CN105404128B - Multiframe phase-shifted digital holographic method and device - Google Patents

Abstract

The invention discloses a multi-frame phase-shift digital holography method and device. The related method includes: using two acousto-optic modulators (AOMs) to generate low-frequency heterodyne by using two acousto-optic modulators (AOMs) respectively corresponding to modulating object light and reference light, and using a high-frame-frequency area array to detect detector to obtain a multi-frame phase-shift hologram; respectively perform time-domain spectral transformation on the signal of each pixel in the multi-frame phase-shift hologram, and extract the amplitude and phase of the spectral value at the heterodyne frequency; from The target amplitude and target phase are extracted from the amplitude and phase of the spectrum value at the heterodyne frequency, thereby realizing holographic reconstruction. By adopting the solution provided by the present invention, not only the influence of twin images can be eliminated, but also the interference of random noise can be effectively reduced, and the imaging precision and environmental adaptability can be improved.

Description

Multi-frame phase shift digital holography method and device

technical field

The invention relates to the technical field of digital holography, in particular to a multi-frame phase-shifting digital holography method and device.

Background technique

Holographic technology uses interference recording and diffraction reconstruction to obtain the amplitude and phase information of the target, so as to realize the three-dimensional imaging of the target. Digital holography technology utilizes digital equipment to realize holographic recording and reconstruction. Compared with other 3D imaging technologies, digital holography has the advantages of non-contact, non-scanning, non-staining, and quantitative, and can be applied to imaging and measurement in the fields of biology and microelectromechanical systems. There are two main problems in digital holography: one is the twin image problem, and the other is the random noise problem (mechanical vibration, airflow disturbance, etc.). These problems affect the imaging accuracy of digital holography and limit the application range. Phase-shift digital holography technology can obtain several frames of holograms (usually 4 frames) with a certain amount of phase shift by introducing a phase shift between the object light and the reference light. After digital processing, the twin image interference can be eliminated, and to a certain extent Mitigates the effects of random noise.

At present, there are several phase-shift holographic devices and corresponding processing algorithms, but due to their respective problems such as phase-shift nonlinearity, limited number of detection frames, and limited noise resistance of processing algorithms, these systems have weak noise immunity. Disadvantages, which affect the imaging accuracy and limit the field of application.

Contents of the invention

The purpose of the present invention is to provide a multi-frame phase-shift digital holography method and device, which can not only eliminate the influence of twin images, but also effectively reduce the interference of random noise, and improve imaging accuracy and environmental adaptability.

The purpose of the present invention is achieved through the following technical solutions:

A multi-frame phase-shifting digital holography method, comprising:

Two acousto-optic modulators (AOM) are used to generate low-frequency heterodyne corresponding to the modulated object light and reference light, and are detected by a high-frame-frequency area array detector to obtain a multi-frame phase-shift hologram;

Carry out time-domain spectral transformation to the signal of each pixel point in the multi-frame phase-shifted hologram respectively, and extract the amplitude and phase of the spectral value at the heterodyne frequency;

The target amplitude and target phase are extracted from the amplitude and phase of the spectral value at the heterodyne frequency, thereby realizing holographic reconstruction.

Further, said obtaining multi-frame phase shift hologram includes:

Obtain the object light U _O and reference light U _R reaching the high frame rate area detector:

In the formula, t is the time, a _O ,ω _O , are the amplitude, angular frequency and phase of the object light respectively; a _R , ω _R , are the amplitude, angular frequency and phase of the reference light, respectively;

The ideal noise-free hologram s produced by the interference of object light and reference light is expressed as:

where ω _H is the heterodyne angular frequency, f _H is the heterodyne frequency of the two AOMs, and their relationship is:

ω _H = ω _O - ω _R = 2πf _H ;

Contains random noise The total signal of the multi-frame phase-shifted hologram is:

In this way, a multi-frame phase shift hologram s _total (t(n)) is obtained, where n is the serial number of each frame, and t(n) is the sampling time corresponding to the nth frame serial number, n=1,2,..., N _total , where N _total is the total number of frames of the obtained hologram.

Further, the step of performing time-domain spectral transformation on the signal of each pixel in the multi-frame phase-shifted hologram, and extracting the amplitude and phase of the spectral value at the heterodyne frequency includes:

The signal of each pixel point in the multi-frame phase-shifted hologram is respectively subjected to time-domain spectrum transformation by using Fast Fourier Transform FFT to obtain the time-domain spectrum S(ω):

In the formula, ω represents the heterodyne angular frequency, s _total is the total signal of the multi-frame phase-shifted hologram, n is the serial number of each frame, t(n) is the sampling time corresponding to the nth frame serial number, and N _total is the obtained hologram Total number of frames;

Extract the value S(ω _H ) of the time-domain spectrum S(ω) at the heterodyne angular frequency ω _H :

In the formula, σ is the standard deviation;

Thus, the amplitude of the spectral value of the time-domain spectrum S(ω) at the heterodyne frequency is obtained and phase

Further, the target amplitude and target phase are extracted from the amplitude and phase of the spectral value at the heterodyne frequency, so as to realize holographic reconstruction and obtain complete target information, which includes:

amplitude middle is a constant, after normalization processing, the normalized amplitude a _norm of the target is obtained:

In the formula, max and min respectively represent the two-dimensional spatial distribution data Take the maximum and minimum values;

Set the position of the reference optical path lens and the objective optical path lens to be symmetrical, then the phase of the object light when the target is not included equal target aspect Expressed as:

And, according to the obtained target amplitude and target phase, the complete target information O _result is obtained, which is expressed as:

A multi-frame phase shift digital holographic device, including: a preset optical path, a high frame rate area array detector and a data processor; wherein:

The preset optical path includes two acousto-optic modulators AOM, which are used to generate low-frequency heterodyne for the corresponding modulation object light and reference light;

The high frame rate area array detector detects the light in the imaging mirror of the preset optical path, obtains a multi-frame phase shift hologram, and transmits it to the data processor;

The data processor performs time-domain spectral transformation on the signal of each pixel point in the multi-frame phase-shifted hologram, and extracts the amplitude and phase of the spectral value at the heterodyne frequency; then from the heterodyne frequency The target amplitude and target phase are extracted from the amplitude and phase of the spectral value, thereby realizing holographic reconstruction.

Further, the preset optical path includes:

One laser, one half-wave plate, three polarization beamsplitters, two mirrors, two acousto-optic modulators (AOM), two apertures, one spatial filter, two 1/4 wave plates, two microscope objectives , a reference mirror and an imaging mirror; where:

The laser light emitted by the laser is divided into two beams of polarized light, object light and reference light, through a half-wave plate and a polarization beam splitter prism 1, the object light is polarized light in the p direction, and the reference light is polarized light in the s direction;

The object light passes through the reflector 1, enters the AOM 1, and the outgoing -1 order and 0 order lights are blocked by the diaphragm 1, and the outgoing +1 diffracted light enters the spatial filter through the polarization beam splitter prism 2, and after filtering The light beam sequentially passes through the polarization beam splitter 3, 1/4 wave plate 1, and the microscope objective lens 1, and then irradiates the target; the reflected light carrying target information passes through the microscope objective lens 1, 1/4 wave plate 1, and becomes polarized in the s direction The light is injected into the imaging mirror after passing through the polarization beam splitter 3 and the polarizer placed at an angle of 45 degrees;

After the reference light enters the AOM 2, the emitted -1st order and 0th order light are blocked by the diaphragm 2, and the outgoing +1st order diffracted light enters the spatial filter after passing through the reflector 2 and the polarization beam splitter prism 2, and the filtered beam passes through the Polarizing beam splitter 3, 1/4 wave plate 2, and microscopic objective lens 2 are irradiated onto the reference mirror; the reflected light turns into p-direction polarized light after passing through microscopic objective lens 2, 1/4 wave plate 2, and then polarized The dichroic prism 3 and the polarizing plate placed at an angle of 45 degrees are injected into the imaging mirror.

Further, the preset optical path includes:

One laser, one half-wave plate, three polarization beamsplitters, two mirrors, two acousto-optic modulators (AOMs), two apertures, one spatial filter, one beamsplitter, two 1/4 wave plates, one Microscope objective lens, a reference mirror and an imaging mirror; wherein:

The laser light emitted by the laser is divided into two beams of polarized light, object light and reference light, through a half-wave plate and a polarization beam splitter prism 1, the object light is polarized light in the p direction, and the reference light is polarized light in the s direction;

The object light passes through the reflector 1, enters the AOM 1, and the outgoing -1 order and 0 order lights are blocked by the diaphragm 1, and the outgoing +1 diffracted light enters the spatial filter through the polarization beam splitter prism 2, and after filtering The light beam sequentially passes through the beam splitter prism, microscope objective lens, polarization beam splitter 3 and 1/4 wave plate 1, and then irradiates the target; the reflected light carrying target information becomes polarized light in the s direction after passing through the 1/4 wave plate 1, and then sequentially After passing through the polarizing beam-splitting prism 3, the microscopic objective lens, the beam-splitting prism and the polarizer placed at an angle of 45 degrees, it enters the imaging mirror;

After the reference light enters the AOM 2, the emitted -1st order and 0th order light are blocked by the diaphragm 2, and the outgoing +1st order diffracted light enters the spatial filter after passing through the reflector 2 and the polarization beam splitter prism 2, and the filtered beam passes through the The beam-splitting prism, microscope objective lens, polarization beam-splitter prism 3 and 1/4 wave plate 2 are then irradiated onto the reference mirror; the reflected light becomes polarized light in the p direction after passing through the 1/4 wave plate 2, and then passes through the polarization beam-splitter prism 3 in turn , a microscope objective lens, a dichroic prism and a polarizer placed at an angle of 45 degrees are injected into the imaging mirror.

Further, the preset optical path includes:

One laser, two half-wave plates, one polarizing beam splitter prism, two mirrors, two acousto-optic modulators AOM, two apertures, two spatial filters, one beam combining prism and one imaging mirror; where:

The laser light emitted by the laser is divided into two beams of polarized light, object light and reference light, through the half-wave plate 1 and the polarization beam splitter prism, the object light is polarized light in the p direction, and the reference light is polarized light in the s direction;

The object light passes through the half-wave plate 2 and the mirror 1 in sequence, and after entering the acousto-optic modulator AOM 1, the emitted -1 order and 0 order lights are blocked by the diaphragm 1, and the outgoing +1 diffracted light enters the spatial filter 1, After filtering, the light beam passes through the beam combining prism and the microscopic objective lens in turn and then irradiates the target; the reflected light carrying target information passes through the microscopic objective lens and the beam combining prism in turn and then enters the imaging mirror;

After the reference light enters the AOM 2, the emitted -1st order and 0th order light are blocked by the diaphragm 2, and the outgoing +1st order diffracted light enters the spatial filter 2 after passing through the mirror 2, and the filtered beam passes through the beam combining prism and then shoots into the imaging mirror.

Further, said obtaining multi-frame phase shift hologram includes:

Obtain the object light U _O and reference light U _R reaching the high frame rate area detector:

In the formula, t is the time, a _O ,ω _O , are the amplitude, angular frequency and phase of the object light respectively; a _R , ω _R , are the amplitude, angular frequency and phase of the reference light, respectively;

The ideal noise-free hologram s produced by the interference of object light and reference light is expressed as:

where ω _H is the heterodyne angular frequency, f _H is the heterodyne frequency of the two AOMs, and their relationship is:

ω _H = ω _O - ω _R = 2πf _H ;

Contains random noise The total signal of the multi-frame phase-shifted hologram is:

In this way, a multi-frame phase shift hologram s _total (t(n)) is obtained, where n is the serial number of each frame, and t(n) is the sampling time corresponding to the nth frame serial number, n=1,2,..., N _total , N _total is the total number of frames of the obtained hologram;

The data processor performs time-domain spectral transformation on the signal of each pixel point in the multi-frame phase-shifted hologram, and extracts the amplitude and phase of the spectral value at the heterodyne frequency, including:

The signal of each pixel point in the multi-frame phase-shifted hologram is respectively subjected to time-domain spectrum transformation by using Fast Fourier Transform FFT to obtain the time-domain spectrum S(ω):

In the formula, ω represents the heterodyne angular frequency, s _total (t(n)) is a multi-frame phase shift hologram, n is the serial number of each frame, t(n) is the sampling time corresponding to the nth frame serial number, and N _total is The total number of hologram frames obtained;

Extract the value S(ω _H ) of the time-domain spectrum S(ω) at the heterodyne angular frequency ω _H :

In the formula, σ is the standard deviation;

Thus, the amplitude of the spectral value of the time-domain spectrum S(ω) at the heterodyne frequency is obtained and phase

Further, the target amplitude and target phase are extracted from the amplitude and phase of the spectral value at the heterodyne frequency, so as to realize holographic reconstruction and obtain complete target information, which includes:

amplitude middle is a constant, after normalization processing, the normalized amplitude a _norm of the target is obtained:

In the formula, max and min respectively represent the two-dimensional spatial distribution data Take the maximum and minimum values;

Set the position of the reference optical path lens and the objective optical path lens to be symmetrical, then the phase of the object light when the target is not included equal target aspect Expressed as:

Obtain the complete information O _result of the target according to the obtained target amplitude and target phase, which is expressed as:

It can be seen from the above-mentioned technical solution provided by the present invention that two acousto-optic modulators (acousto-optic modulator, AOM) are used to modulate object light and reference light to generate low-frequency heterodyne, and a high frame rate area array detector is used for detection, thereby Acquire hundreds or even thousands of frames of accurate linear phase-shifted holograms in a short time. Moreover, unlike the traditional 4-step holographic algorithm, the multi-frame phase-shift holographic algorithm is used to extract the target amplitude and phase information, which can effectively remove the interference of twin images and random noise, and obtain a clear and accurate 3D reconstruction image of the target.

Description of drawings

In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the following will briefly introduce the accompanying drawings that need to be used in the description of the embodiments. Obviously, the accompanying drawings in the following description are only some embodiments of the present invention. For Those of ordinary skill in the art can also obtain other drawings based on these drawings on the premise of not paying creative work.

FIG. 1 is a flowchart of a multi-frame phase-shift digital holography method provided by an embodiment of the present invention;

Fig. 2a is a schematic diagram of a three-dimensional matrix composed of multi-frame phase shift holograms provided by an embodiment of the present invention;

FIG. 2b is a schematic diagram of a time-domain sampling signal at a certain point provided by an embodiment of the present invention;

Fig. 3a is a schematic structural diagram of a multi-frame phase-shift digital holographic device provided by an embodiment of the present invention;

Fig. 3b is a schematic structural diagram of another multi-frame phase-shift digital holographic device provided by an embodiment of the present invention;

Fig. 3c is a schematic structural diagram of another multi-frame phase-shift digital holographic device provided by an embodiment of the present invention;

FIG. 4 is a schematic diagram of time-domain spectrum obtained by processing the signal with FFT according to an embodiment of the present invention;

Fig. 5 is a schematic diagram of a three-dimensional image provided by an embodiment of the present invention.

Detailed ways

The technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only some of the embodiments of the present invention, not all of them. Based on the embodiments of the present invention, all other embodiments obtained by persons of ordinary skill in the art without making creative efforts belong to the protection scope of the present invention.

FIG. 1 is a flowchart of a multi-frame phase-shift digital holography method provided by an embodiment of the present invention. As shown in Figure 1, it mainly includes the following steps:

Step 11: Using two acousto-optic modulators AOMs to generate low-frequency heterodyne for the modulating object light and reference light respectively, and using a high-frame-frequency area array detector for detection to obtain a multi-frame phase-shift hologram.

There are two embodiments of the present invention. The obtaining of multi-frame phase shift holograms includes:

Obtain the object light U _O and reference light U _R reaching the high frame rate area detector:

In the formula, t is the time, a _O ,ω _O , are the amplitude, angular frequency and phase of the object light respectively; a _R , ω _R , are the amplitude, angular frequency and phase of the reference light, respectively;

and The relationship is:

in, is the phase of the object light wave without the target, is the phase change of the object light wave after placing the target.

The ideal noise-free hologram s produced by the interference of object light and reference light is expressed as:

where ω _H is the heterodyne angular frequency, f _H is the heterodyne frequency of the two AOMs, and their relationship is:

ω _H ＝ω _O −ω _R ＝2πf _H ; (5)

Contains random noise The total signal of the multi-frame phase-shifted hologram is:

It can be seen from the above that the amplitude and phase information of the target is modulated and recorded in the third item, and the modulation angular frequency is ω _H , and the twin image is recorded in the second item, with a different modulation angular frequency - ω _H .

In the embodiment of the present invention, the heterodyne frequency f _H , the detector frame frequency f _D , each frame number n, each frame corresponds to the sampling time t(n), and the total recording time t _total , then the total number of hologram frames obtained N _total , the number of phase shift cycles N _period , and the relationship between the number of phase shift steps N _step in each cycle are:

N _period = f _H t _total (7)

N _total ＝ f _D t _total ＝ N _period N _step (9)

For example, if a 10Hz heterodyne frequency and a 200Hz detection frame frequency are used, 200*3=600 frames of holograms can be obtained within 3 seconds, including 10*3=30 cycles, and each cycle has 200/10=20 phase steps shift, the sampling time interval of each frame is 1/200=0.005s.

Thus, a multi-frame phase shift hologram s _total (t(n)) is obtained, n=1, 2, . . . , N _total .

Step 12. Perform time-domain spectral transformation on the signal of each pixel in the multi-frame phase-shifted hologram, and extract the amplitude and phase of the spectral value at the heterodyne frequency.

The obtained multi-frame phase-shifted hologram s _total (t(n)) constitutes a data matrix M[r,c,N _total ] in the computer. This matrix includes two dimensions of space and one dimension of time. r is the number of pixels in the detector row, c is the number of pixels in the detector column, and N _total is the total number of frames of the hologram. Therefore, for any pixel point (x,y) of the detector, the detection signal M[x,y,1:N _total ] of the point can be obtained, and the number of samples is equal to the total number of frames N _total . As shown in Figure 2, Figure 2a is a three-dimensional matrix composed of multiple frames of phase-shifted holograms, and Figure 2b is a time-domain sampling signal at a certain point.

In the embodiment of the present invention, the signal of each pixel point in the multi-frame phase-shift hologram is respectively subjected to time-domain spectrum transformation by using Fast Fourier Transform (FFT) to obtain the time-domain spectrum S(ω):

where ω represents the heterodyne angular frequency. Then extract the value S(ω _H ) of the time-domain spectrum S(ω) at the heterodyne angular frequency ω _H. According to the Nyquist-Shannon sampling law, when the frame frequency f _D satisfies twice and twice the heterodyne frequency f _H When above, the target spectrum and the twin image spectrum are separated. Extract the target information on the third item in the s _total (t) expression (Formula 6) corresponding to the pixel point at the time-domain spectral heterodyne frequency point ω _H , as shown in the above S(ω) expression. Since only the signal of the spectral point ω _H is extracted and the signal of the spectral point -ω _H where the twin image is located is avoided, the influence of the twin image can be eliminated, as shown in the following formula:

When the number of sampling frames N _total is positive infinity, the cumulative factor in the above formula is The mathematical expectation of is as follows:

Usually, random noise is complex, including random vibration, airflow disturbance, etc. If the statistical distribution of the noise satisfies the normal (Gaussian) distribution, and the expected value μ=0, then the probability density function p and the characteristic function F are shown in formulas (14) (15), where σ is the standard deviation.

The result of formula (13) is equal to the value when k=1 in formula (15), as shown in formula (16).

Through the derivation of the above formulas (13), (14), (15), and (16), the result of the formula (12) can be obtained, as shown in the formula (17):

When the number of sampling frames is infinite, the statistical value of random noise is a constant, and the extracted target information is not affected by noise. In practice, increasing the number of sampling frames N _total , the closer the formula (17) is to the equation, thereby reducing the influence of random noise and improving the reconstruction accuracy.

Therefore, by extracting target information in the time-frequency domain, the influence of twin images can be effectively removed. Moreover, by obtaining enough frames of holograms for calculation, the statistical value of random noise can be approached to a constant, thereby alleviating the influence of random noise. For different levels of noise, the corresponding heterodyne frequency and sampling frame number can be set to obtain the required holographic imaging accuracy.

From formula (17), it can be obtained that the amplitude of the value S(ω H ) of the time-domain spectrum S( _ω ) at the heterodyne angular frequency point ω _H is The phase value is

Step 13, extracting the target amplitude and target phase from the amplitude and phase of the spectral value at the heterodyne frequency, so as to realize holographic reconstruction.

amplitude middle is a constant, after normalization processing, the normalized amplitude a _norm of the target is obtained:

In the formula, max and min respectively represent the two-dimensional spatial distribution data Take the maximum and minimum values;

In the embodiment of the present invention, if the position of the reference optical path lens and the objective optical path lens are symmetrical, then the phase of the object light when the target is not included equal target aspect Expressed as:

And since the detector is located at the image plane of the lens, then It is equal to the reconstruction phase of the target, and there is no need for diffraction reconstruction.

Thereby obtaining the normalized amplitude a _norm and phase reconstruction phase of the target Realize holographic reconstruction.

In addition, the complete information O _result of the target is obtained according to the obtained target amplitude and target phase, which is expressed as:

In another embodiment of the present invention, a multi-frame phase-shift digital holographic device is also provided, which mainly includes: a preset optical path, a high frame rate area array detector and a data processor; wherein:

The preset optical path includes two acousto-optic modulators AOM, which are used to generate low-frequency heterodyne for the corresponding modulation object light and reference light;

The high frame rate area array detector detects the light in the imaging mirror of the preset optical path, obtains a multi-frame phase shift hologram, and transmits it to the data processor;

The data processor performs time-domain spectral transformation on the signal of each pixel point in the multi-frame phase-shifted hologram, and extracts the amplitude and phase of the spectral value at the heterodyne frequency; then from the heterodyne frequency The target amplitude and target phase are extracted from the amplitude and phase of the spectral value, thereby realizing holographic reconstruction.

In the embodiment of the present invention, the preset optical path has various structural forms. As shown in Figures 3a-3c, three optical paths with different structures are shown, and the optical paths shown at the same time together with the high frame rate area array detector and data processor constitute a multi-frame phase-shift digital holographic device. The following mainly introduces the optical path except the high frame rate area array detector and the data processor in Fig. 3a-3c.

As shown in Figure 3a, it mainly includes: a laser, a half-wave plate, three polarization beam splitters, two mirrors, two acousto-optic modulators AOM, two apertures, a spatial filter, two 1 /4 wave plate, two microscope objective lenses, a reference mirror and an imaging mirror; where:

The laser light emitted by the laser is divided into two beams of polarized light, object light and reference light, through a half-wave plate and a polarization beam splitter prism 1, the object light is polarized light in the p direction, and the reference light is polarized light in the s direction;

The object light passes through the reflector 1, enters the AOM 1, and the outgoing -1 order and 0 order lights are blocked by the diaphragm 1, and the outgoing +1 diffracted light enters the spatial filter through the polarization beam splitter prism 2, and after filtering The light beam sequentially passes through the polarization beam splitter 3, 1/4 wave plate 1, and the microscope objective lens 1, and then irradiates the target; the reflected light carrying target information passes through the microscope objective lens 1, 1/4 wave plate 1, and becomes polarized in the s direction The light is injected into the imaging mirror after passing through the polarization beam splitter 3 and the polarizer placed at an angle of 45 degrees;

After the reference light enters the AOM 2, the emitted -1st order and 0th order light are blocked by the diaphragm 2, and the outgoing +1st order diffracted light enters the spatial filter after passing through the reflector 2 and the polarization beam splitter prism 2, and the filtered beam passes through the Polarizing beam splitter 3, 1/4 wave plate 2, and microscopic objective lens 2 are irradiated onto the reference mirror; the reflected light turns into p-direction polarized light after passing through microscopic objective lens 2, 1/4 wave plate 2, and then polarized The dichroic prism 3 and the polarizing plate placed at an angle of 45 degrees are injected into the imaging mirror.

As shown in Figure 3b, it mainly includes: a laser, a half-wave plate, three polarization beam splitters, two mirrors, two acousto-optic modulators AOM, two apertures, a spatial filter, and a beam splitter , two 1/4 wave plates, a microscope objective lens, a reference mirror and an imaging mirror; where:

The laser light emitted by the laser is divided into two beams of polarized light, object light and reference light, through a half-wave plate and a polarization beam splitter prism 1, the object light is polarized light in the p direction, and the reference light is polarized light in the s direction;

The object light passes through the reflector 1, enters the AOM 1, and the outgoing -1 order and 0 order lights are blocked by the diaphragm 1, and the outgoing +1 diffracted light enters the spatial filter through the polarization beam splitter prism 2, and after filtering The light beam sequentially passes through the beam splitter prism, microscope objective lens, polarization beam splitter 3 and 1/4 wave plate 1, and then irradiates the target; the reflected light carrying target information becomes polarized light in the s direction after passing through the 1/4 wave plate 1, and then sequentially After passing through the polarizing beam-splitting prism 3, the microscopic objective lens, the beam-splitting prism and the polarizer placed at an angle of 45 degrees, it enters the imaging mirror;

After the reference light enters the AOM 2, the emitted -1st order and 0th order light are blocked by the diaphragm 2, and the outgoing +1st order diffracted light enters the spatial filter after passing through the reflector 2 and the polarization beam splitter prism 2, and the filtered beam passes through the The beam-splitting prism, microscope objective lens, polarization beam-splitter prism 3 and 1/4 wave plate 2 are then irradiated onto the reference mirror; the reflected light becomes polarized light in the p direction after passing through the 1/4 wave plate 2, and then passes through the polarization beam-splitter prism 3 in turn , a microscope objective lens, a dichroic prism and a polarizer placed at an angle of 45 degrees are injected into the imaging mirror.

As shown in Figure 3c, it mainly includes: a laser, two half-wave plates, a polarizing beam splitter, two mirrors, two acousto-optic modulators (AOM), two apertures, two spatial filters, a combining beam prism and an imaging mirror; where:

The laser light emitted by the laser is divided into two beams of polarized light, object light and reference light, through the half-wave plate 1 and the polarization beam splitter prism, the object light is polarized light in the p direction, and the reference light is polarized light in the s direction;

The object light passes through the half-wave plate 2 and the mirror 1 in sequence, and after entering the acousto-optic modulator AOM 1, the emitted -1 order and 0 order lights are blocked by the diaphragm 1, and the outgoing +1 diffracted light enters the spatial filter 1, After filtering, the light beam passes through the beam combining prism and the microscopic objective lens in turn and then irradiates the target; the reflected light carrying target information passes through the microscopic objective lens and the beam combining prism in turn and then enters the imaging mirror;

After the reference light enters the AOM 2, the emitted -1st order and 0th order light are blocked by the diaphragm 2, and the outgoing +1st order diffracted light enters the spatial filter 2 after passing through the mirror 2, and the filtered beam passes through the beam combining prism and then shoots into the imaging mirror.

In the optical path of FIG. 3a above, the same parameter microscopic objective lens 1 and 2 are placed symmetrically in the object optical path and the reference optical path, so as to compensate the phase change introduced by the microscopic objective lens 1. In the optical path device in Fig. 3b, the object light and the reference light pass through the same long working distance microscope objective to compensate for the phase change introduced by the object light passing through the microscope objective. In the light path in Fig. 3c, the reference light is parallel light. Phase changes introduced by the microscope objective need to be removed during digital processing.

Further, said obtaining multi-frame phase shift hologram includes:

Obtain the object light U _O and reference light U _R reaching the high frame rate area detector:

In the formula, t is the time, a _O ,ω _O , are the amplitude, angular frequency and phase of the object light respectively; a _R , ω _R , are the amplitude, angular frequency and phase of the reference light, respectively;

The ideal noise-free hologram s produced by the interference of object light and reference light is expressed as:

where ω _H is the heterodyne angular frequency, f _H is the heterodyne frequency of the two AOMs, and their relationship is:

ω _H = ω _O - ω _R = 2πf _H ;

Contains random noise The total signal of the multi-frame phase-shifted hologram is:

In this way, a multi-frame phase shift hologram s _total (t(n)) is obtained, where n is the serial number of each frame, and t(n) is the sampling time corresponding to the nth frame serial number, n=1,2,..., N _total , where N _total is the total number of frames of the obtained hologram.

Further, the data processor performs time-domain spectral transformation on the signal of each pixel in the multi-frame phase-shifted hologram, and extracts the amplitude and phase of the spectral value at the heterodyne frequency, including:

The signal of each pixel point in the multi-frame phase-shifted hologram is respectively subjected to time-domain spectrum transformation by using Fast Fourier Transform FFT to obtain the time-domain spectrum S(ω):

In the formula, ω represents the heterodyne angular frequency, s _total (t(n)) is a multi-frame phase shift hologram, n is the serial number of each frame, t(n) is the sampling time corresponding to the nth frame serial number, and N _total is The total number of hologram frames obtained;

Extract the value S(ω _H ) of the time-domain spectrum S(ω) at the heterodyne angular frequency ω _H :

In the formula, σ is the standard deviation;

Thus, the amplitude of the spectral value of the time-domain spectrum S(ω) at the heterodyne frequency is obtained and phase

Further, the extracting the target amplitude and target phase from the amplitude and phase of the spectral value at the heterodyne frequency, so as to realize holographic reconstruction includes:

amplitude middle is a constant, after normalization processing, the normalized amplitude a _norm of the target is obtained:

In the formula, max and min respectively represent the two-dimensional spatial distribution data Take the maximum and minimum values;

Set the position of the reference optical path lens and the objective optical path lens to be symmetrical, then the phase of the object light when the target is not included equal target aspect Expressed as:

Further, the complete target information O _result is obtained according to the obtained target amplitude and target phase, which is expressed as:

It should be noted that the specific implementation manners of the functions implemented by the various functional modules included in the above apparatus have been described in detail in the previous embodiments, so details will not be repeated here.

Those skilled in the art can clearly understand that for the convenience and brevity of description, only the division of the above-mentioned functional modules is used as an example for illustration. In practical applications, the above-mentioned function allocation can be completed by different functional modules according to needs. The internal structure of the device is divided into different functional modules to complete all or part of the functions described above.

In order to facilitate the understanding of the present invention, the following experiments were carried out in conjunction with the above-mentioned examples.

In this experiment, the main devices used are: a 532nm single-mode continuous laser, two AOMs with a heterodyne value of 10Hz, a 10x microscope objective lens (NA=0.25) for the lens, and a high frame rate area array detector (1280 *1024pixels, 500fps max) detection frame rate 100Hz, unstained mouse cells were selected as the target.

step 1:

Using the above device, 100 frames of phase-shifted holograms were obtained within 1 s, including 10 periods and each period contained 10 steps of phase shift.

For clear display, a hologram with a size of 120*120 pixels can be cut out from a full field of view of 1280*1024 pixels.

Use 100 frames of holograms to build a 120*120*100 three-dimensional data matrix, and a certain pixel (x, y) on the kth hologram corresponds to the (x, y, k) coordinate point data on the matrix. Therefore, for a certain pixel point on the detector, we obtain 100 sampling data at this point. The sampling signal at a certain point is similar to the previous Figure 2b, and it can be seen that the phase shift is linear and accurate.

Step 2:

Use FFT to process the signal to obtain the time-domain spectrum, as shown in Figure 4.

Step 3:

In the time-domain spectrum, it can be seen that the frequency point where the twin image is located is clearly separated from the target frequency point. The amplitude and phase information are extracted at the heterodyne frequency 10Hz corresponding to the heterodyne angular frequency ω _H , and the amplitude information is normalized to obtain the amplitude information of the target. By performing the above operations on each pixel, two-dimensional target amplitude and phase information can be obtained.

Step 4:

After the above steps, the complete information of the target is obtained. The 3D image is shown in Figure 5, and it can be seen that the reconstruction target is clear.

The above experimental proof demonstrates the scheme, and proves the feasibility and advantages of the scheme.

The solutions provided by the above-mentioned embodiments of the present invention can obtain hundreds or even thousands of frames of phase shift holograms by using low-frequency heterodyne and high-frame-frequency area array detection. Then use the time-domain spectrum analysis algorithm to obtain the amplitude and phase information of each point, so that the three-dimensional image of the target can be obtained. Compared with the traditional technology, the main advantage of this technology is that it can effectively reduce the influence of random noise, so it is more accurate and has better environmental adaptability. Furthermore, under different intensity noises, the heterodyne frequency and detection frame frequency can be flexibly designed to obtain multi-frame phase-shifted holograms with different periods and different synchronous numbers to obtain better noise immunity.

The above is only a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any person familiar with the technical field can easily conceive of changes or changes within the technical scope disclosed in the present invention. Replacement should be covered within the protection scope of the present invention. Therefore, the protection scope of the present invention should be determined by the protection scope of the claims.

Claims (10)

1. A multi-frame phase-shifting digital holography method, characterized in that, comprising: Two acousto-optic modulators (AOM) are used to generate low-frequency heterodyne corresponding to the modulated object light and reference light, and are detected by a high-frame-frequency area array detector to obtain a multi-frame phase-shift hologram; Carry out time-domain spectral transformation to the signal of each pixel point in the multi-frame phase-shifted hologram respectively, and extract the amplitude and phase of the spectral value at the heterodyne frequency; The target amplitude and target phase are extracted from the amplitude and phase of the spectral value at the heterodyne frequency, thereby realizing holographic reconstruction. 2. The method according to claim 1, wherein said obtaining a multi-frame phase shift hologram comprises: Obtain the object light U _O and reference light U _R reaching the high frame rate area detector: In the formula, t is the time, a _O ,ω _O , are the amplitude, angular frequency and phase of the object light respectively; a _R , ω _R , are the amplitude, angular frequency and phase of the reference light, respectively; The ideal noise-free hologram s produced by the interference of object light and reference light is expressed as: where ω _H is the heterodyne angular frequency, f _H is the heterodyne frequency of the two AOMs, and their relationship is: ω _H = ω _O - ω _R = 2πf _H ; Contains random noise The total signal of the multi-frame phase-shifted hologram is: In this way, a multi-frame phase shift hologram s _total (t(n)) is obtained, where n is the serial number of each frame, and t(n) is the sampling time corresponding to the nth frame serial number, n=1,2,..., N _total , where N _total is the total number of frames of the obtained hologram. 3. The method according to claim 1 or 2, characterized in that, the signal of each pixel point in the multi-frame phase-shifted hologram is respectively subjected to time-domain spectrum conversion, and the spectrum at the heterodyne frequency is extracted The magnitude and phase of the values include: The signal of each pixel point in the multi-frame phase-shifted hologram is respectively subjected to time-domain spectrum transformation by using Fast Fourier Transform FFT to obtain the time-domain spectrum S(ω): <mrow><mi>S</mi><mrow><mo>(</mo><mi>&amp;omega;</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><msub><mi>N</mi><mrow><mi>t</mi><mi>o</mi><mi>t</mi><mi>a</mi><mi>l</mi></mrow></msub></mfrac><munderover><mo>&amp;Sigma;</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><msub><mi>N</mi><mrow><mi>t</mi><mi>o</mi><mi>t</mi><mi>a</mi><mi>l</mi></mrow></msub></munderover><msub><mi>S</mi><mrow><mi>t</mi><mi>o</mi><mi>t</mi><mi>a</mi><mi>l</mi></mrow></msub><mi>&amp;omega;</mi><mrow><mo>(</mo><mi>t</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo></mrow><msup><mi>e</mi><mrow><mo>-</mo><mi>j</mi><mrow><mo>(</mo><mi>&amp;omega;</mi><mi>t</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo></mrow></mrow></msup><mo>;</mo></mrow> In the formula, ω represents the heterodyne angular frequency, s _total is the total signal of the multi-frame phase-shifted hologram, n is the serial number of each frame, t(n) is the sampling time corresponding to the nth frame serial number, and N _total is the obtained hologram Total number of frames; Extract the value S(ω _H ) of the time-domain spectrum S(ω) at the heterodyne angular frequency point ω _H : In the formula, σ is the standard deviation; Thus, the amplitude of the spectral value of the time-domain spectrum S(ω) at the heterodyne frequency is obtained and phase 4. method according to claim 3, is characterized in that, extracts target amplitude and target phase from the amplitude and the phase of spectrum value at described heterodyne frequency place, thereby realizes holographic reconstruction, obtains target complete information, and it comprises: amplitude middle is a constant, after normalization processing, the normalized amplitude a _norm of the target is obtained: <mrow><msub><mi>a</mi><mrow><mi>n</mi><mi>o</mi><mi>r</mi><mi>m</mi></mrow></msub><mo>=</mo><mfrac><mrow><msup><mi>e</mi><mrow><mo>-</mo><mfrac><msup><mi>&amp;sigma;</mi><mn>2</mn></msup><mn>2</mn></mfrac></mrow></msup><msub><mi>a</mi><mi>R</mi></msub><msub><mi>a</mi><mn>0</mn></msub><mo>-</mo><mi>min</mi><mrow><mo>(</mo><msup><mi>e</mi><mrow><mo>-</mo><mfrac><msup><mi>&amp;sigma;</mi><mn>2</mn></msup><mn>2</mn></mfrac></mrow></msup><msub><mi>a</mi><mi>R</mi></msub><msub><mi>a</mi><mn>0</mn></msub><mo>)</mo></mrow></mrow><mrow><mi>max</mi><mrow><mo>(</mo><msup><mi>e</mi><mrow><mo>-</mo><mfrac><msup><mi>&amp;sigma;</mi><mn>2</mn></msup><mn>2</mn></mfrac></mrow></msup><msub><mi>a</mi><mi>R</mi></msub><msub><mi>a</mi><mn>0</mn></msub><mo>)</mo></mrow><mo>-</mo><mi>min</mi><mrow><mo>(</mo><msup><mi>e</mi><mrow><mo>-</mo><mfrac><msup><mi>&amp;sigma;</mi><mn>2</mn></msup><mn>2</mn></mfrac></mrow></msup><msub><mi>a</mi><mi>R</mi></msub><msub><mi>a</mi><mn>0</mn></msub><mo>)</mo></mrow></mrow></mfrac><mo>;</mo></mrow> In the formula, max and min respectively represent the two-dimensional spatial distribution data Take the maximum and minimum values; Set the position of the reference optical path lens and the objective optical path lens to be symmetrical, then the phase of the object light when the target is not included equal target aspect Expressed as: And, according to the obtained target amplitude and target phase, the complete target information O _result is obtained, which is expressed as: 5. A multi-frame phase-shift digital holographic device, characterized in that it includes: a preset optical path, a high frame rate area array detector and a data processor; wherein: The preset optical path includes two acousto-optic modulators AOM, which are used to generate low-frequency heterodyne for the corresponding modulation object light and reference light; The high frame rate area array detector detects the light in the imaging mirror of the preset optical path, obtains a multi-frame phase shift hologram, and transmits it to the data processor; The data processor performs time-domain spectral transformation on the signal of each pixel point in the multi-frame phase-shifted hologram, and extracts the amplitude and phase of the spectral value at the heterodyne frequency; then from the heterodyne frequency The target amplitude and target phase are extracted from the amplitude and phase of the spectral value, thereby realizing holographic reconstruction. 6. The device according to claim 5, wherein the preset optical path comprises: One laser, one half-wave plate, three polarization beamsplitters, two mirrors, two acousto-optic modulators (AOM), two apertures, one spatial filter, two 1/4 wave plates, two microscope objectives , a reference mirror and an imaging mirror; where: The laser light emitted by the laser is divided into two beams of polarized light, object light and reference light, through a half-wave plate and a polarization beam splitter prism 1, the object light is polarized light in the p direction, and the reference light is polarized light in the s direction; The object light passes through the reflector 1, enters the AOM 1, and the outgoing -1 order and 0 order lights are blocked by the diaphragm 1, and the outgoing +1 diffracted light enters the spatial filter through the polarization beam splitter prism 2, and after filtering The light beam sequentially passes through the polarization beam splitter 3, 1/4 wave plate 1, and the microscope objective lens 1, and then irradiates the target; the reflected light carrying target information passes through the microscope objective lens 1, 1/4 wave plate 1, and becomes polarized in the s direction The light is injected into the imaging mirror after passing through the polarization beam splitter 3 and the polarizer placed at an angle of 45 degrees; After the reference light enters the AOM 2, the emitted -1st order and 0th order light are blocked by the diaphragm 2, and the outgoing +1st order diffracted light enters the spatial filter after passing through the reflector 2 and the polarization beam splitter prism 2, and the filtered beam passes through the Polarizing beam splitter 3, 1/4 wave plate 2, and microscopic objective lens 2 are irradiated onto the reference mirror; the reflected light turns into p-direction polarized light after passing through microscopic objective lens 2, 1/4 wave plate 2, and then polarized The dichroic prism 3 and the polarizing plate placed at an angle of 45 degrees are injected into the imaging mirror. 7. The device according to claim 5, wherein the preset optical path comprises: One laser, one half-wave plate, three polarization beamsplitters, two mirrors, two acousto-optic modulators (AOMs), two apertures, one spatial filter, one beamsplitter, two 1/4 wave plates, one Microscope objective lens, a reference mirror and an imaging mirror; wherein: The laser light emitted by the laser is divided into two beams of polarized light, object light and reference light, through a half-wave plate and a polarization beam splitter prism 1, the object light is polarized light in the p direction, and the reference light is polarized light in the s direction; The object light passes through the reflector 1, enters the AOM 1, and the outgoing -1 order and 0 order lights are blocked by the diaphragm 1, and the outgoing +1 diffracted light enters the spatial filter through the polarization beam splitter prism 2, and after filtering The light beam sequentially passes through the beam splitter prism, microscope objective lens, polarization beam splitter 3 and 1/4 wave plate 1, and then irradiates the target; the reflected light carrying target information becomes polarized light in the s direction after passing through the 1/4 wave plate 1, and then sequentially After passing through the polarizing beam-splitting prism 3, the microscopic objective lens, the beam-splitting prism and the polarizer placed at an angle of 45 degrees, it enters the imaging mirror; After the reference light enters the AOM 2, the emitted -1st order and 0th order light are blocked by the diaphragm 2, and the outgoing +1st order diffracted light enters the spatial filter after passing through the reflector 2 and the polarization beam splitter prism 2, and the filtered beam passes through the The beam-splitting prism, microscope objective lens, polarization beam-splitter prism 3 and 1/4 wave plate 2 are then irradiated onto the reference mirror; the reflected light becomes polarized light in the p direction after passing through the 1/4 wave plate 2, and then passes through the polarization beam-splitter prism 3 in turn , a microscope objective lens, a dichroic prism and a polarizer placed at an angle of 45 degrees are injected into the imaging mirror. 8. The device according to claim 5, wherein the preset optical path comprises: One laser, two half-wave plates, one polarizing beam splitter prism, two mirrors, two acousto-optic modulators AOM, two apertures, two spatial filters, one beam combining prism and one imaging mirror; where: The laser light emitted by the laser is divided into two beams of polarized light, object light and reference light, through the half-wave plate 1 and the polarization beam splitter prism, the object light is polarized light in the p direction, and the reference light is polarized light in the s direction; The object light passes through the half-wave plate 2 and the mirror 1 in sequence, and after entering the acousto-optic modulator AOM 1, the emitted -1 order and 0 order lights are blocked by the diaphragm 1, and the outgoing +1 diffracted light enters the spatial filter 1, After filtering, the light beam passes through the beam combining prism and the microscopic objective lens in turn and then irradiates the target; the reflected light carrying target information passes through the microscopic objective lens and the beam combining prism in turn and then enters the imaging mirror; After the reference light enters the AOM 2, the emitted -1st order and 0th order light are blocked by the diaphragm 2, and the outgoing +1st order diffracted light enters the spatial filter 2 after passing through the mirror 2, and the filtered beam passes through the beam combining prism and then shoots into the imaging mirror. 9. The device of claim 5, wherein: The obtaining of multi-frame phase-shifted holograms includes: Obtain the object light U _O and reference light U _R reaching the high frame rate area detector: In the formula, t is the time, a _O ,ω _O , are the amplitude, angular frequency and phase of the object light respectively; a _R , ω _R , are the amplitude, angular frequency and phase of the reference light, respectively; The ideal noise-free hologram s produced by the interference of object light and reference light is expressed as: where ω _H is the heterodyne angular frequency, f _H is the heterodyne frequency of the two AOMs, and their relationship is: ω _H = ω _O - ω _R = 2πf _H ; Contains random noise The total signal of the multi-frame phase-shifted hologram is: In this way, a multi-frame phase shift hologram s _total (t(n)) is obtained, where n is the serial number of each frame, and t(n) is the sampling time corresponding to the nth frame serial number, n=1,2,..., N _total , N _total is the total number of frames of the obtained hologram; The data processor performs time-domain spectral transformation on the signal of each pixel point in the multi-frame phase-shifted hologram, and extracts the amplitude and phase of the spectral value at the heterodyne frequency, including: The signal of each pixel point in the multi-frame phase-shifted hologram is respectively subjected to time-domain spectrum transformation by using Fast Fourier Transform FFT to obtain the time-domain spectrum S(ω): <mrow><mi>S</mi><mrow><mo>(</mo><mi>&amp;omega;</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><msub><mi>N</mi><mrow><mi>t</mi><mi>o</mi><mi>t</mi><mi>a</mi><mi>l</mi></mrow></msub></mfrac><munderover><mo>&amp;Sigma;</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><msub><mi>N</mi><mrow><mi>t</mi><mi>o</mi><mi>t</mi><mi>a</mi><mi>l</mi></mrow></msub></munderover><msub><mi>S</mi><mrow><mi>t</mi><mi>o</mi><mi>t</mi><mi>a</mi><mi>l</mi></mrow></msub><mi>&amp;omega;</mi><mrow><mo>(</mo><mi>t</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo></mrow><msup><mi>e</mi><mrow><mo>-</mo><mi>j</mi><mrow><mo>(</mo><mi>&amp;omega;</mi><mi>t</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo></mrow></mrow></msup><mo>;</mo></mrow> In the formula, ω represents the heterodyne angular frequency, s _total (t(n)) is a multi-frame phase shift hologram, n is the serial number of each frame, t(n) is the sampling time corresponding to the nth frame serial number, and N _total is The total number of hologram frames obtained; Extract the value S(ω _H ) of the time-domain spectrum S(ω) at the heterodyne angular frequency point ω _H : In the formula, σ is the standard deviation; Thus, the amplitude of the spectral value of the time-domain spectrum S(ω) at the heterodyne frequency is obtained and phase 10. The device according to claim 9, wherein the target amplitude and target phase are extracted from the amplitude and phase of the spectral value at the heterodyne frequency, so as to realize holographic reconstruction and obtain complete target information, which includes: amplitude middle is a constant, after normalization processing, the normalized amplitude a _norm of the target is obtained: <mrow><msub><mi>a</mi><mrow><mi>n</mi><mi>o</mi><mi>r</mi><mi>m</mi></mrow></msub><mo>=</mo><mfrac><mrow><msup><mi>e</mi><mrow><mo>-</mo><mfrac><msup><mi>&amp;sigma;</mi><mn>2</mn></msup><mn>2</mn></mfrac></mrow></msup><msub><mi>a</mi><mi>R</mi></msub><msub><mi>a</mi><mn>0</mn></msub><mo>-</mo><mi>min</mi><mrow><mo>(</mo><msup><mi>e</mi><mrow><mo>-</mo><mfrac><msup><mi>&amp;sigma;</mi><mn>2</mn></msup><mn>2</mn></mfrac></mrow></msup><msub><mi>a</mi><mi>R</mi></msub><msub><mi>a</mi><mn>0</mn></msub><mo>)</mo></mrow></mrow><mrow><mi>max</mi><mrow><mo>(</mo><msup><mi>e</mi><mrow><mo>-</mo><mfrac><msup><mi>&amp;sigma;</mi><mn>2</mn></msup><mn>2</mn></mfrac></mrow></msup><msub><mi>a</mi><mi>R</mi></msub><msub><mi>a</mi><mn>0</mn></msub><mo>)</mo></mrow><mo>-</mo><mi>min</mi><mrow><mo>(</mo><msup><mi>e</mi><mrow><mo>-</mo><mfrac><msup><mi>&amp;sigma;</mi><mn>2</mn></msup><mn>2</mn></mfrac></mrow></msup><msub><mi>a</mi><mi>R</mi></msub><msub><mi>a</mi><mn>0</mn></msub><mo>)</mo></mrow></mrow></mfrac><mo>;</mo></mrow> In the formula, max and min respectively represent the two-dimensional spatial distribution data Take the maximum and minimum values; Set the position of the reference optical path lens and the objective optical path lens to be symmetrical, then the phase of the object light when the target is not included equal target aspect Expressed as: Obtain the complete information O _result of the target according to the obtained target amplitude and target phase, which is expressed as: