Changeable focused field distribution of double-ring-shaped cylindrical vector beams

Optics Communications 342 (2015) 204–213

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Optics Communications journal homepage: www.elsevier.com/locate/optcom

Changeable focused field distribution of double-ring-shaped cylindrical vector beams Liping Gong a, Zhuqing Zhu a,n, Xiaolei Wang b, Yang Li a, Ming Wang a, Shouping Nie a a Key Laboratory of Optoelectronic Technology of Jiangsu Province, School of Physical Science and Technology, Nanjing Normal University, Nanjing 210023, China b Institute of Modern Optics, Nankai University, Key Laboratory of Optoelectronic Information Science & Technology, Ministry of Education, Tianjin 300071, China

art ic l e i nf o

a b s t r a c t

Article history: Received 1 July 2014 Received in revised form 21 December 2014 Accepted 27 December 2014 Available online 31 December 2014

Based on vector diffraction theory, the focusing features of double-ring-shaped cylindrical vector (DCV) beams are studied in this paper. The intensity pattern in the vicinity of the focus can be tailored by appropriately adjusting the parameters of the ratio of the inner-to-outer ring radius and the polarization state of the incident DCV beams. It is shown that focused field with flattop or optical cage profile is easily obtained. Moreover, we also propose a new approach to generate a flattop focus with extended depth of focus and lower side-lobe levels by adding a diffractive optical element (DOE) which is cleverly loaded on the spatial light modulator (SLM). The results from this study would especially facilitate light manipulation of two types of particles with different refractive indices in only one optical-trap system. & 2014 Elsevier B.V. All rights reserved.

Keywords: Double-ring-shaped cylindrical Vector beams Flattop focus Optical cage

1. Introduction In recent years, more theories and experiments about focusing features of cylindrically vector (CV) beams have been reported [1–4]. It has been shown that radially polarized vector beams can generate a strong axial electric field component, and the azimuthally polarized beams can produce pure axial magnetic field component [5]. Owing to those unique properties, CV beams are widely applied in optical trapping [6], scanning optical microscopy [7], laser cutting of metals [8], determination of the orientation of single molecules [9,10] and particle acceleration [11,12]. For optical engineers and scientists, dynamic regulation of focal filed distribution is always one of the most important topics. To obtain smaller focal spot and diverse focused field distributions, it is often preferable to choose double-ring-shaped cylindrical vector (DCV) beams as the input beams which can be easily generated by a 4-f system with a spatial light modulator (SLM) and a common path interferometric arrangement [13]. Two types of tightly focused field of DCV beams, flattop light field and optical cage, have recently gained considerable research interest. By appropriately adjusting the rotation angle, K. Prabakaran et al. generated flat-topped focus shapes with high NA lens axicon and diffractive optical elements (DOEs) [14]. Similarly, K.B. Rajesh et al. generated a flattop light field by a modified high NA lens axicon with spherical aberration and the depth of focus was effectively extended [15,16]. Yuichi Kozawa et al. produced optical n

Corresponding author. E-mail address: [email protected] (Z. Zhu).

http://dx.doi.org/10.1016/j.optcom.2014.12.077 0030-4018/& 2014 Elsevier B.V. All rights reserved.

cage by changing the ratio of the pupil radius to the beams radius [17]. Bokor and Nándor generated the optical cage with uniform light intensity by changing the polarization of the inner and outer ring of the DCV beams [18]. Based on Bokor's work, Wang Xiling et al. proposed a new method to produce a controllable 3D optical cage [19]. The work listed above can only produce one of the two kinds of focal field at a time in the designed optical system. To obtain both of them, a diffractive optical element (DOE) and special optical elements are needed to be redesigned and manufactured, which are costly and time consuming [20]. However, two kinds of particles often need to be simultaneously manipulated in one optical-trap system [15,21,22]. In this paper a method that can alternatively generate focused field with flattop or optical cage profile by adjusting the polarization sate distribution and the radius radio of the inner and outer ring of DCV beams is proposed. Combining techniques in Ref. [13], a new approach using only a DOE loaded on the spatial light modulator (SLM) is also introduced to further increase the depth of focus of the flattop focal field and to inhibit its side lobe. Furthermore, the evolution of polarization through the focal region of optical cage is also studied in details, which may apply to trapping, manipulation, and orientation analysis.

2. Theoretical basis on tightly focused DCV beams with high NA lens As described in Ref. [13], arbitrary vector beams with different polarization states can be obtained when an appropriate

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additional phase distribution δ is use, which δ can be a function of beam radius ρ and polarization angle ϕ and can be calculated from

⎧ 0 < ρ ≤ Rρ 0 ⎪ ϕ + ϕ1 δ ( ρ , ϕ) = ⎨ ⎪ R ϕ + ϕ ρ0 < ρ ≤ ρ0 ⎩ 2 where ρ0 is the radius of the input beam determined by the aperture of the objective lens. R is the ratio of the inner-to-outer ring radius. ϕ1 and ϕ2 denote the polarization angles with respect to the radial direction for the inner and out ring respectively. Fig. 1 demonstrates an electric field of a DCV beam, in poplar coordinate, propagating in the z direction. The field is described as

Fig. 1. The schematic diagram of the polarization of DCV beams and its convergence through a lens.

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Fig. 2. Intensity patterns in the vicinity of the focus in the X–Z plane when ϕ1 ¼ 0.80π, ϕ2 ¼0, R¼ 0.58. (a) The radial component; (b) the azimuthal component; (c) the axial component; (d) the total intensity distribution; (e) intensity profiles in the transverse and longitudinal direction. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

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numerical aperture (NA), the electric field distribution in the vicinity of the focus can be expressed as

→ ⎯→ ⎯ ⎯⎯→ E (ρ , ϕ) = E ρ e ρ + Eϕ eϕ ⎯ ⎯⎯→ ⎧ cos ϕ ⎯→ ⎪ 1 e ρ + sin ϕ1 eϕ 0 < ρ ≤ Rρo =⎨ ⎯→ ⎯ ⎯⎯→ ⎪ ⎩ cos ϕ2 e ρ + sin ϕ2 eϕ Rρo < ρ ≤ ρo

→ E (r , φ, z) = ⎡⎣ cos ϕ1I1 (0, α1) + cos ϕ2 I1 (α1, α2 ) ⎤⎦ ⎯→ ⎯ er + ⎡⎣ sin ϕ I2 (0, α1) + sin ϕ I2 (α1, α2 ) ⎤⎦

(1)

⎯→ ⎯ ⎯⎯⎯→ where e ρ and eϕ are the unit vectors in the radial and azimuthal directions in the input plane of the polar coordinate respectively. Based on the Richards–Wolf vector diffraction theory [23,24], when the DCV beam is focused by an objective lens with a high

1

2

⎯⎯→ eφ ⎯→ ⎯ + ⎡⎣ cos ϕ1I3 (0, α1) + cos ϕ2 I3 (α1, α2 ) ⎤⎦ ez

(2)

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z(x)/λ Fig. 3. Intensity patterns in the vicinity of the focus in the X–Z plane when ϕ1 ¼ 0.41π, ϕ2 ¼ 0.10π, R¼ 0.78. (a) The radial component; (b) the azimuthal component; (c) the axial component; (d) the total intensity distribution; (e) intensity profiles in the transverse and longitudinal direction. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

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⎯→ ⎯ ⎯→ ⎯ ⎯⎯⎯→ where er , eφ and ez are the unit vectors in the radial, azimuthal and axial directions of the cylindrical coordinate system respectively with the plane z = 0 located at the focal plane. α1 is given by α1 = sin−1(R NA) and α2 is defined as α2 = sin−1NA . The corresponding radial, azimuthal and axial electric field intensity (I1, I2 and I3) are defined as the following:

I1 (ξ , ς) = A

∫ξ

I2 (ξ , ς) = 2A

ς

∫ξ

I3 (ξ , ς) = 2iA

cos (θ) sin (2θ) J1 (kr sin θ) exp (jkz cos θ) dθ ς

∫ξ

cos (θ) sin (θ) J1 (kr sin θ) exp (jkz cos θ) dθ

ς

cos (θ) sin2 (θ) J0 (kr sin θ) exp (jkz cos θ) dθ

where A is a constant that relates to the focal length and the wavelength; J0 and J1 denote the Bessel functions of the first kind of order 0 and 1. As shown in Eq. (2), the intensity distribution in the vicinity of the focus remains the same when the parameters ϕ1 and ϕ2 take complementary values. The different combinations between the polarization state and the ratio of the inner ring and outer ring radius can produce a variety of field structures in the vicinity of the focus. Therefore, by choosing proper parameters we can flexibly control the focal field to satisfy the practical demand, especially in the particle manipulation systems.

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Fig. 4. Intensity patterns in the vicinity of the focus in the X–Z plane when ϕ1 ¼0.70π, ϕ2 ¼0.18π, R¼ 0.59. (a) The radial component, (b) the azimuthal component, (c) the axial component, (d) the total intensity distribution and (e) intensity profiles in the transverse and longitudinal direction. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

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3. Tightly focused field characteristics of DCV beams In this section, the characteristics of the tightly focused field of DCV beams are studied by numerical simulations. 3.1. The generation of flattop light field and its property Firstly, we explored the calculation of flattop light field where the related parameters are set as follows: NA ¼ 0.90, ϕ1 ¼ 0.80π, ϕ2 ¼0, R ¼0.58. The distribution of tightly focused electric field intensity in the radial, azimuthal and axial direction and the total energy density are illustrated in Fig. 2. It can be seen that flattop light field along the longitudinal direction has been obtained in Fig. 2(e) where the length of the uniform intensity reaches 0.70λ as shown in the red rectangle, and the full width at half maximum (FWHM) reaches 3.77λ . Interestingly, if the parameters are selected as follows: ϕ1 ¼0.41π, ϕ2 ¼0.10π, R ¼0.78, the flattop light field along the transverse direction can be also obtained. The corresponding intensity distribution is shown in Fig. 3, wherein the length of the uniform intensity reaches 0.62λ as shown in the red rectangle, and the FWHM reaches 1.72λ. It should be noticed that the intensity uniformity is a key problem in the particle trapping system because focused light field with nonuniform intensity can interfere the study on particles characteristics due to the different gradient forces along the different directions. Fig. 4 shows the flattop light intensity distribution in two dimensions we obtained, where the parameters are chosen as the following: ϕ1 ¼0.70π, ϕ2 ¼ 0.18π and R¼ 0.59 based on the experiment trials. As shown in Fig. 4(e), the area of the uniform intensity as shown in the red rectangle nearly reaches 0.54λ2 and the area of the FWHM almost reaches 7.43λ2 . The calculated results show that the flattop light field in different directions can be achieved by appropriately adjusting the polarization and the radio of the inner ring and outer ring radius. To further improve the quality of the focal field, different approaches have been adopted, such as the annular pupil [25], binary optical element [26] or DOE [27]. High numerical aperture lens are usually introduced in order to further realize the super resolution imaging and to stretch the depth of focus. However, the side-lobe effect will increase when increasing the depth of focus and decreasing the size of focus. Different from the methods

above-mentioned, a DOE loaded on the SLM at the back of the beam is introduced to expand the focus and inhibit the growth of side-lobe. Here, we mainly focused on preventing the side lobe in the transverse and longitudinal direction illustrated in Fig. 2 (e) because it is more serious compared with those shown in Figs. 3(e) and 4(e). As shown in Fig. 5, DCV beams are modulated by DOE which consist of three concentric circles regions. The numerical apertures that correspond to the outer edge of the three regions are NA1 ¼ 0.300, NA2 ¼0.707 and NA ¼0.900 respectively and the local transmittances are 1,  1 and 1. Stripe patterns demonstrate the transmittance of DCV beams through the unmodulated SLM and that from the modulated SLM by DOE respectively. Binary DOE with three concentric regions is introduced where δ (ρ , ϕ) is appropriately chosen as follows:

⎧ ϕ + ϕ1 0 < ρ ≤ 0.333ρ 0 ⎪ ⎪ ϕ + π + ϕ1 0.333ρ 0 < ρ ≤ Rρ 0 δ ( ρ , ϕ) = ⎨ ⎪ ϕ + π + ϕ2 Rρ 0 < ρ ≤ 0.786ρ 0 ⎪ 0.786ρ 0 < ρ ≤ ρ 0 ⎩ ϕ + ϕ2

(4)

Considering the effect of pupil apodization function on the energy distribution, we explored the Helmhotlz condition cos (θ)− (3/2) , which is better for the generation of the flattop light field with extended DOF [28]. When ϕ1 ¼1.00π, ϕ2 ¼0, R ¼0.58, the intensity distribution is shown in Fig. 6. Compared with the FWHM of 3.77λ and the main lobe energy of 22.093% in Fig. 2(e), the FWHM in Fig. 6(e) is about 5.98λ and the main lobe energy is about 28.998%. Therefore the side-lobe effect is effectively inhibited accompanied by obvious increase in the depth of focus. 3.2. The generation of optical cage and its properties For the unique characteristics of non-invasive measurement, the optical cage can be widely used in biological operation, biological imaging and nanometer fabrication [29,30]. Similar to the generation of flattop light field, optical cage can be obtained by simply regulating the polarization and the radius ratio of DCP vector beams and this is demonstrated in this section. According to Eq. (2), the parameters are selected as follows: NA ¼ 0.90, ϕ1 ¼0.70π, ϕ2 ¼0, R ¼0.87, and the calculated intensity

Fig. 5. The diagram of DCV beams modulated by the DOE and the transmittance function on the SLM.

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z(x)/ λ Fig. 6. Intensity patterns in the vicinity of the focus in the X–Z plane when ϕ1 ¼ 1.0π, ϕ2 ¼0π, R ¼0.58 with DOE. (a) The radial component; (b) the azimuthal component; (c) the axial component; (d) the total intensity distribution; (e) intensity profiles in the transverse and longitudinal direction.

distribution is shown in Fig. 7. As illustrated in Fig. 7(e), we find that the intensity distributions of the optical cage along the transverse and longitudinal directions have the same maximum value. The distance between both the maximum values along the transverse direction is about 1.05λ with the depth of 0.2792. Accordingly, the one along the longitudinal direction is about 2.86λ and the depth is 0.2792 too, which indicate optical cage is more uniform by proposed method than that by designed DOE [31,32]. Although the size of optical

cage is slightly smaller, the depth in the transverse direction increases nearly 40 times, while one in the longitudinal direction is nearly 16 times. This more striking contrast demonstrates the effectiveness and the possibility of this method. To make optical cage open in the transverse direction, the intensity in the transverse and azimuth direction should be weakened. Meanwhile, the intensity in the axis direction should be contrarily increased. The intensity distribution in the vicinity of the focus is shown in Fig. 8 as the parameters ϕ1, ϕ2 and R are appropriately

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z(x)/λ Fig. 7. Intensity patterns in the vicinity of the focus in the X–Z plane ϕ1 ¼ 0.70π, ϕ2 ¼ 0, R ¼0.87. (a) The radial component, (b) the azimuthal component, (c) the axial component, (d) the total intensity distribution and (e) intensity profiles in the transverse and longitudinal direction.

chosen. As shown in Fig. 8(e), we can find the peak value in the transverse direction is about half of that in the longitudinal direction which means that the restoring force in the transversal dimension is higher according to Ref. [33]. Therefore, this kind of optical cage can trap the low-refractive-index particle only in the longitudinal direction while impossible in the transverse direction. Accordingly, to make optical cage open in the longitudinal direction, the intensity in the longitudinal direction should be weakened. The intensity distribution in the vicinity of the focus is illustrated in Fig. 9. Similar to the case in the last paragraph, this

optical cage can trap low-refractive-index particles only in the transverse direction. Thus, it is shown that we can control the opening direction of the optical cage by adjusting the parameters such as ϕ1, ϕ2 and R to easily satisfy the need for trapping particles. Polarization, as a modulated parameter, has attracted increasing attention due to its broad potential applications in various areas, such as polarization information encryption [34], multidimensional optical data storage [35], nonlinear optics [36], imaging [37] and surface plasmon-based photonic devices [38]. In

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Fig. 8. Intensity patterns in the vicinity of the focus in the X–Z plane ϕ1 ¼ 1.00π, ϕ2 ¼ 0.10π, R¼ 0.78. (a) The radial component; (b) the azimuthal component; (c) the axial component; (d) the total intensity distribution; (e) intensity profiles in the transverse and longitudinal direction.

order to study the polarization of the electric field in the vicinity of the focus, we investigated the polarization of optical cage based on Fig. 7. However, the dynamic distribution of the field E (r , φ , z) in the vicinity of the focus is complicated. To show the field polarization, we simulated the distribution of the averaged electric field, where the radial component is understood as modulus Er = [Er (r , φ , z) Er⁎ (r , φ , z)]1/2 , the azimuthal component as the modulus

1/2 Eφ = ⎡⎣Eφ (r , φ , z) Eφ⁎ (r , φ , z) ⎤⎦

and

the

longitudinal

component as the modulus E z = [E z (r , φ , z) E z⁎ (r , φ , z)]1/2. Fig. 10 shows the distribution of the time-averaged electric field in the X– Z cross-section. The direction and the length of the arrows in Fig. 10 describe the direction and size of the electric field vector, respectively and positive sign is assigned to the vector direction along z-axis. As shown in Fig.10, the vectors gradually turn to zaxis and tend to be identical in the propagating direction. The polarization direction on the z-axis is consistent along the positive

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Fig. 9. Intensity patterns in the vicinity of the focus in the X–Z plane when ϕ1 ¼ 0.40π, ϕ2 ¼0.60π, R ¼0.70. (a) The radial component; (b) the azimuthal component; (c) the axial component; (d) the total intensity distribution; (e) intensity profiles in the transverse and longitudinal direction.

direction. Interestingly, the field polarization in the X–Z crosssection is similar to that of the radially polarization vector beam modulated by DOE [21]. The main difference is that the polarization of the electric field is nonuniform along the z-axis and reaches the minimum value in the center of optical cage. It further proves that the gradient force of optical cage exists which can capture

particles with refractive indices less than that of the ambient. Based on our study, we find that the polarization is more multitudinous in the vicinity of the focus which may have poten tial applications of polarization information encryption, multidimensional optical data storage.

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to Prof. Bing Gu from Southeast University for his useful discussion and suggestion. We also thank Dr. Lipei Song from Nankai University for her critical reading of the manuscript.

References [1] [2] [3] [4] [5] [6] [7] [8] Fig. 10. Polarization characteristics of time-averaged electric field in the focal region in the x–z cross-section.

[9] [10] [11]

4. Conclusions This paper proposes a method to dynamically control the light field of the tightly focused DCV beams in the vicinity of the focus. Both flattop light field and optical cage can be obtained when the polarization and the radio of the inner ring and outer ring radius are regulated. Numerical simulation results indicate that the uniformity of the flattop light field in transverse and longitudinal direction can be controlled and the FWHM both in the longitudinal and transverse direction can reach about 5.98λ and 1.7λ respectively. Moreover, two-dimensional flattop light field can be easily obtained with the area of the FWHM about 7.43λ2. In addition, we get optical cage with uniform intensity distribution, and the light intensity can be controlled in the transverse or longitudinal direction to switch the optical cage. In a word, the proposed method can flexibly produce diverse light fields such as flattop light field and optical cage, which are the ideal choice for trapping two types of particles.

Acknowledgment This work was supported by the National Natural Science Foundation of China (Grants: 61275133, 61377003), the Scientific Research Innovation Program for the Graduate Students in Institution of Higher Education of Jiangsu Province (Grants: KYLX_0720) and the Priority Academic Program Development of Jiangsu Higher Education Institutions. We are particularly grateful

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