## Abstract

To analyze the various LPFGs with thermal changes, we present how makes the kernel function to translate the information of thermal change into the coupling coefficient and detuning factor changed by temperature. We propose the extended fundamental matrix model with the proposed kernel function. To verify the validity of the proposed model experimentally, we have manufactured the LPFG structures with the thermal changes using the divided coil heater. We have observed that the transmission spectra calculated using the proposed model are close to the corresponding measured spectra in the wavelength band of interest.

© 2004 Optical Society of America

## 1. Introduction

Long period fiber gratings (LPFG) are formed by exposing a single mode fiber segment with a KrF excimer laser (248 nm) through an amplitude mask. In an LPFG, there is one fundamental core mode and multiple cladding modes, which all propagate in the same (forward) direction. Due to their unique features such as low insertion loss, low back-reflection and excellent polarization insensitivity, the LPFG have attracted great interest in the optical telecommunication and sensor applications [1]. Many researchers have studied the various LPFGs like the gain equalizers of the optical fiber amplifier [2, 3, 4], the wavelength-division multiplexing (WDM) isolation fiber filters [5], the band rejection filters [1, 6], and the sensors [8]. In particular, several researchers have presented about the temperature sensors [7, 8] and the various tunable filters using the temperature sensitivity of the LPFG [9, 10]. The central points of their study are proposed about the physical phenomenon of the LPFG varied by the thermal change [7] and the manufacture of the tunable filter system [9, 10]. However to accurately analyze the LPFGs with the thermal changes, we need the kernel function to calculate the the coupling coefficient and detuning factor changed by the thermal change. In general, the kernel function is a weighting function used in non-parametric function estimation. In this paper, we will proposed how to find a thermal kernel function used for the analysis of the LPFGs.

The characteristics of the LPFG can be analyzed based on the coupled mode theory [11, 12] derived from the Maxwell equations, and the fundamental matrix model [11, 13, 14]. In these models, fundamental matrix model have generally used to the analyzing the non-uniform LPFG. However analysis of the various LPFGs with the thermal change using these tools is very difficult, so we proposed the extended fundamental matrix model using the proposed kernel function, which can be analyzed the various LPFGs with the thermal changes.

In this paper, we present the method to determine the kernel function from several spectra measured along thermal changes. Thereafter we propose an accurate and versatile extended fundamental matrix model with the kernel function considering the thermal changes. To verify the validity of the proposed analysis method, we have fabricated the LPFGs with the thermal changes using the divided coil heater, and their experimental results compared with theoretical ones will be presented.

## 2. Thermal kernel function

The behavior of an LPFG can be analyzed using the coupled mode equations [11] which can be derived from the Maxwell equations. Consider the uniform long period gratings which are modulated by the induced index change Δ*n* along the *z*-direction. In this paper, for the sake of simplicity, we model the refractive index *n*(*z*) in the core as being of the form [6, 12]

where *n*_{co}
is the unperturbed refractive index, Λ is the period of the grating, and *L* is the length of the LPFG. The duty cycle in (1) is 50%.

The interaction between the amplitude of the fundamental core mode (*LP*
_{01}) and the ith cladding mode (*LP*
_{0i}) can be represented by the simplified coupled mode equations [11, 12],

$$\frac{dB\left(z\right)}{\mathit{dz}}=-j\delta B\left(z\right)+j{\kappa}^{*}A\left(z\right),$$

where *A*(*z*) and *B*(*z*) are, respectively, the complex amplitudes of the fundamental core mode and of the *i*
^{th} cladding mode, *κ* and *δ* are the *coupling coefficient and detuning factor* between the fundamental core mode and the *i*
^{th} cladding mode, and j≜√-1. For the boundary condition *A*(0)=1 and *B*(0)=0, the solutions of the coupled mode equations are

Many researchers have studied that the spectrum of the LPFG was changed by temperature [7, 8, 9, 10]. The solutions of the coupled mode equations in Eqs. (3) and (4) are the function of *κ* and *δ* which depend on thermal change. To analyze the LPFGs with the thermal changes, we need the kernel function as shown in Fig. 1 between the coupled mode equation and the changed temperature. In Fig. 1, *T* is temperature, and *κ*_{T}
and *δ*_{T}
is the coupling coefficient and detuning factor changed by *T*, respectively. To find the kernel function, we measure the transmission spectra *D*_{i}
(*i*=1,2,⋯11) of *LP*
_{05} mode of a single uniform LPFG with total length *L*=4 cm (grating period Λ=421.15 µm and 95 gratings) along the thermal changes *T*={24.9, 35, 45, 55, 65.3, 75, 85, 95.2, 105.3, 115, 125.1} °*C* as shown in Fig. 2(a). The parameters of the fiber used in this paper are as follows:

where *n*_{cl}
is the refractive index of the cladding, *n*_{air}
is the refractive index of air, *r*_{co}
is the radius of the core, and *r*_{cl}
is the radius of the cladding. The used fiber to fabricate the LPFG is a boron-codoped germanosilicate fiber, which can be significantly enhanced a thermal tuning efficiency as compared with the standard telecommunication fiber in terms of the temperature sensitivity of the LPFG [15].

The *κ* and *δ* for the spectrum *D*
_{1} in Fig. 2(a) without the thermal change (in our case *T*=24.9°*C*) can be calculated as shown in Fig. 2(b) using follows:

where *C* is the overall integral factor between the modes [11], ${n}_{\mathit{\text{eff}}}^{\mathit{\text{co}}}$
is the effective index of the core, and ${n}_{\mathit{\text{eff}}}^{\mathit{\text{cl}}}$
is the effective index of the cladding. The ${C\mathit{,}n}_{\mathit{\text{eff}}}^{\mathit{\text{co}}}$
, and ${n}_{\mathit{\text{eff}}}^{\mathit{\text{cl}}}$
are changed along temperature, but period Λ is ignored because of changing very small along temperature [7]. The kernel functions for *κ* and *δ* changed by temperature are

$$=\frac{\pi}{\lambda}\left[\left(1+\frac{d{C}_{T}}{C}+\frac{d\Delta {n}_{T}}{\Delta n}\right)\Delta nC+d\Delta {n}_{T}d{C}_{T}\right]$$

$$\approx \frac{1}{\lambda}\left(aX+b\right),$$

$$\approx \delta +c,$$

where *dΔn*_{T}*, dC*_{T}, ${\mathit{\text{dn}}}_{\mathit{\text{eff}}\mathit{,}T}^{\mathit{\text{co}}}$
, and ${\mathit{\text{dn}}}_{\mathit{\text{eff}}\mathit{,}T}^{\mathit{\text{cl}}}$
are a small change of Δ${n\mathit{,}C\mathit{,}n}_{\mathit{\text{eff}}}^{\mathit{\text{co}}}$
, and ${n}_{\mathit{\text{eff}}}^{\mathit{\text{cl}}}$
by *T*, respectively. The *κ* is a important factor for the bandwidth and the depth of the spectrum, and the *δ* is a very important factor for the resonance wavelength. Here, *a* is the function to fit the variation of the bandwidth and the depth of the spectrum, *b* is the function to fit the variation of the depth of the spectrum, and *c* is the function to fit the variation of the resonance wavelength of the spectrum. Because the variation of *a, b*, and *c* varied along wavelength in band 1490 nm and 1560 nm are very small changes, we can approximately set that there are the function of temperature as

$$b\triangleq {b}_{0}{T}^{m}+{b}_{2}{T}^{m-1}+\cdots +{b}_{m-1}{T}^{1}+{b}_{m},$$

$$c\triangleq {c}_{0}{T}^{m}+{c}_{2}{T}^{m-1}+\cdots +{c}_{m-1}{T}^{1}+{c}_{m}.$$

The *X* and *δ* for 24.9 °*C* as shown in Fig. 2(b) can be computed from Λ=421.15 *µ*m, Δ*n*=0.0001652, and fiber parameters, where Δ*n* have determined by fitting transmission spectrum in Fig. 2(a) using the generalized simulated method [16]. *X*_{T}*=aX+b* and *δ*_{T}*=d+c* for the measured spectra (*D*
_{2},⋯,*D*
_{11}) with thermal changes in Fig. 2(a) can be calculated by finding the *a, b*, and *c* as rectangles shown in Figs. 3(a), (c), and 3(e). The *a, b*, and *c* are determined by fitting the measured transmission spectra *D*_{i}
(*D*
_{2},⋯,*D*
_{11}) as solid and dotted curves in Fig. 2(a) using the generalized simulated method. The coefficients *a*_{m}
(*m*=0,1,⋯,25), *b*_{m}
(*m*=0,1,⋯,15), and *c*_{m}
(*m*=0,1,⋯,23) in Figs. 3(b), (d), and (f) for *a, b*, and *c* in (9) can determined from rectangles shown in Fig. 3(a), (c), and 3(e) as following steps: (i) To obtain more exact coefficients, there interpolate as dotted lines using cubic spline data interpolation [17, 18] in between rectangles shown in Fig. 3(a), (c), and 3(e). (ii) We can find the coefficients as shown in Fig. 3(b), (d), and (f) from optimally polynomial curve fitting as dashed lines (see Fig. 3(a), (c), and 3(e)) using a least squares method [17, 18]. In this paper, we consider the *LP*
_{05} of the uniform LPFG, but the kernel function to analyze the LPFG with the other mode [19] have to calculated from the measured spectra of that mode. The number of the measured spectra is numerically determined as the complexity of the spectra with thermal change. We have calculated the transmission spectra *S*_{i}
(*i*=1,⋯,10) (see Fig. 2(a)) along temperatures *T*={30,40,50,60,70,80,90,100,110,120} °*C* using computation processing step with the thermal kernel function as shown in Fig. 1.

## 3. Analysis of the LPFGs with the thermal changes

#### 3.1. Extended fundamental matrix model

Consider the gratings which are grouped into *M sections* in such a way that the grating period Λ_{i}, the refractive index *n*_{i}
, and thermal change *T*_{i}
inside each of the *M* sections are uniform (see Fig. 4). The field amplitudes coming into and out from the ith section can be shown [11, 13] to be

The 2×2 complex matrix *F*_{i}
for LPFG with co-directional interactions is given by

where *L*_{i}
is the section length, and ${\Upsilon}_{i}\stackrel{\mathrm{def}}{=}\sqrt{{\left({\kappa}_{{T}_{i}}\right)}^{2}+{\left({\delta}_{{T}_{i}}\right)}^{2}}$ is the *effective detuning*, all for the ith section. The overall field amplitudes of the entire structure having M uniform sections are then

$$F\triangleq {F}_{M}{F}_{M-1}\cdots {F}_{1}\triangleq \left[\begin{array}{cc}{F}_{11}& {F}_{12}\\ {F}_{21}& {F}_{22}\end{array}\right].$$

The overall transmission coefficient for the LPFGs *t*≜*A*_{M}*/A*
_{0} with *B*
_{0}=0, is easily seen to be *t=F*
_{1,1}, which is the (1,1)-element of *F* matrix. Here the transmission coefficient is the function with coupling coefficient and detuning factor.

#### 3.2. Examples for the proposed model

We analytically calculated the transmission spectra for the LPFGs with the thermal changes in the following examples by computing the transmission coefficient *F*
_{1,1} in (12).

### 3.2.1. Example1:

Because Λ=415 *µ*m is used in this example, we have calculated the *δ*
_{24.9} as shown in Fig. 5(a). Here *X* and coefficients of *a, b*, and *c* have used the same ones in section II, because of the independence from L. Figures 5(b), (c), and (d) show the *LP*
_{05} mode transmission spectra in the wavelength range between 1490 nm and 1560 nm for the structure of *Section*
_{1}- *Section*
_{2}-*Section*
_{3} with *T*
_{1}=24.9 °*C*, *T*
_{2}=125 °*C*, *T*
_{3}=24.9 °*C* and *L*
_{1}=50Λ, *L*
_{2}=*N*_{x}
Λ (*N*_{x}
={1,5,10,20,100,300}), and *L*
_{3}=50Λ.

### 3.2.2. Example2:

Figures 6(b), (c), and (d) show the *LP*
_{05} mode transmission spectra in the wavelength range between 1490 nm and 1560 nm for the structure of *Section*
_{1}-*Section*
_{2}-*Section*
_{3} with *T*
_{1}=125 °*C*, *T*
_{2}=24.9 °*C*, *T*
_{3}=125 °*C* and *L*
_{1}=50Λ, *L*
_{2}=*N*_{x}
Λ (*N*_{x}
={1,5,10,20,100,300}), and *L*
_{3}=50Λ, where Λ=427.85 *µ*m. Figure 6(a) shows the calculated *δ*
_{24.9} for Λ.

### 3.2.3. Example3:

Figures 7(a) and (b) show the *LP*
_{05} mode transmission spectra in the wavelength range between 1490 nm and 1560 nm for the structure of *Section*
_{1}-*Section*
_{2}-*Section*
_{3}. The structure of Fig. 7(a) are composed of *T*
_{1}=24.9 °*C*, *T*
_{2}=*t*_{x}
°*C* (*t*_{x}
={24.9,50,100,125}), *T*
_{3}=24.9 °*C* and *L*
_{1}=50Λ, *L*
_{2}=5Λ, and *L*
_{3}=50Λ, where Λ=415 *µ*m. The structure of Fig. 7(b) are composed of *T*
_{1}=125 °*C*, *T*
_{2}=*t*_{x}
°*C* (*t*_{x}
={24.9,50,100,125}), *T*
_{3}=125 °*C* and *L*
_{1}=50Λ, *L*
_{2}=5Λ, and *L*
_{3}=50Λ, where Λ=427.85 *µ*m. We can show that the Figs. 7(a) and 7(b) have a symmetrical spectrum figures.

In these examples, we can see that the spectra shown in Figs. 5, 6, and 7 are the same effect as the phase-shifted and cascaded LPFGs [3, 20]. The spectrum of *Section*
_{2} with large *N*_{x}
shifted by the thermal change have shaped new band-rejection filter as shown in Figs. 5(c) and 6(c). This type of device can be used as a multiband isolation and band rejection filter, and a multiband isolation filter for the WDM communication system.

## 4. Experimental results

#### 4.1. Experimental setup

Figure 8(a) shows a experimental setup for the tunable optical fiber grating filter with the thermal changes using the LPFGs with the divided Ni-Cr coil heater [9, 10]. The divided coil heater is composed of 64 coil heater sections which can be individually controlled until the maximum 64 sections. The length of each coil heater elements is 2300 *µ*m, where the spacing of each heater element is 200 *µ*m, the inner diameter was 300 *µ*m, and the turn number of each coil is 12 turns (see Fig. 8 (b)). The divided coil heater is held on substrate by high temperature silicon as shown in Figs. 8 (b) and (d) normal fiber holder is used to minimize the induced bending of fiber. The controller adjusts the ratio of electric powers of each coil heater section independently to make an appropriate temperature distribution in the LPFG. The refraction index of each LPFG sections are independently modified with the temperature variation of each coil heater and a single uniform LPFG can be tuned as the piecewise-uniform LPFG sections. Each coil heater section can be heated up to 125°*C* in the room temperature, with an electrical power of less than 0.09 W. A fan cooler, above the coil heater, was used for heat sinking and preventing thermal transmission to other section (see Fig. 8(c)).

#### 4.2. Comparison of measured and calculated spectrum Curves

The measured and the theoretically calculated spectrum curves are compared in Figs. 9–17. The parameters of the fiber used to fabricate the LPFGs are those noted in section II. To verify our analysis method, we have used the transmission spectrum of *LP*
_{05} mode of a single uniform LPFG with total length *L*=6.1cm (grating period Λ=423.69 *µ*m and 144 gratings) and induced index change Δ*n*=0.000119 as shown in Fig 9(a). We have used that the *κ* and *δ* are calculated to temperature *T*=24.9 °*C* as shown in Fig. 9(b) and the coefficients *a, b*, and *c* are used the same ones obtained in section II.

Figures 10–17 shows the *LP*
_{05} mode transmission spectra in the wavelength range between 1475 nm and 1575 nm for the structure of the LPFGs with 24 sections formed by thermal changes. The gratings of each section is 6, approximately, so our experimental setup has the control error of thermal change. We have carried out the experiments to measure the spectra as the thermal changes shown in (b) in Figs. 10–17, and then the measured spectra have showed in (a) in Figs. 10–17. Note that the measured (dashed curves) and the calculated transmission spectra (solid curves) (see (a) in Figs. 10–17) match well in the wavelength band.

## 5. Conclusions

To calculate the transmission spectrum of the LPFGs with the thermal changes in the wavelength band of interest, the proposed kernel function can be used in between temperature bound 24.9 °*C* and 125.1 °*C*. We have showed how the proposed analysis method using the kernel function can be applied to analyze the LPFGs with the thermal changes. We have analyzed several types having the effect of the phase-shifted and cascaded LPFGs using a single uniform LPFG with the thermal changes, and showed that analytically calculated results obtained by using the proposed model are well matched with experimentally measured responses.

## Acknowledgments

This works was supported by Brain Korea 21 Project.

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