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Abstract
This paper presents a novel, simple, robust, and very stable polarimeter which is able to measure in a single-shot the state of polarization (SOP) of an optical field. In this proposal, an optical field with an unknown SOP and an optical field with a reference SOP are superposed by using a double aperture common-path interferometer (DACPI), obtaining two interferograms with adjustable carrier frequency in a single shot. Each interferogram has different visibility and phase-step where the information of the unknown SOP is contained. It is shown that under a typical Fourier analysis any SOP can be successfully measured without the necessity of knowing the carrier frequency. The present proposal is justified mathematically and probed experimentally.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Corrections
24 September 2018: Typographical corrections were made to the funding section.
1. Introduction
The measurement of the state of polarization (SOP) of an optical field has been an arduous task during the last decades. This activity has been motived mainly by its great utility in ellipsometry and polarimetry for detecting fractures in metallic structures [1,2], and optical activity [3]. Also, for estimating concentration of sugars [4], for characterization of liquid samples [3,5,6], nuclear polarization [7,8], and applications: medical [9], astronomic [10], and communications [12,13], among others. Many basic methods measure directly the intensity of the light beam that emerges from an optical system, which includes at least a wave plate and an analyzer [14–17]. Each Stokes parameter is obtained for the full spatial field by comparing the intensity of a set of images, acquired by changing lightly the system configuration. A typical method to vary the system configuration is done by rotating some or all the elements in discrete steps. However, these mechanical movements could introduce some drawbacks, for instance: mechanical vibration, system misalignment, and positioning errors. Another significant limitation in this kind of polarimeters is the inability to measure SOP in real time, because usually at least four frames are needed.
To overcome these drawbacks, there are several proposals to estimate the SOP in one-shot by the use of special optical elements such as image sensor microfilters [18–21], stress-engineered optical devices [22–25], scattered materials [26, 27], 2D metasurfaces [12], liquid crystal modulators [21,27], or special prisms [11,28], among others [29]. In some cases, however, the experimental schema can become very sophisticated and complicated, and its cost could be hight. Most of these techniques belong to the division of aperture polarimeters (DoAP) or the division of focal-plane polarimeters (DoFP), and some of them do not measurement the full spatial field.
Among the one-shot polarimeters are those based on interferometry [30–33] which belong to the division amplitude polarimeters (DoAmP), and provide a mapping of the full optical field. These kind of polarimeters are robust, simple, and maintain the advantages of being fast, thus they can be used in real-time measurements. Ohtsuka and Oka [31] proposed first a method for estimating a mapping of the SOP distribution of a test beam by interfering with a polarized reference beam. The optical system consists of a Mach-Zehnder interferometer with a Michelson interferometer coupled to its reference arm. The reference beam is composed of two optical fields with known orthogonal polarizations, and in each field component contains a spatial carrier frequency. Due to the two carrier frequencies, the interferogram Fourier spectrum has four spectral lobes; two for the interference term of the vertical components and two for the horizontal components. The Stokes parameters are estimated by comparing the information from de horizontal and vertical components using the Fourier method analysis. Naik et al. [32,33] complete this idea by including a third known optical field for self-calibration. With the introduction of a third beam, the number of spectral lobes in the Fourier space with information to be filtered amounted to four.
Nonetheless, these proposed interferometers are not of common-path schemes, for this reason they are exposed to errors due to environmental vibrations and optical path decompensations. To overcome these drawbacks, Han et al. [34] recently proposed a method to multiplex two carrier frequencies into a single interferogram by using a triple aperture common path interferometer, which is more stable and resistant to vibrations than Michaelson or Mach-Zehnder interferometers. However, this proposal needs to isolate more than one lobe from the same interferogram spectrum, therefore several band-pass filters are needed, whose positions could be an error source. As well two references and a test are proposed, which need three beams in total that could increment errors because atmospheric turbulence, and inhomogeneities in the illumination. Additionally, this method needs to capture intensities without the object placed in the test window before to do the measurements. It is well known [35,36] that one of the critical issues for Fourier transform analysis is to accurately extract the desired spectral order in the Fourier spectral domain, which requires a proper band-pass filter to retain the most information while keeping separate the background spectrum and the other higher order spectra [37]. An inadequate band-pass filter is a cause of carrier removal error and spectral leakage error, so the increase in spectral lobes number to be filtered also increases the risk of filtering errors, which is undesirable.
In this work, a method for measuring the SOP of an optical field in a single-shot is presented. This proposal is achieved in a modified double-aperture common-path interferometer (DACPI), which is very robust compared to interferometers with separate arms, such as the Michelson or Mach-Zehnder interferometers. The DACPI comprises a telescopic system to which a Ronchi ruling is placed at the Fourier plane, and this allows the two apertures in the input plane to overlap due to diffraction. At the output plane, there are multiple replicates of the interference pattern where two of them are filtered by two polarizers oriented at the horizontal and vertical axes respectively. Then, each interferogram has the information about the polarized field component. The carrier frequency is introduced by moving the Ronchi ruling out of the Fourier plane [38]. Since both interferograms are replicates of the same pattern, it is enough to identify one of the spectral lobes of an interference pattern and apply the same filter for the second interferogram, this allows to reduce errors due to the spatial filter. A great advantage of this proposal is that it is not necessary to know the spatial frequency; only it must be large enough to separate the diffraction orders.
2. Theoretical model
Figure 1 shows the schema of modified DACPI used to implement the proposed method. It consists of a 4 f optical system with two apertures at the input plane, and a grating placed outside the Fourier plane, which produces adjustable carrier fringes as it was demonstrated in [38]. These two apertures consist of rectangles of sides aw and bw separated by x0 on the horizontal direction. The left window is used as the probe window, where a polarized beam with an unknown elliptical SOP is introduced, while the right window is used as a reference window, where a linear SOP at π/4 respect to the x-axis is introduced, which is gotten by placing a polarizer Pπ/4 over the aperture. In order to explain in detail this process, let us suppose that an entrance optical field has a SOP described by the Jones vector, J = i cos σ + j exp(iα)sin σ, where i denotes the imaginary unit, subject to the relation i2 = −1, i and j are unitary vectors, σ is known as the auxiliary angle and α = αy − αx is the phase difference between the field components, αx and αy are their initial phases. This optical field is described in vector form as:
which is a monochromatic plane wave traveling on z-direction, oscillating at ω frequency with wavelength λ and wavenumber k = 2π/λ, where A = AJ is its complex amplitude, and A is its magnitude. If this field crosses a polarizer at π/4, the new complex amplitude is calculated through Ar = Pπ/4A, where Pπ/4 = (1, 1; 1, 1)/2 is the Jones matrix, that is: with where explicitly, and Then, the transmittance function at the input plane can be defined as where for convenience the coordinates (x, y) have been omitted in the amplitudes A, Ar, and the phases ϕp, ϕr, which represent the phase variations accumulated because of beam propagation along the optical system. w(x, y) = rect (x/aw)rect(y/bw) describes a window function, with rect(…) denoting the rectangle function.As known, when the grating is placed outside of the Fourier plane, the complex amplitude at the image plane is given by [38]
where up denotes the grating period with a fill factor of 1/2, n is an integer that denotes the diffraction order, ∆f denotes its axial translation from the Fourier plane, and f denotes the focal length of the lenses. Now by substituting Eq. (6) into Eq. (7), and assuming the matched condition λf/up = x0 for the interference, the optical field is: where the complex coefficient is with cn = sinc(n/2)/2. Now, if the optical field, for n = −1, is observed at the window w(−x0/2 −x′, −y′) through an analyzer at zero radians, as it can be appreciated in Fig. 1, the expression of the field is reduced to and if the optical field, for n = 0, is observed at the window w(x0/2 − x′, −y′) through an analyzer at π/2 radians (see Fig. 1), thenThe irradiance observed by an optical detector from the optical fields described in Eq. (9) and (10) is given by
where ϕ = ϕp − ϕr − αr and μ0 = ∆f/upf. The πλ∆f/up2 term is also known, measurable and constant if ∆f is not changed, it is easily removed through a simple calibration. Then, by applying the Fourier-transform to interferograms given in Eq. (11) and Eq. (12), and assuming that μ0 is large enough to separate the three lobules of the interferogram spectrums, it is possible filtering only one lobule at the time [35,36,38]. Thus we can obtain where dx and dy are complex numbers that have information of SOP, object phase, and carrier frequency, and exponential functions form part of cosine in the Euler formula 2 cos(θ) = exp (iθ) + exp (−iθ). Now, the SOP described by the Jones vector is given in terms of α and σ, then, by combining the expressions given in Eq. (13) and (14) the coefficients c0, c1, the amplitudes A, Ar, the phase ϕ, and the carrier frequency μ0 are missed, which is a simple complex number whose magnitude contains σ and its phase is α, therefore, they are easily computed fromIf the test light beam is partially polarized, the Jones formalism cannot be applied [39,40], in this case, the Stokes parameters will be estimated by
and where A0 = 21/2c0c1AAr/2, and the asterisk ∗ denotes the complex conjugate operator. Equation (16) to (21) are valid except when it becomes an undefined number, that is when dx = dy = 0. In this case, it is easy to see that Ar = 0 and from Eq. (4) can be established and due to σ is defined positive within the range [0, π/2], this expression has a real solution when sin 2σ = 1, and cos α = −1 is that when σ = π/4, and α = π. Then, for this particular case, σ and α can be deduced only from Eq. (22). For the other cases than the particular case, σ and α are computed without ambiguity by using Eq. (16) to (21), which was deduced starting of two interferograms with carrier frequency changed in phase by a phase-step given by α captured in a single-shot. Because this method is built on a common-path interferometer achieved in a single-shot, this experimental setup is robust mechanically, immunity to vibrations, and useful to measure SOP of optical fields that change with the time.3. Experimental verification
To verify the theoretical model of the interferometric polarimeter studied in the previous section, the schema depicted in Fig. 1 was implemented. The illumination source consisted of a semiconductor laser RGB-655/500mW using the green line with λ = 532nm. The built DACPI was composed of two lenses with the same focal length f = 400mm, and a grating with a fill factor of 1/2 and period up = 12.7μm, at the input plane two windows of sides aw = bw = 5mm with separation x0 = λf/up = 16.7mm were made. Three polarizers P0, Pπ/4, and Pπ/2 were disposed of such as is schematized in Fig. 1; Pπ/4 was placed on the reference aperture at the input plane, while P0 and Pπ/2 were placed at the image plane matching with the output optical field n = −1 and n = 0, respectively. In order to calibrate the polarimeter, a set of light beams with known SOP were modulated. The polarized entrance beam was created as shown in Fig. 2, in front of the probe window a polarizer, P, and a quarter-wave plate, Qπ/4, were placed to generate any state of polarization; thus J was obtained experimentally.
Fig. 2 Arrangement of elements for calibration of the polarimeter: a wave-plate Qπ/4 and a polarizer P.
The Ronchi ruling of the DACPI is placed outside the Fourier plane in order to generate a carrier frequency long enough to separate the lobules of the interferogram spectrums in the Fourier space, which is a condition that will allow to filter only one lobule, where the information of SOP is contained. The election of filter position is done by observation, so if the carrier frequency is changed the filter position must be changed by hand. That situation does not represent any problem in the experiment, since in practice the carrier frequency remains fixed during the SOP measurements. It is worth saying that this carrier frequency does not need to be known since the method presented here does not need this value to compute σ and α or the Stokes parameters. At the image plane a CCD camera model: Point Grey Grasshopper3 GS3-U3-23S6M-C was employed to capture and keep the interferograms in the computer.
3.1. Calibrating process
In order to calibrate the polarimetric system, a known light beam with linear polarization at π/4 is used as a test beam, note that it represents the same SOP of the reference beam. Under this condition, the phase shift between I−1,x and I0,y must be zero, and for both interferograms, the object phase, the backlight, and the light modulation must be the same. Because both interferograms are recorded in the same capture, as the first step a digital spatial filter is applied on each interferogram in order to separate them, Fig. 3(a). If there is a constant phase-shift between them, , due to the Ronchi ruling movement, the filter can be adjusted by displacing it a few pixels until both interferograms match. Next, the fast Fourier transform is calculated for each interferogram, Fig. 3(b); notice that because the carrier frequency the spectrum lobules are separated by the same distance in both images. After that, a digital bandpass filter is applied to one of the spectrum lobes in the Fourier spectrum of I−1x (red circle), in this way the information of the horizontal component is isolated. Meanwhile, the same bandpass filter is applied to the Fourier spectrum of I0y, then, the vertical component is isolated (blue circle). Finally, the inverse fast Fourier transform of both filtered spectrums are calculated, and the SOP is estimated by mean of the Eq. (16) to (21). Note that the same digital spatial filters, both in spatial and frequency domain, must be used for measuring other unknown beams in all subsequent experiments if the Ronchi ruling is not moved again.
Fig. 3 Digital spatial filters of the interferograms in (a) spatial domain, and (b) frequency domain.
3.2. Results
To probe the proposed polarimeter we consider a set of homogeneous polarizations; as the first stage, Qπ/4 was removed from the setup shown in Fig. 2, while P was rotated at the angles φp = π/2, ±π/4, ±π/6 and −π/3 to create six linear SOP experimentally. In the Fig. 4(a) there is a representation of the SOP on Poincare’s sphere, this graphic was built with the Stokes parameters information. In Fig. 4(b) there are the parametric plots of the elliptical SOP drawn up from the Jones vector information.
Fig. 4 Representation of the six linear states of polarization on (a) Poincare’s sphere, (b) parametric plots, and (c) values of S1, S2, and S3.
The theoretical values are shown with black points on the Poincare’s sphere in Fig. 4(a), and with black dashed lines in Fig. 4(b). For clarity, in Fig. 4(c) there are shown the plots of the theoretical values of S1, S2, and S3 in contrast to the experimental measurements. Where the theoretical values are represented by asterisks and color solid lines, while the experimental measurements are represented by color diamonds. The experimental measurements were done by evaluating the interferograms as is indicated in the expressions described by Eq. (16) to (21), then the average of each parameter distribution was obtained, these results are represented with the blue points in Fig. 4(a) and the blue dashed lines in Fig. 4(b).
In the second stage, Qπ/4 was rotated and fixed at φQ = −π/3 while P was rotated at the angles φp = π/2, ±π/3, ±π/6 and 0 to create six elliptical SOP as shown with black points (theoretical values) on the Poincare’s sphere in Fig. 5(a), and with black dashed lines in Fig. 5(b). The magenta lines and magenta points in Fig. 5 respectively show the experimental mean of the measured SOP. For this example, the comparisons between the theoretical and experimental values of Stokes parameters are shown in Fig. 5(c) for each polarizer angle.
Fig. 5 Representation of the six eliptical states of polarization on (a) Poincare’s sphere, (b) parametric plots, and (c) Stokes parameters.
In the third stage, we present a different pair of angle values, (φQ, φp), of Qπ/4 and P. These combinations were (4π/9, 0), (−π/3, π/12), (−π/3, −π/3), (−2π/9, π/4), (−2π/9, −π/6), (0, −π/6). The theoretical SOP are shown with black points, and black dashed lines in Fig. 6 while the green lines and green points show the SOP measured experimentally.
Fig. 6 Representation of several states of polarization on (a) Poincare’s sphere, and (b) parametric plots.
Finally, four sets of experimental measurements of the Stokes parameters, S1, S2 and S3 are presented in Fig. 7. In these measurements φp was varied within [−π/2, π/2], while φQ was kept constant: Fig. 7(a) φQ = 0, Fig. 7(b) φQ = −2π/9, Fig. 7(c) φQ = π/6, and Fig. 7(d) φQ = −4π/9. Notice that all experimental measurement (color diamonds) corresponds to theoretical values which are represented by color asterisks and solid lines. Red color means S1, the blue color means S2 while black color corresponds to S3.
Fig. 7 Plots of experimental and theoretical values of S1, S2 and S3 parameters as function of φp. The waveplate is keep on (a) φQ = 0, (b) φQ = −2π/9, (c) φQ = π/6 and (d) φQ = −4π/9.
4. Conclusion and remarks
In this work, there was proposed a novel one-shot interferometric polarimeter, whose theoretical and experimental demonstrations were successfully done.
This proposal was based on a DACPI, which shows at least three relevant contributions to simplify the interferometric polarimeter. First, the practical and simple experimental method used to introduce an adjustable carrier frequency without the necessity of additional optical elements to create it and the reference beam. In the present method only was necessary to translate the grating axially, placing it outside of the Fourier plane. Second, and most importantly, is the ability to obtain two interferograms in a single shot with mechanical stability; this gives the possibility of measuring in real time any SOP, since only one spectral filter is needed. Third, the theoretical model allows to eliminate: the diffraction constants , ; the initial amplitudes of the beams A, Ar; the phase difference ϕ; and the carrier frequency μ0, to estimate the SOP successfully, which are spatial functions that do not need to be known.
To calculate the SOP, the test beam components were compared; the interferograms of the horizontal and vertical components were obtained by placing orthogonal linear polarizers on the interfering output beams with diffraction order n = −1 and n = 0 respectively, as Eq. (11) and Eq. (12) indicates. The information about horizontal and vertical components of the unknown SOP was recovered by using the Fourier analysis. Because both interferograms are replicas of each other, it was only necessary to design a single bandpass filter in the Fourier spectral domain and applied it to both spectra.
The experimental results, shown in Fig. 4 and 6, supports this proposal, where the σ and α values were estimated by Eqs. (16) and (17), and the Stokes parameters were estimated with Eq. (18) to (21). The computation of accuracy is beyond the scope of this paper, but it will be done in future work. Nevertheless, in this point it is important to remark that because in DACPI both beams travel by the same optical components the polarimeter is mechanically stable, robust, tolerate defects in optical components and inhomogeneities in the illumination. However, it could be affected by the bad quality of polarizers P0 and Pπ/2 used to obtain the interferograms, since they could introduce undesirable spatial variations in intensities and phases different in each interferogram, and besides, their bad angular position between them, different from π/2, could be a source of error. The present method needs interference fringes to measure any SOP, therefore the coherence and the polarization between the reference and test beams must be appropriated. Enough coherence beams to produce observable fringes are needed, so it works good for total coherence beams, and for same degree of coherence, but for incoherence beams the method is not able to see interference fringes. On the other hand, parallel SOPs the method works very well, but cuasi-orthogonal SOP also will produce low visibility of the fringes, however this drawback was dropped by the condition given in Eq. (22).
It is worthwhile to comment that some error sources in the calibration process, using a π/4 linear SOP, are the spatial filter position, and the polarizers alignment. α is related to the phase shift between I0y and I−1x interferograms, being crucial that the fringes I0y and I−1x are matched. The visibility difference between interferograms is associated with σ, then, it is essential that the polarizers at the output planes must be orthogonal among them, because in this case, the visibility of both interferograms is the same. Because all this process is calculated for each pixel of the interferogram, we think that the present method is capable of measuring states of polarization of optical fields both homogeneous and inhomogeneous, even those that change over time.
Funding
Vicerrectoria de Investigacion y Estudios de Posgrado of Benemerita Universidad Autonoma de Puebla (100425744-VIEP2018); Consejo Nacional de Ciencia y Tecnologia (257853); Direccion de Investigacion e Innovacion of Universidad de Montemorelos (2017-448).
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