## 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 Optical Society of America

## 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 polarization-selective coupling of circularly polarized modes [1], which is well explained by the coupled-mode 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 (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

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 polarized modes to circularly polarized ones, the coupled-mode equations for circularly polarized modes are expressed as,

where *κ* and $\kappa \text{'}$ 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. *W*
_{co}
* ^{l}*,

*W*

_{cl}

*and*

^{l}*W*

_{co}

*,*

^{r}*W*

_{cl}

*denote the amplitudes of the left and right circularly polarized core and cladding modes, respectively.*

^{r}*β*

_{co}and

*β*

_{cl}are the phase constants of the core mode HE

_{11}and a cladding mode HE

_{1m}in a perfect isotropic fiber, respectively. ${\overrightarrow{e}}_{11}^{x}$,${\overrightarrow{e}}_{11}^{y}$,${\overrightarrow{e}}_{1m}^{x}$and ${\overrightarrow{e}}_{1m}^{y}$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. Δ

*ε*(

_{x}*x,y*) or Δ

*ε*(

_{y}*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 *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,

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

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),

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 *N*
_{1}(*z*) and *N*
_{2}(*z*) by using a mode transformation, and obtain,

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,

where $\tilde{\beta}\left(z\right)=\sqrt{{\delta}^{2}/4+{\kappa}^{2}}$ and $\phi \text{'}\left(z\right)=\text{d}\phi /\text{d}z$.

Then define $\eta \left(z\right)=\phi \text{'}\left(z\right)/\left[2\tilde{\beta}\left(z\right)\right]$ to measure the strength of the coupling [8]. 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*,

where ${\rho}_{0}\left(z\right)=\left({\beta}_{\text{co}}+{\beta}_{\text{cl}}\right)z/2$, $\rho \left(z\right)={\displaystyle {\int}_{0}^{z}\tilde{\beta}\left(z\text{'}\right)}\text{\hspace{0.17em} dz'}$.

An ideal adiabatic polarizer is also a perfect coupler, namely, if the initiate condition is *W*
_{co}
* ^{r}*(0) = 1,

*W*

_{cl}

*(0) = 0 at the input end of the grating, then we will have*

^{l}*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]; 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],

*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,

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

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]. 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}

*(0)| = |*

^{l}*W*

_{co}

*(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.*

^{r}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 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.

To improve the extinction ratio, the parameters of the grating are carefully selected to render the core mode phase-matched with several cladding modes in the working bandwidth. Figure 3(a)
shows the transmission spectrum of this kind of CLPGs. The Panda fiber used in this CLPG is the same as the previous one except that the beat length is 2mm to enhance the mode couplings. The twist pitch of the CLPG varies linearly from 431μm to 718μm, and the total length is 20cm. Seven cladding modes (from HE_{12} to HE_{18}) are considered when the working wavelengths are less than 1.7μm. As seen in Fig. 3(a), the extinction ratio of more than 25dB can be obtained in the bandwidth of 200nm.

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

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