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Letter
Vol. 43, No. 15 / 1 August 2018 / Optics Letters
Twisted Gaussian Schell-model array beams LIPENG WAN
AND
DAOMU ZHAO*
Department of Physics, Zhejiang University, Hangzhou 310027, China *Corresponding author:
[email protected] Received 30 May 2018; accepted 24 June 2018; posted 27 June 2018 (Doc. ID 332985); published 19 July 2018
We introduce, to the best of our knowledge, a new class of twisted partially coherent sources for producing rotating Gaussian array profiles. The general analytical formula for the cross-spectral density function of a beam generated by such a novel source propagating in free space is derived, and its propagation characteristics are analyzed. It is shown that both the irradiance profile of each element of the array and the degree of coherence rotate during propagation, but in opposite directions. Further, the twist effects of the spectral density and the degree of coherence are quantified. It is shown that the direction of rotation is changeable upon propagation. Our results may provide new insight into the twist phase and may be applied in optical trapping. © 2018 Optical Society of America OCIS codes: (030.0030) Coherence and statistical optics; (030.1640) Coherence; (030.6600) Statistical optics; (350.5500) Propagation. https://doi.org/10.1364/OL.43.003554
Twist phase, discovered 25 years ago by Simon and Mukunda [1], is a nonseparable quadratic phase whose presence induces the rotation of the beam spot during propagation. Simon and Mukunda proposed that such a phase could be imposed on the conventional Gaussian Schell-model sources to introduce a class of sources referred to as the twisted Gaussian Schell-model (TGSM). Specifically, the cross-spectral density (CSD) of a resulting TGSM source at points specified by position vectors r 1 and r 2 is given by 2 r 1 r 22 r 1 − r 2 2 exp − W T r 1 , r 2 exp − 4σ 20 2δ2μ × exp−iux 1 y 2 − x 2 y 1 ,
(1)
where σ 0 is the width of the source intensity, δμ is related to the coherence width, and u characterizes the strength of beam twist. For brevity, we omit the dependence on frequency. It was further found by Simon and Mukunda that, in order for such a source to be physically realizable, the strength of the beam twist should be bounded by inequality uδ2μ ≤ 1 [1]. As a result, a completely coherent beam, for which δμ → ∞, cannot be twisted at all. Since the introduction of the pioneering concept described above, twist phase has been a subject of interest because of its distinctive properties [2–13]. However, a long-standing 0146-9592/18/153554-04 Journal © 2018 Optical Society of America
challenge is that the twist phase cannot be arbitrarily imposed on partially coherent sources, because the result can violate the nonnegative definiteness of the corresponding CSD function. Consequently, studies involving twist phase, though they have been going on for 25 years, are restricted to axially symmetric TGSM sources. Recently, considerable efforts have been devoted to addressing this challenge. For instance, Borghi et al. used a model analysis to evaluate a bona fide twisted Schell-model source [14]. Mei and Korotkova proposed a modal representation method for depicting twisted partially coherent sources [15]. Quite recently, Gori and Santarsiero provided a general mathematical description to devise bona fide twisted sources [16], and the necessary and sufficient condition of a bona fide twisted CSD endowed with axial symmetry was derived by Borghi [17]. Undoubtedly, these studies have paved the way towards a new age of exploration involving the twist phase. The majority of publications concerned with twist phase deal with axially symmetric TGSM sources (a notable exception is the twisted flattop beams proposed by Mei and Korotkova [15]), and the lack of nontrivial twisted sources restricts the opportunity to further investigate the twist effect. In the present Letter, we employ the modeling procedure suggested by Gori and Santarsiero [16] to introduce a novel class of twisted sources, which is different from the TGSM one, to the best of our knowledge. Such sources are characterized by the rotating Gaussian array patterns they radiated. To provide a detailed insight into the beams generated by such twisted sources, we shall derive the general analytical formula for the propagated CSD function. Some interesting phenomena regarding the twist phase are shown when the general analytical formula is considered. We start by considering a beam propagating along the z axis. Its second-order correlation properties in the plane z 0 may be depicted by the CSD function W 0 r 1 , r 2 at a pair of points r 1 x 1 , y 1 and r 2 x 2 , y 2 . Suppose that the beam axis is tilted to wave vector kr 0 2παy0 , − x 0 , where α is a real number. The CSD function W 0 r 1 , r 2 should then be correspondingly transformed into W 1 r 1 , r 2 ; r 0 , viz. [11], W 1 r 1 , r 2 ; r 0 W 0 r 1 − r 0 , r 2 − r 0 exp2πiv12 · r 0 , (2) where v12 is equal to αy 2 − y 1 , x 1 − x 2 , which denotes a rotated version of D r 1 − r 2 . Notice that an exponential term is now introduced through wave vector transformation. By superimposing an ensemble of such mutually uncorrelated tilted fields W 1 r 1 , r 2 ; r 0 weighted by the nonnegative function p0 r 0 , and introducing a new variable r 0 r m − r with
Vol. 43, No. 15 / 1 August 2018 / Optics Letters
Letter
2 2 x y τr exp − 2 exp − 2 , 4σ x 4σ y
r m r 1 r 2 ∕2, we obtain the following expression for a bona fide twisted CSD [16]: W r 1 , r 2 Rr 1 , r 2 exp2πiv12 · r m ,
(3)
where v12 · r m is a twist term and equals αx 1 y2 − x 2 y 1 and function R is referred to as the reminder, viz., Rr 1 , r 2 Z p0 r m − rW 0 r D∕2, r − D∕2 exp−2πiv12 · rd2 r: (4)
W 0 r 1 ,r 2 2 r r2 x − x 2 y − y 2 exp − 1 2 2 exp − 1 2 2 exp − 1 2 2 4σ 0 2δx0 2δy0 Q P X 2πny R y0 y 1 − y 2 2πnx R x0 x 1 − x 2 X × cos cos , δy0 δx0 n −P n −Q x
y
(6) where P N x − 1∕2 and Q N y − 1∕2; N x and N y are positive integrals that determine the number of lobes of the array. On substituting from Eqs. (4)–(6) into Eq. (3) and introducing the following simplifications: u −2πα; 1 1 1 u2 σ 20 ; 2 2 2 2 2δj 8σ 0 2δj0 −1∕2 R j0 1 1 2 2 u σ0 , Rj δj0 4σ 20 δ2j0
(7)
P X nx −P
cosC x x 1 − x 2
Q X ny −Q
× exp−iux 1 y 2 expiux 2 y 1 ,
cosC y y 1 − y 2 (8)
where C j 2πnj R j ∕δj , R j and δj are coherence parameters, and u denotes the twist strength. So far, we have not imposed any restrictions on the amplitude of the source; this can be seen from the fact that spectral density Sr W r, r is independent of coordinates. It is therefore necessary to place a proper amplitude mask τr in front of the source as a spatial limitation of the actual source. Without loss of generality, we then assume that
P X nx −P
cosC x x 1 − x 2
Q X ny −Q
× exp−iux 1 y 2 expiux 2 y 1 :
cosC y y1 − y2 (10)
Partially coherent sources defined by Eq. (10) may be referred to as twisted Gaussian Schell-model arrays (TGSMA). It should be emphasized that the nonnegative constraint of the CSD has been automatically satisfied, and the symmetry has been broken in the above derivation. To analyze the paraxial propagation characteristics of beams generated by TGSMA sources, one may turn to the Fresnel diffraction formula [21], ZZ W ρ1 , ρ2 , z W r 1 , r 2 H z ρ1 , r 1 H z ρ2 , r 2 d2 r 1 d2 r 2 , (11) where function H z is known as the propagation kernel that characterizes the propagation in the domain beyond the source. When the propagation takes place in free space, H z takes the form −ik expikρ − r2 ∕2z, (12) 2πz where k is the wavenumber of light. Upon substituting from Eqs. (10) and (12) into Eq. (11) and introducing the new variables of integration x a x 1 x 2 ∕2, x s x 1 − x 2 , y a y 1 y 2 ∕2, and ys y 1 − y 2 , one obtains, after a lengthy calculation, that H z ρ, r
W ρ1 ,ρ2 ,z
with j denoting the x or y component, one obtains x 1 − x 2 2 y 1 − y 2 2 exp − W r 1 , r 2 exp − 2δ2x 2δ2y ×
×
(5)
On substituting from Eq. (5) into Eq. (4), we can easily find that the reminder is only dependent on the position vector difference D. Such a case can thus be classified as being of twisted Schell-model type. To generate a field with a rotating Gaussian array pattern, one may select the following form of W 0 [18–20]:
(9)
where σ x and σ y are the transverse intensity widths along the x and y directions, respectively. Hence the CSD function W r 1 , r 2 becomes 2 2 x x2 y y2 W r 1 , r 2 exp − 1 2 2 exp − 1 2 2 4σ x 4σ y 2 x 1 − x 2 y 1 − y 2 2 exp − × exp − 2δ2x 2δ2y
It is to be noted that the reminder should be a real function to ensure the survival of the twist term in Eq. (3). In order to satisfy this requirement, we choose p0 r m − r 1:
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Q P X X kπσ x 2 ik 2 2 pffiffiffiffiffiffiffiffiffiffiffi ρ exp − − ρ 2 2z 1 2 nx −P ny −Q HΘΔz 2 2 k σx 0 0 2 × exp − 2 x 1 − x 2 2z 2 1 ik 1 2 1 × exp 2m1 y 10 − y 20 γ β2 4Δ z 4H 4Θ 2 1 ik 1 2 1 exp 2m1 y10 − y 20 γ − β2 4Δ z 4H 4Θ 2 1 ik 1 2 1 exp 2m1− y 10 − y 20 γ − β2− 4Δ z 4H 4Θ 2 1 ik 0 0 1 2 1 2 , exp 2m1− y 1 − y 2 γ β 4Δ z 4H −− 4Θ − (13a)
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Vol. 43, No. 15 / 1 August 2018 / Optics Letters
where ik 0 g y y 20 − m2 iC y ; 4Δ 2z 1 ik k2 σ2 β x 10 x 20 2 x x 10 − x 20 iC x ; z 2z iu ukσ 2x β 0 0 − x 1 − x 2 ; β ;m m1 z 2Θ 4Θ 2 1 u2 k2 σ 4x u2 σ 2x 1 iu2 σ 2x k ik 2 H − − ; 2 αy 4Θz 2 4Δ 2Θz z 2 2 iu σ x k ik 2m1 iky 10 − y 20 ∕z; g −2 2Θz z γ
Θ
1 k2 σ 2x 1 u2 1 1 1 ; 2 2 : (13b) 2 ;Δ 2 2z ax 2σ y 4Θ αj 8σ j 2δj
Equations (13a) and (13b) may be regarded as the general analytical expressions for the CSD in any transverse cross section of the beams generated by TGSMA sources. In a special case when N x N y 1, they reduce to propagated CSD functions for TGSM beams. Applying Eqs. (13a) and (13b), one may readily study the evolution of transverse spectral density and transverse spatial coherence of the TGSMA beam on propagation. The spectral density at a point (ρ, z), say Sρ, z, is obtained from Eqs. (13a) and (13b) by setting ρ1 ρ2 , and the degree of coherence (DOC) at a pair of points ρ1 , z and ρ2 , z is generally given by μρ1 , ρ2 , z W ρ1 , ρ2 , z∕Sρ1 , zSρ2 , z1∕2. A stack of images taken in Fig. 1 illustrates the typical evolution of the transverse spectral density as a TGSMA beam propagates in free space. Source parameters are chosen to be
Fig. 1. Changes in the spectral density associated with a typical TGSMA beam on free-space propagation. (See Visualization 1 for an animated representation of the full propagation).
Letter N x N y 3, σ x δx 1 mm, σ y δy 0.3 mm, R x 2R y 3 mm, and u 3 mm−2 , and the wavelength is fixed at 632.8 nm. One sees that the initial Gaussian ellipse progressively splits into a rotating Gaussian array upon propagation. Although each lobe of the Gaussian array rotates as an individual around its respective lobe center, they are all in a synchronous motion during propagation. In this regard, the rotating lattice-like field may act as a powerful tool for dynamic control over multiple particles. Figure 2 presents the modulus of DOC of the field generated by the same source as in Fig. 1. It is demonstrated that DOC rotates counterclockwise around the beam center upon propagation, while simultaneously degenerating into a Gaussian profile. An interesting feature from this rotating field is that the twist phase causes DOC to rotate in the opposite direction of spectral density, as seen by comparing the rotation in Figs. 1 and 2. Quantifying twist effects is important because doing so can provide in-depth insight into the nature of twisted beams. Hereafter, we will quantify the twist effects of the spectral density and the DOC in terms of θs and θc , respectively, where θs denotes the rotation angle of the major axis of the intensity lobe ellipse, θc denotes the rotation angle of the major axis of the coherence ellipse, and both are functions of propagation distance. Positive and negative values of the derivatives of these rotation angles [i.e., the angular velocities of rotation ωs θs0 z and ωc θc0 z] are defined as representing clockwise and counterclockwise rotation, respectively. To measure the rotation angles of spectral density and DOC, it is convenient to
Fig. 2. Modulus of DOC of the TGSMA beam as a function of ρd ρ1 − ρ2 at several propagation distances, where ρ1 x d0 ∕2, y d0 ∕2 and ρ2 −x d0 ∕2, − yd0 ∕2 are two symmetrical points with respect to the origin of the coordinate axis. (See Visualization 2 for an animated representation of the full propagation.)
Letter
Fig. 3. Rotation angles of a TGSMA beam versus the normalized propagation distance z∕z R for u 0.03, 0.3, and 3 mm−2 . (a) Spectral density. (b) DOC.
Fig. 4. Rotation angles versus propagation distance z for different values of u (mm−2 ). (a) Spectral density; Visualization 3 presents the evolution of the spectral density for u 9 mm−2. (b) DOC; Visualization 4 presents the evolution of DOC for u 9 mm−2.
resort to the regionprops function (Image Processing Toolbox) in MATLAB (The MathWorks, Inc., Natick, Massachusetts, USA). Figure 3 shows the behavior of rotation angles for several values of twist parameter u. The other parameters are chosen to be the same as in Fig. 1. In the case of σ x ∕σ y δx ∕δy , the curves of the rotation angles can be presented as functions of normalized propagation distance z∕z R , where the normalized coefficient z R is the Rayleigh range of a single lobe and is given by σ x σ y −1 4σ x σ y −1 δx δy −1 ∕k 2 u2 ∕k 2 −1∕2 [1]. Figure 3(a) shows the twisted behavior of spectral density. As can been seen, the rotation angle increases monotonically from 0° to 90° during propagation, reaching 45 deg at the Rayleigh range z R regardless of the value of u. On the other hand, the twist effect is shown to be dependent on the choice of twist strength u. More specifically, a smaller value of u causes the rotation to become less pronounced. We would like to note that these characteristics are similar to those of TGSM beams described in Ref. [1]. Figure 3(b) shows the twisted behavior of DOC. It is found that DOC is of the same twist effect as spectral density, except that they rotate in opposite directions. It is well known that the twist phase can induce unidirectional rotation of a beam spot upon propagation. However, in the case of σ x ∕σ y ≠ δx ∕δy , the twist effect may greatly differ. Such a case is presented in Fig. 4, where the rotation angles of spectral density and DOC with N x N y 3, σ x δy 0.6 mm, σ y δx 0.1 mm, and R y 2R x 3 mm are plotted as functions of propagation distance for several
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values of u. Figure 4(a) presents the twisted behavior of spectral density. It is shown, in particular, that rotation angle θs cannot saturate at 90 deg upon propagation, while reaching its peak at critical distance z s0 , where ωs z s0 θs0 z s0 0. Once the propagation distance exceeds the critical value, the angular velocity would change its sign, indicating that each lobe of the Gaussian array begins to rotate in the opposite direction. A similar phenomenon can be observed from the twisted behavior of DOC in Fig. 4(b). Unlike in the case depicted in Fig. 3, however, the rotation behaviors of the spectral density and the DOC are not synchronized here. We conclude this Letter by stating that a new class of twisted partially coherent sources, termed TGSMA, has been introduced, to the best of our knowledge. Such sources, which can be viewed as a generalization of axially symmetric TGSM sources, generate beams whose lattice-like average intensities and DOCs rotate upon propagation. Of particular interest may be the fact that the directions of rotation are changeable upon propagation by appropriate source correlations. We foresee a broad range of applications for this new class of twisted beams from optical trapping to sensing and threedimensional laser material processing. Synthesizing multiple mutually uncorrelated, spatially displaced lattice-like beams, as revealed by Eq. (3), is a possible approach for experimental generation of TGSMA beams. In view of the rapidly growing interest in the twist phase, it is hoped that our result will provide a broader basis for further research into the twist effect. Funding. National Natural Science Foundation of China (NSFC) (11274273, 11474253); Fundamental Research Funds for the Central Universities (2018FZA3005).
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