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Degree of paraxiality of cylindrical vector partially coherent Laguerre–Gaussian beams
Optics Communications 333 (2014) 237–242
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Degree of paraxiality of cylindrical vector partially coherent Laguerre–Gaussian beams Yiming Dong a,b, Fei Wang b,n, Yangjian Cai b,n, Min Yao c a
Department of Physics, Shaoxing University, Shaoxing 312000, China College of Physics, Optoelectronics and Energy & Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China c School of Science and Technology, Zhejiang International Studies University, Hangzhou 310012, China b
art ic l e i nf o
a b s t r a c t
Article history: Received 8 June 2014 Received in revised form 28 July 2014 Accepted 30 July 2014 Available online 11 August 2014
We derive the analytical expression for the degree of paraxiality of cylindrical vector (CV) partially coherent Laguerre–Gaussian (LG) beams based on the definition of the degree of paraxiality. Numerical results show that the degree of paraxiality of the CV partially coherent LG beams does not always increase with the increase of the initial beam waist, but oscillates when the initial beam waist and wavelength are comparable. The degree of paraxiality also depends on the beam orders and the initial coherence length of the CV partially coherent LG beams. Our results are useful for determining whether the CV partially coherent LG beams are paraxial or nonparaxial. It may be useful in some applications such as atomic trapping, tight focusing and micro-lithography. & 2014 Elsevier B.V. All rights reserved.
Keywords: Cylindrical vector beams Partially coherence Degree of paraxiality
1. Introduction In the past decades, considerable attention has been paid to laser beams with spatially non-uniform state of polarization. Cylindrical vector (CV) beams such as radially and azimuthally polarized beam are a typical class of non-uniformly polarized beams that have cylindrical symmetry in polarization [1]. Due to the unique polarization characteristic, there has found many applications of the CV beams in optical trapping, dark-field microscopy, optical data storage and free-space optical communication [2–7]. Up to now, several theoretical models (e.g., Laguerre–Gaussian model, Bessel–Gaussian and Modified Bessel– Gaussian models) which are all the solutions of Maxwell's equation have been proposed to describe the CV beams [8–10], and much work has been devoted to experimental achievement of the CV beams including active and passive technique [11–16]. Meanwhile, the study of the paraxial or nonparaxial propagation and tight focusing properties of the CV beams remain of continued interest in optics [17–22]. Cai et al. analyzed the average intensity and polarization properties of radially or azimuthally polarized beams propagating through atmospheric turbulence [17]. The nonparaxial propagation properties of radially polarized LG beams were investigated in [18]. Recently, Dong et al. introduced the
n
Corresponding authors. E-mail addresses: [email protected] (F. Wang), [email protected] (Y. Cai). http://dx.doi.org/10.1016/j.optcom.2014.07.088 0030-4018/& 2014 Elsevier B.V. All rights reserved.
theoretical model for CV partially coherent LG beams as a natural extension of CV LG beams [23]. The statistical properties such as average intensity, degree of polarization and degree of coherence of CV partially coherent LG beams in free space under paraxial and nonparaxial region were explored in [23,24]. More recently, Wang et al. reported experimental generation of a partially coherent radially polarized beam [25] and a partially coherent azimuthally polarized beam [26], and measured the coherence and polarization properties of the partially coherent radially polarized beam [27]. It was found that the beam profile with dark-hollow and fattop in the focal plane can be achieved by controlling the initial spatial coherence of a partially coherent radially polarized beam, which will be useful for material thermal processing and particle trapping. More recently, Zhu et al. reported theoretical and experimental studies of the spectral changes of a polychromatic partially coherent radially polarized beam [28]. When a light beam is focused by a high numerical aperture (NA) lens or the beam width of a laser beam is comparable with its wavelength, it is usually regarded as a nonparaxial beam [29–32]. In recent years, research interest in description of optical beams in nonparaxial regime has been growing increasingly due to the development of nano-optics in which the linear dimension and the spatial scale of the optical structures are comparable with or even smaller than wavelength. Therefore, how to determine quantitively the light beams are nonparaxial or not is an interesting and practical topic in optics. In 2008, Gawhary et al. introduced a parameter named degree of paraxiality to measure the paraxiality of monochromatic light beams [33]. Later, they revised the
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Y. Dong et al. / Optics Communications 333 (2014) 237–242
definition of degree of paraxiality to a more correct form [34], and studied the influence of field correlations and field properties on the degree of paraxiality of monochromatic light beams [34,35]. The behaviors of degree of paraxiality of a Gaussian beam diffracted by a circular aperture were reported in [36]. Wang et al. extended the concept of the degree of paraxialy to partially coherent scalar and vector fields based on the definition of paraxiality introduced by Gawhary et al., and investigated the effect of the spatial coherence and degree of polarization of partially coherent beams on the degree of paraxiality [37,38]. Our motivation in this paper is to study the degree of paraxiality of CV partially coherent LG beams. The analytical expression for the degree of paraxiality of the CV partially coherent LG beams is derived. Then, we investigate the influence of the beam waist, beam orders and initial coherence length of the CV partially coherent LG beams on the degree of paraxiality through numerical examples in detail. Some interesting phenomena are discussed.
2. Degree of paraxiality of CV partially coherent LG beams In this section, we first briefly review the framework for extending the CV LG beams to CV partially coherent LG beams, and then obtain the analytical expression for the degree of paraxiality of the CV partially coherent LG beams. In cylindrical coordinate, the electric field of CV LG beams in the source plane (z¼0) is expressed [9] ! !ðn 7 1Þ=2 ! 2 r2 2r 2 n 7 1 2r E ðr; ϕ; z ¼ 0Þ ¼ exp 2 L p w0 w20 w20 8 9 < cos ðnϕÞ e ϕ 8 sin ðnϕÞ e r = ; ð1Þ : 7 sin ðnϕÞe ϕ þ cos ðnϕÞ e r ; where r and ϕ are the radial and azimuthal coordinates, respectively. Lnp 7 1 denotes the Laguerre polynomial of mode orders p and n 7 1; w0 is the beam waist of the fundamental Gaussian mode; e r and e ϕ are the unit vectors along the radial and azimuthal directions, respectively. If p ¼ n ¼ 0, Eq. (1) reduces to an electric field of a radially (lower part) or azimuthally (upper part) polarized LG01 beam. The relationship of unit vectors between in the cylindrical coordinate and in Cartesian coordinate reads as er
-
-
-
-
-
¼ cos ϕ e x þ sin ϕ e y ; e ϕ ¼ sin ϕ e x þ cos ϕ e y ; -
ð2Þ
-
where e x and e y represent the unit vectors along the x- and ydirection in Cartesian coordinate, respectively. Applying Eqs. (1) and (2), The electric field of the CV LG beams in the source plane takes the following alternative formula: ! !ðn 7 1Þ=2 ! 2 r2 2r 2 n 7 1 2r E 0 ðr; ϕ; 0Þ ¼ exp 2 L p w0 w20 w20 2 - 3 8 sin ðnϕ 7 ϕÞ e x þ cos ðnϕ 7 ϕÞ e y 4 5: ð3Þ cos ðnϕ 7 ϕÞ e x 7 sin ðnϕ 7 ϕÞ e y As shown in Eq. (1) and Eq. (3), there are four sets of the electric field of the CV LG beams, and each set represents the one basis of the CV LG beams. For sake of simplicity and without loss of generality, we only choose one set for analysis in the following context, which is -
-
-
E 0 ðr; ϕ; 0Þ ¼ Ex ðr; ϕ; 0Þ e x þ Ey ðr; ϕ; 0Þ e y ! !ðn þ 1Þ=2 ! 2 r2 2r 2 n þ 1 2r ¼ exp 2 L p w0 w20 w20 -
-
½ cos ðnϕ þ ϕÞ e x þ sin ðnϕ þ ϕÞ e y :
ð4Þ
Based on the unified theory of coherence and polarization, the second-order statistical properties of partially coherent vector beams in the space-frequency domain may be characterized by 2 2 cross spectral density (CSD) matrix at z ¼0 [20] * + ! Enx ðr1 ; 0Þ 2 W ðr1 ; r2 ; 0Þ ¼ ðEx ðr1 ; 0Þ; Ey ðr2 ; 0ÞÞ n Ey ðr2 ; 0Þ ! W xx ðr1 ; r2 ; 0ÞW xy ðr1 ; r2 ; 0Þ ¼ ; ð5Þ W yx ðr1 ; r2 ; 0ÞW yy ðr1 ; r2 ; 0Þ with W αβ ðr1 ; r2 ; 0Þ ¼ 〈Eα ðr1 ; r2 ; 0Þ Enβ ðr1 ; r2 ; 0Þ〉; ðα; β ¼ x; yÞ
ð6Þ
where r1 ðr 1 ; ϕ1 Þ and r2 ðr 2 ; ϕ2 Þ are the two arbitrary points in the source plane. The asterisk and angular brackets denote the complex conjugate and the ensemble average over the source fluctuation. Assume that the light beams are generated by Gaussian Schell2 model sources. For such sources, the elements of the W -matrix have the form [23] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi W αβ ðr1 ; r2 ; 0Þ ¼ Sα ðr1 ; 0ÞSβ ðr2 ; 0Þ g αβ ðr1 r2 ; 0Þ; ðα; β ¼ x; yÞ ð7Þ In Eq. (7), Sα ðr; 0Þ ¼ 〈Eα ðr; 0Þ Enα ðr; 0Þ〉 ðα ¼ x; yÞ denotes the spectral density of x- or y- component of the electric field, respectively. g αβ ðr1 r2 ; 0Þ is the correlation function between the two components of the electric field in the source plane, given by " # ðr1 r2 Þ2 g αβ ðr1 r2 ; 0Þ ¼ Bαβ exp ð8Þ ; ðα; β ¼ x; yÞ 2σ 2αβ where σ xx ,σ yy and σ xy are the r.m.s (or called initial coherence length) of the auto-correlation function of the x-component of the electric field, of the y-component of the electric filed and the mutual-correlation function of x and y field components, respectively. Bxx ¼ Byy ¼ 1, and Bαβ ¼ jBαβ jexpðiφαβ Þ ðα a β Þ are the complex correlation coefficients between the x and y field components with φαβ being the phase difference between the two components, and satisfy the relation Bxy ¼ Bnyx . If σ αβ -1, the CV partially coherent LG beams reduce to the CV LG beams. While σ αβ -0, the light beams become the completely incoherent CV LG beams. The parameters σ xx , σ yy , σ xy , jBxy j, φxy should satisfy several intrinsic constraints named realizability condition [39,40]. According to the analysis in [41], the realizability condition of the CV partially coherent LG beams is Bxy ¼ Byx ¼ 1, σ xx ¼ σ yy ¼ σ xy . In the following context, we use the parameter σ 0 to represent the r.m.s of the correlation function of x–x, y–y or x–y components of the electric field. Combining Eqs. (4)–(8), we obtain the expressions for each element of CSD matrix of the CV partially coherent LG beams, i.e., !ðn þ 1Þ=2 !ðn þ 1Þ=2 2r 21 2r 22 W xx ðr1 ; r2 ; 0Þ ¼ Lnp þ 1 w20 w20 ! ! 2r 21 n þ 1 2r 22 L cos ðnϕ1 þ ϕ1 Þ w20 p w20 " # r21 þ r22 ðr1 r2 Þ2 ; ð9Þ cos ðnϕ2 þ ϕ2 Þexp w20 2σ 20 !ðn þ 1Þ=2 2r 22 Lnp þ 1 W xy ðr1 ; r2 ; 0Þ ¼ w20 ! ! 2r 21 n þ 1 2r 22 L cos ðnϕ1 þ ϕ1 Þ w20 p w20 " # r2 þ r2 ðr1 r2 Þ2 sin ðnϕ2 þ ϕ2 Þexp 1 2 2 : w0 2σ 20 2r 21 w20
!ðn þ 1Þ=2
ð10Þ
Y. Dong et al. / Optics Communications 333 (2014) 237–242
W yx ðr1 ; r2 ; 0Þ ¼ ½W xy ðr1 ; r2 ; 0Þn ;
ð11Þ
!ðn þ 1Þ=2 2r 22 Lnp þ 1 W yy ðr1 ; r2 ; 0Þ ¼ w20 ! ! 2r 21 n þ 1 2r 22 L sin ðnϕ1 þ ϕ1 Þ w20 p w20 " # r2 þr2 ðr1 r2 Þ2 sin ðnϕ2 þ ϕ2 Þexp 1 2 2 : w0 2σ 20 2r 21 w20
j ¼ x;y
ð12Þ
ð13Þ
2
∬p2 o 1=λ2 ∑ Ajj ðp; pÞd p j ¼ x;y
where λ is the wavelength of light beams. Ajj ðp; pÞ; ðj ¼ x; yÞ is the angular correlation function of the elements W jj ðr1 ; r2 ; 0Þ of the partially coherent vector beams with p ðpx ; py Þ in Cartesian coordinate or p ðpr ; φr Þin the cylindrical coordinate. The above definition is derived from the comparison between the total energy flux along the propagation axis associated with the solution of Helmholtz equation and that associated with the solution of a corresponding the paraxial wave equation [33]. It is clear from Eq. (13) that 0 r P r 1 always holds. When P ¼ 1, the light beam can be viewed as a completely paraxial beam, while it is completely nonparaxial if P ¼ 0. For the intermediate value of P, The light beam can be regarded as a partially paraxial beam. The angular correlation function of the elements of CSD matrix is defined as the following integral [42,43]: Z 1 Z 1 2 2 Ajj ðp; pÞ ¼ W jj ðr1 ; r2 0Þ exp½2π ip U ðr1 r2 Þd r1 d r2 ; ðj ¼ x; yÞ 1
1
ð14Þ To evaluate Eq. (14), we introduce the sum and difference coordinate, i.e., r ¼ ðr1 þ r2 Þ=2; Δr ¼ r1 r2 :
ð15Þ
and we rewrite the elements of CSD matrix of CV partially coherent LG beams as follows: W jj ðr1 ; r2 0Þ ¼ Enj ðr1 ; 0ÞEj ðr2 ; 0Þg jj ðr1 r2 ; 0Þ;
ð16Þ
With Ej ðr; 0Þ ¼ exp
r2 w20
!
2r 2 w20
!ðn þ 1Þ=2 Lnp þ 1
2r 2 w20
!(
cos ðnϕ þ ϕÞ; j ¼ x : sin ðnϕ þ ϕÞ; j ¼ y ð17Þ
Substituting Eqs.(15) and (16) into Eq. (14), Eq. (14) is simplified as Z Ajj ðp; pÞ ¼
1 1
Z
1 1
g jj ðΔr;0Þ expði2πΔr U pÞd
2
Δr½Enj ðr þ Δr=2; 0ÞEj ðrΔr=2; 0Þd2 r: ð18Þ
Now, we write the function Enj ðr þ Δr=2; 0Þ and Ej ðrΔr=2; 0Þ in Eq. (18) in terms of their Fourier transform, i.e. R n 2 Enj ðr þ Δr=2; 0Þ ¼ E~ j ðu1 Þ exp½i2π ðr þ Δr=2Þ U u1 d u1 ; Z 2 Ej ðr Δr=2; 0Þ ¼ ð19Þ E~ j ðuÞ exp½ i2π ðrΔr=2Þ U ud u; where E~ j is the Fourier Transform of Ej . Substituting Eq. (19) into Eq. (18), and after some manipulations and integrations, it yields Z 1 2 jE~ j ðuÞj2 g~ jj ½ ðu þ pÞd u; ð20Þ Ajj ðp; pÞ ¼ 1
where g~ jj is the Fourier transform of g jj ðΔrÞ, given by g~ jj ð ðu þ pÞÞ ¼ 2πσ 20 exp½ 2π 2 σ 20 ðu þ pÞ2 :
!ðn þ 1Þ=2
Now, Let us turn to derive the degree of paraxiality of the CV partially coherent LG beams. On the basis of the theoretical analysis in [33,38], The degree of paraxiality of partially coherent vector beams is defined as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ∬p2 o 1=λ2 ∑ Ajj ðp; pÞ 1 λ p2 d p P¼
239
ð21Þ
As shown in Eq. (20), the angular correlation function is represented by the 2D integral, instead of 4D in Eq. (14). Apart from being simple, Analytical expression for jE~ j ðuÞj2 has been found for our practical problem, which have the form pffiffiffi jE~ j ðuÞj2 ¼ π 2 w40 ð 2π uw0 Þ2ðn þ 1Þ expð 2π 2 u2 w20 Þ ( cos 2 ðnφ þ φÞ; j ¼ x ½Lnp þ 1 ð2π 2 u2 w20 Þ2 ð22Þ sin 2 ðnφ þ φÞ; j ¼ y Substituting from Eqs. (21) and (22) into Eq. (20), and after integrating, we obtain the expression for the sum of the angular correlation function of x and y components in the cylindrical coordinate, i.e., ! p p 2π 2 σ 20 w20 p2r 2 2 2 ∑ Ajj ðpr ; pr Þ ¼ π σ 0 w0 exp ∑ ∑ 2 þ σ2 w s ¼ 0 t ¼ 0 j 0 0 ! ! p þ n þ1 p þ n þ 1 ðs þ t þ n þ 1Þ! ð 1Þs þ t ps pt s!t! !s þ t þ n þ 2 ! w2 2π 2 σ 4 p2 L0s þ t þ n þ 1 2 0 2r : 2 0 2 ð23Þ σ 0 þ w0 w0 þ σ 0 Substituting Eq. (23) into Eq. (13), and after tedious integration, we obtain P¼
P1 ; P2
ð24Þ
with P1 ¼
π 5=2 k2 σ 20 w20 8
p
∑ ∑
p
∑ ð 1Þs þ t
s¼0t ¼0m¼0
pþnþ1
p
!
p
ps
!
ðs þ t þ n þ 1Þ! p þ n þ 1 pt s!t!
!
pm
! !m 2 2 k σ 20 w20 k σ 40 5 1 F 1 1 þ m; þ m; 2 2ðw20 þ σ 20 Þ 2ðw20 þ σ 20 Þ !s þ t þ n þ 2 w2 2 0 2 ; σ 0 þ w0 P2 ¼
π w20
2
p
p
p
∑ ð 1Þs þ t
∑ ∑
s¼0t ¼0m¼0
pþnþ1 ps w20 2 σ 0 þ w20
!
p
pm !s þ t þ n þ 1
!
σ 20 w20
ðs þ t þ n þ 1Þ! p þ n þ 1 pt s!t!m!
!m m! Γ 1 þ m;
ð25Þ !
k σ 20 w20 2ðw20 þ σ 20 Þ 2
!! ; ð26Þ
where 1 F 1 ð U Þ denotes a hyperfunction, and Γ ð U Þ stands the incomplete Gama function. k ¼ 2π =λ is the wavenumber of the light beams. Eq. (24) is the main analytical result in our paper, which provides us a convenient way to study the degree of paraxiality of the CV partially coherent LG beams. As shown in Eq. (24), the degree of paraxiality of CV partially coherent LG beams is determined by the beam orders, initial coherence length and the beam waist. Note the degree of paraxiality of CV partially coherent LG beams is also independent of the propagation distance z, which is consistent with conclusion drawn by Gawhary et al. in Ref. [33].
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Y. Dong et al. / Optics Communications 333 (2014) 237–242
3. Numerical results In this section, as numerical examples, we will study the influence of the beam parameters, such as beam orders, initial coherence length and beam waist on the degree of paraxiality of CV partially coherent LG beams based on Eqs. (24)–(26) derived in Section 2. In numerical calculation, the wavelength of light beams is λ ¼ 632:8 nm and fixed throughout the following examples. Fig. 1 illustrates the variation of the degree of paraxiality of the CV partially coherent LG beams with the normalized beam waist w0 =λ for several values of beam order p. The other beam parameters in the calculation are n þ 1 ¼ 1 and σ 0 -1 (completely coherent case). As shown in Fig. 1, Both the beam waist and the beam order p determine the degree of paratiality of the CV partially coherent LG beams. When p ¼0 (Solid line), the degree of paraxiality increases monotonously as the beam waist size increases ð0 o w0 o 4λÞ, and tends to 1 when w0 4 4λ, indicating that the light beams become complete paraxial. For another limiting case ðw0 -0Þ, the light beam reduces to a point source, while the value of degree of paraxiality tends to 0.535, which represents the residual paraxiality the CV partially coherent LG beams remained as described in [34]. For p 40, the value of the degree of paraxiality is almost same to those of p¼ 0 for the two cases: w0 -0 and w0 44λ, respectively, but the degree of paraxiality increases in the way of oscillating as the beam waist increases ð0 ow0 o λÞ, which is much different from the case of p ¼0. Notice that the number of local peaks in the oscillation equals to the beam order p. Fig. 2 shows the dependence of the degree of paraxiality of the CV partially coherent LG beams on the normalized beam waist w0 =λ for different values of beam orders n þ 1 with p ¼ 1 and σ 0 -1. One finds from Fig. 2 that the beam order n þ 1 does not determine the number of local peaks appeared in lines of the variation of the degree of paraxiality with the beam waist, but the values of beam waist corresponding to the position of local peak increase gradually as the beam order n þ 1 increases. In order to explain the results obtained in Fig. 1 and Fig. 2, we calculate the divergence angle of the CV LG light beams ðσ 0 -1Þ beyond paraxial approximation defined by Porras in [44]. According to [44], the divergence angle of CV LG beams can be written as
Fig. 2. Dependence of the degree of paraxiality of CV partially coherent LG beams on normalized beam waist w0 =λ for different values of beam orders n þ 1. The other beam parameters are p ¼ 1, σ 0 -1.
where pz ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1=λ p2x p2y . θx and θy is the divergence angle of CV -E
LG beams along x and y direction, respectively. A ðpx ; py Þ and -H
A ðpx ; py Þ are the angular spectrum of the electric field and -E -Hn magnetic field of light beams, respectively. A A ; α
ðα ¼ x; y; zÞ denotes the angular energy flux along the α direction. -E -Hn R represents z component of Pow ¼ p2 þ p2 o 1=λ2 dpx dpy A A x
y
z
the total angular energy flux carried by the light beams in the region of p2x þ p2y o 1=λ . For an incident electromagnetic plane 2
-E
-H
wave, A ðpx ; py Þand A ðpx ; py Þ satisfy the relation rffiffiffi
-H
A ðpx ; py Þ ¼
ε - -E λ p A ðpx ; py Þ μ
ð28Þ
-E -Hn " -E -Hn -E -Hn # 1 1 þðpy =pz Þ A A B ðpx =pz Þ A A C ! 2 ðpx =pz Þ A A Z B x y x C tan 2 θxy 4 B " C # dp dp ¼ B -E -Hn -E -Hn -E -Hn C; C tan 2 θy Pow p2x þ p2y o 1=λ2 x y B 1 @ ðp =p Þ A A A þðpy =pz Þ A A ðpy =pz Þ A A x z 2 0
tan 2 θx
tan 2 θxy
y
x
ð27Þ
y
-
with p ¼ ðpx ; py ; pz Þ. Substituting from Eq. (23) into Eq. (27) and applying Eq. (28), Eq. (27) reduces to tan 2 θx tan 2 θxy tan 2 θxy tan 2 θy
!
rffiffiffi
ε 4 λ μ Pow
¼
1 p2x ∑ jAEj ðpx ; py jÞ2 dpx dpy 1 p2x þ p2y o 1=λ pz j ¼ x;y
Z
2
1 1
;
ð29Þ with Pow ¼ where
rffiffiffiZ
ε μ
p2x þ p2y o 1=λ
λpz ∑ jAEj ðpx ; py Þj2 dpx dpy ; j ¼ x;y
ð30Þ
∑ jAEj ðpx ; py Þj2 ¼ ∑ Ajj ðp; pÞ. It is clearly seen from Eq.
j ¼ x;y
Fig. 1. Dependence of the degree of paraxiality of CV partially coherent LG beams on normalized beam waist w0 =λ for different values of beam orders p. The other beam parameters are n þ 1 ¼ 1, σ 0 -1.
2
j ¼ x;y
(29) that the divergence angles of CV LG beams in x, y or x–y directions are all equal. In fact, the divergence angles of the CV LG beams in any direction are same because of the cylindrical symmetry of the intensity and polarization distribution. For simplicity, we introduce the angle θ to represent the divergence angle of the CV
Y. Dong et al. / Optics Communications 333 (2014) 237–242
Fig. 3. Dependence of the divergence angle of CV LG beams on the normalized beam waist w0 =λ for different beam orders p, with another beam order nþ 1 ¼ 1.
241
Fig. 5. The degree of paraxiality of CV partially coherent LG beams as a function of normalized beam waist w0 =λ for different values of initial coherence length with p ¼ 1, and n þ 1 ¼ 1.
of the beam along z-axis in homogenous domain ðp2 o 1=λ Þ, and the denominator represents the total energy flow S of the beam in homogenous domain. With the increase of the beam waist, Sz and S display different growth speeds and behavior. The growth speed of Sz displays an oscillatory behavior while the growth speed of S displays a monotonic behavior. Thus, the increase of the degree of paraxiality of a CV partially coherent LG beam also displays an oscillating behavior with the increase of the beam waist. Now we investigate the influence of the initial coherence length of the CV partially coherent LG beams on the degree of paraxiality. Fig. 5 shows the variation of degree of paraxiality of the CV partially coherent LG beams as a function of the normalized beam waist size for different values of σ 0 with beam orders p ¼ 1 and n þ 1 ¼ 1. As illustrated in Fig. 5, the initial coherence length has great influence on the degree of paraxiality. When the beam waist size large enough ðw0 4 3λÞ, the value of degree of paraxiality of the CV LG beams tends to one (Solid line), while the degree of paraxiality of the CV partially coherent LG beams does not tends to one. Moreover, the smaller the initial coherent length is, the more nonparaxiality the beam is. This is because the low coherence length leads to the high divergence angle of the beam. As a results, the degree of paraxiality decreases with the decrease of the initial coherence length. Note that the degree of paraxiality is almost independent of the beam waist when the value of the initial coherence length enough small ðσ 0 ¼ 0:05λ; dash dotted lineÞ. This phenomenon can be explained by the fact that the beam orders, beam waist size and the coherence length determine the degree of paraxiality together. With the decrease of the coherence length, the effect of the coherence length on degree of paraxiality appears gradually. When the coherence length is very small, this parameter plays a dominant role to determine to degree of paraxiality compared to other beam parameters. Therefore, the degree of paraxiality almost remains unchanged with the varying of w0 . 2
Fig. 4. Dependence of the divergence angle of CV LG beams on the normalized beam waist w0 =λ for different values of beam orders n þ 1 with p ¼ 1.
LG beam in any direction. Figs. 3,4 illustrate the variation of θ of CV LG beams with the normalized beam waist for different beam orders p and n þ 1, respectively. The other beam parameters in the calculation are identical to those used in Figs. 1 and 2. One finds from Figs. 3 and 4 that the divergence angle θ is also closely related to the beam orders and the beam waist. When the beam order n þ 1 is fixed (see in Fig. 3), θ decreases with the increase of beam wait size for p¼ 0, which means the directionality of CV LG beams becomes better. Thus, the degree of paraxiality increases. Forp 4 0, θ also exhibits the oscillating behavior when the beam waist starts to increase from the value of zero to about λ (0 ow0 o λ), and thus leading to the fluctuation of degree of paraxiality in this region (see in Fig. 1), which implies that the evolution properties of θ determines those of degree of paraxiality. From numerical calculation, we find that the position of the local dips in Fig. 3 coincides with the position of the local peaks in Fig. 1 for the corresponding lines between Fig. 1 and Fig. 3, indicating that the better the directionality of CV LG beams is, the larger value the degree of paraxiality is. The similar results also can be obtained compared with the figure between Fig. 2 and Fig. 4. In [38], the degree of paraxiality of a stochastic electromagnetic beam (i.e., partially coherent vector beam with space-invariant polarization state) increases monotonically with the increase of the beam waist. Compared to the results in [38], the degree of paraxiality of a CV partially coherent LG beam (i.e., partially coherent beam with space-invariant polarization state) increases in an oscillatory manner due to the impact of vector properties. Besides from the aspect of the divergence of beam, we also can explain the oscillating behavior of the degree of paraxiality in the following way. From the definition of the degree of paraxiality given by Eq. (13), the numerator denotes the total energy flow Sz
4. Conclusion We obtain an explicit expression for the degree of paraxiality of CV partially coherent LG beams based on the recently introduced definition of the degree of paraxiality. Our numerical results have shown that the degree of paraxiality of CV partially coherent LG beams is closely determined by the beam waist, beam orders and the initial coherence length. When the beam order is p ¼0, the degree of paraxiality increases monotonously with the decrease of beam waist if the size of beam waist is comparable to wavelength, and then tends to a stable value when the value of beam waist is
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enough large. For p 40, the degree of paraxiality increases in an oscillating way with the increase of the beam waist when the beam waist is comparable to wavelength. The number of the local peaks in increasing is equal to the beam order p, while the value of degree of paraxiality with w0 -0 and the value of the beam waist corresponding to the local peaks are closely determined by the beam order n þ 1. The initial coherence length of the CV partially coherent LG beams is another key factor to effect on the degree of paraxiality when the value of the coherence length is very small. The smaller the initial coherence length is, the more nonparaxial the CV partially coherent LG beams are. Our results provide a method for determining the CV partially coherent LG beams paraxial or nonparaxial. It may be useful in some applications such as micro-lithography, tight focusing, where nonparaxial beams are usually encountered. Acknowledgments This work is supported by the National Natural Science Foundation of China under Grant nos. 11304204&11274005&11104195&11304287, the National Natural Science Foundation of Zhejiang Province of China under Grant no. LQ13A040003, the Universities Natural Science Research Project of Jiangsu Province under Grant no. 11KJB140007, and the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. References [1] [2] [3] [4]
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Degree of paraxiality of cylindrical vector partially coherent Laguerre–Gaussian beams Yiming Dong a,b, Fei Wang b,n, Yangjian Cai b,n, Min Yao c a
Department of Physics, Shaoxing University, Shaoxing 312000, China College of Physics, Optoelectronics and Energy & Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China c School of Science and Technology, Zhejiang International Studies University, Hangzhou 310012, China b
art ic l e i nf o
a b s t r a c t
Article history: Received 8 June 2014 Received in revised form 28 July 2014 Accepted 30 July 2014 Available online 11 August 2014
We derive the analytical expression for the degree of paraxiality of cylindrical vector (CV) partially coherent Laguerre–Gaussian (LG) beams based on the definition of the degree of paraxiality. Numerical results show that the degree of paraxiality of the CV partially coherent LG beams does not always increase with the increase of the initial beam waist, but oscillates when the initial beam waist and wavelength are comparable. The degree of paraxiality also depends on the beam orders and the initial coherence length of the CV partially coherent LG beams. Our results are useful for determining whether the CV partially coherent LG beams are paraxial or nonparaxial. It may be useful in some applications such as atomic trapping, tight focusing and micro-lithography. & 2014 Elsevier B.V. All rights reserved.
Keywords: Cylindrical vector beams Partially coherence Degree of paraxiality
1. Introduction In the past decades, considerable attention has been paid to laser beams with spatially non-uniform state of polarization. Cylindrical vector (CV) beams such as radially and azimuthally polarized beam are a typical class of non-uniformly polarized beams that have cylindrical symmetry in polarization [1]. Due to the unique polarization characteristic, there has found many applications of the CV beams in optical trapping, dark-field microscopy, optical data storage and free-space optical communication [2–7]. Up to now, several theoretical models (e.g., Laguerre–Gaussian model, Bessel–Gaussian and Modified Bessel– Gaussian models) which are all the solutions of Maxwell's equation have been proposed to describe the CV beams [8–10], and much work has been devoted to experimental achievement of the CV beams including active and passive technique [11–16]. Meanwhile, the study of the paraxial or nonparaxial propagation and tight focusing properties of the CV beams remain of continued interest in optics [17–22]. Cai et al. analyzed the average intensity and polarization properties of radially or azimuthally polarized beams propagating through atmospheric turbulence [17]. The nonparaxial propagation properties of radially polarized LG beams were investigated in [18]. Recently, Dong et al. introduced the
n
Corresponding authors. E-mail addresses: [email protected] (F. Wang), [email protected] (Y. Cai). http://dx.doi.org/10.1016/j.optcom.2014.07.088 0030-4018/& 2014 Elsevier B.V. All rights reserved.
theoretical model for CV partially coherent LG beams as a natural extension of CV LG beams [23]. The statistical properties such as average intensity, degree of polarization and degree of coherence of CV partially coherent LG beams in free space under paraxial and nonparaxial region were explored in [23,24]. More recently, Wang et al. reported experimental generation of a partially coherent radially polarized beam [25] and a partially coherent azimuthally polarized beam [26], and measured the coherence and polarization properties of the partially coherent radially polarized beam [27]. It was found that the beam profile with dark-hollow and fattop in the focal plane can be achieved by controlling the initial spatial coherence of a partially coherent radially polarized beam, which will be useful for material thermal processing and particle trapping. More recently, Zhu et al. reported theoretical and experimental studies of the spectral changes of a polychromatic partially coherent radially polarized beam [28]. When a light beam is focused by a high numerical aperture (NA) lens or the beam width of a laser beam is comparable with its wavelength, it is usually regarded as a nonparaxial beam [29–32]. In recent years, research interest in description of optical beams in nonparaxial regime has been growing increasingly due to the development of nano-optics in which the linear dimension and the spatial scale of the optical structures are comparable with or even smaller than wavelength. Therefore, how to determine quantitively the light beams are nonparaxial or not is an interesting and practical topic in optics. In 2008, Gawhary et al. introduced a parameter named degree of paraxiality to measure the paraxiality of monochromatic light beams [33]. Later, they revised the
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definition of degree of paraxiality to a more correct form [34], and studied the influence of field correlations and field properties on the degree of paraxiality of monochromatic light beams [34,35]. The behaviors of degree of paraxiality of a Gaussian beam diffracted by a circular aperture were reported in [36]. Wang et al. extended the concept of the degree of paraxialy to partially coherent scalar and vector fields based on the definition of paraxiality introduced by Gawhary et al., and investigated the effect of the spatial coherence and degree of polarization of partially coherent beams on the degree of paraxiality [37,38]. Our motivation in this paper is to study the degree of paraxiality of CV partially coherent LG beams. The analytical expression for the degree of paraxiality of the CV partially coherent LG beams is derived. Then, we investigate the influence of the beam waist, beam orders and initial coherence length of the CV partially coherent LG beams on the degree of paraxiality through numerical examples in detail. Some interesting phenomena are discussed.
2. Degree of paraxiality of CV partially coherent LG beams In this section, we first briefly review the framework for extending the CV LG beams to CV partially coherent LG beams, and then obtain the analytical expression for the degree of paraxiality of the CV partially coherent LG beams. In cylindrical coordinate, the electric field of CV LG beams in the source plane (z¼0) is expressed [9] ! !ðn 7 1Þ=2 ! 2 r2 2r 2 n 7 1 2r E ðr; ϕ; z ¼ 0Þ ¼ exp 2 L p w0 w20 w20 8 9 < cos ðnϕÞ e ϕ 8 sin ðnϕÞ e r = ; ð1Þ : 7 sin ðnϕÞe ϕ þ cos ðnϕÞ e r ; where r and ϕ are the radial and azimuthal coordinates, respectively. Lnp 7 1 denotes the Laguerre polynomial of mode orders p and n 7 1; w0 is the beam waist of the fundamental Gaussian mode; e r and e ϕ are the unit vectors along the radial and azimuthal directions, respectively. If p ¼ n ¼ 0, Eq. (1) reduces to an electric field of a radially (lower part) or azimuthally (upper part) polarized LG01 beam. The relationship of unit vectors between in the cylindrical coordinate and in Cartesian coordinate reads as er
-
-
-
-
-
¼ cos ϕ e x þ sin ϕ e y ; e ϕ ¼ sin ϕ e x þ cos ϕ e y ; -
ð2Þ
-
where e x and e y represent the unit vectors along the x- and ydirection in Cartesian coordinate, respectively. Applying Eqs. (1) and (2), The electric field of the CV LG beams in the source plane takes the following alternative formula: ! !ðn 7 1Þ=2 ! 2 r2 2r 2 n 7 1 2r E 0 ðr; ϕ; 0Þ ¼ exp 2 L p w0 w20 w20 2 - 3 8 sin ðnϕ 7 ϕÞ e x þ cos ðnϕ 7 ϕÞ e y 4 5: ð3Þ cos ðnϕ 7 ϕÞ e x 7 sin ðnϕ 7 ϕÞ e y As shown in Eq. (1) and Eq. (3), there are four sets of the electric field of the CV LG beams, and each set represents the one basis of the CV LG beams. For sake of simplicity and without loss of generality, we only choose one set for analysis in the following context, which is -
-
-
E 0 ðr; ϕ; 0Þ ¼ Ex ðr; ϕ; 0Þ e x þ Ey ðr; ϕ; 0Þ e y ! !ðn þ 1Þ=2 ! 2 r2 2r 2 n þ 1 2r ¼ exp 2 L p w0 w20 w20 -
-
½ cos ðnϕ þ ϕÞ e x þ sin ðnϕ þ ϕÞ e y :
ð4Þ
Based on the unified theory of coherence and polarization, the second-order statistical properties of partially coherent vector beams in the space-frequency domain may be characterized by 2 2 cross spectral density (CSD) matrix at z ¼0 [20] * + ! Enx ðr1 ; 0Þ 2 W ðr1 ; r2 ; 0Þ ¼ ðEx ðr1 ; 0Þ; Ey ðr2 ; 0ÞÞ n Ey ðr2 ; 0Þ ! W xx ðr1 ; r2 ; 0ÞW xy ðr1 ; r2 ; 0Þ ¼ ; ð5Þ W yx ðr1 ; r2 ; 0ÞW yy ðr1 ; r2 ; 0Þ with W αβ ðr1 ; r2 ; 0Þ ¼ 〈Eα ðr1 ; r2 ; 0Þ Enβ ðr1 ; r2 ; 0Þ〉; ðα; β ¼ x; yÞ
ð6Þ
where r1 ðr 1 ; ϕ1 Þ and r2 ðr 2 ; ϕ2 Þ are the two arbitrary points in the source plane. The asterisk and angular brackets denote the complex conjugate and the ensemble average over the source fluctuation. Assume that the light beams are generated by Gaussian Schell2 model sources. For such sources, the elements of the W -matrix have the form [23] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi W αβ ðr1 ; r2 ; 0Þ ¼ Sα ðr1 ; 0ÞSβ ðr2 ; 0Þ g αβ ðr1 r2 ; 0Þ; ðα; β ¼ x; yÞ ð7Þ In Eq. (7), Sα ðr; 0Þ ¼ 〈Eα ðr; 0Þ Enα ðr; 0Þ〉 ðα ¼ x; yÞ denotes the spectral density of x- or y- component of the electric field, respectively. g αβ ðr1 r2 ; 0Þ is the correlation function between the two components of the electric field in the source plane, given by " # ðr1 r2 Þ2 g αβ ðr1 r2 ; 0Þ ¼ Bαβ exp ð8Þ ; ðα; β ¼ x; yÞ 2σ 2αβ where σ xx ,σ yy and σ xy are the r.m.s (or called initial coherence length) of the auto-correlation function of the x-component of the electric field, of the y-component of the electric filed and the mutual-correlation function of x and y field components, respectively. Bxx ¼ Byy ¼ 1, and Bαβ ¼ jBαβ jexpðiφαβ Þ ðα a β Þ are the complex correlation coefficients between the x and y field components with φαβ being the phase difference between the two components, and satisfy the relation Bxy ¼ Bnyx . If σ αβ -1, the CV partially coherent LG beams reduce to the CV LG beams. While σ αβ -0, the light beams become the completely incoherent CV LG beams. The parameters σ xx , σ yy , σ xy , jBxy j, φxy should satisfy several intrinsic constraints named realizability condition [39,40]. According to the analysis in [41], the realizability condition of the CV partially coherent LG beams is Bxy ¼ Byx ¼ 1, σ xx ¼ σ yy ¼ σ xy . In the following context, we use the parameter σ 0 to represent the r.m.s of the correlation function of x–x, y–y or x–y components of the electric field. Combining Eqs. (4)–(8), we obtain the expressions for each element of CSD matrix of the CV partially coherent LG beams, i.e., !ðn þ 1Þ=2 !ðn þ 1Þ=2 2r 21 2r 22 W xx ðr1 ; r2 ; 0Þ ¼ Lnp þ 1 w20 w20 ! ! 2r 21 n þ 1 2r 22 L cos ðnϕ1 þ ϕ1 Þ w20 p w20 " # r21 þ r22 ðr1 r2 Þ2 ; ð9Þ cos ðnϕ2 þ ϕ2 Þexp w20 2σ 20 !ðn þ 1Þ=2 2r 22 Lnp þ 1 W xy ðr1 ; r2 ; 0Þ ¼ w20 ! ! 2r 21 n þ 1 2r 22 L cos ðnϕ1 þ ϕ1 Þ w20 p w20 " # r2 þ r2 ðr1 r2 Þ2 sin ðnϕ2 þ ϕ2 Þexp 1 2 2 : w0 2σ 20 2r 21 w20
!ðn þ 1Þ=2
ð10Þ
Y. Dong et al. / Optics Communications 333 (2014) 237–242
W yx ðr1 ; r2 ; 0Þ ¼ ½W xy ðr1 ; r2 ; 0Þn ;
ð11Þ
!ðn þ 1Þ=2 2r 22 Lnp þ 1 W yy ðr1 ; r2 ; 0Þ ¼ w20 ! ! 2r 21 n þ 1 2r 22 L sin ðnϕ1 þ ϕ1 Þ w20 p w20 " # r2 þr2 ðr1 r2 Þ2 sin ðnϕ2 þ ϕ2 Þexp 1 2 2 : w0 2σ 20 2r 21 w20
j ¼ x;y
ð12Þ
ð13Þ
2
∬p2 o 1=λ2 ∑ Ajj ðp; pÞd p j ¼ x;y
where λ is the wavelength of light beams. Ajj ðp; pÞ; ðj ¼ x; yÞ is the angular correlation function of the elements W jj ðr1 ; r2 ; 0Þ of the partially coherent vector beams with p ðpx ; py Þ in Cartesian coordinate or p ðpr ; φr Þin the cylindrical coordinate. The above definition is derived from the comparison between the total energy flux along the propagation axis associated with the solution of Helmholtz equation and that associated with the solution of a corresponding the paraxial wave equation [33]. It is clear from Eq. (13) that 0 r P r 1 always holds. When P ¼ 1, the light beam can be viewed as a completely paraxial beam, while it is completely nonparaxial if P ¼ 0. For the intermediate value of P, The light beam can be regarded as a partially paraxial beam. The angular correlation function of the elements of CSD matrix is defined as the following integral [42,43]: Z 1 Z 1 2 2 Ajj ðp; pÞ ¼ W jj ðr1 ; r2 0Þ exp½2π ip U ðr1 r2 Þd r1 d r2 ; ðj ¼ x; yÞ 1
1
ð14Þ To evaluate Eq. (14), we introduce the sum and difference coordinate, i.e., r ¼ ðr1 þ r2 Þ=2; Δr ¼ r1 r2 :
ð15Þ
and we rewrite the elements of CSD matrix of CV partially coherent LG beams as follows: W jj ðr1 ; r2 0Þ ¼ Enj ðr1 ; 0ÞEj ðr2 ; 0Þg jj ðr1 r2 ; 0Þ;
ð16Þ
With Ej ðr; 0Þ ¼ exp
r2 w20
!
2r 2 w20
!ðn þ 1Þ=2 Lnp þ 1
2r 2 w20
!(
cos ðnϕ þ ϕÞ; j ¼ x : sin ðnϕ þ ϕÞ; j ¼ y ð17Þ
Substituting Eqs.(15) and (16) into Eq. (14), Eq. (14) is simplified as Z Ajj ðp; pÞ ¼
1 1
Z
1 1
g jj ðΔr;0Þ expði2πΔr U pÞd
2
Δr½Enj ðr þ Δr=2; 0ÞEj ðrΔr=2; 0Þd2 r: ð18Þ
Now, we write the function Enj ðr þ Δr=2; 0Þ and Ej ðrΔr=2; 0Þ in Eq. (18) in terms of their Fourier transform, i.e. R n 2 Enj ðr þ Δr=2; 0Þ ¼ E~ j ðu1 Þ exp½i2π ðr þ Δr=2Þ U u1 d u1 ; Z 2 Ej ðr Δr=2; 0Þ ¼ ð19Þ E~ j ðuÞ exp½ i2π ðrΔr=2Þ U ud u; where E~ j is the Fourier Transform of Ej . Substituting Eq. (19) into Eq. (18), and after some manipulations and integrations, it yields Z 1 2 jE~ j ðuÞj2 g~ jj ½ ðu þ pÞd u; ð20Þ Ajj ðp; pÞ ¼ 1
where g~ jj is the Fourier transform of g jj ðΔrÞ, given by g~ jj ð ðu þ pÞÞ ¼ 2πσ 20 exp½ 2π 2 σ 20 ðu þ pÞ2 :
!ðn þ 1Þ=2
Now, Let us turn to derive the degree of paraxiality of the CV partially coherent LG beams. On the basis of the theoretical analysis in [33,38], The degree of paraxiality of partially coherent vector beams is defined as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ∬p2 o 1=λ2 ∑ Ajj ðp; pÞ 1 λ p2 d p P¼
239
ð21Þ
As shown in Eq. (20), the angular correlation function is represented by the 2D integral, instead of 4D in Eq. (14). Apart from being simple, Analytical expression for jE~ j ðuÞj2 has been found for our practical problem, which have the form pffiffiffi jE~ j ðuÞj2 ¼ π 2 w40 ð 2π uw0 Þ2ðn þ 1Þ expð 2π 2 u2 w20 Þ ( cos 2 ðnφ þ φÞ; j ¼ x ½Lnp þ 1 ð2π 2 u2 w20 Þ2 ð22Þ sin 2 ðnφ þ φÞ; j ¼ y Substituting from Eqs. (21) and (22) into Eq. (20), and after integrating, we obtain the expression for the sum of the angular correlation function of x and y components in the cylindrical coordinate, i.e., ! p p 2π 2 σ 20 w20 p2r 2 2 2 ∑ Ajj ðpr ; pr Þ ¼ π σ 0 w0 exp ∑ ∑ 2 þ σ2 w s ¼ 0 t ¼ 0 j 0 0 ! ! p þ n þ1 p þ n þ 1 ðs þ t þ n þ 1Þ! ð 1Þs þ t ps pt s!t! !s þ t þ n þ 2 ! w2 2π 2 σ 4 p2 L0s þ t þ n þ 1 2 0 2r : 2 0 2 ð23Þ σ 0 þ w0 w0 þ σ 0 Substituting Eq. (23) into Eq. (13), and after tedious integration, we obtain P¼
P1 ; P2
ð24Þ
with P1 ¼
π 5=2 k2 σ 20 w20 8
p
∑ ∑
p
∑ ð 1Þs þ t
s¼0t ¼0m¼0
pþnþ1
p
!
p
ps
!
ðs þ t þ n þ 1Þ! p þ n þ 1 pt s!t!
!
pm
! !m 2 2 k σ 20 w20 k σ 40 5 1 F 1 1 þ m; þ m; 2 2ðw20 þ σ 20 Þ 2ðw20 þ σ 20 Þ !s þ t þ n þ 2 w2 2 0 2 ; σ 0 þ w0 P2 ¼
π w20
2
p
p
p
∑ ð 1Þs þ t
∑ ∑
s¼0t ¼0m¼0
pþnþ1 ps w20 2 σ 0 þ w20
!
p
pm !s þ t þ n þ 1
!
σ 20 w20
ðs þ t þ n þ 1Þ! p þ n þ 1 pt s!t!m!
!m m! Γ 1 þ m;
ð25Þ !
k σ 20 w20 2ðw20 þ σ 20 Þ 2
!! ; ð26Þ
where 1 F 1 ð U Þ denotes a hyperfunction, and Γ ð U Þ stands the incomplete Gama function. k ¼ 2π =λ is the wavenumber of the light beams. Eq. (24) is the main analytical result in our paper, which provides us a convenient way to study the degree of paraxiality of the CV partially coherent LG beams. As shown in Eq. (24), the degree of paraxiality of CV partially coherent LG beams is determined by the beam orders, initial coherence length and the beam waist. Note the degree of paraxiality of CV partially coherent LG beams is also independent of the propagation distance z, which is consistent with conclusion drawn by Gawhary et al. in Ref. [33].
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Y. Dong et al. / Optics Communications 333 (2014) 237–242
3. Numerical results In this section, as numerical examples, we will study the influence of the beam parameters, such as beam orders, initial coherence length and beam waist on the degree of paraxiality of CV partially coherent LG beams based on Eqs. (24)–(26) derived in Section 2. In numerical calculation, the wavelength of light beams is λ ¼ 632:8 nm and fixed throughout the following examples. Fig. 1 illustrates the variation of the degree of paraxiality of the CV partially coherent LG beams with the normalized beam waist w0 =λ for several values of beam order p. The other beam parameters in the calculation are n þ 1 ¼ 1 and σ 0 -1 (completely coherent case). As shown in Fig. 1, Both the beam waist and the beam order p determine the degree of paratiality of the CV partially coherent LG beams. When p ¼0 (Solid line), the degree of paraxiality increases monotonously as the beam waist size increases ð0 o w0 o 4λÞ, and tends to 1 when w0 4 4λ, indicating that the light beams become complete paraxial. For another limiting case ðw0 -0Þ, the light beam reduces to a point source, while the value of degree of paraxiality tends to 0.535, which represents the residual paraxiality the CV partially coherent LG beams remained as described in [34]. For p 40, the value of the degree of paraxiality is almost same to those of p¼ 0 for the two cases: w0 -0 and w0 44λ, respectively, but the degree of paraxiality increases in the way of oscillating as the beam waist increases ð0 ow0 o λÞ, which is much different from the case of p ¼0. Notice that the number of local peaks in the oscillation equals to the beam order p. Fig. 2 shows the dependence of the degree of paraxiality of the CV partially coherent LG beams on the normalized beam waist w0 =λ for different values of beam orders n þ 1 with p ¼ 1 and σ 0 -1. One finds from Fig. 2 that the beam order n þ 1 does not determine the number of local peaks appeared in lines of the variation of the degree of paraxiality with the beam waist, but the values of beam waist corresponding to the position of local peak increase gradually as the beam order n þ 1 increases. In order to explain the results obtained in Fig. 1 and Fig. 2, we calculate the divergence angle of the CV LG light beams ðσ 0 -1Þ beyond paraxial approximation defined by Porras in [44]. According to [44], the divergence angle of CV LG beams can be written as
Fig. 2. Dependence of the degree of paraxiality of CV partially coherent LG beams on normalized beam waist w0 =λ for different values of beam orders n þ 1. The other beam parameters are p ¼ 1, σ 0 -1.
where pz ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1=λ p2x p2y . θx and θy is the divergence angle of CV -E
LG beams along x and y direction, respectively. A ðpx ; py Þ and -H
A ðpx ; py Þ are the angular spectrum of the electric field and -E -Hn magnetic field of light beams, respectively. A A ; α
ðα ¼ x; y; zÞ denotes the angular energy flux along the α direction. -E -Hn R represents z component of Pow ¼ p2 þ p2 o 1=λ2 dpx dpy A A x
y
z
the total angular energy flux carried by the light beams in the region of p2x þ p2y o 1=λ . For an incident electromagnetic plane 2
-E
-H
wave, A ðpx ; py Þand A ðpx ; py Þ satisfy the relation rffiffiffi
-H
A ðpx ; py Þ ¼
ε - -E λ p A ðpx ; py Þ μ
ð28Þ
-E -Hn " -E -Hn -E -Hn # 1 1 þðpy =pz Þ A A B ðpx =pz Þ A A C ! 2 ðpx =pz Þ A A Z B x y x C tan 2 θxy 4 B " C # dp dp ¼ B -E -Hn -E -Hn -E -Hn C; C tan 2 θy Pow p2x þ p2y o 1=λ2 x y B 1 @ ðp =p Þ A A A þðpy =pz Þ A A ðpy =pz Þ A A x z 2 0
tan 2 θx
tan 2 θxy
y
x
ð27Þ
y
-
with p ¼ ðpx ; py ; pz Þ. Substituting from Eq. (23) into Eq. (27) and applying Eq. (28), Eq. (27) reduces to tan 2 θx tan 2 θxy tan 2 θxy tan 2 θy
!
rffiffiffi
ε 4 λ μ Pow
¼
1 p2x ∑ jAEj ðpx ; py jÞ2 dpx dpy 1 p2x þ p2y o 1=λ pz j ¼ x;y
Z
2
1 1
;
ð29Þ with Pow ¼ where
rffiffiffiZ
ε μ
p2x þ p2y o 1=λ
λpz ∑ jAEj ðpx ; py Þj2 dpx dpy ; j ¼ x;y
ð30Þ
∑ jAEj ðpx ; py Þj2 ¼ ∑ Ajj ðp; pÞ. It is clearly seen from Eq.
j ¼ x;y
Fig. 1. Dependence of the degree of paraxiality of CV partially coherent LG beams on normalized beam waist w0 =λ for different values of beam orders p. The other beam parameters are n þ 1 ¼ 1, σ 0 -1.
2
j ¼ x;y
(29) that the divergence angles of CV LG beams in x, y or x–y directions are all equal. In fact, the divergence angles of the CV LG beams in any direction are same because of the cylindrical symmetry of the intensity and polarization distribution. For simplicity, we introduce the angle θ to represent the divergence angle of the CV
Y. Dong et al. / Optics Communications 333 (2014) 237–242
Fig. 3. Dependence of the divergence angle of CV LG beams on the normalized beam waist w0 =λ for different beam orders p, with another beam order nþ 1 ¼ 1.
241
Fig. 5. The degree of paraxiality of CV partially coherent LG beams as a function of normalized beam waist w0 =λ for different values of initial coherence length with p ¼ 1, and n þ 1 ¼ 1.
of the beam along z-axis in homogenous domain ðp2 o 1=λ Þ, and the denominator represents the total energy flow S of the beam in homogenous domain. With the increase of the beam waist, Sz and S display different growth speeds and behavior. The growth speed of Sz displays an oscillatory behavior while the growth speed of S displays a monotonic behavior. Thus, the increase of the degree of paraxiality of a CV partially coherent LG beam also displays an oscillating behavior with the increase of the beam waist. Now we investigate the influence of the initial coherence length of the CV partially coherent LG beams on the degree of paraxiality. Fig. 5 shows the variation of degree of paraxiality of the CV partially coherent LG beams as a function of the normalized beam waist size for different values of σ 0 with beam orders p ¼ 1 and n þ 1 ¼ 1. As illustrated in Fig. 5, the initial coherence length has great influence on the degree of paraxiality. When the beam waist size large enough ðw0 4 3λÞ, the value of degree of paraxiality of the CV LG beams tends to one (Solid line), while the degree of paraxiality of the CV partially coherent LG beams does not tends to one. Moreover, the smaller the initial coherent length is, the more nonparaxiality the beam is. This is because the low coherence length leads to the high divergence angle of the beam. As a results, the degree of paraxiality decreases with the decrease of the initial coherence length. Note that the degree of paraxiality is almost independent of the beam waist when the value of the initial coherence length enough small ðσ 0 ¼ 0:05λ; dash dotted lineÞ. This phenomenon can be explained by the fact that the beam orders, beam waist size and the coherence length determine the degree of paraxiality together. With the decrease of the coherence length, the effect of the coherence length on degree of paraxiality appears gradually. When the coherence length is very small, this parameter plays a dominant role to determine to degree of paraxiality compared to other beam parameters. Therefore, the degree of paraxiality almost remains unchanged with the varying of w0 . 2
Fig. 4. Dependence of the divergence angle of CV LG beams on the normalized beam waist w0 =λ for different values of beam orders n þ 1 with p ¼ 1.
LG beam in any direction. Figs. 3,4 illustrate the variation of θ of CV LG beams with the normalized beam waist for different beam orders p and n þ 1, respectively. The other beam parameters in the calculation are identical to those used in Figs. 1 and 2. One finds from Figs. 3 and 4 that the divergence angle θ is also closely related to the beam orders and the beam waist. When the beam order n þ 1 is fixed (see in Fig. 3), θ decreases with the increase of beam wait size for p¼ 0, which means the directionality of CV LG beams becomes better. Thus, the degree of paraxiality increases. Forp 4 0, θ also exhibits the oscillating behavior when the beam waist starts to increase from the value of zero to about λ (0 ow0 o λ), and thus leading to the fluctuation of degree of paraxiality in this region (see in Fig. 1), which implies that the evolution properties of θ determines those of degree of paraxiality. From numerical calculation, we find that the position of the local dips in Fig. 3 coincides with the position of the local peaks in Fig. 1 for the corresponding lines between Fig. 1 and Fig. 3, indicating that the better the directionality of CV LG beams is, the larger value the degree of paraxiality is. The similar results also can be obtained compared with the figure between Fig. 2 and Fig. 4. In [38], the degree of paraxiality of a stochastic electromagnetic beam (i.e., partially coherent vector beam with space-invariant polarization state) increases monotonically with the increase of the beam waist. Compared to the results in [38], the degree of paraxiality of a CV partially coherent LG beam (i.e., partially coherent beam with space-invariant polarization state) increases in an oscillatory manner due to the impact of vector properties. Besides from the aspect of the divergence of beam, we also can explain the oscillating behavior of the degree of paraxiality in the following way. From the definition of the degree of paraxiality given by Eq. (13), the numerator denotes the total energy flow Sz
4. Conclusion We obtain an explicit expression for the degree of paraxiality of CV partially coherent LG beams based on the recently introduced definition of the degree of paraxiality. Our numerical results have shown that the degree of paraxiality of CV partially coherent LG beams is closely determined by the beam waist, beam orders and the initial coherence length. When the beam order is p ¼0, the degree of paraxiality increases monotonously with the decrease of beam waist if the size of beam waist is comparable to wavelength, and then tends to a stable value when the value of beam waist is
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enough large. For p 40, the degree of paraxiality increases in an oscillating way with the increase of the beam waist when the beam waist is comparable to wavelength. The number of the local peaks in increasing is equal to the beam order p, while the value of degree of paraxiality with w0 -0 and the value of the beam waist corresponding to the local peaks are closely determined by the beam order n þ 1. The initial coherence length of the CV partially coherent LG beams is another key factor to effect on the degree of paraxiality when the value of the coherence length is very small. The smaller the initial coherence length is, the more nonparaxial the CV partially coherent LG beams are. Our results provide a method for determining the CV partially coherent LG beams paraxial or nonparaxial. It may be useful in some applications such as micro-lithography, tight focusing, where nonparaxial beams are usually encountered. Acknowledgments This work is supported by the National Natural Science Foundation of China under Grant nos. 11304204&11274005&11104195&11304287, the National Natural Science Foundation of Zhejiang Province of China under Grant no. LQ13A040003, the Universities Natural Science Research Project of Jiangsu Province under Grant no. 11KJB140007, and the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. References [1] [2] [3] [4]
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