Modified Bessel-correlated vortex beams and their propagation properties

Optics and Laser Technology 126 (2020) 106088

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Modified Bessel-correlated vortex beams and their propagation properties

T

Zhangrong Mei Department of Physics, Huzhou University, Huzhou 313000, China Sci-Tech School, Huzhou University Qiuzhen College, Huzhou 313000, China

H I GH L IG H T S

method for modeling modified Bessel-correlated vortex beams is proposed. • AA new notable feature of new beams is that the intensity vortices exist in any coherent state. • Another notable feature of such beams remains stable vortex profile upon propagation. •

A R T I C LE I N FO

A B S T R A C T

Keywords: Partially coherent vortex beams Beam model Atmospheric turbulence Propagation characteristics

A new method for modeling partially coherent vortex beams with modified Bessel-correlated properties is proposed. Analytical formulas for the cross-spectral density function of such beams propagating in free space and atmospheric turbulence are derived and used to investigate the propagation characteristics. The results show that such beams possess unique characteristics different from classical Schell-model type vortex beams. The intensity vortices exist in both high coherence and low coherence cases, and remain stable vortex profile on propagating in free space and weak atmospheric turbulence.

1. Introduction Vortex beams with zero central intensity have attracted extensive interest due to their wide application prospects as optical tweezers in precise and contactless manipulation and control of microscopic particles. Previously, the research on vortex beam was restricted to the domain of fully coherent field. Since Gori et al in 1998 introduced a class of Gaussian correlated partially sources with annular shapes intensity distribution [1], partially coherent vortex beams have received great attention. Various partially coherent vortex beams were introduced by using Wolf’s model theory of coherence [2], and some investigations have also been carried out to their propagation characteristics [3–9]. Compared with fully coherent beams, partially coherent beams have many unique characteristics, such as more uniform transverse intensity distribution, lower beam scintillation and diffusion, better self-repairing ability, and so on [10–13]. Moreover, the spatial correlation structure of partially coherent beams plays an important role in modulating the propagation characteristics of beams to satisfy the demand of different application fields, which makes them more superior in certain applications [14]. Therefore, in recent years, there has been a growing interest in studying the generation and properties of partially coherent vortex beams with special spatial correlation structures.

A general representation for partially coherent sources [15], ensuring the generated fields to have bona fide spatial correlation functions, played a significant role in recent studies of partially coherent fields [16–21]. Some new partially coherent vortex beam models were also devised using the prescriptions of [15], such as Laguerre-Gaussian correlated Schell-mode vortex beams, mutli-Gaussian Schell-mode vortex beams, sine Schell-model vortex beams, etc [22–24]. However, the structure of the correlation functions of these models was limited to the classical Schell-model type. Just like the basic Gaussian Schellmodel vortex beam [4], the limitation of these partially coherent vortex beams is that their intensity dark vortex core usually exists only in the case of high coherence and the core fills with diffuse light in the low coherence cases. Moreover, other statistical properties of such beams are not invariant in propagation. These are not conducive to the application of vortex beams in certain fields. Recently, a class of Besselcorrelated fields was introduced by using the shifted-elementary-beam theory, which has self-Fourier-transforming and approximately shape invariant characteristics under certain conditions [25]. In this paper, we employ the model representation method of Ref. [15] to construct novel classes of partially coherent vortex beams exhibiting Bessel-type correlations, which can be represented as an incoherent superposition of Bessel Gaussian elementary modes weighted by the nonnegative weight function. Such beams are also self-Fourier-transforming, the

E-mail address: [email protected]. https://doi.org/10.1016/j.optlastec.2020.106088 Received 31 October 2019; Received in revised form 9 January 2020; Accepted 20 January 2020 0030-3992/ © 2020 Elsevier Ltd. All rights reserved.

Optics and Laser Technology 126 (2020) 106088

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Eq. (4) into Eq. (8) we obtain the following expression for the Fourier transform of the CSD of the source Eq. (4):

form of correlated function and the shape of intensity vortex remain invariance on paraxial propagation.

∼ W0 (f1 , f2 )

2. Modeling of a class of partially coherent light source and propagation of the fields that they generate.

= [w 2 (4μ)] exp[−(f12 + f22 ) (4μ)] In [w 2f1 f2 (2μδ 2)] exp[in (θ2 − θ1)], (9)

Recall that according to the superposition rule, the CSD of a widesense stationary source can be expressed as [15]

W0 (ρ′1 , ρ′2 ) =

∫ p (v) H ∗ (ρ′1, v) H (ρ′2 , v) d2 v,

where μ = 1 σ 2 + 1 (σ 2 + δ 2) . Eq. (9) has a functional form similar to Eq. (4), so the field radiated by the source Eq. (4) is a self-Fouriertransforming field. On inserting from Eq. (9) into Eq. (7) and integrating, we arrive at the formulas

(1)

where ρ′1 = (ρ1′, ϕ1′) and ρ′2 = (ρ2′, ϕ2′) are position vectors of two source points, p (v) is a non-negative weighting function, H (ρ′, v) is an arbitrary complex-valued function accounting for the correlation class and star denotes complex conjugate. If the kernel H takes form H (ρ′, v) = τ (ρ′) exp(−2πiρ′·v) , where τ (ρ′) is a possibly complex profile function, the CSDs belong to the conventional Schell-like correlations. In recent studies of partially coherent fields, this correlation class has been used to define a huge variety of proper correlation functions by varying weighting function. As a new case of the possible partially coherent fields, we consider a Bessel Gaussian mode for kernel function H (ρ′, v)

H (ρ′, v ) = exp(−ρ′ 2 σ 2) Jn (vρ′) exp(−inϕ′),

W (ρ1 , ρ2 , z ) =

S (ρ , z ) = W (ρ , ρ , z ) =

(3)

(4)

= + where In is the modified Bessel function of the first kind of order n . We will term such a source the modified Bessel-correlated model source. The spectral degree of coherence (SDOC) is defined by the expression δ−2 ,

W0 (ρ′1 , ρ′2 ) W0 (ρ′1 , ρ′1 ) W0 (ρ′2 , ρ′2 )

ω2 (z ) = 4

. (5)

μ (ρ′1 , ρ′2 ) = exp[−in (ϕ′2 − ϕ′1)]

In (2ρ1′ 2 δ 2) In (2ρ2′ 2 δ 2)

2ρ1 ρ2

2

δ2Δ(z )

2

2

ρ +ρ − ρ22 ) ⎤ exp ⎡− 12 2 ⎤ ⎦ ⎣ w Δ(z ) ⎦

− ϕ1)],

(10)

δ−4 )

2ρ2 ⎤ ⎛ 2ρ2 ⎞ 1 . exp ⎡− 2 In 2 ⎢ Δ(z ) ⎣ w Δ(z ) ⎥ ⎦ ⎝ δ Δ(z ) ⎠ ⎜

∫ ρ2S (ρ, z ) d2ρ ∫ S (ρ, z ) d2ρ.

∞

In (x ) =

∑ m=0

. (6)

⎟

(11)

(12)

1 x 2m + n ⎛ ⎞ , m !Γ(m + n + 1) ⎝ 2 ⎠

(13)

where Γ is the Gamma function, we obtain

Fig. 1 shows the modulus of the SDOC of the modified Bessel-correlated model source for different values of parameters n and δ. From Fig. 1, we can see that the modulus of the SDOC are equal to 1 in the region near original point and ρ1′ = ρ2′, and that the greater values of parameters n and δ the slower the decrease of the modulus of SDOC to both sides of the diagonal ρ1′ = ρ2′. The CSD of a beam at a pair of points ρ1 = (ρ1 , ϕ1) and ρ2 = (ρ2 , ϕ2) in any transverse plane of the half-space z > 0 , in the paraxial domain, may be expressed in the form [2]

ω2 (z ) ∞

= 2w 2Δ(z )

∑ m1= 0

2m1+ n

(n + 2m1 + 1)! ⎛ w 2 ⎞ m1 !Γ(n + m1 + 1) ⎝ 2δ 2 ⎠ ⎜

∞

2m2 + n

2 (n + 2m2)! ⎡ ⎛w ⎞ ⎢ ∑ m !Γ(n + m + 1) 2δ 2 2 ⎝ ⎠ ⎣ m2 = 0 2 ⎜

⎟

⎟

−1

⎤ ⎥ . ⎦

(14)

To illustrate the evolutional behavior of the mean-squared beam width, we display in Fig. 3, ω (z ) for the modified Bessel-correlated beam with different values of order n and spatial coherence length δ. As can be seen from Fig. 3 the beam width and the diffusive angle increase with the increase of n. And the directionality of the beams gets worse with the decrease of the spatial coherence length δ. To describe more fully the quality of the novel family of partially coherent beam, one may use the beam quality factor, i.e., the M 2 factor. The M 2 factor for any beam with an axially symmetric intensity distribution can be defined as [27]

W (ρ1 , ρ2 , z ) ∼ = W0 (f1, f2) exp[i (f1·ρ1 + f2·ρ2 )] exp[iz (f12 − f22 ) 2k ] d 2f1 d 2f2 ,

∬

(7) where

∼ W0 (f1, f2) = (2π )−4

2 1

On substituting form Eq. (11) into Eq. (12) and applying the power series expansion of Bessel functions

Then, one finds that the SDOC μ at a pair of points ρ′1 and ρ′2 in the source plane takes the form

In (2ρ′1 ρ′2 δ 2)

) (ρ

Numerical illustrative examples for the evolution of the intensity are shown in Fig. 2 for different values of order n and spatial coherence length δ. The parameters of the source are chosen to be λ = 632.8nm and σ = 0.5mm . It can be seen from Fig. 2 that the intensity patterns of the new beams have propagation invariance in shape (but not size). Except the Gaussian distribution for n = 0, the transverse intensity distribution of the any member of the new family of field always retains a dark hollow pattern throughout the propagating process. The area of dark central region increases with the increase of the order n and spatial coherence length δ. The transport of the mean-squared beam width is an important propagation characteristic for describing the directionality of the beam, which is defined as [26]

W0 (ρ1′, ρ2′) = exp[− (ρ1′ 2 + ρ2′ 2 ) w 2] In (2ρ′1 ρ′2 δ 2) exp[−in (ϕ′2 − ϕ′1)],

μ (ρ1′, ρ2′) =

1 Δ(z )

− + 1. where Δ(z ) = The expression of the intensity at any point (ρ , z ) within the cross section of the beam, which is represented by the diagonal element of the CSD function, follows from Eq. (10)

where δ is a positive real constant. We obtain on substituting Eqs. (2) and (3) into Eq. (1) the CSD of the form

σ −2

(

( ) exp[in (ϕ

4z 2k −2 (w−4

where Jn is the nth-order Bessel function of the first kind, and a Gaussian function for the weighting function

w−2

ik

exp ⎡− 2z 1 − ⎣

× In

(2)

p (v ) = δ 2 exp(−δ 2v 2 4) (4π ),

1 Δ(z )

∬ W0 (ρ′1, ρ′2) exp[−i (f1 ·ρ′1 + f2 ·ρ′2)] d2ρ′1 d2ρ′2 , (8)

M 2 = 2πσ0 σ∞,

f1 = (f1 , θ1) and f2 = (f2 , θ2) are spatial-frequency vectors in the Fourier space, k is the wave number of the field. On substituting from

(15)

where σ0 and σ∞ are the second-order moments of the spectral density 2

Optics and Laser Technology 126 (2020) 106088

Z. Mei

Fig. 1. Modulus of the spectral degree of coherence of the modified Bessel-correlated model source for different values of parameters n and δ. (a) n = 0, δ = 0.5 mm; (b) n = 1, δ = 0.5 mm; (c) n = 2, δ = 0.5 mm; (d) n = 2, δ = 1.0 mm.

The expression for the source field spectral density S0 (ρ) and radiation field spectral density J (f) can be found by Eqs. (4) and (9), they take the following form:

distribution for source field and radiation field respectively, which are defined by the expressions

σ0 =

∫ ρ2S0 (ρ) d2ρ ∫ S0 (ρ) d2ρ,

(16)

σ∞ =

∫ f 2 J (f) d2 f ∫ J (f) d2 f.

(17)

S0 (ρ) = exp(−2ρ2 w 2) In (2ρ2 δ 2),

(18)

J (f ) = [w 2 (4μ)] exp[−f 2 (2μ)] In [w 2f 2 (2μδ 2)].

(19)

Fig. 2. Evolution of the intensity distribution of the modified Bessel-correlated vortex beams with parameters as in Fig. 1. 3

Optics and Laser Technology 126 (2020) 106088

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Fig. 3. Mean-squared beam width w(z) versus propagation distance z. (a) δ = 0.5 mm; (b) n = 1.

On substituting Eqs. (18) and (19) into Eqs. (16) and (17) respectively, applying Eq. (13) and calculating the resulting integral, and then substituting them into Eq. (15), we arrive at the formulas for the M 2 factor ∞

M 2 = 2πμw 2 ∑ k1= 0 ∞

× ∑ l1= 0

(n + 2k1 + 1) ! k1 ! Γ(n + k1 + 1)

(n + 2l1 + 1) ! l1 ! Γ(n + l1 + 1)

w2 2δ2

( )

2k1+ n

2l1+ n

⎡∑ ⎢l = 0 ⎣2

∞

( ) w2 2δ2

∞

⎡∑ ⎢k =0 ⎣2

(n + 2k2) ! k2 ! Γ(n + k2 + 1)

(n + 2l2) ! l2 ! Γ(n + l2 + 1)

( ) w2 2δ2

( ) w2 2δ2

2l2 + n

2k2+ n

W (ρ1 , ρ2 , z ) = × exp

{

ik − 2z

k 2 2πz

( ) ∫ ∫ d ρ′ ∫ ∫ d ρ′ W (ρ′ , ρ′ )

[(ρ1 − ρ′1

2

2

1

2

0

1

2

)2

}

− (ρ2 − ρ′2 )2] exp[ϕ ∗ (ρ1 , ρ′1 , z ) + ϕ (ρ2 , ρ′2 , z )]

−1

⎤ ⎥ ⎦

M

,

(21)

where ϕ denotes the complex phase perturbation induced by the random medium, and … M denotes the ensemble average of random media. Under a quadratic approximation of Rytov's phase structure function, the ensemble average term can be given by the expression [10,11]

−1

⎤ . ⎥ ⎦ (20)

exp[ϕ ∗ (ρ1 , ρ′1 , z ) + ϕ (ρ2 , ρ′2 , z )]

The behavior of the M 2 factor for the modified Bessel-correlated beam is plotted in Fig. 4 as a function of the spatial coherence length δ with n and σ as a parameter respectively. One can see from Fig. 4 that the M 2 factor initially increases with the increase of the spatial coherence length δ, has a turning point at a certain value of δ and reaches the maximum values, then decreases with the increase of δ and finally reaches a stable value. The maximum values of the M 2 factor increase with the increase of n, but they are the same for difference values of σ. However, the values of δ at the turning point increase with the increase of σ, but they are almost the same for different values of n.

M

= exp[−0.5Dϕ (ρ′1 − ρ′2 )]

= exp[−(ρ′1 − ρ′2 ) β02],

(22)

where Dϕ is the phase structure function and β0 = (0.545Cn2 k 2z )−3 5 is the coherence length of a spherical wave propagation in the turbulent medium, Cn2 is the refractive-index structure parameter. Substituting Eqs. (4) and (22) into Eq. (21), we arrive at the CSD function in the form

W (ρ1 , ρ2 , z ) =

k 2 2πz

( ) exp ⎡⎣− + A∗ ρ2′ 2 )] In

3. Propagation of modified Bessel-correlated vortex beams in atmospheric turbulence

(

ik 2z

∞ ∞ 2π 2π

(ρ12 − ρ22 ) ⎤ ∫ ∫ ∫ ∫ exp[−(Aρ1′ 2 ⎦ 0 0 0 0

2ρ′1 ρ′2 δ2

)

× exp[−in (ϕ′1 − ϕ′2)] exp[ikρ1 ρ′1 cos(ϕ1 − ϕ′1) z ] exp[−ikρ2 ρ′2 cos(ϕ2 − ϕ′2) z ]

Now we will analyze the propagation-induced spectral density of the modified Bessel-correlated vortex beam in atmospheric turbulence. According to the extended Huygens-Fresnel integral principle adjusted for propagation in linear random medium, the CSD function at two points ρ1 and ρ1 in the same transverse plane are related to those in the source plane as [28]

× exp[2ρ′1 ρ′2 cos(ϕ′1 − ϕ′2) β02 ] ρ′1 ρ′2 dρ′1 dρ′2 dϕ′1 dϕ′2 , A = 1 w 2 + 1 β02 + ik (2z ). Applying the following formula

Fig. 4. The M 2 factor versus the spatial coherence length δ. (a) σ = 1 mm; (b) n = 1. 4

(23)

Optics and Laser Technology 126 (2020) 106088

Z. Mei ∞

∑

exp[(ikρ1 ρ′1 z ) cos(ϕ1 − ϕ′1)] =

for the modified Bessel-correlated vortex beams with n = 2, σ = 0.5 mm and δ = 1 mm on propagation in atmospheric turbulence for different Cn2 is shown at several selected distances z from the source plane. One can see that from Fig. 5 that, for the weaker turbulence the intensity distribution retains a dark hollow pattern. In this case, the effect of atmospheric turbulence on the beam is not obvious, and the propagation characteristics of the beam are similar to those in free space. While for the strong turbulence, the influence of turbulence on the propagation of the beam becomes more and more obvious, and the profile of the hollow intensity is destroyed gradually with the increase of propagation distance and becomes Gaussian distribution. The larger Cn2 the shorter the propagation distance is required for forming a Gaussian profile.

ilJl (kρ1 ρ′1 z ) exp[il (ϕ1 − ϕ′1)],

l =−∞

(24) and firstly calculating the integral to the angle, Eq. (23) becomes

W (ρ1 , ρ2 , z ) =

k 2 z

( ) exp[−in (ϕ

1

− ϕ2)] exp ⎡ ⎣ ∞

ik

∞ ∞

− 2z (ρ12 − ρ22 ) ⎤ ∑ ∫ ∫ exp[−(Aρ1′ 2 + A∗ ρ2′ 2 )] ⎦ l =−∞ 0 0 × In

(

2ρ′1 ρ′2 δ2

)I

n+l ⎛

2ρ′1 ρ′2

⎝

β02

⎞ Jl ⎠

( ′ ) J ( ′ ) ρ′ ρ′ dρ′ dρ′ . kρ1 ρ 1

l

z

kρ2 ρ 2 z

1

2

1

2

(25)

Applying the power series expansion of Bessel functions (13) and calculating the resulting integral, we obtain the expression of the CSD function for the modified Bessel-correlated beam propagated in atmospheric turbulence:

W (ρ1 , ρ2 , z ) = ∞

∞

∞

× ⎧∑ ∑ ∑ ⎨ ⎩ l = 0 k1= 0 k2= 0 ∞

∞

k 2 z

( ) exp[−in (ϕ

1

4. Conclusions In this paper, we have considered a class of random sources that the radiate field possesses stable vortex form during propagation. Such source is obtained as an incoherent superposition of Bessel Gaussian elementary modes weighted by a Gaussian weight function. Our procedure lent itself to derive the simpler modal structure of modified Bessel-correlated beams in a simpler way. The propagation properties of the beams generated by such source are studied in detail. The results show that such beams possess two notable features which are different from other partially coherent vortex beams. First, the ring-shaped intensity distribution remains stable on propagating in free space and weak atmosphere turbulence. Second, unlike the existing vortex fields, intensity vortex only occur in the case of high coherence, such fields hold a ring-shaped intensity profile in both high coherence and low coherence cases. These features make them have unique advantages for applications involving particle manipulation in cases when the presence of the distance between the source and the particle need to be changed randomly.

ik

− ϕ2)] exp ⎡− 2z (ρ12 − ρ22 )⎤ ⎣ ⎦

l

(k2ρ1 ρ2 z 2) [(a − 1) !]2 1F1 (a; l + 1; y1) 1F1 (a; l + 1; y2 ) 22l + 2 (l !)2k1 ! k2 ! (n + k1) ! (n + l + k2) ! δ2(n + 2k1) β02(n + l + 2k 2) (AA∗ )a

∞

l

(k2ρ1 ρ2 z 2) [(b − 1) !]2 1F1 (b; l + 1; y1) 1F1 (b; l + 1; y2 ) ⎫, 2l + 2 (l !)2k ! k ! (n + k ) ! ( n − l + k ) ! δ2(n + 2k1) β 2( n − l +2k 2) (AA∗ )b ⎬ 1 2 1 2 0 l = 1 k1= 0 k2= 0 2 ⎭

+∑ ∑ ∑

(26) where 1F1 (a; l + 1; y ) is the confluent hyper-geometric function which can be expressed as ∞ 1F1 (a ;

l + 1; y ) =

Γ(a + i)Γ(l + 1) y i

∑ Γ(a)Γ(i + l + 1)Γ(i + 1) , i=0

(27)

a = (2n + 2k1 + 2k2 + 2l + 1) 2,

(28)

b = (n + n − l + 2k1 + 2k2 + l + 1) 2,

(29)

y1 =

−k 2ρ12

(4Az 2),

y2 = −k 2ρ22 (4A∗ z 2).

(30)

Declaration of Competing Interest

(31)

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to

In Fig. 5 the evolution of transverse cross-sections of the intensity

Fig. 5. Evolution of the intensity distribution of the modified Bessel-correlated vortex beams on propagation in atmospheric turbulence for different Cn2 . 5

Optics and Laser Technology 126 (2020) 106088

Z. Mei

influence the work reported in this paper.

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