## Abstract

In a 4Pi focusing system radially polarized laser beams can be focused to a spherical focal spot. For many applications, e.g., for moving trapped particles or for scanning a specimen, one would like to change the position of focal spot along the optical axis without moving lenses or laser beams. We demonstrate how this can be achieved by modulating the phase of the input field at the pupil plane of the lens. The required phase modulation function is determined by spherical wave expansion of the plane wave factors in the Richards–Wolf integral.

©2011 Optical Society of America

## 1. Introduction

When radially polarized laser beams are focused by a lens of high numerical aperture (NA), they exhibit unique focusing properties in comparison to linearly polarized ones: a smaller focus spot and a strong axial electric field component [1, 2]. This made them find applications in many fields, e.g., electron acceleration [3, 4], spectroscopy [5], and particle trapping and manipulation [6–10]. By imposing proper amplitude or/and phase modulations on radially polarized input fields, unusual field distributions can be constructed at the focus. For example, a uniform and non-diffracting axially polarized beam with sub-diffraction beam size was obtained by focusing a radially polarized Bessel–Gaussian beam with a combination of a binary-phase optical element and a lens of high NA [11, 12]. Recently, the possibility of focusing a radially polarized beam to a sharp spherical focal spot was demonstrated theoretically for a 4Pi focusing system by properly choosing the input field at the pupil plane of the lens [13, 14]. Such spherical focal fields provide, e.g., equal axial and transverse resolutions for confocal microscopy. In this paper, we generalize the approach of Refs [13, 14]. in order to achieve a dynamical spherical spot, i.e., a spherical spot that can be shifted along the optical axis in real time.

## 2. Theory

Figure 1
sketches the geometry of a typical 4Pi focusing system. It consists of two objective lenses of high numerical aperture (For the simplicity of mathematic derivation, we assume that NA = 1). Two counter-propagating radially polarized light beams are focused by both lenses such that the foci coincide. The input field at the pupil plane of each lens takes the form of **E**
_{input} = *l*(*θ*)**e**
* _{ρ}*, where

**e**

*(*

_{ρ}*θ*) is the unit vector in radial direction and

*l*(

*θ*) represents the input field. We denote the input fields at the pupil planes by

*l*

_{left}(

*θ*) and

*l*

_{right}(

*θ*) for the left and right lens, respectively, where

*θ*covers the range from 0° to 90° for

*l*

_{left}and from 180° to 90° for

*l*

_{right}, while the single input field

*l*(

*θ*) at the pupil plane of the lenses covers the full range from 0° to 180°. When

*l*

_{left}and

*l*

_{right}have a certain phase relation, the

*z*-component of the electric fields near the focus becomes remarkably strong thanks to constructive interference, while the radial component experiences destructive interference and thus is very weak. Mathematical description of this interference effect can be established by using the well-known Richards–Wolf integral [2, 15]

*E*and

_{ρ}*E*are the radial and axial components of the electric field at an observation point

_{z}**R**(

*ρ*,

*z*) near the focus,

*ρ*and

*z*are cylindrical coordinates, and

*X*(

*θ*) is the pupil apodization function where

*X*(

*θ*) = (cos

*θ*)

^{1/2}for an aplanatic lens and

*X*(

*θ*) = 1 for a Herschel-type lens (used in this paper),

*J*(

_{n}*x*) is the cylindrical Bessel function of first kind of order

*n*,

*h*(

*θ*) =

*k*cos

*θ*is the

*z-*component of

**k**, and

*k*is the wave number.

Choosing the input field *l*(*θ*) to be *l*
_{0}(*θ*) = sin*θ* exp(−2sin^{2}
*θ*) and substituting this into the integrals (1) and (2), one can see that the intensity of the focal field, dominated by the *z*-component, exhibits an almost spherical distribution, i.e., a spherical focal spot, at *z* = 0. Our aim is to determine *l*(*θ*) such that it corresponds to a spherical (focal) spot translated to another position *z* = *z*
_{0} along the optical axis. As mentioned, the by far dominating contribution to the spherical intensity distribution at the focus is coming from the *z*-component. Since |*E _{ρ}*|

^{2}is negligibly weak compared to |

*E*|

_{z}^{2}, [for example, if

*l*(

*θ*) =

*l*

_{0}(

*θ*), max(|

*E*|

_{ρ}^{2})/max(|

*E*|

_{z}^{2}) ≈0.1303 in the focal region] it is sufficient to consider only Eq. (2). As we know, the

*z*-component of the electric field obeys the scalar wave equation, whose spherically symmetrical solution is the first kind of spherical Bessel function of zero order:

*j*

_{0}(

*kR*) ≡ sin(

*kR*)/(

*kR*), where

*R*= |

**R**|. For a fixed vector

**z**=

**u**

_{3}

*z*

_{0}on the optical axis (here

**u**

_{3}is the unit vector in the

*z*direction), the desired solution is

*j*

_{0}(

*k|*

**R**

*−*

**z**

*|*) with an intensity |

*j*

_{0}(

*k|*

**R**

*−*

**z**

*|*)|

^{2}that forms a spherical spot at the shifted position

*z*

_{0}.

To derive the field *l*(*θ*) by which this can be achieved, we transform from cylindrical to spherical coordinates by expressing the factor *J*
_{0}(*kρ*sin*θ*)*e ^{ih}*

^{(}

^{θ}^{)}

*in terms of the spherical wave functions [16]:*

^{z}*P*(

_{n}*x*) denotes the Legendre polynomial of order

*n*,

*α*is the polar angle of the observation point

**R**, and

*j*(

_{n}*x*) is the spherical Bessel function of the first kind with order

*n.*When Eq. (3) is inserted into Eq. (2), one obtains [Note that we use here

*X*(

*θ*) = 1]

*A*by which the right hand side of Eq. (4) transforms into an expression of the form of

_{n}*j*

_{0}(

*k|*

**R**

*−*

**z**

*|*) that represents a spherical spot centered at

*z*

_{0}on the optical axis. Consider the addition theorem of the spherical Bessel functions [16]

*j*(

_{n}*x*) = (−1)

*(*

^{n}j_{n}*x*) and equating the right hand sides of Eqs. (4) and (6), we find thatPutting

*x*= cos

*θ*and

*g*(

*x*) =

*l*(

*θ*), Eq. (5) becomes

*x*

^{2})

^{1/2}

*l*(

*x*). So, we require the input field

*g*(

*x*) to be

*x*

^{2})

^{−1/2}has a divergence at

*x*= 1. To overcome this, we replace (1−

*x*

^{2})

^{−1/2}by

*g*

_{0}(

*x*). Here

*g*

_{0}(

*x*) = (1−

*x*

^{2})

^{1/2}exp[−2(1−

*x*

^{2})] or

*l*

_{0}(

*θ*) = sin

*θ*exp[−2sin

^{2}

*θ*], respectively, denotes the fundamental radial polarization mode. Since Bokor and Davidson use the input field

*l*

_{0}(

*θ*) = sin

*θ*exp(−2sin

^{2}

*θ*) to produce a spherical focal spot [13], we expect it to work in our case, too. So Eq. (9) becomeswith

*l*(

*θ*) can be thought of as a modulation of the fundamental mode

*l*

_{0}(

*θ*) by

*A*(

*θ*).

## 3. Numerical results

In the last section we have derived expression (10) for the input field *l*(*θ*). In this section, we substitute *l*(*θ*) into Eqs. (1) and (2) to check whether this gives the desired result. For *A*(*θ*) = 1, the situation is the same as in [13], where a spherical focal spot is produced with the input field *l*(*θ*) = *l*
_{0}(*θ*) at the origin, but what we want is a spherical spot at another position *z*
_{0} on the optical axis. For numerical demonstration we have chosen *z*
_{0} = 1.5λ. We calculate the coefficients *A _{n}* from Eq. (7), and the input field

*l*(

*θ*) from Eq. (10). In Fig. 2(a) the intensity |

*l*(

*θ*)|

^{2}of the input field is plotted. We find that

*l*

_{left}(

*θ*) and

*l*

_{right}(

*θ*) have exactly the sameintensity distributions and each distribution exhibits the intensity property of a fundamental radially polarized field: an annular intensity distribution with intensity minimum in the center, i.e., both are still fields with radial polarization. In fact, it can be verified that the modulation function has a constant modulus, suggesting that modulation does not change the intensity distribution of the input field. In Fig. 2(b), we present the phase of the input field

*l*(

*θ*). From the phase distribution, a two-belt phase structure is found for

*l*

_{left}(

*θ*). The first belt covers the range from 0 to about 70° and the second one goes from 70° to 90°. In the first belt, the phase increases almost linearly from 0 to 2

*π*. Then, after a transition to 0 at

*θ*= 70°, it increases linearly from 0 to

*π*in the second belt. The phase of

*l*

_{right}(

*θ*) has the same two-belt structure as

*l*

_{left}(

*θ*), but with a difference of sign in the corresponding belts. Figures 2(a) and 2(b) suggest that the input field

*l*(

*θ*) can be obtained from a fundamental radial field mode by a two-belt phase modulation, which can be achieved with a spatial light modulator. Having determined

*l*(

*θ*), we calculate from the integrals (1) and (2) the electric intensities in the focal region. Figure 3 shows line scans of the total intensity

*I*= |

*E*|

_{z}^{2}+ |

*E*|

_{ρ}^{2}in the axial direction (solid line) and in the transversal direction (dashed line), respectively. The maximum of the intensity has been normalized to unity. As can be seen, a nearly spherical intensity spot centered at

*z*

_{0}= 1.5λ can indeed be realized. Axial and transversal spot diameters are nearly identical and have a width of approximately 0.5λ, in agreement with the results in [13] and [14] at

*z*

_{0}= 0.

By changing the parameter *z*
_{0} continuously, it is evident that the spherical spot will move along the optical axis, i.e., the goal of a real time shifting of a spherical spot thus is solved. For illustration, we plot 2D (*XZ* plane) color graphs of the intensity in the vicinity of the focus for different values of *z*
_{0} in Fig. 4
and demonstrate by a movie (Media 1) the evolution of the intensity in the vicinity of the focus when the parameter *z*
_{0} is changed continuously from −2λ to 2λ (The value of *z*
_{0} increases by 0.05λ every 0.05s). Figures 4(a)-(e) are successive frames extracted from the movie at *t* = 0s, 1s, 2s, 3s and 4s (or *z*
_{0} = −2λ, −1λ, 0λ, 1λ and 2λ), respectively. As desired, we obtain a series of spots centered at *z*
_{0} that are excellent approximations to the desired spherical spots of intensity and that all have a radius of approximately 0.5λ. The corresponding movie (Media 1) proves that the intensity distribution keeps its nearly spherical shape over the whole translation range. To quantify the extent of the shape of the spot to spherical shape, we introduce the size mismatch parameter ∆* _{XZ}*, measured by the difference between transverse and axial diameters. We find that the size mismatch ∆

*(slightly elongated along the*

_{XZ}*z*axis) are all roughly −0.036λ for three

*z*

_{0}( = 0λ, 1λ and 2λ), implying quantitatively the almost spherical shape of the spot during the whole translation.

Before leaving this section, we want to make some remarks concerning the practical applications of our design. First, the objective lens used in this paper is a Herschel-type lens, while aplanatic lenses are much more widely used in practice. The Herschel-type lens has a special principal surface (see Fig. 1 in [13]), which can eliminate the first-order axial aberration. Furthermore, for a Herschel-type focusing system the apodizer factor due to the conservation of energy appearing in the Richards–Wolf integral (1) and (2) is simply *X*(*θ*) = 1, for which a fundamental radial polarization mode is focused into a spherical spot centered at the focus. For an aplanatic focusing system, the apodizer factor *X*(*θ*) = (cos*θ*)^{1/2}, the resulting focal spot is not of spherical shape but slightly transversely elongated (see Fig. 2 in [13]). To obtain the spherical focal spot in the aplanatic focusing system, we can simply divide the field amplitude *l*(*θ*) of the incoming beam by (cos*θ*)^{1/2}, which means the introduction of an amplitude modulation, as done by Chen and Zhan in [14]. Second, we have assumed the maximum converging angle *θ*
_{max} to be 90° for the objective lens, which is absolutely impossible for practical objective lens. Meanwhile, when the aplanatic focusing system is used, the introduction of the amplitude modulation [*l*(*θ*)/(cos*θ*)^{1/2}, as discussed above] also requires *θ*
_{max} less than 90°. For two high numerical aperture objective lenses with *θ*
_{max} = 79.6° [NA = 1.49 oil (1.515) immersion objective, Nikon], we find that the focal spot can still be moved along the axis, but the spot is elongated along the transverse direction with the size mismatch ∆* _{XZ}* ≈0.1480λ (

*z*

_{0}= 2λ). However, we can reduce the size mismatch by properly choosing the minimum converging angle

*θ*

_{min}of the focusing system, which can be achieved by blocking the central region of the incoming beam with an opaque disk. For the case discussed here (

*θ*

_{max}= 79.6°), if we put

*θ*

_{min}= 25°, the size mismatch will be ∆

*≈0.082λ. The final comment refers to the translation range of the spot, i.e, the maximum value of*

_{XZ}*z*

_{0}. In our calculation, we find that for larger values of

*z*

_{0}our design still works. For example, when

*z*

_{0}= 10λ, we obtain a nearly spherical spot with ∆

*≈-0.037λ. But the corresponding phase distribution of the input field will become a ten-belt structure, while the phase structure for*

_{XZ}*z*

_{0}= 1.5

*λ*is of only two belt as shown in Fig. 2(b). With further increasing

*z*

_{0}, the phase structure becomes more complex. As a result, we conclude that our design can realize ± 10λ translation of the spherical spot of intensity without complex phase modulation.

## 4. Conclusion

A method is proposed that allows shifting of an on-axis spherical spot in real time in a 4Pi focusing system using a radially polarized beam. We have determined the proper input field *l*(*θ*) that does the trick to produce a spherical intensity spot at any designated position *z*
_{0}. As concerns the intensity, *l*(*θ*) has the same distribution as the fundamental radial polarization mode, but its phase is modulated. In other words, *l*(*θ*) turns out to be simply a phase modulated fundamental mode. We have shown that the position parameter *z*
_{0} can be varied continuously such that the spherical intensity distribution of the focus is maintained during dynamical movement of the focal spot along the optical axis. In conclusion, we have pointed out a way how to move a trapped particle or to scan a specimen without moving objective lenses or laser beams.

## Acknowledgments

This research is supported by the Natural Science Foundation of China (NSFC) (10874240, 61077005) and the Chinese Academy of Sciences (CAS)/State Administration of Foreign Experts Affairs of China (SAFEA) International Partnership Program for Creative Research Teams.

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