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We present the polarization dependence of strong asymmetric long period fiber gratings written on tapered fibers. We found that for off-resonance conditions the spectral response and the output mode strongly depend on the input state of polarization. We utilize this dependence to obtain a mode selective device and demonstrate radially polarized and azimuthally polarized fiber lasers based on these asymmetric long period fiber gratings.
© 2016 Optical Society of America
Corrections
Gilad Masri, Shir Shahal, Avi Klein, Hamootal Duadi, and Moti Fridman, "Polarization dependence of asymmetric off-resonance long period fiber gratings: erratum," Opt. Express 25, 5787-5787 (2017)https://opg.optica.org/oe/abstract.cfm?uri=oe-25-5-5787
1. Introduction
Long period fiber gratings (LPFGs) are fiber gratings with a periodicity much longer than the wavelength [1, 2]. Thanks to their long periodicity, LPFGs are suitable for coupling between different transverse modes in fibers [3–5]. Thus, they are implemented in add/drop devices [6–9] and in laser beam shaping devices [10, 11]. The coupling between modes is sensitive to the resonance condition which makes the LPFG suitable for detecting temperature, chemical compounds and stresses [12–22]. LPFGs also serve for exciting modes which are usually degenerate by tapering the fiber which lifts off the degeneracy [23–28].
Recently, we presented LPFGs resulting from a tapered fiber with periodic diameter oscillations. Such LPFGs show off-resonance frequency response [29] and are more stable and robust. Also, they have wider bandwidth than on-resonance LPFG. Therefore, these LPFG can open an alternate route in mode shaping, fiber detectors and in in-fiber light manipulation. However, since the bandwidth of these LPFGs is wide, it is hard to separate between neighboring modes with similar propagation parameters even when the fiber is tapered which lifts off the degeneracy.
In this manuscript, we present an asymmetric LPFG which couples light between modes according to the light frequency and to the state of polarization. Therefore, it is possible to obtain LPFGs which couple light to a specific high-order mode over a wide bandwidth without exciting other modes. These asymmetric LPFGs have both the stability of strong LPFGs together with the tunability of regular LPFGs. Thus, it can increase the efficiency of LPFG based detectors, it can lead to stable fiber lasers with high order modes, and can be implemented in in-fiber modal manipulating devices. We present calculated and measured results to support our claims and elaborate on our fabrication methods. We also utilized this asymmetric LPFG to demonstrate a unique radially and azimuthally polarized fiber laser with no free-space optics over a broad bandwidth.
2. Cylindrical modes in tapered fibers
The transverse modes in optical fibers are obtained by solving the Maxwell equations in a dielectric cylinder [30]. The modes of the traveling electromagnetic field are expressed by the Bessel function in the core and the Hankel function at the cladding. The propagation parameter, β, of the field is found numerically, by satisfying these two equations:
where u and w are normalized frequencies according to [30], and nc and n are the refraction indexes of the core and the cladding, respectively. These two equations lead to the HE modes. The lowest mode is the HE11 and the first excited modes are the TM01 and TE01 modes which are known as the radial and azimuthal polarized modes [24]. The power distributions of these modes together with the local state of polarization are presented in the insets of Fig. 1(a). The calculated normalized propagation parameter b [30], as a function of the fiber diameter for 1.5 µm wavelength is presented in Fig. 1(a). The results reveal that for a fiber with wide diameter, the propagation parameters of the TM01 and the TE01 modes are identical. As the fiber diameter drops, the degeneracy is lifted and the propagation parameters splits into two separate values.
Fig. 1 (a) Normalized propagation parameter [30], as a function of the fiber diameter for the first three modes: HE11, TE01 and TM01 modes. Insets show the power distribution and the local state of polarization of the three modes. (b) Relative difference between the propagation parameters of TE01 and TM01 as a function of the fiber diameter.
Tapering the fiber lifts the degeneracy in the propagation parameters of the radial and azimuthal states of polarization, namely the TE01 and the TM01 modes, as presented in Fig. 1(a). Once we lift of the degeneracy, we obtain two propagation parameters. One for TE01, denoted as β1, and one for TM01, denoted as β2. The relative difference between them, namely ∆β = 1/2 · (β1 − β2)/(β1 + β2), indicates how high the degeneracy was lifted. We present ∆β as a function of the fiber diameter in Fig. 1(b). The difference between the two propagation parameters grows exponentially when reducing the fiber diameter. Thus, by tapering fibers it is possible to excite one mode and not the other.
3. Symmetric vs asymmetric LPFG
When the LPFG is cylindrical symmetric, it’s spectral response depends only on the light frequency and not on the input state of polarization. However, when the LPFG is not symmetric, exciting to high order modes depends both on the frequency and on the input state of polarization. We investigate asymmetric LPFG in which the fiber diameter oscillates in the x axis while remaining constant in the y axis. Schematics of such asymmetric LPFG are presented in Fig. 2, where inset (b) presents the cross-section in the x − z plane showing the fiber diameter oscillations, and inset (c) presents the cross-section in the y − z plane showing a constant fiber diameter.
Fig. 2 Schematics of the asymmetric LPFG presenting a tapered fiber with diameter oscillations in the x direction and constant diameter in the y direction. (b) cross section of the x − z plane showing the oscillations in the fiber diameter, (c) cross-section of the y −z plane showing a constant fiber diameter. Insets (a) and (d) shows numerical simulation of an HE11 mode propagating in such asymmetric LPFG where inset (a) shows the input electric field distribution and inset (d) shows the output electric field distribution. A video showing the electric field propagation in an asymmetric LPFG is presented in Visualization 1.
We simulated the electric field propagation in the asymmetric LPFG by utilizing the 4’th order Runge-Kutta method on the wave equation in fiber, as
We set the input mode by numerically evaluating the mode distribution in fiber with a diameter of 3 µm and set the first derivative to be 0. Next, we propagate the field along a fiber with an oscillating diameter in the x axis and constant diameter in the y axis. The results are presented in Fig. 2. Inset (a) shows the electric field distribution of the HE11 mode as the input and inset (d) shows the electric field distribution after a few grating cycles. The simulated propagation of the electric field in the asymmetric LPFG is presented in Visualization 1. From this simulation, we deduced that the fiber radiates only along the axis with the oscillating diameter, as presented in inset (d). The polarization of the radiated field is the same as the polarization of the input mode. But since it is only radiating from the x axis it couple to either the TE01 mode or the TM01 mode according to the input state of polarization.
We also evaluated the coupling strength between the HE11 mode and each of the high order modes as a function of the input state of polarization from this simulation. This was done by calculating the overlap between the radiated power from the HE11 mode and the high order modes. The coupling strength was found to be higher than 0.3 and thus, these LPFGs are in the strong coupling regime which has off-resonance frequency response [29].
Next, we denote the HE11 mode with vertical state of polarization as TE11 and the HE11 mode with horizontal state of polarization as TM11. In TE11 input mode, the polarization orientation is transverse to the asymmetric LPFG while in TM11 input mode, the polarization orientation is perpendicular to the asymmetric LPFG. Thus, only the TE11 mode can excite the TE01 mode and only the TM11 mode can excite the TM01 mode. So, the input state of polarization must be tuned in order to excite high order modes. Also, since the propagation parameters of the TE01 mode and the TM01 mode are different in tapered fibers, the spectral response of the LPFG is different, and we obtained a spectral response which is a function of the input states of polarization.
We calculated the spectral response as a function of the input state of polarization by extending the numerical method presented in [30] to include the input state of polarization. This was done by calculating the propagation parameter of all the modes in the fiber as a function of frequency and simulating light propagating in the LPFG while each perturbation in the fiber diameter couples light between modes. The coupling strength depends on the state of polarization and was calculated according to the numerical simulation of propagating the electric field in the grating. Thus, we obtain the power distribution between the fiber modes at the output as a function of the light frequency. Since the output fiber is single mode, all the high order modes radiate outside and the measured transmission spectrum is determined according to the output power in the HE11 mode.
We calculated the transmission spectrum for vertical and horizontal input states of polarization and present them in Fig. 3(a), where red dots denote the transmission spectrum for horizontal state of polarization and blue asterisks denote the transmission spectrum for vertical state of polarization. The two off-resonance responses are in anti-phase, so frequencies with high transmission in one state of polarization have low transmission in the orthogonal state of polarization.
Fig. 3 Transmission spectrum of the LPFG as a function of the input state of polarization. (a) and (b) show calculated and measured transmission spectra for horizontal (blue asterisks) and vertical (red dots) states of polarization. (c) and (d) show calculated and measured transmission spectra as a function of the polarization orientation, where H denotes horizontal, D denotes diagonal and V denotes vertical states of polarization.
Next, we calculated the transmission spectrum as a function of the polarization orientation from horizontal to vertical and present it in Fig. 3(c). Where H denotes horizontal, D diagonal and V vertical states of polarization. As evident, the spectral response gradually transforms from the response for the horizontal polarization to the response for the vertical one.
We expand the analytic evolution of strong LPFG [29,30] to include the state of polarization of the light. We start from the definition of the excited state spectrum, as presented in Eq. (39) in Ref. [30]. We consider two states of polarization with two different propagation parameters as β1 for the TE01 mode and β2 for the TM01 mode. Then, the propagation parameter difference between the HE11 mode and the two excited modes are ∆β1 = β0 − β1, and ∆β2 = β0 −β2, where β0 is the propagation parameter of the HE11 mode. The spectrum of the excited state, t× is defined as:
where κ is the coupling strength, is the self-coupling coefficient, Λ is the LPFG periodicity, is the Stokes vector representing the field state of polarization, and Ŝ1 is a unit Stokes vector denoting horizontal state of polarization. This equation adds two off-resonance transmission spectra, one for horizontal state of polarization and one for vertical state of polarization. By rotating the input state of polarization, the transmission spectrum gradually changes from the response to horizontal state of polarization to the response to vertical state of polarization.4. Fabrication and measured results
The fabrication process of the asymmetric LPFG is similar to the one in [29] and is based on mechanical oscillations of tapered fibers. In order to break the cylindrical symmetry of the fiber and of the grating, we misaligned the CO2 lasers resulting in an asymmetric heating profile which leads to an asymmetric oscillations in the fiber diameter.
We fabricated an asymmetric LPFG and measured the output spectrum as a function of the input state of polarization. The transmission spectra for vertical and horizontal input states of polarization are presented in Fig. 3(b) where red dots denote the transmission spectrum for horizontal state of polarization and blue asterisks denote the transmission spectrum for vertical state of polarization. The measured transmission spectra show the same anti-phase behavior as do the calculated results.
Careful analysis of the measured spectrum presented as the red curve in Fig. 3(b), reveals that the spectral response is composed from the addition of two spectral oscillations, as presented in Fig. 4. The larger amplitude spectral oscillations, presented as the dashed curve, are due to the coupling to the TE01 mode as presented in our model and Eq. (3). The smaller amplitude spectral oscillations, which are presented as the dotted curve, rise from coupling to higher order modes which were not consider in our simplified model. The model presented here can be further extend to include as many modes as needed by adding more terms to Eq. (3).
Fig. 4 Measured spectral response of the asymmetric LPFG as presented by the red dots in Fig. 3(b) separated into two spectral oscillations presented as the dashed and the dotted curves.
We measured the transmission spectrum while rotating the orientation of the input states of polarization from horizontal to vertical. The measured transmission spectrum as a function of the input state of polarization is presented in Fig. 3(d) where H denotes horizontal, D diagonal and V vertical states of polarization. Comparing the experimental results to the calculations shows good agreement including the gradual transition from the response to horizontal polarized light to vertical polarized light.
These results reveal that the asymmetric LPFG has a unique off-resonance spectral response which depends on the input state of polarization. Switching the input state of polarization from horizontal to vertical completely alter the spectral response of the LPFG and can be utilized for tuning the LPFG in real-time and for optical switches.
To demonstrate this, we fabricated an asymmetric LPFG and measured the output mode as a function of the input state of polarization. The experimental scheme is presented in Fig. 5. We start with a tunable laser beam and a controllable state of polarization and send the beam through the asymmetric LPFG. Next, we direct the output in free space toward a Calcite crystal which splits the beam into its two orthogonal states of polarization and detect the intensity distribution of the beams by a CCD camera. By comparing the intensity distributions of the two beams, we reconstruct the local state of polarization of the beam [31, 32]. We set the input state of polarization to a TE11 mode and to a TM11 mode, and the results are presented in Fig. 5. Insets (a), (b) and (c) shows that a TE11 mode couples into an azimuthally polarized mode, namely a TE01 mode. Insets (d), (e) and (f) shows that a TM11 mode couples into a radially polarized mode, namely a TM01 mode.
Fig. 5 Measurement scheme of the output mode from an asymmetric LPFG as a function of the input state of polarization. The calcite crystal splits the output beam into two orthogonal states of polarization where a CCD camera measures the intensity distribution. The local state of polarization is recovered from the camera output. Horizontal input state of polarization: insets (a) and (b) present the output intensity distributions and inset (c) presents the recovered output local state of polarization showing an azimuthal mode. Vertical input state of polarization: inset (d) and (e) present the output intensity distributions and inset (f) presents the recovered output local state of polarization showing a radial mode.
We evaluated the mode purity of the output mode, by projecting the output mode onto an idle mode and calculating the power relation between the projected mode and the original output mode. This is equivalent to calculating the relative power transmitted through a polarizer with the desired polarization distribution. The mode purity of both the radial and the azimuthal polarized beams was over 85%, indicating that the LPFG efficiently couples light from the two HE11 modes to radially or azimuthally polarized modes according to the input state of polarization.
5. Radially and azimuthally polarized lasers
Radially and azimuthally polarized beams are important for microscopy [34], material processing [35,36], trapping and accelerating particles [37,38] and laser light amplification [39]. However, free-space technology for converting the fundamental HE11 mode into radially and azimuthally polarized modes (such as axion, spatial light modulators, etc.) are bulky, sensitive and have high cost [35–41]. Therefore, fiber devices for coupling light to radially polarized beams were developed such as off-axis splicing which requires accurate splicing and on-resonance LPFGs which have narrow bandwidth [23–27,42,43]. While usually LPFG operates in resonance conditions which are sensitive to fluctuations in the grating periodicity and light frequency, we present radially and azimuthally polarized fiber lasers based on our off-resonance asymmetric LPFG which are more robust due to their wider bandwidth response.
Our radially and azimuthally polarized lasers are based on exciting one mode in the cavity and introducing losses to the other mode. This setup is similar to the laser scheme in [44, 45] and is presented in Fig. 6. The laser cavity consists of a few-modes Ytterbium doped fiber with a side diode pump. The LPFG is written on one end of the fiber and is spliced to a single mode fiber which is spliced to a high reflection Bragg grating. On the front end, the light is collimated and directed to the output coupler of the laser cavity. The single mode fiber supports only the fundamental HE11 mode, while the rest of the cavity supports both the HE11 and the radial and azimuthal modes. We fabricated a LPFG which couples the HE11 mode to the radially polarized mode, thus, only the radially polarized mode is supported by this cavity and the azimuthal mode radiates away along the single mode fiber. We fabricated a second LPFG which couples the HE11 mode to the azimuthal polarized mode to obtain a cavity which support only the azimuthal mode and radiating away the radial mode.
Fig. 6 Schematics of the radially or azimuthally polarized modes fiber laser based on asymmetric LPFG. HR - high reflecting, LPFG - long period fiber grating, OC - output coupler. The doped fiber length was 1.5 m with core diameter of 15 µm.
We measured the output modes of the two lasers with the configuration of Fig. 5. The measured output intensity distributions together with the calculated local states of polarization are presented in Fig. 7. The polarization purity of both lasers was over 90%, and we believe that by introducing longer LPFGs it can be improved. The output power was 200 mW and can be increased by resorting to radially polarized fiber amplifiers [45]. We note that while for the results presented in Fig. 5 we had to tune the input wavelength and the state of polarization, the results presented in Fig. 7 were easily obtained thanks to the laser gain competition. The output modes were stable and robust even when the fibers were moved.
Fig. 7 Measured output intensity distribution and the reconstructed polarization distributions of the fiber lasers output. (top) Azimuthally polarized laser; (a) measured output intensity distribution after vertical polarizer and (b) after horizontal polarizer (right), (c) reconstructed local state of polarization showing an azimuthally polarized beam. (bottom) Radially polarized laser; (d) measured output intensity distribution after vertical polarizer and (e) after horizontal polarizer, (f) reconstructed local state of polarization showing a radially polarized beam.
These results demonstrate highly efficient radially and azimuthally polarized fiber lasers based on our off-resonance asymmetric LPFGs. The off-resonance asymmetric LPFGs enable wide bandwidth fiber lasers which are more stable than other schemes.
6. Conclusions
To conclude, we presented asymmetric long period fiber gratings operating in off-resonance conditions. We numerically simulated the electric field evolution in such LPFGs as a function of the input state of polarization and obtained the coupling strength between the different modes. Then, we obtained the spectral response of these LPFGs as a function of the input state of polarization and found them to agree with the experimental results. Finally, We demonstrated radially and azimuthally polarized fiber lasers based on our asymmetric LPFGs. We believe that our asymmetric off-resonance LPFGs can be implemented in high mode fiber lasers, in LPFG based detectors and in in-fiber modal manipulation devices.
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