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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 13 — Jul. 1, 2013
  • pp: 15432–15437
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Experimental study on modulation of Stokes parameters on propagation of a Gaussian Schell model beam in free space

Manish Verma, P. Senthilkumaran, Joby Joseph, and H. C. Kandpal  »View Author Affiliations


Optics Express, Vol. 21, Issue 13, pp. 15432-15437 (2013)
http://dx.doi.org/10.1364/OE.21.015432


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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 δ x x , δ y y , δ x y are equal and the beam width parameters σ x = σ 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

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]

4

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]

], and also in different media [5

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]

7

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]

]. The recently developed Wolf’s unified theory of coherence and polarization [8

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

], 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

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]

, 9

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

11

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]

], 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 δxx, δyy, δxy are equal to each other and the beam width parameters σx=σy. Schell model sources are the sources whose spectral degree of coherence μ(0)(ξ1,ξ2,ω) depends on the location of the two points only through the differenceξ1ξ2, of their position vector ξ1 and ξ2 [12

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

]. 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

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]

].

2. Theory

P(r,z)=(S1S0)2+(S2S0)2+(S3S0)2.
(8)

The Stokes parameters can be determined by putting r1=r2 in the Eq. (7) and can be calculated from experimental measurements [16

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

], 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
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. P1 and P2 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.
. Suppose I(θ,ϕ) represents the intensity of 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 P2 makes an angle ϕ with x-axis. The four Stokes parameters can be given by the following equations [16

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

],

S0=I(0,0)+I(0,90)S1=I(0,0)I(0,90)S2=I(0,45)I(0,135)S3=I(45,45)I(45,135)}.
(9)

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 P1 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 I0x to I0y 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 (FWHMoftheGaussianfit)/8ln2 which equals to 0.248 mm. Using Eq. (9), the Stokes parameters are measured at different points along the propagation direction.

The transverse coherence lengths δxx, δyy and δxy 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

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

]) by keeping the polarization axis of the polarizers P2 and P3 in the (x, x), (y, y) and (x, y) directions respectively as shown in Fig. 3
Fig. 3 Experimental setup for determining the transverse coherence length in the source plane at z = 0. BS is Beam splitter, P1, P2 are polarizers, T1 and T2 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 T1 and T2. 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.
. The beam is divided into two parts with the help of a 50-50 beam splitter. After that two polarizers P2 and P3 are placed in different arms of the HBT setup followed by two single mode fibers T1 and T2 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 δxx=δyy=δxy=δ. The curve showing the coincidence counts with the displacement between the two fiber tips T1 and T2 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 δ used in Eq. (2) is calculated by (FWHMoftheGaussianfit)/8ln2 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)
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 A0x/A0y=(I0x/I0y)1/2=1.45, |Bxy| = 0.91, arg(Bxy)=π/2, σ = 0.248mm, ω=3×1015sec1, c=3×108m/sec, δxx = δyy = δxy = δ = 0.260mm are put in Eqs. (5) and (6) for getting the theoretically expected results. (b) Variation of normalized Stokes parameters S1/S0, S2/S0 and S3/S0 on propagation of the beam in free space. Dots represent the experimentally calculated values while thick curves represent theoretical curves.
. 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 Bxy comes out to ~0.91 (which is nearly equal to one since laser beam is linearly polarized) and arg(Bxy) comes out to π/2 which is obvious because quarter wave plate introduces a phase difference of π/2 between x and y component of the electric field. Slight decrease in the magnitude of Bxy is due to experimental errors. The parameters Bij, δij and σ satisfy the constraints for a Gaussian Schell model beam given in Eqs. (3a) and (3b).

In Fig. 4(b), we see that the magnitudes of normalized Stokes parameters S1/S0, S2/S0 and S3/S0 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

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]

, 12

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

], because the transverse coherence lengths δxx, δyy and δxyare 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 δxx, δyy, δxy are equal and the beam width parameters σx=σy, the normalized Stokes parameters and degree of polarization will not vary with propagation. However in the situation when δxxδyyδxy, the normalized Stokes parameters and degree of polarization will vary with propagation of the beam in free space [12

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

]. 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]

OCIS Codes
(030.0030) Coherence and statistical optics : Coherence and statistical optics
(260.5430) Physical optics : Polarization
(350.5500) Other areas of optics : Propagation

ToC Category:
Coherence and Statistical Optics

History
Original Manuscript: January 7, 2013
Revised Manuscript: February 18, 2013
Manuscript Accepted: February 19, 2013
Published: June 21, 2013

Citation
Manish Verma, P. Senthilkumaran, Joby Joseph, and H. C. Kandpal, "Experimental study on modulation of Stokes parameters on propagation of a Gaussian Schell model beam in free space," Opt. Express 21, 15432-15437 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-13-15432


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References

  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. Express17(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. A11(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 Media14(4), 513–523 (2004). [CrossRef]
  8. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A312(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,” Nature177(4497), 27–29 (1956). [CrossRef]

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