The self-referenced imaging and the RBL
Let us briefly summarize the self-referenced hologram generating method. An SHS generates two waves (a wave pair) with different wavefront curvatures from a light coming from a point of the object.
The interference of the pair of waves produces a Self-referenced Interference Pattern (SIP). The SIP captured by a digital camera is the digital intensity hologram. Several object points will have several SIPs by the SHS, and the camera captured image of their sum also is the digital intensity hologram.
When the light coming from different points of the object are coherent with each other, the complex amplitude of their SHS generated SIPs is added. Otherwise, only the intensities of their SIPs are summed. (Obviously there is a case when the SIPs are partially coherent with each other.) Thus, in the particular case of fluorescent objects, only the intensities of the SIPs are summed, irrespectively of the type of their excitation light.
Let us analyse the formation of a single SIP in the following. A spherical wave emitted (or reflected) by a single point is divided than modulated differently by the SHS into two spherical waves, with a radius of curvatures (R1, R2) at the detector plane. Our calculations pointed out that the interaction of these waves generates an interference intensity pattern, which is the same as the intensity of the interference of a plane wave and a spherical wave with radius of curvature Rd (Eq. (1)), where
$$ {\mathrm{R}}_d=\pm \frac{{\mathrm{R}}_1*{\mathrm{R}}_2}{{\mathrm{R}}_1-{\mathrm{R}}_2}\ . $$
(1)
Rd is used to define the reconstruction distance of the corresponding object point. As the SIPs are incoherently summed up, they do not disturb each other, and therefore, their reconstruction distances remain unbiased. This property is important when the reconstruction is used for object localization. However, the more incoherent summation of the SIPs, the less relative dynamic range of the detector, which results in considerable loss of contrast of the captured digital hologram.
Next, our novel optical solution is presented for creating double coherent waves from a single one for the purpose of self-referenced holography.
RBL hologram generation
At the central point of our SHS stands the Ring-shaped Bifocal Lens that realise the self-referenced hologram with the required beam splitting. This splitting is the result of the division of the aperture of the RBL. The two apertures are different; the central one is circular while the other is a ring around it. Both of them are symmetric to the optical axis. These aperture areas have a focus difference, which in general can be reached with the optical property of the material (e.g. grin lens), or with the geometry. Here, we use an RBL where the geometry of the lens generates the different focuses. To ensure exactly two focuses, only one of the two surfaces (the right one) is diversified, as it can be seen in Fig. 1.
In our experiments, we use a custom made RBL, which consists of a central plano-convex lens (focal length 400 mm), and a “biplane ring-shaped lens” having infinite focus. The outer diameter of the whole RBL is 10 mm while the diameter of the inner lens is 6 mm. The scheme of the hologram generation of the actual RBL is shown in Fig. 2 when the source object is at an infinite distance. One can see that the RBL creates two beams, a central placed cone shaped one, and a hollow one. These beams have a ring-shaped cross section at the plane of detection. The self-referenced hologram of a single point created by the RBL is ring-shaped as it is shown in Fig. 4. Because the middle of this hologram is missing, we call it gappy hologram. The shape of the hologram is determined by the shape of the outer aperture of the RBL, and only the divergences of the beams depend on the focal parameters (Fig. 2).
The RBL modulates the incident U(r,RBL1) complex amplitude beam the following way (Eq. (2)):
$$ U\left(r,RBL2\right)=U\left(r,RBL1\right)*\left({A}_o{L}_o+{A}_c{L}_c\right) $$
(2)
Where U(r, RBL2) denotes the generated beam, Ao (Eq. (3)) and Ac (Eq. (4)) correspond to the ring and the central apertures of the RBL, while Lo (Eq. (5)) and Lc (Eq. (6)) describe the phase modulation property of the lenses in the ring-shaped and the central area.
$$ {A}_o= sign\left(r-R\right) $$
(3)
$$ {A}_c= sign\left(R-r\right) $$
(4)
$$ {L}_o= \exp \Big(i\pi /\left(\lambda *{f}_o\right){r}^2 $$
(5)
$$ {L}_c= \exp \Big(i\pi /\left(\lambda *{f}_c\right){r}^2 $$
(6)
where
$$ sign(x)=\left\{\begin{array}{c}\hfill 0,\kern0.75em x<0\hfill \\ {}\hfill 1,\kern0.5em x\ge 0,\hfill \end{array}\kern0.5em \right. $$
(7)
and r denotes the distance from the optical axis, R the inner radius of the RBL, and λ the applied wavelength, while fo and fc are the focal lengths of the ring and the central areas, respectively.
The optical path difference problem
It is known, that in SHSs the Optical Path Difference (OPD) between their different optical ways have to be smaller than the coherent length of the used light to have interference phenomena and hologram.
First, let us observe the OPD on a convenient Hariharan-Sen [15] interferometer based SHS in the case when a single target point is on the optical axis. It is also shown in Fig. 3, and one can see, that even if the light of that target point is split, modulated independently than united to make interference, the created two beams after their union have zero OPD along the optical axis. Due to the curvature differences of the wavefronts of the two beams their OPD increases with their distance from the optical axis. At the plane of the detector, where the OPD become bigger than the coherent length of the used light the interference phenomena will disappear, and there will be the border of the hologram.
Considering these findings and that the RBL formed hologram is a gappy one, we aimed to design the RBL to achieve zero OPD between the central beam and the hollow beam at the common ring-shaped area at the detector plane not on the optical axis. We illustrate the design on Fig. 3. To have this zero OPD we calculated the required thickness difference between the central and the ring area of the custom made RBL.
As the relative curvatures between the wavefronts change during the propagation, it can be seen that the OPDs at the plane of detection also depend on the actual position of the detector.
The attributes of the RBL
Using an RBL based SHS a single point target generates a ring-shaped (gappy) hologram, which is shown in Fig. 4. In the central region of such hologram, the interference pattern with the low spatial frequency components is missing because the intensity of the hollow beam is zero at that place.
The reconstructed image of the ring-shaped hologram will appear in the middle of the ring, not on the interference fringes. Thus, an even (homogenous) background can have the reconstructed image. Furthermore, the twin image diffractions expanding outwards and never overlap with the image reconstruction itself. However, we should note, that the absence of low-frequency components might result in deterioration of the image.
Even if the image reconstruction is not perfect, we will show that using the proposed method, one can still detect the 3D position of the target objects and will be able to discriminate between them correctly.
Using numerical simulations, we investigated the relationship between the parameters of the ring-shaped hologram and its reconstruction properties. We handled both the depth of focus of the reconstruction and the Strehl ratio (the quotient of the maximum intensities of an actual and a reference reconstruction images) with exceptional care.
First, we simulated the diffraction limited hologram of a single point source. In the simulated case, that contains seventeen concentric rings of interference. Second, starting from the middle one, we mask more and more interference rings, thus simulating gappy holograms. The relationship between the number of missing interference rings and the quality of the reconstructed images were measured.
The reconstruction distance was set 3000 μm while the wavelength and pixel size were 530 nm, and 0.9 μm respectively. The simulated holograms, the reconstructions and the measured parameters like the Strehl ratio and the depth of focus are shown in Fig. 5.
To determine the depth of focus I used the following definition: a point is reconstructed at a special reconstruction distance if the intensity of the reconstructed point is not lower than the maximal intensity of its background. The background intensity is estimated from a surrounding area nine times bigger than the reconstructed point.
The Strehl ratio is defined as the quotient of the maximum intensities of the actual gappy hologram reconstruction and the full hologram reconstruction.
Our results show that by the increase of the missing area, the Strehl ratio decreases and the depth of focus increases. It can also be observed that the depth of focus is increasing when a ring-shaped hologram becomes thinner, but this effect is only apparent when the gap is sufficiently large, in the first few cases when the gap is small, it is unnoticeable. Thus, setting the rate of the aperture areas of the hologram with the aperture parameters of the RBL, these properties of the reconstructed image can be hold. Observing the point spread functions (PSF) of the spot diagram, it can be seen that as the size of the central spot and its maximal intensity decrease while the artefact caused by the surrounding ring increase with the increase of the relative gap size. As these diffraction rings can produce interference with the fringes of the target object reconstruction, the size and shape (spot, line, grid…) parameters of the target will also shape the resolution of the imaging. In section 4.2 the PSF of the built SHS setup is presented.
We have to note that the narrow ring-shaped hologram of a single point source has an elongated, line-like image along the direction of the reconstruction. This is an axicon-like property. The further discussion is of this topic is beyond the scope of the present paper.
The self-referenced holographic microscope
In this section, we discuss an actual implementation of a self-referenced digital holographic microscope based on a commercially available microscope and the RBL discussed so far.
For this purpose, the RBL was placed into an Olympus (IX71) microscope, on the rear surface of the objective (Olympus, 4x Plan, NA = 0.16, f = 45 mm, infinity corrected). The hologram was recorded with a high sensitivity C-MOS camera (ASI120MM-S, 1280x960). The camera was placed 40 mm out of the focal plane of the camera adapter (U-TV0.5XC-3) which was placed on a trinocular tube (U-TR30-2). The excitation light had a wavelength of 405 nm. For the observance of the fluorescent USAF test-target (EO Stock No. 57-792), a dichroic emission filter was applied (10 nm bandwidth and the central wavelength is 530 nm, Thorlabs FB530-10) between the RBL and the trinocular tube. When the algae sample was measured, we also used a longpass filter (Thorlabs FEL 600). The scheme of the measuring setup is illustrated in Fig. 6.
As the RBL generates two different beams from the light of a single point object, it also generates its two real images at the first and the second image planes. The first image is produced by the cone-shaped beam, which is surrounded by the defocused image of the hollow beam. On the other hand, the second image is reproduced from the hollow beam, while the defocused cone shaped beam overlaps with it. The two images appear at different planes. We illustrate such dual-imaging at Fig. 7.