High performance micro-fiber coupler-based polarizer and band-rejection filter

doi:10.1364/OE.20.017258

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High performance micro-fiber coupler-based polarizer and band-rejection filter

Jianhui Yu, Yao Du, Yi Xiao, Haozhi Li, Yanfang Zhai, Jun Zhang, and Zhe Chen »View Author Affiliations

Optics Express, Vol. 20, Issue 15, pp. 17258-17270 (2012)
http://dx.doi.org/10.1364/OE.20.017258

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▸ Article Outline
– Abstract
1. Introduction
2. Theoretical analysis and design
3. Fabrication of the coupler-based polarizer
4. Experimental results and discussions
5. Conclusions
– References and links

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Abstract

Using full vector finite element method and super-mode theory, we analyzed the feasibility to fabricate micro-fiber-coupler-based optical polarizer. Our theoretical analysis showed that there exist a set of optimal pairs of two coupler geometric parameters, i.e. the coupling length and the micro-fiber diameter of the coupler, that can result in high performance polarizers. Experimentally, we fabricated three such coupler-based polarizers using the dual fiber drawing technique and characterized their performance. Our experimental measurement results confirmed our theoretical prediction in several aspects. When the diameter of the coupler-forming micro-fiber is relatively small (~3.5μm), the degree of polarization (DOP) of the fabricated polarizer was found relatively low (~50%) even over some coupling length range. However, when the diameter of the coupler-forming micro-fiber is larger (about 5μm to 9μm), a much higher DOP (>91.4%) and better linear polarization extinction ratio (LPER) of ~60dB could be achieved. The measured geometric parameters of two polarizer samples that showed high polarizing performance agreed very well with our theoretical values. Furthermore, we also demonstrated that such a coupler-based polarizer can be used as an optical filter as well. The filter exhibited an extinction ratio as high as 20dB at the center wavelength and the full width at half maximum (FWHM) was 10nm.

1. Introduction

Micro/nano fibers (MNFs) are special optical fibers with a diameter of several hundred nanometer to several micrometer. They can be fabricated by flame-heating and taper-drawing technique [1

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426(6968), 816–819 (2003). [CrossRef] [PubMed]

X. Xing, Y. Wang, and B. Li, “Nanofibers drawing and nanodevices assembly in poly(trimethylene terephthalate),” Opt. Express 16(14), 10815–10822 (2008). [CrossRef] [PubMed]

]. In recent years, MNFs have received a considerable attention due to several striking optical properties including strong evanescent field, strong light confinement, low loss and high flexibility. MNFs can be made into a low-loss curved waveguide with a very small bending radius of 5μm. Such a small bending radius implies that MNFs can become a promising building block of various micro optical devices. Already reported examples included include Mach-Zehnder interferometer [3

Y. Wang, H. Zhu, and B. Li, “Cascaded Mach-Zehnder interferometers assembled by submicrometer PTT wires,” IEEE Photon. Technol. Lett. 21(16), 1115–1117 (2009). [CrossRef]

Y. Li and L. Tong, “Mach-Zehnder interferometers assembled with optical microfibers or nanofibers,” Opt. Lett. 33(4), 303–305 (2008). [CrossRef] [PubMed]

], optical ring resonators [5

X. Guo and L. Tong, “Supported microfiber loops for optical sensing,” Opt. Express 16(19), 14429–14434 (2008). [CrossRef] [PubMed]

–7

L. Xiao and T. A. Birks, “High finesse microfiber knot resonators made from double-ended tapered fibers,” Opt. Lett. 36(7), 1098–1100 (2011). [CrossRef] [PubMed]

], all-optical tunable resonator [8

Z. Chen, V. K. S. Hsiao, X. Li, Z. Li, J. Yu, and J. Zhang, “Optically tunable microfiber-knot resonator,” Opt. Express 19(15), 14217–14222 (2011). [CrossRef] [PubMed]

], compact filters [9

Y. Chen, Z. Ma, Q. Yang, and L. M. Tong, “Compact optical short-pass filters based on microfibers,” Opt. Lett. 33(21), 2565–2567 (2008). [PubMed]

], all-optical switcher [10

J. Yu, R. Feng, and W. She, “Low-power all-optical switch based on the bend effect of a nm fiber taper driven by outgoing light,” Opt. Express 17(6), 4640–4645 (2009). [CrossRef] [PubMed]

], optical sensors [11

F. Gu, L. Zhang, X. Yin, and L. Tong, “Polymer single-nanowire optical sensors,” Nano Lett. 8(9), 2757–2761 (2008). [CrossRef] [PubMed]

–14

J. Wo, G. Wang, Y. Cui, Q. Sun, R. Liang, P. P. Shum, and D. Liu, “Refractive index sensor using microfiber-based Mach-Zehnder interferometer,” Opt. Lett. 37(1), 67–69 (2012). [CrossRef] [PubMed]

]. In addition, MNFs have also been used to study the fundamental controversy, Abraham-Minkowski controversy on optical momentum in a medium [15

W. She, J. Yu, and R. Feng, “Observation of a push force on the end face of a nanometer silica filament exerted by outgoing light,” Phys. Rev. Lett. 101(24), 243601 (2008). [CrossRef] [PubMed]

Recently optical energy coupling between two MNFs has been investigated theoretically and experimentally by several groups [16

K. Huang, S. Yang, and L. Tong, “Modeling of evanescent coupling between two parallel optical nanowires,” Appl. Opt. 46(9), 1429–1434 (2007). [CrossRef] [PubMed]

–18

Z. Hong, X. Li, L. Zhou, X. Shen, J. Shen, S. Li, and J. Chen, “Coupling characteristics between two conical micro/nano fibers: simulation and experiment,” Opt. Express 19(5), 3854–3861 (2011). [CrossRef] [PubMed]

]. Very compact coupler with coupling length of only 2μm and very high coupling efficiency has been demonstrated [16

K. Huang, S. Yang, and L. Tong, “Modeling of evanescent coupling between two parallel optical nanowires,” Appl. Opt. 46(9), 1429–1434 (2007). [CrossRef] [PubMed]

–18

]. Some new coupling characteristics between two conical micro-fibers were reported by X. Li et al. [18

]. A ultra compact polarizer based on Silicon-on-Insulator (SOI) ridge nanowire directional coupler has also been demonstrated [19

H. Fukuda, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Shinojima, and S. Itabashi, “Ultrasmall polarization splitter based on silicon wire waveguides,” Opt. Express 14(25), 12401–12408 (2006). [CrossRef] [PubMed]

]. However, this SOI polarizer requires complex fabrication and rigorous fabrication accuracy, which results in relative higher cost and lower polarization performance. These shortages motivate us to use low-cost MNF instead of SOI nanowire to realize a coupler-based polarizer with high polarization performance.

An optical polarizer is an important optical device that can select only one polarization of upolarized light and output polarized light. Optical polarizers are widely used in optical systems that need an effective control of the state of polarization (SOP). For example, coherent optical communication systems and interference-based optical sensor systems can benefit from good optical polarizers. Recently, in-fiber polarizers have attracted much attention due to its convenient and seamless connection to other optical fiber devices, and various in-fiber polarizers have been reported. These polarizers are based on long period fiber grating [20

Y. Wang, L. Xiao, D. N. Wang, and W. Jin, “In-fiber polarizer based on a long-period fiber grating written on photonic crystal fiber,” Opt. Lett. 32(9), 1035–1037 (2007). [CrossRef] [PubMed]

], hollow core photonic bandgap fibers [21

H. F. Xuan, W. Jin, J. Ju, Y. P. Wang, M. Zhang, Y. B. Liao, and M. H. Chen, “Hollow-core photonic bandgap fiber polarizer,” Opt. Lett. 33(8), 845–847 (2008). [CrossRef] [PubMed]

,22

W. Qian, C. L. Zhao, Y. Wang, C. C. Chan, S. Liu, and W. Jin, “Partially liquid-filled hollow-core photonic crystal fiber polarizer,” Opt. Lett. 36(16), 3296–3298 (2011). [CrossRef] [PubMed]

], side polished fibers [23

S. Ma and S. Tseng, “High-performance side-polished fibers and applications as liquid crystal clad fiber polarizers,” J. Lightwave Technol. 15(8), 1554–1558 (1997). [CrossRef]

,24

A. Adnreev, B. Pantchev, P. Danesh, B. Zafirova, and E. Karakoleva, “a-Si:H film on side-polished fiber as optical polarizer and narrow-band filter,” Thin Solid Films 330(2), 150–156 (1998). [CrossRef]

] and eccentric core single mode fibers [25

T. Hosaka, K. Okamoto, and J. Noda, “Single-mode fiber-type polarizer,” IEEE Trans. Microw. Theory Tech. 30(10), 1557–1560 (1982). [CrossRef]

]. However, these in-fiber polarizers require a relative complex fabrication process, which results in relative lower linear polarization extinction ratio (LPER) of about 20dB. In this paper, we utilized flame-heating and drawing technique to fabricate an MNF coupler-based polarizer. Compared with the fabrication of those in-fiber polarizers, this fabrication reported here is much simpler and lower cost, since only two low-cost standard single-mode fibers are needed during whole fabrication process. Most important is that this simpler fabrication process can realize a very high performance polarizer with very high LPER of 60dB and with a compact size of 15μm × 9.1mm. In addition, we used full vector finite element method (FVFEM) and super-mode theory (SMT) to analyze such a polarizer. Our analysis resulted in the optimization of the geometrical parameters of the coupler that can provide the best polarizing performance. We fabricated three sample polarizers and measured their polarizing performances, especially the wavelength dependence of their degree of polarization (DOP) and linear polarization extinction ratio (LPER). Two of the three fabricated samples showed very good polarizing performance and the measured geometric parameters of these two samples also agreed well with the theoretically predicted optimal parameters. Furthermore, we also show such a polarizer could be used as an optical band-rejection filter and discussed the characteristic of the filter.

2. Theoretical analysis and design

The best polarizing performance of a coupler-based polarizer can be achieved only with both TE(TM) maximum coupling efficiency and TM(TE) minimum coupling efficiency. According to couple-mode theory [26

K. Okamoto, Fundamental of Optical Waveguides (Elsevier Academic Press, 2006), Chap. 4.

], this condition can only be met when the propagation constants of the two waveguides of the coupler are identical [26

K. Okamoto, Fundamental of Optical Waveguides (Elsevier Academic Press, 2006), Chap. 4.

]. Therefore, two identical micro-fibers are chosen in this study to construct the polarizer for high polarizing performance. When light in an ordinary single-mode fiber is coupled through a conic fiber taper into a micro-fiber [27

S. Lacroix, R. Bourbonnais, F. Gonthier, and J. Bures, “Tapered monomode optical fibers: understanding large power transfer,” Appl. Opt. 25(23), 4421–4425 (1986). [CrossRef] [PubMed]

], if conic taper satisfies the adiabatic condition [28

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983), Chap. 19.

], most of energy of light will be carried by fundamental mode of the micro-fiber, and little will be carried by higher order mode. Therefore, in our analysis below, coupling between high-order modes in the micro-fiber coupler are neglected and only fundamental mode is considered.

Figure 1 is a schematic showing super-mode theory (SMT) of waveguide directional coupling. Based on SMT [26

K. Okamoto, Fundamental of Optical Waveguides (Elsevier Academic Press, 2006), Chap. 4.

], two parallel and clingy micro-fibers in coupling region denoted by red dashed-line rectangle in Fig. 1, where two micro-fibers are parallel and next to each other, is regarded as a new waveguide. Coupling can be interpreted as an interference between even super-mode (ESM) and odd super-mode (OSM); both these super-modes are steadily guided modes of the new waveguide. Once light in TE(TM) polarization is launched in one of inputs of the coupler, TE(TM) ESM and OSM will be excited simultaneously and coupling between the two micro-fibers occurs. When phase difference between ESM and OSM is accumulated to π after a half of beat length, the interference results in that the TE(TM) light in the one input port will be completely cross-coupled to cross output port as shown in Fig. 1. Therefore, the beat-length of TE(TM) mode

L_{i}^{B}

can be written as

L_{i}^{B} (λ) = 2 π / [β {(λ)}_{i, e v e n} - β {(λ)}_{i, o d d}] = λ / [(n {(λ)}_{e f f}^{i, e v e n} - n {(λ)}_{e f f}^{i, o d d})],

(1)

where subscript index i indicates TE(TM) polarization,

β_{i, e v e n}

and

β_{i, o d d}

, which are related to effective refractive index (ERI) of

n_{e f f}^{i, e v e n}

and

n_{e f f}^{i, o d d}

β = (\frac{2 π}{λ}) n_{e f f}

, are respectively propagation constants of even and odd mode in TE(TM) polarization. Notice that the beat lengths, the propagation constants and the ERIs are all function of light wavelength λ. From the SMT, it can be seen that when the coupling length are odd times of half of beat length, light launched in one input port will be cross coupled to a cross port micro-fiber as shown in Fig. 1. When the coupling length is even times of half of beat length, light will be coupled back to the original micro-fiber. Thus, in the best case that the coupling length

L_{C}^{}

of a coupler equals to odd times of half of TE(TM) beat length and at the same time to even times of half of TM(TE) beat length TE and TM polarization can be separated into different outputs of the coupler to render the coupler a polarizer. The condition for this best case, which is what we seek for in the following analysis, can be expressed mathematically as

L_{C}^{} = m L_{T E}^{B} / 2 = (m + n) L_{T M}^{B} / 2 \begin{matrix} (\begin{array}{l} m = 1, 2, 3...... \\ n = 1, 3, 5, ...... \end{array}) \end{matrix},

(2)

where m and n are integer and odd number, respectively.

Fig. 1 Schematic of super-mode theory of waveguide directional coupling

Using full vector finite element method (FVFEM), we calculated two polarization modes (TE and TM) of both ESM and OSM at 1550nm wavelength. Their fields and local direction of polarization are shown in Fig. 1. The E_y profile of TM mode for ESM and OSM is respectively shown in Figs. 2(a) and 2(b). The E_x profile of TE mode is shown in Figs. 2(c) and 2(d), respectively. The two circles in Figs. 2(a)-2(d) indicate cross-section of two next to each other micro-fiber having the same diameter of 3μm. Figures 2(b) and 2(d) show that the ESM field will distort the near contact point of the two micro-fibers and this distortion of the TM ESM is larger than that of the TE ESM.

Fig. 2 Field profiles and local polarization directions of TE and TM super-modes in coupling region, where two identical micro-fibers have 3μm diameter and are clingy together. Figures (a)-(d) are field profiles of TM odd, TM even, TE odd, and TE even modes, respectively.

Figure 3 shows the dependence of the beat lengths and the ERIs on the diameter of the micro fibers. It can be seen that the coupling effect of TM is stronger than that of TE. In other words, the ERI difference between TM ESM and TM OSM is larger than the ERI difference between TE ESM and TE OSM, which results in shorter TE beat length

L_{T E}^{B}

. Thus there can possibly exist a pair of optimal parameters, i.e. the best coupling length

L_{C}^{b e s t}

and the best diameter of micro-fiber D (D,

L_{C}^{b e s t}

), that satisfies the best polarizing condition of Eq. (2), for some given combination of m and n.

Fig. 3 Variations of four super-mode effective refractive indexes with respect to micro-fiber diameter and variations of TE(TM) beat length with respect to the diameter of two identical micro-fibers.

The calculation of the four super-mode ERIs is done using the FVFEM. Their variations with respect to the diameter of the two identical micro-fibers are shown in Fig. 3 with square, circles, upward triangle, and downward triangle. The two TE and TM beat length were numerically calculated by substituting the calculated ERIs into Eq. (1), and the variation of the beat length with respect to the micro-fiber diameter is shown by the dashed and solid lines in Fig. 3. In the calculations, a wavelength of 1550nm and a material refractive index (MRI) of 1.46 for micro-fibers were assumed; the micro-fiber diameter is varied from 1μm up to 13μm with 0.01μm increment. Figure 3 tells us that when diameter of the micro-fiber increases, the ERIs increase and levels off at the MRI of 1.46. Meanwhile, the ERI difference between TE and TM modes decreases. This is rational in physics since when the diameter increases, the evanescent field outside the waveguide in air decreases while mode field inside micro-fiber increases. Accordingly, optical coupling becomes weaker and a longer beat length is expected. In addition, it can be seen from Fig. 3 that the beat length difference between TE and TM modes increases when the diameter increases. The half of beat length of TE mode is longer than that of TM mode as result of a weaker coupling of TE mode as indicated by mode field shown in Fig. 2.

From the plots of TE and TM beat length in Fig. 3, it can also be seen that if a proper diameter of the micro-fiber is chosen and a pair of integer number (m,n) are selected, there can be a best coupling length that fully satisfies the best polarizing condition as given by Eq. (2). Thus for a given pair of (m,n) and device optimization, the micro-fiber and coupling length diameter form a pair of geometric parameters, (D, L_C^best) that can be optimally designed.

Each hollow square in Fig. 4(a) corresponds to a pair of given integer numbers (m, n) and a pair of optimal geometric parameters (D, L_C^best) for the best polarizing performance. As can be seen in Fig. 4(a), each group of hollow squares for a different m number form a curve corresponding to a given n number (n = 1,3,5… from bottom up). For example, three red, blue and green groups of hollow squares as shown in Fig. 4(a) corresponds to the cases of n = 1,3,5 respectively. For each curve, number m will increase when micro-fiber diameter increases. From our previous SMT analysis, it can be inferred that any two adjacent hollow squares on each group curve in Fig. 4 corresponds to two mutually orthogonal polarizations (TE vs TM).

Fig. 4 (a) Optimal geometrical parameters at 1550nm for best polarizing performance. A pair of the optimal parameters, i.e. micro-fiber diameter D and best coupling length L_C^best, corresponds to a pair of integer number (m,n) and is denoted by one hollow square,(D, L_C^best). Each group of hollow squares for a different m number form a curve corresponding to a given n number (n = 1,3,5… from bottom up); (b) enlarged image of ‘zone a’; (c) enlarged image of ‘zone b’

Based on the density of the hollow squares in Fig. 4(a), these optimal parameters can be grouped into three zones. In the first zone or ‘zone a’ that has the highest density, the optimal micro-fiber diameters D lies in the 2.14-2.5μm range. Figure 4(b) shows the enlarged image of ‘zone a’ in Fig. 4(a). As seen from Fig. 4(b), the best coupling lengths lies in range of 2-45mm, and increases abruptly as the micro fiber diameter becomes smaller.

In ‘zone b’ as shown enclosed by a trapezoid in Fig. 4(a) and enlarged image in Fig. 4(c), the best coupling length reaches a minimum of 1.26mm with a micro-fiber diameter of 2.98μm. However, in both ‘zone a’ and ‘zone b’, the polarizing performance will be difficult to control as a result of device fabrication parameter tolerance because the optimal parameters are very dense and the hollow squares are very close to each other on a group curve.

In the last zone or ‘zone c’, better polarizing performance should be more achievable, since the optimal parameters are relatively much more sparse there.

When LPER of more than 20dB is required, maximum tolerance of coupling length ΔL_c^Max can be estimated by [See appendix for detail derivation],

Δ L_{c}^{M a x} = \frac{L_{T M}^{B}}{5 \cdot π} .

(3)

Equation (3) indicates that maximum tolerance of coupling length will decrease when micro-fiber diameter decreases, since the decrease of the beat length leads to the decrease of the micro-fiber diameter as shown in Fig. 3. Therefore, thicker diameter of micro-fiber will be preferred for this type of MNF coupler-based polarizer. For example,

Δ L_{c}^{M a x} = 33.4 μ m

with corresponding

L_{T M}^{B} = 3.6 μ m

when optimal parameters are of D = 3.60μm and L_C^best = 1.51mm.

It is worth noting that the optimal parameters (D, L_C^best) shown as squares in Fig. 4 only mean that the polarizer with these parameters will have higher linear polarization extinction ratio (LPER) and better polarizing capability. This means that while other parameters around but very close to each hollow square (D,L_C^best) will result in a device having a slightly lower LPER and a slightly poorer polarizing capability. Note also that these optimal parameters depend on wavelength of light; the above designs are for a center wavelength of 1550nm. Therefore, if the working wavelength changes, the performance of the polarizer such as the LPER will degrade.

3. Fabrication of the coupler-based polarizer

We fabricated several coupler-based polarizers with two identical micro-fibers using our flame-heating and taper-drawing technique. Figure 5 shows the experimental and fabrication setup, which includes a tunable laser source preset at 1550nm (TLS, ANDO AQ4321D, wavelength range of 1520nm-1620nm), an optical polarization scrambler (OPS, FIBERPRO PS3200), two sets of two-dimensional translation stages, and a state-of-polarization analyzer (SOPA, FIBERPRO SA2000) used for monitoring the fabrication of the polarizer. Two parallel in-contact standard bare single-mode fibers were clamped on the two 2D stages and were pulled in opposite directions when the fibers were being melt by the flame of a burner. During the fabrication, a 1550nm light launched from a tunable laser source was depolarized down to ~4% degree of polarization (DOP) by an optical polarization scrambler(OPS), and was launched into one input port of the coupler to be made. A state-of-polarization analyzer (SOPA) was used to continuously monitor SOP of the output light from a cross-output-port of the coupler. Once the DOP reached over 90% and LPER became greater than 15dB, the draw of two micro-fibers was stopped and fabrication of the polarizer was finished.

Fig. 5 Schematic of experimental and fabrication setup

This dual-fiber-tapering method has several advantages. One advantage is that the method can guarantee the two micro-fiber geometrical parameters of the polarizer are exactly same. As a result, both maximum and minimum coupling efficiency can be reached. In addition, the method can also ensure good parallel contact of the two micro-fibers as a result of their mutual self-sticking nature under van der Waals’ force and electrostatic force during the drawing process. Furthermore, the method can also provide more flexibility in customizing a polarizer with a given central operating wavelength, an important feature that can turn such a polarizer into a band-rejection filter as will be discussed below.

We fabricated three samples of the coupler-based polarizer with different geometrical parameters. In our fabrication, except for the first sample, we continuously monitored the output DOP using a SOPA to ensure both high DOP and high LPER at 1550nm. Figure 6(a) show a micrograph of the coupling region of the first sample coupler. The measured smallest micro-fiber diameter is 3.5μm. The coupling region is 27.8mm and the output DOP is ~50%. Figures 6(b) and 6(c) show, respectively, the coupling region micrographs of the second and the third samples. From these micrographs, the measured parameters for both the second and third sample couplers are: 8.6μm micro-fiber diameter and 10.43mm coupling length for the second sample polarizer, and 5.1μm micro-fiber diameter and 7.87mm coupling length for the third sample polarizer. The second and third sample couplers have a higher DOP of >90% and higher LPER of over 15dB as will be further discussed in next section. Note that the geometric parameters as measured for the second and third sample couplers agreed relatively well with two sets of the predicted optimal geometrical parameters in Fig. 4, i.e. (D = 8.8μm, L_C^best = 12.3mm) and (D = 5.22μm, L_C^best = 9.1mm). The slight departure of the optimal coupling length of fabricated geometric parameters is partly due to the taper shape of micro-fibers in the coupling region. It is likely the taper structure that resulted in a longer coupling length since the diameters were measured at the thinnest portion of the pulled micro-fibers. The micro-fiber diameters at the Y-junction region of two micro-fibers were measured as well, and they are 5μm for the first sample, 13μm for the second sample, and 7.2μm for the third sample. This confirmed the taper structure along the coupling region. Another reason for the departure is that the coupling region as measured by microscope only considered the overlapping length of the coupling section with the two micro-fibers clinging together, and this length should be shorter than the actual coupling length since optical coupling starts from the joining point of the two micro-fibers.

Fig. 6 Coupling region micrographs of three coupler-based polarizer samples. (a)-(c) correspond to the first, second, third polarizer samples. Their measured geometric parameter, i.e. micro-fiber diameter and coupling length, are: 3.5μm and 27.8mm for the first sample, and 8.6μm and 10.43mm for the second sample, and 5.1μm and 7.87mm for the third sample.

4. Experimental results and discussions

The output light DOP from the first polarizer sample as a function of overlapping length was measured and is shown in Fig. 7 . We slowly moved the translation stage to control the overlapping length with examination and measurement of the coupling length under a microscope. In our measurement, depolarized light with 4.45% DOP at 1550nm was launched into the input port of the coupler and the output light DOP was measured by a SOPA at the cross-port of the coupler. The overlapping length was measured using a microscope with a 20x objective lens and the overlapping length was slowly changed from 27.9mm to 26.9mm with a step of 10μm. The result shows that cross-port DOP of the first sample is only around 50% with a little fluctuation of 5% and the polarization enhancement is about 45.55%. The result indicates that this first sample has a relatively weak polarizing capability. We also found that the bar-port DOP of this first sample oscillates with an amplitude of ~20% and with a quasi-period of ~0.34mm when overlapping length of the coupler changes. The DOP of the coupler remains at a relatively low level regardless of whether the optimal geometric parameters are matched or not. The expected optimal parameters are (3.2μm,27.4mm) which might have been matched during scanning coupler length but the experiment showed that the matching was not there. This experiment confirms our prediction as discussed in the section 2 that a smaller diameter of the micro-fiber requires more stringent fabrication tolerance which is more difficult to achieve and hence the fabricated polarizer can have worse polarizing performance.

Fig. 7 Output light DOP of output light from first polarizer sample as a function of overlapping length

The second sample coupler has 8.6μm diameter micro-fibers and 10.43 mm coupling length. these geometric parameters are very close to one pair of the optimal parameter (8.8μm, 12.3 mm) as shown in Fig. 4. Therefore the polarizer sample should have a higher polarizing capability than the first sample. The DOP and LPER dependence on wavelength of the second sample was measured using a SOPA in the optical communication band of 1520nm-1620nm. The measurement results are shown in Figs. 8(a) and 8(b). The solid and dot lines in Fig. 8(a) and 8(b) were experimentally obtained at the bar-port and at the cross-port, respectively. The dashed line in Fig. 8(a) represents the DOP of the input light at different wavelengths. As can be seen, DOP of the input light is less than 5% across the studied communication band, and is very close to that of unpolarized light. From Fig. 8(a), it can also be seen that the DOP at the bar-output-port and the cross-output-port varies over the studied wavelength range without any fixed oscillation period and amplitude. Note that the DOP at the cross-output-port, reached more that 91.4% in wavelength range of 1545nm-1560nm [in between the two arrows in Fig. 8(a)] as would be expected in previous SMT analysis. Very high LPERs of 58.73dB at 1537nm and of 27.79dB at 1547nm were achieved indicating that this second coupler may be used as a polarizer in practical applications. The four small insets in Fig. 8(b) show the SOPs at the cross-output-port at different light wavelengths. As shown in the insets, the SOPs of the output light at the cross-output-port are: from right to left, approximate circular polarization with LPER of 2.96dB at 1555nm, elliptical polarization with LPER of 14.5dB at 1550nm, linear polarizations with LPER of 27.7dB at 1547nm and with LPER of 58.73dB at 1537nm. The polarizer can therefore be used as a polarization filter or a polarization controller when operating wavelength slightly tuned.

Fig. 8 The cross-output-port DOP (a) and LPER (b) dependence of the second polarizer on wavelength in optical communication band. In (a) and (b), the experimental measurement at the bar-output-port and at the cross-output-port are plotted by solid and dot line, respectively. The dashed line in (a) represents DOP of the input light at different wavelengths.

To verify the better predictability in terms of practically achieving the less dense optimal parameters as given in Fig. 4(a), a third polarizer sample as has already been shown in Fig. 6(c) was fabricated and was found to have a micro-fiber diameter of 5.1μm and a overlapping length of 7.87mm. Polarizing property related measurement results obtained using a SOPA showed that both high DOP and high LPER at near 1550nm were obtained. Good polarizing performance is expected because the measured geometric parameters agreed well with a pair of optimal parameter (D = 5.22μm, L_C^best = 9.1mm) shown in Fig. 4(a). The measured cross-port DOP and LPER as a function of wavelength are shown in Fig. 9(a) . Both the DOP and the LPER curves as shown in Fig. 9(a) have similar wavelength dependence characteristics as that of the second sample, i.e. the wavelength dependence behavior is not oscillatory and is without any fixed period or fixed amplitude, but the DOP and LPER values are very high in the desired wavelength band. As can be seen in Fig. 9(a), there are four LPER peaks located at 542nm, 1556nm, 1578nm and 1596nm are with a value of 11.6dB, 60dB, 40dB and 21dB from left to right respectively. The four corresponding SOP are shown in the four insets respectively. The optimal geometric parameters at these four wavelengths as obtained based on the previously discussed theoretical analysis are (5.22μm,8.94mm), (5.22μm,9.1mm), (5.22μm,9.04mm) and (5.22μm,8.94mm), respectively, which are all close to the measured geometric parameters (5.10μm,7.87mm,).

Fig. 9 (a) Cross-output-port DOP and LPER spectrums of the third sample coupler. The four LPER peaks located at 1542nm, 1556nm, 1578.5nm and 1596nm are with a value 11.6dB, 60dB, 40dB and 21dB, from left to right respectively. (b) Cross-output-port transmission spectrums with an additional in-line rotatable polarizer which were rotated at angle 0° and at angle 90 °.

From Figs. 8(a), 8(b) and Fig. 9(a), it can be seen that DOP and LPER at bar-output port and cross-output port reach highest level and lowest level at the same time at most wavelength because the coupler polarizer has separated the TE and TM polarization into bar-output port and cross-output port. However, the above case did not appear at some wavelength, which may be caused by the polarization influence from higher order modes of the micro-fiber. LPER spectrum of the second sample as shown in Fig. 8(b) shows that most LPER peaks are with almost same level of relative higher LPER. On the contrary, the four LPER peaks of third sampler as shown in Fig. 9(a) are with different LPER level. This is because the second sample has thicker diameter and thus larger maximum tolerance of coupling length than that of the third sample. Therefore, relative thinner micro-fiber is preferred for its suppression of the polarization influence from higher order modes, and more accurate fabrication is required to remain high LPER at other wavelength.

In addition to performing polarizing functions, the couplers studied here can also be used as an optical band-rejection filter. Figure 9(b) shows such an application. In demonstrating the filter performance, an ASE (amplified spontaneous emission) light source with a wavelength range of 1520-1620nm is used in place of a TLS (tunable laser source) in Fig. 5 and an optical spectrum analyzer (OSA) is used in place of the SOPA in Fig. 5 to measure transmission spectrum. An additional rotatable polarizer, which can rotate the linear polarization direction, is inserted between the polarizer sample and the OSA. The measured transmission spectrum by OSA is shown in Fig. 9(b). In our measurement, the additional polarizer was rotated at an angle of 90°and 0° such that the transmission at 1556nm was minimum and maximum, respectively. The transmission spectrums in the two cases are plotted as dashed and solid lines in Fig. 9(b), respectively. From Fig. 9(b), it can be seen that in the 90°case, extinction ratio (ER) at 1556nm is about 20dB and full width of half maximum (FWHM) is about 10nm,which shows this coupler-based polarizer can be used as a filter in practice. However, in the 0°case, three relatively shallow troughs appeared at 1542nm, 1578nm and 1596nm. The reason why the troughs are relatively shallow can perhaps be explained from the fact that their linear polarization is approximately orthogonal to that at 1556nm [see SOP insets in Fig. 9(a)]. Their ERs are all of about 6dB, and their FWHMs are all of near 11nm. A comparison between Figs. 9(a) and 9(b) shows that the wavelengths of four relatively shallow troughs in transmission spectrum are in good agreement with that of four LPER peaks indicated by the the arrows in Fig. 9(a). The only exception is the little LPER peak at 1570nm which did not show up in Fig. 9(b) due to its low DOP of 37%.

It can be seen from the LPER spectrums of the last two samples as shown in Fig. 8(a) and Fig. 9(a) that the operating wavelength at maximum LPER departed slightly from the most desired wavelength of 1550nm. This departure can be reduced by monitoring LPER instead of DOP during our fabrication process.

5. Conclusions

In summary, using SMT and FVFEM, we have analyzed the polarizing performance of all-micro-fiber-coupler-based polarizers and figured out the optimal geometric parameters that can lead to best polarizing performance. Dual-fiber-drawing technique was used to fabricate three sample polarizers. Our experimental results confirmed, to a certain degree, our theoretically predictions. Two of the three fabricated polarizers achieved high output DOP of over 97% and a maximum LPER of ~60dB at near 1550nm. Compared with relative lower LEPR of 15dB for SOI polarizer reported in reference 19

, we have realized a high performance coupler-based polarizer by simply heating and drawing standard single-mode fiber. Both theory and experiment show that this type of coupler-based polarizer made with larger diameter micro-fibers can have high polarizing performance. Furthermore, we also demonstrated that the coupler-based polarizer can be used as a filter. Our experimental results show that an ER of 20dB and a FWHM of 10nm can be achieved for such a filter and therefore the filter can be used in practice. More compact MNF polarizer could be realized by optimizing fabrication parameters.

Appendices

Appendix

Using the coupled mode theory [26

K. Okamoto, Fundamental of Optical Waveguides (Elsevier Academic Press, 2006), Chap. 4.

] for an ideal case, the output optical power at bar port (P_bar) and cross port (P_cross) of the micro-fiber coupled-based polarizer can be expressed as,

P_{b a r} = P_{T E} \cos^{2} (π L_{c} / L_{T E}^{B}) + P_{T M} \cos^{2} (π L_{c} / L_{T M}^{B})

(4)

P_{c r o s s} = P_{T E} \sin^{2} (π L_{c} / L_{T E}^{B}) + P_{T M} \sin^{2} (π L_{c} / L_{T M}^{B}),

(5)

where P_TE and P_TM are respectively the input TE and TM power of light, L_c, L_TE^B and L_TM^B are the coupling length of the polarizer, TE and TM beat lengths, respectively. Because the input light is approximately unpolarized light in our experiments, i.e. TE and TM lights are incoherent, we can directly add the two optical powers along TE and TM polarization and have P_TE = P_TM. Since first term and second term in left side of Eq. (4) and Eq. (5) denote output optical power along TE and TM polarization respectively, the linear polarization extinction ratio (LPER) at the bar port and the cross port can be approximately derived from Eq. (4) and Eq. (5),

\begin{array}{l} L P E R_{b a r} = 10 \lg | \frac{\cos^{2} (π L_{c} / L_{T E}^{B})}{\cos^{2} (π L_{c} / L_{T M}^{B})} | \\ L P E R_{c r o s s} = 10 \lg | \frac{\sin^{2} (π L_{c} / L_{T E}^{B})}{\sin^{2} (π L_{c} / L_{T M}^{B})} | \end{array}

(6)

where P_TE = P_TM has been used. If LPER of more than 20dB is required, we can derive the following inequations,

\begin{array}{l} L P E R_{b a r} = 10 \lg | \frac{\cos^{2} (m π / 2 + π Δ L_{c} / L_{T E})}{\cos^{2} ((m + n) π / 2 + π Δ L_{c} / L_{T M})} | \geq 20 \\ L P E R_{c r o s s} = 10 \lg | \frac{\sin^{2} (m π / 2 + π Δ L_{c} / L_{T E})}{\sin^{2} ((m + n) π / 2 + π Δ L_{c} / L_{T M})} | \geq 20 \end{array}

(7)

where ΔL_c is departure from best coupling length. By Taylor’s expanding of cosine and sine functions around mπ/2 and (m + n)π/2, we can approximately derive the maximum tolerance of coupling length ΔL_c^max from the first inequation in Eq. (7),

\begin{array}{l} Δ L_{c}^{\max} = \frac{- 10 π / L_{T M} + 2 \sqrt{{(5 π / L_{T M})}^{2} + {(π / L_{T E})}^{2}}}{{(π / L_{T E})}^{2}} \approx \sqrt{\frac{L_{T M}}{5 π}}, \end{array}

(8)

or derive ΔL_c^max from second inequation in Eq. (7),

Δ L_{c}^{\max} = \frac{- 10 π / L_{T E} + 2 \sqrt{{(5 π / L_{T E})}^{2} + {(π / L_{T M})}^{2}}}{{(π / L_{T M})}^{2}} \approx \sqrt{\frac{L_{T E}}{5 π}} .

(9)

Since L_TM is less than L_TE due to more strong coupling of TM light as shown in Fig. 2, we selected Eq. (8) as maximum tolerance of coupling length.

Acknowledgments

This work is supported by National Nature Science Foundation of China (NSFC) (Grant Nos. 11004086, 61027010, 10776090, 61177075), by the National High Technology Research and Development Program of China (Grant No. 2009AA04Z315), by Foundation for Distinguished Young Talents in Higher Education of Guangdong of China (Grant No. LYM10024), by Fundamental Research Funds for the Central Universities of China (Grant Nos. 21609508, 21609421, 21611602, 216112139, and 21611516) and by Jinan University’s Scientific Research Creativeness Cultivation Project for Outstanding Undergraduates Recommended for Postgraduate Study. Authors thank Dr. Yan Zhou for his language assistance.

References and links

1.	L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426(6968), 816–819 (2003). [CrossRef] [PubMed]
2.	X. Xing, Y. Wang, and B. Li, “Nanofibers drawing and nanodevices assembly in poly(trimethylene terephthalate),” Opt. Express 16(14), 10815–10822 (2008). [CrossRef] [PubMed]
3.	Y. Wang, H. Zhu, and B. Li, “Cascaded Mach-Zehnder interferometers assembled by submicrometer PTT wires,” IEEE Photon. Technol. Lett. 21(16), 1115–1117 (2009). [CrossRef]
4.	Y. Li and L. Tong, “Mach-Zehnder interferometers assembled with optical microfibers or nanofibers,” Opt. Lett. 33(4), 303–305 (2008). [CrossRef] [PubMed]
5.	X. Guo and L. Tong, “Supported microfiber loops for optical sensing,” Opt. Express 16(19), 14429–14434 (2008). [CrossRef] [PubMed]
6.	M. Sumetsky, Y. Dulashko, J. M. Fini, A. Hale, and D. J. DiGiovanni, “The microfiber loop resonator: theory, experiment and application,” J. Lightwave Technol. 24(1), 242–250 (2006). [CrossRef]
7.	L. Xiao and T. A. Birks, “High finesse microfiber knot resonators made from double-ended tapered fibers,” Opt. Lett. 36(7), 1098–1100 (2011). [CrossRef] [PubMed]
8.	Z. Chen, V. K. S. Hsiao, X. Li, Z. Li, J. Yu, and J. Zhang, “Optically tunable microfiber-knot resonator,” Opt. Express 19(15), 14217–14222 (2011). [CrossRef] [PubMed]
9.	Y. Chen, Z. Ma, Q. Yang, and L. M. Tong, “Compact optical short-pass filters based on microfibers,” Opt. Lett. 33(21), 2565–2567 (2008). [PubMed]
10.	J. Yu, R. Feng, and W. She, “Low-power all-optical switch based on the bend effect of a nm fiber taper driven by outgoing light,” Opt. Express 17(6), 4640–4645 (2009). [CrossRef] [PubMed]
11.	F. Gu, L. Zhang, X. Yin, and L. Tong, “Polymer single-nanowire optical sensors,” Nano Lett. 8(9), 2757–2761 (2008). [CrossRef] [PubMed]
12.	G. Brambilla, “Optical fibre nanotaper sensors,” Opt. Fiber Technol. 16(6), 331–342 (2010). [CrossRef]
13.	M. Belal, Z. Q. Song, Y. Jung, G. Brambilla, and T. Newson, “An interferometric current sensor based on optical fiber micro wires,” Opt. Express 18(19), 19951–19956 (2010). [CrossRef] [PubMed]
14.	J. Wo, G. Wang, Y. Cui, Q. Sun, R. Liang, P. P. Shum, and D. Liu, “Refractive index sensor using microfiber-based Mach-Zehnder interferometer,” Opt. Lett. 37(1), 67–69 (2012). [CrossRef] [PubMed]
15.	W. She, J. Yu, and R. Feng, “Observation of a push force on the end face of a nanometer silica filament exerted by outgoing light,” Phys. Rev. Lett. 101(24), 243601 (2008). [CrossRef] [PubMed]
16.	K. Huang, S. Yang, and L. Tong, “Modeling of evanescent coupling between two parallel optical nanowires,” Appl. Opt. 46(9), 1429–1434 (2007). [CrossRef] [PubMed]
17.	X. Xing, H. Zhu, Y. Wang, and B. Li, “Ultracompact photonic coupling splitters twisted by PTT nanowires,” Nano Lett. 8(9), 2839–2843 (2008). [CrossRef] [PubMed]
18.	Z. Hong, X. Li, L. Zhou, X. Shen, J. Shen, S. Li, and J. Chen, “Coupling characteristics between two conical micro/nano fibers: simulation and experiment,” Opt. Express 19(5), 3854–3861 (2011). [CrossRef] [PubMed]
19.	H. Fukuda, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Shinojima, and S. Itabashi, “Ultrasmall polarization splitter based on silicon wire waveguides,” Opt. Express 14(25), 12401–12408 (2006). [CrossRef] [PubMed]
20.	Y. Wang, L. Xiao, D. N. Wang, and W. Jin, “In-fiber polarizer based on a long-period fiber grating written on photonic crystal fiber,” Opt. Lett. 32(9), 1035–1037 (2007). [CrossRef] [PubMed]
21.	H. F. Xuan, W. Jin, J. Ju, Y. P. Wang, M. Zhang, Y. B. Liao, and M. H. Chen, “Hollow-core photonic bandgap fiber polarizer,” Opt. Lett. 33(8), 845–847 (2008). [CrossRef] [PubMed]
22.	W. Qian, C. L. Zhao, Y. Wang, C. C. Chan, S. Liu, and W. Jin, “Partially liquid-filled hollow-core photonic crystal fiber polarizer,” Opt. Lett. 36(16), 3296–3298 (2011). [CrossRef] [PubMed]
23.	S. Ma and S. Tseng, “High-performance side-polished fibers and applications as liquid crystal clad fiber polarizers,” J. Lightwave Technol. 15(8), 1554–1558 (1997). [CrossRef]
24.	A. Adnreev, B. Pantchev, P. Danesh, B. Zafirova, and E. Karakoleva, “a-Si:H film on side-polished fiber as optical polarizer and narrow-band filter,” Thin Solid Films 330(2), 150–156 (1998). [CrossRef]
25.	T. Hosaka, K. Okamoto, and J. Noda, “Single-mode fiber-type polarizer,” IEEE Trans. Microw. Theory Tech. 30(10), 1557–1560 (1982). [CrossRef]
26.	K. Okamoto, Fundamental of Optical Waveguides (Elsevier Academic Press, 2006), Chap. 4.
27.	S. Lacroix, R. Bourbonnais, F. Gonthier, and J. Bures, “Tapered monomode optical fibers: understanding large power transfer,” Appl. Opt. 25(23), 4421–4425 (1986). [CrossRef] [PubMed]
28.	A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983), Chap. 19.

OCIS Codes
(060.1810) Fiber optics and optical communications : Buffers, couplers, routers, switches, and multiplexers
(230.5440) Optical devices : Polarization-selective devices
(230.7408) Optical devices : Wavelength filtering devices

ToC Category:
Optical Devices

History
Original Manuscript: April 24, 2012
Revised Manuscript: June 22, 2012
Manuscript Accepted: July 6, 2012
Published: July 13, 2012

Citation
Jianhui Yu, Yao Du, Yi Xiao, Haozhi Li, Yanfang Zhai, Jun Zhang, and Zhe Chen, "High performance micro-fiber coupler-based polarizer and band-rejection filter," Opt. Express 20, 17258-17270 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-15-17258

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References

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature426(6968), 816–819 (2003). [CrossRef] [PubMed]
X. Xing, Y. Wang, and B. Li, “Nanofibers drawing and nanodevices assembly in poly(trimethylene terephthalate),” Opt. Express16(14), 10815–10822 (2008). [CrossRef] [PubMed]
Y. Wang, H. Zhu, and B. Li, “Cascaded Mach-Zehnder interferometers assembled by submicrometer PTT wires,” IEEE Photon. Technol. Lett.21(16), 1115–1117 (2009). [CrossRef]
Y. Li and L. Tong, “Mach-Zehnder interferometers assembled with optical microfibers or nanofibers,” Opt. Lett.33(4), 303–305 (2008). [CrossRef] [PubMed]
X. Guo and L. Tong, “Supported microfiber loops for optical sensing,” Opt. Express16(19), 14429–14434 (2008). [CrossRef] [PubMed]
M. Sumetsky, Y. Dulashko, J. M. Fini, A. Hale, and D. J. DiGiovanni, “The microfiber loop resonator: theory, experiment and application,” J. Lightwave Technol.24(1), 242–250 (2006). [CrossRef]
L. Xiao and T. A. Birks, “High finesse microfiber knot resonators made from double-ended tapered fibers,” Opt. Lett.36(7), 1098–1100 (2011). [CrossRef] [PubMed]
Z. Chen, V. K. S. Hsiao, X. Li, Z. Li, J. Yu, and J. Zhang, “Optically tunable microfiber-knot resonator,” Opt. Express19(15), 14217–14222 (2011). [CrossRef] [PubMed]
Y. Chen, Z. Ma, Q. Yang, and L. M. Tong, “Compact optical short-pass filters based on microfibers,” Opt. Lett.33(21), 2565–2567 (2008). [PubMed]
J. Yu, R. Feng, and W. She, “Low-power all-optical switch based on the bend effect of a nm fiber taper driven by outgoing light,” Opt. Express17(6), 4640–4645 (2009). [CrossRef] [PubMed]
F. Gu, L. Zhang, X. Yin, and L. Tong, “Polymer single-nanowire optical sensors,” Nano Lett.8(9), 2757–2761 (2008). [CrossRef] [PubMed]
G. Brambilla, “Optical fibre nanotaper sensors,” Opt. Fiber Technol.16(6), 331–342 (2010). [CrossRef]
M. Belal, Z. Q. Song, Y. Jung, G. Brambilla, and T. Newson, “An interferometric current sensor based on optical fiber micro wires,” Opt. Express18(19), 19951–19956 (2010). [CrossRef] [PubMed]
J. Wo, G. Wang, Y. Cui, Q. Sun, R. Liang, P. P. Shum, and D. Liu, “Refractive index sensor using microfiber-based Mach-Zehnder interferometer,” Opt. Lett.37(1), 67–69 (2012). [CrossRef] [PubMed]
W. She, J. Yu, and R. Feng, “Observation of a push force on the end face of a nanometer silica filament exerted by outgoing light,” Phys. Rev. Lett.101(24), 243601 (2008). [CrossRef] [PubMed]
K. Huang, S. Yang, and L. Tong, “Modeling of evanescent coupling between two parallel optical nanowires,” Appl. Opt.46(9), 1429–1434 (2007). [CrossRef] [PubMed]
X. Xing, H. Zhu, Y. Wang, and B. Li, “Ultracompact photonic coupling splitters twisted by PTT nanowires,” Nano Lett.8(9), 2839–2843 (2008). [CrossRef] [PubMed]
Z. Hong, X. Li, L. Zhou, X. Shen, J. Shen, S. Li, and J. Chen, “Coupling characteristics between two conical micro/nano fibers: simulation and experiment,” Opt. Express19(5), 3854–3861 (2011). [CrossRef] [PubMed]
H. Fukuda, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Shinojima, and S. Itabashi, “Ultrasmall polarization splitter based on silicon wire waveguides,” Opt. Express14(25), 12401–12408 (2006). [CrossRef] [PubMed]
Y. Wang, L. Xiao, D. N. Wang, and W. Jin, “In-fiber polarizer based on a long-period fiber grating written on photonic crystal fiber,” Opt. Lett.32(9), 1035–1037 (2007). [CrossRef] [PubMed]
H. F. Xuan, W. Jin, J. Ju, Y. P. Wang, M. Zhang, Y. B. Liao, and M. H. Chen, “Hollow-core photonic bandgap fiber polarizer,” Opt. Lett.33(8), 845–847 (2008). [CrossRef] [PubMed]
W. Qian, C. L. Zhao, Y. Wang, C. C. Chan, S. Liu, and W. Jin, “Partially liquid-filled hollow-core photonic crystal fiber polarizer,” Opt. Lett.36(16), 3296–3298 (2011). [CrossRef] [PubMed]
S. Ma and S. Tseng, “High-performance side-polished fibers and applications as liquid crystal clad fiber polarizers,” J. Lightwave Technol.15(8), 1554–1558 (1997). [CrossRef]
A. Adnreev, B. Pantchev, P. Danesh, B. Zafirova, and E. Karakoleva, “a-Si:H film on side-polished fiber as optical polarizer and narrow-band filter,” Thin Solid Films330(2), 150–156 (1998). [CrossRef]
T. Hosaka, K. Okamoto, and J. Noda, “Single-mode fiber-type polarizer,” IEEE Trans. Microw. Theory Tech.30(10), 1557–1560 (1982). [CrossRef]
K. Okamoto, Fundamental of Optical Waveguides (Elsevier Academic Press, 2006), Chap. 4.
S. Lacroix, R. Bourbonnais, F. Gonthier, and J. Bures, “Tapered monomode optical fibers: understanding large power transfer,” Appl. Opt.25(23), 4421–4425 (1986). [CrossRef] [PubMed]
A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983), Chap. 19.

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