Abstract

Under circularly polarized illumination, a trigonal birefringence pattern produces concentric rings of alternating right and left circular polarization. When a window with trigonal stress is placed in the pupil of an imaging system, the birefringence induces an axial splitting of the focus. We analyze and experimentally test this phenomenon and discuss applications to optical imaging.

© 2007 Optical Society of America

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References

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  1. A. K. Spilman and T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt. 26, 61–66 (2007), http://ao.osa.org/abstract.cfm?id=119864.
    [Crossref]
  2. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7, 77–87 (2000).
    [Crossref] [PubMed]
  3. D. P. Biss and T. G. Brown, “Cylindrical vector beam focusing through a dielectric surface,” Opt. Express 9, 490–497 (2001).
    [Crossref] [PubMed]
  4. M. Dennis, F. Flossmann, M. Maier, and U.T. Schwarz, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
    [Crossref] [PubMed]
  5. J. PawleyHandbook of Biological Confocal Microscopy (IMR Press, Madison, 1989).
  6. Y. Unno, ”Distorted wave front produced by a high-resolution projection optical system having rotationally symmetric birefringence,” Appl. Opt. 37, 7241–7247 (1998).
    [Crossref]

2007 (1)

A. K. Spilman and T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt. 26, 61–66 (2007), http://ao.osa.org/abstract.cfm?id=119864.
[Crossref]

2005 (1)

M. Dennis, F. Flossmann, M. Maier, and U.T. Schwarz, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[Crossref] [PubMed]

2001 (1)

2000 (1)

1998 (1)

Biss, D. P.

Brown, T. G.

Dennis, M.

M. Dennis, F. Flossmann, M. Maier, and U.T. Schwarz, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[Crossref] [PubMed]

Flossmann, F.

M. Dennis, F. Flossmann, M. Maier, and U.T. Schwarz, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[Crossref] [PubMed]

Maier, M.

M. Dennis, F. Flossmann, M. Maier, and U.T. Schwarz, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[Crossref] [PubMed]

Pawley, J.

J. PawleyHandbook of Biological Confocal Microscopy (IMR Press, Madison, 1989).

Schwarz, U.T.

M. Dennis, F. Flossmann, M. Maier, and U.T. Schwarz, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[Crossref] [PubMed]

Spilman, A. K.

A. K. Spilman and T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt. 26, 61–66 (2007), http://ao.osa.org/abstract.cfm?id=119864.
[Crossref]

Unno, Y.

Youngworth, K. S.

Appl. Opt. (2)

A. K. Spilman and T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt. 26, 61–66 (2007), http://ao.osa.org/abstract.cfm?id=119864.
[Crossref]

Y. Unno, ”Distorted wave front produced by a high-resolution projection optical system having rotationally symmetric birefringence,” Appl. Opt. 37, 7241–7247 (1998).
[Crossref]

Opt. Express (2)

Phys. Rev. Lett. (1)

M. Dennis, F. Flossmann, M. Maier, and U.T. Schwarz, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[Crossref] [PubMed]

Other (1)

J. PawleyHandbook of Biological Confocal Microscopy (IMR Press, Madison, 1989).

Supplementary Material (3)

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» Media 2: MOV (386 KB)     
» Media 3: MOV (1644 KB)     

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Figures (10)

Fig. 1.
Fig. 1.

Experimental design. The Air Force target is placed at plane P 1, the front focal plane of the objective L 1, and the trigonally stressed window is placed at plane P 3. L 2 is a Bertrand lens arranged to create an image of the aperture stop (P 2) in the plane of the stressed window and L 3 is a relay lens. An intermediate image is formed between L 2 and P 3. For the optical system, NA=0.06.

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Fig. 2.
Fig. 2.

Rings of equal retardance in a window held under trigonal symmetric stress. (a) Photograph of stressed window held between circular polarizers. (b) An expanded view of the center region of the photograph.

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Fig. 3.
Fig. 3.

(a) Plot of the irradiance of the LHC light color coded with phase. (b) Photograph of the LHC component of the trigonally stressed window between circular polarizers. (c) Pupil image of RHC light. (d) Pupil image of LHC light.

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Fig. 4.
Fig. 4.

Simulation of the PSF slice of the trigonally stressed window for varying values of defocus (file size: 2MB). [Media 1]

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Fig. 5.
Fig. 5.

Axial irradiance as a function of focal shift, measured in microns. The circles represent the experiment and the solid line represents the theoretical curve.

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Fig. 6.
Fig. 6.

(a) Experimental through-focus PSF of the trigonally stressed window in the pupil plane of an NA = 0.06 imaging system. At each stage the PSF is normalized to its peak irradiance (file size: 388 KB). [Media 2] (b) Theoretical simulation of the through-focus PSF of the trigonally stressed window in the pupil plane plotted in normalized units of λ/NA (file size: 1.6 MB). [Media 3]

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Fig. 7.
Fig. 7.

Zone plate with alternating rings of RHC and LHC light. The period of the rings is given as Λ.

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Fig. 8.
Fig. 8.

Schematic illustrating the origin of the stress parameter, c. |2εL | is the separation between the two foci.

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Fig. 9.
Fig. 9.

Relative axial irradiance for various values of the stress parameter, c.

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Fig. 10.
Fig. 10.

(a) Experimental PSFs. (b) Predicted PSFs. (c) Air Force target positioned at each focus (left and right) and midway between (middle image).

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Equations (13)

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[ R ( θ ) ] − 1 [ σ xx σ xy σ yx σ yy ] [ R ( θ ) ] = [ σ 11 0 0 σ 22 ] .
M ρ ϕ = [ cos θ sin θ − sin θ cos θ ] [ e iδ 2 0 0 e − iδ 2 ] [ cos θ − sin θ sin θ cos θ ] .
E out = M ( ρ ) E in .
E out = 1 2 e − iθ [ cos θ e i δ 2 + i sin θ e − i δ 2 − sin θ e i δ 2 + i cos θ e − i δ 2 ] .
E out ∣ RHC = cos ( δ 2 ) .
E out ∣ LHC = ie − 2 iθ sin ( δ 2 ) .
E out ∣ RHC = cos ( cρ 2 )
E out ∣ LHC = ie iϕ sin ( cρ 2 ) .
[ A x x y A y x y ] = ∫ ∫ df x df y [ E x f x f y E y f x f y ] e i 2 π ( f x x + f y y )
∣ 2 ε L ∣ = 2 κt ( NA ) 2 .
∣ 2 ε L ∣ = cλ πNA 2 .
∣ 2 ε L ∣ = 4 ω D NA 2 .
ω D = cλ 4 π .

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