CN114236815B - Efficiency calculation method of optical element with wave band structure - Google Patents
Efficiency calculation method of optical element with wave band structure Download PDFInfo
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- 230000003287 optical effect Effects 0.000 title claims abstract description 44
- 238000004364 calculation method Methods 0.000 title claims abstract description 26
- 238000000034 method Methods 0.000 claims abstract description 14
- 230000010363 phase shift Effects 0.000 claims description 19
- 239000000463 material Substances 0.000 claims description 7
- 230000008802 morphological function Effects 0.000 claims description 4
- 230000004907 flux Effects 0.000 claims description 3
- 230000008569 process Effects 0.000 abstract description 7
- 238000013461 design Methods 0.000 abstract description 4
- 238000004458 analytical method Methods 0.000 abstract description 3
- PCHJSUWPFVWCPO-UHFFFAOYSA-N gold Chemical compound [Au] PCHJSUWPFVWCPO-UHFFFAOYSA-N 0.000 description 9
- 239000010931 gold Substances 0.000 description 9
- 229910052737 gold Inorganic materials 0.000 description 9
- 230000000694 effects Effects 0.000 description 4
- 238000002360 preparation method Methods 0.000 description 4
- 238000012876 topography Methods 0.000 description 4
- 230000014509 gene expression Effects 0.000 description 3
- 238000004519 manufacturing process Methods 0.000 description 3
- VYPSYNLAJGMNEJ-UHFFFAOYSA-N Silicium dioxide Chemical compound O=[Si]=O VYPSYNLAJGMNEJ-UHFFFAOYSA-N 0.000 description 2
- 230000008859 change Effects 0.000 description 2
- 238000010586 diagram Methods 0.000 description 2
- 230000006872 improvement Effects 0.000 description 2
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- 230000009286 beneficial effect Effects 0.000 description 1
- 230000007547 defect Effects 0.000 description 1
- 238000011161 development Methods 0.000 description 1
- 238000000609 electron-beam lithography Methods 0.000 description 1
- 238000011156 evaluation Methods 0.000 description 1
- 238000003384 imaging method Methods 0.000 description 1
- 238000005259 measurement Methods 0.000 description 1
- 238000001000 micrograph Methods 0.000 description 1
- 238000005457 optimization Methods 0.000 description 1
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- 229920002120 photoresistant polymer Polymers 0.000 description 1
- 230000008092 positive effect Effects 0.000 description 1
- 238000012545 processing Methods 0.000 description 1
- 239000000377 silicon dioxide Substances 0.000 description 1
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- G—PHYSICS
- G02—OPTICS
- G02B—OPTICAL ELEMENTS, SYSTEMS OR APPARATUS
- G02B27/00—Optical systems or apparatus not provided for by any of the groups G02B1/00 - G02B26/00, G02B30/00
- G02B27/0012—Optical design, e.g. procedures, algorithms, optimisation routines
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Abstract
The invention belongs to the technical field of optical devices, and particularly relates to an efficiency calculation method of an optical element with a wave band structure. The invention promotes Kirz formulas of thin grating approximation, and can be used for calculating the theoretical efficiency of the optical element with the wave band structure of the X-ray and extreme ultraviolet wave band. The method is suitable for any zone morphology which can be expressed by functions, including rectangular zone plate morphology, triangular Kinoform morphology and the like. Meanwhile, the efficiency of a special wave band structure with the wave band morphology function changed along with the period can be calculated, for example, a non-rectangular wave band plate influenced by process factors. The calculation method provided by the invention carries out theoretical analysis and efficiency calculation on the appearance of the optical element, has the advantages of simple calculation, strong pertinence and wide application scene, and has important guiding function on the design and performance analysis of the optical element with the wave band structure.
Description
Technical Field
The invention belongs to the technical field of optical devices, and particularly relates to an efficiency calculation method of an optical element with a wave band structure.
Background
Focusing optics are a critical component in various types of imaging systems. The design and calculation of the optical lens are the first and key steps for designing and developing the optical lens, and have important guiding effects on design optimization, processing preparation and performance evaluation of the lens. In recent years, with the improvement of the resolution of the lens and the diversification of the manufacturing process, the influence of the micro-nano structure of the lens on the focusing efficiency cannot be ignored. Bao Guangshan the approximate formula (THIN GRATING approximation) is a theoretical formula for calculating the efficiency of the phase type rectangular zone plate of X-rays and extreme ultraviolet, which is proposed by Kirz in 1974, and can conveniently estimate the theoretical focusing efficiency of the phase type rectangular zone plate. However, it can only be used for theoretical efficiency calculation of the phase zone plate, and there is still a limit to other types of zone structure lenses, such as Kinoform lenses, at present. The phase zone plate prepared by the actual process is not ideal rectangular shape, the side wall of the phase zone plate can incline due to the influence of the process defect, and the thickness of each period is not completely consistent, so that the actual measurement efficiency can deviate from the theoretical value estimated by Kirz formula greatly. The efficiency calculation of these actually manufactured lenses is a tedious and huge task, and the conventional optical simulation software such as FDTD, coomsol and the like often has the problems of large calculation amount, low precision, long calculation time and the like when simulating the extreme ultraviolet and X-ray lenses.
Therefore, it is necessary to establish a calculation method capable of rapidly and effectively estimating the optical element with the complex wave band structure, and considering the influence of the micro-nano structure of the lens on the focusing efficiency, so as to improve the prediction accuracy of the optical element with the complex wave band structure on the efficiency of the optical element with the actual wave band structure.
Disclosure of Invention
The invention aims to provide an efficiency calculation method of a band structure optical element with a complex morphology, which is short in time consumption and high in precision.
The method for calculating the efficiency of the optical element with the wave band structure with the complex morphology is based on a thin grating approximation formula and comprises the following specific steps:
(1) Setting an incident light field and optical element parameters; wherein:
the incident light field parameters include wavelength λ, amplitude C;
The optical element parameters include: the number of wave bands N of the optical element, the refractive index n= (1-delta) -iβ, 1-delta and β of the material used for the optical element represent the real part and the imaginary part of the refractive index of the material respectively; the morphology function of the i-th band is t i (θ), which is the optical path length difference to the focal spot over a period, describing the variation of the height t with the optical path difference θ. The optical element parameters are used to calculate the phase shift and amplitude in the lens. i=1, 2, …, N;
(2) Calculate the phase shift function Φ i (θ) of the i-th band:
From the morphology function t i (θ) of the i-th band, the phase shift function of the corresponding i-th band can be expressed as:
(3) Knowing the phase shift function, the amplitude a i of the i-th band is found according to the thin grating approximation formula:
(4) Solving the efficiency of the ith band:
(5) Repeating the calculation of the efficiency according to different morphological functions of 1-N wave bands, wherein the initial luminous flux accepted by each wave band is the same, and the total efficiency of the final optical element is the average value of the efficiencies of all wave bands:
j in the formula (2) represents an imaginary unit.
In the present invention, the optical element has a band structure.
In the invention, the wave band morphology of each period of the optical element is a function of the optical path difference and can be described by different functions respectively; the zone profiles include, but are not limited to, zone plates, kinoform lenses, step lenses, fresnel lenses, and their non-ideal profiles.
In the invention, the phase shift function can be calculated according to the morphology of each period.
In the present invention, the amplitude of each period is the integral of the phase shift function over that period.
In the present invention, the total efficiency of the optical element is an average value of all band efficiencies.
In the invention, the morphology function is t i (theta), which can be an analytical expression or a discrete numerical value, and is used for describing morphology features and finally substituted into an efficiency formula to evaluate. The more complex the corresponding function of the topography, the more complex it is, which requires a decision as to how to model depending on the goal. In example 1, a relatively complex topography is described directly by discrete numerical points as a function. The features of example 2 are relatively simple and the expressions can be parsed, so they are described in terms of parsed piecewise functions.
Compared with the prior art, the method has the beneficial effects that:
firstly, the invention promotes Kirz formulas of thin grating approximation, can be used for calculating the theoretical efficiency of an optical element with a wave band structure, and is applicable to any wave band morphology which can be expressed by functions, including rectangular wave band plate morphology and triangular Kinoform morphology;
Second, the present invention considers the morphology and materials of the actual lens, and can calculate the efficiency of the special band structure with the band morphology function changed along with the period, such as a non-rectangular zone plate affected by the process factors. The design freedom degree of the lens is greatly improved, the lens can be used for designing and developing novel lenses, and the method has important theoretical guiding significance.
Thirdly, the calculation method provided by the invention carries out theoretical analysis and efficiency calculation on the appearance of the optical element, and has the advantages of simple calculation, strong pertinence and wide application scene.
Drawings
FIG. 1 is a scanning electron microscope image of a gold Kinoform lens actually prepared in example 1.
Fig. 2 is a graph showing the morphology function of all the wavebands of the gold Kinoform lens actually fabricated in example 1.
FIG. 3 is a graph showing the focusing efficiency of the actually prepared gold Kinoform lens at an energy of 5 to 15keV in example 1 as a function of energy.
Fig. 4 is a schematic diagram of a phase function of a band structure lens with a peak profile in example 2.
Fig. 5 is a graph showing the calculated efficiency of the band structure lens of the peak profile of example 2 as a function of the structural parameters m and n at 500 eV.
Fig. 6 is a schematic diagram of phase function calculation of an optical element with a band structure having an arbitrary morphology according to the present invention.
Detailed Description
The invention is further described below with reference to the drawings and examples, but the invention is not limited to these examples. All simple changes of the calculation parameters in the embodiment are within the protection scope of the invention.
Example 1: the focusing efficiency (resolution 100nm, diameter 100 μm, band number N125) of the actually prepared gold Kinoform lens at 5-15 keV energy was calculated:
Since the electron beam lithography is affected by the proximity effect, the Kinoform triangle shape can shift to different degrees along with the period increase, and especially the shape near the periphery can become a vertical shape due to the smaller scale. Such deviations may result in each band contributing differently to efficiency. The theoretical efficiency calculation formula provided by the invention is used, the influence of morphology and period on efficiency is considered, and the specific steps are as follows:
(1) The incident light field energy was set to 10keV and was normally incident as a plane wave of unit amplitude (c=1). Setting optical parameters of the lens: according to the actually prepared gold Kinoform lens (shown in fig. 1), the morphological characteristics are extracted, a morphological function t (theta) is established, and the morphological characteristics are described by actually collected data points for convenience, and the refractive index is set as gold as shown in fig. 2.
(2) Calculate the phase shift function Φ 1 (θ) for the 1 st band:
From the profile function t 1 (θ) of the 1 st band, the phase shift function (numerical expression) of the corresponding 1 st band can be expressed as
(3) Knowing the phase shift function, the amplitude a 1 of the 1 st band is found according to the thin grating approximation formula:
(4) Solving for the efficiency of the 1 st band:
(5) Repeating the calculation of the efficiency according to different morphological functions of 1-125 wave bands, wherein the initial luminous flux accepted by each wave band is the same, and the final total efficiency is the average value of the efficiencies of all wave bands:
(6) Repeating the efficiency calculation of the steps (1) - (5) to finally obtain a curve of focusing efficiency of the actually prepared gold Kinoform lens at 5-15 keV along with energy, as shown in figure 3. It can be seen that the focusing efficiency of the actual morphology gold Kinoform lens is lower than the theoretical value, and that an improvement in the process is required during the actual manufacturing process to reduce the adverse effect of structural defects on efficiency.
Example 2: the structural parameters of the band structure lens with the peak morphology under 500keV energy (the resolution is 100nm, the diameter is 100 mu m, the band number N is 125, and the maximum thickness is 700 nm) are optimized:
In the soft X-ray band of 0-1 keV, the traditional gold material has strong absorption and greatly limited focusing efficiency. In addition, the diffraction efficiency of the rectangular zone plate morphology can not be further improved, and the Kinoform morphology has application limitation due to the difficulty in process preparation. In the embodiment, taking an HSQ photoresist material with low absorptivity as an example, a zone plate lens with novel morphology is designed in a soft X-ray wave band, so that the zone plate lens is suitable for an actual preparation process, and high-efficiency and high-resolution focusing can be realized.
The method comprises the following specific steps:
(1) The incident light field energy was set to 500eV and was normally incident as a plane wave of unit amplitude. The refractive index is set to silica and the profile function is established as t (θ) in the radial direction.
(2) Calculate the phase shift function Φ 1 (θ) for the 1 st band: fig. 4 shows the phase shift function of the peaked zone plate. m and n describe the relative positions of the gap and tip in the band (0 < m < n < 1). The sidewalls are arranged non-vertically in consideration of influence of proximity effect in actual manufacturing. While shallower regions typically produce gaps due to over-development or under-exposure dose to calculate its theoretical efficiency, each cycle is represented by a piecewise function.
From the profile function t 1 (θ) for the 1 st band, the phase shift function for the corresponding 1 st band can be expressed as:
Where Φ 0 is the maximum phase shift value of the lens, the segment intervals of the phase shift function correspond to the 3 regions marked in fig. 4, respectively.
(3) Knowing the phase shift function, the amplitude a 1 of the 1 st band is found according to the thin grating approximation formula:
(4) Solving for the efficiency of the 1 st band:
(5) Since the topography will be discussed further in step 6, the set topography here does not change with the period. The overall efficiency is therefore:
FE＝FE1＝FE2＝…＝FEN，
(6) Setting different structural parameters m and n, repeating the efficiency calculation of the steps (1) - (5), and drawing a graph of the efficiency change along with the structural parameters under 500eV, as shown in fig. 5. When n=1, m=0.2, the peripheral sidewall of each period is perfectly vertical and 20% of the gap is left, at which time the lens can achieve the highest efficiency. The calculation result shows that the gap has positive effect on improving the efficiency of the band structure lens, and meanwhile, the difficulty in process preparation is reduced; in addition, to achieve a higher efficiency of focusing, each period of the lens needs to have a strict vertical edge.
Claims (2)
1. The method for calculating the efficiency of the optical element with the wave band structure is characterized by comprising the following specific steps:
(1) Setting an incident light field and optical element parameters; wherein:
the incident light field parameters include wavelength λ, amplitude C;
the optical element parameters include: the number of wave bands N of the optical element, the refractive index n= (1-delta) -iβ, 1-delta and β of the material used for the optical element represent the real part and the imaginary part of the refractive index of the material respectively; the morphology function of the ith band is t i (θ), which is the optical path length difference to the focal spot over a period, describing the variation of height t with optical path difference θ; the optical element parameters are used for calculating the phase shift and the amplitude in the lens; i=1, 2, …, N;
(2) Calculate the phase shift function Φ i (θ) of the i-th band:
From the morphology function t i (θ) of the i-th band, the phase shift function of the corresponding i-th band can be expressed as:
(3) Knowing the phase shift function, the amplitude a i of the i-th band is found according to the thin grating approximation formula:
(4) Solving the efficiency of the ith band:
(5) Repeating the efficiency calculation of the steps (2) to (4) according to different morphological functions of 1 to N wave bands, wherein the initial luminous flux accepted by each wave band is the same, and the total efficiency of the final optical element is the average value of the efficiencies of all wave bands:
in the formula (2), j represents an imaginary unit.
2. The method of claim 1, wherein the zone profile of each period of the optical element is a function of the optical path difference, described by a different function, the zone profile of the optical element comprising a zone plate, kinoform lens, step lens, fresnel lens, and non-ideal profiles thereof.
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| Publication number | Priority date | Publication date | Assignee | Title |
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| IL115295A0 (en) * | 1995-09-14 | 1996-12-05 | Yeda Res & Dev | Multilevel diffractive optical element |
| US6839174B1 (en) * | 1997-03-03 | 2005-01-04 | Olympus Corporation | Relief type diffraction optical element, optical system comprising the same and mold for manufacturing the same |
| TW200502670A (en) * | 2002-11-21 | 2005-01-16 | Konica Minolta Holdings Inc | Objective lens, optical system and optical pickup apparatus |
| WO2010129060A1 (en) * | 2009-05-07 | 2010-11-11 | President And Fellows Of Harvard College | Methods and apparatus for scanning microscopy using one or more fresnel zone plates |
| GB201115124D0 (en) * | 2011-09-01 | 2011-10-19 | Crosby David | Improved adjustable refractive optical device for context specific use |
| EP2823489A2 (en) * | 2012-03-08 | 2015-01-14 | Max-Planck-Gesellschaft zur Förderung der Wissenschaften | Method of producing a fresnel zone plate for applications in high energy radiation |
| JP6071262B2 (en) * | 2012-06-21 | 2017-02-01 | キヤノン株式会社 | Diffractive optical element, optical system and optical instrument |
| CN112885499A (en) * | 2021-01-17 | 2021-06-01 | 复旦大学 | Kinoform medium lens for synchrotron radiation soft X-ray focusing imaging and preparation method thereof |
| CN113281900B (en) * | 2021-05-26 | 2022-03-18 | 复旦大学 | Optical modeling and calculating method based on Hankel transformation and beam propagation method |
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| high-efficiency focusing and imaging by dielectric kinoform zone plate lenses with soft X-rays;xujie tong，yifang chen等;journal of synchrotron radiation;20230331;第30卷;319-326 * |
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