1. INTRODUCTION

The linear properties of twisted birefringent fibers have been well-investigated [1, 2, 3]. However, studies of their nonlinear optical properties have focused mainly on reducing polarization dependence in such third-order processes as self- and cross-phase modulation [4, 5], and four-wave mixing [6]. Here, we report for the first time, to our knowledge, that twisting a birefringent fiber with an artificially induced second-order nonlinearity $\left({\chi }^{\left(2\right)}\right)$ can result in the generation of second-harmonic (SH) phase-matchings not observed in the untwisted fiber.

We demonstrate this in a periodically poled silica fiber (PPSF) [7] that has been quasi-phase-matched (QPM) via periodic ultraviolet (UV) erasure [8, 9] for the second-harmonic generation (SHG) of ${\lambda }_{\text{SH}}\approx 775\text{\hspace{0.17em} nm}$ light in the ${\text{LP}}_{01}$ guiding mode. The induced ${\chi }^{\left(2\right)}$ in this fiber is attributed to the product of a frozen-in direct current (DC) field $E_x^{\text{DC}}$ (taken by convention to be in the x direction) and the third-order optical nonlinearity for amorphous fused silica: ${\chi }_{ijk}^{\left(2\right)}\left(-2{\omega }_0;{\omega }_0,{\omega }_0\right)=3{\chi }_{ijkx}^{\left(3\right)}\left(-2{\omega }_0;{\omega }_0,{\omega }_0,0\right)E_x^{\text{DC}}$. The nonzero ${\chi }^{\left(2\right)}$ tensor elements that arise from this model have been found to be experimentally valid for our fibers [10]: ${\chi }_{xxx}^{\left(2\right)}=3{\chi }_{xyy}^{\left(2\right)}=3{\chi }_{yxy}^{\left(2\right)}=3{\chi }_{yyx}^{\left(2\right)}$. We may then write the ${\chi }^{\left(2\right)}$ tensor as follows:

Consequently, only second-order processes involving the nonzero tensor elements (1) are allowed. When birefringence is not present in the fiber, the resulting SHG signals are degenerate in wavelength. However, birefringence can lift the degeneracy, and the phase-matching condition for the various SHG signals can be written as

For each phase-matched signal, we define the normalized conversion efficiency ${\eta }_{\text{SH}}$, a measure of the fiber’s second-order nonlinearity (3), as ${\eta }_{\text{SH}}\equiv P_i^{\left(2{\omega }_0\right)}/\left(P_j^{\left({\omega }_0\right)}{P}_k^{({\omega }_0}\right)$. In Fig. 1 , ${\eta }_{\text{SH}}$ is plotted against the fundamental wavelength $\left({\lambda }_F\right)$ for each of the three spectrally separated (i.e., non-degenerate) SHG signals obtained in our birefringent poled fiber. Also shown in Fig. 1 (inset) is a schematic of the poled fiber cross-section. The core of the fiber is sandwiched between two large air-holes that were used to accommodate the electrodes during the thermal poling process [7]. The fiber was poled for 75 min at a potential of 4 kV, and the electrodes were removed after a frozen-in DC field $\left(E_x^{\text{DC}}\right)$ was established in the core region. A QPM period of $\Lambda \approx 50\mu \text{m}$ was applied along the length of the fiber through periodic UV erasure [7, 8].

Due to its cross-sectional geometry and the different transverse stresses it experiences during fiber draw, our fiber is weakly birefringent, with its principal axes (x and y) aligned parallel and perpendicular (respectively) to the direction of the DC field, $E_x^{\text{DC}}$, as again shown in the inset of Fig. 1. Hence, the three SHG peaks in Fig. 1, representing the normalized conversion efficiencies of the $y+y\to x$, $x+x\to x$, and $y+x\to y$ signals from left to right, have the expected ratio of 1:9:4 [10]; as ${\eta }_{\text{SH}}\propto {\left|\left(2-{\delta }_{jk}\right){\chi }_{ijk}^{\left(2\right)}\right|}^2$ and ${\left|{\chi }_{xyy}^{\left(2\right)}\right|}^2:{\left|{\chi }_{xxx}^{\left(2\right)}\right|}^2:{\left|{\chi }_{yxy}^{\left(2\right)}+{\chi }_{yyx}^{\left(2\right)}\right|}^2=1:9:4$, this is in agreement with Eq. (1). No other peaks significantly above the noise floor are observed experimentally [11], again in accordance with Eq. (1).

The relevant physical parameters of our poled fiber are given as follows: It is a step-index fiber with core radius of $a=2.0\mu \text{m}$, a numerical aperture of 0.20, and a length of $L=23\text{\hspace{0.17em} cm}$. The tensor element ${\chi }_{xxx}^{\left(2\right)}$ is determined to be 0.067 pm/V from the $x+x\to x$ phase-matching’s peak ${\eta }_{\text{SH}}$ (Fig. 1). The birefringence at the fundamental frequency $\left({\omega }_0\right)$ is determined by a linear-optical technique [12] and found to be $\delta {\beta }^{\left({\omega }_0\right)}\equiv \left({\beta }_y^{\left({\omega }_0\right)}-{\beta }_x^{\left({\omega }_0\right)}\right)=\left(7.4\pm 2.8\right)\times {10}^{-5}\mu {\text{m}}^{-1}$ [$\delta n^{\left({\omega }_0\right)}\equiv n_{\text{eff},y}^{\left({\omega }_0\right)}-n_{\text{eff},x}^{\left({\omega }_0\right)}=\left(1.8\pm 0.7\right)\times {10}^{-5}$, where $n_{\text{eff},y}$ is the effective index of the y-polarized mode]. The birefringence at the SH frequency $\left(2{\omega }_0\right)$ is estimated to be $\delta {\beta }^{\left(2{\omega }_0\right)}=13.6\times {10}^{-5}\mu {\text{m}}^{-1}$ $\left(\delta n^{\left(2{\omega }_0\right)}=1.7\times {10}^{-5}\right)$ from the peak separations in the SHG spectrum (Fig. 1) using

2. TWISTING A BIREFRINGENT POLED FIBER

When a birefringent fiber is twisted, its polarization eigenmodes are no longer aligned to the principal axes $\left(x,y\right)$ but are, in general, elliptical [2]. This is because the x- and y-polarized modes are coupled by the fiber twisting; energy will be transferred back and forth between the two modes as light propagates along the length $\left(z\right)$ of the twisted fiber. Writing the electric field as $\mathbf{E}\left(z\right)=\widehat{x}{\mathcal{E}}_x\left(z\right)+\widehat{y}{\mathcal{E}}_y\left(z\right)$ ($\left\{\widehat{x},\widehat{y}\right\}$ are the principal axes in the rotating frame of the twisted fiber), the coupled-mode equation governing this phenomenon [2, 13] is

We may obtain the polarization eigenmodes $\left(\mathbf{X},\mathbf{Y}\right)$ and associated eigenvalues (the propagation constants ${\beta }_X,{\beta }_Y$) of the twisted fiber by diagonalizing the matrix $\underset{̱}{C}$ in Eq. (5). Without loss of generality, one can make the simplifying assumption ${\beta }_y>{\beta }_x$ so that

Although the new eigenmodes (6a, 7a) hold for slowly varying rates of twist $\left[\left(1/\beta \right)\left(d\Omega \left(z\right)/dz\right)⪡\Omega \left(z\right)\right]$, we will deal only with uniformly twisted fiber $\left[\Omega \left(z\right)=\text{const}\right]$ in the remainder of this paper.

In such a uniformly twisted birefringent poled fiber, the phase-matching condition (2) for SHG and other three-wave-mixing parametric processes remains valid. For small twists $\Omega \left(\Omega ⪡{\beta }_{x,y}\right)$, the QPM period Λ also remains unchanged. However, SHG now involves phase-matchings for the new eigenmodes (X and Y) of the twisted fiber, and $\left\{{\beta }_X,{\beta }_Y\right\}$ are used in place of $\left\{{\beta }_x,{\beta }_y\right\}$. This means that certain phase-matchings between the polarization eigenmodes (that were not permitted in the untwisted fiber) will be allowed in the twisted fiber.

We use the shorthand notation $J+K\to I$ to denote a SHG phase-matching in the twisted fiber; I $\left(J,K\right)$ labels the polarization eigenmode(s) at the SH (fundamental) wavelength and can take on values of X or Y. We may predict which of the new phase-matchings give rise to nonzero conversion efficiencies by writing the ${\chi }^{\left(2\right)}$ tensor (1) in terms of these new eigenmodes:

In the weakly coupled regime $\left(\left|\xi \right|⪡1\right)$, we can write the relative conversion efficiencies of all phase-matchings completely in terms of ${\xi }^{\left({\omega }_0\right)}$ (Table 1 ) by observing that ${\xi }^{\left(2{\omega }_0\right)}\approx \frac{1}{2}{\xi }^{\left({\omega }_0\right)}$ for our fiber (because $\delta {\beta }^{\left(2{\omega }_0\right)}\approx 2\delta {\beta }^{\left({\omega }_0\right)}$). Table 1 gives us a sense of how the strengths of the SHG signals change with twist. The series expansions of the relative conversion efficiencies contain only even powers of ${\xi }^{\left({\omega }_0\right)}$; in fact, the relative strengths of all the signals are even functions of ${\xi }^{\left({\omega }_0\right)}$. Therefore, twisting the fiber in either clockwise or counter-clockwise direction results in the same effect on the SHG spectrum.

The three phase-matchings observed in the untwisted fiber $\left(y+y\to x,x+x\to x,y+x\to y\right)$ are still present in the twisted fiber, but are now relabeled (x has become X, and y changed to Y) to reflect the new polarization eigenmodes. Three new SHG signals ($Y+Y\to Y$, $X+X\to Y$, and $Y+X\to X$) appear due to fiber twisting. Table 1 tells us that the twisting has the greatest effect on the new type II phase-matching $\left(Y+X\to X\right)$; for small twists, its predicted conversion efficiency is an order of magnitude larger than that of the $Y+Y\to Y$ signal, which in turn is 1 order of magnitude larger than the $X+X\to Y$ phase-matching.

We may use Eq. (9) to plot (Fig. 2 ) the (normalized) conversion efficiencies of the various phase-matchings as a function of twist ($\Omega L/2\pi$, measured in number of revolutions, with L being the length of the PPSF). The $Y+X\to X$ signal is found to be enhanced for moderate twist rates $\left(>2\text{\hspace{0.5em} rev}\right)$ to the point where its relative conversion efficiency exceeds that of the $X+X\to X$ and $Y+X\to Y$ phase-matchings.

3. SHG EXPERIMENT

In practice, a twist is effected upon the poled fiber using the apparatus shown in Fig. 3a . Although the length L of the PPSF is 23 cm, it is spliced to Corning SMF-28 pigtails on both ends, and the composite fiber is 161 cm long. A twist is applied to this entire length of fiber by holding fixed one of its connectorized ends and rotating the other end with a rotation mount.

Two erbium-doped fiber-amplified (EDFA) continuous-wave (CW) laser sources tunable in the fundamental wavelength $\left({\lambda }_F\approx 1.55\mu \text{m}\right)$ are used for the SHG experiment [Fig. 3b]. The sources are combined with a polarization beam combiner (PBC). Fiber-based polarization controllers (FPC1, FPC2) are used to align the polarization states of the two beams to the principal axes of the PBC. The result is two orthogonally polarized beams at the output of the PBC. Each time a different rate of twist $\left(\Omega \right)$ is applied to the PPSF, FPC3 is used to align these two orthogonal polarizations to the new principal polarizations $\left[X^{\left({\omega }_0\right)}\left(\Omega \right),Y^{\left({\omega }_0\right)}\left(\Omega \right)\right]$ of that fiber. This is done by keeping only one laser source on, tuning its wavelength to the $X+X\to X$ peak, and using FPC3 to sweep over the entire Poincaré sphere [10] to find the state of polarization $\left(X\right)$ at the fundamental wavelength that maximizes the $X+X\to X$ signal’s conversion efficiency. In this way, we ensure that one laser source launches X-polarized light into the twisted poled fiber, while the other source is Y-polarized.

A wavelength sweep is then performed for the following three input polarization states: $X^{\left({\omega }_0\right)},Y^{\left({\omega }_0\right)}$ (in which cases one of the laser sources is turned off), and $X^{\left({\omega }_0\right)}+Y^{\left({\omega }_0\right)}$ (in which case both sources are switched on). The six SHG phase-matchings (Table 1) are expected to lie in the range 1540–1560 nm for the fundamental wavelength ${\lambda }_F$, and that is the wavelength range swept. At the output of the poled fiber [Fig. 3c], the fundamental and SH beams are separated with a wavelength division multiplexer (WDM); the SH power $P^{\left(2{\omega }_0\right)}$ is measured with a silicon (Si) power meter, while the fundamental power $P^{\left({\omega }_0\right)}$ and polarization are monitored with a polarimeter to ensure that the amount of power coupled into the respective eigenmodes of the poled fiber remains fixed.

It is not always possible to align the input polarization states of the fundamental light perfectly to that of the eigenmodes of the twisted fiber. As a result of this misalignment, there is an error in the measured conversion efficiency for each of the signals. Additionally, because the poled fiber is fusion-spliced to single-mode fiber (SMF), the total length of fiber being rotated is much greater than L; the amount of twist applied to the PPSF itself may be nonuniform along its length $\left[\Omega \left(z\right)\ne \text{const}\right]$. We cannot twist the PPSF by more than 1 rev because the presence of the splices also makes the fiber more fragile toward mechanical stresses (such as twisting).

Figure 4a gives the experimental SHG spectra for three values of twist. Figure 4b plots ${\eta }_{\text{SH}}$ of the SHG signals for various twists; they are in good agreement with the theory (solid lines). The theoretical ${\eta }_{\text{SH}}$ values are calculated using the same parameters as in Fig. 2.

We see that the spectral position of each signal shifts with increasing twist in Fig. 4a. This is a result of the effective birefringence $\delta {\beta }_{XY}^{\left(\omega \right)}\left(\equiv {\beta }_Y^{\left(\omega \right)}-{\beta }_X^{\left(\omega \right)}\right)$ at the fundamental $\left(\omega ={\omega }_0\right)$ and the SH $\left(\omega =2{\omega }_0\right)$ increasing as the amount of twist increases [see Eqs. (6b, 7b)]:

It is also possible to measure $\delta {\beta }^{\left(\omega \right)}$ by observing the spectral separation between the various peaks for different rates of twist. Denoting the fundamental frequency of the $Y+Y\to X$ and $X+X\to X$ peaks by ${\omega }_{YY\to X}$ and ${\omega }_{XX\to X}$, respectively (${\omega }_{JK\to I}$, in general, denotes the fundamental frequency of the $J+K\to I$ peak), equating their phase-matching conditions (2) gives

An expression similar to Eq. (12) can be found at the SH frequency:

Finally, we stress that the experiment demonstrated here would not have been possible had the fiber birefringence $\delta \beta \approx {10}^{-4}\mu {\text{m}}^{-1}$ at ${\omega }_0$ and $2{\omega }_0$ been two to three times smaller or larger. The birefringence was large enough to allow for spectral separation between the three SHG signals of the untwisted fiber, but small enough so that a realistic amount of fiber twisting $\left(\Omega L/2\pi \approx 1\text{\hspace{0.5em} rev}\right)$ effects a marked change in the SHG spectrum of the fiber. Remarkably, twisting allows for two type II phase-matched signals ($Y+X\to X$ and $Y+X\to Y$) of comparable conversion efficiencies to be present in the SHG spectrum (Fig. 4) of our poled fiber.

4. CONCLUSION

We have demonstrated that new SHG phase-matchings in birefringent poled fiber can be generated, with their strengths and spectral positions controlled, by twisting the fiber. This results from the eigenmodes of the twisted fiber evolving from linearly polarized to elliptically polarized, which allows for the intermix of the various ${\chi }^{\left(2\right)}$ tensor elements of the fiber. With moderate amounts of twist, a new type II signal, $Y+X\to X$, can be observed and its conversion efficiency greatly enhanced.

Table 1. Relative Conversion Efficiencies of SHG Signals for Slightly Twisted $\left({\xi }^{\left({\omega }_0\right)}⪡1\right)$ Poled Fiber

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Fig. 1 Experimental SHG spectrum of the untwisted poled fiber is plotted against the fundamental wavelength ${\lambda }_F$, showing the expected 1:9:4 ratio for the peak conversion efficiencies of each signal. See [10, 11] for details. The inset shows the fiber cross-section, where one of the principal axes $\left(x\right)$ is also aligned to the direction of the frozen-in DC field $E_x^{\text{DC}}$.

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Fig. 2 Theoretical efficiencies ${\left|\left(2-{\delta }_{JK}\right){\widehat{\chi }}_{IJK}^{\left(2\right)}\right|}^2$ for the six SHG signals are plotted as a function of twist. The efficiencies are scaled to reflect the experimentally obtained ${\eta }_{\text{SH}}$ values of the untwisted fiber. The $X+X\to Y$ phase-matching is not visible because its estimated efficiency is significantly smaller than the other SHG signals. The parameters used for this plot were the birefringence at the fundamental $\left(\delta {\beta }^{\left({\omega }_0\right)}=7.4\times {10}^{-5}\mu {\text{m}}^{-1}\right)$ and SH $\left(\delta {\beta }^{\left(2{\omega }_0\right)}=13.6\times {10}^{-5}\mu {\text{m}}^{-1}\right)$, the length $\left(L=23\text{\hspace{0.17em} cm}\right)$ of the PPSF, and the elasto-optic coefficient $\left(g=0.15\right)$.

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Fig. 3 Experimental SHG setup for the twisted poled fiber. (a) The poled fiber is fusion-spliced to connectorized standard SMF on both ends; one connectorized end is held fixed, while the other end is placed on a rotation mount, where it is free to rotate. A translation stage is used to keep the fiber taut, so that it does not curl when twisted. (b) A fiber-based PBC combines two orthogonal polarizations $\left(X^{\left({\omega }_0\right)}\updownarrow ,Y^{\left({\omega }_0\right)}\circ \right)$ of light from EDFA-amplified CW tunable laser sources (TLS) in the 1540–1560 nm range. Fiber polarization controllers (FPCs) control the polarization state of light being launched into the PBC and poled fiber, with FPC1 and FPC2 used to align the input beams to the principal axes of the PBC, and FPC3 used to align the two orthogonally polarized beams to the polarization eigenmodes of the twisted PPSF. (c) At the output end of the fiber, the SH $\left({\lambda }_{\text{SH}}\right)$ and fundamental $\left({\lambda }_F\right)$ light are separated with a WDM; the power and polarization at ${\lambda }_F$ is measured with a polarimeter, while a silicon (Si) detector monitors the SH power.

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Fig. 4 (a) Experimental SHG spectrum for the poled fiber at varying twists ($\text{red}=0\text{\hspace{0.5em} rev}$, $\text{white}=0.58$, $\text{black}=1.02$). The inset shows the spectrum for the $Y+Y\to Y$ phase-matching, also for the three values of fiber twist. (b) A scatter plot of the experimental ${\eta }_{\text{SH}}$ for all signals at varying rates of twist. For comparison, the theoretical prediction for each phase-matching is drawn as a solid line; the curves used here are identical to those found in Fig. 2. Note that the $X+X\to Y$ phase-matching is so much weaker than any of the other five signals that it is not observed above the measurement noise floor.

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Fig. 5 Theoretical SHG spectrum for the twisted fiber at the same values of twist ($\text{red}=0\text{\hspace{0.5em} rev}$, $\text{white}=0.58$, $\text{black}=1.02$) as in Fig. 4a. Observe that the spectral separation between the $X+X\to X$ and $Y+Y\to X$ signals increases as the amount of twist changes from 0 to 1 revolutions. The insets show the spectra for the $Y+Y\to Y$ and $X+X\to Y$ phase-matchings, which have significantly smaller conversion efficiencies than the other four SHG signals. Figures 2, 4b, and this figure were created using the same parameters.

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Fig. 6 (Fundamental) Wavelength separation between the $X+X\to X$ and $Y+Y\to X$ peaks is plotted against the amount of twist applied to the fiber. The solid line is the curve of best fit given by Eq. (13); this fitting provides another measure of the birefringence $\left(\delta {\beta }^{\left({\omega }_0\right)}\approx \left(9.2\pm 0.7\right)\times {10}^{-5}\mu {\text{m}}^{-1}\right)$ for the untwisted fiber at the fundamental wavelength. The elasto-optic constant g is once again taken to be $g=0.15$.

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