Abstract

The chromatic behavior of diffractive optical elements, exhibiting 2π-wrapped phase profiles, implemented into liquid crystal spatial light modulators (LC-SLM) is described. A wrapped phase map is only equivalent to the original continuous profile for the design wavelength while at other wavelengths there are unwanted phase jumps and the profile does not correspond to a pure defocus. For those conditions the wrapped profile behaves as a multiple order lens (multi-focal lens). The optical power dispersion for each order is linearly proportional to the wavelength, while the energy of each order depends on the design wavelength and the material dispersion. For practical purposes, for most of the visible range only first order (main defocus) is relevant but two other orders may also be considered depending on the actual PSF of the system. As an application, we demonstrate that the longitudinal chromatic aberration of the eye can be compensated by the diffractive lens dispersion when the appropriate defocus is programmed into the SLM.

© 2017 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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2016 (2)

2015 (1)

2013 (1)

J. Albero, P. García-Martínez, J. L. Martínez, and I. Moreno, “Second order diffractive optical elements in a spatial light modulator with large phase dynamic range,” Opt. Lasers Eng. 51(2), 111–115 (2013).
[Crossref]

2009 (2)

2007 (1)

2005 (1)

2004 (2)

1995 (1)

1990 (1)

Albero, J.

V. Calero, P. García-Martínez, J. Albero, M. M. Sánchez López, and I. Moreno, “Liquid crystal spatial light modulator with very large phase modulation operating in high harmonic orders,” Opt. Lett. 38(22), 4663–4666 (2015).
[Crossref]

J. Albero, P. García-Martínez, J. L. Martínez, and I. Moreno, “Second order diffractive optical elements in a spatial light modulator with large phase dynamic range,” Opt. Lasers Eng. 51(2), 111–115 (2013).
[Crossref]

Artal, P.

Atchinson, D. A.

Calero, V.

Campos, J.

I. Moreno, C. Iemmi, A. Márquez, J. Campos, and M. J. Yzuel, “Modulation light efficiency of diffractive lenses displayed in a restricted phase-mostly modulation display,” Appl. Opt. 43(34), 6278–6284 (2004).
[Crossref]

I. Moreno, C. Iemmi, A. Márquez, J. Campos, and M. J. Yzuel, “Modulation light efficiency of spatial light modulators,” Proceedings of IEEE Conference 10th Euro-American Workshop on Information Optics(IEEE, 2011), pp. 1–4.

Cottrell, D. M.

Davis, J. A.

Espinosa, J.

Fernández, E. J.

Fuentes, J. L. M.

García-Martínez, P.

V. Calero, P. García-Martínez, J. Albero, M. M. Sánchez López, and I. Moreno, “Liquid crystal spatial light modulator with very large phase modulation operating in high harmonic orders,” Opt. Lett. 38(22), 4663–4666 (2015).
[Crossref]

J. Albero, P. García-Martínez, J. L. Martínez, and I. Moreno, “Second order diffractive optical elements in a spatial light modulator with large phase dynamic range,” Opt. Lasers Eng. 51(2), 111–115 (2013).
[Crossref]

Hedman, T. R.

Iemmi, C.

I. Moreno, C. Iemmi, A. Márquez, J. Campos, and M. J. Yzuel, “Modulation light efficiency of diffractive lenses displayed in a restricted phase-mostly modulation display,” Appl. Opt. 43(34), 6278–6284 (2004).
[Crossref]

I. Moreno, C. Iemmi, A. Márquez, J. Campos, and M. J. Yzuel, “Modulation light efficiency of spatial light modulators,” Proceedings of IEEE Conference 10th Euro-American Workshop on Information Optics(IEEE, 2011), pp. 1–4.

Illueca, C.

Lilly, R. A.

Manzanera, S.

Márquez, A.

I. Moreno, C. Iemmi, A. Márquez, J. Campos, and M. J. Yzuel, “Modulation light efficiency of diffractive lenses displayed in a restricted phase-mostly modulation display,” Appl. Opt. 43(34), 6278–6284 (2004).
[Crossref]

I. Moreno, C. Iemmi, A. Márquez, J. Campos, and M. J. Yzuel, “Modulation light efficiency of spatial light modulators,” Proceedings of IEEE Conference 10th Euro-American Workshop on Information Optics(IEEE, 2011), pp. 1–4.

Martínez, J. L.

J. Albero, P. García-Martínez, J. L. Martínez, and I. Moreno, “Second order diffractive optical elements in a spatial light modulator with large phase dynamic range,” Opt. Lasers Eng. 51(2), 111–115 (2013).
[Crossref]

Mas, D.

Menon, R.

P. Wang, N. Mohammad, and R. Menon, “Chromatic-aberration-corrected diffractive lenses for ultra-broadband focusing,” Sci. Rep. 6(1), 21545 (2016).
[Crossref]

Mohammad, N.

P. Wang, N. Mohammad, and R. Menon, “Chromatic-aberration-corrected diffractive lenses for ultra-broadband focusing,” Sci. Rep. 6(1), 21545 (2016).
[Crossref]

Moreno, I.

V. Calero, P. García-Martínez, J. Albero, M. M. Sánchez López, and I. Moreno, “Liquid crystal spatial light modulator with very large phase modulation operating in high harmonic orders,” Opt. Lett. 38(22), 4663–4666 (2015).
[Crossref]

J. Albero, P. García-Martínez, J. L. Martínez, and I. Moreno, “Second order diffractive optical elements in a spatial light modulator with large phase dynamic range,” Opt. Lasers Eng. 51(2), 111–115 (2013).
[Crossref]

I. Moreno, C. Iemmi, A. Márquez, J. Campos, and M. J. Yzuel, “Modulation light efficiency of diffractive lenses displayed in a restricted phase-mostly modulation display,” Appl. Opt. 43(34), 6278–6284 (2004).
[Crossref]

I. Moreno, C. Iemmi, A. Márquez, J. Campos, and M. J. Yzuel, “Modulation light efficiency of spatial light modulators,” Proceedings of IEEE Conference 10th Euro-American Workshop on Information Optics(IEEE, 2011), pp. 1–4.

Perez, J.

Prieto, P. M.

Sánchez López, M. M.

Smith, G.

Sommagren, G. E.

Sweeny, D. W.

Wang, P.

P. Wang, N. Mohammad, and R. Menon, “Chromatic-aberration-corrected diffractive lenses for ultra-broadband focusing,” Sci. Rep. 6(1), 21545 (2016).
[Crossref]

Yzuel, M. J.

I. Moreno, C. Iemmi, A. Márquez, J. Campos, and M. J. Yzuel, “Modulation light efficiency of diffractive lenses displayed in a restricted phase-mostly modulation display,” Appl. Opt. 43(34), 6278–6284 (2004).
[Crossref]

I. Moreno, C. Iemmi, A. Márquez, J. Campos, and M. J. Yzuel, “Modulation light efficiency of spatial light modulators,” Proceedings of IEEE Conference 10th Euro-American Workshop on Information Optics(IEEE, 2011), pp. 1–4.

Appl. Opt. (3)

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

Opt. Express (4)

Opt. Lasers Eng. (1)

J. Albero, P. García-Martínez, J. L. Martínez, and I. Moreno, “Second order diffractive optical elements in a spatial light modulator with large phase dynamic range,” Opt. Lasers Eng. 51(2), 111–115 (2013).
[Crossref]

Opt. Lett. (2)

Sci. Rep. (1)

P. Wang, N. Mohammad, and R. Menon, “Chromatic-aberration-corrected diffractive lenses for ultra-broadband focusing,” Sci. Rep. 6(1), 21545 (2016).
[Crossref]

Other (1)

I. Moreno, C. Iemmi, A. Márquez, J. Campos, and M. J. Yzuel, “Modulation light efficiency of spatial light modulators,” Proceedings of IEEE Conference 10th Euro-American Workshop on Information Optics(IEEE, 2011), pp. 1–4.

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Figures (7)

Fig. 1
Fig. 1

Conceptual scheme comparing the phase profiles of a refractive element (pure defocus) and its equivalent diffractive element obtained by phase wrapping at design wavelength λ0.

Fig. 2
Fig. 2

Behavior of a diffractive lens when illuminated with the design wavelength (top) or a different wavelength (bottom).

Fig. 3
Fig. 3

(a) Intensity efficiency vs. diffraction order, q, for different wavelengths in the visible range. (b) Optical power dispersion curves for a refractive lens (red solid curve) together with the first diffraction order (solid green curve) and the overall predominant dispersion (blue circles) of the corresponding diffractive lens.

Fig. 4
Fig. 4

(a) Phase calibration curves for the tested wavelengths. (b) Experimental refractive index ratio for each wavelength (symbols) and Cauchy-like fitting curve (red line). Reference wavelength λ0 = 532 nm.

Fig. 5
Fig. 5

Experimental setup used for obtaining the spectral response of a phase profile programmed into the SLM. Legend: OBJ, microscope objective; PH, pinhole aperture; Lc, collimating lens; P, polarizer; P1 and P2, conjugated pupil planes; L1-L2 and L3-L4, relay telescopes; and L7, PSF-generating lens.

Fig. 6
Fig. 6

Theoretical (squares) and experimentally determined (circles) diameter of the PSF corresponding to a 1D lens 2π-wrapped for 532 nm.

Fig. 7
Fig. 7

Power dispersion for the naked eye (blue), for a 3.2D diffractive lens generated with the SLM (red), and for the combination of both (green).

Equations (16)

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φ( λ 0 )= 2π λ 0 Δn( λ 0 )·d·g,
φ(λ)= 2π λ Δn(λ)·d·g= λ 0 λ Δn(λ) Δn( λ 0 ) ·φ( λ 0 ).
m( λ )=exp( iφ( λ ) ).
m( λ )= q= η q m ( λ 0 ) q = q= η q exp( iqφ( λ 0 ) ) .
η q = 1 2π 0 2π m( λ )exp( iqφ( λ 0 ) )dφ( λ 0 ) ,
η q =sinc( [ λ 0 λ Δn( λ ) Δn( λ 0 ) q ] )·exp( iπ[ λ 0 λ Δn( λ ) Δn( λ 0 ) q ] ).
m( λ 0 ,r )=exp( i π r 2 λ 0 P 0 ).
m R ( λ,r )=exp( i π r 2 λ P R (λ) ),
m D ( λ,r )= q= η q m ( λ 0 ,r ) q = q= η q exp( i π r 2 λ 0 q P 0 ) = q= η q exp( i π r 2 λ P D,q (λ) ) .
P R ( λ )= Δn(λ) Δn( λ 0 ) · P 0 .
P D,q ( λ )=q· λ λ 0 · P 0 .
Δn( λ ) Δn( λ 0 ) a( λ )·λ a( λ 0 )· λ 0 .
Δn( λ ) Δn( λ 0 ) 0.6985+ 8.125· 10 4 λ 2 + 8.292· 10 5 λ 4 ,
PSF =2ρ· P(λ) P PSF ,
D eye ( λ )=1.81341+ 6.70941· 10 5 λ 2 5.5534· 10 10 λ 4 + 5.59998· 10 15 λ 6 .
D SLM ( λ )= P 0 [ λ λ 0 1 ].

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