## Abstract

Digital inline holographic microscopy using a pinhole for sample illumination allows lensless imaging. To overcome restrictions of the sample size and density in the setup additional reference waves are generated by extending the single pinhole to a regular 2D pinhole array illumination. A technique is presented that uses phase shifting between the pinhole waves. Multiple foci with stable phase differences and a phase error (rms) of 0.027 rad generate pinhole waves which illuminate an undiluted, dense blood smear sample. Amplitude and phase images of the blood sample were sucessfully reconstructed.

© 2012 OSA

## 1. Introduction

Digital inline holographic microscopy (DIHM) allows lensless high-resolution imaging of microscopic objects [1–6]. The inline holographic setup basically consists of a coherent light source illuminating a pinhole and an image sensor capturing the diffracted light and is based on the setup invented by Gabor [1]. The sample is placed between the pinhole and the sensor. The pinhole that creates a spherical wavefront is essential to achieve a high resolution in lensless DIHM. The spherical wavefront magnifies the interference pattern of the sample, so that it can be recorded by an image sensor. Besides the object image (intensity) the quantitative phase distribution is also be recovered and can be used to determine the thickness of a phase object with nanometer precision. The amplitude and phase information of an object can only be recovered reliably for small objects in sparse samples [7]. In this case the phase of the hologram wave is only slightly influenced by the sample and can be well approximated by the undisturbed spherical wavefront created by the pinhole. If this essential approximation does not hold (e.g. for larger objects) the reconstructed objects become increasingly disturbed by artifacts like the twin-image, which is the complex conjugate of the object. First the phase information is lost in the reconstructed image and with increasing number and size of objects the amplitude information degrades, too. To overcome this drawback different approaches are known using multiple measurements at different distances to the sample [8, 9] or introducing an additional reference wave that is not disturbed by the sample [10–13]. With an additional reference wave the phase shifting technique [14], that varies the phase difference between the hologram wave containing the object information and the reference wave, can be applied to recover the phase of the hologram wave. This requires complex and elaborate setups with an additional beam path and is not possible in lensless DIHM using a single pinhole to illuminate the sample. Additional spherical reference waves are introduced by extending the single pinhole to an array of pinholes. The pinholes are illuminated by multiple foci with defined phase differences that are generated by a spatial light modulator (SLM). Phase shifting is done by varying the phase differences between the different foci. Phase fluctuations of liquid crystal on silicon (LCoS) devices that have been reported [15] were also considered in this work.

After a short description of the experimental setup in section 2 the algorithm to calculate the phase masks for the SLM is described in section 3 and the stability of the phase difference between the foci is analyzed in section 4. It is shown that phase differences between the spots are however stable, despite the fluctuations of the SLM device. Additionally the phase shifting technique unsing a pinhole array is applied in section 4 to image an undiluted and dense blood smear sample. Experimental results showing amplitude and phase images of the sample are given.

## 2. Experimental setup

The experimental setup (Fig. 1) includes a Fourier lens (L), an SLM and a beam splitter cube (BSC). A fiber-coupled diode laser (iBeam, TOPTICA Photonics AG, Germany, λ = 661nm) is used as light source. The SLM is a PLUTO VIS model (HOLOEYE Photonics AG, Germany) with a pixel size of 8μm. It was calibrated to realize a dynamic range of 2π at a wavelength of 661nm. To suppress stray light from the SLM frame an aperture (A) of 8 mm diameter was placed after the lens. Placing the BSC between the fiber and the lens saves an additional lens for focusing the spots and reduces the height of the setup, so that it fits under a common light microscope. Here the microscope was helpful for adjustments, especially when matching the spots to the pinholes. The aberrations that are introduced by the BSC can be simulated by scalar wavefield propagation and can be corrected by the SLM, so that the spot size is theoretically limited by diffraction. The spot size depends on the numerical aperture (NA), i.e. the focal length *f* = 40.5mm of the lens and the beam diameter. It is given by 5.8μm, which was calculated by Fourier transforming the intensity distribution on the SLM, a Gaussian beam (*w*_{0} = 8mm) truncated by the aperture (8 mm). The maximum spot separation is dependent on *f* and on the minimum grating period *d _{min}* (twice the pixel size) that can be realized by the SLM. With

*d*= 16μm the maximum spot separation becomes 1.6 mm in this setup. The spots are then truncated by the pinholes to a size of 2μm and simultaneously the NA is increased from 0.1 to 0.2.

_{min}## 3. Phase mask calculation for the SLM

The multi-spot illumination for the pinhole array is generated by a phase mask or computer generated hologram (CGH) displayed on the SLM. The iterative Fourier transform algorithm (IFTA) [16–21] is known to perform well to derive such a CGH. Variants of this algorithm allow free choice of the phase and intensity within a region of interest Ω [18] and can be used to generate spots with defined phase differences. Amplitude and phase freedoms outside this region are used for optimization. Minor improvements were made to this algorithm. To generate spot patterns with a defined phase difference between the spots in Ω the phase must be constrained at least at the spot positions. In general a phase constraint is applied to the whole region Ω and an arbitrary wavefield can be created in this region. For creating a sparse pattern of spots this would be an unnecessary strong constraint and it is more reasonable to apply the phase constraint only at the spot positions where the scalar target amplitude *A _{trg}*(

*x,y*) is greater than a given threshold

*t*. The following Eqs. (1)–(4) show the operations made in one iteration step. The algorithm starts in the focal plane were the spots are located with a complex wavefield

*U*

_{0}(

*x,y*) = 0.

In Eq. (1) the amplitude of the field within Ω is set to *A _{trg}*(

*x,y*) and the phase is set to the target value

*ϕ*(

_{trg}*x,y*) if

*A*(

_{trg}*x,y*) ≥

*t*or left unchanged otherwise. The complex wavefield in the SLM plane

*Ũ*(ξ

_{n}*,*η) is obtained by inverse Fourier transform (FT

^{−}^{1}) as shown in Eq. (2). Its amplitude is replaced by

*Ã*(ξ

_{ill}*,*η), the amplitude of the illuminating beam (Eq. (3)). The Fourier transform (FT) of this field yields the field in the spot plane for the next iteration step. The illumination shape does not necessarily have to be considered during the iteration process, instead

*Ã*(ξ

_{ill}*,*η) = 1 can be used. Then a different illumination in the experiment corresponds to a multiplication of

*Ã*(ξ

_{ill}*,*η) with the actual illumination. In the spot plane this results in a convolution of

*U*(

_{n}*x,y*) with the Fourier transform of this illumination amplitude and is denoted by the dashed line in Fig. 2(b).

Results of the IFTA are presented in Fig. 2, where (a) shows a section of the resulting phase mask for a 2*×*2 spots array with spacing *a* = 240μm. It resembles a two dimensional binary grating as it is expected for 2*×*2 spots. The plain illumination on the SLM and the constraints within Ω lead to modifications of the grating. In Fig. 2(b) a simulation of the resulting intensity distribution is shown. The target region Ω containing the spot array was 1 mm*×*1 mm. Some weak noise spots [22] occur in this region, but they are mainly present in the outside region and will be suppressed by the according pinhole array. To make the noise spots visible the intensity scale in this figure was limited to 10% of the maximum spot intensity.

## 4. Experimental results

#### 4.1. Phase shifting

The appropriate array (*a* = 240μm) with pinholes of 2μm in diameter was placed in the spot plane. Because the phase difference between the spots could not be analyzed directly, interference patterns of the pinhole waves (Fig. 3(a)) were recorded at a distance *z* = 1.4mm above the pinhole array. A microscope with a Carl Zeiss Epiplan 100*×*/0.75 HD objective was used. At this distance and for a regular 2*×*2 array the intensity is approximately sinusoidal in both dimensions with a period *L* = (λ*/*2)[1 + (2*z/a*)^{2}]^{1}^{/}^{2} of 3.86μm.

Intensity fluctuations resulting from phase fluctuations of the SLM were observed, as pointed out in [15]. To investigate their influence on the phase difference between the spots and the stability of the setup, the shortest possible integration time of the camera (AxioCam HRm, Carl Zeiss) of 0.8 ms was chosen to record a series of 20 measurements. The pinhole size of 2μm had to be used to pass enough light through the holes allowing such short integration times. The phase value to be generated by the SLM is in fact a time averaged value, averaged over one frame (*t _{frame}* = 16.67ms, frame rate 60 Hz). Theoretical considerations can however explain that there should be no influence on the phase difference between the spots. Treating the phase mask on the SLM as a two dimensional binary grating and neglecting the modifications to it, the generated spots are the

*±*1st diffraction orders of this grating. The phase fluctuations of the SLM only influence the phase shift of the displayed grating, neither its period nor its lateral position. However, such a varying phase shift does only influence the diffraction efficiency, which causes the intensity fluctuations. This was confirmed by repetitive measurements of the interference pattern with an integration time of 0.8 ms. A sine curve was fitted to a profile line taken from the series of 20 interference patterns and amplitude, period and lateral position were determined. A change in the phase difference between the spots would lead to lateral shift Δ

*x*of the interference pattern. The phase interval from 0 to 2π translates linearly into a lateral shift from 0 to

*L*the period of the interference pattern. The resulting error (rms) was 0.017μm corresponding to 0.027 rad phase error. This is a relative error of only 0.43%, while the amplitude had a relative error of 3.75%, due to the intensity fluctuations.

To analyze phase shifting a series of 24 images was recorded with a linearly increasing phase difference. The shifts of the interference patterns were again determined from a profile line. Five steps with phase shifts of 0, π/2, π, 3π/2 and 2π are shown within Fig. 3(a) (right column from top to bottom). The lateral shifts of the complete scan are shown in Fig. 3(b) (crosses) together with the theoretical values (straight line). The theoretical values were calculated by Δ*x _{q}* =

*qL*/24 (

*q*is the step number). The average error (rms) with respect to these values was 0.046μm or 0.075 rad, corresponding to a relative error with respect to

*L*of 1.2%.

#### 4.2. Application to holographic microscopy

The application of this phase shifting technique is demonstrated by imaging a sample of human blood with the setup described above. Amplitude as well as phase images of the blood cells are given and compared to single pinhole DIHM reconstructions. The holograms were recorded using an appropriate objective lens (Carl Zeiss Epiplan 20×/0.40 HD) to resolve the interference structures. Due to a limited field of view of this objective the hologram was truncated. This led to artifacts in the reconstructed images especially at the borders, but also to high frequency oszillations in the center region. When the full hologram is detected directly by an adequate image sensor, this effect will disappear.

The sample (Fig. 4(a)) was placed 620μm above the pinhole array in a plane, where the pinhole waves did not overlap. The recording distance of all holograms was 1250μm, where the pinhole waves overlaped and interfered. One pinhole wave was left undisturbed, indicated by ref in Fig. 4(a) and was used as reference wave. Three pairs of pinholes having the reference pinhole in common were chosen for phase shifting. For each pair of reference and sample pinhole, three holograms with phase steps of 0, π/2 and π between the two pinholes were recorded. The phase *ϕ* of the sample pinhole wave was calculated similar to [14] using the intensities *I*_{0}, *I*_{π/2} and *I*_{π} of the corresponding phase shifted holograms and Eq. (5).

Additionally the single pinhole holograms of pinhole 1–3 were recorded separately with turning off all other pinholes. Images of the sample were reconstructed using these holograms and the angular spectrum propagation [23] for progating from the hologram plane to the object plane. To the spherical phase used in DIHM for reconstruction the phase *ϕ* calulated from the phase shifted holograms was added. Fig. 4(b) shows the sum of all three reconstructed sample parts. Detailed views according to the dotted boxes in Fig. 4(b) are given in Fig. 5(a).

The areas illuminated by the pinholes 1–3 were covered by the sample to different amounts. The pinhole waves and the holograms are disturbed to different degrees. Pinhole wave 1 is least disturbed, followed by pinhole wave 3, while pinhole wave 2 is completely covered by the sample. For all three pinholes the blood cells are clearly visible in the intensity and phase images (Fig. 5(a)), that were reconstructed using the additional reference pinhole and phase shifting. The reconstruction was also carried out without using the phase obtained from phase shifting, but with an undisturbed spherical phase (Fig. 5(b)). The different degrees of disturbance are also reflected in these reconstructions, where a reconstruction for pinhole 1 is barely possible and the blood cells can still be recognized in the intensity and phase image. For pinhole 2 a reconstruction with only a spherical phase is not possible, due to the high degree of disturbance, caused by the dense sample (Fig. 5(b)). The requirement for small and sparse samples in lensless DIHM is violated.

## 5. Conclusion

A phase shifting technique for an inline holographic microscopy setup was presented using an array of pinholes that illuminates the sample. An undisturbed pinhole acted as additional reference wave and by shifting the phase difference between this pinhole and the disturbed pinholes the hologram phase can be recovered.

The pinholes were illuminated by multiple spots with defined and controllable phase differences that were generated by a SLM. The phase difference between these spots was changed for phase shifting. The influence of phase fluctuations introduced by the LCoS based SLM were discussed and it was shown that the influence on the phase differences between the spots can be neglected. Stable phase differences with a mean phase error (rms) of 0.027 rad (0.43%) were experimentally realized, while the amplitude error was 3.75% due to the SLM fluctuations.

Experimental results showing a successful application of this technique in DIHM with pinhole array were presented. A dense blood smear sample was imaged by three pinholes with different degrees of disturbance by the sample. The reconstruction of intensity and phase images of the blood cells was successfull even in case of a fully disturbed pinhole wave. Only one arbituary pinhole of the array is needed as reference wave. Each of the other pinholes can be entirely covered by the sample. Depending on the count N of pinholes in the array, the sample fill factor can be achieve (N-1)/N. In the present setup using a 2x2 pinhole array, 75% of the imaging field of view can be covered by the sample. Compared to single pinhole illumination the image quality is significantly improved for all three pinholes by reducing artifacts, like the twin image. This technique for switching pinholes could also enable further applications in the field of digital holography using pinhole arrays [24, 25].

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