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Abstract
Diffraction-free electromagnetic beam propagates in free space without change in its two-dimensional transverse profile. Elongating diffraction-free length can benefit the practical application of this beam. Here, we demonstrate that a THz diffraction-free beam with meter-scale length can be achieved by using only one optical element. By circumscribing the line-shape of spherical harmonic function on a traditional axicon, such optical element is designed, and then can be fabricated by 3D-printing technique. Simulated, experimental, and theoretical results all show that the diffraction-free length of generated beam is over 1000 mm. Further analysis based on Fourier optics theory indicates that the spatial frequency of this beam has a comb distribution, which plays a key role during the beam generation process. Moreover, such distribution also demonstrates the beam generated by our invented optical element is not the Bessel beam, but a new diffraction-free beam. It is believed that this meter-scale THz diffraction-free beam can be useful in a non-contact and non-destructive THz imaging system for large objects.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In the past three decades, the structured beams, i.e. the electromagnetic beams with custom intensity, polarization and phase, have attracted a lot of research interests, ranging from the microwave, terahertz (THz) wave, infrared band, visible light to the extreme ultraviolet [1–4]. Due to their unique characteristics, some structured beams have already used in microscopy [5–8], optical communication [9–11], imaging [12–15], manipulation and trapping micro-particles [16–19] (for more implication of structured beams one can refer to Ref. [2] and references therein). The so-called diffraction-free beam (DFB) is probably the most famous one in the zoology of structured beams [20–22]. Many kinds of DFBs, e.g. Bessel beam, Mathieu beam, Airy beam and Weber beam, have been successfully realized at various electromagnetic spectral band [23–29]. For example, a monochromatic THz Bessel beam in free space can be easily obtained nowadays via a simple optical element, i.e. the 3D-printed axicon [30]. Using this DFB, researchers can extend the depth of field (DoF) of THz imaging system to approximately one hundred of millimeters [31,32].
As we know, longer diffraction-free length can benefit the practical application of THz DFB, especially the imaging system for large object. There is a simple method for axicon-based approach to elongate the diffraction-free length of THz Bessel beam, i.e. increasing the size of input Gaussian beam. Theoretical calculation shows that a Gaussian beam with a waist radius of $113.05\textrm{ mm}$ impinged on an axicon with base angle of 10° can get a THz Bessel beam with diffraction-free length of $1000\textrm{ mm}$, which is totally unpractical in reality. On the other hand, although decreasing the base angle of axicon can also elongate the diffraction-free length, the radius of central main lobe of beam is simultaneously increased, which will reduce the transverse resolution of THz imaging system. In 2020, Xiang et al proposed a multiple cascaded lens-axicon doublets to generate a meter-scale THz DFB with small radius of central main lobe [33]. Moreover, this approach also solved a common problem in many generation methods, i.e. the diffraction-free region adheres to the generating element or device. In details, there is a certain distance between their THz DFB and the last axicon. Subsequently, they realized a THz ptychographic system based on this DFB [34].
In 2022, a novel transmitted refractive optical element, named elliptic axicon, has been invented to realize a THz DFB with “focal length”, i.e. there is also a certain distance between the generated DFB and the elliptic axicon [35]. The diffraction-free length of generated beam is about $300\textrm{ mm}$.The design philosophy of such optical element is to circumscribe semi-ellipse on the surface of a traditional axicon. Inspired by this idea, we also design a similar optical element in this paper, of which the surface is circumscribed by the line-shape of spherical harmonics. For simplicity, we temporally name it spherical harmonics axicon (SHA) in the following. Numerical and experimental results both show that the THz DFB generated by such a single optical element has a diffraction-free length over $1000\textrm{ mm}$, and meanwhile it also has so-called “focal length”. More importantly, further analysis based on Fourier optics theory [36] show that such DFB is not a Bessel beam claimed in Ref. [35], but a new DFB since it has a comb spatial-frequency distribution. It is such spatial-frequency distribution leading to its meter-scale diffraction-free length.
2. Design process and simulation
2.1 Design process
The spherical harmonics ${\textrm{Y}_{lm}}(\theta ,\varphi )$ are the solutions to Laplace's equation in the spherical domains [37], and can be written as:
where $\theta$ is the polar angle (which ranges from 0 to $\pi /2$ in our design in the following) and $\varphi$ is the azimuth. The degree $l$ of spherical harmonic is non-negative integer and the order $m$ of ${\textrm{Y}_{lm}}(\theta ,\varphi )$ is integer with $- l \le m \le l$. The $N_{lm}^{}$ is a normalization coefficient varied with respect to $l$ and $m$, and the $\textrm{P}_l^m(cos\theta )$ is an associated Legendre polynomial.Since the spherical harmonic is a complex function, we will use its module square as the line-shape for designing the SHA:
Based on the Eq. (2), the line-shapes of module square of spherical harmonic are also dependent on to $l$ and m. For simplicity, we only consider the situation with $l\textrm{ = }m$ since the module square ${\textrm{W}_{ll}}(\theta )$ (Fig. 1(a)), according to the theory of spherical harmonics [37], can be simply written as:
Fig. 1. (a) Schematic of the module square of a spherical harmonic function; (b) the cross sections of SHA with four l-values; (c) the 3D-models of four designed SHAs.
In Ref. [35], the key designing idea is to circumscribe a function curve on one equal side of an isosceles triangle (i.e., the cross section of a traditional axicon), see Fig. 1(b). The base line and base angle of this triangle are $2R = 101.6\textrm{ mm}$ (4-inch) and $\gamma = {10^ \circ }$ (corresponding to the following experiments), respectively. Correspondingly, the length of equal side is also a fixed value $a = R/\cos \gamma$. Based on such geometrical relationship, the square of normalization constant $N_{ll}^2$ is chosen to be $N_{ll}^2 = a$. Figure 1(a) illustrates a line-shape corresponding to the spherical harmonic with $l = 1$, in which the value of $\textrm{W}_{ll}^{}(\theta )$ is the distance of the point on blue curve to the origin. Then we can circumscribe this line-shape on the isosceles triangles, see Fig. 1(b) for four l-values. By rotating these cross sections ${360^ \circ }$ around the optical axis in Fig. 1(b), the 3D-models of designed SHA can be obtained, as shown in Fig. 1(c).
2.2 Numerical simulation
We use the finite-difference time-domain (FDTD) method (FDTD 2021 R2, Ansys/Lumerical) to simulate the performance of designed SHAs. The 3D-models shown in Fig. 1(c) can be easily imported in this software. The incident beam impinged on the SHA is a 0.1-THz ($\lambda = 3\textrm{ mm}$) collimated Gaussian beam with the full widths at half-maximum (FWHM) of $60\textrm{ mm}$ (corresponding to the following experiments), and the refractive index $n$ is about 1.645. The boundary condition is the perfect matched layer (PML). Figure 2 illustrates the simulation results of the four SHAs shown in Fig. 1(c), respectively. For comparison, we also perform the simulation of a 4-inch-size traditional axicon with base angle of $\gamma = {10^ \circ }$, as also shown in Fig. 2. The position $z = 0\textrm{ mm}$ corresponds to the exit plane of each element.
Fig. 2. Simulation results: (a) the intensity distribution of the generated beams downstream of five elements in $x\textrm{ - }z$ plane, respectively; (b) the intensity distribution with respect to x-coordinate at different z-position in (a1)-(a5), respectively.
Figure 2(a1)-(a5) give the calculated intensity distribution $I(x,z)$ of five generated beams in $x\textrm{ - }z$ plane, in which the white dash lines are the axial evolution of central-peak intensities $I(0,z)$ normalized to the maximum of $I(x,z)$, respectively. Generally, the output beams downstream of the SHAs firstly have two expanding symmetric side lobes close to each element, while the paraxial intensities are relatively weaker, see the black curves in Fig. 2(b1)-(b4). In other words, the generated beams have no main lobe at first. After propagating a certain distance, the beams begin to gradually converge. In other words, the generated beam via an SHA is not adhere to the element, but has a certain distance. Contrarily, the Bessel beam generated by a traditional axicon is near the element, as shown in Fig. 2(a5).
For the output beam behind SHA with $l = 1$, although its main lobe does emerge, the intensities is very weaker than that of the side lobes, see the green curves in Fig. 2(b1). Moreover, the maximum normalized $I(0,z)$ is only about 0.03, as the white dash line shown in Fig. 2(a1). Thus, the ability of SHA with $l = 1$ to converge DFB is very weak. With the increase of l-value, the situation is changed, as shown in Fig. 2(a2) and (b2). For the output beam behind SHA with $l = 10$, the intensities of main lobe behind $z = 300\textrm{ mm}$ are visibly increased, but still weak (the maximum of normalized $I(0,z)$ is about 0.2 at $z = 482\textrm{ mm}$), see the green curve in Fig. 2(b2). After propagating $1000\textrm{ mm}$, these two beams can be considered of disappearing, see the red curves in Fig. 2(b1) and (b2) respectively.
Figure 2(a3)-(a4) illustrate the calculated intensity distribution of the beams generated by SHA with $l = 15$ and $l = 20$ respectively. It can be clearly seen that long distance DFBs have been achieved. For the SHA with $l = 15$ ($l = 20$), the maximum of central-peak normalized intensities $I(x\textrm{ = }0,z)$ is about 0.63 (0.98) at $z = 335\textrm{ mm}$ ($z = 304\textrm{ mm}$) where the intensities of main lobe are obviously stronger than that of two side lobes, see the green curves in Fig. 2(b3) and (b4). Although the main lobe of generated DFB is expanded with propagation, the maximum of central-peak normalized intensity is only attenuated to be about 0.21 (0.26) for the SHA with $l = 15$ ($l = 20$) after propagating $1000\textrm{ mm}$, as red curves in Fig. 2(b3) and (b4). In other words, there is always a main lobe that propagates along the direction in the outgoing beams from SHAs with large l-values. On the contrary, the Bessel beam generated by a traditional axicon cannot propagate such long distance, see Fig. 2(a5) and (b5). Based on these simulation results, we will mainly focus the SHAs with $l = 15$ and $l = 20$ in the following experimental realization.
3. Experimental realization
Thanks to the 3D-printing technique, we can easily fabricate the 4-inch-size SHAs with different l-values (see the inset of Fig. 3 for one example) by importing the designed models shown in Fig. 1(c) into a 3D-printer (Lite450HD, UnionTech, Leyi3D). The refractive index ${n_1}$ and the absorption coefficient $\alpha$ of the polymer 3D-printing material are about 1.645 and $1.5\textrm{ c}{\textrm{m}^{ - 1}}$ at the frequency of 0.1-THz, respectively. The transverse and longitudinal printing resolution of this 3D-printer are $42\,\mathrm{\mu}\textrm{m}$ and $28\,\mathrm{\mu}\textrm{m}$, respectively.
In the experimental setup shown in Fig. 3, the 0.1-THz transmitter (GKa-100, SPACEK LABS) is an InP Gunn diode coupled a horn antenna, delivering an expanding Gaussian beam with the power of $25\textrm{ mW}$ into the free space. Using a high-density polyethylene (HDPE) lens with focal length of $100\textrm{ mm}$, we can get a collimated Gaussian beam impinged on the SHA, of which the FWHM is about $60\textrm{ mm}$, see the inset in Fig. 3. The THz receiver is a broadband high sensitivity Schottky diode (WR-10ZBD, Virginia Diode Inc.) mounted on a three-dimensional translation stage to measure the 0.1THz signals. The strokes of x, y and z-axis are $200\textrm{ mm}$, $200\textrm{ mm}$ and $800\textrm{ mm}$, respectively. Figure 4 illustrates the measured results of the output DFBs downstream of the SHA with $l = 15$ and $l = 20$, respectively.
Fig. 4. Experimental results: the measured intensity distribution of generated beam downstream of SHAs with (a) l = 15 and (b) l = 20, respectively. Number 1 corresponds to the measured intensities in the x-z plane, and Number 2-8 are the intensities in the x-y plane at different z-coordinates, respectively.
Figure 4(a1) and (b1) illustrate the measured intensities in $x\textrm{ - }z$ plane of output beam behind two fabricated SHAs, respectively. It should be noted that the measured data are spliced because the stroke of our translation stage of z-direction is only $800\textrm{ mm}$. Generally speaking, these experimental results are matched with simulation shown in Fig. 2(a3) and (a4). Taking the SHA with $l = 15$ as an example, one can clearly see that the output beam firstly has an annular intensity distribution, see Fig. 4(a2). When it propagates along z-direction to the position $z = 346\textrm{ mm}$, the generated beam gradually converges to form a main lobe localized near the z-axis and few of weak side lobes, e.g., the measured transvers intensity distribution at $z = 450\textrm{ mm}$ in Fig. 4(a3). With the propagation distance increased, the generated beam will slowly spread in free space, as illustrated in Fig. 4(a3)-(a8). The calculation shows that the FWHM of main lobe for $l = 15$ is $18.91\textrm{ mm}$ at $z = 450\textrm{ mm}$, and increased to be $58.60\textrm{ mm}$ at $z = 1450\textrm{ mm}$. For the SHA with $l = 20$, the FWHM of output beam is increased from $19.14\textrm{ mm}$ to $68.60\textrm{ mm}$ after propagating $1000\textrm{ mm}$, see Fig. 4(b3)-(b8). Theoretical calculation shows that a focusing 0.1-THz Gaussian beam with FWHM of $20\textrm{ mm}$ will expand to $160\textrm{ mm}$ beam-width after propagating $1000\textrm{ mm}$. On the other hand, the Bessel beam generated by an axicon, as calculated before, does not propagate such long distance. Thus, the output beam downstream of our invented SHA can be considered as diffraction-free.
4. Results analysis
In Ref. [35], the authors analyzed their designed elliptic axicon based on geometric optics theory, especially giving the reason why there is a distance between the localized DFB and the element. Although such geometric optics analysis can also explain our partial results, it fails to give the physical mechanism behind such long diffraction-free length of the DFB generated by SHA. Next, we will analyze this result by using Fourier optics theory [36].
Based on the above results, our designed SHA is actually a beam converter, i.e. transforming the input collimated Gaussian beam into an output long distance DFB. Generally, the relation between the input and output scalar electric field for a beam converter is:
where $\varDelta \phi$ is the phase-shifting of beam converter. So the key point is to find the $\varDelta \phi$ of designed SHA. Figure 5(a) display the cross section of an SHA. Thus, the optical path between the entrance and exit surface can be written as ${n_0}{d_0}(\xi ) + {n_1}{d_1}(\xi )$ at an arbitrary $\xi$-coordinate, in which ${n_0} = 1$ is the refractive index of air. As shown in Fig. 5(a), the ${d_0}(\xi )$ and ${d_1}(\xi )$ are the thicknesses of the air layer and SHA between the entrance and exit surface of element, respectively. Thus, the phase-shifting is $\Delta \phi (\xi ) = {n_0}k{d_0}(\xi ) + {n_1}k{d_1}(\xi )$, in which $k = 2\pi /\lambda$. Since ${d_1}(\xi )$ is the $\eta$-value of green curve $f(\xi ,\eta ) = 0$ in the first quadrant, the problem is changed to how to rewrite the module square ${W_{ll}}(\theta )$ of spherical harmonics to $f(\xi ,\eta ) = 0$ in the Cartesian coordinate $\xi \textrm{ - }\eta$ system shown in Fig. 5(a).Fig. 5. (a) The geometry of an SHA’s cross section in Cartesian coordinate $\xi \textrm{ - }\eta$ system; (b) the phase-shifting distribution of designed SHAs with l = 15 and l = 20 respectively; (c) the calculated intensity distribution in x-z plane based on the phase-shifting shown in (b).
After some derivation (the detailed process is shown in Supplement 1), the $f(\xi ,\eta ) = 0$ is written as:
Taking the first derivative of Eq. (5) with respect to $\xi$, one can get:
According to the Eq. (7), the phase-shifting $\Delta \phi$ of two designed SHAs shown in Fig. 1(c), i.e., $l = 15$ and $l = 20$, can be calculated by rotating $\Delta \phi (\xi )$ 360° around $\eta$-axis, see Fig. 5(b1) and (b2). By taking the phase distribution into Eq. (4), the E-fields of output beam, i.e.${E_{\textrm{out}}}$ in Eq. (4), downstream of SHA can calculated based on angular spectrum theory [36]. Figure 5(c1) and (c2) illustrate the calculated ${|{{E_{\textrm{out}}}} |^2}$ in $x\textrm{ - }z$ plane of output beam behind these two SHAs, respectively. It can be clearly seen that these theoretical results are matched with that of FDTD simulation and experiment.
Given that the E-field of an electromagnetic beam is known, the spatial frequency distribution (i.e. angular spectrum) of ${E_{\textrm{out}}}$ in Eq. (4) can be calculated by Fourier transform. Figure 6 displays the spatial frequency distribution of DFB generated by SHAs with $l = 15$ (blue curve) and $l = 20$ (red curve). For comparison, the spatial frequency distribution of the beam behind SHA with $l = 1$ (black curve) and the Bessel beam behind a traditional axicon (green curve) [23,31] are also calculated. Note that the areas below all curves in Fig. 6 are equal, which means all of generated beams have same energy.
Fig. 6. Calculated angular spectrums of the output beams downstream of three SHAs and an axicon, respectively.
For the Bessel beam, its angular spectrum is an annular distribution, i.e. two sets of symmetric ${f_x} \ne 0$ components, see the green curve in Fig. 6. Such nonzero spatial frequency component corresponds to the plane wave whose wave vector direction isn’t parallel to the z-direction. In principle, these plane waves will overlap in the free space, leading to the diffraction-free region of Bessel beam [23]. On the other hand, the Bessel beam’s zero-spatial-frequency ${f_x} = 0$ component (corresponding to the plane wave whose wave vector direction is along the z-direction) has no or very low energy. Thus, Bessel beam rapidly spreads and lose the main lobe behind its diffraction-free region, see Fig. 2(a5) and (b5). For the beam behind SHA with $l = 1$, its angular spectrum is a comb distribution, see the black curve in Fig. 6. One can clearly see that the ${f_x} = 0$ component of this beam has certain energies, not like the Bessel beam behind an axicon. Calculation shows that the proportion of total energy for main-lobe is about 27%. While the comb side frequencies, especially the high-frequency ${f_x} \ne 0$ components, have more energies. Thus, the beam generated by SHA with $l = 1$ does always have a main lobe, but with very low intensities, see Fig. 2(a1) and (b1). In other words, the ability of SHA with $l = 1$ to converge DFB is weak, as we mentioned before. For the SHAs with large l-values, the energies of generated DFB’s ${f_x} = 0$ component is significantly increased, see the blue and red curves in Fig. 6. The proportion of total energy for main-lobe corresponding to $l = 15$ ($l = 20$) is about 73% (77%). These results reveal that there is always a plane wave with high intensity in generated DFB that propagates along the z-direction, leading to the meter-scale diffraction-free length. On the contrary, the energies of comb side frequency components are reduced, even some high frequency components disappear, as shown in Fig. 6.
5. Conclusion
To conclude, we have realized a single optical element to obtain THz diffraction-free beam with meter-scale length. The key point for designing such optical element is to circumscribe the line-shape of spherical harmonic function on the surface of a traditional axicon. The 3D-printing technique offers us a powerful tool to fabricate the optical element working at THz band. Based on Fourier optical theory, we remark that its comb spatial frequency distribution plays a key role in constructing our long-distance diffraction-free beam, especially the zero-spatial-frequency component. Since our designing idea is very simple and general, one can get a series of similar optical elements by circumscribing different function line-shapes. In addition, such elements can be fabricated for working at any frequency band if the relevant processing techniques are available.
Funding
Fundamental Research Funds for the Central Universities (2017KFYXJJ029).
Acknowledgments
K. W. conceived of the idea. S. T. performed the numerical simulation. S. T. and J. P. carried out the experimental measurements under the supervision of K. W. K. W. and S. T. did the major part of the theoretical derivation and wrote the paper. All the authors discussed the results and contributed to the writing of the paper.
Disclosures
The authors declare no conflicts of interest.
Data availability
The main data supporting the results in this study are available within the main text and Supplement 1. The raw numerical, experimental, and theoretical data are too large to be publicly shared, yet they are available for research purposes from the corresponding author upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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