Coherence and polarization properties of a radially polarized beam with variable spatial coherence

In a recent publication [Appl. Phys. Lett, 100, 051108 (2012)], a radially polarized (RP) beam with variable spatial coherence (i.e., partially coherent RP beam) was generated experimentally. In this paper, we derive the realizability conditions for a partially coherent RP beam, and we carry out theoretical and experimental study of the coherence and polarization properties of a partially coherent RP beam. It is found that after passing through a thin lens, both the degree of coherence and the degree of polarization of a partially coherent RP beam varies on propagation, while the state of polarization of the completely polarized part of such beam remains invariant. The variations of the degree of coherence and the degree of polarization depend closely on the initial spatial coherence. Our experimental results agree well with the theoretical predictions. ©2012 Optical Society of America OCIS codes: (030.0030) Coherence and statistical optics; (260.5430) Polarization; (350.5500) Propagation. References and links 1. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1(1), 1–57 (2009). 2. Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002). 3. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000). 4. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). 5. D. P. Biss and T. G. Brown, “Cylindrical vector beam focusing through a dielectric interface,” Opt. Express 9(10), 490–497 (2001). 6. P. Wróbel, J. Pniewski, T. J. Antosiewicz, and T. Szoplik, “Focusing radially polarized light by a concentrically corrugated silver film without a hole,” Phys. Rev. Lett. 102(18), 183902 (2009). 7. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001). 8. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004). 9. D. P. Biss, K. S. Youngworth, and T. G. Brown, “Dark-field imaging with cylindrical-vector beams,” Appl. Opt. 45(3), 470–479 (2006). 10. H. Wang, L. Shi, B. Lukyanchuk, C. J. R. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008). 11. W. Chen, D. C. Abeysinghe, R. L. Nelson, and Q. Zhan, “Plasmonic lens made of multiple concentric metallic rings under radially polarized illumination,” Nano Lett. 9(12), 4320–4325 (2009). 12. M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. B 86, 329–334 (2007). 13. Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010). 14. K. P. Singh and M. Kumar, “Electron acceleration by a radially polarized laser pulse during ionization of low density gases,” Phys. Rev. ST Accel. Beams 14(3), 030401 (2011). 15. J. Li, Y. Salamin, B. J. Galow, and C. Keitel, “Acceleration of proton bunches by petawatt chirped radially polarized laser pulses,” Phys. Rev. A 85(6), 063832 (2012). 16. A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre–Gaussian laser beams,” J. Opt. Soc. Am. A 15(10), 2705–2711 (1998). 17. D. Deng, “Nonparaxial propagation of radially polarized light beams,” J. Opt. Soc. Am. B 23(6), 1228–1234 (2006). #176399 $15.00 USD Received 18 Sep 2012; revised 29 Oct 2012; accepted 27 Nov 2012; published 6 Dec 2012 (C) 2012 OSA 17 December 2012 / Vol. 20, No. 27 / OPTICS EXPRESS 28301 18. D. Deng and Q. Guo, “Analytical vectorial structure of radially polarized light beams,” Opt. Lett. 32(18), 2711– 2713 (2007). 19. Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008). 20. W. Cheng, J. W. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express 17(20), 17829–17836 (2009). 21. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995). 22. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007). 23. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(56), 263–267 (2003). 24. F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998). 25. F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schellmodel beams,” J. Opt. A, Pure Appl. Opt. 3(1), 1–9 (2001). 26. O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. 29(11), 1173–1175 (2004). 27. O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30(2), 198–200 (2005). 28. O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1-3), 35–43 (2005). 29. T. Shirai and E. Wolf, “Correlation between intensity fluctuations in stochastic electromagnetic beams of any state of coherence and polarization,” Opt. Commun. 272(2), 289–292 (2007). 30. J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003). 31. J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248(4-6), 333–337 (2005). 32. O. Korotkova and E. Wolf, “Spectral degree of coherence of a random three-dimensional electromagnetic field,” J. Opt. Soc. Am. A 21(12), 2382–2385 (2004). 33. F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schellmodel sources,” J. Opt. Soc. Am. A 25(5), 1016–1021 (2008). 34. L. Zhang, F. Wang, Y. Cai, and O. Korotkova, “Degree of paraxiality of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Commun. 284(5), 1111–1117 (2011). 35. T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7(5), 232–237 (2005). 36. M. Santarsiero, R. Borghi, and V. Ramirez-Sanchez, “Synthesis of electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 26(6), 1437–1443 (2009). 37. B. Kanseri, S. Rath, and H. C. Kandpal, “Determination of the beam coherence-polarization matrix of a random electromagnetic beam,” IEEE J. Quantum Electron. 45(9), 1163–1167 (2009). 38. F. Wang, G. Wu, X. Liu, S. Zhu, and Y. Cai, “Experimental measurement of the beam parameters of an electromagnetic Gaussian Schell-model source,” Opt. Lett. 36(14), 2722–2724 (2011). 39. M. Salem and G. P. Agrawal, “Coupling of stochastic electromagnetic beams into optical fibers,” Opt. Lett. 34(18), 2829–2831 (2009). 40. C. Zhao, Y. Dong, G. Wu, F. Wang, Y. Cai, and O. Korotkova, “Experimental demonstration of coupling of an electromagnetic Gaussian Schell-model beam into a single-mode optical fiber,” Appl. Phys. B, doi:10.1007/s00340-012-5176-5. 41. O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281(9), 2342–2348 (2008). 42. M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “Evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Lett. 33(19), 2266–2268 (2008). 43. Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16(20), 15834–15846 (2008). 44. C. Zhao, Y. Cai, and O. Korotkova, “Radiation force of scalar and electromagnetic twisted Gaussian Schellmodel beams,” Opt. Express 17(24), 21472–21487 (2009). 45. Z. Tong, Y. Cai, and O. Korotkova, “Ghost imaging with electromagnetic stochastic beams,” Opt. Commun. 283(20), 3838–3845 (2010). 46. Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82(3), 033836 (2010). 47. M. Yao, Y. Cai, O. Korotkova, Q. Lin, and Z. Wang, “Spatio-temporal coupling of random electromagnetic pulses interacting with reflecting gratings,” Opt. Express 18(21), 22503–22514 (2010). 48. G. Wu and Y. Cai, “Modulation of spectral intensity, polarization and coherence of a stochastic electromagnetic beam,” Opt. Express 19(9), 8700–8714 (2011). 49. M. Salem and E. Wolf, “Coherence-induced polarization changes in light beams,” Opt. Lett. 33(11), 1180–1182 (2008). 50. I. Vidal, E. J. S. Fonseca, and J. M. Hickmann, “Light polarization control during free-space propagation using coherence,” Phys. Rev. A 84(3), 033836 (2011). 51. S. Sahin, Z. Tong, and O. Korotkova, “Sensing of semi-rough targets embedded in atmospheric turbulence by means of stochastic electromagnetic beams,” Opt. Commun. 283(22), 4512–4518 (2010). 52. Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express 19(7), 5979–5992 (2011). #176399 $15.00 USD Received 18 Sep 2012; revised 29 Oct 2012; accepted 27 Nov 2012; published 6 Dec 2012 (C) 2012 OSA 17 December 2012 / Vol. 20, No. 27 / OPTICS EXPRESS 28302 53. Y. Dong, F. Feng, Y. Chen, C. Zhao, and Y. Cai, “Statistical properties of a nonparaxial cylindrical vector partially coherent field in free space,” Opt. Express 20(14), 15908–15927 (2012). 54. Y. Luo and B. Lu, “Spectral stokes singularities of partially coherent radially polarized beams focused by a high numerical aperture objective,” J. Opt. 12(11), 115703 (2010). 55. H. Wang, D. Liu, and Z. Zhou, “The propaga

Coherence is one of the important characteristics of light beams [21].Recently, partially coherent vector beam attracts more and more attentions [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40].In 2003, Wolf proposed a unified theory of coherence and polarization for partially coherent vector beam [23], which can be applied to study the changes of the statistical properties such as degree of coherence, degree of polarization and sate of polarization of such beam upon propagation.Partially coherent vector beam with spatially uniform state of polarization usually is called stochastic electromagnetic beam in spatial frequency domain or partially coherent and partially polarized beam in space-time domain [22][23][24].Characterization, generation and propagation of stochastic electromagnetic beam have been studied extensively due to its important applications in free-space optical communications, optical imaging, active laser radar systems and remote sensing .
Partially coherent vector beam with spatially non-uniform state of polarization called cylindrical vector partially coherent beam was introduced recently [52,53].Partially coherent RP beam can be regarded as a special case of cylindrical vector partially coherent beam [54][55][56][57].More recently, partially coherent RP beam with variable spatial coherence was generated in experiment [57], and it was found that we can shape the beam profile of the focused RP beam by varying its initial spatial coherence, which is useful for material thermal processing and particle trapping.
For a stochastic electromagnetic beam, it has been shown that its spectral degree of coherence, spectral degree of polarization and state of polarization change on propagation .Coherence-induced polarization changes of a stochastic electromagnetic beam were demonstrated both theoretically [49] and experimentally [50].More importantly, it was found that the stochastic electromagnetic beam could be used for sensing semi-rough target by comparing coherence and polarization properties of the source beam and of the return beam in turbulent atmosphere [51].In this paper, our aim is to explore the coherence and polarization properties of a partially coherent RP beam both theoretically and experimentally.Our results show that the degree of coherence and the degree of polarization of a focused partially coherent RP beam varies on propagation, while the state of polarization of the completely polarized part of such beam remains invariant.

Theory
We first outline briefly the theoretical model for a partially coherent RP beam, then we study its coherence and polarization properties.The vector electric field of a coherent RP beam can be expressed as the superposition of orthogonally polarized Hermite Gaussian HG 10 and HG 01 modes [1] ( ) ( ) ( ) where 0 ω denotes the beam waist size of a fundamental Gaussian mode.x e  and y e  represent the unit vectors in the x and y directions, respectively.Based on the unified theory of coherence and polarization, the second-order correlation properties of a partially coherent vector beam at z = 0, in space-frequency domain, can be characterized by the 2 2  × cross-spectral density (CSD) matrix of the electric field, defined by the formula [22,23] , , , ,0 , , , ,0 , , , ,0 , , , , ,0 , , , ,0 where ( ) , ) Here 1 1 ( , ) x y and 2 2 ( , ) x y denote the coordinates of two arbitrary points at the source plane, E x and E y denote the components of the random electric vector, along two mutually orthogonal x and y directions perpendicular to the z-axis.The angular brackets denote ensemble average and the asterisk denotes the complex conjugate.
The element of the CSD matrix of a partially coherent RP beam radiated from a Schellmodel source [21,22] can be expressed as [52,53] ( ) ( ) , , , ,0 , , , ,0 , where ( ) is the correlation coefficient between the E x and E y field components, Within the validity of the paraxial approximation, propagation of the elements of the CSD matrix of a partially coherent vector beam through ABCD optical system can be studied with the help of the following generalized Collins formula [58] 2 2 , dx dy dy (8) where 2 / k π λ = is the wave number with λ being the wavelength, A, B, C and D are the transfer matrix elements of optical system and the asterisk is required for a general optical system with loss or gain, although it does not appear in Eq. ( 13) of [58].Equation ( 8) is also valid for the propagation of the elements of the CSD matrix of an electromagnetic Gaussian Schell-model (GSM) beam.With the help of Eq. ( 8), the propagation properties of an electromagnetic GSM beam through paraxial ABCD optical system, resonator and turbulent atmosphere have been studied in [42,43,47,48].In fact, Eq. ( 8) also can be used to treat the propagation properties of an electromagnetic GSM beam in free space by setting 1, , 0, 8) [28].It has been found that the degree of polarization of the electromagnetic GSM beam varies on propagation even in free space [28,42,43,47,48].
The degree of polarization of the partially coherent vector beam at point ( ) where Det stands for the determinant of the matrix.
The state of polarization of the completely polarized beam can be characterized by the polarization ellipse.The major and minor semi-axes of the polarization ellipse, 1 A and 2 A , as well as its degree of ellipticity, ε, and its orientation angle, θ, are related to the elements of the CSD matrix W  by the following formulas [28] 2 2 1,2 ( , , , , ) ( , , ) , ( , , , , ) , , , )  ( , , , , )  λ ω = = From Fig. 1, one finds that the degree of polarization of a coherent RP beam remains invariant on propagation, while the degree of polarization of a partially coherent RP beam decreases with the propagation distance increases, which means the completely polarized partially coherent RP beam is depolarized on propagation as expected [52].Furthermore, the variation of the degree of polarization on propagation is closely determined by the correlation radii, and the degree of polarization decreases more rapidly as the correlation radii decreases (i.e., the coherence of the beam decreases).We may physically explain this phenomenon by the fact that the degree of polarization denotes the ratio of the intensity of the polarized part of the beam to the total intensity of the beam for a fixed point, while the correlation radii represent the strength of the correlations of the statistical field.Decreasing the values of correlation radii (i.e., decreasing the coherence of the beam) will enhance the relative intensity of the unpolarized part for the fixed point on propagation.Now we analyze the state of polarization of a focused partially coherent RP beam.Substituting Eq. ( 33) into Eqs.( 9)-( 12) and ( 25)-( 27), we obtain ( ) 2 ( , , , , ) 0, ( , , , , ) 0, ( , , , , ) arctan / From Eq. ( 34), we can come to the conclusion that the completely polarized part of the partially coherent RP beam remains radially polarized on propagation although the completely unpolarized part appears, in other words, the state of polarization of the completely polarized part of the partially coherent RP beam remains invariant on propagation.We calculate in Fig. 2 and Fig. 3 the intensity distributions of a partially coherent RP beam, its completely polarized part and its completely unpolarized part for 1mm 3 that the dark hollow beam profile of a focused partially coherent RP beam disappears on propagation as expected [57], and the dark hollow beam profile disappears more rapidly as the correlation radii decreases.The completely polarized part keeps its dark hollow beam profile invariant on propagation, while the completely unpolarized part gradually appears on propagation and has a Gaussian beam profile.Furthermore, the contribution of the completely unpolarized part to the total intensity becomes larger on propagation, thus leading to the change of the degree of polarization of the partially coherent RP beam.To learn more about the power transition from the completely polarized part to the completely unpolarized part, we now study variation of the normalized power of the completely polarized part or unpolarized part, which is defined as , , , , , , , the partially coherent RP beam at the focal plane can be regarded as a completely unpolarized beam.Furthermore, we also find an interesting phenomenon that when ( ) ( ) , the total intensity distribution of the focused partially coherent RP beam has a flat-topped beam profile (see Fig. 5).

Experimental results
In this section, we carry out experimental study of the coherence and polarization properties of a focused partially coherent RP beam. Figure 8 shows the experimental setup for generating and measuring the coherence properties of a partially coherent RP beam.Similar to Ref [57], a linearly polarized He-Ne laser beam focused by the thin lens L1 is reflected by a reflecting mirror and then illuminates a rotating ground-glass plate (RGGP), producing a partially coherent beam with Gaussian statistics [59,60].After passing through a collimation thin lens L2 and a Gaussian amplitude filter (GAF), the transmitted beam becomes a linearly polarized Gaussian Schell-model (GSM) beam characterized by the cross-spectral density The transmission function of the GAF determines the value of 0 ω , and 0 ω is equal to 1.05mm in our experiment.The transverse coherence width δ is determined by the focused beam spot size on the RGGP and the roughness of the RGGP together.In our experiment, the roughness of the RGGP is fixed and we mainly modulate the value of δ by varying the focused beam spot on the RGGP (i.e., the distance between L 1 and RGGP).We adopt the method proposed in [59] to measure the coherence width δ of the GSM beam.In our experiment, we choose two different values of δ ( 0.20mm,0.55mmδ = ) to generate a RP beam with variable spatial coherence.Figure 9 shows the experimental results (dotted curves) of the modulus of the square of the degree of coherence of the generated GSM beam and the corresponding Gaussian fit (solid curves) for two different focused beam spot sizes on the RGGP.One finds from Fig. 9 that the degree of coherence of the beam produced from the RGGP indeed satisfy Gaussian distribution as expected.In order to obtain the information of the degree of polarization of a focused partially coherent RP beam, we need to measure the elements of its cross-spectral density matrix.Here we use the method proposed in [61] to measure the degree of polarization of a focused partially coherent RP beam, and this method has been adopted successfully to measure the degree of polarization of an electromagnetic GSM beam [62].To measure the elements xx W and yy W , we put a linear polarizer whose transmission axis forms an angle θ with the xaxis just before the thin lens L 3 (see Fig. 11), and the charge-coupled device (CCD) is used to measure the average intensity at the focal plane.The average intensity distribution of the partially coherent RP beam at the focal plane is written as where / 2 φ π = represents the phase difference between the x-component and y component of the vector field induced by the quarter-wave plate.In Eq. (39), 0 φ = means the quarter-wave plate is removed.From Eq. ( 39), we can obtain the following expressions for the elements   Figure 12 shows our experimental results of the degree of polarization of a focused partially coherent RP beam at the focal plane versus the transverse coordinate u (v = 0) for two different values of the coherence width δ .For the convenience of comparison, the corresponding results calculated by the theoretical formulas are also shown in Fig. 12.One finds from Fig. 12 that the focused partially coherent RP beam at the focal plane indeed is depolarized, and the depolarization is more serious as δ decreases.Furthermore, with the increase of the transverse coordinate u, the degree of polarization increases.Our experimental results are also consistent with the theoretical predictions.Note that with the increase of the transverse coordinate, the degree of polarization decrease at about u = 0.2mm, which may be due to the unexpected fluctuation from the beam source.

Summary
We have studied the coherence and polarization properties of a new class of partially coherent vector beam with spatially non-uniform state of polarization named partially coherent radially polarized beam both theoretically and experimentally.Our results show that the degree of coherence of focused partially coherent RP beam becomes of non-Gaussian distribution, and the focused partially coherent RP beam is depolarized, while the state of polarization of its completely polarized part remains invariant, which is much different from that of an electromagnetic GSM beam.We have found that the changes of the degree of coherence and the degree of polarization of a partially coherent RP beam on propagation are controlled by

Fig. 1 .Figure 1
Fig. 1.Degree of polarization of a focused partially coherent RP beam at point ( ) 0.05mm 0.05mm versus the propagation distance z for different values of the correlation radii , , .xx yy xy δ δ δ Figure 1 shows the degree of polarization of a focused partially coherent RP beam at point ( ) 0.05mm 0.05mm versus the propagation distance z for different values of the correlation radii , , xx y y x y δ δ δ with

Fig. 4 .Fig. 5 .
Fig. 4. Variation of the normalized powers of the completely polarized part and the completely unpolarized part of a focused partially coherent RP beam (a) versus the propagation distance with 0.20mm xx yy xy δ δ δ = = = , (b) versus the correlation coefficient xx δ with xx yy xy δ δ δ = = at are given by Eq.(28).Figure 4(a) shows the variation of the normalized powers of the completely polarized part and the completely unpolarized part of a focused partially coherent RP beam versus the propagation distance with 0b) shows the variation of the normalized powers at the focal plane versus the correlation coefficient xx δ with finds from Fig. 4(a) that the normalized power ( ) p η of the completely polarized part decreases on propagation, #176399 -$15.00USD Received 18 Sep 2012; revised 29 Oct 2012; accepted 27 Nov 2012; published 6 Dec 2012 (C) 2012 OSA while the normalized power ( ) u η the completely unpolarized part increases.At the focal plane, ( ) u η approaches to 0.99, which means the partially coherent RP beam almost becomes completely unpolarized at the focal plane.One finds from Fig. 4(b) that ( ) p η and ( ) u η at the focal plane are closely determined by the correlation radii.With the decrease of the correlation radii, ( )

Fig. 6 .
Fig. 6.Square of the degree of coherence of a focused partially coherent RP beam at several propagation distances with 0.20mm xx yy xy δ δ δ = = = Fig.6that the degree of coherence of the partially coherent RP beam at z = 0 has a Gaussian profile, while it varies on propagation and it gradually becomes of non-Gaussian profile.The distribution of the degree of coherence is also closely related with the correlation radii , , , xx y y x y δ δ δ and the side robes become more significant as the correlation radii increases.

Fig. 7 .
Fig. 7. Square of the degree of coherence of a focused partially coherent RP beam at the focal plane z = 400mm for different values of the correlation radii , , .xx yy xy δ δ δ

δ
being the beam waist size and the transverse coherence width.The radial polarization converter (RPC) located just behind the GAF converts the generated GSM beam into a partially coherent RP beam.The RPC just alter the polarization state of the GSM beam, while it does not alter its spatial coherence and beam spot size, thus the beam waist size of the partially coherent RP beam is approximately equal to that of the GSM beam, and the correlation radii of the partially coherent RP beam are approximated as xx yy xy

Fig. 9 .
Fig. 9. Experimental results (dotted curves) of the modulus of the square of the degree of coherence of the generated GSM beam and the corresponding Gaussian fit (solid curves) for two different focused beamspot sizes on the RGGP.

Fig. 10 .
Fig. 10.Experimental results (dotted curves) of the modulus of the square of the degree of coherence of a focused partially coherent RP beam at the focal plane for two different values of the coherence width δ .The solid curves are calculated by theoretical formulas.After passing through a thin lens L 3 located just behind the RPC, the focused partially coherent RP beam is split into two beams by a beam splitter.The transmitted and reflected beams arrive at D 1 and D 2 , which scan the transverse plane u 1 and u 2 , respectively.Both the distances from the L 3 to D 1 and from L 3 to D 2 are f.The electronic coincidence circuit is used to measure the fourth-order correlation function (i.e., intensity correlation function) between two detectors.By measuring the fourth-order correlation function, we can obtain the information of the degree of coherence of the focused partially coherent RP beam at the focal

Fig. 11 .
Fig. 11.Experimental setup for measuring the degree of polarization of a focused partially coherent RP beam at the focal plane.LP, linear polarizer; QWP, quarter-wave plate; CCD, charge-coupled device.

2 I
π π , 3 /4, /2 I π π at the focal plane, we can obtain the information of xy W and yx W .

Fig. 12 .
Fig. 12. Experimental results of the degree of polarization of a focused partially coherent RP beam at the focal plane versus the transverse coordinate u (v = 0) for two different values of the coherence width δ .The solid curves are calculated by theoretical formulas.