CN111238363B - Multi-wave radial shearing interferometer based on Fresnel zone plate - Google Patents
Multi-wave radial shearing interferometer based on Fresnel zone plate Download PDFInfo
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- CN111238363B CN111238363B CN201811432862.6A CN201811432862A CN111238363B CN 111238363 B CN111238363 B CN 111238363B CN 201811432862 A CN201811432862 A CN 201811432862A CN 111238363 B CN111238363 B CN 111238363B
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- G01—MEASURING; TESTING
- G01B—MEASURING LENGTH, THICKNESS OR SIMILAR LINEAR DIMENSIONS; MEASURING ANGLES; MEASURING AREAS; MEASURING IRREGULARITIES OF SURFACES OR CONTOURS
- G01B9/00—Measuring instruments characterised by the use of optical techniques
- G01B9/02—Interferometers
- G01B9/02041—Interferometers characterised by particular imaging or detection techniques
Abstract
The invention provides a Fresnel zone plate-based multi-wave radial shearing interferometer which comprises a Fresnel zone plate and an image detector, wherein a light beam to be measured enters the Fresnel zone plate, the zone plate is provided with a plurality of focuses, a plurality of light beams are formed through diffraction of the Fresnel zone plate, after the plurality of light beams are transmitted for a distance, radial shearing interference is formed due to different apertures of the light beams, and interference fringes are recorded by the image detector. The phase difference can be extracted through a single interference pattern, the wavefront distortion information to be measured is reconstructed by utilizing a wavefront restoration algorithm, and the dynamic range of measurement can be adjusted by adjusting the distance from the image detector to the wave zone plate. The invention can realize wavefront measurement only by using one Fresnel zone plate and one image detector, has simple structure, stability and reliability, and is suitable for transient wavefront and large dynamic range wavefront detection.
Description
Technical Field
The invention belongs to the technical field of optical information measurement, relates to an interferometer for measuring the wavefront of an incident beam, and particularly relates to a multi-wave radial shearing interferometer based on a Fresnel zone plate.
Background
The radial shearing interferometer is one of the main methods of wavefront detection technology, and is widely applied to the fields of optical element detection, adaptive optics, laser beam diagnosis and corneal topography measurement. Different from the lateral shearing interferometer, partial information does not participate in interference and the problem of information loss exists, and the reduced light beam participating in shearing interference in the radial shearing interferometer contains all information of the light beam to be measured and has no information loss, so that the radial shearing interferometer has advantages.
At present, the most commonly used radial shearing interferometer is a loop type radial shearing interferometer, which has high measurement accuracy and good repeatability. However, it needs two lenses, two mirrors and a beam splitter to implement radial shear interference, has a complex structure and constant adjustment, and needs to introduce tilt or phase shift in the optical path in order to extract phase information from the interferogram, which easily destroys the stability of the interferometer. Another commonly used radial shearing interferometer is a dual fresnel zone plate radial shearing interferometer, which requires two fresnel zone plates and a pinhole to filter out the redundant diffraction orders of the zone plates, but the size and position of the pinhole affect the measurement accuracy and are difficult to adjust, and meanwhile, an inclination needs to be introduced into the optical path to extract the phase. Other less common radial shearing interferometers also suffer from the above problems or more serious disadvantages. Therefore, it is urgent to design a compact radial shearing interferometer which is easy to adjust.
Disclosure of Invention
The technical problem to be solved by the invention is as follows: 1. the traditional radial shearing interferometer is complex in structure, needs a plurality of optical elements to realize radial shearing interference, and is inconvenient to adjust. 2. In order to extract phase information from an interferogram, a conventional radial shearing interferometer needs to introduce phase shift or tilt into an optical path, which is easy to destroy the stability of an interference system.
The technical scheme adopted by the invention is as follows:
a multi-wave radial shearing interferometer based on a Fresnel zone plate is disclosed, wherein the interferometer consists of a Fresnel zone plate and an image detector; the light beam to be measured is diffracted by the Fresnel zone plate to form a plurality of light beams, wherein the light beams comprise approximate plane light, convergent light and divergent light, after the light beams are spread for a distance, the distance is smaller than the focal length of the Fresnel zone plate, radial shearing interference occurs due to different apertures of the light beams, and multi-wave radial shearing interference fringes are generated on the target surface of the image detector; the single interference fringe comprises a plurality of shearing phase differences, the radial slope information of the wavefront to be detected can be extracted by using a certain phase extraction algorithm, and then the wavefront to be detected is reconstructed; the radial shear ratio of the radial shear interferometer is adjustable, the adjustment is convenient, only the distance z from the image detector to the Fresnel zone plate needs to be changed, and for the shear formed by +/-1-order diffracted light, the radial shear ratio is (f-z)/(f + z), and f is the focal length of the Fresnel zone plate.
The Fresnel zone plate can be of an amplitude type or a phase type; for the amplitude type, the fresnel zone plate may be a positive fresnel zone plate, a negative fresnel zone plate, or a fresnel zone plate whose radius satisfies a cosine function.
The image detector may be a CCD, CMOS, or other array type detector, among others.
Compared with the prior art, the invention has the following advantages:
(1) according to the invention, through the single Fresnel zone plate, the radial shearing interference is realized, the structure of the radial shearing interferometer is simplified, and the application field of the radial shearing interferometer is enlarged;
(2) the radial shear ratio of the invention can be adjusted by changing the distance from the image detector to the Fresnel zone plate, thus being suitable for large dynamic range wave-front detection;
(3) the method can reconstruct the wavefront to be detected only by a single radial shearing interference pattern, and can be applied to dynamic wavefront detection;
(4) the optical path of the invention naturally contains circular carrier frequency, phase shift or inclination is not required to be introduced into the optical path, and a pinhole is not required to be added into the optical path, so that the phase information can be extracted, and the system is more stable and reliable.
Drawings
FIG. 1 is a schematic structural diagram of a Fresnel zone plate-based multi-wave radial shearing interferometer according to the present invention;
FIG. 2 is a schematic diagram of a complex amplitude transmittance structure of an amplitude type positive Fresnel zone plate;
fig. 3 is a diagram (unit is wavelength) of the wavefront reconstruction simulation result of the fresnel zone plate-based multi-wave radial shearing interferometer, wherein fig. 3(a) is the wavefront to be measured, fig. 3(b) is the interference diagram of the numerically-simulated multi-wave radial shearing interferometer, fig. 3(c) is the reconstructed wavefront, and fig. 3(d) is the difference between the reconstructed wavefront and the original wavefront.
Detailed Description
The invention is further illustrated by the following figures and examples.
The invention provides a radial shearing interferometer composed of a Fresnel zone plate and an image detector, which has a simple structure and is convenient to adjust. Although the optical path of the interferometer comprises a plurality of diffracted beams, phase information can be directly extracted from a single interference pattern by a certain mathematical method without introducing phase shift or inclination in the optical path or adding a pinhole in the optical path, and the system is stable and reliable.
The complex amplitude transmittance function of the amplitude type Fresnel zone plate is as follows:
where r is the radial variable of the polar coordinate system, j is the imaginary unit, d is a constant, and d is f × λ, where f is the focal length of the zone plate, λ is the wavelength of the light beam to be measured, cnIs a coefficient of each order, wherein c0＝1/2，
c±2＝0，c±3＝±1/3π。
Let the measured complex amplitude be:
Ui(r,θ)＝Ai(r,θ)exp[jkW(r,θ)] (2)
where θ is the angular variation of the polar coordinate system, Ai(r, θ) is the amplitude of the measuring beam, W (r, θ) is the wavefront of the measuring beam, and k ═ 2 π/λ is the wavevector. After passing through the Fresnel zone plate, the light beam to be measured diffracts a plurality of beams of light, and the complex amplitude is changed into the following form:
and (3) propagating a distance z, and assuming that z is less than f and z is not equal to f/n, receiving a target surface complex amplitude expression at an image detector:
wherein, cnAn(r, θ, z) is the amplitude distribution of the nth order, αnIs a coordinate transformation coefficient to be used for the coordinate transformation,
and further obtaining an expression of an interference pattern received by the image detector, wherein the expression is as follows:
wherein the content of the first and second substances,
is UzComplex conjugation of (r, θ, z);
to extract the radial shear phase difference from the interferogram, first, the interferogram expression (6) can be rewritten as:
wherein the content of the first and second substances,
in equation (8), when n is 1 and m is-1, W (α)1r, θ) is the reduced wavefront in a multi-wave radial shearing interferometer, W (α)-1r, θ) is the expanding wavefront in a multi-wave radial shear interferometer, so the radial shear ratio is s ═ r/α1r)/(r/α-1r) — (f-z)/(f + z), the radial shear ratio can be changed by changing the distance from the image detector to the fresnel zone plate, thereby changing the measurement dynamic range to accommodate the detection of different wavefronts. For a specific measuring system, Fresnel waveThe focal length f of the zone plate and the distance z of the image detector to the Fresnel zone plate are both known, so alpha1And alpha-1Also known, a simulated reference interferogram can be generated according to the following equation,
wherein
Is a constant, 1,2,3,4, multiplying equation (7) by equation (9) yields equation (10)
In the formula (10), the first two terms are low-frequency terms, and the other terms are high-frequency terms, so that the first two terms can be filtered out by designing a low-pass filter,
where H (r, θ) is a low pass filter,
is a sign of the convolution of the symbols,
order to
Then there is
Therefore, the radial shearing phase difference can be extracted from the single interference image of the multi-wave radial shearing interferometer, and the wavefront to be measured can be reconstructed by combining the wavefront reconstruction algorithm of the radial shearing interferometer.
A single radial shearing interference pattern is obtained on an image detector, the phase can be demodulated from the obtained interference fringes by using the method to obtain radial wavefront slope information, and then the wavefront to be detected is reconstructed by combining a radial shearing interference mode wavefront restoration algorithm. The shearing ratio of the radial shearing interferometer is adjustable and convenient to adjust, and only the distance z from the image detector to the Fresnel zone plate needs to be changed, wherein the radial shearing ratio s is (f-z)/(f + z). The Fresnel zone plate can be of an amplitude type or a phase type; for the amplitude type, it may be a positive fresnel zone plate or a negative fresnel zone plate. The image detector may be a CCD, CMOS, or other array type detector.
As shown in fig. 1 (only 0, ± 1, ± 3 diffraction orders are drawn in the figure), the fresnel zone plate-based multi-wave radial shearing interferometer in the embodiment of the present invention is composed of a fresnel zone plate 1 and a CCD imaging detector 2, and as shown in fig. 2, is a schematic diagram of an amplitude type positive fresnel zone plate, and the focal length of the fresnel zone plate is f ═ 500 mm; the diameter of the beam aperture is 8 mm; the wavelength of the light beam is 632.8 nm; the distance from the CCD imaging detector to the Fresnel zone plate is z which is 55 mm; the radial shear ratio is then s-0.8. The numerically simulated interferogram was sampled 512 x 512 at a sampling interval of 15.6 μm.
Fig. 3 is a numerical simulation result of wavefront reconstruction using an embodiment of the present invention, in units of wavelength. Fig. 3(a) is a test wavefront generated within a unit circle, which is a random wavefront generated by the first 21 st order zernike polynomial in combination with randomly generated coefficients, with PV of 1.4159 λ and RMS of 0.2513 λ, with uniformly distributed amplitudes. Fig. 3(b) is an interferogram generated by angular spectrum theory simulation, and a radial shear phase difference can be extracted by combining the phase extraction method in the invention content, and then the wavefront to be measured can be reconstructed by using a radial shear interference wavefront reconstruction algorithm. Fig. 3(c) is a distribution diagram of the reconstructed wavefront, whose PV is 1.4301 λ and RMS is 0.2516 λ, and fig. 3(d) is a difference between the reconstructed wavefront and the wavefront, whose PV is 0.0274 λ and RMS is 0.0046 λ. The difference between the two is very small, and the simulation result shows that the invention can realize the accurate detection of the wave front.
The invention has not been described in detail and is within the skill of the art. The above description is only an embodiment of the present invention, but the scope of the present invention is not limited thereto, and any person skilled in the art can understand that the modifications or substitutions within the technical scope of the present invention are included in the scope of the present invention.
Claims (1)
1. A multi-wave radial shearing interferometer based on a Fresnel zone plate is characterized in that: the interferometer consists of a Fresnel zone plate and an image detector; the light beam to be measured is diffracted by the Fresnel zone plate to form a plurality of light beams, wherein the light beams comprise approximate plane light, convergent light and divergent light, after the light beams are spread for a distance, the distance is smaller than the focal length of the Fresnel zone plate, radial shearing interference occurs due to different apertures of the light beams, and multi-wave radial shearing interference fringes are generated on the target surface of the image detector; the single interference fringe comprises a plurality of shearing phase differences, the radial slope information of the wavefront to be detected can be extracted by using a certain phase extraction algorithm, and then the wavefront to be detected is reconstructed; the radial shearing ratio of the radial shearing interferometer is adjustable and convenient to adjust, only the distance z from the image detector to the Fresnel zone plate needs to be changed, the radial shearing ratio s is (f-z)/(f + z), and f is the focal length of the Fresnel zone plate;
the Fresnel zone plate can be of an amplitude type or a phase type; for the amplitude type, the Fresnel zone plate can be a positive Fresnel zone plate, a negative Fresnel zone plate or a Fresnel zone plate with the radius meeting the cosine function;
the image detector may be a CCD, CMOS, or other array type detector;
the complex amplitude transmittance function of the amplitude type Fresnel zone plate is as follows:
where r is the radial variable of the polar coordinate system, j is the imaginary unit, d is a constant, and d is f × λ, where f is the focal length of the zone plate, λ is the wavelength of the light beam to be measured, cnIs a coefficient of each order, wherein c0＝1/2，
c±2＝0，c±3＝±1/3π；
Let the measured complex amplitude be:
Ui(r,θ)＝Ai(r,θ)exp[jkW(r,θ)] (2)
where θ is the angular variation of the polar coordinate system, Ai(r, θ) is the amplitude of the measuring beam, W (r, θ) is the wavefront of the measuring beam, k is 2 pi/λ is the wavevector, the measuring beam diffracts multiple beams after passing through the fresnel zone plate, and the complex amplitude becomes as follows:
and (3) propagating a distance z, and assuming that z is less than f and z is not equal to f/n, receiving a target surface complex amplitude expression at an image detector:
wherein, cnAn(r, θ, z) is the amplitude distribution of the nth order, αnIs a coordinate transformation coefficient to be used for the coordinate transformation,
and further obtaining an expression of an interference pattern received by the image detector, wherein the expression is as follows:
wherein the content of the first and second substances,
is UzComplex conjugation of (r, θ, z);
to extract the radial shear phase difference from the interferogram, first, the interferogram expression (6) can be rewritten as:
wherein the content of the first and second substances,
in equation (8), when n is 1 and m is-1, W (α)1r, θ) is the reduced wavefront in a multi-wave radial shearing interferometer, W (α)- 1r, θ) is the expanding wavefront in a multi-wave radial shear interferometer, so the radial shear ratio is s ═ r/α1r)/(r/α-1r) ((f-z)/(f + z), the radial shear ratio can be changed by changing the distance from the image detector to the fresnel zone plate, thereby changing the dynamic range of measurement to accommodate the detection of different wavefronts, and for a given measurement system, the focal length f of the fresnel zone plate and the distance z from the image detector to the fresnel zone plate are both known, so that α is1And alpha-1Also known, a simulated reference interferogram can be generated according to the following equation,
wherein
Is a constant, 1,2,3,4, multiplying equation (7) by equation (9) yields equation (10)
In the formula (10), the first two terms are low-frequency terms, and the other terms are high-frequency terms, so that the first two terms can be filtered out by designing a low-pass filter,
where H (r, θ) is a low pass filter,
is a sign of the convolution of the symbols,
order to
Then there is
At this moment, extracting a radial shearing phase difference from a single interference image of the multi-wave radial shearing interferometer, and reconstructing the wavefront to be measured by combining a wavefront recovery algorithm of the radial shearing interferometer;
obtaining a single radial shearing interferogram on an image detector, demodulating a phase from the obtained interference fringe to obtain radial wavefront slope information, and further reconstructing a wavefront to be detected by combining a radial shearing interference mode wavefront restoration algorithm, wherein the shearing ratio of the radial shearing interferometer is adjustable and convenient to adjust, namely, only the distance z from the image detector to a Fresnel zone plate needs to be changed, wherein the radial shearing ratio s is (f-z)/(f + z), and the Fresnel zone plate can be in an amplitude type or a phase type; for the amplitude type, it can be either positive or negative fresnel zone plate, and the image detector can be CCD, CMOS, or other array type detector.
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