All-fiber spatial rotation manipulation for radially asymmetric modes

We propose and experimentally demonstrate spatial rotation manipulation for radially asymmetric modes based on two kinds of polarization maintaining few-mode fibers (PM-FMFs). Theoretical finding shows that due to successful suppression of both polarization and spatial mode coupling, the spatial rotation of radially asymmetric modes has an excellent linear relationship with the twist angle of PM-FMF. Both elliptical core and panda type FMFs are fabricated, in order to realize manageable spatial rotation of LP11 mode within ±360° range. Finally, we characterize individual PM-FMF based spatial orientation rotator and present comprehensive performance comparison between two PM-FMFs in terms of insertion loss, temperature sensitivity, linear polarization maintenance, and mode scalability.


Results
We experimentally examine the characteristics of arbitrary mode rotation by two PM-FMFs. The experimental setup is presented in Fig. 1. The output of a distributed feedback (DFB) laser diode (LD) (Yenista Optics OSICS-DFB), whose operation wavelength is 1550 nm, is coupled to one input port of the mode selective PL (Phoenix Photonics 3PL-0160153). The PL is a three-input mode excitation device which can be used to selectively convert the input fundamental modes to LP 01 , LP 11a , and LP 11b modes, respectively. After the orientation optimization, the PM-FMF is fusion spliced to the output end of the PL by the polarization maintaining fiber fusion splicer (Fujikura FSM 100 P+). As a result, LP 11a mode is selectively excited. The two ends of the PM-FMF are respectively fixed to fiber rotator (FR, Thorlabs HFR007) whose angle increment is 5°. The total length of PM-FMF between two FRs is only 12 cm. By manually adjusting the second FR, we are able to manage the fiber twist angle with 2° precision. Mode pattern at the PM-FMF output is captured, in order to carry out both spatial distribution characterization and power measurement. Specifically, spatial distribution is recorded by a collimation lens and an infrared CCD (OPHIR Photonics SP620U-1550). The power measurement is accomplished by a collimation lens and integrating sphere (New Port Power Meter 1918-R with detector 819D-IG-2-CAL). The rotation device including two FRs and a section of PM-FMF is put into a self-fabricated temperature controlled box for temperature-dependent performance investigation. In order to realize pure mode rotation, only single mode is allowed to be launched into PM-FMFs each time, because the power spatial distribution of the mode basis possesses certain orientation with respect to the symmetrical axis of PM-FMF, as shown in Fig. 2. The propagation constants of those modes differ from each other. Otherwise, there will be an evolution of output mode pattern 20 .
Arbitrary rotation of LP 11 mode. First we launch LP 11 mode into two PM-FMFs, respectively, and implement the fiber twisting with the help of the FR2. To make sure only one mode is launched, polarization maintaining fiber fusion splicer is used to rotate the PM-FMF input end to optimize orientation before splicing. When only one mode is launched and the PM-FMF is manually perturbed, no distortion of the captured mode pattern is observed. Next, the rotation angle of FR 2 is recorded, together with the captured output mode field. For the ease of comparison, LP 11 mode rotation by twisting the commercial two-mode fiber (TMF, OFS) is also recorded. The results are summarized in Fig. 3.
Performance comparison between two PM-FMFs. In order to provide a quantitative comparison of spatial mode rotation between the commercial TMF (c-TMF) and two PM-FMFs, rotation efficiency 21 between the captured mode profile at PM-FMF output and the corresponding ideal LP mode profile with designated orientation angle are given by The output electric field of PM-FMF E capture is obtained by the combination of the captured mode intensity distribution with fixed phase difference between adjacent lobes. The ideal LP mode distribution E pure can be achieved according to the diffraction theory 22 . Then, the overlap integral between E capture and E pure can be a good indicator of rotating efficiency after spatial rotation manipulation. As shown in Fig. 4, the mode rotating efficiency of LP 11 mode by using two types of PM-FMFs is above 0.9. We believe that the little degradation is mainly due to the imperfect mode excitation at the input of PM-FMF. However, the mode rotating efficiency by using the commercial TMF degrades to 0.7. Because of severe mode coupling between degenerate modes arising in commercial TMF, the spatial rotation manipulation by commercial TMF is impossible. In order to investigate the temperature-dependent performance of all-fiber spatial mode manipulation, spatial rotation of LP 11 modes using two PM-FMFs at four specific twist angles (−360°, −180°, 180°, and 360°) are individually measured, when the temperature is varied from 20 °C to 140 °C. In each twist angle measurement, only the environmental temperature is varied and all the other components are kept stable. The variations of rotation angle as a comparison with initial results at 20 °C are recorded, as shown in Fig. 5. All fluctuations of rotation angles are below ±0.01°, indicating of temperature-insensitive operation of the proposed all-fiber spatial orientation rotator. Meanwhile, we characterize the IL of the proposed spatial mode rotator. We monitor the maximum variation of optical power with respect to the operation wavelength, when the LP 11 mode are rotated within 360° range. The IL in comparison with the situation without mode rotation operation is summarized in Fig. 6. The IL is less than 0.45 dB for all modes within 360° range, when the operation wavelength is varied from 1540 nm to 1560 nm. Finally, the state of polarization (SOP) at the PM-FMF output is determined, because both elliptical core and panda-type fiber are commonly used for the purpose of polarization maintaining in the SMF design 23 . Thus, we carry out accurate measurement of the Stokes vector during the spatial rotation of LP 11 mode with interval of 10°, based on a manually rotatable quarter-wave plate and a fixed linear polarizer 24 , as shown in Fig. 1(d). When the e-FMF is rotated, the polarization extinction ratio are 15.1 dB for LP 01 mode and 14.0 dB for LP 11 mode. When panda-FMF is used for spatial mode rotation, the corresponding polarization extinction ratio are 26.8 dB for LP 01 mode and 21.0 dB for LP 11 mode. The CCD-captured LP 11 mode pattern, when the polarizer located at the PM-FMF output with two orthogonal linear polarizations, is shown in Fig. 7. For the ease of understanding the SOP characterization results, we also plot the results on Poincare sphere, as shown in Fig. 8. We can conclude that the rotated LP 11 mode is still linearly polarized, and its linear SOP can also rotate the same value as that of spatial pattern.

Discussion
As shown in Fig. 3, for both e-FMF and panda-FMF, the rotation of LP 11 mode has an excellent linear relationship with respect to the twist angle of PM-FMF, indicating a flexible manipulation of arbitrary mode profile rotation. Performance comparison of the two PM-FMF has also been carried out in several aspects. By calculating overlap integral between the captured mode profile at PM-FMF output and the corresponding ideal LP 11 mode profile with designated orientation angle, rotation efficiency of LP 11 by two PM-FMF are obtained. The LP 11 mode rotation efficiency for two PM-FMF are almost the same, showing a good advantage over commercial TMF. When environmental temperature is varied, the variations of rotation angle of LP 11 mode by two PM-FMFs is negligible, indicating of temperature-insensitive operation for two types of PM-FMFs. The IL over an operation wavelength from 1540 nm to 1560 nm are also characterized. For both PM-FMF based mode rotators, the IL are less than Figure 2. Axis of radially asymmetric modes agree with the symmetrical axis of the PM-FMFs 0.45 dB. However, the capability of linear polarization maintenance of panda-FMF is better than that of e-FMF. The linear polarization maintenance of e-FMF can be further improved by increasing the ellipticity 25 . However, considering the issue of fiber coupling and mode scalability, the mode profile of panda-FMF is more compatible with that of c-FMF because of its circular core. In a summary, all-fiber spatial rotation performance between e-FMF and panda-FMF are almost the same except the polarization maintenance capability. Thus, we do recommend the use of panda-FMF for the purpose of arbitrary mode rotation manipulation. In particular, the rotation of other radially asymmetric modes is expected, when the designated mode is solely guided over the proposed specialty FMFs. Normally, the increment of core size is helpful to guide more high order radially asymmetric modes.

Conclusions
In conclusion, we are able to arbitrarily rotate LP 11 mode orientation within ±360° range by two types of PM-FMFs. Theoretical finding shows that spatial rotation manipulation of high-order radially asymmetric modes  is possible due to the suppression of both spatial and polarization mode coupling, when the symmetric axis of input radially asymmetric mode is aligned to the symmetric axis of PM-FMF. Finally, we fabricate both e-FMF and panda-FMF and carry out a thorough performance comparison between two PM-FMFs in many aspects, including insertion loss over different operation wavelength, temperature sensitivity and linear polarization maintenance capability. Finally, we recommend the use of panda-FMF for the purpose of all-fiber spatial rotation manipulation for arbitrary radially asymmetric modes.

Methods
Generally, mode propagation in the FMF can be described by coupled mode theory (CMT). Coupled mode equation can be solved analytically for a length of z in Jones matrix form 26 : where E p (0) and E q (0) are the electrical fields of two input adjacent vector modes. Mode coupling may occur between arbitrary two vector modes, but coupling between adjacent modes is usually the strongest. β 0 = (β p + k pp + β q + k qq )/2 is the common propagation constant, δ = (β p + k pp − β q − k qq )/2 is the detuning factor while β = 2 π/λ is propagation constants in vacuum, β p = 2 πn/λ is propagation constant of mode p arising in the FMF, n is the mode effective refractive index, and λ is operation wavelength.
is the coupling strength, k pq is inter-coupling coefficient and k pp is the self-coupling coefficient under the designated LP mode group. Normally, the transmission matrix in Eq. (2) must be in close proximity to a unit matrix, in order to preserve the same output mode profile as the input one. Therefore, the off-diagonal elements must be small enough, regardless of the environmental perturbation, especially when the fiber is twisted. As a result, the coupling coefficient k pq and coupling strength S need to be well managed by specifically enlarging the detuning factor δ and reducing the coupling coefficient k pq . Under the condition of fiber twisting, k pq is determined by the following procedures. Generally, the transverse electric field of vector modes in the FMF can be described as: where A p (z) = a p exp(iβ p z) is the mode amplitude of the FMF and a p is the amplitude coefficient of mode p with propagation constant β p . When the FMF is twisted, the coupling coefficients becomes 27 where фt is twisting rate. Hence, the electric fields at the stationary coordinate can be obtained, when the FMF is twisted. It is obvious that if the off-diagonal coefficients are small and Eq. (2) can be approximated as a unit matrix, there exists a prospect of manageable mode rotation of LP mode, when the FMF is twisted. In order to minimize off-diagonal coefficients, small coupling coefficient k pq and a large detuning factor δ between propagated modes are required. In other words, mode spacing between propagated modes should be enlarged. However, for c-FMF, the coupling between adjacent vector modes within the designated LP mode group is severe 29 , so that corresponding transmission matrix in Eq. (2) cannot be approximated as a unit matrix. Consequently, the output mode profile after propagation over a section of c-FMF is random 6,18 . Since e-FMF has been proposed for MIMO-less transmission with the mode spacing substantially enlarged 30,31 , it might be a promising candidate for realizing mode rotation manipulation. We start to design and fabricate e-FMF first. The circular fiber preform is successfully fabricated by a method of plasma chemical vapor deposition (PCVD). Then, we carry out rod in tube (RIT) process after symmetrically grinding in order to obtain an elliptical shape. After that, the preform is drawn at the drawing speed of 200 m/min with a tension of 140 g. Finally, the e-FMF is ready for further characterization. The major and minor axes of elliptical core are 23 μm and 16 μm, respectively. The diameter of cladding is  121 μm while the refractive index (RI) difference between core and cladding is 3.9 × 10 −3 . The scanning electron microscope (SEM) figure is shown in Fig. 9(a). Meanwhile, since the panda-type structure is commonly used to suppress the polarization mode coupling arising in the polarization maintaining single mode fiber (SMF), we also design PM-FMF with panda-type structure. Panda-FMF is fabricated by a multi-step process, involving drilling holes into a solid preform and inserting pre-fabricated borosilicated stress rod. With almost the same fiber drawing technique, the panda-FMF with core cladding RI difference of 5.0 × 10 −3 can be obtained, while the corresponding SEM figure is shown in Fig. 9(b). The core and cladding diameters of panda-FMF are 21 μm and 125 μm, respectively. The distance between the centre of stress-applied parts and fiber core is 30 μm. We are able to take advantage of finite element mode solver (COMSOL5.2) to obtain the field distribution and its corresponding mode effective index. For the e-FMF, there are total 10 guided spatial and polarization modes, including LP 01x , LP 01y , LP 11ax , LP 11ay , LP 11bx , LP 11by , LP 21ax , LP 21ay , LP 21bx , and LP 21by . Similarly, for the panda-FMF, the 11 guided spatial and polarization modes are LP 01x , LP 01y , LP 11ax , LP 11bx , LP 11by , LP 11ay , LP 21x , LP 21bx , LP 02x , LP 31ax , and LP 31bx , respectively. For the ease of understanding, the spatial distribution of LP 11 and LP 21 modes with the corresponding mode spacing to the LP 01x mode are shown in the Fig. 10. When the fiber structure varies from conventional circular core, mode degeneration is broken and LP modes become true modes in PM-FMFs, in contrast with c-FMF 25 . When twisting of these PM-FMF happens, two kinds of coupling can be determined. Specifically, spatial mode coupling means the coupling between LP 11a and LP 11b , while polarization mode coupling occurs between LP 11x and LP 11y . Consequently, for the e-FMF, the off-diagonal coefficients in Eq. (1) are of about 0.09 for spatial mode coupling and 0.71 for polarization mode coupling, respectively. Obviously, the spatial mode coupling is trivial enough, while the polarization mode coupling is non-negligible. When the e-FMF is twisted, spatial distribution of the launched LP 11 mode can be maintained, but the linear polarization may vary from time to time. Similarly, spatial and polarization mode coupling of panda-FMF are calculated as 6.9 × 10 −4 and 9.2 × 10 −8 . Both couplings are extremely weak, indicating that the spatial pattern can be well preserved with linear polarization, when the panda-FMF is twisted.