CN110161679B - Design method for enlarging diffraction image - Google Patents

Design method for enlarging diffraction image Download PDF

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CN110161679B
CN110161679B CN201910343399.6A CN201910343399A CN110161679B CN 110161679 B CN110161679 B CN 110161679B CN 201910343399 A CN201910343399 A CN 201910343399A CN 110161679 B CN110161679 B CN 110161679B
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doe
diffraction image
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CN110161679A (en
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谭峭峰
孔哲
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Tsinghua University
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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
    • 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 design method for enlarging diffraction images, belonging to the field of optical design. Firstly, setting a virtual middle surface, and calculating the light field distribution on an output surface in two steps; at the input surface, i.e. the diffractive optical element DOE: zero padding is carried out on the plane where the optical fiber is located; a low-pass filter is arranged on the middle surface for filtering, and image aliasing is eliminated; then optimizing the DOE phase by combining an optimization algorithm, and finally taking the phase of the non-zero-filling region as the design phase of the DOE to finally achieve the purpose of amplifying the diffraction image; the invention can effectively enlarge the maximum size of the diffraction image which can be calculated on the output surface, and does not generate image aliasing. The method can be used in the fields of laser manufacturing, dot matrix generation, holographic two-dimensional/three-dimensional display, games, entertainment and the like.

Description

Design method for enlarging diffraction image
Technical Field
The invention belongs to the field of optical design, and particularly relates to a design method for enlarging a diffraction image.
Background
When a Diffractive Optical Element (DOE) is illuminated with parallel light, discretization of the diffractive optical element results in a limited size of the diffraction image on the output face. How to obtain a larger diffraction image without reducing the sampling interval of the diffraction optical element and generating image aliasing has been a difficult problem in the optical field.
Assuming that a Diffractive Optical Element (DOE) is illuminated with a beam of parallel light, the optical field distribution just after the beam has passed through the DOE can be given by u (x), for a one-dimensional example0) And (4) showing. By discretizing the DOE phase, the light field can then be further represented as
Figure BDA0002041462330000011
Where comb represents a comb function; rect represents a rectangular function; x is the number of0Are coordinates on the input surface; Δ x0Is the sampling interval of the DOE; is the convolution symbol; the output surface of the diffraction image is arranged in the Fresnel diffraction zone, and the light field distribution at the output surface can be expressed as
Figure BDA0002041462330000012
If order
Figure BDA0002041462330000013
Figure BDA0002041462330000014
Then there is
Figure BDA0002041462330000021
Where λ is the wavelength of the incident light, z is the distance from the DOE to the output face, k-2 pi/λ is the wave vector,
Figure BDA0002041462330000022
represents a fourier transform; x is the coordinate on the output face. The above formula can be further expressed as
Figure BDA0002041462330000023
Wherein G (x/λ z), comb (Δ x)0x/λz)，sinc(Δx0x/λ z) are each g (x)0)，comb(x0/Δx0)，rect(x0/Δx0) Form after Fourier transform, fxX/λ z represents the output face upper corner frequency.
FIG. 1(a) is a schematic diagram of an arbitrary function C (x) G (x/λ z); FIG. 1(b) shows the above function and the comb function comb (Δ x)0x/λ z) convolution, from which it can be seen that the function C (x) G (x/λ z) is extended periodically and with a period λ z/Δ x0(ii) a FIG. 1(c) is a diagram of the optical field distribution u (x) at the output face, i.e. using sinc (Δ x)0x/λ z) to perform amplitude modulation on the periodically extended function; in the case of parallel light incidence, the size of the maximum diffraction image that can be calculated by the conventional method is L ═ λ z/Δ x0. As can be seen from FIG. 1(c), if the size of the calculated diffraction image exceeds the period λ z/Δ x0Image aliasing occurs and the size of the diffraction image is limited, so that a larger diffraction image needs to be obtained without changing the DOE sampling interval.
Disclosure of Invention
The invention aims to provide a design method for enlarging diffraction images, which is characterized by comprising the following steps:
step one, setting a virtual middle surface, and calculating the light field distribution on an output surface in two steps;
in the calculation process, zero padding is carried out on an input surface, namely a plane where the DOE is located;
step three, a low-pass filter is arranged on the middle surface for filtering so as to eliminate image aliasing;
and step four, optimizing the phase of the DOE by combining an optimization algorithm, taking the phase of the non-zero-filling area as the design phase of the DOE, and finally achieving the purpose of amplifying the diffraction image.
The two steps of the first step calculate the light field distribution on the output surface, taking one dimension as an example, the specific operation is as follows:
first, Fresnel diffraction from the DOE to an intermediate surface, where the light field distribution over the intermediate surface is represented as
Figure BDA0002041462330000031
Wherein x is0，x1Representing coordinates on the input surface and the intermediate surface, u, respectively0(x0) Is the distribution of the optical field, z, just after passing through the DOE1Is the distance from the input surface to the intermediate surface, λ is the wavelength of the incident parallel light, k 2 pi/λ is the wave vector,
Figure BDA0002041462330000032
representing a fourier transform. In many cases, it is difficult to obtain an analytical solution of equation (1), and numerical calculation is necessary. In this case, it is necessary to write the formula (1) in a discrete form
Figure BDA0002041462330000033
Wherein, Δ x0Is the sampling interval, Δ x, of the DOE1Is the sampling interval on the middle plane, m0，m1Is the interval [ -N/2, N/2-1]And N is the number of sampling points of the input surface and the middle surface in discrete operation. If the equation (2) is calculated using fast Fourier transform, it should be satisfied
Figure BDA0002041462330000034
In order to satisfy the Nyquist sampling theorem, the distance z from the plane where the DOE is located to the middle plane1Should satisfy
Figure BDA0002041462330000035
Wherein z isT＝NΔx0 2And/λ is the characteristic distance.
In the second step, the Fresnel diffraction from the intermediate surface to the output surface is similar to the first calculation, and if the light field distribution on the output surface is calculated by using the fast Fourier transform, it is satisfied
Figure BDA0002041462330000041
Where Δ x is the sampling interval on the output face, LzIs the maximum size of the diffraction image, z, that can be calculated on the output face2Is the distance from the intermediate face to the output face; from the formulae (3) and (5), it is possible to obtain
Figure BDA0002041462330000042
Wherein z is1+z2Is the distance from the input face to the output face.
The distance between the intermediate surface and the DOE, which is provided here, is exactly the characteristic distance N Δ x0 2λ, the maximum size of the diffraction image is
Figure BDA0002041462330000043
Zero padding is carried out on the input surface, namely the surface where the DOE is located; zero padding is carried out on the periphery of an input surface, namely a plane where the DOE is located, namely when numerical calculation is carried out, 0 element is padded on the periphery of a matrix representing the optical field distribution of the input surface, so that the sampling interval of the middle surface can be reduced, and the sampling range on the output surface, namely the maximum size of a diffraction image, is enlarged; assuming that the ratio of the sampling range of the input surface after zero padding to the preceding is α, when α >1, the maximum size of the diffraction image is
Figure BDA0002041462330000044
Therefore, the purpose of adjusting the size of the diffraction image on the output surface can be achieved by adjusting the value of alpha.
And step three, arranging a low-pass filter on the middle surface to achieve the purpose of eliminating image aliasing.
The low-pass filter is a square diaphragm, a light through hole is arranged in the middle of the diaphragm, and the shape of the light through hole is square, rectangular, polygonal or circular; the shape and size of the light through hole are determined according to the specific conditions of the DOE, and preferably, the light through hole is square and has the side length of N delta x0。
The method has the advantages that compared with the existing method, the maximum size of the diffraction image which can be calculated on the output surface can be effectively enlarged, and image aliasing does not occur.
Drawings
FIG. 1 is a schematic representation of the optical field distribution at the output face.
Fig. 2 is a light path diagram of the method.
FIG. 3 is a schematic diagram of the desired distribution (10X 10 lattice) on the output face.
Fig. 4 is a schematic diagram of a simulation result of a lattice of square light-passing holes when α is 1.6.
FIG. 5 is a schematic diagram of the desired distribution (circles) on the output face.
FIG. 6 is a schematic diagram of a computer simulation result of a ring obtained by the method.
Fig. 7 is a photographic image of the experimental result of a circle without low pass filter filtering.
Fig. 8 is a photograph of the experimental result of the ring after filtering.
Detailed Description
Fig. 2 shows the optical path diagram of the present solution, where 1 is parallel light, 2 is a Diffractive Optical Element (DOE), 3 is a low-pass filter (the position where the low-pass filter is disposed is the middle plane), and 4 is an output plane. Taking one dimension as an example, the specific scheme is as follows:
the first step of Fresnel diffraction from the DOE to the intermediate surface, the distribution of the light field on the intermediate surface can be expressed as
Figure BDA0002041462330000051
Wherein x is0，x1Representing coordinates on the input surface and the intermediate surface, u, respectively0(x0) Is the distribution of the optical field, z, just after passing through the DOE1Is the distance from the input surface to the intermediate surface, λ is the wavelength of the incident parallel light, k 2 pi/λ is the wave vector,
Figure BDA0002041462330000052
representing a fourier transform. In many cases, it is difficult to obtain an analytical solution of equation (1), and numerical calculation is necessary. In this case, it is necessary to write the formula (1) in a discrete form
Figure BDA0002041462330000061
Wherein, Δ x0Is the sampling interval, Δ x, of the DOE1Is the sampling interval on the middle plane, m0，m1Is the interval [ -N/2, N/2-1]And N is the number of sampling points of the input surface and the middle surface in discrete operation. If the equation (2) is calculated using fast Fourier transform, it should be satisfied
Figure BDA0002041462330000062
In order to satisfy the Nyquist sampling theorem, the distance z from the plane where the DOE is located to the middle plane1Should satisfy
Figure BDA0002041462330000063
Wherein z isT＝NΔx0 2And/λ is the characteristic distance.
The second step is the Fresnel diffraction from the intermediate surface to the output surface, which, like the first calculation, is satisfied if the light field distribution on the output surface is calculated using the fast Fourier transform
Figure BDA0002041462330000064
Where Δ x is the sampling interval on the output face, LzIs the maximum size of the diffraction image, z, that can be calculated on the output face2Is the distance from the intermediate face to the output face; from the formulae (3) and (5), it is possible to obtain
Figure BDA0002041462330000065
Wherein z is1+z2Is the distance from the input face to the output face.
The distance between the intermediate surface and the DOE, which is provided here, is exactly the characteristic distance N Δ x0 2λ, the maximum size of the diffraction image is
Figure BDA0002041462330000066
In the second step, since the first step itself cannot achieve the purpose of enlarging the diffraction image, in the calculation process, zero padding is needed to be performed around the input surface (i.e., the plane where the DOE is located), so that the sampling interval of the middle surface can be reduced, and the sampling range (the maximum size of the diffraction image) on the output surface can be enlarged. Assuming that the ratio of the sampling range of the input surface after zero padding to the preceding is α (α >1), the maximum size of the diffraction image is
Figure BDA0002041462330000071
Therefore, the purpose of adjusting the size of the diffraction image on the output surface can be achieved by adjusting the value of alpha.
And step three, arranging a low-pass filter on the middle surface to achieve the purpose of eliminating image aliasing. The low-pass filter is a square diaphragm, a light through hole (as shown in fig. 3) is arranged in the middle of the diaphragm, the shape and the size of the light through hole need to be adjusted according to specific design requirements, the shape of the light through hole shown here is square (as shown in fig. 4), and the side length is N Δ x0However, in practical applications, the light passing hole may be rectangular, circular, polygonal, etc.
And step four, the DOE phase can be optimized by combining an optimization algorithm, at this time, the obtained phase is an alpha N multiplied by alpha N matrix, the middle N multiplied by N matrix is taken as the finally designed phase distribution of the DOE, and the purpose of amplifying the diffraction image can be finally achieved according to the light path shown in figure 2.
The above method is simulated and experimentally verified, and λ 633nm, N1080, Δ x are assumed0＝8μm，z1＝NΔx0 2/λ≈110mm，z2330 mm. Fig. 3 shows the desired distribution (10 × 10 lattice) on the output surface, fig. 4 shows the simulation result of the lattice of square light passing holes when α is 1.6, fig. 5 shows the desired distribution (ring) on the output surface, fig. 6 shows the simulation result of the ring when α is 1.6, fig. 7 shows the experiment result of the ring which is not filtered by the low-pass filter when α is 1.6, and fig. 8 shows the experiment result of the ring which is filtered when α is 1.6. The size of the diffraction image obtained by computer simulation is 41.5mm multiplied by 41.5mm, and the size of the diffraction image obtained by experiment is 42.0mm multiplied by 42.0mm, which shows that the experimental result and the simulation result are consistent. Other methods can produce the largest diffraction image with the same parameters and without image aliasing with dimensions 34.8mm x 34.8 mm. When α is 1.6, the size of the maximum diffraction image of the method is enlarged by 19.3% relative to other methods. In practical application, the size of the diffraction image on the output surface can be adjusted by adjusting the value of alpha. However, the size of the maximum diffraction image is limited by the sinc function envelope shown in fig. 1 (c).

Claims (3)

1. A design method for enlarging a diffraction image, comprising the steps of:
step one, setting a virtual middle surface, and calculating the light field distribution on an output surface in two steps;
in the calculation process, zero padding is carried out on an input surface, namely a plane where the DOE is located;
step three, a low-pass filter is arranged on the middle surface for filtering so as to eliminate image aliasing;
optimizing the DOE phase by combining an optimization algorithm, taking the phase of the non-zero-filling area as the design phase of the DOE, and finally achieving the purpose of amplifying the diffraction image; it is characterized in that the preparation method is characterized in that,
the two steps of the first step calculate the light field distribution on the output surface, taking one dimension as an example, the specific operation is as follows:
first step, Fresnel diffraction from the DOE to an intermediate surface, on which the light field distribution u is1(x1) Is shown as
Figure FDA0002581839100000011
Wherein x is0，x1Representing coordinates on the input surface and the intermediate surface, u, respectively0(x0) Is the distribution of the optical field, z, just after passing through the DOE1Is the distance from the input surface to the intermediate surface, λ is the wavelength of the incident parallel light, k 2 pi/λ is the wave vector,
Figure FDA0002581839100000012
represents a fourier transform; in many cases, it is difficult to find an analytical solution of the formula (1), and therefore, it is necessary to perform numerical calculation, and in this case, it is necessary to write the formula (1) in a discrete form
Figure FDA0002581839100000013
Wherein, Δ x0Is the sampling interval, Δ x, of the DOE1Is the sampling interval on the middle plane, m0，m1Is the interval [ -N/2, N/2-1]Internal integer, N being the input during discrete operationsSampling points of the inlet surface and the middle surface; if the equation (2) is calculated using fast Fourier transform, it should be satisfied
Figure FDA0002581839100000014
In order to satisfy the Nyquist sampling theorem, the distance z from the plane where the DOE is located to the middle plane1Should satisfy
Figure FDA0002581839100000015
Wherein z isT＝NΔx0 2λ is the characteristic distance;
the second step is the Fresnel diffraction from the intermediate surface to the output surface, which, like the first calculation, is satisfied if the light field distribution on the output surface is calculated using the fast Fourier transform
Figure FDA0002581839100000021
Where Δ x is the sampling interval on the output face, LzIs the maximum size of the diffraction image, z, that can be calculated on the output face2Is the distance from the intermediate face to the output face; from the formulae (3) and (5), it is possible to obtain
Figure FDA0002581839100000022
Wherein z is1+z2Is the distance from the input face to the output face;
the distance between the intermediate surface and the DOE, which is provided here, is exactly the characteristic distance N Δ x0 2λ, the maximum size L of the diffraction image is
Figure FDA0002581839100000023
2. The design method for enlarging diffraction images according to claim 1, wherein the second step is to perform zero padding on the input surface, i.e. the surface where the DOE is located; zero filling is carried out on the periphery of an input surface, namely a plane where the DOE is located, namely when numerical calculation is carried out, 0 element is filled on the periphery of a matrix representing the optical field distribution of the input surface; the sampling interval of the middle surface can be reduced, so that the sampling range on the output surface, namely the maximum size of the diffraction image, is enlarged; assuming that the ratio of the input surface sampling range after zero padding to the preceding is α, which is α >1, the maximum size of the diffraction image is
Figure FDA0002581839100000024
Therefore, the purpose of adjusting the size of the diffraction image on the output surface can be achieved by adjusting the value of alpha.
3. The design method for enlarging diffraction images of claim 1, wherein the low pass filter is a square diaphragm, and a light-passing hole is formed in the middle of the diaphragm, and the light-passing hole has a shape of a square, a rectangle, a polygon or a circle; the shape and size of the light through hole are determined according to the specific conditions of the DOE, and when the light through hole is square, the side length of the light through hole is N delta x0。
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