1. Introduction

Nondiffracting Airy and parabolic beams [1,2] demonstrate an invariant shape whose transverse position moves quadratically with distance over substantial ranges. They are most usually generated experimentally [3–6] using Fourier masks with amplitude and phase or with phase-only information. There have been attempts to implant vortices onto these beams with limited success [7,8].

By using radial symmetry, abruptly autofocusing beams have been predicted [9] and experimentally generated [10–12] that form a tightly focused beam. It is expected that these radial beams will have numerous applications because they can deliver high intensity with small spot size at substantial distances from the optical generating system. However the design parameters that define the focus distance and numerical aperture for these beams have not been discussed. In addition, the capability to encode vortices onto these beams would substantially increase the number of applications.

In this work, we present a simple approach for the experimental generation of radially symmetric phase-only masks representing the Fourier transform for these beams and derive equations that easily allow the calculation of the focusing distance and numerical aperture. Next we form vortex patterns at the focus of these beams. We experimentally study the numerical aperture of these beams and obtain subdiffraction limited focus spot sizes. Finally we show how to move the focus point in 3 dimensions by encoding additional optical elements onto the phase pattern.

The capability for using abruptly autofocusing beams to generate high intensity vortex beams will be useful for optical trapping and machining applications. The ability to further adjust both the axial and transverse position of the focused spot will also be extremely useful.

2. Theory and Fourier mask design

The nondiffracting Airy beam is generated using a cubic phase mask of the form ${\varphi }_{AIRY}=\beta x_1^3$ where $x_1$ is a parameter normalized to the width W of the LCD and defined as $\left(-1\le x_1\le 1\right)$ and $\beta$ is an adjustable parameter that controls the strength of the acceleration. Note that this definition is different from Ref. [2]. where ${\varphi }_{AIRY}={\kappa }^3{k}_x^3/3$. In this notation,$k_{xMAX}=(kW)/(2f)$ is the maximum spatial frequency, $k=2\pi /\lambda$ and is the focal length of the Fourier transform lens. While a Gaussian amplitude is usually applied, successful results are achieved with phase-only patterns [6]. Usually the mask is encoded onto a pixelated liquid crystal display (LCD) and the Fourier transform is taken optically to create the Airy beam. The usual optical system [3–6] consists of a lens having a focal length separated from the input and output planes by the distance . In previous work, techniques for shortening the physical length have been demonstrated [6].

The resulting Airy beam travels with a trajectory given [2] by $x_2=[1/(4k^2{\kappa }^3)]z^2$ where $x_2$ is the transverse displacement in the region past the Fourier plane, and is the propagation distance. We can derive a more useful expression where we set $\beta ={\kappa }^3{k}_x^3/3$ and obtain $x_2=[\pi W^3/48\lambda f^3\beta ]z^2=\alpha z^2$, and is the width of the LCD having pixels of size .

We can add a linear phase term having a linear period of $d=n\Delta$ pixels as ${\varphi }_{LIN}=k_x/d=k_x/n\Delta$ to the cubic phase term. Consequently the trajectory of the Airy beam is represented by $x_2=x_{02}-\alpha z^2$ where $x_{02}=f\lambda /n\Delta$ and $\alpha$ is defined above.

In this work, we simply formed a rotationally symmetric version of this mask following the same prescription as for the linear case. We begin with a radial cubic phase term as ${\varphi }_{AIRY}=\beta r_1^3$. Here the radial mask is defined as $0<r_1<1$ and is normalized to half of the size of the LCD. This phase pattern is added to a radial phase term ${\varphi }_{LIN}=2\pi r_1/d=2\pi r_1/n\Delta$ (note that this pattern is actually a diffractive axicon). Consequently the total radial phase is written as $\varphi =\beta r_1^3+2\pi r_1/n\Delta$. Finally, because this is a radially symmetric phase mask, we can add an azimuthal phase that is commonly used for creating vortex patterns as ${\varphi }_{VORTEX}=2\pi \ell \varphi$ where $\ell$ represents the charge of the vortex and can be positive or negative and can have fractional values. Consequently the total phase that is encoded onto the LCD is given as

It has been noted by a reviewer that this phase mask (without the vortex pattern) represents the phase of the argument of the Bessel function that is the Fourier transform of the Airy ring [12]. It is interesting that the phase-only version of this Bessel function mask is so effective.

Figure 1 shows the details of this mask design. Figure 1(a) shows the $512x512$ radial cubic phase term as ${\varphi }_{AIRY}=\beta r_1^3$ formed with a rather small value of $\beta =50$ in order to clearly see the pattern. This pattern is multiplied by a circular aperture to remove the rectangular symmetry for the mask. Figure 1(b) shows a radial axicon phase term ${\varphi }_{LIN}=2\pi r_1/d=2\pi r_1/n\Delta$ – again with a rather large period of $n=20$ pixels. Figure 1(c) shows the sum of these two phase functions. Finally Fig. 1(d) shows the sum of an azimuthal phase having a charge of $\ell =10$ with the sum from Fig. 1(c).

 

Fig. 1 Steps in building the mask for a radially autofocusing vortex beam. (a) cubic phase for β = 50, (b) linear radial phase for period of n = 20 pixels, (c) sum of the two, (d) sum with azimuthal phase with charge of l = 10.

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The Fourier transform of this phase term is formed using the optical system as before. In this case, the radial trajectory of the beam in the region past the Fourier plane can be described by the equation

Here, $\rho$ is measured after the Fourier plane, the parameter $\alpha$ has been previously defined and ${\rho }_0=f\lambda /n\Delta$. We note that the initial radius ${\rho }_0$ increases as the number of pixels in the period of the radial axicon pattern decreases. We also see that the acceleration increases as the parameter $\beta$ decreases.

Using Eq. (2), we can define the distance $z_f$ from the output Fourier plane to the focus point where $\rho =0$ as

Equation (3) shows that the focal length of the Fourier transform lens, the parameter $\beta$, the size $W=N\Delta$ of the LCD and the pixel size drastically affect this focus distance.

We can also define an approximate numerical aperture for the beam at its focus by evaluating $d\rho /dz$ at the point $z=z_f$. This gives an expression for the numerical aperture as

The numerical aperture for these abruptly focusing beams (given in Eq. (4) can be compared with that for a conventional lens. In making this comparison, the focal length $f_1$ of the lens is chosen to be equal to the distance $z_f$ for the abruptly focusing lens. Consequently, the ratio of the numerical apertures is given by

This surprisingly simple result shows that the focus size of the abruptly focusing beam will decrease relative to that of a conventional lens as the focal length of the Fourier transform lens increases while the pixel size of the SLM and the period of the axicon decrease.

3. Experimental setup

In our experiments, linearly polarized light from an Argon laser is spatially filtered, expanded, and collimated. The optical elements are encoded onto a parallel-aligned LCD manufactured by Seiko Epson with 640x480 pixels and pixel spacing of $\Delta =42\mu m$ [13,14] where the phase of each pixel can be controlled over a range of$2\pi$ radians. The focal length of the Fourier transform lens was varied during the course of the experiments. The output images were recorded using a Wincam D camera witha pixel size of $6.7\mu m$. The high resolution images were obtained using a microscope objective mounted in front of the camera.

4. Experimental results - focusing

We first control the focusing dynamics for these beams. Figure 2 shows experimental results at various distances of $z=0,10,20,30,40cm$ from the focal plane of the Fourier transform system where we vary the period of the radial pattern and the strength of the acceleration. In this case, the focal length of the Fourier transform lens was $f=100cm$ and $W=2cm$ corresponding to 480 pixels of the LCD (the maximum radius is 240 pixels). Each image shows a 512x512 array of detector pixels with no magnification. The top row shows results for a radial period of $n=12$ pixels and $\beta =40rad$. A circular pattern is formed at the focal plane of the lens with a diameter basically determined by the value of $n$ and the beam forms a sharp focus at a distance of $z_f=20cm$. The middle row shows results for a radial period of $n=6$ pixels and $\beta =40rad$. As expected, the initial diameter is larger and the beam now forms a sharp focus at a distance of $z_f=30cm$. Finally the lower row shows results for a radial period again of $n=6$ pixels, but with a much larger value of $\beta =95rad$. Now the beam forms a sharp focus at a longer distance of $z_f=40cm$. In each case, the image at the focal plane is not well defined and shows the various rays that are focused at different distances from the focal plane. All of these results are in excellent agreement with theory and show the versatility of the encoding procedure.

 

Fig. 2 Images taken at distances from the focal plane of (columns) $z=0,10,20,30,40cm$ for (top row) $n=12,\beta =40rad$, (middle row) $n=6,\beta =40rad$, (bottom row) $n=6,\beta =95rad$.

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However, the success of this approach depends the capability of the SLM to encode the desired phase pattern and depends on several factors. First, we have a Nyquist limit of $n=2$ pixels/period for the axicon. For lower values of $\beta$ and smaller values for the axicon period, the axicon pattern dominates. On the other hand, higher values of $\beta$ can cause additional Nyquist encoding errors. The Nyquist limit for the cubic phase can be estimated by examining the case where the phase of the $Nth$ pixel differs from the phase of the $N-1$ pixel by π radians. Based on this analysis, we find that the Nyquist limit on the cubic phase is given by ${\varphi }_N=\pi N/3$ where $N$ is the number of pixels for the maximum radius of the pattern. In our case where $N=240$ pixels, the maximum Nyquist phase is ${\varphi }_N=250rad$.

Some of these effects are shown in Fig. 3 where we plot the effective focal distance $z_f$ for values of $n=2.28,5.10,15,20,25,30,35,40$ (from right to left) and for different values of $\beta$ as a function of$\sqrt{1/n}$. The circles and solid line show results for $\beta =200rad$ and the agreement between theory and experiment is excellent. The diamonds and the dashed line show results for $\beta =100rad$. Here the agreement is good until the lower values for the axicon period. Finally the squares and lower dashed line show results for $\beta =50rad$. Nowthe experimental results deviate significantly from theory at lower values for the axicon period. We see these same results using a smaller focal length of $f=50cm$ for the Fourier transform lens. As mentioned above, we attribute these results to the fact that the axicon can dominate the cubic phase term at lower values for the cubic phase and smaller axicon periods. Some of these problems could be alleviated by encoding these cubic phase masks using submicron etching techniques (14).

 

Fig. 3 Effective focal length as a function of $\sqrt{1/n}$ for values of$\beta =50rad$ (squares), $\beta =100rad$ (diamonds) and $\beta =200rad$ (circles).

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In addition, several experimental factors are important. First, the Fourier transform optical system must be correctly aligned where input and output planes are separated from the Fourier lens by the distance . Otherwise the experimental results in Fig. 3 will not agree with theory. For example, if the first distance is longer than , then the slope in Fig. 3 will be higher while, if the first distance is less than , then the slope will be lower. These effects have been experimentally confirmed and similar effects have been reported elsewhere [6].

Secondly, the diameter of the Fourier transform lens should be large enough so that low pass spatial filtering does not occur. This is particularly important when using large values for the Fourier lens focal length and for small values of the axicon parameter $n$ and the pixel size $\Delta$ in Eq. (3).

5. Experimental results – vortex focus points

Next, we examined the image in the plane where $z=z_f$ when we encoded various circular harmonics onto the mask as shown in Fig. 1(d). In this case, we used a radial period of $n=2.28$ pixels, a value of $\beta =195rad$, a Fourier transform lens with a focal length of $f=100cm$ and an aperture of $W=2cm$. These values predicted a focal distance of $z_f=100cm$ in agreement with experiment. Results from left to right in Fig. 4 show circular harmonic values of $\ell =0,10,20,30$. In each case, the image represents an area of 172x172pixels of the detector with no magnification. As expected the diameter of the vortex spot increases as the value of the circular harmonic increases. At distances before this focal plane, the beams look similar to those shown in Fig. 2.

 

Fig. 4 Images taken at $z=z_f=100cm$ for circular harmonic values of $\ell =0,10,20,30$ using $n=2.28,\beta =195rad$.

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6. Numerical aperture

Next we compared the size of the focused spot for the abruptly focusing beam with that from a glass lens having the same focal length. From Eq. (5), we want a large focal length for the Fourier transform lens and a small axicon period. We again adjusted the parameters ($n=2.28,\beta =195rad$) for the abruptly focusing beam so that it focused at a distance of $z=z_f=100cm$from the output Fourier plane. In Fig. 5 , we used a 40x objective lens mounted onto the camera. The images show an area of 384x384 pixels and the magnification is $M_T=8.3x$. First we examined the focused spot size for a glass lens having a focal length of $f_1=100cm$ and a lens aperture size of $D=2cm$ and the results are shown on the left. The focused spot diameter is$D_{Lens}=72\mu m$ in good agreement with the predicted Airy spot size. The right image shows the results for the abruptly focusing beam and the focused spot diameter now is $D_{Airy}=45\mu m$. These data show that we can exceed the diffraction limited spot size using these abruptly focusing beams and agree with the predictions of Eq. (5).

 

Fig. 5 Images of the focused spot for (left) a glass lens having a focal length of 100 cm and (right) for the abruptly focusing Airy beam at a focus distance $z=z_f=100cm$.

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Obvious improvements would be to use a LCD with a smaller pixel size or to use submicron-etching techniques for the cubic phase masks (14). It would be interesting to see how small the focused spot size can be made.

7. Manipulation of the focus position

Finally, the position of the focused image can be adjusted by encoding a lens function onto the cubic phase pattern. Here we again define the focal length of the Fourier transform lens as $f$ and the focal length of the added lens as $F$. Using a ray matrix analysis, the Fourier transform plane (as well as the entire transverse position of the abruptly focusing beam) is moved by a distance z where $z=-f^2/F$. Next we prove this.

The normal ray matrix (15) for a Fourier transform system is given by the following

Note that the $"A"$ matrix element has a value of zero corresponding to the Fourier transform condition. Now, we consider the matrix system corresponding to the case where a lens having a focal length of $F$ is placed in front of the Fourier transform system and followed by a translation matrix.

The Fourier transform condition that $A=0$ is satisfied when $z=-f^2/F$.

Figure 6 shows experimental results where an additional lens is encoded onto the cubic phase pattern. Note that small axial shift distances require large focal lengths encoded onto the cubic phase and this is no problem for the LCD [15]. Here we again use $f=100cm$ for the Fourier lens and encode lenses having focal lengths of $F=\pm 100m,\pm 33.3m,\pm 20m$ onto the patterns on the LCD. The axial movement of the focal plane is in excellent agreement with the theory.

 

Fig. 6 Movement of the axial position of the focused spot by encoding an additional lens function having a focal length of $F$ onto the cubic phase mask.

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Finally, the position of the focused spot can be moved in the lateral direction by changing the center location of the added lens. Figure 7 shows experimental results where the encoded lens is offset by $-5,0,+5$ pixels. Again we use parameters of $n=2.28$, and $\beta =195rad$ with $f=100cm$ and $F=20m$. We show a detector size of 128x128 pixels with no magnification. The lateral position of the focused spot changes by $\pm 220\mu m$.

 

Fig. 7 Movement of the transverse position of the focused spot by moving the center of the additional lens function by$+5pixels,0,-5pixels$. The focused spot moves by $+220\mu m,0,-220\mu m$.

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These results show that we can dynamically control the position of the focused spot in three dimensions.

8. Conclusions

We show explicit equations that show the relationship between a cubic phase mask and the focusing properties of the abruptly focusing Airy beam. We show that vortex patterns can be encoded onto these beams to produce focus spots having angular momentum. We show that the focused spot size can be made smaller than the Airy disc spot size for a conventional lens. Finally, we can adjust the position of the focused spot in 3 dimensions by adding an additional lens function onto the cubic phase pattern. We expect that the best performance can be obtained by a submicron etched cubic phase mask combined with a LCD for control of the vortex charge and for 3-D control of the location of the focused spot.