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  • Editor: Xi-Cheng Zhang
  • Vol. 39, Iss. 4 — Feb. 15, 2014
  • pp: 821–824
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Accuracy of the capillary approximation for gas-filled kagomé-style photonic crystal fibers

M. A. Finger, N. Y. Joly, T. Weiss, and P. St.J. Russell  »View Author Affiliations


Optics Letters, Vol. 39, Issue 4, pp. 821-824 (2014)
http://dx.doi.org/10.1364/OL.39.000821


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Abstract

Precise knowledge of the group velocity dispersion in gas-filled hollow-core photonic crystal fiber is essential for accurate modeling of ultrafast nonlinear dynamics. Here we study the validity of the capillary approximation commonly used to calculate the modal refractive index in kagomé-style photonic crystal fibers. For area-preserving core radius aAP and core wall thickness t, measurements and finite element simulations show that the approximation has an error greater than 15% for wavelengths longer than 0.56(aAPt), independently of the gas-filling pressure. By introducing an empirical wavelength-dependent core radius, the range of validity of the capillary approximation is extended out to a wavelength of at least 0.98(aAPt).

© 2014 Optical Society of America

The wide applicability of kagomé PCF in nonlinear optics is apparent from recent publications on the generation of tunable UV radiation [5

5. K. F. Mak, J. C. Travers, P. Hölzer, N. Y. Joly, and P. St.J. Russell, Opt. Express 21, 10942 (2013). [CrossRef]

], plasma-influenced nonlinear optics [6

6. P. Hölzer, W. Chang, J. C. Travers, A. Nazarkin, J. Nold, N. Y. Joly, M. F. Saleh, F. Biancalana, and P. St.J. Russell, Phys. Rev. Lett. 107, 203901 (2011). [CrossRef]

], Raman-free nonlinear optics in very high pressure systems [7

7. M. Azhar, G. K. L. Wong, W. Chang, N. Y. Joly, and P. St.J. Russell, Opt. Express 21, 4405 (2013). [CrossRef]

], and low threshold high-harmonic generation [8

8. O. H. Heckl, C. R. E. Baer, C. Kränkel, S. V. Marchese, F. Schnapper, M. Holler, T. Südmeyer, J. S. Robinson, J. W. G. Tisch, F. Couny, P. Light, F. Benabid, and U. Keller, Appl. Phys. B 97, 369 (2009). [CrossRef]

]. Furthermore, simulations have shown that such systems have the potential, given the appropriate dispersion, to generate supercontinua extending far into the deep-UV, covering the whole wavelength range from 140 to 1000 nm [4

4. J. C. Travers, W. Chang, J. Nold, N. Y. Joly, and P. St.J. Russell, J. Opt. Soc. Am. B 28, A11 (2011). [CrossRef]

]. Kagomé PCF is also of great interest in the infrared spectral region.

All these applications rely on an accurate knowledge of the group velocity dispersion (GVD), in particular its dependence on wavelength. In this Letter we perform a detailed experimental and theoretical study of the GVD of evacuated and gas-filled kagomé PCFs, concentrating in particular on the visible to IR spectral ranges, where the capillary-based Marcatili–Schmelzer model (MSM) is found to deviate significantly from experimental measurements.

In gas-filled kagomé PCF the total GVD arises from two competing effects: the anomalous dispersion of the empty waveguide and the normal dispersion of the filling gas [3

3. J. Nold, P. Hölzer, N. Y. Joly, G. K. L. Wong, A. Nazarkin, A. Podlipensky, M. Scharrer, and P. St.J. Russell, Opt. Lett. 35, 2922 (2010). [CrossRef]

]. The MSM predicts that the refractive index nmp (m being the azimuthal and p the radial mode order) of the modes guided in a capillary can be written in the approximate form [9

9. E. A. J. Marcatili and R. A. Schmeltzer, Bell Syst. Tech. J. 43, 1783 (1964). [CrossRef]

]
nmp(λ)=ngas2(λ)ump2k02a2,
(1)
where ump is the pth zero of the (m1)th order Bessel function of the first kind, k0 the vacuum wavevector, a the radius of the capillary, and ngas the refractive index of the filling gas, which itself depends on pressure and temperature [10

10. A. Börzsönyi, Z. Heiner, M. P. Kalashnikov, A. P. Kovács, and K. Osvay, Appl. Opt. 47, 4856 (2008). [CrossRef]

]. In this Letter we explore the accuracy of this expression as a predictor of GVD over a broad range of wavelengths.

The GVD of the kagomé PCFs was measured over the wavelength range 525–1375 nm using a white-light Mach–Zehnder interferometer [11

11. M. Tateda, N. Shibata, and S. Seikai, IEEE J. Quantum Electron. 17, 404 (1981). [CrossRef]

]. Since the presence of water can strongly influence the dispersion measurements, the fiber samples were placed in a vacuum oven for at least 24 h at 100°C to remove all remaining water without damaging the coating. Figure 2(a) shows the result of measurements on a 725 mm length of five-ring kagomé PCF (Fig. 1) with an area-preserving (AP) core diameter of 20.4 μm; the thickness of the core wall was 190 nm and the interhole spacing 13.7 μm. It can be seen that the data points become increasingly scattered at longer wavelengths. We attribute this to increasing penetration of the core light into the cladding, where it couples to localized inhomogeneous resonances that cause the GVD to fluctuate rapidly with wavelength. In order to extract the general trend of the dispersion, we therefore fitted a third-order polynomial using the least-square method.

Fig. 1. Generic set-up for ultrafast nonlinear optical experiments using gas-filled kagomé-PCF. In most cases fiber lengths of less than 1 m are sufficient. The observed dynamics depend on the filling gas, the pressure and the pump pulse characteristics.

Figure 2(b) compares the measured GVD curve with the values from different theoretical models. First, the GVD of an idealized one-ring kagomé PCF [inset of Fig. 2(b)] was computed using the finite element (FEM) method (JCM wave). The experimentally observed GVD fluctuations were also seen in the FEM simulations and have been reported previously [12

12. S. J. Im, A. Husakou, and J. Herrmann, Opt. Express 17, 13052 (2009).

]. Since however it is the overall trend of the GVD curve (not these fiber-dependent fluctuations) that is important in ultrafast experiments, the calculated β values were smoothed using a fifth-order polynomial and the GVD derived by differentiation.

Fig. 2. (a) Measured relative group delay, fitted to a third-order polynomial, for a 725 mm length of five-ring kagomé PCF. (b) 1: GVD calculated using the MSM for an area-preserving (AP) diameter of 21.4 μm and a flat-to-flat (FF) diameter of 20.4 μm. 2: GVD for a thin capillary (inner radius 10.2 μm, wall thickness 190 nm, calculated analytically). 3: a fifth-order polynomial fit to FEM simulations of the simplified kagomé PCF (structure shown in the inset). 4: measured GVD using (a).

Next, we investigated the influence of finite core wall thickness (here 190 nm) on the dispersion of a silica capillary of inner diameter 20.4 μm. The dispersion of the thin capillary [Fig. 2(b)] was calculated using a transfer matrix method [17

17. C. W. Tee and S. F. Yu, J. Lightwave Technol. 21, 3379 (2003).

]. In contrast to FEM, this has the advantage that the computation time is reduced to a few seconds or even less. In Fig. 2(b), it may be seen that the dispersion of the thin-wall capillary matches the FEM simulation perfectly up to 1100 nm, the major deviations to the MSM coming from the finite thickness of the core wall. Beyond 1100 nm the deviations between the thin-wall capillary model and FEM become larger, indicating that the outer cladding structure is beginning to play a role. The thin-wall capillary provides an attractive trade-off between speed of calculation and accuracy, occupying the middle ground between FEM (accurate but slow) and the MSM (limited accuracy but very fast). It is much faster than FEM, while remaining accurate out to longer wavelengths than is true of the MSM. Additionally, it models the main loss windows of kagomé PCF (caused by resonances in the core wall) and their influence on the GVD [13

13. G. J. Pearce, G. S. Wiederhecker, C. G. Poulton, S. Burger, and P. St.J. Russell, Opt. Express 15, 12680 (2007). [CrossRef]

].

We now introduce an empirical scaling law that can be used to increase the range of validity of the MSM. It is based on replacing the core radius in Eq. (1) with an effective radius that depends on the wavelength and the core wall thickness:
a(λ)=aAP1+sλ2/(aAPt),
(3)
where s is a dimensionless empirical parameter. From a least-squares fit to the calculated effective refractive indices we find that the value s=0.047 works well for all the FEM simulations. The measured GVD is however slightly more anomalous than that predicted by FEM for the simplified structure in Fig. 2(b). A value of s=0.065 was found to work best for the experimental data points. We checked this modified MSM (mMSM) experimentally for two five-ring kagomé PCFs with core diameters of 20.4 and 26.6 μm, core-wall thicknesses of 190 and 160 nm, and interhole spacings of 13.7 μm. The mMSM works well for the fundamental mode of both fibers out to a wavelength of 1400 nm—the limit of our measurement range. In Fig. 3 the GVDs predicted by FEM, the mMSM, and the AP MSM are compared with the experimental values. For the correct value of s, the mMSM fits the measurements and simulations much more closely, merging with the MSM for wavelengths shorter than 600 nm. In conclusion, it seems that the range of validity of the MSM can be extended out to a wavelength of 0.98(aAPt) by introducing the wavelength-dependent radius in Eq. (3).

Fig. 3. Comparison between the MSM [Eq. (1)], measurement, FEM, and the modified MSM, including a(λ) [Eq. (3)] for the FEM determined and experimental s-parameter.

Finally, we investigated how the GVD behaves if the fiber is filled with argon, as in many experiments [3

3. J. Nold, P. Hölzer, N. Y. Joly, G. K. L. Wong, A. Nazarkin, A. Podlipensky, M. Scharrer, and P. St.J. Russell, Opt. Lett. 35, 2922 (2010). [CrossRef]

,6

6. P. Hölzer, W. Chang, J. C. Travers, A. Nazarkin, J. Nold, N. Y. Joly, M. F. Saleh, F. Biancalana, and P. St.J. Russell, Phys. Rev. Lett. 107, 203901 (2011). [CrossRef]

,7

7. M. Azhar, G. K. L. Wong, W. Chang, N. Y. Joly, and P. St.J. Russell, Opt. Express 21, 4405 (2013). [CrossRef]

,18

18. N. Y. Joly, J. Nold, W. Chang, P. Hölzer, A. Nazarkin, G. K. L. Wong, F. Biancalana, and P. St.J. Russell, Phys. Rev. Lett. 106, 203901 (2011). [CrossRef]

]. Figure 4 shows the difference between the measured GVD at 5 bars and 10 bars and that measured in the evacuated fiber. GVD curves at 10 bars, obtained from both the MSM and the mMSM, are also shown. It is clear that the MSM for the pressurized fibers shows the same long-wavelength deviations as observed in the evacuated fiber, while the predictions of the mMSM match the measurement much closer at NIR wavelengths. Nevertheless, for gas-filled kagomé PCF the MSM predicts the correct GVD for wavelengths below λcrit, which in this case is 800 nm.

Fig. 4. Measured GVD of argon-filled kagomé PCF minus the GVD of the evacuated fiber. The MSM and mMSM (s=0.065) were calculated for 10 bars argon filling and an area-preserving radius of 10.71 μm.

In summary, an empirical scaling law allows one to distinguish two GVD regimes: a short-wavelength regime (λ<λcrit) where the total GVD is predominantly determined by the size of the hexagonal core and a long-wavelength regime (λ>λcrit) where the finite core-wall thickness and outer cladding structure become more and more important. The transfer-matrix method applied to a thin-wall capillary represents a compromise between very time-consuming FEM calculations and the MSM, although it must still be solved numerically. The most attractive model (which has the advantage of being analytical and accurate) is however the empirically modified MSM, in which the effective core radius is assumed wavelength dependent. Note that, despite these very helpful tools, it will still be necessary to carry out full FEM simulations, or measure the dispersion directly, if a highly accurate dispersion curve is needed, including the intrinsic GVD jittering caused by anticrossings with cladding states.

References

1.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St.J. Russell, P. J. Roberts, and D. C. Allan, Science 285, 1537 (1999). [CrossRef]

2.

F. Benabid, J. C. Knight, G. Antonopolous, and P. St.J. Russell, Science 298, 399 (2002). [CrossRef]

3.

J. Nold, P. Hölzer, N. Y. Joly, G. K. L. Wong, A. Nazarkin, A. Podlipensky, M. Scharrer, and P. St.J. Russell, Opt. Lett. 35, 2922 (2010). [CrossRef]

4.

J. C. Travers, W. Chang, J. Nold, N. Y. Joly, and P. St.J. Russell, J. Opt. Soc. Am. B 28, A11 (2011). [CrossRef]

5.

K. F. Mak, J. C. Travers, P. Hölzer, N. Y. Joly, and P. St.J. Russell, Opt. Express 21, 10942 (2013). [CrossRef]

6.

P. Hölzer, W. Chang, J. C. Travers, A. Nazarkin, J. Nold, N. Y. Joly, M. F. Saleh, F. Biancalana, and P. St.J. Russell, Phys. Rev. Lett. 107, 203901 (2011). [CrossRef]

7.

M. Azhar, G. K. L. Wong, W. Chang, N. Y. Joly, and P. St.J. Russell, Opt. Express 21, 4405 (2013). [CrossRef]

8.

O. H. Heckl, C. R. E. Baer, C. Kränkel, S. V. Marchese, F. Schnapper, M. Holler, T. Südmeyer, J. S. Robinson, J. W. G. Tisch, F. Couny, P. Light, F. Benabid, and U. Keller, Appl. Phys. B 97, 369 (2009). [CrossRef]

9.

E. A. J. Marcatili and R. A. Schmeltzer, Bell Syst. Tech. J. 43, 1783 (1964). [CrossRef]

10.

A. Börzsönyi, Z. Heiner, M. P. Kalashnikov, A. P. Kovács, and K. Osvay, Appl. Opt. 47, 4856 (2008). [CrossRef]

11.

M. Tateda, N. Shibata, and S. Seikai, IEEE J. Quantum Electron. 17, 404 (1981). [CrossRef]

12.

S. J. Im, A. Husakou, and J. Herrmann, Opt. Express 17, 13052 (2009).

13.

G. J. Pearce, G. S. Wiederhecker, C. G. Poulton, S. Burger, and P. St.J. Russell, Opt. Express 15, 12680 (2007). [CrossRef]

14.

Y. Y. Wang, N. V. Wheeler, F. Couny, P. J. Roberts, and F. Benabid, Opt. Lett. 36, 669 (2011). [CrossRef]

15.

S. Février, B. Beaudou, and P. Viale, Opt. Express 18, 5142 (2010). [CrossRef]

16.

F. Yu, W. J. Wadsworth, and J. C. Knight, Opt. Express 20, 11153 (2012). [CrossRef]

17.

C. W. Tee and S. F. Yu, J. Lightwave Technol. 21, 3379 (2003).

18.

N. Y. Joly, J. Nold, W. Chang, P. Hölzer, A. Nazarkin, G. K. L. Wong, F. Biancalana, and P. St.J. Russell, Phys. Rev. Lett. 106, 203901 (2011). [CrossRef]

OCIS Codes
(060.2270) Fiber optics and optical communications : Fiber characterization
(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers
(060.5295) Fiber optics and optical communications : Photonic crystal fibers

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: December 6, 2013
Manuscript Accepted: December 14, 2013
Published: February 6, 2014

Virtual Issues
February 21, 2014 Spotlight on Optics

Citation
M. A. Finger, N. Y. Joly, T. Weiss, and P. St.J. Russell, "Accuracy of the capillary approximation for gas-filled kagomé-style photonic crystal fibers," Opt. Lett. 39, 821-824 (2014)
http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-39-4-821


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References

  1. R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St.J. Russell, P. J. Roberts, and D. C. Allan, Science 285, 1537 (1999). [CrossRef]
  2. F. Benabid, J. C. Knight, G. Antonopolous, and P. St.J. Russell, Science 298, 399 (2002). [CrossRef]
  3. J. Nold, P. Hölzer, N. Y. Joly, G. K. L. Wong, A. Nazarkin, A. Podlipensky, M. Scharrer, and P. St.J. Russell, Opt. Lett. 35, 2922 (2010). [CrossRef]
  4. J. C. Travers, W. Chang, J. Nold, N. Y. Joly, and P. St.J. Russell, J. Opt. Soc. Am. B 28, A11 (2011). [CrossRef]
  5. K. F. Mak, J. C. Travers, P. Hölzer, N. Y. Joly, and P. St.J. Russell, Opt. Express 21, 10942 (2013). [CrossRef]
  6. P. Hölzer, W. Chang, J. C. Travers, A. Nazarkin, J. Nold, N. Y. Joly, M. F. Saleh, F. Biancalana, and P. St.J. Russell, Phys. Rev. Lett. 107, 203901 (2011). [CrossRef]
  7. M. Azhar, G. K. L. Wong, W. Chang, N. Y. Joly, and P. St.J. Russell, Opt. Express 21, 4405 (2013). [CrossRef]
  8. O. H. Heckl, C. R. E. Baer, C. Kränkel, S. V. Marchese, F. Schnapper, M. Holler, T. Südmeyer, J. S. Robinson, J. W. G. Tisch, F. Couny, P. Light, F. Benabid, and U. Keller, Appl. Phys. B 97, 369 (2009). [CrossRef]
  9. E. A. J. Marcatili and R. A. Schmeltzer, Bell Syst. Tech. J. 43, 1783 (1964). [CrossRef]
  10. A. Börzsönyi, Z. Heiner, M. P. Kalashnikov, A. P. Kovács, and K. Osvay, Appl. Opt. 47, 4856 (2008). [CrossRef]
  11. M. Tateda, N. Shibata, and S. Seikai, IEEE J. Quantum Electron. 17, 404 (1981). [CrossRef]
  12. S. J. Im, A. Husakou, and J. Herrmann, Opt. Express 17, 13052 (2009).
  13. G. J. Pearce, G. S. Wiederhecker, C. G. Poulton, S. Burger, and P. St.J. Russell, Opt. Express 15, 12680 (2007). [CrossRef]
  14. Y. Y. Wang, N. V. Wheeler, F. Couny, P. J. Roberts, and F. Benabid, Opt. Lett. 36, 669 (2011). [CrossRef]
  15. S. Février, B. Beaudou, and P. Viale, Opt. Express 18, 5142 (2010). [CrossRef]
  16. F. Yu, W. J. Wadsworth, and J. C. Knight, Opt. Express 20, 11153 (2012). [CrossRef]
  17. C. W. Tee and S. F. Yu, J. Lightwave Technol. 21, 3379 (2003).
  18. N. Y. Joly, J. Nold, W. Chang, P. Hölzer, A. Nazarkin, G. K. L. Wong, F. Biancalana, and P. St.J. Russell, Phys. Rev. Lett. 106, 203901 (2011). [CrossRef]

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