1 Introduction

In physical optics, when a light beam from a monochromatic source is divided into two or more beams to be superposed again at any point in space, the intensity in the superposition area varies from maxima to minima, which is known as interference patterns or interferogram. The intensity of the interferogram consists of a cosine of the phase difference between the two waves (two-beam interference) or more (multiple-beam interference) and a factor given mainly by the product of waves’ amplitudes known as a light modulation. Many scientists, including Thomas Young, Newton, Fizeau, Michelson, and others designed many optical arrangements, known as interferometers, in order to generate wave light interference to measure, with high accuracy, the changes in the optical path difference or deformation in the wavefront by measuring the phase differences (see, e.g., [1]). Generally, in two-beam interferometer, one beam, called a reference beam, has often flat wavefront and the other beam, which is called a probe beam, has deformed wavefront due to the tested sample.

Optical interferometry is an effective technique for precision measurements such as surface topography, displacement, temperature, pressure, and gas flows measurement, visualization, and analysis of structural vibrations, and structural parameters and optical properties of transparent materials. The interferometer provides us with many quantitative and qualitative information encoded in the form of two-dimensional interference patterns named interferograms. In the past, this interferogram must be photographed and enlarged to a suitable magnification, and then the required information is obtained from the magnified photograph. With aid of computer and image capture devices, the interferogram analysis has become faster, easier, and more accurate in obtaining the desired information [2,3,4]. There are many techniques used to analyze the interferograms automatically. In general, these techniques can be classified into two basic categories, the first of which depends on the determination of the contour line of the interference fringe [5,6,7,8] and the second which depends on the calculation of phase distribution in the interferogram [9,10,11,12,13,14,15,16,17,18,19,20]. The phase obtained techniques can be divided into two basic categories: electronic and analytical techniques [see, e.g., 21, 22, and the references therein]. Electronic techniques need hardware such as zero-crossing detectors, phase-lock loops, and up–down counters to monitor interferogram intensity data as the phase is modulated. In analytical techniques, while the phase temporally modulates, the intensity of fringes has been recorded and analyzed with computer aid to obtain the phase distribution [23]. These techniques depend on inducing a known or unknown phase shift to one arm of the interferometer with respect to the other arm. There are many methods for phase modulation process, such as moving grating or mirror using a piezoelectric transducer (PZT), tilting a glass plate, rotating an analyzer or a half-wave plate [see, e.g., 24 and the references therein]. It is worth to be mentioned that these techniques require capturing of three or more interferograms of the tested object with inducing a known or unknown phase step for extracting phase distribution that is related to object deformation.

As well known, a majority of phase measurement methods return results of the wrapped phase limited in the range (− π, π). Actually, the phase ambiguities are due to the modulo 2π calculation caused by the fundamental properties of the mathematical arctangent function. Therefore, unwrapping methods are necessary to restore the real values of desired continuous phase distributions. In other words, the phase unwrapping techniques are carried out to detect 2π discontinuities based on their neighboring phase values and remove them by adding or subtracting multiples of 2π from individual pixels until the phase difference between the adjacent pixels is less than π. In recent years, a lot of phase unwrapping algorithms have been proposed, which differ in the solution method, application area, and robustness [25,26,27,28,29,30]. These algorithms can classify into three main groups: path-following algorithms [31,32,33,34], minimum norm methods [35, 36], and optimization estimation algorithms [37, 38]. Because of the phase unwrapping process is usually very time-consuming as it involves a lot of computations as well as iterations, and numerous authors have proposed several algorithms for extracting phase distributions without phase unwrapping step [39,40,41,42,43,44,45,46,47,48,49,50]. The expansion and development of these algorithms have led to the development of many optical microscopy techniques, such as quantitative phase imaging [51, 52] and digital holographic microscopy [53, 54].

Although there are many algorithms to calculate the phase distribution, there is, in fact, still a need to develop new sophisticated methods for calculating the phase distribution. In this paper, a method to obtain directly the phase distribution from one shot of a two-dimensional fringe pattern interferogram is proposed. The method overcomes the difficulties that face our previous algorithm [55]. The proposed method does not need the phase unwrapping process, and there is no need to induce phase to the interferogram. Besides, it reduces the number of captures needed for interferograms to extract the phase distributions, and only one interferogram needed to deduce the real values of desired continuous phase distributions.

1.1 Theoretical considerations

Phase-shifting interferometry is one of the stages of the development of the interferometry, which came as a result of the development in computers and image capture machines [56,57,58]. The intensity of the one-dimensional interference fringe pattern can be expressed as

$$I_{K} \left( x \right) = I_{\text{o}} \left( x \right)\left[ {1 + \gamma \left( x \right)\cos \left( {2\pi f_{\text{o}} x + \varphi \left( x \right) + K\delta } \right)} \right],$$
(1)

where Io(x) is the dc irradiance or background intensity, γ(x) is the fringe contrast or fringe visibility, fo is the frequency carriers of the interference fringe, φ(x) is the wanted phase that is deformed or modulated by a tested object, K is the number of frames that needing to calculate the phase distribution (K = 0,1,2,3…), and δ is a known induced phase, traditionally employs phase steps that are typically multiples of 0.5π. For simplicity, consider that the background Io(x) of the interference fringe is constant in all frames. In most cases, Io(x) and γ(x) vary very slowly compared to fo, in addition, they appear in both the numerator and denominator of phase calculation equation, so we can say that the obtained phase φ(x) is not affected by the variations in Io(x) and γ(x) from pixel to pixel.

To determine φ(x) using Eq. (1), it is necessary to take at least three interferograms with different values of δ. The form of the phase equation depends on the number of capture interferograms (K) and the induced phase δ. For instance, in the case of three interferograms and if the induced phases δ = 0°, 60°, and 120°, the phase φ(x) is given by

$$\varphi (x) = \tan^{ - 1} \left[ {\frac{{\left( {2I_{1} - 3I_{2} + I_{3} } \right)}}{{\sqrt 3 \left( {I_{2} - I_{3} } \right)}}} \right]$$
(2)

While if the induced phases δ = 0°, 90°, and 180°, the phase φ(x) is given by

$$\varphi (x) = \tan^{ - 1} \left[ {\frac{{\left( {I_{1} - 2I_{2} + I_{3} } \right)}}{{\left( {I_{1} - I_{3} } \right)}}} \right]$$
(3)

In the case of four interferograms and induced phase δ = 0°, 90°, 180°, and 270°, the phase φ(x) is given by

$$\varphi (x) = \tan^{ - 1} \left[ {\frac{{\left( {I_{4} - I_{2} } \right)}}{{\left( {I_{1} - I_{3} } \right)}}} \right]$$
(4)

And in the case of five interferograms and induced phase δ = 0°, 90°, 180°, 270°, and 360°, the phase φ(x) is given by

$$\varphi (x) = \tan^{ - 1} \left[ {\frac{{7\left( {I_{4} - I_{2} } \right)}}{{\left( {4I_{1} - I_{2} - 6I_{3} - I_{4} + 4I_{5} } \right)}}} \right]$$
(5)

From Eqs. 2–5, it is clear that the obtained phase φ(x) is wrapped and limited within the interval (− π, π) due to the fundamental properties of the mathematical arctangent function. The unwrapping phase \(\varphi \,(x)\,\) can be determined from the equation

$$\phi (x) = \varphi (x) + n(2\pi )$$
(6)

where n is an integer number referring to the fringe orders.

What worries researchers most are the sources of error that affect the accuracy of calculations. The main sources of error in the phase shift interferometry are the noise, the miscalibration in PZT, the detector nonlinearity, and the random error due to air turbulence or mechanical vibration. These sources of error make us skeptical in the exact constant of phase shift induced to interferograms (at least three interferograms), which need to extract the phase distribution. Here appears the importance of algorithms that can calculate the phase distribution using a one-shot interferogram [59, 60]. The sub-fringe integration method was proposed to avoid these sources of error [55, 61]. The sub-fringe integration method has a major advantage such as the ability of whole-field measurement and calculation speed because it requires one shot interferogram, and it does not need processing of transform, filtering, and inverse transform as in Fourier transform method. Even in case of poor contrast of interferograms, it provides good results, and furthermore, it has a high resistance to noise and error caused by the errors in carrier frequency, but unfortunately, it still needs unwrapping steps. The aspects of the method and the robustness in the face of the noises were investigated in detail by El-Morsy et al. [55]. Figure 1 illustrates the basic idea of the sub-fringe integration algorithm which can be summarized as follows: the period of the sinusoidal signal is divided into at least three or more buckets and then each bucket is integrated. In the following part of the paper, we will present the theoretical consideration to develop this method to be able to calculate the phase distribution without unwrapping process. Consider the intensity distribution of the one-dimensional interference fringe is given by

$$I(x) = a(x)\left[ {1 + b(x)\text{Cos} \left( {\frac{2\pi x}{T} + \varphi (x)} \right)} \right]$$
(7)

where a(x) and b(x) are the background illumination and the amplitude of the intensity distribution of the interference fringe, respectively, T is the fringe period, φ(x) is the phase of the object that we have to analyze at any point (x) in the interferogram. According to the sub-fringe method in case of four buckets, the four-bucket intensity distribution over the finite space is given by

Fig. 1
figure 1

Illustration of the sub-fringe method one complete period divided into: a exactly four bucket, b four bucket with separation ΔT, c four bucket with overlapping ΔT, I1, I2, I3, and I4 are the four intensities integration bucket over a periodic space T

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$$\begin{aligned} & I(x) = \int\limits_{0}^{{\frac{T}{4}}} {I_{1} (x){\text{d}}x} + \int\limits_{{\frac{T}{4}}}^{{\frac{T}{2}}} {I_{2} (x){\text{d}}x} + \int\limits_{{\frac{T}{2}}}^{{\frac{3\,T}{4}}} {I_{3} (x){\text{d}}x} + \int\limits_{{\frac{3\,T}{4}}}^{T} {I_{2} (x){\text{d}}x} \\ & I{}_{1}(x) = A(x) + B(x)\left[ {\text{Cos} \varphi (x) - \text{Sin} \varphi (x)} \right] \\ & I{}_{2}(x) = A(x) + B(x)\left[ { - \text{Cos} \varphi (x) - \text{Sin} \varphi (x)} \right] \\ & I{}_{3}(x) = A(x) + B(x)\left[ { - \text{Cos} \varphi (x) + \text{Sin} \varphi (x)} \right] \\ & I{}_{4}(x) = A(x) + B(x)\left[ {\text{Cos} \varphi (x) + \text{Sin} \varphi (x)} \right] \\ \end{aligned}$$
(8)

where

$$A(x) = \frac{T}{4}a(x),\quad B(x) = \frac{T}{2\pi }b(x)$$

Suppose that

$$I_{B} (x) = \left( {I_{1} (x) + I_{2} (x)} \right) - \left( {I_{3} (x) + I_{4} (x)} \right) = - 4B\text{Sin} \varphi (x)$$
(9)
$$I_{M} (x) = 2\left( {I_{2} (x) - I_{1} (x)} \right) = - 4B\text{Cos} \varphi (x)$$
(10)

The derivative of Eqs. 9 and 10 with respect to x is:

$$I_{B}^{\backslash } (x) = \frac{{\partial I_{B} (x)}}{\partial (x)} = - 4B(x)\varphi^{\backslash } (x)\text{Cos} \varphi (x) - 4B^{\backslash } (x)\text{Sin} \varphi (x)$$
(11)
$$I_{M}^{\backslash } (x) = \frac{{\partial I_{M} (x)}}{\partial (x)} = 4B(x)\varphi^{\backslash } (x)\text{Sin} \varphi (x) - 4B^{\backslash } (x)\text{Cos} \varphi (x)$$
(12)

From Eq. 11

$$\begin{aligned} & \varphi^{\backslash } (x) = - \frac{{I_{B}^{\backslash } (x) + 4B^{\backslash } (x)\text{Sin} \varphi (x)}}{{4B(x)\text{Cos} \varphi (x)}} \\ & \varphi^{\backslash } (x) = - \frac{{I_{B}^{\backslash } (x)}}{{4B(x)\text{Cos} \varphi (x)}} - \frac{{B^{\backslash } (x)\text{Sin} \varphi (x)}}{{B(x)\text{Cos} \varphi (x)}} \\ \end{aligned}$$
(13)

From Eq. 12

$$\begin{aligned} B^{\backslash } (x) & = - \frac{{I_{M}^{\backslash } (x)}}{{4\text{Cos} \varphi (x)}} + \frac{{4B(x)\varphi^{\backslash } (x)\text{Sin} \varphi (x)}}{{4\text{Cos} \varphi (x)}} \\ & = - \frac{{I_{M}^{\backslash } (x)}}{{4\text{Cos} \varphi (x)}} + \frac{{B(x)\varphi^{\backslash } (x)\text{Sin} \varphi (x)}}{{\text{Cos} \varphi (x)}} \\ \end{aligned}$$
(14)

From Eqs. 13 and 14, we will get

$$\begin{aligned} & \varphi^{\backslash } (x) = - \frac{{I_{B}^{\backslash } (x)}}{{4B(x)\text{Cos} \varphi (x)}} - \frac{{\text{Sin} \varphi (x)}}{{B(x)\text{Cos} \varphi (x)}}\left[ { - \frac{{I_{M}^{\backslash } (x)}}{{4\text{Cos} \varphi (x)}} + \frac{{4B(x)\varphi^{\backslash } (x)\text{Sin} \varphi (x)}}{{4\text{Cos} \varphi (x)}}} \right] \\ & \quad = - \frac{{I_{B}^{\backslash } (x)}}{{4B(x)\text{Cos} \varphi (x)}} + \frac{{I_{M}^{\backslash } (x)B(x)\text{Sin} \varphi (x)}}{{4(B(x)\text{Cos} \varphi (x))^{2} }} - \frac{{4\varphi^{\backslash } (x)(B(x)\text{Sin} \varphi (x))^{2} }}{{4(B(x)\text{Cos} \varphi (x))^{2} }} \\ & \varphi^{\backslash } (x) + \frac{{4\varphi^{\backslash } (x)(B(x)\text{Sin} \varphi (x))^{2} }}{{4\,\,(B(x)\,\,\,\text{Cos} \,\phi \,(x))^{2} }} = - \frac{{I_{B}^{\backslash } (x)}}{{4B(x)\text{Cos} \varphi (x)}} + \frac{{I_{M}^{\backslash } (x)B(x)\text{Sin} \varphi (x)}}{{4(B(x)\text{Cos} \varphi (x))^{2} }} \\ & \varphi^{\backslash } (x)\left[ {\frac{{4(B(x)\text{Cos} \varphi (x))^{2} + 4(B(x)\text{Sin} \varphi (x))^{2} }}{{4(B(x)\text{Cos} \varphi (x))^{2} }}} \right] = - \frac{{I_{B}^{\backslash } (x)}}{{4B(x)\text{Cos} \varphi (x)}} + \frac{{I_{M}^{\backslash } (x)B(x)\text{Sin} \varphi (x)}}{{4(B(x)\text{Cos} \varphi (x))^{2} }} \\ & \varphi^{\backslash } (x) = \frac{{ - \frac{{4I_{B}^{\backslash } (x)(B(x)\text{Cos} \varphi (x))^{2} }}{{4B(x)\text{Cos} \varphi (x)}} + \frac{{4I_{M}^{\backslash } (x)B(x)(B(x)\text{Cos} \varphi (x))^{2} \text{Sin} \varphi (x)}}{{4(B(x)\text{Cos} \varphi (x))^{2} }}}}{{4(B(x)\text{Cos} \varphi (x))^{2} + 4(B(x)\text{Sin} \varphi (x))^{2} }} \\ & \therefore \varphi^{\backslash } (x) = \frac{{ - I_{B}^{\backslash } (x)(B(x)\text{Cos} \varphi (x)) + I_{M}^{\backslash } (x)B(x)\text{Sin} \varphi (x)}}{{4(B(x)\text{Cos} \varphi (x))^{2} + 4(B(x)\text{Sin} \varphi (x))^{2} }} \\ \end{aligned}$$
(15)

From Eqs. 9, 10 and 15, we will get

$$\begin{aligned} & \varphi^{\backslash } (x) = \frac{{ - I_{B}^{\backslash } (x)\left( { - \frac{{I_{M} (x)}}{4}} \right) + I_{M}^{\backslash } (x)\left( { - \frac{{I_{B} (x)}}{4}} \right)}}{{4\left( { - \frac{{I_{M} (x)}}{4}} \right)^{2} + 4\left( { - \frac{{I_{B} (x)}}{4}} \right)^{2} }} \\ & \varphi^{\backslash } \,(x) = \frac{{I_{B}^{\backslash } (x)I_{M} (x) - I_{M}^{\backslash } (x)I_{B} (x)}}{{I_{M}^{2} (x) + I_{B}^{2} (x)}} \\ \end{aligned}$$
(16)

The variables in the right-hand side of Eq. 16 can be obtained numerically from the microinterferogram. The phase distribution can be obtained by performing the mathematical technique of line integration of Eq. 16, as follow

$$\begin{aligned} & \varphi_{i} (x_{i} ) = \int\limits_{{x_{0} }}^{{x_{i} }} {\frac{{I_{B}^{\backslash } (x)I_{M} (x) - I_{M}^{\backslash } (x)I_{B} (x)}}{{\left( {I_{M} (x)} \right)^{2} + \left( {I_{B} (x)} \right)^{2} }}{\text{d}}} x \\ & \therefore \varphi_{i} (x_{i} ) = \varphi_{0} (0) + \sum\limits_{i\, = 1}^{i} {\varphi_{i} (x_{i} )} \\ \end{aligned}$$
(17)

After presenting the theoretical considerations and deriving the basic equations, we can summarize the main steps required to implement the phase unwrapping algorithm as follows:

  • Step 1 Read a raw fringe data I(x) from the interferogram and convert it into a one-dimensional array

  • Step 2 Determine the period of the sinusoidal signal (T) and divide it into four buckets, bucket1 from 0 to T/4, bucket2 from T/4 to T/2, bucket3 from T/2 to 3T/4, and bucket4 from 3T/4 to T.

  • Step 3 Integrate each bucket, using the line integral technique, in order to obtain the intensities I1(x), I2(x), I3(x), and I4(x), as shown in Fig. 1.

  • Step 4 Calculate the variables IB(x) and IM(x) using Eqs. 9 & 10, respectively.

  • Step 5 Calculate the derivatives of the variables \(I_{B}^{\backslash } \,(x)\,\) and \(\,I_{M}^{\backslash } \,(x)\) using the following equation

    $$\begin{aligned} & I_{B}^{\backslash } (x) = I_{B} (x + 1) - I_{B} (x) \\ & I_{M}^{\backslash } (x) = I_{M} (x + 1) - I_{M} (x) \\ \end{aligned}$$
    (18)

  • Step 6 Calculate the phase derivatives φ\(x) using Eq. 16.

  • Step 7 Calculate the unwrapping phase distribution using Eq. 17 which based on the mathematical technique of line integration of Eq. 16

  • Step 8 Repeat steps 3–7 for each pixel in the interferogram.

All previous steps are illustrated in the flowchart shown in Fig. 2.

Fig. 2
figure 2

The flowchart of the principal steps of the proposed method for direct unwrapping phase reconstruction from interference interferogram, I1, I2, I3, I4, IM, IB, \(I^{\backslash }_{M}\), \(I^{\backslash }_{B}\), φ\ and φ were explained in the main text

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2 Simulation and experimental results

In the remaining part of the article, we will go for testing the ability of our method in direct phase retrieval from the intensities of only one interferogram, without the need of the unwrapping process. The test will include several simulations interferograms (400 × 600 pixels) and the pixel size is 0.625 µm. Also, for more complete verification, the proposed method will be tested by experimental interferograms. Equation (7) used to produce simulated interferograms with different phase modulations. The phase modulation is given by:

$$\varphi (x) = \sum\limits_{i = 0}^{i = 10} {A_{i} x^{i} } ,\quad 0 \le x \le r$$
(19)

where r is the fiber radius and Ai are the polynomial coefficients, and its values are shown in Table 1.

Table 1 The polynomial coefficients and the basic information used to simulate the interferograms
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First, the proposed algorithm is verified by simulated interferograms. Figure 3a shows the simulated interferometric pattern of the two-beam interference technique (for further details on the optical fiber investigation using two-beam interference techniques, see Ch. 3, pp. 33–53 in Ref. [62] and the references therein). The interferogram includes three parts, as shown in Fig. 3a. Part A represents the interference region of liquid, while parts B and C represent the interference regions of the clad and core of the optical fiber, respectively. Part A is a reference for parts B and C. In other words, the shift inside parts B and C is measured with respect to part A. As we can see, the shift inside region of liquid (part A) is zero, and the fringes are appearing quite straight, so we can consider that the phase within that region is zero (φo(0) = 0 in Eq. 17). The diameter of the optical fiber is 125 µm and the optical fiber (parts B and C) acts as phase modulation. The difference in the shift inside the fiber is due to the difference in the optical path difference, which is due to the difference in refractive indices of the fiber (clad (ncl) and core (nco)) and the liquid (nL).

Fig. 3
figure 3

a Simulated two-beam interference fringe showing the phase modulation due to optical fiber immersion in a liquid (A, B, and C refer to the fringe shift in the immersion liquid, clad fiber, and core fiber, respectively), b the recovered phase distribution maps with normalized background, c the two-dimensional (2D) phase map

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Figure 3a shows the simulated interferogram of optical fiber in the case of the refractive index of liquid is far from the refractive index of fiber (clad and core). In other words, the refractive index of liquid (nL) less than the refractive indices of fiber clad and fiber core, i.e., nL < ncl < nco. In this case, the shift inside the fiber (clad and core) is positive. The fringe shift inside the clad is around one order of the interfringe spacing (the period of the sinusoidal signal) with respect to the liquid, and the fringe shift inside the core is around two orders with respect to fiber clad or three orders with respect to the liquid. Figure 3b represents the successfully recovered phase distribution with a normalized background for interferogram shown in Fig. 3a. The estimated two-dimensional (2D) phase map is shown in Fig. 3c.

As it is known that, the greater the difference between the refractive indices of fiber and liquid, the greater the fringe shift order. Also, as long as the fringe shift is less than one interference order, there is no need for an unwrapping process. If the shift exceeds one interference order, the unwrapping process is needed. The greater the fringe shift, the more complex the unwrapping process becomes. So, we are going to test our method with interferograms where the fringe shift is large (four orders or more).

Figure 4a is the same as Fig. 3a, but the fringe shift inside the clad is two orders and inside the core is five orders with respect to the liquid. Our method succeeded in extracting the wanted phase without needing the unwrapped process, as shown in Fig. 4b and c. Figure 5 represents the phase distribution for one line of the interferograms shown in Figs. 3a and 4a).

Fig. 4
figure 4

Simulated interferograms similar Fig. 3 but with high fringe shift order (five orders)

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Fig. 5
figure 5

The 1D recovered phase distribution for one line of the interferograms shown in Figs. 3 and 4

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Figures 6 and 7 are similar to Fig. 3, but in the case of liquid refractive index in between the refractive index of fiber clad and fiber core, i.e., ncl < nL < nco. Figures 6b and 7b represent the reconstructed phase distribution with a normalized background for interferogram shown in Figs. 6 and 7, respectively. The estimated two-dimensional (2D) phase map is shown in Figs. 6c and 7c, while Fig. 8 represents the phase distribution for one line of the interferograms shown in Figs. 6 and 7.

Fig. 6
figure 6

a  Simulated interferograms similar Fig. 3 but with negative shift in fiber clad, b the recovered phase distribution maps with normalized background, c the two-dimensional (2D) phase map

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Fig. 7
figure 7

Simulated interferograms similar Fig. 6 but with different fringe shift order in fiber clad and core

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Fig. 8
figure 8

The 1D recovered phase distribution for one line of the interferograms shown in Figs. 6 and 7

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We added another challenge to our method by testing its ability to analyze an interferogram containing positive and negative shifts (for further details on the Pluta’s interference polarizing microscope, see the reference [62, Ch. 7, pp. 120–123]). As is in the case of the fiber which has a double refractive indices, one of the refractive indices in the parallel direction (left shift), and the other in the perpendicular direction (right shift), as shown in Fig. 9a. The results confirmed the ability of the proposed method to estimate the phase in this kind of interferograms, as shown in Fig. 9b and c and Fig. 10.

Fig. 9
figure 9

a Simulated interferogram with phase modulation in positive and negative direction, b the recovered phase distribution maps with normalized background, c the 2D recovered phase distribution maps

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Fig. 10
figure 10

The 1D recovered phase distribution for one line of the interferogram shown in Fig. 9a

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For further investigation, experimental microinterferograms are carried out to confirm the performance of our method. The experimental microinterferograms have other challenges to the proposed method. These challenges are the noise in the microinterferograms, the irregularity of the radius of the fiber, and the irregularity of the fringe shift along the fiber. Figures 11a and 12a represent an experimental interference fringe of polypropylene (PP) fibers during the drawing process, using monochromatic light of wavelength 546 nm and immersing liquid of refractive index 1.495 [63]. From these figures, at first glance, it is easy to notice that the fringes have different displacements along the fiber, and we will notice that the radius of the fiber changes over the interferograms due to the necking phenomena which produced during to the drawing process. Also, our method succeeded in recovering the phase distribution, and the results are shown in Figs. 11 and 12.

Fig. 11
figure 11

a Experimental interference fringe of polypropylene (PP) fiber during drawing process using the non-duplicated position of Pluta microscope, monochromatic light of wavelength 546 nm, and immersing liquid of refractive index 1.495, b the recovered phase distribution maps with normalized background, c the 2D recovered phase distribution maps

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Fig. 12
figure 12

Experimental interferograms similar Fig. 11 but with different necking position along the fiber

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A final and important point is the degree of precision between the values of the original phase and the retrieved phase distributions. Figure 13a shows the comparison between the original phase modulation of the fringes in the interferogram of Fig. 3a and the retrieved phase which calculated by our new method. Figure 13b represents the difference between the original phase and the calculated phase along the fiber. From Fig. 13, it is clear that there is a good agreement in the results and the maximum difference between the two phases is 0.0305 and the standard deviation is about 5.7 × 10−3.

Fig. 13
figure 13

Comparison between the original phase modulation of the fringes in the interferogram of Fig. 5a and the retrieved phase which calculated by our new method

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From the previous results, we can be assured of the ability of the proposed method to extract the unwrapping phase distribution directly using only one interferogram and without needing to unwrapping process. This method requires neither complex programming nor advanced image processing steps (such as image enhancement or image filtering), as well as its ability to resist the noise.

3 Conclusion

Sure, there is no one algorithm capable of analyzing all kinds of interferograms. Some of the methods presented in the studies may be prepared to analyze a specific kind of interferogram. We have proposed a novel algorithm of direct unwrapping phase reconstruction to improve our previous sub-fringe integration method [55]. The algorithm is suitable to analyze the parallel fringes, like two-beam interference fringe and equispaced Fizeau fringes. The new algorithm is based on the mathematical technique of line integration. Besides, it gives a solution to the phase ambiguity problem for the sub-fringe method. The new algorithm adds strength to the power of the sub-fringe method. Now the sub-fringe method does not need unwrapping process and does not need hardware to introduce phase shift, and also does not need more than one interferogram only. Furthermore, it is fast and easy to implement, and insensitive to the inhomogeneous of the intensity distribution of the interferogram. The results obtained, whether from experimental or simulated interferograms, demonstrated the usefulness of the new method in analyzing two-beam interference fringes and demonstrated the flexibility of the method in dealing with the different situations of fringe shifts resulting from the immersion the fiber in the liquid, whether the refractive index of liquid is close to or far from the refractive index of the fiber. The new method also succeeded to estimate the phase for interference fringes where the fringe shifts are variable along the fiber and with the irregularity of the fiber radius across the interferogram. The experimental interferometric fringes used as the input are the interferogram type recorded during investigating the textile fiber using the two-beam Pluta interference technique. Finally, the two-dimensional reconstructed phase can be used to investigate the optical properties of the fiber in two dimensions, not only along the fiber radius but also along the fiber length.