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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 17 — Aug. 18, 2008
  • pp: 12967–12972
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Special optical fiber for temperature sensing based on cladding-mode resonance

Fufei Pang, Wenchao Xiang, Hairun Guo, Na Chen, Xianglong Zeng, Zhenyi Chen, and Tingyun Wang  »View Author Affiliations


Optics Express, Vol. 16, Issue 17, pp. 12967-12972 (2008)
http://dx.doi.org/10.1364/OE.16.012967


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Abstract

A fiber-optic temperature sensor by using a multi-cladding special fiber is presented. It works on the basis of leaky mode resonance from fiber core to outer cladding. With the thin-thickness inner cladding, the cladding mode is strongly excited and the resonant spectrum is very sensitive to the refractive index variation of coating material. By coating the special fiber with temperature-sensitive silicone, the temperature response was investigated experimentally from -20°C to 80°C. The results show high temperature sensitivity (240pm/°C at 20°C) and good repeatability.

© 2008 Optical Society of America

1. Introduction

Fiber-optic temperature sensor is very attractive in many fields for its well-known advantages, such as fast response, high sensitivity, anti-EM interference, corrosion resistance, remote sensing capability and multiplexing ability [1

1. B. Culshaw, “Fiber-optic sensors: applications and advances,” Opt. Photonics News16, 24–29(2005). [CrossRef]

]. A considerable number of fiber temperature sensors with different structures and principles have been developed. Most of them were exploited by using special micro-structure in single mode fiber such as fiber Bragg gratings (FBGs) [2

2. Y. -G. Han, X. Dong, J. H. Lee, and S. B. Lee, “Simultaneous measurement of bending and temperature based on a single sampled chirped fiber Bragg grating embedded on a flexible cantilever beam,” Opt. Lett. 31, 2839–2841 (2006). [CrossRef] [PubMed]

], long period gratings (LPGs) [3

3. Y. -J. Rao, Z. -l. Ran, X. Liao, and H. -y. Deng, “Hybrid LPFG/MEFPI sensor for simultaneous measurement of high-temperature and strain,” Opt. Express 15, 14936–14941 (2007). [CrossRef] [PubMed]

, 4

4. Y. -G. Han, S. Lee, C. -S. Kim, J. Kang, Y. Chung, and U. -C. Paek, “Simultaneous measurement of temperature and strain using dual long-period fiber gratings with controlled temperature and strain sensitivities,” Opt. Express 11, 476–481 (2003). [CrossRef] [PubMed]

], fiber Fabry-Perot interferometer (F-P) [5

5. Z. Huang, Y. Zhu, X. Chen, and A. Wang, “Intrinsic Fabry-Perot fiber sensor for temperature and strain measurements,” IEEE Photon. Technol. Lett. 18, 1879–1881(2006).

, 6

6. L. Cheng, A. J. Steckl, and J. Scofield, “SiC thin-film Fabry-Perot interferometer for fiber-optic temperature sensor,” IEEE Trans. Electron Devices 50, 2159–2164 (2003). [CrossRef]

], tapered fiber [7

7. H. Bao, T. Wang, and Y. Shen, “High sensitive coupling evanescent wave temperature sensor,” Proc. SPIE 5634, 558–562 (2005). [CrossRef]

], D-shape fiber [8

8. S. M. Chandani and N. A. F. Jaeger, “Fiber-optic temperature sensor using evanescent fields in D fibers,” IEEE Photon. Technol. Lett. 17, 2706–2708 (2005). [CrossRef]

] etc. These micro-structures are post-fabricated by using various techniques such as UV-laser writing, laser machining, mechanical polishing, fusion tapering, chemical etching etc. Generally these techniques require complicated processes. In recent years, many fiber optic sensors have been proposed based on the special optical fiber, such as photonic crystal fiber, multi-core fiber, hollow fiber, polarization-maintaining fiber, multimode fiber, which are very convenient to be integrated into conventional single mode fiber sensing system. The special structure in the fiber is formed during the fabrication. Thus, it doesn’t need extra post processing for fabricating fiber sensing head. Enbang Li [9

9. E. Li, X. Wang, and C. Zhang, “Fiber-optic temperature sensor based on interference of selective higher-order modes,” Appl. Phys. Lett. 89, 091119-1-3 (2006). [CrossRef]

] demonstrated a multimode fiber interference temperature sensor. Jian Ju [10

10. J. Ju, Z. Wang, W. Jin, and M. S. Demokan, “Temperature sensitivity of a two-mode photonic crystal fiber interferometric sensor,” IEEE Photon. Technol. Lett. 18, 2168–2170(2006). [CrossRef]

] studied the temperature sensitivity of two-mode photonic crystal fiber. D. S. Moon [11

11. D. S. Moon, B. H. Kim, A. Lin, G. Sun, Y. -G. Han, W. -T. Han, and Y. Chung, “The temperature sensitivity of Sagnac loop interferometer based on polarization maintaining side-hole fiber,” Opt. Express 15, 7962–7967 (2007). [CrossRef] [PubMed]

] researched a temperature sensor based on the Sagnac interferometer made of polarization-maintaining fiber.

In this paper, a novel fiber temperature sensor is proposed and demonstrated based on a multi-cladding special fiber whose core and inner cladding are made of pure silica and fluorine-doped silica deposited with the conventional modified chemical vapor deposition (MCVD) technique. Attributed to the thin thickness of the inner cladding, the core mode will leak to the outer cladding and propagates as cladding modes. And a resonant spectrum is presented, which is very sensitive to the refractive index change of the fiber coating. By coating the special fiber with the temperature-sensitive silicone, a fiber-optic temperature sensor with high sensitivity is demonstrated.

Fig. 1. Refractive index distribution of the special fiber.

2. Theory

As shown in Fig. 1, the proposed special fiber comprises four-layer cylindrical structure: core, inner cladding, outer cladding and coating. The core and the outer cladding are with the same refractive index which is higher than that of the inner cladding and the coating. Since there is no difference in refractive index between the core and the outer cladding, the effective index of the core mode is absolutely lower than the refractive index of the outer cladding. Thus the core mode is a leaky one, and the optical power propagating in the fiber core will leak out into the outer cladding [12

12. L. G. Cohen, D. Marcuse, and W. L. Mammel, “Radiating leaky-mode losses in single-mode lightguides with depressed-index claddings,” IEEE J. Quantum Electron. QE-18, 1467–1472(1982). [CrossRef]

]. The outer cladding, which is sandwiched between the inner cladding and the coating, can also support guiding mode. Based on the coupled mode theory [13

13. D. Marcuse, Light Transmission Optics, (Van Nostrand Reinhold, New York, 1972), Chap. 10.

], the optical power exchanging between the fiber core and the outer cladding can be treated equivalently as modes coupling to each other through the evanescent wave, namely, the core (rod waveguide) mode and the cladding (ring waveguide) mode. According to the coupled mode theory, the two coupled modes must satisfy the phase-matching condition:

βcore=βclad,
(1)

where βcore=k 0 neff-co and βclad=k 0 neff-cl are propagation constants for the core mode and the cladding mode respectively. k 0 is the wave vector in vacuum, neff-co and neff-cl are the effective indexes of the core mode and the cladding mode respectively. According to the transcendental equations given by F. D. Nunes [14

14. F. D. Nunes, C. Adriana, d. S. Melo, H. Filomeno, and D. S. Filho, “Theoretical study of coaxial fibers,” Appl. Opt. 35, 388–399 (1996). [CrossRef] [PubMed]

], we calculated the normalized propagation constant B=(neff 2-n 2 2)/(n 2 1-n 2 2) for the core mode and various the cladding modes of the special fiber with a=4.639µm, b=13µm, c=63µm, n1=n3=1.456, n2=1.453875, n4=1.4317. Their dispersion curves are depicted in Fig. 2(a). We can see that the curves for the core mode HE11 and the cladding mode HE15 cross over at λ=1.5µm, namely the resonant wavelength.

Fig. 2. (a). Dispersion curves of the rod waveguide fundamental mode, the cladding modes, (b) normalized field distribution of supermodes, and the dispersion curves of supermodes HE15 and HE16 shown in the insert drawing

With this multi-cladding fiber structure, the optical power exchanges between the core and the outer cladding. The optical power propagating in the core can be calculated as follows [15

15. R. V. Schmidt and R. C. Alferness, “Directional coupler switches, modulators and filters using alternating Δ β techniques,” IEEE Trans. Circuits Syst. CAS-26, 1099–1108(1979). [CrossRef]

]:

Pr=1sin2[κL1+(Δβ2κ)2]1+(Δβ2κ)2,
(2)

where κ is the coupling coefficient which is calculated from the fields overlap integral of the core mode and the cladding mode [13

13. D. Marcuse, Light Transmission Optics, (Van Nostrand Reinhold, New York, 1972), Chap. 10.

]. L is the length of the special fiber and Δβ=|βcore-βclad| is the difference in propagation constants between the core mode and the cladding mode. The transmission is normalized with initial power launched into the fiber core. By using Eq. (2), the resonant spectrum is shown in Fig. 3. Here the length of the special fiber is equal to Lo which satisfies the condition of κ·Lo=π/2. For this length, the optical power at the resonant wavelength is transferred from the core to the outer cladding completely.

The light power transferring from the core to the outer cladding can also be explained by the supermodes method which is to consider the four-layer structure as an entire waveguide [16

16. P. L. Frangois and C. Vassallo, “Finite cladding effects in W fibers: a new interpretation of leaky modes,” Appl. Opt. 221, 3109–3120(1983). [CrossRef]

]. With a conventional single mode fiber output as exciting field, all circular symmetric modes of the entire fiber can be excited. For the adopted fiber parameters previously, there exist only six order guiding supermodes. Their field distributions are plotted in Fig. 2(b). Here the field distributions are normalized to carry a unit power. It can be seen that HE15 and HE16 contain major light in the core region, whereas other lower order modes contain very little in core region. Hence the HE15 and HE16 modes are excited preferentially because their fields distribution within the core are similar to the exciting field. The dispersion curves of these two excited supermodes are plotted in insert drawing of Fig. 2(b). To comparing with the coupled mode method, the dispersion curves of the core mode and the ring waveguide mode HE15 are also plotted together. The HE15 and HE16 modes have different propagation constant and beat to each other as propagating along the special fiber. Thus the light power at the phase-matching wavelength exchanges between the core and the outer cladding.

Fig. 3. Calculated resonant spectrum of the special fiber.
Fig. 4. The relationship between the resonant wavelength and the refractive index of the coating.
Fig. 5. Shift of the phase-matching wavelength with different n4.

Although the supermodes method is more accurate than the coupled mode method for the analysis of special fiber, the latter method is clearer and more intuitive for explaining the sensor operation principle. So the coupled mode theory is applied to explain sensing principle. When temperature is varied, the refractive index of the coating changes according to the thermo-optic effect. As a result, the propagation constant of the outer cladding mode varies, and the resonant wavelength shifts accordingly. Figure 4 shows the relationship between the resonant wavelength and the refractive index of the coating. With the refractive index of the coating increasing, the resonant wavelength shifts toward a longer wavelength. To explain the impact of the coating straightforward, Fig. 5 depicts the variation of the phase-matching crosspoint. For the outer cladding, a lower refractive index of the coating will decrease the effective refractive index and shift the dispersion curve to the left side. For the core, the core and the inner cladding are made of pure silica and F-doped silica respectively, which have very low thermo-optical coefficient [17

17. A. Koike and N. Sugimoto, “Temperature dependences of optical path length in fluorine-doped silica glass and bismuthate glass,” Proc. SPIE 6116, 61160Y1–61160Y8 (2006).

]. Thus the effective index of the core mode can be treated as an unchangeable curve approximately. In this way, the temperature variation can be measured by the shifts of phase-matching wavelength.

3. Fabrication of the special fiber temperature sensor

The cladding-mode resonant special fiber was fabricated based on the MCVD technique. A high-purity fused silica tube (diameter, 20mm; wall thickness, 2mm) was as substrate. Firstly, the fluorine-doped layer was deposited with O2, Freon and SiCl4 as the starting reactants. Then the pure silica layer was deposited with O2 and SiCl4. The deposited temperature was 1500°C. After the deposition, the supporting tube was collapsed around 2000°C to a preform. At last, the preform was placed onto drawing tower and drawn into a fiber at 1800 °C. The refractive index profile around the core and inner cladding region of the special fiber was tested by the optical fiber analyzer (EXFO NR9200). As shown in Fig. 6, the special fiber core is about 9.3µm in diameter. And the refractive index difference between the core and the inner cladding is about 0.2%. Thus the fiber core as a rod waveguide only supports the fundamental mode. The outer cladding whose diameter is 125µm is a multi-modes ring waveguide. Through the thin-thickness inner cladding, the core mode and the cladding modes are coupled with each other based on the evanescent waves.

Fig. 6. Refractive index profile tested by optical fiber analyzer (EXFO NR9200).
Fig. 7. Experimental setup for temperature sensing.

To apply the special fiber to temperature sensing, a segment of the special fiber was spliced into conventional single mode (SM) fiber forming the structure of SM fiber-special fiber-SM fiber by using fusion splicer (FITEL S177), as shown in Fig. 7. Then silicone gel with negative thermo-optic coefficient was dip-coated around the sensor fiber and dried in room temperature for 24h. Additionally, for aging purpose, the silicone-coated special fiber was thermally annealed at 100°C for 10 hours. One SM fiber inputs light wave into the special fiber at one end, and the other SM fiber collects output light wave from the special fiber core at the other end. In this way, the resonant spectrum of the special fiber and its temperature sensing characteristics can be observed and studied.

Fig. 8. Spectra measured by OSA with temperature increasing.

4. Experiments and discussions

Figure 7 shows the experimental setup for investigating temperature sensing characteristics. An 1550nm-wavelength superluminescent diode (SLED, SL3200-C42) and an optical spectrum analyzer (OSA, ANDO AQ-6315A) were used to record resonant spectrum. Temperature-controlled chamber (ESL-04KA) was utilized to adjust temperature. The length of the spliced special fiber was about 14mm. To avoid the impact of fiber bending or strain on temperature measuring results, the sensing fiber was hung vertically with a load.

The dependence of the resonant spectrum on temperature was investigated experimentally within the range from -20°C to +80°C. As shown in Fig. 8, with temperature increasing, the center wavelength of the spectrum shifts to shorter wavelength and the total wavelength shift is up to 21.17nm. Due to the negative thermo-optic coefficient of the silicone coating material, the refractive index of the coating decreases when temperature increases. According to the analysis in section 2, the resonant wavelength satisfying the phase-matching condition shifts toward shorter wavelength. Figure 9 shows experimental curve of the dependence of the resonant wavelength on temperature, which is consistent with the theory result in Fig. 4.

To investigate repeatability of temperature response, the resonant spectra were also tested with temperature decreasing from 80°C to -20°C. Compared with the response in temperature increasing process, the sensor demonstrates a good repeatability, as shown in Fig. 9.

Fig. 9. Measurement performance of the cladding-mode resonant temperature sensor.
Fig. 10. Transmission variation of the sensor head at resonant wavelength

From the experimental results shown in Fig. 8, we can also see that the optical power transmission at the resonant wavelength varies. Within the temperature range from -20°C to +80°C, the transmission at resonant wavelength becomes larger monotonically from -27dB to -22.5dB, as shown in Fig. 10. As far as a length-known special fiber is concerned, as indicated by Eq. (2), the transmission at resonant wavelength depends on the coupling coefficient κ which is calculated by the field overlap integral of the coupled modes [13

13. D. Marcuse, Light Transmission Optics, (Van Nostrand Reinhold, New York, 1972), Chap. 10.

]. According to the study by A. Cusano [18

18. A. Cusano, A. Iadicicco, P. Pilla, L. Contessa, S. Campopiano, A. Cutolo, and M. Giordano, “Mode transition in high refractive index coated long period gratings,” Opt. Express 14, 19–34 (2006). [CrossRef] [PubMed]

], a decreasing refractive index of coating can shift the transverse field profile of the cladding modes to the inner cladding so that the evanescent wave is enhanced in the fiber core region. In this case, the fields overlap integral between the cladding and core mode becomes larger and the coupling coefficient also increases accordingly. As a result, the variation of the transmission at resonant wavelength can be observed in our experiment, shown in Fig. 10. From the monotone relationship, we can also conclude that the length of the spliced special fiber makes itself work at over coupling region, namely κ·L>π/2. The monotone response is quite important to sensing signal process in the practice application.

5. Conclusion

A novel fiber-optic temperature sensor based on SM fiber-special fiber-SM fiber structure is presented in this paper. The sensor utilizes the leaky mode resonance between the core and the outer cladding through the inner cladding. Theoretical studies were carried out and the shift of resonant wavelength versus refractive index of coating was calculated numerically. To achieve temperature sensor, the special fiber was coated with the temperature-sensitive silicone. The experimental result shows high temperature sensitivity (240pm/°C at 20°C). By testing the resonant wavelength shift from -20°C to 80°C, the total shift up to 21.17nm is achieved, which is much larger than that of the conventional FBGs sensor. And after the anneal processing, the sensor presented a good repeatability. With the attractive performance, the cladding-mode special fiber temperature sensor can be found wide applications.

Acknowledgment

This work was funded by the Natural Science Foundation of China (60677031 and 60577043), the Specialized Research Fund for the Doctoral Program of Higher Education of China (20060280001), Chenguang Program (2007CG54), Science and Technology Commission of Shanghai Municipality (STCSM) (07dz22024 and 075307017) and nano-technology special project of Shanghai (0652nm046), Shanghai Municipal Education Commission (06AZ032).

References and links

1.

B. Culshaw, “Fiber-optic sensors: applications and advances,” Opt. Photonics News16, 24–29(2005). [CrossRef]

2.

Y. -G. Han, X. Dong, J. H. Lee, and S. B. Lee, “Simultaneous measurement of bending and temperature based on a single sampled chirped fiber Bragg grating embedded on a flexible cantilever beam,” Opt. Lett. 31, 2839–2841 (2006). [CrossRef] [PubMed]

3.

Y. -J. Rao, Z. -l. Ran, X. Liao, and H. -y. Deng, “Hybrid LPFG/MEFPI sensor for simultaneous measurement of high-temperature and strain,” Opt. Express 15, 14936–14941 (2007). [CrossRef] [PubMed]

4.

Y. -G. Han, S. Lee, C. -S. Kim, J. Kang, Y. Chung, and U. -C. Paek, “Simultaneous measurement of temperature and strain using dual long-period fiber gratings with controlled temperature and strain sensitivities,” Opt. Express 11, 476–481 (2003). [CrossRef] [PubMed]

5.

Z. Huang, Y. Zhu, X. Chen, and A. Wang, “Intrinsic Fabry-Perot fiber sensor for temperature and strain measurements,” IEEE Photon. Technol. Lett. 18, 1879–1881(2006).

6.

L. Cheng, A. J. Steckl, and J. Scofield, “SiC thin-film Fabry-Perot interferometer for fiber-optic temperature sensor,” IEEE Trans. Electron Devices 50, 2159–2164 (2003). [CrossRef]

7.

H. Bao, T. Wang, and Y. Shen, “High sensitive coupling evanescent wave temperature sensor,” Proc. SPIE 5634, 558–562 (2005). [CrossRef]

8.

S. M. Chandani and N. A. F. Jaeger, “Fiber-optic temperature sensor using evanescent fields in D fibers,” IEEE Photon. Technol. Lett. 17, 2706–2708 (2005). [CrossRef]

9.

E. Li, X. Wang, and C. Zhang, “Fiber-optic temperature sensor based on interference of selective higher-order modes,” Appl. Phys. Lett. 89, 091119-1-3 (2006). [CrossRef]

10.

J. Ju, Z. Wang, W. Jin, and M. S. Demokan, “Temperature sensitivity of a two-mode photonic crystal fiber interferometric sensor,” IEEE Photon. Technol. Lett. 18, 2168–2170(2006). [CrossRef]

11.

D. S. Moon, B. H. Kim, A. Lin, G. Sun, Y. -G. Han, W. -T. Han, and Y. Chung, “The temperature sensitivity of Sagnac loop interferometer based on polarization maintaining side-hole fiber,” Opt. Express 15, 7962–7967 (2007). [CrossRef] [PubMed]

12.

L. G. Cohen, D. Marcuse, and W. L. Mammel, “Radiating leaky-mode losses in single-mode lightguides with depressed-index claddings,” IEEE J. Quantum Electron. QE-18, 1467–1472(1982). [CrossRef]

13.

D. Marcuse, Light Transmission Optics, (Van Nostrand Reinhold, New York, 1972), Chap. 10.

14.

F. D. Nunes, C. Adriana, d. S. Melo, H. Filomeno, and D. S. Filho, “Theoretical study of coaxial fibers,” Appl. Opt. 35, 388–399 (1996). [CrossRef] [PubMed]

15.

R. V. Schmidt and R. C. Alferness, “Directional coupler switches, modulators and filters using alternating Δ β techniques,” IEEE Trans. Circuits Syst. CAS-26, 1099–1108(1979). [CrossRef]

16.

P. L. Frangois and C. Vassallo, “Finite cladding effects in W fibers: a new interpretation of leaky modes,” Appl. Opt. 221, 3109–3120(1983). [CrossRef]

17.

A. Koike and N. Sugimoto, “Temperature dependences of optical path length in fluorine-doped silica glass and bismuthate glass,” Proc. SPIE 6116, 61160Y1–61160Y8 (2006).

18.

A. Cusano, A. Iadicicco, P. Pilla, L. Contessa, S. Campopiano, A. Cutolo, and M. Giordano, “Mode transition in high refractive index coated long period gratings,” Opt. Express 14, 19–34 (2006). [CrossRef] [PubMed]

OCIS Codes
(060.2340) Fiber optics and optical communications : Fiber optics components
(060.2370) Fiber optics and optical communications : Fiber optics sensors

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: May 23, 2008
Revised Manuscript: July 31, 2008
Manuscript Accepted: July 31, 2008
Published: August 11, 2008

Citation
Fufei Pang, Wenchao Xiang, Hairun Guo, Na Chen, Xianglong Zeng, Zhenyi Chen, and Tingyun Wang, "Special optical fiber for temperature sensing based on cladding-mode resonance," Opt. Express 16, 12967-12972 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-12967


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References

  1. B. Culshaw, "Fiber-optic sensors: applications and advances," Opt. Photonics News 16, 24-29(2005). [CrossRef]
  2. Y. -G. Han, X. Dong, J. H. Lee, and S. B. Lee, "Simultaneous measurement of bending and temperature based on a single sampled chirped fiber Bragg grating embedded on a flexible cantilever beam," Opt. Lett. 31, 2839-2841 (2006). [CrossRef] [PubMed]
  3. Y. -J. Rao, Z. -l. Ran, X. Liao, and H. -y. Deng, "Hybrid LPFG/MEFPI sensor for simultaneous measurement of high-temperature and strain," Opt. Express 15, 14936-14941 (2007). [CrossRef] [PubMed]
  4. Y. -G. Han, S. Lee, C. -S. Kim, J. Kang, Y. Chung, and U. -C. Paek, "Simultaneous measurement of temperature and strain using dual long-period fiber gratings with controlled temperature and strain sensitivities," Opt. Express 11, 476-481 (2003). [CrossRef] [PubMed]
  5. Z. Huang, Y. Zhu, X. Chen, and A. Wang, "Intrinsic Fabry-Perot fiber sensor for temperature and strain measurements," IEEE Photon. Technol. Lett. 18, 1879-1881(2006).
  6. L. Cheng, A. J. Steckl, and J. Scofield, "SiC thin-film Fabry-Perot interferometer for fiber-optic temperature sensor," IEEE Trans. Electron Devices 50, 2159-2164 (2003). [CrossRef]
  7. H. Bao, T. Wang, and Y. Shen, "High sensitive coupling evanescent wave temperature sensor," Proc. SPIE 5634, 558-562 (2005). [CrossRef]
  8. S. M. Chandani, and N. A. F. Jaeger, "Fiber-optic temperature sensor using evanescent fields in D fibers," IEEE Photon. Technol. Lett. 17, 2706-2708 (2005). [CrossRef]
  9. E. Li, X. Wang, and C. Zhang, "Fiber-optic temperature sensor based on interference of selective higher-order modes," Appl. Phys. Lett. 89, 091119-1-3 (2006). [CrossRef]
  10. J. Ju, Z. Wang, W. Jin, and M. S. Demokan, "Temperature sensitivity of a two-mode photonic crystal fiber interferometric sensor," IEEE Photon. Technol. Lett. 18, 2168-2170(2006). [CrossRef]
  11. D. S. Moon, B. H. Kim, A. Lin, G. Sun, Y. -G. Han, W. -T. Han, and Y. Chung, "The temperature sensitivity of Sagnac loop interferometer based on polarization maintaining side-hole fiber," Opt. Express 15, 7962-7967 (2007). [CrossRef] [PubMed]
  12. L. G. Cohen, D. Marcuse and W. L. Mammel, "Radiating leaky-mode losses in single-mode lightguides with depressed-index claddings," IEEE J. Quantum Electron. QE-18, 1467-1472(1982). [CrossRef]
  13. D. Marcuse, Light Transmission Optics, (Van Nostrand Reinhold, New York, 1972), Chap. 10.
  14. F. D. Nunes, C. Adriana, d. S. Melo, H. Filomeno, and D. S. Filho, "Theoretical study of coaxial fibers," Appl. Opt. 35, 388-399 (1996). [CrossRef] [PubMed]
  15. R. V. Schmidt and R. C. Alferness, "Directional coupler switches, modulators and filters using alternating ?β techniques," IEEE Trans. Circuits Syst. CAS-26, 1099-1108(1979). [CrossRef]
  16. P. L. Frangois and C. Vassallo, "Finite cladding effects in W fibers: a new interpretation of leaky modes," Appl. Opt. 221, 3109-3120(1983). [CrossRef]
  17. A.  Koike and N.  Sugimoto, "Temperature dependences of optical path length in fluorine-doped silica glass and bismuthate glass," Proc. SPIE 6116, 61160Y1-61160Y8 (2006).
  18. A. Cusano, A. Iadicicco, P. Pilla, L. Contessa, S. Campopiano, A. Cutolo, and M. Giordano, "Mode transition in high refractive index coated long period gratings," Opt. Express 14, 19-34 (2006). [CrossRef] [PubMed]

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