Linearly polarized orbital angular momentum mode ... - OSA Publishing

1990

Research Article

Vol. 56, No. 7 / March 1 2017 / Applied Optics

Linearly polarized orbital angular momentum mode purity measurement in optical fibers YOUCHAO JIANG,1,2 GUOBIN REN,1,2,* HAISU LI,1,2 MIN TANG,1,2 YU LIU,1,2 YUE WU,1,2 WEI JIAN,1,2 AND SHUISHENG JIAN1,2 1

Key Laboratory of All Optical Network & Advanced Telecommunication Network of EMC, Beijing Jiaotong University, Beijing 100044, China Institute of Lightwave Technology, Beijing Jiaotong University, Beijing 100044, China *Corresponding author: [email protected]

2

Received 14 November 2016; revised 24 January 2017; accepted 7 February 2017; posted 8 February 2017 (Doc. ID 280706); published 1 March 2017

We presented a simple method for measuring the mode purity of linearly polarized orbital angular momentum (OAM) modes in optical fibers. The method is based on the analysis of OAM beam projections filtered by a polarizer. The amplitude spectrum and phase spectrum of a data ring derived from the beam pattern are obtained by Fourier transform. Then the coefficients of the mixed electric field expression can be determined and the mode purity can be obtained. The proposed method is validated and it is experimentally demonstrated in a two-mode fiber. © 2017 Optical Society of America OCIS codes: (050.4865) Optical vortices; (060.2310) Fiber optics; (060.2400) Fiber properties. https://doi.org/10.1364/AO.56.001990

1. INTRODUCTION An optical vortex carrying orbital angular momentum (OAM) is characterized by a helical phase front of expil ϕ, in which ϕ refers to the azimuth angle and l is the topological charge number [1]. Because of the helical phase structure, the light wave cancels out at the axis, resulting in a phase singularity at which the phase is indeterminate and the amplitude is zero. The optical vortices have been studied in a plurality of areas including motion sensing [2], higher dimensional optical communication [3,4], optical manipulation and trapping [5], and microscopy [6]. There have been many attempts to generate and manipulate OAM beams, including using cylindrical lens, mode converters, q-plates, spiral phase plates, spatial light modulators, metamaterials-based phase plates, and silicon integrated devices [1,7–11]. Meanwhile, the OAM generation and propagation based on fibers has also been studied, especially in recent years [12–18]. To carry OAM states, a fiber supporting higher-order modes is required. Generally, methods to generate OAM in fiber can be classified into two types according to the source modes. For one type, the OAM modes can be generated by combining two linearly polarized (LP) modes where the generated vortex has no spin angular momentum (SAM) [12,19]. For the other type, the OAM modes can be generated by combining two vector modes, where the generated vortex has SAM characterized by 1 or −1 [13,15]. A circularly polarized OAM mode can be regarded as a combination of two linearly polarized OAM modes. In addition, 1559-128X/17/071990-06 Journal © 2017 Optical Society of America

the result of filtering a circularly polarized OAM mode by a polarizer in any direction is a linearly polarized OAM mode. Moreover, several methods to generate tunable OAM in optical fibers have been reported, such as combining two LP modes [19], or two different vector modes [20]. The mode purity as one characteristic of OAM beams indicates the share of all power that the OAM mode accounts for. Quantitative measurements of OAM typically require complicated or custom apparatuses, such as Dove prism arrays, holograms, Shack–Hartmann wavefront sensors, or custom phase masks. These techniques can reveal more detailed information about the beam, but the size, complexity, and fragility of the required setups make them undesirable in some situations [19,21,22]. In 2012, Bozinovic et al. presented a ring method to measure the mode purity of circularly polarized OAM beams in a two-mode fiber [15]. In the ring method, they introduced a so-called vortex basis set to analyze the purity of OAM modes. By taking the Fourier series of left or right circular projections, the mode purity is determined. However, due to the circular polarization of the vortex basis set, such a method is only suitable for circularly polarized OAM light, and some additional setups are also needed in the method [15]. In this paper, we propose a method called the scalar intensity analysis method (SIAM) to determine the purity of LP OAM modes in optical fibers. This method is based on the analysis of OAM beam projections that are filtered by a polarizer. Consequently, no additional setup is needed when implementing this technique, which means that the LP OAM mode purity

Research Article


can be obtained without any other apparatuses in some experiments and applications, such as an all-fiber OAM mode converter, few-mode polarization-maintaining fiber OAM generator, and tunable OAM generator [17,19,20,23]. In our method, the amplitude spectrum and phase spectrum can be obtained by taking the Fourier transform of a data ring derived from the filtered electric field intensity. Then the coefficients of the mixed electric field expression can be achieved through the corresponding amplitude spectrum and phase spectrum. Finally, the mode purity is calculated by the coefficients. The proposed method is verified through a simulated mixed mode. Then we experimentally demonstrated this method for measuring the OAM mode purity in a two-mode fiber (TMF). 2. LP OAM MODE PURITY CALCULATION In few-mode or multimode optical fibers, the transverse electric field distributions of higher-order LP modes can be described as cosine or sine functions of the azimuth angle. With considering the polarization directions, the LP modes are nominated as cs;xy in the Cartesian coordinate system, where the subLPl ;m scripts l and m denote transverse and radial indices, the superscript cs denotes cosine (sine) function, and x (y) denotes polarized direction. According to the electric field distribution and polarized directions, LP modes are classified as even and odd modes with both x and y polarization directions. As Fig. 1 shows, in a two-mode fiber there are four kinds of patterns with considering the different polarization directions: s;y c;y s;x (a) LPc;x 1;1 , (d) LP1;1 are even modes, (b) LP1;1 , (c) LP1;1 are odd modes. It should be noted that the polarization directions of LP modes are generally neglected when the LP modes are just classified as even or odd modes. But in this paper, both the polarization directions and odevity should be considered. The higher-order vector modes in optical fibers could be classified as HEe;o , EHe;o , TE, and TM modes, where e and o refer to the even and odd modes, respectively. In the Cartesian coordinate system, using the weakly guiding approximation, these transverse electric field distributions of vector modes can be expressed as superposition of two LP modes [24]: HEel1;m xˆ cosl ϕ − yˆ sinl ϕ F l ;m r l ≥ 1; HEol1;m xˆ sinl ϕ yˆ cosl ϕ EHel−1;m xˆ cosl ϕ yˆ sinl ϕ l > 1; F l ;m r EHol−1;m xˆ sinl ϕ − yˆ cosl ϕ TM0;m xˆ cosϕ yˆ sinϕ l 1; F 1;m r TE0;m xˆ sinϕ − yˆ cosϕ

1991

where F l ;m represents solution of the Bessel equation (the radial field distribution of the corresponding LP mode), and ϕ is the azimuthal coordinate. The vector modes expressed in Eq. (1) filtered by a polarizer can be described as FHEel1;m pˆ cosl ϕ θ l ≥ 1; F l ;m r FHEol1;m pˆ sinl ϕ θ FEHel−1;m pˆ cosl ϕ − θ l > 1; F l ;m r FEHol−1;m pˆ sinl ϕ − θ FTM0;m pˆ cosϕ − θ F 1;m r l 1; (2) FTE0;m pˆ sinϕ − θ where pˆ represents the polarization direction of polarizer, θ is the angle between pˆ and the positive direction of the x-axix ranging between 0 and π. The filtering process is schematically shown in Fig. 2 and the results are shown in Fig. 3. From Eq. (2) and Fig. 3, it is known that LP modes can be obtained by filtering the corresponding vector mode. As Fig. 3 shows, with constant intensity and different θ, the intensity pattern rotates clockwise when the sign of θ is positive or rotates counterclockwise when the sign is negative.

Fig. 2. Filter the vector mode by using a polarizer with different polarization directions.

(1)

Fig. 1. Modal patterns of LP11 mode. (a) Even in x-direction, (b) odd in x-direction, (c) odd in y-direction, and (d) even in y-direction.

Fig. 3. Results of filtering the vector modes (TM0;1 , HEe2;1 , HEo2;1 , and TE0;1 ) by using a polarizer. The arrows in the first row represent different polarization directions.

1992

Research Article


The scalar field of a typical LP OAM mode V l ;m l ≥ 1 in the x-polarization direction is expressed as V l ;m xˆ F l ;m rfcosl ϕ i sinl ϕg xˆ F l ;m r expil ϕ;

l ≥ 1:

(3)

Equation (3) illustrates that the OAM can be generated by combining two LP modes of type even and type odd in the same polarization direction. The same as shown in Fig. 2, this LP OAM beam in the x-polarization direction filtered by a polarizer can be described as F V l ;m pˆ F l ;m r cosθfcosl ϕ i sinl ϕg pˆ F l ;m r cosθ expil ϕ;

l ≥ 1; (4)

where pˆ represents the polarization direction of the polarizer, θ is the angle between pˆ and the positive direction of the x-axis ranging between 0 and π. By observing Eq. (3), it is known that there are three main factors that influence the LP OAM mode purity, including the amplitude difference between cos and sin components, the phase mismatch between the cos and sin components, and other possible mode components. In a TMF, there are six vector modes: HEe1;1 , HEo1;1 , HEe2;1 , HEo2;1 , TM0;1 , and TE0;1 (corresponding to two LP mode groups). As mentioned in the introduction, combining two LP modes can generate LP OAM modes [12,17,19], and the generated OAM modes can be described as Eq. (3). No matter how complex the mode components in the TMF are, based on Eqs. (2) and (4), we can define the total output electric field (including the LP OAM mode) after passing through a polarizer as IOAM pˆ F1;1 r a cosϕ − ϕ0 b sinϕ − ϕ0 π φ1 × exp i 2 pˆ F 0;1 rc expiφ2 ;

(5)

where ϕ0 is the original azimuthal position, positive a and b represent amplitudes of the cos and sin components, c represents amplitude of HE11 mode, φ1 is the phase mismatch, φ2 is the phase difference between HE11 mode and the cos component (we set the phase of the cos component as reference). In Eq. (5), if the amplitude of the cos and sin components are equal, the phase difference between them is π∕2 and there are no other mode components (no fundamental mode in the TMF), the equation degenerate into Eq. (3). By observing Eq. (5), for generating the LP OAM mode, the amplitude of the LP OAM mode must be the smaller one of the cos and sin components. Because of the phase mismatch φ1 , the actual amplitude of the sin component is b cosφ1 . Then the mode purity MP can be calculated from MP

∯ F1;1 r2 min fa; b cosφ1 g2 ds ; ∯ jIOAMj2 ds

3. SCALAR INTENSITY ANALYSIS METHOD In order to obtain the coefficients in Eq. (5), we present the SIAM. For validating the method, we first construct an electric field intensity pattern using Eq. (5), so that the coefficients are known and the mode purity can be obtained directly. Then we calculate the mode purity again through the SIAM. Finally, by comparing the mode purity obtained from the SIAM and the preset value, we can assess the reliability of the method. An electric field is constructed with the parameters a 1, b 0.8, c 0.1, φ1 0.3, φ2 0.2, and l 1 so that the preset value of the LP OAM mode purity can be calculated from Eq. (6) and the preset value is 70.52%. The constructed OAM beam and the interference pattern are shown in Figs. 4(a) and 4(b), respectively. Now, the mode purity is calculated again by using the SIAM and the process can be divided into four steps. First, for calculating the mode purity, an appropriate coordinate system should be chosen to change the value of ϕ0 to zero. Second, we confine attention to the points on a ring with radius of r0 [see Fig. 4(c)], for which F 0;1 r 0 ≈ F 1;1 r 0 F r 0 . The r0 conveniently corresponds to the radius of the projection ring [15], about in the center between the inner boundary and outer boundary. The intensity of IOAM at r 0 derived from Eq. (5) can be expressed as jIOAMj2r 0 F r 0 2

a2 b2 c2 2 2

2ac cosφ2 cosϕ

2 2 π a b − cos2ϕ 2bc sinφ1 φ2 cos ϕ 2 2 2 π : (7) ab sinφ1 cos 2ϕ 2 Third, the amplitude spectrum and phase spectrum shown in Figs. 4(d) and 4(e) are obtained by taking the Fourier transform of jIOAMj2r 0 with respect to ϕ. Because Eq. (7) only includes

(6)

where minfa; b cosφ1 g means the smaller one of a and b cosφ1 , d s is the area element of the whole pattern area. Now, the key to calculate the OAM mode purity is how to get these coefficients.

Fig. 4. (a) Constructed mixed OAM mode, (b) interference pattern, (c) data ring of radius r0 , (d) amplitude spectrum of data ring, and (e) phase spectrum of data ring.

Research Article direct current, cosϕ, and cos2ϕ components, coefficients a, b, c, φ1 , and φ2 can be obtained from the amplitude spectrum and phase spectrum. It should be noted that if the ϕ0 is not zero, the corresponding amplitude spectrum is constant, while the phase spectrum varies with different ϕ0 . In this case, these coefficients cannot be obtained. Finally, the mode purity can be calculated by Eq. (6), and the calculated mode purity is 69.55%. The deviation of OAM purity between the preset value (70.53%) and the SIAM calculated value is 1.39%. The reason could be the center offset and/or azimuthal offset when building the coordinate system, the selection of r0 , and the computational error in computing the integral. In order to figure out how much the uncertainty is caused by choosing the data ring radius, the mode purities are calculated with different r 0 and the results are shown in Fig. 5. The optimal r 0 locates at about the center point between the inner radius and outer radius. So the uncertainty can be reduced by averaging the values that are calculated at several different points around the center point. For studying the sensitivity of the method to the OAM mode purity, the LP OAM mode purities are calculated with different parameters that are used in the construction of mode pattern, respectively. According to the results of calculations, the change of the mode purity that can be distinguished is less than 0.3%, and the deviation between the preset mode purity and the calculated mode purity is about 1%. In a nutshell, the proposed SIAM is very applicable for LP OAM mode purity evaluation in TMFs, especially in some applications such as an all-fiber OAM mode converter, few-mode polarization-maintaining fiber OAM generator, and tunable OAM generator. Furthermore, when most of the power is confined to second-order modes, with neglecting the first-order mode, the OAM mode purity is mainly related to the amplitudes of two cos and sin components (two LP modes) and the phase relationship between them. Therefore, Eq. (7) can be simplified as


1993

In contrast to Eq. (7), the number of unknown coefficients is down to three so that the coefficients a, b, and φ1 can be obtained more easily. Because only the second-order LP mode group exists in a TMF, therefore, the calculated mode purity of the LP OAM mode is accurate. For few-mode fibers (FMFs), the method is not applicable when more than one higher-order LP mode group exists and no one of them is dominated. Nonetheless, the purity can be approximately calculated with considering only one higher-order LP mode group in FMFs. In other words, when one dominated higher-order LP mode group and fundamental mode exist, or only one dominated higherorder LP mode group exists in FMFs, the method is applicable. This means that the mode purity can be calculated by Eq. (7) or (8) with replacing ϕ with l ϕ. 4. EXPERIMENT DEMONSTRATION The SIAM is experimentally demonstrated in a homemade multilayer core TMF that is shown in Fig. 6. The core radius is 9.5 μm and the cutoff wavelength of the third-order LP mode group is 1465 nm. The experimental setup for the generation and detection of a LP OAM beam is sketched in Fig. 7. The output beam from a tunable laser is divided into two paths by an optical coupler with a proportion of 1:1. One path is used to generate OAM beams, and the other is the reference in the context of interfering with generated OAM beams. Two adjustable attenuators are inserted in both paths to balance their power to

2 2 a2 b2 a b − cos2ϕ 2 2 2 2 π ab sinφ1 cos 2ϕ : (8) 2

jIOAMj2r 0 F r 0 2

Fig. 5. Calculated mode purity changes with different location of r0 , 0 point presents that r0 locates at the middle point between the inner radius and outer radius of the projection ring. The inset shows the location range of r0 , the diameter of the pattern is about 80 (pixel).

Fig. 6. Fiber refractive index profile of the TMF, the inset is an image of the fiber cross section.

Fig. 7. Experiment setup for the generation and detection of tunable OAM beams. PC, polarization controller; TMF, two-mode fiber; MS, mode stripper; LPG, long-period grating; FBG, fiber brag grating; Rot., rotator; MPS, metal parallel slab; Obj., objective; Pol., polarizer; BS, beam splitter; HWP, half-wave plate; Col., collimator; and SMF, single-mode fiber.

1994

Research Article


obtain an optimal interference. A mechanical long-period grating (LPG) is applied on the TMF as a mode converter. A mode stripper realized by the tightly bending TMF is used to ensure a pure fundamental mode (first-order mode) launching. In addition, a fiber Bragg grating written in the TMF is used to mitigate the effects of the first-order mode after the LPG. The TMF output is collimated using a 20× objective lens. A polarizer after the 20× objective lens is used to adjust the polarization directions. The polarization of the reference beam can be adjusted by the half-wave plate. The LP OAM beams and circularly polarized OAM beams can both be generated with our setup. Therefore, by varying the pressure applied to the mechanical LPG and metal parallel slab, adjusting the input polarization and rotating the fiber carefully, the LP OAM beams can be obtained. Then the output beam is collimated and the beam intensity is imaged using a CCD camera. The experiment results with different polarized directions are shown in Fig. 8. The variation with different polarization degrees indicates that this OAM beam is a LP OAM beam, because the circularly polarized OAM beam remains a constant intensity when filtered by a polarizer with different polarization directions. So the SIAM is applicable in this experiment. One key to calculate the mode purity is the selection of an appropriate coordinate system in which the ϕ0 is zero. It is known from Eq. (4) that the intensity of the electric field is maximum when θ is zero. Hence, the coordinate system can be built with setting the direction in which the intensity is maximum during turning the polarizer as positive x-axis, such that the value of ϕ0 is zero. From Figs. 8(a1)–8(a3) and 8(b1)– 8(b3), the intensity changes from maximum to minimum while the polarizer rotates 90° and the polarization mode purities in Figs. 8(a) and 8(b) are 99.7% and 96.8%, respectively. The change of intensity reflects that the OAM beam is linearly polarized and we can build the coordinate system when the intensity is maximum. The mode purities of Figs. 8(a1) and 8(b1) are calculated, respectively, and the processes are shown in Figs. 9 and 10. The obtained coefficients are summarized in

Fig. 9. (a) Intensity of Fig. 8(a1), (b) data ring of radius r0 , (c) amplitude spectrum of data ring, and (d) phase spectrum of data ring.

Fig. 10. (a) Intensity of Fig. 8(b1), (b) data ring of radius r0 , (c) amplitude spectrum of data ring, and (d) phase spectrum of data ring.

Table 1 and the mode purity values are about 59.0% and 94.7% in Figs. 8(a1) and 8(b1), respectively. In addition, the LP OAM purity can be corrected by multiplying the LP OAM mode purity by the previously calculated polarization mode purity with considering the polarization mode purity. However, the method is not applicable when the linear polarization purity is too low, because the mode may be a circularly polarized OAM mode in this case, as mentioned in [15].

Table 1. Calculated Coefficients Fig. 8. Experiment results, the first row is intensity and the second row is interference patterns in (a) and (b), respectively. Double-sided arrows denote the polarizer directions. Inten., intensity and Inter., interference.

Fig. 8(a1) Fig. 8(b1)

a

b

c

φ1

φ2

Mode Purity (%)

0.61 0.52

0.43 0.52

0.05 0.06

−0.32 0.17

0.02 2.51

59.0 94.7

Research Article In order to improve the accuracy of the measurement, the following points should be noticed. The signal-to-noise ratio of images should be reduced during the experiment for obtaining a clear pattern. An appropriate size of the image is indispensable for reducing the error in choosing the center and r0 . A suitable level of the intensity of pattern should be chosen to reduce the influence of image noise and avoid the saturation distortion in the meanwhile. An accurate choice of x-axix is also very crucial when establishing the coordinate system in which the value of ϕ0 is zero. 5. CONCLUSIONS In summary, we presented a method called SIAM based on the analysis of OAM beam projections filtered by a polarizer to measure the mode purity of LP OAM beams in optical fibers. The method is validated by comparing the value obtained from the SIAM and the value that we preset. This method is suitable for the LP OAM beams in a TMF, and we have experimentally demonstrated this method in a TMF. The technique also works in FMFs with considering only one higher-order LP mode group. The setup is simple and the method is easy to realize. Considering many applications about OAM beams are realized in TMFs, our method will be a useful tool in the mode purity calculation of such beams. In the future, the method can be modified for measuring the LP mode purity in few-mode fiber couplers and few-mode fiber lasers. Funding. National Natural Science Foundation of China (NSFC) (61178008, 61275092). REFERENCES 1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of LaguerreGaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). 2. N. Cvijetic, G. Milione, E. Ip, and T. Wang, “Detecting lateral motion using light’s orbital angular momentum,” Sci. Rep. 5, 15422 (2015). 3. H. Huang, G. Milione, M. P. Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. An Nguyen, D. A. Nolan, M. J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 14931 (2015). 4. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013). 5. M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5, 343–348 (2011). 6. S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral interferometry,” Opt. Lett. 30, 1953–1955 (2005).


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