OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 9 — Apr. 25, 2011
  • pp: 8700–8714
« Show journal navigation

Modulation of spectral intensity, polarization and coherence of a stochastic electromagnetic beam

Gaofeng Wu and Yangjian Cai  »View Author Affiliations


Optics Express, Vol. 19, Issue 9, pp. 8700-8714 (2011)
http://dx.doi.org/10.1364/OE.19.008700


View Full Text Article

Acrobat PDF (1459 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

Analytical formula for the cross-spectral density matrix of a stochastic electromagnetic Gaussian Schell-model (EGSM) beam truncated by a circular phase aperture propagating in free space is derived with the help of a tensor method, which provides a reliable and fast way for studying the propagation and transformation of a truncated EGSM beam. Statistics properties, such as the spectral intensity, the degree of coherence, the degree of polarization and the polarization ellipse of a truncated EGSM beam in free space are studied numerically. The propagation factor of a truncated EGSM beam is also analyzed. Our numerical results show that we can modulate the spectral intensity, the polarization, the coherence and the propagation factor of an EGSM beam by a circular phase aperture. It is found that the phase aperture can be used to shape the beam profile of an EGSM beam and generate electromagnetic partially coherent dark hollow or flat-topped beam, which is useful in some applications, such as optical trapping, material processing, free-space optical communications.

© 2011 OSA

1. Introduction

2. Statistics properties of an EGSM beam truncated by a circular phase aperture in free space

Figure 1
Fig. 1 Schematic diagram of an EGSM beam truncated by a phase aperture propagating in free space.
shows the schematic diagram of an EGSM beam truncated by a circular phase aperture whose phase delay is ϕ and radius is a propagating in free space. Within the validity of the paraxial approximation, the propagation of a partially coherent beam truncated by a circular phase aperture in free space can be studied with the help of the following generalized Huygens-Fresnel integral [55

55. 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]

]
W(ρx1,ρy1,ρx2,ρy2,z)=(1λz)2T(x1,y1)T*(x2,y2)W(x1,y1,x2,y2,0)                                      ×exp[ik2z(x12x222x1ρx1+2x2ρx2+ρx12ρx22)]                                      ×exp[ik2z(y12y222y1ρy1+2y2ρx2+ρy12ρy22)]dx1dx2dy1dy2,
(1)
where k=2π/λ is the wave number with λ being the wavelength, W(x1,y1,x2,y2,0) and W(ρx1,ρy1,ρx2,ρy2,z) are the cross-spectral densities in the input and output planes, xi,yi and ρxi,ρyiare the position coordinates in the input and output planes. T(x1,y1) is the transmission function of the phase aperture with T(x1,y1)=exp(iϕ) for x12+y12a2 and T(x1,y1)=1 for x12+y12>a2. T(x1,y1) can be expressed in the following alternative form
T(x1,y1)=1+[exp(iϕ)1]H(x1,y1),
(2)
where H(x1,y1) is the transmission function of a circular hard aperture with H(x1,y1)=1 for x12+y12a2 and H(x1,y1)=0 for x12+y12>a2. Then the termT(x1,y1)T*(x2,y2)in Eq. (1) equals to
T(x1,y1)T*(x2,y2)=1+[exp(iϕ)1]H(x1,y1)+[exp(iϕ)1]H*(x2,y2)                                 +[exp(iϕ)1][exp(iϕ)1]H(x1,y1)H*(x2,y2),
(3)
H(x1,y1)can be expanded as the following finite sum of complex Gaussian functions [56

56. J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83(5), 1752–1756 (1988). [CrossRef]

,57

57. 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]

]
H(x1,y1)=m=1MAmexp[Bma2(x12+y12)],
(4)
whereAm and Bm are the expansion and Gaussian coefficients, which can be obtained by optimization computation, a table ofAmandBmcan be found in Ref [56

56. J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83(5), 1752–1756 (1988). [CrossRef]

]. For a hard aperture, M = 10 assures a very good description of the diffracted beam [56

56. J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83(5), 1752–1756 (1988). [CrossRef]

,57

57. 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]

]. Although the method of complex Gaussian expansion of a circular hard aperture is an approximately method, both numerical results [56

56. J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83(5), 1752–1756 (1988). [CrossRef]

,57

57. 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]

] and experimental results [58

58. F. Wang, Y. Cai, and Q. Lin, “Experimental observation of truncated fractional Fourier transform for a partially coherent Gaussian Schell-model beam,” J. Opt. Soc. Am. A 25(8), 2001–2010 (2008). [CrossRef]

] have shown that this method is a reliable and fast method for treating the propagation of a truncated beam.

After some arrangement, Eq. (1) can be expressed in the following alternative tensor form
W(ρ˜,z)=1λ2(detB˜)1/2T(r1)T*(r2)W(r˜,0)exp[ik2(r˜TB˜1r˜2r˜TB˜1ρ˜+ρ˜TB˜1ρ˜)]dr˜,
(5)
where r˜T=(r1Tr2T)=(x1y1x2y2), ρ˜T=(ρ1Tρ2T)=(ρx1ρy1ρx2ρy2), dr˜=dx1dx2dy1dy2, and
B˜=(zI0I0IzI).
(6)
Equation (3) can be expressed in the following alternative form
T(r1)T*(r2)=1+[exp(iϕ)1]m=1MAmexp(ik2r˜TB˜mr˜)+[exp(iϕ)1]p=1MAp*exp[ik2r˜TB˜pr˜]                     +[exp(iϕ)1][exp(iϕ)1]m=1Mp=1MAmAp*exp[ik2r˜TB˜mpr˜],
(7)
where
B˜m=(2iBmka2I0I0I0I), B˜p=(0I0I0I2iBp*ka2I),    B˜mp=(2iBmka2I0I0I2iBp*ka2I). 
(8)
here I is a 2×2 identity matrix,

The second-order statistical properties of an EGSM beam is generally characterized by the 2 × 2 cross-spectral density (CSD) matrix W(r1,r2,0) specified at any two points with position vectors r1and r2in the source plane with elements [1

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

4

4. 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]

,26

26. 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]

]
Wαβ(r˜,0)=AαAβBαβexp[ik2r˜TM0αβ1r˜],  (α=x,y;β=x,y),
(9)
where Aαis the square root of the spectral density of electric field component Eα, Bαβ=|Bαβ|exp(iφ) is the correlation coefficient between the ExandEyfield components, T stands for vector transposition and M0αβ1is the 4×4 matrix of the form
M0αβ1=(1ik(12σa2+1δαβ2)Iikδαβ2Iikδαβ2I1ik(12σβ2+1δαβ2)I),
(10)
where σα is the r.m.s width of the spectral density along α direction, δxx, δyy and δxy are the r.m.s 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. In Eqs. (9) and (10) as well as in all the formulas below the explicit dependence of the parameters σx, σy, δxx, δyy and δxyon the frequency was omitted for simplicity.

Substituting Eqs. (7) and (9) into Eq. (5), we obtain (after tedious vector integration and tensor operation) the following expression for the elements of the cross-spectral density matrix of the truncated EGSM beam in the output plane
Wαβ(ρ˜,z)=AαAβBαβ{det[I˜+B˜M0αβ1]1/2exp[ik2ρ˜TM1αβ1ρ˜]               +[exp(iϕ)1]m=1MAmdet[I˜+B˜M0αβ1+B˜B˜m]1/2×exp[ik2ρ˜TM2αβ1ρ˜]               +[exp(iϕ)1]p=1MAp*det[I˜+B˜M0αβ1+B˜B˜p]1/2exp[ik2ρ˜TM3αβ1ρ˜]+                [exp(iϕ)1][exp(iϕ)1]m=1Mp=1MAmAp*det[I˜+B˜M0αβ1+B˜B˜mp]1/2exp[ik2ρ˜TM4αβ1ρ˜]},
(11)
where I˜ is a 4×4 identity matrix
M1αβ1=(M0αβ+B˜)1,  M2αβ1=[(M0αβ1+B˜m)1+B˜]1M3αβ1=[(M0εβ1+B˜p)1+B˜]1, M4αβ1=[(M0αβ1+B˜mp)1+B˜]1.
(12)
The spectral density and the degree of polarization of an EGSM beam at point ρ are defined by the expressions [1

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

]
I(ρ,z)=TrW(ρ,ρ,z),
(13)
and
P(ρ,l)=14DetW(ρ,ρ,z)[TrW(ρ,ρ,z)]2.
(14)
The degree of coherence of the EGSM beam at a pair of transverse points ρ1 and ρ2 is defined by the formula [1

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

]

μ(ρ1,ρ2,z)=TrW(ρ1,ρ2,z)TrW(ρ1,ρ1,z)TrW(ρ2,ρ2,z)
(15)

Figure 2
Fig. 2 Normalized spectral intensity (contour graph) and the corresponding cross line (y = 0) of an EGSM beam truncated by a circular phase aperture at several propagation distances in free space.
shows the normalized spectral intensity (contour graph) and the corresponding cross line (y = 0) of an EGSM beam truncated by a circular phase aperture at several propagation distances in free space with a=1mm, δxx=3mm, δyy=3.5mm, ϕ=6.15π/4. One finds from Fig. 2 that the evolution properties of the intensity distribution of the truncated EGSM beam on propagation in free space are quite interesting. The Gaussian beam profile of the EGSM beam disappears gradually on propagation in the near field, and the central intensity decreases gradually on propagation. At certain propagation, electromagnetic partially coherent DH beam can be formed. With the further increase of the propagation distance, the hollow beam profile disappears gradually, and in the far field the truncated EGSM beam retrieves its Gaussian beam profile again. To understand this propagation phenomenon, we start from the fact that the phase of the cross region of the truncated EGSM beam within the circular phase aperture was different from that outside the circular phase aperture, and the total field of the truncated EGSM beam on propagation can be regarded as the superposition of the light from the region within the phase aperture and that from the region outside the phase aperture. There is interference between the light from the region within the phase aperture and that from the region outside the phase aperture due to the phase difference, and we may call this self-interference. Self-interference of a truncated EGSM beam leads to its interesting propagation properties as shown in Fig. 2. Figure 3
Fig. 3 Normalized spectral intensity (cross line y = 0) of a truncated EGSM beam at z = 6m for different values of the phase delay ϕ and correlation coefficients δxx and δyy.
shows the normalized spectral intensity (cross line y = 0) of a truncated EGSM beam at z = 6m for different values of the phase delay ϕ and correlation coefficients δxx and δyy with a=1mm. As shown in Fig. 3, at fixed propagation distance, the beam profile of a truncated EGSM beam varies with the change of the phase delay of the phase aperture or the change of the correlation coefficients δxx and δyy of the input beam. For fixed values of the correlation coefficientsδxx and δyy, we can generate electromagnetic partially coherent DH or FT beam by varying the phase delay of the phase aperture. For fixed value of the phase delay, we can generate electromagnetic partially coherent DH or FT beam by varying the correlation coefficients δxx and δyy of the input beam. From Eq. (13), it is known that the spectral intensity of the truncated EGSM beam is expressed as the sum of two components Wxx(ρ˜,z) and Wyy(ρ˜,z). Wxx(ρ˜,z) and Wyy(ρ˜,z) are closely determined with δxx and δyy, respectively. Different values of δxx and δyylead to different evolution properties of Wxx(ρ˜,z) and Wyy(ρ˜,z). Thus, by varying δxx and δyy, the superposition of Wxx(ρ˜,z) and Wyy(ρ˜,z)varies, and we can modulate the spectral intensity of an truncated EGSM beam through controlling the values of δxx and δyy. What’s more, we know that we can trap the Rayleigh dielectric particle with the refractive index smaller than the ambient by a DH beam [33

33. J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, Vol. 44, E. Wolf, ed., North-Holland, Amsterdam, 2003, pp.119–204.

], and trap the particle with the refractive index larger than the ambient by a Gaussian or FT beam [59

59. C. 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]

]. Thus, our results in this paper may be applied for trapping two kinds of Rayleigh dielectric particles.

Figure 6
Fig. 6 Degree of polarization (cross line y = 0) of a truncated EGSM beam at several propagation distances for different values of the phase delay ϕ.
shows the degree of polarization (cross line y = 0) of a truncated EGSM beam at several propagation distances for different values of the phase delay ϕ with a=1mm, |Bxy|=0.25, φ=π/6, δxx=3mm, δyy=3.5mm, δxy=4mm. The polarization properties the truncated EGSM beam in the source plane are uniform across the source plane. As shown in Fig. 6, after propagation in free space, the initial uniformly polarized EGSM beam becomes non-uniformly polarized. The distribution of the degree of polarization of a truncated EGSM beam on propagation is much different from that of an EGSM beam without truncation, and is closely determined by the phase delay ϕ of the circular phase aperture. To learn about the behavior of the on-axis polarization ellipse of the truncated EGSM beam on propagation. We calculate in Fig. 7
Fig. 7 On-axis polarization ellipse of the truncated EGSM beam at several propagation distances in free space.
the polarization ellipse of the truncated EGSM beam at several propagation distances in free space with a=1mm, φ=π/6,  |Bxy|=0.25, δxx=3mm, δyy=3.5mm,  δxy=4mm, ϕ=6.15π/4. Figure 8
Fig. 8 Polarization ellipse of the truncated EGSM beam at z = 6m for different values of the phase delay ϕ in free space.
shows the on-axis polarization ellipse of the truncated EGSM beam at z = 6m for different values of the phase delay ϕ in free space with a=1mm, φ=π/6, |Bxy|=0.25, δxx=3mm,δyy=3.5mm,δxy=4mm. As shown in Figs. 7 and 8, the evolution properties of the on-axis polarization ellipse of the truncated EGSM beam are quite interesting and different from the evolution properties of an EGSM beam without truncation [14

14. 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]

]. Both the orientation angle and degree of ellipticity vary on propagation, and are closely determined by the phase delay of the circular phase aperture. From above discussions, we can come to conclusion that we can modulate the degree of polarization and the polarization ellipse of a truncated EGSM beam by a circular phase aperture.

3. Propagation factor of an EGSM beam truncated by a circular phase aperture

The propagation factor (also known as the M2-factor) proposed by Siegman [60

60. A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2C14 (1990).

] is a particularly useful property of an optical laser beam, and plays an important role in the characterization of beam behavior on propagation. The definition of M2-factor was extended to the partially coherent beams in [61

61. M. Santarsiero, F. Gori, R. Borghi, G. Cincotti, and P. Vahimaa, “Spreading properties of beams radiated by partially coherent Schell-model sources,” J. Opt. Soc. Am. A 16(1), 106–112 (1999). [CrossRef]

]. The propagation factor of an EGSM beam without truncation has been studied in [20

20. S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express 18(12), 12587–12598 (2010). [CrossRef] [PubMed]

], and the evolution properties of the propagation factor of an EGSM beam in a Gaussian cavity were analyzed in [29

29. S. Zhu and Y. Cai, “M2-factor of a stochastic electromagnetic beam in a Gaussian cavity,” Opt. Express 18(26), 27567–27581 (2010). [CrossRef]

]. In this section, we study the propagation factor of an EGSM beam truncated by a circular phase aperture.

TheM2-factor of a partially coherent beam is defined as follows [29

29. S. Zhu and Y. Cai, “M2-factor of a stochastic electromagnetic beam in a Gaussian cavity,” Opt. Express 18(26), 27567–27581 (2010). [CrossRef]

]
Mx2=4πΔxΔpx,     My2=4πΔyΔpy,
(23)
where
Δx=1J(xx¯)2W(x,y,x,y)dxdy,
(24)
Δy=1J(yy¯)2W(x,y,x,y)dxdy,
(25)
Δpx=1J(pxp¯x)2W˜(px,py,px,py)dpxdpy,
(26)
Δpy=1J(pyp¯y)2W˜(px,py,px,py)dpxdpy,
(27)
and the normalization factor J is given by:
J=W(x,y,x,y)dxdy=W˜(px,py,px,py)dpxdpy,
(28)
W˜is the Fourier transform of W. The mean valuesx¯, p¯ are defined as follows

x¯=1JxW(x,y,x,y)dxdy,
(29)
p¯x=1JpxW˜(px,py,px,py)dpxdpy.
(30)

As shown in section 2, the elements of the cross-spectral density matrix of an EGSM beam truncated by a phase aperture at z = 0 can be expressed as follows
WTαβ(r1,r2,0)=T(r1)T*(r2)Wαβ(r1,r2,0),(α=x,y;β=x,y)
(31)
where T(r1)T*(r2) are given by Eq. (7) and Wαβ(r˜,0) are given by Eq. (9). The trace of the cross-spectral density matrix of the truncated EGSM beam at z = 0 is expressed as
WTr(r1,r2,0)=T(r1)T*(r2)Wxx(r1,r2,0)+T(r1)T*(r2)Wyy(r1,r2,0).
(32)
To calculate the M2-factor of a truncated EGSM beam, we should replace W in Eqs. (24)(30) with WTr(r,r,0). After tedious operation, we obtain the following expression for the M2-factor of a truncated EGSM beam
M2=Mx2=My2=2J(Gxx+Gyy)(Qxx+Qyy),
(33)
where
Gxx=Ax2{2πσ4+[exp(iϕ)1]m=1MAmπ2αxx12+[exp(iϕ)1]p=1MAp*π2αxx22         +[exp(iϕ)1][exp(iϕ)1]m=1Mp=1MAmAp*π2αxx32},
(34)
Gyy=Ay2{2πσ4+[exp(iϕ)1]m=1MAmπ2αyy12+[exp(iϕ)1]p=1MAp*π2αyy22         +[exp(iϕ)1][exp(iϕ)1]m=1Mp=1MAmAp*π2αyy32},
(35)
Qxx=Ax2{π(2βxx0δxx2+1)2(2βxx0δxx21)+(exp(iϕ)1)m=1MAmπ(4βxx0βxx1δxx41)2(βxx0δxx2+βxx1δxx21)2+(exp(iϕ)1)×p=1MAp*π(4βxx0βxx2δxx41)2(βxx0δxx2+βxx2δxx21)2+[exp(iϕ)1][exp(iϕ)1]m=1Mp=1MAmAp*π(4βxx1βxx2δxx41)2(βxx1δxx2+βxx2δxx21)2,
(36)
Qyy=Ay2{π(2βyy0δyy2+1)2(2βyy0δyy21)+(exp(iϕ)1)m=1MAmπ(4βyy0βyy1δyy41)2(βyy0δyy2+βyy1δyy21)2+(exp(iϕ)1)×p=1MAp*π(4βyy0βyy2δyy41)2(βyy0δyy2+βyy2δyy21)2+[exp(iϕ)1][exp(iϕ)1]m=1Mp=1MAmAp*π(4βyy1βyy2δyy41)2(βyy1δyy2+βyy2δyy21)2,
(37)
J=Ax2{2πσ2+[exp(iϕ)1]m=1MAmπαxx1+[exp(iϕ)1]p=1MAp*παxx2      +[exp(iϕ)1][exp(iϕ)1]m=1Mp=1MAmAp*παxx3}+Ay2{2πσ2+[exp(iϕ)1]m=1MAmπαyy1      +[exp(iϕ)1]p=1MAp*παyy2+[exp(iϕ)1][exp(iϕ)1]m=1Mp=1MAmAp*παyy3},
(38)
with
ααβ1=12σ2+Bma2,ααβ2=12σ2+Bp*a2,ααβ3=12σ2+Bma2+Bp*a2,βαβ0=14σ2+12δαβ2,βαβ1=14σ2+12δαβ2+Bma2,βαβ2=14σ2+12δαβ2+Bp*a2.
(39)
With the help of Eqs. (32)(38), we calculate in Fig. 9
Fig. 9 Dependence of the M2-factor of a truncated EGSM beam on the phase delay ϕ.
the dependence of the M2-factor of a truncated EGSM beam on the phase delay ϕ with a=1mm, δxx=1.2mm, δyy=1mm.. Figure 10
Fig. 10 Dependence of the M2-factor of a truncated EGSM beam on the radius a.
shows the dependence of the M2-factor of a truncated EGSM beam on the radius a with ϕ=3, δxx=1.2mm, δyy=1mm. One finds from Figs. 9 and 10 that the self-interference of a truncated EGSM beam also induces the change of its propagation factor, and we can modulate the M2-factor of a truncated EGSM beam by a circular phase aperture with suitable parameters of the phase delay and the radius of the aperture.

4. Conclusion

We have obtained the analytical formula for an EGSM beam truncated by a circular phase aperture propagating in free, and studied the statistics properties the truncated EGSM beam on propagation in free space. We have found that the circular phase aperture can be used to modulate the spectral intensity, the degree of coherence, the degree of polarization and the polarization ellipse of an EGSM beam. Electromagnetic partially coherent DH or FT beam can be formed with the help of the circular phase aperture. The propagation factor of a truncated EGSM beam is also investigated, and it is found that the phase aperture can be used to modulate the propagation factor of an EGSM beam. Our results may be useful in some applications, such as optical trapping, material processing and free-space optical communications.

Acknowledgments

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, the Natural Science 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 and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

References and links

1.

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

2.

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]

3.

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

4.

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]

5.

T. Saastamoinen, J. Turunen, J. Tervo, T. Setälä, and A. T. Friberg, “Electromagnetic coherence theory of laser resonator modes,” J. Opt. Soc. Am. A 22(1), 103–108 (2005). [CrossRef]

6.

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]

7.

H. Wang, X. Wang, A. Zeng, and K. Yang, “Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation,” Opt. Lett. 32(15), 2215–2217 (2007). [CrossRef] [PubMed]

8.

X. Ji, E. Zhang, and B. Lu, “Changes in the spectrum and polarization of polychromatic partially coherent electromagnetic beams in the turbulent atmosphere,” Opt. Commun. 275(2), 292–300 (2007). [CrossRef]

9.

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

10.

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]

11.

L. Pan, M. Sun, C. Ding, Z. Zhao, and B. Lü, “Effects of astigmatism on spectra, coherence and polarization of stochastic electromagnetic beams passing through an astigmatic optical system,” Opt. Express 17(9), 7310–7321 (2009). [CrossRef] [PubMed]

12.

L. Pan, Z. Zhao, C. Ding, and B. Lu, “Effect of polarization on spectral switches in the diffraction of stochastic electromagnetic beams,” Appl. Phys. Lett. 95(18), 181112 (2009). [CrossRef]

13.

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]

14.

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]

15.

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]

16.

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52(11), 1611–1618 (2005). [CrossRef]

17.

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15(3), 353–364 (2005). [CrossRef]

18.

Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 9(12), 1123–1130 (2007). [CrossRef]

19.

H. Wang, X. Wang, A. Zeng, and W. Lu, “Changes in the coherence of quasi-monochromatic electromagnetic Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 276(2), 218–221 (2007). [CrossRef]

20.

S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express 18(12), 12587–12598 (2010). [CrossRef] [PubMed]

21.

O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 2342–2348 (2008).

22.

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]

23.

O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94(4), 681–690 (2009). [CrossRef]

24.

C. Zhao, Y. Cai, and O. Korotkova, “Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams,” Opt. Express 17(24), 21472–21487 (2009). [CrossRef] [PubMed]

25.

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

26.

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]

27.

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

28.

M. Yao, Y. Cai, and O. Korotkova, “Spectral shift of a stochastic electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Commun. 283(22), 4505–4511 (2010). [CrossRef]

29.

S. Zhu and Y. Cai, “M2-factor of a stochastic electromagnetic beam in a Gaussian cavity,” Opt. Express 18(26), 27567–27581 (2010). [CrossRef]

30.

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]

31.

C. Ding, L. Pan, and B. Lu, “Changes in the state of polarization of apertured stochastic electromagnetic modified Bessel-Gauss beams in free-space propagation,” Appl. Phys. B 99(1-2), 307–315 (2010). [CrossRef]

32.

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]

33.

J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, Vol. 44, E. Wolf, ed., North-Holland, Amsterdam, 2003, pp.119–204.

34.

Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006). [CrossRef] [PubMed]

35.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53(11), 1057–1060 (1984). [CrossRef]

36.

N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka, N. Miyanaga, and M. Nakatsuka, “Two-dimensional multi-lens array with circular aperture spherical lens for flat-top irradiation of inertial confinement fusion target,” Opt. Rev. 7(3), 216–220 (2000). [CrossRef]

37.

Y. Cai and S. He, “Partially coherent flattened Gaussian beam and its paraxial propagation properties,” J. Opt. Soc. Am. A 23(10), 2623–2628 (2006). [CrossRef]

38.

F. Wang, Y. Cai, and X. Ma, “Circular partially coherent flattened Gaussian beam,” Opt. Lasers Eng. 49(4), 481–489 (2011). [CrossRef]

39.

Y. Cai and L. Zhang, “Coherent and partially coherent dark hollow beams with rectangular symmetry and paraxial propagation,” J. Opt. Soc. Am. B 23(7), 1398–1407 (2006). [CrossRef]

40.

X. Lu and Y. Cai, “Partially coherent circular and elliptical dark hollow beams and their paraxial propagations,” Phys. Lett. A 369(1-2), 157–166 (2007). [CrossRef]

41.

X. Li, F. Wang, and Y. Cai, “An alternative model for a partially coherent elliptical dark hollow beam,” Opt. Laser Technol. 43(3), 577–585 (2011). [CrossRef]

42.

C. Zhao, Y. Cai, F. Wang, X. Lu, and Y. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. 33(12), 1389–1391 (2008). [CrossRef] [PubMed]

43.

F. Wang and Y. Cai, “Experimental generation of a partially coherent flat-topped beam,” Opt. Lett. 33(16), 1795–1797 (2008). [CrossRef] [PubMed]

44.

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (2009). [CrossRef] [PubMed]

45.

Y. Qiu, Z. Chen, and L. Liu, “Partially coherent dark hollow beams propagating through real ABCD optical systems in a turbulent atmosphere,” J. Mod. Opt. 57(8), 662–669 (2010). [CrossRef]

46.

M. Alavinejad and B. Ghafary, “Turbulence-induced degradation properties of partially coherent flat-topped beams,” Opt. Lasers Eng. 46(5), 357–362 (2008). [CrossRef]

47.

F. Cheng and Y. Cai, “Propagation factor of a truncated partially coherent flat-topped beam in turbulent atmosphere,” Opt. Commun. 284(1), 30–37 (2011). [CrossRef]

48.

Y. Dan and B. Zhang, “Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere,” Opt. Express 16(20), 15563–15575 (2008). [CrossRef] [PubMed]

49.

X. Ji, X. Chen, S. Chen, X. Li, and B. Lü, “Influence of atmospheric turbulence on the spatial correlation properties of partially coherent flat-topped beams,” J. Opt. Soc. Am. A 24(11), 3554–3563 (2007). [CrossRef]

50.

X. Chu, Z. Li, and Y. Wu, “Relay propagation of partially coherent flattened Gaussian beam in turbulent atmosphere,” Opt. Commun. 282(23), 4486–4489 (2009). [CrossRef]

51.

K. Ait-Ameur, “Effects of a phase aperture on the fundamental mode of a hard-apertured cavity,” J. Mod. Opt. 49(7), 1157–1168 (2002). [CrossRef]

52.

V. A. Zenkin, V. R. Kushnir, and L. V. Tarasov, “Phase aperture as means for selecting the lowest transverse mode,” Sov. J. Quantum Electron. 11(2), 243–245 (1981). [CrossRef]

53.

Z. Jiang, Q. Lu, and Z. Liu, “Relative phase shifts of apertured Gaussian beams and transformation of a Gaussian beam through a phase aperture,” Appl. Opt. 36(4), 772–778 (1997). [CrossRef] [PubMed]

54.

G. Zhou, “Far-field structure property of a Gaussian beam diffracted by a phase aperture,” Opt. Commun. 284(1), 8–14 (2011). [CrossRef]

55.

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]

56.

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83(5), 1752–1756 (1988). [CrossRef]

57.

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]

58.

F. Wang, Y. Cai, and Q. Lin, “Experimental observation of truncated fractional Fourier transform for a partially coherent Gaussian Schell-model beam,” J. Opt. Soc. Am. A 25(8), 2001–2010 (2008). [CrossRef]

59.

C. 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]

60.

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2C14 (1990).

61.

M. Santarsiero, F. Gori, R. Borghi, G. Cincotti, and P. Vahimaa, “Spreading properties of beams radiated by partially coherent Schell-model sources,” J. Opt. Soc. Am. A 16(1), 106–112 (1999). [CrossRef]

OCIS Codes
(030.0030) Coherence and statistical optics : Coherence and statistical optics
(050.1220) Diffraction and gratings : Apertures
(140.3300) Lasers and laser optics : Laser beam shaping
(260.5430) Physical optics : Polarization
(350.5500) Other areas of optics : Propagation

ToC Category:
Coherence and Statistical Optics

History
Original Manuscript: February 28, 2011
Revised Manuscript: April 13, 2011
Manuscript Accepted: April 14, 2011
Published: April 19, 2011

Citation
Gaofeng Wu and Yangjian Cai, "Modulation of spectral intensity, polarization and coherence of a stochastic electromagnetic beam," Opt. Express 19, 8700-8714 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-9-8700


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. E. Wolf, Introduction to the theory of coherence and polarization of light (Cambridge U. Press, 2007).
  2. 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]
  3. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003). [CrossRef]
  4. 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]
  5. T. Saastamoinen, J. Turunen, J. Tervo, T. Setälä, and A. T. Friberg, “Electromagnetic coherence theory of laser resonator modes,” J. Opt. Soc. Am. A 22(1), 103–108 (2005). [CrossRef]
  6. 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]
  7. H. Wang, X. Wang, A. Zeng, and K. Yang, “Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation,” Opt. Lett. 32(15), 2215–2217 (2007). [CrossRef] [PubMed]
  8. X. Ji, E. Zhang, and B. Lu, “Changes in the spectrum and polarization of polychromatic partially coherent electromagnetic beams in the turbulent atmosphere,” Opt. Commun. 275(2), 292–300 (2007). [CrossRef]
  9. F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schellmodel sources,” J. Opt. Soc. Am. A 25(5), 1016–1021 (2008). [CrossRef]
  10. 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]
  11. L. Pan, M. Sun, C. Ding, Z. Zhao, and B. Lü, “Effects of astigmatism on spectra, coherence and polarization of stochastic electromagnetic beams passing through an astigmatic optical system,” Opt. Express 17(9), 7310–7321 (2009). [CrossRef] [PubMed]
  12. L. Pan, Z. Zhao, C. Ding, and B. Lu, “Effect of polarization on spectral switches in the diffraction of stochastic electromagnetic beams,” Appl. Phys. Lett. 95(18), 181112 (2009). [CrossRef]
  13. 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]
  14. 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]
  15. 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]
  16. H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52(11), 1611–1618 (2005). [CrossRef]
  17. O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15(3), 353–364 (2005). [CrossRef]
  18. Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 9(12), 1123–1130 (2007). [CrossRef]
  19. H. Wang, X. Wang, A. Zeng, and W. Lu, “Changes in the coherence of quasi-monochromatic electromagnetic Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 276(2), 218–221 (2007). [CrossRef]
  20. S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express 18(12), 12587–12598 (2010). [CrossRef] [PubMed]
  21. O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 2342–2348 (2008).
  22. 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]
  23. O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94(4), 681–690 (2009). [CrossRef]
  24. C. Zhao, Y. Cai, and O. Korotkova, “Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams,” Opt. Express 17(24), 21472–21487 (2009). [CrossRef] [PubMed]
  25. Z. Tong, Y. Cai, and O. Korotkova, “Ghost imaging with electromagnetic stochastic beams,” Opt. Commun. 283(20), 3838–3845 (2010). [CrossRef]
  26. 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]
  27. O. Korotkova, M. Yao, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “State of polarization of a stochastic electromagnetic beam in an optical resonator,” J. Opt. Soc. Am. A 25(11), 2710–2720 (2008). [CrossRef]
  28. M. Yao, Y. Cai, and O. Korotkova, “Spectral shift of a stochastic electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Commun. 283(22), 4505–4511 (2010). [CrossRef]
  29. S. Zhu and Y. Cai, “M2-factor of a stochastic electromagnetic beam in a Gaussian cavity,” Opt. Express 18(26), 27567–27581 (2010). [CrossRef]
  30. 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]
  31. C. Ding, L. Pan, and B. Lu, “Changes in the state of polarization of apertured stochastic electromagnetic modified Bessel-Gauss beams in free-space propagation,” Appl. Phys. B 99(1-2), 307–315 (2010). [CrossRef]
  32. 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]
  33. J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, Vol. 44, E. Wolf, ed., North-Holland, Amsterdam, 2003, pp.119–204.
  34. Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006). [CrossRef] [PubMed]
  35. Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53(11), 1057–1060 (1984). [CrossRef]
  36. N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka, N. Miyanaga, and M. Nakatsuka, “Two-dimensional multi-lens array with circular aperture spherical lens for flat-top irradiation of inertial confinement fusion target,” Opt. Rev. 7(3), 216–220 (2000). [CrossRef]
  37. Y. Cai and S. He, “Partially coherent flattened Gaussian beam and its paraxial propagation properties,” J. Opt. Soc. Am. A 23(10), 2623–2628 (2006). [CrossRef]
  38. F. Wang, Y. Cai, and X. Ma, “Circular partially coherent flattened Gaussian beam,” Opt. Lasers Eng. 49(4), 481–489 (2011). [CrossRef]
  39. Y. Cai and L. Zhang, “Coherent and partially coherent dark hollow beams with rectangular symmetry and paraxial propagation,” J. Opt. Soc. Am. B 23(7), 1398–1407 (2006). [CrossRef]
  40. X. Lu and Y. Cai, “Partially coherent circular and elliptical dark hollow beams and their paraxial propagations,” Phys. Lett. A 369(1-2), 157–166 (2007). [CrossRef]
  41. X. Li, F. Wang, and Y. Cai, “An alternative model for a partially coherent elliptical dark hollow beam,” Opt. Laser Technol. 43(3), 577–585 (2011). [CrossRef]
  42. C. Zhao, Y. Cai, F. Wang, X. Lu, and Y. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. 33(12), 1389–1391 (2008). [CrossRef] [PubMed]
  43. F. Wang and Y. Cai, “Experimental generation of a partially coherent flat-topped beam,” Opt. Lett. 33(16), 1795–1797 (2008). [CrossRef] [PubMed]
  44. Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (2009). [CrossRef] [PubMed]
  45. Y. Qiu, Z. Chen, and L. Liu, “Partially coherent dark hollow beams propagating through real ABCD optical systems in a turbulent atmosphere,” J. Mod. Opt. 57(8), 662–669 (2010). [CrossRef]
  46. M. Alavinejad and B. Ghafary, “Turbulence-induced degradation properties of partially coherent flat-topped beams,” Opt. Lasers Eng. 46(5), 357–362 (2008). [CrossRef]
  47. F. Cheng and Y. Cai, “Propagation factor of a truncated partially coherent flat-topped beam in turbulent atmosphere,” Opt. Commun. 284(1), 30–37 (2011). [CrossRef]
  48. Y. Dan and B. Zhang, “Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere,” Opt. Express 16(20), 15563–15575 (2008). [CrossRef] [PubMed]
  49. X. Ji, X. Chen, S. Chen, X. Li, and B. Lü, “Influence of atmospheric turbulence on the spatial correlation properties of partially coherent flat-topped beams,” J. Opt. Soc. Am. A 24(11), 3554–3563 (2007). [CrossRef]
  50. X. Chu, Z. Li, and Y. Wu, “Relay propagation of partially coherent flattened Gaussian beam in turbulent atmosphere,” Opt. Commun. 282(23), 4486–4489 (2009). [CrossRef]
  51. K. Ait-Ameur, “Effects of a phase aperture on the fundamental mode of a hard-apertured cavity,” J. Mod. Opt. 49(7), 1157–1168 (2002). [CrossRef]
  52. V. A. Zenkin, V. R. Kushnir, and L. V. Tarasov, “Phase aperture as means for selecting the lowest transverse mode,” Sov. J. Quantum Electron. 11(2), 243–245 (1981). [CrossRef]
  53. Z. Jiang, Q. Lu, and Z. Liu, “Relative phase shifts of apertured Gaussian beams and transformation of a Gaussian beam through a phase aperture,” Appl. Opt. 36(4), 772–778 (1997). [CrossRef] [PubMed]
  54. G. Zhou, “Far-field structure property of a Gaussian beam diffracted by a phase aperture,” Opt. Commun. 284(1), 8–14 (2011). [CrossRef]
  55. 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]
  56. J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83(5), 1752–1756 (1988). [CrossRef]
  57. 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]
  58. F. Wang, Y. Cai, and Q. Lin, “Experimental observation of truncated fractional Fourier transform for a partially coherent Gaussian Schell-model beam,” J. Opt. Soc. Am. A 25(8), 2001–2010 (2008). [CrossRef]
  59. C. 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]
  60. A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2C14 (1990).
  61. M. Santarsiero, F. Gori, R. Borghi, G. Cincotti, and P. Vahimaa, “Spreading properties of beams radiated by partially coherent Schell-model sources,” J. Opt. Soc. Am. A 16(1), 106–112 (1999). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited