CN110376731B - Construction method of broadband achromatic super-structure lens group based on multilayer super-structure surface - Google Patents

Construction method of broadband achromatic super-structure lens group based on multilayer super-structure surface Download PDF

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CN110376731B
CN110376731B CN201910632239.3A CN201910632239A CN110376731B CN 110376731 B CN110376731 B CN 110376731B CN 201910632239 A CN201910632239 A CN 201910632239A CN 110376731 B CN110376731 B CN 110376731B
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super
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structure lens
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CN110376731A (en
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俞叶峰
李帅帅
李猛猛
陈如山
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Nanjing University of Science and Technology
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Abstract

The invention discloses a construction method of a broadband achromatic super-structure lens group based on a multilayer super-structure surface. The method comprises the following steps: firstly, constructing a super unit, and simulating to obtain a transmission coefficient and a phase change curve of a uniform array corresponding to the super unit; then constructing a focusing super-structure lens and a diverging super-structure lens, generating focusing super-structure lens and diverging super-structure lens samples, and calculating the Abelian constant of each focusing super-structure lens sample and each diverging super-structure lens sample according to the dispersion characteristics; then solving an optimization problem to obtain a combination of a focusing super-structure lens and a diverging super-structure lens with minimum dispersion; and finally, obtaining the dispersion characteristic of the broadband achromatic super-structure lens group by adopting full-wave simulation, comparing the dispersion characteristic with the single-layer chromatic dispersion super-structure lens, and evaluating the chromatic aberration improvement condition of the broadband achromatic super-structure lens group. The invention reduces the complexity of processing the broadband achromatic device and improves the integration level and the reliability of the broadband achromatic device.

Description

Construction method of broadband achromatic super-structure lens group based on multilayer super-structure surface
Technical Field
The invention belongs to the field of electromagnetic wave metamaterial, and particularly relates to a construction method of a broadband achromatic metamaterial lens group based on a multilayer metamaterial surface.
Background
Various lenses are widely applied to multiple fields of radar imaging, optical imaging, photoelectric sensing and the like. The recently developed planar type super-structure lens adopts a planar type super-structure surface to replace a curved surface, compared with the traditional lens, the spherical aberration can be effectively eliminated, and the lens has the advantages of small volume, strong integration and batch processing and preparation by utilizing a semiconductor process, so that the lens is widely concerned in various application fields. Planar-type, super-structured lens designs have generally focused on only one of convergence efficiency, numerical aperture, or operating bandwidth in the early days, and current designs have begun to focus on two or more parameters. Recently, there has been much interest in improving the operating bandwidth of planar type super-structured lenses, i.e., achromatization. However, the existing achromatic plane type super-structure lens is mainly based on a single-layer super-structure plane, and has the problems of complex super-structure unit construction, relatively small numerical aperture, low convergence efficiency and the like.
Disclosure of Invention
The invention aims to provide a convenient, quick, efficient and accurate construction method of a broadband achromatic super-structure lens group based on a multilayer super-structure surface.
The technical solution for realizing the purpose of the invention is as follows: a construction method of a broadband achromatic super-structure lens group based on a multilayer super-structure surface comprises the following steps:
step 1, constructing a super unit, and simulating to obtain a transmission coefficient and a phase change curve of a uniform array corresponding to the super unit;
step 2, constructing a focusing super-structure lens, generating focusing super-structure lens samples, and calculating the Abelian constant of each focusing super-structure lens sample according to the dispersion characteristic;
step 3, constructing a divergent super-structure lens, generating divergent super-structure lens samples, and calculating the Abelian constant of each divergent super-structure lens sample according to the dispersion characteristics;
step 4, solving an optimization problem to obtain a combination of a focusing super-structure lens and a diverging super-structure lens with minimum chromatic dispersion;
and 5, obtaining the dispersion characteristic of the broadband achromatic super-structure lens group by adopting full-wave simulation, comparing the dispersion characteristic with a single-layer chromatic dispersion super-structure lens, and evaluating the chromatic aberration improvement condition of the broadband achromatic super-structure lens group.
Further, the constructing of the superunit in step 1, and the simulation result shows that the transmission coefficient and the phase change curve of the uniform array corresponding to the superunit are as follows:
step 1.1, constructing a super unit, so that the corresponding uniform array of the super unit meets the conditions that the transmission coefficient is greater than 0.9 and the phase covers 0-2 pi in the whole wavelength range;
and step 1.2, simulating to obtain the transmission coefficient and the phase change curve of the uniform array corresponding to the superunit.
Further, the step 2 of constructing a focusing super-structure lens to generate focusing super-structure lens samples, and calculating abelian constants of each focusing super-structure lens sample according to the dispersion characteristics specifically as follows:
step 2.1, constructing a focusing super-structure lens, and generating focusing super-structure lens samples with focal lengths f11,f12,f13,…,f1NWherein N is the total number of samples, and the compensation required by the focusing super-structure lens with different focal lengths is calculated according to the geometric optical path differencePhase position
Figure BDA0002129059360000021
Figure BDA0002129059360000022
Wherein r is the distance between any point on the focusing super-structure lens and the center of the focusing super-structure lens, and r is the distance between any point on the focusing super-structure lens and the center of the focusing super-structure lens under a rectangular coordinate system
Figure BDA0002129059360000023
Figure BDA0002129059360000024
For the compensation phase of the point correspondence, f1In order to focus the focal length of the super-structured lens, λ is the wavelength,
Figure BDA0002129059360000025
the initial phase of each unit of the focusing super-structure lens is also the compensation phase of the center of the focusing super-structure lens;
step 2.2, obtaining a compensation phase required by each unit on the focusing super-structure lens according to the compensation phase formula (1), interpolating from the phase characteristic curve to obtain units meeting conditions, and arranging the focusing super-structure lens meeting construction requirements;
the correspondence between the focal length of the thin lens in vacuum and its geometry is as follows:
Figure BDA0002129059360000026
where f is the focal length of the thin lens and n1Is the refractive index of the thin lens material, r1And r2The curvature radius of the spherical surfaces on the left side and the right side of the thin lens respectively;
step 2.3, performing full-wave simulation on the focusing super-structure lenses with different focal lengths respectively, and calculating a relation curve of the actual focal position of the focusing super-structure lens along with the change of the wavelength, namely a dispersion characteristic curve;
because the plane type focusing super-structure lens is approximately composed of a spherical surface and a plane r at the optical axis2A plano-convex thin lens constructed ∞, so the thin lens formula for a focusing super-structured lens is written as:
Figure BDA0002129059360000027
wherein f is1For focusing the focal length of the superstructural lens, p1Is a curvature of n1Is the equivalent refractive index of the focusing super-structured lens;
step 2.4, according to the dispersion characteristic curve obtained by full-wave simulation, the Abelian constant V of the focusing super-structure lens is solved1The formula is as follows:
Figure BDA0002129059360000028
wherein f is1C、f1D、f1FThe focal lengths of the focusing super-structure lens at low frequency, central frequency and high frequency are respectively obtained from the dispersion characteristic curve.
Further, the abelian constant of each focusing metamaterial lens sample is calculated according to the dispersion characteristics in step 2, which is specifically as follows:
obtaining the focal length f of the focusing super-structure lens according to the formula (3)1Curvature rho1With equivalent refractive index n1The relationship between them is as follows:
Figure BDA0002129059360000031
thereby obtaining the corresponding equivalent refractive index n of the focusing super-structure lens under three different frequencies1C、n1D、n1FComprises the following steps:
Figure BDA0002129059360000032
in-focus superstructuresIn the lens, the source of dispersion and the equivalent refractive index are directly related together, and the curvatures corresponding to different frequencies are set to be equal, namely rho1C＝ρ1D＝ρ1F＝ρ1According to the definition of Abelian constant, the following results are obtained:
Figure BDA0002129059360000033
further, constructing the diverging metamaterial lens in step 3, generating diverging metamaterial lens samples, and calculating abelian constants of each diverging metamaterial lens sample according to dispersion characteristics, specifically as follows:
step 3.1, constructing a divergent super-structure lens to generate divergent super-structure lens samples with focal lengths f21,f22,f23,…,f2NWherein N is the total number of samples, and the compensation phase required by the divergent super-structure lens with different focal lengths is calculated according to the geometric optical path difference
Figure BDA0002129059360000034
Figure BDA0002129059360000035
Wherein r is the distance between any point on the diverging super-structure lens and the center of the diverging super-structure lens, and r is the distance between any point on the diverging super-structure lens and the center of the diverging super-structure lens under a rectangular coordinate system
Figure BDA0002129059360000036
Figure BDA0002129059360000037
For the compensation phase of the point correspondence, f2Is the focal length of the divergent super-structure lens, D is the diameter of the divergent super-structure lens array, lambda is the wavelength,
Figure BDA0002129059360000038
for the initial phase of each cell of the diverging meta-lens, i.e. the compensation phase corresponding to the edge super-cell of the diverging meta-lensA bit;
step 3.2, obtaining the compensation phase required by each unit on the divergent super-structure lens according to the compensation phase formula (7), interpolating from the phase characteristic curve to obtain the units meeting the conditions, and arranging the divergent super-structure lens meeting the construction requirements;
3.3, performing full-wave simulation on the divergent super-structure lenses with different focal lengths respectively, calculating electric field distribution at a wavelength above the divergent super-structure lens array, extracting phases of electric field components to obtain phase curves with different wavelengths, and fitting the phase curves to obtain a relation curve of the focal lengths of the divergent super-structure lenses changing along with the wavelength, namely a dispersion characteristic curve;
since the planar diverging nanostructured lens is approximated at the optical axis to be a plano-concave thin lens composed of one spherical surface and one plane surface, the thin lens formula of the diverging nanostructured lens is written as:
Figure BDA0002129059360000041
wherein f is2To diverge the focal length of the super-structured lens, p2Is a curvature of n2Is the equivalent refractive index of the diverging super-structured lens;
step 3.4, according to the dispersion characteristic curve obtained by full-wave simulation, the Abelian constant V of the divergent super-structure lens is solved2The formula is as follows:
Figure BDA0002129059360000042
wherein f is2C、f2D、f2FThe focal lengths of the divergent super-structure lens corresponding to the low frequency, the central frequency and the high frequency are respectively obtained from the dispersion characteristic curve.
Further, the abelian constant of each diverging metamaterial lens sample is calculated according to the dispersion characteristics in step 3, which is as follows:
obtaining the focal length f of the divergent super-structured lens according to the formula (8)2Curvature rho2With equivalent refractive index n2The relationship between them is as follows:
Figure BDA0002129059360000043
thereby obtaining the equivalent refractive index n of the divergent super-structure lens corresponding to three different frequencies2C、n2D、n2FComprises the following steps:
Figure BDA0002129059360000044
in a diverging meta-lens, the source of dispersion and the equivalent refractive index are directly related together, and the curvatures corresponding to different frequencies are set to be equal, i.e. p2C＝ρ2D＝ρ2F＝ρ2According to the definition of Abelian constant, the following results are obtained:
Figure BDA0002129059360000051
further, the optimization problem is solved in step 4, and a combination of the focusing super-structure lens and the diverging super-structure lens with minimum chromatic dispersion is obtained, specifically as follows:
the constraint condition for the optimization problem is the focal length formula of the thin lens group with the set distance:
Figure BDA0002129059360000052
wherein f isgIs the focal length of the super-constituent lens group, f1To focus the focal length of the superstructural lens, f2D is the distance between the focusing super-structure lens and the diverging super-structure lens;
the objective function of the optimization problem is:
Figure BDA0002129059360000053
the whole optimization problem thus becomes: selecting the focal length f of the focusing super-structure lens under the condition of satisfying the focal length formula (12) of the thin lens group1Diverging super-structured lens focal length f2And the distance d between the two, so that the value of the objective function (13) tends to zero;
the objective function of the optimization problem is derived by combining the thin lens group focal length formula (12) with the requirement of achromatism, and the formula (3) and the formula (8) are substituted into the formula (12) to obtain:
Figure BDA0002129059360000054
to achieve the elimination of chromatic aberration, the focal length fgMust be independent of wavelength, i.e.
Figure BDA0002129059360000055
The dispersion is due to the wavelength dependence of the equivalent refractive index, i.e. the equivalent refractive index n1、n2Both are functions of wavelength, so the left and right sides of equation (14) are separately biased for wavelength to obtain:
Figure BDA0002129059360000056
wherein the equivalent refractive index n1、n2The partial derivative with respect to wavelength is approximated by the dispersion characteristics, or the slope of the dispersion characteristic, at low and high frequencies, as shown in the following equation:
Figure BDA0002129059360000057
Figure BDA0002129059360000061
then substituting the formulas (15b) and (15c) into the formula (15a), and finishing to obtain:
Figure BDA0002129059360000062
introducing an Abelian constant:
Figure BDA0002129059360000063
combining equation (3) and equation (8), equation (16) is simplified as:
Figure BDA0002129059360000064
the left side of the equation is the objective function (13), even if the objective function (13) tends to zero.
Compared with the prior art, the invention has the following remarkable advantages: (1) the planar super-structure lens is adopted to replace the traditional geometric curved surface lens, a novel planar electromagnetic wave device is constructed, the processing complexity is reduced, and the efficiency and convenience of simulation calculation and production processing are improved; (2) the multi-layer planar super-structure lens combination is adopted, various combination modes are provided, the multi-layer planar super-structure lens combination can adapt to different application scenes, the adaptability and the reliability are high, and the multi-layer planar super-structure lens combination can be widely applied to the fields of optical imaging, radar imaging, holographic imaging, photoelectric sensing and the like.
Drawings
FIG. 1 is a flow chart of a construction method of a broadband achromatic super lens set based on a multilayer super structure surface.
Fig. 2 is a schematic structural diagram of a super-structure lens and a super-structure lens group in the present invention, wherein (a) is a schematic structural diagram of a phase compensation principle (lower half) and dispersion (upper half) of a focusing super-structure lens, (b) is a schematic structural diagram of a phase compensation principle (lower half) and dispersion (upper half) of a diverging super-structure lens, (c) is a schematic structural diagram of an achromatic double-layer super-structure lens group 2D, and (D) is a schematic structural diagram of an achromatic double-layer super-structure lens group 3D.
FIG. 3 is a schematic diagram of a superstructure unit and its optical characteristics according to the present invention, wherein (a) is a schematic diagram of a gallium nitride cylinder on a sapphire substrate, (b) is a phase compensation of the gallium nitride cylinder on the sapphire substrate, and (c) is a relationship between a transmission coefficient of the gallium nitride cylinder on the sapphire substrate and a wavelength and a unit radius; (d) a schematic diagram of the gallium nitride cylinder wrapped inside the sapphire substrate, (e) phase compensation of the gallium nitride cylinder wrapped inside the sapphire substrate, and (f) a relation between a transmission coefficient of the gallium nitride cylinder wrapped inside the sapphire substrate and a wavelength and a cell radius.
Fig. 4 is a comparison diagram of the double-layer achromatic super-structure lens group and the single-layer dispersive super-structure lens in example 1 of the present invention, in which (a) is a comparison diagram of focal lengths of the double-layer achromatic super-structure lens group and the single-layer dispersive super-structure lens, (b) is a comparison diagram of focusing efficiencies of the double-layer achromatic super-structure lens group and the single-layer dispersive super-structure lens, (c) is a comparison diagram of half-power beam widths of the double-layer achromatic super-structure lens group and the single-layer dispersive super-structure lens, (d) is a result diagram of a change of focal length of the single-layer dispersive focusing super-structure lens with a change of wavelength, and (e) is a result diagram of a change of focal length of the double-layer achromatic super-structure lens group with a change of wavelength.
Fig. 5 is a comparison diagram of the double-layer achromatic super-structure lens group and the single-layer dispersive super-structure lens in example 2 of the present invention, in which (a) is a focal length comparison diagram of the double-layer achromatic super-structure lens group and the single-layer dispersive super-structure lens, (b) is a focusing efficiency comparison diagram of the double-layer achromatic super-structure lens group and the single-layer dispersive super-structure lens, and (c) is a half-power beam width comparison diagram of the double-layer achromatic super-structure lens group and the single-layer dispersive super-structure lens; (d) is a schematic diagram showing the result that the focal length of a single-layer dispersive focusing super-structure lens changes along with the change of the wavelength.
FIG. 6 is a graph of focal length versus focusing efficiency for two-layer achromatic super-lens of different numerical aperture in example 3 of the present invention, wherein (a) is a graph of focal length versus optimal two-layer achromatic super-lens group of different NA, and (b) is a graph of focusing efficiency versus optimal two-layer achromatic super-lens group of different NA; (c) the focal length contrast diagram of the double-layer achromatic super lens group with different NA when the focusing super lens is not changed, and the focal length contrast diagram of the double-layer achromatic super lens group with different NA when the focusing super lens is not changed.
Detailed Description
The invention is described in further detail below with reference to the figures and specific embodiments.
With reference to fig. 1, the method for constructing the broadband achromatic super lens group based on the multilayer super structure surface includes the following steps:
step 1, constructing a super unit, and simulating to obtain a transmission coefficient and a phase change curve of a uniform array corresponding to the super unit, wherein the transmission coefficient and the phase change curve are as follows:
step 1.1, constructing a super unit, so that the corresponding uniform array of the super unit meets the conditions that the transmission coefficient is greater than 0.9 and the phase covers 0-2 pi in the whole wavelength range;
step 1.2, obtaining a transmission coefficient and a phase change curve of the uniform array corresponding to the superunit through simulation;
step 2, constructing a focusing super-structure lens, generating focusing super-structure lens samples, and calculating an Abelian constant of each focusing super-structure lens sample according to dispersion characteristics, wherein the Abelian constant is as follows by combining with the graph 2:
step 2.1, constructing a focusing super-structure lens, and generating focusing super-structure lens samples with focal lengths f11,f12,f13,…,f1NWherein N is the total number of samples, and the compensation phase required by the focusing super-structure lens with different focal lengths is calculated according to the geometric optical path difference
Figure BDA0002129059360000071
Figure BDA0002129059360000072
Wherein r is the distance between any point on the focusing super-structure lens and the center of the focusing super-structure lens, and r is the distance between any point on the focusing super-structure lens and the center of the focusing super-structure lens under a rectangular coordinate system
Figure BDA0002129059360000081
Figure BDA0002129059360000082
For the compensation phase of the point correspondence, f1In order to focus the focal length of the super-structured lens, λ is the wavelength,
Figure BDA0002129059360000083
the initial phase of each unit of the focusing super-structure lens is also the compensation phase of the center of the focusing super-structure lens;
step 2.2, obtaining a compensation phase required by each unit on the focusing super-structure lens according to the compensation phase formula (1), interpolating from the phase characteristic curve to obtain units meeting conditions, and arranging the focusing super-structure lens meeting construction requirements;
the correspondence between the focal length of the thin lens in vacuum and its geometry is as follows:
Figure BDA0002129059360000084
where f is the focal length of the thin lens and n1Is the refractive index of the thin lens material, r1And r2The curvature radius of the spherical surfaces on the left side and the right side of the thin lens respectively;
step 2.3, performing full-wave simulation on the focusing super-structure lenses with different focal lengths respectively, and calculating a relation curve of the actual focal position of the focusing super-structure lens along with the change of the wavelength, namely a dispersion characteristic curve;
the planar focusing super-structure lens can be approximated to be composed of a spherical surface and a plane (r) near the optical axis2∞) of a thin plano-convex lens, the thin lens formula for a focusing super-structured lens can therefore be written as:
Figure BDA0002129059360000085
wherein f is1For focusing the focal length of the superstructural lens, p1Is a curvature of n1Is the equivalent refractive index of the focusing super-structured lens;
step 2.4, obtaining the focal length f of the focusing super-structure lens according to the formula (3)1Curvature rho1With equivalent refractive index n1The relationship between them is as follows:
Figure BDA0002129059360000086
thereby obtaining the corresponding equivalent refractive index n of the focusing super-structure lens under three different frequencies1C、n1D、n1FComprises the following steps:
Figure BDA0002129059360000087
in the traditional geometric refractive lens, the dispersion is derived from the fact that the refractive index of a material changes along with the frequency and is not related to the curvature, therefore, in the focusing super-structure lens, the source of the dispersion and the equivalent refractive index are directly related together, and the curvatures corresponding to different frequencies are set to be equal, namely rho1C＝ρ1D＝ρ1F＝ρ1According to the definition of Abelian constant, the following results are obtained:
Figure BDA0002129059360000091
wherein f is1C、f1D、f1FThe focal lengths of the focusing super-structure lens at low frequency, center frequency and high frequency are obtained from the dispersion characteristic curve.
Step 3, constructing a divergent super-structure lens, generating divergent super-structure lens samples, calculating the Abelian constant of each divergent super-structure lens sample according to the dispersion characteristics, and combining the Abelian constants with the graphs in the steps (a) to (d) of the graph in FIG. 2 as follows:
step 3.1, constructing a divergent super-structure lens to generate divergent super-structure lens samples with focal lengths f21,f22,f23,…,f2NWherein N is the total number of samples, and the compensation phase required by the divergent super-structure lens with different focal lengths is calculated according to the geometric optical path difference
Figure BDA0002129059360000092
Figure BDA0002129059360000093
Wherein r is the distance between any point on the diverging super-structure lens and the center of the diverging super-structure lens, and r is the distance between any point on the diverging super-structure lens and the center of the diverging super-structure lens under a rectangular coordinate system
Figure BDA0002129059360000094
Figure BDA0002129059360000095
For the compensation phase of the point correspondence, f2Is the focal length of the divergent super-structure lens, D is the diameter of the divergent super-structure lens array, lambda is the wavelength,
Figure BDA0002129059360000096
the initial phase of each unit of the divergent super-structure lens is also the compensation phase of the edge of the divergent super-structure lens;
step 3.2, obtaining the compensation phase required by each unit on the divergent super-structure lens according to the compensation phase formula (7), interpolating from the phase characteristic curve to obtain the units meeting the conditions, and arranging the divergent super-structure lens meeting the construction requirements;
3.3, performing full-wave simulation on the divergent super-structure lenses with different focal lengths respectively, calculating electric field distribution at a wavelength above the divergent super-structure lens array, extracting phases of electric field components to obtain phase curves with different wavelengths, and fitting the phase curves to obtain a relation curve of the focal lengths of the divergent super-structure lenses changing along with the wavelength, namely a dispersion characteristic curve;
since the planar diverging metamorphic lens can be approximated near the optical axis to be a plano-concave thin lens composed of one spherical surface and one plane, the thin lens formula of the diverging metamorphic lens can be written as:
Figure BDA0002129059360000097
wherein f is2To diverge the focal length of the super-structured lens, p2Is a curvature of n2Is the equivalent refractive index of the diverging super-structured lens;
step 3.4, obtaining the focal length f of the divergent super-structure lens according to the formula (8)2Curvature rho2With equivalent refractive index n2The relationship between them is as follows:
Figure BDA0002129059360000101
thereby obtaining the equivalent refractive index n of the diverging super-structure lens corresponding to three different frequencies2C、n2D、n2FComprises the following steps:
Figure BDA0002129059360000102
in a diverging meta-lens, the source of dispersion and the equivalent refractive index are directly related together, and the curvatures corresponding to different frequencies are set to be equal, i.e. p2C＝ρ2D＝ρ2F＝ρ2According to the definition of Abelian constant, the following results are obtained:
Figure BDA0002129059360000103
wherein f is2C、f2D、f2FThe focal lengths of the divergent super-structure lens corresponding to the low frequency, the central frequency and the high frequency are respectively obtained from the dispersion characteristic curve.
And 4, solving an optimization problem to obtain a combination of the focusing super-structure lens and the diverging super-structure lens with minimum dispersion, wherein the combination is as follows:
solving the optimization problem, wherein the construction focal length of the super-structure lens group is fgThe focal lengths of the focusing and diverging super-constituent lenses forming the super-constituent lens group need to satisfy the relationship of the focal lengths in the thin lens group, and the combination of the focusing and diverging super-constituent lenses with minimum chromatic dispersion is obtained by utilizing mutual dispersion cancellation in combination with the characteristic that the chromatic dispersion characteristics of the focusing and diverging super-constituent lenses are opposite.
The constraint condition for the optimization problem is the focal length formula of the thin lens group with a certain distance:
Figure BDA0002129059360000104
wherein f isgIs the focal length of the super-constituent lens group, f1To focus the focal length of the superstructural lens, f2D is the distance between the focusing super-structure lens and the diverging super-structure lens;
the objective function of the optimization problem is:
Figure BDA0002129059360000105
the whole optimization problem thus becomes: under the condition of satisfying the focal length formula (12) of the thin lens group, selecting a proper focal length f of the focusing super-structure lens1Diverging super-structured lens focal length f2And the distance d between the two, so that the value of the objective function (13) tends to zero as much as possible;
the objective function of the optimization problem is derived by combining the thin lens group focal length formula (12) with the requirement of achromatism, and the formula (3) and the formula (8) are substituted into the formula (12) to obtain:
Figure BDA0002129059360000111
to achieve the elimination of chromatic aberration, the focal length fgMust be independent of wavelength, i.e.
Figure BDA0002129059360000112
The dispersion being caused by the wavelength dependence of the equivalent refractive index, i.e. the equivalent refractive index n1、n2Both are functions of wavelength, so the left and right sides of equation (14) are separately biased for wavelength to obtain:
Figure BDA0002129059360000113
wherein the equivalent refractive index n1、n2The partial derivative with respect to wavelength can be approximated by the dispersion characteristic, or the slope of the dispersion characteristic, at low and high frequencies, as shown in the following equation:
Figure BDA0002129059360000114
Figure BDA0002129059360000115
then substituting the formulas (15b) and (15c) into the formula (15a), and finishing to obtain:
Figure BDA0002129059360000116
introducing an Abelian constant:
Figure BDA0002129059360000117
combining equation (3) and equation (8), equation (16) is simplified as:
Figure BDA0002129059360000118
the left side of the formula is an objective function (13);
the actual achromatization does not usually guarantee a formula
Figure BDA0002129059360000119
It holds true strictly at any wavelength, so the simultaneous solution of equations (12) and (18) translates into an optimization problem, even if the objective function (13) is as close to zero as possible.
And 5, obtaining the dispersion characteristic of the broadband achromatic super-structure lens group by adopting full-wave simulation, comparing the dispersion characteristic with a single-layer chromatic dispersion super-structure lens, and evaluating the chromatic aberration improvement condition of the broadband achromatic super-structure lens group, wherein the method specifically comprises the following steps:
obtaining the dispersion characteristic of the constructed broadband achromatic super-structure lens group by adopting full wave simulation (FDTD), namely the variation relation of the focal length along with the wavelength, and constructing the focal length fgThe single-layer chromatic dispersion super-structure lens is used for comparing with the broadband achromatic super-structure lens group and evaluating the chromatic aberration improvement condition of the broadband achromatic super-structure lens group.
Example 1
1. Unit construction
Constructing superunits working between 400nm and 700nm, the units having two structures: (1) a dielectric gallium nitride (GaN) cylinder in a dielectric sapphire (Al)2O3) Above the substrate, as shown in fig. 3 (a); (2) the dielectric GaN cylinder is wrapped in the dielectric Al2O3Inside the substrate, as shown in fig. 3 (d). The construction requirements are as follows: the transmission coefficient is more than 0.9 in the whole frequency band, and the compensation phase covers 0-2 pi. A transmission coefficient of more than 0.9 is not a necessary condition for the present construction method, but in order to improve the transmission efficiency of the entire device, it is recommended to select a superunit having a relatively high transmission coefficient. Two unit structures meeting the requirements are finally constructed by optimizing the period of the unit, the height of the medium column and the radius of the medium column: the period P corresponding to the unit structure (1) is 195nm, the height H is 900nm, the radius variation range is 25nm-75nm, the curve of the phase variation with the cylinder radius at different wavelengths is shown in fig. 3(b), it can be seen that the compensation phase satisfies the full-band coverage of 0-2 pi, the curve of the transmission coefficient variation with the cylinder radius at different wavelengths is shown in fig. 3(c), and the full-band coverage is greater than 0.9; the period P corresponding to the unit structure (2) is 195nm, the height H is 1400nm, the radius variation range is 25nm-75nm, the curve of the phase variation with the cylinder radius at different wavelengths is shown in fig. 3(e), it can be seen that the phase satisfies the full band coverage of 0-2 pi, the curve of the transmission coefficient variation with the cylinder radius at different wavelengths is shown in fig. 3(f), and the full band coverage is greater than 0.9.
2. Supertexture lens sample library generation
The two-dimensional and three-dimensional schematic diagrams of the broadband achromatic super-structure lens group are respectively shown in fig. 2(c) and 2(d), wherein the refractive index n of fig. 2(c)1、n2、n3、n4、n5The refractive index of the material representing the dielectric layer can be freely selected according to the construction requirement. The yellow box represents a medium cylinder, and the medium cylinder is contained in n1、n2In the dielectric layer. Refractive index n of five medium layers1、n2、n3、n4、n5The free combination can construct various forms of super-structure lens combination. For example, n is selected2、n5Is air, n1、n3、n4As medium Al2O3Layer of n such that1The GaN cylinder as the medium is wrapped in Al as the medium2O3In a layer of n2Exposing the dielectric GaN cylinder to air, and selecting n1The cylinder of the medium represents a confocal super-structured lens, n2The medium cylinder at the position represents a divergent super-structure lens to form a double-layer super-structure lens group. N can also be selected1-n5Are all media, and since focusing is ultimately to be achieved in air, n can be specified5Has a certain thickness, so that the surfaces of the two layers of super structures are embedded in the medium Al by adopting the medium GaN cylinder2O3Structure in a layer. Aiming at the two different combination modes, the structure of the focusing super-structure lens is not changed, and the two structural forms of the diverging super-structure lens are adopted, so that three sample libraries are needed, Group I is the focusing super-structure lens sample library, Group II is the diverging super-structure lens sample library formed by the units of the medium GaN cylinder exposed in the air, and Group III is also the diverging super-structure lens sample library.
Generating a focusing super-structure lens sample library Group I: the units are selected as shown in fig. 3(d), the phase compensation formula (1) can be obtained by derivation according to the schematic diagram of the phase compensation principle of the focusing super-structure lens shown in fig. 2(a), the phase compensation of the focusing super-structure lens is calculated, then the sizes of the units at different positions of the focusing super-structure lens are obtained by interpolation values in the phase curves 3(b) of the corresponding units, and the array arrangement is completed (the selected central wavelength is 550 nm). Focal length range is 8um-30um, one convergent superstructured lens is taken every 1 μm, array diameter D is 19.5 μm, and each convergent superstructureFull-wave simulation is carried out on the surface to obtain corresponding dispersion characteristics, and the Abelian constant V of each focusing super-structure lens is calculated by using a formula (6)11,V12,V13,…,V1NAnd generating a confocal super-structure lens sample library Group I.
Generating a divergent super-structure lens sample library Group II: the units are units shown in fig. 3(a), the phase compensation formula (7) can be obtained by derivation according to the schematic diagram of the phase compensation principle of the divergent super-structure lens shown in fig. 2(b), the phase compensation of the divergent super-structure lens is calculated, then the sizes of the units at different positions of the divergent super-structure lens are obtained by interpolation values in the phase curve 3(b) of the corresponding units, and the central wavelength is selected to be 550nm to complete array arrangement. Focal length range-60 mu m and f2And (2) taking a divergent super-structure lens every 1 mu m with the array diameter D being 19.5 mu m, carrying out full-wave simulation on each divergent super-structure lens to obtain the corresponding dispersion characteristic, and calculating the Abel constant V of each focusing super-structure lens by using a formula (11)21,V22,V23,…,V2NAnd generating a focusing hyper-structural lens sample library Group II.
Generating a divergent super-structure lens sample library Group III: the units are units shown in fig. 3(d), the phase compensation formula (7) can be obtained by derivation according to the schematic diagram of the phase compensation principle of the divergent super-structure lens shown in fig. 2(b), the phase compensation of the divergent super-structure lens is calculated, then the sizes of the units at different positions of the divergent super-structure lens are obtained by interpolation values in the phase curves 3(e) of the corresponding units, and the central wavelength is selected to be 550nm to complete array arrangement. Focal length range-60 mu m and f2And (3) taking one divergence super-structure lens every 1 mu m, wherein the array diameter D is 19.5 mu m, carrying out full-wave simulation on each divergence super-structure lens to obtain the corresponding dispersion characteristic, and calculating the Abel constant V 'of each focusing super-structure lens by using a formula (11)'21,V′22,V′23,…,V′2NAnd generating a confocal super-structure lens sample library Group III.
Example 2
Ideal focal length f of double-layer achromatic lens group to be constructedg36 μm. Selecting focusing super-structure lens sample library Group I and diverging super-structureAnd determining the distance change range between the optimized focusing super-structure lens and the diverging super-structure lens to be d which is more than or equal to 0.5 mu m and less than or equal to 5 mu m by the lens sample library Group II, and then solving the optimal value of the lens Group meeting the achromatic relation according to the step 4, namely the combination of the optimal focusing super-structure lens and the diverging super-structure lens meeting the achromatic dispersion.
Programming to complete the optimization solving process of the formula (14) and finally obtaining the focal length f of the focusing super-structure lens115.6 μm, focal length f of the diverging metamaterial lens2-26.7 μm, the distance d between the diverging and focusing lenses being 2 μm. And then combining the focusing super-structure lens and the diverging super-structure lens to form a double-layer broadband achromatic super-structure lens group, and calculating parameters such as dispersion characteristics, focusing efficiency and the like of the super-structure lens group by full-wave simulation. In addition, f is also requiredgIn contrast to a 36 μm single layer dispersive super lens, the focal length pair of single layer dispersive super lens and double layer achromatic super lens set is shown in fig. 4(a), and it can be seen that the double layer achromatic super lens set satisfies the requirement in the range of 460nm to 700nm
Figure BDA0002129059360000141
Namely, the focal length is not greatly changed along with the wavelength, and the chromatic dispersion elimination phenomenon is obvious. In addition, the focusing efficiency and half-power beam width of the double-layer achromatic super-structure lens group and the single-layer dispersive super-structure lens are compared, as shown in fig. 4(b) and 4(c), the focusing efficiency of the double-layer achromatic super-structure lens group can exceed 50% in the range of 460 plus 700nm, and can even reach 60% in a part of frequency bands. Fig. 4(d) shows the focusing effect of the single-layer chromatic dispersion metamaterial lens at different wavelengths, and the focusing effect can be seen as the focal length is reduced along with the increase of the wavelength and the chromatic dispersion is stronger, while fig. 4(e) shows the focusing effect of the double-layer achromatic metamaterial lens group at different wavelengths, the focal length is independent of the wavelength, and the chromatic dispersion is eliminated.
Example 3
Ideal focal length f of double-layer achromatic lens group to be constructedg36 μm. Selecting a focusing super-structure lens sample library Group I and a diverging super-structure lens sample library Group III, and determining that the distance variation range between the optimized focusing super-structure lens and the diverging super-structure lens is 0.5 muD is more than or equal to m and less than or equal to 5 mu m, and then the optimal value of the lens group meeting the achromatic relation is solved according to the step 4, namely the combination of the optimal focusing super-structure lens and the diverging super-structure lens meeting the achromatic color.
Programming to complete the optimization solving process of the formula (14) and finally obtaining the focal length f of the focusing super-structure lens115.6 μm, focal length f of the diverging metamaterial lens226.7 μm, the distance d between the diverging and focusing lenses is 2 μm, n1-n5The dielectric layer is made of Al2O3，n5The thickness of the dielectric layer was 1 μm. And then combining the focusing super-structure lens and the diverging super-structure lens to form a double-layer broadband achromatic super-structure lens group, and calculating parameters such as dispersion characteristics, focusing efficiency and the like of the super-structure lens group by full-wave simulation. In addition, f is also requiredgIn contrast to the original focusing super lens of 36 μm, the focal length pair of the original focusing super lens and the double-layer achromatic super lens group is shown in fig. 5(a), and it can be seen that the double-layer achromatic super lens group satisfies the requirement in the range of 400nm to 700nm
Figure BDA0002129059360000142
Namely, the focal length does not change along with the side length, is approximately a certain value, and has obvious chromatic dispersion elimination phenomenon. In addition, the focusing efficiency and half-power beam width of the double-layer achromatic super-structure lens group and the original focusing super-structure lens are compared, as shown in fig. 5(b) and 5(c), the average focusing efficiency of the double-layer achromatic super-structure lens group in the range of 400 and 700nm can exceed 50%, and even can reach 60% in a part of frequency bands. Fig. 5(d) shows the focusing effect of the double-layer achromatic super-lens set at different wavelengths, and it can be seen that the focal length is independent of the wavelength and the chromatic dispersion has been substantially eliminated.
Example 4
The construction scheme disclosed by the invention can construct double-layer super-structure lens groups with different NA, if the height and the material of a medium cylinder of the unit shown in figure 2(a) are changed, units with different dispersion characteristics can be obtained, the change range of the height is 450nm-900nm usually, the change range of the refractive index of the medium material is 2.56-4.0, and the principle of selecting the height of the cylinder and the refractive index of the material is to ensure the unitAt a wavelength of 700nm, the phase can cover 0-2 π, and generally if the refractive index of the dielectric cylinder is increased, the height is correspondingly decreased, and vice versa. Considering the ratio of the height and the refractive index of the material, the cell shown in fig. 2(a) can be expanded into cells of various forms, such as: 1) height H1900nm, refractive index n 2.56; 2) height H1600nm, refractive index n 3.5; 3) height H1500nm, refractive index n 3.75; 4) height H1450nm, and a refractive index n of 4. For the fixed focusing super-structure lens, if the units are respectively used for array arrangement, the super-structure lenses with different dispersion characteristics under the same focal length can be obtained, the principle is that dispersion cancellation is realized by utilizing the characteristic that the dispersion characteristics of the divergent super-structure lens and the focusing super-structure lens are opposite, and therefore, the super-structure lenses with different dispersion characteristics under the same focal length provide more choices for the whole construction process. By using these units to generate a divergent-metamaterial lens sample library, and then using the method of embodiment 2 or embodiment 3 to match the focusing and diverging-metamaterial lenses, different double-layer achromatic-metamaterial lens sets with NA from 0.18 to 0.53 can be constructed, the focal length and focusing efficiency of which are shown in fig. 6, and the construction parameters of the corresponding focusing and diverging-metamaterial lenses are shown in the following table:
TABLE 1 construction parameters of corresponding focusing and diverging super-constituent lenses in different NA double-layer achromatic super-constituent lens group
Figure BDA0002129059360000151
As shown in fig. 6(a), the variation curves of the focal length with the wavelength are given for NA 0.18, NA 0.26, NA 0.44 and NA 0.53, and it can be seen that the dispersion cancellation is better for NA 0.18, NA 0.26 and NA 0.44, and NA 0.53 is less effective, because NA 0.53 tends to be the limit of the construction method, and the wide band is significantly narrowed. Fig. 6(b) shows the corresponding focusing efficiencies under different NA conditions, because the focusing efficiencies are arranged according to the central frequency, the highest focusing efficiencies of different double-layer achromatic super-structure lens groups all appear near the central frequency, and the trend is decreasing toward high frequency and low frequency. The focusing efficiency corresponding to NA 0.53 is reduced rapidly towards two sides, the available range of bandwidth is further limited, and how to apply the method to construct the achromatic broadband high-efficiency double-layer super-structural lens group with NA 0.5 needs further exploration. The three NA contrasts shown in fig. 6(c) are that the focusing super-structure lens is unchanged, only the diverging super-structure lens is changed, and the coverage of NA from 0.24 to 0.35 is realized, that is, the method disclosed by the invention can construct a double-layer achromatic super-structure lens group with different NA under the condition of fixing the focusing super-structure lens and only changing the diverging super-structure lens, although the optimal construction scheme is not realized, that is, the working bandwidth is slightly narrowed, and it can be seen that NA equal to 0.24 and NA equal to 0.35 both fluctuate to some extent in the focal length of the high-frequency part, and the corresponding focusing efficiency pair is shown in fig. 6(d), for example.
The method for constructing the broadband achromatic super-structure lens group based on the multilayer super-structure surface uses the planar super-structure surface to replace a geometric curved surface to construct a novel planar electromagnetic wave device, reduces the processing complexity, improves the integration level and the reliability, breaks through the problems of strong dispersion, narrow frequency band, low efficiency and the like of a planar device based on a conventional single-layer super-structure surface, and can be widely applied to the fields of optical imaging, radar imaging, holographic imaging, photoelectric sensing and the like.

Claims (5)

1. A construction method of a broadband achromatic super-structure lens group based on a multilayer super-structure surface is characterized by comprising the following steps:
step 1, constructing a super unit, and simulating to obtain a transmission coefficient and a phase change curve of a uniform array corresponding to the super unit;
step 2, constructing a focusing super-structure lens, generating focusing super-structure lens samples, and calculating the Abelian constant of each focusing super-structure lens sample according to the dispersion characteristic;
step 3, constructing a divergent super-structure lens, generating divergent super-structure lens samples, and calculating the Abelian constant of each divergent super-structure lens sample according to the dispersion characteristics;
step 4, solving an optimization problem to obtain a combination of a focusing super-structure lens and a diverging super-structure lens with minimum chromatic dispersion;
step 5, obtaining the dispersion characteristic of the broadband achromatic super-structure lens group by adopting full-wave simulation, comparing the dispersion characteristic with a single-layer chromatic dispersion super-structure lens, and evaluating the chromatic aberration improvement condition of the broadband achromatic super-structure lens group;
constructing a super cell in the step 1, and obtaining a transmission coefficient and a phase change curve of a uniform array corresponding to the super cell through simulation, wherein the transmission coefficient and the phase change curve are as follows:
step 1.1, constructing a super unit, so that the corresponding uniform array of the super unit meets the conditions that the transmission coefficient is greater than 0.9 and the phase covers 0-2 pi in the whole wavelength range;
step 1.2, obtaining a transmission coefficient and a phase change curve of the uniform array corresponding to the superunit through simulation;
constructing a focusing super-structure lens in the step 2, generating focusing super-structure lens samples, and calculating the Abelian constant of each focusing super-structure lens sample according to the dispersion characteristic, wherein the Abelian constant is as follows:
step 2.1, constructing a focusing super-structure lens, and generating focusing super-structure lens samples with focal lengths f11,f12,f13,…,f1NWherein N is the total number of samples, and the compensation phase required by the focusing super-structure lens with different focal lengths is calculated according to the geometric optical path difference
Figure FDA0003148001520000011
Figure FDA0003148001520000012
Wherein r is the distance between any point on the focusing super-structure lens and the center of the focusing super-structure lens, and r is the distance between any point on the focusing super-structure lens and the center of the focusing super-structure lens under a rectangular coordinate system
Figure FDA0003148001520000013
For the compensation phase of the point correspondence, f1In order to focus the focal length of the super-structured lens, λ is the wavelength,
Figure FDA0003148001520000014
the initial phase of each unit of the focusing super-structure lens is also the compensation phase of the center of the focusing super-structure lens;
step 2.2, obtaining a compensation phase required by each unit on the focusing super-structure lens according to the compensation phase formula (1), interpolating from the phase characteristic curve to obtain units meeting conditions, and arranging the focusing super-structure lens meeting construction requirements;
the correspondence between the focal length of the thin lens in vacuum and its geometry is as follows:
Figure FDA0003148001520000015
where f is the focal length of the thin lens and n1Is the refractive index of the thin lens material, r1And r2The curvature radius of the spherical surfaces on the left side and the right side of the thin lens respectively;
step 2.3, performing full-wave simulation on the focusing super-structure lenses with different focal lengths respectively, and calculating a relation curve of the actual focal position of the focusing super-structure lens along with the change of the wavelength, namely a dispersion characteristic curve;
because the plane type focusing super-structure lens is approximately composed of a spherical surface and a plane r at the optical axis2A plano-convex thin lens constructed ∞, so the thin lens formula for a focusing super-structured lens is written as:
Figure FDA0003148001520000021
wherein f is1For focusing the focal length of the superstructural lens, p1Is a curvature of n1Is the equivalent refractive index of the focusing super-structured lens;
step 2.4, according to the dispersion characteristic curve obtained by full-wave simulation, the Abelian constant V of the focusing super-structure lens is solved1The formula is as follows:
Figure FDA0003148001520000022
wherein f is1C、f1D、f1FThe focal lengths of the focusing super-structure lens at low frequency, central frequency and high frequency are respectively obtained from the dispersion characteristic curve.
2. The method for constructing a broadband achromatic metamaterial lens group based on multilayer metamaterial surfaces as claimed in claim 1, wherein the abelian constant of each focusing metamaterial lens sample is calculated according to dispersion characteristics in step 2, specifically as follows:
obtaining the focal length f of the focusing super-structure lens according to the formula (3)1Curvature rho1With equivalent refractive index n1The relationship between them is as follows:
Figure FDA0003148001520000023
thereby obtaining the corresponding equivalent refractive index n of the focusing super-structure lens under three different frequencies1C、n1D、n1FComprises the following steps:
Figure FDA0003148001520000024
in a focusing super-structured lens, the source of chromatic dispersion and the equivalent refractive index are directly related together, and the curvatures corresponding to different frequencies are set to be equal, namely rho1C＝ρ1D＝ρ1F＝ρ1According to the definition of Abelian constant, the following results are obtained:
Figure FDA0003148001520000031
3. the method for constructing a broadband achromatic metamorphic lens group based on multilayer metamorphic surfaces as claimed in claim 1 or 2, wherein said constructing a diverging metamorphic lens in step 3 generates diverging metamorphic lens samples, and the abelian constant of each diverging metamorphic lens sample is calculated according to the dispersion characteristics as follows:
step 3.1, constructing a divergent super-structure lens to generate divergent super-structure lens samples with focal lengths f21,f22,f23,…,f2NWherein N is the total number of samples, and the compensation phase required by the divergent super-structure lens with different focal lengths is calculated according to the geometric optical path difference
Figure FDA0003148001520000032
Figure FDA0003148001520000033
Wherein r is the distance between any point on the diverging super-structure lens and the center of the diverging super-structure lens, and r is the distance between any point on the diverging super-structure lens and the center of the diverging super-structure lens under a rectangular coordinate system
Figure FDA0003148001520000034
For the compensation phase of the point correspondence, f2Is the focal length of the divergent super-structure lens, D is the diameter of the divergent super-structure lens array, lambda is the wavelength,
Figure FDA0003148001520000035
the initial phase of each unit of the divergent super-structure lens, namely the compensation phase corresponding to the edge super-unit of the divergent super-structure lens;
step 3.2, obtaining the compensation phase required by each unit on the divergent super-structure lens according to the compensation phase formula (7), interpolating from the phase characteristic curve to obtain the units meeting the conditions, and arranging the divergent super-structure lens meeting the construction requirements;
3.3, performing full-wave simulation on the divergent super-structure lenses with different focal lengths respectively, calculating electric field distribution at a wavelength above the divergent super-structure lens array, extracting phases of electric field components to obtain phase curves with different wavelengths, and fitting the phase curves to obtain a relation curve of the focal lengths of the divergent super-structure lenses changing along with the wavelength, namely a dispersion characteristic curve;
since the planar diverging nanostructured lens is approximated at the optical axis to be a plano-concave thin lens composed of one spherical surface and one plane surface, the thin lens formula of the diverging nanostructured lens is written as:
Figure FDA0003148001520000036
wherein f is2To diverge the focal length of the super-structured lens, p2Is a curvature of n2Is the equivalent refractive index of the diverging super-structured lens;
step 3.4, according to the dispersion characteristic curve obtained by full-wave simulation, the Abelian constant V of the divergent super-structure lens is solved2The formula is as follows:
Figure FDA0003148001520000041
wherein f is2C、f2D、f2FThe focal lengths of the divergent super-structure lens corresponding to the low frequency, the central frequency and the high frequency are respectively obtained from the dispersion characteristic curve.
4. The method for constructing a broadband achromatic metamaterial lens group based on multilayer metamaterial surfaces as claimed in claim 3, wherein the Abel constant of each divergent metamaterial lens sample is calculated according to dispersion characteristics in step 3, specifically as follows:
obtaining the focal length f of the divergent super-structured lens according to the formula (8)2Curvature rho2With equivalent refractive index n2The relationship between them is as follows:
Figure FDA0003148001520000042
thereby obtaining the equivalent refractive index n of the divergent super-structure lens corresponding to three different frequencies2C、n2D、n2FComprises the following steps:
Figure FDA0003148001520000043
in a diverging meta-lens, the source of dispersion and the equivalent refractive index are directly related together, and the curvatures corresponding to different frequencies are set to be equal, i.e. p2C＝ρ2D＝ρ2F＝ρ2According to the definition of Abelian constant, the following results are obtained:
Figure FDA0003148001520000044
5. the method for constructing a broadband achromatic metamaterial lens group based on multilayer metamaterial surfaces as claimed in claim 3, wherein the step 4 of solving the optimization problem to obtain the combination of the focusing and diverging metamaterial lenses with minimum chromatic dispersion is as follows:
the constraint condition for the optimization problem is the focal length formula of the thin lens group with the set distance:
Figure FDA0003148001520000045
wherein f isgIs the focal length of the super-constituent lens group, f1To focus the focal length of the superstructural lens, f2D is the distance between the focusing super-structure lens and the diverging super-structure lens;
the objective function of the optimization problem is:
Figure FDA0003148001520000051
the whole optimization problem thus becomes: selecting the focal length f of the focusing super-structure lens under the condition of satisfying the focal length formula (12) of the thin lens group1Divergent super-structured lens focusDistance f2And the distance d between the two, so that the value of the objective function (13) tends to zero;
the objective function of the optimization problem is derived by combining the thin lens group focal length formula (12) with the requirement of achromatism, and the formula (3) and the formula (8) are substituted into the formula (12) to obtain:
Figure FDA0003148001520000052
to achieve the elimination of chromatic aberration, the focal length fgMust be independent of wavelength, i.e.
Figure FDA0003148001520000053
The dispersion is due to the wavelength dependence of the equivalent refractive index, i.e. the equivalent refractive index n1、n2Both are functions of wavelength, so the left and right sides of equation (14) are separately biased for wavelength to obtain:
Figure FDA0003148001520000054
wherein the equivalent refractive index n1、n2The partial derivative with respect to wavelength is approximated by the dispersion characteristics, or the slope of the dispersion characteristic, at low and high frequencies, as shown in the following equation:
Figure FDA0003148001520000055
Figure FDA0003148001520000056
then substituting the formulas (15b) and (15c) into the formula (15a), and finishing to obtain:
Figure FDA0003148001520000057
introducing an Abelian constant:
Figure FDA0003148001520000058
combining equation (3) and equation (8), equation (16) is simplified as:
Figure FDA0003148001520000059
the left side of the equation is the objective function (13), even if the objective function (13) tends to zero.
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