## Abstract

We report a new method to create high purity longitudinally polarized field with extremely long depth of focus in the focal volume of a high numerical aperture (NA) objective lens. Through reversing the radiated field from an electric dipole array situated near the focus of the high-NA lens, the required incident field distribution in the pupil plane for the creation of an ultra-long optical needle field can be found. Numerical examples demonstrate that an optical needle field with a depth of focus up to 8λ is obtainable. Throughout the depth of focus, this engineered focal field maintains a diffraction limited transverse spot size (<0.43λ) with high longitudinal polarization purity. From the calculated pupil plane distribution, a simplified discrete complex pupil filter can be designed and significant improvements over the previously reported complex filters are clearly demonstrated.

©2010 Optical Society of America

## 1. Introduction

With the recent advances in the spatially variant polarization and particularly in the cylindrical vector beams [1], vectorial distribution of the focused electromagnetic fields has drawn significant amount of attentions. Low NA element, such as axicon [2,3] or axicon polarizer [4], has been used to create radially or azimuthally polarized non-diffracting fields. But the achieved longitudinal component usually was very weak. Focused with a high NA objective lens, vector field distributions in the focal volume [5] can be exploited for focal intensity shaping [6,7]. Furthermore, controllable polarization characteristics within the focal volume can be realized with spatial variant polarized illuminations [8–11]. One example that attracted recent interest is the so-called optical needle field that is substantially polarized in the longitudinal direction with long depth of focus (*DOF*) created through the focusing of filtered radial polarization [12–16]. These engineered optical vector focal fields pave the way to the applications in polarization sensitive orientation imaging [10,17], particle manipulation and acceleration [18,19], light-matter interaction on the nanoscale and so on [20].

The goal of this paper is to provide a systematic method to create optical needle field with high purity longitudinal polarization and long uniform axial intensity. A spatial spectrum method utilizing Gaussian amplitude envelope around the spatial spectrum center was reported recently to produce Bessel beams with long uniform axial intensity [21]. Ultra-long high resolution beam also was generated with a multi-zone rotationally symmetrical complex pupil filter [22]. But both papers dealt with engineered scalar fields and neither of them considered the vectorial characteristic, which is the key difference from our method descried in this paper. It has been shown experimentally that radially polarized beam can be focused into smaller spot size compared with linearly polarized or circularly polarized beams [23]. This superior focusing capability can be understood as the reversing of the field radiated by an electric dipole collected by an objective at the pupil plane [1]. This observation has been extended and applied to create diffraction limited focal spot with arbitrary 3-dimensional optical polarization in the focal volume by reversing the radiation field of a single electric dipole or two co-located electric dipole with appropriate relative phase and strength [11]. In this paper, we introduce a new method that allows us to obtain a high purity ultra-long optical needle field by reversing the radiation pattern of an electric dipole array.

## 2. Description of method

From the popular antenna pattern synthesis method for discrete linear dipole array [24], we can compute the radiation pattern of a dipole array collected by a high-NA lens in its pupil plane. Although infinitesimal dipole array (length of dipole *l _{0}* <<λ) is not very practical at the optical wavelength, its radiation pattern can guide us to find the pupil illumination in order to engineer the optical field characteristics in the focal volume. The schematic of this pupil illumination synthesis method is illustrated in Fig. 1
. In Fig. 1(a), identical infinitesimal electric dipoles are located along the optical axis to form a dipole array in the focal volume. The detailed conceptual diagram of a dipole array with 2

*N*electric dipoles is shown in Fig. 1(b). These dipoles are located in mirror-symmetric with respect to the focal plane and have a common oscillating direction along the z direction. Here the oscillating frequency of the dipole will not influence the distribution of the pupil field and needle field. Our proposed method is universal for electromagnetic field in all wave range under tight focusing condition.

The electric radiation of the dipole array at point A on the spherical surface Ω is

*z*is the intrinsic impedance of air,

_{air}*I*and

_{0}*l*is standard current and dipole length,

_{0}*f*is the focal length of the objective lens. Without loss of generality,

*C*is normalized to 1 in our calculation. The

*AF*is array factor (AF) relating to the phase delay caused by the spacing distance d

_{N}_{n}and initial phase difference β

_{n}of each pair of the dipole that are mirror symmetric with respect to the focal plane

*A*is the ratio of the radiation amplitude between the

_{n}*n*dipole pair and the standard dipole pair with normalized amplitude. The radiation only has a radial component on Ω [11] and it can also be written as${\stackrel{\rightharpoonup}{E}}_{0}(\theta )={E}_{DA}(\theta ){\stackrel{\rightharpoonup}{a}}_{{\rho}_{0}}$. For an objective lens that obeys the sine condition, the radiation pattern collected at the pupil plane

^{th}*P*can be expressed as

_{i}*ρ*=

_{i}*f*sinθ. If we start with this field as the incident in the pupil plane and reverse its propagation, the electric field reconstructed in the focal volume can be computed with the vectorial Debye theory as

*DOF*is mainly determined by the total number of dipole elements 2

*N*. Higher

*N*generally leads to longer depth of focus. However, the corresponding pupil plane distribution may become increasingly complicated and the number of parameters for optimization will increase accordingly. Two dipole array examples (

*N*= 2 and

*N*= 3) are illustrated here to realize optical needle with extended

*DOF*. In order to obtain the desired field distribution,

*A*,

_{n}*d*and

_{n}*β*need to be optimized. Here we adopt an empirical direct search procedure for the optimization of these parameters to reconstruct an optical needle field that is substantially polarized in the longitudinal direction with uniform axial (

_{n}*r = 0*) intensity distribution. At first, initial positions of the dipole elements are set such that the spacing between adjacent dipole elements is 1.5λ with initial

*A*= 1 and

_{n}*β*= 0. Then we slightly modify the

_{n}*d*’s to adjust the axial intensity profile such that the overall axial distribution has a long depth of focus and relative steep slope at the edge. Too large dipole spacing creates large modulation of the interacted dipole field and uniform intensity cannot be realized in the followed search. On the other hand, too small spacing will lead to shorter depth of focus and less steep slope at the edge of the axial intensity. After this step, the

_{n}*A*’s are manually adjusted (between 0~1) to control the intensity near the positions of the dipole elements such that the peak intensities around each dipole position are approximately equal. However, the overall profile of the axial intensity may be not flat. To fine tune the intensity modulation for the flatness at the top of the axial intensity distribution,

_{n}*β*’s are adjusted (between 0~π) to control the initial phase difference between two neighboring dipoles. A few manual iterations of the above procedure may be necessary to fine tune these parameters for the overall flatness within the entire depth of focus. We would like to point out that though, for each given

_{n}*N*, the group of

*A*’s,

_{n}*d*’s and

_{n}*β*’s for uniform axial intensity found in this way may not be unique and the optimized parameters may not be global due to the complicated interactions among these dipole elements and the large number of parameters involved. For globally optimized parameters, more systematic studies using methods such as genetic algorithm or simulated annealing are necessary. The optimized results are summarized in Table 1 .

_{n}The corresponding optical focal field distributions are shown in Fig. 2(a)
–2(d) to illustrate the capability of achieving optical needle with extended *DOF*. Here${I}_{total}={\left|{E}_{r}(r,z)\right|}^{2}+{\left|{E}_{z}(r,z)\right|}^{2}$ is the total intensity. From Fig. 2(c) and 2(d), the *DOF* defined as the axial full width of above 80% maximum intensity can reach 5λ and 8λ for *N* = 2 and *N* = 3 respectively. We also calculated the beam purity η defined as the percentage of the longitudinal component intensity of total field [12], $\eta ={\Phi}_{z}/({\Phi}_{z}+{\Phi}_{r}),{\Phi}_{z(r)}={{\displaystyle {\int}_{0}^{{r}_{0}}\left|{E}_{z(r)}(r,z)\right|}}^{2}rdr$, where *r _{0}* is the first zero point in the distribution of radial component intensity. In the focal plane (z = 0), we obtained η

_{N = 2}= 86% and η

_{N = 3}= 87%. A very important advantage of our method is that the η remains nearly constant throughout the entire

*DOF*(>81% for both cases). This demonstrates that very high longitudinal polarization component purity has been achieved in the entire

*DOF*. Nearly flat top axial distribution in focal volume can be achieved. With increasing of

*DOF*from 5λ to 8λ, the intensity near the edge will also decrease more steeply, a very desirable property for focal fields with extended

*DOF*. A

*FWHM*spot size of the total field intensity of 0.408λ and 0.405λ is realized for

*N*= 2 and

*N*= 3, respectively. The later case leads to a spot size of 0.129λ

^{2}and the FWHM is restricted below 0.43λ for −3.5λ<z <3.5λ.

The corresponding distributions of the required incident field in the pupil plane P* _{i}* are calculated with Eq. (3) and shown in Fig. 2(e) and 2(f). For dipole arrays with 2

*N*elements, the computed incident distributions can be divided into 2

*N*annular bright zones separated by dark rings. The characteristic of this distribution can be expressed as

*P(ρ*where

_{i})exp (-jψ(ρ_{i})),*P(ρ*denotes a continuous amplitude distribution. The innermost core of the pupil field is dark, indicating a polarization singularity point. The peak amplitude of the bright zones grows monotonically from the center to the outermost zone. The phase

_{i})*ψ(ρ*takes binary values that alternates between 0 and π from one zone to another. The pupil field is radially polarized. Due to the alternating phase, the polarization actually flips its directions between adjacent zones.

_{i})## 3. Discrete complex pupil filter

The desired incident pupil field calculated in the previous section can be considered as a radially polarized field filtered by a complex pupil filter, where *P(ρ _{i})* is the amplitude transmittance and

*ψ(ρ*is the phase of the complex filter. In the above design,

_{i})*P(ρ*is a continuous function while

_{i})*ψ(ρ*takes binary values. Liquid crystal spatial light modulators (SLM) have been demonstrated for the control of the spatial distribution of the light field in the objective’s pupil [11, 25, 26]. However, precise control of continuous transmittance is very challenging due to the needs of extremely careful microscale adjustment and the pixelated format of the SLM, especially when rapid amplitude transmittance spatial variation occurs. In order to simplify the pupil filter design and draw a comparison between our pupil filter and conventional binary DOE design presented previously [12–14, 22], we introduce a discrete complex filter design with non-continuous gray scale amplitude transmittance.

_{i})An example of this new type of filter is designed based on the continuous pupil field distribution of a four-dipole array (*N* = 2) shown in Fig. 2(e). The structure and parameters of this filter are summarized in Fig. 3
. Zones with zero transmittance are designed according to the dark central core and the three dark rings in Fig. 2(e). An amplitude threshold of 5% of the maximal amplitude in the pupil plane shown in Fig. 2(e) is chosen. The transmittances for those regions with amplitude lower than this threshold are set to zero. This allows us to determine the transition points for those zones with zero transmittance. In the other parts, we reduce the continuous amplitude transmittance to discrete transmittance through averaging. Five annular zones with discrete transmittance ranging from 0.07 to 1 and alternating phase 0 or π are shown in Fig. 3. The transition points and radial widths of these annular transmitted zones are optimized and given in Fig. 3. Each transmitted zone corresponds to one bright ring in Fig. 2(e) except for the outermost zone. Owing to the sharp amplitude variation in the outermost zone, we divide it into two zones with the same phase and different average transmittance. Note that the parameters of the outermost annular zone strongly influence the transverse distribution of the focal field while the inner zones help to flatten the axial field intensity distribution to achieve extended *DOF*.

The corresponding total field intensity distributions in the transverse plane and along the z axis are illustrated in Fig. 4(a)
and 4(b) respectively. Similar to the continuous pupil filer, this discrete filter also gives rise to longitudinally polarized field with extended *DOF*. The results are comparable with those shown in Fig. 2(a) and 2(c). This demonstrates that such a discrete complex pupil filter can be used as a proper substitute of the continuous pupil filter.

The performance of this discrete filter demonstrates significant advantages over previously reported binary DOE based methods. Firstly, the constructed focal field is almost purely longitudinal field in the entire focal volume instead of only in the center of the focus. The variation of beam quality η in the main *DOF* is calculated and shown in Fig. 4(c). It can be seen that η is above 85% at the geometric focus and varies slightly in the main axial lobe of the focus in our design. The corresponding results of the reported binary DOE filter with five [12] and four [14] belts are also illustrated in Fig. 4(c) for comparison. Clearly the optical needle obtained with our discrete pupil filter has higher beam purity throughout the focal volume. The second improvement is in the lateral confinement. From Fig. 4(d), *FWHM* spot size of 0.41λ at the focal plane is achieved and maintained below 0.43λ in the main *DOF.* For comparison, the *FWHM* spot size varies between 0.43λ and 0.57λ for$0\le \left|z\right|\le 2\lambda $ in the five belts binary DOE design [12], and varies between 0.44λ and 0.53λ for$0\le \left|z\right|\le 1.5\lambda $ in the four belts binary DOE design [14]. Obviously, a much tighter lateral confinement of the optical needle field can be realized with our discrete complex filter design. Lastly, we’d like to point out that in the previous optical needle designs [12, 14], strong axial intensity sidelobes (above 90% and 50% of the peak intensity respectively) will appear for$\left|z\right|>3\lambda $, which may severely limit their practical applications. In contrary, this problem can be avoided as the axial sidelobe is very well restrained in our design [less than 4% as shown in Fig. 4 (b)].

In practical applications, imperfections of the designed filter will influence the quality of the generated optical needle field. In general, the deviation of radius, transmitted amplitude and phase in each zone will influence the optical needle field quality to some extent. Considering the large number of parameters, a complete tolerance analysis is beyond the scope of the current manuscript. Thus we choose the outer-most annular zone as an example to demonstrate the tolerances for practical implementation. This zone has the largest incident angle and the highest field amplitude, thus the quality of the optical needle field should be most sensitive to imperfections of this zone. For +/−2.5% error in transition radius or +/−10% error in transmittance of this zone, we found that the axial intensity peak-to-valley fluctuations can be controlled below 5% of the peak intensity with the *DOF* greater than 4λ. The corresponding polarization purity still remains above 80% through the total depth of focus. And for +/−0.2π phase deviations from π, axial intensity fluctuations can be controlled below 2% of the peak intensity while maintaining the long DOF (>4λ) and high polarization purity (>80%) throughout the depth of focus.

## 4. Discussions and conclusions

In summary, we presented a method to engineer optical focal field through reversing the radiation pattern of optical dipole array. It is demonstrated that optical focal field with unique properties can be obtained by controlling the pupil field distribution. Optical needle field that consists of dominant longitudinal field component is chosen as an example to illustrate the proposed method. A continuous pupil plane filter design can be found through the reversing of the dipole array radiation. Discrete complex pupil filter is also designed as an approximation of the continuous filter for easier implementation. Numerical examples demonstrate that an optical needle field with a depth of focus of 8λ is obtainable. Throughout the depth of focus, this engineered focal field maintains a diffraction limited transverse spot size (<0.43λ) with very high longitudinal polarization purity (as high as 87%). It is possible to further increase the *DOF* if dipole array with more elements is used. In addition, such a reversing method may be further extended in the reconstruction of other desired field in the focal volume. These specially engineered focal fields have important potential applications including optical microscopy, optical manipulations, optical micromachining and photolithography.

## Acknowledgements

Jiming Wang thanks for the support of the Research Funding No. NS2010200 and Overseas Training Program of Nanjing University of Aeronautics and Astronautics.

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