Abstract

Diffractive Optical Elements (DOE), that generate a propagation-invariant transverse intensity pattern, can be used for metrology and imaging application because they provide a very wide depth of focus. However, exact implementation of such DOE is not easy, so we generally code the transmittance by a binary approximation. In this paper, we will study the influence of the binary approximation of Continuously Self-Imaging Gratings (CSIG) on the propagated intensity pattern, for amplitude or phase coding. We will thus demonstrate that under specific conditions, parasitic effects due to the binarization disappear and we retrieve the theoretical non-diffracting property of CSIG’s.

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2010 (1)

S. López-Aguayo, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Nondiffracting light on-demand,” Opt. Photonics News 21(12), 43 (2010).
[CrossRef]

2008 (1)

2007 (2)

G. Druart, N. Guérineau, R. Haïdar, J. Primot, P. Chavel, and J. Taboury, “Non-paraxial analysis of continuous self-imaging gratings in oblique illumination,” J. Opt. Soc. Am. A 24(10), 3379–3387 (2007).
[CrossRef]

G. Druart, N. Guérineau, R. Haïdar, J. Primot, A. Kattnig, and J. Taboury, “Image formation by use of continuously self-imaging gratings and diffractive axicons,” Proc. SPIE 6712, 1–11 (2007).

2004 (1)

2003 (1)

N. Guérineau, S. Rommeluere, E. Di Mambro, I. Ribet, and J. Primot, “New techniques of characterization,” C. R. Phys. 4(10), 1175–1185 (2003).

2001 (1)

2000 (1)

1999 (1)

1998 (2)

1997 (1)

1996 (2)

1990 (1)

1987 (1)

1971 (1)

1967 (1)

W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. A 57(6), 772–778 (1967).
[CrossRef]

1954 (1)

Bandres, M. A.

Carcole, E.

Chavel, P.

Chávez-Cerda, S.

Cottrell, D. M.

Dallas, W. J.

Davis, J. A.

Di Mambro, E.

N. Guérineau, S. Rommeluere, E. Di Mambro, I. Ribet, and J. Primot, “New techniques of characterization,” C. R. Phys. 4(10), 1175–1185 (2003).

Druart, G.

Durnin, J.

Friberg, A. T.

Guérineau, N.

Gutiérrez-Vega, J. C.

Haïdar, R.

Harchaoui, B.

Heggarty, K.

Jaroszewicz, Z.

Kartashov, Y. V.

S. López-Aguayo, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Nondiffracting light on-demand,” Opt. Photonics News 21(12), 43 (2010).
[CrossRef]

Kattnig, A.

G. Druart, J. Taboury, N. Guérineau, R. Haïdar, H. Sauer, A. Kattnig, and J. Primot, “Demonstration of image-zooming capability for diffractive axicons,” Opt. Lett. 33(4), 366–368 (2008).
[CrossRef] [PubMed]

G. Druart, N. Guérineau, R. Haïdar, J. Primot, A. Kattnig, and J. Taboury, “Image formation by use of continuously self-imaging gratings and diffractive axicons,” Proc. SPIE 6712, 1–11 (2007).

Kettunen, V.

Lanzl, T.

López-Aguayo, S.

S. López-Aguayo, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Nondiffracting light on-demand,” Opt. Photonics News 21(12), 43 (2010).
[CrossRef]

Maier, M.

McLeod, J. H.

Montgomery, W. D.

W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. A 57(6), 772–778 (1967).
[CrossRef]

Morales, J.

Niggl, L.

Primot, J.

Ribet, I.

N. Guérineau, S. Rommeluere, E. Di Mambro, I. Ribet, and J. Primot, “New techniques of characterization,” C. R. Phys. 4(10), 1175–1185 (2003).

Rommeluere, S.

N. Guérineau, S. Rommeluere, E. Di Mambro, I. Ribet, and J. Primot, “New techniques of characterization,” C. R. Phys. 4(10), 1175–1185 (2003).

Sauer, H.

Taboury, J.

Torner, L.

S. López-Aguayo, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Nondiffracting light on-demand,” Opt. Photonics News 21(12), 43 (2010).
[CrossRef]

Turunen, J.

Vysloukh, V. A.

S. López-Aguayo, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Nondiffracting light on-demand,” Opt. Photonics News 21(12), 43 (2010).
[CrossRef]

Wyrowski, F.

Appl. Opt. (3)

C. R. Phys. (1)

N. Guérineau, S. Rommeluere, E. Di Mambro, I. Ribet, and J. Primot, “New techniques of characterization,” C. R. Phys. 4(10), 1175–1185 (2003).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (8)

Opt. Lett. (4)

Opt. Photonics News (1)

S. López-Aguayo, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Nondiffracting light on-demand,” Opt. Photonics News 21(12), 43 (2010).
[CrossRef]

Proc. SPIE (1)

G. Druart, N. Guérineau, R. Haïdar, J. Primot, A. Kattnig, and J. Taboury, “Image formation by use of continuously self-imaging gratings and diffractive axicons,” Proc. SPIE 6712, 1–11 (2007).

Other (1)

J. W. Goodman, “Foundations of Scalar Diffraction Theory,” in Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005), pp. 31 - 62.

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Figures (8)

Fig. 1
Fig. 1

Illustration of one elementary cell of size a0a0 of tCSIG (a) and of the PSF (b) for a CSIG of parameter η2 = 9425. TCSIG (c) and the MTF (d) are also represented. For a better visualization, TCSIG is zoomed of a factor × 2 compared to the MTF.

Fig. 2
Fig. 2

The transmittance tCSIG (a) can be considered as the product of its two levels sign function (b) by its absolute value (c) function (in gray scale). Only a part of the elementary cell of tCSIG is represented for a CSIG of parameter η2 = 9425.

Fig. 3
Fig. 3

3D representation of TCSIG for an ideal CSIG of parameter η2 = 9425 (a), for the associated binary amplitude CSIG (b), and for the binary phase CSIG coded at λcoding = 532nm and illuminated at λ = 532nm (c) or λ = 600nm (d).

Fig. 4
Fig. 4

Illustration in the Fourier domain of the FT of the propagated intensity pattern (a). Small dots represent propagation-invariant Dirac peaks located at MTF’s spatial frequencies and big dots represent the propagation-dependant peaks located at the N spatial frequencies of TCSIG. (b) represents the FT of an experimental image recorded at 20mm of the amplitude CSIG of parameters η2 = 9425 and a0 = 7.5mm.

Fig. 5
Fig. 5

Evolution along the z axis of the peaks of the MTF for the binary CSIG of parameter η2 = 9425 and a0 = 7.5mm. On the vertical axis, z is ranging from 17 to 110mm. On the horizontal axis all 1201 peaks sorted, from left to right, by polar coordinates. Black lines localize peaks n°390 and n°665.

Fig. 6
Fig. 6

Scheme of the experimental set-up containing a LED of wavelength λLED = 635nm located at a distance d = 4m of a CSIG of parameters η2 = 9425 and a0 = 7.5mm. A matrix detector is mounted on a translation stage.

Fig. 7
Fig. 7

(a) and (c) illustrate the evolution versus z of the peaks n° 390 and n° 665 of the MTF for the CSIG of parameter η2 = 9425. This evolution is given for three different coding: ideal, binary amplitude and binary phase coded at 532 nm. A comparison between simulation and experimental data obtained with the phase CSIG is given for the peaks n° 390 (b) and n° 665 (d).

Fig. 8
Fig. 8

(a) and (c) illustrate the evolution versus z of the peaks n° 390 and n° 665 of the MTF for the CSIG of parameter η2 = 9425. This evolution is given for a monochromatic and a broadband source. A comparison between simulation and experimental data obtained with the phase CSIG is given for the peaks n° 390 (b) and n° 665 (d).

Tables (2)

Tables Icon

Table 1 Ratio R of Energy of N Main Peaks over Total Energy for CSIG of Parameter η2 = 9425

Tables Icon

Table 2 Ratio R’ of Energy of Useful Peaks over Total Energy of MTF for CSIG of Parameter η2 = 9425.

Equations (28)

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k[ 1,N ]: p k 2 a 0 2 + q k 2 a 0 2 = η k 2 a 0 2 = α 2 ,
{ t CSIG ( r )= k=1 N c k exp( i2π p k x+ q k y a 0 ) T CSIG ( σ )= k=1 N c k δ( σ x p k a 0 , σ y q k a 0 ) .
{ PSF( r )= | t CSIG ( r ) | 2 MTF( σ )=| FT[ PSF ]( σ ) |=| T CSIG ( σ ) T CSIG * ( σ ) | ,
MTF( σ )= k=1 N' D k δ( σ x p ' k a 0 , σ y q ' k a 0 ),
N'=1+N+( N2 )× N 2 = N 2 2 +1.
{ t amplitude ( r )= sign[ t CSIG ( r ) ]+1 2 t phase,λ ( r )=exp( iπ λ coding λ ( 1sign[ t CSIG ( r ) ] 2 ) ) .
{ t binary ( r )= ( k,l ) 2 c k,l exp( i2π kx+ly a 0 ) T binary ( σ )= ( k,l ) 2 c k,l δ( σ x k a 0 , σ y l a 0 ) ,
a( λ )= pattern t phase,λ ( r )dr= cos( π λ coding 2λ )exp( iπ λ coding 2λ ).
{ t binary ( r )=a+b t CSIG ( r )+ k,l c k,l exp( i2π kx+ly a 0 ) T binary ( σ )=aδ( σ )+b T CSIG ( σ )+ k,l c k,l δ( σ x k a 0 , σ y l a 0 ) ,
R= N | b | 2 | a | 2 +N | b | 2 + k,l | c k,l | 2
u trans ( r )= I 0 × t binary ( r ).
u prop ( r,z )= I 0 [ aexp( i2π z λ )+bexp( i2π z λ 1 λ 2 η 2 a 0 2 ) t CSIG ( r ) ],
{ I prop ( r,z )= I 0 [ a 2 +2abcos( 2π z Z T ) t CSIG ( r )+ b 2 PSF( r ) ] I ˜ prop ( σ,z )= I 0 [ a 2 δ( σ )+2abcos( 2π z Z T ) T CSIG ( σ )+ b 2 MTF( σ ) ] .
Z T = λ 1 1 λ 2 η 2 a 0 2 2 a 0 2 λ η 2 .
{ I prop ( r,z )= I 0 ( k,l,k',l' ) 4 c k,l c k',l' * exp( i2π z Z k,l,k',l' )×exp( i2π ( kk' )x+( ll' )y a 0 ) I ˜ prop ( σ,z )= I 0 ( k,l,k',l' ) 4 c k,l c k',l' * exp( i2π z Z k,l,k',l' )×δ( σ x kk' a 0 , σ y ll' a 0 ) ,
Z k,l,k',l' = λ 1 λ 2 k 2 + l 2 a 0 2 1 λ 2 k ' 2 +l ' 2 a 0 2 2 a 0 2 λ( k ' 2 +l ' 2 k 2 l 2 ) .
I ˜ prop ( σ )= I 0 | k 2 + l 2 =k ' 2 +l ' 2 c k,l c k',l' * δ( σ x kk' a 0 , σ y ll' a 0 ) + 2 k 2 + l 2 k ' 2 +l ' 2 ρ k,l,k',l' cos( 2π z Z k,l,k',l' + φ k,l,k',l' )×δ( σ x kk' a 0 , σ y ll' a 0 ) ,
c k,l c k',l' * = ρ k,l,k',l' exp( i φ k,l,k',l' ).
I prop,λ ( r,z )= I λ | k 2 + l 2 =k ' 2 +l ' 2 c k,l c k',l' * × e i2π ( kk' )x+( ll' )y a 0 + 2 k 2 + l 2 k ' 2 +l ' 2 ρ k,l,k',l' cos( 2π z Z k,l,k',l' + φ k,l,k',l' )× e i2π ( kk' )x+( ll' )y a 0 .
I panchro ( r,z )= s( λ ) I prop,λ ( r,z )dλ .
I panchro ( r,z )=| I 0 k 2 + l 2 =k ' 2 +l ' 2 s( λ ) c k,l ( λ ) c k',l' * ( λ )dλ × e i2π ( kk' )x+( ll' )y a 0 + 2 k 2 + l 2 k ' 2 +l ' 2 s( λ ) ρ k,l,k',l' ( λ )cos( 2π z Z k,l,k',l' + φ k,l,k',l' ( λ ) )dλ × e i2π ( kk' )x+( ll' )y a 0
{ s( λ )= I 0 1 σ 2π exp( ( λ λ 0 ) 2 2 σ 2 ) S( Λ )= I 0 exp( i2π λ 0 Λ )exp( 2 π 2 σ 2 Λ 2 ) ,
FWHM=2 2ln( 2 )σ .
f( z )=2 ρ k,l,k',l' s( λ )cos( 2π ( k ' 2 +l ' 2 k 2 l 2 )z 2 a 0 2 λ+ φ k,l,k',l' )dλ .
f( z )= I 0 2 ρ k,l,k',l' exp( 2 π 2 z 2 Z ' 2 )cos( 2π z Z T + φ k,l,k',l' ),
{ Z'= 2 a 0 2 σ( k ' 2 +l ' 2 k 2 l 2 ) Z T = 2 a 0 2 λ 0 ( k ' 2 +l ' 2 k 2 l 2 ) .
I ˜ panchro ( σ,z )=| I 0 k 2 + l 2 =k ' 2 +l ' 2 s( λ ) c k,l ( λ ) c k',l' * ( λ )dλ×δ( σ x kk' a 0 , σ y ll' a 0 ).
R'= ( N 2 2 N )× 2 2 +N× 1 2 1× N 2 +( N 2 2 N )× 2 2 +N× 1 2 = 2 3 N 3 3 N .

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