1. Introduction

A conventional transmission imaging microscope has an inherent limitation in terms of a small depth of focus stemmed from the use of an objective lens with high numerical aperture and magnification factor. Thus, mechanical scanning is required in order to measure 3-D volumetric micro objects. At the early stage of holography, Gabor proposed that the 3-D object information could be recorded in a hologram and optically reconstructed [1]. However, it was not until the rapid progress of CCD sensors and computer technology has been made that holography became a practical solution for the quantitative 3-D object measurement in microscopy. In digital holography, holograms are directly recorded by a CCD camera instead of photographic plates and the 3-D reconstruction of the object is performed numerically by transforming the measured digital hologram [2–9]. In on-axis digital holography, however, undesired DC terms and a conjugate image term are superimposed upon a reconstructed object image. A simple solution to this problem is to tilt the reference beam with respect to the object which generates a spatial carrier frequency enough to separate the real object from the undesired terms [2–3]. However, the off-axis scheme prohibits the effective use of the total pixel number of the CCD due to the carrier fringes. The resolution of the reconstructed image is also restricted by the presence of the undesired terms. In order to remove them in on-axis scheme, phase shifting digital holography based on PZT actuation has been commonly used as a solution to the superposition problem of the on-axis approach [8].

As an alternative approach to the PZT based phase shifting, spectral phase shifting scheme was used [11–14]. The spectral phase shifting has been conducted both in heterodyne interferometry and in digital holography which use the wavelength tunability of a laser diode by modulating the injection current or the temperature of the laser diode. However, the use of a coherent laser source for spectral phase shifting causes an inherent speckle problem. In digital holographic microscopy, partially coherent light source can be used to make speckle free conditions [10, 14].

In this paper, we propose a white light on-axis digital holographic microscopy based on spectral phase shifting. The proposed spectral phase shifting approach has no mechanical moving part for inducing the phase shifting and can provide a speckle-free capability. It has also the capability of providing full color 3-D spectral section imaging information since it employs a white light source and a spectral tunable filter. Although it is beyond the subject of this paper, the proposed microscopic system has the potential that it may be applied for 3-D full spectral imaging of transparent micro objects such as biological sample.

2. On-axis digital holographic microscopy based on spectral phase shifting

Figure 1 shows the coordinate used for the proposed white light on-axis digital holographic microscopy. For the on-axis digital holographic microscopy, the CCD is located in the Fresnel plane in order to generate a slightly defocused image. A sharply focused object image can be numerically reconstructed by using this approach. The exact knowledge on complex optical field in a plane permits computing the optical fields in other parallel planes by use of the Kirchhoff-Fresnel equation [2–3]. The capability of the 3-D section imaging of micro objects can be achieved by applying an interferometric setup to a microscope configuration in which the microscope object lens magnifies the object into the image plane. In this paper, a spectrally induced 5 bucket phase shifting method is demonstrated for the white light on-axis digital holographic microscopy.

The schematic diagram of the white light on-axis digital holographic microscopic system is depicted in Fig. 2. As pointed out, an incoherent white light is employed as a light source in combination with a spectral tunable filter. For the spectral tunability, an acousto-optic tunable filter (AOTF) is employed in this study. By combining the two, partially coherent light with the coherence length of hundreds of microns can be generated. This amount of coherence length can be sufficiently used in digital holographic microscopy by employing a conventional interferometric setup as shown in Fig. 2.

 

Fig. 1. Coordinates used for the spectral phase shifting based white light on-axis digital holographic microscopy.

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The path difference between the object and reference wave should be within this coherence length in order to apply the spectral phase shifting method for the 3-D reconstruction of micro objects.

 

Fig. 2. Schematic diagram of the proposed AOTF based white light on-axis digital holographic microscopic system.

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The spectrally induced 5 bucket phase shifting method can be applied to Fresnel analysis by slightly defocusing the object by moving the CCD from the image plane to the Fresnel plane. The complex optical field of the object in the Fresnel plane can be obtained by applying the 5 bucket spectral phase shifting method. Once the complex field in holographic plane is obtained, the 3-D section images can be numerically reconstructed by using Kirchhoff-Fresnel propagation integral. Inducing a spectral carrier frequency h₀ is required for applying the 5-bucket spectral phase shifting method. Equation (1) represents an approximately estimated intensity distribution in the holographic plane. It contains a high spectral carrier frequency phase term of 2kh₀.

Here, O(x,y,k) and R(x,y,k) represent object and reference wave, respectively. The complex field generated by the reference and the object at the Fresnel plane can also be written as |R(x,y,k)|exp[iϕ_R(x,y,k)] and |O(x,y,k)|exp[iϕ_o(x,y,k)], respectively. The digital hologram I(x,y,k) in the Fresnel plane is containing two DC terms, object and conjugate term of the object. Also, the ϕ(x,y,k) corresponds to phase difference between the object and reference wave, i.e., ϕ(x,y,k)= ϕ_o(x,y,k)-ϕ_R(x,y,k). By inducing spectral carrier frequency h₀, the measured intensity distribution versus wavenumber k becomes dominantly dependent upon the high frequency phase term 2kh₀ in the spectral domain. The relatively low frequency terms ϕ(x,y,k), O(x,y,k) and R(x,y,k) can be treated as invariants versus wavenumber k. Also, let us redefine selected wavenumber k=k_c + δk. This enables Eq. (1) to be re-written as follows:

Here, k_c and δ_k is the central wavenumber and the scanned wavenumber deviation, respectively. k_c is set as an arbitrary known value. In order to obtain the phase value ϕ(x,y) for each coordinate (x,y) at the central wavenumber k_c, five intensity values corresponding to each five wavenumbers are required as follows:

With these five intensity values at the central wavenumber k_c,, the total phase ϕ(x,y) and amplitude i₀ of the object in Fresnel plane can be obtained as follows.

Here, I₀ indicates the mean value of digital holograms. By using the complex wave information of the object in the Fresnel plane, 3-D section images on any parallel plane can be numerically reconstructed without suffering from DC terms or the conjugate object image problem. For an animated 3-D imaging, successive section image reconstruction can be executed by computing the following Kirchhoff Fresnel integral by varying the distance d.

Here, u_i(x,y) represents the complex field in Fresnel plane, which can be written as i₀(x,y) exp[iϕ(x,y)]. u₀(x’,y’) is the complex optical field in image plane. [ℑ_x,yf(α,β](γ,ξ) denotes the direct or inverse two dimensional Fourier transformations as follows.

3. Experimental results

An experiment using a semiconductor patterned object was carried out. Spectral imaging must be conducted with consideration for the AOTF characteristic of image shift depending on diffracted wavelength. In order to apply the spectral carrier frequency h₀, the measured sample surface is positioned such that the distance between the two arms was around 30 μm. Initially the CCD is positioned to get d equal to around 1mm. With this initial setup of the measured object and the CCD, five spectral phase shifting with the wavenumber increment Δk of 0.017 cm^-1 at k_c of 11.977 cm^-1 has been carried out in order to obtain the five digital holograms I(k)s described in Eq.(3).

 

Fig. 3. Movie (1.1M) of the reconstruction result of the white light on-axis digital holographic microscopy for d=1 mm: (a) defocused image and (b) reconstructed focused image of micro patterned object. [Media 1]

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Fig. 4. Movie (0.7M) of the reconstruction result of the white light on-axis digital holographic microscopy for d= 10 mm: (a) defocused image and (b) reconstructed focused image of micro patterned object. [Media 2]

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Figure 3(a) shows a slightly defocused image data and Fig. 3(b) represents a sharply reconstructed focused object image by using the proposed spectral phase shifting method. As can be expected from the theoretical view, the proposed system performs very well. Although there has not been executed any detail comparison between the PZT based phase shifting and the spectral phase shifting approach, the result implies that the current method can provides a possible alternative approach to the conventional PZT based digital holographic microscopy.

Figure 4 shows the reconstruction result for d=10 mm. It shows that the reconstructed image quality is not as sharp as the first case of d=1 mm. Figure 4(b) shows the best focused object image that can be reconstructed in this setup. The proposed system has also a limited reconstructed depth as discussed in Ref. 10. This is estimated to be an inherent limitation of the digital holographic microscopy which uses near field Fresnel transformation.

4. Conclusions

A white light on-axis digital holographic microscopic system using spectral phase shifting is described. The feasibility of the AOTF based spectral phase shifting approach for on-axis digital holographic microscopy has been showed. Likewise the PZT based phase shifting digital holographic microscopy, it has an inherent limited reconstructed depth. The proposed spectral phase shifting based on-axis digital holographic scheme has a benefit in terms of system robustness due to no moving part and speckle-free capability stemmed from the use of the partial coherent source produced by combining a white light source and a spectral tunable filter. Although the proposed spectral phase shifting method has a limitation in terms of a limited reconstruction depth of possibly permissible defocused distance d, the 3-D section images can be reconstructed numerically accurately by using the proposed 5-bucket spectral phase shifting method. The proposed microscopic system has the potential that may be applied for 3-D full spectral imaging of transparent micro objects such as biological samples.