Generating Radially and Azimuthally Polarized Beams by ... - MDPI

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Generating Radially and Azimuthally Polarized Beams by Using a Pair of Lateral Displacement Beamsplitters Chien-Yuan Han 1 , Zhi-Hao Wei 2 , Yung Hsu 3 , Kun-Huang Chen 4 , Chien-Hung Yeh 2 , Wei-Xuan Wu 2 and Jing-Heng Chen 2, * 1 2 3 4

*

Department of Electro-Optical Engineering, National United University, No. 2, Lienda, Nanshi Li, Miaoli 36063, Taiwan; [email protected] Department of Photonics, Feng Chia University, No. 100, Wenhwa Rd., Seatwen, Taichung 40724, Taiwan; [email protected] (Z.-H.W.); [email protected] (C.-H.Y.); [email protected] (W.-X.W.) Institute of Electro-Optical Engineering, National Chiao Tung University, No. 1001, University Rd., Hsinchu 30010, Taiwan; [email protected] Department of Electrical Engineering, Feng Chia University, No. 100, Wenhwa Rd., Seatwen, Taichung 40724, Taiwan; [email protected] Correspondence: [email protected]; Tel.: +886-4-2451-7250 (ext. 5093); Fax: +886-4-2451-0182

Academic Editors: Malte C. Kaluza and Shoou-Jinn Chang Received: 27 May 2016; Accepted: 23 August 2016; Published: 27 August 2016

Abstract: This paper proposes a modified polarization converter for generating radially and azimuthally polarized beams. Based on a Mach–Zehnder-like interferometric structure, the device consists of a pair of lateral displacement beamsplitters (LDBs) and two half-wave plates that manipulate the states of polarization of incident linearly polarized light. Through the coherent superposition of two orthogonal Hermite–Gaussian modes in the far field, radially and azimuthally polarized beams can be obtained simultaneously. A prototype was assembled to demonstrate the feasibility of the design. The proposed design has the advantage of having a simple, symmetric, compact, and robust structure. In addition, the introduction of LDBs substantially reduces the cost of this device. Moreover, the design can be applied to a broadband pulsed laser with considerable potential in high-power applications. Keywords: beam splitters; waveplates; coherence; interference; polarizations

1. Introduction Radially and azimuthally polarized beams have been increasingly studied in recent years because of their unique characteristic of axial polarization symmetry, and they can break the diffraction limit with a strong longitudinal electromagnetic field in focus [1–3]. The applications of these beams include high-resolution microscopy, surface plasmon excitation, optical trapping, material processing, and high-density data storage [4–6]. Various generation theories and methods for these beams have been proposed. The generation methods can be categorized as intracavity and extracavity methods. For the intracavity method, a specially designed birefringent crystal, an adjustable aperture, conical and axicon prisms, or a non-periodic sub-wavelength grating is set inside the resonant cavity of a laser to generate the radially or azimuthally polarized beam [7,8]. However, laser cavity structures are typically complicated. This complicated structuring could cause difficulty for alignment and thus additional power loss in laser manufacturing. Additionally, limited by the form of the laser cavity, the intracavity method is not suitable for fiber lasers, lithography ultraviolet (UV) lasers, or semiconductor lasers. By contrast, the extracavity method (which refers to two types of techniques), achieves radially and azimuthally polarized beams outside of the laser cavity. The first type of technique uses a cut half-wave Appl. Sci. 2016, 6, 241; doi:10.3390/app6090241

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plate, polarization converter, or liquid-crystal spatial light modulator to convert a linearly or circularly polarized beam into a radially or azimuthally polarized beam [9,10]. The second type of technique uses various interferometric structures to obtain a radially or azimuthally polarized beam [11,12]. The implementation of these methods could be restricted by the requirements of a highly coherent light source and environmental stability. This possibility makes the implementation of these methods potentially difficult. In addition, the generated beam is typically limited by low power. Phua and Lai proposed a simple manipulation method that uses birefringent crystals and half-wave plates to coherently generate a radially polarized beam [13]. In their method, the structure fulfills the demands of compactness, mechanical stability, and robustness for practical high power applications. However, because of the birefringence of the crystal polarization mode, dispersion occurs during the manipulation of different states of polarization. To eliminate the temporal walk-off problem, a second birefringent crystal is required to compensate the optical path difference between two orthogonal polarized modes. Therefore, the cost of the method could be high. In this paper, we propose a modified method for the generation of radially and azimuthally polarized beams. Based on a Mach–Zehnder-like interferometric arrangement, a pair of lateral displacement beamsplitters (LDBs) is used to replace the expensive birefringent crystals. Half-size and full-size half-wave plates (h-HWP and f-HWP) are applied to manipulate the states of polarization of the incident beam. The operation principles of this method, accompanied with a Jones matrix calculus, are described. With coherent superposition, radially and azimuthally polarized beams can be simultaneously achieved in the far-field region. To demonstrate the feasibility of the method, a prototype of the proposed polarization converter was successfully assembled and tested. Because of the application of LDBs, the design has a simple, symmetric, compact, and robust structure; furthermore, this design reduces the cost of the device substantially. In addition, the device can operate with a broadband pulsed laser and has considerable potential for high-power applications. 2. Principles Figure 1 illustrates the design of the optical structure for the generation of radially and azimuthally polarized beams. It consists of a pair of confocal cylindrical lenses with a focal length ratio of 1:2, a lateral displacement polarizing beamsplitter (LDPB), h-HWP and f-HWP, and an LDB. To facilitate an understanding of the principle, an orthogonal x-y-z coordinate system is introduced. The cross-section of the laser beam (viewing against the direction of light propagation) and the associated states of polarization after passing through each component are depicted in the four quadrants of Q1 , Q2 , Q3 , and Q4 , as shown in Figure 1a–g. In Figure 1, the Jones column matrix for a linearly polarized beam propagating in the +z-direction can be expressed as [14]:   cosθ E (θ) = (1) sinθ where θ indicates the direction of polarization with respect to the x-axis; and cosθ and sinθ represent the horizontal and vertical components of the polarization, respectively. In Figure 1a, a circular Gaussian beam with a 45◦ linear polarization (with respect to the x-axis) is incident into a pair of cylindrical lenses in the +z-direction. Consequently, its beam cross-section is transformed into an elliptical Gaussian beam, as illustrated in Figure 1b. After passing through the LDPB, the 45◦ linearly polarized elliptical Gaussian beam is separated into two orthogonally polarized components, as illustrated in Figure 1c. These components continue to pass through the h-HWP and the f-HWP. The optical axes of the h-HWP and f-HWP are set at 45◦ and 22.5◦ with respect to the x-axis, respectively. Therefore, as shown in Figure 1d, the states of polarization in the right-half regions (quadrants Q1 and Q4 ) for these elliptical beams are rotated by 90◦ . In Figure 1e, the states of polarizations at the four quadrants of these elliptical

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beams are rotated by 45◦ . Accordingly, the Jones column vector EQi of each polarized beam at the i-th quadrant (i = 1 to 4) can be expressed as: EQi = J f − HWP (Θ)J h− HWP (Θ) E(θ), (for i = 1 then θ = 0◦ ; for i = 4 then θ = 270◦ ) and Appl. Sci. 2016, 6, 241  EQi = J f − HWP (Θ) E(θ), (for i = 2 then θ = 180◦ ; for i = 3 then θ = 90◦ )

(2) 3 of 8 

(3)

and 

where J represents the Jones matrix of the half-wave plate, and Θ represents the direction of the optical EQi  Jwith ) E (  ) , (for i = 2 then θ = 180°; for i = 3 then θ = 90°)  f  HWP (  axis of the half-wave plate respect to the x-axis. Consequently, the Jones column vectors(3)  of the polarized beams at the four quadrants can be written as  represents the direction of the  where J represents the Jones matrix of the half‐wave plate, and          optical axis of the half‐wave plate with respect to the x‐axis. Consequently, the Jones column vectors  1 −1 −1 1 of the polarized beams at the four quadrants can be written as  EQ1 = , EQ2 = , EQ3 = , and EQ4 =

1

1

−1

−1

1  1   1 1 E Q 1    , E Q 2    , E Q 3    , and E Q 4    1 1  Finally, the two polarized beams1are concentrated after passing through        1and reflecting off the LDB,

respectively. In Figure 1f,g, the beams are two sets of 45◦ frame-rotated Hermite–Gaussian TEM10 Finally, the two polarized beams are concentrated after passing through and reflecting off the LDB,  and respectively. In Figure 1f,g, the beams are two sets of 45° frame‐rotated Hermite–Gaussian TEM TEM01 modes. Based on the image inversion caused by a right-angle prism and the reflection of 10 and  polarized light, the Jones column vectors of the polarized beam (illustrated in Figure 1g) with a new TEM01  modes.  Based  on  the  image  inversion  caused  by  a  right‐angle  prism  and  the  reflection  of  set ofpolarized light, the Jones column vectors of the polarized beam (illustrated in Figure 1g) with a new  quadrants Q1 *, Q2 *, Q3 *, and Q4 *, can be written as set of quadrants Q1*, Q2*, Q  3*, and Q4*, can be written as    

   −1 1 1 −1 EQ1 ∗ = , E , and EQ4 , 1∗ = 1 ∗ = Q2 1, EQ3 ∗ = 1  1 E Q1*    , E Q 2* 1   , E Q 3*   − −1  ,1and E Q 4*   1 , 1 1  1        where the direction of polarization occurs with respect to the − z-axis. Accordingly, approximately where the direction of polarization occurs with respect to the −z‐axis. Accordingly, approximately  radially and azimuthally polarized beams can be obtained in the far field with coherent radially and azimuthally polarized beams can be obtained in the far field with coherent superposition  superposition [15] by using a positive imaging lens (IL). In Figure 1, the radially and azimuthally [15] by using a positive imaging lens (IL). In Figure 1, the radially and azimuthally polarized beams  are checked with an analyzer (AN) and a charge‐coupled device (CCD) camera.  polarized beams are checked with an analyzer (AN) and a charge-coupled device (CCD) camera.

  Figure 1. Schematic representation of the generation of radially and azimuthally polarized beams. Figure  1.  Schematic  representation  of the  generation  of  radially  and  azimuthally  polarized  beams.  The The diagrams illustrate the cross‐section of the laser beam and the associated states of polarizations  diagrams illustrate the cross-section of the laser beam and the associated states of polarizations caused by passing through (a) the laser; (b) a pair of cylindrical lenses; (c) the lateral displacement  caused by passing through (a) the laser; (b) a pair of cylindrical lenses; (c) the lateral displacement polarizing beamsplitter (LDPB); (d) the half‐size half‐wave plate (h‐HWP); (e) the full‐size half‐wave  polarizing beamsplitter (LDPB); (d) the half-size half-wave plate (h-HWP); (e) the full-size half-wave plateplate (f‐HWP); (f) the lateral displacement beamsplitter (LDB); and by reflecting off (g) the LDB. The  (f-HWP); (f) the lateral displacement beamsplitter (LDB); and by reflecting off (g) the LDB. The coherent superposition of orthogonal polarizations in the far field is achieved with a positive imaging  coherent superposition of orthogonal polarizations in the far field is achieved with a positive lens (IL) and the beam profiles are checked using an analyzer (AN) and a charge‐coupled device (CCD)  imaging lens (IL) and the beam profiles are checked using an analyzer (AN) and a charge-coupled camera.  device (CCD) camera. 3. Results and Discussion  To show the feasibility of the design, a prototype of the proposed polarization converter was  assembled  that  incorporated  a  pair  of  confocal  cylindrical  lenses  (LJ1878L1‐A;  LJ1960L1‐A,  THORLABS) with  focal lengths  of 10 mm  and 20 mm,  respectively, a  pair  of  lateral displacement 

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3. Results and Discussion To show the feasibility of the design, a prototype of the proposed polarization converter was Appl. Sci. 2016, 6, 241  4 of 8  assembled that incorporated a pair of confocal cylindrical lenses (LJ1878L1-A; LJ1960L1-A, THORLABS) (polarizing  and  nonpolarizing)  (PBS251;  PS911,  THORLABS),  and  and two  with focal lengths of 10 mm and 20beamsplitters  mm, respectively, a pairBS013;  of lateral displacement (polarizing multiorder half‐wave plates (MOWP‐633‐1/2‐10, BVO). The dimensions of the device were 125 × 50.8  nonpolarizing) beamsplitters (PBS251; BS013; PS911, THORLABS), and two multiorder half-wave × 25.4 mm. A 632.8 nm helium‐neon laser (AH100M, ONSET) was used as the light source, with an  plates (MOWP-633-1/2-10, BVO). The dimensions of the device were 125 × 50.8 × 25.4 mm. A 632.8 nm output power of 10 mW. The phase condition between each quadrant was maintained as in‐phase  helium-neon laser (AH100M, ONSET) was used as the light source, with an output power of 10 mW. through angle adjustments of the h‐HWP on the y‐axis. A positive IL with a focal length of 100 mm  The phase condition between each quadrant was maintained as in-phase through angle adjustments was used to generate the radially and azimuthally polarized beams in the far field. The donut‐shaped  of the h-HWP on the y-axis. A positive IL with a focal length of 100 mm was used to generate the patterns are shown in Figures 2 and 3. The size of the beams illustrated in Figures 2a and 3a was  radially and azimuthally polarized beams in the far field. The donut-shaped patterns are shown in approximately 200 μm. To verify the radial and azimuthal characteristic of these beams, an analyzer  Figures 2 and 3. The size of the beams illustrated in Figures 2a and 3a was approximately 200 µm. was applied with different azimuthal angles prior to the CCD camera (XCD‐U100CR, SONY) and a  To verify the radial and azimuthal characteristic of these beams, an analyzer was applied with different laser beam profiler (BP109‐VIS, THORLABS). In Figure 2b–e, the beam profiles were observed with  azimuthal angles prior to the CCD camera (XCD-U100CR, SONY) and a laser beam profiler (BP109-VIS, the transmitting axis of the analyzer being 0°, 45°, 90°, and 135° with respect to the x‐axis. In Figure  THORLABS). In Figure 2b–e, the beam profiles were observed with the transmitting axis of the 3b–e, the beam profiles were observed with the transmitting axis of the analyzer being 0°, 45°, 90°,  analyzer being 0◦ , 45◦ , 90◦ , and 135◦ with respect to the x-axis. In Figure 3b–e, the beam profiles and 135° with respect to the −z‐axis. To examine the polarization quality of these beams, an additional  were observed with the transmitting axis of the analyzer being 0◦ , 45◦ , 90◦ , and 135◦ with respect to quarter‐wave  (WPQ05M‐633,  THORLABS)  was  inserted  to  the quarter-wave analyzer  for  the  the −z-axis. Toplate  examine the polarization quality of these beams, anprior  additional plate demonstration  of  the  spatial was distributions  of  Stokes parameters [14,16], as  shown  in Figure  4.  The  (WPQ05M-633, THORLABS) inserted prior to the analyzer for the demonstration of the spatial Stokes parameter S 0 describes the total intensity of the optical beam, which was scaled between 0 and  distributions of Stokes parameters [14,16], as shown in Figure 4. The Stokes parameter S0 describes the 2.  The  parameter  1  is  the  intensity  difference  between  and  vertical  polarization.  The  total intensity of theSoptical beam, which was scaled betweenhorizontal  0 and 2. The parameter S1 is the intensity parameter S 2 represents the intensity difference between linear polarization oriented at 45° and 135°.  difference between horizontal and vertical polarization. The parameter S2 represents the intensity Therefore, the values of S  and S2 were between +1 and −1. The parameter S  describes the intensity  difference between linear 1polarization oriented at 45◦ and 135◦ . Therefore, 3the values of S1 and S2 difference  between  right  and  left  circular  polarization,  which  was  scaled  between  were between +1 and −1. The parameter S3 describes the intensity difference between+0.5  rightand  and−0.5.  left Ideally, polarization, the  values  of  S3  should  be  much  smaller  0,  S1Ideally, ,  and  Sthe 2.  Although  the  intensity  circular which was scaled between +0.5 than  and −S0.5. values of S should be 3 distributions did not show a good cylindrical symmetry, they could be improved by applying highly  much smaller than S0 , S1 , and S2 . Although the intensity distributions did not show a good cylindrical accurate translation and rotation stages for alignment. These results successfully demonstrated the  symmetry, they could be improved by applying highly accurate translation and rotation stages for feasibility of the proposed design.  alignment. These results successfully demonstrated the feasibility of the proposed design.

  Figure 2. 2. Radially Radially polarized polarized  lights lights in in the the transmission transmission output output converted converted from from aa linearly linearly polarized polarized  Figure light: (a) donut‐shaped beam profile and intensity distribution in the x‐direction observed without an  light: (a) donut-shaped beam profile and intensity distribution in the x-direction observed without ◦;   analyzer,  and  beam  profiles  observed  with  45°;  an analyzer, and beam profiles observed withthe  thetransmitting  transmittingaxis  axisof  ofan  ananalyzer  analyzer at  at (b)  (b) 0°;  0◦ ; (c)  (c) 45 ◦ ◦ (d) 90°; and (e) 135° with respect to the x‐axis.  (d) 90 ; and (e) 135 with respect to the x-axis.

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    Figure 3. Azimuthally polarized lights in the reflection output converted from a linearly polarized  Figure 3. Azimuthally polarized lights in the reflection output converted from a linearly polarized Figure 3. Azimuthally polarized lights in the reflection output converted from a linearly polarized  light: (a) donut‐shaped beam profile and intensity distribution in the z‐direction observed without an  light: (a) donut-shaped beam profile and intensity distribution in the z-direction observed without light: (a) donut‐shaped beam profile and intensity distribution in the z‐direction observed without an  and  the  transmitting  axis  of of an an analyzer  at  (b)  0°; 0(c)  45°;  ◦ ; (c) ananalyzer,  analyzer, andbeam  beamprofiles  profilesobserved  observedwith  with the transmitting axis analyzer at (b) 45 ◦ ; analyzer,  and  beam  profiles  observed  with  the  transmitting  axis  of  an  analyzer  at  (b)  0°;  (c)  45°;    (d) 90°; and (e) 135° with respect to the −z‐axis.  ◦ ; and (e) 135◦ with respect to the −z-axis. (d) 90(d) 90°; and (e) 135° with respect to the −z‐axis. 

(a) (a)

(b) (b) Figure 4. Spatial distributions of Stokes vectors of S0, S1, S2, and S3: (a) radially polarized light and    Figure 4. Spatial distributions of Stokes vectors of S   Figure 4. Spatial distributions of Stokes vectors of S00, S , S11, S , S22, and S , and S3: (a) radially polarized light and  3 : (a) radially polarized light and (b) azimuthally polarized light.  (b) azimuthally polarized light.  (b) azimuthally polarized light.

The proposed polarization converter has a simple and symmetric structure that facilitates the 

The proposed polarization converter has a simple and symmetric structure that facilitates the  The proposed hasthe  a simple andof  symmetric structure that facilitates maintenance  of polarization the  in‐phase  converter condition  for  generation  radially  and  azimuthally  polarized  the maintenance  of  the  in‐phase  condition  for  the  generation  of  radially  and  azimuthally  polarized  beams.  In  the  condition device  can  radially  of and  azimuthally  polarized  beams  maintenance of addition,  the in-phase forgenerate  the generation radially and azimuthally polarized beams.  In  addition,  the  device  can  generate  radially  and  azimuthally  polarized  beams  simultaneously. In an application, a linear superposition of radially polarized mode and azimuthally  beams. In addition, the device can generate radially and azimuthally polarized beams simultaneously. simultaneously. In an application, a linear superposition of radially polarized mode and azimuthally  In anpolarized mode generates cylindrical vector (CV) beams [3], as shown in Figure 5. Because two beams  application, a linear superposition of radially polarized mode and azimuthally polarized polarized mode generates cylindrical vector (CV) beams [3], as shown in Figure 5. Because two beams  are always produced simultaneously, it causes 50% losses if only one beam mode is needed. In this  mode generates cylindrical vector (CV) beams [3], as shown in Figure 5. Because two beams are are always produced simultaneously, it causes 50% losses if only one beam mode is needed. In this  condition, an additional reflection prism (RP) could be applied to transform one mode into another.  always produced simultaneously, it causes 50% losses if only one beam mode is needed. In this condition, an additional reflection prism (RP) could be applied to transform one mode into another.  In Figure 6a, a reflection prism is set next to the LDB, which transforms the azimuthally polarized  condition, an additional reflection prism (RP) could be applied to transform one mode into another. In Figure 6a, a reflection prism is set next to the LDB, which transforms the azimuthally polarized  mode into a radially polarized mode by reflection. In Figure 6b, a reflection prism is set after the LDB,  In mode into a radially polarized mode by reflection. In Figure 6b, a reflection prism is set after the LDB,  Figure 6a, a reflection ispolarized  set next to the into  LDB, transforms the mode  azimuthally polarized which  transforms  the prism radially  mode  an which azimuthally  polarized  by  reflection.  mode into a radiallythe  polarized by reflection. 6b, a polarized  reflection mode  prismby  is reflection.  set after the Therefore, two outputs of radially polarized beam in +x‐direction and two outputs of azimuthally  which  transforms  radially  mode polarized  mode  into  In an Figure azimuthally  LDB, which transforms radially polarized mode intoA an azimuthally polarized mode byin  reflection. polarized  beam  in  the +y‐direction  could  be  achieved.  continuous  wave  laser  was  used  the  Therefore, two outputs of radially polarized beam in +x‐direction and two outputs of azimuthally  experiment for demonstration. Because of the symmetric structure, the two orthogonally polarized  Therefore, two outputs of radially polarized beam in +x-direction and two outputs of azimuthally polarized  beam  in  +y‐direction  could  be  achieved.  A  continuous  wave  laser  was  used  in  the  modes  have  same  optical  path  (OPL).  Having  the wave same  laser OPL  was implies  that  no  polarized beam inthe  +y-direction could belength  achieved. A continuous used in ideally,  the experiment experiment for demonstration. Because of the symmetric structure, the two orthogonally polarized  temporal walk‐off problem exists (polarization mode dispersion) between the orthogonally polarized  have  the  same  optical  path  length  (OPL).  Having  OPL  implies  that  ideally,  formodes  demonstration. Because of the symmetric structure, the the  twosame  orthogonally polarized modes no  have thetemporal walk‐off problem exists (polarization mode dispersion) between the orthogonally polarized  same optical path length (OPL). Having the same OPL implies that ideally, no temporal walk-off problem exists (polarization mode dispersion) between the orthogonally polarized modes in the design.

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Accordingly, the device can operate with a broadband pulsed laser. In [13], an additional crystal-type modes in the design. Accordingly, the device can operate with a broadband pulsed laser. In [13], an  modes in the design. Accordingly, the device can operate with a broadband pulsed laser. In [13], an  spatial walk-off polarizer was required to compensate for the OPL difference between two orthogonal additional crystal‐type spatial walk‐off polarizer was required to compensate for the OPL difference  additional crystal‐type spatial walk‐off polarizer was required to compensate for the OPL difference  polarized modes. By contrast, the cost of the proposed device can be substantially reduced by applying between  two  polarized  By  the  proposed  can  two  orthogonal  orthogonal  polarized  modes.  modes.  By  contrast,  contrast,  the  cost  cost  of  of  the  the  proposed  device  device  can  be  be  thebetween  lateral displacement beamsplitters. However, lateral displacement beamsplitters are commonly substantially  reduced  by  applying  the  lateral  displacement  beamsplitters.  However,  lateral  substantially  reduced  by  applying  the  lateral  displacement  beamsplitters.  However,  lateral  constructed out of a cube and a prism. These separate components could prism.  slightly complicate the displacement  displacement  beamsplitters  beamsplitters  are  are  commonly  commonly  constructed  constructed  out  out  of  of  a  a  cube  cube  and  and  a  a  prism.  These  These  separate  separate  device. Figure 7 illustrates that a proper shift should occur between the cube and the prism of the components could slightly complicate the device. Figure 7 illustrates that a proper shift should occur  components could slightly complicate the device. Figure 7 illustrates that a proper shift should occur  LDB to concentrate the polarized beams at four quadrants from a long distance L to a short distance between the cube and the prism of the LDB to concentrate the polarized beams at four quadrants  between the cube and the prism of the LDB to concentrate the polarized beams at four quadrants  l. When smaller components are used and glued together, a more compact and robust device can be from a long distance L to a short distance l. When smaller components are used and glued together,  from a long distance L to a short distance l. When smaller components are used and glued together,  achieved. Because of the application of bulk solid materials, the device has a high damage threshold of a more compact and robust device can be achieved. Because of the application of bulk solid materials,  a more compact and robust device can be achieved. Because of the application of bulk solid materials,  2 designed at an operation wavelength of at least 50 W/cm2 . With the same principle, the device can be the device has a high damage threshold of at least 50 W/cm the device has a high damage threshold of at least 50 W/cm2. With the same principle, the device can  . With the same principle, the device can  1064 nm. For high-power applications, zero-order half-wave plates should be used to eliminate the be designed at an operation wavelength of 1064 nm. For high‐power applications, zero‐order half‐ be designed at an operation wavelength of 1064 nm. For high‐power applications, zero‐order half‐ wave plates should be used to eliminate the influence of the thermal effect.  influence of the thermal effect. wave plates should be used to eliminate the influence of the thermal effect. 

   Figure 5. Spatial distributions of instantaneous electric vector field for (a) radially polarized mode;  Figure 5. Spatial distributions of instantaneous electric vector field for (a) radially polarized mode;  Figure 5. Spatial distributions of instantaneous electric vector field for (a) radially polarized mode; (b)  azimuthally  polarized  mode;  and  (c)  generalized  vector  (CV)  beams  as  a  (b)  azimuthally  polarized mode; mode; and and  (c) (c)  generalized generalized  cylindrical  cylindrical  a  linear  linear  (b) azimuthally polarized cylindricalvector  vector(CV)  (CV)beams  beamsas as a linear superposition of (a) and (b).  superposition of (a) and (b).  superposition of (a) and (b).

(a) (a)

(b) (b)

  

Figure  6.  A  prism  (RP)  is  to  one  into  (a)  two  Figure  6. reflection A  reflection  reflection  prism  (RP)  is  applied  applied  to  transform  transform  one  mode  mode  into  another:  another:  (a) output two  output  output  Figure 6. A prism (RP) is+x‐direction,  applied to transform one output  mode into another: (a) two radially radially  polarized  beams  in  the  and  (b)  two  azimuthally  polarized  radially  polarized  beams  in  the  +x‐direction,  and  (b)  two  output  azimuthally  polarized  beams  beams  +y‐ +y‐ polarized beams in the +x-direction, and (b) two output azimuthally polarized beams +y-direction. direction.  direction. 

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Figure 7. Lateral view of the alignment and assembly considerations of the design (not to scale). Figure 7. Lateral view of the alignment and assembly considerations of the design (not to scale). 

4. Conclusions 4. Conclusions  This paper proposed an improved design for a polarization converter that can generate radially This paper proposed an improved design for a polarization converter that can generate radially  and azimuthally polarized beams simultaneously. The device applies a pair of lateral displacement and azimuthally polarized beams simultaneously. The device applies a pair of lateral displacement  beamsplitters to replace expensive crystal-type spatial walk-off polarizers. Two half-wave plates beamsplitters to replace expensive crystal‐type spatial walk‐off polarizers. Two half‐wave plates are  are incorporated to manipulate the states of polarization of the incident light. The design enables incorporated  manipulate  the  states  of polarized polarization  of  the  incident  light.  approximatelyto  radially and azimuthally beams to be obtained in The  the design  far fieldenables  with approximately radially and azimuthally polarized beams to be obtained in the far field with coherent  coherent superposition. A prototype of the polarization converter was successfully demonstrated, superposition. A prototype of the polarization converter was successfully demonstrated, and several  and several of its advantages were discussed. The proposed device has considerable potential for of  its advantages  were  discussed.  The proposed  device  has  considerable  potential for  high‐power  high-power applications. applications.  Acknowledgments: This work was supported by the Ministry of Science and Technology of the Republic of China under the grant numbers andMinistry  MOST 105-2221-E-035-045. Acknowledgments:  This MOST work  104-2221-E-035-064 was  supported  by  the  of  Science  and  Technology  of  the  Republic  of  China under the grant numbers MOST 104‐2221‐E‐035‐064 and MOST 105‐2221‐E‐035‐045.  Author Contributions: Jing-Heng Chen and Chien-Yuan Han conceived and designed the experiments; Zhi-Hao Wei, Yung Hsu and Wei-Xuan Wu performed the experiments; Jing-Heng Chen and Chien-Yuan Han Author  Contributions:  Jing‐Heng  Chen  and  Chien‐Yuan  Han  conceived  and  designed the  experiments;  Zhi‐ analyzed the data; Jing-Heng Chen, Chien-Yuan Han, Kun-Huang Chen and Chien-Hung Yeh contributed the Hao  Wei, and Yung  Hsu  and  Wei‐Xuan  Wu  performed  Jing‐Heng  Chen  and  Chien‐Yuan  Han  materials; Jing-Heng Chen and Zhi-Hao Wei wrotethe  theexperiments;  paper. analyzed the data; Jing‐Heng Chen, Chien‐Yuan Han, Kun‐Huang Chen and Chien‐Hung Yeh contributed the  Conflicts of Interest: The authors declare no conflict of interest. materials; and Jing‐Heng Chen and Zhi‐Hao Wei wrote the paper. 

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