Experimental study on modulation of Stokes parameters on propagation of a Gaussian Schell model beam in free space

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 xx δ , yy δ , xy δ are equal and the beam width parameters x y σ σ = . 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 Optical Society of America OCIS codes: (030.0030) Coherence and statistical optics; (260.5430) Polarization; (350.5500) Propagation. 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). 2. E. Wolf, “Polarization invariance in beam propagation,” Opt. Lett. 32(23), 3400–3401 (2007). 3. O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30(2), 198–200 (2005). 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). 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). 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). 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). 8. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(56), 263–267 (2003). 9. O. Korotkova, “Sufficient condition for polarization invariance of beams generated by quasi-homogeneous sources,” Opt. Lett. 36(19), 3768–3770 (2011). 10. E. Wolf, “Invariance of the Spectrum of Light on Propagation,” Phys. Rev. Lett. 56(13), 1370–1372 (1986). 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). 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). 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). 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). 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). #182512 $15.00 USD Received 7 Jan 2013; revised 18 Feb 2013; accepted 19 Feb 2013; published 21 Jun 2013 (C) 2013 OSA 1 July 2013 | Vol. 21, No. 13 | DOI:10.1364/OE.21.015432 | OPTICS EXPRESS 15432


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][2][3][4], and also in different media [5][6][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][10][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 xx δ , yy δ , xy δ are equal to each other and the beam width parameters x y σ σ = .Schell model sources are the sources whose spectral degree of coherence ( ) ( ) μ ω ξ ξ depends on the location of the two points only through the difference 1 2 ξ − ξ , of their position vector 1 ξ and 2 ξ [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].

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, ( ) where , i x y = denotes the two orthogonal polarizations, 0i A is the amplitude, ω is the frequency, ξ is the position vector in the source plane and the parameter i σ is related to the full width at half maximum of the beam.Here for simplicity it is assumed that x y σ σ σ = = .For a Gaussian Schell Model beam, the spectral degree of coherence in the plane at z = 0, has the form [12], where , , i j x y = .The coefficients 0i A , ij B and the parameter ij δ are independent of position but may depend on frequency.In addition, these parameters also have to satisfy some constraints [12,14,15], i.e. , 1 for , , 1 The parameter B ij is related to the degree of polarization [5,12].The cross spectral density matrix in the plane at z = 0 is given by, , ; , , , , .
From the beam propagation law [12], the cross spectral density matrix at any two points r 1 and r 2 in the plane at a distance 'z' perpendicular to the direction of propagation is given by, ( , , ; exp exp exp , 8 8 2 where 2 k is the wave propagation constant and the quantities ( ) The generalized Stokes parameters in terms of the elements of cross spectral density matrix in the observation plane (r, z) are given by [12], The degree of polarization in terms of the Stokes parameters in the observation plane (r, z) is given by [12], ( ) The Stokes parameters can be determined by putting 1 2 = r r 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 θ φ represents the intensity of #182512 -$15.00USD the beam recorded at CCD camera when the fast axis of the quarter wave plate makes an angle θ and polarization axis of the polarizer P 2 makes an angle φ with x-axis.The four Stokes parameters can be given by the following equations [16],

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 0y 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 σ used in Eq. ( 1) is calculated by ( )8 ln 2 FW HM of the Gaussian fit which equals to 0.248 mm.Using Eq. ( 9), the Stokes parameters are measured at different points along the propagation direction.The parameter δ used in Eq. ( 2) is calculated by ( )8 ln 2 FW HM of the Gaussian fit which comes out to be 0.260mm.Fig. 3. Experimental setup for determining the transverse coherence length in the source plane at z = 0. BS is Beam splitter, P 1 , P 2 are polarizers, T 1 and T 2 are tips of two single mode fibers, APD refers to Avalanche photodiode, TAC is time to amplitude converter, MCA is multichannel analyzer.Curve in the inset shows the plot of coincidence counts with the displacement between the two fiber tips T 1 and T 2 .Square dots represent the experimental values while the red curve represents its Gaussian fit.The FWHM of the Gaussian fit (red curve) is 0.613 mm.

Results and discussion
The effect of propagation of the beam on Stokes parameter is shown in Fig. 4 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 xx δ , yy δ and xy δ are equal.

Conclusion
In , 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.

Fig. 1 .
Fig. 1.Gaussian beam propagating in free space along the z-axis.The planes at ( ξ , z = 0) and ( r , z) are source plane and observation plane respectively.Here ξ and r are two dimensional vectors.

Fig. 2 .
Fig. 2. Schematics of the experimental setup.M, NDF, QWP, CCD refers to the mirror, Neutral density filter, Quarter wave plate and CCD camera respectively.P 1 and P 2 are polarizers.The curve in the inset shows the intensity profile of cross section of the beam in the plane at z = 0. Black dots represent the experimental values while red curve represents its Gaussian fit.The FWHM of the Gaussian fit (red curve) is 0.583 mm.The transverse coherence lengths xx δ , yy δ and xy δ 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 (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 xy ) comes out to 2 π which is obvious because quarter wave plate introduces a phase difference of 2 π between x and y component of the electric field.Slight decrease in the magnitude of B xy is due to experimental errors.The parameters ij B , ij δ and σ satisfy the constraints for a Gaussian Schell model beam given in Eqs.(3a) and (3b).

Fig. 4 .
Fig. 4. (a) Variation in the magnitude of the Stokes parameters on propagation of the Gaussian Schell model beam in free space.Dots represent the experimentally calculated values while thick curves represent theoretical curves obtained from Eqs. (9) and (7) respectively.The experimental parameters 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 xx δ , yy δ , xy δ are equal and the beam width parameters x y σ σ = , the normalized Stokes parameters and degree of polarization will not vary with propagation.However in the situation when