Abstract

Rotatable retarder fixed polarizer (RRFP) Stokes polarimeters are most commonly used to measure the state of polarization (SOP) of an electromagnetic (EM) wave. Most of commercialized RRFP Stokes polarimeters realize the SOP measurements by rotating a 90° retarder to N(N5)uniformly spaced angles over 360° and performing a discrete Fourier transform of data. In this paper, we address the noise properties of such uniformly-rotating RRFP Stokes polarimeters employing a retarder with an arbitrary retardance. The covariance matrices on the measurement noises of four Stokes parameters are derived for Gaussian noise and Poisson noise, respectively. Based on these covariance matrices, it can be concluded that 1) the measurement noises of Stokes parameters seriously depend on the retardance of the retarder in use. 2) for Gaussian noise dominated RRFP Stokes polarimeters, the retardance 130.48° leads to the minimum overall measurement noises when the sum of the measurement noises of four Stokes parameters (viz., the trace of the covariance matrix) is used as the criterion. The retardance in the range from 126.06° to 134.72° can have a nearly-minimum measurement noise which is only 1% larger than the minimum. On the other hand, the retardance 126.87° results in the equalized noises of the last three Stokes parameters. 3) for Poisson noise dominated RRFP Stokes polarimeters, the covariance matrix is also a fuction of the SOP of the incident EM wave. Even so, the retardance in the range from 126.06° to 134.72° can also result in nearly-minimum measurement noise for Poisson noise. 4) in the case of Poisson noise, N=5,10,12uniformly spaced angles over 360° have special covariance matrices that depend on the initial angle (the first angle in use). Finally, simulations are performed to verify these theoretical findings.

© 2013 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Goldstein, Polarized Light (Marcel Dekker, Inc., 2003), Chap. 27.
  2. H. Ito, K. Suizu, T. Yamashita, A. Nawahara, and T. Sato, “Radom frequency accessible broad tunable terahertz-wave source using phase-matched 4-dimethylamino-N-methyl-4-stilbazolium Tosylate crystal,” J. Appl. Phys.93, 7321–7324 (2007).
  3. A. Galtarossa and L. Palmieri, “Reflectometric measurements of polarization properties in optical-fiber links,” IEEE Trans. Instrum. Meas.53(1), 86–94 (2004).
    [CrossRef]
  4. L. Giudicotti and M. Brombin, “Data analysis for a rotating quarter-wave, far-infrared Stokes polarimeter,” Appl. Opt.46(14), 2638–2648 (2007).
    [CrossRef] [PubMed]
  5. Manual of Thorlabs PAX Series polarimeters, http://www.thorlabs.com/NewGroupPage9.cfm?ObjectGroup_ID=1564
  6. A. Ambirajan and D. C. Look., “Optimum angles for a polarimeter: part I,” Opt. Eng.34(6), 1651–1655 (1995).
    [CrossRef]
  7. D. S. Sabatke, M. R. Descour, E. L. Dereniak, W. C. Sweatt, S. A. Kemme, and G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett.25(11), 802–804 (2000).
    [CrossRef] [PubMed]
  8. G. P. Agrawal, Fiber-Optic Communication Systems (John Wiley & Sons, Inc., 2002), Chap. 4.
  9. R. Q. Hui and M. O’Sullivan, Fiber Optics Measurement Techniques (Elsevier Academic, 2009), Chap. 1.
  10. F. Goudail, “Noise minimization and equalization for Stokes polarimeters in the presence of signal-dependent Poisson shot noise,” Opt. Lett.34(5), 647–649 (2009).
    [CrossRef] [PubMed]
  11. H. Dong, M. Tang, and Y. D. Gong, “Measurement errors induced by deformation of optical axes of achromatic waveplate retarders in RRFP Stokes polarimeters,” Opt. Express20(24), 26649–26666 (2012).
    [CrossRef] [PubMed]
  12. F. Goudail and A. Bénière, “Estimation precision of the degree of linear polarization and of the angle of polarization in the presence of different sources of noise,” Appl. Opt.49(4), 683–693 (2010).
    [CrossRef] [PubMed]
  13. H. Dong, Y. D. Gong, V. Paulose, P. Shum, and M. Olivo, “Effect of input states of polarization on the measurement error of Mueller matrix in a system having small polarization-dependent loss or gain,” Opt. Express17(15), 13017–13030 (2009).
    [CrossRef] [PubMed]
  14. J. S. Tyo, “Noise equalization in Stokes parameter images obtained by use of variable-retardance polarimeters,” Opt. Lett.25(16), 1198–1200 (2000).
    [CrossRef] [PubMed]

2012

2010

2009

2007

H. Ito, K. Suizu, T. Yamashita, A. Nawahara, and T. Sato, “Radom frequency accessible broad tunable terahertz-wave source using phase-matched 4-dimethylamino-N-methyl-4-stilbazolium Tosylate crystal,” J. Appl. Phys.93, 7321–7324 (2007).

L. Giudicotti and M. Brombin, “Data analysis for a rotating quarter-wave, far-infrared Stokes polarimeter,” Appl. Opt.46(14), 2638–2648 (2007).
[CrossRef] [PubMed]

2004

A. Galtarossa and L. Palmieri, “Reflectometric measurements of polarization properties in optical-fiber links,” IEEE Trans. Instrum. Meas.53(1), 86–94 (2004).
[CrossRef]

2000

1995

A. Ambirajan and D. C. Look., “Optimum angles for a polarimeter: part I,” Opt. Eng.34(6), 1651–1655 (1995).
[CrossRef]

Ambirajan, A.

A. Ambirajan and D. C. Look., “Optimum angles for a polarimeter: part I,” Opt. Eng.34(6), 1651–1655 (1995).
[CrossRef]

Bénière, A.

Brombin, M.

Dereniak, E. L.

Descour, M. R.

Dong, H.

Galtarossa, A.

A. Galtarossa and L. Palmieri, “Reflectometric measurements of polarization properties in optical-fiber links,” IEEE Trans. Instrum. Meas.53(1), 86–94 (2004).
[CrossRef]

Giudicotti, L.

Gong, Y. D.

Goudail, F.

Ito, H.

H. Ito, K. Suizu, T. Yamashita, A. Nawahara, and T. Sato, “Radom frequency accessible broad tunable terahertz-wave source using phase-matched 4-dimethylamino-N-methyl-4-stilbazolium Tosylate crystal,” J. Appl. Phys.93, 7321–7324 (2007).

Kemme, S. A.

Look, D. C.

A. Ambirajan and D. C. Look., “Optimum angles for a polarimeter: part I,” Opt. Eng.34(6), 1651–1655 (1995).
[CrossRef]

Nawahara, A.

H. Ito, K. Suizu, T. Yamashita, A. Nawahara, and T. Sato, “Radom frequency accessible broad tunable terahertz-wave source using phase-matched 4-dimethylamino-N-methyl-4-stilbazolium Tosylate crystal,” J. Appl. Phys.93, 7321–7324 (2007).

Olivo, M.

Palmieri, L.

A. Galtarossa and L. Palmieri, “Reflectometric measurements of polarization properties in optical-fiber links,” IEEE Trans. Instrum. Meas.53(1), 86–94 (2004).
[CrossRef]

Paulose, V.

Phipps, G. S.

Sabatke, D. S.

Sato, T.

H. Ito, K. Suizu, T. Yamashita, A. Nawahara, and T. Sato, “Radom frequency accessible broad tunable terahertz-wave source using phase-matched 4-dimethylamino-N-methyl-4-stilbazolium Tosylate crystal,” J. Appl. Phys.93, 7321–7324 (2007).

Shum, P.

Suizu, K.

H. Ito, K. Suizu, T. Yamashita, A. Nawahara, and T. Sato, “Radom frequency accessible broad tunable terahertz-wave source using phase-matched 4-dimethylamino-N-methyl-4-stilbazolium Tosylate crystal,” J. Appl. Phys.93, 7321–7324 (2007).

Sweatt, W. C.

Tang, M.

Tyo, J. S.

Yamashita, T.

H. Ito, K. Suizu, T. Yamashita, A. Nawahara, and T. Sato, “Radom frequency accessible broad tunable terahertz-wave source using phase-matched 4-dimethylamino-N-methyl-4-stilbazolium Tosylate crystal,” J. Appl. Phys.93, 7321–7324 (2007).

Appl. Opt.

IEEE Trans. Instrum. Meas.

A. Galtarossa and L. Palmieri, “Reflectometric measurements of polarization properties in optical-fiber links,” IEEE Trans. Instrum. Meas.53(1), 86–94 (2004).
[CrossRef]

J. Appl. Phys.

H. Ito, K. Suizu, T. Yamashita, A. Nawahara, and T. Sato, “Radom frequency accessible broad tunable terahertz-wave source using phase-matched 4-dimethylamino-N-methyl-4-stilbazolium Tosylate crystal,” J. Appl. Phys.93, 7321–7324 (2007).

Opt. Eng.

A. Ambirajan and D. C. Look., “Optimum angles for a polarimeter: part I,” Opt. Eng.34(6), 1651–1655 (1995).
[CrossRef]

Opt. Express

Opt. Lett.

Other

G. P. Agrawal, Fiber-Optic Communication Systems (John Wiley & Sons, Inc., 2002), Chap. 4.

R. Q. Hui and M. O’Sullivan, Fiber Optics Measurement Techniques (Elsevier Academic, 2009), Chap. 1.

D. Goldstein, Polarized Light (Marcel Dekker, Inc., 2003), Chap. 27.

Manual of Thorlabs PAX Series polarimeters, http://www.thorlabs.com/NewGroupPage9.cfm?ObjectGroup_ID=1564

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

The schematic structure of a RRFP Stokes polarimeter.

Fig. 2
Fig. 2

The relationship between the EWV and the retardance when Gaussian noise dominates in a RRFP Stokes polarimeter employing N uniformly spaced angles. The inset shows the “zoom in” views of the same data.

Fig. 3
Fig. 3

The relationships between the meansurement noises of Stokes parameters Δ s i 2 ,i=0,1,3 and the retardance. The inset shows the “zoom in” views of the same data.

Fig. 4
Fig. 4

The relationship between C0, C1 defined in Eq. (23) and the retardance when Poisson noise dominates in a RRFP Stokes polarimeter employing N uniformly spaced angles. The inset shows the “zoom in” views of the same data.

Fig. 5
Fig. 5

Simulation and theoretical results of Δ s i 2 ,i=0,1,2,3 for Gaussian noise and the five uniformly spaced angles (−90°, −54°, −18°, 18°, 54°). The inset shows the “zoom in” views of the same data.

Fig. 6
Fig. 6

Simulation and theoretical results of Δ s i 2 ,i=0,1,2,3 for Poisson noise and the five uniformly spaced angles (−90°, −54°, −18°, 18°, 54°). The inset shows the “zoom in” views of the same data.

Fig. 7
Fig. 7

Simulation and theoretical results of Δ s i 2 ,i=0,1,2,3 for Poisson noise and the five uniformly spaced angles (−81°, −45°, −9°, 27°, 63°). The inset shows the “zoom in” views of the same data.

Fig. 8
Fig. 8

Simulation and theoretical results of Δ s i 2 ,i=0,1,2,3 for Poisson noise and the twelve uniformly spaced angles (0°, 30°, 60°, …, 300°, 330°). The inset shows the “zoom in” views of the same data.

Equations (38)

Equations on this page are rendered with MathJax. Learn more.

I =W S
W= 1 2 ( 1 cos 2 2 θ 1 +cosδ sin 2 2 θ 1 sin 2 ( δ/2 )sin4 θ 1 sinδsin2 θ 1 1 cos 2 2 θ 2 +cosδ sin 2 2 θ 2 sin 2 ( δ/2 )sin4 θ 2 sinδsin2 θ 2 1 cos 2 2 θ N +cosδ sin 2 2 θ N sin 2 ( δ/2 )sin4 θ N sinδsin2 θ N )
Δ S = W + Δ I
W + = ( W T W ) 1 W T
Γ=( Δ s 0 2 Δ s 0 Δ s 1 Δ s 0 Δ s 2 Δ s 0 Δ s 3 Δ s 0 Δ s 1 Δ s 1 2 Δ s 1 Δ s 2 Δ s 1 Δ s 3 Δ s 0 Δ s 2 Δ s 1 Δ s 2 Δ s 2 2 Δ s 2 Δ s 3 Δ s 0 Δ s 3 Δ s 1 Δ s 3 Δ s 2 Δ s 3 Δ s 3 2 )= W + ( Δ I 1 2 Δ I 1 Δ I 2 Δ I 1 Δ I N Δ I 1 Δ I 2 Δ I 2 2 Δ I 2 Δ I N Δ I 1 Δ I N Δ I 2 Δ I N Δ I N 2 ) ( W + ) T
Γ= W + ( Δ I 1 2 00 0 Δ I 2 2 0 00 Δ I N 2 ) ( W + ) T
Δ I 1 2 = Δ I 2 2 == Δ I N 2 = σ G 2
Γ G = σ G 2 W + I ( W + ) T = σ G 2 [ ( W T W ) 1 W T ]I[ W ( W T W ) 1 ] = σ G 2 ( W T W ) 1 ( W T W ) ( W T W ) 1 = σ G 2 ( W T W ) 1
( W T W ) 1 = 4 N A
A=( 3+2cosδ+3 cos 2 δ ( 1cosδ ) 2 4( 1+cosδ ) ( 1cosδ ) 2 00 4( 1+cosδ ) ( 1cosδ ) 2 8 ( 1cosδ ) 2 00 00 8 ( 1cosδ ) 2 0 000 2 sin 2 δ )
{ i=1 N sin2 θ i = i=1 N sin4 θ i = i=1 N sin 3 2 θ i = i=1 N sin4 θ i cos 2 2 θ i = i=1 N sin4 θ i sin 2 2 θ i = i=1 N sin2 θ i cos 2 2 θ i = i=1 N sin2 θ i sin4 θ i = 0 i=1 N cos 2 2 θ i = i=1 N sin 2 2 θ i = i=1 N sin 2 4 θ i =N/2 i=1 N cos 4 2 θ i = i=1 N sin 4 2 θ i =3N/8 i=1 N sin 2 2 θ i cos 2 2 θ i =N/8
i=0 3 Δ s i 2 =Tr[ Γ G ]= σ G 2 Tr[ ( W T W ) 1 ]=EWV σ G 2
EWV= 4 N 3 cos 3 δ+5 cos 2 δ+19cosδ+21 cos 3 δ cos 2 δcosδ+1
Γ G = σ G 2 2N ( 9500 52500 00250 00025 )
Δ I i 2 =k I i ,i=1,2,,N
Δ I i 2 = k 2 { s 0 +( cos 2 2 θ 2 +cosδ sin 2 2 θ 2 ) s 1 +[ sin 2 ( δ/2 )sin4 θ 2 ] s 2 sinδsin2 θ 2 s 3 },i=1,2,,N
Γ P = W + Γ I ( W + ) T = ( W T W ) 1 ( W T Γ I W ) ( W T W ) 1
W T Γ I W= Nk 8 ( B s 0 +C )
B=( 1 1+cosδ 2 00 1+cosδ 2 3+2cosδ+3 cos 2 δ 8 00 00 sin 4 ( δ/2 ) 2 0 000 sin 2 δ 2 )
C=( 1+cosδ 2 s 1 3+2cosδ+3 cos 2 δ 8 s 1 sin 4 ( δ/2 ) 2 s 2 sin 2 δ 2 s 3 3+2cosδ+3 cos 2 δ 8 s 1 5+3cosδ+3 cos 2 δ+5 cos 3 δ 16 s 1 sin 4 ( δ/2 )( 1+cosδ ) 4 s 2 sin 2 δ( 1+3cosδ ) 8 s 3 sin 4 ( δ/2 ) 2 s 2 sin 4 ( δ/2 )( 1+cosδ ) 4 s 2 sin 4 ( δ/2 )( 1+cosδ ) 4 s 1 0 sin 2 δ 2 s 3 sin 2 δ( 1+3cosδ ) 8 s 3 0 sin 2 δ( 1+3cosδ ) 8 s 1 )
{ i=1 N cos 6 2 θ i = i=1 N sin 6 2 θ i =5N/16( 20.1 ) i=1 N cos 4 2 θ i sin 2 2 θ i = i=1 N cos 2 2 θ i sin 4 2 θ i =N/16(20.2) i=1 N cos 2 2 θ i sin 2 4 θ i = i=1 N sin 2 2 θ i sin 2 4 θ i =N/4(20.3) i=1 N cos 4 2 θ i sin4 θ i = i=1 N sin 4 2 θ i sin4 θ i = i=1 N cos 2 2 θ i sin 2 2 θ i sin4 θ i = i=1 N sin 3 4 θ i =0(20.4) i=1 N cos 4 2 θ i sin2 θ i = i=1 N cos 2 2 θ i sin 3 2 θ i = i=1 N sin 2 4 θ i sin2 θ i = i=1 N sin 5 2 θ i = i=1 N cos 2 2 θ i sin4 θ i sin2 θ i = i=1 N sin 2 2 θ i sin4 θ i sin2 θ i =0( 20.5 )
Γ P = 2k N ( A s 0 +D )
D=( 1+7cosδ+7 cos 2 δ+ cos 3 δ 2 ( 1cosδ ) 2 s 1 1+6cosδ+ cos 2 δ ( 1cosδ ) 2 s 1 s 2 2 s 3 1cosδ 1+6cosδ+ cos 2 δ ( 1cosδ ) 2 s 1 4( 1+cosδ ) ( 1cosδ ) 2 s 1 0 2 s 3 1cosδ s 2 0 4( 1+cosδ ) ( 1cosδ ) 2 s 1 0 2 s 3 1cosδ 2 s 3 1cosδ 0 1+3cosδ 2 sin 2 δ s 1 )
i=0 3 Δ s i 2 =Tr( Γ P )= 2k N ( C 0 s 0 + C 1 s 1 )
C 0 = 3 cos 3 δ+5 cos 2 δ+19cosδ+21 cos 3 δ cos 2 δcosδ+1 , C 1 = cos 4 δ+8 cos 3 δ+27 cos 2 δ+42cosδ+18 2( cos 3 δ cos 2 δcosδ+1 )
Γ P = k 4N ( 9 s 0 1.4 s 1 5 s 0 +7 s 1 s 2 10 s 3 5 s 0 +7 s 1 25 s 0 +5 s 1 010 s 3 s 2 025 s 0 +5 s 1 0 10 s 3 10 s 3 025 s 0 5 s 1 )
Γ P = k 2 ( s 0 s 1 s 2 s 3 s 1 3 s 0 s 2 2 s 3 2 s 2 s 3 s 2 s 2 2 s 3 3 s 0 + s 1 2 s 1 s 3 2 s 2 s 3 2 s 1 3 s 0 s 1 )
{ i=1 N cos 4 2 θ i sin2 θ i = i=1 N sin 5 2 θ i = Nπsin( 10 θ 1 )/50 i=1 N cos 2 2 θ i sin 3 2 θ i =Nπsin( 10 θ 1 )/50 i=1 N cos 2 2 θ i sin4 θ i sin2 θ i =Nπcos( 10 θ 1 )/25 i=1 N sin 2 2 θ i sin4 θ i sin2 θ i =Nπcos( 10 θ 1 )/25 i=1 N sin 2 4 θ i sin2 θ i =2Nπsin( 10 θ 1 )/25
W T Γ I W= Nk 8 ( B s 0 +C+E )
E= π 50 sinδ ( 1cosδ ) 2 ( 0000 0 e 1 s 3 e 2 s 3 e 2 s 2 e 1 s 1 0 e 2 s 3 e 1 s 3 e 2 s 1 + e 1 s 2 0 e 2 s 2 e 1 s 1 e 2 s 1 + e 1 s 2 0 )
Γ P = 2k N ( A s 0 +D+F )
F= 16πsinδ 25 ( ( 1+cosδ ) 2 e 1 2 ( 1cosδ ) 2 s 3 ( 1+cosδ ) e 1 ( 1cosδ ) 2 s 3 ( 1+cosδ ) e 2 ( 1cosδ ) 2 s 3 e 1 s 1 e 2 s 2 4( 1cosδ ) ( 1+cosδ ) e 1 ( 1cosδ ) 2 s 3 2 e 1 ( 1cosδ ) 2 s 3 2 e 2 ( 1cosδ ) 2 s 3 e 2 s 2 e 1 s 1 2 sin 2 δ ( 1+cosδ ) e 2 ( 1cosδ ) 2 s 3 2 e 2 ( 1cosδ ) 2 s 3 2 e 1 ( 1cosδ ) 2 s 3 e 2 s 1 + e 1 s 2 2 sin 2 δ e 1 s 1 e 2 s 2 4( 1cosδ ) e 2 s 2 e 1 s 1 2 sin 2 δ e 2 s 1 + e 1 s 2 2 sin 2 δ 0 )
{ i=1 N cos 6 2 θ i =5N/16+3cos( 12 θ 1 )/8 i=1 N sin 6 2 θ i =5N/163cos( 12 θ 1 )/8 i=1 N cos 4 2 θ i sin 2 2 θ i =N/163cos( 12 θ 1 )/8 i=1 N cos 2 2 θ i sin 4 2 θ i =N/16+3cos( 12 θ 1 )/8 i=1 N cos 2 2 θ i sin 2 4 θ i =N/43cos( 12 θ 1 )/2 i=1 N sin 2 2 θ i sin 2 4 θ i =N/4+3cos( 12 θ 1 )/2 i=1 N cos 4 2 θ i sin4 θ i = i=1 N sin 4 2 θ i sin4 θ i =3sin( 12 θ 1 )/4 i=1 N cos 2 2 θ i sin 2 2 θ i sin4 θ i =3sin( 12 θ 1 )/4 i=1 N sin 3 4 θ i =3sin( 12 θ 1 )
W T Γ I W= Nk 8 ( B s 0 +C+G )
G= ( 1cosδ ) 2 32 ( 0000 0 g 1 g 2 0 0 g 2 g 1 0 0000 )
g 1 =cos( 12 θ 1 ) s 1 +sin( 12 θ 1 ) s 2 , g 2 =sin( 12 θ 1 ) s 1 cos( 12 θ 1 ) s 2
Γ P = 2k N ( A s 0 +D+H )
H=( ( 1+cosδ ) 2 g 1 2( 1cosδ ) ( 1+cosδ ) g 1 1cosδ ( 1+cosδ ) g 2 1cosδ 0 ( 1+cosδ ) g 1 1cosδ 2 g 1 1cosδ 2 g 2 1cosδ 0 ( 1+cosδ ) g 2 1cosδ 2 g 2 1cosδ 2 g 1 1cosδ 0 0000 )

Metrics