Abstract

Star test polarimetry is an imaging polarimetry technique in which an element with spatially-varying birefringence is placed in the pupil plane to encode polarization information into the point-spread function (PSF) of an imaging system. In this work, a variational calculation is performed to find the optimal birefringence distribution that effectively encodes polarization information while producing the smallest possible PSF, thus maximizing the resolution for imaging polarimetry. This optimal solution is found to be nearly equivalent to the birefringence distribution that results from a glass window being subjected to three uniformly spaced stress points at its edges, which has been used in previous star test polarimetry setups.

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References

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  1. C. F. LaCasse, R. A. Chipman, and J. S. Tyo, “Band limited data reconstruction in modulated polarimeters,” Opt. Express 19(16), 14976–14989 (2011).
    [Crossref]
  2. R. Azzam, “Stokes-vector and Mueller-matrix polarimetry,” J. Opt. Soc. Am. A 33(7), 1396–1408 (2016).
    [Crossref]
  3. E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, New York, 1993).
  4. J. Chang, N. Zeng, H. He, Y. He, and H. Ma, “Single-shot spatially modulated Stokes polarimeter based on a GRIN lens,” Opt. Lett. 39(9), 2656–2659 (2014).
    [Crossref]
  5. R. Chipman, “Chapter 22: Polarimetry,” in Handbook of Optics, Vol. II, M. Bass, ed. (McGraw-Hill, New York, 1995).
  6. R. Ramkhalawon, A. M. Beckley, and T. G. Brown, “Star test polarimetry using stress-engineered optical elements,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XIX, vol. 8227 (International Society for Optics and Photonics, 2012), p. 82270Q.
  7. R. D. Ramkhalawon, T. G. Brown, and M. A. Alonso, “Imaging the polarization of a light field,” Opt. Express 21(4), 4106–4115 (2013).
    [Crossref]
  8. S. Sivankutty, E. R. Andresen, G. Bouwmans, T. G. Brown, M. A. Alonso, and H. Rigneault, “Single-shot polarimetry imaging of multicore fiber,” Opt. Lett. 41(9), 2105–2108 (2016).
    [Crossref]
  9. V. Curcio, T. G. Brown, S. Brasselet, and M. A. Alonso, “Birefringent Fourier filtering for single molecule Coordinate and Height super-resolution Imaging with Dithering and Orientation (CHIDO),” arXiv preprint arXiv:1907.05828 (2019).
  10. B. G. Zimmerman, R. Ramkhalawon, M. Alonso, and T. G. Brown, “Pinhole array implementation of star test polarimetry,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XXI, vol. 8949 (International Society for Optics and Photonics, 2014), p. 894912.
  11. B. G. Zimmerman and T. G. Brown, “Star test image-sampling polarimeter,” Opt. Express 24(20), 23154–23161 (2016).
    [Crossref]
  12. A. Vella and M. A. Alonso, “Poincaré sphere representation for spatially varying birefringence,” Opt. Lett. 43(3), 379–382 (2018).
    [Crossref]
  13. A. Vella, “Description and applications of space-variant polarization states and elements,” Ph.D. thesis, University of Rochester (2018).
  14. J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45(22), 5453–5469 (2006).
    [Crossref]
  15. A. Vella, H. Dourdent, L. Novotny, and M. A. Alonso, “Birefringent masks that are optimal for generating bottle fields,” Opt. Express 25(8), 9318–9332 (2017).
    [Crossref]
  16. A. Vella, “Tutorial: Maximum likelihood estimation in the context of an optical measurement,” arXiv preprint arXiv:1806.04503 (2018).

2018 (1)

2017 (1)

2016 (3)

2014 (1)

2013 (1)

2011 (1)

2006 (1)

Alonso, M.

B. G. Zimmerman, R. Ramkhalawon, M. Alonso, and T. G. Brown, “Pinhole array implementation of star test polarimetry,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XXI, vol. 8949 (International Society for Optics and Photonics, 2014), p. 894912.

Alonso, M. A.

Andresen, E. R.

Azzam, R.

Beckley, A. M.

R. Ramkhalawon, A. M. Beckley, and T. G. Brown, “Star test polarimetry using stress-engineered optical elements,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XIX, vol. 8227 (International Society for Optics and Photonics, 2012), p. 82270Q.

Bouwmans, G.

Brasselet, S.

V. Curcio, T. G. Brown, S. Brasselet, and M. A. Alonso, “Birefringent Fourier filtering for single molecule Coordinate and Height super-resolution Imaging with Dithering and Orientation (CHIDO),” arXiv preprint arXiv:1907.05828 (2019).

Brown, T. G.

S. Sivankutty, E. R. Andresen, G. Bouwmans, T. G. Brown, M. A. Alonso, and H. Rigneault, “Single-shot polarimetry imaging of multicore fiber,” Opt. Lett. 41(9), 2105–2108 (2016).
[Crossref]

B. G. Zimmerman and T. G. Brown, “Star test image-sampling polarimeter,” Opt. Express 24(20), 23154–23161 (2016).
[Crossref]

R. D. Ramkhalawon, T. G. Brown, and M. A. Alonso, “Imaging the polarization of a light field,” Opt. Express 21(4), 4106–4115 (2013).
[Crossref]

B. G. Zimmerman, R. Ramkhalawon, M. Alonso, and T. G. Brown, “Pinhole array implementation of star test polarimetry,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XXI, vol. 8949 (International Society for Optics and Photonics, 2014), p. 894912.

R. Ramkhalawon, A. M. Beckley, and T. G. Brown, “Star test polarimetry using stress-engineered optical elements,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XIX, vol. 8227 (International Society for Optics and Photonics, 2012), p. 82270Q.

V. Curcio, T. G. Brown, S. Brasselet, and M. A. Alonso, “Birefringent Fourier filtering for single molecule Coordinate and Height super-resolution Imaging with Dithering and Orientation (CHIDO),” arXiv preprint arXiv:1907.05828 (2019).

Chang, J.

Chenault, D. B.

Chipman, R.

R. Chipman, “Chapter 22: Polarimetry,” in Handbook of Optics, Vol. II, M. Bass, ed. (McGraw-Hill, New York, 1995).

Chipman, R. A.

Collett, E.

E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, New York, 1993).

Curcio, V.

V. Curcio, T. G. Brown, S. Brasselet, and M. A. Alonso, “Birefringent Fourier filtering for single molecule Coordinate and Height super-resolution Imaging with Dithering and Orientation (CHIDO),” arXiv preprint arXiv:1907.05828 (2019).

Dourdent, H.

Goldstein, D. L.

He, H.

He, Y.

LaCasse, C. F.

Ma, H.

Novotny, L.

Ramkhalawon, R.

B. G. Zimmerman, R. Ramkhalawon, M. Alonso, and T. G. Brown, “Pinhole array implementation of star test polarimetry,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XXI, vol. 8949 (International Society for Optics and Photonics, 2014), p. 894912.

R. Ramkhalawon, A. M. Beckley, and T. G. Brown, “Star test polarimetry using stress-engineered optical elements,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XIX, vol. 8227 (International Society for Optics and Photonics, 2012), p. 82270Q.

Ramkhalawon, R. D.

Rigneault, H.

Shaw, J. A.

Sivankutty, S.

Tyo, J. S.

Vella, A.

A. Vella and M. A. Alonso, “Poincaré sphere representation for spatially varying birefringence,” Opt. Lett. 43(3), 379–382 (2018).
[Crossref]

A. Vella, H. Dourdent, L. Novotny, and M. A. Alonso, “Birefringent masks that are optimal for generating bottle fields,” Opt. Express 25(8), 9318–9332 (2017).
[Crossref]

A. Vella, “Description and applications of space-variant polarization states and elements,” Ph.D. thesis, University of Rochester (2018).

A. Vella, “Tutorial: Maximum likelihood estimation in the context of an optical measurement,” arXiv preprint arXiv:1806.04503 (2018).

Zeng, N.

Zimmerman, B. G.

B. G. Zimmerman and T. G. Brown, “Star test image-sampling polarimeter,” Opt. Express 24(20), 23154–23161 (2016).
[Crossref]

B. G. Zimmerman, R. Ramkhalawon, M. Alonso, and T. G. Brown, “Pinhole array implementation of star test polarimetry,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XXI, vol. 8949 (International Society for Optics and Photonics, 2014), p. 894912.

Appl. Opt. (1)

J. Opt. Soc. Am. A (1)

Opt. Express (4)

Opt. Lett. (3)

Other (7)

E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, New York, 1993).

R. Chipman, “Chapter 22: Polarimetry,” in Handbook of Optics, Vol. II, M. Bass, ed. (McGraw-Hill, New York, 1995).

R. Ramkhalawon, A. M. Beckley, and T. G. Brown, “Star test polarimetry using stress-engineered optical elements,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XIX, vol. 8227 (International Society for Optics and Photonics, 2012), p. 82270Q.

V. Curcio, T. G. Brown, S. Brasselet, and M. A. Alonso, “Birefringent Fourier filtering for single molecule Coordinate and Height super-resolution Imaging with Dithering and Orientation (CHIDO),” arXiv preprint arXiv:1907.05828 (2019).

B. G. Zimmerman, R. Ramkhalawon, M. Alonso, and T. G. Brown, “Pinhole array implementation of star test polarimetry,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XXI, vol. 8949 (International Society for Optics and Photonics, 2014), p. 894912.

A. Vella, “Description and applications of space-variant polarization states and elements,” Ph.D. thesis, University of Rochester (2018).

A. Vella, “Tutorial: Maximum likelihood estimation in the context of an optical measurement,” arXiv preprint arXiv:1806.04503 (2018).

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Figures (7)

Fig. 1.
Fig. 1. System layout for a star test polarimetry measurement of an unknown input field $\mathbf {E}_0$, illustrated for the case in which a single circular polarization component is imaged. The pupil plane is described by a radial coordinate $u=\sin \theta$ and an azimuthal angle $\phi$.
Fig. 2.
Fig. 2. Value of Eq. (22) as a function of $b_m$ (times a scaling factor) for several values of $|m|$. Note that this quantity can vanish only for $1\leq |m|\leq 4$. The inset shows the corresponding values of $\Delta r$ for the eight values of $|m|$ and $b_m$ for which the condition $\langle \beta _3\rangle _A=0$ is satisfied.
Fig. 3.
Fig. 3. Radial retardance distribution $\bar {\delta }(v)$ for the optimal BM solution ignoring the boundary conditions, the true optimum, and an SEO with stress coefficient $c=1.166/\mathrm {NA}$, plotted as functions of the normalized radial pupil coordinate $v$.
Fig. 4.
Fig. 4. For the optimal BM solution: complex fields $G$ and $H$ (left, with amplitude cross-sections through the center shown on the right), and PSF contributions $\mathcal {I}_n^{(j)}(\mathbf {x})$ to the PSFs of the $\mathbf {e}_1$ (top row) and $\mathbf {e}_2$ (bottom row) output polarization components. The plots are shown over a square region with half-width $1.25\lambda /\mathrm {NA}$.
Fig. 5.
Fig. 5. Horizontal slices through $y=0$ of the intensity contributions $\mathcal {I}_n^{(1)}$. The solid and dashed curves correspond to the optimal birefringence distribution and an SEO, respectively.
Fig. 6.
Fig. 6. Two-dimensional cross-sections of the error ellipsoids for incident polarization states in the Poincaré sphere’s $s_1$-$s_2$ and $s_1$-$s_3$ cross-sections (shown only for $s_1\ge 0$ due to symmetry) for (a-b) the optimal BM and (c-d) an SEO, when only the $\mathbf {e}_1$ output polarization component is measured with $\mathcal {N}=1500$ detected photons (red) and when both polarization components are measured with $\mathcal {N}=3000$ detected photons (blue). The error ellipses for unpolarized light are magnified in the inset of each plot.
Fig. 7.
Fig. 7. Histograms of the power coverage of different values of $s_3$ (characterized by $\mu _3$) over the extent of the PSFs, for the cases when (a) only the PSF for the $\mathbf {e}_1$ component is used, and (b) the PSFs for both components are used.

Tables (1)

Tables Icon

Table 1. Performance metrics for the PSFs resulting from using three BM distributions characterized by the half-retardance functions in the second column, as well as for a diffraction-limited system, when the incident field is unpolarized.

Equations (49)

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J ( u ) = e i Γ ( u ) [ q 0 ( u ) + i q 3 ( u ) q 2 ( u ) + i q 1 ( u ) q 2 ( u ) + i q 1 ( u ) q 0 ( u ) i q 3 ( u ) ] ,
q 0 ( u ) = cos δ ( u ) ,
q 1 ( u ) = sin δ ( u ) cos Θ ( u ) cos Φ ( u ) ,
q 2 ( u ) = sin δ ( u ) cos Θ ( u ) sin Φ ( u ) ,
q 3 ( u ) = sin δ ( u ) sin Θ ( u ) ,
I ( 1 , 2 ) ( x ) = | A ( u ) e 1 , 2 J ( u ) E 0 exp [ i k ( u x ) ] d 2 u | 2 T ,
g ( u ) = A ( u ) [ q 0 ( u ) + i q 3 ( u ) ] , h ( u ) = A ( u ) [ q 2 ( u ) + i q 1 ( u ) ]
G ( x ) = g ( u ) exp [ i k ( u x ) ] d 2 u , H ( x ) = h ( u ) exp [ i k ( u x ) ] d 2 u .
I ( 1 , 2 ) ( x ) = 1 2 n = 0 3 S n I n ( 1 , 2 ) ( x ) ,
I 0 ( 1 , 2 ) ( x ) = | G ( ± x ) | 2 + | H ( ± x ) | 2 ,
I 1 ( 1 , 2 ) ( x ) = ± 2 R e { G ( ± x ) H ( ± x ) } ,
I 2 ( 1 , 2 ) ( x ) = ± 2 I m { G ( ± x ) H ( ± x ) } ,
I 3 ( 1 , 2 ) ( x ) = ± ( | G ( ± x ) | 2 | H ( ± x ) | 2 ) .
[ N F ( s ) ] m n = N 1 + μ ¯ s [ ( μ m μ n 1 + μ s ) ¯ μ m ¯ μ n ¯ 1 + μ ¯ s ] ,
μ ( x ) = 1 I 0 ( 1 ) ( x ) [ I 1 ( 1 ) ( x ) I 2 ( 1 ) ( x ) I 3 ( 1 ) ( x ) ]
Ψ ( 1 , 2 ) = 1 2 [ S 0 ( | g | 2 + | h | 2 ) ± 2 S 1 R e { g h } ± 2 S 2 I m { g h } ± S 3 ( | g | 2 | h | 2 ) ] d 2 u = 1 2 S 0 A 2 ( 1 ± β s ) d 2 u ,
r 2 = | x | 2 I ( 1 ) ( x ) d 2 x I ( 1 ) ( x ) d 2 x = 1 κ ( | A | 2 + A 2 q 2 ) d 2 u ,
Δ r 2 = 1 κ A 2 q 2 d 2 u = 1 κ n = 0 3 A 2 q n q n d 2 u
ϵ 0 = ϵ q q 0 .
Δ r 2 Δ r 2 + 2 κ n = 0 3 A 2 q n ϵ n d 2 u = Δ r 2 2 κ n = 0 3 ϵ n ( A 2 q n ) d 2 u + [ 2 κ edge A 2 ϵ q d u ] ,
β 1 A β 1 A + 2 A 2 ( ϵ 0 q 2 + ϵ 1 q 3 ϵ 2 q 0 + ϵ 3 q 1 ) d 2 u ,
β 2 A β 2 A + 2 A 2 ( ϵ 0 q 1 + ϵ 1 q 0 + ϵ 2 q 3 + ϵ 3 q 2 ) d 2 u ,
β 3 A β 3 A + 2 A 2 ( ϵ 0 q 0 ϵ 1 q 1 ϵ 2 q 2 + ϵ 3 q 3 ) d 2 u .
q 1 ( A 2 q 0 ) q 0 ( A 2 q 1 ) + A 2 [ Λ 1 ( q 0 q 3 + q 1 q 2 ) + Λ 2 ( q 0 2 q 1 2 ) 2 Λ 3 q 0 q 1 ] = 0 ,
q 2 ( A 2 q 0 ) q 0 ( A 2 q 2 ) + A 2 [ Λ 1 ( q 2 2 q 0 2 ) + Λ 2 ( q 0 q 3 q 1 q 2 ) 2 Λ 3 q 0 q 2 ] = 0 ,
q 3 ( A 2 q 0 ) q 0 ( A 2 q 3 ) + A 2 [ Λ 1 ( q 0 q 1 + q 2 q 3 ) + Λ 2 ( q 0 q 2 q 1 q 3 ) ] = 0.
q 0 q | edge = q q 0 | edge .
q 0 ( A 2 q ) = q ( A 2 q 0 )
( A 2 q 0 ) q 0 = ( A 2 q 1 ) q 1 = ( A 2 q 2 ) q 2 = ( A 2 q 3 ) q 3 .
A ( u ) = { 1 , u N A , 0 otherwise .
q ( u , ϕ ) = 2 b m u | m | 1 + b m 2 u 2 | m | [ cos ( m ϕ ) , sin ( m ϕ ) , 0 ]
β 3 A 2 π N A 2 = 1 N A 2 0 N A b m 4 u 4 | m | 6 b m 2 u 2 | m | + 1 ( 1 + b m 2 u 2 | m | ) 2 u d u = 0 1 ( b m N A | m | v | m | ) 4 6 ( b m N A | m | v | m | ) 2 + 1 [ 1 + ( b m N A | m | v | m | ) 2 ] 2 v d v ,
Δ r = 1 2 π λ N A [ 1 π n = 0 3 0 2 π 0 1 ( v q n v q n ) v d v d ϕ ] 1 / 2 = 1 2 π λ N A ( 2 0 1 { [ δ ¯ ( v ) ] 2 + m 2 v 2 sin 2 δ ¯ ( v ) } v d v ) 1 / 2 ,
δ ¯ ( 1 ) = 0.
δ ¯ ( v ) + δ ¯ ( v ) v + ( N A 2 Λ 3 m 2 2 v 2 ) sin [ 2 δ ¯ ( v ) ] = 0
β 3 A 2 π N A 2 = 1 N A 2 0 N A cos ( 2 c u ) u d u = [ cos ( c N A ) 1 2 s i n c ( c N A ) ] s i n c ( c N A ) = 0 ,
[ N F ( s ) ] m n = N ( μ m μ n 1 + μ s ) ¯ .
g ± ( u ) = A ( u ) [ q 0 ( u ) ± i q 3 ( u ) ] , h ± ( u ) = A ( u ) [ q 2 ( u ) ± i q 1 ( u ) ]
G ± ( x ) = g ± ( u ) exp [ i k ( u x ) ] d 2 u , H ± ( x ) = h ± ( u ) exp [ i k ( u x ) ] d 2 u .
I ( 1 , 2 ) ( x ) = | G ± ( x ) E 1 , 2 ± H ± ( x ) E 2 , 1 | 2 T ,
I ( 1 , 2 ) ( x ) = | G ± | 2 | E 1 , 2 | 2 + | H ± | 2 | E 2 , 1 | 2 ± 2 R e { G ± H ± E 1 , 2 E 2 , 1 } T = | G ± | 2 | E 1 , 2 | 2 T + | H ± | 2 | E 2 , 1 | 2 T = ± 2 R e { G ± H ± } R e { E 1 , 2 E 2 , 1 } T 2 I m { G ± H ± } I m { E 1 , 2 E 2 , 1 } T = 1 2 [ | G ± | 2 ( S 0 ± S 3 ) + | H ± | 2 ( S 0 S 3 ) ± 2 R e { G ± H ± } S 1 + 2 I m { G ± H ± } S 2 ] ,
S 0 = | E 1 | 2 T + | E 2 | 2 T , S 1 = 2 R e { E 2 E 1 } T , S 2 = 2 I m { E 2 E 1 } T , S 3 = | E 1 | 2 T | E 2 | 2 T .
I ( 1 , 2 ) ( x ) = 1 2 n = 0 3 S n I n ( 1 , 2 ) ( x ) ,
I 0 ( 1 , 2 ) ( x ) = | G ± ( x ) | 2 + | H ± ( x ) | 2 , I 1 ( 1 , 2 ) ( x ) = ± 2 R e { G ± ( x ) H ± ( x ) } , I 2 ( 1 , 2 ) ( x ) = 2 I m { G ± ( x ) H ± ( x ) } , I 3 ( 1 , 2 ) ( x ) = ± ( | G ± ( x ) | 2 | H ± ( x ) | 2 ) .
P ( x | s ) = I ( 1 ) ( x ) I ( 1 ) ( x ) d 2 x .
I 0 ( 1 ) ( 1 + n = 1 3 s n I n ( 1 ) / I 0 ( 1 ) ) I 0 ( 1 ) ( 1 + n = 1 3 s n I n ( 1 ) / I 0 ( 1 ) ) d 2 x = w ( x ) 1 + μ ( x ) s 1 + μ ¯ s ,
( s | x ) = ln [ w ( x ) 1 + μ ( x ) s 1 + μ ¯ s ] .
s n = μ n 1 + μ s μ n ¯ 1 + μ ¯ s , 2 s m s n = μ m μ n ( 1 + μ s ) 2 + μ m ¯ μ n ¯ ( 1 + μ ¯ s ) 2 .
[ F ( s ) ] m n = ( 2 s m s n ( s | x ) ) P ( x | s ) d 2 x = w ( x ) 1 + μ s 1 + μ ¯ s [ μ m μ n ( 1 + μ s ) 2 + μ m ¯ μ n ¯ ( 1 + μ ¯ s ) 2 ] d 2 x = ( 1 1 + μ ¯ s μ m μ n 1 + μ s w ( x ) d 2 x ) ( μ m ¯ μ n ¯ ( 1 + μ ¯ s ) 3 ( 1 + μ s ) w ( x ) d 2 x ) .

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