Abstract

The strong chromatic distortion associated with diffractive optical elements is fully exploited to achieve an achromatic optical Fourier transformation under broadband point-source illumination by means of an air-spaced diffractive lens doublet. An analysis of the system is carried out by use of the Fresnel diffraction theory, and the residual secondary spectrum (both axial and transversal) is evaluated. We recognize that the proposed optical architecture allows us to tune the scale factor of the achromatic Fraunhofer diffraction pattern of the input by simply moving the diffracting screen along the optical axis of the system. The performance of our proposed optical setup is verified by several laboratory results.

© 1998 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. L. Rogers, Noncoherent Optical Processing (Wiley, New York, 1977).
  2. F. T. S. Yu, White-Light Optical Signal Processing (Wiley, New York, 1985).
  3. G. M. Morris, D. A. Zweig, “White-light Fourier transformations,” in Optical Signal Processing, J. L. Horner, ed. (Academic, New York, 1987), pp. 23–71.
  4. C. G. Wyne, “Extending the bandwidth of speckle interferometry,” Opt. Commun. 28, 21–25 (1979).
    [CrossRef]
  5. C. Brophy, “Design of an all-glass achromatic Fourier transform lens,” Opt. Commun. 47, 364–368 (1983).
    [CrossRef]
  6. G. M. Morris, “An ideal achromatic Fourier processor,” Opt. Commun. 39, 143–147 (1981).
    [CrossRef]
  7. G. M. Morris, “Diffraction theory for an achromatic Fourier transformation,” Appl. Opt. 20, 2017–2025 (1981).
    [CrossRef] [PubMed]
  8. R. H. Katyl, “Compensating optical systems. Part 3: achromatic Fourier transformation,” Appl. Opt. 11, 1255–1260 (1972).
    [CrossRef] [PubMed]
  9. S. Leon, E. N. Leith, “Optical processing and holography with polychromatic point source illumination,” Appl. Opt. 11, 1255–1260 (1972).
  10. R. Ferrière, J. P. Goedgebuer, “Achromatic systems for far-field diffraction with broadband illumination,” Appl. Opt. 22, 1540–1545 (1983).
    [CrossRef]
  11. P. Andrés, J. Lancis, W. D. Furlan, “White-light Fourier transformer with low chromatic aberration,” Appl. Opt. 23, 4682–4687 (1992).
    [CrossRef]
  12. J. Lancis, P. Andrés, W. D. Furlan, A. Pons, “All-diffractive achromatic Fourier transform setup,” Opt. Lett. 19, 402–404 (1994).
    [PubMed]
  13. V. N. Mahajan, Aberration Theory Made Simple, Vol. TT06 of SPIE Tutorial Text Series (SPIE Press, Bellingham, Wash., 1991).
  14. W. B. Yun, M. R. Howells, “High-resolution Fresnel zone plates for x-ray applications by spatial-frequency multiplication,” J. Opt. Soc. Am. A 4, 34–40 (1987).
    [CrossRef]

1994

1992

P. Andrés, J. Lancis, W. D. Furlan, “White-light Fourier transformer with low chromatic aberration,” Appl. Opt. 23, 4682–4687 (1992).
[CrossRef]

1987

1983

C. Brophy, “Design of an all-glass achromatic Fourier transform lens,” Opt. Commun. 47, 364–368 (1983).
[CrossRef]

R. Ferrière, J. P. Goedgebuer, “Achromatic systems for far-field diffraction with broadband illumination,” Appl. Opt. 22, 1540–1545 (1983).
[CrossRef]

1981

1979

C. G. Wyne, “Extending the bandwidth of speckle interferometry,” Opt. Commun. 28, 21–25 (1979).
[CrossRef]

1972

Andrés, P.

J. Lancis, P. Andrés, W. D. Furlan, A. Pons, “All-diffractive achromatic Fourier transform setup,” Opt. Lett. 19, 402–404 (1994).
[PubMed]

P. Andrés, J. Lancis, W. D. Furlan, “White-light Fourier transformer with low chromatic aberration,” Appl. Opt. 23, 4682–4687 (1992).
[CrossRef]

Brophy, C.

C. Brophy, “Design of an all-glass achromatic Fourier transform lens,” Opt. Commun. 47, 364–368 (1983).
[CrossRef]

Ferrière, R.

Furlan, W. D.

J. Lancis, P. Andrés, W. D. Furlan, A. Pons, “All-diffractive achromatic Fourier transform setup,” Opt. Lett. 19, 402–404 (1994).
[PubMed]

P. Andrés, J. Lancis, W. D. Furlan, “White-light Fourier transformer with low chromatic aberration,” Appl. Opt. 23, 4682–4687 (1992).
[CrossRef]

Goedgebuer, J. P.

Howells, M. R.

Katyl, R. H.

Lancis, J.

J. Lancis, P. Andrés, W. D. Furlan, A. Pons, “All-diffractive achromatic Fourier transform setup,” Opt. Lett. 19, 402–404 (1994).
[PubMed]

P. Andrés, J. Lancis, W. D. Furlan, “White-light Fourier transformer with low chromatic aberration,” Appl. Opt. 23, 4682–4687 (1992).
[CrossRef]

Leith, E. N.

Leon, S.

Mahajan, V. N.

V. N. Mahajan, Aberration Theory Made Simple, Vol. TT06 of SPIE Tutorial Text Series (SPIE Press, Bellingham, Wash., 1991).

Morris, G. M.

G. M. Morris, “An ideal achromatic Fourier processor,” Opt. Commun. 39, 143–147 (1981).
[CrossRef]

G. M. Morris, “Diffraction theory for an achromatic Fourier transformation,” Appl. Opt. 20, 2017–2025 (1981).
[CrossRef] [PubMed]

G. M. Morris, D. A. Zweig, “White-light Fourier transformations,” in Optical Signal Processing, J. L. Horner, ed. (Academic, New York, 1987), pp. 23–71.

Pons, A.

Rogers, G. L.

G. L. Rogers, Noncoherent Optical Processing (Wiley, New York, 1977).

Wyne, C. G.

C. G. Wyne, “Extending the bandwidth of speckle interferometry,” Opt. Commun. 28, 21–25 (1979).
[CrossRef]

Yu, F. T. S.

F. T. S. Yu, White-Light Optical Signal Processing (Wiley, New York, 1985).

Yun, W. B.

Zweig, D. A.

G. M. Morris, D. A. Zweig, “White-light Fourier transformations,” in Optical Signal Processing, J. L. Horner, ed. (Academic, New York, 1987), pp. 23–71.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Commun.

C. G. Wyne, “Extending the bandwidth of speckle interferometry,” Opt. Commun. 28, 21–25 (1979).
[CrossRef]

C. Brophy, “Design of an all-glass achromatic Fourier transform lens,” Opt. Commun. 47, 364–368 (1983).
[CrossRef]

G. M. Morris, “An ideal achromatic Fourier processor,” Opt. Commun. 39, 143–147 (1981).
[CrossRef]

Opt. Lett.

Other

V. N. Mahajan, Aberration Theory Made Simple, Vol. TT06 of SPIE Tutorial Text Series (SPIE Press, Bellingham, Wash., 1991).

G. L. Rogers, Noncoherent Optical Processing (Wiley, New York, 1977).

F. T. S. Yu, White-Light Optical Signal Processing (Wiley, New York, 1985).

G. M. Morris, D. A. Zweig, “White-light Fourier transformations,” in Optical Signal Processing, J. L. Horner, ed. (Academic, New York, 1987), pp. 23–71.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Under white-light point source illumination the Fraunhofer diffraction pattern of an arbitrary transparency provided by a DL is chromatically dispersed both axial and laterally. The different monochromatic versions form a frustum of a right-hand cone whose apex coincides with the point source S.

Fig. 2
Fig. 2

Air-separated DL doublet arranged for producing an achromatic Fourier transformation.

Fig. 3
Fig. 3

Plot of the functional dependence on σ of the secondary spectrum associated with the Fraunhofer diffraction pattern provided by (a) our proposed DL doublet (solid curve) and (b) a conventional nondispersive objective (dashed line). In both cases we assume white-light illumination.

Fig. 4
Fig. 4

Graphical representation of the maximum value CA M of the function CA(σ) versus the spectral bandwidth Δσ of the illuminating source for (a) the DL doublet (solid curve) and (b) an achromatic objective (dashed line). The extreme wave numbers were chosen in such a way that σ0, given by Eq. (24), is in all cases equal to 1.75 μm-1.

Fig. 5
Fig. 5

Irradiance distribution at the achromatic Fraunhofer plane of the setup shown in Fig. 2 when the input, a 2-D square diffraction grating, is located in two different axial positions: (a) Gray-level picture of the Fraunhofer irradiance distribution when z = -283 mm. (b) Irradiance profile along a horizontal line in (a) for each RGB component of the incident light. (c) Gray-level picture of the Fraunhofer irradiance distribution when z = -200 mm. (d) RGB irradiance profile along a horizontal line in (c).

Fig. 6
Fig. 6

White-light Fraunhofer diffraction pattern of a 2-D diffraction grating provided by a conventional achromatic refractive objective: (a) gray-tone representation and (b) RGB irradiance profile along a horizontal line.

Fig. 7
Fig. 7

Same as for Fig. 5 but for three different axial positions of a double circular aperture as the input object: (a) z = -283 mm, (c) z = -200 mm, and (e) z = -160 mm.

Fig. 8
Fig. 8

Same as for Fig. 6 but for a double circular aperture as a diffracting screen.

Equations (28)

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

h x ,   y = exp - i π σ 0 Z 0 x 2 + y 2 .
U 0 x ,   y ;   σ = exp i π σ d - z - σ 0 Z 0 x 2 + y 2 × -     t x ,   y × exp i π σ 1 z + 1 d - z x 2 + y 2 × exp - i 2 π   σ d - z xx + yy d x d y ,
U 1 x ,   y ;   σ ,   d = exp i π   σ d 1 - σ Ad x 2 + y 2 × -     t x ,   y × exp i π   σ d - z d z - σ A d - z × x 2 + y 2 × exp - i 2 π   σ 2 Ad d - z × xx + yy d x d y ,
A = σ d - z + σ d - σ 0 Z 0 .
d z - σ A d - z = 0 ,
d σ = Z 0 d σ σ 0 d - Z 0 σ .
x σ u = y σ v = z σ d σ d = Z 0 z σ 0 d - Z 0 σ ,
U 2 x ,   y ;   σ = exp - i π σ d + σ 2 A d 2 + σ 0 Z 0 x 2 + y 2 × -     t x ,   y × exp i π   σ d - z × d z - σ A d - z x 2 + y 2 × exp i 2 π   σ 2 A d d - z xx + yy d x d y ,
A = σ z d d - z - σ 0 Z 0 .
U 3 x ,   y ;   σ ,   D = exp - i π   σ D 1 + σ BD x 2 + y 2 × -     t x ,   y exp i π   σ d - z × d z - σ A d - z + σ 3 BA 2 d 2 d , - z × x 2 + y 2 × exp - i 2 π   σ 3 BA dD d - z × xx + yy d x d y .
B = σ d + σ 2 A d 2 + σ 0 Z 0 - σ D .
d z - σ A d - z + σ 3 BA 2 d 2 d - z = 0 .
D σ = 1 1 d + σ 0 Z 0 σ - Z 0 σ σ 0 d 2 .
x σ u = y σ v = zZ 0 σ 0 d 2   D σ .
D ( σ ) σ σ = σ 0 = χ ( σ ) σ σ = σ 0 = γ ( σ ) σ σ = σ 0 .
Z 0 = - d 2 Z 0 ,
D 0 = d 2 d - 2 Z 0 .
U 4 r ,   θ ;   σ ,   D 0 = exp - i 2 π   E σ b 2   r 2 0 0 2 π   t r ,   θ × exp i 2 π   F σ a 2   r 2 × exp - i 2 π   G σ ab   r   r × cos θ - θ   r d r d θ .
E σ = 1 2 b D 0 2 × D 0 σ + 1 d - z dz σ + d - z Z 0 σ 0 - 1 Z 0 2 σ - σ 0 σ 2 , F σ = - 1 2 a z 2 σ d - z dz - 1 Z 0 σ 2 σ - σ 0 σ - σ 0 2 , G σ = - a z b D 0 Z 0 σ 0 1 - σ - σ 0 2 σ 2 d - z dz Z 0 σ 0 σ + 1 .
DCA σ = F σ .
TCA σ = G σ - G σ 0 .
DCA σ = - 1 2 a d 2 Z 0 σ 0 CA σ , TCA σ = - a d b D 0   Z 0 σ 0 CA σ ,
CA σ = σ - σ 0 2 σ 0 2 σ - σ 0 .
σ 0 = 2   σ 1 σ 2 σ 1 + σ 2 .
TCA σ = - a f b f   f σ 0 CA σ ,
CA σ = σ - σ 0 σ 0 .
CA M = Δ σ σ 0 2 1 + 1 + Δ σ σ 0 2 1 / 2 ,
CA M = CA σ 2 - CA σ 1 = Δ σ σ 0 .

Metrics