A holographic multiplexing imaging method based on kramers-kroning relation

Abstract

The application discloses a holographic multiplexing imaging method based on Kramers-Kroning relation, relates to the field of high-throughput multi-channel multiplexing digital holographic imaging technology, and is based on the quantitative phase measurement and the improvement of the space-bandwidth product; the holographic multiplexing imaging method based on Kramers-Kroning relation directly detects the complex amplitude field without any constraint; in addition, the holographic multiplexing imaging method can allow the target spectrum and other spectra to overlap without sacrificing the space-bandwidth and the time resolution, can realize the quantitative optical imaging without scanning, and provides a new way for significantly improving the application of digital holography and quantitative optical imaging.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of high-throughput multi-channel multiplexed digital holographic imaging, and particularly relates to a holographic multiplexed imaging method based on Kramers-Kroning relationship. BACKGROUND

[0002] Digital holographic imaging technology, as a non-invasive and quantitative phase measurement optical imaging method, has attracted extensive attention since its inception. It quantifies the interaction between light and matter, not only detecting the intensity of light, but also detecting the phase map profile generated by the phase delay, thereby realizing the measurement of the three-dimensional morphology of the object, and characterizing important physical parameters such as the volume, dry mass and refractive index of biological cells, and the surface shape, waviness and roughness of industrial devices.

[0003] However, the wave nature of light will cause the recorded hologram to include not only the required target light field complex amplitude information, but also zero-order diffraction direct current term noise and twin conjugate image noise of the conjugate light field complex amplitude distribution. In addition, in order to successfully record the interference hologram, the conventional digital holographic microscopy usually uses a single-frequency laser for illumination, and the highly coherent time and spatial characteristics and the surface defects of optical devices in the optical system (such as lenses and beam splitting prisms) will introduce parasitic fringes and other coherent noise, which will inevitably reduce the imaging quality of the reconstructed intensity image and phase image. For the zero-order diffraction and twin image noise problem, two methods of on-axis phase shifting and off-axis holography are usually used. In on-axis phase shifting, the object light and the reference light propagate in the same direction, and by introducing different phase shift amounts between the object light and the reference light in turn and using simple numerical operations to suppress the zero-order diffraction and the twin image, high-precision and non-contact measurement of the original light field complex amplitude information can be realized. On-axis phase shifting can fully utilize the high-frequency information of the target light field, and has high imaging resolution, but it needs to collect multiple phase-shifted holographic images in time, and the imaging speed is slow, which cannot be used for observation of moving targets. Compared with on-axis phase shifting, off-axis holography can completely separate the object light spectrum and its conjugate spectrum in the frequency domain by introducing a tilted reference light to interfere with the object light, thereby realizing single-image quantitative phase reconstruction and having the ability of real-time imaging, but the frequency spectrum utilization rate is low and the imaging resolution is not high. In order to suppress parasitic fringes and other coherent noise, low-coherence illumination light source, partially coherent light source or LED illumination method can be used. Since the coherence length of the partially coherent light source is small, when a tilted reference light path is used in digital holographic technology to introduce a high-frequency carrier to suppress the interference of zero-order diffraction and twin image, the phase difference between the object light and the reference light is inevitably increased, which causes the imaging area corresponding to the object light phase difference meeting the coherence length requirement of the partially coherent light source to be reduced to 10% to 25% of the detector photosensitive area, making it difficult to realize full-field imaging.

[0004] With the development of modern optical technology, the demand for fast and high space-bandwidth product imaging is increasing. How to balance the spatial resolution, temporal resolution and field flux is a problem that needs to be solved in the field of holographic imaging. In 2004, Rodenburg et al. proposed a synthetic aperture technology of spatial scanning and splicing. This method uses a local illumination probe to scan the sample sub-aperture in space, and records a series of diffraction images in the far field with a camera in the back end. The optimal solution that meets the frequency domain and spatial domain support constraints is reconstructed through iterative phase recovery algorithm. Compared with GS (Gerchberg-Saxton) algorithm, it can achieve faster convergence and more accurate imaging. In 2012, Brady team proposed a multi-scale camera array imaging system, which used a combination of main lens and auxiliary lens to overcome the problem of off-axis aberration. In this system, the object forms a curved relay focal plane through a large main lens, and multiple small lens arrays adjust the images on the relay focal plane to their respective focal points for physical compensation and physical splicing, finally forming a large field of view image. In 2013, Zheng et al. proposed a computational imaging technology named Fourier ptychographic imaging. This technology combines with multi-angle illumination, uses low numerical aperture objective to realize frequency scanning imaging, realizes the movement of the sample spectrum in the frequency domain, and provides amplitude information in the spatial domain. In the frequency domain, the limited transfer function provides frequency support. Through alternating iterative solution, the optimal solution that meets the spatial and frequency support constraints is obtained, so that the frequency domain bandwidth of imaging is expanded, the numerical aperture of the imaging system is improved, and high-resolution reconstruction of the sample is realized without losing the size of the imaging field of view. In 2019, Park et al. used the half-space bandwidth of the sensor and applied the Kramers-Kronig relationship to the high space-bandwidth product imaging of micro-off-axis holography. For diffraction-limited optical systems, the real and imaginary parts of the object function constitute the causality law. Given one of them, the calculation result of the dual quantity can be obtained according to the spatial Kramers-Kronig relationship. This method does not require an iterative process and does not require strict constraints. It can convert the change of intensity in space to the change of phase, so as to realize three-dimensional refractive index tomography of micro objects, and provides a new way for non-interferometric holographic imaging. In 2022, Lee et al. proposed a single-frame full-field topography measurement based on the spatial domain Kramers-Kronig relationship. This method realizes high-resolution phase conversion to sample height information through single-frame imaging of a color camera.

[0005] At present, there are problems such as complex structure, large amount of calculation, low imaging efficiency, fuzzy sampling standard and poor dynamic observation effect in holographic microscopic imaging technology, which seriously restricts its development and application. Therefore, seeking a low-cost, large field of view, high spatial resolution, high temporal resolution and quantitative phase imaging method is the development trend of current holographic microscopic imaging. SUMMARY

[0006] In view of the problems of complex system structure, low imaging resolution, long iteration time, and low space-bandwidth utilization in the phase measurement process, the application provides a holographic multiplexing imaging method based on Kramers-Kroning relationship.

[0007] The technical problem to be solved by the application is solved by the following technical scheme.

[0008] A holographic multiplexing imaging method based on Kramers-Kroning relationship comprises the following steps.

[0009] S1, a multi-path holographic multiplexing interference imaging system is built, and each optical element is installed according to a predetermined position; a light source is expanded and collimated to generate a plane wave after passing through a collimating lens composed of a lens L1, a pinhole Pin and a lens L2, and a beam splitter BS1 divides the plane wave into illumination light and reference light; the illumination light forms object light after irradiating a sample, and the object light passes through a transmission bright field microscopic imaging system composed of a microscope objective MO and a lens L3 and a field multiplexing system composed of a beam splitter BS2, a mirror M2 and a mirror M3; a beam combiner BS3 combines the two paths of object light reflected by the mirror M2 and the mirror M3 and the reference light passing through the mirror M1, and maps them into a CCD camera through a polarizer P. If the incident directions of the two paths of object light are different, the positions of the two paths of object light distributed in the frequency domain are different when the intensity map is formed by interference of the reference light and the object light in the CCD camera plane, which can provide a basis for separation and reconstruction of the object light.

[0010] S2, the exit directions of the two paths of object light are adjusted by rotating the mirror M2 and the mirror M3, and the optical information of the two paths of object light mapped into the CCD camera plane corresponds to different areas of the sample to achieve the multiplexing effect of expanding the field of view.

[0011] S3, after the CCD camera collects the multiplexed hologram, a reconstruction algorithm based on Kramers-Kroning relationship is used to obtain the amplitude and phase maps of the sample.

[0012] Further, the light source in step S1 is the light emitted by a helium-neon laser with a wavelength of 632.8 nm.

[0013] Further, the sample in step S1 is located at the focal plane of the microscope objective MO.

[0014] Further, the magnification of the field multiplexing system in step S1 is 1.

[0015] Further, step S2 specifically comprises the following steps:

[0016] The reference light and the two paths of object light form a double-channel multiplexed image on the CCD camera plane, and the intensity I(r) of the hologram can be expressed as:

[0017] I(r) = R(r) + S1(r) + S2(r) 2 (1)

[0018] where R(r) is the reference light function, which is a circular function modulated by the coherent transfer function of the optical system, and its radius is determined by the numerical aperture NA of the system target; S1(r) and S2(r) are two object light functions, which carry the information of different fields of view in the system. For an ideal imaging system with a numerical aperture of NA and a lateral magnification of M, the target spectrum can be isolated in the frequency domain. An auxiliary function T(r) is introduced here, which is expressed as follows:

[0019]

[0020] Further derivation from equation (2) gives equation (3):

[0021]

[0022] where Re and Im represent the real part and the imaginary part of T(r), respectively, and i is the imaginary unit. When the object light signals of the two channels are complex amplitude distributions, T(r) should be a complex number, which contains the complex amplitude distribution on the hologram plane. Re[T(r)] can be expressed as:

[0023]

[0024] The real part of T(r) can be directly obtained from the hologram. The Kramers-Kroning relationship describes the mathematical relationship between the real part and the imaginary part of a causal square-integrable function, so the imaginary part of T(r) can be obtained according to the Kramers-Kroning relationship, and its relationship with the real part is:

[0025]

[0026] where p.v. represents the Cauchy principal value, τ is an intermediate variable, and r || The above method is based on the assumption that the auxiliary function introduced is continuous. In fact, since the hologram collected by the detector is pixelated, the analyticity in discrete space is not clear, and the discrete Hilbert transform extends this theory to discrete images composed of discrete analytic signals. The Hilbert transform of the real and imaginary parts is equivalent to applying the Hilbert transform to one variable of the function. Therefore, the contour integral can be replaced by:

[0027]

[0028] where sgn is the sign function, For two-dimensional Fourier transform, k = k1k || + k2k ⊥ denotes the coordinate in spatial frequency domain, k1and k2are the horizontal and vertical coordinate axes on the Fourier plane, respectively, k || and k ⊥ are the projections of the illumination wave vector k on the horizontal and vertical coordinate axes, respectively. The calculation of formula (6) is based on an analytically discrete signal, and in combination with formula (5), the imaginary part of T(r) can be derived.

[0029] In order to meet the applicability condition of Hilbert transform, T(r) needs to meet the requirement of analyticity, and formula (3) is further derived to obtain formula (7):

[0030] S1(r) + S2(r) = R(r) {e Re[T(r)]+iIm[T(r)] -1} (7)

[0031] Taylor expansion is performed on formula (2) to obtain formula (8):

[0032]

[0033] wherein n represents the highest order of the term contained in the expansion of formula (8) at a certain point. It is derived from formula (8) that the intensity of the reference light needs to exceed that of the object light to realize Taylor expansion, and formula (1) indicates that enhancing the intensity of the reference light helps to enhance the contrast of the object light in the frequency domain.

[0034] From the above theoretical analysis, it can be seen that two prerequisite conditions need to be met for the system to realize holographic multiplexing of intensity patterns: one is that the bandwidth of the sample is limited. Since a general optical system can be regarded as a system with limited bandwidth, the first condition is easy to meet; the second is that when the reference wave is introduced in the off-axis device, the complete spectrum of the sample is moved to the positive half-axis in the spatial frequency domain, and its real part and imaginary part are analytically, and are Hilbert transforms of each other. Since the sum of the analytic functions is still analytically, the holographic multiplexing imaging method based on the Kramers-Kroning relationship can break through the multiplexing limit of a single hologram compared with the traditional off-axis multiplexing method, and relaxes the restriction condition in the off-axis multiplexing.

[0035] Further, the step S3 specifically comprises the following steps:

[0036] (1) dividing the intensity I(r) of the hologram obtained in step S2 by the intensity of the reference light |R(r)| 2 ;

[0037] (2) performing logarithmic transformation on the result obtained in step (1), so that the real part Re[T(r)] of the auxiliary function T(r) is obtained;

[0038] (3) Hilbert transform Re[T(r)] to obtain the imaginary part Im[T(r)] of the auxiliary function T(r);

[0039] (4) Using the auxiliary function T(r) to solve the complex amplitude field of the sample from formula (7).

[0040] The beneficial effects of the present application are: the present application starts from the aspects of realizing quantitative phase measurement and improving the space-bandwidth product, and directly detects the complex amplitude field without applying any constraints by using the holographic multiplexing imaging method based on the Kramers-Kroning relationship; in addition, the holographic multiplexing imaging method can allow the target spectrum and other spectra to overlap without sacrificing the spatial bandwidth and time resolution, and can realize quantitative optical imaging without scanning, thereby providing a new way for significantly improving the application of digital holography and quantitative optical imaging. BRIEF DESCRIPTION OF DRAWINGS

[0041] Figure 1 It is the optical principle diagram of the multiplexing holographic interference imaging system described in the present application;

[0042] Figure 2 It is the multiplexing hologram and spectrum diagram of the multiplexing holographic interference imaging system described in the present application; wherein, figure (a) is a multiplexing hologram model, and figure (b) is a spectrum diagram of figure (a);

[0043] Figure 3 It is an effect diagram for improving the image quality by using the holographic multiplexing imaging method described in the present application; wherein, figure (a1) and figure (a2) are respectively an ideal amplitude graph and a phase graph in input object light 1; figure (b1) and figure (b2) are respectively an amplitude graph and a phase graph obtained by reconstructing object light 1 by using the holographic multiplexing imaging method described in the present application; figure (c1) and figure (c2) are respectively an amplitude graph and a phase graph obtained by reconstructing object light 1 by using a traditional filtering method;

[0044] Figure 4 It is an effect diagram for improving the image quality by using the holographic multiplexing imaging method described in the present application; wherein, figure (a1) and figure (a2) are respectively an ideal amplitude graph and a phase graph in input object light 2; figure (b1) and figure (b2) are respectively an amplitude graph and a phase graph obtained by reconstructing object light 2 by using the holographic multiplexing imaging method described in the present application; figure (c1) and figure (c2) are respectively an amplitude graph and a phase graph obtained by reconstructing object light 2 by using a traditional filtering method;

[0045] Figure 5 It is a peak signal-to-noise ratio comparison diagram of the amplitude and phase of object 1 (corresponding to object light 1) and object 2 (corresponding to object light 2). DETAILED DESCRIPTION

[0046] In order to make the technical means, creative features, purposes and effects of the present application easy to understand, the present application is further described below in combination with specific examples and drawings.

[0047] The present application provides a holographic multiplexing imaging method based on Kramers-Kroning relation (hereinafter referred to as the present method), which adopts a multi-channel holographic multiplexing interference imaging system as shown in Figure 1 The light source is a helium-neon laser with a wavelength of 632.8 nm, which is expanded and collimated to generate a plane wave after passing through a collimating lens composed of a lens L1, a pinhole Pin and a lens L2. The plane wave is divided into illumination light and reference light by a beam splitter BS1. After the illumination light irradiates the sample, object light is formed. The object light passes through a transmission bright field microscopic imaging system composed of a microscope objective MO (the sample is located at the focal plane of the microscope objective MO) and a lens L3, and a field multiplexing system composed of a beam splitter BS2, a mirror M2 and a mirror M3. The magnification of the field multiplexing system is 1. The two paths of object light reflected by the mirror M2 and the mirror M3 and the reference light passing through the mirror M1 are combined by a beam combiner BS3 and mapped into a CCD camera through a polarizer P. The two paths of object light and the reference light interfere to form a multiplexed hologram model in the camera plane, as shown in Figure 2 (a). Figure 2 (b) is the Fourier spectrum of the multiplexed hologram, in which the object spectrum overlaps with other spectra.

[0048] The reference light formed by the mirror M1 and the two paths of object light form a dual-channel multiplexed image on the CCD camera plane. The intensity I(r) of the hologram can be expressed as:

[0049] I(r) = R(r) + S1(r) + S2(r) 2 (1)

[0050] Wherein, R(r) is the reference light function, the coherent transfer function of the optical system modulates the outgoing sample wave, and the shape is generally circular, and the radius is determined by the numerical aperture NA of the system target; S1(r) and S2(r) are two paths of object light functions, which carry different field information in the system. For an ideal imaging system with a numerical aperture of NA and a lateral magnification of M, the target spectrum can be isolated in the frequency domain. An auxiliary function T(r) is introduced here, and its expression is as follows:

[0051]

[0052] Further deduced from formula (2), formula (3) is obtained:

[0053]

[0054] where Re and Im represent the real and imaginary parts of T(r), respectively, and i is the imaginary unit. When both channels of the object light signal are complex-amplitude distributions, T(r) should be a complex number, containing the complex-amplitude distribution on the hologram plane. Re[T(r)] can be expressed as:

[0055]

[0056] The real part of T(r) can be directly obtained from the hologram. The Kramers-Kroning relation describes the mathematical connection between the real and imaginary parts of a causal, square-integrable function, so the imaginary part of T(r) can be obtained according to the Kramers-Kroning relation, which is related to the real part as:

[0057]

[0058] where p.v. represents the Cauchy principal value, τ is an intermediate variable, and r || is the projection of the position vector r on the horizontal coordinate axis. The above method is valid on the premise that the auxiliary function introduced is continuous. In fact, since the hologram collected by the detector is pixelated, the analyticity in the discrete space is not clear, and the discrete Hilbert transform extends this theory to a discrete image composed of discrete analytic signals. The Hilbert transform of the real and imaginary parts is equivalent to applying the Hilbert transform to one variable of the function. Therefore, the contour integral can be replaced by:

[0059]

[0060] where sgn is the sign function, is the two-dimensional Fourier transform, k = k1k || +k2k ⊥ represents the coordinates in the spatial frequency domain, k1 and k2 are the horizontal and vertical coordinate axes on the Fourier plane, respectively, k || and k ⊥ are the projections of the illumination wave vector k on the horizontal and vertical coordinate axes, respectively. The calculation of formula (6) is based on an analytically discrete signal, and in combination with formula (5), the imaginary part of T(r) can be derived. Thus, based on the obtained auxiliary function T(r), the complex-amplitude distribution of the object light field is directly reconstructed using formula (2), removing the constraint condition that multiple object light needs to be completely separated from its autocorrelation spectrum or cross-correlation spectrum, and only needs to satisfy the applicable condition of the Hilbert transform.

[0061] Therefore, T(r) needs to satisfy the requirement of analyticity, and formula (3) is further derived to obtain formula (7):

[0062] S1(r) + S2(r) = R(r) {e Re[T(r)]+iIm[T(r)]-1} (7)

[0063] The T(r) analyticity condition can be rewritten after Taylor expansion of equation (2) to get equation (8):

[0064]

[0065] From equation (8), it can be seen that the intensity of the reference light must exceed that of the object light to achieve Taylor expansion, and equation (1) shows that increasing the intensity of the reference light helps to increase the contrast of the object light in the frequency domain.

[0066] From the above theoretical analysis, it can be seen that two prerequisite conditions are required for the system to achieve holographic multiplexing of intensity patterns: one is that the bandwidth of the sample is limited. Since a general optical system can be regarded as a system with limited bandwidth, the first condition is easy to meet; the second is that when the reference wave is introduced in the off-axis device, the complete spectrum of the sample is moved to the positive half-axis in the spatial frequency domain, and its real part and imaginary part are analytic and are the Hilbert transform of each other. Since the sum of analytic functions is still analytic, the holographic multiplexing imaging method based on the Kramers-Kroning relationship can break through the multiplexing limit of a single hologram and relax the restriction conditions in off-axis multiplexing compared with the traditional off-axis multiplexing method.

[0067] The CCD camera is used to collect the multiplexed hologram, and the amplitude and phase images of the two channels in the sample are obtained using the reconstruction algorithm based on the Kramers-Kroning relationship, as shown in Figure 3 and Figure 4 .

[0068] Figure 3 (a1) and (a2) are ideal high-resolution amplitude and phase distribution maps of the object light 1. Figure 3 (b1) and (b2) show the reconstruction results of the amplitude and phase of the object light 1 using the method, while the results of the reconstruction of the amplitude and phase of the object light 1 using the traditional filtering method are shown in Figure 3 (c1) and (c2). It can be seen from Figure 3 that the resolution of the reconstructed amplitude and phase maps using the method is better than that of the traditional filtering method. Figure 4 (b1) and (b2) show the reconstruction results of the amplitude and phase of the object light 2 using the method, while the results of the reconstruction of the amplitude and phase of the object light 2 using the traditional filtering method are shown in Figure 4 (c1) and (c2). It can be seen from Figure 4 that the clarity of the reconstructed amplitude and phase maps using the method is higher than that of the traditional filtering method

[0069] In order to visually compare the image reconstruction effect, the peak signal-to-noise ratio is used as an evaluation index of image quality, and the amplitude and phase of object 1 (corresponding to object light 1) and the amplitude and phase of object 2 (corresponding to object light 2) are calculated respectively, and the calculation results are as shown in Figure 5 The higher the peak signal-to-noise ratio of the image is, the smaller the distortion degree of the image is, and the better the reconstruction quality is. Figure 5 It can be concluded that, in the case of overlap between the target spectrum and other spectra, the method can obviously improve the image reconstruction quality compared with the traditional filtering method, and realize the quantitative optical imaging without scanning; and compared with the traditional off-axis holography, the method can realize the double multiplexing of single-frame image information, and improve the space-bandwidth utilization of the system.

[0070] The basic principles and main features of the present application and the advantages of the present application are shown and described above. It should be understood by those skilled in the art that the present application is not limited by the above examples, and the above examples and descriptions in the specification are only to illustrate the principles of the present application. Without departing from the spirit and scope of the present application, various changes and improvements can be made to the present application, and these changes and improvements all fall within the scope of the present application. The scope of protection of the present application is defined by the appended claims and their equivalents.

Claims

1. A holographic multiplexing imaging method based on Kramers-Kroning relation, characterized in that, The method comprises the following steps: S1, a multi-path holographic multiplexing interference imaging system is built, and each optical element is installed according to a predetermined position; a light source is expanded and collimated to generate a plane wave after passing through a collimating lens composed of a lens L1, a pinhole Pin and a lens L2, and the plane wave is divided into illumination light and reference light by a beam splitter BS1; the illumination light irradiates a sample to form object light, and the object light passes through a transmission bright field microscopic imaging system composed of a microscope objective MO and a lens L3 and a field multiplexing system composed of a beam splitter BS2, a mirror M2 and a mirror M3; the two paths of object light reflected by the mirror M2 and the mirror M3 and the reference light passing through the mirror M1 are combined by a beam combiner BS3 and mapped into a CCD camera through a polarizer P; S2, the exit directions of the two paths of object light are adjusted by rotating the mirror M2 and the mirror M3, and the optical information in the CCD camera plane corresponding to different areas of the sample is obtained by mapping the two paths of object light, so that the multiplexing effect of expanding the field of view is achieved; S3, after the CCD camera collects the multiplexed hologram, a reconstruction algorithm based on the Kramers-Kroning relationship is used to obtain the amplitude and phase maps of the sample.

2. The Kramers-Kroning relation based holographic multiplexing imaging method of claim 1, characterized in that: In the step S1, the light source is a helium-neon laser with a wavelength of 632.8 nm.

3. The Kramers-Kroning relation based holographic multiplexing imaging method of claim 1, characterized in that: In the step S1, the sample is located at the focal plane of the microscope objective MO.

4. The Kramers-Kroning relation based holographic multiplexing imaging method of claim 1, characterized in that: In the step S1, the magnification of the field multiplexing system is 1.

5. The Kramers-Kroning relation based holographic multiplexing imaging method of claim 1, wherein, The step S2 specifically comprises the following steps: The reference light and the two paths of object light form a double-channel multiplexed image on the CCD camera plane, and the intensity I(r) of the hologram is expressed as: I(r) = R(r) + S1(r) + S2(r) 2 (1) Wherein, R(r) is a reference light function; S1(r) and S2(r) are two paths of object light functions; An auxiliary function T(r) is introduced, and the expression is as follows: Further derivation from formula (2) obtains formula (3): Wherein, Re and Im represent the real part and the imaginary part of T(r) respectively, and i is the imaginary unit; Re[T(r)] is expressed as: The real part of T(r) is directly obtained from the hologram; the imaginary part of T(r) is obtained according to the Kramers-Kroning relationship, and the relationship between the real part and the imaginary part is as follows: where p.v. denotes the principal value of the Cauchy principal value, τ is an intermediate variable, r || is the projection of the position vector r on the horizontal coordinate axis; The contour integral is replaced by: where sgn is the sign function, is the two-dimensional Fourier transform, k = k1k || +k2k ⊥ denotes the coordinates in the spatial frequency domain, k1and k2are the horizontal and vertical coordinate axes on the Fourier plane, respectively, k || and k ⊥ are the projections of the illumination wave vector k on the horizontal and vertical coordinate axes, respectively. Further derivation of formula (3) obtains formula (7): S1(r) + S2(r) = R(r) {e Re[T(r)]+iIm[T(r)] -1} (7) Taylor expansion is performed on formula (2) to obtain formula (8): Wherein, n represents the highest order of the term contained in the expansion formula of formula (8) at a certain point.

6. The Kramers-Kroning relation based holographic multiplexing imaging method of claim 1, wherein, The step S3 specifically comprises the following steps: (1) dividing the intensity I(r) of the hologram obtained in step S2 by the intensity |R(r) of the reference light 2 ; (2) logarithmic transformation is performed on the result obtained in step (1), so that the real part Re[T(r)] of the auxiliary function T(r) is obtained; (3) the real part Re[T(r)] is subjected to Hilbert transformation to obtain the imaginary part Im[T(r)] of the auxiliary function T(r); (4) the complex amplitude field of the sample is solved from formula (7) by using the auxiliary function T(r).

Images