## Abstract

A novel in-fiber modal interferometer based on a long period grating (LPG) inscribed in a two-mode all-solid photonic bandgap fiber (AS-PBGF) is presented. After inserting a small piece of the AS-PBGF into two sections of standard single-mode fiber (SMF) via being spliced slight core offset, LPG is inscribed in the AS-PBGF. The LPG is especially designed to realize the coupling between two core modes of LP_{01} and LP_{11} in the AS-PBGF. Two core modes LP_{01} and LP_{11} of the AS-PBGF are excited firstly at the input spliced point and actualized energy exchange when they pass through the LPG. Then the two beams will interfere at the output spliced point to form a high-contrast in-fiber modal interferometer. The proposed interferometer has some advantages such as configuration compact, high interference contrast and the wavelength spacing well controlled by changing the position of the LPG without changing the total length of AS-PBGF.

©2011 Optical Society of America

## 1. Introduction

The development of fiber modal interferometers has been a recent and active research topic nowadays. Based on the high sensitivity and high resolution due to narrow width of interference fringes, they have been widely applied in WDM communications [1], wavelength filters and sensors for measurement of many physical parameters including strain, temperature, refractive index and curvature [2–9].

Several types of in-fiber interferometers have been reported such as interferometers made of a pair of identical LPGs [1, 10] or a fiber LPG with a reflector [11], and the intermodal interferometers based on SMF–special fiber–SMF structures [2–9]. The interferometers based on fiber LPG pair have a series of interference fringes in each stop band of a conventional LPG's spectrum. The interferometer’s two optical paths are the transmission core mode and one of the cladding modes. A drawback of fiber LPG pair interferometer is that it is difficult to fabricate two absolutely identical LPGs along the fiber. The intermodal interferometers are based on the modal interference effect in special fibers, especially photonic crystal fibers (PCFs). By core-offsetting splice joints of a compact SMF–special fiber–SMF structure, the excited cladding modes [3, 4] or higher order core mode [9] can re-couple with the fundamental mode propagating in the special fiber at the output splice joint. Some methods to excite cladding modes or higher order core modes like collapsing air holes in a piece of PCF [8, 12] or using a tapered fiber [7] were also proposed. The intermodal interferometers have the advantages of simple and compact structure, small size. However, large insert loss of 2-5 dB has been induced in those interferometers because of the large core off-setting splicing, the air hole collapsing or tapering the fiber [7, 8]. Moreover, the free spectrum range (FSM) in those interferometers is dependent on the used special fiber length and thus most of them have to using a very short fiber length like millimeters in order to get a wide FSM filter or sensor, which is not easy in experimental operation.

In this paper, a novel LPG assistant in-fiber modal interferometer is proposed and demonstrated. The interferometer consists of a specially designed LPG inscribed in a two-mode all-solid photonic bandgap fiber (AS-PBGF), whose both ends are fusing spliced to SMF with slight core offset. An interference contrast of 9.1 dB is achieved at the resonant wavelength of the LPG. Compared to the intermodal interferometers reported in [3, 8], our interferometer have lower insert loss and can achieve high extinction ratio without large lateral offset at the fusion spliced points.

## 2. Interferometer structure and operation principle

As shown in Fig. 1
, the proposed modal interferometer consists of a strong resonance LPG inscribed in a two-mode fiber, whose both ends are fusion spliced with SMF with slight core offset. L_{1}/L_{2} is the distance between the middle of the grating and the incident spliced point/output spliced point. The LPG is designed to realize the coupling between two core modes of LP_{01} and LP_{11} in the two-mode fiber. At the incidence spliced point, a small fraction of propagating power from the SMF goes to LP_{11} core mode and most goes to the fundamental mode in the two-mode fiber. LP_{11} core mode propagates a distance of L_{1} before it couples to the fundamental mode (LP_{01} core mode) by the LPG and continues to propagate a distance of L_{2} as the fundamental mode. Meanwhile the fundamental mode of the two-mode fiber propagates L_{1} distance from the incidence spliced point to the LPG, then couples to the LP_{11} core mode by the LPG and continues to propagate a distance of L_{2} as LP_{11} mode reaching the output end. Because only the fundamental mode can propagate in SMF losslessly, some power of LP_{11} core mode couples to the fundamental mode in the SMF and another portion is lost when LP_{11} core mode passes through the output point, where interference obtains. The LP_{01}-LP_{11} and LP_{11}-LP_{01} paths are the two arms of this interferometer.

The interference depends on the optical path difference (OPD) between the two arms. Since the LP_{11} core mode has a lower effective index than the LP_{01} core mode, when the beams pass through the two-core fiber, index difference induces an optical path difference:

_{01}and LP

_{11}core modes, respectively. $\Delta \text{n}={\text{n}}_{\text{eff}}^{01}-{\text{n}}_{\text{eff}}^{11}$, $\Delta \text{L}={\text{L}}_{2}-{\text{L}}_{1}$.

If we approximate the wavelengths of two adjacent intensity minima near the resonance wavelength of the LP in the interference spectrum to the resonance wavelength λ and ignore the effect of dispersion, based on the Eq. (1), we deduce that the interference wavelength spacing is:

We can see the OPD and the wavelength spacing $\Delta \text{S}$ depends on $\Delta \text{L}$ rather than the total fiber length (L_{1} + L_{2}). If we fix a wavelength λ, the effective index difference can be calculated by phase matching condition, and ΔS is in inverse proportion to ΔL, so we can control the wavelength spacing by changing the position of LPG rather than the total length of the two-mode fiber. When the two beams interference, the extreme value of obtained intensities can be mathematically described by:

The contrast of interference fringes can be expressed as:

We suppose that the input intensity of LP_{01} in SMF is *I*, the energy coupling efficiencies of the input and output spliced points are ${\alpha}_{1}$ and ${\alpha}_{2}$for LP_{11}, ${\beta}_{1}$ and ${\beta}_{2}$for LP_{01}, respectively, ${\alpha}_{1}\text{+}{\beta}_{1}\le 1,{\alpha}_{2}\text{+}{\beta}_{2}\le 1$, and that of the grating is $\gamma \left(90\text{\%}\le \gamma \le 1\right)$ . Therefore, after passing through the output spliced point, the intensities of LP_{11} and LP_{01} are $\gamma {\alpha}_{2}{\beta}_{1}\text{I}$ and$\gamma {\alpha}_{1}{\beta}_{2}\text{I}$ , respectively. According to Eq. (3)-(5), ${\text{I}}_{\text{max}}=\gamma \left({\alpha}_{1}{\beta}_{2}+{\alpha}_{2}{\beta}_{1}+2\sqrt{{\alpha}_{1}{\alpha}_{2}{\beta}_{1}{\beta}_{2}}\right)I$, ${\text{I}}_{\text{min}}=\gamma \left({\alpha}_{1}{\beta}_{2}+{\alpha}_{2}{\beta}_{1}-2\sqrt{{\alpha}_{1}{\alpha}_{2}{\beta}_{1}{\beta}_{2}}\right)I$ and $V=\frac{2\sqrt{{\alpha}_{1}{\alpha}_{2}{\beta}_{1}{\beta}_{2}}}{{\alpha}_{1}{\beta}_{2}+{\alpha}_{2}{\beta}_{1}}$ are obtained. Obviously, V is independent of *γ* and V = 1 as long as ${\alpha}_{1}{\beta}_{2}={\alpha}_{2}{\beta}_{1}$ or ${\alpha}_{1}={\alpha}_{2}$, ${\beta}_{1}={\beta}_{2}$. However, if we don’t introduce the LPG in this interferometer, at the output spliced point the intensities of LP_{11} and LP_{01} are ${\alpha}_{1}{\alpha}_{2}\text{I}$ and${\beta}_{1}{\beta}_{2}\text{I}$, respectively. We get${\text{I}}_{\mathrm{max}}\text{=(}{\alpha}_{1}{\alpha}_{2}+{\beta}_{1}{\beta}_{2}+2\sqrt{{\alpha}_{1}{\alpha}_{2}{\beta}_{1}{\beta}_{2}}\text{)I}$, ${\text{I}}_{\mathrm{min}}\text{=(}{\alpha}_{1}{\alpha}_{2}+{\beta}_{1}{\beta}_{2}-2\sqrt{{\alpha}_{1}{\alpha}_{2}{\beta}_{1}{\beta}_{2}}\text{)I}$and$V=\frac{2\sqrt{{\alpha}_{1}{\alpha}_{2}{\beta}_{1}{\beta}_{2}}}{{\alpha}_{1}{\alpha}_{2}+{\beta}_{1}{\beta}_{2}}$. Thus V = 1 as long as the condition ${\alpha}_{1}{\alpha}_{2}={\beta}_{1}{\beta}_{2}$is satisfied. That means sufficient core offset at the fusion spliced points is needed to excite enough great power of LP_{11} in order to achieve a large V, for example${\alpha}_{1}={\alpha}_{2}={\beta}_{1}={\beta}_{2}=50\%$. The large lateral core offset will not only weaken the strength of the spliced point but also introduce large insert loss in the interferometer.

If we assume that ${\alpha}_{1}=5\text{\%,}{\alpha}_{2}\text{=}10\%,$ and ${\beta}_{1}=85\text{\%,}{\beta}_{2}\text{=8}0\%,$calculated value of the V is 0.933 and 0.17 for the interferometers with and without the LPG. It is obvious that the use of the LPG immensely heightens contrast of the interference spectrum.

## 3. Experimental results

The two-mode fiber used in this work is an all solid photonic bandgap fiber (AS-PBGF), whose cross section is shown in the inset of Fig. 2 , fabricated by Yangtze Optical Fiber and Cable Corporation. In the fiber a high index rod surrounded by a low index ring lattice of five layers is embedded in pure silica background, the core is formed by omitting a high index rod and a low index ring. The diameter of the fiber is about 125 µm, and the pitch between adjacent rods Λ is 9.24 µm. The outer radii of the high index rod and the low index ring are 0.181Λ and 0.3786Λ, respectively. Compared with the pure silica background, the average refractive index differences of the high index rods and the low index rings are approximately 0.028 and −0.008.

By means of plane wave expansion method and a commercial finite element method, the bandgaps, the effective refractive indices of fundamental core mode and LP_{11}core mode are calculated, which are shown in Fig. 2. It can be seen that the AS-PBGF only supports two core modes in its photonic bandgaps.

Figure 1 shows the schematic configuration of the experimental setup. The AS-PBGF is spliced with SMFs at both ends. Light from a supercontinuum source (650nm ~1750nm) is launched into the input SMF and the transmission spectra are measured with an ANDO AQ6317B optical spectrum analyzer (OSA).

The higher-order core modes can be excited by lateral core-offset splicing between AS-PBGF and SMF. Unlike some early reports [4, 6, 9], we don’t need to excite great power of LP_{11} mode to improve interference depth. In this structure, a small quantity of LP_{11} mode excited at the input spliced point is sufficient to obtain intense interference after mode coupled by the LPG. Before LPG inscription, obvious interference patterns can be observed as shown as the black curve shown in Fig. 3(a)
, with a fringe contrast of ~1 dB, which indicates LP_{11} core mode is not excited much. The formation of those shallow, intensive interference fringes outside the LPG resonant wavelength is similar with that in Ref [9].

In experiment, we inscribe an LPG with a period of 610 μm in the AS-PBGF through a point by point side illumination process using a CO_{2} inscription platform which is similar to that in Ref [13]. The red curve shown in Fig. 3 is the measured spectrum of the interferometer. We can observe a resonant wavelength near 1609nm of LPG,and distinct interference peaks at ii:1582.20 nm, iii:1597.27nm, iv:1609.15 nm, v:1621.48 nm and vi:1633.75nm. The minimum 3 dB bandwidth of those peaks is about 2.97 nm. It is notable that far away from resonant wavelength of the LPG, the interference ripples are similar to that before the LPG inscription. The maximum extinction ratio reaches 9.1 dB while the extinction ratio is only ~1dB before the LPG inscription. Therefore, as forecast in the theoretic analysis, the interference contrast is greatly enhanced by introducing the LPG in the interferometer.

We use infrared CCD to observe near field images of peak ii-iv shown in the inset of Fig. 3(a). Evidently, the ii-iv peaks all come from the coupling between fundamental core mode and LP_{11} core mode, which also gives evidence of high coupling efficiency of the LPG.

Moreover, we also did a series of AS-PBGF cutback experiments to validate the interference effect. Figure 3(b) shows two typical transmission spectra with different Δ*L*. We assume that the mode coupling is finished at the middle point of the LPG. We measure *L*
_{1} and *L*
_{2} based on this assumption, and obtain two arm length differences, Δ*L*
_{1} = 8.3 cm (*L*
_{1} = 10.4 cm, *L*
_{2} = 2.1 cm) and Δ*L*
_{2} = 7.0 cm (*L*
_{1} = 9.1 cm, *L*
_{2} = 2.1 cm). In an LPG, the phase match condition is Δ*n* = λ/Λ, where λ represents resonant wavelength of the LPG and Λ represents the period of grating. According to the calculated phase-matching curve in Fig. 4(a)
the resonant wavelength of the LPG is 1605 nm if Λ = 610 μm, and the corresponding index difference around resonant wavelength, Δ*n* = 0.0026311. According to Eq. (2), when Δ*L* is 8.3 cm and 7.0 cm, the calculated values of ΔS is 11.8 nm and 14.0 nm, respectively. Contrastively, the experimentally measured values are about 12.2 nm and 13.2 nm, respectively. Thus the theoretical results agree well with the experimental data. The slight deviation between the calculating results and experimental results is due to several reasons such as the assumption that modes coupling is finished at the middle point of the LPG, the estimated resonant center wavelength of the LPG is not very accurate and the Δ*n* is a approximate value.

We note that as Δ*L* changes from 8.3 cm to 7.0 cm, the wavelength spacing increases from 12.2 nm to 13.2 nm. So we can control wavelength spacing by adjusting the length difference of *L*
_{1} and *L*
_{2}.We can get a wide FSM as long as choosing an appropriate Δ*L* value even if the total length of the interferometer is large.

We also fabricated an interferometer including an LPG with a grating pitch of 620 μm. With the period increasing, the resonant wavelength and the interference peaks move to the shorter wavelength. According to the dispersion curve of the two core modes, we calculate the phase-matching curve of LPG as shown in Fig. 4 and the resonance wavelength *λ* of the LPG deceases with the grating pitch Λ increases. So we can control the wavelength of the interference peaks by fabricating the LPGs with different periods.

Compared with the inferferometer as shown in Ref [3, 8], our interferometer have some advantages of higher interference contrast without obvious core off-setting splicing or taping and more flexible design to get different interference spacing without changing total length. In addition, our interferometer can also be applied in the measurements of strain, temperature and curvature like the interferometers proposed in the previous reports [2–9]. Furthermore, the wavelength spacing ΔS and the LPG resonant wavelength λ are both affected by Δ*n*, so the proposed interferometer has a promising future for dual parameter sensing measurement.

## 4. Conclusion

In summary, a novel modal interferometer based on a specially designed LPG inscribed in a two-mode all-solid PCF is proposed and demonstrated. The resonance peak of the LPG arises from coupling between the fundamental core mode and the LP_{11} core mode. OPD between the two modes before and after the LPG causes interference fringes. We can change the distance between splicing points and the LPG to control the interference wavelength spacing. The theoretical results agree well with the experimental results. The interferometer has advantages of high interference contrast and structure compact. It may be applied in sensing and wavelength filter system.

## Acknowledgments

This work was supported by the National Key Basic Research and Development Program of China under grant No. 2010CB327605, the National Natural Science Foundation of China under grants No. 50802044, 60736039, and 11004100, Program for New Century Excellent Talents in University (NCET-09-0483) and National Undergraduate Innovative Test Program (091005554).

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