Hydrostatic Pressure and Temperature

sensors Article

Hydrostatic Pressure and Temperature Measurements Using an In-Line Mach-Zehnder Interferometer Based on a Two-Mode Highly Birefringent Microstructured Fiber Gabriela Statkiewicz-Barabach 1, *, Jacek Olszewski 1 , Pawel Mergo 2 and Waclaw. Urbanczyk 1 1

2

*

Department of Optics and Photonics, Faculty of Fundamental Problems of Technology, Wroclaw University of Science and Technology, 50-370 Wroclaw, Poland; [email protected] (J.O.); [email protected] (W.U.) Laboratory of Optical Fiber Technology, Maria Curie-Sklodowska University, 20-031 Lublin, Poland; [email protected] Correspondence: [email protected]; Tel.: +48-71-320-4117

Received: 7 June 2017; Accepted: 14 July 2017; Published: 18 July 2017

Abstract: We present a comprehensive study of an in-line Mach-Zehnder intermodal interferometer fabricated in a boron-doped two-mode highly birefringent microstructured fiber. We observed different interference signals at the output of the interferometer, related to the intermodal interference of the fundamental and the first order modes of the orthogonal polarizations and a beating of the polarimetric signal related to the difference in the group modal birefringence between the fundamental and the first order modes, respectively. The proposed interferometer was tested for measurements of hydrostatic pressure and temperature for different alignments of the input polarizer with no analyzer at the output. The sensitivities to hydrostatic pressure of the intermodal interference signals for x- and y-polarizations had an opposite sign and were equal to 0.229 nm/MPa and −0.179 nm/MPa, respectively, while the temperature sensitivities for both polarizations were similar and equal 0.020 nm/◦ C and 0.019 nm/◦ C. In the case of pressure, for the simultaneous excitation of both polarization modes, we observed a displacement of intermodal fringes with a sensitivity depending on the azimuth of the input polarization state, as well as on the displacement of their envelope with a sensitivity of 2.14 nm/MPa, accompanied by a change in the fringes visibility. Such properties of the proposed interferometer allow for convenient adjustments to the pressure sensitivity of the intermodal fringes and possible applications for the simultaneous interrogation of temperature and pressure. Keywords: intermodal temperature measurement

interferometer;

microstructured

fiber;

hydrostatic

pressure;

1. Introduction In recent years, traditional fiber-optic Mach-Zehnder interferometers (MZIs) with two separated reference and sensing arms have been replaced in many applications by simpler in-line fiber interferometers. Fiber-optic couplers, necessary to split and recombine a guided light, have been replaced in the in-line MZIs by coupling points splitting or recombining light guided in the fundamental core mode and the cladding, or higher order modes. One can distinguish between two different arrangements of this type of in-line interferometer, which are made on a single piece of fiber or spliced from several pieces of fibers of different modes. One of the possible designs of the MZI in one piece of fiber is based on a pair of long period gratings (LPGs) acting as modal couplers [1–4]. However, in such cases, both LPGs should be identical to obtain the maximum performance of the MZI [3]. Fiber tapering, or collapsing air holes in the microstructured fiber is, compared to LPGs, Sensors 2017, 17, 1648; doi:10.3390/s17071648

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a relatively simple and fast method of making coupling points [5–8]. Additionally, in such an approach, the interference fringes are not limited to the spectral range of the LPGs. Some drawbacks to these methods are weaker fiber strength at the tapered regions and high insertion loss at the collapsing points [7]. Another cost-effective and fast method of making coupling points is to splice fibers of different core sizes, for example, a fiber with a smaller mode field diameter is inserted between two conventional single mode fibers (SMFs) [9]. One can also splice a piece of a fiber between two different (or the same fibers) with lateral offset [7,10,11]. In this case, the number of excited cladding modes can be controlled by adjusting the fibers’ lateral shift. However, to obtain a high excitation coefficient of the cladding modes, the core-offset has to be significant, which results in relatively high insertion loss [11]. In the literature, one can also find studies of the formation of in-line MZIs by combining two of the abovementioned methods, for example, devices based on a taper and an LPG [12], a core-offset and an LPG [13], or a core-offset and a taper [14]. Due to flexible configurations, small size, and high sensitivity and resolution, a possible range of applications of in-line intermodal Mach-Zehnder interferometers cover various types of devices for measuring different physical parameters, such as temperature [6,8,9,14], strain [8,14], or refractive index change [5,6]. Simultaneous measurements of temperature and strain were reported by using an in-line MZI with a pair of LPGs made in double cladding fiber [4], an interferometer fabricated in a boron-doped highly birefringent microstructured fiber with two tapered regions [8], and an MZI with a core-offset and taper in a single mode fiber [14]. Moreover, simultaneous measurements of temperature and refractive index using the MZI made in a single mode fiber were reported in Reference [6]. In this paper, we present for the first time a comprehensive study of the sensitivity to hydrostatic pressure and temperature of the in-line Mach-Zehnder fiber interferometer fabricated in a boron-doped two-mode highly birefringent microstructured optical fiber (HB MOF) using a fusion splicer. The characterized sensor was comprised of an HB MOF segment spliced with two single mode leading fibers and a linear polarizer placed at the sensor input. A small lateral shift of the HB MOF and the leading-in single mode fiber allowed for the excitation of the fundamental and the first order mode guided in the HB MOF, and resulted in the intermodal interference observed in the spectral domain at the sensor output. Spectral interferograms were acquired for the different azimuths of the input polarization state with no polarizer at the sensor output and analyzed for different values of applied temperature and hydrostatic pressure. The proposed sensor showed several unusual features such as the opposite sign of sensitivity to hydrostatic pressure for x- and y-polarized intermodal interference signals and the possibility of tuning the intermodal sensitivity to hydrostatic pressure by adjusting the input polarization state. Such properties of the proposed sensor allowed for simultaneous measurements of temperature and hydrostatic pressure, or other parameters by interrogating the visibility and displacement of the intermodal interference fringes and their envelope. 2. The Principle of the Sensor Operation For the fabrication of the proposed in-line Mach-Zehnder interferometer, we used a boron-doped birefringent microstructured fiber as shown in Figure 1, which was fabricated in the Laboratory of Optical Fiber Technology, University of Marie-Curie Sklodowska (UMCS) in Lublin, Poland. The birefringence in this fiber was induced by two large holes located symmetrically with respect to the core. The fiber geometrical parameters averaged for the first two layers of holes surrounding the core were as follows: pitch distance 3.9 µm; diameter of cladding holes 1.75 µm; size of large holes 4.5 µm × 4.0 µm and 3.6 µm × 3.0 µm; external diameter of the fiber 127 µm; and the size of the boron-doped inclusion was 0.68 µm × 0.92 µm. The boron concentration in the inclusion was equal to 13 mol %. It is worth mentioning that a similar construction fiber has been used previously to fabricate an intermodal interferometer for simultaneous strain and temperature measurements [8].

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Figure 1. Scanning Electron Microscope (SEM) image of the birefringent microstructured optical fiber

Figure 1. Scanning Electron Microscope (SEM) image of the birefringent microstructured optical fiber with boron-doped inclusion in the center of the core (darker spot) used for the fabrication of the with boron-doped inclusion in the center of the core (darker spot) used for the fabrication of the in-line in-line interferometer. Figure Mach-Zehnder 1. Scanning Electron Microscope (SEM) image of the birefringent microstructured optical fiber Mach-Zehnder interferometer. with boron-doped inclusion in the center of the core (darker spot) used for the fabrication of the

Deeper insight into the operation principle of the intermodal interferometer requires knowledge in-line Mach-Zehnder interferometer. of the properties the boron-doped which was used for the sensorrequires fabrication. We Deepermodal insight into the of operation principlefiber, of the intermodal interferometer knowledge numerically simulated the fiber properties by means of the finite-element method (FEM) Deeperproperties insight intoof thethe operation principle of the intermodal interferometer requires knowledge of the modal boron-doped fiber, which was used for the sensor fabrication. implemented in a commercially available COMSOL Multiphysics software In fabrication. our model, We the of the modal simulated properties ofthe the fiber boron-doped fiber,by which was of used the [15]. sensor We numerically properties means theforfinite-element method (FEM) mesh was generated using the geometry obtained by the digital processing of an SEM image of the numerically the fiber properties by means of the finite-element (FEM) the implemented in asimulated commercially available COMSOL Multiphysics softwareindices [15].method In our model, investigated 1). COMSOL It was assumed that thesoftware refractive of model, pure silica implementedfiber in a cross-section commercially(Figure available Multiphysics [15]. In our the meshglass was generated usingglass the satisfied geometry by the digital processing of an SEM of the theobtained dispersionby relations given in References Byimage solving mesh and wasboron-doped generated using the geometry obtained the digital processing of an [16,17]. SEM image of the investigated fiber cross-section 1). It was assumed thatwere the able refractive indices of pure silica the fully-vector wave equation(Figure for the defined fiberassumed structure,that we to determine complex investigated fiber cross-section (Figure 1). It was the refractive indices ofthe pure silica glasseffective and boron-doped glass satisfied the dispersion given in References [16,17]. By solving refractive indices for fundamental LP 01 and relations the first high-order LP 11 polarization modes as glass and boron-doped glass satisfied the dispersion relations given in References [16,17]. By solving athe function of wavelength. In the next step, we calculated the spectral dependences of the group modal the fully-vector wave equation for the defined fiber structure, we were able to determine the complex fully-vector wave equation for the defined fiber structure, we were able to determine the complex birefringence and the difference between the effective indices of the first ordermodes and the effective refractive indices forforfundamental LP0101group andthethe first high-order LP modes effective refractive indices fundamental LP and first high-order LP11 polarization as 11 polarization fundamental modes. We found that the simulation results depended on the procedure used a functionof of wavelength. wavelength. InIn thethe next step,step, we calculated the spectral dependences of the group modal as a function next we calculated the spectral dependences of thetogroup process an image ofthe thedifference cross-section, namely theeffective selectionindices of a binarization threshold which birefringence andand between the group of theoffirst orderorder and the modal birefringence thefiber difference between the group effective indices the first and the influenced the diameter of air-holes and the boron-doped inclusion in the fiber model. In the SEM fundamental modes. We found that the simulation results depended on the procedure used to fundamental modes. We found that the simulation results depended on the procedure used to process image, the edges of the air-holes and inclusion werethe slightly blurred; therefore, itthreshold was difficult to process an image of the fiber cross-section, namely selection of a binarization which an image of the fiber cross-section, namely the selection of a binarization thresholdof which influenced the correctly setthe thediameter thresholdofvalue. We adjusted threshold by matching the simulations influenced air-holes and the the boron-doped inclusion in the the results fiber model. In the SEM diameter air-holesvalues. and the boron-doped inclusion in the fiber model. In the SEM image, the edges of to theofmeasured image, the edges of the air-holes and inclusion were slightly blurred; therefore, it was difficult to the air-holes and inclusion were of slightly blurred;interferometer therefore, it was difficult to illustrated correctly set the threshold The operation principle schematically in Figure 2, correctly set the threshold value.the Weproposed adjusted the threshold byismatching the results of the simulations value.and We adjusted the threshold by matching the results of the simulations to the measured values. was composed of a 56-mm long HB MOF spliced with a lateral shift with two pieces of Corning to the measured values. SMF-28 The splices were made using aninterferometer Ericsson fusion splicer. We controlled theincontrast of The operation principle ofof the schematically illustrated in Figure 2, Thefiber. operation principle theproposed proposed interferometer isis schematically illustrated Figure 2, the interference when the fibers’with lateral shift shift before making each splice. and was composed of aof56-mm long HB spliced with lateral shift with pieces of The Corning and intermodal was composed a 56-mm longaligning HBMOF MOF spliced aalateral with twotwo pieces of Corning maximum contrast in the spectral range ofan 1400–1700 nm was splicer. obtained forcontrolled the lateral offset of about SMF-28 The splices were made using an Ericsson fusion splicer. We We controlled the ofof the SMF-28 fiber.fiber. The splices were made using Ericsson fusion thecontrast contrast 2the μm. Due to the lateral shiftwhen and aaligning slight collapse of the microstructured region ateach both splice. splices,The an intermodal interference the fibers’ lateral shift before making intermodal interference when aligning the fibers’ lateral shift before making each splice. The maximum insertion loss slightly higher than 3 dB was observed at obtained each splice. At lateral the first splice, the maximum contrast in the spectral range of 1400–1700 nm was for the offset about contrast in the spectral range of 1400–1700 nm was obtained for the lateral offset of about 2ofµm. Due to fundamental mode was to the LP11 higher mode, and region at the second both 2 μm. Due to LP the01 lateral shift coupled and a slight collapse of theorder microstructured at bothsplice splices, an the lateral shift and a slight collapse of the microstructured region atdepended both splices, an insertion loss modes producing signal, which the polarizer insertionwere loss recombined slightly higher than 3 the dB interference was observed at each splice. At theon first splice, the slightly higher than 3 dB was observed at each splice. At the first splice, the fundamental LP mode was 01 alignment at the of was the leading-in fundamental LP01input mode coupled tofiber. the LP11 higher order mode, and at the second splice both

coupled to the LP higher order mode, and at the second splice both modes were recombined producing modes were 11recombined producing the interference signal, which depended on the polarizer the interference which depended the polarizer alignment at the input of the leading-in fiber. alignment atsignal, the input of the leading-inon fiber.

Figure 2. Microscope image and schematic modes distribution in the proposed in-line interferometric sensor composed of 56 mm of HB MOF spliced with leading-in SMF-28 fibers. Figure 2. Microscope image and schematic modes distribution in the proposed in-line interferometric

Figure 2. Microscope image and schematic modes distribution in the proposed in-line interferometric sensor composed of 56 mm of HB MOF spliced with leading-in SMF-28 fibers. sensor composed of 56 mm of HB MOF spliced with leading-in SMF-28 fibers.

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If the linear polarization state (controlled by the polarizer at the input of the leading-in fiber) is If polarization state (controlled byofthe at the inputfiber, of thewe leading-in is alignedthe in linear parallel to one of the symmetry axes thepolarizer highly birefringent excite infiber) the HB aligned in parallel to one of the symmetry axes of the highly birefringent fiber, we excite in the HB MOF only one pair of polarization modes and as a result, the interference signal produced by the pair x MOF only one polarization modes asbe a result, the interference signal produced by the pair LP01x pair LP01y and LP11yand of modes andofLP can expressed by: 11 or y y x x of modes LP01 and LP11 or LP01 and LP11 can be expressed by: x  2 π(n x − n 11 )L  x − nx )L I x ( λ ) = I 0 cos 2α  1 + γ cos 2π(n01 (1) 01λ 11  Ix (λ) = I0 cos2 α 1+ γ cos (1)  λ and and ! y y y y   2π ( n − n ) L π − 2 (n n )L 2 01 11 11 2 01 I 0 sin 1 +γγcos cos Iy (λI)y = α α1+ (2)  ( λI)0=sin (2) λλ   where where α α is is the the azimuth azimuth of of the the input input polarization state and γγisisthe thefringe fringe contrast contrast depending depending on on the the fibers’ fibers’ lateral lateral offset offset at at splices; splices; finally, finally,LLisisthe thelength lengthof ofthe theHB HBMOF. MOF. The The periodicity periodicity of of the the interference interference fringes fringes observed observed in in the the output output spectrogram spectrogram depends depends on on the the difference difference in in the the group group refractive refractive indices indices of of the the fundamental fundamental and the first order order modes modes for for xx- and and y-polarization, y-polarization, respectively respectively (Figure (Figure 3a). 3a). The The phase phase shift shift ∆ϕ(λ) ∆ϕ(λ) between the fundamental fundamental and and the the first first order modes can be expressed as: order modes can be expressed as: x,y 2π(nx,y − nx,yx,y)L ) =2π(n0101 λ− n1111 )L ∆ϕΔϕ (λ()λ= λ

(3) (3)

After the thedifferentiation differentiation ofabove the above equation and performing rather straightforward After of the equation and performing rather straightforward calculations, calculations, can an easily obtain an for thedifference group index difference of theand fundamental one can easilyone obtain expression forexpression the group index of the fundamental first order and first order mode, respectively, for xor y-polarization: mode, respectively, for x- or y-polarization: λ 2 dΔϕ x,y Nx,y − N x,y x,y = − λ d∆ϕ N01 01 − N1111 = −2πL dλ 2πL dλ 2

(4) (4)

In Figure 4, we compared the calculated and the measured difference in the group refractive In Figure 4, we compared the calculated and the measured difference in the group x,y refractive x,y N11 indices of the fundamental and the first order modes for both polarizations, i.e., N01 −x,y . The x,y indices of the fundamental and the first order modes for both polarizations, i.e., N 01 − N11 . x,y x,y N − N x,y x,y measured andand the the calculated values of of01N 11− N showed very good agreement, which confirms the The measured calculated values 01 11 showed very good agreement, which confirms validity of our numerical model. the validity of our numerical model. (b) -50

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1500 1550 1600 Wavelength [nm]

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Figure 3. 3. Spectral Spectral interference interferencefringes fringesatatthe theoutput outputofofan anin-line in-lineintermodal intermodalinterferometer interferometer based Figure based onon a a 56-mm long two-mode HB MOF for excited x-polarization (black line) and y-polarization 56-mm long two-mode HB MOF for excited x-polarization (black line) and y-polarization (red line) (a) (redboth line)polarization (a) and bothmodes polarization modes by setting azimuth of the input polarization state at and by setting the azimuth of the the input polarization state at α = 34◦ (b). α = 34° (b).

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N x,01y-N x,11y

0.004

0.003 x-polarization (calculation) x-polarization (experiment) y-polarization (calculation) y-polarization (experiment)

0.002

0.001

800

1200 Wavelength [nm]

1600

Figure 4. 4. Calculated Calculated and and measured measured difference differencein ingroup groupeffective effectiveindices indicesofofthe theLP LP0101and andLP LP1111 modes modes Figure for both polarizations. for both polarizations.

If the azimuth of the input polarization state is set at a certain angle α to the polarization axes of If the azimuth of the input polarization state is set at a certain angle α to the polarization axes of the the HB MOF, both polarization modes are excited in this fiber. As a result, we observed the HB MOF, both polarization modes are excited in this fiber. As a result, we observed the superposition superposition of two interference signals produced by two pairs of the orthogonally polarized modes of two interference signals produced by two pairs of the orthogonally polarized modes at the output at the output of the leading out fiber, which can be represented by the following formula: of the leading out fiber, which can be represented by the following formula: y y x x   2π(n 01  2 π(n 01 )L  )L   !# − n11 − n11 " 2 I ( λ ) = I 0 1 + γ cos 2 αcos (nx − nx )L  + γ sin α cos  2π(ny − ny )L 2π 2 2 01 01 11 λ λ   + γ sin α cos I(λ) = I0 1 + γcos α cos  11 

(5) (5)

  2 π(n 01 − n11 )L   π(B01 − B11 )L   2π(n 01 − n11 )L   π(B01 − B11 )L   I ( λ ) =hI 0 1 + γ cos   i  − γ cos 2α sin   a −na ) a −na )L   cos π(B − Lsin 2π ( n 2π ( n B ) L λ λ λ 01 11 01 11  01 11 cos   − γ cos 2α sin   π(Bλ01 −B11 )L sin I(λ) = I0 1 + γ cos 

(6) (6)

λ After performing rather straightforward derivations, one can write: After performing rather astraightforward derivations, one can write: a a a λ

a

λ

λ

λ

λ

a

where n 01 and n11 are the effective refractive indices of the fundamental and the first order modes where na01 and na11 are the effective refractive indices of the fundamental and the first order modes averaged over over polarization: polarization: averaged y y y y n xnx + nnx01x + aa 11 + nn01 01 n11 + n11 11 a 01 na01n= , n (7) = (7) = , n11 = 01 11 22 2 2 and B01 and B11 are the phase modal birefringences of the fundamental and the first order mode: and B01 and B11 are the phase modal birefringences of the fundamental and the first order mode: y

y

xx xx y , B y B01B = − =nn −n n01 , 11 B 11== nn11 − nn11 0101− 11 01 01 11

(8) (8)

If the the azimuth azimuth of of the the input input polarization polarization state state is is aligned aligned at at αα == 45 45°◦ with respect to the the fiber fiber If polarization axis, axis, Equation Equation (6) (6) takes takes the the following followingform: form: polarization a a  2 π(n  )L   11 B11 )L 2π (na01 01−−nna1111))L L  cos  π(B π(01B−01B− I I 1 cos λ = + γ  ( )   I(λ) = I0 1 +0 γ cos   cos λ λ       λλ

(9) (9)

The first firstcosine cosine term in above the above formula represents the intermodal interference fringes The term in the formula represents the intermodal interference fringes averaged averaged over polarization, while term the second termthe represents envelopebydetermined by the over polarization, while the second represents envelope the determined the differences in differences in birefringence of the fundamental and the first order modes. The corresponding terms birefringence of the fundamental and the first order modes. The corresponding terms are indicated areblack indicated by black red lines Figure 3b, which showspectrogram the output spectrogram by by and red lines and in Figure 3b, in which show the output produced byproduced the 56-mm the 56-mm long HB MOF. As per Equation (6), the visibility of the envelope depends on the azimuth long HB MOF. As per Equation (6), the visibility of the envelope depends on the azimuth of the of thepolarization input polarization and the reaches thevalue highest = 45°.5, In weexperimental present the input and reaches highest at α value = 45◦ . at In α Figure weFigure present5,the experimental spectrograms registered at the sensor outputα.for different Onethat should note that the spectrograms registered at the sensor output for different One shouldα.note the alignment of alignment of the input polarization state assuring the maximum visibility of the envelope is different the input polarization state assuring the maximum visibility of the envelope is different from the 45◦ ◦ , which does from the in 45°Equation predicted Figure we display the spectrogram for predicted (6).inInEquation Figure 5e,(6). we In display the5e, spectrogram registered for α = 48registered α = differ 48°, which much from the spectrogram α = 45°, showing a relatively not muchdoes fromnot thediffer spectrogram obtained for α = 45◦ obtained , showingfor a relatively low contrast of the low contrast of thevisible envelope. Clearly Figure setting the state inputazimuth polarization state envelope. Clearly in Figure 5c,visible settinginthe input5c, polarization at α = 30◦ azimuth yielded at α = 30° yielded the maximum modulation depth of the envelope in the spectral range 1600–1650 nm. The discrepancy between the theoretical model and the experimental observations is not fully

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the maximum modulation depth of the envelope in the spectral range 1600–1650 nm. The discrepancy Sensors 17, 1648 6 of 14 be between the2017, theoretical model and the experimental observations is not fully understood, and may caused by several factors, including: the unequal excitation of x- and y-polarized LP01 and LP11 modes understood, and may be caused by several factors, including: the unequal excitation of x- and yat the laterally shifted splices, the polarization-dependent loss (PDL) of the used HB MOF [18], or the polarized LP01 and LP11 modes at the laterally shifted splices, the polarization-dependent loss (PDL) spectral dependence of the coupling coefficients at theof splices. of the used HB MOF [18], or the spectral dependence the coupling coefficients at the splices. (b) -50

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Figure 5. Spectral interference fringes at the output of the in-line intermodal interferometer based on

Figure 5. Spectral interference fringes at the output of the in-line intermodal interferometer based on the 56-mm long HB MOF registered for different azimuth of the input polarization state: α = 10° (a); the 56-mm long HB MOF registered for different azimuth of the input polarization state: α = 10◦ (a); α = 20° (b); α = 30° (c); α = 38° (d); α = 48° (e); α = 58° (f); α = 68° (g); α = 78° (h). α = 20◦ (b); α = 30◦ (c); α = 38◦ (d); α = 48◦ (e); α = 58◦ (f); α = 68◦ (g); α = 78◦ (h).

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We also also compared compared the the difference difference in in group group modal modal birefringence birefringence between between the the fundamental fundamental and and We the first first order order modes modes G G01–G 11 calculated by the FEM method and determined from the spectrogram the 01 –G11 calculated by the FEM method and determined from the spectrogram shown in Figure 3b. The positions of the the minima minima and and maxima maxima of of the the interference interference pattern pattern envelope envelope shown in Figure 3b. The positions of were approximated by the red curve shown in Figure 3b. In this case, the phase shift ∆ϕ(λ) in the the were approximated by the red curve shown in Figure 3b. In this case, the phase shift ∆ϕ(λ) in registered spectrogram spectrogram can can be be expressed expressed as: as: registered 2π(B01 − B11 )L (10) − B11 )L Δϕ ( λ ) =2π(B01 ∆ϕ(λ) = (10) λ λ For the experimental evaluation of the group birefringence difference, we used the following For the experimental evaluation of the group birefringence difference, we used the expression: following expression: λλ2 2 dd∆ϕ Δϕ . −G = − (11) G01G− . (11) 01 G1111= − πL dλdλ 22πL

The measured and and calculated calculatedvalues valuesof of –G(presented in Figure 6) showed very 0111 11 (presented The measured G01G –G in Figure 6) showed very good good agreement. agreement. 0.0000

G01-G11

-0.0004

-0.0008

-0.0012

calculation experiment 800

1200 Wavelength [nm]

1600

Figure 6.6.Comparison Comparisonofof measured calculated difference the modal group birefringence modal birefringence Figure measured andand calculated difference in the in group G01 –G11 G01the –G11 01 and LP 11 modes in the investigated HB MOF with boron-doped inclusion. of LPof01the andLP LP modes in the investigated HB MOF with boron-doped inclusion. 11

3. Sensing 3. Sensing Characteristics Characteristics To measure measure the the sensitivity sensitivity to to temperature, temperature, the the proposed proposed interferometer interferometer was was placed placed in in aa water water To heating system systemequipped equipped a thermocouple and exposed to increasing and temperature decreasing heating withwith a thermocouple and exposed to increasing and decreasing ◦ temperature cycles in a range of 20–100 °C. The ambient temperature was measured by the the cycles in a range of 20–100 C. The ambient temperature was measured by the thermocouple with ◦ thermocouple with the accuracy of 0.1 °C. To determine pressure sensitivity, the interferometer was accuracy of 0.1 C. To determine pressure sensitivity, the interferometer was installed in a specially installed in a specially designed filled with oil and subjected to several pressure designed pressure chamber filledpressure with oilchamber and subjected to several pressure cycles in a range from cycles in a range from 0.1 (atmospheric pressure) to 10 MPa with an accuracy of 0.1 MPa. To avoid 0.1 (atmospheric pressure) to 10 MPa with an accuracy of 0.1 MPa. To avoid the possible influence thebending possible on influence of bending on the sensitivity fiber was keptall straight during of the sensitivity measurements, the measurements, fiber was kept the straight during experiments. all experiments. Depending onthe thepolarization azimuth of the polarization state at the input of the interferometer, Depending on the azimuth of state at the input of the interferometer, we observed a we observed a variation of the interference fringe visibility, the fringe displacement, anddisplacement the envelope variation of the interference fringe visibility, the fringe displacement, and the envelope displacement response changes in hydrostatic pressure and inapplied temperature to the in response toinchanges in to hydrostatic pressure and in temperature to the applied interferometer. interferometer. The displacement of the interference fringes was caused by changes in the effective The displacement of the interference fringes was caused by changes in the effective refractive indices xof x ,y ,y x,y both polarizations x,y n 01 refractive indicesand of the andfor theboth firstpolarizations order mode nfor and n11 the fundamental the fundamental first order mode and n and in the interferometer 01

11

and in Lthe [8], while therelated shift oftothe related to the variation of length [8],interferometer while the shiftlength of the Lenvelope was theenvelope variationwas of the birefringence difference thethe birefringence difference of the fundamental the first order of mode. As the displacement of the of fundamental and the first order mode. Asand the displacement the intermodal fringes is scaled intermodal fringes is scaled proportionally length L while spectral proportionally with the interferometer length Lwith whilethe theinterferometer spectral separation of the fringesthe is inversely separation oftothe fringes is be inversely proportional to L, care has to be taken when would selecting the proportional L, care has to taken when selecting the interferometer length L, which assure interferometer fringe lengthtracking L, whichinwould assure unambiguous tracking inand the pressure. full measurement unambiguous the full measurement rangefringe of temperature For this range ofthe temperature and length pressure. For set thisatreason, interferometer reason, interferometer L was 56 mm the in our experiments.length L was set at 56 mm in our experiments. If the input polarization state is in parallel alignment to one of the symmetry axes of the HB MOF, we observed in the output spectrogram the interference fringes with periodicity depending on

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If the input polarization state is in parallel alignment to one of the symmetry axes of the HB MOF, we observed in the output spectrogram the interference fringes with periodicity depending on the difference in the group refractive indices of the fundamental and the first order mode for x- or the difference in the group refractive indices of the fundamental and the first order mode for x- or y-polarization. As a consequence, the variation of the measurand applied to the HB MOF resulted y-polarization. As a consequence, the variation of the measurand applied to the HB MOF resulted only only in a shift of the intermodal fringes. As shown in Figure 7a,b, the intermodal fringes moved in a shift of the intermodal fringes. As shown in Figure 7a,b, the intermodal fringes moved towards towards shorter wavelengths for x-polarized light and towards longer wavelengths for y-polarized shorter wavelengths for x-polarized light and towards longer wavelengths for y-polarized light in light in response to increasing pressure. The displacement of the fringe with an intensity minimum response to increasing pressure. The displacement of the fringe with an intensity minimum close to close to 1563 nm was linear in the investigated pressure range with no trace of hysteresis. The 1563 nm was linear in the investigated pressure range with no trace of hysteresis. The sensitivities to sensitivities to pressure had an opposite signx and were equal to Kypx = 0.229 nm/MPa and pressure had an opposite sign and were equal to Kp = 0.229 nm/MPa and Kp = −0.179 nm/MPa for x−0.179 nm/MPa for x- and (Figure y-polarization, (Figure 8a; black and red points). Kpy =y-polarization, and respectively 8a; blackrespectively and red points). (a)

-48

α = 0° (x-polarization)

P [dB]

-52

-56

-60 1560


-50

0.1 MPa 1 MPa 2 MPa 3 MPa 4 MPa 5 MPa 6 MPa 7 MPa 8 MPa 9 MPa 10 MPa

-56

-60

1568

1556 1558 1560 1562 1564 1566 Wavelength [nm]

α = 34° 0.1 MPa 10 MPa

(c)

α = 90° (y-polarization)

-52 P [dB]

0.1 MPa 1 MPa 2 MPa 3 MPa 4 MPa 5 MPa 6 MPa 7 MPa 8 MPa 9 MPa 10 MPa

(b)

(d) -52

P [dB]

P [dB]

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-55

-56 -58

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-60

1500

1520


1558 1560 1562 1564 1566 1568 Wavelength [nm]

1600 -50

α = 58° 0.1 MPa 10 MPa

(e) -50

(f)

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P [dB]

P [dB]

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1500

1520


1600

-62 1556 1558 1560 1562 1564 1566 Wavelength [nm]

α = 34° 0.1 MPa 1 MPa 2 MPa 3 MPa 4 MPa 5 MPa 6 MPa 7 MPa 8 MPa 9 MPa 9.5 MPa 10 MPa α = 58° 0.1 MPa 1 MPa 2 MPa 3 MPa 4 MPa 5 MPa 6 MPa 7 MPa 8 MPa 9 MPa 10 MPa

Figure 7. Evolution of the intermodal interference fringe with the intensity minimum close to 1563 Figure 7. Evolution of the intermodal interference fringe with the intensity minimum close to 1563 nm nm registered for increasing hydrostatic pressure in the range of 0.1–10 MPa for the azimuth of the registered for increasing hydrostatic pressure in the range of 0.1–10 MPa for the azimuth of the input input polarization state set at α = 0° (a); α = 90° (b); α = 34° (d); and α = 58° (f); and output spectrograms polarization state set at α = 0◦ (a); α = 90◦ (b); α = 34◦ (d); and α = 58◦ (f); and output spectrograms registered in the the range range from from 1490 1490 nm nm to to 1610 1610 nm nm for for selected selected values values of of hydrostatic hydrostatic pressure pressure showing showing registered in the displacement of the envelope for the azimuth of the input polarization state set at α = 34° (c); and and ◦ the displacement of the envelope for the azimuth of the input polarization state set at α = 34 (c); α = 58° (e). α = 58◦ (e).

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Δλ [nm]

1 0 -1 -2

0

2

4 6 p [MPa]

8

10

α= 0° 15° 20° 25° 32° 34° 36° 48° 58° 78° 90°

0.0 -0.1

-30

0

1573.8

(d)

30

60 α [°]

90

120

150

1573.6 λav [nm]

15 Δλ [nm]

experiment calculation

0.1

-0.2

(c) sensitivity (envelope) 2.14 nm/MPa

20

(b)

0.2 Δλ/Δp [nm/MPa]

(a)

2

10 α = 34° 0.1-10 MPa 10-0.1 MPa

5 0 0

2

4 6 p [MPa]

8

1573.4 1573.2 1573.0

experiment calculation -30

10

0

30

60 α [°]

90

120

150

Figure Displacement of of the fringe withwith an intensity minimum close toclose 1563 to Figure 8. 8.Displacement the intermodal intermodalinterference interference fringe an intensity minimum nm versus the applied pressure for different azimuths of the input polarization state (a) and a 1563 nm versus the applied pressure for different azimuths of the input polarization state (a) and comparison of the experimental and theoretical value of the sensitivity to pressure upon the azimuth a comparison of the experimental and theoretical value of the sensitivity to pressure upon the azimuth of the input polarization state (b). Displacement of the envelope versus applied pressure (c) and of the input polarization state (b). Displacement of the envelope versus applied pressure (c) and experimental and theoretical dependence of the position of the interference minimum at zero pressure experimental and theoretical dependence of the position of the interference minimum at zero pressure and the intermodal sensitivity to pressure upon the azimuth of the input polarization state (d). and the intermodal sensitivity to pressure upon the azimuth of the input polarization state (d).

If the input polarization state is aligned at a certain angle α to the symmetry axes of the HB MOF, the input polarization is aligned at a of certain angle α to theAs symmetry axes ofwe the HB MOF, weIfexcited both polarizationstate modes at the input the interferometer. a consequence, observed we(as excited both polarization the input of the interferometer. a consequence, we observed no analyzer was used)modes at the at sensor output superposition ofAstwo intermodal interference produced two pairs the orthogonally LP01 and LP11 intermodal modes. We performed (assignals no analyzer wasby used) at theofsensor output thepolarized superposition of two interference sensitivity measurements of the intermodal interference fringes for different azimuths of input signals produced by two pairs of the orthogonally polarized LP01 and LP11 modes. We the performed polarization state α (Figure 8a). intermodal For this purpose, we monitored displacement sensitivity measurements of the interference fringesthe forpressure-induced different azimuths of the input of the intermodal interference fringes the highest Asthe shown in Figure 8b, the value of the of polarization state α (Figure 8a). For thisofpurpose, we contrast. monitored pressure-induced displacement was fringes stronglyofdependent oncontrast. the azimuth the input polarization theintermodal intermodalsensitivity interference the highest As of shown in Figure 8b, thestate, valuewhile of the the sensitivity of the envelope remained unchanged and equaled 2.14 nm/MPa (Figure 8c). is worth intermodal sensitivity was strongly dependent on the azimuth of the input polarization Itstate, while mentioning that to determine the envelope shift, we used the procedure presented in Reference [19]. the sensitivity of the envelope remained unchanged and equaled 2.14 nm/MPa (Figure 8c). It is worth The dependence of the intermodal sensitivity upon the azimuth of the input polarization state can be mentioning that to determine the envelope shift, we used the procedure presented in Reference [19]. explained using the approach outlined below. In the vicinity of the interference minimum, a parabolic The dependence of the intermodal sensitivity upon the azimuth of the input polarization state can be approximation of intensity can be used to describe the fringe position versus applied pressure: explained using the approach outlined below. In the vicinity of the interference minimum, a parabolic 2 x ,y x ,y approximation of intensity can be Iused I 0 λ − λ 0x ,ythe − Kfringe p γposition + const versus applied pressure:(12) ( λ , pto) =describe p

(

)

2 x,y for the interference of x- and 11 x,y modes, respectively. In the above equation, Ix,yy-polarized λ 01 −and λ0 LP −K p γ+ const (12) (λ, p) = I0 LP p x,y is the sensitivity λ0 indicates the initial position of the intensity minimum at zero pressure, K x,y p forto the interference of x-inand y-polarized LP11 modes, In the above equation, γ 01is and pressure expressed nm/MPa, while LP the contrast of the respectively. intermodal interference fringes. As x,y x,y λ0 the indicates the initial position of the intensity minimum at zero pressure, K is the sensitivity to p sensor output of the two polarizations sum-up, the overall intensity can be expressed as: pressure expressed in nm/MPa, while γ is the contrast of 2the intermodal interference fringes. As the I ( λ ,sum-up, p ) = I 0 ( λ −the λ av − K av p ) intensity γ + const can be expressed as: (13) sensor output of the two polarizations 0 overall p 2 av I(λ, p) = I0 λ − λav − K p γ + const p 0

(13)

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where λav and Kav indicate the position of the interference minimum at zero pressure and the p av 0 where λ0 and Kav indicate the position of the interference minimum at zero pressure and the sensitivity p sensitivity to pressure for interference summative signal, respectively, and are expressed by the to pressure for interference summative signal, respectively, and are expressed by the following relations: following relations: x x2 2 2 y av λav α +α λ+yλsin λ cos sin 2αα 0 =λ λ0=cos 0

(14) (14)

0

y

y 2 2 Kav Kpx cos αK +K sin2 2αα Kav Kxp=cos α+ p psin p = p

(15) (15)

Equation (14) explains thethe dependence of of thethe position of of thethe interference minimum at at zero Equation (14) explains dependence position interference minimum zero pressure upon the azimuth of the input polarization (Figure 8d). Moreover, according to Equation (15), pressure upon the azimuth of the input polarization (Figure 8d). Moreover, according to theEquation sensitivity to the pressure of thetointermodal fringes summed-up versus polarizations (15), sensitivity pressure ofinterference the intermodal interference fringes summed-up versus canpolarizations be significantly modified by changing the azimuth of the input polarization state, which is in can be significantly modified by changing the azimuth of the input polarization state, qualitative agreement with the experimental observations presented in Figure 8b. In particular, as per which is in qualitative agreement with the experimental observations presented in Figure 8b. In ◦ yields zero sensitivity to pressure, Equation (15),as theper azimuth of the input α = 48polarization particular, Equation (15), thepolarization azimuth of state the input state α = 48° yields zero which is in qualitative agreement with the experimental observations. In the experiment, the zero sensitivity to pressure, which is in qualitative agreement with the experimental observations. In the ◦ . The discrepancies between the experimental result and sensitivity to pressure was obtained for α = 34 experiment, the zero sensitivity to pressure was obtained for α = 34°. The discrepancies between the theexperimental prediction based Equation (15) were moston probably related to different contrasts of intermodal resultonand the prediction based Equation (15) were most probably related to different interference fringes for xand y-polarization caused by the unequal excitation of polarization modes at contrasts of intermodal interference fringes for x- and y-polarization caused by the unequal excitation laterally shifted splices. of polarization modes at laterally shifted splices. Different values of the intermodal fringes and thethe envelope sensitivity caused a variation of of thethe Different values of the intermodal fringes and envelope sensitivity caused a variation interference fringes visibility versus applied pressure. If we selected a high contrast fringe (in our interference fringes visibility versus applied pressure. If we selected a high contrast fringe (in our case thethe fringe with an an intensity minimum close to to 1563 nm; Figure 7d), thethe dependence of of thethe case fringe with intensity minimum close 1563 nm; Figure 7d), dependence visibility versus applied pressure in the range of 0.1–10 MPa was linear and strongly depended onon visibility versus applied pressure in the range of 0.1–10 MPa was linear and strongly depended thethe azimuth of the input polarization state (Figure 9b). The greatest visibility variation was observed azimuth of the input polarization state (Figure 9b). The greatest visibility variation was observed ◦ . Additionally, as an example, we present in forfor thethe azimuth of of thethe input polarization state α =α34 azimuth input polarization state = 34°. Additionally, as an example, we present in ◦. Figure 7e,f the interferometer response for the azimuth of the input polarization state setset at at α =α 58 Figure 7e,f the interferometer response for the azimuth of the input polarization state = 58°. ForFor such an excitation, the intermodal fringes fringes moved towards while the envelope such an excitation, the intermodal moved shorter towardswavelengths, shorter wavelengths, while the towards longer wavelengths was in response to increasing pressure. The displacement of the envelope envelope towards longer wavelengths was in response to increasing pressure. The displacement of as the wellenvelope as the intermodal fringe was linear in was the investigated range of pressure changes with no as well as the intermodal fringe linear in the investigated range of pressure changes trace of no hysteresis. sensitivity were significantly different anddifferent equal toand 2.14equal nm/MPa with trace of The hysteresis. The coefficients sensitivity coefficients were significantly to 2.14 and − 0.137 nm/MPa for the envelope and the intermodal fringe of the highest contrast, respectively. nm/MPa and −0.137 nm/MPa for the envelope and the intermodal fringe of the highest contrast, Forrespectively. α = 58◦ , we also a linear of the intermodal visibility versus the applied For observed α = 58°, we also dependence observed a linear dependencefringe of the intermodal fringe visibility − 1 pressure a sensitivity coefficient to ∆V/∆p = −0.019 MPa . versuswith the applied pressure with a equal sensitivity coefficient equal to ∆V/∆p = −0.019 MPa−1. (a)

V

0.8

0.6

0.4

0.2

0

2

4 6 p [MPa]

8

10

α= 0° 10° 20° 30° 32° 34° 36° 38° 48° 58° 68° 78° 90°

(b) 0.00 ΔV/Δp [1/MPa]

1.0

-0.02

-0.04

-0.06 0

20

40 α [°]

60

80

Figure 9. Variation of the intermodal fringe visibility versus applied pressure corresponding Figure 9. Variation of the intermodal fringe visibility versus applied pressure (a) (a) andand corresponding sensitivity different azimuths of the input polarization state. sensitivity (b)(b) forfor different azimuths of the input polarization state.

Similar experiments were performed for changes in temperature. As shown in Figure 10a,b,d, Similar experiments were performed for changes in temperature. As shown in Figure 10a,b,d, for for increasing temperature, the intermodal interference fringes moved towards longer wavelengths increasing temperature, the intermodal interference fringes moved towards longer wavelengths for xfor x- and y-polarized modes. A similar response to temperature was observed for the input and y-polarized modes. A similar response to temperature was observed for the input polarization polarization state set at α = 30°, for which the intermodal sensitivity to pressure was close to null. state set at α = 30◦ , for which the intermodal sensitivity to pressure was close to null. Moreover, the

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Moreover, the fringes envelope, determined the differenceofinthe birefringence thethe fundamental fringes envelope, determined by the differenceby in birefringence fundamentalofand first order and themoved first order mode, moved towards longer wavelengths for increasing temperature (Figure 10c). mode, towards longer wavelengths for increasing temperature (Figure 10c). (a)

(b)

(x-polarization)

α = 0° 23.8°C 34.6°C 42.6°C 51.5°C 60.0°C 71.6°C 79.2°C 87.9°C 98.5°C

P [dB]

-54 -56 -58 -60 1560


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1568

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α = 34° 24.1°C 100°C

(c)

(y-polarization)

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α = 34° 24.1°C 33.1°C 40.1°C 47.3°C 53.9°C 60.9°C 67.5°C 74.1°C 80.6°C 87.4°C 93.0°C 100°C

(d)

-52

P [dB]

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1480 1500 1520 1540 1560 1580 Wavelength [nm]

1.5

(e) Δλ/ΔT = 0.019 nm/°C

Δλ [nm]

Δλ [nm]

0.5 α = 0° 20-100 °C 100-20 °C

0.0 20

40

60 T [°C]

(f) Δλ/ΔT = 0.020 nm/°C

1.5

1.0

80

100

α = 90° 22.9°C 32.4°C 39.4°C 47.9°C 55.4°C 62.8°C 70.0°C 76.7°C 82.1°C 89.2°C 94.1°C 99.9°C

1.0

0.5

α = 90° 20-100 °C 100-20 °C

0.0 20

40

60 T [°C]

80

100

Figure 10. Displacement of the intermodal interference fringe with the intensity minimum close to Figure 10. Displacement of the intermodal interference fringe with the intensity minimum close to 1563 nm nm for for increasing increasing temperature temperature in in the the range range of of20–100 20–100 ◦°C and azimuth azimuth of of the the input input polarization polarization 1563 C and state set at α = 0° (a); α = 90° (b); and α = 34° (d). Output spectrograms registered for the lowest ◦ ◦ ◦ state set at α = 0 (a); α = 90 (b); and α = 34 (d). Output spectrograms registered for the lowest andand the the highest values of temperature showing the displacement of the envelope for α = 34° (c). ◦ highest values of temperature showing the displacement of the envelope for α = 34 (c). Displacement Displacement of the intermodal interference fringe with an intensity close to 1563 nmand for of the intermodal interference fringe with an intensity minimum closeminimum to 1563 nm for increasing increasing and decreasing temperature for the azimuth of the input polarization state set at α = 0° (e); ◦ ◦ decreasing temperature for the azimuth of the input polarization state set at α = 0 (e); and α = 90 (f). and α = 90° (f).

Similarly, for hydrostatic pressure, the displacement of the intermodal fringes was linear in the Similarly, for hydrostatic pressure, the displacement of the intermodal fringes was linear in the investigated range of temperature with no trace of hysteresis. The temperature sensitivity coefficients investigated range of temperature with no trace of hysteresis. The temperature sensitivity coefficients for x- and y-polarized modes had the same sign and almost the same values, KxTx = 0.019 nm/◦ C and y x- and y-polarized modes had the same sign and almost the same values, K = 0.019 nm/°C and for T KT = 0.020 nm/◦ C, Figure 10e,f, respectively. For the azimuth of the input polarization state set at y ◦ = 0.020 nm/°C, Figure 10e,f, respectively. For thewavelengths azimuth of the polarization state equal set at αK=T 30 , the intermodal fringes moved towards longer withinput a sensitivity coefficient av ◦ C. Similarly, the envelope moved towards longer wavelengths with a sensitivity to K = 0.0195 nm/ α = 30°, the intermodal fringes moved towards longer wavelengths with a sensitivity coefficient equal T ◦ av nm/ C. In the case of temperature, the intermodal sensitivity was almost independent of of 0.02 to K = 0.0195 nm/°C. Similarly, the envelope moved towards longer wavelengths with a sensitivity T

of 0.02 nm/°C. In the case of temperature, the intermodal sensitivity was almost independent of the

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the azimuth of the input polarization state. Moreover, changes in the fringe visibility caused by the temperature increase in a range from 20–100 ◦ C were practically undetectable. The performed measurements showed that for hydrostatic pressure, the sensitivity coefficients for the intermodal fringes and their visibility strongly depended on the azimuth of the input polarization state. This was in contrast to temperature sensitivity, which was practically independent of the azimuth of the input polarization state. The measured values of the sensitivities to temperature and pressure for different azimuths of the input polarization state are shown in Table 1. Table 1. Measured values of the sensitivities to temperature and pressure for different azimuths of the input polarization state. Hydrostatic Pressure, Range 0.1–10 MPa

Temperature, Range 20–100 ◦ C

α

Intermodal Sensitivity (nm/Mpa)

Envelope Sensitivity (nm/Mpa)

Fringes Visibility Change (1/Mpa)

Intermodal Sensitivity (nm/◦ C)

Envelope Sensitivity (nm/◦ C)

Fringes Visibility Change (1/◦ C)

0◦ 90◦ 34◦ 58◦

0.229 –0.179 0 –0.137

— — 2.14 2.14

0 0 −0.059 −0.019

0.019 0.020 0.0195 0.0195

— — 0.02 0.02

0 0 0 0

The experimental data in Table 1 clearly show that for an appropriate azimuth of the input polarization state, for example for α = 58◦ , one could obtain significantly different sensitivity values of the intermodal fringes and the envelope in the case of hydrostatic pressure, and the same values in the case of temperature. As the sensitivity coefficients for pressure and temperature were significantly different, the proposed sensor provided the possibility of the simultaneous measurement of these parameters by interrogating two out of three possible quantities, i.e., the displacement and visibility of the intermodal fringes and the displacement of their envelope. As explained in Reference [20], two-parameter sensing could be accomplished by solving a set of linear equations describing the changes in the interrogated quantities observed in response to a variation in the measured physical parameters. To obtain good accuracy in the two-parameter measurements, the sensitivity coefficient 2 × 2 matrix must be well-conditioned. The data presented in Table 1 show that this requirement was fulfilled by our sensor in both the temperature and the hydrostatic pressure measurements. 4. Conclusions In this work, we studied the sensitivity to hydrostatic pressure and to temperature of the in-line Mach-Zehnder interferometer fabricated in a two-mode highly birefringent boron-doped microstructured fiber. The operations principle of the proposed sensor exploits the effects of intermodal interference arising between the fundamental LP01 and the first order LP11 mode. Due to high fiber birefringence, the intermodal interference fringes were modulated in visibility for differences in the azimuth of the input polarization state from 0◦ and 90◦ . The proposed interferometer was tested for measurements of hydrostatic pressure and temperature for different alignments of the input polarizer. Due to asymmetric fiber geometry, the sensitivities to hydrostatic pressure of the intermodal interference fringes for x- and y-polarizations had an opposite sign and were equal to 0.229 nm/MPa and −0.179 nm/MPa, respectively. We further demonstrated that in the case of hydrostatic pressure, the sensitivity of the intermodal fringes and their visibility was strongly dependent on the azimuth of the input polarization state, while the sensitivity of the envelope remained constant. For the azimuth of the input polarization state equal to α = 34◦ , the sensitivity of the envelope equaled 2.14 nm/MPa, the sensitivity of the intermodal fringes was equal to zero, while the visibility of the fringes varied in response to pressure changes at a rate of −0.059 MPa−1 . For α = 58◦ , the sensitivities of the intermodal fringes and their visibility were significantly different and equaled −0.137 nm/MPa and −0.019 MPa−1 , respectively.

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For temperature, the intermodal sensitivities for x- and y-polarized modes had almost the same values, and were equal to 0.019 nm/◦ C and 0.020 nm/◦ C for x- and y-polarization, respectively. Consequently, the average sensitivity Kav T was weakly dependent on the azimuth of the input polarization state. Moreover, temperature-induced changes in the fringes visibility were negligible for any alignment of the input polarization state. As in the case of hydrostatic pressure, the sensitivity of the intermodal fringes and their visibility were strongly dependent on the azimuth of the input polarization α and was practically independent of α in the case of temperature. Furthermore, one could adjust the sensor response to allow for simultaneous measurements of temperature and pressure by interrogating two out of three possible parameters, i.e., the displacement and the visibility of intermodal fringes and the displacement of their envelope. Acknowledgments: This work was supported by the Polish Ministry of Science and Higher Education as a research project No. IP2014044573 (Iuventus Plus). Author Contributions: Gabriela Statkiewicz-Barabach conceived and performed the experiments and wrote a major part of the manuscript. Jacek Olszewski performed the numerical simulations and contributed to writing the manuscript. Pawel Mergo fabricated the HB microstructured fiber. Waclaw Urbanczyk revised the manuscript. Conflicts of Interest: The authors declare no conflict of interest.

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