Semi-empirical multi-port lattice model for long-period
fiber grating analysis under arbitrary
temperature distributions

doi:10.1364/OE.16.000598

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Semi-empirical multi-port lattice model for long-period fiber grating analysis under arbitrary temperature distributions

Jun Kye Bae, Namkyoo Park, Jinho Bae, Dongyeon Koh, Sang Hyuck Kim, and Sang Bae Lee »View Author Affiliations

Optics Express, Vol. 16, Issue 2, pp. 598-606 (2008)
http://dx.doi.org/10.1364/OE.16.000598

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▸ Article Outline
– Abstract
1. Introduction
2. Formulation of the principle
3. Determination of parameters
4. Application of the semi-empirical LPFG analysis model
6. Conclusion
– References and links

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Abstract

We propose a semi-empirical model for the complete analysis (spectrum, bandwidth, and wavelength/phase shifts) of a temperature-tuned long-period fiber grating (LPFG) filter. By applying the multi-port lattice model to LPFGs, while deriving and utilizing the empirically determined temperature-dependence of core-to-cladding intermodal dispersions, we achieve a precise, practical means of spectrum analysis. Excellent agreement of the model with the experimental results was obtained over wide spectral ranges.

1. Introduction

Long-period fiber gratings (LPFGs) have been a topic of continuous research interest, with importance in practical applications [1–4

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long-period fiber gratings as band rejection filters,” J. Lightwave Technol. 14, 58–65 (1996). [CrossRef]

] such as; band-rejection filters, optical fiber polarizer, sensors, and optical switches. With their low insertion-loss/back-reflection and polarization insensitivity [5

A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, P. J. Lemaire, N. S. Bergano, and C. R. Davidson, “Long-period fiber-grating-based gain equalizers,” Opt. Lett. 21, 336–338 (1996). [CrossRef] [PubMed]

], LPFGs also provide an attractive solution to the gain equalization problem of fiber amplifiers. In this regard, various approaches have been suggested to custom engineer the spectrum of an LPFG filter to that of the gain-flattened optical amplifier, for example concatenated LPFGs [5

], phase-shifted LPFGs [6

Y. Liu, J. A. Willims, L. Zhang, and I. Bennion, “Phase shifted and cascaded long period fiber gratings,” Opt. Commun. 164, 27–31 (1999). [CrossRef]

] or non-uniform LPFGs [7

M. Harurnoto, M. Shigehara, and H. Suganurna, “Gain-flattening filter using long-period fiber gratings,” J. Lightwave Technol. 20, 1027–1033 (2002). [CrossRef]

]. The tuning of the LPFG spectrum has also been achieved by adopting various control techniques, including the control of the temperature [8

X. Shu, T. Allsop, B. Gwandu, and L. Zhang, “High-temperature sensitivity of long-period gratings in B—Ge codoped fiber,” Photon. Technol. Lett. 13, 818–820 (2001). [CrossRef]

, 9

J. K. Bae, J. Bae, S. H. Kim, N. Park, and S. B. Lee, “Dynamic EDFA gain-flattening filter using two LPFGs with divided coil heaters,” Photon. Technol. Lett. 17, 1226–1228 (2005). [CrossRef]

] or strain [3

V. Grubsky and J. Feinberg, “Long-period fiber gratings with variable coupling for real-time sensing applications,” Opt. Lett. 25, 203–205 (2000). [CrossRef]

] over the grating (whole or part of), with proper distribution functions. For all of these applications, it is crucial to be able to precisely understand, predict and engineer, at will, the spectral response of the LPFG - as well having a method of analysis for the evolutions/couplings of the propagation modes.

To provide a reliable and convenient tool supporting LPFG analysis, various approaches have been suggested in the past, including spectral summation [5

] and fundamental matrix models [6

Y. Liu, J. A. Willims, L. Zhang, and I. Bennion, “Phase shifted and cascaded long period fiber gratings,” Opt. Commun. 164, 27–31 (1999). [CrossRef]

M. Harurnoto, M. Shigehara, and H. Suganurna, “Gain-flattening filter using long-period fiber gratings,” J. Lightwave Technol. 20, 1027–1033 (2002). [CrossRef]

]. Further elaboration of the LPFG model was also achieved recently, including: 1) the multi-port lattice model [10

J. Bae, J. Chun, and S. B. Lee, “Synthesis of long-period fiber gratings with the inverted Erbium gain spectrum using the multiport lattice filter model,” J. Lightwave Technol. 22, 1976–1986 (2004). [CrossRef]

] for the analysis of multiple cladding modes, and 2) the extended fundamental matrix model (considering only one cladding mode) for the inclusion of a temperature dependent mode coupling term in the analysis [11

J. Bae, J. K. Bae, S. H. Kim, S. B. Lee, and J. Chun, “Analysis for long period fiber gratings using thermal kernel function,” Opt. Exp. 12, 797–810 (2004). [CrossRef]

]. Nevertheless, as regards the practical application of these techniques, there has not been a complete and simple method of non-uniform LPFG analysis enabling the estimation of the spectral characteristics under arbitrary temperature distributions, while accommodating the interactions of multiple cladding modes. Employing - for the determination of the inter-modal dispersion relations - either full modal analysis [12

T. Erdogan, “Cladding-mode resonances in short- and long- period fiber grating filters,” J. Opt. Soc. Am. A 14, 1760–1773 (1997). [CrossRef]

] requiring a full series of detailed, precise fiber parameters, or the high-order fitting method [11

J. Bae, J. K. Bae, S. H. Kim, S. B. Lee, and J. Chun, “Analysis for long period fiber gratings using thermal kernel function,” Opt. Exp. 12, 797–810 (2004). [CrossRef]

] with its fitting function limited to the peak wavelength while neglecting phase relations, these approaches also suffer from inherent limitations, often failing to derive the exact filter spectral shape (especially for asymmetric filters, e.g. for [11

J. Bae, J. K. Bae, S. H. Kim, S. B. Lee, and J. Chun, “Analysis for long period fiber gratings using thermal kernel function,” Opt. Exp. 12, 797–810 (2004). [CrossRef]

]) or leading to erroneous results from the imperfectly assumed fiber parameters (note that as many as several 10’s of related fiber parameters have to be used to calculate the precise dispersion relations with the full-modal analysis).

In this paper, we develop a simplified form of the multi-port lattice model by introducing semi-empirical mode-coupling functionals, in order to provide a convenient means of analysis for a general, multiple grating LPFG, having multiple cladding modes under arbitrary temperature distributions. By using the empirically determined intermodal dispersion functions, we achieve a precise, complete and convenient method of spectral shape analysis, allowing for the prediction of the transmission spectra, wavelength shift, bandwidth/phase changes of the grating over wide spectral ranges (1000–1700nm). The comparison of the model to the experimental results obtained with a temperature-tuned (independently controlled, 64-segment coil heater) LPFG shows excellent agreement in terms of its transmission spectrum.

2. Formulation of the principle

Keeping in mind that the transmission spectrum of the LPFG is determined by the interaction between the core mode and multiple cladding modes, without any loss of generality we start our discussion with the 2×2 fundamental matrix model describing the interaction (intermodal dispersion) between the core mode (0) and one of the cladding modes (p). Focusing on the temperature dependences of the key parameters in the LPFG analysis, viz. the detuning factor δ(λ, T) and coupling coefficient κ(λ, T) of the cladding mode, we first focus our attention on κ. Employing the notation in [3

V. Grubsky and J. Feinberg, “Long-period fiber gratings with variable coupling for real-time sensing applications,” Opt. Lett. 25, 203–205 (2000). [CrossRef]

, 11

J. Bae, J. K. Bae, S. H. Kim, S. B. Lee, and J. Chun, “Analysis for long period fiber gratings using thermal kernel function,” Opt. Exp. 12, 797–810 (2004). [CrossRef]

] and noting that the temperature dependencies for κ can be ignored for all practical purposes (reported to be <1% [11

J. Bae, J. K. Bae, S. H. Kim, S. B. Lee, and J. Chun, “Analysis for long period fiber gratings using thermal kernel function,” Opt. Exp. 12, 797–810 (2004). [CrossRef]

] - mostly depending on the dielectric structure of the fiber [12

T. Erdogan, “Cladding-mode resonances in short- and long- period fiber grating filters,” J. Opt. Soc. Am. A 14, 1760–1773 (1997). [CrossRef]

]) we will assume the following expression of κ for all later discussions,

κ (λ, T) = \frac{2 π \cdot σ}{λ} \cdot C (λ, T) \approx \frac{2 π \cdot σ}{λ} \cdot C (λ)

(1)

where σ is the normalized induced-index change, and C(λ,T) is the overlap integral between the core and cladding mode filed over the fiber cross section[10

], [12

T. Erdogan, “Cladding-mode resonances in short- and long- period fiber grating filters,” J. Opt. Soc. Am. A 14, 1760–1773 (1997). [CrossRef]

]. Next, for the convenient handling/manipulation of the detuning factor δ(λ,T), without any loss of generality we now employ the expression in [3

V. Grubsky and J. Feinberg, “Long-period fiber gratings with variable coupling for real-time sensing applications,” Opt. Lett. 25, 203–205 (2000). [CrossRef]

] and then rewrite the equation - in terms of the detuning factor at fixed temperature δ₀(λ,T ₀), and the differential dispersion function ΔΦ(λ, T-T ₀) - as follows,

δ (λ, T) = \frac{1}{2} (β^{(0)} - β^{(p)} - \frac{2 π}{Λ}) = \frac{1}{2} (Φ (λ, T) - \frac{2 π}{Λ})

= \frac{1}{2} (Φ (λ, T_{0}) + Δ Φ (λ, T - T_{0}) - \frac{2 π}{Λ}) \equiv δ_{0} (λ, T_{0}) + \frac{Δ Φ (λ, T - T_{0})}{2}

(2)

where Φ is the intermodal (core-to-cladding) dispersion function, Λ the pitch of the grating, β the propagation constant of the corresponding modes, and λ the vacuum wavelength. Worth to note, the temperature dependence of grating period Λ was ignored considering the order of thermal expansion coefficients (0.55×10^-6/°C) of silica-based single mode fiber [13

K. Shima, K. Himeno, T. Sakai, S. Okude, A. Wada, and R. Yamauchi, “A novel temperature-insensitive long-period fiber grating using a boron-codoped-germanosilicate-core fiber,” Proc. Optical Fiber Communication Conf. Dallas, TX , 347–348, (1997). [CrossRef]

Meanwhile, also utilizing the definition of Φ(λ, T)=β⁽⁰⁾-β^(p), we note that ΔΦ(λ, T-T ₀) can be alternatively expressed in terms of the effective index of the core and cladding modes.

Δ Φ (λ, T - T_{0}) = \frac{2 π}{λ} (Δ n_{eff}^{(0)} (λ, T - T_{0}) - Δ n_{eff}^{(p)} (λ, T - T_{0}))

(3)

Now, in order to express the above Eq. (3) in terms of empirically accessible parameters, we use the well-known formula for the normalized effective index of core mode, b [14

G. P. Agrawal, Fiber-Optic Communications Systems (New York: Wiley, 1997), Chap. 2.

b = \frac{n_{eff}^{(0)} - n_{cl}}{n_{co} - n_{cl}} ≅ {(1.1428 - \frac{0.9960}{V})}^{2}

(4)

expressed in terms of the normalized frequency, V [14

G. P. Agrawal, Fiber-Optic Communications Systems (New York: Wiley, 1997), Chap. 2.

V = \frac{2 π a}{λ} \cdot \sqrt{{n_{co}}^{2} - {n_{cl}}^{2}}

(5)

to write n ⁽⁰⁾ _eff =b(n_co -n_cl )+n_cl and then Δn ⁽⁰⁾ _eff =Δ[b(n_co -n_cl )+n_cl ], where a is the core radius. Now, rewriting ΔΦ(λ, T-T₀) in Eq. (3) in terms of b, and its differential, Δb, we get,

Δ Φ = \frac{2 π}{λ} (Δ n_{eff}^{(0)} - Δ n_{eff}^{(p)}) = \frac{2 π}{λ} [Δ b (n_{co} - n_{cl}) + b Δ (n_{co} - n_{cl}) + Δ n_{cl} - Δ n_{eff}^{(p)}]

(6)

Further simplification of the above expression can be accomplished in a straightforward manner from the definition of V and also by using (4) for the calculation of ΔV and Δb;

Δ V = \frac{2 π a}{λ} \cdot \frac{n_{co} {Δ n}_{co} - n_{cl} {Δ n}_{cl}}{\sqrt{{n_{co}}^{2} - {n_{cl}}^{2}}} ≅ V \cdot \frac{Δ (n_{co} - n_{cl})}{2 (n_{co} - n_{cl})} ∵ n_{co} ≅ n_{cl}

(7)

Δ b = 2 \cdot (1.1428 - \frac{0.9960}{V}) \cdot \frac{0.9960}{V^{2}} \cdot Δ V = [1.1428 \cdot \frac{0.9960}{V} - {(\frac{0.9960}{V})}^{2}] \cdot \frac{Δ (n_{co} - n_{cl})}{(n_{co} - n_{cl})}

(8)

Finally, with V=2.405 λ_cutoff.T0/λ and using Δn_cl ≅Δn ^(p) _eff (as all the cladding modes ~ up to 20^th - are confined almost entirely within the cladding material), we derive the interim expression of ΔΦ(λ, T-T ₀), which can be used in Eq. (2); expressed as the product of the wavelength dependent part and temperature dependent component,

Δ Φ (λ, T - T_{0}) ≅ \frac{2 π}{λ} \cdot ({1.1428}^{2} - \frac{1.1428 \cdot 0.9960}{2.405 \cdot λ_{cutoff, T 0}} \cdot λ) \cdot Δ (n_{co} - n_{cl})

(9)

where λ_cutoff,T0 is the cutoff wavelength of the fiber at fixed temperature.

With the above expression in Eq. (9), the LPFG analysis problem reduces to a much simpler form, requiring the empirical characterization of Δ(n _co-n _cl) as a function of temperature. For this purpose, we first obtain Φ(λ,T) and its differential, ΔΦ(λ,T), from the measurement, and then use the fitting Eq. (9) to get Δ(n _co-n _cl).

3. Determination of parameters

Fig. 1. Intermodal dispersion Φ(λ,T) for 1^st~7^th cladding modes extrapolated from experimental data, utilizing phase matching condition [3

V. Grubsky and J. Feinberg, “Long-period fiber gratings with variable coupling for real-time sensing applications,” Opt. Lett. 25, 203–205 (2000). [CrossRef]

] and resonance peak wavelength of each grating.

Fig. 2. Experimentally obtained (a) difference of intermodal dispersion ΔΦ(λ,T-T ₀) and (b) core-to-cladding refractive index difference Δ(n_co-n_cl) values at different temperatures (reference temperature=25°C, 80°C, 140°C), for 1^st to 7^th cladding modes.

Figure 1 shows the plot of the Φ(λ,T) values obtained from the spectrum measurement on the different LPFGs (with Λ=625, 540, 496, 440, 400 and 344um for the 1^st to 6^th cladding modes and Λ=344, 348, 352, 356, and 360um for the 7th cladding mode. T was varied between 26 and 140°C. The Φ(λ,T) values were calculated from the measured peak wavelengths and grating period, Λ, at each λ and T [3

V. Grubsky and J. Feinberg, “Long-period fiber gratings with variable coupling for real-time sensing applications,” Opt. Lett. 25, 203–205 (2000). [CrossRef]

]). Figure 2 also shows the plot of the ΔΦ(λ,T-T ₀) values derived from Fig. 1 for the 1^st~7^th cladding modes (wavelengths between 1000nm to 1650nm). It is worth mentioning that the measured ΔΦ(λ,T-T ₀) and Δ(n _co-n _cl) values for the different cladding modes overlap in a perfect manner, consistent with the result based on the full-model analysis [8

X. Shu, T. Allsop, B. Gwandu, and L. Zhang, “High-temperature sensitivity of long-period gratings in B—Ge codoped fiber,” Photon. Technol. Lett. 13, 818–820 (2001). [CrossRef]

In this regard, it can be said that the temperature variation of the core-to-cladding index difference is indeed the dominant fundamental parameter in the determination of the temperature dependence of the intermodal dispersion, irrespective of the order of the cladding modes, implying that one does not have to experimentally characterize ΔΦ(λ,T-T ₀) for each and every cladding mode. Especially, by taking advantage of the linear relationship observed in Fig. 2(b) between Δ(n _co-n _cl) and the temperature variation, one can rewrite the Eq. (9), at least within the tuning range of interest (25~140°C), as follows;

δ (λ, T) ≅ δ_{0} (λ, T_{0}) + \frac{π}{λ} \cdot ({1.1428}^{2} - \frac{1.1428 \cdot 0.9960}{2.405 \cdot λ_{cutoff, T 0}} \cdot λ) \cdot Δ (n_{co} - n_{cl})

≅ δ_{0} (λ, T_{0}) + (\frac{4.103}{λ} - \frac{1.487}{λ_{cutoff, 0}}) \cdot α_{T} \cdot (T - T_{0})

(10)

with a cutoff wavelength of the fiber λ_{cutoff, T0} at fixed temperature, and a fitting constant α_T, which can be determined from simple measurements of the uniform LPFG spectrum at several different temperatures (focusing on only one of the cladding modes).

4. Application of the semi-empirical LPFG analysis model

Fig. 3. Experimental setup for the verification of the semi-empirical model. Independently tunable divided coil heater array with LPFG.

Figure 3 shows the experimental setup employed to synthesize the filter spectrum, used for the verification of the LPFG analysis method based on our semi-empirical construction. Tunable LPFG with 64 sections of independent thermal controllers (coil heaters for temperature tuning between 26 to 126°C) were used [9

J. K. Bae, J. Bae, S. H. Kim, N. Park, and S. B. Lee, “Dynamic EDFA gain-flattening filter using two LPFGs with divided coil heaters,” Photon. Technol. Lett. 17, 1226–1228 (2005). [CrossRef]

]. With the above experimental setup, independent controls of the coupling properties at different positions of the non-uniform LPFG sections were possible. It is worth mentioning that the fiber used in the LPFG construction was Fibercore photosensitive fiber (Boron co-doped Ge-Si fiber) with a cutoff wavelength λ_cutoff,0=1153nm. For this specific fiber, the measured value of α_T was 1.66×10^-6/°C.

For the analysis, we use the multiport lattice filter model [10

] to simultaneously consider the couplings of multiple cladding modes over the whole bandwidth of interest (note that conventional 2×2 models with core mode+1 cladding mode are limited in their wavelength range or in the treatment of the concatenated LPFGs with different grating properties having multiple cladding modes in the same wavelength range). Dividing the general non-uniform LPFG with arbitrary temperature distributions into k uniform LPFG sections, we now build the transfer matrix Q in (11) from the product of the M_k (T_k )’s, viz. the transfer matrix of the uniform LPFGs at temperature T_k , with its elements calculated from (1) and (7), as follows :

[\begin{matrix} E^{(0)} (L) \\ E^{(1)} (L) \\ ⋮ \\ E^{(p - 1)} (L) \\ E^{(p)} (L) \end{matrix}] = Q \cdot [\begin{matrix} E^{(0)} (0) \\ E^{(1)} (0) \\ ⋮ \\ E^{(p - 1)} (0) \\ E^{(p)} (0) \end{matrix}], Q = M_{k} (T_{k}) \cdot M_{k - 1} (T_{k - 1}) \dots M_{1} (T_{1}),

(11)

where E ⁽⁰⁾(L) and E ^(p)(L) are the resulting electric fields in the core and cladding, respectively.

4.1 Example 1

Fig. 4. cascaded LPFGs with same cladding modes(1450~1650nm range).

As a first application example, we tested a cascaded LPFG composed of two uniform LPFGs with different grating periods (Λ₁=348µm and Λ₂=360µm for LPFG1 and LPFG2, respectively, as shown in Fig. 4), but utilizing the same cladding mode (7^th). For the prediction of the LPFG spectral shape, given the inherent limitations in the conventional, lightweight approaches such as the 2×2 matrix model or higher order fitting method, one can employ the full modal analysis [12

T. Erdogan, “Cladding-mode resonances in short- and long- period fiber grating filters,” J. Opt. Soc. Am. A 14, 1760–1773 (1997). [CrossRef]

] with a series of precisely characterized fiber parameters or alternatively the proposed semi-empirical analysis method whose results are shown in detail below.

For the analysis, the wavelength dependence of the dispersion function Φ (λ, T ₀) obtained in the previous section was used. With the wavelength dependent curvature of the intermodal dispersion function of the 7^th cladding mode Φ⁽⁷⁾ (Fig. 5(b), data from Fig. 1), the intermodal dispersion comes to have a different slope for the different grating periods, Λ1 and Λ2, resulting in different spectral shapes at λ1_res and λ2_res. Utilizing Eqs. (1) and (10) to get κ and δ, respectively, and then using (11) to get the final spectrum, the analytically obtained spectrum peak wavelength λ_res and FWHM (full width half maximum) bandwidth Δλ_FWHM were 1506nm and 46nm for LPFG1, and 1628.4nm and 25nm for LPFG2, respectively, exactly overlapping with the experimentally determined values [Fig. 5(a)].

Fig. 5. (a). measured/calculated spectra and (b). intermodal dispersion curve of 7th cladding mode for the concatenated LPFG with different grating periods.

4.2 Example 2

Fig. 6. Cascaded LPFGs with different cladding modes.

To examine the spectral changes of the LPFG under temperature distribution control, a cascaded non-uniform LPFG composed of three uniform LPFGs was used (Fig. 6 grating period Λ₁=Λ₃=356µm, Λ₂=433µm). For the analysis, the dispersion function Φ(λ, T ₀) and its temperature differential ΔΦ(T-T ₀) obtained in the previous section were used. Since the core mode of LPFG1/LPFG3 mainly couples with the 7^th cladding mode, and the core mode of LPFG2 couples with the 6^th cladding mode, all the other cladding modes were ignored in the calculation within the wavelength range of interest.

Fig. 7. (a), (c), (e) Measured/calculated LPFG spectra and (b), (d), (f) variations of intermodal dispersion under various temperature distributions (inset).

In Figs. 7(a) and 7(b), we show the transmission spectra and intermodal dispersion of the LPFG held at room temperature (26°C, see the inset of the figure). Since LPFG1 and LPFG3 have identical grating properties (length and period), the coupling of LPFG1’s 7^th cladding mode to LPFG3’s 7^th cladding mode was observed, exhibiting interference fringes within the resonance band of the composite grating (with the free spectral range of the comb FSR_comb=12nm). Figure 7(c) also shows the spectrum of the LPFG at a temperature of 120°C. Not only a shift in the resonance wavelength, but also changes in FSR_comb of the 7^th cladding mode (12nm to 9nm) were predicted from the model and were confirmed by experiment. It should be mentioned that the changes in FSR_comb are mainly due to the temperature induced vertical shift in the dispersion curve Φ⁽⁷⁾ (for LPFGs 1 and 3), and corresponding changes in the intermodal dispersion slope at the resonance frequency. It is also worth noting that even with almost identical ΔΦ(or Δ(n _co-n _cl)) values for the 6^th and 7^th cladding modes, the resonance wavelength shift of the 7^th cladding mode is much larger than that of the 6^th cladding mode, because of the steeper intermodal dispersion slope of the former.

Finally, setting the temperature of LPFG2 to 126°C (LPFG1, 3 at 26°C), a blue shift in the resonance peak λ2_res of LPFG2’s 6^th cladding mode was observed which, it is important to note, was accompanied by spectral changes of the 7^th cladding mode [Fig. 7(e)].

We attribute this to the temperature-induced phase changes in LPFG2 affecting the coupling of the 7^th cladding modes between LPFG1 and LPFG3 which, although minor, led to changes in the resonance peaks and spectral profiles. It is important to note that the above example clearly demonstrates the successful treatment of multiple cladding modes and the couplings in-between them using our simplistic, semi-empirical format. Accurate analysis on the temperature dependencies of LPFGs were also achieved (which has not been practically feasible with the full-modal analysis or high-order fitting method) for our approach, using external parameter sets measured with minimal efforts.

4.3 Example 3

Fig. 8. LPFG unit for tunable EDFA gain equalization filter; cascaded LPFGs with different grating lengths and different grating periods.

As a final application example, we analyze the dynamic spectrum control of the EDFA gain equalization filter. For this purpose, two concatenated, thermally controlled LPFGs were used. The grating period and length were 356µm and 11cm for LPFG1, and 433µm and 5cm for LPFG2, respectively.

By independently applying the currents to each of the 64 segments of the coil heaters distributed over the whole grating, piecewise-uniform control/tuning of the LPFG temperature distribution was achieved. Figure 9 shows the measured (pink dotted line), and theoretically calculated spectrum curves (black line). The initial LPFG spectra [at room temperature, 28°C, in Fig. 9(a)] are shown in Fig. 9(b). Also shown in the figure are the temperature distributions and achieved (experimental, theoretical) optimized transmission spectra of the LPFGs for the EDFA ASE spectrum, in the case where its pump is driven with currents of 51mA and 71mA. Excellent agreement in the transmission spectrum was observed, revealing the contributions of each cladding mode (6^th:dash, 7^th:dot).

Fig. 9. LPFG based dynamic gain equalization filter ; Applied temperature distributions (a, c, e) and their LPFG spectra (b, d, f).

6. Conclusion

We proposed and experimentally verified a simplified method of LPFG analysis based on a semi-empirical formulation. Empirically determined intermodal dispersion functions were introduced and used to handle the spectral shape of non-uniform gratings possessing arbitrary temperature distributions. The application of our empirical formulation to a multi-port lattice matrix framework allowed for the successful treatment of multiple cladding modes and the couplings in-between them to be achieved. The analysis on the transmission spectra of general, multiple-cladding mode LPFGs under an arbitrary temperature distribution and grating periods was demonstrated, without relying on the rigorous full model analysis or higher-order fitting methods. Excellent agreement with the experimental results was observed in the analysis of the peak wavelength shape/shift, phase difference and FSR variations. Given its simple and general formulation based on the key fundamental fiber characteristics (temperature variations of the core-to-cladding index difference, thus requiring the characterization of the LPFG at only a few subsets of the cladding modes), we believe that our approach offers an excellent but convenient means of general, dynamic LPFG analysis.

References and links

1.	A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long-period fiber gratings as band rejection filters,” J. Lightwave Technol. 14, 58–65 (1996). [CrossRef]
2.	B. Ortega, L. Dong, W. F. Liu, J. P. de Sandro, L. Reekie, S. I. Tsypina, V. N. Bagratashvili, and R. I. Laming, “High-performance optical fiber polarizers based on long-period grating in birefringent optical fibers,” Photon. Technol. Lett. 9, 1370–1372 (1997). [CrossRef]
3.	V. Grubsky and J. Feinberg, “Long-period fiber gratings with variable coupling for real-time sensing applications,” Opt. Lett. 25, 203–205 (2000). [CrossRef]
4.	B. J. Eggleton, R. E. Slusher, J. B. Judkins, J. B. Stark, and A. M. Vengsarkar, “All-optical switching in long period fiber gratings,” Opt. Lett. 22, 883–885 (1997). [CrossRef] [PubMed]
5.	A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, P. J. Lemaire, N. S. Bergano, and C. R. Davidson, “Long-period fiber-grating-based gain equalizers,” Opt. Lett. 21, 336–338 (1996). [CrossRef] [PubMed]
6.	Y. Liu, J. A. Willims, L. Zhang, and I. Bennion, “Phase shifted and cascaded long period fiber gratings,” Opt. Commun. 164, 27–31 (1999). [CrossRef]
7.	M. Harurnoto, M. Shigehara, and H. Suganurna, “Gain-flattening filter using long-period fiber gratings,” J. Lightwave Technol. 20, 1027–1033 (2002). [CrossRef]
8.	X. Shu, T. Allsop, B. Gwandu, and L. Zhang, “High-temperature sensitivity of long-period gratings in B—Ge codoped fiber,” Photon. Technol. Lett. 13, 818–820 (2001). [CrossRef]
9.	J. K. Bae, J. Bae, S. H. Kim, N. Park, and S. B. Lee, “Dynamic EDFA gain-flattening filter using two LPFGs with divided coil heaters,” Photon. Technol. Lett. 17, 1226–1228 (2005). [CrossRef]
10.	J. Bae, J. Chun, and S. B. Lee, “Synthesis of long-period fiber gratings with the inverted Erbium gain spectrum using the multiport lattice filter model,” J. Lightwave Technol. 22, 1976–1986 (2004). [CrossRef]
11.	J. Bae, J. K. Bae, S. H. Kim, S. B. Lee, and J. Chun, “Analysis for long period fiber gratings using thermal kernel function,” Opt. Exp. 12, 797–810 (2004). [CrossRef]
12.	T. Erdogan, “Cladding-mode resonances in short- and long- period fiber grating filters,” J. Opt. Soc. Am. A 14, 1760–1773 (1997). [CrossRef]
13.	K. Shima, K. Himeno, T. Sakai, S. Okude, A. Wada, and R. Yamauchi, “A novel temperature-insensitive long-period fiber grating using a boron-codoped-germanosilicate-core fiber,” Proc. Optical Fiber Communication Conf. Dallas, TX , 347–348, (1997). [CrossRef]
14.	G. P. Agrawal, Fiber-Optic Communications Systems (New York: Wiley, 1997), Chap. 2.

OCIS Codes
(060.2330) Fiber optics and optical communications : Fiber optics communications
(350.2770) Other areas of optics : Gratings

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: November 14, 2007
Revised Manuscript: December 28, 2007
Manuscript Accepted: January 3, 2008
Published: January 8, 2008

Citation
Jun Kye Bae, Namkyoo Park, Jinho Bae, Dongyeon Koh, Sang Hyuck Kim, and Sang Bae Lee, "Semi-empirical multi-port lattice model for long-period fiber grating analysis under arbitrary temperature distributions," Opt. Express 16, 598-606 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-2-598

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References

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, "Long-period fiber gratings as band rejection filters," J. Lightwave Technol. 14, 58-65 (1996). [CrossRef]
B. Ortega, L. Dong, W. F. Liu, J. P. de Sandro, L. Reekie, S. I. Tsypina, V. N. Bagratashvili, and R. I. Laming, "High-performance optical fiber polarizers based on long-period grating in birefringent optical fibers," Photon. Technol. Lett. 9, 1370-1372 (1997). [CrossRef]
V. Grubsky and J. Feinberg, "Long-period fiber gratings with variable coupling for real-time sensing applications," Opt. Lett. 25, 203-205 (2000). [CrossRef]
B. J. Eggleton, R. E. Slusher, J. B. Judkins, J. B. Stark and A. M. Vengsarkar, "All-optical switching in long period fiber gratings," Opt. Lett. 22, 883-885 (1997). [CrossRef] [PubMed]
A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, P. J. Lemaire, N. S. Bergano and, C. R. Davidson, "Long-period fiber-grating-based gain equalizers," Opt. Lett. 21, 336-338 (1996). [CrossRef] [PubMed]
Y. Liu, J. A. Willims, L. Zhang, and I. Bennion, "Phase shifted and cascaded long period fiber gratings," Opt. Commun. 164, 27-31 (1999). [CrossRef]
M. Harurnoto, M. Shigehara, and H. Suganurna, "Gain-flattening filter using long-period fiber gratings," J. Lightwave Technol. 20, 1027 - 1033 (2002). [CrossRef]
X. Shu, T. Allsop, B. Gwandu and L. Zhang, "High-temperature sensitivity of long-period gratings in B-Ge codoped fiber," Photon. Technol. Lett. 13, 818-820 (2001). [CrossRef]
J. K. Bae, J. Bae, S. H. Kim, N. Park and S. B. Lee, "Dynamic EDFA gain-flattening filter using two LPFGs with divided coil heaters," Photon. Technol. Lett. 17, 1226-1228 (2005). [CrossRef]
J. Bae, J. Chun, and S. B. Lee, "Synthesis of long-period fiber gratings with the inverted Erbium gain spectrum using the multiport lattice filter model," J. Lightwave Technol. 22, 1976-1986 (2004). [CrossRef]
J. Bae, J. K. Bae, S. H. Kim, S. B. Lee, and J. Chun, "Analysis for long period fiber gratings using thermal kernel function," Opt. Express 12, 797-810 (2004). [CrossRef]
T. Erdogan, "Cladding-mode resonances in short- and long- period fiber grating filters," J. Opt. Soc. Am. A 14, 1760 - 1773 (1997). [CrossRef]
K. Shima, K. Himeno, T. Sakai, S. Okude, A. Wada, and R. Yamauchi, "A novel temperature-insensitive long-period fiber grating using a boron-codoped-germanosilicate-core fiber," Proc. Optical Fiber Communication Conf. Dallas, TX, 347-348, (1997). [CrossRef]
G. P. Agrawal, Fiber-Optic Communications Systems (New York: Wiley, 1997), Chap. 2.

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