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Propagation of a Laguerre–Gaussian correlated Schell-model beam beyond the paraxial approximation
Optics Communications 352 (2015) 127–134
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Propagation of a Laguerre–Gaussian correlated Schell-model beam beyond the paraxial approximation Lina Guo a,b, Yahong Chen b, Lin Liu b, Yangjian Cai b,n a
School of Electronics and Information, GuangDong Polytechnic Normal University, Guangzhou, Guangdong 510665, China School of Physical Science and Technology and Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China
b
art ic l e i nf o
a b s t r a c t
Article history: Received 29 March 2015 Received in revised form 28 April 2015 Accepted 29 April 2015 Available online 4 May 2015
Recently, Laguerre–Gaussian correlated Schell-model (LGCSM) beam was introduced theoretically [Mei and Korotkova, Opt. Lett. 38(2), (2013), 91–93] and generated experimentally [Wang et al., Opt. Lett. 38 (11), (2013), 1814–1816]. In this paper, we treat the propagation of a LGCSM beam beyond the paraxial approximation. Based on the generalized Rayleigh–Sommerfeld diffraction integral, analytical expressions for the intensity and spectral degree of coherence of a nonparaxial LGCSM beam propagating in free space are obtained, and the corresponding results of a paraxial LGCSM beam are also derived as a special case. Our numerical results show that the nonparaxial propagation properties of a LGCSM beam are closely related to the initial beam parameters, such as the beam waist width, the coherence width and the mode order. Furthermore, in the far field, flat-topped, hollow intensity profiles can be formed through varying the initial beam parameters, which is different from that of a nonparaxial Gaussian Schell-model beam. & 2015 Elsevier B.V. All rights reserved.
Keywords: Laguerre–Gaussian correlated Schell-model beam Nonparaxial Propagation
1. Introduction Recently, modulating the correlation function of partially coherent beam has become a subject of rapidly growing interest [1– 14]. As an additional freedom, correlation function can be used to modulate light field, such as creation of partially coherent beams with nonconventional correlation functions. Recent studies have shown that partially coherent beams with nonconventional correlation functions have displayed many extraordinary properties, e.g., partially coherent beam with nonuniform correlated function displays the self-focusing property during propagation [5], partially coherent beam with Hermite–Gaussian correlated function displays the self-splitting effect [6] and partially coherent beams with Multi-Gaussian and Laguerre–Gaussian correlated functions exhibit prescribed beam profiles in the far-field [7–14]. Due to these extraordinary properties, partially coherent beams with nonconventional correlation functions have potential applications in nonlinear optics, atom optics and free-space optical communications. Partially coherent beam with Laguerre–Gaussian correlated function (also named Laguerre–Gaussian correlated Schell-model n
Corresponding author. E-mail address: [email protected] (Y. Cai).
http://dx.doi.org/10.1016/j.optcom.2015.04.082 0030-4018/& 2015 Elsevier B.V. All rights reserved.
beam) was firstly introduced in [10] and experimental generated in [11]. It was found that the far-field intensity distribution of a Laguerre–Gaussian correlated Schell-model (LGCSM) beam exhibits ring-shaped beam profile, although it has the same intensity distribution with that of a Gaussian Schell-model beam in the source plane. Partially coherent beam with nonconventional correlation function can be generated through the method based on the modulation of the intensity in the Fourier plane, and we reported experimental generation of a controllable optical cage by focusing a LGCSM beam, which will be useful for trapping particles or atoms in [12]. In [15], we showed that LGCSM beam has advantages over partially coherent beam with conventional correlation function (i.e. Gaussian Schell-model beam) to overcome turbulence-induced degeneration when passing through turbulent atmosphere, which is useful in free-space optical communications. Evolution properties of a LGCSM beam in uniaxial crystals were explored in [16] and the experimental generation of a LGCSM vortex beam was reported in [17]. When a laser beam has a small spot size that is comparable with the wavelength or a large far-field divergence angle, it is usually regarded as a nonparaxial beam [18]. For a partially coherent beam, weak coherence will result in a large divergence angle, which also plays an important role in determining the nonparaxiality of a laser beam [19]. Nonparaxial partially coherent
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beams can be produced from multimode oscillation diode laser or tight focusing of a paraxial partially coherent beam with high numerical aperture [19,20]. Many approaches have been developed to treat the propagation of nonparaxial partially coherent beams, such as generalized Rayleigh–Sommerfeld diffraction integrals, Wigner distribution function, angular spectrum representation and the vectorial moment theory [21–24]. Based on these methods, propagation properties of nonparaxial scalar and vector GSM beams with various beam profiles have been extensively studied in free space [25–34]. Recently, Zhang et al. performed the nonparaxial propagation of a twisted anisotropic Gaussian Schell-model beam based on the generalized Rayleigh– Sommerfeld diffraction integral with the help of a tensor method [35], and they also examined the nonparaxial propagation of a partially coherent beam in uniaxial crystals [36]. However, up to now, most previous researches are confined to nonparaxial properties of a partially coherent beam with conventional correlation function. In this paper, our aim is to study the nonparaxial propagation properties of a LGCSM beam in free space. With the help of the generalized Rayleigh–Sommerfeld integral method, the analytical nonparaxial propagation formula for a LGCSM beam in free space is derived. The evolution properties of the intensity and spectral degree of coherence of a nonparaxial LGCSM beam in free space are illustrated numerically.
⎡ ρ2 + ρ2 (ρ10 − ρ20 )2 ⎤ ⎥ W ρ10 , ρ20 , 0 = exp ⎢− 10 2 20 − ⎢⎣ ⎥⎦ 4σ 0 2δ 02
(
)
⎡ (ρ − ρ )2 ⎤ 10 20 ⎥ L n0 ⎢ ⎢⎣ ⎥⎦ 2δ 02
where ρj0≡(xj0, yj0) (j¼ 1, 2) is the position vector, s0 is the transverse beam waist width of the corresponding Gaussian beam, δ0 is the transverse coherence width, and Ln0 is the generalized Laguerre polynomial with mode orders n and 0, respectively. When n¼ 0, Eq. (1) reduces to a conventional Gaussian Schell-model beam. The spectral degree of coherence of a LGCSM beam is modulated by mode order n and coherence width δ0, and is consists of several side robes with a central bright spot, as shown in Fig. 1. According to the generalized Rayleigh–Sommerfeld diffraction integral, the propagation of CSD of a LGCSM beam at two points (ρ1, z) and (ρ2, z) in the free space can be expressed as [19,37]
(
W ρ1 , ρ2 , z
)
⎛ zk ⎞2 =⎜ ⎟ ⎝ 2π ⎠
∞
∞
∞
∞
∫−∞ ∫−∞ ∫−∞ ∫−∞ W (ρ10 , ρ20 , 0)
exp ⎡⎣ik (R2 − R1) ⎤⎦ R12R22
d2ρ10 d2ρ20 ,
(2)
where k ¼2π/λ is the wave number related to the wavelength
λ, Rj = rj − ρj0 = ρ j20 + ρ j2 − 2ρj0 ⋅ρj + z 2 , (j¼ 1,2), and
(
2. Theory
(1)
)
rj ≡ xj , yj , z is the position vector at the z plane. Under weak
The second-order spatial correlation properties of a partially coherent beam are generally characterized by the cross-spectral density (CSD) in the space-frequency domain. The CSD of a LGCSM beam at a pair of arbitrary points in the source plane at z¼ 0 can be expressed as [10–12]:
nonparaxial approximation, Rj can be retained the zero and firstorder expansions
(
)
R j = rj + ρ j20 − 2ρj ⋅ρj0 /2rj
(3)
with rj = rj . Replacing Rj in the exponential term of Eq. (2) by
Fig. 1. The spectral degree of coherence distribution of a LGCSM beam in the source plane for different values of n with s0 ¼ 1.5λ and δ0 ¼ 5λ.
L. Guo et al. / Optics Communications 352 (2015) 127–134
Eq. (3), and that in denominator term by rj, Eq. (2) can be rewritten as.
(
W ρ1 , ρ2 , z
)
⎛ zk ⎞2 exp ⎡⎣ik (r2 − r1) ⎤⎦ =⎜ ⎟ ⎝ 2π ⎠ r12r22
∞
∞
(
∞
For the convenience of integration, the “sum” and “difference” coordinates are adopted
(5)
Substituting Eqs. (1) and (5) into Eq. (4), we obtained the following alternative form
⎛ zk ⎞2 exp ⎡⎣ik (r2 − r1) ⎤⎦ W ρ1 , ρ2 , z = ⎜ ⎟ ⎝ 2π ⎠ r12r22
)
∞
∞
∞
∞
∫−∞ ∫−∞ ∫∞ ∫−∞ P*
⎛ Δρ0 ⎞ ⎛ Δρ0 ⎞ ⎜ρs0 + ⎟ P ⎜ρs0 − ⎟ g Δρ0 ⎝ 2 ⎠ ⎝ 2 ⎠ ⎡ ik ⎛ Δρ0 ⎞ ik ⎛ Δρ0 ⎞ ⎤ × exp ⎢ ρ1⋅⎜ρs0 + ⎟ − ρ2 ⋅⎜ρs0 − ⎟⎥ 2 ⎠ r2 ⎝ 2 ⎠⎦ ⎣ r1 ⎝
(
)
d2ρs0 d2Δρ0 ,
)
⎡ r1 2 ⎢ ⎛ 2 σ *2 (r1) ρ1 + σ (r2 ) ρ2 r2 k ⎞ × Ln ⎢ −⎜ ⎟ ⎢ ⎝ 2r1 ⎠ σ *2 (r1) + σ 2 (r2 ) + 2δ 2 0 ⎢ ⎣
(
(7)
⎡⎛ Δρ0 ⎞2⎤ ik 1 ⎞⎛ ⎟⎟ ⎜ρs0 + P * ρs0 + Δρ0 /2 = exp ⎢ ⎜⎜ − − ⎟⎥ ⎢⎣ ⎝ 2r1 2 ⎠ ⎥⎦ 4σ02 ⎠ ⎝
(
)
⎡⎛ Δρ0 ⎞2⎤ ik 1 ⎞⎛ ⎟⎟ ⎜ρs0 − − Δρ0 /2 = exp ⎢ ⎜⎜ − ⎟ ⎥. ⎢⎣ ⎝ 2r2 2 ⎠ ⎥⎦ 4σ02 ⎠ ⎝
)
(
)
(
( ) (
)
with σ rj = 1/4σ02 − ik /2rj , (j¼ 1, 2). Eq. (12) is the basic analytical result obtained in this paper, which can be used to describe the intensity and spectral degree of coherence of a nonparaxial LGCSM beam propagating in free space. On placing ρ1 ¼ ρ2 ¼ ρ and r1 ¼ r2 ¼r into Eq. (12), the intensity distribution of a nonparaxial LGCSM beam at the z plane is given by
⎛ kz ⎞2 I (ρ, z ) = 2n − 1 ⎜ ⎟ σ *2σ 2δ 02n+ 2 σ *2 + σ 2 + 2δ 02 ⎝ r2 ⎠ ⎤ ⎡ 2 ⎛ k ⎞2 2δ 0 σ *2 + σ 2 2 ⎥ exp ⎢−⎜ ⎟ ρ ⎢ ⎝ 2r ⎠ σ * 2 + σ 2 + 2δ 2 ⎥ 0 ⎦ ⎣
⎛
⎛ Δρ0 ⎞ 1 P ⎜ρs0 − ⎟= ⎝ 2 ⎠ (λr2 )2
⎛
(
(
(9)
⎞
∫−∞ ∫−∞ P∼* ⎜⎝ λur11 ⎟⎠
⎡ ik ⎛ Δρ0 ⎞ ⎤ 2 exp ⎢ ⎜ρs0 + ⎟⋅u1⎥ d u1 2 ⎠ ⎦ ⎣ r1 ⎝
)
∞
.
(
W ρ1 , ρ2 , z
(
)
)(
I ρ1 , z I ρ2 , z
)
(14)
obtains the following paraxial CSD expression of a LGCSM beam at the z plane,
(10)
⎛ k ⎞2 Wp ρ1 , ρ2 , z = 2n − 1 ⎜ ⎟ σ p*2 σ p2 δ 02n +2 σ p*2 + σ p2 + 2δ 02 ⎝z⎠
(
(
)
−n − 1
)
⎡ ⎛ ⎞⎤ ⎤ ⎢ ⎜ ⎟⎥ ⎥ × exp⎢ − ik ⎜ρ 2 − ρ 2 ⎟⎥ 1 2 ⎥⎦ ⎢ 2z ⎜ ⎟⎥ ⎣ ⎝ ⎠⎦ ⎡ 2⎤ 2 2 * ⎢ ⎛ k ⎞2 σ p ρ1 + σ p ρ2 ⎥ × exp ⎢ ⎜ ⎟ ⎥ 2 2 2 ⎢ ⎝ 2z ⎠ σ p* + σ p + 2δ 0 ⎥ ⎣ ⎦
⎞
⎡ ik ⎛ Δρ0 ⎞ ⎤ 2 exp ⎢ − ⎜ρs0 − ⎟⋅u2 ⎥ d u2 2 ⎠ ⎦ ⎣ r2 ⎝
−1/2
)
(13)
where W(ρ1, ρ2, z) and I(ρj, z) are given by Eqs. (12) and (13), respectively. The paraxial result can be regarded as a special case of the nonparaxial result. Under the condition of paraxial approximation, rj can be expanded into series and keeping the first and second terms, i.e., rj ≈ z + ρ j2 /2z . Replacing the rj in Eq. (12) with z, one
⎡ ⎛ ⎞2 k exp ⎢−⎜ ⎟ σ p*2 ρ12 + σ p2 ρ22 ⎢⎣ ⎝ 2z ⎠
∫−∞ ∫−∞ P∼ ⎜⎝ λur22 ⎟⎠ ∞
)
)
with σ = 1/4σ02 − ik /2r
μ ρ1 , ρ2 , z =
)
∞
−n − 1
)
⎤ ⎡ 2 2 2 ⎥ ⎢ ⎛ k ⎞2 σ * + σ 2 , ρ × Ln ⎢−⎜ ⎟ ⎝ 2r ⎠ σ *2 + σ 2 + 2δ 02 ⎥ ⎥⎦ ⎢⎣
(8)
∼ ∼ their Fourier transform P * (u1/λr1) and P (u2/λr2 ) as follows: ∞
(12)
−1/2
The spectral degree of coherence between two arbitrary points r1 ¼(ρ1, z) and r2 ¼(ρ2, z) of a nonparaxial LGCSM beam at the z plane is determined by [38]
P * ρs0 + Δρ0 /2 and P ρs0 − Δρ0 /2 can be expressed in terms of
⎛ Δρ0 ⎞ 1 P * ⎜ρs0 + ⎟= ⎝ 2 ⎠ (λr1)2
⎥ ⎥ ⎦
(
(6)
⎡ Δρ 2 ⎤ ⎡ Δρ 2 ⎤ 0 ⎥ L n0 ⎢ 0 ⎥ = exp ⎢ − 2δ 02 ⎦ ⎣ 2δ 02 ⎦ ⎣
2⎤
) ⎥⎥,
(
g (Δρ 0 )
(
(
(
with
P ρs0
2⎤ r1 2 2 ⎛ k ⎞2 σ * (r1) ρ1 + r 2 σ (r2 ) ρ2 ⎥ ⎥ +⎜ ⎟ ⎝ 2r1 ⎠ σ *2 (r1) + σ 2 (r2 ) + 2δ 02 ⎥ ⎥ ⎦
(4)
ρ10 + ρ20 , Δρ 0 = ρ10 − ρ20 . 2
−n − 1
)
⎡ ⎢ ⎛ 2 ⎞ r12 2 k ⎞ ⎛⎜ 2 σ (r2 ) ρ22 ⎟⎟ × exp ⎢ −⎜ ⎟ σ * (r1) ρ12 + ⎢ ⎝ 2r1 ⎠ ⎜⎝ 2 r2 ⎠ ⎢ ⎣
∫−∞ ∫−∞ ∫−∞ ∫−∞ W (ρ10 , ρ20 , 0)
d2ρ10 d2ρ20 .
(
⎛ kz ⎞2 W (ρ1 , ρ2 , z ) = 2n − 1 ⎜ ⎟ exp ⎡⎣ik (r2 − r1) ⎤⎦ σ *2 (r1) σ 2 (r2 ) ⎝ r1r2 ⎠ δ 02n +2 σ *2 (r1) + σ 2 (r2 ) + 2δ 02
∞
⎛ ⎡ ik ⎞ ⎛ ik ⎞⎤ ik ik × exp ⎜⎜ ⎢ ρ202 − ρ20 ⋅ρ2 ⎟⎟−⎜ ρ102 − ρ10 ⋅ρ1 ⎟ ⎥ 2 r r 2 r r ⎣ ⎝ ⎠⎦ ⎝ ⎠ 2 2 1 1
ρs0 =
129
(
(
(11)
∼ where P (⋅) is the Fourier transform of P (⋅) and the asterisk denotes the complex conjugate. After tedious but straightforward integral calculations, we obtain the following analytical expression for CSD of a nonparaxial LGCSM beam at the z plane.
)
)
⎡ 2⎤ 2 2 ⎢ ⎛ k ⎞2 σ p* ρ1 + σ p ρ2 ⎥ Ln ⎢−⎜ ⎟ ⎥, 2 2 2 ⎢ ⎝ 2z ⎠ σ p* + σ p + 2δ 0 ⎥ ⎣ ⎦
(
(
where σp = 1/4σ02 − ik /2z
)
−1/2
)
.
(15)
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L. Guo et al. / Optics Communications 352 (2015) 127–134
On placing ρ1 ¼ ρ2 ¼ ρ into Eq. (15), the paraxial intensity of a LGCSM beam at the z plane is given by
Ip (ρ, z ) =
⎛ k ⎞2 2n − 1 ⎜ ⎟ σ p*2 σ p2 δ 02n+ 2
(
σ p*2
σ p2
+ + ⎝z⎠ ⎤ ⎡ 2 2 2 ⎛ k ⎞2 2δ 0 σ p* + σ p 2 ⎥ exp ⎢−⎜ ⎟ ρ ⎢ ⎝ 2z ⎠ σ *2 + σ 2 + 2δ 2 ⎥ p p 0 ⎦ ⎣
(
−n − 1 2δ 02
)
)
⎤ ⎡ 2 2 2 ⎥ ⎢ ⎛ k ⎞2 σ p* + σ p 2 ρ . × Ln ⎢−⎜ ⎟ ⎝ 2z ⎠ σ p*2 + σ p2 + 2δ 02 ⎥ ⎥⎦ ⎢⎣
(
)
(16)
Eq. (16) is identical to Eq. (18) with B ¼z in Ref. [17], which is derived from another alternative method. The spectral degree of coherence of a paraxial LGCSM beam between two arbitrary points r1 ¼(ρ1, z) and r2 ¼(ρ2, z) in the z plane is determined by
(
)
μp ρ1 , ρ2 , z = where Wp(ρ1,
(
Wp ρ1 , ρ2 , z
(
) (
)
Ip ρ1 , z Ip ρ2 , z
)
(17)
ρ2, z) and Ip(ρj, z) are given by Eqs. (15) and (16).
3. Numerical results In this section, we study numerically the evolution properties of the intensity and spectral degree of coherence of a nonparaxial LGCSM beam in free space by applying the formulae derived in Section 2. In the following numerical examples, n ¼ 5, δ0 ¼5λ, unless other values are indicated in the figures. The propagation distance is normalized to the Rayleigh distance z/zR (zR ¼ πs02/λ). In Fig. 2, the intensity distribution of a nonparaxial LGCSM beam at several propagation distances z with different values of the beam waist width s0 (solid curve) is shown and normalized to the maximum intensity I/Imax. For the convenience of comparison, the corresponding normalized paraxial results (dotted curve) are plotted together. One can see from Fig. 1 that the nonparaxiality of the LGCSM beam is closely related with the initial beam waist width s0 and propagation distance z. For the case of a large value of s0 ¼ 20λ, there is little difference between nonparaxial and paraxial results in both the near and far fields, which means that the paraxial approximation is valid. However, for the case of a small value s0, for example, s0 ¼1.5λ and 0.7λ, the discrepancy between the paraxial and nonparaxial results in the near field becomes noticeable, which means that the paraxial approximation
Fig. 2. Normalized intensity distribution I/Imax of a LGCSM beam at several propagation distances for various values of the initial beam waist width s0 with δ0 ¼ 1.5λ and n¼ 5.
L. Guo et al. / Optics Communications 352 (2015) 127–134
131
Fig. 3. The normalized intensity distribution I/Imax of a LGCSM beam at z¼ 5zR for various values of the initial transverse coherence width with s0 ¼ 1.5λ and n¼ 5.
Fig. 4. Normalized intensity distribution I/Imax of a LGCSM beam at z ¼ 5zR for various mode orders of n with s0 ¼ 1.5λ and δ0 ¼ 2λ.
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L. Guo et al. / Optics Communications 352 (2015) 127–134
is instable, however, the two calculated results are almost the same in the far field. Therefore, the nonparaxial behavior should be taken into consideration when describing the intensity properties of a LGCSM beam with a beam waist width that is comparable to λ. What is more, the evolution properties of the intensity distribution of nonparaxial LGCSM beams are closed related to the beam waist width s0. It can be seen from Fig. 1 that, for a large value of s0 ¼ 20λ, the central intensity of Gaussian beam profile decreases gradually on propagation, and the intensity distribution finally becomes hollow beam profile in the far field. For a moderate value of s0 ¼1.5λ, the intensity distribution evolves gradually into a central-dark intensity pattern in the far field. The depth of central-dark profile becomes much shallow with the decrease of s0. For a small value of s0 ¼0.7λ, the LGCSM beam retains Gaussian beam profile in the near and far field. To investigate the effect of spatial coherence on the intensity distribution of a LGCSM beam, we calculate in Fig. 3 the normalized intensity distribution at z¼ 5zR for different values of spatial coherence width δ0 with s0 ¼ 1.5λ. The corresponding paraxial results are also plotted together for comparison in Fig. 2. As shown in Fig. 3, for a large value of δ0 ¼ 2λ, 5λ and 20λ, there is little difference between nonparaxial and paraxial results (see Fig. 3(b)– (d)). However, it is not the case for a small value of δ0 ¼0.5λ, an appreciable discrepancy between the nonparaxial and paraxial results appear (see Fig. 3(a)). It can be explained that a small coherence width leads to a larger divergence angle, thus the beam will become “nonparaxial”. In this case, the nonparaxial approach is necessary. Fig. 3 also shows that, by selecting proper spatial
coherence of the LGCSM beam, the flattened, hollow or centraldark intensity profiles can be obtained, however, the GSM beams do not have these characteristics. Fig. 4 plots the normalized intensity distribution of the LGCSM beam at z ¼5zR for different values of the mode orders n with s0 ¼ 1.5λ and δ0 ¼2λ. One finds that the discrepancy between the nonparaxial and paraxial results increases as the beam order n increases. This can be interpreted physically as follows: with the increase of n, the spectral degree of coherence of a LGCSM beam in the source plane decreases, which enhances the nonparaxiality of the beams. It can be seen from Figs. 3 and 4 that the effect of n on the intensity is similar to that of δ0. Next, let us turn to study the spectral coherence properties of a nonparaxial LGCSM beam on propagation, we are going to study the evolution of the spectral degree of coherence at two contrary spatial positions r1 ¼(ρ, z) and r2 ¼ ( ρ, z). Fig. 5 shows the modulus of spectral degree of coherence μ (ρ, − ρ, z ) of a nonparaxial LGCSM beam at several propagation distances for s0 ¼0.7λ, 1.5λ and 20λ. The corresponding paraxial results (dotted curve) are plotted together for comparison. One sees clearly that, the nonparaxial and paraxial μ (ρ, − ρ, z ) remain the non-Gaussian distribution with several side robes around the central bright spot in the near field. As the propagation distance increases, the side robes gradually diminish, only the central bright spot exists which degenerates into Gaussian distribution in the far field. The degenerated speed becomes more quickly for a smaller s0 on propagation. On the other hand, for a large value of s0 ¼ 20λ, the nonparaxial μ (ρ, − ρ, z ) coincides with paraxial μ (ρ, − ρ, z ) in the
Fig. 5. Modulus of the spectral degree of coherence μ (ρ, − ρ, z ) of a LGCSM beam at several propagation distances for various values of beam waist width s0 with δ0 ¼ 5λ and n¼ 5.
L. Guo et al. / Optics Communications 352 (2015) 127–134
133
Fig. 6. Modulus of the spectral degree of coherence μ (ρ, − ρ, z ) of a LGCSM beam at several propagation distances for various values of transverse coherence width δ0 with s0 ¼ 1.5λ and n¼ 5.
Fig. 7. Modulus of the spectral degree of coherence μ (ρ, − ρ, z ) of a LGCSM beam at several propagation distances for various values of mode orders n with s0 ¼ 1.5λ and δ0 ¼ 2λ.
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L. Guo et al. / Optics Communications 352 (2015) 127–134
near and far field. For a small value of s0 ¼1.5λ and 0.7λ, an appreciable discrepancy between the nonparaxial and paraxial results appears and decreases with the propagation distance, which means that the paraxial approach is no longer valid. Additionally, μ (ρ, − ρ, z ) divergences more quickly in the transverse plane on propagation with a larger value of s0, although they have the same spectral degree of coherence distribution in the source plane. Fig. 6 shows the μ (ρ, − ρ, z ) of the nonparaxial LGCSM beam at several propagation distances for different values of δ0 with s0 ¼ 1.5λ. Fig. 7 shows the μ (ρ, − ρ, z ) of the nonparaxial LGCSM beam at several propagation distances for different values of n with s0 ¼1.5λ and δ0 ¼2λ. One finds that the degeneration of μ (ρ, − ρ, z ) from non-Gaussian distribution to Gaussian distribution in Figs. 6 and 7 are similar to that in Fig. 5(e)–(h), however, the degeneration speed on propagation is related to δ0 and n. The μ (ρ, − ρ, z ) diverges more quickly in the transverse plane on propagation with a larger value of δ0 or a smaller value of n.
4. Conclusion We have investigated the propagation of a nonparaxial LGCSM beam in free space. Analytical expressions for the intensity and the spectral degree of coherence formula of a nonparaxial LGCSM beam on propagation have been obtained by using the generalized Rayleigh–Sommerfeld diffraction integral. The comparisons between the nonparaxial and paraxial results showed that the initial beam parameters s0, δ0, n and the propagation distance z together determined the nonparaxiality of a LGCSM beam and effect on the propagation properties of the intensity and spectral degree of coherence. Our results also show that flat-topped, hollow and central-dark beam profiles can be formed in the far field by varying the initial beam parameters, which will be useful in some applications, such as particle trapping, holographic microscopy and micro-lithography, where nonparaxial partially coherent beams are commonly encountered.
Acknowledgments This work is supported by the National Natural Science Foundation of China under Grant nos. 11274005, 11404067 and 11404234, and the Innovation Plan for Graduate Students in the Universities of Jiangsu Province under Grant no. KYLX-1218.
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Propagation of a Laguerre–Gaussian correlated Schell-model beam beyond the paraxial approximation Lina Guo a,b, Yahong Chen b, Lin Liu b, Yangjian Cai b,n a
School of Electronics and Information, GuangDong Polytechnic Normal University, Guangzhou, Guangdong 510665, China School of Physical Science and Technology and Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China
b
art ic l e i nf o
a b s t r a c t
Article history: Received 29 March 2015 Received in revised form 28 April 2015 Accepted 29 April 2015 Available online 4 May 2015
Recently, Laguerre–Gaussian correlated Schell-model (LGCSM) beam was introduced theoretically [Mei and Korotkova, Opt. Lett. 38(2), (2013), 91–93] and generated experimentally [Wang et al., Opt. Lett. 38 (11), (2013), 1814–1816]. In this paper, we treat the propagation of a LGCSM beam beyond the paraxial approximation. Based on the generalized Rayleigh–Sommerfeld diffraction integral, analytical expressions for the intensity and spectral degree of coherence of a nonparaxial LGCSM beam propagating in free space are obtained, and the corresponding results of a paraxial LGCSM beam are also derived as a special case. Our numerical results show that the nonparaxial propagation properties of a LGCSM beam are closely related to the initial beam parameters, such as the beam waist width, the coherence width and the mode order. Furthermore, in the far field, flat-topped, hollow intensity profiles can be formed through varying the initial beam parameters, which is different from that of a nonparaxial Gaussian Schell-model beam. & 2015 Elsevier B.V. All rights reserved.
Keywords: Laguerre–Gaussian correlated Schell-model beam Nonparaxial Propagation
1. Introduction Recently, modulating the correlation function of partially coherent beam has become a subject of rapidly growing interest [1– 14]. As an additional freedom, correlation function can be used to modulate light field, such as creation of partially coherent beams with nonconventional correlation functions. Recent studies have shown that partially coherent beams with nonconventional correlation functions have displayed many extraordinary properties, e.g., partially coherent beam with nonuniform correlated function displays the self-focusing property during propagation [5], partially coherent beam with Hermite–Gaussian correlated function displays the self-splitting effect [6] and partially coherent beams with Multi-Gaussian and Laguerre–Gaussian correlated functions exhibit prescribed beam profiles in the far-field [7–14]. Due to these extraordinary properties, partially coherent beams with nonconventional correlation functions have potential applications in nonlinear optics, atom optics and free-space optical communications. Partially coherent beam with Laguerre–Gaussian correlated function (also named Laguerre–Gaussian correlated Schell-model n
Corresponding author. E-mail address: [email protected] (Y. Cai).
http://dx.doi.org/10.1016/j.optcom.2015.04.082 0030-4018/& 2015 Elsevier B.V. All rights reserved.
beam) was firstly introduced in [10] and experimental generated in [11]. It was found that the far-field intensity distribution of a Laguerre–Gaussian correlated Schell-model (LGCSM) beam exhibits ring-shaped beam profile, although it has the same intensity distribution with that of a Gaussian Schell-model beam in the source plane. Partially coherent beam with nonconventional correlation function can be generated through the method based on the modulation of the intensity in the Fourier plane, and we reported experimental generation of a controllable optical cage by focusing a LGCSM beam, which will be useful for trapping particles or atoms in [12]. In [15], we showed that LGCSM beam has advantages over partially coherent beam with conventional correlation function (i.e. Gaussian Schell-model beam) to overcome turbulence-induced degeneration when passing through turbulent atmosphere, which is useful in free-space optical communications. Evolution properties of a LGCSM beam in uniaxial crystals were explored in [16] and the experimental generation of a LGCSM vortex beam was reported in [17]. When a laser beam has a small spot size that is comparable with the wavelength or a large far-field divergence angle, it is usually regarded as a nonparaxial beam [18]. For a partially coherent beam, weak coherence will result in a large divergence angle, which also plays an important role in determining the nonparaxiality of a laser beam [19]. Nonparaxial partially coherent
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L. Guo et al. / Optics Communications 352 (2015) 127–134
beams can be produced from multimode oscillation diode laser or tight focusing of a paraxial partially coherent beam with high numerical aperture [19,20]. Many approaches have been developed to treat the propagation of nonparaxial partially coherent beams, such as generalized Rayleigh–Sommerfeld diffraction integrals, Wigner distribution function, angular spectrum representation and the vectorial moment theory [21–24]. Based on these methods, propagation properties of nonparaxial scalar and vector GSM beams with various beam profiles have been extensively studied in free space [25–34]. Recently, Zhang et al. performed the nonparaxial propagation of a twisted anisotropic Gaussian Schell-model beam based on the generalized Rayleigh– Sommerfeld diffraction integral with the help of a tensor method [35], and they also examined the nonparaxial propagation of a partially coherent beam in uniaxial crystals [36]. However, up to now, most previous researches are confined to nonparaxial properties of a partially coherent beam with conventional correlation function. In this paper, our aim is to study the nonparaxial propagation properties of a LGCSM beam in free space. With the help of the generalized Rayleigh–Sommerfeld integral method, the analytical nonparaxial propagation formula for a LGCSM beam in free space is derived. The evolution properties of the intensity and spectral degree of coherence of a nonparaxial LGCSM beam in free space are illustrated numerically.
⎡ ρ2 + ρ2 (ρ10 − ρ20 )2 ⎤ ⎥ W ρ10 , ρ20 , 0 = exp ⎢− 10 2 20 − ⎢⎣ ⎥⎦ 4σ 0 2δ 02
(
)
⎡ (ρ − ρ )2 ⎤ 10 20 ⎥ L n0 ⎢ ⎢⎣ ⎥⎦ 2δ 02
where ρj0≡(xj0, yj0) (j¼ 1, 2) is the position vector, s0 is the transverse beam waist width of the corresponding Gaussian beam, δ0 is the transverse coherence width, and Ln0 is the generalized Laguerre polynomial with mode orders n and 0, respectively. When n¼ 0, Eq. (1) reduces to a conventional Gaussian Schell-model beam. The spectral degree of coherence of a LGCSM beam is modulated by mode order n and coherence width δ0, and is consists of several side robes with a central bright spot, as shown in Fig. 1. According to the generalized Rayleigh–Sommerfeld diffraction integral, the propagation of CSD of a LGCSM beam at two points (ρ1, z) and (ρ2, z) in the free space can be expressed as [19,37]
(
W ρ1 , ρ2 , z
)
⎛ zk ⎞2 =⎜ ⎟ ⎝ 2π ⎠
∞
∞
∞
∞
∫−∞ ∫−∞ ∫−∞ ∫−∞ W (ρ10 , ρ20 , 0)
exp ⎡⎣ik (R2 − R1) ⎤⎦ R12R22
d2ρ10 d2ρ20 ,
(2)
where k ¼2π/λ is the wave number related to the wavelength
λ, Rj = rj − ρj0 = ρ j20 + ρ j2 − 2ρj0 ⋅ρj + z 2 , (j¼ 1,2), and
(
2. Theory
(1)
)
rj ≡ xj , yj , z is the position vector at the z plane. Under weak
The second-order spatial correlation properties of a partially coherent beam are generally characterized by the cross-spectral density (CSD) in the space-frequency domain. The CSD of a LGCSM beam at a pair of arbitrary points in the source plane at z¼ 0 can be expressed as [10–12]:
nonparaxial approximation, Rj can be retained the zero and firstorder expansions
(
)
R j = rj + ρ j20 − 2ρj ⋅ρj0 /2rj
(3)
with rj = rj . Replacing Rj in the exponential term of Eq. (2) by
Fig. 1. The spectral degree of coherence distribution of a LGCSM beam in the source plane for different values of n with s0 ¼ 1.5λ and δ0 ¼ 5λ.
L. Guo et al. / Optics Communications 352 (2015) 127–134
Eq. (3), and that in denominator term by rj, Eq. (2) can be rewritten as.
(
W ρ1 , ρ2 , z
)
⎛ zk ⎞2 exp ⎡⎣ik (r2 − r1) ⎤⎦ =⎜ ⎟ ⎝ 2π ⎠ r12r22
∞
∞
(
∞
For the convenience of integration, the “sum” and “difference” coordinates are adopted
(5)
Substituting Eqs. (1) and (5) into Eq. (4), we obtained the following alternative form
⎛ zk ⎞2 exp ⎡⎣ik (r2 − r1) ⎤⎦ W ρ1 , ρ2 , z = ⎜ ⎟ ⎝ 2π ⎠ r12r22
)
∞
∞
∞
∞
∫−∞ ∫−∞ ∫∞ ∫−∞ P*
⎛ Δρ0 ⎞ ⎛ Δρ0 ⎞ ⎜ρs0 + ⎟ P ⎜ρs0 − ⎟ g Δρ0 ⎝ 2 ⎠ ⎝ 2 ⎠ ⎡ ik ⎛ Δρ0 ⎞ ik ⎛ Δρ0 ⎞ ⎤ × exp ⎢ ρ1⋅⎜ρs0 + ⎟ − ρ2 ⋅⎜ρs0 − ⎟⎥ 2 ⎠ r2 ⎝ 2 ⎠⎦ ⎣ r1 ⎝
(
)
d2ρs0 d2Δρ0 ,
)
⎡ r1 2 ⎢ ⎛ 2 σ *2 (r1) ρ1 + σ (r2 ) ρ2 r2 k ⎞ × Ln ⎢ −⎜ ⎟ ⎢ ⎝ 2r1 ⎠ σ *2 (r1) + σ 2 (r2 ) + 2δ 2 0 ⎢ ⎣
(
(7)
⎡⎛ Δρ0 ⎞2⎤ ik 1 ⎞⎛ ⎟⎟ ⎜ρs0 + P * ρs0 + Δρ0 /2 = exp ⎢ ⎜⎜ − − ⎟⎥ ⎢⎣ ⎝ 2r1 2 ⎠ ⎥⎦ 4σ02 ⎠ ⎝
(
)
⎡⎛ Δρ0 ⎞2⎤ ik 1 ⎞⎛ ⎟⎟ ⎜ρs0 − − Δρ0 /2 = exp ⎢ ⎜⎜ − ⎟ ⎥. ⎢⎣ ⎝ 2r2 2 ⎠ ⎥⎦ 4σ02 ⎠ ⎝
)
(
)
(
( ) (
)
with σ rj = 1/4σ02 − ik /2rj , (j¼ 1, 2). Eq. (12) is the basic analytical result obtained in this paper, which can be used to describe the intensity and spectral degree of coherence of a nonparaxial LGCSM beam propagating in free space. On placing ρ1 ¼ ρ2 ¼ ρ and r1 ¼ r2 ¼r into Eq. (12), the intensity distribution of a nonparaxial LGCSM beam at the z plane is given by
⎛ kz ⎞2 I (ρ, z ) = 2n − 1 ⎜ ⎟ σ *2σ 2δ 02n+ 2 σ *2 + σ 2 + 2δ 02 ⎝ r2 ⎠ ⎤ ⎡ 2 ⎛ k ⎞2 2δ 0 σ *2 + σ 2 2 ⎥ exp ⎢−⎜ ⎟ ρ ⎢ ⎝ 2r ⎠ σ * 2 + σ 2 + 2δ 2 ⎥ 0 ⎦ ⎣
⎛
⎛ Δρ0 ⎞ 1 P ⎜ρs0 − ⎟= ⎝ 2 ⎠ (λr2 )2
⎛
(
(
(9)
⎞
∫−∞ ∫−∞ P∼* ⎜⎝ λur11 ⎟⎠
⎡ ik ⎛ Δρ0 ⎞ ⎤ 2 exp ⎢ ⎜ρs0 + ⎟⋅u1⎥ d u1 2 ⎠ ⎦ ⎣ r1 ⎝
)
∞
.
(
W ρ1 , ρ2 , z
(
)
)(
I ρ1 , z I ρ2 , z
)
(14)
obtains the following paraxial CSD expression of a LGCSM beam at the z plane,
(10)
⎛ k ⎞2 Wp ρ1 , ρ2 , z = 2n − 1 ⎜ ⎟ σ p*2 σ p2 δ 02n +2 σ p*2 + σ p2 + 2δ 02 ⎝z⎠
(
(
)
−n − 1
)
⎡ ⎛ ⎞⎤ ⎤ ⎢ ⎜ ⎟⎥ ⎥ × exp⎢ − ik ⎜ρ 2 − ρ 2 ⎟⎥ 1 2 ⎥⎦ ⎢ 2z ⎜ ⎟⎥ ⎣ ⎝ ⎠⎦ ⎡ 2⎤ 2 2 * ⎢ ⎛ k ⎞2 σ p ρ1 + σ p ρ2 ⎥ × exp ⎢ ⎜ ⎟ ⎥ 2 2 2 ⎢ ⎝ 2z ⎠ σ p* + σ p + 2δ 0 ⎥ ⎣ ⎦
⎞
⎡ ik ⎛ Δρ0 ⎞ ⎤ 2 exp ⎢ − ⎜ρs0 − ⎟⋅u2 ⎥ d u2 2 ⎠ ⎦ ⎣ r2 ⎝
−1/2
)
(13)
where W(ρ1, ρ2, z) and I(ρj, z) are given by Eqs. (12) and (13), respectively. The paraxial result can be regarded as a special case of the nonparaxial result. Under the condition of paraxial approximation, rj can be expanded into series and keeping the first and second terms, i.e., rj ≈ z + ρ j2 /2z . Replacing the rj in Eq. (12) with z, one
⎡ ⎛ ⎞2 k exp ⎢−⎜ ⎟ σ p*2 ρ12 + σ p2 ρ22 ⎢⎣ ⎝ 2z ⎠
∫−∞ ∫−∞ P∼ ⎜⎝ λur22 ⎟⎠ ∞
)
)
with σ = 1/4σ02 − ik /2r
μ ρ1 , ρ2 , z =
)
∞
−n − 1
)
⎤ ⎡ 2 2 2 ⎥ ⎢ ⎛ k ⎞2 σ * + σ 2 , ρ × Ln ⎢−⎜ ⎟ ⎝ 2r ⎠ σ *2 + σ 2 + 2δ 02 ⎥ ⎥⎦ ⎢⎣
(8)
∼ ∼ their Fourier transform P * (u1/λr1) and P (u2/λr2 ) as follows: ∞
(12)
−1/2
The spectral degree of coherence between two arbitrary points r1 ¼(ρ1, z) and r2 ¼(ρ2, z) of a nonparaxial LGCSM beam at the z plane is determined by [38]
P * ρs0 + Δρ0 /2 and P ρs0 − Δρ0 /2 can be expressed in terms of
⎛ Δρ0 ⎞ 1 P * ⎜ρs0 + ⎟= ⎝ 2 ⎠ (λr1)2
⎥ ⎥ ⎦
(
(6)
⎡ Δρ 2 ⎤ ⎡ Δρ 2 ⎤ 0 ⎥ L n0 ⎢ 0 ⎥ = exp ⎢ − 2δ 02 ⎦ ⎣ 2δ 02 ⎦ ⎣
2⎤
) ⎥⎥,
(
g (Δρ 0 )
(
(
(
with
P ρs0
2⎤ r1 2 2 ⎛ k ⎞2 σ * (r1) ρ1 + r 2 σ (r2 ) ρ2 ⎥ ⎥ +⎜ ⎟ ⎝ 2r1 ⎠ σ *2 (r1) + σ 2 (r2 ) + 2δ 02 ⎥ ⎥ ⎦
(4)
ρ10 + ρ20 , Δρ 0 = ρ10 − ρ20 . 2
−n − 1
)
⎡ ⎢ ⎛ 2 ⎞ r12 2 k ⎞ ⎛⎜ 2 σ (r2 ) ρ22 ⎟⎟ × exp ⎢ −⎜ ⎟ σ * (r1) ρ12 + ⎢ ⎝ 2r1 ⎠ ⎜⎝ 2 r2 ⎠ ⎢ ⎣
∫−∞ ∫−∞ ∫−∞ ∫−∞ W (ρ10 , ρ20 , 0)
d2ρ10 d2ρ20 .
(
⎛ kz ⎞2 W (ρ1 , ρ2 , z ) = 2n − 1 ⎜ ⎟ exp ⎡⎣ik (r2 − r1) ⎤⎦ σ *2 (r1) σ 2 (r2 ) ⎝ r1r2 ⎠ δ 02n +2 σ *2 (r1) + σ 2 (r2 ) + 2δ 02
∞
⎛ ⎡ ik ⎞ ⎛ ik ⎞⎤ ik ik × exp ⎜⎜ ⎢ ρ202 − ρ20 ⋅ρ2 ⎟⎟−⎜ ρ102 − ρ10 ⋅ρ1 ⎟ ⎥ 2 r r 2 r r ⎣ ⎝ ⎠⎦ ⎝ ⎠ 2 2 1 1
ρs0 =
129
(
(
(11)
∼ where P (⋅) is the Fourier transform of P (⋅) and the asterisk denotes the complex conjugate. After tedious but straightforward integral calculations, we obtain the following analytical expression for CSD of a nonparaxial LGCSM beam at the z plane.
)
)
⎡ 2⎤ 2 2 ⎢ ⎛ k ⎞2 σ p* ρ1 + σ p ρ2 ⎥ Ln ⎢−⎜ ⎟ ⎥, 2 2 2 ⎢ ⎝ 2z ⎠ σ p* + σ p + 2δ 0 ⎥ ⎣ ⎦
(
(
where σp = 1/4σ02 − ik /2z
)
−1/2
)
.
(15)
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L. Guo et al. / Optics Communications 352 (2015) 127–134
On placing ρ1 ¼ ρ2 ¼ ρ into Eq. (15), the paraxial intensity of a LGCSM beam at the z plane is given by
Ip (ρ, z ) =
⎛ k ⎞2 2n − 1 ⎜ ⎟ σ p*2 σ p2 δ 02n+ 2
(
σ p*2
σ p2
+ + ⎝z⎠ ⎤ ⎡ 2 2 2 ⎛ k ⎞2 2δ 0 σ p* + σ p 2 ⎥ exp ⎢−⎜ ⎟ ρ ⎢ ⎝ 2z ⎠ σ *2 + σ 2 + 2δ 2 ⎥ p p 0 ⎦ ⎣
(
−n − 1 2δ 02
)
)
⎤ ⎡ 2 2 2 ⎥ ⎢ ⎛ k ⎞2 σ p* + σ p 2 ρ . × Ln ⎢−⎜ ⎟ ⎝ 2z ⎠ σ p*2 + σ p2 + 2δ 02 ⎥ ⎥⎦ ⎢⎣
(
)
(16)
Eq. (16) is identical to Eq. (18) with B ¼z in Ref. [17], which is derived from another alternative method. The spectral degree of coherence of a paraxial LGCSM beam between two arbitrary points r1 ¼(ρ1, z) and r2 ¼(ρ2, z) in the z plane is determined by
(
)
μp ρ1 , ρ2 , z = where Wp(ρ1,
(
Wp ρ1 , ρ2 , z
(
) (
)
Ip ρ1 , z Ip ρ2 , z
)
(17)
ρ2, z) and Ip(ρj, z) are given by Eqs. (15) and (16).
3. Numerical results In this section, we study numerically the evolution properties of the intensity and spectral degree of coherence of a nonparaxial LGCSM beam in free space by applying the formulae derived in Section 2. In the following numerical examples, n ¼ 5, δ0 ¼5λ, unless other values are indicated in the figures. The propagation distance is normalized to the Rayleigh distance z/zR (zR ¼ πs02/λ). In Fig. 2, the intensity distribution of a nonparaxial LGCSM beam at several propagation distances z with different values of the beam waist width s0 (solid curve) is shown and normalized to the maximum intensity I/Imax. For the convenience of comparison, the corresponding normalized paraxial results (dotted curve) are plotted together. One can see from Fig. 1 that the nonparaxiality of the LGCSM beam is closely related with the initial beam waist width s0 and propagation distance z. For the case of a large value of s0 ¼ 20λ, there is little difference between nonparaxial and paraxial results in both the near and far fields, which means that the paraxial approximation is valid. However, for the case of a small value s0, for example, s0 ¼1.5λ and 0.7λ, the discrepancy between the paraxial and nonparaxial results in the near field becomes noticeable, which means that the paraxial approximation
Fig. 2. Normalized intensity distribution I/Imax of a LGCSM beam at several propagation distances for various values of the initial beam waist width s0 with δ0 ¼ 1.5λ and n¼ 5.
L. Guo et al. / Optics Communications 352 (2015) 127–134
131
Fig. 3. The normalized intensity distribution I/Imax of a LGCSM beam at z¼ 5zR for various values of the initial transverse coherence width with s0 ¼ 1.5λ and n¼ 5.
Fig. 4. Normalized intensity distribution I/Imax of a LGCSM beam at z ¼ 5zR for various mode orders of n with s0 ¼ 1.5λ and δ0 ¼ 2λ.
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is instable, however, the two calculated results are almost the same in the far field. Therefore, the nonparaxial behavior should be taken into consideration when describing the intensity properties of a LGCSM beam with a beam waist width that is comparable to λ. What is more, the evolution properties of the intensity distribution of nonparaxial LGCSM beams are closed related to the beam waist width s0. It can be seen from Fig. 1 that, for a large value of s0 ¼ 20λ, the central intensity of Gaussian beam profile decreases gradually on propagation, and the intensity distribution finally becomes hollow beam profile in the far field. For a moderate value of s0 ¼1.5λ, the intensity distribution evolves gradually into a central-dark intensity pattern in the far field. The depth of central-dark profile becomes much shallow with the decrease of s0. For a small value of s0 ¼0.7λ, the LGCSM beam retains Gaussian beam profile in the near and far field. To investigate the effect of spatial coherence on the intensity distribution of a LGCSM beam, we calculate in Fig. 3 the normalized intensity distribution at z¼ 5zR for different values of spatial coherence width δ0 with s0 ¼ 1.5λ. The corresponding paraxial results are also plotted together for comparison in Fig. 2. As shown in Fig. 3, for a large value of δ0 ¼ 2λ, 5λ and 20λ, there is little difference between nonparaxial and paraxial results (see Fig. 3(b)– (d)). However, it is not the case for a small value of δ0 ¼0.5λ, an appreciable discrepancy between the nonparaxial and paraxial results appear (see Fig. 3(a)). It can be explained that a small coherence width leads to a larger divergence angle, thus the beam will become “nonparaxial”. In this case, the nonparaxial approach is necessary. Fig. 3 also shows that, by selecting proper spatial
coherence of the LGCSM beam, the flattened, hollow or centraldark intensity profiles can be obtained, however, the GSM beams do not have these characteristics. Fig. 4 plots the normalized intensity distribution of the LGCSM beam at z ¼5zR for different values of the mode orders n with s0 ¼ 1.5λ and δ0 ¼2λ. One finds that the discrepancy between the nonparaxial and paraxial results increases as the beam order n increases. This can be interpreted physically as follows: with the increase of n, the spectral degree of coherence of a LGCSM beam in the source plane decreases, which enhances the nonparaxiality of the beams. It can be seen from Figs. 3 and 4 that the effect of n on the intensity is similar to that of δ0. Next, let us turn to study the spectral coherence properties of a nonparaxial LGCSM beam on propagation, we are going to study the evolution of the spectral degree of coherence at two contrary spatial positions r1 ¼(ρ, z) and r2 ¼ ( ρ, z). Fig. 5 shows the modulus of spectral degree of coherence μ (ρ, − ρ, z ) of a nonparaxial LGCSM beam at several propagation distances for s0 ¼0.7λ, 1.5λ and 20λ. The corresponding paraxial results (dotted curve) are plotted together for comparison. One sees clearly that, the nonparaxial and paraxial μ (ρ, − ρ, z ) remain the non-Gaussian distribution with several side robes around the central bright spot in the near field. As the propagation distance increases, the side robes gradually diminish, only the central bright spot exists which degenerates into Gaussian distribution in the far field. The degenerated speed becomes more quickly for a smaller s0 on propagation. On the other hand, for a large value of s0 ¼ 20λ, the nonparaxial μ (ρ, − ρ, z ) coincides with paraxial μ (ρ, − ρ, z ) in the
Fig. 5. Modulus of the spectral degree of coherence μ (ρ, − ρ, z ) of a LGCSM beam at several propagation distances for various values of beam waist width s0 with δ0 ¼ 5λ and n¼ 5.
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Fig. 6. Modulus of the spectral degree of coherence μ (ρ, − ρ, z ) of a LGCSM beam at several propagation distances for various values of transverse coherence width δ0 with s0 ¼ 1.5λ and n¼ 5.
Fig. 7. Modulus of the spectral degree of coherence μ (ρ, − ρ, z ) of a LGCSM beam at several propagation distances for various values of mode orders n with s0 ¼ 1.5λ and δ0 ¼ 2λ.
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near and far field. For a small value of s0 ¼1.5λ and 0.7λ, an appreciable discrepancy between the nonparaxial and paraxial results appears and decreases with the propagation distance, which means that the paraxial approach is no longer valid. Additionally, μ (ρ, − ρ, z ) divergences more quickly in the transverse plane on propagation with a larger value of s0, although they have the same spectral degree of coherence distribution in the source plane. Fig. 6 shows the μ (ρ, − ρ, z ) of the nonparaxial LGCSM beam at several propagation distances for different values of δ0 with s0 ¼ 1.5λ. Fig. 7 shows the μ (ρ, − ρ, z ) of the nonparaxial LGCSM beam at several propagation distances for different values of n with s0 ¼1.5λ and δ0 ¼2λ. One finds that the degeneration of μ (ρ, − ρ, z ) from non-Gaussian distribution to Gaussian distribution in Figs. 6 and 7 are similar to that in Fig. 5(e)–(h), however, the degeneration speed on propagation is related to δ0 and n. The μ (ρ, − ρ, z ) diverges more quickly in the transverse plane on propagation with a larger value of δ0 or a smaller value of n.
4. Conclusion We have investigated the propagation of a nonparaxial LGCSM beam in free space. Analytical expressions for the intensity and the spectral degree of coherence formula of a nonparaxial LGCSM beam on propagation have been obtained by using the generalized Rayleigh–Sommerfeld diffraction integral. The comparisons between the nonparaxial and paraxial results showed that the initial beam parameters s0, δ0, n and the propagation distance z together determined the nonparaxiality of a LGCSM beam and effect on the propagation properties of the intensity and spectral degree of coherence. Our results also show that flat-topped, hollow and central-dark beam profiles can be formed in the far field by varying the initial beam parameters, which will be useful in some applications, such as particle trapping, holographic microscopy and micro-lithography, where nonparaxial partially coherent beams are commonly encountered.
Acknowledgments This work is supported by the National Natural Science Foundation of China under Grant nos. 11274005, 11404067 and 11404234, and the Innovation Plan for Graduate Students in the Universities of Jiangsu Province under Grant no. KYLX-1218.
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