CN110161679A - A kind of design method expanding diffraction image - Google Patents
A kind of design method expanding diffraction image Download PDFInfo
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- CN110161679A CN110161679A CN201910343399.6A CN201910343399A CN110161679A CN 110161679 A CN110161679 A CN 110161679A CN 201910343399 A CN201910343399 A CN 201910343399A CN 110161679 A CN110161679 A CN 110161679A
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- G—PHYSICS
- G02—OPTICS
- G02B—OPTICAL ELEMENTS, SYSTEMS OR APPARATUS
- G02B27/00—Optical systems or apparatus not provided for by any of the groups G02B1/00 - G02B26/00, G02B30/00
- G02B27/0012—Optical design, e.g. procedures, algorithms, optimisation routines
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- G—PHYSICS
- G02—OPTICS
- G02B—OPTICAL ELEMENTS, SYSTEMS OR APPARATUS
- G02B27/00—Optical systems or apparatus not provided for by any of the groups G02B1/00 - G02B26/00, G02B30/00
- G02B27/42—Diffraction optics, i.e. systems including a diffractive element being designed for providing a diffractive effect
- G02B27/4205—Diffraction optics, i.e. systems including a diffractive element being designed for providing a diffractive effect having a diffractive optical element [DOE] contributing to image formation, e.g. whereby modulation transfer function MTF or optical aberrations are relevant
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Abstract
The invention discloses a kind of design methods for the expansion diffraction image for belonging to optical design arts.This method is that a virtual median surface is arranged first, calculates the optical field distribution in output face in two steps；In input face, that is, diffraction optical element DOE: carrying out zero padding in the plane at place；A low-pass filter is arranged on median surface to be filtered, eliminates image aliasing；Then in conjunction with optimization algorithm, optimization DOE phase, designed phase of the phase in last negated zero padding region as DOE is finally reached the purpose of amplification diffraction image；The present invention can effectively expand the full-size of computable diffraction image in output face, and image aliasing does not occur.The fields such as this method can be used for laser manufacture, dot matrix generates, holographic two-dimensional/three-dimensional is shown, game, amusement.
Description
Technical field
The invention belongs to optical design arts, in particular to a kind of design method for expanding diffraction image.
Background technique
When irradiating diffraction optical element (DOE) with directional light, the discretization of diffraction optical element leads to spreading out in output face
Penetrate the size-constrained of image.How in the case where not reducing the diffraction optical element sampling interval and not generating image aliasing, obtain
It is always the problem of optical field to bigger diffraction image.
Assuming that with a branch of directional light irradiate diffraction optical element (DOE), for one-dimensional, light beam just passed through DOE it
Optical field distribution afterwards can use u (x0) indicate.It is discrete to the progress of DOE phase, then the light field can be further represented as
Here comb represents comb function；Rect represents rectangular function；x0It is the coordinate on input face；Δx0It is adopting for DOE
Sample interval；It * is convolution symbol；The output face of diffraction image is arranged in Fresnel region of diffraction, then the light field in output face point
Cloth can be expressed as
If enabled
Then have
Wherein, λ is the wavelength of incident light, and z is distance of the DOE to output face, and k=2 π/λ is wave vector,It represents in Fu
Leaf transformation；X is the coordinate in output face.Above-mentioned formula can be further represented as
Wherein, G (x/ λ z), comb (Δ x0X/ λ z), sinc (Δ x0X/ λ z) it is g (x respectively0), comb (x0/Δx0),
rect(x0/Δx0) by the form after Fourier transformation, fx=x/ λ z represents angular frequency in output face.
Fig. 1 (a) is the schematic diagram of arbitrary function C (x) G (x/ λ z)；Fig. 1 (b) is above-mentioned function and comb function comb
(Δx0X/ λ z) after convolution as a result, function C (x) G (x/ λ z) has been carried out cyclic extension as can be seen from this figure, and
And the period is λ z/ Δ x0；Fig. 1 (c) is the image of optical field distribution u (x) in output face, that is, with sinc (Δ x0X/ λ z) it is right
Function after above-mentioned cyclic extension carries out amplitude modulation again；In the case where directional light incidence, existing method is computable
The size of maximum diffraction image is L=λ z/ Δ x0.If from Fig. 1 (c) as can be seen that the size of calculated diffraction image
More than period λ z/ Δ x0, then the aliasing of image can occur, be restricted the size of diffraction image, therefore, it is necessary to not change
In the case where the DOE sampling interval, a bigger diffraction image is obtained.
Summary of the invention
The object of the present invention is to provide a kind of design methods for expanding diffraction image, which comprises the steps of:
A virtual median surface is arranged in step 1, calculates the optical field distribution in output face in two steps；
Step 2 in calculating process, carries out zero padding in the plane where input face, that is, DOE；
Step 3 is arranged a low-pass filter on median surface and is filtered to eliminate image aliasing；
Step 4, in conjunction with optimization algorithm, the phase of optimization DOE, design phase of the phase in negated zero padding region as DOE
Position can finally achieve the purpose that amplify diffraction image.
Two steps of the step 1 calculate the optical field distribution in output face, and for one-dimensional, concrete operations are:
The first step, the fresnel diffraction from DOE to median surface, the optical field distribution on in-between face are expressed as
Wherein, x0, x1Respectively represent the coordinate on input face and median surface, u0(x0) it is just by the light field after DOE
Distribution, z1It is distance of the input face to median surface, λ is the wavelength of incident parallel light, and k=2 π/λ is wave vector,It represents in Fu
Leaf transformation.In most cases, it is difficult to find out the analytic solutions of formula (1), therefore needs to carry out numerical value calculating.In this case,
Wushu (1) is just needed to be write as discrete form
Wherein, Δ x0It is the sampling interval of DOE, Δ x1It is the sampling interval on median surface, m0, m1It is section [- N/2, N/
2-1] in integer, the sampling number of input face and median surface when N is discrete computing.If with Fast Fourier Transform (FFT) to formula
(2) it is calculated, then should be met
In order to meet the requirement of nyquist sampling theorem, the distance z of plane where DOE to median surface1It should meet
Wherein, zT=N Δ x0 2/ λ is characteristic distance.
Second step, the fresnel diffraction from median surface to output face is similar with first step calculating, if in quick Fu
Leaf transformation calculates the optical field distribution in output face, then should meet
Wherein, Δ x is the sampling interval in output face, LzIt is the full-size of computable diffraction image in output face, z2
It is the distance from median surface to output face；By formula (3) and formula (5), can obtain
Wherein, z=z1+z2It is the distance from input face to output face.
Here the distance between median surface and DOE for being arranged should just be characteristic distance N Δ x0 2/ λ, then diffraction image is most
Large scale is
The step 2 carries out zero padding on the face where input face, that is, DOE；Around the plane where input face, that is, DOE
The periphery for representing the matrix of input face optical field distribution is filled 0 element, can reduced in this way by zero padding that is, when carrying out numerical value calculating
The sampling interval of median surface, so that the sample range i.e. full-size of diffraction image in output face be made to be amplified；Assuming that filling up
Input face sample range and ratio before after zero are α, and as α > 1, then the full-size of diffraction image is
Thus it can achieve the purpose that the size for adjusting diffraction image in output face by adjusting the value of α.
A low-pass filter is arranged on median surface to reach the mesh for eliminating image aliasing in the step 3.
The low-pass filter is exactly a rectangular diaphragm, and light hole, the shape of light hole are arranged among the diaphragm
For rectangular, rectangle, polygon or round；Shape, the size of its light hole should be determined according to the concrete condition of the DOE used, be made
To be preferred, it is N Δ x that light hole, which is selected as rectangular and side length,0。
The invention has the advantages that compared with the existing methods, can effectively expand computable in output face spread out
The full-size of image is penetrated, and image aliasing does not occur.
Detailed description of the invention
Fig. 1 is the schematic diagram of optical field distribution in output face.
Fig. 2 is the index path of this method.
Fig. 3 is desired distribution (10 × 10 dot matrix) schematic diagram in output face.
Fig. 4 is the dot matrix simulation result schematic diagram of the when side α=1.6 light hole.
Fig. 5 is desired distribution (annulus) schematic diagram in output face.
Fig. 6 is the computer artificial result schematic diagram of the annulus obtained with this method.
Fig. 7 is the experimental result real shooting photo figure of the annulus without low-pass filter filtering.
Fig. 8 is the experimental result real shooting photo figure by filtered annulus.
Specific embodiment
The index path of the shown this programme of Fig. 2,1 is directional light, and 2 be diffraction optical device (DOE), and 3 be low pass filtered
Wave device (position set by low-pass filtering is exactly median surface), 4 be output face.For one-dimensional, specific scheme is as follows:
Fresnel diffraction of the first step from DOE to median surface, the optical field distribution on median surface can be expressed as
Wherein, x0, x1Respectively represent the coordinate on input face and median surface, u0(x0) it is just by the light field after DOE
Distribution, z1It is distance of the input face to median surface, λ is the wavelength of incident parallel light, and k=2 π/λ is wave vector,It represents in Fu
Leaf transformation.In most cases, it is difficult to find out the analytic solutions of formula (1), therefore needs to carry out numerical value calculating.In this case,
Wushu (1) is just needed to be write as discrete form
Wherein, Δ x0It is the sampling interval of DOE, Δ x1It is the sampling interval on median surface, m0, m1It is section [- N/2, N/
2-1] in integer, the sampling number of input face and median surface when N is discrete computing.If with Fast Fourier Transform (FFT) to formula
(2) it is calculated, then should be met
In order to meet the requirement of nyquist sampling theorem, the distance z of plane where DOE to median surface1It should meet
Wherein, zT=N Δ x0 2/ λ is characteristic distance.
Second step is the fresnel diffraction from median surface to output face, similar with first step calculating, if with quick Fu
In leaf transformation calculate output face on optical field distribution, then should meet
Wherein, Δ x is the sampling interval in output face, LzIt is the full-size of computable diffraction image in output face, z2
It is the distance from median surface to output face；By formula (3) and formula (5), can obtain
Wherein, z=z1+z2It is the distance from input face to output face.
Here the distance between median surface and DOE for being arranged should just be characteristic distance N Δ x0 2/ λ, then diffraction image is most
Large scale is
Step 2 is needed since first step itself can not achieve the purpose that increase diffraction image, therefore in calculating process
By input face (plane where namely DOE) surrounding zero padding, the sampling interval of median surface can be reduced in this way, to make to export
Sample range (full-size of diffraction image) amplification on face.Assuming that filling up the input face sample range after zero and ratio before
Example is α (α > 1), then the full-size of diffraction image is
Thus it can achieve the purpose that the size for adjusting diffraction image in output face by adjusting the value of α.
A low-pass filter is arranged on median surface to achieve the purpose that eliminate image aliasing in step 3.Here low pass
Low-pass filter described in filter is exactly a rectangular diaphragm, and light hole (as shown in Figure 3) is arranged among the diaphragm, light passing
The shape and size needs in hole are adjusted according to specific design requirement, and the shape of light hole illustrated herein is rectangular
(as shown in Figure 4), and side length is N Δ x0, but light hole may be rectangle, circle, polygon etc. in practical application.
Step 4, in conjunction with optimization algorithm, can with optimization DOE phase, at this point, obtained phase be α N × α N square
Battle array takes the phase distribution of the DOE of the matrix of intermediate N × N out as final design finally can according to optical path shown in Fig. 2
Achieve the purpose that amplify diffraction image.
Emulation and experimental verification are carried out to the above method, it is assumed that λ=633nm, N=1080, Δ x0=8 μm, z1=N Δ
x0 2/ λ ≈ 110mm, z2=330mm.Fig. 3 is the desired distribution (10 × 10 dot matrix) in output face, and Fig. 4 is the Shi Fangtong of α=1.6
The dot matrix simulation result of unthreaded hole, Fig. 5 are the desired distribution (annulus) in output face, the emulation knot of annulus when Fig. 6 is α=1.6
Fruit, the experimental result of the annulus without low-pass filter filtering when Fig. 7 is α=1.6, Fig. 8 filtered circle when being α=1.6
The experimental result of ring.The size for the diffraction image that Computer Simulation obtains is 41.5mm × 41.5mm, the diffraction pattern tested
The size of picture is 42.0mm × 42.0mm, illustrates that experimental result is consistent with simulation result.In identical parameter and do not occur
In the case where image aliasing, the size for the maximum diffraction image that other methods can generate is 34.8mm × 34.8mm.When α=
When 1.6, dimension enlargement 19.3% of this method relative to other methods maximum diffraction images.It can lead in practical applications
The value for crossing adjustment α achievees the purpose that the size for adjusting diffraction image in output face.But the size of maximum diffraction image
It to be limited by sinc function envelope shown in Fig. 1 (c).
Claims (5)
1. a kind of design method for expanding diffraction image, which comprises the steps of:
A virtual median surface is arranged in step 1, calculates the optical field distribution in output face in two steps；
Step 2 in calculating process, carries out zero padding in the plane where input face, that is, DOE；
Step 3 is arranged a low-pass filter on median surface and is filtered to eliminate image aliasing；
Step 4, in conjunction with optimization algorithm, optimization DOE phase, designed phase of the phase in negated zero padding region as DOE, most
It can achieve the purpose that amplify diffraction image eventually.
2. a kind of design method for expanding diffraction image according to claim 1, which is characterized in that two steps of the step 1
The optical field distribution in output face is calculated, for one-dimensional, concrete operations are:
The first step, the fresnel diffraction from DOE to median surface, the optical field distribution on in-between face can be expressed as
Wherein, x0, x1Respectively represent the coordinate on input face and median surface, u0(x0) it is the optical field distribution just passed through after DOE,
z1It is distance of the input face to median surface, λ is the wavelength of incident parallel light, and k=2 π/λ is wave vector,Represent Fourier's change
It changes；In most cases, it is difficult to find out the analytic solutions of formula (1), therefore needs to carry out numerical value calculating, in this case, it is necessary to
Wushu (1) is write as discrete form
Wherein, Δ x0It is the sampling interval of DOE, Δ x1It is the sampling interval on median surface, m0, m1It is in section [- N/2, N/2-1]
Integer, the sampling number of input face and median surface when N is discrete computing；If with Fast Fourier Transform (FFT) to formula (2) into
Row calculates, then should meet
In order to meet the requirement of nyquist sampling theorem, the distance z of plane where DOE to median surface1It should meet
Wherein, zT=N Δ x0 2/ λ is characteristic distance；
Second step is the fresnel diffraction from median surface to output face, similar with first step calculating, if with fast Fourier
Optical field distribution in transformation calculations output face should then meet
Wherein, Δ x is the sampling interval in output face, LzIt is the full-size of computable diffraction image in output face, z2Be from
Distance of the median surface to output face；By formula (3) and formula (5), can obtain
Wherein, z=z1+z2It is the distance from input face to output face；
Here the distance between median surface and DOE for being arranged should just be characteristic distance N Δ x0 2/ λ, the then maximum ruler of diffraction image
It is very little to be
3. a kind of design method for expanding diffraction image according to claim 1, which is characterized in that the step 2 is inputting
Zero padding is carried out on face where face, that is, DOE；By zero padding around the plane where input face, that is, DOE, i.e., when carrying out numerical value calculating,
0 element is filled on the periphery for representing the matrix of input face optical field distribution；The sampling interval that median surface can be reduced in this way, to make
Sample range, that is, diffraction image full-size in output face is amplified；Assuming that input face sample range after zero padding with before
Ratio be α, α > 1, then the full-size of diffraction image be
Thus it can achieve the purpose that the size for adjusting diffraction image in output face by adjusting the value of α.
4. a kind of design method for expanding diffraction image according to claim 1, which is characterized in that the step 3 is in centre
One low-pass filter is set on face to reach the mesh for eliminating image aliasing.
5. a kind of design method for expanding diffraction image according to claim 4, which is characterized in that the low-pass filter is just
It is a rectangular diaphragm, and light hole is set among the diaphragm, the shape of light hole is rectangular, rectangle, polygon or round；
Shape, the size of its light hole should be determined according to the concrete condition of the DOE used, when light hole is selected as rectangular, light hole
Side length is N Δ x0。
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| CN114137645A (en) * | 2021-12-03 | 2022-03-04 | 浙江水晶光电科技股份有限公司 | Diffractive optical element, preparation method thereof and design method of master diffraction pattern |
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| Publication number | Priority date | Publication date | Assignee | Title |
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| CN114137645A (en) * | 2021-12-03 | 2022-03-04 | 浙江水晶光电科技股份有限公司 | Diffractive optical element, preparation method thereof and design method of master diffraction pattern |
| CN114137645B (en) * | 2021-12-03 | 2023-10-24 | 浙江水晶光电科技股份有限公司 | Diffraction optical element, preparation method thereof and design method of master plate diffraction pattern |
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