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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 3 — Jan. 31, 2011
  • pp: 2251–2256
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Adiabatic circular polarizer based on chiral fiber grating

Li Yang, Lin-Lin Xue, Cheng Li, Jue Su, and Jing-Ren Qian  »View Author Affiliations


Optics Express, Vol. 19, Issue 3, pp. 2251-2256 (2011)
http://dx.doi.org/10.1364/OE.19.002251


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Abstract

Based on the adiabatic coupling principle, a new scheme of a broadband circular polarizer formed by twisting a high-birefringence (Hi-Bi) 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 OSA

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

1. 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). [CrossRef] [PubMed]

5

5. 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). [CrossRef]

]. The most salient property may be the polarization-selective coupling of circularly polarized modes [1

1. 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). [CrossRef] [PubMed]

], which is well explained by the coupled-mode analysis with the view of local normal modes [6

6. 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). [CrossRef]

,7

7. 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 toJ. Lightwave Technol.

]. 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

3. 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). [CrossRef]

]. In a chiral fiber long-period grating (CLPG) with a certain twist handedness (right-handed as shown in Fig. 1
Fig. 1 CLPG formed by twisting a panda fiber.
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

1. 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). [CrossRef] [PubMed]

,3

3. 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). [CrossRef]

]. 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

5. 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). [CrossRef]

]. 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

8. W. H. Louisell, “Analysis of the single tapered mode coupler,” Bell Syst. Tech. J. 34(4), 853–870 (1955).

10

10. X. Sun, H. C. Liu, and A. Yariv, “Adiabaticity criterion and the shortest adiabatic mode transformer in a coupled-waveguide system,” Opt. Lett. 34(3), 280–282 (2009). [CrossRef] [PubMed]

] 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

11. H. C. Huang, “Fiber-optic analogs of bulk-optic wave plates,” Appl. Opt. 36(18), 4241–4258 (1997). [CrossRef] [PubMed]

] 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

Coupled-mode equations for linearly polarized modes in a fiber with an anisotropic core were formulated early in 1986 [12

12. J.-R. Qian and W.-P. Huang, “Coupled-mode theory for LP modes,” J. Lightwave Technol. 4(6), 619–625 (1986). [CrossRef]

]. 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

6. 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). [CrossRef]

]. 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

6. 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). [CrossRef]

,7

7. 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 toJ. Lightwave Technol.

]. To make the coupling vanishing, by using mode transformations from linearly polarized modes to circularly polarized ones, the coupled-mode equations for circularly polarized modes are expressed as,

ddz[WcolWcorWcllWclr]=j[βco+τ(z)0κ'jκ0βcoτ(z)jκκ'κ'jκβcl+τ(z)0jκκ'0βclτ(z)][WcolWcorWcllWclr]
(1)
κ=ωε0(Δεxe11xe1mxΔεye11ye1my)ds/2κ'=ωε0(Δεxe11xe1mx+Δεye11ye1my)ds/2
(2)

Since β co>β cl for all cladding modes and τ is positive for right handed structures, only the following phase matching condition can be fulfilled,

βcoτ(z)=βcl+τ(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

ddz[WcorWcll]=j[βcoτ(z)jκjκβcl+τ(z)][WcorWcll]
(4)

which is essentially the same as that obtained by Shvets et al by using a general coupled-mode perturbation theory [5

5. 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). [CrossRef]

], 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),

δ(z)=βcoβcl2τ(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

8. W. H. Louisell, “Analysis of the single tapered mode coupler,” Bell Syst. Tech. J. 34(4), 853–870 (1955).

], we first introduce two tapered modes with the amplitudes of N 1(z) and N 2(z) by using a mode transformation, and obtain,

[N1(z)N2(z)]=[cos[φ(z)/2]jsin[φ(z)/2]jsin[φ(z)/2]cos[φ(z)/2]][Wcor(z)Wcll(z)]
(6)
φ(z)=arctan[2κ/δ(z)]
(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,

ddz[N1(z)N2(z)]=[j[(βco+βcl)/2+β˜(z)]jφ'(z)/2jφ'(z)/2j[(βco+βcl)/2β˜(z)]][N1(z)N2(z)]
(8)

where β˜(z)=δ2/4+κ2 and φ'(z)=dφ/dz.

Then define η(z)=φ'(z)/[2β˜(z)] to measure the strength of the coupling [8

8. W. H. Louisell, “Analysis of the single tapered mode coupler,” Bell Syst. Tech. J. 34(4), 853–870 (1955).

]. If τ varies so slowly that makes η<<1, and then the coupling will be very weak. The amplitudes of two tapered modes at the final end z = L are thus obtained within the first-order approximation in η,

N1(L)exp[jρ0(L)+jρ(L)]{N1(0)+jN2(0)0Ldzφ'(z)exp[2jρ(z)]/2}N2(L)exp[jρ0(L)jρ(L)]{N2(0)+jN1(0)0Ldzφ'(z)exp[2jρ(z)]/2}
(9)

where ρ0(z)=(βco+βcl)z/2, ρ(z)=0zβ˜(z')  dz'.

An ideal adiabatic polarizer is also a perfect coupler, namely, if the initiate condition is W co r(0) = 1, W cl l(0) = 0 at the input end of the grating, then we will have W co r(L) = 0, W cl l(L) = 1 at the output end z = L. In order to realize this, we require: 1. φ(0) = 0 at the input end. Thus, as seen from Eq. (6), only the first tapered mode is excited by the right circularly polarized core mode at the input end, i.e. N 1(0) = 1, N 2(0) = 0. 2.τ varies slowly, then φ′(z) is very small and the cross-coupling is negligible. Thus, only the tapered mode excited at the input end propagates along the grating, i.e. N 1(z) = 1, N 2(z) = 0, at any point along z. Although the amplitude remains constant, tapered modes are tapered slowly by the variation of τ or φ, and the first tapered mode varies from the right circularly polarized core mode at the input end to the left circularly polarized cladding mode at the output end. 3. φ(L) = π, then as seen from Eq. (6) that N 1(L) = W cl l(L) = 1, N 2(L) = W co r(L) = 0, the output mode is the left circularly polarized cladding mode at the output end. According to Eq. (7), the above three requirements can be nearly fulfilled in practice if τ(z) or δ(z) is well managed as follows: 1. κ/δ is positive and much smaller than unity at one end; 2. τ varies so slowly along z that η is much smaller than unity [8

8. W. H. Louisell, “Analysis of the single tapered mode coupler,” Bell Syst. Tech. J. 34(4), 853–870 (1955).

]; 3. κ/δ is negative and its absolute value is much smaller than unity at the other end. Since κ is assumed to be positive and is a constant independent of z [7

7. 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 toJ. Lightwave Technol.

], δ should be positive and much larger than κ at one end, then it reduces monotonically and slowly through the zero point and becomes negative, finally, its absolute value is much larger than κ at the other end. This is the sufficient condition of δ or τ for achieving a practical adiabatic polarizer. Since δ changes with working wavelength, the above condition needs to be satisfied at any wavelength within the required bandwidth, except for those at or near the upper and lower limits of the band. That is why an adiabatic polarizer is capable for broadband use. Moreover, the bandwidth of the polarizer can be also easily managed by selecting the twist rates at the two ends, τ(0) and τ(L). Usually, the phase matching condition is set to be satisfied at the middle point of the grating at the central wavelength. For a CLPG with a monotonically increasing τ(z), the point where the phase matching occurs moves backward or forward along z when the working wavelength increases or decreases, respectively, from the central wavelength. When the phase matching occurs at the input or the output end of the grating, the wavelength approaches to the upper or lower limit of the bandwidth (λ max or λ min), respectively, and the following relationships are satisfied at the two ends of the polarizer,

βco(λmax)βcl(λmax)=2τ(0)
(10a)
2τ(L)[βco(λmax)βcl(λmax)]>>κ
(10b)
βco(λmin)βcl(λmin)=2τ(L)
(11a)
βco(λmin)βcl(λmin)2τ(0)>>κ
(11b)

Then from Eq. (7) we have φ(0) = π/2, φ(L)π or φ(0)0, φ(L) = π/2 at the wavelength of λ max or λ min, respectively. It implies that at these two wavelengths the adiabatic condition is fulfilled only at one end of the grating, respectively. It is obvious from Eq. (6) that half of the power of the incident right circularly polarized core mode will pass through the polarizer at the two wavelengths. Therefore, according to Eq. (10a) and Eq. (11a), the 3 dB bandwidth of an adiabatic polarizer is able to be determined easily by selecting the twist rates at two ends of the grating. Inserting Eq. (10a) and Eq. (11a) into Eq. (10b) and Eq. (11b), we have

2[τ(L)τ(0)]>>κ
(12)

This is the necessary condition for realizing an adiabatic broadband polarizer. Obviously, with the increase of the grating length, a broad bandwidth will be resulted from a large difference of the twist rates at the two ends. Besides, the amplitude of the residual right-handed circular polarized core mode at the output end W co r(L) will be remarkably reduced [8

8. W. H. Louisell, “Analysis of the single tapered mode coupler,” Bell Syst. Tech. J. 34(4), 853–870 (1955).

]. As a result, another important performance specification of the polarizer, the extinction ratio, which equals to 20log[|W co l(L)|/|W co r(L)|] where |W co l(L)| = |W co l(0)| = |W co r(0)| = 1 will be remarkably improved. It also indicates how high the extinction ratio can be achieved strongly depends on whether the adiabatic conditions are well fulfilled.

Actually, in a CLPG with a varying pitch, the phase matching condition is fulfilled for two or more cladding modes if the working bandwidth is sufficiently broad. Since the phase matchings occur at different places along the grating for the different cladding mode, this broadband polarizer can be considered as a cascade of several adiabatic polarizers. For each individual one, only a cladding mode is taken into account and the number of the individual ones is equal to that of cladding modes which are phase-matched with the core mode in the bandwidth. The resultant transmission spectrum is approximately the superposition of those of individual ones. Sometimes, the influence of the near-by cladding modes needs to be taken into account to acquire more precise individual spectra. The extinction ratio of such a broadband polarizer will be significantly improved, if the superposition is well designed, as seen in the following simulation.

3. Simulation results

We simulated the transmission spectra of the right circularly polarized core modes (RCPCM) in right-handed CLPGs by directly solving the coupled-mode equations in Eq. (4) with the transformation matrix method combined with piecewise uniform technique. The simulated results for two mode couplings agree well with those obtained by the approximate analytical expressions in Eq. (9).

Figure 2
Fig. 2 Transmission spectra of right-circular polarization core modes in CLPGs.
gives an example based on a two-mode coupling. The CLPG is formed by twisting a section of Panda fiber with a beat length of 4.0mm, a numerical aperture of 0.1525 and a length of 41.5cm. A twist pitch of 516μm at the middle point of the grating is selected to meet the phase matching condition between the core mode and the fifth order cladding mode at the central wavelength of 1.55μm. The twist pitch varies linearly from 496μm to 537μm, corresponding to the 3dB bandwidth of about 100nm according to Eq. (10a) and Eq. (11a). A resonant dip with 3 dB bandwidth of 12.5nm for a CLPG with a constant pitch of 515μm is also shown by dashed line for comparison. In the bandwidth of about 32nm around 1.55μm, the extinction ratio is better than 15 dB, which is not really sufficient for most applications.

The extinction ratio can also be improved by using the coupling to a mode with a slight and suitable loss [13

13. 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.

]. For the same circular polarizer described in Fig. 2, two lossy cladding modes with different imaginary parts of the effective refractive indices of 4 × 10−6 and 3.7 × 10−5 are considered respectively. The simulated transmission spectra are shown by solid and dashed line, respectively, in Fig. 3(b). It is found that the extinction ratio is better than 26 dB and 29 dB, respectively, in the bandwidth of 50 nm. It indicates that the adiabatic condition shown in Eq. (12) is relaxed by using the coupling to a lossy mode in a properly designed structure. Furthermore, the length of the circular polarizer can be shortened by this way.

6. Conclusion

In conclusion, it has been shown from the theoretical analysis and simulations that a broadband circular polarizer is achieved in a CLPG by twisting a Hi-Bi fiber with a slowly varying twist rate. Its bandwidth is determined by the twist rates at the two ends of the grating. Both the difference of the two twist rates and the length of gratings need to be sufficiently large to ensure the adiabatic conditions to be fulfilled. Fortunately, by achieving a multi-mode phase-matching along the grating or by using the couplings to lossy modes in the grating, the adiabatic conditions are remarkably relaxed, which makes the circular polarizer more practical for uses.

The work is supported by the National Natural Science Foundation of China under grant 60807023, the Fundamental Research Funds for the Central Universities, and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

References and links

1.

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). [CrossRef] [PubMed]

2.

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). [CrossRef]

3.

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). [CrossRef]

4.

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). [CrossRef] [PubMed]

5.

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). [CrossRef]

6.

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). [CrossRef]

7.

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 toJ. Lightwave Technol.

8.

W. H. Louisell, “Analysis of the single tapered mode coupler,” Bell Syst. Tech. J. 34(4), 853–870 (1955).

9.

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). [CrossRef]

10.

X. Sun, H. C. Liu, and A. Yariv, “Adiabaticity criterion and the shortest adiabatic mode transformer in a coupled-waveguide system,” Opt. Lett. 34(3), 280–282 (2009). [CrossRef] [PubMed]

11.

H. C. Huang, “Fiber-optic analogs of bulk-optic wave plates,” Appl. Opt. 36(18), 4241–4258 (1997). [CrossRef] [PubMed]

12.

J.-R. Qian and W.-P. Huang, “Coupled-mode theory for LP modes,” J. Lightwave Technol. 4(6), 619–625 (1986). [CrossRef]

13.

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.

OCIS Codes
(050.2770) Diffraction and gratings : Gratings
(060.2340) Fiber optics and optical communications : Fiber optics components
(230.5440) Optical devices : Polarization-selective devices

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: December 3, 2010
Revised Manuscript: January 14, 2011
Manuscript Accepted: January 14, 2011
Published: January 24, 2011

Citation
Li Yang, Lin-Lin Xue, Cheng Li, Jue Su, and Jing-Ren Qian, "Adiabatic circular polarizer based on chiral fiber grating," Opt. Express 19, 2251-2256 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-3-2251


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References

  1. 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). [CrossRef] [PubMed]
  2. 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). [CrossRef]
  3. 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). [CrossRef]
  4. 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). [CrossRef] [PubMed]
  5. 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). [CrossRef]
  6. 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). [CrossRef]
  7. 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 toJ. Lightwave Technol.
  8. W. H. Louisell, “Analysis of the single tapered mode coupler,” Bell Syst. Tech. J. 34(4), 853–870 (1955).
  9. 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). [CrossRef]
  10. X. Sun, H. C. Liu, and A. Yariv, “Adiabaticity criterion and the shortest adiabatic mode transformer in a coupled-waveguide system,” Opt. Lett. 34(3), 280–282 (2009). [CrossRef] [PubMed]
  11. H. C. Huang, “Fiber-optic analogs of bulk-optic wave plates,” Appl. Opt. 36(18), 4241–4258 (1997). [CrossRef] [PubMed]
  12. J.-R. Qian and W.-P. Huang, “Coupled-mode theory for LP modes,” J. Lightwave Technol. 4(6), 619–625 (1986). [CrossRef]
  13. 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.

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