Abstract

The design of a lens that modulates the geometric phase of an optical beam by manipulating its polarization is presented. To produce such a geometric phase element with a spatially varying phase function, one needs a wave plate with varying orientation. One can use subwavelength grooves to produce form birefringence, but the variation in orientation generally leads to branch points in the groove pattern. These branch points do not affect the phase of the traversing beam directly because the grooves are subwavelength. However, they do produce errors in the groove orientation, which indirectly leads to errors in the geometric phase function that is implemented. A design procedure is provided to compute the groove pattern for such a rotationally symmetric geometric phase element; and, with the aid of a numerical simulation, the effect of the branch points in the groove pattern on its performance is investigated.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. Pancharatnam, "Generalized theory of interference, and its application," Proc. Indian Acad. Sci., Sect. A 44, 247-262 (1956).
  2. M. V. Berry, "Quantal phase factors accompanying adiabatic changes," Proc. R. Soc. London, Ser. A 392, 45-57 (1984).
    [CrossRef]
  3. F. S. Roux, "Coupling of noncanonical optical vortices," J. Opt. Soc. Am. B 21, 664-670 (2004).
    [CrossRef]
  4. E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, "Geometrical phase associated with mode transformations of optical beams bearing orbital angular momentum," Phys. Rev. Lett. 90, 203901 (2003).
    [CrossRef] [PubMed]
  5. R. Bhandari, "Polarization of light and topological phases," Phys. Rep. 182, 1-64 (1997).
    [CrossRef]
  6. M. W. Farn, "Binary gratings with increased efficiency," Appl. Opt. 31, 4453-4458 (1992).
    [CrossRef] [PubMed]
  7. J. Tervo and J. Turunen, "Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings," Opt. Lett. 25, 785-786 (2000).
    [CrossRef]
  8. Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, "Space-variant Pancharatnam-Berry phase optical elements with computer-generated subwavelength gratings," Opt. Lett. 27, 1141-1143 (2002).
    [CrossRef]
  9. J. Tervo, V. Kettunen, M. Honkanen, and J. Turunen, "Design of space-variant diffractive polarization elements," J. Opt. Soc. Am. B 20, 282-289 (2003).
    [CrossRef]
  10. E. Hasman, V. Kleiner, G. Biener, and A. Niv, "Polarization dependent focusing lens by use of quantized Pancharatnam-Berry phase diffractive optics," Appl. Phys. Lett. 82, 328-330 (2003).
    [CrossRef]
  11. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Chap. 14.
  12. P. Lalanne, "Effective properties and band structures in lamellar subwavelength crystals: plane-wave method revisited," Phys. Rev. B 58, 9801-9807 (1998).
    [CrossRef]
  13. F. S. Roux, "Branch-point diffractive optics," J. Opt. Soc. Am. A 11, 2236-2243 (1994).
    [CrossRef]
  14. F. S. Roux, "Single-element diffractive optical system for real-time processing of synthetic aperture radar data," Appl. Opt. 34, 5045-5052 (1995).
    [CrossRef] [PubMed]

2004 (1)

2003 (3)

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, "Geometrical phase associated with mode transformations of optical beams bearing orbital angular momentum," Phys. Rev. Lett. 90, 203901 (2003).
[CrossRef] [PubMed]

J. Tervo, V. Kettunen, M. Honkanen, and J. Turunen, "Design of space-variant diffractive polarization elements," J. Opt. Soc. Am. B 20, 282-289 (2003).
[CrossRef]

E. Hasman, V. Kleiner, G. Biener, and A. Niv, "Polarization dependent focusing lens by use of quantized Pancharatnam-Berry phase diffractive optics," Appl. Phys. Lett. 82, 328-330 (2003).
[CrossRef]

2002 (1)

2000 (1)

1998 (1)

P. Lalanne, "Effective properties and band structures in lamellar subwavelength crystals: plane-wave method revisited," Phys. Rev. B 58, 9801-9807 (1998).
[CrossRef]

1997 (1)

R. Bhandari, "Polarization of light and topological phases," Phys. Rep. 182, 1-64 (1997).
[CrossRef]

1995 (1)

1994 (1)

1992 (1)

1984 (1)

M. V. Berry, "Quantal phase factors accompanying adiabatic changes," Proc. R. Soc. London, Ser. A 392, 45-57 (1984).
[CrossRef]

1956 (1)

S. Pancharatnam, "Generalized theory of interference, and its application," Proc. Indian Acad. Sci., Sect. A 44, 247-262 (1956).

Berry, M. V.

M. V. Berry, "Quantal phase factors accompanying adiabatic changes," Proc. R. Soc. London, Ser. A 392, 45-57 (1984).
[CrossRef]

Bhandari, R.

R. Bhandari, "Polarization of light and topological phases," Phys. Rep. 182, 1-64 (1997).
[CrossRef]

Biener, G.

E. Hasman, V. Kleiner, G. Biener, and A. Niv, "Polarization dependent focusing lens by use of quantized Pancharatnam-Berry phase diffractive optics," Appl. Phys. Lett. 82, 328-330 (2003).
[CrossRef]

Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, "Space-variant Pancharatnam-Berry phase optical elements with computer-generated subwavelength gratings," Opt. Lett. 27, 1141-1143 (2002).
[CrossRef]

Bomzon, Z.

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Chap. 14.

Crawford, P. R.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, "Geometrical phase associated with mode transformations of optical beams bearing orbital angular momentum," Phys. Rev. Lett. 90, 203901 (2003).
[CrossRef] [PubMed]

Farn, M. W.

Galvez, E. J.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, "Geometrical phase associated with mode transformations of optical beams bearing orbital angular momentum," Phys. Rev. Lett. 90, 203901 (2003).
[CrossRef] [PubMed]

Haglin, P. J.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, "Geometrical phase associated with mode transformations of optical beams bearing orbital angular momentum," Phys. Rev. Lett. 90, 203901 (2003).
[CrossRef] [PubMed]

Hasman, E.

E. Hasman, V. Kleiner, G. Biener, and A. Niv, "Polarization dependent focusing lens by use of quantized Pancharatnam-Berry phase diffractive optics," Appl. Phys. Lett. 82, 328-330 (2003).
[CrossRef]

Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, "Space-variant Pancharatnam-Berry phase optical elements with computer-generated subwavelength gratings," Opt. Lett. 27, 1141-1143 (2002).
[CrossRef]

Honkanen, M.

J. Tervo, V. Kettunen, M. Honkanen, and J. Turunen, "Design of space-variant diffractive polarization elements," J. Opt. Soc. Am. B 20, 282-289 (2003).
[CrossRef]

Kettunen, V.

J. Tervo, V. Kettunen, M. Honkanen, and J. Turunen, "Design of space-variant diffractive polarization elements," J. Opt. Soc. Am. B 20, 282-289 (2003).
[CrossRef]

Kleiner, V.

E. Hasman, V. Kleiner, G. Biener, and A. Niv, "Polarization dependent focusing lens by use of quantized Pancharatnam-Berry phase diffractive optics," Appl. Phys. Lett. 82, 328-330 (2003).
[CrossRef]

Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, "Space-variant Pancharatnam-Berry phase optical elements with computer-generated subwavelength gratings," Opt. Lett. 27, 1141-1143 (2002).
[CrossRef]

Lalanne, P.

P. Lalanne, "Effective properties and band structures in lamellar subwavelength crystals: plane-wave method revisited," Phys. Rev. B 58, 9801-9807 (1998).
[CrossRef]

Niv, A.

E. Hasman, V. Kleiner, G. Biener, and A. Niv, "Polarization dependent focusing lens by use of quantized Pancharatnam-Berry phase diffractive optics," Appl. Phys. Lett. 82, 328-330 (2003).
[CrossRef]

Pancharatnam, S.

S. Pancharatnam, "Generalized theory of interference, and its application," Proc. Indian Acad. Sci., Sect. A 44, 247-262 (1956).

Pysher, M. J.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, "Geometrical phase associated with mode transformations of optical beams bearing orbital angular momentum," Phys. Rev. Lett. 90, 203901 (2003).
[CrossRef] [PubMed]

Roux, F. S.

Sztul, H. I.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, "Geometrical phase associated with mode transformations of optical beams bearing orbital angular momentum," Phys. Rev. Lett. 90, 203901 (2003).
[CrossRef] [PubMed]

Tervo, J.

J. Tervo, V. Kettunen, M. Honkanen, and J. Turunen, "Design of space-variant diffractive polarization elements," J. Opt. Soc. Am. B 20, 282-289 (2003).
[CrossRef]

J. Tervo and J. Turunen, "Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings," Opt. Lett. 25, 785-786 (2000).
[CrossRef]

Turunen, J.

J. Tervo, V. Kettunen, M. Honkanen, and J. Turunen, "Design of space-variant diffractive polarization elements," J. Opt. Soc. Am. B 20, 282-289 (2003).
[CrossRef]

J. Tervo and J. Turunen, "Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings," Opt. Lett. 25, 785-786 (2000).
[CrossRef]

Williams, R. E.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, "Geometrical phase associated with mode transformations of optical beams bearing orbital angular momentum," Phys. Rev. Lett. 90, 203901 (2003).
[CrossRef] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Chap. 14.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

E. Hasman, V. Kleiner, G. Biener, and A. Niv, "Polarization dependent focusing lens by use of quantized Pancharatnam-Berry phase diffractive optics," Appl. Phys. Lett. 82, 328-330 (2003).
[CrossRef]

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

J. Opt. Soc. Am. B (2)

F. S. Roux, "Coupling of noncanonical optical vortices," J. Opt. Soc. Am. B 21, 664-670 (2004).
[CrossRef]

J. Tervo, V. Kettunen, M. Honkanen, and J. Turunen, "Design of space-variant diffractive polarization elements," J. Opt. Soc. Am. B 20, 282-289 (2003).
[CrossRef]

Opt. Lett. (2)

Phys. Rep. (1)

R. Bhandari, "Polarization of light and topological phases," Phys. Rep. 182, 1-64 (1997).
[CrossRef]

Phys. Rev. B (1)

P. Lalanne, "Effective properties and band structures in lamellar subwavelength crystals: plane-wave method revisited," Phys. Rev. B 58, 9801-9807 (1998).
[CrossRef]

Phys. Rev. Lett. (1)

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, "Geometrical phase associated with mode transformations of optical beams bearing orbital angular momentum," Phys. Rev. Lett. 90, 203901 (2003).
[CrossRef] [PubMed]

Proc. Indian Acad. Sci., Sect. A (1)

S. Pancharatnam, "Generalized theory of interference, and its application," Proc. Indian Acad. Sci., Sect. A 44, 247-262 (1956).

Proc. R. Soc. London, Ser. A (1)

M. V. Berry, "Quantal phase factors accompanying adiabatic changes," Proc. R. Soc. London, Ser. A 392, 45-57 (1984).
[CrossRef]

Other (1)

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Chap. 14.

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

Closed path on the Poincaré sphere. The starting point is denoted by A for linearly polarized light at 45°. A quarter-wave plate turns this into right-handed circular polarized light represented by point B. A half-wave plate oriented at an angle α turns this into left-handed circular polarized light represented by point C. (Note that the azimuthal angle on the Poincaré sphere is double the orientation angle.) A final quarter-wave plate turns this back into linearly polarized light at 45°. The geometric phase that is picked up through this whole process is given by half of the orientation angle of the half-wave plate.

Fig. 2
Fig. 2

Form birefringent layer is produced by a subwavelength grating, which produces different effective refractive indices for the s-polarization (TM) and the p-polarization (TE) states.

Fig. 3
Fig. 3

Part of the subwavelength groove pattern for the geometric phase lens, showing several branch points. The phase of the lens element is given by twice the orientation angle of the grooves. This part of the transmission function corresponds to the part of the effective phase function in the square shown in Fig. 4.

Fig. 4
Fig. 4

Effective phase function of the geometric phase lens. Black represents π rad and white represents π rad. The square represents the part of the effective phase function that corresponds to the part of the transmission function shown in Fig. 3. Note that the deviation from the rotational symmetry beyond the 512 - pixel radius is a result of the fact that the branch points end at this radius.

Fig. 5
Fig. 5

RMS wavefront error of the effective phase function as a function of the aperture radius measured in pixels. The maximum radius is 512   pixels .

Fig. 6
Fig. 6

(a) Input amplitude distributions and (b) the output amplitude distributions as they, respectively, appear in the front and back focal planes of the numerical simulation of the geometric phase lens. Black represents a large amplitude (normalized to 1) and white represents a small amplitude (zero).

Fig. 7
Fig. 7

Phase function of the output distribution as obtained from the simulation of the geometric phase lens. Black represents π rad and white represents π rad.

Fig. 8
Fig. 8

RMS wavefront error for the output distribution as a function of the aperture radius, measured in terms of the 1 e 2 beam radius ω 0 .

Equations (20)

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

E ( x , y , z ) = E 0 ( x , y , z ) exp ( i k z ) 1 2 ( x ̂ + y ̂ ) ,
E = E 0 exp ( i k x d ) 1 2 ( x ̂ i y ̂ ) ,
η R = 1 2 ( 1 i ) = [ 1 0 0 i ] 1 2 ( 1 1 ) = Q η in ,
H = [ 1 0 0 1 ]
R = [ cos α sin α sin α cos α ] , R 1 = [ cos α sin α sin α cos α ] ,
Q R 1 H R Q = [ cos 2 α i sin 2 α i sin 2 α cos 2 α ] .
n o = ( 1 + ϵ 2 ) 1 2 , n e = ( 1 + ϵ 2 ϵ ) 1 2 ,
F = F 0 { cos [ α ( ρ ) ] x ̂ + sin [ α ( ρ ) ] y ̂ } = F 0 { cos [ α ( ρ ) ϕ ] ρ ̂ + sin [ α ( ρ ) ϕ ] ϕ ̂ } ,
D ( ρ , ϕ ) = z ̂ ( × F ) = F 0 cos [ α ( ρ ) ϕ ] d α ( ρ ) d ρ .
F ( ρ , ϕ ) = Φ ( ρ , ϕ ) ,
2 π [ F x ( s ) ( x , y ) + i F y ( s ) ( x , y ) ] = i D ( x , y ) ( x x ) i ( y y ) d x d y .
2 π [ F x ( s ) + i F y ( s ) ] = 2 π F 0 { exp ( i 2 ϕ ) 0 ρ exp [ i α ( ρ ) ] ρ ρ 2 d ρ + i exp ( i ϕ ) sin [ α ( ρ ) ϕ ] 1 2 exp [ i α ( R ) ] }
Φ c = 2 π F 2 π F ( s ) .
( x + i y ) Φ c = 2 π F 0 exp ( i α ) 2 π [ F x ( s ) + i F y ( s ) ] ,
( x + i y ) = exp ( i ϕ ) ( ρ + i ρ ϕ ) ,
( ρ + i ρ ϕ ) Φ c ( ρ , ϕ ) = 2 π F 0 exp [ i ( α ϕ ) ] 2 π exp ( i ϕ ) [ F x ( s ) + i F y ( s ) ] .
Φ c ( ρ , ϕ ) = 2 π F 0 { ρ 2 cos [ α ( R ) ϕ ] + 0 ρ cos [ α ( ρ ) ϕ ] ρ ρ d ρ }
θ ( ρ ) = 2 α ( ρ ) = π ρ 2 λ f ,
Φ c ( ρ , ϕ ) = π F 0 [ ρ cos ( A R 2 ϕ ) + sin ( A ρ 2 ϕ ) A ρ + sin ( ϕ ) A ρ ]
D ( ρ , ϕ ) = 2 A F 0 ρ exp ( A ρ 2 ϕ ) ,

Metrics