Abstract

Tolerance in angles of continuously self-imaging gratings (CSIGs) is explored. The degradation in angle of the shape of the point-spread function is theoretically investigated and illustrated by simulations and experiments. The formalism presented is inspired by the one used for classical lenses and can be easily generalized to diffraction gratings. It turns out that well-designed CSIGs could be used for scanning optical systems requiring a large field of view.

© 2007 Optical Society of America

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  1. J. Durnin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. 4, 651-654 (1987).
    [CrossRef]
  2. J. H. McLoed, "The axicon: a new type of optical element," J. Opt. Soc. Am. 14, 592-597 (1954).
    [CrossRef]
  3. L. M. Soroko, "Axicons and meso-optical imaging devices," in Progress in Optics, Vol. 27, E.Wolf, ed. (Elsevier, 1989), pp. 111-127.
  4. J. Sochacki, A. Kolodziejczyk, Z. Jaroszewicz, and S. Bara, "Nonparaxial design of generalized axicons," Appl. Opt. 31, 5326-5330 (1992).
    [CrossRef] [PubMed]
  5. A. T. Friberg, "Stationary-phase analysis of generalized axicons," J. Opt. Soc. Am. A 13, 743-750 (1996).
    [CrossRef]
  6. Z. Jaroszewicz and J. Morales, "Lens axicons: systems composed of a diverging aberrated lens and a perfect converging lens," J. Opt. Soc. Am. A 15, 2383-2390 (1998).
    [CrossRef]
  7. C. J. Zapata-Rodríguez and A. Sánchez-Losa, "Three-dimensional field distribution in the focal region of low-Fresnel-number axicons," J. Opt. Soc. Am. A 23, 3016-3026 (2006).
    [CrossRef]
  8. R. Arimoto, C. Saloma, T. Tanaka, and S. Kawata, "Imaging properties of axicon in a scanning optical system," Appl. Opt. 31, 6653-6657 (1992).
    [CrossRef] [PubMed]
  9. T. Grosjean, F. Baida, and D. Courjon, "Conical optics: the solution to confine light," Appl. Opt. 46, 1994-2000 (2007).
    [CrossRef] [PubMed]
  10. Z. Bin and L. Zhu, "Diffraction property of an axicon in oblique illumination," Appl. Opt. 37, 2563-2568 (1998).
    [CrossRef]
  11. T. Tanaka and S. Yamamoto, "Comparison of aberration between axicon and lens," Opt. Commun. 184, 113-118 (2000).
    [CrossRef]
  12. A. Thaning, Z. Jaroszewicz, and A. T. Friberg, "Diffractive axicons in oblique illumination: analysis and experiments and comparaison with elliptical axicons," Appl. Opt. 42, 9-17 (2003).
    [CrossRef] [PubMed]
  13. A. Burvall, K. Kolacz, A. V. Goncharov, Z. Jaroszewicz, and C. Dainty, "Lens axicons in oblique illumination," Appl. Opt. 46, 312-318 (2007).
    [CrossRef] [PubMed]
  14. N. Guérineau and J. Primot, "Nondiffracting array generation using an N-wave interferometer," J. Opt. Soc. Am. 16, 293-298 (1999).
    [CrossRef]
  15. N. Guérineau, B. Harchaoui, J. Primot, and K. Heggarty, "Generation of achromatic and propagation-invariant spot arrays by use of continuously self-imaging gratings," Opt. Lett. 26, 411-413 (2001).
    [CrossRef]
  16. J. Primot and N. Guérineau, "Extended Hartmann test based on the pseudoguiding property of a Hartmann mask completed by a phase chessboard," Appl. Opt. 39, 5715-5720 (2000).
    [CrossRef]
  17. N. Guérineau, S. Rommeluere, E. Di Mambro, I. Ribet, and J. Primot, "New techniques of characterization," C. R. Phys. 4, 1175-1185 (2003).
    [CrossRef]
  18. R. F. Edgar, "The Fresnel diffraction images of periodic structures," Opt. Acta 16, 281-287 (1969).
    [CrossRef]
  19. M. Testorf, J. Jahns, N. A. Khilo, and A. M. Goncharenko, "Talbot effect for oblique angle of light propagation," Opt. Commun. 129, 167-172 (1996).
    [CrossRef]
  20. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968), p. 101.
  21. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1989), Chap. V, p. 203.
  22. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1989), Chap. IX, p. 468.
  23. E. di Mambro, R. Haïdar, N. Guérineau, and J. Primot, "Sharpness limitations in the projection of thin lines by use of the Talbot experiment," J. Opt. Soc. Am. A 21, 2276-2282 (2004).
    [CrossRef]

2007 (2)

2006 (1)

2004 (1)

2003 (2)

2001 (1)

2000 (2)

1999 (1)

1998 (2)

1996 (2)

A. T. Friberg, "Stationary-phase analysis of generalized axicons," J. Opt. Soc. Am. A 13, 743-750 (1996).
[CrossRef]

M. Testorf, J. Jahns, N. A. Khilo, and A. M. Goncharenko, "Talbot effect for oblique angle of light propagation," Opt. Commun. 129, 167-172 (1996).
[CrossRef]

1992 (2)

1987 (1)

J. Durnin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. 4, 651-654 (1987).
[CrossRef]

1969 (1)

R. F. Edgar, "The Fresnel diffraction images of periodic structures," Opt. Acta 16, 281-287 (1969).
[CrossRef]

1954 (1)

J. H. McLoed, "The axicon: a new type of optical element," J. Opt. Soc. Am. 14, 592-597 (1954).
[CrossRef]

Arimoto, R.

Baida, F.

Bara, S.

Bin, Z.

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1989), Chap. IX, p. 468.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1989), Chap. V, p. 203.

Burvall, A.

Courjon, D.

Dainty, C.

di Mambro, E.

E. di Mambro, R. Haïdar, N. Guérineau, and J. Primot, "Sharpness limitations in the projection of thin lines by use of the Talbot experiment," J. Opt. Soc. Am. A 21, 2276-2282 (2004).
[CrossRef]

N. Guérineau, S. Rommeluere, E. Di Mambro, I. Ribet, and J. Primot, "New techniques of characterization," C. R. Phys. 4, 1175-1185 (2003).
[CrossRef]

Durnin, J.

J. Durnin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. 4, 651-654 (1987).
[CrossRef]

Edgar, R. F.

R. F. Edgar, "The Fresnel diffraction images of periodic structures," Opt. Acta 16, 281-287 (1969).
[CrossRef]

Friberg, A. T.

Goncharenko, A. M.

M. Testorf, J. Jahns, N. A. Khilo, and A. M. Goncharenko, "Talbot effect for oblique angle of light propagation," Opt. Commun. 129, 167-172 (1996).
[CrossRef]

Goncharov, A. V.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968), p. 101.

Grosjean, T.

Guérineau, N.

Haïdar, R.

Harchaoui, B.

Heggarty, K.

Jahns, J.

M. Testorf, J. Jahns, N. A. Khilo, and A. M. Goncharenko, "Talbot effect for oblique angle of light propagation," Opt. Commun. 129, 167-172 (1996).
[CrossRef]

Jaroszewicz, Z.

Kawata, S.

Khilo, N. A.

M. Testorf, J. Jahns, N. A. Khilo, and A. M. Goncharenko, "Talbot effect for oblique angle of light propagation," Opt. Commun. 129, 167-172 (1996).
[CrossRef]

Kolacz, K.

Kolodziejczyk, A.

McLoed, J. H.

J. H. McLoed, "The axicon: a new type of optical element," J. Opt. Soc. Am. 14, 592-597 (1954).
[CrossRef]

Morales, J.

Primot, J.

Ribet, I.

N. Guérineau, S. Rommeluere, E. Di Mambro, I. Ribet, and J. Primot, "New techniques of characterization," C. R. Phys. 4, 1175-1185 (2003).
[CrossRef]

Rommeluere, S.

N. Guérineau, S. Rommeluere, E. Di Mambro, I. Ribet, and J. Primot, "New techniques of characterization," C. R. Phys. 4, 1175-1185 (2003).
[CrossRef]

Saloma, C.

Sánchez-Losa, A.

Sochacki, J.

Soroko, L. M.

L. M. Soroko, "Axicons and meso-optical imaging devices," in Progress in Optics, Vol. 27, E.Wolf, ed. (Elsevier, 1989), pp. 111-127.

Tanaka, T.

Testorf, M.

M. Testorf, J. Jahns, N. A. Khilo, and A. M. Goncharenko, "Talbot effect for oblique angle of light propagation," Opt. Commun. 129, 167-172 (1996).
[CrossRef]

Thaning, A.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1989), Chap. V, p. 203.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1989), Chap. IX, p. 468.

Yamamoto, S.

T. Tanaka and S. Yamamoto, "Comparison of aberration between axicon and lens," Opt. Commun. 184, 113-118 (2000).
[CrossRef]

Zapata-Rodríguez, C. J.

Zhu, L.

Appl. Opt. (7)

C. R. Phys. (1)

N. Guérineau, S. Rommeluere, E. Di Mambro, I. Ribet, and J. Primot, "New techniques of characterization," C. R. Phys. 4, 1175-1185 (2003).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Durnin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. 4, 651-654 (1987).
[CrossRef]

J. H. McLoed, "The axicon: a new type of optical element," J. Opt. Soc. Am. 14, 592-597 (1954).
[CrossRef]

N. Guérineau and J. Primot, "Nondiffracting array generation using an N-wave interferometer," J. Opt. Soc. Am. 16, 293-298 (1999).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Acta (1)

R. F. Edgar, "The Fresnel diffraction images of periodic structures," Opt. Acta 16, 281-287 (1969).
[CrossRef]

Opt. Commun. (2)

M. Testorf, J. Jahns, N. A. Khilo, and A. M. Goncharenko, "Talbot effect for oblique angle of light propagation," Opt. Commun. 129, 167-172 (1996).
[CrossRef]

T. Tanaka and S. Yamamoto, "Comparison of aberration between axicon and lens," Opt. Commun. 184, 113-118 (2000).
[CrossRef]

Opt. Lett. (1)

Other (4)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968), p. 101.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1989), Chap. V, p. 203.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1989), Chap. IX, p. 468.

L. M. Soroko, "Axicons and meso-optical imaging devices," in Progress in Optics, Vol. 27, E.Wolf, ed. (Elsevier, 1989), pp. 111-127.

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Figures (12)

Fig. 1
Fig. 1

Construction in the Fourier plane of a 24-order CSIG by selecting orders from a Cartesian grid with a ring of radius η.

Fig. 2
Fig. 2

PSFs of a 24-order CSIG ( η 2 = 650 ) and a 48-order CSIG ( η 2 = 9425 ) .

Fig. 3
Fig. 3

Diffraction by a 2D periodic object.

Fig. 4
Fig. 4

Notation used to determine the expressions of the off-axis aberrations.

Fig. 5
Fig. 5

Variation of the aberrant phase of a 48-order CSIG for different oblique illuminations.

Fig. 6
Fig. 6

Evolution of the Strehl ratio according to various incidence angles for two configurations of CSIGs.

Fig. 7
Fig. 7

Biperiodic array of bright spots.

Fig. 8
Fig. 8

PSF spectrum produced by a 48-order CSIG ( η 2 = 9425 ) illuminated at normal incidence.

Fig. 9
Fig. 9

Theoretical evolution of the degradation factor D versus incidence angle α 1 for two different configurations of CSIGs.

Fig. 10
Fig. 10

Presentation of the experimental setup.

Fig. 11
Fig. 11

PSF of a 24-order CSIG ( η 2 = 650 ) obtained by either simulation (a) or experimentation (b) for various incidence angles.

Fig. 12
Fig. 12

Comparison of the degradation factor obtained by simulation and experimentation in the case of pinhole masks with a diameter of (a) 1 mm or (b) 2 mm .

Equations (45)

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ν = p a 0 = η a 0 cos ( φ ) ,
μ = q a 0 = η a 0 sin ( φ ) ,
η = p 2 + q 2 .
t ( x , y ) = p , q C p q exp ( 2 i π ( p x + q y ) a 0 ) ,
U ( x , y , z ) = U 0 p , q C p , q exp ( 2 i π λ z 1 λ 2 ( p 2 + q 2 ) a 0 2 ) exp ( 2 i π ( p x + q y ) a 0 ) ,
U ( x , y , z ) = U 0 exp [ 2 i π λ z 1 λ 2 η 2 a 0 2 ] t ( x , y ) .
PSF ( x , y ) = U ( x , y , z ) 2 = U 0 2 t ( x , y ) 2 .
r 0 0.38 a 0 η .
U ( x , y , z ) = U 0 exp ( 2 i π λ x sin ( α 1 ) ) p , q C p , q exp ( i ϕ p , q ( z ) ) exp ( 2 i π ( p x + q y ) a 0 ) ,
ϕ p , q ( z ) = 2 π λ z 1 ( sin ( α 1 ) + λ p a 0 ) 2 λ 2 q 2 a 0 2 .
ϕ p q ( 0 ) = 2 π λ cos ( α 1 ) z ,
ϕ p q ( 1 ) = 2 π λ λ a 0 p tan ( α 1 ) z ,
ϕ p q ( 2 ) = 2 π λ 1 2 λ 2 a 0 2 ( p 2 cos 3 ( α 1 ) + q 2 cos ( α 1 ) ) z ,
ϕ p q ( 3 ) = 2 π λ 1 2 λ 3 a 0 3 ( sin ( α 1 ) p 3 cos 5 ( α 1 ) + sin ( α 1 ) p q 2 cos 3 ( α 1 ) ) z ,
ϕ p q ( 4 ) = 2 π λ 1 2 λ 4 a 0 4 ( 4 sin 2 ( α 1 ) + 1 4 cos 7 ( α 1 ) p 4 + 2 sin 2 ( α 1 ) + 1 2 cos 5 ( α 1 ) p 2 q 2 + q 4 4 cos 3 ( α 1 ) ) z .
ϕ p q ( 1 ) = 2 π λ λ a 0 ( p α 1 + 1 3 p α 1 3 ) z ,
ϕ p q ( 2 ) = 2 π λ 1 2 λ 2 a 0 2 ( η 2 + 1 2 η 2 α 1 2 + p 2 α 1 2 ) z ,
ϕ p q ( 3 ) = 2 π λ 1 2 λ 3 a 0 3 p η 2 α 1 z ,
ϕ p q ( 4 ) = 2 π λ 1 8 λ 4 a 0 4 η 4 z .
W ( ρ , φ : R ) = 1 2 b 1 ρ 2 + b 2 R ρ cos ( φ ) + 1 4 c 1 ρ 4 + c 2 R 2 ρ 2 cos 2 ( φ ) + 1 2 c 3 R 2 ρ 2 + c 4 R 3 ρ cos ( φ ) + c 5 R ρ 3 cos ( φ ) ,
R = z tan ( α 1 ) ,
ρ cos ( φ ) z = λ p a 0 ,
ρ sin ( φ ) z = λ q a 0 .
ϕ p q ( 1 ) = 2 π λ 1 z R ρ cos ( φ ) 2 π λ 1 3 z 3 R 3 ρ cos ( φ ) ,
ϕ p q ( 2 ) = 2 π λ 1 2 z ρ 2 2 π λ 1 4 z 3 R 2 ρ 2 2 π λ 1 2 z 3 R 2 ρ 2 cos 2 ( φ ) ,
ϕ p q ( 3 ) = 2 π λ 1 2 z 3 R ρ 3 cos ( φ ) ,
ϕ p q ( 4 ) = 2 π λ 1 8 z 3 ρ 4 .
b 1 = b 2 = 1 z ,
c 1 = c 2 = c 3 = c 5 = 1 2 z 3 ,
c 4 = 1 3 z 3 .
ϕ aberr ( ρ , φ ) = 2 π λ 1 2 z 3 R 2 ρ 2 cos ( φ ) 2 .
PSF ( x , y , α 1 ) = U 0 2 p , q C p , q exp [ 2 i π a 0 ( p ( x Δ x ) + q y ) ] exp [ i ϕ aberr ( p , q ) ] 2 ,
Δ x = ( 1 + 1 3 α 1 2 + 1 2 λ 2 a 0 2 η 2 ) α 1 z .
Δ PV = max ( ϕ aberr ) min ( ϕ aberr ) .
Δ PV ( α ) = π λ a 0 2 η 2 α 2 z .
α c = a 0 η ( 1 2 λ z ) 1 2 2 r 0 ( 1 λ z ) 1 2 ,
Δ RMS = φ ( ϕ aberr ( φ ) ϕ ¯ aberr ) 2 N ,
S = max ( PSF α 1 ) max ( PSF 0 ° ) .
PSF ( x , y , α 1 ) = U 0 2 p , q C p , q exp [ 2 i π a 0 ( p ( x Δ x ) + q y ) ] ( 1 + i ϕ aberr ( p , q ) ϕ aberr 2 ( p , q ) 2 ) 2 U 0 2 t ( x Δ x , y ) 2 1 + i ϕ ¯ ϕ 2 ¯ 2 2 PSF ( x Δ x , y , 0 ° ) ( 1 σ ϕ 2 ) ,
S 1 Δ RMS 2 .
f = 1 d 2 d 2 d 2 d 2 d 2 I ( x , y ) 2 d x d y AVG 2 ,
AVG = 1 d 2 d 2 d 2 d 2 d 2 I ( x , y ) d x d y .
f = d 2 π ( Δ 2 ) 2 .
f = p q D p q 2 D 00 2 ,
D = ( ( p , q ) ( 0 , 0 ) D p q 2 ) α 1 ( ( p , q ) ( 0 , 0 ) D p q 2 ) 0 ° .

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