## Abstract

The effect on the Stokes parameters of a Gaussian Schell model beam on propagation in free
space is studied experimentally and results are matched with the theory [X. H. Zhao, et al.
Opt. Express **17**, 17888 (2009)] that in general the degree of polarization of a
Gaussian Schell model beam doesn’t change on propagation if the three spectral
correlation widths ${\delta}_{xx}$, ${\delta}_{yy}$, ${\delta}_{xy}$ are equal and the beam width parameters ${\sigma}_{x}={\sigma}_{y}$. It is experimentally shown that all the four Stokes parameters
at the center of the beam decrease on propagation while the magnitudes of the normalized Stokes
parameters and the spectral degree of polarization at the center of the beam remain constant
for different propagation distances.

© 2013 OSA

## 1. Introduction

In recent years, a lot of research has shown that the spectral degree of coherence and spectral degree of polarization may change on propagation of an electromagnetic Gaussian Schell model beam in free space [1–4], and also in different media [5–7]. The recently developed Wolf’s unified theory of coherence and polarization [8], provides an intimate relationship between coherence and polarization properties of an electromagnetic beam and can predict changes in the coherence and polarization properties of the beam as it propagates.

In this paper we have verified experimentally the theoretical prediction [1, 9–11], that in general the degree of polarization of a Gaussian Schell model beam doesn’t change on propagation in free space if the spectral correlation lengths ${\delta}_{xx}$, ${\delta}_{yy}$, ${\delta}_{xy}$ are equal to each other and the beam width parameters ${\sigma}_{x}={\sigma}_{y}$. Schell model sources are the sources whose spectral degree of coherence ${\mu}^{\left(0\right)}\left({\xi}_{1},{\xi}_{2},\omega \right)$ depends on the location of the two points only through the difference${\xi}_{1}-{\xi}_{2}$, of their position vector ${\xi}_{1}$ and ${\xi}_{2}$ [12]. It is shown that the magnitudes of the four Stokes parameters at the center of the beam change with the distance of propagation but the magnitude of the normalized stokes parameters and the degree of polarization remain unchanged. The study might be helpful in the area of free space optical communication (FSO) technology in which various polarization shift keying (PolSK) modulation schemes are used [13].

## 2. Theory

To prove the prediction made we would like to have a glimpse of some important results. Consider an electromagnetic Gaussian Schell model beam generated by a source in the plane at z = 0, propagating along the z-axis into the half-space z>0 as shown in Fig. 1. The spectral density for the x and y components of the beam in the plane at z = 0 are given by,

For a Gaussian Schell Model beam, the spectral degree of coherence in the plane at z = 0, has the form [12],

_{,}${B}_{ij}$ and the parameter ${\delta}_{ij}$ are independent of position but may depend on frequency. In addition, these parameters also have to satisfy some constraints [12, 14, 15], i.e.

*B*is related to the degree of polarization [5, 12]. The cross spectral density matrix in the plane at z = 0 is given by,

_{ij}_{1}and r

_{2}in the plane at a distance ‘z’ perpendicular to the direction of propagation is given by,

The Stokes parameters can be determined by putting ${r}_{1}={r}_{2}$ in the Eq. (7) and
can be calculated from experimental measurements [16],
using a combination of a quarter wave plate (QWP) and a linear polarizer in series with
appropriate orientations of the fast axis and polarization axis respectively as shown in Fig. 2. Suppose $I\left(\theta ,\text{\hspace{0.17em}}\varphi \right)$ represents the intensity of the beam recorded at CCD camera when
the fast axis of the quarter wave plate makes an angle $\theta $ and polarization axis of the polarizer P_{2} makes an
angle $\varphi $ with x-axis. The four Stokes parameters can be given by the
following equations [16],

## 3. Experimental details

We have used a single mode linearly polarized intensity stabilized 632.8 nm He-Ne laser
(Mellus Griot) as shown in Fig. 2. Since the laser is
vertically polarized, the polarization axis of the polarizer P_{1} is oriented
arbitrarily so that the experiment can be generalized for any orientation of the plane of
polarization of the linearly polarized light. The ratio of peak intensities
*I _{0x}* to

*I*in the plane at z = 0 is equal to 2.1 and the intensity profile of the beam in the plane is shown in the inset of Fig. 2. Black dots represent the experimental values while thick red curve represents its Gaussian fit. The full width at half maximum (FWHM) of the Gaussian fit (thick red curve) comes out to be 0.583 mm. The parameter $\sigma $ used in Eq. (1) is calculated by $\left(FWHM\text{\hspace{0.17em}}of\text{\hspace{0.17em}}the\text{\hspace{0.17em}}Gaussian\text{\hspace{0.17em}}fit\right)/\sqrt{8\text{\hspace{0.17em}}\mathrm{ln}2}$ which equals to 0.248 mm. Using Eq. (9), the Stokes parameters are measured at different points along the propagation direction.

_{0y}The transverse coherence lengths ${\delta}_{xx}$**, **${\delta}_{yy}$ and ${\delta}_{xy}$ are measured using the HBT setup (Hanbury Brown and Twiss type
setup, in which intensity-intensity correlations are measured rather than measuring the
amplitude correlations [17]) by keeping the polarization
axis of the polarizers P_{2} and P_{3} in the (x, x), (y, y) and (x, y)
directions respectively as shown in Fig. 3. The beam is divided into two parts with the help of a 50-50 beam splitter. After that
two polarizers P_{2} and P_{3} are placed in different arms of the HBT setup
followed by two single mode fibers T_{1} and T_{2} connected to two avalanche
photodiodes (APDs). One of the outputs of these APDs is fed to the ‘Start’
terminal and the other one is fed to the ‘Stop’ terminal (with a delay of 110
microsec) of a time to amplitude converter (TAC). The coincidence counts are measured by keeping
one of the fiber tips fixed and other moving with the help of a computerized translational
stage. The coincidence counts are recorded with a multichannel analyzer (MCA) connected to the
computer. The counts are taken for an integration time of 10s. The individual detector counts
are kept low enough to eliminate accidental coincidence counts. For the laser beam the
parameters ${\delta}_{xx}={\delta}_{yy}={\delta}_{xy}=\delta $. The curve showing the coincidence counts with the displacement
between the two fiber tips T_{1} and T_{2} is shown in the inset in Fig. 3. Black dots represent the experimental values while
thick red curve represents its Gaussian fit. The FWHM of the Gaussian fit (red curve) in the
inset of Fig. 3, is 0.613mm. The parameter
$\delta $ used in Eq. (2) is
calculated by $\left(FWHM\text{\hspace{0.17em}}of\text{\hspace{0.17em}}the\text{\hspace{0.17em}}Gaussian\text{\hspace{0.17em}}fit\right)/\sqrt{8\text{\hspace{0.17em}}\mathrm{ln}2}$ which comes out to be 0.260mm.

## 4. Results and discussion

The effect of propagation of the beam on Stokes parameter is shown in Fig. 4(a). The experimental values (black dots) fit reasonably well with the theoretically expected
values (thick curves), within experimental uncertainty, with the parameters used in the
experiment. After fitting theoretical curves the magnitude of *B _{xy}*
comes out to ~0.91 (which is nearly equal to one since laser beam is linearly polarized) and
arg(

*B*) comes out to $\pi /2$ which is obvious because quarter wave plate introduces a phase difference of $\pi /2$ between x and y component of the electric field. Slight decrease in the magnitude of

_{xy}*B*is due to experimental errors. The parameters ${B}_{ij}$, ${\delta}_{ij}$ and $\sigma $ satisfy the constraints for a Gaussian Schell model beam given in Eqs. (3a) and (3b).

_{xy}In Fig. 4(b), we see that the magnitudes of normalized
Stokes parameters S_{1}/S_{0}, S_{2}/S_{0} and
S_{3}/S_{0} remain essentially constant and the experimental data agrees well
with the theoretical prediction (thick curves) within experimental uncertainty. Since the
spectral degree of polarization is square root of the squared sum of these normalized Stokes
parameters as shown in Eq. (8), therefore it
remains constant also. The constant behavior of the normalized Stokes parameters and the
spectral degree of polarization on propagation of the beam is predicted [1, 12], because the transverse coherence
lengths ${\delta}_{xx}$**, **${\delta}_{yy}$ and ${\delta}_{xy}$are equal.

## 5. Conclusion

In summary, the modulation of the Stokes parameters at the center of a Gaussian Schell model beam on propagation is studied experimentally. It is shown that the magnitudes of the Stokes parameters decrease and the magnitudes of the normalized Stokes parameters and spectral degree of polarization remain unchanged on propagation of a Gaussian Schell model beam in free space. The decrease in the magnitude of the Stokes parameters is due to the diffraction effects. The experiment has been verified by putting the experimental parameters in the theory given above. It is shown that if the conditions that the three spectral correlation widths ${\delta}_{xx}$, ${\delta}_{yy}$, ${\delta}_{xy}$ are equal and the beam width parameters ${\sigma}_{x}={\sigma}_{y}$, the normalized Stokes parameters and degree of polarization will not vary with propagation. However in the situation when ${\delta}_{xx}\ne {\delta}_{yy}\ne {\delta}_{xy}$, the normalized Stokes parameters and degree of polarization will vary with propagation of the beam in free space [12]. These results might have applications in free space optical (FSO) communication technologies where polarization shift keying (PolSK) modulation schemes are used.

## Acknowledgments

The author M. Verma thanks the Council of Scientific and Industrial Research (CSIR), New Delhi for providing financial support as a Senior Research Fellowship. The authors also thank the Director, NPL and the Director, IIT Delhi for giving permission to publish this paper.

## References and links

**1. **X. H. Zhao, Y. Yao, Y. Sun, and C. Liu, “Condition for Gaussian Schell-model beam to maintain
the state of polarization on the propagation in free space,”
Opt. Express **17**(20), 17888–17894
(2009). [CrossRef] [PubMed]

**2. **E. Wolf, “Polarization invariance in beam
propagation,” Opt. Lett. **32**(23), 3400–3401
(2007). [CrossRef] [PubMed]

**3. **O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic
beams,” Opt. Lett. **30**(2), 198–200
(2005). [CrossRef] [PubMed]

**4. **D. F. V. James, “Change of polarization of light beams on propagation in
free space,” J. Opt. Soc. Am. A **11**(5), 1641–1643
(1994). [CrossRef]

**5. **O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of
electromagnetic beams propagating through atmospheric turbulence,”
Opt. Commun. **233**(4-6), 225–230
(2004). [CrossRef]

**6. **H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent
electromagnetic beams propagating through the turbulent atmosphere,”
J. Mod. Opt. **52**, 1611–1618 (2005). [CrossRef]

**7. **M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent
electromagnetic beams propagating through turbulent atmosphere,”
Waves Random Media **14**(4), 513–523
(2004). [CrossRef]

**8. **E. Wolf, “Unified theory of coherence and polarization of random
electromagnetic beams,” Phys. Lett. A **312**(5-6), 263–267
(2003). [CrossRef]

**9. **O. Korotkova, “Sufficient condition for polarization invariance of
beams generated by quasi-homogeneous sources,” Opt.
Lett. **36**(19), 3768–3770
(2011). [CrossRef] [PubMed]

**10. **E. Wolf, “Invariance of the Spectrum of Light on
Propagation,” Phys. Rev. Lett. **56**(13), 1370–1372
(1986). [CrossRef] [PubMed]

**11. **J. Pu, O. Korotkova, and E. Wolf, “Invariance and noninvariance of the spectra of
stochastic electromagnetic beams on propogation,” Opt.
Lett. **31**(14), 2097–2099
(2006). [CrossRef] [PubMed]

**12. **E. Wolf, *Introduction to the Theory of Coherence
and Polarization of Light* (Cambridge University, 2007).

**13. **X. Zhao, Y. Yao, Y. Sun, and C. Liu, “Circle polarization shift keying with direct detection
for free-space optical communication,” J. Opt. Commun.
Netw. **1**(4), 307–312
(2009). [CrossRef]

**14. **O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an
electromagnetic Gaussian Schell-model source,” Opt.
Lett. **29**(11), 1173–1175
(2004). [CrossRef] [PubMed]

**15. **F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model
beams,” J. Opt. A, Pure Appl. Opt. **3**(1), 1–9
(2001). [CrossRef]

**16. **L. Mandel and E. Wolf, *Optical Coherence and
Quantum Optics* (Cambridge University, 1995).

**17. **R. H. Brown and R. Twiss, “Correlation between photons in two coherent beams of
light,” Nature **177**(4497), 27–29
(1956). [CrossRef]