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Optics Express

  • Editor: Michael Duncan
  • Vol. 10, Iss. 15 — Jul. 29, 2002
  • pp: 699–706
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Stokes parameters of a Gaussian beam in a calcite crystal

Damiano Provenziani, Alessandro Ciattoni, Gabriella Cincotti, Claudio Palma, Francesco Ravaccia, and Carmine Sapia  »View Author Affiliations


Optics Express, Vol. 10, Issue 15, pp. 699-706 (2002)
http://dx.doi.org/10.1364/OE.10.000699


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Abstract

We derive the analytical expression of the Stokes parameters corresponding to a Gaussian beam propagating along the optical axis of a uniaxial crystal, pointing the simultaneous effects of anisotropy and diffraction out. The theoretical results are compared with experimental measurements at the output of a calcite crystal, showing a good agreement.

© 2002 Optical Society of America

1 Introduction

Optical technologies using liquid crystals, electro-optic devices, polarimetric sensors, anisotropic fibers and fiber amplifiers focus the research interest on the evolution of the state of polarization of light through anisotropic media [1

1. R. M. A. Azzam, B. E. Merrill, and N. M. Bashara, “Trajectories describing the evolution of polarized light in homogeneous anisotropic media and liquid crystals,” Appl. Opt. 12, 4, 764–771 (1973). [CrossRef] [PubMed]

]–[6

6. M. J. Bloemer and J. W. Haus, “Broadband waveguide polarizers based on the anisotropic optical constants of nanocomposite films,” IEEE J. Lightwave Tech. 14, 6, 1534–1540 (1996). [CrossRef]

]. Generally, the change of the polarization state is discussed for plane-waves, as within the Jones formalism or the Mueller calculus. The evolution of the Stokes parameters through an anisotropic medium is a well-understood phenomenon and a general treatment is given in literature for both monochromatic [7

7. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

]–[14

14. L. Dettwiller, “General expression of light intensity emerging from a linear anisotropic device using Stokes parameters,” J. Mod. Opt. 42, 4, 841–848 (1995). [CrossRef]

] and quasi-monochromatic (partially polarized) plane waves [15

15. C. Brosseau, “Evolution of the Stokes parameters in optically anisotropic media” Opt. Lett. 20, 11, 1221–1223 (1995). [CrossRef] [PubMed]

],[16

16. J. F. Mosiño, O. Barbosa-García, A. Starodumov, L. A. Díaz-Torres, M. A. Meneses-Nava, and J. T. Vega-Durán, “Evolution of partially polarized light through non-depolarizing anisotropic media the Stokes parameters in optically anisotropic media” Opt. Commun. 173, 57–71 (2000). [CrossRef]

]. But the plane-wave analysis does not take the diffraction effects into account and it could be a serious shortcoming when the beam width is comparable with the wavelength. Many numerical and analytical methods have been proposed to evaluate the propagation of an optical beam through an unbounded anisotropic media [17

17. J.J. Stamnes and G.C. Sherman, “Radiation of electromagnetic fields in uniaxially anisotropic media,” J. Opt. Soc. Am. A 66, 780–788 (1976). [CrossRef]

],[21

21. R. Martínez-Herrero, J. M. Movilla, and P. M. Mejías, “Radiation of electromagnetic fields in uniaxially anisotropic media”, J. Opt. Soc. Am. A 18, 8, 2009–2014 (2001). [CrossRef]

], but a comparison between computational and experimental results is lacking in literature.

In the present paper, we theoretically and experimentally investigate the evolution of the Stokes parameters associated with a laser beam propagating along the optical axis of a calcite crystal; due to the simultaneous effects of anisotropy and diffraction, the propagating field is not uniformly polarized, and the Stokes parameters are functions of the position vector at any transverse plane. The Stokes parameters are computed in Sec. 2, starting from the expression of the fundamental Gaussian beam inside a uniaxial medium given in Ref. [22

22. G. Cincotti, A. Ciattoni, and C. Palma, “Hermite-Gauss beams in uniaxially anisotropic crystals,” IEEE J. Quantum Electron. 37, 12, 1517–1524 (2001). [CrossRef]

], whereas the experimental setup and results are described in Sec. 3. The agreement between theory and experiment is such to confirm the validity of the theoretical approach.

The case of light propagation at any angle to the oprical axis can be treated in a similar way, but the algebra is somewhat more tedius.

2 Stokes parameters

The use of the Stokes parameters is a standard method to characterize the state of polarization of an optical field, by means of simple measurements of the intensity distribution. Referring to the electric field E (x, y, z) = Ex (x, y, z) êx + Ey (x, y, z) êy, the four Stokes parameters are defined as

s0(x,y,z)=Ex(x,y,z)2+Ey(x,y,z)2
s1(x,y,z)=Ex(x,y,z)2Ey(x,y,z)2
s2(x,y,z)=Ex(x,y,z)Ey*(x,y,z)+Ey(x,y,z)Ex*(x,y,z)
s3(x,y,z)=i[Ey(x,y,z)Ex*(x,y,z)Ex(x,y,z)Ey*(x,y,z)],
(1)

where asterisk denotes the complex conjugate. The parameters s 0 and s 1 are straightforwardly evaluated by summing or subtracting the intensities of light transmitted by two polarizers which accept linear polarization along the x- or y-axis, respectively; the parameter s 2 is analogously obtained rotating the polarizer of 45o and 135o with respect to the x-axis, whereas s 3 is measured at the output of the same polarizer, when a further phase difference among Ex and Ey has been introduced with a π/2 compensator, i.e. a λ/4 plate with the slow axis lying along the y-axis [9

9. M. Born and E. Wolf, Principles of optics (Pergamon Press, Oxford1993).

].

We consider the propagation of a Gaussian beam along the optical axis z of a uniaxial crystal with ordinary no and extraordinary ne refractive indices; if the input beam is linearly polarized along the x-axis, the Cartesian components of the field inside the crystal have the following expressions [22

22. G. Cincotti, A. Ciattoni, and C. Palma, “Hermite-Gauss beams in uniaxially anisotropic crystals,” IEEE J. Quantum Electron. 37, 12, 1517–1524 (2001). [CrossRef]

]

Ex(x,y,z)=E̅w02x2+y2exp(ik0noz)[(y2Qo(z)+y2x22(x2+y2))exp(x2+y2Qo(z))
+(x2Qe(z)+x2y22(x2+y2))exp(x2+y2Qe(z))]
Ey(x,y,z)=E̅w02x2+y2eik0noz[(xyQo(z)+xyx2+y2)exp(x2+y2Qo(z))
+(xyQe(z)+xyx2+y2)exp(x2+y2Qe(z))],
(2)

where Qo (z) = w02 + i(2z)/(k 0 no ) and Qe (z) = w02 + i(2z no )/(k 0 ne2) are the complex propagation parameters associated to the ordinary and extraordinary components, respectively. Substituting Eqs. (2) into (1), we obtain a compact expression of the four Stokes parameters

sn(x,y,z)=E̅2w04(x2+y2)2{fn(o)(x,y,z)exp(2w02(x2+y2)Qo(z)2)
+fn(e)(x,y,z)exp(2w02(x2+y2)Qe(z)2)
+fn(oe)(x,y,z)exp[w02(x2+y2)(1Qo(z)2+1Qe(z)2)]}
(n=0,1,2,3),
(3)

with

fn(oe)(x,y,z)=fn(s)(x,y,z)sin[2zk0no(1Qo(z)2no2ne21Qe(z)2)(x2+y2)]
fn(c)(x,y,z)cos[2zk0no(1Qo(z)2no2ne21Qe(z)2)(x2+y2)],
(4)

and

f0(o)(x,y,z)=14+(w02+x2+y2)y2Qo(z)2
f0(e)(x,y,z)=14+(w02+x2+y2)x2Qe(z)2
f0(s)(x,y,z)=2zk0no(y2Qo(z)2no2ne2x2Qe(z)2)
f0(c)(x,y,z)=(12+x2w02Qe(z)2+y2w02Qo(z)2)
f1(o)(x,y,z)=x4+y46x2y24(x2+y2)2+y2(y4x4+(y23x2)w02)(x2+y2)Qo(z)2
f1(e)(x,y,z)=x4+y46x2y24(x2+y2)2+x2(x4y4+(x23y2)w02)(x2+y2)Qe(z)2
f1(s)(x,y,z)=2zk0no(x2(x23y2)x2+y2no2ne21Qe(z)2y2(y23x2)x2+y21Qo(z)2
+4x2y2ne2no2ne2w02Qo(z)2Qe(z)2)
f1(c)(x,y,z)=12+4x2y2(1(x2+y2)2+4z2k02ne2+w04Qo(z)2Qe(z)2)+y2(y23x2)x2+y2w02Qo(z)2
+x2(x23y2)x2+y2w02Qe(z)2
f2(o)(x,y,z)=xy(x2y2(x2+y2)22y2Qo(z)23y2x2x2+y2w02Qo(z)2)
f2(e)(x,y,z)=xy(x2y2(x2+y2)2+2x2Qe(z)2+3x2y2x2+y2w02Qe(z)2)
f2(s)(x,y,z)=2zxyk0n0(3y2x2x2+y21Qo(z)2+3x2y2x2+y2no2ne21Qe(z)2
+2(x2y2)no2ne2ne2w02Qo(z)2Qe(z)2)
f2(c)(x,y,z)=xy[3y2x2x2+y2w02Qo(z)23x2y2x2+y2w02Qe(z)2
+(y2x2)(1(x2+y2)2+w04+4z2k02ne2Qo(z)2Qe(z)2)]
f3(o)(x,y,z)=xy2zk0n01Qo(z)2
f3(e)(x,y,z)=xy2zk0n0no2ne21Qe(z)2
f3(s)(x,y,z)=xy(w02Qo(z)2+w02Qe(z)2+2(x2+y2)w044z2k02ne2Qo(z)2Qe(z)2)
f3(c)(x,y,z)=xy2zk0no(no2ne21Qe(z)21Qo(z)2+2(x2+y2)w02Qo(z)2Qe(z)2)
(5)

The movie sequences of Fig. 1 show the evolution of the Stokes parameters for propagation distances up to z = 4zRo , being zRo = πw02 no /λ the Rayleigh distance in a homogeneous medium, with refraction index no . From an inspection of s 0 and s 1, the combined effects of anisotropy and diffraction is evident, as the beam loses its boundary cylindrical symmetry due to different diffraction lengths of the ordinary and extraordinary components [23

23. A. Ciattoni, G. Cincotti, and C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A , 19, 792–796 (2002). [CrossRef]

]. On the other hand, from the animation of s 2 and s 3 one can notice that for z = zRe , being zRe = πw02 ne /λ the Rayleigh distance in a homogeneous medium with refraction index ne , the phase difference between Ex and Ey practically vanishes. In fact, both the moduli of s 2 and s 3 (that are zero at the input plane z = 0) at first grow with z and then decrease: in the plane z = zRe , s 3 vanishes almost everywhere, whereas | s 2 | is maximum, so that the beam is almost linearly polarized. Of course, due to the anisotropy, the polarization direction is different at any position.

Fig. 1. Animations of the evolution of the Stokes parameters associated with a Gaussian beam propagating along the optical axis of a calcite crystal, from the z = 0 to the z = 4zRo planes; the transverse coordinates are normalized with respect to the input spot size w 0. [Media 1] [Media 2] [Media 3] [Media 4]

3 Experiment

Fig. 2. Laboratory setup.

We report in Fig. 3 the experimental and the numerical results for the case of a Gaussian beam with input spot-size w 0 = 10μm, propagated for a distance z = 20zRo ; analogously, Fig. 4 shows the Stokes parameter for the case w 0 = 15μm and z = 8.5zRo . It is evident that the agreement between theory and measurements is pretty good. To have a deeper insight of the results achieved, in Fig. 5 we contrast the numerical and experimental intensity profiles of the Stokes parameters evaluated at the x = y plane (for s 0, s 1 and s 3) or at the x = 6w 0 plane (for s 2 that vanishes along the x, y bisector).

4 Summary

The combined effects of anisotropy and diffraction on a Gaussian beam propagating along the optical axis of a calcite crystal have been investigated, by means of numerical and experimental studies of the evolution of the Stokes parameters. The agreement between theory and experiment is good. We also show that the propagating beam is almost linearly polarized at the plane z = πw02 ne /λ, even though the polarization direction varies with the position vector.

Fig. 3. Numerical (left hand) and experimental (right hand) results for Stokes parameters of a Gaussian beam with input spot size w 0 = 10μm propagated for a distance z = 20zRo along the optical axis of a calcite crystal.
Fig. 4. The same as Fig. 3 for w 0 = 15μm and z = 8.5zR0 .
Fig. 5. Comparison between numerical and experimental results for the Stokes parameters of Fig. 3 evaluated for x = y or x = 6w 0. Solid line refers to experimental values and dashed line to numerical values.
Fig. 6. Comparison between numerical and experimental results for the Stokes parameters of Fig. 4 evaluated for x = y or x = 6w 0. Solid line refers to experimental values and dashed line to numerical values.

References and links

1.

R. M. A. Azzam, B. E. Merrill, and N. M. Bashara, “Trajectories describing the evolution of polarized light in homogeneous anisotropic media and liquid crystals,” Appl. Opt. 12, 4, 764–771 (1973). [CrossRef] [PubMed]

2.

M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21, 23, 1948–1950 (1996). [CrossRef] [PubMed]

3.

J. L. Wagener, D. G. Falquier, J. J. F. Digonnet, and H. J. Shaw, “A Mueller matrix formalism for modelling polarization effects in erbium-doped fiber,” IEEE J. Lightwave Tech. 16, 2, 200–206 (1998). [CrossRef]

4.

W. M. Shute, C. S. Brown, and J. Jarzynski, “Polarization model for a helically wound optical fiber,” J. Opt. Soc. Am. A 14, 12, 3251–3261 (1997). [CrossRef]

5.

Z. K. Ioannidis, R. Kadiwar, and I. Giles, “Anisotropic polarization maintaining optical fiber ring resonators,” IEEE J. Lightwave Tech. 14, 3, 377–384 (1996). [CrossRef]

6.

M. J. Bloemer and J. W. Haus, “Broadband waveguide polarizers based on the anisotropic optical constants of nanocomposite films,” IEEE J. Lightwave Tech. 14, 6, 1534–1540 (1996). [CrossRef]

7.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

8.

W. A. Shurcliff, Polarized light (Harvard Univ. Press, Cambridge, MA, 1962).

9.

M. Born and E. Wolf, Principles of optics (Pergamon Press, Oxford1993).

10.

E. Collett, Polarized light (Marcel Dekker, New York, 1992).

11.

R. M. Azzam and N. M. Bashara, Ellipsometry and polarized light (North-Holland, Amsterdam, 1977).

12.

R. C. Jones, “New calculus for the treatment of optical systems” J. Opt. Soc. Am. A 31, 488–450 (1941). [CrossRef]

13.

C. Brosseau, Fundamentals of polarized light (Wiley, New York, 1998).

14.

L. Dettwiller, “General expression of light intensity emerging from a linear anisotropic device using Stokes parameters,” J. Mod. Opt. 42, 4, 841–848 (1995). [CrossRef]

15.

C. Brosseau, “Evolution of the Stokes parameters in optically anisotropic media” Opt. Lett. 20, 11, 1221–1223 (1995). [CrossRef] [PubMed]

16.

J. F. Mosiño, O. Barbosa-García, A. Starodumov, L. A. Díaz-Torres, M. A. Meneses-Nava, and J. T. Vega-Durán, “Evolution of partially polarized light through non-depolarizing anisotropic media the Stokes parameters in optically anisotropic media” Opt. Commun. 173, 57–71 (2000). [CrossRef]

17.

J.J. Stamnes and G.C. Sherman, “Radiation of electromagnetic fields in uniaxially anisotropic media,” J. Opt. Soc. Am. A 66, 780–788 (1976). [CrossRef]

18.

J.A. Fleck Jr. and M. D. Feit, “Beam propagation in uniaxial anisotropic media,” J. Opt. Soc. Am. A 73, 920–926 (1983). [CrossRef]

19.

J. M. Liu and L. Gomelsky, “Vectorial beam propagation method,” J. Opt. Soc. Am. A 9, 9, 1574–1585 (1992). [CrossRef]

20.

S. Selleri, L. Vincetti, and M. Zoboli, “Full-vector finite-element beams propagation method for anisotropic optical device analysis,” IEEE J. Quantum Electron. 36, 1392–1401 (2000). [CrossRef]

21.

R. Martínez-Herrero, J. M. Movilla, and P. M. Mejías, “Radiation of electromagnetic fields in uniaxially anisotropic media”, J. Opt. Soc. Am. A 18, 8, 2009–2014 (2001). [CrossRef]

22.

G. Cincotti, A. Ciattoni, and C. Palma, “Hermite-Gauss beams in uniaxially anisotropic crystals,” IEEE J. Quantum Electron. 37, 12, 1517–1524 (2001). [CrossRef]

23.

A. Ciattoni, G. Cincotti, and C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A , 19, 792–796 (2002). [CrossRef]

OCIS Codes
(260.1180) Physical optics : Crystal optics
(260.1960) Physical optics : Diffraction theory
(260.5430) Physical optics : Polarization

ToC Category:
Research Papers

History
Original Manuscript: May 23, 2002
Revised Manuscript: July 24, 2002
Published: July 29, 2002

Citation
Damiano Provenziani, Alessandro Ciattoni, Gabriella Cincotti, Claudio Palma, Francesco Ravaccia, and Carmine Sapia, "Stokes parameters of a Gaussian beam in a calcite crystal," Opt. Express 10, 699-706 (2002)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-15-699


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References

  1. R. M. A. Azzam, B. E. Merrill, and N. M. Bashara, ???Trajectories describing the evolution of polarized light in homogeneous anisotropic media and liquid crystals,??? Appl. Opt. 12, 764-771 (1973). [CrossRef] [PubMed]
  2. M. Stalder, and M. Schadt, ???Linearly polarized light with axial symmetry generated by liquidcrystal polarization converters,??? Opt. Lett. 21, 1948-1950 (1996). [CrossRef] [PubMed]
  3. J. L. Wagener, D. G. Falquier, J. J. F. Digonnet, and H. J. Shaw, ???A Mueller matrix formalism for modelling polarization effects in erbium-doped fiber,??? IEEE J. Lightwave Tech. 16, 200-206 (1998). [CrossRef]
  4. W. M. Shute, C. S. Brown, and J. Jarzynski, ???Polarization model for a helically wound optical fiber,??? J. Opt. Soc. Am. A 14, 3251-3261 (1997). [CrossRef]
  5. Z. K. Ioannidis, R. Kadiwar, and I. Giles, ???Anisotropic polarization maintaining optical fiber ring resonators,??? IEEE J. Lightwave Tech. 14, 377-384 (1996). [CrossRef]
  6. M. J. Bloemer, and J. W. Haus, ???Broadband waveguide polarizers based on the anisotropic optical constants of nanocomposite films,??? IEEE J. Lightwave Tech. 14, 1534-1540 (1996). [CrossRef]
  7. A. Yariv, and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).
  8. W. A. Shurcliff, Polarized light (Harvard Univ. Press, Cambridge, MA, 1962).
  9. M. Born, and E. Wolf, Principles of optics (Pergamon Press, Oxford 1993).
  10. E. Collett, Polarized light (Marcel Dekker, New York, 1992).
  11. R. M. Azzam, and N. M. Bashara, Ellipsometry and polarized light (North-Holland, Amsterdam, 1977).
  12. R. C. Jones, ???New calculus for the treatment of optical systems,??? J. Opt. Soc. Am. A 31, 488-450 (1941). [CrossRef]
  13. C. Brosseau, Fundamentals of polarized light (Wiley, New York, 1998).
  14. L. Dettwiller, ???General expression of light intensity emerging from a linear anisotropic device using Stokes parameters,??? J. Mod. Opt. 42, 841-848 (1995). [CrossRef]
  15. C. Brosseau, ???Evolution of the Stokes parameters in optically anisotropic media,??? Opt. Lett. 20, 1221-1223 (1995). [CrossRef] [PubMed]
  16. J. F. Mosi??no, O. Barbosa-Garcýa, A. Starodumov, L. A. Dýaz-Torres, M. A. Meneses-Nava, and J. T. Vega-Duran, ???Evolution of partially polarized light through non-depolarizing anisotropic media the Stokes parameters in optically anisotropic media,??? Opt. Commun. 173, 57-71 (2000). [CrossRef]
  17. J.J. Stamnes, and G.C. Sherman, ???Radiation of electromagnetic .elds in uniaxially anisotropic media,??? J. Opt. Soc. Am. A 66, 780-788 (1976). [CrossRef]
  18. J.A. Fleck Jr., and M. D. Feit, ???Beam propagation in uniaxial anisotropic media,??? J. Opt. Soc. Am. A 73, 920-926 (1983). [CrossRef]
  19. J. M. Liu, and L. Gomelsky, ???Vectorial beam propagation method,??? J. Opt. Soc. Am. A 9, 1574-1585 (1992). [CrossRef]
  20. S. Selleri, L. Vincetti, and M. Zoboli, ???Full-vector finite-element beams propagation method for anisotropic optical device analysis,??? IEEE J. Quantum Electron. 36, 1392-1401 (2000). [CrossRef]
  21. R. Martýnez-Herrero, J. M. Movilla, and P. M. Mejýas, ???Radiation of electromagnetic fields in uniaxially anisotropic media,??? J. Opt. Soc. Am. A 18, 2009-2014 (2001). [CrossRef]
  22. G. Cincotti, A. Ciattoni, and C. Palma, ???Hermite-Gauss beams in uniaxially anisotropic crystals,??? IEEE J. Quantum Electron. 37, 1517-1524 (2001). [CrossRef]
  23. A. Ciattoni, G. Cincotti, and C. Palma, ???Propagation of cylindrically symmetric fields in uniaxial crystals,??? J. Opt. Soc. Am. A 19, 792-796 (2002). [CrossRef]

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