Evolution Properties of a Partially Coherent Flat-topped Vortex Hollow Beam Propagating in Uniaxial Crystals Orthogonal to the Optical Axis

Liu
Dajun; Wang
Guiqiu; Luo
Xixian; Yin
Hongming; Wang
Yaochuan

doi:10.3807/JOSK.2016.20.6.686

OA학술지
Journal of the optical society of Korea

Evolution Properties of a Partially Coherent Flat-topped Vortex Hollow Beam Propagating in Uniaxial Crystals Orthogonal to the Optical Axis

DOI : 10.3807/JOSK.2016.20.6.686
Author: Liu Dajun, Wang Guiqiu, Luo Xixian, Yin Hongming, Wang Yaochuan
Publish: Journal of the optical society of Korea Volume 20, Issue6, p686~693, 25 Dec 2016

ABSTRACT

The analytical expressions for a partially coherent flat-topped vortex hollow beam propagating in uniaxial crystals orthogonal to the optical axis are derived, and the intensity and coherent vortex properties of partially coherent flat-topped vortex hollow beam propagation in uniaxial crystals orthogonal to the optical axis are analyzed by numerical examples. The influence of beam order parameter N, topological charge M, the coherence length and the ratio of refractive indices n_e/n_o of uniaxial crystals on the normalized intensity distribution and coherent vortex of a partially coherent flat-topped vortex hollow beam propagating in uniaxial crystals are discussed in detail.

KEYWORD

Partially coherent flat-topped vortex hollow beam , Uniaxial crystal , Laser propagation , Propagation properties

본문

Collapse all

I. INTRODUCTION

Uniaxial crystals have been widely studied in the design of wave plates, polarizers, compensators, and optical modulation devices [1]. The propagation properties of laser beams in uniaxial crystals have been treated by solving the boundary value problems of Maxwell’s equations in uniaxial crystals [2-4]. Since the vectorial theory of laser beam propagation in uniaxial crystals was constructed, propagation of various laser beams [5-18], such as Laguerre-Gauss and Bessel-Gauss beams, Hermite-Gauss beams, dark hollow beams, flat-topped beams, elliptical Gaussian beams, elliptical Gaussian vortex beams and beams generated by a Gaussian mirror resonator in uniaxial crystals, partially coherent flat-topped beams, partially polarized and partially coherent beams, Laguerre-Gaussian correlated Schell model beams, coherent and partially coherent four-petal Gaussian vortex beams, has been widely investigated.

On the other hand, optical vortex beams have been widely studied due to their applications in free-space optical communication. Among the optical vortex beams, the properties of the elliptical Gaussian vortex beam [12], partially coherent four-petal Gaussian vortex beam [18], four-petal Gaussian vortex beam [19], vortex beam carried by an Airy beam [20], vortex beams [21], and partially coherent hollow vortex Gaussian beam [22], have been studied. However, to our knowledge, there have been no reports about partially coherent flat-topped vortex hollow beams propagating in uniaxial crystals orthogonal to the optical axis. And the studies of partially coherent flat-topped vortex hollow beams propagating in birefringent crystals have the potential application in the design of electro-optic devices and in trapping of particles. In this work, we mainly investigate the evolution of the beam’s intensity and vortex distributions of a partially coherent flat-topped vortex hollow beam propagating in uniaxial crystals orthogonal to the optical axis.

II. THEORY ANALYSIS

In our analysis, assume that the beam propagation direction in uniaxial crystals orthogonal to the optical axis is along the z axis, and the optical axis of uniaxial crystals coincides with the x-axis, then the relative dielectric tensor of uniaxial crystals can be expressed as

with n_o and n_e represent the ordinary and extraordinary refractive indices, respectively. In this case, we mainly consider a laser beam propagating in uniaxial crystals orthogonal to the optical axis. Within the framework of paraxial approximation, the components of the laser beam propagating in uniaxial crystals orthogonal to the optical axis can be treated by the following formulas [2-4]:

with k=2π/λ is the wave number, r= (x, y) and r₀ = (x₀, y₀) represent the position vectors at the output and input planes, respectively; E_α (r, z) and E_α (r₀, 0) (α = x, y) represent the electric fields in the output and input planes, respectively. It can be seen from Eqs. (2) that the x component of the laser beam (Eq. (2a)) undergoes an asymmetry diffraction spreading its shape in x and y directions, and the y component of the laser beam (Eq. (2b)) is the same as that which propagates in isotropic media. In this work, we focus on the propagation properties of an x-polarized laser beam in uniaxial crystals orthogonal to the optical axis.

Based on the theory of coherence, the second-order correlation properties of the x-polarized laser beam can be characterized by the cross-spectral density function [23]:

Then the cross-spectral density function of the x-polarized laser beam propagating in uniaxial crystals orthogonal to the optical axis can be obtained as:

where W(r₁, r₂, z) and W(r₁₀, r₂₀, 0) represent the cross-spectral density functions of the laser beam in the output and input planes, respectively.

Assume that the flat-topped vortex hollow beam considered in this work is x-polarized and is incident into uniaxial crystals at the plane z=0, then the flat-topped vortex hollow beam at the source plane z=0 can be written as [24]:

where N is the order of the flat-topped vortex hollow beam, M is the topological charge, w_x and w_y are the beam waist width in x and y directions, respectively. denotes the binomial coefficient; and if w_x = w_y, the beam represents the circular flat-topped vortex hollow beam.

The cross-spectral density function for a partially coherent flat-topped vortex hollow beam by using the coherence theory can be obtained as

with σ is the coherence length.

By utilizing the following formulas [25]:

Substituting Eq. (6) into Eq. (4), we obtain

with

with

Eqs. (10)-(13) are the main analytical results for a partially coherent flat-topped vortex hollow beam propagating in uniaxial crystals orthogonal to the optical axis.

Based on the theory of coherence, the degree of coherence for the laser beam can be expressed as [23]:

The position of coherent vortices for the partially coherent flat-topped vortex hollow beam at the propagation distance z can be expressed as [26]:

where Re and Im are the real and imaginary parts of μ(r₁, r₂, z), respectively.

III. NUMERICAL EXAMPLES

In this section, we study the evolution properties of a partially coherent four-petal Gaussian vortex beam propagating in uniaxial crystals orthogonal to the optical axis. The parameters w_x , n_o and λ in the whole paper are chosen as w_x = 30 μm, n_o = 2.616 (rutile crystal) and λ = 532 nm.

Figs. 1 and 2 show the contour graphs of the normalized intensity for a partially coherent flat-topped vortex hollow beam propagating in uniaxial crystals orthogonal to the optical axis with N = 3, M= 1 and σ = 15 μm for the different n_e/n_o, respectively. From Figs. 1 and 2, it is shown that the beam can keep its original dark hollow center pattern in the short propagation distance (Figs. 1(a) and 2(a)); and the beam will lose its initial dark hollow center pattern as the propagation distance increases (Figs. 1(b) and 2(b)), then the beam will evolve into the desired flat-topped beam (Figs. 2(c) and 2(b)); finally, the beam will evolve into the elliptical Gaussian-like beam (Figs. 1(d) and 2(d)) due to the influence of uniaxial crystals; it is also found that the beam propagating in uniaxial crystals with n_e / n_o = 1.5 spreads faster along the x direction than the y direction due to the refractive index n_e > n_o of uniaxial crystals.

[FIG. 1.] The contour graphs of normalized intensity for partially coherent flat-topped vortex hollow beam propagating in uniaxial crystals with ne / no = 1.1. (a) z = 500 μm, (b) z = 2000 μm, (c) z = 6000 μm, (d) z = 10000 μm.

[FIG. 2.] The contour graphs of normalized intensity for partially coherent flat-topped vortex hollow beam propagating in uniaxial crystals with ne / no = 1.5, (a) z = 500 μm, (b) z = 2000 μm, (c) z = 6000 μm, (d) z = 10000 μm.

In order to investigate the influence of coherence length, Fig. 3 shows the contour graphs of the normalized intensity for fully coherent flat-topped vortex hollow beam propagating in uniaxial crystals with n_e / n_o = 1.1, N = 3, M = 1 and σ = ∞ By comparing Figs. 1-3, it can be seen that the fully coherent beam propagating in uniaxial crystals can keep its initial dark hollow center as the propagating distance increases. And the partially coherent beam evolves into the Gaussian-like beam due to the influence of coherence length.

[FIG. 3.] The contour graphs of normalized intensity for fully coherent flat-topped vortex hollow beam propagating in uniaxial crystals with ne / no = 1.1, (a) z = 500 μm, (b) z = 2000 μm, (c) z = 6000 μm, (d) z = 10000 μm.

From Fig. 4, it can be found that the partially coherent flat-topped vortex hollow beam (N = 2, M= 2) propagating in uniaxial crystals with n_e / n_o = 1.5 have similar evolution properties with the beam with the different M and N (Figs. 1 and 2), and the beam will evolve into a Gaussian-like beam as the beam propagation distance increases.

[FIG. 4.] Figure 4 The contour graphs of normalized intensity for partially coherent flat-topped vortex hollow beam propagating in uniaxial crystals with ne / no = 1.5, (a) z = 500 μm, (b) z = 2000 μm, (c) z = 6000 μm, (d) z = 10000 μm.

Figs. 5 and 6 give the curves of Re μ(r₁, r₂, z) = 0 and Im μ(r₁, r₂, z) = 0 for a partially coherent flat-topped vortex hollow beam propagating in uniaxial crystals orthogonal to the optical axis with r₂ = (2 μm, 5 μm), N = 2 and M= 1 for the different n_e / n_o, respectively. From Figs. 5(a) and 6(a), it can be seen that the beam propagating in uniaxial crystals at the distance z = 500 μm has a coherent vortex, and as the propagation distance increases, the beam at the distance z = 1000 μm has more coherent vortices than for the propagation distance z = 500 μm. Then the partially coherent flat-topped vortex hollow beam will have the changed number of coherent vortices during the beam propagation in uniaxial crystals. And we also found that the beam propagating in uniaxial crystals with higher n_e / n_o becomes narrower in the x direction than in the y direction.

[FIG. 5.] The curves of Re μ = 0 and Im μ = 0 for partially coherent flat-topped vortex hollow beam propagating in uniaxial crystal with ne / no = 1.1, (a) z = 500 μm, (b) z = 1000 μm.

[FIG. 6.] The curves of Re μ = 0 and Im μ = 0 for partially coherent flat-topped vortex hollow beam propagating in uniaxial crystal with ne / no = 1.5, (a) z = 500 μm, (b) z = 1000 μm.

Fig. 7 gives the curves of Re μ(r₁, r₂, z) = 0 and Im μ(r₁, r₂, z) = 0 for partially coherent flat-topped vortex hollow beam propagating in uniaxial crystals with n_e / n_o = 1.1, r₂ = (2 μm, 5 μm), N = 3 and M = 2. We found that the beam has similar evolution properties, the beam will have a changed number of coherent vortices with the propagation distance increasing.

[FIG. 7.] The curves of Re μ = 0 and Im μ = 0 for partially coherent flat-topped vortex hollow beam propagating in uniaxial crystal with ne / no = 1.1, (a) z = 500 μm, (b) z = 1000 μm.

Fig. 8 gives the contour graphs of the normalized intensity for a partially coherent elliptical flat-topped vortex hollow beam propagating in uniaxial crystals with n_e / n_o = 1.1, w_y = 40 μm N = 3, M= 1 and σ = 15 μm. We found that the elliptical beam has the similar evolution properties with the circular beam with the propagation distance increasing, and the beam will evolve into the elliptical Gaussian-like beam.

[FIG. 8.] The contour graphs of normalized intensity for partially coherent elliptical flat-topped vortex hollow beam propagating in uniaxial crystals with ne / no = 1.1, (a) z = 500 μm, (b) z = 2000 μm, (c) z = 6000 μm, (d) z = 10000 μm.

IV. CONCLUSION

In this paper, the analytical expressions of a partially coherent flat-topped vortex hollow beam propagating in uniaxial crystals is derived and analyzed. We found that the partially coherent flat-topped vortex hollow beam will keep its original dark hollow center pattern in the short propagation distance, and the beam will evolve into an elliptical Gaussian-like beam with the propagation distance increasing due to the influence of uniaxial and the coherence length; and the beam will spread faster in the x direction than in the y direction with the larger n_e / n_o. And the partially coherent flat-topped vortex hollow beam will have a changed number of coherent vortices as the propagation distance increases.

참고문헌

1. Born M., Wolf E. 1999 Principle of Optics
2. Ciattoni A., Crosignani B., Di Porto P. (2001) “Vectorial theory of propagation in uniaxially anisotropic media,” [J. Opt. Soc. Am. A] Vol.18 P.1656-1661
3. Ciattoni A., Cincotti G., Palma C. (2002) “Nonparaxial description of reflection and transmission at the interface between an isotropic media and a uniaxial crystal,” [J. Opt. Soc. Am. A] Vol.19 P.1422-1431
4. Ciattoni A., Palma C. (2003) “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” [J. Opt. Soc. Am. A] Vol.20 P.2163-2171
5. Cincotti G., Ciattoni A., Palma C. (2002) “Laguerre-Gauss and Bessel-Gauss beams in uniaxial crystals,” [J. Opt. Soc. Am. A] Vol.19 P.1680-1688
6. Cincotti G., Ciattoni A., Palma C. (2001) “Hermite-Gauss beams in uniaxially anisotropic crystals,” [IEEE J. Quantum Electron.] Vol.37 P.1517-1524
7. Deng D., Yu H., Xu S., Shao J., Fan Z. (2008) “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” [Opt. Commun.] Vol.281 P.202-209
8. Liu D., Zhou Z. (2008) “Various dark hollow beams propagating in uniaxial crystals orthogonal to the optical axis,” [J. Opt. A: Pure Appl. Opt.] Vol.10 P.095005
9. Lu B., Luo S. (2004) “Propagation properties of three-dimensional flattened Gaussian beams in uniaxially anisotropic crystals,” [Opt. Laser Technol.] Vol.36 P.51-56
10. Liu D., Zhou Z. (2010) “Propagation and the kurtosis parameter of Gaussian flat-topped beams in uniaxial crystals orthogonal to the optical axis,” [Opt. Laser Eng.] Vol.48 P.58-63
11. Liu D., Wang H., Wang Y., Yin Y. (2015) “Propagation properties of elliptical Gaussian beam in uniaxial crystals along the optical axis,” [Opt. Laser Technol.] Vol.73 P.12-18
12. Wang X., Liu Z., Zhao D. (2014) “Nonparaxial propagation of elliptical Gaussian vortex beams in uniaxial crystal orthogonal to the optical axis,” [J. Opt. Soc. Am. A] Vol.31 P.2268-2274
13. Deng D., Shen J., Tian Y., Shao J., Fan Z. (2007) “Propagation properties of beams generated by Gaussian mirror resonator in uniaxial crystals,” [Optik] Vol.118 P.547-551
14. Liu D., Zhou Z. (2009) “Propagation of Partially Coherent Flat-topped Beams in Uniaxial Crystals Orthogonal to the Optical Axis,” [J. Opt. Soc. Am. A] Vol.26 P.924-930
15. Liua D. J., Zhou Z. X. (2009) “Propagation of Partially Polarized, Partially Coherent Beams？in Uniaxial Crystals Orthogonal to the Optical Axis,” [Eur. Phys. J. D] Vol.54 P.95-101
16. Zhu Z., Liu L., Wang F., Cai Y. (2015) “Evolution properties of a Laguerre-Gaussian correlated Schell-model beam propagating in uniaxial crystals orthogonal to the optical axis,” [J. Opt. Soc. Am. A] Vol.32 P.374-380
17. Liu D., Wang H., Wang Y., Yin H. (2015) “Evolution properties of four-petal Gaussian vortex beams propagating in uniaxial crystals orthogonal to the optical axis,” [Eur. Phys. J. D] Vol.69 P.218
18. Liu D., Wang Y., Yin H. (2015) “Evolution properties of partially coherent four-petal Gaussian vortex beams propagating in uniaxial crystals orthogonal to the optical axis,” [J. Opt. Soc. Am. A] Vol.32 P.1683-1690
19. Guo L., Tang Z., Wan W. (2014) “Propagation of four-petal Gaussian vortex beam through a paraxial ABCD optical system,” [Optik] Vol.125 P.5542-5545
20. Dai H., Liu Y., Luo D., Sun X. (2011) “Propagation properties of an optical vortex carried by an Airy beam: experimental implementation,” [Opt. Lett.] Vol.36 P.1617-1619
21. Lukin V., Konyaev P., Sennikov V. (2012) “Beam spreading of vortex beams propagating in turbulent atmosphere,” [Appl. Opt.] Vol.51 P.C84-C87
22. Zhou G., Cai Y., Chu X. (2012) “Propagation of a partially coherent hollow vortex Gaussian beam through a paraxial ABCD optical system in turbulent atmosphere,” [Opt. Express] Vol.20 P.9897-9910
23. Wolf E. (2003) “Unified theory of coherence and polarization of random electromagnetic beams,” [Phys. Lett. A] Vol.312 P.263-267
24. Liu H., Lu Y., Xia J., Pu X., Zhang L. (2015) “Flat-topped vortex hollow beam and its propagation properties,” [J. Opt,] Vol.17 P.075606
25. Jeffrey A., Dai H. 2008 Handbook of Mathematical Formulas and Integrals
26. Gbur G., Visser T. (2003) “Coherence vortices in partially coherent beams,” [Opt. Commun.] Vol.222 P.117-125

이미지 / 테이블

[ ]
[ ]
[ ]
[ ]
[ ]
[ ]
[ ]
[ ]
[ ]
[ ]
[ ]
[ ]
[ ]
[ ]
[ ]
[ ]
[ ]
[ ]
[ ]
[ ]
[ ]
[ ]
[ FIG. 1. ] The contour graphs of normalized intensity for partially coherent flat-topped vortex hollow beam propagating in uniaxial crystals with ne / no = 1.1. (a) z = 500 μm, (b) z = 2000 μm, (c) z = 6000 μm, (d) z = 10000 μm.
[ FIG. 2. ] The contour graphs of normalized intensity for partially coherent flat-topped vortex hollow beam propagating in uniaxial crystals with ne / no = 1.5, (a) z = 500 μm, (b) z = 2000 μm, (c) z = 6000 μm, (d) z = 10000 μm.
[ FIG. 3. ] The contour graphs of normalized intensity for fully coherent flat-topped vortex hollow beam propagating in uniaxial crystals with ne / no = 1.1, (a) z = 500 μm, (b) z = 2000 μm, (c) z = 6000 μm, (d) z = 10000 μm.
[ FIG. 4. ] Figure 4 The contour graphs of normalized intensity for partially coherent flat-topped vortex hollow beam propagating in uniaxial crystals with ne / no = 1.5, (a) z = 500 μm, (b) z = 2000 μm, (c) z = 6000 μm, (d) z = 10000 μm.
[ FIG. 5. ] The curves of Re μ = 0 and Im μ = 0 for partially coherent flat-topped vortex hollow beam propagating in uniaxial crystal with ne / no = 1.1, (a) z = 500 μm, (b) z = 1000 μm.
[ FIG. 6. ] The curves of Re μ = 0 and Im μ = 0 for partially coherent flat-topped vortex hollow beam propagating in uniaxial crystal with ne / no = 1.5, (a) z = 500 μm, (b) z = 1000 μm.
[ FIG. 7. ] The curves of Re μ = 0 and Im μ = 0 for partially coherent flat-topped vortex hollow beam propagating in uniaxial crystal with ne / no = 1.1, (a) z = 500 μm, (b) z = 1000 μm.
[ FIG. 8. ] The contour graphs of normalized intensity for partially coherent elliptical flat-topped vortex hollow beam propagating in uniaxial crystals with ne / no = 1.1, (a) z = 500 μm, (b) z = 2000 μm, (c) z = 6000 μm, (d) z = 10000 μm.