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Design, fabrication and characterization of subwavelength computer-generated holograms for spot array generation

Uriel Levy, Chia-Ho Tsai, Hyo-Chang Kim, and Yeshaiahu Fainman

Open Access Open Access

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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    [Crossref]
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    [Crossref]
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2004 (2)

U. Levy and Y. Fainman, “Dispersion properties of inhomogeneous nanostructures,” J. Opt. Soc. Am. A. 21, 881–889 (2004).
[Crossref]

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]

2003 (2)

J. Tervo, V. Kettunen, M. Honkanen, and J. Turunen, “Design of space-variant diffractive polarization elements,” J. Opt. Soc. Am. A,  20, 282–289 (2003).
[Crossref]

G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Near-field Fourier transform polarimetry by use of a discrete space-variant subwavelength grating,” J. Opt. Soc. Am. A. 20, 1940–1948 (2003).
[Crossref]

2002 (2)

2000 (3)

1999 (2)

1998 (1)

1996 (2)

1995 (3)

1992 (1)

1983 (1)

S. Kirpatrick, C. D. Gellat, and M. P. Vecchi, “Optimization by simulated annealing,” Science,  220, 671–680 (1983).
[Crossref]

1982 (1)

1956 (1)

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

Astilean, S.

Biener, G.

Bomzon, Z.

Born, M.

M. Born and E. Wolf, Principles of Optics, (Cambridge university press1980), Chap. 14.

Bryngdahl, O.

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]

Cambril, E.

Chavel, P.

Chen, F. T.

Cheng, C.

Craighead, H. G.

Deguzman, P.

Dial, O.

Fainman, Y.

Gao, X.

Gaylord, T. K.

Gellat, C. D.

S. Kirpatrick, C. D. Gellat, and M. P. Vecchi, “Optimization by simulated annealing,” Science,  220, 671–680 (1983).
[Crossref]

Gori, F.

Gu, C.

Hasman, E.

Honkanen, M.

J. Tervo, V. Kettunen, M. Honkanen, and J. Turunen, “Design of space-variant diffractive polarization elements,” J. Opt. Soc. Am. A,  20, 282–289 (2003).
[Crossref]

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).

Kettunen, V.

J. Tervo, V. Kettunen, M. Honkanen, and J. Turunen, “Design of space-variant diffractive polarization elements,” J. Opt. Soc. Am. A,  20, 282–289 (2003).
[Crossref]

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).

Kirpatrick, S.

S. Kirpatrick, C. D. Gellat, and M. P. Vecchi, “Optimization by simulated annealing,” Science,  220, 671–680 (1983).
[Crossref]

Kleiner, V.

Krackhardt, U.

Lalanne, P.

Launois, H.

Levy, U.

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]

U. Levy and Y. Fainman, “Dispersion properties of inhomogeneous nanostructures,” J. Opt. Soc. Am. A. 21, 881–889 (2004).
[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]

Mait, J. N.

Marom, E.

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]

Mendlovic, D.

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]

Moharam, M. G.

Niv, A.

G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Near-field Fourier transform polarimetry by use of a discrete space-variant subwavelength grating,” J. Opt. Soc. Am. A. 20, 1940–1948 (2003).
[Crossref]

Nordin, G.

Pang, L.

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

Prather, D. W.

Richter, I.

Rytov, S. M.

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

Scherer, A.

Streibl, N.

Sun, P. C.

Tervo, J.

Tsai, C. H.

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

Turunen, J.

Tyan, R.

Vecchi, M. P.

S. Kirpatrick, C. D. Gellat, and M. P. Vecchi, “Optimization by simulated annealing,” Science,  220, 671–680 (1983).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, (Cambridge university press1980), Chap. 14.

Wyrowski, F.

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]

Xu, F.

Yeh, P.

Appl. Opt. (2)

J. Mod. Opt. (1)

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. (1)

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

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

G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Near-field Fourier transform polarimetry by use of a discrete space-variant subwavelength grating,” J. Opt. Soc. Am. A. 20, 1940–1948 (2003).
[Crossref]

U. Levy and Y. Fainman, “Dispersion properties of inhomogeneous nanostructures,” J. Opt. Soc. Am. A. 21, 881–889 (2004).
[Crossref]

Opt. Commun. (1)

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 (1)

Opt. Lett. (10)

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

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]

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]

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, and 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]

Science (1)

S. Kirpatrick, C. D. Gellat, and M. P. Vecchi, “Optimization by simulated annealing,” Science,  220, 671–680 (1983).
[Crossref]

Sov. Phys. JETP (1)

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

Other (3)

M. Born and E. Wolf, Principles of Optics, (Cambridge university press1980), Chap. 14.

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]

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

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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).

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Fig. 2.
Fig. 2. Typical SEM cross section of the fabricated F-BCGH 1X3 element

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Fig. 3.
Fig. 3. Experimentally obtained image of the Fourier transform of the F-BCGH element illuminated by a linearly polarized beam.

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Fig. 4.
Fig. 4. Cross-section of Fig 3. The cross section was calculated by integrating Fig. 3 along the vertical axis.

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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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