1. Introduction

Optical fibers, have been used for a broad range of applications ranging from medical illuminators to high-bandwidth fiber optical communications, fiber sensors, fiber lasers and amplifiers. For many of the latter applications, the state of polarization (SOP) of the light propagating in the optical fiber [1, 2] needs to be manipulated or controlled for optimum operation of the fiber-optic devices. Low- and high-birefringence optical fibers [3] with different intrinsic structures have been developed for a range of applications. For example, polarization-maintaining fibers (PMFs) [4] possessing high-birefringence resulting from elliptical-shaped cores or intrinsic stresses, have been introduced to alleviate environmental effects and to sustain one linearly polarized mode without coupling energy to the orthogonal mode; likewise twisted and spun fibers [5–10] have been proposed and demonstrated to reduce linear birefringence and polarization mode dispersion (PMD) for specific applications.

Controlled transformation of the state of polarization of light is very important for many laser and fiber optic device applications, and is usually accomplished with bulk-optic wave plates which are inherently narrowband, bulky, and often require careful manual tuning and adjustments. Huang first proposed and demonstrated [11, 12] a novel all-fiber polarization transformer (AFPT), fabricated by spinning a birefringent fiber preform with a slowly varying spin rate from very large to zero (or conversely from zero to very large) while the fiber is drawn. Such an AFPT is not only a compact all-fiber device, but depicts precisely-controllable “optical waveplate” type of properties over a remarkably broad range of wavelengths, limited only by the “practical single-mode bandwidth” of the fiber, as demonstrated recently in independent work by several researchers [13–15]. Moreover, an AFPT with a similar intrinsic structure -- and similar polarization transformation properties -- has been fabricated successfully by post-draw twisting at the softening temperature of the glass fiber [15].

In this paper, power coupling between local eigenstates has been studied to gain insight on -- and to understand detailed characteristics of -- the polarization transformation behavior. This has been graphically illustrated via plots of the relative power in these local eigenstates as a function of distance along the length of the fiber and plots of the extinction ratio of the output state of polarization (SOP) as a function of the normalized spin rate. Deeper understanding of such polarization transformers has been further elucidated by quantitative determination and clear specification of two crucial requirements for fabricating practical AFPT devices. Our calculations have also indicated that the polarization mode dispersion behaviour of the AFPT is much smaller than that of the original birefringent fiber. Finally, a specific AFPT was experimentally investigated at two widely-separated wavelengths of interest in telecommunications systems (1310 nm and 1550 nm) applications, further demonstrating and elucidating the broadband character of such AFPTs.

2. Basic Physics of AFPTs

2.1 Mode coupling in AFPTs

 

Fig. 1. “Basics” of our AFPT

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Fig. 2. Coordinate systems for the AFPT

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In an AFPT, the orthogonal high-stress and low-stress directions or birefringence axes (depicted by the purple and blue lines in Fig. 1 and by the co-ordinate systems in Fig. 2) rotate along the fiber core as a function of z (the optical axis of the fiber) with a slowly increasing rotation rate per unit length. At the unspun end (U) the rate of rotation (per unit length) of these birefringence axes is zero, while at the fast-spun end (F) there are several rotations of the internal axes in a distance corresponding to the beat length of the “original” or “unspun” birefringent fiber. Such an AFPT can be made by applying a slowly varying spin to a high-birefringence (Hi-Bi) preform during the drawing of the fiber [11–14], or by post-draw twisting of a Hi-Bi fiber at the softening temperature [15]. Both methods alter the original birefringent fiber significantly by a geometric rotation of the local birefringence axes. We focus here on the coupling between two linearly-polarized local modes caused by variations in the orientation of the two local birefringence axes along the fiber, using the formalism of Refs [16] and [17]. If we assume that the rotation of the birefringence axes between z and z + Δz is described by ΔΩ, as illustrated schematically in Fig. 2, the electric field of local modes E_x,y(z+Δz) at z+Δz can be expressed in terms of ΔΩ and the electric field E_x,y(z) at z as:

Dividing both sides of Eq. (1) by Δz, and keeping the first order terms in the limit Δz → 0, we arrive at the coupled-mode equations:

If the spin rate varies smoothly (without any singularity) and the AFPT is divided into many infinitesimally small slices (Δz → 0), then the mode-coupling coefficient can be assumed to be constant for each small slice and given by the spin rate dΩ/dz, which can be defined [11] for simplicity as ξ(z). The coupled-mode equations can then be rewritten as:

We see that the effect of rotating the birefringence axes leads to a position-dependent mode-coupling coefficient, given essentially by the spin rate ξ(z). Thus, according to Equation (3), the electric fields of the two local modes are coupled to each other and the coupling rate is determined simply by a mode-coupling coefficient equal to the spin rate. As will be shown in Section 3 below, when linearly polarized light is launched along one birefringence axis (i.e., into one local mode, E_x or E_y) at the unspun end U of the AFPT, its power will be coupled to the other local mode (E_y or E_x) gradually as light propagates along the axis of the fiber, and the powers in the two modes become equal when the light reaches the fast-spun end F of the AFPT. On the other hand, due to the gradual rotation of the birefringence in the intrinsic structure, the phase difference between the two local modes changes from π at the unspun end to π/2 at the fast-spun end. Consequently, linearly polarized light launched at the unspun end gets transformed into circularly polarized light. On the flip side, when circularly polarized light is launched at the fast-spun end of the AFPT, the power in the excited circularly-polarized eigenmode transforms gradually into one linearly-polarized eigenmode; in other words, the power in one local linearly-polarized mode (P_x or P_y) is coupled to the other linearly-polarized mode (P_y or P_x) gradually as the light propagates, so that the total power gradually accumulates in just one single linearly-polarized local mode (P_y or P_x) when light reaches the unspun end, causing circularly polarized light to be transformed into linearly polarized light. The above physical description provides intuitive insight into the basic operation of AFPT; more rigorous analytical details are described below.

2.2 Eigenstates or Normal Modes in AFPTs

The simplest eigenstates, or “normal modes” of a Hi-Bi fiber are two orthogonal linear polarization states (i.e., two local modes, E_x and E_y) aligned parallel to the birefringence axes; these eigenstates are also solutions to the coupled-mode equations when the mode-coupling coefficient ξ(z) is zero. The preferred eigenstates of a fast-spun fiber are the two orthogonal circular polarization states (i.e., the right and left circular polarization states) which are solutions to the coupled-mode equations when ξ(z) is much larger than the linear birefringence Δβ = β_y - β_x. The mode-coupling coefficient ξ(z) in the coupled-mode equations of an AFPT is a slowly varying function of position. Thus there are no invariable polarization eigenstates for the entire device, but “local” eigenstates of polarization that vary as a function of position. Solving Eq. (3) above, the two eigenvalues β_±(z) are as follows:

The corresponding eigenstates E_± (z) are:

As expected, the eigenstates at the unspun end (Δβ/2ξ(0) → ∞) are linear polarization states along the birefringence axes, E₊ = E_y = [0, 1] and E_- = E_x = [1, 0]; likewise, the eigenstates at the fast-spun end (Δβ/2ξ(z_L) → 0) are the two orthogonal circular polarization states, E₊ = E_L = [1/√2 , -i/√2] and E_-= E_R = [1/√2, i/√2]. The eigenstates at other positions (where Δβ/2ξ(z) is a non-zero number of modest magnitude) are orthogonal elliptical polarization states. Therefore, the varying eigenstates of an AFPT are a direct consequence of the slowly varying intrinsic structure. At the unspun end, the spin rate ξ(0) is zero and the fiber is highly birefringent, so the eigenstates are two linear polarization states along the birefringence axes. At the fast-spun end, the spin rate is so large that the rotation of the birefringence axes totally negates the effect of linear birefringence, resulting effectively in circular anisotropy on a macroscopic scale, and thus the natural eigenstates are the two orthogonal circular polarization states. At other positions in the AFPT, the spin rate is not large enough to reduce the effect of linear birefringence completely, resulting in elliptical anisotropy on a macroscopic scale, and thus to two orthogonal elliptical polarization states as the natural eigenstates.

In other words, the physical structure of the AFPT, which varies from linear anisotropy at the unspun end to circular anisotropy at the fast-spun end, causes natural evolution of polarization eigenstates from those of linear character to those of circular character, as illustrated in Fig. 3, thereby enabling the AFPT to perform a “natural” linear ↔ circular SOP transformation, similar to the properties of a bulk-optic quarter-wave plate, except that for an AFPT this transformation occurs independently of wavelength. To obtain wavelength-independent SOP transformation, one simply launches only one of the two orthogonal eigenstates at the appropriate end of the AFPT, i.e., a linear polarization state along a principal (birefringence) axis at the unspun end or a circular polarization state at the fast-spun end. This operation of wavelength-independent SOP transformation is thus simply one of controlled transformation of a single eigenstate from one end of the AFPT to the other.

 

Fig. 3. Evolution of pol. eigenstates in an AFPT.

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One can also use the “effective linear birefringence”, Δβ_E:

defined as the retardance [6, 8] of a small slice Δz, to analyze the effect of the varying intrinsic structure of an AFPT on the polarization. Fig. 4 depicts the effective linear birefringence Δβ_E as a function of position in a 20 cm long AFPT in which the monotonically increasing spin rate (starting from an initial value of zero) is described by the mathematical function ξ(z) = ξ_max[0.5-0.5cos(πz/z_L)]². Near the unspun end (ξ ≪ Δβ), Δβ_E ≈ Δβ; this part of the AFPT can thus be treated as a linear retarder (R ≈ Δβz) whose eigenstates are the two orthogonal linear polarization states along the birefringence axes. When the spin rate is very large (ξ ≫ Δβ), Δβ_E ≈ 0; this part of the AFPT can be treated as a circularly anisotropic medium whose eigenstates are the two orthogonal circular polarization states. At values of Δβ_E that lie in between these two extremes, i.e., at intermediate positions of the AFPT, such that Δβ > Δβ_E > 0, the AFPT can be treated as an elliptically anisotropic medium whose eigenstates are two specific orthogonal elliptical polarization states. Also, as seen in the red curve of Fig. 4, when the AFPT has a larger value for the final normalized spin rate Q_max = ξ_max/Δβ, the effective linear birefringence Δβ_E decreases to a small value much faster (i.e., over a much shorter distance), and is much closer to zero at the fast-spun end. In other words, the intrinsic structure in an AFPT with a much larger change in the spin rate changes from linear anisotropy to circular anisotropy more rapidly, and the circular anisotropy at the fast-spun end is “more perfect” for such an AFPT.

 

Fig. 4. Effective linear birefringence as a fn. of position in a 20 cm long AFPT with Δβ = 10³ rad/m, ξ_max = 100Δβ and 1000Δβ respectively.

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3. SOP Evolution and Other Characteristics of AFPTs

As described above, AFPTs appear to perform an SOP transformation that is similar to that of a bulk-optic quarter-wave plate, at least as far as the two end points are concerned, except for the fact that the SOP transformation performed by an AFPT is inherently independent of wavelength. However, several additional details can be understood better by carefully looking at the evolution of the SOP of the light as it propagates within the AFPT. In this section, key characteristics of AFPTs, including details of the SOP evolution, will be elucidated by specific parametric plots obtained via numerical calculations. This is followed by discussion on the design of practical AFPTs, analysis of polarization mode dispersion in AFPTs, and description of a few key experiments using a 20 cm AFPT and polarized laser sources at the telecommunication-relevant wavelengths of 1310 nm and 1550 nm.

3.1 Power coupling between two local modes and evolution of the SOP

The AFPT coupled-mode equations (3) and (4) can be rewritten as [11]:

There is no analytical solution to these coupled-mode equations because the coupling coefficient is a function of the position. In order to investigate the power coupling between two local modes and the evolution of SOP, we look at the zeroth-order approximation solution to these coupled-mode equations provided by Huang [11]:

where

and Q(z) = ξ(z)/Δβ is the normalized spin rate. As done earlier in Section 2 (Fig. 4), we will again use the relatively simple monotonously increasing function ξ(z) = ξ_max[0.5-0.5cos(πz/z_L)]² to describe the spin rate. The evolution of the relative power in the two orthogonal local modes (P_x,y = |E_x,y|²) subsequent to a single-eigenstate excitation process is shown in Fig. 5. In Fig. 5(a), when light that is polarized linearly along the x axis is launched at the unspun end, the power in the E_x mode is coupled to the E_y mode gradually as light propagates along the AFPT. P_x and P_y become equal at the fast-spun end when Q_max = 100 and 1000 (green and red lines, respectively). As expected for the case of a “modest” final spin rate, i.e., when Q_max = 10 (the blue line), there is still some difference between the values of P_x and P_y at the fast-spun end. In Fig. 5(b), when circularly polarized light is launched at the fast-spun end, the power in the E_y mode is eventually coupled totally to the E_x mode, and no power is left in the E_y mode, although the polarization state “wanders” back and forth a lot more in the case of a “modest” final spin rate, i.e., when Q_max = 10 (the blue line). The output light at the unspun end is thus linearly polarized along the x axis. Comparing the power coupling of these two single-eigenstate processes, we see that Q_max does not need to be as large for “near-perfect” SOP transformation from circular to linear as it does for “near-perfect” SOP transformation from linear to circular. By using Eqs. (8) and (9), evolution of SOPs for two single-eigenstate processes are simulated for Q_max= 10, 100 and 1000, as seen in corresponding animations.

When light that is linearly polarized along the x axis is input at the unspun end, the larger the final normalized spin rate Q_max, the earlier the SOP changes from linear to circular, and the more purely circular the output SOP. This is because when the AFPT has a larger Q_max, its intrinsic structure changes from linear anisotropy to circular anisotropy in a shorter length and the eigenstate at the fast-spun end will be a more purely circular SOP. When circularly polarized light is input at the fast-spun end, the larger the Q_max, the later the SOP changes from circular to linear, and the more purely linear will the output SOP be. This is because the AFPT with a larger Q_max keeps the intrinsic structure in a near-circular anisotropic state over a longer length, and the input circular SOP matches the eigenstate at the fast-spun end more closely. Although evolution of SOPs along the fiber with different values of Q_max appear significantly different, the output SOPs are nearly the same and perfect linear ↔ circular SOP transformations can be achieved whenever the value of Q_max is sufficiently large; the transformation just occurs faster when the final spin rates are larger. This also explains the ultra-broad operation bandwidth of AFPTs with sufficiently large values of final spin rates, because different operating wavelengths correspond effectively to different maximum normalized spin rates Q_max.

 

Fig. 5. Evolution of relative power in two local modes, P_x (solid line) and P_y (dash-dot line) as a function of the normalized distance, z_R (= z/z_L), for different values of the normalized final spin rate Q_max [1000 (red), 100 (green), 10 (blue)], assuming single-eigenstate excitation processes. (a) Linearly polarized (along the x axis) light is launched at the unspun end, U (light shading); (b) Circularly polarized light is launched at the fast-spun end, F (dark shading; note that the grayscale in the inset pictorially denotes the spin rate in the AFPT).

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3.2 Role of basic fabrication requirements on optimal operation of AFPTs

There are two basic constraints [11] that must be met for the fabrication of practical AFPTs. These are: (1) the spin rate ξ_max at the fast-spun end must be large enough (ξ_max ≫ Δβ); and (2) the variation of the spin rate from zero to fast should be slow, i.e., dξ/dz ≈ 0. In this subsection, we will quantitatively evaluate the importance of these two factors.

 

Fig. 6. The extinction ratio of the output SOP as a function of the maximum normalized spin rate Q_max when the AFPT operates in the single-eigenstate excitation process. (a) Light linearly polarized along the x-axis is launched at the unspun end; (b) Circularly polarized light is launched at the fast-spun end (the inset shows details for smaller values of Q_max , i.e., it “magnifies” the 0 to 10 range of the horizontal scale).

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The first constraint results in a sufficiently-high circular anisotropy at the fast-spun end to assure local end-point eigenstates that are two orthogonal circular polarization states. The extinction ratio of the output SOP as a function of the maximum normalized spin rate Q_max for the two basic single-eigenstate excitation processes are plotted in Fig. 6. For these plots the normalized spin rate is Q(z) = Q_max[0.5-0.5cos(πz/z_L)]² and the linear birefringence Δβ is set as 10³ rad/m. In Fig. 6(a), linearly polarized light is launched at the unspun end and the output SOP at the fast-spun end is circular. The larger the value of Q_max, the smaller is the extinction ratio of the circularly polarized output. When Q_max is 10, the extinction ratio of the output SOP is ~ 0.43 dB, and when Q_max is 100, the extinction ratio of the output SOP is ~ 0.043 dB. In Fig. 6(b), circularly polarized light is launched at the fast-spun end and the output SOP at the unspun end is essentially linear (large extinction ratio between the 2 orthogonal linearly polarized components). The larger the Q_max, the larger the extinction ratio of the linear output SOP; when Q_max is 10, the extinction ratio of the output SOP is ~ 30 dB, and when Q_max is 100, the extinction ratio of the output SOP is over 50 dB. The ripples in Fig. 6(b) may be caused by the remaining power coupling between the two local modes even when Q_max is large; this theoretical behaviour of rapidly changing extinction ratio has not been discussed by any of the previous researchers, but has very little practical significance for most applications, because small imperfections in device fabrication or non-ideal launching of the input polarization could dominate over these interesting but “hard to achieve” large polarization extinction ratios.

Because the zeroth-order approximation solution to the coupling-mode equations assumes that the rate of variation of the spin rate as a function of length is negligible (dξ/dz ≈ 0) [11], we used the 4th order Runge-Kutta method to numerically solve the coupled-mode equations in order to better understand how this second requirement may affect the operation of practical AFPTs. Since the power in the two local modes (P_x and P_y) and the phase difference between two local modes (ΔΦ) determine the SOP of the light propagating in the fiber, evolution of P_x, P_y, and ΔΦ in the AFPTs with different lengths and different maximum spin rates max were investigated. Fig. 7 shows the evolution of P_x, P_y, and ΔΦ as a function of distance in a 0.2 m long AFPT with Δβ = 10³ rad/m for the case that light that is linearly polarized along the x axis is launched into the unspun end, with parametric variation of the maximum spin rates ξ_max to values of 10⁵ rad/m, 5×10⁴ rad/m, and 10⁴ rad/m. Fig. 8 depicts the same parameters for a shorter (0.02 m long) AFPT. In Fig. 7, when the fiber length is 0.2 m, the condition dξ/dz ≈ 0 is much more easily satisfied. P_x couples to P_y gradually as light propagates along the AFPT and these 2 components become equal at the fast-spun end. ΔΦ changes from π at the unspun end to π/2 at the fast-spun end. Consequently, the output SOP is circular and the AFPT can transform the SOP from linear to circular to a very high level of precision in this case. In Fig. 8, when the fiber length is 0.02 m, the condition dξ/dz ≈ 0 is not satisfied and the dξ/dz term disrupts the power coupling. P_x and P_y oscillate wildly around a mean value whose amplitude is proportional to dξ/dz when light approaches the fast-spun end, implying that coupling between P_x and P_y is still large at the fast-spun end. The phase difference, ΔΦ, between the two local modes also oscillates around π/2. Therefore, when dξ/dz is not small enough, the AFPT is not able to properly perform the SOP transformation from linear at the unspun end to circular at the fast-spun end.

 

Fig. 7. Evolution of (a) P_x, P_y, and (b) ΔΦ (the phase difference between the two local modes) as a function of distance in a 0.2 m long AFPT with Δβ = 10³ rad/m when linearly-polarized light (along the x-axis) is launched into the unspun end (U), for maximum spin rates ξ_max of 10⁵ (red curves), 5×10⁴ (green curves), and 10⁴ rad/m(blue curves).

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Fig. 8. Evolution of (a) P_x, P_y, and (b) ΔΦ (phase difference between the two local modes) as a function of distance in a 0.02 m long AFPT with Δβ = 10³ rad/m when linearly-polarized light (along the x-axis) is launched into the unspun end (U), for maximum spin rates ξ_max of 10⁵ (red curves), 5×10⁴ (green curves), and 10⁴ rad/m(blue curves).

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Figure 9 shows the evolution of P_x, P_y, and ΔΦ as a function of distance in a 0.2 m long AFPT with Δβ = 10³ rad/m for the case that circularly-polarized light is launched at the fast-spun end, with parametric variation of the maximum spin rates ξ_max to values of 105 rad/m, 5×10⁴ rad/m, and 10⁴ rad/m. Fig. 10 depicts the same parameters for a shorter (0.02 m long) AFPT. When the fiber length is 0.2 m, P_y is totally coupled to P_x and no power is left in the E_y mode at the unspun end. ΔΦ is changed from π/2 to ±π. In this case, the AFPT can perform an excellent SOP transformation from circular at the fast-spun end to linear at the unspun end. Note however that the phase shift changes rapidly from + π to - π for small changes in the propagation distance, corresponding to rapid changes in the “sense” of the circular polarization, a property that is interesting, but may lead to polarization instabilities for the linear to circular polarization transformation process.

When the fiber length is reduced to 0.02 m, the constraint of dξ/dz ≈ 0 is not satisfied. P_y is only partially coupled to P_x and some power is left in the E_y mode at the unspun end, as shown in Fig. 10 (a). Therefore, the FTP is not able to adequately perform the SOP transformation from circular at the fast-spun end to linear at the unspun end. An intuitive explanation for this lack of a perfect linear-to-circular SOP transformation under these conditions is that controlled coupling between the two local modes is not allowed when the spin rate dξ/dz is large.

 

Fig. 9. Evolution of (a) P_x, P_y, and (b) ΔΦ (the phase difference between the two local modes) as a function of distance in a 0.2 m long AFPT with Δβ = 10³ rad/m when circularly polarized is launched at the fast-spun end (F) for maximum spin rates ξ_max of 10⁵ (red curves), 5×10⁴ (green curves), and 10⁴ rad/m(blue curves).

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Fig. 10. Evolution of (a) P_x, P_y, and (b) ΔΦ (the phase difference between the two local modes) as a function of distance in a 0.02 m long AFPT with Δβ = 10³ rad/m when circularly polarized is launched at the fast-spun end (F) for maximum spin rates ξ_max of 10⁵ (red curves), 5× 10⁴ (green curves), and 10⁴ rad/m(blue curves).

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3.3 Polarization transformation behavior under dual-eigenstate excitation in AFPTs

Although AFPTs possess the ability to perform linear ↔ circular SOP transformations similar to those performed by bulk-optic quarter-wave plates, a high-quality SOP transformation -one that is independent of wavelength -- can be achieved optimally when the AFPT is used in a manner such that only a single eigenstate is excited, i.e., the SOP of the input light is identical to one of the “natural” eigenstates at the corresponding end. Briefly, when an AFPT operates under single-eigenstate excitation, light that is linearly polarized along the birefringence axes can be transformed into circularly polarized light when launched at the unspun end, and circularly polarized light launched at the fast-spun end can be transformed into light that is linearly polarized along one of the two birefringence axes. For any other input polarization, the operation of the AFPT is described by a dual-eigenstate excitation process. In this part, we describe a few key polarization characteristics resulting from operation of the AFPT via the dual-eigenstate excitation process.

When linearly polarized light at an arbitrary azimuthal angle (other than 0° or 90°), or circularly/elliptically polarized light is launched at the unspun end, two eigenstates are excited, as seen by treating any arbitrary polarization as a linear combination of linear polarizations along the slow and fast birefringence axes, leading to excitation of both linearly polarized eigenstates at the unspun end of the AFPT. When the input SOP is circular or linear at an azimuthal angle of 45° with respect to the birefringence axes, the initial amplitudes of these two eigenstates are equal. Consequently, two opposite circular polarization states of equal amplitudes are obtained at the fast-spun end and the output SOP at the fast-spun end is thus linear, as shown in Figs. 11 and 12, as expected for a linear combination of two orthogonal circular polarization states with equal amplitude. The azimuth of the output linear SOP simply depends on the phase difference between the two eigenstates at the fast-spun end. When the input SOP is linear (at 45° wrt x-axis, Fig. 11) and Q_max = 10, 100, and 1000, the output SOP is linear, with azimuthal angles of 75.9°, 147.9° and 28.5° respectively. In Fig. 12, when the input SOP is right circular and Q_max = 10, 100 and 1000, the output SOP is linear with azimuthal angles of 120.9°, 12.9° and 73.5°, respectively. The small deviation from linearity for the blue curves correspond to the deviation from ideality of this AFPT (due to the relatively small value, 10, of the maximum normalized spin rate).

Note again that when the input SOP is neither circular nor linearly polarized either along the birefringence axes nor at 45° wrt the birefringence axes, the two excited eigenstates have different amplitudes. Consequently, the output SOP at the fast-spun end is elliptical, resulting from the combination of two opposite circular polarization states with unequal amplitudes.

 

Fig. 11. Dual-eigenstate excitation process in an AFPT for the case when light polarized linearly at 45° wrt the birefringence axes is launched at the unspun end. The output polarizations observed correspond to maximum normalized spin rates of 1000 (red), 100 (green) and 10 (blue) respectively.

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Fig. 12. Dual-eigenstate excitation process in an AFPT when right circularly-polarized light is launched at the unspun end. The output polarizations observed correspond to maximum normalized spin rates of 1000 (red), 100 (green) and 10 (blue) respectively. [Media 2]

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When linearly polarized light is launched at the fast-spun end, both the circularly-polarized eigenstates are always excited (as stated above) with equal initial amplitudes, since one linear polarization state can be treated as the combination of two opposite circular polarization states with equal amplitude. Thus, the output SOP can be any polarization state, depending on the phase difference between the two eigenstates at the unspun end. Applying a matrix analysis similar to that described in Ref. [11], we know that when light linearly polarized at a specific angle θ to the local birefringence axes is input at the fast-spun end, the output SOP is circular. The angle θ is related to the normalized spin rate of the AFPT

When the light is linearly polarized at a specific angle ρ + 45° with respect to the local birefringence axes, the output SOP is linear. Figs. 13 and 14 graphically illustrate these two special cases. As seen in Fig. 13, in order to obtain circular polarization outputs when light is input on the fast-spun end of AFPTs whose Q_max values are 10, 100 and 1000, the input SOPs should be linear and at azimuthal angles θ of 149.2°, 77°, and 16.4° respectively. As seen in Fig. 14, for AFPTs with Q_max values of 10, 100, and 1000, linearly polarized outputs will be obtained at the unspun end when the linear incident polarizations are launched into the fast-spun end at azimuthal angles of 14.2°, 122°, and 61.4° respectively.

 

Fig. 13. Generation of circularly polarized light at the unspun end of an AFPT by using a dual-eigenstate excitation process and linearly polarized light at the fast-spun end. The different input polarizations maximum correspond to normalized spin rates of 1000 (red), 100 (green) and 10 (blue), respectively. [Media 1]

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Fig. 14. Dual-eigenstate process of the AFPT when linearly polarized light with azimuth of ρ + 45° is launched at the fast-spun end. The maximum normalized spin rates are 1000 (red), 100 (green) and 10 (blue), respectively.

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3.4 Polarization Mode Dispersion in AFPTs

Polarization mode dispersion (PMD) in a “normal” high birefringent (Hi-Bi) fiber is defined by the delay in the transit times of the two orthogonal linear polarizations [3]:

where L is the fiber length, c is the velocity of light, B_G is the shape birefringence, and BS is the stress birefringence. As expected, the PMD in a spun birefringent fiber decreases relative to that in the original unspun birefringent fiber, and this decreased PMD is related to the spin rate [3]. The PMD experienced by light transiting an AFPT can be easily seen to be given by

where Q(z) = ξ(z)/Δβ. For L_B = 1.5 mm, λ = 1.5 μm, L = 0.3 m, and B_S = 10^-3 rad/m, Δτ, the PMD of such a Hi-Bi fiber is 1 ps, however that of the corresponding AFPT with a maximum spin rate ξ_max of 106 rad/m is only 5.423×10^-2 ps , and is 0.17 ps if the maximum spin rate ξ_max is only 10⁵ rad/m.

3.5 Experimental Studies on a specific 20 cm long AFPT

 

Fig. 15. Schematic experimental setup used to characterize our AFPT.

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We performed a set of experiments to measure the ellipticity, ε_out, of the output SOP at one end of the AFPT as a continuous function of the orientation of the linearly polarized light launched at the other end of the AFPT. A schematic experimental setup for the set of experiments is shown in Fig. 15. These experiments employed a 20 cm long AFPT fabricated by Huang [see Ref. 11 and acknowledgements], detailed characteristics of which were not known, except for the fact that the fabrication process was optimized to satisfy the basic criteria needed for a practical AFPT (see Section 3.2). The laser source consisted of either a laser diode at 1310 nm or at 1550 nm, and was followed by a “polarization processor,” which consisted of a set of bulk-optic linear polarizers and bulk-optic quarter-wave plates arranged in such a way as to create either purely linear polarization or purely circular polarization, as needed for the chosen experiments, and monitored carefully with a commercial polarimeter. A 2 m piece of polarization maintaining fiber (PMF) of birefringent refractive index difference ~ 7.5×10^-4 (Newport F-SPPC-15) was connected to the AFPT by means of a “home-made” rotary splice. This PMF enables rotation of the input linear SOP into the AFPT while minimizing variations in the coupling and avoiding cladding mode excitation, both of which may occur when unmonitored continuous orientation of the linear polarization is done by means of an external polarization rotator (such as a bulk-optic half-wave plate or a combination of a quarter-wave plate and a linear polarizer) followed by a coupling lens. The desired linear polarization state at the input end of the AFPT was obtained by careful alignment of the principal axes of the PMF with the linear input polarization (as verified by the polarimeter), and by rotating the output end of the PMF to obtain the desired linear polarization (as verified again via “calibration run” experiments with the polarimeter). Circularly polarized light used at the input end of the AFPT also was obtained by using the PMF as a quarter-wave plate, i.e., by aligning one of its principal axes at 45° with respect to the input polarization, and then cleaving the PMF carefully by small increments (≾ 0.5 mm each) until a “nearly perfect” circular polarization state (ellipticity ≥ 0.9, extinction ratio ≤ 0.5 dB) was obtained at the output, as verified by direct measurements with the polarimeter. For our experiments, the ellipticity of the polarization state [defined as the ratio of the minor axis to the major axis of the polarization ellipse, 0 for linear polarization, +1 for right-handed circular polarization (RCP), and -1 for left-handed circular polarization (LCP)]was obtained directly from the polarimeter measurements.

Figure 16 shows results for experiments performed with linearly polarized light from the PMF entering the unspun end U and exiting the fast-spun end F of the AFPT. Experimental tests were performed on the ellipticity ε_{out, F} of the output state as a function of the orientation of the input linear polarization θ_{in, U} over an input azimuthal angle range of 90° for both the 1310 nm and 1510 nm input wavelengths, with θ_{in, U} = 0° chosen to correspond to the largest practical ellipticity (ie a near-linear SOP) for the output state (closest to ε_{out, F} = 1); for this choice, θ_{in, U} = 90° for the input polarization is seen to correspond to the largest value of negative ellipticity for the output state, as seen in Fig. 16. A theoretical plot of the anticipated ellipticity ε_{out, F} of the output SOP as a function of θ_{in, U} (the orientation of the input linear SOP) is also depicted (by the solid line) in Fig. 16. For the theoretical plot, we assumed a value of 10³ rad/m for Δβ and a value of 10⁶ rad/m for the spin rate ξ_max at the fast-spun end, and that the variable spin rate can be approximated by ξ_max [0.5-0.5cos(πz/z_L)]², since it was not possible to get specific data on these values from the AFPT manufacturer. According to the above discussion, for input polarization states of _{θin, U} = 0°, 45° and 90°, the output polarization was expected to be described by RCP, linear, and LCP, corresponding to ellipticities ε_{out, F} of +1, 0, and -1, respectively, as confirmed in the experimental data of Fig. 16. The strong general agreement of our data with the theoretical plots, and their nearly-perfect overlap for 1310 nm and 1550 nm, provides extremely good experimental verification of the theory on the operating principle of these AFPTs, and of their operability over a very broad range of wavelengths (from 1310 nm to 1550 nm in the present case) with negligible change in the performance characteristics. Although, the dual-eigenstate process of the AFPT (θ_{in, U} ≠ 0° and 90°) is wavelength-dependent, our experiments indicated that the output ellipticities at different input orientations are nearly equal from 1310 nm to 1550 nm due to the small difference of Q_max over the wavelength range. The deviation of the maximum ellipticities of ± 0.9 (corresponding to an extinction ration of ~0.46) from the anticipated theoretical values of ± 1 is presumably due to the fact that successive “trimming” of the AFPT caused it to deviate from the optimal conditions (of ξ₀ = 0 and ξ_max ≫ 1). When circularly polarized light was launched at the unspun end, the ellipticity of the output polarization state was measured as 0.026 (extinction ratio 25.9 dB) at 1310 nm and 0.029 (extinction ratio 25.6 dB) at 1550 nm, respectively, within experimental error of the theoretically expected values of 0 (linear polarization).

 

Fig. 16. Ellipticity of the output SOP at the fast-spun end as a function of the azimuthal angle of the input linear polarization state launched at the unspun end.

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The experiments were repeated by flipping the AFPT’s input and output ends, i.e., the output end of the PMF in Fig. 15 was connected to the fast-spun end “F” (making the fast-spun end as the input end of the AFPT for the following set of measurements), with the “output light” now exiting the unspun end “U.” Figure 17 shows the ellipticity ε_out,U of the polarization of the new output end as a function of the orientation of θ_{in, F} of the input linear polarization, which now was varied over a range of nearly 180°. Setting of the reference “value” for the azimuthal angle (with respect to the x-axis) for this data was established by matching the condition for a linearly polarized output with the theoretical curves, which were observed to occur at input polarizations corresponding to θ_{in, F} = 35° and 125° (with respect to the principal axes of birefringence in the AFPT), as shown in Fig. 17. For these cases, the output at the unspun end “U” was still linearly polarized, because the phase difference between the two linear principal states was zero or an integral multiple of π. For input polarizations displaced by 45° from the above values (namely θ_{in, F} = 80° and 170°), the output light is expected to be circularly polarized. In our experiments, the measured values ε_out,U were ~ 0.8 (extinction ratio 0.97 dB) and - 0.9 (extinction ratio 0.46 dB), respectively, which are reasonably close to the expected theoretical values of +1 and -1, respectively. When circularly polarized light was input at the fast-spun end, the polarization output at the unspun end was nearly linear, as expected, with measured ellipticities of 0.0017 (extinction ratio 27.7 dB) and 0.021 (extinction ratio 26.8 dB) at 1310 nm and 1550 nm respectively.

 

Fig. 17. Ellipticity of the output SOP at the unspun end as a function of the orientation of the input linear polarization state launched at the fast-spun end.

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4. Summary

We have provided quantitative characterization and new physical insights to the operation of an all-fiber polarization transformer (AFPT), which can perform linear → circular SOP transformations similar to those performed by bulk-optic quarter-wave plates, but with he added advantage that the AFPT exhibits these polarization transformation properties over an ultra-broad range limited only by the single-mode range of the fiber. This unique characteristic of the AFPT is attributed to the gradual evolution of local eigenstates, from linear at the unspun end to circular at the fast-spun end, induced by the special intrinsic structure which slowly varies from linear anisotropy to circular anisotropy. Such AFPTs can perform high extinction ratio linear → circular SOP transformations only if the spin rate ξ_max at the fast-spun end is large sufficiently large (ξ_max ≫ Δβ) and the rate of variation of the spin rate follows an adiabatic approximation (dξ/dz ≈ 0). Additionally, we showed that the performance of a practical AFPT is independent of wavelength only if the AFPT is utilized in the single-eigenstate excitation mode, i.e., the input SOP should match the intrinsic “local eigenstate” at the input end of the AFPT. Because spinning reduces the linear birefringence of the original Hi-Bi fiber, the PMD exhibited by AFPTs is much smaller than that of the original Hi-Bi fiber. Our experimental results using a 20 cm long AFPT showed strong agreement with theoretical predictions, and demonstrated the intrinsic broadband character of such AFPTs.