Abstract

Measurements of the group-birefringence dispersion of a microstructured optical fiber using the wavelength-scanning technique are reported. Resulting interferograms are processed using the continuous wavelet transform. We discuss application of this approach for determination of birefringence of few-mode fibers and show that with careful analysis it is possible to determine birefringence dispersion of the higher-order modes in optical fibers.

© 2012 OSA

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    [CrossRef]
  8. Y. Ruan, E. P. Schartner, H. Ebendorff-Heidepriem, P. Hoffmann, and T. M. Monro, “Detection of quantum-dot labelled proteins using soft glass microstructured optical fibers,” Opt. Express15(26), 17819–17826 (2007).
    [CrossRef] [PubMed]
  9. M. Grabka, B. Wajnchold, S. Pustelny, W. Gawlik, K. Skorupski, and P. Mergo, “Experimental and theoretical study of light propagation in suspended-core optical fiber,” Acta Phys. Pol. A118, 1127–1132 (2010).
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [PubMed]
  18. A. Ortigosa-Blanch, J. C. Knight, W. J. Wadsworth, J. Arriaga, B. J. Mangan, T. A. Birks, and P. St. J. Russell, “Highly birefringent photonic crystal fibers,” Opt. Lett.25(18), 1325–1327 (2000).
    [CrossRef] [PubMed]
  19. J. Wojcik, P. Mergo, M. Makara, K. Poturaj, L. Czyzewska, J. Klimek, and A. Walewski, “Technology of suspended core microstructured optical fibers for evanescent wave and plasmon resonance optical fiber sensor,” Proc. SPIE6990, 6990T (2008).
  20. C. S. Burrus, R. A. Gopinath, and H. Guo, Introduction to Wavelets and Wavelet Transforms: A Primer, (Prentice Hall, 1997)
  21. O. Koysal, S. E. San, S. Ozder, and F. N. Ecevit “A novel approach for the determination of birefringence dispersion in nematic liquid crystals by using the continuous wavelet transform,” Meas. Sci. Technol.14(6), 790–795 (2003).
    [CrossRef]

2010

T. M. Monro, S. Warren-Smith, E. P. Schartner, A. Francois, S. Heng, H. Ebendorff-Heidepriem, and V. S. Afshar, “Sensing with suspended-core optical fibers,” Opt. Fiber Technol.16(6), 343–356 (2010).
[CrossRef]

M. Grabka, B. Wajnchold, S. Pustelny, W. Gawlik, K. Skorupski, and P. Mergo, “Experimental and theoretical study of light propagation in suspended-core optical fiber,” Acta Phys. Pol. A118, 1127–1132 (2010).

2009

P. Hlubina, D. Ciprian, and M. Kadulova, “Wide spectral range measurement of modal birefringence in polarization-maintaining fibres,” Meas. Sci. Technol.20(2), 025301 (2009).
[CrossRef]

2008

J. Wojcik, P. Mergo, M. Makara, K. Poturaj, L. Czyzewska, J. Klimek, and A. Walewski, “Technology of suspended core microstructured optical fibers for evanescent wave and plasmon resonance optical fiber sensor,” Proc. SPIE6990, 6990T (2008).

L. Fu, B. K. Thomas, and L. Dong, “Efficient supercontinuum generations in silica suspended core fibers,” Opt. Express16(24), 19629–19642 (2008).
[CrossRef] [PubMed]

2007

Y. Ruan, E. P. Schartner, H. Ebendorff-Heidepriem, P. Hoffmann, and T. M. Monro, “Detection of quantum-dot labelled proteins using soft glass microstructured optical fibers,” Opt. Express15(26), 17819–17826 (2007).
[CrossRef] [PubMed]

T. Martynkien, M. Szpulak, G. Statkiewicz, G. Golojuch, J. Olszewski, W. Urbanczyk, J. Wojcik, P. Mergo, M. Makara, T. Nasilowski, F. Berghmans, and H. Thienpont, “Measurements of sensitivity to hydrostatic pressure and temperature in highly birefringent photonic crystal fibers,” Opt. Quantum Electron.39(4-6), 481–489 (2007).
[CrossRef]

2006

2005

2003

2001

2000

1998

P. Hlubina, “Measuring dispersion between modes of an optical fibre using time-domain and spectral-domain low-coherence interferometry,” J. Mod. Opt.45(9), 1767–1774 (1998).
[CrossRef]

1983

1979

S. R. Norman, D. N. Payne, M. J. Adams, and A. M. Smith, “Fabrication of single-mode fibers exhibiting extremely low polarization birefringence,” Electron. Lett.15(11), 309–311 (1979).
[CrossRef]

Adams, M. J.

S. R. Norman, D. N. Payne, M. J. Adams, and A. M. Smith, “Fabrication of single-mode fibers exhibiting extremely low polarization birefringence,” Electron. Lett.15(11), 309–311 (1979).
[CrossRef]

Afshar, V. S.

T. M. Monro, S. Warren-Smith, E. P. Schartner, A. Francois, S. Heng, H. Ebendorff-Heidepriem, and V. S. Afshar, “Sensing with suspended-core optical fibers,” Opt. Fiber Technol.16(6), 343–356 (2010).
[CrossRef]

Allott, R.

Andres, M.

Andres, P.

Arriaga, J.

Berghmans, F.

T. Martynkien, M. Szpulak, G. Statkiewicz, G. Golojuch, J. Olszewski, W. Urbanczyk, J. Wojcik, P. Mergo, M. Makara, T. Nasilowski, F. Berghmans, and H. Thienpont, “Measurements of sensitivity to hydrostatic pressure and temperature in highly birefringent photonic crystal fibers,” Opt. Quantum Electron.39(4-6), 481–489 (2007).
[CrossRef]

Birks, T. A.

Botten, L. C.

Bouwmans, G.

Ciprian, D.

P. Hlubina, D. Ciprian, and M. Kadulova, “Wide spectral range measurement of modal birefringence in polarization-maintaining fibres,” Meas. Sci. Technol.20(2), 025301 (2009).
[CrossRef]

Czyzewska, L.

J. Wojcik, P. Mergo, M. Makara, K. Poturaj, L. Czyzewska, J. Klimek, and A. Walewski, “Technology of suspended core microstructured optical fibers for evanescent wave and plasmon resonance optical fiber sensor,” Proc. SPIE6990, 6990T (2008).

Dong, L.

Ebendorff-Heidepriem, H.

T. M. Monro, S. Warren-Smith, E. P. Schartner, A. Francois, S. Heng, H. Ebendorff-Heidepriem, and V. S. Afshar, “Sensing with suspended-core optical fibers,” Opt. Fiber Technol.16(6), 343–356 (2010).
[CrossRef]

Y. Ruan, E. P. Schartner, H. Ebendorff-Heidepriem, P. Hoffmann, and T. M. Monro, “Detection of quantum-dot labelled proteins using soft glass microstructured optical fibers,” Opt. Express15(26), 17819–17826 (2007).
[CrossRef] [PubMed]

Ecevit, F. N.

O. Koysal, S. E. San, S. Ozder, and F. N. Ecevit “A novel approach for the determination of birefringence dispersion in nematic liquid crystals by using the continuous wavelet transform,” Meas. Sci. Technol.14(6), 790–795 (2003).
[CrossRef]

Ferrando, A.

Flanagan, J. C.

Francois, A.

T. M. Monro, S. Warren-Smith, E. P. Schartner, A. Francois, S. Heng, H. Ebendorff-Heidepriem, and V. S. Afshar, “Sensing with suspended-core optical fibers,” Opt. Fiber Technol.16(6), 343–356 (2010).
[CrossRef]

Fu, L.

Gawlik, W.

M. Grabka, B. Wajnchold, S. Pustelny, W. Gawlik, K. Skorupski, and P. Mergo, “Experimental and theoretical study of light propagation in suspended-core optical fiber,” Acta Phys. Pol. A118, 1127–1132 (2010).

Golojuch, G.

T. Martynkien, M. Szpulak, G. Statkiewicz, G. Golojuch, J. Olszewski, W. Urbanczyk, J. Wojcik, P. Mergo, M. Makara, T. Nasilowski, F. Berghmans, and H. Thienpont, “Measurements of sensitivity to hydrostatic pressure and temperature in highly birefringent photonic crystal fibers,” Opt. Quantum Electron.39(4-6), 481–489 (2007).
[CrossRef]

Grabka, M.

M. Grabka, B. Wajnchold, S. Pustelny, W. Gawlik, K. Skorupski, and P. Mergo, “Experimental and theoretical study of light propagation in suspended-core optical fiber,” Acta Phys. Pol. A118, 1127–1132 (2010).

Grunewald, P.

Hayes, J. R.

Heng, S.

T. M. Monro, S. Warren-Smith, E. P. Schartner, A. Francois, S. Heng, H. Ebendorff-Heidepriem, and V. S. Afshar, “Sensing with suspended-core optical fibers,” Opt. Fiber Technol.16(6), 343–356 (2010).
[CrossRef]

Hlubina, P.

P. Hlubina, D. Ciprian, and M. Kadulova, “Wide spectral range measurement of modal birefringence in polarization-maintaining fibres,” Meas. Sci. Technol.20(2), 025301 (2009).
[CrossRef]

P. Hlubina, T. Martynkien, and W. Urbańczyk, “Dispersion of group and phase modal birefringence in elliptical-core fiber measured by white-light spectral interferometry,” Opt. Express11(22), 2793–2798 (2003).
[PubMed]

P. Hlubina, “Measuring dispersion between modes of an optical fibre using time-domain and spectral-domain low-coherence interferometry,” J. Mod. Opt.45(9), 1767–1774 (1998).
[CrossRef]

Hoffmann, P.

Kadulova, M.

P. Hlubina, D. Ciprian, and M. Kadulova, “Wide spectral range measurement of modal birefringence in polarization-maintaining fibres,” Meas. Sci. Technol.20(2), 025301 (2009).
[CrossRef]

Klimek, J.

J. Wojcik, P. Mergo, M. Makara, K. Poturaj, L. Czyzewska, J. Klimek, and A. Walewski, “Technology of suspended core microstructured optical fibers for evanescent wave and plasmon resonance optical fiber sensor,” Proc. SPIE6990, 6990T (2008).

Knight, J. C.

Koshiba, M.

Koysal, O.

O. Koysal, S. E. San, S. Ozder, and F. N. Ecevit “A novel approach for the determination of birefringence dispersion in nematic liquid crystals by using the continuous wavelet transform,” Meas. Sci. Technol.14(6), 790–795 (2003).
[CrossRef]

Makara, M.

J. Wojcik, P. Mergo, M. Makara, K. Poturaj, L. Czyzewska, J. Klimek, and A. Walewski, “Technology of suspended core microstructured optical fibers for evanescent wave and plasmon resonance optical fiber sensor,” Proc. SPIE6990, 6990T (2008).

T. Martynkien, M. Szpulak, G. Statkiewicz, G. Golojuch, J. Olszewski, W. Urbanczyk, J. Wojcik, P. Mergo, M. Makara, T. Nasilowski, F. Berghmans, and H. Thienpont, “Measurements of sensitivity to hydrostatic pressure and temperature in highly birefringent photonic crystal fibers,” Opt. Quantum Electron.39(4-6), 481–489 (2007).
[CrossRef]

Mangan, B. J.

Martijn de Sterke, C.

Martynkien, T.

T. Martynkien, M. Szpulak, G. Statkiewicz, G. Golojuch, J. Olszewski, W. Urbanczyk, J. Wojcik, P. Mergo, M. Makara, T. Nasilowski, F. Berghmans, and H. Thienpont, “Measurements of sensitivity to hydrostatic pressure and temperature in highly birefringent photonic crystal fibers,” Opt. Quantum Electron.39(4-6), 481–489 (2007).
[CrossRef]

P. Hlubina, T. Martynkien, and W. Urbańczyk, “Dispersion of group and phase modal birefringence in elliptical-core fiber measured by white-light spectral interferometry,” Opt. Express11(22), 2793–2798 (2003).
[PubMed]

McPhedran, R. C.

Mergo, P.

M. Grabka, B. Wajnchold, S. Pustelny, W. Gawlik, K. Skorupski, and P. Mergo, “Experimental and theoretical study of light propagation in suspended-core optical fiber,” Acta Phys. Pol. A118, 1127–1132 (2010).

J. Wojcik, P. Mergo, M. Makara, K. Poturaj, L. Czyzewska, J. Klimek, and A. Walewski, “Technology of suspended core microstructured optical fibers for evanescent wave and plasmon resonance optical fiber sensor,” Proc. SPIE6990, 6990T (2008).

T. Martynkien, M. Szpulak, G. Statkiewicz, G. Golojuch, J. Olszewski, W. Urbanczyk, J. Wojcik, P. Mergo, M. Makara, T. Nasilowski, F. Berghmans, and H. Thienpont, “Measurements of sensitivity to hydrostatic pressure and temperature in highly birefringent photonic crystal fibers,” Opt. Quantum Electron.39(4-6), 481–489 (2007).
[CrossRef]

Miret, J.

Monro, T. M.

Mortensen, N. A.

Nasilowski, T.

T. Martynkien, M. Szpulak, G. Statkiewicz, G. Golojuch, J. Olszewski, W. Urbanczyk, J. Wojcik, P. Mergo, M. Makara, T. Nasilowski, F. Berghmans, and H. Thienpont, “Measurements of sensitivity to hydrostatic pressure and temperature in highly birefringent photonic crystal fibers,” Opt. Quantum Electron.39(4-6), 481–489 (2007).
[CrossRef]

Norman, S. R.

S. R. Norman, D. N. Payne, M. J. Adams, and A. M. Smith, “Fabrication of single-mode fibers exhibiting extremely low polarization birefringence,” Electron. Lett.15(11), 309–311 (1979).
[CrossRef]

Olszewski, J.

T. Martynkien, M. Szpulak, G. Statkiewicz, G. Golojuch, J. Olszewski, W. Urbanczyk, J. Wojcik, P. Mergo, M. Makara, T. Nasilowski, F. Berghmans, and H. Thienpont, “Measurements of sensitivity to hydrostatic pressure and temperature in highly birefringent photonic crystal fibers,” Opt. Quantum Electron.39(4-6), 481–489 (2007).
[CrossRef]

Ortigosa-Blanch, A.

Ozder, S.

O. Koysal, S. E. San, S. Ozder, and F. N. Ecevit “A novel approach for the determination of birefringence dispersion in nematic liquid crystals by using the continuous wavelet transform,” Meas. Sci. Technol.14(6), 790–795 (2003).
[CrossRef]

Payne, D. N.

S. R. Norman, D. N. Payne, M. J. Adams, and A. M. Smith, “Fabrication of single-mode fibers exhibiting extremely low polarization birefringence,” Electron. Lett.15(11), 309–311 (1979).
[CrossRef]

Percival, R.

Poturaj, K.

J. Wojcik, P. Mergo, M. Makara, K. Poturaj, L. Czyzewska, J. Klimek, and A. Walewski, “Technology of suspended core microstructured optical fibers for evanescent wave and plasmon resonance optical fiber sensor,” Proc. SPIE6990, 6990T (2008).

Pustelny, S.

M. Grabka, B. Wajnchold, S. Pustelny, W. Gawlik, K. Skorupski, and P. Mergo, “Experimental and theoretical study of light propagation in suspended-core optical fiber,” Acta Phys. Pol. A118, 1127–1132 (2010).

Rashleigh, S. C.

Richardson, D. J.

Ruan, Y.

Russell, P. St. J.

Saitoh, K.

San, S. E.

O. Koysal, S. E. San, S. Ozder, and F. N. Ecevit “A novel approach for the determination of birefringence dispersion in nematic liquid crystals by using the continuous wavelet transform,” Meas. Sci. Technol.14(6), 790–795 (2003).
[CrossRef]

Schartner, E. P.

T. M. Monro, S. Warren-Smith, E. P. Schartner, A. Francois, S. Heng, H. Ebendorff-Heidepriem, and V. S. Afshar, “Sensing with suspended-core optical fibers,” Opt. Fiber Technol.16(6), 343–356 (2010).
[CrossRef]

Y. Ruan, E. P. Schartner, H. Ebendorff-Heidepriem, P. Hoffmann, and T. M. Monro, “Detection of quantum-dot labelled proteins using soft glass microstructured optical fibers,” Opt. Express15(26), 17819–17826 (2007).
[CrossRef] [PubMed]

Silvestre, E.

Skorupski, K.

M. Grabka, B. Wajnchold, S. Pustelny, W. Gawlik, K. Skorupski, and P. Mergo, “Experimental and theoretical study of light propagation in suspended-core optical fiber,” Acta Phys. Pol. A118, 1127–1132 (2010).

Smith, A. M.

S. R. Norman, D. N. Payne, M. J. Adams, and A. M. Smith, “Fabrication of single-mode fibers exhibiting extremely low polarization birefringence,” Electron. Lett.15(11), 309–311 (1979).
[CrossRef]

Statkiewicz, G.

T. Martynkien, M. Szpulak, G. Statkiewicz, G. Golojuch, J. Olszewski, W. Urbanczyk, J. Wojcik, P. Mergo, M. Makara, T. Nasilowski, F. Berghmans, and H. Thienpont, “Measurements of sensitivity to hydrostatic pressure and temperature in highly birefringent photonic crystal fibers,” Opt. Quantum Electron.39(4-6), 481–489 (2007).
[CrossRef]

Steel, M. J.

Szpulak, M.

T. Martynkien, M. Szpulak, G. Statkiewicz, G. Golojuch, J. Olszewski, W. Urbanczyk, J. Wojcik, P. Mergo, M. Makara, T. Nasilowski, F. Berghmans, and H. Thienpont, “Measurements of sensitivity to hydrostatic pressure and temperature in highly birefringent photonic crystal fibers,” Opt. Quantum Electron.39(4-6), 481–489 (2007).
[CrossRef]

Thienpont, H.

T. Martynkien, M. Szpulak, G. Statkiewicz, G. Golojuch, J. Olszewski, W. Urbanczyk, J. Wojcik, P. Mergo, M. Makara, T. Nasilowski, F. Berghmans, and H. Thienpont, “Measurements of sensitivity to hydrostatic pressure and temperature in highly birefringent photonic crystal fibers,” Opt. Quantum Electron.39(4-6), 481–489 (2007).
[CrossRef]

Thomas, B. K.

Tsuchida, Y.

Urbanczyk, W.

T. Martynkien, M. Szpulak, G. Statkiewicz, G. Golojuch, J. Olszewski, W. Urbanczyk, J. Wojcik, P. Mergo, M. Makara, T. Nasilowski, F. Berghmans, and H. Thienpont, “Measurements of sensitivity to hydrostatic pressure and temperature in highly birefringent photonic crystal fibers,” Opt. Quantum Electron.39(4-6), 481–489 (2007).
[CrossRef]

P. Hlubina, T. Martynkien, and W. Urbańczyk, “Dispersion of group and phase modal birefringence in elliptical-core fiber measured by white-light spectral interferometry,” Opt. Express11(22), 2793–2798 (2003).
[PubMed]

Wadsworth, W. J.

Wajnchold, B.

M. Grabka, B. Wajnchold, S. Pustelny, W. Gawlik, K. Skorupski, and P. Mergo, “Experimental and theoretical study of light propagation in suspended-core optical fiber,” Acta Phys. Pol. A118, 1127–1132 (2010).

Walewski, A.

J. Wojcik, P. Mergo, M. Makara, K. Poturaj, L. Czyzewska, J. Klimek, and A. Walewski, “Technology of suspended core microstructured optical fibers for evanescent wave and plasmon resonance optical fiber sensor,” Proc. SPIE6990, 6990T (2008).

Warren-Smith, S.

T. M. Monro, S. Warren-Smith, E. P. Schartner, A. Francois, S. Heng, H. Ebendorff-Heidepriem, and V. S. Afshar, “Sensing with suspended-core optical fibers,” Opt. Fiber Technol.16(6), 343–356 (2010).
[CrossRef]

White, T. P.

Wojcik, J.

J. Wojcik, P. Mergo, M. Makara, K. Poturaj, L. Czyzewska, J. Klimek, and A. Walewski, “Technology of suspended core microstructured optical fibers for evanescent wave and plasmon resonance optical fiber sensor,” Proc. SPIE6990, 6990T (2008).

T. Martynkien, M. Szpulak, G. Statkiewicz, G. Golojuch, J. Olszewski, W. Urbanczyk, J. Wojcik, P. Mergo, M. Makara, T. Nasilowski, F. Berghmans, and H. Thienpont, “Measurements of sensitivity to hydrostatic pressure and temperature in highly birefringent photonic crystal fibers,” Opt. Quantum Electron.39(4-6), 481–489 (2007).
[CrossRef]

Acta Phys. Pol. A

M. Grabka, B. Wajnchold, S. Pustelny, W. Gawlik, K. Skorupski, and P. Mergo, “Experimental and theoretical study of light propagation in suspended-core optical fiber,” Acta Phys. Pol. A118, 1127–1132 (2010).

Electron. Lett.

S. R. Norman, D. N. Payne, M. J. Adams, and A. M. Smith, “Fabrication of single-mode fibers exhibiting extremely low polarization birefringence,” Electron. Lett.15(11), 309–311 (1979).
[CrossRef]

J. Mod. Opt.

P. Hlubina, “Measuring dispersion between modes of an optical fibre using time-domain and spectral-domain low-coherence interferometry,” J. Mod. Opt.45(9), 1767–1774 (1998).
[CrossRef]

Meas. Sci. Technol.

P. Hlubina, D. Ciprian, and M. Kadulova, “Wide spectral range measurement of modal birefringence in polarization-maintaining fibres,” Meas. Sci. Technol.20(2), 025301 (2009).
[CrossRef]

O. Koysal, S. E. San, S. Ozder, and F. N. Ecevit “A novel approach for the determination of birefringence dispersion in nematic liquid crystals by using the continuous wavelet transform,” Meas. Sci. Technol.14(6), 790–795 (2003).
[CrossRef]

Opt. Express

Opt. Fiber Technol.

T. M. Monro, S. Warren-Smith, E. P. Schartner, A. Francois, S. Heng, H. Ebendorff-Heidepriem, and V. S. Afshar, “Sensing with suspended-core optical fibers,” Opt. Fiber Technol.16(6), 343–356 (2010).
[CrossRef]

Opt. Lett.

Opt. Quantum Electron.

T. Martynkien, M. Szpulak, G. Statkiewicz, G. Golojuch, J. Olszewski, W. Urbanczyk, J. Wojcik, P. Mergo, M. Makara, T. Nasilowski, F. Berghmans, and H. Thienpont, “Measurements of sensitivity to hydrostatic pressure and temperature in highly birefringent photonic crystal fibers,” Opt. Quantum Electron.39(4-6), 481–489 (2007).
[CrossRef]

Proc. SPIE

J. Wojcik, P. Mergo, M. Makara, K. Poturaj, L. Czyzewska, J. Klimek, and A. Walewski, “Technology of suspended core microstructured optical fibers for evanescent wave and plasmon resonance optical fiber sensor,” Proc. SPIE6990, 6990T (2008).

Other

C. S. Burrus, R. A. Gopinath, and H. Guo, Introduction to Wavelets and Wavelet Transforms: A Primer, (Prentice Hall, 1997)

A. Kumar and A. Ghatak, Polarization of Light with Applications in Optical Fibers (SPIE Press, 2011)

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Figures (6)

Fig. 1
Fig. 1

Experimental setup for birefringence measurements using the wavelength scanning method.

Fig. 2
Fig. 2

Scanning electron microscope images of the cross section of the examined fiber: a) whole fiber (80 μm diameter), b) core region (800 nm diameter).

Fig. 3
Fig. 3

Typical transmission spectrum obtained for a 290 mm long piece of a suspended-core fiber.

Fig. 4
Fig. 4

(a) Real part of the wavelet spectrum of the signal shown in Fig. 3. Three distinct visible traces are marked as A, B, C, (b) FFT spectrum of the same signal. Besides of a large zero-frequency component, only the primary birefringence mode is visible (at frequency close to 0.04 nm−1). Other features relevant for CWT trances B and C should be located in the 0.1-0.5 nm−1 frequency range, but are not visible.

Fig. 5
Fig. 5

(a) Local extremes of the real part of the wavelet spectrum without filtering, (b) Signal filtered out using the criteria described in the text. Each point of the plots is related to an extremum in the CWT spectrum in Fig. 4(a) and defines a pair (λ, Δλ). The pairs group into bent lines which reflect the systematic wavelength dependence of the group birefringence of the resolved modes

Fig. 6
Fig. 6

Group birefringence values collected from measurements done for different arrangement of polarizers and different fiber lengths. Dashed lines limit the birefringence range accessible in measurements using our spectrometer.

Equations (3)

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G(λ)= λ 2 LΔλ ,
CWT(a,b)= 1 a + ϕ( xb a )f(x)dx.
ϕ(x)= 1 π 4 exp( i z 0 x x 2 /2 ),

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