recent res. devel. optics, 8 (2013): 11-30 isbn: 978-81-308-0478-1 2. experimental ... · ·...
TRANSCRIPT
Research Signpost
37/661 (2), Fort P.O.
Trivandrum-695 023
Kerala, India
Original Article
Recent Res. Devel. Optics, 8 (2013): 11-30 ISBN: 978-81-308-0478-1
2. Experimental polarimetric properties of
long-period fiber gratings
Karla M. Salas-Alcántara, Rafael Espinosa-Luna and Ismael Torres-Gómez Centro de Investigaciones en Óptica, A. C., Loma del Bosque 115, Colonia Lomas del
Campestre, 37150 León, Guanajuato, México
Abstract. A Mueller-Stokes analysis of the polarization characteristics of both mechanical and UV-induced long-period fiber gratings is presented. An explicit methodology for the experimental determination of the complete Mueller matrix is reported. Some scalar polarimetric metrics are applied for the determination of the diattenuation (absorption), PDL, gain, attenuation, depolarization degree of anisotropy, among others. The results show a clear dependence on the incident polarization states, which could be employed to design and control the output signal from these fibers or from potential polarization-based devices.
Introduction
Long-period fiber gratings (LPFGs) are in-fiber optic components, which
have found attractive applications in optical communication networks and
optical measurement systems [1]. In optical telecommunication, this
component has been used in the equalization spectral gain of erbium doped
fiber amplifiers (EDFAs) and fiber lasers also band-rejection filters [2-3].
LPFGs have found application as polarization-dependent multi-wavelength
filter, polarization dependent loss compensation. In optical measurement systems,
Correspondence/Reprint request: Dr. Rafael Espinosa-Luna, GIPYS Lab, Centro de Investigaciones en Óptica
León, Guanajuato, México. E-mail: [email protected]
Karla M. Salas-Alcántara et al. 12
LPFGs have shown applications as optical transducers to sense physical and
chemical parameters, such as temperature, torsion, bending, stress, refractive
index, O2 concentration and pH, between others [4-5].
Long-period gratings are periodic structures that couple co-propagating
modes in optical fibers. Since the coupling ratio changes with wavelength,
the LPFG acts as a wavelength-dependent loss element. Phase matching
condition occurs at a range of different wavelengths associated to the
excitation of specific higher order cladding modes. The specific cladding
mode through which the core mode is coupled is determined by the following
resonance condition:
)( m
cladcoreresm nn (1)
where ncore and nclad are the effective refraction indices of the fundamental
guided mode and the m-th order cladding mode, Λ is the period of the
grating, and λm res is the resonance wavelength. The resonant wavelengths
depend on the fiber characteristics through the effective indices of the core
and cladding. Hence, the change of the grating period Λ and the effective
refraction index of the cladding mode will affect the resonant wavelength of
the corresponding cladding mode. This property of the LPFGs makes
attractive for applications in optical, biological and chemical sensing and
many more [6-8]. According to the LPFG fabrication technique, the
respective fabrication parameters, and also on the fiber type, several
mechanisms may contribute simultaneously to the grating formation [9].
Consequently, the polarization optical response depends on the characteristics
of the grating. Several studies about the polarization dependent loss (PDL)
have been published; however, these studies are focused on the response to
only two states of polarization mutually orthogonal [10].
This chapter is dedicated to the determination of the Mueller matrix
associated to both a mechanical- and an ultraviolet (UV)- induced long-
period fiber gratings. The Mueller matrix, MM, is determined through the
Stokes vectors, which are measured by using an incomplete, commercially
available, Stokes polarimeter. The 4-method has been used for the
determination of the MM associated to a commercially available ultraviolet
long-period fiber grating (UV-LPFG) in a H2 pre-loading fiber (all fiber
measurement) and a mechanically induced long-period fiber grating
(M-LPFG) in a photonic crystal fiber, in an open space measurement. Results
show an increasing in the birefringence when the LPFG is present in the
fiber. The PDL values of the UV-LPFG are intrinsically low in comparison to
the grating produced by mechanically induced technique. It should be noted
that the method employed here to determine the scalar depolarization metrics
Experimental polarimetric properties of long-period fiber gratings 13
provides more accurate information than the usually reported, where only two
orthogonal linear polarizations are used.
1. Long-period fiber gratings fabrication technique
The fabrication of LPFGs implies the introduction of a periodic index
modulation of the core in the optical fiber. This can be obtained by
permanent modification of the refractive index of the fiber core or by
temporal physical deformation of the fiber, respectively. Techniques like UV
irradiation, laser CO2 exposure, electric arc discharge, mechanical
microbends, etched corrugations, ion beam implantation, and femtosecond
laser exposure, among others [11-13]. In the following section, a brief
overview of the manufacture of LPFGs by UV induced and mechanical
methods will be described. In sections 8 and 9 we will determine the Mueller
matrix and calculated some scalar polarimetric metrics for LPFGs
manufactured by these two techniques.
2. LPFGs manufactured by UV radiation
The ultraviolet (UV) irradiation was the first method used to manufacture
gratings in 1989 by Meltz et al., who used holographic interference between
two coherent beams directed to the fiber core [14]. The UV exposure method
requires that the fiber first be made receptive to UV irradiation, i.e. the fiber
must be made photosensitive prior to writing the grating. It can be obtained
by doping the fiber core (usually of silica) with impurity atoms (such as
germanium, boron or a combination of these elements through the fabrication
process) or by hydrogen loading exposing the fiber to high-pressure H2 gas at
elevated temperatures for a long period of time so that hydrogen diffusion
into the core material takes place. The later method is preferable because
hydrogen loading can be achieved in standard fibers, providing a cheaper and
simpler way to obtain UV photosensitivity fibers [15].
The UV inscription method for LPFGs can be made in two other ways.
One of them consists on the irradiation of the fiber through an amplitude
mask with the desired period [16]. The inscription occurs when the UV
irradiation is exposed in front of the mask where alternate bands of maximum
intensity are transmitted. The exposure is repeated until the index modulation
has reached a sufficient level to provide the desired attenuation depth in the
LPFG transmission spectrum. This technique implies a permanent
modification of the refractive index of the fiber core. A second way to
generate gratings with UV irradiation is based on a periodic point by point
modulation of the core refractive index until the desired grating length is
Karla M. Salas-Alcántara et al. 14
obtained. In general, both techniques above described are considered
expensive.
3. Mechanically induced long-period fiber gratings
The optical refractive index of glass can be modulated when the glass is exposed to stress. Because the index modulation period in a LPFG can be as large as hundreds of micrometers, it is possible to induce it mechanically via the photoelastic effect. Different techniques of mechanically induced LPFGs (M-LPFGs) have been reported in standard fibers and microstructured fibers, where corrugated plates, strings, and springs have been used to apply periodic mechanical stress on the optical fiber in order to induce the effective index modulation to obtain the coupling of light from the fundamental mode to cladding modes [17-19]. One of the most relevant characteristics that share these techniques is their tunability by simple adjustment of the mechanical stress period. Under this concept, mechanically induced LPFGs offer a tuning range at least one order of magnitude wider than other methods reported in tunable permanent recorded LPFGs with similar isolation loss and line-width.
4. Mueller matrix method
Polarization effects play a key role in the operation of many important optical devices. The complete characterization of scattered light is described by Stokes vectors and Mueller matrices. The most general form of the scattering matrix coupled with polarizer and quarter wave plates demonstrates the physical relationships among the matrix elements and polarization measurements. The Mueller matrix method is most commonly suited for describing irradiance-measuring instruments, including most polarimeter, radiometers and spectrometers. In the Mueller matrix method (MMM), the Stokes vector S is used to describe the polarization state of a light beam, and the Mueller matrix M to describe the polarization-altering characteristics from a sample. This sample may be a surface, a polarization element, an optical system, biological tissues, and any other light-matter interaction, which produces a reflected, refracted, diffracted, absorption, or scattering of the light beam. Popular theories of polarized light interaction with optical elements or scattering media may be divided into two groups: Jones matrix, which assumes a coherent addition of waves; and the Stokes-Mueller matrix, which assumes an incoherent addition of waves. In both approaches, one usually starts a theoretical analysis with severely restrictive assumptions. In the Jones matrix, one starts out with Maxwell´s equations, whereas in the Mueller matrix, it starts by postulating a linear relation between the input Stokes vector and the
Experimental polarimetric properties of long-period fiber gratings 15
output Stokes vector emerging from the optical medium. The development of the Mueller analysis is heuristic and has an advantage in that it deals with intensities rather than field vectors. The Mueller matrix represents the polarization rotation characteristics of
an optical device such as an optical fiber, which is determined by the
relationship between a set of input polarization vectors and their
corresponding output polarization vectors. The linear response of a physical
system can be expressed in terms of the intensities, through the relation [20]
iiii
iiii
iiii
iiii
i
i
i
i
o
o
o
o
io
smsmsmsm
smsmsmsm
smsmsmsm
smsmsmsm
s
s
s
s
mmmm
mmmm
mmmm
mmmm
s
s
s
s
MSS
333232131030
323222121020
313212111010
303202101000
3
2
1
0
33323130
23222120
13121110
03020100
3
2
1
0
(2)
where M is called the Mueller matrix of the system, represented as a 4 4
matrix with real elements, and S is the Stokes vector. S represents the
polarization state of light, defined in terms of the orthogonal components of
the electric field vector (Ep, Es) and its phase difference.
)**(
**
**
**
3
2
1
0
a
p
a
s
a
s
a
p
a
p
a
s
a
s
a
p
a
s
a
s
a
p
a
p
a
s
a
s
a
p
a
p
a
a
a
a
a
EEEEi
EEEE
EEEE
EEEE
s
s
s
s
S
(3)
where a = incidence (i) or output (o). Angular brackets represent temporal
averages and * indicates complex conjugation of a
spE , , 12i is the complex
number. The upper (lower) sign in the right-hand side of as3 corresponds to a
description of polarization states as looking to the source (propagation
direction).
The normalized Stokes parameters can be displayed in a real three
dimensional space as a function of the azimuth 0 and the ellipticity
angles 44
of the polarization ellipse, respectively.
)2sin(
)2sin()2cos(
)2cos()2cos(
1
0sS
(4)
Karla M. Salas-Alcántara et al. 16
Where 0s represents the average intensity associated to the Stokes parameters;
usually, it is fixed to the unitary value.
5. Ideal polarimeter arrangement (IPA)
A typical polarimeter arrangement consists of a Source + PSG + Sample
+ PSA + Detector. See figure 1. The incident Stokes vectors are generated by
a Polarizer State Generator, PSG, and a source; while the scattered Stokes
vector from the sample under study is analyzed or “filtered” by a Polarizer
State Analyzer, PSA. The exiting intensity from the PSA is measured by a
sensor or detector device. Usually a PSA is named as a Polarization State
Detector, PSD, when the analyzer is a system composed of a PSA plus a
sensor (detector) of intensity. It is common to find a mirror-symmetry order
between the optical components that constitute a PSG (linear polarizer +
quarter-wave retardation plate) and a PSA (quarter-wave retardation plate +
linear polarizer), for a complete polarimeter arrangement. For a passive Ideal
Polarimeter Arrangement, IPA, the output Stokes vector reaching the detector
is given as
3
0
3
0
det
k
i
kjk
j
ij
oi smaASAMSS
(5)
Where A is the Mueller matrix that describes the action of the PSA on the
Stokes vector scattered (So) from the sample (M). An IPA arrangement is
formed by classical, passive, optical polarization elements, like linear
polarizers (LP) based in calcite crystal (of the Glan-Thompson, Glan-Laser,
Glan-Taylor, Rochon, or of the Wollaston-type), wave-plate retarders made
of mica, quartz, polarizing thin-films, among many other possibilities, and
where the theoretical linear response is supposed to be ideal. Indeed, there is
not any formal restriction if the passive retarders of an IPA are changed by
Liquid Crystal Variable Retarders, LCVR, set to a fixed voltage.
Figure 1. A typical complete polarimeter arrangement (IPA).
Experimental polarimetric properties of long-period fiber gratings 17
6. Mueller matrix determination
For an arbitrary optical system and by considering an IPA setup, it has
been shown that the 16 Mueller elements can be obtained from a set of 4
incident Stokes vectors (PSG) and 4 states filtered or analyzed (PSA) [21].
The Stokes vectors correspond to linear polarization states parallel (p),
perpendicular (s), and to +45 degrees (+) respect to the incidence plane,
respectively, and to a right-hand (r) polarization state. The polarized states
mentioned above, form a tetrahedron inscribed into/within the Poincaré
sphere, whose vertexes are contained on the surface. The 16 arbitrary Mueller
elements can be determined from the following 16 intensity-measurements
taken from the so detected Stokes element [22]:
)(2
1);(
2
1
)(2
1);(
2
1
)(2
1);(
2
1
)(2
1);(
2
1
)(2
1);(
2
1
)(2
1);(
2
1
)(2
1);(
2
1
)(2
1);(
2
1
3330030023200300
1310030013100300
3230020022200200
1210020012100200
3101300021012000
1001110011011000
3130010021200100
1110010011100100
mmmmImmmmI
mmmmImmmmI
mmmmImmmmI
mmmmImmmmI
mmmmImmmmI
mmmmImmmmI
mmmmImmmmI
mmmmImmmmI
rrr
rsrp
r
sp
srs
sssp
prp
pspp
(6)
where Imn = (s0)mn denotes the intensity measured when an m-polarized state
illuminates the sample and an n-polarized state is analyzed. The Mueller
matrix is obtained by solving the equation (6) for each of the 16 mij elements.
On the other hand, we have used a commercially available deterministic
polarizer generator and an incomplete polarimeter analyzer (PAX) for the
determination of the Mueller matrix, where the normalized output is registered
as:
3
2
1
´
´
´
1
s
s
sS PAX
(7)
Karla M. Salas-Alcántara et al. 18
From Eq. (2), the scattered Stokes vector incident onto the PSD analyzer is
closely related to the incident Stokes vector on the sample. For the set of 4
incident polarization states considered here, the scattered Stokes vectors from
the sample (represented as sio
) are expressed by [22, 25]
3330
2320
1310
0300
3230
2220
1210
0200
3130
2120
1110
0100
3130
2120
1110
0100
,,,
mm
mm
mm
mm
S
mm
mm
mm
mm
S
mm
mm
mm
mm
S
mm
mm
mm
mm
S roosopo
(8)
The “power” detected is given by the total intensity associated to each
scattered Stokes vector, according to
03000020000100001000 ,,, mmsmmsmmsmms rddsdpd
(9)
On the other hand, the detected normalized Stokes vectors (Sid
) are displayed
as
3130
2120
1110
0100
0
3
2
1
3130
2120
1110
0100
0
3
2
1 1
1
,1
1
mm
mm
mm
mm
s
s
s
sS
mm
mm
mm
mm
s
s
s
sS
sd
sd
sd
sd
sd
pd
pd
pd
pd
pd
(10a)
And
3330
2320
1310
0300
0
3
2
1
3230
2220
1210
0200
0
3
2
1 1
1
,1
1
mm
mm
mm
mm
s
s
s
sS
mm
mm
mm
mm
s
s
s
sS
rd
rd
rd
rd
rd
d
d
d
d
d
(10b)
The 16 elements of the Mueller matrix are obtained from eqs. (9 and 10)
300333300332030331030330
2002232002220202210202120
100113100112010111010110
000030000200010000
,),(2
1),(
2
1
,),(2
1),(
2
1
,),(2
1),(
2
1
,),(2
1),(
2
1
mssmmssmssssmssssm
mssmmssmssssmssssm
mssmmssmssssmssssm
msmmsmssmssm
rdrdddsdsdpdpdsdsdpdpd
rdrdddsdsdpdpdsdsdpdpd
rdrdddsdsdpdpdsdsdpdpd
rddsdpdsdpd
(11)
Experimental polarimetric properties of long-period fiber gratings 19
7. Polarimetric scalar metrics
By using the following depolarization scalar metrics (see Table 1) it is
possible to analyze the polarimetric properties associated to the long-period
fiber grating and its capability to depolarize light, in order to predict and
understand the LPFGs behavior [22-24].
Table 1. Polarimetric scalar metrics applied for the analysis of the obtained Mueller
matrix.
Depolarization index,
DI(M). 13/)(0 00
2/13
0,
2
00
2 mmmMDIkj
jk
Degree of polarization,
DoP(M,S). 1
)()()()(
),(0303202101000
2
1
3
1
2
33221100
0
2
3
2
2
2
1
iiii
j
i
j
i
j
i
j
i
j
o
ooo
smsmsmsm
smsmsmsm
s
sssSMDoP
Anisotropic degree of
depolarization, Add. 1
)()(
)()(0
min
min
oo
Max
oo
Max
DoPDoP
DoPDoPAdd
Diattenuation parameter,
D(M).
1/)(0 00
2
03
2
02
2
01 mmmmMD
Polarizance parameter,
P(M).
1/)(0 00
2
30
2
20
2
10 mmmmMP
The Q(M) metric.
3)]([1
)]([/
)]([1
)]([)]([3)(0
2
22
00
3
1,
2
2
22
3
0
2
0
3
0,1
2
MD
MPmm
MD
MDMDI
m
m
MQkj
jk
k
k
kj
jk
The Gil-Bernabeu
theorem.
2
004)( mMMTr t
The polarization
dependent loss, PDL.
2/12
03
2
02
2
0100
2/12
03
2
02
2
0100
)(
)(log10min)max/log(10
mmmm
mmmmTTPDL
Gain, g. 10
0
303202101000
0
0
i
iiii
i
o
s
smsmsmsm
s
sg
8. MM of the LPFG induced by UV radiation
In Section 2, we have described briefly the technique used to fabricate
the UV-long-period fiber grating. Now, we present the Mueller matrix and
the polarimetric characterization associated to a commercially available UV
Karla M. Salas-Alcántara et al. 20
induced long-period fiber in a H2 pre-loading fiber, when the PSG and the
PSA are contained into an enclosed space (all optical fiber). In order to
determine the MM of the UV-LPFG, the setup shown in figure 2 was used.
The source is a Tunable Laser at 1450-1590 nm, from Anritsu, Tunics Plus
SC. This laser is connected to a Deterministic Polarization Controller
(Thorlabs, model DPC5500) input. The output signal from the DPC is used as
a PSG for the fiber under study, which is connected directly to the polarizer
state analyzer PSA (Thorlabs, model PAX5710/IR3) and the measurements
are taken for the 4 incident polarization states p, s, +45, and r, respectively. A
laptop computer controls the PSG and the PSA and a software program
provides the results obtained.
Figure 3 shows the transmission spectra of the UV-LPFG studied here.
The UV-LPFG transmission spectra have been measured using an un-polarized
Figure 2. Experimental setup applied for the determination of the Mueller matrix
associated to a UV-LPFG in a H2 pre-loading fiber.
Figure 3. Transmission wavelength response of UV long-period fiber grating with
resonance centered at 1543 nm.
Experimental polarimetric properties of long-period fiber gratings 21
source (white-light source AQ4305) and an optical spectrum analyzer
(AQ6315A) with a wavelength resolution of 2 nm.
For the analysis of the polarimetric properties, we have determined the
Mueller matrix of the fiber with and without the UV-LPFG, at the main
resonance (1543 nm). The normalized Mueller matrix obtained for the fiber
without grating, at 1543 nm, is given by
1343.04444.08884.00007.0
8235.04294.01339.00483.0
4540.07746.04351.00166.0
0000.00000.00000.00000.1
FiberM
(12)
We can obtain some insight about the polarimetric characteristics of a given
system, by analyzing the form of its associated Mueller matrix. The form of
this matrix, Eq. (12), suggests a depolarization effect (the presence of non-
zero values on the upper and the lower side of the main diagonal) and phase
retardation also. It should be noted that this information could not be obtained
if only the p- and the s-polarization states were employed as the incident
Stokes states.
To show the general potential from the MM obtained, Eq. (12), we have
calculated the polarimetric response for some characteristic parameters (table 2
and figure 4). The gain has the unitary value for all the incident polarization
states, according to Fig. 4(a). The degree of polarization DoP, Fig 4(b), has
some kind of anisotropy behavior dependent of the incident polarization
states. The deterioration mechanism in the polarization degree is based on the
assumption that an arbitrary incident polarized light is split into two separate
eigen-polarization modes, which propagate at different group velocity values.
Calculating the anisotropic degree of depolarization, a value of 0.022 is
obtained. The Poincaré output response, Figure 4(c), has not a spherical
symmetry, which can be interpreted as the presence of depolarization effects
due to the reduction of the degree of polarization of light at the propagation
through the optical fiber. As a consequence of the birefringence, due to possible
deviations of the core from the circular cross section, as well as transverse
stress, the poles of the sphere rotate.
In order to investigate the influence of UV-LPFG on the fiber, at 1543 nm,
the corresponding (normalized) Mueller matrix obtained is given by
0971.02047.09611.00092.0
3487.08472.02451.00421.0
8735.03532.01139.00331.0
0067.00069.00023.00000.1
1543LPFGM
(13)
Karla M. Salas-Alcántara et al. 22
Figure 4. Analysis of the Mueller matrix associated to H2 pre-loading fiber. Fig. (4a)
gain, (4b) degree of polarization, (4c) the Poincaré output sphere, and (4d) attenuation
(dB).
The form of this matrix, suggests the presence of diattenuation effects,
polarizance and a marked depolarization effect that can be attributed to the
presence of UV-LPFG in the fiber (the exposure of the fiber to the UV beam
creates an index change in the core). The grating coupling strength occurs at
1543 nm for un-polarized broadband white light source. In this way, the
transmission is reduced; that is, the diattenuation increases also. The results
obtained are shown in figure 5 and table 2.
According to figure 5(a), the gain depends strongly of the polarization
state incident in the optical fiber (basal plane); the variation is due to the
intrinsic birefringence of the fiber. On the other hand, the output degree of
polarization, figure 5(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, figure 5(c), confirms the
presence of depolarization effects and an increase of birefringence induced in
Experimental polarimetric properties of long-period fiber gratings 23
Figure 5. Analysis of the Mueller matrix associated to the UV-LPFG fiber. Fig. (5a)
gain, (5b) output degree of polarization, (5c) the Poincaré output sphere, (5d)
attenuation (dB).
Table 2. Polarimetric data obtained from the Mueller matrix associated to the fiber
without and with UV-LPFG.
D(M) P(M) DI(M) Q(M) Tr PDL Add
Fiber 0.0000 0.0511 0.9804 2.8834 0.9709 0.0000 0.0222
UV-LPFG 0.0671 0.0543 0.9631 2.7658 0.9547 1.3449 0.0574
the fiber core due to the UV radiation exposure. The attenuation, figure 5(d),
shows a strong dependence of the polarization state incident in the optical
fiber, which is caused because the transmission is reduced due to the presence
of the UV-LPFG.
Additional information about the behavior of the pre-loading H2 fiber,
and the fiber, with and without a UV-LPFG can be obtained from the
polarimetric data (table 2).
From previous results (table 2) we can build the following conclusions:
The diattenuation parameter, D(M), indicates the UV-LPFG generates a
Karla M. Salas-Alcántara et al. 24
diattenuation effect, not present without the induced grating. In this sense the
diattenuation, D(M), increases greatly with the presence of the UV-LPFG.
This behavior can be understood as follows: The period of the LPFG shows a
resonance wavelength at 1543 nm. In this way, the transmission is reduced
because the resonance occurs; that is, the diattenuation is increased. The
LPFG affects the response to the transmitting polarization also. The physical
mechanism responsible of this behavior must be associated to a dichroic,
highly birefringent change in the core due to the UV-LPFG. The
depolarization index, DI(M), the Q(M) depolarization scalar metric, and the
theorem of Gil-Bernabeu, all of them provide consistent results that indicate
effect of depolarization. We can also observe that the PDL parameter value
just indicates that the fiber is affected by the presence of the UV-LPFG as a
consequence of the birefringence present in the grating structure. The PDL
values are intrinsically low in comparison to gratings produced by other
techniques used to fabricate them like through mechanical stress [22, 26],
electric arc discharges, gratings produced by CO2 laser radiation, among
others. Even more when the response to each of the 4-polarization incident
states generates a high degree of polarization output states (see table 1), the
arithmetic average behavior shows a tendency to a loss of the DoP for any
PSG when the UV-LPFG is on the fiber.
9. MM of the mechanically induced long-period fiber grating
In this section the Mueller matrix associated to a mechanically induced
long-period fiber grating is analyzed. The MM measurements have been
performed in the fiber before and after grating inscription, respectively. For
the measurements, we have used the experimental setup depicted schematically
in figure 6. It consists of a source (continuous-wave Ytterbium Doped Fiber
Laser at 1064 nm, from IPG Photonics, PYL-10LP). This laser illuminates
the polarizer state generator, PSG, which consists of a linear polarizer of the
Glan-Laser type (Thorlabs, model GL10), a half-wave plate (Thorlabs,
WPMH05M-1064) and a quarter-wave plate (Thorlabs, WPMQ05M-1064).
The fiber used in the experiment is a commercial photonic crystal fiber
(PCF) denominated F-SM10, figure 7. The Mechanically induced LPFG
(M-LPFG) was generated by pressing a section of the Photonic Crystal Fiber
between two corrugated grooved plates (CGPs). In figure (7a) it can be
observed the spectrum of a LPFG with a period of 480 µm; the period of the
M-LPFG has been designed to show resonance at the wavelength employed
(1064 nm). The PCF is connected directly to the polarizer state analyzer, PSA
(Thorlabs Polarimeter, PAX5710/IR2) and the measurements are taken for
Experimental polarimetric properties of long-period fiber gratings 25
Figure 6. Experimental setup employed for the determination of the Mueller matrix
associated to a photonic crystal fiber and to a photonic crystal fiber with a
mechanically induced long-period fiber grating, respectively [22].
Figure 7. a) The transversal structure of the F-SM10. b) Transmission wavelength
response of the M-LPFG in the PCF.
the 4-incident polarization states p, s, +45 and r respectively. In order to show
the quality of the polarization states generated and detected by our
polarimetric system. We have self-calibrated our equipment with respect to
the air [22].
Using the PCF as the calibration reference, the normalized Mueller
matrix obtained for the PCF, is given by
9651.03109.00250.00220.0
1641.09183.00669.00128.0
1641.02206.09970.00014.0
0000.00000.00000.00000.1
PCFM
(14)
where we can observe that the value of elements of the main diagonal
Mueller matrix are close to unitary values. An important result we have found
Karla M. Salas-Alcántara et al. 26
here, is just that the PCF has a tendency to maintain the incident polarization
states. The first row indicates there is not diattenuation (light travels through
the fiber without perturbation). However, the presence of non-zero values on
the upper and the lower side of the main diagonal, indicates the presence of
phase retardation effects. The following information can be deduced of the
graphical elements of the Mueller matrix (see figure 8).
According to figure (8a), the gain has the unitary value for all the
incident polarization states. The output degree of polarization varies around
the unitary value and the PCF has a strongly dependence on the input
polarization states, figure (8b). The anisotropic degree of depolarization
provides a value of 0.1718. We believe this value reflects an intrinsic
anisotropic depolarization behavior associated to the PCF, which can not be
identified if only the orthogonal polarizations p and s are used. Finally, the
Poincaré output sphere suffers a slightly deformation with respect to the
spherically symmetric Poincaré sphere associated to the input polarization
states. This is a consequence of the non-zero value associated to Eq. (14).
Figure 8. Analysis of the Mueller matrix associated to the PCF. Figures (8a) show the
gain, (8b) the output degree of polarization, (8c) the Poincaré output sphere and (8d)
the attenuation.
Experimental polarimetric properties of long-period fiber gratings 27
Finally, the attenuation in the fiber is practically zero due to the length of the
PCF is short (140 cm), see figure (8d).
In order to know the polarimetric response of a LPFG formed by
mechanical-induced pressure technique, the normalized Mueller matrix was
obtained also [22]
3266.03759.08056.00077.0
0762.05138.04730.00070.0
4761.05622.03471.00812.0
3463.02004.00187.00000.1
LPFGPCFM
(15)
The M-LPFG on the PCF fiber produces evident effects such as diattenuation (first row) and a marked depolarization effect (first row and the main diagonal values). In this case there is an increase of phase retardation and depolarization effects also, according to the values shown in the upper and the lower diagonal parts of the matrix given by Eq. (15). The transmission is reduced because the phase matching condition is fulfilled; that is, the diattenuation is increased. The polarizance, P(M), increases 320% with the presence of the M-LPFG (see table 3). The Add value increases with the presence of the M-LPFG on the PCF, which means the system becomes more anisotropic for some incident polarization states. By drawing the Mueller elements in a real three dimensional space as a function of the azimuth and the ellipticity angles of the polarization ellipse, the following information was obtained. The gain, figure (9a) depends strongly of the polarization state incident in the photonic crystal fiber. On the other hand, the output degree of polarization, figure (9b), shows a tendency to polarize light linearly and to depolarize with the presence of the M-LPFG. Finally, the Poincaré sphere, figure (9c) confirms the previous arguments: a reduction of the unitary radius is associated to anisotropic depolarization effects, and a deviation of the spherical shape is associated to an anisotropic degree of depolarization, Eq. (15), and the rotation of the axis, is associated to the presence of birefringence. These effects increase with the presence of the M-LPFG in the PCF. Experimentally it is very hard to obtain a perfect generation and analysis of the polarization states, mainly for circular polarization. This is the reason why the output degree of polarization is slightly up to its physical limit. However, the results we have reported here are as close as possible within the precision values considered by the manufacturer of our equipment. Additional information about the behavior of the UV-LPFG can be
obtained from the polarimetric data, through their respective Mueller matrices,
see table 3.
Karla M. Salas-Alcántara et al. 28
Figure 9. Analysis of the Mueller matrix associated to the PCF with M-LPFG. Figures
(9a) show the gain, (9b) the output degree of polarization, (9c) the Poincaré output
sphere and (9d) the attenuation.
Table 3. Polarimetric data obtained from the Mueller matrix associated to the PCF and to the PCF+M-LPFG.
D(M) P(M) DI(M) Q(M) Tr PDL(dB) Add
PCF 0.0000 0.0255 0.9914 2.9489 0.9872 0.0000 0.1718
PCF+LPFG 0.4006 0.0818 0.8604 1.7755 0.8052 8.4864 0.1751
The diattenuation parameter, D(M), indicates the M-LPFG generates a
diattenuation effect, not present without the induced grating (which is used
here as the reference). The depolarization index, DI(M), the Q(M)
depolarization scalar metric, and the theorem of Gil-Bernabeu, all of them
provide consistent results that indicate the presence of the LPFG increases in
a 15% the effect of depolarization. This numerical result is coherent with the
qualitative behavior presented with the deformation spherical shape of the
Experimental polarimetric properties of long-period fiber gratings 29
Poincaré output sphere, Fig. (9c). We can also observe that the PDL
parameter value just indicates that the PCF is affected by the presence of the
M-LPFG. Even more when the response to each of the 4-polarization incident
states generates a high degree of polarization output states (see table 3), the
arithmetic average behavior shows a tendency to a loss of the DoP for any
PSG when the M-LPFG is in the PCF.
10. Conclusions
We have presented the explicit relationships for the intensity
measurements required for the experimental determination of the Mueller
matrix associated to an arbitrary optical system. The 4-method has been used
in the determination of the Mueller matrix associated to both UV-and
mechanical- induced and long-period fiber gratings. Because the performance
of the optical fiber is directly related to its polarization properties, we have
calculated the depolarization index, DI(M), the Q(M) depolarization scalar
metric, the theorem of Gil-Bernabeu, the degree of polarization, DoP, and
the anisotropic depolarization degree, Add. These metrics provide consistent
results that indicate an increasing in the birefringence when the LPFG is
presented in the fiber. The PDL values of the UV-LPFG are intrinsically low
in comparison to gratings produced by mechanically induced technique. It
should be noted that the method employed here to determine the scalar
depolarization metrics provides more accurate information than the usually
reported where only two orthogonal linear polarizations are used, which
could be used to design and control the output signal from these fibers or
from potential polarization-based-devices such as optical detectors, optical
components, optical testing sets, and optical fiber sensing.
Acknowledgements
K. M. Salas-Alcántara and R. Espinosa-Luna acknowledge to CONACYT
for the economical support provided for the realization of this work, under
projects 100361 and Bisnano.
References
1. Vengsarkar, A. M., Pedrazzani, J. R., Judkins, J. B., Lemaire, P. J. 1996, Opt.
Lett., 21, 336.
2. Pandit, M. K., Chiang, K. S., Chen Z. H., Li., S. P. 2000, Microwave Opt.
Technol. Lett., 25, 181.
Karla M. Salas-Alcántara et al. 30
3. Eggleton, B. J., Slusher, R. E., Judkins, J. B., Stark, J. B., Vengsarkar, A. M.
1997, Opt. Lett., 22, 883.
4. Yan, M., Luo S., Zhan L., Zhang, Z., Xia, Y. 2007, Opt. Express, 15, 3685.
5. Ceballos-Herrera, D. E., Torres-Gómez, I., Martínez-Rios, A., Sanchez-
Mondragon J. J. 2010, IEEE Sens. J., 10, 1200.
6. Tang, J. L, Wang J. N. 2008, Sensors, 8, 171.
7. Rindorf L., Jensen J. B. 2006, Opt. Express, 14, 8224.
8. Tatam J. S. 2003, Meas. Sci. Technol., 14, R49.
9. Martinez-Rios A., Monzon-Hernández D., Torres-Gomez, I., Salceda-Delgado G.
2012, Fiber Optic Sensors, InTech. 11, 275.
10. Rego G., Melo M, Santos J. L., Salgado H.M. 2006, Opt. Commun., 262, 152.
11. Davis, D. D., Gaylod, T. K., Glytsis, E. N., Kosinski, S. G., Mettler S. C.,
Vengsarkar, A. M. 1998, Electron. Lett., 26, 61.
12. Hwang, I. K., Yun, S. H., Kim, B. Y. 1999, Opt. Lett., 24, 1263.
13. Rego G., Dianov E., Sulimov, V. 2001, Lightwave Technol., 19, 1547.
14. Meltz G., Morey, W., Glenn W. 1989, Opt. Lett., 14, 823.
15. Hill, K., Fuji, Y., Johnson, D., Kawasaki, B. 1987, Appl. Phys. Lett., 62, 647.
16. Kashyap R. 1999, Optics and Photonics, 3, 85.
17. Savin, S., Digonnet, M. J. F., Kino, G. S., Shaw, H. 2000, Opt. Lett., 25, 710.
18. Rego, G., Fernandez, J. R. A., Santos, J. L., Salgado, H. M., Marques, P. V. S.
2003, Opt. Commun., 220, 111.
19. Yokouchi, T., Suzaki, Y., Nakagawa, K., Yamauchi, M., Kimura, M., Mizutani,
Y., Kimura, S., Ejima, S. 2005, Appl. Opt., 44, 5024.
20. Goldstein D. 2003, Polarized Light, second ed., Marcel Dekker.
21. Espinosa-Luna R., Mendoza-Suárez, A., Atondo-Rubio, G., Hinojosa, S., Rivera-
Vázquez, J. O, Guillén-Bonilla, J. T. 2006, Opt. Commun., 259, 60.
22. Salas-Alcántara, K., Espinosa-Luna, R., Torres-Gómez, I. 2012, Opt. Eng.,
51, 085005.
23. Peinado, A., Lizana, A., Vidal, J., Iemmi, C., Campos, J. 2010, Opt. Express,
18, 9815.
24. Gil, J. J., Bernabeu, E. 1985, Opt. Acta 32, 259.
25. Atondo-Rubio, G., Espinosa-Luna, R., Mendoza-Suárez, A. 2005, Opt. Commun.,
244,7.
26. Efimov, T. A., Bock, W. J., Chen J., Mikulic P. 2009, Lightwave Tech., 27, 3759.