Adiabatic circular polarizer based on chiral fiber ... - OSA Publishing

Adiabatic circular polarizer based on chiral fiber grating Li Yang,* Lin-Lin Xue, Cheng Li, Jue Su, and Jing-Ren Qian Department of Electronic Engineering and Information Science, University of Science and Technology of China, Hefei, Anhui, 230027, China *[email protected]

Abstract: Based on the adiabatic coupling principle, a new scheme of a broadband circular polarizer formed by twisting a high-birefringence (HiBi) fiber with a slowly varying twist rate is proposed. The conditions of adiabatic coupling for the adiabatic polarizer are first identified through analytical derivations. These conditions are easily realized by choosing a reasonable variation of the twist rate. Moreover, the bandwidth of the polarizer is able to be directly determined by the twist rates at the two ends. Finally, the broadband characteristics of the polarizer are demonstrated by simulations. It is also shown that the performance of the polarizer can be remarkably improved by accomplishing a multi-mode phase-matching along the grating or by using of the couplings of the core mode to lossy modes. ©2011 Optical Society of America OCIS codes: (060.2340) Fiber optics components; (230.5440) Polarization-selective devices; (050.2770) Gratings.

References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

V. I. Kopp, V. M. Churikov, J. Singer, N. Chao, D. Neugroschl, and A. Z. Genack, “Chiral fiber gratings,” Science 305(5680), 74–75 (2004). V. I. Kopp, V. M. Churikov, G. Y. Zhang, J. Singer, C. W. Draper, N. Chao, D. Neugroschl, and A. Z. Genack, “Single- and double-helix chiral fiber sensors,” J. Opt. Soc. Am. B 24(10), A48–A52 (2007). D. Neugroschl, V. I. Kopp, J. Singer, and G. Y. Zhang, “„Vanishing-core‟ tapered coupler for interconnect applications,” Proc. SPIE 7221, 72210G, 72210G-8 (2009). V. I. Kopp, V. M. Churikov, and A. Z. Genack, “Synchronization of optical polarization conversion and scattering in chiral fibers,” Opt. Lett. 31(5), 571–573 (2006). G. Shvets, S. Trendafilov, V. I. Kopp, D. Neugroschl, and A. Z. Genack, “Polarization properties of chiral fiber gratings,” J. Opt. A, Pure Appl. Opt. 11(7), 074007 (2009). J. R. Qian, Q. Guo, and L. Li, “Spun linear birefringence fibers and their sensing mechanism in current sensors with temperature compensations,” IEE Proc., Optoelectron. 141(6), 373–380 (1994). J. R. Qian, J. Su, L. L. Xue, and L. Yang, “Coupled-mode analysis for chiral fiber long-period gratings using local mode approach” submitted to J. Lightwave Technol. W. H. Louisell, “Analysis of the single tapered mode coupler,” Bell Syst. Tech. J. 34(4), 853–870 (1955). J. D. Love, W. M. Henry, W. J. Stewart, R. J. Black, S. Lacroix, and F. Gonthier, “Tapered single-mode fibres and devices. Part 1: Adiabaticity criteria,” IEE Proc., J Optoelectron. 138(5), 343–354 (1991). X. Sun, H. C. Liu, and A. Yariv, “Adiabaticity criterion and the shortest adiabatic mode transformer in a coupledwaveguide system,” Opt. Lett. 34(3), 280–282 (2009). H. C. Huang, “Fiber-optic analogs of bulk-optic wave plates,” Appl. Opt. 36(18), 4241–4258 (1997). J.-R. Qian and W.-P. Huang, “Coupled-mode theory for LP modes,” J. Lightwave Technol. 4(6), 619–625 (1986). L. L. Xue, L. Yang, H. X. Xu, J. Su, and J. R. Qian, “A novel all-fiber circular polarizer PGC2010,” presented at Photonics Global Conference, Singapore, Dec.14–16, 2010.

1. Introduction Double-helix chiral fiber gratings formed by twisting Hi-Bi fibers with pitches of hundreds of microns were reported by Kopp et al; their salient properties and multiple promising applications were demonstrated later [1–5]. The most salient property may be the polarizationselective coupling of circularly polarized modes [1], which is well explained by the coupledmode analysis with the view of local normal modes [6,7]. Moreover, the mode coupling mechanism of chiral fiber gratings with different orders of pitches are similar to that of conventional fiber gratings with long or short periods [3]. In a chiral fiber long-period grating #139068 - $15.00 USD

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Received 3 Dec 2010; revised 14 Jan 2011; accepted 14 Jan 2011; published 24 Jan 2011

31 January 2011 / Vol. 19, No. 3 / OPTICS EXPRESS 2251

(CLPG) with a certain twist handedness (right-handed as shown in Fig. 1 or left-handed), the co-handed (right or left) circularly polarized core mode will couple to the co-propagating cross-handed (left or right) circularly polarized cladding modes and suffer from loss at certain resonant wavelengths, while the cross-handed circularly polarized core mode will pass through. By using one of these resonant couplings, a cross-handed circularly polarized filter would thus be developed. The bandwidth of the filter is about ten nanometers, which is too narrow for a circular polarizer. Kopp et al. have demonstrated a broadband circular polarizer by decreasing the twist pitch to tens of microns and making use of the coupling with radiation modes in chiral intermediate period gratings (CIPG) [1,3]. However, this scheme may be harder to implement because of the shorter period controlling. Shvets et al. have proposed a perfect circular polarizer regardless of its precise length by chirping the period of the CLPG [5]. In fact, their perfect polarizer may also have broadband properties if it is well designed as expounded below. In this paper, based on CLPGs, we utilize the adiabatic coupling principle [8–10] to achieve a broadband circular polarizer. This principle was used successfully in microwaves to achieve broadband couplers. In the coupler only one local mode or tapered mode was excited, it was then tapered from the mode at the input end to the other wanted mode at the output end by slowly tapering a parameter of the coupler, while the coupling between the tapered modes was suppressed by the slow variation of the tapering parameter. Based on the principle, a circular polarizer of CLPG with a slowly varying twist rate is proposed. Kopp et al. have experimentally demonstrated a broadband in-fiber linear polarizer, where the adiabatic twist proposed by Huang [11] was used to transform the polarization state of the core mode from the circular state to the linear one, while we will use the adiabatic coupling to convert the input power from the core mode to a certain cladding mode. The conditions of the adiabatic coupling between modes are not so simple as those for the adiabatic conversion between polarization states, which will be presented in this paper. Moreover, analytic relations among performance specifications such as the bandwidth and the extinction ratio of a polarizer and structure parameters such as the variation of twist rate, the fiber birefringence and the grating length are obtained, which provides a foundation for the optimization of the polarizer. From the simulated transmission spectra of the polarizers, we have found the grating length is too long to achieve a practical device if the adiabatic condition is fulfilled strictly. This means the performance of the device will be degraded if its length is in a reasonable length. Two approaches are thus proposed to relax the adiabatic condition, and their effectivities are confirmed by the simulations. 2. Theoretical analysis

Fig. 1. CLPG formed by twisting a panda fiber.

Coupled-mode equations for linearly polarized modes in a fiber with an anisotropic core were formulated early in 1986 [12]. Modes propagating along a spun Hi-Bi fiber were described by the coupling between two orthogonal linearly polarized modes, and circularly polarized modes were proved to be the eigenmodes in the fiber [6]. Based on these, for the CLPG with a right-handed twist structure and a high twist rate as shown in Fig. 1, coupled-mode equations for x- and y-polarized core and cladding modes are formulated directly in local coordinates, where the coupling between a pair of x- and y-polarized core (or cladding) modes is due to the twist [6,7]. To make the coupling vanishing, by using mode transformations from linearly #139068 - $15.00 USD

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polarized modes to circularly polarized ones, the coupled-mode equations for circularly polarized modes are expressed as, W lco    co    z  0 ' j  W lco   r   r  0  co    z  j  '  W co d W co      j   W lcl  '  j  cl    z  0 dz W lcl       ' 0  cl    z   W clr    j W clr 

   0    x e11x  e1xm   y e11y  e1ym ds 2

 '   0    x e11x  e1xm   y e11y  e1ym ds 2

(1)

(2)

where  and  ' are the coupling coefficients. τ(z) is the twist rate that varies slowly along the fiber axis z. τ(z) = 2π/p(z), where p(z) is the twist pitch in the order of 100μm. Wcol, Wcll and Wcor, Wclr denote the amplitudes of the left and right circularly polarized core and cladding modes, respectively. βco and βcl are the phase constants of the core mode HE11 and a cladding mode HE1m in a perfect isotropic fiber, respectively. e11x , e11y , e1xm and e1ym are the corresponding distribution of modal fields. The superscript x or y denotes that the dominant transverse component of the mode is x- or y-polarization. Δεx(x,y) or Δεy(x,y) is anisotropic perturbation for x- or y-polarized lights to the dielectric constant distribution in the cross section of a perfect isotropic fiber, induced by the birefringence. In the derivation, the pair of circularly polarized core modes and a certain pair of circularly polarized cladding modes are taken into account. Though in strict sense, all the cladding modes are possibly coupled to the core mode, only the one whose phase constant is matched with that of the core mode is considered here. Since βco>βcl for all cladding modes and τ is positive for right handed structures, only the following phase matching condition can be fulfilled,

co    z    cl    z 

(3)

which enables the interaction between the right circularly polarized core and the left circularly polarized cladding mode stronger than that between other modes. Thus, Eq. (1) becomes r r   co    z  j  W co   d W co  l   l    j   j    z dz W cl    W cl  cl 

(4)

which is essentially the same as that obtained by Shvets et al by using a general coupled-mode perturbation theory [5], except that the present twist rate τ varies with z. We define δ(z) as the phase constant difference between the two modes discussed in Eq. (4),

  z   co  cl  2  z 

(5) Thus, the phase matching condition can be simply reduced to δ(z) = 0. Since τ slowly varies along z and δ(z) is easy to become zero somewhere in the grating, the adiabatic coupling principle can be used to achieve an adiabatic polarizer as it was used in an adiabatic coupler. Following a similar procedure in [8], we first introduce two tapered modes with the amplitudes of N1(z) and N2(z) by using a mode transformation, and obtain,

 N1 ( z )   cos   z  2  N ( z)    2   jsin   z  2

r jsin   z  2  W co ( z)  l  cos   z  2  W cl( z ) 

  z   arctan  2   z  

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(6)

(7)



It is easy to see from Eq. (6) that tapered modes are composite modes composed of the core and the cladding mode, the relative ratio of the compositions changes with φ along z. Inserting Eq. (6) into Eq. (4), coupled-mode equations for two tapered modes are obtained as,    N ( z)  j '  z  2 d  N1 ( z )    j   co   cl  2    z     1  (8)  dz  N 2 ( z )     j '  z  2  j   co   cl  2    z     N 2 ( z )  

where   z    2 4   2 and  '  z   d dz . Then define   z    '  z   2  z   to measure the strength of the coupling [8]. If τ varies so slowly that makes η