Non-iterative holographic axial localization using complex amplitude of diffraction-free vortices

Petr Bouchal; Zdeněk Bouchal

doi:10.1364/OE.22.030200

1. Introduction

During the last two decades, the development of light microscopy has been significantly influenced by implementation of the principles of digital holography enabled by a progress in the technical facilities. In this way, the microscopy techniques have been enriched by a possibility to operate with the phase information lost in the traditional lens imaging [1]. The compact and flexible digital holographic modules have been integrated into the systems of modern microscopy, allowing 3D imaging without mechanical scanning, visualization and phase quantification of the transparent objects, or numerical refocusing and compensation of aberrations [2, 3]. An exceptional optical performance of digital holographic microscopy with a possibility of real-time phase measurement and imaging with a high axial resolution opened up numerous applications in material and life sciences, medical research and biophotonics [4].

Digital holographic microscopy has also been successfully applied to the single particle localization and tracking, which is important for many biological investigations [5]. It overcomes the limitations of the traditional optical imaging, in which the localization is possible only in a narrow region given by the depth of focus of the used microscope objective. In the techniques of digital holography, the 3D image field is numerically reconstructed from the 2D holographic record, so that the localization of the particles distributed throughout a volume can be performed by a single CCD snapshot. Because the holographic reconstruction does not result in the true 3D imaging, the reconstructed signal must be further processed to obtain precise positions of individual particles. In recent years, many algorithms have been proposed for analysis of the micro-particle holograms. In the common implementation, the holographic record is successively reconstructed in a number of planes being placed repeatedly with a small step behind the hologram. In the next stage, each particle is localized in the calculated 3D diffracted field using appropriate criteria. The algorithms based on the peak searching [6] or the quantification of the image sharpness [7] have been proposed to detect the best focusing plane of the particles. These approaches suffer from various constraints, such as a weak depth accuracy and time-consuming operation [8]. The accuracy of the axial localization can be increased in advanced methods based on the 3D deconvolution [9] or the inverse problem approach [8, 10]. These techniques are even more time-consuming than the methods of direct reconstruction, especially in the case of a large number of particles, and their applicability is limited to the particles modeled by few geometrical parameters. In the localization methods of digital holography, a compromise between the accuracy and the processing time is crucial.

In optical microscopy, the axial localization can be advantageously performed by the rotation of the out of focus image [11]. The method operates with the optical vortices whose interference results in the rotational effects induced by a defocusing [12–14]. The axial position of the particles is determined from the angular rotation of the point spread function (PSF) used as an extremely sensitive indicator of the defocusing [12]. The method has been well established in optical microscopy and successfully applied to a nanoscale localization and tracking of point-like objects and single-molecule fluorescent imaging beyond the diffraction limit [15, 16]. The disadvantage of this technique is a narrow range of the axial localization restricted by the depth of focus of the used lens. The method is also less flexible in controlling the PSF and has a lower energy efficiency.

In this paper, a concept of digital vortex holography is for the first time proposed and implemented to demonstrate a dynamic localization and tracking of particles. It combines advantages of the holographic imaging with an advanced depth estimation using all-digital rotating vortices. In the presented technique, the image rotation is achieved by virtual operations applied to the holographic image reconstructed at an arbitrary target plane. In contrary to the optical methods operating with the intensity detection, only the real part of the complex amplitude can be processed in digital holography. In this case, the defocusing rotation does not need interference and can be monitored in a single vortex mode. The vortex field used for the axial localization is obtained by a spiral phase modulation of a single radial component of the digitally established spatial spectrum. Such spectral filtering results in the image structures analogous to the diffraction-free beams [17–19]. This approach provides an exceptional variability, which can be utilized for a full control of the rotation rate, extension of the localization range and optimal shaping and scaling of the image spot. The positions of all particles distributed in a sample volume are directly determined from the hologram reconstruction at a single image plane. In comparison with the known localization techniques of digital holography, the method is flexible and simple to implement, because it avoids a time-consuming multiplane hologram reconstruction. In addition, the method ensures elimination of the zero-order term and the twin holographic image, so the particles can be localized by a single in-line hologram without the phase shifting processing [20]. This is advantageous for a real-time localization and tracking of particles.

The paper is organized as follows. First, the basic principles of the vortex image rotation and the axial localization are outlined in a unified approach involving both coherent and incoherent methods of digital holography. Using the computational model, the relationship between the rotating PSF and the parameters of the digital spiral modulation is examined and benefits of the vortex axial localization discussed. Finally, the performance of the method is experimentally demonstrated by a common-path holographic microscope based on the FINCH principle [21–25]. The PSF engineering and analysis of the localization accuracy is performed by the processing of the correlation record of a single point emitter. The application potential of the method is shown by a defocusing image rotation of 5μm fluorescent microspheres, dynamic tracking of 0.5μm polystyrene beads and 3D trajectory reconstruction of a selected polystyrene bead moving by Brownian motion.

2. Concept of the rotating vortex imaging in digital holography

The proposed holographic localization of the point-like objects operates with vortex images whose angular rotation serves as a sensitive indicator of the axial position. A conversion of the conventional point images to the rotating vortex structures is implemented in a digital holographic system shown in Fig. 1. In optical part, the point objects to be localized are holographically recorded on a CCD. As will be shown later, either coherent or incoherent light can be used for hologram acquisition. The following operations are fully digital. The hologram is first numerically reconstructed at a selected image plane using the Fresnel transform. At the used reconstruction plane, only the point sources located at the optically conjugate object plane are imaged sharply. The remaining point emitters distributed throughout the volume are out of focus at the reconstruction plane. In standard localization methods, repeated numerical reconstructions of the hologram must be carried out to achieve a successive focusing of all point objects and to find their axial position by the peak searching algorithm. In the proposed method, the position of all point objects is simultaneously determined from the holographic image obtained at a single reconstruction plane. The axial localization is carried out by the digital spiral processing which ensures transformation of the reconstructed point images to the diffraction-free vortex spots. Information about the longitudinal position of the individual point objects is encoded in the in-plane angular rotation of the vortices. The situation is schematically illustrated in Fig. 1. The point images that are in focus at the reconstruction plane become vortices with unturned profile. The out of focus point images are transformed to the vortex structures with the angular rotation proportional to the defocusing shift Δz′. The positive and negative defocusing is converted to the clockwise and counter-clockwise vortex rotation, respectively.

Fig. 1 Block diagram with three basic parts of the holographic localization of point-like objects by the diffraction-free vortices: optical hologram acquisition, digital hologram reconstruction and conversion of the holographic images to the vortex spots whose angular rotation indicates axial position.

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3. Digital implementation of the rotating diffraction-free vortices

For applications, a computational model providing a relationship between the angular rotation of the vortices and the axial position of the examined point-like objects is required. Since the vortex localization is based on the processing of the holographic image, the recording and digital reconstruction of a set of point emitters must be included into the modeling. The hologram is proportional to the intensity of the interference pattern, T ∝ |E_R + E_S|², created by the signal and reference waves with the scalar complex amplitudes E_R and E_S, respectively. Besides information on the image to be reconstructed, $E_{S} E_{R}^{*}$ , the hologram contains also the zero-order term, |E_R|² + |E_S|², and the twin image, $E_{S}^{*} E_{R}$ . To eliminate an influence of the undesirable hologram terms, the off-axis recording geometry must be used in optical holography. It ensures the desired spatial separation of the reconstructed image but deteriorates the space-bandwidth product [26]. In digital holography, the advantageous in-line geometry can be used. The separation of the reconstructed image is then obtained by the numerical processing of three phase-shifted records of the object [20]. In the proposed axial localization, the zero-order term and the twin holographic image are eliminated by the filtering of the spatial spectrum, so that the time-consuming phase shifting is not needed. The reconstructed image is obtained by the Fresnel transform, E ∝ FrT{T}, where $T = E_{S} E_{R}^{*}$ is the recorded hologram after elimination of the undesirable terms by any of the methods mentioned above.

In digital holography, the recording of the object is performed using coherent light, as shown in Fig. 2(a). A coherent reference wave illuminates the point scatterers which create divergent spherical waves. Neglecting mutual interferences of weak scattered waves, the hologram arises due to interference of the individual spherical waves with the initial reference wave. In this case, the complex amplitudes E_R and E_S can be written as E_R = e_R exp(iφ_R) and E_S = ∑_m e_{S_m} exp(iφ_{S_m}), where e_R, φ_R and e_{S_m}, φ_{S_m} are amplitudes and phases of the reference wave and the spherical waves emitted by the point scatterers, respectively. The hologram to be reconstructed is then given as

T \propto \sum_{m} e_{S_{m}} e_{R}^{*} exp [i (φ_{S_{m}} - φ_{R})] .

The proposed axial localization is also applicable to incoherent digital holography based on a self-interference of waves. The principle of the hologram recording is obvious from Fig. 2(b). The point objects are illuminated by spatially incoherent light, so that the scattered spherical waves are mutually uncorrelated and their interference with the illuminating wave is also impossible. In order to achieve light interference needed in holography, each spherical wave coming from the object is divided into two waves with different shapes of wavefront [21]. As the created waves originate from the same object point, they are spatially correlated. If the optical path difference does not exceed the coherence length, the waves interfere. Unlike coherent holography, any global reference wave does not exist. For each point of the object, the local signal and reference waves are created enabling an acquisition of the holographic records with incoherent light. The splitting of the waves can be advantageously done using a spatial light modulator (SLM), which also allows to adjust amplitudes and phases of the individual reference waves. The complex amplitude of the m-th reference wave then can be written as E_{R_m} = e_{R_m} exp(iφ_{R_m}). The overall record of the object is created by summing the intensities of the individual point holograms. After elimination of the undesirable terms, the reconstructed hologram can be written as

T \propto \sum_{m} e_{S_{m}} e_{R_{m}}^{*} exp [i (φ_{S_{m}} - φ_{R_{m}})] .

As is obvious, the holograms Eqs. (1) and (2) have a formally identical form, regardless of whether they were recorded in coherent or incoherent light.

Fig. 2 Digital conversion of the holographic point images to the diffraction-free vortices with angular rotation depending on the defocusing. The vortex axial localization is applicable to both (a) coherent holography with a global reference wave and (b) incoherent self-interference holography using local reference waves created by a microscope objective (MO) and a SLM.

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The principle of the axial localization is demonstrated by a single point emitter placed on the optical axis to simplify mathematical relations. In this case, the record is represented by only one term of the hologram Eq. (1). In a paraxial approximation using paraboloidal waves, the hologram can be written as the transmission function of a lens with the focal length f_L,

T \propto e_{S} e_{R}^{*} exp (i \frac{k {| r_{⊥} |}^{2}}{2 f_{L}}),

where r_⊥ = (x, y) is the position vector at a CCD used for the hologram detection, k = 2π/λ, and λ denotes the wavelength. The focal length of the holographic lens is determined by the geometry of interfering waves and can be written as 1/f_L = 1/z_S − 1/z_R, where z_S and z_R denote the distance of the center of the signal and reference waves from the CCD. The object space defocusing then can be written as Δz = z_S − z₀, where z₀ determines the position of a reference object point. In incoherent self-interference holography, the waves emitted by the point scatterers must be transformed by the microscope objective and the SLM to obtain distances z_S and z_R and to find the dependence of the focal length f_L on the defocusing Δz [22]. In the standard numerical reconstruction of the image, the hologram T is illuminated by a plane wave and the transmitted light propagated to a selected target plane by the Fresnel transform. To reach the image rotation needed for the axial localization, a digital spiral processing is applied. Its operation includes the direct and inverse Fourier transform of the reconstructed image with the spiral phase modulation and the radial amplitude filtering applied in between. If the transmission of the spiral filter is denoted as S, and the direct and inverse Fourier transform as ℱ and ℱ⁻¹, respectively, the complex amplitude of the signal after the spiral image processing can be written as

E^{'} \propto ℱ^{- 1} {S \cdot ℱ {F r T (T)}} .

To examine the image rotation, the full form of Eq. (4) is written for the reconstruction plane at a distance z′ behind the CCD,

\begin{array}{r} E^{'} \propto \int_{- \infty}^{\infty} S (F_{⊥}) \int_{- \infty}^{\infty} exp [- i k \frac{{| R_{⊥} |}^{2}}{2 z^{'}}] \int_{- \infty}^{\infty} exp [i k \frac{{| r_{⊥} |}^{2}}{2} Δ κ] exp [i k \frac{r_{⊥} \cdot R_{⊥}}{z^{'}}] d r_{⊥} \\ \times exp [i k F_{⊥} \cdot (\frac{R_{⊥}}{f_{1}} + \frac{r_{⊥}^{'}}{f_{2}})] d R_{⊥} d F_{⊥}, \end{array}

where Δκ = 1/f_L − 1/z′, f₁ and f₂ are the focal lengths of the virtual lenses L₁ and L₂ used in the direct and inverse Fourier transform, and the vectors R_⊥ = (X, Y), F_⊥ = (F_x, F_y) and r′_⊥ = (x′, y′) represent the transverse Cartesian coordinates at the reconstruction plane, the Fourier plane and the output plane of the 4-f system, respectively. Performing the Fourier transform of the quadratic phase functions, we can write

E^{'} \propto \int_{- \infty}^{\infty} S (F_{⊥}) exp [i k \frac{{| F_{⊥} |}^{2} Δ z^{'}}{2 f_{1}^{2}}] exp [i k \frac{F_{⊥} \cdot r_{⊥}^{'}}{f_{2}}] d F_{⊥},

where Δz′ = z′ − f_L.

In optical implementation of the axial localization by the double-helix PSF [11–13, 15, 16], the depth information is extracted from the angular rotation of the image recorded by a CCD. When detecting the image intensity, the rotational effects occur due to interference of vortices. In experiments, a superposition of several vortex beams with appropriately designed parameters must be created. In the proposed method, the localization of particles is implemented via digital operations, so the real part of the complex amplitude can be processed when examining the rotation of the image. In this case, a single vortex mode is sufficient for determination of the axial position. Accepting the principles of a diffraction-free propagation of light, the vortex beam can be advantageously formed by the spiral phase modulation of a single radial component of the spatial spectrum,

S (F_{⊥}) = δ (| F_{⊥} | - F_{0}) exp (i l φ),

where δ is the Dirac delta function and F₀ is the radius of a narrow annular filter at the Fourier plane. The chosen radial spectral component is modulated by the helical phase determined by the topological charge l and the azimuthal coordinate φ = arctan(F_y/F_x). Substituting Eq. (7) into Eq. (6) and assuming f₁ = f₂, we obtain

E^{'} \propto J_{l} (α r^{'}) exp (i l φ_{0}^{'} + i l φ^{'} + i β Δ z^{'}),

where r′ = |r′_⊥|, φ′ = arctan(y′/x′), φ′₀ = π/2 and J_l denotes the Bessel function of the first kind and l-th order. The parameters α and β are specified by the radial frequency ν₀ = F₀/(λf₁) used in the spatial filtering, and can be written as

α = 2 π ν_{0},

β = π λ ν_{0}^{2} .

By the spiral hologram processing given by Eqs. (4) and (7), the image complex amplitude Eq. (8) analogous to the diffraction-free vortex beam is obtained. Specifically, the expression is formally identical with the Bessel beam, in which the propagation coordinate is replaced by the image defocusing Δz′. The optical intensity of the image I′ = |E′|² creates a rotationally symmetrical spot. This is a reason why the single vortex beam can not be adopted in the optical localization methods based on the image rotation. If the real part of the complex amplitude E′ is used, e′ = (E′ + E′^*)/2, the image spot is azimuthally modulated and rotates under defocusing,

e^{'} \propto J_{l} (α r^{'}) cos (l φ_{0}^{'} + l φ^{'} + β Δ z^{'}) .

The Fourier filtering given by the Dirac delta function has been used to simplify mathematical description. In the practical hologram processing, the annular filter with a finite width and a Gaussian apodization of the slit may be used, resulting in the PSF profile with an advantageous damping of oscillations of the Bessel function. The localization demonstrated for the axial point object can be easily generalized to any lateral position of the point emitter using the concept described in [18,27].

Fig. 3 Experimental set-up for the particle localization by the rotating diffraction-free vortices in digital holographic microscopy using incoherently illuminated and fluorescent samples (TM-transmission module for LED illumination, FM-fluorescence excitation module, ML-mercury lamp, MO-microscope objective, RL₁, RL₂-relay lenses, EF₁-excitation filter, EF₂-emission filter, P-polarizer, BS-beam splitter, CGH-computer generated hologram and SLM-spatial light modulator).

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4. Benefits of the holographic axial localization using the diffraction-free vortices

4.1. Shaping, scaling and controlling the rotation rate of the image spot

When determining the position of the point objects, the absolute value |e′| of the image field Eq. (11) is processed. In this case, the image spot has 2l lobes with the maxima oriented at the angles

φ_{\max}^{'} = \frac{m π - l φ_{0}^{'} - β Δ z^{'}}{l}, m = 1, 2, \dots .

The orientation of the maxima depends on Δz′ so that the image rotates due to a defocusing. If the defocusing is changed by Λ′ representing a longitudinal period, the image spot is rotated by the angle π/l. This means that the image profiles obtained for Δz′ and Δz′ + Λ′ are in the indistinguishable angular positions. The longitudinal period is determined as Λ′ = π/β and can be adjusted by the radial spatial frequency ν₀ related to β by Eq. (10). Its value can be changed in the interval Λ′ ∈ < Λ′_min, ∞) that is limited by the shortest attainable period,

Λ_{\min}^{'} = \frac{1}{λ ν_{\max}^{2}},

where ν_max is the highest applicable radial spatial frequency. It depends on the geometry used for the hologram recording and can be written as ν_max = NA_L/λ, where NA_L is the numerical aperture of the holographic lens Eq. (3) given by the overlapping of interfering waves [24]. The rotation rate of the image Ω depends on the parameters of the digital spiral filter Eq. (7),

Ω \equiv \frac{d φ^{'}}{d Δ z^{'}} = - \frac{β}{l},

and can be simply driven by the radial spatial frequency ν₀. The spot size of the image can also be controlled by the spiral filtering. If the spiral phase with a topological charge l is used, the radius of the image spot can be estimated as

r_{0}^{'} = \frac{q_{l}}{2 π ν_{0}},

where q_l denotes the first root of the Bessel function J_l. The lateral scaling and controlling the rotation rate of the image is demonstrated in Fig. 5 of the experimental section.

Fig. 4 Experimental and computational facilities used in demonstrations of the PSF engineering: (a) point holographic record taken in the set-up shown in Fig. 3, (b) digital annular filter with a spiral phase modulation.

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Fig. 5 Resizing of the vortex image and control of the longitudinal period by the radial spatial frequency ν₀ of the narrow annular filter (Δν = 0.05ν₀, l = 1). In demonstrations, the point correlation record shown in Fig. 4(a) was processed.

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4.2. Controlling the depth of localization

By the digital spiral processing Eq. (4) used with the Dirac delta filter, an ideal shape-invariant image spot analogous to the diffraction-free beam is obtained. Its lateral profile remains exactly the same and only the angular rotation indicates the image defocusing Δz′. In this case, the significant side oscillations of the Bessel profile appear making it difficult to localize nearby objects. If the annular filter with a Gaussian apodization of the slit is used instead of the Dirac delta filter, the undesirable oscillations are effectively damped. With such the filtering, an analogy with the ideal diffraction-free beam is lost and the lateral image profile changes with Δz′. The concept of the pseudo-nondiffracting beam with a limited propagation range [27] may be used to examine a lateral spreading of the image profile under defocusing. In practical cases, the width of the annular filter is chosen in a compromise between the damping of oscillations and the applicable localization depth. The damping of oscillations performed with the experimentally acquired point hologram is demonstrated in Fig. 6.

Fig. 6 Comparison of the sharp and the out of focus PSF obtained by the annular filters with the narrow and wide slit: (a) narrow slit and the defocusing-invariant PSF analogous to the Bessel beam, (b) wide slit with the Gaussian apodization and the PSF analogous to the Bessel-Gauss beam.

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4.3. Localization by the single in-line hologram

The particle localization based on the diffraction-free vortex imaging brings additional benefits for the in-line holographic techniques. The radial spiral filtering allows to eliminate the zero-order term and the twin holographic image, so that the sharp image can be reconstructed without phase shifting just from a single in-line hologram. The elimination of the twin image is efficient only if the radial frequency ν₀ of the applied digital filter sufficiently exceeds the width of the spatial spectrum of the image to be removed. This is assured by the inequality

ν_{0} > ν_{\max} (1 - \frac{f_{L}}{f_{L} + 0.5 f_{1}}) .

The possibility of using a single holographic record is important for localization of moving particles. All experimental demonstrations presented in this paper have been carried out by a single holographic record acquired in the in-line geometry.

4.4. Volumetric localization by a single plane hologram reconstruction

In standard methods of digital holography, the axial localization is performed by means of the peak searching algorithms requiring a time-consuming multiplane hologram reconstruction. In the proposed method, a single plane hologram reconstruction is sufficient for localization of point emitters distributed in a volume. In the digital spiral processing, both focused and out of focus point images reconstructed at the selected target plane are converted to the diffraction-free vortex spots of the same lateral size. The different axial positions of the examined point-like objects are determined directly from the in-plane angular rotation of the individual vortex images. The simplicity and flexibility of the proposed positioning provide options for a dynamic particle tracking.

4.5. Precision of the axial localization

The achievable accuracy of the axial localization can be estimated using the information content provided by the method applied. In the seminal paper on the depth estimation from diffracted rotation [11], a significant increase in the Fisher information was demonstrated for the rotating double-helix PSF in comparison with the depth estimation by a common image defocusing. This guarantees that the positioning by the rotating diffraction-free vortices is beneficial compared to the peak searching methods frequently used in digital holography.

4.6. Options for increasing the sensitivity of localization in the object space

The longitudinal period Λ′ and the spot size r′₀ given by Eqs. (13) and (15) determine the accuracy of the axial localization and the lateral resolution in the image space. To assess the performance of the method, adequate parameters specified for the object space are crucial. Basic information about the axial and lateral resolution in the object space is obtained by scaling the image parameters, r₀ = r′₀/m, Λ = Λ′/M, where m and M represent the lateral and longitudinal magnification of the holographic imaging, respectively. The scaling of the holographic image is particularly important for the FINCH techniques, where the violation of the Lagrange invariant was discovered [22, 28] and used for demonstration of the sub-diffraction resolution [23]. In this paper, the image scaling will be demonstrated for a basic FINCH geometry operating with plane reference waves. A detailed analysis of other FINCH configurations is beyond the scope of this paper and is a challenge for further research.

5. Experimental results

5.1. Implementation of the method

The proposed technique of the depth estimation is applicable to both coherent and incoherent holography. In this paper, the correlation holographic records taken in the FINCH setup [21–23] were used. The system was adapted to demonstrate options of the vortex particle localization in digital microscopy using incoherently illuminated and fluorescent specimens. The used experimental set-up is shown in Fig. 3. It was designed as a combination of the optical microscope with the common-path interferometer aided by a SLM. In the illumination path, the transmission and fluorescence modules were used enabling both a direct LED illumination and a mercury lamp fluorescent excitation of microscopic specimens. In the experiments, spatially distributed polystyrene beads or fixed fluorescent microspheres were used as the samples and placed near the focal plane of the microscope objective MO. Light transformed by the microscope objective was directed through the relay lenses RL₁ and RL₂ toward the reflective SLM (Hamamatsu X10468, 16mm × 12mm, 800 × 600 pixels). The SLM was operated as a divider providing splitting of each incident wave into two interfering waves with a desired shape of the wavefront. The computer generated hologram resulting in the interference configuration referred as standard was used. In this geometry, the correlation records of individual object points are created by interference of the plane and spherical waves. The relay lenses RL₁ and RL₂ and the polarizer P were used to achieve a perfect overlapping of interfering waves [24] and an optimal phase operation of the SLM. The correlation records of the examined samples were captured by the CCD (Retiga 4000 R, 15mm × 15mm, 2048 × 2048 pixels).

5.2. Demonstration of the PSF engineering

In the experiments, the theoretically discovered methods for the 3D shaping of the vortex image were verified. All demonstrations of the PSF engineering were prepared with the same point holographic record. The used point source was approximately realized by a single mode optical fiber illuminated by the LED (Thorlabs, 625nm) and placed near the object focal plane of the microscope objective (Olympus, NA = 0.25). The point hologram is shown in Fig. 4(a). It was recorded in the set-up shown in Fig. 3, where the optical fiber was placed instead of the sample. When processing the hologram, the reconstruction by the Fresnel transform was carried out, followed by the radial filtering and the spiral phase modulation Eq. (4) implemented in a virtual 4-f Fourier system. To separate the selected radial components of the spatial spectrum, an annular filter with the radius F₀ and the width of the slit ΔF was used. In the transmitting area of the filter, a spiral phase modulation with the topological charge l = 1 was applied, as shown in Fig. 4(b). By changing F₀, while retaining the focal length of the first Fourier lens f₁, the radial frequency ν₀ was varied resulting in the change of the longitudinal rotation period Λ.

As follows from Eqs. (13) and (15), the longitudinal period is shortened and the image spot size reduced when the radial frequency of the filtering is increased. This trend is still evident from Fig. 5, where the effect of a numerical refocusing is demonstrated for different values of the radial spatial frequency ν₀. When examining the longitudinal period, the PSF rotation was carried out by the processing of the holographic image obtained by the reconstruction of the point hologram 4(a) at the target planes with different image defocusing Δz′. The related object space defocusing Δz was obtained by the known longitudinal magnification of the system used for the hologram recording [22]. For l = 1, the defocusing that causes the PSF rotation by the angle π represents the longitudinal period. For ν₀ = 7.9 l/mm, the object space period Λ = 47μm was obtained [Fig. 5(a)]. When the radial frequency ν₀ = 6.6l/mm and ν₀ = 4.0l/mm was used, the period was enlarged to Λ = 68μm and Λ = 190μm, respectively, and the image spots became larger [Fig. 5(b) and Fig. 5(c)]. These results agree well with the theory. In the processing of the holographic image, the narrow width of the annular filter Δν = 0.05ν₀ was applied that well approximates the Dirac delta filter used in the calculations. The topological charge of the spiral phase modulation l = 1 was used. To achieve a better compactness of the PSF lobes, the image given as |e′_N|^a was processed, where e′_N is the normalized image given by Eq. (11) and a > 1 is a constant. The results in Fig. 5 and Fig. 6 were obtained with a = 2.5.

5.3. Damping of the side oscillations of the vortex PSF

Changing the width of the annular filter and using a Gaussian apodization in the slit, the vortex image spots acquire properties analogous to the higher-order Bessel-Gauss beams. Unwanted side oscillations of the Bessel function are apparently damped when the increasing width of the slit is used. Unfortunately, the available range of the axial localization is shortened at the same time. The damping of side oscillations is evident from Fig. 6. The demonstrated vortex PSF was obtained by the processing of the point correlation record Fig. 4(a) using a = 2.5. In Fig. 6(a), the annular filter with a narrow slit was applied (Δν = 0.05ν₀, ν₀ = 4l/mm), resulting in the PSF with distinct oscillations of the Bessel function. With the narrow slit, the image analogy with the diffraction-free beam was maintained, so that the PSF for Δz′ = 0 and Δz′ = Λ′ remained almost unchanged. Results obtained by the annular filter with the wider slit (Δν = 0.3ν₀, ν₀ = 4l/mm) and the Gaussian apodization are shown in Fig. 6(b). The side oscillations were removed for the sharp image with Δz′ = 0. When the defocusing Δz′ = Λ′ was applied, a weak spiral deformation of the PSF lobes occurred.

5.4. Evaluation of the localization accuracy

The proposed axial localization deals with a linear dependence of the PSF rotation on the image space defocusing, φ′ = ΩΔz′. To assess the accuracy of the method, the theoretically discovered dependence was verified experimentally. As shown by Eq. (14), the rotation rate changes with the longitudinal period, Ω = −π/(lΛ′). In the experimental analysis, the dependence of the rotation angle on the ratio Δz′/Λ′ was assessed. It enabled a unified evaluation of the results obtained by the digital filters with the different radial spatial frequencies related to the longitudinal period as ν₀ = 1/(λΛ′)^1/2. The results obtained with the point hologram recorded in the FINCH set-up are shown in Fig. 7. The hologram Fig. 4(a) was processed using the digital spiral filters with l = 1, Δν = 0.2ν₀ and the radial spatial frequencies ν₀ = 3.52, 3.96, 4.52 and 5.27l/mm, respectively. The rotation of the vortex image was evaluated for 12 selected ratios Δz′/Λ′. The angular position of the image corresponding to the theoretical reconstruction plane was taken as a reference angle. For each defocusing, the standard deviation was evaluated using 4 angle measurements, with minimal and maximal values 0.59° and 2.39° within two longitudinal periods. Using all 48 evaluated angular positions of the image, the overall standard deviation $\bar{Δ φ^{'}} = 1.27 °$ was obtained.

Fig. 7 The theoretical dependence of the angular image rotation on the normalized defocusing (solid line) and experimental values of the angles obtained for 12 selected ratios Δz′/Λ′. The listed standard deviations were obtained from four independent measurements carried out by the digital filters with l = 1, Δν = 0.2ν₀ and ν₀ = 3.52, 3.96, 4.52 and 5.27l/mm. The overall standard deviation $\bar{Δ φ^{'}} = 1.27 °$ allows to estimate the accuracy of the axial localization.

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If the standard deviation of the angular image rotation is known, the accuracy of the axial localization may be estimated using the dependence of the rotation angle on the object space defocusing, φ′ = ΩMΔz. In the realized FINCH experiment, the microscope objective NA = 0.25 was used. In combination with the set-up parameters, in particular the focal length of the SLM lens, f_m = 400mm, and the distance between the SLM and the CCD, d = 600mm, the longitudinal magnification M = 370 was obtained [22]. The accuracy of the axial localization in the object space Δz can be written as $Δ z = \bar{Δ φ^{'}} / (π λ M ν_{0}^{2})$ . With $\bar{Δ φ^{'}} = 0.022 rad$ , λ = 625nm, M = 370 and ν₀ = 3.52, 3.96, 4.52 and 5.27l/mm, the localization accuracy can be estimated as Δz = 2.5, 2, 1.5 and 1μm. These values seem to be acceptable taking into account that the depth of field determined by the Strehl ratio exceeding the value of 0.8 corresponds to 10μm for the used microscope objective with NA = 0.25.

5.5. Testing of the defocusing image rotation by fixed fluorescent microspheres

To verify the functionality of the proposed axial localization technique, the experiment with fixed fluorescent microspheres was prepared. The rotating imaging of the fluorescent spheres induced by the image space defocusing is demonstrated in Fig. 8. As a sample, 5μm fluorescent microspheres mounted on a slide were used (Invitrogen Focal Check Microspheres). The hologram recording was carried out in the FINCH set-up shown in Fig. 3 using the fluorescence module with a mercury lamp and the excitation and emission filters at 543nm and 582nm, respectively. The fluorescent radiation emitted by the microspheres was collimated by the microscope objective (Melles Griot, NA = 0.55) and after transformation by the relay lenses modulated by the SLM. The correlation records of the individual microspheres were created with the plane reference wave and the spherical signal wave generated by the SLM lens with the focal length f_m = 400mm. The distance between the SLM and the CCD was d = 600mm. A portion of the resulting CCD record is shown in Fig. 8(a). To examine the PSF rotation, the image was numerically defocused. This was achieved by a repeated reconstruction of the hologram at the target planes with different Δz′. The standard hologram reconstruction performed at one of the selected planes by the Fresnel transform is shown in Fig. 8(b). The obtained images of the spheres then were processed using the spiral phase modulation and the radial spatial filtering and converted to the diffraction-free image spots illustrated in Fig. 8(c). In the processing, the spiral annular filter with l = 1, ν₀ = 5l/mm and Δν₀ = 0.35ν₀ was used together with the coefficient a = 2.5. In Media 1, the common defocusing spreading of the image is compared with the defocusing rotation of the diffraction-free vortex spots. Although the applied object space defocusing exceeded the depth of focus of the used microscope lens nearly nine times, Δz ∈ <−8.6μm, 8.6μm>, the vortex images remained almost shape-invariant. As the microspheres were mounted on a slide, exactly the same rotation of all image spots was observed proving that the applied depth estimation was free of artifacts.

Fig. 8 Defocusing spreading and rotation of the image demonstrated by 5μm fluorescent microspheres (Invitrogen Focal Check Microspheres) recorded in the set-up shown in Fig. 3: (a) portion of the holographic correlation record of the sample, (b) standard image reconstruction of the hologram, (c) rotating vortex images obtained by the spiral processing of the hologram given by Eq. (4) ( Media 1).

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5.6. Tracking of moving polystyrene beads and reconstruction of 3D trajectory

A possibility to use the rotating diffraction-free vortex imaging for a dynamic localization and tracking of particles was verified using a suspension of 0.5μm polystyrene beads. The transparent polystyrene beads freely moving in a capillary tube were illuminated by the collimated LED (Thorlabs, 625nm). The correlation record of the sample was made in the FINCH set-up shown in Fig. 3 using the same parameters as in the fluorescence experiment. To get information on the particle motion, the holograms were taken by a CCD with the frequency of 10Hz. The captured holograms and the vortex images obtained by the spiral processing Eq. (4) of the standard holograms are shown in Fig. 9(a) and Fig. 9(b), respectively. A dynamic localization of the selected particle by the rotating vortex image is presented in Fig. 9(c). Determining the actual positions and angular rotations of the vortex spot, the spatial trajectory of the particle was found. The trajectory is shown in Fig. 9(d) together with the range of spatial coordinates specifying the volume, where the particle was monitored. In Media 2, the recorded holograms, the rotating vortex images and the particular positions of the polystyrene bead in the sample volume are shown.

Fig. 9 Dynamic localization of moving 0.5μm polystyrene beads by the rotating diffraction-free vortex imaging: (a) sequence of the correlation records taken with the frequency of 10Hz, (b) vortex images obtained by the spiral processing of the standard holograms given by Eq. (4), (c) tracking of the selected particle by the rotating vortex image, (d) 3D trajectory of the tracked particle ( Media 2).

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The localization accuracy can be estimated by the experimental parameters (λ = 625nm, NA = 0.55, f_m = 400mm, d = 600mm) and the radial frequency of the digital filter (ν₀ = 5l/mm) using the methodology presented in Sec. 5.4. With the previously determined standard deviation of the angle $\bar{Δ φ^{'}} = 0.022 rad$ , the axial accuracy of the object space localization Δz = 60nm was obtained. The accuracy analysis was restricted to the evaluation of the position in the image space to assess the performance of the proposed method. The accuracy of the object space positioning was recalculated using a constant longitudinal magnification. In practice, the longitudinal magnification varies slightly through the area of longitudinal localization. This problem is specific for a particular configuration used for the hologram recording and must be solved by the system calibration.

6. Conclusions

We have presented a novel technique of the axial localization in digital holography, advantageously operating with the complex amplitude of the all-digital diffraction-free vortex images. The method is based on the spiral processing of the holograms recorded in either coherent or incoherent holographic systems. In this way, the common holographic images are transformed into the vortex spots analogous to the diffraction-free beams, whose angular rotation provides a depth information. To evaluate the position of the point-like objects deployed in a volume, a single plane reconstruction of only one in-line hologram is sufficient. This option is important for a dynamic localization of particles and represents the main advantage over the known methods of digital holography, where a time-consuming multiplane hologram reconstruction or the use of iterative numerical methods is necessary. Besides the implementation simplicity and versatility, the method has an exceptional variability in the PSF engineering and allows to control the sensitivity and the range of the axial localization.

In the experimental part, the performance of the method has been demonstrated using a common-path correlation microscope allowing the holographic recording of incoherently illuminated or fluorescent specimens by the FINCH principle. The theoretically discovered PSF engineering and the control of the image rotation have been experimentally verified by the correlation record of the point emitter realized by a single-mode fiber. By the point hologram, a linear dependence of the angular image rotation on the defocusing has been verified and the accuracy of the axial localization assessed. The application potential of the method has been demonstrated by the defocusing image rotation of 5μm fluorescent microspheres and the 3D localization of 0.5μm polystyrene beads freely moving in a capillary tube. A dynamic tracking of a selected particle and the reconstruction of the 3D trajectory have also been successfully carried out.

Acknowledgments

This work was supported by the project CEITEC - ”Central European Institute of Technology” No. CZ.1.05/1.1.00/02.0068 from European Regional Development Fund, the project No. TE01020229 of the Technology Agency of the Czech Republic and the IGA project of the Palacký University PrF 2014-14.

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Non-iterative holographic axial localization using complex amplitude of diffraction-free vortices

Abstract

1. Introduction

2. Concept of the rotating vortex imaging in digital holography

3. Digital implementation of the rotating diffraction-free vortices

4. Benefits of the holographic axial localization using the diffraction-free vortices

4.1. Shaping, scaling and controlling the rotation rate of the image spot

4.2. Controlling the depth of localization

4.3. Localization by the single in-line hologram

4.4. Volumetric localization by a single plane hologram reconstruction

4.5. Precision of the axial localization

4.6. Options for increasing the sensitivity of localization in the object space

5. Experimental results

5.1. Implementation of the method

5.2. Demonstration of the PSF engineering

5.3. Damping of the side oscillations of the vortex PSF

5.4. Evaluation of the localization accuracy

5.5. Testing of the defocusing image rotation by fixed fluorescent microspheres

5.6. Tracking of moving polystyrene beads and reconstruction of 3D trajectory

6. Conclusions

Acknowledgments

References and links

Supplementary Material (2)

Cited By

Figures (9)

Equations (16)

Optics Express