Aberration resistant axial localization using a self-imaging of vortices

Michal Baránek; Petr Bouchal; Martin Šiler; Zdeněk Bouchal

doi:10.1364/OE.23.015316

1. Introduction

Optical microscopy is essentially a 2D imaging technique providing multitude of biological applications. In the studies of biological tissues and cell processes, the volumetric information has also been of growing importance. Its benefits are particularly valuable in techniques of fluorescence microscopy, where a non-destructive tracking and investigation of single molecules can be realized. Several methods using astigmatism [1], multi-plane detection [2] or two-photon processes [3] have been proposed to overcome limitations of standard imaging and to get information from volume samples. The specific vortex beams, whose intensity profile rotates during free-space propagation [4, 5], can also be effectively used for the point spread function (PSF) engineering in 3D imaging. The vortex-like PSF exhibits a very high rotation sensitivity to defocusing and can be uniquely applied to an accurate depth estimation. In the seminal paper on depth from diffracted rotation [6], a superposition of Gauss-Laguerre beams was used to generate a rotating DH PSF. A low energy efficiency of the method was later improved using iterative optimization algorithms [7]. An annular vortex PSF was generated by a high-efficiency technique based on the Fourier spiral filtering and applied to the edge-contrast enhancement in the spiral phase contrast microscopy [8]. Optically and digitally implemented vortices were deployed in the Fresnel incoherent correlation holography [9], where a coherent image reconstruction [10] was used for a selective edge enhancement in 3D imaging [11]. Recently, a quantized spiral phase mask was used to create an azimuthally modulated PSF with the rotational sensitivity to defocusing [12, 13, 14]. In [12], a general concept of the sampled spiral phase modulation was outlined with a particular attention focused on the image rotation achieved by a spiral mask with discrete phase levels in separate azimuthal sections. A sampling of the spiral phase into the radial Fresnel zones was examined in [13], where generation of a single-lobe PSF was demonstrated by numerical simulations. In [14], the single-lobe PSF using the Fresnel zones was implemented in two polarization channels and the DH PSF advantageously obtained by merging the orthogonally polarized images on a computer. The localization and tracking of particles were also realized in digital holography, where the nondiffracting beams were used for generation of the DH PSF [15].

The rotating PSF was applied to a variety of precise localization experiments. This technique was successfully used for 3D tracking of fluorescent microparticles in a photon-limited double-helix response system [16]. By a corkscrew PSF, the 3D localization of 0.2μm fluorescent beads was demonstrated with nanometer precision [17]. In the double-helix microscope, the extended biological structures were super-resolved by localizing single blinking molecules [18]. The 3D single-molecule fluorescence imaging beyond the diffraction limit was also demonstrated [19]. In most studies, diffraction-limited optics was considered and little attention was paid to optical aberrations. Recently, the degradation effects of the spherical aberration were investigated in computational optical sectioning microscopy using the rotating DH PSF generated by a superposition of Gauss-Laguerre beams [20].

In this paper, the axial localization based on the DH PSF is originally implemented by the effects that are analogical to the SI of nondiffracting vortices. The method is examined in detail both theoretically and experimentally and its benefits over other known methods are outlined and demonstrated. The method can be adapted to the localization of conventional fluorescent objects, but the high contrast DH PSF can also be generated when using transparent weakly scattering objects. In the proposed technique, the phase objects illuminated by the transmitted light can be localized with the suppression of a bright background reminding conditions of a dark-field imaging. Other benefits resulting from a nondiffracting nature of the vortex modes are demonstrated by an aberration resistant axial localization, an extension of the working range and an operation with the shape-invariant PSF.

The paper is organized as follows. First, the vortex SI implemented in imaging systems is presented and examined in an appropriate computational model. The rotation of the DH PSF is explained as a result of different phase shifts of the nondiffracting vortices, which sensitively indicate wavefront deformations caused by the defocusing and aberrations. By the theoretical analysis, the basic experimental parameters for the SI localization are determined and the conditions for an optimal PSF shaping and controlling the localization sensitivity discussed. Subsequently, an influence of the spherical aberration on the image rotation is quantitatively investigated and the axial localization proposed, in which the aberration effects are completely eliminated. Finally, the verification experiments are performed using the spiral modulation implemented by a phase-only SLM. The practical usability of the proposed method is demonstrated by a defocusing induced rotation of the fixed and freely moving polystyrene beads.

2. Concept of the vortex SI in optical imaging

The axial localization presented in this paper is achieved by modifying the SI effect previously examined in a free propagation of light [21, 22]. The SI is typically manifested by a periodic reproduction of the intensity profile, which occurs due to interference of nondiffracting beams with suitable propagation constants [23]. In the proposed axial localization, the nondiffracting vortex beams with appropriately chosen topological charges are preferably used. The vortices create a rotationally asymmetric interference pattern that continuously rotates during defocusing, while retaining its shape unchanged. This behavior is different from the conventional SI effect, in which the transverse beam profile restores only in periodically repeating planes and disappears between them. In the manuscript, the additional condition ensuring a continuous rotation of the stable image profile is found and applied in the experiments. The defocusing rotation has its origin in different phase shifts embedded on the nondiffracting modes in the Fourier space. The image rotation caused by the phase shifts is used as a sensitive indicator of the defocusing and aberrations imposed on the examined wavefront.

To investigate a new concept of the axial localization and its aberration resistance and applicability, a general calculation model of the vortex SI was proposed. In an approximate analytical treatment, basic connections between the shape of the examined wavefront, parameters of the spiral phase modulation and the rotation rate of the image were found. Using numerical calculations, effects appearing in experiments were simulated and an optimal design of the spiral SLM mask investigated.

2.1. SI of nondiffracting vortices in a 4-f optical system

The aberration resistant axial localization using the SI effect can be advantageously implemented in the 4-f optical system [24] shown in Fig. 1(a). To simplify modeling of the system, the imaging of the axial object point of the plane r _⊥ = (x, y) located at a distance z in front of a Fourier lens FL₁ is examined. The lens has the focal length f ₁ and exhibits the spherical wave aberration W_S. The spiral mask with the transmission function S is placed in the Fourier plane R _⊥ = (X, Y) of the 4-f system and the image is detected in the plane r′ _⊥ = (x′, y′) placed at a distance z′ behind an ideal lens FL₂ with the focal length f ₂. The complex amplitude of the image can then be written as

U^{'} \propto {FrT}_{z^{'}} {P_{2} {FrT}_{f_{2}} {S {FrT}_{f_{1}} {U_{0} P_{1}}}},

where FrT_p denotes the Fresnel transform describing light propagation through the distance p = z′, f ₁ and f ₂, respectively, U ₀ is the complex amplitude of a divergent wave emanating from the axial point of the object plane, and P ₁ and P ₂ are the pupil functions of the lenses FL₁ and FL₂. For a collimated light beam, the spherical wave aberration in the focal plane of the lens FL₁ is just the same as behind the lens. In this case we can write

U^{'} \propto {FrT}_{z^{'}} {exp (i Φ_{2}) {FrT}_{f_{2}} {S exp [- i Φ_{1} (1 - z / f_{1}) + i k W_{S}]}},

where W_S ≡ W_S(|R _⊥|), Φ_p = k|r _⊥p|²/2f_p and r _⊥p, p = 1, 2, are the position vectors in the transverse plane of the Fourier lenses. The kernel of the Fresnel transform can be rewritten by the quadratic phase functions and the Fourier transform kernel [24]. If the sequence of the Fourier transforms used in FrT_f₂ and FrT_z′ is reversed, the complex amplitude of the image at a distance z′ behind the lens L ₂ can be determined as

U^{'} \propto {FT}_{f_{2}} {S exp [i k (W_{D} + W_{S})]},

where the Fourier transform FT is performed in the spatial frequencies r′ _⊥/λf ₂. The wave f₂ aberration includes the defocusing and the third-order spherical aberration,

W_{D} = A_{020} ρ^{2},

W_{S} = A_{040} ρ^{4},

where ρ denotes the radial coordinate R = |R _⊥| normalized by the maximal radius of the spiral mask R_max, ρ = R/R_max. The defocusing coefficient is given as

A_{020} = \frac{R_{\max}^{2}}{2} (\frac{Δ z}{f_{1}^{2}} + \frac{Δ z^{'}}{f_{2}^{2}}),

where Δz = z − f ₁, Δz′ = z′ − f ₂. The coefficient of the third-order spherical aberration can be written in the form

A_{040} = \frac{R_{\max}^{2}}{4 f_{2}^{2}} Δ z_{S}^{'},

where Δz′_S is the maximal value of the longitudinal component of the geometrical spherical aberration.

Fig. 1 Axial localization by the rotating PSF based on the SI of vortices implemented by the spiral mask SM. (a) SM is placed in the Fourier plane of the 4-f system so that the nondiffracting vortices are created by interference of plane waves behind the lens FL₂. (b) SM is placed in the exit pupil of the lens L and the divergent vortices are created by interference of spherical waves.

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In optical microscopy, the various spiral modulation techniques were proposed for the rotation of the defocused image and successfully applied to the localization and tracking of particles. In this paper, the shape invariant rotating imaging is examined enabling a large scale axial localization resistant to the spherical aberration. The imaging is implemented by a spiral modulation satisfying the conditions of the vortex SI. The proposed technique is examined in a simplified description obtained for an ideal spiral mask composed of a set of thin annular zones given by the Dirac delta-function,

S (R, ψ) = \sum_{m = 1}^{M} \frac{δ (R - ρ_{m} R_{\max})}{ρ_{m} R_{\max}} exp (i l_{m} ψ + i κ_{m}),

where ψ = arctan(Y/X). When using the mask, the light passes through M concentric rings with the radii ρ_mR_max, in which the spiral phase modulation with the different topological charges l_m is performed. The constant phase shifts κ_m added in the individual rings are used for an optimal shaping of the image spot. Substituting (8) into (3) and using integrals containing Bessel functions [25], the Fourier transform results in

U^{'} \propto \sum_{m = 1}^{M} J_{l_{m}} (\frac{k r^{'} ρ_{m} R_{\max}}{f_{2}}) exp [i l_{m} (φ^{'} + \frac{π}{2}) + i κ_{m} + i k (A_{020} ρ_{m}^{2} + A_{040} ρ_{m}^{4})],

where r′ = (x′ ² + y′ ²)^1/2 and φ′ = arctan(y′/x′) are the cylindrical coordinates in the image plane. The complex amplitude of the resulting image is formed by the superposition of the partial images with the helical phase specified by the topological charge l_m and the transverse profile given by the higher-order Bessel function. The constituents of the SI appearing in the summation (9) represent the nondiffracting beams [26, 27], because their amplitude given by the Bessel function is independent of the defocusing and the coefficient A ₀₂₀ appears only in the phase term.

In the axial localization, the DH PSF is required, whose lobes rotate linearly with defocusing. To realize such a rotating imaging, the additional conditions must be applied to the topological charges l_m and the radii of the mask rings ρ_m. These conditions can be established by examining the intensity of the image spot, I′ = |U′|², calculated using (9) and written as

I^{'} \propto \sum_{m = 1}^{M} J_{l_{m}}^{2} + \sum_{m = 1}^{M} \sum_{m^{'} = m + 1}^{M} 2 J_{| l_{m} |} J_{| l_{m^{'}} |} cos Ω_{m, m^{'}},

where the arguments of the Bessel functions are omitted and Ω_m,m′ is given by

Ω_{m, m^{'}} = (l_{m} - l_{m^{'}}) φ^{'} + (κ_{m} - κ_{m}^{'}) + \frac{π}{2} (| l_{m} | - | l_{m^{'}} |) + k A_{020} (ρ_{m}^{2} - ρ_{m^{'}}^{2}) + k A_{040} (ρ_{m}^{4} - ρ_{m^{'}}^{4}) .

The image rotation is first studied for aberration-free system, A ₀₄₀ = 0, then disturbances caused by the spherical aberration are discussed. The image consists of a set of annular Bessel spots azimuthally modulated by the interference terms. The cosine modulation creates the lobes rotating with the defocusing coefficient A ₀₂₀. The defocusing rotation can be demonstrated by changing the position of the detector, Δz′ ≠ 0, while keeping the location of the object unchanged, Δz = 0. On the contrary, in the axial localization applications, a change of the object position, Δz ≠ 0, is monitored by the image rotation in a fixed detection plane, Δz′ = 0. In order to achieve a rigid rotation of the image spot during defocusing, the rotation rate of the individual interference terms must be constant,

\frac{d φ^{'}}{d A_{020}} = - \frac{k (ρ_{m}^{2} - ρ_{m^{'}}^{2})}{l_{m} - l_{m^{'}}} = const .

This condition is met, if the normalized radii of the mask rings ρ_m and the topological charges l_m are chosen as

ρ_{m} = \sqrt{\frac{m}{M}},

l_{m} = l_{0} + m Δ l,

where l ₀ and Δl keep constant values. The condition (13) ensures the SI effect, in which the image spot reproduces its shape for specific values of A ₀₂₀ repeating periodically. If the conditions (13) and (14) are fulfilled simultaneously, the image has Δl lobes that rotate smoothly with the continuous change of A ₀₂₀. The rotation rate is constant and can be written as dφ′/dA ₀₂₀ = −k/(MΔl). For Δz = 0, the rotation rate can be rewritten to the form

\frac{d φ^{'}}{d Δ z^{'}} = - \frac{k N A^{' 2}}{2 M Δ l},

where NA′ = R_max/f ₂. The axial shift Δz′ ≡ Λ′ causing the angular rotation 2π is defined as the image space period. It follows from the condition dφ′/dΔz′ = 2π/Λ′ and can be written as

Λ^{'} = \frac{2 λ M Δ l}{N A^{' 2}} .

Both the rotation rate and the rotation period depend on the image aperture NA′, which can be controlled by the choice of the maximal mask radius R_max.

If the detector is placed in the paraxial image plane, Δz′ = 0, and the defocusing is performed in the object space, Δz ≠ 0, the rotational period can be determined as

Λ = \frac{2 λ M Δ l}{N A^{2}},

where NA = R_max/f ₁. The ratio of the rotation periods in the image and object space, γ = Λ′/Λ, corresponds to the longitudinal magnification of the 4-f system. It can be expressed as γ = NA ²/NA′ ². The principle of the image rotation (10) is explained in Fig. 2 for the simplest case of the SI composed of two vortex modes with the difference of the topological charges Δl = 2 and Δl = 3, respectively.

Fig. 2 Graphical illustration of the rotating image (10) demonstrating two mode vortex SI (M = 2) with difference of the topological charges Δl = 2 and Δl = 3. The image rotation occurs due to an azimuthal displacement of the interference maxima during defocusing.

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In the coefficient A ₀₂₀, the defocusing performed either in the object or image space is included. In both cases a rotation of the PSF occurs, but the reasons and ways of the rotation are different. If the object is in a fixed position (Δz = 0) and the detector moves (Δz′ ≠ 0), the image rotation is a result of the propagation form of the SI. In the propagation SI, the nondiffracting beams generated by the mask composed of a set of thin transparent annular zones are used. Divergent spherical waves emanating from the m-th ring are collimated by the lens, so that plane waves are obtained, whose wave vectors form a conical surface with the vertex angle θ_m. Due to interference of the plane waves, a nondiffracting beam appears, whose propagation constant is given as a projection of the wave vectors to the beam axis, β_m = k cos θ_m. If the radii of the rings and the topological charges of the spiral phase changing around the rings satisfy the conditions (13) and (14), the vortex SI can be observed. This effect manifests itself as a periodic rotation of the intensity spot during a movement of the detector. It is caused by the interference of nondiffracting beams with the different propagation constants β_m.

If the detector is fixed (Δz′ = 0) and the object changes its position (Δz ≠ 0), the PSF rotation is still caused by interference of the nondiffracting beams, but instead of the propagation rotation, the in-plane rotation of the image occurs. In this case, the rotation is not affected by the different propagation constants β_m, but it appears due to different initial phases $Δ β_{m} = k (A_{020} ρ_{m}^{2} + A_{040} ρ_{m}^{4})$ of the nondiffracting beams in the mask plane (Fig. 1(a)). The initial phase shift in the individual radial zones is determined by a shape of the incident wave, so that the image rotation depends on both the defocusing and the wave aberrations. In the developed computation model, the rotational effect of the spherical aberration can be quantified and its consequences fully eliminated when determining the axial position.

2.2. SI of divergent vortices in a lens imaging

The vortex SI examined in the 4-f system can also be implemented in a lens imaging shown in Fig. 1(b). In this case, the spiral mask (8) is placed in the exit pupil of the lens L with the focal length f and the spherical wave aberration W_S. The position of the point object and its sharp image is specified by the distances z ₀ and z′ ₀ satisfying the lens equation. The defocusing shifts in the object and image space are denoted as Δz and Δz′, respectively. The complex amplitude of the out of focus image can be calculated by applying the Fresnel transform,

U^{'} = {FrT}_{z^{'}} {U_{0} S},

where U ₀ is the complex amplitude of a wave emanating from the axial object point at the distance z = z ₀ + Δz in front of the lens, S is the transmission function of the spiral mask (8), and FrT_z′ describes the propagation of light to a distance z′ behind the lens. After integration (18) performed using [25], the image complex amplitude is obtained. It can again be written in the form (9), but f ₂ must be replaced by z′ = z′ ₀ + Δz′ in the argument of the Bessel function and the defocusing coefficient calculated as

A_{020} = \frac{R_{\max}^{2}}{2} [\frac{Δ z}{z_{0} (z_{0} + Δ z)} + \frac{Δ z^{'}}{z_{0}^{'} (z_{0}^{'} + Δ z^{'})}] .

The resulting image is given as a superposition of the partial images described by the higher-order Bessel functions. In this case, the image size depends on the defocusing and the image spot spreads if Δz′ increases. The components of the image correspond to divergent Bessel beams arising due to interference of spherical waves emitted by individual points of the rings of the spiral mask [28]. If the image is detected in the fixed image plane and the defocusing in the object space is performed, the image spot remains unchanged and only rotates when Δz varies. For an aberration-free case with W_S = 0, the image rotation analysis carried out for the 4-f system remains valid. When used with the numerical apertures NA′ = R_max/z′ ₀ and NA = R_max/z ₀, the rotation rate (15) and the image and object space rotation periods (16) and (17) remain applicable.

3. Experimental verification of the aberration resistant axial localization

3.1. Implementation of the method

In experimental part of the paper, the benefits of the proposed method were validated and its applicability demonstrated. The experiments were carried out in the setup shown in Fig. 3. It works with two input imaging paths connected via a beam splitter BS₁. In the optical path Fig. 3(a), the required complex modulation was ensured by the SLM and used to investigate the axial localization in the presence of aberrations. The optical path Fig. 3(b) combines the amplitude modulation in the exit pupil of the microscope objective with the spiral phase-only modulation by the SLM. In this optical path, the tracking of polystyrene beads with an improved signal to noise ratio was demonstrated.

Fig. 3 Experimental setup for aberration resistant axial localization by the vortex SI (SF -spatial filter, MO - microscope objective, C - capillary tube with a suspension of polystyrene beads, AM - amplitude mask, BS₁, BS₂ - beam splitters, TL - tube lens, FL₁, FL₂ – Fourier lenses, P - polarizer, SLM - spatial light modulator): (a) optical path for calibration and testing of the axial localization, (b) optical path for tracking of polystyrene beads.

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In the verification experiments, the spherical aberration was examined and the measurement technique proposed, which enabled an efficient elimination of aberration effects in the long-range axial localization. The aberration resistant axial localization was examined by the He-Ne laser beam (20mW, 632.8nm) spatially filtered in the imaging path Fig. 3(a). The pinhole of the spatial filter SF with the radius of 1μm was used as a point source emitting light into the numerical aperture NA ≈ 0.1. Light from the pinhole was collimated by the microscope objective MO (Newport, 20×, NA = 0.4, f ₀ = 9mm) and by the beam splitter BS₁ diverted toward a tube lens TL. The collimated beam was focused by the TL with the focal length f_T = 200mm to the front focal plane of the 4-f system. The 4-f system was composed of the lenses FL₁ and FL₂ (f ₁ = 200mm, f ₂ = 400mm) and operated with the phase-only SLM (Hamamatsu X10469, 16mm×12mm, 800×600 pixels) located in the Fourier plane of the system. The SLM was used for an approximate realization of the spiral mask (8). The rings defined by the Dirac delta function were created as thin annular zones with a width of tens of pixels. The radial zones were azimuthally modulated by a spiral phase with the topological charge l_m. In multiple-mode experiments, the constant phase shifts κ_m were added in the individual zones to optimize the shape of the image spot. As the SLM provided a phase-only operation, a blazed phase grating was applied in the area outside the annular zones to ensure the deflection of unwanted light replacing its amplitude attenuation required therein. The created rotating image spot was captured by the CCD placed in the focal plane of the lens FL₂ performing the inverse Fourier transform of the light reflected from the SLM. In the experiments, the vortex SI was advantageously applied to ensure the elimination of aberrations and extending the working area in the axial localization.

3.2. SI composed of two nondiffracting vortices

As was shown theoretically, the rotation rate (15) and the rotation periods (16) and (17) are proportional to the number of the used nondiffracting modes M and the difference of the topological charges Δl in the neighboring annular zones of the spiral mask. In order to achieve the highest sensitivity in the axial localization, the SI composed of two vortices with the smallest difference of the topological charges must be used. Since the determination of the angular rotation is difficult for a single-lobe PSF with Δl = 1, the DH PSF obtained for M = 2 and Δl = 2 seems to be an optimal choice. With regard to the condition (14) used for the choice of the topological charges in the individual rings of the mask, l ₀ remains a free parameter affecting the lateral resolution. As is evident from Fig. 2, the PSF lobes represent azimuthal slices of the ring formed by the Bessel functions. The radius of the ring increases with the index of the Bessel function given by the topological charge of the related helical phase [25]. Thus, the smallest DH PSF and the best image resolution are achieved with the lowest topological charges of the used vortex modes. As follows from (14), this requirement corresponds to the choice l ₀ = −3 and Δl = 2 that results in l ₁ = −1 and l ₂ = +1.

In the demonstrated experiments, the SI composed of two vortices was proved to be beneficial due to a shape invariant image spot, a long localization region and a resistance to the spherical aberration. The axial localization was examined using the pinhole mounted on a precise translation stage and illuminated by the focused laser beam, as shown in Fig. 3(a). The mask composed of two annular zones with the azimuthal spiral phase modulation given by the topological charges l _1,2 = ±1 and the phase shifts κ _1,2 = 0 was displayed on the SLM to generate the nondiffracting vortex beams. When the pinhole was placed exactly in the focal plane of the microscope objective, the individual radial zones of the mask were illuminated by a collimated light beam with the constant phase. In this case, the SI formed a double-lobe image spot, whose angular orientation was used as a reference position in the axial localization. When the pinhole was moved out of the focal plane of the microscope objective, the mask was illuminated by a convergent or divergent spherical wave, respectively, so that the radial zones acquired a different initial phase. Consequently, the generated nondiffracting vortex beams received different phase shifts resulting in the rotation of the detected image due to the SI effect.

The image rotation caused by the negative and positive object space defocusing is illustrated in Fig. 4 for the optical system with a well-corrected spherical aberration simulated by A ₀₄₀ = 0 in the performed calculations. The image spots shown in Fig. 4(a) were obtained by the spiral mask displayed on the SLM and providing a nearly perfect SI. The mask was composed of two narrow rings with the radii R ₁ = 0.64mm and R ₂ = 0.9mm and the width of 0.1mm. Along the rings, a spiral phase modulation specified by the topological charges l _1,2 = ±1 was applied. The image rotation was evaluated for the defocusing Δz ₀ measured in front of the microscope objective. It can be written as $Δ z_{0} = Δ z f_{0}^{2} / f_{T}^{2}$ , where Δz is the object space defocusing of the 4-f system used in (6). The theory is in a good agreement with the experiment and the results obtained demonstrate the robustness of the rotating image in a large axial range. Due to a nearly perfect SI, the shape and size of the DH PSF were retained even though the large defocusing ±120μm was applied. The object space aperture of the microscope objective was NA ₀ = 0.1 (R_max = 0.9mm, f ₀ = 9mm, f_T = f ₁), so that the depth of field was exceeded almost four times.

Fig. 4 Defocusing rotation of the theoretical and experimental DH PSF created in an aberration-corrected system by interference of M = 2 vortices with the topological charges l ₁ = −1 and l ₂ = 1: (a) exact SI of nondiffracting vortices generated by the mask with the narrow rings, (b) approximate SI of vortices generated by the mask with fully transparent zones.

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With the narrow rings of the mask, the SI was realized in a nearly exact form, but only a small portion of the input light energy was utilized. In the case demonstrated in Fig. 4(a), the efficiency was about 35%. The easiest way to increase the efficiency is to extend the width of the rings. This may be done successively so that in an extreme case a fully transparent phase mask can be applied. The image spots shown in Fig. 4(b) were created in an aberration-corrected system using a phase-only mask, whose active area was divided into two zones with the spiral phase modulation specified by the topological charges l _1,2 = ±1. Although the exact SI condition (13) was violated and only the topological charges were correctly selected according to (14), the image spots maintained an acceptable quality in the monitored area.

To demonstrate the robustness of the axial localization based on the vortex SI, the third-order spherical aberration was added on the defocused wavefront. In order to achieve a full control of the spherical aberration, it was created by the SLM together with the spiral mask (8). An influence of the spherical aberration given by A ₀₄₀ = 0.4λ is evident from Fig. 5 obtained with the same masks that were used in Fig. 4. The rotating image spots obtained by a mask with the narrow rings ensuring an accurate SI are shown in Fig. 5(a). The shape and size of the DH PSF remained the same as in the aberration-corrected case illustrated in Fig. 4(a). The only manifestation of the spherical aberration was an additional angular rotation of the image, which was constant in all defocusing positions. If the mask with two fully transmitting zones was used, the SI condition (13) was not exactly fulfilled, and consequently an apparent deformation of the image spots occurred. As shown, by changing the width of the rings, a compromise between the energy efficiency and the resistance against aberrations can be found.

Fig. 5 The same as in Fig. 4, but for the third-order spherical aberration with the coefficient A ₀₄₀ = 0.4λ.

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In the experiments using the accurate SI, the additional rotation due to the spherical aberration can be evaluated quantitatively and its influence in the axial localization eliminated. Dependence of the PSF rotation on the defocusing and the spherical aberration stems from (10). For given A ₀₂₀ and A ₀₄₀, the shape and the angular position of the PSF are given by the interference terms 2J _{|l_m|} J _{|l_m′|} cosΩ_m,m′. The Bessel functions have a rotational symmetry and cause the additional constant phase shift π, when the negative topological charges given by an odd number are used. The azimuthal intensity distribution has a cosine modification with maxima given by Ω_m,m′ = 2qπ, where q = 0, ±1, ±2···. By evaluating (11) using (13), (14) and with κ_m = κ′_m = 0, the angles with respect to the x′-axis can be found, which determine the azimuthal position of the intensity maxima of the interference terms in dependence on the defocusing and the spherical aberration,

φ_{m, m^{'}}^{'} = \frac{π}{(m - m^{'}) Δ l} [2 q - \frac{1}{2} (| l_{m} | - | l_{m}^{'} |)] - \frac{k}{M Δ l} [A_{020} + \frac{(m + m^{'})}{M} A_{040}] .

In the SI composed of two vortices with the topological charges l ₁ = −1 and l ₂ = 1, the azimuthal maximum of the DH PSF corresponding to q = 0 follows directly from (20) and can be written as

Δ φ^{'} \equiv φ_{1, 2}^{'} = Δ φ_{D}^{'} + Δ φ_{S A}^{'},

where Δφ′_D and Δφ′_SA denote the rotations caused by the defocusing and the spherical aberration,

Δ φ_{D}^{'} = - \frac{π}{2 λ} A_{020},

Δ φ_{S A}^{'} = - \frac{3 π}{4 λ} A_{040} .

The image rotates linearly with the defocusing and the spherical aberration causes a constant additional rotation. This property of the SI technique is useful to attain the aberration resistant axial localization. It can be implemented using a reference object position allowing a relative evaluation of the angular rotation. The measurement procedure is as follows. First, a point object in an unknown reference position Δz _{0_ref} is recorded. For Δl = 2, the image spot with two lobes is obtained, whose first azimuthal intensity maximum is oriented at the angle Δφ′_ref. Subsequently, the point object in a measured position Δz _{0_meas} is processed, whose image spot has the intensity maximum at the angle Δφ′_meas. Finally, the relative rotation of the image, Δφ′ = Δφ′_meas − Δφ′_ref, is determined applying a correlation of the recorded intensity spots. Assuming f ₁ = f_T, the change of the position of the point object in front of the microscope objective can be evaluated by

Δ z_{0} = Δ z_{0_{meas}} - Δ z_{0_{ref}} = - \frac{4 λ}{π N A_{0}^{2}} Δ φ^{'} .

In the proposed technique of the axial localization, the change of the object position is exactly specified by the relative angular rotation of the image, which remains unaffected by the spherical aberration. Experimental verification of the axial localization insensitive to the spherical aberration was successfully carried out in the setup shown in Fig. 3(a). The pinhole representing a point source was placed to the reference position and its image recorded on a CCD. Subsequently, the point source was moved to 4 different axial positions and the DH PSF repeatedly recorded. The individual records were affected by the spherical aberration, which was added to the transmission function of the spiral mask displayed on the SLM. The relative rotation of the individual records was determined by evaluating the correlation overlapping with the reference record. The dependence of Δφ′ on Δz ₀ was then evaluated and compared with the theoretical relation (24). The results obtained are shown in Fig. 6. The standard deviations in the measured positions were obtained from 40 recordings made with the third-order spherical aberration, whose coefficient was randomly chosen from the interval <−0.4λ, 0.4λ>.

Fig. 6 Experimental verification of Eq. (24) demonstrating aberration robustness of the axial localization: theoretical dependence of the rotation angle Δφ′ on the axial position Δz ₀ (full line), experimental evaluation of the rotation angle - the average values and the standard deviations were obtained by the processing of 40 image spots recorded in the system with different values of the spherical aberration, A ₀₄₀ ∈ <−0.4λ, 0.4λ> (error bars).

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3.3. SI composed of multiple nondiffracting vortices

The efficiency of the proposed localization technique can be apparently improved even when a nearly perfect SI is maintained. In this case, the mask with a higher number of narrow rings is used to operate with the multi-mode SI. The rotating PSF is given by (10), in which the topological charges l_m and the radii of the rings ρ_m fulfill (13) and (14). With these conditions, the PSF still remains stable during rotation, but its shape is deformed when compared with the related two-mode case using the same Δl. The individual interference patterns created by different pairs of vortices have their maxima at different azimuthal positions. When added together, a spreading and deformation of the PSF lobes appear. In order to align the angular positions of the intensity maxima of the interference terms, the factors κ_m are used. They represent constant phase shifts of the individual SI modes. By the optimal choice of κ_m, the well defined lobes of the PSF are obtained with the quality comparable to the two-mode SI. The experimental results demonstrating the DH PSF generated by the SI consisting of M = 4 vortex modes with the topological charges l ₁ = −3, l ₂ = −1, l ₃ = 1 and l ₄ = 3 are shown in Fig. 7. To achieve a sharp localization of the lobes, the constant phase shifts κ ₁ = π/2, κ ₂ = 0, κ ₃ = π/2 and κ ₄ = 0 were applied. The values of the phase shifts were determined by the conditions ensuring an optimal angular overlapping of the individual interference terms. The image spots in the upper row of Fig. 7 were created in a well-corrected optical system, while the records demonstrated in the bottom row were captured under the influence of the third-order spherical aberration (A ₀₄₀ = 0.4λ) introduced by the SLM. The radius of the largest ring of the mask was R_max = 0.9mm resulting into the object space numerical aperture NA ₀ = 0.1. The narrow rings with the width of 0.1mm provided a nearly perfect SI. As shown in Fig. 7, the shape and dimension of the PSF remained unchanged during defocusing not only for the aberration-corrected system (upper row) but also for the system with aberrations (bottom row). As verified, the additional angular rotation caused by the spherical aberration can be fully eliminated in measurement using a reference PSF record and the axial localization resistant against aberrations realized. When using the SI composed of four vortex modes, the energy efficiency of 63% was reached for NA = 0.1. It should be noted that when a high number of modes is used, the rotation rate is reduced and the lateral size of the image spot enlarged. This means that the optimal number of modes should be found in a compromise between the efficiency and the transverse and longitudinal resolution.

Fig. 7 Experimental demonstration of the DH PSF obtained by the SI composed of M = 4 vortex modes with the topological charges l ₁ = −3, l ₂ = −1, l ₃ = 1 and l ₄ = 3 and the constant phase shifts κ ₁ = π/2, κ ₂ = 0, κ ₃ = π/2, and κ ₄ = 0: aberration-corrected system with A ₀₄₀ ≈ 0 (upper row), system with the spherical aberration given by A ₀₄₀ = 0.4λ (bottom row).

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In the paper, the main attention is focused on the new technique of the PSF rotation and its benefits are only briefly outlined. Specifically, the demonstration of the aberration resistance of the axial localization is restricted to the third-order spherical aberration. As shown, the proposed method is based on the change of the initial phase of the nondiffracting vortices in the Fourier plane, where the spatial spectrum of the individual modes corresponds to the narrow annular rings. Modulating the rings by a quadratic phase changing due to the object space defocusing, the PSF rotation occurs as a result of the interference of the vortices. The rotationally symmetrical spherical aberration causes only an additional rotation of the PSF without affecting its shape. If the asymmetric aberration such as coma was added to the wavefront incident on the rings, the shape of PSF remained nearly unchanged, even when the coma coefficient λ/2 was used. The aberration stability of the proposed method significantly exceeds the standard imaging, because the rotating PSF arises from the radial samples of the spatial spectrum that are weakly affected by aberrations. The comprehensive analysis of the aberration effects is beyond the scope of the paper and will be presented elsewhere.

4. Demonstration of defocusing rotation of fixed and moving microspheres

The proposed 3D localization based on the vortex SI was also tested using samples with fixed and moving polystyrene beads. The experiment was performed in the setup shown in Fig. 3(b). The capillary tube with a suspension of 1μm polystyrene beads was illuminated by the LED (Thorlabs, 625nm, FWHM 10nm) and the transmitted light captured by a specially adapted long working distance microscope objective (Melles Griot 10×, NA=0.28, f ₀ = 20mm, infinity corrected). Light beams collimated by the microscope objective were directed toward the system that was previously used in the examination of the aberration resistant axial localization. Configuration of the setup behind the beam splitter BS₁ remained the same, but the mask composed of a set of rings with a spiral phase modulation inside was implemented in a completely different way. The SLM was operated as a phase only device creating the spiral phase modulation with the topological charge l ₁ = −1 in the central circular region of the display, and l ₂ = 1 in the outer annular zone. The amplitude modulation by narrow rings required for generation of the nondiffracting vortex modes was achieved by modifying the microscope objective. The amplitude modulation was performed by a mask transmitting light through two narrow concentric rings. The rings were engraved in the layer of a black lacquer deposited on a microscope slide cover. The mask was then inserted into the back focal plane of the microscope objective and projected on the SLM by the lenses TL and FL₁. In combination with the spiral modulation provided by the SLM, the required complex mask was implemented. This realization of the mask was advantageous in comparison with the previously used amplitude modulation based on the deflection of unwanted light by a grating displayed on the SLM. Combining the amplitude modulation in the exit pupil of the microscope objective with the SLM phase modulation, the nondiffracted light that occurred when using diffraction grating was completely removed and the signal to noise ratio significantly improved. In Fig. 8, the experimental results obtained by this technique are presented. The results demonstrate the capabilities of the proposed SI technique in the localization and tracking of fixed and moving polystyrene beads. The accessible field of view and its magnified portion are shown in Fig. 8(a) and Fig. 8(b), respectively. In Media 1, the robust rotating imaging of the polystyrene beads is presented in an extended defocusing range. In this experiment, the fixed polystyrene beads were relocated by a precise linear stage in the axial range ±150μm that exceeded the depth of field of the microscope objective more than 10×. The rotating image spots were sequentially recorded with the frequency of 5Hz to demonstrate a stable DH PSF, whose size and shape remained almost unchanged during defocusing. In Media 2, the polystyrene beads suspended in the capillary tube and freely moving by the Brownian motion were observed and the image spots successively recorded during 10s lasting interval with the frequency of 5Hz. The single frames selected from Media 1 and Media 2 are shown in Fig. 8(c) and Fig. 8(d).

Fig. 8 Demonstration of the two-mode SI (l ₁ = −1, l ₂ = 1) by defocusing rotation of 1μm polystyrene beads: (a) accessible field of view, (b) enlarged portion of the field of view, (c) rotation of the PSF during defocusing of fixed beads realized by a precise linear stage ( Media 1), (d) defocusing rotation of movable polystyrene beads suspended in a capillary tube ( Media 2).

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5. Conclusions

We have presented a versatile technique of the PSF engineering, in which the SI of nondiffracting vortices has been used for a sensitive depth estimation in the rotating imaging. The experiments can be adapted to the localization of the conventional fluorescent particles but also transparent weakly scattering objects can be used. The proposed technique ensures the suppression of a bright background in the transmitted light illumination, resulting in an improved signal to noise ratio. In contrast to the known methods, our approach allows a long range axial localization with the full elimination of the spherical aberration. In the theoretical part, the general conditions of the SI were found and the basic characteristics and parameters examined and quantified in order to optimize experiments. The experiments verifying the benefits of the proposed method has been carried out in the setup involving a SLM based microscope supplemented by a 4-f optical system. The various modulation techniques resulting in the SI composed of two or multiple vortices have been designed allowing to strike a trade-off between the energy efficiency, the size of the working area and the resistance against aberrations. The feasibility and applicability of the method has been demonstrated by the defocusing induced rotation of the fixed and moving 1μm polystyrene beads. Simplification of the experimental system allowing the implementation of the vortex SI in a conventional optical microscope without using the SLM is a challenge for further research.

Acknowledgments

This work was supported by the project No. 15-14612S of the Grant Agency of the Czech Republic, the project CEITEC - ”Central European Institute of Technology” No. CZ.1.05/1.1.00/02.0068 from European Regional Development Fund, the project POST-UP II, No. CZ.1.07/2.3.00/30.0041 from European Social Fund in the Czech Republic and the IGA project of the Palacký University PrF 2015 002.

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Aberration resistant axial localization using a self-imaging of vortices

Abstract

1. Introduction

2. Concept of the vortex SI in optical imaging

2.1. SI of nondiffracting vortices in a 4-f optical system

2.2. SI of divergent vortices in a lens imaging

3. Experimental verification of the aberration resistant axial localization

3.1. Implementation of the method

3.2. SI composed of two nondiffracting vortices

3.3. SI composed of multiple nondiffracting vortices

4. Demonstration of defocusing rotation of fixed and moving microspheres

5. Conclusions

Acknowledgments

References and links

Supplementary Material (2)

Cited By

Figures (8)

Equations (24)

Optics Express