Control of the multifocal properties of composite vector beams in tightly focusing systems

Hanming Guo; Guorong Sui; Xiaoyu Weng; Xiangmei Dong; Qi Hu; Songlin Zhuang

doi:10.1364/OE.19.024067

1. Introduction

High-resolution and high-speed imaging is urgently needed for the observation of biological phenomena that change spatiotemporally [1] and the mass micronano-fabrication of microstructures [2]. In the last few decades, the most common way to achieve high-resolution is to use pupil filters, which have been very difficult to be further optimized. Recently, the applications of the polarization of light in enhancing resolution get much attention [3–6]. This method is based on the unique focusing properties of various polarized beams in optical system with high numerical aperture (NA). For example, Serrels and coauthors utilize the polarization-sensitive imaging to realize sub-100nm resolution for the illumination light with the wavelength of 1530nm [3], which is based on the asymmetric focal spot caused by a linearly polarized illumination [7, 8]. Due to the unique focusing properties with a resolution higher than that for a linearly polarized illumination [9], a radially polarized beam was used for microscopy [4–6]. In 2009, Sheppard et al. present a Bessel beam with special polarization that can give a central lobe width 9% narrower than for radially polarized illumination [10]. In 2011, Huang et al. also investigate the multifocal properties of a vector-vortex Bessel–Gauss beam (VVBGB), which makes it become possible that VVBGB be used in optical tweezers with multitraps and high-speed multifocal micronano-fabrication [11]. Therefore, it is interesting to investigate focusing properties of a beam with special polarization, which will open a new way to enhance resolution.

To our knowledge, up to now, the common characteristic of those polarized beams who can enhance resolution is that the axial field is bigger than transverse fields in the focal spot [4–6, 9–11] and the contribution of the axial field is to decrease the size of the focal spot. This common characteristic is helpful for micronano-fabrication, optical tweezers and particle acceleration, whereas it might be awful for imaging. The reason is that the index of refraction (IR) mismatch between the objective coupling medium with low IR and the sample with high IR will reduce the contribution of the axial field [4,6] so that the resolution and the focal spot shape deteriorate [12]. An approach to solve this problem is to utilize three-zone complex pupil filters [4] that, due to the multiple beam interference, are difficult to be designed and fabricated successfully. Therefore, in order to apply the polarization of light to realize high resolution in imaging system with IR mismatch between the objective coupling medium and the sample, one of the best ways is to make the contribution of the transverse fields instead of the axial field decrease the size of the focal spot.

As we know, in theory, arbitrary polarized beams may be considered as the coherent superposition of two orthogonally polarized beams, which can be called the polarization modulation, such as cylindrical vector beams [13] and vector-vortex beams [11, 14]. Recently, we present a composite vector beam (CVB) composed of two orthogonally linearly polarized beams with inhomogeneous polarization modulation so that a multifocus with excellent quality is obtained [12]. This CVB is based on the combination of the polarization and amplitude modulations. However, the axial field still plays a key role in the focal spot [12].

In this paper, considering the vortex modulation factor n [11], we first extend the imaging model of the CVB used in Ref. 12 into a more universal imaging model where the focusing properties similar to those of vector-vortex beams [11] can be obtained. Finally, by optimizing the two modulation factors that will be given later, the regulars to control the focal numbers and the dominant field (i.e., whether the axial field or the transverse fields play dominant role in the focusing field) are discussed in detail. For the various applications, some important conclusions and constructive advices are given. These works provide a method for achieving the multifocus with high resolution in the imaging besides the regions of the micronano-fabrication, optical tweezers and particle acceleration.

2. Imaging model of the composite vector beam (CVB)

Here, we assume that the time dependence is $\exp (j ω t)$ and the incident light propagating along +z axis is focused into a sample by an imaging system $L$ obeying the sine condition (Fig. 1 ). $O^{'}$ is the actual focusing point of the ideal focus $O$ that is $d_{s}$ distant from the interface of the sample and is also the origin of rectangular axes $x - y - z$ . $\sum$ is the sphere with centre at $O$ and with radius f equal to the focal length of $L$ . $n_{1}$ and $n_{2}$ are the refractive indices of the image space and the sample. θ is the angle of the emergent ray with the negative z axis. The electric field of the CVB at the incident pupil is

E_{o} = [x e_{x} \cos^{n} (m φ + φ_{0}) + y e_{y} \sin^{n} (m φ + φ_{0})] B (h) A_{o} (h),

with

e_{x}, e_{y} = \pm 1, 0, \pm j

and a constant phase

φ_{0}

, where

A_{o} (h) = \exp (- h^{2} / w^{2})

denotes a Gaussian field with waist radius

w

in a cylindrical coordinate system

(h, φ)

.

n

is called the amplitude modulation factor that determines the power of the sin&cos amplitude modulation. As

m

causes the vortex polarization of the incident beam [11], it is hereafter called the vortex modulation factor. The modulation term

B (h)

denotes a pupil filter with cylindrical symmetry. If

B (h)

is a real, pure imaginary, or complex number, it represents an additional amplitude, phase, or complex pupil filter, respectively. For

n = 0

,

E_{o}

is linearly polarized for

e_{x} / e_{y}

or

e_{y} / e_{x} = \pm 1

and circularly polarized for

e_{x} / e_{y} = \pm j

. For

n = 1

and

e_{x} = e_{y} = 1

, it is radially polarized for

φ_{0} = 0, π

and azimuthally polarized for

φ_{0} = 0.5 π

,

1.5 π

[13]. So, radially and azimuthally polarized beams can be viewed CVBs. For various order

n

,

m

,

e_{x}

,

e_{y}

and

φ_{0}

, more complicated inhomogeneous polarization may be obtained from Eq. (1).

Fig. 1 Geometry of imaging of an aplanatic system.

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In terms of the series expansions of cosine functions with $n$ order, when $n = 2 \tilde{n} + 1$ is odd,

\cos^{n} (m φ + φ_{0}) = \frac{1}{2^{2 \tilde{n}}} \sum_{k = 0}^{\tilde{n}} C_{n}^{k} \cos [(2 \tilde{n} - 2 k + 1) (m φ + φ_{0})],

\sin^{n} (m φ + φ_{0}) = \frac{1}{2^{2 \tilde{n}}} \sum_{k = 0}^{\tilde{n}} {(- 1)}^{\tilde{n} + k} C_{n}^{k} \sin [(2 \tilde{n} - 2 k + 1) (m φ + φ_{0})],

with a combination function

C_{n}^{k} = n! / [k! (n - k)!]

. When

n = 2 \tilde{n}

is even, the similar series expansions exit, namely,

\cos^{n} (m φ + φ_{0}) = \frac{1}{2^{2 \tilde{n} - 1}} {\sum_{k = 0}^{\tilde{n} - 1} C_{n}^{k} \cos [(2 \tilde{n} - 2 k) (m φ + φ_{0})] + \frac{1}{2} C_{n}^{\tilde{n}}},

\sin^{n} (m φ + φ_{0}) = \frac{1}{2^{2 \tilde{n} - 1}} {\sum_{k = 0}^{\tilde{n} - 1} {(- 1)}^{\tilde{n} + k} C_{n}^{k} \cos [(2 \tilde{n} - 2 k) (m φ + φ_{0})] + \frac{1}{2} C_{n}^{\tilde{n}}} .

When the above series expansions are used and the illumination of $L$ is the CVB [Eq. (1)], by a well-known procedure described in prior publications [7, 8, 15], the following expressions for the electric fields of an arbitrary point $P (ρ, ϕ, z)$ in the image space of the imaging system $L$ are obtained easily, for the case of the odd number $n = 2 \tilde{n} + 1$ ,

\begin{array}{l} E_{x} (s_{i}) = - j A \sum_{k = 0}^{\tilde{n}} j^{ℓ} C_{n}^{k} [e_{x} (2 A_{ℓ} \cos Θ^{0 +} + D_{ℓ + 2} \cos Θ^{2 +} + D_{ℓ - 2} \cos Θ^{2 -}) \\ \begin{matrix}  \end{matrix} - e_{y} (^{- 1) \tilde{n} + k} (D_{ℓ + 2} \cos Θ^{2 +} - D_{ℓ - 2} \cos Θ^{2 -})], \end{array}

\begin{array}{l} E_{y} (s_{i}) = - j A \sum_{k = 0}^{\tilde{n}} j^{ℓ} C_{n}^{k} [e_{x} (D_{ℓ + 2} \sin Θ^{2 +} - D_{ℓ - 2} \sin Θ^{2 -}) \\ \begin{matrix}  \end{matrix} + e_{y} (^{- 1) \tilde{n} + k} (2 A_{ℓ} \sin Θ^{0 +} - D_{ℓ + 2} \sin Θ^{2 +} - D_{ℓ - 2} \sin Θ^{2 -})], \end{array}

\begin{array}{l} E_{z} (s_{i}) = 2 A \sum_{k = 0}^{\tilde{n}} j^{ℓ} C_{n}^{k} [e_{x} (B_{ℓ + 1} \cos Θ^{1 +} - B_{ℓ - 1} \cos Θ^{1 -}) \\ \begin{matrix}  \end{matrix} + e_{y} (^{- 1) \tilde{n} + k - 1} (B_{ℓ + 1} \cos Θ^{1 +} + B_{ℓ - 1} \cos Θ^{1 -})], \end{array}

with

Θ^{i \pm} = (n - 2 k) (m ϕ + φ_{0}) \pm i ϕ

,

i = 0, 1, 2

and

A_{ℓ} = \int_{0}^{Φ} \cos^{\frac{1}{2}} θ_{1} \sin θ_{1} (T^{″} + T^{'} \cos θ_{1}) ℜ d θ_{1},

B_{ℓ} = \int_{0}^{Φ} \cos^{\frac{3}{2}} θ_{1} \sin θ_{1} \cos^{- 1} θ_{2} \sin θ_{2} T^{'} ℜ d θ_{1},

D_{ℓ} = \int_{0}^{Φ} \cos^{\frac{1}{2}} θ_{1} \sin θ_{1} (T^{″} - T^{'} \cos θ_{1}) ℜ d θ_{1},

with

ℜ = J_{ℓ} (k_{1} ρ \sin θ_{1}) A_{o} (h) \exp (j Ψ)

,

ℓ = (n - 2 k) m

,

Ψ =

k_{1} \cos θ_{1} d_{s} - k_{2} \cos θ_{2} (z + d_{s})

,

T^{″}

2 n_{1} \cos θ_{1} {(n_{1} \cos θ_{2} + n_{2} \cos θ_{1})}^{- 1}

,

T^{'} = 2 n_{1} \cos θ_{1} {(n_{1} \cos θ_{1} + n_{2} \cos θ_{2})}^{- 1}

,

A = π f λ_{1}^{- 1} 2^{- n} B (h)

, and

J_{ℓ} (\cdot)

being the Bessel function of the first kind and order

ℓ

.

Φ

is the maximum aperture angle.

λ_{1}

,

k_{1}

, and

k_{2}

are the wavelength and wave numbers in image space and sample, respectively. When

m = 1

, formulas (4) will be identical to formulas (2) of image fields in Ref. 12. We are very sorry that there are two misprints in formula (2b) and the definition for

Θ^{i \pm}

in Ref. 12. The correct forms refer to formula (4b) and

Θ^{i \pm}

in this paper.

For the case of the even number $n = 2 \tilde{n}$ ,

\begin{array}{l} E_{x} (s_{i}) = - j A C_{n}^{\tilde{n}} [e_{x} (A_{0} + D_{2} \cos 2 ϕ) + e_{y} D_{2} \sin 2 ϕ] \\ \begin{matrix}  \end{matrix} - j A \sum_{k = 0}^{\tilde{n} - 1} {(- 1)}^{(\tilde{n} - k) m} C_{n}^{k} [e_{x} (2 A_{ℓ} \cos Θ^{0 +} + D_{2 (\tilde{n} - k) m + 2} \cos Θ^{2 +} + D_{ℓ - 2} \cos Θ^{2 -}) \\ \begin{matrix}  \end{matrix} + (^{- 1) \tilde{n} + k} e_{y} (D_{ℓ + 2} \sin Θ^{2 +} - D_{ℓ - 2} \sin Θ^{2 -})], \end{array}

\begin{array}{l} {\tilde{E}}_{y} (s_{i}) = - j A C_{n}^{\tilde{n}} [e_{x} D_{2} \sin 2 ϕ + e_{y} (A_{0} - D_{2} \cos 2 ϕ)] \\ \begin{matrix}  \end{matrix} - j A \sum_{k = 0}^{\tilde{n} - 1} {(- 1)}^{(\tilde{n} - k) m} C_{n}^{k} [e_{x} (D_{ℓ + 2} \sin Θ^{2 +} - D_{ℓ - 2} \sin Θ^{2 -}) \\ \begin{matrix}  \end{matrix} + (^{- 1) \tilde{n} + k} e_{y} (2 A_{ℓ} \cos Θ^{0 +} - D_{ℓ + 2} \cos Θ^{2 +} - D_{ℓ - 2} \cos Θ^{2 -})], \end{array}

\begin{array}{l} {\tilde{E}}_{z} (s_{i}) = 2 A C_{n}^{\tilde{n}} B_{1} (e_{x} \cos ϕ + e_{y} \sin ϕ) \\ \begin{matrix}  \end{matrix} + 2 A \sum_{k = 0}^{\tilde{n} - 1} {(- 1)}^{(\tilde{n} - k) m} C_{n}^{k} [e_{x} (B_{ℓ + 1} \cos Θ^{1 +} - B_{ℓ - 1} \cos Θ^{1 -}) \\ \begin{matrix}  \end{matrix} + (^{- 1) \tilde{n} + k} e_{y} (B_{ℓ + 1} \sin Θ^{1 +} + B_{ℓ - 1} \sin Θ^{1 -})] . \end{array}

In formulas (6), the forms of and the definitions for all of parameters are identical to those used in formulas (4) and (5) except for the fact that one is the even number $n = 2 \tilde{n}$ and the other is the odd number $n = 2 \tilde{n} + 1$ .

In terms of Eq. (1), $n = 0$ means that there are not polarization modulations. Meanwhile, the polarization of the incident beam is linearly or circularly polarized. As $n = 0$ is even and the combination function $C_{0}^{0}$ is equal to 1, the focusing fields described by formulas (6) are identical to formulas (27) of image fields in Ref. 8. Therefore, formulas (4)-(6) describe a universal imaging model of aplanatic systems for various polarized beams, such as linearly, circularly, radially, azimuthally, and vortex polarized beams. Moreover, the sin&cos amplitude modulation with arbitrary orders and the pupil filters with cylindrical symmetry described by the additional modulation term $B (h)$ are also involved.

3. Simulations and discussions

In terms of the sine condition, it is easy to derive $A_{o} (h) = \exp [- {(β \sin θ_{1} / \sin Φ)}^{2}]$ and $h = b \sin θ_{1} / \sin Φ$ , where $β = b / w$ and $b$ is the radius of the entrance pupil.

Next, focusing properties of the CVB [Eq. (1)] will be investigated in detail. In all calculations, $B (h) = 1$ and $\sin Φ = 0.95$ . In Subsection 3.1, $d_{s} = 10 μ m$ , whereas $d_{s} = 0.1 μ m$ in Subsection 3.2. The reason for different values of $d_{s}$ used in simulation calculations will be given in Subsection 3.2. It is noted that, when an annular beam is used, $A_{o} (h)$ is equivalent to 1 for $Ω < \sin θ_{1} < \sin Φ$ and 0 for others, where $Ω$ is the normalized inner radius of the annular beam. In other cases, $A_{o} (h)$ is the Gaussian field defined before and $β = 0.2$ .

3.1 Multifocus with large axial field

Multifocus with large axial field is very helpful for the parallel micronano-fabrication with multispots and the optical tweezers with multitraps. In our previous paper [12], on the basis of the polarization modulation factor $x \cos^{n} φ - y \sin^{n} φ$ , a multifocus with excellent quality including high resolution is obtained. However, in Ref [12], only one or four focal spots can be obtained. The realization of other numbers of focal spots, such as two, six or ten focal spots, needs to revise the factors $\cos^{n} (φ + φ_{0})$ and $\sin^{n} (φ + φ_{0})$ in Eq. (1) of Ref [12]. into $\cos^{n} (m φ + φ_{0})$ and $\sin^{n} (m φ + φ_{0})$ with proper values of n and m in Eq. (1) of this paper. In the following, some conclusions for the case of multifocus with large axial field have been given in Ref. 12, but more extensive conclusions will be firstly discussed.

In terms of the simulation calculations, the numbers of focal spots are 2(m-1) for $m > 1$ and 2(|m| + 1) for $m < 0$ when n is odd and $e_{x} = e_{y} = 1$ . The exponent n mainly determines the intensity ratio $I_{z} / I_{t}$ of the axial to the transverse components. Actually, $n > 1$ means that the sin&cos amplitude modulation is involved in the CVBs. Moreover, as indicated in Ref. 12, the function of the exponent n can be substituted by the normalized inner radius $Ω$ of an annular beam formed by negative cones Therefore, by selecting proper values of n or $Ω$ so as to control $I_{z} / I_{t}$ , the distributions of each focal spots may become nearly circular symmetry [12], which cannot be achieved by the vector-vortex Bessel–Gauss beams [11]. Simulation calculations in Ref. 12 show that the optimal values of n or $Ω$ is to realize $I_{z} / I_{t}$ of about 2.5. Only $I_{z} / I_{t}$ of about 2.5 will make each spot of the multi-focus be nearly circular.

As an annular beam may be formed by blocking the center part of a beam, equal as in results, the functions of n or $Ω$ can be viewed the application of a high-pass amplitude pupil filter for the incident light $A_{o} (h)$ . In Ref. 4, a three-zone complex pupil filter is used for a radially polarized beam, but, due to the doughnut distribution of a radially polarized beam and the block of the second zone, the beam after transmitting through the three-zone complex pupil filter [4] is nearly equivalent to an annular beam. Therefore, for any vector beam whose focusing fields are dominated by the axial field, such as the CVB of this paper and Ref. 12, and the vector-vortex Bessel–Gauss beams [11], the method for increasing $I_{z} / I_{t}$ in order to deal with the problem of the poor transmission through the low–high refractive index interface is to use a high-pass amplitude pupil filter or make the annular radius of the annular beam be more narrow (i.e., increase $Ω$ ), which is also the basis of the application of radially polarized illumination to imaging systems.

As the CVB formed by an annular beam can decrease the power loss of the incident beam [12], the annular beam used in this subsection. Figure 2 describes the light intensities of the CVB shown in Eq. (1) at the focal plane for different values of $Ω$ , which are completely similar to the focusing properties of the vector-vortex Bessel–Gauss beams [See Fig. 3(c) in Ref. 11]. In terms of Eq. (1), the CVB in this paper can be obtained through the coherent superposition of the two annular beams with orthogonal polarization whose electric fields are $A_{o} (h)$ and utilize sin&cos amplitude modulation, which is easily realized by spatial light modulator and Mach-Zehnder interferometer. In addition, it is easy to see from the maximum values of the colorbars in Fig. 2 that $I_{z} / I_{t}$ increases with the increase of $Ω$ , namely one can control $I_{z} / I_{t}$ in order to adjust the contrast of the multifocus by changing the value of $Ω$ , which is impossible for the vector-vortex Bessel–Gauss beams [11]. Therefore, for the vector vortex beams that have been paid much attention during the past few years [11], due to the difficulties of the generation of vortex beams [14, 16, 17] and the flexible characteristic of the CVB given in Eq. (1), this CVB will be the best substitute for the vector vortex beams.

Fig. 2 (Color online) Light intensities of the transverse [(a) and (d)] and axial [(b) and (e)]components and the total light intensity [(c) and (f)] at the focal plane for the case of n = 1, m = 2, $e_{x} = e_{y} = 1$ , and $n_{1} = n_{2} = 1$ , where $Ω = 0.5$ [(a)~(c)] and $Ω = 0.8$ [(d)~(f)].

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Fig. 3 (Color online) The total focusing light intensity at the focal plane for the case of m = 1, $e_{x} = e_{y} = 1$ , $n_{1} = n_{2} = 1$ , $Ω = 0.8$ , (a) n = 4, (b) n = 6, and (c) n = 8.

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Shown in Fig. 2(d) and (e), although $Ω$ is as big as 0.8, the intensity ratio $I_{z} / I_{t}$ is only 1.35. It is so small that the contrast of the two main bright spots on the light intensity of the elliptically annular background spot [See Fig. 2(f)] is very low. Moreover, this phenomenon is true for all the case of $m > 1$ , such as $m = 3$ [See Fig. 3(d) in Ref. 11]. In order to obtain multifocus with high contrast, $m < 0$ with $e_{x} = e_{y} = 1$ are necessary, such as the examples shown in Ref. 12 and Fig. 3(a) in Ref. 11. As the numbers of focal spots are 2(m-1) for $m > 1$ and 2(|m| + 1) for $m < 0$ when n is odd and $e_{x} = e_{y} = 1$ , the two focal spots with high contrast cannot be obtained for the case of odd number n and the vector-vortex Bessel–Gauss beams given by Ref. 11.

In terms of the simulation calculations, when n is even, the focusing properties are unfavorable except for the case of $m = 1$ . We find that, for an arbitrarily even number n, the two focal spots with high contrast can be obtained as long as $m = 1$ (See Fig. 3). These two focal spots arrange in the 45° direction with the x axis. Moreover, as $I_{z} / I_{t}$ increases with the increase of n, the two focal spots are gradually separated and always are nearly circular symmetry. It is needed to indicate that, the two focal spots are separated, but the center position of each focal spot is fixed, which is determined by the position of the focal spots formed by the axial field.

Figure 4 describes the normalized intensities of the total focusing light intensity at the focal plane for various $Ω$ at the case of n = 4, m = 1 and $e_{x} = e_{y} = 1$ . As indicated before, the exponent n has the function the normalized inner radius $Ω$ . Figure 4 also shows that the two focal spots are gradually separated with the increase of $Ω$ and the center position of each focal spot is fixed. Moreover, the FWHM for each focal spot is slightly reduced.

Fig. 4 (Color online) Normalized intensities of the total focusing light intensity at the focal plane for the case of n = 4, m = 1, $e_{x} = e_{y} = 1$ , $n_{1} = n_{2} = 1$ , and $Ω = 0.8$ (solid curve), 0.7 (dashed curve), and 0.6 (dotted curve) along the 45° direction with the x axis.

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3.2 Multifocus with large transverse fields

In Subsection 3.1, the realization of multifocus with large axial field is discussed in detail. In order to utilize multifocus imaging to improve the imaging speed [12, 18, 19], the realization of multifocus with large transverse field is very important.

In this subsection, the annular beam advised in Subsection 3.1 is not applicable to the realization of multifocus with large transverse field. The reason is that the electric fields of the center part of the incident light contribute to the transverse fields of the focal spot. Therefore, The Gaussian field $A_{o} (h) = \exp [- {(β \sin θ_{1} / \sin Φ)}^{2}]$ is used in the following discussions.

In terms of the simulation calculations, we find that, for the CVB defined by Eq. (1), only when $e_{x} = 0$ and $e_{y} = 1$ or $e_{x} = 1$ and $e_{y} = 0$ , the focusing fields are dominated by the transverse fields. Moreover, the numbers of focal spots are 2|m| for $m \neq 0$ when n is odd. For the even number n, the multifocus cannot be achieved.

Figure 5 describes the light intensity distribution of the focusing field at the focal plane for the case of n = 3, m = 2, $e_{x} = 0$ , $e_{y} = 1$ , $n_{1} = 1$ and $n_{2} = 1.5$ . Shown in the colorbars of Fig. 5, the transverse components are far bigger than the axial component. The light intensity distribution of the multifocus is almost the same with that of the transverse components, whose major reason is the mismatch between the objective coupling medium with low IR and the sample with high IR reduces obviously the contribution of the axial field.

Fig. 5 (Color online) Light intensities of the (a) transverse and (b) axial components and (c) the total light intensity at the focal plane for the case of n = 3, m = 2, $e_{x} = 0$ , $e_{y} = 1$ , $n_{1} = 1$ and $n_{2} = 1.5$ .

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At this time, the normalized intensities of the left focus in the first row of Fig. 5(c) (dashed and solid curves) along the x and y axes are shown in Fig. 6 . The four focal spots are completely separated and their FWHMs along the two orthogonal direction are very near. Moreover, shown in Fig. 6, the sizes of each focus of the multifocus are approximately equivalent to that of the single focus (i.e., the dotted and dashed dotted curves) formed by the linearly polarized illumination.

Fig. 6 (Color online) Normalized intensities of the left focus in the first row of Fig. 5(c) (dashed and solid curves) and the focus of the linearly polarized beam (dotted and dashed dotted curves) along the x and y axes.

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However, if $n_{1} = n_{2}$ , namely the refractive index of the objective coupling medium and the sample are matched, although the other parameters are the same with those used in Fig. 5 except for the refractive index $n_{1}$ and $n_{2}$ , the multifocus cannot be completely separated [See Fig. 7(a) ] because the axial component in the focal spots is too big. In terms of simulation calculations, this phenomenon is also true for other various m. Fortunately, this phenomenon may be resolved for the case of $n_{1} < n_{2}$ . Moreover, the sizes of each focal spot almost are not reduced for the case of small $d_{s}$ , where $d_{s} = 0.1 μ m$ . The reason for $d_{s}$ using small value instead of big value (i.e., $d_{s} = 10 μ m$ ) used in Subsection 3.1 is that $d_{s}$ is bigger, the defocus and the sizes of the focus are bigger for the case of the focus with large transverse fields, whereas it is not true for the case of the focus with large axial field, where the defocus is the distance between the actual focusing point $O^{'}$ and the ideal focus $O$ (See Fig. 1). So, the effects of the refractive index mismatch on the field distribution of the focus for the case of the focus with large transverse fields are far more serious than the case of the focus with large axial field. These subjects including the resolved methods are reserved for the future.

Fig. 7 (Color online) The total focusing light intensity at the focal plane for the case of m = 2, $e_{x} = 0$ , $e_{y} = 1$ , and $n_{1} = 1$ . (a) n = 3 and $n_{2} = 1$ . (b) n = 1 and $n_{2} = 1.5$ .

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For other various odd number n and the fixed value m = 2, the similar multifocus shown in Fig. 5 may be obtained. Actually, when $e_{x} = 0$ and $e_{y} = 1$ or $e_{x} = 1$ and $e_{y} = 0$ , the CVB become a linear polarized beam with n order sin&cos amplitude modulation [See Eq. (1)]. Therefore, unlike the function of n introduced in Subsection 3.1, $n$ mainly affects the shape of the intensity distribution of the transverse fields in each focal spot and might make each focal spot be nearly circular symmetry. For example, shown in Fig. 5(c) and Fig. 7(b), the former is nearly circular symmetry, whereas the latter is a little triangular for the case of n = 1.

4. Conclusion

In conclusion, we have derived the imaging model of the CVB given by Eq. (1), which describes a universal imaging model of aplanatic systems for various polarized beams, such as linearly, circularly, radially, azimuthally, and vortex polarized beams. Moreover, the sin&cos amplitude modulation with arbitrary orders and the pupil filters with cylindrical symmetry are also involved in this imaging model.

On the basis of the above imaging model, the regulars to control the numbers of the focal spots and the dominant fields, namely, whether the axial field or the transverse fields play dominant role in the focusing field, are discussed in detail. In addition, some important conclusions are drawn.

First, for the case of the focus with large axial field, when n is odd and $e_{x} = e_{y} = 1$ in the CVB given by Eq. (1), the numbers of focal spots are 2(m-1) for $m > 1$ and 2(|m|+1) for $m < 0$ . However, in order to obtain multifocus with high contrast, $m < 0$ with $e_{x} = e_{y} = 1$ are necessary. For all the case of $m > 1$ , the multifocus with high contrast cannot be obtained. In addition, the two focal spots with high contrast cannot be obtained for the case of odd number n and the vector-vortex Bessel–Gauss beams given by Ref. 11. Moreover, when n is even, the focusing properties are unfavorable except for the case of $m = 1$ . We find that, for an arbitrarily even number n, the two focal spots with high contrast can be obtained as long as $m = 1$ . Finally, by using a high-pass amplitude pupil filter or making the annular radius of the annular beam be more narrow (i.e., increase $Ω$ ), one can control $I_{z} / I_{t}$ in order to adjust the contrast of the multifocus and deal with the problem of the poor transmission through the low–high refractive index interface. In our opinions, for the vector vortex beams that have been paid much attention during the past few years [11], due to the difficulties of the generation of vortex beams [14, 16, 17] and the flexible characteristic of the CVB given in Eq. (1), this CVB is the best substitute for the vector vortex beams.

Second, for the case of the focus with large transverse fields, only when $e_{x} = 0$ and $e_{y} = 1$ or $e_{x} = 1$ and $e_{y} = 0$ , the focusing fields are dominated by the transverse fields. Moreover, the numbers of focal spots are 2|m| for $m \neq 0$ when n is odd. For the even number n, the multifocus cannot be achieved. Unlike the function of n in the case of the focus with large axial field, for the case of the focus with large transverse fields, $n$ mainly affects the shape of the intensity distribution of the transverse fields in each focal spot and might make each focal spot be nearly circular symmetry. In addition, in order to realize the multifocus separated completely, the mismatch between the objective coupling medium with low IR and the sample with high IR is helpful. Moreover, the sizes of each focus of the multifocus are approximately equivalent to that of the single focus formed by the linearly polarized illumination.

The work in this paper is important for high-resolution and high-speed imaging in biology and micro-/nanofabrication and the optical tweezers with multitraps.

Acknowledgments

This work was supported by the National Basic Research Program of China (2011CB707504), the Shanghai Leading Academic Discipline Project (S30502), the National Natural Science Foundation of China (60807007 and 61178079), the Shanghai Foundation for Development of Science and Technology (08QA14051), the Fok Ying-Tong Education Foundation, China (121010), and the Foundation for the Author of National Excellent Doctoral Dissertation of P. R. China (201033).

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Control of the multifocal properties of composite vector beams in tightly focusing systems

Abstract

1. Introduction

2. Imaging model of the composite vector beam (CVB)

3. Simulations and discussions

3.1 Multifocus with large axial field

3.2 Multifocus with large transverse fields

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (7)

Equations (14)

Optics Express