Optimizing three-dimensional point spread function in lensless holographic microscopy

Jaromír Běhal; Zdeněk Bouchal

doi:10.1364/OE.25.029026

1. Introduction

In optical imaging theory, the basic imaging functions and criteria for evaluation of the imaging performance were preferably defined for two-dimensional objects. The three-dimensional (3D) imaging of volume samples became possible after the invention of confocal microscopy [1], and it has been further developed in modern imaging techniques including digital holographic microscopy [2] or super-resolution localization microscopy [3,4]. Properties of the 3D imaging have been of particular importance also in laser manipulation [5] and particle tracking [6]. The point spread function (PSF) representing the blurred image of a point object was traditionally used as an indicator of the optical performance of imaging systems. To determine the 3D PSF in lens imaging, diffraction of a convergent spherical wave bounded by lens aperture was solved. In exact treatment, Fresnel-Kirchhoff and Rayleigh-Sommerfeld diffraction theories were formulated and the paraxial and Debye approximations proposed for the investigation of optical imaging. To apply these approximations correctly, the specific imaging conditions must be carefully considered. In optical systems with a low numerical aperture, where apodization, polarization and aberration effects are not essential, the paraxial approximation is well applicable. In high-aperture optical systems, the Debye approximation is used successfully. It is based on the representation of the focused light field by a superposition of plane waves whose propagation vectors fall inside the cone given by the aperture angle. A paraxial elaboration of the Debye approximation has also been proposed. It provides the 3D PSF with a symmetrical distribution of the axial intensity. This result is correct for higher apertures but fails in systems with a low Fresnel number, where the axial intensity exhibits asymmetry and its maximum is shifted away from the geometric focus [7, 8]. These effects are not apparent in lens imaging implemented by optical systems with commonly used parameters but can significantly affect the holographic imaging. In optical systems with higher numerical apertures, the axial asymmetry caused by diffraction is suppressed but the attention must be devoted to optical aberrations deteriorating the image quality and reducing the image resolution [9].

Here we examine the 3D PSF reconstructed from phase-shifted point holograms recorded in the experimental setup for digital in-line holographic microscopy [10]. The main attention is paid to the distortion of the axial profile of the PSF whose intensity peak is shifted out of the paraxial image plane. This undesirable effect is studied as a result of the diffraction or holographic aberrations and its connection with the geometric parameters of experiments and the lateral image magnification is clarified. To implement quantitative criteria for the assessment of the diffraction and aberration deterioration of the 3D PSF, suitable approximations are used whose validity is verified by precise numerical simulations and experiments. The analysis provides an optimal combination of experimental parameters and an applicable range of the lateral magnifications for which the diffraction and aberration deterioration of the PSF is maintained at an acceptable level. Theoretical findings are verified in experiments using a spatial light modulator (SLM) that changes the tested parameters flexibly and permits implementation of the phase-shifting.

2. Methods

2.1. Three-dimensional PSF in digital in-line holography

The performance of optical imaging realized by digital in-line holography is studied using Fig. 1(a), illustrating both recording and numerical reconstruction of a point hologram. Laser light of the wavelength λ is focused on a pinhole of few microns in diameter, which acts as a source from which a reference wave ψ_r emanates. This wave illuminates a point-like scatterer that is located just behind the pinhole and generates a divergent signal wave ψ_s. The interference pattern captured by the CCD represents a point hologram given as H = |ψ_s + ψ_r|². To determine the 3D PSF, the hologram is illuminated by a virtual reconstruction wave and the image is obtained by calculating free propagation of light diffracted at the hologram. In the exact treatment, Fresnel-Kirchhoff and Rayleigh-Sommerfeld diffraction integrals are solved numerically to reconstruct the image spot [11]. For small numerical apertures, the Fresnel approximation can be used to solve the diffraction integral with a sufficient accuracy [12].

Fig. 1 Optical scheme for evaluation of the 3D PSF in lensless holographic microscopy: (a) in-line recording geometry, (b) reconstruction geometry, (c) asymmetric axial profile of the PSF with geometrical parameters used in the quantitative evaluation of the focal shift and the axial PSF asymmetry.

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In digital holography, the reconstruction is carried out using a virtual monochromatic wave ψ_c having the same wavelength as that used in the recording. The light diffracted on the hologram placed in the plane x, y can be described using the first Rayleigh-Sommerfeld diffraction integral [13]. The complex amplitude of the image ψ evaluated at the lateral position x_i, y_i located in the distance z_i from the hologram can be written as

ψ \propto \frac{z_{i}}{i λ} \int \int_{- \infty}^{\infty} t (r_{⊥}) \frac{exp (i k | r_{i} |)}{{| r_{i} |}^{2}} d r_{⊥},

where k = 2π/λ, r_⊥ = (x, y), r_i = (x − x_i, y − y_i, z_i) and t is the complex function representing transformation of the reconstructing wave ψ_c by the transmission function of the hologram,

t (r_{⊥}) = T (r_{⊥}) ψ_{c} {| ψ_{s} + ψ_{r} |}^{2} .

Because ψ_s and ψ_r correspond to divergent spherical waves, the aperture function T is introduced, which defines a limited area of the hologram. In real experiments, this area is determined by an overlapping of interfering waves or as an active detector area. The interference pattern created by the signal and reference waves |ψ_s + ψ_r|² includes three terms, which are known as dc term |ψ_s|² + |ψ_r|², and real and virtual images given by

ψ_{s}^{*} ψ_{r}

and

ψ_{s} ψ_{r}^{*}

, respectively. When the virtual or real image is reconstructed from the hologram (2), the remaining terms cause undesirable blurring. In optical holography, this problem has been solved by off-axis arrangement ensuring sufficient spatial separation of the dc term and the holographic twin image from the reconstructed image. In digital in-line holography various techniques have been proposed, in which the dc term was subtracted [14], or both the dc term and the twin image were completely eliminated by processing three or more holograms with different phase shifts between signal and reference waves [15]. In experiments, the required phase shifts were precisely implemented using a piezoelectric transducer [15], a spiral phase plate [16], geometric phase optical elements [17], a liquid crystal variable retarder [18] or a spatial light modulator [19–22]. Applying the phase-shifting technique [15], the complex hologram is created allowing to reconstruct either virtual or real image. Here the 3D PSF is examined for a virtual image reconstructed from the complex function t given as

t (r_{⊥}) = T (r_{⊥}) A_{s} A_{r} A_{c} exp [i (Φ_{s} - Φ_{r} + Φ_{c})],

where A_j and Φ_j, j = s, r, c, are amplitudes and phases of the signal, reference and reconstruction waves. When these waves are considered as ideal spherical waves emanating from a pinhole, an off-axis point scatterer and a virtual source, respectively, located in the positions x_j, y_j, z_j, j = s, r, c, the amplitudes and phases within the hologram plane relative to the phase at the origin can be written as

A_{j} = \frac{a_{j}}{| r_{0 j} |}, Φ_{j} = - k (| r_{j} | - | r_{0 j} |), j = s, r, c,

where r_j = (x − x_j, y − y_j, z_j), r_0j = (x_j, y_j, z_j) and a_j are constant amplitudes. In this paper, the diffraction integral (1) is solved numerically for the complex point hologram (3) to obtain the 3D PSF characterizing the reconstructed image. The axial asymmetry and the focal shift of the diffraction-limited PSF and its deterioration caused by aberrations are studied in appropriate approximations introduced in recording and reconstruction of the holograms.

2.2. Focal shift and axial asymmetry of the diffraction-limited PSF

Description of the experiments implemented with a low numerical aperture can be successfully performed using the paraxial approximation. The signal, reference and reconstruction waves then become paraboloidal waves whose phase (4) is simplified to the form

Φ_{j} = k (\frac{{| r_{⊥} |}^{2}}{2 z_{j}} - \frac{r_{⊥} \cdot r_{⊥_{0 j}}}{z_{j}}), j = s, r, c,

where r_{⊥_0j} = (x_j, y_j). In the phase term of the diffraction integral (1) the Fresnel approximation is used, while in the amplitude term a less accurate replacement |r_i| ≈ |z_i| is applied. In the calculations, the optical sign convention is respected, in which z_s < 0, z_r < 0, z_c < 0 and z_i < 0 are assigned to divergent signal, reference and reconstruction waves and a virtual reconstructed image, respectively. To simplify the discussion, the pinhole located on the optical axis is considered, |r_{⊥_0r}| = 0. Neglecting terms that do not affect the shape of the 3D PSF, the complex amplitude Eq. (1) using the complex function Eq. (3) can be rewritten as the Fourier transform of the hologram aperture function T multiplied by a quadratic phase term,

ψ \propto \frac{1}{z_{i}} \int \int_{- \infty}^{\infty} T (r_{⊥}) exp (- i k \frac{Ω {| r_{⊥} |}^{2}}{2}) exp (i 2 π r_{⊥} \cdot R_{⊥}) d r_{⊥},

where R_⊥ = (X, Y) and Ω and the spatial frequencies X, Y are given as

Ω = \frac{1}{z_{i}} - \frac{1}{z_{s}} + \frac{1}{z_{r}} - \frac{1}{z_{c}},

X = \frac{1}{λ} (\frac{x_{i}}{z_{i}} - \frac{x_{s}}{z_{s}} - \frac{x_{c}}{z_{c}}), Y = \frac{1}{λ} (\frac{y_{i}}{z_{i}} - \frac{y_{s}}{z_{s}} - \frac{y_{c}}{z_{c}}) .

When the paraboloidal waves are used in the calculation model, holographic aberrations do not occur and the image is perfectly focused provided the quadratic phase term is eliminated, Ω = 0. This is achieved if the image is reconstructed at a distance z_i = f determined as

\frac{1}{f} = \frac{1}{z_{c}} - \frac{Δ}{z_{r} (z_{r} + Δ)},

where Δ = z_s − z_r. Assuming that the hologram is laterally unbounded (T = const.), the geometric image given by the Dirac delta function is reconstructed, ψ ∝ δ(X, Y). The coordinates of this point image are x_i = βx_s and y_i = βy_s, where β = dx_i/dx_s is the lateral magnification given by

β = \frac{f}{z_{r} + Δ} .

In order to examine the axial asymmetry of the PSF, the reconstruction of the image is performed at positions shifted away from the focal plane, z_i = f + Δz_i. When the hologram is bounded by a circular aperture of radius ρ_h, its aperture function is given by a circle function, T = circ(|r_⊥|/ρ_h). In this case, the intensity distribution providing the 3D PSF must be calculated numerically. If the axial profile of the PSF is investigated for a point scatterer positioned on the optical axis (|R_⊥| = 0), the diffraction integral has an analytical solution providing the axial intensity defined as I = |ψ (|R_⊥| = 0, Δz_i)|². The normalized axial intensity that is unitary in the paraxial focal plane, I(Δz_i = 0) = 1, then can be written as

I (q) = {(1 - \frac{q}{π n_{i}})}^{2} {sinc}^{2} (\frac{q}{2}),

where

q = \frac{π n_{i} Δ z_{i}}{f + Δ z_{i}}, n_{i} = \frac{ρ_{h}^{2}}{λ f} .

The term in the bracket leads to the shift of the intensity maximum out of the geometric focus, while the sinc function causes asymmetry of the intensity profile shown in Fig. 1(b). These effects are influenced by the geometry of experiments and for their quantification the Fresnel number can be used that is defined as N_i = |n_i|. The axial intensity varies periodically and takes zero values when q = 2mπ, where m = ±1, ±2, ···. The asymmetry of the axial intensity profile is assessed by the positions of the nearest zero points

Δ z_{i}^{+}

and

Δ z_{i}^{-}

related to m = ±1 [Fig. 1(b)],

Δ z_{i}^{+} = \frac{2 f}{n_{i} - 2}, Δ z_{i}^{-} = - \frac{2 f}{n_{i} + 2} .

The peak of the axial intensity is not located at the paraxial focal plane but is shifted towards the reconstructed hologram. Its position is determined as the root of a transcendental equation which follows from dI/dq = 0 [8],

tan (\frac{q}{2}) = (\frac{q}{2}) (1 - \frac{q}{π n_{i}}) .

The root q_max of the equation (11) must be determined numerically. The position of the intensity maximum

Δ z_{i}^{\max}

is then given by

Δ z_{i}^{\max} = \frac{f q_{\max}}{π n_{i} - q_{\max}} .

The half-widths of the central intensity peak

Λ_{i}^{+}

and

Λ_{i}^{-}

are given by the first zero points of the oscillating axial intensity that are closest to the intensity maximum,

Λ_{i}^{\pm} = Δ z_{i}^{\pm} - Δ z_{i}^{\max} .

The geometric meaning of the used symbols is clear from Fig. 1(b). When examining the axial asymmetry of the PSF in the image space, the coefficient Q_i can be used, which is defined as

Q_{i} = | \frac{Λ_{i}^{+}}{Λ_{i}^{-}} | .

The axial PSF can also be analyzed in the object space by transforming the corresponding longitudinal positions by the longitudinal magnification given as γ = dz_i/dz_s. When the constant longitudinal magnification determined for the paraxial image plane is used, the asymmetry coefficients in the object and image spaces are equal, Q = Q_i. In the exact analysis, the positions of the intensity maximum and the nearest zero points of the oscillating axial intensity are transformed into the object space with different values of γ, hence Q and Q_i are slightly different in the object and image spaces.

The optical performance achieved in digital in-line holography is influenced by three basic geometric parameters, ρ_p, Δ and z_r, which denote the pinhole radius and distances from the pinhole to the sample and CCD, respectively. The pixel size of the CCD, p_CCD, determines the conditions of the discretization of the record and is also an important parameter affecting the image quality. The lateral image magnification is dependent on Δ and z_r but is also influenced by the reconstruction geometry. If the reconstruction wave is exactly matched to the reference wave, z_c = z_r, an aberration-free image is reconstructed at the object position with the unitary lateral magnification, f = z_s and β = 1 [Eqs. (7) and (8)]. To get the magnified image, the reconstruction wave originating from a virtual source placed at the distances z_c ∈ (−∞, z_r) from the hologram is used. For the specified parameters Δ and z_r, the largest lateral magnification is achieved for a plane wave reconstruction (z_c → ∞). The mismatch between the reference and reconstruction waves results in optical aberrations and diffraction effects causing the axial asymmetry of the PSF. Manifestations of these effects vary with the lateral image magnification and will be examined both theoretically and experimentally.

When the point hologram is recorded using a low numerical aperture, the diffraction-limited image is reconstructed. The radius of the central Airy spot is given as Δr_i = αλ/NA_i, where α = 0.61 for the circular hologram aperture and NA_i ≈ ρ_h/|f|. The lateral resolution in the object space then can be written as Δr = Δr_i/|β|, hence Δr being inversely proportional to NA_i|β|. As this product is independent of z_c, NA_i|β| = ρ_h/(z_r − Δ), the theoretical lateral resolution in the object space remains unchanged when the reconstruction wave is changed. If the hologram aperture is defined by the light spot created by the reference wave on the CCD, the radius ρ_h is determined by the pinhole radius ρ_p as ρ_h = αλ|z_r|/ρ_p. The resolution in the object space can then be rewritten to the form

Δ r = ρ_{p} (1 - \frac{Δ}{| z_{r} |}),

where Δ acquires only positive values in the examined in-line holographic geometry. The Fresnel number of the image space depends on the basic parameters of the experiment, but also changes with the reconstruction geometry. For limiting cases of the reconstruction it can be written as

N_{i} = \frac{α^{2} λ κ}{ρ_{p} Δ r},

where κ = |z_r| for the reconstruction wave exactly matched to the reference wave, while κ = Δ for the plane wave reconstruction. In experiments, |z_r| >> Δ and the high N_i is reached with common geometrical parameters when reconstructing with z_c = z_r. Hence, in the matched hologram reconstruction providing the unitary lateral magnification β = 1, the diffraction-limited image is obtained that possesses the axial symmetry of the PSF guaranteed by the high Fresnel number. If the zooming is applied by the mismatch of the reconstruction and reference waves, the holographic aberrations and the axial asymmetry of the PSF appear even for the recording made with perfect spherical waves.

In this paper, the image deterioration is investigated for the plane wave reconstruction providing the highest lateral magnification for given Δ and z_r. With the increasing distance between the pinhole and the sample Δ, both the object space resolution and the axial symmetry of the PSF are improved [Eqs. (15) and (16)], but the detection requirements increase strongly. Considering the Nyquist-Shannon sampling theorem for the reconstruction using z_c → ∞, the largest specimen to pinhole distance Δ permissible for the pixel size p_CCD is determined by the condition

Δ \leq \frac{| z_{r} | Δ r}{2 α p_{CCD}} .

To examine how the actual imaging performance is approaching the theoretical object space resolution limit (15), the holographic aberrations and diffraction effects causing the axial asymmetry of the PSF must be included in the analysis.

2.3. Influence of holographic aberrations

The virtual image is reconstructed from complex function (3) determined by the product of complex amplitudes of signal, reference and reconstruction waves, whose amplitudes and phases are given by (4). If the hologram is reconstructed by a monochromatic wave of the same wavelength as that used in the recording, and the paraxial approximation operating with paraboloidal waves is applied, the phase of the reconstructed hologram matches the phase of the wave converging at the paraxial image point with the coordinates x_i = βx_s, y_i = βy_s and z_i = f. When the experiment is performed with parameters beyond the paraxial approximation, the hologram phase determined by the product of the complex amplitudes of the spherical waves no longer corresponds to the phase of the spherical wave converging at the paraxial image point. The differences that appear are holographic aberrations. The third-order holographic aberrations are described by Taylor expansion applied to the phase Φ_j given by Eq. (4), in which only first three terms are included [23]. To simplify the discussion, the pinhole and the source of the reconstruction wave located on the optical axis are considered, x_j = y_j = 0, j = r, c. Following the procedure specified in Appendix, the third-order holographic aberrations are obtained in the form

W = S ρ_{N}^{4} + C ρ_{N}^{3} cos φ + A ρ_{N}^{2} {cos}^{2} φ + F ρ_{N}^{2} + D ρ_{N} cos φ,

where ρ_N = ρ/ρ_h, (ρ, φ) denote the polar coordinates in the hologram plane and the coefficients S, C, A, F and D represent spherical aberration, coma, astigmatism, field curvature and distortion, respectively, and can be written as

\begin{array}{l} S = \frac{ρ_{h}^{4}}{8} (\frac{1}{z_{i}^{3}} + \frac{1}{z_{r}^{3}} - \frac{1}{z_{s}^{3}} - \frac{1}{z_{c}^{3}}), & C = - \frac{x_{s} ρ_{h}^{3}}{2} (\frac{β}{z_{i}^{3}} - \frac{1}{z_{s}^{3}}), \\ A = \frac{x_{s}^{2} ρ_{h}^{2}}{2} (\frac{β^{2}}{z_{i}^{3}} - \frac{1}{z_{s}^{3}}), & F = \frac{1}{2} A, & D = - \frac{x_{s}^{3} ρ_{h}}{2} (\frac{β^{3}}{z_{i}^{3}} - \frac{1}{z_{s}^{3}}) . \end{array}

Using the reconstruction wave that is exactly matched to the reference wave (z_c = z_r), the phases of the waves cancel each other in the reconstructed complex function Eq. (3) and the signal wave is fully recovered. In this case, the aberration-free image with the unitary magnification is obtained (z_i = z_s, β = 1, hence S = C = A = F = D = 0).

When optimizing the magnified lensless holographic imaging achieved by the mismatch of the reconstruction and reference waves, the focus shift and the axial asymmetry of the PSF caused by diffraction are taken into account together with the PSF deterioration due to the spherical aberration. The off-axis aberrations are then assessed to determine the acceptable field of view. The analysis is performed for the plane wave reconstruction (z_c → ∞) providing the highest lateral magnification. If the spherical aberration is evaluated in the paraxial image plane (z_i = f), the coefficient S can be written as

S = - \frac{3}{8} \frac{ρ_{h}^{4}}{z_{s} z_{r} f} .

Depending on the pinhole to sample distance Δ, the coefficient of the spherical aberration can be rewritten as

S = \frac{3}{8} \frac{ρ_{h}^{4} Δ}{z_{r}^{2} {(z_{r} + Δ)}^{2}} .

As previously shown, higher Δ values are favorable for improvement of the theoretical lateral resolution (15) and result in the increased Fresnel number (16) that ensures elimination of the axial asymmetry and the focal shift of the diffraction-limited PSF. Unfortunately, since Δ is positive and z_r negative, the spherical aberration increases significantly with larger Δ as shown in (20). An accurate assessment of the effect of the holographic spherical aberration can be done numerically but a good estimate is even possible with the Strehl ratio K accessible by simplified calculations [24]. Assuming a circular aperture with the radius ρ_h and a holographic wave aberration W, the Strehl ratio is defined as

K = {| \frac{1}{π} \int_{0}^{2 π} \int_{0}^{1} exp (i k W) ρ_{N} d ρ_{N} d φ |}^{2} .

By expanding the complex exponential in a power series and retaining the first three terms, the approximate expression of the Strehl ratio valid for small aberrations is

K = 1 - k^{2} σ_{W}^{2},

where

σ_{W}^{2} = 〈 W^{2} 〉 - {〈 W 〉}^{2}

is the variance of the wave aberration defined by

〈 W^{2} 〉 = \frac{1}{π} \int_{0}^{2 π} \int_{0}^{1} W^{2} ρ_{N} d ρ_{N} d φ, 〈 W 〉 = \frac{1}{π} \int_{0}^{2 π} \int_{0}^{1} W ρ_{N} d ρ_{N} d φ .

When assessing the effect of the third-order spherical aberration in the paraxial image plane,

W = S ρ_{N}^{4}

, the Strehl ratio (22) results in

K = 1 - \frac{4}{45} k^{2} S^{2} .

The quality of the reconstructed image is significantly improved applying a numerical refocusing. To perform optimal focusing for a spherical aberration with the coefficient S, the wave aberration

W = S ρ_{N}^{4} + R ρ_{N}^{2}

is considered, where R is the refocusing coefficient. An optimal focusing is obtained when the Strehl ratio K is calculated as a function of R and the condition dK/dR = 0 providing its extremum used. The Strehl ratio calculated for the image deteriorated by the spherical aberration S and reconstructed in the optimal image plane is then determined as

K = 1 - \frac{1}{180} k^{2} S^{2} .

As follows from (23) and (24), the decrease in Strehl ratio caused by spherical aberration S is after optimal refocusing approximately 16× smaller than in the paraxial image plane. As an indicator of the high imaging performance, the Strehl ratio K ≥ 0.8 is applied resulting in the requirement of a weak holographic spherical aberration limited by S ≤ 6/k. To achieve the desired values of K in the optimal image plane, the condition for the combination of basic experimental parameters can be obtained using (20),

z_{r}^{2} Δ {(1 + \frac{z_{r}}{Δ})}^{2} \geq \frac{2 ρ_{h}^{4}}{5 λ} .

If the radius of the hologram is determined by the light spot created by the reference wave diffracted on the pinhole with the radius ρ_p, the combination of the parameters ensuring the imaging with K ≥ 0.8 can be rewritten as

\frac{Δ}{z_{r}^{2}} {(1 + \frac{z_{r}}{Δ})}^{2} \geq \frac{λ^{3}}{18 ρ_{p}^{4}} .

By specifying the position and the size of the pinhole, z_r and ρ_p, or the hologram radius ρ_h, the sample to pinhole distances Δ obtained by applying the conditions (25) and (26) ensure that the reconstructed image is not deteriorated by the spherical aberration at an optimal image plane.

3. Results and discussion

The magnified lensless holographic imaging realized by the mismatch of the reconstruction and reference waves is significantly affected by the pinhole radius ρ_p and the distances z_r and Δ determining the pinhole position relative to the CCD and the sample, respectively. By combining these parameters inappropriately, the axial asymmetry and the focal shift or the holographic aberrations deteriorate the 3D PSF. Theoretical predictions of these effects resulting in an optimal choice of geometrical parameters were verified in experiments exploiting versatility of a SLM.

3.1. Experimental setup

The optical setup used is shown in Fig. 2. The light beam from He-Ne laser (10 mW, 632.8 nm) is spatially filtered by the single-mode fiber (Thorlabs P1-630A-FC-2, mode field diameter 3.6 – 5.3 μm) and collimated by the lens L₁ (achromatic doublet Thorlabs AC254-150-A, diameter 25.4 mm, focal length 150 mm). The created beam passes through a beam splitter BS towards the reflective SLM (Hamamatsu X10468, 800 × 600, pixel size 20 μm). By a random pixel selection [19, 20], two different phase functions are simultaneously displayed on the SLM generating converging and diverging lenses with the focal lengths ± f_m. Three different constant phase shifts ϑ_j are added to the quadratic phase of the converging lens so the phase functions t_j sequentially sent to the SLM are given as

t_{j} = exp (\frac{i k r_{m}^{2}}{2 f_{m}} + i ϑ_{j}) + exp (\frac{- i k r_{m}^{2}}{2 f_{m}}), j = 1, 2, 3,

where r_m is the radial polar coordinate at the SLM plane and ϑ₁ = 0, ϑ₂ = 2π/3 and ϑ₃ = 4π/3. Using the SLM, the incident collimated beam is divided into two slightly convergent/divergent waves which are further transformed by a lens L₂ with the focal length f_d. Since the focal lengths f_m ≫ f_d are used, two focal points F_r and F_s are formed behind the lens L₂ whose mutual distance Δf is approximately determined as

Δ f = 2 f_{d}^{2} / f_{m}

. Using the lens L₂ with f_d = 38.1 mm (achromatic doublet Edmund #49–775, diameter 12.7 mm), the distance between the focal points Δf may be varied in the range 1–10 mm when the focal lengths of the converging/diverging lenses created on the SLM are changed from ±2900 mm to ±290 mm. The focal points F_r and F_s represent sources of reference and signal waves that interfere and form a point hologram captured by a CCD (QImaging Retiga 4000R, 2048 × 2048 pixels, chip size 15 × 15 mm²). The spacing of the focal points F_r and F_s then corresponds to the distance between the pinhole and the sample used in the calculations, Δ = Δf. To ensure experimental conditions closely related to the lensless digital in-line holography examined theoretically, almost perfect reference and signal waves must emerge from the focal points F_r and F_s. This requirement was achieved by the optimized design of the optical system carried out using Oslo Premium software. With the optical components used, the diffraction-limited light spots with the diameter of about 5 μm were obtained at the focal points F_r and F_s. For the apertures used, the optical system worked as aberration-free and the geometric image of the fiber face created with the magnification determined by the lenses L₁ and L₂ was almost three times smaller than the diffraction spot. In the experiments carried out, the versatility of the SLM was fully exploited. By controlling the SLM, the distance Δ was operatively altered and the phase shifts of recorded holograms were performed without any mechanical movement. By processing three phase shifted records [15,25,], a complex hologram corresponding to the virtual image was obtained and the 3D PSF reconstructed numerically. When reconstructing image of the resolution target, only one convergent lens was displayed on the SLM that created the reference wave in combination with lens L₂. The signal wave was created by diffraction of the reference wave on the resolution target.

Fig. 2 In-line holographic setup used for reconstruction of the 3D PSFs and image of the resolution target in testing an optimal design of geometric parameters: SMF ···single-mode fiber, L₁ ··· collimating lens, BS··· beam splitter, SLM··· spatial light modulator creating converging/diverging lenses with varying focal lengths ± f_m, L₂ ··· imaging lens.

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3.2. Experimental results

In preparing the measurement, a theoretical dependence of the axial asymmetry and the aberration deterioration of the PSF on geometric parameters of the setup was examined using established quantitative indicators. In all demonstrated results, the monochromatic plane wave was used to reconstruct the recorded holograms. In Fig. 3, the deterioration of the reconstructed image of a point object caused by spherical aberration is demonstrated by the color-coded Strehl ratio evaluated in the image plane obtained by an optimal refocusing. The changes in the Strehl ratio are evaluated for different pinhole settings z_r relative to the CCD and different positions of the point object from the pinhole Δ. In Fig. 3(a), the pinhole radius ρ_p = 2.75 μm is used, and for monitored settings z_r, the radius of the detected point hologram is determined by the size of the diffraction spot created by the reference wave, ρ_h = αλ|z_r|/ρ_p. In this case, the image deterioration does not change significantly with z_r and the unacceptable decrease in the Strehl ratio occurs for Δ > 4 mm. In Fig. 3(b), the Strehl ratio is evaluated for the constant radius of the hologram ρ_h = 5.6 mm, corresponding to the diffraction spot created by the pinhole at the closest position in front of the CCD.

Fig. 3 Image deterioration due to spherical aberration demonstrated by the color-coded Strehl ratio (image reconstructed by the plane wave and evaluated at the optimal image plane): (a) pinhole radius ρ_p = 2.75 μm, hologram radius ρ_h = αλ|z_r|/ρ_p, (b) constant hologram radius ρ_h = 5.6 mm.

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The dependence of the Strehl ratio K on Δ is shown in Fig. 4 (black line) for the constant pinhole position z_r = −43.5 mm and the pinhole radius ρ_p = 2.75 μm, together with the diffraction-limited resolution in the object space Δr (green line), the resolution limit given by sampling conditions (thin black line), the coefficient of asymmetry Q (red line) and the coefficient of spherical aberration S (blue line). The coefficient Q is evaluated in the object space using a varying longitudinal magnification to transform the longitudinal distances from the image to object space. The dependence of K on Δ corresponds to the cross section of the color-coded graph in Fig. 3(a) indicated by a white dashed line. The range of the evaluated distances between the pinhole and the sample Δ ∈ 〈0, 10〉 mm is limited by the sampling conditions determined for CCD QImaging Retiga 4000R. For the greatest value of Δ, the best lateral resolution Δr ≈ 3.3λ is obtained. The lateral resolution is demonstrated by the radius of the diffraction spot Δr determined in the object space. While Δr decreases linearly with increasing Δ, the asymmetry and aberration coefficients Q and S show just opposite changes. For small values of Δ, the coefficient Q rapidly decreases indicating a strong asymmetry and shift of the axial PSF profile. In this case, the spherical aberration given by the coefficient S is very low so the Strehl ratio K is almost unitary. The situation is just opposite for the large distances Δ. The axial symmetry of the PSF with the coefficient Q ≈ 1 is achieved, while the spherical aberration increases and the Strehl ratio drops sharply. Axial symmetry of the PSF and nearly aberration-free imaging are guaranteed if both the asymmetry coefficient and the Strehl ratio are close to unit value. With the conditions Q ≥ 0.8 and K ≥ 0.8, the range of applicable positions Δ can be determined for the fixed parameters z_r and ρ_p defining three different areas in Fig. 4. The areas identified as I, II and III determine the parameters resulting in diffraction-limited imaging with strong axial asymmetry of the PSF, low-aberration imaging with the nearly symmetrical 3D PSF and imaging strongly deteriorated by the spherical aberration, respectively. For ρ_p = 2.75 μm and z_r = −43.5 mm, the applicable area II is defined by the limit values of the sample to pinhole distance Δ = 0.17 mm and 3.5 mm. When the position of the sample is changed within this range, the lateral magnification varies from β = 257 to 12.5, respectively. The maximum usable value of Δ is determined by the decrease of the Strehl ratio to K = 0.8 and follows from Eq. (26). The minimum value of Δ is given by the decrease of the asymmetry coefficient to Q = 0.8 and must be determined numerically. The experiments carried out with the parameters belonging to the individual areas of Fig. 4 (marked by violet points) provide results which closely match the theoretical predictions.

Fig. 4 Dependence of the indicators of imaging performance on the pinhole to object distance Δ (evaluation performed for z_r = −43.5 mm and ρ_p = 2.75 μm): object space diffraction-limited lateral resolution in multiples of λ, Δr (green line), resolution limit in multiples of λ given by sampling conditions for CCD QImaging Retiga 4000R (thin black line), asymmetry coefficient, Q (red line), Strehl ratio for optimal image plane, K (black line) and coefficient of spherical aberration in multiples of λ, S (blue line). Ranges of parameters: I-axially asymmetric PSF, II-symmetric nearly diffraction-limited PSF (Q ≥ 0.8, K ≥ 0.8), III-strong spherical aberration.

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Verification measurements were performed in the setup shown in Fig. 2, in which the light spots created at the focal points F_r and F_s replaced the role of the pinhole and the point object in the lensless holographic configuration in Fig. 1. The radius of the diffraction spot created at the focal point F_r was approximately equal to the pinhole radius used in the analysis presented in Fig. 4. The distance Δ was operatively altered by displaying various converging and diverging lenses on the SLM. Each point hologram was recorded repeatedly with three different phase shifts between the signal and reference waves (0, 2π/3, 4π/3) that were set by the SLM. By processing the records using the phase-shifting technique [15], a complex hologram was created allowing to reconstruct the PSF undisturbed by the nondiffracted light and the holographic twin image.

The results obtained are demonstrated in Figs. 5, 6 and 7. In all three cases, the same position of the reference focal spot z_r = −43.5 mm was used and the plane wave reconstruction of the PSF was carried out using the point holograms recorded with the same radius ρ_h = 6.1 mm. In the individual cases, the distance between the reference and signal focal spots F_r and F_s was set to Δ = 0.1 mm, 1.5 mm and 4.9 mm, respectively. The settings were carried out by the SLM and resulted in different focal lengths of the reconstructed complex hologram. With the parameters used in the individual cases, nearly the same diffraction-limited lateral resolution was achieved in the object space, specified by the theoretical values Δr = 2.7 μm, 2.65 μm and 2.4 μm, respectively. If the axial PSF profile is calculated using the paraxial Debye approximation, a symmetrical shape described by the sinc function is obtained. The half-width of the longitudinal spot determined by the first zero point of the sinc function remains approximately the same for all three cases provided the axial PSFs are compared in the object space. The theoretical values obtained for the examined cases are Δz = 64 μm, 60 μm and 56 μm. Although the demonstrated image reconstructions provide nearly the same theoretical values of the lateral and longitudinal resolution in the paraxial Debye approximation, significant differences are found when the 3D PSFs are experimentally implemented and correctly evaluated. The 3D PSFs shown in Figs. 5, 6 and 7 can be assigned, according to the effects influencing their shape, to the areas I, II and III in Fig. 4. Significant differences in the reconstructed PSFs are due to the fact that nearly the same lateral and longitudinal resolutions were achieved in a different manner in each case.

Fig. 5 Diffraction-limited axially asymmetric PSF reconstructed by the plane wave from three phase-shifted point holograms taken with the geometric parameters belonging to the area I in Fig. 4 (ρ_h = 6.1 mm, z_r = −43.5 mm, Δ = 0.1 mm). The axial and radial PSF profiles obtained from experimental data were transformed into the object space and compared with numerical simulations of the hologram recording and reconstruction.

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Fig. 6 Symmetric low-aberration PSF reconstructed by the plane wave from three phase-shifted point holograms taken with the geometric parameters belonging to the area II in Fig. 4 (ρ_h = 6.1 mm, z_r = −43.5 mm, Δ = 1.5 mm).

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Fig. 7 PSF deteriorated by a strong holographic aberration and reconstructed by the plane wave from three phase-shifted point holograms taken with the geometric parameters belonging to the area III in Fig. 4 (ρ_h = 6.1 mm, z_r = −43.5 mm, Δ = 4.9 mm).

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The PSF illustrated in Fig. 5 was reconstructed from the holograms recorded with ρ_h = 6.1 mm, z_r = −43.5 mm and Δ = 0.1 mm. The used parameters correspond to the left violet dot in Fig. 4. With these parameters, the paraxial lateral and longitudinal resolutions Δr = 2.7 μm and Δz = 64 μm were obtained by combining a very low numerical aperture of the reconstructed hologram (NA_i = 0.0003) with the extremely large lateral and longitudinal magnifications (β = 435, γ = β²). As the numerical aperture was low, the image was not impaired by aberrations, but the diffraction effects caused a strong asymmetry of the axial PSF due to the low Fresnel number (N_i = 3.1). The asymmetric axial PSF (blue line) was reconstructed from experimentally acquired holograms by algorithms based on the Kirchhoff transform [26]. To demonstrate a good consistency between experiment and theory, a numerical model was created in which both hologram recording and reconstruction were simulated with parameters used in the real experiment. Results obtained in the precise nonparaxial approach are illustrated by a red line while the use of the paraxial approximation corresponds to a black dashed line.

The PSF shown in Fig. 6 belongs to the area II in Fig. 4 and was obtained by reconstructing the records acquired with optimally selected parameters ρ_h = 6.1 mm, z_r = −43.5 mm and Δ = 1.5 mm (middle violet dot in Fig. 4). In this case, the paraxial lateral and longitudinal resolutions Δr = 2.65 μm and Δz = 60 μm were achieved by a balanced combination of the hologram aperture (NA_i = 0.005) and the lateral and longitudinal magnifications (β = 29, γ = β²). The Fresnel number N_i = 48.3 was high enough for the axial PSF to be nearly symmetric and the aberrations were so low that the imaging was still almost diffraction-limited. The combination of parameters used complied with the condition (25), hence the Strehl ratio K ≥ 0.8 was ensured in the optimal image plane.

The PSF in Fig. 7 belongs to the area III in Fig. 4 (ρ_h = 6.1 mm, z_r = −43.5 mm, Δ = 4.9 mm, right violet dot in Fig. 4), and corresponds to the paraxial lateral and longitudinal resolutions Δr = 2.4 μm and Δz = 56 μm obtained with the high image space aperture (NA_i = 0.017) and the low lateral and longitudinal magnifications (β = 8.9, γ = β²). The Fresnel number is extremely high (N_i = 172) so the diffraction image asymmetry does not appear, but the lateral and longitudinal PSF profiles are strongly impaired by the spherical aberration due to the high image space aperture. The experimental PSF (blue line) is again in a good agreement with the PSF obtained in the numerical model (red line) in which both the recording and reconstruction of the hologram were simulated in the nonparaxial approach.

The impact of the geometric parameters on the imaging performance was also investigated using a resolution target. Experiments were carried out in the setup shown in Fig. 2, in which the distance of the reference focus F_r from the CCD was maintained constant, z_r = −46.4 mm. In the examined cases, the resolution target was placed in three different positions relative to the reference focus given by Δ = 0.7 mm, 3 mm and 10 mm, respectively. The reconstruction of acquired holograms was performed using the monochromatic plane wave. The various positions of the target resulted in significantly different combinations of the image space aperture and the magnification. The image space apertures change as a result of different distances in which the holograms are reconstructed. In the settings under consideration, the reconstruction distances and the lateral magnifications get values z_i = −3029 mm, −671 mm and −169 mm, and β = 66.3, 15.5 and 4.6, respectively. The imaging performance of individual settings can be assessed using the object space resolution estimated as the radius of the Airy diffraction spot divided by the lateral magnification, Δr = 0.61λ|z_i|/(ρ_hβ). Hence, the resolution in the object space is improved with reduced |z_i|/β ratio for constant ρ_h. This ratio decreases in each of the examined cases, giving values |z_i|/β = 45.7, 43.4 and 36.7, respectively.

In the reconstructions of the resolution target in Fig. 8, the trend predicting improvement of the object space resolution with the increasing Δ is not fully confirmed because the aberration effects occur in these settings, as previously shown by the PSF reconstructions. The experimental results shown in Fig. 8 were obtained with a positive resolution target in which the line width in the finest group was 2.5 μm. In each reconstruction, an enlarged cut of the finest target group is shown along with the cross profiles and the average visibility.

Fig. 8 Plane wave reconstructions of the resolution target verifying an optimal choice of geometric parameters for hologram recording: (a) aberration-free imaging with reduced diffraction-limited lateral resolution (z_r = −46.4 mm, Δ = 0.7 mm), (b) low-aberration imaging with optimal lateral resolution approaching the limit allowed by the sampling conditions (z_r = −46.4 mm, Δ = 3 mm), (c) imaging with resolution reduced by holographic aberrations (z_r = −46.4 mm, Δ = 10 mm).

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In the image reconstruction in Fig. 8(a), performed with the low image space aperture and the high lateral magnification, aberration effects are negligible, but the resolution is limited by the |z_i|/β ratio, which is highest of all three cases evaluated. In addition, axial asymmetry of the PSF occurs with these parameters.

The target reconstruction shown in Fig. 8(b) represents an optimal combination of geometric parameters resulting in imaging only weakly affected by holographic aberrations and providing resolution approaching the resolution limit determined by the sampling conditions applied to the CCD used. The PSF assigned to this case provides almost perfect axial symmetry.

Although the parameters used for the target reconstruction shown in Fig. 8(c) give the smallest |z_i|/β ratio, the best resolution is not achieved in this case. The reason for the lower image contrast is spherical aberration which deteriorates both the transverse and axial profiles of the PSF as previously demonstrated.

4. Conclusion

We present theoretical and experimental analysis of the lensless holographic microscopy focused on a diffraction asymmetry of the axial PSF, conditions of correct sampling of experimental data and effects of holographic aberrations that occur when the magnified image is created by the mismatch of reconstruction and reference waves. We show in the imaging using the plane wave reconstruction that the same theoretical resolution can be achieved with various combinations of experimental parameters having significantly different impact on the imaging performance. In extreme cases of the design, the same theoretical lateral and longitudinal resolutions are achieved by combining a very low numerical aperture of the reconstructed hologram with an extremely high lateral and longitudinal magnifications, or conversely, a high hologram aperture is used with low magnifications. While in the first case an extremely asymmetric axial PSF arises with its maximum shifted outside the paraxial focus, in the latter case the spherical aberration unacceptably deteriorates both lateral and longitudinal profiles of the PSF. In suitably designed experiments, a symmetrical, almost diffraction-limited PSF is obtained by a balanced combination of the numerical aperture of the reconstructed hologram and the magnification. The choice of the geometric parameters that ensure optimal imaging performance is governed by the conditions that are found and verified in this paper.

Effects examined theoretically are verified experimentally in the setup utilizing the versatility of a SLM. The experimental results presented by reconstructions of the 3D PSFs and the resolution target are in a good agreement with theoretical predictions.

Appendix

To examine the holographic aberrations, the square roots in the phase of signal and reference waves (4) are expanded to Taylor series. In the same way, the phase of the wave corresponding to the paraxial image is expanded for the point of convergence (x_i, y_i, z_i). Considering only the first three terms of the series and respecting the introduced sign convention, the phase of waves can be written as

Φ_{j} = Φ_{j}^{(1)} - Φ_{j}^{(3)}, j = s, r, c, i,

Φ_{j}^{(1)} = q_{j} k \frac{{| r_{⊥_{j}} |}^{2} - {| r_{⊥ 0 j} |}^{2}}{2 z_{j}}, Φ_{j}^{(3)} = q_{j} k \frac{{| r_{⊥_{j}} |}^{4} - {| r_{⊥ 0 j} |}^{4}}{8 z_{j}^{3}},

where q_j = −1 for j = s, r, c, q_j = 1 for j = i, r_{⊥ _j} = (x − x_j, y − y_j) and r_⊥0j = (x_j, y_j). By the condition

Φ_{i}^{(1)} = Φ_{r}^{(1)} - Φ_{s}^{(1)} - Φ_{c}^{(1)}

ensuring that the first-order terms in z_j are matched, the paraxial imaging is described. From this condition, the focal length f and the lateral magnification β given by (7) and (8) can be obtained. Holographic wave aberrations defined as the optical path difference between the reference wavefront and the equiphase surface given by the complex hologram are determined by the terms of the third order in z_j,

W = \frac{1}{k} (Φ_{i}^{3} - Φ_{r}^{(3)} + Φ_{s}^{(3)} + Φ_{c}^{(3)}) .

To simplify the description of holographic aberrations, the axial positions of the pinhole and the source of the reconstruction wave are used together with the assumption that the signal wave source is located on the x-axis (x_j = y_j = 0, for j = r, c and y_j = 0 for j = s, i). Using the polar coordinates in the hologram plane, x = ρ cosφ, y = ρ sinφ, the third-order terms can be rewritten to the form

\begin{array}{l} \frac{q_{j}}{k} Φ_{j}^{(3)} & = & A_{j_{040}} ρ^{4} + A_{j_{131}} x_{j} ρ^{3} cos φ + A_{j_{222}} x_{j}^{2} ρ^{2} {cos}^{2} φ \\ + & A_{j_{220}} x_{j}^{2} ρ^{2} + A_{j_{311}} x_{j}^{3} ρ cos φ, j = s, r, c, i, \end{array}

where the used coefficients have numerical indices indicating the power of the parameters x_j, ρ and cosφ, respectively, and are given as

A_{j_{040}} = \frac{1}{8 z_{j}^{3}}, A_{j_{131}} = - 4 A_{j_{040}}, A_{j_{222}} = 4 A_{j_{040}}, A_{j_{220}} = 2 A_{j_{040}}, A_{j_{311}} = - 4 A_{j_{040}} .

Substituting (30) into (29) and using the normalized radial coordinate ρ_N = ρ/ρ_h, where ρ_h is the radius of the circular hologram aperture, the total holographic aberration given by Eq. (18) is obtained.

Funding

Grant Agency of the Czech Republic (No. 15-14612S); Palacký University in Olomouc (IGA-PrF-2017-002).

Acknowledgments

P. Bouchal from Brno University of Technology, the Czech Republic, is acknowledged for his contribution to the analysis of axially asymmetric holographic imaging made at the early stage of the research.

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Optimizing three-dimensional point spread function in lensless holographic microscopy

Abstract

1. Introduction

2. Methods

2.1. Three-dimensional PSF in digital in-line holography

2.2. Focal shift and axial asymmetry of the diffraction-limited PSF

2.3. Influence of holographic aberrations

3. Results and discussion

3.1. Experimental setup

3.2. Experimental results

4. Conclusion

Appendix

Funding

Acknowledgments

References and links

Cited By

Figures (8)

Equations (37)

Optics Express