Abstract

In this work, we present a theoretical and experimental study of the response of a lossless polarizer to a signal beam with a time-varying state of polarization (SOP). By lossless polarizer, we mean a nonlinear conservative medium (e.g., an optical fiber) that is counterpumped by an intense and fully polarized pump beam. Such a medium transforms input uniform or random distributions of the SOP of an intense signal beam into output distributions that are tightly localized around a well-defined SOP. We introduce and characterize an important parameter of a lossless polarizer—its response time. Whenever the fluctuations of the SOP of the input signal beam are slower than its response time, a lossless polarizer provides an efficient repolarization of the beam at its output. Otherwise, if input polarization fluctuations are faster than the response time, the polarizer is not able to repolarize light.

© 2011 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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2011 (1)

2010 (4)

2009 (3)

2008 (2)

2005 (1)

S. Pitois, A. Picozzi, G. Millot, H. R. Jauslin, and M. Haelterman, “Polarization and modal attractors in conservative counterpropagating four-wave interaction,” Europhys. Lett. 70, 88-94(2005).
[CrossRef]

2001 (1)

2000 (1)

1993 (1)

B. Daino and S. Wabnitz, “Polarization domains and instabilities in nonlinear optical fibers,” Phys. Lett. A 182, 289-293 (1993).
[CrossRef]

1991 (1)

Amorim, A. A.

Assémat, E.

Bennink, R. S.

Bernardo, L. M.

Boyd, R. W.

Chaudhari, C.

Crespo, H. M.

Daino, B.

B. Daino and S. Wabnitz, “Polarization domains and instabilities in nonlinear optical fibers,” Phys. Lett. A 182, 289-293 (1993).
[CrossRef]

de Sterke, C. M.

Fatome, J.

Fisher, R. A.

Haelterman, M.

S. Pitois, A. Picozzi, G. Millot, H. R. Jauslin, and M. Haelterman, “Polarization and modal attractors in conservative counterpropagating four-wave interaction,” Europhys. Lett. 70, 88-94(2005).
[CrossRef]

Heebner, J. E.

Jackson, K. R.

Jauslin, H. R.

S. Lagrange, D. Sugny, A. Picozzi, and H. R. Jauslin, “Singular tori as attractors of four-wave-interaction systems,” Phys. Rev. E 81, 016202 (2010).
[CrossRef]

E. Assémat, S. Lagrange, A. Picozzi, H. R. Jauslin, and D. Sugny, “Complete nonlinear polarization control in an optical fiber system,” Opt. Lett. 35, 2025 (2010).
[CrossRef] [PubMed]

D. Sugny, A. Picozzi, S. Lagrange, and H. R. Jauslin, “Role of singular tori in the dynamics of spatiotemporal nonlinear wave systems,” Phys. Rev. Lett. 103, 034102 (2009).
[CrossRef] [PubMed]

S. Pitois, A. Picozzi, G. Millot, H. R. Jauslin, and M. Haelterman, “Polarization and modal attractors in conservative counterpropagating four-wave interaction,” Europhys. Lett. 70, 88-94(2005).
[CrossRef]

Kärtner, F. X.

Kozlov, V. V.

Lagrange, S.

E. Assémat, S. Lagrange, A. Picozzi, H. R. Jauslin, and D. Sugny, “Complete nonlinear polarization control in an optical fiber system,” Opt. Lett. 35, 2025 (2010).
[CrossRef] [PubMed]

S. Lagrange, D. Sugny, A. Picozzi, and H. R. Jauslin, “Singular tori as attractors of four-wave-interaction systems,” Phys. Rev. E 81, 016202 (2010).
[CrossRef]

D. Sugny, A. Picozzi, S. Lagrange, and H. R. Jauslin, “Role of singular tori in the dynamics of spatiotemporal nonlinear wave systems,” Phys. Rev. Lett. 103, 034102 (2009).
[CrossRef] [PubMed]

Liao, M.

Millot, G.

Morin, P.

Nuño, J.

Ohishi, Y.

Oliveira, P.

Picozzi, A.

E. Assémat, S. Lagrange, A. Picozzi, H. R. Jauslin, and D. Sugny, “Complete nonlinear polarization control in an optical fiber system,” Opt. Lett. 35, 2025 (2010).
[CrossRef] [PubMed]

S. Lagrange, D. Sugny, A. Picozzi, and H. R. Jauslin, “Singular tori as attractors of four-wave-interaction systems,” Phys. Rev. E 81, 016202 (2010).
[CrossRef]

D. Sugny, A. Picozzi, S. Lagrange, and H. R. Jauslin, “Role of singular tori in the dynamics of spatiotemporal nonlinear wave systems,” Phys. Rev. Lett. 103, 034102 (2009).
[CrossRef] [PubMed]

A. Picozzi, “Spontaneous polarization induced by natural thermalization of incoherent light,” Opt. Express 16, 17171-17185(2008).
[CrossRef] [PubMed]

S. Pitois, A. Picozzi, G. Millot, H. R. Jauslin, and M. Haelterman, “Polarization and modal attractors in conservative counterpropagating four-wave interaction,” Europhys. Lett. 70, 88-94(2005).
[CrossRef]

Pitois, S.

Qin, G. I.

Robert, B. D.

Silva, J. L.

Sugny, D.

E. Assémat, S. Lagrange, A. Picozzi, H. R. Jauslin, and D. Sugny, “Complete nonlinear polarization control in an optical fiber system,” Opt. Lett. 35, 2025 (2010).
[CrossRef] [PubMed]

S. Lagrange, D. Sugny, A. Picozzi, and H. R. Jauslin, “Singular tori as attractors of four-wave-interaction systems,” Phys. Rev. E 81, 016202 (2010).
[CrossRef]

D. Sugny, A. Picozzi, S. Lagrange, and H. R. Jauslin, “Role of singular tori in the dynamics of spatiotemporal nonlinear wave systems,” Phys. Rev. Lett. 103, 034102 (2009).
[CrossRef] [PubMed]

Suzuki, T.

Tognetti, M. V.

Wabnitz, S.

Yan, X.

Europhys. Lett. (1)

S. Pitois, A. Picozzi, G. Millot, H. R. Jauslin, and M. Haelterman, “Polarization and modal attractors in conservative counterpropagating four-wave interaction,” Europhys. Lett. 70, 88-94(2005).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Express (4)

Opt. Lett. (4)

Phys. Lett. A (1)

B. Daino and S. Wabnitz, “Polarization domains and instabilities in nonlinear optical fibers,” Phys. Lett. A 182, 289-293 (1993).
[CrossRef]

Phys. Rev. E (1)

S. Lagrange, D. Sugny, A. Picozzi, and H. R. Jauslin, “Singular tori as attractors of four-wave-interaction systems,” Phys. Rev. E 81, 016202 (2010).
[CrossRef]

Phys. Rev. Lett. (1)

D. Sugny, A. Picozzi, S. Lagrange, and H. R. Jauslin, “Role of singular tori in the dynamics of spatiotemporal nonlinear wave systems,” Phys. Rev. Lett. 103, 034102 (2009).
[CrossRef] [PubMed]

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Figures (14)

Fig. 1
Fig. 1

Stokes parameters of the signal beam at z = L as a function of time: S 1 + (black solid curve), S 2 + (red dashed curve), and S 3 + (green dotted curve). Time is measured in units T N L . Stokes parameters are normalized to S 0 + ; S 0 = 3 S 0 + .

Fig. 2
Fig. 2

Stokes parameters of the signal beam inside the medium at four instants of time: (a)  t = 30 , (b)  t = 36 , (c)  t = 42 , and (d)  t = 48 . S 1 + (black solid curve), S 2 + (red dashed curve), and S 3 + (green dotted curve). Time is measured in units T N L . All Stokes parameters are normalized to S 0 + . Parameters are as in Fig. 1. The four snapshots cover exactly one period of the oscillations that are shown in Fig. 1.

Fig. 3
Fig. 3

(a) Period (in units of T N L ) of the Stokes parameter oscillations at z = L versus pump power (in units of S 0 + ). (b) DOP versus time for the same case as in Figs. 1, 2.

Fig. 4
Fig. 4

Components of the mean Stokes vector and the DOP of the output signal beam as a function of the relative power of the backward beam: S 1 + (black solid curve), S 2 + (red dashed curve), S 3 + (green dotted curve), and DOP (blue dashed–dotted curve), for the input SOP of the pump beam: ( 0.99 , 0.1 , 0.1 ) . Thick (thin) curves are calculated with Approach S (U). The Stokes parameters are normalized with respect to S 0 + ( z , t ) . The observation time in Approach U is T = 50 ω m 1 = 250000 T N L and ω m T N L = 0.0002 . The observation time in Approach S is T = 10000 T N L and N = 110 .

Fig. 5
Fig. 5

The first component S 1 + of the input and output Stokes vector of the signal beam as a function of time: (a), (b) input; (c), (d) output. (b) and (d) are the exploded views of (a) and (c). The first Stokes component is normalized with respect to S 0 + ( z , t ) . The observation time T = 50 ω m 1 ; ω m T N L = 0.0002 ; S 0 = 3 S 0 + .

Fig. 6
Fig. 6

DOP of the output signal beam as function of the pump power for different values of the product ω m T N L : 0.0002 (black solid curve), 0.02 (red dashed curve), 0.2 (green dotted curve), and 1 (blue dashed–dotted curve). The observation time is T = 250000 T N L .

Fig. 7
Fig. 7

The first Stokes component of the signal beam at the input (red dashed curve) and output (black solid curve) of the medium as a function of time. The polarization burst imposed on the otherwise steady-state input SOP is described by Eqs. (14, 15). S 0 = 3 S 0 + .

Fig. 8
Fig. 8

Illustration of the burst annihilation: the first Stokes component of the signal beam at the output of the medium as the function of time for increasing values of the signal power (from bottom to top). T 0 = L / v .

Fig. 9
Fig. 9

Experimental setup.

Fig. 10
Fig. 10

(a) SOP of the input signal. (b) SOP of the output signal.

Fig. 11
Fig. 11

(a) SOP of the input signal. (b) SOP of the output signal. (c) Stokes parameters as a function of time (dashed curve, input; solid curve, output).

Fig. 12
Fig. 12

Evolution of a signal polarization burst as a function of sig nal average power and detected at the output of the fiber behind a polarizer: along (a) the 0 ° axis and (b) the orthogonal axis.

Fig. 13
Fig. 13

Ratio of energy on 0 ° / 90 ° axes contained in the polarization burst.

Fig. 14
Fig. 14

Two examples of polarization burst evolution as a function of pump power and detected at the output of the fiber behind a po larizer. In contrast to the rest of the paper, here the nonlinear length L N L is defined in terms of the pump power.

Equations (23)

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D = 1 S 0 i = 1 3 S i 2
ξ S + = S + × J s S + + S + × J x S ,
η S - = S × J s S + S × J x S + .
S 0 ± = ( S 1 ± ) 2 + ( S 2 ± ) 2 + ( S 3 ± ) 2 .
S i + ( z = 0 , t ) = S i + ( z , t = 0 ) ,
S i - ( z = L , t ) = S i - ( z , t = 0 ) ,
S 1 + ( z = 0 , t ) = S 0 + sin α 1 ( t ) cos α 2 ( t ) ,
S 2 + ( z = 0 , t ) = S 0 + sin α 1 ( t ) sin α 2 ( t ) ,
S 3 + ( z = 0 , t ) = S 0 + cos α 1 ( t ) .
α 1 = α 0 + 2.53 ω m t ,
α 2 = α 0 3.53 ω m t .
D U ( T ) = 1 T 1 S 0 + i = 1 3 [ 0 T d t S i + ( L , t ) ] 2 .
S i + ( L , T ) T = 1 T 0 T d t S i + ( L , t ) ,
α 1 = α 0 + π sech [ 2.53 ( ω m t 125 ) ] ,
α 2 = α 0 π 2 sech [ 3.53 ( ω m t - 125 ) ] ,
S i + ( L ) = 1 N j = 1 N [ S i + ( L ) ] j ,
θ 0 = arccos ( S 3 S 1 + 2 + S 2 + 2 + S 3 + 2 ) ,
ϕ 0 = a tan 2 ( S 2 + , S 1 + ) .
a tan 2 ( x , y ) = { arctan ( y / x ) x > 0 π + arctan ( y / x ) y 0 , x < 0 - π + arctan ( y / x ) y < 0 , x < 0 π / 2 y > 0 , x = 0 π / 2 y < 0 , x = 0 undefined y = 0 , x = 0 .
S ¯ 1 = S 0 + sin θ 0 cos ϕ 0 ,
S ¯ 2 = S 0 + sin θ 0 sin ϕ 0 ,
S ¯ 3 = S 0 + cos θ 0 .
D S = S 1 + 2 + S 2 + 2 + S 3 + 2 / S 0 + .

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