CN113433689A - Achromatic superlens design method based on effective medium theory - Google Patents
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Abstract
The invention discloses an achromatic superlens design method based on an effective medium theory, which comprises the following steps of firstly, calculating the height required by a sub-wavelength structural unit according to the phase profile distribution and the phase-optical path relation of a super surface required by focusing; then, according to an effective medium theory, converting the structural units with the same width and different heights at each radial position into the structural units with the same height and different widths by using a filling coefficient invariant principle, thereby obtaining the integral structure of the planarization super lens; and finally, directly cutting and forming the optical material substrate in a focused ion beam mode. The invention adopts the transparent material with low dispersion in the target working waveband to prepare the superlens based on the effective medium theory, and can realize high efficiency, insensitive polarization and continuous achromatization in a wide frequency range. The invention has simple design and simple configuration, can be formed in one step, simplifies the preparation process of the traditional optical super lens, improves the design freedom of the achromatic super lens and widens the application scene of the achromatic super lens.
Description
Technical Field
The invention relates to the technical field of optical lenses, in particular to an achromatic superlens design method based on an effective medium theory.
Background
In modern optical systems, dispersion regulation plays a crucial role in achieving the performance of high-quality precision instruments. Optical fibers from microscopes and lithography machines required in laboratories to the communication field, even mobile phones, cameras, video cameras and the like which are market products for the public, are not designed with achromatism. In traditional engineering optics, achromatism is based on the elimination of chromatic aberration by combining a plurality of lenses to form a group of bulky lens groups. This results in limited device miniaturization and integration, especially for portable and wearable devices. In addition, fine processing of aspheric lenses and optical alignment of complex lens groups are difficult, and the resulting achromatization performance is usually limited to three discrete wavelengths, red, green and blue, making it difficult to achieve continuous bandwidth achromatization.
With the development of semiconductor micro-nano processing technology and the combination of super surfaces which are widely concerned in recent years, a two-dimensional planarization medium super lens constructed based on a sub-wavelength nano structure unit is produced, can realize achromatic focusing of incident light with different wavelengths, and has the remarkable advantages of small volume, light weight, easiness in integration and the like. Conventional achromatic superlens design methods (Light sci. appl.2018,7,85.) typically manipulate the phase at different spatial locations based on the resonance effect of high index materials. The specific design process comprises the following steps: firstly, carrying out a large number of configurations and simulation on a sub-wavelength structure array to generate a dispersion space with a phase coverage range larger than or equal to 2 pi; then expanding and encrypting a dispersion space by a machine learning method; and finally, searching a structural configuration which simultaneously meets the required phase-group delay dispersion at different radial positions. Therefore, the design method has the disadvantages of complex principle, complicated process and low efficiency.
In addition, achromatic lenses prepared by using the resonance effect have problems in performance. For example, (1) the limited dispersion space also limits the overall size of the achromatic lens due to the different dispersion conditions required at different radial positions. (2) Since a phase jump is generated by using a resonance effect of a high refractive index material, high-order resonance is often accompanied in a high frequency band, making it difficult to achieve continuous high transmission in a wide frequency range. (3) Most achromatic superlenses are sensitive to the polarization of the incident light, further limiting their range of applications.
Disclosure of Invention
The invention provides an achromatic superlens design method based on an effective medium theory, and aims to solve the technical problems that the existing superlens design method is complex in principle, tedious in process and low in efficiency, an achromatic lens prepared by utilizing a resonance effect is not ideal in performance, and the application range is limited.
In order to solve the technical problems, the invention provides the following technical scheme:
an achromatic superlens design method based on an effective medium theory, wherein the achromatic superlens is composed of sub-wavelength structural units obtained according to an equivalent medium theory, and the method comprises the following steps:
s1, setting a target working waveband and determining the adopted optical material; wherein the optical material is a transparent material having low dispersion characteristics within the target operating band;
s2, determining the period, the whole size and the focal length of the sub-wavelength structural unit of the achromatic superlens;
s3, solving phase changes required by the sub-wavelength structure units to focus the light field at different radial positions according to a Fresnel hyperbolic phase modulation formula;
s4, according to the phase-optical path relation, the height required by the sub-wavelength structure unit at different radial positions is obtained;
s5, converting sub-wavelength structural units with the same width and different heights into sub-wavelength structural units with the same height and different widths in the period of the sub-wavelength structural units according to an effective medium theory and based on a filling coefficient invariant principle to obtain a two-dimensional cross-sectional structure configuration of the achromatic superlens; rotating the two-dimensional profile structure along the central symmetry axis to obtain a three-dimensional overall structure of the achromatic superlens;
and S6, processing and preparing the achromatic superlens according to the three-dimensional overall structure configuration.
Further, the phase change required by the sub-wavelength structure unit for focusing the light field at different radial positions is solved according to a fresnel hyperbolic phase modulation formula, and the expression is as follows:
wherein r is a radial coordinate, λ0Is the center wavelength, F is the design focal length, C0Is the initial phase constant.
Further, the required heights of the sub-wavelength structural units at different radial positions are obtained according to the phase-optical path relation, and the expression is as follows:
wherein n is0For optical materials at the operating wavelength lambda0Refractive index of (d).
Further, in S2, the period p of the sub-wavelength structural unit of the achromatic superlens is smaller than the incident wavelength.
Further, in S3, the radial position coordinate r is a distance between the center of the subwavelength structural unit and the center of the achromatic superlens, and is denoted as r ═ mp, where m is an integer.
Further, in S4, the height d (r). ltoreq.d (0) of the subwavelength structural units, where d (0) is the height of the subwavelength structural units at the center position of the achromatic superlens.
Further, in S5, the periodic size of the subwavelength structural unit remains unchanged before and after the transition, with a width p and a height d (0).
Further, the effective medium theory is described by a Bruggeman model or Maxwell Garnett model in S5.
Further, in S5, the principle of the fill factor invariance is that f is within the unit period before and after the structural unit conversion1(r) andf2(r) the value is unchanged, and the width w of the converted sub-wavelength structural unit is represented by the following formula:
w(r)＝f2(r)·p
wherein f is1(r) ═ 1-d (r)/d (0) and f2(r) ═ d (r)/d (0) are the fill factors of air and optical material in the unit period, respectively.
Further, in S6, the achromatic superlens is formed by directly cutting and shaping the optical material substrate with a focused ion beam.
The technical scheme provided by the invention has the beneficial effects that at least:
1. the invention has simple design principle and high efficiency;
2. the sub-wavelength structure adopted by the invention has simple configuration and strong robustness, can be prepared and formed in one step by focusing ion beams, and has short processing period and low cost;
3. the invention can realize the excellent characteristics of high transmission, insensitive polarization and continuous achromatization in a wide frequency range;
4. the material selected by the invention is a low-scattering optical material, so that the requirements of high refractive index and extremely low extinction coefficient required by the traditional achromatic lens can be avoided, the freedom degree of material selection is widened, and the compatibility with semiconductor manufacturing is improved.
Drawings
In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the drawings needed to be used in the description of the embodiments will be briefly introduced below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and it is obvious for those skilled in the art to obtain other drawings based on these drawings without creative efforts.
FIG. 1 is a flow chart of an achromatic superlens design method based on effective medium theory provided by the present invention;
FIG. 2 is a schematic diagram of a sub-wavelength structural unit based on effective medium theory according to the present invention;
FIG. 3 is a two-dimensional cross-sectional structural configuration of a superlens provided by the present invention;
FIG. 4 is a three-dimensional overall structural configuration of a superlens provided by the present invention;
FIG. 5 is a diagram illustrating a distribution of incident light intensity modulated by a superlens according to a first embodiment of the present invention;
FIG. 6 is a graph of focusing efficiency versus wavelength for a superlens provided in accordance with a first embodiment of the present invention;
FIG. 7 is a diagram illustrating a distribution of incident light intensity modulated by a superlens according to a second embodiment of the present invention;
FIG. 8 is a graph of focusing efficiency versus wavelength for a superlens according to a second embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, embodiments of the present invention will be described in detail with reference to the accompanying drawings.
First embodiment
The embodiment provides an achromatic superlens design method based on an effective medium theory, wherein the achromatic superlens is composed of sub-wavelength structural units obtained according to an equivalent medium theory, and the method combines a super-surface design method with the effective medium theory to design the achromatic superlens with a simple principle, convenience in processing and excellent performance, and specifically, as shown in fig. 1, the method comprises the following steps:
s1, setting a target working waveband and determining the adopted optical material; wherein the optical material is a transparent material having low dispersion characteristics within the target operating band; specifically, the present embodiment is designed for an incident wavelength band in the visible light range of 400-667 nm, taking 532nm as an example of the central wavelength, and the adopted low-dispersion optical material is CaF2Imaginary part of refractive index k<0.01, and dispersion Δ n is 0.01 within the operating bandwidth.
S2, determining the period, the whole size and the focal length of the sub-wavelength structural unit of the achromatic superlens; wherein the period p of the sub-wavelength structural unit of the achromatic superlens is smaller than the incident wavelength so as to avoid the diffraction effect. Specifically, in the present embodiment, the sub-wavelength structural unit period p of the superlens is 400nm, the overall size is 19.6 μm, and the focal length is 110.0 μm.
S3, according to the Fresnel hyperbolic phase modulation formula, the phase change required by the sub-wavelength structure unit to focus the light field at different radial positions r is solved, and the expression is as follows:
wherein r is a radial coordinate, represents the distance between the center of the sub-wavelength structural unit and the center of the achromatic superlens, and is represented as mp, and m is an integer; lambda [ alpha ]0Is the center wavelength, F is the design focal length, C0Is the initial phase constant.
S4, according to the phase-optical path relation, the height required by the sub-wavelength structural unit at different radial positions r is obtained, and the expression is as follows:
wherein n is0For optical materials at the operating wavelength lambda0A refractive index of; the height d (r) of the subwavelength structural unit is less than or equal to d (0), and d (0) is 973nm, which is the height of the subwavelength structural unit at the center position of the superlens.
S5, converting sub-wavelength structure units with the same width and different heights into sub-wavelength structure units with the same height and different widths as shown in FIG. 2 in a unit period according to an effective medium theory and based on a filling coefficient invariance principle, so as to obtain a two-dimensional cross-sectional structure configuration of the achromatic superlens as shown in FIG. 3; further rotating the two-dimensional cross-sectional structure configuration along the central symmetry axis thereof to finally obtain the three-dimensional overall structure configuration of the achromatic superlens shown in FIG. 4; wherein the periodic size of the sub-wavelength structural unit is kept unchanged before and after the conversion, the width is p, and the height is d (0). The effective medium theory is described by the Bruggeman model (BG model) or Maxwell Garnett model (MG model):
BG model:
MG model:
wherein epsilon1And ε2The dielectric constants of air and optical material, respectively; f. of1(r) ═ 1-d (r)/d (0) and f2(r) ═ d (r)/d (0) are the fill factors of air and the optical material in the unit period, respectively; ε (r) is the equivalent dielectric constant of the unit period at the radial coordinate r position.
The principle of unchanged filling coefficient is that f in unit period before and after structural unit conversion1(r) and f2(r) the value is unchanged, and the width w of the converted sub-wavelength structural unit is represented by the following formula:
w(r)＝f2(r)·p
and S6, processing and preparing the achromatic superlens according to the three-dimensional overall structure configuration.
Specifically, in this embodiment, the processing method of the achromatic superlens is as follows: and according to the three-dimensional integral structure configuration, directly cutting and forming the optical material substrate by using a focused ion beam.
In the embodiment, finite element electromagnetic simulation software is adopted to model the parametric superlens; and a light intensity distribution diagram in a visible light range under a vertical incidence condition is obtained through simulation by adopting plane waves as an excitation source. Fig. 5 shows the light intensity distribution of the incident light modulated by the superlens, and the focal positions of the incident light at different wavelengths are substantially the same, which shows that the superlens has an excellent achromatic effect in the operating band (due to the limitation of computational resources, this embodiment explains the focusing condition of the complete lens by simulating the vertical cross section passing through the center of the bottom surface of the lens).
Fig. 6 shows the focusing efficiency of the incident light after being modulated by the superlens, and the average efficiency is as high as 82%, so that the high-efficiency broadband achromatic effect is realized. In addition, since the sub-wavelength structural units have four-fold rotational symmetry, the sub-wavelength structural units are insensitive to incident polarized light.
Second embodiment
The embodiment provides an achromatic superlens design method based on an effective medium theory, wherein the achromatic superlens is composed of sub-wavelength structural units obtained according to an equivalent medium theory, and the method combines a super-surface design method with the effective medium theory to design the achromatic superlens with a simple principle, convenience in processing and excellent performance, and specifically, as shown in fig. 1, the method comprises the following steps:
s1, setting a target working waveband and determining the adopted optical material; wherein the optical material is a transparent material having low dispersion characteristics within the target operating band; specifically, the present embodiment is designed for an incident wavelength band in the visible light range of 400 to 667nm, taking 532nm as an example of the central wavelength, and the adopted low-dispersion optical material is SiO2Imaginary part of refractive index k<0.01, and dispersion Δ n is 0.015 over the operating bandwidth.
S2, determining the period, the whole size and the focal length of the sub-wavelength structural unit of the achromatic superlens; wherein the period p of the sub-wavelength structural unit of the achromatic superlens is smaller than the incident wavelength so as to avoid the diffraction effect. Specifically, in the present embodiment, the sub-wavelength structural unit period p of the superlens is 400nm, the overall size is 7.6 μm, and the focal length is 15.0 μm.
S3, according to the Fresnel hyperbolic phase modulation formula, the phase change required by the sub-wavelength structure unit to focus the light field at different radial positions r is solved, and the expression is as follows:
wherein r is a radial coordinate, represents the distance between the center of the sub-wavelength structural unit and the center of the achromatic superlens, and is represented as mp, and m is an integer; lambda [ alpha ]0Is the center wavelength, F is the design focal length, C0Is the initial phase constant.
S4, according to the phase-optical path relation, the height required by the sub-wavelength structural unit at different radial positions r is obtained, and the expression is as follows:
wherein n is0For optical materials at the operating wavelength lambda0A refractive index of; the height d (r) of the subwavelength structural unit is less than or equal to d (0), and d (0) is 829nm which is the height of the subwavelength structural unit at the center position of the superlens.
S5, converting sub-wavelength structure units with the same width and different heights into sub-wavelength structure units with the same height and different widths as shown in FIG. 2 in a unit period according to an effective medium theory and based on a filling coefficient invariance principle, so as to obtain a two-dimensional cross-sectional structure configuration of the achromatic superlens as shown in FIG. 3; further rotating the two-dimensional cross-sectional structure configuration along the central symmetry axis thereof to finally obtain the three-dimensional overall structure configuration of the achromatic superlens shown in FIG. 4; wherein the periodic size of the sub-wavelength structural unit is kept unchanged before and after the conversion, the width is p, and the height is d (0). The effective medium theory is described by the Bruggeman model (BG model) or Maxwell Garnett model (MG model):
BG model:
MG model:
wherein epsilon1And ε2The dielectric constants of air and optical material, respectively; f. of1(r) ═ 1-d (r)/d (0) and f2(r) ═ d (r)/d (0) are the fill factors of air and the optical material in the unit period, respectively; ε (r) is the equivalent dielectric constant of the unit period at the radial coordinate r position.
Fill factorThe principle of invariance is that f in unit period before and after structural unit conversion1(r) and f2(r) the value is unchanged, and the width w of the converted sub-wavelength structural unit is represented by the following formula:
w(r)＝f2(r)·p
and S6, processing and preparing the achromatic superlens according to the three-dimensional overall structure configuration.
Specifically, in this embodiment, the processing method of the achromatic superlens is as follows: and according to the three-dimensional integral structure configuration, directly cutting and forming the optical material substrate by using a focused ion beam.
In the embodiment, finite element electromagnetic simulation software is adopted to model the parametric superlens; and a light intensity distribution diagram in a visible light range under a vertical incidence condition is obtained through simulation by adopting plane waves as an excitation source. Fig. 7 shows the light intensity distribution of the incident light modulated by the superlens, and the focal positions of the incident light at different wavelengths are substantially the same, which shows that the superlens has an excellent achromatic effect in the operating band (due to the limitation of computational resources, this embodiment explains the focusing condition of the complete lens by simulating the vertical cross section passing through the center of the bottom surface of the lens).
Fig. 8 shows the focusing efficiency of the incident light modulated by the superlens, the average efficiency is up to 87%, and the broadband achromatic effect with high efficiency is realized. In addition, since the sub-wavelength structural units have four-fold rotational symmetry, the sub-wavelength structural units are insensitive to incident polarized light.
In summary, the present invention can determine the sub-wavelength structural unit size parameters at different radial positions according to the relationship between the phase profile of the super-surface required for focusing and the effective medium theory. The sub-wavelength structural units are all rectangular structures without complex configuration, the preparation is simple, the robustness is strong, and the one-step preparation molding can be realized through focused ion beams. The designed achromatic superlens can realize high efficiency, insensitivity to polarization and continuous achromatization to incident light within a target bandwidth. The achromatic superlens can be processed and prepared by adopting a low-refractive-index material, so that the freedom degree of material selection is widened, and the manufacturing compatibility of a semiconductor is improved. The design method has the advantages of simple principle, convenience and quickness in preparation, low cost, high integration level and the like, and has very wide potential application in the industrial industry and the scientific technology field, such as mobile phones, cameras, photoetching machines, endoscopes, virtual and augmented reality applications and the like.
Further, it should be noted that, in this document, the terms "comprises," "comprising," or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or terminal apparatus that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or terminal apparatus. Without further limitation, an element defined by the phrase "comprising an … …" does not exclude the presence of other like elements in a process, method, article, or terminal that comprises the element.
Finally, it should be noted that while the above describes a preferred embodiment of the invention, it will be appreciated by those skilled in the art that, once the basic inventive concepts have been learned, numerous changes and modifications may be made without departing from the principles of the invention, which shall be deemed to be within the scope of the invention. Therefore, it is intended that the appended claims be interpreted as including preferred embodiments and all such alterations and modifications as fall within the scope of the embodiments of the invention.
Claims (10)
1. An achromatic superlens design method based on an effective medium theory, wherein the achromatic superlens is composed of sub-wavelength structural units obtained according to an equivalent medium theory, and the method comprises the following steps:
s1, setting a target working waveband and determining the adopted optical material; wherein the optical material is a transparent material having low dispersion characteristics within the target operating band;
s2, determining the period, the whole size and the focal length of the sub-wavelength structural unit of the achromatic superlens;
s3, solving phase changes required by the sub-wavelength structure units to focus the light field at different radial positions according to a Fresnel hyperbolic phase modulation formula;
s4, according to the phase-optical path relation, the height required by the sub-wavelength structure unit at different radial positions is obtained;
s5, converting sub-wavelength structural units with the same width and different heights into sub-wavelength structural units with the same height and different widths in the period of the sub-wavelength structural units according to an effective medium theory and based on a filling coefficient invariant principle to obtain a two-dimensional cross-sectional structure configuration of the achromatic superlens; rotating the two-dimensional profile structure along the central symmetry axis to obtain a three-dimensional overall structure of the achromatic superlens;
and S6, processing and preparing the achromatic superlens according to the three-dimensional overall structure configuration.
2. The method for designing an achromatic superlens based on an effective medium theory according to claim 1, wherein the phase change required by the sub-wavelength structural unit for focusing the light field at different radial positions is obtained according to a fresnel hyperbolic phase modulation formula, and the expression is as follows:
wherein r is a radial coordinate, λ0Is the center wavelength, F is the design focal length, C0Is the initial phase constant.
3. The method of claim 2, wherein the required height of the sub-wavelength structure unit at different radial positions is determined according to the phase-path relationship, and the expression is:
wherein n is0For optical materials at the operating wavelength lambda0Refractive index of (d).
4. The method of claim 1, wherein in S2, the period p of the subwavelength structural elements of the achromatic superlens is less than the incident wavelength.
5. The method of claim 4, wherein in S3, the radial position coordinate r is the distance between the center of the subwavelength structure unit and the center of the achromatic superlens, where m is an integer, and is denoted as r ═ mp.
6. The method of claim 4, wherein in S4, the height d (r) of the subwavelength structural elements is ≦ d (0), wherein d (0) is the height of the subwavelength structural elements at the center of the achromatic superlens.
7. The effective medium theory-based achromatic superlens design method of claim 6, wherein in S5, the periodic size of the sub-wavelength structural unit remains unchanged before and after the transition, and has a width p and a height d (0).
8. The method of claim 1, wherein the effective medium theory is described in S5 by a Bruggeman model or a Maxwell Garnett model.
9. The method of claim 4, wherein in S5, the principle of constant fill factor is f in the unit period before and after the structural unit is transformed1(r) and f2(r) the value is unchanged, and the width w of the converted sub-wavelength structural unit is represented by the following formula:
w(r)＝f2(r)·p
wherein f is1(r) ═ 1-d (r)/d (0) and f2(r) ═ d (r)/d (0) are the fill factors of air and optical material in the unit period, respectively.
10. The method of designing an achromatic superlens according to an effective medium theory as claimed in any one of claims 1 to 9, wherein in S6, the achromatic superlens is formed by directly cutting and shaping an optical material substrate by a focused ion beam.
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| CN113885106A (en) * | 2021-11-09 | 2022-01-04 | 深圳迈塔兰斯科技有限公司 | Design method and device of super-lens antireflection film and electronic equipment |
| CN114967127A (en) * | 2022-06-16 | 2022-08-30 | 曹桂源 | Design method of multi-wavelength achromatic ultrathin planar lens |
| CN115586642A (en) * | 2022-12-12 | 2023-01-10 | 北京航空航天大学 | Design method of broadband achromatic superlens |
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