1. Introduction.

Phase-shifting interferometry is a reliable technique to extract phase information from interferograms [1]. The technique is based on a system of linear equations which uses at least three different interferograms obtained from the same phase distribution [2]. In order to achieve these interferograms, certain phase shift values are introduced between the signal wave and the reference wave. Among other variants, using four interferograms has been proved to be very useful for the case of interferograms having an adequate signal-to-noise ratio and good contrast [3]. A constant phase shift value of α=90° is widely used for a case of N=4 interferograms [4]. In this communication, we will focus ourselves on this variant.

For N interferograms, N-1 shifts must be carried out. For static phase distributions, this means that, for the case of N=4, four shots in sequence must be done. However, when a time-varying phase distribution is to be extracted, a suitable technique able to get four shifted interferograms at the same time is therefore needed. Some approaches to perform this task have been already demonstrated. One of them is a spatial phase stepping method with a holographic element, used for transient deformation measurements with electronic speckle pattern interferometry (ESPI) [5-6]. A screen filter with an array of polarizing modulators under circular polarization has been also shown for optical testing applications [7], but these polarizers have to be so small that its size is as tiny as a camera pixel size; the authors have named it as micropolarizer. Other similar case that uses micropolarizers is described in the patents from the references 8 and 9. In both cases, it is necessary the high precision technology and a reference arm interference. However, although these methods are versatile, they need rather special components.

In this communication, a simplified one-shot phase-shifting experimental method using grating interferometry is proposed and shown. It is based on a phase grating placed as the pupil of a 4f Fourier optical system.

With two windows in the object plane with a proper spacing in between, so that superposition of diffraction orders with no shear are achieved in the image plane [10]. Such a system performs as a common-path interferometer when the optical phase associated to one window can be taken as reference and the other one as the signal, thus having great mechanical stability [11]. With the addition of wave plates for each window and linear polarizing filters for each diffraction order of interest, one-shot four interferograms 90° phase apart can be proven feasible. Other possible variants are also briefly discussed.

Two windows placed in the object plane of an imaging optical system were early employed in all-optical character recognition applications, where a space modulated Young interference fringe pattern is to be expected in the frequency plane when the fields within each of the windows in the object plane have the same relative distribution. An inspection after appropriate average of the resulting fringes of two similar or dissimilar fields has been reported to give useful information [12]. Placing a grating with its fundamental spatial frequency matching that of the Young interference pattern would form also a moiré pattern, as is the case in an interferometer reported for testing planarity [13]. In this interferometer, shifting of such moiré fringes can be achieved by displacements of the grating and can be used for tuning the reference phase [13]. Two windows are also used in the optical joint transform correlator [14]. In this case, however, a hologram is recorded with the two characters to be compared and placed in the Fourier plane of some lens for reconstruction. Another predecessor of grating interferometry with two windows is the system for image subtraction to perform subtraction of both photographs from a stereoscopic pair in order to generate contours of equal elevation to obtain topography maps, as described in ref.15 and references therein. The conveniences of using phase gratings were also remarked there. Polarizing filters in the Fourier plane to extract 180° phase images contained in grating diffraction orders for image subtraction are also described in [16]. Probably due to the applications they were aimed to, none of these systems have incorporated the benefits of phase shifting. To our knowledge, the first attempts to explicitly suggest such a possibility can be drawn to ref. 10, 11, 17.

In the following sections, a double-window grating-interferometer featuring retardation-wave plates under linearly polarized light is first analyzed. Departures from the exact retardation values are considered in order to introduce an experimental set-up. Finally, some experimental results using it are shown.

2. Basic considerations

The Fig. 1 shows the arrangement of an ideal one-shot phase-shifting grating interferometer incorporating modulation of polarization. A combination of a quarter-wave plate Q and a linear polarizing filter P generates linearly polarized light at an appropriate azimuth angle (45°) entering the interferometer. Two quarter-wave plates (Q_L and Q_R) with their fast axes setted each other orthogonally are placed in front of the two windows of the common-path interferometer so as to generate left and right circularly polarized light as the corresponding beam leaves each window. A phase grating is placed at the system’s Fourier plane as the pupil. In the image plane, superimposition of diffraction orders, causing replicated images to interfere.

 

Fig. 1. One-shot phase-shifting grating interferometer with modulation of polarization. A, B: windows. Side view: ψ_i: polarizing angles (with i=1,2,3,4 to obtain phase-shifts ξ_i=0°, 90°, 180°, 270° respectively). Involved interference order superpositions are indicated from -2 to +2.

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The phase shifting ξ_i, i=1…4, results after placing a linear polarizer to each one of the interference patterns generated on each diffracting orders in the exit plane (P₁, P₂, P₃, P₄). Each polarizing filter transmission axis is adjusted at different angle ψ_i, so we obtain the desired phase shift ξ_i for each pair of orders. For a 90° phase-shift ξ_i between interfering fields, the polarization angles ψ_i in each diffraction order must be 0°, 45°, 90° and 135° for the case of ideal quarter-wave retardation (α′=90°). In the next sections, some particularities arising from the optical components available for our set-up are discussed. Among these, the calculation of ψ_i for the case of a non exact quarter-wave retardation is considered.

2.1. Interference patterns with polarizing filters and retarding plate.

When using birefringent plates which do not perform exactly as quarter-wave plates for the wavelength employed, the polarization angles of the linear polarizing filters to obtain 90° phase-shifts must change. To calculate the phase shifts induced in a more general polarization states by linear polarizers, consider two fields whose Jones vectors are described respectively by

${\overrightarrow{J}}_L(x,y)=\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ e^{i\alpha \prime }\end{array}\right){\overrightarrow{J}}_R(x,y)=\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ e^{-i\alpha \prime }\end{array}\right)e^{i\varphi (x,y)}$ .

These vectors represent the polarization states of two beams emerging from a retarding plate with phase retardation ±α′. Each beam enters the plate with linear polarization at ±45° with respect to the plate fast axis. Due to their orientations, the beams rotate in opposite directions, thereby the indices L and R are used. The beam with subscript R is supposed to carry a phase distribution ϕ(x, y). When each field is observed through a linear polarizing filter whose transmission axis is at an angle ψ, the new polarization states are

where the spatial dependence has been dropped for simplicity and with the linear polarizing transmission matrix given by

If the new fields are left to interfere, the resulting irradiance can be written as

where

and

Plots of ξ(ψ, α′) and A(ψ, α′) A are shown in Fig. 2 for several values of α′.

 

Fig. 2. (a) Phase shift ξ(ψ, α′) as a function of ψ for several values of α′. Insert: α′ for ideal retardation and experimental retardation. (b) Amplitude A(ψ,α′) as a function of ψ for several values of α′.

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For the special ideal case of $\alpha \prime =\frac{\pi }{2}$ (quarter-wave plate), it is readily found that

which can be verified in Fig. 2. Then, the irradiance of Eq. (3) reduces to

which is a well known expression used already for phase shifting [9]. In fact, by denoting four irradiances at four different angles as

with i=1…4, the relative phase can be calculated as [5]

where ‖J⃗ ₁‖², ‖J⃗ ₂‖², ‖J⃗ ₃‖² and ‖J⃗ ₄‖² are the intensity measurements with the values of ψ given by ψ ₁=0, ψ ₂=π/4, ψ ₃=π/2, ψ ₄=3π/4. When the phase retardation is different from π/2, Eq.3 to Eq.5 must be used. In those cases, the value of ψ can be determined from Eq.4 looking for ξ=0,π/2,π,3π/2. For ξ=0 it is easy to see from the Fig. 2(a) that ψ ₁=0. Cases ξ=π/2,3π/2 lead to the condition

which can be transformed in the following second degree equation for cos² ψ

with two solutions ψ_a,b given by

which enables the choosing of

where ψ_a and ψ_b are two meaningful different solutions arising from Eq. (12.a) and |ψ_b|<ψ_a. For the case ξ=π, it is found that the following condition must be fulfilled

sin2ψ·sinα′+sin² ψ·sin2α′=0

which leads to

The value with n=1 can be chosen.

2.2. Interferometer with phase-grating.

The phase object under test is placed in one of the two windows in the object plane. Thus, the Jones vector in the object plane can be written as

where x ₀ is the separation of center to center between the two windows, the object phase being described with the function ϕ(x, y) included in J⃗_R(x, y). Here, two different polarizations are related to the field of each window and the corresponding Jones vectors are denoted by J⃗_i(x, y), i=R, L as before. The rectangular aperture w(x, y) can be written as w(x, y)=rect[x/a_w]·rect[y/b_w], where a_w and b_w represent the widths of the window.

A grating of spatial period d=λf/X ₀ is placed in the Fourier plane. Then, the corresponding transmittance is given by

with δ(μ) denoting the Dirac delta function and * the convolution operation. μ=u/λf and ν=v/λf are the frequency coordinates scaled to the relevant wavelength λ and the focal length f. The actual frequency coordinates are thus u and v. The grating’s profile for a period is given by G_P(μ). The point spread function of a system with such a pupil can be obtained with the inverse Fourier transform of G(μ,ν), which results in

 

Fig. 3. Relative positions of diffraction orders from windows A and B. The case of order +1 out of phase by π with respect to the others is shown. Two interferograms with inverse contrast result. X ₀=x ₀.

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The system’s image plane is the convolution of O⃗(x, y) and G̃(x, y), which is basically the replication of each window at distances X ₀. The replications of each window are displaced by $\pm \frac{1}{2}{x}_0$ with respect to the origin, so they are superimposed if Nx ₀=X ₀. Figure 3 shows the case of N=1. The case of similar amplitudes for each spectral diffraction orders (from order -2 to order +2) is presented, a similar situation can be found in phase gratings [15]. Also, in the Fig. 3 one of the four orders shown, the +1, is depicted π out of phase with respect to the others. This effect can be obtained with phase gratings because odd order amplitudes are proportional to Bessel functions of odd orders, which have in turn odd parity, whereas even order amplitudes follow Bessel functions of even parity. Thus, diffraction orders as described are expected to be obtained with phase gratings due to their particular distribution G̃(x, y).

Assuming that no additional changes of polarization occurs other than the one imposed by the retarding plates, the diffraction order superposition is expected to follow the description outlined in sec.2.1. So, in order to obtain four interferograms with a phase shift of π/2, the previous sections justify the use of an interferometer consisting of two birefringent windows separated by x ₀ in the object plane and a phase grating in the Fourier plane. The interference of the fields of each window is obtained in the image plane when superposing itself the appropriate orders of diffraction. Linear polarizers in front of each order at the proper angle ψ would give the values of phase shifts according with Eq. (3). In order to superpose orders +1 +2, 0+1, -1 0 and -2 -1 it is necessary to fulfill the condition d=λf/x ₀.

3. Experimental set-up

A green laser light with λ=532 nm was employed to illuminate the system of Fig. 1. Figure 4 shows four interferograms from the system as a preliminary observation. They were obtained before placing retardation plates and polarizing filters. The patterns show the relative phases of the diffraction orders as discussed in Sec.2.2. Two rightmost interferograms have the same fringe contrast. Such contrast appears to be the complementary one of the remaining leftmost pair of interferograms. The Fourier spectrum of the grating behaves as the one of Fig. 3 (with a π phase difference between even and odd single orders). In fact, the contrast of the remaining patterns (not shown) follows changes that can be explained in agreement with the parity properties of the Bessel functions.

 

Fig. 4. Image plane from a system as depicted in Fig. 1. Neither plate retardation plates nor polarizing filters were used. Two opposite fringe contrast can be seen. Compare with Fig. 3.

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To complete the set-up, off-the-shelf retarding plates designed as quarter-wave plates for λ_a=514.5 nm were used in the windows. Thus, a nominal retardation of

is calculated. Introducing this value in the solutions described in sec.2.1, the results ψ ₁=0 ψ ₂=46.577° ψ ₃=92.989° ψ ₄=136.42° were obtained. In the experimental setup, these values must be changed to ψ′₁, ψ′₂, ψ′₃, and ψ′₄ due to the additional 180° phase difference, as described later on.

The phase grating employed (110 ln/mm) generates five diffraction orders of similar but not equal average irradiance (Fig. 4), as expected. Because the respective irradiances do vary due both to the diffraction order amplitude and the variations of pattern amplitude (Eq. (5)), each interferogram was subject to a normalization process to each maximum of its irradiance before using Eq. (9). The separation between window centers was of x ₀≈10mm. Other parameters used were focal lengths of f≈160mm, a_w=6mm and b_w=10mm.

4. Experimental results

An object phase has placed in window A and the window B is the reference. The transmission axes of the polarizing filters attached to interferograms of same contrast were adjusted according with the first two values calculated as explained in previous sections (ψ′₁=ψ ₁ and ψ′₂=ψ ₂) to achieve mutual phase differences of Δξ=ξ(ψ′₂, α′)-ξ(ψ′₁, α′)=π/2. The remaining two axes were adjusted taking into account the additional phase shift of 180°. Using the same angles ψ′₃=ψ ₁ and ψ′₄=ψ ₂, the two required phase differences Δξ=ξ(ψ′₃, α′)+π-ξ(ψ′₁, α′)=π and Δξ=ξ(ψ′₄, α′)+π-ξ(ψ′₁, α′)=3π/2 can be obtained.

4.1. Static distributions

Two test objects were prepared evaporating magnesium fluoride (MgF₂) on a glass substrate: a disk or phase dot and a phase step. When each object was placed separately in one of the windows using the interferometer of Fig. 1 with polarizers P₁, P₂, P₃ and P₄, using the previously calculated angles ψ′₁, ψ′₂, ψ′₃, and ψ′₄, the interferograms of Fig. 5 were obtained. For each object, the four interferograms are shown together with the unwrapped phase calculated with Eq.9 at the right (in 256 grey levels). Examples, some typical raster lines for each unwrapped phase are shown in Fig. 6 (in arbitrary phase units).

 

Fig. 5. Upper row: phase dot. Four 90° phase-shifted interferograms and unwrapped phase. Lower row: phase step. Four 90° phase-shifted interferograms and unwrapped phase.

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Fig. 6. Unwrapped calculated phases along typical raster lines of each object of Fig. 5. Scale factor: 0.405 rad.

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These observations suggest a further simplification for the polarizing filters array for the case of the phase shift of π from a phase grating. It consists of using only one enough big filter to cover the two central interferograms at the same angle of ψ ₂′ (ψ ₂′=ψ ₂=ψ ₃) instead of two separate filters at ψ ₂, ψ ₃ respectively. Thus, only three linear polarizing filters have to be used. The transmission axes of the filters P₁ and P₄ can be both horizontally oriented (ψ ₁′=ψ ₁=ψ ₄), see Fig. 7.

 

Fig. 7. Simplify polarizer filters array

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This only alters the order of the shifted interferograms. Because the departure from an ideal case for the experimental conditions is relatively small, this configuration could be experimentally tested with the described set-up resulting in qualitative good results.

 

Fig. 8. Typical four 90° phase-shifted interferograms from oil flowing (Media 1)

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4.2. Moving distributions

Immersion oil was applied to a glass microscope slide and allowed to flow under the effect of gravity by tilting the slide slightly. The slide was put in front of one of the object windows of the system of Fig. 1. Figure 8 shows a typical sequence of four shifted interferograms from the oil flow in arbitrary units. Figure 9 shows the resulting unwrapped phase evolution of another oil flow.

 

Fig. 9. typical unwrapped phase from interferograms of oil flowing (Media 2).

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5. Final remarks

The experimental set-up for a polarizing two-window phase-grating common-path interferometer has been described. This system is able to obtain four interferograms 90° phase-apart in only one shot. Therefore, it is suitable to carry out phase extraction using phase shifting techniques. Phase evolving in time can then be calculated and displayed. The system is considerably simpler than previous proposals to attain four interferograms with only one shot. In its present form, it is, however, best suited to relative small objects which do not introduce polarization changes. Because it works with interferograms placed relatively far from the optical axes, experimental results suggest that some method has to be introduced to compensate mainly for distortion, among other off-axis aberrations. This compensation could be optical (as a better design of the optical imaging system) or digital (fringe distortion compensation by inverse transformation). In the experiments, the use of is described retarding plates which are not quarter-wave plates. Although they can perform well enough in principle, it seems better to use quarter-wave plates because no additional variations of the interferogram amplitude arise. Also in this case, a simpler polarization filter array can be used taking advantage of the diffraction properties of a phase grating. Some special phase gratings design could optimize the interferometric system described.