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Optics Express

Optics Express

  • Editor: C. Martijin de Sterke
  • Vol. 19, Iss. 7 — Mar. 28, 2011
  • pp: 6253–6259
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Mechanically-induced π-shifted long-period fiber gratings

Xiaojun Zhou, Shenghui Shi, Zhiyao Zhang, Jun Zou, and Yong Liu  »View Author Affiliations


Optics Express, Vol. 19, Issue 7, pp. 6253-6259 (2011)
http://dx.doi.org/10.1364/OE.19.006253


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Abstract

A band-pass filter based on mechanically-induced multi-π-shifted long-period fiber gratings is proposed. The pass band width of the filter depends on the number N of the sub-gratings divided by π-shifts in the long-period fiber grating. The depth of the two lateral rejection bands can be changed by the amount of pressure applied to the fiber. This paper demonstrates a multi-π-shifted long-period fiber grating created by pressing a fiber between two periodically grooved plates. For N = 7 and LP12 mode coupling, the extinction ratio is 22.22 dB, and the pass band loss is 0.85 dB. For LP12 mode coupling, the pass band width varies from 14.23 nm to 39.31 nm when N increases from 2 to 10.

© 2011 OSA

1. Introduction

Long-period fiber gratings (LPFGs) offer coupling from core fundamental mode to cladding modes in a fiber, which can be used as spectral filters with the merits of low back-reflection and low insertion loss [1

1. A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14(1), 58–65 (1996). [CrossRef]

]. Thanks to the band-rejection filtering characteristic, LPFGs had been applied in spectral shaping [2

2. J. K. Bae, S. H. Kim, J. H. Kim, J. Bae, S. B. Lee, and J.-M. Jeong, “Spectral shape tunable band-rejection filter using a long-period fiber grating with divided coil heaters,” IEEE Photon. Technol. Lett. 15(3), 407–409 (2003). [CrossRef]

] and mode conversion [3

3. S. Ramachandran, Z. Wang, and M. Yan, “Bandwidth control of long-period grating-based mode converters in few-mode fibers,” Opt. Lett. 27(9), 698–700 (2002). [CrossRef]

]. Phase-shifted LPFGs were also investigated and used as gain-flattening filters in erbium-doped fiber amplifiers [4

4. J. R. Qian and H. F. Chen, “Gain flattening fibre filters using phase-shifted long period fiber gratings,” Electron. Lett. 34(11), 1132–1133 (1998). [CrossRef]

] and fiber sensors [5

5. V. Bhatia, “Applications of long-period gratings to single and multi-parameter sensing,” Opt. Express 4(11), 457–466 (1999). [CrossRef] [PubMed]

]. The π-shifted LPFGs which introduce a π-shift in the middle of a LPFG or multi-π-shifts in a LPFG are interesting. In these gratings, the destructive mode coupling is converted to a constructive one, with the consequence that the pass band is opened within the resonant notches of the LPFG. This band-pass filter is very useful for actively mode-locked erbium-doped fiber lasers [6

6. O. Deparis, R. Kiyan, O. Pottiez, M. Blondel, I. G. Korolev, S. A. Vasiliev, and E. M. Dianov, “Bandpass filters based on π-shifted long-period fiber gratings for actively mode-locked erbium fiber lasers,” Opt. Lett. 26(16), 1239–1241 (2001). [CrossRef]

], high order temporal differentiators [7

7. M. Kulishov, D. Krcmarík, and R. Slavík, “Design of terahertz-bandwidth arbitrary-order temporal differentiators based on long-period fiber gratings,” Opt. Lett. 32(20), 2978–2980 (2007). [CrossRef] [PubMed]

] and sensors [8

8. R. Falate, O. Frazão, G. Rego, J. L. Fabris, and J. L. Santos, “Refractometric sensor based on a phase-shifted long-period fiber grating,” Appl. Opt. 45(21), 5066–5072 (2006). [CrossRef] [PubMed]

]. The π-shifted LPFGs are usually fabricated by exposing an optical fiber to an ultraviolet laser through an amplitude mask with half a period missed in the center. However, this kind of special amplitude mask is very expensive, and the phase-shift in the grating cannot be changed after fabrication. The π-shifted LPFGs can also be realized through point-by-point writing on a fiber using a CO2-laser [9

9. Y. Gu, K. S. Chiang, and Y. J. Rao, “Writing of apodized phase-shifted long-period fiber gratings with a computer-controlled CO2 laser,” IEEE Photon. Technol. Lett. 21(10), 657–659 (2009). [CrossRef]

] or electric-arc technique [10

10. G. Humbert and A. Malki, “High performance bandpass filters based on electric arc-induced π-shifted long-period fiber gratings,” Electron. Lett. 39(21), 1506–1507 (2003). [CrossRef]

], wherein a complicated controlling method is needed, especially in the case of multi-π-shifted LPFGs. On the other hand, electrically-induced long-period grating in a waveguide with an electro-optic core is applied to a design of reconfigurable band-pass filters through the introduction of π-shift which can be easily switched ON and OFF or moved along the grating [11

11. M. Kulishov and X. Daxhelet, “Reconfigurable π-shifted and Mach–Zehnder bandpass filters on the basis of electrooptically induced long-period gratings in a planar waveguide,” J. Lightwave Technol. 21(3), 854–861 (2003). [CrossRef]

].

In this paper, we present the mechanically-induced multi-π-shifted LPFGs which exhibit filtering characteristics similar to those fabricated through other methods. In our experiment, the maximum number of π-shifts induced in a grating is nine, which is the largest number up to the present to our best knowledge. The mechanically-induced multi-π-shifted LPFGs are tunable, reconfigurable, as well as inexpensive.

2. Mechanicaly-induced multi-π-shifted LPFGs

It is well known that the periodical index modulation of a single mode fiber can be realized by applying a pressure on the fiber through the photo-elastic effect and the micro-bending effect if the period of the applied force is as large as hundreds of micrometers. The mechanically-induced LPFGs (MI-LPFGs) can be generated by pressing the fiber with a periodically grooved plate and a flat plate or two periodically grooved plates [12

12. S. Savin, M. J. F. Digonnet, G. S. Kino, and H. J. Shaw, “Tunable mechanically induced long-period fiber gratings,” Opt. Lett. 25(10), 710–712 (2000). [CrossRef]

].

We found in experiment that out-of-band loss of the MI-LPFGs by pressing a jacketed fiber with two periodically grooved plates is smaller than that generated with a periodically grooved plate and a flat plate. Therefore, in our experiments, π-shifted MI-LPFGs are induced through imposing a pressure on the fiber through two periodically grooved plates which have a small offset between them (half a grating pitch), as shown in the cross section in Fig. 1
Fig. 1 Experimental setup of π-shifted MI-LPFG. P: polarizer, PC: polarization controller.
. It can be seen from Fig. 1 that a jacketed fiber is placed between two periodically grooved plates, where the plate below has a V-shape groove to fix the fiber. The π-shifted section in the MI-LPFG is actualized by extending a groove length from b to (b + Λ/2) where Λ is the grating pitch. The pressure on the fiber is induced through placing metal cylinders with different weights on the plate.

In general, π-shifted LPFGs have some ripples in transmission spectra, which can be suppressed by index-apodization (with equal length of subgratings) or length-apodization (with equal index distribution in fiber) [13

13. H. Ke, K. S. Chiang, and J. H. Peng, “Analysis of phase-shifted long-period fiber gratings,” IEEE Photon. Technol. Lett. 10(11), 1596–1598 (2009). [CrossRef]

15

15. L. R. Chen, “Design of flat top bandpass filters based on symmetric multiple phaseshifted long-period fiber gratings,” Opt. Commun. 205, 271–276 (2002).

]. The ripples can be suppressed using the index-apodization more than when using the length-apodization in multi-π-shifts LPFGs [9

9. Y. Gu, K. S. Chiang, and Y. J. Rao, “Writing of apodized phase-shifted long-period fiber gratings with a computer-controlled CO2 laser,” IEEE Photon. Technol. Lett. 21(10), 657–659 (2009). [CrossRef]

]. Therefore, the index-apodization induced by pressure along a fiber with the heaviest weight in the middle and lighter ones on both sides was adopted (see Fig. 1). The coupling from the core fundamental mode to the cladding modes is controlled by changing the total weight on the plate, due to the fact that the coupling coefficient is directly proportional to the bending angle of the fiber and the refractive-index changes caused by stress in the bent fiber [16

16. J. N. Blake, B. Y. Kim, H. E. Engan, and H. J. Shaw, “Analysis of intermodal coupling in a two-mode fiber with periodic microbends,” Opt. Lett. 12(4), 281–283 (1987). [CrossRef] [PubMed]

]. When the total weight is increased, index modulation in both the core and cladding of the fiber increases. Therefore, the mode coupling from the core fundamental mode to the cladding modes is enhanced.

The grating is intersected as N uniform sections by (N-1) π-shifts. To compare the filtering characteristics of the π-shifted MI-LPFGs with different sub-grating numbers, several gratings were fabricated, whose parameters are listed in Table 1

Table 1. π-shifted MI-LPFGs with different parameters

table-icon
View This Table
.

3. Experimental results and discussion

The experimental setup for the π-shifted MI-LPFG is shown in Fig. 1. The un-polarized light, launched from a super-continuum source (Koheras: Superk Compact), is polarized using a polarizer, and the polarization direction is controlled by a polarization controller. The transmission spectra of the MI-LPFGs are measured by an optical spectrum analyzer (Yokogawa: AQ6370B).

First, the transmission of the phase-shift-free MI-LPFG under different pressures was measured in order to compare with that of the π-shifted MI-LPFGs. The grooved plate in the setup consists of 200 periods having grating pitch Λ = 600 µm, groove width b = 400 µm and bulge width a = 200 µm for both the phase-shift-free MI-LPFG and the π-shifted ones. The fiber used in the experiment is the jacketed single-mode fiber (Corning SMF-28), and the transmission spectra of the phase-shift-free MI-LPFG under different pressures are given in Fig. 2
Fig. 2 Transmission of the MI-LPFG with the grating pitch of 600µm.
. The notches at the resonant wavelengths indicate the coupling from the core fundamental mode to the cladding modes (1430 nm for LP11, 1483.8 nm for LP12 and 1551.73 nm for LP13). As the MI-LPFG is induced by pressing the fiber through two grooved plates, couplings from the core fundamental mode towards the anti-symmetric cladding modes of LP1m are created. The notches at the resonant wavelengths, which indicate these couplings, are deeper for greater pressure because the index modulation in the fiber increases with the pressure. The loss induced by fiber bending rises with increasing pressure, which can be seen from the variation of the out-of-band loss. It is shown in Fig. 2 that the out-of-band loss is 0.03 dB, 0.017 dB, 0.045 dB at 1460 nm and 0.24 dB, 0.52 dB, 0.63 dB at 1520 nm for the increasing pressure from p1 to p3, respectively. The bent fiber under pressure can be considered as a fiber with a curvature radius of R. When pressure is increased, the curvature radius of the fiber decreases and more optical field of the core fundamental mode extends to the jacketed layer of the fiber where it is absorbed. Therefore the curvature loss of the bent fiber increases with pressure, especially at longer wavelengths, which is in agreement with the analysis of curvature loss for fibers [17

17. D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. 66(3), 216–220 (1976). [CrossRef]

]. The cladding mode loss around the resonant wavelength, which is much larger than the core fundamental mode loss, can affect the transmission characteristics of the LPFGs as predicted in Ref [18

18. X. Daxhelet and M. Kulishov, “Theory and practice of long-period gratings: when a loss becomes a gain,” Opt. Lett. 28(9), 686–688 (2003). [CrossRef] [PubMed]

]. The depth of the notch in the transmission spectrum depends on the coupling coefficient and the cladding mode loss, both of which are determined by the applied pressure. The resonant wavelengths of λres for all mode couplings depend on the size of the grating pitch which provides the phase-matching in the couplings. Relation between λres and the grating pitch is investigated, and the results are given in Fig. 3
Fig. 3 Relation between λres and the grating pitch.
.

Next, the MI-LPFGs with N sub-gratings, whose parameters are listed in Table 1, are investigated. The length of the π-shifted sections in the experimental setup is 700 µm and the other parameters are the same as the phase-shift-free MI-LPFG. The transmission spectra of the π-shifted MI-LPFG for N = 7 are shown in Fig. 4
Fig. 4 Transmission spectra of the π-shifted MI-LPFG for N = 7 under increasing pressure of (a) P1-P4, (b) P5.
, where the pressure is increased from P1 to P4. The resonant notches for the phase-shift-free MI-LPFG (see Fig. 2) become the pass bands centered at the resonant wavelengths, with a rejection band on each side for the π-shifted MI-LPFG (see Fig. 4). From Fig. 4 (a), we know that the extinction ratio of the pass band to the rejection bands rises as pressure is increased from P1 to P4. Under the pressure P4, for LP12 and LP13 mode couplings, the extinction ratio is 22.22 dB and 21.07 dB, and the pass band loss is 0.85 dB and 2.72 dB, respectively. Figure 4 (b) shows the transmission spectrum of the grating for pressure P5 larger than P4. It can be seen that, for the couplings to LP12 and LP13 modes, the depth of the notches is smaller than that for pressure P4, and the losses in the pass bands and out-of-bands become larger than that for pressure P4. In this case, the fiber bend under pressure P5 induces not only a larger index modulation but also a more serious curvature loss in the fiber. This increasing loss and coupling strength are responsible for the transmission spectrum evolution as analyzed in Ref [18

18. X. Daxhelet and M. Kulishov, “Theory and practice of long-period gratings: when a loss becomes a gain,” Opt. Lett. 28(9), 686–688 (2003). [CrossRef] [PubMed]

,19

19. M. Kulishov, V. Grubsky, J. Schwartz, X. Daxhelet, and D. V. Plant, “Tunable waveguide transmission gratings based on active gain control,” IEEE J. Quantum Electron. 40(12), 1715–1724 (2004). [CrossRef]

]. In our experiments, no ripples appear in either the pass bands of the π-shifted MI-LPFGs or the out-of-bands of the phase-shift-free MI-LPFG for all the pressures. Cladding mode loss and index-apodization account for this ripple-free phenomenon [19

19. M. Kulishov, V. Grubsky, J. Schwartz, X. Daxhelet, and D. V. Plant, “Tunable waveguide transmission gratings based on active gain control,” IEEE J. Quantum Electron. 40(12), 1715–1724 (2004). [CrossRef]

,9

9. Y. Gu, K. S. Chiang, and Y. J. Rao, “Writing of apodized phase-shifted long-period fiber gratings with a computer-controlled CO2 laser,” IEEE Photon. Technol. Lett. 21(10), 657–659 (2009). [CrossRef]

].

Finally, the relation between the pass band width Δλ and the sub-grating number N is investigated, and the results are given in Fig. 5
Fig. 5 Relation between the pass band width Δλ and the sub-grating number N.
. For the three cladding-mode couplings, the pass band widths increase with N, as predicted in the theoretical analysis [13

13. H. Ke, K. S. Chiang, and J. H. Peng, “Analysis of phase-shifted long-period fiber gratings,” IEEE Photon. Technol. Lett. 10(11), 1596–1598 (2009). [CrossRef]

]. For the LP12 and LP13 mode couplings, the pass band width increases from 14.23 nm to 39.31 nm and from 19.2 nm to 51.54 nm, respectively, when N increases from 2 to 10. As shown in Fig. 6
Fig. 6 Relation between the central wavelength λc and the sub-grating number N.
, the central wavelengths of the pass bands are nearly the same for different sub-grating number N.

It should be mentioned that the mode coupling for the π-shifted MI-LPFGs can be modified by the pressure applied. Therefore, the depth of the notches in the transmission spectrum is variable, which is convenient for adjusting the performance of the filters. When the π-shifted MI-LPFG is used as a filter in application, the pass band loss should be as small as possible. It can be decreased through less bending of the fiber, which can be implemented by reducing the pressure. At the same time, the length of the grating must be increased in order to keep the same mode coupling strength (κL). The π-shifted MI-LPFGs are polarization-dependent, which can be beneficial for producing a polarized fiber laser and for filtering in a polarization-division-multiplexing communication system. However, in other polarization-insensitive applications, the pressure-induced birefringence needs to be removed. Several proposed birefringence-compensating methods [20

20. J. Y. Cho and K. S. Lee, “A birefringence compensation method for mechanically induced long-period fiber gratings,” Opt. Commun. 213(4-6), 281–284 (2002). [CrossRef]

,21

21. S. Ramachandran, S. Golowich, M. F. Yan, E. Monberg, F. V. Dimarcello, J. Fleming, S. Ghalmi, and P. Wisk, “Lifting polarization degeneracy of modes by fiber design: a platform for polarization-insensitive microbend fiber gratings,” Opt. Lett. 30(21), 2864–2866 (2005). [CrossRef] [PubMed]

] can be used in the π-shifted MI-LPFGs.

4. Conclusion

This work demonstrates an index-apodized π-shifts MI-LPFG filter whose pass band width depends on the sub-grating number N. The depth of the two lateral rejection bands is determined by the amount of pressure applied to the fiber. The central wavelengths of the pass bands do not vary with the sub-grating number. The proposed method is flexible and simple for fabricating a band-pass filter through appropriately choosing the grating pitch, the π-shift number, the sub-grating length and the applied pressure. It is believed that the π-shifted MI-LPFGs will have extensive applications in fiber-optic communications and fiber-optic sensing.

Acknowledgements

This work was supported by the National Nature Science Foundation of China (No. 60925019, 61090393). This work was also partially supported by 973 Program under Grant No. 2011CB301705.

References and links

1.

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14(1), 58–65 (1996). [CrossRef]

2.

J. K. Bae, S. H. Kim, J. H. Kim, J. Bae, S. B. Lee, and J.-M. Jeong, “Spectral shape tunable band-rejection filter using a long-period fiber grating with divided coil heaters,” IEEE Photon. Technol. Lett. 15(3), 407–409 (2003). [CrossRef]

3.

S. Ramachandran, Z. Wang, and M. Yan, “Bandwidth control of long-period grating-based mode converters in few-mode fibers,” Opt. Lett. 27(9), 698–700 (2002). [CrossRef]

4.

J. R. Qian and H. F. Chen, “Gain flattening fibre filters using phase-shifted long period fiber gratings,” Electron. Lett. 34(11), 1132–1133 (1998). [CrossRef]

5.

V. Bhatia, “Applications of long-period gratings to single and multi-parameter sensing,” Opt. Express 4(11), 457–466 (1999). [CrossRef] [PubMed]

6.

O. Deparis, R. Kiyan, O. Pottiez, M. Blondel, I. G. Korolev, S. A. Vasiliev, and E. M. Dianov, “Bandpass filters based on π-shifted long-period fiber gratings for actively mode-locked erbium fiber lasers,” Opt. Lett. 26(16), 1239–1241 (2001). [CrossRef]

7.

M. Kulishov, D. Krcmarík, and R. Slavík, “Design of terahertz-bandwidth arbitrary-order temporal differentiators based on long-period fiber gratings,” Opt. Lett. 32(20), 2978–2980 (2007). [CrossRef] [PubMed]

8.

R. Falate, O. Frazão, G. Rego, J. L. Fabris, and J. L. Santos, “Refractometric sensor based on a phase-shifted long-period fiber grating,” Appl. Opt. 45(21), 5066–5072 (2006). [CrossRef] [PubMed]

9.

Y. Gu, K. S. Chiang, and Y. J. Rao, “Writing of apodized phase-shifted long-period fiber gratings with a computer-controlled CO2 laser,” IEEE Photon. Technol. Lett. 21(10), 657–659 (2009). [CrossRef]

10.

G. Humbert and A. Malki, “High performance bandpass filters based on electric arc-induced π-shifted long-period fiber gratings,” Electron. Lett. 39(21), 1506–1507 (2003). [CrossRef]

11.

M. Kulishov and X. Daxhelet, “Reconfigurable π-shifted and Mach–Zehnder bandpass filters on the basis of electrooptically induced long-period gratings in a planar waveguide,” J. Lightwave Technol. 21(3), 854–861 (2003). [CrossRef]

12.

S. Savin, M. J. F. Digonnet, G. S. Kino, and H. J. Shaw, “Tunable mechanically induced long-period fiber gratings,” Opt. Lett. 25(10), 710–712 (2000). [CrossRef]

13.

H. Ke, K. S. Chiang, and J. H. Peng, “Analysis of phase-shifted long-period fiber gratings,” IEEE Photon. Technol. Lett. 10(11), 1596–1598 (2009). [CrossRef]

14.

F. Y. M. Chan and K. S. Chiang, “Analysis of apodized phase-shifted long-period fiber gratings,” Opt. Commun. 244(1-6), 233–243 (2005). [CrossRef]

15.

L. R. Chen, “Design of flat top bandpass filters based on symmetric multiple phaseshifted long-period fiber gratings,” Opt. Commun. 205, 271–276 (2002).

16.

J. N. Blake, B. Y. Kim, H. E. Engan, and H. J. Shaw, “Analysis of intermodal coupling in a two-mode fiber with periodic microbends,” Opt. Lett. 12(4), 281–283 (1987). [CrossRef] [PubMed]

17.

D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. 66(3), 216–220 (1976). [CrossRef]

18.

X. Daxhelet and M. Kulishov, “Theory and practice of long-period gratings: when a loss becomes a gain,” Opt. Lett. 28(9), 686–688 (2003). [CrossRef] [PubMed]

19.

M. Kulishov, V. Grubsky, J. Schwartz, X. Daxhelet, and D. V. Plant, “Tunable waveguide transmission gratings based on active gain control,” IEEE J. Quantum Electron. 40(12), 1715–1724 (2004). [CrossRef]

20.

J. Y. Cho and K. S. Lee, “A birefringence compensation method for mechanically induced long-period fiber gratings,” Opt. Commun. 213(4-6), 281–284 (2002). [CrossRef]

21.

S. Ramachandran, S. Golowich, M. F. Yan, E. Monberg, F. V. Dimarcello, J. Fleming, S. Ghalmi, and P. Wisk, “Lifting polarization degeneracy of modes by fiber design: a platform for polarization-insensitive microbend fiber gratings,” Opt. Lett. 30(21), 2864–2866 (2005). [CrossRef] [PubMed]

OCIS Codes
(050.2770) Diffraction and gratings : Gratings
(060.2310) Fiber optics and optical communications : Fiber optics
(060.2340) Fiber optics and optical communications : Fiber optics components

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: January 4, 2011
Revised Manuscript: February 24, 2011
Manuscript Accepted: February 24, 2011
Published: March 18, 2011

Citation
Xiaojun Zhou, Shenghui Shi, Zhiyao Zhang, Jun Zou, and Yong Liu, "Mechanically-induced π-shifted long-period fiber gratings," Opt. Express 19, 6253-6259 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-7-6253


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References

  1. A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14(1), 58–65 (1996). [CrossRef]
  2. J. K. Bae, S. H. Kim, J. H. Kim, J. Bae, S. B. Lee, and J.-M. Jeong, “Spectral shape tunable band-rejection filter using a long-period fiber grating with divided coil heaters,” IEEE Photon. Technol. Lett. 15(3), 407–409 (2003). [CrossRef]
  3. S. Ramachandran, Z. Wang, and M. Yan, “Bandwidth control of long-period grating-based mode converters in few-mode fibers,” Opt. Lett. 27(9), 698–700 (2002). [CrossRef]
  4. J. R. Qian and H. F. Chen, “Gain flattening fibre filters using phase-shifted long period fiber gratings,” Electron. Lett. 34(11), 1132–1133 (1998). [CrossRef]
  5. V. Bhatia, “Applications of long-period gratings to single and multi-parameter sensing,” Opt. Express 4(11), 457–466 (1999). [CrossRef] [PubMed]
  6. O. Deparis, R. Kiyan, O. Pottiez, M. Blondel, I. G. Korolev, S. A. Vasiliev, and E. M. Dianov, “Bandpass filters based on π-shifted long-period fiber gratings for actively mode-locked erbium fiber lasers,” Opt. Lett. 26(16), 1239–1241 (2001). [CrossRef]
  7. M. Kulishov, D. Krcmarík, and R. Slavík, “Design of terahertz-bandwidth arbitrary-order temporal differentiators based on long-period fiber gratings,” Opt. Lett. 32(20), 2978–2980 (2007). [CrossRef] [PubMed]
  8. R. Falate, O. Frazão, G. Rego, J. L. Fabris, and J. L. Santos, “Refractometric sensor based on a phase-shifted long-period fiber grating,” Appl. Opt. 45(21), 5066–5072 (2006). [CrossRef] [PubMed]
  9. Y. Gu, K. S. Chiang, and Y. J. Rao, “Writing of apodized phase-shifted long-period fiber gratings with a computer-controlled CO2 laser,” IEEE Photon. Technol. Lett. 21(10), 657–659 (2009). [CrossRef]
  10. G. Humbert and A. Malki, “High performance bandpass filters based on electric arc-induced π-shifted long-period fiber gratings,” Electron. Lett. 39(21), 1506–1507 (2003). [CrossRef]
  11. M. Kulishov and X. Daxhelet, “Reconfigurable π-shifted and Mach–Zehnder bandpass filters on the basis of electrooptically induced long-period gratings in a planar waveguide,” J. Lightwave Technol. 21(3), 854–861 (2003). [CrossRef]
  12. S. Savin, M. J. F. Digonnet, G. S. Kino, and H. J. Shaw, “Tunable mechanically induced long-period fiber gratings,” Opt. Lett. 25(10), 710–712 (2000). [CrossRef]
  13. H. Ke, K. S. Chiang, and J. H. Peng, “Analysis of phase-shifted long-period fiber gratings,” IEEE Photon. Technol. Lett. 10(11), 1596–1598 (2009). [CrossRef]
  14. F. Y. M. Chan and K. S. Chiang, “Analysis of apodized phase-shifted long-period fiber gratings,” Opt. Commun. 244(1-6), 233–243 (2005). [CrossRef]
  15. L. R. Chen, “Design of flat top bandpass filters based on symmetric multiple phaseshifted long-period fiber gratings,” Opt. Commun. 205, 271–276 (2002).
  16. J. N. Blake, B. Y. Kim, H. E. Engan, and H. J. Shaw, “Analysis of intermodal coupling in a two-mode fiber with periodic microbends,” Opt. Lett. 12(4), 281–283 (1987). [CrossRef] [PubMed]
  17. D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. 66(3), 216–220 (1976). [CrossRef]
  18. X. Daxhelet and M. Kulishov, “Theory and practice of long-period gratings: when a loss becomes a gain,” Opt. Lett. 28(9), 686–688 (2003). [CrossRef] [PubMed]
  19. M. Kulishov, V. Grubsky, J. Schwartz, X. Daxhelet, and D. V. Plant, “Tunable waveguide transmission gratings based on active gain control,” IEEE J. Quantum Electron. 40(12), 1715–1724 (2004). [CrossRef]
  20. J. Y. Cho and K. S. Lee, “A birefringence compensation method for mechanically induced long-period fiber gratings,” Opt. Commun. 213(4-6), 281–284 (2002). [CrossRef]
  21. S. Ramachandran, S. Golowich, M. F. Yan, E. Monberg, F. V. Dimarcello, J. Fleming, S. Ghalmi, and P. Wisk, “Lifting polarization degeneracy of modes by fiber design: a platform for polarization-insensitive microbend fiber gratings,” Opt. Lett. 30(21), 2864–2866 (2005). [CrossRef] [PubMed]

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