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The propagation and focusing properties of partially coherent vector beams including radially polarized and azi- muthally polarized (AP) beams are theoretically ...
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J. Opt. Soc. Am. A / Vol. 29, No. 11 / November 2012

Hu et al.

Investigation on partially coherent vector beams and their propagation and focusing properties Kelei Hu, Ziyang Chen, and Jixiong Pu* College of Information Science and Engineering, Huaqiao University, Xiamen, Fujian 361021, China *Corresponding author: [email protected] Received August 13, 2012; revised September 18, 2012; accepted September 19, 2012; posted September 19, 2012 (Doc. ID 174170); published October 11, 2012 The propagation and focusing properties of partially coherent vector beams including radially polarized and azimuthally polarized (AP) beams are theoretically and experimentally investigated. The beam profile of a partially coherent radially or AP beam can be shaped by adjusting the initial spatial coherence length. The dark hollow, flattopped, and Gaussian beam spots can be obtained, which will be useful in trapping particles. The experimental observations are consistent with the theoretical results. © 2012 Optical Society of America OCIS codes: 030.1670, 260.5430, 030.1640.

1. INTRODUCTION In recent years, vector beams have attracted significant interest due to the unique feature compared with homogeneously polarized beams [1–5]. Two extreme cases of vector beams are radially polarized (RP) and azimuthally polarized (AP) beams. An RP beam can be focused to generate a strong longitudinal and nonpropagating electric field at the focal plane, resulting in a sharper focal spot than a homogeneously polarized beam [1,2]. Moreover, an AP beam can be focused into a hollow dark spot [3]. These peculiar properties are useful for many applications such as optical data storage, particle trapping and acceleration, high-resolution microscopy, material processing, and determination of single fluorescent molecule orientation, etc. [6–9]. RP beams can be generated either inside a laser resonator, in which a conical mirror or a conical Brewster element is placed inside the resonator, or outside a laser cavity, e.g., by using a space-invariant dielectric subwavelength grating, a dual conical prism, or an interferometric technique [10–12]. The focusing properties and paraxial and nonparaxial propagation properties through paraxial optical system or free space have been widely studied [1–7, 12, 13]. To our knowledge, litter attention has been paid to the experimental study of the propagation and focusing properties of partially coherent radially or AP beams. In this study, based on the extended Huygens–Fresnel method by adopting a beam coherence-polarization (BCP) matrix, the propagation and focusing properties of a partially coherent radially or AP beam are studied. We use a liquid crystal polarization converter (Arcoptix, Switzerland) to convert the linearly polarized beam emerging from a 632.8 nm He– Ne laser into a radially or an AP beam. Analytical formulae are derived, and numerical examples and experimental results are illustrated.

2. THEORETICAL ANALYSIS Within the framework of the paraxial approximation, the vectorial electric field of a radially polarized doughnut beam (PDB) is expressed as the coherent superposition of a 1084-7529/12/112300-07$15.00/0

TEM01 with a polarization direction parallel to the x-axis and a TEM10 with a polarization direction parallel to the y-axis [2,6]  Er x; y; 0  E0

 2  2  x r y r exp − 2 ex  exp − 2 ey ; w0 w w w 0 (1)

where r 2  x2  y2 , w0 denotes the beam waist size of a Gaussian beam; E0 is a constant. In a similar way, the vectorial electric field of an azimuthally PDB is expressed as follows:   2  2  y r x r Eθ x; y;0  E0 − exp − 2 ex  exp − 2 ey : (2) w0 w w w 0 The longitudinal electric field component is neglected under the paraxial condition [12,14]. The BCP matrix provides the information of polarization and spatial correlation, and the elements of the BCP matrix for a vectorial electric field across a typical plane z  constant (here z is the propagation axis) are defined as follows [15–17]: W αβ r1 ; r2 ; z  hE α r1 ; zEβ r2 ; zi

α; β  x; y;

(3)

where E x and Ey are the components of the vectorial electric field in the x and y directions, respectively, and the angle brackets denote an ensemble average over the medium statistics. The equivalent irradiance distribution of a polarized beam is given by [15] Ir; z  W xx r; r; z  W yy r; r; z:

(4)

By applying Eqs. (1)–(4), the BCP matrices for a partially coherent RP beam and an AP partially coherent beam at source plane (z  0) are expressed as follows: W rαβ r10 ; r20 ; 0 

  2 E 20 r10  r220 gαβ r10 ; r20 α1 β2 ; (5) exp − w2 w2

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W θαβ r10 ; r20 ; 0 

  2 E20 r10  r220 gαβ r10 ; r20 α1 β2 : (6) exp − w2 w2

Gxx 

The complex degree of spatial coherence length gαβ r10 ; r20  is supposed to be [18]  x − x 2 y − y 2 gαβ r10 ; r20   exp − 10 2 20 − 10 2 20 ; σ αβ σ αβ 

Z Z Z Z

(7)

1 1 ikA ;   2B w2 σ 2xx

11a

1 1 ikA 1 1 −  − ; w2 σ 2xx 2B ax σ 4xx

(11b)

ax 

bx 

 2  2  1 π2 r1 ik G exp yy 4ay B λ2 B2 w2 ay by  2 2  S y1 r1 S 2y2 r22 S y1 S y2 × exp   r1 r2 ; 4by 4by 2by

W ryy r1 ; r2 ; z  E20

(12)

where ∞

W αβ r10 ; r20 ; 0 −∞  ik × exp − Ax210 − x220  y210 − y220  2B

S y1 

Gyy  where k  2π=λ is the wave number and λ is the wavelength of the light. By substituting Eq. (5) into Eq. (8), we can obtain the following expressions for the elements of the cross-spectral density matrix for a partially coherent RP beam propagating in free space:  2  2  1 π2 r1 ik W rxx r1 ; r2 ; z  G exp xx 2 2a b 4ax B λ B w x x   2 2 S x1 r1 S 2x2 r22 S x1 S x2   r1 r2 ; × exp 4bx 4bx 2bx

(9)

by 

where

S x2  −

ik ; B

(10b)

13a

(13b)

(13c)

with 1 1 ikA ;   2B w2 σ 2yy

14a

1 1 ikA 1 1 −  − ; w2 σ 2yy 2B ay σ 4yy

(14b)

ay 

(10a)

ik ; B

  1 1 1 2 1  S y  S y  y2 2 2ay by σ 2yy 2by y1 1   1 1 iky1  S y1 y1  S y2 y2 ; 2ay 2by B

E 20 2

1 1 ik ;  ax σ 2xx B

1 1 ik ; ay σ 2yy B

S y2  −

 Dx21 − x22  y21 − y22  − 2x1 x10 − x2 x20  y1 y10  − y2 y20  dx10 dy10 dx20 dy20 ; (8)

S x1

(10c)

with

where σ αβ is the mutual coherence length [when (α ≠ β)] or auto-coherent length [when (α  β)]. One finds from Eqs. (4)– (6) that a partially coherent RP beam and an AP partially coherent beam have the same irradiance distribution at z  0. Figure 1 shows the propagation geometry of a partially coherent radially or AP beam through an ABCD optical system in free space. The paraxial propagation of a laser beam through an ABCD optical system in free space can be treated with the well-known extended Huygens–Fresnel integral formula, and the elements of BCP matrix W αβ r; r; z at the output plane are given as follows [19–21]: 1 W αβ r1 ; r2 ; z  2 2 λ B

  1 1 1 2 1  S x  S x  x1 1 x2 2 2ax bx σ 2xx 2bx   1 1 ikx1 S x1 x1  S x2 x2 ;  2ax 2bx B

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 2  2  1 π2 r1 ik G exp xy 4axy B λ2 B2 w2 axy bxy  2 2  S xy1 rxy1 S 2xy2 r22 S xy1 S xy2 × exp   r1 r2 ; 4bxy 4bxy 2bxy

W rxy r1 ; r2 ;z  E 20

(15)

where S xy1 

1 1 ik ; axy σ 2xy B

S xy2  −

ik ; B

16a (16b)

 Gxy 

Fig. 1. (Color online) Propagation geometry of a partially coherent radially or AP beam through ABCD optical system in free space.

with

1 1 S x  S xy2 x2  2axy bxy σ 2xy xy1 1   1 ikx1 1  S y  S y2 y2 ; 2axy B 2bxy y1 1

(16c)

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axy 

bxy 

Hu et al.

1 1 ikA ;   2B w2 σ 2xy

17a

1 1 ikA 1 1 −  − ; w2 σ 2xy 2B axy σ 4xy

(17b)

I r r; z  W rxx r; zcos2 ϕ  W ryy r; zsin2 ϕ  W rxy r; z sin 2ϕ;

I θ r; z  W θxx r; zcos2 ϕ  W θyy r; zsin2 ϕ  W θxy r; z sin 2ϕ:

 2  2  1 π2 r1 ik G exp yx 4ayx B λ2 B2 w2 ayx byx  2 2  S yx1 ryx1 S 2yx2 r22 S yx1 S yx2 × exp   r1 r2 ; (18) 4byx 4byx 2byx

(22)

(23)

W ryx r1 ; r2 ; z  E20

One finds from Eqs. (9)–(23) that we can distinguish a partially coherent RP beam from a partially coherent AP beam by measuring their irradiances behind the linear polarizer with proper ϕ.

where

3. RESULTS AND DISCUSSION S yx1 

1 1 ik ; ayx σ 2yx B

S yx2  −

 Gyx  ×

ik ; B

19a

(19b)

  1 1 1 iky1 S y  S y   xy2 2 2ayx byx σ 2yx xy1 1 2ayx B 1 S y  S yx2 y2 ; 2byx yx1 1

(19c)

with 1 1 ikA ;   2B w2 σ 2yx

(20a)

1 1 ikA 1 1 −  − : w2 σ 2yx 2B ayx σ 4yx

(20b)

ayx 

byx 

By using Eqs. (9)–(20), we can investigate the propagation and focusing properties of the partially coherent RP beam propagating in free space. The intensity of a partially coherent RP beam propagating in free space can be written as Ir; z  W rxx r; r; z  W ryy r; r; z, where we suppose σ xx  σ yy  σ 0 in our theoretical calculation. In a similar way, we obtain the following expressions for the elements of a BCP matrix of an AP partially coherent beam:

In this section, the propagation and focusing properties of the radially and AP partially coherent beam in free space are investigated based on the formulae derived in the previous section. Part 1 of Fig. 2 shows the experimental setup for generating a radially and an AP Gaussian Schell-model (GSM) beam. An He– Ne laser beam is expended by the telescope system, which consists of two thin lenses L1 and L2 with focal length of f 1  15 cm and f 2  30 cm, respectively. A rotating ground-glass plate (RGGP) is inserted near the focus of L1 and L2 . The correlation length of the laser beam can be modulated by adjusting the position of the RGGP. The power loss caused by the RGGP is around 50%. P1 is a linear polarizer to generate a linear polarized beam. An aperture is located before the polarization converter to obtain a certain width beam. The linearly polarized beam emerging from a 632.8 nm He–Ne laser is converted into a radially or an AP beam by a liquid crystal polarization converter (Arcoptix, Switzerland). The intensity of the generated partially coherent radially or AP beam just behind the polarization converter can be measured by the beam profile analyzer (BPA) directly. Part 2 of Fig. 2 shows our experimental setup for measuring the intensity of the generated GSM beam. P2 is an analyzer that can be measured by the polarization of the generated partially coherent beams from the liquid crystal polarization converter. The experimental result shows that the beam width w0 in our experiment is 0.35 mm. The coherent length of the generated partially coherent radially or AP beam is measured by Young’s double-slit experiment. The generated GSM beams pass through a thin lens L3 with focal length f  30 cm, and then arrive at the BPA, which is used to measure the focused intensity distribution. The thin lens L3 will be

W θxx r; z  W ryy r; z; W θyy r; z  W rxx r; z; W θxy r; z  W θyx r; z  −W rxy r; z:

(21)

One finds from Eqs. (4)–(6) and (9)–(21) that a partially coherent RP beam and an AP partially coherent beam have the same irradiance distribution in free space. To distinguish a partially coherent RP beam and an AP partially coherent beam in free space, it is necessary to introduce some linear polarizer. Suppose that a linear polarizer is located at a position z, whose transmission axis forms an angle ϕ with the x-axis. Then the irradiances of a partially coherent RP beam and an AP partially coherent beam at z become

Fig. 2. (Color online) Experimental setup for generating a GSM RP beam or an AP beam and measuring its intensity. L1 , L2 , and L3 , thin lenses; RGGP, rotating ground-glass plate; P1 and P2 , linear polarizer; BPA, beam profile analyzer; PC-1 and PC-2, personal computer.

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Fig. 3. Generated radially partially coherent beam with σ 0  1.166 mm at z  20 cm. (a)–(e) are theoretical calculations; (f)–(j) are corresponding experimental results. The black arrows represent the transmission axis forms an angle ϕ with the x-axis, 0, π=4, π=2, and 3π=4, respectively. The dimensions in the first and second rows are 0.5 mm × 0.5 mm.

Fig. 4. Generated azimuthally partially coherent beam with σ 0  1.166 mm at z  20 cm. (a)–(e) are numerical calculations; (f)–(j) are corresponding experimental results. The black arrows represent the transmission axis of the polarizer forms an angle ϕ with the x-axis, ϕ  0, π=4, π=2, and 3π=4. The dimensions in the first and second rows are 0.5 mm × 0.5 mm.

removed when we study the propagation properties of the partially coherent radially and AP beam in free space. The transfer matrix between polarization converter and BPA reads as 

A B C D



 

1 f 0 1



1 0 −1=f 1



1 f 0 1



 

 1 f : (24) −1=f 1

Figure 3 shows theoretical calculation and the experimental results of a partially coherent RP beam with σ 0  1.166 mm at z  20 cm. Figures 3(b)–3(e) show the intensity distributions of the partially coherent RP beam, after being transmitted through an analyzer whose orientation is indicated by the arrows. Figures 3(g)–3(j) are corresponding experimental results. The fanlike extinction pattern appears in

Fig. 5. Propagation properties of partially coherent RP beam in free space with σ 0  1.166 mm at several propagation distances with z  20, 40, 60, 80, 100, and 120 cm. (a)–(f) are numerical calculations; (g)–(l) are corresponding experimental results. The dimensions of (a)–(g) are 0.5 mm × 0.5 mm, and (h)–(l) are 0.75 mm × 0.75 mm.

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Fig. 6. Propagation properties of partially coherent RP beam in free space with σ 0  0.441 mm at several propagation distances with z  20, 40, 60, 80, and 100 cm. (a)–(f) are numerical calculations; (g)–(l) are corresponding experimental results. The dimensions of (a)–(g) are 0.5 mm × 0.5 mm, and (h)–(l) are 0.75 mm × 0.75 mm.

Fig. 7. Intensity distributions for partially coherent RP beam at the same propagation distances z  140 cm with different coherent length in free space. (a)–(d) are numerical calculations, and the coherent lengths of the first rows are 1.166, 0.903, 0.607, and 0.491 mm, respectively; (e)–(h) are corresponding experimental results. The dimensions in the first and second rows are 0.5 mm × 0.5 mm and 0.75 mm × 0.75 mm, respectively.

intensity distribution, owing to the cylindrical-symmetry polarization distribution in the beam cross section. The black arrows represent the transmission axis forms an angle ϕ with the x-axis, 0, π=4, π=2, and 3π=4, respectively.

Furthermore, the theoretical calculation and the experimental results of an azimuthally partially coherent beam with σ 0  1.166 mm at z  20 cm are presented in Fig. 4. In the same way, when an analyzer is used, the fanlike extinction pattern

Fig. 8. Polarization distributions of the FT beam. The black arrows represent the transmission axis of the polarizer forms an angle ϕ with the x-axis, ϕ  0, π=4, π=2, and 3π=4, respectively. (a)–(e) are numerical calculations, and (f)–(i) are corresponding experimental results, respectively. The dimensions in the first and second rows are 0.5 mm × 0.5 mm and 0.75 mm × 0.75 mm.

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Fig. 9. Propagation properties of radially and AP partially coherent beam with the same coherent length at several propagation distances. (a)–(d) are numerical calculations, and (e)–(h) and (i)–(l) are corresponding experimental results of the radially and AP partially coherent beam. The dimensions in the first, second, and third rows are 0.5 mm × 0.5 mm, 0.5 mm × 0.5 mm, and 0.75 mm × 0.75 mm, respectively.

appears in intensity distribution, also owing to the cylindricalsymmetry polarization distribution in the beam cross section. The black arrows represent the transmission axis forms an angle ϕ with the x-axis, 0, π=4, π=2, and 3π=4, respectively. It should be noted that the fanlike extinction pattern that appears in the intensity distribution is just vertical to that of Fig. 3. This is because the states of polarization of radially and azimuthally partially coherent beams are orthogonal to each other. One finds from Figs. 3 and 4 that the experimental results agree well with the theoretical calculation. The propagation properties of partially coherent RP beams in free space are presented in Fig. 5. For a long propagation distance, the partially coherent RP beam propagating in free space can keep its initial beam profile almost invariant. In this situation, we can obtain the beam profile with a dark hollow beam. The experimental results are compared with the theoretical counterparts. It should be noted that the size of experimental results is larger than that of theoretical calculation. The reason that there is an aperture located before the polarization converter, making the generated partially coherent RP beam diffraction faster. Figure 6 shows the intensity distribution of theoretical calculation and experimental results for σ 0  0.441 mm at several propagation distances (z  20, 40, 60, 80, and 100 cm, respectively). For a short propagation distance, one also finds that the partially coherent RP beam propagating in free space can keep its initial beam profile almost invariant. As propagation distance increases further, the initial beam profile gradually disappears, and the on-axis intensity increase during propagation within certain propagation distance and the beam pattern with flat-topped (FT) profile can be formed at certain propagation distance (z  100 cm). The experimental results are compared with the theoretical counterparts. Figure 7 gives the intensity distributions for partially coherent RP beam at the same propagation distances z  140 cm

with different coherent length in free space. The coherent length of the first rows is 1.166, 0.906, 0.607, and 0.491 mm, respectively. As coherent length decreases further, the initial beam profile gradually disappears and the on-axis intensity increase during propagation within certain propagation distance. When σ 0  0.491 mm, the FT beam can be obtained. Next, the state of polarization when the intensity profile of the partially coherent RP beam becomes FT beam is studied. As shown in Fig. 8, when σ 0  0.41 mm at z  80 cm, we can obtain the beam pattern with FT profile. When an analyzer is used, one finds that the obtained intensity pattern varies from a constant intensity with almost zero contrast along the azimuthal direction. It can be seen that the intensity along the direction of the polarizer is not null; thus polarization is no longer RP completely. The experimental results agree well with the theoretical results. The propagation properties of radially and AP partially coherent beams with the same coherent length at several propagation distances are investigated as well. In the theoretical analysis section, the propagation properties of radially and AP partially coherent beams with the same intensity distribution can be found as well. As shown in Fig. 9, in the first rows are theoretical calculations for z  50, 70, 90, and 120 cm, respectively. In the second and third rows are experimental results of the radially and AP partially coherent beams, respectively. It can be seen that the radially and AP partially coherent beams propagate in free space with the same intensity distribution. The experimental results are compared with the theoretical ones. The focusing properties of partially coherent RP beams with different coherent length (1.166, 0.494, and 0.441 mm, respectively) are investigated. Figure 10 shows the intensity distribution of the focusing with different coherent length. The phenomenon is similar to the propagation in free space. We can shape the beam profile by varying their initial spatial

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Fig. 10. Focusing properties of partially coherent RP beam with different coherent lengths. (a)–(c) are numerical calculation, and (d)–(f) are corresponding experimental results. The dimensions in the first, second, and third rows are 0.25 mm × 0.25 mm, 0.25 mm × 0.25 mm, and 0.375 mm × 0.375 mm, respectively.

coherence length. The dark hollow, FT, and Gaussian beam spots in our experiment can be obtained, which will be useful for trapping particles.

4. CONCLUSIONS In conclusion, based on the extended Huygens–Fresnel method by adopting a BCP matrix, the propagation and focusing properties of partially coherent vector beams including RP and AP beams were investigated experimentally and theoretically. It is found that we can shape the beam profile of the partially coherent radially or AP beams by varying their initial spatial coherence length. The dark hollow, FT, and Gaussian beam spots can be obtained in our experiment, which will be useful in trapping particles. The experimental results are consistent with the theoretical results.

ACKNOWLEDGMENTS This research is supported by National Natural Science Foundation of China (Grant Nos. 60977068 and 61178015).

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