Abstract

Optical fields whose coherence and/or polarization properties appear to change under propagation have intrigued researchers for many years. We describe and experimentally demonstrate a class of optical fields whose polarization content at any transverse plane spans a disk-like region within the Poincaré sphere. When examined through a paraxial focal region, the disk rotates under propagation, spanning all possible states of polarization. We map the change in Stokes parameters through focus for each case, comparing experiment with the theoretical predictions.

© 2012 OSA

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References

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  1. A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express18, 10777–10785 (2010).
    [CrossRef] [PubMed]
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    [CrossRef]
  3. H. Chen, J. Hao, B-F Zhang, J. Xu, J. Ding, and H-T Wang, “Generation of vector beam with space-variant distribution of both polarization and phase,” Opt. Lett.36, 3179–3181 (2011)
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  6. I. Mokhun, R. Brandel, and Ju. Viktorovskaya, “Angular momentum of electromagnetic field in areas of polarization singularities,” Ukr. J. Phys. Opt.7, 63–73 (2006).
    [CrossRef]
  7. I. Mokhun and R. Khrobatin, “Shift of application point of angular momentum in the area of elementary polarization singularity,” J. Opt. A: Pure and Appl. Opt.10, 1–10 (2008).
    [CrossRef]
  8. A. K. Spilman and T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt.26, 61–66 (2007).
    [CrossRef]
  9. A. K. Spilman, A. M. Beckley, and T. G. Brown, “Focal splitting and optical vortex structure induced by stress birefringence” Proc. SPIE6667, 666701 (2007).
  10. A. K. Spilman and T. G. Brown, “Stress-induced focal splitting,” Opt. Express15, 8411–8421 (2007).
    [CrossRef] [PubMed]
  11. T. G. Brown and Q. Zhan, “Introduction: Unconventional Polarization States of Light Focus Issue,” Opt. Express18, 10775–10776 (2010).
    [CrossRef] [PubMed]

2011

2010

2008

I. Mokhun and R. Khrobatin, “Shift of application point of angular momentum in the area of elementary polarization singularity,” J. Opt. A: Pure and Appl. Opt.10, 1–10 (2008).
[CrossRef]

2007

A. K. Spilman and T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt.26, 61–66 (2007).
[CrossRef]

A. K. Spilman, A. M. Beckley, and T. G. Brown, “Focal splitting and optical vortex structure induced by stress birefringence” Proc. SPIE6667, 666701 (2007).

A. K. Spilman and T. G. Brown, “Stress-induced focal splitting,” Opt. Express15, 8411–8421 (2007).
[CrossRef] [PubMed]

2006

I. Mokhun, R. Brandel, and Ju. Viktorovskaya, “Angular momentum of electromagnetic field in areas of polarization singularities,” Ukr. J. Phys. Opt.7, 63–73 (2006).
[CrossRef]

Alonso, M. A.

Beckley, A. M.

A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express18, 10777–10785 (2010).
[CrossRef] [PubMed]

A. K. Spilman, A. M. Beckley, and T. G. Brown, “Focal splitting and optical vortex structure induced by stress birefringence” Proc. SPIE6667, 666701 (2007).

Brandel, R.

I. Mokhun, R. Brandel, and Ju. Viktorovskaya, “Angular momentum of electromagnetic field in areas of polarization singularities,” Ukr. J. Phys. Opt.7, 63–73 (2006).
[CrossRef]

Brown, T. G.

T. G. Brown and Q. Zhan, “Introduction: Unconventional Polarization States of Light Focus Issue,” Opt. Express18, 10775–10776 (2010).
[CrossRef] [PubMed]

A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express18, 10777–10785 (2010).
[CrossRef] [PubMed]

A. K. Spilman, A. M. Beckley, and T. G. Brown, “Focal splitting and optical vortex structure induced by stress birefringence” Proc. SPIE6667, 666701 (2007).

A. K. Spilman and T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt.26, 61–66 (2007).
[CrossRef]

A. K. Spilman and T. G. Brown, “Stress-induced focal splitting,” Opt. Express15, 8411–8421 (2007).
[CrossRef] [PubMed]

Chen, H.

Cheng, W.

Ding, J.

Gutiérrez-Vega, J. C.

Han, W.

Hao, J.

Khrobatin, R.

I. Mokhun and R. Khrobatin, “Shift of application point of angular momentum in the area of elementary polarization singularity,” J. Opt. A: Pure and Appl. Opt.10, 1–10 (2008).
[CrossRef]

Lerman, G. M.

Levy, U.

Mokhun, I.

I. Mokhun and R. Khrobatin, “Shift of application point of angular momentum in the area of elementary polarization singularity,” J. Opt. A: Pure and Appl. Opt.10, 1–10 (2008).
[CrossRef]

I. Mokhun, R. Brandel, and Ju. Viktorovskaya, “Angular momentum of electromagnetic field in areas of polarization singularities,” Ukr. J. Phys. Opt.7, 63–73 (2006).
[CrossRef]

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).
[CrossRef]

A. K. Spilman and T. G. Brown, “Stress-induced focal splitting,” Opt. Express15, 8411–8421 (2007).
[CrossRef] [PubMed]

A. K. Spilman, A. M. Beckley, and T. G. Brown, “Focal splitting and optical vortex structure induced by stress birefringence” Proc. SPIE6667, 666701 (2007).

Stern, L.

Viktorovskaya, Ju.

I. Mokhun, R. Brandel, and Ju. Viktorovskaya, “Angular momentum of electromagnetic field in areas of polarization singularities,” Ukr. J. Phys. Opt.7, 63–73 (2006).
[CrossRef]

Wang, H-T

Xu, J.

Zhan, Q.

Zhang, B-F

Appl. Opt.

A. K. Spilman and T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt.26, 61–66 (2007).
[CrossRef]

J. Opt. A: Pure and Appl. Opt.

I. Mokhun and R. Khrobatin, “Shift of application point of angular momentum in the area of elementary polarization singularity,” J. Opt. A: Pure and Appl. Opt.10, 1–10 (2008).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. SPIE

A. K. Spilman, A. M. Beckley, and T. G. Brown, “Focal splitting and optical vortex structure induced by stress birefringence” Proc. SPIE6667, 666701 (2007).

Ukr. J. Phys. Opt.

I. Mokhun, R. Brandel, and Ju. Viktorovskaya, “Angular momentum of electromagnetic field in areas of polarization singularities,” Ukr. J. Phys. Opt.7, 63–73 (2006).
[CrossRef]

Supplementary Material (2)

» Media 1: MOV (180 KB)     
» Media 2: MOV (231 KB)     

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Figures (3)

Fig. 1
Fig. 1

The theoretical (left ( Media 1)) and experimental (right ( Media 2)) Poincaré sphere coverage of beams that span the sphere. Clicking on each figure will play a movie that shows the through-focus rotation of each disk about the S3 axis. In the example shown, the Rayleigh range is about 90 mm

Fig. 2
Fig. 2

The experimental setup for partially polarized FP beam combinations. Illumination laser light is filtered and collimated, polarized (P), phase apodized by the stress-engineered optical element (SEOE), focused by a slow lens (f = 300mm), and analyzed (A) then measured on a CCD. For beams that span the sphere, two half-wave liquid crystal retarders (LC-1,2) convert the beam and vortex states. For beams of a changing degree of polarization, LC-2 is removed.

Fig. 3
Fig. 3

The experimental (left) and simulated (right) realizations of beams that span the sphere at (a) −1.1zR, (b) the focal plane and (c) 1.1zR. The simulation is matched to the beam size, value of γ, Rayleigh range and ϕ̄ = ϕ′ = −π/4.

Equations (13)

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U 00 ( r ) = u 0 ξ ( z ) exp [ i k z ρ 2 w 0 2 ξ ( z ) ] ,
ξ ( z ) = 1 + i 2 z k w 0 2 = 1 + i z z R ,
U 0 ± 1 ( r ) = 2 x ± i y w 0 ξ ( z ) U 00 ( r ) = 2 u 0 ρ w ( z ) exp { i [ ± ϕ ϕ ξ ( z ) ] } exp [ i k z ρ 2 w 0 2 ξ ( z ) ] ,
E P ± ( r ; γ , ϕ 0 , e ^ 1 , e ^ 2 ) = cos γ e ^ 1 U 00 ( r ) + exp ( i ϕ 0 ) sin γ e ^ 2 U 0 ± 1 ( r ) = ( e ^ 1 + exp { i [ ± ϕ + ϕ 0 ϕ ξ ( z ) ] } ρ ¯ e ^ 2 ) cos γ U 00 ( r ) ,
| E P ± ( r , γ , ϕ 0 , e ^ 1 , e ^ 2 ) | 2 = cos 2 γ | U 00 ( r ) | 2 + sin 2 γ | U 0 ± 1 ( r ) | 2 = u 0 2 cos 2 γ | ξ ( z ) | 2 ( 1 + ρ ¯ 2 ) exp ( ρ ¯ 2 tan 2 γ ) .
S 1 S 0 = 2 ρ ¯ cos [ ϕ ± ( ϕ 0 ϕ ξ ) ] 1 + ρ ¯ 2 , S 2 S 0 = 2 ρ ¯ sin [ ϕ ± ( ϕ 0 ϕ ξ ) ] 1 + ρ ¯ 2 , S 3 S 0 = 1 ρ ¯ 2 1 + ρ ¯ 2 ,
E FP ( r ; γ , ϕ 1 , ϕ 2 , e ^ 1 , e ^ 2 ) = a 1 E P + ( r ; γ , ϕ 1 , e ^ 1 , e ^ 2 ) + a 2 E P ( r ; γ , ϕ 2 , e ^ 1 , e ^ 2 ) ,
S 1 S 0 = 2 ρ ¯ cos ( ϕ ϕ ) cos ( ϕ ξ ϕ ¯ ) 1 + ρ ¯ 2 , S 2 S 0 = 2 ρ ¯ cos ( ϕ ϕ ) sin ( ϕ ξ ϕ ¯ ) 1 + ρ ¯ 2 , S 3 S 0 = 1 ρ ¯ 2 1 + ρ ¯ 2 ,
J = ( ρ ) = cos ( c ρ 2 ) I = + i sin ( c ρ 2 ) [ cos ( ϕ ) sin ( ϕ ) sin ( ϕ ) cos ( ϕ ) ]
E t ( ρ ) = g ( ρ ) [ cos ( c ρ 2 ) e ^ 1 + i e i ϕ sin ( c ρ 2 ) e ^ 2 ] E P + ( r ; γ , π / 2 , e ^ 1 , e ^ 2 ) .
E t ( ρ ) = g ( ρ ) [ cos ( c ρ 2 ) e ^ 1 + i e i( ϕ + 4 ϕ H ) sin ( c ρ 2 ) e ^ 2 ] E P ( r ; γ , π / 2 4 ϕ H , e ^ 1 , e ^ 2 ) ,
E var ( r ; γ , ϕ 1 , ϕ 2 , e ^ 1 , e ^ 2 ) = a 1 E P + ( r ; γ , ϕ 1 , e ^ 1 , e ^ 2 ) + a 2 E P ( r ; γ , ϕ 2 , e ^ 2 , e ^ 1 ) ,
S 1 S 0 = 2 ρ ¯ cos ( ϕ + ϕ ) cos ( ϕ ξ ϕ ¯ ) 1 + ρ ¯ 2 , S 2 S 0 = 2 ρ ¯ sin ( ϕ + ϕ ) cos ( ϕ ξ ϕ ) 1 + ρ ¯ 2 , S 3 S 0 = 0 ,

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