Sub-wavelength sized transversely polarized optical needle with exceptionally suppressed side-lobes

Zhongsheng Man; Changjun Min; Luping Du; Yuquan Zhang; Siwei Zhu; Xiaocong Yuan

doi:10.1364/OE.24.000874

1. Introduction

In recent years, the nondiffraction beam with ultra-long depth of focus (DOF), also called optical needle, has attracted significant interest due to its attractive applications including particle acceleration [1], high-resolution optical imaging [2], second-harmonic generation [3], Raman spectroscopy [4], etc. Great developments have been achieved for generating such long DOF beams in free space through ingenious designing the optical characters of the light focused by a high numerical aperture (NA) lens. For example, a longitudinally polarized optical needle can be created by focusing a radially polarized beam with the combination of binary phase elements and a high NA lens [5,6 ], and the effective limit on the transverse width of focus can also be achieved [7]. Similar results can also be obtained by using a narrow angular aperture [8,9 ], conical mirrors [10], simple spherical mirrors [11], or other amplitude filtering mask [12]. In all above cases, the total field of focal spot is dominated by the prominently enhanced longitudinal electric field component, thus resulting in a strong longitudinally polarized optical needle. However, as a counterpart, the generation of transversely polarized optical needle at a sub-wavelength scale is rarely involved, even though it is highly desired for some applications such as ultrahigh density magnetic storage and atomic trapping [13–15 ]. Though the azimuthally polarized beam with total transversely polarized focus could be a candidate, its donut-shaped pattern only gives rise to a long dark channel rather than a bright needle after amplitude modulation [16]. Very recently, we generated a needle of transversally polarized beam by means of focusing a multi-belt spiral phase hologram modulated azimuthally polarized beam with a high NA lens [17], however, it suffers from the strong sidelobes and not long enough DOF. Apart from the well-known radially and azimuthally polarized beams, a new type of vector beams with hybrid state of polarization including linear, elliptical, and circular polarizations in the beam cross section are proposed and demonstrated recently [18–20 ]. Such hybrid state of polarization provides additional optical degree of freedom in manipulating the focal field distribution, and therefore offers a new chance for the transversely polarized optical needle generation in free space.

In this paper, generation of an ultra-long transversely polarized optical needle with exceptionally weak sidelobes is introduced based on a hybridly polarized (HP) beam modulated by a binary phase mask. Firstly, we build up a universal analytical model for calculating the three-dimensional focal field of the HP beams based on the vector diffraction theory [21]. It is found that the radial index of the incident HP beams greatly affects the DOF after tightly focused and the optimum incident polarization mode is obtained in considering both the uniformity and length of the focal spot, which has a DOF of 2.78λ, nearly 1.9 times longer than that of the well-known radially polarized beams (1.46λ) under the same focusing conditions. Based on that, the relative DOF is further extended to 5.83λ by four-belt binary phase mask modulation, and the peak of the sidelobe is suppressed to only 9.9% of the principal lobe. The polarization evolution of such non-diffraction beam is also investigated in details. Though the polarization is spatially variant within the non-diffraction range, it keeps almost invariant in the mainlobe at each observation plane. Furthermore, it is found that more belts can further extend the needle length but accompanying with degraded intensity uniformity and higher sidelobes.

2. Building of the analytical model for focusing HP beams

The schematic configuration is shown in Fig. 1(a) . An HP beam transmits through a binary phase mask and is subsequently focused by a high NA lens. The binary phase mask consists of multiple belts in the radial direction and has a π phase shift between adjacent belts as shown in Fig. 1(b). Its transmission function T(θ) equals to 1 and −1 for odd and even belts, respectively.

Fig. 1 The schematic configuration. (a) Focusing system composed of a binary phase plate and a high NA lens. The focal plane of the focusing lens is at z = 0; (b) Phase structure of a seven-belt binary element (left) and its transmission function T(θ) (right); (c) Polarization and intensity distributions of four radial-variant HP beams with m = 0.2, 0.5, 0.59, and 1, respectively.

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Mathematically, the HP beam can be represented as [18,22,23 ]:

\begin{array}{l} E = \frac{A_{0} (r, 0)}{\sqrt{2}} [(\cos ψ {\hat{e}}_{x} + \sin ψ {\hat{e}}_{y}) \exp [i δ (r, φ)] \\ + (- \sin ψ {\hat{e}}_{x} + \cos ψ {\hat{e}}_{y}) \exp [- i δ (r, φ)]], \end{array}

where A ₀ is the relative amplitude, the angle ψ determines which pair of orthogonal linear polarizations as the base vectors (without loss of generality, we set ψ = -π/4 in the following calculations). δ is a function of r and φ determining the polarization mode, where r and φ are the polar radius and azimuthal angle, and

{\hat{e}}_{x}

and

{\hat{e}}_{y}

are the unit vectors along x and y axis, respectively.

Generally, an optical field should be characterized by at least four parameters including amplitude, phase, polarization, and wavelength. Under tight focusing conditions, the contribution of polarization becomes dominant in determining the focused field. Richards and Wolf firstly introduced the contributions of incident polarization for calculating the focal field of a high NA lens [21], where they considered a linearly polarized beam, and was later adopted for the cases of radially and azimuthally polarized beams [24]. Based on the Richards and Wolf diffraction theory, we further expand the incident optical field to HP beam described by Eq. (1), and derive the corresponding electric field components near the focus in cylindrical coordinate as:

\begin{array}{l} E_{ρ} (r_{s}, φ_{s}, z_{s}) = \frac{- i B}{π} \int_{0}^{2 π} \int_{0}^{α} \sin θ A (θ) l (θ) \exp {i k [z_{s} \cos θ + ρ_{s} \sin θ \cos (φ - φ_{s})]} \\ \times {[\cos (ψ - φ) \cos θ \cos (φ - φ_{s}) + \sin (ψ - φ) \sin (φ_{s} - φ)] \exp (j δ) \\ - [\sin (ψ - φ) \cos θ \cos (φ - φ_{s}) + \cos (ψ - φ) \sin (φ - φ_{s})] \exp (- j δ)} d φ d θ, \end{array}

\begin{array}{l} E_{φ} (r_{s}, φ_{s}, z_{s}) = \frac{- i B}{π} \int_{0}^{2 π} \int_{0}^{α} \sin θ A (θ) l (θ) \exp {i k [z_{s} \cos θ + ρ_{s} \sin θ \cos (φ - φ_{s})]} \\ \times {[\cos (ψ - φ) \cos θ \sin (φ - φ_{s}) + \sin (ψ - φ) \cos (φ - φ_{s})] \exp (j δ) \\ + [\sin (ψ - φ) \cos θ \sin (φ_{s} - φ) + \cos (ψ - φ) \cos (φ - φ_{s})] \exp (- j δ)} d φ d θ, \end{array}

\begin{array}{l} E_{φ} (r_{s}, φ_{s}, z_{s}) = \frac{- i B}{π} \int_{0}^{2 π} \int_{0}^{α} \sin θ A (θ) l (θ) \exp {i k [z_{s} \cos θ + ρ_{s} \sin θ \cos (φ - φ_{s})]} \\ \times [- \cos (ψ - φ) \sin θ \exp (j δ) + \sin (ψ - φ) \sin θ \exp (- j δ)] d φ d θ . \end{array}

where the maximum aperture angle α = arcsin(NA/n), n is the refractive index in image space, k is the wave number, B is a constant, A(θ) is the apodization factor of the focusing lens, and l(θ) is the relative amplitude of the incident beam at the pupil plane. For the Bessel-Gaussian beam considered throughout this Letter, l(θ) is given in the following form [5]:

l (θ) = \exp [- β_{0}^{2} {(\frac{\sin θ}{\sin α})}^{2}] J_{1} (2 β_{0} \frac{\sin θ}{\sin α}) .

Here, J ₁(⋅) is the first kind Bessel function of order 1, and β ₀ is a size parameter of incident beam defined as the ratio of the pupil radius to the beam waist. In this Letter, all length measurements are in units of wavelength, and the refractive index n = 1, NA = 0.95, β ₀ = 1,

A (θ) = \sqrt{\cos θ}

, and B = 1 are used in our calculations.

Despite that the function δ has arbitrary spatial distribution in Eqs. (2)-(4) , for obtaining an optical needle with long DOF in free space, we consider the case of δ = 2mπr/r ₀ (r ₀ is the radius of the incident vector beam, m is the radial index), which is a function of r while independent of φ. Such HP beams can be experimentally generated by a common path interferometric method [19]. As examples, Fig. 1(c) gives polarization distributions of four HP beams with different m. We can see that the polarization distributions here are much different from the well-known radially and azimuthally polarized beams. Firstly, it is not only composed of linear polarization, but also includes linear, circular and elliptical polarizations in the beam cross section, thus providing more optical degree of freedom for modulating the focal field. Secondly, the polarization distributions are not azimuthally-variant but radially-variant. It is identical for any point with the same radius r, but varies dramatically between the two adjacent rings. It should be pointed out that different from the radially and azimuthally polarized beams, the four radial-variant HP beams have no polarization singularity in the beam center, all of which are linearly polarized on the optical axis, as shown in Fig. 1(c).

3. Tight focusing properties of the radially-variant HP beams

We find that the radial index m affects the DOF strongly after tight focusing (without binary phase plate), as shown in Fig. 2 . A longer DOF can be obtained with a large radial index, while at the expense of the uniformity of intensity distribution along the optical axis. For example, when the radial index increases from m = 0.2 to m = 1, the focal spot firstly gets longer as seen in Figs. 2(a)-2(c), but then splits to two spots along the optical axis as depicted in Fig. 2(d). Therefore, a trade-off between the DOF and intensity uniformity has to be made. The optimum result in considering the trade-off is obtained at m = 0.59, as shown in Fig. 2(c), from which we can see that such beam exhibits both long DOF and uniform intensity distribution along optical axis.

Fig. 2 Electric energy density in the focal region. (a−d), Total intensity distributions of focus in the x-z plane for the HP beams with m = 0.2, 0.5, 0.59, and 1 respectively. (e), Normalized intensity profiles along the optical axis for the above four HP beams compared to the results of linearly, circularly, and radially polarized beams under the same focusing conditions.

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In Fig. 2(e) we compare the focal field intensity profiles of the above four radially-variant HP beams along the optical axis to the results of well-known linearly, circularly and radially polarized beams under the same focusing conditions. It is clear that the HP beams possess longer focal spots than the other polarized beams. For the case of m = 0.59, the full-width at half-maximum (FWHM) values of the focal spot in the axial direction is calculated to be about 2.78λ, nearly 1.9 times larger than that of the radially polarized beam (1.46λ). The fascinating properties of longer DOF of tight focusing HP beam is due to the phase difference between any two different concentric rings in the beam cross section as described in Eq. (1).

4. Achievement of the transversely polarized optical needle

The above results reveal that the HP beam with m = 0.59 has the potential to achieve a longer DOF assisted by further phase or amplitude modulations. Since the amplitude modulation usually blocks some incident light, which inevitably will weaken the energy of the focal field and reduce the conversion efficiency. Here we only consider the phase modulation by introducing a multi-belt binary phase element before focusing lens. As an example, we use a four-belt binary optical element and optimize the structural parameters based on the traditional global-search-optimization algorithm [5]. The optimized set of angles is given by:

θ_{1} = 11.489 °, θ_{2} = 50.838 °, θ_{3} = 66.061 ° .

The transverse, longitudinal and total electric field intensity distributions of the focal field after four-belt phase modulation are shown in Figs. 3(a)-3(c) . It is noted that side-lobes are usually considered as background noise that need to be suppressed. In Fig. 3 the transverse component is much stronger and dominates the total field, which is also very uniform with very weak side-lobes over a long range, and the longitudinal component locating at the outside of focus is so weak that it nearly has no effect on the side-lobes of the total field; in other words it is a non-diffracting beam. The field intensity profiles along the optical axis are plotted in Fig. 3(d), from which we can easily obtain the FWHM value of the focal spot, which is 5.83λ in the axial direction, much longer than previous result achieved by using five-belt spiral phase hologram modulated azimuthally polarized beam (4.84λ) [17]. Further, it is inspiring that the curve of the total field is a flattop-shaped pattern, which is contributed by radial and azimuthal components. The total electric field intensity distribution in the focal plane is depicted in the inset of Fig. 3(d). The maximum FWHM value of the focal spot is calculated to be about 0.78λ in the transverse direction and the peak of sidelobe is only 9.9% of the maximum of the main lobe, also much weaker than the results in Ref [17].

Fig. 3 Generation of the transversely polarized optical needle. (a−c), The electric intensity distributions of transverse, longitudinal, and the total (sum of transverse and longitudinal components) in the x-z plane for the radial-variant HP beam with m = 0.59 after modulation of a four-belt binary phase element respectively, (d), Normalized intensity profile along the optical axis. Inset shows the total intensity distribution in the focal plane, and the size is 3λ × 3λ.

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The polarization evolution of this non-diffraction beam is studied by calculating the Stokes polarization parameters [25,28 ]. The ellipticities of the local polarization ellipses at different z are shown in Fig. 4(a) . The polarization is spatially variant within the DOF range. For example, the beam on the axis is right-hand elliptically polarized when z = 2λ and evolved to left-hand elliptically polarized when z = 3λ. However, it is nearly a flattop-shaped pattern in the mainlobe at each observation plane. At the same time, we find that the azimuthal angle (orientation of the long axis) of the local polarization ellipse plane is nearly identical in the main lobe. As an example, Fig. 4(b) gives the azimuthal angle distributions in the focal plane, which are all oriented along the y-axis directions in the mainlobe. The above results reveal that the polarization of the nondiffraction beam keeps almost invariant in the mainlobe at each observation plane.

Fig. 4 Revealing the polarization evolution. (a), Cross section of the ellipticity of local polarization ellipses at different observation planes, (b), Azimuthal angle distribution at the focal plane.

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5. Effect of belt number of binary phase plate to the focus

To reveal the effect of belt number of binary phase plate to the focus, in Fig. 5 we give the comparison of optimized electric field distributions by different modulations in terms of two-belt (θ ₁ = 66.061°), three-belt (θ ₁ = 16.515°, θ ₂ = 57.444°), aforementioned four-belt (θ ₁ = 11.489°, θ ₂ = 50.838°, θ ₃ = 66.061°) and five-belt (θ ₁ = 15.079°, θ ₂ = 43.801°, θ ₃ = 55.289°, θ ₄ = 61.034°) binary phase mask on the incident beam. Obviously, the DOF can be improved by use of more belts as seen in Fig. 5(b) at the expense of the transverse beam size depicted in Fig. 5(a). For example, when the four-belt binary phase mask is replaced by the five-belt one the DOF of the beam increases from 5.83λ to 8.94λ while the FWHM changes from 0.78λ to 0.84λ. However, it should be noted that the uniformity of the axial intensity becomes weak for the case of five-belt mask since there appears several peaks in the axial direction and the maximum intensity is no longer located on the exactly focus point of the lens as shown in Fig. 5(b). Further, the peak of the sidelobe grows greatly for the case of five-belt mask in comparison with the four-belt one as depicted in Fig. 5(a). Such results confirm the trade-off between the needle length and beam qualities.

Fig. 5 Revealing the effect of belt number of binary phase plate to the focus. (a), (b), Intensity profiles in the focal plane and along the optical axis for the two-belt (dotted curve), three-belt (dashed-dotted curve), four-belt (dashed curve) and five-belt (solid curve) binary phase modulated HP beam with m = 0.59 respectively.

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6. Discussion

It is well-known that the tightly focused field is mainly determined by the incident state of polarization of light. For instance, radially polarized beam can produce a strong longitudinally polarized focused field due to the radial orientation of electric field vibration at each point in the beam cross section. Radial-variant HP beam, as a new kind of vector beams, has more complicated polarization distributions that much different from other common vector beams. For one hand, it includes linear, circular and elliptical polarizations in the beam cross section thus has more flexible polarization degree of freedom. For another hand, its polarization is radial-variant rather than commonly azimuthal-variant, which is identical for electric field vibration at the same radius but varies greatly for any adjacent rings. Therefore, a radial-variant HP beam is similar to a composite vector beam composed of series of concentric ring-shaped homogeneously polarized beams with different ellipticities for different rings. We all known that homogeneously polarized beam is transversely dominated after tight focusing [26–31 ], thus the radial-variant HP can also generate a transversely polarized focal field. Furthermore, when the radial index is increased, the number of the concentric ring-shaped beams will increase, and the effect of quasi multiple-beam interference will become more obvious, giving rise to a gradually elongated focus. It is important to point out that there is a trade-off between the length and uniformity of the focus. However, when introducing a binary phase optical element, the DOF can be extended markedly on the promise of preservation of uniformity. In addition, the sidelobes are suppressed concomitantly, due to the fact that the binary phase mask works as a special polarization filter enhancing the transverse component and suppressing the sidelobes.

7. Summary

Based on the Richards and Wolf diffraction theory, we have built a universal analytical model for calculating the three-dimensional focal field of HP beams. Based on such model, a strong transversely polarized optical needle with long DOF and exceptionally suppressed side-lobes was achieved by focusing a radial-variant HP beam with a high NA lens and a multi-belt binary phase filter. The relative DOF can extend to 5.83λ with a four-belt phase filter and the peak of the sidelobe is only 9.9% of the maximum of the principal lobe. The demonstrated strongly transversely optical needle holds great potential in ultrahigh density magnetic recording as well as atomic trap and switches. We believe that this work not only provides a new method for generating transversely polarized optical needle, but also delivers an important contribution towards a comprehensive understanding and application of the hybrid state of polarization of vector beams.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (NSFC) under Grant Nos.61138003, 61427819, 61422506, and 61405121; National Key Basic Research Program of China (973) under grant No.2015CB352004; Science and Technology Innovation Commission of Shenzhen under grant Nos.KQCS2015032416183980, KQCS201532416183981, JCYJ20140418091413543, and the start-up funding at Shenzhen University.

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Sub-wavelength sized transversely polarized optical needle with exceptionally suppressed side-lobes

Abstract

1. Introduction

2. Building of the analytical model for focusing HP beams

3. Tight focusing properties of the radially-variant HP beams

4. Achievement of the transversely polarized optical needle

5. Effect of belt number of binary phase plate to the focus

6. Discussion

7. Summary

Acknowledgments

References and links

Cited By

Figures (5)

Equations (6)

Optics Express