## Abstract

The paper presents a novel algorithm based on digital holographic interferometry and being promising for evaluation of phase variations from highly noisy or modulated by speckle-structures digital holograms. The suggested algorithm simulates an interferogram in finite width fringes, by analogy with classical double exposure holographic interferometry. Thus obtained interferogram is then processed as a digital hologram. The advantages of the suggested approach are demonstrated in numerical experiments on calculations of differences in phase distributions of wave fronts modulated by speckle structure, as well as in a physical experiment on the analysis of laser-induced heating dynamics of an aqueous solution of a photosensitizer. It is shown that owing to the inherent capability of the approach to perform adjustable smoothing of compared wave fronts, the resulting difference undergoes noise filtering. This capability of adjustable smoothing may be used to minimize losses in spatial resolution. Since the method allows to vary an observation angle of compared wave fields, an opportunity to compensate misalignment of optical axes of these wave fronts arises. This feature can be required, for example, when using two different setups in comparative digital holography or for compensation of recording system displacements during a set of exposures in studies of dynamic processes.

© 2014 Optical Society of America

## 1. Introduction

Due to the rapid progress in computer technologies and development of digital cameras digital methods of hologram processing gain avalanche attention becoming in many aspects a worthy alternative to traditional holographic techniques. One of the important directions in the development of digital holographic technologies is digital holographic interferometry. Its applications are widespread and cover such fields as nondestructive testing: deformation analysis [1], comparison of topological geometries of manufactured industrial components with standards [2], vibrometry [3]; medicine: observation of biological processes in organelles and live cell cultures [4, 5], to name only a few.

In most of the above mentioned applications dynamic changes of an object under study are analyzed from the phase retardation of the wave front gained over time. The straightforward processing procedure for digital holograms is as follows: First the reconstruction of amplitude and phase distributions is performed for two object waves which differ in a sought quantity depending upon the introduced disturbance, deformation or shape difference. Then the recorded phase distributions undergo the phase unwrapping procedure in the vicinity of −*π* and *π* magnitudes [6]. The phase difference is calculated from the absolute phase distributions. Note that this approach [4, 7] gives results depending heavily on the success of phase unwrapping procedure. If one of spatial distributions is unwrapped, the phase difference can be calculated without phase unwrapping on the other distribution. However such an opportunity is not always available. If the intereference fringe contrast is low, intense noise is present, or the wave front shape is highly non-uniform, as e.g. when studying rough surfaces, phase unwrapping becomes challenging. To bypass this problem in phase-stepping speckle interferometry various filtration procedures [8,9] are applied first. That provides speckle noise suppression and removal of background illumination. However it should be taken into account that these transformations reduce resolution of the resulting phase distribution and may be applied only if the modulation frequency of the disturbance is much lower than that of the speckle structure. In vibrometry it was suggested to increase the frequency of hologram recording so that the disturbance value would not exceed 2*π* [3]. In this case it becomes possible to calculate the phase difference omitting the unwrapping procedure.

Several alternative approaches also have been developed for calculation of phase differences. One of them is based on the technique of classical double-exposure holographic interferometry where an interferogram is formed physically by two optical wave fronts, may be recorded by a digital camera and reconstructed using the algorithm based on fringe shift measurements [10]. In this case the absolute phase difference is obtained directly from the interferogram omitting the phase unwrapping procedure. In classical holographic interferometry two holograms of the two states of the object under study are registered on the same place of the recording material. Then the material undergoes processing, most commonly chemical. Two interfering wave fronts are formed at optical reconstruction of the recorded holograms. Spatial light modulators may be used as an alternative to conventional holographic materials [2, 11]. Various methods for calculation of phase difference were also suggested for in-line phase shifting interferometer, e.g. method of the single phase step [12]. However they require recording of additional holograms. Finally, having an information on spatial distributions of the wave field in two states, one may divide them by each other [13], or, which is equivalent, express them through the real and imaginary parts and then apply a tangent of the difference formula to directly calculate the phase difference [5, 14, 15]. One can also perform subtraction of unwrapped phase differences, but in this case a comparison of their magnitudes and adding of 2*π* according to the result of comparison are required [16, 17]. These approaches apparently are the simplest, but in cases where the original phase distributions have a deviation from each other as a result of vibration or noise presence the final phase difference distributions still need to be filtered [18–20], in order to eliminate singular points in the spatial phase distribution that prevent its unwrapping. Thus, none of the above methods in general can provide the possibility to obtain a difference of two arbitrary wave fields that are ready for further processing.

In this paper we suggest an alternative approach for holographic interferometry of arbitrary wave fields. The algorithm relies on the basic technology of interferogram formation and reconstruction but shows several meaningful distinctions from other methods:

- First, an interferogram is obtained numerically from spatial distributions of amplitude and phase reconstructed from digital holograms. Both the holograms reconstruction and interferogram analysis are performed via the complex wave retrieval method proposed by Liebling et al. [21] in the recording plane. This algorithm provides high-quality wave fields reconstruction from off-axis holograms even at very small angles between the object and reference waves, in contrast to commonly used algorithm based on the Fourier transformation [1, 5, 7, 11, 15, 22] limited by overlapping of the diffraction orders. Our reconstruction algorithm also provides advantages over the well-known phase-step technique [8, 9, 12, 23], which demands for precise monitoring of phase retardation, a crucial parameter affecting the accuracy of the reconstructed phase distribution.
- Second, for calculations of a digital interferogram from two object waves a new hologram is simulated first. It is based on one of these object waves but with a reference wave phase being inclined at an angle somewhat bigger than that used at the recording. That allows us to turn from infinite to finite width fringes at the calculated interferogram; moreover, these finite width fringes provide an opportunity to consider the interferogram as an off-axis hologram and to reconstruct it by means of the algorithm applied to reconstruction of optically recorded holograms.

Therefore the suggested approach provides the following important advantages. The difference of any phase distributions may be obtained regardless of existing phase wraps and singularities. The technique facilitates processing of phase retardation of any value (including those exceeding 2*π*), it allows to avoid usage of spatial-temporal light modulators and holographic materials requiring chemical processing. Moreover, as it will be shown below, the algorithm removes singularities in the obtained phase distribution, which may appear due to a low signal-to-noise ratio or optical system misalignment between the exposures. That all considerably simplifies the processing procedure. The construction of the experimental setup is thus simplified and the data processing procedure is automatically controlled.

Our paper is organized as follows. Section 2 describes the two approaches for calculation of phase differences: the methods of direct wavefronts dividing and the proposed method of simulated digital interferogram. The analogy of the latter with classical holographic interferometry, ability to smoothen singularities and suppress noise as well as specific features of its application are also considered in this section. Examples of off-axis holographic optical schemes for which the suggested method can be applied in studies of macro- and microscopic objects are discussed in the section 3. The algorithm peculiarities are considered in the presence of quadratic phase factor which appears often in digital holographic microscopy. In the section 4 the calculations results for the difference of modeled speckle patterns are demonstrated. Potential advantages of the proposed algorithm in case of noisy observations, unstable wave fields and misalignments are discussed. In the section 5 the suggested algorithm is tested experimentally in highly noisy conditions and is compared with another method used for difference calculation and phase distributions filtering. Finally, Section 6 presents the conclusions.

## 2. Algorithms providing calculation of wave fronts difference without prior unwrapping

Suppose there are two arbitrary object waves *A* = |*A*|exp(*iφ*), *B* = |*B*|exp(*iϕ*), for instance, corresponding to wave fields of two component parts to be compared or two object states, before and after a disturbance. Let us determine a difference between the two wave fronts *X* = *A/B* = |*X*|exp *i*(*ϕ* − *φ*) = |*X*|exp(*iε*). Hereinafter symbols denoting wave amplitudes and phases imply spatial distributions of these values, in particular |*X*| is the spatial difference of amplitudes and *ε* is the spatial phase difference. Thus, the problem reduces to finding |*X*| and *ε*. Obviously the determination of a difference of amplitude distributions is trivial. Thus in this paper we focus on the subtraction of phase distributions of wave fronts.

#### 2.1. Methods of direct wavefronts dividing

Consider two object waves

reconstucted from the holograms*I*

_{1}and

*I*

_{2}, respectively. For simplicity, we take |

*A*| = |

*B*| = 1, then

*X*=

*A/B*= exp(

*iε*), and [13] Similarly, we can calculate [5, 14, 15]

*π*/2,

*π*/2). To find an argument of a complex number one should take into account a number of the quarter on the coordinate plane where the complex number is located [16]. In this case we obtain a ramped phase within the interval (−

*π*,

*π*).

#### 2.2. Method of simulated digital interferogram

First it is worth noting that the suggested method can be applied regardless of the shape of interfering waves. The hologram may be recorded as a result of interference of two plane waves or a plane and a spherical wave, as is often the case in holographic microscopy (see section 3 for details). In this section we describe the suggested method as a sequence of six operations, with a major attention being paid to the interference of two plane waves. However in Fig. 1, illustrating each of the six operations, diagonal insertions are provided being denoted by primed letters, which correspond to the cases of interference of a plane and a spherical wave. To describe the suggested method we introduce also a stationary plane reference wave or spherical wave *C* = |*C*|exp(*iγ _{α}*) incoming to the recording plane at an angle

*α*. The intensity distributions of the recorded holograms formed as a result of interference of the waves

*A*and

*C*, and

*B*and

*C*, at an angle

*α*will appear in the form [Fig. 1(a) and 1(b)]:

- Reconstruction of two physically recorded digital holograms
*I*_{1}and*I*_{2}aimed to obtain amplitude distributions of the object and reference waves (|*A*| [Fig. 1(c)], |*B*| [Fig. 1(g)] and |*C*| [Fig. 1(e)] respectively) and a phase distribution of the object wave*φ*[Fig. 1(d)]. When using the method [21] for reconstruction of the plane (or spherical) reference wave the phase*γ*is synthesized basing on a priori information on the angle_{α}*α*of wave incidence onto the photodetector. The uncertainty of the angle alpha (and that of a spherical wavefront curvature) does not affect the result of calculation of a phase difference. The main requirement here is the usage of the same angle in reconstruction of both the holograms. Then the additional phase retardation gained due to parameter mismatch will be eliminated as a result of phase subtraction. This aspect will be discussed in more detail at the end of section 4. - Simulation of a plane [Fig. 1(j)] (or spherical [Fig. 1(j’)]) reference wave
*C*= |*C*|exp (*i*(*γ*+_{α}*θ*)), incoming onto photodetector at an angle_{β}*α*+*β*, somewhat bigger than the angle*α*formed by the object and reference waves during the hologram recording. As a result an extra phase retardation*θ*is added to the initial phase_{β}*γ*of the reference wave_{α}*C*. - Simulation of the digital hologram
*I*_{3}, basing on the synthesized reference wave |*C*|exp (*i*(*γ*+_{α}*θ*)) and object wave before the disturbance |_{β}*A*|exp(*iφ*) [Fig. 1(i)]: - Multiplication of trigonometric functions describing interference fringes in the intensity distributions
*I*_{2}and*I*_{3}, governed by the Eqs. (5b) and (6). Multiplication yields the following distribution [Fig. 1(k)]:$$\text{cos}(\varphi -{\gamma}_{\alpha})\text{cos}(\phi -({\gamma}_{\alpha}+{\theta}_{\beta}))=\frac{\text{cos}\left((\varphi -\phi )+{\theta}_{\beta}\right)}{2}+\frac{\text{cos}(\varphi +\phi -2{\gamma}_{\alpha}-{\theta}_{\beta})}{2},$$where$$\text{cos}(\varphi -{\gamma}_{\alpha})=\frac{{I}_{2}-{\left|B\right|}^{2}-{\left|C\right|}^{2}}{2\left|B\right|\left|C\right|}$$and$$\text{cos}\left(-({\gamma}_{\alpha}+{\theta}_{\beta})\right)=\frac{{I}_{3}-{\left|A\right|}^{2}-{\left|C\right|}^{2}}{2\left|A\right|\left|C\right|}.$$The suggested algorithm operates in the approximation of a rapidly oscillating reference wave, that may be achieved by selection of a sufficiently wide angle*α*between the object and reference waves. Then the reference wave phase*γ*in the photodetector plane [Fig. 1(f)] will change much faster than the object waves phase_{α}*φ*and*ϕ*before [Fig. 1(d)] and after [Fig. 1(h)] the disturbance, respectively. Here, the second summand on the right side of the Eq. (7) oscillates much faster than the first one. - Removal of the rapidly oscillating second summand in the right side of the Eq. (7), being realized by operation of an automated smoothing over the rectangular area with the side equal to the full oscillation size (in practice, 3–6 pixels). A spatial distribution of the value of cos (
*ϕ*−*φ*) +*θ*= cos(_{β}*ε*+*θ*) is thus obtained [Fig. 1(l)]. It can be presented both as a normalized interferogram of the two object waves_{β}*A*and*B*, recorded at the angle*β*and a normalized interferogram of the object wave with the phase*ϕ*−*φ*and reference wave with the phase*θ*. To clarify the latter statement consider the two-dimensional function cos(_{β}*ε*+*θ*) obtained as a result of algorithm application procedure. It can be easily shown that it represents a normalized interference summand of a hologram formed as a result of interference of the object |_{β}*A*+*B*|exp (*i*(*ϕ*−*φ*)) and reference |*C*|exp(*iθ*) waves:_{β}$${I}_{4}={\left|A+B\right|}^{2}+{\left|C\right|}^{2}+2\left|A+B\right|\left|C\right|\text{cos}(\epsilon +{\theta}_{\beta})$$The angle between them obviously equals to the preassigned value*β*, and the object wave phase (*ϕ*−*φ*) represents the sought phase difference of the two object wave fields*A*and*B*. The intensity distribution is modulated by relatively thick (compared to the initial holograms) fringes, due to the smallness of the angle*β*. It will be shown below that in general the selected magnitude of the angle*β*affects also the filtering degree of noise and singularities. Thus, the distribution obtained according to the Eq. (7), represents, after the smoothing of the fast oscillating summand, a digital analogue of a classical double-exposure interferogram. It should be noted that despite the fact that smoothing at this step is aimed to remove the rapidly oscillating second summand in the right side of the Eq. (7), it also suppresses low noise, similarly to other filtering algorithms [18, 19]. However with the rise of noise level more defects remain which can not be removed by simple averaging over small regions. - Reconstruction of the sought phase distributions difference from the obtained interferogram, based on the analysis of finite width fringes [Fig. 1(m)]. Herewith since the addend to the reference wave introduced at the step 3 is always a plane wave incoming at the angle
*β*, the same wave should be used for reconstruction of the interferograms obtained. Thus the suggested algorithm reproduces an interferogram in finite width fringes which would be recorded by means of classical holographic interferometry. Therefore this interferogram may be analyzed using the traditional algorithm of holographic interferometry utilizing measurements of carrier fringe shifts. Except for classical methods the interferogram may be processed and analyzed using any other algorithms providing reconstruction of the complex field amplitude. We suggest to apply the algorithm described in [21]. The specific feature of this approach is an opportunity to reconstruct interferograms recorded at small angles between the object and reference waves. Without going into details mention that the algorithm principle consists in the solution of an overdetermined system of equations within the area of processing window basically similar to that applied in the phase-stepping method. However, as distinct from the latter one, when calculating phase for each pixel intensity magnitudes are taken not from several independent interferograms in infinite width fringes, but from neighboring pixels around the sought one (processing window), located at the same interferogram in finite width fringes. Thereby for successful usage of a concept of spatial phase steps it is assumed that the processing window covers at least one interference fringe. The construction of such an overdetermined system of equations requires a premise on the slow variation of the object wave phase, that results in its smoothing and noise filtering. Thus adjusting the processing window size one can improve filtering properties sustaining at the same time the required spatial resolution. Further to the above it should be noted that (as it will be shown below) the best filtering result is achieved at the step 6 of the suggested algorithm when using the procedure [21] for reconstruction of the modeled interferogram rather than holograms recorded at the step 1. The filtering effect therewith is related also to the choice of angle*β*magnitude. When the fringe width increases (which happens when the angle*β*decreases), one should just increase the width of the processing window determining the number of utilized pixels and filtering degree. For example at angles*β*= 0.9° the acceptable quality of the reconstructed image from noisy experimental observations was attained with the processing window size containing 20–32 pixels.

Let us clarify in more detail the necessity of artificial broadening of an angle between the object and reference waves by the magnitude *β* in the synthesized hologram. It is stimulated by the necessity to create an off-axis interference pattern analogous to an interefernce pattern in finite width fringes in terms of classical holographic interferometry. At *β* = 0° we obtain an intereference pattern that would be formed as a result of intereference of two object waves *A* and *B* in an in-line optical system, for example at the recording of interefernce pattern in infinite width fringes in the classical holographic arrangement. The condition of relatively slow change of the first summand in the right side of Eq. (7) assumes a relatively small magnitude of the angle *β*. On the other hand when using some methods of wave front reconstruction from an interference pattern *I*_{4} higher magnitudes of the angle are required to achieve higher quality of processing. We have empirically obtain that *β* magnitudes of the order of 0.4° − 1.1° satisfy both requirements.

It should be emphasized that reconstruction of a simulated hologram at an angle different than *β* causes a change of the observation angle. That may be useful if optical axes of the two wave fronts *A* and *B* do not coincide, as for instance if some misalignment occurs between the exposures due to, say, vibrations or usage of different optical setups located at the different places for sample object comparison with a master [2, 11]. Note also that due to averaging over rectangular areas at the step 5 the algorithm causes slight decrease of spatial resolution in accordance with the area size.

## 3. Experimental setups

Figure 2 presents basic schematics of typical optical setups used to record digital off-axis holograms of macro- (a) and microscopic (b) objects.

Due to finite dimensions of optical elements it is often difficult in practice to align object and reference beams in the image plane at a small angle providing resolution of interference fringes in accordance with the Nyquist’s theorem. That is why the scheme of Mach-Zehnder interferometer with beam splitters becomes very popular. In this arrangement a reference wave is added directly in front of a recording array; and the rotation of the second beamsplitter within fairly narrow limits allows to adjust the frequency of interference fringes in the hologram plane.

A problem of object scaling arises when studying macroscopic objects [Fig. 2(a)]. Various approaches may be used to solve this problem, such as, for instance, usage of a system of two concave lenses [24]. In our experiments we applied a 4 *f* bitelecentric system which, in addition to proper scaling, transfers the object image onto the recording plane [25,26]. Within the framework of the applied algorithm of digital hologram reconstruction it saves from the necessity to apply additional procedure of numerical propagation of the wave front from the recording plane back to the object plane. It should be noted also that requirements to the optical system alignment are not too strict in the suggested algorithm since the double-exposure interferometric technique allows to compensate any inherent wavefront curvature caused by the optical system misalignment [13].

When working with microscopic objects [Fig. 2(b)], the microlens is placed in the reference beam [27]. This leads to the appearance of a quadratic term in the reconstructed spatial distribution of the object phase describing a spherical wave. As shown in [21], a hologram resulting as a sum of an object wave containing a quadratic phase factor with a plane object wave can be interpreted as the interference of an object wave without spherical aberration with a nonplanar reference wave [Fig. 1(f’)]. It was also shown in [21] that an algorithm with the variable processing window size may be applied to recover the object wave phase in this case. Thus holograms recorded in the scheme of digital holographic microscopy may be reconstructed following recommendations given in [21]; this is to be done on the step 1 of our algorithm. Depending upon the experimental setup geometry the refocusing algorithm can be used for numerical propagation of the object wave front. Since the reference wave *C* [Fig. 1(f’)] is considered nonplanar, then addition of a summand *θ _{β}* to its phase at the step 2 also gives a nonplanar wave [Fig. 1(j’)]. This will affect the new hologram

*I*

_{3}, being synthesized at the step 3. At the last key steps starting from the 4

*, we deal with an interferogram which, after filtering at the step 5, can be considered as a hologram being recorded with a plane reference wave incoming at the angle*

^{th}*β*. In a similar way an interference pattern formed by two spherical waves in the off-axis configuration had been recorded in [28] aimed to compensate the quadratic phase term in digital holographic microscopy.

## 4. Numerical investigation of proposed algorithm

Several examples have been prepared to demonstrate the advantages of the suggested approach by means of numerical simulations.

#### 4.1. Performance of the algorithm for speckled wave fields

The first example corresponds to the determination of a disturbance [Fig. 3(c)] introduced in the speckle structure [Fig. 3(a) and 3(b)]. The most indicative is an example of sought disturbance containing fast oscillating distribution. Its modulation frequency is comparable to the mean speckle size (as indicated on the right side of [Fig. 3(c)]). In this case interferometric methods applying filtration before unwrapping (e.g. [8, 12]) fail.

Two off-axis holograms, 512 × 512 pixels in size, were simulated basing on phase distributions [Fig. 3(a) and 3(b)]. The following parameters were used for calculation: an angle between the object and reference waves in the horizontal axis direction *α* = 1.8°, pixel size 4.65 *μ*m, laser wavelength 632.8 nm. Simulation of the interferogram [Fig. 3(d)] was performed for the angle *β* = 0.7°, size of the averaging region window 6 × 6 pixels. Note that the size of the averaging region depends on how fast the object wave phase *γ _{α}* is changing. At wide angles

*α*between the object and reference waves this magnitude changes faster, thus averaging over smaller region is required. For the specified parameters

*α*= 1.8° and pixel size 4.65

*μ*m the reasonable averaging region is a square, 36 pixels in area.

The processing of the obtained interference pattern yields the phase difference of wave front distributions modulated by speckle structures and allows to estimate the disturbance magnitude [Fig. 3(e)]. When comparing thus obtained phase distribution with the initial one it is hard to overlook that lines of phase wrap on it are positioned differently. This is due to the fact that phase retardation distribution reconstructed by the algorithm [21] may differ by an arbitrary constant. As follows from [Fig. 3(g)], where over-the-center cross-sections of phase distributions of the prescribed and reconstructed disturbances are shown, the algorithm provides good accuracy. The error to a lesser extent depends upon the size of a square region used at smoothing, as compared to the processing window size employed at the step 6 of the reconstruction procedure.

#### 4.2. Performance of the algorithm in case of noisy observations

The results shown in Fig. 3 demonstrate the algorithm capability in the cases when other algorithms mentioned in the section 1 fail. Note that similar results may be obtained using the methods of wavefronts dividing. Let us emphasize advantages of the suggested algorithm over them.

To demonstrate the ability of the suggested method to noise filtration two digital holograms were modeled from phase distributions [Fig. 3(a) and 3(b)]. Gaussian noise was added to the modeled holograms to achieve the signal-to-noise ratio of the order of 0.4. [Fig. 4(a)] presents results of phase calculations using direct wavefront dividing followed by sin-cos filtration [18], being applied 35 times, with the processing window equal to 3 × 3. [Fig. 4(b) and 4(c)] present results of phase calculations by means of direct wavefront dividing where two holograms were reconstructed prior by the algorithm [21] with the processing window size equal to 8 and 24, respectively. Finally, [Fig. 4(d)] presents results of reconstruction of the simulated interferogram by means of the suggested algorithm with noise filtration achieved due to the usage of the averaging window 3 × 3 at the step 5 and reconstruction by the algorithm [21] at the step 6 with processing window size equal to 24. Comparing these 4 spatial distributions of phase difference obtained by different methods one can notice that the suggested algorithm possesses better capability of noise suppression.

Consider more thoroughly each of the results obtained. In [Fig. 4(a)], after multiple application of the filtering procedure not all the noise is suppressed, however, as shown in the insertion [Fig. 4(a)]), the fast oscillating component is completely smoothed by multiply repeated averaging procedure. The prior application of the algorithm [21] as a filter before the direct wavefronts dividing is also not very efficient: even though it results in adjustable (depending upon the processing window size) smoothing of phase difference, but after the operation of direct wavefronts dividing some singularities still remain on the resulting phase difference [Fig. 4(a) and 4(b)]. This is due to the fact that both filtering and smoothing performed during the reconstruction by the algorithm [21] when hitting in the vicinity of singular points cause their displacement. Since the velocity of phase variations in the vicinity of singular points is relatively high the displacement of singular points in one direction on the two compared speckle patterns is unlikely. Thus they do not compensate each other in calculations of phase difference. The application of smoothing procedure at the step 6 of the interferogram reconstruction allows to avoid this issue [Fig. 4(d)]. Obviously it is also valid for other algorithms of noise filtering based on the smoothing procedure. Taking into account that the vast majority of real objects more or less exhibit elements of speckle structures it is most efficient to perform the filtering procedure after subtraction of phase distributions.

#### 4.3. Performance of the algorithm in case of unstable wave fields and misalignment

Any instability of an optical system in a real physical experiment may cause misalignment of optical axes [Fig. 5(a)–5(c)]. When studying smoothly varying wave fronts it results in some distortion of the phase distribution obtained that does not hinder its unwrapping. But the obtained difference of phase distributions will be inclined at an angle corresponding to the optical axes tilt [Fig. 5(c)].

The suggested algorithm allows to vary the angle *β* which is used at the 6* ^{th}* step of reconstruction procedure, and thus to compensate the system displacement. In the case which is presented at the [Figs. 5(a) and 5(b)] the angle variation in the course of usage of the algorithm allows to obtain plane wavefront (not shown in the figure). In the case of strongly distorted wave fields modulated by speckle structure [Fig. 5(d)] such a displacement causes considerable difficulties in processing of digital holograms by the algorithm of tangent arguments difference since singularities in the phase distribution are not canceled due to subtraction [Fig. 5(e)]. An interference fringe bending at one of the exposures on a distance of only two pixels results in the appearance of unwanted artifacts on the obtained distribution. The algorithm of simulated digital interferogram eliminates such inaccuracy facilitating the unwrapping procedure [Fig. 5(f)].

Therefore, the creation of an interference pattern with a small angle between the object and reference waves at the 5* ^{th}* step of the suggested algorithm and its further reconstruction at adjustable angle results not only in subtraction of phase distributions but also provides elimination of errors caused by imperfections of the optical system.

## 5. Experimental validation of the algorithm

The algorithm had been experimentally validated in the setup shown in [Fig. 2(a)]. The water cell filled with an aqueous
solution of Radachlorin® photosensitizer was used as an object under study.
The solution was illuminated by the diode laser (405 nm, 50 mW) at the wavelength
within the photosensitizer absorption band [Fig. 6(c)]. The experiment aimed to reconstruct spatial
distributions of phase variations gained due to heat-induced refractive index
variations. Digital holograms were recorded before illumination by the excitation
laser [Fig. 6(a)] and after
the 8-second illumination [Fig.
6(b)]. In [Fig.
6(b)] the green frame designates the working region used in the
processing. The efficiency of the suggested algorithm had been evaluated in the
conditions of low contrast interference fringes [Fig. 6(b)], resulting in highly noisy phase
distributions obtained at the reconstruction [Fig. 6(e)]. The recording He-Ne laser wavelength
(633 nm) was out of the photosensitizer absorption band. The exciting laser
radiation was directed onto the solution surface perpendicularly to the optical axis
of the recording beam. Holograms were recorded by means of the digital camera
`Videoscan-2-205` (pixel size 4.65
*μ*m; frame rate up to 7.7 frames per second). The
central perpendicular cross-section of the experimental cell containing solution
under study was projected onto the camera photodetector array using a bitelecentric
4 *f* system providing magnification M = 1/3. A plane wave
front was used as a reference beam, it was combined by the beam-splitting cube with
the object beam at the angles 0.55° and 2.05° horizontally and
vertically, respectively.

The hologram reconstruction was performed using the algorithm [21]. Due to non-ideal alignment of the bitelecentric 4 *f* system the reconstructed wave front passed through the solution before the excitation was not plane [Fig. 6(d)]. However, as it was shown in the section 3 the wave front curvature could be neglected in this experiment since only a differential phase retardation accumulated due to thermal gradient was to be measured. Applying the suggested algorithm with the averaging region of 3 × 3 pixels and processing window of 32 × 32 pixels, the phase distribution in the disturbance under study had been found [Fig. 6(f)]. Automatic smoothing of singular points in the course of algorithm execution allows further straightforward processing of the image obtained for the calculation of refractive index variations [Fig. 6(k)]. Note that direct dividing of wave fronts being used to calculate the disturbance causes an increase of a number of singular points attributable to noise and renders difficulties in unwrapping procedure [Fig. 6(g)]. The procedure of sin-cos filtration [18] with the same averaging region as that at the step 5 of the suggested algorithm, being repeated 30 times, results in efficient noise suppression although leaves some phase defects that hinder its further processing. Comparison of the results presented in [Fig. 6(f)–(h)] in cross sections is given in [Fig. 6(j)]. As it was already mentioned above the reconstructing algorithm [21] provides itself some filtering properties depending upon the processing window size. However if this method with the same processing window, as at [Fig. 6(f)] is applied prior to direct division of obtained wavefronts [Fig. 6(i)] the quality of the obtained phase distribution decreases due to a mismatch of similar singularities mentioned in the section 4.2.

## 6. Conclusion

The paper presents a new algorithm for digital hologram processing aimed for calculations of a difference of two arbitrary phase distributions without prior unwrapping. The algorithm provides elimination of most of singular points caused by low signal-to-noise ratio and optical system vibrations and allows processing of the recorded image without additional filtration. The interferogram simulation with wide interference fringes enables application of the methods of classical holographic interferometry for digital holograms processing. The interferogram reconstruction at an angle different from the angle of recording offers an opportunity to choose the angle of observation of a wave front obtained as a result of subtraction. A short review of currently existing algorithms is given. It is demonstrated that neither of them provides the whole combination of advantages being offered by the suggested approach.

## Acknowledgments

The authors thank V. Katkovnik for valuable remarks. The work was performed under the financing from Russian Science Foundation (project # 14-13-00266). Authors are grateful to RadaPharma company for rendering Radachlorin® photosensitizer for the experiments.

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