We show theoretically and experimentally that the polarization sensitivity of asymmetric nondegenerate fiber four-wave mixing can be eliminated by using circularly polarized pump waves in a twisted fiber. By twisting a fiber at 15 turns/m and aligning the pump waves to a circular state of polarization, we successfully reduce the polarization sensitivity from 5.8 dB to 0.9 dB. Although the polarization-mode dispersion (PMD) of the twisted fiber sets the limitation to the conversion bandwidth, its effect is relatively small owing to the small PMD of the twisted fiber. The demonstrated scheme should be a simple and efficient way of realizing all-optical tunable wavelength converters and wavelength-exchange devices without polarization dependence.
©2005 Optical Society of America
Four-wave-mixing (FWM) in an optical fiber has been studied extensively as an attractive method of realizing ultra-fast all-optical signal processing devices. The commonly studied FWM configuration is the case where the idler wave is generated symmetrically with respect to either one pump (partially degenerate FWM) or two pumps (symmetric nondegenerate FWM). On the other hand, there has been special interest in the asymmetric nondegenerate FWM, where the idler wave is generated asymmetrically with respect to the two pumps through a parametric process of ω 1 - ω 2 + ω 3 → ω 4 as shown schematically in Fig. 1. This asymmetric FWM offers various attractive functions that cannot be realized by the symmetric configuration. For example, wavelength conversion of a signal light at an arbitrary wavelength to another arbitrary idler wavelength is possible by tuning the two-pump wavelengths so that the phase-matching condition is satisfied [1,2]. In a similar setup, it can also offer the wavelength-exchange operation, where the modulated data on the signal and idler wavelengths are exchanged [3,4]. The asymmetric FWM also has a practical advantage over the symmetric configuration that the stimulated Brillouin scattering (SBS) can be suppressed without influencing the idler wave by simply co-modulating two-pump phases using a single external phase-modulator [5,2].
One common problem of the FWM-based signal processing is that its efficiency depends on the state of polarization (SOP) of the signal wave relative to those of the pump waves. In the symmetric FWM, we can eliminate the polarization sensitivity by using two orthogonally polarized pumps [6,7]. This scheme, however, is not valid in the asymmetric configuration. More precisely, it has been proved theoretically that in a realistic low-birefringence fiber where the birefringence axis varies randomly with a period shorter than the nonlinear length, the polarization sensitivity of asymmetric FWM becomes as large as 6 dB when the pumps are co-polarized and larger when they are cross-polarized . To the best of our knowledge, only the possible scheme of realizing polarization-insensitive asymmetric FWM demonstrated to date is to employ the polarization-diversity architecture . This scheme, however, requires additional optical components as well as precise control of the polarization states inside the loop, which may cause an instability problem when a long fiber is employed.
In this paper, we show both theoretically and experimentally that polarization-insensitive asymmetric FWM is possible in a simple straight-line configuration by using two circularly polarized pumps in a circular-birefringence fiber, which is fabricated by twisting a fiber. While a twisted fiber has been proved effective in realizing polarization-insensitive signal processing based on cross-phase modulation [9,10], FWM in twisted fiber has not been studied in detail, to our knowledge. By twisting a 200-m-long dispersion-shifted fiber (DSF) at a rate of 15 turns/m and aligning the pump SOP to a circular mode, we achieve effective reduction of the polarization sensitivity from 5.8 dB to as small as 0.9 dB. Although the conversion bandwidth is limited by the circular polarization-mode dispersion (PMD) of the twisted fiber, the polarization sensitivity is kept below 1.8 dB within an idler wavelength tuning range of ±10 nm, owing to the relatively small PMD of the twisted fiber. Several schemes of mitigating the effect of PMD are discussed.
2. Theory of polarization-insensitive asymmetric four-wave mixing
Assuming that the third-order nonlinearity of fiber is dominated by the instantaneous electronic response (i.e., ignoring the Raman effect), the nonlinear polarization vector P (3)(r, t) induced inside a fiber is expressed as
where ε 0 is the vacuum permittivity, χ (3) (= ) is the diagonal tensor component of the third-order susceptibility, and E(r, t) is the real electric-field vector . In the case of the FWM process as described in Fig. 1, E(r, t) is composed of four frequency components:
where E 1(r), E 2(r), E 3(r), and E 4(r) are the complex amplitudes of the two pump, signal, and idler electric fields, respectively. Following similar analyses in [12,13], nonlinear-polarization vectors at respective wavelengths P (3) m(r) (m = 1, 2, 3, 4) are obtained by inserting Eq. (2) to Eq. (1) and assuming |E 1|2, |E 2|2 ≫ |E 3|2, |E 4|2:
where e j (j = 1, 2, 3, or 4) are the unit complex vectors of E j, representing the SOPs of respective waves (E j = E j e j). The last terms in Eqs. (5) and (6) are responsible for the FWM interaction. Due to the presence of inner vector products in the FWM terms, their magnitudes depend on the relative SOPs of four waves, which results in the strong polarization dependence of the FWM process.
We now consider a special case where E 1 and E 2 are circularly polarized with the same handedness: for example, right-handed [e 1 = e 2 = 1/√2 (1, i)T]. Then, Eqs. (3) and (4) reduce to scalar expressions (the nonlinear-polarization-rotation factor vanishes ):
We see that the magnitude of the FWM-coupling term in Eq. (10) [Eq. (9)] is now independent of the signal (idler) SOP and it is co-polarized with the signal (idler) wave. Equations (9) and (10), therefore, suggest that polarization-insensitive FWM is possible, provided that two pumps are kept circularly polarized along the fiber. In fact, a standard low-birefringence fiber can be easily made circular-polarization maintaining by applying a twist, so that a sufficient circular birefringence is induced [14,15]. It is therefore expected that polarization-insensitive FWM can be achieved by injecting circularly polarized pumps into a twisted fiber.
The evolution of four optical fields in a circular-birefringence fiber is studied by inserting Eqs. (7)-(10) into the Maxwell wave equation [12,13]. The transverse dependence of E j is factored out by writing E j(r) = F(x,y)A j(z)exp(ikjz) = F(x,y)Aj (z)exp(ikjz)ej , where kj are the propagation constants of respective waves, which include both the group-velocity dispersion (GVD) and the circular birefringence of the fiber. Under the slowly-varying-envelope approximation, we obtain a set of coupled-mode equations, described as
where γ=3k 0 χ (3)/(8nAeff ) (k 0: the average propagation constant, n: the linear refractive index, and Aeff : the effective core area) is the nonlinear coefficient of fiber and Δk (≡ k 1 - k 2 + k 3 - k 4) is the phase-mismatching factor. When deriving Eqs. (11)-(14), we ignore the residual linear birefringence of the fiber, which should have negligible effect in a circular-birefringence fiber.
Since Eqs. (11)-(14) do not contain any vector product, they can be solved in the same manner as for the scalar case presented in . The idler conversion efficiency at the fiber output (z = L) is thus written as
where Pj ≡ |Aj (0)|2 (j = 1 or 2) and g is given by
Note that Eqs. (15) and (16) are identical to those for the scalar case , except that the nonlinear coefficient is reduced by a factor of 2/3 for our case of circularly polarized pumps. When P 1 = P 2 ≡ P and γPL ≪ 1 (small conversion regime), Eq. (15) reduces to
When the signal wave is co-polarized with pumps, i.e. right-handed, Δk is given by
where β 2 (=∂ 2 k/∂ω 2) is the second-order GVD at the center wavelength ω 0 and Δωj follow the notations in Fig. 1. We assume that the higher-order GVD is negligible. From Eq. (18), the phase-matching condition (Δk = 0) is satisfied for a broad wavelength range by setting ω 0 to the zero-dispersion wavelength of the fiber .
On the other hand, when the signal is orthogonally polarized with the pumps, kj are written as
where k 0R (k 0L) and 1/β 1R (1/β 1L) are the propagation constant and group velocity of the right-handed (left-handed) wave, respectively. The polarization dependence of β 2 is ignored. From Eqs. (19)-(22), Δk becomes
where τ (=β 1R-β 1L) is the circular PMD of the fiber. Note that the large circular birefringence, k 0R - k 0L, is absent in Eq. (23) and only the PMD of the circular-birefringence fiber affects the phase-matching condition.
From Eqs. (15), (16), (18) and (23), the FWM efficiency becomes dependent on the signal SOP when τ(Δω 1 - Δω 2)L is large, which would consequently limit the wavelength-tuning range of the signal and idler waves. Fortunately, PMD of a twisted fiber is relatively small (≈ 1 ps/km when twisted at 15 turns/m ) compared to a linear-polarization-maintaining fiber with an equal amount of birefringence, so that a reasonably wide tuning range should be achieved by using a moderate length of fiber. From Eqs. (17) and (23), we estimate that the polarization sensitivity should be less than 1 dB for the idler (or signal) wavelength range of ±10 nm, if we use a 200-m DSF twisted at 15 turns/m.
3. Fabrication and characterization of a twisted fiber
A 200-m-long DSF was twisted at a rate of 15 turns/m and wound on a bobbin with a diameter of 16 cm. The twist introduces a torsional stress, which via the photoelastic effect, leads to high circular birefringence (optical activity) . Since this is much larger compared to the intrinsic birefringence of the fiber as well as the bend-induced linear birefringence that arises when wound on the bobbin, the fiber becomes circular-polarization maintaining . We should note that, for the proposed application, the twisted fiber is greatly advantageous over a spun fiber, because the circular birefringence of a spun fiber is significantly small due to the absence of optical activity , and it would be difficult to preserve circular polarization when wound on a bobbin.
For comparison, we prepared another spool of 200-m DSF, which was drawn from the same preform but wound on a bobbin without imposing a twist. Other than the twist rate and PMD, both fibers had identical properties: the zero-dispersion wavelength at 1556 nm, dispersion slope of 0.07 ps/nm2/km, propagation loss of 0.2 dB/km, and the nonlinear coefficient γ= 3.0 /W/km (The effective γ in the non-twisted DSF is 8γ/9 due to the polarization scrambling effect ).
To confirm the circular-polarization-maintaining property of the twisted DSF, we measured the polarimetric optical time-domain reflectometry (P-OTDR) traces. The P-OTDR setup and the measured traces are shown in Figs. 2(a) and 2(b), respectively. Because of the polarizer inserted at the OTDR output/input port, the received optical power becomes a function of the polarization ellipticity at the Rayleigh-scattered point of the fiber . Smooth P-OTDR traces without any ripple in the twisted-DSF section clearly indicates that the ellipticity of polarization is maintained excellently along the twisted DSF. We found well-defined eigen-polarization modes by injecting a continuous-wave (CW) light into the twisted DSF and observing the wavelength dependence of the output SOP with a polarimeter. The PMD of the twisted DSF was measured to be 1.1 ps/km by the fixed-analyzer method.
4. Experiment of polarization-insensitive asymmetric four-wave mixing
Figure 3 shows the experimental setup, which is similar to that in . Three external-cavity CW lasers (ECL) generated pump 1, pump 2, and the signal at wavelengths of 1557.5 nm, 1559.5 nm, and 1554.5 nm, respectively. Two pumps were combined with an identical SOP and phase-modulated with electrically combined RF-sinusoidal signals (149 and 450 MHz) to increase the SBS threshold. The signal was intensity-modulated with a 10-Gb/s pseudorandom bit sequence (PRBS) and injected into a polarization scrambler, which could be either enabled or disabled. The pump and signal waves were combined by using a 1:9 directional coupler and launched on either the twisted or non-twisted 200-m-long DSF. Two pumps had an identical power of 20.5 dBm at the input of DSF, while the input signal power was 4.5 dBm. The input SOPs of the pump and signal waves were controlled via PC1 and PC2, respectively. We also inserted a polarizer (P) followed by a half-wave plate (HWP) at the each arm of the pump and signal transmitters to study the polarization dependence quantitatively. The pump SOP was monitored with a polarimeter at the output of DSF.
Figure 4 shows eye diagrams of the output idler wave for various cases. In the case of non-twisted DSF, the eyes are degraded seriously as we turn the polarization scrambler ON [Fig. 4(a)]. In contrast, Fig. 4(b) shows the case when we employ the twisted DSF and adjust the pump SOP to one of the circular eigen-polarization modes of the twisted fiber. Excellent eyes are observed even when the polarization scrambler is enabled, indicating small polarization sensitivity of the FWM efficiency.
We then turn the polarization scrambler OFF and measure the polarization sensitivity of the idler conversion efficiency [(output idler power) / (input signal power)] by first adjusting PC2 to maximize the idler power and then rotating HWP2. It is cofirmed that when HWP2 is rotated by 45 deg, the idler power coincides with the minimum power obtainable by adjusting PC2. Figure 5 shows the polarization sensitivity of the conversion efficiency when we employ the twisted DSF (circles) and non-twisted DSF (dots). While the polarization sensitivity is as large as 5.8 dB in the case of non-twisted DSF (in agreement with the theoretical value of 6 dB ), it is reduced to 0.9 dB by employing the twisted DSF.
We next measure the polarization sensitivity as a function of the pump SOP. We first adjust PC1 so that the pumps are circularly polarized inside the twisted fiber, and then rotate HWP1. Due to the unitary property of PC1, the ellipticity of pump SOP in the twisted fiber is directly projected onto the angle of HWP1. (The pump SOP moves along a longitude of the Poincaré sphere with a rotation of HWP1.) In Fig. 6, dots and circles show the maximum and minimum conversion efficiency, respectively, measured as we vary the signal SOP with PC2. Triangles show the ratio of those two, representing the polarization sensitivity of the conversion efficiency as a function of the pump SOP. As expected, the polarization sensitivity is minimized when the pumps are circularly polarized, and becomes as large as 8 dB when they are linearly polarized.
Finally, the wavelength tunability of the idler wave is examined by tuning the wavelength of pump 2 while fixing pump 1 to 1557.5 nm. Figure 7 shows the maximum (dots) and minimum (circles) conversion efficiency and their ratios (triangles) as functions of the wavelength separation between two pumps. The theoretical curves are calculated from Eqs. (17) and (23). Although the measured values are somehow degraded from the theory, the polarization sensitivity is kept below 1.8 dB even at a large pump wavelength separation of 10 nm. The difference between the theory and experiment can be attributed to the Raman effect and the polarization dependence of β 2, which are ignored in the theory. Since the phase-mismatching effect scales with the fiber length, we can increase the wavelength tuning range by reducing the fiber length at the cost of conversion efficiency. Twisting a short length of a highly nonlinear fiber (HNLF) or periodically inverting the direction of the twist is expected to solve this problem.
In summary, we have shown for the first time that the efficiency of asymmetric nondegenerate fiber FWM becomes insensitive to the signal SOP when the two pumps are kept circularly polarized along the fiber. The theory was verified experimentally by using a twisted fiber. By twisting a 200-m DSF at 15 turns/m and aligning the pump waves to a circular SOP, we achieved efficient reduction of the polarization sensitivity from 5.8 dB to 0.9 dB. Although the conversion bandwidth is limited by the PMD of the twisted DSF, it can be increased by using a shorter length of the twisted HNLF or periodically inverted twisted fiber. The proposed scheme is expected to be a simple and efficient way of realizing tunable wavelength converters and wavelength-exchange devices without polarization dependence.
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