Abstract

A compact in-line modal interferometer based on a long period grating (LPG) inscribed in water-filled photonic crystal fiber (PCF) is proposed and demonstrated. The interferometer works from the interference between fundamental core mode and different vector components of LP11 core mode. The LPG is especially inscribed to realize the energy exchange between the fundamental core mode and different vector components of LP11 core mode in the PCF. We build a complete theoretical model and systematically analyze the multi-component-intermodal-interference mechanism of the interferometer based on coupled-mode theory. Due to the asymmetric index distribution over the cross section of the PCF caused by CO2-laser side illumination, the dispersion curves and temperature sensitivities referring to different vector components of LP11 core mode are quite different. Thus the interferometer is polarization-dependent and the adjacent interference fringes according to different components of LP11 mode show greatly discrimination in sensitivities of temperature and strain, making it a good candidate for multiple physics parameters measurements.

© 2014 Optical Society of America

1. Introduction

Modal interferometers have been explored in sensing and communication area largely due to their outstanding advantages over conventional sensors, such as high sensitivity, high interference contrast and compact structure [19]. Different fiber in-line modal interferometers (MI) based on photonic crystal fibers (PCFs) have been proposed such as an in-line comb filter with flat-top spectral response based on a cascaded all-solid photonic band-gap fiber interferometer [2], modal interferometers based on simple SMF-PCF-SMF structures [310], and interferometric sensors based on a long period fiber grating pair [11,12]. However, those interferometers have relatively large insert loss of 2-5 dB and each fringe in the interference region exhibits almost the same sensing property excluding the influence of material dispersion [110].

To reduce the insert loss, a long period grating (LPG) assistant modal interferometer has been presented in our previous work, and the preliminary results of the experimental characteristics of the interferometer have also been demonstrated [13]. With the assistant of LPG to exchange the energy between LP01 and LP11 core modes in the PCF, this interferometer realizes lower insert loss and high extinction ratio without large core-offset at the splice regions. However, we didn’t consider the influence of multi-component interference of LP11 core mode, and the PCF used in our previous work is all-solid photonic band-gap fiber, whose refractive index is insensitive to external conditions, and the difference of sensing property among interference fringes is hard to distinguish.

In this paper, a compact LPG assistant fluid-filled photonic crystal fiber modal interferometer with high temperature sensitivity is proposed and demonstrated. We build a complete theoretical model and systematically analyze the mechanism of the interferometer including the propagation and interaction of the core modes through the LPG based on coupled-mode theory. The interferometer works from the interference between fundamental core mode and different polarization components of LP11 core mode, which are more stable and reliable than cladding modes. The LPG is especially inscribed to realize the energy exchange between the fundamental core mode and different vector components of LP11 core mode in the PCF. Due to the asymmetric index distribution over the cross section of the PCF caused by CO2-laser side illumination, the interferometer is polarization-dependent and different fringes in the interference region with reference to different vector components of LP11 core mode show discrimination in sensitivities of temperature and strain, making it a good candidate for bio-sensing and multiple parameters measurement.

2. Principle and structure

A two-mode pure silica PCF, produced by Yangtze Optical Fiber and Cable Company Ltd, is employed to fabricate the modal interferometer. The PCF includes 5 rings of air holes running along the axis, whose optical microscopic picture of cross section is shown in Fig. 1(a). The average diameter of the air holes is 3.9 μm and the average inter-hole spacing is 5.75 μm. In order to enhance the temperature sensitivity of the interferometer, we infiltrated water into the micro holes of the PCF, for water is the most basic component of all organisms and has a relatively high thermo-optic coefficient. Because the refractive index of silica is higher than water, the water-filled PCF is still refractive index guiding PCF.

 

Fig. 1 Schematic configuration of the proposed interferometer. The inset (a) is the cross section of the PCF.

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As shown in Fig. 1(b), light from a supercontinuum source (650 nm ~1750 nm) is launched into the single-mode fiber (SMF) and the output transmission spectra is measured by an optical spectrum analyzer (OSA) with the highest resolution of 0.02 nm. A polarizer and a polarization controller are inserted between the supercontinuum source and the interferometer for polarization analysis. The proposed modal interferometer is fabricated using a piece of water-filled two-mode PCF directly spliced between two sections of SMF with slight core offset. Next a strong resonance LPG is inscribed in the water-filled PCF to realize the energy exchange between fundamental core mode and LP11 core mode in the PCF. L1 and L2 are the distance between the LPG and the two spliced points, and Lc is the length of the LPG.

We fabricated the LPG by use of a focused CO2-laser beam to notch periodically on one side of the PCF. Due to the high-energy laser beam irradiation, the incident side of the PCF was deeply grooved while the shadow side is intact, and part of the water in the air holes was evaporated at the irradiation side. Since antisymmetric LP11 mode is degenerate linear combinations of different distinct, polarization-sensitive modes (TE01, TM01, HE21(1) and HE21(2)), the asymmetric index distribution over the cross section of the LPG results in differences among the dispersion of different components of LP11 modes. Thus these modes exhibit different characteristics in sensing field and the interferometer is also polarization-dependent.

When the incident light propagates through the first spliced region, LP01 core mode and different degenerate LP11 modes are excited. We suppose that the intensity of input light in SMF is I, the initial phases of the core modes at the first spliced region are zero, and the energy coupling coefficients of the input and output spliced points are b1 and b2 for LP11 mode, a1 and a2 for LP01 mode, respectively, a1+b11, a2+b21. As the modes are orthogonal in an ideal waveguide, the transverse component of the electric field can be expressed as a superposition of the ideal modes. If we consider only LP01 mode and one of the different vector components of LP11 mode exist in the PCF, after propagating a distance of L1, the transverse component of the electric field can be expressed as:

E(x,y,z)=E1(x,y,z)+E2(x,y,z)=a1Ieiβ01L1+b1Ieiβ11L1
Where β01 and β11 are the propagation constants of the LP01 mode and one of the different components of LP11 core mode, and E1(x,y,z) and E2(x,y,z) are the electric field of the LP01 and one of the different components of LP11 mode, respectively.

Due to the dielectric perturbation caused by CO2-laser pulses, the co-propagating core modes can be coupled to each other through the LPG and the electric field varies along the LPG. Since the evolutions of the two modes along the LPG are similar to each other, we just analyze the coupling process of LP01 mode for example. The electric field of the initial excited LP01 mode along the LPG can be expressed as:

E1(x,y,z)=A(z)a1Ieiβ01L1+B(z)a1Ieiβ01L1
Where the coefficient B(z) represents the proportion of the energy which is coupled to LP11 core mode and coefficient A(z) represents the rest energy of the LP01 mode, respectively. According to the coupled-mode theory [14], the coefficients A(z) and B(z) satisfy the condition:
dA(z)dz=iκ*B(z)ei2Δz,dB(z)dz=iκA(z)ei2Δz
In Eq. (3), Δ is the phase mismatching coefficient, Δ=12[β11(β01+2πΛ)]. κ* is the conjugation of κ, and κ is the coupling mode coefficient between the two core modes and is given by:
κμν(z)exp(i2πzΛ)ω4Eμ*(x,y)Δε(x,y)Eν(x,y)dxdy
Where Λ is the period of the grating, Eμ(x,y) and Eν(x,y) are the electric field of the coupled modes, and Δε(x,y) is the perturbation to the permittivity. In our work, we only investigate the case of strong resonance, and we also consider the length of the LPG to be the coupling length. So the coefficient κ satisfies the equation Lc=π/(2|κ|) at the resonance wavelength of the LPG. Since the coupling mode coefficient κ varies slightly with the wavelengths in the interference region, we consider the coefficient κ as a constant approximately.

From Eq. (3), we can obtain the coefficients:

A(z)=eiΔz[cosγz+iΔγsinγz],B(z)=eiΔziκγsinγz
γ=|κ|2+Δ2. From Eq. (1), Eq. (2) and Eq. (5), we can calculate the transverse component of the electric field in the PCF after propagating a distance of L1+Lc:

E(x,y,z)=eiΔLc[cosγLc+iΔγsinγLc]a1Ieiβ01L1+eiΔLciκγsinγLca1Ieiβ01L1+eiΔLc[cosγLc+iΔγsinγLc]b1Ieiβ11L1+eiΔLciκγsinγLcb1Ieiβ11L1

After passing through the LPG, the coupled LP01 core mode continues to propagate a distance of L2 as LP11 mode while the coupled LP11 mode propagates the same distance as the fundamental mode in the PCF. The two core modes accumulate different optical paths and interfere at the second spliced region. The electric field here can be expressed as:

E(x,y,z)=eiΔLc[cosγLc+iΔγsinγLc]a1a2Ieiβ01(L1+L2)+eiΔLciκγsinγLca1b2Iei(β01L1+β11L2)+eiΔLc[cosγLc+iΔγsinγLc]b1b2Ieiβ11(L1+L2)+eiΔLciκγsinγLcb1a2Iei(β11L1+β01L2)
The interference light intensity at the output is:
Iout=E(x,y)E*(x,y)
To simplify the calculation, we assume that the input and output spliced points are identical, i.e. a=a1=a2, b=b1=b2. The resulting equation of the light intensity at the output can be expressed as:
Iout=(a2+b2)IA1+abIB1+aabIC1+babID1
A1=cos2γLc+Δ2γ2sin2γLc
B1=2[cos2γLc+Δ2γ2sin2γLc]cos[(β11β01)(L1+Lc)]+π22γ2Lc2sin2γLc[1+cos[(β11β01)(L1L2)]]
C1=2πcosγLcsinγLcγLc[sin[GLc2ΔLc(β01β11)L1+L22]cos[(β01β11)L1L22]]2Δπsin2γLcγ2Lc[cos[GLc2ΔLc(β01β11)L1+L22]cos[(β01β11)L1L22]]
D1=2πcosγLcsinγLcγLc[sin[GLc2ΔLc(β11β01)L1+L22]cos[(β11β01)L1L22]]2Δπsin2γLcγ2Lc[cos[GLc2ΔLc(β11β01)L1+L22]cos[(β11β01)L1L22]]
In Eq. (9), the term aabIC1 represents the superposition of the interference between the LP01 mode which is coupled to LP11 mode and the rest part of LP01 mode which isn’t coupled to LP11 mode, and the interference between the LP11 mode which is coupled to LP01 mode and the part of LP01 mode which isn’t coupled to LP11 mode. The term babID1 represents the superposition of the interference between the LP11 mode which is coupled to LP01 mode and the rest part of LP11 mode which isn’t coupled to LP01 mode, and the interference between the LP01 mode which is coupled to LP11 mode and the part of LP11 mode which isn’t coupled to LP01 mode. As we only consider the case of strong resonance in the LPG, the terms aabIC1 and babID1 in Eq. (9) have little effect on the entire interference and can be ignored. So Eq. (9) is simplified to be:
Iout=(a2+b2)I(cos2γLc+Δ2γ2sin2γLc)+abIπ22Lc2γ2sin2γLc[1+cos[(β11β01)(L1L2)]]+2abI(cos2γLc+Δ2γ2sin2γLc)cos[(β11β01)(L1+L2)]
In our theoretical calculations, Eq. (14) is a general formula for the interference between LP01 core mode and any one of the different degenerate modes (TE01, TM01, HE21(1) and HE21(2)) if we replace β11 with the corresponding propagation constant.

However, without the LPG, the excited LP01 mode and LP11 mode will propagate through the PCF without any energy exchange and interfere at the output spliced point. The interference light intensity at the output without the LPG can be expressed as:

Iout'=(a2+b2)I+2abIcos[(β11β01)(L1+Lc+L2)]

Due to the slight core offset at the spliced points between the SMF and the PCF, almost all the energy of light from the SMF is coupled to LP01 mode in the PCF while only a small part of the energy is coupled to LP11 mode at the first spliced point. Thus the energy coupling efficiency a is much larger than b.

To simulate the transmission spectrum of the interferometer, a commercial finite element code (Comsol) was employed for the calculation of the propagation constants of the coupled core modes. Figure 2(a) shows the phase-matching curves and theoretical intensity distributions of different vector components of LP11 core mode in the PCF. The inset image in Fig. 2 is equivalent model for the cross-section of the PCF after inscribing LPG, and the red part of the inset image represent the filled water. Due to the asymmetric index distribution along the LPG caused by side illumination, different vector components of the LP11 mode are separated in resonant wavelengths.

 

Fig. 2 (a) Phase-matching curves (the curves of HE21(1) mode and HE21(2) mode almost coincide) and theoretical intensity distributions of different vector components of LP11 core mode in PCF, and the inset is equivalent model for the cross-section of the LPG; (b) Calculated transmission spectra of the interferometers with (red line) and without (blue line) the LPG.

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Figure 2(b) shows the interference between HE11 mode and TM01 mode, the energy coupling efficiencies of the spliced points are assumed to be 0.95 for HE11 mode and 0.05 for TM01 mode. L1 = 90 mm, Lc = 7.6 mm, L2 = 10 mm. With the assistant of the LPG, the extinction ratio of interference fringes (red line) is much higher than the interferometer without a LPG (blue line).

The interference spacing of the fringes depends on the optical path difference (OPD) between the two core modes:

OPD=[n1(λ,T)×L1+n2(λ,T)×L2][n2(λ,T)×L1+n1(λ,T)×L2]=Δn(λ,T)×ΔL
Where n1(λ,T) and n2(λ,T) are the effective refractive indices of the LP01 and LP11 core modes, respectively. Δn(λ,T)=n1(λ,T)n2(λ,T), ΔL=L1L2.

Equation (16) shows that the OPD depends on ΔL rather than the whole length of the PCF, so we can control the free spectrum range simply by adjusting the position of the LPG.

The wavelengths of the interference dips satisfy the phase matching condition:

Δn(λ,T)×ΔLλ(T)=m+12
Where m is an integer.

Then we calculate the derivative of two sides in Eq. (17) to T and get the temperature sensitivity:

S=dλdT=Δn(λ,T)T×λ(T)ΔnΔn(λ,T)λ×λ(T)=Δn(λ,T)T×λ(T)Ng
Where Ng is the group refractive index difference of two core modes:

Ng=ΔnΔn(λ,T)λ×λ(T)

Equation (18) shows that the temperature sensitivity depends on Ng and thermo-optic coefficient of the water-filled PCF. In our calculation, the coefficient Ng changes little after filling the PCF with water. So the thermo-optic coefficient of the water-filled PCF dominates in temperature sensitivity. As the thermo-optic coefficient of water (~8.0×105/K) is much higher than that of pure silica (~8.6×106/K), the temperature sensitivity can be greatly increased.

We calculated the temperature sensitivity for different theoretical equivalent models. Figure 3 shows two typical examples of the theoretical simulations, the inset images are the theoretical models of the cross section of the LPG to simulate, and the red part of the inset image represents the filled water. The inset images of the theoretical equivalent models directly show the exact different values of the degree of evaporation of water and grooves in the PCF in our simulations. Due to the dispersion effect, the temperature sensitivity decreases as wavelength increases.

 

Fig. 3 Calculated temperature sensitivity of the interferometer for different theoretical models, the inset images are the theoretical equivalent models for the cross-section of the LPG.

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As shown in Fig. 3, different degrees of evaporation of water and depths of the grooves in the PCF lead to different polarization mode dispersion and temperature sensitivity of the interferometers. Since the two interference modes are both core modes, the closer the grooves to the fiber core, the more water close to the fiber core is evaporated at the irradiation side, and the differences in sensitivity between different polarization modes will be larger.

As the temperature sensitivities of the corresponding vector components of the LP11 mode are different, the temperature response coefficients of the adjacent interference fringes may be extremely different. To prove the conclusion, we simulated the case when two vector components of the LP11 modes exist, and all the theoretical analysis is based on the precondition of strong resonance.

As shown in Fig. 4, we select the equivalent model in Fig. 3(a) and set L1 = 90 mm, Lc = 7.6 mm, L2 = 50 mm to control the number of the fringes. If we consider the case of only TM01 mode or HE21 mode existing [the inset of Fig. 4(a)], the temperature responses will be −190.65 pm/°C for peak I and −368.78 pm/°C for peak II, respectively.

 

Fig. 4 (a) Calculated transmission spectrum when TM01 mode and HE21 mode exist simultaneously, the inset images is the spectra when only TM01 or HE21 mode exists; (b) Theoretical temperature responses of peak ① and peak ②.

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When both TM01 mode and HE21 mode exist, as shown in Fig. 4(a), there are two obvious peaks in the transmission of the interferometer. The sensitivity of peak ① is mainly determined by HE21 mode while peak ② is mainly determined by TM01 mode. We calculated the temperature responses of the two peaks, as shown in Fig. 4(b). The sensitivities of the two peaks are quite different. When temperature changes from 15°C to 35°C, the temperature responses are −242 pm/°C for peak ① and −322 pm/°C for peak ②, respectively.

The wavelength interval between peak ① and ② is about 10 nm, and from Fig. 3(a) we can see the difference in sensitivities caused by single-mode dispersion is less than 15 pm/°C. Thus the big difference in the temperature response coefficients among the adjacent interference fringes is mainly due to the characteristics of different vector components of LP11 mode. The same applies to the situation when more than two polarizations modes exist.

3. Experimental results

In experiment, we fabricated several different interferometers including an LPG with different grating periods through a point by point side illumination process using a CO2 inscription platform. Figure 5 shows the spectra of interferometers for the LPGs with a period of 180 μm and 190 μm. With the period increasing, the resonant wavelength and interference fringes move to shorter wavelength. So we can control the wavelengths of the interference fringes simply by fabricating the LPGs with different periods. The depth of grooves in the PCF caused by CO2-laser pulses is shown in Fig. 5(c).

 

Fig. 5 Measured transmission spectra of the interferometers for the LPGs with a period of (a) 180 μm and (b) 190 μm; (c) Microscopic image of the LPG area.

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We measured the length of the interferometer in Fig. 5(b), L1 = 50 mm, Lc = 7.2 mm, L2 = 15 mm. The calculated value of fringe spacing at 1560 nm is 6.7 nm, and the experimentally measured value which is shown in Fig. 6(a) is about 7.0 nm. This indicates that the theoretical analysis agrees well with the experimental result.

 

Fig. 6 Spectra characteristics of the proposed interferometer with regard to the state of polarization of incident light.

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As the wavelengths of the interference fringes are around 1550 nm when the period is 180 μm, we investigated the characteristics of the interferometer in Fig. 5(a).

Before measuring the temperature sensitivity of the proposed modal interferometer, we investigated the polarization dependency of the interferometer. Figure 6 shows the spectra of interferometer at different polarization states. The interference fringes are greatly changed with the rotation of the polarizer, which indicated that there is more than one polarization mode of LP11 involved in the interference. The deepest peaks in the spectra of the three polarization states are located at 1560.9 nm, 1567.9 nm, 1555.3 nm, and 1547.1 nm, respectively.

Next we investigate the temperature responses of the interference peaks at the three polarization states. In this experiment, the fiber interferometer is straightened with a constant axial strain provided by a preset load in order to avoid the bending influence on the results.

As shown in Fig. 7, we obtain large temperature tuning coefficients. The temperature sensing coefficients are −197.5 pm/°C for dip i, −315 pm/°C for dip ii, −170 pm/°C for dip iii, and −225 pm/°C for dip iv, respectively. Since the thermo-optic coefficient of the water is below zero, heating the fiber induces a blue-shift of the interference fringes.

 

Fig. 7 Wavelength responses of the interferometer at different polarization states.

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The wavelength differences among the four peaks are less than 20 nm, thus from Fig. 3 we can see the differences in sensitivities caused by single-mode dispersion are less than 15 pm/°C. Thus all the experiment results indicated that the four peaks come from different vector components of LP11 mode respectively. The big differences in the temperature coefficients among the dips are mainly due to the characteristics of different polarization modes. The experimental results agree well with the theoretical analysis.

Moreover, we kept the temperature to be 25°C and investigated the strain characteristics of the interferometer. We fabricate another interferometer including an LPG with a grating pitch of 180 μm, and monitor the shift of the interference fringes as strain varies. As shown in Fig. 8, the fringes are linearly shifted toward shorter wavelengths as strain increases, with slopes of −1.33 pm/με, −1.45 pm/με and −1.14 pm/με, respectively. The refractive index of silica decreases as strain increases, thus the effective refractive index of the core modes decreases. The blue-shift of the fringes indicates that the effective index of the fundamental core mode varies more slowly with the wavelength than that of the LP11 modes.

 

Fig. 8 Wavelength responses of the interferometer for variations of applied strain.

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Since the interference fringes show discrimination in sensitivities of temperature and strain, this interferometer has potential applications in multiple physics parameters measurements [15,16].

4. Conclusion

In summary, we fabricated and demonstrated a fluid-filled photonic crystal fiber modal interferometer based on long period grating with low insert loss and high extinction ratio. We built a complete theoretical model and systematically analyzed the physical mechanism of this interferometer based on coupled-mode theory. We specially inscribed the LPG in the interferometer to realize the coupling between fundamental core mode and different vector components of LP11 core mode. Due to the asymmetric index distribution over the cross section of the LPG caused by side illumination, the interferometer is polarization-dependent and the big difference in the temperature coefficients among interference fringes is mainly affect by the characteristics of different vector components of LP11 mode. Moreover, we investigated the strain characteristics of the interferometer. The interference fringes show discrimination in sensitivities of temperature and strain, so it can be applied to measure temperature and strain simultaneously. Compared with the general interferometers, our interferometer has advantages of simplicity and compactness, and the fringe spacing can be controlled simply by adjusting the position of the LPG.

Acknowledgments

This work was supported by the National Key Basic Research and Development Program of China (Grant No. 2010CB327605 and 2011CB301701), the National Natural Science Foundation of China (Grant Nos. 61322510, 11174154 and 11174155), and the Tianjin Natural Science Foundation (Grant No. 12JCZDJC20600).

References and links

1. S. C. Gao, W. G. Zhang, P. C. Ge, X. L. Xue, H. Zhang, and Z. Y. Bai, “Highly sensitive in-fiber refractive index sensor based on down-bitaper seeded up-bitaper pair,” IEEE Photonics Technol. Lett. 24(20), 1878–1881 (2012). [CrossRef]  

2. Y. Geng, X. Li, X. Tan, Y. Deng, and Y. Yu, “In-line flat-top comb filter based on a cascaded all-solid photonic bandgap fiber intermodal interferometer,” Opt. Express 21(14), 17352–17358 (2013). [CrossRef]   [PubMed]  

3. H. Gong, X. Li, Y. Jin, and X. Dong, “Hollow-core photonic crystal fiber based modal interferometer for strain measurement,” Sens. Actuators A Phys. 187, 95–97 (2012). [CrossRef]  

4. W. J. Zhou, W. C. Wong, C. C. Chan, L. Y. Shao, and X. Y. Dong, “Highly sensitive fiber loop ringdown strain sensor using photonic crystal fiber interferometer,” Appl. Opt. 50(19), 3087–3092 (2011). [CrossRef]   [PubMed]  

5. G. A. Cárdenas-Sevilla, V. Finazzi, J. Villatoro, and V. Pruneri, “Photonic crystal fiber sensor array based on modes overlapping,” Opt. Express 19(8), 7596–7602 (2011). [CrossRef]   [PubMed]  

6. S. Wang, Y. G. Liu, Z. Wang, T. Han, W. Xu, Y. Wang, and S. Wang, “Intermodal interferometer based on a fluid-filled two-mode photonic crystal fiber for sensing applications,” Appl. Opt. 52(14), 3166–3171 (2013). [CrossRef]   [PubMed]  

7. G. Coviello, V. Finazzi, J. Villatoro, and V. Pruneri, “Encapsulated and coated photonic crystal fibre sensor for temperature measurements up to 1000°C,” in European Conference on Lasers and Electro-Optics 2009 and the European Quantum Electronics Conference. CLEO Europe - EQEC, IEEE (2009).

8. B. Dong and E. J. Hao, “Temperature-insensitive and intensity-modulated embedded photonic-crystal-fiber modal-interferometer-based micro displacement sensor,” J. Opt. Soc. Am. B 28(10), 2332–2336 (2011). [CrossRef]  

9. P. J. Huang, C. H. Hung, P. Y. Tai, and J. M. Hsu, “Sensitivity enhanced photonic crystal fiber interferometers with material dispersion engineering,” in Opto-Electronics and Communications Conference Technical Digest (2012), pp. 403–404.

10. H. T. Hu, C. H. Chen, Y. C. Chen, J. N. Wang, J. L. Tang, and W. T. Wu, “Investigations of refractive index sensing with a photonic crystal fiber interferometer,” in Photonics Global Conference (2010), pp. 1–3. [CrossRef]  

11. J. H. Lim, H. S. Jang, K. S. Lee, J. C. Kim, and B. H. Lee, “Mach-Zehnder interferometer formed in a photonic crystal fiber based on a pair of long-period fiber gratings,” Opt. Lett. 29(4), 346–348 (2004). [CrossRef]   [PubMed]  

12. G. A. Cárdenas-Sevilla, D. Monzon-Hernandez, I. Torres-Gomez, and A. Martınez-Rıos, “Tapered Mach–Zehnder interferometer based on two mechanically induced long period fiber gratings as refractive index sensor,” Opt. Laser Technol. 44(5), 1516–1520 (2012). [CrossRef]  

13. Z. L. Sun, Y. G. Liu, Z. Wang, B. Y. Tai, T. T. Han, B. Liu, W. T. Cui, H. F. Wei, and W. J. Tong, “Long period grating assistant photonic crystal fiber modal interferometer,” Opt. Express 19(14), 12913–12918 (2011). [CrossRef]   [PubMed]  

14. J. Liu, Photonic Devices (Cambridge University, 2005), Chap. 4.

15. T. Han, Y. G. Liu, Z. Wang, Z. Wu, S. Wang, and S. Li, “Simultaneous temperature and force measurement using Fabry-Perot interferometer and bandgap effect of a fluid-filled photonic crystal fiber,” Opt. Express 20(12), 13320–13325 (2012). [CrossRef]   [PubMed]  

16. Z. Wu, Y. Liu, Z. Wang, T. Han, S. Li, M. Jiang, P. P. Shum, and X. Q. Dinh, “In-line Mach–Zehnder interferometer composed of microtaper and long-period grating in all-solid photonic bandgap fiber,” Appl. Phys. Lett. 101(14), 141106 (2012). [CrossRef]  

References

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  1. S. C. Gao, W. G. Zhang, P. C. Ge, X. L. Xue, H. Zhang, and Z. Y. Bai, “Highly sensitive in-fiber refractive index sensor based on down-bitaper seeded up-bitaper pair,” IEEE Photonics Technol. Lett. 24(20), 1878–1881 (2012).
    [Crossref]
  2. Y. Geng, X. Li, X. Tan, Y. Deng, and Y. Yu, “In-line flat-top comb filter based on a cascaded all-solid photonic bandgap fiber intermodal interferometer,” Opt. Express 21(14), 17352–17358 (2013).
    [Crossref] [PubMed]
  3. H. Gong, X. Li, Y. Jin, and X. Dong, “Hollow-core photonic crystal fiber based modal interferometer for strain measurement,” Sens. Actuators A Phys. 187, 95–97 (2012).
    [Crossref]
  4. W. J. Zhou, W. C. Wong, C. C. Chan, L. Y. Shao, and X. Y. Dong, “Highly sensitive fiber loop ringdown strain sensor using photonic crystal fiber interferometer,” Appl. Opt. 50(19), 3087–3092 (2011).
    [Crossref] [PubMed]
  5. G. A. Cárdenas-Sevilla, V. Finazzi, J. Villatoro, and V. Pruneri, “Photonic crystal fiber sensor array based on modes overlapping,” Opt. Express 19(8), 7596–7602 (2011).
    [Crossref] [PubMed]
  6. S. Wang, Y. G. Liu, Z. Wang, T. Han, W. Xu, Y. Wang, and S. Wang, “Intermodal interferometer based on a fluid-filled two-mode photonic crystal fiber for sensing applications,” Appl. Opt. 52(14), 3166–3171 (2013).
    [Crossref] [PubMed]
  7. G. Coviello, V. Finazzi, J. Villatoro, and V. Pruneri, “Encapsulated and coated photonic crystal fibre sensor for temperature measurements up to 1000°C,” in European Conference on Lasers and Electro-Optics 2009 and the European Quantum Electronics Conference. CLEO Europe - EQEC, IEEE (2009).
  8. B. Dong and E. J. Hao, “Temperature-insensitive and intensity-modulated embedded photonic-crystal-fiber modal-interferometer-based micro displacement sensor,” J. Opt. Soc. Am. B 28(10), 2332–2336 (2011).
    [Crossref]
  9. P. J. Huang, C. H. Hung, P. Y. Tai, and J. M. Hsu, “Sensitivity enhanced photonic crystal fiber interferometers with material dispersion engineering,” in Opto-Electronics and Communications Conference Technical Digest (2012), pp. 403–404.
  10. H. T. Hu, C. H. Chen, Y. C. Chen, J. N. Wang, J. L. Tang, and W. T. Wu, “Investigations of refractive index sensing with a photonic crystal fiber interferometer,” in Photonics Global Conference (2010), pp. 1–3.
    [Crossref]
  11. J. H. Lim, H. S. Jang, K. S. Lee, J. C. Kim, and B. H. Lee, “Mach-Zehnder interferometer formed in a photonic crystal fiber based on a pair of long-period fiber gratings,” Opt. Lett. 29(4), 346–348 (2004).
    [Crossref] [PubMed]
  12. G. A. Cárdenas-Sevilla, D. Monzon-Hernandez, I. Torres-Gomez, and A. Martınez-Rıos, “Tapered Mach–Zehnder interferometer based on two mechanically induced long period fiber gratings as refractive index sensor,” Opt. Laser Technol. 44(5), 1516–1520 (2012).
    [Crossref]
  13. Z. L. Sun, Y. G. Liu, Z. Wang, B. Y. Tai, T. T. Han, B. Liu, W. T. Cui, H. F. Wei, and W. J. Tong, “Long period grating assistant photonic crystal fiber modal interferometer,” Opt. Express 19(14), 12913–12918 (2011).
    [Crossref] [PubMed]
  14. J. Liu, Photonic Devices (Cambridge University, 2005), Chap. 4.
  15. T. Han, Y. G. Liu, Z. Wang, Z. Wu, S. Wang, and S. Li, “Simultaneous temperature and force measurement using Fabry-Perot interferometer and bandgap effect of a fluid-filled photonic crystal fiber,” Opt. Express 20(12), 13320–13325 (2012).
    [Crossref] [PubMed]
  16. Z. Wu, Y. Liu, Z. Wang, T. Han, S. Li, M. Jiang, P. P. Shum, and X. Q. Dinh, “In-line Mach–Zehnder interferometer composed of microtaper and long-period grating in all-solid photonic bandgap fiber,” Appl. Phys. Lett. 101(14), 141106 (2012).
    [Crossref]

2013 (2)

2012 (5)

G. A. Cárdenas-Sevilla, D. Monzon-Hernandez, I. Torres-Gomez, and A. Martınez-Rıos, “Tapered Mach–Zehnder interferometer based on two mechanically induced long period fiber gratings as refractive index sensor,” Opt. Laser Technol. 44(5), 1516–1520 (2012).
[Crossref]

H. Gong, X. Li, Y. Jin, and X. Dong, “Hollow-core photonic crystal fiber based modal interferometer for strain measurement,” Sens. Actuators A Phys. 187, 95–97 (2012).
[Crossref]

T. Han, Y. G. Liu, Z. Wang, Z. Wu, S. Wang, and S. Li, “Simultaneous temperature and force measurement using Fabry-Perot interferometer and bandgap effect of a fluid-filled photonic crystal fiber,” Opt. Express 20(12), 13320–13325 (2012).
[Crossref] [PubMed]

Z. Wu, Y. Liu, Z. Wang, T. Han, S. Li, M. Jiang, P. P. Shum, and X. Q. Dinh, “In-line Mach–Zehnder interferometer composed of microtaper and long-period grating in all-solid photonic bandgap fiber,” Appl. Phys. Lett. 101(14), 141106 (2012).
[Crossref]

S. C. Gao, W. G. Zhang, P. C. Ge, X. L. Xue, H. Zhang, and Z. Y. Bai, “Highly sensitive in-fiber refractive index sensor based on down-bitaper seeded up-bitaper pair,” IEEE Photonics Technol. Lett. 24(20), 1878–1881 (2012).
[Crossref]

2011 (4)

2004 (1)

Bai, Z. Y.

S. C. Gao, W. G. Zhang, P. C. Ge, X. L. Xue, H. Zhang, and Z. Y. Bai, “Highly sensitive in-fiber refractive index sensor based on down-bitaper seeded up-bitaper pair,” IEEE Photonics Technol. Lett. 24(20), 1878–1881 (2012).
[Crossref]

Cárdenas-Sevilla, G. A.

G. A. Cárdenas-Sevilla, D. Monzon-Hernandez, I. Torres-Gomez, and A. Martınez-Rıos, “Tapered Mach–Zehnder interferometer based on two mechanically induced long period fiber gratings as refractive index sensor,” Opt. Laser Technol. 44(5), 1516–1520 (2012).
[Crossref]

G. A. Cárdenas-Sevilla, V. Finazzi, J. Villatoro, and V. Pruneri, “Photonic crystal fiber sensor array based on modes overlapping,” Opt. Express 19(8), 7596–7602 (2011).
[Crossref] [PubMed]

Chan, C. C.

Chen, C. H.

H. T. Hu, C. H. Chen, Y. C. Chen, J. N. Wang, J. L. Tang, and W. T. Wu, “Investigations of refractive index sensing with a photonic crystal fiber interferometer,” in Photonics Global Conference (2010), pp. 1–3.
[Crossref]

Chen, Y. C.

H. T. Hu, C. H. Chen, Y. C. Chen, J. N. Wang, J. L. Tang, and W. T. Wu, “Investigations of refractive index sensing with a photonic crystal fiber interferometer,” in Photonics Global Conference (2010), pp. 1–3.
[Crossref]

Cui, W. T.

Deng, Y.

Dinh, X. Q.

Z. Wu, Y. Liu, Z. Wang, T. Han, S. Li, M. Jiang, P. P. Shum, and X. Q. Dinh, “In-line Mach–Zehnder interferometer composed of microtaper and long-period grating in all-solid photonic bandgap fiber,” Appl. Phys. Lett. 101(14), 141106 (2012).
[Crossref]

Dong, B.

Dong, X.

H. Gong, X. Li, Y. Jin, and X. Dong, “Hollow-core photonic crystal fiber based modal interferometer for strain measurement,” Sens. Actuators A Phys. 187, 95–97 (2012).
[Crossref]

Dong, X. Y.

Finazzi, V.

Gao, S. C.

S. C. Gao, W. G. Zhang, P. C. Ge, X. L. Xue, H. Zhang, and Z. Y. Bai, “Highly sensitive in-fiber refractive index sensor based on down-bitaper seeded up-bitaper pair,” IEEE Photonics Technol. Lett. 24(20), 1878–1881 (2012).
[Crossref]

Ge, P. C.

S. C. Gao, W. G. Zhang, P. C. Ge, X. L. Xue, H. Zhang, and Z. Y. Bai, “Highly sensitive in-fiber refractive index sensor based on down-bitaper seeded up-bitaper pair,” IEEE Photonics Technol. Lett. 24(20), 1878–1881 (2012).
[Crossref]

Geng, Y.

Gong, H.

H. Gong, X. Li, Y. Jin, and X. Dong, “Hollow-core photonic crystal fiber based modal interferometer for strain measurement,” Sens. Actuators A Phys. 187, 95–97 (2012).
[Crossref]

Han, T.

Han, T. T.

Hao, E. J.

Hu, H. T.

H. T. Hu, C. H. Chen, Y. C. Chen, J. N. Wang, J. L. Tang, and W. T. Wu, “Investigations of refractive index sensing with a photonic crystal fiber interferometer,” in Photonics Global Conference (2010), pp. 1–3.
[Crossref]

Jang, H. S.

Jiang, M.

Z. Wu, Y. Liu, Z. Wang, T. Han, S. Li, M. Jiang, P. P. Shum, and X. Q. Dinh, “In-line Mach–Zehnder interferometer composed of microtaper and long-period grating in all-solid photonic bandgap fiber,” Appl. Phys. Lett. 101(14), 141106 (2012).
[Crossref]

Jin, Y.

H. Gong, X. Li, Y. Jin, and X. Dong, “Hollow-core photonic crystal fiber based modal interferometer for strain measurement,” Sens. Actuators A Phys. 187, 95–97 (2012).
[Crossref]

Kim, J. C.

Lee, B. H.

Lee, K. S.

Li, S.

Z. Wu, Y. Liu, Z. Wang, T. Han, S. Li, M. Jiang, P. P. Shum, and X. Q. Dinh, “In-line Mach–Zehnder interferometer composed of microtaper and long-period grating in all-solid photonic bandgap fiber,” Appl. Phys. Lett. 101(14), 141106 (2012).
[Crossref]

T. Han, Y. G. Liu, Z. Wang, Z. Wu, S. Wang, and S. Li, “Simultaneous temperature and force measurement using Fabry-Perot interferometer and bandgap effect of a fluid-filled photonic crystal fiber,” Opt. Express 20(12), 13320–13325 (2012).
[Crossref] [PubMed]

Li, X.

Y. Geng, X. Li, X. Tan, Y. Deng, and Y. Yu, “In-line flat-top comb filter based on a cascaded all-solid photonic bandgap fiber intermodal interferometer,” Opt. Express 21(14), 17352–17358 (2013).
[Crossref] [PubMed]

H. Gong, X. Li, Y. Jin, and X. Dong, “Hollow-core photonic crystal fiber based modal interferometer for strain measurement,” Sens. Actuators A Phys. 187, 95–97 (2012).
[Crossref]

Lim, J. H.

Liu, B.

Liu, Y.

Z. Wu, Y. Liu, Z. Wang, T. Han, S. Li, M. Jiang, P. P. Shum, and X. Q. Dinh, “In-line Mach–Zehnder interferometer composed of microtaper and long-period grating in all-solid photonic bandgap fiber,” Appl. Phys. Lett. 101(14), 141106 (2012).
[Crossref]

Liu, Y. G.

Martinez-Rios, A.

G. A. Cárdenas-Sevilla, D. Monzon-Hernandez, I. Torres-Gomez, and A. Martınez-Rıos, “Tapered Mach–Zehnder interferometer based on two mechanically induced long period fiber gratings as refractive index sensor,” Opt. Laser Technol. 44(5), 1516–1520 (2012).
[Crossref]

Monzon-Hernandez, D.

G. A. Cárdenas-Sevilla, D. Monzon-Hernandez, I. Torres-Gomez, and A. Martınez-Rıos, “Tapered Mach–Zehnder interferometer based on two mechanically induced long period fiber gratings as refractive index sensor,” Opt. Laser Technol. 44(5), 1516–1520 (2012).
[Crossref]

Pruneri, V.

Shao, L. Y.

Shum, P. P.

Z. Wu, Y. Liu, Z. Wang, T. Han, S. Li, M. Jiang, P. P. Shum, and X. Q. Dinh, “In-line Mach–Zehnder interferometer composed of microtaper and long-period grating in all-solid photonic bandgap fiber,” Appl. Phys. Lett. 101(14), 141106 (2012).
[Crossref]

Sun, Z. L.

Tai, B. Y.

Tan, X.

Tang, J. L.

H. T. Hu, C. H. Chen, Y. C. Chen, J. N. Wang, J. L. Tang, and W. T. Wu, “Investigations of refractive index sensing with a photonic crystal fiber interferometer,” in Photonics Global Conference (2010), pp. 1–3.
[Crossref]

Tong, W. J.

Torres-Gomez, I.

G. A. Cárdenas-Sevilla, D. Monzon-Hernandez, I. Torres-Gomez, and A. Martınez-Rıos, “Tapered Mach–Zehnder interferometer based on two mechanically induced long period fiber gratings as refractive index sensor,” Opt. Laser Technol. 44(5), 1516–1520 (2012).
[Crossref]

Villatoro, J.

Wang, J. N.

H. T. Hu, C. H. Chen, Y. C. Chen, J. N. Wang, J. L. Tang, and W. T. Wu, “Investigations of refractive index sensing with a photonic crystal fiber interferometer,” in Photonics Global Conference (2010), pp. 1–3.
[Crossref]

Wang, S.

Wang, Y.

Wang, Z.

Wei, H. F.

Wong, W. C.

Wu, W. T.

H. T. Hu, C. H. Chen, Y. C. Chen, J. N. Wang, J. L. Tang, and W. T. Wu, “Investigations of refractive index sensing with a photonic crystal fiber interferometer,” in Photonics Global Conference (2010), pp. 1–3.
[Crossref]

Wu, Z.

T. Han, Y. G. Liu, Z. Wang, Z. Wu, S. Wang, and S. Li, “Simultaneous temperature and force measurement using Fabry-Perot interferometer and bandgap effect of a fluid-filled photonic crystal fiber,” Opt. Express 20(12), 13320–13325 (2012).
[Crossref] [PubMed]

Z. Wu, Y. Liu, Z. Wang, T. Han, S. Li, M. Jiang, P. P. Shum, and X. Q. Dinh, “In-line Mach–Zehnder interferometer composed of microtaper and long-period grating in all-solid photonic bandgap fiber,” Appl. Phys. Lett. 101(14), 141106 (2012).
[Crossref]

Xu, W.

Xue, X. L.

S. C. Gao, W. G. Zhang, P. C. Ge, X. L. Xue, H. Zhang, and Z. Y. Bai, “Highly sensitive in-fiber refractive index sensor based on down-bitaper seeded up-bitaper pair,” IEEE Photonics Technol. Lett. 24(20), 1878–1881 (2012).
[Crossref]

Yu, Y.

Zhang, H.

S. C. Gao, W. G. Zhang, P. C. Ge, X. L. Xue, H. Zhang, and Z. Y. Bai, “Highly sensitive in-fiber refractive index sensor based on down-bitaper seeded up-bitaper pair,” IEEE Photonics Technol. Lett. 24(20), 1878–1881 (2012).
[Crossref]

Zhang, W. G.

S. C. Gao, W. G. Zhang, P. C. Ge, X. L. Xue, H. Zhang, and Z. Y. Bai, “Highly sensitive in-fiber refractive index sensor based on down-bitaper seeded up-bitaper pair,” IEEE Photonics Technol. Lett. 24(20), 1878–1881 (2012).
[Crossref]

Zhou, W. J.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

Z. Wu, Y. Liu, Z. Wang, T. Han, S. Li, M. Jiang, P. P. Shum, and X. Q. Dinh, “In-line Mach–Zehnder interferometer composed of microtaper and long-period grating in all-solid photonic bandgap fiber,” Appl. Phys. Lett. 101(14), 141106 (2012).
[Crossref]

IEEE Photonics Technol. Lett. (1)

S. C. Gao, W. G. Zhang, P. C. Ge, X. L. Xue, H. Zhang, and Z. Y. Bai, “Highly sensitive in-fiber refractive index sensor based on down-bitaper seeded up-bitaper pair,” IEEE Photonics Technol. Lett. 24(20), 1878–1881 (2012).
[Crossref]

J. Opt. Soc. Am. B (1)

Opt. Express (4)

Opt. Laser Technol. (1)

G. A. Cárdenas-Sevilla, D. Monzon-Hernandez, I. Torres-Gomez, and A. Martınez-Rıos, “Tapered Mach–Zehnder interferometer based on two mechanically induced long period fiber gratings as refractive index sensor,” Opt. Laser Technol. 44(5), 1516–1520 (2012).
[Crossref]

Opt. Lett. (1)

Sens. Actuators A Phys. (1)

H. Gong, X. Li, Y. Jin, and X. Dong, “Hollow-core photonic crystal fiber based modal interferometer for strain measurement,” Sens. Actuators A Phys. 187, 95–97 (2012).
[Crossref]

Other (4)

P. J. Huang, C. H. Hung, P. Y. Tai, and J. M. Hsu, “Sensitivity enhanced photonic crystal fiber interferometers with material dispersion engineering,” in Opto-Electronics and Communications Conference Technical Digest (2012), pp. 403–404.

H. T. Hu, C. H. Chen, Y. C. Chen, J. N. Wang, J. L. Tang, and W. T. Wu, “Investigations of refractive index sensing with a photonic crystal fiber interferometer,” in Photonics Global Conference (2010), pp. 1–3.
[Crossref]

G. Coviello, V. Finazzi, J. Villatoro, and V. Pruneri, “Encapsulated and coated photonic crystal fibre sensor for temperature measurements up to 1000°C,” in European Conference on Lasers and Electro-Optics 2009 and the European Quantum Electronics Conference. CLEO Europe - EQEC, IEEE (2009).

J. Liu, Photonic Devices (Cambridge University, 2005), Chap. 4.

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Figures (8)

Fig. 1
Fig. 1 Schematic configuration of the proposed interferometer. The inset (a) is the cross section of the PCF.
Fig. 2
Fig. 2 (a) Phase-matching curves (the curves of HE 21 (1) mode and HE 21 (2) mode almost coincide) and theoretical intensity distributions of different vector components of LP11 core mode in PCF, and the inset is equivalent model for the cross-section of the LPG; (b) Calculated transmission spectra of the interferometers with (red line) and without (blue line) the LPG.
Fig. 3
Fig. 3 Calculated temperature sensitivity of the interferometer for different theoretical models, the inset images are the theoretical equivalent models for the cross-section of the LPG.
Fig. 4
Fig. 4 (a) Calculated transmission spectrum when TM01 mode and HE21 mode exist simultaneously, the inset images is the spectra when only TM01 or HE21 mode exists; (b) Theoretical temperature responses of peak ① and peak ②.
Fig. 5
Fig. 5 Measured transmission spectra of the interferometers for the LPGs with a period of (a) 180 μm and (b) 190 μm; (c) Microscopic image of the LPG area.
Fig. 6
Fig. 6 Spectra characteristics of the proposed interferometer with regard to the state of polarization of incident light.
Fig. 7
Fig. 7 Wavelength responses of the interferometer at different polarization states.
Fig. 8
Fig. 8 Wavelength responses of the interferometer for variations of applied strain.

Equations (19)

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E ( x , y , z ) = E 1 ( x , y , z ) + E 2 ( x , y , z ) = a 1 I e i β 01 L 1 + b 1 I e i β 11 L 1
E 1 ( x , y , z ) = A ( z ) a 1 I e i β 01 L 1 + B ( z ) a 1 I e i β 01 L 1
d A ( z ) d z = i κ * B ( z ) e i 2 Δ z , d B ( z ) d z = i κ A ( z ) e i 2 Δ z
κ μ ν ( z ) exp ( i 2 π z Λ ) ω 4 E μ * ( x , y ) Δ ε ( x , y ) E ν ( x , y ) d x d y
A ( z ) = e i Δ z [ cos γ z + i Δ γ sin γ z ] , B ( z ) = e i Δ z i κ γ sin γ z
E ( x , y , z ) = e i Δ L c [ cos γ L c + i Δ γ sin γ L c ] a 1 I e i β 01 L 1 + e i Δ L c i κ γ sin γ L c a 1 I e i β 01 L 1 + e i Δ L c [ cos γ L c + i Δ γ sin γ L c ] b 1 I e i β 11 L 1 + e i Δ L c i κ γ sin γ L c b 1 I e i β 11 L 1
E ( x , y , z ) = e i Δ L c [ cos γ L c + i Δ γ sin γ L c ] a 1 a 2 I e i β 01 ( L 1 + L 2 ) + e i Δ L c i κ γ sin γ L c a 1 b 2 I e i ( β 01 L 1 + β 11 L 2 ) + e i Δ L c [ cos γ L c + i Δ γ sin γ L c ] b 1 b 2 I e i β 11 ( L 1 + L 2 ) + e i Δ L c i κ γ sin γ L c b 1 a 2 I e i ( β 11 L 1 + β 01 L 2 )
I o u t = E ( x , y ) E * ( x , y )
I o u t = ( a 2 + b 2 ) I A 1 + a b I B 1 + a a b I C 1 + b a b I D 1
A 1 = cos 2 γ L c + Δ 2 γ 2 sin 2 γ L c
B 1 = 2 [ cos 2 γ L c + Δ 2 γ 2 sin 2 γ L c ] cos [ ( β 11 β 01 ) ( L 1 + L c ) ] + π 2 2 γ 2 L c 2 sin 2 γ L c [ 1 + cos [ ( β 11 β 01 ) ( L 1 L 2 ) ] ]
C 1 = 2 π cos γ L c sin γ L c γ L c [ sin [ G L c 2 Δ L c ( β 01 β 11 ) L 1 + L 2 2 ] cos [ ( β 01 β 11 ) L 1 L 2 2 ] ] 2 Δ π sin 2 γ L c γ 2 L c [ cos [ G L c 2 Δ L c ( β 01 β 11 ) L 1 + L 2 2 ] cos [ ( β 01 β 11 ) L 1 L 2 2 ] ]
D 1 = 2 π cos γ L c sin γ L c γ L c [ sin [ G L c 2 Δ L c ( β 11 β 01 ) L 1 + L 2 2 ] cos [ ( β 11 β 01 ) L 1 L 2 2 ] ] 2 Δ π sin 2 γ L c γ 2 L c [ cos [ G L c 2 Δ L c ( β 11 β 01 ) L 1 + L 2 2 ] cos [ ( β 11 β 01 ) L 1 L 2 2 ] ]
I o u t = ( a 2 + b 2 ) I ( cos 2 γ L c + Δ 2 γ 2 sin 2 γ L c ) + a b I π 2 2 L c 2 γ 2 sin 2 γ L c [ 1 + cos [ ( β 11 β 01 ) ( L 1 L 2 ) ] ] + 2 a b I ( cos 2 γ L c + Δ 2 γ 2 sin 2 γ L c ) cos [ ( β 11 β 01 ) ( L 1 + L 2 ) ]
I o u t ' = ( a 2 + b 2 ) I + 2 a b I cos [ ( β 11 β 01 ) ( L 1 + L c + L 2 ) ]
O P D = [ n 1 ( λ , T ) × L 1 + n 2 ( λ , T ) × L 2 ] [ n 2 ( λ , T ) × L 1 + n 1 ( λ , T ) × L 2 ] = Δ n ( λ , T ) × Δ L
Δ n ( λ , T ) × Δ L λ ( T ) = m + 1 2
S = d λ d T = Δ n ( λ , T ) T × λ ( T ) Δ n Δ n ( λ , T ) λ × λ ( T ) = Δ n ( λ , T ) T × λ ( T ) N g
N g = Δ n Δ n ( λ , T ) λ × λ ( T )

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