Virtual interferogram-generation algorithm for robust complex amplitude measurement using two interferograms

Jin Nozawa; Atsushi Okamoto; Kazuhisa Ogawa; Akihisa Tomita

doi:10.1364/OE.24.024002

1. Introduction

As optical complex amplitude measurements were first used in the early 1980s for adaptive optical systems [1], they have been applied in many fields, including optical microscopy [2,3], surface shape measurement [4,5], object recognition [6,7], compensation of aberrations [8], wavefront curvature [9], phase conjugation in optical communication channels [10], and particle measurement [11,12]. The most commonly used complex amplitude measurement is the Shack-Hartmann sensor [13], which estimates the wavefront from the deviation of focus spots generated by a lenslet array. The spatial resolution of this technique is much lower than that of an image sensor due to the distance between the focus spots. Phase-shifting digital holography (PSDH) [14], which is the combination of digital holography [15] and phase-shifting interferometer [16], was proposed in 1997 as a high-spatial-resolution complex amplitude measurement. Although various PSDH methods [17–19] have been proposed in a single decade, PSDH typically requires three or four phase-shifted interference patterns between a signal beam and a reference beam. These interference patterns are captured by an image sensor, such as a charge-coupled device (CCD), to calculate the complex amplitude of a signal beam in a computer.

Of particular interest, there are several PSDH methods [20–23] that use only two phase-shifted interference patterns for calculating the complex amplitude of a signal beam. These methods often solve the quadratic formula of the dc component of two interference patterns. In terms of the quadratic formula, these methods have an operating condition that the square root of the quadratic formula must be positive. Previous studies [20,23] show this condition indicates that the intensity of a reference beam is much greater than that of a signal beam. However, the measurement under this condition leads to the consumption of a dynamic range and the shortage of the signal’s amplitude is caused by the degradation of the signal’s visibility. Moreover, most conventional algorithms assume the phase shift of interferograms is equal to π/2. Therefore, conventional algorithms cannot deal with the phase shift error.

In this paper, we propose the Virtual Interferogram-Generation Algorithm (VIGA). Unlike conventional algorithms, VIGA utilizes the fact that a π phase-shifted hologram can be generated from (1) the intensity distribution of a signal beam, (2) the intensity distribution of reference beam, and (3) the interferograms between them. In fact, by calculating two virtual holograms from two phase-shifted interferograms captured by an image sensor, four phase-shifted interference patterns can be obtained. Phase calculation algorithms using four interference patterns [24] have no limitations in terms of the amplitude of the captured intensity distribution. Therefore, VIGA can operate normally even when the intensity distribution of a reference beam is less than that of the signal beam. In addition, this algorithm can easily change the phase shift of two interferograms to an arbitrary value. Thus, VIGA can also choose the most suitable phase shift when the amount of the phase shift is given.

In Sect. 2, the basic principle of VIGA is described. In Sect. 3, we describe four numerical simulations performed to confirm the effectiveness of VIGA. In Sect. 4, we describe two experiments performed using Holographic Diversity Interferometry [19], one of the PSDH, to show that VIGA is independent to the magnitude of captured intensities and can choose the most suitable phase shift when calculating the complex amplitude.

2. Virtual interferogram-generation algorithm

Figure 1 shows a conceptual diagram of VIGA. This algorithm obtains four phase-shifted interference patterns by calculating a π phase-shifted hologram from three intensity distributions, which consist of a signal beam, a reference beam, and an interferogram between them. After obtaining four phase-shifted interference patterns, the phase distribution of a signal beam can be easily evaluated. Here, we assume the amplitude of signal beam is A_o, the amplitude of a reference beam is A_r, and the phase of a signal beam is ϕ. The two phase-shifted interference patterns H₁ and H₂, with a γ phase shift, are described as Eqs. (1) and (2).

Fig. 1 Conceptual diagram of the VIGA.

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H_{1} = A_{o}^{2} + A_{r}^{2} + 2 A_{o} A_{r} \cos φ

H_{2} = α A_{o}^{2} + β A_{r}^{2} + 2 \sqrt{α β} A_{o} A_{r} \cos (φ - γ)

Captured by an image sensor, the variables, α and β, are the intensity ratios of the signal and reference beams, respectively. Trigonometric functions of Eqs. (1) and (2) are as follows.

2 A_{o} A_{r} \cos φ = H_{1} - (A_{o}^{2} + A_{r}^{2})

2 \sqrt{α β} A_{o} A_{r} \cos (φ - γ) = H_{2} - (α A_{o}^{2} + β A_{r}^{2})

From Eqs. (3) and (4), H_v₁ and H_v₂, which are π phase-shifted virtual holograms of H₁ and H₂, are given by Eqs. (5) and (6).

H_{v 1} = A_{o}^{2} + A_{r}^{2} - 2 A_{o} A_{r} \cos ϕ = 2 (A_{o}^{2} + A_{r}^{2}) - H_{1}

H_{v 2} = α A_{o}^{2} + β A_{r}^{2} - 2 \sqrt{α β} A_{o} A_{r} \cos (ϕ - γ) = 2 (α A_{o}^{2} + β A_{r}^{2}) - H_{2}

From Eqs. (5) and (6), H_v₁ and H_v₂ are described as the functions of intensity ratios, α and β, the intensity of signal beam, A_o², the intensity of reference beam, A_r², and two interferograms, H₁ and H₂. These six variables can be obtained directly from the image sensor. Thus, the two four phase-shifted interference patterns can be calculated. The complex amplitude of a signal beam, described in Eq. (7), is the function of the four phase-shifted interference pattern, with phase shifts denoted by 0, γ, π, π + γ.

\begin{matrix} A_{o} \exp (i ϕ) = A_{o} \cos ϕ + i A_{o} \sin ϕ \\ = \frac{H_{1} - H_{v 1}}{4 A_{r}} + i \frac{(H_{2} - H_{v 2}) - \sqrt{α β} \cos γ (H_{1} - H_{v 1})}{4 \sqrt{α β} A_{r} \sin γ} \end{matrix}

From Eq. (7), capturing two signal intensities, two reference intensities, and two interferograms with an arbitrary phase shift, the complex amplitude of a signal beam can be reconstructed. In order to compute VIGA, we need to first capture two signals and two reference beams and calculate the signal intensity ratio, α, and the reference intensity ratio, β. Therefore, compared to conventional algorithms [20,23], the measuring speed of VIGA is degraded by this step. However, VIGA has no limitation in terms of the amplitude of the computed intensities from the captured signals and can compensate the phase shift error for the measuring accuracy. Additionally, the robustness of PSDH using two interferograms is significantly improved compared to conventional algorithms.

3. Simulation

To evaluate the effectiveness of VIGA, we carried out four numerical simulations. In these simulations, we regard a similar algorithm, the two-channel algorithm [23], which is considered the optimal PSDH algorithm that uses two image sensors, as “the conventional algorithm”. Initially, we verified that VIGA can generate the ideal π phase shifted hologram H_v1 from the signal intensity A_o², the reference intensity A_r², and the interferograms H₁. Next, for verifying the effectiveness of the proposed VIGA, we assumed the following two measuring conditions: (1) the intensity of the reference beam is less than that of the signal beam and (2) the phase shift of interferograms γ is not equal to π/2. We then compared the images calculated by the two-channel algorithm and VIGA. Finally, by introducing a quantization error, we compared VIGA with a four-step algorithm [24]. In these simulations, to evaluate the measuring accuracy, we calculated the Root Mean Square Error (RMSE), which is commonly used as an index for error evaluation. The RMSE is defined as follows:

R M S E = \sqrt{\frac{1}{N_{x} N_{y}} \sum_{i = 0}^{N_{x} - 1} \sum_{j = 0}^{N_{y} - 1} {(ϕ_{i j} - {\hat{ϕ}}_{i j})}^{2}}

where N_x is the number of pixels the along horizontal direction, N_y is the number of pixels along the transverse direction,

ϕ_{i j}

is the actual measured value,

{\hat{ϕ}}_{i j}

is the value of the original image. If RMSE = 0.0, the measurement error is assumed to be equal to zero.

Figures 2(a) and 2(b) show the signal and reference intensities used in the first numerical simulation. Two-dimensional intensity images were used for generating the virtual hologram and each image consists of 200 × 200 data pixels, which was the ideal Gaussian for both signal and reference intensities. The data pixels in the signal phase were modulated using the bmp image, well known as Lena, and those in the reference phase were modulated to 0, which is assumed to be the ideal plane wave. Figure 2(c) shows the interferogram between the signal and reference beam and the interference fringe followed by the change of signal phase can be observed. Figures 2(d) and 2(e) show the virtual hologram calculated by (1) the signals in Figs. 2(a)–2(c) and (2) the ideal hologram, which has a phase shift of π. Compared to Figs. 2(d) and 2(e), the virtual hologram coincided with the ideal π phase shifted hologram and the RMSE between them is almost 0.0. Moreover, compared to Figs. 2(c) and 2(d), the brightest and darkest portion of Fig. 2(c) are reversed to that of Fig. 2(d). This characteristic is in good agreement with the relationship between two holograms, with an ideal phase shift of π. These results suggest that VIGA can generate the ideal π phase shifted hologram from three intensities, which consists of the signal intensity, the reference intensity, and interferograms between them.

Fig. 2 Comparison of the ideal hologram and the calculated virtual hologram. (A) the signal intensity image (B) the reference intensity image. (C) the interferogram image. (D) the virtual hologram calculated by VIGA. (E) the ideal hologram, with phase shift of π.

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Figures 3(a) and 3(b) show the original intensity and phase images used in the second numerical simulation. The two-dimensional intensity and phase images were used as a measuring object, with each image consisting of 200 × 200 data pixels. The data pixels in the intensity image were kept constant, and the data pixels in the phase image were modulated from left to right in a staircase pattern at intervals of π/200. The intensity ratio of the object beam α and the intensity ratio of the reference beam β were set to 1.2 and the phase shift of interferograms γ was set to π/2. Figures 3(c) and 3(d) show the intensity and phase images calculated by the two-channel algorithm. Both the intensity and phase images were blurred and differed significantly from the original images. This is because the two-channel algorithm cannot work well when the intensity of reference beam is less than that of the signal beam. Figures 3(e) and 3(f) show the intensity and phase images measured using VIGA. Compared with the two-channel algorithm, both the intensity and phase image measured were almost identical to the original images. This result suggests that VIGA can operate accurately even when the intensity of a reference beam is less than that of a signal beam.

Fig. 3 Reconstructed complex amplitude when the intensity of a reference beam was less than that of a signal beam. (A) the original intensity image (B) the original phase image. (C) the intensity image calculated by the two-channel algorithm. (D) the phase image calculated by the two-channel algorithm. (E) the intensity image calculated by VIGA. (F) the phase image calculated by VIGA.

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Figure 4 shows the RMSE between the original phase image and the calculated phase image when the intensity of the reference beam is changed from 1 μW to 20 μW and the intensity of the signal beam was set to 10 μW. The x-axis is the intensity of the reference beam and the y-axis is the RMSE. In this figure, the number of calculation points is 20. The RMSE of VIGA was calculated to be approximately equal to 0 for all reference intensities. In contrast, the RMSE of the two-channel algorithm was increased when the intensity of the reference beam was less than that of the signal beam. Thus, by introducing VIGA to the PSDH using two interferograms, highly accurate phase measurements can be achieved when the intensity of a reference beam is less than that of a signal beam.

Fig. 4 RMSE between the original phase image and the calculated phase image when the intensity of the reference beam was changed from 1 μW to 20 μW. The intensity of signal beam was 10 μW.

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Figures 5(a) and 5(b) show the original intensity and phase images used in the third numerical simulation. Original intensity and phase images are identical to that from the first simulation, shown in Figs. 3(a) and 3(b). The intensity ratio of the signal beam, α, and the intensity ratio of the reference beam, β, were set to 1.2 and the phase shift of the inteferograms, γ, was set to π/4. Figures 5(c) and 5(d) show the intensity and phase images calculated by the two-channel algorithm. Both the intensity and the phase images were completely different from the original images and have calculation errors surrounded by the rectangle area. This is because the two-channel algorithm can work well only when the phase shift of interferogram is π/2. Figures 5(e) and 5(f) show the intensity and phase images measured using VIGA. Compared to the two-channel algorithm, both the intensity and phase images measured using VIGA were almost identical to the original images. This result suggests that VIGA can operate accurately even when the phase shift of interferograms is not π/2.

Fig. 5 Reconstructed complex amplitude when the phase shift of interferograms was π/4. (A) The original intensity image. (B) The original phase image. (C) the intensity image calculated by the two-channel algorithm (D) the phase image calculated by the two-channel algorithm. (E) the intensity image calculated by VIGA. (F) the phase image calculated by VIGA

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Moreover, we evaluated the performance of VIGA using an RMSE between the original phase image and the calculated phase image. Figure 6 shows the RMSE between the original phase image and the calculated phase image when the phase shift of interferograms was changed from -π to π at intervals of π/100. In the figure, the x-axis is the phase shift of inteferograms and the y-axis is the RMSE. The number of calculation points used was 200. If we cannot calculate the RMSE owing to the calculation errors shown in Fig. 5, we denote these points as RMSE = −1.0. The RMSE of VIGA was approximately equal to 0 except when the phase shift is –π, 0, or π. This suggests that VIGA cannot deal with the Nπ phase shift, when N is −1, 0, or 1. This is because the phase shift of the calculated virtual hologram is π. In contrast, the RMSE of the two-channel algorithm was significantly increased except when the phase shift is π/2. Therefore, by introducing VIGA to the PSDH using two interferograms, the most suitable phase shift can be selected if the phase shift of the interferograms is given.

Fig. 6 RMSE between the original phase image and the calculated phase image when phase shift of interferograms was changed from –π to π Calculation error points were denoted as RMSE = −1.0.

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Figures 7(a) and 7(b) show the original intensity and phase images used in the fourth numerical simulation. The original intensity and phase images are identical to those from the second simulation, shown in Figs. 3(a) and 3(b). The intensity ratio of the signal beam, α, and the intensity ratio of the reference beam, β, were set to 1.0. The phase shift of the inteferograms, γ, was set to π/2. The phase shift of the four-step algorithm was set to 0, π/2, π, and 3π/2, respectively.

Fig. 7 Reconstructed complex amplitude when the grayscale of the image sensor is 8 bit. (A) Original intensity image. (B) Original phase image. (C) Intensity image calculated by four-step algorithm. (D) Phase image calculated by four-step algorithm. (E) Intensity image calculated by VIGA. (F) Phase image calculated by VIGA

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The gray level of the image sensor is assumed to be 8 bit. Figures 7(c) and 7(d) show the intensity and phase images calculated using the four-step algorithm. Both the intensity and the phase images were almost the same as the original images. Figures 7(e) and 7(f) show the intensity and phase images measured using VIGA. Compared with the four-step algorithm, both the intensity and phase images measured using VIGA were almost the same as those of the four-step algorithm. This means that the measurement accuracy of the VIGA was almost the same as that of the four-step algorithm.

Moreover, we compared VIGA with the four-step algorithm using an RMSE between the original phase image and the calculated phase image. Figure 8 shows the RMSE between the original phase image and the calculated phase image when the gray level of the image sensor was changed from 8 bit to 16 bit. In the figure, the x-axis is the gray level, and the y-axis is the RMSE. The number of calculation points was 8. The RMSE of VIGA was slightly increased compared with that of the four-step algorithm. This is because the quantization error of the calculated virtual hologram is greater than that of the ideal hologram. In order to improve the accuracy of VIGA, sufficient gray level must be ensured.

Fig. 8 RMSE between original phase image and calculated phase image when gray level of image sensor is changed from 8 bit to 16 bit.

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4. Experiment

In this section, we describe two experiments to verify the effectiveness of VIGA. Figure 9 shows the first experimental setup for demonstrating the Holographic Diversity Interferometry (HDI) [18,22] based on the proposed algorithm. In this experiment, a reflective liquid crystal on silicon-spatial light modulator (LCOS-SLM: Hamamatsu, X10468-01), which can create a precise phase profile, was used as the measurement object. A beam emitted from a Diode Pumped Solid State (DPSS) laser with a wavelength of 532 nm is divided into a signal path and reference path using a Polarizing Beam Splitter (PBS1) with an extinction ratio of 1/400 after passing through the isolator, Neutral Density filter, beam expander consisting of an objective lens, pinhole, achromatic lens, and Half Wave Plate (HWP1). The signal beam was linearly polarized at 45° using HWP2, which illuminated the measuring object. After passing through the SLM, it is delivered to the two image sensors through the 4-f system constructed from four lenses (Lens2 - Lens5). Here, the Charge Coupled Device (CCD: Allied Vision Technology F-125B, with a bit depth of 10) was used as the image sensor. Moreover, for achieving highly accurate measurements, the magnification of the signal beam was set to 3/8 for matching the pixel size of the SLM, 20.00 × 20.00 μm², and that of the CCD, 3.75 × 3.75 μm². Under this condition, a single pixel of the SLM corresponded to 2 × 2 pixels of the CCD. For example, when an image with 100 × 100 data pixels was displayed on the SLM, the captured image on the CCD is 200 × 200. The reference beam was circularly polarized by a Quarter Wave Plate (QWP), and propagated to two CCDs via the Beam Splitter (BS) and the PBS2. Since circularly polarized light is composed of orthogonal polarization components with a phase difference of π/2, the reference beam is divided into two components with orthogonal polarization by PBS2. Thus, the HDI system can be used to obtain two phase-shifted interference patterns simultaneously between the signal beam and reference beam on two CCD imagers.

Fig. 9 Experimental setup for complex amplitude measurements. A LCOS-SLM (Hamamatsu, X10468-01) was used as the measuring object. The signal beam is delivered to two CCDs using relay lens constructed from Lens2 to Lens5. HWP(Sigma Koki, WPQ-5320-4M). QWP(Sigma Koki, WPQ-5320-4M), BS(Sigma Koki, NPCH-20-5320), SF(Sigma Koki, SFB-16RO-OBL10-25), PBS(Sigma Koki, PBS-20-5320).

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Figure 10 shows the phase image displayed on the SLM and the measured intensity images used for the complex amplitude measurement by HDI. Figure 10(a) shows the original phase image displayed on the SLM that consists of 200 × 200 data pixels and 20 × 20 data symbols. The data symbols in the phase image were randomly assigned to 0 or π. Figures 10(b) and 10(c) show the intensity images of the signal beam captured by CCD1 and CCD2, respectively. The signal intensity was approximately constant expect for the discontinuity regions of the phase image. Figures 10(d) and 10(e) show the intensity images of the reference beam captured by CCD1 and CCD2, respectively. Although the slight intensity distortion was caused by the different sensitivities of each image sensor, almost same intensity images can be observed. This difference can be compensated for calculating the intensity ratio of the signal beam, α, and the intensity ratio of the reference beam, β. Figures 10(f) and 10(g) show the intensity images of interferograms captured by CCD1 and CCD2, respectively. The interference images are divided into two parts: white and black. This suggests that the signal beam is accurately modulated in the two-valued phase signal. Figures 10(h) and 10(i) show the virtual holograms calculated by Figs. 10(b), 10(d), 10(g) and Figs. 10(c), 10(e), 10(g), respectively. Compared to Figs. 10(f) and 10(g), the white part and the black part are reversed in a similar way to the numerical simulation, shown in Fig. 2. This suggests that the π phase shifted two virtual holograms are accurately generated by VIGA.

Fig. 10 Phase image displayed on the SLM and intensity distributions captured by two image sensors in first experiment. (A)Phase image displayed on the SLM (B) Signal intensity captured by CCD1. (C) Signal intensity captured by CCD2. (D) Reference intensity captured by CCD1. (E) Reference intensity captured by CCD2. (F) Interference pattern captured by CCD1. (G) Interference pattern captured by CCD2. (H) virtual hologram calculated by Fig. 10(f). (I) virtual hologram calculated by Fig. 10(g).

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Figures 11(a) and 11(b) show the signal intensity images measured using the two-channel algorithm and VIGA, respectively. Both intensity images show a profile that is approximately identical to that of the original shown in Figs. 10(b) or 10(c). However, the intensity image of Fig. 11(a) has many calculation error points, which are shown as black dots, and contrast to the intensity image of Fig. 11(b). This is because the two-channel algorithm cannot work well in principle when the reference intensity is less than the signal intensity. Figures 11(c) and 11(d) show phase images measured by the two-channel algorithm and VIGA, respectively. Both phase images show a two-dimensional phase image that has approximately an identical profile as that of the original shown in Fig. 10(a) and the same tendency of measured intensity images can be observed. This suggests that VIGA suppressed the calculation error of the two-channel algorithm significantly. In order to clarify the error suppression effect of VIGA, we also counted the number of calculation error points and generated the calculation error maps. Figures 11(e) and 11(f) show calculation error maps of the two-channel algorithm and VIGA, respectively. In Figs. 11(e) and 11(f), error points are represented as white points. In the two-channel algorithm, calculation errors occur when the calculated value of the square root of the algorithm is negative. In contrast, in VIGA, the complex amplitude cannot be calculated when the denominators of Eq. (7) are zero. Thus, we discriminated the error points from the good points by evaluating the inside of the square root and the denominators of Eq. (7). These figures clearly indicate that VIGA can suppress most calculation errors. The number of error points from VIGA was 16, whereas the two-channel algorithm had 31232. This result quantitatively confirms that VIGA can significantly improve the robustness of the PSDH using two interferograms.

Fig. 11 measured complex amplitudes and calculation error maps when the reference intensity was 0.96 μW and the signal intensity was 3.21 μW. (A) intensity image measured by two-channel algorithm. (B) Phase image measured by two-channel algorithm. (C) Intensity image measured by VIGA. (D) Phase image measured by VIGA. (E) Calculation error map of two-channel algorithm (F) Calculation error map of VIGA

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In addition, for evaluating the measurement accuracy of VIGA, we also calculated the RMSE between the phase image displayed on the SLM and the phase image measured by the two-channel algorithm and VIGA. Figure 12 shows the RMSE when the signal intensity was 3.21 μW and the reference intensity was gradually changed from 0.43 μW to 9.86 μW. The yellow vertical line shown in Fig. 10 indicates the signal intensity and the number of measuring points is 8. To the right of the vertical line, the RMSE of the two-channel algorithm was almost the same as that of the proposed algorithm when the reference intensity was greater than the signal intensity. To the left of the vertical line, the RMSE of the two-channel algorithm was much greater than that of the proposed algorithm when the reference intensity was less than the signal intensity. Although the measuring point was small compared to the numerical simulation, this result was in good agreement with the numerical simulation shown in Fig. 4. Therefore, this result confirmed that the highly accurate phase measurement independent to the reference intensity can be achieved by VIGA with PSDH using two interferograms.

Fig. 12 RMSE between phase image displayed on SLM and measured phase image when intensity of reference beam is changed and intensity of signal beam is 3.21 μW.

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Moreover, we estimated the influence of the intensity ratios α and β. Figure 13 shows the calculated intensity ratios and a comparison of the calculated phase image with and without the intensity ratios. Figure 13(a) and 13(b) show the signal intensity ratio α and the reference intensity ratio β, calculated using Figs. 10(b)–10(e). With regard to the signal intensity, except for certain parts of the image, the intensity difference was very small. However, with regard to the reference intensity, the intensity difference captured by the two image sensors was observed. This difference causes a degradation in the measured phase image.

Fig. 13 Measured intensity ratios and influence of intensity ratios. (A) Calculated signal intensity ratio α. (B) Calculated reference intensity ratio β. (C) Calculated phase image with intensity ratios. (D) Calculated phase image without intensity ratios

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Figure 13(c) and 13(d) show the calculated phase image with and without the intensity ratios. Comparing the original image with the calculated phase image without introducing intensity ratios, it can be seen that the calculated phase image was slightly degraded. Moreover, the RMSEs of the calculated phase images with and without the intensity ratios are 0.87 and 0.89, respectively. These results confirmed that the difference between captured intensity images causes degradation of the phase image, and that this difference can be compensated by introducing the intensity ratios α and β.

Figure 14 shows the second experimental setup, which demonstrates the compensation of the phase shift error using the proposed algorithm. It is similar to the first experimental setup, except for two differences. The first is the wavelength of the DPSS laser, which was set to 593 nm to provide for the given phase shift error. In HDI, the phase shift of interferograms is generated by the phase difference of orthogonal components between the circular polarization and the linear polarization. Thus, when the wavelength of light source is different to the supported wavelength of HWP and QWP, the actual phase shift is changed to the ideal phase shift. Here, the supported wavelength of HWP and QWP and the wavelengths of light sources are defined as λ_s and λ, respectively. The actual phase shift δ is derived as

Fig. 14 Experimental setup for compensation of phase shift error. The wavelength of laser was changed from 532nm to 593nm for providing the given phase shift error. HWP(Sigma Koki, WPQ-5320-2M). QWP(Sigma Koki, WPQ-5320-4M), BS(Sigma Koki, HBCH-20-550), SF(Sigma Koki, SFB-16RO-OBL10-25), PBS(Sigma Koki, PBSW-20-550)

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δ = \frac{2 π}{λ} • \frac{λ_{s}}{2} - \frac{2 π}{λ} • \frac{λ_{s}}{4} = \frac{λ_{s}}{λ} • \frac{π}{2}

The second is the intensity of the reference beam. Because for this experiment, the beam was used only to evaluate the phase shift error compensation of VIGA, the reference intensity was much greater than the signal intensity. It should be noted that this condition is the preferable condition of the two-channel algorithm. For this experiment, we compared the complex amplitude image measured using VIGA to that measured with the two-channel algorithm when the phase shift γ is not equal to π/2 and evaluated VIGA’s effectiveness.

Figure 15 shows the phase image displayed on the SLM and the measured intensity images. Figure 15(a) shows the original phase image displayed on the SLM. Although the setup of data pixels and the data symbol is identical to that of the first experiment, the arrangement of data symbols is different. This is because the phase modulated amount of SLM is changed by the wavelength of the light source and the rebuilding of the phase image is needed. Figures 15(b) and 15(c) show the intensity images of the signal beam captured by CCD1 and CCD2. The signal intensity was approximately constant expect for the discontinuity regions of the phase image. Figures 15(d) and 15(e) show the intensity images of the reference beam captured by CCD1 and CCD2. Although the slight intensity fluctuation was caused by the different sensitivity between two image sensors, almost same intensity images can be observed. Figures 15(f) and 15(g) show the intensity images of interferograms captured by CCD1 and CCD2, respectively. The interference images are divided into two parts: white and black. However, the contrast of the image is diminished. This occurred because the reference intensity was much greater than the signal intensity for stabilizing the two-channel algorithm and the visibility of interferograms was degraded. Figures 15(h) and 15(i) show virtual holograms calculated by Figs. 15(b)–15(g), respectively. Compared to Figs. 10(f) and 10(g), the white part and the black part are reversed in a similar manner to the numerical simulation, shown in Figs. 2.

Fig. 15 Phase image displayed on the SLM and intensity distributions captured by two image sensors in first experiment. (A)Phase image displayed on the SLM (B) Signal intensity captured by CCD1. (C) Signal intensity captured by CCD2. (D) Reference intensity captured by CCD1. (E) Reference intensity captured by CCD2. (F) Interference pattern captured by CCD1. (G) Interference pattern captured by CCD2. (H) virtual hologram calculated by Fig. 15(f). (I) virtual hologram calculated by Fig. 15(g).

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Figures 16(a) and 16(b) show the intensity images measured using the conventional two-channel algorithm and VIGA, respectively. Both intensity images show a similar profile as that of the original shown in Figs. 15(b) or 15(c). Figures 16(c) and 16(d) show the phase images measured by the two-channel algorithm and VIGA, respectively. Both phase images show a two-dimensional phase image that has a similar profile to Fig. 15(a). However, the contrast of the phase image calculated by the two-channel algorithm is degraded compared to that of the proposed algorithm. In order to clarify this point, we quantitatively evaluated these phase images in terms of two parameters. The first parameter is the RMSE. In this experiment, the RMSE of the VIGA was 0.90 and that of the two-channel algorithm was 1.07. Thus, the measuring accuracy of VIGA was greater than that of two-channel algorithm. The second parameter involved the profile of the measured phase image. We compared the phase value of the original image with that of the measured image. Figure 17 shows the phase values along the yellow horizontal line in Fig. 16. Both algorithms can definitely measure the two phase value. However, both slightly shifted from the original phase values. This occurred because PSDH measured the relative phase value, and not the absolute, so that the calculated phase value can be relatively shifted in each measurement. In addition, the phase contrast of two-channel algorithm was deteriorated compared to that of VIGA, shown in Fig. 16. Two reasons can be considered. The first is the phase shift of interferograms. The phase shift of two-channel algorithm was assumed to be equal to π/2, in spite of the actual phase shift of inteferograms not being equal to π/2. Therefore, the calculation error due to the phase shift error occurred. The second is the degradation of the visibility of interferogram. The contrast of the captured interferograms was small, as shown in Fig. 15, and the dynamic range of image sensor was significantly consumed. This led to the deterioration of the measuring accuracy. In terms of these facts, the calculation error caused by the consumption of visibility and the phase shift error of interferograms can be compensated by VIGA.

Fig. 16 measured complex amplitudes when the phase shift of interferograms is not π/2. (A) intensity image measured by two-channel algorithm. (B) Intensity image measured by VIGA. (C) Phase image measured by two-channel algorithm. (D) Phase image measured by VIGA.

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Fig. 17 Phase values along the horizontal yellow line of Fig. 16. The phase contrast of VIGA was greater than that of two-channel algorithm.

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To summarize, capturing two interferograms with an arbitrary phase shift and calculating two virtual holograms in the computer, the calculation error of two-channel algorithm caused by the shortage of the reference intensity and the phase shift error can be drastically suppressed. Although the acquisition time of VIGA is inferior to that of the two-channel algorithm, the measurement accuracy and robustness of the PSDH using two interferograms were greatly enhanced by the proposed VIGA.

5. Conclusion

VIGA was developed to compute the complex amplitude of a signal beam by capturing two-phase shifted interferograms in the optical system and generating two phase-shifted holograms in the computer. Simulation results confirmed that the VIGA is independent of the intensity of the reference beam, unlike the two-channel algorithm, and can choose the most suitable phase shift, except for the Nπ phase shift. Experimental results verified that the HDI with VIGA can measure the complex amplitude of signal beam without calculation errors when the intensity of reference beam is less than that of the signal beam and can compensate the given phase shift error caused by the difference of the supported wavelength between the light source and the polarization elements, HWP and QWP. We believe that the HDI with VIGA will contribute to robust and accurate dynamic measurement of 3D objects, such as a biological specimen, fluid physics, and damage measurement.

Funding

Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number 25289110.

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Virtual interferogram-generation algorithm for robust complex amplitude measurement using two interferograms

Abstract

1. Introduction

2. Virtual interferogram-generation algorithm

3. Simulation

4. Experiment

5. Conclusion

Funding

References and links

Cited By

Figures (17)

Equations (9)

Optics Express