## Abstract

The technique for generating the partially coherent and partially polarized source starting from the completely coherent and completely polarized laser source is proposed and analyzed. This technique differs from the known ones by the simplicity of its physical realization. The efficiency of the proposed technique is illustrated with the results of physical experiment in which an original technique for characterizing the coherence and polarization properties of the generated source is employed.

©2010 Optical Society of America

## 1. Introduction

During the last decade the substantial efforts were made in the developing the vector coherence theory of electromagnetic fields (for a brief analysis of the state of art in this area see, e.g., Ref. 1). To realize any experimental investigation on this subject, one needs a genuine partially coherent and partially polarized electromagnetic source with known statistical properties. Nevertheless, the source commonly used in optical practice, is the gas laser, whose radiation is almost completely coherent and completely polarized. Thus, the problem of the controlled altering the coherence and polarization properties of the laser radiation arises. In Ref. 2 it was shown that such an alteration can be done by means of the computer controlled liquid crystal (LC) spatial light modulator (SLM). Somewhat later this technique has been improved with using two LC SLMs placed at the opposite arms of the Mach-Zehnder interferometer [3]. Recently we have proposed an alternative technique employing two crossed parallel aligned LC SLMs [4]. Unfortunately, the mentioned techniques have not been yet realized in practice because of the lack of commercial LC SLMs with the desired characteristics.

In the present paper we propose a very simple technique for modulating the coherence and polarization of laser radiation which does not need any LC SLM. This technique bears some resemblance with the one reported in Ref. 5, but differs from it as by another action principle so by the simplicity. Due to the last and the wide-scale availability of the required optical components we have succeeded in physical realization of the proposed technique and have demonstrated its efficiency in physical experiment.

Once the desired secondary partially coherent and partially polarized source has been created, it must be subject to experimental characterization. The idea of such a determination is well known [6–8], but its physical realization has not been yet reported. Here we propose also a rather simple technique for characterizing the coherence of electromagnetic source, which was used in our experiments.

## 2. Background

As well known, the second-order statistical properties of a random planar (primary or secondary) electromagnetic source can be completely described by the cross-spectral density matrix (for brevity we omit the explicit dependence of the considered quantities on frequency $\text{\nu}$) given by the formula [9–11]

In Eqs. (2)–(4) Tr stands for the trace and Det denotes the determinant.

It is not out of place to mention here that, equally with Wolf’s definition of the degree of coherence, there are known the other definitions [12–14]. But, for our following analysis, the definition given by Eq. (3) proves to be quite sufficient.

## 3. Technique for generation

As a primary source (PS) we consider the single-mode gas laser whose radiation is linearly polarized and in some plane normal to the direction of propagation, under certain conditions, can be characterized by the cross-spectral density matrix

*x*axis. On substituting from Eq. (5) into Eqs. (3) and (4), it may be readily verified that in this case $\text{\hspace{0.17em}}\left|{\mu}_{\text{PS}}\text{\hspace{0.05em}}({x}_{1},{x}_{2})\right|=1$ and ${P}_{\text{PS}}(x)=1\text{\hspace{0.17em}},$ i.e. that such a source is completely coherent and completely polarized. Let us assume that the radiation from this source passes through a Mach-Zehnder interferometer sketched schematically in Fig. 1 . The polarizing beam splitter PBS separates the orthogonal field components ${E}_{x}(x)$ and ${E}_{y}(x)$ so that each of them can be independently changed by means of two rotating ground glass plates GGP

_{1}and GGP

_{2}placed at the opposite arms of the interferometer.

Disregarding the negligible changes of the coherence and polarization properties of the electrical field induced by a free space propagation within the interferometer, one can represent the considering system as a thin polarization dependent phase screen with an amplitude transmittance given by the Jones matrix

Moreover, taking into account that both variables ${\phi}_{1}\text{(}x\text{)}$ and ${\phi}_{2}\text{(}x\text{)}$ are originated by two different ground glass plates, we can assume that they are statistically independent, i.e.

The cross-spectral density matrix of the secondary source (SS) created just behind the interferometer can be calculated as follows:

We note here that for a ground glass plate the root-mean-square phase retardation σ is much greater than $2\text{\pi}$ rad. Hence, the following approximations can be accepted (see, e.g., Ref. 2):

andThen substituting this result into definitions given by Eqs. (3) and (4), we obtain respectively,

Equations (15) and (16) show that the generated source is partially coherent and partially polarized. The transverse coherence length $\eta \text{\hspace{0.17em}}$of this source depends on the diffusion properties of the used ground glass plates characterized by parameter γ, and its degree of polarization can change in the range from 1 to 0 with a proper choice of polarization angle *θ*.

## 4. Technique for characterization

Once the desired secondary partially coherent and partially polarized source has been created, it must be subject to experimental characterization, i.e. the elements ${W}_{ij}^{\text{SS}}$ of the matrix ${W}_{\text{SS}}$ have to be experimentally determined. Taking into account the symmetry of the cross-spectral density given by Eq. (11), for this purpose the technique sketched schematically in Fig. 2 can be used.

This technique represents a modified version of well known two-pinhole Young’s experiment. The Mach-Zehnder interferometer is employed here to extend the effective distance between the pinholes so that the radiation emerged from each pinhole can be processed independently by finite size optical components. The polarizers P_{1} and P_{2} serve to cut off only one of the orthogonal field components. The removable rotators R_{1} and R_{2} serve to produce the rotation of one of the transmitted field component through 90°. For such a purpose a suitably oriented half-wave birefringent plate can be used. The operation description of the technique is given bellow.

The determination of the elements ${W}_{ij}^{\text{SS}}$ of the matrix ${W}_{\text{SS}}$can be realized by means of the following four experiments. In the first experiment the polarizers P_{1} and P_{2} are aligned to transmit only *x* component of the incident field without any subsequent rotation of the plane of polarization. In the second experiment P_{1} and P_{2} are aligned to transmit only *y* components of the incident field again without any subsequent rotation of the plane of polarization. In the third and the fourth experiments the polarizers P_{1} and P_{2} cut off the different orthogonal components of the incident field and the corresponding polarization rotator R_{1} or R_{2} serves to allow the interference of these components.

The power spectrum of the field observed at the output of the interference system in each experiment can be described by the spectral interference law [8]

*k*is the wave number, ${z}_{0}$is the geometrical path between the pinhole plane and the observation plane, and ${\alpha}_{ij}=\mathrm{arg}\text{\hspace{0.17em}}{W}_{ij}^{\text{SS}}\text{\hspace{0.17em}}.$ As well known, the measure of the contrast of the interference fringes is the so-called visibility coefficient defined as

On substituting from Eq. (17) with $\mathrm{cos}(\text{\hspace{0.17em}}.\text{\hspace{0.17em}})=\pm 1$ into Eq. (18), we find that

The spectra ${S}_{i}^{\text{SS}}$ and ${S}_{j}^{\text{SS}}$ can be easily measured when one of the pinholes is covered by an opaque screen. The phase ${\alpha}_{ij}$can be measured by determining the location of maxima in the interference pattern. Hence, measuring in each experiment the visibility ${V}_{ij}^{}\text{\hspace{0.17em}},$ power spectra ${S}_{i(j)}^{\text{SS}},$and phase ${\alpha}_{ij}$, one can determine all the elements ${W}_{ij}^{\text{SS}}$ of the matrix ${W}_{\text{SS}}$. The degree of coherence and the degree of polarization of the generated source can be then calculated using definitions given by Eqs. (3) and (4).

## 5. Experiments and results

To verify the efficiency of the proposed technique in practice, we realized a physical experiment sketched in Fig. 3 . The principal part of the experimental setup was composed of two interferometers shown in Figs. 1 and 2. Besides, to reduce the power loss, we used the output beam splitter of the first interferometer as the input beam splitter of the second interferometer. As the primary source we used the expanded well collimated linearly polarized beam generated by the He-Ne laser. To generate the secondary source with two different values of the transverse coherence length, we used two pairs of ground glass plates with the diffusion angles of 10° and 30°. The interference pattern at the first output of the second interferometer was registered by the CCD camera connected to the computer and the corresponding power spectra were measured by the photodiode with optical power meter located at the second output.

The results of the experiments for different pairs of ground glass plates and polarization angle $\text{\theta}={\text{45}}^{\circ}$are plotted in Fig. 4 together with their theoretical interpolations in accordance with Eq. (15). The calculated spectral degree of polarization changed in the full range from 1 to 0 when varying the polarization angle θ from 0° to ${\text{45}}^{\circ}$in both experiments. As can be seen, the obtained experimental results are in a good correspondence with the theoretical predictions.

## 6. Conclusions

We have proposed a rather simple technique for generating the partially coherent and partially polarized electromagnetic source using two rotating ground glass plates placed at the opposite arms of a Mach-Zehnder interferometer. We would like to stress particularly that such a source represents a special case of the well known Gaussian Schell-model uniformly polarized electromagnetic source [10], which appears here not as some handy mathematical construction but as a true product of physical experiment. Besides, the diagonal form of the cross-spectral density matrix given by Eq. (14), permits to find in a closed form the coherent-mode structure [15], a fact that results in considerable simplification when analyzing optical systems with such an illumination [16].

The efficiency of the proposed technique has been confirmed with the physical experiment. We consider that the proposed technique for generating the partially coherent and partially polarized electromagnetic source, as well as the employed technique of its characterization, can be easily reproduced in any advanced optics laboratory and, hence, will serve for subsequent developing the experimental researches on coherence and polarization of electromagnetic fields.

## Appendix: To derivation of Eq. (11)

Taking into account Eq. (7), one can state the relation

Then, making use of well known Fourier-transform relation

we find

Now we will introduce a random variable

Employing Eq. (7), one can write the probability distribution of this variable as

where

Calculating the square in Eq. (A6) and applying Eq. (8), we find

Then, by analogy with derivation of Eq. (A3), but this time for argument ψ, we find

Finally, on making use of Eq. (A3) and the assumption given by Eq. (9) in the main text, we find

Equations (A8) and (A9) are the required results to obtain Eq. (11) in the main text.

## Acknowledgements

The authors gratefully acknowledge the financial support from the Autonomous University of Puebla under project OSA-EXT-10-G.

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