Determination of the Mueller matrix of UV-inscribed ... - OSA Publishing

Determination of the Mueller matrix of UV-inscribed long-period fiber grating Karla M. Salas-Alcántara,1 R. Espinosa-Luna,1,* I. Torres-Gómez,2 and Yuri O. Barmenkov2 1

GIPYS, Centro de Investigaciones en Óptica, A. C., Loma del Bosque 115, Colonia Lomas del Campestre, C. P. 37150 León, Guanajuato, Mexico 2

Centro de Investigaciones en Óptica, A. C., Loma del Bosque 115, Colonia Lomas del Campestre, C. P. 37150 León, Guanajuato, Mexico *Corresponding author: [email protected] Received 27 August 2013; revised 19 November 2013; accepted 30 November 2013; posted 5 December 2013 (Doc. ID 196493); published 9 January 2014

An explicit method for determination of the Mueller matrix elements of a commercial long-period fiber grating inscribed with ultraviolet CW laser irradiation (UV-LPFG) is presented. From the Mueller matrix obtained for such UV-LPFG, the full polarimetric response of the grating was found. Our polarimetric analysis was focused mainly on the polarization-dependent loss and other polarimetric properties, such as the polarizance, the depolarization index, and the diattenuation parameters. The full polarimetric analysis allows us to obtain more complete information than the usually reported ones, in which only two orthogonal linear polarizations are considered; for example, with our analysis, we prove that a small depolarization effect is inherent in UV-LPFG and that attenuation depends on the polarization state. This additional polarimetric information could be useful to control the output LPFG signal, for instance, for the realization of wavelength switchable or Q-switched fiber lasers, among other applications. © 2014 Optical Society of America OCIS codes: (060.2310) Fiber optics; (060.2400) Fiber properties; (060.2270) Fiber characterization. http://dx.doi.org/10.1364/AO.53.000269

1. Introduction

Long-period fiber gratings (LPFGs) are versatile components used in multiple applications. Although LPFGs were originally proposed for such purposes as band-rejection filters [1], nowadays, they are capable of a much broader scope of operation in all-fiber devices. LPFGs have been studied for applications in optical fiber polarizers [2], optical sensors [3–5], and optical switching [6,7], among others. They have been used in the reshaping gain spectra of active fiber devices like erbium-doped fiber amplifiers (EDFAs) and fiber lasers. LPFGs have found practical application in the equalization spectral gain of EDFAs, and tunable and selective elements in fiber 1559-128X/14/020269-09$15.00/0 © 2014 Optical Society of America

lasers [8–10]. On the other hand, with the use of two concatenated identical gratings, it is possible to obtain very narrow optical filters, which are useful devices in wavelength division multiplexing [11,12]. The optimal performance of the LPFGs in most of the applications is determined, in part, by the polarization properties, such as the polarizationdependent loss (PDL), and the birefringence. Such parameters are critical for some specific applications of LPFGs and also for improving the LPFG fabrication process [13–15]. In this sense, it is important to obtain a full description of the polarization properties of LPFGs. LPFGs couple light from the fundamental fiber core mode into a series of forward propagating cladding modes, which are attenuated due to absorption, scattering, or radiation mechanisms [1]. Since the coupling from the fundamental mode to cladding modes is 10 January 2014 / Vol. 53, No. 2 / APPLIED OPTICS

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wavelength-dependent, spectrally selective loss filters, based on LPFGs, can be produced. Long-period gratings are fabricated in optical fibers by applying the point-by-point technique or interferometric pattern using UV, NIR, or CO2 CW or pulsed laser irradiation, or by the periodic application of heat, pressure, or geometric modulation, among other methods [16]. According to the chosen LPFG fabrication technique, and the fiber type, various mechanisms may contribute simultaneously to the grating formation, affecting the optical polarization response of the induced grating. Several studies on PDL have been reported. Most of these studies focused on reducing or suppressing the PDL of LPFG to the low level required in high-performance communication systems. However, these studies only examined the behaviors of two orthogonal polarization states [17,18]. Recently, we have proposed an explicit method for the complete determination of the Mueller matrix associated with a mechanically induced LPFG in a photonic crystal fiber (PCF) [19]. In Ref. [19], 4incident polarization states (linear parallel, vertical,
45 degrees, and right-hand circular) have been coupled externally to the fiber grating, using a CWillumination at 1064 nm. In the present work, the Mueller matrix associated with a commercial UVLPFG, written in a H2 -loaded single-mode fiber, is presented. The analysis is focused on a study of the polarization effects, including PDL, and some other important polarimetric scalar metrics, such as the polarizance and the diattenuation parameters. The Mueller matrix elements were obtained from a set of six incident Stokes vectors, where we have added the −45 degrees linear polarization and also the lefthand circular polarization state to the 4-incident polarization method [19]. In practice, we have found the method of 6-incident polarization states provides more stable results and less noise than the 4-incident method. A similar tendency to least sensitivity with regard to flux noise has also been found for the scattering of light by one-dimensional, randomly rough surfaces [20,21], where an ideal polarimeter arrangement (IPA) has been reported, to compare results generated with the 4- and the 6-reduced methods for the determination of the Mueller matrix. It seems that experimental noise decreases with an increase of measurements in the determination of the Mueller matrix. Several authors have analyzed the efficiency of polarimeters and the figures of merit versus number of measurements for several measurement methods [22–24]. As our results show, UV-LPFG slightly depolarizes the incident light, which appears in the decreasing polarization degree of the output light beam as a function of the incident polarization state. It is also shown that scalar polarimetric metrics, such as the diattenuation and the polarizance parameters, are wavelength-dependent, i.e., the values of these metrics decrease at detuning from the main LPFG resonance. The attenuation, the PDL, and the gain or transmittance also depend on the incident polarization and the wavelength. One of the 270

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main contributions of our work is the experimental method used to obtain the Mueller matrix of UVLPFG when the fiber polarizer state generator (PSG) and the bulk polarizer state analyzer (PSA) are utilized under an all-fiber configuration. 2. Mathematical Mueller–Stokes Model

The input and output Stokes parameters from a system under study are related by the linear response to light as [25]: 0

so0

1

2

m00 m01 m02 m03

30

si0

1

B oC 6 7B C B s1 C 6 m10 m11 m12 m13 7B si1 C C6 7B C So  MSi ⇒ B B so C 6 m 7B C @ 2 A 4 20 m21 m22 m23 5@ si2 A so3 m30 m31 m32 m33 si3 1 0 m00 si0
m01 si1
m02 si2
m03 si3 C B B m10 si0
m11 si1
m12 si2
m13 si3 C C; (1) B B C @ m20 si0
m21 si1
m22 si2
m23 si3 A m30 si0
m31 si1
m32 si2
m33 si3 where M is called the Mueller matrix of the system and S is the Stokes vector. Si;o represents the polarization state of the incident and the output (superscript symbols “i” and “o,” respectively) light beams, defined in terms of the orthogonal components of the electric field vector (Ep, Es) and their phase differences. The normalized Stokes parameters can be displayed in a real three-dimensional space as a function of the azimuth 0 ≤ ψ ≤ π and the ellipticity angles −π∕4 ≤ χ ≤ π∕4 of the polarization ellipse, respectively [25]: 0

1

1

B C B cos2χ cos2ψ C C S  hs0 iB B cos2χ sin2ψ C; @ A sin2χ

(2)

where hs0 i represents the average light intensity associated to the Stokes parameters; usually, this is fixed to the unitary value. By using the following depolarization scalar metrics, it is possible to analyze the polarimetric properties associated with the UV-LPFG and its capability to depolarize light (see Table 1) [19,26]. 3. Experimental Determination of the UV-LPFG Mueller Matrix

To experimentally determine the Mueller matrix elements of the UV-LPFG, we propose the use of a set of six incident Stokes vectors (PSG) and the same polarization states analyzed (PSA). Both, the PSG and the PSA are controlled by a computer. The incident and the output Stokes vectors correspond to linear polarization states that are parallel (p), perpendicular (s),
45 degrees (
), and −45 degrees (−) with respect to the horizontal plane of the working optical table,

Table 1.

Polarization-Dependent Parameters Attainable from the Mueller Matrix

(

Depolarization index, DIM,

0 ≤ DIM 

3 X

)1∕2

m2jk

−

m200

p


∕ 3m00 ≤ 1

j;k0

Degree of polarization, DoPM; S,

0 ≤ DoPM; S 

q












































so1 2
so2 2
so3 2 so0

0 ≤ Add 

Anisotropic degree of depolarization, Add





P3

1

j1

mj0 si0
mj1 si1
mj2 si2
mj3 si3 2 2


m00 si0

m01 si1




0 ≤ DM 

q







































m201
m202
m203 ∕m00 ≤ 1

Polarizance parameters, PM,

0 ≤ PM 

q







































m210
m220
m230 ∕m00 ≤ 1

P3 0 ≤ QM 

2 j1;k0 mjk P3 2 k0 m0k

3DIM2 − DM2   1
DM2

Polarization-dependent loss, PDL

PDL  10 logT max ∕T min  10 log

Gain, G

0≤g

respectively, and to a right-hand (r) and a left-hand polarization state (l). The total intensity associated with each incident polarization state (S0 ) is denoted as the “power” and is measured in dBm. From Eq. (1), the scattered Stokes vector incident onto the PSA is closely related to the Stokes vector incident on the sample. For the set of incident polarization states (i  p, s, 45, r, and l), the scattered Stokes vectors from the sample (Sio ) are expressed, respectively, as: m00
m01

1

B C B m10
m11 C po B C; S B C @ m20
m21 A m30
m31 0 1 m00
m02 B C B m10
m12 C C S
o  B B m
m C; @ 20 22 A m30
m32 1 0 m00
m03 C B B m10
m13 C ro C; B S B C @ m20
m23 A m30
m33

≤1

nP 3

j;k1

m2jk

o. m200
PM2

1
DM2

≤3

TrM t M  4m200

Gil–Bernabeu theorem

0




m03 si3

DoPoMax − DoPomin ≤1 DoPoMax
DoPomin

Diattenuation, DM,

The QM metric

m02 si2

0

m00 − m01

1

B C B m10 − m11 C so B C; S B C @ m20 − m21 A m30 − m31 0 1 m00 − m02 B C B m10 − m12 C C S−o  B Bm − m C @ 20 22 A m30 − m32 1 0 m00 − m03 C B B m10 − m13 C lo C: B S B C @ m20 − m23 A m30 − m33

" # m00
m201
m202
m203 1∕2 m00 − m201
m202
m203 1∕2

so0 m00 si0
m01 si1
m02 si2
m03 si3  ≤1 si0 si0

The “power” detected by the equipment is given by the total intensity associated to each scattered Stokes vector, according to:

spd 0  m00
m01 ;

ssd 0  m00 − m01 ;

s
d 0  m00
m02 ;

s−d 0  m00 − m02 ;

srd 0  m00
m03 ;

sld 0  m00 − m03 :

(4)

On the other hand, the detected normalized Stokes vectors (Sid ) are displayed as: 0 Spd

Ssd

1

0

m00
m01

1

B pd C B C B s1 C C 1 B B C B m10
m11 C  B pd C  pd B C; B s2 C s B m20
m21 C 0 @ @ A A m30
m31 1 1 m00 − m01 B C B sd C C Bs C 1 B B m10 − m11 C B 1 C  B sd C  sd B C; B s2 C s0 B m20 − m21 C A @ A @ m30 − m31 ssd 3 0

(3)

1

spd 3

1

0

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(5a)

271

0 S
d

1

0

m00
m02

1

B s
d C B C B 1 C m10
m12 C 1 B B C B C;  B
d C 
d B C @ s2 A s0 @ m20
m22 A s
d 3

m30
m32 1 1 m00 − m02 B −d C B C B s1 C B m − m12 C C  1 B 10 C B B s−d C s−d B m − m C; @ 2 A 22 A 0 @ 20 m30 − m32 s−d 3 0

S−d

1

1

0

(5b)

4. Experimental Results and Discussion

and, 0

1

0

1

1 m00
m03 B rd C B C B s1 C B m
m13 C C  1 B 10 C Srd  B B srd C srd B m
m C; @ 2 A 20 23 A 0 @ m30
m33 srd 3 0 1 0 1 1 m00 − m03 B ld C B C B s1 C m10 − m13 C 1 B ld B C B C: S  B ld C  ld B C @ s2 A s0 @ m20 − m23 A m30 − m33 sld 3

(5c)

The 16 elements of the Mueller matrix are obtained from Eq. (5) : 1 1 sd m01  spd − ssd m00  spd 0 ; 0
s0 ; 2 2 0 1 1 − s−d m03  srd − sld m02  s
d 0 ; 0  2 0 2 0 1 1 pd sd sd sd m11  spd spd − ssd m10  spd 1 s0 ; 1 s0
s1 s0 ; 2 2 1 0 1 1 −d ld s
d − s−d m13  srd srd − sld m12  s
d 1 s0 ; 1 s0  2 1 0 2 1 0 1 1 sd sd spd
ssd m21  spd spd − ssd m20  spd 2 s0 ; 2 s0 ; 2 2 0 2 2 0 1 1
d −d −d ld m23  srd srd − sld m22  s
d 2 s0  2 s0 − s2 s0 ; 2 2 2 0 1 1 sd sd spd
ssd m31  spd spd − ssd m30  spd 3 s0 ; 3 s0 ; 2 3 0 2 3 0 1 1
d −d −d ld m32  s
d m33  srd srd − sld 3 s0 − s3 s0 ; 3 s0 : 2 2 3 0 (6) The above development has been realized under the consideration that the experimental setup has been calibrated with respect to the incident intensity associated to each polarization state. It is essential to assign a physical sense to the measured Mueller matrix. It means that, in the absence of any polarization-sensitive effects in the optical medium placed between the PSG and the PSA, any polarization state detected corresponds to the same polarization state generated. 272

We calibrated the experimental setup using the following procedure. We considered the horizontal plane of the optical table as the reference for both the PSG and the PSA, and found and fixed a value for the “azimuth angle” in the setup option of the PSG software when a horizontal linear polarization, p, had been generated and analyzed. We have verified that the experimental setup, with a fixed value of the “azimuth angle,” permits us to generate and analyze the remaining five polarization states.

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The main goal in this work is the full determination of the Mueller matrix associated with a commercial UV-LPFG. For the measurements of polarization properties of the grating, we used the experimental setup sketched in Fig. 1. The light source is a semiconductor laser, tunable within 1450–1590 nm range (Anritsu, Tunics Plus SC). This laser is connected to a deterministic polarization controller (DPC) (Thorlabs, model DPC5500) input. The optical signal with an arbitrary input polarization state enters on the DPC, while at its output we obtain a signal with a fixed and predetermined polarization state. The output signal from the DPC is used as a PSG for the fiber with the UV-LPFG being under study. The UV-LPFG is connected directly to the PSG (Thorlabs, model PAX5710/IR3) and the measurements are taken for the six incident polarization states p, s,
45, −45, r, and l. A computer controls the PSG and the PSA, and a computer program provides the results obtained. For each polarization state generated, the polarimeter analyzes the Stokes vector of the light beam leaving the system under study. Then, six measurements provide the 16 Mueller parameters required, according to Eqs. (5) and (6). The UV-LPFG studied in this work has a transmission wavelength response centered at 1543 nm. The UV-LPFG was manufactured by O/E Land Inc., model OEFBG-100 inv. Figure 2 shows the transmission spectrum of the UV-LPFG, which was measured using an unpolarized white light source (Ando, AQ4305) and an optical spectrum analyzer (Ando, AQ6315A) with a wavelength resolution of 2 nm.

Fig. 1. Experimental setup applied for the determination of the Mueller matrix associated with a UV-LPFG.

2

1.0000

0.0000

0.0000

0.0000

3

6 7 6 0.0166 0.4351 −0.7641 0.4664 7 6 7: (7) M Fiber  6 7 4 −0.0483 0.1339 −0.4744 −0.8704 5 −0.0007 0.8884 0.4365 −0.1548

Fig. 2. Transmission wavelength response of the UV-LPFG under study, the resonance is centered at 1543 nm.

To obtain the UV-LPFG response and make an analysis of its polarimetric properties, we have determined the Mueller matrix of the fiber with and without the grating at two symmetrically spaced wavelengths around 1543 nm. The normalized Mueller matrix obtained for the fiber without grating, at 1543 nm, is given by:

The form of the matrix Eq. (7) suggests a depolarization effect (the presence of nonzero values out of the main diagonal), and also the phase retardation. From the Mueller matrix obtained, we calculated the polarimetric response for the most important characteristic parameters. The results obtained are shown in Figs. 3(a)–3(d). The gain has the unitary value for all the incident polarization states, according to Fig. 3(a). The degree of polarization DoP [see Fig. 3(b)] has some kind of anisotropic behavior, dependent of the incident polarization states. Calculating the anisotropic degree of depolarization, a value of 0.0179 is obtained. The found Poincaré sphere [see Fig. 3(c)] does not have a spherical symmetry, which can be interpreted as small depolarization effects due to the reduction of the degree of polarization of light propagated through the core of the optical fiber without UV-LPFG inscribed. As a consequence

Fig. 3. Mueller matrix associated to the fiber studied here at 1543 nm with (a) unnormalized gain, (b) output degree of polarization, (c) the Poincaré output sphere, and (d) attenuation. 10 January 2014 / Vol. 53, No. 2 / APPLIED OPTICS

273

Fig. 4. Mueller matrix associated with the UV-LPFG with (a) unnormalized gain, (b) the output degree of polarization, (c) the Poincaré output sphere, (d) the attenuation, (e) the total, linear, and circular diattenuation and polarizance parameters, (f) the total diattenuation and polarizance for the fiber with and without the grating, (g) PDL for the fiber with and without the UV-LPFG, and (h) the depolarization index for the fiber with and without the grating.

of the birefringence arising from possible geometrical deviations of the core from the circular cross section, from uncontrollable transverse stress, as well as from some other achievable reasons, the poles of 274

APPLIED OPTICS / Vol. 53, No. 2 / 10 January 2014

the sphere are rotated. We should note that the fiber attenuation is practically zero due to the short length of the fiber (a few centimeters long) [27] [see Fig. 3(d)].

To understand the UV-LPFG behavior, we have determined the normalized Mueller matrix at 1543 nm: 2

1.0000

0.0023

0.0214 −0.0288

3

6 7 6 0.0331 0.1139 0.3761 −0.8831 7 6 7: M UV-LPFG1543  6 7 4 0.0421 −0.2451 0.8766 0.3569 5 0.0092 0.9611 0.1828 0.1241 (8) The matrix form suggests the presence of diattenuation effects (nonzero values in the first row) and polarizance (nonzero values in the first column). Moreover, we observe a marked depolarization effect that can be attributed to the presence the UV-LPFG on the fiber core. According to this, we can infer that the exposure of the fiber to the UV light beam, used for the grating writing, creates a slight birefringence in the fiber core, a result found and reported by other authors [28–32]. The wavelength of resonance of the UV-LPFG occurs at 1543 nm for an unpolarized broadband white light source. In this way, the grating transmission decreases correspond to increasing diattenuation. The results obtained are shown in Figs. 4(a)–4(h). The following information can be deduced from the graphical elements of the Mueller matrix. According to Fig. 4(a), the gain depends strongly on the polarization state incident in the UV-LPFG optical fiber, and the gain variation is due to the intrinsic birefringence of the fiber. On the other hand, the output degree of polarization [see Fig. 4(b)] shows a tendency to polarize light linearly and to depolarize slightly for some incident polarization states. The Poincaré sphere representation of the output polarization states, shown in Fig. 4(c), confirms the presence of depolarization effects and an increase of birefringence induced in the fiber core, due to the UV radiation exposure. The attenuation presented in Fig. 4(d) shows a strong dependence on the polarization state incident in the optical fiber, which is caused primarily by the existence of the UV-LPFG in the fiber core. Figure 4(e) shows the total, linear, and circular diattenuations and the polarizance contributions for incident light for various wavelengths of tunable laser (1533, 1538, 1543, 1548, and 1553 nm). The main contribution in the total diattenuation D depends of the linear diattenuation DL, which is

Table 2.

greater for the main resonance (1543 nm) and decreases at detuning from this. The observed circular diattenuation DC, has smaller, but nonnegligible values. Another parameter calculated in this work is the polarizance; the total polarizance, P, increases with the presence of the UV-LPFG, and the linear and the circular polarizance, PL and PC, are also wavelength-dependent [see Fig. 4(e)]. The linear polarizance is bigger than the circular polarizance; however, its contribution also decreases with wavelength detuning from the grating main resonance. Figure 4(f) shows the total diattenuation D, and the total polarizance P of the fiber, with and without the grating. When the UV-LPFG is presented on the fiber, the polarizance increases to 106%; from this figure, we can see that the contribution of the circular polarizance is greater in the fiber with the grating, although the contribution of the linear polarizance is dominant. Figure 4(g) shows the PDL for the fiber, with and without the grating. The PDL results as a consequence of the birefringence presented in the grating structure, and it is stronger at the principal resonance and diminishes with the wavelength at the ends of the UV-LPFG. Bearing in mind that the value of PDL was null in the fiber without LPFG [see Fig. 4(g)], the PDL values are intrinsically low in comparison with those found for gratings produced by other techniques; for instance, through mechanical stress [19,30], electric arc discharges, and gratings produced by CO2 laser radiation, among others. Finally, we calculated the depolarization index that appears strongly with the presence of the UV-LPFG, according to Fig. 4(h). From the calculus, we can conclude that the depolarization effect is stronger at the grating resonance wavelength, and it symmetrically decreases at detuning to both sides from the resonance wavelength (1543 nm). Additional information about the behavior of the UV-LPFG can be obtained from the polarimetric data, through their respective Mueller matrices [19,26]. Unlike commonly reported results obtained by considering solely two orthogonal polarization states, s and p, which correspond, somehow, to an approximation when the system under study does not have a simple symmetry, we are able to provide more precise information such as the circular diattenuation, the circular polarizance, among others. All of them depend on the incident polarization state and could be used to design and control the output signal from these fibers or from potential polarization-based

Polarimetric Data Obtained from the Mueller Matrix Associated with the UV-LPFG

λ (nm)

DM

DLM

DCM

PM

PL

PC

DI

QM

Tr

PDL (dBm)

Add

1553 1548 1543 1538 1533

0.0276 0.0345 0.0359 0.0113 0.0190

0.0196 0.0277 0.0214 0.0113 0.0189

0.0194 0.0205 0.0288 0.0000 0.0011

0.0242 0.0451 0.0543 0.0541 0.0952

0.0241 0.0432 0.0536 0.0498 0.0951

0.0023 0.0011 0.0092 0.0211 0.0026

0.9957 0.9941 0.9776 0.9939 0.9978

2.9714 2.9600 2.8621 2.9631 2.9854

0.9936 0.9912 0.9668 0.9909 0.9967

0.5512 0.6895 0.7175 0.2266 0.3797

0.0103 0.2940 0.0339 0.0153 0.0555

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devices. Table 2 shows the results obtained for several scalar depolarization metrics, where all of them have physically realizable values. The depolarization index DIM, the QM depolarization scalar metric, and the theorem of Gil–Bernabeu show consistent results, offering proof that the UV-LPFG shows a small depolarization effect at the resonance wavelength. This means, the Jones formalism can be employed to describe it there, once the small depolarization effects are taken into account, and at other wavelengths, where the effect of depolarization occurs in a smaller proportion. 5. Conclusion

We have presented the explicit relationships of the intensity measurements required for the experimental determination of the Mueller matrix associated to an arbitrary optical system. A set of six incident Stokes vectors has been used for the determination of the Mueller matrix associated to a commercial UV-LPFG and has also been applied to the fiber without the UV-LPFG, for comparison. We have calculated its depolarization metrics, because the performance of the optical fiber is directly related to its polarization properties. These metrics provide consistent results that indicate an increase in the birefringence induced in the fiber core, due to the exposure of the fiber to UV radiation, a behavior found and reported by other authors [13,28–32]. The anisotropic depolarization degree, Add, indicates the polarization direction at which this behavior occurs. The obtained PDL values are intrinsically low in comparison with those in gratings produced by other fabrication techniques. It should be noted that the method employed here to determine the scalar depolarization metrics provides more accurate information than what is usually reported, when only two orthogonal linear polarizations are considered. In this sense, a full polarimetric analysis could be used to design and control the output signal from these LPFGs or from potential polarization-based devices to realize the wavelength switchable fiber laser, among other applications. K. M. Salas-Alcántara and R. Espinosa-Luna acknowledge to CONACYT (México) for the financial support, under projects 100361 and Bisnano. References 1. A. M. Vengsarkar, P. J. Lemaire, J. R. Pedrazzani, J. B. Judkins, and V. Bhatia, “Long-period fiber-grating as bandrejection filters,” J. Lightwave Technol. 14, 58–65 (1996). 2. A. S. Kurkov, M. Douay, O. Duhem, B. Leleu, J. F. Henninot, J. F. Bayon, and L. Rivoallan, “Long-period fiber grating as a wavelength selective polarization element,” Electron. Lett. 33, 616–617 (1997). 3. C. Y. Lin, L. A. Wang, and G. W. Chern, “Corrugated longperiod fiber gratings as strain, torsion, and bending sensors,” J. Lightwave Technol. 19, 1159–1168 (2001). 4. V. Bhatia and A. M. Vengsarkar, “Optical fiber long-period grating sensors,” Opt. Lett. 21, 692–694 (1996). 5. D. E. Ceballos-Herrera, I. Torres-Gómez, A. Martínez-Rios, and J. J. Sanchez-Mondragon, “Torsion sensing characteristics of mechanically induced long-period holey fiber gratings,” IEEE Sensors J. 10, 1200–1205 (2010). 276

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