Magnetically controllable wavelength-division-multiplexing fiber coupler

Wei Lin; Hao Zhang; Binbin Song; Yinping Miao; Bo Liu; Donglin Yan; Yange Liu

doi:10.1364/OE.23.011123

1. Introduction

As one of the most efficient methods to expand the transmission spectral window, wavelength division multiplexing (WDM) is one of the key technologies for optical communications as well as optical sensing systems [1]. Due to the great demands of data exchange applications in multi-channel optical networks, much efforts have been put on the development of multiplexing/demultiplexing components, including diffraction grating [2], dielectric film filter [3], fiber Bragg grating [4], long period fiber grating [5], and fiber coupler [6]. Amongst these components, fiber coupler plays a particularly significant role in building up all-fiber optical systems and have attracted increasing research interests in the past few decades [7~12]. The wavelength tunability of fiber couplers provides convenient, fiber-compatible and hence low loss approaches for wavelength channel selection and add/drop in fiber-optic systems, and they have found various applications in such ever-growing fields as fiber laser, fiber sensor, and fiber-optic communications technologies. In the past few years, wavelength tunability of fiber couplers has been fulfilled by employing the micro-mechanical platform or thermo-optical medium [13–17]. Fiber-optic couplers could be generally classified into two categories: the mechanically tunable [13, 14] and evanescent-field-assisted ones [15–17]. The latter would be of special attraction as the evanescent field is highly sensitive to the environmental medium. This property makes it possible to achieve a tunable WDM coupler with high performances by integrating the fiber-optic coupler with various functional materials.

As an attractive functional material, the magnetic fluid (MF) possesses many intriguing magneto-optical properties such as tunable refractive index [18], tunable transmission [19], tunable birefringence and dichroism [20], etc. By using these properties, a good variety of magneto-optical devices have been proposed, including magnetic field sensors [21–24], optical switches [25, 26], and optical modulators [27, 28], etc. The variation in refractive index of the MF could normally reach a magnitude of 10⁻² when external magnetic field is applied [29]. This property could be exploited to tune the coupling coefficient of fiber-optic couplers.

Actually some works have been engaged on the design of MF-based tunable couplers [30, 31]. These works aim to theoretically design the magnetically controlled coupler operating at one particular wavelength. In the work presented in this paper, we have theoretically proposed and experimentally validated a magnetically controllable wavelength-selective fiber coupler for WDM applications. Its wavelength tunability is achieved by integrating the fiber coupler with the MF. A theoretical model has been set up to analyze the operation principle of the proposed WDM coupler, which is verified by our experimental observation on the transmission spectral evolution in response to the applied magnetic field intensity. Our proposed magnetically controllable WDM coupler has high channel wavelength tunability and splitting ratio with high isolation, which ensures its applicability for potential applications in fiber laser and fiber-optic communications systems, as well as fiber sensing occasions. Furthermore, our proposed schemes also support the magnetically controllable mode division multiplexing by using the MF-integrated fiber coupler.

2. Theory

2.1 Fundamental principle

Several schemes have been proposed to analyze the coupling mathematically of tapered fiber couplers [10–12]. According to the fusion degree, the coupling region of the fiber coupler can be classified into weakly coupling and strong coupling regions, as shown in Fig. 1. Under weakly coupling condition, the two fibers are not fused together. When the light propagating through the two fibers is constrained inside the core area, the coupling between the two fibers is rather weak and could be actually neglected. As the fiber radius further reduces during the pulling process and the fiber core would no longer thoroughly constrain the light, and the fiber cladding would turn to serve as the waveguide core with ambient medium working as the waveguide cladding. In this case the coupling coefficient could be expressed as [12]:

C_{W C} = \frac{2}{r} \cdot {(\frac{Δ}{2 π D})}^{1 / 2} \cdot \frac{U_{\infty}}{V^{5 / 2} e^{V (2 D - 2)}} = \frac{λ}{2 π r^{2} n_{2} \sqrt{π D}} U_{\infty} V^{- \frac{3}{2}} e^{- V (2 D - 2)}

where, Δ = (n₂²-n₃²)/2n₂² and V = 2πr(n₂²-n₃²)^1/2/λ refer to the relative refractive index difference and normalized propagation constant, respectively. Here n₂ and n₃ represent the refractive indices of the cladding and the external medium, respectively, and r is the fiber radius which depends on the fusion degree and length of the tapering region. The eigenvalue U_∞, is 2.405 when the fundamental core mode is far from cut-off region. D is defined as D = d/2r to describe the fusion degree, where d is the inter-core distance.

Fig. 1 Schematic diagram of a symmetrical 2 × 2 tapered coupler.

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For the strong coupling case, the fibers are fused together and Eq. (1) is no longer valid. Under this condition, the coupler could be considered as a hybrid waveguide and the coupling coefficient can be calculated by half of the propagation constant difference between the even and odd modes. In other words, the fiber fusion region is simplified as a rectangular waveguide and the coupling coefficient can be described as [12]:

C_{S C} = \frac{β_{0} - β_{1}}{2} = \frac{3 π λ}{32 n_{2} r^{2}} \cdot \frac{1}{{(1 + 1 / V)}^{2}}

Therefore, the phase difference between the light at the two output ports can be calculated by integrating the coupling efficient along z-axis over the coupling region, as expressed by the following equation [12]:

\begin{array}{l} φ (λ, n_{3}) = \int_{W C} C_{W C} (λ, n_{3}, z) d z + \int_{S C} C_{S C} (λ, n_{3}, z) d z \\ \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} = φ_{W C} (λ, n_{3}) + φ_{S C} (λ, n_{3}) \end{array}

Thus the normalized powers of the output ports could be acquired using P_C (λ) = 1- sin² φ (λ, n₃) and P_D (λ) = sin² φ (λ, n₃) [12]. It should be noted that the transmission loss is neglected in the above calculation. However, with the reduction of fiber radius, stronger evanescent field would be excited and will be absorbed or scattered by ambient medium, causing considerable transmission loss when the light propagates though the fiber fusion region. Besides, the modal phase difference between the two fibers should be also taken into account as it is rather difficult to maintain identical fiber geometry during the pulling process. Considering these factors, the output powers normalized to the input power should be modified as:

P_{C} (λ) = 10^{- α} (1 - F \sin^{2} φ (λ, n_{3}))

P_{D} (λ) = 10^{- α} F \sin^{2} φ (λ, n_{3})

where, α = α₀ + α_ext (H) is the transmission loss introduced during the pulling process and the ambient medium, and F≡|κ|²/φ² (0≤F≤1) is the phase matching degree. Here, κ is the coupling coefficients between the two fibers. For an ideal fiber coupler, the phase is well matched and in practical applications it is reasonable to assume that F≈1. The performance of the coupler can be theoretically evaluated by the following parameters, including insertion loss L, additional loss L_add, and splitting ratio SR, which could be expressed below:

L_{X} = - 10 \lg \frac{P_{X}}{P_{i n}} = - 10 \lg P_{X} (λ); X = C or D

\begin{array}{l} L_{a d d} = - 10 \lg \frac{P_{o u t, t o t a l}}{P_{i n}} = - 10 \lg (P_{C} (λ) + P_{D} (λ)) \\ \begin{matrix}  \end{matrix} = α = - 10 \lg [10^{(- L_{C} / 10)} + 10^{(- L_{D} / 10)}] \end{array}

S R_{X} = \frac{P_{X}}{P_{o u t, t o t a l}} \times 100 % = 10^{- (L - L_{a d d}) / 10} \times 100 %; X = C or D

When sin² φ (λ, n₃) = 1, the minimum P_C (λ) could be reached while the maximum could be acquired when sin² φ (λ, n₃) = 0. Thus, for port C, the band-pass or band-rejection operations can be achieved when the following conditions are respectively satisfied:

φ (λ, n_{3}) = {\begin{matrix} π / 2 + k π \\ k π \end{matrix} \begin{matrix} \begin{matrix} \begin{matrix} band-rejection channel \\ band-pass channel \end{matrix} \end{matrix} & ; k is integer \end{matrix}

S R_{C - c h a n n e l} = {\begin{matrix} (1 - F) \times 100 % \\ 1 \end{matrix} \begin{matrix} \begin{matrix} band-rejection channel \\ band-pass channel \end{matrix} \end{matrix}

2.2 Theoretical analysis

As the applied magnetic field intensity varies, the refractive index and transmission loss of the magnetic fluid would change accordingly [18, 19]. And therefore, the phase difference of the coupler φ (λ, n₃), the phase matching degree F as well as the transmission loss α would change accordingly. Since the transmission loss increases with the increment of the magnetic field intensity, the additional loss would increase according to Eq. (7) [19]. While, the insertion losses are affected by the transmission loss α, the phase difference φ (λ, n₃) and the phase matching degree F. However, the channel splitting ratio SR only depends on the phase matching degree F.

Additionally, the channel wavelength would also shift with the variation in the refractive index of the magnetic fluid, and the corresponding sensitivity can be expressed as:

\frac{\partial λ}{\partial n_{3}} = - \frac{\frac{\partial φ}{\partial n_{3}}}{\frac{\partial φ}{\partial λ}} = \frac{- (\frac{\partial φ_{W C}}{\partial n_{3}} + \frac{\partial φ_{S C}}{\partial n_{3}})}{\frac{\partial φ_{W C}}{\partial λ} + \frac{\partial φ_{S C}}{\partial λ}}

According to Eqs. (1) and (2), it is easy to deduce that ∂φ_SC/∂n₃ <0, ∂φ_WC/∂λ>0 and ∂φ_WC/∂n₃>0. And in order to ensure good light constraint ability, the eigenvalues V of the tapered fiber should be larger than 2.045, which means that ∂φ_SC/∂λ >0. Thus, if weak coupling is the major factor that account for the coupling mechanism, ∂λ/∂n₃<0 and the channel wavelength would exhibit blue shift with the increment of the external refractive index. While when strong coupling plays more significant role, the channel wavelength would show red shift.

The splitting ratios (SR) as functions of operation wavelength for port A ~D have been calculated for different ambient refractive indices n₃, as shown in Fig. 2. It should be mentioned that in our calculation process, the strong coupling region is considered as a cylindrical waveguide with a waist radius of r_min and the fusion degree D as well as tapered fiber radius r are assumed to increase linearly with the increment of the distance from the taper waist. The lengths of weak coupling and strong coupling regions are l_wc = 23 mm and l_sc = 5mm, respectively. The waist radius r_min = 20 μm. The maximum and minimum of D are 2 and 1, respectively. The diameter of the un-tapered SMF is 125μm. The transmission loss α = 0 and the phase matching degree F = 1.

Fig. 2 Calculated splitting ratio as functions of operation wavelength for port A ~D when n3 = 1.4, 1.405, 1.41, 1.415 and 1.42.

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As shown in Fig. 2, the operation channel wavelength move towards longer wavelength region with the increment of n₃. According to [32], Chen et al…, the refractive index of the MF would increase with the increment of the applied magnetic field intensity. And consequently, as the applied magnetic field intensity increases, the operation channel wavelength would exhibit some red shift.

3. Experimental setup

Figure 3 shows the schematic diagram of the proposed magnetically controllable WDM coupler. It is fabricated by a homemade fiber coupler immersed into the MF (EMG605, Ferrotec) though a capillary. The fiber coupler is fabricated using a fiber taper machine (produced by E-Otron, Shanghai, China). Single-mode fibers (SMFs) with coating stripped off are wound together and stretched using the oxyhydrogen flame scanning method until the tapered section length is about 23 mm. The flame is about 5 mm in width and stands still during the tapering process. The coupler is kept straightly inside a capillary, and the MF is infiltrated into the capillary to completely immerse the WDM coupler. To prevent the MF from leaking off the silica tube, the capillary is sealed with paraffin at its both ends.

Fig. 3 Schematic diagram of the proposed magnetically controllable WDM coupler.

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A test system is employed to investigate the performances of the magnetically controllable WDM coupler under a room temperature of 19 °C. As shown in Fig. 4(a), the SMF-based magnetically controllable WDM coupler is spliced between a supercontinuum broadband light source (SBLS) and an optical spectrum analyzer (OSA: Yokogawa AQ6370C, operation wavelength ranges from 600 nm to 1700 nm) using a commercially available fiber fusion splicer. A supercontinuum broadband source based on the nonlinear spectral expansion of a segment of ~20 m photonic crystal fiber pumped by a 1060 nm pulsed laser is employed in out experiment. The SBLS is able to provide a broadband wavelength range from 600nm to 1700nm and its optical spectrum is shown in Fig. 4(b). A pair of electromagnets is used to apply external magnetic field perpendicular to the fiber axis. The electromagnets are placed oppositely with a distance of about 4 cm in order to apply uniform magnetic field between the electromagnets. The proposed WDM coupler is placed around the central region and its environmental temperature fluctuates less than 1.5 °C as the electromagnets keep operating for 60 mins. The intensity of the external magnetic field is measured by a Tesla meter with a resolution of 0.1 Oe. Figure 4(c) shows the additional loss and insertion loss of the WDM before and after immersion into the MF. It can be seen that four channels are present within the spectral range of 975nm to 1700nm. When the fiber coupler is immersed into the MF, all of these operation channels would move toward longer wavelength region and the insertion loss increases in the meanwhile, which implies that strong coupling should be the major factor that accounts for the coupling mechanism. The additional loss of the initial WDM α₀ introduced during the pulling process is around 0.6 dB, and it could be reduced by further improving the pulling technique. The additional loss increases when the coupler is immersed into the MF and its variation becomes larger in the longer wavelength region.

Fig. 4 (a) Experimental setup of the test system (b) Optical spectrum of the SBLS (c) Additional loss of the WDM before and after immersion into the MF; the inset shows the wavelength-dependent insertion loss of the WDM before and after the WDM is immersed into the MF for port A to port C and D, respectively.

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4. Experimental results and discussion

The magnetic field intensity could be controlled by tuning the voltage applied on the electromagnets. While the applied magnetic field intensity increases from 0 Oe to 500 Oe, all of the insertion losses for port A to port C and D (L_C and L_D), increase gradually with the increment of the magnetic field intensity, as shown in Fig. 5. The spectral channels of the magnetically controllable WDM coupler move toward longer wavelength region, as shown in inset (a) and (b) of Fig. 5, respectively. According to Eq. (7), the additional loss L_add can be calculated. This figure indicates that L_add increases with the increment of applied magnetic field intensity which is in agreement of our theoretical analysis.

Fig. 5 Additional loss as functions of wavelength for different applied magnetic field intensities; insets show the insertion loss in response to the applied magnetic field intensity for (a) port A to port C, (b) port A to port D, respectively.

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The splitting ratio SR could be calculated according to Eq. (8) and it is obvious that SR_C + SR_D = 1. Figure 6 gives the calculated SRs under different magnetic field intensities. It can be seen that the SRs of the magnetically controllable WDM coupler exhibits gradual red shift behavior when the applied magnetic field intensity increases, which is in agreement with the above theoretical analysis. The difference between the calculated SR as functions of operation wavelength and the experimental measurement results may be caused by the above assumptions and the deviation of the calculation parameters from the actual fabricated WDM coupler geometry.

Fig. 6 (a), (b) Wavelength-dependent splitting ratio of the magnetically controllable WDM coupler under different magnetic field intensities for port A to C and port A to D respectively

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Four channels from port A to port C are selected to evaluate the wavelength tunability of the magnetically controllable WDM coupler. Channel 1 and 3 are band-rejection channels, while Channel 2 and 4 correspond to band-pass channels. The dependences of the selected channel wavelengths on the applied magnetic field intensity have been experimentally investigated, as illustrated in Fig. 7. For band-rejection channels, as the applied magnetic intensity increases from 0 Oe to 500 Oe, the channel wavelength moves from 1069.9 nm to 1080.3 nm for Channel 1and its SR varies from 0.349% to 0.766%. And for Channel 3, the channel wavelength increases from 1400.5 nm to 1417.9nm and its SR increases from 0.283% to 5.6%. The respectively wavelength of the band-pass channels increase by 13.6 nm and 24 nm within an applied magnetic field intensity range from 0 Oe to 500 Oe for Channel 2 and 4. And their SRs decrease form 99.875% to 96.981% and 98.242% to 82.153% respectively. The channel wavelengths show red shift and exhibit the Langevin-function-governed nonlinear behavior, which is similar to the results in in Hong’s Work [33]. The channel wavelengths show little shift when the applied magnetic field intensity is lower than 25 Oe or larger than 250 Oe, while exhibit considerable shift as the applied magnetic field intensity increases from 25 Oe to 250 Oe. This phenomenon could attribute to the initial magnetization and saturation magnetization of the MF. According to the above theoretical analysis, the shift directions of the channel wavelength indicate that strong coupling plays a significant role in the WDM coupling mechanism. In addition, the asymmetry of the coupler increases in a high external magnetic field owing to the influence of the birefringence of MF, which induces the reduction of the phase matching degree F. And as a result, according to Eq. (10), the SR of the band-rejection channels for port C increases with the increment of applied magnetic field intensity. However, for band-pass channels, the SRs would decrease with the increment of the magnetic field intensity, which reflects the influence of the polarization-dependent property of our proposed coupler. When no magnetic field is applied, the polarization property of the coupler is very low and SR of port C only depends on sin² φ (λ, n₃). When the applied external magnetic field increases, the increasing birefringence of the surrounding MF would enhance the polarization-dependent property of the coupler, causing the increment of the phase differences along two polarization directions: Δφ (λ, n₃) = φ_x (λ, n₃)- φ_y (λ, n₃). As a result, according to Lacroix’s work, SRs of the band-pass channels would decrease with the increment of the applied magnetic field intensity [34].

Fig. 7 Wavelength and Splitting ratio as functions of the applied magnetic field intensity for the four selected channels

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Considering the influences of additional losses on its practical performances, the proposed tunable WDM could operate with high spectral quality for a magnetic field intensity ranging from 25 Oe to 125 Oe. Compared with the results shown in Figs. 7(c) and 7(d), the SRs of Channel 1 and 2 would be more stable than those of Channel 3 and 4. Thus, Channel 1 and 2 are selected to evaluate the wavelength tunability of the proposed WDM within the above range, as shown in Fig. 8.

Fig. 8 Channel wavelength as applied magnetic field intensity for the two selected channels within a magnetic field intensity ranging from 25 Oe to 125 Oe; Inset (a) shows the additional losses of the two channels within this range, black symbol: channel 1, red symbol: channel 2; Inset (b) shows the isolations for these two channels, black symbol: channel 1, red symbol: channel 2.

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Figure 8 illustrates the wavelength tunability of the magnetically controllable WDM coupler for a magnetic field intensity range of 25 Oe to 125 Oe. As shown in Fig. 8, the wavelength shifts increase from 0.6 nm to 5.3 nm and from 0.3 nm to 7.6 nm for Channel 1 and 2, respectively. And their wavelength sensitivities reach 0.05 nm/Oe and 0.0744 nm/Oe, respectively. The magnetic field sensitivity and the wavelength tuning range of the WDM coupler could be further improved by using the MF with higher concentration whose refractive index is closer to silica. Inset (a) illustrates the additional loss as functions of the applied magnetic field intensity for the two selected channels. Within this range, the additional losses increase from 2.263 dB to 2.840 and 1.616 dB to 2.578 dB with the increment of magnetic field intensity for Channel 1 and Channel 2, respectively. The additional losses of the channels are lower than 3 dB and could be further reduced by using the MF with lower concentration. However, this method may sacrifice the wavelength tunability of the proposed tunable WDM. Inset (b) of Fig. 8 shows the isolations between the selected channels, which could be calculated by Iso = 10∙|lg (SR/(1-SR))|. From 25 Oe to 125 Oe, the isolation varies only a little around 24.2 dB for Channel 1, while it decreases from 29.168 dB to 24.089 dB for Channel 2. These results indicate that the channels of the tunable WDM could operation with high isolation within this particular magnetic field intensity range.

The temperature effect on the proposed WDM coupler has also been experimentally investigated, as shown in Fig. 9. It could be seen that the operation channel wavelength shifts a little with the increment of environmental temperature. As temperature increases from 20 °C to 40 °C, a wavelength shift of only 1 nm occurs. And in considerations of the experimentally measured magnetic field sensitivities of 0.05 nm/Oe and 0.0744 nm/Oe for Channel 1 and Channel 2 and the environmental temperature fluctuation of less than 1.5 °C, the temperature-induced uncertainty of the applied magnetic field intensity should be no more than 1.5 Oe. This indicates that the proposed WDM coupler could operate with low temperature effect.

Fig. 9 Temperature response of the selected two channels, black squares: Channel 1, red circles: Channel 2; the inset shows the wavelength-dependent splitting ratios of the WDM coupler for port A to port C under different temperatures.

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5. Conclusion

In this paper, a magnetically controllable WDM fiber coupler has been theoretical analyzed and experimentally demonstrated. The WDM coupler is fabricated by immersing the fiber fused coupler into the MF. The tunable properties of the WDM coupler have been demonstrated for a magnetic field intensity from 0 Oe to 500 Oe. The tunable WDM could particularly operate with high spectral quality from 25 Oe to 125 Oe. The high wavelength sensitivities of 0.05 nm/Oe and 0.0744 nm/Oe have been achieved for Channel 1 and Channel 2, respectively, and the channel isolations are higher than 24.089 dB in this range. Besides, the number of operation channels and the wavelength tunability of the WDM coupler could be controlled by adjusting the tapering length and the flame scanning length during the fiber tapering process. As an important photonic component, our proposed tunable WDM coupler would find potential applications in fiber laser and optical communications systems as well as fiber sensing occasions. And moreover, the employed tunable coupling approach presented in this work would be valuable for the design of magnetically controllable fiber couplers for mode-division-multiplexing applications.

Appendix

An analysis on the contributions of weakly as well as strong couplings in the proposed fiber fused coupler to the channel wavelength shift would be presented in the following section. The channel wavelength tunability in response to ambient refractive index can be calculated according to Eq. (11). Before that, it is necessary to acquire ∂φ_WC/∂λ, ∂φ_SC/∂λ, ∂φ_WC/∂n₃ and ∂φ_SC/∂n₃. According to Eqs. (1) and (2), their respective derivations could be described as following equations:

\begin{array}{l} \frac{\partial φ_{W C}}{\partial λ} = \frac{\partial}{\partial λ} \int_{W C} C_{W C} (λ, n_{3}, z) d z = \int_{W C} \frac{\partial C_{W C} (λ, n_{3}, z)}{\partial λ} d z \\ = \int_{W C} {\frac{U_{\infty}}{2 π r^{2} n_{2} \sqrt{π D}} V^{- \frac{3}{2}} e^{- V (2 D - 2)} \\ + \frac{U_{\infty} λ}{2 π r^{2} n_{2} \sqrt{π D}} [- \frac{3}{2} V^{- \frac{5}{2}} e^{- V (2 D - 2)} - V^{- \frac{3}{2}} (2 D - 2) e^{- V (2 D - 2)}] \frac{\partial V}{\partial λ}} d z \\ = \int_{W C} \frac{U_{\infty}}{2 π r^{2} n_{2} \sqrt{π D}} [\frac{5}{2} V^{- \frac{3}{2}} e^{- V (2 D - 2)} + V^{- \frac{1}{2}} (2 D - 2) e^{- V (2 D - 2)}] d z \end{array}

\begin{array}{l} \frac{\partial φ_{S C}}{\partial λ} = \frac{\partial}{\partial λ} \int_{S C} C_{S C} (λ, n_{3}, z) d z = \int_{S C} \frac{\partial C_{S C} (λ, n_{3}, z)}{\partial λ} d z \\ = \int_{S C} [\frac{3 π}{32 n_{2} r^{2}} \cdot \frac{V^{2}}{{(V + 1)}^{2}} + \frac{3 π λ}{32 n_{2} r^{2}} [2 V {(V + 1)}^{- 2} - 2 V^{2} {(V + 1)}^{- 3}] (\frac{\partial V}{\partial λ})] d z \\ = \int_{S C} [\frac{3 π}{32 n_{2} r^{2}} \cdot \frac{V^{2} (V - 1)}{{(V + 1)}^{3}}] d z \end{array}

\begin{array}{l} \frac{\partial φ_{W C}}{\partial n_{3}} = \frac{\partial}{\partial n_{3}} \int_{W C} C_{W C} (λ, n_{3}, z) d z = \int_{W C} \frac{\partial C_{W C} (λ, n_{3}, z)}{\partial n_{3}} d z \\ = \int_{W C} \frac{λ}{2 π r^{2} n_{2} \sqrt{π D}} U_{\infty} [- \frac{3}{2} V^{- \frac{5}{2}} - V^{- \frac{1}{2}} (2 D - 2)] e^{- V (2 D - 2)} \cdot (\frac{\partial V}{\partial n_{3}}) d z \\ = \int_{W C} \frac{2 π n_{3} U_{\infty}}{λ n_{2} \sqrt{π D}} [\frac{3}{2} V^{- \frac{7}{2}} + V^{- \frac{3}{2}} (2 D - 2)] e^{- V (2 D - 2)} d z \end{array}

\begin{array}{l} \frac{\partial φ_{S C}}{\partial n_{3}} = \frac{\partial}{\partial n_{3}} \int_{S C} C_{S C} (λ, n_{3}, z) d z = \int_{S C} \frac{\partial C_{S C} (λ, n_{3}, z)}{\partial n_{3}} d z \\ = \int_{S C} \frac{3 π λ}{32 n_{2} r^{2}} [2 V {(V + 1)}^{- 2} - 2 V^{2} {(V + 1)}^{- 3}] (\frac{\partial V}{\partial n_{3}}) d z \\ = - \int_{S C} \frac{3 π^{3} n_{3}}{4 n_{2} λ} \frac{1}{{(V + 1)}^{3}} d z \end{array}

It is easy to find that ∂φ_SC/∂n₃ is less than 0. In addition, D should be higher than 1 because the two fibers are spatially separate. Therefore, we have ∂φ_WC/∂λ>0 and ∂φ_WC/∂n₃>0. Under the strong coupling condition, in order to maintain good light constraint ability, the eigenvalue V of the tapered fiber should be larger than 2.045, and ∂φ_SC/∂λ is larger than 0. Hence according to Eqs. (11)-(15), the channel wavelength would shift toward longer wavelength region with the increment of ambient refractive index when -∂φ_SC/∂n₃>∂φ_WC/∂n₃, while it will turn out blue shift when -∂φ_SC/∂n₃<∂φ_WC/∂n₃. In addition, the tapered fiber radius can be calculated according to Yang’s work reported in [10], Yang et al…, where both of the strong coupling and weakly coupling lengths were measured. And thus the wavelength tunability of the WDM coupler can be completely calculated.

Acknowledgments

This work was jointly supported by the National Natural Science Foundation of China under Grant Nos. 61377095, 61322510, 11274182, and 11204212, the 863 National High Technology Program of China under Grant No. 2013AA014201, Key Natural Science Foundation Project of Tianjin (Grant No.13JCZDJC26100), China Postdoctoral Science Foundation Funded Project under Grant No. 2012M520024, and the Fundamental Research Funds for the Central Universities.

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Magnetically controllable wavelength-division-multiplexing fiber coupler

Abstract

1. Introduction

2. Theory

2.1 Fundamental principle

2.2 Theoretical analysis

3. Experimental setup

4. Experimental results and discussion

5. Conclusion

Appendix

Acknowledgments

References and links

Cited By

Figures (9)

Equations (15)

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