Creation of identical multiple focal spots with three-dimensional arbitrary shifting

Xiaolei Wang; Liping Gong; Zhuqing Zhu; Bing Gu; Qiwen Zhan

doi:10.1364/OE.25.017737

1. Introduction

Over the past few years, the peculiar focusing properties of cylindrical vector (CV) beams [1] have drawn much attention due to their intriguing applications in optical microlithography [2], particle trapping and manipulation [3], and spectroscopy [4]. Through modulating the incident CV beam with diffractive optical elements (DOEs), vortex phase and other special filters in a high numerical aperture (NA) focusing system, researchers have obtained a variety of unique focal fields such as optical needle [5, 6], flattop focus [7], optical bubble or cage [8, 9], optical dark-hole [10] and spherical spot [11]. Up to now, many works related to the generation of single focal spot with different intensity profile have been reported [7–9, 11]. Besides, bi-focus or multi-focus with identical intensity profiles has potential applications in laser direct writing, optical microscopy and multiple-particles trapping, especially for efficiency improvement. Importantly, multi-focal spots have been created by many approaches, such as by a concentric multi-belt pure phase filter [12], modulating radial-variant vector beam [13], by complex phase mask [14], and solving the inverse problem of antenna radiation [15]. It should be noted that the above-generated static focal spots were limitedly located along the optical axis. Recently, B. Yao’s group has demonstrated the continuous dynamic shifting of a single spherical focal spot along the optical axis [16] and in three-dimensional space [17] in a 4π focusing system by modulating the phase of the input field. Although these works had demonstrated the generation of bi-focus or multi-focus and the limited shifting ability of single focal spot, it could be more desirable to control the three-dimensional arbitrary shifting of multiple focal spots for optical engineers and biologists.

In this paper, we propose a simple and flexible method to obtain identical multiple focal spots in a single lens focusing system and realize three-dimensional arbitrary shifting of the generated multiple focal spots without moving lens or laser beams. The incident CV beam is modulated by both phase and amplitude and then tightly focused by a single lens. The functions of phase and amplitude modulations on the incident CV beam are the key of the proposed method and are detailed in section 2 of this paper for revealing how to find them. The multiple focal spots with predetermined number and positions are generated based on the method and the identical intensity distribution as well as the polarization distribution for each individual focal spot is demonstrated. We also demonstrate the three-dimensional shifting ability of the multiple focal spots in real-time with prescribed trajectory by continuously regulating the phase and amplitude modulations on the incident CV beam. Furthermore, the method also can generate multiple focal spots with unique intensity profile when proper diffractive optical element (DOE) is applied. Because the proposed method provides identical multiple focal spot with three-dimensional arbitrary shifting ability and desired intensity profile, it may find potential applications in laser three-dimensional printing, moving multi-particles trapping and biological molecules manipulation.

2. Theoretical analysis

Firstly, we illustrate the shifting of single focal spot and Fig. 1 shows the schematic of single focal spot shifting system illuminated with a generalized CV beam. Compared with the conventional single lens focusing system, the incident CV beam is purposefully modulated by phase and amplitude and then passes through a high numerical aperture (NA) objective lens, where the original focal spot (marked O) is shifted to a new position (marked O').

Fig. 1 Schematic diagram of single focal spot shifting system under the illumination of a generalized CV beam. V(r,ϕ, z)/ V'(r',ϕ', z') is an observational point in the focal plane before /after focal shifting.

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The electric field in the pupil plane of the focal lens can be expressed as:

{\vec{E}}_{0} (θ, ϕ) = l_{0} circ(\sin θ / \sin α)(\cos ϕ_{0} {\vec{e}}_{ρ} + \sin ϕ_{0} {\vec{e}}_{ϕ}),

Here l₀ is a constant related to the amplitude of the field,

circ(\sin θ / \sin α)= {\begin{matrix} 1 \\ 0 \end{matrix} \begin{matrix} θ \leq α \\ θ > α \end{matrix}

, which denotes the input beam within a circular aperture, where α = arcsin(NA) is the maximal angle determined by the numerical aperture (NA) of the objective lens that obeys sine condition [18], ϕ₀ is the initial polarization azimuth angle of the incident CV beam.

{\vec{e}}_{ρ}

and

{\vec{e}}_{ϕ}

are the unit vectors of radial and azimuthal directions in the spherical coordinate system (θ, ϕ) respectively:

{\vec{e}}_{ρ} = \cos ϕ {\vec{e}}_{x} + \sin ϕ {\vec{e}}_{y},

{\vec{e}}_{ϕ} = - \sin ϕ {\vec{e}}_{x} + \cos ϕ {\vec{e}}_{y} .

Obviously, when ϕ₀ = 0 or π/2 in Eq. (1), the incident generalized CV beam turns into the radially or azimuthally polarized beams, respectively.

As illustrated in Fig. 1, the refraction of the objective lens also changes the polarization unit vectors and the corresponding ones are

{\vec{e}}_{r}^{'} = \cos θ (\cos ϕ {\vec{e}}_{x} + \sin ϕ {\vec{e}}_{y}) + \sin θ {\vec{e}}_{z},

{\vec{e}}_{ϕ}^{'} = - \sin ϕ {\vec{e}}_{x} + \cos ϕ {\vec{e}}_{y} .

According to the Richards–Wolf vector diffraction theory, the electric field near focus is given by the diffraction integral over the vector field [19, 20]:

\vec{E} (r, φ, z) = \frac{- i k}{2 π} \int_{0}^{α} d θ \int_{0}^{2 π} {\vec{E}}_{1} (θ, ϕ) e^{i k (\vec{s} \cdot \vec{r})} \sin θ d ϕ,

where k is the wavevector,

\vec{s}

is the propagation direction of the ray through the objective lens with the pupil plane apodization function

A (θ) = \sqrt{\cos θ}

, and the field strength factor

{\vec{E}}_{1} (θ, ϕ)

is given by

{\vec{E}}_{1} (θ, ϕ) = l_{0} circ(\sin θ / \sin α) A (θ)(\cos ϕ_{0} {\vec{e}}_{r}^{'} + \sin ϕ_{0} {\vec{e}}_{ϕ}^{'}) .

For an observational point V in the vicinity of the focus, we have

\vec{s} \cdot \vec{r} = z \cos θ + r \sin θ \cos (ϕ - φ) .

If the focal point $O (0, 0, 0)$ is shifted to another position $O^{'} (x_{0}, y_{0}, z_{0})$ , the corresponding radius vector $\vec{O O^{'}}$ is given as ${\vec{r}}_{0} = (x_{0}, y_{0}, z_{0})$ . Related to new focus $O^{'}$ , the characteristics of observational point $V^{'}$ is expected to be the same as equivalent point $U$ in the x-y-z coordinate system, i.e., $\vec{O^{'} V^{'}} = \vec{O U}$ , which indicates the spatial translation invariance of the focus. Thus, the $\vec{s} \cdot \vec{r}$ item in Eq. (6) can be modified as:

\vec{s} \cdot (\vec{r} - {\vec{r}}_{0}) = (z - z_{0}) \cos θ + r \sin θ \cos (φ - ϕ) - \sin θ (\cos ϕ x_{0} + \sin ϕ y_{0})

Substituting Eq. (9) into Eq. (6), we can find that the incident CV beam is multiplied by an additional phase modulation, which, in fact, is just the predesigned phase modulation leading to the focus shifting in Fig. 1. The corresponding phase modulation function P(θ, ϕ) can be expressed as:

P (θ, ϕ) = e^{- i k (\vec{s} \cdot {\vec{r}}_{0})} = e^{- i k (\sin θ \cos ϕ x_{0} + \sin θ \sin ϕ y_{0} + \cos θ z_{0})} .

Then, we can obtain the 3D electric field distribution in the focal region of a high-NA objective lens in cylindrical coordinates as:

\begin{matrix} E_{r} (r, φ, z) = 2 \int_{0}^{α} l_{0} \sqrt{\cos θ} e^{i k z^{'} \cos θ} \sin θ [\cos θ \cos ϕ_{0} \cos (φ^{'} - φ) J_{1} (k r^{'} \sin θ) \\ - \sin ϕ_{0} \sin (φ^{'} - φ) J_{1} (k r^{'} \sin θ)] d θ, \end{matrix}

\begin{matrix} E_{φ} (r, φ, z) = 2 \int_{0}^{α} l_{0} \sqrt{\cos θ} e^{i k z^{'} \cos θ} \sin θ [\cos θ \cos ϕ_{0} \sin (φ^{'} - φ) J_{1} (k r^{'} \sin θ) \\ + \sin ϕ_{0} \cos (φ^{'} - φ) J_{1} (k r^{'} \sin θ)] d θ, \end{matrix}

E_{z} (r, φ, z) = - 2 i \int_{0}^{α} l_{0} \sqrt{\cos θ} e^{i k z^{'} \cos θ} \cos ϕ_{0} \sin^{2} θ J_{0} (k r^{'} \sin θ) d θ,

Here

r^{'} = \sqrt{{(r \cos φ - x_{0})}^{2} + {(r \sin φ - y_{0})}^{2}},

\cos φ^{'} = (r \cos φ - x_{0}) / r^{'},

\sin φ^{'} = (r \sin φ - y_{0}) / r^{'},

z^{'} = z - z_{0},

and

J_{n} (x)

is the nth order Bessel function of the first kind.

For the simple case of ${\vec{r}}_{0} = 0$ and ϕ₀=0, i.e., the focal spot is located at $(x_{0} = 0, y_{0} = 0, z_{0} = 0)$ , Eqs. (11)-(13) give the 3D focal field distribution of a radially polarized beam [20]. When ${\vec{r}}_{0} \neq 0$ , the focal spot is shifted under the modulated CV beam illumination, which has the identical intensity and polarization distributions with that before phase modulation. A slight difference between them only lies in the coordinate transformation. This particular characteristic provides the possibility of shifting the focal spot as prescribed trajectory. When the phase modulation on incident CV beam is manipulated in real-time, we can realize the three-dimensional continuous shifting of the single focal spot in a stationary coordinate system.

For potential applications in multi-particle trapping and biological molecules manipulation, we introduce a simple approach for the creation of identical multiple focal spots with three-dimensional arbitrary shifting based on phase superposition. Supposing that m identical focal spots near the focal region locate in ${\vec{r}}_{1}$ , ${\vec{r}}_{2}$ ,…, ${\vec{r}}_{m}$ , the required individual phase modulation f_i(θ, ϕ), similar with Eq. (10), can be described as:

\begin{array}{l} f_{i} (θ, ϕ) = e^{- i k (\vec{s} \cdot {\vec{r}}_{i})} \\ = e^{- i k (\sin θ \cos ϕ x_{i} + \sin θ \sin ϕ y_{i} + \cos θ z_{i})}, (i = 1 \dots m) . \end{array}

To realize the same results of multiple phase interaction, we deduce an equivalent amplitude modulation T(θ, ϕ) and phase modulation S(θ, ϕ) as:

T (θ, ϕ) S (θ, ϕ) {\vec{E}}_{0} (θ, ϕ) = \frac{1}{m} {\vec{E}}_{0} (θ, ϕ) \sum_{i = 1}^{m} f_{i} (θ, ϕ),

where

T (θ, ϕ) = \frac{\sqrt{m + 2 \sum_{j, i = 1, j \neq i}^{m} \cos [k (\vec{s} \cdot {\vec{r}}_{j} - \vec{s} \cdot {\vec{r}}_{i})]}}{m},

S (θ, ϕ) = \exp {- i arc \tan \frac{\sum_{j = 1}^{m} \sin [k (\vec{s} \cdot {\vec{r}}_{j})]}{\sum_{j = 1}^{m} \cos [k (\vec{s} \cdot {\vec{r}}_{j})]}} .

As described by the right-hand side of the Eq. (19), the superposition of the individual phase modulation means that the distribution of the focal field is composed of multiple focal spots, and the position of each focus is only dependent on the respective phase modulation function f_i (θ, ϕ). Therefore, the position of each focus can be manipulated independently. Meanwhile, each focus has identical intensity distribution with its value reduced to 1/m² times in comparison with that of single focal spot. Of course, we can increase the intensity of every focus by increasing the power of incident laser beam to satisfy the practical requirement such as multiple particles trapping.

3. Numerical results and discussions

3.1 Multiple focal spots with three-dimensional shifting

In this section, the intensity of electric fields within the focal region is illustrated under the tightly focusing modulated CV beam. The fixed parameters for the numerical calculations are l₀ = 1 and NA = 0.9. The wavelength is set as unit of coordinates in all figures. The radially polarized beam is adopted as the incident CV beam. Based on the above-mentioned equations, Matlab codes are developed to perform the numerical simulations.

To demonstrate the feasibility of both the generation and three-dimensional shifting of multiple focal spots, numerical calculations have been performed for three cases of focal spots number: one focal spot, two focal spots and three focal spots. For the case of one focal spot shifting, the location of focal spot is at ${\vec{r}}_{0}$ = (3λ, 3λ, 4λ) as an example. The required phase modulation S(θ, ϕ) and amplitude modulation T(θ, ϕ) can be obtained by calculating Eqs. (20) and (21) with parameters of m = 1 and ${\vec{r}}_{0}$ = (3λ, 3λ, 4λ), as shown in Figs. 2(a) and 2(b), respectively. Specially, the amplitude modulation function is taken as T(θ, ϕ) = 1 in this case. By addressing the pure phase modulation on the incident CV beam, the one focal spot has been shifted to the new position (3λ, 3λ, 4λ) as displayed in Fig. 2(c). The normalized intensity distribution of electric field at z = 4λ in the focal region is shown in the inset of Fig. 2(c). Moreover, the polarizations of transverse electric field in X-Y plane are shown by the arrows aligned in the radial direction which indicate the focal spot is predominantly radially polarized. Clearly, the shifted focal spot has the identical characteristics with and without additional phase modulation, which is in agreement with the reported ones previously [20]. It should be noted that the pure phase modulation only changes the position of the focal spot, but does not affect the intensity profile and the polarization distribution of the focal spot.

Fig. 2 Schematic of one focal spot shifting with (a) phase modulation, (b) amplitude modulation and (c) the position of the shifted one focal spot. The insert of (c) shows the distributions of both normalized intensity and polarization in the cross-section of the focal plane.

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The locations at ${\vec{r}}_{1}$ = (−3λ, 3λ, 3λ) and ${\vec{r}}_{2}$ = (−3λ, −3λ, 6λ) for two focal spots and ${\vec{r}}_{1}$ = (3λ, 3λ, 2λ), ${\vec{r}}_{2}$ = (−3λ, 3λ, 4λ) and ${\vec{r}}_{3}$ = (0, −3λ, 6λ) for three focal spots are chosen to illustrate the generation of multiple focal spots with shifting. The phase modulation and amplitude modulation, which strongly depend on the number and positions of focal spots described by Eqs. (20) and (21), become more complex and have unique structure distributions as shown in Figs. 3(a) and 3(b) for two focal spots and in Figs. 3(d) and 3(e) for three focal spots. Figures 3(c) and 3(f) show that the two and three focal spots have been exactly shifted to the preconceived positions by the use of corresponding phase and amplitude modulations. For each case, the generated multiple focal spots have the same intensity distribution and polarization distribution. Moreover, intensity evolutions of the multiple focal spots propagating in free space also can be observed in Figs. 3(c) and 3(f). Besides, when the number of generated focal spots is more than one, the amplitude modulation is not equal to 1 which causes energy loss. Numerical calculation indicates that the relative focal energy reduces to 1/m for m focal spots. Hence, the energy loss caused by amplitude modulation is (m-1)/m.

Fig. 3 Schematic of multiple focal spots shifting. (a)-(c) phase modulation, amplitude modulation and positions as well as normalized intensity distribution in the cross-section of the focal plane of two focal spots and (d)-(f) of three focal spots.

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According to the desired shifting trajectory, we can obtain real-time phase and amplitude modulations by continuously changing the radius vector ${\vec{r}}_{0}$ in Eqs. (20) and (21), which determine the dynamic moving of the multiple focus spots. Figure 4 shows one frame of the motion of multiple focal spots along a Pyramid-like trajectory in three-dimensional space. Under continuous phase modulation S(θ, ϕ) and amplitude modulation T(θ, ϕ) at the pupil plane, the shifting of four focal spots are also simultaneously presented in the movie (see Visualization 1). From the movie, we can see that the original one focal spot located at (0, 0, 0) evolves into four focal spots respectively located at (3λ, 3λ, 3λ),(3λ, −3λ, 3λ),(−3λ, −3λ, 3λ) and (−3λ, 3λ, 3λ). At the same time, these four focal spots rotate around one square with the length 3λ of each side. When four focal spots go back to the four corners of one square, they continue to move to another layer at z = 6λ and the corresponding positions are (6λ, −6λ, 6λ), (−6λ, −6λ, 6λ), (−6λ, 6λ, 6λ) and (6λ, 6λ, 6λ) respectively. Finally, the four focal spots rotate around square in the layer at z = 9λ. The demonstration in Fig. 4 indicates that when the number and positions of multiple focus spots are carefully engineered in each layer of the additive manufacturing, laser 3D printing with higher efficiency can be flexibly realized [21].

Fig. 4 One frame of four focal spots shifting along a Pyramid-like trajectory in three-dimensional space. Under continuous phase and amplitude modulations, four focal spots are moved in real time (see Visualization 1).

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In practical application, it is a preferable choice to use liquid crystal spatial light modulators (SLMs) to modulate both the phase and amplitude simultaneously on the incident CV beam. Importantly, phase-only and intensity-only SLMs should be used for phase and amplitude modulations, respectively. For this type of devices, 8-bit (256 level) gray level is typically adopted. The other important parameters include the pixel size (typically a few microns), fill factor, and the SLM display format. To generate more focal spots, finer phase modulation with higher spatial resolution will be required. Thus smaller pixel size will be preferred. The accuracy and range of focal spot shifting are restricted by the quantizing noises caused by the limited resolution and gray level of SLM. In addition, the quantizing noises and the fill factor also induce the generation of high-order diffraction that reduces the energy within expected focal spots. All of these factors clearly need to be considered very carefully in practical system design. In order to realize the shifting of the multiple focal spots in real-time applications, the tuning speed will be very critical and is determined by the frame rate and response time of SLMs, which are typically on the order of 100Hz and few millisecond, respectively. A full study of these parameters and their optimization is beyond the scope of the current work and will be subject for future studies.

3.2 Multiple focal spots with flattop intensity profile

The above-mentioned results indicate that, without affecting the regulation of other optical components, the proposed method can generate identical multiple focal spots with three-dimensional shifting. Besides, the method is also suitable for unique focal field generation system, such as flattop profile [7], optical cage [22] and so on. For instance, we take flattop focus for demonstrating the variety of intensity profile of the multiple focal spots generated by our method.

Similar to the work reported previously [23], the flattop focus can be generated by a tightly focused CV beam under modulation of diffractive optical element (DOE) with three concentric circles regions. The numerical apertures corresponding to the outer edge of the three regions are NA₁, NA₂ and NA and the local transmittances are 1, −1 and 1, respectively. Here, we choose the optimized parameters NA₁ = 0.31, NA₂ = 0.56, NA = 0.9 and ϕ₀ = 38° from [7] and assume that the multiple focal spots are located at (10λ, 0, −10λ), (0, 0, 0) and (−10λ, 0, 10λ), respectively. In this case, the DOE together with phase modulation and amplitude modulation, which are shown in Figs. 5(a) and 5(b) respectively, is used to comprehensively modulate the incident CV beam. The tightly focusing performance is shown in Fig. 5(c), in which the generated three focal spots are located at predetermined positions and have identical flattop profile near the focus, in agreement with that of single focus reported in [7]. Because the number and positions of the multiple focal spots are determined by the phase modulation and amplitude modulation described by Eqs. (20) and (21), various intensity profiles can be achieved when a proper DOE with specific structure and parameters is associated in the tightly focusing system.

Fig. 5 Schematic of three focal spots shifting with the same flattop profile. (a) phase modulation, (b) amplitude modulation and (c) normalized total intensity distributions at focus and through focus. Τhe focal spot is located at (10λ, 0, −10λ), (0, 0, 0) and (−10λ, 0, 10λ), respectively.

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4. Conclusions

In summary, we have demonstrated that multiple focal spots with a prescribed number and positions in three-dimensional space can be easily engineered with the predesigned phase and amplitude modulations. The generated multiple focal spots have identical intensity profile and the same polarization distribution as those of the original incident beam. The generation and three-dimensional shifting of multiple focal spots with unique fields such as flattop profile are also demonstrated. Without optimized procedure and with only single focal lens, this approach is simpler and more flexible than other methods reported previously. We believed that this approach can find potential in many applications, such as 3D laser printing and engraving, moving multiple particles trapping, manipulating, alignment and transportation.

Funding

National Natural Science Foundation of China (NSFC) (No. 61275133, 11474052)

Acknowledgments

We are particularly grateful to undergraduate student Zhao Li (School of Applied Mathmatics, Nanjing University of Finace & Economics, Nanjing, China) for his help with the animation production for the Pyramid-like 3D continuous shifting with four focal spots.

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Creation of identical multiple focal spots with three-dimensional arbitrary shifting

Abstract

1. Introduction

2. Theoretical analysis

3. Numerical results and discussions

3.1 Multiple focal spots with three-dimensional shifting

3.2 Multiple focal spots with flattop intensity profile

4. Conclusions

Funding

Acknowledgments

References and links

Supplementary Material (1)

Cited By

Figures (5)

Equations (21)

Optics Express