計畫成果自評

本子計畫欲完成之目標主要分成兩大項：一、980nm/1300-1600nm 之光纖耦合器。二、

寬頻增益平坦技術。第一項在理論模擬分析及豐富之實作經驗支援下順利達成目標：訊號 光源在 300nm 的頻寬下，穿透損耗(IL)小於 0.5dB 同時極化效應影響因子—極化相關損失 (PDL)亦小於 0.5dB；同時 980nm 之光源之耦合率亦達 75%以上。第二項寬頻增益平坦技術 係利用長週期光纖光柵技術透過理論分析可成功將增益頻譜平坦化，理論部分已臻完善唯 實驗上之實作有待未來實行。

以下為在本計畫支持下所得成果發表支論文列表同時附於本成果報告書之附錄中。

1. Tzong-Lin Wu and Hung-Jiun Ou,"Rigorous Analysis of Polarization Dependence Loss (PDL) for Equilateral 3×3 Fused Fiber Couplers," IEEE Photonics Technology Letters (SCI, NSC92-2215-E-110-005), Vol. 16, No.

1, pp. 165 - 167, Jan. 2004.

2. Tzong-Lin Wu and Hung-Jiun Ou,"A Vector Power Coupling Model for Analyzing Polarization Dependent Loss (PDL) of Equilateral Triangular 3x3 Weakly Fused Fiber Couplers,"Optics Communications (SCI, NSC 91-2215-E-110-018), Vol. 224, pp. 81-88, Aug. 2003.

3. Tzong-Lin Wu and Chia-Hsin Chao,"3D Electromagnetic Modeling of Polarization-dependent Coupling Characteristics of 1×3 Linear Array Weakly Fused Fiber Couplers," Fiber and Integrated Optics (SCI, NSC91-2215-E-110-018), Vol. 22, No. 6, pp. 415-432, Nov. 2003.

成果論文

附錄一

Rigorous Analysis of Polarization Dependence Loss (PDL) for Equilateral 3×3 Fused Fiber Couplers

Tzong-Lin Wu, Member, IEEE, and Hung-Jiun Ou

Abstract -- A rigorous power coupling model for weakly fused 3×3 triangular fiber couplers is proposed to investigate the polarization dependent loss (PDL) of the couplers. The accuracy of the power coupling model is checked by comparing with the experimental results. The agreement between them is reasonably good. The effect of fabricating parameters of the coupler, fusion degree and heated length, on the PDL of the coupler is investigated by combining the Mueller matrix method into the proposed model. It is found the fusion degree is the dominant factor to influent the PDL performance of the coupler. The PDL significantly increases as the coupler is fused in weakly fused condition and the fusion degree is suggested to be less than 0.95 for designing a low-PDL 3×3 coupler.

Index Terms – Equilateral 3 × 3 fused fiber coupler, polarization effect, polarization-dependent-loss, optical waveguide theory, optical fibers.

This work is supported by the National Science Council of the Republic of China under the grand NSC92-2215-E-110-005.

T. L. Wu and H. J. Ou are with the Department of Electrical of Engineering, National Sun Yat-Sen University, Kaohsiung 80424, Taiwan, R.O.C.

I. Introduction

Fused fiber-optic couplers are the key passive components in many communication and sensor applications. The 3× 3 fused couplers with three fibers in equilateral triangular configurations have attracted much attention because of their 120 degree of rotational symmetry.

Besides employed in the application of gyroscopes and interferometers [1], [2], the equilateral triangular 3×3 coupler can also be used in the design of power splitters, all-fiber switches, or ring resonators/filters for wavelength division multiplexing (WDM) applications [3]. With the trend of increasing transmission capacity in dense WDM system, the PDL of the fiber-optic components is becoming one of the major sources of pulse distortion that can increase the system bit-error-rate.

It has been noticed that the fusion degree of the fused type coupler plays an important role on the polarization-dependent coupling characteristics [4], [5]. Therefore, developing a theoretical model, which can predict the PDL of the equilateral 3×3 fused coupler in different fusion degree, is important for designing such kind of couplers with low polarization effects.

Several works have discussed the propagation constants and mode field patterns of the vector normal modes of the equilateral 3×3 couplers either based on perturbation method [6] or circular harmonic expansion method [7], but the power coupling behavior for the couplers are not considered. To our best knowledge, most investigations on power coupling characteristics are based on couple-mode theory [3] and both the state of the polarization (SOP) of input light and the birefringence of the couplers are not considered. The purpose of this paper is to provide a simple but rigorous power coupling model to analyze the polarization-dependent coupling characteristics of the couplers. Based on the surface integral equations method (SIEM) [4], the propagation characteristics of six vector coupling modes are solved. By considering the mode interference between those six coupling modes, the power coupling formulations are derived with the SOP of input light being concerned. The accuracy of the proposed power coupling model is reasonably good compared with the measured results presented in [2]. Furthermore, by employing the Mueller matrix method [8], the PDL of the 3×3 couplers can be simulated and the effect of the designing parameters, such as fusion degree and heated length, on the PDL can be studied.

II. POWER COUPLING MODLE

Fig. 1 shows a sketch of a 3×3 weakly fused coupler with equilateral cross-section, where

n1 and n₂ are the refractive indices of the fiber-cladding and the surrounding medium, r is the radius of the fiber cladding, d is the distance between two fiber centers, and z is the direction of the power propagation. The fusion degree of the coupler is defined asD d r≡ 2 . According to the full-wave numerical analysis of SIEM, the six electrical field patterns of the coupling normal modes and their corresponding propagation constants can be rigorously solved. To establish a power coupling model with the SOP of input light being considered, it is assumed that the power coupling behavior for these normal modes is dominantly decided by the vectorial fields at the centers of the fibers. Fig. 2(a) and 2(b) show the simplified field patterns at the center of each fiber. The arrows represent the orientation and relative strength of the electric fields at the centers of fibers. These six modes are classified to two types. One type consists of three modes, as shown in Fig. 2(a), with x-polarized E-field in fiber 1 and their corresponding propagation constants are β_m^x (m = 1, 2, or 3). The other type, as shown in Fig. 2(b), is the modes with y -polarized E-field in fiber 1 and their propagation constants are denoted as β_m^y (m = 1, 2, or 3). All six modes are excited as an arbitrary SOP of input light is launched into the coupler. The higher-order modes are neglected because it is assumed that the effect of these modes on the power coupling behavior is small.

Due to the rotational symmetry and the orthogonality between the normal modes of the

equilateral 3 × 3 coupler, it can be shown

thatθ₁₁^x = θ₁₂^x =θ₁₃^x =θ₂₁^x =θ₃₁^x = 0 ,θ22^x = −

( )

2 3 π ,θ32^x =

( )

2 3 π ,θ23^x =

( )

2 3 π ,θ33^x = −

( )

2 3 π ,

11

θy =θ₁₂^y =θ₁₃^y =θ₂₁^y =θ₃₁^y =π 2,θ22^y = −

( )

5 6 π ,θ32^y = −

( )

1 6 π ,θ23^y = −

( )

1 6 π ,θ33^y = −

( )

5 6 π . Consider a 3×3 triangular fiber coupler that is launched by a polarized light into fiber 1. The input light of arbitrary SOP can be expressed in the form of electric field at the center of fiber 1 as

(1) , where C and _a C are amplitude of x- and y-polarized components, respectively, and _b δ is the phase difference between these two components. Four kinds of polarization state, x-polarization, y-polarization, linear 45° polarization, and right-hand circular polarization, are needed in Mueller matrix method for obtaining the PDL of the coupler [8]. These four states can be represented by (1) withC_b = and0 δ =0, 0C_a = andδ =0, C_a =C_b andδ =0, C_a =C_b andδ = −π / 2, respectively. The PDL of the coupler is defined as the ratio between the maximum and minimum transmitted power at the output port for all possible SOP of input light and is normally expressed in dB. By employing the Mueller matrix method, the PDL can be calculated with only knowing the output power behavior of those four polarization states.

According to the discussion of source excitation in the previous paragraph, three x-polarized modes (β_m^x-modes) and y-polarized modes (β_m^y-modes) will be excited by the input light (1) at z

= 0 and propagate along z with different propagation velocity. Theβ_m^x-modes with excited amplitude C has phase delay _a δ from theβ_m^y- modes with amplitudeC . The power guided by _b the individual fiber is defined by the real part of the Poynting power at the center of the fiber. By the definition of the Poynting power, the normalized output power at fiber n (n = 1, 2, or 3) could be derived as

(2a)

1xy

ψ =^cos

(

^∆φ12xy+δ

)

^+cos

(

^∆φ21xy+δ

)

^+cos

(

^∆φ33xy+δ

)

^{, ψ =2xy cos}

(

^{∆φ13xy +δ}

)

⁺

(

²²

) (

³¹

)

cos ∆φ^xy +δ +cos ∆φ^xy+δ , (∆φ_ij^pq =

∫

β_i^p−β^q_j)dz^, p and q = x or y , i and j = 1, 2, or 3. In the deviation of (2), the symmetry and degeneracy properties of the six vector modes, i.e. β₁^x =β₁^y and β₂^x =β₃^y, is considered [7]. As shown in (2), the accumulated phase differences between any two vector modes are the dominant factor to influence the coupling behavior of the coupler. It is notable that the power conservation at output fibers is still valid in the proposed model, i.e., P P₁+ ₂+P₃ = . The exponential taper profile of the fused couplers 1 with a waist of uniform dimension is used in our 3D modeling [5].

III. RESULTS AND DISCUSSIONS

To understand the accuracy of the proposed model, the measured power coupling behavior of the weakly fused 3×3 coupler fabricated by Birks [2] is compared with our simulation results. The coupler was designed for 1/3 power divider at wavelength about 1290nm. Fig. 3 shows both measured and simulated output power ratio for wavelength from 1200nm to 1600nm at three output ports of the equilateral coupler.

The coupler parameters of n₁ = 1.45, n₂ = 1.0, D = 0.99, and heated length 6mm are used in the model. The input light is assumed unpolarized. As shown in Fig. 3, the agreement between the simulation and experiment is favorably good. It is seen that the model well predict the output power behavior of port1 within the 400nm wavelength range, but the theoretical wavelength responses of output power at port2 and port3 has some discrepancy with the measured results. As the model predicted and shown in Fig. 3, the output power ratios for these two ports have to be identical for the unpolarized input light due to the reflection symmetry of the equilateral three fiber couplers. However, it is seen that measured output power at port2 and port3 are not identical. As explained in [2], the unequal power coupling ratio between these two ports could be explained by the asymmetry of the cross-section or the helicity (or twist) of the whole component in the manufacturing process [2].

Fig. 4 shows the PDL variation versus heated length of the coupler for different fusion degreeD d r≡ 2 . Each point on the curves represents a 1/3 power divider with different coupler design parameters, fusion degree and heated length. It is seen that the PDL decreases as the heated length increases for all the fusion degrees, but the variation is more significant (about 0.1dB) for the coupler with larger D (weaker fused condition, 0.99) than for the one with smaller D. Another point worth to noting is that PDL is quite dependent on the fusion degree of the coupler. As shown in Fig. 4, the PDL is above 0.5dB for the coupler of D = 0.99 but well below 0.2dB for the coupler in stronger fused condition (D = 0.95). The fusion degree has to be, for

example, less than 0.95 if the coupler should be designed with PDL lower than 0.15dB.

Ⅳ. CONCLUSION

A rigorous power coupling model for weakly-fused 3×3 triangular fiber couplers has been established to investigate the power transfer characteristics of the couplers with considering the SOP of the input light. The accuracy of the proposed model is checked by comparing with the measured wavelength response in previous literature. The agreement between them is reasonably good. The PDL of the triangular 3×3 coupler has been studied by combining the Mueller matrix method into the power coupling model. It has been found the fusion degree is the dominant factor to influent the PDL performance of the coupler. The PDL significantly increases as the coupler is fused in weaker fusion condition (larger D). From designing point of view, it is suggested to design the fusion degree less than 0.95 for achieving low-PDL equal power divider.

REFERENCES

[1] G., Trommer, "Wavelength dependence of 3×3 fibre couplers for gyroscope applications,"

Electron Lett., vol. 25, pp. 944-945, 1989.

[2] T. A. Birks, "Effect of twist in 3×3 fused tapered couplers," Appl. Opt., vol. 31, pp.

3004-3014, 1992.

[3] Y. H. Ja, "Performance parameters of a wavelength-division demultiplexer made with a single 3×3 coupler optical fiber ring or loop resonator," J. Lightwave Technol., vol. 11, pp.

1337-1343, 1993.

[4] T. L. Wu, and H. C. Chang, "Rigorous analysis of form birefringence of weakly fused fiber-optic couplers," J. Lightwave Technol., vol. 13, pp. 687-691, 1995.

[5] T. L. Wu, "Three-dimensional electromagnetic modeling of fiber-core effects on the coupling characteristics of weakly fused tapered fiber-optic couplers," J. Lightwave Technol., vol. 18, pp. 1024-1030, 2000.

[6] A. J. Stevenson, and J. D. Love, "Vector modes of six-port couplers," Electron. Lett., vol. 23, pp. 1011-1013, 1987.

[7] H. S. Huang, and H. C. Chang, "Analysis of equilateral three-core fibers by circular harmonics expansion method," J. Lightwave Technol., vol. 8, pp. 945-952, 1990.

[8] Y. Zhu, E. Simova, P. Berini, and C. P. Grover, "A comparison of wavelength dependent polarization dependent loss measurements in fiber gratings," IEEE Trans. Instrumentation and Measurement, vol. 49, pp.1231-1239, 2000.

FIGURE CAPTIONS

Figure 1 The sketch of a 3×3 weakly fused coupler with equilateral cross-section.

Figure 2 The simplified field patterns of (a) three modes with _x-polarized E-field in fiber 1, and (b) three modes with y -polarized E-field in fiber 1.

Figure 3 Comparison of the wavelength responses between the simulated results and the measured results.

Figure 4 The PDL variation versus heated length of the coupler for different fusion degree.

Fig.1

Fig.2(a)

Fig.2(b)

Fig.3

Fig.4

附錄二

I. A Vector Power Coupling Model for Analyzing Polarization Dependent Loss (PDL) of Equilateral Triangular 3

×

3 Weakly Fused Fiber Couplers

1. Tzong-Lin Wu*, Hung-Jiun Ou

Department of Electrical Engineering, National Sun Yat-sen University, 70 Lien-Hai Rd., Kaohsiung, 80424, Taiwan.

Abstract

A rigorous power coupling model for weakly fused 3×3 triangular fiber couplers is proposed based on a full-wave surface integral equation method. In addition to the birefringence effect of the coupler considered in the proposed model, the influence of the state of the polarization (SOP) of the input light on the coupling behavior of the 3×3 triangular couplers can also be simulated.

The accuracy of the power coupling model is checked by comparing with the experimental results.

The agreement between them is reasonably good. The effect of fabricating parameters of the coupler, fusion degree, and heated length on the polarization dependent loss (PDL) of the coupler is investigated by combining the Mueller matrix method into the power coupling model. It is found that the fusion degree is the dominant factor to influence the PDL performance of the coupler. The PDL significantly increases as the coupler is fused in weakly fused condition, and the low-PDL coupler can be achieved by stronger fused couplers.

Author Keywords: Polarization dependent loss (PDL); 3×3 triangular fused fiber coupler;

polarization effect; optical waveguide theory; optical fibers.

PACS classification codes: 42.25.Ja; 42.82.Et

1. Introduction

Fused fiber-optic couplers are the key passive components in many communication and sensor applications [1]. The 3×3 fused couplers with three fibers in equilateral triangular configurations have attracted much attention because of their 120 degree of rotational symmetry. In addition to being employed in the application of gyroscopes and interferometers [2]–[4], the equilateral triangular 3×3 coupler can also be used in the design of power splitters, all-fiber switches, or ring resonators/filters for wavelength division multiplexing (WDM) applications [5]-[8]. With the trend of increasing transmission capacity in dense WDM system, the polarization dependent loss (PDL) of the fiber-optic components is becoming one of the major sources of pulse distortion that can increase the system bit-error-rate. It has been noticed that the degree of fusion of the fused type coupler plays a dominant role on the polarization-dependent coupling characteristics [9]-[11].

Therefore, developing a theoretical model which can predict the PDL of the equilateral 3×3 fused coupler in different fusion degree, is important for designing such couplers with low polarization effects.

Several works have discussed the propagation constants and mode field patterns of the vector normal modes of the equilateral 3×3 couplers, based on either the perturbation method [12] or the circular harmonic expansion method [13], but the power coupling behavior for the couplers are not considered. To our best knowledge, most investigations on power coupling characteristics are based on couple-mode theory [6], [7] and both the polarization state of input light and the polarization effect of the couplers are not considered. The purpose of this paper is to provide a simple yet rigorous power coupling model to analyze the polarization-dependent coupling characteristics of the 3×3 couplers. Based on the surface integral equations method (SIEM) [9], [14], the propagation characteristics of six vector coupling modes are solved. By considering the mode interference between those six coupling modes, the power coupling formulations are derived with the input state of polarization (SOP) being concerned. The accuracy of the proposed power coupling model is reasonably good compared with the measured results presented in [5]. Furthermore, by employing the Mueller matrix method [15], [16], the PDL of the 3×3 couplers can be simulated.

The theoretical details of this power coupling model are given in Section II. In Section III, the simulated results are presented and discussed. The conclusion is drawn in Section IV.

2. 3-D Power coupling formulations

(A) Vector coupling modes and excitations

Fig. 1 shows a sketch of the cross-section of the 3×3 triangular fiber couplers, where

1 1.45

n = and n₂ =1.0 are the refractive indices of the fiber-cladding and the surrounding medium, r is the radius of the fiber cladding, d is the distance between two fiber centers, and z is the direction of the power propagation. Because the diameters of both the fiber-core and fiber-cladding are decreased in the waist region due to the tapering process of fused coupler, the fiber-cores could be ignored, and power is strongly guided between the fiber-cladding and the surrounding medium, which is air unless some packaging material is used [17]. According to the full-wave numerical analysis of the SIEM, the six electrical field patterns of the coupling normal modes are shown in Fig. 2 with the degree of fusion D d r≡ 2 =0.99 and the normalized frequencyV ≡

(

2π λr

)

n1²−n2² =80. The arrows represent the orientation and relative strength of the electrical fields at the roots of the arrows. These six modes are classified into two types. Type Ⅰ consists of three modes, as shown in Fig. 2(a), with x-polarized E-field in fiber 1, and their corresponding propagation constants are denoted as β_m^x (m = 1, 2, or 3). Type Ⅱ, as shown in Fig. 2(b), is the modes with y -polarized E-field in fiber 1, and their propagation constants are denoted as β_m^y (m = 1, 2, or 3). As shown in Fig. 3(a), the x-polarized input light at fiber 1 could be considered as the superposition of equal amplitude of

those three modes of Type I. Similarly, as shown in Fig. 3(b), the y -polarized input light at fiber 1 could be considered as the superposition of equal amplitude of those three modes of Type Ⅱ.

(B) Power coupling formulations

To establish a power coupling model with the polarized input light being considered for 3×3 triangular fiber coupler, it is assumed that the power coupling behavior for these normal modes is dominantly decided by the vectorial fields at the centers of the fibers. Therefore, as shown in Fig. 4(a) and 4(b), the simplified field patterns only describes the direction and amplitude of the E-field at the center of each fiber are used to represent the mode patterns in Fig.

To establish a power coupling model with the polarized input light being considered for 3×3 triangular fiber coupler, it is assumed that the power coupling behavior for these normal modes is dominantly decided by the vectorial fields at the centers of the fibers. Therefore, as shown in Fig. 4(a) and 4(b), the simplified field patterns only describes the direction and amplitude of the E-field at the center of each fiber are used to represent the mode patterns in Fig.

在文檔中 高速寬頻高密度分波多工系統模組之研製與應用---子計畫III：摻鉻光纖放大器增益平坦化及寬頻耦合器最佳化設計及研製(III)The Optimum Design and Fabrication of Gain Flattened Chromium Doped Fiber Amplifier and Broadband Coupler (III) (頁 37-77)