations


Optics Express, Vol. 13, Issue 24, pp. 9721-9728 (2005)
http://dx.doi.org/10.1364/OPEX.13.009721


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Abstract

We measure the propagation properties of a highly nonlinear photonic crystal fiber (PCF). The spatial, temporal and frequency dependent properties of the propagating modes are measured under conditions of high power, seven picosecond excitation, white light continuum generation. The experimentally determined multi-mode nature of the white light continuum is found to be in good agreement with numerical simulations.

© 2005 Optical Society of America

1. Introduction

In a highly non-linear photonic crystal fiber (PCF), in which the light is guided through a small solid silica core, surrounded by a periodic arrangement of air holes running along the full length of the fiber, strong confinement of the guided light has been demonstrated along with a very large nonlinear coefficient. Much recent interest has been generated in these fibers due to their unique properties. (For a review, see [1

1 . P. Russell , “ Photonic Crystal Fibers ,” Science 299 , 358 – 362 ( 2003 ). [CrossRef] [PubMed]

].) The high nonlinearity leads to efficient supercontinuum generation [2

2 . S. Coen , A. Chau , R. Leonhardt , J. Harvey , J. Knight , W. Wadsworth , and P. Russell , “ White-light supercontinuum with 60 ps pump pulses in a photonic crystal fiber ,” Opt. Lett. 26 , 1356 – 1358 ( 2001 ). [CrossRef]

,3

3 . S. Coen , A. Chau , R. Leonhardt , J. Harvey , J. Knight , W. Wadsworth , and P. Russell , “ Supercontinuum generation by stimulated Raman scattering and parametric four-wave mixing in photonic crystal fibers ,” J. Opt. Soc. Am. B3 19 , 753 – 764 ( 2002 ). [CrossRef]

], while the zero dispersion wavelength of such a fiber can be significantly modified, over a wide range in the visible and near infrared spectrum, by selecting the appropriate core diameter. By adjusting the ratio of the air hole diameter, d, to the hole spacing, Λ, (known as the pitch), “endlessly single mode” fibers have been reported [4–6

4 . W. Wadsworth , N. Joly , J. Knight , T. Birks , F. Biancalana , and P. Russell , “ Supercontinuum and four-wave mixing with Q-swiched pulses in endlessly single-mode photonic crystal fibers ,” Opt. Express 12 , 299 – 309 ( 2004 ). [CrossRef] [PubMed]

]. In such fibers “all” wavelengths (within the transparency window of the fiber) propagate with the fundamental mode characteristics of the fiber and the range of wavelengths is only limited by bending losses of the fiber.

According to recently published results [7–12

7 . J. Knight , T. Birks , and P. Russell , “ Properties of photonic crystal fiber and the effective index model ,” J. Opt. Soc. Am. A 15 , 748 – 752 ( 1998 ). [CrossRef]

] when the air hole diameter to hole spacing ratio, d/Λ, exceeds 0.41, PCF fibers are not strictly single mode at all wavelengths, though excitation of high order modes is believed to be only significant when the excitation pulses into the fiber are not properly matched to the fiber parameters. Here we demonstrate that for a typical PCF fiber, with about 2 micron core diameter, and a d/Λ~0.8, the propagation for all visible and infrared wavelengths is multimode in nature and the high order modes are excited nonlinearly at high intensities. We find that at sufficiently high intensities the multimode propagation is impossible to avoid no matter how one couples into the PCF fiber.

Propagation of light in a multimode PCF can be difficult to predict theoretically because each mode is randomly perturbed by fiber bends, fiber defects and thermal effects along the propagation length. It is also difficult to experimentally separate the various modes [13–15

13 . C. Kerbage , B. Eggleton , P. Westbrook , and R. Windeler , “ Experimental and scalar beam propagation analysis of an air-silica microstructure fiber ,” Opt. Express 7 , 113 – 122 ( 2000 ). [CrossRef] [PubMed]

] as each mode has its own temporal dependence, spatial structure and nonlinearly generated frequency spectrum. Accurate experimental examination of the various fiber modes, however, can reveal many interesting PCF properties. In this paper we demonstrate that high order spatial modes can be examined experimentally and simultaneously resolved in the wavelength, spatial and time domains. Furthermore, we note that the measured propagation time delay between different modes is several ps per meter of PCF fiber, and in many applications of white light generation this can lead to significant temporal broadening of the white light pulse.

2. Theory

Step index fibers are parameterized by the normalized frequency parameter V,

V=2πρλ[nc2ncl2]12
(1)

where ρ is the fiber core diameter, λ, the propagating wavelength, and nc and ncl are the refractive indicies (generally assumed to be wavelength independent) of the fiber core and cladding respectively. The V parameter determines the number of modes at any given wavelength that can propagate through the fiber. Single mode wave guiding requires a low V (V<2.405), and as the wavelength decreases and V increases, increasingly higher order modes can propagate through the fiber core. In a PCF the V parameter is complicated by the wavelength dependent effective index of the cladding and is in general larger than in step index fibers, which means that a PCF can support a greater number of high order modes at a given wavelength.

A number of researchers have shown, both theoretically and experimentally [7–12

7 . J. Knight , T. Birks , and P. Russell , “ Properties of photonic crystal fiber and the effective index model ,” J. Opt. Soc. Am. A 15 , 748 – 752 ( 1998 ). [CrossRef]

] that one can define the V parameter for a PCF as

VPCF=2πΛλ[nc2(λ)ncl2(λ)]12
(2)

where nc(λ)=cβ/ω and ncl(λ) are the effective index of the fundamental mode in the core and in the air hole structure, respectively, β is the propagation constant and ω is the angular frequency. The requirement for single mode wave guiding has been shown to be VPCF≤π and for a PCF fiber with a given geometry the wavelength, λ, where the value of V decreases to be below π, is the second order cutoff wavelength.

From multiple solutions of Maxwell’s equations, it was numerically found [9–11

9 . N. Mortensen , J. Folkenberg , M. Nielsen , and K. Hansen , “ Modal cutoff and the V parameter in photonic crystal fibers ,” Opt. Lett. 28 , 1879 – 1881 ( 2003 ). [CrossRef] [PubMed]

] that a phase diagram, as shown in Fig. 1, for single-mode and multi-mode propagation regimes can be plotted based on the expression

λΛα(dΛd*Λ)γ
(3)

where α=2.80±0.12, γ=0.89±0.02, and d*/Λ≈0.406. The “endlessly single mode” fibers reported in the literature are those for which the ratio d/Λ<d*/Λ. For d/Λ>d*/Λ, the PCF supports higher order modes, whose number increases as the wavelength gets longer. In Fig. 1 we plot the log(λ/Λ) vs d/Λ from Eq. 4 using the constants as given above, and also show the limits in the curve due to the uncertainty of the constants. The arrow in the figure indicates the value of d/Λ ~ 0.8, for the fiber used in our experiments while the shaded area represents the wavelength range of the generated white light continuum (λ/Λ between 0.15 and 0.54). Clearly for all the wavelengths generated in the fiber used in our experiments, the propagation is well within the multi-mode regime.

Fig. 1. Single mode vs multimode phase diagram as predicted by Eq. (4). The dashed lines indicate the uncertainty of the constants used. The arrow indicates the value of d/Λ for the PCF fiber used in the experiments while the shaded region represents the range of observed wavelengths in the white light continuum generated by the PCF fiber.

The multi-mode propagation produces a time delay between the various spatial modes which is analogues to propagation speeds in a step index fiber, where the time delay between spatial modes is due to the different propagation constants βq which decrease from ≈ n1k0 for the fundamental mode, q=1, to n2k0, for the highest order propagating mode, where n1 is the core index, n2 is the cladding index and k0 is the free space wave number. Between these two extremes the modes are bound and can propagate through the fiber. Those modes whose propagation constant falls outside this range are not guided and radiate out of the fiber. The group velocity of the modes, defined as Vq=dω/dβq, decreases from ≈c0/n1 to ≈c0/n1(1-Δ) where Δ=(n1-n2)/n1, where we have ignored material dispersion, which can be justified if working near the zero GVD regime.

A highly non-linear PCF may be compared to traditional step index fibers operating by total internal reflection. This can be seen from the effective indices of the guided modes (Fig. 2), which have values between the core and cladding indices [15–18

15 . J. Ranka , R. Windeler , and A. Stentz , “ Optical properties of high-delta air-silica microstructure optical fibers ,” Opt. Lett. 25 , 796 – 798 ( 2000 ). [CrossRef]

](indicated by the dashed lines in Fig. 2). Modes that have an effective index lower then the cladding index can not propagate through the fiber. For the fiber used in our experiments, we calculate that the first 12 higher order modes are guided and all higher order modes are radiated out. In addition, the 12 guided modes, are clustered in four different groups with four different effective indecies as shown in Fig. 2.

Fig. 2. Effective indices of the spatial modes as calculated by simulation, the blue dashed lines represent the effective indices extremes of the core and cladding.

3. Experimental setup and results

In the experiments we used a mode-locked Ti:Sapphire laser operating at a wavelength, λ= 790nm, producing 1.5 Watt average power with a pulse length of 7-ps at an 82 MHz mode-locking frequency. The pulses were coupled into a 6-m-long PCF, with a 2.3-μm diameter silica core surrounded by a cobweb structure of air holes, as shown in a SEM image supplied by the PCF manufacturer in Fig. 3. The zero dispersion wavelength of this fiber is at 790±5nm and the nonlinear coefficient is 80 W -1 · km 1, enabling the generation of a very broad and high power supercontinuum (SC) as shown in Fig. 4. High order mode generation was non-linear in input intensity and could be eliminated by reducing the input power to ~0.15 Watt.

The experimental setup for measuring the mode structure of the PCF output is shown in Fig. 5. After collimation, part of the SC light was filtered through a monochromator with 8 nm bandwidth and measured using an optical sampling oscilloscope with time resolution of ~20 ps, and an optical spectrum analyzer. A part of the SC light was filtered with narrow-band interference and absorption filters and the far field image was recorded by an intensified, gated, CCD camera [20] with ~100 ps wide, movable gate. We could therefore simultaneously observe the relative timing of the light emanating from the PCF, its spectral composition and its spatial mode structure as a function of time. The data presented here were taken near 790 nm, where the chromatic dispersion of the propagation through 6 m of fiber did not affect the results, while at other wavelengths the propagation delays of the higher order modes were also affected by this dispersion.

Fig. 3. SEM image of PCF showing the core surrounded by air holes as supplied by the manufacturer (Thorlabs NL 2.3).
Fig. 4. Broadband SC generation in 6 m of fiber pumped by 7 ps long pulses at 790 nm from a 82 MHz, 1.5 W average power, mode-locked Ti:Sapphire laser.
Fig. 5. Experimental setup for measuring the spectral, temporal and spatial structure of the supercontinuum generated in the PCF.

For numerically simulating the higher order mode propagation in the PCF we used Lumerical mode solutions software [21] which is a fully-vectorial optical mode solver based upon the finite difference time domain [22

22 . A. Ward 1 and J. Pendry , “ A program for calculating photonic band structures, Green’s functions and transmission/reflection coefficients using a non-orthogonal FDTD method ,” Comput. Phys. Commun. 28 , 590 – 621 ( 2000 ). [CrossRef]

]. The input to the software specifies the geometry of the PCF used and the code then determines the mode intensity profiles and group delay for each of the modes. The geometric structure simulated was designed to match as closely as possible the real structure of the PCF, as shown in Fig. 6, though differences between the real structure and the simulation can be seen. In the simulated structure the core size was taken as 2.3-μm and the air hole pitch was taken as 2.6 μm. The silica forming the core and the web was assumed to have an index, n = 1.44 , and the air holes were assigned an index, n = 1.

Fig. 6. PCF structure in simulation (a) manufacturer supplied SEM image of PCF core and cladding structure (b).

We calculated the intensity distribution of the first 20 transverse modes in the far field at 790 nm as shown in Fig. 7 as well as the time delay between the modes that one would expect for propagation through 6 m of PCF. This time delay for the first 12 modes is shown in Fig. 8. The first two eigenmodes are the degenerate fundamental modes of the PCF corresponding to the two orthogonal polarization states of the fundamental mode. Because the PCF has sixfold rotational symmetry, these modes are degenerate [23

23 . T. White , C. Martijn de Sterke , and R. McPhedran , “ Symmetry and degeneracy in microstructured optical fibers ,” Opt. Lett. 26 , 488 – 490 ( 2001 ). [CrossRef]

] and cannot be separated either spatially or temporally. The group delay difference of 1.7 ps between the modes as calculated by the simulation is a result of numerical error in the simulation.

Fig. 7. Intensity distribution of the first 20 spatial modes.

In Fig. 9 we show the observed temporal structure of the light emanating from the PCF. This figure, which has a 20 ps instrumental time resolution, represents the temporal structure when the wavelength is near 790 nm. No light could be seen coming from the fiber with a delay greater than 1.1 ns. From a comparison with the calculated results, it appears that we are observing up to a maximum mode number of 12, though not all of the 12 modes can be individually resolved in time.

The first peak (at zero time delay) corresponds to the two degenerate fundamental eigenmodes of the PCF. The third and fourth modes are 308 ps and 364 ps delayed, in very good agreement with the calculation. The calculated group delay between modes 4 and 5 is 2 ps, which again is produced by numerical error. Modes 5 and 6 are separated by 8 ps which is too small to be resolved in the measurement, and these three modes make up the third peak in the figure. The seventh, eight and ninth modes emerge 456 ps later as the fourth peak in the figure, the tenth mode 168 ps later, and finally modes eleven and twelve emerge 50 ps later.

Fig. 8. Group delay of the first 12 modes as calculated by the simulation
Fig. 9. The lowest order modes of the PCF in the time domain at a wavelength near 790nm.

The measured group delays do not correspond exactly to the calculated ones for a number of reasons. Fiber bending and defects in the cladding hole array is expected to affect the various modes differently and this is not taken into account in the simulation. [24

24 . The effects of small disorder in the holes nearest the core has been shown to break the mode symmetry. N. Mortensen , M. Nielsen , J. Folkenberg , K. Hansen , and J. Lægsgaard , “ Small-core photonic crystal fibers with weakly disordered air-hole claddings ,” J. Opt. A: Pure Appl. Opt. 6 , 221 – 223 ( 2004 ). [CrossRef]

] Furthermore, the simulated PCF structure does not perfectly match the geometry of the real structure.

The spatial structure of the propagating modes was determined using the fast gated camera which could resolve modes with a propagation delay difference of less than 100 ps. (The camera gate time is ~100 ps but the gate rise and fall times are faster and the gate position time can be controlled with 5 ps accuracy. We could therefore spatially resolve modes with even less than 100 ps time delay between them.) The observed spatial mode profiles, which are shown in Fig. 10, were measured near 790 nm with 100 ps gate time. The time difference between the frames a and b is 300 ps, between b and c, 450 ps and between c and d, 150 ps. In between these times, there was no detectable light emanating form the fiber. Based on the temporal measurements shown in Fig. 9, frame (a) corresponds to the two first order degenerate modes, frame (b) to the third through the sixth mode, frame (c) to the seventh through the ten mode and frame (d) to modes eleven and twelve. A comparison to the calculated mode structure shown in Fig. 7 gives very reasonable agreement.

Fig. 10. Far field images of the spatial modes after spectral and intensity filtering. The images were obtained on a gated camera with 300 ps between images (a) and (b), 450 ps between (b) and (c) and 150 ps time difference between (c) and (d).

The excitation of the various modes within the cluster of degenerate or nearly degenerate modes is strongly dependent on the input coupling of the light into the PCF fiber. Thus for example, the excitation of any one of the modes in the group 6 through 10, or 11 and 12, could be selectively enhanced by the slightest displacement of the PCF with respect to the incoming laser beam, without this displacement affecting the coupling efficiency of the laser into the fiber.

4. Conclusion

High order transverse modes of white light continuum, generated in a highly non-linear PCF have been observed in both the spatial and temporal domain and have been compared, with good agreement, to a numerical simulation which predicts the spatial intensity distribution and group delays of the modes. The inevitable excitation of higher order modes in a PCF leads to temporal broadening of the generated white light continuum. The significantly different group velocities of the modes eventually lead to a temporal separation of the emerging light into multiple pulses, provided sufficient fiber lengths are used. In our experiments, with 6 m of PCF, maximum pulse separations of about 1 ns were observed.

References and links

1 .

P. Russell , “ Photonic Crystal Fibers ,” Science 299 , 358 – 362 ( 2003 ). [CrossRef] [PubMed]

2 .

S. Coen , A. Chau , R. Leonhardt , J. Harvey , J. Knight , W. Wadsworth , and P. Russell , “ White-light supercontinuum with 60 ps pump pulses in a photonic crystal fiber ,” Opt. Lett. 26 , 1356 – 1358 ( 2001 ). [CrossRef]

3 .

S. Coen , A. Chau , R. Leonhardt , J. Harvey , J. Knight , W. Wadsworth , and P. Russell , “ Supercontinuum generation by stimulated Raman scattering and parametric four-wave mixing in photonic crystal fibers ,” J. Opt. Soc. Am. B3 19 , 753 – 764 ( 2002 ). [CrossRef]

4 .

W. Wadsworth , N. Joly , J. Knight , T. Birks , F. Biancalana , and P. Russell , “ Supercontinuum and four-wave mixing with Q-swiched pulses in endlessly single-mode photonic crystal fibers ,” Opt. Express 12 , 299 – 309 ( 2004 ). [CrossRef] [PubMed]

5 .

T. A. Birks , J. Knight , and P. Russell , “ Endlessly single-mode photonic crystal fibre ,” Opt. Lett. 22 , 961 – 963 ( 1997 ). [CrossRef] [PubMed]

6 .

J. Knight , T. Birks , P. Russell , and D. Atkin , “ All-silica single-mode fiber with photonic crystal cladding ,” Opt. Lett. 21 , 1547 – 1549 ( 1996 ); Errata, Opt. Lett. 22 , 484 – 485 ( 1997 ). [CrossRef] [PubMed]

7 .

J. Knight , T. Birks , and P. Russell , “ Properties of photonic crystal fiber and the effective index model ,” J. Opt. Soc. Am. A 15 , 748 – 752 ( 1998 ). [CrossRef]

8 .

A. Ferrando , E. Silvestre , J. Miret , P. Andres , and M. Andres , “ Full-vector analysis of a realistic photonic crystal fiber ,” Opt. Lett. 24 , 276 – 278 ( 1999 ). [CrossRef]

9 .

N. Mortensen , J. Folkenberg , M. Nielsen , and K. Hansen , “ Modal cutoff and the V parameter in photonic crystal fibers ,” Opt. Lett. 28 , 1879 – 1881 ( 2003 ). [CrossRef] [PubMed]

10 .

J. Folkenberg , N. Mortensen , K. Hansen , T. Hansen , H. Simonsen , and C. Jakobsen , “ Experimental investigation of cutoff phenomena in nonlinear photonic crystal fibers ,” Opt. Lett. 28 , 1882 – 1884 ( 2003 ). [CrossRef] [PubMed]

11 .

B. Kuhlmey , “ Modal cutoff in microstructured optical fibers ,” Opt. Lett. 27 , 1684 – 1686 ( 2002 ). [CrossRef]

12 .

M. Nielsen and N. Mortensen , “ Photonic crystal fiber design based on the V-parameter ,” Opt. Express 11 , 2762 – 2768 ( 2003 ). [CrossRef] [PubMed]

13 .

C. Kerbage , B. Eggleton , P. Westbrook , and R. Windeler , “ Experimental and scalar beam propagation analysis of an air-silica microstructure fiber ,” Opt. Express 7 , 113 – 122 ( 2000 ). [CrossRef] [PubMed]

14 .

A. Efimov and A. Taylor , “ Nonlinear generation of very high-order UV modes in microstructured fibers ,” Opt. Express 11 , 910 – 918 ( 2003 ). [CrossRef] [PubMed]

15 .

J. Ranka , R. Windeler , and A. Stentz , “ Optical properties of high-delta air-silica microstructure optical fibers ,” Opt. Lett. 25 , 796 – 798 ( 2000 ). [CrossRef]

16 .

G. Agrawal , Nonlinear fiber optics , ( Academic Press , 2nd edition, 1995 ).

17 .

R. Guobin , W. Zhi , L. Shuqin , and J. Shuisheng , “ Mode classification and degeneracy in photonic crystal fibers ,” Opt. Express 11 , 1310 – 1321 ( 2003 ). [CrossRef] [PubMed]

18 .

R. Rokitski and S. Fainman , “ Propagation of ultrashort pulses in multimode fiber in space and time ,” Opt. Express 11 , 1497 – 1502 ( 2003 ). [CrossRef] [PubMed]

19 .

N. A. Mortensen , M. Stach , J. Broeng , A. Petersson , H. R. Simonsen , and R. Michalzik ,“ Multi-mode photonic crystal fibers for VCSEL based data transmission ,” Opt. Express 11 , 1953 – 1959 ( 2003 ). [CrossRef] [PubMed]

20 .

http://www.stanfordcomputeroptics.com

21 .

http://www.lumerical.com

22 .

A. Ward 1 and J. Pendry , “ A program for calculating photonic band structures, Green’s functions and transmission/reflection coefficients using a non-orthogonal FDTD method ,” Comput. Phys. Commun. 28 , 590 – 621 ( 2000 ). [CrossRef]

23 .

T. White , C. Martijn de Sterke , and R. McPhedran , “ Symmetry and degeneracy in microstructured optical fibers ,” Opt. Lett. 26 , 488 – 490 ( 2001 ). [CrossRef]

24 .

The effects of small disorder in the holes nearest the core has been shown to break the mode symmetry. N. Mortensen , M. Nielsen , J. Folkenberg , K. Hansen , and J. Lægsgaard , “ Small-core photonic crystal fibers with weakly disordered air-hole claddings ,” J. Opt. A: Pure Appl. Opt. 6 , 221 – 223 ( 2004 ). [CrossRef]

OCIS Codes
(060.2270) Fiber optics and optical communications : Fiber characterization
(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers
(060.7140) Fiber optics and optical communications : Ultrafast processes in fibers

ToC Category:
Research Papers

History
Original Manuscript: October 18, 2005
Revised Manuscript: October 11, 2005
Published: November 28, 2005

Citation
Y. Vidne and M. Rosenbluh, "Spatial modes in a PCF fiber generated continuum," Opt. Express 13, 9721-9728 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-24-9721


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References

  1. P. Russell, "Photonic Crystal Fibers," Science 299, 358-362 (2003). [CrossRef] [PubMed]
  2. S. Coen, A. Chau, R. Leonhardt, J. Harvey, J. Knight, W. Wadsworth, and P. Russell, “White-light supercontinuum with 60 ps pump pulses in a photonic crystal fiber,” Opt. Lett. 26, 1356-1358 (2001). [CrossRef]
  3. S. Coen, A. Chau, R. Leonhardt, J. Harvey, J. Knight, W. Wadsworth, and P. Russell, “Supercontinuum generation by stimulated Raman scattering and parametric four-wave mixing in photonic crystal fibers,” J. Opt. Soc. Am. B 19, 753-764 (2002). [CrossRef]
  4. W. Wadsworth, N. Joly, J. Knight, T. Birks, F. Biancalana, P. Russell, “Supercontinuum and four-wave mixing with Q-swiched pulses in endlessly single-mode photonic crystal fibers,” Opt. Express 12, 299-309 (2004). [CrossRef] [PubMed]
  5. T. A. Birks, J. Knight, and P. Russell, “Endlessly single-mode photonic crystal fibre,” Opt. Lett. 22, 961-963 (1997). [CrossRef] [PubMed]
  6. J. Knight, T. Birks, P. Russell, and D. Atkin, “All-silica single-mode fiber with photonic crystal cladding,” Opt. Lett. 21, 1547-1549 (1996); Errata, Opt. Lett. 22, 484-485 (1997). [CrossRef] [PubMed]
  7. J. Knight, T. Birks, and P. Russell, "Properties of photonic crystal fiber and the effective index model," J. Opt. Soc. Am. A 15, 748-752 (1998). [CrossRef]
  8. A. Ferrando, E. Silvestre, J. Miret, P. Andres and M. Andres, “Full-vector analysis of a realistic photonic crystal fiber,” Opt. Lett. 24, 276-278 (1999). [CrossRef]
  9. N. Mortensen, J. Folkenberg, M. Nielsen and K. Hansen, "Modal cutoff and the V parameter in photonic crystal fibers," Opt. Lett. 28, 1879-1881 (2003). [CrossRef] [PubMed]
  10. J. Folkenberg, N. Mortensen, K. Hansen, T. Hansen, H. Simonsen, C. Jakobsen, “Experimental investigation of cutoff phenomena in nonlinear photonic crystal fibers,” Opt. Lett. 28, 1882-1884 (2003). [CrossRef] [PubMed]
  11. B. Kuhlmey, "Modal cutoff in microstructured optical fibers," Opt. Lett. 27, 1684-1686 (2002). [CrossRef]
  12. M. Nielsen and N. Mortensen, “Photonic crystal fiber design based on the V-parameter,” Opt. Express 11, 2762-2768 (2003). [CrossRef] [PubMed]
  13. C. Kerbage, B. Eggleton, P. Westbrook and R. Windeler, "Experimental and scalar beam propagation analysis of an air-silica microstructure fiber," Opt. Express 7, 113-122 (2000). [CrossRef] [PubMed]
  14. A. Efimov, A. Taylor, " Nonlinear generation of very high-order UV modes in microstructured fibers," Opt. Express 11, 910-918 (2003). [CrossRef] [PubMed]
  15. J. Ranka, R. Windeler, and A. Stentz, “Optical properties of high-delta air-silica microstructure optical fibers,” Opt. Lett. 25, 796-798 (2000). [CrossRef]
  16. G. Agrawal, Nonlinear fiber optics, (Academic Press, 2nd edition, 1995).
  17. R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, “Mode classification and degeneracy in photonic crystal fibers,” Opt. Express 11, 1310-1321 (2003). [CrossRef] [PubMed]
  18. R. Rokitski and S. Fainman, "Propagation of ultrashort pulses in multimode fiber in space and time," Opt. Express 11, 1497-1502 (2003). [CrossRef] [PubMed]
  19. N. A. Mortensen, M. Stach, J. Broeng, A. Petersson, H. R. Simonsen, and R. Michalzik," Multi-mode photonic crystal fibers for VCSEL based data transmission," Opt. Express 11, 1953-1959 (2003). [CrossRef] [PubMed]
  20. <a href= "http://www.stanfordcomputeroptics.com">http://www.stanfordcomputeroptics.com</a>
  21. <a href= "http://www.lumerical.com">http://www.lumerical.com</a>
  22. A. Ward 1, J. Pendry, "A program for calculating photonic band structures, Green’s functions and transmission/reflection coefficients using a non-orthogonal FDTD method," Comput. Phys. Commun. 28, 590–621 (2000). [CrossRef]
  23. T. White, C. Martijn de Sterke, and R. McPhedran, "Symmetry and degeneracy in microstructured optical fibers," Opt. Lett. 26, 488-490 (2001). [CrossRef]
  24. The effects of small disorder in the holes nearest the core has been shown to break the mode symmetry. N Mortensen, M. Nielsen, J. Folkenberg, K. Hansen, and J. Lægsgaard, “Small-core photonic crystal fibers with weakly disordered air-hole claddings,” J. Opt. A: Pure Appl. Opt. 6, 221–223 (2004). [CrossRef]

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