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.
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
[5

7]
, 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 [
5

7
,
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
The LPFG structure with core mode blocker.
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
[10

12]
. Here,
D
_{1}
and
D
_{2}
are the delay of the diagonal matrix,
CB
is matrix for core mode blocker,
E
_{co}
and
E
_{cl}
^{(i)}
are the Efields (electric fields) of the fundamental core mode and the ith cladding mode, respectively. And
β
_{co}
and
β
^{cl}
_{(i)}
are the propagation constant of the fundamental core mode and the ith cladding mode for the LPFGs,
L
_{1}
and
L
_{2}
are the length of LPFG
_{1}
and LPFG
_{2}
, and
d_{1}
,
d_{2}
, and d
_{b}
are the length for
D_{1}
,
D_{2}
, and
CB
.
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 matrix
M
_{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 from
F
^{(p)}
in
Fig. 2
(b). The relation between
G
^{(p)}
and
F
^{(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, and
κ
_{p}
is the coupling coefficient between the fundamental core mode and the pth cladding mode
[10

13]
.
Note that
M
_{k}
in (2) can be regarded as a multiport lattice section. Because of the decoupling property of modes, the matrices
G
^{(p)}
in
M
_{k}
commute with each other so that the
(a) An equivalent block diagram for Fig. 1. (b) Multiport lattice filter model.
2×2 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)
with
E
_{cl}
^{(i)}
(in)
= 0, (1≤i≤N) is easily seen to be t =
Q
_{1,l}
, which is the (1,1)element of
Q
.
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㎛ 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.
 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 1300
nm
and 1580
nm
, where LPFG
_{1}
and LPFG
_{2}
have uniform gratings.Their lengths are
L
_{1}
= 100Λ
_{1}
and
L
_{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 Δ
n
_{1}
=
n
_{2}
= 0.00011,respectively. The length of the D
_{1}
and the D
_{2}
are
d
_{1}
= 400 Λ
_{1}
and
d
_{2}
= 400Λ
_{1}
, respectively. The length of the
CB
is assumed to be
d
_{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}
and
LP
_{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
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 ㎛).
(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.).
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 is
d_{1}
=
d_{2}
= 0 and the core mode blocker is
CB
=
diag
(0,1,1┅,1), ideally. The 3dB bandwidth of the LPFG
_{1}
and LPFG
_{2}
is 13.9
nm
. In the case, we can control the bandwidth of the flattop bandpass filter in the wavelength range less than 13.9
nm
± ε (where
ε
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 14.2
nm
(
ε
= 0.3
nm
) and 8
nm
, 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}
and
κ
_{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 14.2
nm
(ε = 0.1
nm
) and 8
nm
, 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.
Acknowledgements
This work was supported by the research grant from the Chuongbong Academic Research Fund of Jeju National University in 2010.
Othonos A
,
Kalli K
1999
Fiber Bragg Gratings  Fundamentals and Applications in Telecommunications and Sensing
Artech House
Boston USA
Kim S
,
Kim G. H
,
Hwang K
,
Lim S. D
,
Lee K
,
Kim S. H
,
Lee S. B
,
Jeong J
2009
Investigation of an arcinduced long period fiber grating inscribed in a photonic crystal fiber with two large air holes
J. Opt. Soc. Korea
13
428 
433
Han Y. G
,
Kim S. H
,
Lee S. B
,
Paek U. C
,
Chung Y
2003
Development of core mode blocker with H2loaded GeB codoped fibres
Electron. Lett.
39
1107 
1108
Lim S. D
,
Kim J. G
,
Lee K
,
Lee S. B
,
Kim B. Y
2009
Fabrication of a highly efficient coremode blocker using a femtosecond laser ablation technique
Opt. Express
17
18449 
18454
Choi S
,
Eom T. J
,
Yu J. W
,
Lee B. H
,
Oh K
2002
Novel allfiber bandpass filter based on hollow optical fiber
IEEE Photon. Technol. Lett.
14
1701 
1703
Choi S
,
Eom T. J
,
Yu J. W
,
Lee B. H
2005
Broadband tunable allfiber bandpass filter based on hollow optical fiber and longperiod grating pair
IEEE Photon. Technol. Lett.
17
115 
117
Sakata H
,
Takata Y
,
Suzuki S
2007
Singlechannel bandpass filter based on vernieraligned longperiod fiber gratings
IEEE Photon. Technol. Lett.
19
1661 
1663
Starodubov D. S
,
Grubsky V
,
Feinberg J
1998
Allfiber bandpass filter with adjustable transmission using claddingmode coupling
IEEE Photon. Technol. Lett.
10
1590 
1592
Han Y
,
Kim S. H
,
Lee S. B
2004
Flexibly tunable multichannel filter and bandpass filter based on longperiod fiber gratings
Opt. Express
12
1902 
1907
Bae J
,
Chun J
,
Lee S. B
2004
Synthesis of longperiod fiber gratings with the inverted erbium gain spectrum using the multiport lattice filter model
J. Ligthwave Technol.
22
1976 
1986
Kim H
,
Bae J
,
Chun J
2007
Synthesis of flattop bandpass filters using twoband rejection longperiod fiber gratings저 IEEE Photon. Technol. Lett.
IEEE Photon. Technol. Lett.
19
1466 
1468
Kim H
,
Bae J
,
Lee J. W
,
Chun J
,
Lee S. B
2003
Analysis of concatenated longperiod fiber gratings having phaseshifted and cascaded effects
Jpn. J. Appl. Phys. Part 1
42
5098 
5101
Yi M
,
Kim Y
,
Yee D
,
Ahn J
2009
Terahertz frequency spreading filter via onedimensional dielectric multilayer structures저 J. Opt. Soc. Korea
13
398 
402
Song G
,
Shin S
1985
Design of corrugated waveguide filters by the Gel’'fandLevitanMarchenko inversescattering method
J. Opt. Soc. Am. A
2
1905 
1915
Peral E
,
Capmany J
,
Marti J
1996
Iterative solution to the Gel’'fandLevitanMarchenko coupled equations and application to synthesis of fiber gratings
J. Quantum Electron.
32
2078 
2084