A ray-transfer-matrix model for hybrid fiber Fabry-Perot sensor based on graded-index multimode fiber

Yuan Gong; Tian Zhao; Yun-Jiang Rao; Yu Wu; Yu Guo

doi:10.1364/OE.18.015844

1. Introduction

Fiber-optic refractive-index (RI) sensors have been widely investigated in recent years for the applications in biomedical, chemical and environmental fields. These sensors are based on surface plasmon resonance (SPR) [1], long-period fiber gratings [2,3], fiber Bragg gratings [4], photonic crystal fibers [5], fiber Fabry-Perot cavities [6–9], etc., or by combining two of them [10,11]. The SPR-based fiber-optic RI sensors can offer high sensitivity, but it is relatively complex for depositing a homogeneous thin film on an optical fiber and the stability of these films is not good. The grating-based fiber-optic RI sensors have the inherent cross-talk problem between the refractive-index and temperature measurement and also require expensive fabrication equipments. Fiber-optic Fabry-Perot (FFP) RI sensors is one of the most important branches of RI sensors. Generally speaking, the refractive index and the temperature can be determined by measuring the intensity variation (or the maximum fringe contrast between the adjacent resonant peak and notch) and wavelength shift of the reflective spectrum of the FFP sensor, respectively. The cross-talk between the refractive index and temperature measurement can be easily eliminated by simultaneous intensity and phase demodulation [7]. By using the 157nm excimer laser micromachining technology, Ran et al. [9] fabricated a FFP-RI sensor with a RI sensitivity of ~27dB/RIU (Refractive Index Unit) over the range of 1.33-1.44. Recently, FFP-RI sensors with hybrid structures were fabricated to realize simultaneous measurement of refractive index and temperature [7].The RI sensitivity was about 16dB/RIU.

In this paper a novel graded-index multimode fiber (GI-MMF) based hybrid-structured fiber Fabry-Perot (GI-FFP) sensor using the self-focusing effect is fabricated and a maximum RI sensitivity of ~160dB/RIU is obtained, which is highest to date among the FFP-RI sensors, to our knowledge. This kind of RI sensor has the advantages of easy fabrication, low cost and high sensitivity. A theoretical model based on the ray-transfer-matrix (RTM) method is developed for explaining the principle of the sensor. A good agreement between the theory and experiment is also achieved.

2. Theoretical model

The RTM theory [12] is used to describe the principle of the GI-FFP sensor and to investigate the influence of experimental parameters on the performance of the sensor. A Gaussian beam can be expressed as

E (r, z) = \frac{A_{0}}{w} \exp (- i Φ - \frac{i n k r^{2}}{2 q}) .

A_{0}

remains constant for energy conservation.

k = 2 π / λ

is the wavenumber with λ the free-space wavelength.

Φ = n k z

denotes the phase shift of the light beam as it propagates and n is the refractive index of the medium in which the light propagates. The complex beam parameter, q, is given by

\frac{1}{q} = \frac{1}{ρ} - i \frac{λ}{n π w^{2}} .

ρ and w are the radius of curvature and the beam radius of the Gaussian beam, respectively. If the transfer matrix from an input plane to an output plane is M, the transformation of the complex beam parameter is represented by

q^{'} = \frac{A q + B}{C q + D} .

Here

A = M (1, 1), B = M (1, 2), C = M (2, 1), D = M (2, 2) .

q and

q^{'}

are the complex beam parameters at the input and output planes, respectively.

As shown in Fig. 1(a) , the hybrid GI-FFP sensor is fabricated by cascading an air gap and a short section of GI-MMF to a single mode fiber (SMF). The air gap is formed by fusion splicing the SMF with a chemically etched micronotch on the GI-MMF end. As well-known, light rays propagate periodically in GI-MMF and follow sinusoidal path if the index profile is parabolic. In Fig. 1(a), the GI-MMF length is selected with its end at node of the propagation path. The propagation of the light beam reflected by the GI-MMF end (surface III) is described by dashed lines in Fig. 1(a). The schematic principle of the three-beam interference model and the microscopic image of the GI-FFP sensor are shown in Figs. 1(b) and 1(c), respectively.

Fig. 1 (a) Beam propagation in the hybrid GI-FFP cavity; (b) schematic diagram of the three-beam interference model and (c) the microscopic image of the GI-FFP sensor.

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The electronic amplitude of the incident beam at location 0 is $E_{0} (r)$ and corresponding complex parameter is $q_{0} = i n_{s} π w_{s}^{2} / λ$ . $n_{s}$ is the refractive index of the SMF core and $w_{s}$ is the beam radius in the SMF. The reflectance of surface I, $R_{I}$ , is determined by the Fresnel equation, $R_{I} = {{(n_{s} - n_{0})}^{2} / (n_{s} + n_{0})}^{2}$ , and the electronic amplitude of the light beam reflected by surface I is $E_{I} (r) = \sqrt{R_{I}} E_{0} (r)$ . $n_{0}$ is the refractive index of the ambient medium and in the air gap it is approximately 1. The reflectance of the etched micronotch (surface II) is given by

R_{I I} = \int_{0}^{w_{I I r}} {| E_{I I r} (r) |}^{2} R (r) r d r / \int_{0}^{w_{I I i}} {| E_{I I i} (r) |}^{2} r d r,

w_{I I i}

and

w_{I I r}

are the beam radii of the light beam incident on and reflected by surface II. The reflectance,

R (r)

, is given by the radial distribution,

R (r) = {[n (r) - n_{0}]}^{2} / {[n (r) + n_{0}]}^{2} .

n (r)

is the refractive index profile of the GI-MMF core and can be expressed as

n (r) = n_{1} \sqrt{1 - g^{2} r^{2}}, r < a .

Here

n_{1}

is the maximum index at

r = 0

, a is the radius of the GI-MMF core, g is a factor that determines the index profile of the core. The ABCD matrices corresponding to

E_{I I r}

and

E_{I I i}

are given by

M_{I I i} = M_{12} M_{01}

and

M_{I I r} = M_{2} M_{12} M_{01},

where

M_{i j}

is the matrix describing the transformation of the complex beam parameters between locations i and j. The elementary matrices are given by

M_{01} = [\begin{matrix} 1 & 0 \\ 0 & n_{s} / n_{0} \end{matrix}],

M_{12} = [\begin{matrix} 1 & L_{0} \\ 0 & 1 \end{matrix}],

M_{2} = [\begin{matrix} 1 & 0 \\ - 2 / ρ_{1} & 1 \end{matrix}] .

ρ_{1}

is the radius of curvature of the etched micronotch on the GI-MMF end and

L_{0}

is the effective cavity length of the air gap. Considering the radius of curvature of the etched micronotch,

L_{0}

is smaller than the spacing between the SMF end and the bottom of the micronotch. The transformation of the complex beam parameters is determined by (3). The electronic amplitude of the light beam reflected by surface II is

E_{I I} (r) = \sqrt{{R^{'}}_{I I}} {E^{'}}_{I I} (r)

with

{R^{'}}_{I I} = T_{I}^{2} (1 - A_{I}) R_{I I}

.

T_{I}

is the transmittance of surface I and

A_{I}

is the propagation losses in the air gap. The additional phase in

E_{I I}

is

Φ_{I I} = 2 n_{0} k L_{0}

. The ABCD matrix corresponding to

{E^{'}}_{I I}

is given by

M_{I I} = M_{1' 0'} M_{2' 1'} M_{2} M_{12} M_{01},

with

M_{2' 1'} = M_{12},

M_{1' 0'} = [\begin{matrix} 1 & 0 \\ 0 & n_{0} / n_{s} \end{matrix}] .

The effective reflectance of surface II can be expressed as

R_{I I e f f} = \frac{\int_{0}^{a_{s}} {| E_{I I} (r) |}^{2} r d r}{\int_{0}^{a_{s}} {| E_{0} (r) |}^{2} r d r}

with

2 a_{s}

the mode field diameter of the SMF.

Similar to the reflection from surface II, the reflectance of surface III is given by

R_{I I I} = \int_{0}^{w_{I I I i}} {| E_{I I I r} (r) |}^{2} R (r) r d r / \int_{0}^{w_{I I I i}} {| E_{I I I i} (r) |}^{2} r d r .

The electronic amplitude of the light beam reflected by surface III is

E_{I I I} (r) = \sqrt{{R^{'}}_{I I I}} {E^{'}}_{I I I} (r)

with

{R^{'}}_{I I I} = T_{I}^{2} T_{I I}^{2} (1 - A_{I I}) R_{I I I}

.

T_{I I}

is the transmittance of surface II and

A_{I I}

is the propagation losses in the hybrid FFP. The additional phase in

E_{I I I}

is

Φ_{I I I} = 2 k (n_{0} L_{0} + n_{1} L)

. The ABCD matrices corresponding to

E_{I I I i}

,

E_{I I I r}

and

E_{I I I}

are given by

M_{I I I i} = M_{34} M_{23} M_{12} M_{01},

M_{I I I r} = M_{I I I i},

M_{I I I} = M_{1' 0'} M_{2' 1'} M_{3' 2'} M_{33'} M_{23} M_{12} M_{01},

respectively, with

M_{23} = [\begin{matrix} 1 & 0 \\ - (n_{0} - n_{1}) / ρ_{1} n_{1} & n_{0} / n_{1} \end{matrix}], M_{34} = [\begin{matrix} \cos (g L) & \sin (g L) / g \\ - g \sin (g L) & \cos (g L) \end{matrix}],

M_{33'} = [\begin{matrix} cos (2 g L) & \sin (2 g L) / g \\ - g \sin (2 g L) & cos (2 g L) \end{matrix}], M_{3' 2'} = [\begin{matrix} 1 & 0 \\ (n_{1} - n_{0}) / ρ_{1} n_{0} & n_{1} / n_{0} \end{matrix}] .

Considering the coupling losses of the light beam reflected from surface III into the SMF, the effective reflectance of surface III can be expressed as

R_{I I I e f f} = \int_{0}^{a_{s}} {| E_{I I I} (r) |}^{2} r d r / \int_{0}^{a_{s}} {| E_{0} (r) |}^{2} r d r .

R_{I I I e f f}

is mainly dependent on the coupling coefficient of the light beam into the SMF as the GI-MMF length changes. Generally speaking,

R_{I I I e f f}

is smaller than

R_{I I I}

, due to the coupling losses and propagation losses. The reflective signal considering the three beam interference is given by

I (λ) = \int_{0}^{a_{s}} {| E_{I} + E_{I I} + E_{I I I} |}^{2} r d r

The maximum fringe contrast can be easily obtained from the reflective spectrum of the hybrid FFP cavity by

V = - 10 \log (\frac{I_{\max} (λ)}{I_{\min} (λ)}) .

Here

I_{\max} (λ)

and

I_{\min} (λ)

are the maximum and minimum values of adjacent resonant peak and notch, respectively.

By using the effective reflectance of the three reflective surfaces of the hybrid GI-FFP sensor, the above RTM theory can be simplified. For the case of $n_{0} < n_{1}$ , the normalized intensity of the three-beam interference as shown in Fig. 1(b) can be expressed as

I (λ) = R_{I} + R_{I I e f f} + R_{I I I e f f} - 2 \sqrt{R_{I} R_{I I e f f}} \cos Φ_{I I} - 2 \sqrt{R_{I I e f f} R_{I I I e f f}} \cos (Φ_{I I I} - Φ_{I I}) + 2 \sqrt{R_{I} R_{I I I e f f}} \cos Φ_{I I I} .

As well-known, the fringe contrast of the two-beam interference reaches its maximum value when the reflectance of the two surfaces equals, which is a strict constraint. The corresponding constraint condition on the effective reflectance of the three surfaces for the three-beam interference can be deduced by (11) and is given by

- 1 \leq \frac{R_{I} + R_{I I e f f} - R_{I I I e f f}}{2 \sqrt{R_{I} R_{I I e f f}}} \leq 1

Unlike the two-beam interference, the constraint condition for the three-beam interference to obtain the optimal fringe contrast is an inequality, i.e., the requirement on the reflectance is a relatively wide range rather than a decided value. This makes easier for the three-surface based hybrid GI-FFP sensor to obtain high performance than the conventional two-surface FFP sensors.

3. Numerical simulation and comparison with experiment

Based on the theoretical model presented in Section 2, the principle of the hybrid GI-FFP sensor is numerically investigated in the following. The parameters used in the simulations are listed in Table 1 . As shown in Fig. 2(a) , both the beam radius at the GI-MMF end and the reflectance of surface III change periodically with the GI-MMF length. The beam radius at the GI-MMF end is calculated by (2) according to the complex beam parameter at location 4 in Fig. 1(a). The reflectivity of the GI-MMF end is higher than that of the common SMFs as the refractive index of GI-MMF core is larger than that of the common SMFs. On the other hand, because the refractive index of the GI-MMF decreases gradually along the radial direction, the reflectance of surface III becomes higher as the beam radius decreases. With the GI-MMF end at nodes, the optical power is self-focused to a region with higher refractive index, leading to the maximum values of $R_{I I I}$ .

Table 1. Values of parameters used in the simulations

View Table

Fig. 2 (a) Beam radius at the GI-MMF end and $R_{I I I}$ , and (b) the maximum fringe contrast of the reflective spectra of the GI-FFP sensors in air (dots) and calculated $R_{I I I e f f}$ (line), as a function of GI-MMF length.

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The effective reflectance of surface III, $R_{I I I e f f}$ , changes periodically with the GI-MMF length but the period is twice larger than that of $R_{I I I}$ , as shown in Fig. 2(b). In the simulation, the propagation losses in the hybrid FFP, $A_{I I}$ , is set to be zero and in the experiment the non-zero $A_{I I}$ leads to smaller $R_{I I I e f f}$ but do not affect on the period. When the GI-MMF length is selected with its end at node or antinode of the propagation path, each light ray is reversibly reflected by the GI-MMF end. The coupling losses of light reflected into the lead-in SMF are small and $R_{I I I e f f}$ is high.

GI-FFP sensors with various GI-MMF lengths were experimentally fabricated and corresponding values of maximum fringe contrast of the reflective spectra of the sensor in air are shown in Fig. 2(b). High fringe contrast of >20 dB is observed with periodic length of the GI-MMF. The period is in good agreement with that of the calculated $R_{I I I e f f}$ . Even and odd numbers in Fig. 2(b) represent multiples of quarter-pitch, corresponding to the GI-MMF length with its end at node or antinode.

The reproducibility of the chemical etching process is high as the material of the optical fiber is homogeneous and the etching rate difference between the fiber core and cladding remains almost constant. The influence of the splicing process is greatly reduced by decreasing the discharge intensity to ~4% of the standard discharge intensity of the SMF-to-SMF splicing. The reflective spectrum, thus the performance, of the sensor is greatly influenced by $R_{I I I e f f}$ , which is dependent on the GI-MMF length, as shown in Fig. 2(b). Therefore, it is important to control the GI-MMF length. The reproducibility of the sensor length is not good if the GI-MMF is cleaved arbitrarily but can be greatly improved by cleaving under an optical microscope. It is worth noting that high fringe contrast is relatively more difficult to be obtained with the GI-MMF end at antinode than at node. In the antinode case, the influence of the residual cleaved angle on the coupling losses is larger than that in the node case, as corresponding beam radii at antinode are ~15.5μm, much larger than that at node ~5.3μm.

The reflective spectra of the GI-FFP sensors with GI-MMF length of ~515μm and ~610μm are calculated and shown in Figs. 3(a) and 3(b), respectively. For the former case, the experimental parameters are L = 515μm, $L_{0}$ = 24.35μm, $A_{I}$ = 0.887, $A_{I I}$ = 0.3, $R_{I I}$ = 0.428% and $R_{I I I e f f}$ = 2.213%. For the latter case, the experimental parameters are L = 610μm, $L_{0}$ = 22.94μm, $A_{I}$ = 0.84, $A_{I I}$ = 0.82, $R_{I I}$ = 0.608% and $R_{I I I e f f}$ = 0.195%. The simulated spectra are in good agreement with the experimental spectra, which are shown in Figs. 3(c) and 3(d). The GI-FFP sensors with GI-MMF length of ~515μm and ~610μm are typical for obtaining the reflective spectra with high fringe contrast and low fringe contrast. The effective reflectance of surface III, $R_{I I I e f f}$ , is much higher, 2.213%, in the former case than 0.195% in the latter. It is observed in Fig. 3(c) that the minimum resonant notch of the three-beam interference, $λ_{I I I}$ , do not coincide with that of the two-beam interference, $λ_{I I}$ . We investigated the resonance condition of the three-beam interference and found that the deviation between the two resonant wavelength, $Δ λ = λ_{I I I} - λ_{I I}$ , became larger with higher $R_{I I I e f f}$ . In Fig. 3(c), $Δ λ$ is determined to be ~5.7nm. Furthermore, It is indicated by the simulations that the high fringe contrast can be obtained even if $R_{I I}$ is small. Therefore, the performance of the hybrid FFP sensor can be further enhanced by reducing the roughness and improving the geometry of the etched micronotch on the GI-MMF end.

Fig. 3 Calculated and experimental reflective spectra of the GI-FFP snesors with GI-MMF length of ~515μm and ~610μm. The reflective spectra from the air gap and the SMF end are also given.

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4. Performance of GI-FFP sensor

The fabrication process of this kind of hybrid FFP sensor included chemical etching, fusion splicing and cleaving. First, the cleaved GI-MMF was dipped into the 40% hydrofluoric acid and chemically etched for 3 minutes. A micronotch was fabricated on the GI-MMF end and the etched GI-MMF was fusion spliced to a common single mode fiber, forming surface I and surface II of the air gap. By cleaving the GI-MMF, the surface III was obtained and the hybrid GI-FFP sensor was achieved. In order to obtain a high sensitivity to refractive index, the GI-MMF length was precisely controlled by cleaving the GI-MMF to be ~505μm under an optical microscope. The microscopic image of the sensor is shown in Fig. 1(c). The spacing between the SMF end and the bottom of the micronotch, ${L^{'}}_{0}$ , was measured to be 35μm by an optical microscope. The effective cavity length of the air-gap, $L_{0}$ , is shorter than ${L^{'}}_{0}$ due to the curved surface II. The reflective spectra of the sensor in air and deionized water are shown in Fig. 4 . The maximum fringe contrast of the spectrum of the sensor in deionized water is 18.21dB near 1564.26nm. It is worth noting that the wavelength deviation, $Δ λ$ ~17.6nm, as shown in Fig. 4, is much larger than the sensor with GI-MMF length of ~515μm, indicating higher $R_{I I I e f f}$ with GI-MMF length of ~505μm.

Fig. 4 The reflective spectra of the sensor in air and deionized water.

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The refractive index responses of the sensor were investigated by dipping it into glucose solutions with different concentrations. The refractive indices of glucose solutions were calibrated by an Abbe refractometer. The fringe contrast of the reflective spectrum, V, near 1564.26 nm as a function of the external refractive index, $n_{0}$ , is shown in Fig. 5(a) . The $V - n_{0}$ curve is fitted by a quadratic function and the coefficient of determination, R ², is ~0.99942. The sensitivity of the refractive index, η, is determined by the first derivative of the $V - n_{0}$ curve, i.e., $η = 554.92 n_{0} - 899.46$ dB/RIU. The sensitivity of the refractive index increases as the refractive index of aqueous solutions decreases and reaches its maximum value, −159.8dB/RIU, in the deionized water, which is more sensitive than that of the FFP-RI sensors recently reported in [7] and [9].

Fig. 5 (a) Refractive index and (b) temperature responses of the GI-FFP sensor. The measurement sensitivity of the refractive index is also given in (a).

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The temperature responses of the sensor were investigated by putting it into the deionized water with different temperatures. The wavelength shifts of the resonant peak near 1563.5nm and resonant notch near 1564.5nm as a function of temperature are shown in Fig. 5(b). The temperature sensitivity is determined to be 10.4pm/°C. Good linearity of ~0.99951 and ~0.99936 are obtained, respectively. When the GI-FFP sensor was dipped into glucose solutions with different indices at constant temperature, the wavelength of the resonant notch remained constant, indicating that the temperature measurement is not influenced by the external refractive index changes. The influence of the temperature variation on the refractive index measurement can be compensated by measuring the temperature via the wavelength shift and the cross-talk can be eliminated. The compensation method for this kind of GI-FFP sensor was similar to [7].

5. Conclusion

A hybrid GI-FFP sensor with GI-MMF length of 505μm has been fabricated and a high sensitivity of refractive index, ~160dB/RIU, has been obtained. A theoretical model based on the ray-transfer-matrix method has also been developed for understanding the principle of the sensor. Good agreements between simulated and experimental results have been obtained.

Acknowledgment

This work was supported by National Natural Science Foundation of China under Grant 60537040.

References and links

1. J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance sensors: review,” Sens. Actuators B Chem. 54(1-2), 3–15 (1999). [CrossRef]

2. V. Bhatia and A. M. Vengsarkar, “Optical fiber long-period grating sensors,” Opt. Lett. 21(9), 692–694 (1996). [CrossRef] [PubMed]

3. A. Kapoor and E. K. Sharma, “Long period grating refractive-index sensor: optimal design for single wavelength interrogation,” Appl. Opt. 48(31), G88–G94 (2009). [CrossRef] [PubMed]

4. W. Liang, Y. Huang, Y. Xu, R. K. Lee, and A. Yariv, “Highly sensitive fiber Bragg grating refractive index sensors,” Appl. Phys. Lett. 86(15), 151122 (2005). [CrossRef]

5. D. K. C. Wu, B. T. Kuhlmey, and B. J. Eggleton, “Ultrasensitive photonic crystal fiber refractive index sensor,” Opt. Lett. 34(3), 322–324 (2009). [CrossRef] [PubMed]

6. Z. Ran, Y. J. Rao, J. Zhang, Z. Liu, and B. Xu, “A miniature fiber-optic refractive-index sensor based on laser-machined Fabry-Perot interferometer tip,” J. Lightwave Technol. 27(23), 5426–5429 (2009). [CrossRef]

7. H. Y. Choi, G. Mudhana, K. S. Park, U. C. Paek, and B. H. Lee, “Cross-talk free and ultra-compact fiber optic sensor for simultaneous measurement of temperature and refractive index,” Opt. Express 18(1), 141–149 (2010). [CrossRef] [PubMed]

8. T. Wei, Y. Han, Y. Li, H. L. Tsai, and H. Xiao, “Temperature-insensitive miniaturized fiber inline Fabry-Perot interferometer for highly sensitive refractive index measurement,” Opt. Express 16(8), 5764–5769 (2008). [CrossRef] [PubMed]

9. Z. L. Ran, Y. J. Rao, W. J. Liu, X. Liao, and K. S. Chiang, “Laser-micromachined Fabry-Perot optical fiber tip sensor for high-resolution temperature-independent measurement of refractive index,” Opt. Express 16(3), 2252–2263 (2008). [CrossRef] [PubMed]

10. L. Mosquera, D. Sáez-Rodriguez, J. L. Cruz, and M. V. Andrés, “In-fiber Fabry-Perot refractometer assisted by a long-period grating,” Opt. Lett. 35(4), 613–615 (2010). [CrossRef] [PubMed]

11. Y. Lu, W. P. Huang, and S. S. Jian, “Influence of mode loss on the feasibility of grating-assisted optical fiber surface plasmon resonance refractive index sensors,” J. Lightwave Technol. 27(21), 4804–4808 (2009). [CrossRef]

12. W. L. Emkey and C. A. Jack, “Analysis and evaluation of graded-index fiber-lenses,” J. Lightwave Technol. 5(9), 1156–1164 (1987). [CrossRef]

Parameter	Value	Unit
$N A_{s}$	0.14	-
$N A_{m}$	0.275	-
g	5518.6	-
$2 a$	62.5	μm
$2 a_{s}$	10.4	μm
$L_{0}$	22.94, 24.35	μm
$ρ_{1}$	100	μm
L	515, 610	μm

A ray-transfer-matrix model for hybrid fiber Fabry-Perot sensor based on graded-index multimode fiber

Abstract

1. Introduction

2. Theoretical model

3. Numerical simulation and comparison with experiment

4. Performance of GI-FFP sensor

5. Conclusion

Acknowledgment

References and links

Cited By

Figures (5)

Tables (1)

Equations (14)

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