1. Introduction

For digital holography (DH), a camera (CCD or CMOS) is used to record the holograms and object wave-front is reconstructed numerically by simulating the propagation of the complex amplitude of the optical beam using a computer. The combination of its singular optical features and computational advantages makes it especially convenient for microscopy in biology [1], characterization of silicon MEMS [2], three dimensional (3D) imaging [3], particle image analysis [4], etc. However, some disadvantages also exist in DH. In fact, it is noticed that the resolution of reconstructed image is limited by the numerical aperture (NA), which is directly related to the size of the solid-state array sensor. So the resolution achieved in DH is too low and restricted for some practical applications, especially in the case of a long working distance required.

Recently, various strategies or approaches based on the synthetic aperture have been proposed to increase the resolution of DH. In some of the approaches [5–7], the NA of DH can effectively increase by recording multiple holograms with a relative movement between the sample and the camera, and then composed them to a larger digital hologram for reconstruction. However, the resulting reconstruction strongly depended on the measuring accuracy of relative distance between the holograms. C. Liu [8], M. Paturzo [9] proposed a method for collecting high frequency components of the object spectrum with an appropriate diffractive optical element inserted in optical path, which raised complication of the system. Recently, Mico et al. [10] presented an aperture synthesis approach, where a series of sub-holograms were recorded by tilted transmission illumination with a VCSEL. However, in this system a special mask was needed to prevent incorrect spectrum overlapping, which was difficult to be fabricated and adjusted in the application. Similarly, Caojin Yuan et al. [11] recorded successively multiple sub-holograms with angular multiplexing in pulsed digital holography, in which the time delay between three pulse pairs would accurately adjust to ensure incoherent overlapping of sub-holograms. A different approach was presented by Alexandrov et al. [12], where a set of Fourier holograms covered different regions of the spatial frequency spectrum are recorded by rotating the object. However their approach has failed to use to in-situ microscopy imaging.

In this paper, we introduce and demonstrate a new synthetic aperture digital holography method for microscopy imaging at long working distance. First, a series of Fresnel off-axis holograms are recorded with different tilted plane wave illuminations, while no lenses were set between the object and camera for pre-magnifying. Then the corresponding complex amplitudes are reconstructed and magnified by using digital arithmetic. Finally, the synthesized image with the resolution enhanced is obtained by incoherent superposition of the individual reconstructed complex fields. Concurrently, the speckle noises can be well suppressed in the reconstructed image. The major advantage of the proposed method is that the imaging system isn’t limited by the working distance and the effect of lens aberrations on reconstruction can be prevented. Moreover, this approach can simplify the structure of set-up and the procedure of aperture synthesis.

2. Fresnel Off-axis digital holography and Synthetic Aperture

In Fresnel off-axis digital hologram, the information about the phase and the amplitude of object is captured in only one shot and the quality of reconstructed image doesn’t be degraded by the unfocused virtual image and the dc term, with compared to on-axis digital holography and phase shifting holography. Furthermore, the Fresnel transformation method (FTM) is more appropriate for longer working distances according to the paraxial approximation. On the other hand, in the FTM processing, the pixel of reconstructed image increases with the reconstruction distance and so the size of the reconstructed image is reduced for longer distance, which limited the resolution of reconstructed image. Nevertheless, some solutions were proposed that adopt a magnification algorithm based on the FTM to control the pixel of the reconstructed images, independent of the recording distance [13,14]. In this way, the reconstructed image can be appropriately magnified according to the effective areas of the array sensor.

The schematic diagram of the Fresnel off-axis digital hologram is shown in Fig. 1, where an opaque object is observed and a tilted plane wave used to illuminate the object. In this case, the reflective complex field in the object plane (x, y) is written as

where b(x,y) is the complex reflectance profile, i.e., representing the surface reflectivity and shape, of the object, and A_i = A ₀ exp [-j2π(γ_ix+ζ_iy)] is the complex amplitude of the illumination plane light, where A ₀ is a complex constant, γ_i = sin θ_i cos ϕ_i/λ and ζ_i = sin ϕ_i/λ represent the spatial frequencies of horizontal and vertical direction respectively. The quantity θ_i, ϕ_i are the polar and azimuthal angle of illumination light and λ is the wavelength of laser.

 

Fig. 1. Coordination system of object and hologram

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According to Fourier optics, the object spectrum can be obtained by computing the Fourier transform of U_i (x, y), that is

where F represents the Fourier transform, b̃ is the Fourier transform of b, f_x and f_y are the spatial frequency of horizontal and vertical direction in the spatial frequency domain (γ, ζ).

The off-axis digital hologram is originated in the hologram plane (η, ξ), but only a portion of the hologram can be recorded because of the limited aperture of the solid-state array sensor. That is to say, only a part of the object spectrum is transmitted to the diffraction limited imaging system, and the digital hologram covered finite spatial frequency range of object spectrum. The process of Fresnel off-axis digital holography can be interpreted as a coherent optical system, and the point spread function (PSF) of this special imaging system is represented by the Fraunhofer diffraction pattern of the aperture defined by the solid-state array sensor dimensions [15]. Consequently, in the image plane (x′, y′), the spatial frequency spectrum of the reconstructed image can be written as

where rect represents the rectangle function, C is a complex function, L and H are the length and height of the solid-state array sensor, respectively. From Eq. (3) it is easy to see that the spatial frequency spectrum of reconstructed image can be regarded as the object spatial frequency spectrum with a shift of the value of (γ_i, ξ_i) related to the incident angle of illumination plane light multiplied by the rectangle function defined by the size of the array sensor (L,H). So, if using plane wave illuminations with different incident angle and keeping the object unchanged in the recording procedure, the object information of different frequency ranges of object spectrum can be recorded and reconstructed in the case of constrained NA. Now, a series of off-axis holograms are recorded with different tilted plane wave illuminations. Then, by using the Fresnel double-step reconstruction algorithm [15], the real images are reconstructed and magnified from these holograms. Afterward, the synthesized image can be obtained by incoherent addition of the reconstructed real images. Consequently, the spatial frequency spectrum of the synthesized image can be expressed as

where

Equation (5) represents a synthetic aperture which composed by the shifted version of the rectangular aperture of the array sensor, and so the NA of the imaging system is increased effectively with comparison with Eq. (3). That is to say, the digital holography system can get more spatial frequency components of object spectrum, as a result, the resolution of the reconstructed image can be improved.

3. Experiments and results

The experimental set-up is illustrated in Fig. 2. A solid-state laser with a power of 20 mW and a wavelength of 532 nm is used as a light source. A polarized beam splitter (PBS) is adopted to split and adjust the intensity ratio of the illumination light to reference light with the help of two λ/2 retarded wave-plates. Both the two lights are collimated as plane waves by beam expanders (BE₁ and BE₂ composed by spatial filters and collimating lens). Incident angle of illumination light can be adjusted by a couple of mirrors (M₃ and another mirror not shown in Fig. 2 for the sake of clarity). Consequently, the illumination plane waves can transmit from 0, R1, L1, U1, D1, R2, L2, U2, D2. A CMOS camera is used to record holograms with pixel array of 1024×1024 and a pixel size of 6.7μm×6.7μm. Two kinds of objects have been used: a die of the size 10mm×10mm and a negative USAF-1951 resolution test target with the distance of 650 mm between the object and CMOS. In the recording procedure, multiple holograms are recorded with different titled illuminations, and the object, the camera and the reference light remain unchanged. Afterward, the holograms are processed by a frequency filtering followed by an inverse Fourier transform to suppress the effect of zero-order image. Finally, the magnified real images are reconstructed by using the Fresnel double-step reconstruction algorithm.

 

Fig. 2. Schematic diagram of experimental set-up, M₁, M₂, M₃, M₄: mirror; BE₁, BE₂: optical beam expanders; BS: beam splitter; PBS: polarized beam splitter.

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Figures 3(a), 3(b), 3(c) and 3(d) show the reconstructed intensity images of a die from the holograms of R2, L2, U2 and D2 illumination, respectively. It can be seen clearly that the reconstructed images shown different object information according to the particular geometry of the die. This demonstrates that each digital hologram originated from the rays scattered from the object covered different spatial frequency ranges of the object spectrum and so possessed unique information. Furthermore, from Figs. 3(a)-3(d) it is possible to found out that the coherent speckle in the reconstructed images can be well seen because the reflecting waves has stochastic phase due to the roughness of the surface and superposed coherently in the hologram plane. Moreover these speckle noise have different distribution, according to the recording condition changing, namely altering the object illumination directions in the recording process.

Figure 4(a) shows the reconstructed intensity image obtained from only one hologram with 0 illumination. Figure 4(b) and Fig. 4(c) show the synthesized intensity images from the holograms with 0, R1, L1, U1, D1 illuminations and with 0, R1, L1, U1, D1, R2, L2, U2, D2 illuminations, respectively. In addition, the partial intensity images with the dimensions of 169pixels×169pixels are shown in Fig. 4(d), 4(e) and 4(f), which cut out from Figs. 4(a), 4(b) and 4(c) where indicated by the white rectangle. The results show that the spatial resolution and definition of the intensity reconstructed image are improve effectively according to numerical aperture increasing and the speckle noises is well suppressed for superposing the speckle fields with different distribution.

 

Fig. 3. (a), (b), (c), (d) Reconstructed intensity images with illumination from R2, L2, U2, D2 respectively.

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Fig. 4. (a). Reconstructed intensity image from one hologram with illumination wave from 0; (b). Synthetic intensity image from five holograms with illumination waves from 0, R1, L1, U1 and D1; (c). Synthetic intensity image from nine holograms with illumination waves from 0, R1, L1, U1, D1, R2, L2, U2, D2; (d), (e), (f). Regional magnified intensity images corresponding to selected region of (a), (b), (c) respectively.

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To demonstrate the resolution enhancement of the phase image, a USAF-1951 resolution test target is used as observed object in the experiment. Figure 5(a) shows the two dimensional (2D) phase image reconstructed from only a hologram with 0 illumination, and Figs. 5(c) and 5(e) presents the 2D phase images synthesized from the holograms with 0, R1, L1, U1, D1 illuminations and with 0, R1, L1, U1, D1, R2, L2, U2, D2 illuminations, respectively. The corresponding three dimensional (3D) phase images are shown in Figs. 5(b), 5(d) and 5(f). All of the phase image have the size of 795pixels×795pixels. Moreover, for quantifying the resolution enhancement, the smallest resolved element found in the ellipse part of Fig. 5(a) is element 6 in group 2 (7.13 line-pairs/mm), that of Fig. 5(c) is element 3 in group 3 (10.1 line-pairs/mm), and that of Fig. 5(e) is element 5 in group 3 (12.7 line-pairs/mm), respectively. It is necessary to point out that the phase unwrapping is carried out by means of least-squares algorithm based on the Fast Fourier Transform [16] without any compensation of phase aberrations. In the incoherent superposing of the phase surfaces, the ratio of amplitude to sum of amplitudes from the different reconstructed image acts as weighted factor for each pixel in image plane, in order to suppress the unwrapped phase errors due to low signal to noise ratio. These results show the effectiveness of the proposed method.

 

Fig. 5 (a), (b) 2D and 3D reconstructed phase images from one hologram with object illumination from 0; (c), (d) 2D and 3D synthetic phase images from five holograms with object illuminations from 0, R1, L1, U1 and D1; (e), (f) 2D and 3D synthetic phase images from nine holograms with object illuminations from 0, R1, L1, U1, D1, R2, L2, U2, D2 respectively.

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4. Conclusions

In this paper, we present a synthetic aperture method for digital holography. The main advantages of the proposed approach are that the digital holography imaging system isn’t limited by the working distance and the reconstructed image doesn’t subject to lens aberrations, while the set-up structure and aperture synthesis procedure are simple in comparison of other methods. Furthermore, the experimental results indicate that not only the spatial resolution of synthesis image can be improved effectively, but also the speckle noises of the synthesized image can be well suppressed. In the future, the proposed method can be easily used to in-situ three dimensional microscope at a long working distance.