^{1}

^{2}

^{2}

^{1}

^{2}

^{1}

^{1}

^{1}

^{2}

A digital hologram-optimizing method was proposed to improve the imaging quality of dual-wavelength digital holographic microscopy (DDHM) by reducing the phase noise level. In our previous work, phase noise reduction was achieved by dual-wavelength digital image-plane holographic microscopy (DDIPHM). In the optimization method in this paper, the phase noise was further reduced by enhancing the real-image term and suppressing effects of the zero-order term in the frequency spectrum of a digital hologram. Practically, the carrier frequency of the real-image term has the correspondence with interference fringes in the hologram. Mathematically, the first order intrinsic mode function (IMF1) in bidimensional empirical mode decomposition (BEMD) has similar characteristics to the grayscale values of ideal interference fringes. Therefore, with the combination of DDIPHM and BEMD, by utilizing the characteristics of IMF1, the digital hologram was optimized with purified interference fringes, enhancing the real-image term simultaneously. Finally, the validity of the proposed method was verified by experimental results on a microstep.

With various advantages such as the real-time performance, noninvasive property, and easy processing by mathematical computing, digital holographic microscopy (DHM) has experienced substantial development in surface profile measurement of microstructures [

However, the phase noise, especially in the recording process, is amplified when the measurement height range is amplified simultaneously, resulting in a loss of axial resolution in the measurement [

Except for image processing methods [

The empirical mode decomposition (EMD) method has been used in digital holography. EMD directly performs the task of particle sizing and axial locating from in-line digital holograms rather than reconstructing the optical field [

In this paper, different from [

The experimental setup for DDHM is depicted in Figure

Experimental setup for DDHM. (a) The illustration of the DDHM system. (b) The apparatus of DDHM. NF1 and NF2: variable neutral filters; BS1–BS4: beam splitters; M1–M5: mirrors; BE1–BE3: beam expanders; MO: microscope objective with magnification 50x and numerical aperture

The imaging noise of DDHM originates from coherent recording and the finite size of the pixels in the CCD camera. The temperature variation in media and visible blemishes on any window where light passes through can also cause diffraction and reflection. The above-mentioned disturbing factors should be removed at the stage of hologram processing; otherwise, they would introduce phase noises in the measurement for surface profiling of microstructures.

The intensity of the digital hologram recorded in single-wavelength DHM can be written as

However, the carrier frequency of

EMD decomposes a complex time series into the sum of a limited number of IMFs. Each IMF needs to satisfy the following two conditions:

The number of extreme points should be equal to or larger than the number of zero points in the entire time series

At any point, the mean value of the envelopes formed by the local maximum point and the local minimum point is zero

Figure

The interference fringes in grayscale value of Young’s double-slit interference.

Therefore, the information of interference fringes can be obtained by the sifting process of the hologram. Since the hologram is two-dimensional, the BEMD sifting process is applied and described as follows [

Identify sets of minima (

Connect all the local maxima of

If the subtraction result meets the IMF condition, save

By analyzing the frequency spectrum of the hologram, the hologram is decomposed by BEMD, and IMF1 can be remained as the optimized hologram with the processed frequency spectrum to be calculated in the reconstruction.

The intensity distribution of the hologram of DDHM can be written as

Experimental hologram and frequency spectrum. (a) Image-plane hologram. (b) IMF1 of the image-plane hologram. (c) Frequency spectrum of the hologram. (d) Frequency spectrum of IMF1. The boxes in figures are the magnified parts.

By using DDIPHM, the phase and amplitude of the sample can be straightforwardly extracted:

The height of the sample is

The experimental results should be discussed from the perspectives of previous studies and working hypotheses. The findings and their implications should be discussed in the broadest context. Future research directions may also be highlighted.

To assess the validity, a microstep (surface gold-plated, a testing sample of Lyncee tec) was measured by the setup of DDHM, and a stylus profilometer (KLA-Tencor, P-16+/P-6) with the force of 1 mg for comparison. In this part, the experimental results of DDIPHM, DDIPHM with BEMD, and DDHM are compared to demonstrate that BEMD can achieve a lower phase noise level.

The image-plane hologram of the microstep is presented in Figure

Figure

The experimental results of DDIPHM with BEMD. (a) The phase image of microstep measured by DDIPHM with BEMD. (b) The surface profile of the microstep measured by DDIPHM with BEMD. (c) The height profile lines plotted along the black line in Figure

Since the precision of DHM can be 0.1 nm, the calculated height value was kept one decimal digit. The average height of multiple profile lines is the experimental results (Figure

The microstep height experimental results.

Step number | 1 | 2 | 3 | 4 |
---|---|---|---|---|

DDIPHM | 4048.1 ± 18.8 nm | 2021.1 ± 16.8 nm | 988.5 ± 15.9 nm | 89.4 ± 20.6 nm |

DDIPHM with BEMD | 4043.1 ± 12.1 nm | 2027.1 ± 10.2 nm | 986.5 ± 9.3 nm | 83.5 ± 10.3 nm |

Stylus profilometer | 4030.8 ± 17.2 nm | 1980.0 ± 13.5 nm | 963.7 ± 10.7 nm | 130.0 ± 9.1 nm |

Classical DDHM | — | 2032.4 ± 54.8 nm | 994.3 ± 52.5 nm | 89.6 ± 42.5 nm |

Two points can be concluded from the experimental results: first, compared to DDIPHM, the noise is obviously suppressed in the measurement of DDIPHM with BEMD; second, compared to stylus profilometry, the measuring correctness of DDIPHM with BEMD is verified by the good accordance of the two measurement results. Since BEDM is used to enhance the contrast of interference fringes, DDIPHM with BEMD is especially suitable for the reconstruction of holograms acquired in the environment with speckle noises. The refractive index difference between biological cells or tissues and environment can be quite large. Therefore, DDIPHM with BEMD was meant to be the appropriate method to retrieve the phase of biological samples. Though the phase range of the measurement was enlarged, the lateral resolution was maintained.

In this paper, a hologram-optimizing method was proposed. By using the DDIPHM with BEMD method, the interference fringes were extracted, resulting in the enhancement of the real-image term and suppression of the zero-order term in the frequency spectrum of the hologram. The affection of disturbing factors in the recording process was suppressed simultaneously. According to the experimental results, the measured noise level of the DDIPHM with BEMD method can be further reduced compared to DDIPHM. The validity of the proposed method was verified compared to stylus profilometer measurement.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

The research work was supported by National Natural Science Foundation of China (NSFC) (51775381), Education Commission Research Project of Tianjin (2017KJ182 and JWK1612), State Key Laboratory of Precision Measuring Technology and Instruments (Tianjin University) Foundation (pilab1704), and Tianjin Nature Science Foundation (18JCQNJC05600).