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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 25 — Dec. 7, 2009
  • pp: 22366–22379
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Partially coherent standard and elegant Laguerre-Gaussian beams of all orders

Fei Wang, Yangjian Cai, and Olga Korotkova  »View Author Affiliations


Optics Express, Vol. 17, Issue 25, pp. 22366-22379 (2009)
http://dx.doi.org/10.1364/OE.17.022366


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Abstract

Partially coherent standard and elegant Laguerre-Gaussian (LG) beams of all orders are introduced as a natural extension of coherent standard and elegant LG beams to the stochastic domain. By expanding the LG modes into a finite sum of Hermite-Gaussian modes, the analytical formulae are obtained for the cross-spectral densities of partially coherent standard and elegant LG beams in the source plane and after passing through paraxial ABCD optical system, based on the generalized Collins integral formula. A comparative study of the propagation properties of the partially coherent standard and elegant LG beams in free space is carried out via a set of numerical examples. Our results indicate that the intensity and spreading properties of partially coherent standard and elegant LG beams are closely related to their initial coherence states, and are very different from the corresponding results for the coherent standard and elegant LG beams. In particular, an elegant LG beam spreads slower than a standard LG beam, while this advantage disappears when their initial coherences are very small. Our results may find applications in connection with laser beam shaping, singular optics and astrophysical measurements of angular momentum of radiation.

© 2009 OSA

1. Introduction

Standard Hermite-Gaussian (HG) and LG modes are eigenmodes of the paraxial wave equation. The Gaussian part of the standard HG or LG modes has a complex argument, but the Hermite or Laguerre part is purely real. Siegman was the first to introduce new HG solutions named elegant HG modes that satisfy the paraxial wave equation but have a more symmetrical form [26

26. A. E. Siegman, “Hermite-Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63(9), 1093–1094 ( 1973). [CrossRef]

]. Later, Takenaka et al. proposed the elegant LG beam as an extension of standard LG beam [27

27. T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation of light beams beyond the paraxial approximate,” J. Opt. Soc. Am. A 2(6), 826–829 ( 1985). [CrossRef]

29

29. S. Saghafi and C. J. R. Sheppard, “Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes,” J. Mod. Opt. 45, 1999–2009 ( 1998). [CrossRef]

]. Relationship between elegant LG and Bessel-Gauss beams was studied in [30

30. M. A. Porras, R. Borghi, and M. Santarsiero, “Relationship between elegant Laguerre-Gauss and Bessel-Gauss beams,” J. Opt. Soc. Am. A 18(1), 177–184 ( 2001). [CrossRef]

]. Elegant LG beams were also proposed as a tool for describing axisymmetric flattened Gaussian beams [31

31. R. Borghi, “Elegant Laguerre-Gauss beams as a new tool for describing axisymmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 18(7), 1627–1633 ( 2001). [CrossRef]

]. Paraxial and non-paraxial propagation of elegant LG beams have been carried out in [32

32. M. A. Bandres and J. C. Gutiérrez-Vega, “Higher-order complex source for elegant Laguerre-Gaussian waves,” Opt. Lett. 29(19), 2213–2215 ( 2004). [CrossRef] [PubMed]

37

37. Z. Mei and J. Gu, “Comparative studies of paraxial and nonparaxial vectorial elegant Laguerre-Gaussian beams,” Opt. Express 17(17), 14865–14871 ( 2009). [CrossRef] [PubMed]

].

Coherence is one important properties of a laser beam. Laser beams with low coherence named partially coherent beams have advantages over coherent beam in many applications, such as free-space optical communication [38

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

], inertial confinement fusion [39

39. 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]

], harmonic generation [40

40. M. S. Zubairy and J. K. McIver, “Second-harmonic generation by a partially coherent beam,” Phys. Rev. A 36(1), 202–206 ( 1987). [CrossRef] [PubMed]

,41

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

], optical trapping [42

42. 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]

], optical projection [43

43. G. Dente and J. S. Osgood, “Some observations of the effects of partial coherence on projection system imagery,” Opt. Eng. 22, 720–724 ( 1983).

], photography [44

44. C. Cheng, W. Liu, and W. Gui, “Diffraction halo function of partially coherent speckle photography,” Appl. Opt. 38(32), 6687–6691 ( 1999). [CrossRef] [PubMed]

], optical imaging [45

45. T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93(6), 068103 ( 2004). [CrossRef] [PubMed]

47

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

].

Several attempts were previously made in the literature to extend the coherent LG beams to stochastic domain. Ponamorenko [58

58. S. A. Ponamorenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. 18(1), 150–156 ( 2001). [CrossRef]

] used the LG modes for construction, by means of the coherent mode decomposition [59

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

], of the partially coherent fields carrying separable vortexes. This has led to the class of stochastic fields with the degree of coherence in a form of a modified Bessel function. Later this class of beams was further generalized to dark and anti-dark beams with a degree of coherence being a linear combination of Bessel and modified Bessel beams [60

60. S. A. Ponomarenko, W. Huang, and M. Cada, “Dark and antidark diffraction-free beams,” Opt. Lett. 32(17), 2508–2510 ( 2007). [CrossRef] [PubMed]

].

An alternative way of obtaining a class of partially coherent beams on the basis of the LG modes is to assume that the cross-spectral density function has the Gaussian Schell-model form [59

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

], in which the intensity part is based on the LG modes but the degree of coherence can be chosen simply Gaussian. To our knowledge, only few papers were dedicated to beams belonging to this class, e.g [61

61. T. van Dijk and T. D. Visser, “Evolution of singularities in a partially coherent vortex beam,” J. Opt. Soc. Am. A 26(4), 741–744 ( 2009). [CrossRef]

63

63. Y. Qiu, J. Liu, and Z. Chen, “Propagation properties of radially polarized partially coherent LG(0,1) beams,” Opt. Commun. 282(1), 69–73 ( 2009). [CrossRef]

]. These studies, however, were aimed at investigating issues like dynamics of optical singularities and of radial polarization and discussed only particular representatives of this rich beam class. No studies were performed so far on partially coherent elegant LG beams.

To our knowledge, up to now, partially coherent standard or elegant LG beam of all orders has not been formulated. Thus it is of practical importance to formulate partially standard or elegant LG beam of all orders and study its propagation properties. Partially coherent vortex beams [61

61. T. van Dijk and T. D. Visser, “Evolution of singularities in a partially coherent vortex beam,” J. Opt. Soc. Am. A 26(4), 741–744 ( 2009). [CrossRef]

,62

62. T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47(3), 036002 ( 2008). [CrossRef]

] and partially coherent LG(0,1) beams [63

63. Y. Qiu, J. Liu, and Z. Chen, “Propagation properties of radially polarized partially coherent LG(0,1) beams,” Opt. Commun. 282(1), 69–73 ( 2009). [CrossRef]

] can be regarded as special cases of partially coherent LG beams. Recent research has shown that partially coherent beams with special profiles have advantages over corresponding coherent beams for reducing turbulence-induced intensity fading in laser communication systems [50

50. X. Ji, X. Chen, and B. Lu, “Spreading and directionality of partially coherent Hermite-Gaussian beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 25(1), 21–28 ( 2008). [CrossRef]

,52

52. H. T. Eyyuboğlu and Y. Baykal, “Transmittance of partially coherent cosh-Gaussian, cos-Gaussian and annular beams in turbulence,” Opt. Commun. 278(1), 17–22 ( 2007). [CrossRef]

,55

55. H. T. Eyyuboğlu, “Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. 40(1), 156–166 ( 2008). [CrossRef]

57

57. G. Zhou and X. Chu, “Propagation of a partially coherent cosine-Gaussian beam through an ABCD optical system in turbulent atmosphere,” Opt. Express 17(13), 10529–10534 ( 2009). [CrossRef] [PubMed]

,62

62. T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47(3), 036002 ( 2008). [CrossRef]

], and for optical trapping [42

42. 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]

]. We expect to apply partially coherent standard or elegant LG beam of all orders to free-space optical communication and optical trapping.

The goal of this paper is to establish the unified theoretical model for both standard and elegant stochastic LG beams of all orders (with LG intensity distribution and Gaussian degree of coherence). The paper is organized as follows. We will first derive the formulas for the cross-spectral density function of the new class of beams in the source plane and in a transverse plane after passing through a linear optical ABCD system (Section 2). We will then carefully study, via numerous examples, the free-space propagation of standard and elegant partially coherent LG beams, which is just a particular case of the general passage of a beam through the ABCD system (Section 3). Finally, the summary of our results will be given (Section 4).

2. Theory

Let us begin by recalling that the electric field of a standard or elegant LG beam in the plane of the source, z = 0, is expressed as follows [1

1. A. E. Siegman, Lasers (Mill Valley, CA: University Science Books, 1986)

35

35. Z. Mei and D. Zhao, “The generalized beam propagation factor of truncated standard and elegant Laguerre-Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6(11), 1005–1011 ( 2004). [CrossRef]

]
Epl(r,φ;0)=(qrω0)lLpl(q2r2ω02)exp(r2ω02)exp(ilφ),
(1)
where r and φ are the radial and azimuthal (angle) coordinates, Lpldenotes the Laguerre polynomial with mode orders p and l, ω0 is the beam width of the fundamental Gaussian mode. For q=2, Eq. (1) reduces to the electric field of a standard LG beam; for q=1, it gives the electric field of an elegant LG beam; also for p=0 and l=0, Eq. (1) degenerates to the electric field of a fundamental Gaussian beam.

By use of the following relation between an LG mode and an HG mode [64

64. K. Sidoro and R. E. Luis, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2563–2567 ( 1993).

]
eilφρlLpl(ρ2)=(1)p22p+lp!m=0pn=0lin(pm)(ln)H2m+ln(x)H2p2m+n(y),
(2)
withH(x) being the Hermite polynomial,(pm)and (ln)being binomial coefficients, Eq. (1) can be expressed in following alternative form in Cartesian coordinates

Epl(x,y;0)=(1)p22p+lp!m=0ps=0lis(pm)(ls)H2m+ls(qxω0)H2p2m+s(qyω0)exp(x2+y2ω02).
(3)

We will now extend standard and elegant LG beams to the partially coherent case. The second-order statistical properties of a partially coherent beam are generally characterized by the cross-spectral density W(x1,y1,x2,y2;z)=E*(x1,y1;z)E(x2,y2;z), where denotes the ensemble average and “*” is the complex conjugate [59

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

]. The intensity distribution, of a partially coherent beam at any position (x, y) in plane z, z ≥ 0, can be determined from the relation I(x,y;z)=W(x,y,x,y;z). For a partially coherent beam generated by a Schell-model source (at z = 0), the cross-spectral density can be expressed in the following well-known form [59

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

]
W(x1,y1,x2,y2;0)=I(x1,y1;0)I(x2,y2;0)g(x1x2;y1y2;0),
(4)
where g(x1x2;y1y2;0) is the spectral degree of coherence which we will assume to have Gaussian profile, i.e.
g(x1x2;y1y2;0)=exp[(x1x2)2+(y1y2)22σg2],
(5)
whereσgis the transverse coherence width. We now assume that the intensity distribution of our the Gaussian Schell-model source can be represented by a standard or elegant LG beam. ThenI(xi,yi;0)can be written as

I(x,y;0)=|(1)p22p+lp!m=0ps=0lis(pm)(ls)H2m+ls(qxω0)H2p2m+s(qyω0)exp(x2+y2ω02)|2.
(6)

Substituting Eq. (6) into Eq. (4), we can express the cross-spectral density of a partially coherent standard or elegant LG beam as follows

W(x1,y1,x2,y2;0)=124p+2l(p!)2m=0pn=0lh=0ps=0l(in)*is(pm)(ln)(ph)(ls)                               ×H2m+ln(qx1ω0)H2h+ls(qx2ω0)H2p2m+n(qy1ω0)H2p2h+s(qy2ω0)                               ×exp(x12+y12+x22+y22ω02)exp((x1x2)2+(y1y2)22σg2).
(7).

Under the condition of σg, Eq. (7) reduces to the expression for a coherent standard or elegant LG beam. Under the condition of p=0 and l=0, Eq. (7) reduces to the expression for a partially coherent Gaussian Schell-model beam [65

65. 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]

70

70. E. Tervonen, A. T. Friberg, and J. Turunen, “Gaussian Schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A 9(5), 796–803 ( 1992). [CrossRef]

]. By transforming Eq. (1) into Eq. (3) with the help of Eq. (2) and expressing the cross-spectral density of a partially coherent standard or elegant LG beam of all orders in the form of Eq. (7) in the Cartesian coordinates, we can obtain analytical propagation formula for the cross-spectral density and analytical expression for the effective beam size of a partially coherent standard or elegant LG beam in an easy way as shown later. In the cylindrical coordinates, it is very difficult for us to obtain analytical propagation formula of a partially coherent standard or elegant LG beam.

Now we study the propagation of beams generated by a partially coherent standard or elegant LG source (7) through a paraxial ABCD optical system. Within the validity of the paraxial approximation, the propagation of the cross-spectral density of a partially coherent beam through an aligned ABCD optical system in free space can be studied with the help of the following generalized Collins formula [49

49. Y. Cai and C. Chen, “Paraxial propagation of a partially coherent Hermite-Gaussian beam through aligned and misaligned ABCD optical systems,” J. Opt. Soc. Am. A 24(8), 2394–2401 ( 2007). [CrossRef]

,71

71. 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]

]
W(u1,v1,u2,v2,z)=(1λ|B|)2W(x1,y1,x2,y2;0)          ×exp[ik2B*(A*x122x1u1+D*u12)ik2B*(A*y122y1v1+D*v12)]          ×exp[ik2B(Ax222x2u2+Du22)+ik2B(Ay222y2v2+Dy22)]dx1dx2dy1dy2,
(8)
wherexi,yi and ui,vi are the position coordinates in the input and output planes, A,B,C and D are the transfer matrix elements of optical system, k=2π/λ is the wave number withλbeing the wavelength.

Substituting from Eq. (7) into Eq. (8), we obtain the formula
W(u1,v1,u2,v2,z)=(1λ|B|)2124p+2l(p!)2m=0pn=0lh=0ps=0l(in)*is(pm)(ln)(ph)(ls)                           H2m+ln(qx1ω0)H2h+ls(qx2ω0)exp(x12+x22ω02)exp((x1x2)22σg2)                             ×exp[ik2B*(A*x122x1u1+D*u12)+ik2B(Ax222x2u2+Du22)]dx1dx2                         H2p2m+n(qy1ω0)H2p2h+s(qy2ω0)exp(y12+y22ω02)exp((y1y2)22σg2)                           ×exp[ik2B*(A*y122y1v1+D*v12)+ik2B(Ay222y2v2+Dy22)]dy1dy2.
(9)
After integration over x1,x2,y1,y2, Eq. (9) becomes
W(u1,u2,v1,v2,z)=(1λ|B|)2124p+2l(p!)2π2M1M2(12M2q22M1M2ω02)(2p+l)/2(2qω0)2p+l                        ×exp[ikD*2B*u12+ikD2Bu22)]exp(k2u224M1B2)exp[k24M2(u1B*u22M1σg2B)2]                        ×exp[ikD*2B*v12+ikD2Bv22)]exp(k2v224M1B2)exp[k24M2(v1B*v22M1σg2B)2]                        ×m=0pn=0lc1=0[(2m+ln)/2]e1=0[(2p2m+n)/2]h=0ps=0ld=02h+lsc2=0[d/2]d1=02p2h+se2=0[d1/2](in)*is(pm)(ln)                        ×(ph)(ls)(2h+lsd)(2p2h+sd1)(1)c1+c2+e1+e2(2m+ln)!c1!(2m+ln2c1)!                        ×d!c2!(d2c2)!(2p2m+n)!e1!(2p2m+n2e1)!d1!e2!(d12e2)!(2i)2c1+2c2+2e1+2e2dd12pl                        ×(1M2)d+d12c12c22e12e2(2qω0)2c12e1(2q2σg2M12ω02q2M1)d+d12c22e2                        ×H2h+lsd(iqku22BM12ω02q2M1)H2m+ln+d2c12c2(ku24M1M2σg2Bku12M2B*)                        ×H2p2h+sd1(iqkv22BM12ω02q2M1)H2p2m+n+d12e12e2(kv24M1M2σg2Bkv12M2B*),
(10)
with M1=1/ω02+1/(2σg2)ikA/(2B),M2=1/ω02+1/(2σg2)+ikA*/(2B*)1/(4M1σg4). In above derivations, we have used the following integral and expansion formulae [72

72. A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

,73

73. M. Abramowitz, and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U. S. Department of Commerce, 1970).

]

exp[(xy)2]Hn(ax)dx=π(1a2)n2Hn(ay(1a2)1/2)
(11)
xnexp[(xβ)2]dx=(2i)nπHn(iβ),
(12)
Hn(x+y)=12n/2k=0n(nk)Hk(2x)Hnk(2y),
(13)
Hn(x1)=m=0[n/2](1)mn!m!(n2m)!(2x1)n2m.
(14)

Thus, Eq. (10) is the analytical formulae for the cross-spectral density of a partially coherent standard or elegant LG beam passing a paraxial ABCD optical system. The intensity of the partially coherent standard or elegant LG beam at the output plane can be obtained by setting, in Eq. (10), u1=u2andv1=v2.

The effective beam size is a useful parameter for characterizing the spreading properties of a beam. According to [74

74. W. H. Carter, “Spot size and divergence for Hermite-Gaussian beams of any order,” Appl. Opt. 19(7), 1027–1029 ( 1980). [CrossRef] [PubMed]

], by use of twice the variance of x or y, the effective beam size of a partially coherent standard or elegant LG beam at plane z is defined as

Wsz(z)=2s2I(x,y,z)dxdyI(x,y,z)dxdy      (s=x,y) .
(15)

On substituting from Eq. (10) into Eq. (15), we obtain the following expression for the effective beam size of a partially coherent standard or elegant LG beam after propagation
Wxz=Wyz=A1(z)A2(z)
(16)
where
A1(z)=(1λ|B|)2124p+2l(p!)2π3M1M2M3(12M2q22M1M2ω02)(2p+l)/2(2qω0)2p+l         ×m=0pn=0lc1=0[(2m+ln)/2]e1=0[(2p2m+n)/2]h=0ps=0ld=02h+lsc2=0[d/2]d1=02p2h+se2=0[d1/2]f1=0[(2h+lsd)/2]         ×f2=0[(2m+ln+d2c12c2)/2]g1=0[(2p2h+sd1)/2]g2=0[(2p2m+n+d12e12c2)/2](in)*is(pm)(ln)(ph)         ×(ls)(2h+lsd)(2p2h+sd1)(1)c1+c2+e1+e2+f1+f2+g1+g2(2m+ln)!c1!(2m+ln2c1)!d!c2!(d2c2)!         ×(2p2m+n)!e1!(2p2m+n2e1)!d1!e2!(d12e2)!(2m+ln+d2c12c2)!f2!(2m+ln+d2c12c22f2)!         ×(2h+lsd)!f1!(2h+lsd2f1)!(2p2m+n+d12e12e2)!g2!(2p2m+n+d12e12e22g2)!(2p2h+sd1)!g1!(2p2h+sd12g1)!         ×(2i)4c1+4c2+4e1+4e22f12f22g12g2dd16p3l2(1M2)d+d12c12c22e12e2(2qω0)2c12e1         ×(2q2σg2M12ω02q2M1)d+d12c22e2(2iqk2BM12ω02q2M1)2h+lsd2f1         ×(k2M1M2σg2BkM2B*)2m+ln+d2c12c22f2(2iqk2BM12ω02q2M1)2p2h+sd12g1         ×(k2M1M2σg2BkM2B*)2p2m+n+d12e12e22g2(1M3)2p+lc1c2f1f2e1e2g1g2+1         ×H2h+2m+2lsn2c12c22f12f2+2(0)H4p2h2m+s+n2e12e22g12g2(0),                                       
(17)
and
A2(z)=(1λ|B|)2124p+2l(p!)2π3M1M2M3(12M2q22M1M2ω02)(2p+l)/2(2qω0)2p+l               ×m=0pn=0lc1=0[(2m+ln)/2]e1=0[(2p2m+n)/2]h=0ps=0ld=02h+lsc2=0[d/2]d1=02p2h+se2=0[d1/2]f1=0[(2h+lsd)/2]            ×f2=0[(2m+ln+d2c12c2)/2]g1=0[(2p2h+sd1)/2]g2=0[(2p2m+n+d12e12c2)/2](in)*is(pm)(ln)(ph)(ls)           ×(2h+lsd)(2p2h+sd1)(1)c1+c2+e1+e2+f1+f2+g1+g2(2m+ln)!c1!(2m+ln2c1)!d!c2!(d2c2)!           ×(2p2m+n)!e1!(2p2m+n2e1)!d1!e2!(d12e2)!(2m+ln+d2c12c2)!f2!(2m+ln+d2c12c22f2)!           ×(2h+lsd)!f1!(2h+lsd2f1)!(2p2m+n+d12e12e2)!g2!(2p2m+n+d12e12e22g2)!(2p2h+sd1)!g1!(2p2h+sd12g1)!           ×(2i)4c1+4c2+4e1+4e2+2g1+2g2+2f1+2f2dd16p3l(1M2)d+d12c12c22e12e2(2qω0)2c12e1           ×(2q2σg2M12ω02q2M1)d+d12c22e2(2iqk2BM12ω02q2M1)2h+lsd2f1          ×(k2M1M2σg2BkM2B*)2m+ln+d2c12c22f2(2iqk2BM12ω02q2M1)2p2h+sd12g1          ×(k2M1M2σg2BkM2B*)2p2m+n+d12e12e22g2(1M3)2p+le1e2g1g2c1c2f1f2          ×H2h+2m+2lsn2c12c22f12f2(0)H4p2h2m+s+n2e12e22g12g2(0),                                             
(18)
with M3=k24M1B2+k24M2(1B*12M1σg2B)2+ikD2BikD*2B*. Equations (10) and (16)-(18) are the main analytical results of the paper. Although Eqs. (10) and (16)-(18) involve up to 14 nested sums of binomial coefficients and Hermite function, it takes only several minutes to calculate each line of Figs. 1
Fig. 1 Normalized intensity distribution (cross line v = 0) of a partially coherent standard LG beam for different values of the initial coherence width σg with p = 1, l = 1 at several propagation distances in free space. (a) z = 0, (b) z = 3m, (c) z = 10m, (d) z = 30m
-4
Fig. 4 Effective beam sizes of partially coherent standard LG beams and elegant LG beams versus the propagation distance z in free space for different values of mode orders p and l and coherence widthσg
. If we want to calculate each line of Figs. 1-4 by Eqs. (7) and (8) through direct numerical integration, it is very time consuming due to four integrals, usually its take several hours or more to do this. More comparisons between calculation by analytical formula and by direct numerical calculation can be found in [75

75. 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]

].

3. Propagation properties of partially coherent standard and elegant LG beams in free space

In this section we will carry out a comparative study of the properties of partially coherent standard and elegant LG beams propagating in free space by using the formulae derived in Section 2.

The ray transfer matrix relating to free-space propagation between the source plane (z = 0) and output plane (z ≥ 0) takes the form

(ABCD)=(1z01).
(19)

For a convenient comparison, we calculate in Fig. 2
Fig. 2 Normalized intensity distribution (cross line v = 0) of a partially coherent elegant LG beam for different values of the initial coherence width σg with p = 1, l = 1 at several propagation distances in free space. (a) z = 0, (b) z = 3m, (c) z = 10m, (d) z = 30m
the normalized intensity distribution (cross line v = 0) of a partially coherent elegant LG beam for different values of the initial coherence width σg at several propagation distances in free space with p=1, l=1,ω0=2mmand λ=632.8nm. We find from Fig. 2 that the propagation properties of an elegant LG beam are also largely determined by its initial degree of coherence. The beam profile of a coherent elegant LG beam transforms into a dark hollow beam profile in the far field, as expected [27

27. T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation of light beams beyond the paraxial approximate,” J. Opt. Soc. Am. A 2(6), 826–829 ( 1985). [CrossRef]

37

37. Z. Mei and J. Gu, “Comparative studies of paraxial and nonparaxial vectorial elegant Laguerre-Gaussian beams,” Opt. Express 17(17), 14865–14871 ( 2009). [CrossRef] [PubMed]

]. Similar to a partially coherent standard LG beam, the beam profile of a partially coherent elegant LG beam also gradually disappears on propagation and eventually takes a Gaussian shape. As the initial coherence decreases, the conversion from an elegant LG beam into a Gaussian beam occurs more quickly and the beam spreads more rapidly. From Figs. 1 and 2, one comes to the conclusion that by degrading the coherence of a standard or elegant LG beam it possible to perform beam shaping in the far field.

In Fig. 4, we calculate the effective beam sizes of partially coherent standard LG beams and elegant LG beams versus the propagation distance z in free space for different values of mode orders p and l. For the convenience of comparison, in Fig. 4(a)-(c), we have chosen ω0=10.0mm for p=1, l=1,ω0=10.54mm for p=2, l=1,ω0=10.80mm for p=3,l=1,ω0=8.165mm for p=1, l=2,ω0=6.90mm for p=1, l=3, respectively, so that all partially coherent elegant LG beams have the same effective beam sizes on the source plane. In Fig. 4 (d)-(f), we have chosen ω0=10.0mm for p=1, l=1,ω0=8.165mm for p=2, l=1,ω0=7.07mm for p=3, l=1,ω0=8.944mm for p=1, l=2,ω0=8.166mm for p=1, l=3 respectively. It is evident from Fig. 4 that the mode orders p and l of partially coherent standard and elegant LG beams affect their spreading properties strongly when the initial degree of coherence is high. Both partially coherent standard and elegant LG beams spread more rapidly as their mode orders p and l increase when the initial degree of coherence is high (see Fig. 4(a), (b), (d) and (e))). When the initial coherence is small, the partially coherent LG beams, both standard and elegant, with different mode orders exhibit almost the same spreading features (see Fig. 4 (c) and (f)).

4. Summary

We have proposed theoretical model to describe partially coherent standard and elegant LG beams, and have derived the analytical formulae for the cross-spectral densities of such beams propagating through paraxial ABCD optical systems. By numerical examples, we have studied the intensity and spreading properties of partially coherent standard and elegant LG beams in free space, comparatively. We have found that the properties of standard and elegant LG beams on free-space propagation are much different from those pertaining to coherent standard and elegant LG beams. As a general rule, a partially coherent elegant LG beam spreads more slowly in free space than a partially coherent standard LG beam. The advantage of a partially coherent elegant LG beam over a partially coherent standard LG beam disappears when the initial coherence in the source plane is very low. Thus, coherent or partially coherent elegant LG beams have some advantages over the corresponding standard LG beams, the result which can be employed in applications, such as free-space optical communications and remote sensing.

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 and the Natural Science of Jiangsu Province under Grant No. BK2009114. O. Korotkova’s research is funded by the AFOSR (grant FA 95500810102).

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A. A. Ishaaya, N. Davidson, and A. A. Friesem, “Very high-order pure Laguerre-Gaussian mode selection in a passive Q-switched Nd:YAG laser,” Opt. Express 13(13), 4952–4962 ( 2005). [CrossRef] [PubMed]

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V. Jarutis, R. Paskauskas, and A. Stabinis, “Focusing of Laguerre-Gaussian beams by axicon,” Opt. Commun. 184(1-4), 105–112 ( 2000). [CrossRef]

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R. Simon and G. S. Agarwal, “Wigner representation of Laguerre--Gaussian beams,” Opt. Lett. 25(18), 1313–1315 ( 2000). [CrossRef] [PubMed]

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OCIS Codes
(030.1670) Coherence and statistical optics : Coherent optical effects
(140.3300) Lasers and laser optics : Laser beam shaping
(350.5500) Other areas of optics : Propagation

ToC Category:
Coherence and Statistical Optics

History
Original Manuscript: September 23, 2009
Revised Manuscript: November 3, 2009
Manuscript Accepted: November 13, 2009
Published: November 23, 2009

Citation
Fei Wang, Yangjian Cai, and Olga Korotkova, "Partially coherent standard and elegant Laguerre-Gaussian beams of all orders," Opt. Express 17, 22366-22379 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-25-22366


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References

  1. A. E. Siegman, Lasers (Mill Valley, CA: University Science Books, 1986)
  2. N. Matsumoto, T. Ando, T. Inoue, Y. Ohtake, N. Fukuchi, and T. Hara, “Generation of high-quality higher-order Laguerre-Gaussian beams using liquid-crystal-on-silicon spatial light modulators,” J. Opt. Soc. Am. A 25(7), 1642–1651 (2008). [CrossRef]
  3. A. A. Ishaaya, N. Davidson, and A. A. Friesem, “Very high-order pure Laguerre-Gaussian mode selection in a passive Q-switched Nd:YAG laser,” Opt. Express 13(13), 4952–4962 (2005). [CrossRef] [PubMed]
  4. C. Tamm, “Frequency locking of two transverse optical modes of a laser,” Phys. Rev. A 38(11), 5960–5963 (1988). [CrossRef] [PubMed]
  5. C. Tamm and C. Weiss, “Bistability and optical switching of spatial patterns in a laser,” J. Opt. Soc. Am. B 7(6), 1034–1038 (1990). [CrossRef]
  6. M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A 43(9), 5090–5113 (1991). [CrossRef] [PubMed]
  7. Y. Chen, Y. Lan, and S. Wang, “Generation of Laguerre-Gaussian modes in fiber-coupled laser diode end-pumped lasers,” Appl. Phys. B 72, 167–170 (2001).
  8. T. Hasegawa and T. Shimizu, “Frequency-doubled Hermite-Gaussian beam and the mode conversion to the Laguerre–Gaussian beam,” Opt. Commun. 160(1-3), 103–108 (1999). [CrossRef]
  9. H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42(1), 217–223 (1995). [CrossRef]
  10. C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence-induced beam spreading of higher-order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002). [CrossRef]
  11. Y. Gu, O. Korotkova, and G. Gbur, “Scintillation of nonuniformly polarized beams in atmospheric turbulence,” Opt. Lett. 34(15), 2261–2263 (2009). [CrossRef] [PubMed]
  12. T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, “Novel Optical Trap of Atoms with a Doughnut Beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997). [CrossRef]
  13. J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71(4), 549–554 (2000). [CrossRef]
  14. Z. Wang, Z. Zhang, and Q. Lin, “Atom interferometers manipulated through the toroidal trap realized by the interference patterns of Laguerre-Gaussian beams,” Eur. Phys. J. D 53(2), 127–131 (2009). [CrossRef]
  15. X. Zhang, W. Wang, Y. Xie, P. Wang, Q. Kong, and Y. Ho, “Field properties and vacuum electron acceleration in a laser beam of high-order Laguerre-Gaussian mode,” Opt. Commun. 281(15-16), 4103–4108 (2008). [CrossRef]
  16. D. S. Bradshaw and D. L. Andrews, “Interactions between spherical nanoparticles optically trapped in Laguerre-Gaussian modes,” Opt. Lett. 30(22), 3039–3041 (2005). [CrossRef] [PubMed]
  17. J. Arlt, R. Kuhn, and K. Dholakia, “Spatial transformation of Laguerre-Gaussian laser modes,” J. Mod. Opt. 48, 783–787 (2001).
  18. V. Jarutis, R. Paskauskas, and A. Stabinis, “Focusing of Laguerre-Gaussian beams by axicon,” Opt. Commun. 184(1-4), 105–112 (2000). [CrossRef]
  19. R. Simon and G. S. Agarwal, “Wigner representation of Laguerre--Gaussian beams,” Opt. Lett. 25(18), 1313–1315 (2000). [CrossRef] [PubMed]
  20. S. R. Seshadri, “Virtual source for a Laguerre-Gauss beam,” Opt. Lett. 27(21), 1872–1874 (2002). [CrossRef] [PubMed]
  21. G. Cincotti, A. Ciattoni, and C. Palma, “Laguerre-Gauss and Bessel-Gauss beams in uniaxial crystals,” J. Opt. Soc. Am. A 19(8), 1680–1688 (2002). [CrossRef]
  22. S. Orlov and A. Stabinis, “Free-space propagation of light field created by Bessel-Gauss and Laguerre-Gauss singular beams,” Opt. Commun. 226(1-6), 97–105 (2003). [CrossRef]
  23. Y. Cai and S. He, “Propagation of a Laguerre–Gaussian beam through a slightly misaligned paraxial optical system,” Appl. Phys. B 84(3), 493–500 (2006). [CrossRef]
  24. G. Zhou, “Propagation of a vectorial Laguerre–Gaussian beam beyond the paraxial approximation,” Opt. Laser Technol. 40(7), 930–935 (2008). [CrossRef]
  25. C. J. R. Sheppard, “Beam duality, with application to generalized Bessel-Gaussian, and Hermite- and Laguerre- Gaussian beams,” Opt. Express 17(5), 3690–3697 (2009). [CrossRef] [PubMed]
  26. A. E. Siegman, “Hermite-Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63(9), 1093–1094 (1973). [CrossRef]
  27. T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation of light beams beyond the paraxial approximate,” J. Opt. Soc. Am. A 2(6), 826–829 (1985). [CrossRef]
  28. E. Zauderer, “Complex argument Hermite-Gaussian and Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 3(4), 465–469 (1986). [CrossRef]
  29. S. Saghafi and C. J. R. Sheppard, “Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes,” J. Mod. Opt. 45, 1999–2009 (1998). [CrossRef]
  30. M. A. Porras, R. Borghi, and M. Santarsiero, “Relationship between elegant Laguerre-Gauss and Bessel-Gauss beams,” J. Opt. Soc. Am. A 18(1), 177–184 (2001). [CrossRef]
  31. R. Borghi, “Elegant Laguerre-Gauss beams as a new tool for describing axisymmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 18(7), 1627–1633 (2001). [CrossRef]
  32. M. A. Bandres and J. C. Gutiérrez-Vega, “Higher-order complex source for elegant Laguerre-Gaussian waves,” Opt. Lett. 29(19), 2213–2215 (2004). [CrossRef] [PubMed]
  33. Z. Mei, D. Zhao, and J. Gu, “Propagation of elegant Laguerre-Gaussian beams through an annular apertured paraxial ABCD optical system,” Opt. Commun. 240(4-6), 337–343 (2004). [CrossRef]
  34. Z. Mei and D. Zhao, “Propagation of Laguerre-Gaussian and elegant Laguerre-Gaussian beams in apertured fractional Hankel transform systems,” J. Opt. Soc. Am. A 21(12), 2375–2381 (2004). [CrossRef]
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