Abstract

We report the analysis, design, fabrication and experimental characterization of novel subwavelength computer-generated holograms that produce uniform symmetric spot array. We distinguish between a polarization-sensitive and polarization-insensitive far-field reconstruction and show that a linearly polarized incident illumination is required in the former case in order to generate a symmetric reconstruction. The polarization-insensitive case generates a symmetric response independent of the illumination polarization. We show that this response is equivalent to that of a scalar-based computer-generated hologram but with an additional, independent, term that describes the undiffracted zeroth order. These findings simplify the design and optimization of form birefringent computer-generated holograms (F-BCGH) significantly. We present experimental results that verify our analysis and highlight the advantage of these novel elements over scalar-designed elements.

© 2004 Optical Society of America

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References

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Appl. Opt.

J. Mod. Opt.

M. Honkanen, V. Kettunen, J. Tervo, and J. Turunen, �??Fourier array illuminators with 100% efficiency: analytical Jones-matrix construction,�?? J. Mod. Opt. 47, 2351-2359 (2000).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. A.

U. Levy and Y. Fainman, �??Dispersion properties of inhomogeneous nanostructures,�?? J. Opt. Soc. Am. A. 21, 881-889 (2004).
[CrossRef]

Opt. Commun.

U. Levy, E. Marom and D. Mendlovic, �??Thin element approximation for the analysis of blazed gratings: simplified model and validity limits,�?? Opt. Commun. 229, 11-21 (2004)
[CrossRef]

Opt. Express

Opt. Lett.

Z. Bomzon, G. Biener, V. Kleiner and E. Hasman, �??Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings,�?? Opt. Lett. 27, 285-287 (2002).
[CrossRef]

U. Levy, C. H. Tsai, L. Pang and Y. Fainman, �??Engineering space-variant inhomogeneous media for polarization control,�?? Opt. Lett. 29, (2004).
[CrossRef] [PubMed]

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

F. Xu, R. Tyan, P. C. Sun, Y. Fainman, C. Cheng and A. Scherer, �??Form-birefringent computer-generated holograms,�?? Opt. Lett. 21, 1513-1515 (1996).
[CrossRef] [PubMed]

F. T. Chen and H. G. Craighead, �??Diffractive phase elements based on two-dimensional artificial dielectrics,�?? Opt. Lett. 20, 121-123 (1995).
[CrossRef] [PubMed]

P. Lalanne, S. Astilean, P. Chavel, E. Cambril, H. Launois, �??Blazed binary subwavelength gratings with efficiencies larger than those of conventional echelette gratings,�?? Opt. Lett. 23, 1081-1083 (1998).
[CrossRef]

J. N. Mait, A. Scherer, O. Dial, D. W. Prather and X. Gao, �??Diffractive lens fabricated with binary features less than 60nm,�?? Opt. Lett. 25, 381-383 (2000).
[CrossRef]

F. Gori, �??Measuring Stokes parameters by means of polarization gratings,�?? Opt. Lett. 24, 584-586 (1999).
[CrossRef]

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

F. Xu, R. Tyan, P. C. Sun, C. Cheng, A. Scherer and Y. Fainman, �??Fabrication, modeling, and characterization of form-birefringent nanostructures,�?? Opt. Lett. 20, 2457-2459, (1995).
[CrossRef] [PubMed]

C. Gu and P. Yeh, �??Form birefringence dispersion in periodic layered media,�?? Opt. Lett. 21, 504-506 (1996).
[CrossRef] [PubMed]

Progress in Optics

O. Bryngdahl and F. Wyrowski, �??Digital holography �?? computer-generated holograms,�?? in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1990), Vol 28, Chap. 1.
[CrossRef]

Science

S. Kirpatrick, C. D. Gellat, Jr., M. P. Vecchi, �??Optimization by simulated annealing,�?? Science, 220, 671-680 (1983).
[CrossRef]

Sov. Phys. JETP

S. M. Rytov, "Electromagnetic properties of a finely stratified medium," Sov. Phys. JETP, 2, 466-475, (1956).

Other

M. Born and E. Wolf, Principles of Optics, (Cambridge university press 1980), Chap. 14.

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Figures (4)

Fig. 1.
Fig. 1.

Schematic diagram of diffractive optical element using subwavelength grating based phase modulation: subwavelngth grating with a period Λ is introduced into each cell with oriantation of the grating θ(x,y).

Fig. 2.
Fig. 2.

Typical SEM cross section of the fabricated F-BCGH 1X3 element

Fig. 3.
Fig. 3.

Experimentally obtained image of the Fourier transform of the F-BCGH element illuminated by a linearly polarized beam.

Fig. 4.
Fig. 4.

Cross-section of Fig 3. The cross section was calculated by integrating Fig. 3 along the vertical axis.

Equations (30)

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E ¯ T ( x , y , z = 0 + ) = R = ( x , y ) 1 G = R = ( x , y ) V ¯ in
E ¯ TR ( x , y , z = 0 + ) = cos ( ϕ / 2 ) [ 1 j ] j sin ( ϕ / 2 ) exp [ + j 2 θ ( x , y ) ] [ 1 j ]
E ¯ TL ( x , y , z = 0 + ) = cos ( ϕ / 2 ) [ 1 j ] j sin ( ϕ / 2 ) exp [ + j 2 θ ( x , y ) ] [ 1 j ]
E ˜ R ( x , y ) = + E ¯ TR ( x , y ) exp [ j 2 π ( xx + yy ) ] dxdy =
cos ( ϕ / 2 ) δ d ( x , y ) [ 1 j ] j sin ( ϕ / 2 ) + exp [ j 2 θ ( x , y ) ] exp [ j 2 π ( xx + yy ) dxdy [ 1 j ] ]
E ˜ L ( x , y ) = + E ¯ TL ( x , y ) exp [ j 2 π ( xx + yy ) ] dxdy = cos ( ϕ / 2 ) δ d ( x , y ) [ 1 j ]
j sin ( ϕ / 2 ) + exp [ j 2 θ ( x , y ) ] exp [ j 2 π ( xx + yy ) dxdy [ 1 j ] = cos ( ϕ / 2 ) δ ( x , y ) [ 1 j ]
j sin ( ϕ / 2 ) { + exp [ j 2 θ ( x , y ) ] exp [ j 2 π ( x ( x ) + y ( y ) ) ] dxdy } * [ 1 j ]
E ˜ R ( x , y ) = E ˜ L ( x , y )
V ¯ in = [ cos ( χ ) exp ( j δ / 2 ) sin ( χ ) exp ( + j δ / 2 ) ] ,
V ¯ in = α [ 1 j ] + β [ 1 j ]
2 α = cos ( χ δ / 2 ) j sin ( χ + δ / 2 ) , 2 β = cos ( χ + δ / 2 ) + j sin ( χ δ / 2 )
I ( x , y ) = α 2 E ˜ R ( x , y ) 2 + β 2 E ˜ L ( x , y ) 2 = α 2 E ˜ R ( x , y ) 2 + β 2 E ˜ R ( x , y ) 2 ,
cos 2 ( χ δ / 2 ) cos 2 ( χ + δ / 2 ) = 0 ,
χ = n π / 2 or δ = m π
I L ( x , y ) = E ˜ L ( x , y ) 2 = E ˜ R ( x , y ) 2 = E ˜ R ( x , y ) 2 = I R ( x , y )
E ¯ s ( x , y , z = 0 + ) = exp [ j Φ ( x , y ) ] ,
E ¯ TR ( x , y , z = 0 + ) = E ¯ s ( x , y , z = 0 + ) ,
E ˜ R ( x , y ) = E ˜ S ( x , y ) ,
I ( x , y ) = α 2 E ˜ R ( x , y ) 2 + β 2 E ˜ L ( x , y ) 2 = E ˜ R ( x , y ) 2 = E ˜ S ( x , y ) 2 = I S ,
E ˜ R 2 = a n = 1 N δ d ( x n Δ x ) + b n = 1 N δ d ( x + n Δ x ) ,
E ˜ L 2 = b n = 1 N δ d ( x n Δ x ) + a n = 1 N δ d ( x + n Δ x )
I = E ˜ R 2 + E ˜ L 2 = ( a + b ) [ 1 N δ ( x Δ x ) + 1 N δ ( x + Δ x ) ]
E ¯ T ( x , y , z = 0 + ) = j sin ( ϕ / 2 ) sin [ 2 θ ( x , y ) ] x ̂ { cos ( ϕ / 2 ) + j sin ( ϕ / 2 ) cos [ 2 θ ( x , y ) ] } y ̂
I ( x , y ) = E ˜ x ( x , y ) 2 + E ˜ y ( x , y ) 2 = cos 2 ( ϕ / 2 ) δ d ( x , y ) + sin 2 ( ϕ / 2 ) ×
× { { sin [ 2 θ ( x , y ) ] exp ( j 2 π ( xx + yy ) ) dxdy } 2 + { cos [ 2 θ ( x , y ) ] exp ( j 2 π ( xx + yy ) ) dxdy } 2 }
I ( x = 0 , y = 0 ) = cos 2 ( ϕ / 2 ) + sin 2 ( ϕ / 2 ) { { sin [ 2 θ ( x , y ) ] dxdy } 2 + { cos [ 2 θ ( x , y ) ] } 2 }
cos 2 ( ϕ / 2 ) I ( 0 , 0 )
ϕ min = 2 cos 1 ( 1 2 N + 1 )
e = x , y ROI η I desired ( x , y ) I obtained ( x , y )

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