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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 24 — Nov. 23, 2009
  • pp: 21472–21487
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Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams

Chengliang Zhao, Yangjian Cai, and Olga Korotkova  »View Author Affiliations


Optics Express, Vol. 17, Issue 24, pp. 21472-21487 (2009)
http://dx.doi.org/10.1364/OE.17.021472


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Abstract

Radiation force of a focused scalar twisted Gaussian Schell-model (TGSM) beam on a Rayleigh dielectric sphere is investigated. It is found that the twist phase affects the radiation force and by raising the absolute value of the twist factor it is possible to increase both transverse and longitudinal trapping ranges at the real focus where the maximum on-axis intensity is located. Numerical calculations of radiation forces induced by a focused electromagnetic TGSM beam on a Rayleigh dielectric sphere are carried out. It is found that radiation force is closely related to the twist phase, degree of polarization and correlation factors of the initial beam. The trapping stability is also discussed.

© 2009 OSA

1. Introduction

A Gaussian Schell-model (GSM) beam is a stochastic beam whose spectral degree of coherence and the intensity distribution are Gaussian functions [1

1. E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,” Opt. Commun. 25(3), 293–296 ( 1978). [CrossRef]

6

6. Q. S. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67(4), 245–250 ( 1988). [CrossRef]

]. Beams of this class can be generated by scattering a coherent laser beam with a rotating grounded glass and then by transforming the intensity distribution of the scattered light into profile with a Gaussian amplitude [2

2. F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34(3), 301–305 ( 1978). [CrossRef]

4

4. 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]

]. GSM beams can also be generated with the help of specially synthesized rough surfaces, spatial light modulators and synthetic acousto-optic holograms [5

5. A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 ( 1982). [CrossRef]

]. They have found wide applications in imaging [6

6. Q. S. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67(4), 245–250 ( 1988). [CrossRef]

, 7

7. Y. Cai and S. Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5), 056607 ( 2005). [CrossRef] [PubMed]

], free space optical communications [8

8. J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19(9), 1794–1802 ( 2002). [CrossRef]

], nonlinear optics [9

9. N. A. Ansari and M. S. Zubairy, “Second-harmonic generation by a Gaussian Schell-model source,” Opt. Commun. 59(5-6), 385–390 ( 1986). [CrossRef]

], etc [10

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

]. As an extension of a conventional GSM beam, Simon and Mukunda introduced a twisted GSM beam [11

11. R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10(1), 95–109 ( 1993). [CrossRef]

], whose phase, called twist phase, differs in many respects from the ordinary quadratic phase factor [11

11. R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10(1), 95–109 ( 1993). [CrossRef]

15

15. A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11(6), 1818–1826 ( 1994). [CrossRef]

]. The twist phase is bounded in strength due to the positive semi-definiteness requirement on the cross-spectral density function and disappears in the limit of a coherent Gaussian beam. It has intrinsically a two-dimensional spatial dependence, i.e. it cannot be separated into a sum of one-dimensional contributions. The twist phase rotates the beam spot on propagation due to its intrinsic chiral (handedness) property and increases the beam divergence on propagation. The first experimental observation of a twisted GSM beam was reported by Friberg et al. [15

15. A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11(6), 1818–1826 ( 1994). [CrossRef]

]. Their realization by superposition, the coherent-mode decomposition, the orbital angular momentum and their propagation effects have been later extensively studied [16

16. D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41(7), 1391–1399 ( 1994). [CrossRef]

30

30. Y. Cai, Q. Lin, and O. Korotkova, “Ghost imaging with twisted Gaussian Schell-model beam,” Opt. Express 17(4), 2453–2464 ( 2009). [CrossRef] [PubMed]

]. The conventional method for treating the propagation of GSM beam and twisted GSM beam is Wigner distribution function [22

22. M. J. Bastiaans, “Wigner-distribution function applied to twisted Gaussian light propagating in first-order optical systems,” J. Opt. Soc. Am. A 17(12), 2475–2480 ( 2000). [CrossRef]

, 31

31. M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3(8), 1227–1238 ( 1986). [CrossRef]

]. Lin and Cai have recently introduced a convenient alternative tensor method for treating propagation of GSM and twisted GSM beams [24

24. 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]

]. The tensor method has proved to be reliable for studying the passage of such beams through paraxial optical systems, fractional Fourier transform optical systems, dispersive media and apertured optical systems [4

4. 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]

, 25

25. Q. Lin and Y. Cai, “Fractional Fourier transform for partially coherent Gaussian-Schell model beams,” Opt. Lett. 27(19), 1672–1674 ( 2002). [CrossRef] [PubMed]

28

28. Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 ( 2006). [CrossRef]

]. More recently, Cai and associates applied the tensor method for the analysis of the effects of the twist phase on the second harmonic generation and on the ghost imaging [29

29. Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express 15(23), 15480–15492 ( 2007). [CrossRef] [PubMed]

, 30

30. Y. Cai, Q. Lin, and O. Korotkova, “Ghost imaging with twisted Gaussian Schell-model beam,” Opt. Express 17(4), 2453–2464 ( 2009). [CrossRef] [PubMed]

].

Polarization state is another important property of the beam arising from its vectorial nature. Conventionally, coherence and polarization states of light were studied separately [10

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

,32

32. C. Brosseau, Fundamentals of polarized light-a statistical approach (Wiley, New York, 1998).

], until James found, in 1994, that the degree of polarization of a stochastic electromagnetic beam may change on propagation in vacuum, such changes being dependent on the coherence properties of the source of the beam [33

33. D. F. V. James, “Changes of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11(5), 1641–1643 ( 1994). [CrossRef]

]. Since then such correlation-induced polarization modification of stochastic electromagnetic beams have been explored in depth [34

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

46

46. B. Kanseri and H. C. Kandpal, “Experimental determination of electric cross-spectral density matrix and generalized Stokes parameters for a laser beam,” Opt. Lett. 33(20), 2410–2412 ( 2008). [CrossRef] [PubMed]

] in general and in particular for the important class of beams called electromagnetic Gaussian Schell-model [EGSM] beams, introduced by Gori et al. [36

36. 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), 301 ( 2001). [CrossRef]

]. Cai et al. later applied the tensor method for the analysis of the evolution of the EGSM beams in resonators and in radar systems operating through the turbulent atmosphere [47

47. M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “The 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]

49

49. 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 ( 2008). [CrossRef] [PubMed]

]. More recently, Cai and Korotkova introduced twisted EGSM beams, and studied their propagation in free space [50

50. Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B 96(2-3), 499–507 ( 2009). [CrossRef]

].

2. Focusing properties of scalar and electromagnetic twisted Gaussian Schell-model beams

The second-order statistical properties of a twisted EGSM beam can be characterized by the 2×2 cross-spectral density matrixW(r1,r2;0) specified at any two points with position vectors r1 and r2 in the source plane with elements [33

33. D. F. V. James, “Changes of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11(5), 1641–1643 ( 1994). [CrossRef]

50

50. Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B 96(2-3), 499–507 ( 2009). [CrossRef]

]
Wαβ(r1,r2;0)=Wαβ(r˜;0)=Eα*(r1;0)Eβ(r2;0)=AαAβBαβexp[ik2r˜TM0αβ1r˜],(α=x,y;β=x,y)
(7)
whereAαis the square root of the spectral density of electric field component Eα, Bαβ=|Bαβ|exp(iϕ) is the correlation coefficient between the ExandEy field components, satisfying the relation Bαβ=Bβα*, M0αβ1 is a 4×4 matrix of the form [50

50. Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B 96(2-3), 499–507 ( 2009). [CrossRef]

]
M0αβ1=(1ik(12σaβ2+1δαβ2)Iikδαβ2I+μαβJikδαβ2I+μαβJT1ik(12σαβ2+1δαβ2)I),
(8)
whereσαβandδαβ denote the widths of the spectral density and correlation coefficient, respectively. μαβ represents the twist factor and is limited by μαβ21/(k2δαβ4) if α = β due to the non-negativity requirement of the cross-spectral density [Eq. (7)] [11

11. R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10(1), 95–109 ( 1993). [CrossRef]

,50

50. Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B 96(2-3), 499–507 ( 2009). [CrossRef]

]. Aα,Bαβ,σαβ,δαβandμαβ are independent of position but, in general, depend on the frequency. We note here that if off-diagonal elements are to be included in calculation the realizability condition of the source should relate the on- and off-diagonal twist factors and the coefficient B (and, perhaps, some other source parameters). But these conditions are not known so far and, moreover, are not needed in this work since only the effect of the twist phases of diagonal matrix elements is important on the focusing and trapping ability of the beam, since the later depend only on the beam intensity.

The spectral density of a twisted EGSM beam at point r of the source plane is defined by the expression [33

33. D. F. V. James, “Changes of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11(5), 1641–1643 ( 1994). [CrossRef]

50

50. Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B 96(2-3), 499–507 ( 2009). [CrossRef]

]
I(r;0)=TrW(r,r;0)=Axexp[ik2r˜TM0xx1r˜]+Ayexp[ik2r˜TM0yy1r˜].
(9)
Assuming that Ay=ηAx and that the total transmitted power of the twisted EGSM beam is Q, we find that

Q=Axπ(1+η)(σxx2+σyy2)2.
(10)

The degree of polarization of an electromagnetic beam at pointris defined by the expression [33

33. D. F. V. James, “Changes of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11(5), 1641–1643 ( 1994). [CrossRef]

50

50. Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B 96(2-3), 499–507 ( 2009). [CrossRef]

]
P0(r;0)=14DetW(r,r;0)[TrW(r,r;0)]2,
(11)
where Tr denotes the trace of the matrix.

For the conciseness of the analysis, in this paper we will only consider the twisted EGSM beams that are generated by sources whose cross-spectral density matrices are diagonal, i.e., of the form
W(r1,r2;0)=(Wxx(0)(r1,r2;0)00Wyy(0)(r1,r2;0)).
(12)
Furthermore, we will assume that σxx=σyy, exploring the effect of difference between correlations δx and δy alone. For such restricted type of beams the degree of polarization across the source is given by the formula

P0(r;0)=|1η|1+η.
(13)

In the rest of this section we will study the focusing properties of scalar and electromagnetic twisted GSM beams passing through a thin lens. The power Q of all considered beams in this paper at the input plane is set to be 1W and the wavelength is set to beλ=632.8nm. The schematic diagram of a focusing optical system is shown in Fig. 1
Fig. 1 Schematic diagram of a focusing optical system
, where a thin lens with focal length f is located at the input plane (z=0), and the output plane is located at z. Then the transfer matrix between the input and output planes can be expressed as follows

(ABCD)=(IzI0II)(I0I(1/f)II)=((1z/f)IzI(1/f)II).
(16)

Substituting from Eq. (16) into Eqs. (4) - (6), we calculate in Fig. 2(a)
Fig. 2 (a) Intensity distribution of a scalar twisted GSM beam at the real focal plane for different absolute values of the initial twist factor normalized with γ=[k2δ4]1/2=0.1m1. The inset shows scaled down intensity distribution for the case |μ0/γ|=1. (b) Focus shift fzmaxof a scalar twisted GSM beam behind the thin lens versus the normalized initial twist factor|μ0/γ|
the intensity distribution of a scalar twisted GSM beam at the real focal plane for different absolute values of the initial twist factor normalized with γ=[k2δ4]1/2=0.1m1. Here the real focal plane is defined as the plane transverse to direction of propagation z, where the maximum on-axis intensity is located. We calculate in Fig. 2(b) the focus shift fzmaxof a scalar twisted GSM beam behind the thin lens versus the normalized initial twist factor|μ0/γ|. zmax is the position of the real focal plane. We have chosen the other parameters to be σ=10mm,δ=1mmandf=5mm. One finds from Fig. 2(a) that the twist phase has strong influence on the focusing properties of a scalar twisted GSM beam. As the absolute value of the twist factor increases, the focused beam spot becomes larger and the maximum intensity decreases. Thus the twist phase is expected to affect the radiation force, mainly determined by the focused intensity, as will be shown later. As was shown in [15

15. A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11(6), 1818–1826 ( 1994). [CrossRef]

] (see Eq. (8)) as the absolute value of the twist factor increases, the effective degree of coherence decreases, the focused beam spot becomes larger and the maximum intensity decreases. Thus the twist phase is expected to affect the radiation forces (mainly determined by the focused intensity) induced by the focused beam as shown later.

It is important to note that the beam waist (minimum spot size and maximum intensity) is not located in the focal plane (i.e., at z = f), but rather closer to the lens due to the focus shift [see Fig. 2(b)]. The focus shift is dependent on the intensity and coherence widths, as well as on the twist parameter which decreases the effective coherence width (see, for instance, Ref. 6

6. Q. S. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67(4), 245–250 ( 1988). [CrossRef]

).

Substituting from Eq. (16) into Eqs. (14) and (15), we calculate in Fig. 3
Fig. 3 Intensity distribution of a twisted EGSM beam at the real focal plane for different absolute values of the initial twist factorsμxxand μyy normalized with γxx=[k2δxx4]1/2=0.1m1and γyy=[k2δyy4]1/2=0.025m1, respectively.
the intensity distribution of a twisted EGSM beam at the real focal plane for different absolute values of the normalized initial twist factors |μxx/γxx|and |μyy/γyy|with σxx=σyy=10mm,δxx=1mm,δyy=2mm,P0=0.333,γxx=[k2δxx4]1/2=0.1m1andγyy=[k2δyy4]1/2=0.025m1. Figure 4
Fig. 4 Intensity distribution of a twisted EGSM beam at the real focal plane for different values of the initial degree of polarization
shows the intensity distribution of a twisted EGSM beam at the real focal plane for different values of the initial degree of polarization P 0 with σxx=σyy=10mm,δxx=1mm, δyy=2mm. Figure 5
Fig. 5 Intensity distribution of an electromagnetic twisted GSM beam at the real focal plane for different values of the initial correlation coefficients δxxandδyy. In Fig. 5 The inset shows the scaled down intensity distribution for the case δxx=1mmandδyy=2mm
shows the intensity distribution of a twisted EGSM beam at the real focal plane for different values of the initial correlation coefficients δxxandδyywith σxx=σyy=10mm. In calculation of Figs. 4 and 5, we have chosen the twist factors to be μxx/γxx=0.04, μyy/γyy=0.1 withγxx=[k2δxx4]1/2=0.1m1 and γyy=[k2δyy4]1/2=0.025m1. One finds from Figs. 2 and 3 that the influence of the twist factors on the focused intensities of scalar and electromagnetic twisted GSM beams is similar, i.e., the width of the focused beam spot increases with the increase of the absolute value of the twist factors. From Fig. 4, it is clear that the focused intensity is also determined by the initial degree of polarization of the electromagnetic twisted GSM beam. A partially polarized twisted EGSM beam whose cross-spectral density matrices only have on-diagonal elements can be focused more tightly than a completely polarized one. If the off-diagonal elements are included, we need to consider the state of polarization, then this conclusion may not always be valid, and we leave this for future study. From Fig. 5, we find that the initial correlation coefficients δxxandδyyalso affect the focusing properties of a twisted EGSM beam, and the twisted EGSM beam with larger values of correlation coefficients can be focused more tightly.

3. Radiation force induced by focused scalar and electromagnetic twisted GSM beams on a Rayleigh particle

In this section we determine the magnitudes of radiation forces induced by focused scalar and electromagnetic twisted GSM beams on a Rayleigh dielectric sphere with radius a, (a<<λ) and refractive index np. The reader can refer to Fig. 1 where the schematic diagram is given for trapping a Rayleigh dielectric sphere placed near the focus of the beam.

The radiation force is a combination of the scattering force and the gradient force. The former force component, which is caused by scattering of light by the sphere, is proportional to the beam intensity and acts along its direction of propagation. It can be expressed as [54

54. Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5-6), 529–541 ( 1996). [CrossRef]

]
FScat(r;z)=eznmαI(r;z)/c,
(17)
whereI(r;z) is the intensity of the focused beam at the output plane, ez is a unity vector along the direction of propagation, α=(8/3)π(ka)4a2[(χ21)/(χ2+2)]2, χ=np/nmwith nmbeing the refractive index of the ambient. In the following text, we choose a=50 nm, np=1.59 (glass) and nm=1.33 (water). The gradient force, produced by a non-uniform field, acts along the gradient of light intensity, and can be expressed as [54

54. Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5-6), 529–541 ( 1996). [CrossRef]

]
FGrad(r;z)=2πnmβI(r;z)/c,
(18)
whereβ=a3(χ21)/(χ2+2).

Figure 6 can also be of use for analyzing trapping ranges and trapping stability. Trapping range along a certain direction is the distance from the equilibrium position to the position at which trapping starts to occur. By trapping stability along a certain direction we mean the difference between the magnitudes of the force at the real focal plane and at the boundary of the trapping region. Since the absolute values of the scattering force and of the gradient force generally decrease as the absolute value of the twist factor increases, the trapping stability of the beam is deteriorated. From Fig. 6(b) and 6(c), we see that one stable equilibrium point exists at the real focus which is not located at z=f (i.e., geometrical focus), which implies that we can use focused scalar twisted GSM beam to trap a Rayleigh particle whose refractive index is larger than the ambient at the real focus. Furthermore, with increase of the absolute value of the twist factor, although the trapping stability becomes worse, as shown by Fig. 6(b) and 6(c), both transverse trapping range and longitudinal trapping range becomes larger (i.e., the positions of peak values deviate away from the focus). The scattering and gradient forces decrease and the trapping range increases as the twist parameter increases follow directly from the property that increased twist parameter implies reduced effective spatial coherence of the beam and therefore a more broadly distributed focal spot. Tables 1

Table 1. The trapping ranges from Fig. 6(b) for different absolute values of μ0

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and 2

Table 2. The trapping ranges from Fig. 6(c) for different absolute values of μ0

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show the trapping ranges corresponding to Fig. 6(b) and 6(c) for different absolute values of μ0, from which we find that both transverse and longitudinal trapping ranges increase remarkably as the absolute values of μ0 increases. We should point out that the Rayleigh particle will be diffused instead of being trapped when the absolute value of μ0 is very large because the radiation forces becomes comparable, in this regime, to the Brownian force, as will be shown later. Thus for particle trapping the absolute values of μ0must be limited from above.

Applying Eqs. (14)-(18), we calculate in Fig. 8(a)
Fig. 8 (a) Scattering force (cross-section y = 0) at the real focal plane, (b) transverse gradient force (cross-section y=0) at the real focal plane, and (c) longitudinal gradient force at r=0 of an electromagnetic twisted GSM beam for different absolute values of the initial twist factorsμxxand μyy normalized with γxx=[k2δxx4]1/2=0.1m1and γyy=[k2δyy4]1/2=0.025m1, respectively. The inset in Fig. 8(c) shows the zoomed region where the crossing of the radiation forces with zero occurs.
the scattering force (cross-section y=0) at real focal plane, in Fig. 8(b) the transverse gradient force (cross-section y=0) at real focal plane, and in Fig. 8(c) the longitudinal gradient force at r=0 of a twisted EGSM beam for different absolute values of the initial twist factors μxxand μyywith σxx=σyy=10mm,δxx=1mm, δyy=2mm and P0=0.333 and f=5mm. By comparing Figs. 6 and 8, we come to the conclusion that the effect of the twist phase on the radiation force induced by a twisted EGSM beam is similar to that induced by a scalar twisted GSM beam, i.e., the radiation force decreases with the increase of the twist factors but both transverse and longitudinal trapping increases. We calculate in Fig. 9(a)
Fig. 9 (a) Scattering force (cross-section y=0) at the real focal plane, (b) transverse gradient force (cross-section y=0) at the real focal plane, and (c) longitudinal gradient force at r=0 of an electromagnetic twisted GSM beam for different values of the initial degree of polarization. In Fig. 9(c) the inset shows the zoomed region where the crossing of the radiation forces with zero occurs.
the scattering force (cross-section y=0) at real focal plane, in Fig. 9(b) the transverse gradient force (cross-section y=0) at real focal plane, and in Fig. 9(c) the longitudinal gradient force at r=0 of a twisted EGSM beam for different values of the initial degree of polarization with σxx=σyy=5mm, δxx=1mm andδyy=2mm. Further, we calculate in Fig. 10(a)
Fig. 10 (a) Scattering force (cross-section y=0) at the real focal plane, (b) transverse gradient force (cross-section y=0) at the real focal plane, and (c) longitudinal gradient force at r=0 of an electromagnetic twisted GSM beam for different values of the initial correlation coefficients δxxandδyy. In Fig. 10(c), the inset shows the zoomed region where the crossing of the radiation forces with zero occurs.
the scattering force (cross-section y=0) at real focal plane, in Fig. 10(b) the transverse gradient force (cross-section y = 0) at real focal plane, and in Fig. 10(c) the longitudinal gradient force at r=0 of an electromagnetic twisted GSM beam for different values of the initial correlation coefficients δxxandδyywith σxx=σyy=5mm. In calculation of Figs. 9 and 10, we have chosen the twist factors to be μxx/γxx=0.04, μyy/γyy=0.1 withγxx=[k2δxx4]1/2=0.1m1 and γyy=[k2δyy4]1/2=0.025m1. From Figs. 9 and 10 one finds that the radiation force induced by a twisted EGSM beam is also closely determined by its initial degree of coherence and correlation coefficients. As the values of the initial degree of polarization or correlation coefficients decrease, the radiation force decreases (i.e., the trapping stability becomes worse), while the positions of peak values of the gradient forces deviate away from the focus (i.e., trapping ranges become larger). From above discussions, we come to the conclusion that we can control the trapping stability and the trapping ranges by choosing suitable values of the twist phase, degree of polarization and correlation coefficients of the partially coherent beam at the input plane.

4. Analysis of the trapping stability

Now we analyze the trapping stability in greater detail by taking into consideration the Brownian motion of the trapped particle. The magnitude of the Brownian force is expressed as |FB|=(12πκakBT)1/2 according to the fluctuation-dissipation theorem of Einstein [66

66. K. Okamoto and S. Kawata, “Radiation force exerted on subwavelength particles near a nanoaperture,” Phys. Rev. Lett. 83(22), 4534–4537 ( 1999). [CrossRef]

], here κis the viscosity of the ambient (in our case, for water [67

67. J. Kestin, M. Sokolov, and W. A. Wakeham, “Viscosity of Liquid Water in the Range - 8°C to 150° C,” J. Phys. Chem. Ref. Data 7, 941–948 ( 1978). [CrossRef]

], κ=7.977×104Pas at T=300K), a is the radius of the particle and kBis the Boltzmann constant. Thus, the magnitude of the Brownian force becomes FB=2.5×103pN. Comparing this value with the values of the scattering and the gradient components of the radiation force in Fig. 6 we can find that both components of the radiation force are much larger than the Brownian force, provided the absolute value of the twist factor is small. We calculate in Fig. 11
Fig. 11 Dependence of the radiation forcesFScatMax, FGrad-xMax andFGrad-zMax induced by a scalar twisted GSM beam on the absolute value of the normalized initial twist factor |μ0/γ| at the real focal plane. FBis the Brownian force
the dependence of the radiation forces FScatMax, FGrad-xMax andFGrad-zMax induced by a scalar twisted GSM beam on the absolute value of the twist factor μ0at the real focal plane. For comparison, Brownian forceFBis also shown in Fig. 11. From Fig. 11, one finds that both scattering force and gradient force decrease as |μ0| increases, which is consistent with Fig. 6. When |μ0|is very large, then the scalar twisted GSM beam may no longer be used for trapping a Rayleigh particle because the radiation forces becomes smaller than the Brownian force. The line Q in Fig. 11 can be regarded as the critical line. Our numerical results show that the dependence of the radiation forces of a twisted EGSM beam on the twist factors is similar to that of a scalar twisted GSM beam (not present here to save space). We illustrate in Fig. 12(a)
Fig. 12 (a) Dependence of the radiation forcesFScatMax, FGrad-xMax andFGrad-zMaxa induced by an electromagnetic twisted GSM beam on the initial degree of polarization at z 1 = 0, (b) dependencies of the radiation forcesFScatMax, FGrad-xMax andFGrad-zMaxon the initial correlation coefficients with ξ=δxx/σxxand δyy=2.5δxx. In Fig. 12 (b) the inset shows the zoomed region where the crossing of the three radiation forces with the Brownian force occurs.
the dependence of the radiation forcesFScatMax, FGrad-xMax andFGrad-zMax induced by an electromagnetic twisted GSM beam on the initial degree of polarization at the real focal plane with σxx=σyy=5mm, δxx=1mm, δyy=2.5mm, and in Fig. 12(b) the dependence of the radiation forcesFScatMax, FGrad-xMax andFGrad-zMaxon the initial correlation coefficients with η=δxx/σxx, δyy=2.5δxx, σxx=σyy=5mm. In calculation of Figs. 12(a) and 12(b), we have chosen the twist factors to be μxx/γxx=0.04, μyy/γyy=0.1 withγxx=[k2δxx4]1/2=0.1m1 and γyy=[k2δyy4]1/2=0.025m1. One finds from Fig. 12(a) that although the radiation force decreases as the degree of polarization increases, it remains larger than the Brownian force. Hence, by tuning the degree of polarization of the input partially coherent beam it is possible to control particle trapping. From Fig. 12(b), we see that the radiation force decreases as the values of the correlation coefficients decrease. If the correlation coefficients are smaller than certain values, the radiation force becomes smaller than the Brownian force, and the particle cannot be trapped. The line Q in Fig. 12(b) also represents critical line. From above discussion, we conclude that it is necessary to choose suitable values of the twist phase, degree of polarization and correlation coefficients of a partially coherent beam for particle trapping.

5. Conclusion

Acknowledgements

Yangjian Cai acknowledges the support by the National Natural Science Foundation of China under Grant No. 10904102, the Foundation for the Author of National Excellent Doctoral Dissertation of PR China under Grant No. 200928 and the Natural Science of Jiangsu Province under Grant No. BK2009114. O. Korotkova’s research is funded by the AFOSR (grant FA 95500810102).

References and links

1.

E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,” Opt. Commun. 25(3), 293–296 ( 1978). [CrossRef]

2.

F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34(3), 301–305 ( 1978). [CrossRef]

3.

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

4.

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]

5.

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 ( 1982). [CrossRef]

6.

Q. S. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67(4), 245–250 ( 1988). [CrossRef]

7.

Y. Cai and S. Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5), 056607 ( 2005). [CrossRef] [PubMed]

8.

J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19(9), 1794–1802 ( 2002). [CrossRef]

9.

N. A. Ansari and M. S. Zubairy, “Second-harmonic generation by a Gaussian Schell-model source,” Opt. Commun. 59(5-6), 385–390 ( 1986). [CrossRef]

10.

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

11.

R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10(1), 95–109 ( 1993). [CrossRef]

12.

R. Simon, K. Sundar, and N. Mukunda, “Twisted Gaussian Schell-model beams. I. Symmetry structure and normal-mode spectrum,” J. Opt. Soc. Am. A 10(9), 2008–2016 ( 1993). [CrossRef]

13.

K. Sundar, R. Simon, and N. Mukunda, “Twisted Gaussian Schell-model beams. II. Spectrum analysis and propagation characteristics,” J. Opt. Soc. Am. A 10(9), 2017–2023 ( 1993). [CrossRef]

14.

R. Simon and N. Mukunda, “Twist phase in Gaussian-beam optics,” J. Opt. Soc. Am. A 15(9), 2373–2382 ( 1998). [CrossRef]

15.

A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11(6), 1818–1826 ( 1994). [CrossRef]

16.

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41(7), 1391–1399 ( 1994). [CrossRef]

17.

K. Sundar, N. Mukunda, and R. Simon, “Coherent-mode decomposition of general anisotropic Gaussian Schell-model beams,” J. Opt. Soc. Am. A 12(3), 560–569 ( 1995). [CrossRef]

18.

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5(3), 331–343 ( 1996). [CrossRef]

19.

R. Simon and N. Mukunda, “Gaussian Schell-model beams and general shape invariance,” J. Opt. Soc. Am. A 16(10), 2465–2475 ( 1999). [CrossRef]

20.

S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(3), 036618 ( 2001). [CrossRef] [PubMed]

21.

J. Serna and J. M. Movilla, “Orbital angular momentum of partially coherent beams,” Opt. Lett. 26(7), 405–407 ( 2001). [CrossRef] [PubMed]

22.

M. J. Bastiaans, “Wigner-distribution function applied to twisted Gaussian light propagating in first-order optical systems,” J. Opt. Soc. Am. A 17(12), 2475–2480 ( 2000). [CrossRef]

23.

P. Östlund and A. T. Friberg, “Radiometry and Radiation Efficiency of Twisted Gaussian Schell-Model Sources,” Opt. Rev. 8(1), 1–8 ( 2001). [CrossRef]

24.

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]

25.

Q. Lin and Y. Cai, “Fractional Fourier transform for partially coherent Gaussian-Schell model beams,” Opt. Lett. 27(19), 1672–1674 ( 2002). [CrossRef] [PubMed]

26.

Y. Cai, Q. Lin, and D. Ge, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams in dispersive and absorbing media,” J. Opt. Soc. Am. A 19(10), 2036–2042 ( 2002). [CrossRef]

27.

Y. Cai and L. Hu, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system,” Opt. Lett. 31(6), 685–687 ( 2006). [CrossRef] [PubMed]

28.

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 ( 2006). [CrossRef]

29.

Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express 15(23), 15480–15492 ( 2007). [CrossRef] [PubMed]

30.

Y. Cai, Q. Lin, and O. Korotkova, “Ghost imaging with twisted Gaussian Schell-model beam,” Opt. Express 17(4), 2453–2464 ( 2009). [CrossRef] [PubMed]

31.

M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3(8), 1227–1238 ( 1986). [CrossRef]

32.

C. Brosseau, Fundamentals of polarized light-a statistical approach (Wiley, New York, 1998).

33.

D. F. V. James, “Changes of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11(5), 1641–1643 ( 1994). [CrossRef]

34.

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

35.

G. P. Agrawal and E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A 17(11), 2019–2023 ( 2000). [CrossRef]

36.

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), 301 ( 2001). [CrossRef]

37.

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]

38.

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

39.

Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 5(5), 453–459 ( 2003). [CrossRef]

40.

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). [CrossRef] [PubMed]

41.

E. Wolf, Introduction to the theory of coherence and polarization of light (Cambridge U. Press, 2007).

42.

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 ( 2004). [CrossRef]

43.

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). [CrossRef]

44.

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4-6), 379–385 ( 2005). [CrossRef]

45.

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]

46.

B. Kanseri and H. C. Kandpal, “Experimental determination of electric cross-spectral density matrix and generalized Stokes parameters for a laser beam,” Opt. Lett. 33(20), 2410–2412 ( 2008). [CrossRef] [PubMed]

47.

M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “The 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]

48.

O. Korotkova, M. Yao, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “The state of polarization of a stochastic electromagnetic beam in an optical resonator,” J. Opt. Soc. Am. A 25(11), 2710–2720 ( 2008). [CrossRef]

49.

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 ( 2008). [CrossRef] [PubMed]

50.

Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B 96(2-3), 499–507 ( 2009). [CrossRef]

51.

A. Ashkin, “Acceleration and trapping of particles by radiation forces,” Phys. Rev. Lett. 24(4), 156–159 ( 1970). [CrossRef]

52.

A. Ashkin, “Trapping of atoms by resonance radiation pressure,” Phys. Rev. Lett. 40(12), 729–732 ( 1978). [CrossRef]

53.

A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11(5), 288–290 ( 1986). [CrossRef] [PubMed]

54.

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5-6), 529–541 ( 1996). [CrossRef]

55.

S. M. Block, L. S. Goldstein, and B. J. Schnapp, “Bead movement by single kinesin molecules studied with optical tweezers,” Nature 348(6299), 348–352 ( 1990). [CrossRef] [PubMed]

56.

C. Day, “Optical trap resolves the stepwise transfer of genetic information from DNA to RNA,” Phys. Today 59(1), 26–27 ( 2006). [CrossRef]

57.

C. H. Chen, P. T. Tai, and W. F. Hsieh, “Bottle beam from a bare laser for single-beam trapping,” Appl. Phys. Lett. 43, 6001–6006 ( 2004).

58.

J. Y. Ye, G. Chang, T. B. Norris, C. Tse, M. J. Zohdy, K. W. Hollman, M. O’Donnell, and J. R. Baker Jr., “Trapping cavitation bubbles with a self-focused laser beam,” Opt. Lett. 29(18), 2136–2138 ( 2004). [CrossRef] [PubMed]

59.

Q. Lin, Z. Wang, and Z. Liu, “Radiation forces produced by standing wave trapping of non-paraxial Gaussian beams,” Opt. Commun. 198(1-3), 95–100 ( 2001). [CrossRef]

60.

Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 ( 2004). [CrossRef] [PubMed]

61.

L. G. Wang, C. L. Zhao, L. Q. Wang, X. H. Lu, and S. Y. Zhu, “Effect of spatial coherence on radiation forces acting on a Rayleigh dielectric sphere,” Opt. Lett. 32(11), 1393–1395 ( 2007). [CrossRef] [PubMed]

62.

C. L. Zhao, Y. Cai, X. Lu, and H. T. Eyyuboğlu, “Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle,” Opt. Express 17(3), 1753–1765 ( 2009). [CrossRef] [PubMed]

63.

E. Wolf, “Coherence and polarization properties of electromagnetic laser modes,” Opt. Commun. 265(1), 60–62 ( 2006). [CrossRef]

64.

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

65.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 ( 1997). [CrossRef]

66.

K. Okamoto and S. Kawata, “Radiation force exerted on subwavelength particles near a nanoaperture,” Phys. Rev. Lett. 83(22), 4534–4537 ( 1999). [CrossRef]

67.

J. Kestin, M. Sokolov, and W. A. Wakeham, “Viscosity of Liquid Water in the Range - 8°C to 150° C,” J. Phys. Chem. Ref. Data 7, 941–948 ( 1978). [CrossRef]

OCIS Codes
(030.1640) Coherence and statistical optics : Coherence
(030.1670) Coherence and statistical optics : Coherent optical effects
(140.7010) Lasers and laser optics : Laser trapping
(260.5430) Physical optics : Polarization

ToC Category:
Optical Trapping and Manipulation

History
Original Manuscript: August 10, 2009
Revised Manuscript: October 7, 2009
Manuscript Accepted: October 28, 2009
Published: November 10, 2009

Citation
Chengliang Zhao, Yangjian Cai, and Olga Korotkova, "Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams," Opt. Express 17, 21472-21487 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-24-21472


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References

  1. E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,” Opt. Commun. 25(3), 293–296 (1978). [CrossRef]
  2. F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34(3), 301–305 (1978). [CrossRef]
  3. P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29(3), 256–260 (1979). [CrossRef]
  4. 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]
  5. A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982). [CrossRef]
  6. Q. S. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67(4), 245–250 (1988). [CrossRef]
  7. Y. Cai and S. Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5), 056607 (2005). [CrossRef] [PubMed]
  8. J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19(9), 1794–1802 (2002). [CrossRef]
  9. N. A. Ansari and M. S. Zubairy, “Second-harmonic generation by a Gaussian Schell-model source,” Opt. Commun. 59(5-6), 385–390 (1986). [CrossRef]
  10. L. Mandel, and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  11. R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10(1), 95–109 (1993). [CrossRef]
  12. R. Simon, K. Sundar, and N. Mukunda, “Twisted Gaussian Schell-model beams. I. Symmetry structure and normal-mode spectrum,” J. Opt. Soc. Am. A 10(9), 2008–2016 (1993). [CrossRef]
  13. K. Sundar, R. Simon, and N. Mukunda, “Twisted Gaussian Schell-model beams. II. Spectrum analysis and propagation characteristics,” J. Opt. Soc. Am. A 10(9), 2017–2023 (1993). [CrossRef]
  14. R. Simon and N. Mukunda, “Twist phase in Gaussian-beam optics,” J. Opt. Soc. Am. A 15(9), 2373–2382 (1998). [CrossRef]
  15. A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11(6), 1818–1826 (1994). [CrossRef]
  16. D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41(7), 1391–1399 (1994). [CrossRef]
  17. K. Sundar, N. Mukunda, and R. Simon, “Coherent-mode decomposition of general anisotropic Gaussian Schell-model beams,” J. Opt. Soc. Am. A 12(3), 560–569 (1995). [CrossRef]
  18. R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5(3), 331–343 (1996). [CrossRef]
  19. R. Simon and N. Mukunda, “Gaussian Schell-model beams and general shape invariance,” J. Opt. Soc. Am. A 16(10), 2465–2475 (1999). [CrossRef]
  20. S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(3), 036618 (2001). [CrossRef] [PubMed]
  21. J. Serna and J. M. Movilla, “Orbital angular momentum of partially coherent beams,” Opt. Lett. 26(7), 405–407 (2001). [CrossRef] [PubMed]
  22. M. J. Bastiaans, “Wigner-distribution function applied to twisted Gaussian light propagating in first-order optical systems,” J. Opt. Soc. Am. A 17(12), 2475–2480 (2000). [CrossRef]
  23. P. Östlund and A. T. Friberg, “Radiometry and Radiation Efficiency of Twisted Gaussian Schell-Model Sources,” Opt. Rev. 8(1), 1–8 (2001). [CrossRef]
  24. 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]
  25. Q. Lin and Y. Cai, “Fractional Fourier transform for partially coherent Gaussian-Schell model beams,” Opt. Lett. 27(19), 1672–1674 (2002). [CrossRef] [PubMed]
  26. Y. Cai, Q. Lin, and D. Ge, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams in dispersive and absorbing media,” J. Opt. Soc. Am. A 19(10), 2036–2042 (2002). [CrossRef]
  27. Y. Cai and L. Hu, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system,” Opt. Lett. 31(6), 685–687 (2006). [CrossRef] [PubMed]
  28. Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006). [CrossRef]
  29. Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express 15(23), 15480–15492 (2007). [CrossRef] [PubMed]
  30. Y. Cai, Q. Lin, and O. Korotkova, “Ghost imaging with twisted Gaussian Schell-model beam,” Opt. Express 17(4), 2453–2464 (2009). [CrossRef] [PubMed]
  31. M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3(8), 1227–1238 (1986). [CrossRef]
  32. C. Brosseau, Fundamentals of polarized light-a statistical approach (Wiley, New York, 1998).
  33. D. F. V. James, “Changes of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11(5), 1641–1643 (1994). [CrossRef]
  34. F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998). [CrossRef] [PubMed]
  35. G. P. Agrawal and E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A 17(11), 2019–2023 (2000). [CrossRef]
  36. 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), 301 (2001). [CrossRef]
  37. 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]
  38. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003). [CrossRef]
  39. Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 5(5), 453–459 (2003). [CrossRef]
  40. 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). [CrossRef] [PubMed]
  41. E. Wolf, Introduction to the theory of coherence and polarization of light (Cambridge U. Press, 2007).
  42. O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004). [CrossRef]
  43. 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). [CrossRef]
  44. H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4-6), 379–385 (2005). [CrossRef]
  45. 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]
  46. B. Kanseri and H. C. Kandpal, “Experimental determination of electric cross-spectral density matrix and generalized Stokes parameters for a laser beam,” Opt. Lett. 33(20), 2410–2412 (2008). [CrossRef] [PubMed]
  47. M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “The 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]
  48. O. Korotkova, M. Yao, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “The state of polarization of a stochastic electromagnetic beam in an optical resonator,” J. Opt. Soc. Am. A 25(11), 2710–2720 (2008). [CrossRef]
  49. 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 (2008). [CrossRef] [PubMed]
  50. Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B 96(2-3), 499–507 (2009). [CrossRef]
  51. A. Ashkin, “Acceleration and trapping of particles by radiation forces,” Phys. Rev. Lett. 24(4), 156–159 (1970). [CrossRef]
  52. A. Ashkin, “Trapping of atoms by resonance radiation pressure,” Phys. Rev. Lett. 40(12), 729–732 (1978). [CrossRef]
  53. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11(5), 288–290 (1986). [CrossRef] [PubMed]
  54. Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5-6), 529–541 (1996). [CrossRef]
  55. S. M. Block, L. S. Goldstein, and B. J. Schnapp, “Bead movement by single kinesin molecules studied with optical tweezers,” Nature 348(6299), 348–352 (1990). [CrossRef] [PubMed]
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