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Reflectivity characteristics of the fiber loop mirror with a polarization controller
Optics Communications 277 (2007) 322–328 www.elsevier.com/locate/optcom
Reflectivity characteristics of the fiber loop mirror with a polarization controller Sujuan Feng a, Qinghe Mao b
a,*
, Liang Shang a, John W.Y. Lit
b
a Anhui Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Hefei, Anhui 230031, China Department of Physics and Computer Science, Wilfrid Laurier University, Waterloo, Ontario, Canada N2L 3C5
Received 12 December 2006; received in revised form 1 May 2007; accepted 9 May 2007
Abstract In this paper, reflectivity characteristics of a fiber loop mirror (FLM), which is formed by inserting a fiber polarization controller (PC) into the fiber loop of an ordinary FLM, are investigated in detail. A theoretical model for determining the reflectivity characteristics of the FLM is present by using the equivalent optical path technique, and the reflectivity characteristics of the FLM are then simulated with the model. The simulation results show that, when the FLM is based on a 3 dB optical coupler (OC), the reflectivity of the FLM may be continuously adjusted to any value between 1 and 0 by changing the PC state, i.e. by either changing the fast axis orientation or the birefringence intensity of the PC alone, as well as both of them; the reflectivity spectra of the FLM are wide and flattened for any PC state, mainly limited by the operating bandwidth of the OC used. The reflectivity characteristics of the FLM are further tested experimentally. The results verify that the reflectivity of the FLM may truly be continuously adjusted between its maximum and minimum values by changing the PC state. The obtainable maximum and minimum reflectivities of the FLM are measured to be 93% and 2%, respectively. Moreover, the experimental results are in agreement with those of the simulations. 2007 Elsevier B.V. All rights reserved. Keywords: Fiber loop mirror (FLM); Polarization controller (PC); Optical coupler (OC); Reflectivity
1. Introduction Reflectors are very important elements for almost all optical systems. Fiber-type reflectors are desirable to construct all-fiber optical systems for telecommunications, remote sensing and monitoring, and optical signal processing [1–3]. Currently, two kinds of all-fiber reflectors are used for these applications, including fiber Bragg gratings (FBGs) and various fiber loops based on optical couplers (OCs) [4,5]. For an FBG, the central wavelength, the reflection bandwidth, and the reflectivity are mainly determined by the Bragg period, the refractivity modulation index, and length of the grating [4]. Although the central reflection wavelength and the bandwidth of an FBG may be tuned and varied by stretching, compressing, applying lateral *
Corresponding author. Tel.: +86 551 5593375; fax: +86 551 5591054. E-mail address: [email protected] (Q. Mao).
0030-4018/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.05.016
stress and magnetic attractive force, and thermal tuning [6], the reflectivity of the FBG may be difficult to adjust. For fiber loops, the reflection characteristics may be designed specifically by utilizing various topology configurations, for instance, compound loops [7,8]. However, the reflectivities of such fiber loops are also difficult to be adjusted because the reflectivities of almost all fiber loops are given by the coupling coefficients of the OCs used [5,7,8]. Therefore, it is still significant to design and implement all-fiber reflectors with adjustable reflectivity. Fiber loops based on OCs have been developed to various topology configurations for different applications. Among them, the fiber loop mirror (FLM) is the basic and simplest, and has been studied and put into practical applications for many years [5,7–11]. One of the most important results in the previous work regarding the FLM is that the birefringence may influence the reflectivity of an FLM [11]. This suggests that the reflectivity of the
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FLM may be controlled and adjusted by the polarization effect. By making use of this idea, we most recently inserted a fiber polarization controller (PC) into the fiber loop of an ordinary FLM to induce a controllable birefringence, and the FLM was then used as a cavity mirror for a dual-wavelength erbium-doped fiber laser [12,13]. Our results show that the reflectivity of the FLM could be adjusted by changing the PC state only, and this property could further be used for mapping the bistability and switching the dualwavelength of the laser [14], and further for obtaining wide wavelength tuning by combining with the tuning mechanism in [13]. However, the reflectivity characteristics of such an FLM were not studied in detail in [14]. Although the operation principle of the FLM with birefringent elements inserted into the loop has been studied theoretically and experimentally [11], the reflectivity characteristics of the FLM with a PC in the loop are still worthy investigating in detail because both the birefringence intensity and fast axis orientation of the PC could be adjusted. In this paper, we investigate the characteristics of the FLM with a PC inserted into the fiber loop. By introducing the phenomenological rotators (ROTs) to describe the polarization orientation changes of the two orthogonally polarized components in the counter-propagating directions in the loop of the FLM, a theoretical model of the FLM is directly given with the help of the equivalent optical path technique. Some new results, such as the effect of the birefringence intensity and the fast axis orientation of the PC, and their combined effect on the reflectivity of the FLM, the wide and flattened reflection spectral properties of the FLM, are obtained theoretically and experimentally. The paper is organized as follows. In Section 2, we describe the basic configuration and give a brief discussion for the operation principle of the FLM. Based on it, in Section 3, by using the equivalent optical path technique and the matrix optics method, we present a theoretical model for the analysis of the reflectivity characteristics of the FLM. Section 4 presents the simulation results of the reflectivity characteristics, in particular, the relation between the reflectivity of the FLM and the fast axis orientation and birefringence intensity of the PC, the influences of the coupling coefficient of the OC on the reflectivity, and the reflectivity spectral characteristics of the FLM. We also show that for the FLM based on a 3 dB OC the reflectivity of the FLM may be continuously adjusted to be at any value between 0 and 1 by only changing the PC state. To test the validity of the simulation results, in Section 5, we further investigate the reflectivity of the FLM experimentally, which is formed with the commercial 3 dB OC and fiber-type PC. Finally, the main results are summarized in Section 6.
Fig. 1. Schematic configuration of the FLM.
ary FLM, as is well known, the input light is divided into two waves by the OC and the two waves then propagate in the clockwise and counterclockwise directions in the fiber loop. If the input optical field is Ei, the optical fields of the two divided waves right after the OC in the clockwise and counterclockwise directions are Ecw and Eccw with a phase difference of p/2 between them, and become Ecw 0 and Eccw after a round-trip back to the OC, respectively, 0 the phase difference between Ecw and Eccw are still p/2 0 0 because the optical paths for the counter-propagating waves in the loop are the same. Thus, the reflectivity of an ordinary FLM is only determined by the coupling ratio of the OC, and the ordinary FLM with a 3 dB OC has a complete reflection property [5]. When a PC is inserted into the loop of an ordinary FLM shown as Fig. 1, the birefringence is induced into the FLM. For the fields in the counter-propagating directions, although the birefringence intensities of the PC are identical, the angles of their polarization orientations to the fast axis of the PC at their entrances may be different. This may cause the phase delays for the counter-propagating waves to be different during their passing through the PC, giving rise to an extra phase difference between the counter-propagating waves after a round-trip. Thus, the reflectivity of the FLM may be determined by both the OC coupling coefficient and the birefringence of the PC. Since the fast axis orientation and the birefringence intensity of the PC are controllable and adjustable (see Appendix), the reflectivity of the FLM shown as Fig. 1 may be varied accordingly. 3. Modelling By analyzing the routings of the counter-propagating waves in the fiber loop, we can obtain the equivalent optical path of the FLM as Fig. 2. Here the phenomenological ROTs are introduced to describe the polarization orientation changes of the two orthogonally polarized components in the counter-propagating directions due to the special topology of the FLM [15]. As shown in the figure, the waves in the counter-propagating directions travel independently, with the routing of single mode fiber 1ðSMFcw 1 Þ !
2. Configuration and principle Fig. 1 schematically shows the configuration of the FLM studied in this paper, which is formed by inserting a fiber PC into the loop of an ordinary FLM. For an ordin-
323
Fig. 2. Equivalent optical path of the FLM.
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cw rotator 1ðROTcw ! rotator 2ðROTcw 1 Þ ! PC 2 Þ ! single cw mode fiber 2ðSMF2 Þ, and single mode fiber 2ðSMFccw 2 Þ! ccw rotator 2ðROTccw ! rotator 1ðROTccw 2 Þ ! PC 1 Þ ! single mode fiber 1ðSMFccw 1 Þ, for the clockwise and the counter clockwise, respectively. Ei;x cw ccw By respectively writing Ei, E and E as Ei ¼ , Ei;y cw ccw Ex Ex and Eccw ¼ , we have Ecw ¼ Ecw Eccw y y cw En Ei;n ¼ T ð1Þ OC Eccw 0 n " # 1=2 1=2 1 ð1 K Þ jK n n where T OC ¼ ð1 cÞ2 is the 1=2 jK n1=2 ð1 K n Þ
nent, respectively. Note that the transfer matrixes of the PC cw T have the property of T ccw PC ¼ ðT PC Þ [9,11]. The reflected light Er and transmitted light Et of the FLM may respectively be given by " ccw # E0;x Er;x ð4Þ ¼ T OC Ecw Et;x 0;x " ccw # E0;y Er;y ¼ T OC ð5Þ Ecw Et;y 0;y
transfer matrix of the OC, with Kn and c being the coupling coefficient and the insertion loss of the OC (c has been assumed to be polarization-independent), respectively; n = x, y stands for the two orthogonally polarized components. According to the equivalent optical path in Fig. 2, we obtain " cw # " cw # E0;x Ex cw cw cw cw cw ¼ T SMF2 T ROT2 T PC T ROT1 T SMF1 cw ð2Þ cw E0;y Ey " ccw # " ccw # Ex E0;x ccw ccw ccw ccw ¼ T ccw ð3Þ SMF1 T ROT1 T PC T ROT2 T SMF2 Eccw Eccw 0;y y
ð7Þ
1 0 jðkl1;2 þjal1;2 Þ where ¼ ¼ e are the 0 1 transfer matrixes of SMF 1 and 2, with k being the wave vector, l1,2 the fiber lengths, and a the fiber loss coefficient, ccw respectively; T cw ROT1;2 and T ROT1;2 are the transfer matrixes of ROT 1 and 2 for the waves propagating in clockwise and counterclockwise, respectively, and they satisfy the T cw SMF1;2
T ccw SMF1;2
by using (2)–(5), we obtain Er;x A B Ei;x ¼ ð1 cÞej½kðl1 þl2 Þþjaðl1 þl2 Þ Er;y C D Ei;y Ei;x Et;x E F ¼ ð1 cÞej½kðl1 þl2 Þþjaðl1 þl2 Þ Et;y G H Ei;y where 8 A ¼ 2ðsin d cos 2X j cos dÞK x1=2 ð1 K x1=2 Þ > > > > 1=2 1=2 > B ¼ C ¼ sin d sin 2X½K y1=2 ð1 K 1=2 > x Þ K x ð1 K y Þ > > > > 1=2 1=2 > > < D ¼ 2ðsin d cos 2X þ j cos dÞK y ð1 K y Þ E ¼ ðcos d þ j sin d cos 2XÞð2K x 1Þ > > > F ¼ j sin d sin 2X½K 1=2 K 1=2 þ ð1 K x Þ1=2 ð1 K y Þ1=2 > > x y > > > 1=2 > > G ¼ j sin d sin 2X½K x K y1=2 þ ð1 K x Þ1=2 ð1 K y Þ1=2 > > : H ¼ ðcos d j sin d cos 2XÞð1 2K y Þ
ð8Þ Using the definitions of the reflectivity and transmittivity of R ¼ Er Er =Ei Ei and T ¼ Et Et =Ei Ei , after straightforward matrix manipulation and having neglected the cross terms of the orthogonal polarization components, we arrive at
R¼
ðAEi;x þ BEi;y ÞðAEi;x þ BEi;y Þ þ ðCEi;x þ DEi;y ÞðCEi;x þ DEi;y Þ 2 ð1 cÞ e2aðl1 þl2 Þ Ei;x Ei;x þ Ei;y Ei;y
T ¼
ðEEi;x þ FEi;y ÞðEEi;x þ FEi;y Þ þ ðGEi;x þ HEi;y ÞðGEi;x þ HEi;y Þ 2 ð1 cÞ e2aðl1 þl2 Þ Ei;x Ei;x þ Ei;y Ei;y
T ðT ccw ROT1 Þ
ð9Þ
1 0 ¼ and T cw ROT2 ¼ 0 1 [9], with the superscript T stand-
T cw ¼ ROT1
ð6Þ
ð10Þ
relations of 1 0 T ccw ðT ROT2 Þ ¼ 0 1 ing for the transposing manipulation; T cw PC ¼ cos d þ j sin d cos 2X j sin d sin 2X is the transj sin d sin 2X cos d j sin d cos 2X fer matrix of the PC for the field propagating in the clockwise direction, with 2d being the birefringence intensity and defined as the phase difference between the fast and the slow axes of the PC, and X the angle of the fast axis orientation of the PC with respect to the x polarization compo-
Further assuming that the coupling coefficient of the OC is polarization-independent, i.e. Kx = Ky = K, we finally obtain the following expressions: 2
R ¼ 4Kð1 KÞð1 cÞ e2aðl1 þl2 Þ ðcos2 d þ sin2 d cos2 2XÞ ð11Þ 2 2aðl1 þl2 Þ
T ¼ ð1 cÞ e 2
2
2
2
½sin d sin 2X þ ð2K 1Þ
ðcos d þ sin d cos2 2XÞ
2
ð12Þ
Eqs. (11) and (12) show that the reflectivity and transmittivity of the FLM are not only determined by the coupling
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coefficient of the OC, but also by the fast axis orientation X and the birefringence intensity 2d of the PC. 4. Simulation results In this section we simulate the influences of the PC state on the reflectivity characteristics of the FLM with Eq. (11). For simplicity and without the loss of generality, the coupling coefficient of the OC is chosen to be 0.5, and both the transmission loss of the fiber and the insertion losses of the OC are neglected in the simulations. Fig. 3a shows the reflectivity of the FLM as a function of X for different 2d. As seen in the figure, when 2d = 0, 2p, the reflectivity of the FLM does not change with X, keeping the maximum reflectivity of 1. When 2d 5 0, 2p, the reflectivity of the FLM oscillates periodically as X varies, with the maximum reflectivity of 1 at X = 0, p/2, p, and the minimum at X = p/4, 3p/4, respectively. Note that for different 2d the minimum reflectivities are different, only when 2d = p, the minimum reflectivity could reach zero. In fact, X and 2d respectively stand for the fast axis orientation and the birefringence intensity of the PC. When 2d = 0, 2p, corresponding to a birefringence intensity of zero, the FLM shown in Fig. 1 then degenerates to an ordinary FLM for any X, resulting in the complete reflection of the FLM. When X = 0, p/2, p, the birefringence fast axis of the PC is parallel or perpendicular to the orthogonally polarized compo-
325
nents, which causes the phase delays of both orthogonally polarized components in the counter-propagating directions to be the same whatever 2d is, thus, the FLM also degenerates to an ordinary FLM, exhibiting the complete reflection property. When X = p/4, 3p/4, the angles of the fast axis to the two orthogonally polarized components are the same and equal to p/4, giving the strongest influence of 2d on the reflectivity of the FLM. Fig. 3b shows the reflectivity of the FLM as a function of 2d for different X. We can see that when X = 0, 2p, at which the birefringence fast axis of the PC is parallel or perpendicular to the two orthogonally polarized components, the reflectivity of the FLM is independent of 2d, being a constant of 1. When X 5 0, 2p, the reflectivity of the FLM also varies periodically with 2d, with the maximum reflectivity of 1 at 2d = 0, 2p and the minimum at 2d = p where the birefringence intensity of the PC is the minimum and the maximum, respectively. Similarly, the value of the minimum reflectivity at 2d = p depends on X, and could reach zero with X being p/4 or 3p/4. To further show the influences of the PC state on the reflectivity of the FLM, the reflectivity as functions of X and 2d is illustrated in Fig. 3c. This figure clearly shows the effect of X and 2d, and their combined effect on the reflectivity of the FLM. Meanwhile, the figure also shows that the reflectivity of the FLM may be more flexible adjusted to expected value by varying both X and 2d.
Fig. 3. Influences of X and 2d on the reflectivity of the FLM. (a) The reflectivity as a function of X for a 2d of 0 or 2p (curve 1), p/3 or 5p/3 (curve 2), 3p/5 or 7p/5 (curve 3), and p (curve 4); (b) the reflectively as a function of 2d for a X of 0 or p/2 or p (curve 1), p/10 or 2p/5 or p/5 or 9p/10 (curve 2), p/7 or 5p/14 or 9p/14 or p/6 (curve 3), and p/4 or 3p/4 (curve 4); and (c) the reflectivity as functions of X and 2d.
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As in the case of an ordinary FLM, the OC coupling coefficient, K, also affects the reflectivity of the FLM. Fig. 4a and b shows the reflectivity as functions of K and X for 2d = p, and 2d for X = p/4, respectively. As seen, for any X or 2d, the reflectivity of the FLM reaches the maximum when K = 0.5, and decreases gradually as K deviates from 0.5, which is the same as that for the common FLM. When 2d = p, or when X = p/4 or 3p/4, however, the reflectivity of the FLM keeps being zero for any value of K, which is different from the case of an ordinary FLM. Now, we turn to the reflectivity spectral characteristics of the FLM. In general, the coupling coefficient of an OC is a function of the operating wavelength, and may be approximately expressed as K(k) = [cos(xk u)]/2. On the other hand, X is the birefringence fast axis orientation of the PC and is independent of the wavelength. However, 2d may be expressed as a function of wavelength by 2d(k) = 2pLPCDn/k, where LPC is the effective length of the PC; Dn is the refractive index difference between the fast and the slow axes of the PC. Using K(k) and 2d(k) into Eq. (11), and ignoring the dispersion of Dn, we can obtain the reflectivity spectral characteristics of the FLM. Fig. 5a and b shows the simulation results for Dn = 0, 1.5 · 107, 2.5 · 107 and 3.95 · 107 when X is fixed at p/4 and for X = 0, p/12, p/8 and p/4 when Dn is fixed at 3.95 · 107, respectively. Here LPC is assumed to be 2m, and the OC is assumed with a coupling coefficient of 50:50 at the central operating wavelength of 1570 nm and the FWHM of
Fig. 5. Reflectivity spectral characteristics of the FLM with a coupling coefficient of 50:50 at the central operating wavelength of 1570 nm. (a) X = p/4, Dn = 0 (curve 1), 1.5 · 107 (curve 2), 2.5 · 107 (curve 3) and 3.95 · 107 (curve 4); (b) Dn = 3.95 · 107, X = 0 (curve 1), p/8 (curve 2), p/6 (curve 3) and p/4 (curve 4).
60 nm, which gives x = p/90 and x = 157p/9 for K(k). As seen, the reflectivity of the FLM has wide and flattened spectral property over a range of 60 nm with the peak reflectivity right at the central operating wavelength of the OC, implying that the reflectivity characteristic is mainly determined by the coupling coefficient spectrum of the OC. Moreover, the reflectivity may be adjusted to any value between 0 and 1 by varying X or 2d. 5. Experiments
Fig. 4. Influence of K on the reflectivity of the FLM. (a) 2d = p and (b) X = p/4.
In this section, we further investigate, experimentally, the reflectivity characteristics of the FLM. The FLM is formed by an OC with a nominal coupling coefficient of 50:50 at the central wavelength of 1570 nm and an operating bandwidth of C plus L bands, and with an ordinary PC, which consists of three rigid discs with the single mode fiber being coiled, respectively functioning as k/4, k/2 and k/4 wave-plates (see Fig. 9 in Appendix), being inserted into the fiber loop. Both the OC and the PC are made with Corning SMF-28. The fiber length of the FLM is about 2 m. The testing scheme is illustrated as Fig. 6. A tunable laser and a broadband ASE source based on an erbiumdoped fiber amplifier (EDFA) are respectively used as the signal sources. The signal light is launched into the FLM with a three-port optical circulator (CIR), which functions as an optical isolator to eliminate the possible reflections in
S. Feng et al. / Optics Communications 277 (2007) 322–328
327
Fig. 6. Experimental setup. CIR: optical circular; ISO: optical isolator.
the input end of the FLM, for the same purpose, an optical isolator (ISO) is also inserted in another end of the FLM. The reflected and the transmitted lights of the FLM are measured with an optical spectrum analyzer and a power meter, respectively. The reflectivity of the FLM is first measured as a function of X or 2d with the tunable laser. Note that the values of X and 2d may be determined by the values of h1, h2 and h3, according to (A.6) given in the Appendix. For an example, when h1 = h3 = 0, we obtain X = p/4, 2d = 2p 4h2 by using (A.6), meaning that if h1 and h3 are fixed at 0, then X keeps being p/4, and 2d may be determined when h2 is adjusted from 0 to p/2 or from p/2 to p. In such an adjustment, the measured reflectivity of the FLM is plotted as a function of 2d for the wavelength at 1570 nm, as illustrated in Fig. 7a. The solid and the open circles are the data points measured when h2 is adjusted from 0 to p/2, and from p/2 to p, respectively. As shown in the figure, the reflectivity may be set to any value between its maximum and minimum by adjusting 2d (or h2). Moreover, the measured reflectivity is in agreement with the simulation results by taking the insertion losses with c = 0.05 into account, which is also shown with a solid line in Fig. 7a. Fig. 7b shows the measured reflectivity as a function of X for the wavelength of 1570 nm. To obtain the data points, this time h3 is fixed at p/2, and h2 is kept being half of h1 and/or (h1 + p)/2 while h1 is being adjusted from 0 to p, corresponding to the condition that 2d is fixed at p and X is adjusted with h2 according to (A.6). Here we can see that the measured reflectivity may also be set to any value between its maximum and minimum by adjusting X, and is
Fig. 7. Measured reflectivity of the FLM. (a) X = p/4, solid and the open circles are the data points measured when h2 is respectively adjusted from 0 to p/2 and from p/2 to p to change 2d; (b) 2d = p. Solid lines in both figures are given by the simulations.
Fig. 8. Typical reflectivity spectra of the FLM measured in the condition the same as that in Fig. 7a.
again in agreement with the simulation results (solid line). It is worthwhile to point out that the reflectivity characteristics of the FLM with various sets of adjustments in terms of h1, h2 and h3 are also tested and the measured results are in agreement with those obtained from the simulations. The reflectivity spectral characteristics of the FLM are then investigated with the broadband ASE source. Fig. 8 shows a set of typical experimental results obtained with the condition the same as that for Fig. 7a, i.e. X = p/4, 2d is adjusted from 0 to p/2. Here we can see clearly that the measured reflectivity spectra are wide and flattened, the fluctuation of the reflectivity is about 4% in the wavelength range of 60 nm; the reflectivity can be adjusted to any value between the maximum of 93% and the minimum of 2% by changing 2d (or h2). In addition, the measured reflectivity spectra with different sets of h1, h2 and h3 also verify that the reflectivity spectrum is wide and flattened. Obviously, such a wide and flattened all-fiber reflector with adjustable reflectivity will play an important role in the cavity optimization of fiber lasers, fiber sensing and optical component characterizations. 6. Conclusions We have investigated the reflectivity characteristics of the FLM formed by inserting a PC into the fiber loop based on an OC. With the equivalent optical path technique and the matrix optics approach, a theoretical model has been proposed for determining the reflectivity characteristics of the FLM. The simulation results obtained with the model show that the reflectivity of the FLM may be continuously adjusted by either changing the fast axis orientation or the birefringence intensity of the PC. The adjustable range is between 0 and 1 if the OC coupling coefficient is 0.5 and the loss is neglected, and decreases if the OC coupling coefficient offsets from 0.5. The simulation results also show that the reflectivity spectrum is wide and flattened, mainly determined by the coupling coefficient spectrum of the OC. The reflectivity characteristics of the FLM have further been investigated experimentally to prove the theoretical results. With an FLM formed by a 3-dB broadband OC and an ordinary PC, the experimental results verify that, the
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reflectivity spectrum of the FLM is truly wide and flattened, and the reflectivity can be adjusted continuously between its maximum and minimum which are measured to be 93% and 2%, respectively, by changing the PC state. Moreover, the experimental results are in agreement with those of the simulation. Obviously, such a FLM with the broadband continuously adjustable reflectivity can play an important role in the fiber devices and systems. Acknowledgement This work is supported by the ‘‘Hundreds of Talents Program’’ of the Chinese Academy of Sciences and by the National Natural Science Foundation of China (Grant No. 60677050). The authors would like to thank Mr. Qing Sun for his help. Appendix Fig. 9 shows schematically the structure of an ordinary PC, which consists of three rigid discs with the single mode fiber being coiled. According to [16], the discs with label of 1, 2 and 3 function as k/l, k/m and k/n wave-plates with l, m and n being real numbers, respectively. The combined effect of the three wave-plates allows continuous, reset-free polarization transformations from an input arbitrary polarization state to a specific output polarization state by rotating the three rigid discs. If the angular orientations of the three wave-plates are h1, h2 and h3 relative to the xoz plane, the transfer matrixes of the three wave-plate are [17] " # cos u2i þ j sin u2i cos 2hi j sin u2i sin 2hi Ti ¼ j sin u2i sin 2hi cos u2i j sin u2i cos 2hi ði ¼ 1; 2 and 3Þ
ðA:1Þ
where u1 = 2p/l, u2 = 2p/m and u3 = 2p/n, respectively. Then the transfer matrix of the PC can be derived as T PC ðl; m; n; h1 ; h2 ; h3 Þ ¼ T 3 T 2 T 1
ðA:2Þ
Obviously, the PC transmission characteristic is determined by both the values of l, m and n, and the orientations of the three wave-plates to the horizontal direction. On the other hand, the transfer matrix of the PC can also be expressed with the fast axis orientation X and the birefringence intensity 2d of a linear birefringence component [11] cos d þ j sin d cos 2X j sin d sin 2X T PC ðd; XÞ ¼ j sin d sin 2X cos d j sin d cos 2X ðA:3Þ
Fig. 9. Schematic view of ordinary commercial PC.
By comparing (A.2) to (A.3), we can obtain the relations between X and d and h1, h2 and h3 for a specific PC with the values of l, m and n having been given. For most commercial PCs, l, m and n are respectively designed to be 4, 2 and 4, then the transfer matrix given in (A.2) becomes j M N T PC ðh1 ; h2 ; h3 Þ ¼ T 3 T 2 T 1 ¼ ðA:4Þ 2 O P where 8 M ¼ cos 2h2 þ j cosð2h1 2h2 Þ þ j cosð2h2 2h3 Þ > > > > > cosð2h1 2h2 þ 2h3 Þ > > > > > N ¼ sin 2h2 þ j sinð2h1 2h2 Þ þ j sinð2h2 2h3 Þ > > > < sinð2h1 2h2 þ 2h3 Þ > O ¼ sin 2h2 j sinð2h1 2h2 Þ j sinð2h2 2h3 Þ > > > > > sinð2h1 2h2 þ 2h3 Þ > > > > > P ¼ cos 2h2 þ j cosð2h1 2h2 Þ þ j cosð2h2 2h3 Þ > > : þ cosð2h1 2h2 þ 2h3 Þ ðA:5Þ Comparing (A.4) to (A.3), we can obtain 8 cos 2h2 cosð2h1 2h2 þ 2h3 Þ ¼ 2 sin d cos 2X > > > < cosð2h 2h Þ þ cosð2h 2h Þ ¼ 2 cos d 1 2 2 3 > sinð2h 2h Þ þ sinð2h 2h 1 2 2 3Þ ¼ 0 > > : sin 2h2 sinð2h1 2h2 þ 2h3 Þ ¼ 2 sin d sin 2X
ðA:6Þ
Then X and d are functions of h1, h2 and h3, and thus, the PC fast axis orientation and intensity can be achieved by varying the orientations of the three wave-plates. References [1] G. Rossi, O. Jerphagnon, B.E. Olsson, D.J. Blumenthal, IEEE Photon. Technol. Lett. 12 (2000) 897. [2] B. Dong, Q.D. Zhao, L.J. Feng, T. Guo, L.F. Xue, Sh.H. Li, H. Gu, Appl. Opt. 45 (2006) 7767. [3] S. Bigo, O. Lecelrc, E. Desurvire, IEEE J. Select. Topics Quantum Electron. 3 (1997) 1208. [4] T.P. Lee, C.E. Zah, R. Bhat, W.C. Young, B. Pathak, F. Favire, P.S.D. Lin, N.C. Andreadakis, C. Caneau, A.W. Rahjel, M. Koza, J.K. Gamelin, L. Curtis, D.D. Mahoney, A. Lepore, J. Lightwave Technol. 14 (1996) 967. [5] T.A. Birks, P. Morkel, Appl. Opt. 27 (1988) 3107. [6] B.O. Guan, H.Y. Tam, H.L.W. Chan, X.Y. Dong, C.L. Choy, M.S. Demokan, Opt. Commun. 202 (2002) 331. [7] J.L. Zhang, J.W.Y. Lit, J. Opt. Soc. Amer. A 11 (1994) 1867. [8] S.W. Kim, J.U. Kang, IEEE Photon. Technol. Lett. 16 (2004) 494. [9] A. Yu, A.S. Siddiqui, IEE Proc. Optoelectron. 142 (1995) 165. [10] E.A. Kuzin, J.A. Andrade-Lucio, B. Ibarra Escamilla, R. RojacLaguna, J. Sanchez-Mondragon, Opt. Commun. 144 (1997) 60. [11] D.B. Mortimore, J. Lightwave Technol. 6 (1988) 1217. [12] Q.H. Mao, J.W.Y. Lit, IEEE Photon. Technol. Lett. 14 (2002) 1252. [13] Q.H. Mao, J.W.Y. Lit, Appl. Phys. Lett. 82 (2003) 1335. [14] Q.H. Mao, S.J. Feng, W.Q. Liu, J.W.Y. Lit, IEEE Photon. Technol. Lett. 18 (2006) 1973. [15] R. Kiyan, Y.K. Byoung, IEEE Photon. Technol. Lett. 10 (1998) 340. [16] H.C. Lefevre, Electron. Lett. 16 (1980) 778. [17] F. Heismann, J. Lightwave Technol. 12 (1994) 690.
Reflectivity characteristics of the fiber loop mirror with a polarization controller Sujuan Feng a, Qinghe Mao b
a,*
, Liang Shang a, John W.Y. Lit
b
a Anhui Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Hefei, Anhui 230031, China Department of Physics and Computer Science, Wilfrid Laurier University, Waterloo, Ontario, Canada N2L 3C5
Received 12 December 2006; received in revised form 1 May 2007; accepted 9 May 2007
Abstract In this paper, reflectivity characteristics of a fiber loop mirror (FLM), which is formed by inserting a fiber polarization controller (PC) into the fiber loop of an ordinary FLM, are investigated in detail. A theoretical model for determining the reflectivity characteristics of the FLM is present by using the equivalent optical path technique, and the reflectivity characteristics of the FLM are then simulated with the model. The simulation results show that, when the FLM is based on a 3 dB optical coupler (OC), the reflectivity of the FLM may be continuously adjusted to any value between 1 and 0 by changing the PC state, i.e. by either changing the fast axis orientation or the birefringence intensity of the PC alone, as well as both of them; the reflectivity spectra of the FLM are wide and flattened for any PC state, mainly limited by the operating bandwidth of the OC used. The reflectivity characteristics of the FLM are further tested experimentally. The results verify that the reflectivity of the FLM may truly be continuously adjusted between its maximum and minimum values by changing the PC state. The obtainable maximum and minimum reflectivities of the FLM are measured to be 93% and 2%, respectively. Moreover, the experimental results are in agreement with those of the simulations. 2007 Elsevier B.V. All rights reserved. Keywords: Fiber loop mirror (FLM); Polarization controller (PC); Optical coupler (OC); Reflectivity
1. Introduction Reflectors are very important elements for almost all optical systems. Fiber-type reflectors are desirable to construct all-fiber optical systems for telecommunications, remote sensing and monitoring, and optical signal processing [1–3]. Currently, two kinds of all-fiber reflectors are used for these applications, including fiber Bragg gratings (FBGs) and various fiber loops based on optical couplers (OCs) [4,5]. For an FBG, the central wavelength, the reflection bandwidth, and the reflectivity are mainly determined by the Bragg period, the refractivity modulation index, and length of the grating [4]. Although the central reflection wavelength and the bandwidth of an FBG may be tuned and varied by stretching, compressing, applying lateral *
Corresponding author. Tel.: +86 551 5593375; fax: +86 551 5591054. E-mail address: [email protected] (Q. Mao).
0030-4018/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.05.016
stress and magnetic attractive force, and thermal tuning [6], the reflectivity of the FBG may be difficult to adjust. For fiber loops, the reflection characteristics may be designed specifically by utilizing various topology configurations, for instance, compound loops [7,8]. However, the reflectivities of such fiber loops are also difficult to be adjusted because the reflectivities of almost all fiber loops are given by the coupling coefficients of the OCs used [5,7,8]. Therefore, it is still significant to design and implement all-fiber reflectors with adjustable reflectivity. Fiber loops based on OCs have been developed to various topology configurations for different applications. Among them, the fiber loop mirror (FLM) is the basic and simplest, and has been studied and put into practical applications for many years [5,7–11]. One of the most important results in the previous work regarding the FLM is that the birefringence may influence the reflectivity of an FLM [11]. This suggests that the reflectivity of the
S. Feng et al. / Optics Communications 277 (2007) 322–328
FLM may be controlled and adjusted by the polarization effect. By making use of this idea, we most recently inserted a fiber polarization controller (PC) into the fiber loop of an ordinary FLM to induce a controllable birefringence, and the FLM was then used as a cavity mirror for a dual-wavelength erbium-doped fiber laser [12,13]. Our results show that the reflectivity of the FLM could be adjusted by changing the PC state only, and this property could further be used for mapping the bistability and switching the dualwavelength of the laser [14], and further for obtaining wide wavelength tuning by combining with the tuning mechanism in [13]. However, the reflectivity characteristics of such an FLM were not studied in detail in [14]. Although the operation principle of the FLM with birefringent elements inserted into the loop has been studied theoretically and experimentally [11], the reflectivity characteristics of the FLM with a PC in the loop are still worthy investigating in detail because both the birefringence intensity and fast axis orientation of the PC could be adjusted. In this paper, we investigate the characteristics of the FLM with a PC inserted into the fiber loop. By introducing the phenomenological rotators (ROTs) to describe the polarization orientation changes of the two orthogonally polarized components in the counter-propagating directions in the loop of the FLM, a theoretical model of the FLM is directly given with the help of the equivalent optical path technique. Some new results, such as the effect of the birefringence intensity and the fast axis orientation of the PC, and their combined effect on the reflectivity of the FLM, the wide and flattened reflection spectral properties of the FLM, are obtained theoretically and experimentally. The paper is organized as follows. In Section 2, we describe the basic configuration and give a brief discussion for the operation principle of the FLM. Based on it, in Section 3, by using the equivalent optical path technique and the matrix optics method, we present a theoretical model for the analysis of the reflectivity characteristics of the FLM. Section 4 presents the simulation results of the reflectivity characteristics, in particular, the relation between the reflectivity of the FLM and the fast axis orientation and birefringence intensity of the PC, the influences of the coupling coefficient of the OC on the reflectivity, and the reflectivity spectral characteristics of the FLM. We also show that for the FLM based on a 3 dB OC the reflectivity of the FLM may be continuously adjusted to be at any value between 0 and 1 by only changing the PC state. To test the validity of the simulation results, in Section 5, we further investigate the reflectivity of the FLM experimentally, which is formed with the commercial 3 dB OC and fiber-type PC. Finally, the main results are summarized in Section 6.
Fig. 1. Schematic configuration of the FLM.
ary FLM, as is well known, the input light is divided into two waves by the OC and the two waves then propagate in the clockwise and counterclockwise directions in the fiber loop. If the input optical field is Ei, the optical fields of the two divided waves right after the OC in the clockwise and counterclockwise directions are Ecw and Eccw with a phase difference of p/2 between them, and become Ecw 0 and Eccw after a round-trip back to the OC, respectively, 0 the phase difference between Ecw and Eccw are still p/2 0 0 because the optical paths for the counter-propagating waves in the loop are the same. Thus, the reflectivity of an ordinary FLM is only determined by the coupling ratio of the OC, and the ordinary FLM with a 3 dB OC has a complete reflection property [5]. When a PC is inserted into the loop of an ordinary FLM shown as Fig. 1, the birefringence is induced into the FLM. For the fields in the counter-propagating directions, although the birefringence intensities of the PC are identical, the angles of their polarization orientations to the fast axis of the PC at their entrances may be different. This may cause the phase delays for the counter-propagating waves to be different during their passing through the PC, giving rise to an extra phase difference between the counter-propagating waves after a round-trip. Thus, the reflectivity of the FLM may be determined by both the OC coupling coefficient and the birefringence of the PC. Since the fast axis orientation and the birefringence intensity of the PC are controllable and adjustable (see Appendix), the reflectivity of the FLM shown as Fig. 1 may be varied accordingly. 3. Modelling By analyzing the routings of the counter-propagating waves in the fiber loop, we can obtain the equivalent optical path of the FLM as Fig. 2. Here the phenomenological ROTs are introduced to describe the polarization orientation changes of the two orthogonally polarized components in the counter-propagating directions due to the special topology of the FLM [15]. As shown in the figure, the waves in the counter-propagating directions travel independently, with the routing of single mode fiber 1ðSMFcw 1 Þ !
2. Configuration and principle Fig. 1 schematically shows the configuration of the FLM studied in this paper, which is formed by inserting a fiber PC into the loop of an ordinary FLM. For an ordin-
323
Fig. 2. Equivalent optical path of the FLM.
324
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cw rotator 1ðROTcw ! rotator 2ðROTcw 1 Þ ! PC 2 Þ ! single cw mode fiber 2ðSMF2 Þ, and single mode fiber 2ðSMFccw 2 Þ! ccw rotator 2ðROTccw ! rotator 1ðROTccw 2 Þ ! PC 1 Þ ! single mode fiber 1ðSMFccw 1 Þ, for the clockwise and the counter clockwise, respectively. Ei;x cw ccw By respectively writing Ei, E and E as Ei ¼ , Ei;y cw ccw Ex Ex and Eccw ¼ , we have Ecw ¼ Ecw Eccw y y cw En Ei;n ¼ T ð1Þ OC Eccw 0 n " # 1=2 1=2 1 ð1 K Þ jK n n where T OC ¼ ð1 cÞ2 is the 1=2 jK n1=2 ð1 K n Þ
nent, respectively. Note that the transfer matrixes of the PC cw T have the property of T ccw PC ¼ ðT PC Þ [9,11]. The reflected light Er and transmitted light Et of the FLM may respectively be given by " ccw # E0;x Er;x ð4Þ ¼ T OC Ecw Et;x 0;x " ccw # E0;y Er;y ¼ T OC ð5Þ Ecw Et;y 0;y
transfer matrix of the OC, with Kn and c being the coupling coefficient and the insertion loss of the OC (c has been assumed to be polarization-independent), respectively; n = x, y stands for the two orthogonally polarized components. According to the equivalent optical path in Fig. 2, we obtain " cw # " cw # E0;x Ex cw cw cw cw cw ¼ T SMF2 T ROT2 T PC T ROT1 T SMF1 cw ð2Þ cw E0;y Ey " ccw # " ccw # Ex E0;x ccw ccw ccw ccw ¼ T ccw ð3Þ SMF1 T ROT1 T PC T ROT2 T SMF2 Eccw Eccw 0;y y
ð7Þ
1 0 jðkl1;2 þjal1;2 Þ where ¼ ¼ e are the 0 1 transfer matrixes of SMF 1 and 2, with k being the wave vector, l1,2 the fiber lengths, and a the fiber loss coefficient, ccw respectively; T cw ROT1;2 and T ROT1;2 are the transfer matrixes of ROT 1 and 2 for the waves propagating in clockwise and counterclockwise, respectively, and they satisfy the T cw SMF1;2
T ccw SMF1;2
by using (2)–(5), we obtain Er;x A B Ei;x ¼ ð1 cÞej½kðl1 þl2 Þþjaðl1 þl2 Þ Er;y C D Ei;y Ei;x Et;x E F ¼ ð1 cÞej½kðl1 þl2 Þþjaðl1 þl2 Þ Et;y G H Ei;y where 8 A ¼ 2ðsin d cos 2X j cos dÞK x1=2 ð1 K x1=2 Þ > > > > 1=2 1=2 > B ¼ C ¼ sin d sin 2X½K y1=2 ð1 K 1=2 > x Þ K x ð1 K y Þ > > > > 1=2 1=2 > > < D ¼ 2ðsin d cos 2X þ j cos dÞK y ð1 K y Þ E ¼ ðcos d þ j sin d cos 2XÞð2K x 1Þ > > > F ¼ j sin d sin 2X½K 1=2 K 1=2 þ ð1 K x Þ1=2 ð1 K y Þ1=2 > > x y > > > 1=2 > > G ¼ j sin d sin 2X½K x K y1=2 þ ð1 K x Þ1=2 ð1 K y Þ1=2 > > : H ¼ ðcos d j sin d cos 2XÞð1 2K y Þ
ð8Þ Using the definitions of the reflectivity and transmittivity of R ¼ Er Er =Ei Ei and T ¼ Et Et =Ei Ei , after straightforward matrix manipulation and having neglected the cross terms of the orthogonal polarization components, we arrive at
R¼
ðAEi;x þ BEi;y ÞðAEi;x þ BEi;y Þ þ ðCEi;x þ DEi;y ÞðCEi;x þ DEi;y Þ 2 ð1 cÞ e2aðl1 þl2 Þ Ei;x Ei;x þ Ei;y Ei;y
T ¼
ðEEi;x þ FEi;y ÞðEEi;x þ FEi;y Þ þ ðGEi;x þ HEi;y ÞðGEi;x þ HEi;y Þ 2 ð1 cÞ e2aðl1 þl2 Þ Ei;x Ei;x þ Ei;y Ei;y
T ðT ccw ROT1 Þ
ð9Þ
1 0 ¼ and T cw ROT2 ¼ 0 1 [9], with the superscript T stand-
T cw ¼ ROT1
ð6Þ
ð10Þ
relations of 1 0 T ccw ðT ROT2 Þ ¼ 0 1 ing for the transposing manipulation; T cw PC ¼ cos d þ j sin d cos 2X j sin d sin 2X is the transj sin d sin 2X cos d j sin d cos 2X fer matrix of the PC for the field propagating in the clockwise direction, with 2d being the birefringence intensity and defined as the phase difference between the fast and the slow axes of the PC, and X the angle of the fast axis orientation of the PC with respect to the x polarization compo-
Further assuming that the coupling coefficient of the OC is polarization-independent, i.e. Kx = Ky = K, we finally obtain the following expressions: 2
R ¼ 4Kð1 KÞð1 cÞ e2aðl1 þl2 Þ ðcos2 d þ sin2 d cos2 2XÞ ð11Þ 2 2aðl1 þl2 Þ
T ¼ ð1 cÞ e 2
2
2
2
½sin d sin 2X þ ð2K 1Þ
ðcos d þ sin d cos2 2XÞ
2
ð12Þ
Eqs. (11) and (12) show that the reflectivity and transmittivity of the FLM are not only determined by the coupling
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coefficient of the OC, but also by the fast axis orientation X and the birefringence intensity 2d of the PC. 4. Simulation results In this section we simulate the influences of the PC state on the reflectivity characteristics of the FLM with Eq. (11). For simplicity and without the loss of generality, the coupling coefficient of the OC is chosen to be 0.5, and both the transmission loss of the fiber and the insertion losses of the OC are neglected in the simulations. Fig. 3a shows the reflectivity of the FLM as a function of X for different 2d. As seen in the figure, when 2d = 0, 2p, the reflectivity of the FLM does not change with X, keeping the maximum reflectivity of 1. When 2d 5 0, 2p, the reflectivity of the FLM oscillates periodically as X varies, with the maximum reflectivity of 1 at X = 0, p/2, p, and the minimum at X = p/4, 3p/4, respectively. Note that for different 2d the minimum reflectivities are different, only when 2d = p, the minimum reflectivity could reach zero. In fact, X and 2d respectively stand for the fast axis orientation and the birefringence intensity of the PC. When 2d = 0, 2p, corresponding to a birefringence intensity of zero, the FLM shown in Fig. 1 then degenerates to an ordinary FLM for any X, resulting in the complete reflection of the FLM. When X = 0, p/2, p, the birefringence fast axis of the PC is parallel or perpendicular to the orthogonally polarized compo-
325
nents, which causes the phase delays of both orthogonally polarized components in the counter-propagating directions to be the same whatever 2d is, thus, the FLM also degenerates to an ordinary FLM, exhibiting the complete reflection property. When X = p/4, 3p/4, the angles of the fast axis to the two orthogonally polarized components are the same and equal to p/4, giving the strongest influence of 2d on the reflectivity of the FLM. Fig. 3b shows the reflectivity of the FLM as a function of 2d for different X. We can see that when X = 0, 2p, at which the birefringence fast axis of the PC is parallel or perpendicular to the two orthogonally polarized components, the reflectivity of the FLM is independent of 2d, being a constant of 1. When X 5 0, 2p, the reflectivity of the FLM also varies periodically with 2d, with the maximum reflectivity of 1 at 2d = 0, 2p and the minimum at 2d = p where the birefringence intensity of the PC is the minimum and the maximum, respectively. Similarly, the value of the minimum reflectivity at 2d = p depends on X, and could reach zero with X being p/4 or 3p/4. To further show the influences of the PC state on the reflectivity of the FLM, the reflectivity as functions of X and 2d is illustrated in Fig. 3c. This figure clearly shows the effect of X and 2d, and their combined effect on the reflectivity of the FLM. Meanwhile, the figure also shows that the reflectivity of the FLM may be more flexible adjusted to expected value by varying both X and 2d.
Fig. 3. Influences of X and 2d on the reflectivity of the FLM. (a) The reflectivity as a function of X for a 2d of 0 or 2p (curve 1), p/3 or 5p/3 (curve 2), 3p/5 or 7p/5 (curve 3), and p (curve 4); (b) the reflectively as a function of 2d for a X of 0 or p/2 or p (curve 1), p/10 or 2p/5 or p/5 or 9p/10 (curve 2), p/7 or 5p/14 or 9p/14 or p/6 (curve 3), and p/4 or 3p/4 (curve 4); and (c) the reflectivity as functions of X and 2d.
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As in the case of an ordinary FLM, the OC coupling coefficient, K, also affects the reflectivity of the FLM. Fig. 4a and b shows the reflectivity as functions of K and X for 2d = p, and 2d for X = p/4, respectively. As seen, for any X or 2d, the reflectivity of the FLM reaches the maximum when K = 0.5, and decreases gradually as K deviates from 0.5, which is the same as that for the common FLM. When 2d = p, or when X = p/4 or 3p/4, however, the reflectivity of the FLM keeps being zero for any value of K, which is different from the case of an ordinary FLM. Now, we turn to the reflectivity spectral characteristics of the FLM. In general, the coupling coefficient of an OC is a function of the operating wavelength, and may be approximately expressed as K(k) = [cos(xk u)]/2. On the other hand, X is the birefringence fast axis orientation of the PC and is independent of the wavelength. However, 2d may be expressed as a function of wavelength by 2d(k) = 2pLPCDn/k, where LPC is the effective length of the PC; Dn is the refractive index difference between the fast and the slow axes of the PC. Using K(k) and 2d(k) into Eq. (11), and ignoring the dispersion of Dn, we can obtain the reflectivity spectral characteristics of the FLM. Fig. 5a and b shows the simulation results for Dn = 0, 1.5 · 107, 2.5 · 107 and 3.95 · 107 when X is fixed at p/4 and for X = 0, p/12, p/8 and p/4 when Dn is fixed at 3.95 · 107, respectively. Here LPC is assumed to be 2m, and the OC is assumed with a coupling coefficient of 50:50 at the central operating wavelength of 1570 nm and the FWHM of
Fig. 5. Reflectivity spectral characteristics of the FLM with a coupling coefficient of 50:50 at the central operating wavelength of 1570 nm. (a) X = p/4, Dn = 0 (curve 1), 1.5 · 107 (curve 2), 2.5 · 107 (curve 3) and 3.95 · 107 (curve 4); (b) Dn = 3.95 · 107, X = 0 (curve 1), p/8 (curve 2), p/6 (curve 3) and p/4 (curve 4).
60 nm, which gives x = p/90 and x = 157p/9 for K(k). As seen, the reflectivity of the FLM has wide and flattened spectral property over a range of 60 nm with the peak reflectivity right at the central operating wavelength of the OC, implying that the reflectivity characteristic is mainly determined by the coupling coefficient spectrum of the OC. Moreover, the reflectivity may be adjusted to any value between 0 and 1 by varying X or 2d. 5. Experiments
Fig. 4. Influence of K on the reflectivity of the FLM. (a) 2d = p and (b) X = p/4.
In this section, we further investigate, experimentally, the reflectivity characteristics of the FLM. The FLM is formed by an OC with a nominal coupling coefficient of 50:50 at the central wavelength of 1570 nm and an operating bandwidth of C plus L bands, and with an ordinary PC, which consists of three rigid discs with the single mode fiber being coiled, respectively functioning as k/4, k/2 and k/4 wave-plates (see Fig. 9 in Appendix), being inserted into the fiber loop. Both the OC and the PC are made with Corning SMF-28. The fiber length of the FLM is about 2 m. The testing scheme is illustrated as Fig. 6. A tunable laser and a broadband ASE source based on an erbiumdoped fiber amplifier (EDFA) are respectively used as the signal sources. The signal light is launched into the FLM with a three-port optical circulator (CIR), which functions as an optical isolator to eliminate the possible reflections in
S. Feng et al. / Optics Communications 277 (2007) 322–328
327
Fig. 6. Experimental setup. CIR: optical circular; ISO: optical isolator.
the input end of the FLM, for the same purpose, an optical isolator (ISO) is also inserted in another end of the FLM. The reflected and the transmitted lights of the FLM are measured with an optical spectrum analyzer and a power meter, respectively. The reflectivity of the FLM is first measured as a function of X or 2d with the tunable laser. Note that the values of X and 2d may be determined by the values of h1, h2 and h3, according to (A.6) given in the Appendix. For an example, when h1 = h3 = 0, we obtain X = p/4, 2d = 2p 4h2 by using (A.6), meaning that if h1 and h3 are fixed at 0, then X keeps being p/4, and 2d may be determined when h2 is adjusted from 0 to p/2 or from p/2 to p. In such an adjustment, the measured reflectivity of the FLM is plotted as a function of 2d for the wavelength at 1570 nm, as illustrated in Fig. 7a. The solid and the open circles are the data points measured when h2 is adjusted from 0 to p/2, and from p/2 to p, respectively. As shown in the figure, the reflectivity may be set to any value between its maximum and minimum by adjusting 2d (or h2). Moreover, the measured reflectivity is in agreement with the simulation results by taking the insertion losses with c = 0.05 into account, which is also shown with a solid line in Fig. 7a. Fig. 7b shows the measured reflectivity as a function of X for the wavelength of 1570 nm. To obtain the data points, this time h3 is fixed at p/2, and h2 is kept being half of h1 and/or (h1 + p)/2 while h1 is being adjusted from 0 to p, corresponding to the condition that 2d is fixed at p and X is adjusted with h2 according to (A.6). Here we can see that the measured reflectivity may also be set to any value between its maximum and minimum by adjusting X, and is
Fig. 7. Measured reflectivity of the FLM. (a) X = p/4, solid and the open circles are the data points measured when h2 is respectively adjusted from 0 to p/2 and from p/2 to p to change 2d; (b) 2d = p. Solid lines in both figures are given by the simulations.
Fig. 8. Typical reflectivity spectra of the FLM measured in the condition the same as that in Fig. 7a.
again in agreement with the simulation results (solid line). It is worthwhile to point out that the reflectivity characteristics of the FLM with various sets of adjustments in terms of h1, h2 and h3 are also tested and the measured results are in agreement with those obtained from the simulations. The reflectivity spectral characteristics of the FLM are then investigated with the broadband ASE source. Fig. 8 shows a set of typical experimental results obtained with the condition the same as that for Fig. 7a, i.e. X = p/4, 2d is adjusted from 0 to p/2. Here we can see clearly that the measured reflectivity spectra are wide and flattened, the fluctuation of the reflectivity is about 4% in the wavelength range of 60 nm; the reflectivity can be adjusted to any value between the maximum of 93% and the minimum of 2% by changing 2d (or h2). In addition, the measured reflectivity spectra with different sets of h1, h2 and h3 also verify that the reflectivity spectrum is wide and flattened. Obviously, such a wide and flattened all-fiber reflector with adjustable reflectivity will play an important role in the cavity optimization of fiber lasers, fiber sensing and optical component characterizations. 6. Conclusions We have investigated the reflectivity characteristics of the FLM formed by inserting a PC into the fiber loop based on an OC. With the equivalent optical path technique and the matrix optics approach, a theoretical model has been proposed for determining the reflectivity characteristics of the FLM. The simulation results obtained with the model show that the reflectivity of the FLM may be continuously adjusted by either changing the fast axis orientation or the birefringence intensity of the PC. The adjustable range is between 0 and 1 if the OC coupling coefficient is 0.5 and the loss is neglected, and decreases if the OC coupling coefficient offsets from 0.5. The simulation results also show that the reflectivity spectrum is wide and flattened, mainly determined by the coupling coefficient spectrum of the OC. The reflectivity characteristics of the FLM have further been investigated experimentally to prove the theoretical results. With an FLM formed by a 3-dB broadband OC and an ordinary PC, the experimental results verify that, the
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reflectivity spectrum of the FLM is truly wide and flattened, and the reflectivity can be adjusted continuously between its maximum and minimum which are measured to be 93% and 2%, respectively, by changing the PC state. Moreover, the experimental results are in agreement with those of the simulation. Obviously, such a FLM with the broadband continuously adjustable reflectivity can play an important role in the fiber devices and systems. Acknowledgement This work is supported by the ‘‘Hundreds of Talents Program’’ of the Chinese Academy of Sciences and by the National Natural Science Foundation of China (Grant No. 60677050). The authors would like to thank Mr. Qing Sun for his help. Appendix Fig. 9 shows schematically the structure of an ordinary PC, which consists of three rigid discs with the single mode fiber being coiled. According to [16], the discs with label of 1, 2 and 3 function as k/l, k/m and k/n wave-plates with l, m and n being real numbers, respectively. The combined effect of the three wave-plates allows continuous, reset-free polarization transformations from an input arbitrary polarization state to a specific output polarization state by rotating the three rigid discs. If the angular orientations of the three wave-plates are h1, h2 and h3 relative to the xoz plane, the transfer matrixes of the three wave-plate are [17] " # cos u2i þ j sin u2i cos 2hi j sin u2i sin 2hi Ti ¼ j sin u2i sin 2hi cos u2i j sin u2i cos 2hi ði ¼ 1; 2 and 3Þ
ðA:1Þ
where u1 = 2p/l, u2 = 2p/m and u3 = 2p/n, respectively. Then the transfer matrix of the PC can be derived as T PC ðl; m; n; h1 ; h2 ; h3 Þ ¼ T 3 T 2 T 1
ðA:2Þ
Obviously, the PC transmission characteristic is determined by both the values of l, m and n, and the orientations of the three wave-plates to the horizontal direction. On the other hand, the transfer matrix of the PC can also be expressed with the fast axis orientation X and the birefringence intensity 2d of a linear birefringence component [11] cos d þ j sin d cos 2X j sin d sin 2X T PC ðd; XÞ ¼ j sin d sin 2X cos d j sin d cos 2X ðA:3Þ
Fig. 9. Schematic view of ordinary commercial PC.
By comparing (A.2) to (A.3), we can obtain the relations between X and d and h1, h2 and h3 for a specific PC with the values of l, m and n having been given. For most commercial PCs, l, m and n are respectively designed to be 4, 2 and 4, then the transfer matrix given in (A.2) becomes j M N T PC ðh1 ; h2 ; h3 Þ ¼ T 3 T 2 T 1 ¼ ðA:4Þ 2 O P where 8 M ¼ cos 2h2 þ j cosð2h1 2h2 Þ þ j cosð2h2 2h3 Þ > > > > > cosð2h1 2h2 þ 2h3 Þ > > > > > N ¼ sin 2h2 þ j sinð2h1 2h2 Þ þ j sinð2h2 2h3 Þ > > > < sinð2h1 2h2 þ 2h3 Þ > O ¼ sin 2h2 j sinð2h1 2h2 Þ j sinð2h2 2h3 Þ > > > > > sinð2h1 2h2 þ 2h3 Þ > > > > > P ¼ cos 2h2 þ j cosð2h1 2h2 Þ þ j cosð2h2 2h3 Þ > > : þ cosð2h1 2h2 þ 2h3 Þ ðA:5Þ Comparing (A.4) to (A.3), we can obtain 8 cos 2h2 cosð2h1 2h2 þ 2h3 Þ ¼ 2 sin d cos 2X > > > < cosð2h 2h Þ þ cosð2h 2h Þ ¼ 2 cos d 1 2 2 3 > sinð2h 2h Þ þ sinð2h 2h 1 2 2 3Þ ¼ 0 > > : sin 2h2 sinð2h1 2h2 þ 2h3 Þ ¼ 2 sin d sin 2X
ðA:6Þ
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