We theoretically demonstrate that the doughnut focal spot can continuously be manipulated by synthetically using various beam modulation techniques. Comparatively, a more evident effect can be expected by different orders of phase modulation, while accurate manipulation stems from changing the phase diversity between two arms in an image inverting interferometer (III). The size of central dark spot can thus be continuously adjusted in a theoretically infinite scale, although it may actually be limited by resolution of Spatial Light Modulator (SLM). This approach brings additional flexibility to many applications, such as optical tweezers.
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Since the vectorial diffraction theory was proposed, humans have gradually realized that it is not precise to describe the focal spot of a collimated beam as an Airy disk  when the numerical aperture (NA) of objective lens utilized is high enough. Multiple aspects, including the phase , beam shape , polarization of incident light  and aberration  can potentially change the shape of focal spot. This natural property of light, although makes it more difficult to forecast the focal phenomenon, indeed brings additional flexibility in optical design. Based on the explicit design to the whole system, some special, but useful focal spot patterns are created, and the corresponding design methods are proposed. However, once all devices in the optical path are fixed, it is not easy to adjust the final pattern.
Doughnut focal spot, with a dark center surrounded by a high-intensity ring, is widely used in many research domains: such as nonlinear optics , bio-optics  and excitation of surface wave . It is not surprising that the performance of corresponding system highly relies on the intensity distribution of doughnut focal spot, especially the size of central dark area. Generally, the doughnut focal spot can be generated by using incident beam phase-encoded by a vortex 0~2π phase plate , azimuthally polarized beam , or some micro/nano-structures . To any of the current approaches, changing the size of central dark area is a task hard to achieve, which severely limits the flexibility in practical applications. This problem can partially be solved by using high-order vortex phase plate . However, because a perfect doughnut focal spot can only be expected when the value of order is even, it is still impossible to continuously manipulate the size of central dark area. Theoretical simulation has confirmed that if the incident light is elliptically polarized, the size can continuously be manipulated in one direction by changing the ellipticility of polarization . However, a following question is that the uniformity of intensity around the central dark area will be destroyed. By using a hollow incident beam and defocusing the objective plane, Du et al.  proposed a simple way to create the doughnut spot, and the size of central dark area can readily be manipulated in a large scale by modifying the defocusing distance. However, the diffractive effect makes it is unrealistic to shrink the size to subwavelength scale, otherwise, the central intensity can never maintain at zero.
Recently, Hao et al.  represented a method to solve this problem. Based on image inverting interference (III), combined with 0~2π vortex phase modulation, they theoretically demonstrated that the doughnut focal spot could readily be manipulated, and either shrinkage or expansion of size of central dark area is possible in a scale of 0.555λ-0.830λ, where λ is the wavelength of incident beam. Although the continuous manipulation of the doughnut focal spot has been realized, the manipulation scale is still small. In practical system, such as optical tweezers, a larger manipulation scale will result in more optimized performance by supplying the capability to trap particles with diverse sizes or weights.
In this paper, we develop previous works, and propose a new approach to continuously manipulate the size of doughnut focal spot in a much larger scale. We firstly notice the fact that the size of doughnut focal spot can evidently be enlarged by higher order of vortex phase modulation, and further confirm that III, combined with phase diversity adjustment, can universally be adopted to change size of doughnut focal spot in a finite interval no matter what the vortex order is. To overlay the manipulation gap between two adjacent intervals to realize continuous manipulation in the whole scale, the incident beam is re-shaped to expand each single interval. The whole system works like a transmission in a road-going vehicle, that the “gear” is decided by the order of vortex phase modulation, while the “throttle” is the phase diversity between two arms of III. The final result is simulated by vectorial diffractive theory. We further discuss its feasibility in both forward and 4Pi structures and forecast its potential in practical systems.
Although scalar one is easier and more effective mathematically, the vectorial diffractive theory is more accurate when NA is high. Based on the pioneer work of Richards et al., the Debye integral  can thus be written as
A straightforward way to modify the shape of focal spot is introducing the Spatial Light Modulator (SLM), or the phase plate, into the optical path. Generally, the doughnut focal spot can be attained by vortex phase modulation, and the corresponding phase delay parameter can be expressed as:Eq. (2) into Eq. (1), the amplitude distribution of doughnut focal spot can be calculated without difficulty. A larger doughnut focal spot will stem from higher order of vortex phase modulation, and the detailed data is listed in Table 1 . The utilization of the image inverting interference (III) makes the distribution a bit more complex. The scheme of an III  is a two-beam interferometer with an image inverting optics included in one of the arms. In this situation, the amplitude distribution of final focal spot will be the interference result of the two spots independently generated by the noninverted and inverted beams, respectively, and can be written as:
3. Results and discussion
To realize the continuous manipulation of doughnut focal spot in a large scale, we set up a system as shown in Fig. 1 . After radiated from the laser source, the incident beam will propagate through the Beam Shape Modulator (BSM), SLM and III, and finally be focused by the high NA objective lens to generate a doughnut focal spot. The function of BSM will be explicitly discussed in later sections. Many forthcoming devices are available to realize image inverting, such as 4f telescope , confocal concave mirror pairs , double Porro prism , UZ-interferometer , or modified Michelson one . It should be emphasized that the mirror based solutions, especially the double Porro prism, is probably very essential to the system working with femtosecond lasers, because it is simple and more compact, while the phase diversity between two arms can be modulated by adjusting the gap between two single prisms along the optical axis. Furthermore, the flat mirror is very insensitive to chromatic dispersion. To simplify the discussion, some conditions are listed as the premises without loss of generality. The incident light is a collimated beam with Gauss intensity distribution and circular polarization. The objective lens is aberration-free  and oil-immersed, with NA = 1.4. The detection plane is placed exactly at the ideal foci of objective lens. Based on the assumptions above, all parameters in Eq. (1) can thus be clarified, that
Considering the optimization to final focal spot, we make the intensity ratio between two arms of III as I1:I2 = 1:1, although it may not be the unique option. Thus, N and α will be the only variable parameters which can influence the amplitude distribution of focal spot. By substituting Eq. (4) into Eq. (3), the amplitude distribution can be derived, and the intensity can further be calculated by the formula . If defining the size of central dark spot s as the distance between two intensity peaks along the radial direction, s versus N and α can thus be presented as shown in Fig. 2 , and the manipulation scales are concisely summarized in Table 2 . Comparatively, a more evident effect can be expected by different orders of phase modulation (we name these values as the “Standard Cases”), while changing the phase diversity between two arms in III can modify the size s in both positive and negative directions around the “Standard Case”, and a finite manipulation interval can thus be created. Hence, a whole manipulation scale will be the mosaic result of the all intervals, and can be created by comprehensively adjusting two values. The whole system works like a transmission in a road-going vehicle, that the “gear” is decided by the order of vortex phase plate, while the “throttle” is the phase diversity between two arms of III. However, it is notable that although it is very small and can even be neglected, a gap indeed exists between two adjacent intervals, which make the whole manipulation actually discontinuous.
To overcome this defect, it is necessary to supply additional parameters to improve the manipulation flexibility, and that is the reason why BSM is used. The BSM includes two separate optical paths and can be switched to either one by the rotatable mirrors. The beam can either maintain the Gauss intensity distribution in cross section, or be reshaped as the hollow one. The hollow beam can simply be generated by making the incident beam propagating through an annular aperture, or by using some specific devices like axicon , adaptive optics  or optical fiber . The hollow distribution in cross section can approximately be described as the Bessel-Gauss function , thatTable 2. Compared with the data of using Gauss Beam alone, the gap between two intervals can now be filled by switching the optical paths of BSM when necessary.
By using the method mentioned above, the expansion of size of central dark area is theoretically infinite in upper limit. A larger size stems from a higher order of vortex phase modulation. However, in practical experiment, the limited resolution of SLM will confine the acquisition of high order vortex phase modulation, which indirectly limits the attainable maximum size. Moreover, BSM can be abandoned if the 4Pi  structure can be utilized. In a 4Pi structure, what is different from the forward structure mentioned above is that two opposing objective lenses which both are focused to the same geometrical location are introduced. The inverted and non-inverted arm of STED beam will propagate along opposite directions and be focused by two objective lenses, as shown in Fig. 3(a) . The amplitude distribution around the ideal foci can be expressed asFig. 3(b). In this case, the overlap can be achieved by switching between the forward and 4Pi structures.
However, even we deliberately ignore the additional cost and complexity induced, the introduction of 4Pi structure will worsen the intensity uniformity around the central dark area of focal spot (as shown in Fig. 4(a) ), and this situation becomes more severe when the order N is enlarged. To give a quantitative evaluation, we define the uniformity parameter asFig. 4(b). The curve presents an apparent downward trend of uniformity following with the increasing of N when 4Pi is used. On the other hand, similar influence can never be detected in the forward structure.
As mentioned above, the doughnut focal spot possesses a lot of potential applications, while the variable size of central dark area will brings additional flexibility to them. Taking the optical tweezers for example, the origin of the trapping force is the gradient of field intensity. Generally, the gradient can be expressed as :
In conclusion, this paper proposes a new approach to expand the continuous manipulation scale of doughnut focal spot. This target is achieved by synthetically using various beam modulation techniques, including intensity and phase modulation of incident beam, and III. The whole system works like a transmission in a road-going vehicle, that the “gear” is decided by the order of vortex phase modulation, and the “throttle” is the phase diversity between two arms of III. The manipulation gaps of size between two adjacent orders cannot be overlaid unless the beam shape is switched between Gaussian and Bessel-Gaussian modes. The result is simulated based on the vectorial diffraction theory. Although in theory the continuous manipulation scale can infinitely be broadened by this method, but the actual upper limit will be decided by the resolution of SLM. The introduction of 4Pi will optimize the performance of system by increasing the overlap of manipulation scale between orders of vortex, but at the expense of the worse intensity uniformity. This method will significantly enhance the performance of optical tweezers by enabling trapping a large variety of particles in the same optical system.
This work was financially supported by grants from the Qianjiang Talent Project (Grant No.2011R10010), the Doctoral Fund of Ministry of Education of China (Grant No.20110101120061), the Scholarship Award for Excellent Doctoral Student granted by Ministry of Education, and the Fundamental Research Funds for the Central Universities (Grant No.2012FZA5004).
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