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

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References

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Appl. Opt.

IEEE J. Lightwave Tech.

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]

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

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]

IEEE J. Quantum Electron.

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]

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

J. Mod. Opt.

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

J. Opt. Soc. Am. A

Opt. Commun.

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]

Opt. Lett.

Other

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

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

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

M. Born, and E. Wolf, Principles of optics (Pergamon Press, Oxford 1993).

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

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

Supplementary Material (4)

» Media 1: MPG (321 KB)     
» Media 2: MPG (319 KB)     
» Media 3: MPG (307 KB)     
» Media 4: MPG (308 KB)     

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Figures (6)

Fig. 1.
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]

Fig. 2.
Fig. 2.

Laboratory setup.

Fig. 3.
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.
Fig. 4.

The same as Fig. 3 for w 0 = 15μm and z = 8.5zR0 .

Fig. 5.
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.
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.

Equations (34)

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s 0 ( x , y , z ) = E x ( x , y , z ) 2 + E y ( x , y , z ) 2
s 1 ( x , y , z ) = E x ( x , y , z ) 2 E y ( x , y , z ) 2
s 2 ( x , y , z ) = E x ( x , y , z ) E y * ( x , y , z ) + E y ( x , y , z ) E x * ( x , y , z )
s 3 ( x , y , z ) = i [ E y ( x , y , z ) E x * ( x , y , z ) E x ( x , y , z ) E y * ( x , y , z ) ] ,
E x ( x , y , z ) = E ̅ w 0 2 x 2 + y 2 exp ( i k 0 n o z ) [ ( y 2 Q o ( z ) + y 2 x 2 2 ( x 2 + y 2 ) ) exp ( x 2 + y 2 Q o ( z ) )
+ ( x 2 Q e ( z ) + x 2 y 2 2 ( x 2 + y 2 ) ) exp ( x 2 + y 2 Q e ( z ) ) ]
E y ( x , y , z ) = E ̅ w 0 2 x 2 + y 2 e i k 0 n o z [ ( x y Q o ( z ) + x y x 2 + y 2 ) exp ( x 2 + y 2 Q o ( z ) )
+ ( x y Q e ( z ) + x y x 2 + y 2 ) exp ( x 2 + y 2 Q e ( z ) ) ] ,
s n ( x , y , z ) = E ̅ 2 w 0 4 ( x 2 + y 2 ) 2 { f n ( o ) ( x , y , z ) exp ( 2 w 0 2 ( x 2 + y 2 ) Q o ( z ) 2 )
+ f n ( e ) ( x , y , z ) exp ( 2 w 0 2 ( x 2 + y 2 ) Q e ( z ) 2 )
+ f n ( o e ) ( x , y , z ) exp [ w 0 2 ( x 2 + y 2 ) ( 1 Q o ( z ) 2 + 1 Q e ( z ) 2 ) ] }
( n = 0,1,2,3 ) ,
f n ( o e ) ( x , y , z ) = f n ( s ) ( x , y , z ) sin [ 2 z k 0 n o ( 1 Q o ( z ) 2 n o 2 n e 2 1 Q e ( z ) 2 ) ( x 2 + y 2 ) ]
f n ( c ) ( x , y , z ) cos [ 2 z k 0 n o ( 1 Q o ( z ) 2 n o 2 n e 2 1 Q e ( z ) 2 ) ( x 2 + y 2 ) ] ,
f 0 ( o ) ( x , y , z ) = 1 4 + ( w 0 2 + x 2 + y 2 ) y 2 Q o ( z ) 2
f 0 ( e ) ( x , y , z ) = 1 4 + ( w 0 2 + x 2 + y 2 ) x 2 Q e ( z ) 2
f 0 ( s ) ( x , y , z ) = 2 z k 0 n o ( y 2 Q o ( z ) 2 n o 2 n e 2 x 2 Q e ( z ) 2 )
f 0 ( c ) ( x , y , z ) = ( 1 2 + x 2 w 0 2 Q e ( z ) 2 + y 2 w 0 2 Q o ( z ) 2 )
f 1 ( o ) ( x , y , z ) = x 4 + y 4 6 x 2 y 2 4 ( x 2 + y 2 ) 2 + y 2 ( y 4 x 4 + ( y 2 3 x 2 ) w 0 2 ) ( x 2 + y 2 ) Q o ( z ) 2
f 1 ( e ) ( x , y , z ) = x 4 + y 4 6 x 2 y 2 4 ( x 2 + y 2 ) 2 + x 2 ( x 4 y 4 + ( x 2 3 y 2 ) w 0 2 ) ( x 2 + y 2 ) Q e ( z ) 2
f 1 ( s ) ( x , y , z ) = 2 z k 0 n o ( x 2 ( x 2 3 y 2 ) x 2 + y 2 n o 2 n e 2 1 Q e ( z ) 2 y 2 ( y 2 3 x 2 ) x 2 + y 2 1 Q o ( z ) 2
+ 4 x 2 y 2 n e 2 n o 2 n e 2 w 0 2 Q o ( z ) 2 Q e ( z ) 2 )
f 1 ( c ) ( x , y , z ) = 1 2 + 4 x 2 y 2 ( 1 ( x 2 + y 2 ) 2 + 4 z 2 k 0 2 n e 2 + w 0 4 Q o ( z ) 2 Q e ( z ) 2 ) + y 2 ( y 2 3 x 2 ) x 2 + y 2 w 0 2 Q o ( z ) 2
+ x 2 ( x 2 3 y 2 ) x 2 + y 2 w 0 2 Q e ( z ) 2
f 2 ( o ) ( x , y , z ) = xy ( x 2 y 2 ( x 2 + y 2 ) 2 2 y 2 Q o ( z ) 2 3 y 2 x 2 x 2 + y 2 w 0 2 Q o ( z ) 2 )
f 2 ( e ) ( x , y , z ) = xy ( x 2 y 2 ( x 2 + y 2 ) 2 + 2 x 2 Q e ( z ) 2 + 3 x 2 y 2 x 2 + y 2 w 0 2 Q e ( z ) 2 )
f 2 ( s ) ( x , y , z ) = 2 zxy k 0 n 0 ( 3 y 2 x 2 x 2 + y 2 1 Q o ( z ) 2 + 3 x 2 y 2 x 2 + y 2 n o 2 n e 2 1 Q e ( z ) 2
+ 2 ( x 2 y 2 ) n o 2 n e 2 n e 2 w 0 2 Q o ( z ) 2 Q e ( z ) 2 )
f 2 ( c ) ( x , y , z ) = xy [ 3 y 2 x 2 x 2 + y 2 w 0 2 Q o ( z ) 2 3 x 2 y 2 x 2 + y 2 w 0 2 Q e ( z ) 2
+ ( y 2 x 2 ) ( 1 ( x 2 + y 2 ) 2 + w 0 4 + 4 z 2 k 0 2 n e 2 Q o ( z ) 2 Q e ( z ) 2 ) ]
f 3 ( o ) ( x , y , z ) = xy 2 z k 0 n 0 1 Q o ( z ) 2
f 3 ( e ) ( x , y , z ) = xy 2 z k 0 n 0 n o 2 n e 2 1 Q e ( z ) 2
f 3 ( s ) ( x , y , z ) = xy ( w 0 2 Q o ( z ) 2 + w 0 2 Q e ( z ) 2 + 2 ( x 2 + y 2 ) w 0 4 4 z 2 k 0 2 n e 2 Q o ( z ) 2 Q e ( z ) 2 )
f 3 ( c ) ( x , y , z ) = xy 2 z k 0 n o ( n o 2 n e 2 1 Q e ( z ) 2 1 Q o ( z ) 2 + 2 ( x 2 + y 2 ) w 0 2 Q o ( z ) 2 Q e ( z ) 2 )

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