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Abstract
Using 3D-printed n-faced pyramid lenses, we generate a series of structured diffraction free terahertz (THz) beams. Based on angular spectrum theory, analytical solutions of the output THz beams from these lenses can be obtained. Furthermore, we experimentally realize these non-diffraction THz beams, showing that the measured results are consistent with theory. It is believed that our structured non-diffraction THz beams can be used in THz imaging with large depth of focus.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In recent decades, non-diffraction beams [1–8], each with transverse intensity distribution unchanged after a long-distance propagation, have already been employed in diverse papers, e.g., Bessel beams [2,3], Airy beams [4,5], and Mathieu beams [6–8]. Owing to the intriguing properties, they are widely applied in microscopy [9–11], particle manipulation [12], light sectioning [13], optical trapping [14], imaging [15], microwave communication [16,17] and laser machining [18,19]. Similarly, non-diffraction beams with distinctive features have attracted a great deal of interest in the THz domain. In 2009, Shaukat et al. used a polytetrafluoroethylene axicon and the continuous wave (CW) THz source to achieve the generation of the Bessel beam [20]. In 2015, Monnai et al. proposed a periodically corrugated metal surface readily integrated with solid-state THz sources to brought out the THz Bessel beam [21]. In 2016, He et al. utilized a meta-hologram composed of gold C-shaped slot antennas to achieve the generation of abruptly autofocusing ring-Airy beam in the THz domain [22]. In 2016, Liu et al. designed 3D-printed diffractive elements, which imprinted the desired phase pattern and performed the required Fourier transform, induced accelerating THz Airy beam [23]. With the generation of THz non-diffraction beams, they are gradually applied in improving the performances of current THz systems. In 2012, a broad THz quasi Bessel beam was demonstrated to enhance the imaging depth of a THz raster-scan imaging system [24]. In our previous work, several popular non-diffraction beams were selected in remote imaging and THz CT systems due to their self-healing and non-diffractive features. Particularly, THz Bessel beams were more favored in imaging applications because of their easy production [25,26].
There are numerous methods reported in the literature for generating the THz Bessel beams. Since the first realization of a THz Bessel beam, axicon has mostly been the first choice to generate the THz Bessel beam [27,28]. In our previous work, we employed 3D printed axicon to generate arbitrary order Bessel beam, which is proved to be more convenient and efficient compared with the mechanical methods [29]. Besides the axicon which is based on pure refraction, metasurfaces provide another way for the generation of Bessel beams, which are formed subwavelength-spaced phase shifters. By controlling the geometry of the micro-structures, the phase profile of metasurfaces is tailored while maintaining a high reflectivity [30,31]. In THz domain, we show great interest to 3D printing technology due to its easily introducing the required phase patterns.
As a special central symmetry optical device, the axicon can be used to generate a THz Bessel beam with a concentric structure. When the axicons change into the rotationally symmetric pyramids, what structure does THz Bessel beam evolve into? Yao et al. proposed a method for producing optical structures with multi-beam interference by using a pyramid-like lens [32]. Compared with the principal methods generating the optical lattices, the pyramid-like lens had an edge in simple operation, low cost, ease of integration, good stability, high damage threshold and apply to all kinds of CW and pulse lasers. In addition, the pyramids were expected to be used for the generation of multi-beam optical tweezers, cell sorting, and Lab-on-a-chip technology. In their work, the limitation of the base angle was the least of perfection. In order to neglect the influence of the polarization state on refraction and interference, they only studied that the crossing angle of the beams should be limited to less than 10 degrees. In this paper, we investigate symmetric pyramids in the THz by using the angular spectrum method. Therefore, the angle ${\gamma _2}$ between the pyramid base and one of its side faces can be casually designed via 3D printing technology without considering the influence of the polarization state. In addition, we obtain the complex phase profiles of the n-faceted pyramid. Based on angular spectrum theory, analytical solutions of the output THz beams from these pyramids can be obtained. Our observations are in good with the numerical simulations.
2. Theory and simulation
In our previous work, the light from the incident Gaussian beam refracts toward optical axis after passing through the axicon, and its interference comes into quasi-Bessel beam. Such a beam propagates diffraction-free over a distance ${z_{max}}$, starting from the tip of the axicon. When the waist of the incident beam ${\omega _0}$ is a known value, ${z_{max}}$ can remarkably increase as the decrease of the bottom angle of the axicon $\gamma$ [29]. According to the symmetry of the n-faced pyramid, they similarly generate one-dimensional (${n}$ is an odd.) or two-dimensional (${n}$ is an even.) diffraction free beams along the direction of light propagation. Figure 1(a) sketches the experimental situation to be analyzed for the four-faced pyramid. In a plane composed of the side edges of the pyramid (${OM}$ and ${OM}^{\prime}$) and the projection of them on the bottom surface (see Figure 1(b)), the four-faced pyramid can be regarded as the axicon with a base angle of ${\gamma _1}$. A Gaussian beam passing through the pyramid can generate a certain distance diffraction-free beam on this plane. In a plane consisting of the slant height of the pyramid (${ON}$ and ${ON}^{\prime}$) and the projection of them on the bottom surface (see Figure 1(c)), the four-faced pyramid can be treated as the axicon with a base angle of ${\gamma _2}$. The distance of the diffraction-free beam generated in this plane is the shortest, because ${\gamma _2}$ is mathematically larger than ${\gamma _1}$, its corresponding ${z_{max}}$ is smaller for the same incident spot. As one would expect, ${\gamma _2}$ determines the diffraction-free distance that the n-faced pyramid can generate, which is similar to the Wooden Bucket Theory. However, the shape of the transverse light field generated by the n-faced pyramid is not the same concentric structure. This is the biggest difference between the n-faced pyramid and the axicon.
Fig. 1. (a)Schematic diagram of the four-faced symmetric pyramid. The geometric optics analysis in (b) $OMM^{\prime}$ plane and in (c) $ONN^{\prime}$ plane
To simplify the generation and overcome the previous drawbacks, we treat 3D-printed pyramids as phase plates to imprint phase structure onto the THz beam. Then the electric field distribution of the Gaussian beam passing through the symmetry pyramids can be calculated via the angular spectrum method.
When passing through a material with a thickness of ${h}$, the phase shift of the light can be calculated as $\Delta \phi = {{2\pi ({n_0} - 1)h} / \lambda }$, where ${{n}_{0}}$ is the refractive index of the material, $\lambda$ is the wavelength of the light. Therefore, the phase profiles of symmetry pyramids can be obtained after their height profiles have been derived. N-faced pyramids mentioned in this paper have the same symmetrical structure as the corresponding regular polygon. Suggested by the symmetry of the problem the best choice for analysis are cylindrical coordinates with ${z}$ along the symmetry direction, $\rho \textrm{ = }\sqrt {{{x}^2} + {y^2}}$ normal to it, and $\mathrm{\theta }$ being the polar angle with reference to the ${x}$ axis. In the Fig. 1(a), the central height of the pyramid $OO^{\prime}$ is preset to a certain value, the bottom surface of the pyramid is a regular polygon inscribed inside a circle with a radius of 50.8 $\textrm{mm}$. In mathematics, the central angle corresponding to each side of the polygon is ${{2\pi } / n}$, the apothem length is given by the formula:
Where r is the radius of the circumscribed circle, ${O^{\prime}}$ which is arranged as the origin of the coordinate system is the projection of origin ${O}$ in the $(\rho ,\theta )$ plane. This case is also used in ${PP^{\prime}}$, where ${P}$ is an arbitrary point on one side of the pyramid. The extension cord connecting points ${O^{\prime}}$ and ${P^{\prime}}$ intersects the bottom edge at point ${Q}$. The length of ${O^{\prime}Q}$ can be expressed utilizing the characteristics of the right triangle $\Delta O^{\prime}NQ$:
Obviously, the height profile of the pyramid is a periodic function with a period of $T = {{2\pi } / n}$. If the expression in the period ${T}$ is available, the entire periodic function can be obtained by repeating the operation. The height profile of any point ${P}$ on one side of the n-faced pyramid can be derived using the mathematical similar triangle principle $\Delta PP^{\prime}Q \simeq \Delta OO^{\prime}Q$:
Where ${{h}_{0}}$ is the central height of the pyramid. More remarkable, the independent variable $\mathrm{\rho }$ should be limited to r when the value of $\theta$ ranging from $0$ to $2\pi$. The phase profile of the pyramid can be given by
Using the angular spectrum method, the electric field distribution of Gaussian beam passing through the pyramid is derived as
Figure 2 shows the phase of the n-faced pyramid $(n = 3,4,5,6,40)$ derived from Eq. (4). As a periodic function, the phase can be wrapped into ${2}{\pi }$. The phase of the 40-faced pyramid brings into line with the phase of the axicon calculated from different method.
According to the theory of angular spectrum transmission, the light field distribution on the input or output surface can be regarded as the combination of plane waves of different spatial frequencies. The Fourier transform of $u(x,y)$ can be described as
Where $u(x,y)$ is the expression of $u(\rho ,\theta )$ in cartesian coordinates. ${exp}[ - i2\pi ({f_x}{x + }{f_y}{y})]$ can be regarded as the expression of space plane waves, and each spatial frequency $({f_x},{f_y})$ represents plane waves with different propagation directions. In the simulation, the spectrum function can be rightfully obtained by the Fourier transform of the light field distribution $u(x,y)$. The real part of the spectrum function is the amplitude distribution of the light field in the frequency domain.
Figure 3 illustrates the spatial spectrum distribution of the light field along the direction of light propagation after the Gaussian beam passing through the n-faced pyramid $(n = 3,4,5,6,40)$ and the axicon. The spectral distribution of the n-faced pyramid appears to be oscillating along the ${f_x}$ axis until $n$ increases to a certain value. As the number of edges of the pyramid increases, the main lobe peak of the amplitude distribution decreases, but the side lobe one increases. Finally, the spatial amplitude spectrum of the 40-faced pyramid keeps pace with the simulation result of the axicon. In short, the theory is verified from another perspective, and at the same time, the changes in the spatial spectrum further explain how the n-faced pyramid progressively evolved into the axicon.
3. Experimental setup and results
The schematic diagram of the experimental setup for generating diffraction free beams with pyramids is illustrated in Fig. 4. The transmitter is made up of a 0.1-THz Gunn diode (Spacek Labs Inc., GW-102P) and a horn antenna (Virginia Diode Inc., WR-3.4CH). The former is utilized as the THz source, which generates 0.1-THz continuous wave with an output power of 20 mW. The latter is devoted to emit the THz wave into the free space. The original THz beam is collimated by a high-density polyethylene (HDPE) lens and then is directed onto the pyramid. In our experiment, the pyramid is fixed on a circular base plate with a diameter of 101.6 mm, as shown in Fig. 5. The diameter of the incident spot is 60.8 mm when the pyramid is placed behind the HDPE lens $(f = 100\textrm{ }\textrm{mm})$ with a certain distance (L). Detection is executed by a receiver that consists of a Schottky diode (Virginia Diode Inc., WR-3.4ZBD) and a horn antenna. The signal from the receiver is measured by a lock-in amplifier referenced to the 530 Hz bias modulation frequency. The THz receiver is mounted on a three-dimensional translation stage for detection at various positions.
Fig. 4. Schematic diagram of the experimental setup for generating diffraction free beams with complex structures
The normalized intensity distribution behind the n-faced pyramid $(n = 3,4,5,6,40)$ and the axicon illustrated in Fig. 6. As indicated on the right side of the figure, the detection area in the xy-plane is $100\textrm{ }\textrm{mm} \times 100\textrm{ }\textrm{mm}$ with the pixel number of $101 \times 101$. The detecting area and pixel size of the images in the xz-plane are $100\textrm{ }\textrm{mm} \times 300\textrm{ }\textrm{mm}$ and $101 \times 301$, respectively. As the number of edges of the pyramid increases, the structure of the intensity distribution in the xy-plane becomes more complex after the collimated beam passing through the n-faced pyramid, the maximum diffraction-free distance where the intensity distribution remains unchanged gets longer. On the whole, the center spot of the non-diffracted beam is smaller than the incident Gaussian spot.
Fig. 6. Simulation (left) and Experimental (right) results of the normalized intensity distribution behind n-faced pyramid and axicon
Combining the preceding conditions, we verify the theoretical validity by using MATLAB software. The calculation was performed for the following parameters: $\lambda \textrm{ = 3 mm}$, ${\omega _\textrm{0}}\textrm{ = 30}\textrm{.4 mm}$, ${h_0} = 13.612\textrm{ mm}$, $r\textrm{ = 50}\textrm{.8 mm}$. As shown on the left side of Fig. 6, the top row is the normalized intensity distribution of the n-faced pyramid $(n = 3,4,5,6,40)$ and the axicon in the xz-plane, the bottom row is the normalized intensity distribution of the n-faced pyramid $(n = 3,4,5,6,40)$ and the axicon in the xy-plane with the square area. As the number of edges of the pyramid increases, the structure of the intensity distribution in the xy-plane is numerically and experimentally getting more complicated. As illustrated in the first row, the simulated maximum diffraction-free distance change with the experiment, the law of change validates a fact that the angle ${\gamma _2}$ determines the diffraction-free distance of the n-faced pyramid. It is worth mentioning that the intensity distribution of the 3-faced pyramid can be affected by stray light with strong intensity in the experiment. In essence, the theory is feasible by contrasting the simulation data with the test data.
4. Conclusion
In this work, we both theoretically and experimentally demonstrate the intensity distribution of the n-faced pyramid $(n = 3,4,5,6,40)$. As a phase plate, the phase profile of the n-faced pyramid is calculated and verified. Linear with the phase shift, the height of the n-faced pyramid can be wrapped into ${\lambda / {({n_0} - 1)}}$. In our future work, the n-faced pyramid can be designed into the wrapped diffractive element and fabricated by 3D printing technology. The low cost of these diffractive elements can promote the application THz non-diffraction beam generated by the pyramid. At the same time, the experimental intensity distribution of the pyramid can be optimized when the collimated beam with a smaller wavelength and lower power goes through these diffractive elements. Non-diffraction beam generated by the pyramid will be used in THz imaging techniques.
Funding
Fundamental Research Funds for the Central Universities (2017KFYXJJ029); National Natural Science Foundation of China (11574105, 61905232); National Defense Pre-Research Foundation of China (61422160107).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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