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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 20 — Sep. 26, 2011
  • pp: 19582–19590
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Fiber based polarization filter for radially and azimuthally polarized light

Christoph Jocher, Cesar Jauregui, Christian Voigtländer, Fabian Stutzki, Stefan Nolte, Jens Limpert, and Andreas Tünnermann  »View Author Affiliations


Optics Express, Vol. 19, Issue 20, pp. 19582-19590 (2011)
http://dx.doi.org/10.1364/OE.19.019582


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Abstract

We demonstrate a new fiber based concept to filter azimuthally or radially polarized light. This concept is based on the lifting of the modal degeneracy that takes place in high numerical aperture fibers. In such fibers, the radially and azimuthally polarized modes can be spectrally separated using a fiber Bragg grating. As a proof of principle, we filter azimuthally polarized light in a commercially available fiber in which a fiber Bragg grating has been written by a femtosecond pulsed laser.

© 2011 OSA

1. Introduction

Azimuthally and radially polarized beams have attracted significant attention recently. The reasons for this interest are to be found in the wide application field of these polarizations, which include efficient material processing [1

1. V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D Appl. Phys. 32(13), 1455–1461 (1999). [CrossRef]

], microscopy [2

2. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001). [CrossRef] [PubMed]

], excitation of plasmons [3

3. Q. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance excitation by a radially polarized beam,” Opt. Lett. 31(11), 1726–1728 (2006). [CrossRef] [PubMed]

], optical trapping [4

4. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004). [CrossRef] [PubMed]

] and electron acceleration [5

5. W. D. Kimura, G. H. Kim, R. D. Romea, L. C. Steinhauer, I. V. Pogorelsky, K. P. Kusche, R. C. Fernow, X. Wang, and Y. Liu, “Laser acceleration of relativistic electrons using the inverse Cherenkov effect,” Phys. Rev. Lett. 74(4), 546–549 (1995). [CrossRef] [PubMed]

]. A stable source with high polarization purity is required for these applications. The common approach to generate azimuthally and radially polarized beams is based on converters, which transform a linearly polarized beam in these polarization states. Three converter types are often used: liquid-crystal-based polarization converters [6

6. M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21(23), 1948–1950 (1996). [CrossRef] [PubMed]

], segmented half-wave plates [7

7. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef] [PubMed]

], and interferometric setups [8

8. S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. 29(15), 2234–2239 (1990). [CrossRef] [PubMed]

]. The disadvantage of these methods is the sophisticated experimental setup with their free-space beam propagation that makes them sensitive to misalignments which negatively affect the polarization purity of the converted light.

2. Theory

2.1 Strongly guiding optical fibers

An important characteristic of the weakly guiding fibers which have been preferentially used in the last decade is that many HOMs are degenerated (i.e. they share the same intensity profile and have nearly identical effective indices) [14

14. A. W. Snyder, “Asymptotic expressions for eigenfunctions and eigenvalues of a dielectric or optical waveguide,” IEEE Trans. Microw. Theory Tech. 17(12), 1130–1138 (1969). [CrossRef]

]. In practice this means that they cannot be individually excited with conventional means (or only with extreme difficulty), and only a degenerated mode formed by the overlap of several HOMs will be observed. As shown in [15

15. D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10(10), 2252–2258 (1971). [CrossRef] [PubMed]

] these degenerated modes are, in a first approximation, linearly polarized. Because of this characteristic, i.e. their state of polarization, the modes of weakly guiding fibers are called linearly polarized (LP)-Modes [15

15. D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10(10), 2252–2258 (1971). [CrossRef] [PubMed]

]. However, it can be demonstrated that cylindrical waveguides support the propagation of other polarization states in HOMs, such as the radially and azimuthally polarized modes [16

16. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Kluwer Academic, 1983).

]. Thus, in cylindrical optical fibers one group of interesting HOMs sharing a similar intensity profile are the TE01, TM01 and two HE21 modes. The TE01 and the TM01 being particularly interesting, because they intrinsically show outstanding polarization characteristics: they are azimuthally (TE01) and radially (TM01) polarized, respectively, as seen in Fig. 1
Fig. 1 Intensity patterns together with the intrinsic polarization characteristics for the TE01, TM01 and two HE21 modes. In weakly guiding fibers this set of modes is degenerated giving rise to the LP11-modes, which are shown on the right-hand side.
. However, if the optical fiber has a low NA, these interesting modes are guided with nearly the same effective refractive index, i.e. they are degenerated. Consequently, as mentioned before they cannot be individually excited using conventional techniques and, additionally, mode coupling can easily occur at any imperfection and, as a result, the observed mode is predominantly linearly polarized and not radially or azimuthally any more.

The differences in effective refractive index between the TE01, TM01 and the two HE21 modes in a strongly guiding step index fiber when varying the NA (from 0.35 to 0.65) and the core size (radius ranging from 0.5 to 4 µm) have been numerically calculated for a wavelength of 1030 nm. Figure 2 (a)
Fig. 2 Effective index difference (a) between the TE01 and its nearest neighbor mode and (b) between the TM01 and the two HE21 modes in a strongly guiding step index fiber as a function of the core radius and the NA, shown in logarithmic scale. The wavelength is 1030 nm.
shows the effective index difference between the azimuthally polarized TE01 mode and its nearest neighbor in logarithmic scale (base 10). It should be noted that the “nearest neighbor” denomination can refer to different modes, as in some cases the TM01 has a higher effective refractive index than the two quasi-degenerated HE21 modes and vice-versa. In Fig. 2 (b), on the other hand, the absolute value of the effective index difference between the TM01 and the two almost degenerated HE21 modes is shown. It should be mentioned that this difference is negative for fibers operating near the single mode region (represented by the white areas in Fig. 2), which implies that the TM01 mode has a higher effective index than the HE21 modes. Additionally, for some combinations of NA and core size the TM01 mode becomes degenerated with the two HE21 modes, a situation that is indicated by a zero distance (dark blue stripe) between these modes in Fig. 2 (b).

The calculation also shows that the highest effective index separation for any given NA is not reached for the smallest core size. Additionally, if azimuthally and radially polarized beams should be filtered using the same fiber, this fiber should lift the polarization degeneracy in such a degree that these two modes exhibit a large separation in effective index to one another and to the other modes. Thus, the optimum fiber core radius can be read from Fig. 2 for any given NA.

From the figure above it is clear that in order to filter an azimuthally and/or radially polarized beam a fiber with a higher NA is more suitable than a fiber with a lower NA. Furthermore, smaller cores are normally advantageous (provided that they are not too small). Such high NA small core fibers are commercially available since they are widely used for enhancing non-linear effects, e.g. for supercontinuum generation.

2.2 Filtering of radially and/or azimuthally polarized light using fiber Bragg gratings in strongly guiding fibers

A fiber Bragg grating (FBG) is a periodic or quasi-periodic modulation of the refractive index along the fiber. If the modulation period is Λ, a fiber mode with an effective refractive index neff and a wavelength λ will be reflected by the FBG in the diffraction order m, if the Bragg condition (1) is fulfilled [17

17. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997). [CrossRef]

]:

λ=2neffΛm.
(1)

Thus, the spectral separation Δλ between the grating resonances corresponding to two transversal modes with effective refractive index neff,1 and neff,2 can be calculated by Eq. (2). This way, the FBG maps the effective index differences of all propagating modes into the spectrum:

Δλ=2(neff,1neff,2)Λm.
(2)

Additionally, if two modes should be clearly separated, the reflection bandwidth of the FBG has to be smaller than the spectral separation between two adjacent grating resonances. Otherwise, if the reflection bandwidth is too large, the spectral wings of one grating resonance will overlap with the neighboring one and, therefore, the polarization purity of the reflected light will be reduced. In fact, the ideal situation is that the reflection bandwidth of the FBG is narrower than the spectral separation between the modes of interest. Thus, each mode is reflected at its own wavelength and they do not overlap. In this situation a radially or azimuthally polarized beam can be obtained by addressing the FBG with the appropriate wavelength (assuming that the corresponding mode has been excited at this wavelength). Of course the drawback is that the radially or azimuthally polarized modes have to exist in the fiber first in order to be reflected by the homogenous FBG. However, in a passive fiber this can be easily achieved by playing with the coupling optics (in free space setups) or by using, for example, offset splices. The expected stability and polarization purity offered by this approach is high, because only one mode is reflected at each wavelength.

One additional advantage of this FBG-based approach to filter radially or azimuthally polarized light is that the Bragg condition allows converting one mode into another at some specific wavelengths. However, in order to do this an inhomogeneous FBG is required [18

18. J. U. Thomas, C. Voigtländer, S. Nolte, A. Tünnermann, N. Jovanovic, G. D. Marshall, M. J. Withford, and M. Steel, “Mode selective fibre Bragg grating,” Proc. SPIE 7589, 75890J, 75890J-9 (2010). [CrossRef]

]. For example the fundamental mode HE11 can be converted into a radially or azimuthally polarized mode. If the complete reflection spectrum is considered, the conversion peaks are spectrally equidistant to the resonances of the fundamental mode and to the resonance of their corresponding counterpart TE01, TM01 or HE21. However, this mode conversion is a two-way road, which implies that any radially or azimuthally polarized mode incident on the FBG will be converted into a linearly polarized fundamental mode. Therefore, for an efficient conversion it has to be ensured that only the fundamental mode is excited in the incoming direction.

3. Experiment

3.1 Experimental setup

For the proof of principle of this technique we use a commercially available step index high NA fiber (Nufern UHNA7). The cut-off wavelength of this fiber is 1470 nm and its core radius is ~1.5 µm. However, the most important parameter of this fiber is its very high NA, with a nominal value of 0.41. Thus, according to Fig. 2 or Fig. 3, it can be expected that in this fiber the effective index difference between the azimuthally polarized mode and its closest neighbor is 5.4x10−4 at a wavelength of 1030 nm. On the other hand, for the radially polarized mode this effective index difference is 8x10−6. The fiber grating (FBG) was written using a femtosecond laser and the phase mask technique described in [19

19. J. Thomas, E. Wikszak, T. Clausnitzer, U. Fuchs, U. Zeitner, S. Nolte, and A. Tünnermann, “Inscription of fiber Bragg gratings with femtosecond pulses using a phase mask scanning technique,” Appl. Phys., A Mater. Sci. Process. 86(2), 153–157 (2006). [CrossRef]

]. The FBG was 15 mm long with a uniform profile and had a photo-induced index change of around 1x10−3 (peak-to-peak). During the fabrication process the transmission spectrum of the fundamental mode was measured to obtain an estimation of the grating reflectivity, which was found to be ~23 dB. Taking into account that the period is 712.5 nm and in the setup its second diffraction order (m = 2 in Eq. (1)) is used, we can expect a spectral separation of 400 pm for the azimuthally polarized mode and 6 pm for the radially polarized one. As can be seen, and also as previously discussed, in this fiber it is not possible to fully resolve the radially polarized mode in this wavelength range since the bandwidth of the FBG is ~150 pm. As a proof of principle, though, it is still possible to filter the azimuthally polarized mode.

The experimental setup is illustrated in Fig. 4
Fig. 4 Experimental setup used for the proof-of-principle filtering of an azimuthally polarized beam. PC: polarizing cube, BS: beam splitter.
. A home-made unpolarized single mode ASE source is used as the light source. It is coupled into the high-NA fiber using a polarization independent 50% dichroic beam splitter. In order to excite the azimuthally polarized beam, the focusing lens is purposefully misaligned. Two etalons selectively filter the ASE source leaving just a 0.2 nm bandwidth range around the desired resonance wavelength of the FBG. The beam reflected by the FBG is analyzed after the beam splitter, where a second dichroic band-pass filter with a transmission band of 10 nm is used to filter out spurious wavelengths coming from the spectrally periodic transfer function of the etalons. The polarization is analyzed by using a polarization cube, a rotating half-wave plate and imaging the near-field on a camera.

3.2 Experimental results

With the above described diagnostic setup, the reflection peaks coming from the FBG are individually spectrally filtered and analyzed. The modes TE01 and TM01 (and eventually even the HE21) can be clearly identified using a camera and a polarizer. However, in discerning between modes, the transmission direction of the polarizer is of capital importance. The two lobes shown in Fig. 6
Fig. 6 Theoretical azimuthally polarized beam (above) together with the corresponding measurement (below) at 1045.2 nm. (a) Intensity distribution without PC and half-wave plate. (b-e) Polarization filtered beam with the PC and half-wave plate at different transmission directions as indicated by the arrow. A video showing the rotation of the polarization filtered beam with the half-wave plate is presented in Media 1.
appear for each of these modes after going through the polarizer, but for a TE01 the zero-intensity line separating the lobes should be oriented parallel to the transmission direction of the polarizer (transmission direction indicated by the white arrows in Fig. 6). On the other hand, for a TM01 that zero-intensity line should be oriented orthogonal to the transmission direction of the polarization cube. Additionally, if the half-wave plate is turned, the rotation direction of the two lobes helps to distinguish between the TE01/TM01 (they rotate in the direction of the half-wave plate) modes and the HE21 modes (they rotate in the opposite direction of the half-wave plate). In the paper the measured beam shows all the characteristics of a TE01 mode. Figure 6 shows the theoretically expected azimuthally polarized beam for this fiber (above) together with the experimental measurement at 1045.2 nm (below). The column (a) in Fig. 6 illustrates the full intensity pattern of the azimuthally polarized mode. Without any polarization analysis, it exhibits the characteristic doughnut shape. In Fig. 6 column (b-e) the results of the polarization analysis with different transmission directions of the polarizer (indicated by the white arrows) is shown in false-color. It should be pointed out that all the false-color pictures have been normalized to the maximum intensity of column (a). A video obtained by rotating the half-wave plate can also be seen in Media 1. It is worth mentioning that in this setup, due to the mentioned birefringence of the FBG, no perfectly azimuthally polarized beam could be filtered. This explains the non-perfectly symmetric polarization results shown in Figs. 6 (c) and (e). Additionally, the polarization of the analyzed beam is further affected by the dichroic beam splitter used (which exhibits a slight polarization dependence). The impact of these effects is particularly clear in Fig. 6 (e), which shows a nearly linearly polarized doughnut shape. In spite of these experimental caveats, it can be seen that the polarization purity of the azimuthally polarized beam is reasonably high. In future work we expect to overcome the described problems and to further enhance the polarization purity.

4. Conclusion

In conclusion we have presented a new fiber based mode filter for azimuthally and radially polarized light. This simple concept uses only a strongly guiding fiber with a FBG. Therefore, this approach makes compact all-fiber setups possible.

This concept has been experimentally verified in a proof-of-principle experiment that uses a commercially available strongly guiding fiber. In this setup the azimuthally polarized mode has been successfully separated. The results of this proof-of-principle experiment are very encouraging.

Acknowledgments

We acknowledge financial support by the German Federal Ministry of Education and Research (BMBF) under contract No. 13N9687 and the German Research Foundation (DFG) within the Leibniz program.

References and links

1.

V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D Appl. Phys. 32(13), 1455–1461 (1999). [CrossRef]

2.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001). [CrossRef] [PubMed]

3.

Q. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance excitation by a radially polarized beam,” Opt. Lett. 31(11), 1726–1728 (2006). [CrossRef] [PubMed]

4.

Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004). [CrossRef] [PubMed]

5.

W. D. Kimura, G. H. Kim, R. D. Romea, L. C. Steinhauer, I. V. Pogorelsky, K. P. Kusche, R. C. Fernow, X. Wang, and Y. Liu, “Laser acceleration of relativistic electrons using the inverse Cherenkov effect,” Phys. Rev. Lett. 74(4), 546–549 (1995). [CrossRef] [PubMed]

6.

M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21(23), 1948–1950 (1996). [CrossRef] [PubMed]

7.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef] [PubMed]

8.

S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. 29(15), 2234–2239 (1990). [CrossRef] [PubMed]

9.

T. G. Euser, M. A. Schmidt, N. Y. Joly, C. Gabriel, C. Marquardt, L. Y. Zang, M. Förtsch, P. Banzer, A. Brenn, D. Elser, M. Scharrer, G. Leuchs, and P. St. J. Russell, “Birefringence and dispersion of cylindrically polarized modes in nanobore photonic crystal fiber,” J. Opt. Soc. Am. B 28(1), 193–198 (2011). [CrossRef]

10.

S. Ramachandran, P. Kristensen, and M. F. Yan, “Generation and propagation of radially polarized beams in optical fibers,” Opt. Lett. 34(16), 2525–2527 (2009). [CrossRef] [PubMed]

11.

F. Stutzki, C. Jauregui, C. Voigtländer, J. U. Thomas, J. Limpert, S. Nolte, and A. Tünnermann, “Passively stabilized 215 W monolithic CW LMA-fiber laser with innovative transversal mode filter,” Proc. SPIE 7580, 75801K, 75801K-10 (2010). [CrossRef]

12.

F. Stutzki, C. Jauregui, J. Limpert, and A. Tünnermann, “Real-time characterisation of modal content in monolithic few-mode fiber lasers,” Electron. Lett. 47(4), 274–275 (2011). [CrossRef]

13.

J. M. O. Daniel, J. S. P. Chan, J. W. Kim, J. K. Sahu, M. Ibsen, and W. A. Clarkson, “Novel technique for mode selection in a multimode fiber laser,” Opt. Express 19(13), 12434–12439 (2011). [CrossRef] [PubMed]

14.

A. W. Snyder, “Asymptotic expressions for eigenfunctions and eigenvalues of a dielectric or optical waveguide,” IEEE Trans. Microw. Theory Tech. 17(12), 1130–1138 (1969). [CrossRef]

15.

D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10(10), 2252–2258 (1971). [CrossRef] [PubMed]

16.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Kluwer Academic, 1983).

17.

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997). [CrossRef]

18.

J. U. Thomas, C. Voigtländer, S. Nolte, A. Tünnermann, N. Jovanovic, G. D. Marshall, M. J. Withford, and M. Steel, “Mode selective fibre Bragg grating,” Proc. SPIE 7589, 75890J, 75890J-9 (2010). [CrossRef]

19.

J. Thomas, E. Wikszak, T. Clausnitzer, U. Fuchs, U. Zeitner, S. Nolte, and A. Tünnermann, “Inscription of fiber Bragg gratings with femtosecond pulses using a phase mask scanning technique,” Appl. Phys., A Mater. Sci. Process. 86(2), 153–157 (2006). [CrossRef]

OCIS Codes
(060.2280) Fiber optics and optical communications : Fiber design and fabrication
(060.2310) Fiber optics and optical communications : Fiber optics
(060.2400) Fiber optics and optical communications : Fiber properties
(060.2420) Fiber optics and optical communications : Fibers, polarization-maintaining
(060.3735) Fiber optics and optical communications : Fiber Bragg gratings

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: July 29, 2011
Revised Manuscript: August 29, 2011
Manuscript Accepted: August 29, 2011
Published: September 22, 2011

Citation
Christoph Jocher, Cesar Jauregui, Christian Voigtländer, Fabian Stutzki, Stefan Nolte, Jens Limpert, and Andreas Tünnermann, "Fiber based polarization filter for radially and azimuthally polarized light," Opt. Express 19, 19582-19590 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-20-19582


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References

  1. V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D Appl. Phys.32(13), 1455–1461 (1999). [CrossRef]
  2. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett.86(23), 5251–5254 (2001). [CrossRef] [PubMed]
  3. Q. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance excitation by a radially polarized beam,” Opt. Lett.31(11), 1726–1728 (2006). [CrossRef] [PubMed]
  4. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express12(15), 3377–3382 (2004). [CrossRef] [PubMed]
  5. W. D. Kimura, G. H. Kim, R. D. Romea, L. C. Steinhauer, I. V. Pogorelsky, K. P. Kusche, R. C. Fernow, X. Wang, and Y. Liu, “Laser acceleration of relativistic electrons using the inverse Cherenkov effect,” Phys. Rev. Lett.74(4), 546–549 (1995). [CrossRef] [PubMed]
  6. M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett.21(23), 1948–1950 (1996). [CrossRef] [PubMed]
  7. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett.91(23), 233901 (2003). [CrossRef] [PubMed]
  8. S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt.29(15), 2234–2239 (1990). [CrossRef] [PubMed]
  9. T. G. Euser, M. A. Schmidt, N. Y. Joly, C. Gabriel, C. Marquardt, L. Y. Zang, M. Förtsch, P. Banzer, A. Brenn, D. Elser, M. Scharrer, G. Leuchs, and P. St. J. Russell, “Birefringence and dispersion of cylindrically polarized modes in nanobore photonic crystal fiber,” J. Opt. Soc. Am. B28(1), 193–198 (2011). [CrossRef]
  10. S. Ramachandran, P. Kristensen, and M. F. Yan, “Generation and propagation of radially polarized beams in optical fibers,” Opt. Lett.34(16), 2525–2527 (2009). [CrossRef] [PubMed]
  11. F. Stutzki, C. Jauregui, C. Voigtländer, J. U. Thomas, J. Limpert, S. Nolte, and A. Tünnermann, “Passively stabilized 215 W monolithic CW LMA-fiber laser with innovative transversal mode filter,” Proc. SPIE7580, 75801K, 75801K-10 (2010). [CrossRef]
  12. F. Stutzki, C. Jauregui, J. Limpert, and A. Tünnermann, “Real-time characterisation of modal content in monolithic few-mode fiber lasers,” Electron. Lett.47(4), 274–275 (2011). [CrossRef]
  13. J. M. O. Daniel, J. S. P. Chan, J. W. Kim, J. K. Sahu, M. Ibsen, and W. A. Clarkson, “Novel technique for mode selection in a multimode fiber laser,” Opt. Express19(13), 12434–12439 (2011). [CrossRef] [PubMed]
  14. A. W. Snyder, “Asymptotic expressions for eigenfunctions and eigenvalues of a dielectric or optical waveguide,” IEEE Trans. Microw. Theory Tech.17(12), 1130–1138 (1969). [CrossRef]
  15. D. Gloge, “Weakly guiding fibers,” Appl. Opt.10(10), 2252–2258 (1971). [CrossRef] [PubMed]
  16. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Kluwer Academic, 1983).
  17. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol.15(8), 1277–1294 (1997). [CrossRef]
  18. J. U. Thomas, C. Voigtländer, S. Nolte, A. Tünnermann, N. Jovanovic, G. D. Marshall, M. J. Withford, and M. Steel, “Mode selective fibre Bragg grating,” Proc. SPIE7589, 75890J, 75890J-9 (2010). [CrossRef]
  19. J. Thomas, E. Wikszak, T. Clausnitzer, U. Fuchs, U. Zeitner, S. Nolte, and A. Tünnermann, “Inscription of fiber Bragg gratings with femtosecond pulses using a phase mask scanning technique,” Appl. Phys., A Mater. Sci. Process.86(2), 153–157 (2006). [CrossRef]

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