IOP PUBLISHING

MEASUREMENT SCIENCE AND TECHNOLOGY

doi:10.1088/0957-0233/18/9/026

Meas. Sci. Technol. 18 (2007) 2943–2948

Application of an artificial neural network for simultaneous measurement of temperature and strain by using a photonic crystal fiber long-period grating J Sun1,3 , C C Chan1, X Y Dong2 and P Shum2 1

School of Chemical and Biomedical Engineering, Nanyang Technological University, Singapore 639798 2 School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798 E-mail: [email protected]

Received 19 March 2007, in final form 3 June 2007 Published 10 August 2007 Online at stacks.iop.org/MST/18/2943 Abstract A general analysis of an inserted long-period grating in an air-clad photonic crystal fiber for temperature and strain measurement is presented. The temperature and strain can be detected simultaneously by using an artificial neural network. A simulation study was carried out with the data set generated by using theoretical strain and temperature sensitivities of the long-period gratings. It indicates that the maximum temperature error is 0.04 ◦ C in the temperature range from 35 ◦ C to 120 ◦ C. At the same time, the maximum strain error is 2.7 µε in the strain range from 0 to 3000 µε. Keywords: photonic crystal fiber (PCF), long-period grating (LPG), artificial neural network (ANN), temperature measurement, strain measurement

(Some figures in this article are in colour only in the electronic version)

1. Introduction Long-period gratings (LPG) have shown tremendous potential for applications in communication systems as band rejection filters or EDFA gain-flattening filters due to their inherent small size and low back reflection [1]. Considerable research effort has also been devoted to the applications in sensing areas because of their high sensitivity to external perturbations, such as temperature, strain and bending [2]. In the past decades, the thermal responsivity of LPG has been extensively explored for implementing widely tunable notch filters or super high sensitivity temperature sensors. In 1996, Bhatia and Vengsarkar demonstrated a LPG optical sensor capable of measuring temperature and found that the average temperature sensitivity of LPG was much larger than that of fiber Bragg 3 Address for correspondence: Network Technology Research Centre, Nanyang Technological University, Research TechnoPlaza, #04-12, XFrontiers Block, 50 Nanyang Drive, Singapore 639798.

0957-0233/07/092943+06$30.00

© 2007 IOP Publishing Ltd

gratings (FBG) [3]. Shu et al reported that the temperature sensitivity could be dramatically enhanced by writing the LPG in B/Ge fiber and by choosing appropriate cladding modes [4]. The temperature sensitivity could also be enhanced by surrounding the LPG with a material with a high thermo-optic coefficient [5]. LPG notch filters were also demonstrated with an enhanced thermal responsivity which was realized in a photonic crystal fiber (PCF) with air holes in the cladding, filled with a special polymer whose refractive index was sensitive to the temperature [6]. However, the intrinsic responses of the LPG to both strain and temperature impose serious implications for the temperature measurements. Therefore, a lot of research efforts have been made to recover temperature and strain without ambiguity. In general, the strain and temperature information may be recovered simultaneously by a standard matrix inversion method [7]. If two optical parameters X and Y are perturbed simultaneously by both temperature and strain,

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same time, the period of grating will also be changed by the strain and temperature. By consideration of the thermo-optic effects and photoelastic effects, the refractive indices of core (nGe core), cladding (nsilica) and polymer (npolymer) are thus given by

Polymer Ge doped core

nGe-core = 1.455 − 0.3208ε nsilica = 1.45 − 0.316ε . npolymer = 1.45 − 0.004T − 0.4103ε Silica

Figure 1. Cross section of the air-clad photonic crystal fiber.

then X = AT + B ε

Y = C T + Dε

(1)

where A, B, C and D are regarded as constant values and can be determined by measuring the temperature and strain responses separately for X and Y. The temperature and strain variations can thus be evaluated by a standard matrix inversion method. However, this method will introduce large errors due to nonlinear and/or cross-sensitivity problems since the four coefficients are no longer constant values, but complex functions of temperature and strain variations in some situations. In this paper, the shifts of the resonance dips in the transmission spectrum of an LPG by the effects of temperature and strain are investigated when the LPG is inscribed in an airclad PCF with hybrid cladding. An artificial neural network approach [8, 9] for the discrimination of these two effects is also presented. The principle of operation is described in section 2; simulation results are presented in section 3 and a conclusion is given in section 4.

2. Principle of operation The hybrid polymer silica holey fiber is shown in figure 1. Six air holes with a diameter of about 40 µm form a hexagonal ring around a central silica region where its diameter is about 32 µm. The core of the fiber in the center silica region is germanium (Ge) doped which has a diameter of 8 µm and the relative refractive index () is about 0.35%. An acrylatebased polymer with a refractive index of 1.45, which is similar to silica, is introduced into the air regions of the fiber. The polymer is then UV cured to form a hybrid waveguide [6]. The external medium is air. The period of the LPG was chosen to be 270 µm, and the LPG is inscribed into this fiber for sensing applications. The phase-matching condition that governs the LPG operation can be given by [2] eff (2) λm = neff co − ncl,m where λm is the wavelength of the mth order of transmission dip, neff co is the effective index of the fundamental core mode, neff cl,m is the effective index of the mth cladding mode and is the period of the grating. The variation of the strain (temperature) will change the effective indices of the core and cladding modes by photo-elastic (thermo-optic) effect. At the 2944

(3)

The thermo-optic coefficients and photo-elastic coefficients were referred to [2] and [10]. Their values may change with the nature and concentration of the dopants. Here, the thermo effects on refractive indices of the core and cladding are ignored due to its relatively small thermo-optic coefficient compared to that of the polymer. The thermoexpansion can be approximated by that of silica and hence 1 d = 4.1 × 10−7 ◦ C−1 . dT The fiber structure can be modeled as a fiber which is composed of a Ge doped core of diameter DGe core with refractive index of nGe core and a circular cladding of diameter Deff with the refractive index of nsilica. The fiber is surrounded by an infinite polymer medium of refractive index npolymer. The effective indices of the core and the mth cladding modes can be evaluated by using the plane wave approximation, and can thus be given by [6] 2 − π/2 (2πDGe core /λ) n2Ge core − neff co 2 2 eff n core − nco = 2 cos−1 Ge 2 2 nGe core − nsilica 2 − (m − 3/4)2π (2πDeff /λ) n2silica − neff cl,m 2 eff 2 nsilica − ncl,m = 2 cos−1 2 . nsilica − n2polymer

(4)

The temperature and strain effect on the shift of resonance wavelength can be viewed from equations (2)–(4). From equation (3), it is clear that any temperature and/or strain variation can change the refractive indices of core (nGe core), cladding (nsilica) and polymer (npolymer). The changes in nGe core, nsilica and npolymer can modify the effective indices of the core mode and the mth cladding mode according to equation (4). This finally results in the wavelength shift as is clear from equation (2). Since the temperature dependence on the shifts of two resonant wavelength dips (λ1 and λ2) behaves differently, it renders the possibility for simultaneous measurement of multiple parameters by exploitation of the differential modulation of these two resonant bands. Figure 2 shows the structure of the ANN, which has been used in our investigation to recover the temperature and strain variations. The network is configured to have three layers: an input layer, a hidden layer and an output layer. The hidden layer consists of three to ten neurons, with nonlinear transfer functions which allow the network to learn the nonlinear and linear relationships between input and output layers. This network can be used as a general function approximator. It can approximate any function with a finite number of discontinuities, given sufficient neurons in the hidden layer [8]. Therefore, it is well suited for recovering

Application of an artificial neural network for simultaneous measurement of temperature and strain

Figure 2. Structure of the artificial neural network.

the temperature and strain effect for the nonlinear and cross sensitivity problems. Each neuron in the hidden layer behaves as an activation function f (x). It is an S-shape monotonic increasing function that has the general form f (x) = (1 + e−cx )−1 , where c is a constant that determines the steepness of the S-shape curve. The output of the Rth neuron, in the hidden layer, i.e. HR, can thus be given by HR = f (λ1 · wR,λ1 + λ2 · wR,λ2 )

(5)

where λ1 and λ2 are the wavelength shifts of two resonance dips, and wR,λ1 and wR,λ2 are the weight values between HR and λ1 and λ2, respectively. The values of temperature variation T and strain variation ε in the output layer are related to the Rth neuron in the hidden layer by

(vT ,R · HR ) T = f

ε = f

R

(6)

(vε,R · HR )

R

where vT ,R and vε,R are the weight values between HR and T and ε, respectively. The weight values in equations (5) and (6) can be obtained by using a back propagation method [8] during the training process. The network is trained by using a training data set {(λ1 train , λ2 train ), (Ttrain , εtrain )}. In the experiment, the training data will be obtained from the experimental

results. In our simulation, the training data set is obtained from equations (2)–(4). Here, Ttrain and ε train are chosen randomly and inserted into equation (3) to obtain nGe core, nsilica and npolymer, which were then inserted into equation (4) to eff eff eff obtain neff co and ncl,m . After that, nco and ncl,m were inserted into equation (2) to obtain λ1 train and λ2 train. The data set of {(λ1 train, λ2 train), (Ttrain, ε train)} will be used as ‘experimental’ results in our simulation. The training process starts with some small random initial weight values. For each training cycle, the estimated values (Test, ε est) can be obtained by feeding forward (λ1 train, λ2 train) into equations (5) and (6). The errors between the estimated results (Test, ε est) and the ‘experimental results’ (Ttrain, ε train) can thus be used to update the weights by a gradient-descent method [8]. According to the gradient-descent algorithm, the weights in the input-to-hidden connections are updated by [9] ∂E + αwR,λ1 (t − 1) ∂wR,λ1 ∂E + αwR,λ2 (t − 1). wR,λ2 (t) = −η ∂wR,λ2

wR,λ1 (t) = −η

(7)

Similarly, the weights in the hidden-to-output connections are updated by ∂E + αvT ,R (t − 1) ∂vT ,R ∂E vε,R (t) = −η + αvε,R (t − 1) ∂vε,R

vT ,R (t) = −η

(8)

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where E is the error between the estimated value and the experimental value, η is the learning rate coefficient, and α is the momentum coefficient. The momentum coefficient α is added to improve the normal gradient-descent method because the speed of convergence can be very slow if the learning rate is either too small or too large. The updated weights can thus be given by wR,λ1 (t + 1) = wR,λ1 (t) + wR,λ1 (t − 1) wR,λ2 (t + 1) = wR,λ2 (t) + wR,λ2 (t − 1) . vT ,R (t + 1) = vT ,R (t) + vT ,R (t − 1) vε,R (t + 1) = vε,R (t) + vε,R (t − 1)

(9)

The values of the weights obtained from equations (7) and (8) can be inserted into equations (5) and (6) to evaluate the updated T and ε. This procedure will not stop until the errors between the estimated values of Test and ε est and ‘experimental’ values of Ttrain and ε train are below a specific value. After the training process, the nonlinear function between the input (λ1, λ2) and output (T, ε) is generated. It can thus be used to recover the temperature and strain information through the inputs of the wavelength variations of the two transmission dips (λ1 and λ2).

(a)

3. Simulation results and discussion The shifts of two resonant dips with temperature and strain effects are shown in figure 3. The measurable resonant wavelengths λ1 and λ2 manifest as two loss dips in the LPG transmission spectrum, and each of them is perturbed simultaneously by both temperature and strain effects. However, for the large values of T, the shifts of two resonant wavelengths with the temperature may not be linear, i.e. A or C in equation (1) are functions of T, respectively. In addition, the strain sensitivity is also changed with respect to the temperature, which means that B and D in equation (1) are also functions of T. This introduces complexity on the matrix inversion method that imposes difficulties and limitations for recovering the strain and temperature information, simultaneously. The nonlinearity and cross-sensitivity issues can, however, be minimized by adapting the implement of the artificial neural network, which is able to recover the temperature and strain information for given pairs of λ1 and λ2. The training data set consists 200 pairs of {(λ1 train, λ2 train), (Ttrain, ε train)}, where Ttrain and ε train were generated randomly within a range of 35–120 ◦ C and 0– 3000 µε, and λ1 train and λ2 train were evaluated by equations (2)–(4). The learning rate coefficient η and the momentum coefficient α in equations (7) and (8) were chosen in an informed manner to be 0.95 and 0.85, respectively. The specific values of the errors for the temperature and strain were 0.001 ◦ C and 0.001 µε, respectively. The number of neurons in the hidden layer was decided by making several tests with R = 5, 8, 10, 20 and 36. It was found that the accuracy increases with the number of hidden layers. However, when R > 10 the accuracy increases slightly with the number of neurons. Considering these results, it was decided to use ten hidden layer neurons for the ANN to be trained. In this instance, the number of training cycle is less than 1500. After 2946

(b)

Figure 3. Wavelength shifts of (a) first-order and (b) second-order transmission dips of the LPG.

training, the weight values of wR,λ1 , wR,λ2 , vT ,R and vε,R were obtained and listed in table 1. The network was then tested by using another set consisting of 500 new pairs of {(λ1 test, λ2 test), (Ttest, ε test)}. Ttest and ε test were supposed to be the ‘accurate’ values of temperature variation and strain variation. λ1 test and λ2 test were the temperature and strain induced wavelength shifts which were evaluated by inserting Ttest and ε test into equation (4). The ‘estimate’ values, i.e. Test and ε est, were those evaluated by the trained ANN where the inputs were the calculated values of λ1 test and λ2 test. The errors for the strain and temperature (i.e. the difference between the ‘accurate’ values of Ttest and ε test and estimate values of Test and ε est) are shown in figures 4(a) and (b), respectively. The maximum absolute values of errors are 0.04 ◦ C and 2.7 µε, respectively. To evaluate the performance, the results have also been compared to the matrix inversion method to show the improvement in the strain and temperature measurement. The matrix coefficients in equation (1) were derived by using the same training data set and the least-squares algorithm [11]. The ‘estimate’ values were those evaluated by the matrix inversion method where the inputs were the calculated values of λ1 test and λ2 test. The errors for the strain and temperature by using the matrix inversion method are shown in figures 5(a) and (b). The maximum absolute values of errors

Application of an artificial neural network for simultaneous measurement of temperature and strain

Table 1. Weight values of wR,λ1 , wR,λ2 , vT ,R and vε,R after training. R 1

2

3

4

5

wR,λ1 1.7146 1.4017 1.9922 2.2373 2.5052 wR,λ2 3.2271 0.5285 2.1683 2.3533 2.6425 vT ,R 0.4976 0.5012 0.4186 0.4842 0.2343 vε,R −3.242 −3.161 −7.790 −3.370 −3.516

(a)

6

7

8

9

10

6.3701 6.1629 0.4580 3.4297

2.9703 4.7968 0.5090 3.1990

0.3478 2.1149 0.3884 12.8625

0.6435 0.0215 2.8102 4.1480

1.2901 0.2034 1.8539 6.2899

(a)

(b)

Figure 4. Errors of (a) temperature and (b) strain within the temperature range 35–120 ◦ C and strain range of 0–3000 µε by using the artificial neural network approach.

are 9.3 ◦ C and 150 µε, respectively, which is much larger than those obtained by the ANN method. The demonstrated errors are relatively small compared to those reported in the literature, including the results by Chan et al [9] (0.4 ◦ C, 4.9 µε for small cross-sensitivity; 2 ◦ C, 6.4 µε for large cross-sensitivity) and by Gusmeroli et al [12] (1 ◦ C, 10 µε) and by Patrick et al [13] (1.5 ◦ C, 9 µε). The main reason for this is that the temperature and strain response of the LPG exhibit relatively small nonlinearity and cross-sensitivity over a large temperature and strain range. Therefore, the ANN method shows good performance in recovering the temperature and strain information. However, the small errors might not be repeated in the experiment. The primary concern is the signal to noise ratio which may deteriorate the performance of the ANN method. However, this does not prevent its application, since the performance of ANN is much better than the matrix inversion method. The

(b)

Figure 5. Errors of (a) temperature and (b) strain within the temperature range 35–120 ◦ C and strain range of 0–3000 µε by using the matrix inversion method.

experimental demonstration of the performance of ANN may need further investigation. It has been demonstrated that the temperature and strain sensitivities of the conventional LPG change with the surrounding medium, arising from the dependence of effective indices of the cladding modes on the surrounding medium [14]. However, for the hybrid polymer–silica structure, it has been demonstrated that LPG resonances are insensitive to the external refractive index, which is for the reason that the cladding modes are mainly confined by polymer cladding [15]. The sensitivity is thus expected to be independent of the surrounding medium, which may need further investigation. Although the ANN method is applied in a similar way in [9], the difference in the performance indicates that a sensor with small nonlinearity and cross-sensitivity 2947

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could yield a better performance with the ANN method. Therefore, the PCF-based sensor should be more appropriate for simultaneous temperature and strain measurement due to its relatively small nonlinearity and cross-sensitivity.

[5] [6]

4. Conclusion In conclusion, an artificial neural network approach is proposed for simultaneous measurement of temperature and strain by using a LPG which is inscribed in a photonic crystal fiber with hybrid cladding. The method is based on the measurement of the shifts of two resonant wavelengths which behave differently with the temperature and strain. By using the ANN approach, the temperature and strain effects can be determined simultaneously. The maximum temperature and strain errors of 0.04 ◦ C and 2.7 µε are obtained within the temperature range from 35 ◦ C to 120 ◦ C and strain range from 0 to 3000 µε. Compared to the standard matrix inversion method, the ANN method is well suited for implementation in sensors which suffer nonlinearity and cross-sensitivity.

References [1] Vengsarkar A M, Lemaire P J, Judkins J B, Bhatia V, Erdogan T and Sipe J E 1996 Long-period fiber gratings as band-rejection filters J. Lightwave Technol. 14 58–64 [2] James S W and Tatam R P 2003 Optical fibre long-period grating sensors: characteristics and application Meas. Sci. Technol. 14 49–61 [3] Bhatia V and Vengsarkar A M 1996 Optical fiber long-period grating sensors Opt. Lett. 21 692–4 [4] Shu X, Allsop T, Zhang L and Bennion I 2001 High temperature sensitivity of long period gratings in

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[7] [8] [9]

[10] [11] [12]

[13]

[14]

[15]

B/Ge co-doped fiber IEEE Photon. Technol. Lett. 13 818–20 Khaliq S, James S and Tatam R P 2002 Enhanced sensitivity fibre optic long period grating temperature sensor Meas. Sci. Technol. 13 792–5 Westbrook P S, Eggleton B J, Windeleer R S, Hale A, Strasser T A and Burdge G L 2000 Cladding-mode resonances in hybrid polymer–silica microstructured optical fiber gratings IEEE Photon. Tech. Lett. 12 495–7 Jin W, Michie C, Thursby G, Konstantaki M and Culshaw B 1997 Simultaneous measurement of strain and temperature: error analysis Opt. Eng. 36 589–609 Patterson D W 1996 Artificial Neural Network: Theory and Applications (Singapore: Prentice-Hall) Chan C C, Jin W, Rad A B and Demokan M S 1998 Simultaneous measurement of temperature and strain: an artificial neural network approach IEEE Photon. Tech. Lett. 10 854–6 Chu P L 2005 Polymer optical fiber Bragg gratings Opt. Photonics News 16 (7) 52–6 Wellstead P E and Zarrop M B 1991 Self-Tuning Systems: Control and Signal Processing (New York: Wiley) Gusmeroli V and Martinelli M 1994 Non-incremental interferometric fiber-optic measurement method for simultaneous detection of temperature and strain Opt. Lett. 19 2164–6 Patrick H J, Williams G M, Kersey A D, Pedrazzani J R and Vengsarkar A M 1996 Hybrid fiber Bragg grating/long period grating sensor for strain/temperature discrimination IEEE Photon. Technol. Lett. 8 1223–5 Kamikawachi R C, Possetti G R C, Falate R, Muller M and Fabris J L 2007 Influence of surrounding media refractive index on the thermal and strain sensitivities of long-period gratings Appl. Opt. 46 2831–7 Espindola R P, Windeler R S, Abramov A A, Eggleton B J, Strasser T A and DiGiovanni D J 1999 External refractive index insensitive air-clad long period fiber grating Electron. Lett. 35 327–8

MEASUREMENT SCIENCE AND TECHNOLOGY

doi:10.1088/0957-0233/18/9/026

Meas. Sci. Technol. 18 (2007) 2943–2948

Application of an artificial neural network for simultaneous measurement of temperature and strain by using a photonic crystal fiber long-period grating J Sun1,3 , C C Chan1, X Y Dong2 and P Shum2 1

School of Chemical and Biomedical Engineering, Nanyang Technological University, Singapore 639798 2 School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798 E-mail: [email protected]

Received 19 March 2007, in final form 3 June 2007 Published 10 August 2007 Online at stacks.iop.org/MST/18/2943 Abstract A general analysis of an inserted long-period grating in an air-clad photonic crystal fiber for temperature and strain measurement is presented. The temperature and strain can be detected simultaneously by using an artificial neural network. A simulation study was carried out with the data set generated by using theoretical strain and temperature sensitivities of the long-period gratings. It indicates that the maximum temperature error is 0.04 ◦ C in the temperature range from 35 ◦ C to 120 ◦ C. At the same time, the maximum strain error is 2.7 µε in the strain range from 0 to 3000 µε. Keywords: photonic crystal fiber (PCF), long-period grating (LPG), artificial neural network (ANN), temperature measurement, strain measurement

(Some figures in this article are in colour only in the electronic version)

1. Introduction Long-period gratings (LPG) have shown tremendous potential for applications in communication systems as band rejection filters or EDFA gain-flattening filters due to their inherent small size and low back reflection [1]. Considerable research effort has also been devoted to the applications in sensing areas because of their high sensitivity to external perturbations, such as temperature, strain and bending [2]. In the past decades, the thermal responsivity of LPG has been extensively explored for implementing widely tunable notch filters or super high sensitivity temperature sensors. In 1996, Bhatia and Vengsarkar demonstrated a LPG optical sensor capable of measuring temperature and found that the average temperature sensitivity of LPG was much larger than that of fiber Bragg 3 Address for correspondence: Network Technology Research Centre, Nanyang Technological University, Research TechnoPlaza, #04-12, XFrontiers Block, 50 Nanyang Drive, Singapore 639798.

0957-0233/07/092943+06$30.00

© 2007 IOP Publishing Ltd

gratings (FBG) [3]. Shu et al reported that the temperature sensitivity could be dramatically enhanced by writing the LPG in B/Ge fiber and by choosing appropriate cladding modes [4]. The temperature sensitivity could also be enhanced by surrounding the LPG with a material with a high thermo-optic coefficient [5]. LPG notch filters were also demonstrated with an enhanced thermal responsivity which was realized in a photonic crystal fiber (PCF) with air holes in the cladding, filled with a special polymer whose refractive index was sensitive to the temperature [6]. However, the intrinsic responses of the LPG to both strain and temperature impose serious implications for the temperature measurements. Therefore, a lot of research efforts have been made to recover temperature and strain without ambiguity. In general, the strain and temperature information may be recovered simultaneously by a standard matrix inversion method [7]. If two optical parameters X and Y are perturbed simultaneously by both temperature and strain,

Printed in the UK

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same time, the period of grating will also be changed by the strain and temperature. By consideration of the thermo-optic effects and photoelastic effects, the refractive indices of core (nGe core), cladding (nsilica) and polymer (npolymer) are thus given by

Polymer Ge doped core

nGe-core = 1.455 − 0.3208ε nsilica = 1.45 − 0.316ε . npolymer = 1.45 − 0.004T − 0.4103ε Silica

Figure 1. Cross section of the air-clad photonic crystal fiber.

then X = AT + B ε

Y = C T + Dε

(1)

where A, B, C and D are regarded as constant values and can be determined by measuring the temperature and strain responses separately for X and Y. The temperature and strain variations can thus be evaluated by a standard matrix inversion method. However, this method will introduce large errors due to nonlinear and/or cross-sensitivity problems since the four coefficients are no longer constant values, but complex functions of temperature and strain variations in some situations. In this paper, the shifts of the resonance dips in the transmission spectrum of an LPG by the effects of temperature and strain are investigated when the LPG is inscribed in an airclad PCF with hybrid cladding. An artificial neural network approach [8, 9] for the discrimination of these two effects is also presented. The principle of operation is described in section 2; simulation results are presented in section 3 and a conclusion is given in section 4.

2. Principle of operation The hybrid polymer silica holey fiber is shown in figure 1. Six air holes with a diameter of about 40 µm form a hexagonal ring around a central silica region where its diameter is about 32 µm. The core of the fiber in the center silica region is germanium (Ge) doped which has a diameter of 8 µm and the relative refractive index () is about 0.35%. An acrylatebased polymer with a refractive index of 1.45, which is similar to silica, is introduced into the air regions of the fiber. The polymer is then UV cured to form a hybrid waveguide [6]. The external medium is air. The period of the LPG was chosen to be 270 µm, and the LPG is inscribed into this fiber for sensing applications. The phase-matching condition that governs the LPG operation can be given by [2] eff (2) λm = neff co − ncl,m where λm is the wavelength of the mth order of transmission dip, neff co is the effective index of the fundamental core mode, neff cl,m is the effective index of the mth cladding mode and is the period of the grating. The variation of the strain (temperature) will change the effective indices of the core and cladding modes by photo-elastic (thermo-optic) effect. At the 2944

(3)

The thermo-optic coefficients and photo-elastic coefficients were referred to [2] and [10]. Their values may change with the nature and concentration of the dopants. Here, the thermo effects on refractive indices of the core and cladding are ignored due to its relatively small thermo-optic coefficient compared to that of the polymer. The thermoexpansion can be approximated by that of silica and hence 1 d = 4.1 × 10−7 ◦ C−1 . dT The fiber structure can be modeled as a fiber which is composed of a Ge doped core of diameter DGe core with refractive index of nGe core and a circular cladding of diameter Deff with the refractive index of nsilica. The fiber is surrounded by an infinite polymer medium of refractive index npolymer. The effective indices of the core and the mth cladding modes can be evaluated by using the plane wave approximation, and can thus be given by [6] 2 − π/2 (2πDGe core /λ) n2Ge core − neff co 2 2 eff n core − nco = 2 cos−1 Ge 2 2 nGe core − nsilica 2 − (m − 3/4)2π (2πDeff /λ) n2silica − neff cl,m 2 eff 2 nsilica − ncl,m = 2 cos−1 2 . nsilica − n2polymer

(4)

The temperature and strain effect on the shift of resonance wavelength can be viewed from equations (2)–(4). From equation (3), it is clear that any temperature and/or strain variation can change the refractive indices of core (nGe core), cladding (nsilica) and polymer (npolymer). The changes in nGe core, nsilica and npolymer can modify the effective indices of the core mode and the mth cladding mode according to equation (4). This finally results in the wavelength shift as is clear from equation (2). Since the temperature dependence on the shifts of two resonant wavelength dips (λ1 and λ2) behaves differently, it renders the possibility for simultaneous measurement of multiple parameters by exploitation of the differential modulation of these two resonant bands. Figure 2 shows the structure of the ANN, which has been used in our investigation to recover the temperature and strain variations. The network is configured to have three layers: an input layer, a hidden layer and an output layer. The hidden layer consists of three to ten neurons, with nonlinear transfer functions which allow the network to learn the nonlinear and linear relationships between input and output layers. This network can be used as a general function approximator. It can approximate any function with a finite number of discontinuities, given sufficient neurons in the hidden layer [8]. Therefore, it is well suited for recovering

Application of an artificial neural network for simultaneous measurement of temperature and strain

Figure 2. Structure of the artificial neural network.

the temperature and strain effect for the nonlinear and cross sensitivity problems. Each neuron in the hidden layer behaves as an activation function f (x). It is an S-shape monotonic increasing function that has the general form f (x) = (1 + e−cx )−1 , where c is a constant that determines the steepness of the S-shape curve. The output of the Rth neuron, in the hidden layer, i.e. HR, can thus be given by HR = f (λ1 · wR,λ1 + λ2 · wR,λ2 )

(5)

where λ1 and λ2 are the wavelength shifts of two resonance dips, and wR,λ1 and wR,λ2 are the weight values between HR and λ1 and λ2, respectively. The values of temperature variation T and strain variation ε in the output layer are related to the Rth neuron in the hidden layer by

(vT ,R · HR ) T = f

ε = f

R

(6)

(vε,R · HR )

R

where vT ,R and vε,R are the weight values between HR and T and ε, respectively. The weight values in equations (5) and (6) can be obtained by using a back propagation method [8] during the training process. The network is trained by using a training data set {(λ1 train , λ2 train ), (Ttrain , εtrain )}. In the experiment, the training data will be obtained from the experimental

results. In our simulation, the training data set is obtained from equations (2)–(4). Here, Ttrain and ε train are chosen randomly and inserted into equation (3) to obtain nGe core, nsilica and npolymer, which were then inserted into equation (4) to eff eff eff obtain neff co and ncl,m . After that, nco and ncl,m were inserted into equation (2) to obtain λ1 train and λ2 train. The data set of {(λ1 train, λ2 train), (Ttrain, ε train)} will be used as ‘experimental’ results in our simulation. The training process starts with some small random initial weight values. For each training cycle, the estimated values (Test, ε est) can be obtained by feeding forward (λ1 train, λ2 train) into equations (5) and (6). The errors between the estimated results (Test, ε est) and the ‘experimental results’ (Ttrain, ε train) can thus be used to update the weights by a gradient-descent method [8]. According to the gradient-descent algorithm, the weights in the input-to-hidden connections are updated by [9] ∂E + αwR,λ1 (t − 1) ∂wR,λ1 ∂E + αwR,λ2 (t − 1). wR,λ2 (t) = −η ∂wR,λ2

wR,λ1 (t) = −η

(7)

Similarly, the weights in the hidden-to-output connections are updated by ∂E + αvT ,R (t − 1) ∂vT ,R ∂E vε,R (t) = −η + αvε,R (t − 1) ∂vε,R

vT ,R (t) = −η

(8)

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where E is the error between the estimated value and the experimental value, η is the learning rate coefficient, and α is the momentum coefficient. The momentum coefficient α is added to improve the normal gradient-descent method because the speed of convergence can be very slow if the learning rate is either too small or too large. The updated weights can thus be given by wR,λ1 (t + 1) = wR,λ1 (t) + wR,λ1 (t − 1) wR,λ2 (t + 1) = wR,λ2 (t) + wR,λ2 (t − 1) . vT ,R (t + 1) = vT ,R (t) + vT ,R (t − 1) vε,R (t + 1) = vε,R (t) + vε,R (t − 1)

(9)

The values of the weights obtained from equations (7) and (8) can be inserted into equations (5) and (6) to evaluate the updated T and ε. This procedure will not stop until the errors between the estimated values of Test and ε est and ‘experimental’ values of Ttrain and ε train are below a specific value. After the training process, the nonlinear function between the input (λ1, λ2) and output (T, ε) is generated. It can thus be used to recover the temperature and strain information through the inputs of the wavelength variations of the two transmission dips (λ1 and λ2).

(a)

3. Simulation results and discussion The shifts of two resonant dips with temperature and strain effects are shown in figure 3. The measurable resonant wavelengths λ1 and λ2 manifest as two loss dips in the LPG transmission spectrum, and each of them is perturbed simultaneously by both temperature and strain effects. However, for the large values of T, the shifts of two resonant wavelengths with the temperature may not be linear, i.e. A or C in equation (1) are functions of T, respectively. In addition, the strain sensitivity is also changed with respect to the temperature, which means that B and D in equation (1) are also functions of T. This introduces complexity on the matrix inversion method that imposes difficulties and limitations for recovering the strain and temperature information, simultaneously. The nonlinearity and cross-sensitivity issues can, however, be minimized by adapting the implement of the artificial neural network, which is able to recover the temperature and strain information for given pairs of λ1 and λ2. The training data set consists 200 pairs of {(λ1 train, λ2 train), (Ttrain, ε train)}, where Ttrain and ε train were generated randomly within a range of 35–120 ◦ C and 0– 3000 µε, and λ1 train and λ2 train were evaluated by equations (2)–(4). The learning rate coefficient η and the momentum coefficient α in equations (7) and (8) were chosen in an informed manner to be 0.95 and 0.85, respectively. The specific values of the errors for the temperature and strain were 0.001 ◦ C and 0.001 µε, respectively. The number of neurons in the hidden layer was decided by making several tests with R = 5, 8, 10, 20 and 36. It was found that the accuracy increases with the number of hidden layers. However, when R > 10 the accuracy increases slightly with the number of neurons. Considering these results, it was decided to use ten hidden layer neurons for the ANN to be trained. In this instance, the number of training cycle is less than 1500. After 2946

(b)

Figure 3. Wavelength shifts of (a) first-order and (b) second-order transmission dips of the LPG.

training, the weight values of wR,λ1 , wR,λ2 , vT ,R and vε,R were obtained and listed in table 1. The network was then tested by using another set consisting of 500 new pairs of {(λ1 test, λ2 test), (Ttest, ε test)}. Ttest and ε test were supposed to be the ‘accurate’ values of temperature variation and strain variation. λ1 test and λ2 test were the temperature and strain induced wavelength shifts which were evaluated by inserting Ttest and ε test into equation (4). The ‘estimate’ values, i.e. Test and ε est, were those evaluated by the trained ANN where the inputs were the calculated values of λ1 test and λ2 test. The errors for the strain and temperature (i.e. the difference between the ‘accurate’ values of Ttest and ε test and estimate values of Test and ε est) are shown in figures 4(a) and (b), respectively. The maximum absolute values of errors are 0.04 ◦ C and 2.7 µε, respectively. To evaluate the performance, the results have also been compared to the matrix inversion method to show the improvement in the strain and temperature measurement. The matrix coefficients in equation (1) were derived by using the same training data set and the least-squares algorithm [11]. The ‘estimate’ values were those evaluated by the matrix inversion method where the inputs were the calculated values of λ1 test and λ2 test. The errors for the strain and temperature by using the matrix inversion method are shown in figures 5(a) and (b). The maximum absolute values of errors

Application of an artificial neural network for simultaneous measurement of temperature and strain

Table 1. Weight values of wR,λ1 , wR,λ2 , vT ,R and vε,R after training. R 1

2

3

4

5

wR,λ1 1.7146 1.4017 1.9922 2.2373 2.5052 wR,λ2 3.2271 0.5285 2.1683 2.3533 2.6425 vT ,R 0.4976 0.5012 0.4186 0.4842 0.2343 vε,R −3.242 −3.161 −7.790 −3.370 −3.516

(a)

6

7

8

9

10

6.3701 6.1629 0.4580 3.4297

2.9703 4.7968 0.5090 3.1990

0.3478 2.1149 0.3884 12.8625

0.6435 0.0215 2.8102 4.1480

1.2901 0.2034 1.8539 6.2899

(a)

(b)

Figure 4. Errors of (a) temperature and (b) strain within the temperature range 35–120 ◦ C and strain range of 0–3000 µε by using the artificial neural network approach.

are 9.3 ◦ C and 150 µε, respectively, which is much larger than those obtained by the ANN method. The demonstrated errors are relatively small compared to those reported in the literature, including the results by Chan et al [9] (0.4 ◦ C, 4.9 µε for small cross-sensitivity; 2 ◦ C, 6.4 µε for large cross-sensitivity) and by Gusmeroli et al [12] (1 ◦ C, 10 µε) and by Patrick et al [13] (1.5 ◦ C, 9 µε). The main reason for this is that the temperature and strain response of the LPG exhibit relatively small nonlinearity and cross-sensitivity over a large temperature and strain range. Therefore, the ANN method shows good performance in recovering the temperature and strain information. However, the small errors might not be repeated in the experiment. The primary concern is the signal to noise ratio which may deteriorate the performance of the ANN method. However, this does not prevent its application, since the performance of ANN is much better than the matrix inversion method. The

(b)

Figure 5. Errors of (a) temperature and (b) strain within the temperature range 35–120 ◦ C and strain range of 0–3000 µε by using the matrix inversion method.

experimental demonstration of the performance of ANN may need further investigation. It has been demonstrated that the temperature and strain sensitivities of the conventional LPG change with the surrounding medium, arising from the dependence of effective indices of the cladding modes on the surrounding medium [14]. However, for the hybrid polymer–silica structure, it has been demonstrated that LPG resonances are insensitive to the external refractive index, which is for the reason that the cladding modes are mainly confined by polymer cladding [15]. The sensitivity is thus expected to be independent of the surrounding medium, which may need further investigation. Although the ANN method is applied in a similar way in [9], the difference in the performance indicates that a sensor with small nonlinearity and cross-sensitivity 2947

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could yield a better performance with the ANN method. Therefore, the PCF-based sensor should be more appropriate for simultaneous temperature and strain measurement due to its relatively small nonlinearity and cross-sensitivity.

[5] [6]

4. Conclusion In conclusion, an artificial neural network approach is proposed for simultaneous measurement of temperature and strain by using a LPG which is inscribed in a photonic crystal fiber with hybrid cladding. The method is based on the measurement of the shifts of two resonant wavelengths which behave differently with the temperature and strain. By using the ANN approach, the temperature and strain effects can be determined simultaneously. The maximum temperature and strain errors of 0.04 ◦ C and 2.7 µε are obtained within the temperature range from 35 ◦ C to 120 ◦ C and strain range from 0 to 3000 µε. Compared to the standard matrix inversion method, the ANN method is well suited for implementation in sensors which suffer nonlinearity and cross-sensitivity.

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B/Ge co-doped fiber IEEE Photon. Technol. Lett. 13 818–20 Khaliq S, James S and Tatam R P 2002 Enhanced sensitivity fibre optic long period grating temperature sensor Meas. Sci. Technol. 13 792–5 Westbrook P S, Eggleton B J, Windeleer R S, Hale A, Strasser T A and Burdge G L 2000 Cladding-mode resonances in hybrid polymer–silica microstructured optical fiber gratings IEEE Photon. Tech. Lett. 12 495–7 Jin W, Michie C, Thursby G, Konstantaki M and Culshaw B 1997 Simultaneous measurement of strain and temperature: error analysis Opt. Eng. 36 589–609 Patterson D W 1996 Artificial Neural Network: Theory and Applications (Singapore: Prentice-Hall) Chan C C, Jin W, Rad A B and Demokan M S 1998 Simultaneous measurement of temperature and strain: an artificial neural network approach IEEE Photon. Tech. Lett. 10 854–6 Chu P L 2005 Polymer optical fiber Bragg gratings Opt. Photonics News 16 (7) 52–6 Wellstead P E and Zarrop M B 1991 Self-Tuning Systems: Control and Signal Processing (New York: Wiley) Gusmeroli V and Martinelli M 1994 Non-incremental interferometric fiber-optic measurement method for simultaneous detection of temperature and strain Opt. Lett. 19 2164–6 Patrick H J, Williams G M, Kersey A D, Pedrazzani J R and Vengsarkar A M 1996 Hybrid fiber Bragg grating/long period grating sensor for strain/temperature discrimination IEEE Photon. Technol. Lett. 8 1223–5 Kamikawachi R C, Possetti G R C, Falate R, Muller M and Fabris J L 2007 Influence of surrounding media refractive index on the thermal and strain sensitivities of long-period gratings Appl. Opt. 46 2831–7 Espindola R P, Windeler R S, Abramov A A, Eggleton B J, Strasser T A and DiGiovanni D J 1999 External refractive index insensitive air-clad long period fiber grating Electron. Lett. 35 327–8