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Abstract
Phase-shifting holography has been widely applied in the field of non-destructive testing and interference imaging. Compared to the previous single-shot phase-shifting holography, here tri-focal Fibonacci-Billet split lens was introduced into Mach-Zehnder interferometer, in which the upper half of the Fibonacci-Billet split lens can realize three phase-locking copies of the planar reference wave and the lower half is used to generate three identical copies of object. The interference pattern is recorded by a detector in one single exposure. The test object can be reconstructed by three-step phase-shifting interferometry. The corresponding experiment is carried out to verify the effectiveness of this method. With advantages of real-time reconstruction and amplitude-only diffraction lens, it is very useful for fast imaging and optical element detection.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Gabor proposed the holography to improve the resolution of the electron microscope by recording the interference pattern between the reference and object waves [1]. Two decades later, digital holography was successfully invented to reconstruct the test object by post processing [2–4]. Since then, holography has been paid much more attention and made considerable progress in many research field, such as information processing [5–7], interferometry [8–10], three-dimensional display [11–13], microscopy imaging [14,15], and so on. Holographic recording mainly contains in-line and off-axis holography. Although off-axis holography can eliminate the undesired terms by spatial filtering in frequency plane, it cannot take full advantage of the space -bandwidth product of the optical detector [16]. As for in-line holography, there exists cross-talk in the reconstructed image plane due to the conjugate and zero-order terms. Until the advent of phase-shifting interferometry [17], the above shortcoming was unsolved. The key device of phase-shifting interferometry is the phase shifter, such as piezoelectric ceramics transducer (PZT) [18], wave plate [19], grating [20], and liquid crystal retarder or spatial light modulator (SLM) [21].
Generally, phase-shifting holography needs multiple frames of interference pattern to reconstruct the test object. In order to improve the imaging speed and imaging quality, many kinds of single-shot phase-shifting holography are suggested [22]. Holographic grating is used to generate multiple object waves by beam splitting [23]. In this case, multiple detectors must be required for synchronous detection so as to extract multi-frames of hologram as required. Thus inevitably it is nearly impossible to control the zero-order diffraction intensity, which causes the optical path more and more complex and difficult [24]. Random-phase reference beam produced by point diffraction interference can also be applied to single exposure, but the reconstruction solution is limited by the pixel size of optical detector and minimum structure size of optical device [25,26]. As for diffuse light in one single exposure, these two images seriously reduce the image quality [27]. Talbot effect had been suggested to be used in reference beam path in order to record multi-copies simultaneously on condition that the minimum structure size of array holes should be the same as the pixel size of the detector [28]. Besides, parallel quasi-phase-shifting holography was proposed for single-exposure recording, in which four-step phase-shifting interferometry was used to reconstruct the test object by using array devices in the reference beam to achieve spatial partition multiplexing [29].
Based on the concept of Billet split lens, a kind of tri-focal Fibonacci-Billet split lens was introduced into Mach-Zehnder interferometer, in which the upper half of the Fibonacci-Billet split lens can realize three phase-locking copies of the planar reference wave and the lower half can generate three identical copies of object. Next, the three reference and object waves then form a frame of interference pattern on the detector screen. Then three independent phase-shifting sub-holograms can be extracted from the single-exposure hologram, and be applied for reconstruction by three-step phase-shifting interferometry. In experiment, a tri-focal Fibonacci-Billet split lens was fabricated by lithography and was split into two parts. One of which was placed in the reference beam and the other in the object beam. The experimental results verify the effectiveness of our proposed method. With advantages of real-time reconstruction and amplitude-only diffraction lens, it has great potential in the field of fast imaging and optical element detection.
2. Principle of phase-shifting holography with tri-focal Fibonacci-Billet split lens
Based on the previous work on photon sieves [30–32], a Fibonacci sieve is fabricated with a diameter of 9.9 mm and three focal lengths of 75.0 mm, 90.0 mm and 112.5 mm, in which there are about 464 thousands of transparent pinholes. Figure 1 shows the schematic of the tri-focal Fibonacci-Billet split sieve and its imaging properties. When a tiny test object is in the front of multi-focal sieve, the multiple images will be not only distributed at different image planes but also separated in the lateral direction. Figure 2 indicates the telescope with a tri-focal Fibonacci-Billet filter. The tri-focal sieve is placed at the focal plane of the telescope consisting of two condenser lens with the same focal length of 60 mm. Suppose that the center of the tri-focal sieve is defined as the original coordinate, the test object is placed in front of telescope, for example, at 48 mm. According to Gaussian imaging formula [33], when the object passes through the first lens, the image distance is −240 mm and lateral magnification is 5. For the shortest focal length of Fibonacci-Billet split lens, the object distance is −240 mm −60 mm=−300 mm. According to the Gaussian formula again, the image distance is 100 mm, and the lateral magnification is −1/3. Similarly, for the second lens of the telescope, the object distance is 100 mm −60 mm=40 mm, the image distance is 24 mm, and the lateral magnification is 3/5. The total magnification of the system is 5*−1/3*3/5=−1. It is should be noted that no matter where the test object is placed, the image spacing between any two adjacent images equals to 8 mm forever. The imaging properties of the telescope with focal-plane Fibonacci-Billet split sieve are described in Table 1.
Fig. 1. Tri-focal Fibonacci-Billet split sieve and its imaging path. (a) Tri-focal lens. (b) Fibonacci-Billet split lens. (c) Reference beam. (d) Object beam.
Fig. 2. Telescope with a focal-plane Fibonacci-Billet split lens, reference (blue line) and object paths (black line).
Table 1. Imaging of telescope with tri-focal Fibonacci-Billet split sieve (Unit: mm)
For the more general case, the first and the second lens in telescope have their own focal lengths f1 and f2, and the photon sieve has the focal length of F, when the object distance is denoted by p, the image distance q can be given by
Fig. 3. Schematic of single-shot phase-shifting image-plane holography with Fibonacci-Billet split lens.
Suppose that the phase difference of the three copies is $\beta$, the second interference pattern can be expressed as
3. Experiment and discussion
In experiment, a plane wave laser beam is split into two parts by a beam splitter (BS1). In the reference beam path, the beam passes through a Kepler telescope with the upper half of tri-focal Fibonacci-Billet split lens. Three copies of plane reference wave are produced and distributed in the different image planes. Note that the three plane reference waves have the same amplitude but different phase retarder, which meet the requirement of phase-shifting interferometry. The second part, object beam, passes through a test object and then through another Kepler telescope with the bottom half of tri-focal Fibonacci-Billet split lens. Three copies of object with different phase are produced simultaneously. In order to obtain three identical copies in the same plane, an image relay system is introduced between the telescope and CCD screen. The three phase-locking plane reference waves and the three identical copies of object are combined by the beam combiner (BS2) and then form a frame of single-exposure hologram on the CCD (GT3300, 3296×2472, 5.5 µm × 5.5 µm size). The whole process of recording does not require mechanical phase-shifting operation. Finally, the object is then can be reconstructed by three-step phase-shifting holography [34].
The test object is a letter “A” fabricated on a chrome plate with its line-width of 50µm. The Fig. 4 illustrates the single-exposure hologram. Based on this, we can extract three frames of phase-shifting sub-hologram, and reconstruct the test object. The reconstructed images are shown in Fig. 5. The intensity distribution is represented in Fig. 5(a) and the reconstructed phase distribution can be resolved clearly, as described in Fig. 5(b). The last point to be emphasized is that the signal region has the same wave-front distribution because of amplitude-only object. The experimental results verify the validity of the proposed single-shot phase-shifting image-plane digital holography.
4. Conclusion
In conclusion, tri-focal Fibonacci-Billet split lens was proposed based on the concept of the traditional Billet split lens, and then used for multi-focal photon-sieve filter in telescope to generate multi-copies of reference or object waves. A single-exposure hologram consisting of three frames of phase-shifting sub-hologram was recorded by the optical detector. Three-step phase-shifting holography can be used to reconstruct the test object in one single exposure. The experimental results in the optical region are consistent with the theoretical analysis, which verifies the effectiveness of the proposed method. Single-shot technique greatly improves the system stability during the process of the recording or monitoring. The proposed single-shot phase-shifting digital holography with tri-focal Fibonacci-Billet split lens can be applied in the fields of bio-microscopy, multiple identical imaging, array pre-amplified holography, and so on. As a kind of amplitude-only diffractive lens, multi-focal photon sieve can be suitable for focusing and imaging in the region of soft X-ray and extreme ultraviolet.
Funding
National Natural Science Foundation of China (61775222); Youth Innovation Promotion Association (2017292).
Disclosures
The authors declare no conflicts of interest.
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