To obtain a phase distribution without the use of an optical path besides an object beam, a reference-free holographic diversity interferometry (RF-HDI) has been proposed. Although the RF-HDI can generate an internal reference beam from the object beam, the method has a problem of measurement accuracy due to insufficient power of the internal reference beam. To solve the problem, we newly propose a RF-HDI via iterative measurements. Our method improves the measurement accuracy by utilizing iterative measurements and feedback of each obtained phase image to the measurement system. In the experiment, the phase image, which has a random pattern, can be measured as an object beam with a higher accuracy than in the conventional RF-HDI. To support this result, we also evaluated the wavefront accuracy and optical power efficiency of an internal reference beam in this method. As a result, we verified that our method enables us to generate an internal reference beam that has the wavefront of a near single plane wave and a higher power efficiency than the conventional RF-HDI. In addition, our method can be applied to measurement for the modal content in an optical fiber, atmosphere turbulence, etc., where it is difficult to prepare an external reference beam with a high coherency.
© 2016 Optical Society of America
Phase-shifting digital holography (PSDH) [1, 2], which is based on digital holography [3–5] and phase-shifting interferometry , is a technique for detecting a complex optical amplitude distribution, including intensity and phase information. In particular, since the PSDH enables us to perform high accuracy measurements of the phase distribution in the complex optical amplitude, it has been used in many fields such as adaptive optics [7, 8], surface measurements [9, 10], and optical microscopy [11–13]. In PSDH, two methods have been proposed: a sequential method [14, 15] and a parallel method [16, 17]. Recently, a new PSDH method, holographic diversity interferometry (HDI) , is proposed. HDI allows us to achieve a spatial resolution equivalent to that of the image sensor and single-shot detection of multiple phase-shifted interference fringes.
In the case of a measurement system via the long-distance object such as an optical fiber [19–21] and atmosphere [22, 23], it is difficult to acquire a reference beam having a high temporal coherence with the object beam. Therefore, measuring the complex amplitude distribution of the object beam is also difficult in this case. To detect the beam transmitted through the optical fiber, an additional transmission path for the reference beam along with the signal transmission path  is used to measure the complex amplitude distribution of the object beam. The problem with this method is that the system consumes transmission channel resources. Hence, a reference-free HDI (RF-HDI) was proposed for solving this problem [25, 26]. In RF-HDI, an object beam is divided into two via a beam splitter and one of them is used as an internal reference beam. Then, a spatial filter composed of lenses and a pinhole extracts a component of a single plane wave (DC-component) in the internal reference beam. Thus, the complex amplitude of the object beam can be measured without an additional reference path that has a high temporal coherence with the desired spatial mode.
However, in RF-HDI, when the DC-component is not well contained in the object beam, the optical power of the internal reference beam markedly reduces because of the spatial filter. Conversely, if the pinhole diameter is set to obtain adequate optical power in the internal reference beam, a highly accurate DC-component cannot be obtained because the high-frequency component of the object beam remains in the internal reference beam. The upshot is that the internal reference beam in RF-HDI has a tradeoff between the optical power and wavefront accuracy. Therefore, it is difficult to achieve a high measurement accuracy in conventional RF-HDI because the interference fringe is not entirely obtained.
In the field of microscopy, several methods for the generation of a reference beam from an object beam that passes through the sample have been proposed. Digital holographic microscopy [27,28], in which an object beam is divided into two via a beam splitter—one of them is used as an internal reference beam in the same way as RF-HDI (causing these methods to have a problem similar to RF-HDI)—has been reported. Moreover, another form of digital holographic microscopy that does not require spatial filtering of the reference beam, has also been reported [29, 30]. These methods restrict the effective imaging area because a part of the beam divided from the object beam, which contains no object information, is used as the internal reference beam. In contrast, although phase contrast microscopy  has been discussed, it could lead to the restriction of the degrees of freedom in the incident beam because of the presence of a special ring-mask called condenser annulus. In addition, quantitative phase microscopy using shearing interferometry and modified Shack-Hartmann mask has also been discussed upon . This method restricts the range of spatial frequency of the phase distribution due to phase retrieval by Fourier analysis with spatial filtering for the object beam. Thus, in these methods of microscopy, although phase distribution can be obtained more easily, they are difficult to apply in specific situations that require precision phase distribution analysis, e.g., measurement of modal contents in optical fiber supporting a number of spatial modes.
In this paper, to overcome this problem, we propose a high measurement accuracy RF-HDI via iterative measurements. In our method, first, the phase distribution of the object beam is measured the same way as conventional RF-HDI. Then, through a phase-type spatial light modulator (PSLM), the phase distribution of internal reference beam is modulated in dependence upon the measured image so that its DC-component is increased. By iteratively performing these processes, the internal reference beam, which has a higher optical power and higher wavefront accuracy than conventional RF-HDI, can be generated. As a result, our method enables a reference-free phase measurement with high measurement accuracy. In addition, iterative measurements and feedback of each measured image reduces the optical power loss in the measurement system.
We will describe the basic operation of our method in Sect. 2. In Sect. 3.1 and 3.2, we will verify our method and compare phase images obtained using our method and conventional RF-HDI. Then, in Sect. 3.3 we will evaluate the accuracy and optical power of the internal reference beam in our method.
2. Basic operation
In this section, we describe the basic operation of our method. The schematic is shown in Fig. 1. A beam to be measured E (r) is described as:
In the first measurement, a blank is displayed on the PSLM. Thus, the internal reference beam R(r) is:
Next, during the second measurement, the sign of the phase term in U(r) is inverted using the computer, and the inverted phase distribution (feedback image) is displayed on the PSLM. In this method, displaying the feedback image on the PSLM is referred to as “feedback”. The internal reference beam performing the feedback by the PSLM is:Eq. (2)] and that at the second measurement R′(r) [Eq. (5)], the phase term of R′(r) contains only the component filtered at ν. Hence, the power of the beam passing through the pinhole can be improved by increasing the DC-component in the internal reference beam.
In the case that above operation is iteratively implemented while reducing the cutoff frequency ν of the pinhole, the internal reference beam can approach an absolute DC-component that does not include a high-frequency component. As a result, the obtained complex amplitude distribution can be measured with a higher accuracy than without feedback, similar to the case of conventional RF-HDI because the phase-shifting error caused by an incorrect DC-component can be avoided. Therefore, our method enables us to solve the problem with the trade-off relationship between the optical power efficiency and the measurement accuracy in the conventional RF-HDI.
Although the constitution of our method seems to be more complex than conventional methods, a measurement accuracy similar to that in the case of using an external reference beam can be achieved. In addition, as our method is based on PSDH, the range of spatial frequency is not restricted, and the lateral resolution of the image corresponding to pixels of the image sensor can be obtained.
In the experiment, a two-channel type HDI (2ch-HDI) and a two-channel algorithm , which is a calculation algorithm for the complex amplitude distribution of the 2ch-HDI, are used to acquire the complex amplitude of the object beam. Figure 2 shows the setup for the experiment. The wavelength of the light source (DPSS laser, CW) is 532 nm and other experimental parameters are shown in the caption of Fig. 2. As shown in Fig. 2, the beam to be measured is modulated by a random phase distribution, including 0 and π. In addition, the feedback number is set to five; hence, the number of iterative measurements is six. The diameter of the variable pinhole in each iteration is arbitrarily set to 200.50 μm, 153.90 μm, 105.30 μm, 72.90 μm, and 48.60 μm. Furthermore, an optical power meter (OPM) is used to determine the optical power of the internal reference beam passing through the variable pinhole. In addition, CCD3 and BS3 are used to determine the accuracy of the internal reference beam in Sect. 3.3.
In the following, we explain the experimental protocol. First, the object beam is divided by polarizing beam splitter (PBS1). Here, half wave plate (HWP1) determines the branch ratio of the divided intensities. In the optical path of the transmission side of PBS1, the object beam is imaged on CCDs via lenses (L1 and L2). In the optical path of the reflection side of PBS1, the internal reference beam is imaged on the the phase-type spatial light modulator (PSLM) via lenses (L3 and L4). In the first measurement using our method and conventional RF-HDI, a white blank is displayed on PSLM2 instead of a feedback image. Then, the feedback image is displayed on the PSLM after a second measurement using our method. Next, the beam reflected from the PSLM is transmitted through lenses (L5 and L6). Then, this beam is focused onto the the intensity-type spatial light modulator (ISLM) through lens (L7). Here, by displaying the pinhole pattern, which is a circular aperture, the ISLM can be used as a variable pinhole. Finally, after filtering using the variable pinhole, the internal reference beam is combined and interfered with the object beam by beam splitter (BS2) via lenses (L8 and L9). Note that the half wave plate (HWP2) and the quarter wave plate (QWP) are used to generate two phase-shifted interference fringes in 2ch-HDI. The phase of the object beam is obtained from two interference fringes using the two-channel algorithm. In addition, in our method, the final phase image can be obtained by iterative measurements while displaying a feedback image on the PSLM.
Figures 3(a) and 3(b) show the original phase distributions of the input object beam. Table 1 shows the feedback images displayed on PSLM and phase images obtained by 2ch-HDI for each iterative measurement when a random phase distribution, including 0 and ÏǍ, as shown in Fig 3(a), is measured. Table 2 shows each feedback and corresponding images obtained when the Lenna image, as shown in Fig 3(b), is measured. According to Table 1, the quality of the obtained phase images seems to improve with an increasing number of iterations; in particular, the part enclosed by the orange circle in the final obtained phase image seems better than the first obtained phase image, which has an inverted phase difference. Similarly, in the case of measurement of the Lenna image, as shown in Table 2, the quality of the obtained phase images seems to improve with increasing number of iterations. Then, the final detected phase image, as shown in the lower right end of Table 1 and Table 2, are in good agreement with the original phase distributions of the input object beam, as shown in Figs. 3(a) and 3(b), respectively.
Furthermore, the signal-to-noise ratio (SNR) is calculated to evaluate the phase distribution obtained as a gray-scaled image. The SNR of the image data is defined as:Eq. (6) is the mean square error in the obtained phase image. Figures 4(a) and 4(b) show the SNRs of the obtained phase images for each number of iterations when the random phase and Lenna image are measured, respectively. In Fig. 4, the SNRs increased evidently with the increasing number of iterations in the case of both the random phase and Lenna image. In particular, when the random phase distribution was measured, the SNR of the obtained phase image in the sixth measurement was better at 8.46 dB than that of the first measurement. This is because the wavefront accuracy, which is capable of increasing the DC-component in the internal reference beam, is improved with the increasing number of iterations. The detailed wavefront accuracy of the internal reference beam will be discussed in Sect. 3.3.
Here, we compared the measurement accuracy in our method with that of the conventional RF-HDI when a random phase distribution, as shown in Fig 3(a), is measured. The pinhole size for conventional RF-HDI is set to 48.60 μm, which corresponds to that at the 6th iteration in our method. Here, the spot size of the internal reference beam focused by L7 is 37.63 μm in the case of an absolute DC-component. Moreover, the optical power irradiating PBS1 and the branch ratio of the divided beams are set to the same values for our method and conventional RF-HDI so that the optical power of the internal reference beam before filtering is maintained constant. Table 3 summarizes the intensity distribution images of the object beam, reference beam, and interference fringe, which were detected by CCD1, and the calculated phase image. The upper and lower parts of Table 3 correspond to the conventional and proposed methods, respectively. According to Table 3, the contrast of the interference fringe in conventional RF-HDI is partially deteriorated, as shown in the part enclosed by the orange circle. The same part in the obtained phase image is also partially inverted due to the phase difference. This is because the pinhole diameter in RF-HDI is larger than the spot size, and the internal reference beam does not approach the complete DC-component. In contrast, in our method, the contrast in the interference fringe and the obtained phase image can be clearly seen. In fact, to evaluate the phase images obtained with conventional RF-HDI and our method, we calculated their SNRs. As a result, the SNRs for conventional RF-HDI and our method are 5.70 dB and 14.47 dB, respectively.
Moreover, Figs. 5(a) and 5(b) show histograms of the normalized intensity of the internal reference beam for conventional RF-HDI and our method when the number of iterations is 6, respectively. By comparing both histograms, it is observed that the median of the normalized intensity in the proposed method is higher than that for conventional RF-HDI. Therefore, the internal reference beam of our method is brighter than that for conventional RF-HDI. In fact, the optical powers of the internal reference beam passing through the variable pinhole in conventional RF-HDI and our method are 0.15 μW and 1.57 μW, respectively. The above results show that our method enables us to achieve a more accurate measurement using the improved optical power efficiency of the internal reference beam.
3.3. Evaluation for the internal reference beam
In order to support the above results, we discuss the wavefront accuracy and power efficiency of the internal reference beam in our method when a random phase distribution, as shown in Fig 3(a), is measured. Here, “wavefront accuracy” means the amount of the DC-component included in the internal reference beam. In particular, in the case where the internal reference beam does not contain high-frequency components, it is referred to as highly accuracy.
First, we verify that the DC-component in the internal reference beam is improved when increasing the number of iterative measurements by observing the spatial frequency distribution of the internal reference beam using CCD3. Figures 6(a)–6(f) show the intensity of each spatial frequency distribution observed by CCD3 when the feedback image is displayed on PSLM. Then, Fig. 6(g) shows the intensity of the spatial frequency distribution observed by CCD3 when the blank image is displayed on PSLM, which can be assumed as the ideal internal reference beam including only the absolute DC-component. In Figs. 6(a)–6(f), the intensity distribution was further focused on the spot in proportion to the number of iterations, whereas the intensity distribution without feedback was scattered by the high-frequency component in the original phase distribution of the object beam. Compared with Fig. 6(g) and Fig. 6(f), the focal spot of the internal reference beam in the 5th feedback was focused in the same way as that of the absolute DC-component. In addition, Fig. 6(h) plots the intensity profile at y = 0.0 μm of Fig. 6(f) (represented by the blue line) and Fig. 6(g) (represented by the black line). According to Fig. 6(h), the full-width at half-maximum (FWHM) of the focal spot in the absolute DC-component shown in Fig. 6(g) is 37.63 μm and that of the 5th feedback internal reference beam shown in Fig. 6(f) is 39.65 μm. The FWHM in the 5th feedback was in good agreement with that in the case of the absolute DC-component. The centers of the peaks are shifted slightly to the right because the feedback image generated from the obtained phase image contains an aberration in the object beam path. In other words, these results show that the accuracy of the internal reference beam can be improved even if the accuracy of the alignment of the optical system is slightly lower.
Next, by observing the optical power of the beam passing through the variable pinhole using the OPM located between the PBS and the QWP, as shown in Fig. 2, we verify that the optical power efficiency of the internal reference beam increases with the number of iterations. In this part, the pinhole diameter is fixed to 48.60 μm, and the optical power of the beam passing through the pinhole is observed by the OPM in sequence when each feedback image is displayed on PSLM. Figure 7 shows the optical power of the beam passing through the variable pinhole for each iteration. According to Fig. 7, the power efficiency of the beam passing through the pinhole is clearly improved by approximately 10-fold in accordance with the feedback images displayed on PSLM. This is because the optical power of the internal reference beam passes through the spatial filter by increasing the DC-component. Thus, we verified that the wavefront accuracy and the optical power efficiency of the internal reference beam are improved simultaneously in accordance with the number of iterations. As the result, the improvement of the internal reference beam brings improvement to the measurement accuracy.
We proposed a high-measurement-accuracy RF-HDI via iterative measurements, in which the phase distribution of a complex optical amplitude can be obtained with a higher accuracy than the RF-HDI. In the experiment, we verified that the phase distribution of the object beam could be obtained at 8.77 dB with a higher accuracy than that for conventional RF-HDI. In addition, in order to support this result, we evaluated the wavefront accuracy and the optical power efficiency of the internal reference beam in our method by observing the spot size of the focused internal reference beam and the optical power passing through the pinhole. As a result, the wavefront accuracy and the optical power efficiency were simultaneously improved in accordance with the number of iterative measurements. In future work, we will reduce the iterative number of feedback by optimizing the pinhole diameter and verify that this method can measure the modal content in an optical fiber, atmosphere turbulence, etc. with higher accuracy than conventional RF-HDI.
Japan Society for the Promotion of Science (JSPS) KAKENHI (JP25289110).
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