CN106199957B - A kind of design method of concavees lens - Google Patents
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Abstract
The present invention relates to a kind of design methods of concavees lens, former concavees lens with conventional structure are designed to new concavees lens by this method, the new concavees lens are connected in sequence by three layers of Meta Materials, two end faces of new concavees lens are convex surface or plane, new concavees lens and former concavees lens have equivalent optical effect, method is the following steps are included: S01, the space reflection relationship of determining original concavees lens to new concavees lens；S02 chooses the coordinate transform function for being able to achieve above-mentioned spatial alternation；S03 calculates the material parameter of arbitrary structures concavees lens each section using transform optics principle.Compared with prior art, the present invention, which designs one kind, not only has non-concave structure, but also the new concavees lens with concavees lens optical characteristics, design method simple possible, the new concavees lens flexible structure designed is easily achieved, is easily installed, debugs and integrates, to reduce the processing cost of concavees lens.
Description
Technical field
The present invention relates to a kind of design methods of concavees lens, more particularly, to a kind of design of concavees lens based on Meta Materials
Method.
Background technique
Lens are basic electromagnetism, optical device, and concavees lens are one of important kinds.Due to concavees lens diverging and
Imaging characteristic, it is widely used in all kinds of imaging devices and system.
The diverging and imaging of concavees lens mainly generate what refraction was formed in lens surface by light.By geometric optics it is found that
They are showed themselves in that
The light for being parallel to primary optical axis will dissipate after concavees lens, and the reverse extending line along divergent rays is in throw light
The same side meets at virtual focus F '；Original by the light of lens virtual focus (focus with beam incident optical heteropleural), after refraction with key light
Axis is parallel；It is constant by concavees lens rear direction by the light of optical center.
If object is u at a distance from concavees lens, as being set as v at a distance from concavees lens.Focal length is f.
When object is material object, u>2f, v<f are ipsilateral at upright, the diminution virtual image in object；U=2f, v < f object it is ipsilateral at
Upright, the diminution virtual image；F <u < 2f, v < f are ipsilateral at upright, the diminution virtual image in object；U=f, v < f are ipsilateral at upright, contracting in object
The small virtual image；U < f, v < f are ipsilateral at upright, the diminution virtual image in object.
When object is virtual object, and the distance of concavees lens to virtual object is within one times of focal length (referring to absolute value), at upright, amplification
Real image, as with object lens ipsilateral (u < f)；When the distance of concavees lens to virtual object is one times of focal length (referring to absolute value), imaging
In infinity (u=f)；The distance of concavees lens to virtual object is when (referring both to absolute value) within two focus length other than one times of focal length, at
Stand upside down, amplification the virtual image, as with object lens heteropleural (f <u < 2f)；The distance of concavees lens to virtual object is that two focus length (refers to absolutely
To value) when, at an equal amount of virtual image of object, as with object lens heteropleural (u=2f)；The distance of concavees lens to virtual object is
When (referring to absolute value) other than two focus length, at stand upside down, reduce the virtual image, as with object lens heteropleural (u > 2f).
Concavees lens make them have different characteristics from the difference of convex lens and planar lens in structure, thus also have
There is respectively different applications.Spectacles eyeglass, telescope, opal peephole etc. are typical case of the concavees lens in life.
It needs to have it in use using the optical characteristics but its single concave structure of concavees lens in some applications many
Inconvenience, at this moment design surface texture needed for both having specific occasion has now but also with the lens of conventional concavees lens optical characteristics
Net price value.However from the point of view of the production angle of conventional lenses, effective design method is not found still at present.
Summary of the invention
It is an object of the present invention to overcome the above-mentioned drawbacks of the prior art and provide a kind of gained concavees lens both
With any required structure but also with the design method of the concavees lens based on Meta Materials of conventional concavees lens optical characteristics.
The purpose of the present invention can be achieved through the following technical solutions:
A kind of design method of the concavees lens based on Meta Materials, comprising the following steps:
S01, according to the space reflection relationship of required design concavees lens structure determination original concavees lens to new concavees lens.Specifically
Are as follows: former concave lens surface is set as surface C KD, GLH as shown in Fig. 1, and the space that it is occupied is CKDHLG, with s (x, y, z) table
Show；The surface of required design concavees lens is arbitrary surface (containing plane, as surface A B in attached drawing 1, IJ are indicated)；On the right side of concavees lens certainly
It is ABDKC by space, uses sr(x, y, z) is indicated；Free space is IJHLG on the left of concavees lens, uses sl(x, y, z) is indicated.Pass through sky
Between convert the left and right concave surface CKD of former concavees lens, GLH be mapped to plane GH and CD respectively, then the left side space of former concavees lens
GEFHLG is extended to GEFHG, and right side space ECKDFE is extended to ECDFE.While former concavees lens are extended, lens
The free space of two sides is compressed, i.e., free space s on the right side of former concavees lensr(x, y, z) is compressed to sr' (x, y, z) (in attached drawing 1
ABDC is indicated), former concavees lens left side free space sl(x, y, z) is compressed to sl' (x, y, z) (IJHG is indicated in attached drawing 1).From
Vacuum is referred to by Spatial General 6 R, is herein air space.
S02 chooses the coordinate transform function for being able to achieve above-mentioned spatial alternation, specifically:
The coordinate transform along primary optical axis (being x-axis in attached drawing 1) is taken, due to the symmetry in left and right sides space, this sentences the right side
Side spatial alternation is illustrated.The coordinate transform that the right side space ECKDFE of former concavees lens is extended to ECDFE indicates are as follows:
X'=f1(x), y'=y, z'=z (1)
The boundary condition that need to meet are as follows:
The transformation that free space ABDKC is compressed to ABDC on the right side of former concavees lens is expressed as:
X'=f2(x), y'=y, z'=z (3)
The boundary condition that need to meet is
Wherein, (x, y, z) indicates that the coordinate in former space, (x', y', z') indicate the coordinate in space after transformation；x1, x2It is flat
The abscissa of face EF and CD；f1, f2For the arbitrary function for meeting boundary condition；Γ1, Γ2, Γ3Indicate boundary EF, CKD, AB.
If f1, f2Using linear coordinate transformation function, then formula (1) is represented by
Formula (3) is represented by
Wherein xΓ2, xΓ3For boundary Γ2, Γ3On abscissa.The coordinate transform of leftward space can be obtained by symmetry.
f1, f2Nonlinear coordinate transformation function can also be used.
S03 calculates the material parameter of arbitrary structures concavees lens each section using transform optics principle.Boundary condition will be met
Formula (1), formula (2) substitutes into following calculation formula, calculates the relative dielectric constant ε ' and relative permeability μ ' of each section material:
ε '=A ε AT/ det (A), μ '=A μ AT/det(A) (7)
A is Jacobi transformation matrix in formula, and ε and μ are respectively the relative dielectric constant and relative permeability in former space.
Light can be reflected in arbitrary structures concave lens surface and lens interior, and light propagation path may occur curved
Folding；The propagation path that light is pierced by propagation path arrival same position after light passes through conventional concavees lens after lens is identical.
Therefore from outer, light is seemingly that have passed through a conventional concavees lens, i.e. designed arbitrary structures lens and often
It is equivalent to advise concavees lens.
Illustrate so that parallel rays irradiates concavees lens as an example.For conventional concavees lens (imaginary curve indicates in Fig. 2), put down
The incident light of row occurs to reflect and dissipate on two surfaces of concavees lens, and the reverse extending line of divergent rays meets at focus F ',
Its path is as shown in straight dashed line with the arrow in Fig. 2.When parallel rays irradiates the arbitrary structures concavees lens in the design, in first layer
In medium IJHG, the light direction of propagation is constant, still in the horizontal direction.Light is in the interface HG for reaching dielectric layer IJHG and GHDC
Place is since refraction effect, the direction of propagation are bent, and light can be biased to primary optical axis and be no longer along original level side at this time
To propagation.Continue to bend at the interface DC of light arrival dielectric layer GHDC and ABDC, can slightly deviate primary optical axis side at this time
To.When light propagates at lens outer boundary AB, bending occurs again for the light direction of propagation slightly toward key light axis direction, rolls over
Direction and the light for penetrating light (being pierced by the light of lens) are identical across the direction of propagation that conventional concavees lens reach at the position,
And then the reverse extending line of light is still assembled in conventional concavees lens focal point.Propagation path of the light in arbitrary structures concavees lens
As with the arrow shown in solid in Fig. 2.Therefore, from outer lens and have not been changed the propagation path of light, it and it is conventional
Concavees lens are equivalent to the effect of light.
Compared with prior art, the invention has the following advantages that
(1) prior art blank is compensated for, designs and unconventional concave lens shape is presented but also with routine in a kind of shape
" concavees lens " of concavees lens optical characteristics, designed " concavees lens " both ends end face can be plane or any convex surface so that
Concavees lens can be processed according to existing process equipment, flexible structure, be suitable for microwave, Terahertz and optics frequency range.With it is common
Concavees lens are compared, the concavees lens using flexible of arbitrary structures, with strong points, at low cost, long service life.
(2) can be used for and spherical surface concavees lens, ellipsoid concavees lens, paraboloid of revolution concavees lens or hyperboloid of revolution concavees lens
Equivalent arbitrary structures concavees lens, it is applied widely.
(3) design method simple possible, it is easy to accomplish.
Detailed description of the invention
Fig. 1 is the spatial alternation schematic diagram of arbitrary structures concavees lens of the present invention；
Fig. 2 is the propagation path schematic diagram that parallel rays passes through arbitrary structures concavees lens；
Average energy distribution map when Fig. 3 is horizontal beam irradiated plane structure concavees lens；
Fig. 4 is average energy distribution map when horizontal beam irradiates conventional structure concavees lens；
Fig. 5 is average energy distribution map when horizontal beam irradiates spherical structure concavees lens；
Fig. 6 is distribution map of three kinds of lens midfields along optical axis.
Specific embodiment
The present invention is described in detail with specific embodiment below in conjunction with the accompanying drawings.The present embodiment is with technical solution of the present invention
Premised on implemented, the detailed implementation method and specific operation process are given, but protection scope of the present invention is not limited to
Following embodiments.
Embodiment 1
A kind of design method of concavees lens, for former concavees lens to be converted to arbitrary structures concavees lens, the original is recessed
The concave surface of mirror is spherical surface, ellipsoid, the paraboloid of revolution or the hyperboloid of revolution.Former concavees lens are converted into planar junction in the present embodiment
Structure lens, i.e. AB, IJ are planar structure, at this time xΓ3For the abscissa x of plane AB3.Design method the following steps are included:
It (a) is spherical surface (x-c) for concave surface2+y2+z2=a2Concavees lens,Then formula (5) has
Body is taken as:
Formula (6) is specifically taken as:
In formula (6), formula (7), c is the centre of sphere x coordinate of sphere where spherical surface, and a is the radius of sphere where spherical surface.
It (b) is ellipsoid (x-c) for concave surface2/a2+(y2+z2)/b2=1 concavees lens,Then formula (5) is specifically taken as:
Formula (6) is specifically taken as:
In formula (8), formula (9), c is the centre of sphere x coordinate of spheroid where ellipsoid, and a, b are respectively ellipsoid where ellipsoid
Half axial length of x-axis, half axial length of y-axis of body.
It (c) is paraboloid of revolution 2p (x-c)=(y for concave surface2+z2) concavees lens, xΓ2=c+ (y2+z2)/(2p), then
Formula (5) is specifically taken as:
Formula (6) is specifically taken as:
In formula (10), formula (11), c is the vertex x coordinate of the paraboloid of revolution, and p is the focal length of the paraboloid of revolution.
It (d) is the hyperboloid of revolution (x-c) for concave surface2/a2-(y2+z2)/b2=1 concavees lens,Then formula (5) is specifically taken as:
Formula (6) is specifically taken as:
In formula (12), formula (13), a, b are respectively half axial length of real axis in standard Hyperbolic Equation, the imaginary axis, and c is hyperboloid
Center x coordinate.
Above coordinate transform formula substitution formula (7) can be calculated separately into out the material parameter in corresponding CDFE and ABDC.
Coordinate transform and the calculating of material parameter and the calculating of right half part for former concavees lens left-half is symmetrical.
It is below that a specific planar structure puts down-concavees lens (former concavees lens concave surface is spherical surface) simulating, verifying.Lens are set
Count parameter are as follows: a=0.18m, c=0.19m, x1=0m, x2=0.04m, x3=0.065m.The Gaussian beam of frequency f=12GHz
The average energy distribution in space is as shown in Figure 3 when horizontal irradiation planar lens from left to right.As seen from the figure, wave beam passes through planar junction
It is dissipated after structure concavees lens.In order to do intuitive comparison with conventional concavees lens, after Fig. 4 gives wave beam horizontal irradiation routine concavees lens
Energy distribution situation.By comparison as it can be seen that wave beam is by planar structure concavees lens designed in the present embodiment and by conventional
Concavees lens are equivalent.In order to more accurately compare two kinds of lens, Fig. 6 gives the quantitative distribution situation that optical axis is entered the court, wherein grey body
Heavy line represents conventional concavees lens, and black matrix zone circle solid line represents planar structure concavees lens designed in embodiment.Shown result
Show that two kinds of lens midfields are almost consistent along the distribution of primary optical axis.The above design embodiment and numerical experiment demonstrate design side
The correctness of method and design result.
Compared with conventional concavees lens, the concavees lens processing cost of planar structure is low.
Embodiment 2
A kind of design method of concavees lens, for former concavees lens to be converted to arbitrary structures concavees lens, the original is recessed
The concave surface of mirror is spherical surface, ellipsoid, the paraboloid of revolution or the hyperboloid of revolution.Former concavees lens are converted into spherical surface knot in the present embodiment
Structure lens, i.e. AB, IJ are spherical structure, if its spherical equation is (x-x0)2+y2+z2=r0 2, wherein x0The sphere where spherical surface
Centre of sphere x coordinate, r0The radius of sphere where spherical surface.X at this timeΓ3For the abscissa put on spherical surface, it is represented byDesign method the following steps are included:
It (a) is spherical surface (x-c) for concave surface2+y2+z2=a2Concavees lens,Then formula (5) has
Body is taken as:
Formula (6) is specifically taken as:
In formula (14), formula (15), c is the centre of sphere x coordinate of sphere where spherical surface, and a is the radius of sphere where spherical surface.
It (b) is ellipsoid (x-c) for concave surface2/a2+(y2+z2)/b2=1 concavees lens,Then formula (5) is specifically taken as:
Formula (6) is specifically taken as:
In formula (16), formula (17), c is the centre of sphere x coordinate of spheroid where ellipsoid, and a, b are respectively that ellipsoid place is ellipse
Half axial length of x-axis, half axial length of y-axis of sphere.
It (c) is paraboloid of revolution 2p (x-c)=(y for concave surface2+z2) concavees lens, xΓ2=c+ (y2+z2)/(2p), then
Formula (5) is specifically taken as:
Formula (6) is specifically taken as:
In formula (18), formula (19), c is the vertex x coordinate of the paraboloid of revolution, and p is the focal length of the paraboloid of revolution.
It (d) is the hyperboloid of revolution (x-c) for concave surface2/a2-(y2+z2)/b2=1 concavees lens,Then formula (5) is specifically taken as:
Formula (6) is specifically taken as:
In formula (20), formula (21), a, b are respectively half axial length of real axis in standard Hyperbolic Equation, the imaginary axis, and c is hyperboloid
Center x coordinate.
Above coordinate transform formula substitution formula (5) can be calculated separately into out the material parameter in corresponding CDFE and ABDC.
Coordinate transform and the calculating of material parameter and the calculating of right half part for former concavees lens left-half is symmetrical.
It is below that a specific spherical structure puts down-concavees lens (former concavees lens concave surface is spherical surface) simulating, verifying.Lens are set
Count parameter are as follows: r0=a=0.18m, c=0.19m, x0=-0.085m, x1=0m, x2=0.04m.The Gauss of frequency f=12GHz
Wave beam from left to right horizontal irradiation spherical lens when space average energy distribution it is as shown in Figure 5.As seen from the figure, wave beam passes through ball
It is dissipated after the structure lens of face.The comparison of effect picture 4 with conventional concavees lens is as it can be seen that wave beam passes through spherical surface designed in the present embodiment
Structure lens and the conventional concavees lens of process are equivalent.In order to more accurately compare three kinds of lens, what Fig. 6 gave that optical axis enters the court is quantified
Distribution situation, wherein grey body heavy line represents conventional concavees lens, and black matrix zone circle solid line represents planar junction designed in embodiment 1
Structure concavees lens, black matrix represent spherical structure concavees lens designed in embodiment 2 with triangle solid line.It is shown the result shows that, three kinds
Lens midfield is almost consistent along the distribution of primary optical axis.The above design embodiment and numerical experiment demonstrate design method and design knot
The correctness of fruit.
Compared with conventional concavees lens, the concavees lens long service life of convex configuration, shock resistance is good.
Claims (10)
1. a kind of design method of concavees lens, which is characterized in that the former concavees lens with conventional structure are designed to newly by this method
Concavees lens, the new concavees lens are connected in sequence by three layers of Meta Materials, and two end faces of new concavees lens are convex surface or plane,
New concavees lens and former concavees lens have equivalent optical effect, the method the following steps are included:
S01, the space reflection relationship of determining original concavees lens to new concavees lens, specifically:
The primary optical axis of former concavees lens is set to x-axis, the space that former concavees lens occupy is s (x, y, z), including left side and right half
Portion, the left side free space of former concavees lens are sl(x, y, z), right side free space are sr(x, y, z), will be former by spatial alternation
The left and right concave surface of concavees lens is each mapped to plane, and space s (x, y, z) is made to be extended to s'(x, y, z), as new concavees lens
The thickness of middle layer, the left and right end surface shapes of new concavees lens, left layer and right layer designs according to demand, make on the right side of former concavees lens from
By space sr(x, y, z) is compressed to sr' (x, y, z), former concavees lens left side free space sl(x, y, z) is compressed to sl'(x,
y,z)；
S02 chooses the coordinate transform function for being able to achieve above-mentioned spatial alternation, specifically:
The coordinate transform along x-axis is taken, the coordinate transform in the right side space of former concavees lens indicates are as follows:
X'=f1(x), y'=y, z'=z (1)
The boundary condition that need to meet are as follows:
The coordinate transform of free space indicates on the right side of former concavees lens are as follows:
X'=f2(x), y'=y, z'=z (3)
The boundary condition that need to meet is
Wherein, (x, y, z) indicates that the coordinate in former space, (x', y', z') indicate the coordinate in space after transformation；x1For in former concavees lens
The x coordinate of heart point, x2The coordinate maximum value projected for the right concave surface of former concavees lens in x-axis；f1, f2To meet appointing for boundary condition
Meaning function；Γ1Indicate the boundary between former concavees lens left side and right side, Γ2Indicate the right margin of former concavees lens, Γ3It indicates
The right margin of new concavees lens；
The coordinate transform in the left side space of former concavees lens and the coordinate transform in right side space are symmetrical；
S03 calculates the material parameter of arbitrary structures concavees lens each section using transform optics principle.
2. a kind of design method of concavees lens according to claim 1, which is characterized in that in the step S02, f1, f2
Using linear coordinate transformation function, formula (1) is expressed as
Formula (3) is expressed as
Wherein xΓ2, xΓ3For boundary Γ2, Γ3On abscissa.
3. a kind of design method of concavees lens according to claim 2, which is characterized in that two of the new concavees lens
End face is plane, xΓ3For the abscissa x of right margin3, it is spherical surface (x-c) for concave surface2+y2+z2=a2Concavees lens,Formula (5) is specifically taken as:
Formula (6) is specifically taken as:
In formula (6), formula (7), c is the centre of sphere x coordinate of sphere where spherical surface, and a is the radius of sphere where spherical surface.
4. a kind of design method of concavees lens according to claim 2, which is characterized in that two of the new concavees lens
End face is plane, xΓ3For the abscissa x of right margin3, it is ellipsoid (x-c) for concave surface2/a2+(y2+z2)/b2=1 it is recessed
Mirror,Formula (5) is specifically taken as:
Formula (6) is specifically taken as:
In formula (8), formula (9), c is the centre of sphere x coordinate of spheroid where ellipsoid, and a, b are respectively the x of spheroid where ellipsoid
Half axial length of axis, half axial length of y-axis.
5. a kind of design method of concavees lens according to claim 2, which is characterized in that two of the new concavees lens
End face is plane, xΓ3For the abscissa x of right margin3, it is paraboloid of revolution 2p (x-c)=(y for concave surface2+z2) concavees lens,
xΓ2=c+ (y2+z2)/(2p), formula (5) is specifically taken as:
Formula (6) is specifically taken as:
In formula (10), formula (11), c is the vertex x coordinate of the paraboloid of revolution, and p is the focal length of the paraboloid of revolution.
6. a kind of design method of concavees lens according to claim 2, which is characterized in that two of the new concavees lens
End face is plane, xΓ3For the abscissa x of right margin3, it is the hyperboloid of revolution (x-c) for concave surface2/a2-(y2+z2)/b2=1
Concavees lens,Formula (5) is specifically taken as:
Formula (6) is specifically taken as:
In formula (12), formula (13), a, b are respectively half axial length of real axis in standard Hyperbolic Equation, the imaginary axis, and c is hyperboloid center x
Coordinate.
7. a kind of design method of concavees lens according to claim 2, which is characterized in that two of the new concavees lens
End face is spherical surface, and spherical equation is (x-x0)2+y2+z2=r0 2, wherein x0The centre of sphere x coordinate of sphere, r where spherical surface0For spherical surface
The radius of place sphere, at this time xΓ3For the abscissa put on spherical surface, it is expressed asIt is for concave surface
Spherical surface (x-c)2+y2+z2=a2Concavees lens,Formula (5) is specifically taken as:
Formula (6) is specifically taken as:
In formula (14), formula (15), c is the centre of sphere x coordinate of sphere where spherical surface, and a is the radius of sphere where spherical surface.
8. a kind of design method of concavees lens according to claim 2, which is characterized in that two of the new concavees lens
End face is spherical surface, and spherical equation is (x-x0)2+y2+z2=r0 2, wherein x0The centre of sphere x coordinate of sphere, r where spherical surface0For spherical surface
The radius of place sphere, at this time xΓ3For the abscissa put on spherical surface, it is expressed asIt is for concave surface
Ellipsoid (x-c)2/a2+(y2+z2)/b2=1 concavees lens,Formula (5) is specifically taken as:
Formula (6) is specifically taken as:
In formula (16), formula (17), c is the centre of sphere x coordinate of spheroid where ellipsoid, and a, b are respectively spheroid where ellipsoid
Half axial length of x-axis, half axial length of y-axis.
9. a kind of design method of concavees lens according to claim 2, which is characterized in that two of the new concavees lens
End face is spherical surface, and spherical equation is (x-x0)2+y2+z2=r0 2, wherein x0The centre of sphere x coordinate of sphere, r where spherical surface0For spherical surface
The radius of place sphere, at this time xΓ3For the abscissa put on spherical surface, it is expressed asIt is for concave surface
Paraboloid of revolution 2p (x-c)=(y2+z2) concavees lens, xΓ2=c+ (y2+z2)/(2p), formula (5) is specifically taken as:
Formula (6) is specifically taken as:
In formula (18), formula (19), c is the vertex x coordinate of the paraboloid of revolution, and p is the focal length of the paraboloid of revolution.
10. a kind of design method of concavees lens according to claim 2, which is characterized in that the two of the new concavees lens
A end face is spherical surface, and spherical equation is (x-x0)2+y2+z2=r0 2, wherein x0The centre of sphere x coordinate of sphere, r where spherical surface0For ball
The radius of sphere where face, at this time xΓ3For the abscissa put on spherical surface, it is expressed asFor concave surface
For the hyperboloid of revolution (x-c)2/a2-(y2+z2)/b2=1 concavees lens,Then formula (5)
Specifically it is taken as:
Formula (6) is specifically taken as:
In formula (20), formula (21), a, b are respectively half axial length of real axis in standard Hyperbolic Equation, the imaginary axis, and c is hyperboloid center x
Coordinate.
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| CN102460268A (en) * | 2009-05-14 | 2012-05-16 | 秦内蒂克有限公司 | Reflector assembly and beam forming |
| CN103558655A (en) * | 2013-11-18 | 2014-02-05 | 上海师范大学 | Design method of conical surface reflector of full-plane structure on basis of metamaterial |
| CN105305096A (en) * | 2015-09-28 | 2016-02-03 | 上海师范大学 | Design method of compact plane structure parabolic reflector antenna based on meta-material |
| CN105223689A (en) * | 2015-10-20 | 2016-01-06 | 上海师范大学 | A kind of method for designing of the whole plane structure convex lens based on Meta Materials |
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