Abstract

Understanding and modeling the propagation of polarized light through thick, space variant birefringent media is important in both fundamental and applied optics. We present and experimentally evaluate two methods to model the off axis propagation of polarized light through a thick stress-engineered optic (SEO). First, we use a differential equation solving method, which utilizes the analytic expression for the Jones matrix of the SEO leading to a numerical solution for the output electric field. Then we present a geometric method to obtain similar results with much less computational complexity. Finally, a comparison is done between the data and the simulations.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. L. Mollenauer, D. Downie, H. Engstrom, and W. Grant, “Stress plate optical modulator for circular dichroism measurements,” Appl. optics 8, 661–665 (1969).
    [Crossref]
  2. A. K. Spilman and T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt. 46, 61–66 (2007).
    [Crossref]
  3. A. K. Spilman and T. G. Brown, “Stress-induced focal splitting,” Opt. Express 15, 8411–8421 (2007).
    [Crossref] [PubMed]
  4. A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full poincaré beams,” Opt. Express 18, 10777–10785 (2010).
    [Crossref] [PubMed]
  5. T. G. Brown and A. M. Beckley, “Stress engineering and the applications of inhomogeneously polarized optical fields,” Front. Optoelectronics 6, 89–96 (2013).
    [Crossref]
  6. R. D. Ramkhalawon, T. G. Brown, and M. A. Alonso, “Imaging the polarization of a light field,” Opt. Express 21, 4106–4115 (2013).
    [Crossref] [PubMed]
  7. B. G. Zimmerman, R. Ramkhalawon, M. Alonso, and T. G. Brown, “Pinhole array implementation of star test polarimetry,” (2014).
  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, 2105–2108 (2016).
    [Crossref] [PubMed]
  9. R. Chipman, “Polarization ray tracing,” in “Proceedings of SPIE - The International Society for Optical Engineering,” , vol. 766 C. Londono and R. Fischer, eds. (SPIE, 1987), vol. 766, pp. 61–68.
  10. R. Chipman, “Mechanics of polarization ray tracing,” Opt. Eng. 34, 1636–1645 (1995).
    [Crossref]
  11. G. Yun, K. Crabtree, and R. Chipman, “Three-dimensional polarization ray-tracing calculus i: Definition and diattenuation,” Appl. Opt. 50, 2855–2865 (2011).
    [Crossref] [PubMed]
  12. G. Yun, S. McClain, and R. Chipman, “Three-dimensional polarization ray-tracing calculus ii: Retardance,” Appl. Opt. 50, 2866–2874 (2011).
    [Crossref] [PubMed]
  13. R. Chipman, “The polarization ray tracing calculus,” in “Frontiers in Optics, FiO 2014,” (Optical Society of America (OSA), 2014).
  14. M. R. Dennis, K. O’Holleran, and M. J. Padgett, Chapter 5 Singular Optics: Optical Vortices and Polarization Singularities (Elsevier, 2009), vol. 53 of Progress in Optics, pp. 293–363.
  15. E. J. Galvez and S. Khadka, “Poincare modes of light,” in “Proc.SPIE,” , vol. 8274 (2012), vol. 8274.
  16. E. J. Galvez and S. Zhang, “Multitwist mobius polarization in crossed complex light beams,” in “Proc.SPIE,” , vol. 10549 (2018), vol. 10549.
  17. T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization möbius strips,” Science. 347, 964–966 (2015).
    [Crossref] [PubMed]

2016 (1)

2015 (1)

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization möbius strips,” Science. 347, 964–966 (2015).
[Crossref] [PubMed]

2013 (2)

T. G. Brown and A. M. Beckley, “Stress engineering and the applications of inhomogeneously polarized optical fields,” Front. Optoelectronics 6, 89–96 (2013).
[Crossref]

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

2011 (2)

2010 (1)

2007 (2)

1995 (1)

R. Chipman, “Mechanics of polarization ray tracing,” Opt. Eng. 34, 1636–1645 (1995).
[Crossref]

1969 (1)

L. Mollenauer, D. Downie, H. Engstrom, and W. Grant, “Stress plate optical modulator for circular dichroism measurements,” Appl. optics 8, 661–665 (1969).
[Crossref]

Alonso, M.

B. G. Zimmerman, R. Ramkhalawon, M. Alonso, and T. G. Brown, “Pinhole array implementation of star test polarimetry,” (2014).

Alonso, M. A.

Andresen, E. R.

Banzer, P.

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization möbius strips,” Science. 347, 964–966 (2015).
[Crossref] [PubMed]

Bauer, T.

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization möbius strips,” Science. 347, 964–966 (2015).
[Crossref] [PubMed]

Beckley, A. M.

T. G. Brown and A. M. Beckley, “Stress engineering and the applications of inhomogeneously polarized optical fields,” Front. Optoelectronics 6, 89–96 (2013).
[Crossref]

A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full poincaré beams,” Opt. Express 18, 10777–10785 (2010).
[Crossref] [PubMed]

Bouwmans, G.

Boyd, R. W.

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization möbius strips,” Science. 347, 964–966 (2015).
[Crossref] [PubMed]

Brown, T. G.

Chipman, R.

G. Yun, K. Crabtree, and R. Chipman, “Three-dimensional polarization ray-tracing calculus i: Definition and diattenuation,” Appl. Opt. 50, 2855–2865 (2011).
[Crossref] [PubMed]

G. Yun, S. McClain, and R. Chipman, “Three-dimensional polarization ray-tracing calculus ii: Retardance,” Appl. Opt. 50, 2866–2874 (2011).
[Crossref] [PubMed]

R. Chipman, “Mechanics of polarization ray tracing,” Opt. Eng. 34, 1636–1645 (1995).
[Crossref]

R. Chipman, “Polarization ray tracing,” in “Proceedings of SPIE - The International Society for Optical Engineering,” , vol. 766 C. Londono and R. Fischer, eds. (SPIE, 1987), vol. 766, pp. 61–68.

R. Chipman, “The polarization ray tracing calculus,” in “Frontiers in Optics, FiO 2014,” (Optical Society of America (OSA), 2014).

Crabtree, K.

Dennis, M. R.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, Chapter 5 Singular Optics: Optical Vortices and Polarization Singularities (Elsevier, 2009), vol. 53 of Progress in Optics, pp. 293–363.

Downie, D.

L. Mollenauer, D. Downie, H. Engstrom, and W. Grant, “Stress plate optical modulator for circular dichroism measurements,” Appl. optics 8, 661–665 (1969).
[Crossref]

Engstrom, H.

L. Mollenauer, D. Downie, H. Engstrom, and W. Grant, “Stress plate optical modulator for circular dichroism measurements,” Appl. optics 8, 661–665 (1969).
[Crossref]

Galvez, E. J.

E. J. Galvez and S. Khadka, “Poincare modes of light,” in “Proc.SPIE,” , vol. 8274 (2012), vol. 8274.

E. J. Galvez and S. Zhang, “Multitwist mobius polarization in crossed complex light beams,” in “Proc.SPIE,” , vol. 10549 (2018), vol. 10549.

Grant, W.

L. Mollenauer, D. Downie, H. Engstrom, and W. Grant, “Stress plate optical modulator for circular dichroism measurements,” Appl. optics 8, 661–665 (1969).
[Crossref]

Karimi, E.

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization möbius strips,” Science. 347, 964–966 (2015).
[Crossref] [PubMed]

Khadka, S.

E. J. Galvez and S. Khadka, “Poincare modes of light,” in “Proc.SPIE,” , vol. 8274 (2012), vol. 8274.

Leuchs, G.

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization möbius strips,” Science. 347, 964–966 (2015).
[Crossref] [PubMed]

Marrucci, L.

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization möbius strips,” Science. 347, 964–966 (2015).
[Crossref] [PubMed]

McClain, S.

Mollenauer, L.

L. Mollenauer, D. Downie, H. Engstrom, and W. Grant, “Stress plate optical modulator for circular dichroism measurements,” Appl. optics 8, 661–665 (1969).
[Crossref]

O’Holleran, K.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, Chapter 5 Singular Optics: Optical Vortices and Polarization Singularities (Elsevier, 2009), vol. 53 of Progress in Optics, pp. 293–363.

Orlov, S.

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization möbius strips,” Science. 347, 964–966 (2015).
[Crossref] [PubMed]

Padgett, M. J.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, Chapter 5 Singular Optics: Optical Vortices and Polarization Singularities (Elsevier, 2009), vol. 53 of Progress in Optics, pp. 293–363.

Ramkhalawon, R.

B. G. Zimmerman, R. Ramkhalawon, M. Alonso, and T. G. Brown, “Pinhole array implementation of star test polarimetry,” (2014).

Ramkhalawon, R. D.

Rigneault, H.

Rubano, A.

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization möbius strips,” Science. 347, 964–966 (2015).
[Crossref] [PubMed]

Santamato, E.

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization möbius strips,” Science. 347, 964–966 (2015).
[Crossref] [PubMed]

Sivankutty, S.

Spilman, A. K.

Yun, G.

Zhang, S.

E. J. Galvez and S. Zhang, “Multitwist mobius polarization in crossed complex light beams,” in “Proc.SPIE,” , vol. 10549 (2018), vol. 10549.

Zimmerman, B. G.

B. G. Zimmerman, R. Ramkhalawon, M. Alonso, and T. G. Brown, “Pinhole array implementation of star test polarimetry,” (2014).

Appl. Opt. (3)

Appl. optics (1)

L. Mollenauer, D. Downie, H. Engstrom, and W. Grant, “Stress plate optical modulator for circular dichroism measurements,” Appl. optics 8, 661–665 (1969).
[Crossref]

Front. Optoelectronics (1)

T. G. Brown and A. M. Beckley, “Stress engineering and the applications of inhomogeneously polarized optical fields,” Front. Optoelectronics 6, 89–96 (2013).
[Crossref]

Opt. Eng. (1)

R. Chipman, “Mechanics of polarization ray tracing,” Opt. Eng. 34, 1636–1645 (1995).
[Crossref]

Opt. Express (3)

Opt. Lett. (1)

Science. (1)

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization möbius strips,” Science. 347, 964–966 (2015).
[Crossref] [PubMed]

Other (6)

R. Chipman, “The polarization ray tracing calculus,” in “Frontiers in Optics, FiO 2014,” (Optical Society of America (OSA), 2014).

M. R. Dennis, K. O’Holleran, and M. J. Padgett, Chapter 5 Singular Optics: Optical Vortices and Polarization Singularities (Elsevier, 2009), vol. 53 of Progress in Optics, pp. 293–363.

E. J. Galvez and S. Khadka, “Poincare modes of light,” in “Proc.SPIE,” , vol. 8274 (2012), vol. 8274.

E. J. Galvez and S. Zhang, “Multitwist mobius polarization in crossed complex light beams,” in “Proc.SPIE,” , vol. 10549 (2018), vol. 10549.

R. Chipman, “Polarization ray tracing,” in “Proceedings of SPIE - The International Society for Optical Engineering,” , vol. 766 C. Londono and R. Fischer, eds. (SPIE, 1987), vol. 766, pp. 61–68.

B. G. Zimmerman, R. Ramkhalawon, M. Alonso, and T. G. Brown, “Pinhole array implementation of star test polarimetry,” (2014).

Supplementary Material (2)

NameDescription
» Visualization 1       Comparison between irradiance patterns from experimental images and the simulations involving oblique propagation of light through stress-engineered optic (SEO). The irradiance contour lines obtained from the numerical method have been overlaid on bo
» Visualization 2       Polarization map computed after propagating through a stress-engineered optic (SEO) at angles 0 through 24 degrees. The ellipses are overlaid on the intensity pattern after the beam goes through the circular analyzer, which blocks the RHC polarized l

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Figures (8)

Fig. 1
Fig. 1 Steps of creating an stress engineered optic. (1) material is removed from inside of a metal ring in a trigonal symmetry. (2) an optically flat glass window is inserted into the heated ring. (3) when cooled, the contact points compress the window creating the required stress pattern.
Fig. 2
Fig. 2 Geometry of oblique propagation of light through the SEO
Fig. 3
Fig. 3 Illustration of discrete retarder method. The propagation of the electric field inside the SEO can be modeled as a discrete propagation of continuously shifting electric field. The total shift of the field is equal to Δ, which is determined by the refraction geometry
Fig. 4
Fig. 4 Experimental Setup. Image of the SEO (m = 3) irradiance pattern is also shown in the inset, in which the dashed circle indicates the region of the SEO covered by the pupil. An aperture after the rotating diffuser served to limit the range of angles.
Fig. 5
Fig. 5 Comparison between the experimental images and the simulations from the continuous method (CM) and the discrete retarder method (DRM). The two images on the right show the irradiance contour lines obtained from the CM overlaid on the corresponding simulated image and the measured image. ( Visualization 1).
Fig. 6
Fig. 6 Polarization map computed after propagating through the SEO at angles 0°, 10°, and 20°. The ellipses are overlaid on the intensity pattern after the beam goes through the circular analyzer, which blocks the RHC polarized light. The handedness of each polarization ellipse is color coded (green: right handed, red: left handed). Bottom panel illustrates the contour lines of equal ellipticity overlaid on the color coded azimuthal angle of each polarization ellipse on the Poincaré sphere. Thick lines correspond to S3 = ±0.95 and they encircle the points of circular polarization (S3 = ±1). Thick dashed lines show the contours of linear polarization where S3 = 0 (see Visualization 2).
Fig. 7
Fig. 7 Top row: Simulated results for different incident angles showing contours of the third component of the Stokes vector, S3. Bottom row: Measured S3 contours.
Fig. 8
Fig. 8 Root sum squared error (E) as a fraction of the total energy in the irradiance profile from the continuous model. The total energy has been normalized to unity. Inset depicts the behavior of E as a function of 1/N for incident angle of 24°.

Tables (1)

Tables Icon

Table 1 Analytic expressions for electric field components in circular basis for various input polarizations: horizontal (H), vertical (V), +45° (P), −45° (M), left hand circular (LHC), and right hand circular (RHC). The Jones vector of these inputs at z = 0 in circular basis, (eL(z = 0), eR(z = 0))T , are given in the second column and the output electric field components are given in the last two columns. Here, ρ′ = ρm−2 and ϕ′ = (m − 2)ϕ.

Equations (20)

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E out = 𝕁 E in ,
𝕁 = cos ( δ 2 ) 𝕀 + i sin ( δ 2 ) 𝕇 ( 2 α ) .
𝕇 l ( 2 α ) = ( cos 2 α sin 2 α sin 2 α cos 2 α ) , 𝕇 c ( 2 α ) = ( 0 e i 2 α e i 2 α 0 ) .
𝕁 = cos ( c ρ m 2 2 ) 𝕀 + i sin ( c ρ m 2 2 ) 𝕇 ( ( 2 m ) ϕ ) ,
𝕁 = cos ( c ρ 2 ) 𝕀 + i sin ( c ρ 2 ) 𝕇 ( ϕ ) .
𝕁 δ z = cos ( c ρ m 2 δ z 2 L ) 𝕀 + i sin ( c ρ m 2 δ z 2 L ) 𝕇 ( ( 2 m ) ϕ ) ,
E ( z + δ z ) = e i k δ z 𝕁 δ z E ( z ) ,
e ( z + δ z ) e i k ( z + δ z ) = e i k δ z 𝕁 δ e ( z ) e i k z ,
e ( z + δ z ) = 𝕁 δ z e ( z ) ,
= ( cos ( c ρ m 2 δ z 2 L ) 𝕀 + i sin ( c ρ m 2 δ z 2 L ) 𝕇 ( ( 2 m ) ϕ ) ) e ( z ) ) .
e ( z + δ z ) = ( 𝕀 + i c ρ m 2 δ z 2 L 𝕇 ( ( 2 m ) ϕ ) ) e ( z ) ) ,
e ( z + δ z ) e ( z ) δ z = i c ρ m 2 2 L 𝕇 ( ( 2 m ) ϕ ) e ( z ) .
d d z e ( z ; ρ , ϕ ) = i c ρ m 2 2 L ( 0 e i ( 2 m ) ϕ e i ( 2 m ) ϕ 0 ) e ( z ; ρ , ϕ ) .
x ( z ) = x ( 0 ) + z tan θ ,
y ( z ) = y ( 0 ) ,
ρ ( z ) = x 2 ( z ) + y 2 ( z ) ,
ϕ ( z ) = tan 1 ( y ( z ) / x ( z ) ) .
𝕁 n = cos ( c ρ 2 N ) 𝕀 + i sin ( c ρ 2 N ) 𝕇 ( ϕ ) .
E ( z + L ) = e i k L 𝕁 n N E ( z ) .
E = pixels ( I C M I D R M ) 2 .

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