## Abstract

We propose a technique to implement digital holography by means of a point diffraction interferometer. The device uses a liquid crystal television display and works under partially coherent illumination which makes it useful for 3D microscopy. We present the theory on which the method is based and the obtained results.

© 2005 Optical Society of America

## 1. Introduction

Since the introduction of digital holography by Schnars *et al.* [1] this technique has been widely applied [2–6]. It is well known that conventional holography is able to record 3D information of an object but photographic recording requires time and chemical processing; on the other hand mechanical focusing is needed in order to analyze different depths of the optically reconstructed image.

In digital holography the interference pattern is recorded onto a charge-coupled device (CCD) and it is numerically reconstructed by using the discrete Kirchhoff-Fresnel propagation equations. As resolution of a CCD is much lower than standard film materials the angle between the reference wave and the object wave must be of few degrees. This fact causes that the reconstructed image, the zero order and the conjugated image are generated very near one to each other. To avoid this drawback Yamaguchi *et al.* [7] proposed a method based on the phase-shifting interferometry technique. The employed optical set up was a Mach Zehnder type interferometer in which the object beam and the reference beam formed an in-line interference pattern onto the CCD plane. The phase of the reference wave was changed stepwise, and the resulting four interferograms were processed by a computer to yield the complex amplitude of the object wave at the CCD plane. Numerically propagating this complex amplitude allows reconstructing the object wave avoiding the zero order and the twin image. Most of the subsequent works employ similar interferometric set ups to combine digital holography with phase shifting techniques.

Among the different applications of digital holography 3D microscopy is one of the most attractive. Conventional optical microscopes suffer from small depth of focus and mechanical movements of the optical system or the specimen are necessary for acquisition of 3D images. Digital holography allows overcoming this problem by reconstructing numerically the object wave at different planes. Conventional holographic microscopes require lens systems to enlarge the studied object; although this is not a problem under incoherent illumination, when coherent beams are used it could be a source of noise. Some efforts have been devoted to solve this problem by replacing the coherent source by one of partial spatial coherence [5], nevertheless in these cases the optical paths in the interferometer must be carefully controlled.

In this work we propose the use of a point diffraction interferometer (PDI) to implement digital holography. This common path interferometer allows the use of partially coherent sources, requires relatively few optical elements and its alignment is very simple. The PDI was introduced by Linnik [8] and has been used as a robust device for the measurement of optical wave fronts. However the common path design that gives place to its robustness makes difficult the implementation of phase shifting techniques that constitute an accurate and effective method for phase measurement. Among the different modifications proposed to add phase shifting capability to this instrument [9,10] the liquid crystal point diffraction interferometer (LCPDI) [11] is particularly attractive because it permits to carry out a flexible phase stepping interferometry while retaining the fully common path optical nature of the device. In this architecture the wave front to be tested is focused onto a filter that consists of a glass or plastic microsphere embedded within a nematic liquid crystal host. The light that propagates around the microsphere forms the object beam while the light diffracted by the microsphere forms the reference wave. A voltage applied to the liquid crystal cell causes the phase shift of the object beam relative to the reference one. Actually many microspheres are embedded within the liquid crystal and care must be taken to avoid multi-reference beams generated by the illumination of more than a single microsphere.

In a previous paper [12] we proposed a different approach by using a liquid crystal Television display (LCTV). In that case the phase shifts were introduced by changing the voltage applied to the central pixel ensuring, in this way, a single reference beam. The device was used to evaluate and correct *in situ* an arbitrary phase distribution caused by aberrations introduced by the various elements that constitute an optical correlator.

In this work we show the capability of this architecture to carry out digital holography with partially coherent sources. To this end we use a laser beam impinging onto a rotating diffuser and a spectrally filtered mercury lamp. Phase shifting digital holography is sensitive to vibrations because several frames of data are taken sequentially in time. One way of reducing the effects of vibrations is to use a common-path interferometer, in which the same vibration is present in the object and reference beams, and cancel each other. It is for this reason that the point diffraction interferometer is especially suitable for this application. The reconstruction of the object beam will be less affected by the environmental vibrations if a common path interferometer is used to obtain the digital hologram.

The proposed PDI does not use microspheres, and then the problem of multiple reference beams is not present. It has phase shifting capabilities, and then it allows measurement of the object beam phase. It is an on-axis interferometer and then reduces the space band width needed in the CCD camera. The advantage with respect to other common path interferometer like the shear interferometer is that the amplitude and phase of the object beam is directly obtained from the measurements reducing the time consumed to evaluate them.

The content of the paper is divided as follows. In Section 2 we describe the proposed method to obtain the digital hologram and we review the refocusing algorithms. In Section 3 we show the obtained results. Finally, some comments and discussions are given in Section 4.

## 2. Description of the method

#### 2.1 LCTV point diffraction interferometer

LCTV’s have been widely used in image processing during the last years because of their capability to display images at video rates and to act as programmable spatial light modulators. It is well known that these twisted nematic displays produce coupled, amplitude and phase modulation, when they are used between linear polarizers but, pure phase or amplitude modulation can be achieved under elliptically polarized illumination [13].

Let us describe the proposed optical system following the sketch shown in Fig. 1. A source *S* illuminates an object *O* and onto the conjugate plane of the source by means of lens *L1* the Fourier transform of *O* is obtained. In that plane is placed a *LCTV* sandwiched between linear polarizers *P*
^{’}
*s* and wave plates *WP*’*s* that, combined, act as a pure phase filter. This phase modulator can be programmed pixel by pixel offering phase retardations in the range 0–2π. A second convergent lens images the object O onto the final plane Π.

When no object is present a bright central spot corresponding to the Fourier transform of the ideal entrance pupil will be focused onto the LCTV. The pixel on which this sharp function is centered is smaller than the focused spot and can be used as perturbation to generate a spherical wave by diffraction. The centering process is not difficult and can be performed following a procedure similar to that described in Ref. 14. When an object is present most of the light diffracted by it will not be affected by this central pixel so, after the filter two waves will be present, an object wave and a reference wave that will interfere producing an in-line hologram. As we can see only the central pixel of the LCTV is used in the set-up. We have used a LCTV because it is easily available off the self, but a plate with only one small pixel (smaller than the image of the source S through the lens L1) at the center would be more convenient.

The complex amplitude of the object wave in the hologram plane can be obtained by means of the phase stepping interferometry technique. To this end the phase of the reference wave is changed by applying different voltages to the central pixel.

Mathematically, the filter transfer function is

Where *δ*(*u*) is the Kronecker function that takes the value 1 when *u*=0 and the value 0 elsewhere, and 2*πn*/*N* is the phase shift introduced in each of the *N* steps applied to obtain the unknown phase. For simplicity, we carry out 1-D analysis. The intensity transmission of this filter is uniform but the phase of the pixel at the origin is shifted with respect to the other pixels by 2*πn*/*N* radians. The amplitude at the final plane is given by the expression

Where *O*(*x*)=*o*(*x*) *e*
^{iΨ(x)} is the unknown complex amplitude of the object wave, *h*(*x*) is the filter impulse response, the symbol * denote a convolution and the complex constant *K*=|*K*|*e*
^{iµ} is the mean value of *O*(*x*).

In the final plane the square modulus of the amplitude *A _{n}*(

*x*) is detected by a CCD. To obtain the phase information (

*ψ*(

*x*)) several interference patterns |

*A*|

_{n}^{2}with

*n*=

*0*,…,

*N*-

*1*(

*N*>

*3*) are needed. Following the synchronous detection of interference fringes [15], the measured intensities |

*A*|

_{n}^{2}are multiplied by $\mathrm{cos}\left(\frac{2\pi n}{N}\right)$ and $\mathrm{sin}\left(\frac{2\pi n}{N}\right)$ according to the following expressions:

According to the orthogonality properties of the sinusoidal functions and by replacing in Eq. (3) the expression of *A _{n}* given in Eq. (2) we obtain:

Finally, the unknown phase Ψ(*x*) can be obtained as

Where µ is a constant phase and *C _{o}*=

*N*|

*K*|

^{2}. The value of

*C*is obtained by evaluating expressions (4) at those points in which

_{o}*o*(

*x*)=

*0*as it is explained in Ref. 12. This implies that the object O should be limited by a field stop, to assure that outside the stop

*o*(

*x*)=

*0*. In the final image plane, the CCD camera should capture part of the field stop image. In this way the constant C

_{0}can be evaluated.

The amplitude of the optical field o(*x*) is calculated as the square root of the object wave intensity.

## 2.2 Refocusing algorithms

In digital holography the knowledge of the complex amplitude in a plane allows, by numerical propagation, to evaluate the optical intensity in another parallel plane. As we previously explained, the proposed LCTV based point diffraction interferometer allows measurement the complex field of the object wave in the CCD plane. Subsequently, by use of the Kirchhoff-Fresnel equation, it is possible to know the optical perturbation in another plane. If the optical system used to image the object onto the CCD has a small depth of focus, as is the case of microscopes, only a slice of a bulky object will be focused, nevertheless by computing the propagation equation the blurred slices can be refocused.

As we previously established *O*(*x*) represents the object wave in plane Π, i.e., in the input plane of the CCD. We can evaluate the complex amplitude *O*(*x*
^{′}) in a plane Π’ parallel to Π and separated by a distance *d* by means of the Kirchhoff-Fresnel propagation integral. This integral can be expressed in two different forms:

Where *λ* is the wavelength, *k*=2*π*/*λ*, *x* is the spatial variable, *v _{x}* is the spatial frequency and

*𝓕*and

*𝓕*

^{-1}indicate the continuous Fourier transform and the inverse Fourier transform respectively. Although these two expressions are equivalent, Eq. (7) is more convenient for near field refocusing because of its dependence with the distance

*d*in the quadratic phase factors [5]. For numerical computation the continuous Fourier transform has to be replaced by the discrete transformation, so Eq. (7) takes the form:

Where Δ is the sampling distance, *N* is the number of pixels and *m*, *m*
^{’} and *l* are integer numbers that vary from *0* to *N*-1.

## 3. Results

In our experiments we used as partially coherent light source a spectrally filtered (λ=436 nm) mercury short arc lamp. The wavelength was selected to achieve 2π phase modulation with the liquid crystal display. The used LCTV was a Sony LCX012BL with VGA resolution (640×480 pixels) extracted from a commercial video projector and the phase only modulation was obtained by following the procedure described in Ref. [13]. In this procedure the LCTV has to be illuminated with an appropriate elliptically polarized beam which is generated by introducing between the source and the LCTV a wave plate and a polarizer. In the set-up shown in Fig. 1 these elements (P_{1}, WP_{1}) were placed before the object O. After the LCTV an elliptically polarization state must be detected, then a wave plate and a polarizer must be introduced between the LCTV and the detector. In Fig. 1 these elements (P_{2}, WP_{2}) were placed just after the LCTV. The used detector was a CCD of 512×512 pixels and with 11 µm pixel size and the object was a graduated scale etched in a piece of glass from an eye piece. The distance between two consecutive numbers of the scale is 1 mm.

As our purpose, at least in this first stage, is to show the suitability of the method to be applied in digital holographic microscopy and not to construct a holographic microscope, the lenses, object size and distances used in our device are not those typical of a microscope system. Nevertheless it should consider just as a scale factor.

Figures 2(a) and 2(b) are movies that show refocusing of the object from holograms obtained by applying the proposed method. The size of the original object is marked in the lower left part of these figures. The CCD plane, i.e the plane in which each hologram was registered corresponds to the most blurred image. In the case of Fig. 2(a) this plane was placed -2 cm out of focus and the different frames that constitute the movie have been calculated each 1.4 mm. The CCD plane for Fig. 2(b) was 6cm out of focus and each frame has been calculated each 6 mm.

It is noticeable in both figures the absence of speckle noise and the capability of the method to perform a refocusing process. In the digital refocusing algorithm the distance between pixels is maintained, so the scale of the figure is maintained. Nevertheless Figs. 2(a) and 2(b) have different magnification because they have been obtained in different image planes, and consequently the optical magnification changes.

## 4. Final discussion

We have shown that it is possible to perform digital holography with a point diffraction interferometer. The common path nature of the device makes it robust, easy to align and especially suitable to use with partially coherent light source because no equalization of optical paths are needed. Moreover the phase stepping interferometry method is of simple implementation by means of a LCTV working as a pure phase modulator. Refocusing of images was calculated from digital holograms obtained under a spectrally filtered mercury lamp.

We have used a pixelated LCTV in the Fourier plane of the object because it is easily available, but in fact, only one pixel is used in the method. The pixilated nature of the LCTV creates replicas in image plane, decreasing both the light efficiency and the field of view (the size of the image should be small enough to avoid the overlapping of the different orders). All these drawbacks can be avoided by designing a plate with only one pixel with a size smaller than the image of the source through the lens L1.

The architecture of the set-up can be modified, depending on the type of illumination used. The essential part of the method is to have access to the Fourier transform of the object, which is formed in the image plane of the light source. In the zero frequency a liquid crystal pixel should be introduced to allow phase shift interferometry. For instance, in a microscope objective under Köhler illumination the Fourier plane is in the rear focal plane of the objective. To calculate the constant C_{0} in Eq (5) part of the field stop should be captured by the CCD. This field stop or an intermediate image should be place on the object.

Once the amplitude and phase of the object beam is obtained from the interferograms, it can be propagated to different distances. Then the visualization of 3D objects or the refocusing of defocused beam can be performed in the computer. The computer time consumed by these operations depends on the computer capacity. Typical frame rates are about 30 frames per second. The beam propagation involves the computation of two Fourier transforms. For images of 512×512 pixels this operations takes a fraction of second. Faster operations can be performed by using especially dedicated processors. Then the visualization of 3D objects could be performed almost in real time.

## Acknowledgments

This work has been partially financed by the Ministerio de Ciencia y Tecnología under project BFM2000-0036-C02-01.C. Iemmi acknowledge the support of CONICET and UBA of Argentina, and the Generalitat de Catalunya, project ACI 2003-42

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