Abstract

The paper presents the polychromatic analysis of two diffractive optical elements with extended depth of focus: the linear axicon and the light sword optical element. Chromatic aberration produces axial displacement of the focal segment line. Thus, we explore the possibility of extending the focal depth of these elements to permit superposition of the chromatic foci. In the case of an axicon, we achieve an achromatic zone where focusing is produced. In the case of the light sword element, we show that the focusing segment is out of axis. Therefore a superposition of colors is produced, but not on axis overlapping. Instead, three colored and separated foci are simultaneously obtained in a single plane. Three dimensional structures of the propagated beams are analyzed in order to provide better understanding of the properties and applications of such elements.

© 2007 Optical Society of America

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    [CrossRef] [PubMed]
  2. A. Kolodziejczyk, S. Bara, Z. Jaroszewicz, and M. Sypek, "The light sword optical element -a new diffraction structure with extended depth of focus," J. Mod. Opt. 37, 1283-1286 (1990).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  28. D. Mas, J. Espinosa, J. Pérez, and C. Illueca, "Scale corrections for faster evaluation of convergent Fresnel patterns," J. Mod. Opt. 53, 259-266 (2006).
    [CrossRef]

2007

2006

2005

G. Mikula, A. Kolodziejczyk, M. Makowski, C. Prokopowicz and M. Sypek, "Diffractive elements for imaging with extended depth of focus," Opt. Eng. 44, 58001 (2005).
[CrossRef]

2004

2003

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C Illueca, "Fast numerical calculation of Fresnel patterns in convergent systems," Opt. Commun. 227, 245-258 (2003).
[CrossRef]

1999

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, "Fast algorithms for free-space diffraction patterns calculation," Opt. Commun. 164, 233-245 (1999).
[CrossRef]

Z. Jaroszewicz and J. Morales, "Lens axicons: systems composed of a diverging aberrated lens and a converging aberrated lens," J. Opt. Soc. Am. A 16, 191-197 (1999).
[CrossRef]

1998

1996

1992

1990

A. Kolodziejczyk, S. Bara, Z. Jaroszewicz, and M. Sypek, "The light sword optical element -a new diffraction structure with extended depth of focus," J. Mod. Opt. 37, 1283-1286 (1990).
[CrossRef]

1989

Ahluwalia, B. P. S.

Ahouzi, E.

J. L. Martínez, I. Moreno, and E. Ahouzi, "Diffraction and signal processing experiments with a liquid crystal microdisplay," Eur. J. Phys. 27, 1221-1231 (2006).
[CrossRef]

Alió,, J. L.

C. Illueca, D. Mas, J. Perez, J. Espinosa, D. Ortiz, J. L. Alió, and E. Sala, "Comparative analysis of visual performance and pseudoaccommodation between presbylasik corneas and multifocal IOL implants," J. Refract. Surg., in press (2007).
[PubMed]

Andres,, P.

Bara, S.

J. Sochacki, A. Kolodziejczyk, Z. Jaroszewicz, and S. Bara, "Non paraxial design of generalized axicons," Appl. Opt. 31, 5326-5330 (1992).
[CrossRef] [PubMed]

A. Kolodziejczyk, S. Bara, Z. Jaroszewicz, and M. Sypek, "The light sword optical element -a new diffraction structure with extended depth of focus," J. Mod. Opt. 37, 1283-1286 (1990).
[CrossRef]

Ben-Yaish, S.

Bernardo,, L. M.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, "Fast algorithms for free-space diffraction patterns calculation," Opt. Commun. 164, 233-245 (1999).
[CrossRef]

Bu, J.

Burvall, A.

Campos, J.

Campos,, J.

Cheong, W. C.

Connely, S. W.

Cottrell, D. M.

Dainty, C.

Davis, J. A.

Escalera, J. C.

Espinosa, J.

C. Illueca, D. Mas, J. Perez, J. Espinosa, D. Ortiz, J. L. Alió, and E. Sala, "Comparative analysis of visual performance and pseudoaccommodation between presbylasik corneas and multifocal IOL implants," J. Refract. Surg., in press (2007).
[PubMed]

D. Mas, J. Espinosa, J. Pérez, and C. Illueca, "Scale corrections for faster evaluation of convergent Fresnel patterns," J. Mod. Opt. 53, 259-266 (2006).
[CrossRef]

Exposito, E.

Ferreira, C.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, "Fast algorithms for free-space diffraction patterns calculation," Opt. Commun. 164, 233-245 (1999).
[CrossRef]

Flores, A.

Friberg, A. T.

S. Y. Popov and A. T. Friberg, "Apodization of generalized axicons to produce uniform axial line images," Pure Appl. Opt 7, 537-548 (1998).
[CrossRef]

Friberg,, A. T.

Furlan, W.

Garcia, J.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, "Fast algorithms for free-space diffraction patterns calculation," Opt. Commun. 164, 233-245 (1999).
[CrossRef]

Gibson, G. M.

Gimeno,, R.

Girkin, J. M.

Golub, I.

Gomez Reino, C.

Goncharov, A.

Hernández, C.

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C Illueca, "Fast numerical calculation of Fresnel patterns in convergent systems," Opt. Commun. 227, 245-258 (2003).
[CrossRef]

Iemmi, C.

Illueca, C

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C Illueca, "Fast numerical calculation of Fresnel patterns in convergent systems," Opt. Commun. 227, 245-258 (2003).
[CrossRef]

Illueca, C.

C. Illueca, D. Mas, J. Perez, J. Espinosa, D. Ortiz, J. L. Alió, and E. Sala, "Comparative analysis of visual performance and pseudoaccommodation between presbylasik corneas and multifocal IOL implants," J. Refract. Surg., in press (2007).
[PubMed]

D. Mas, J. Espinosa, J. Pérez, and C. Illueca, "Scale corrections for faster evaluation of convergent Fresnel patterns," J. Mod. Opt. 53, 259-266 (2006).
[CrossRef]

Jaroszewicz, Z.

Jaroszewicz,, Z.

Kolacz, K.

Kolodziejczyk, A.

G. Mikula, Z. Jaroszewicz, A. Kolodziejczyk, K. Petelczyk, and M. Sypek, "Imaging with extended focal depth by means of lenses with radial and angular modulation," Opt. Express 15, 9184-9193 (2007).
[CrossRef] [PubMed]

G. Mikula, A. Kolodziejczyk, M. Makowski, C. Prokopowicz and M. Sypek, "Diffractive elements for imaging with extended depth of focus," Opt. Eng. 44, 58001 (2005).
[CrossRef]

J. Sochacki, A. Kolodziejczyk, Z. Jaroszewicz, and S. Bara, "Non paraxial design of generalized axicons," Appl. Opt. 31, 5326-5330 (1992).
[CrossRef] [PubMed]

A. Kolodziejczyk, S. Bara, Z. Jaroszewicz, and M. Sypek, "The light sword optical element -a new diffraction structure with extended depth of focus," J. Mod. Opt. 37, 1283-1286 (1990).
[CrossRef]

Lancis, J.

Leach, J.

Lilly, R. A.

Lopez-Coronado, O.

Lu, Z.

Z. Lu. H. Liu, R. Wang. F. Li, and Y. Liu, "Diffractive axicons fabricated by laser direct writer on curved surface", J. Opt. A: Pure Appl. Opt,  9, 160-164 (2007).
[CrossRef]

Makowski, M.

G. Mikula, A. Kolodziejczyk, M. Makowski, C. Prokopowicz and M. Sypek, "Diffractive elements for imaging with extended depth of focus," Opt. Eng. 44, 58001 (2005).
[CrossRef]

Marinho, F.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, "Fast algorithms for free-space diffraction patterns calculation," Opt. Commun. 164, 233-245 (1999).
[CrossRef]

Martínez, J. L.

J. L. Martínez, I. Moreno, and E. Ahouzi, "Diffraction and signal processing experiments with a liquid crystal microdisplay," Eur. J. Phys. 27, 1221-1231 (2006).
[CrossRef]

Mas, D.

C. Illueca, D. Mas, J. Perez, J. Espinosa, D. Ortiz, J. L. Alió, and E. Sala, "Comparative analysis of visual performance and pseudoaccommodation between presbylasik corneas and multifocal IOL implants," J. Refract. Surg., in press (2007).
[PubMed]

D. Mas, J. Espinosa, J. Pérez, and C. Illueca, "Scale corrections for faster evaluation of convergent Fresnel patterns," J. Mod. Opt. 53, 259-266 (2006).
[CrossRef]

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C Illueca, "Fast numerical calculation of Fresnel patterns in convergent systems," Opt. Commun. 227, 245-258 (2003).
[CrossRef]

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, "Fast algorithms for free-space diffraction patterns calculation," Opt. Commun. 164, 233-245 (1999).
[CrossRef]

McConell, G.

Mikula, G.

G. Mikula, Z. Jaroszewicz, A. Kolodziejczyk, K. Petelczyk, and M. Sypek, "Imaging with extended focal depth by means of lenses with radial and angular modulation," Opt. Express 15, 9184-9193 (2007).
[CrossRef] [PubMed]

G. Mikula, A. Kolodziejczyk, M. Makowski, C. Prokopowicz and M. Sypek, "Diffractive elements for imaging with extended depth of focus," Opt. Eng. 44, 58001 (2005).
[CrossRef]

Miret,, J. J.

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C Illueca, "Fast numerical calculation of Fresnel patterns in convergent systems," Opt. Commun. 227, 245-258 (2003).
[CrossRef]

Monsoriu, J.

Morales, J.

Moreno, I.

J. L. Martínez, I. Moreno, and E. Ahouzi, "Diffraction and signal processing experiments with a liquid crystal microdisplay," Eur. J. Phys. 27, 1221-1231 (2006).
[CrossRef]

Ortiz, D.

C. Illueca, D. Mas, J. Perez, J. Espinosa, D. Ortiz, J. L. Alió, and E. Sala, "Comparative analysis of visual performance and pseudoaccommodation between presbylasik corneas and multifocal IOL implants," J. Refract. Surg., in press (2007).
[PubMed]

Padgett, M.

Perez, J.

C. Illueca, D. Mas, J. Perez, J. Espinosa, D. Ortiz, J. L. Alió, and E. Sala, "Comparative analysis of visual performance and pseudoaccommodation between presbylasik corneas and multifocal IOL implants," J. Refract. Surg., in press (2007).
[PubMed]

Pérez, J.

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C Illueca, "Fast numerical calculation of Fresnel patterns in convergent systems," Opt. Commun. 227, 245-258 (2003).
[CrossRef]

Pérez,, J.

D. Mas, J. Espinosa, J. Pérez, and C. Illueca, "Scale corrections for faster evaluation of convergent Fresnel patterns," J. Mod. Opt. 53, 259-266 (2006).
[CrossRef]

Petelczyk, K.

Popov, S. Y.

S. Y. Popov and A. T. Friberg, "Apodization of generalized axicons to produce uniform axial line images," Pure Appl. Opt 7, 537-548 (1998).
[CrossRef]

S. Y. Popov and A. T. Friberg, "Design of diffractive axicons for partially coherent light," Opt. Lett. 23, 1639-1641 (1998).
[CrossRef]

Prokopowicz, C.

G. Mikula, A. Kolodziejczyk, M. Makowski, C. Prokopowicz and M. Sypek, "Diffractive elements for imaging with extended depth of focus," Opt. Eng. 44, 58001 (2005).
[CrossRef]

Roman Dopazo,, J. F.

Sala, E.

C. Illueca, D. Mas, J. Perez, J. Espinosa, D. Ortiz, J. L. Alió, and E. Sala, "Comparative analysis of visual performance and pseudoaccommodation between presbylasik corneas and multifocal IOL implants," J. Refract. Surg., in press (2007).
[PubMed]

Sánchez-Losa, A.

Sedukhin, A. G.

Sochacki, J.

Sypek, M.

G. Mikula, Z. Jaroszewicz, A. Kolodziejczyk, K. Petelczyk, and M. Sypek, "Imaging with extended focal depth by means of lenses with radial and angular modulation," Opt. Express 15, 9184-9193 (2007).
[CrossRef] [PubMed]

G. Mikula, A. Kolodziejczyk, M. Makowski, C. Prokopowicz and M. Sypek, "Diffractive elements for imaging with extended depth of focus," Opt. Eng. 44, 58001 (2005).
[CrossRef]

A. Kolodziejczyk, S. Bara, Z. Jaroszewicz, and M. Sypek, "The light sword optical element -a new diffraction structure with extended depth of focus," J. Mod. Opt. 37, 1283-1286 (1990).
[CrossRef]

Tao, S. -H.

Tuvey, C.

Vázquez, C.

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C Illueca, "Fast numerical calculation of Fresnel patterns in convergent systems," Opt. Commun. 227, 245-258 (2003).
[CrossRef]

Wang, H.

Wang,, M.

Wright,, A. J.

Yang, J. J.

Yuan, X.-C.

Yzuel, M. J.

Zalevski, Z.

Zapata-Rodríguez, C. J.

Zhang, L.-S.

Appl. Opt.

Eur. J. Phys.

J. L. Martínez, I. Moreno, and E. Ahouzi, "Diffraction and signal processing experiments with a liquid crystal microdisplay," Eur. J. Phys. 27, 1221-1231 (2006).
[CrossRef]

J. Mod. Opt.

A. Kolodziejczyk, S. Bara, Z. Jaroszewicz, and M. Sypek, "The light sword optical element -a new diffraction structure with extended depth of focus," J. Mod. Opt. 37, 1283-1286 (1990).
[CrossRef]

D. Mas, J. Espinosa, J. Pérez, and C. Illueca, "Scale corrections for faster evaluation of convergent Fresnel patterns," J. Mod. Opt. 53, 259-266 (2006).
[CrossRef]

J. Opt. A: Pure Appl. Opt

Z. Lu. H. Liu, R. Wang. F. Li, and Y. Liu, "Diffractive axicons fabricated by laser direct writer on curved surface", J. Opt. A: Pure Appl. Opt,  9, 160-164 (2007).
[CrossRef]

J. Opt. Soc. Am. A

J. Refract. Surg.

C. Illueca, D. Mas, J. Perez, J. Espinosa, D. Ortiz, J. L. Alió, and E. Sala, "Comparative analysis of visual performance and pseudoaccommodation between presbylasik corneas and multifocal IOL implants," J. Refract. Surg., in press (2007).
[PubMed]

Opt. Commun.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, "Fast algorithms for free-space diffraction patterns calculation," Opt. Commun. 164, 233-245 (1999).
[CrossRef]

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C Illueca, "Fast numerical calculation of Fresnel patterns in convergent systems," Opt. Commun. 227, 245-258 (2003).
[CrossRef]

Opt. Eng.

G. Mikula, A. Kolodziejczyk, M. Makowski, C. Prokopowicz and M. Sypek, "Diffractive elements for imaging with extended depth of focus," Opt. Eng. 44, 58001 (2005).
[CrossRef]

Opt. Express

Opt. Lett.

Pure Appl. Opt

S. Y. Popov and A. T. Friberg, "Apodization of generalized axicons to produce uniform axial line images," Pure Appl. Opt 7, 537-548 (1998).
[CrossRef]

Other

Image Processing and Analysis in Java, http://rsb.info.nih.gov/ij/.

Supplementary Material (3)

» Media 1: MPG (1300 KB)     
» Media 2: MPG (2512 KB)     
» Media 3: MPG (1066 KB)     

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Figures (8)

Fig. 1.
Fig. 1.

Numerical simulation of on axis irradiance for a linear axicon illuminated with red, green and blue light.. The DOE has been designed for focusing in the region [f 0=550 mm, f 1=700mm] when illuminated with λG wavelength.

Fig. 2.
Fig. 2.

Numerical simulation of on axis irradiance for a linear axicon illuminated with red, green and blue light. The DOE has been designed for green light focusing in an extended region [f 0=550 mm, f 1’=940mm] when illuminated with λG wavelength. Notice that superposition of colors is obtained from 650 mm to 780 mm.

Fig. 3.
Fig. 3.

Radial MTF of the linear axicon. Notice widening of the function on the region of light focalization.

Fig. 4.
Fig. 4.

Radial MTF for a linear axicon at the beginning (618 mm), middle (673 mm) and end (732 mm) of the region of interest. Notice that balanced MTF is obtained for three colors.

Fig. 5.
Fig. 5.

Animation with the three dimensional structure of the axicon focusing segment. Notice the presence of the achromatic segment. [Media 1]

Fig. 6.
Fig. 6.

Numerical simulation of on axis (a) and three dimensional structure [Media 2] (b) of the LSOE focusing segment [Media 3]. Notice the presence of three separated colored spots at a single plane.

Fig. 7.
Fig. 7.

Radial MTF of the linear axicon. Notice widening of the function on the region of light focalization

Fig. 8.
Fig. 8.

Radial MTF for a LSOE at the beginning, middle and end of the region of interest.

Tables (1)

Tables Icon

Table 1. Constants used for numerical calculations of propagated fields from DOES: λG corresponds to design wavelength. f 0 and f 1 are extreme distances of the focal segment, N is the number of samples used for discrete calculations. R 1 and Rp are the inner and the aperture radii of the DOE. We also show the focusing range.

Equations (21)

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

u ( r , z , λ 0 ) = exp ( i 2 π z λ ) i λ z exp ( i π λ 0 z ρ 2 ) ×
× + u 0 ( r 0 ) exp ( i π λ 0 z ρ 0 2 ) exp ( i 2 π λ 0 z r · r 0 ) d r 0
u ( r , z , λ ) = exp ( i 2 π z λ ) i λ z exp ( i π λ z ρ 2 ) ×
× + u 0 ( r 0 ) exp ( i π λ z ρ 0 2 ) exp ( i 2 π λ z r · r 0 ) d r 0
= exp ( i 2 π ( λ 0 λ ) z λ 0 ) i λ 0 z ( λ λ 0 ) exp ( i π λ 0 z ( λ λ 0 ) ρ 2 ) ×
× + u 0 ( r 0 ) exp ( i π λ 0 z ( λ λ 0 ) ρ 0 2 ) exp ( i 2 π λ 0 z ( λ λ 0 ) r · r 0 ) d r 0
exp ( i 2 π z ( λ 2 λ 0 2 λ 0 λ 2 ) ) u ( r , z , λ 0 )
z = z λ λ 0
u ( ρ , θ , z ) = exp ( i 2 π z λ ) i λ z exp ( i 2 π λ z ρ 2 ) 0 2 π 0 R p u 0 ( ρ 0 , θ 0 ) ×
× exp [ i 2 π λ ( ρ 0 2 2 z ρ ρ 0 cos ( θ θ ) z ) ] ρ 0 d ρ 0 d θ 0
2 π R 1 R p I in ( ρ ) ρ d ρ = f 0 f 1 I out ( z ) dz
T ( ρ 0 ) = exp ( i 2 π λ 0 ( 1 + a ) ρ 0 2 + f 0 2 a R 1 2 1 + a ) ρ 0 [ R 1 , R p ] T ( ρ 0 ) = 0 elsewhere
a = f 1 2 f 0 2 R p 2 R 1 2 ,
f 1 λ G λ R > f 0 λ G λ B f 1 f 0 > λ R λ B
T ( ρ 0 , θ 0 ) = exp ( i 2 π λ 0 ρ 0 2 f 0 + ( f 1 f 0 ) θ 0 2 π ) ρ 0 [ R 1 , R p ] , θ 0 [ 0 , 2 π ] T ( ρ 0 , θ 0 ) = 0 elsewhere
T ( ρ 0 , θ 0 ) = exp ( i 2 π λ 0 ρ 0 2 f ( θ 0 ) )
f ( θ 0 ) = f 0 + ( f 1 f 0 ) θ 0 2 π ,
z 1 = f 0 + ( f 1 f 0 ) θ 0 2 π .
z 1 = λ 0 λ ( f 0 + ( f 1 f 0 ) θ 0 2 π ) .
λ 0 λ ( f 0 + ( f 1 f 0 ) θ 0 2 π ) = ( f 0 + ( f 1 f 0 ) θ 2 π )
θ = f 0 f 1 f 0 ( λ λ 0 1 ) + λ λ 0 θ 0

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