Optical-fiber surface-plasmon-resonance sensor ... - OSA Publishing

He et al.

Vol. 23, No. 5 / May 2006 / J. Opt. Soc. Am. B

801

Optical-fiber surface-plasmon-resonance sensor employing long-period fiber gratings in multiplexing Yue-Jing He Department of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan

Yu-Lung Lo* Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan

Jen-Fa Huang Department of Electrical Engineering, National Chen Kung University, Tainan, Taiwan Received August 8, 2005; revised November 30, 2005; accepted December 2, 2005; posted December 8, 2005 (Doc. ID 63973) A new type of optical-fiber surface-plasmon-resonance (SPR) sensor based on a thin metallic film and longperiod fiber gratings for measuring small changes of refractive index of analyte is presented. This sensor simply employs a long-period fiber grating with a proper period to couple a core mode 共HE11兲 to the copropagating cladding mode that can excite a surface-plasmon wave (SPW). The mainly theoretical base used to analyze this new structure is the unconjugated form of coupled-mode equations. In this new SPR sensor, the variation of the refractive index of analyte is determined by monitoring the change of the transmitted core mode power, which is calculated by unconjugated two-mode coupled-mode equations at a fixed wavelength. The numerical results have demonstrated that this new and simple configuration may be used as a highly sensitive amplitude sensor. As far as the excitation of SPW, the model of numerical simulation, and the complexity of measurement equipment are concerned, this new structure is superior to the proposed sensor, consisting of a bent polished singlemode SPR optical fiber. Furthermore, the structure can be easily adapted for a SPR fiber optical probe if a mirror is deposited on the fiber tip. © 2006 Optical Society of America OCIS codes: 240.6680, 240.6690, 050.2770.

1. INTRODUCTION Over the past decade, in order to minimize the size of surface-plasmon-resonance (SPR) sensors, the configuration using a bent, polished, single-mode optical fiber and a thin metallic film has been proposed.1–4 According to geometric optic theory, the trajectory of a core mode, solved by Maxwell’s equations, forms a ray angle at the corecladding interface. Therefore it is obvious that this structure is similar to the Kretschmann configuration in the excitation of a surface-plasmon wave (SPW).5–8 In the bent, polished structure, the variations in the refractive index of analyte are determined by monitoring the changes in the transmitted optical power at a fixed wavelength or by monitoring the changes in the wavelength at which the resonance occurs. Furthermore, in order to adjust the range of the measurable refractive index, a buffer layer with appropriate refractive index and thickness is used. This kind of structure has become common, since it is strongly dependent on the refractive index of the surrounding medium. Most notably, the cylindrical symmetry of the fiber in this configuration is not real owing to the presence of the side-polished layer. Therefore it is very difficult to model the waveguide properties rigorously. All of the theoretical models that have been proposed use the pseudo-TM mode of the unpolished fiber to 0740-3224/06/050801-11/$15.00

analyze the bent, polished structure.1–4 As a result, these models make some degree of error in analysis. In this paper, a new type of optical-fiber SPR sensor based on a thin metallic film, which is deposited averagely on the cladding of an optical fiber, and a long-period fiber grating is presented. This sensor simply employs long-period fiber gratings with proper period to couple a core mode to the copropagating cladding mode that can excite a SPW and monitors the change of the transmitted core mode power at a fixed wavelength to sense the variation of the refractive index of analyte. In comparison with the bent, polished configuration, the SPR sensor proposed in this paper has three excellent characteristics. First, for operating at a fixed wavelength, this new SPR sensor does not need any buffer layer to shift the effective refractive index of the SPW to the phase-matched condition, because there are as many as 76 cladding modes in the sensor, which may excite SPW. In other words, the excitation of SPW in this new configuration is inherent. In addition, for operating at a fixed wavelength, the multiple measuring ranges can be simply accomplished by writing a number of long-period fiber gratings with various periods into the same fiber. Second, as far as the model of the numerical simulation is concerned, we can clearly find that the fiber used in this new structure is no longer broken (bent © 2006 Optical Society of America

802

J. Opt. Soc. Am. B / Vol. 23, No. 5 / May 2006

He et al.

and polished), and all of the used guiding modes are exact solutions.9 We believe that the simulation result for this new structure is more accurate. Third, it is obvious that this new SPR sensor does not need any polarization controller, since the thin metallic film is deposited averagely on the cladding of an optical fiber. The remainder of this paper is constructed as follows. Section 2 describes the guiding modes in this four-layer SPR sensor by the dispersion relation and the unconjugated form of coupled-mode equations by the unconjugated form of the reciprocity theorem. Section 3 describes the principle used to detect the variation of the analyte and the simulation result by unconjugated two-mode, coupled-mode equations. Otherwise, in order to ensure that the analytic result is meaningful, the unconjugated two-mode coupling equations are checked by the unconjugated four-mode coupled-mode equations. Finally, the conclusion is given in Section 4.

U2

␨co =

␨co = ␨⬘co ,

J1⬘ 共u1a1兲 UJ1共u1a1兲

␤co

+

K1⬘ 共w1a1兲 WK1共w1a1兲

冉 冊

册

V2

共2兲

,

w⑀1 W2

冉 冊

␤co V2 ␨⬘co = U

2

冋

w␮ W2 J1⬘ 共u1a1兲 UJ1共u1a1兲

+

⑀2 K1⬘ 共w1a1兲 ⑀1 WK1共w1a1兲

册

.

共3兲

The following definitions have been used in Eqs. (2) and (3):

2. UNCONJUGATED FORM OF COUPLED-MODE EQUATIONS As shown in Fig. 1, a four-layer cylindrical optical fiber is used to design the new type of SPR sensor, where n1, n2, n3, and n4 indicate the refractive indices for the core layer, the cladding layer, the metal layer, and the analyte layer, respectively. Among this configuration, a metallic film with a thickness d is averagely deposited on the cladding of an optical fiber, and two long-period fiber gratings (LPGs) with various periods, ⌳1 for LPG 1 and ⌳2 for LPG 2, are written into the core layer of an optical fiber in order to sense two various ranges of analyte. Throughout this paper all electromagnetic field components are assumed to have the form as exp关i共k0neffz-wt兲兴, and the z direction is assumed to be coincident with the axis of an optical fiber. Even though the linearly polarized approximation should be sufficient to describe a mode guided by the fiber core, since we are mainly interested in low ⌬ = 共n1 − n2兲 / n1 fiber, the exact fields and dispersion for the HE11 core mode are still used in this paper, so as to enhance the accuracy of simulation result. After substituting the tangential fields of the HE11 mode, as shown in Appendix A, into the boundary continuity conditions at the radii r = a1, the exact dispersion relation that we solve to obtain the effective refractive index of the HE11 mode, which is co ⬍ n1, is given by defined by n2 ⬍ neff

冋

U = u 1a 1 ,

共4兲

W = w 2a 1 ,

共5兲

co 2 1/2 兲 , u1 = 共2␲/␭兲共n12 − neff

共6兲

co 2 w1 = 共2␲/␭兲共neff − n22兲1/2 .

共7兲

where

In Eqs. (2) and (6), the prime notation indicates differentiation with respect to the total argument. In this paper, the long-period fiber grating is assumed to be a circularly symmetric index perturbation in any transverse plane of the fiber, so that the coupling interaction only occurs between HE11 and the cladding modes with azimuthal order l = 1.10 For the remainder of the analysis, therefore, only the cladding modes with azimuthal order l = 1 are concerned. For this absorbing SPR sensor, the cladding-mode electromagnetic fields in every layer are listed in Appendix A. After substituting the tangential fields of the cladding mode into the boundary continuity conditions at the radii r = a1, r = a2, and r = a3, the exact and complex dispersion relation is given by

⬘, ␨cl = ␨cl

共8兲

共1兲

where

where

␨cl =

⬘= ␨cl Fig. 1. Scheme of an optical-fiber SPR sensor based on two longperiod fiber gratings and a deposited metallic film in multiplexing.

− R 1H 1 + T 1H 2 + U 1H 3 − V 1H 4 R 1M 1 − T 1M 2 − U 1M 3 + V 1M 4

− R 2H 3 + T 2H 4 + U 2H 1 − V 2H 2 R 2M 3 − T 2M 4 − U 2M 1 + V 2M 2

,

共9兲

.

共10兲

The following definitions have been used in Eqs. (9) and (10):

He et al.

H1 =

冋

Vol. 23, No. 5 / May 2006 / J. Opt. Soc. Am. B

␤cl w ⑀ 4u 3a 3

冋 冋 冋 冋 冋 冋 冋

⫻

−

⫻

+

⫻

−

⫻

u3 u2

K1共w4a3兲K1共u3a3兲

I1⬘ 共u3a2兲K1共u2a2兲 −

␤cl w ⑀ 4u 3a 3 u3 u2

w ⑀ 3u 2a 2 w4 u3

w ⑀ 3u 2a 2 w4 u3

u32

I 共u a 兲K1⬘ 共u2a2兲 2 1 3 2

u2

K1共w4a3兲I1共u3a3兲 u32 u22

冉 冊册 w42 u32

−

w42⑀3 u32⑀4

+

冉 冊册 u22

u32⑀4

w ␮ u 2a 2

冋 冋 冋 冋 冋 冋 冋

⫻

册

K1共w4a3兲K1⬘ 共u3a3兲

u32

w42⑀3

册

−1

u22

冋

␤cl

⫻

−1

冉 冊册 u32

K1共u3a2兲K1共u2a2兲

K1⬘ 共w4a3兲I1共u3a3兲 −

册

K1共u3a2兲K1⬘ 共u2a2兲

I1共u3a2兲K1共u2a2兲

K1⬘ 共w4a3兲K1共u3a3兲 −

␤cl

H3 =

−1

u32

K1⬘ 共u3a2兲K1共u2a2兲 −

␤cl

冉 冊册 w42

⫻

−1

−

册

K1共w4a3兲I1⬘ 共u3a3兲 ,

⫻

I1共w3a2兲K1共u2a2兲

␤cl w ⑀ 4u 3a 3

␤cl w ␮ u 2a 2

w ⑀ 4u 3a 3 u3 u2

u3 u3 u2

u22⑀3

u3

K1⬘ 共w4a3兲I1共u3a3兲 −

−1

u32

册

冉 冊册 冉 冊册

u32⑀2

K1⬘ 共u3a2兲K1共u2a2兲 −

w4

w42

u32

u22

−1

w42

K1共w4a3兲I1共u3a3兲

K1⬘ 共w4a3兲K1共u3a3兲 −

−1

u22

K1共w4a3兲K1共u3a3兲

I1⬘ 共u3a2兲K1共u2a2兲 −

w4

冉 冊册 u32

K1共u3a2兲K1共u2a2兲

␤cl

803

u32

−1

I1共u3a2兲K1⬘ 共u2a2兲

w42⑀3 u32⑀4

u32⑀2 u22⑀3

册

K1共w4a3兲K1⬘ 共u3a3兲

K1共u3a2兲K1⬘ 共u2a2兲

w42⑀3 u32⑀4

K1共w4a3兲I1⬘ 共u3a3兲 ,

共11兲

H2 =

冋

␤cl w ⑀ 4u 3a 3

冋 冋 冋 冋 冋 冋 冋

⫻

−

⫻

+

⫻

−

⫻

u3 u2

K1共w4a3兲K1共u3a3兲

I1⬘ 共u3a2兲I1共u2a2兲 −

␤cl w ⑀ 4u 3a 3 u3 u2

w ⑀ 3u 2a 2 w4 u3

w ⑀ 3u 2a 2 w4 u3

u32

I 共u a 兲I⬘ 共u a 兲 2 1 3 2 1 2 2

u2

K1共w4a3兲I1共u3a3兲 u32 u22

w42 u32

册

u32

u22

w42⑀3 u32⑀4

u22

u32⑀4

册

w ␮ u 2a 2

冋 冋 冋 冋 冋 冋 冋

⫻

+

K1共w4a3兲K1⬘ 共u3a3兲

冉 冊册

w42⑀3

−

−1

u32

冋

␤cl

⫻

−1

冉 冊册

K1共u3a2兲I1共u2a2兲

K1⬘ 共w4a3兲I1共u3a3兲 −

冉 冊册

共13兲

H4 =

K1共u3a2兲I1⬘ 共u2a2兲

I1共u3a2兲I1共u2a2兲

K1⬘ 共w4a3兲K1共u3a3兲 −

␤cl

−1

u32

K1⬘ 共u3a2兲I1共u2a2兲 −

␤cl

冉 冊册 w42

册

−1

⫻

−

册

K1共w4a3兲I1⬘ 共u3a3兲 , 共12兲

册 册

册

⫻

I1共u3a2兲I1共u2a2兲

␤cl w ⑀ 4u 3a 3

␤cl w ␮ u 2a 2

w ⑀ 4u 3a 3

u2

u3 u3 u2

u3

w42 u32

u32

u22

u32⑀2 u22⑀3

K1⬘ 共w4a3兲K1共u3a3兲 −

u32

−1

I1共u3a2兲I1⬘ 共u2a2兲

w42⑀3 u32⑀4

u32⑀2 u22⑀3

K1⬘ 共w4a3兲I1共u3a3兲 −

−1

−1

w42

K1共w4a3兲I1共u3a3兲

K1⬘ 共u3a2兲I1共u2a2兲 −

w4

−1

K1共w4a3兲K1共u3a3兲

I1⬘ 共u3a2兲I1共u2a2兲 −

w4

u22

K1共u3a2兲I1共u2a2兲

␤cl

u3

冉 冊册 冉 冊册 冉 冊册 冉 冊册 u32

册

K1共w4a3兲K1⬘ 共u3a3兲

K1共u3a2兲I1⬘ 共u2a2兲

w42⑀3 u32⑀4

册

册

册

K1共w4a3兲I1⬘ 共u3a3兲 , 共14兲

804

M1 =

J. Opt. Soc. Am. B / Vol. 23, No. 5 / May 2006

冋

␤cl w ⑀ 3u 2a 2

冋 冋 冋 冋 冋 冋 冋

⫻

−

⫻

+

⫻

−

⫻

M2 =

冋 −

冋 冋 冋 冋 冋 冋 冋

⫻

−

w ␮ u 3a 3

w ⑀ 3u 2a 2

␤cl w ␮ u 3a 3 u3 u2

u3 u3 u2

u3

K1⬘ 共w4a3兲I1共u3a3兲 −

⫻

w ⑀ 3u 2a 2

w ␮ u 3a 3

u3 u3 u2

u3

−

册

u3

K 共u a 兲K1⬘ 共u2a2兲 2 1 3 2

u2

w42

+

册 册

册

⫻

−

K 共w4a3兲I1⬘ 共u3a3兲 ,共15兲 2 1

u3

冉 冊册 冉 冊册 冉 冊册 冉 冊册 w42 u32

u22

w42 u32

K1⬘ 共w4a3兲K1共u3a3兲 −

w42 u32

u32 u22

−

−1

册

u32

+

K1共w4a3兲K1⬘ 共u3a3兲

册

册

冋 冋 冋 冋 冋 冋 冋

⫻

−

册

K1共w4a3兲I1⬘ 共u3a3兲 ,共16兲

u2

⫻

K1共w4a3兲K1共u3a3兲

I1⬘ 共u3a2兲K1共u2a2兲 −

␤cl w ␮ u 3a 3 u3 u2

␤cl w ␮ u 2a 2 w4 u3

w ␮ u 2a 2 w4 u3

u3 u2

w ␮ u 3a 3

u2

w ␮ u 2a 2 w4 u3

w ␮ u 2a 2 w4 u3

u32

K1共w4a3兲K1⬘ 共u3a3兲

u32

u32

冉 冊册 u32

u22

−1

冉 冊册 I1共u3a2兲I1⬘ 共u2a2兲

冉 冊册 w42 u32

u32⑀2 u22⑀3

K1共u3a2兲I1⬘ 共u2a2兲

u32

u22

w42 u32

u32

册

−1

K1共w4a3兲K1⬘ 共u3a3兲

冉 冊册

w42

册

−1

冉 冊册

K1共u3a2兲I1共u2a2兲

册

−1

u32

K1共w4a3兲I1共u3a3兲

册

K1共w4a3兲I1⬘ 共u3a3兲 ,共17兲

w42

u22⑀3

册

−1

u22

u32⑀2

K1⬘ 共w4a3兲I1共u3a3兲 −

−1

冉 冊册

w42

I1共u3a2兲I1共u2a2兲

册

K1共u3a2兲K1⬘ 共u2a2兲

u32

w42

K1⬘ 共w4a3兲K1共u3a3兲 −

␤cl

w42

K1共u3a2兲K1共u2a2兲

K1⬘ 共u3a2兲I1共u2a2兲 −

␤cl

冉 冊册

u22⑀3

I1共u3a2兲K1共u2a2兲

I1⬘ 共u3a2兲I1共u2a2兲 −

I1共u3a2兲K1⬘ 共u2a2兲

u32⑀2

K1共w4a3兲K1共u3a3兲

␤cl

u3

K1共w4a3兲I1共u3a3兲

K1⬘ 共w4a3兲I1共u3a3兲 −

−1

u32

u22⑀3

K1⬘ 共w4a3兲K1共u3a3兲 −

␤cl

冉 冊册 w42

u32⑀2

K1⬘ 共u3a2兲K1共u2a2兲 −

w ␮ u 3a 3

⫻

−1

K1共u3a2兲I1⬘ 共u2a2兲

w42

冋

u3

␤cl

⫻

−1

I1共u3a2兲I1⬘ 共u2a2兲 u22

⫻

M4 =

−1

u32

u32

K1⬘ 共w4a3兲I1共u3a3兲 −

⫻

−1

K 共w4a3兲K1⬘ 共u3a3兲 2 1

u22

冋 冋 冋 冋 冋 冋 冋

−1

w42

u32

w ␮ u 3a 3

⫻

I1共u3a2兲K1⬘ 共u2a2兲

K1共w4a3兲I1共u3a3兲

K1⬘ 共u3a2兲I1共u2a2兲 −

w4

u22

u32

K1共u3a2兲I1共u2a2兲

I1⬘ 共u3a2兲I1共u2a2兲 −

w4

u32

w42

K1共w4a3兲K1共u3a3兲

␤cl ␤cl

u32

冋

␤cl

−1

u32

I1共u3a2兲I1共u2a2兲

w ␮ u 3a 3

u2

u22

K1⬘ 共w4a3兲K1共u3a3兲 −

␤cl

u3

u32

M3 =

−1

w42

K1共w4a3兲I1共u3a3兲

K1⬘ 共u3a2兲K1共u2a2兲 −

w4

u22

K1共u3a2兲K1共u2a2兲

I1⬘ 共u3a2兲K1共u2a2兲 −

w4

冉 冊册 冉 冊册 冉 冊册 冉 冊册 u32

K1共w4a3兲K1共u3a3兲

␤cl

w ⑀ 3u 2a 2

⫻

+

␤cl

␤cl

⫻

I1共u3a2兲K1共u2a2兲

He et al.

u32

u22

册

−1

册

K1共w4a3兲I1⬘ 共u3a3兲 ,共18兲

He et al.

R1 =

T1 =

U1 =

V1 =

R2 =

T2 =

U2 =

V2 =

Vol. 23, No. 5 / May 2006 / J. Opt. Soc. Am. B

u22

I1共u2a1兲I1⬘ 共u1a1兲 u12

−

u2 u1

I1⬘ 共u2a1兲I1共u1a1兲,

u22

u2

u1

u1

K 共u a 兲I⬘ 共u a 兲 − 2 1 2 1 1 1 1

␤cl w ⑀ 2u 1a 1

␤cl w ⑀ 2u 1a 1

I1共u2a1兲I1共u1a1兲

I1共u2a1兲I1⬘ 共u1a1兲 u12⑀2

u12⑀2

−

w ␮ u 1a 1

␤cl w ␮ u 1a 1

u22

u1

u22

u12

−1 ,

I1⬘ 共u2a1兲I1共u1a1兲,

u2 u1

I1共u2a1兲I1共u1a1兲

−1 ,

u12

u2

K1共u2a1兲I1⬘ 共u1a1兲 −

␤cl

冉 冊 冉 冊

K1共u2a1兲I1共u1a1兲

u22⑀1

u22⑀1

K1⬘ 共u2a1兲I1共u1a1兲,

K1⬘ 共u2a1兲I1共u1a1兲,

冉 冊 冉 冊 u22

u12

K1共u2a1兲I1共u1a1兲

−1 ,

u22

u12

−1 ,

共19兲

共20兲

共21兲

共22兲

共23兲

共24兲

共25兲

共26兲

where u1 =

cl2 共2␲/␭兲共neff

u2 =

cl2 共2␲/␭兲共neff

−

n12兲1/2 ,

−

n22兲1/2 ,

共28兲

cl u3 = 共2␲/␭兲共neff − n32兲1/2 ,

2

共29兲

2

共30兲

cl w4 = 共2␲/␭兲共neff − n42兲1/2 .

805

共27兲

In Eqs. (11)–(26), the prime notation indicates differentiation with respect to the total argument. In Eqs. cl cl , which is defined by n4 ⬍ Re共neff 兲 ⬍ n2, is the (27)–(30), neff effective refractive index of the cladding mode. After solving the dispersion relation of the cladding mode with the parameters n1 = 1.4602, n2 = 1.4443, n3 = 0.00298 + 13.4857i, n4 = 1.36061, a1 = 2.25 ␮m, a2 = 62.5 ␮m, d = 36 nm, and ␭ = 1550 nm, there are as many as 76 cladding modes in this new SPR structure. To conveniently interpret the characteristic of cladding modes, we call the cladding mode possessing the largest electromagnetic field on the metal surface as SPW, and the other cladding modes are renumbered according to the magnitude of cl cl Re共neff 兲 [the cladding mode having the largest Re共neff 兲 is cl 兲 is numnumbered 1 and that having the smallest Re共neff bered 75]. Figures 2(a) and 2(b) show the azimuthal magnetic fields for the odd and even cladding modes, respectively. Notice that each mode is set to carry a power of 1 W. From Fig. 2(a), we can clearly see that on the metal surface, the surface plasmon resonance for some odd cladding modes is very strong. In contrast to the Kretchman configuration that SPW exists only if the incident right is

Fig. 2. Plot of the azimuthal magnetic fields for the renumbered cladding modes: (a) odd modes, (b) even modes.

TM polarization,5–8 the weak SPR arises for some even cladding modes, as shown in Fig. 2(b), mainly because the cladding modes are hybrid. When the SPR sensor is designed to operate at a fixed wavelength, namely, a fixed core mode, the setup of this system is potentially simpler and less expensive. However, in the bent, polished configuration, SPW could hardly be excited for most analyte, and it is necessary to use a buffer layer with appropriate refractive index and thickness to shift the effective refractive index of SPW to the phase-matched condition. According to geometric optic theory, 76 cladding modes in this new SPR sensor represent 76 optical rays propagating with various angles. In other words, there are as many as 76 optical rays to excite SPW simultaneously at a fixed wavelength. The presence of SPW is inherent, and all we have to do is solve the dispersion relation of the cladding mode. When an uniform long-period fiber grating is induced in the fiber core, the refractive index of the core layer can be described as n共z兲 = n1 + n1␴关1 + cos共2␲z/⌳兲兴,

共31兲

where n1 is the unperturbed core index, ⌳ is the period of the grating, and ␴ is the UV-induced variation. Owing to the presence of the absorbing medium, the conjugated form of coupled-mode equations is no more suitable to describe the mode coupling in this new SPR sensor. Utilizing the unconjugated form of the reciprocity theorem and mode orthogonality,11 the unconjugated form of coupled-

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He et al.

mode equations that describe the copropagating interaction in a long-period fiber grating can be derived as shown in Appendix B, and the result is obtained as dA␮ dz

=i

兺A k ␯

␯ ␯−␮关1

+ cos共2␲z/⌳兲兴exp关i共␤␯ − ␤␮兲z兴, 共32兲

with

冕冕 冕冕 2␲

k␯−␮ = w⑀0n12␴

a1

0

r共E␯r E␮r + E␸␯E␸␮ − Ez␯Ez␮兲drd␸

0

2␲

0

,

⬁

r共E␮r H␸␮ −

E␸␮H␮r 兲drd␸

0

共33兲 where A␮共z兲 is the amplitude for the transverse mode field and k␯−␮ is the coupling constant between mode ␯ and mode ␮. In Eq. (33), we assume that the UV-induced variation is sufficiently small [⌬⑀ Ⰶ ⑀ or 共⌬⑀ ⑀兲 / 共⌬⑀ + ⑀兲 ⬵ ⌬⑀], and the relation between coupling coefficient and coupling constant is K␯−␮ = k␯−␮关1 + cos共2␲z/⌳兲兴.

共34兲

Note that according to Eq. (32), we can know that the occurrence of coupling between two copropagating modes must simultaneously meet two essential conditions, that is, a phase-matched condition and a proper cross coupling constant 共k␯-␮ , ␯ ⫽ ␮兲. After properly designing the period of LPG (according to the phase-matched condition) and the length of LPG (according to the magnitude of the cross-coupling constant), the core mode propagating in this new SPR sensor can be coupled to the cladding mode that we desire. However, it seems possible for some cladding modes to match the same phase-matched condition, even if in an ideal symmetrical geometry. Therefore it seems necessary to consider multimode coupling when we use a coupled-mode equation to analyze this new SPR sensor. In addition, this phenomenon will also seriously influence the performance of the SPR sensor, and it is almost impossible for us to solve the multimode coupledmode equations. To confirm whether the coupling among these 76 copropagating cladding modes really exists in this structure, we calculate all of the coupling constants

Fig. 4. Cross-coupling constants for 75 renumbered cladding modes and SPW at a wavelength of 1550 nm: (a) 兩k␯-co兩, and (b) 兩kco-␮兩.

among these 77 propagating modes, including 76 cladding modes and one core mode, with ␴ = 1 ⫻ 10−3, and the result is shown in Fig. 3. In Fig. 3, we use the color map to represent the magnitudes of all the coupling constants. From Fig. 3, we can clearly find kco-co Ⰷ k␯-co, kco-␮, and k␯-␮ (␯ ⫽ ␮, and ␯, ␮苸 cladding mode). This arises mainly because the index perturbation ⌬⑀ exists only in the fiber core, and the magnitude of the core-mode field in the core layer is much larger than that of the cladding-mode field. Obviously, it is almost impossible to meet simultaneously the same phase-matched condition and the same cross coupling constant for the coupling among various cladding modes, and the approximation to ignore the coupling among various cladding modes is reasonable when we use synchronous approximation to simplify the unconjugated form of coupled-mode equations. Finally, if only two modes (core mode and SPW) are involved at the operating wavelength, the unconjugated form of two-mode coupledmode equations can be written as follows: dAco共z兲 dz dAspw共z兲

Fig. 3. Coupling constants for all of the propagating modes at a wavelength of 1550 nm. The color of the colormap is used to represent the magnitudes of all of the coupling constants.

dz

= ikco−coAco共z兲 + ikspw−co/2Aspw共z兲exp共− i2␦兲,

= ikco−spw/2Aco共z兲exp共i2␦兲 + ikspw−spwAspw共z兲 共35兲

He et al.

Vol. 23, No. 5 / May 2006 / J. Opt. Soc. Am. B

with

␦=

1 2

冉

␤co − ␤spw −

2␲ ⌳

冊

,

807

共36兲

where Aco共z兲 is the amplitude for the core mode and Aspw共z兲 is the amplitude for the SPW. Figure 4 shows the magnitude of cross-coupling constants for 75 renumbered cladding modes and SPW at a wavelength of 1550 nm. Note that k␯-co and kco-␮ are different according to Eqs. (33), though they seem identical in Fig. 4.

3. ANALYSIS OF THE INDEX VARIATION OF THE ANALYTE It is well known that SPW is highly sensitive to the surrounding media. Therefore we can anticipate that the effective refractive index of SPW will strongly depend on the refractive index of the analyte adjacent to the gold film. So as to confirm this characteristic, Fig. 5 shows the relation between the effective refractive index of SPW and the refractive index of analyte. From Fig. 5, we can clearly find that except SPW and some odd cladding modes, the deposited metallic film exhibits a strong shielding effect. Based on the above statement, we utilize long-period fiber gratings to couple a core mode to a copropagating SPW and monitor the change of the transmitted core mode power, which is calculated by unconjugated two-mode, coupled-mode equations at a fixed wavelength, to sense the variation of the refractive index of analyte. Utilizing the boundary conditions of a long-

Fig. 6. Plot of the power transmission versus the length of a long-period fiber grating for (a) LPG1 and (b) LPG2.

period fiber grating, Aco共z = 0兲 = 1 and Aspw共z = 0兲 = 0, the bar and cross power transmission can be written as T=共z兲 = 兩Aco共z兲兩2/兩Aco共0兲兩2 ,

共37兲

T⫻共z兲 = 兩Aspw共z兲exp共i␤spwz兲兩2/兩Aco共0兲兩2 . 共38兲 In unconjugated two-mode, coupled-mode equations, the long-period fiber grating’s period and length are two important parameters for maximizing the power transmission. First, in order to couple the core mode to the SPW, the period of the long-period fiber grating should be properly designed by

冉

⌳ = Re

Fig. 5. Plot of the effective refractive indices of the cladding modes versus the refractive index of the analyte: (a) odd modes and (b) even modes.

2␲ kco−co − kspw−spw + ␤co − ␤spw

冊

.

共39兲

In this new SPR sensor, LPG 1 and LPG 2, as shown in Fig. 1, are designed to sense two ranges of analyte (1.35619艋 n4 艋 1.35659 and 1.36041艋 n4 艋 1.36081) by using ⌳1 = 17.620⫻ 10−6 and ⌳2 = 18.521⫻ 10−6. Note that above, ⌳1 and ⌳2 are designed at the center of range from 1.35619 to 1.35659 as n4 = 1.35639 and the center of range from 1.36041 to 1.36081 as n4 = 1.36061, respectively. For these two long-period fiber gratings, the relations between power transmission (including bar and cross power transmission) and length of the long-period fiber grating can be obtained by solving Eqs. (37) and (38), and the results are illustrated in Figs. 6(a) and 6(b), respectively. In

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He et al.

Fig. 6, we find that the presence of metallic film makes the power gradually decay with an increase of the grating’s length. Finally, we obtain the length in 6.925 cm for LPG 1 and 8.320 cm for LPG 2. Utilizing the above parameters for LPG 1 and LPG 2, the relations between the power transmission and the refractive index of analyte are shown in Fig. 7(a) for the range from 1.35619 to 1.35659 and in Fig. 7(b) for the range from 1.36041 to 1.36081. What we must emphasize is that so far, all the analyses are based on the assumption that only two modes (core mode and SPW) are involved at the operating wavelength 共␭ = 1550 nm兲. From Fig. 5, we can clearly find that as far as SPW is concerned, the variation in the effective refractive index is rather sensitive regarding the change in the refractive index of analyte. It implies that Fig. 7 is certainly very sharp and for some sensing range, the use of unconjugated two-mode coupled-mode equations may be insufficient, since SPW is very close to some cladding mode. To check the validity in Fig. 7, we will compare the result of unconjugated two-mode coupled-mode equations with that of unconjugated four-mode coupled-mode equations. According to Eq. (32), the unconjugated four-mode coupled-mode equations can be written as follows: dAco共z兲 dz

= ikco−coAco共z兲关1 + cos共2␲z/⌳兲兴 + ik71−coA71共z兲关1

Note that the unconjugated four-mode coupled-mode equations include the core mode, SPW, and the two cladding modes that are closest to SPW in the real part of effective refractive index (as shown in Fig. 5, the modes closest to SPW are mode 71 and mode 72). From Fig. 8, obviously, it is reasonable to use unconjugated two-mode coupled-mode equations to analyze the range from 1.35619 to 1.35659 and the range from 1.36041 to 1.36081. However, we must again emphasize that, for arbitrary sensing range, the use of unconjugated threemode coupled-mode equations is necessary, since SPW may be very close to some cladding mode.

4. CONCLUSIONS We have presented a new type of optical-fiber SPR sensor for measuring small changes of refractive index of analyte. This sensor utilizes a long-period fiber grating to couple a core mode to a copropagating SPW and monitors the change of the transmitted core-mode power, which is calculated by unconjugated two-mode coupled-mode equations, at a fixed wavelength to sense the variation of the refractive index of analyte. The unbent and unpolished characteristics make this new SPR sensor superior to the conventional configuration. In this paper, we derived the exact dispersion relation and the unconjugated form of coupled-mode equations to analyze this four-layer absorbing SPR sensor. By properly designing the grating’s pe-

+ cos共2␲z/⌳兲兴exp关i共␤71 − ␤co兲z兴 + ikspw−coAspw共z兲关1 + cos共2␲z/⌳兲兴exp关i共␤spw − ␤co兲z兴 + ik72−coA72共z兲关1 + cos共2␲z/⌳兲兴exp关i共␤72 − ␤co兲z兴, dA71共z兲 dz

共40兲

= ikco−71Aco共z兲关1 + cos共2␲z/⌳兲兴exp关i共␤co − ␤71兲z兴 + ik71−71A71共z兲关1 + cos共2␲z/⌳兲兴 + ikspw−71Aspw共z兲 ⫻关1 + cos共2␲z/⌳兲兴exp关i共␤spw − ␤71兲z兴 + ik72−71A72共z兲关1 + cos共2␲z/⌳兲兴exp关i共␤72 − ␤71兲z兴, 共41兲

dAspw共z兲 dz

= iKco−spwAco共z兲关1 + cos共2␲z/⌳兲兴exp关i共␤co − ␤spw兲z兴 + iK71−spwA71共z兲关1 + cos共2␲z/⌳兲兴exp关i共␤71 − ␤spw兲z兴 + iKspw−spwAspw共z兲关1 + cos共2␲z/⌳兲兴 + iK72−spwA72共z兲关1 + cos共2␲z/⌳兲兴exp关i共␤72 − ␤spw兲z兴,

dA72共z兲 dz

共42兲

= iKco−72Aco共z兲关1 + cos共2␲z/⌳兲兴exp关i共␤co − ␤72兲z兴 + iK71−72A71共z兲关1 + cos共2␲z/⌳兲兴exp关i共␤71 − ␤72兲z兴 + iKspw−72Aspw共z兲关1 + cos共2␲z/⌳兲兴exp关i共␤spw − ␤72兲z兴 + iK72−72A72共z兲关1 + cos共2␲z/⌳兲兴.

共43兲

Fig. 7. Plot of the power transmission versus the refractive index of analyte for range (a) from 1.35619 to 1.35659 and (b) from 1.36041 to 1.36081.

He et al.

Vol. 23, No. 5 / May 2006 / J. Opt. Soc. Am. B

冋

809

册

1 ␤co Hr = − C1coJ1共u1cor兲 + A1cou1coJ1⬘ 共u1cor兲 sin共␾兲, r w␮ 共A4兲

冋

H␾ = − C1cou1coJ1⬘ 共u1cor兲 −

␤co 1 w␮ r

册

A1coJ1共u1cor兲 cos共␾兲, 共A5兲

Hz =

u1co

2

A1coJ1共u1cor兲sin共␾兲

jw␮

共A6兲

inside the fiber core layer 共0 艋 r 艋 a1兲 and Er =

冋

1 r

B2coK1共w2cor兲 −

␤co w⑀2

册

D2cow2coK1⬘ 共w2cor兲 cos共␾兲, 共A7兲

冋

E␾ = − B2cow2coK1⬘ 共w2cor兲 −

␤co 1 w⑀2 r

册

D2coK1共w2cor兲 sin共␾兲, 共A8兲

Ez =

Fig. 8. Comparison of unconjugated two-mode coupled-mode equations with unconjugated four-mode coupled-mode equations for range (a) from 1.35619 to 1.35659 and (b) from 1.36041 to 1.36081.

riod and length, the numerical results have demonstrated that this new and simple SPR sensor possesses rather excellent performance. Most notably, the excitation of SPW in this new configuration is inherent, and the multiple measuring ranges can be simply accomplished by writing a number of long-period fiber gratings with various periods into the same fiber. Furthermore, it can be easily adapted for a SPR fiber optical probe by depositing a mirror on the fiber tip.

APPENDIX A: ELECTROMAGNETIC FIELDS OF THE CLADDING MODES The electromagnetic fields of a core mode propagating in a four-layer absorbing SPR sensor are given as Er =

冋

1 r

A1coJ1共u1cor兲 −

冋

␤co w⑀1

E␾ = − A1cou1coJ1⬘ 共u1cor兲 −

册

C1cou1coJ1⬘ 共u1cor兲 cos共␾兲, 共A1兲

␤co 1 w⑀1 r

册

C1coJ1共u1cor兲 sin共␾兲,

w2co

2

jw⑀2

D2coK1共w2cor兲cos共␾兲,

冋

册

1 ␤co B2cow2coK1⬘ 共w2cor兲 sin共␾兲, Hr = − D2coK1共w2cor兲 + r w␮ 共A10兲

冋

H␾ = − D2cow2coK1⬘ 共w2cor兲 −

␤co 1 w␮ r

u1co

Hz = −

w2co

2

jw␮

B2coK1共w2cor兲sin共␾兲

jw⑀1

共A3兲

共A12兲

inside the fiber cladding layer 共a1 艋 r 艋 a2兲. In Eqs. (A1)–(A12), C1co, A1co, D2co, and B2co are field coefficients of the core mode, and J1 and K1 are the Bessel function. The electromagnetic fields of the cladding mode propagating in a four-layer absorbing SPR sensor are given as Er =

冋

1 r

C1clI1共u1clr兲 −

冋

␤cl w⑀1

E␾ = − C1clu1clI1⬘ 共u1clr兲 −

册

A1clu1clI1⬘ 共u1clr兲 cos共␾兲, 共A13兲

␤cl 1 w⑀1 r

册

A1clI1共u1clr兲 sin共␾兲, 共A14兲

2

C1coJ1共u1cor兲cos共␾兲,

册

B2coK1共w2cor兲 cos共␾兲, 共A11兲

共A2兲

Ez = −

共A9兲

Ez =

u1cl

2

jw⑀1

A1clI1共u1clr兲cos共␾兲,

共A15兲

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J. Opt. Soc. Am. B / Vol. 23, No. 5 / May 2006

冋

He et al.

册

1 ␤cl Hr = − A1clI1共u1clr兲 + C1clu1clI1⬘ 共u1clr兲 sin共␾兲, r w␮

再

H␾ = − A1clu1clI1⬘ 共u1r兲 −

u1cl

Hz = −

␤cl 1 w␮ r

Ez = 共A16兲

冎

u3cl

jw⑀3

C1clI1共u1r兲 cos共␾兲, 共A17兲

C1clI1共u1clr兲sin共␾兲

再

再

1

关C2clI1共u2clr兲 + D2clK1共u2clr兲兴 −

共A18兲

r

冎

␤cl w⑀2

u2cl关A2clI1⬘ 共u2clr兲

+ B2clK1⬘ 共u2clr兲兴 cos共␾兲,

再

共A19兲

−

Ez =

␤cl 1 w⑀2 r

u2cl

再

+

再

Hr = −

r

关A2clI1共u2clr兲

B2clK1共u2clr兲兴

D2clK1⬘ 共u2clr兲兴

再

冎

+

B2clK1共u2clr兲兴

冎

sin共␾兲,

+

␤cl w␮

共A20兲

共A21兲

共A22兲

␤cl 1 w␮ r

关C2clI1共u2clr兲 共A23兲

关C2clI1共u2clr兲 + D2clK1共u2clr兲兴sin共␾兲

再

1 r

关C3clI1共u3clr兲 + D3clK1共u3clr兲兴 −

冎

␤cl w⑀3

E␾ = − u3cl关C3clI1⬘ 共u3clr兲 + D3clK1⬘ 共u3clr兲兴 −

冎

+ B3clK1共u3clr兲兴 sin共␾兲,

2

jw␮

冋

关C3clI1共u3clr兲 + D3clK1共u3clr兲兴sin共␾兲

1 r

D4clK1共w4clr兲 −

␤cl w⑀4

冋

E␾ = − D4clw4clK1⬘ 共w4clr兲 −

w4cl

共A30兲

册

B4clw4clK1⬘ 共w4clr兲 cos共␾兲,

␤cl 1 w⑀4 r

册

B4clK1共w4clr兲 sin共␾兲,

共A25兲

␤cl 1 w⑀3 r

2

jw⑀4

B4clK1共w4clr兲cos共␾兲,

关A3clI1共u3clr兲 共A26兲

共A33兲

册

冋

1 ␤cl Hr = − B4clK1共w4clr兲 + D4clw4clK1⬘ 共w4clr兲 sin共␾兲, r w␮ 共A34兲

冋

H␾ = − B4clw4clK1⬘ 共w4clr兲 −

␤cl 1 w␮ r

册

D4clK1共w4clr兲 cos共␾兲, 共A35兲

Hz = −

u3cl关A3clI1⬘ 共u3clr兲

+ B3clK1⬘ 共u3clr兲兴 cos共␾兲,

再

共A29兲

共A24兲

inside the fiber cladding layer 共a1 艋 r 艋 a2兲, Er =

u3cl

2

jw␮

关C3clI1共u3clr兲

共A32兲

sin共␾兲,

冎

w␮ r

共A31兲

u2cl关C2clI1⬘ 共u2clr兲

+ D2clK1共u2clr兲兴 cos共␾兲,

Hz = −

Hz = −

Ez =

H␾ = − u2cl关A2clI1⬘ 共u2clr兲 + B2clK1⬘ 共u2clr兲兴 −

u2cl

冎

␤cl 1

+ D3clK1共u3clr兲兴 cos共␾兲,

Er =

关A2clI1共u2clr兲 + B2clK1共u2clr兲兴cos共␾兲,

1

共A28兲

H␾ = − u3cl关A3clI1⬘ 共u3clr兲 + B3clK1⬘ 共u3clr兲兴 −

2

jw⑀2

+

关A2clI1共u2clr兲

冎

inside the metal layer 共a2 艋 r 艋 a3兲, and

E␾ = − u2cl关C2clI1⬘ 共u2clr兲 + D2clK1⬘ 共u2clr兲兴

共A27兲

+ D3clK1⬘ 共u3clr兲兴 sin共␾兲,

inside the fiber core layer 共0 艋 r 艋 a1兲, Er =

关A3clI1共u3clr兲 + B3clK1共u3clr兲兴cos共␾兲,

␤cl 1 Hr = − 关A3clI1共u3clr兲 + B3clK1共u3clr兲兴 + u3cl关C3clI1⬘ 共u3clr兲 r w␮

2

jw␮

2

w4cl

2

jw␮

D4clK1共w4clr兲sin共␾兲

共A36兲

inside the analyte layer 共a3 艋 r兲. In Eqs. (A13)–(A36), A1cl, C1cl, A2cl, B2cl, C2cl, D2cl, A3cl, B3cl, C3cl, D3cl, B4cl, D4cl are field coefficients of cladding mode, and I1 and K1 are the modified Bessel function.

APPENDIX B: UNCONJUGATED FORM OF COUPLED-MODE EQUATIONS For an absorbing waveguide, the unconjugated form of the reciprocity theorem can be written as11

He et al.

⳵ ⳵z

冕

Vol. 23, No. 5 / May 2006 / J. Opt. Soc. Am. B

with 共E1 ⫻ H2 − E2 ⫻ H1兲 · zˆdA

K ␯−␮

A⬁

=

冕

ⵜ · 共E1 ⫻ H2 − E2 ⫻ H1兲dA,

共B1兲

w

A⬁

=

where the fields labeled 1 and 2 are two arbitrary electromagnetic fields that satisfy Maxwell’s equation. Further, if we regard these two arbitrary fields as two guided modes satisfying the boundary condition of the four-layer absorbing SPW sensor and follow a similar procedure of derivation as given in Ref. 11, the unconjugated form of orthogonality relation is obtained as

冕

Et␯ ⫻ Ht␮ · zˆdA =

A⬁

冕

Et␮ ⫻ Ht␯ · zˆdA = 0

A⬁

for ␤␯ ⫽ ± ␤␮ . 共B2兲 When the four-layer absorbing SPW sensor is broken, since a long-period fiber grating is written periodically by ultraviolet light into the core layer of an optical fiber, the guided mode in the broken structure can be expanded in terms of all the guided mode in unbroken fiber, viz., E1 =

兺 A 共z兲E ␯

␯

+

H1 =

⌬⑀ · ⑀ ⌬⑀ + ⑀

t␯共r, ␾兲exp共i␤vz兲

兺 ␯

兺 A 共z兲H ␯

␯

z␯共r, ␾兲exp共i␤␯z兲.

共B3兲

2. 3.

E2 = 关Et␮共r, ␾兲 − Ez␮共r, ␾兲兴exp共− i␤␮z兲

共B5兲

H2 = 关− Ht␮共r, ␾兲 + Hz␮共r, ␾兲兴exp共− i␤␮z兲.

共B6兲

6.

After substituting Eqs. (B3)–(B6) into Eqs. (B1) and using the unconjugated form of the orthogonality relation, Eqs. (B2), the unconjugated form of coupled-mode equations can be obtained as follows: =i

兺A K ␯

␯

␯−␮

冕

r共Ez␯Ez␮兲dA

A⬁

.

⬁

r共E␮r H␾␮ −

E␾␮H␮r 兲drd␾

0

*Corresponding author Y. L. Lo may be reached by e-mail at [email protected].

5.

dA␮

冕冕 2␲

共⌬⑀ + ⑀兲

The authors gratefully acknowledge the support provided to this study in part by the Ministry of Education’s Program for Promoting Academic Excellence of Universities under Grant A-91-E-FA08-1-4 and by the Advanced Optoelectronic Technology Center, National Cheng Kung University, under projects from the Ministry of Education and the National Science Council (NSC94-218-M-008009) of Taiwan.

For deriving the unconjugated form of the coupled-mode equations, we set

␯

A⬁

2

⌬⑀

Note that Eqs (B7) and (B8) are valid for both absorbing and nonabsorbing waveguides.

4.

␯

r共E␯r E␮r + E␾␯E␾␮兲dA −

REFERENCES

t␯共r, ␾兲exp共i␤vz兲

兺 A 共z兲H

冕

共B8兲

1.

A␯共r, ␾兲Ez␯共r, ␾兲exp共i␤vz兲,

⌬⑀

0

共B4兲

+

dz

811

7. 8. 9. 10.

exp关i共␤␯ − ␤␮兲z兴

共␯ and ␮ are any guiding modes in the fiber兲, 共B7兲

11.

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