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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 27 — Dec. 17, 2012
  • pp: 28301–28318
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Coherence and polarization properties of a radially polarized beam with variable spatial coherence

Gaofeng Wu, Fei Wang, and Yangjian Cai  »View Author Affiliations


Optics Express, Vol. 20, Issue 27, pp. 28301-28318 (2012)
http://dx.doi.org/10.1364/OE.20.028301


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Abstract

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 OSA

1. Introduction

Coherence is one of the important characteristics of light beams [21

21. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

]. Recently, partially coherent vector beam attracts more and more attentions [22

22. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).

40

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:. [CrossRef]

]. In 2003, Wolf proposed a unified theory of coherence and polarization for partially coherent vector beam [23

23. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003). [CrossRef]

], 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

22. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).

24

24. F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998). [CrossRef] [PubMed]

]. 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 [25

25. F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 3(1), 1–9 (2001). [CrossRef]

51

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). [CrossRef]

].

Partially coherent vector beam with spatially non-uniform state of polarization called cylindrical vector partially coherent beam was introduced recently [52

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). [CrossRef] [PubMed]

, 53

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). [CrossRef] [PubMed]

]. Partially coherent RP beam can be regarded as a special case of cylindrical vector partially coherent beam [54

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). [CrossRef]

57

57. F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012). [CrossRef]

]. More recently, partially coherent RP beam with variable spatial coherence was generated in experiment [57

57. F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012). [CrossRef]

], 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 [22

22. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).

48

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). [CrossRef] [PubMed]

]. Coherence-induced polarization changes of a stochastic electromagnetic beam were demonstrated both theoretically [49

49. M. Salem and E. Wolf, “Coherence-induced polarization changes in light beams,” Opt. Lett. 33(11), 1180–1182 (2008). [CrossRef] [PubMed]

] and experimentally [50

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). [CrossRef]

]. 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

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). [CrossRef]

]. 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.

2. 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 HG10 and HG01 modes [1

1. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1(1), 1–57 (2009). [CrossRef]

]
E(x,y)=Ex(x,y)ex+Ey(x,y)ey=xω0exp(x2+y2ω02)ex+yω0exp(x2+y2ω02)ey,
(1)
whereω0denotes the beam waist size of a fundamental Gaussian mode. exandeyrepresent 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

22. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).

, 23

23. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003). [CrossRef]

]
W(x1,y1,x2,y2,0)=(Wxx(x1,y1,x2,y2,0)Wxy(x1,y1,x2,y2,0)Wyx(x1,y1,x2,y2,0)Wyy(x1,y1,x2,y2,0)),
(2)
where
Wαβ(x1,y1,x2,y2,0)=Eα(x1,y1,0)Eβ*(x2,y2,0).(α=x,y;β=x,y)
(3)
Here (x1,y1) and (x2,y2) denote the coordinates of two arbitrary points at the source plane, Ex and Ey 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 Schell-model source [21

21. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

,22

22. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).

] can be expressed as [52

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). [CrossRef] [PubMed]

,53

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). [CrossRef] [PubMed]

]
Wxx(x1,y1,x2,y2,0)=x1x2ω02exp[x12+y12+x22+y22ω02(x1-x2)2+(y1-y2)22δxx2],
(4)
Wxy(x1,y1,x2,y2,0)=Bxyx1y2ω02exp[x12+y12+x22+y22ω02(x1-x2)2+(y1-y2)22δxy2],
(5)
Wyx(x1,y1,x2,y2,0)=Wxy*(x2,y2,x1,y1,0),
(6)
Wyy(x1,y1,x2,y2,0)=y1y2ω02exp[x12+y12+x22+y22ω02(x1-x2)2+(y1-y2)22δyy2],
(7)
whereBxy=|Bxy|exp(iϕ)is the correlation coefficient between the Ex and Ey field components, ϕ is the phase difference between the x-and y-components of the field, δxx, δyy and δxy are the widths of auto-correlation functions of the x component of the field, of the y component of the field and of the mutual correlation function of x and y field components, respectively. The parameters Bxy, δxx, δyy and δxysatisfy the realizability conditions Bxy=1, δxx=δyy=δxy(see Appendix A).

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

58. Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002). [CrossRef] [PubMed]

]
Wαβ(u1,v1,u2,v2,z)=(1λ|B|)2Wαβ(x1,y1,x2,y2,0)×exp[ik2B(Ax122x1u1+Du12)ik2B(Ay122y1v1+Dv12)]×exp[ik2B*(A*x222x2u2+D*u22)+ik2B*(A*y222y2v2+D*y22)]dx1dx2dy1dy2,
(8)
where k=2π/λ is the wave number withλbeing 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

58. Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002). [CrossRef] [PubMed]

]. 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

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). [CrossRef] [PubMed]

, 43

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). [CrossRef] [PubMed]

, 47

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). [CrossRef] [PubMed]

, 48

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). [CrossRef] [PubMed]

]. In fact, Eq. (8) also can be used to treat the propagation properties of an electromagnetic GSM beam in free space by setting A=1,B=z,C=0,D=1in Eq. (8) [28

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). [CrossRef]

]. It has been found that the degree of polarization of the electromagnetic GSM beam varies on propagation even in free space [28

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). [CrossRef]

, 42

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). [CrossRef] [PubMed]

, 43

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). [CrossRef] [PubMed]

, 47

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). [CrossRef] [PubMed]

, 48

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). [CrossRef] [PubMed]

].

Substituting from Eqs. (4)-(7) into Eq. (8), after some tedious integration and operations, we obtain the following expressions for the elements of the CSD matrix of a partially coherent RP beam after propagating through ABCD optical system
Wxx(u1,v1,u2,v2,z)==π28δxx2M1xx2M2xx3ω02(1λ|B|)2exp[ikD2B(u12+v12)ikD*2B*(u22+v22)]×exp[k2(u12+v12)4M1xxB24k2M1xx2B2δxx4(u22+v22)+k2(B*)2(u12+v12)4k2M1xx|B|2δxx2(u1u2+v1v2)16M1xx2M2xx|B|4δxx4]×[k2(2M1xxBδxx2u2B*u12M1xx|B|2δxx2)2+2M2xxk2δxx2u1B(2M1xxBδxx2u2B*u12M1xx|B|2δxx2)+2M2xx],
(9)
Wxy(u1,v1,u2,v2,z)=Bxyk2π28M1xy2M2xy3δxy2w02(1λ|B|)2exp[ikD2B(u12+v12)ikD*2B*(u22+v22)]×exp[k2(u22+v22)4M1xyB*24k2M1xy2B*2δxy4(u12+v12)+k2B2(u22+v22)4k2M1xy|B|2δxy2(u1u2+v1v2)16M1xy2M2xy|B|4δxy4]×(u22M1xyB*δxy2u1B)(v1B1+4M1xyM2xyδxy42M1xyB*δxy2v2),
(10)
Wyx(u1,v1,u2,v2,z)=Wxy*(u2,v2,u1,v1,z),
(11)
Wyy(u1,v1,u2,v2,z)=π28δyy2M1yy2M2yy3ω02(1λ|B|)2exp[ikD2B(u12+v12)ikD*2B*(u22+v22)]×exp[k2(u12+v12)4M1yyB24k2M1yy2B2δyy4(u22+v22)+k2(B*)2(u12+v12)4k2M1yy|B|2δyy2(u1u2+v1v2)16M1yy2M2yy|B|4δyy4]×[k2(2M1yyBδyy2v2B*v12M1yy|B|2δyy2)2+2M2yyk2δyy2v1B(2M1yyBδyy2v2B*v12M1yy|B|2δyy2)+2M2yy],
(12)
Where

M1xx=1/ω02+1/(2δxx2)ikA/(2B),M2xx=1/ω02+1/(2δxx2)+ikA*/(2B*)1/(4M1xxδxx4),
(13)
M1xy=1/ω02+1/(2δxy2)+ikA*/(2B*),M2xy=1/ω02+1/(2δxy2)ikA/(2B)1/(4M1xyδxy4),
(14)
M1yy=1/ω02+1/(2δyy2)ikA/(2B),M2yy=1/ω02+1/(2δyy2)+ikA*/(2B*)1/(4M1yyδyy4).
(15)

The average intensity of the partially coherent RP beam at the output plane is expressed as [22

22. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).

]
I(u,v,z)=TrW(u1,v1,u2,v2,z)=Wxx(u,v,u,v,z)+Wyy(u,v,u,v,z),
(16)
where Tr denotes the trace of the matrix.

The degree of polarization of the partially coherent vector beam at point(uv)is defined by the expression [22

22. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).

]
P(u,v,z)=14DetW(u,v,u,v,z)[TrW(u,v,u,v,z)]2,
(17)
where Det stands for the determinant of the matrix.

It was shown in [52

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). [CrossRef] [PubMed]

] that a completely polarized cylindrical vector partially coherent beam is depolarized (i.e., becomes partially polarized) after propagation, thus it is important to analyze its state of polarization. The CSD matrix of a partially polarized partially coherent vector beam at point (uv)can be locally represented as a sum of completely polarized beam and a completely unpolarized beam [22

22. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).

, 28

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). [CrossRef]

]
W(u,v,u,v,z)=W(u)(u,v,u,v,z)+W(p)(u,v,u,v,z),
(18)
where
W(u)(u,v,u,v,z)=(A(u,v,u,v,z)00A(u,v,u,v,z)),
(19)
W(p)(u,v,u,v,z)=(B(u,v,u,v,z)D(u,v,u,v,z)D*(u,v,u,v,z)C(u,v,u,v,z)),
(20)
With

A(u,v,u,v,z)=12[Wxx(u,v,u,v,z)+Wyy(u,v,u,v,z)[Wxx(u,v,u,v,z)Wyy(u,v,u,v,z)]2+4|Wxy(u,v,u,v,z)|2],
(21)
B(u,v,u,v,z)=12[Wxx(u,v,u,v,z)Wyy(u,v,u,v,z)+[Wxx(u,v,u,v,z)Wyy(u,v,u,v,z)]2+4|Wxy(u,v,u,v,z)|2,
(22)
C(u,v,u,v,z)=12[Wyy(u,v,u,v,z)Wxx(u,v,u,v,z)+[Wxx(u,v,u,v,z)Wyy(u,v,u,v,z)]2+4|Wxy(u,v,u,v,z)|2],
(23)
D(u,v,u,v,z)=Wxy(u,v,u,v,z).
(24)

The average intensity of the completely unpolarized beam and the intensity of the completely polarized beam are expressed as

I(u)(u,v,z)=TrW(u)(u,v,u,v,z),I(p)(u,v,z)=TrW(p)(u,v,u,v,z).
(28)

There are two definitions of the degree of coherence for a paraxial partially coherent vector beam [22

22. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).

24

24. F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998). [CrossRef] [PubMed]

, 30

30. J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003). [CrossRef] [PubMed]

]. For the convenience of experimental measurement, in this paper we adopt the definition introduced by Tervo et al., according to [30

30. J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003). [CrossRef] [PubMed]

], the degree of coherence μ of the partially coherent vector beam at a pair of transverse points with position vectors (u1v1) and (u2v2)is defined as
μ2(u1,v1,u2,v2,z)=Tr[W(u1,v1,u2,v2,z)W(u2,v2,u1,v1,z)]I(u1,v1,z)I(u2,v2,z).
(29)
where I(ui,vi,z) is the average intensity of the beam at point (uivi).

If the partially coherent vector beam obeys the Gaussian statistics [21

21. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

], the numerator in right hand of Eq. (29) can be simplified as [30

30. J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003). [CrossRef] [PubMed]

]
Tr[W(u1,v1,u2,v2,z)W(u2,v2,u1,v1,z)]=α,β|Wαβ(u1,v1,u2,v2,z)|2.(α,β=x,y)
(30)
Furthermore, the fourth-order correlation function between two points (u1v1) and (u2v2)can be expanded as [45

45. Z. Tong, Y. Cai, and O. Korotkova, “Ghost imaging with electromagnetic stochastic beams,” Opt. Commun. 283(20), 3838–3845 (2010). [CrossRef]

]
G(2)(u1,v1,u2,v2,z)=I(u1,v1,z)I(u2,v2,z)+α,β|Wαβ(u1,v1,u2,v2,z)|2.
(31)
Then the normalized fourth-order correlation function can be expressed in terms of the degree of coherence as follows
g(2)(u1,v1,u2,v2,z)=1+μ2(u1,v1,u2,v2,z).
(32)
Thus, we can obtain the information of the degree of coherence by measuring the fourth-order correlation function in experiment as shown later in section 3.

Now applying the above derived formulas, we study the polarization and coherence properties of a partially coherent RP beam after passing through a thin lens with focal length f=400mmwhich is located at z = 0. The transfer matrix between the source plane at z = 0 and the output plane at z reads as

(ABCD)=(1z01)(101/f1)=(1z/fz1/f1).
(33)

Figure 1
Fig. 1 Degree of polarization of a focused partially coherent RP beam at point (0.05mm0.05mm)versus the propagation distance z for different values of the correlation radii δxx,δyy,δxy.
shows the degree of polarization of a focused partially coherent RP beam at point (0.05mm0.05mm)versus the propagation distance z for different values of the correlation radii δxx,δyy,δxywithλ=632.8nm,ω0=1.05mm.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

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). [CrossRef] [PubMed]

]. 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
A2(u,v,u,v,z)=0,ε(u,v,u,v,z)=0,θ(u,v,u,v,z)=arctan(v/u).
(34)
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
Fig. 2 Intensity distributions of a focused partially coherent RP beam, its completely polarized part and its completely unpolarized part with δxx=δyy=δxy=1mm.
and Fig. 3
Fig. 3 Intensity distributions of a focused partially coherent RP beam, its completely polarized part and its completely unpolarized part with .δxx=δyy=δxy=0.20mm
the intensity distributions of a partially coherent RP beam, its completely polarized part and its completely unpolarized part for δxx=δyy=δxy=1mmand δxx=δyy=δxy=0.20mm, respectively. One finds from Figs. 2 and 3 that the dark hollow beam profile of a focused partially coherent RP beam disappears on propagation as expected [57

57. F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012). [CrossRef]

], 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
η(l)=I(l)(u,v,z)dudvI(u,v,z)dudv,(l=p,u)
(35)
where I(u)(u,v,z) and I(p)(u,v,z) are given by Eq. (28). Figure 4(a)
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 δxx=δyy=δxy=0.20mm, (b) versus the correlation coefficient δxxwith δxx=δyy=δxyat the focal plane z = 400mm.
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 δxx=δyy=δxy=0.20mm. Figure 4(b) shows the variation of the normalized powers at the focal plane versus the correlation coefficient δxxwith δxx=δyy=δxy. One finds from Fig. 4(a) that the normalized power η(p) of the completely polarized part decreases on propagation, while the normalized power η(u) the completely unpolarized part increases. At the focal plane, η(u)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 η(p)and η(u)at the focal plane are closely determined by the correlation radii. With the decrease of the correlation radii, η(u)increases while η(p)decreases. For the case of δxx<0.1mm, 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 η(u)=η(p), the total intensity distribution of the focused partially coherent RP beam has a flat-topped beam profile (see Fig. 5
Fig. 5 Normalized intensity (cross line, v = 0) of a focused partially coherent RP beam under the condition of η(u)=η(p).
).

3. 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
Fig. 8 Experimental setup for generating a partially coherent RP beam and measuring its coherence properties. L1, L2, L3, thin lenses; RM, reflecting mirror; RGGP, rotating ground-glass plate; GAF, Gaussian amplitude filter; RPC, radial polarization converter; beam splitter; D1, D2, single photon detectors.
shows the experimental setup for generating and measuring the coherence properties of a partially coherent RP beam. Similar to Ref [57

57. F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012). [CrossRef]

], 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

59. P. De Santis, F. Gori, G. Guattari, and C. Palma, “an example of a collectt-wolf source,” Opt. Commun. 29(3), 256–260 (1979). [CrossRef]

, 60

60. F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24(7), 1937–1944 (2007). [CrossRef] [PubMed]

]. 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 W(x1,y1,x2,y2)=exp[(x12+y12+x22+y22)/ω02(x1x2)2/2δ2(y1y2)2/2δ2],with ω0 and 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 δxx=δyy=δxy=δ. The transmission function of the GAF determines the value of ω0, and ω0 is equal to 1.05mm in our experiment. The transverse coherence widthδ 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 L1 and RGGP). We adopt the method proposed in [59

59. P. De Santis, F. Gori, G. Guattari, and C. Palma, “an example of a collectt-wolf source,” Opt. Commun. 29(3), 256–260 (1979). [CrossRef]

] to measure the coherence width δ of the GSM beam. In our experiment, we choose two different values of δ(δ=0.20mm,0.55mm) to generate a RP beam with variable spatial coherence. Figure 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.
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.

After passing through a thin lens L3 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 D1 and D2, which scan the transverse plane u1 and u2, respectively. Both the distances from the L3 to D1 and from L3 to D2 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 plane through the relation given by Eq. (32). Figure 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.
shows our experimental results 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. For the convenience of comparison, the corresponding results calculated by the theoretical formulas with the measured values of ω0and δare also shown in Fig. 10. One finds from Fig. 10 that the degree of coherence of the focused partially coherent RP beam at the focal plane indeed becomes of non-Gaussian distribution, and our experimental results agree well with the theoretical results.

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

61. M. Born and E. Wolf, Principles of optics, seventh ed. (Cambridge University Press, Cambridge, 1999).

] 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

62. G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002). [CrossRef]

]. To measure the elements Wxxand Wyy, we put a linear polarizer whose transmission axis forms an angle θ with the x-axis just before the thin lens L3 (see 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.
), 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
Iθ(u,v,f)=cos2θWxx(u,v,u,v,f)+sin2θWyy(u,v,u,v,f)+[Wxy(u,v,u,v,f)+Wyx(u,v,u,v,f)]sinθcosθ.
(36)
For the case of θ=0, we obtain
Wxx(u,v,u,v,f)=I0(u,v,f).
(37)
For the case of θ=π/2, we obtain

Wyy(u,v,u,v,f)=Iπ/2(u,v,f).
(38)

To measure the elements Wxyand Wyx, we put a quarter-wave plate and a linear polarizer together just before the thin lens L3 (see Fig. 10), then the CCD is used to measure the average intensity at the focal plane. The fast axis and slow axis are along y-axis and x-axis, respectively. In this case, the average intensity distribution at the focal plane is written as
Iθ,ϕ(u,v,f)=sin2θWyy(u,v,u,v,f)+cos2θWxx(u,v,u,v,f)+exp(iϕ)sinθ[Wxy(u,v,u,v,f)+exp(2iϕ)Wyx(u,v,u,v,f)]cosθ,
(39)
where ϕ=π/2 represents the phase difference between the x-component and y component of the vector field induced by the quarter-wave plate. In Eq. (39), ϕ=0 means the quarter-wave plate is removed. From Eq. (39), we can obtain the following expressions for the elements Wxyand Wyx
Wxy(u,v,u,v,f)=12[Iπ/4,0(u,v,f)I3π/4,0(u,v,f)]+i12[Iπ/4,π/2(u,v,f)I3π/4,π/2(u,v,f)],
(40)
Wyx(u,v,u,v,f)=Wxy*(u,v,u,v,f).
(41)
Thus by measuring the average intensities Iπ/4,0, I3π/4,0, Iπ/4,π/2, I3π/4,π/2at the focal plane, we can obtain the information of Wxyand Wyx.

Figure 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.
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 δdecreases. 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.

4. 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 the correlation radii of the source (i.e., degree of coherence of source field), which means that we can modulate the coherence and polarization properties of a partially coherent RP beam by varying its spatial coherence. Our experimental results agree well with theoretical results.

Appendix A. Derivation of the realizability conditions for a partially coherent radially polarized beam

It is known that a partially coherent vector beam generated by a Schell-model source should satisfy the semi-positive definiteness condition [21

21. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

, 22

22. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).

]. Based on this condition, Gori et al. derived the following general realizability conditions for the beam parameters [33

33. F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25(5), 1016–1021 (2008). [CrossRef] [PubMed]

]

δxx2+δyy22δxyδxxδyy|Bxy|,|Bxy|2δxx/δyy+δyy/δxx.
(A1)

For a partially coherent RP beam generated by a Schell-model source, besides the restrictions on the parameters Bxy,δxx,δyy,δxy shown in Eq. (A1), following two additional conditions should be satisfied

  • (1) Any point of the beam in the source plane is linearly polarized.
  • (2) The orientation angle of the polarization at any point in the source plane should satisfyθ(x,y,0)=arctan(y/x).

According to Eqs. (25)-(27) in the text, the major and minor semi-axes of the polarization ellipse, A1 andA2, as well as its degree of ellipticity, ε, and its orientation angle, θ, of the completely polarized part of the partially coherent RP beam in the source plane are expressed as

A1,2(x,y,0)=12[[Wxx(x,y,x,y,0)Wyy(x,y,x,y,0)]2+4|Wxy(x,y,x,y,0)|2±[Wxx(x,y,x,y,0)Wyy(x,y,x,y,0)]2+4[ReWxy(x,y,x,y,0)]2]1/2,
(A2)
ε(x,y,0)=A2(x,y,x,y,0)A1(x,y,x,y,0),
(A3)
θ(x,y,0)=12arctan[2ReWxy(x,y,x,y,0)Wxx(x,y,x,y,0)Wyy(x,y,x,y,0)].
(A4)

The additional condition (1) requires that A2equals to zero. With the requirements of the additional conditions (1) and (2), and applying Eqs. (4)-(7), one can obtain the following restriction on Bxy,
Im[Bxy]=0,Re[Bxy]=1,
(A5)
where Im and Re denote the imaginary and real parts of Bxy, respectively. From Eq. (A5), one can easily obtain Bxy=1. Under the condition of Bxy=1, one can easily obtain from Eq. (A1) that δxx=δyy=δxy. Thus, the final realizability conditions for a partially coherent RP beam should be Bxy=1andδxx=δyy=δxy.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant Nos. 11274005 & 10904102 &11104195, the Foundation for the Author of National Excellent Doctoral Dissertation of PR China under Grant No. 200928, the Natural Science Foundation of Jiangsu Province under Grant No. BK2009114, the Huo Ying Dong Education Foundation of China under Grant No. 121009, the Key Project of Chinese Ministry of Education under Grant No. 210081, the Universities Natural Science Research Project of Jiangsu Province under grant 11KJB140007, and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

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57.

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58.

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59.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “an example of a collectt-wolf source,” Opt. Commun. 29(3), 256–260 (1979). [CrossRef]

60.

F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24(7), 1937–1944 (2007). [CrossRef] [PubMed]

61.

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62.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002). [CrossRef]

OCIS Codes
(030.0030) Coherence and statistical optics : Coherence and statistical optics
(260.5430) Physical optics : Polarization
(350.5500) Other areas of optics : Propagation

ToC Category:
Coherence and Statistical Optics

History
Original Manuscript: September 18, 2012
Revised Manuscript: October 29, 2012
Manuscript Accepted: November 27, 2012
Published: December 6, 2012

Citation
Gaofeng Wu, Fei Wang, and Yangjian Cai, "Coherence and polarization properties of a radially polarized beam with variable spatial coherence," Opt. Express 20, 28301-28318 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-27-28301


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References

  1. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon.1(1), 1–57 (2009). [CrossRef]
  2. Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express10(7), 324–331 (2002). [CrossRef] [PubMed]
  3. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express7(2), 77–87 (2000). [CrossRef] [PubMed]
  4. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett.91(23), 233901 (2003). [CrossRef] [PubMed]
  5. D. P. Biss and T. G. Brown, “Cylindrical vector beam focusing through a dielectric interface,” Opt. Express9(10), 490–497 (2001). [CrossRef] [PubMed]
  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). [CrossRef] [PubMed]
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