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Abstract
The axial focus number (the number of focal spots along the axial direction) and focus intensity of a micro-Fresnel zone plate (FZP) are analyzed from deep ultraviolet to infrared using the Fourier decomposition, the vectorial angular spectrum (VAS) theory, and the three-dimensional finite-difference time-domain (FDTD) method. For a low-numerical aperture (NA) micro-FZP (NA<0.1), there are multiple axial high-order foci, and the intensity of each focus decreases slowly. However, the intensity of each high-order focus decreases rapidly with NA increasing. For a relatively high-NA micro-FZP (NA>0.3), the axial high-order foci are suppressed and there is one single focus. A fast, precise, and cost-efficient additive manufacturing method, i.e. two-photon polymerization, is used to fabricate high-NA phase-type micro-FZPs. The experiment has validated the phenomenon of linear negative focal shift of a high-NA micro-FZP. This property can be particularly applied in precise measurement of micro-displacement, film thickness, micro/nano step height, and wavelength.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The Fresnel Zone Plate (FZP) is a typical binary optical diffraction element and has been widely used in a variety of applications, such as nanolithography [1,2], X-ray nanoscopy [3–5], holographic microscopy [6,7], and microwave antennas [8,9]. As it is well-know, a FZP has many high-order foci besides the main focus along the axial direction. Some researchers believe that the relative intensities of high-order foci to the main focus would be 1/9, 1/25, 1/49, …etc. [2,10], while some other researchers argue that the intensities of the main focus and high-order foci are basically unchanged [11]. In recent years, many results show that one single focus appears for a relatively high-numerical-aperture (NA) FZP [12–14]. So far, for a FZP, there is no uniform rule about the focus number and the intensity of each focus, especially for micro-FZPs.
In this paper, we use the Fourier analysis method to study the number of axial focus and the intensity of each focus behind a micro-FZP (a micro-FZP is a small FZP, generally with a diameter less than 500 µm). The vectorial angular spectrum (VAS) theory is used to calculate the axial intensity distribution of a micro-FZP. The three-dimensional finite-difference time-domain (FDTD) method, a strict numerical calculation method of the electromagnetic field via directly solving Maxwell's equations, is then used to rigorously test the results of the VAS theory. The rule of the number of axial focus and each focus intensity under different NA is summarized. Besides, for a micro-FZP at arbitrary NA, the main focus shifts approximately linearly along the negative axial direction with the wavelength increasing. A high-NA phase-type micro-FZP is fabricated by two-photon polymerization (2PP) method that is a powerful direct fabrication method with high efficiency and precision [15,16], and the linear focal shift phenomenon is verified by experiment. This work finally gives a rule for the axial intensity distribution of a micro-FZP at arbitrary numerical aperture, which expands the common understandings of the well-known axial focusing intensity distribution of a FZP and enriches the practical applications of FZPs.
2. Fourier analysis method
For a binary FZP, the radius coordinates of each annulus can be calculated as [13,14,17]
where f0 is the focal length of the main focus and N is the total annulus number. λ = λ0/η with λ0 being the illumination light wavelength and η the refractive index of the immersion medium. The NA of a FZP is defined by NA = ηsinα. α is the maximum focusing semi-angle, satisfying tanα = rN/ f0.For a binary amplitude-type FZP, the transmission function t(r) is mathematically described as
For a FZP, when f0≫n·λ/4, ${r_n} \approx \sqrt {n\lambda {f_0}}$, the transmission function can be expanded by Fourier series, which is equivalent to the sum of sine functions, as shown in Eq. (3) [2],
where cm = sin(mπ/2)/mπ, and γ = π/λf0. A FZP can be regarded as a circular grating with a rectangular profile, composed of many sinusoidal circular gratings. Each sinusoidal grating produces only one diffraction focus along the axial direction, such that a FZP can generate multi-order diffraction foci. And Eq. (2) can be further written asFor a linearly polarized beam (LPB) normally illuminating a FZP, as shown in Fig. 1, based on the Rayleigh–Sommerfeld diffraction integral, any point $P(x^{\prime},y^{\prime},z)$ in the observation plane (z > 0) is described as [18,19]
Fig. 1. Schematic diagram of the focusing geometry of a FZP based on Rayleigh–Sommerfeld diffraction.
Considering the circular symmetry of the FZP, the amplitude distribution along the axial direction ($r^{\prime} = 0$) can be expressed as [19]
By inserting Eq. (4), Eq. (7) becomes
Then we calculate the first term ${u_0}$ and any other terms ${u_m}$ in Eq. (8).
Equation (9) shows the zero-order diffracted light transmitting directly, which corresponds to a plane wave diffraction of a circular aperture.
Equation (10) shows that when f0/z ± m = 0, um takes its maximum value. It means that the focus is generated at z = ±f0/m (m is odd). At the same time, Eq. (10) illustrates that when the total annulus number N is a constant, the intensity of each focus is only related to the obliquity factor cos θ0. The obliquity factor is closely related to the numerical aperture NA, as shown in Fig. 2, from which we can see that when the NA is low, the obliquity factor changes little, and the intensities of high-order foci are basically unchanged or slightly reduced, while when the NA is high, the obliquity factor changes significantly, causing the high-order foci intensity to decrease rapidly.
Since many approximations are used in the derivation using that Fourier analysis, the axial focusing intensity distribution of a micro-FZP cannot be accurately calculated, especially for high-NA micro-FZPs; only qualitative analysis can be performed. To further verify the results obtained, the VAS theory is used to derive the axial intensity distribution of a micro-FZP, and the FDTD method is adopted to verify the results in the next section further.
3. Axial intensity distribution based on the vectorial angular spectrum theory
The angular spectrum theory is an effective method to calculate electromagnetic wave propagation. Its essence is to use Fourier transform and inverse transform to calculate the light field distribution of any plane behind the diffraction screen in the frequency domain. It is divided into scalar angular spectrum theory and VAS theory. It is essential to use the VAS theory when the size of a diffractive element equals to the subwavelength scale. If the electric field immediately behind the mask or aperture plane of a metasurface is known, the light field in any observation plane away from the surface can be determined by the VAS theory [20,21]. For a sufficiently thin mask (e.g., FZPs or super-oscillatory lens), the electric field behind the mask plane is approximated by the multiplication of the illumination electric field in the mask plane and the scalar transmission function of the mask [22]. According to the VAS theory, the integral representation for the electric field behind a FZP has been derived for a linearly polarized beam (LPB) in [23]. For the circularly and radially polarized beams, the derivations in detail can be found in [23–25]. Recently, a modified VAS formula for propagation of non-paraxial beams is proposed to avoid the singular points for longitudinal angular spectrum component at kz = 0 [26].
For an x-polarized LPB normally illuminating a FZP, components of the electric field E for any point in the observation plane (z > 0) are described as [23]
When calculating the axial intensity distribution (r=0), we can get J0(2πψr) = 1, J1(2πψr) = 0, and ${A_{x,0}}(\psi ) = \int_0^\infty {t(r)g(r){J_0}(2\pi \psi r)2\pi rdr} \textrm{ = }\sum\limits_{n = 1}^N {{t_n}\pi (r_n^2 - r_{n - 1}^2)}$. Thus, the axial intensity distribution can be expressed as
Taking the amplitude-type FZPs as an example, the FZPs with different NA are designed. The typical wavelengths are selected from deep ultraviolet to infrared as shown in Table 1. Considering the limited computer memory, it is difficult to verify the results by the FDTD method if the value of N is too large, thus the FZPs with small values of N are chosen. Different NA are obtained by setting different focal lengths f0.
Table 1. Parameters of FZPs
According to Eq. (12), normalized axial intensity distributions of the FZPs with different NA under different wavelengths are calculated. The results are plotted in Fig. 3–6. Besides, Eq. (12) is rigorously validated using the three-dimensional FDTD method [23,24]. The FDTD method was first proposed by Kane Yee in 1966 [27]. This method describes the evolution process of the electromagnetic field in space with time by directly solving Maxwell's equations and has been widely used in this area. The focusing property of a micro-FZP can be rigorously calculated using the FDTD method [17]. The total-field scattered-field (TFSF) boundary and perfectly matched layer (PML) absorbing boundary condition are applied. For an amplitude-type micro-FZP in the visible spectrum, a 100 nm-thick aluminum film is coated on the glass substrate to absorb the incident light [28]. According to our previous experience of FZPs simulation, the accuracy of simulation results can be ensured when the size of mesh grid is less than λ0/20. The sizes of mesh grid in three dimensions are 30 nm for 633 nm, 8 nm for 193 nm, 20 nm for 532 nm and 50 nm for 1550 nm. To further illustrate the problem, the size of mesh grid is reduced to 15 nm (λ0/42) when λ0 = 633 nm, the parameters of the FZP are N = 8, η = 1 (air), f0 = 20 µm, the comparison results are shown in Fig. 7. The calculated results of the mesh grid size of 15 nm are consistent with the results of 30 nm (λ0/21). And then the comparisons between the FDTD simulation results (red dashed line) and the VAS theory calculation results (solid black line) are shown in Fig. 3(e) and (f), Fig. 4(d), Fig. 5(d) and Fig. 6(d). It can be found that the FDTD results are in good agreement with the calculation results of Eq. (12) based on the VAS theory. For phase-type micro-FZPs, we can obtain similar results. However, the efficiency of a phase-type FZP is higher than that of an amplitude-type FZP by replacing the opaque rings with transparent rings of a groove depth equivalent to a phase change of π [3].
Fig. 3. Normalized axial intensity distribution of the FZPS1 with different NA at 633 nm. The NA of (a)-(f) is 0.041, 0.1, 0.202, 0.307, 0.461 and 0.603, respectively.
Fig. 4. Normalized axial intensity distribution of the FZPS2 with different NA at 193 nm. The NA of (a)-(d) is 0.068, 0.204, 0.306 and 0.631, respectively.
Fig. 5. Normalized axial intensity distribution of the FZPS3 with different NA at 532 nm. The NA of (a)-(d) is 0.068, 0.200, 0.304 and 0.613, respectively.
Fig. 6. Normalized axial intensity distribution of the FZPS4 with different NA at 1550 nm. The NA of (a)-(d) is 0.096, 0.200, 0.302 and 0.613, respectively.
Fig. 7. Comparison of mesh grid sizes. (a) and (b) for 30 nm; (c) and (d) for 15 nm; (a) and (c) are the intensity distributions in the x-z plane; (b) and (d) are the intensity distributions of the focal planes; (e) the on-axis intensities; (f) the focal plane intensities in x and y directions.
Figure 3(a) and (b) show that when NA<0.1, there are multiple high-order foci along the axial direction, and the intensity of each high-order focus decreases slowly. With NA increasing, the intensity of each high-order focus decreases rapidly, as shown in Fig. 3(c) and (d). When NA>0.3, the ratio of the maximum intensity of the high-order foci to the intensity of the main focus is less than 20%, as shown in Fig. 3(e) and (f), which means that for a relatively high-NA micro-FZP, the axial high-order foci are basically suppressed and there is only one main focus along the axial direction. Furthermore, one single focus of a high-NA FZP will improve the light intensity and contrast of FZPs imaging. Similar results can be obtained with other wavelengths from deep ultraviolet to infrared, as shown in Fig. 4–6.
4. Linear focal shift of micro-FZPs
According to the VAS theory and the FDTD method, we calculate the variation of the axial main focus shift of low-NA and high-NA micro-FZPs with wavelength deviation, and the results show that the main focus of a micro-FZP is approximately shifting linearly along negative axial direction with the wavelength increasing. However, only one focus is required in most applications, and there is only one focus with a high-NA micro-FZP based on the above conclusion. Therefore, in this section, a high-NA, single focus phase-type micro-FZP (FZPS5, NA=0.6027) is designed to investigate the wavelength dispersion property. The parameters are λ0 = 633 nm, η = 1 (air), N = 20 and f0 = 25 µm. The width of the outermost annulus is 530 nm. The practical illumination wavelength is λi, and the wavelength difference is Δλ = λi - λ0. Equation (12) is used to calculate the practical focal length Zp under different illumination wavelengths. A fast, precise and cost-efficient additive manufacturing method (Two-Photon Polymerization) is used to fabricate FZPS5. Finally, the axial focusing intensity distributions experiment with FZPS5 is carried out with our self-made device.
4.1 Fabrication and characterization of the phase-type micro-FZP
Nowadays, Electron Beam Lithography (EBL) and Focused Ion Beam etching (FIB) are the common methods for the fabrication of micro-Zone-Plate [29–31]. These methods have many technological processes, such as coating and etching which usually take several hours or more. However, a phase-type micro-FZP can be completed in one step using Two-Photon Polymerization (2PP), and the resolution in fabrication can be better than 160 nm (Photonic Professional GT, Nanoscribe GmbH, Germany). 2PP method is a locally confined polymerization reaction by two-photon absorption and is considered as a promising approach for the processing of the micro-optical elements [32–35]. The fabrication steps of FZPS5 are as follows: Prepare clean quartz glass as the substrate and drip 2-3 drops of IP-I780 photoresist (refractive index 1.519@589 nm) in the center of the glass substrate. Then put the substrate on the sample stage and put it into the processing model. The processing rate is 2 mm/s, the pulse laser power is 70 mW, and the wavelength is 780 nm. The whole process needs only several minutes, and then the sample is immersed in acetone for 10 minutes and isopropanol for 5 minutes to remove the uncured photoresist. For a phase-type micro-FZP, the phase modulation φ is determined by the groove depth h of the dielectric material, which satisfies the formula
where λ0 is the incident wavelength, ${n_{FZP}}$ and ${n_{air}}$ are the refractive index of the FZP dielectric material and working medium, respectively. When the phase change is π, the focusing light intensity is the largest, so the groove depth of FZPS5 is set to be 610 nm [36]. The surface topography of FZPS5 was measured via atomic force microscope (AFM, NT-MDT, Russia) and shown in Fig. 8. The groove depth of FZPS5 outer rings can't be accurately measured, mainly because the width of the outer rings is too small. However, from the groove depth of the central ring, we can see that the groove depth (about 530 nm in Fig. 8) agrees well with the design requirement.Furthermore, the FDTD method is used to explore the effect of groove depth on focusing properties. The phase modulation changes from 0.4π to 1.6π, and the corresponding groove depth is 244 nm, 366 nm, 488 nm, 610 nm, 732 nm, 854 nm and 976 nm. The results are shown in Fig. 9. It can be seen from Fig. 9(a) that when the phase modulation is π, the central focusing intensity is the largest; when the phase modulation deviates from π, the central focusing intensity decreases. Therefore, to ensure high focusing efficiency of micro-FZPs, the phase modulation deviation should be within ± 0.2π (0.8π-1.2π). The results in Fig. 9(b) show that with the increase of groove depth, the ratio of the stray light to the central focusing intensity becomes larger, which means that the stray light is more serious with the increase of groove depth. At the same time, larger phase modulation needs deeper groove depth, which causes difficulties in the processing. In addition, the groove depth does not affect the position of the focus. In summary, the groove depth of 0.8π-π phase modulation is the best choice for phase-type micro-FZPs processing. According to the measurement results in Fig. 8, FZPS5 meets the requirement of the focal shift experiment of high-NA micro-FZPs.
Fig. 9. Phase modulation and focused central intensity varying with the groove depth (a) and the normalized axial intensities with different phase modulation (b).
4.2 Experimental results and discussion
The self-made device is used to measure the focusing intensity distributions at different wavelengths. The schematic diagram of the experimental optical setup is shown in Fig. 10. A supercontinuum laser source (400-2400 nm, Rock 400, LEUKOS, France) and a filter (400-850 nm, Bebop, LEUKOS, France) are used to select the desired visible wavelength. The practical illumination wavelengths are 553 nm, 573 nm, 593 nm, 613 nm, 633 nm, 653 nm, 673 nm, 693 nm, 713 nm, and the spectral bandwidth of each wavelength is 5 nm. The different illumination wavelengths focus at a specific position through the FZPS5, and then the focusing light field is recorded by a CCD camera through a 100× objective (NA=0.95, Nikon, Japan) and a tube lens (focal length of 200 mm). The axial focusing intensity distributions in different illumination wavelengths is obtained by scanning the piezoelectric ceramic transducer (PZT, P-611.ZS, PI, Germany) nano-positioning stage (100 nm/one step).
The experimental results for the practical focal distance Zp and the calculation results of the VAS theory and the FDTD method are plotted in Fig. 11. The solid black line depicts the calculation results of the VAS theory, the blue dashed line and the red dashed line are the FDTD method results and the experiment results, respectively. The red error bars only represent the focal depth of the foci in the experiment. The insets illustrate the focus spots with illumination wavelengths of 553 nm, 593 nm, 633 nm, 673 nm, 713 nm, respectively (from left to right). The detailed data are shown in Table 2. The intensity distributions in the x-z planes of the insets are shown on the right of Fig. 12, and the FDTD simulation results are shown on the left of Fig. 12. The axial intensity distributions and the focal plane intensity distributions in x and y directions of the insets are plotted in Fig. 13. It can be seen that when the practical illumination wavelength deviates from the design wavelength, the focus is approximately shifting linearly along the negative axial direction. And the phenomenon of linear focal shift can be explained qualitatively by the grating equation. According to the Fourier series expansion for a FZP, for normal incidence, the grating equation can be expressed as [37]
Fig. 11. Comparisons of the practical focal length for VAS theory results (solid black line), FDTD results (blue dashed line), and experiment results (red dashed line) with different illumination wavelengths. The red error bars only represent the focal depth of the foci in the experiment. The insets illustrate the focus spots with illumination wavelengths of 553 nm, 593 nm, 633 nm, 673 nm, 713 nm, respectively (from left to right).
Fig. 12. The intensity distributions in the x-z plane of FDTD (left) and experiment (right) at different wavelengths (553 nm, 593 nm, 633 nm, 673 nm, 713 nm). The units of the color bars on the left and right are normalized intensity and intensity (arb. units), respectively.
Fig. 13. The axial intensity distributions and the focal plane intensity distributions in x and y directions of the experiment at different wavelengths (553 nm, 593 nm, 633 nm, 673 nm, 713 nm).
Table 2. Results of the focal distance with wavelength dispersion
The focal spot size is analyzed in dependence on the focal shift, and the results are shown in Fig. 14. The full width at half maximum (FWHM) represents the focal spot size in each direction. With the increase of Δλ (from -80 nm to 40 nm, λi is from 553 nm to 673 nm), the axial focal spot size (FWHMz) is gradually decreasing, the focal transverse spot sizes in x and y directions (FWHMx and FWHMy) are slightly increasing. The changing trend of the experimental results is consistent with the results of the VAS theory and the FDTD method. However, the experiment results have a large deviation from the VAS theory and the FDTD results. This may be caused by the large spectral bandwidth of the laser wavelength in the experiment. Besides, when Δλ changes from 40 nm to 80 nm (λi is from 673 nm to 713 nm), the focal spot sizes in the experimental results have a remarkable decrease. The possible reason is that the power density of the supercontinuum laser source in the experiment is too low in the wavelength range of 673-713 nm. The optical spectrum of the supercontinuum laser source is shown in Fig. 15. The low power density leads to the small focal spot size. And the focal spot sizes will affect the measurement accuracy of FZPs in the practical applications. Therefore, if the linear focal shift of high-NA micro-FZPs is accurately calibrated by the interferometer, an adequate high-NA micro-FZP is designed, and a laser with narrow spectral bandwidth is selected, the linear focal shift phenomenon is suitable for precise measurement of micro-displacements, film thickness, and micro/nano step height by the axial wavelength scanning in confocal microscopy. Meanwhile, it can also be used for precise measurement of laser wavelengths, and further measurement experiments will be analyzed in the next article.
5. Conclusions
The Fourier analysis method is used to analyze the axial focusing intensity distribution of a micro-FZP, which shows that when the total annulus number N is a constant, the intensity of the axial focus is only related to the obliquity factor (cosine of diffraction angle). Rigorous results calculated from deep ultraviolet to infrared by the vectorial angular spectrum (VAS) theory and the finite-difference time-domain (FDTD) method indicate that when NA<0.1, there are multiple pronounced high-order foci along the axial direction, and the intensity of each high-order focus slowly decreases; the intensity of each high-order focus decreases rapidly with NA increasing; when NA>0.3, the ratio of the maximum intensity of the high-order foci to the intensity of the main focus is less than 20%, and there is only one single focus. In addition, a high-NA phase-type FZP is fabricated by the two-photon polymerization (2PP) method, and the linear focal shift property of high-NA FZPs is experimentally validated. The potential applications of this property can be found in chromatic confocal microscopy for the precise measurement of micro-displacement, film thickness, micro/nano step height, and wavelength. At the same time, the advanced theory (the VAS theory and the FDTD method) and the processing method (2PP) in this study are promising for in-depth studies of high-NA micro-FZPs.
Funding
National Key Research and Development Program of China (2017YFB1104700); National Science Fund for Excellent Young Scholars (51722509); Program for Science and Technology Innovation Group of Shaanxi Province (2019TD-011); Key Research and Development Program of Shaanxi Province (2020ZDLGY04-02, 2021ZDLGY12-06).
Disclosures
The authors declare no conflicts of interest.
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