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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 7 — Apr. 3, 2006
  • pp: 3000–3006
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Experimental demonstration of the frequency shift of bandgaps in photonic crystal fibers due to refractive index scaling

G. Antonopoulos, F. Benabid, T. A. Birks, D. M. Bird, J. C. Knight, and P. St. J. Russell  »View Author Affiliations


Optics Express, Vol. 14, Issue 7, pp. 3000-3006 (2006)
http://dx.doi.org/10.1364/OE.14.003000


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Abstract

Experimental demonstration of the frequency shift of photonic bandgaps due to refractive index scaling using D2O-filled hollow-core photonic crystal fibers is presented. The results confirm a simple scaling law for bandgaps in fibers in which the low-index medium is varied.

© 2006 Optical Society of America

1. Introduction

Hollow-core photonic crystal fibers (HC-PCF) [1

1. R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air.,” Science 285,1537–1539 (1999) [CrossRef] [PubMed]

] comprise an air-core surrounded by a cladding consisting of a periodic array of air-holes in silica. The cladding acts as a photonic crystal that forbids light in a range of frequencies to propagate. Light within these bandgaps is confined and guided in the air-core. Therefore, in HC-PCF’s light propagates in air, in a well-confined good-quality mode over very long distances. This makes them very attractive hosts for enhanced interactions between light and matter in applications that range from gas-based nonlinear optics [2

2. F. Benabid, J. C. Knight, G. Antonopoulos, and P. St. J. Russell, “Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,” Science 298, 399–402 (2002) [CrossRef] [PubMed]

, 3

3. F. Benabid, G. Antonopoulos, J. C. Knight, and P. St. J. Russell, “Stokes amplification regimes in quasi-CW pumped hydrogen-filled hollow-core PCF,” Phys. Rev. Lett. 95, 213903–1–4 (2005) [CrossRef]

] to laser-induced guidance of atoms and particles [4

4. F. Benabid, J. C. Knight, and P. S. J. Russell, “Particle levitation and guidance in hollow-core photonic crystal fiber,” Opt. Express 10, 1195–1203 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-21-1195 [PubMed]

]. In general, such applications require filling the fiber with materials other than air. This brings about a change in the refractive index contrast of the fiber structure with a subsequent change in the transmission properties of the HC-PCF. In order to study these changes, it would be very useful to obtain scaling laws for the refractive index of the fiber structure — similar to the well-known and widely used scaling laws that describe the shift in the bandgap frequency when the size scale of the fiber is altered.

Until recently such refractive index scaling laws did not exist, mainly because they cannot be derived from the vector wave equation that describes the propagation of light in a photonic bandgap fiber. However, under the scalar-wave approximation, simple index scaling laws were derived [5

5. T. A. Birks, D. M. Bird, T. D. Hedley, J. M. Pottage, and P. S. J. Russell, “Scaling laws and vector effects in bandgap-guiding fibers,” Opt. Express 12, 69–74 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-69 [CrossRef] [PubMed]

] that predict the way the photonic states of the fiber scale with refractive index contrast. Apart from the above mentioned applications, such laws can be useful in other fields of fiber technology: there has recently been an increasing interest in bandgap fibers made from materials other than silica and air, the usual materials of most common HC-PCF’s. For example, high index glasses can be used for guidance in the infrared region of the spectrum [6

6. G. J. Pearce, J. M. Pottage, D. M. Bird, P. J. Roberts, J. C. Knight, and P. S. Russell, “Hollow-core PCF for guidance in the mid to far infra-red,” Opt. Express 13, 6937–6946 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-18-6937 [CrossRef] [PubMed]

, 7

7. J. Riished, J. Broeng, and A. Bjarklev, “All silica photonic bandgap fiber,” Presented at Conference on Lasers and Electro-Optics,” Paper CTuC5. Baltimore (2003)

], while tunable fiber devices can be made by filling the holes of a photonic crystal fiber (PCF) with liquid crystals [8

8. R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, and D. J. Trevor, “Tunable photonic bandgap fiber. In Proceedings of Optical Fiber Communication Conference, pages 466–468 (2002)

]. All these applications require fibers of different index contrasts between the high and low refractive index regions of the photonic crystal cladding of the fiber.

2. Theoretical background: Refractive index scaling laws

The wave equation for the scalar field distribution in a photonic bandgap fiber is given by:

2Ψxy+(k2n02β2)Ψxy=0,
(1)

where k is the free-space wavenumber, n 0 the transverse distribution of the refractive index of the structure, β the propagation constant of the mode and ∇ the transverse Laplacian operator. The scalar equation is strictly valid for very small index contrasts; however it can still approximately describe propagation in a silica/air HC-PCF [5

5. T. A. Birks, D. M. Bird, T. D. Hedley, J. M. Pottage, and P. S. J. Russell, “Scaling laws and vector effects in bandgap-guiding fibers,” Opt. Express 12, 69–74 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-69 [CrossRef] [PubMed]

]. In the scalar case, for a photonic bandgap structure consisting of a material with high index n 1 and a material with low index n 2 with pitch Λ, it was found [5

5. T. A. Birks, D. M. Bird, T. D. Hedley, J. M. Pottage, and P. S. J. Russell, “Scaling laws and vector effects in bandgap-guiding fibers,” Opt. Express 12, 69–74 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-69 [CrossRef] [PubMed]

] that the photonic states scale so that the quantities:

v2=k2Λ2(n12n22)
w2=Λ2(β2k2n22),
(2)

remain invariant with any change of the parameters k, Λ, n 1 and n 2. These equations yield useful scaling laws that can describe the shift in frequency of the photonic states of the fiber when the index contrast of the latter is altered. In particular, from the first of Eqs. (2), when the low index material n 2 of the PCF is varied while the high index n 1 remains unchanged, so that the initial index contrast N 0 =n 1/n 2 becomes N, any bandgaps found originally at a wavelength λ0 will shift to a new wavelength λ given by [9

9. G. Antonopoulos, F. Benabid, T. A. Birks, D. M. Bird, G. Bouwmans, J. C. Knight, and P. St. J. Russell, “Experimental demonstration of refractive index scaling in photonic bandgap fibers,” In Proceedings of Conference on Lasers and Electro-Optics, Paper CThHH1. San Francisco (2004)

]:

λ=λ0[1N21N02]12.
(3)

This equation can be a very useful tool for the study of bandgap structures with varying low index materials mainly because it avoids the numerical complexities of solving the vector wave equation. This scaling law is also particularly relevant to any application that requires filling the entire air region (core and cladding) of HC-PCF with gases or liquids.

3. Experimental results and discussion

Here we experimentally demonstrate the bandgap shift in two HC-PCF’s when air is replaced with liquid deuterium oxide or heavy water (D2O). D2O has a refractive index of about 1.33 in the visible part of the spectrum. By replacing the air in the HC-PCF with heavy water, the index contrast of the fiber structure was reduced from about 1.46 to 1.10. D2O was preferred to ordinary water because it is much less lossy in the near infrared [10

10. A. C. Tam and K. N. Patel, “Optical absorption of light and heavy water by laser optoacoustic spectroscopy”, App. Opt , 18, 3348–3358 (1979) [CrossRef]

]. White light and supercontinuum transmission spectra were taken before and after the fibers were filled with D2O. It was, therefore, possible to record the changes in the transmitted spectra due to the change in the index contrast alone, while using the same piece of fiber and keeping, thus, the pitch and symmetry of the lattice unchanged. This eliminated any spectral changes that would be due to structural differences between different fiber samples of varying index contrast.

The experimental set-up is shown in Fig. 1(a). The two HC-PCF ends were mounted into specially designed cells and the core and cladding holes were filled along the whole length of the fiber with liquid D2O. Two samples of HC-PCF were used: one with a bandgap centered at 1060 nm and another with a bandgap centered at 1550 nm. The 1060 nm fiber has a cladding pitch of 3 μm, a core diameter of 8.2 μm, while the length of the sample was 40 cm. The respective figures for the 1550 nm fiber are 3.75 μm, 10 μm and 70 cm. The fiber cells were specially designed to replace the air in the HC-PCF’s with D2O, while allowing the fibers to be optically accessed via butt-coupling between the liquid-immersed HC-PCF and ordinary solid-core fibers.

Fig. 1. Experimental set-up for the acquisition of (a) transmission spectra and (b) near-field images of the liquid-filled fibers.
Fig. 2. (a) Transmission spectra of the 1060 nm fiber using a SC source and (b) of the 1550 nm fiber using a tungsten lamp. The spectra were taken before and after filling the holes of the HC-PCF with liquid D2O (light and dark grey areas respectively). The vertical lines define the location of the new shifted bandgaps as predicted by the index scaling law. The arrows mark the position of the respective transmission peaks.

As a light source, both a supercontinuum (SC) source and a tungsten lamp were used. The SC source comprises a solid-core PCF with a core diameter of about 5 μm pumped by a passively Q-switched Nd:YAG microchip laser [10

10. A. C. Tam and K. N. Patel, “Optical absorption of light and heavy water by laser optoacoustic spectroscopy”, App. Opt , 18, 3348–3358 (1979) [CrossRef]

] generating a SC covering a spectrum ranging from 400 nm to 1460 nm. For the 1060-nm-fiber the SC source was used, while the tungsten lamp was employed for the 1550-nm-fiber (since the SC source is weak at 1550 nm in wavelength). In the experiments the SC-generating solid-core PCF was directly butt-coupled to the input end of the D2O-filled PCF. In the case of the tungsten lamp, white light was collected from the source using a single-mode fiber SMF-28 which in turn was butt-coupled to the liquid-filled PCF. The transmission spectra taken before and after the filling process are plotted together for each fiber and are shown in Fig. 2. The transmitted spectra –normalized to the input source spectrum – are plotted both in terms of wavelength and normalized frequency kΛ. The vertical lines show the edges of the shifted bandgaps as predicted by the index-scaling law. These edges were estimated by applying Eq. (3) to the experimentally observed edges of the pass band of the original silica/air fiber. For the 1060 nm fiber the original pass band extended over the region 900–1200 nm; using Eq. (3) this band should shift to about 510–680 nm. Similarly for the other fiber, the original pass band between 1400 nm and 1800 nm shifts to the region 790–1020 nm. Finally, the arrow marks the peak of the initial pass band and the wavelength to which this peak is predicted to shift. In Fig. 2, there is very good agreement between the experimental spectra and the predictions of the index scaling law. The shifted spectra for both fibers peak within the bandgap edges given by Eq. (3). Light recorded outside these edges is believed to have propagated in the cladding. By comparing Figs. 2(a) and 2(b) it is made obvious that any discrepancies in the tungsten lamp transmission spectra from the index scaling law are mainly due to cladding contributions.

Fig. 3. CCD camera images of the near-field intensity distribution at the output end of the D2O-filled PCF. In (a) the input light is the supercontinuum light, while in (b) the input light is that of a laser diode at 633nm

The transmitted signal of the D2O-filled fibers was found to be weaker than that of the unfilled ones. Without taking into account the intrinsic fiber loss rate (about 70–75 dB/km, practically negligible for such short pieces of fiber), the power drop is due to a combination of effects. The first is the higher intrinsic absorption coefficient of D2O compared to that of air and to scattering impurities in the liquid that fills the entire length of the fiber (e.g. suspended impurities coming from the inner surface of the cells or air bubbles inside the fiber holes). Bright scattering points were occasionally seen on the monitor screen during these experiments. The second is the expected increase in confinement and bending losses due to index contrast decrease. Finally, the butt-coupling losses were measured to be in the range 3–5 dB before the fiber was filled with D2O. Inside the liquid, the coupling losses could increase due to the scattering impurities between the end-faces of the fibers. Since the loss rate of liquid-filled fiber-which, fundamentally exhibits different confinement loss as that of the hosting HC-PCF is not known, these coupling losses cannot be directly estimated.

The near-field intensity distribution at the output fiber end was imaged for the 1060 nm D2O-filled PCF. The experimental set-up is that shown in Fig. 1(b). A long working distance ×20 microscope objective collected the light at the fiber output and imaged it onto a color CCD camera. Initially, the supercontinuum light was used as a source. The transmitted core modes had an orange/red color that matches very well with the location of the shifted bandgap of the liquid-filled fiber (see Fig. 3) and provides a visual demonstration of refractive index scaling. Moreover, it is worth noting that this is, to our knowledge, the first demonstration of bandgap guidance in a D2O/silica photonic crystal fiber. The same procedure was repeated using a laser diode operating at 633 nm as a source. This wavelength lies well within the shifted bandgap of the liquid-filled fiber. The same core modes as the ones observed with the white light source were excited at the output of the fiber.

Fig. 4. Full-vector calculations for the density of states for the photonic crystal shown at the top. The background index of the crystal was taken to be that of silica (1.46), while the index in the holes was equal to that of air (bottom left plot) and that of heavy water (bottom right)

Full-vector calculations were also performed for one of the fiber structure used in the experiments (the 1550 nm fiber). The modeled structure is shown in Fig. 4 (top). Density of states (DOS) [13

13. T. D. Hedley, D. M. Bird, F. Benabid, J. C. Knight, and P. St. J. Russell, “Modelling a novel hollow-core photonic crystal fiber,” QLES 2003, QTuL4, Baltimore (2003)

] maps were produced for the structure with a background index of 1.45 (silica), while for the index of the holes both the values of 1.00 (air) and 1.33 (D2O) were taken. The method used was the same as in [12

12. J. M. Pottage, D. Bird, T. D. Hedley, J. C. Knight, T. A. Birks, P. S. J. Russell, and P. J. Roberts, “Robust photonic band gaps for hollow core guidance in PCF made from high index glass,” Opt. Express 11, 2854–2861 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2854 [CrossRef] [PubMed]

, 13

13. T. D. Hedley, D. M. Bird, F. Benabid, J. C. Knight, and P. St. J. Russell, “Modelling a novel hollow-core photonic crystal fiber,” QLES 2003, QTuL4, Baltimore (2003)

]. The DOS plots for silica/air and silica/D2O are also shown in Fig. 4 (bottom). The initial (silica/air) bandgap lies between 13.8 and 16.9 units of normalized frequency. When air is replaced by D2O the bandgap shifts to the region between 23.1 and 28.9 units of normalized frequency. This agrees well with the experimental transmission spectra of 1550 nm fiber before and after D2O filling respectively (Fig. 2(b)). It was also observed, both in the experiments and in the full-vector calculations, that the shifted bandgaps occurring for smaller index contrasts are wider in frequency than the bandgaps of structures with higher index contrasts. This is due to the following [5

5. T. A. Birks, D. M. Bird, T. D. Hedley, J. M. Pottage, and P. S. J. Russell, “Scaling laws and vector effects in bandgap-guiding fibers,” Opt. Express 12, 69–74 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-69 [CrossRef] [PubMed]

]: the overall bandgap is produced by the overlap of the bandgaps of each of the two states of polarization. For small index contrasts (scalar regime) the two polarization states are degenerate and their bandgaps completely overlap producing wider bandgaps. As the index contrast increases (vector regime), splitting of the two polarization states occur, the bandgap overlap is smaller and so a narrower bandgap is generated.

4. Conclusions

In these experiments the usefulness of the index-scaling laws as, at least, a preliminary tool for the study of photonic bandgap fibers was demonstrated. There was good agreement between the observed transmission bands of the liquid-filled PCF’s and the predictions of both a full-vector model and the scaling law. Discrepancies between experiment and the index scaling laws are mostly due to the fact the latter are strictly valid for the smallest index contrasts and also due to experimental factors, such as contribution from light in the fiber cladding. Some widening of the shifted bandgaps in the low-index-contrast fibers was observed in accordance with theory. These results can be of use in the study of bandgap fibers made from materials of varying index contrast and also in applications of HC-PCF’s that require replacing the air in the fiber with different materials. Moreover, the experiments demonstrated bandgap guidance of light in liquid with many potential applications (e.g. UV transmission, laser guidance of liquid-suspended particles and tunable bandgap fibers using liquid crystals).

Acknowledgments

The authors would like to thank W. Wadsworth for suggesting the heavy water as a suitable filling liquid.

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, “Single-mode photonic band gap guidance of light in air.,” Science 285,1537–1539 (1999) [CrossRef] [PubMed]

2.

F. Benabid, J. C. Knight, G. Antonopoulos, and P. St. J. Russell, “Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,” Science 298, 399–402 (2002) [CrossRef] [PubMed]

3.

F. Benabid, G. Antonopoulos, J. C. Knight, and P. St. J. Russell, “Stokes amplification regimes in quasi-CW pumped hydrogen-filled hollow-core PCF,” Phys. Rev. Lett. 95, 213903–1–4 (2005) [CrossRef]

4.

F. Benabid, J. C. Knight, and P. S. J. Russell, “Particle levitation and guidance in hollow-core photonic crystal fiber,” Opt. Express 10, 1195–1203 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-21-1195 [PubMed]

5.

T. A. Birks, D. M. Bird, T. D. Hedley, J. M. Pottage, and P. S. J. Russell, “Scaling laws and vector effects in bandgap-guiding fibers,” Opt. Express 12, 69–74 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-69 [CrossRef] [PubMed]

6.

G. J. Pearce, J. M. Pottage, D. M. Bird, P. J. Roberts, J. C. Knight, and P. S. Russell, “Hollow-core PCF for guidance in the mid to far infra-red,” Opt. Express 13, 6937–6946 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-18-6937 [CrossRef] [PubMed]

7.

J. Riished, J. Broeng, and A. Bjarklev, “All silica photonic bandgap fiber,” Presented at Conference on Lasers and Electro-Optics,” Paper CTuC5. Baltimore (2003)

8.

R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, and D. J. Trevor, “Tunable photonic bandgap fiber. In Proceedings of Optical Fiber Communication Conference, pages 466–468 (2002)

9.

G. Antonopoulos, F. Benabid, T. A. Birks, D. M. Bird, G. Bouwmans, J. C. Knight, and P. St. J. Russell, “Experimental demonstration of refractive index scaling in photonic bandgap fibers,” In Proceedings of Conference on Lasers and Electro-Optics, Paper CThHH1. San Francisco (2004)

10.

A. C. Tam and K. N. Patel, “Optical absorption of light and heavy water by laser optoacoustic spectroscopy”, App. Opt , 18, 3348–3358 (1979) [CrossRef]

11.

W. J. Wadsworth, N. Joly, J. C. Knight, T. A. Birks, F. Biancalana, and P. S. J. Russell, “Supercontinuum and four-wave mixing with Q-switched pulses in endlessly single-mode photonic crystal fibers,” Opt. Express 12, 299–309 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-2-299 [CrossRef] [PubMed]

12.

J. M. Pottage, D. Bird, T. D. Hedley, J. C. Knight, T. A. Birks, P. S. J. Russell, and P. J. Roberts, “Robust photonic band gaps for hollow core guidance in PCF made from high index glass,” Opt. Express 11, 2854–2861 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2854 [CrossRef] [PubMed]

13.

T. D. Hedley, D. M. Bird, F. Benabid, J. C. Knight, and P. St. J. Russell, “Modelling a novel hollow-core photonic crystal fiber,” QLES 2003, QTuL4, Baltimore (2003)

OCIS Codes
(060.2280) Fiber optics and optical communications : Fiber design and fabrication

ToC Category:
Photonic Crystal Fibers

History
Original Manuscript: January 6, 2006
Revised Manuscript: March 13, 2006
Manuscript Accepted: March 16, 2006
Published: April 3, 2006

Citation
G. Antonopoulos, F. Benabid, T. A. Birks, D. M. Bird, J. C. Knight, and P. St. J. Russell, "Experimental demonstration of the frequency shift of bandgaps in photonic crystal fibers due to refractive index scaling," Opt. Express 14, 3000-3006 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-7-3000


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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, "Single-mode photonic band gap guidance of light in air.," Science 285,1537-1539 (1999) [CrossRef] [PubMed]
  2. F. Benabid, J. C. Knight, G. Antonopoulos, and P. St. J. Russell, "Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber," Science 298, 399-402 (2002) [CrossRef] [PubMed]
  3. F. Benabid, G. Antonopoulos, J. C. Knight, and P. St. J. Russell, "Stokes amplification regimes in quasi-CW pumped hydrogen-filled hollow-core PCF," Phys. Rev. Lett. 95, 21390314 (2005) [CrossRef]
  4. F. Benabid, J. C. Knight, and P. S. J. Russell, "Particle levitation and guidance in hollow-core photonic crystal fiber," Opt. Express 10,1195-1203 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-21-1195 [PubMed]
  5. T. A. Birks, D. M. Bird, T. D. Hedley, J. M. Pottage, and P. S. J. Russell, "Scaling laws and vector effects in bandgap-guiding fibers," Opt. Express 12,69-74 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-69 [CrossRef] [PubMed]
  6. G. J. Pearce, J. M. Pottage, D. M. Bird, P. J. Roberts, J. C. Knight, and P. S. Russell, "Hollow-core PCF for guidance in the mid to far infra-red," Opt. Express 13,6937-6946 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-18-6937 [CrossRef] [PubMed]
  7. J. Riished, J. Broeng, and A. Bjarklev, "All silica photonic bandgap fiber," Presented at Conference on Lasers and Electro-Optics," Paper CTuC5. Baltimore (2003)
  8. R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, and D. J. Trevor, "Tunable photonic bandgap fiber. In Proceedings of Optical Fiber Communication Conference, pages 466-468 (2002)
  9. G. Antonopoulos, F. Benabid, T. A. Birks, D. M. Bird, G. Bouwmans, J. C. Knight, and P. St. J. Russell, "Experimental demonstration of refractive index scaling in photonic bandgap fibers," In Proceedings of Conference on Lasers and Electro-Optics, Paper CThHH1. San Francisco (2004)
  10. A. C. Tam and K. N. Patel, "Optical absorption of light and heavy water by laser optoacoustic spectroscopy", App.Opt,  18, 3348-3358 (1979) [CrossRef]
  11. W. J. Wadsworth, N. Joly, J. C. Knight, T. A. Birks, F. Biancalana, and P. S. J. Russell, "Supercontinuum and four-wave mixing with Q-switched pulses in endlessly single-mode photonic crystal fibers," Opt. Express 12,299-309 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-2-299 [CrossRef] [PubMed]
  12. J. M. Pottage, D. Bird, T. D. Hedley, J. C. Knight, T. A. Birks, P. S. J. Russell, and P. J. Roberts, "Robust photonic band gaps for hollow core guidance in PCF made from high index glass," Opt. Express 11,2854-2861 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2854 [CrossRef] [PubMed]
  13. T. D. Hedley, D. M. Bird, F. Benabid, J. C. Knight, and P. St. J. Russell, "Modelling a novel hollow-core photonic crystal fiber," QLES 2003, QTuL4, Baltimore (2003)

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