Inverse method to engineer uniform-intensity focal fields with arbitrary shape

Guan-Lin Zhang; Xu-Zhen Gao; Yue Pan; Meng-Dan Zhao; Dan Wang; Hai-Han Zhang; Yongnan Li; Chenghou Tu; Hui-Tian Wang

doi:10.1364/OE.26.016782

1. Introduction

Focal field shaping becomes increasingly important due to the abundant applications in various realms [1–12], in details, fabrication of surface microstructures [2–4], optical trapping of small particles [5–7], surface plasmon polariton [8], super-resolution imaging [9], femtosecond laser filamentation [10], second-harmonic imaging [11], optical storage [12], and so on. Obviously, the shaping of focal field, including amplitude, phase, and polarization states, is of great sense.

To achieve a desired focal field, we need to design the amplitude, phase, and polarization states of the input optical field. An idea is to design some diffractive optical elements [13–17], or design directly the complicated vector optical fields (VOFs) [1,18–29]. Optimization methods based on cylindrical vector beams is of particular importance [15–20, 29]. Specific distributions of the focal fields, such as the H-like shape [21], optical bubbles [16], and the optical needle [25], have been obtained. Optimized Gerchberg-Saxton algorithm also provides an approximate solution to the desired focal field [14], but it is time-consuming. Several non-iterative methods have also been proposed, but they are limited to the case of low numerical aperture (NA) [28,29]. Therefore, engineering the focal field of lens with a moderate and even high NA fast and precisely is a significant issue.

In this article, we presented an inverse method to engineer the focal field, using the Richards-Wolf diffraction integral [30] and the Fourier transform, which will be introduced in Section 2. In Section 3, we will offer an example to engineer uniform-intensity focal field with arbitrary shape and size. Simulated focal fields and the corresponding input fields will be shown here. In Section 4, we will show the experimental results and the relevant discussions. The major difference of this work from Ref. [14] is reflected in that we used all the three dimensions of amplitude, phase and polarization states to achieve a non-iterative inverse calculation method.

2. Basic principle

As is well known, the focusing process of an optical field by an aplanatic lens has two steps: the refraction of lens and the propagation in free space [31]. After passing through the lens, an input field Eⁱ is transformed into the transmitted field E^t at the exit pupil and then propagates to the focal plane, forming the focal field E^f. There exist two coordinate systems, (xⁱ, yⁱ, zⁱ) and (rⁱ, ϕⁱ, zⁱ) correspond to the input plane, and (x^f, y^f, z^f) and (r^f, ϕ^f, z^f) correspond to the focal plane. The schematic of the focusing process is shown in Fig. 1.

Fig. 1 Schematic of the focusing process.

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In this section, we will show the details of our inverse calculation method in both cylindrical and Cartesian coordinate systems. We will also make some discussions on the difference and the relation between them.

2.1. Inverse calculation method in the cylindrical coordinate system

An input field in the cylindrical coordinate system can be written as $E^{i} = E_{r}^{i} {\hat{e}}_{r}^{i} + E_{ϕ}^{i} {\hat{e}}_{ϕ}^{i} + E_{z}^{i} {\hat{e}}_{z}^{i}$ , in particular, $E_{z}^{i}$ will always be zero due to the requirement of transversality condition. Upon the refraction of the lens, the radial component is deflected by θ, but the azimuthal compoment is unaffected, so the transmitted field E^t has a nonzero component $E_{z}^{t}$ , we can obtain $E_{r}^{t} = E_{r}^{i} cos θ$ , $E_{z}^{t} = E_{r}^{i} sin θ$ , and $E_{ϕ}^{t} = E_{ϕ}^{i}$ [30]. This process can be expressed by a polarization transformation matrix as

E^{t} = [\begin{matrix} E_{r}^{t} \\ E_{ϕ}^{t} \\ E_{z}^{t} \end{matrix}] = \sqrt{cos θ} M_{p} [\begin{matrix} E_{r}^{i} \\ E_{ϕ}^{i} \\ E_{z}^{i} \end{matrix}],

with

M_{p} = [\begin{matrix} cos θ & 0 & - sin θ \\ 0 & 1 & 0 \\ sin θ & 0 & cos θ \end{matrix}],

where

\sqrt{cos θ}

is the apodization function of lens. The polarization transformation matrix M_p is supplemented by some elements to guarantee the energy conservation, which means the determinant of M_p must be one. After that, the focal field E^f is obtained by integrating the transmitted field E^t [31] as follows

\begin{array}{l} E^{f} & = - \frac{i f}{λ_{0}} \iint_{Ω} E^{t} e^{i (k_{z}^{t} z^{f} - k_{x}^{t} x^{f} - k_{y}^{t} y^{f})} d Ω \\ = - \frac{i f}{λ_{0}} \iint E^{t} e^{i (k_{0} cos θ z^{f} - k_{0} cos ϕ^{i} sin θ x^{f} - k_{0} sin ϕ^{i} sin θ y^{f})} sin θ d θ d ϕ^{i}, \end{array}

where f is the focal length of the lens, λ₀ is the wave length, k₀ = 2π/λ₀ is the wave vector. Considering the projection function rⁱ / f = sin θ, Eq. (3) can be further simplified as

\begin{array}{l} E^{f} & = - \frac{i f}{λ_{0}} \iint E^{t} e^{i [k_{0} cos θ z^{f} - k_{0} cos ϕ^{i} (r^{i} / f) x^{f} - k_{0} sin ϕ^{i} (r^{i} / f) y^{f}]} \frac{r^{i}}{f cos θ} d r^{i} d ϕ^{i} \\ = - \frac{i f}{λ_{0}} \iint E^{t} e^{i [k_{0} cos θ z^{f} - k_{0} (r^{i} / f) (cos ϕ^{i} r^{f} cos φ^{f} + sin ϕ^{i} r^{f} sin ϕ^{f})]} \frac{r^{i}}{f cos θ} d r^{i} d ϕ^{i} \\ = - \frac{i f}{λ_{0}} \iint [E^{t} e^{i k_{0} cos θ z^{f}} / (f cos θ)] e^{- i k_{0} r^{i} r^{f} cos (ϕ^{i} - ϕ^{f}) / f} r^{i} d r^{i} d ϕ^{i} . \end{array}

When we study on the electric field E^f in the focal plane, z^f = 0 and f = 1 will always be chosen. So we can obtain

E^{f} (z^{f} = 0) = - \frac{i}{λ_{0}} \iint (E^{t} / cos θ) e^{- i k_{0} r^{i} r^{f} cos (ϕ^{i} - ϕ^{f})} r^{i} d r^{i} d ϕ^{i} .

Obviously, Eq. (5) is the form of two-dimensional (2D) Fourier transform in the polar coordinate system. Now we can integrate the apodization function of the lens and the polarization transformation matrix to implement the direct calculation from the input field to the focal field [31].

E^{f} (z^{f} = 0) = [\begin{matrix} E_{r}^{f} (z^{f} = 0) \\ E_{ϕ}^{f} (z^{f} = 0) \\ E_{z}^{f} (z^{f} = 0) \end{matrix}] = - \frac{i}{λ_{0}} ℱ_{p} {M_{p} [\begin{matrix} E_{r}^{i} \\ E_{ϕ}^{i} \\ E_{z}^{i} \end{matrix}] / \sqrt{cos θ}},

where ℱ_p is the polar Fourier transform [32,33]. With Eq. (5), we can easily obtain the relation from the focal field to the transmitted field using the inverse Fourier transform as follows

E^{t} = - \frac{i λ_{0} cos θ}{k_{0}^{2}} \iint E^{f} (z^{f} = 0) e^{i k_{0} r^{i} r^{f} cos (ϕ^{i} - ϕ^{f})} r^{f} d r^{f} d ϕ^{f},

and the direct calculation from the focal field to the input field as

E^{i} = [\begin{matrix} E_{r}^{i} \\ E_{ϕ}^{i} \\ E_{z}^{i} \end{matrix}] = i λ_{0} \sqrt{cos θ} M_{p}^{- 1} ℱ_{p}^{- 1} {\begin{matrix} E_{r}^{f} (z^{f} = 0) \\ E_{ϕ}^{f} (z^{f} = 0) \\ E_{z}^{f} (z^{f} = 0) \end{matrix}},

with

M_{p}^{- 1} = [\begin{matrix} cos θ & 0 & sin θ \\ 0 & 1 & 0 \\ - sin θ & 0 & cos θ \end{matrix}],

where

ℱ_{p}^{- 1}

is the inverse polar Fourier transform [32, 33]. With Eq. (8), we can find

E_{z}^{i} = \frac{i λ_{0}^{3} \sqrt{cos θ}}{4 π^{2}} [- sin θ ℱ_{p}^{- 1} (E_{r}^{f}) + cos θ ℱ_{p}^{- 1} (E_{z}^{f})]

. As mentioned above, a real input field must have no longitudinal component such that the transversality condition of optical field is guaranteed, so

E_{z}^{i}

will always be zero. Finally, we obtain a significant relation between

E_{r}^{f}

and

E_{z}^{f}

:

ℱ_{p}^{- 1} {E_{z}^{f} (z^{f} = 0)} = tan θ ℱ_{p}^{- 1} {E_{r}^{f} (z^{f} = 0)} .

Equations (8) and (10) are the basic formulae of the inverse calculation method in the cylindrical coordinate system. In this case, we should use the polar Fourier transform.

2.2. Inverse calculation method in the Cartesian coordinate system

An input field in the Cartesian coordinate system can be written as $E^{i} = E_{x}^{i} {\hat{e}}_{x}^{i} + E_{y}^{i} {\hat{e}}_{y}^{i} + E_{z}^{i} {\hat{e}}_{z}^{i}$ , in particular, $E_{z}^{i}$ will always be zero. The first step of the focusing process is the refraction of lens, which has been shown in Eq. (2). By applying E_r = E_x cos ϕ + E_y sin ϕ and E_ϕ = E_y cos ϕ − E_x sin ϕ [30], Eq. (2) can be rewritten as

E^{t} = [\begin{matrix} E_{x}^{t} \\ E_{y}^{t} \\ E_{z}^{t} \end{matrix}] = \sqrt{cos θ} M_{c} [\begin{matrix} E_{x}^{i} \\ E_{y}^{i} \\ E_{z}^{i} \end{matrix}],

with

M_{c} = [\begin{matrix} {cos}^{2} ϕ^{i} cos θ + {sin}^{2} ϕ^{i} & (cos θ - 1) sin ϕ^{i} cos ϕ^{i} & - sin θ cos ϕ^{i} \\ (cos θ - 1) sin ϕ^{i} cos ϕ^{i} & {cos}^{2} ϕ^{i} + {sin}^{2} ϕ^{i} cos θ & - sin θ sin ϕ^{i} \\ sin θ cos ϕ^{i} & sin θ sin ϕ^{i} & cos θ \end{matrix}] .

The focal field E^f is obtained by integrating the transmitted field E^t, the projection function rⁱ / f = sin θ is used repeatedly for simplification, we can obtain

\begin{array}{l} E^{f} & = - \frac{i f}{λ_{0}} \iint_{Ω} E^{t} e^{i (k_{z}^{t} z^{f} - k_{x}^{t} x^{f} - k_{y}^{t} y^{f})} d Ω \\ = - \frac{i f}{λ_{0}} \iint E^{t} e^{i (k_{0} cos θ z^{f} - k_{0} cos ϕ^{i} sin θ x^{f} - k_{0} sin ϕ^{i} sin θ y^{f})} sin θ d θ d ϕ^{i} \\ = - \frac{i f}{λ_{0}} \iint E^{t} e^{i [k_{0} cos θ z^{f} - k_{0} cos ϕ^{i} (r^{i} / f) x^{f} - k_{0} sin ϕ^{i} (r^{i} / f) y^{f}]} \frac{r^{i}}{f cos θ} d r^{i} d ϕ^{i} \\ = - \frac{i f}{λ_{0}} \iint E^{t} e^{i [k_{0} cos θ z^{f} - (k_{0} / f) x^{i} x^{f} - (k_{0} / f) y^{i} y^{f}]} \frac{1}{f cos θ} d x^{i} d y^{i} \\ = - \frac{i f}{λ_{0}} \iint [E^{t} e^{i k_{0} cos θ z^{f}} / (f cos θ)] e^{- i k_{0} (x^{i} x^{f} + y^{i} y^{f}) / f} d x^{i} d y^{i} . \end{array}

Choose the same condition as above : z^f = 0 and f = 1, we can give

E^{f} (z^{f} = 0) = - \frac{i}{λ_{0}} \iint (E^{t} / cos θ) e^{- i k_{0} (x^{i} x^{f} + y^{i} y^{f})} d x^{i} d y^{i} .

Obviously, Eq. (14) is the form of 2D Fourier transform in the Cartesian coordinate system. Now we can derive the direct calculation from the input field to the focal field [31].

E^{f} (z^{f} = 0) = [\begin{matrix} E_{x}^{f} (z^{f} = 0) \\ E_{y}^{f} (z^{f} = 0) \\ E_{z}^{f} (z^{f} = 0) \end{matrix}] = - \frac{i}{λ_{0}} ℱ_{c} {M_{c} [\begin{matrix} E_{x}^{i} \\ E_{y}^{i} \\ E_{z}^{i} \end{matrix}] / \sqrt{cos θ}},

where ℱ_c is the Fourier transform in the Cartesian coordinate system. With Eq. (14), we can also obtain the relation from the focal field to the transmitted field using the inverse Fourier transform as follows

E^{t} = - \frac{i λ_{0} cos θ}{k_{0}^{2}} \iint E^{f} (z^{f} = 0) e^{i k_{0} (x^{i} x^{f} + y^{i} y^{f})} d x^{f} d y^{f},

and the direct calculation from the focal field to the input field as

E^{i} = [\begin{matrix} E_{x}^{i} \\ E_{y}^{i} \\ E_{z}^{i} \end{matrix}] = i λ_{0} \sqrt{cos θ} M_{c}^{- 1} ℱ^{- 1} {\begin{matrix} E_{x}^{f} (z^{f} = 0) \\ E_{y}^{f} (z^{f} = 0) \\ E_{z}^{f} (z^{f} = 0) \end{matrix}},

with

M_{c}^{- 1} = [\begin{matrix} {cos}^{2} ϕ^{i} cos θ + {sin}^{2} ϕ^{i} & (cos θ - 1) sin ϕ^{i} cos ϕ^{i} & sin θ cos ϕ^{i} \\ (cos θ - 1) sin ϕ^{i} cos ϕ^{i} & {cos}^{2} ϕ^{i} + {sin}^{2} ϕ^{i} cos θ & sin θ sin ϕ^{i} \\ - sin θ cos ϕ^{i} & - sin θ sin ϕ^{i} & cos θ \end{matrix}],

where

ℱ_{c}^{- 1}

is the inverse Fourier transform in the Cartesian coordinate system. With Eq. (18), we can find

E_{z}^{i} = \frac{i λ_{0}^{3} \sqrt{cos θ}}{4 π^{2}} [- sin θ cos ϕ^{i} ℱ_{c}^{- 1} (E_{x}^{f}) - sin θ sin ϕ^{i} ℱ_{c}^{- 1} (E_{y}^{f}) + cos θ ℱ_{c}^{- 1} (E_{z}^{f})]

. Since

E_{z}^{i} = 0

and rⁱ = sin θ, we can obtain the relations among

E_{x}^{f}

,

E_{y}^{f}

and

E_{z}^{f}

as follows

ℱ_{c}^{- 1} {E_{z}^{f} (z^{f} = 0)} = tan θ [cos ϕ^{i} ℱ_{c}^{- 1} (E_{x}^{f} (z^{f} = 0)) + sin ϕ^{i} ℱ_{c}^{- 1} (E_{y}^{f} (z^{f} = 0))] .

Equations. (17) and (19) are the basic formulae of the inverse calculation method in the Cartesian coordinate system.

2.3. Procedure of the inverse calculation method

The inverse calculation methods in the two different coordinate systems are essentially the same one, because they describe the same physical process. However, they are different in form, because they correspond to different components of optical field with different orthogonal unit vectors. Equations (8) and (10) are the basic formulae of the inverse calculation method with (ê_r + ê_ϕ + ê_z), corresponding to radial, azimuthal and longitudinal components of optical field. Equations (17) and (19) are the basic formulae of the inverse calculation method with (ê_x, ê_y, ê_z), corresponding to horizontal, vertical and longitudinal components of optical field.

Another difference on those two methods is the Fourier transform process. The operator $ℱ_{p}^{- 1}$ in Eqs. (8) and (10) is the polar Fourier transform, which means that the sampling frequency points are equispaced in the polar coordinates [32]. The operator $ℱ_{c}^{- 1}$ in Eqs. (17) and (19) is the common Fourier transform in the Cartesian coordinate system, which means that the sampling frequency points are equispaced in the x- and y-dimensions. Those two kinds of the Fourier transform are identical in the ideal condition, but they have some difference in actual discrete sampling process [32].

As is well known, under the tightly focusing case, the longitudinal field component is related to the radial component only, while has nothing to the azimuthal component. Only when the radial and longitudinal components of the desired focal field satisfy Eq. (10), the input field may be truely exist, and the focal field obtained is a true electromagnetic field; or only when the horizontal, vertical and longitudinal components of the desired focal field satisfy Eq. (19), the input field may be truely exist.

The two kinds of the inverse calculation methods are both important and useful, beucase any one has its own advantages. For the first one, the form is the most concise and explicit, and we can obtain a direct computing relation between $E_{r}^{f}$ and $E_{z}^{f}$ , thus $E_{ϕ}^{f}$ is an independent variable, but programing a fast and accurate algorithm in the polar Fourier transform is an unsolved problem [32,33]. For the second one, the form is more complicated, but it is widespreadly used, because we can use the fast Fourier transform (FFT) method.

The inverse calculation method for the focal field can be divided into the following procedure:

Prescribe the distributions of amplitude, phase and polarization states of the desired focal field, and the components must satisfy Eq. (10) [Eq. (19)].
Use Eq. (8) [Eq. (17)] to calculate the input field.
Generate the input field experimentally, which is actually vector optical field (VOF).
Produce the focal field through the objective illuminated by the input field.

3. Engineering the uniform-intensity focal fields with arbitrary shape

With Eqs. (8) and (10) [Eqs. (17) and (19)], we can obtain the desired focal fields with structured amplitude, phase and polarization states. In many applications, however, the total intensity is more important, so we give several examples of uniform-intensity focal fields. Our method is based on the Richard-Wolf vector diffraction integral, which is an extension of scalar diffraction theory, and it contains both the low and high NA cases. Considering the universality of the theory, we choose parameter NA = 0.9 in the following discussion in this section.

The longitudinal component is the most uncertain factor especially in the case of high NA, so we make a pre-procedure based on Eq. (10) before performing the inverse calculation. Here we choose the simplest way, the corresponding pre-procedure can be divided into the following steps:

Select $E_{z}^{f}$ (z^f = 0) = A, where A = A(x, y) is the function of desired uniform shape.
Calculate the corresponding $E_{r}^{f}$ (z^f = 0) by Eq. (10).
Set $E_{ϕ}^{f}$ (z^f = 0) that acts as a supplement, making the total intensity uniform.

In this case, the polar Fourier transform should be used in Step 2, but we find that we can also obtain a kind of uniform-intensity focal field when using the FFT.

The sampling points in the FFT process is set to be 1080 × 1080, corresponding to the 1920 × 1080 resolution of the spatial light modulator (SLM) used in our experiments. Those sampling points are padded with zeros to 2¹⁵ × 2¹⁵ (see Section 3.4 and Ref. [31]), for the accuracy of FFT algorithm.

3.1. Simulations under the ideal condition

The ideal condition means that we can achieve an ideal modulation of the amplitude, phase and polarization states of light, that is to say, we can generate arbitrary perfect VOF. As the simulated example shown in Fig. 2, the focal field is a uniform-intensity rhombus pattern. For this case, we have A(x, y) = rect(x + y)rect(x − y), where rect(x/a) is the so-called rectangular function as $rect (x / a) = {\begin{array}{l} 1, & | x | \leq a / 2 \\ 0, & | x | > a / 2 \end{array}$ . Radial, azimuthal and longitudinal components are designed to lead to a uniform total intensity with NA = 0.9. The longitudinal component dominately contributes in the center of rhombus pattern, which is exactly located at the polarization singularity of the transverse components. There is a spiral distribution of polarization states in the focal field, on account of the equal proportions of $I_{r}^{f}$ and $I_{ϕ}^{f}$ . The distribution of polarization states can be changed by adjusting the relative ratio of $E_{r}^{f}$ and $E_{ϕ}^{f}$ in the pre-procedure, what is more, can be more complicated so long as Eq. (10) [Eq. (19)] is satisfied. The global phase is uniform and we can add the desired phase distribution during the inverse calculation process easily.

Fig. 2 Simulated focal field with a shape of rhombus with NA = 0.9 under the ideal condition. (a) shows the total intensity, and (b), (c) and (d) show the radial, azimuthal and longitudinal components, respectively. Any picture has dimensions of 45λ × 45λ.

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Figure 3 shows the simulated input field corresponding to the focal field in Fig. 2. We can find that the total intensity exhibits a four-fold rotational symmetry in Fig. 3(a), the distribution of polarization states has a two-fold rotational symmetry in Fig. 3(b), and the distribution of phase has also a two-fold rotational symmetry with a change of π in Fig. 3(c), respectively. Those specific distributions lead to a phenomenon of destructive interference, which corresponds to the dark point of the transverse components, as shown in Figs. 2(b) and 2(c).

Fig. 3 Simulated input field corresponding to the focal field in Fig. 2. (a) the total intensity, (b) the distribution of polarization states, (c) the distribution of phase.

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3.2. Simulations under the existing experimental condition

Since the space domain and the frequency domain in the Fourier transformation are relative, we can treat the input field as the equivalent frequency domain. As shown in Fig. 3(a), the input field has quite a lot of low-frequency information, because the focal field is uniform. Under the existing experimental condition, we are hard to achieve such an extreme amplitude modulation, and we will obtain the edge-enhanced images [34–36]. However, we can multiply a random phase hologram to avoid this edge-enhancement effect [36], which means the function A(x, y) is replaced by A(x, y)Rand(x, y) in Step 1 of the pre-procedure, where Rand(x, y) is a random function. Simulated example is shown in Fig. 4 and the corresponding input field is shown in Fig. 5.

Fig. 4 Simulated rhombus-shaped focal field with NA = 0.9 under the existing experimental condition. (a) shows the total intensity, and (b), (c) and (d) show the radial, azimuthal and longitudinal components, respectively. Any picture has dimensions of 45λ × 45λ.

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Fig. 5 Simulated input field corresponding to the focal field in Fig. 4 under the existing experimental condition. (a) the total intensity, (b) the distribution of polarization states, (c) the distribution of phase.

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Compared Fig. 5 with Fig. 3, we can find that the low-frequency information is restrained, implying that the operation is effective. We also find some changes of phase and polarization states, and partial loss of symmetry. Due to the introdution of the random phase, the resultant focal fields will have some difference for each simulation.

The population Pearson correlation coefficient (CC) between the result and the designed one is used to evaluate the shape and uniformity of focal field, simultaneously. This definition is widely used to characterize the distinction of two functions or two images. The formula for the CC definition is

CC = \frac{cov (X, Y)}{σ_{X} σ_{Y}},

where cov is the covariance between X and Y, and σ is the standard deviation, respectively. The closer the CC value is to 1, the better the result will be. Usually, the CC value of 0.5∼1.0 is regarded as the strong correlation. In fact, CC between the focal field in Fig. 4 and the designed one is 0.9814, indicating the feasibility of our pre-procedure.

3.3. Control on the shape of focal fields

The shape of the focal field can be controlled by changing the shape-function A in Step 1 of the pre-procedure. Focal fields with the other shapes are shown in Fig. 6, the corresponding input fields are shown in Fig. 7. The triangle-shaped focal field in Figs. 6(a)–6(d), with $A (x, y) = step (\sqrt{3} x - y + 2) step (- \sqrt{3} x - y - 2) step (y + 1)$ , has CC = 0.9775. The red-corss-shaped focal field in Figs. 6(e)–6(h), with A(x, y) = rect(x)rect(10y) + rect(10x)rect(y) − rect(x)rect(y), has CC = 0.9448. Due to the same pre-procedure, the polarization states of focal fields shown in Fig. 6 is similar to those shown in Fig. 4, and the global phase is also uniform.

Fig. 6 Simulated focal fields in shapes of triangle [(a)–(d)] and red-cross [(e)–(h)] with NA = 0.9. (a) and (e), the total intensity; (b) and (f), the radial component; (c) and (g), the azimuthal component; (d) and (h), the longitudinal component. Any picture has dimensions of 45λ × 45λ.

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Fig. 7 Simulated input fields corresponding to the focal fields in Fig. 6. (a)–(c) correspond to the triangle shape focal field in Figs. 6(a)–6(d), and (d)–(f) correspond to the red-cross shape focal field in Figs. 6(e)–6(h). (a) and (d) show the total intensity, (b) and (e) show the distribution of polarization states, (c) and (f) show the distribution of phase.

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3.4. Control on the size of focal fields

As is well known, zero-padding is widely used in the FFT by adjusting the sampling positions aiming to amplifying the frequency domain, and it can also be used in the focal field calculations for changing the size of focal field image [31]. To investigate the influence of zero-padding before, we give another simulated example in Fig. 8. Total intensity of tightly focused fields with NA = 0.9 is shown, all of them have a rhombus shape but different dimensions. Images in Figs. 8(a)–8(d), with different sizes of zero-padding operations, have different dimensions of 60λ × 60λ, 30λ × 30λ, 15λ × 15λ, and 10λ × 10λ, correspondingly, with CC = 0.9865, 0.9762, 0.9561, and 0.9382, respectively.

Fig. 8 Simulated total intensity of rhombus-shaped focal fields with NA = 0.9. From (a) to (d), the dimensions are 60λ × 60λ, 30λ × 30λ, 15λ × 15λ, and 10λ × 10λ, respectively.

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Figure 9 shows the simulated input fields corresponding to the focal fields in Fig. 8. We can find that, as the size of desired focal field decreases, both amplitude and phase variation will be lesser. More significantly, the extreme value of the amplitude becomes normal, leading to an easier realization in actual experiments. What is more, the CC value has a decreasing tendency as the size of desired focal field decreases, meaning that the uniformity of total intensity becomes inferior. The dominant reason is that in our inverse calculation process, the zero-padding implies the information loss in frequency domain, so we hope it to be as little as possible in experiments. The nonuniform and blurry reason is due to the diffraction limited system in our simulation. When the size becomes comparable to the diffraction limit, the difficulty of achieving a target distribution in focal field can be understood as a manifestation of the uncertainty principle [14].

Fig. 9 Simulated input fields corresponding to the focal fields in Fig. 8. (a1)–(a3) correspond to the focal field in Fig. 8(a), (b1)–(b3) correspond to the focal field in Fig. 8(b), (c1)–(c3) correspond to the focal field in Fig. 8(c), and (d1)–(d3) correspond to the focal field in Fig. 8(d). (a1)–(d1) show total intensity, (a2)–(d2) show the distribution of polarization states, and (a3)–(d3) show the distribution of phase.

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3.5. Discussions on the parameter NA

According to the relation rⁱ / f = sin θ and $NA = r_{\max}^{i} / f$ , after normalizing rⁱ with $r_{\max}^{i} = 1$ , we can get sin θ = rⁱ NA. So we have $cos θ = \sqrt{1 - {(r^{i} NA)}^{2}}$ , and Eqs. (10) and (19) can be expressed by NA as

ℱ_{p}^{- 1} [E_{z}^{f} (z^{f} = 0)] = \frac{r^{i}}{\sqrt{1 - {(r^{i} N A)}^{2}}} ℱ_{p}^{- 1} [E_{r}^{f} (z^{f} = 0)],

ℱ_{c}^{- 1} [E_{z}^{f} (z^{f} = 0)] = \frac{r^{i}}{\sqrt{1 - {(r^{i} N A)}^{2}}} {cos ϕ^{i} ℱ_{c}^{- 1} [E_{x}^{f} (z^{f} = 0)] + sin ϕ^{i} ℱ_{c}^{- 1} [E_{y}^{f} (z^{f} = 0)]} .

We can adjust the parameter NA in the inverse calculation process by using Eq. (21) or Eq. (22). Figure 10 shows the simulated example with different NA. Images in Figs. 10(a)–10(d), with different NA and the same size of zero-padding operations to Fig. 4, have different dimensions of 405λ × 405λ, 135λ × 135λ, 81λ × 81λ, and 58λ × 58λ, correspondingly, with CC = 0.9847, 0.9849, 0.9854, and 0.9856, respectively.

Fig. 10 Simulated total intensity of rhombus-shaped focal fields with different NA. All of them have the same size of zero-padding operations to Figs. 4(a)–4(d) correspond to NA = 0.1, 0.3, 0.5, 0.7, with the dimensions of 405λ × 405λ, 135λ × 135λ, 81λ × 81λ, and 58λ × 58λ, respectively.

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The longitudinal component can be easliy weakened by phase modulation (reduce the relative NA) and polarization setting (reduce the radial component), but difficulty enhanced due to the limit of NA. Since the specific pre-procedure, there will exist a transverse polarization singularity in the focal field, and the longitudinal component may be not enough to supplement, resulting in a center dark as shown in Fig. 10(a). However, in the case of NA < 0.3, the longitudinal component can be neglected, so we can change or discard the pre-procedure to obtain another ordinary polarization distribution and a better result.

4. Experimental analysis

In this section, we will perform experiments for demonstrating the capability of our approach for engineering the focal field with the uniform intensity for a given shape. Firstly, we will show the experimental arrangement for generating the calculated input fields. Then we will show the measured results of the focal fields, which verify the capability of our method to achieve such a focal field with a moderate NA. Finally, we will analyze the reason for the imperfect experimental results, we will also give some advice and guesses on the experiments with the high-NA.

4.1. Experimental arrangement for generating the input VOFs

We can generate the input optical field after figuring out the components we need. There are many ways to modulate amplitude and phase simultaneously, in this work we use the macro-pixel encoding method by a SLM [37]. The experimental arrangement is the typical generator of VOFs based on the 4f system [37], as shown in Fig. 11. After the complex amplitude modulation by the SLM, the polarization conversion by the λ/4 plates, and the interferometric superposition by the Ronchi grating, the input optical field we desired can be generated. We use objectives with different NA to focus and a CCD camera to detect. The CCD camera has a size of 1280 × 1024 pixels with a pixel pitch of 4.6 μm. The SLM has a size of 1920 × 1080 pixels with a pixel pitch of 8.0 μm. The light source used in experiment is a laser at a wavelength of 532 nm. The original input beam in experiment is a top-hat beam.

Fig. 11 Schematic of experimental setup. SLM—spatial light modulator, L1 and L2—a pair of lenses, λ/4—1/4 wave plates, SF—spatial filter, G—Ronchi phase grating, O—objective, CCD—charge-coupled device.

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In experiment, when using the grating encoded with hologram directly, we try to generate the input field shown in Fig. 3, but we obtain only a hollow shape, which is called the edge-enhancement effect [36]. As mentioned in Section 3.2, therefore, the introduction of random phase hologram is not only necessary but also effective.

4.2. Experimental results

Figure 12 shows the measured total intensity of focal fields. We generate the input fields and detect their focal fields with different NA. From the first to third column, images have different dimensions of 1500 × 1500 μm², 250 × 250 μm², and 150 × 150 μm², correspondingly, with NA = 0.1, 0.4, and 0.65, respectively. Images in Figs. 12(a)–12(c) show the rhombus-shaped focal fields, with CC = 0.9536, 0.9495, and 0.8863, respectively. Images in Figs. 12(d)–12(f) show the triangle-shaped focal fields, with CC = 0.9519, 0.9493, and 0.8367, respectively. Images in Figs. 12(g)–12(i) show the red-cross-shaped focal fields, with CC = 0.9197, 0.9383, and 0.8270, respectively. Measured intensity patterns of components are also in agreement with the simulated results, meaning the correctness of polarization.

Fig. 12 Measured total intensity of focused fields. Images in (a)–(c) show the focal fields in shape of rhombus. Images in (d)–(f) show the focal fields in shape of triangle. Images in (g)–(i) show the focal fields in shape of red-cross. Images in the first column have dimensions of 1500 × 1500 μm², with NA = 0.1. Images in the second column have dimensions of 250 × 250 μm², with NA = 0.4. Images in the third column have dimensions of 150 × 150 μm², with NA = 0.65.

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4.3. Discussions and suggestions for experiments

Experimental results in Fig. 12 suggest that we can well engineer the shape of focal field with a moderate NA, which has a uniform total intensity. Focal fields with NA = 0.9 fails to be detected, because of the tiny working distance of objectives (∼0.2 mm) and the thick protective glass of CCD (∼1 mm).

We cannot get the arbitrary large focal field, unless using another experimental arrangement. In the actual experiment, the ±1-orders in the plane of spatial filter (Fig. 11) become the rhombus shape. Here we need two pinholes for filtering, and the two rhombus-shaped spots cannot overlap. Considering the period of the Ronchi grating, the interval between the ±1-orders in the plane of SF is ∼6 mm, so the interval between those two rhombuses is ∼4.2 mm. Hence the maximal size of the rhombus in the SF plane is ∼2.9 mm × 2.9 mm for the lens L1 with NA = 0.035. Thus we can achieve the maximal size of the focal field to be ∼100/NA μm. What is more, the period of the Ronchi grating and the NA of lens L1 can be changed to obtain a larger size.

In the detecting process by the CCD, the focal field is in fact composed of many speckles, due to the use of random phase hologram (see Section 3.2). When increasing the power of laser, a desired shape can be seen in the focal plane, and the total intensity is uniform. The center singularity is mingled with light specks, becoming undistinguished.

The imperfect experimental results in Fig. 12 are due to the reasons as follows: (i) To avoid the edge-enhancement effect [36], a random function is multiplied on the initial desired components of the focal field, which may influence the uniformity of the focal field in the actual experiment. (ii) We use a phase-only SLM to change the amplitude and phase of the light simultaneously, the coding method contains of one macro-pixel and four sub-pixels [37], and the size of macro-pixel is 16 μm × 16 μm. In this work, the polarization states of the input VOF are quite complicated, such a big size of pixel may influence the clarity of boundary in the focal field. (iii) The pixel size of the CCD camera (4.6 μm) is smaller than the macro-pixel of the SLM (16 μm). This may magnify the defects of focal field to get a bad detection. (iv) The working distance of the objectives with NA = 0.65 is 0.6 mm, which approximates to the thickness of the protective glass of the CCD, so images taken by the CCD [see Figs. 11(c), 11(f) and 11(i)] may not the precise image plane of the focal field. (v) Replacement of the Fourier integrals by discreet transformations is known to difficultly approximate these integrals [38], so the FFT process may be inaccurate. An optimized interpolation operation may improve the accuracy to get a better experimental result.

For the experiments with the higher NA, to our knowledge, the difficulties mainly come from two points: the short working distance of the objectives and the longitudinal components. To solve the problem caused by the former, an imaging system (relay optical system) may be used to magnify the target images, a complicated equivalent lens system with long working distance may also be useful. As to the latter, it is indeed difficult to measure the intensity profiles for a high-NA lens, because the objectives cannot map the longitudinal electric fields, and Novotny et al. made an attempt on that [39]. For our case, however, the longitudinal field is not so strong meanwhile it occupies a small region as shown in Fig. 4(d), which provides a convenience in the actual experiment. What is more, if the longitudinal component in experiment is strong, we can adjust the pre-setted NA of the backward calculation process bigger than the observed NA, which may be effective.

5. Conclusions

We presented a non-iterative inverse method to engineer a focal field with uniform intensity. The shape and size can be precisely controlled, by adjusting the phase and polarization states in the pre-procedure and the inverse calculation process. Our theory and simulations are verified by the experiments with the moderate NA, in fact, which requires VOFs. Experiments with the higher NA are also discussed. We have demonstrated the feasibility and validity of our method at least under the moderate NA, which should have wide applications in optical storage, optical micromachining, optical coherent tomograghy, optical trapping and so on.

The simulations in Section 3 correspond to an input top-hat beam in actual experiment. For the input cases of Gaussian, Laguerre-Gaussian or Bessel-Gaussian beams, the amplitude modulation needs to be changed, by dividing the preceding amplitude by the function of a given beam. The phase adjustment is similar.

Although our method has been demonstrated to be successful for engineering the uniform-intensity focal fields, it should also have important reference for achieving the nonuniform-intensity focal fields. Besides, focusing shaping with aberrations may be a new research direction [26].

Funding

National Key R&D Program of China (2017YFA0303800, 2017YFA0303700); National Natural Science Foundation of China (11534006, 11774183, 11674184); Natural Science Foundation of Tianjin (16JCZDJC31300); 111 Project (B07013).

Acknowledgments

We acknowledge the support by Collaborative Innovation Center of Extreme Optics.

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Inverse method to engineer uniform-intensity focal fields with arbitrary shape

Abstract

1. Introduction

2. Basic principle

2.1. Inverse calculation method in the cylindrical coordinate system

2.2. Inverse calculation method in the Cartesian coordinate system

2.3. Procedure of the inverse calculation method

3. Engineering the uniform-intensity focal fields with arbitrary shape

3.1. Simulations under the ideal condition

3.2. Simulations under the existing experimental condition

3.3. Control on the shape of focal fields

3.4. Control on the size of focal fields

3.5. Discussions on the parameter NA

4. Experimental analysis

4.1. Experimental arrangement for generating the input VOFs

4.2. Experimental results

4.3. Discussions and suggestions for experiments

5. Conclusions

Funding

Acknowledgments

References and links

Cited By

Figures (12)

Equations (22)

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