Design of Tunable Flattop Bandpass Filter Based on Two Longperiod Fiber Gratings and Core Mode Blocker
 Author: Bae Jinho, Bae Junkye, Lee Sang Bae
 Organization: Bae Jinho; Bae Junkye; Lee Sang Bae
 Publish: Journal of the Optical Society of Korea Volume 15, Issue2, p202~206, 25 June 2011

ABSTRACT
We propose a tunable flattop bandpass filter to pass light in a customized wavelength band by using longperiod fiber gratings (LPFG) structure. The LPFG structure is composed of a core mode blocker in between two LPFGs. The bandpass spectrum of the proposed structure is obtained in overlapped wavelength band of two LPFGs operating on the same modes. To analyze the properties, we introduce a mathematical matrix model for the structure. We theoretically demonstrate flexibility of the flattop bandpass filter with various bandwidths.

KEYWORD
Flattop bandpass , Longperiod fiber gratings , Core mode blocker , Matrix model , (350.2770) Gratings , (350.2460) Filters , interference , (230.2285) Fiber devices and optical amplifiers , (070.1170) Analog optical signal processing

I. INTRODUCTION
In longperiod fiber gratings (LPFG), a fundamental core mode and the multiple cladding modes are coupled,which all propagate in the same (forward) direction. Due to unique features of low insertion loss, low backreflection and excellent polarization insensitivity, LPFGs have attracted great interest in the optical communications and sensor applications [1, 2]. Some researchers have studied the manufacture of the core mode blocker [3, 4], band pass filter using the core mode blocker [57], and tunable bandpass filter using coil heater [8, 9].
The core mode blocker is a device to block off the propagated light in the core, and pass the propagated light in the cladding. Some researchers have developed bandpass filters using LPFGs, and some of them have studied the topics based on a core mode blocker as shown in Fig. 1. In Fig. 1, the LPFG_{1} is used to interact between core and cladding at resonant wavelength, and light existing in the core is extinguished by the core mode blocker. The light in the resonant wavelength is propagated through the cladding to the second LPFG. The resonant light existing in the cladding by the LPFG_{2} is transferred back into the core.The central points of the research are to study the physical phenomenon [3, 4] and the manufacture of the bandpass filter [57, 9]. However the design of the flattop bandpass filter [11] with specific customized wavelength band using an accurate analysis model was not proposed yet.
In the paper, we present the matrix model to analyze the
matrix model to analyze the LPFG structures with the core mode blocker. The proposed matrix model is a useful mathematical tool to describe the properties of the structure. We analyze the physical properties of the LPFG structure with a core mode blocker by using the proposed model. By using specially coupled feature of only the same modes in overlapped wavelength region, we demonstrate the design of the flattop bandpass filter with the customized bandwidth.
II. THE PROPOSED MATHEMATICAL MATRIX MODEL
The structure in Fig. 1 is modeled as the FIR (Finite Impulse Response) filter block diagram as shown in Fig.2(a). Figure 2(b) is the signal flow graph to analyze both LPFG_{1}and LPFG_{2}, which is called the multiport lattice filter model [1012]. Here,
D _{1} andD _{2} are the delay of the diagonal matrix,CB is matrix for core mode blocker,E _{co} andE _{cl} ^{(i)} are the Efields (electric fields) of the fundamental core mode and the ith cladding mode, respectively. Andβ _{co} andβ ^{cl} _{(i)} are the propagation constant of the fundamental core mode and the ith cladding mode for the LPFGs,L _{1} andL _{2} are the length of LPFG_{1} and LPFG_{2}, andd_{1} ,d_{2} , and d_{b} are the length forD_{1} ,D_{2} , andCB .In Fig. 2(a), the Efields coming into and out from the structure can be written to be
where
are diagonal matrices and
is a diagonal matrix with zero determinant.
By assuming the LPFG_{k}, (
k = 1, 2) are uniform long period fiber gratings, we can easily get the matrix form of the (N+1)×(N+1) complex matrixM _{k}, (k = 1, 2) for the LPFG_{k} with codirectional interactions since the LPFG_{k} has the multiport lattice filter structure in [10] as follows:where
G ^{(p)} is (N+1)×(N+1) identity matrix except for the four entries adopted fromF ^{(p)} in Fig. 2(b). The relation betweenG ^{(p)} andF ^{(p)} are as follows:(Here, B_{k,l} denotes the (
k, l ) element of the matrix B.) The matrix DG represents the phase shift of the LPFG,and
and
, L_{k} is the length of the LPFG_{k}, δ
_{p} is the detuning factor for the pth cladding mode, andκ _{p} is the coupling coefficient between the fundamental core mode and the pth cladding mode [1013].Note that
M _{k} in (2) can be regarded as a multiport lattice section. Because of the decoupling property of modes, the matricesG ^{(p)} inM _{k} commute with each other so that the2×2 subsections may be interchanged freely. If LPFG_{1} and LPFG_{2} are extended as piecewise uniform LPFGs, we can also easily get the Mk using the multiport lattice filter model [10]. Now, the overall transmission coefficient
t ≡E _{co}(out) ？E _{co}(in) withE _{cl}^{(i)}(in) = 0, (1≤i≤N) is easily seen to be t =Q _{1,l}, which is the (1,1)element ofQ .As shown in Fig. 2(a), after the coupled light on the LPFG_{1} is absorbed at CB, and then in the LPFG_{2}, the propagated light in the cladding is coupled with the same modes from cladding to core, due to the orthogonality of modes as shown in Fig. 2(b) [10]. Because the overall transmission is calculated as the complex multiplying of two bandpass filters, we cannot get the spectrum in the core if two filters have entirely different bandpass regions.
III. EXAMPLES
We analytically calculated the transmission spectrum curves for the concatenated LPFGs with a core mode blocker in the following examples by computing the transmission coefficient
t =Q _{1,l} in (1). The parameters of the fiber used in these examples are as follows:n _{co} = 1.449,n _{cl} = 1.444,n _{ai} = 1,r _{co} = 4.5㎛,r _{cl} = 62.5㎛ wheren _{cl} is the refractive index of the cladding,n _{air} is the refractive index of air,r _{co} is the radius of the core, andr _{cl} is the radius of the cladding.3.1. Example 1 for Uniform LPFG with Both Core Mode Blocker and Delay
Figure 3 shows the transmission for the modes
LP _{0i} , (i = 1,？,4) in the wavelength range between 1300nm and 1580nm , where LPFG_{1} and LPFG_{2} have uniform gratings.Their lengths areL _{1} = 100Λ_{1} andL _{2} = 100Λ_{2}, respectively,the grating period for LPFG_{1} is Λ_{1} = 441.24 ㎛, Λ_{2} for LPFG_{2} is varied (441.24 ㎛ for Fig. 3(a), 443.44 ㎛ for Fig. 3(b), 445.65 ㎛ for Fig. 3(c), and 501.28 ㎛ for Fig.3(d)) and their induced index changes are Δn _{1} =n _{2} = 0.00011,respectively. The length of the D_{1} and the D_{2} ared _{1} = 400 Λ_{1} andd _{2} = 400Λ_{1}, respectively. The length of theCB is assumed to bed _{b} = 0.To analyze the properties of the concatenated LPFGs with the core mode blocker, we have calculated the transmission spectra along the change of the Λ_{2} as shown in Fig 3. Figures 3(a)3(c) show the transmission along the overlapped region of the spectrum band of the same modes. We can only get the bandpass spectra by coupling the same modes in the overlapped region. Figure 3(d) shows the transmission for the coupling with the different modes (
LP _{04} of LPFG_{1} andLP _{03} for LPFG_{2}). From the result, we know that the bandpass filter cannot be obtained from coupling the different modes.3.2. Example 2 for Flattop Bandpass Filters
We can design a flattop bandpass filter with the customized bandwidth using the property of coupling with the same modes and the overlapped band. We have synthesized
the tunable flattop filter by considering a single cladding mode as shown in Fig. 4. We have utilized the coupling coefficients
κ and the detuning factors δ, respectively, as shown in the Figs. 4(a) and 4(b). The coupling coefficients in Fig. 4(a) can be found by the Gel’fandLevitanMarchenko coupled equations [14, 15].The
κ _{1} in Fig. 4(a) is used for the LPFG_{1} and LPFG_{2} of Figs. 4(c)4(f). In Fig. 4(b), the δ_{0} is used for the LPFG_{1} and LPFG_{2} is used the δ_{0} for Fig. 4(c), the δ_{0}50 for Fig. 4(d), the δ0100 for Fig. 4(e), and the δ0150 for Fig. 4(f). We have assumed the delay isd_{1} =d_{2} = 0 and the core mode blocker isCB =diag (0,1,1？,1), ideally. The 3dB bandwidth of the LPFG_{1} and LPFG_{2} is 13.9nm . In the case, we can control the bandwidth of the flattop bandpass filter in the wavelength range less than 13.9nm ± ε (whereε is a small variation by calculating the equation (1).). The 3dB bandwidth of the obtained flattop filter in Figs. 4(c) and 4(d) are 14.2nm (ε = 0.3nm ) and 8nm , respectively.From the results, we can observe that the bandwidth of the bandpass is changed along the overlapped band region of the flattop bandrejection of both LPFG_{1} and LPFG_{2}. If we can make the ideal bandrejection filter, we can synthesize a sharp bandpass filter with the customized bandwidth as well as the bandpass filters with a tunable bandwidth. Especially,if the bandwidth of LPFG_{1} and LPFG_{2} aren’t overlapped as shown in Fig. 4(f), we cannot synthesize the desired filter.
We also demonstrated the transmission along the overlapped region, when the coupling coefficients are different as shown in Figs. 4(g) and 4(h). The coupling coefficients
κ _{1} for the LPFG_{1} andκ _{2} the LPFG_{2} are utilized as shown in Fig.4(a). The detuning factors δ for Figs. 4(g) and 4(h) are the same ones as for Figs. 4(c) and 4(d). We have also used the delay and the core mode blocker as above examples.From the Fig. 4(g) and 4(h), we have obtained similar results to Figs. 4(c) and 4(d) although the LPFG_{1} and LPFG_{2} have different magnitude levels. The 3 dB bandwidth of the LPFG_{2} is 14.1
nm . The 3 dB bandwidth of the obtained flattop filter in Figs. 4(c) and 4(d) are 14.2nm (ε = 0.1nm ) and 8nm , respectively. The results show that the magnitude level of the flattop bandpass filter is dependent on the magnitudes of the LPFG_{1} and LPFG_{2}.IV. CONCLUSION
The analytical model for the LPFG with the core mode blocker is proposed. We have also analyzed and described the physical phenomenon of the structure by using the proposed model. We have proposed a design method of the flattop bandpass filter with customized bandwidth. We have also demonstrated, through computer simulations, the bandwidth of the flattop filters controlled by tuning the overlapped band region.

[FIG. 1.] The LPFG structure with core mode blocker.

[FIG. 2.] (a) An equivalent block diagram for Fig. 1. (b) Multiport lattice filter model.

[fig. 3.] Λ1 = 441.24 ㎛ is fixed and Λ2 is changed ((a) Λ2 = 441.24 ㎛ (b) Λ2 = 443.44 ㎛ (c) Λ2 = 445.65㎛ and (d) Λ2 = 501.28 ㎛).

[FIG. 4.] (a) Coupling coefficients. (b) Detuning factors. (c) The κ 1 and the δ0 are utilized for the LPFG1 and the LPFG2. (d) LPFG1 (κ 1 and δ0 is used.) and LPFG2 (κ 1 and δ050 is used.). (e) LPFG1 (κ 1 and δ0 is used.) and LPFG2 (κ 1 and δ0100 is used.). (f) LPFG1 (κ 1 and δ0 is used.) and LPFG2 (κ 1 and δ0150 is used.). (g) LPFG1 (κ 1 and δ0 is used.) and LPFG2 (κ 2 and δ0 is used.). (h) LPFG1 (κ 1 and δ0 is used.) and LPFG2 (κ 2 and δ050 is used.).